Finite Group Databasebeta

← #29718838

Subgroups — #29718838

Order 15616.

944 conjugacy classes of subgroups.

The subgroup lattice has 718 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839925C211yesmaximal, characteristic
2C2×#16839822C211yesmaximal, characteristic, complemented
3#21451621C211yesmaximal, characteristic, complemented
4#21448409C211yesmaximal, characteristic
5#21447211C211yesmaximal, characteristic, complemented
6#21447142C211yesmaximal, characteristic, complemented
7#21447136C211yesmaximal, characteristic, complemented
8#21446701C211yesmaximal, characteristic, complemented
9#21446678C211yesmaximal, characteristic, complemented
10#21446507C211yesmaximal, characteristic, complemented
11#21446499C211yesmaximal, characteristic, complemented
12#21446125C211yesmaximal, characteristic, complemented
13#21446005C211yesmaximal, characteristic, complemented
14#21445756C211yesmaximal, characteristic
15C61×#2067C211yesmaximal, characteristic, complemented
16C22×#11758224C2211yescharacteristic
17C22×Dic244C2211yescharacteristic
18C22×#11758200C2211yescharacteristic
19C2×#14519757C2211yescharacteristic
20C2×#14519734C2211yescharacteristic
21C2×C4⋊C4×D61C2211yescharacteristic
22#16839925—24no—
23#16839822—24no—
24C2×#14519710C2211yescharacteristic
25C2×C4⋊#92418C2211yescharacteristic
26C2×#14519702C2211yescharacteristic
27C2×#14519702C2211yescharacteristic
28C2×#14519700C2211yescharacteristic
29C2×#14519700C2211yescharacteristic
30C2×#14519700C2211yescharacteristic
31C2×#14519698C2211yescharacteristic
32C2×#14519689C2211yescharacteristic
33C2×#14519689C2211yescharacteristic
34C2×#14519687C2211yescharacteristic
35C2×C8×Dic61C2211yescharacteristic
36#16839704—21no—
37#16839695—21no—
38#16839691—21no—
39#16839688—21no—
40#16839668C2211yescharacteristic
41#16839668C2211yescharacteristic
42#16839667C2211yescharacteristic
43#16839667C2211yescharacteristic
44#16839665C2211yescharacteristic
45#16839658C2211yescharacteristic
46#16839638—21no—
47#16839638—21no—
48#16839620—21no—
49#16839620—21no—
50#16839619C2211yescharacteristic
51#16839525C2211yescharacteristic
52#16839524C2211yescharacteristic
53#16839524C2211yescharacteristic
54C2×#107×C61C2211yescharacteristic
55C2×C4⋊C8×C61C2211yescharacteristic
56C2×#102×C61C2211yescharacteristic
57C2×#102×C61C2211yescharacteristic
58C61×#390C2211yescharacteristic
59C61×#389C2211yescharacteristic
60C61×#340C2211yescharacteristic
61C2×#11758224—22no—
62C2×#11758224—22no—
63C2×#11758224—22no—
64C2×#11758224—22no—
65C2×#11758224—41no—
66C2×#11758224—41no—
67C22×Dic122C2311yescharacteristic
68C22×Dic122C2311yescharacteristic
69C22×C22⋊D61—21no—
70C23×Dic61—21no—
71C22×C4×D61—21no—
72C22×C4×D61C2311yescharacteristic
