Finite Group Databasebeta

← #37584094

Subgroups — #37584094

Order 24448.

472 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#20259354C211yesmaximal, characteristic, complemented
2C22×#15690277C211yesmaximal, characteristic, complemented
3#27133326C211yesmaximal, characteristic, complemented
4#27133324C211yesmaximal, characteristic
5#27133056C211yesmaximal, characteristic, complemented
6#27132920C211yesmaximal, characteristic, complemented
7#27132894C211yesmaximal, characteristic, complemented
8#27132888C211yesmaximal, characteristic, complemented
9#27132871C211yesmaximal, characteristic, complemented
10#27132828C211yesmaximal, characteristic, complemented
11#27132817C211yesmaximal, characteristic, complemented
12#27132804C211yesmaximal, characteristic, complemented
13#27132767C211yesmaximal, characteristic, complemented
14#27132731C211yesmaximal, characteristic, complemented
15#451×C191C211yesmaximal, characteristic, complemented
16C2×#15690288C2211yescharacteristic
17C22×Dic382C2211yescharacteristic
18C2×Q8×D191C2211yescharacteristic
19#20259354—24no—
20C2×#15690277—24no—
21C2×#15690277C2211yescharacteristic
22C2×#15690277C2211yescharacteristic
23C2×#15690277C2211yescharacteristic
24C2×#15690277C2211yescharacteristic
25C2×#15690276C2211yescharacteristic
26C2×#15690276C2211yescharacteristic
27C2×#15690274C2211yescharacteristic
28C2×#15690270C2211yescharacteristic
29C22×C191⋊C8C2211yescharacteristic
30#20259321—21no—
31#20259320C2211yescharacteristic
32#20259311—21no—
33#20259311C2211yescharacteristic
34#20259302C2211yescharacteristic
35#20259300—21no—
36#20259299—21no—
37#20259298—21no—
38#20259298C2211yescharacteristic
39#20259296—21no—
40#20259296C2211yescharacteristic
41Q8×Dic191C2211yescharacteristic
42#20259287C2211yescharacteristic
43C4×C22⋊D191—21no—
44C4×C22⋊D191C2211yescharacteristic
45(C22⋊D191)⋊C4—21no—
46#20259253C2211yescharacteristic
47Dic191⋊C8C2211yescharacteristic
48#20259247C2211yescharacteristic
49#20259247C2211yescharacteristic
50#20259246C2211yescharacteristic
51#20259242C2211yescharacteristic
52(C2×D191)⋊C8C2211yescharacteristic
53#20259232C2211yescharacteristic
54C22×Q8×C191C2211yescharacteristic
55C2×Q16×C191C2211yescharacteristic
56C2×Q16×C191C2211yescharacteristic
57C22⋊Q8×C191C2211yescharacteristic
58#103×C191C2211yescharacteristic
59#103×C191C2211yescharacteristic
60C22⋊C8×C191C2211yescharacteristic
61#15690288—22no—
62#15690288—41no—
63C2×Dic382—21no—
64C2×Dic382—22no—
65C2×Dic382—21no—
66C2×Dic382—41no—
67C2×Dic382C2311yescharacteristic
68C2×Dic382C2311yescharacteristic
69Q8×D191—22no—
70Q8×D191—41no—
71C2×D764—21no—
72C2×D764C2311yescharacteristic
73C2×C22⋊D191—21no—
74C2×C22⋊D191D411yescharacteristic
75C22×Dic191—21no—
76C2×C4×D191—21no—
77C2×C4×D191—21no—
78C2×C4×D191C2311yescharacteristic
79#15690277—22no—
80#15690277—22no—
81#15690277—22no—
82#15690277D414yescomplemented
83#15690276—22no—
84#15690276—41no—
85#15690274—22no—
86#15690270—22no—
87C2×C191⋊C8—22no—
88C2×C191⋊C8C2311yescharacteristic
89C2×C191⋊C8C2311yescharacteristic
90C4⋊Dic191—21no—
91C4⋊Dic191—21no—
92C4⋊Dic191C2311yescharacteristic
93Dic191⋊C4—21no—
94Dic191⋊C4—21no—
95Dic191⋊C4—21no—
