Finite Group Databasebeta

← #37584126

Subgroups — #37584126

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×#20259349C211yesmaximal, characteristic, complemented
2C2×#20259346C211yesmaximal, characteristic, complemented
3#27133308C211yesmaximal, characteristic, complemented
4#27133306C211yesmaximal, characteristic, complemented
5#27133045C211yesmaximal, characteristic, complemented
6#27132912C211yesmaximal, characteristic, complemented
7#27132908C211yesmaximal, characteristic, complemented
8#27132879C211yesmaximal, characteristic, complemented
9#27132828C211yesmaximal, characteristic, complemented
10#27132822C211yesmaximal, characteristic, complemented
11#27132798C211yesmaximal, characteristic, complemented
12#27132795C211yesmaximal, characteristic, complemented
13C4×#15690272C211yesmaximal, characteristic, complemented
14#27132740C211yesmaximal, characteristic, complemented
15#463×C191C211yesmaximal, characteristic, complemented
16C2×#15690288C2211yescharacteristic
17C2×#15690286C2211yescharacteristic
18C2×D4×D191C2211yescharacteristic, complemented
19C2×D4×D191C2211yescharacteristic, complemented
20#20259349—24no—
21#20259346—24no—
22C2×#15690277C2211yescharacteristic, complemented
23C2×#15690276C2211yescharacteristic, complemented
24C2×#15690275C2211yescharacteristic, complemented
25C2×#15690274C2211yescharacteristic, complemented
26C2×D1528C2211yescharacteristic
27C2×#15690272C2211yescharacteristic
28C2×#15690272C2211yescharacteristic
29C2×#15690270C2211yescharacteristic, complemented
30C2×#15690270C2211yescharacteristic, complemented
31#20259310C2211yescharacteristic, complemented
32#20259306—21no—
33#20259304—21no—
34#20259302C2211yescharacteristic, complemented
35#20259298C2211yescharacteristic, complemented
36#20259297C2211yescharacteristic
37#20259296—21no—
38#20259295—21no—
39#20259293C2211yescharacteristic
40#20259292—21no—
41#20259292—21no—
42C4×D764C2211yescharacteristic
43C4×D764C2211yescharacteristic, complemented
44C2×(C2×D191)⋊C4—21no—
45C22⋊C4×D191—21no—
46#20259255C2211yescharacteristic, complemented
47#20259247C2211yescharacteristic, complemented
48#20259244C2211yescharacteristic, complemented
49#20259243C2211yescharacteristic, complemented
50#20259242C2211yescharacteristic, complemented
51(C2×D191)⋊C8C2211yescharacteristic, complemented
52(C2×D191)⋊C8C2211yescharacteristic, complemented
53C4×C191⋊C8C2211yescharacteristic, complemented
54C2×SD16×C191C2211yescharacteristic, complemented
55C2×D8×C191C2211yescharacteristic
56#124×C191C2211yescharacteristic, complemented
57C4×D4×C191C2211yescharacteristic, complemented
58C4⋊C8×C191C2211yescharacteristic, complemented
59#103×C191C2211yescharacteristic, complemented
60#102×C191C2211yescharacteristic, complemented
61C23×D191—21no—
62C23×D191—21no—
63#15690288—22no—
64#15690288—41no—
65#15690286—22no—
66#15690286—41no—
67C2×Dic382C2311yescharacteristic
68C2×D764—21no—
69C2×D764—21no—
70C2×D764D411yescharacteristic
71C2×D764C2311yescharacteristic
72C2×D764C2311yescharacteristic
73C2×C22⋊D191—21no—
74C2×C22⋊D191—21no—
75C2×C22⋊D191—21no—
76C2×C22⋊D191—21no—
77D4×D191—22no—
78D4×D191—22no—
79D4×D191—41no—
80D4×D191—41no—
81C22×Dic191—21no—
82C2×C4×D191—21no—
83C2×C4×D191—21no—
84C2×C4×D191C2311yescharacteristic
85C2×C4×D191C2311yescharacteristic
86#15690277—41no—
87#15690276—22no—
88#15690275—22no—
89#15690274—22no—
90D1528—22no—
91#15690272—41no—
92#15690272D414yescomplemented
93#15690270—22no—
94#15690270—22no—
95C2×C191⋊C8C2311yescharacteristic
96C2×C191⋊C8C2311yescharacteristic
97C4⋊Dic191—21no—
98C4⋊Dic191D411yescharacteristic
99C4⋊Dic191C2311yescharacteristic
100Dic191⋊C4—21no—
101C22⋊Dic191—21no—
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
109(C2×D191)⋊C4D411yescharacteristic
110C2×Q8×C191C2311yescharacteristic
111C2×D4×C191C2311yescharacteristic
112C2×D4×C191C2311yescharacteristic
113C22×C4×C191—21no—
114SD16×C191—22no—
115D8×C191—22no—
116C2×C8×C191C2311yescharacteristic
117C2×C8×C191C2311yescharacteristic
118C4⋊C4×C191C2311yescharacteristic
119C22⋊C4×C191—21no—
120C22⋊C4×C191—21no—
121C22⋊C4×C191—21no—
122C42×C191C2311yescharacteristic
123C22×D191—21no—
124C22×D191—21no—
125C22×D191—42no—
126C22×D191—42no—
127C22×D191—42no—
128C22×D191—42no—
129C22×D191—81no—
130C22×D191—81no—
131C22×D191C2×D411yescharacteristic
132C22×D191C2×D411yescharacteristic
133Dic382—21no—
134Dic382—21no—
135D764—21no—
136D764—21no—
137D764—21no—
138D764—41no—
139D764—42no—
140D764—81no—
141D764C2×D412yes—
142C22⋊D191—42no—
