Finite Group Databasebeta

← #24914283

Subgroups — #24914283

Order 10624.

450 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14581787C211yesmaximal, characteristic
2#19127639C211yesmaximal, characteristic, complemented
3#19127631C211yesmaximal, characteristic, complemented
4#19127627C211yesmaximal, characteristic
5#19127623C211yesmaximal, characteristic
6#19127612C211yesmaximal, characteristic
7#19127612C211yesmaximal, characteristic
8#19127608C211yesmaximal, characteristic, complemented
9#19127608C211yesmaximal, characteristic, complemented
10#19127592C211yesmaximal, characteristic
11#19127591C211yesmaximal, characteristic
12#19127587C211yesmaximal, characteristic, complemented
13#19127585C211yesmaximal, characteristic, complemented
14#19127582C211yesmaximal, characteristic, complemented
15C2×C4⋊Q8×C83C211yesmaximal, characteristic, complemented
16#14581787—24no—
17C2×C4⋊Dic83C2211yescharacteristic
18C2×C4⋊Dic83C2211yescharacteristic
19C2×Dic83⋊C4C2211yescharacteristic
20C2×Dic83⋊C4C2211yescharacteristic
21C2×Dic83⋊C4C2211yescharacteristic
22C2×(C2×D83)⋊C4C2211yescharacteristic
23C2×(C2×D83)⋊C4C2211yescharacteristic
24C2×(C2×D83)⋊C4C2211yescharacteristic
25C2×(C2×D83)⋊C4C2211yescharacteristic
26C2×(C2×D83)⋊C4C2211yescharacteristic
27C2×(C2×D83)⋊C4C2211yescharacteristic
28C2×(C2×D83)⋊C4C2211yescharacteristic
29C2×C4×Dic83C2211yescharacteristic
30C2×C4×Dic83C2211yescharacteristic
31C83⋊#95C2211yescharacteristic
32C83⋊#95C2211yescharacteristic
33C83⋊#95C2211yescharacteristic
34C83⋊#95C2211yescharacteristic
35C83⋊#95C2211yescharacteristic
36C83⋊#95C2211yescharacteristic
37C83⋊#95C2211yescharacteristic
38C83⋊#95C2211yescharacteristic
39C83⋊#95C2211yescharacteristic
40C83⋊#95C2211yescharacteristic
41C83⋊#95C2211yescharacteristic
42C83⋊#95C2211yescharacteristic
43C83⋊#95C2211yescharacteristic
44C83⋊#95C2211yescharacteristic
45C22×Q8×C83C2211yescharacteristic
46C22×Q8×C83C2211yescharacteristic
47C4⋊Q8×C83—24no—
48C2×C4⋊C4×C83C2211yescharacteristic
49C2×C4⋊C4×C83C2211yescharacteristic
50C2×C4⋊C4×C83C2211yescharacteristic
51C2×C4⋊C4×C83C2211yescharacteristic
52C2×C42×C83C2211yescharacteristic
53C23×D83C2311yescharacteristic
54C22×Dic83C2311yescharacteristic
55C22×Dic83C2311yescharacteristic
56C22×Dic83C2311yescharacteristic
57C22×Dic83C2311yescharacteristic
58C22×Dic83C2311yescharacteristic
59C22×Dic83C2311yescharacteristic
60C22×Dic83C2311yescharacteristic
61C4⋊Dic83—22no—
62C4⋊Dic83—41no—
63Dic83⋊C4—22no—
64Dic83⋊C4—22no—
65Dic83⋊C4—22no—
66(C2×D83)⋊C4—22no—
67(C2×D83)⋊C4—22no—
68(C2×D83)⋊C4—22no—
69(C2×D83)⋊C4—22no—
70(C2×D83)⋊C4—22no—
71(C2×D83)⋊C4—41no—
72(C2×D83)⋊C4—41no—
73C4×Dic83—22no—
74C4×Dic83—41no—
75C2×Q8×C83—22no—
76C2×Q8×C83—22no—
77C2×Q8×C83—22no—
78C2×Q8×C83—22no—
79C2×Q8×C83—41no—
80C2×Q8×C83—41no—
81C22×C4×C83C2311yescharacteristic
82C22×C4×C83C2311yescharacteristic
83C22×C4×C83C2311yescharacteristic
84C22×C4×C83C2311yescharacteristic
85C22×C4×C83C2311yescharacteristic
86C22×C4×C83C2311yescharacteristic
87C22×C4×C83C2311yescharacteristic
