Finite Group Databasebeta

← #37584009

Subgroups — #37584009

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×SD16×D191C211yesmaximal, characteristic, complemented
2#27133320C211yesmaximal, characteristic, complemented
3D4×C22⋊D191C211yesmaximal, characteristic, complemented
4#27133051C211yesmaximal, characteristic, complemented
5#27132909C211yesmaximal, characteristic, complemented
6#27132896C211yesmaximal, characteristic, complemented
7#27132876C211yesmaximal, characteristic, complemented
8#27132861C211yesmaximal, characteristic, complemented
9#27132814C211yesmaximal, characteristic, complemented
10#27132808C211yesmaximal, characteristic, complemented
11#27132800C211yesmaximal, characteristic, complemented
12#27132795C211yesmaximal, characteristic, complemented
13#27132764C211yesmaximal, characteristic, complemented
14#27132732C211yesmaximal, characteristic, complemented
15C2×AGL(1,ℤ/8)×C191C211yesmaximal, characteristic, complemented
16C2×Q8×D191C2211yescharacteristic, complemented
17C22×C22⋊D191—21no—
18C2×D4×D191C2211yescharacteristic
19C2×#15690276C2211yescharacteristic, complemented
20C2×#15690275C2211yescharacteristic
21C2×#15690275C2211yescharacteristic, complemented
22C2×#15690274C2211yescharacteristic, complemented
23SD16×D191—24no—
24C2×#15690272C2211yescharacteristic, complemented
25C2×C8×D191C2211yescharacteristic, complemented
26#20259318C2211yescharacteristic
27#20259309—21no—
28#20259309C2211yescharacteristic, complemented
29#20259301—21no—
30#20259301C2211yescharacteristic, complemented
31#20259296C2211yescharacteristic
32#20259294—21no—
33#20259293—21no—
34#20259293C2211yescharacteristic
35#20259293C2211yescharacteristic, complemented
36#20259291—21no—
37#20259290—21no—
38Q8×Dic191C2211yescharacteristic, complemented
39D4×Dic191C2211yescharacteristic
40C4×C22⋊D191—21no—
41C4×C22⋊D191C2211yescharacteristic
42#20259256C2211yescharacteristic, complemented
43Dic191⋊C8C2211yescharacteristic, complemented
44#20259247C2211yescharacteristic, complemented
45#20259246C2211yescharacteristic, complemented
46#20259244C2211yescharacteristic, complemented
47#20259243C2211yescharacteristic
48#20259243C2211yescharacteristic, complemented
49#20259243C2211yescharacteristic, complemented
50C22⋊(C191⋊C8)C2211yescharacteristic, complemented
51(C2×D191)⋊C8C2211yescharacteristic, complemented
52#20259232C2211yescharacteristic, complemented
53C2×#41×C191C2211yescharacteristic, complemented
54C22×D4×C191C2211yescharacteristic
55AGL(1,ℤ/8)×C191—24no—
56C2×SD16×C191C2211yescharacteristic
57C2×SD16×C191C2211yescharacteristic, complemented
58C2×D8×C191C2211yescharacteristic
59C2×D8×C191C2211yescharacteristic, complemented
60C2×M16×C191C2211yescharacteristic, complemented
61C23×D191—21no—
62C2×Dic382—21no—
63C2×Dic382C2311yescharacteristic
64Q8×D191—22no—
65Q8×D191—41no—
