Finite Group Databasebeta

← #29725124

Subgroups — #29725124

Order 15616.

638 conjugacy classes of subgroups.

The subgroup lattice has 548 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840027C211yesmaximal, characteristic
2#21454674C211yesmaximal, characteristic, complemented
3#21447287C211yesmaximal, characteristic, complemented
4#21447277C211yesmaximal, characteristic, complemented
5#21447092C211yesmaximal, characteristic
6#21446723C211yesmaximal, characteristic, complemented
7#21446606C211yesmaximal, characteristic, complemented
8#21446315C211yesmaximal, characteristic, complemented
9#21446303C211yesmaximal, characteristic
10#21445940C211yesmaximal, characteristic, complemented
11#21445931C211yesmaximal, characteristic, complemented
12#21445929C211yesmaximal, characteristic, complemented
13#21445871C211yesmaximal, characteristic, complemented
14#21445622C211yesmaximal, characteristic, complemented
15C61×#2049C211yesmaximal, characteristic, complemented
16C2×#14519802C2211yescharacteristic
17C22×#11758201C2211yescharacteristic
18C2×#14519784C2211yescharacteristic
19#16840240C2211yescharacteristic
20#16840236C2211yescharacteristic
21#16840027—22no—
22#16840027—22no—
23#16839865—22no—
24#16839847—22no—
25#16839810C2211yescharacteristic
26C2×#14519712C2211yescharacteristic
27C2×#14519712C2211yescharacteristic
28#16839798C2211yescharacteristic
29#16839779C2211yescharacteristic
30#16839771C2211yescharacteristic
31#16839768C2211yescharacteristic
32#16839768C2211yescharacteristic
33C2×#14519696C2211yescharacteristic
34C2×#14519693C2211yescharacteristic
35C2×#14519693C2211yescharacteristic
36#16839736C2211yescharacteristic
37#16839720C2211yescharacteristic
38#16839720C2211yescharacteristic
39#16839534C2211yescharacteristic
40#16839534C2211yescharacteristic
41#16839534C2211yescharacteristic
42#16839533C2211yescharacteristic
43#16839533C2211yescharacteristic
44#16839533C2211yescharacteristic
45#16839532C2211yescharacteristic
46#16839519C2211yescharacteristic
47#16839518C2211yescharacteristic
48C2×#137×C61C2211yescharacteristic
49C61×#430C2211yescharacteristic
50C61×#421C2211yescharacteristic
51C61×#419C2211yescharacteristic
52C61×#413C2211yescharacteristic
53C2×#101×C61C2211yescharacteristic
54C61×#344C2211yescharacteristic
55C22×D244C2311yescharacteristic
56#14519802—22no—
57#14519802—41no—
58C2×#11758201—21no—
59C2×#11758201—21no—
60C2×#11758201—21no—
61C2×#11758201—21no—
62C2×#11758201—21no—
63C2×#11758201D412yes—
64C2×#11758201D412yes—
65C2×#11758199—21no—
66#14519784—21no—
67#14519784—21no—
68#14519784—21no—
69#14519784—21no—
70C2×#11758194—21no—
71C2×#11758194C2311yescharacteristic
72C2×#11758194C2311yescharacteristic
73C2×#11758194C2311yescharacteristic
74C22×#92418—21no—
75C22×#92418C2311yescharacteristic
76Q8×Dic61—22no—
77Q8×Dic61—41no—
78C4×D244—41no—
79#14519739—22no—
80#14519736—22no—
81#14519736C2311yescharacteristic
82#14519736C2311yescharacteristic
83#14519736C2311yescharacteristic
84C22⋊C4×D61—22no—
85#14519712—21no—
86#14519712—21no—
87#14519712—21no—
88#14519712—21no—
89#14519712—41no—
90#14519710—21no—
91#14519708—21no—
92#14519704—41no—
93#14519701—41no—
94#14519699—22no—
95#14519699—41no—
96#14519696—22no—