96Dic191⋊C4—21no—
97Dic191⋊C4D411yescharacteristic
98C22⋊Dic191D411yescharacteristic
99(C2×D191)⋊C4—21no—
100(C2×D191)⋊C4—21no—
101(C2×D191)⋊C4—21no—
102(C2×D191)⋊C4—21no—
103(C2×D191)⋊C4—21no—
104(C2×D191)⋊C4D411yescharacteristic
105C4×Dic191—21no—
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×D191C2×D411yescharacteristic
125Dic382—21no—
126Dic382—21no—
127Dic382—21no—
128Dic382—42no—
129Dic382—42no—
130Dic382C2×D412yes—
131D764—21no—
132D764—21no—
133D764—42no—
134C22⋊D191—22no—
135C22⋊D191—42no—
136C2×Dic191—21no—
137C2×Dic191—21no—
138C2×Dic191—21no—
139C2×Dic191—42no—
140C2×Dic191C2×D411yescharacteristic
141C2×Dic191C2×D411yescharacteristic
142C2×Dic191C2×D411yescharacteristic
143C4×D191—21no—
144C4×D191—21no—
145C4×D191—42no—
146C4×D191—42no—
147C191⋊C8—21no—
148C191⋊C8C2×D412yes—
149C23×C191C2×D411yescharacteristic
150Q8×C191—21no—
151Q8×C191—21no—
152Q8×C191—21no—
153Q8×C191—21no—
154Q8×C191—21no—
155Q8×C191—42no—
156Q8×C191C2×D412yes—
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—42no—
164C2×C4×C191C2×D411yescharacteristic
165C2×C4×C191C2411yescharacteristic
166C8×C191—21no—
167C8×C191—21no—
168C22×C191—21no—
169C22×C191—21no—
170C22×C191#13712yescomplemented
171C22×C191C22×D411yescharacteristic
172C2×D191—21no—
173C2×D191—21no—
174C2×D191—21no—
175C2×D191—41no—
176C2×D191—41no—
177C2×D191—41no—
178C4×C191—21no—
179C4×C191—22no—
180C4×C191—41no—
181C4×C191—41no—
182C4×C191—41no—
183C4×C191C22×D411yescharacteristic
184C4×C191H(5,𝔽2)11yescharacteristic
185Dic191—21no—
186Dic191—22no—
187Dic191—21no—
188Dic191—21no—
189Dic191—41no—
190Dic191—41no—
191C2×C191—22no—
192C2×C191D4211yescharacteristic
193C2×C191C2×#13711yescharacteristic
194C2×C191#57811yescharacteristic
195D191—41no—
196D191—81no—
197C191#327011yescharacteristic, complemented, Sylow
198#3270—1911nomaximal, Sylow
199C2×#137—1911no—
200C22×Q16—1911no—
201D4×Q8—1911no—
202#548—1911no—
203#497—1911no—
204#470—1911no—
205#465—1911no—
206#464—1911no—
207#462—1911no—
208#452—1911no—
209#451—1911no—
210#451D19111yescharacteristic, complemented
211#448—1911no—
212#441—1911no—
213#435—1911no—
214C2×#41—1911no—
215C22×Q8—1911no—
216C22×Q8—1911no—
217C22×Q8C2×D19111yescharacteristic
218#137—3824no—
219C2×Q16—3824no—
220C2×Q16—1911no—
221C2×Q16—1911no—
222C2×Q16—1911no—
223C2×Q16—1911no—
224C2×Q16C2×D19111yescharacteristic
225C2×Q16C2×D19111yescharacteristic
226C2×SD16—1911no—
227C2×SD16—1911no—
228C2×SD16—1911no—
229C2×M16—1911no—
230C22×C8—1911no—
231C4⋊Q8—3821no—
232C4⋊Q8—1911no—
233#124—3821no—
234#124—1911no—
235C22⋊Q8—3821no—
236C22⋊Q8—3821no—
237C22⋊Q8—3821no—
238C22⋊Q8—1911no—
239C22⋊Q8—1911no—
240C22⋊Q8C2×D19111yescharacteristic
241#121—3821no—
242#121—1911no—
243C4×Q8—1911no—
244C4×Q8—1911no—
245C4×D4—3821no—
246C4×D4—3821no—
247C4×D4—1911no—
248#106—1911no—
249C4⋊C8—1911no—
250#103—1911no—
251#103—1911no—
252#103—1911no—
253#103C2×D19111yescharacteristic
254#103C2×D19111yescharacteristic
255#102—1911no—
256C22⋊C8—1911no—
257C22⋊C8C2×D19111yescharacteristic
258#97—1911no—
259#41—3822no—
260#41—7641no—
261C2×Q8—21no—
262C2×Q8—22no—
263C2×Q8—41no—
264C2×Q8—3821no—
265C2×Q8—3822no—
266C2×Q8—3821no—
