Finite Group Databasebeta

← #37583335

Subgroups — #37583335

Order 24448.

466 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#27133176C211yesmaximal, characteristic, complemented
2#121×D191C211yesmaximal, characteristic, complemented
3#27132992C211yesmaximal, characteristic, complemented
4#27132990C211yesmaximal, characteristic, complemented
5#27132986C211yesmaximal, characteristic, complemented
6#27132978C211yesmaximal, characteristic, complemented
7#27132916C211yesmaximal, characteristic, complemented
8#27132914C211yesmaximal, characteristic, complemented
9#27132911C211yesmaximal, characteristic, complemented
10#27132812C211yesmaximal, characteristic, complemented
11#27132811C211yesmaximal, characteristic, complemented
12#27132807C211yesmaximal, characteristic, complemented
13#27132798C211yesmaximal, characteristic, complemented
14#27132626C211yesmaximal, characteristic, complemented
15#468×C191C211yesmaximal, characteristic, complemented
16C2×#15690288C2211yescharacteristic, complemented
17C2×D4×D191—21no—
18C2×D4×D191C2211yescharacteristic
19C22×C4×D191C2211yescharacteristic
20C2×#15690274C2211yescharacteristic, complemented
21C2×D1528C2211yescharacteristic, complemented
22C2×#15690272C2211yescharacteristic, complemented
23C2×#15690271C2211yescharacteristic, complemented
24#20259307—21no—
25#20259298C2211yescharacteristic
26#20259297C2211yescharacteristic
27#20259297C2211yescharacteristic
28#20259296C2211yescharacteristic
29#20259296C2211yescharacteristic, complemented
30#20259295—21no—
31#20259294—21no—
32#20259293C2211yescharacteristic
33#20259292—21no—
34#20259284—21no—
35(C22⋊D191)⋊C4—21no—
36#20259280C2211yescharacteristic, complemented
37C4⋊C4×D191C2211yescharacteristic
38C22⋊C4×D191—21no—
39#20259255C2211yescharacteristic, complemented
40#20259254C2211yescharacteristic, complemented
41#20259253C2211yescharacteristic, complemented
42#20259246C2211yescharacteristic, complemented
43#20259244C2211yescharacteristic, complemented
44#20259244C2211yescharacteristic, complemented
45#20259243C2211yescharacteristic, complemented
46#20259242C2211yescharacteristic, complemented
47#20259242C2211yescharacteristic, complemented
48(C2×D191)⋊C8C2211yescharacteristic
49(C2×D191)⋊C8C2211yescharacteristic
50(C2×D191)⋊C8C2211yescharacteristic, complemented
51(C2×D191)⋊C8C2211yescharacteristic, complemented
52C2×SD16×C191C2211yescharacteristic, complemented
53C2×M16×C191C2211yescharacteristic, complemented
54C22⋊Q8×C191C2211yescharacteristic, complemented
55#121×C191C2211yescharacteristic
56#107×C191C2211yescharacteristic, complemented
57#103×C191C2211yescharacteristic, complemented
58#102×C191C2211yescharacteristic, complemented
59C23×D191—21no—
60C23×D191D411yescharacteristic
61#15690288—41no—
62#15690288—41no—
63C2×D764—21no—
64C2×D764—21no—
65C2×D764—21no—
66C2×D764C2311yescharacteristic
67C2×D764C2311yescharacteristic
68C2×C22⋊D191—21no—
69C2×C22⋊D191—21no—
70C2×C22⋊D191—21no—
71C2×C22⋊D191—21no—
72D4×D191—41no—
73D4×D191—41no—
74D4×D191—44no—
75C22×Dic191D411yescharacteristic
76C2×C4×D191—21no—
77C2×C4×D191—21no—
78C2×C4×D191—41no—
79C2×C4×D191—41no—
80C2×C4×D191D411yescharacteristic
81C2×C4×D191D411yescharacteristic
82C2×C4×D191C2311yescharacteristic
83C2×C4×D191C2311yescharacteristic
84#15690274—41no—
85D1528—41no—
86#15690272—41no—
87#15690271—41no—
