Finite Group Databasebeta

← #37584066

Subgroups — #37584066

Order 24448.

510 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#20259346C211yesmaximal, characteristic, complemented
2C2×#20259341C211yesmaximal, characteristic, complemented
3#27133333C211yesmaximal, characteristic, complemented
4#27133308C211yesmaximal, characteristic, complemented
5#27133050C211yesmaximal, characteristic, complemented
6#27132908C211yesmaximal, characteristic, complemented
7#27132898C211yesmaximal, characteristic, complemented
8#27132886C211yesmaximal, characteristic, complemented
9#27132826C211yesmaximal, characteristic, complemented
10#27132809C211yesmaximal, characteristic, complemented
11#27132805C211yesmaximal, characteristic, complemented
12#27132795C211yesmaximal, characteristic, complemented
13#27132763C211yesmaximal, characteristic, complemented
14#27132733C211yesmaximal, characteristic, complemented
15#459×C191C211yesmaximal, characteristic, complemented
16C2×#15690286C2211yescharacteristic
17C2×#15690286C2211yescharacteristic
18C2×#15690286C2211yescharacteristic, complemented
19C2×D4×D191C2211yescharacteristic
20#20259346—24no—
21#20259341—24no—
22C2×Dic764C2211yescharacteristic, complemented
23C2×#15690276C2211yescharacteristic
24C2×#15690275C2211yescharacteristic
25C2×#15690275C2211yescharacteristic, complemented
26C2×#15690275C2211yescharacteristic, complemented
27C2×#15690272C2211yescharacteristic
28C2×#15690272C2211yescharacteristic
29C2×#15690270C2211yescharacteristic
30C2×C8×D191C2211yescharacteristic, complemented
31#20259318—21no—
32#20259310C2211yescharacteristic
33#20259306—21no—
34#20259302C2211yescharacteristic
35#20259297C2211yescharacteristic
36#20259296—21no—
37#20259294—21no—
38#20259294—21no—
39#20259293C2211yescharacteristic
40#20259293C2211yescharacteristic, complemented
41#20259292—21no—
42C4×Dic382C2211yescharacteristic, complemented
43C4×D764C2211yescharacteristic
44D4×Dic191—21no—
45C2×(C2×D191)⋊C4—21no—
46#20259255C2211yescharacteristic, complemented
47#20259247C2211yescharacteristic, complemented
48#20259245C2211yescharacteristic, complemented
49#20259244C2211yescharacteristic
50#20259243C2211yescharacteristic
51(C2×D191)⋊C8C2211yescharacteristic
52(C2×D191)⋊C8C2211yescharacteristic, complemented
53#20259233C2211yescharacteristic, complemented
54C2×D8×C191C2211yescharacteristic
55C2×D8×C191C2211yescharacteristic
56#127×C191C2211yescharacteristic, complemented
57C4×D4×C191C2211yescharacteristic
58C4⋊C8×C191C2211yescharacteristic, complemented
59#102×C191C2211yescharacteristic, complemented
60#102×C191C2211yescharacteristic, complemented
61C23×D191—21no—
62#15690286—22no—
63#15690286—22no—
64#15690286—22no—
65#15690286—41no—
66#15690286—41no—
67#15690286—41no—
68C2×Dic382C2311yescharacteristic
69C2×Dic382C2311yescharacteristic
70C2×D764—21no—
71C2×D764D411yescharacteristic
72C2×D764C2311yescharacteristic
73C2×C22⋊D191—21no—
74C2×C22⋊D191—21no—
75C2×C22⋊D191—21no—
76C2×C22⋊D191—21no—
77C2×C22⋊D191—21no—
78C2×C22⋊D191—21no—
79D4×D191—22no—
80D4×D191—41no—
81C22×Dic191—21no—
82C22×Dic191—21no—
83C22×Dic191—21no—
84C2×C4×D191C2311yescharacteristic
85C2×C4×D191C2311yescharacteristic
86Dic764—22no—
87#15690276—22no—
88#15690275—22no—
89#15690275—41no—
90#15690275D414yescomplemented
91#15690272—22no—
92#15690272—41no—
93#15690270—22no—
94C2×C191⋊C8C2311yescharacteristic
95C2×C191⋊C8C2311yescharacteristic
