Finite Group Databasebeta

← #29730267

Subgroups — #29730267

Order 15616.

904 conjugacy classes of subgroups.

The subgroup lattice has 720 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840041C211yesmaximal, characteristic, complemented
2C2×#16839836C211yesmaximal, characteristic, complemented
3#21450989C211yesmaximal, characteristic, complemented
4#21450722C211yesmaximal, characteristic
5#21447251C211yesmaximal, characteristic, complemented
6#21447011C211yesmaximal, characteristic, complemented
7#21447002C211yesmaximal, characteristic, complemented
8#21446790C211yesmaximal, characteristic, complemented
9#21446590C211yesmaximal, characteristic, complemented
10#21446255C211yesmaximal, characteristic, complemented
11#21446246C211yesmaximal, characteristic, complemented
12#21445850C211yesmaximal, characteristic, complemented
13#21445845C211yesmaximal, characteristic, complemented
14#21445643C211yesmaximal, characteristic, complemented
15C61×#2075C211yesmaximal, characteristic, complemented
16C22×#11758194C2211yescharacteristic
17C22×C8×D61C2211yescharacteristic
18C2×#14519776C2211yescharacteristic
19C2×#14519752C2211yescharacteristic
20C2×#14519751C2211yescharacteristic
21C2×#14519739C2211yescharacteristic
22C2×C4×C22⋊D61C2211yescharacteristic
23#16840041—24no—
24#16839836—24no—
25C2×#14519711C2211yescharacteristic
26C2×#14519710C2211yescharacteristic
27C2×#14519708C2211yescharacteristic
28C2×#14519706C2211yescharacteristic
29C2×#14519699C2211yescharacteristic
30C2×#14519699C2211yescharacteristic
31C2×#14519697C2211yescharacteristic
32C2×#14519697C2211yescharacteristic
33C2×#14519689C2211yescharacteristic
34C2×#14519687C2211yescharacteristic
35#16839691—21no—
36#16839675—21no—
37#16839674—21no—
38#16839671—21no—
39#16839662—21no—
40#16839640—21no—
41#16839638—21no—
42#16839627C2211yescharacteristic
43#16839626—21no—
44#16839622C2211yescharacteristic
45#16839619C2211yescharacteristic
46#16839619C2211yescharacteristic
47#16839619C2211yescharacteristic
48C4⋊C4×Dic61C2211yescharacteristic
49#16839525C2211yescharacteristic
50#16839524C2211yescharacteristic
51#16839524C2211yescharacteristic
52#16839524C2211yescharacteristic
53#16839524C2211yescharacteristic
54C2×#107×C61C2211yescharacteristic
55C2×C22⋊C8×C61C2211yescharacteristic
56C61×#380C2211yescharacteristic
57C61×#380C2211yescharacteristic
58C61×#340C2211yescharacteristic
59C61×#340C2211yescharacteristic
60C61×#340C2211yescharacteristic
61C22×Dic122—21no—
62C22×D244—21no—
63C22×D244C2311yescharacteristic
64C22×C22⋊D61—21no—
65C22×C22⋊D61D411yescharacteristic
66C22×C4×D61—21no—
67C22×C4×D61C2311yescharacteristic
68C2×#11758194—22no—
69C2×#11758194—22no—
70C2×#11758194—41no—
71C22×#92418C2311yescharacteristic
