Finite Group Databasebeta

← #37582781

Subgroups — #37582781

Order 24448.

384 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#27133282C211yesmaximal, characteristic, complemented
2#27133145C211yesmaximal, characteristic, complemented
3#27132997C211yesmaximal, characteristic, complemented
4#27132975C211yesmaximal, characteristic, complemented
5#27132958C211yesmaximal, characteristic, complemented
6#27132944C211yesmaximal, characteristic, complemented
7#27132875C211yesmaximal, characteristic, complemented
8#27132824C211yesmaximal, characteristic, complemented
9#27132807C211yesmaximal, characteristic
10#27132766C211yesmaximal, characteristic, complemented
11#27132764C211yesmaximal, characteristic, complemented
12Q16×Dic191C211yesmaximal, characteristic, complemented
13#27132743C211yesmaximal, characteristic, complemented
14#27132723C211yesmaximal, characteristic, complemented
15#464×C191C211yesmaximal, characteristic, complemented
16C2×#15690288C2211yescharacteristic
17C2×#15690274C2211yescharacteristic
18C2×#15690272C2211yescharacteristic
19#20259322C2211yescharacteristic
20#20259315—21no—
21#20259308—21no—
22#20259306—21no—
23#20259299—21no—
24#20259298C2211yescharacteristic
25#20259293C2211yescharacteristic
26Q8×Dic191C2211yescharacteristic
27Q8×Dic191C2211yescharacteristic
28C4×D764—21no—
29C4×D764C2211yescharacteristic
30D4×Dic191C2211yescharacteristic
31#20259284—21no—
32#20259284C2211yescharacteristic
33#20259279—21no—
34#20259279C2211yescharacteristic
35C4⋊C4×D191—21no—
36C4⋊C4×D191C2211yescharacteristic
37#20259256C2211yescharacteristic
38#20259256C2211yescharacteristic
39#20259255C2211yescharacteristic
40#20259255C2211yescharacteristic
41C4⋊(C191⋊C8)C2211yescharacteristic
42#20259246C2211yescharacteristic
43#20259246C2211yescharacteristic
44#20259246C2211yescharacteristic
45#20259244C2211yescharacteristic
46#20259244C2211yescharacteristic
47#20259243C2211yescharacteristic
48(C2×D191)⋊C8C2211yescharacteristic
49(C2×D191)⋊C8C2211yescharacteristic
50#20259232C2211yescharacteristic
51C8×Dic191C2211yescharacteristic
52C2×Q16×C191C2211yescharacteristic
53C2×SD16×C191C2211yescharacteristic
54#124×C191C2211yescharacteristic
55C4×Q8×C191C2211yescharacteristic
56C4⋊C8×C191C2211yescharacteristic
57#103×C191C2211yescharacteristic
58#102×C191C2211yescharacteristic
59#15690288—22no—
60#15690288—22no—
61C2×Dic382—21no—
62C2×D764—21no—
63C2×D764C2311yescharacteristic
64C2×C22⋊D191—21no—
65C22×Dic191—21no—
66C2×C4×D191—21no—
67C2×C4×D191—21no—
68C2×C4×D191—21no—
69C2×C4×D191—21no—
70C2×C4×D191C2311yescharacteristic
71#15690274—22no—
72#15690272—22no—
73C2×C191⋊C8C2311yescharacteristic
74C2×C191⋊C8C2311yescharacteristic
75C4⋊Dic191—21no—
76C4⋊Dic191—21no—
77C4⋊Dic191—21no—
78C4⋊Dic191C2311yescharacteristic
79C4⋊Dic191C2311yescharacteristic
80C4⋊Dic191C2311yescharacteristic
81Dic191⋊C4—21no—
82Dic191⋊C4—21no—
83Dic191⋊C4D411yescharacteristic
84Dic191⋊C4D411yescharacteristic
85C22⋊Dic191—21no—
86C22⋊Dic191—21no—
87(C2×D191)⋊C4—21no—
88(C2×D191)⋊C4—21no—
89(C2×D191)⋊C4—21no—
90(C2×D191)⋊C4D411yescharacteristic
