Finite Group Databasebeta

← #24914337

Subgroups — #24914337

Order 10624.

472 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14581782C211yesmaximal, characteristic, complemented
2#19127639C211yesmaximal, characteristic, complemented
3#19127637C211yesmaximal, characteristic
4#19127632C211yesmaximal, characteristic, complemented
5#19127624C211yesmaximal, characteristic
6#19127620C211yesmaximal, characteristic, complemented
7#19127620C211yesmaximal, characteristic, complemented
8#19127597C211yesmaximal, characteristic, complemented
9#19127595C211yesmaximal, characteristic
10#19127589C211yesmaximal, characteristic
11#19127587C211yesmaximal, characteristic, complemented
12#19127574C211yesmaximal, characteristic
13#19127574C211yesmaximal, characteristic
14#19127570C211yesmaximal, characteristic, complemented
15C2×C22⋊Q8×C83C211yesmaximal, characteristic
16#14581782—24no—
17C2×C4⋊Dic83C2211yescharacteristic
18C2×C4⋊Dic83C2211yescharacteristic
19C2×Dic83⋊C4C2211yescharacteristic
20C2×Dic83⋊C4C2211yescharacteristic
21C2×Dic83⋊C4C2211yescharacteristic
22C2×Dic83⋊C4C2211yescharacteristic
23C2×Dic83⋊C4C2211yescharacteristic
24C2×Dic83⋊C4C2211yescharacteristic
25C2×Dic83⋊C4C2211yescharacteristic
26C2×Dic83⋊C4C2211yescharacteristic
27C2×Dic83⋊C4C2211yescharacteristic
28C2×Dic83⋊C4C2211yescharacteristic
29C2×C22⋊Dic83C2211yescharacteristic
30C2×C22⋊Dic83C2211yescharacteristic
31C2×C22⋊Dic83C2211yescharacteristic
32C2×C22⋊Dic83C2211yescharacteristic
33C2×C4×Dic83C2211yescharacteristic
34C2×C4×Dic83C2211yescharacteristic
35C83⋊#95C2211yescharacteristic
36C83⋊#95C2211yescharacteristic
37C83⋊#95C2211yescharacteristic
38C83⋊#95C2211yescharacteristic
39C83⋊#95C2211yescharacteristic
40C83⋊#95C2211yescharacteristic
41C83⋊#95C2211yescharacteristic
42C83⋊#95C2211yescharacteristic
43C83⋊#95C2211yescharacteristic
44C83⋊#95C2211yescharacteristic
45C22×Q8×C83C2211yescharacteristic
46C23×C4×C83C2211yescharacteristic
47C22⋊Q8×C83—24no—
48C2×C4⋊C4×C83C2211yescharacteristic
49C2×C4⋊C4×C83C2211yescharacteristic
50C2×C4⋊C4×C83C2211yescharacteristic
51C2×C22⋊C4×C83C2211yescharacteristic
52C2×C22⋊C4×C83C2211yescharacteristic
53C22×Dic83C2311yescharacteristic
54C22×Dic83C2311yescharacteristic
55C22×Dic83C2311yescharacteristic
56C22×Dic83C2311yescharacteristic
57C22×Dic83C2311yescharacteristic
58C22×Dic83C2311yescharacteristic
59C22×Dic83C2311yescharacteristic
60C22×Dic83C2311yescharacteristic
61C4⋊Dic83—22no—
62C4⋊Dic83—22no—
63Dic83⋊C4—22no—
64Dic83⋊C4—22no—
65Dic83⋊C4—22no—
66Dic83⋊C4—22no—
67Dic83⋊C4—41no—
68Dic83⋊C4—41no—
69Dic83⋊C4—41no—
70Dic83⋊C4—41no—
71Dic83⋊C4—41no—
72Dic83⋊C4—41no—
73C22⋊Dic83—22no—
74C22⋊Dic83—41no—
75C22⋊Dic83—41no—
76C22⋊Dic83Q814yescomplemented
77C4×Dic83—22no—
78C4×Dic83—22no—
79C24×C83C2311yescharacteristic
80C2×Q8×C83—22no—
81C2×Q8×C83—41no—
82C2×Q8×C83—41no—
