Finite Group Databasebeta

← #37576228

Subgroups — #37576228

Order 24448.

492 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#20259312C211yesmaximal, characteristic, complemented
2C2×#20259278C211yesmaximal, characteristic
3#27132599C211yesmaximal, characteristic
4#27132599C211yesmaximal, characteristic
5#27132586C211yesmaximal, characteristic
6#27132562C211yesmaximal, characteristic, complemented
7#27132560C211yesmaximal, characteristic, complemented
8#27132558C211yesmaximal, characteristic, complemented
9#27132558C211yesmaximal, characteristic, complemented
10#27132545C211yesmaximal, characteristic, complemented
11#27132544C211yesmaximal, characteristic, complemented
12#27132532C211yesmaximal, characteristic, complemented
13#27132521C211yesmaximal, characteristic
14C22⋊C4×Dic191C211yesmaximal, characteristic
15#396×C191C211yesmaximal, characteristic
16C23×Dic191C2211yescharacteristic
17#20259312—24no—
18#20259278—24no—
19C2×C4⋊Dic191C2211yescharacteristic
20C2×C4⋊Dic191C2211yescharacteristic
21C2×C4⋊Dic191C2211yescharacteristic
22C2×Dic191⋊C4C2211yescharacteristic
23C2×Dic191⋊C4C2211yescharacteristic
24C2×Dic191⋊C4C2211yescharacteristic
25C2×Dic191⋊C4C2211yescharacteristic
26C2×Dic191⋊C4C2211yescharacteristic
27C2×Dic191⋊C4C2211yescharacteristic
28C2×Dic191⋊C4C2211yescharacteristic
29C2×Dic191⋊C4C2211yescharacteristic
30C2×Dic191⋊C4C2211yescharacteristic
31C2×Dic191⋊C4C2211yescharacteristic
32C2×C22⋊Dic191C2211yescharacteristic
33C2×C22⋊Dic191C2211yescharacteristic
34C2×C22⋊Dic191C2211yescharacteristic
35C2×C4×Dic191C2211yescharacteristic
36C2×C4×Dic191C2211yescharacteristic
37C2×C4×Dic191C2211yescharacteristic
38C2×C4×Dic191C2211yescharacteristic
39C191⋊#95C2211yescharacteristic
40C191⋊#95C2211yescharacteristic
41C191⋊#95C2211yescharacteristic
42C191⋊#95C2211yescharacteristic
43C191⋊#95C2211yescharacteristic
44C191⋊#95C2211yescharacteristic
45C191⋊#95C2211yescharacteristic
46C2×C4⋊C4×C191C2211yescharacteristic
47C2×C4⋊C4×C191C2211yescharacteristic
48C2×C4⋊C4×C191C2211yescharacteristic
49C2×C22⋊C4×C191C2211yescharacteristic
50C2×C22⋊C4×C191C2211yescharacteristic
51C2×C22⋊C4×C191C2211yescharacteristic
52#95×C191C2211yescharacteristic
53C22×Dic191—22no—
54C22×Dic191—22no—
55C22×Dic191Q814yes—
56C22×Dic191C2311yescharacteristic
57C22×Dic191C2311yescharacteristic
58C22×Dic191C2311yescharacteristic
59C22×Dic191C2311yescharacteristic
60C22×Dic191C2311yescharacteristic
61C22×Dic191C2311yescharacteristic
62C22×Dic191C2311yescharacteristic
63C22×Dic191C2311yescharacteristic
64C4⋊Dic191—22no—
65C4⋊Dic191—41no—
66C4⋊Dic191—41no—
67Dic191⋊C4—22no—
68Dic191⋊C4—22no—
69Dic191⋊C4—22no—
70Dic191⋊C4—22no—
71Dic191⋊C4—22no—
72Dic191⋊C4—22no—
73Dic191⋊C4—22no—
74Dic191⋊C4—22no—
75Dic191⋊C4—41no—
76Dic191⋊C4D414yes—
77C22⋊Dic191—22no—
78C22⋊Dic191—22no—
79C22⋊Dic191—41no—
80C4×Dic191—22no—
81C4×Dic191—22no—
82C4×Dic191—22no—
83C4×Dic191—22no—
84C24×C191C2311yescharacteristic
85C22×C4×C191C2311yescharacteristic
86C22×C4×C191C2311yescharacteristic
87C22×C4×C191C2311yescharacteristic
88C22×C4×C191C2311yescharacteristic
89C22×C4×C191C2311yescharacteristic
90C22×C4×C191C2311yescharacteristic