66C2×D764C2311yescharacteristic
67C2×C22⋊D191—21no—
68C2×C22⋊D191—21no—
69C2×C22⋊D191—21no—
70C2×C22⋊D191—21no—
71C2×C22⋊D191—21no—
72C2×C22⋊D191—21no—
73C2×C22⋊D191—44no—
74C2×C22⋊D191D411yescharacteristic
75D4×D191—22no—
76D4×D191—41no—
77C22×Dic191—21no—
78C2×C4×D191—21no—
79C2×C4×D191C2311yescharacteristic
80#15690276—22no—
81#15690275—22no—
82#15690275—41no—
83#15690274—22no—
84#15690272—41no—
85C2×C191⋊C8C2311yescharacteristic
86C2×C191⋊C8C2311yescharacteristic
87C8×D191—22no—
88C4⋊Dic191—21no—
89C4⋊Dic191C2311yescharacteristic
90C4⋊Dic191C2311yescharacteristic
91Dic191⋊C4—21no—
92Dic191⋊C4D411yescharacteristic
93C22⋊Dic191—21no—
94C22⋊Dic191—21no—
95C22⋊Dic191—21no—
96C22⋊Dic191—21no—
97C22⋊Dic191—21no—
98C22⋊Dic191—21no—
99C22⋊Dic191D411yescharacteristic
100(C2×D191)⋊C4—21no—
101(C2×D191)⋊C4D411yescharacteristic
102C4×Dic191—21no—
103C4×Dic191C2311yescharacteristic
104C24×C191—21no—
105#41×C191—22no—
106#41×C191—41no—
107C2×Q8×C191C2311yescharacteristic
108C2×D4×C191—22no—
109C2×D4×C191—21no—
110C2×D4×C191—21no—
111C2×D4×C191—41no—
112C2×D4×C191C2311yescharacteristic
113C2×D4×C191C2311yescharacteristic
114C2×D4×C191C2311yescharacteristic
115C22×C4×C191—21no—
116C22×C4×C191C2311yescharacteristic
117SD16×C191—22no—
118SD16×C191D414yescomplemented
119D8×C191—22no—
120D8×C191—22no—
121M16×C191—22no—
122C2×C8×C191C2311yescharacteristic
123C2×C8×C191C2311yescharacteristic
124C22×D191—21no—
125C22×D191—42no—
126C22×D191—42no—
127C22×D191—81no—
128C22×D191C2×D411yescharacteristic
129Dic382—21no—
130Dic382—21no—
131Dic382—42no—
132D764—21no—
133D764—21no—
134C22⋊D191—22no—
135C22⋊D191—42no—
136C22⋊D191—42no—
137C22⋊D191—42no—
138C22⋊D191—42no—
139C22⋊D191—42no—
140C22⋊D191—81no—
141C2×Dic191—21no—
142C2×Dic191—21no—
143C2×Dic191—21no—
144C2×Dic191—42no—
145C2×Dic191C2×D411yescharacteristic
146C2×Dic191C2×D411yescharacteristic
147C2×Dic191C2×D411yescharacteristic
148C4×D191—21no—
149C4×D191—21no—
150C4×D191—42no—
151C191⋊C8—21no—
152C191⋊C8—21no—
153C23×C191—21no—
154C23×C191—21no—
155C23×C191—21no—
156C23×C191—42no—
157C23×C191—42no—
158C23×C191—81no—
159C23×C191C2×D411yescharacteristic
160Q8×C191—21no—
161Q8×C191C2×D412yescomplemented
162D4×C191—21no—
163D4×C191—21no—
164D4×C191—21no—
165D4×C191—21no—
166D4×C191—21no—
167D4×C191—42no—
168D4×C191—42no—
169D4×C191C2×D412yes—
170C2×C4×C191—21no—
171C2×C4×C191—21no—
172C2×C4×C191—21no—
173C2×C4×C191—42no—
174C2×C4×C191C2×D411yescharacteristic
175C2×C4×C191C2411yescharacteristic
176C8×C191—21no—
177C8×C191C2×D412yescomplemented
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—41no—
188C22×C191—41no—
189C22×C191—41no—
190C22×C191—84no—
191C22×C191C22×D411yescharacteristic
192C2×D191—21no—
193C2×D191—21no—
194C2×D191—41no—
195C2×D191—41no—
196C2×D191—41no—
197C2×D191—84no—
198C2×D191AGL(1,ℤ/8)12yescomplemented
199C4×C191—21no—
