Finite Group Databasebeta

← #29718849

Subgroups — #29718849

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×#16839922C211yesmaximal, characteristic
2C2×#16839820C211yesmaximal, characteristic, complemented
3#21451617C211yesmaximal, characteristic, complemented
4#21448412C211yesmaximal, characteristic
5#21447215C211yesmaximal, characteristic, complemented
6#21447208C211yesmaximal, characteristic, complemented
7#21447140C211yesmaximal, characteristic, complemented
8#21447133C211yesmaximal, characteristic, complemented
9#21446697C211yesmaximal, characteristic, complemented
10#21446677C211yesmaximal, characteristic, complemented
11#21446507C211yesmaximal, characteristic, complemented
12#21446500C211yesmaximal, characteristic, complemented
13#21446007C211yesmaximal, characteristic, complemented
14#21445760C211yesmaximal, characteristic
15C61×#1872C211yesmaximal, characteristic, complemented
16C22×#11758222C2211yescharacteristic
17C22×#11758202C2211yescharacteristic
18C22×#11758201C2211yescharacteristic
19C2×#14519736C2211yescharacteristic
20#16839922—24no—
21#16839820—24no—
22C2×#14519710C2211yescharacteristic
23C2×#14519710C2211yescharacteristic
24C2×#14519706C2211yescharacteristic
25C2×#14519706C2211yescharacteristic
26C2×#14519702C2211yescharacteristic
27C2×#14519702C2211yescharacteristic
28C2×#14519700C2211yescharacteristic
29C2×#14519699C2211yescharacteristic
30C2×#14519699C2211yescharacteristic
31C2×#14519697C2211yescharacteristic
32C2×C22⋊#92418C2211yescharacteristic
33C2×#14519687C2211yescharacteristic
34C2×C8×Dic61C2211yescharacteristic
35#16839701—21no—
36#16839692—21no—
37#16839690—21no—
38#16839690—21no—
39#16839669C2211yescharacteristic
40#16839669C2211yescharacteristic
41#16839667C2211yescharacteristic
42#16839667C2211yescharacteristic
43#16839663C2211yescharacteristic
44#16839663C2211yescharacteristic
45#16839658C2211yescharacteristic
46#16839658C2211yescharacteristic
47#16839641—21no—
48#16839641—21no—
49#16839622C2211yescharacteristic
50#16839619—21no—
51#16839619—21no—
52#16839524C2211yescharacteristic
53#16839524C2211yescharacteristic
54C2×C22⋊Q8×C61C2211yescharacteristic
55C22×C4⋊C4×C61C2211yescharacteristic
56C2×#103×C61C2211yescharacteristic
57C2×#103×C61C2211yescharacteristic
58C2×C22⋊C8×C61C2211yescharacteristic
59C61×#340C2211yescharacteristic
60C61×#340C2211yescharacteristic
61C2×#11758222—22no—
62C2×#11758222—22no—
63C2×#11758222—22no—
64C2×#11758222—22no—
65C2×#11758222—41no—
66C2×#11758222—41no—
67C22×Dic122C2311yescharacteristic
68C22×D244C2311yescharacteristic
69C22×C22⋊D61—21no—
70C22×C4×D61—21no—
71C2×#11758202—22no—
72C2×#11758202—22no—
73C2×#11758202—41no—
74C2×#11758201—22no—
75C2×#11758201—22no—
76C2×#11758201—41no—
77C22×#92418C2311yescharacteristic
78C22×#92418C2311yescharacteristic
79#14519736—22no—
80#14519736—22no—
81C2×C4⋊Dic61D411yescharacteristic
82C2×C4⋊Dic61D411yescharacteristic
83C2×C4⋊Dic61C2311yescharacteristic
84C2×C4⋊Dic61C2311yescharacteristic
85C2×#11758196—21no—
86C2×#11758196—21no—
87C2×C22⋊Dic61—21no—
