Finite Group Databasebeta

← #37575229

Subgroups — #37575229

Order 24448.

612 conjugacy classes of subgroups.

The subgroup lattice has 412 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#20259308C211yesmaximal, characteristic
2C2×C4×Dic382C211yesmaximal, characteristic, complemented
3C2×(C22⋊D191)⋊C4C211yesmaximal, characteristic
4#27132567C211yesmaximal, characteristic, complemented
5#27132563C211yesmaximal, characteristic
6#27132555C211yesmaximal, characteristic, complemented
7#27132554C211yesmaximal, characteristic, complemented
8#27132551C211yesmaximal, characteristic, complemented
9#27132538C211yesmaximal, characteristic, complemented
10#27132537C211yesmaximal, characteristic, complemented
11#27132535C211yesmaximal, characteristic, complemented
12#27132534C211yesmaximal, characteristic, complemented
13#27132532C211yesmaximal, characteristic, complemented
14#27132508C211yesmaximal, characteristic, complemented
15#388×C191C211yesmaximal, characteristic, complemented
16C22×Dic382C2211yescharacteristic
17C22×C22⋊D191C2211yescharacteristic
18C23×Dic191C2211yescharacteristic
19C22×C4×D191C2211yescharacteristic
20#20259308C418yescomplemented
21C4×Dic382—24no—
22(C22⋊D191)⋊C4—24no—
23C2×C4⋊Dic191C2211yescharacteristic
24C2×C4⋊Dic191C2211yescharacteristic
25C2×Dic191⋊C4C2211yescharacteristic
26C2×Dic191⋊C4C2211yescharacteristic
27C2×Dic191⋊C4C2211yescharacteristic
28C2×Dic191⋊C4C2211yescharacteristic
29C2×Dic191⋊C4C2211yescharacteristic
30C2×C22⋊Dic191C2211yescharacteristic
31C2×C22⋊Dic191C2211yescharacteristic
32C2×(C2×D191)⋊C4C2211yescharacteristic
33C2×(C2×D191)⋊C4C2211yescharacteristic
34C2×(C2×D191)⋊C4C2211yescharacteristic
35C2×(C2×D191)⋊C4C2211yescharacteristic
36C2×(C2×D191)⋊C4C2211yescharacteristic
37C2×C4×Dic191C2211yescharacteristic
38C2×C4×Dic191C2211yescharacteristic
39C2×C4×Dic191C2211yescharacteristic
40C2×C4×Dic191C2211yescharacteristic
41C191⋊#95C2211yescharacteristic
42C191⋊#95C2211yescharacteristic
43C191⋊#95C2211yescharacteristic
44C191⋊#95C2211yescharacteristic
45C191⋊#95C2211yescharacteristic
46C191⋊#95C2211yescharacteristic
47C2×C4⋊C4×C191C2211yescharacteristic
48C2×C22⋊C4×C191C2211yescharacteristic
49C2×C22⋊C4×C191C2211yescharacteristic
50C2×C22⋊C4×C191C2211yescharacteristic
51C2×C42×C191C2211yescharacteristic
52#95×C191C2211yescharacteristic
53#95×C191C2211yescharacteristic
54C23×D191C2311yescharacteristic
55C2×Dic382—22no—
56C2×Dic382—22no—
57C2×Dic382C2×C414yescomplemented
58C2×C22⋊D191—22no—
59C2×C22⋊D191—22no—
60C2×C22⋊D191C2×C414yes—
61C22×Dic191—22no—
62C22×Dic191—41no—
63C22×Dic191—41no—
64C22×Dic191C2311yescharacteristic