267C2×Q8—3822no—
268C2×Q8—1911no—
269C2×Q8—1911no—
270C2×Q8—7641no—
271C2×Q8—7641no—
272C2×Q8C22×D19111yescharacteristic
273C2×Q8C22×D19111yescharacteristic
274C2×Q8C22×D19111yescharacteristic
275C2×D4—3821no—
276C2×D4—3821no—
277C2×D4—1911no—
278C2×D4—1911no—
279C22×C4—21no—
280C22×C4—3821no—
281C22×C4—3821no—
282C22×C4—3821no—
283C22×C4—1911no—
284C22×C4C22×D19111yescharacteristic
285Q16—22no—
286Q16—41no—
287Q16—3822no—
288Q16—3822no—
289Q16—3822no—
290Q16—1914no—
291SD16—3822no—
292SD16—3822no—
293SD16—7641no—
294M16—3822no—
295C2×C8—3822no—
296C2×C8—1911no—
297C2×C8—1911no—
298C2×C8C22×D19111yescharacteristic
299C2×C8C22×D19111yescharacteristic
300C4⋊C4—21no—
301C4⋊C4—3821no—
302C4⋊C4—3821no—
303C4⋊C4—3821no—
304C4⋊C4—3821no—
305C4⋊C4—3821no—
306C4⋊C4—3821no—
307C4⋊C4—1911no—
308C4⋊C4—1911no—
309C4⋊C4C22×D19111yescharacteristic
310C22⋊C4—21no—
311C22⋊C4—3821no—
312C22⋊C4—3821no—
313C22⋊C4—3821no—
314C22⋊C4—3821no—
315C22⋊C4—3821no—
316C22⋊C4—1911no—
317C22⋊C4—1911no—
318C42—3821no—
319C42—3821no—
320C42—1911no—
321C23—3821no—
322C23—1911no—
323C23D4×D19111yescharacteristic
324Q8—21no—
325Q8—21no—
326Q8—21no—
327Q8—21no—
328Q8—21no—
329Q8—42no—
330Q8—3821no—
331Q8—3821no—
332Q8—3821no—
333Q8—1912no—
334Q8—7642no—
335Q8—7642no—
336Q8D4×D19112yes—
337D4—3822no—
338D4—3821no—
339D4—3821no—
340D4—7642no—
341D4—7642no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—42no—
349C2×C4—3821no—
350C2×C4—3821no—
351C2×C4—3821no—
352C2×C4—3821no—
353C2×C4—3821no—
354C2×C4—1911no—
355C2×C4—1911no—
356C2×C4—1911no—
357C2×C4—7642no—
358C2×C4—7642no—
359C2×C4—7642no—
360C2×C4D4×D19111yescharacteristic
361C2×C4C23×D19111yescharacteristic
362C8—21no—
363C8—21no—
364C8—3821no—
365C8—1912no—
366C22—21no—
367C22—21no—
368C22—3821no—
369C22—3821no—
370C22—3821no—
371C22—7641no—
372C22—7641no—
373C22—7641no—
374C22#2025935412yescomplemented
375C22C2×D4×D19111yescharacteristic
376C4—21no—
377C4—22no—
378C4—41no—
379C4—41no—
380C4—41no—
381C4—3821no—
382C4—3821no—
383C4—3821no—
384C4—3822no—
385C4—7641no—
386C4—7641no—
387C4C2×D4×D19111yescharacteristic
388C4#2025936711yescharacteristic
389C2—22no—
390C2—15281no—
391C2—7641no—
392C2#2713329511yescharacteristic
393C2C2×#2025935411yescharacteristic
394C2#2713347911yescharacteristic
subgroupG/H
centerC22C2×D4×D191
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
Fitting#451×C191C2
socleC22×C191C22×D4
prime radical#27133324C2
hypercenter#451D191
hypocenterC191#3270
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×#15690277C2semidirect
3. #27133326C2semidirect
5. #27133056C2semidirect
6. #27132920C2semidirect
7. #27132894C2semidirect
8. #27132888C2semidirect
9. #27132871C2semidirect
10. #27132828C2semidirect
11. #27132817C2semidirect
12. #27132804C2semidirect
13. #27132767C2semidirect
14. #27132731C2semidirect
15. #451×C191C2semidirect
82. #15690277D4semidirect
170. C22×C191#137semidirect
197. C191#3270semidirect
210. #451D191semidirect
374. 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×#156902772
#271333262
#271333242
#271330562
#271329202
#271328942
#271328882
#271328712
#271328282
#271328172
#271328042
#271327672
#271327312
#451×C1912
#3270191