73C2×Dic244—22no—
74C2×Dic244—22no—
75C2×Dic244—41no—
76C2×#11758200—22no—
77C2×#11758200—22no—
78C2×#11758200—41no—
79C22×#92418C2311yescharacteristic
80C22×#92418C2311yescharacteristic
81#14519757—22no—
82#14519757—22no—
83#14519734—22no—
84#14519734—22no—
85C2×C4⋊Dic61—21no—
86C2×C4⋊Dic61C2311yescharacteristic
87C2×C4⋊Dic61C2311yescharacteristic
88C2×#11758196—21no—
89C2×#11758196—21no—
90C2×#11758196D411yescharacteristic
91C2×#11758196D411yescharacteristic
92C4⋊C4×D61—41no—
93C4⋊C4×D61—41no—
94C2×C22⋊Dic61—21no—
95C2×C22⋊Dic61—21no—
96C2×#11758198—21no—
97C2×#11758198—21no—
98C2×#11758198D411yescharacteristic
99C2×#11758198D411yescharacteristic
100C2×C4×Dic61C2311yescharacteristic
101#14519710—41no—
102C4⋊#92418—22no—
103#14519702—22no—
104#14519702—41no—
105#14519700—22no—
106#14519700—41no—
107#14519700—41no—
108#14519698—22no—
109#14519689—22no—
110#14519689D414yes—
111#14519687—22no—
112C8×Dic61—22no—
113#14519684—21no—
114#14519684—21no—
115#14519684—21no—
116#14519684—21no—
117#14519684—21no—
118#14519684—21no—
119#14519684—21no—
120#14519684—21no—
121#14519684—21no—
122#14519684—21no—
123C22×D4×C61C2311yescharacteristic
124C22×C8×C61C2311yescharacteristic
125C22×C8×C61C2311yescharacteristic
126C2×C4⋊C4×C61—21no—
127C2×C4⋊C4×C61C2311yescharacteristic
128C2×C4⋊C4×C61C2311yescharacteristic
129C2×C4⋊C4×C61C2311yescharacteristic
130C2×C22⋊C4×C61—21no—
131C2×C22⋊C4×C61—21no—
132C2×C42×C61C2311yescharacteristic
133#107×C61—41no—
134C4⋊C8×C61—22no—
135#102×C61—22no—
136#102×C61—41no—
137#95×C61—21no—
138C24×C61—21no—
139C23×D61C2×D411yescharacteristic
140C2×D4×C61—21no—
141C2×D4×C61—21no—
142C2×D4×C61—21no—
143C2×D4×C61—21no—
144C2×D4×C61—21no—
145C2×D4×C61#4112yes—
146C22×C4×C61—21no—
147C22×C4×C61—21no—
148C22×C4×C61—21no—
149C22×C4×C61C2×D411yescharacteristic
150C22×C4×C61C2×D411yescharacteristic
151C22×C4×C61C2411yescharacteristic
152C2×C22⋊D61—24no—
153C2×C22⋊D61—42no—
154C2×C22⋊D61—42no—
155C22×Dic61—24no—
156C22×Dic61—21no—
157C22×Dic61—21no—
158C22×Dic61—21no—
159C22×Dic61—21no—
160C22×Dic61—42no—
161C22×Dic61—42no—
162C22×Dic61C2×D411yescharacteristic
163C22×Dic61C2×D411yescharacteristic
164C22×Dic61C2×D411yescharacteristic
165#11758224—44no—
166#11758224—44no—
167#11758224—44no—
168#11758224—44no—
169C2×C4×D61—21no—
170C2×C4×D61—21no—
171C2×C4×D61—21no—
172C2×C4×D61—21no—
173C2×C4×D61—42no—
174C2×C4×D61—81no—
175C2×C4×D61—81no—
176C2×C4×D61C2×D412yes—
177C2×C4×D61#4112yes—
178C2×Dic122—21no—
179C2×Dic122—21no—
180C2×Dic122—21no—
181C2×Dic122—21no—
182C2×Dic122—21no—
183C2×Dic122—21no—
184C2×Dic122—21no—
185C2×Dic122—21no—
186C2×Dic122—21no—
187C2×Dic122—21no—
188C2×Dic122—21no—
189C2×Dic122#4112yes—
190C2×C8×C61—21no—
