Finite Group Databasebeta

← #37584031

Subgroups — #37584031

Order 24448.

494 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#20259353C211yesmaximal, characteristic, complemented
2C22×#15690274C211yesmaximal, characteristic, complemented
3#27133309C211yesmaximal, characteristic
4#27133055C211yesmaximal, characteristic, complemented
5#27132911C211yesmaximal, characteristic, complemented
6#27132902C211yesmaximal, characteristic, complemented
7#27132885C211yesmaximal, characteristic, complemented
8#27132865C211yesmaximal, characteristic, complemented
9#27132823C211yesmaximal, characteristic, complemented
10#27132819C211yesmaximal, characteristic, complemented
11#27132813C211yesmaximal, characteristic, complemented
12#27132802C211yesmaximal, characteristic, complemented
13#27132761C211yesmaximal, characteristic, complemented
14#27132735C211yesmaximal, characteristic, complemented
15D4×Q8×C191C211yesmaximal, characteristic, complemented
16C2×#15690287C2211yescharacteristic
17C22×D764C2211yescharacteristic
18#20259353—24no—
19C2×#15690277C2211yescharacteristic, complemented
20C2×#15690277C2211yescharacteristic, complemented
21C2×#15690277C2211yescharacteristic, complemented
22C2×#15690274—24no—
23C2×#15690274C2211yescharacteristic
24C2×#15690274C2211yescharacteristic
25C2×#15690274C2211yescharacteristic
26C2×#15690274C2211yescharacteristic
27C2×#15690274C2211yescharacteristic
28C2×#15690274C2211yescharacteristic
29C2×#15690271C2211yescharacteristic, complemented
30C22×C191⋊C8C2211yescharacteristic
31#20259310C2211yescharacteristic
32#20259307—21no—
33#20259301C2211yescharacteristic
34#20259296C2211yescharacteristic
35#20259295—21no—
36#20259290—21no—
37C4×D764C2211yescharacteristic
38C22⋊C4×D191—21no—
39#20259255C2211yescharacteristic, complemented
40C4⋊(C191⋊C8)C2211yescharacteristic, complemented
41#20259246C2211yescharacteristic, complemented
42#20259246C2211yescharacteristic, complemented
43#20259246C2211yescharacteristic, complemented
44#20259245C2211yescharacteristic, complemented
45#20259242C2211yescharacteristic
46#20259242C2211yescharacteristic
47C22⋊(C191⋊C8)C2211yescharacteristic
48C22⋊(C191⋊C8)C2211yescharacteristic, complemented
49#20259233C2211yescharacteristic, complemented
50C22×Q8×C191C2211yescharacteristic
51C22×Q8×C191C2211yescharacteristic, complemented
52C4⋊Q8×C191—21no—
53C4⋊Q8×C191C2211yescharacteristic, complemented
54C22⋊Q8×C191—21no—
55C22⋊Q8×C191—21no—
56C22⋊Q8×C191C2211yescharacteristic
57C22⋊Q8×C191C2211yescharacteristic, complemented
58C4×Q8×C191C2211yescharacteristic, complemented
59C4×D4×C191—21no—
60C4×D4×C191C2211yescharacteristic
61C23×D191—21no—
62#15690287—22no—
63#15690287—41no—
64C2×Dic382C2311yescharacteristic
65C2×D764—22no—
66C2×D764—21no—
67C2×D764—41no—
68C2×D764C2311yescharacteristic
69C2×D764C2311yescharacteristic
70C2×C22⋊D191—21no—
71C2×C22⋊D191—21no—
72C2×C4×D191—21no—
73C2×C4×D191—21no—
74#15690277—22no—
75#15690277—22no—
76#15690277—41no—
77#15690274—22no—
78#15690274—22no—
79#15690274—22no—
80#15690274—22no—
81#15690274—41no—
82#15690274D414yescomplemented
83#15690271—22no—
84C2×C191⋊C8—22no—
85C2×C191⋊C8C2311yescharacteristic
86C2×C191⋊C8C2311yescharacteristic
87C2×C191⋊C8C2311yescharacteristic
88C2×C191⋊C8C2311yescharacteristic
89C4⋊Dic191C2311yescharacteristic
90Dic191⋊C4—21no—
91C22⋊Dic191—21no—
