Finite Group Databasebeta

← #29725116

Subgroups — #29725116

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×#16840023C211yesmaximal, characteristic
2#21454681C211yesmaximal, characteristic, complemented
3#21447284C211yesmaximal, characteristic, complemented
4#21447280C211yesmaximal, characteristic, complemented
5#21447099C211yesmaximal, characteristic, complemented
6#21446719C211yesmaximal, characteristic, complemented
7#21446606C211yesmaximal, characteristic, complemented
8#21446310C211yesmaximal, characteristic, complemented
9#21446308C211yesmaximal, characteristic
10#21445940C211yesmaximal, characteristic, complemented
11#21445932C211yesmaximal, characteristic, complemented
12#21445928C211yesmaximal, characteristic, complemented
13#21445871C211yesmaximal, characteristic, complemented
14#21445622C211yesmaximal, characteristic, complemented
15C61×#2048C211yesmaximal, characteristic
16C2×#14519807C2211yescharacteristic
17C22×#11758200C2211yescharacteristic
18C2×#14519784C2211yescharacteristic
19#16840239C2211yescharacteristic
20#16840237C2211yescharacteristic
21#16840023—22no—
22#16840023—22no—
23#16839859—22no—
24#16839853—22no—
25#16839810C2211yescharacteristic
26C2×#14519712C2211yescharacteristic
27C2×#14519712C2211yescharacteristic
28#16839798C2211yescharacteristic
29#16839779C2211yescharacteristic
30#16839772C2211yescharacteristic
31#16839772C2211yescharacteristic
32#16839767C2211yescharacteristic
33C2×#14519695C2211yescharacteristic
34C2×#14519695C2211yescharacteristic
35C2×#14519694C2211yescharacteristic
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×AGL(1,ℤ/8)×C61C2211yescharacteristic
49C61×#430C2211yescharacteristic
50C61×#421C2211yescharacteristic
51C61×#418C2211yescharacteristic
52C61×#413C2211yescharacteristic
53C2×#100×C61C2211yescharacteristic
54C61×#344C2211yescharacteristic
55C22×Dic122C2311yescharacteristic
56#14519807—22no—
57#14519807—41no—
58C2×#11758202—21no—
59C2×#11758200—21no—
60C2×#11758200—21no—
61C2×#11758200—21no—
62C2×#11758200—21no—
63C2×#11758200—21no—
64C2×#11758200D412yes—
65C2×#11758200D412yes—
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
76C4×Dic122—41no—
77#14519742—22no—
78D4×Dic61—22no—
79D4×Dic61—41no—
80#14519736C2311yescharacteristic
81#14519736C2311yescharacteristic
82#14519736C2311yescharacteristic
83#14519733—22no—
84C2×C22⋊Dic61—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#14519702—22no—
94#14519702—41no—
95#14519698—41no—
96#14519695—22no—
97#14519695—41no—
98#14519694—22no—
99#14519692—22no—
100#14519687—21no—
101#14519687—21no—
102C8×Dic61—21no—
103C8×Dic61—21no—
104C2×#41×C61C2311yescharacteristic
105C22×D4×C61C2311yescharacteristic
106AGL(1,ℤ/8)×C61—22no—
107AGL(1,ℤ/8)×C61—41no—
108C2×SD16×C61—21no—
109C2×D8×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#102×C61—22no—
118#101×C61—21no—
119#100×C61—21no—
120#100×C61—41no—
121C24×C61—21no—
122#41×C61—21no—
123#41×C61—21no—
124#41×C61—21no—
125#41×C61—21no—
126C2×Q8×C61#4111yescharacteristic
127C2×D4×C61—21no—
128C2×D4×C61—21no—
129C2×D4×C61—21no—
130C2×D4×C61—21no—
131C2×D4×C61—21no—
132C2×D4×C61—21no—
133C2×D4×C61C2×D412yes—
134C2×D4×C61#4111yescharacteristic
135C22×C4×C61—21no—
136C22×C4×C61C2411yescharacteristic
137C22×Dic61—21no—
138C22×Dic61—42no—
139C2×Dic122—21no—
140C2×Dic122—21no—
141C2×Dic122—21no—
142C2×Dic122—21no—
143C2×Dic122—21no—
144C2×Dic122C2×D412yes—
145SD16×C61—42no—
146D8×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—42no—
165C22⋊Dic61#4112yes—
166#11758202—42no—
167#11758200—22no—
168#11758200—22no—
169#11758200—22no—
170#11758200—22no—
171#11758200—42no—
172C4⋊Dic61—21no—
173C4⋊Dic61—21no—
174C4⋊Dic61—21no—
175#11758196—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×C61—22no—
201C23×C61—41no—
202C23×C61—41no—
203C23×C61—161no—
204C23×C61C2×#4111yescharacteristic
205Q8×C61—21no—
206Q8×C61—21no—
207D4×C61—22no—
208D4×C61—21no—
209D4×C61—22no—
210D4×C61—21no—
211D4×C61—41no—
212D4×C61—41no—
213D4×C61—41no—
214D4×C61—41no—
215C2×C4×C61—21no—
216C2×C4×C61—41no—
217C2×C4×C61—41no—
218C2×C4×C61—41no—
219C2×C4×C61C22×D411yescharacteristic
220C2×C4×C61C2×#4111yescharacteristic
221C2×C4×C61C2×#4111yescharacteristic
222C2×C4×C61C2×#4111yescharacteristic
