Finite Group Databasebeta

← #37584037

Subgroups — #37584037

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×#20259355C211yesmaximal, characteristic, complemented
2C2×SD16×D191C211yesmaximal, characteristic, complemented
3#27133326C211yesmaximal, characteristic, complemented
4#27133310C211yesmaximal, characteristic
5#27133054C211yesmaximal, characteristic, complemented
6#27132905C211yesmaximal, characteristic, complemented
7#27132884C211yesmaximal, characteristic, complemented
8#27132867C211yesmaximal, characteristic, complemented
9#27132823C211yesmaximal, characteristic, complemented
10#27132820C211yesmaximal, characteristic, complemented
11#27132815C211yesmaximal, characteristic, complemented
12#27132801C211yesmaximal, characteristic, complemented
13#27132762C211yesmaximal, characteristic, complemented
14#27132732C211yesmaximal, characteristic, complemented
15#468×C191C211yesmaximal, characteristic, complemented
16C2×#15690287C2211yescharacteristic
17C22×Dic382C2211yescharacteristic, complemented
18C2×Q8×D191C2211yescharacteristic
19C2×D4×D191C2211yescharacteristic
20#20259355—24no—
21C2×Dic764C2211yescharacteristic, complemented
22C2×Dic764C2211yescharacteristic, complemented
23C2×#15690277C2211yescharacteristic, complemented
24C2×#15690276C2211yescharacteristic
25C2×#15690276C2211yescharacteristic
26C2×#15690275C2211yescharacteristic
27C2×#15690275C2211yescharacteristic
28C2×#15690274C2211yescharacteristic
29SD16×D191—24no—
30C2×C8×D191C2211yescharacteristic
31#20259321—21no—
32#20259320C2211yescharacteristic, complemented
33#20259309C2211yescharacteristic
34#20259304—21no—
35#20259302C2211yescharacteristic
36#20259302C2211yescharacteristic
37#20259300—21no—
38#20259299—21no—
39#20259293—21no—
40#20259290—21no—
41#20259287C2211yescharacteristic, complemented
42#20259284C2211yescharacteristic
43C4×C22⋊D191C2211yescharacteristic
44(C22⋊D191)⋊C4—21no—
45C22⋊C4×D191—21no—
46Dic191⋊C8C2211yescharacteristic, complemented
47#20259245C2211yescharacteristic, complemented
48#20259245C2211yescharacteristic, complemented
49#20259245C2211yescharacteristic, complemented
50#20259242C2211yescharacteristic
51C22⋊(C191⋊C8)C2211yescharacteristic, complemented
52(C2×D191)⋊C8C2211yescharacteristic
53#20259232C2211yescharacteristic, complemented
54C2×SD16×C191C2211yescharacteristic
55C2×M16×C191C2211yescharacteristic, complemented
56C22⋊Q8×C191C2211yescharacteristic, complemented
57#121×C191C2211yescharacteristic
58#107×C191C2211yescharacteristic, complemented
59#103×C191C2211yescharacteristic, complemented
60#102×C191C2211yescharacteristic
61C23×D191—21no—
62#15690287—22no—
63#15690287—41no—
64C2×Dic382—21no—
65C2×Dic382—22no—
66C2×Dic382—21no—
67C2×Dic382—41no—
68C2×Dic382C2311yescharacteristic
69C2×Dic382C2311yescharacteristic
70C2×Dic382C2311yescharacteristic
71Q8×D191—22no—
72Q8×D191—41no—
73C2×D764C2311yescharacteristic
74C2×C22⋊D191—21no—
75C2×C22⋊D191—21no—
76C2×C22⋊D191D411yescharacteristic
77D4×D191—22no—
78D4×D191—41no—
79C22×Dic191—21no—
80C2×C4×D191—21no—
81C2×C4×D191—21no—
82C2×C4×D191—21no—
83C2×C4×D191C2311yescharacteristic
84Dic764—22no—
85Dic764—22no—
86#15690277—41no—
87#15690276—22no—
88#15690276D414yescomplemented
89#15690275—22no—
90#15690275—41no—
91#15690274—22no—
92C2×C191⋊C8C2311yescharacteristic
93C2×C191⋊C8C2311yescharacteristic
94C8×D191—22no—
95C4⋊Dic191—21no—
96C4⋊Dic191—21no—
97Dic191⋊C4—21no—
98Dic191⋊C4—21no—
99Dic191⋊C4—21no—
100Dic191⋊C4D411yescharacteristic
101C22⋊Dic191—21no—
102C22⋊Dic191—21no—
103C22⋊Dic191D411yescharacteristic
104(C2×D191)⋊C4—21no—
105(C2×D191)⋊C4—21no—
106(C2×D191)⋊C4—21no—
107(C2×D191)⋊C4D411yescharacteristic
108C4×Dic191—21no—
109C4×Dic191C2311yescharacteristic
110C2×Q8×C191C2311yescharacteristic
111C2×D4×C191—21no—
112C2×D4×C191C2311yescharacteristic
113C22×C4×C191C2311yescharacteristic
114SD16×C191—22no—
115M16×C191—22no—
116C2×C8×C191C2311yescharacteristic
117C2×C8×C191C2311yescharacteristic
118C4⋊C4×C191—21no—
119C4⋊C4×C191C2311yescharacteristic
120C4⋊C4×C191C2311yescharacteristic
121C22⋊C4×C191—21no—
122C22⋊C4×C191—21no—
123C22×D191—21no—
124C22×D191—42no—
125C22×D191—42no—
126C22×D191—81no—
127C22×D191C2×D411yescharacteristic
128Dic382—21no—
129Dic382—21no—
130Dic382—21no—
131Dic382—21no—
132Dic382—21no—
133Dic382—42no—
134Dic382—42no—
135Dic382C2×D412yes—
136D764—21no—
137D764C2×D412yes—
138C22⋊D191—22no—
139C22⋊D191—42no—
140C22⋊D191—42no—
141C2×Dic191—21no—
142C2×Dic191—21no—
143C2×Dic191—21no—
144C2×Dic191—42no—
145C2×Dic191C2×D411yescharacteristic
146C2×Dic191C2×D411yescharacteristic
147C2×Dic191C2×D411yescharacteristic
148C4×D191—21no—
149C4×D191—21no—
150C4×D191—42no—
151C4×D191—42no—
152C4×D191—81no—
153C191⋊C8—21no—
154C191⋊C8—21no—
