Finite Group Databasebeta

← #37584023

Subgroups — #37584023

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×#20259354C211yesmaximal, characteristic, complemented
2C22×#15690274C211yesmaximal, characteristic, complemented
3Q8×C22⋊D191C211yesmaximal, characteristic, complemented
4#27133304C211yesmaximal, characteristic
5#27133056C211yesmaximal, characteristic, complemented
6#27132912C211yesmaximal, characteristic, complemented
7#27132902C211yesmaximal, characteristic, complemented
8#27132887C211yesmaximal, characteristic, complemented
9#27132867C211yesmaximal, characteristic, complemented
10#27132824C211yesmaximal, characteristic, complemented
11#27132816C211yesmaximal, characteristic, complemented
12#27132803C211yesmaximal, characteristic, complemented
13#27132766C211yesmaximal, characteristic, complemented
14#27132731C211yesmaximal, characteristic, complemented
15#451×C191C211yesmaximal, characteristic, complemented
16C2×#15690288C2211yescharacteristic
17C2×Q8×D191C2211yescharacteristic, complemented
18C22×D764C2211yescharacteristic
19#20259354—24no—
20C2×#15690277C2211yescharacteristic, complemented
21C2×#15690276C2211yescharacteristic
22C2×#15690276C2211yescharacteristic
23C2×#15690274—24no—
24C2×#15690274C2211yescharacteristic
25C2×#15690274C2211yescharacteristic
26C2×#15690274C2211yescharacteristic
27C2×#15690274C2211yescharacteristic
28C2×#15690270C2211yescharacteristic, complemented
29C22×C191⋊C8C2211yescharacteristic
30#20259320—21no—
31#20259320C2211yescharacteristic, complemented
32#20259311C2211yescharacteristic
33#20259307—21no—
34#20259301—21no—
35#20259301C2211yescharacteristic
36#20259298—21no—
37#20259298C2211yescharacteristic, complemented
38#20259297—21no—
39#20259297C2211yescharacteristic
40#20259292—21no—
41Q8×Dic191C2211yescharacteristic, complemented
42#20259284C2211yescharacteristic
43C4×C22⋊D191—21no—
44C4×C22⋊D191C2211yescharacteristic
45C22⋊C4×D191—21no—
46#20259255C2211yescharacteristic, complemented
47Dic191⋊C8C2211yescharacteristic, complemented
48#20259246C2211yescharacteristic, complemented
49#20259245C2211yescharacteristic, complemented
50#20259244C2211yescharacteristic
51#20259244C2211yescharacteristic
52(C2×D191)⋊C8C2211yescharacteristic, complemented
53#20259232C2211yescharacteristic, complemented
54C22×Q8×C191C2211yescharacteristic
55C2×Q16×C191C2211yescharacteristic, complemented
56C2×Q16×C191C2211yescharacteristic, complemented
57C22⋊Q8×C191C2211yescharacteristic
58#103×C191C2211yescharacteristic, complemented
59#103×C191C2211yescharacteristic, complemented
60C22⋊C8×C191C2211yescharacteristic
61C23×D191—21no—
62#15690288—22no—
63#15690288—41no—
64C2×Dic382—21no—
65C2×Dic382C2311yescharacteristic
66Q8×D191—22no—
67Q8×D191—41no—
68C2×D764—22no—
69C2×D764—21no—
70C2×D764—21no—
71C2×D764—21no—
72C2×D764—41no—
73C2×D764C2311yescharacteristic
74C2×D764C2311yescharacteristic
75C2×C22⋊D191D411yescharacteristic
76C2×C4×D191—21no—
77C2×C4×D191—21no—
78C2×C4×D191—21no—
79C2×C4×D191C2311yescharacteristic
80#15690277—22no—
81#15690276—22no—
82#15690276—41no—
83#15690274—22no—
84#15690274—22no—
85#15690274—22no—
86#15690274D414yescomplemented
87#15690270—22no—
88C2×C191⋊C8—22no—
89C2×C191⋊C8C2311yescharacteristic
90C2×C191⋊C8C2311yescharacteristic
91C4⋊Dic191—21no—
92C4⋊Dic191C2311yescharacteristic
93Dic191⋊C4—21no—
94Dic191⋊C4—21no—
95Dic191⋊C4—21no—
96Dic191⋊C4—21no—
97Dic191⋊C4D411yescharacteristic
98C22⋊Dic191—21no—
99C22⋊Dic191D411yescharacteristic
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)⋊C4D411yescharacteristic
106C4×Dic191—21no—
107C4×Dic191C2311yescharacteristic
108C2×Q8×C191—21no—
109C2×Q8×C191—22no—
110C2×Q8×C191—41no—
111C2×Q8×C191C2311yescharacteristic
112C2×Q8×C191C2311yescharacteristic
113C2×Q8×C191C2311yescharacteristic
114C22×C4×C191—21no—
115C22×C4×C191C2311yescharacteristic
116Q16×C191—22no—
117Q16×C191—41no—
118C2×C8×C191C2311yescharacteristic
119C2×C8×C191C2311yescharacteristic
120C4⋊C4×C191—21no—
121C4⋊C4×C191C2311yescharacteristic
122C22⋊C4×C191—21no—
123C22×D191—21no—
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—
134D764—21no—
135D764—42no—
136D764—42no—
137D764—81no—
138D764C2×D412yes—
139C22⋊D191—22no—
140C2×Dic191—21no—
141C2×Dic191—21no—
142C2×Dic191C2×D411yescharacteristic
143C2×Dic191C2×D411yescharacteristic
144C2×Dic191C2×D411yescharacteristic
145C4×D191—21no—
146C4×D191—21no—
147C4×D191—42no—
148C4×D191—42no—
149C4×D191—81no—
150C191⋊C8—21no—
151C191⋊C8C2×D412yes—
152C23×C191C2×D411yescharacteristic
153Q8×C191—21no—
154Q8×C191—21no—
155Q8×C191—21no—
156Q8×C191—21no—
157Q8×C191—21no—
158Q8×C191—42no—
159Q8×C191C2×D412yes—
160C2×C4×C191—21no—