143C22⋊D191—42no—
144C22⋊D191—42no—
145C22⋊D191—81no—
146C2×Dic191—21no—
147C2×Dic191—21no—
148C2×Dic191—42no—
149C2×Dic191C2×D411yescharacteristic
150C2×Dic191C2×D411yescharacteristic
151C4×D191—21no—
152C4×D191—21no—
153C4×D191—21no—
154C4×D191—21no—
155C4×D191—42no—
156C4×D191—81no—
157C191⋊C8—21no—
158C191⋊C8C2×D412yes—
159C23×C191—21no—
160C23×C191—21no—
161Q8×C191—21no—
162Q8×C191—21no—
163D4×C191—21no—
164D4×C191—21no—
165D4×C191—21no—
166D4×C191C2×D412yes—
167C2×C4×C191—21no—
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×C191C22×D411yescharacteristic
182C2×D191—21no—
183C2×D191—21no—
184C2×D191—22no—
185C2×D191—21no—
186C2×D191—21no—
187C2×D191—21no—
188C2×D191—21no—
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—21no—
198C4×C191—41no—
199C4×C191—41no—
200C4×C191C22×D411yescharacteristic
201C4×C191H(5,𝔽2)11yescharacteristic
202Dic191—21no—
203Dic191—21no—
204Dic191—41no—
205Dic191—41no—
206C2×C191—42no—
207C2×C191—81no—
208C2×C191D4211yescharacteristic
209C2×C191#57511yescharacteristic
210C2×C191#57611yescharacteristic
211D191—41no—
212D191—41no—
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#468—1911no—
224#463—1911no—
225#463D19111yescharacteristic, complemented
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×D8C2×D19111yescharacteristic
247C2×M16—1911no—
248C2×M16—1911no—
249#124—1911no—
250#124C2×D19111yescharacteristic, complemented
251#123—3821no—
252#123—3821no—
253C22⋊Q8—1911no—
254C22⋊Q8—1911no—
255#121—3821no—
256#121—3821no—
257#121—1911no—
258#121—1911no—
259#120—3821no—
260#120—3821no—
261C4×D4—1911no—
262C4×D4—1911no—
263C4×D4C2×D19111yescharacteristic, complemented
264C2×C22⋊C4—3821no—
265C2×C22⋊C4—3821no—
266#107—1911no—
267C4⋊C8C2×D19111yescharacteristic, complemented
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—3821no—
286C2×D4—3821no—
287C2×D4—3822no—
288C2×D4—3822no—
289C2×D4—3821no—
290C2×D4—3821no—
291C2×D4—3821no—
292C2×D4—3821no—
293C2×D4—1911no—
294C2×D4—1911no—
295C2×D4—1911no—
296C2×D4—7641no—
297C2×D4—7641no—
298C2×D4C22×D19111yescharacteristic
299C2×D4C22×D19111yescharacteristic
300C22×C4—21no—
301C22×C4—3821no—
302C22×C4—3821no—
303C22×C4—3821no—
304C22×C4—1911no—
305C22×C4—1911no—
306Q16—7641no—
307SD16—22no—
308SD16—3822no—
309SD16—3822no—
310SD16—3822no—
311D8—22no—
312D8—3822no—
313D8—1914no—
314D8—7641no—
315M16—3822no—
316M16—3822no—
317C2×C8—1911no—
318C2×C8—1911no—
319C2×C8C22×D19111yescharacteristic
320C2×C8C22×D19111yescharacteristic
321C4⋊C4—3821no—
322C4⋊C4—3821no—
323C4⋊C4—1911no—
324C4⋊C4—1911no—
325C4⋊C4C22×D19111yescharacteristic
326C22⋊C4—21no—
327C22⋊C4—21no—
328C22⋊C4—21no—
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—
338C42C22×D19111yescharacteristic
339C23—21no—
340C23—21no—
341C23—3821no—
342C23—3821no—
343C23—1911no—
344C23—1911no—
345C23—15281no—
346C23—15281no—
347C23—7642no—
348C23—7642no—
349C23—7642no—
350C23—7642no—
351Q8—21no—
352Q8—21no—
353Q8—3821no—
354Q8—3821no—
355D4—21no—
356D4—21no—
357D4—21no—
358D4—3821no—
359D4—3821no—
360D4—3821no—
361D4—1912no—
362D4—15281no—
363D4—15281no—
364D4—7641no—
365D4—7642no—
366D4—7642no—
367D4—7642no—
368D4—7642no—
369D4D4×D19112yes—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—81no—
373C2×C4—3821no—
374C2×C4—3821no—
375C2×C4—3821no—
376C2×C4—3821no—
377C2×C4—3821no—
378C2×C4—3821no—
379C2×C4—1911no—
380C2×C4—1911no—
381C2×C4—15281no—
382C2×C4—7642no—
383C2×C4—7642no—
384C2×C4D4×D19111yescharacteristic
385C2×C4D4×D19111yescharacteristic
386C2×C4C23×D19111yescharacteristic
387C8—21no—
388C8—21no—
389C8—3821no—
390C8—1912no—
391C22—22no—
392C22—41no—
393C22—41no—
394C22—41no—
395C22—41no—
396C22—41no—
397C22—3821no—
398C22—3821no—
399C22—3821no—
400C22—3821no—
401C22—3821no—
402C22—3821no—
403C22—3822no—
404C22—15284no—
405C22—15284no—
406C22—7641no—
407C22—7641no—
408C22—7641no—
409C22—7641no—
410C22—7641no—
411C22C2×D4×D19111yescharacteristic
412C4—21no—
413C4—21no—
414C4—41no—
415C4—41no—
416C4—3821no—
417C4—3821no—
418C4—7641no—
419C4—7641no—
420C4C2×D4×D19111yescharacteristic
421C4#2025936811yescharacteristic
422C2—42no—
423C2—81no—
424C2—15281no—
425C2—7641no—
426C2—7641no—
427C2—7642no—
428C2#2713329711yescharacteristic
429C2#2713346011yescharacteristic
430C2#2713346711yescharacteristic
subgroupG/H
centerC22C2×D4×D191
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
Fitting#463×C191C2
socleC22×C191C22×D4
hypercenter#463D191
hypocenterC191#3273
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C22 ◁ C42 ◁ #463

