Finite Group Databasebeta

← #37584033

Subgroups — #37584033

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×#20259353C211yesmaximal, characteristic, complemented
2C2×SD16×D191C211yesmaximal, characteristic, complemented
3Q8×C22⋊D191C211yesmaximal, characteristic, complemented
4#27133310C211yesmaximal, characteristic
5#27133054C211yesmaximal, characteristic, complemented
6#27132907C211yesmaximal, characteristic, complemented
7#27132903C211yesmaximal, characteristic, complemented
8#27132883C211yesmaximal, characteristic, complemented
9#27132867C211yesmaximal, characteristic, complemented
10#27132827C211yesmaximal, characteristic, complemented
11#27132820C211yesmaximal, characteristic, complemented
12#27132812C211yesmaximal, characteristic, complemented
13#27132766C211yesmaximal, characteristic, complemented
14#27132737C211yesmaximal, characteristic, complemented
15#448×C191C211yesmaximal, characteristic, complemented
16C2×#15690287C2211yescharacteristic
17C2×Q8×D191C2211yescharacteristic
18C2×D4×D191C2211yescharacteristic
19#20259353—24no—
20C2×Dic764C2211yescharacteristic, complemented
21C2×#15690277C2211yescharacteristic, complemented
22C2×#15690277C2211yescharacteristic, complemented
23C2×#15690276C2211yescharacteristic
24C2×#15690275C2211yescharacteristic
25C2×#15690274C2211yescharacteristic
26C2×#15690274C2211yescharacteristic
27SD16×D191—24no—
28C2×#15690271C2211yescharacteristic, complemented
29C2×C8×D191C2211yescharacteristic
30#20259320—21no—
31#20259320C2211yescharacteristic, complemented
32#20259309C2211yescharacteristic
33#20259304—21no—
34#20259302C2211yescharacteristic
35#20259301—21no—
36#20259301C2211yescharacteristic, complemented
37#20259298—21no—
38#20259298C2211yescharacteristic
39#20259293—21no—
40#20259290—21no—
41Q8×Dic191C2211yescharacteristic, complemented
42#20259284C2211yescharacteristic
43C4×C22⋊D191—21no—
44C4×C22⋊D191C2211yescharacteristic
45C22⋊C4×D191—21no—
46#20259255C2211yescharacteristic, complemented
47Dic191⋊C8C2211yescharacteristic, complemented
48#20259247C2211yescharacteristic, complemented
49#20259245C2211yescharacteristic, complemented
50#20259244C2211yescharacteristic
51#20259243C2211yescharacteristic
52(C2×D191)⋊C8C2211yescharacteristic
53#20259232C2211yescharacteristic, complemented
54C22×Q8×C191C2211yescharacteristic, complemented
55C2×SD16×C191C2211yescharacteristic
56C2×SD16×C191C2211yescharacteristic
57#121×C191C2211yescharacteristic
58#103×C191C2211yescharacteristic, complemented
59#103×C191C2211yescharacteristic, complemented
60C22⋊C8×C191C2211yescharacteristic, complemented
61C23×D191—21no—
62#15690287—22no—
63#15690287—41no—
64C2×Dic382—21no—
65C2×Dic382C2311yescharacteristic
66C2×Dic382C2311yescharacteristic
67Q8×D191—22no—
68Q8×D191—41no—
69C2×D764C2311yescharacteristic
70C2×C22⋊D191—21no—
71C2×C22⋊D191—21no—
72C2×C22⋊D191D411yescharacteristic
73D4×D191—22no—
74D4×D191—41no—
75C2×C4×D191—21no—
76C2×C4×D191—21no—
77C2×C4×D191—21no—
78C2×C4×D191C2311yescharacteristic
79Dic764—41no—
80#15690277—22no—
81#15690277—22no—
82#15690276—22no—
83#15690275—22no—
84#15690274—22no—
85#15690274D414yescomplemented
86#15690271—22no—
87C2×C191⋊C8C2311yescharacteristic
88C2×C191⋊C8C2311yescharacteristic
89C8×D191—22no—
90C4⋊Dic191—21no—
91C4⋊Dic191—21no—
92C4⋊Dic191C2311yescharacteristic
93Dic191⋊C4—21no—
94Dic191⋊C4—21no—
95Dic191⋊C4—21no—
96Dic191⋊C4—21no—
97Dic191⋊C4D411yescharacteristic
98C22⋊Dic191—21no—
99C22⋊Dic191—21no—
100C22⋊Dic191—21no—
101C22⋊Dic191D411yescharacteristic
102(C2×D191)⋊C4—21no—
103(C2×D191)⋊C4—21no—
104(C2×D191)⋊C4—21no—
105(C2×D191)⋊C4D411yescharacteristic
106C4×Dic191—21no—
107C4×Dic191C2311yescharacteristic
108C2×Q8×C191—21no—
109C2×Q8×C191—22no—
110C2×Q8×C191—41no—
111C2×Q8×C191C2311yescharacteristic
112C2×Q8×C191C2311yescharacteristic
113C2×D4×C191—21no—
114C2×D4×C191C2311yescharacteristic
115C22×C4×C191—21no—
116C22×C4×C191C2311yescharacteristic
117SD16×C191—22no—
118SD16×C191—41no—
119C2×C8×C191C2311yescharacteristic
120C2×C8×C191C2311yescharacteristic
