Finite Group Databasebeta

← #29718953

Subgroups — #29718953

Order 15616.

874 conjugacy classes of subgroups.

The subgroup lattice has 560 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455760C211yesmaximal, characteristic
2C2×#16839901C211yesmaximal, characteristic
3#21454479C211yesmaximal, characteristic, complemented
4C2×#16839771C211yesmaximal, characteristic, complemented
5#21447182C212yesmaximal, complemented
6#21447078C212yesmaximal, complemented
7#21446723C212yesmaximal, complemented
8#21446476C212yesmaximal, complemented
9#21446280C211yesmaximal, characteristic
10#21446023C211yesmaximal, characteristic, complemented
11C61×#1784C211yesmaximal, characteristic, complemented
12C22×#11758201C2212yes—
13C22×#11758194C2211yescharacteristic
14C2×#14519751C2211yescharacteristic
15C2×#14519736C2211yescharacteristic
16#16840087—21no—
17#16840087—21no—
18#16840081—21no—
19#16840076—21no—
20#16839901—21no—
21#16839901—21no—
22#16839901—22no—
23#16839820—22no—
24#16839816—21no—
25#16839812—21no—
26C2×#14519712C2212yes—
27#16839804C2211yescharacteristic
28#16839804C2211yescharacteristic
29#16839790C2212yes—
30#16839771—21no—
31#16839771—21no—
32#16839771C2212yes—
33#16839771C2212yes—
34#16839768C2211yescharacteristic
35#16839768C2211yescharacteristic
36C2×#14519701—21no—
37C2×#14519701—21no—
38C2×#14519701C2212yes—
39C2×#14519693C2212yes—
40C2×C22⋊#92418C2211yescharacteristic
41#16839720C2212yes—
42#16839530C2212yes—
43#16839526C2212yes—
44C23×Q8×C61C2211yescharacteristic
45C2×#101×C61C2212yes—
46C2×#101×C61C2212yes—
47C61×#407C2211yescharacteristic
48C61×#407C2211yescharacteristic
49C22×D244C2311yescharacteristic
50C22×C22⋊D61—21no—
51C2×#11758201—24no—
52C2×#11758201—22no—
53C2×#11758201—22no—
54C2×#11758201—42no—
55C2×#11758194—21no—
56C2×#11758194—42no—
57C2×#11758194C2312yes—
58C22×#92418—21no—
59C22×#92418C2312yes—
60#14519751—21no—
61#14519751—21no—
62#14519751—21no—
63#14519751—21no—
64#14519736—21no—
65#14519736—21no—
66#14519736D411yescharacteristic
67#14519736D411yescharacteristic
68#14519736C2311yescharacteristic
69#14519736C2311yescharacteristic
70C2×C4⋊Dic61D411yescharacteristic
71C2×C4⋊Dic61D411yescharacteristic
72C2×C4⋊Dic61C2311yescharacteristic
73C2×C22⋊Dic61—21no—
74C2×#11758198—21no—
75C2×C4×Dic61—21no—
76#14519712—42no—
77#14519711—21no—
78#14519711—21no—
79#14519709—21no—
80#14519709—21no—
81#14519706—22no—
82#14519706—22no—
83#14519701—22no—
84#14519701—22no—
85#14519701—22no—
86#14519701—22no—
87#14519701—22no—
88#14519701—22no—
89#14519699—21no—
90#14519699—21no—
91#14519699—21no—
92#14519699—21no—
93#14519693—42no—
94C22⋊#92418D411yescharacteristic
95C22⋊#92418D411yescharacteristic
96C22⋊#92418D412yes—
97#14519687—22no—
98C8×Dic61—22no—
99C22×Q8×C61—21no—
100C22×Q8×C61—24no—
101C22×Q8×C61—21no—
102C22×Q8×C61—21no—
103C22×Q8×C61—21no—
104C22×Q8×C61—21no—
105C22×Q8×C61—42no—
106C22×Q8×C61C2312yes—
107C23×C4×C61—21no—
108C23×C4×C61C2311yescharacteristic
109C2×M16×C61C2312yes—
110C2×M16×C61C2311yescharacteristic
111C2×M16×C61C2311yescharacteristic
112#101×C61—24no—
113#101×C61—42no—
114C22⋊C8×C61—21no—
