Finite Group Databasebeta

← #37575689

Subgroups — #37575689

Order 24448.

520 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×#20259308C211yesmaximal, characteristic
2#27132569C211yesmaximal, characteristic, complemented
3#27132567C211yesmaximal, characteristic, complemented
4#27132554C211yesmaximal, characteristic, complemented
5#27132553C211yesmaximal, characteristic, complemented
6#27132551C211yesmaximal, characteristic, complemented
7#27132540C211yesmaximal, characteristic
8#27132539C211yesmaximal, characteristic
9#27132538C211yesmaximal, characteristic, complemented
10#27132537C211yesmaximal, characteristic, complemented
11#27132537C211yesmaximal, characteristic, complemented
12#27132535C211yesmaximal, characteristic, complemented
13#27132528C211yesmaximal, characteristic, complemented
14#27132527C211yesmaximal, characteristic, complemented
15#388×C191C211yesmaximal, characteristic, complemented
16C22×Dic382C2211yescharacteristic
17C22×C22⋊D191C2211yescharacteristic
18C23×Dic191C2211yescharacteristic
19C22×C4×D191C2211yescharacteristic
20#20259308C418yescomplemented
21C2×C4⋊Dic191C2211yescharacteristic
22C2×C4⋊Dic191C2211yescharacteristic
23C2×Dic191⋊C4C2211yescharacteristic
24C2×Dic191⋊C4C2211yescharacteristic
25C2×Dic191⋊C4C2211yescharacteristic
26C2×C22⋊Dic191C2211yescharacteristic
27C2×C22⋊Dic191C2211yescharacteristic
28C2×(C2×D191)⋊C4C2211yescharacteristic
29C2×(C2×D191)⋊C4C2211yescharacteristic
30C2×(C2×D191)⋊C4C2211yescharacteristic
31C2×(C2×D191)⋊C4C2211yescharacteristic
32C2×(C2×D191)⋊C4C2211yescharacteristic
33C2×C4×Dic191C2211yescharacteristic
34C2×C4×Dic191C2211yescharacteristic
35C191⋊#95C2211yescharacteristic
36C191⋊#95C2211yescharacteristic
37C191⋊#95C2211yescharacteristic
38C191⋊#95C2211yescharacteristic
39C191⋊#95C2211yescharacteristic
40C191⋊#95C2211yescharacteristic
41C191⋊#95C2211yescharacteristic
42C191⋊#95C2211yescharacteristic
43C191⋊#95C2211yescharacteristic
44C191⋊#95C2211yescharacteristic
45C2×C4⋊C4×C191C2211yescharacteristic
46C2×C22⋊C4×C191C2211yescharacteristic
47C2×C22⋊C4×C191C2211yescharacteristic
48C2×C22⋊C4×C191C2211yescharacteristic
49C2×C42×C191C2211yescharacteristic
50#95×C191C2211yescharacteristic
51#95×C191C2211yescharacteristic
52C23×D191C2311yescharacteristic
53C2×Dic382—41no—
54C2×Dic382—41no—
55C2×Dic382C2×C414yescomplemented
56C2×C22⋊D191—41no—
57C2×C22⋊D191—41no—
58C2×C22⋊D191C2×C414yes—
59C22×Dic191—41no—
60C22×Dic191—41no—
61C22×Dic191—41no—
62C22×Dic191C2311yescharacteristic
63C22×Dic191C2311yescharacteristic
64C22×Dic191C2311yescharacteristic
65C22×Dic191C2311yescharacteristic
66C22×Dic191C2311yescharacteristic