96C8×D191—22no—
97C4⋊Dic191—21no—
98C4⋊Dic191D411yescharacteristic
99C4⋊Dic191C2311yescharacteristic
100Dic191⋊C4—21no—
101C22⋊Dic191—21no—
102C22⋊Dic191—21no—
103(C2×D191)⋊C4—21no—
104(C2×D191)⋊C4—21no—
105(C2×D191)⋊C4—21no—
106(C2×D191)⋊C4—21no—
107(C2×D191)⋊C4D411yescharacteristic
108(C2×D191)⋊C4D411yescharacteristic
109C4×Dic191—21no—
110C2×D4×C191—21no—
111C2×D4×C191—21no—
112C2×D4×C191C2311yescharacteristic
113C2×D4×C191C2311yescharacteristic
114C2×D4×C191C2311yescharacteristic
115C22×C4×C191—21no—
116D8×C191—22no—
117D8×C191—22no—
118C2×C8×C191C2311yescharacteristic
119C2×C8×C191C2311yescharacteristic
120C4⋊C4×C191C2311yescharacteristic
121C22⋊C4×C191—21no—
122C42×C191C2311yescharacteristic
123C22×D191—21no—
124C22×D191—42no—
125C22×D191—42no—
126C22×D191—81no—
127C22×D191C2×D411yescharacteristic
128C22×D191C2×D411yescharacteristic
129Dic382—21no—
130Dic382—21no—
131Dic382—21no—
132Dic382C2×D412yes—
133D764—21no—
134D764—21no—
135D764—41no—
136D764—81no—
137C22⋊D191—42no—
138C22⋊D191—42no—
139C22⋊D191—42no—
140C22⋊D191—42no—
141C22⋊D191—81no—
142C22⋊D191—81no—
143C2×Dic191—21no—
144C2×Dic191—21no—
145C2×Dic191—21no—
146C2×Dic191—42no—
147C2×Dic191—42no—
148C2×Dic191—42no—
149C2×Dic191C2×D411yescharacteristic
150C2×Dic191C2×D411yescharacteristic
151C4×D191—21no—
152C4×D191—21no—
153C4×D191—21no—
154C4×D191—21no—
155C191⋊C8—21no—
156C191⋊C8C2×D412yes—
157C23×C191—21no—
158C23×C191—21no—
159C23×C191—21no—
160D4×C191—21no—
161D4×C191—21no—
162D4×C191—21no—
163D4×C191—21no—
164D4×C191—21no—
165D4×C191—42no—
166D4×C191—81no—
167D4×C191C2×D412yes—
168C2×C4×C191—21no—
169C2×C4×C191—81no—
170C2×C4×C191C2×D411yescharacteristic
171C2×C4×C191C2×D411yescharacteristic
172C2×C4×C191C2411yescharacteristic
173C8×C191—21no—
174C8×C191—21no—
175C22×C191—22no—
176C22×C191—41no—
177C22×C191—41no—
178C22×C191—41no—
179C22×C191—41no—
180C22×C191—41no—
181C22×C191—41no—
182C22×C191—41no—
183C22×C191—41no—
184C22×C191C22×D411yescharacteristic
185C2×D191—21no—
186C2×D191—21no—
187C2×D191—21no—
188C2×D191—21no—
189C2×D191—21no—
190C2×D191—21no—
191C2×D191—41no—
192C2×D191—41no—
193C2×D191—41no—
194C2×D191—84no—
195C4×C191—21no—
196C4×C191—21no—
197C4×C191—41no—
198C4×C191C22×D411yescharacteristic
199C4×C191H(5,𝔽2)11yescharacteristic
200Dic191—21no—
201Dic191—21no—
202Dic191—22no—
203Dic191—41no—
204Dic191—41no—
205C2×C191—42no—
206C2×C191—81no—
207C2×C191—81no—
208C2×C191D4211yescharacteristic
209C2×C191#57511yescharacteristic
210C2×C191#57611yescharacteristic
211D191—41no—
212D191—41no—
213D191—81no—
214C191#326811yescharacteristic, complemented, Sylow
215#3268—1911nomaximal, Sylow
216C2×AGL(1,ℤ/8)—1911no—
217C2×#135—1911no—
218#550—1911no—
219#546—1911no—
220#496—1911no—
221#468—1911no—
222#466—1911no—
223#464—1911no—
224#459D19111yescharacteristic, complemented
225#452—1911no—
226#449—1911no—
227#449—1911no—
228#447—1911no—
229#440—1911no—
230#435—1911no—
231C2×#41—1911no—
232C2×#41—1911no—
233C2×#41—1911no—
234C22×D4—1911no—
235AGL(1,ℤ/8)—3824no—
236#135—3824no—
237C2×Q16—1911no—
238C2×SD16—1911no—
239C2×SD16—1911no—
240C2×SD16—1911no—
241C2×SD16—1911no—
242C2×D8—1911no—
243C2×D8—1911no—
244C2×D8C2×D19111yescharacteristic
245C2×D8C2×D19111yescharacteristic
246C2×M16—1911no—
247C22×C8—1911no—
248#127—3821no—
249#127C2×D19111yescharacteristic, complemented
250#124—1911no—
251#123—3821no—
252C22⋊Q8—1911no—
