Finite Group Databasebeta

← #37581362

Subgroups — #37581362

Order 24448.

470 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#20259354C211yesmaximal, characteristic
2#27133145C211yesmaximal, characteristic
3#27133139C211yesmaximal, characteristic, complemented
4#27132767C211yesmaximal, characteristic, complemented
5#27132766C211yesmaximal, characteristic, complemented
6#27132765C211yesmaximal, characteristic, complemented
7Q16×Dic191C211yesmaximal, characteristic, complemented
8C4×#15690277C211yesmaximal, characteristic, complemented
9C4×#15690274C211yesmaximal, characteristic, complemented
10#27132750C211yesmaximal, characteristic, complemented
11#27132702C211yesmaximal, characteristic, complemented
12#27132658C211yesmaximal, characteristic
13#103×D191C211yesmaximal, characteristic, complemented
14#27132620C211yesmaximal, characteristic, complemented
15#441×C191C211yesmaximal, characteristic, complemented
16C2×#15690288C2211yescharacteristic
17C2×Q8×D191C2211yescharacteristic
18#20259354C418yescomplemented
19C2×#15690277C2211yescharacteristic
20C2×#15690276C2211yescharacteristic
21C2×#15690274C2211yescharacteristic
22C2×#15690270C2211yescharacteristic
23C2×C8×D191C2211yescharacteristic
24Q8×Dic191C2211yescharacteristic
25Q8×Dic191C2211yescharacteristic
26C4×Dic382—21no—
27C4×Dic382C2211yescharacteristic
28#20259287—21no—
29#20259287C2211yescharacteristic
30C4×D764—21no—
31C4×D764C2211yescharacteristic
32#20259284—21no—
33#20259284C2211yescharacteristic
34#20259280—21no—
35#20259280C2211yescharacteristic
36#20259279—21no—
37#20259279—21no—
38#20259279C2211yescharacteristic
39C4⋊C4×D191—21no—
40C4⋊C4×D191C2211yescharacteristic
41#20259256C2211yescharacteristic
42#20259255C2211yescharacteristic
43#20259253C2211yescharacteristic
44#20259247C2211yescharacteristic
45#20259246C2211yescharacteristic
46#20259246C2211yescharacteristic
47#20259245C2211yescharacteristic
48#20259244C2211yescharacteristic
49#20259242C2211yescharacteristic
50#20259232C2211yescharacteristic
51C4×C191⋊C8C2211yescharacteristic
52C8×Dic191C2211yescharacteristic
53C2×Q16×C191C2211yescharacteristic
54C4×Q8×C191C2211yescharacteristic
55C4×Q8×C191C2211yescharacteristic
56#106×C191C2211yescharacteristic
57#103×C191C2211yescharacteristic
58#103×C191C2211yescharacteristic
59#97×C191C2211yescharacteristic
60#15690288—22no—
61#15690288C2×C414yes—
62C2×Dic382—21no—
63C2×Dic382C2311yescharacteristic
64Q8×D191—22no—
65Q8×D191C2×C414yescomplemented
66C2×D764—21no—
67C2×D764C2311yescharacteristic
68C2×C4×D191—21no—
69C2×C4×D191—21no—
70C2×C4×D191—21no—
71C2×C4×D191—21no—
72C2×C4×D191—21no—
73C2×C4×D191C2311yescharacteristic
74#15690277C2×C414yescomplemented
75#15690276C2×C414yescomplemented
76#15690274C2×C414yescomplemented
77#15690270C2×C414yescomplemented
78C2×C191⋊C8C2311yescharacteristic
79C2×C191⋊C8C2311yescharacteristic
80C8×D191—22no—
81C4⋊Dic191—21no—
82C4⋊Dic191—21no—
83C4⋊Dic191C2311yescharacteristic
84C4⋊Dic191C2311yescharacteristic
85Dic191⋊C4—21no—
86Dic191⋊C4—21no—
87Dic191⋊C4—21no—
88Dic191⋊C4D411yescharacteristic
89Dic191⋊C4D411yescharacteristic
90(C2×D191)⋊C4—21no—
91(C2×D191)⋊C4—21no—
92(C2×D191)⋊C4—21no—
93(C2×D191)⋊C4D411yescharacteristic
94(C2×D191)⋊C4D411yescharacteristic
