Finite Group Databasebeta

← #29718951

Subgroups — #29718951

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#21455759C211yesmaximal, characteristic
2C2×#16839901C211yesmaximal, characteristic
3#21454477C211yesmaximal, characteristic, complemented
4#21447186C212yesmaximal, complemented
5#21447079C212yesmaximal, complemented
6#21446721C212yesmaximal, complemented
7#21446477C212yesmaximal, complemented
8#21446282C211yesmaximal, characteristic
9#21446024C211yesmaximal, characteristic, complemented
10#21445743C211yesmaximal, characteristic, complemented
11C2×C61×#419C211yesmaximal, characteristic, complemented
12C22×#11758201C2212yes—
13C2×#14519751C2211yescharacteristic
14#16840086—21no—
15#16840086—21no—
16#16840080—21no—
17#16840075—21no—
18#16839901—21no—
19#16839901—21no—
20#16839901—22no—
21#16839823—22no—
22#16839819—21no—
23#16839815—21no—
24C2×#14519712C2212yes—
25#16839805C2211yescharacteristic
26#16839805C2211yescharacteristic
27#16839789C2212yes—
28#16839766C2211yescharacteristic
29#16839766C2211yescharacteristic
30C2×#14519701C2212yes—
31C2×#14519696C2212yes—
32C2×#14519696C2212yes—
33C2×#14519693C2212yes—
34#16839726C2211yescharacteristic
35#16839726C2211yescharacteristic
36C2×C22⋊#92418C2211yescharacteristic
37#16839718C2212yes—
38#16839530C2212yes—
39#16839526C2212yes—
40C23×Q8×C61C2211yescharacteristic
41C22×M16×C61C2211yescharacteristic
42C2×#117×C61C2211yescharacteristic
43C61×#419—21no—
44C61×#419—21no—
45C61×#419C2212yes—
46C61×#419C2212yes—
47C2×#103×C61—21no—
48C2×#103×C61—21no—
49C22×D244C2311yescharacteristic
50C22×C22⋊D61—21no—
51C2×#11758201—24no—
52C2×#11758201—22no—
53C2×#11758201—22no—
54C2×#11758201—42no—
55C2×#11758194C2312yes—
56C2×#11758194C2311yescharacteristic
57C2×#11758194C2311yescharacteristic
58C22×#92418C2312yes—
59#14519751—21no—
60#14519751—21no—
61#14519751—21no—
62#14519751—21no—
63C2×C4⋊Dic61C2311yescharacteristic
64C2×#11758198—21no—
65#14519712—42no—
66#14519710—21no—
67#14519710—21no—
68#14519708—21no—
69#14519708—21no—
70C4⋊#92418—22no—
71C4⋊#92418—22no—
72#14519701—22no—
73#14519701—22no—
74#14519697—21no—
75#14519697—21no—
76#14519697—21no—
77#14519697—21no—
78#14519696—24no—
79#14519696—42no—
80#14519693—42no—
81C22⋊#92418—21no—
82C22⋊#92418—21no—
83C22⋊#92418—21no—
84C22⋊#92418—21no—
85C22⋊#92418D411yescharacteristic
86C22⋊#92418D411yescharacteristic
87C22⋊#92418D412yes—
88#14519688—22no—
89C4×#92418—22no—
90C22×Q8×C61—21no—
91C22×Q8×C61—24no—
92C22×Q8×C61—21no—
93C22×Q8×C61—21no—
94C22×Q8×C61—21no—
95C22×Q8×C61—21no—
96C22×Q8×C61—42no—
97C22×Q8×C61C2312yes—
98C23×C4×C61—21no—
99C23×C4×C61C2311yescharacteristic
100C2×M16×C61—21no—
101C2×M16×C61—42no—
102C2×M16×C61C2312yes—
103C22×C8×C61—21no—
104#117×C61—21no—
105#117×C61—21no—
106#117×C61D411yescharacteristic
107#117×C61D411yescharacteristic
108#117×C61C2311yescharacteristic
109#117×C61C2311yescharacteristic
110C2×C4⋊C4×C61D411yescharacteristic
111C2×C4⋊C4×C61D411yescharacteristic
112C2×C22⋊C4×C61—21no—
113C2×C42×C61—21no—
114#103×C61—22no—
115#103×C61—22no—
116#103×C61—22no—
117#103×C61—22no—
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—21no—
138C22×C4×C61—22no—
139C22×C4×C61—21no—
140C22×C4×C61—44no—
141C22×C4×C61—41no—
142C22×C4×C61—41no—
143C22×C4×C61C2×D411yescharacteristic
144C22×C4×C61C2×D411yescharacteristic