83C22×C4×C83—22no—
84C22×C4×C83—41no—
85C22×C4×C83—41no—
86C22×C4×C83C2311yescharacteristic
87C22×C4×C83C2311yescharacteristic
88C22×C4×C83C2311yescharacteristic
89C22×C4×C83C2311yescharacteristic
90C22×C4×C83C2311yescharacteristic
91C22×C4×C83C2311yescharacteristic
92C4⋊C4×C83—22no—
93C4⋊C4×C83—22no—
94C4⋊C4×C83D414yes—
95C22⋊C4×C83—22no—
96C22⋊C4×C83—22no—
97C23×C83—21no—
98C23×C83—21no—
99C23×C83—21no—
100C23×C83—21no—
101C23×C83—21no—
102C23×C83—21no—
103C23×C83C2×Q812yes—
104C23×C83C2411yescharacteristic
105Q8×C83—44no—
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×C83—21no—
120C2×C4×C83—21no—
121C2×C4×C83—44no—
122C2×C4×C83C2×D412yes—
123C2×C4×C83C2×D412yes—
124C2×C4×C83C2×D412yes—
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—
145C2×Dic83—21no—
146C2×Dic83C2×Q812yes—
147C2×Dic83C2×Q812yes—
148C2×Dic83#4112yes—
149C22×C83—22no—
150C22×C83—22no—
151C22×C83—22no—
152C22×C83—41no—
153C22×C83—41no—
154C22×C83—41no—
155C22×C83—41no—
156C22×C83C22×D411yescharacteristic
157C22×C83C22×Q811yescharacteristic
158C22×C83C2×#4111yescharacteristic
159C22×C83H(5,𝔽2)11yescharacteristic
160C22×C83H(5,𝔽2)11yescharacteristic
161C22×C832−1+411yescharacteristic
162C22×C832−1+411yescharacteristic
163C4×C83—22no—
164C4×C83—22no—
165C4×C83—22no—
166C4×C83—41no—
167C4×C83—41no—
168C4×C83—41no—
169Dic83—22no—
170Dic83—22no—
171Dic83—22no—
172Dic83—41no—
173Dic83—41no—
174Dic83—41no—
175Dic83—41no—
176Dic83—41no—
177C2×C83—42no—
178C2×C83#53511yescharacteristic
179C2×C83#53911yescharacteristic
180C2×C83#54411yescharacteristic
181C2×C83#54711yescharacteristic
182C2×C83D4×Q811yescharacteristic
183C2×C83#55411yescharacteristic
184C2×C83#56311yescharacteristic
185#2720—831nomaximal, Sylow
186C83#272011yescharacteristic, complemented, Sylow
187C2×#125—831no—
188C2×C22⋊Q8D8311yescharacteristic
189#400—831no—
190#399—831no—
191#398—831no—
192#396—831no—
193#395—831no—
194#395—831no—
195#389—831no—
196#388—831no—
197#388—831no—
198#387—831no—
199#385—831no—
200#385—831no—
201#384—831no—
202C22×Q8C2×D8311yescharacteristic
203C23×C4C2×D8311yescharacteristic
204#125—1664no—
205C22⋊Q8—24no—
206C2×C4⋊C4—831no—
207C2×C4⋊C4—831no—
208C2×C4⋊C4—831no—
209C2×C4⋊C4—831no—
210C2×C4⋊C4—831no—
211C2×C4⋊C4—831no—
212C2×C4⋊C4—831no—
213C2×C4⋊C4—831no—
214C2×C4⋊C4—831no—
215C2×C4⋊C4—831no—
216C2×C4⋊C4—831no—
217C2×C4⋊C4—831no—
218C2×C4⋊C4C2×D8311yescharacteristic
219C2×C4⋊C4C2×D8311yescharacteristic
220C2×C4⋊C4C2×D8311yescharacteristic
221C2×C22⋊C4—831no—
222C2×C22⋊C4—831no—
223C2×C22⋊C4—831no—
224C2×C22⋊C4—831no—
225C2×C22⋊C4C2×D8311yescharacteristic
226C2×C22⋊C4C2×D8311yescharacteristic
227C2×C42—831no—
228C2×C42—831no—
229#95—831no—
230#95—831no—
231#95—831no—
232#95—831no—
233#95—831no—
234#95—831no—
235#95—831no—
236#95—831no—
237#95—831no—
238#95—831no—
239C24C22×D8311yescharacteristic
240C2×Q8—22no—