91(C2×D191)⋊C4D411yescharacteristic
92C4×Dic191—21no—
93C4×Dic191—21no—
94C4×Dic191C2311yescharacteristic
95C2×Q8×C191C2311yescharacteristic
96C2×Q8×C191C2311yescharacteristic
97C2×D4×C191C2311yescharacteristic
98Q16×C191—22no—
99SD16×C191—22no—
100C2×C8×C191C2311yescharacteristic
101C2×C8×C191C2311yescharacteristic
102C4⋊C4×C191—21no—
103C4⋊C4×C191C2311yescharacteristic
104C22⋊C4×C191—21no—
105C22⋊C4×C191—21no—
106C42×C191—21no—
107C42×C191C2311yescharacteristic
108C22×D191—21no—
109C22×D191C2×D411yescharacteristic
110Dic382—81no—
111D764—24no—
112D764—21no—
113D764#4112yes—
114C22⋊D191—81no—
115C2×Dic191—21no—
116C2×Dic191—21no—
117C2×Dic191—21no—
118C2×Dic191—81no—
119C2×Dic191C2×D411yescharacteristic
120C2×Dic191C2×D411yescharacteristic
121C2×Dic191C2×D411yescharacteristic
122C4×D191—24no—
123C4×D191—21no—
124C4×D191—21no—
125C4×D191—81no—
126C4×D191—81no—
127C4×D191—81no—
128C191⋊C8—21no—
129C191⋊C8—21no—
130C23×C191—21no—
131Q8×C191—21no—
132Q8×C191—21no—
133Q8×C191—21no—
134Q8×C191#4112yes—
135D4×C191—21no—
136D4×C191—21no—
137C2×C4×C191—21no—
138C2×C4×C191—21no—
139C2×C4×C191—21no—
140C2×C4×C191C2×D411yescharacteristic
141C2×C4×C191C2×D411yescharacteristic
142C2×C4×C191C2411yescharacteristic
143C8×C191—21no—
144C8×C191—21no—
145C22×C191—41no—
146C22×C191—41no—
147C22×C191—41no—
148C22×C191C22×D411yescharacteristic
149C2×D191—21no—
150C2×D191—22no—
151C2×D191—21no—
152C2×D191—21no—
153C2×D191—41no—
154C2×D191—41no—
155C4×C191—22no—
156C4×C191—21no—
157C4×C191—21no—
158C4×C191—41no—
159C4×C191—41no—
160C4×C191C2×#4111yescharacteristic
161C4×C191C2×#4111yescharacteristic
162Dic191—21no—
163Dic191—21no—
164Dic191—21no—
165Dic191—41no—
166Dic191—41no—
167Dic191—41no—
168C2×C191—81no—
169C2×C191#52511yescharacteristic
170C2×C191#57611yescharacteristic
171C2×C191#57811yescharacteristic
172D191—41no—
173D191—42no—
174C191#312411yescharacteristic, complemented, Sylow
175#3124—1911nomaximal, Sylow
176#542—1911no—
177#520—1911no—
178#484—1911no—
179#480—1911no—
180#477—1911no—
181#474—1911no—
182#464D19111yescharacteristic, complemented
183#463—1911no—
184#452—1911no—
185#449—1911no—
186#440—1911no—
187#440—1911no—
188C4×Q16—1911no—
189C4×D8—1911no—
190#433—1911no—
191C2×#41—1911no—
192C2×Q16C2×D19111yescharacteristic
193C2×SD16—1911no—
194C2×SD16C2×D19111yescharacteristic
195C2×D8—1911no—
196C4⋊Q8—1911no—
197#126—3821no—
198#124—3821no—
199#124C2×D19111yescharacteristic
200#123—3821no—
201C22⋊Q8—3821no—
202C22⋊Q8—1911no—
203#121—1911no—
204C4×Q8—1911no—
205C4×Q8—1911no—
206C4×Q8C2×D19111yescharacteristic
207C4×D4—3821no—
208C4×D4—3821no—
209C4×D4—1911no—
210C4×D4—1911no—
211C4×D4—1911no—
212#117—3821no—
213#117—1911no—
214C2×C4⋊C4—3821no—
215C2×C4⋊C4—1911no—
216#107—1911no—
217#107—1911no—
218#107—1911no—
219#107—1911no—
220C4⋊C8—1911no—
221C4⋊C8C2×D19111yescharacteristic
222#103—1911no—
223#103—1911no—
224#103—1911no—
225#103C2×D19111yescharacteristic
226#102—1911no—
227#102—1911no—
228#102—1911no—
229#102C2×D19111yescharacteristic
230C22⋊C8—1911no—
231C22⋊C8—1911no—
232#97—1911no—
233C4×C8—1911no—
234#41—3822no—
235#41—3822no—