191C2×C8×C61—21no—
192C2×C8×C61—21no—
193C2×C8×C61—21no—
194C2×C8×C61—21no—
195C2×C8×C61C2×D412yes—
196C4⋊C4×C61—21no—
197C4⋊C4×C61—21no—
198C4⋊C4×C61—21no—
199C4⋊C4×C61—21no—
200C4⋊C4×C61—21no—
201C4⋊C4×C61—21no—
202C4⋊C4×C61—81no—
203C22⋊C4×C61—81no—
204C22⋊C4×C61—81no—
205C42×C61—21no—
206C42×C61—21no—
207C22⋊Dic61—81no—
208C22⋊Dic61—81no—
209#11758200—44no—
210#11758198—22no—
211#11758198—22no—
212#11758198—42no—
213#11758198—81no—
214C4⋊Dic61—21no—
215C4⋊Dic61—21no—
216C4⋊Dic61—21no—
217C4⋊Dic61—21no—
218C4⋊Dic61—42no—
219#11758196—22no—
220#11758196—22no—
221#11758196—42no—
222#11758196—81no—
223C4×Dic61—21no—
224C4×Dic61—21no—
225C2×#92418—21no—
226C2×#92418—21no—
227C2×#92418—21no—
228C2×#92418—21no—
229C2×#92418—21no—
230C2×#92418C2×D412yes—
231Dic244—44no—
232C23×C61—22no—
233C23×C61—41no—
234C23×C61—41no—
235C23×C61—41no—
236C23×C61—41no—
237C23×C61—41no—
238C23×C61—41no—
239C23×C61C22×D411yescharacteristic
240C22×D61—21no—
241C22×D61—21no—
242C22×D61—21no—
243C22×D61—21no—
244C22×D61—21no—
245C22×D61—21no—
246C22×D61#12112yes—
247D4×C61—22no—
248D4×C61—22no—
249D4×C61—41no—
250D4×C61—41no—
251C2×C4×C61—21no—
252C2×C4×C61—21no—
253C2×C4×C61—21no—
254C2×C4×C61—21no—
255C2×C4×C61—21no—
256C2×C4×C61—21no—
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×C61—41no—
266C2×C4×C61C22×D411yescharacteristic
267C2×C4×C61C2×#4111yescharacteristic
268C2×C4×C61C2×#4111yescharacteristic
269C2×C4×C61C2×#4111yescharacteristic
270C2×C4×C61H(5,𝔽2)11yescharacteristic
271C2×C4×C61H(5,𝔽2)11yescharacteristic
272C22⋊D61—48no—
273C2×Dic61—21no—
274C2×Dic61—21no—
275C2×Dic61—21no—
276C2×Dic61—21no—
277C2×Dic61—22no—
278C2×Dic61—21no—
279C2×Dic61—21no—
280C2×Dic61—21no—
281C2×Dic61—21no—
282C2×Dic61—48no—
283C2×Dic61—41no—
284C2×Dic61—41no—
285C2×Dic61—41no—
286C2×Dic61—41no—
287C2×Dic61—41no—
288C2×Dic61—41no—
289C2×Dic61—41no—
290C2×Dic61—41no—
291C2×Dic61—41no—
292C2×Dic61—41no—
293C2×Dic61—41no—
294C2×Dic61#12112yes—
295C4×D61—22no—
296C4×D61—22no—
297C4×D61—22no—
298C4×D61—22no—
299C4×D61—84no—
300Dic122—22no—
301Dic122—22no—
302Dic122—41no—
303Dic122—41no—
304Dic122—41no—
305Dic122—41no—
306Dic122—41no—
307Dic122—41no—
308C8×C61—22no—
309C8×C61—41no—
310#92418—22no—
311#92418—41no—
312#24515—611nomaximal, Sylow
313C22×C61—42no—
314C22×C61—42no—
315C22×C61—42no—
316C22×C61—81no—
317C22×C61—81no—
318C22×C61—81no—
319C22×C61—81no—
320C22×C61C2×#12111yescharacteristic
321C22×C61#52511yescharacteristic
322C22×C61#53511yescharacteristic
323C22×C61#57611yescharacteristic
324C22×C61#57711yescharacteristic
325C22×C61#57811yescharacteristic
326C22×C61#57811yescharacteristic
327C2×D61—22no—
328C2×D61—22no—
329C2×D61—22no—
330C2×D61—41no—
331C2×D61—41no—
332C2×D61—41no—
