Finite Group Databasebeta

← #37575868

Subgroups — #37575868

Order 24448.

908 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#15690288C211yesmaximal, characteristic, complemented
2C2×#20259298C211yesmaximal, characteristic, complemented
3C2×#20259297C211yesmaximal, characteristic, complemented
4C2×C4×D764C211yesmaximal, characteristic, complemented
5#27132581C212yesmaximal, complemented
6#27132573C212yesmaximal, complemented
7#27132565C211yesmaximal, characteristic, complemented
8#27132541C212yesmaximal, complemented
9#27132537C212yesmaximal, complemented
10#27132520C211yesmaximal, characteristic, complemented
11#391×C191C211yesmaximal, characteristic, complemented
12C2×#15690288—24no—
13C2×#15690288—24no—
14C2×#15690288—24no—
15C22×D764C2212yes—
16C22×D764C2211yescharacteristic
17C22×D764C2212yes—
18C22×D764C2211yescharacteristic
19C22×C4×D191C2212yes—
20C22×C4×D191C2211yescharacteristic
21C22×C4×D191C2211yescharacteristic
22#20259298—24no—
23#20259297—24no—
24C4×D764—24no—
25C2×C4⋊Dic191C2211yescharacteristic
26C2×C4⋊Dic191C2211yescharacteristic
27C2×Dic191⋊C4C2212yes—
28C2×(C2×D191)⋊C4C2212yes—
29C2×(C2×D191)⋊C4C2212yes—
30C2×(C2×D191)⋊C4C2211yescharacteristic
31C2×(C2×D191)⋊C4C2212yes—
32C2×(C2×D191)⋊C4C2211yescharacteristic
33C2×(C2×D191)⋊C4C2212yes—
34C2×(C2×D191)⋊C4C2212yes—
35C191⋊#95C2212yes—
36C22×Q8×C191C2211yescharacteristic
37C2×C4⋊C4×C191C2211yescharacteristic
38C2×C42×C191C2211yescharacteristic
39#95×C191C2212yes—
40#95×C191C2212yes—
41C23×D191C2312yes—
42C23×D191C2311yescharacteristic
43C23×D191C2311yescharacteristic
44#15690288—232no—
45C2×D764—24no—
46C2×D764—24no—
47C2×D764—22no—
48C2×D764—22no—
49C2×D764—24no—
50C2×D764—24no—
51C2×D764—22no—
52C2×D764—24no—
53C2×D764—22no—
54C2×D764—22no—
55C2×D764—42no—
56C2×D764D414yescomplemented
57C22×Dic191C2311yescharacteristic
58C22×Dic191C2312yes—
59C22×Dic191C2311yescharacteristic
60C2×C4×D191—24no—
61C2×C4×D191—24no—
62C2×C4×D191—22no—
63C2×C4×D191—24no—
64C2×C4×D191—22no—
65C2×C4×D191—22no—
66C2×C4×D191—22no—
67C2×C4×D191—22no—
68C2×C4×D191D414yescomplemented
69C4⋊Dic191—22no—
70C4⋊Dic191—22no—
71Dic191⋊C4—24no—
72(C2×D191)⋊C4—22no—
73(C2×D191)⋊C4—24no—
74(C2×D191)⋊C4—22no—
75(C2×D191)⋊C4—24no—
76(C2×D191)⋊C4—24no—
77(C2×D191)⋊C4—42no—
78(C2×D191)⋊C4—42no—
79C2×Q8×C191—22no—
80C2×Q8×C191—22no—
81C2×Q8×C191D414yescomplemented
82C22×C4×C191C2312yes—
83C22×C4×C191C2311yescharacteristic
84C22×C4×C191C2312yes—
85C22×C4×C191C2311yescharacteristic
86C22×C4×C191C2311yescharacteristic
87C4⋊C4×C191—22no—
88C42×C191—22no—
89C22×D191—21no—
90C22×D191—21no—
91C22×D191—22no—
92C22×D191—21no—
93C22×D191—21no—
94C22×D191—22no—
95C22×D191—21no—
96C22×D191—22no—
97C22×D191—21no—
98C22×D191—22no—
99C22×D191—21no—
100C22×D191—21no—
101C22×D191—21no—
102C22×D191—22no—
103C22×D191—21no—
104C22×D191—21no—
105C22×D191—22no—
106C22×D191—21no—
107C22×D191C2×D414yes—
108C22×D191C2×D412yes—
109C22×D191#4112yescomplemented
110D764—28no—
111D764—216no—
112D764—28no—
113D764—48no—
114C2×Dic191—21no—
115C2×Dic191—21no—
116C2×Dic191—21no—
117C2×Dic191—22no—
118C2×Dic191—21no—
119C2×Dic191—21no—