72C22×#92418C2311yescharacteristic
73C2×C8×D61—22no—
74C2×C8×D61—22no—
75C2×C8×D61—41no—
76#14519776—22no—
77#14519776—22no—
78#14519752—22no—
79#14519752—41no—
80#14519751—22no—
81#14519751—41no—
82#14519739—22no—
83#14519739—22no—
84C4×C22⋊D61—22no—
85C4×C22⋊D61—22no—
86C2×C4⋊Dic61C2311yescharacteristic
87C2×C4⋊Dic61C2311yescharacteristic
88C2×C4⋊Dic61C2311yescharacteristic
89C2×#11758196—21no—
90C2×#11758196D411yescharacteristic
91C2×C22⋊Dic61—21no—
92C2×C22⋊Dic61D411yescharacteristic
93C2×#11758198—21no—
94C2×#11758198—21no—
95C2×#11758198—21no—
96C2×#11758198—21no—
97C2×#11758198—21no—
98C2×#11758198—21no—
99C2×#11758198D411yescharacteristic
100C2×C4×Dic61—21no—
101C2×C4×Dic61C2311yescharacteristic
102#14519711—41no—
103#14519710—22no—
104#14519708—41no—
105#14519706D414yescomplemented
106#14519699—22no—
107#14519699—22no—
108#14519697—22no—
109#14519697—22no—
110#14519689—22no—
111#14519687—22no—
112#14519684—21no—
113#14519684—21no—
114#14519684—21no—
115#14519684—21no—
116#14519684—21no—
117#14519684—21no—
118#14519684—21no—
119C23×C4×C61C2311yescharacteristic
120C22×C8×C61C2311yescharacteristic
121C22×C8×C61C2311yescharacteristic
122C2×C4⋊C4×C61C2311yescharacteristic
123C2×C4⋊C4×C61C2311yescharacteristic
124C2×C4⋊C4×C61C2311yescharacteristic
125C2×C4⋊C4×C61C2311yescharacteristic
126C2×C22⋊C4×C61—21no—
127C2×C22⋊C4×C61—21no—
128#107×C61—22no—
129C22⋊C8×C61—22no—
130#95×C61—21no—
131#95×C61—21no—
132C24×C61C2×D411yescharacteristic
133C23×D61—21no—
134C23×D61C2×D411yescharacteristic
135C22×C4×C61—21no—
136C22×C4×C61—21no—
137C22×C4×C61—21no—
138C22×C4×C61—21no—
139C22×C4×C61—21no—
140C22×C4×C61—21no—
141C22×C4×C61—21no—
142C22×C4×C61—21no—
143C22×C4×C61—21no—
144C22×C4×C61—21no—
145C22×C4×C61C2×D411yescharacteristic
146C22×C4×C61C2411yescharacteristic
147C2×C22⋊D61—22no—
148C2×C22⋊D61—22no—
149C2×C22⋊D61—41no—
150C2×C22⋊D61—42no—
151C2×C22⋊D61—42no—
152C2×C22⋊D61—81no—
153C22×Dic61—21no—
154C22×Dic61—21no—
155C22×Dic61—21no—
156C22×Dic61C2×D411yescharacteristic
157C22×Dic61C2×D411yescharacteristic
158C22×Dic61C2×D411yescharacteristic
159C2×D244—21no—
160C2×D244—21no—
161C2×D244—21no—
162C2×D244—21no—
163C2×D244—21no—
164C2×D244—42no—
165C2×D244—42no—
166C2×D244—81no—
167C2×D244#4112yes—
168C2×C4×D61—21no—
169C2×C4×D61—21no—
170C2×C4×D61—21no—
171C2×C4×D61—21no—
172C2×C4×D61—21no—
173C2×C4×D61—21no—
174C2×C4×D61—42no—
175C2×C4×D61—81no—
176C2×C4×D61—81no—
177C2×Dic122—42no—
178C2×Dic122—81no—
179C2×Dic122—81no—
180C2×C8×C61—21no—
181C2×C8×C61—21no—
182C2×C8×C61—21no—
183C2×C8×C61—21no—
184C2×C8×C61—21no—
185C2×C8×C61C2×D412yes—
186C4⋊C4×C61—21no—
187C4⋊C4×C61—21no—
188C4⋊C4×C61—21no—
189C4⋊C4×C61—21no—
190C4⋊C4×C61—21no—
191C4⋊C4×C61—21no—
192C4⋊C4×C61—21no—