333C2×D61—41no—
334C4×C61—41no—
335C4×C61—41no—
336C4×C61—81no—
337C4×C61—81no—
338C4×C61—81no—
339C4×C61#54611yescharacteristic
340C4×C61#54611yescharacteristic
341C4×C61#55111yescharacteristic
342C4×C61#55111yescharacteristic
343Dic61—22no—
344Dic61—42no—
345Dic61—41no—
346Dic61—41no—
347Dic61—81no—
348Dic61—81no—
349Dic61—81no—
350C2×#452—611no—
351C2×#433—611no—
352#2625—611no—
353#2311—611no—
354#2068—611no—
355#2067D6111yescharacteristic, complemented
356#2056—611no—
357#2055—611no—
358#1973—611no—
359#1969—611no—
360#1935—611no—
361#1934—611no—
362#1872—611no—
363#1852—611no—
364#1788—611no—
365C2×C61—82no—
366C2×C61#241111yescharacteristic
367C2×C61#306111yescharacteristic
368C2×C61#306311yescharacteristic
369C2×C61#312011yescharacteristic
370C2×C61#312411yescharacteristic
371C2×C61#322311yescharacteristic
372C2×C61#322411yescharacteristic
373D61—42no—
374C22×#41—611no—
375C22×Q16—611no—
376C22×SD16—611no—
377C2×C22⋊Q8—611no—
378C2×#117—611no—
379C22×C4⋊C4—611no—
380#452—1224no—
381#433—1224no—
382C2×#107—611no—
383C2×#107C2×D6111yescharacteristic
384C2×C4⋊C8—611no—
385C2×C4⋊C8C2×D6111yescharacteristic
386C2×#103—611no—
387C2×#103—611no—
388C2×#103—611no—
389C2×#103—611no—
390C2×#103—611no—
391C2×#102—611no—
392C2×#102C2×D6111yescharacteristic
393C2×#102C2×D6111yescharacteristic
394C2×C22⋊C8—611no—
395C2×C22⋊C8—611no—
396C2×#97—611no—
397C2×C4×C8—611no—
398#400—1221no—
399#398—1221no—
400#397—1221no—
401#397—1221no—
402#391—611no—
403#391—611no—
404#391—611no—
405#391—611no—
406#390—611no—
407#390C2×D6111yescharacteristic
408#389—611no—
409#389C2×D6111yescharacteristic
410#386—1221no—
411#386—1221no—
412#380—611no—
413#380—1221no—
414#380—1221no—
415#340—611no—
416#340—611no—
417#340—611no—
418#340C2×D6111yescharacteristic
419C61#2451511yescharacteristic, complemented, Sylow
420C2×#41—1222no—
421C2×#41—1222no—
422C2×#41—1222no—
423C2×#41—1222no—
424C2×#41—2441no—
425C2×#41—2441no—
426C22×Q8—611no—
427C22×Q8—611no—
428C22×D4—1221no—
429C22×D4C22×D6111yescharacteristic
430C23×C4—611no—
431C23×C4—1221no—
432C23×C4—1221no—
433C2×Q16—1222no—
434C2×Q16—1222no—
435C2×Q16—2441no—
436C2×SD16—1222no—
437C2×SD16—1222no—
438C2×SD16—2441no—
439C22×C8—611no—
440C22×C8—611no—
441C22×C8C22×D6111yescharacteristic
442C22×C8C22×D6111yescharacteristic
443C22⋊Q8—1222no—
444C22⋊Q8—1222no—
445#117—1222no—
446#117—1222no—
447C2×C4⋊C4—21no—
448C2×C4⋊C4—611no—
449C2×C4⋊C4—611no—
450C2×C4⋊C4—611no—
451C2×C4⋊C4—611no—
452C2×C4⋊C4—1221no—
453C2×C4⋊C4—1221no—
454C2×C4⋊C4—1221no—
455C2×C4⋊C4—2441no—
456C2×C4⋊C4—2441no—
457C2×C4⋊C4C22×D6111yescharacteristic
458C2×C4⋊C4C22×D6111yescharacteristic
459C2×C4⋊C4C22×D6111yescharacteristic
460C2×C22⋊C4—21no—
461C2×C22⋊C4—21no—