155C23×C191—21no—
156C23×C191C2×D411yescharacteristic
157Q8×C191—21no—
158Q8×C191—21no—
159D4×C191—21no—
160D4×C191—21no—
161D4×C191—81no—
162C2×C4×C191—21no—
163C2×C4×C191—21no—
164C2×C4×C191—21no—
165C2×C4×C191—21no—
166C2×C4×C191—21no—
167C2×C4×C191C2×D411yescharacteristic
168C2×C4×C191C2411yescharacteristic
169C8×C191—21no—
170C8×C191C2×D412yes—
171C22×C191—21no—
172C22×C191—21no—
173C22×C191—21no—
174C22×C191—41no—
175C22×C191—41no—
176C22×C191—41no—
177C22×C191C22×D411yescharacteristic
178C2×D191—21no—
179C2×D191—21no—
180C2×D191—22no—
181C2×D191—41no—
182C2×D191—41no—
183C2×D191—84no—
184C2×D191#13712yescomplemented
185C4×C191—21no—
186C4×C191—41no—
187C4×C191—41no—
188C4×C191—41no—
189C4×C191C22×D411yescharacteristic
190C4×C191H(5,𝔽2)11yescharacteristic
191Dic191—21no—
192Dic191—22no—
193Dic191—21no—
194Dic191—21no—
195Dic191—41no—
196Dic191—41no—
197C2×C191—41no—
198C2×C191—81no—
199C2×C191D4211yescharacteristic
200C2×C191C2×#13711yescharacteristic
201C2×C191#57711yescharacteristic
202D191—22no—
203D191—42no—
204C191#326611yescharacteristic, complemented, Sylow
205#3266—1911nomaximal, Sylow
206C2×#137—1911no—
207C22×SD16—1911no—
208D4×Q8—1911no—
209#546—1911no—
210#497—1911no—
211#468D19111yescharacteristic, complemented
212#467—1911no—
213#464—1911no—
214#461—1911no—
215#452—1911no—
216#451—1911no—
217#450—1911no—
218#448—1911no—
219#440—1911no—
220#435—1911no—
221C2×#41—1911no—
222C22×Q8—1911no—
223C22×Q8—1911no—
224C22×D4—1911no—
225#137—3824no—
226C2×Q16—1911no—
227C2×Q16—1911no—
228C2×Q16—1911no—
229C2×SD16—3824no—
230C2×SD16—1911no—
231C2×SD16—1911no—
232C2×SD16—1911no—
233C2×SD16—1911no—
234C2×SD16—1911no—
235C2×SD16C2×D19111yescharacteristic
236C2×M16C2×D19111yescharacteristic, complemented
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, complemented
247#121—3821no—
248#121C2×D19111yescharacteristic
249#120—3821no—
250C4×Q8—1911no—
251C4×D4—3821no—
252C4×D4—1911no—
253C4×D4—1911no—
254C2×C22⋊C4—3821no—
255#107C2×D19111yescharacteristic, complemented
256C4⋊C8—1911no—
257#103—1911no—
258#103—1911no—
259#103—1911no—
260#103C2×D19111yescharacteristic, complemented
261#102—1911no—
262#102C2×D19111yescharacteristic
263C22⋊C8—1911no—
264C22⋊C8—1911no—
265#97—1911no—
266C24—3821no—
267#41—3822no—
268#41—7641no—
269C2×Q8—3821no—
270C2×Q8—3822no—
271C2×Q8—3821no—
272C2×Q8—3822no—
273C2×Q8—1911no—
274C2×Q8—1911no—
275C2×Q8—1911no—
276C2×Q8—7641no—
277C2×Q8—7641no—
278C2×Q8C22×D19111yescharacteristic
279C2×D4—21no—
280C2×D4—3822no—
281C2×D4—3821no—
282C2×D4—3821no—
283C2×D4—1911no—
284C2×D4—1911no—
285C2×D4—7641no—
286C2×D4C22×D19111yescharacteristic
287C22×C4—3821no—
288C22×C4—3821no—
289C22×C4—3821no—
290C22×C4—3821no—
291C22×C4—1911no—
292C22×C4C22×D19111yescharacteristic
293Q16—3822no—
294Q16—3822no—
295Q16—7641no—
296SD16—22no—
297SD16—3822no—
298SD16—3822no—
299SD16—3822no—
300SD16—1914no—
301SD16—7641no—
302M16—22no—
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⋊C4C22×D19111yescharacteristic
316C4⋊C4C22×D19111yescharacteristic
317C22⋊C4—21no—
318C22⋊C4—21no—
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—21no—
329C23—3821no—
330C23—1911no—
331C23—15281no—
332C23—7642no—
333C23—7642no—
334C23D4×D19111yescharacteristic
335Q8—21no—
336Q8—21no—
337Q8—3821no—
338Q8—3821no—
339Q8—3821no—
340Q8—3821no—
341Q8—3821no—
342Q8—1912no—
343Q8—7642no—
344Q8—7642no—
345D4—21no—
346D4—21no—
347D4—81no—
348D4—3822no—
349D4—3821no—
350D4—1912no—
351D4—7642no—
352D4—7642no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—3821no—
359C2×C4—3821no—
360C2×C4—3821no—
361C2×C4—3821no—
362C2×C4—3821no—
363C2×C4—1911no—
364C2×C4—1911no—
365C2×C4—1911no—
366C2×C4—15281no—
367C2×C4—7642no—
368C2×C4—7642no—
369C2×C4—7642no—
370C2×C4D4×D19111yescharacteristic
371C2×C4C23×D19111yescharacteristic
372C8—21no—
373C8—3821no—
374C8—3821no—
375C8D4×D19112yes—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—41no—
380C22—41no—
381C22—41no—
382C22—3821no—
383C22—3821no—
384C22—3822no—
385C22—1912no—
386C22—15284no—
387C22—7641no—
388C22—7641no—
389C22C2×D4×D19111yescharacteristic
390C4—21no—
391C4—41no—
392C4—41no—
393C4—41no—
394C4—3821no—
395C4—3821no—
396C4—3821no—
397C4—3822no—
398C4—7641no—
399C4—7641no—
400C4C2×D4×D19111yescharacteristic
401C4#2025936811yescharacteristic
402C2—41no—
403C2—81no—
404C2—3822no—
405C2—7642no—
406C2#2713329711yescharacteristic
407C2#137×D19111yescharacteristic
408C2#2713347311yescharacteristic
subgroupG/H
centerC22C2×D4×D191
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
Fitting#468×C191C2
socleC22×C191C22×D4
prime radical#27133310C2
hypercenter#468D191
hypocenterC191#3266
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C22 ◁ C22×C4 ◁ #468