193C4⋊C4×C61—21no—
194C22⋊C4×C61—81no—
195C22⋊C4×C61—81no—
196C22⋊Dic61—22no—
197C22⋊Dic61—81no—
198#11758198—22no—
199#11758198—42no—
200#11758198—42no—
201#11758198—42no—
202#11758198—81no—
203#11758198—81no—
204#11758198—81no—
205C4⋊Dic61—21no—
206C4⋊Dic61—21no—
207C4⋊Dic61—21no—
208C4⋊Dic61—21no—
209C4⋊Dic61—21no—
210C4⋊Dic61—21no—
211#11758196—22no—
212#11758196—42no—
213C4×Dic61—42no—
214C4×Dic61C2×D412yes—
215C4×Dic61#4112yes—
216#11758194—44no—
217C2×#92418—21no—
218C2×#92418—21no—
219C2×#92418—21no—
220C2×#92418—21no—
221C2×#92418C2×D412yes—
222C2×#92418#4112yes—
223C8×D61—44no—
224C23×C61—21no—
225C23×C61—21no—
226C23×C61—21no—
227C23×C61—21no—
228C23×C61—21no—
229C23×C61—21no—
230C23×C61—21no—
231C23×C61C22×D411yescharacteristic
232C22×D61—21no—
233C22×D61—21no—
234C22×D61—21no—
235C22×D61—21no—
236C22×D61—21no—
237C22×D61—21no—
238C22×D61—22no—
239C22×D61—41no—
240C22×D61—41no—
241C22×D61—41no—
242C22×D61—41no—
243C22×D61—41no—
244C22×D61—41no—
245C22×D61#13712yes—
246C2×C4×C61—21no—
247C2×C4×C61—21no—
248C2×C4×C61—21no—
249C2×C4×C61—41no—
250C2×C4×C61—41no—
251C2×C4×C61—41no—
252C2×C4×C61—41no—
253C2×C4×C61—41no—
254C2×C4×C61—41no—
255C2×C4×C61—41no—
256C2×C4×C61—41no—
257C2×C4×C61—41no—
258C2×C4×C61—41no—
259C2×C4×C61—41no—
260C2×C4×C61—41no—
261C2×C4×C61—41no—
262C2×C4×C61—41no—
263C2×C4×C61—41no—
264C2×C4×C61—41no—
265C2×C4×C61C22×D411yescharacteristic
266C2×C4×C61C2×#4111yescharacteristic
267C2×C4×C61C2×#4111yescharacteristic
268C2×C4×C61C2×#4111yescharacteristic
269C2×C4×C61H(5,𝔽2)11yescharacteristic
270C2×C4×C612−1+411yescharacteristic
271C22⋊D61—44no—
272C22⋊D61—84no—
273C2×Dic61—21no—
274C2×Dic61—21no—
275C2×Dic61—21no—
276C2×Dic61—21no—
277C2×Dic61—21no—
278C2×Dic61—21no—
279C2×Dic61—21no—
280C2×Dic61—41no—
281C2×Dic61—41no—
282C2×Dic61—41no—
283C2×Dic61—41no—
284C2×Dic61—41no—
285C2×Dic61—41no—
286C2×Dic61—41no—
287C2×Dic61—41no—
288C2×Dic61—41no—
289C2×Dic61#12112yes—
290C2×Dic61#12112yes—
291D244—22no—
292D244—22no—
293D244—41no—
294D244—41no—
295D244—84no—
296C4×D61—41no—
297C4×D61—41no—
298C4×D61—41no—
299C4×D61—41no—
300C4×D61—84no—
301Dic122—84no—
302C8×C61—22no—
303C8×C61—41no—
304#92418—22no—
305#92418—22no—
306#25350—611nomaximal, Sylow
307C22×C61—41no—
308C22×C61—41no—
309C22×C61—41no—
310C22×C61—41no—
311C22×C61—41no—
312C22×C61—41no—
313C22×C61—41no—
314C22×C61C2×#12111yescharacteristic
315C22×C61#54611yescharacteristic
316C22×C61#54711yescharacteristic
317C22×C61C2×#13711yescharacteristic
318C22×C61#57511yescharacteristic
319C22×C61#57611yescharacteristic
320C22×C61#57611yescharacteristic
321C2×D61—22no—
322C2×D61—22no—
323C2×D61—22no—
324C2×D61—41no—
325C2×D61—41no—
326C2×D61—41no—
327C2×D61—41no—
328C2×D61—42no—
329C2×D61—42no—
330C2×D61—42no—
