Finite Group Databasebeta

← #37575263

Subgroups — #37575263

Order 24448.

460 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#20259320C211yesmaximal, characteristic
2#27132570C211yesmaximal, characteristic
3#27132567C211yesmaximal, characteristic
4#27132567C211yesmaximal, characteristic
5#27132560C211yesmaximal, characteristic
6#27132558C211yesmaximal, characteristic
7#27132550C211yesmaximal, characteristic
8#27132547C211yesmaximal, characteristic
9#27132545C211yesmaximal, characteristic
10#27132544C211yesmaximal, characteristic
11#27132527C211yesmaximal, characteristic
12C4×Dic191⋊C4C211yesmaximal, characteristic
13#27132514C211yesmaximal, characteristic
14#27132508C211yesmaximal, characteristic
15#391×C191C211yesmaximal, characteristic
16C22×Dic382C2211yescharacteristic
17#20259320C418yescomplemented
18C2×C4⋊Dic191C2211yescharacteristic
19C2×Dic191⋊C4C2211yescharacteristic
20C2×Dic191⋊C4C2211yescharacteristic
21C2×Dic191⋊C4C2211yescharacteristic
22C2×Dic191⋊C4C2211yescharacteristic
23C2×Dic191⋊C4C2211yescharacteristic
24C2×Dic191⋊C4C2211yescharacteristic
25C2×Dic191⋊C4C2211yescharacteristic
26C2×Dic191⋊C4C2211yescharacteristic
27C2×C4×Dic191C2211yescharacteristic
28C2×C4×Dic191C2211yescharacteristic
29C2×C4×Dic191C2211yescharacteristic
30C2×C4×Dic191C2211yescharacteristic
31C2×C4×Dic191C2211yescharacteristic
32C2×C4×Dic191C2211yescharacteristic
33C191⋊#95C2211yescharacteristic
34C191⋊#95C2211yescharacteristic
35C191⋊#95C2211yescharacteristic
36C191⋊#95C2211yescharacteristic
37C191⋊#95C2211yescharacteristic
38C191⋊#95C2211yescharacteristic
39C191⋊#95C2211yescharacteristic
40C191⋊#95C2211yescharacteristic
41C191⋊#95C2211yescharacteristic
42C191⋊#95C2211yescharacteristic
43C191⋊#95C2211yescharacteristic
44C191⋊#95C2211yescharacteristic
45C22×Q8×C191C2211yescharacteristic
46C2×C4⋊C4×C191C2211yescharacteristic
47C2×C42×C191C2211yescharacteristic
48#95×C191C2211yescharacteristic
49#95×C191C2211yescharacteristic
50#95×C191C2211yescharacteristic
51#95×C191C2211yescharacteristic
52C2×Dic382—41no—
53C2×Dic382—41no—
54C2×Dic382C2×C414yes—
55C22×Dic191C2311yescharacteristic
56C22×Dic191C2311yescharacteristic
57C22×Dic191C2311yescharacteristic
58C22×Dic191C2311yescharacteristic
59C22×Dic191C2311yescharacteristic
60C22×Dic191C2311yescharacteristic
61C22×Dic191C2311yescharacteristic
62C22×Dic191C2311yescharacteristic
63C4⋊Dic191—41no—
64Dic191⋊C4—22no—
65Dic191⋊C4—22no—
66Dic191⋊C4—41no—
67Dic191⋊C4—41no—
68Dic191⋊C4C2×C414yes—
69Dic191⋊C4C2×C414yes—
70Dic191⋊C4C2×C414yes—
71Dic191⋊C4C2×C414yes—
72C4×Dic191—22no—
73C4×Dic191—41no—
74C4×Dic191—41no—
75C4×Dic191—41no—
76C4×Dic191C2×C414yes—
77C4×Dic191Q814yescomplemented
78C2×Q8×C191—41no—
79C2×Q8×C191—41no—
80C2×Q8×C191C2×C414yes—
81C22×C4×C191C2311yescharacteristic
82C22×C4×C191C2311yescharacteristic
83C22×C4×C191C2311yescharacteristic