65C22×Dic191C2311yescharacteristic
66C22×Dic191C2311yescharacteristic
67C22×Dic191C2311yescharacteristic
68C22×Dic191C2311yescharacteristic
69C22×Dic191C2311yescharacteristic
70C22×Dic191C2311yescharacteristic
71C2×C4×D191—22no—
72C2×C4×D191—41no—
73C2×C4×D191—41no—
74C4⋊Dic191—22no—
75C4⋊Dic191—41no—
76Dic191⋊C4—22no—
77Dic191⋊C4—22no—
78Dic191⋊C4—22no—
79Dic191⋊C4—41no—
80Dic191⋊C4—41no—
81C22⋊Dic191—41no—
82C22⋊Dic191C2×C414yescomplemented
83(C2×D191)⋊C4—22no—
84(C2×D191)⋊C4—41no—
85(C2×D191)⋊C4—41no—
86(C2×D191)⋊C4C2×C414yescomplemented
87(C2×D191)⋊C4C2×C414yescomplemented
88C4×Dic191—22no—
89C4×Dic191—22no—
90C4×Dic191C2×C414yescomplemented
91C4×Dic191D414yes—
92C24×C191C2311yescharacteristic
93C22×C4×C191C2311yescharacteristic
94C22×C4×C191C2311yescharacteristic
95C22×C4×C191C2311yescharacteristic
96C22×C4×C191C2311yescharacteristic
97C22×C4×C191C2311yescharacteristic
98C22×C4×C191C2311yescharacteristic
99C4⋊C4×C191—41no—
100C22⋊C4×C191—22no—
101C22⋊C4×C191—41no—
102C22⋊C4×C191C2×C414yescomplemented
103C42×C191—22no—
104C22×D191—21no—
105C22×D191—21no—
106C22×D191—21no—
107C22×D191—21no—
108C22×D191—21no—
109C22×D191—21no—
110C22×D191C22×C412yes—
111Dic382—28no—
112C22⋊D191—28no—
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—21no—
126C2×Dic191—21no—
127C2×Dic191—21no—
128C2×Dic191—44no—
129C2×Dic191C22×C412yes—
130C2×Dic191C22×C412yes—
131C2×Dic191C22×C412yes—
132C2×Dic191C2×D412yes—
133C2×Dic191C2×D412yes—
134C2×Dic191#4112yes—
135C4×D191—44no—
136C23×C191—21no—
137C23×C191—21no—
138C23×C191—21no—
139C23×C191—21no—
140C23×C191—21no—
141C23×C191—21no—
142C23×C191C22×C412yes—
143C23×C191C2411yescharacteristic
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×C191—21no—
155C2×C4×C191—21no—
156C2×C4×C191—21no—
157C2×C4×C191—21no—
158C2×C4×C191—21no—
159C2×C4×C191C22×C412yes—
160C2×C4×C191C22×C412yes—
161C2×C4×C191C2×D412yes—
162C22×C191—22no—
163C22×C191—22no—
164C22×C191—22no—
165C22×C191—41no—
166C22×C191—41no—
167C22×C191—41no—
168C22×C191—41no—
169C22×C191C23×C411yescharacteristic
170C22×C191C22×D411yescharacteristic
171C22×C191C2×#4111yescharacteristic
172C22×C191H(5,𝔽2)11yescharacteristic
173C22×C191H(5,𝔽2)11yescharacteristic
174C22×C1912−1+411yescharacteristic
175C22×C1912−1+411yescharacteristic
176C2×D191—22no—
177C2×D191—22no—
178C2×D191—22no—
179C2×D191—41no—
180C2×D191—41no—
181C2×D191—41no—
182C2×D191—41no—
183C4×C191—22no—
184C4×C191—22no—
185C4×C191—22no—
186C4×C191—41no—
187C4×C191—41no—
188C4×C191—41no—
189Dic191—22no—
190Dic191—22no—
191Dic191—22no—
192Dic191—41no—
193Dic191—41no—
194Dic191—41no—
195Dic191C4×D414yes—
196C2×C191—42no—
