Finite Group Databasebeta

← #24912411

Subgroups — #24912411

Order 10624.

676 conjugacy classes of subgroups; only the 171 normal subgroups are shown.

normal subgroups only

This group has 676 conjugacy classes of subgroups — too many to draw. Only its 171 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14581791C211yesmaximal, characteristic, complemented
2C2×#14581770C212yesmaximal, complemented
3C2×#14581769C212yesmaximal, complemented
4C2×(C22⋊D83)⋊C4C211yesmaximal, characteristic
5#19127638C211yesmaximal, characteristic, complemented
6#19127637C211yesmaximal, characteristic, complemented
7#19127636C211yesmaximal, characteristic, complemented
8#19127601C212yesmaximal
9#19127570C212yesmaximal, complemented
10#19127552C211yesmaximal, characteristic, complemented
11#399×C83C211yesmaximal, characteristic, complemented
12C22×Dic166C2212yes—
13C22×C22⋊D83C2211yescharacteristic
14C23×Dic83C2211yescharacteristic
15C22×C4×D83C2211yescharacteristic
16#14581791—24no—
17#14581770—28no—
18#14581769—28no—
19(C22⋊D83)⋊C4—24no—
20C2×C4⋊Dic83C2212yes—
21C2×C4⋊Dic83C2211yescharacteristic
22C2×Dic83⋊C4C2211yescharacteristic
23C2×Dic83⋊C4C2212yes—
24C2×Dic83⋊C4C2212yes—
25C2×Dic83⋊C4C2212yes—
26C2×Dic83⋊C4C2212yes—
27C2×C22⋊Dic83C2212yes—
28C2×(C2×D83)⋊C4C2212yes—
29C2×(C2×D83)⋊C4C2212yes—
30C2×(C2×D83)⋊C4C2211yescharacteristic
31C2×C4×Dic83C2212yes—
32C2×C4×Dic83C2211yescharacteristic
33C83⋊#95C2211yescharacteristic
34C2×C4⋊C4×C83C2212yes—
35C2×C4⋊C4×C83C2211yescharacteristic
36C2×C22⋊C4×C83C2212yes—
37C2×C22⋊C4×C83C2211yescharacteristic
38#95×C83C2211yescharacteristic
39C23×D83C2311yescharacteristic
40C2×Dic166—24no—
41C2×Dic166—24no—
42C2×Dic166—24no—
43C2×C22⋊D83—22no—
44C2×C22⋊D83—22no—
45C2×C22⋊D83Q814yescomplemented
46C22×Dic83—24no—
47C22×Dic83—22no—
48C22×Dic83C2311yescharacteristic
49C22×Dic83C2312yes—
50C22×Dic83C2312yes—
51C22×Dic83C2311yescharacteristic
52C22×Dic83C2311yescharacteristic
53C2×C4×D83—24no—
54C2×C4×D83—22no—
55C4⋊Dic83—24no—
56C4⋊Dic83—22no—
57Dic83⋊C4—24no—
58Dic83⋊C4—24no—
59Dic83⋊C4—24no—
60Dic83⋊C4—22no—
61Dic83⋊C4D418yescomplemented
62C22⋊Dic83—24no—
63(C2×D83)⋊C4—24no—
64(C2×D83)⋊C4—22no—
65(C2×D83)⋊C4—24no—
66C4×Dic83—24no—
67C4×Dic83D414yes—
68C24×C83C2311yescharacteristic
69C22×C4×C83C2311yescharacteristic
70C22×C4×C83C2312yes—
71C22×C4×C83C2311yescharacteristic
72C22×C4×C83C2312yes—
73C4⋊C4×C83—24no—
74C4⋊C4×C83—22no—
75C22⋊C4×C83—24no—
76C22⋊C4×C83—22no—
77C23×C83—22no—
78C23×C83—21no—
79C23×C83—21no—
80C23×C83—22no—
81C23×C83C2×Q812yes—
82C23×C83C2411yescharacteristic
83C22×D83—22no—
84C22×D83—21no—
85C22×D83—21no—
86C22×D83—22no—
87C22×D83C2×Q812yes—
88C2×C4×C83—22no—
89C2×C4×C83—22no—
90C2×C4×C83—22no—
91C2×C4×C83—21no—
92C2×C4×C83—22no—
93C2×C4×C83—22no—
94C2×C4×C83—22no—
95C2×C4×C83—22no—
96C2×C4×C83C2×D412yes—
97C2×C4×C83C2×D414yes—
98C22⋊D83—28no—
99C2×Dic83—22no—
100C2×Dic83—22no—
101C2×Dic83—21no—
102C2×Dic83—22no—
103C2×Dic83—22no—
104C2×Dic83—22no—
105C2×Dic83—22no—
106C2×Dic83—44no—
107C2×Dic83C2×D414yes—
108C2×Dic83C2×D412yes—
109C2×Dic83C2×D412yes—
110C2×Dic83C2×D414yes—
111C2×Dic83C2×Q812yes—
112C2×Dic83#4112yes—
113C4×D83—44no—
114Dic166—48no—
115C22×C83—22no—
116C22×C83—24no—
117C22×C83—41no—
118C22×C83—41no—
119C22×C83—42no—
120C22×C83C22×D411yescharacteristic
121C22×C83C22×D412yes—
122C22×C83C22×Q811yescharacteristic
123C22×C83C2×#4111yescharacteristic
124C22×C832−1+412yes—
125C2×D83—22no—
126C2×D83—24no—
127C2×D83—41no—
128C2×D83—41no—
