Finite Group Databasebeta

← #37584128

Subgroups — #37584128

Order 24448.

520 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#20259350C211yesmaximal, characteristic, complemented
2C2×#20259349C211yesmaximal, characteristic, complemented
3#27133310C211yesmaximal, characteristic, complemented
4#27133304C211yesmaximal, characteristic, complemented
5#27133044C211yesmaximal, characteristic, complemented
6#27132910C211yesmaximal, characteristic, complemented
7#27132881C211yesmaximal, characteristic, complemented
8#27132877C211yesmaximal, characteristic, complemented
9#27132828C211yesmaximal, characteristic, complemented
10#27132823C211yesmaximal, characteristic, complemented
11#27132798C211yesmaximal, characteristic, complemented
12#27132796C211yesmaximal, characteristic, complemented
13#27132745C211yesmaximal, characteristic, complemented
14#27132739C211yesmaximal, characteristic, complemented
15#468×C191C211yesmaximal, characteristic, complemented
16C2×#15690288C2211yescharacteristic
17C2×#15690287C2211yescharacteristic
18C22×D764C2211yescharacteristic, complemented
19C2×D4×D191C2211yescharacteristic, complemented
20#20259350—24no—
21#20259349—24no—
22C2×#15690277C2211yescharacteristic, complemented
23C2×#15690276C2211yescharacteristic, complemented
24C2×#15690276C2211yescharacteristic, complemented
25C2×#15690274C2211yescharacteristic, complemented
26C2×D1528C2211yescharacteristic
27C2×D1528C2211yescharacteristic
28C2×#15690272C2211yescharacteristic
29C2×#15690272C2211yescharacteristic
30C2×#15690270C2211yescharacteristic, complemented
31#20259311C2211yescharacteristic, complemented
32#20259309C2211yescharacteristic, complemented
33#20259307—21no—
34#20259304—21no—
35#20259302C2211yescharacteristic, complemented
36#20259297—21no—
37#20259297C2211yescharacteristic
38#20259293—21no—
39#20259292—21no—
40#20259290—21no—
41#20259284C2211yescharacteristic, complemented
42#20259284C2211yescharacteristic, complemented
43C4×C22⋊D191C2211yescharacteristic
44C22⋊C4×D191—21no—
45C22⋊C4×D191—21no—
46Dic191⋊C8C2211yescharacteristic, complemented
47#20259245C2211yescharacteristic, complemented
48#20259242C2211yescharacteristic, complemented
49#20259242C2211yescharacteristic, complemented
50#20259242C2211yescharacteristic, complemented
51C22⋊(C191⋊C8)C2211yescharacteristic, complemented
52(C2×D191)⋊C8C2211yescharacteristic, complemented
53C8×Dic191C2211yescharacteristic, complemented
54C2×SD16×C191C2211yescharacteristic, complemented
55C2×M16×C191C2211yescharacteristic, complemented
56C22⋊Q8×C191C2211yescharacteristic, complemented
57#121×C191C2211yescharacteristic
58#107×C191C2211yescharacteristic, complemented
59#103×C191C2211yescharacteristic, complemented
60#102×C191C2211yescharacteristic, complemented
61C23×D191—21no—
62C23×D191—21no—
63#15690288—22no—
64#15690288—41no—
65#15690287—22no—
66#15690287—41no—
67C2×Dic382C2311yescharacteristic
68C2×D764—22no—
69C2×D764—21no—
70C2×D764—21no—