88C4⋊C4×C83—22no—
89C4⋊C4×C83—22no—
90C4⋊C4×C83—22no—
91C4⋊C4×C83D414yes—
92C42×C83—22no—
93C23×C83C2411yescharacteristic
94C22×D83—21no—
95C22×D83—21no—
96C22×D83—21no—
97C22×D83—21no—
98C22×D83—21no—
99C22×D83—21no—
100C22×D83—21no—
101Q8×C83—44no—
102Q8×C83—44no—
103C2×C4×C83—21no—
104C2×C4×C83—21no—
105C2×C4×C83—21no—
106C2×C4×C83—21no—
107C2×C4×C83—21no—
108C2×C4×C83—21no—
109C2×C4×C83—21no—
110C2×C4×C83—21no—
111C2×C4×C83—21no—
112C2×C4×C83—21no—
113C2×C4×C83—21no—
114C2×C4×C83—21no—
115C2×C4×C83—21no—
116C2×C4×C83—21no—
117C2×C4×C83—21no—
118C2×C4×C83—21no—
119C2×C4×C83C2×D412yes—
120C2×C4×C83C2×D412yes—
121C2×C4×C83C2×D412yes—
122C2×C4×C83#4112yes—
123C2×C4×C83#4112yes—
124C2×Dic83—21no—
125C2×Dic83—21no—
126C2×Dic83—21no—
127C2×Dic83—21no—
128C2×Dic83—21no—
129C2×Dic83—21no—
130C2×Dic83—21no—
131C2×Dic83—21no—
132C2×Dic83—21no—
133C2×Dic83—21no—
134C2×Dic83—21no—
135C2×Dic83—21no—
136C2×Dic83—21no—
137C2×Dic83—21no—
138C2×Dic83—21no—
139C2×Dic83—21no—
140C2×Dic83—21no—
141C2×Dic83—21no—
142C2×Dic83—21no—
143C2×Dic83—21no—
144C2×Dic83—21no—
145C22×C83C22×D411yescharacteristic
146C22×C83C2×#4111yescharacteristic
147C22×C83C2×#4111yescharacteristic
148C22×C83H(5,𝔽2)11yescharacteristic
149C22×C832−1+411yescharacteristic
150C22×C832−1+411yescharacteristic
151C22×C832−1+411yescharacteristic
152C2×D83—41no—
153C2×D83—41no—
154C2×D83—41no—
155C2×D83—41no—
156C2×D83—41no—
157C2×D83—41no—
158C2×D83—41no—
159C4×C83—22no—
160C4×C83—22no—
161C4×C83—22no—
162C4×C83—22no—
163C4×C83—22no—
164C4×C83—41no—
165C4×C83—41no—
166Dic83—41no—
167Dic83—41no—
168Dic83—41no—
169Dic83—41no—
170Dic83—41no—
171Dic83—41no—
172Dic83—41no—
173C2×C83#53611yescharacteristic
174C2×C83#54111yescharacteristic
175C2×C83#54211yescharacteristic
176C2×C83#54611yescharacteristic
177C2×C83#54811yescharacteristic
178C2×C83#55611yescharacteristic
179C2×C83#56311yescharacteristic
180D83—81no—
181#2716—831nomaximal, Sylow
182C83#271611yescharacteristic, complemented, Sylow
183C2×C4⋊Q8D8311yescharacteristic, complemented
184C2×#126—831no—
185#400—831no—
186#398—831no—
187#397—831no—
188#396—831no—
189#393—831no—
190#393—831no—
191#391—831no—
192#391—831no—
193#388—831no—
194#388—831no—
195#387—831no—
196#387—831no—
197#387—831no—
198C22×Q8C2×D8311yescharacteristic
199C22×Q8C2×D8311yescharacteristic
200C4⋊Q8—24no—
201#126—1664no—
202C2×C4⋊C4—831no—
203C2×C4⋊C4—831no—
204C2×C4⋊C4—831no—
205C2×C4⋊C4—831no—
206C2×C4⋊C4—831no—
207C2×C4⋊C4C2×D8311yescharacteristic
208C2×C4⋊C4C2×D8311yescharacteristic
209C2×C4⋊C4C2×D8311yescharacteristic
210C2×C4⋊C4C2×D8311yescharacteristic
211C2×C22⋊C4—831no—
212C2×C22⋊C4—831no—
213C2×C22⋊C4—831no—
214C2×C22⋊C4—831no—
215C2×C22⋊C4—831no—
216C2×C22⋊C4—831no—
217C2×C22⋊C4—831no—
218C2×C42—831no—
219C2×C42—831no—
220C2×C42C2×D8311yescharacteristic
221#95—831no—
222#95—831no—
223#95—831no—
224#95—831no—
225#95—831no—
226#95—831no—
227#95—831no—
228#95—831no—
229#95—831no—
230#95—831no—
231#95—831no—
232#95—831no—
233#95—831no—
234#95—831no—
235C24—831no—
236C2×Q8—22no—
237C2×Q8—22no—