115C22⋊C8×C61—21no—
116C22⋊C8×C61—21no—
117C22⋊C8×C61—21no—
118C24×C61#4111yescharacteristic
119C23×D61—21no—
120C2×Q8×C61—22no—
121C2×Q8×C61—24no—
122C2×Q8×C61—24no—
123C2×Q8×C61—22no—
124C2×Q8×C61—22no—
125C2×Q8×C61—22no—
126C2×Q8×C61—22no—
127C2×Q8×C61—44no—
128C2×Q8×C61—44no—
129C2×Q8×C61—44no—
130C2×Q8×C61—42no—
131C2×Q8×C61—42no—
132C2×Q8×C61—41no—
133C2×Q8×C61—41no—
134C2×Q8×C61#4114yes—
135C22×C4×C61—22no—
136C22×C4×C61—22no—
137C22×C4×C61—22no—
138C22×C4×C61—44no—
139C22×C4×C61—41no—
140C22×C4×C61—41no—
141C22×C4×C61C2×D411yescharacteristic
142C22×C4×C61C2×D411yescharacteristic
143C22×C4×C61C2×D411yescharacteristic
144C22×C4×C61C2×D411yescharacteristic
145C22×C4×C61#4111yescharacteristic
146C22×C4×C61C2411yescharacteristic
147C2×C22⋊D61—41no—
148C2×C22⋊D61—41no—
149C2×C22⋊D61—82no—
150C22×Dic61—21no—
151C22×Dic61—21no—
152C22×Dic61—21no—
153C2×D244—21no—
154C2×D244—21no—
155C2×D244—22no—
156C2×D244—22no—
157M16×C61—22no—
158M16×C61—22no—
159M16×C61—22no—
160M16×C61—22no—
161C2×C8×C61—22no—
162C2×C8×C61—21no—
163C2×C8×C61—21no—
164C22⋊Dic61—21no—
165C22⋊Dic61—21no—
166C22⋊Dic61—21no—
167C22⋊Dic61—21no—
168#11758201—44no—
169#11758201—44no—
170#11758198—41no—
171#11758198—41no—
172C4⋊Dic61—21no—
173C4⋊Dic61—21no—
174C4⋊Dic61—21no—
175C4⋊Dic61—21no—
176C4⋊Dic61C2×D411yescharacteristic
177C4⋊Dic61C2×D411yescharacteristic
178C4⋊Dic61C2×D411yescharacteristic
179C4⋊Dic61C2×D411yescharacteristic
180C4×Dic61—21no—
181C4×Dic61—21no—
182C4×Dic61—21no—
183C4×Dic61—21no—
184#11758194—22no—
185#11758194—22no—
186#11758194—42no—
187C2×#92418—22no—
188C2×#92418—22no—
189C2×#92418—22no—
190C2×#92418—42no—
191C2×#92418C2×D412yes—
192C2×#92418C2×D412yes—
193C23×C61—22no—
194C23×C61—22no—
195C23×C61—41no—
196C23×C61#12111yescharacteristic
197C23×C61#12111yescharacteristic
198C23×C61C2×#4111yescharacteristic
199C22×D61—41no—
200C22×D61—42no—
201C22×D61—81no—
202C22×D61—81no—
203Q8×C61—24no—
204Q8×C61—24no—
205Q8×C61—44no—
206Q8×C61—44no—
207Q8×C61—42no—
208Q8×C61—42no—
209C2×C4×C61—22no—
210C2×C4×C61—21no—
211C2×C4×C61—21no—
212C2×C4×C61—21no—
213C2×C4×C61—21no—
214C2×C4×C61—22no—
215C2×C4×C61—22no—
216C2×C4×C61—24no—
217C2×C4×C61—44no—
218C2×C4×C61—44no—
219C2×C4×C61—42no—
220C2×C4×C61—42no—
221C2×C4×C61#12111yescharacteristic
222C2×C4×C61#12111yescharacteristic
223C2×C4×C61C22×D411yescharacteristic
224C2×C4×C61C22×D411yescharacteristic
225C2×C4×C61C2×#4112yes—
226C2×C4×C61H(5,𝔽2)12yes—
227C22⋊D61—82no—
228C22⋊D61—82no—
229C2×Dic61—21no—
230C2×Dic61—21no—
231C2×Dic61—21no—
232C2×Dic61—21no—
233C2×Dic61—42no—
234C2×Dic61—41no—
235C2×Dic61—81no—
236C2×Dic61—81no—
237D244—41no—
238D244—41no—
239D244—42no—
240C8×C61—41no—
241C8×C61—41no—
242C8×C61—42no—
243#92418—22no—
244#92418—22no—
245#92418—42no—
246#24523—611nomaximal, Sylow
247C22×C61—21no—
248C22×C61—21no—
249C22×C61—21no—
250C22×C61—21no—
251C22×C61—21no—
252C22×C61—21no—
253C22×C61—21no—
254C22×C61—44no—
255C22×C61#47112yes—
256C22×C61C2×#12111yescharacteristic
257C22×C61#55112yes—
258C2×D61—81no—
259C2×D61—81no—
260C2×D61—81no—
261C2×D61—162no—
262C4×C61—21no—