95C4×Dic191—21no—
96C4×Dic191—21no—
97C4×Dic191—21no—
98C4×Dic191—21no—
99C4×Dic191—21no—
100C4×Dic191C2311yescharacteristic
101C2×Q8×C191C2311yescharacteristic
102C2×Q8×C191C2311yescharacteristic
103Q16×C191C2×C414yescomplemented
104C2×C8×C191C2311yescharacteristic
105C2×C8×C191C2311yescharacteristic
106C4⋊C4×C191—21no—
107C4⋊C4×C191—21no—
108C4⋊C4×C191C2311yescharacteristic
109C4⋊C4×C191C2311yescharacteristic
110C42×C191—21no—
111C42×C191—21no—
112C42×C191C2311yescharacteristic
113C22×D191—21no—
114C22×D191C2×D411yescharacteristic
115Dic382—21no—
116Dic382—24no—
117Dic382C22×C412yes—
118D764—24no—
119D764—21no—
120D764C22×C412yes—
121C2×Dic191—21no—
122C2×Dic191—21no—
123C2×Dic191—21no—
124C2×Dic191C2×D411yescharacteristic
125C2×Dic191C2×D411yescharacteristic
126C2×Dic191C2×D411yescharacteristic
127C4×D191—24no—
128C4×D191—24no—
129C4×D191—81no—
130C4×D191—81no—
131C4×D191—81no—
132C4×D191C22×C412yes—
133C4×D191#4112yes—
134C191⋊C8—21no—
135C191⋊C8C22×C412yes—
136Q8×C191—21no—
137Q8×C191—21no—
138Q8×C191C22×C412yes—
139Q8×C191C22×C412yes—
140C2×C4×C191—21no—
141C2×C4×C191—21no—
142C2×C4×C191—21no—
143C2×C4×C191—21no—
144C2×C4×C191C2×D411yescharacteristic
145C2×C4×C191C2×D411yescharacteristic
146C2×C4×C191C2411yescharacteristic
147C8×C191—21no—
148C8×C191C22×C412yes—
149C22×C191C22×D411yescharacteristic
150C2×D191—21no—
151C2×D191—21no—
152C2×D191—22no—
153C2×D191—41no—
154C2×D191—41no—
155C2×D191C4×D412yes—
156C4×C191—22no—
157C4×C191—22no—
158C4×C191—21no—
159C4×C191—21no—
160C4×C191—41no—
161C4×C191—41no—
162C4×C191C23×C411yescharacteristic
163C4×C191C2×#4111yescharacteristic
164Dic191—22no—
165Dic191—21no—
166Dic191—21no—
167Dic191—41no—
168Dic191—41no—
169Dic191C4×D412yes—
170C2×C191C2×C4×D411yescharacteristic
171C2×C191#57711yescharacteristic
172C2×C191#57811yescharacteristic
173D191—22no—
174D191—42no—
175C191#294711yescharacteristic, complemented, Sylow
176#2947—1911nomaximal, Sylow
177C2×#137—1911no—
178#520—1911no—
179#519—1911no—
180#441—1911no—
181#441D19111yescharacteristic, complemented
182#440—1911no—
183#440—1911no—
184C4×Q16—1911no—
185C4×Q16—1911no—
186C4×SD16—1911no—
187C4×SD16—1911no—
188#428—1911no—
189#416—1911no—
190C2×#103—1911no—
191#405—1911no—
192C2×#41—1911no—
193C22×Q8—1911no—
194#137—1918no—
195C2×Q16—1911no—
196C2×Q16C2×D19111yescharacteristic
197C2×SD16—1911no—
198C2×SD16—1911no—
199C2×M16—1911no—
200C22×C8—1911no—
201C4×Q8—3821no—
202C4×Q8—3821no—
203C4×Q8—1911no—
204C4×Q8—1911no—
205C4×Q8—1911no—
206C4×Q8—1911no—
207C4×Q8C2×D19111yescharacteristic
208C4×Q8C2×D19111yescharacteristic
209C4×D4—3821no—
210C4×D4—3821no—
211C4×D4—1911no—
212C4×D4—1911no—
213#117—3821no—
214#117—3821no—
215#117—3821no—
216#117—1911no—
217#117—1911no—
218C2×C4⋊C4—3821no—
219C2×C4⋊C4—1911no—
220#107—1911no—
221#107—1911no—
222#106—1911no—
223#106C2×D19111yescharacteristic
224#103—1911no—
225#103—1911no—
226#103—1911no—
227#103—1911no—
228#103C2×D19111yescharacteristic
229#103C2×D19111yescharacteristic
230#102—1911no—
231#102—1911no—
232#97—1911no—
233#97C2×D19111yescharacteristic
234C4×C8—1911no—
235C4×C8—1911no—
236#41—3822no—
237#41—1914no—
238C2×Q8—3821no—
239C2×Q8—3822no—
240C2×Q8—1914no—
241C2×Q8—1911no—
242C2×Q8C22×D19111yescharacteristic
