Finite Group Databasebeta

← #37583124

Subgroups — #37583124

Order 24448.

364 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#27133217C211yesmaximal, characteristic, complemented
2#27133200C211yesmaximal, characteristic
3#27133071C211yesmaximal, characteristic, complemented
4#27133056C211yesmaximal, characteristic, complemented
5#27133030C211yesmaximal, characteristic, complemented
6#27133024C211yesmaximal, characteristic, complemented
7#27132920C211yesmaximal, characteristic, complemented
8#27132918C211yesmaximal, characteristic, complemented
9#27132912C211yesmaximal, characteristic
10#27132910C211yesmaximal, characteristic
11C4×#15690277C211yesmaximal, characteristic, complemented
12C4×#15690276C211yesmaximal, characteristic, complemented
13C4×#15690274C211yesmaximal, characteristic, complemented
14C4×#15690270C211yesmaximal, characteristic, complemented
15C4×Q16×C191C211yesmaximal, characteristic, complemented
16C2×#15690277C2211yescharacteristic
17C2×#15690276C2211yescharacteristic
18C2×#15690274C2211yescharacteristic
19C2×#15690270C2211yescharacteristic
20C2×#15690270C2211yescharacteristic
21#20259321C2211yescharacteristic
22#20259320C2211yescharacteristic
23#20259317—21no—
24#20259313C2211yescharacteristic
25#20259312—21no—
26#20259311C2211yescharacteristic
27#20259307—21no—
28#20259302C2211yescharacteristic
29#20259300—21no—
30#20259298C2211yescharacteristic
31#20259298C2211yescharacteristic
32#20259297C2211yescharacteristic
33C4×Dic382—21no—
34C4×Dic382C2211yescharacteristic
35C4×D764—21no—
36C4×D764C2211yescharacteristic
37#20259279—21no—
38#20259279—21no—
39C42×D191C2211yescharacteristic
40#20259255C2211yescharacteristic
41#20259254C2211yescharacteristic
42#20259253C2211yescharacteristic
43#20259247C2211yescharacteristic
44#20259246C2211yescharacteristic
45#20259246C2211yescharacteristic
46#20259245C2211yescharacteristic
47#20259244C2211yescharacteristic
48#20259242C2211yescharacteristic
49#20259232C2211yescharacteristic
50#20259232C2211yescharacteristic
51C4×C191⋊C8C2211yescharacteristic
52C2×Q16×C191C2211yescharacteristic
53C4×Q8×C191C2211yescharacteristic
54C4×Q8×C191C2211yescharacteristic
55#107×C191C2211yescharacteristic
56#103×C191C2211yescharacteristic
57#103×C191C2211yescharacteristic
58C4×C8×C191C2211yescharacteristic
59C2×Dic382—21no—
60C2×Dic382C2311yescharacteristic
61C2×D764—21no—
62C2×D764C2311yescharacteristic
63C2×C4×D191—21no—
64C2×C4×D191D411yescharacteristic
65C2×C4×D191D411yescharacteristic
66C2×C4×D191C2311yescharacteristic
67#15690277—22no—
68#15690276—22no—
69#15690274—22no—
70#15690270—22no—
71#15690270—22no—
72C2×C191⋊C8C2311yescharacteristic
73C2×C191⋊C8C2311yescharacteristic
74C4⋊Dic191—21no—
75C4⋊Dic191—21no—
76C4⋊Dic191C2311yescharacteristic
77C4⋊Dic191C2311yescharacteristic
78Dic191⋊C4—21no—
79Dic191⋊C4—21no—
80Dic191⋊C4—21no—
81Dic191⋊C4—21no—
82Dic191⋊C4—21no—
83Dic191⋊C4—21no—
84Dic191⋊C4—21no—
85(C2×D191)⋊C4—21no—
86(C2×D191)⋊C4—21no—
87(C2×D191)⋊C4—21no—
88(C2×D191)⋊C4—21no—
89(C2×D191)⋊C4—21no—
