Finite Group Databasebeta

← #37584020

Subgroups — #37584020

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×#20259350C211yesmaximal, characteristic, complemented
2C22×#15690276C211yesmaximal, characteristic, complemented
3#27133322C211yesmaximal, characteristic, complemented
4D4×D764C211yesmaximal, characteristic, complemented
5#27133052C211yesmaximal, characteristic, complemented
6#27132909C211yesmaximal, characteristic, complemented
7#27132899C211yesmaximal, characteristic, complemented
8#27132879C211yesmaximal, characteristic, complemented
9#27132864C211yesmaximal, characteristic, complemented
10#27132816C211yesmaximal, characteristic, complemented
11#27132810C211yesmaximal, characteristic, complemented
12#27132804C211yesmaximal, characteristic, complemented
13#27132799C211yesmaximal, characteristic, complemented
14#27132765C211yesmaximal, characteristic, complemented
15#435×C191C211yesmaximal, characteristic, complemented
16C2×#15690287C2211yescharacteristic, complemented
17C22×Dic382C2211yescharacteristic, complemented
18C22×D764C2211yescharacteristic
19C22×D764C2211yescharacteristic
20C2×D4×D191—21no—
21#20259350—24no—
22C2×#15690276—24no—
23C2×#15690276C2211yescharacteristic
24C2×#15690276C2211yescharacteristic
25C2×#15690276C2211yescharacteristic, complemented
26C2×#15690276C2211yescharacteristic, complemented
27C2×#15690276C2211yescharacteristic, complemented
28C2×#15690276C2211yescharacteristic, complemented
29C2×D1528C2211yescharacteristic
30C2×D1528C2211yescharacteristic, complemented
31C2×D1528C2211yescharacteristic, complemented
32#20259319C2211yescharacteristic
33#20259310C2211yescharacteristic, complemented
34#20259308—21no—
35#20259301C2211yescharacteristic, complemented
36#20259300—21no—
37#20259297—21no—
38#20259296C2211yescharacteristic
39#20259296C2211yescharacteristic
40#20259296C2211yescharacteristic, complemented
41#20259294—21no—
42#20259292—21no—
43#20259292—21no—
44C4×Dic382C2211yescharacteristic, complemented
45C4×D764C2211yescharacteristic
46(C22⋊D191)⋊C4—21no—
47#20259256C2211yescharacteristic, complemented
48#20259247C2211yescharacteristic, complemented
49#20259247C2211yescharacteristic, complemented
50#20259244C2211yescharacteristic
51#20259244C2211yescharacteristic, complemented
52#20259244C2211yescharacteristic, complemented
53#20259244C2211yescharacteristic, complemented
54C2×M16×C191C2211yescharacteristic, complemented
55C22×C8×C191C2211yescharacteristic, complemented
56C4×D4×C191C2211yescharacteristic
57C4⋊C8×C191C2211yescharacteristic, complemented
58C22⋊C8×C191C2211yescharacteristic, complemented
59C22⋊C8×C191C2211yescharacteristic, complemented
60#97×C191C2211yescharacteristic, complemented
61C23×D191—21no—
62C23×D191—21no—
63#15690287—22no—
64#15690287—41no—
65C2×Dic382—22no—
66C2×Dic382—21no—
67C2×Dic382—41no—
68C2×Dic382C2311yescharacteristic
69C2×Dic382C2311yescharacteristic
70C2×D764—22no—
71C2×D764—22no—
72C2×D764—21no—
73C2×D764—21no—
74C2×D764—21no—
75C2×D764—21no—
76C2×D764—41no—
77C2×D764—41no—
78C2×D764C2311yescharacteristic
79C2×D764C2311yescharacteristic
80C2×D764C2311yescharacteristic
81C2×D764C2311yescharacteristic
82C2×C22⋊D191—21no—
83C2×C22⋊D191—21no—
84C2×C22⋊D191—21no—
85C2×C22⋊D191—21no—
86D4×D191—44no—
87C22×Dic191—21no—
88C2×C4×D191—21no—
89C2×C4×D191—21no—
90#15690276—22no—
91#15690276—22no—
92#15690276—22no—
93#15690276—22no—
94#15690276—41no—
95#15690276D414yescomplemented
96D1528—22no—
97D1528—22no—
98D1528—41no—
99C4⋊Dic191C2311yescharacteristic
100C4⋊Dic191C2311yescharacteristic
101Dic191⋊C4—21no—
102Dic191⋊C4—21no—
103C22⋊Dic191—21no—
104(C2×D191)⋊C4—21no—
105(C2×D191)⋊C4—21no—
106(C2×D191)⋊C4—21no—
107(C2×D191)⋊C4—21no—
108(C2×D191)⋊C4—21no—
109(C2×D191)⋊C4—21no—
110C4×Dic191—21no—
111C2×D4×C191D411yescharacteristic
112C22×C4×C191C2311yescharacteristic
113C22×C4×C191C2311yescharacteristic
114M16×C191—22no—
115C2×C8×C191—22no—
116C2×C8×C191C2311yescharacteristic
117C2×C8×C191C2311yescharacteristic
118C2×C8×C191C2311yescharacteristic
119C2×C8×C191C2311yescharacteristic
120C4⋊C4×C191D411yescharacteristic
121C22⋊C4×C191D411yescharacteristic
122C22⋊C4×C191D411yescharacteristic
123C42×C191C2311yescharacteristic
124C22×D191—21no—
125C22×D191—21no—
126C22×D191—21no—
127C22×D191—21no—
128C22×D191—42no—
129C22×D191—42no—
130C22×D191—42no—
131C22×D191—42no—
132C22×D191—81no—
133C22×D191—81no—
134Dic382—21no—
135Dic382—21no—
136Dic382—21no—
137Dic382—42no—
138Dic382C2×D412yescomplemented
139D764—21no—
140D764—21no—
141D764—21no—
142D764—21no—
143D764—21no—
144D764—21no—
145D764—21no—
146D764—42no—
147D764—42no—
148D764—42no—
149D764—81no—
150D764C2×D412yes—
151C22⋊D191—42no—
152C22⋊D191—42no—
153C22⋊D191—42no—
154C22⋊D191—42no—
155C2×Dic191—21no—
156C2×Dic191—21no—
157C2×Dic191—21no—
158C2×Dic191—21no—
159C2×Dic191—42no—
160C4×D191—42no—
