Finite Group Databasebeta

← #37575684

Subgroups — #37575684

Order 24448.

464 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#20259302C211yesmaximal, characteristic
2#27132567C211yesmaximal, characteristic, complemented
3#27132567C211yesmaximal, characteristic, complemented
4#27132561C211yesmaximal, characteristic, complemented
5#27132559C211yesmaximal, characteristic, complemented
6#27132555C211yesmaximal, characteristic
7#27132554C211yesmaximal, characteristic
8#27132553C211yesmaximal, characteristic
9#27132552C211yesmaximal, characteristic
10#27132549C211yesmaximal, characteristic, complemented
11#27132548C211yesmaximal, characteristic, complemented
12#27132546C211yesmaximal, characteristic, complemented
13#27132525C211yesmaximal, characteristic
14#27132522C211yesmaximal, characteristic
15#387×C191C211yesmaximal, characteristic, complemented
16C22×Dic382C2211yescharacteristic
17C22×C4×D191C2211yescharacteristic
18#20259302C418yescomplemented
19C2×C4⋊Dic191C2211yescharacteristic
20C2×C4⋊Dic191C2211yescharacteristic
21C2×C4⋊Dic191C2211yescharacteristic
22C2×Dic191⋊C4C2211yescharacteristic
23C2×Dic191⋊C4C2211yescharacteristic
24C2×Dic191⋊C4C2211yescharacteristic
25C2×Dic191⋊C4C2211yescharacteristic
26C2×(C2×D191)⋊C4C2211yescharacteristic
27C2×(C2×D191)⋊C4C2211yescharacteristic
28C2×(C2×D191)⋊C4C2211yescharacteristic
29C2×(C2×D191)⋊C4C2211yescharacteristic
30C2×(C2×D191)⋊C4C2211yescharacteristic
31C2×(C2×D191)⋊C4C2211yescharacteristic
32C2×C4×Dic191C2211yescharacteristic
33C2×C4×Dic191C2211yescharacteristic
34C2×C4×Dic191C2211yescharacteristic
35C191⋊#95C2211yescharacteristic
36C191⋊#95C2211yescharacteristic
37C191⋊#95C2211yescharacteristic
38C191⋊#95C2211yescharacteristic
39C191⋊#95C2211yescharacteristic
40C191⋊#95C2211yescharacteristic
41C191⋊#95C2211yescharacteristic
42C191⋊#95C2211yescharacteristic
43C191⋊#95C2211yescharacteristic
44C191⋊#95C2211yescharacteristic
45C2×C4⋊C4×C191C2211yescharacteristic
46C2×C4⋊C4×C191C2211yescharacteristic
47C2×C42×C191C2211yescharacteristic
48#95×C191C2211yescharacteristic
49#95×C191C2211yescharacteristic
50#95×C191C2211yescharacteristic
51#95×C191C2211yescharacteristic
52C23×D191C2311yescharacteristic
53C2×Dic382—41no—
54C2×Dic382—41no—
55C2×Dic382C2×C414yes—
56C22×Dic191C2311yescharacteristic
57C22×Dic191C2311yescharacteristic
58C22×Dic191C2311yescharacteristic
59C22×Dic191C2311yescharacteristic
60C22×Dic191C2311yescharacteristic
61C22×Dic191C2311yescharacteristic
62C22×Dic191C2311yescharacteristic
63C2×C4×D191—41no—
64C2×C4×D191—41no—
65C2×C4×D191C2×C414yes—
66C4⋊Dic191—41no—
67C4⋊Dic191C2×C414yes—
68C4⋊Dic191C2×C414yes—
69Dic191⋊C4—41no—
70Dic191⋊C4—41no—
71Dic191⋊C4—41no—
72Dic191⋊C4—41no—
73(C2×D191)⋊C4—41no—
74(C2×D191)⋊C4—41no—
75(C2×D191)⋊C4—41no—
76(C2×D191)⋊C4—41no—
77(C2×D191)⋊C4C2×C414yes—
78(C2×D191)⋊C4C2×C414yes—
79C4×Dic191—41no—
80C4×Dic191—41no—
81C4×Dic191—41no—
82C22×C4×C191C2311yescharacteristic
83C22×C4×C191C2311yescharacteristic
84C22×C4×C191C2311yescharacteristic