91C4⋊C4×C191—22no—
92C4⋊C4×C191—22no—
93C4⋊C4×C191—41no—
94C22⋊C4×C191—22no—
95C22⋊C4×C191—22no—
96C22⋊C4×C191—41no—
97C2×Dic191—21no—
98C2×Dic191—21no—
99C2×Dic191—21no—
100C2×Dic191—21no—
101C2×Dic191—21no—
102C2×Dic191—21no—
103C2×Dic191—28no—
104C2×Dic191—21no—
105C2×Dic191—21no—
106C2×Dic191—21no—
107C2×Dic191—21no—
108C2×Dic191—21no—
109C2×Dic191—21no—
110C2×Dic191—21no—
111C2×Dic191—21no—
112C2×Dic191—21no—
113C2×Dic191—21no—
114C2×Dic191—21no—
115C2×Dic191—21no—
116C2×Dic191C2×D412yes—
117C2×Dic191C2×D412yes—
118C2×Dic191C2×Q812yes—
119C2×Dic191C2×Q812yes—
120C2×Dic191#4112yes—
121C2×Dic191#4112yes—
122C23×C191—21no—
123C23×C191—21no—
124C23×C191—21no—
125C23×C191—21no—
126C23×C191—21no—
127C23×C191—21no—
128C23×C191C2×Q812yes—
129C23×C191C2411yescharacteristic
130C2×C4×C191—21no—
131C2×C4×C191—21no—
132C2×C4×C191—21no—
133C2×C4×C191—21no—
134C2×C4×C191—21no—
135C2×C4×C191—21no—
136C2×C4×C191—21no—
137C2×C4×C191—21no—
138C2×C4×C191—21no—
139C2×C4×C191—21no—
140C2×C4×C191—21no—
141C2×C4×C191—21no—
142C2×C4×C191—21no—
143C2×C4×C191—21no—
144C2×C4×C191—21no—
145C2×C4×C191—21no—
146C2×C4×C191C2×D412yes—
147C2×C4×C191#4112yes—
148C22×C191—22no—
149C22×C191—22no—
150C22×C191—22no—
151C22×C191—41no—
152C22×C191—41no—
153C22×C191—41no—
154C22×C191—41no—
155C22×C191C22×D411yescharacteristic
156C22×C191C22×Q811yescharacteristic
157C22×C191C2×#4111yescharacteristic
158C22×C191C2×#4111yescharacteristic
159C22×C191C2×#4111yescharacteristic
160C22×C191H(5,𝔽2)11yescharacteristic
161C22×C1912−1+411yescharacteristic
162C4×C191—22no—
163C4×C191—22no—
164C4×C191—41no—
165C4×C191—41no—
166C4×C191—41no—
167C4×C191—41no—
168Dic191—22no—
169Dic191—22no—
170Dic191—22no—
171Dic191—41no—
172Dic191—41no—
173Dic191—41no—
174Dic191—41no—
175Dic191C22⋊Q814yes—
176C2×C191—42no—
177C2×C191C2×C22⋊Q811yescharacteristic
178C2×C191#53311yescharacteristic
179C2×C191#54411yescharacteristic
180C2×C191#54711yescharacteristic
181C2×C191#55011yescharacteristic
182C2×C191#55211yescharacteristic
183C2×C191#55311yescharacteristic
184C191#244111yescharacteristic, complemented, Sylow
185#2441—1911nomaximal, Sylow
186C2×#125—1911no—
187C2×#117—1911no—
188#399—1911no—
189#399—1911no—
190#396—1911no—
191#396D19111yescharacteristic
192#389—1911no—
193#389—1911no—
194#389—1911no—
195#389—1911no—
196#387—1911no—
197#387—1911no—
198#384—1911no—
199#380—1911no—
200C4×C22⋊C4—1911no—
201C23×C4—1911no—
202#125—3824no—
203#117—3824no—
204C2×C4⋊C4—1911no—
205C2×C4⋊C4—1911no—
206C2×C4⋊C4—1911no—
207C2×C4⋊C4—1911no—
208C2×C4⋊C4—1911no—
209C2×C4⋊C4—1911no—
210C2×C4⋊C4—1911no—
211C2×C4⋊C4—1911no—
212C2×C4⋊C4—1911no—
213C2×C4⋊C4—1911no—
214C2×C4⋊C4—1911no—
215C2×C4⋊C4—1911no—
216C2×C4⋊C4—1911no—
217C2×C4⋊C4C2×D19111yescharacteristic
218C2×C4⋊C4C2×D19111yescharacteristic
219C2×C4⋊C4C2×D19111yescharacteristic
220C2×C22⋊C4—1911no—
221C2×C22⋊C4—1911no—
222C2×C22⋊C4—1911no—
223C2×C22⋊C4C2×D19111yescharacteristic
224C2×C22⋊C4C2×D19111yescharacteristic
225C2×C22⋊C4C2×D19111yescharacteristic
226C2×C42—1911no—
227C2×C42—1911no—
228C2×C42—1911no—
229C2×C42—1911no—
230#95—1911no—
231#95—1911no—
232#95—1911no—
233#95—1911no—
234#95—1911no—