197C2×C191C2×C4×D411yescharacteristic
198C2×C191#52011yescharacteristic
199C2×C191#52011yescharacteristic
200C2×C191#53511yescharacteristic
201C2×C191#53611yescharacteristic
202C2×C191#54211yescharacteristic
203C2×C191#54211yescharacteristic
204D191—42no—
205C191#232611yescharacteristic, complemented, Sylow
206#2326—1911nomaximal, Sylow
207C2×#124—1911no—
208C2×C4×Q8—1911no—
209C2×C4×D4—1911no—
210#391—1911no—
211#390—1911no—
212#388—1911no—
213#388—1911no—
214#388—1911no—
215#388D19111yescharacteristic, complemented
216#385—1911no—
217#385—1911no—
218#385—1911no—
219#385—1911no—
220#384—1911no—
221#376—1911no—
222C22×Q8—1911no—
223C22×D4—1911no—
224C23×C4—1911no—
225C23×C4—1911no—
226#124—1918no—
227C4×Q8—3824no—
228C4×D4—3824no—
229C2×C4⋊C4—1911no—
230C2×C4⋊C4—1911no—
231C2×C4⋊C4—1911no—
232C2×C4⋊C4—1911no—
233C2×C4⋊C4—1911no—
234C2×C4⋊C4—1911no—
235C2×C4⋊C4—1911no—
236C2×C4⋊C4C2×D19111yescharacteristic
237C2×C22⋊C4—1911no—
238C2×C22⋊C4—1911no—
239C2×C22⋊C4—1911no—
240C2×C22⋊C4—1911no—
241C2×C22⋊C4—1911no—
242C2×C22⋊C4—1911no—
243C2×C22⋊C4—1911no—
244C2×C22⋊C4C2×D19111yescharacteristic
245C2×C22⋊C4C2×D19111yescharacteristic
246C2×C22⋊C4C2×D19111yescharacteristic
247C2×C42—1911no—
248C2×C42—1911no—
249C2×C42—1911no—
250C2×C42—1911no—
251C2×C42C2×D19111yescharacteristic
252#95—1911no—
253#95—1911no—
254#95—1911no—
255#95—1911no—
256#95—1911no—
257#95—1911no—
258#95C2×D19111yescharacteristic
259#95C2×D19111yescharacteristic
260C24—1911no—
261C24C22×D19111yescharacteristic
262C2×Q8—3822no—
263C2×Q8—3822no—
264C2×Q8—1914no—
265C2×D4—3822no—
266C2×D4—3822no—
267C2×D4—1914no—
268C22×C4—3822no—
269C22×C4—3822no—
270C22×C4—1911no—
271C22×C4—1911no—
272C22×C4—1911no—
273C22×C4—1911no—
274C22×C4—1911no—
275C22×C4—1911no—
276C22×C4—1911no—
277C22×C4—7641no—
278C22×C4—7641no—
279C22×C4—7641no—
280C22×C4—7641no—
281C22×C4C22×D19111yescharacteristic
282C22×C4C22×D19111yescharacteristic
283C22×C4C22×D19111yescharacteristic
284C22×C4C22×D19111yescharacteristic
285C22×C4C22×D19111yescharacteristic
286C22×C4C22×D19111yescharacteristic
287C4⋊C4—41no—
288C4⋊C4—3822no—
289C4⋊C4—3822no—
290C4⋊C4—3822no—
291C4⋊C4—3822no—
292C4⋊C4—7641no—
293C4⋊C4—7641no—
294C4⋊C4—7641no—
295C22⋊C4—22no—
296C22⋊C4—41no—
297C22⋊C4—3822no—
298C22⋊C4—1914no—
299C22⋊C4—1914no—
300C22⋊C4—1914no—
301C22⋊C4—7641no—
302C22⋊C4—7641no—
303C22⋊C4—7641no—
304C22⋊C4C4×D19114yescomplemented
305C42—22no—
306C42—3822no—
307C42—3822no—
308C42—1914no—
309C42—1914no—
310C23—21no—
311C23—21no—
312C23—21no—
313C23—21no—
314C23—21no—
315C23—21no—
316C23—3821no—
317C23—3821no—
318C23—3821no—
319C23—3821no—
320C23—3821no—