71C2×D764—21no—
72C2×D764—41no—
73C2×D764C2311yescharacteristic
74C2×D764C2311yescharacteristic
75C2×D764C2311yescharacteristic
76C2×C22⋊D191—21no—
77C2×C22⋊D191—21no—
78C2×C22⋊D191D411yescharacteristic
79D4×D191—22no—
80D4×D191—41no—
81C2×C4×D191—21no—
82C2×C4×D191—21no—
83C2×C4×D191—21no—
84C2×C4×D191—21no—
85C2×C4×D191C2311yescharacteristic
86#15690277—41no—
87#15690276—22no—
88#15690276—22no—
89#15690274—22no—
90D1528—22no—
91D1528D414yescomplemented
92#15690272—22no—
93#15690272—41no—
94#15690270—22no—
95C2×C191⋊C8C2311yescharacteristic
96C2×C191⋊C8C2311yescharacteristic
97C4⋊Dic191—21no—
98Dic191⋊C4D411yescharacteristic
99C22⋊Dic191—21no—
100C22⋊Dic191—21no—
101C22⋊Dic191D411yescharacteristic
102(C2×D191)⋊C4—21no—
103(C2×D191)⋊C4—21no—
104(C2×D191)⋊C4—21no—
105(C2×D191)⋊C4—21no—
106(C2×D191)⋊C4—21no—
107(C2×D191)⋊C4—21no—
108(C2×D191)⋊C4D411yescharacteristic
109C4×Dic191C2311yescharacteristic
110C2×Q8×C191C2311yescharacteristic
111C2×D4×C191—21no—
112C2×D4×C191C2311yescharacteristic
113C22×C4×C191C2311yescharacteristic
114SD16×C191—22no—
115M16×C191—22no—
116C2×C8×C191C2311yescharacteristic
117C2×C8×C191C2311yescharacteristic
118C4⋊C4×C191—21no—
119C4⋊C4×C191C2311yescharacteristic
120C4⋊C4×C191C2311yescharacteristic
121C22⋊C4×C191—21no—
122C22⋊C4×C191—21no—
123C22×D191—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—
135D764—21no—
136D764—21no—
137D764—21no—
138D764—21no—
139D764—42no—
140D764—42no—
141D764—81no—
142D764C2×D412yes—
143D764C2×D412yes—
144C22⋊D191—41no—
145C22⋊D191—42no—
146C22⋊D191—42no—
147C2×Dic191—21no—
148C2×Dic191C2×D411yescharacteristic
149C2×Dic191C2×D411yescharacteristic
150C2×Dic191C2×D411yescharacteristic
151C4×D191—21no—
152C4×D191—21no—
153C4×D191—42no—
154C4×D191—42no—
155C4×D191—81no—
156C4×D191—81no—
157C191⋊C8—21no—
158C191⋊C8—21no—
159C23×C191—21no—
160C23×C191C2×D411yescharacteristic
161Q8×C191—21no—
162Q8×C191—21no—
163D4×C191—21no—
164D4×C191—21no—
165D4×C191—81no—
166C2×C4×C191—21no—
167C2×C4×C191—21no—
168C2×C4×C191—21no—
169C2×C4×C191—21no—
170C2×C4×C191—21no—
171C2×C4×C191C2×D411yescharacteristic
172C2×C4×C191C2411yescharacteristic
173C8×C191—21no—
174C8×C191C2×D412yes—
175C22×C191—21no—
176C22×C191—21no—
177C22×C191—21no—
178C22×C191—41no—
179C22×C191—41no—
180C22×C191—41no—
181C22×C191C22×D411yescharacteristic
182C2×D191—21no—
183C2×D191—22no—
184C2×D191—22no—
185C2×D191—21no—
186C2×D191—21no—
187C2×D191—41no—
188C2×D191—41no—
189C2×D191—41no—
190C2×D191—41no—
191C2×D191—41no—
192C2×D191—41no—
193C2×D191—41no—
194C2×D191—84no—
195C2×D191—84no—
196C4×C191—21no—
197C4×C191—41no—
198C4×C191—41no—
199C4×C191—41no—
200C4×C191C22×D411yescharacteristic
201C4×C191H(5,𝔽2)11yescharacteristic
202Dic191—21no—
203Dic191—21no—
204Dic191—21no—
205Dic191—41no—
206C2×C191—41no—
207C2×C191—81no—
208C2×C191D4211yescharacteristic
209C2×C191#57511yescharacteristic
210C2×C191#57611yescharacteristic
211D191—41no—
212D191—42no—
213D191—42no—