88C2×C191⋊C8C2311yescharacteristic
89C2×C191⋊C8C2311yescharacteristic
90C4⋊Dic191C2311yescharacteristic
91C4⋊Dic191C2311yescharacteristic
92Dic191⋊C4—21no—
93Dic191⋊C4—21no—
94C22⋊Dic191—21no—
95(C2×D191)⋊C4—21no—
96(C2×D191)⋊C4—21no—
97(C2×D191)⋊C4—21no—
98(C2×D191)⋊C4—21no—
99(C2×D191)⋊C4—21no—
100(C2×D191)⋊C4—21no—
101C4×Dic191—21no—
102C2×Q8×C191C2311yescharacteristic
103C2×D4×C191—21no—
104C2×D4×C191C2311yescharacteristic
105C22×C4×C191C2311yescharacteristic
106SD16×C191—41no—
107M16×C191—41no—
108C2×C8×C191C2311yescharacteristic
109C2×C8×C191C2311yescharacteristic
110C4⋊C4×C191—21no—
111C4⋊C4×C191C2311yescharacteristic
112C4⋊C4×C191C2311yescharacteristic
113C22⋊C4×C191—21no—
114C22⋊C4×C191—21no—
115C22×D191—22no—
116C22×D191—21no—
117C22×D191—21no—
118C22×D191—41no—
119C22×D191—41no—
120C22×D191—42no—
121C22×D191—42no—
122C22×D191—42no—
123C22×D191C2×D411yescharacteristic
124C22×D191C2×D411yescharacteristic
125D764—21no—
126D764—21no—
127D764—21no—
128D764—21no—
129D764—42no—
130D764—42no—
131D764—42no—
132C22⋊D191—42no—
133C22⋊D191—42no—
134C22⋊D191—42no—
135C22⋊D191—42no—
136C2×Dic191—21no—
137C2×Dic191—21no—
138C2×Dic191—41no—
139C2×Dic191C2×D411yescharacteristic
140C2×Dic191C2×D411yescharacteristic
141C4×D191—21no—
142C4×D191—21no—
143C4×D191—21no—
144C4×D191—21no—
145C4×D191—41no—
146C4×D191—41no—
147C4×D191—42no—
148C4×D191—42no—
149C191⋊C8—21no—
150C191⋊C8—21no—
151C23×C191—21no—
152C23×C191C2×D411yescharacteristic
153Q8×C191—21no—
154Q8×C191—21no—
155D4×C191—21no—
156D4×C191—21no—
157D4×C191—42no—
158C2×C4×C191—21no—
159C2×C4×C191—21no—
160C2×C4×C191—21no—
161C2×C4×C191—21no—
162C2×C4×C191—21no—
163C2×C4×C191C2×D411yescharacteristic
164C2×C4×C191C2411yescharacteristic
165C8×C191—21no—
166C8×C191—21no—
167C22×C191—21no—
168C22×C191—21no—
169C22×C191—21no—
170C22×C191—41no—
171C22×C191—41no—
172C22×C191—41no—
173C22×C191C22×D411yescharacteristic
174C2×D191—21no—
175C2×D191—21no—
176C2×D191—21no—
177C2×D191—21no—
178C2×D191—21no—
179C2×D191—44no—
180C2×D191—41no—
181C2×D191—41no—
182C2×D191—41no—
183C2×D191—41no—
184C2×D191—41no—
185C2×D191—41no—
186C2×D191—84no—
187C2×D191AGL(1,ℤ/8)12yes—
188C4×C191—21no—
189C4×C191—41no—
190C4×C191—41no—
191C4×C191—41no—
192C4×C191H(5,𝔽2)11yescharacteristic
193C4×C191H(5,𝔽2)11yescharacteristic
194Dic191—21no—
195Dic191—21no—
196Dic191—41no—
197Dic191—41no—
198C2×C191—41no—
199C2×C191—81no—
200C2×C191#53411yescharacteristic
201C2×C191C2×AGL(1,ℤ/8)11yescharacteristic
202C2×C191#57511yescharacteristic
203D191—22no—
204D191—41no—
205D191—81no—
206D191—81no—
207C191#319411yescharacteristic, complemented, Sylow
208#3194—1911nomaximal, Sylow
209#525—1911no—
210C2×#121—1911no—
211#483—1911no—
212#482—1911no—
213#482—1911no—
214#480—1911no—
215#469—1911no—
216#469—1911no—
217#468—1911no—
218#468D19111yescharacteristic, complemented
219#450—1911no—
220#449—1911no—
221#449—1911no—
222#447—1911no—
223#407—1911no—
224C2×#41—1911no—
225C22×D4—3821no—
226C22×D4—1911no—
227C23×C4—1911no—
228C2×SD16—1911no—
229C2×SD16C2×D19111yescharacteristic, complemented
230C2×D8—1911no—
231C2×D8—1911no—
232C2×M16—1911no—
233C2×M16C2×D19111yescharacteristic, complemented
234#123—3821no—
235C22⋊Q8—1911no—
236C22⋊Q8C2×D19111yescharacteristic, complemented
237#121—3821no—
238#121—3821no—
239#121—1911no—
240#121—1911no—
241#121—1911no—
242#121—1911no—