236C2×Q8—3821no—
237C2×Q8C22×D19111yescharacteristic
238C2×Q8C22×D19111yescharacteristic
239C2×D4—3821no—
240C2×D4—3821no—
241C2×D4—1911no—
242C2×D4C22×D19111yescharacteristic
243C22×C4—3821no—
244C22×C4—3821no—
245C22×C4—3821no—
246C22×C4—3821no—
247C22×C4—3821no—
248C22×C4—1911no—
249Q16—22no—
250SD16—22no—
251SD16—3822no—
252D8—3822no—
253C2×C8—1911no—
254C2×C8—1911no—
255C2×C8C22×D19111yescharacteristic
256C2×C8C22×D19111yescharacteristic
257C4⋊C4—21no—
258C4⋊C4—3821no—
259C4⋊C4—3821no—
260C4⋊C4—3821no—
261C4⋊C4—3821no—
262C4⋊C4—3821no—
263C4⋊C4—1911no—
264C4⋊C4—1911no—
265C4⋊C4—1911no—
266C4⋊C4—1911no—
267C4⋊C4—1911no—
268C4⋊C4C22×D19111yescharacteristic
269C22⋊C4—21no—
270C22⋊C4—21no—
271C22⋊C4—3821no—
272C22⋊C4—3821no—
273C22⋊C4—3821no—
274C22⋊C4—3821no—
275C22⋊C4—3821no—
276C22⋊C4—1911no—
277C22⋊C4—1911no—
278C42—21no—
279C42—3821no—
280C42—3821no—
281C42—1911no—
282C42C22×D19111yescharacteristic
283C23—21no—
284C23—3821no—
285C23—1911no—
286Q8—21no—
287Q8—21no—
288Q8—21no—
289Q8—15281no—
290Q8#1569028612yes—
291D4—21no—
292D4—21no—
293D4—3824no—
294D4—3821no—
295D4—1912no—
296D4—15281no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—3824no—
301C2×C4—3821no—
302C2×C4—3821no—
303C2×C4—3821no—
304C2×C4—3821no—
305C2×C4—3821no—
306C2×C4—1911no—
307C2×C4—1911no—
308C2×C4—1911no—
309C2×C4—15281no—
310C2×C4—15281no—
311C2×C4—15281no—
312C2×C4—15281no—
313C2×C4D4×D19111yescharacteristic
314C2×C4D4×D19111yescharacteristic
315C2×C4C23×D19111yescharacteristic
316C8—21no—
317C8—21no—
318C8—3821no—
319C8—3821no—
320C22—41no—
321C22—41no—
322C22—41no—
323C22—3821no—
324C22—3821no—
325C22—3821no—
326C22—3822no—
327C22—7641no—
328C22—7641no—
329C22C2×D4×D19111yescharacteristic
330C4—21no—
331C4—21no—
332C4—22no—
333C4—41no—
334C4—41no—
335C4—3821no—
336C4—3821no—
337C4—3821no—
338C4—7641no—
339C4—7641no—
340C4—7641no—
341C4#41×D19111yescharacteristic
342C4C2×#1569028611yescharacteristic
343C2—81no—
344C2—7641no—
345C2—7642no—
346C2#2713317311yescharacteristic
347C2#2713346511yescharacteristic
348C2#2713347911yescharacteristic
subgroupG/H
centerC22C2×D4×D191
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
Fitting#464×C191C2
socleC22×C191C22×D4
prime radical#27132807C2
hypercenter#464D191
hypocenterC191#3124
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C22 ◁ C42 ◁ #464

Complements

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

NH ≅ G/Ntype
1. #27133282C2semidirect
2. #27133145C2semidirect
3. #27132997C2semidirect
4. #27132975C2semidirect
5. #27132958C2semidirect
6. #27132944C2semidirect
7. #27132875C2semidirect
8. #27132824C2semidirect
10. #27132766C2semidirect
11. #27132764C2semidirect
12. Q16×Dic191C2semidirect
13. #27132743C2semidirect
14. #27132723C2semidirect
15. #464×C191C2semidirect
174. C191#3124semidirect
182. #464D191semidirect

Maximal subgroups

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

subgroupindex
#271332822
#271331452
#271329972
#271329752
#271329582
#271329442
#271328752
#271328242
#271328072
#271327662
#271327642
Q16×Dic1912
#271327432
#271327232
#464×C1912
#3124191