200C4×C191—22no—
201C4×C191C22×D411yescharacteristic
202C4×C191H(5,𝔽2)11yescharacteristic
203Dic191—21no—
204Dic191—21no—
205Dic191—21no—
206Dic191—41no—
207Dic191—41no—
208Dic191—41no—
209C2×C191—42no—
210C2×C191—41no—
211C2×C191—81no—
212C2×C191—81no—
213C2×C191D4211yescharacteristic
214C2×C191C2×AGL(1,ℤ/8)11yescharacteristic
215C2×C191#57711yescharacteristic
216D191—22no—
217D191—81no—
218C191#326511yescharacteristic, complemented, Sylow
219#3265—1911nomaximal, Sylow
220C2×AGL(1,ℤ/8)D19111yescharacteristic, complemented
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#435—1911no—
235C2×#41C2×D19111yescharacteristic, complemented
236C22×Q8—1911no—
237C22×D4—3821no—
238C22×D4—1911no—
239C22×D4C2×D19111yescharacteristic
240AGL(1,ℤ/8)—24no—
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×D8C2×D19111yescharacteristic
250C2×D8C2×D19111yescharacteristic, complemented
251C2×M16C2×D19111yescharacteristic, complemented
252C22×C8—1911no—
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×D4—1911no—
269#107—1911no—
270C4⋊C8—1911no—
271#103—1911no—
272#103—1911no—
273#102—1911no—
274#102—1911no—
275#102—1911no—
276#102—1911no—
277C22⋊C8—1911no—
278C22⋊C8—1911no—
279#97—1911no—
280C24—21no—
281C24—3821no—
282#41—22no—
283#41—41no—
284C2×Q8—3821no—
285C2×Q8—3822no—
286C2×Q8—1911no—
287C2×Q8—7641no—
288C2×Q8C22×D19111yescharacteristic
289C2×D4—22no—
290C2×D4—21no—
291C2×D4—21no—
292C2×D4—41no—
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—7641no—
303C2×D4—7644no—
304C2×D4C22×D19111yescharacteristic
305C2×D4C22×D19111yescharacteristic
306C2×D4C22×D19111yescharacteristic
307C22×C4—21no—
308C22×C4—3821no—
309C22×C4—3821no—
310C22×C4—1911no—
311C22×C4C22×D19111yescharacteristic
312SD16—22no—
313SD16—3822no—
314SD16—3822no—
315SD16—3822no—
316SD16—7641no—
317SD16C22⋊D19114yescomplemented
318D8—22no—
319D8—22no—
320D8—7641no—
321M16—22no—
322C2×C8—3822no—
323C2×C8—1911no—
324C2×C8—1911no—
325C2×C8C22×D19111yescharacteristic
326C2×C8C22×D19111yescharacteristic
327C4⋊C4—3821no—
328C4⋊C4—3821no—
329C4⋊C4—1911no—
330C4⋊C4—1911no—
331C4⋊C4—1911no—
332C22⋊C4—3821no—
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—21no—
346C23—42no—
347C23—42no—
348C23—81no—
349C23—3821no—
350C23—1911no—
351C23—15281no—
352C23—7642no—
353C23—7642no—
354C23D4×D19111yescharacteristic
355Q8—21no—
356Q8—3821no—
357Q8—3821no—
358Q8—7642no—
359Q8C2×C22⋊D19112yescomplemented
360D4—21no—
361D4—21no—
362D4—21no—
363D4—21no—
364D4—21no—
365D4—42no—
366D4—42no—
367D4—3822no—
368D4—3821no—
369D4—3821no—
370D4—15281no—
371D4—7642no—
372D4—7642no—
373D4—7642no—
374D4—7642no—
375D4—7642no—
376D4C2×C22⋊D19112yes—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—42no—
381C2×C4—3821no—
382C2×C4—3821no—
383C2×C4—3821no—
384C2×C4—3821no—
385C2×C4—3821no—
386C2×C4—1911no—
387C2×C4—1911no—