223C2×C4×C61H(5,𝔽2)11yescharacteristic
224C2×C4×C61H(5,𝔽2)11yescharacteristic
225C2×Dic61—22no—
226C2×Dic61—22no—
227C2×Dic61—41no—
228C2×Dic61—41no—
229C2×Dic61—41no—
230C2×Dic61—41no—
231C2×Dic61—162no—
232Dic122—22no—
233Dic122—22no—
234Dic122—41no—
235Dic122—41no—
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—42no—
251C22×C61—82no—
252C22×C61—81no—
253C22×C61—161no—
254C22×C61—161no—
255C22×C61#52911yescharacteristic
256C22×C61#54611yescharacteristic
257C22×C61#55511yescharacteristic
258C4×C61—41no—
259C4×C61—81no—
260C4×C61C2×#12311yescharacteristic
261C4×C61#54011yescharacteristic
262C4×C61#54611yescharacteristic
263C4×C61#55311yescharacteristic
264Dic61—42no—
265Dic61—42no—
266Dic61—81no—
267Dic61—81no—
268C2×#471—611no—
269#2949—611no—
270#2082—611no—
271#2081—611no—
272#2049—611no—
273#2048D6111yescharacteristic
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—82no—
288C2×C61#242011yescharacteristic
289C2×#137—611no—
290C2×AGL(1,ℤ/8)C2×D6111yescharacteristic
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#419—611no—
308#418—611no—
309#418C2×D6111yescharacteristic
310#413C2×D6111yescharacteristic
311C2×#101—611no—
312C2×#101—611no—
313C2×#100—611no—
314C2×#100C2×D6111yescharacteristic
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×Q8—611no—
331C22×D4C22×D6111yescharacteristic
332#137—1222no—
333#137—2441no—
334AGL(1,ℤ/8)—22no—
335AGL(1,ℤ/8)—41no—
336C2×Q16—1221no—
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—21no—
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—1222no—
381#103—2441no—
382#102—22no—
383#102—2441no—
384#101—21no—
385#101—1222no—
386#101—2441no—
387#100—21no—
388#100—41no—
389#100—1222no—
390#99—1222no—
391#97—1221no—
392#97—1221no—
393C4×C8—1221no—
394C4×C8—1221no—
395C24—21no—
396#41—21no—
397#41—21no—
398#41—21no—
399#41—21no—
400C2×Q8—612no—
401C2×Q8—1221no—
402C2×Q8—1221no—
403C2×Q8—1221no—
404C2×Q8—1221no—
405C2×Q8—1221no—
406C2×Q8#1175822411yescharacteristic
407C2×D4—21no—
408C2×D4—21no—
409C2×D4—21no—
410C2×D4—21no—
411C2×D4—21no—
412C2×D4—21no—
413C2×D4D4×D6112yes—
414C2×D4#1175822411yescharacteristic
415C22×C4—21no—
416C22×C4—1221no—
417C22×C4—2442no—
418C22×C4C23×D6111yescharacteristic
419Q16—2442no—
420SD16—42no—
421SD16—1222no—
422SD16—1222no—
423SD16—1222no—
424SD16—1222no—
425SD16—2442no—
426D8—42no—
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—22no—
475C23—41no—
476C23—41no—
477C23—161no—
478C23#41×D6111yescharacteristic
479Q8—21no—
480Q8—21no—
481Q8—1222no—
482Q8—1222no—
483Q8—2441no—
484Q8—2441no—
485D4—21no—
486D4—22no—
487D4—21no—
488D4—22no—
489D4—41no—
490D4—41no—
491D4—41no—
492D4—41no—
493C2×C4—21no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—1222no—
498C2×C4—1222no—
499C2×C4—9762no—
500C2×C4—2441no—
501C2×C4—2441no—
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—42no—
524C22—82no—
525C22—81no—
526C22—161no—
527C22—161no—
528C22#1684028911yescharacteristic
529C22#1684040111yescharacteristic
530C22#1684047411yescharacteristic
531C4—41no—
532C4—81no—
533C4—4881no—
534C4—4881no—
535C4—2442no—
536C4—2442no—
537C4#123×D6111yescharacteristic
538C4#1684036211yescharacteristic
539C4#1684039911yescharacteristic
540C4#1684045511yescharacteristic
541C2—21no—
542C2—21no—
543C2—21no—
544C2—82no—
545C2—81no—
546C2#2144930411yescharacteristic
subgroupG/H
centerC2#21449304
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2048C2
socleC2×C61#2420
prime radicalC2×AGL(1,ℤ/8)×C61C22
hypercenter#2048D61
hypocenterC61#24987
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C22×C4 ◁ #2048

Complements

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

NH ≅ G/Ntype
2. #21454681C2semidirect
3. #21447284C2semidirect
4. #21447280C2semidirect
5. #21447099C2semidirect
6. #21446719C2semidirect
7. #21446606C2semidirect
8. #21446310C2semidirect
10. #21445940C2semidirect
11. #21445932C2semidirect
12. #21445928C2semidirect
13. #21445871C2semidirect
14. #21445622C2semidirect
328. C61#24987semidirect

Maximal subgroups

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

subgroupindex
C2×#168400232
#214546812
#214472842
#214472802
#214470992
#214467192
#214466062
#214463102
#214463082
#214459402
#214459322
#214459282
#214458712
#214456222
C61×#20482
#2498761