88C2×C22⋊Dic61D411yescharacteristic
89C2×C22⋊Dic61D411yescharacteristic
90C2×#11758198—21no—
91C2×#11758198—21no—
92C2×#11758198—21no—
93C2×#11758198—21no—
94C2×C4×Dic61C2311yescharacteristic
95C2×C4×Dic61C2311yescharacteristic
96#14519710—41no—
97#14519710—41no—
98#14519706—22no—
99#14519706—22no—
100#14519702—22no—
101#14519702—41no—
102#14519700—22no—
103#14519699—22no—
104#14519699—41no—
105#14519697—22no—
106C22⋊#92418D414yes—
107#14519687—22no—
108C8×Dic61—22no—
109#14519684—21no—
110#14519684—21no—
111#14519684—21no—
112#14519684—21no—
113#14519684—21no—
114#14519684—21no—
115#14519684—21no—
116#14519684—21no—
117#14519684—21no—
118#14519684—21no—
119#14519684—21no—
120C22×Q8×C61C2311yescharacteristic
121C23×C4×C61—21no—
122C23×C4×C61C2311yescharacteristic
123C22×C8×C61C2311yescharacteristic
124C22×C8×C61C2311yescharacteristic
125C22⋊Q8×C61—22no—
126C22⋊Q8×C61—22no—
127C2×C4⋊C4×C61—21no—
128C2×C4⋊C4×C61—21no—
129C2×C4⋊C4×C61—41no—
130C2×C4⋊C4×C61—41no—
131C2×C4⋊C4×C61C2311yescharacteristic
132C2×C4⋊C4×C61C2311yescharacteristic
133C2×C4⋊C4×C61C2311yescharacteristic
134C2×C22⋊C4×C61—21no—
135#103×C61—41no—
136#103×C61—41no—
137C22⋊C8×C61—22no—
138C24×C61C2×D411yescharacteristic
139C23×D61—21no—
140C2×Q8×C61—21no—
141C2×Q8×C61—21no—
142C2×Q8×C61—21no—
143C2×Q8×C61—21no—
144C2×Q8×C61—21no—
145C2×Q8×C61—21no—
146C22×C4×C61—21no—
147C22×C4×C61—21no—
148C22×C4×C61—21no—
149C22×C4×C61—21no—
150C22×C4×C61—21no—
151C22×C4×C61—21no—
152C22×C4×C61—21no—
153C22×C4×C61—21no—
154C22×C4×C61—42no—
155C22×C4×C61—81no—
156C22×C4×C61—81no—
157C22×C4×C61C2×D411yescharacteristic
158C22×C4×C61C2×D412yes—
159C22×C4×C61#4112yes—
160C22×C4×C61C2411yescharacteristic
161C2×C22⋊D61—24no—
162C2×C22⋊D61—42no—
163C2×C22⋊D61—42no—
164C22×Dic61—21no—
165C22×Dic61—21no—
166C22×Dic61—21no—
167C22×Dic61C2×D411yescharacteristic
168C22×Dic61C2×D411yescharacteristic
169C22×Dic61C2×D411yescharacteristic
170C22×Dic61C2×D411yescharacteristic
171#11758222—44no—
172#11758222—44no—
173#11758222—44no—
174#11758222—44no—
175C2×D244—21no—
176C2×D244—21no—
177C2×D244—21no—
178C2×D244—21no—
179C2×D244—21no—
180C2×D244#4112yes—
181C2×C4×D61—24no—
182C2×C4×D61—42no—
183C2×C4×D61—42no—
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×C61—21no—
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—42no—
203C4⋊C4×C61—42no—
204C22⋊C4×C61—42no—
205C22⋊Dic61—22no—
206C22⋊Dic61—22no—
207C22⋊Dic61—81no—
208#11758202—44no—
209#11758201—44no—
210#11758198—81no—
211#11758198—81no—
212#11758198—81no—
213#11758198—81no—
214C4⋊Dic61—22no—
215C4⋊Dic61—22no—
216C4⋊Dic61—21no—
217C4⋊Dic61—21no—
218C4⋊Dic61—21no—
219C4⋊Dic61—21no—
220#11758196—81no—
221#11758196—81no—
222C4×Dic61—21no—
223C4×Dic61—21no—
224C4×Dic61—21no—
225C4×Dic61—21no—
226C2×#92418—21no—
227C2×#92418—21no—
228C2×#92418—21no—
229C2×#92418—21no—
230C2×#92418C2×D412yes—
231C2×#92418C2×D412yes—
232C23×C61—21no—
233C23×C61—21no—
234C23×C61—21no—
235C23×C61—21no—
236C23×C61—21no—
237C23×C61—21no—
238C23×C61#12112yes—
239C23×C61C22×D411yescharacteristic