161C2×C4×C191—21no—
162C2×C4×C191—21no—
163C2×C4×C191—21no—
164C2×C4×C191—21no—
165C2×C4×C191—21no—
166C2×C4×C191—42no—
167C2×C4×C191C2×D411yescharacteristic
168C2×C4×C191C2411yescharacteristic
169C8×C191—21no—
170C8×C191—21no—
171C22×C191—21no—
172C22×C191—21no—
173C22×C191#13712yescomplemented
174C22×C191C22×D411yescharacteristic
175C2×D191—21no—
176C2×D191—22no—
177C2×D191—21no—
178C2×D191—21no—
179C2×D191—41no—
180C2×D191—41no—
181C2×D191—41no—
182C2×D191—41no—
183C2×D191—41no—
184C2×D191—84no—
185C4×C191—21no—
186C4×C191—22no—
187C4×C191—41no—
188C4×C191—41no—
189C4×C191—41no—
190C4×C191C22×D411yescharacteristic
191C4×C191H(5,𝔽2)11yescharacteristic
192Dic191—21no—
193Dic191—21no—
194Dic191—21no—
195Dic191—41no—
196Dic191—41no—
197C2×C191—22no—
198C2×C191D4211yescharacteristic
199C2×C191C2×#13711yescharacteristic
200C2×C191#57711yescharacteristic
201D191—41no—
202D191—42no—
203D191—81no—
204C191#326611yescharacteristic, complemented, Sylow
205#3266—1911nomaximal, Sylow
206C2×#137—1911no—
207C22×SD16—1911no—
208D4×Q8—1911no—
209#546—1911no—
210#497—1911no—
211#468—1911no—
212#467—1911no—
213#464—1911no—
214#461—1911no—
215#452—1911no—
216#451D19111yescharacteristic, complemented
217#450—1911no—
218#448—1911no—
219#440—1911no—
220#435—1911no—
221C2×#41—1911no—
222C22×Q8—1911no—
223C22×Q8C2×D19111yescharacteristic
224C22×D4—1911no—
225#137—3824no—
226C2×Q16—1911no—
227C2×Q16C2×D19111yescharacteristic, complemented
228C2×Q16C2×D19111yescharacteristic, complemented
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—3821no—
239C4⋊Q8—1911no—
240#124—1911no—
241#123—3821no—
242C22⋊Q8—3821no—
243C22⋊Q8—3821no—
244C22⋊Q8—1911no—
245C22⋊Q8—1911no—
246C22⋊Q8C2×D19111yescharacteristic
247#121—3821no—
248#121—1911no—
249#120—3821no—
250C4×Q8—1911no—
251C4×D4—3821no—
252C4×D4—1911no—
253C4×D4—1911no—
254C2×C22⋊C4—3821no—
255#107—1911no—
256C4⋊C8—1911no—
257#103—1911no—
258#103—1911no—
259#103C2×D19111yescharacteristic, complemented
260#103C2×D19111yescharacteristic, complemented
261#102—1911no—
262#102—1911no—
263C22⋊C8—1911no—
264C22⋊C8C2×D19111yescharacteristic
265#97—1911no—
266C24—3821no—
267#41—3822no—
268#41—7641no—
269C2×Q8—21no—
270C2×Q8—22no—
271C2×Q8—41no—
272C2×Q8—3821no—
273C2×Q8—3822no—
274C2×Q8—1911no—
275C2×Q8—7641no—
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—1911no—
286C2×D4—7641no—
287C22×C4—21no—
288C22×C4—3821no—
289C22×C4—3821no—
290C22×C4—3821no—
291C22×C4—1911no—
292C22×C4C22×D19111yescharacteristic
293Q16—22no—
294Q16—41no—
295Q16—3822no—
296SD16—3822no—
297SD16—3822no—
298SD16—3822no—
299SD16—3822no—
300SD16—1914no—
301SD16—7641no—
302M16—3822no—
303C2×C8—3822no—
304C2×C8—1911no—
305C2×C8—1911no—
306C2×C8C22×D19111yescharacteristic
307C2×C8C22×D19111yescharacteristic
308C4⋊C4—21no—
309C4⋊C4—3821no—
310C4⋊C4—3821no—
311C4⋊C4—3821no—
312C4⋊C4—3821no—
313C4⋊C4—3821no—
314C4⋊C4—1911no—
315C4⋊C4—1911no—
316C4⋊C4C22×D19111yescharacteristic
317C22⋊C4—21no—
318C22⋊C4—3821no—
319C22⋊C4—3821no—
320C22⋊C4—3821no—
321C22⋊C4—3821no—
322C22⋊C4—3821no—
323C22⋊C4—3821no—
324C22⋊C4—1911no—
325C22⋊C4—1911no—
326C42—3821no—
327C42—1911no—
328C23—3821no—
329C23—3821no—
330C23—1911no—
331C23—15281no—
332C23—7642no—
333C23—7642no—
334C23D4×D19111yescharacteristic
335Q8—21no—
336Q8—21no—
337Q8—21no—
338Q8—21no—
339Q8—21no—
340Q8—42no—
341Q8—3821no—
342Q8—3821no—
343Q8—7642no—
344Q8D4×D19112yes—
345D4—3822no—
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—42no—
360C2×C4—3821no—
361C2×C4—3821no—
362C2×C4—3821no—
363C2×C4—3821no—
364C2×C4—1911no—
365C2×C4—1911no—
366C2×C4—1911no—
367C2×C4—15281no—
368C2×C4—7642no—
369C2×C4—7642no—
370C2×C4D4×D19111yescharacteristic
371C2×C4C23×D19111yescharacteristic
372C8—21no—
373C8—21no—
374C8—3821no—
375C8—1912no—
376C22—21no—
377C22—21no—
378C22—3821no—
379C22—3821no—
380C22—3821no—
381C22—3822no—
382C22—15284no—
383C22—7641no—
384C22—7641no—
385C22—7641no—
386C22—7641no—
387C22—7641no—
388C22#2025935412yescomplemented
389C22C2×D4×D19111yescharacteristic
390C4—21no—
391C4—22no—
392C4—41no—
393C4—41no—
394C4—41no—
395C4—3821no—
396C4—3821no—
397C4—3821no—
398C4—7641no—
399C4—7641no—
400C4C2×D4×D19111yescharacteristic
401C4#2025936711yescharacteristic
402C2—22no—
403C2—15281no—
404C2—7641no—
405C2—7642no—
406C2#2713329511yescharacteristic
407C2C2×#2025935411yescharacteristic
408C2#2713347211yescharacteristic
subgroupG/H
centerC22C2×D4×D191
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
Fitting#451×C191C2
socleC22×C191C22×D4
prime radical#27133304C2
hypercenter#451D191
hypocenterC191#3266
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C22 ◁ C22×C4 ◁ #451