121C4⋊C4×C191C2311yescharacteristic
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—42no—
133D764—21no—
134D764C2×D412yes—
135C22⋊D191—22no—
136C22⋊D191—42no—
137C22⋊D191—42no—
138C2×Dic191—21no—
139C2×Dic191—21no—
140C2×Dic191—21no—
141C2×Dic191C2×D411yescharacteristic
142C2×Dic191C2×D411yescharacteristic
143C2×Dic191C2×D411yescharacteristic
144C4×D191—21no—
145C4×D191—21no—
146C4×D191—42no—
147C4×D191—42no—
148C4×D191—81no—
149C191⋊C8—21no—
150C191⋊C8C2×D412yes—
151C23×C191—21no—
152C23×C191C2×D411yescharacteristic
153Q8×C191—21no—
154Q8×C191—21no—
155Q8×C191—21no—
156Q8×C191—42no—
157Q8×C191C2×D412yes—
158D4×C191—21no—
159D4×C191—21no—
160D4×C191—81no—
161C2×C4×C191—21no—
162C2×C4×C191—21no—
163C2×C4×C191—21no—
164C2×C4×C191—21no—
165C2×C4×C191—21no—
166C2×C4×C191—42no—
167C2×C4×C191C2×D411yescharacteristic
168C2×C4×C191C2411yescharacteristic
169C8×C191—21no—
170C8×C191—21no—
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—22no—
187C4×C191—41no—
188C4×C191—41no—
189C4×C191C22×D411yescharacteristic
190C4×C191H(5,𝔽2)11yescharacteristic
191Dic191—21no—
192Dic191—21no—
193Dic191—21no—
194Dic191—41no—
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#468—1911no—
212#467—1911no—
213#464—1911no—
214#461—1911no—
215#452—1911no—
216#451—1911no—
217#450—1911no—
218#448D19111yescharacteristic, complemented
219#440—1911no—
220#435—1911no—
221C2×#41—1911no—
222C22×Q8—1911no—
223C22×Q8C2×D19111yescharacteristic, complemented
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×SD16C2×D19111yescharacteristic
235C2×SD16C2×D19111yescharacteristic
236C2×M16—1911no—
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⋊Q8—1911no—
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#107—1911no—
256C4⋊C8—1911no—
257#103—1911no—
258#103—1911no—
259#103C2×D19111yescharacteristic, complemented
260#103C2×D19111yescharacteristic, complemented
261#102—1911no—
262#102—1911no—
263C22⋊C8—1911no—
264C22⋊C8C2×D19111yescharacteristic, complemented
265#97—1911no—
266C24—3821no—
267#41—3822no—
268#41—7641no—
269C2×Q8—21no—
270C2×Q8—22no—
271C2×Q8—41no—
272C2×Q8—3821no—
273C2×Q8—3822no—
274C2×Q8—1911no—
275C2×Q8—1911no—
276C2×Q8—7641no—
277C2×Q8C22×D19111yescharacteristic
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—21no—
288C22×C4—3821no—
289C22×C4—3821no—
290C22×C4—3821no—
291C22×C4—1911no—
292C22×C4C22×D19111yescharacteristic
293Q16—3822no—
294Q16—3822no—
295Q16—7641no—
296SD16—22no—
297SD16—41no—
298SD16—3822no—
299SD16—3822no—
300SD16—3822no—
301SD16—1914no—
302M16—3822no—
303C2×C8—3822no—
304C2×C8—1911no—
305C2×C8—1911no—
306C2×C8C22×D19111yescharacteristic
307C2×C8C22×D19111yescharacteristic
308C4⋊C4—3821no—
309C4⋊C4—3821no—
310C4⋊C4—3821no—
311C4⋊C4—3821no—
312C4⋊C4—3821no—
313C4⋊C4—3821no—
314C4⋊C4—1911no—
315C4⋊C4—1911no—
316C4⋊C4C22×D19111yescharacteristic
317C22⋊C4—21no—
318C22⋊C4—3821no—
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—21no—
338Q8—42no—
339Q8—3821no—
340Q8—3821no—
341Q8—3821no—
342Q8—3821no—
343Q8—7642no—
344Q8D4×D19112yes—
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—42no—
359C2×C4—3821no—
360C2×C4—3821no—
361C2×C4—3821no—
362C2×C4—3821no—
363C2×C4—3821no—
364C2×C4—1911no—
365C2×C4—1911no—
366C2×C4—1911no—
367C2×C4—15281no—
368C2×C4—7642no—
369C2×C4—7642no—
370C2×C4D4×D19111yescharacteristic
371C2×C4C23×D19111yescharacteristic
372C8—21no—
373C8—21no—
374C8—3821no—
375C8—1912no—
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—22no—
392C4—41no—
393C4—41no—
394C4—3821no—
395C4—3821no—
396C4—3821no—
397C4—7641no—
398C4—7641no—
399C4—7641no—
400C4C2×D4×D19111yescharacteristic
401C4#2025936711yescharacteristic
402C2—41no—
403C2—81no—
404C2—3822no—
405C2—7642no—
406C2#2713329511yescharacteristic
407C2#137×D19111yescharacteristic
408C2#2713346911yescharacteristic
subgroupG/H
centerC22C2×D4×D191
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
Fitting#448×C191C2
socleC22×C191C22×D4
prime radical#27133310C2
hypercenter#448D191
hypocenterC191#3266
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C22 ◁ C22×C4 ◁ #448