253#121—3821no—
254#121—3821no—
255#121—3821no—
256#121—1911no—
257#121—1911no—
258#121—1911no—
259#120—3821no—
260C4×Q8—1911no—
261C4×D4—3821no—
262C4×D4—1911no—
263C4×D4C2×D19111yescharacteristic
264C2×C22⋊C4—3821no—
265#107—1911no—
266C4⋊C8C2×D19111yescharacteristic, complemented
267#103—1911no—
268#103—1911no—
269#102—1911no—
270#102—1911no—
271#102C2×D19111yescharacteristic, complemented
272#102C2×D19111yescharacteristic, complemented
273C22⋊C8—1911no—
274C22⋊C8—1911no—
275#97—1911no—
276C24—3821no—
277#41—3822no—
278#41—3822no—
279#41—3822no—
280#41—7641no—
281#41—7641no—
282#41—7641no—
283C2×Q8—1911no—
284C2×Q8—1911no—
285C2×D4—21no—
286C2×D4—21no—
287C2×D4—3821no—
288C2×D4—3822no—
289C2×D4—3821no—
290C2×D4—3821no—
291C2×D4—3821no—
292C2×D4—3821no—
293C2×D4—3821no—
294C2×D4—3821no—
295C2×D4—1911no—
296C2×D4—1911no—
297C2×D4—7641no—
298C2×D4C22×D19111yescharacteristic
299C2×D4C22×D19111yescharacteristic
300C2×D4C22×D19111yescharacteristic
301C22×C4—21no—
302C22×C4—3821no—
303C22×C4—3821no—
304C22×C4—3821no—
305C22×C4—1911no—
306C22×C4—1911no—
307Q16—3822no—
308SD16—3822no—
309SD16—3822no—
310SD16—1914no—
311SD16—7641no—
312D8—22no—
313D8—22no—
314D8—3822no—
315D8—7641no—
316M16—3822no—
317C2×C8—3822no—
318C2×C8—1911no—
319C2×C8—1911no—
320C2×C8C22×D19111yescharacteristic
321C2×C8C22×D19111yescharacteristic
322C4⋊C4—3821no—
323C4⋊C4—3821no—
324C4⋊C4—1911no—
325C4⋊C4—1911no—
326C4⋊C4C22×D19111yescharacteristic
327C22⋊C4—21no—
328C22⋊C4—3821no—
329C22⋊C4—3821no—
330C22⋊C4—3821no—
331C22⋊C4—3821no—
332C22⋊C4—3821no—
333C22⋊C4—3821no—
334C22⋊C4—1911no—
335C22⋊C4—1911no—
336C42—3821no—
337C42C22×D19111yescharacteristic
338C23—21no—
339C23—21no—
340C23—21no—
341C23—3821no—
342C23—1911no—
343C23—1911no—
344C23—15281no—
345C23—7642no—
346C23—7642no—
347Q8—3821no—
348Q8—3821no—
349Q8—3821no—
350Q8—1912no—
351D4—21no—
352D4—21no—
353D4—21no—
354D4—21no—
355D4—21no—
356D4—42no—
357D4—81no—
358D4—3821no—
359D4—3821no—
360D4—15281no—
361D4—15281no—
362D4—15281no—
363D4—7641no—
364D4—7642no—
365D4—7642no—
366D4—7642no—
367D4—7642no—
368D4D4×D19112yes—
369C2×C4—21no—
370C2×C4—81no—
371C2×C4—3821no—
372C2×C4—3821no—
373C2×C4—3821no—
374C2×C4—3821no—
375C2×C4—3821no—
376C2×C4—3821no—
377C2×C4—3821no—
378C2×C4—1911no—
379C2×C4—1911no—
380C2×C4—7642no—
381C2×C4—7642no—
382C2×C4—7642no—
383C2×C4D4×D19111yescharacteristic
384C2×C4D4×D19111yescharacteristic
385C2×C4C23×D19111yescharacteristic
386C8—21no—
387C8—21no—
388C8—3821no—
389C8—1912no—
390C22—22no—
391C22—41no—
392C22—41no—
393C22—41no—
394C22—41no—
395C22—41no—
396C22—41no—
397C22—41no—
398C22—41no—
399C22—3821no—
400C22—3821no—
401C22—3821no—
402C22—3821no—
403C22—3821no—
404C22—3821no—
405C22—15284no—
406C22—7641no—
407C22—7641no—
408C22—7641no—
409C22C2×D4×D19111yescharacteristic
410C4—21no—
411C4—21no—
412C4—41no—
413C4—3821no—
414C4—3821no—
415C4—3822no—
416C4—7641no—
417C4—7641no—
418C4C2×D4×D19111yescharacteristic
419C4#2025936811yescharacteristic
420C2—42no—
421C2—81no—
422C2—81no—
423C2—15281no—
424C2—7641no—
425C2—7641no—
426C2#2713329711yescharacteristic
427C2#2713345911yescharacteristic
428C2#2713346411yescharacteristic
subgroupG/H
centerC22C2×D4×D191
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
Fitting#459×C191C2
socleC22×C191C22×D4
hypercenter#459D191
hypocenterC191#3268
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C22 ◁ C42 ◁ #459