Complements

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

NH ≅ G/Ntype
1. C2×#20259355C2semidirect
2. C2×SD16×D191C2semidirect
3. #27133326C2semidirect
5. #27133054C2semidirect
6. #27132905C2semidirect
7. #27132884C2semidirect
8. #27132867C2semidirect
9. #27132823C2semidirect
10. #27132820C2semidirect
11. #27132815C2semidirect
12. #27132801C2semidirect
13. #27132762C2semidirect
14. #27132732C2semidirect
15. #468×C191C2semidirect
17. C22×Dic382C22semidirect
21. C2×Dic764C22semidirect
22. C2×Dic764C22semidirect
23. C2×#15690277C22semidirect
32. #20259320C22semidirect
41. #20259287C22semidirect
46. Dic191⋊C8C22semidirect
47. #20259245C22semidirect
48. #20259245C22semidirect
49. #20259245C22semidirect
51. C22⋊(C191⋊C8)C22semidirect
53. #20259232C22semidirect
55. C2×M16×C191C22semidirect
56. C22⋊Q8×C191C22semidirect
58. #107×C191C22semidirect
59. #103×C191C22semidirect
88. #15690276D4semidirect
184. C2×D191#137semidirect
204. C191#3266semidirect
211. #468D191semidirect
236. C2×M16C2×D191semidirect
246. C22⋊Q8C2×D191semidirect
255. #107C2×D191semidirect
260. #103C2×D191semidirect

Maximal subgroups

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

subgroupindex
C2×#202593552
C2×SD16×D1912
#271333262
#271333102
#271330542
#271329052
#271328842
#271328672
#271328232
#271328202
#271328152
#271328012
#271327622
#271327322
#468×C1912
#3266191