Finite Group Databasebeta

← #37584013

Subgroups — #37584013

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×#20259348C211yesmaximal, characteristic, complemented
2C2×SD16×D191C211yesmaximal, characteristic, complemented
3#27133323C211yesmaximal, characteristic, complemented
4D4×C22⋊D191C211yesmaximal, characteristic, complemented
5#27133049C211yesmaximal, characteristic, complemented
6#27132911C211yesmaximal, characteristic, complemented
7#27132898C211yesmaximal, characteristic, complemented
8#27132880C211yesmaximal, characteristic, complemented
9#27132862C211yesmaximal, characteristic, complemented
10#27132810C211yesmaximal, characteristic, complemented
11#27132800C211yesmaximal, characteristic, complemented
12#27132798C211yesmaximal, characteristic, complemented
13#27132760C211yesmaximal, characteristic, complemented
14#27132737C211yesmaximal, characteristic, complemented
15#450×C191C211yesmaximal, characteristic, complemented
16C2×#15690287C2211yescharacteristic, complemented
17C2×Q8×D191C2211yescharacteristic, complemented
18C22×C22⋊D191—21no—
19C2×D4×D191C2211yescharacteristic
20#20259348—24no—
21C2×#15690276C2211yescharacteristic, complemented
22C2×#15690275C2211yescharacteristic
23C2×#15690275C2211yescharacteristic, complemented
24C2×#15690274C2211yescharacteristic, complemented
25SD16×D191—24no—
26C2×D1528C2211yescharacteristic, complemented
27C2×#15690272C2211yescharacteristic
28C2×#15690272C2211yescharacteristic, complemented
29C2×#15690271C2211yescharacteristic, complemented
30C2×C8×D191C2211yescharacteristic, complemented
31#20259318C2211yescharacteristic
32#20259311C2211yescharacteristic, complemented
33#20259308—21no—
34#20259300—21no—
35#20259297C2211yescharacteristic, complemented
36#20259296C2211yescharacteristic
37#20259295—21no—
38#20259294—21no—
39#20259293C2211yescharacteristic
40#20259291—21no—
41#20259290—21no—
42#20259287C2211yescharacteristic, complemented
43D4×Dic191C2211yescharacteristic
44#20259284—21no—
45C4×C22⋊D191C2211yescharacteristic
46#20259255C2211yescharacteristic, complemented
47Dic191⋊C8C2211yescharacteristic, complemented
48#20259247C2211yescharacteristic, complemented
49#20259246C2211yescharacteristic, complemented
50#20259244C2211yescharacteristic, complemented
51#20259242C2211yescharacteristic, complemented
52(C2×D191)⋊C8C2211yescharacteristic, complemented
53#20259232C2211yescharacteristic, complemented
54C22×D4×C191C2211yescharacteristic
55C2×SD16×C191C2211yescharacteristic
56C2×SD16×C191C2211yescharacteristic, complemented
57C22⋊Q8×C191C2211yescharacteristic, complemented
58#102×C191C2211yescharacteristic
59#102×C191C2211yescharacteristic, complemented
60C22⋊C8×C191C2211yescharacteristic, complemented
61C23×D191—21no—
62#15690287—22no—
63#15690287—41no—
64C2×Dic382—21no—
65C2×Dic382C2311yescharacteristic
66Q8×D191—22no—
67Q8×D191—41no—
68C2×D764—21no—
69C2×D764C2311yescharacteristic
70C2×D764C2311yescharacteristic
71C2×C22⋊D191—21no—
72C2×C22⋊D191—21no—
73C2×C22⋊D191—21no—
74C2×C22⋊D191—21no—
75C2×C22⋊D191—21no—
76C2×C22⋊D191—21no—
77C2×C22⋊D191—44no—
78C2×C22⋊D191D411yescharacteristic
79D4×D191—22no—
80D4×D191—41no—
81C22×Dic191—21no—
82C2×C4×D191—21no—
83C2×C4×D191—21no—
84C2×C4×D191C2311yescharacteristic
85#15690276—22no—
86#15690275—22no—
87#15690275D414yescomplemented
88#15690274—22no—
89D1528—41no—
90#15690272—22no—
91#15690272—22no—
92#15690271—22no—
93C2×C191⋊C8C2311yescharacteristic
94C2×C191⋊C8C2311yescharacteristic
95C8×D191—22no—
96C4⋊Dic191C2311yescharacteristic