Complements

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

NH ≅ G/Ntype
1. C2×#20259349C2semidirect
2. C2×#20259346C2semidirect
3. #27133308C2semidirect
4. #27133306C2semidirect
5. #27133045C2semidirect
6. #27132912C2semidirect
7. #27132908C2semidirect
8. #27132879C2semidirect
9. #27132828C2semidirect
10. #27132822C2semidirect
11. #27132798C2semidirect
12. #27132795C2semidirect
13. C4×#15690272C2semidirect
14. #27132740C2semidirect
15. #463×C191C2semidirect
18. C2×D4×D191C22semidirect
19. C2×D4×D191C22semidirect
22. C2×#15690277C22semidirect
23. C2×#15690276C22semidirect
24. C2×#15690275C22semidirect
25. C2×#15690274C22semidirect
29. C2×#15690270C22semidirect
30. C2×#15690270C22semidirect
31. #20259310C22semidirect
34. #20259302C22semidirect
35. #20259298C22semidirect
43. C4×D764C22semidirect
46. #20259255C22semidirect
47. #20259247C22semidirect
48. #20259244C22semidirect
49. #20259243C22semidirect
50. #20259242C22semidirect
51. (C2×D191)⋊C8C22semidirect
52. (C2×D191)⋊C8C22semidirect
53. C4×C191⋊C8C22semidirect
54. C2×SD16×C191C22semidirect
56. #124×C191C22semidirect
57. C4×D4×C191C22semidirect
58. C4⋊C8×C191C22semidirect
59. #103×C191C22semidirect
60. #102×C191C22semidirect
92. #15690272D4semidirect
215. C191#3273semidirect
225. #463D191semidirect
242. C2×SD16C2×D191semidirect
250. #124C2×D191semidirect
263. C4×D4C2×D191semidirect
267. C4⋊C8C2×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×#202593492
C2×#202593462
#271333082
#271333062
#271330452
#271329122
#271329082
#271328792
#271328282
#271328222
#271327982
#271327952
C4×#156902722
#271327402
#463×C1912
#3273191