243C2×Q8C22×D19111yescharacteristic
244C2×D4—3821no—
245C2×D4—1911no—
246C22×C4—3821no—
247C22×C4—3821no—
248C22×C4—3821no—
249C22×C4—3821no—
250C22×C4—3821no—
251C22×C4—1911no—
252Q16—1914no—
253Q16C4×D19114yescomplemented
254SD16—1914no—
255SD16—1914no—
256M16—1914no—
257C2×C8—3822no—
258C2×C8—1911no—
259C2×C8—1911no—
260C2×C8C22×D19111yescharacteristic
261C2×C8C22×D19111yescharacteristic
262C4⋊C4—21no—
263C4⋊C4—21no—
264C4⋊C4—3821no—
265C4⋊C4—3821no—
266C4⋊C4—3821no—
267C4⋊C4—3821no—
268C4⋊C4—3821no—
269C4⋊C4—1911no—
270C4⋊C4—1911no—
271C4⋊C4—1911no—
272C4⋊C4—1911no—
273C4⋊C4C22×D19111yescharacteristic
274C4⋊C4C22×D19111yescharacteristic
275C22⋊C4—3821no—
276C22⋊C4—3821no—
277C22⋊C4—3821no—
278C22⋊C4—1911no—
279C22⋊C4—1911no—
280C42—21no—
281C42—21no—
282C42—3821no—
283C42—3821no—
284C42—3821no—
285C42—3821no—
286C42—3821no—
287C42—1911no—
288C42C22×D19111yescharacteristic
289C23—3821no—
290C23—1911no—
291Q8—21no—
292Q8—21no—
293Q8—3824no—
294Q8—3821no—
295Q8—1912no—
296Q8C2×C4×D19112yes—
297Q8C2×C4×D19112yes—
298D4—3824no—
299D4—3821no—
300D4—1912no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—3824no—
306C2×C4—3824no—
307C2×C4—3821no—
308C2×C4—3821no—
309C2×C4—3821no—
310C2×C4—1911no—
311C2×C4—1911no—
312C2×C4—1911no—
313C2×C4—1912no—
314C2×C4—1912no—
315C2×C4—15281no—
316C2×C4—15281no—
317C2×C4—15281no—
318C2×C4D4×D19111yescharacteristic
319C2×C4D4×D19111yescharacteristic
320C2×C4C23×D19111yescharacteristic
321C8—21no—
322C8—3821no—
323C8—1912no—
324C8C2×C4×D19112yes—
325C22—3821no—
326C22—3821no—
327C22—3822no—
328C22—1912no—
329C22—7641no—
330C22—7641no—
331C22C2×D4×D19111yescharacteristic
332C4—21no—
333C4—21no—
334C4—22no—
335C4—22no—
336C4—41no—
337C4—41no—
338C4—3821no—
339C4—3821no—
340C4—3822no—
341C4—1912no—
342C4—7641no—
343C4—7641no—
344C4C22×C4×D19111yescharacteristic
345C4#41×D19111yescharacteristic
346C2—3822no—
347C2—7642no—
348C2C4×D4×D19111yescharacteristic
349C2#2713347211yescharacteristic
350C2#2713347911yescharacteristic
subgroupG/H
centerC22C2×D4×D191
commutatorC4×C191C23×C4
FrattiniC2×C4C23×D191
Fitting#441×C191C2
socleC22×C191C22×D4
prime radicalC2×#15690288C22
cosocleC2×C4×C191C24
hypercenter#441D191
hypocenterC191#2947
LCSG ▷ C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C22 ◁ C42 ◁ #441

Complements

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

NH ≅ G/Ntype
3. #27133139C2semidirect
4. #27132767C2semidirect
5. #27132766C2semidirect
6. #27132765C2semidirect
7. Q16×Dic191C2semidirect
8. C4×#15690277C2semidirect
9. C4×#15690274C2semidirect
10. #27132750C2semidirect
11. #27132702C2semidirect
13. #103×D191C2semidirect
14. #27132620C2semidirect
15. #441×C191C2semidirect
18. #20259354C4semidirect
65. Q8×D191C2×C4semidirect
74. #15690277C2×C4semidirect
75. #15690276C2×C4semidirect
76. #15690274C2×C4semidirect
77. #15690270C2×C4semidirect
103. Q16×C191C2×C4semidirect
175. C191#2947semidirect
181. #441D191semidirect
253. Q16C4×D191semidirect

Maximal subgroups

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

subgroupindex
C2×#202593542
#271331452
#271331392
#271327672
#271327662
#271327652
Q16×Dic1912
C4×#156902772
C4×#156902742
#271327502
#271327022
#271326582
#103×D1912
#271326202
#441×C1912
#2947191