92(C2×D191)⋊C4—21no—
93(C2×D191)⋊C4—21no—
94(C2×D191)⋊C4—21no—
95(C2×D191)⋊C4—21no—
96C2×Q8×C191—21no—
97C2×Q8×C191—21no—
98C2×Q8×C191—22no—
99C2×Q8×C191—22no—
100C2×Q8×C191—41no—
101C2×Q8×C191—41no—
102C2×Q8×C191C2311yescharacteristic
103C2×Q8×C191C2311yescharacteristic
104C2×Q8×C191C2311yescharacteristic
105C2×D4×C191D411yescharacteristic
106C22×C4×C191—21no—
107C22×C4×C191—21no—
108C22×C4×C191C2311yescharacteristic
109C22×C4×C191C2311yescharacteristic
110C4⋊C4×C191—21no—
111C4⋊C4×C191—21no—
112C4⋊C4×C191—21no—
113C4⋊C4×C191—21no—
114C4⋊C4×C191—21no—
115C4⋊C4×C191D411yescharacteristic
116C4⋊C4×C191C2311yescharacteristic
117C22⋊C4×C191—21no—
118C22⋊C4×C191—21no—
119C22⋊C4×C191D411yescharacteristic
120C22⋊C4×C191D411yescharacteristic
121C42×C191—21no—
122C42×C191C2311yescharacteristic
123C22×D191—21no—
124C22×D191—21no—
125C22×D191—42no—
126C22×D191—42no—
127C22×D191—81no—
128Dic382—21no—
129Dic382—21no—
130D764—21no—
131D764—21no—
132D764—21no—
133D764—42no—
134D764C2×D412yes—
135C22⋊D191—42no—
136C22⋊D191—81no—
137C2×Dic191—21no—
138C2×Dic191—21no—
139C4×D191—42no—
140C4×D191—81no—
141C191⋊C8—21no—
142C191⋊C8—21no—
143C191⋊C8—21no—
144C191⋊C8C2×D412yes—
145C23×C191C2×D411yescharacteristic
146C23×C191C2×D411yescharacteristic
147Q8×C191—21no—
148Q8×C191—21no—
149Q8×C191—21no—
150Q8×C191—21no—
151Q8×C191—21no—
152Q8×C191—42no—
153Q8×C191—42no—
154Q8×C191C2×D412yes—
155D4×C191—22no—
156C2×C4×C191—21no—
157C2×C4×C191—21no—
158C2×C4×C191—21no—
159C2×C4×C191—21no—
160C2×C4×C191—21no—
161C2×C4×C191—21no—
162C2×C4×C191—21no—
163C2×C4×C191—21no—
164C2×C4×C191—42no—
165C2×C4×C191—42no—
166C2×C4×C191C2×D411yescharacteristic
167C2×C4×C191C2×D411yescharacteristic
168C2×C4×C191C2×D411yescharacteristic
169C2×C4×C191C2×D411yescharacteristic
170C2×C4×C191C2411yescharacteristic
171C22×C191—21no—
172C22×C191—21no—
173C22×C191—21no—
174C22×C191—21no—
175C22×C191—21no—
176C22×C191#13712yescomplemented
177C22×C191C22×D411yescharacteristic
178C2×D191—22no—
179C2×D191—41no—
180C2×D191—41no—
181C2×D191—41no—
182C2×D191—41no—
183C2×D191—41no—
184C2×D191—84no—
185C4×C191—21no—
186C4×C191—21no—
187C4×C191—22no—
188C4×C191—21no—
189C4×C191—21no—
190C4×C191—41no—
191C4×C191—41no—
192C4×C191—41no—
193C4×C191C22×D411yescharacteristic
194C4×C191H(5,𝔽2)11yescharacteristic
195Dic191—41no—
196Dic191—41no—
197C2×C191—22no—
198C2×C191—41no—
199C2×C191D4211yescharacteristic
200C2×C191C2×#13711yescharacteristic
201C2×C191#57711yescharacteristic
202D191—42no—
203D191—81no—
204C191#326611yescharacteristic, complemented, Sylow
205#3266—1911nomaximal, Sylow
206C2×#137—1911no—
207C22×SD16—1911no—
208D4×Q8D19111yescharacteristic, complemented
209#546—1911no—
210#497—1911no—
211#468—1911no—
212#467—1911no—
213#464—1911no—
214#461—1911no—
215#452—1911no—
216#451—1911no—
217#450—1911no—
218#448—1911no—
219#440—1911no—
220#435—1911no—
221C2×#41—1911no—
222C22×Q8C2×D19111yescharacteristic
223C22×Q8C2×D19111yescharacteristic, complemented
224C22×D4—1911no—
225#137—3824no—
226C2×Q16—1911no—
227C2×Q16—1911no—
228C2×Q16—1911no—
229C2×SD16—3824no—
230C2×SD16—1911no—
231C2×SD16—1911no—
232C2×SD16—1911no—
233C2×SD16—1911no—
234C2×SD16—1911no—
235C2×SD16—1911no—
236C2×M16—1911no—
237C22×C8—1911no—
238C4⋊Q8—21no—
239C4⋊Q8C2×D19111yescharacteristic, complemented
240#124—1911no—
241#123—3821no—
242C22⋊Q8—21no—
243C22⋊Q8—21no—