97Dic191⋊C4—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)⋊C4D411yescharacteristic
107C4×Dic191—21no—
108C4×Dic191C2311yescharacteristic
109C24×C191—21no—
110C2×Q8×C191C2311yescharacteristic
111C2×D4×C191—22no—
112C2×D4×C191—21no—
113C2×D4×C191—41no—
114C2×D4×C191C2311yescharacteristic
115C2×D4×C191C2311yescharacteristic
116C22×C4×C191C2311yescharacteristic
117SD16×C191—22no—
118SD16×C191—41no—
119C2×C8×C191C2311yescharacteristic
120C2×C8×C191C2311yescharacteristic
121C4⋊C4×C191—21no—
122C4⋊C4×C191C2311yescharacteristic
123C22⋊C4×C191—21no—
124C22×D191—21no—
125C22×D191—21no—
126C22×D191—42no—
127C22×D191—42no—
128C22×D191—81no—
129C22×D191C2×D411yescharacteristic
130Dic382—21no—
131Dic382—42no—
132Dic382C2×D412yescomplemented
133D764—21no—
134D764—21no—
135D764—21no—
136D764—21no—
137D764—42no—
138C22⋊D191—22no—
139C22⋊D191—42no—
140C22⋊D191—42no—
141C22⋊D191—42no—
142C22⋊D191—42no—
143C22⋊D191—42no—
144C22⋊D191—81no—
145C2×Dic191—21no—
146C2×Dic191—21no—
147C2×Dic191—42no—
148C2×Dic191C2×D411yescharacteristic
149C2×Dic191C2×D411yescharacteristic
150C2×Dic191C2×D411yescharacteristic
151C4×D191—21no—
152C4×D191—21no—
153C4×D191—42no—
154C4×D191—42no—
155C191⋊C8—21no—
156C191⋊C8C2×D412yescomplemented
157C23×C191—21no—
158C23×C191—21no—
159C23×C191—42no—
160C23×C191—42no—
161C23×C191—81no—
162C23×C191C2×D411yescharacteristic
163Q8×C191—21no—
164Q8×C191—21no—
165D4×C191—21no—
166D4×C191—21no—
167D4×C191—21no—
168D4×C191—42no—
169D4×C191C2×D412yes—
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×C191—21no—
178C22×C191—21no—
179C22×C191—22no—
180C22×C191—21no—
181C22×C191—21no—
182C22×C191—41no—
183C22×C191—41no—
184C22×C191—41no—
185C22×C191—41no—
186C22×C191—41no—
187C22×C191—84no—
188C22×C191C22×D411yescharacteristic
189C2×D191—21no—
190C2×D191—21no—
191C2×D191—41no—
192C2×D191—41no—
193C2×D191—41no—
194C2×D191—41no—
195C2×D191—41no—
196C2×D191—41no—
197C2×D191—84no—
198C2×D191AGL(1,ℤ/8)12yescomplemented
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—42no—
210C2×C191—41no—
211C2×C191—81no—
212C2×C191D4211yescharacteristic
213C2×C191C2×AGL(1,ℤ/8)11yescharacteristic
214C2×C191#57711yescharacteristic
215D191—22no—
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#450D19111yescharacteristic, complemented
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×D4C2×D19111yescharacteristic
240AGL(1,ℤ/8)—3824no—
241C2×SD16—3824no—
242C2×SD16—1911no—
243C2×SD16—1911no—
244C2×SD16—1911no—
245C2×SD16—1911no—
246C2×SD16C2×D19111yescharacteristic
247C2×SD16C2×D19111yescharacteristic, complemented
248C2×D8—1911no—
249C2×D8—1911no—
250C2×D8—1911no—
251C2×M16—1911no—
252C22×C8—1911no—
253#127—1911no—
254#124—3821no—
255#124—1911no—
256C22⋊Q8—3821no—
257C22⋊Q8C2×D19111yescharacteristic, complemented
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×D4—1911no—
269#107—1911no—
270C4⋊C8—1911no—
271#103—1911no—
272#103—1911no—
273#102—1911no—
274#102—1911no—
275#102C2×D19111yescharacteristic
276#102C2×D19111yescharacteristic, complemented
277C22⋊C8—1911no—
278C22⋊C8C2×D19111yescharacteristic, complemented
279#97—1911no—
280C24—21no—
281C24—3821no—
282#41—3822no—
283#41—7641no—
284C2×Q8—3821no—
285C2×Q8—3822no—
286C2×Q8—1911no—
287C2×Q8—7641no—
288C2×Q8C22×D19111yescharacteristic
289C2×D4—22no—
290C2×D4—21no—
291C2×D4—41no—
292C2×D4—3821no—
293C2×D4—3822no—
294C2×D4—3821no—
295C2×D4—3821no—
296C2×D4—3821no—
297C2×D4—3821no—
298C2×D4—3821no—
299C2×D4—3821no—
300C2×D4—1911no—
301C2×D4—1911no—
302C2×D4—1911no—
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—22no—