161C4×D191—42no—
162C23×C191C2×D411yescharacteristic
163C23×C191C2×D411yescharacteristic
164D4×C191—22no—
165C2×C4×C191—21no—
166C2×C4×C191—21no—
167C2×C4×C191—21no—
168C2×C4×C191—21no—
169C2×C4×C191C2×D411yescharacteristic
170C2×C4×C191C2×D411yescharacteristic
171C2×C4×C191C2×D411yescharacteristic
172C2×C4×C191C2×D411yescharacteristic
173C2×C4×C191C2411yescharacteristic
174C8×C191—21no—
175C8×C191—21no—
176C8×C191—21no—
177C8×C191C2×D412yescomplemented
178C22×C191—21no—
179C22×C191—21no—
180C22×C191—21no—
181C22×C191—21no—
182C22×C191—21no—
183C22×C191AGL(1,ℤ/8)12yescomplemented
184C22×C191C22×D411yescharacteristic
185C2×D191—22no—
186C2×D191—41no—
187C2×D191—41no—
188C2×D191—41no—
189C2×D191—41no—
190C2×D191—41no—
191C2×D191—41no—
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—21no—
201C4×C191—21no—
202C4×C191—21no—
203C4×C191C22×D411yescharacteristic
204C4×C191H(5,𝔽2)11yescharacteristic
205Dic191—22no—
206Dic191—41no—
207Dic191—41no—
208Dic191—41no—
209C2×C191—22no—
210C2×C191—41no—
211C2×C191D4211yescharacteristic
212C2×C191C2×AGL(1,ℤ/8)11yescharacteristic
213C2×C191#57711yescharacteristic
214D191—42no—
215D191—81no—
216D191—81no—
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#466—1911no—
227#463—1911no—
228#460—1911no—
229#450—1911no—
230#449—1911no—
231#448—1911no—
232#447—1911no—
233#440—1911no—
234#435D19111yescharacteristic, complemented
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×D8—1911no—
251C2×M16C2×D19111yescharacteristic, complemented
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#121—1911no—
262#121—1911no—
263#120—3821no—
264#120—3821no—
265C4×Q8—1911no—
266C4×D4—3821no—
267C4×D4—1911no—
268C4×D4C2×D19111yescharacteristic
269#107—1911no—
270C4⋊C8C2×D19111yescharacteristic, complemented
271#103—1911no—
272#103—1911no—
273#102—1911no—
274#102—1911no—
275#102—1911no—
276#102—1911no—
277C22⋊C8C2×D19111yescharacteristic, complemented
278C22⋊C8C2×D19111yescharacteristic, complemented
279#97C2×D19111yescharacteristic, complemented
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—3822no—
290C2×D4—3822no—
291C2×D4—3821no—
292C2×D4—3821no—
293C2×D4—3821no—
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—1911no—
303C2×D4—7641no—
304C2×D4—7641no—
305C2×D4—7644no—
306C2×D4D76411yescharacteristic
307C22×C4—3821no—
308C22×C4—3821no—
309C22×C4—3821no—
310C22×C4C22×D19111yescharacteristic
311C22×C4C22×D19111yescharacteristic
312SD16—3822no—
313SD16—3822no—
314SD16—3822no—
315SD16—3822no—
316SD16—1914no—
317SD16—7641no—
318D8—3822no—
319D8—3822no—
320D8—7641no—
321M16—22no—
322C2×C8—22no—
323C2×C8C22×D19111yescharacteristic
324C2×C8C22×D19111yescharacteristic
325C2×C8C22×D19111yescharacteristic
326C2×C8C22×D19111yescharacteristic
327C4⋊C4—3821no—
328C4⋊C4—3821no—
329C4⋊C4—1911no—
330C4⋊C4—1911no—
331C4⋊C4D76411yescharacteristic
332C22⋊C4—3821no—
333C22⋊C4—3821no—
334C22⋊C4—3821no—
335C22⋊C4—3821no—
336C22⋊C4—3821no—
337C22⋊C4—3821no—
338C22⋊C4—3821no—
339C22⋊C4D76411yescharacteristic
340C22⋊C4D76411yescharacteristic
341C42—3821no—
342C42C22×D19111yescharacteristic
343C23—3821no—
344C23—3821no—
345C23—3821no—
346C23—3821no—
347C23—15281no—
348C23—15281no—
349C23—7642no—
350C23—7642no—
351C23—7642no—
352C23—7642no—
353C23C2×D76411yescharacteristic
354C23C2×D76411yescharacteristic
355Q8—3821no—
356Q8—3821no—
357Q8—3821no—
358Q8—1912no—
359Q8—7642no—
360D4—22no—
361D4—3821no—
362D4—3821no—
363D4—3821no—
364D4—3821no—
365D4—3821no—
366D4—3821no—
367D4—3821no—
368D4—1912no—
369D4—15281no—
370D4—7642no—
371D4—7642no—
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—7642no—
386C2×C4—7642no—
387C2×C4—7642no—
388C2×C4C2×D76411yescharacteristic
389C2×C4C2×D76411yescharacteristic
390C2×C4C2×D76411yescharacteristic
391C2×C4C2×D76411yescharacteristic
392C2×C4C23×D19111yescharacteristic
393C8—21no—
394C8—21no—
395C8—21no—
396C8D4×D19112yescomplemented
397C22—21no—
398C22—21no—
399C22—21no—
400C22—21no—
401C22—21no—
402C22—3822no—
403C22—15284no—
404C22—15284no—
405C22—7641no—
406C22—7641no—
407C22—7641no—
408C22—7641no—
409C22—7641no—
410C22—7641no—
411C22—7641no—
412C22—7641no—
413C22—7641no—
414C22—7641no—
415C22—7641no—
416C22#2025935012yescomplemented
417C22C22×D76411yescharacteristic
418C4—21no—
419C4—21no—
420C4—21no—
421C4—21no—
422C4—3822no—
423C4—7641no—
424C4—7641no—
425C4—7641no—
426C4C2×D4×D19111yescharacteristic
427C4#2025936811yescharacteristic
428C2—22no—
429C2—41no—
430C2—15281no—
431C2—15281no—
432C2—15281no—
433C2—7642no—
434C2D4×D76411yescharacteristic
435C2C2×#2025935011yescharacteristic
436C2#2713347511yescharacteristic
subgroupG/H
centerC22C22×D764
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
Fitting#435×C191C2
socleC22×C191C22×D4
hypercenter#435D191
hypocenterC191#3265
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C22 ◁ C4×D4 ◁ #435

