Finite Group Databasebeta

← #37575201

Subgroups — #37575201

Order 24448.

556 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#20259300C211yesmaximal, characteristic
2C2×C4×Dic382C211yesmaximal, characteristic, complemented
3C2×#20259280C211yesmaximal, characteristic
4#27132567C211yesmaximal, characteristic, complemented
5#27132562C211yesmaximal, characteristic, complemented
6#27132560C211yesmaximal, characteristic, complemented
7#27132560C211yesmaximal, characteristic, complemented
8#27132555C211yesmaximal, characteristic
9#27132555C211yesmaximal, characteristic
10#27132554C211yesmaximal, characteristic
11#27132552C211yesmaximal, characteristic
12#27132544C211yesmaximal, characteristic, complemented
13#27132522C211yesmaximal, characteristic
14#27132514C211yesmaximal, characteristic, complemented
15#387×C191C211yesmaximal, characteristic, complemented
16C22×Dic382C2211yescharacteristic
17C22×C4×D191C2211yescharacteristic
18#20259300C418yescomplemented
19C4×Dic382—24no—
20#20259280—24no—
21C2×C4⋊Dic191C2211yescharacteristic
22C2×C4⋊Dic191C2211yescharacteristic
23C2×C4⋊Dic191C2211yescharacteristic
24C2×Dic191⋊C4C2211yescharacteristic
25C2×Dic191⋊C4C2211yescharacteristic
26C2×Dic191⋊C4C2211yescharacteristic
27C2×Dic191⋊C4C2211yescharacteristic
28C2×Dic191⋊C4C2211yescharacteristic
29C2×Dic191⋊C4C2211yescharacteristic
30C2×(C2×D191)⋊C4C2211yescharacteristic
31C2×(C2×D191)⋊C4C2211yescharacteristic
32C2×(C2×D191)⋊C4C2211yescharacteristic
33C2×(C2×D191)⋊C4C2211yescharacteristic
34C2×(C2×D191)⋊C4C2211yescharacteristic
35C2×(C2×D191)⋊C4C2211yescharacteristic
36C2×C4×Dic191C2211yescharacteristic
37C2×C4×Dic191C2211yescharacteristic
38C2×C4×Dic191C2211yescharacteristic
39C2×C4×Dic191C2211yescharacteristic
40C2×C4×Dic191C2211yescharacteristic
41C191⋊#95C2211yescharacteristic
42C191⋊#95C2211yescharacteristic
43C191⋊#95C2211yescharacteristic
44C191⋊#95C2211yescharacteristic
45C191⋊#95C2211yescharacteristic
46C191⋊#95C2211yescharacteristic
47C2×C4⋊C4×C191C2211yescharacteristic
48C2×C4⋊C4×C191C2211yescharacteristic
49C2×C42×C191C2211yescharacteristic
50#95×C191C2211yescharacteristic
51#95×C191C2211yescharacteristic
52#95×C191C2211yescharacteristic
53#95×C191C2211yescharacteristic
54C23×D191C2311yescharacteristic
55C2×Dic382—22no—
56C2×Dic382—22no—
57C2×Dic382C2×C414yes—
58C22×Dic191C2311yescharacteristic
59C22×Dic191C2311yescharacteristic
60C22×Dic191C2311yescharacteristic
61C22×Dic191C2311yescharacteristic
62C22×Dic191C2311yescharacteristic
63C22×Dic191C2311yescharacteristic
64C22×Dic191C2311yescharacteristic
65C2×C4×D191—22no—
66C2×C4×D191—22no—
67C2×C4×D191C2×C414yes—
68C4⋊Dic191—22no—
69C4⋊Dic191—22no—
70C4⋊Dic191—41no—
71Dic191⋊C4—22no—
72Dic191⋊C4—22no—
73Dic191⋊C4—22no—
74Dic191⋊C4—41no—
75Dic191⋊C4C2×C414yes—
76Dic191⋊C4C2×C414yes—
77(C2×D191)⋊C4—22no—
78(C2×D191)⋊C4—22no—
79(C2×D191)⋊C4—41no—
80(C2×D191)⋊C4—41no—
81(C2×D191)⋊C4C2×C414yes—
82(C2×D191)⋊C4C2×C414yes—
83C4×Dic191—22no—
84C4×Dic191—22no—
85C4×Dic191—41no—