90(C2×D191)⋊C4—21no—
91(C2×D191)⋊C4—21no—
92C4×Dic191—21no—
93C4×Dic191D411yescharacteristic
94C4×Dic191D411yescharacteristic
95C4×Dic191C2311yescharacteristic
96C2×Q8×C191C2311yescharacteristic
97C2×Q8×C191C2311yescharacteristic
98Q16×C191—22no—
99C2×C8×C191C2311yescharacteristic
100C2×C8×C191C2311yescharacteristic
101C4⋊C4×C191—21no—
102C4⋊C4×C191—21no—
103C4⋊C4×C191C2311yescharacteristic
104C4⋊C4×C191C2311yescharacteristic
105C42×C191—21no—
106C42×C191—21no—
107C42×C191C2311yescharacteristic
108C22×D191—21no—
109C22×D191C2×D411yescharacteristic
110Dic382—21no—
111Dic382—21no—
112Dic382—42no—
113D764—21no—
114D764—21no—
115D764—42no—
116C2×Dic191—21no—
117C2×Dic191—21no—
118C2×Dic191—21no—
119C2×Dic191C2×D411yescharacteristic
120C2×Dic191C2×D411yescharacteristic
121C2×Dic191C2×D411yescharacteristic
122C4×D191—22no—
123C4×D191—21no—
124C4×D191—21no—
125C4×D191—41no—
126C4×D191—42no—
127C191⋊C8—21no—
128C191⋊C8#4112yes—
129Q8×C191—21no—
130Q8×C191—21no—
131Q8×C191—21no—
132Q8×C191—21no—
133C2×C4×C191—21no—
134C2×C4×C191—21no—
135C2×C4×C191—21no—
136C2×C4×C191—21no—
137C2×C4×C191C2×D411yescharacteristic
138C2×C4×C191C2×D411yescharacteristic
139C2×C4×C191C2411yescharacteristic
140C8×C191—21no—
141C8×C191#4112yes—
142C22×C191C22×D411yescharacteristic
143C2×D191—21no—
144C2×D191—21no—
145C2×D191—21no—
146C2×D191—41no—
147C2×D191—41no—
148C2×D191—41no—
149C4×C191—21no—
150C4×C191—41no—
151C4×C191—41no—
152C4×C191—41no—
153C4×C191—41no—
154C4×C191#13712yes—
155C4×C191C2×#4111yescharacteristic
156C4×C191C2×#4111yescharacteristic
157Dic191—21no—
158Dic191—21no—
159Dic191—21no—
160Dic191—41no—
161Dic191—41no—
162Dic191—41no—
163C2×C191#53211yescharacteristic
164C2×C191C2×#13711yescharacteristic
165C2×C191#57511yescharacteristic
166D191—41no—
167D191—81no—
168C191#317211yescharacteristic, complemented, Sylow
169#3172—1911nomaximal, Sylow
170#533—1911no—
171#529—1911no—
172#501—1911no—
173#497—1911no—
174#491—1911no—
175#489—1911no—
176#470—1911no—
177#470—1911no—
178#468—1911no—
179#468—1911no—
180C4×Q16—1911no—
181C4×Q16D19111yescharacteristic, complemented
182C4×SD16—1911no—
183C4×SD16—1911no—
184C4×M16—1911no—
185C2×Q16—1911no—
186C2×Q16C2×D19111yescharacteristic
187C2×SD16—1911no—
188C2×SD16—1911no—
189C2×M16—1911no—
190C2×M16—1911no—
191C4⋊Q8—1911no—
192C4⋊Q8—1911no—
193#126—3821no—
194#125—3821no—
195#125—1911no—
196#124—1911no—
197#123—3821no—
198C22⋊Q8—3821no—
199C22⋊Q8—1911no—
200C22⋊Q8—1911no—
201C22⋊Q8—1911no—
202#121—1911no—
203C4×Q8—3821no—
204C4×Q8—1911no—
205C4×Q8C2×D19111yescharacteristic
206C4×Q8C2×D19111yescharacteristic
207C4×D4—3821no—
208C4×D4—1911no—
209#117—3821no—
210#117—3821no—
211C2×C42—1911no—
212#107—1911no—
213#107C2×D19111yescharacteristic
214#106—1911no—
215#106—1911no—
216#103—1911no—
217#103—1911no—
218#103—1911no—
219#103—1911no—
220#103C2×D19111yescharacteristic
221#103C2×D19111yescharacteristic
222#102—1911no—
223#102—1911no—
224#97—1911no—
225#97—1911no—
226C4×C8—1911no—
227C4×C8C2×D19111yescharacteristic