462C2×C22⋊C4—611no—
463C2×C22⋊C4—611no—
464C2×C22⋊C4—1221no—
465C2×C22⋊C4—1221no—
466C2×C22⋊C4—1221no—
467C2×C22⋊C4—1221no—
468C2×C42—611no—
469C2×C42C22×D6111yescharacteristic
470#107—41no—
471#107—2441no—
472C4⋊C8—22no—
473C4⋊C8—1222no—
474#103—1222no—
475#103—1222no—
476#103—2441no—
477#103—2441no—
478#103—2441no—
479#102—22no—
480#102—41no—
481#102—1222no—
482C22⋊C8—614no—
483C22⋊C8—1222no—
484#97—1222no—
485C4×C8—1222no—
486#95—21no—
487#95—1221no—
488#95—1221no—
489#95—1221no—
490#95—1221no—
491#95—1221no—
492#95—1221no—
493#95—1221no—
494#95—1221no—
495#95—1221no—
496#95—1221no—
497C24—21no—
498C24—611no—
499#41—2444no—
500#41—2444no—
501#41—2444no—
502#41—2444no—
503C2×Q8—612no—
504C2×Q8—1221no—
505C2×Q8—1221no—
506C2×Q8—1221no—
507C2×Q8—1221no—
508C2×Q8—1221no—
509C2×Q8—1221no—
510C2×Q8—1221no—
511C2×Q8—1221no—
512C2×Q8—1221no—
513C2×Q8—1221no—
514C2×Q8—1221no—
515C2×D4—21no—
516C2×D4—21no—
517C2×D4—21no—
518C2×D4—21no—
519C2×D4—21no—
520C2×D4—1224no—
521C2×D4—2442no—
522C2×D4—2442no—
523C2×D4#1175822612yes—
524C22×C4—21no—
525C22×C4—21no—
526C22×C4—21no—
527C22×C4—611no—
528C22×C4—612no—
529C22×C4—612no—
530C22×C4—611no—
531C22×C4—611no—
532C22×C4—1221no—
533C22×C4—1221no—
534C22×C4—1221no—
535C22×C4—1221no—
536C22×C4—1224no—
537C22×C4—1221no—
538C22×C4—1221no—
539C22×C4—1221no—
540C22×C4—1221no—
541C22×C4—4881no—
542C22×C4—4881no—
543C22×C4—2442no—
544C22×C4—2442no—
545C22×C4—2442no—
546C22×C4D4×D6111yescharacteristic
547C22×C4D4×D6111yescharacteristic
548C22×C4C23×D6111yescharacteristic
549Q16—2444no—
550SD16—2444no—
551C2×C8—21no—
552C2×C8—21no—
553C2×C8—21no—
554C2×C8—21no—
555C2×C8—21no—
556C2×C8—612no—
557C2×C8—1221no—
558C2×C8—1221no—
559C2×C8—1221no—
560C2×C8—1221no—
561C2×C8—1221no—
562C2×C8D4×D6112yes—
563C4⋊C4—21no—
564C4⋊C4—21no—
565C4⋊C4—21no—
566C4⋊C4—21no—
567C4⋊C4—21no—
568C4⋊C4—21no—
569C4⋊C4—81no—
570C4⋊C4—1222no—
571C4⋊C4—1222no—
572C4⋊C4—1221no—
573C4⋊C4—1221no—
574C4⋊C4—1221no—
575C4⋊C4—1221no—
576C4⋊C4—4881no—
577C4⋊C4—2442no—
578C4⋊C4—2442no—
579C22⋊C4—81no—
580C22⋊C4—81no—
581C22⋊C4—1222no—
582C22⋊C4—1222no—
583C22⋊C4—4881no—
584C22⋊C4—4881no—
585C22⋊C4—4881no—
586C22⋊C4—2442no—
587C42—21no—
588C42—21no—
589C42—1221no—
590C42—1221no—
591C23—22no—
592C23—41no—
593C23—41no—
594C23—41no—
595C23—41no—
596C23—41no—
597C23—41no—
598C23—612no—
599C23—1221no—
600C23—1221no—
601C23—1221no—
602C23—1221no—
603C23—1221no—
604C23—1221no—
605C23C2×D4×D6111yescharacteristic
606Q8—1222no—
607Q8—1222no—
608Q8—2441no—
609Q8—2441no—
610Q8—2441no—
611Q8—2441no—
612Q8—2441no—
613Q8—2441no—
614D4—22no—
615D4—22no—
616D4—41no—
617D4—41no—
618D4—2448no—
619C2×C4—21no—
620C2×C4—21no—
621C2×C4—21no—
622C2×C4—21no—
623C2×C4—21no—