86C4×Dic191—41no—
87C4×Dic191D414yes—
88C22×C4×C191C2311yescharacteristic
89C22×C4×C191C2311yescharacteristic
90C22×C4×C191C2311yescharacteristic
91C22×C4×C191C2311yescharacteristic
92C22×C4×C191C2311yescharacteristic
93C22×C4×C191C2311yescharacteristic
94C22×C4×C191C2311yescharacteristic
95C4⋊C4×C191—22no—
96C4⋊C4×C191C2×C414yes—
97C42×C191—22no—
98C22×D191—21no—
99C22×D191—21no—
100C22×D191—21no—
101C22×D191—21no—
102C22×D191—21no—
103C22×D191—21no—
104C22×D191C22×C412yes—
105Dic382—28no—
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×Dic191—21no—
119C2×Dic191—21no—
120C2×Dic191—21no—
121C2×Dic191C22×C412yes—
122C2×Dic191C22×C412yes—
123C2×Dic191C22×C412yes—
124C2×Dic191C2×D412yes—
125C2×Dic191C2×D412yes—
126C2×Dic191#4112yes—
127C4×D191—28no—
128C23×C191C2411yescharacteristic
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×C191—21no—
142C2×C4×C191—21no—
143C2×C4×C191—21no—
144C2×C4×C191—21no—
145C2×C4×C191—21no—
146C2×C4×C191C22×C412yes—
147C2×C4×C191C22×C412yes—
148C2×C4×C191C22×C412yes—
149C2×C4×C191C2×D412yes—
150C22×C191C23×C411yescharacteristic
151C22×C191C22×D411yescharacteristic
152C22×C191C2×#4111yescharacteristic
153C22×C191H(5,𝔽2)11yescharacteristic
154C22×C1912−1+411yescharacteristic
155C22×C1912−1+411yescharacteristic
156C22×C1912−1+411yescharacteristic
157C2×D191—22no—
158C2×D191—22no—
159C2×D191—22no—
160C2×D191—41no—
161C2×D191—41no—
162C2×D191—41no—
163C2×D191—41no—
164C4×C191—22no—
165C4×C191—22no—
166C4×C191—22no—
167C4×C191—22no—
168C4×C191—41no—
169C4×C191—41no—
170C4×C191—41no—
171Dic191—22no—
172Dic191—22no—
173Dic191—22no—
174Dic191—41no—
175Dic191—41no—
176Dic191—41no—
177Dic191C4×D414yes—
178C2×C191C2×C4×D411yescharacteristic
179C2×C191#51911yescharacteristic
180C2×C191#52011yescharacteristic
181C2×C191#53611yescharacteristic
182C2×C191#53711yescharacteristic
183C2×C191#54111yescharacteristic
184C2×C191#54211yescharacteristic
185D191—42no—
186C191#232311yescharacteristic, complemented, Sylow
187#2323—1911nomaximal, Sylow
188C2×C22⋊Q8—1911no—
189C2×C4×Q8—1911no—
190C2×#117—1911no—
191#391—1911no—
192#389—1911no—
193#389—1911no—
194#389—1911no—
195#388—1911no—
196#388—1911no—
197#388—1911no—
198#388—1911no—
199#387—1911no—
200#387D19111yescharacteristic, complemented
201#380—1911no—
202C4×C4⋊C4—1911no—
203C22×Q8—1911no—
204C23×C4—1911no—
205C22⋊Q8—1918no—
206C4×Q8—3824no—
207#117—3824no—
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×C22⋊C4—1911no—
220C2×C22⋊C4—1911no—
221C2×C22⋊C4—1911no—
222C2×C22⋊C4—1911no—
223C2×C22⋊C4—1911no—
224C2×C22⋊C4—1911no—
225C2×C42—1911no—
226C2×C42—1911no—
227C2×C42—1911no—
228C2×C42—1911no—
229C2×C42—1911no—
230C2×C42C2×D19111yescharacteristic
231#95—1911no—
232#95—1911no—
233#95—1911no—
234#95—1911no—
235#95—1911no—
236#95—1911no—
237#95C2×D19111yescharacteristic
238#95C2×D19111yescharacteristic
239#95C2×D19111yescharacteristic
240#95C2×D19111yescharacteristic
241C24—1911no—
242C2×Q8—3822no—
243C2×Q8—3822no—
244C2×Q8—1914no—
245C22×C4—3822no—
246C22×C4—3822no—