244C22⋊Q8—1911no—
245C22⋊Q8C2×D19111yescharacteristic
246C22⋊Q8C2×D19111yescharacteristic, complemented
247#121—3821no—
248#121—1911no—
249#120—3821no—
250C4×Q8C2×D19111yescharacteristic, complemented
251C4×D4—21no—
252C4×D4—1911no—
253C4×D4C2×D19111yescharacteristic
254C2×C22⋊C4—3821no—
255#107—1911no—
256C4⋊C8—1911no—
257#103—1911no—
258#103—1911no—
259#103—1911no—
260#103—1911no—
261#102—1911no—
262#102—1911no—
263C22⋊C8—1911no—
264C22⋊C8—1911no—
265#97—1911no—
266C24—3821no—
267#41—3822no—
268#41—7641no—
269C2×Q8—21no—
270C2×Q8—21no—
271C2×Q8—22no—
272C2×Q8—22no—
273C2×Q8—41no—
274C2×Q8—41no—
275C2×Q8—1911no—
276C2×Q8C22×D19111yescharacteristic
277C2×Q8C22×D19111yescharacteristic
278C2×Q8C22×D19111yescharacteristic
279C2×D4—3822no—
280C2×D4—3821no—
281C2×D4—3821no—
282C2×D4—3821no—
283C2×D4—1911no—
284C2×D4—1911no—
285C2×D4—7641no—
286C2×D4C22⋊D19111yescharacteristic
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—3821no—
290C22×C4—3821no—
291C22×C4C22×D19111yescharacteristic
292C22×C4C22×D19111yescharacteristic
293Q16—3822no—
294Q16—3822no—
295Q16—7641no—
296SD16—3822no—
297SD16—3822no—
298SD16—3822no—
299SD16—3822no—
300SD16—1914no—
301SD16—7641no—
302M16—3822no—
303C2×C8—3822no—
304C2×C8—1911no—
305C2×C8—1911no—
306C2×C8—1911no—
307C2×C8—1911no—
308C4⋊C4—21no—
309C4⋊C4—21no—
310C4⋊C4—21no—
311C4⋊C4—21no—
312C4⋊C4—21no—
313C4⋊C4—3821no—
314C4⋊C4—1911no—
315C4⋊C4C22⋊D19111yescharacteristic
316C4⋊C4C22×D19111yescharacteristic
317C22⋊C4—21no—
318C22⋊C4—21no—
319C22⋊C4—3821no—
320C22⋊C4—3821no—
321C22⋊C4—3821no—
322C22⋊C4—3821no—
323C22⋊C4—3821no—
324C22⋊C4C22⋊D19111yescharacteristic
325C22⋊C4C22⋊D19111yescharacteristic
326C42—21no—
327C42C22×D19111yescharacteristic
328C23—3821no—
329C23—3821no—
330C23—15281no—
331C23—7642no—
332C23—7642no—
333C23C2×C22⋊D19111yescharacteristic
334C23C2×C22⋊D19111yescharacteristic
335Q8—21no—
336Q8—21no—
337Q8—21no—
338Q8—21no—
339Q8—21no—
340Q8—42no—
341Q8—42no—
342Q8—3821no—
343Q8—3821no—
344Q8D4×D19112yes—
345D4—22no—
346D4—3821no—
347D4—3821no—
348D4—3821no—
349D4—1912no—
350D4—15281no—
351D4—7642no—
352D4—7642no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—3821no—
364C2×C4—3821no—
365C2×C4—15281no—
366C2×C4—7642no—
367C2×C4C2×C22⋊D19111yescharacteristic
368C2×C4C2×C22⋊D19111yescharacteristic
369C2×C4C2×C22⋊D19111yescharacteristic
370C2×C4C2×C22⋊D19111yescharacteristic
371C2×C4C23×D19111yescharacteristic
372C8—3821no—
373C8—3821no—
374C8—3821no—
375C8—1912no—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—21no—
380C22—21no—
381C22—3822no—
382C22—15284no—
383C22—7641no—
384C22—7641no—
385C22—7641no—
386C22—7641no—
387C22—7641no—
388C22#2025935312yescomplemented
389C22C22×C22⋊D19111yescharacteristic
390C4—21no—
391C4—21no—
392C4—21no—
393C4—21no—
394C4—22no—
395C4—41no—
396C4—41no—
397C4—41no—
398C4—7641no—
399C4—7641no—
400C4C2×D4×D19111yescharacteristic
401C4#2025936711yescharacteristic
402C2—22no—
403C2—41no—
404C2—15281no—
405C2—7642no—
406C2D4×C22⋊D19111yescharacteristic
407C2C2×#2025935311yescharacteristic
408C2#2713347111yescharacteristic
subgroupG/H
centerC22C22×C22⋊D191
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
FittingD4×Q8×C191C2
socleC22×C191C22×D4
prime radical#27133309C2
hypercenterD4×Q8D191
hypocenterC191#3266
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C22 ◁ C4×D4 ◁ D4×Q8