214D191—81no—
215C191#327311yescharacteristic, complemented, Sylow
216#3273—1911nomaximal, Sylow
217C2×AGL(1,ℤ/8)—1911no—
218C2×AGL(1,ℤ/8)—1911no—
219#546—1911no—
220#546—1911no—
221#495—1911no—
222#468—1911no—
223#468D19111yescharacteristic, complemented
224#463—1911no—
225#463—1911no—
226#452—1911no—
227#452—1911no—
228#447—1911no—
229#447—1911no—
230C4×D8—1911no—
231#436—1911no—
232C2×#41—1911no—
233C2×#41—1911no—
234C22×D4—1911no—
235C22×D4—1911no—
236AGL(1,ℤ/8)—3824no—
237AGL(1,ℤ/8)—3824no—
238C2×Q16—1911no—
239C2×SD16—1911no—
240C2×SD16—1911no—
241C2×SD16—1911no—
242C2×SD16C2×D19111yescharacteristic, complemented
243C2×D8—1911no—
244C2×D8—1911no—
245C2×D8—1911no—
246C2×D8—1911no—
247C2×M16—1911no—
248C2×M16C2×D19111yescharacteristic, complemented
249#124—1911no—
250#124—1911no—
251#123—3821no—
252#123—3821no—
253C22⋊Q8—1911no—
254C22⋊Q8C2×D19111yescharacteristic, complemented
255#121—3821no—
256#121—3821no—
257#121—1911no—
258#121C2×D19111yescharacteristic
259#120—3821no—
260#120—3821no—
261C4×D4—1911no—
262C4×D4—1911no—
263C4×D4—1911no—
264C2×C22⋊C4—3821no—
265C2×C22⋊C4—3821no—
266#107C2×D19111yescharacteristic, complemented
267C4⋊C8—1911no—
268#103—1911no—
269#103C2×D19111yescharacteristic, complemented
270#102—1911no—
271#102—1911no—
272#102—1911no—
273#102C2×D19111yescharacteristic, complemented
274C22⋊C8—1911no—
275C22⋊C8—1911no—
276C4×C8—1911no—
277C24—3821no—
278C24—3821no—
279#41—3822no—
280#41—3822no—
281#41—7641no—
282#41—7641no—
283C2×Q8—1911no—
284C2×Q8C22×D19111yescharacteristic
285C2×D4—21no—
286C2×D4—3822no—
287C2×D4—3821no—
288C2×D4—3821no—
289C2×D4—3822no—
290C2×D4—3821no—
291C2×D4—3821no—
292C2×D4—3821no—
293C2×D4—1911no—
294C2×D4—1911no—
295C2×D4—1911no—
296C2×D4—1911no—
297C2×D4—7641no—
298C2×D4—7641no—
299C2×D4C22×D19111yescharacteristic
300C22×C4—3821no—
301C22×C4—3821no—
302C22×C4—3821no—
303C22×C4—3821no—
304C22×C4—1911no—
305C22×C4C22×D19111yescharacteristic
306Q16—7641no—
307SD16—22no—
308SD16—3822no—
309SD16—3822no—
310SD16—3822no—
311D8—3822no—
312D8—3822no—
313D8—1914no—
314D8—7641no—
315M16—22no—
316M16—3822no—
317C2×C8—1911no—
318C2×C8—1911no—
319C2×C8C22×D19111yescharacteristic
320C2×C8C22×D19111yescharacteristic
321C4⋊C4—21no—
322C4⋊C4—3821no—
323C4⋊C4—1911no—
324C4⋊C4C22×D19111yescharacteristic
325C4⋊C4C22×D19111yescharacteristic
326C22⋊C4—21no—
327C22⋊C4—21no—
328C22⋊C4—3821no—
329C22⋊C4—3821no—
330C22⋊C4—3821no—
331C22⋊C4—3821no—
332C22⋊C4—3821no—
333C22⋊C4—3821no—
334C22⋊C4—3821no—
335C22⋊C4—3821no—
336C22⋊C4—1911no—
337C22⋊C4—1911no—
338C42—1911no—
339C23—21no—
340C23—3821no—
341C23—3821no—
342C23—3821no—
343C23—1911no—
344C23—15281no—
345C23—15281no—
346C23—7642no—
347C23—7642no—
348C23—7642no—
349C23—7642no—
350C23D4×D19111yescharacteristic
351Q8—21no—
352Q8—21no—
353Q8—3821no—
354Q8—3821no—
355D4—21no—
356D4—21no—
357D4—81no—
358D4—3821no—
359D4—3821no—
360D4—3821no—
361D4—3821no—
362D4—1912no—
363D4—1912no—
364D4—15281no—
365D4—7642no—
