Finite Group Databasebeta

← #37584012

Subgroups — #37584012

Order 24448.

546 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#20259346C211yesmaximal, characteristic, complemented
2C22×#15690276C211yesmaximal, characteristic, complemented
3#27133319C211yesmaximal, characteristic, complemented
4#27133297C211yesmaximal, characteristic, complemented
5#27133051C211yesmaximal, characteristic, complemented
6#27132910C211yesmaximal, characteristic, complemented
7#27132877C211yesmaximal, characteristic, complemented
8#27132862C211yesmaximal, characteristic, complemented
9#27132815C211yesmaximal, characteristic, complemented
10#27132809C211yesmaximal, characteristic, complemented
11#27132801C211yesmaximal, characteristic, complemented
12#27132796C211yesmaximal, characteristic, complemented
13#27132762C211yesmaximal, characteristic, complemented
14#27132734C211yesmaximal, characteristic, complemented
15#466×C191C211yesmaximal, characteristic, complemented
16C2×#15690286C2211yescharacteristic, complemented
17C22×Dic382C2211yescharacteristic, complemented
18C22×D764C2211yescharacteristic
19C2×D4×D191—21no—
20C2×D4×D191C2211yescharacteristic
21#20259346—24no—
22C2×#15690276—24no—
23C2×#15690276C2211yescharacteristic
24C2×#15690276C2211yescharacteristic
25C2×#15690276C2211yescharacteristic, complemented
26C2×#15690276C2211yescharacteristic, complemented
27C2×#15690275C2211yescharacteristic, complemented
28C2×#15690275C2211yescharacteristic, complemented
29C2×#15690272C2211yescharacteristic
30C2×#15690272C2211yescharacteristic, complemented
31C2×#15690270C2211yescharacteristic, complemented
32#20259318C2211yescharacteristic
33#20259309C2211yescharacteristic, complemented
34#20259308—21no—
35#20259302C2211yescharacteristic, complemented
36#20259299—21no—
37#20259297C2211yescharacteristic
38#20259295—21no—
39#20259294—21no—
40#20259292—21no—
41#20259290—21no—
42#20259287C2211yescharacteristic, complemented
43D4×Dic191—21no—
44#20259284C2211yescharacteristic
45C4×C22⋊D191C2211yescharacteristic
46Dic191⋊C8C2211yescharacteristic, complemented
47#20259245C2211yescharacteristic, complemented
48#20259245C2211yescharacteristic, complemented
49#20259242C2211yescharacteristic
50#20259242C2211yescharacteristic, complemented
51C22⋊(C191⋊C8)C2211yescharacteristic, complemented
52(C2×D191)⋊C8C2211yescharacteristic, complemented
53#20259232C2211yescharacteristic, complemented
54C2×D8×C191C2211yescharacteristic, complemented
55C22×C8×C191C2211yescharacteristic, complemented
56#121×C191C2211yescharacteristic
57#121×C191C2211yescharacteristic, complemented
58#107×C191C2211yescharacteristic, complemented
59#102×C191C2211yescharacteristic, complemented
60#102×C191C2211yescharacteristic, complemented
61C23×D191—21no—
62C23×D191—21no—
63#15690286—22no—
64#15690286—41no—
65C2×Dic382—22no—
66C2×Dic382—21no—
67C2×Dic382—41no—
68C2×Dic382C2311yescharacteristic
69C2×Dic382C2311yescharacteristic
70C2×D764—22no—
71C2×D764—21no—
72C2×D764—21no—
73C2×D764—41no—
74C2×D764C2311yescharacteristic
75C2×D764C2311yescharacteristic
76C2×C22⋊D191—21no—
77C2×C22⋊D191—21no—
78C2×C22⋊D191—21no—
79C2×C22⋊D191—21no—
80C2×C22⋊D191D411yescharacteristic
81D4×D191—22no—
82D4×D191—41no—
83D4×D191—44no—
84C22×Dic191—21no—
85C22×Dic191—21no—
86C2×C4×D191—21no—
87C2×C4×D191C2311yescharacteristic
88#15690276—22no—
89#15690276—22no—
90#15690276—22no—
91#15690276D414yescomplemented
92#15690275—22no—
93#15690275—41no—
94#15690272—22no—
95#15690272—41no—
96#15690270—22no—