228C2×Q8—3821no—
229C2×Q8—1911no—
230C2×Q8C22×D19111yescharacteristic
231C2×Q8C22×D19111yescharacteristic
232C2×D4—3821no—
233C2×D4—1911no—
234C22×C4—3821no—
235C22×C4—1911no—
236C22×C4—1911no—
237C22×C4—1911no—
238Q16—22no—
239Q16—3822no—
240SD16—3822no—
241SD16—3822no—
242M16—3822no—
243M16—3822no—
244C2×C8—1911no—
245C2×C8—1911no—
246C2×C8C22×D19111yescharacteristic
247C2×C8C22×D19111yescharacteristic
248C4⋊C4—21no—
249C4⋊C4—21no—
250C4⋊C4—3821no—
251C4⋊C4—3821no—
252C4⋊C4—3821no—
253C4⋊C4—3821no—
254C4⋊C4—3821no—
255C4⋊C4—3821no—
256C4⋊C4—3821no—
257C4⋊C4—3821no—
258C4⋊C4—3821no—
259C4⋊C4—1911no—
260C4⋊C4—1911no—
261C4⋊C4C22×D19111yescharacteristic
262C4⋊C4C22×D19111yescharacteristic
263C22⋊C4—3821no—
264C22⋊C4—3821no—
265C22⋊C4—3821no—
266C22⋊C4—3821no—
267C22⋊C4—3821no—
268C22⋊C4—3821no—
269C22⋊C4—3821no—
270C42—21no—
271C42—21no—
272C42—3821no—
273C42—1911no—
274C42—1911no—
275C42—1911no—
276C42C22×D19111yescharacteristic
277C23—3821no—
278C23—1911no—
279Q8—21no—
280Q8—21no—
281Q8—21no—
282Q8—21no—
283Q8—3821no—
284Q8—3821no—
285Q8—7642no—
286D4—3821no—
287D4—3821no—
288D4—7642no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—3822no—
294C2×C4—3821no—
295C2×C4—3821no—
296C2×C4—3821no—
297C2×C4—3821no—
298C2×C4—3821no—
299C2×C4—1911no—
300C2×C4—1911no—
301C2×C4—1911no—
302C2×C4—7641no—
303C2×C4—7642no—
304C2×C4D4×D19111yescharacteristic
305C2×C4D4×D19111yescharacteristic
306C2×C4C23×D19111yescharacteristic
307C8—21no—
308C8—3821no—
309C8—1912no—
310C8#1569028712yes—
311C22—3821no—
312C22—3821no—
313C22—3821no—
314C22—7641no—
315C22—7641no—
316C22—7641no—
317C22C2×D4×D19111yescharacteristic
318C4—21no—
319C4—41no—
320C4—41no—
321C4—41no—
322C4—41no—
323C4—3821no—
324C4—3821no—
325C4—3821no—
326C4—7641no—
327C4—7641no—
328C4—7641no—
329C4#2025935412yes—
330C4#41×D19111yescharacteristic
331C4C2×#1569028711yescharacteristic
332C2—15281no—
333C2—7641no—
334C2#2713321011yescharacteristic
335C2C2×#2025935411yescharacteristic
336C2#2713346011yescharacteristic
subgroupG/H
centerC2×C4D4×D191
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
FittingC4×Q16×C191C2
socleC22×C191C22×D4
prime radical#20259297C22
hypercenterC4×Q16D191
hypocenterC191#3172
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C2×C4 ◁ C42 ◁ C4×Q16

Complements

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

NH ≅ G/Ntype
1. #27133217C2semidirect
3. #27133071C2semidirect
4. #27133056C2semidirect
5. #27133030C2semidirect
6. #27133024C2semidirect
7. #27132920C2semidirect
8. #27132918C2semidirect
11. C4×#15690277C2semidirect
12. C4×#15690276C2semidirect
13. C4×#15690274C2semidirect
14. C4×#15690270C2semidirect
15. C4×Q16×C191C2semidirect
168. C191#3172semidirect
181. C4×Q16D191semidirect

Maximal subgroups

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

subgroupindex
#271332172
#271332002
#271330712
#271330562
#271330302
#271330242
#271329202
#271329182
#271329122
#271329102
C4×#156902772
C4×#156902762
C4×#156902742
C4×#156902702
C4×Q16×C1912
#3172191