366D4—7642no—
367D4—7641no—
368D4—7642no—
369D4—7642no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—3821no—
376C2×C4—3821no—
377C2×C4—3821no—
378C2×C4—1911no—
379C2×C4—1911no—
380C2×C4—1911no—
381C2×C4—15281no—
382C2×C4—15281no—
383C2×C4—7642no—
384C2×C4—7642no—
385C2×C4D4×D19111yescharacteristic
386C2×C4C23×D19111yescharacteristic
387C8—21no—
388C8—3821no—
389C8—3821no—
390C8D4×D19112yes—
391C22—21no—
392C22—21no—
393C22—21no—
394C22—41no—
395C22—41no—
396C22—41no—
397C22—3821no—
398C22—3821no—
399C22—3821no—
400C22—3822no—
401C22—3822no—
402C22—15284no—
403C22—15284no—
404C22—7641no—
405C22—7641no—
406C22—7641no—
407C22—7641no—
408C22—7641no—
409C22—7641no—
410C22—7641no—
411C22C2×D4×D19111yescharacteristic
412C4—21no—
413C4—41no—
414C4—41no—
415C4—41no—
416C4—3821no—
417C4—3821no—
418C4—3821no—
419C4—7641no—
420C4C2×D4×D19111yescharacteristic
421C4#2025936811yescharacteristic
422C2—41no—
423C2—81no—
424C2—15281no—
425C2—7641no—
426C2—7642no—
427C2—7642no—
428C2#2713329711yescharacteristic
429C2#2713346211yescharacteristic
430C2#2713346711yescharacteristic
subgroupG/H
centerC22C2×D4×D191
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
Fitting#468×C191C2
socleC22×C191C22×D4
hypercenter#468D191
hypocenterC191#3273
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. C2×#20259350C2semidirect
2. C2×#20259349C2semidirect
3. #27133310C2semidirect
4. #27133304C2semidirect
5. #27133044C2semidirect
6. #27132910C2semidirect
7. #27132881C2semidirect
8. #27132877C2semidirect
9. #27132828C2semidirect
10. #27132823C2semidirect
11. #27132798C2semidirect
12. #27132796C2semidirect
13. #27132745C2semidirect
14. #27132739C2semidirect
15. #468×C191C2semidirect
18. C22×D764C22semidirect
19. C2×D4×D191C22semidirect
22. C2×#15690277C22semidirect
23. C2×#15690276C22semidirect
24. C2×#15690276C22semidirect
25. C2×#15690274C22semidirect
30. C2×#15690270C22semidirect
31. #20259311C22semidirect
32. #20259309C22semidirect
35. #20259302C22semidirect
41. #20259284C22semidirect
42. #20259284C22semidirect
46. Dic191⋊C8C22semidirect
47. #20259245C22semidirect
48. #20259242C22semidirect
49. #20259242C22semidirect
50. #20259242C22semidirect
51. C22⋊(C191⋊C8)C22semidirect
52. (C2×D191)⋊C8C22semidirect
53. C8×Dic191C22semidirect
54. C2×SD16×C191C22semidirect
55. C2×M16×C191C22semidirect
56. C22⋊Q8×C191C22semidirect
58. #107×C191C22semidirect
59. #103×C191C22semidirect
60. #102×C191C22semidirect
91. D1528D4semidirect
215. C191#3273semidirect
223. #468D191semidirect
242. C2×SD16C2×D191semidirect
248. C2×M16C2×D191semidirect
254. C22⋊Q8C2×D191semidirect
266. #107C2×D191semidirect
269. #103C2×D191semidirect
273. #102C2×D191semidirect

Maximal subgroups

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

subgroupindex
C2×#202593502
C2×#202593492
#271333102
#271333042
#271330442
#271329102
#271328812
#271328772
#271328282
#271328232
#271327982
#271327962
#271327452
#271327392
#468×C1912
#3273191