97C2×C191⋊C8C2311yescharacteristic
98C2×C191⋊C8C2311yescharacteristic
99C4⋊Dic191—21no—
100Dic191⋊C4—21no—
101Dic191⋊C4D411yescharacteristic
102C22⋊Dic191—21no—
103C22⋊Dic191—21no—
104C22⋊Dic191D411yescharacteristic
105(C2×D191)⋊C4—21no—
106(C2×D191)⋊C4—21no—
107(C2×D191)⋊C4—21no—
108(C2×D191)⋊C4D411yescharacteristic
109C4×Dic191—21no—
110C4×Dic191C2311yescharacteristic
111C2×D4×C191—21no—
112C2×D4×C191—21no—
113C2×D4×C191C2311yescharacteristic
114C2×D4×C191C2311yescharacteristic
115C22×C4×C191C2311yescharacteristic
116D8×C191—22no—
117C2×C8×C191—22no—
118C2×C8×C191C2311yescharacteristic
119C2×C8×C191C2311yescharacteristic
120C4⋊C4×C191C2311yescharacteristic
121C4⋊C4×C191C2311yescharacteristic
122C22⋊C4×C191—21no—
123C22⋊C4×C191—21no—
124C22×D191—21no—
125C22×D191—21no—
126C22×D191—42no—
127C22×D191—42no—
128C22×D191—42no—
129C22×D191—42no—
130C22×D191—81no—
131C22×D191—81no—
132C22×D191C2×D411yescharacteristic
133Dic382—21no—
134Dic382—21no—
135Dic382—21no—
136Dic382—42no—
137Dic382C2×D412yescomplemented
138D764—21no—
139D764—21no—
140D764—21no—
141D764—42no—
142D764—42no—
143D764C2×D412yes—
144C22⋊D191—22no—
145C22⋊D191—42no—
146C22⋊D191—42no—
147C22⋊D191—42no—
148C22⋊D191—81no—
149C2×Dic191—21no—
150C2×Dic191—21no—
151C2×Dic191—42no—
152C2×Dic191—42no—
153C2×Dic191C2×D411yescharacteristic
154C2×Dic191C2×D411yescharacteristic
155C2×Dic191C2×D411yescharacteristic
156C4×D191—21no—
157C4×D191—21no—
158C4×D191—42no—
159C191⋊C8—21no—
160C191⋊C8—21no—
161C23×C191—21no—
162C23×C191—21no—
163C23×C191C2×D411yescharacteristic
164D4×C191—21no—
165D4×C191—21no—
166D4×C191—21no—
167D4×C191—21no—
168D4×C191—42no—
169D4×C191—42no—
170C2×C4×C191—21no—
171C2×C4×C191—21no—
172C2×C4×C191—21no—
173C2×C4×C191—21no—
174C2×C4×C191C2×D411yescharacteristic
175C2×C4×C191C2411yescharacteristic
176C8×C191—21no—
177C8×C191C2×D412yescomplemented
178C22×C191—21no—
179C22×C191—21no—
180C22×C191—41no—
181C22×C191—41no—
182C22×C191—41no—
183C22×C191—41no—
184C22×C191—41no—
185C22×C191—41no—
186C22×C191AGL(1,ℤ/8)12yescomplemented
187C22×C191C22×D411yescharacteristic
188C2×D191—21no—
189C2×D191—22no—
190C2×D191—21no—
191C2×D191—21no—
192C2×D191—41no—
193C2×D191—41no—
194C2×D191—41no—
195C2×D191—41no—
196C2×D191—41no—
197C2×D191—84no—
198C2×D191—84no—
199C4×C191—21no—
200C4×C191—41no—
201C4×C191—41no—
202C4×C191C22×D411yescharacteristic
203C4×C191H(5,𝔽2)11yescharacteristic
204Dic191—21no—
205Dic191—22no—
206Dic191—21no—
207Dic191—21no—
208Dic191—41no—
209C2×C191—22no—
210C2×C191—81no—
211C2×C191—81no—
212C2×C191D4211yescharacteristic
213C2×C191C2×AGL(1,ℤ/8)11yescharacteristic
214C2×C191#57711yescharacteristic
215D191—41no—
216D191—42no—
217D191—81no—
218C191#326511yescharacteristic, complemented, Sylow
219#3265—1911nomaximal, Sylow
220C2×AGL(1,ℤ/8)—1911no—
221C22×SD16—1911no—
222#548—1911no—
223D42—1911no—
224#496—1911no—
225#468—1911no—
226#466D19111yescharacteristic, complemented
227#463—1911no—
228#460—1911no—
229#450—1911no—
230#449—1911no—
231#448—1911no—
232#447—1911no—
233#440—1911no—
234#435—1911no—
235C2×#41—1911no—
236C22×Q8—1911no—
237C22×D4—3821no—
238C22×D4—1911no—
239C22×D4—1911no—
240AGL(1,ℤ/8)—3824no—
241C2×SD16—3824no—
242C2×SD16—1911no—
243C2×SD16—1911no—
244C2×SD16—1911no—
245C2×SD16—1911no—
246C2×SD16—1911no—
247C2×SD16—1911no—
248C2×D8—1911no—
249C2×D8—1911no—
250C2×D8C2×D19111yescharacteristic, complemented
251C2×M16—1911no—
252C22×C8C2×D19111yescharacteristic, complemented
253#127—1911no—
254#124—3821no—
255#124—1911no—
256C22⋊Q8—3821no—