321C23—3821no—
322C23—1912no—
323C23C2×C4×D19112yes—
324C23C23×D19111yescharacteristic
325Q8—3828no—
326D4—3828no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—3821no—
343C2×C4—3821no—
344C2×C4—3821no—
345C2×C4—3821no—
346C2×C4—3821no—
347C2×C4—3821no—
348C2×C4—3821no—
349C2×C4—3821no—
350C2×C4—3821no—
351C2×C4—3821no—
352C2×C4—3821no—
353C2×C4—3821no—
354C2×C4—3821no—
355C2×C4—3821no—
356C2×C4—3821no—
357C2×C4—1912no—
358C2×C4—1912no—
359C2×C4—1912no—
360C2×C4—1912no—
361C2×C4—1912no—
362C2×C4—1912no—
363C2×C4—7644no—
364C2×C4—7644no—
365C2×C4C2×C4×D19112yes—
366C2×C4C2×C4×D19112yes—
367C2×C4D4×D19112yes—
368C22—22no—
369C22—22no—
370C22—22no—
371C22—41no—
372C22—41no—
373C22—41no—
374C22—41no—
375C22—3822no—
376C22—3822no—
377C22—3822no—
378C22—7641no—
379C22—7641no—
380C22—7641no—
381C22—7641no—
382C22C22×C4×D19111yescharacteristic
383C22C2×D4×D19111yescharacteristic
384C22#41×D19111yescharacteristic
385C22#2025936711yescharacteristic
386C22#2025936811yescharacteristic
387C22#2025936911yescharacteristic
388C22#2025936911yescharacteristic
389C4—22no—
390C4—22no—
391C4—22no—
392C4—41no—
393C4—41no—
394C4—41no—
395C4—3822no—
396C4—3822no—
397C4—3822no—
398C4—1914no—
399C4—7641no—
400C4—7641no—
401C4—7641no—
402C2—42no—
403C2—7642no—
404C2C4×D4×D19111yescharacteristic
405C2#2713314211yescharacteristic
406C2#2713314311yescharacteristic
407C2#2713322911yescharacteristic
408C2#2713323511yescharacteristic
409C2#2713327711yescharacteristic
410C2#2713327911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D191
commutatorC22×C191C23×C4
Fitting#388×C191C2
socle, cosocleC23×C191C24
prime radicalC22×C22⋊D191C22
hypercenter#388D191
hypocenterC191#2326
LCSG ▷ C22×C191 ▷ C191
UCSC1 ◁ C23 ◁ #388

Complements

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

NH ≅ G/Ntype
2. C2×C4×Dic382C2semidirect
4. #27132567C2semidirect
6. #27132555C2semidirect
7. #27132554C2semidirect
8. #27132551C2semidirect
9. #27132538C2semidirect
10. #27132537C2semidirect
11. #27132535C2semidirect
12. #27132534C2semidirect
13. #27132532C2semidirect
14. #27132508C2semidirect
15. #388×C191C2semidirect
20. #20259308C4semidirect
57. C2×Dic382C2×C4semidirect
82. C22⋊Dic191C2×C4semidirect
86. (C2×D191)⋊C4C2×C4semidirect
87. (C2×D191)⋊C4C2×C4semidirect
90. C4×Dic191C2×C4semidirect
102. C22⋊C4×C191C2×C4semidirect
205. C191#2326semidirect
215. #388D191semidirect
304. 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
C2×C4×Dic3822
C2×(C22⋊D191)⋊C42
#271325672
#271325632
#271325552
#271325542
#271325512
#271325382
#271325372
#271325352
#271325342
#271325322
#271325082
#388×C1912
#2326191