97#14519693—22no—
98#14519693—41no—
99#14519692—22no—
100#14519687—21no—
101#14519687—21no—
102C8×Dic61—21no—
103C8×Dic61—21no—
104C2×#41×C61C2311yescharacteristic
105C22×Q8×C61C2311yescharacteristic
106#137×C61—22no—
107#137×C61—41no—
108C2×Q16×C61—21no—
109C2×SD16×C61—21no—
110C2×M16×C61C2311yescharacteristic
111C2×M16×C61C2311yescharacteristic
112C2×M16×C61C2311yescharacteristic
113C2×M16×C61C2311yescharacteristic
114#117×C61C2311yescharacteristic
115#108×C61—41no—
116#104×C61—41no—
117#103×C61—22no—
118#101×C61—21no—
119#101×C61—41no—
120#100×C61—21no—
121C23×D61—21no—
122#41×C61—21no—
123#41×C61—21no—
124#41×C61—21no—
125#41×C61—21no—
126C2×Q8×C61—21no—
127C2×Q8×C61—21no—
128C2×Q8×C61—21no—
129C2×Q8×C61—21no—
130C2×Q8×C61—21no—
131C2×Q8×C61C2×D412yes—
132C2×Q8×C61#4111yescharacteristic
133C2×D4×C61—21no—
134C2×D4×C61#4111yescharacteristic
135C22×C4×C61—21no—
136C22×C4×C61—21no—
137C22×C4×C61C2411yescharacteristic
138C2×D244—21no—
139C2×D244—21no—
140C2×D244—21no—
141C2×D244—21no—
142C2×D244—21no—
143C2×D244C2×D412yes—
144C2×C4×D61—42no—
145Q16×C61—42no—
146SD16×C61—42no—
147M16×C61—21no—
148M16×C61—21no—
149M16×C61—21no—
150M16×C61—21no—
151M16×C61—21no—
152M16×C61—21no—
153M16×C61—21no—
154M16×C61—21no—
155C2×C8×C61—21no—
156C2×C8×C61—21no—
157C2×C8×C61—21no—
158C2×C8×C61—21no—
159C4⋊C4×C61—21no—
160C22⋊C4×C61—21no—
161C42×C61—21no—
162C22⋊Dic61—21no—
163C22⋊Dic61—21no—
164C22⋊Dic61#4112yes—
165#11758201—22no—
166#11758201—22no—
167#11758201—22no—
168#11758201—22no—
169#11758201—42no—
170#11758199—42no—
171#11758198—42no—
172C4⋊Dic61—21no—
173C4⋊Dic61—21no—
174C4⋊Dic61—21no—
175C4⋊Dic61—42no—
176C4×Dic61—21no—
177C4×Dic61—21no—
178C4×Dic61—21no—
179C4×Dic61—42no—
180#11758194—21no—
181#11758194—21no—
182#11758194—21no—
183#11758194—21no—
184#11758194—21no—
185#11758194—21no—
186#11758194—41no—
187#11758194—41no—
188C2×#92418—21no—
189C2×#92418—21no—
190C2×#92418—21no—
191C2×#92418—41no—
192C2×#92418—41no—
193C2×#92418C2×D411yescharacteristic
194C2×#92418C2×D411yescharacteristic
195C2×#92418#4111yescharacteristic
196C2×#92418#4111yescharacteristic
197C2×#92418#4111yescharacteristic
198C2×#92418#4111yescharacteristic
199C23×C61—21no—
200C23×C61C2×#4111yescharacteristic
201C22×D61—22no—
202C22×D61—41no—
203C22×D61—41no—
204C22×D61—161no—
205Q8×C61—22no—
206Q8×C61—21no—
207Q8×C61—22no—
208Q8×C61—21no—
209Q8×C61—41no—
210Q8×C61—41no—
211D4×C61—21no—
212D4×C61—21no—
213D4×C61—41no—
214D4×C61—41no—
215C2×C4×C61—21no—
216C2×C4×C61—22no—
217C2×C4×C61—41no—
218C2×C4×C61—41no—
219C2×C4×C61—41no—
220C2×C4×C61—41no—
221C2×C4×C61—41no—
222C2×C4×C61C22×D411yescharacteristic
223C2×C4×C61C2×#4111yescharacteristic
224C2×C4×C61C2×#4111yescharacteristic
225C2×C4×C61C2×#4111yescharacteristic
226C2×C4×C61H(5,𝔽2)11yescharacteristic
227C2×C4×C61H(5,𝔽2)11yescharacteristic
228C2×Dic61—22no—
229C2×Dic61—41no—
230C2×Dic61—41no—
231D244—22no—
232D244—22no—
233D244—41no—
234D244—41no—
235C4×D61—162no—
236C8×C61—41no—
237C8×C61—41no—
238C8×C61—41no—
239C8×C61—41no—
240#92418—41no—
241#92418—41no—
242#92418—41no—
243#92418#12311yescharacteristic
244#92418#12311yescharacteristic
245#92418#12311yescharacteristic
246#92418#12311yescharacteristic
247#24987—611nomaximal, Sylow