67C22×Dic191C2311yescharacteristic
68C22×Dic191C2311yescharacteristic
69C2×C4×D191—41no—
70C2×C4×D191—41no—
71C2×C4×D191—41no—
72C4⋊Dic191—41no—
73C4⋊Dic191—41no—
74Dic191⋊C4—41no—
75Dic191⋊C4—41no—
76Dic191⋊C4—41no—
77C22⋊Dic191—41no—
78C22⋊Dic191C2×C414yescomplemented
79(C2×D191)⋊C4—41no—
80(C2×D191)⋊C4—41no—
81(C2×D191)⋊C4—41no—
82(C2×D191)⋊C4C2×C414yescomplemented
83(C2×D191)⋊C4C2×C414yescomplemented
84C4×Dic191—41no—
85C4×Dic191C2×C414yescomplemented
86C24×C191C2311yescharacteristic
87C22×C4×C191C2311yescharacteristic
88C22×C4×C191C2311yescharacteristic
89C22×C4×C191C2311yescharacteristic
90C22×C4×C191C2311yescharacteristic
91C22×C4×C191C2311yescharacteristic
92C22×C4×C191C2311yescharacteristic
93C4⋊C4×C191—41no—
94C22⋊C4×C191—41no—
95C22⋊C4×C191—41no—
96C22⋊C4×C191C2×C414yescomplemented
97C42×C191—41no—
98C22×D191—21no—
99C22×D191—21no—
100C22×D191—21no—
101C22×D191—21no—
102C22×D191—21no—
103C22×D191—21no—
104C22×D191C22×C412yes—
105Dic382—44no—
106C22⋊D191—44no—
107C2×Dic191—21no—
108C2×Dic191—21no—
109C2×Dic191—21no—
110C2×Dic191—21no—
111C2×Dic191—21no—
112C2×Dic191—21no—
113C2×Dic191—21no—
114C2×Dic191—21no—
115C2×Dic191—21no—
116C2×Dic191—21no—
117C2×Dic191—21no—
118C2×Dic191—21no—
119C2×Dic191—21no—
120C2×Dic191—21no—
121C2×Dic191—21no—
122C2×Dic191—21no—
123C2×Dic191—21no—
124C2×Dic191—21no—
125C2×Dic191—44no—
126C2×Dic191C22×C412yes—
127C2×Dic191C22×C412yes—
128C2×Dic191C22×C412yes—
129C4×D191—44no—
130C23×C191—21no—
131C23×C191—21no—
132C23×C191—21no—
133C23×C191—21no—
134C23×C191—21no—
135C23×C191—21no—
136C23×C191C22×C412yes—
137C23×C191C2411yescharacteristic
138C2×C4×C191—21no—
139C2×C4×C191—21no—
140C2×C4×C191—21no—
141C2×C4×C191—21no—
142C2×C4×C191—21no—
143C2×C4×C191—21no—
144C2×C4×C191—21no—
145C2×C4×C191—21no—
146C2×C4×C191—21no—
147C2×C4×C191—21no—
148C2×C4×C191—21no—
149C2×C4×C191—21no—
150C2×C4×C191—21no—
151C2×C4×C191—21no—
152C2×C4×C191—21no—
153C2×C4×C191—21no—
154C2×C4×C191C22×C412yes—
155C2×C4×C191C22×C412yes—
156C22×C191—22no—
157C22×C191—22no—
158C22×C191—22no—
159C22×C191—41no—
160C22×C191—41no—
161C22×C191—41no—
162C22×C191—41no—
163C22×C191C23×C411yescharacteristic
164C22×C191H(5,𝔽2)11yescharacteristic
165C22×C191H(5,𝔽2)11yescharacteristic
166C22×C191H(5,𝔽2)11yescharacteristic
167C22×C191H(5,𝔽2)11yescharacteristic
168C22×C1912−1+411yescharacteristic
169C22×C1912−1+411yescharacteristic
170C2×D191—22no—
171C2×D191—22no—
172C2×D191—22no—
173C2×D191—41no—
174C2×D191—41no—
175C2×D191—41no—
176C2×D191—41no—
177C4×C191—22no—
178C4×C191—22no—
179C4×C191—41no—
180C4×C191—41no—
181C4×C191—41no—
182C4×C191—41no—
183Dic191—22no—