388C2×C4—1911no—
389C2×C4—7642no—
390C2×C4—7642no—
391C2×C4D4×D19111yescharacteristic
392C2×C4C23×D19111yescharacteristic
393C8—21no—
394C8—3821no—
395C8—3821no—
396C8C2×C22⋊D19112yescomplemented
397C22—21no—
398C22—21no—
399C22—21no—
400C22—22no—
401C22—41no—
402C22—41no—
403C22—41no—
404C22—41no—
405C22—41no—
406C22—41no—
407C22—41no—
408C22—41no—
409C22—84no—
410C22—3821no—
411C22—3821no—
412C22—1912no—
413C22—15284no—
414C22—7641no—
415C22—7641no—
416C22—7641no—
417C22C2×D4×D19111yescharacteristic
418C4—21no—
419C4—22no—
420C4—3821no—
421C4—3821no—
422C4—3821no—
423C4—7641no—
424C4—7641no—
425C4—7641no—
426C4C22×C22⋊D19111yescharacteristic
427C4#2025936711yescharacteristic
428C2—41no—
429C2—42no—
430C2—81no—
431C2—81no—
432C2—3822no—
433C2—15281no—
434C2D4×C22⋊D19111yescharacteristic
435C2AGL(1,ℤ/8)×D19111yescharacteristic
436C2#2713347311yescharacteristic
subgroupG/H
centerC22C2×D4×D191
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
FittingC2×AGL(1,ℤ/8)×C191C2
socleC22×C191C22×D4
hypercenterC2×AGL(1,ℤ/8)D191
hypocenterC191#3265
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×SD16×D191C2semidirect
2. #27133320C2semidirect
3. D4×C22⋊D191C2semidirect
4. #27133051C2semidirect
5. #27132909C2semidirect
6. #27132896C2semidirect
7. #27132876C2semidirect
8. #27132861C2semidirect
9. #27132814C2semidirect
10. #27132808C2semidirect
11. #27132800C2semidirect
12. #27132795C2semidirect
13. #27132764C2semidirect
14. #27132732C2semidirect
15. C2×AGL(1,ℤ/8)×C191C2semidirect
16. C2×Q8×D191C22semidirect
19. C2×#15690276C22semidirect
21. C2×#15690275C22semidirect
22. C2×#15690274C22semidirect
24. C2×#15690272C22semidirect
25. C2×C8×D191C22semidirect
28. #20259309C22semidirect
30. #20259301C22semidirect
35. #20259293C22semidirect
38. Q8×Dic191C22semidirect
42. #20259256C22semidirect
43. Dic191⋊C8C22semidirect
44. #20259247C22semidirect
45. #20259246C22semidirect
46. #20259244C22semidirect
48. #20259243C22semidirect
49. #20259243C22semidirect
50. C22⋊(C191⋊C8)C22semidirect
51. (C2×D191)⋊C8C22semidirect
52. #20259232C22semidirect
53. C2×#41×C191C22semidirect
57. C2×SD16×C191C22semidirect
59. C2×D8×C191C22semidirect
60. C2×M16×C191C22semidirect
118. SD16×C191D4semidirect
161. Q8×C191C2×D4semidirect
177. C8×C191C2×D4semidirect
198. C2×D191AGL(1,ℤ/8)semidirect
218. C191#3265semidirect
220. C2×AGL(1,ℤ/8)D191semidirect
235. C2×#41C2×D191semidirect
247. C2×SD16C2×D191semidirect
250. C2×D8C2×D191semidirect
251. C2×M16C2×D191semidirect
317. SD16C22⋊D191semidirect
359. Q8C2×C22⋊D191semidirect
396. C8C2×C22⋊D191semidirect

Maximal subgroups

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

subgroupindex
C2×SD16×D1912
#271333202
D4×C22⋊D1912
#271330512
#271329092
#271328962
#271328762
#271328612
#271328142
#271328082
#271328002
#271327952
#271327642
#271327322
C2×AGL(1,ℤ/8)×C1912
#3265191