247C22×C4—1911no—
248C22×C4—1911no—
249C22×C4—1911no—
250C22×C4—1911no—
251C22×C4—1911no—
252C22×C4—1911no—
253C22×C4—1914no—
254C22×C4—1911no—
255C22×C4C22×D19111yescharacteristic
256C22×C4C22×D19111yescharacteristic
257C22×C4C22×D19111yescharacteristic
258C22×C4C22×D19111yescharacteristic
259C22×C4C22×D19111yescharacteristic
260C22×C4C22×D19111yescharacteristic
261C22×C4C22×D19111yescharacteristic
262C4⋊C4—22no—
263C4⋊C4—3822no—
264C4⋊C4—3822no—
265C4⋊C4—3822no—
266C4⋊C4—3822no—
267C4⋊C4—3822no—
268C4⋊C4—1914no—
269C4⋊C4—1914no—
270C4⋊C4—7641no—
271C4⋊C4—7641no—
272C4⋊C4C4×D19114yes—
273C22⋊C4—3822no—
274C22⋊C4—3822no—
275C22⋊C4—1914no—
276C22⋊C4—1914no—
277C22⋊C4—7641no—
278C22⋊C4—7641no—
279C42—22no—
280C42—3822no—
281C42—3822no—
282C42—1914no—
283C42—7641no—
284C42—7641no—
285C23—3821no—
286C23—3821no—
287C23—3821no—
288C23—3821no—
289C23—3821no—
290C23—3821no—
291C23—1912no—
292C23C23×D19111yescharacteristic
293Q8—3828no—
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—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—3828no—
312C2×C4—3821no—
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—3821no—
326C2×C4—3821no—
327C2×C4—1912no—
328C2×C4—1912no—
329C2×C4—1912no—
330C2×C4—1912no—
331C2×C4—1912no—
332C2×C4—1912no—
333C2×C4C2×C4×D19112yes—
334C2×C4C2×C4×D19112yes—
335C2×C4C2×C4×D19112yes—
336C2×C4D4×D19112yes—
337C22—3822no—
338C22—3822no—
339C22—3822no—
340C22—7641no—
341C22—7641no—
342C22—7641no—
343C22—7641no—
344C22C22×C4×D19111yescharacteristic
345C22C2×D4×D19111yescharacteristic
346C22#41×D19111yescharacteristic
347C22#2025936811yescharacteristic
348C22#2025936911yescharacteristic
349C22#2025936911yescharacteristic
350C22#2025937011yescharacteristic
351C4—22no—
352C4—22no—
353C4—22no—
354C4—22no—
355C4—41no—
356C4—41no—
357C4—41no—
358C4—3822no—
359C4—3822no—
360C4—3822no—
361C4—1914no—
362C4—7641no—
363C4—7641no—
364C4—7641no—
365C2—7642no—
366C2C4×D4×D19111yescharacteristic
367C2#2713313911yescharacteristic
368C2#2713314311yescharacteristic
369C2#2713323511yescharacteristic
370C2#2713324311yescharacteristic
371C2#2713327211yescharacteristic
372C2#2713327911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D191
commutatorC22×C191C23×C4
Fitting#387×C191C2
socle, cosocleC23×C191C24
prime radicalC23×D191C23
hypercenter#387D191
hypocenterC191#2323
LCSG ▷ C22×C191 ▷ C191
UCSC1 ◁ C23 ◁ #387

Complements

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

NH ≅ G/Ntype
2. C2×C4×Dic382C2semidirect
4. #27132567C2semidirect
5. #27132562C2semidirect
6. #27132560C2semidirect
7. #27132560C2semidirect
12. #27132544C2semidirect
14. #27132514C2semidirect
15. #387×C191C2semidirect
18. #20259300C4semidirect
186. C191#2323semidirect
200. #387D191semidirect

Maximal subgroups

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

subgroupindex
C2×#202593002
C2×C4×Dic3822
C2×#202592802
#271325672
#271325622
#271325602
#271325552
#271325542
#271325522
#271325442
#271325222
#271325142
#387×C1912
#2323191