Complements

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

NH ≅ G/Ntype
1. C2×#20259353C2semidirect
2. C22×#15690274C2semidirect
4. #27133055C2semidirect
5. #27132911C2semidirect
6. #27132902C2semidirect
7. #27132885C2semidirect
8. #27132865C2semidirect
9. #27132823C2semidirect
10. #27132819C2semidirect
11. #27132813C2semidirect
12. #27132802C2semidirect
13. #27132761C2semidirect
14. #27132735C2semidirect
15. D4×Q8×C191C2semidirect
19. C2×#15690277C22semidirect
20. C2×#15690277C22semidirect
21. C2×#15690277C22semidirect
29. C2×#15690271C22semidirect
39. #20259255C22semidirect
40. C4⋊(C191⋊C8)C22semidirect
41. #20259246C22semidirect
42. #20259246C22semidirect
43. #20259246C22semidirect
44. #20259245C22semidirect
48. C22⋊(C191⋊C8)C22semidirect
49. #20259233C22semidirect
51. C22×Q8×C191C22semidirect
53. C4⋊Q8×C191C22semidirect
57. C22⋊Q8×C191C22semidirect
58. C4×Q8×C191C22semidirect
82. #15690274D4semidirect
176. C22×C191#137semidirect
204. C191#3266semidirect
208. D4×Q8D191semidirect
223. C22×Q8C2×D191semidirect
239. C4⋊Q8C2×D191semidirect
246. C22⋊Q8C2×D191semidirect
250. C4×Q8C2×D191semidirect
388. C22#20259353semidirect

Maximal subgroups

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

subgroupindex
C2×#202593532
C22×#156902742
#271333092
#271330552
#271329112
#271329022
#271328852
#271328652
#271328232
#271328192
#271328132
#271328022
#271327612
#271327352
D4×Q8×C1912
#3266191