120C2×Dic191—22no—
121C2×Dic191—22no—
122C2×Dic191C2×D412yes—
123C4×D191—216no—
124C4×D191—28no—
125C4×D191—44no—
126C23×C191C2411yescharacteristic
127Q8×C191—28no—
128C2×C4×C191—21no—
129C2×C4×C191—22no—
130C2×C4×C191—22no—
131C2×C4×C191—21no—
132C2×C4×C191—21no—
133C2×C4×C191—22no—
134C2×C4×C191—22no—
135C2×C4×C191—21no—
136C2×C4×C191—22no—
137C2×C4×C191—21no—
138C2×C4×C191C2×D412yes—
139C2×C4×C191C2×D412yes—
140C2×C4×C191C2×D414yes—
141C2×C4×C191C2×D412yes—
142C2×C4×C191#4112yescomplemented
143C22×C191C22×D411yescharacteristic
144C22×C191C22×D411yescharacteristic
145C22×C191C22×D411yescharacteristic
146C22×C191C2×#4111yescharacteristic
147C22×C191C2×#4111yescharacteristic
148C22×C191H(5,𝔽2)11yescharacteristic
149C22×C1912−1+411yescharacteristic
150C2×D191—24no—
151C2×D191—22no—
152C2×D191—22no—
153C2×D191—22no—
154C2×D191—22no—
155C2×D191—22no—
156C2×D191—24no—
157C2×D191—24no—
158C2×D191—22no—
159C2×D191—41no—
160C2×D191—42no—
161C2×D191—41no—
162C2×D191—42no—
163C2×D191—41no—
164C2×D191—41no—
165C2×D191—41no—
166C2×D191—41no—
167C2×D191—42no—
168C2×D191—41no—
169C2×D191—42no—
170C2×D191—41no—
171C4×C191—24no—
172C4×C191—22no—
173C4×C191—41no—
174C4×C191—42no—
175C4×C191#12014yes—
176Dic191—22no—
177Dic191—41no—
178Dic191—42no—
179C2×C191C2×#12011yescharacteristic
180C2×C191#52511yescharacteristic
181C2×C191#53711yescharacteristic
182C2×C191#54611yescharacteristic
183C2×C191#54711yescharacteristic
184C2×C191#54811yescharacteristic
185C2×C191#55011yescharacteristic
186D191—42no—
187D191—44no—
188D191—42no—
189C191#240711yescharacteristic, complemented, Sylow
190#2407—1911nomaximal, Sylow
191C22×#41—1911no—
192C2×C22⋊Q8—1911no—
193C2×#121—1911no—
194C2×C4×D4—1911no—
195#394—1912no—
196#392—1912no—
197#391D19111yescharacteristic, complemented
198#390—1911no—
199#386—1912no—
200#385—1912no—
201#380—1911no—
202C2×#41—3824no—
203C2×#41—3824no—
204C2×#41—3824no—
205C22×Q8C2×D19111yescharacteristic
206C22×D4—1911no—
207C22×D4—1912no—
208C22×D4—1911no—
209C22×D4—1912no—
210C23×C4—1911no—
211C23×C4—1912no—
212C23×C4—1911no—
213C22⋊Q8—3824no—
214#121—3824no—
215C4×D4—3824no—
216C2×C4⋊C4—1911no—
217C2×C4⋊C4—1911no—
218C2×C4⋊C4—1912no—
219C2×C4⋊C4C2×D19111yescharacteristic
220C2×C22⋊C4—1911no—
221C2×C22⋊C4—1911no—
222C2×C22⋊C4—1912no—
223C2×C22⋊C4—1912no—
224C2×C22⋊C4—1912no—
225C2×C22⋊C4—1912no—
226C2×C22⋊C4—1912no—
227C2×C42C2×D19111yescharacteristic
228#95—1912no—
229#95C2×D19112yes—
230#95C2×D19112yes—
231C24—1911no—
232C24—1912no—
233C24—1911no—
234#41—38232no—
235C2×Q8—22no—
236C2×Q8—22no—
237C2×Q8D76414yescomplemented
238C2×D4—3822no—
239C2×D4—3822no—
240C2×D4—3824no—
241C2×D4—3824no—
242C2×D4—3824no—
243C2×D4—3824no—
244C2×D4—3822no—
245C2×D4—3824no—
246C2×D4—3822no—
247C2×D4—3822no—
248C2×D4—1914no—
249C2×D4—7642no—
250C22×C4—3824no—
251C22×C4—3824no—
252C22×C4—3822no—
253C22×C4—3824no—
254C22×C4—3822no—
255C22×C4—3822no—
256C22×C4—3822no—
257C22×C4—3822no—
258C22×C4—1912no—
259C22×C4—1911no—
260C22×C4—1911no—
261C22×C4—1914no—
262C22×C4C22×D19112yes—
263C22×C4C22×D19111yescharacteristic
264C22×C4C22×D19112yes—