248C22×C61—41no—
249C22×C61—41no—
250C22×C61—81no—
251C22×C61#52911yescharacteristic
252C22×C61#54611yescharacteristic
253C22×C61#55511yescharacteristic
254C2×D61—42no—
255C2×D61—82no—
256C2×D61—161no—
257C2×D61—161no—
258C4×C61—41no—
259C4×C61—42no—
260C4×C61—81no—
261C4×C61C2×#12311yescharacteristic
262C4×C61#54011yescharacteristic
263C4×C61#54611yescharacteristic
264C4×C61#55311yescharacteristic
265Dic61—42no—
266Dic61—81no—
267Dic61—81no—
268C2×#471—611no—
269#2949—611no—
270#2082—611no—
271#2081—611no—
272#2049D6111yescharacteristic, complemented
273#2048—611no—
274#1977—611no—
275#1953—611no—
276#1905—611no—
277#1904—611no—
278#1833—611no—
279#1830—611no—
280#1829—611no—
281#1814—611no—
282#1757—611no—
283C2×C61—21no—
284C2×C61—21no—
285C2×C61—21no—
286C2×C61—81no—
287C2×C61#242011yescharacteristic
288D61—82no—
289C2×#137C2×D6111yescharacteristic
290C2×AGL(1,ℤ/8)—611no—
291C22×SD16—611no—
292C2×#131—611no—
293#519—611no—
294#518—611no—
295#471—1222no—
296#471—1222no—
297#440—1222no—
298C4×SD16—1222no—
299#430—611no—
300#430C2×D6111yescharacteristic
301C2×#108—611no—
302C2×#108—611no—
303#427—611no—
304#421—611no—
305#421C2×D6111yescharacteristic
306#419—611no—
307#419C2×D6111yescharacteristic
308#418—611no—
309#418—611no—
310#413C2×D6111yescharacteristic
311C2×#101—611no—
312C2×#101C2×D6111yescharacteristic
313C2×#100—611no—
314C2×#100—611no—
315#410—611no—
316#405—611no—
317#405—611no—
318#344—611no—
319#344—611no—
320#344—611no—
321#344—611no—
322#344—611no—
323#344—611no—
324#344—611no—
325#344C2×D6111yescharacteristic
326H(3,ℤ/4)—611no—
327H(3,ℤ/4)—611no—
328C61#2498711yescharacteristic, complemented, Sylow
329C2×#41C22×D6111yescharacteristic
330C22×Q8C22×D6111yescharacteristic
331C22×D4—611no—
332#137—22no—
333#137—41no—
334AGL(1,ℤ/8)—1222no—
335AGL(1,ℤ/8)—2441no—
336C2×Q16—21no—
337C2×SD16—21no—
338C2×SD16—612no—
339C2×SD16—612no—
340C2×SD16—1221no—
341C2×SD16—1221no—
342C2×SD16—1221no—
343C2×SD16—1221no—
344C2×SD16—1221no—
345C2×D8—1221no—
346#131—1221no—
347#131—1221no—
348#131—1221no—
349#131—1221no—
350C2×M16—611no—
351C2×M16—611no—
352C2×M16—611no—
353C2×M16—1221no—
354C2×M16C22×D6111yescharacteristic
355C2×M16C22×D6111yescharacteristic
356C2×M16C22×D6111yescharacteristic
357C2×M16C22×D6111yescharacteristic
358C22×C8—611no—
359C22×C8—1221no—
360C4×Q8—1222no—
361C4×Q8—2441no—
362C4×D4—1222no—
363C4×D4—2441no—
364#117—611no—
365#117—611no—
366#117—611no—
367#117—1222no—
368#117C22×D6111yescharacteristic
369C2×C22⋊C4—1222no—
370#108—41no—
371#108—1221no—
372#108—1221no—
373#108—1221no—
374#108—1221no—
375#108—2441no—
376#107—1221no—
377#106—1221no—
378#104—41no—
379#104—2441no—
380#103—22no—
381#103—2441no—
382#102—1222no—
383#102—2441no—
384#101—21no—
385#101—41no—
386#101—1222no—
387#100—21no—
388#100—1222no—
389#100—2441no—
390#99—1222no—
391#97—1221no—
392#97—1221no—
393C4×C8—1221no—
394C4×C8—1221no—
395C24—1221no—
396#41—21no—
397#41—21no—
398#41—21no—
399#41—21no—
400C2×Q8—21no—
401C2×Q8—21no—
402C2×Q8—21no—
403C2×Q8—21no—
404C2×Q8—21no—
405C2×Q8D4×D6112yes—
406C2×Q8#1175822411yescharacteristic
407C2×D4—21no—
408C2×D4—612no—
409C2×D4—1221no—
410C2×D4—1221no—
411C2×D4—1221no—
412C2×D4—1221no—
413C2×D4—1221no—