331C2×D61—81no—
332C2×D61—81no—
333C2×D61—81no—
334C2×D61—81no—
335C4×C61—41no—
336C4×C61—81no—
337C4×C61—81no—
338C4×C61—81no—
339C4×C61—81no—
340C4×C61#52911yescharacteristic
341C4×C61#54211yescharacteristic
342C4×C61#54611yescharacteristic
343C4×C61#54811yescharacteristic
344Dic61—22no—
345Dic61—22no—
346Dic61—41no—
347Dic61—81no—
348Dic61—81no—
349Dic61—81no—
350C2×#474—611no—
351C2×#435—611no—
352#2563—611no—
353#2535—611no—
354#2076—611no—
355#2075D6111yescharacteristic, complemented
356#2034—611no—
357#2033—611no—
358#1993—611no—
359#1949—611no—
360#1895—611no—
361#1893—611no—
362#1809—611no—
363#1808—611no—
364#1762—611no—
365C2×C61—81no—
366C2×C61#242711yescharacteristic
367C2×C61#305611yescharacteristic
368C2×C61#305811yescharacteristic
369C2×C61#331411yescharacteristic
370C2×C61#331911yescharacteristic
371C2×C61#334611yescharacteristic
372C2×C61#334711yescharacteristic
373D61—42no—
374D61—82no—
375C22×M16—611no—
376C23×C8—611no—
377C2×C4⋊Q8—611no—
378C2×#121—611no—
379C2×#121—611no—
380C2×C4×D4—611no—
381C2×C4×D4—611no—
382#474—1224no—
383#435—1224no—
384C2×#107—611no—
385C2×#107—611no—
386C2×#107C2×D6111yescharacteristic
387C2×#106—611no—
388C2×C4⋊C8—611no—
389C2×#102—611no—
390C2×#102—611no—
391C2×#102—611no—
392C2×#102—611no—
393C2×C22⋊C8—611no—
394C2×C22⋊C8C2×D6111yescharacteristic
395C2×#97—611no—
396#397—1221no—
397#393—1221no—
398#393—1221no—
399#392—1221no—
400#390—1221no—
401#386—1221no—
402#386—1221no—
403#382—611no—
404#382—1221no—
405#380—611no—
406#380—611no—
407#380—611no—
408#380—611no—
409#380C2×D6111yescharacteristic
410#380C2×D6111yescharacteristic
411C4×C4⋊C4—611no—
412#340—611no—
413#340—611no—
414#340—611no—
415#340—611no—
416#340—611no—
417#340C2×D6111yescharacteristic
418#340C2×D6111yescharacteristic
419#340C2×D6111yescharacteristic
420C61#2535011yescharacteristic, complemented, Sylow
421C22×Q8—1221no—
422C22×D4—611no—
423C22×D4—611no—
424C22×D4—1221no—
425C22×D4—1221no—
426C23×C4—611no—
427C23×C4—1221no—
428C23×C4C22×D6111yescharacteristic
429C2×M16—1222no—
430C2×M16—1222no—
431C2×M16—2441no—
432C22×C8—611no—
433C22×C8—611no—
434C22×C8—1222no—
435C22×C8—1222no—
436C22×C8—2441no—
437C22×C8C22×D6111yescharacteristic
438C22×C8C22×D6111yescharacteristic
439C4⋊Q8—1222no—
440C4⋊Q8—1222no—
441#121—1222no—
442#121—1222no—
443#121—2441no—
444#121—2441no—
445C4×D4—1222no—
446C4×D4—1222no—
447C4×D4—1222no—
448C4×D4—1222no—
449C2×C4⋊C4—611no—
450C2×C4⋊C4—611no—
451C2×C4⋊C4—611no—
452C2×C4⋊C4—611no—
453C2×C4⋊C4—1221no—
454C2×C4⋊C4C22×D6111yescharacteristic
455C2×C4⋊C4C22×D6111yescharacteristic
456C2×C4⋊C4C22×D6111yescharacteristic
457C2×C4⋊C4C22×D6111yescharacteristic
458C2×C22⋊C4—21no—
459C2×C22⋊C4—21no—
460C2×C22⋊C4—611no—
461C2×C22⋊C4—611no—