241C2×Q8—41no—
242C2×Q8—41no—
243C22×C4—22no—
244C22×C4—41no—
245C22×C4—41no—
246C22×C4—831no—
247C22×C4—831no—
248C22×C4—831no—
249C22×C4—831no—
250C22×C4—831no—
251C22×C4—831no—
252C22×C4—831no—
253C22×C4—831no—
254C22×C4C22×D8311yescharacteristic
255C22×C4C22×D8311yescharacteristic
256C22×C4C22×D8311yescharacteristic
257C22×C4C22×D8311yescharacteristic
258C22×C4C22×D8311yescharacteristic
259C22×C4C22×D8311yescharacteristic
260C4⋊C4—22no—
261C4⋊C4—22no—
262C4⋊C4—3321no—
263C4⋊C4—3321no—
264C4⋊C4—3321no—
265C4⋊C4—3321no—
266C4⋊C4—3321no—
267C4⋊C4—3321no—
268C4⋊C4—1662no—
269C4⋊C4—1662no—
270C4⋊C4—1662no—
271C4⋊C4—1662no—
272C4⋊C4—1662no—
273C4⋊C4—1662no—
274C4⋊C4C22⋊D8314yes—
275C22⋊C4—22no—
276C22⋊C4—22no—
277C22⋊C4—3321no—
278C22⋊C4—3321no—
279C22⋊C4—834no—
280C22⋊C4—1662no—
281C42—1662no—
282C42—1662no—
283C23—21no—
284C23—21no—
285C23—21no—
286C23—21no—
287C23—21no—
288C23—21no—
289C23Q8×D8312yes—
290C23C23×D8311yescharacteristic
291Q8—44no—
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—21no—
307C2×C4—44no—
308C2×C4—832no—
309C2×C4—832no—
310C2×C4—832no—
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×C4—1661no—
323C2×C4—1661no—
324C2×C4—1661no—
325C2×C4—1661no—
326C2×C4—1661no—
327C2×C4—1661no—
328C2×C4—1661no—
329C2×C4—1661no—
330C2×C4—1661no—
331C2×C4—1661no—
332C2×C4C2×C22⋊D8312yes—
333C2×C4C2×C22⋊D8312yes—
334C2×C4C2×C22⋊D8312yes—
335C22—22no—
336C22—22no—
337C22—22no—
338C22—41no—
339C22—41no—
340C22—41no—
341C22—41no—
342C22C22×C22⋊D8311yescharacteristic
343C22C2×Q8×D8311yescharacteristic
344C22#41×D8311yescharacteristic
345C22#1458183711yescharacteristic
346C22#1458183711yescharacteristic
347C22#1458183911yescharacteristic
348C22#1458184011yescharacteristic
349C4—22no—
350C4—22no—
351C4—22no—
352C4—41no—
353C4—41no—
354C4—41no—
355C4—3321no—
356C4—3321no—
357C4—3321no—
358C4—3321no—
359C4—3321no—
360C4—1662no—
361C4—1662no—
362C4—1662no—
363C2—42no—
364C2#1912826811yescharacteristic
365C2#1912829711yescharacteristic
366C2#1912833011yescharacteristic
367C2#1912835011yescharacteristic
368C2Q8×C22⋊D8311yescharacteristic
369C2#1912840511yescharacteristic
370C2#1912845211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
FittingC2×C22⋊Q8×C83C2
prime radicalC24×C83C23
hypercenterC2×C22⋊Q8D83
hypocenterC83#2720
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C2×#14581782C2semidirect
2. #19127639C2semidirect
4. #19127632C2semidirect
6. #19127620C2semidirect
7. #19127620C2semidirect
8. #19127597C2semidirect
11. #19127587C2semidirect
14. #19127570C2semidirect
76. C22⋊Dic83Q8semidirect
186. C83#2720semidirect

Maximal subgroups

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

subgroupindex
C2×#145817822
#191276392
#191276372
#191276322
#191276242
#191276202
#191275972
#191275952
#191275892
#191275872
#191275742
#191275702
C2×C22⋊Q8×C832
#272083