Finite Group Databasebeta

← #37582147

Subgroups — #37582147

Order 24448.

496 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#20259354C211yesmaximal, characteristic, complemented
2C22×#15690270C211yesmaximal, characteristic, complemented
3#27133176C211yesmaximal, characteristic
4C22⋊Q8×D191C211yesmaximal, characteristic, complemented
5#27132920C211yesmaximal, characteristic, complemented
6#27132918C211yesmaximal, characteristic, complemented
7#27132912C211yesmaximal, characteristic, complemented
8#27132910C211yesmaximal, characteristic, complemented
9#27132902C211yesmaximal, characteristic, complemented
10#27132894C211yesmaximal, characteristic, complemented
11#27132892C211yesmaximal, characteristic, complemented
12#27132704C211yesmaximal, characteristic, complemented
13#27132670C211yesmaximal, characteristic, complemented
14#27132663C211yesmaximal, characteristic, complemented
15#467×C191C211yesmaximal, characteristic, complemented
16C2×#15690288C2211yescharacteristic
17C2×Q8×D191C2211yescharacteristic
18C22×C4×D191C2211yescharacteristic
19#20259354—24no—
20C2×#15690277C2211yescharacteristic, complemented
21C2×#15690276C2211yescharacteristic, complemented
22C2×#15690274C2211yescharacteristic, complemented
23C2×#15690270—24no—
24C2×#15690270C2211yescharacteristic
25C2×#15690270C2211yescharacteristic
26C2×#15690270C2211yescharacteristic
27C2×#15690270C2211yescharacteristic
28C22×C191⋊C8C2211yescharacteristic
29#20259307—21no—
30#20259302C2211yescharacteristic
31#20259301C2211yescharacteristic
32#20259300—21no—
33#20259299—21no—
34#20259298C2211yescharacteristic
35#20259298C2211yescharacteristic
36#20259297C2211yescharacteristic
37#20259296C2211yescharacteristic
38#20259292—21no—
39#20259284—21no—
40(C22⋊D191)⋊C4—21no—
41#20259280C2211yescharacteristic
42C4⋊C4×D191—21no—
43C4⋊C4×D191C2211yescharacteristic
44C22⋊C4×D191—21no—
45#20259255C2211yescharacteristic, complemented
46#20259254C2211yescharacteristic, complemented
47#20259253C2211yescharacteristic, complemented
48#20259247C2211yescharacteristic, complemented
49#20259246C2211yescharacteristic, complemented
50#20259246C2211yescharacteristic, complemented
51#20259245C2211yescharacteristic, complemented
52#20259244C2211yescharacteristic, complemented
53#20259242C2211yescharacteristic, complemented
54C2×Q16×C191C2211yescharacteristic, complemented
55C22×C8×C191C2211yescharacteristic
56C22⋊Q8×C191C2211yescharacteristic
57C22⋊Q8×C191C2211yescharacteristic
58#107×C191C2211yescharacteristic, complemented
59#103×C191C2211yescharacteristic, complemented
60#103×C191C2211yescharacteristic, complemented
61C23×D191D411yescharacteristic
62#15690288—22no—
63#15690288—22no—
64C2×Dic382—21no—
65C2×Dic382C2311yescharacteristic
66Q8×D191—22no—
67Q8×D191—22no—
68C2×D764—21no—
69C2×D764—21no—
70C2×D764C2311yescharacteristic
71C2×C22⋊D191—21no—
72C22×Dic191D411yescharacteristic
73C2×C4×D191—22no—
74C2×C4×D191—22no—
75C2×C4×D191—21no—
76C2×C4×D191—21no—
77C2×C4×D191—21no—
78C2×C4×D191D411yescharacteristic
79C2×C4×D191D411yescharacteristic
80C2×C4×D191C2311yescharacteristic
81C2×C4×D191C2311yescharacteristic
82#15690277—22no—
83#15690276—22no—
84#15690274—22no—
85#15690270—22no—
86#15690270—22no—
87#15690270—22no—