184Dic191—22no—
185Dic191—22no—
186Dic191—41no—
187Dic191—41no—
188Dic191—41no—
189Dic191—41no—
190C2×C191—42no—
191C2×C191#51811yescharacteristic
192C2×C191#52011yescharacteristic
193C2×C191#52011yescharacteristic
194C2×C191H(3,𝔽4)11yescharacteristic
195C2×C191#56211yescharacteristic
196C2×C191#56211yescharacteristic
197C2×C191#56311yescharacteristic
198D191—42no—
199C191#237911yescharacteristic, complemented, Sylow
200#2379—1911nomaximal, Sylow
201C2×#124—1911no—
202#391—1911no—
203#391—1911no—
204#388—1911no—
205#388—1911no—
206#388—1911no—
207#388D19111yescharacteristic, complemented
208#386—1911no—
209#386—1911no—
210#385—1911no—
211#385—1911no—
212#385—1911no—
213#385—1911no—
214#382—1911no—
215#382—1911no—
216C22×Q8—1911no—
217C22×D4—1911no—
218C23×C4—1911no—
219C23×C4—1911no—
220#124—1918no—
221C2×C4⋊C4—1911no—
222C2×C4⋊C4—1911no—
223C2×C4⋊C4—1911no—
224C2×C4⋊C4—1911no—
225C2×C4⋊C4—1911no—
226C2×C4⋊C4C2×D19111yescharacteristic
227C2×C22⋊C4—1911no—
228C2×C22⋊C4—1911no—
229C2×C22⋊C4—1911no—
230C2×C22⋊C4—1911no—
231C2×C22⋊C4—1911no—
232C2×C22⋊C4—1911no—
233C2×C22⋊C4—1911no—
234C2×C22⋊C4C2×D19111yescharacteristic
235C2×C22⋊C4C2×D19111yescharacteristic
236C2×C22⋊C4C2×D19111yescharacteristic
237C2×C42—1911no—
238C2×C42—1911no—
239C2×C42C2×D19111yescharacteristic
240#95—1911no—
241#95—1911no—
242#95—1911no—
243#95—1911no—
244#95—1911no—
245#95—1911no—
246#95—1911no—
247#95—1911no—
248#95—1911no—
249#95—1911no—
250#95C2×D19111yescharacteristic
251#95C2×D19111yescharacteristic
252C24—1911no—
253C24C22×D19111yescharacteristic
254C2×Q8—1914no—
255C2×Q8—7641no—
256C2×Q8—7641no—
257C2×D4—1914no—
258C2×D4—7641no—
259C2×D4—7641no—
260C22×C4—1911no—
261C22×C4—1911no—
262C22×C4—1911no—
263C22×C4—1911no—
264C22×C4—1911no—
265C22×C4—1911no—
266C22×C4—1911no—
267C22×C4—7641no—
268C22×C4—7641no—
269C22×C4—7641no—
270C22×C4—7641no—
271C22×C4—7641no—
272C22×C4—7641no—
273C22×C4C22×D19111yescharacteristic
274C22×C4C22×D19111yescharacteristic
275C22×C4C22×D19111yescharacteristic
276C22×C4C22×D19111yescharacteristic
277C22×C4C22×D19111yescharacteristic
278C22×C4C22×D19111yescharacteristic
279C4⋊C4—41no—
280C4⋊C4—7641no—
281C4⋊C4—7641no—
282C4⋊C4—7641no—
283C4⋊C4—7641no—
284C4⋊C4—7641no—
285C22⋊C4—41no—
286C22⋊C4—41no—
287C22⋊C4—1914no—
288C22⋊C4—1914no—
289C22⋊C4—1914no—
290C22⋊C4—7641no—
291C22⋊C4—7641no—
292C22⋊C4—7641no—
293C22⋊C4—7641no—
294C22⋊C4C4×D19114yescomplemented
295C42—41no—
296C42—1914no—
297C42—7641no—
298C23—21no—
299C23—21no—
300C23—21no—
301C23—21no—
302C23—21no—
303C23—21no—
304C23—3821no—
305C23—3821no—
306C23—3821no—
307C23—3821no—
308C23—3821no—
309C23—3821no—