Complements

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

NH ≅ G/Ntype
1. C2×#20259354C2semidirect
2. C22×#15690274C2semidirect
3. Q8×C22⋊D191C2semidirect
5. #27133056C2semidirect
6. #27132912C2semidirect
7. #27132902C2semidirect
8. #27132887C2semidirect
9. #27132867C2semidirect
10. #27132824C2semidirect
11. #27132816C2semidirect
12. #27132803C2semidirect
13. #27132766C2semidirect
14. #27132731C2semidirect
15. #451×C191C2semidirect
17. C2×Q8×D191C22semidirect
20. C2×#15690277C22semidirect
28. C2×#15690270C22semidirect
31. #20259320C22semidirect
37. #20259298C22semidirect
41. Q8×Dic191C22semidirect
46. #20259255C22semidirect
47. Dic191⋊C8C22semidirect
48. #20259246C22semidirect
49. #20259245C22semidirect
52. (C2×D191)⋊C8C22semidirect
53. #20259232C22semidirect
55. C2×Q16×C191C22semidirect
56. C2×Q16×C191C22semidirect
58. #103×C191C22semidirect
59. #103×C191C22semidirect
86. #15690274D4semidirect
173. C22×C191#137semidirect
204. C191#3266semidirect
216. #451D191semidirect
227. C2×Q16C2×D191semidirect
228. C2×Q16C2×D191semidirect
259. #103C2×D191semidirect
260. #103C2×D191semidirect
388. 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×#156902742
Q8×C22⋊D1912
#271333042
#271330562
#271329122
#271329022
#271328872
#271328672
#271328242
#271328162
#271328032
#271327662
#271327312
#451×C1912
#3266191