88#15690270D414yescomplemented
89C2×C191⋊C8—22no—
90C2×C191⋊C8C2311yescharacteristic
91C2×C191⋊C8C2311yescharacteristic
92C4⋊Dic191—21no—
93C4⋊Dic191C2311yescharacteristic
94C4⋊Dic191C2311yescharacteristic
95Dic191⋊C4—21no—
96Dic191⋊C4—21no—
97Dic191⋊C4—21no—
98Dic191⋊C4—21no—
99C22⋊Dic191—21no—
100(C2×D191)⋊C4—21no—
101(C2×D191)⋊C4—21no—
102(C2×D191)⋊C4—21no—
103(C2×D191)⋊C4—21no—
104(C2×D191)⋊C4—21no—
105(C2×D191)⋊C4—21no—
106C4×Dic191—21no—
107C2×Q8×C191C2311yescharacteristic
108C2×Q8×C191C2311yescharacteristic
109C22×C4×C191C2311yescharacteristic
110Q16×C191—22no—
111C2×C8×C191—22no—
112C2×C8×C191C2311yescharacteristic
113C2×C8×C191C2311yescharacteristic
114C4⋊C4×C191—21no—
115C4⋊C4×C191—21no—
116C4⋊C4×C191C2311yescharacteristic
117C4⋊C4×C191C2311yescharacteristic
118C22⋊C4×C191—21no—
119C22⋊C4×C191—21no—
120C22×D191—22no—
121C22×D191—22no—
122C22×D191—21no—
123C22×D191—41no—
124C22×D191C2×D411yescharacteristic
125C22×D191C2×D411yescharacteristic
126Dic382—21no—
127Dic382—21no—
128Dic382—42no—
129D764—21no—
130D764—21no—
131D764—42no—
132D764—42no—
133C22⋊D191—42no—
134C2×Dic191—22no—
135C2×Dic191—21no—
136C2×Dic191—21no—
137C2×Dic191—21no—
138C2×Dic191C2×D411yescharacteristic
139C2×Dic191C2×D411yescharacteristic
140C4×D191—22no—
141C4×D191—21no—
142C4×D191—21no—
143C4×D191—22no—
144C4×D191—42no—
145C4×D191—42no—
146C4×D191—42no—
147C4×D191C2×D412yes—
148C4×D191#4112yes—
149C191⋊C8—21no—
150C191⋊C8C2×D412yes—
151C23×C191C2×D411yescharacteristic
152Q8×C191—21no—
153Q8×C191—21no—
154Q8×C191—21no—
155Q8×C191—21no—
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×C191C2×D411yescharacteristic
163C2×C4×C191C2411yescharacteristic
164C8×C191—21no—
165C8×C191C2×D412yes—
166C22×C191—21no—
167C22×C191—21no—
168C22×C191#13712yescomplemented
169C22×C191C22×D411yescharacteristic
170C2×D191—21no—
171C2×D191—21no—
172C2×D191—21no—
173C2×D191—21no—
174C2×D191—21no—
175C2×D191—44no—
176C2×D191—41no—
177C2×D191—41no—
178C2×D191—41no—
179C2×D191#12112yes—
180C4×C191—21no—
181C4×C191—41no—
182C4×C191—41no—
183C4×C191—41no—
184C4×C191—41no—
185C4×C191C22×D411yescharacteristic
186C4×C191C2×#4111yescharacteristic
187Dic191—21no—
188Dic191—41no—
189Dic191—41no—
190Dic191—41no—
191Dic191#12112yes—
192C2×C191—22no—
193C2×C191C2×#12111yescharacteristic
194C2×C191C2×#13711yescharacteristic
195C2×C191#57511yescharacteristic
196D191—22no—
197D191—41no—
198D191—81no—
199C191#305611yescharacteristic, complemented, Sylow
200#3056—1911nomaximal, Sylow
201C2×#137—1911no—
202C22×M16—1911no—
203#525—1911no—
204C2×C22⋊Q8—1911no—
205#470—1911no—
206#470—1911no—
207#468—1911no—
208#468—1911no—
209#467—1911no—
210#467D19111yescharacteristic, complemented
211#465—1911no—
212#465—1911no—
213#428—1911no—
214#419—1911no—
215#417—1911no—
216C2×#41—1911no—
217C22×Q8—1911no—
218C23×C4—1911no—
219#137—3824no—
220C2×Q16—1911no—
221C2×Q16C2×D19111yescharacteristic, complemented
222C2×SD16—1911no—
223C2×SD16—1911no—
224C2×M16—3824no—
225C2×M16—1911no—
226C2×M16—1911no—
227C2×M16—1911no—
228C2×M16—1911no—
229C22×C8—1911no—
230C22×C8C2×D19111yescharacteristic
231#123—3821no—
232C22⋊Q8—3821no—
233C22⋊Q8—3821no—
234C22⋊Q8—1911no—
235C22⋊Q8—1911no—
236C22⋊Q8—1911no—
237C22⋊Q8—1911no—