310C23—1912no—
311C23C2×C4×D19112yes—
312C23C23×D19111yescharacteristic
313Q8—7644no—
314D4—7644no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—3821no—
332C2×C4—3821no—
333C2×C4—3821no—
334C2×C4—3821no—
335C2×C4—3821no—
336C2×C4—3821no—
337C2×C4—3821no—
338C2×C4—3821no—
339C2×C4—3821no—
340C2×C4—3821no—
341C2×C4—3821no—
342C2×C4—3821no—
343C2×C4—3821no—
344C2×C4—3821no—
345C2×C4—3821no—
346C2×C4—3821no—
347C2×C4—3821no—
348C2×C4—3821no—
349C2×C4—1912no—
350C2×C4—1912no—
351C2×C4—1912no—
352C2×C4—7644no—
353C2×C4—7644no—
354C2×C4C2×C4×D19112yes—
355C2×C4C2×C4×D19112yes—
356C22—22no—
357C22—22no—
358C22—22no—
359C22—41no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—3822no—
364C22—3822no—
365C22—3822no—
366C22—7641no—
367C22—7641no—
368C22—7641no—
369C22—7641no—
370C22C22×C4×D19111yescharacteristic
371C22#2025936711yescharacteristic
372C22#2025936711yescharacteristic
373C22#2025936811yescharacteristic
374C22#2025936811yescharacteristic
375C22#2025936911yescharacteristic
376C22#2025936911yescharacteristic
377C4—22no—
378C4—22no—
379C4—41no—
380C4—41no—
381C4—41no—
382C4—41no—
383C4—3822no—
384C4—3822no—
385C4—3822no—
386C4—7641no—
387C4—7641no—
388C4—7641no—
389C4—7641no—
390C2—42no—
391C2—7642no—
392C2#2713313511yescharacteristic
393C2#2713314211yescharacteristic
394C2#2713314311yescharacteristic
395C2#2713340511yescharacteristic
396C2#2713340811yescharacteristic
397C2#2713341011yescharacteristic
398C2#2713341511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D191
commutatorC22×C191C23×C4
Fitting#388×C191C2
socle, cosocleC23×C191C24
prime radicalC22×C22⋊D191C22
hypercenter#388D191
hypocenterC191#2379
LCSG ▷ C22×C191 ▷ C191
UCSC1 ◁ C23 ◁ #388

Complements

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

NH ≅ G/Ntype
2. #27132569C2semidirect
3. #27132567C2semidirect
4. #27132554C2semidirect
5. #27132553C2semidirect
6. #27132551C2semidirect
9. #27132538C2semidirect
10. #27132537C2semidirect
11. #27132537C2semidirect
12. #27132535C2semidirect
13. #27132528C2semidirect
14. #27132527C2semidirect
15. #388×C191C2semidirect
20. #20259308C4semidirect
55. C2×Dic382C2×C4semidirect
78. C22⋊Dic191C2×C4semidirect
82. (C2×D191)⋊C4C2×C4semidirect
83. (C2×D191)⋊C4C2×C4semidirect
85. C4×Dic191C2×C4semidirect
96. C22⋊C4×C191C2×C4semidirect
199. C191#2379semidirect
207. #388D191semidirect
294. C22⋊C4C4×D191semidirect

Maximal subgroups

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

subgroupindex
C2×#202593082
#271325692
#271325672
#271325542
#271325532
#271325512
#271325402
#271325392
#271325382
#271325372
#271325352
#271325282
#271325272
#388×C1912
#2379191