85C22×C4×C191C2311yescharacteristic
86C22×C4×C191C2311yescharacteristic
87C22×C4×C191C2311yescharacteristic
88C22×C4×C191C2311yescharacteristic
89C4⋊C4×C191—41no—
90C4⋊C4×C191C2×C414yes—
91C42×C191—41no—
92C22×D191—21no—
93C22×D191—21no—
94C22×D191—21no—
95C22×D191—21no—
96C22×D191—21no—
97C22×D191—21no—
98C22×D191C22×C412yes—
99Dic382—44no—
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×Dic191—21no—
109C2×Dic191—21no—
110C2×Dic191—21no—
111C2×Dic191—21no—
112C2×Dic191—21no—
113C2×Dic191—21no—
114C2×Dic191—21no—
115C2×Dic191—21no—
116C2×Dic191—21no—
117C2×Dic191—21no—
118C2×Dic191C22×C412yes—
119C2×Dic191C22×C412yes—
120C2×Dic191C22×C412yes—
121C4×D191—44no—
122C23×C191C2411yescharacteristic
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×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×C191C22×C412yes—
142C2×C4×C191C22×C412yes—
143C2×C4×C191C22×C412yes—
144C22×C191C23×C411yescharacteristic
145C22×C191H(5,𝔽2)11yescharacteristic
146C22×C191H(5,𝔽2)11yescharacteristic
147C22×C191H(5,𝔽2)11yescharacteristic
148C22×C1912−1+411yescharacteristic
149C22×C1912−1+411yescharacteristic
150C22×C1912−1+411yescharacteristic
151C2×D191—22no—
152C2×D191—22no—
153C2×D191—22no—
154C2×D191—41no—
155C2×D191—41no—
156C2×D191—41no—
157C2×D191—41no—
158C4×C191—22no—
159C4×C191—22no—
160C4×C191—22no—
161C4×C191—41no—
162C4×C191—41no—
163C4×C191—41no—
164C4×C191—41no—
165Dic191—22no—
166Dic191—22no—
167Dic191—22no—
168Dic191—41no—
169Dic191—41no—
170Dic191—41no—
171Dic191—41no—
172C2×C191#51811yescharacteristic
173C2×C191#51911yescharacteristic
174C2×C191#52011yescharacteristic
175C2×C191#56211yescharacteristic
176C2×C191#56211yescharacteristic
177C2×C191#56311yescharacteristic
178C2×C191#56311yescharacteristic
179D191—42no—
180C191#237811yescharacteristic, complemented, Sylow
181#2378—1911nomaximal, Sylow
182C2×C22⋊Q8—1911no—
183#391—1911no—
184#391—1911no—
185#389—1911no—
186#389—1911no—
187#388—1911no—
188#388—1911no—
189#388—1911no—
190#388—1911no—
191#387—1911no—
192#387—1911no—
193#387—1911no—
194#387D19111yescharacteristic, complemented
195#381—1911no—
196#380—1911no—
197C22×Q8—1911no—
198C23×C4—1911no—
199C22⋊Q8—1918no—
200C2×C4⋊C4—1911no—
201C2×C4⋊C4—1911no—
202C2×C4⋊C4—1911no—
203C2×C4⋊C4—1911no—
204C2×C4⋊C4—1911no—
205C2×C4⋊C4—1911no—
206C2×C4⋊C4—1911no—
207C2×C4⋊C4C2×D19111yescharacteristic
208C2×C4⋊C4C2×D19111yescharacteristic
209C2×C22⋊C4—1911no—
210C2×C22⋊C4—1911no—
211C2×C22⋊C4—1911no—
212C2×C22⋊C4—1911no—
213C2×C22⋊C4—1911no—
214C2×C22⋊C4—1911no—
215C2×C42—1911no—
216C2×C42—1911no—
217C2×C42—1911no—
218C2×C42C2×D19111yescharacteristic
219#95—1911no—
220#95—1911no—
221#95—1911no—
222#95—1911no—
223#95—1911no—
224#95—1911no—
225#95—1911no—
226#95—1911no—
227#95—1911no—
228#95—1911no—
229#95C2×D19111yescharacteristic
230#95C2×D19111yescharacteristic
231#95C2×D19111yescharacteristic
232#95C2×D19111yescharacteristic
233C24—1911no—
234C2×Q8—1914no—
235C2×Q8—7641no—
236C2×Q8—7641no—
237C22×C4—1911no—
238C22×C4—1911no—
239C22×C4—1911no—
240C22×C4—1911no—