257C22⋊Q8—1911no—
258#121—3821no—
259#121—3821no—
260#121—1911no—
261#121C2×D19111yescharacteristic
262#121C2×D19111yescharacteristic, complemented
263#120—3821no—
264#120—3821no—
265C4×Q8—1911no—
266C4×D4—3821no—
267C4×D4—1911no—
268C4×D4—1911no—
269#107C2×D19111yescharacteristic, complemented
270C4⋊C8—1911no—
271#103—1911no—
272#103—1911no—
273#102—1911no—
274#102—1911no—
275#102C2×D19111yescharacteristic, complemented
276#102C2×D19111yescharacteristic, complemented
277C22⋊C8—1911no—
278C22⋊C8—1911no—
279#97—1911no—
280C24—3821no—
281C24—3821no—
282#41—3822no—
283#41—7641no—
284C2×Q8—3822no—
285C2×Q8—3821no—
286C2×Q8—1911no—
287C2×Q8—1911no—
288C2×Q8—7641no—
289C2×D4—21no—
290C2×D4—21no—
291C2×D4—3822no—
292C2×D4—3821no—
293C2×D4—3822no—
294C2×D4—3821no—
295C2×D4—3821no—
296C2×D4—3821no—
297C2×D4—3821no—
298C2×D4—3821no—
299C2×D4—1911no—
300C2×D4—1911no—
301C2×D4—1911no—
302C2×D4—7641no—
303C2×D4—7641no—
304C2×D4—7644no—
305C2×D4C22×D19111yescharacteristic
306C2×D4C22×D19111yescharacteristic
307C22×C4—3821no—
308C22×C4—3821no—
309C22×C4—3821no—
310C22×C4—1911no—
311C22×C4C22×D19111yescharacteristic
312SD16—3822no—
313SD16—3822no—
314SD16—3822no—
315SD16—3822no—
316SD16—1914no—
317SD16—7641no—
318D8—22no—
319D8—3822no—
320D8—7641no—
321M16—3822no—
322C2×C8—22no—
323C2×C8—1911no—
324C2×C8—1911no—
325C2×C8C22×D19111yescharacteristic
326C2×C8C22×D19111yescharacteristic
327C4⋊C4—3821no—
328C4⋊C4—3821no—
329C4⋊C4—1911no—
330C4⋊C4C22×D19111yescharacteristic
331C4⋊C4C22×D19111yescharacteristic
332C22⋊C4—21no—
333C22⋊C4—21no—
334C22⋊C4—3821no—
335C22⋊C4—3821no—
336C22⋊C4—3821no—
337C22⋊C4—3821no—
338C22⋊C4—3821no—
339C22⋊C4—1911no—
340C22⋊C4—1911no—
341C42—3821no—
342C42—1911no—
343C23—21no—
344C23—21no—
345C23—3821no—
346C23—3821no—
347C23—1911no—
348C23—15281no—
349C23—15281no—
350C23—7642no—
351C23—7642no—
352C23—7642no—
353C23—7642no—
354C23D4×D19111yescharacteristic
355Q8—3821no—
356Q8—3821no—
357Q8—3821no—
358Q8—1912no—
359Q8—7642no—
360D4—21no—
361D4—21no—
362D4—21no—
363D4—21no—
364D4—42no—
365D4—42no—
366D4—3822no—
367D4—3821no—
368D4—3821no—
369D4—3821no—
370D4—1912no—
371D4—15281no—
372D4—7642no—
373D4—7642no—
374D4—7642no—
375D4—7642no—
376D4—7642no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—3821no—
382C2×C4—3821no—
383C2×C4—3821no—
384C2×C4—3821no—
385C2×C4—1911no—
386C2×C4—1911no—
387C2×C4—1911no—
388C2×C4—7642no—
389C2×C4—7642no—
390C2×C4—7642no—
391C2×C4D4×D19111yescharacteristic
392C2×C4C23×D19111yescharacteristic
393C8—21no—
394C8—3821no—
395C8—3821no—
396C8D4×D19112yescomplemented
397C22—21no—
398C22—21no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22—41no—
403C22—41no—
404C22—41no—
405C22—3821no—
406C22—3821no—
407C22—3821no—
408C22—3822no—
409C22—15284no—
410C22—15284no—
411C22—7641no—
412C22—7641no—
413C22—7641no—
414C22—7641no—
415C22—7641no—
416C22#2025934612yescomplemented
417C22C2×D4×D19111yescharacteristic
418C4—21no—
419C4—41no—
420C4—41no—
421C4—3821no—
422C4—3821no—
423C4—3821no—
424C4—3822no—
425C4—7641no—
426C4C2×D4×D19111yescharacteristic
427C4#2025936811yescharacteristic
428C2—22no—
429C2—81no—
430C2—81no—
431C2—15281no—
432C2—7641no—
433C2—7642no—
434C2#2713329711yescharacteristic
435C2C2×#2025934611yescharacteristic
436C2#2713347011yescharacteristic
subgroupG/H
centerC22C2×D4×D191
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
Fitting#466×C191C2
socleC22×C191C22×D4
hypercenter#466D191
hypocenterC191#3265
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C22 ◁ C22×C4 ◁ #466