Complements

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

NH ≅ G/Ntype
1. C2×#20259353C2semidirect
2. C2×SD16×D191C2semidirect
3. Q8×C22⋊D191C2semidirect
5. #27133054C2semidirect
6. #27132907C2semidirect
7. #27132903C2semidirect
8. #27132883C2semidirect
9. #27132867C2semidirect
10. #27132827C2semidirect
11. #27132820C2semidirect
12. #27132812C2semidirect
13. #27132766C2semidirect
14. #27132737C2semidirect
15. #448×C191C2semidirect
20. C2×Dic764C22semidirect
21. C2×#15690277C22semidirect
22. C2×#15690277C22semidirect
28. C2×#15690271C22semidirect
31. #20259320C22semidirect
36. #20259301C22semidirect
41. Q8×Dic191C22semidirect
46. #20259255C22semidirect
47. Dic191⋊C8C22semidirect
48. #20259247C22semidirect
49. #20259245C22semidirect
53. #20259232C22semidirect
54. C22×Q8×C191C22semidirect
58. #103×C191C22semidirect
59. #103×C191C22semidirect
60. C22⋊C8×C191C22semidirect
85. #15690274D4semidirect
184. C2×D191#137semidirect
204. C191#3266semidirect
218. #448D191semidirect
223. C22×Q8C2×D191semidirect
259. #103C2×D191semidirect
260. #103C2×D191semidirect
264. C22⋊C8C2×D191semidirect

Maximal subgroups

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

subgroupindex
C2×#202593532
C2×SD16×D1912
Q8×C22⋊D1912
#271333102
#271330542
#271329072
#271329032
#271328832
#271328672
#271328272
#271328202
#271328122
#271327662
#271327372
#448×C1912
#3266191