238C22⋊Q8C2×D19111yescharacteristic
239C22⋊Q8C2×D19111yescharacteristic
240#121—1911no—
241#121—1911no—
242#120—3821no—
243C4×D4—3821no—
244C4×D4—3821no—
245#117—1911no—
246C2×C4⋊C4—3821no—
247C2×C4⋊C4—1911no—
248C2×C22⋊C4—3821no—
249#107—1911no—
250#107C2×D19111yescharacteristic, complemented
251#106—1911no—
252#106—1911no—
253#103—1911no—
254#103—1911no—
255#103—1911no—
256#103—1911no—
257#103C2×D19111yescharacteristic, complemented
258#103C2×D19111yescharacteristic, complemented
259#102—1911no—
260#102—1911no—
261C24—1911no—
262#41—3822no—
263#41—3822no—
264C2×Q8—3821no—
265C2×Q8—3822no—
266C2×Q8—3822no—
267C2×Q8—1911no—
268C2×Q8C22×D19111yescharacteristic
269C2×Q8C22×D19111yescharacteristic
270C2×D4—3821no—
271C2×D4—3821no—
272C2×D4—3821no—
273C2×D4—1911no—
274C22×C4—3822no—
275C22×C4—3822no—
276C22×C4—3821no—
277C22×C4—3821no—
278C22×C4—3821no—
279C22×C4—1911no—
280C22×C4—1911no—
281C22×C4—1911no—
282C22×C4—1911no—
283C22×C4—1911no—
284C22×C4C22×D19111yescharacteristic
285Q16—22no—
286Q16—3822no—
287SD16—3822no—
288SD16—3822no—
289M16—3822no—
290M16—3822no—
291M16—3822no—
292M16—1914no—
293C2×C8—22no—
294C2×C8—3822no—
295C2×C8—1911no—
296C2×C8—1911no—
297C2×C8C22×D19111yescharacteristic
298C2×C8C22×D19111yescharacteristic
299C4⋊C4—21no—
300C4⋊C4—21no—
301C4⋊C4—3821no—
302C4⋊C4—3821no—
303C4⋊C4—3821no—
304C4⋊C4—3821no—
305C4⋊C4—3821no—
306C4⋊C4—1911no—
307C4⋊C4—1911no—
308C4⋊C4C22×D19111yescharacteristic
309C4⋊C4C22×D19111yescharacteristic
310C22⋊C4—21no—
311C22⋊C4—21no—
312C22⋊C4—3821no—
313C22⋊C4—3821no—
314C22⋊C4—3821no—
315C22⋊C4—3821no—
316C22⋊C4—3821no—
317C22⋊C4—3821no—
318C22⋊C4—3821no—
319C42—3821no—
320C23—3822no—
321C23—3822no—
322C23—3821no—
323C23—1911no—
324C23—1911no—
325C23—7641no—
326C23D4×D19111yescharacteristic
327Q8—21no—
328Q8—21no—
329Q8—21no—
330Q8—21no—
331Q8—3821no—
332Q8—3821no—
333Q8—7642no—
334D4—3821no—
335D4—3821no—
336D4—7642no—
337D4—7642no—
338D4—7642no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—3822no—
346C2×C4—3822no—
347C2×C4—3822no—
348C2×C4—3821no—
349C2×C4—3821no—
350C2×C4—3821no—
351C2×C4—3821no—
352C2×C4—3821no—
353C2×C4—1911no—
354C2×C4—1911no—
355C2×C4—1912no—
356C2×C4—1912no—
357C2×C4—7642no—
358C2×C4—7642no—
359C2×C4—7642no—
360C2×C4D4×D19111yescharacteristic
361C2×C4C23×D19111yescharacteristic
362C8—21no—
363C8—3821no—
364C8—1912no—
365C8D4×D19112yes—
366C22—21no—
367C22—21no—
368C22—3821no—
369C22—3821no—
370C22—3821no—
371C22—3821no—
372C22—3821no—
373C22—1912no—
374C22—7644no—
375C22—7641no—
376C22—7641no—
377C22—7641no—
378C22#2025935412yescomplemented
379C22C2×D4×D19111yescharacteristic
380C4—21no—
381C4—41no—
382C4—41no—
383C4—41no—
384C4—41no—
385C4—3821no—
386C4—1912no—
387C4—7641no—
388C4—7641no—
389C4—7641no—
390C4C2×D4×D19111yescharacteristic
391C4#41×D19111yescharacteristic
392C2—22no—
393C2—3822no—
394C2—15281no—
395C2—7641no—
396C2#121×D19111yescharacteristic
397C2C2×#2025935411yescharacteristic
398C2#2713346011yescharacteristic
subgroupG/H
centerC22C2×D4×D191
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
Fitting#467×C191C2
socleC22×C191C22×D4
prime radical#27133176C2
hypercenter#467D191
hypocenterC191#3056
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C22 ◁ C22×C4 ◁ #467