265C22×C4C22×D19111yescharacteristic
266C22×C4C22×D19111yescharacteristic
267C4⋊C4—22no—
268C4⋊C4—3822no—
269C4⋊C4—3824no—
270C4⋊C4—3822no—
271C22⋊C4—3822no—
272C22⋊C4—3824no—
273C22⋊C4—3822no—
274C22⋊C4—3824no—
275C22⋊C4—3824no—
276C22⋊C4—7642no—
277C22⋊C4—7642no—
278C42—22no—
279C23—3821no—
280C23—3821no—
281C23—3822no—
282C23—3821no—
283C23—3821no—
284C23—3822no—
285C23—3821no—
286C23—3822no—
287C23—3821no—
288C23—3822no—
289C23—3821no—
290C23—3821no—
291C23—3821no—
292C23—3822no—
293C23—3821no—
294C23—3821no—
295C23—3822no—
296C23—3821no—
297C23—1914no—
298C23—1912no—
299C23—1912no—
300C23C23×D19111yescharacteristic
301Q8—28no—
302D4—38216no—
303D4—3828no—
304D4—3828no—
305D4—7648no—
306C2×C4—21no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—21no—
314C2×C4—22no—
315C2×C4—21no—
316C2×C4—38216no—
317C2×C4—3821no—
318C2×C4—3821no—
319C2×C4—3828no—
320C2×C4—3821no—
321C2×C4—3821no—
322C2×C4—3822no—
323C2×C4—3821no—
324C2×C4—3822no—
325C2×C4—3822no—
326C2×C4—1912no—
327C2×C4—7644no—
328C2×C4D4×D19112yes—
329C2×C4D4×D19112yes—
330C2×C4C2×D76414yes—
331C2×C4C2×D76412yes—
332C2×C4#1569028812yescomplemented
333C22—3822no—
334C22—3822no—
335C22—3824no—
336C22—3824no—
337C22—3822no—
338C22—3824no—
339C22—3822no—
340C22—3822no—
341C22—3822no—
342C22—7641no—
343C22—7642no—
344C22—7641no—
345C22—7641no—
346C22—7641no—
347C22—7641no—
348C22—7642no—
349C22—7641no—
350C22—7642no—
351C22—7641no—
352C22—7641no—
353C22—7642no—
354C22C2×D4×D19111yescharacteristic
355C22C2×D4×D19111yescharacteristic
356C22C22×D76411yescharacteristic
357C22#41×D19111yescharacteristic
358C22C2×#1569028811yescharacteristic
359C22#2025936811yescharacteristic
360C22#2025937011yescharacteristic
361C4—24no—
362C4—22no—
363C4—41no—
364C4—42no—
365C4—3822no—
366C4—7642no—
367C4—7641no—
368C4#2025929214yes—
369C2—7642no—
370C2—7642no—
371C2—7644no—
372C2C2×#2025929211yescharacteristic
373C2#2713317611yescharacteristic
374C2#2713324311yescharacteristic
375C2#2713330611yescharacteristic
376C2#2713331611yescharacteristic
377C2#2713332511yescharacteristic
378C2#2713333511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D191
commutator, socle, cosocleC23×C191C24
Fitting#391×C191C2
hypercenter#391D191
hypocenterC191#2407
LCSG ▷ C23×C191 ▷ C191
UCSC1 ◁ C23 ◁ #391

Complements

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

NH ≅ G/Ntype
1. C22×#15690288C2semidirect
2. C2×#20259298C2semidirect
3. C2×#20259297C2semidirect
4. C2×C4×D764C2semidirect
5. #27132581C2semidirect
6. #27132573C2semidirect
7. #27132565C2semidirect
8. #27132541C2semidirect
9. #27132537C2semidirect
10. #27132520C2semidirect
11. #391×C191C2semidirect
56. C2×D764D4semidirect
68. C2×C4×D191D4semidirect
81. C2×Q8×C191D4semidirect
109. C22×D191#41semidirect
142. C2×C4×C191#41semidirect
189. C191#2407semidirect
197. #391D191semidirect
237. C2×Q8D764semidirect
332. C2×C4#15690288semidirect

Maximal subgroups

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

subgroupindex
C22×#156902882
C2×#202592982
C2×#202592972
C2×C4×D7642
#271325812
#271325732
#271325652
#271325412
#271325372
#271325202
#391×C1912
#2407191