414C2×D4#1175822411yescharacteristic
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—2442no—
418C22×C4C23×D6111yescharacteristic
419Q16—42no—
420SD16—42no—
421SD16—1222no—
422SD16—1222no—
423SD16—1222no—
424SD16—1222no—
425SD16—2442no—
426D8—2442no—
427M16—21no—
428M16—21no—
429M16—21no—
430M16—21no—
431M16—21no—
432M16—21no—
433M16—21no—
434M16—21no—
435M16—1221no—
436M16—1221no—
437M16—1221no—
438M16—1221no—
439M16—1221no—
440M16—1221no—
441M16—2441no—
442M16—2441no—
443C2×C8—21no—
444C2×C8—21no—
445C2×C8—21no—
446C2×C8—21no—
447C2×C8—611no—
448C2×C8—611no—
449C2×C8—611no—
450C2×C8—611no—
451C2×C8—611no—
452C2×C8—611no—
453C2×C8—1221no—
454C2×C8—1221no—
455C2×C8—1221no—
456C2×C8—2441no—
457C2×C8—2441no—
458C4⋊C4—21no—
459C4⋊C4—1221no—
460C4⋊C4—1221no—
461C4⋊C4—1221no—
462C4⋊C4—2442no—
463C22⋊C4—21no—
464C22⋊C4—612no—
465C22⋊C4—1221no—
466C22⋊C4—1221no—
467C22⋊C4—2442no—
468C42—21no—
469C42—1221no—
470C42—1221no—
471C42—1221no—
472C42—2442no—
473C23—21no—
474C23—1222no—
475C23—9761no—
476C23—2441no—
477C23—2441no—
478C23#41×D6111yescharacteristic
479Q8—21no—
480Q8—22no—
481Q8—22no—
482Q8—21no—
483Q8—41no—
484Q8—41no—
485D4—21no—
486D4—21no—
487D4—41no—
488D4—41no—
489D4—1222no—
490D4—1222no—
491D4—2441no—
492D4—2441no—
493C2×C4—21no—
494C2×C4—22no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—1222no—
501C2×C4—9762no—
502C2×C4—2441no—
503C2×C4—2441no—
504C2×C4C2×D4×D6111yescharacteristic
505C2×C4#41×D6111yescharacteristic
506C2×C4#41×D6111yescharacteristic
507C2×C4C2×#1175822411yescharacteristic
508C2×C4#1451982211yescharacteristic
509C2×C4#1451982211yescharacteristic
510C8—41no—
511C8—41no—
512C8—41no—
513C8—41no—
514C8—611no—
515C8—611no—
516C8—611no—
517C8—611no—
518C8—2441no—
519C8—2441no—
520C8—2441no—
521C22—41no—
522C22—41no—
523C22—81no—
524C22—9761no—
525C22—9761no—
526C22—4882no—
527C22—2442no—
528C22#1684028911yescharacteristic
529C22#1684040111yescharacteristic
530C22#1684047411yescharacteristic
531C4—41no—
532C4—42no—
533C4—81no—
534C4—4881no—
535C4—4881no—
536C4—2442no—
537C4#123×D6111yescharacteristic
538C4#1684036211yescharacteristic
539C4#1684039911yescharacteristic
540C4#1684045511yescharacteristic
541C2—21no—
542C2—21no—
543C2—21no—
544C2—81no—
545C2—4882no—
546C2#2144930411yescharacteristic
subgroupG/H
centerC2#21449304
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2049C2
socleC2×C61#2420
prime radicalC2×#14519802C22
hypercenter#2049D61
hypocenterC61#24987
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C22×C4 ◁ #2049

Complements

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

NH ≅ G/Ntype
2. #21454674C2semidirect
3. #21447287C2semidirect
4. #21447277C2semidirect
6. #21446723C2semidirect
7. #21446606C2semidirect
8. #21446315C2semidirect
10. #21445940C2semidirect
11. #21445931C2semidirect
12. #21445929C2semidirect
13. #21445871C2semidirect
14. #21445622C2semidirect
15. C61×#2049C2semidirect
272. #2049D61semidirect
328. C61#24987semidirect

Maximal subgroups

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

subgroupindex
C2×#168400272
#214546742
#214472872
#214472772
#214470922
#214467232
#214466062
#214463152
#214463032
#214459402
#214459312
#214459292
#214458712
#214456222
C61×#20492
#2498761