Complements

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

NH ≅ G/Ntype
1. C2×#20259354C2semidirect
2. C22×#15690270C2semidirect
4. C22⋊Q8×D191C2semidirect
5. #27132920C2semidirect
6. #27132918C2semidirect
7. #27132912C2semidirect
8. #27132910C2semidirect
9. #27132902C2semidirect
10. #27132894C2semidirect
11. #27132892C2semidirect
12. #27132704C2semidirect
13. #27132670C2semidirect
14. #27132663C2semidirect
15. #467×C191C2semidirect
20. C2×#15690277C22semidirect
21. C2×#15690276C22semidirect
22. C2×#15690274C22semidirect
45. #20259255C22semidirect
46. #20259254C22semidirect
47. #20259253C22semidirect
48. #20259247C22semidirect
49. #20259246C22semidirect
50. #20259246C22semidirect
51. #20259245C22semidirect
52. #20259244C22semidirect
53. #20259242C22semidirect
54. C2×Q16×C191C22semidirect
58. #107×C191C22semidirect
59. #103×C191C22semidirect
60. #103×C191C22semidirect
88. #15690270D4semidirect
168. C22×C191#137semidirect
199. C191#3056semidirect
210. #467D191semidirect
221. C2×Q16C2×D191semidirect
250. #107C2×D191semidirect
257. #103C2×D191semidirect
258. #103C2×D191semidirect
378. C22#20259354semidirect

Maximal subgroups

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

subgroupindex
C2×#202593542
C22×#156902702
#271331762
C22⋊Q8×D1912
#271329202
#271329182
#271329122
#271329102
#271329022
#271328942
#271328922
#271327042
#271326702
#271326632
#467×C1912
#3056191