243#121—1911no—
244#121C2×D19111yescharacteristic
245#120—3821no—
246C4×D4—3821no—
247C4×D4—3821no—
248#117—1911no—
249C2×C4⋊C4—1911no—
250C2×C22⋊C4—3821no—
251#107—1911no—
252#107C2×D19111yescharacteristic, complemented
253#106—1911no—
254#106—1911no—
255#103—1911no—
256#103C2×D19111yescharacteristic, complemented
257#102—1911no—
258#102—1911no—
259#102—1911no—
260#102—1911no—
261#102—1911no—
262#102C2×D19111yescharacteristic, complemented
263C22⋊C8—1911no—
264C22⋊C8—1911no—
265C22⋊C8—1911no—
266C22⋊C8—1911no—
267C24—3821no—
268C24—1911no—
269#41—7641no—
270#41—7641no—
271C2×Q8C22×D19111yescharacteristic
272C2×D4—21no—
273C2×D4—3821no—
274C2×D4—3821no—
275C2×D4—3821no—
276C2×D4—3821no—
277C2×D4—3821no—
278C2×D4—3821no—
279C2×D4—3821no—
280C2×D4—1911no—
281C2×D4—1911no—
282C2×D4—7641no—
283C2×D4—7641no—
284C2×D4—7644no—
285C2×D4C22×D19111yescharacteristic
286C22×C4—3821no—
287C22×C4—3821no—
288C22×C4—1911no—
289C22×C4—1911no—
290C22×C4—1911no—
291C22×C4—1911no—
292C22×C4—1911no—
293C22×C4—7641no—
294C22×C4—7641no—
295C22×C4C22×D19111yescharacteristic
296SD16—41no—
297SD16—7641no—
298D8—7641no—
299D8—7641no—
300M16—41no—
301M16—7641no—
302C2×C8—1911no—
303C2×C8—1911no—
304C2×C8C22×D19111yescharacteristic
305C2×C8C22×D19111yescharacteristic
306C4⋊C4—21no—
307C4⋊C4—3821no—
308C4⋊C4—3821no—
309C4⋊C4—1911no—
310C4⋊C4—1911no—
311C4⋊C4C22×D19111yescharacteristic
312C4⋊C4C22×D19111yescharacteristic
313C22⋊C4—21no—
314C22⋊C4—21no—
315C22⋊C4—3821no—
316C22⋊C4—3821no—
317C22⋊C4—3821no—
318C22⋊C4—3821no—
319C22⋊C4—3821no—
320C22⋊C4—3821no—
321C22⋊C4—3821no—
322C42—3821no—
323C23—21no—
324C23—3822no—
325C23—3821no—
326C23—3821no—
327C23—1911no—
328C23—1911no—
329C23—7641no—
330C23—7641no—
331C23—7642no—
332C23—7642no—
333C23—7642no—
334C23D4×D19111yescharacteristic
335Q8—21no—
336Q8—21no—
337D4—21no—
338D4—21no—
339D4—42no—
340D4—3821no—
341D4—3821no—
342D4—3821no—
343D4—3821no—
344D4—7642no—
345D4—7642no—
346D4—7642no—
347D4—7642no—
348D4—7642no—
349D4—7642no—
350D4—7642no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—3821no—
357C2×C4—3821no—
358C2×C4—3821no—
359C2×C4—3821no—
360C2×C4—3821no—
361C2×C4—3821no—
362C2×C4—1911no—
363C2×C4—1911no—
364C2×C4—7641no—
365C2×C4—7641no—
366C2×C4—7642no—
367C2×C4—7642no—
368C2×C4—7641no—
369C2×C4D4×D19111yescharacteristic
370C2×C4C23×D19111yescharacteristic
371C8—21no—
372C8—21no—
373C8—3821no—
374C8—3821no—
375C22—21no—
376C22—21no—
377C22—21no—
378C22—41no—
379C22—41no—
380C22—41no—
381C22—3821no—
382C22—3821no—
383C22—3821no—
384C22—3821no—
385C22—3821no—
386C22—1912no—
387C22—15284no—
388C22—7644no—
389C22—7641no—
390C22—7641no—
391C22—7641no—
392C22—7641no—
393C22—7641no—
394C22—7641no—
395C22C2×D4×D19111yescharacteristic
396C4—21no—
397C4—41no—
398C4—41no—
399C4—41no—
400C4—3821no—
401C4—3821no—
402C4—7641no—
403C4—7641no—
404C4#2025936711yescharacteristic
405C4#2025936811yescharacteristic
406C2—41no—
407C2—81no—
408C2—3822no—
409C2—15281no—
410C2—15281no—
411C2—7641no—
412C2#2713322111yescharacteristic
413C2AGL(1,ℤ/8)×D19111yescharacteristic
414C2#2713346211yescharacteristic
subgroupG/H
centerC22C2×D4×D191
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
Fitting#468×C191C2
socleC22×C191C22×D4
hypercenter#468D191
hypocenterC191#3194
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C22 ◁ C22×C4 ◁ #468