240C22×D61—22no—
241C22×D61—41no—
242C22×D61—41no—
243C22×D61—41no—
244C22×D61—41no—
245C22×D61—41no—
246C22×D61—41no—
247Q8×C61—41no—
248Q8×C61—41no—
249Q8×C61—41no—
250Q8×C61—41no—
251C2×C4×C61—21no—
252C2×C4×C61—21no—
253C2×C4×C61—22no—
254C2×C4×C61—22no—
255C2×C4×C61—22no—
256C2×C4×C61—22no—
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×C61—41no—
267C2×C4×C61—41no—
268C2×C4×C61—41no—
269C2×C4×C61—84no—
270C2×C4×C61#12112yes—
271C2×C4×C61C22×D411yescharacteristic
272C2×C4×C61C2×#4111yescharacteristic
273C2×C4×C61C2×#4111yescharacteristic
274C2×C4×C61C2×#4111yescharacteristic
275C2×C4×C61H(5,𝔽2)11yescharacteristic
276C2×C4×C61H(5,𝔽2)11yescharacteristic
277C22⋊D61—48no—
278C2×Dic61—21no—
279C2×Dic61—21no—
280C2×Dic61—21no—
281C2×Dic61—21no—
282C2×Dic61—22no—
283C2×Dic61—21no—
284C2×Dic61—21no—
285C2×Dic61—21no—
286C2×Dic61—21no—
287C2×Dic61—21no—
288C2×Dic61—21no—
289C2×Dic61—21no—
290C2×Dic61—21no—
291C2×Dic61—41no—
292C2×Dic61—41no—
293C2×Dic61—41no—
294C2×Dic61—41no—
295C2×Dic61—41no—
296C2×Dic61—41no—
297C2×Dic61—41no—
298C2×Dic61—41no—
299D244—22no—
300D244—22no—
301D244—41no—
302D244—41no—
303C4×D61—48no—
304Dic122—22no—
305Dic122—22no—
306Dic122—41no—
307Dic122—41no—
308C8×C61—41no—
309C8×C61—41no—
310#92418—22no—
311#92418—22no—
312#24515—611nomaximal, Sylow
313C22×C61—22no—
314C22×C61—22no—
315C22×C61—22no—
316C22×C61—41no—
317C22×C61—41no—
318C22×C61—41no—
319C22×C61—41no—
320C22×C61C2×#12111yescharacteristic
321C22×C61#52511yescharacteristic
322C22×C61#53511yescharacteristic
323C22×C61#57611yescharacteristic
324C22×C61#57711yescharacteristic
325C22×C61#57811yescharacteristic
326C22×C61#57811yescharacteristic
327C2×D61—42no—
328C2×D61—42no—
329C2×D61—42no—
330C2×D61—81no—
331C2×D61—81no—
332C2×D61—81no—
333C2×D61—81no—
334C4×C61—22no—
335C4×C61—81no—
336C4×C61—81no—
337C4×C61—81no—
338C4×C61—81no—
339C4×C61#54611yescharacteristic
340C4×C61#54611yescharacteristic
341C4×C61#55111yescharacteristic
342C4×C61#55111yescharacteristic
343Dic61—42no—
344Dic61—41no—
345Dic61—41no—
346Dic61—41no—
347Dic61—41no—
348Dic61—81no—
349Dic61—81no—
350C2×#452—611no—
351C2×#433—611no—
352#2625—611no—
353#2311—611no—
354#2068—611no—
355#2067—611no—
356#2056—611no—
357#2055—611no—
358#1973—611no—
359#1969—611no—
360#1935—611no—
361#1934—611no—
362#1872D6111yescharacteristic, complemented
363#1852—611no—
364#1788—611no—
365C2×C61—42no—
366C2×C61#241111yescharacteristic
367C2×C61#306111yescharacteristic
368C2×C61#306311yescharacteristic
369C2×C61#312011yescharacteristic
370C2×C61#312411yescharacteristic
371C2×C61#322311yescharacteristic
372C2×C61#322411yescharacteristic
373D61—82no—
374C22×#41—611no—
375C22×Q16—611no—
376C22×SD16—611no—
377C2×C22⋊Q8C2×D6111yescharacteristic
378C2×#117—611no—
379C22×C4⋊C4C2×D6111yescharacteristic
380#452—1224no—
381#433—1224no—
382C2×#107—611no—
383C2×#107—611no—
384C2×C4⋊C8—611no—
385C2×C4⋊C8—611no—
386C2×#103—611no—
387C2×#103—611no—
388C2×#103—611no—
389C2×#103C2×D6111yescharacteristic
390C2×#103C2×D6111yescharacteristic
