Finite Group Databasebeta

← #37584090

Subgroups — #37584090

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
2Q8×C22⋊D191C211yesmaximal, characteristic, complemented
3#27133324C211yesmaximal, characteristic
4#27133056C211yesmaximal, characteristic, complemented
5#27132918C211yesmaximal, characteristic, complemented
6#27132892C211yesmaximal, characteristic, complemented
7#27132887C211yesmaximal, characteristic, complemented
8#27132871C211yesmaximal, characteristic, complemented
9#27132824C211yesmaximal, characteristic, complemented
10#27132819C211yesmaximal, characteristic, complemented
11#27132817C211yesmaximal, characteristic, complemented
12#27132802C211yesmaximal, characteristic, complemented
13#27132770C211yesmaximal, characteristic, complemented
14#27132734C211yesmaximal, characteristic, complemented
15C22×Q16×C191C211yesmaximal, characteristic, complemented
16C2×#15690288C2211yescharacteristic
17C2×Q8×D191C2211yescharacteristic
18#20259354—24no—
19C2×#15690277C2211yescharacteristic
20C2×#15690277C2211yescharacteristic
21C2×#15690276C2211yescharacteristic
22C2×#15690274C2211yescharacteristic
23C2×#15690274C2211yescharacteristic
24C2×#15690270C2211yescharacteristic
25#20259320—21no—
26#20259320C2211yescharacteristic
27#20259311—21no—
28#20259311C2211yescharacteristic
29#20259301—21no—
30#20259301C2211yescharacteristic
31#20259298—21no—
32#20259298—21no—
33#20259298C2211yescharacteristic
34#20259298C2211yescharacteristic
35#20259296—21no—
36#20259296C2211yescharacteristic
37Q8×Dic191C2211yescharacteristic
38Q8×Dic191C2211yescharacteristic
39C4×C22⋊D191—21no—
40C4×C22⋊D191—21no—
41C4×C22⋊D191C2211yescharacteristic
42#20259254C2211yescharacteristic
43Dic191⋊C8C2211yescharacteristic
44#20259247C2211yescharacteristic
45#20259246C2211yescharacteristic
46#20259246C2211yescharacteristic
47#20259246C2211yescharacteristic
48#20259246C2211yescharacteristic
49#20259244C2211yescharacteristic
50C22⋊(C191⋊C8)C2211yescharacteristic
51(C2×D191)⋊C8C2211yescharacteristic
52#20259232C2211yescharacteristic
53C22×Q8×C191C2211yescharacteristic
54C22×Q8×C191C2211yescharacteristic
55C2×Q16×C191—24no—
56C2×Q16×C191C2211yescharacteristic
57C2×Q16×C191C2211yescharacteristic
58C2×Q16×C191C2211yescharacteristic
59C2×Q16×C191C2211yescharacteristic
60C22×C8×C191C2211yescharacteristic
61#15690288—22no—
62#15690288—41no—
63C2×Dic382—21no—
64C2×Dic382C2311yescharacteristic
65Q8×D191—22no—
66Q8×D191—41no—
67C2×D764—21no—
68C2×D764C2311yescharacteristic
69C2×C22⋊D191—21no—
70C2×C22⋊D191D411yescharacteristic
71C2×C4×D191—21no—
72C2×C4×D191—21no—
73C2×C4×D191C2311yescharacteristic
74#15690277—22no—
75#15690277—41no—
76#15690276—22no—
77#15690274—22no—
78#15690274—41no—
79#15690270—22no—
80C2×C191⋊C8C2311yescharacteristic
81C2×C191⋊C8C2311yescharacteristic
82C4⋊Dic191—21no—
83C4⋊Dic191—21no—
84C4⋊Dic191C2311yescharacteristic
85C4⋊Dic191C2311yescharacteristic
86Dic191⋊C4—21no—
87Dic191⋊C4—21no—
88Dic191⋊C4—21no—
89Dic191⋊C4—21no—
90Dic191⋊C4—21no—
91Dic191⋊C4D411yescharacteristic
92C22⋊Dic191—21no—
93C22⋊Dic191D411yescharacteristic
94(C2×D191)⋊C4—21no—
95(C2×D191)⋊C4—21no—
96(C2×D191)⋊C4—21no—
97(C2×D191)⋊C4—21no—
98(C2×D191)⋊C4—21no—
99(C2×D191)⋊C4D411yescharacteristic
100C4×Dic191—21no—
101C4×Dic191—21no—
102C4×Dic191C2311yescharacteristic
103C2×Q8×C191—21no—
104C2×Q8×C191—21no—
105C2×Q8×C191—22no—
106C2×Q8×C191—22no—
107C2×Q8×C191—41no—
108C2×Q8×C191—41no—
109C2×Q8×C191C2311yescharacteristic
110C2×Q8×C191C2311yescharacteristic
111C2×Q8×C191C2311yescharacteristic
112C2×Q8×C191C2311yescharacteristic
113C22×C4×C191—21no—
114C22×C4×C191—21no—
115C22×C4×C191C2311yescharacteristic
116Q16×C191—22no—
117Q16×C191—22no—
118Q16×C191—22no—
119Q16×C191D414yescomplemented
120C2×C8×C191—22no—
121C2×C8×C191C2311yescharacteristic
122C2×C8×C191C2311yescharacteristic
123C22×D191—21no—
124C22×D191C2×D411yescharacteristic
125Dic382—21no—
126Dic382—21no—
127Dic382—42no—
128D764—21no—
129D764—21no—
130D764—42no—
131C22⋊D191—22no—
132C22⋊D191—42no—
133C2×Dic191—21no—
134C2×Dic191—21no—
135C2×Dic191—21no—
136C2×Dic191C2×D411yescharacteristic
137C2×Dic191C2×D411yescharacteristic
138C2×Dic191C2×D411yescharacteristic
139C4×D191—21no—
140C4×D191—21no—
141C4×D191—42no—
142C4×D191—42no—
143C191⋊C8—21no—