145C22×C4×C61C2×D411yescharacteristic
146C22×C4×C61C2×D411yescharacteristic
147C22×C4×C61#4111yescharacteristic
148C22×C4×C61C2411yescharacteristic
149C2×C22⋊D61—41no—
150C2×C22⋊D61—41no—
151C2×C22⋊D61—82no—
152C22×Dic61—21no—
153C2×D244—21no—
154C2×D244—21no—
155C2×D244—22no—
156C2×D244—22no—
157M16×C61—22no—
158M16×C61—22no—
159M16×C61—42no—
160C2×C8×C61—22no—
161C2×C8×C61—42no—
162C4⋊C4×C61—21no—
163C4⋊C4×C61—21no—
164C4⋊C4×C61C2×D411yescharacteristic
165C4⋊C4×C61C2×D411yescharacteristic
166C4⋊C4×C61C2×D411yescharacteristic
167C4⋊C4×C61C2×D411yescharacteristic
168C22⋊C4×C61—21no—
169C22⋊C4×C61—21no—
170C22⋊C4×C61—21no—
171C22⋊C4×C61—21no—
172C42×C61—21no—
173C42×C61—21no—
174C42×C61—21no—
175C42×C61—21no—
176#11758201—44no—
177#11758201—44no—
178#11758198—41no—
179#11758198—41no—
180C4⋊Dic61—21no—
181C4⋊Dic61—21no—
182#11758194—22no—
183#11758194—22no—
184#11758194—22no—
185#11758194—22no—
186C2×#92418—22no—
187C2×#92418—22no—
188C2×#92418—22no—
189C2×#92418—21no—
190C2×#92418—21no—
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—21no—
217C2×C4×C61—21no—
218C2×C4×C61—21no—
219C2×C4×C61—21no—
220C2×C4×C61—24no—
221C2×C4×C61—44no—
222C2×C4×C61—44no—
223C2×C4×C61—42no—
224C2×C4×C61—42no—
225C2×C4×C61—42no—
226C2×C4×C61—81no—
227C2×C4×C61#12111yescharacteristic
228C2×C4×C61#12111yescharacteristic
229C2×C4×C61C22×D411yescharacteristic
230C2×C4×C61C22×D411yescharacteristic
231C2×C4×C61C2×#4112yes—
232C2×C4×C61H(5,𝔽2)12yes—
233C22⋊D61—82no—
234C22⋊D61—82no—
235C2×Dic61—41no—
236C2×Dic61—81no—
237D244—41no—
238D244—41no—
239D244—42no—
240C8×C61—42no—
241#92418—22no—
242#92418—22no—
243#92418—41no—
244#92418—41no—
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—41no—
266C4×C61—41no—
267C4×C61—41no—
268C4×C61—41no—
269C4×C61#52511yescharacteristic
270C4×C61#53511yescharacteristic
271C4×C61#54612yes—
272Dic61—81no—
273#3094—611no—
274C2×#448—611no—
275#2924—611no—
276C2×#419D6111yescharacteristic, complemented
277#2064—612no—
278#2046—612no—
279#1977—612no—
280#1930—612no—
281#1900—611no—
282#1855—611no—
283#1784—611no—
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×M16C2×D6111yescharacteristic
294C2×#121—611no—
295C2×#117C2×D6111yescharacteristic
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—21no—
311#419—21no—
312#419C2×D6112yes—
313#419C2×D6112yes—
314#418—611no—
315#418—611no—
316C2×#103—21no—
317C2×#103—21no—
318C2×#103—612no—
319C2×#101—612no—
320C2×#101—612no—
321C2×#100—612no—
322#407—611no—
323#407—611no—
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—21no—
346C2×M16—42no—
347C2×M16—611no—
348C2×M16—612no—
349C2×M16—611no—
350C2×M16C22×D6112yes—
351C22×C8—21no—
352C22×C8—612no—
353#121—1221no—
354#121—1221no—
355#121—1221no—
356#121—1221no—
357#117—21no—
358#117—21no—
359#117C22⋊D6111yescharacteristic
360#117C22⋊D6111yescharacteristic
361#117C22×D6111yescharacteristic
362#117C22×D6111yescharacteristic
363C2×C4⋊C4—611no—
364C2×C4⋊C4C22⋊D6111yescharacteristic
365C2×C4⋊C4C22⋊D6111yescharacteristic
366C2×C22⋊C4—21no—
367C2×C22⋊C4—1221no—
368C2×C42—21no—
369#108—2442no—
370#107—1221no—
371#107—1221no—
372#106—1221no—
373#106—1221no—
374C4⋊C8—1222no—
375C4⋊C8—1222no—
376#103—22no—
377#103—22no—
378#103—22no—
379#103—22no—
380#103—1222no—
381#103—1222no—
382#102—1221no—
383#102—1221no—
384#102—1221no—
385#102—1221no—
386#101—1224no—