391C2×#102—611no—
392C2×#102—611no—
393C2×#102—611no—
394C2×C22⋊C8—611no—
395C2×C22⋊C8C2×D6111yescharacteristic
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#390—611no—
408#389—611no—
409#389—611no—
410#386—1221no—
411#386—1221no—
412#380—611no—
413#380—1221no—
414#380—1221no—
415#340—611no—
416#340—611no—
417#340C2×D6111yescharacteristic
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×Q8C22×D6111yescharacteristic
428C22×D4—611no—
429C22×D4—1221no—
430C23×C4—21no—
431C23×C4—1221no—
432C23×C4C22×D6111yescharacteristic
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—22no—
444C22⋊Q8—22no—
445#117—1222no—
446#117—1222no—
447C2×C4⋊C4—21no—
448C2×C4⋊C4—21no—
449C2×C4⋊C4—41no—
450C2×C4⋊C4—41no—
451C2×C4⋊C4—611no—
452C2×C4⋊C4—611no—
453C2×C4⋊C4—611no—
454C2×C4⋊C4—611no—
455C2×C4⋊C4—1221no—
456C2×C4⋊C4—1221no—
457C2×C4⋊C4C22×D6111yescharacteristic
458C2×C4⋊C4C22×D6111yescharacteristic
459C2×C4⋊C4C22×D6111yescharacteristic
460C2×C22⋊C4—21no—
461C2×C22⋊C4—611no—
462C2×C22⋊C4—611no—
463C2×C22⋊C4—1221no—
464C2×C22⋊C4—1221no—
465C2×C22⋊C4—1221no—
466C2×C22⋊C4—1221no—
467C2×C22⋊C4—1221no—
468C2×C42—611no—
469C2×C42—611no—
470#107—2441no—
471#107—2441no—
472C4⋊C8—1222no—
473C4⋊C8—1222no—
474#103—41no—
475#103—41no—
476#103—1222no—
477#103—1222no—
478#103—2441no—
479#102—1222no—
480#102—1222no—
481#102—2441no—
482C22⋊C8—22no—
483C22⋊C8—614no—
484#97—1222no—
485C4×C8—1222no—
486#95—1221no—
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—1221no—
498C24D4×D6111yescharacteristic
499#41—2444no—
500#41—2444no—
501#41—2444no—
502#41—2444no—
503C2×Q8—21no—
504C2×Q8—21no—
505C2×Q8—21no—
506C2×Q8—21no—
507C2×Q8—21no—
508C2×Q8—21no—
509C2×Q8—612no—
510C2×Q8—1221no—
511C2×Q8—1221no—
512C2×Q8—1221no—
513C2×Q8—1221no—
514C2×Q8—1221no—
515C2×D4—612no—
516C2×D4—1224no—
517C2×D4—1221no—
518C2×D4—1221no—
519C2×D4—1221no—
520C2×D4—1221no—
521C2×D4—1221no—
522C2×D4—2442no—
523C2×D4—2442no—
524C22×C4—21no—
525C22×C4—21no—
526C22×C4—21no—
527C22×C4—21no—
528C22×C4—21no—
529C22×C4—21no—
530C22×C4—21no—
531C22×C4—21no—
532C22×C4—42no—
533C22×C4—81no—
534C22×C4—81no—
535C22×C4—611no—
536C22×C4—611no—
537C22×C4—611no—
538C22×C4—611no—
539C22×C4—1224no—
540C22×C4—1221no—
541C22×C4—1221no—
542C22×C4—1221no—
543C22×C4—2442no—
544C22×C4—2442no—
545C22×C4#1175822212yes—
546C22×C4D4×D6112yes—
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—21no—
557C2×C8—612no—
558C2×C8—612no—
559C2×C8—1221no—
560C2×C8—1221no—
561C2×C8—1221no—
562C2×C8—1221no—
563C4⋊C4—21no—
564C4⋊C4—21no—
565C4⋊C4—21no—
566C4⋊C4—21no—
567C4⋊C4—21no—
568C4⋊C4—21no—
569C4⋊C4—42no—
570C4⋊C4—42no—
571C4⋊C4—1222no—
572C4⋊C4—1222no—
573C4⋊C4—1221no—
574C4⋊C4—1221no—
575C4⋊C4—1221no—
576C4⋊C4—1221no—
577C4⋊C4—4881no—
578C4⋊C4—4881no—
579C22⋊C4—42no—
580C22⋊C4—1222no—
581C22⋊C4—1222no—
582C22⋊C4—4881no—
583C22⋊C4—4881no—
584C22⋊C4—4881no—
585C22⋊C4—4881no—
586C22⋊C4—4881no—
587C42—1221no—
588C42—1221no—
589C42—1221no—
590C42—1221no—
591C23—21no—
592C23—21no—
593C23—21no—
594C23—21no—
595C23—21no—
596C23—21no—