84C22×C4×C191C2311yescharacteristic
85C22×C4×C191C2311yescharacteristic
86C22×C4×C191C2311yescharacteristic
87C22×C4×C191C2311yescharacteristic
88C4⋊C4×C191—41no—
89C42×C191—41no—
90Dic382—44no—
91C2×Dic191—21no—
92C2×Dic191—21no—
93C2×Dic191—21no—
94C2×Dic191—21no—
95C2×Dic191—21no—
96C2×Dic191—21no—
97C2×Dic191—21no—
98C2×Dic191—21no—
99C2×Dic191—21no—
100C2×Dic191—21no—
101C2×Dic191—21no—
102C2×Dic191—21no—
103C2×Dic191—21no—
104C2×Dic191—21no—
105C2×Dic191—21no—
106C2×Dic191—21no—
107C2×Dic191—21no—
108C2×Dic191C22×C412yes—
109C2×Dic191C22×C412yes—
110C2×Dic191C22×C412yes—
111C2×Dic191C22×C412yes—
112C2×Dic191C2×Q812yes—
113C2×Dic191C2×Q812yes—
114C2×Dic191#4112yes—
115C23×C191C2411yescharacteristic
116Q8×C191—44no—
117C2×C4×C191—21no—
118C2×C4×C191—21no—
119C2×C4×C191—21no—
120C2×C4×C191—21no—
121C2×C4×C191—21no—
122C2×C4×C191—21no—
123C2×C4×C191—21no—
124C2×C4×C191—21no—
125C2×C4×C191—21no—
126C2×C4×C191—21no—
127C2×C4×C191—21no—
128C2×C4×C191—21no—
129C2×C4×C191—21no—
130C2×C4×C191—21no—
131C2×C4×C191—21no—
132C2×C4×C191—21no—
133C2×C4×C191—21no—
134C2×C4×C191C22×C412yes—
135C2×C4×C191C22×C412yes—
136C2×C4×C191C22×C412yes—
137C2×C4×C191C2×Q812yes—
138C22×C191C23×C411yescharacteristic
139C22×C191C22×Q811yescharacteristic
140C22×C191C2×#4111yescharacteristic
141C22×C191H(5,𝔽2)11yescharacteristic
142C22×C191H(5,𝔽2)11yescharacteristic
143C22×C1912−1+411yescharacteristic
144C22×C1912−1+411yescharacteristic
145C4×C191—22no—
146C4×C191—22no—
147C4×C191—22no—
148C4×C191—22no—
149C4×C191—41no—
150C4×C191—41no—
151C4×C191—41no—
152Dic191—22no—
153Dic191—22no—
154Dic191—22no—
155Dic191—22no—
156Dic191—41no—
157Dic191—41no—
158Dic191—41no—
159Dic191C4×Q814yes—
160C2×C191C2×C4×Q811yescharacteristic
161C2×C191#51811yescharacteristic
162C2×C191#51911yescharacteristic
163C2×C191#54211yescharacteristic
164C2×C191#54211yescharacteristic
165C2×C191#54411yescharacteristic
166C2×C191#54411yescharacteristic
167C191#233111yescharacteristic, complemented, Sylow
168#2331—1911nomaximal, Sylow
169C2×C4⋊Q8—1911no—
170#391—1911no—
171#391—1911no—
172#391—1911no—
173#391D19111yescharacteristic
174#389—1911no—
175#389—1911no—
176#387—1911no—
177#387—1911no—
178#387—1911no—
179#387—1911no—
180#382—1911no—
181C4×C4⋊C4—1911no—
182C4×C4⋊C4—1911no—
183#376—1911no—
184C22×Q8—1911no—
185C22×Q8C2×D19111yescharacteristic
186C4⋊Q8—1918no—
187C2×C4⋊C4—1911no—
188C2×C4⋊C4—1911no—
189C2×C4⋊C4—1911no—
190C2×C4⋊C4—1911no—
191C2×C4⋊C4—1911no—
192C2×C4⋊C4—1911no—
193C2×C4⋊C4—1911no—
194C2×C4⋊C4—1911no—
195C2×C4⋊C4—1911no—
196C2×C4⋊C4C2×D19111yescharacteristic
197C2×C42—1911no—
198C2×C42—1911no—
199C2×C42—1911no—
200C2×C42—1911no—
201C2×C42—1911no—
202C2×C42—1911no—
203C2×C42C2×D19111yescharacteristic
204#95—1911no—
205#95—1911no—
206#95—1911no—
207#95—1911no—
208#95—1911no—
209#95—1911no—
210#95—1911no—
211#95—1911no—
212#95—1911no—
213#95—1911no—