624C2×C4—21no—
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—41no—
634C2×C4—612no—
635C2×C4—1221no—
636C2×C4—1221no—
637C2×C4—1222no—
638C2×C4—1222no—
639C2×C4—1222no—
640C2×C4—1222no—
641C2×C4—1221no—
642C2×C4—1221no—
643C2×C4—1222no—
644C2×C4—1221no—
645C2×C4—1221no—
646C2×C4—1221no—
647C2×C4—1221no—
648C2×C4—4884no—
649C2×C4—2448no—
650C2×C4—2441no—
651C2×C4—2441no—
652C2×C4—2441no—
653C2×C4—2441no—
654C2×C4—2441no—
655C2×C4—2441no—
656C2×C4—2441no—
657C2×C4—2441no—
658C2×C4—2441no—
659C2×C4—2441no—
660C2×C4—2441no—
661C2×C4C2×D4×D6111yescharacteristic
662C2×C4#41×D6111yescharacteristic
663C2×C4#41×D6111yescharacteristic
664C2×C4C2×#1175822611yescharacteristic
665C2×C4#1451982211yescharacteristic
666C2×C4#1451982311yescharacteristic
667C8—22no—
668C8—41no—
669C8—1222no—
670C8—2441no—
671C22—42no—
672C22—42no—
673C22—42no—
674C22—81no—
675C22—81no—
676C22—81no—
677C22—81no—
678C22—1222no—
679C22—1222no—
680C22—1222no—
681C22—2441no—
682C22—2441no—
683C22—2441no—
684C22—2441no—
685C22#121×D6111yescharacteristic
686C22#1684027511yescharacteristic
687C22#1684032811yescharacteristic
688C22#1684056311yescharacteristic
689C22#1684057211yescharacteristic
690C22#1684057611yescharacteristic
691C22#1684057811yescharacteristic
692C4—41no—
693C4—41no—
694C4—81no—
695C4—81no—
696C4—81no—
697C4—1222no—
698C4—4881no—
699C4—4881no—
700C4—4881no—
701C4—2441no—
702C4—2441no—
703C4—2442no—
704C4#1684040111yescharacteristic
705C4#1684040311yescharacteristic
706C4#1684043711yescharacteristic
707C4#1684043911yescharacteristic
708C2—82no—
709C2—2442no—
710C2#2144919911yescharacteristic
711C2#2145548411yescharacteristic
712C2#2145550511yescharacteristic
713C2#2145603811yescharacteristic
714C2#2145608611yescharacteristic
715C2#2145689911yescharacteristic
716C2#2145691011yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2067C2
socleC23×C61C22×D4
prime radicalC22×#11758224C22
hypercenter#2067D61
hypocenterC61#24515
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2067

Complements

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

NH ≅ G/Ntype
2. C2×#16839822C2semidirect
3. #21451621C2semidirect
5. #21447211C2semidirect
6. #21447142C2semidirect
7. #21447136C2semidirect
8. #21446701C2semidirect
9. #21446678C2semidirect
10. #21446507C2semidirect
11. #21446499C2semidirect
12. #21446125C2semidirect
13. #21446005C2semidirect
15. C61×#2067C2semidirect
355. #2067D61semidirect
419. C61#24515semidirect

Maximal subgroups

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

subgroupindex
C2×#168399252
C2×#168398222
#214516212
#214484092
#214472112
#214471422
#214471362
#214467012
#214466782
#214465072
#214464992
#214461252
#214460052
#214457562
C61×#20672
#2451561