263C4×C61—21no—
264C4×C61—44no—
265C4×C61#52511yescharacteristic
266C4×C61#53511yescharacteristic
267C4×C61#54612yes—
268Dic61—41no—
269Dic61—41no—
270Dic61—41no—
271Dic61—41no—
272Dic61—81no—
273#3094—611no—
274C2×#448—611no—
275#2924—611no—
276C2×#419—611no—
277#2064—612no—
278#2046—612no—
279#1977—612no—
280#1930—612no—
281#1900—611no—
282#1855—611no—
283#1784D6111yescharacteristic, complemented
284C2×C61—22no—
285C2×C61—22no—
286C2×C61—41no—
287C2×C61#241111yescharacteristic
288C2×C61C2×#47111yescharacteristic
289C2×C61#306911yescharacteristic
290D61—161no—
291C23×Q8C2×D6111yescharacteristic
292C22×SD16—612no—
293C22×M16—611no—
294C2×#121—611no—
295C2×#117—611no—
296#482—1221no—
297#482—1221no—
298#481—1221no—
299#480—1221no—
300#448—1221no—
301#448—1221no—
302#448—1222no—
303#433—1222no—
304#432—1221no—
305#431—1221no—
306C2×#108—612no—
307#428—611no—
308#428—611no—
309#424—612no—
310#419—612no—
311#419—612no—
312#419—1221no—
313#419—1221no—
314#418—611no—
315#418—611no—
316C2×#103—612no—
317C2×#103—1221no—
318C2×#103—1221no—
319C2×#101C2×D6112yes—
320C2×#101C2×D6112yes—
321C2×#100—612no—
322#407C2×D6111yescharacteristic
323#407C2×D6111yescharacteristic
324C2×C22⋊C8—611no—
325#405—612no—
326#343—612no—
327#341—612no—
328C61#2452311yescharacteristic, complemented, Sylow
329C22×Q8—21no—
330C22×Q8—24no—
331C22×Q8—21no—
332C22×Q8—21no—
333C22×Q8—21no—
334C22×Q8—21no—
335C22×Q8—42no—
336C22×Q8C22×D6112yes—
337C22×D4—611no—
338C22×D4—1221no—
339C23×C4—21no—
340C23×C4C22×D6111yescharacteristic
341C2×SD16—1222no—
342C2×SD16—1222no—
343C2×SD16—1224no—
344C2×SD16—2442no—
345C2×M16—612no—
346C2×M16—1221no—
347C2×M16—2442no—
348C2×M16C22×D6111yescharacteristic
349C2×M16C22×D6112yes—
350C2×M16C22×D6111yescharacteristic
351C22×C8—612no—
352C22×C8—1221no—
353#121—1221no—
354#121—1221no—
355#121—1221no—
356#121—1221no—
357#117—611no—
358#117—611no—
359#117—611no—
360#117—611no—
361#117—1221no—
362#117—1221no—
363C2×C4⋊C4—611no—
364C2×C4⋊C4—611no—
365C2×C4⋊C4—611no—
366C2×C22⋊C4—1221no—
367C2×C22⋊C4—1221no—
368C2×C42—1221no—
369#108—2442no—
370#107—1221no—
371#107—1221no—
372#106—1221no—
373#106—1221no—
374C4⋊C8—1222no—
375C4⋊C8—1222no—
376#103—1222no—
377#103—1222no—
378#103—1222no—
379#103—1222no—
380#103—1222no—
381#103—1222no—
382#102—1221no—
383#102—1221no—
384#102—1221no—
385#102—1221no—
386#101—24no—
387#101—42no—
388#100—2442no—
389C22⋊C8—21no—
390C22⋊C8—21no—
391C22⋊C8—21no—
392C22⋊C8—21no—
393C22⋊C8—611no—
394C22⋊C8—611no—
395C22⋊C8—612no—
396#97—1222no—
397C4×C8—1222no—
398C24—1221no—
399C24#1175822211yescharacteristic
400C2×Q8—22no—
401C2×Q8—24no—
402C2×Q8—24no—
403C2×Q8—22no—
404C2×Q8—22no—
405C2×Q8—22no—
406C2×Q8—22no—
407C2×Q8—44no—
408C2×Q8—44no—
409C2×Q8—44no—
410C2×Q8—42no—
411C2×Q8—42no—
412C2×Q8—41no—
413C2×Q8—41no—
414C2×Q8#1175822414yes—
415C2×D4—1221no—
416C2×D4—1221no—
417C2×D4—1222no—
418C2×D4—1222no—
419C2×D4—4882no—
420C2×D4—2441no—
421C2×D4—2441no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—44no—
426C22×C4—41no—
427C22×C4—41no—
428C22×C4—1221no—
429C22×C4—1221no—
430C22×C4—1221no—