Complements

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

NH ≅ G/Ntype
1. C2×#20259346C2semidirect
2. C22×#15690276C2semidirect
3. #27133319C2semidirect
4. #27133297C2semidirect
5. #27133051C2semidirect
6. #27132910C2semidirect
7. #27132877C2semidirect
8. #27132862C2semidirect
9. #27132815C2semidirect
10. #27132809C2semidirect
11. #27132801C2semidirect
12. #27132796C2semidirect
13. #27132762C2semidirect
14. #27132734C2semidirect
15. #466×C191C2semidirect
16. C2×#15690286C22semidirect
17. C22×Dic382C22semidirect
25. C2×#15690276C22semidirect
26. C2×#15690276C22semidirect
27. C2×#15690275C22semidirect
28. C2×#15690275C22semidirect
30. C2×#15690272C22semidirect
31. C2×#15690270C22semidirect
33. #20259309C22semidirect
35. #20259302C22semidirect
42. #20259287C22semidirect
46. Dic191⋊C8C22semidirect
47. #20259245C22semidirect
48. #20259245C22semidirect
50. #20259242C22semidirect
51. C22⋊(C191⋊C8)C22semidirect
52. (C2×D191)⋊C8C22semidirect
53. #20259232C22semidirect
54. C2×D8×C191C22semidirect
55. C22×C8×C191C22semidirect
57. #121×C191C22semidirect
58. #107×C191C22semidirect
59. #102×C191C22semidirect
60. #102×C191C22semidirect
91. #15690276D4semidirect
137. Dic382C2×D4semidirect
177. C8×C191C2×D4semidirect
186. C22×C191AGL(1,ℤ/8)semidirect
218. C191#3265semidirect
226. #466D191semidirect
250. C2×D8C2×D191semidirect
252. C22×C8C2×D191semidirect
262. #121C2×D191semidirect
269. #107C2×D191semidirect
275. #102C2×D191semidirect
276. #102C2×D191semidirect
396. C8D4×D191semidirect
416. C22#20259346semidirect

Maximal subgroups

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

subgroupindex
C2×#202593462
C22×#156902762
#271333192
#271332972
#271330512
#271329102
#271328772
#271328622
#271328152
#271328092
#271328012
#271327962
#271327622
#271327342
#466×C1912
#3265191