Complements

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

NH ≅ G/Ntype
1. C2×#20259346C2semidirect
2. C2×#20259341C2semidirect
3. #27133333C2semidirect
4. #27133308C2semidirect
5. #27133050C2semidirect
6. #27132908C2semidirect
7. #27132898C2semidirect
8. #27132886C2semidirect
9. #27132826C2semidirect
10. #27132809C2semidirect
11. #27132805C2semidirect
12. #27132795C2semidirect
13. #27132763C2semidirect
14. #27132733C2semidirect
15. #459×C191C2semidirect
18. C2×#15690286C22semidirect
22. C2×Dic764C22semidirect
25. C2×#15690275C22semidirect
26. C2×#15690275C22semidirect
30. C2×C8×D191C22semidirect
40. #20259293C22semidirect
42. C4×Dic382C22semidirect
46. #20259255C22semidirect
47. #20259247C22semidirect
48. #20259245C22semidirect
52. (C2×D191)⋊C8C22semidirect
53. #20259233C22semidirect
56. #127×C191C22semidirect
58. C4⋊C8×C191C22semidirect
59. #102×C191C22semidirect
60. #102×C191C22semidirect
90. #15690275D4semidirect
214. C191#3268semidirect
224. #459D191semidirect
249. #127C2×D191semidirect
266. C4⋊C8C2×D191semidirect
271. #102C2×D191semidirect
272. #102C2×D191semidirect

Maximal subgroups

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

subgroupindex
C2×#202593462
C2×#202593412
#271333332
#271333082
#271330502
#271329082
#271328982
#271328862
#271328262
#271328092
#271328052
#271327952
#271327632
#271327332
#459×C1912
#3268191