462C2×C22⋊C4—1221no—
463C2×C22⋊C4—1221no—
464C2×C22⋊C4—1221no—
465C2×C22⋊C4—1221no—
466C2×C22⋊C4—1221no—
467C2×C22⋊C4—1221no—
468C2×C22⋊C4—1221no—
469C2×C42—611no—
470C2×C42—1221no—
471#107—22no—
472#107—1222no—
473#107—2441no—
474#106—2441no—
475C4⋊C8—614no—
476#102—1222no—
477#102—1222no—
478#102—1222no—
479#102—1222no—
480C22⋊C8—22no—
481C22⋊C8—1222no—
482#97—1222no—
483#95—21no—
484#95—21no—
485#95—1221no—
486#95—1221no—
487#95—1221no—
488#95—1221no—
489#95—1221no—
490#95—1221no—
491#95—1221no—
492C24—611no—
493C24—1221no—
494C24D4×D6111yescharacteristic
495C2×Q8—4881no—
496C2×Q8—4881no—
497C2×Q8—2442no—
498C2×D4—612no—
499C2×D4—1222no—
500C2×D4—1222no—
501C2×D4—1221no—
502C2×D4—1221no—
503C2×D4—1221no—
504C2×D4—1221no—
505C2×D4—1221no—
506C2×D4—4881no—
507C2×D4—4881no—
508C2×D4—2442no—
509C2×D4—2442no—
510C2×D4—2441no—
511C2×D4—2442no—
512C2×D4—2442no—
513C22×C4—21no—
514C22×C4—21no—
515C22×C4—21no—
516C22×C4—21no—
517C22×C4—21no—
518C22×C4—21no—
519C22×C4—21no—
520C22×C4—21no—
521C22×C4—21no—
522C22×C4—21no—
523C22×C4—611no—
524C22×C4—611no—
525C22×C4—611no—
526C22×C4—1221no—
527C22×C4—1221no—
528C22×C4—1221no—
529C22×C4—1221no—
530C22×C4—1221no—
531C22×C4—1221no—
532C22×C4—1221no—
533C22×C4—1221no—
534C22×C4—1221no—
535C22×C4—4881no—
536C22×C4—4881no—
537C22×C4—2442no—
538C22×C4D4×D6111yescharacteristic
539C22×C4C23×D6111yescharacteristic
540M16—2444no—
541C2×C8—21no—
542C2×C8—21no—
543C2×C8—21no—
544C2×C8—21no—
545C2×C8—21no—
546C2×C8—612no—
547C2×C8—612no—
548C2×C8—1221no—
549C2×C8—1221no—
550C2×C8—1221no—
551C2×C8—1221no—
552C2×C8—2444no—
553C2×C8D4×D6112yes—
554C4⋊C4—21no—
555C4⋊C4—21no—
556C4⋊C4—21no—
557C4⋊C4—21no—
558C4⋊C4—21no—
559C4⋊C4—21no—
560C4⋊C4—21no—
561C4⋊C4—21no—
562C4⋊C4—1222no—
563C4⋊C4—1221no—
564C4⋊C4—1221no—
565C4⋊C4—1221no—
566C4⋊C4—1221no—
567C4⋊C4—1221no—
568C4⋊C4—1221no—
569C4⋊C4—2442no—
570C22⋊C4—81no—
571C22⋊C4—81no—
572C22⋊C4—1222no—
573C22⋊C4—1222no—
574C22⋊C4—4881no—
575C22⋊C4—4881no—
576C22⋊C4—4881no—
577C22⋊C4—4881no—
578C22⋊C4—2442no—
579C22⋊C4—2442no—
580C22⋊C4—2442no—
581C42—612no—
582C42—612no—
583C42—2442no—
584C23—21no—
585C23—21no—
586C23—21no—
587C23—21no—
588C23—21no—
589C23—21no—
590C23—21no—
591C23—612no—
592C23—1221no—
593C23—1221no—
594C23—1221no—
595C23—1221no—
596C23—1221no—
597C23—1221no—
598C23—1222no—
599C23—2441no—
600C23—2441no—
601C23—2441no—
602C23—2441no—
603C23—2441no—
604C23—2441no—
605C23C2×D4×D6111yescharacteristic
606Q8—4884no—
607D4—1222no—
608D4—1222no—
609D4—4884no—
610D4—4884no—
611D4—2444no—
612D4—2441no—
613D4—2441no—
614C2×C4—21no—
615C2×C4—21no—
616C2×C4—21no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—41no—
624C2×C4—41no—
625C2×C4—41no—
626C2×C4—41no—
627C2×C4—41no—
628C2×C4—41no—