Complements

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

NH ≅ G/Ntype
1. #27133176C2semidirect
2. #121×D191C2semidirect
3. #27132992C2semidirect
4. #27132990C2semidirect
5. #27132986C2semidirect
6. #27132978C2semidirect
7. #27132916C2semidirect
8. #27132914C2semidirect
9. #27132911C2semidirect
10. #27132812C2semidirect
11. #27132811C2semidirect
12. #27132807C2semidirect
13. #27132798C2semidirect
14. #27132626C2semidirect
15. #468×C191C2semidirect
16. C2×#15690288C22semidirect
20. C2×#15690274C22semidirect
21. C2×D1528C22semidirect
22. C2×#15690272C22semidirect
23. C2×#15690271C22semidirect
29. #20259296C22semidirect
36. #20259280C22semidirect
39. #20259255C22semidirect
40. #20259254C22semidirect
41. #20259253C22semidirect
42. #20259246C22semidirect
43. #20259244C22semidirect
44. #20259244C22semidirect
45. #20259243C22semidirect
46. #20259242C22semidirect
47. #20259242C22semidirect
50. (C2×D191)⋊C8C22semidirect
51. (C2×D191)⋊C8C22semidirect
52. C2×SD16×C191C22semidirect
53. C2×M16×C191C22semidirect
54. C22⋊Q8×C191C22semidirect
56. #107×C191C22semidirect
57. #103×C191C22semidirect
58. #102×C191C22semidirect
207. C191#3194semidirect
218. #468D191semidirect
229. C2×SD16C2×D191semidirect
233. C2×M16C2×D191semidirect
236. C22⋊Q8C2×D191semidirect
252. #107C2×D191semidirect
256. #103C2×D191semidirect
262. #102C2×D191semidirect

Maximal subgroups

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

subgroupindex
#271331762
#121×D1912
#271329922
#271329902
#271329862
#271329782
#271329162
#271329142
#271329112
#271328122
#271328112
#271328072
#271327982
#271326262
#468×C1912
#3194191