235#95—1911no—
236#95—1911no—
237#95C2×D19111yescharacteristic
238C24C22×D19111yescharacteristic
239C22×C4—3822no—
240C22×C4—3822no—
241C22×C4—1911no—
242C22×C4—1911no—
243C22×C4—1911no—
244C22×C4—1911no—
245C22×C4—1914no—
246C22×C4—1911no—
247C22×C4—1911no—
248C22×C4—1911no—
249C22×C4—1911no—
250C22×C4C22×D19111yescharacteristic
251C22×C4C22×D19111yescharacteristic
252C22×C4C22×D19111yescharacteristic
253C22×C4C22×D19111yescharacteristic
254C22×C4C22×D19111yescharacteristic
255C22×C4C22×D19111yescharacteristic
256C4⋊C4—22no—
257C4⋊C4—22no—
258C4⋊C4—41no—
259C4⋊C4—3822no—
260C4⋊C4—3822no—
261C4⋊C4—3822no—
262C4⋊C4—3822no—
263C4⋊C4—3822no—
264C4⋊C4—3822no—
265C4⋊C4—3822no—
266C4⋊C4—3822no—
267C4⋊C4—3822no—
268C4⋊C4—1914no—
269C4⋊C4—7641no—
270C4⋊C4—7641no—
271C4⋊C4—7641no—
272C22⋊C4—22no—
273C22⋊C4—22no—
274C22⋊C4—41no—
275C22⋊C4—3822no—
276C22⋊C4—3822no—
277C22⋊C4—7641no—
278C42—3822no—
279C42—3822no—
280C42—3822no—
281C42—3822no—
282C23—21no—
283C23—21no—
284C23—21no—
285C23—21no—
286C23—21no—
287C23—21no—
288C23Q8×D19112yes—
289C23C23×D19111yescharacteristic
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—3821no—
307C2×C4—3821no—
308C2×C4—3821no—
309C2×C4—3821no—
310C2×C4—3821no—
311C2×C4—3821no—
312C2×C4—3828no—
313C2×C4—3821no—
314C2×C4—3821no—
315C2×C4—3821no—
316C2×C4—3821no—
317C2×C4—3821no—
318C2×C4—3821no—
319C2×C4—3821no—
320C2×C4—3821no—
321C2×C4—3821no—
322C2×C4—3821no—
323C2×C4—3821no—
324C2×C4—3821no—
325C2×C4—1912no—
326C2×C4—1912no—
327C2×C4—1912no—
328C2×C4—1912no—
329C2×C4—1912no—
330C2×C4—1912no—
331C2×C4D4×D19112yes—
332C2×C4#1569028612yes—
333C22—22no—
334C22—22no—
335C22—22no—
336C22—41no—
337C22—41no—
338C22—41no—
339C22—41no—
340C22C2×D4×D19111yescharacteristic
341C22C2×Q8×D19111yescharacteristic
342C22#41×D19111yescharacteristic
343C22#41×D19111yescharacteristic
344C22C2×#1569028611yescharacteristic
345C22#2025936711yescharacteristic
346C22#2025936911yescharacteristic
347C4—22no—
348C4—22no—
349C4—41no—
350C4—41no—
351C4—41no—
352C4—41no—
353C4—3822no—
354C4—3822no—
355C4—3822no—
356C4—1914no—
357C4—7641no—
358C4—7641no—
359C4—7641no—
360C4—7641no—
361C2—42no—
362C2C22⋊Q8×D19111yescharacteristic
363C2#2713321511yescharacteristic
364C2#2713329211yescharacteristic
365C2#2713331311yescharacteristic
366C2#2713333111yescharacteristic
367C2#2713334911yescharacteristic
368C2#2713335611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D191
commutator, socle, cosocleC23×C191C24
Fitting#396×C191C2
prime radicalC24×C191C23
hypercenter#396D191
hypocenterC191#2441
LCSG ▷ C23×C191 ▷ C191
UCSC1 ◁ C23 ◁ #396

Complements

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

NH ≅ G/Ntype
1. C2×#20259312C2semidirect
6. #27132562C2semidirect
7. #27132560C2semidirect
8. #27132558C2semidirect
9. #27132558C2semidirect
10. #27132545C2semidirect
11. #27132544C2semidirect
12. #27132532C2semidirect
184. C191#2441semidirect

Maximal subgroups

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

subgroupindex
C2×#202593122
C2×#202592782
#271325992
#271325862
#271325622
#271325602
#271325582
#271325452
#271325442
#271325322
#271325212
C22⋊C4×Dic1912
#396×C1912
#2441191