144C191⋊C8—21no—
145C23×C191C2×D411yescharacteristic
146Q8×C191—21no—
147Q8×C191—21no—
148Q8×C191—21no—
149Q8×C191—21no—
150Q8×C191—21no—
151Q8×C191—21no—
152Q8×C191—42no—
153Q8×C191—42no—
154Q8×C191C2×D412yes—
155Q8×C191C2×D412yes—
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×C191—42no—
163C2×C4×C191—42no—
164C2×C4×C191C2×D411yescharacteristic
165C2×C4×C191C2411yescharacteristic
166C8×C191—21no—
167C8×C191C2×D412yes—
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—22no—
181C4×C191—41no—
182C4×C191—41no—
183C4×C191C22×D411yescharacteristic
184C4×C191H(5,𝔽2)11yescharacteristic
185Dic191—21no—
186Dic191—21no—
187Dic191—21no—
188Dic191—41no—
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×Q16D19111yescharacteristic, complemented
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#451—1911no—
211#448—1911no—
212#441—1911no—
213#435—1911no—
214C2×#41—1911no—
215C22×Q8—1911no—
216C22×Q8C2×D19111yescharacteristic
217C22×Q8C2×D19111yescharacteristic
218#137—3824no—
219C2×Q16—24no—
220C2×Q16—1911no—
221C2×Q16—1911no—
222C2×Q16C2×D19111yescharacteristic
223C2×Q16C2×D19111yescharacteristic
224C2×Q16C2×D19111yescharacteristic
225C2×Q16C2×D19111yescharacteristic
226C2×SD16—1911no—
227C2×SD16—1911no—
228C2×SD16—1911no—
229C2×M16—1911no—
230C22×C8C2×D19111yescharacteristic
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⋊Q8—1911no—
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#103—1911no—
254#103—1911no—
255#102—1911no—
256C22⋊C8—1911no—
257C22⋊C8—1911no—
258#97—1911no—
259#41—3822no—
260#41—7641no—
261C2×Q8—21no—
262C2×Q8—21no—
263C2×Q8—22no—
264C2×Q8—22no—
265C2×Q8—41no—
266C2×Q8—41no—
267C2×Q8—3821no—
268C2×Q8—3822no—
269C2×Q8—1911no—
270C2×Q8—7641no—
271C2×Q8C22×D19111yescharacteristic
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—21no—
281C22×C4—3821no—
282C22×C4—3821no—
283C22×C4—1911no—
284C22×C4C22×D19111yescharacteristic
285Q16—22no—
286Q16—22no—
287Q16—22no—
288Q16—3822no—
289Q16—7641no—
290Q16C22⋊D19114yescomplemented
291SD16—3822no—
292SD16—3822no—
293SD16—7641no—
294M16—3822no—
295C2×C8—22no—
296C2×C8—1911no—
297C2×C8—1911no—
298C2×C8C22×D19111yescharacteristic
299C2×C8C22×D19111yescharacteristic
300C4⋊C4—3821no—
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⋊C4—1911no—
310C22⋊C4—3821no—
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—21no—
330Q8—42no—
331Q8—42no—
332Q8—3821no—
333Q8—3821no—
334Q8—7642no—
335Q8C2×C22⋊D19112yes—
336Q8C2×C22⋊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—42no—
350C2×C4—3821no—
351C2×C4—3821no—
352C2×C4—3821no—
353C2×C4—3821no—
354C2×C4—3821no—
355C2×C4—1911no—
356C2×C4—1911no—
357C2×C4—1911no—
358C2×C4—7642no—
359C2×C4—7642no—
360C2×C4D4×D19111yescharacteristic
361C2×C4C23×D19111yescharacteristic
362C8—21no—
363C8—3821no—
364C8—3821no—
365C8C2×C22⋊D19112yes—
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—22no—
379C4—41no—
380C4—41no—
381C4—3821no—
382C4—3821no—
383C4—3821no—
384C4—7641no—
385C4—7641no—
386C4—7641no—
387C4C22×C22⋊D19111yescharacteristic
388C4#2025936711yescharacteristic
389C2—22no—
390C2—15281no—
391C2—7641no—
392C2D4×C22⋊D19111yescharacteristic
393C2C2×#2025935411yescharacteristic
394C2#2713347611yescharacteristic
subgroupG/H
centerC22C2×D4×D191
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
FittingC22×Q16×C191C2
socleC22×C191C22×D4
prime radical#27133324C2
hypercenterC22×Q16D191
hypocenterC191#3270
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×Q16

Complements

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

NH ≅ G/Ntype
1. C2×#20259354C2semidirect
2. Q8×C22⋊D191C2semidirect
4. #27133056C2semidirect
5. #27132918C2semidirect
6. #27132892C2semidirect
7. #27132887C2semidirect
8. #27132871C2semidirect
9. #27132824C2semidirect
10. #27132819C2semidirect
11. #27132817C2semidirect
12. #27132802C2semidirect
13. #27132770C2semidirect
14. #27132734C2semidirect
15. C22×Q16×C191C2semidirect
119. Q16×C191D4semidirect
170. C22×C191#137semidirect
197. C191#3270semidirect
200. C22×Q16D191semidirect
290. Q16C22⋊D191semidirect
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
Q8×C22⋊D1912
#271333242
#271330562
#271329182
#271328922
#271328872
#271328712
#271328242
#271328192
#271328172
#271328022
#271327702
#271327342
C22×Q16×C1912
#3270191