238C2×Q8—22no—
239C2×Q8—22no—
240C2×Q8—41no—
241C2×Q8—41no—
242C22×C4—831no—
243C22×C4—831no—
244C22×C4—831no—
245C22×C4—831no—
246C22×C4—831no—
247C22×C4—831no—
248C22×C4—831no—
249C22×C4C22×D8311yescharacteristic
250C22×C4C22×D8311yescharacteristic
251C22×C4C22×D8311yescharacteristic
252C22×C4C22×D8311yescharacteristic
253C22×C4C22×D8311yescharacteristic
254C22×C4C22×D8311yescharacteristic
255C22×C4C22×D8311yescharacteristic
256C4⋊C4—22no—
257C4⋊C4—22no—
258C4⋊C4—22no—
259C4⋊C4—3321no—
260C4⋊C4—1662no—
261C4⋊C4—1662no—
262C4⋊C4—1662no—
263C4⋊C4—1662no—
264C4⋊C4C22⋊D8314yes—
265C22⋊C4—3321no—
266C22⋊C4—3321no—
267C22⋊C4—1662no—
268C22⋊C4—1662no—
269C22⋊C4—1662no—
270C22⋊C4—1662no—
271C22⋊C4—1662no—
272C42—22no—
273C42—3321no—
274C42—1662no—
275C23—1661no—
276C23—1661no—
277C23—1661no—
278C23—1661no—
279C23—1661no—
280C23—1661no—
281C23—1661no—
282C23C23×D8311yescharacteristic
283Q8—44no—
284Q8—44no—
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—21no—
301C2×C4—1661no—
302C2×C4—1661no—
303C2×C4—1661no—
304C2×C4—1661no—
305C2×C4—1661no—
306C2×C4—1661no—
307C2×C4—1661no—
308C2×C4—1661no—
309C2×C4—1661no—
310C2×C4—1661no—
311C2×C4—1661no—
312C2×C4—1661no—
313C2×C4—1661no—
314C2×C4—1661no—
315C2×C4—1661no—
316C2×C4—1661no—
317C2×C4—1661no—
318C2×C4—1661no—
319C2×C4—1661no—
320C2×C4—1661no—
321C2×C4—1661no—
322C2×C4C2×C22⋊D8312yes—
323C2×C4C2×C22⋊D8312yes—
324C2×C4C2×C22⋊D8312yes—
325C2×C4#1305058912yes—
326C2×C4#1305059112yes—
327C22—3321no—
328C22—3321no—
329C22—3321no—
330C22—3321no—
331C22—3321no—
332C22—3321no—
333C22—3321no—
334C22C22×C22⋊D8311yescharacteristic
335C22C2×#1305058911yescharacteristic
336C22C2×#1305059111yescharacteristic
337C22#1458183711yescharacteristic
338C22#1458184011yescharacteristic
339C22#1458184011yescharacteristic
340C22#1458184011yescharacteristic
341C4—22no—
342C4—22no—
343C4—22no—
344C4—22no—
345C4—22no—
346C4—41no—
347C4—41no—
348C4—3321no—
349C4—3321no—
350C4—3321no—
351C4—3321no—
352C4—3321no—
353C4—3321no—
354C4—3321no—
355C2—6641no—
356C2#1912827511yescharacteristic
357C2#1912831211yescharacteristic
358C2#1912832511yescharacteristic
359C2#1912834511yescharacteristic
360C2#1912836211yescharacteristic
361C2#1912842511yescharacteristic
362C2#1912845011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
FittingC2×C4⋊Q8×C83C2
prime radicalC23×D83C23
hypercenterC2×C4⋊Q8D83
hypocenterC83#2716
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ C2×C4⋊Q8

Complements

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

NH ≅ G/Ntype
2. #19127639C2semidirect
3. #19127631C2semidirect
8. #19127608C2semidirect
9. #19127608C2semidirect
12. #19127587C2semidirect
13. #19127585C2semidirect
14. #19127582C2semidirect
15. C2×C4⋊Q8×C83C2semidirect
182. C83#2716semidirect
183. C2×C4⋊Q8D83semidirect

Maximal subgroups

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

subgroupindex
C2×#145817872
#191276392
#191276312
#191276272
#191276232
#191276122
#191276082
#191275922
#191275912
#191275872
#191275852
#191275822
C2×C4⋊Q8×C832
#271683