431C22×C4#1175822211yescharacteristic
432C22×C4D4×D6111yescharacteristic
433C22×C4D4×D6111yescharacteristic
434C22×C4D4×D6111yescharacteristic
435C22×C4D4×D6111yescharacteristic
436C22×C4C23×D6111yescharacteristic
437SD16—2444no—
438SD16—2444no—
439M16—22no—
440M16—22no—
441M16—22no—
442M16—22no—
443M16—1222no—
444M16—1222no—
445M16—2442no—
446C2×C8—22no—
447C2×C8—21no—
448C2×C8—21no—
449C2×C8—612no—
450C2×C8—612no—
451C2×C8—1222no—
452C2×C8—1222no—
453C2×C8—1222no—
454C2×C8—2442no—
455C4⋊C4—611no—
456C4⋊C4—611no—
457C4⋊C4—611no—
458C4⋊C4—611no—
459C4⋊C4—1221no—
460C4⋊C4—1221no—
461C4⋊C4—1221no—
462C4⋊C4—1221no—
463C22⋊C4—1221no—
464C22⋊C4—1221no—
465C22⋊C4—1221no—
466C22⋊C4—1221no—
467C22⋊C4—2441no—
468C22⋊C4—2441no—
469C42—1221no—
470C42—1221no—
471C42—1221no—
472C42—1221no—
473C23—22no—
474C23—22no—
475C23—41no—
476C23—4881no—
477C23—4881no—
478C23—2442no—
479C23—2441no—
480C23#1451975011yescharacteristic
481C23#1451975011yescharacteristic
482C23C2×#1175822211yescharacteristic
483Q8—24no—
484Q8—24no—
485Q8—44no—
486Q8—44no—
487Q8—42no—
488Q8—42no—
489D4—4882no—
490D4—4882no—
491D4—2441no—
492D4—2441no—
493D4—2442no—
494C2×C4—22no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—24no—
502C2×C4—44no—
503C2×C4—44no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—1221no—
507C2×C4—1221no—
508C2×C4—1221no—
509C2×C4—1221no—
510C2×C4—4881no—
511C2×C4—4881no—
512C2×C4—2441no—
513C2×C4—2442no—
514C2×C4#1451975011yescharacteristic
515C2×C4#1451975011yescharacteristic
516C2×C4C2×D4×D6111yescharacteristic
517C2×C4C2×D4×D6111yescharacteristic
518C2×C4C2×#1175822412yes—
519C2×C4#1451982212yes—
520C8—41no—
521C8—41no—
522C8—42no—
523C8—1222no—
524C8—1222no—
525C8—2442no—
526C22—21no—
527C22—21no—
528C22—21no—
529C22—21no—
530C22—21no—
531C22—21no—
532C22—21no—
533C22—44no—
534C22—9762no—
535C22—4881no—
536C22—4881no—
537C22—4881no—
538C22#1684002812yes—
539C22C2×#1451975011yescharacteristic
540C22#1684043812yes—
541C4—21no—
542C4—21no—
543C4—44no—
544C4—4881no—
545C4—2441no—
546C4—2441no—
547C4—2441no—
548C4—2441no—
549C4#1684026811yescharacteristic
550C4#1684032711yescharacteristic
551C4#1684039912yes—
552C2—22no—
553C2—22no—
554C2—41no—
555C2—9761no—
556C2#2144920611yescharacteristic
557C2C2×#1684002811yescharacteristic
558C2#2145555511yescharacteristic
subgroupG/H
centerC22C2×#14519750
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1784C2
socleC22×C61C2×#121
prime radicalC2×#14519751C22
hypercenter#1784D61
hypocenterC61#24523
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1784

Complements

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

NH ≅ G/Ntype
3. #21454479C2semidirect
4. C2×#16839771C2semidirect
5. #21447182C2semidirect
6. #21447078C2semidirect
7. #21446723C2semidirect
8. #21446476C2semidirect
10. #21446023C2semidirect
11. C61×#1784C2semidirect
283. #1784D61semidirect
328. C61#24523semidirect

Maximal subgroups

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

subgroupindex
#214557602
C2×#168399012
#214544792
C2×#168397712
#214471822
#214470782
#214467232
#214464762
#214462802
#214460232
C61×#17842
#2452361