214#95—1911no—
215#95—1911no—
216#95C2×D19111yescharacteristic
217#95C2×D19111yescharacteristic
218#95C2×D19111yescharacteristic
219#95C2×D19111yescharacteristic
220C2×Q8—41no—
221C2×Q8—41no—
222C2×Q8—1914no—
223C2×Q8—7641no—
224C2×Q8—7641no—
225C2×Q8C4×D19114yes—
226C22×C4—1911no—
227C22×C4—1911no—
228C22×C4—1911no—
229C22×C4—1911no—
230C22×C4—1911no—
231C22×C4—1911no—
232C22×C4—1911no—
233C22×C4—1911no—
234C22×C4C22×D19111yescharacteristic
235C22×C4C22×D19111yescharacteristic
236C22×C4C22×D19111yescharacteristic
237C22×C4C22×D19111yescharacteristic
238C22×C4C22×D19111yescharacteristic
239C22×C4C22×D19111yescharacteristic
240C22×C4C22×D19111yescharacteristic
241C4⋊C4—41no—
242C4⋊C4—3822no—
243C4⋊C4—3822no—
244C4⋊C4—1914no—
245C4⋊C4—1914no—
246C4⋊C4—1914no—
247C4⋊C4—1914no—
248C4⋊C4—7641no—
249C4⋊C4—7641no—
250C4⋊C4—7641no—
251C42—41no—
252C42—3822no—
253C42—1914no—
254C42—1914no—
255C42—7641no—
256C42—7641no—
257C42—7641no—
258C23C23×D19111yescharacteristic
259Q8—44no—
260Q8—7644no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—3821no—
279C2×C4—3821no—
280C2×C4—3821no—
281C2×C4—3821no—
282C2×C4—3821no—
283C2×C4—3821no—
284C2×C4—3821no—
285C2×C4—3821no—
286C2×C4—3821no—
287C2×C4—3821no—
288C2×C4—3821no—
289C2×C4—3821no—
290C2×C4—3821no—
291C2×C4—3821no—
292C2×C4—3821no—
293C2×C4—3821no—
294C2×C4—3821no—
295C2×C4—1912no—
296C2×C4—1912no—
297C2×C4—1912no—
298C2×C4—1912no—
299C2×C4—1912no—
300C2×C4—1912no—
301C2×C4—1912no—
302C2×C4C2×C4×D19112yes—
303C2×C4C2×C4×D19112yes—
304C2×C4C2×C4×D19112yes—
305C2×C4Q8×D19112yes—
306C22C22×C4×D19111yescharacteristic
307C22C2×Q8×D19111yescharacteristic
308C22#41×D19111yescharacteristic
309C22#2025936711yescharacteristic
310C22#2025936711yescharacteristic
311C22#2025936911yescharacteristic
312C22#2025937011yescharacteristic
313C4—22no—
314C4—22no—
315C4—22no—
316C4—22no—
317C4—41no—
318C4—41no—
319C4—41no—
320C4—3822no—
321C4—3822no—
322C4—3822no—
323C4—3822no—
324C4—1914no—
325C4—7641no—
326C4—7641no—
327C4—7641no—
328C2C4×Q8×D19111yescharacteristic
329C2#2713313611yescharacteristic
330C2#2713313911yescharacteristic
331C2#2713327711yescharacteristic
332C2#2713328311yescharacteristic
333C2#2713329011yescharacteristic
334C2#2713329211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D191
commutatorC22×C191C23×C4
Fitting#391×C191C2
socle, prime radical, cosocleC23×C191C24
hypercenter#391D191
hypocenterC191#2331
LCSG ▷ C22×C191 ▷ C191
UCSC1 ◁ C23 ◁ #391

Complements

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

NH ≅ G/Ntype
17. #20259320C4semidirect
77. C4×Dic191Q8semidirect
167. C191#2331semidirect

Maximal subgroups

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

subgroupindex
C2×#202593202
#271325702
#271325672
#271325602
#271325582
#271325502
#271325472
#271325452
#271325442
#271325272
C4×Dic191⋊C42
#271325142
#271325082
#391×C1912
#2331191