313SD16—41no—
314SD16—3822no—
315SD16—3822no—
316SD16—3822no—
317SD16—1914no—
318D8—3822no—
319D8—3822no—
320D8—7641no—
321M16—3822no—
322C2×C8—3822no—
323C2×C8—1911no—
324C2×C8—1911no—
325C2×C8C22×D19111yescharacteristic
326C2×C8C22×D19111yescharacteristic
327C4⋊C4—21no—
328C4⋊C4—3821no—
329C4⋊C4—1911no—
330C4⋊C4—1911no—
331C4⋊C4C22×D19111yescharacteristic
332C22⋊C4—21no—
333C22⋊C4—3821no—
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—42no—
346C23—42no—
347C23—81no—
348C23—3821no—
349C23—3821no—
350C23—1911no—
351C23—15281no—
352C23—7642no—
353C23—7642no—
354C23D4×D19111yescharacteristic
355Q8—21no—
356Q8—21no—
357Q8—3821no—
358Q8—1912no—
359Q8—7642no—
360D4—21no—
361D4—21no—
362D4—21no—
363D4—42no—
364D4—3822no—
365D4—3821no—
366D4—3821no—
367D4—3821no—
368D4—3821no—
369D4—15281no—
370D4—7642no—
371D4—7642no—
372D4—7642no—
373D4—7642no—
374D4—7642no—
375D4—7642no—
376D4D4×D19112yes—
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—21no—
395C8—3821no—
396C8—1912no—
397C22—21no—
398C22—21no—
399C22—21no—
400C22—22no—
401C22—41no—
402C22—41no—
403C22—41no—
404C22—41no—
405C22—41no—
406C22—84no—
407C22—3821no—
408C22—3821no—
409C22—1912no—
410C22—15284no—
411C22—7641no—
412C22—7641no—
413C22—7641no—
414C22—7641no—
415C22—7641no—
416C22—7641no—
417C22C2×D4×D19111yescharacteristic
418C4—21no—
419C4—41no—
420C4—41no—
421C4—3821no—
422C4—3821no—
423C4—3821no—
424C4—3822no—
425C4—7641no—
426C4C2×D4×D19111yescharacteristic
427C4#2025936711yescharacteristic
428C2—41no—
429C2—42no—
430C2—81no—
431C2—3822no—
432C2—15281no—
433C2—15281no—
434C2#2713329511yescharacteristic
435C2AGL(1,ℤ/8)×D19111yescharacteristic
436C2#2713346911yescharacteristic
subgroupG/H
centerC22C2×D4×D191
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
Fitting#450×C191C2
socleC22×C191C22×D4
hypercenter#450D191
hypocenterC191#3265
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. C2×#20259348C2semidirect
2. C2×SD16×D191C2semidirect
3. #27133323C2semidirect
4. D4×C22⋊D191C2semidirect
5. #27133049C2semidirect
6. #27132911C2semidirect
7. #27132898C2semidirect
8. #27132880C2semidirect
9. #27132862C2semidirect
10. #27132810C2semidirect
11. #27132800C2semidirect
12. #27132798C2semidirect
13. #27132760C2semidirect
14. #27132737C2semidirect
15. #450×C191C2semidirect
16. C2×#15690287C22semidirect
17. C2×Q8×D191C22semidirect
21. C2×#15690276C22semidirect
23. C2×#15690275C22semidirect
24. C2×#15690274C22semidirect
26. C2×D1528C22semidirect
28. C2×#15690272C22semidirect
29. C2×#15690271C22semidirect
30. C2×C8×D191C22semidirect
32. #20259311C22semidirect
35. #20259297C22semidirect
42. #20259287C22semidirect
46. #20259255C22semidirect
47. Dic191⋊C8C22semidirect
48. #20259247C22semidirect
49. #20259246C22semidirect
50. #20259244C22semidirect
51. #20259242C22semidirect
52. (C2×D191)⋊C8C22semidirect
53. #20259232C22semidirect
56. C2×SD16×C191C22semidirect
57. C22⋊Q8×C191C22semidirect
59. #102×C191C22semidirect
60. C22⋊C8×C191C22semidirect
87. #15690275D4semidirect
132. Dic382C2×D4semidirect
156. C191⋊C8C2×D4semidirect
198. C2×D191AGL(1,ℤ/8)semidirect
218. C191#3265semidirect
229. #450D191semidirect
247. C2×SD16C2×D191semidirect
257. C22⋊Q8C2×D191semidirect
276. #102C2×D191semidirect
278. C22⋊C8C2×D191semidirect

Maximal subgroups

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

subgroupindex
C2×#202593482
C2×SD16×D1912
#271333232
D4×C22⋊D1912
#271330492
#271329112
#271328982
#271328802
#271328622
#271328102
#271328002
#271327982
#271327602
#271327372
#450×C1912
#3265191