597C23—1222no—
598C23—2441no—
599C23—2441no—
600C23—2441no—
601C23—2441no—
602C23—2441no—
603C23—2441no—
604C23#1451975012yes—
605C23C2×D4×D6111yescharacteristic
606Q8—41no—
607Q8—41no—
608Q8—41no—
609Q8—41no—
610Q8—1222no—
611Q8—1222no—
612Q8—2441no—
613Q8—2441no—
614D4—1222no—
615D4—1222no—
616D4—2448no—
617D4—2441no—
618D4—2441no—
619C2×C4—21no—
620C2×C4—21no—
621C2×C4—22no—
622C2×C4—22no—
623C2×C4—22no—
624C2×C4—22no—
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—41no—
635C2×C4—41no—
636C2×C4—41no—
637C2×C4—84no—
638C2×C4—1221no—
639C2×C4—1221no—
640C2×C4—1221no—
641C2×C4—1221no—
642C2×C4—1222no—
643C2×C4—1221no—
644C2×C4—1221no—
645C2×C4—1221no—
646C2×C4—1221no—
647C2×C4—1221no—
648C2×C4—1221no—
649C2×C4—1221no—
650C2×C4—1221no—
651C2×C4—2448no—
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#1451975012yes—
661C2×C4C2×D4×D6111yescharacteristic
662C2×C4#41×D6111yescharacteristic
663C2×C4#41×D6111yescharacteristic
664C2×C4C2×#1175822211yescharacteristic
665C2×C4#1451982211yescharacteristic
666C2×C4#1451982211yescharacteristic
667C8—41no—
668C8—41no—
669C8—1222no—
670C8—1222no—
671C22—22no—
672C22—22no—
673C22—22no—
674C22—41no—
675C22—41no—
676C22—41no—
677C22—41no—
678C22—4881no—
679C22—4881no—
680C22—4881no—
681C22—4881no—
682C22—2442no—
683C22—2442no—
684C22—2442no—
685C22C2×#1451975011yescharacteristic
686C22#1684027011yescharacteristic
687C22#1684032711yescharacteristic
688C22#1684056611yescharacteristic
689C22#1684056811yescharacteristic
690C22#1684057511yescharacteristic
691C22#1684057811yescharacteristic
692C4—22no—
693C4—81no—
694C4—81no—
695C4—81no—
696C4—81no—
697C4—4881no—
698C4—4881no—
699C4—2441no—
700C4—2441no—
701C4—2441no—
702C4—2441no—
703C4—2442no—
704C4#1684040111yescharacteristic
705C4#1684040111yescharacteristic
706C4#1684044111yescharacteristic
707C4#1684044111yescharacteristic
708C2—42no—
709C2—4882no—
710C2#2144920711yescharacteristic
711C2#2145548811yescharacteristic
712C2#2145550811yescharacteristic
713C2#2145603611yescharacteristic
714C2#2145608411yescharacteristic
715C2#2145689811yescharacteristic
716C2#2145691711yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1872C2
socleC23×C61C22×D4
prime radicalC22×#11758222C22
hypercenter#1872D61
hypocenterC61#24515
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1872

Complements

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

NH ≅ G/Ntype
2. C2×#16839820C2semidirect
3. #21451617C2semidirect
5. #21447215C2semidirect
6. #21447208C2semidirect
7. #21447140C2semidirect
8. #21447133C2semidirect
9. #21446697C2semidirect
10. #21446677C2semidirect
11. #21446507C2semidirect
12. #21446500C2semidirect
13. #21446007C2semidirect
15. C61×#1872C2semidirect
362. #1872D61semidirect
419. C61#24515semidirect

Maximal subgroups

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

subgroupindex
C2×#168399222
C2×#168398202
#214516172
#214484122
#214472152
#214472082
#214471402
#214471332
#214466972
#214466772
#214465072
#214465002
#214460072
#214457602
C61×#18722
#2451561