629C2×C4—41no—
630C2×C4—41no—
631C2×C4—41no—
632C2×C4—41no—
633C2×C4—612no—
634C2×C4—612no—
635C2×C4—1221no—
636C2×C4—1221no—
637C2×C4—1221no—
638C2×C4—1221no—
639C2×C4—1221no—
640C2×C4—1221no—
641C2×C4—1221no—
642C2×C4—4884no—
643C2×C4—2441no—
644C2×C4—2441no—
645C2×C4—2441no—
646C2×C4—2441no—
647C2×C4—2441no—
648C2×C4—2441no—
649C2×C4—2441no—
650C2×C4—2441no—
651C2×C4—2441no—
652C2×C4—2441no—
653C2×C4—2441no—
654C2×C4—2441no—
655C2×C4—2441no—
656C2×C4C2×D4×D6111yescharacteristic
657C2×C4#41×D6111yescharacteristic
658C2×C4#41×D6111yescharacteristic
659C2×C4#41×D6111yescharacteristic
660C2×C4#1451982311yescharacteristic
661C2×C4#1451982411yescharacteristic
662C8—22no—
663C8—41no—
664C8—1222no—
665C8—1222no—
666C22—41no—
667C22—41no—
668C22—41no—
669C22—41no—
670C22—41no—
671C22—41no—
672C22—41no—
673C22—1222no—
674C22—1222no—
675C22—1222no—
676C22—4881no—
677C22—4881no—
678C22—4881no—
679C22—4881no—
680C22—2442no—
681C22—2442no—
682C22—2442no—
683C22—2441no—
684C22—2441no—
685C22—2441no—
686C22—2441no—
687C22#121×D6111yescharacteristic
688C22#1684040411yescharacteristic
689C22#1684041011yescharacteristic
690C22#137×D6111yescharacteristic
691C22#1684056111yescharacteristic
692C22#1684056311yescharacteristic
693C22#1684056411yescharacteristic
694C4—41no—
695C4—81no—
696C4—81no—
697C4—81no—
698C4—81no—
699C4—1222no—
700C4—1222no—
701C4—4881no—
702C4—4881no—
703C4—4881no—
704C4—2441no—
705C4#1684029511yescharacteristic
706C4#1684037811yescharacteristic
707C4#1684040411yescharacteristic
708C4#1684042011yescharacteristic
709C2—81no—
710C2—4882no—
711C2—2442no—
712C2#2144939111yescharacteristic
713C2#2145544811yescharacteristic
714C2#2145546211yescharacteristic
715C2#2145790311yescharacteristic
716C2#2145797811yescharacteristic
717C2#2145829711yescharacteristic
718C2#2145832211yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2075C2
socleC23×C61C22×D4
prime radical#21450722C2
hypercenter#2075D61
hypocenterC61#25350
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2075

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×#16840041C2semidirect
2. C2×#16839836C2semidirect
3. #21450989C2semidirect
5. #21447251C2semidirect
6. #21447011C2semidirect
7. #21447002C2semidirect
8. #21446790C2semidirect
9. #21446590C2semidirect
10. #21446255C2semidirect
11. #21446246C2semidirect
12. #21445850C2semidirect
13. #21445845C2semidirect
14. #21445643C2semidirect
15. C61×#2075C2semidirect
105. #14519706D4semidirect
355. #2075D61semidirect
420. C61#25350semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#168400412
C2×#168398362
#214509892
#214507222
#214472512
#214470112
#214470022
#214467902
#214465902
#214462552
#214462462
#214458502
#214458452
#214456432
C61×#20752
#2535061