387#101—2442no—
388#100—2442no—
389C22⋊C8—611no—
390C22⋊C8—611no—
391C22⋊C8—612no—
392C22⋊C8—1221no—
393C22⋊C8—1221no—
394C22⋊C8—1221no—
395C22⋊C8—1221no—
396#97—1222no—
397C4×C8—1222no—
398C24—1221no—
399C24#1175822411yescharacteristic
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#1175822214yes—
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—21no—
426C22×C4—21no—
427C22×C4—44no—
428C22×C4—41no—
429C22×C4—41no—
430C22×C4—1221no—
431C22×C4D4×D6111yescharacteristic
432C22×C4D4×D6111yescharacteristic
433C22×C4#1175822411yescharacteristic
434C22×C4C2×C22⋊D6111yescharacteristic
435C22×C4C2×C22⋊D6111yescharacteristic
436C22×C4C23×D6111yescharacteristic
437SD16—2444no—
438SD16—2444no—
439M16—22no—
440M16—22no—
441M16—42no—
442M16—1222no—
443M16—1222no—
444M16—1222no—
445M16—1222no—
446C2×C8—22no—
447C2×C8—42no—
448C2×C8—612no—
449C2×C8—612no—
450C2×C8—1222no—
451C2×C8—1222no—
452C2×C8—1222no—
453C2×C8—1221no—
454C2×C8—1221no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—1221no—
458C4⋊C4—1221no—
459C4⋊C4C2×C22⋊D6111yescharacteristic
460C4⋊C4C2×C22⋊D6111yescharacteristic
461C4⋊C4C2×C22⋊D6111yescharacteristic
462C4⋊C4C2×C22⋊D6111yescharacteristic
463C22⋊C4—21no—
464C22⋊C4—21no—
465C22⋊C4—21no—
466C22⋊C4—21no—
467C22⋊C4—2441no—
468C22⋊C4—2441no—
469C42—21no—
470C42—21no—
471C42—21no—
472C42—21no—
473C23—22no—
474C23—22no—
475C23—41no—
476C23—4881no—
477C23—4881no—
478C23—2442no—
479C23—2441no—
480C23#1451974911yescharacteristic
481C23#1451974911yescharacteristic
482C23C2×#1175822411yescharacteristic
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—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—24no—
506C2×C4—44no—
507C2×C4—44no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—81no—
512C2×C4—4881no—
513C2×C4—2441no—
514C2×C4#1451974911yescharacteristic
515C2×C4#1451974911yescharacteristic
516C2×C4C2×D4×D6111yescharacteristic
517C2×C4C22×C22⋊D6111yescharacteristic
518C2×C4C2×#1175822212yes—
519C2×C4#1451982212yes—
520C8—42no—
521C8—1222no—
522C8—1222no—
523C8—2441no—
524C8—2441no—
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#1684002712yes—
539C22C2×#1451974911yescharacteristic
540C22#1684043812yes—
541C4—21no—
542C4—21no—
543C4—41no—
544C4—41no—
545C4—41no—
546C4—41no—
547C4—44no—
548C4—4881no—
549C4#1684027311yescharacteristic
550C4#1684032911yescharacteristic
551C4#1684040212yes—
552C2—22no—
553C2—22no—
554C2—41no—
555C2—9761no—
556C2#2144920411yescharacteristic
557C2C2×#1684002711yescharacteristic
558C2#2145555311yescharacteristic
subgroupG/H
centerC22C2×#14519749
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#419C2
socleC22×C61C2×#121
prime radicalC2×#14519751C22
hypercenterC2×#419D61
hypocenterC61#24523
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#419

Complements

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

NH ≅ G/Ntype
3. #21454477C2semidirect
4. #21447186C2semidirect
5. #21447079C2semidirect
6. #21446721C2semidirect
7. #21446477C2semidirect
9. #21446024C2semidirect
10. #21445743C2semidirect
11. C2×C61×#419C2semidirect
276. C2×#419D61semidirect
328. C61#24523semidirect

Maximal subgroups

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

subgroupindex
#214557592
C2×#168399012
#214544772
#214471862
#214470792
#214467212
#214464772
#214462822
#214460242
#214457432
C2×C61×#4192
#2452361