241C22×C4—1911no—
242C22×C4—1911no—
243C22×C4—1911no—
244C22×C4—1914no—
245C22×C4—7641no—
246C22×C4—7641no—
247C22×C4C22×D19111yescharacteristic
248C22×C4C22×D19111yescharacteristic
249C22×C4C22×D19111yescharacteristic
250C22×C4C22×D19111yescharacteristic
251C22×C4C22×D19111yescharacteristic
252C22×C4C22×D19111yescharacteristic
253C22×C4C22×D19111yescharacteristic
254C4⋊C4—41no—
255C4⋊C4—1914no—
256C4⋊C4—1914no—
257C4⋊C4—7641no—
258C4⋊C4—7641no—
259C4⋊C4—7641no—
260C4⋊C4—7641no—
261C4⋊C4—7641no—
262C4⋊C4C4×D19114yes—
263C22⋊C4—1914no—
264C22⋊C4—1914no—
265C22⋊C4—7641no—
266C22⋊C4—7641no—
267C22⋊C4—7641no—
268C22⋊C4—7641no—
269C42—41no—
270C42—7641no—
271C42—7641no—
272C42—7641no—
273C23—3821no—
274C23—3821no—
275C23—3821no—
276C23—3821no—
277C23—3821no—
278C23—3821no—
279C23—1912no—
280C23C23×D19111yescharacteristic
281Q8—7644no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
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—3821no—
301C2×C4—3821no—
302C2×C4—3821no—
303C2×C4—3821no—
304C2×C4—3821no—
305C2×C4—3821no—
306C2×C4—3821no—
307C2×C4—3821no—
308C2×C4—3821no—
309C2×C4—3821no—
310C2×C4—3821no—
311C2×C4—3821no—
312C2×C4—3821no—
313C2×C4—3821no—
314C2×C4—3821no—
315C2×C4—3821no—
316C2×C4—3821no—
317C2×C4—3821no—
318C2×C4—1912no—
319C2×C4—1912no—
320C2×C4—1912no—
321C2×C4—7644no—
322C2×C4C2×C4×D19112yes—
323C2×C4C2×C4×D19112yes—
324C2×C4C2×C4×D19112yes—
325C22—3822no—
326C22—3822no—
327C22—3822no—
328C22—7641no—
329C22—7641no—
330C22—7641no—
331C22—7641no—
332C22C22×C4×D19111yescharacteristic
333C22#2025936711yescharacteristic
334C22#2025936811yescharacteristic
335C22#2025936811yescharacteristic
336C22#2025936911yescharacteristic
337C22#2025936911yescharacteristic
338C22#2025937011yescharacteristic
339C4—22no—
340C4—22no—
341C4—22no—
342C4—41no—
343C4—41no—
344C4—41no—
345C4—41no—
346C4—3822no—
347C4—3822no—
348C4—3822no—
349C4—7641no—
350C4—7641no—
351C4—7641no—
352C4—7641no—
353C2—7642no—
354C2#2713313711yescharacteristic
355C2#2713313911yescharacteristic
356C2#2713314211yescharacteristic
357C2#2713340911yescharacteristic
358C2#2713341011yescharacteristic
359C2#2713341511yescharacteristic
360C2#2713341711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D191
commutatorC22×C191C23×C4
Fitting#387×C191C2
socle, cosocleC23×C191C24
prime radicalC23×D191C23
hypercenter#387D191
hypocenterC191#2378
LCSG ▷ C22×C191 ▷ C191
UCSC1 ◁ C23 ◁ #387

Complements

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

NH ≅ G/Ntype
2. #27132567C2semidirect
3. #27132567C2semidirect
4. #27132561C2semidirect
5. #27132559C2semidirect
10. #27132549C2semidirect
11. #27132548C2semidirect
12. #27132546C2semidirect
15. #387×C191C2semidirect
18. #20259302C4semidirect
180. C191#2378semidirect
194. #387D191semidirect

Maximal subgroups

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

subgroupindex
C2×#202593022
#271325672
#271325612
#271325592
#271325552
#271325542
#271325532
#271325522
#271325492
#271325482
#271325462
#271325252
#271325222
#387×C1912
#2378191