Complements

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

NH ≅ G/Ntype
1. C2×#20259350C2semidirect
2. C22×#15690276C2semidirect
3. #27133322C2semidirect
4. D4×D764C2semidirect
5. #27133052C2semidirect
6. #27132909C2semidirect
7. #27132899C2semidirect
8. #27132879C2semidirect
9. #27132864C2semidirect
10. #27132816C2semidirect
11. #27132810C2semidirect
12. #27132804C2semidirect
13. #27132799C2semidirect
14. #27132765C2semidirect
15. #435×C191C2semidirect
16. C2×#15690287C22semidirect
17. C22×Dic382C22semidirect
25. C2×#15690276C22semidirect
26. C2×#15690276C22semidirect
27. C2×#15690276C22semidirect
28. C2×#15690276C22semidirect
30. C2×D1528C22semidirect
31. C2×D1528C22semidirect
33. #20259310C22semidirect
35. #20259301C22semidirect
40. #20259296C22semidirect
44. C4×Dic382C22semidirect
47. #20259256C22semidirect
48. #20259247C22semidirect
49. #20259247C22semidirect
51. #20259244C22semidirect
52. #20259244C22semidirect
53. #20259244C22semidirect
54. C2×M16×C191C22semidirect
55. C22×C8×C191C22semidirect
57. C4⋊C8×C191C22semidirect
58. C22⋊C8×C191C22semidirect
59. C22⋊C8×C191C22semidirect
60. #97×C191C22semidirect
95. #15690276D4semidirect
138. Dic382C2×D4semidirect
177. C8×C191C2×D4semidirect
183. C22×C191AGL(1,ℤ/8)semidirect
218. C191#3265semidirect
234. #435D191semidirect
251. C2×M16C2×D191semidirect
252. C22×C8C2×D191semidirect
270. C4⋊C8C2×D191semidirect
277. C22⋊C8C2×D191semidirect
278. C22⋊C8C2×D191semidirect
279. #97C2×D191semidirect
396. C8D4×D191semidirect
416. C22#20259350semidirect

Maximal subgroups

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

subgroupindex
C2×#202593502
C22×#156902762
#271333222
D4×D7642
#271330522
#271329092
#271328992
#271328792
#271328642
#271328162
#271328102
#271328042
#271327992
#271327652
#435×C1912
#3265191