129C2×D83—42no—
130C4×C83—22no—
131C4×C83—24no—
132C4×C83—42no—
133C4×C83—41no—
134Dic83—24no—
135Dic83—42no—
136Dic83—41no—
137Dic83C22⋊Q814yescomplemented
138Dic83#12714yes—
139C2×C83—42no—
140C2×C83C2×C22⋊Q811yescharacteristic
141C2×C83C2×#12711yescharacteristic
142C2×C83#53611yescharacteristic
143C2×C83#54712yes—
144C2×C83D4×Q812yes—
145D83—42no—
146#2542—831nomaximal, Sylow
147C83#254211yescharacteristic, complemented, Sylow
148C2×C4⋊Q8—831no—
149C2×C22⋊Q8—832no—
150C2×C22⋊Q8—832no—
151C2×C4×D4—831no—
152#399—831no—
153#399—831no—
154#399—831no—
155#399D8311yescharacteristic, complemented
156#390—832no—
157#384—832no—
158C4×C4⋊C4—831no—
159C22×Q8—832no—
160C22×D4—831no—
161C23×C4—831no—
162C23×C4—831no—
163C4⋊Q8—1664no—
164C22⋊Q8—1668no—
165C22⋊Q8—1668no—
166C4×D4—1664no—
167C2×C4⋊C4—832no—
168C2×C4⋊C4—831no—
169C2×C4⋊C4—832no—
170C2×C4⋊C4—832no—
171C2×C4⋊C4—832no—
172C2×C4⋊C4—832no—
173C2×C4⋊C4—831no—
174C2×C4⋊C4C2×D8312yes—
175C2×C4⋊C4C2×D8311yescharacteristic
176C2×C22⋊C4—832no—
177C2×C22⋊C4—831no—
178C2×C22⋊C4—832no—
179C2×C22⋊C4—832no—
180C2×C22⋊C4C2×D8311yescharacteristic
181C2×C22⋊C4C2×D8312yes—
182C2×C42—831no—
183C2×C42—832no—
184#95—831no—
185#95C2×D8311yescharacteristic
186C24—831no—
187C24C22×D8311yescharacteristic
188C2×Q8—1664no—
189C2×Q8—1664no—
190C2×Q8—1664no—
191C2×D4—834no—
192C2×D4—1662no—
193C2×D4—1662no—
194C22×C4—831no—
195C22×C4—831no—
196C22×C4—832no—
197C22×C4—832no—
198C22×C4—831no—
199C22×C4—1662no—
200C22×C4—1664no—
201C22×C4—1662no—
202C22×C4—1664no—
203C22×C4C22×D8312yes—
204C22×C4C22×D8311yescharacteristic
205C22×C4C22×D8311yescharacteristic
206C22×C4C22×D8312yes—
207C4⋊C4—24no—
208C4⋊C4—22no—
209C4⋊C4—838no—
210C4⋊C4—1662no—
211C4⋊C4—1664no—
212C4⋊C4—1664no—
213C4⋊C4—1664no—
214C4⋊C4—1664no—
215C4⋊C4—1662no—
216C22⋊C4—22no—
217C22⋊C4—24no—
218C22⋊C4—1664no—
219C22⋊C4—1664no—
220C22⋊C4—1662no—
221C22⋊C4—1664no—
222C42—834no—
223C42—1664no—
224C23—22no—
225C23—21no—
226C23—21no—
227C23—22no—
228C23—832no—
229C23—1662no—
230C23—1661no—
231C23—1661no—
232C23—1662no—
233C23Q8×D8312yes—
234C23C23×D8311yescharacteristic
235Q8—3328no—
236D4—1668no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—21no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—3324no—
246C2×C4—3324no—
247C2×C4—832no—
248C2×C4—832no—
249C2×C4—834no—
250C2×C4—832no—
251C2×C4—832no—
252C2×C4—834no—
253C2×C4—1661no—
254C2×C4—1662no—
255C2×C4—1662no—
256C2×C4—1662no—
257C2×C4—1662no—
258C2×C4—1662no—
259C2×C4—1662no—
260C2×C4D4×D8312yes—
261C2×C4D4×D8314yes—
262C22—24no—
263C22—22no—
264C22—42no—
265C22—41no—
266C22—41no—
267C22—3322no—
268C22—3321no—
269C22—3321no—
270C22—1664no—
271C22—1662no—
272C22C2×D4×D8312yes—
273C22C2×D4×D8311yescharacteristic
274C22C2×Q8×D8311yescharacteristic
275C22#41×D8311yescharacteristic
276C22#1458183912yes—
277C4—24no—
278C4—22no—
279C4—41no—
280C4—42no—
281C4—3321no—
282C4—3322no—
283C4—834no—
284C4—834no—
285C4—1664no—
286C2—42no—
287C2—3322no—
288C2C22⋊Q8×D8311yescharacteristic
289C2#127×D8311yescharacteristic
290C2#1912827311yescharacteristic
291C2#1912834912yes—
292C2#1912836412yes—
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
Fitting#399×C83C2
prime radicalC22×C22⋊D83C22
hypercenter#399D83
hypocenterC83#2542
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ #399

Complements

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

NH ≅ G/Ntype
1. C2×#14581791C2semidirect
2. C2×#14581770C2semidirect
3. C2×#14581769C2semidirect
5. #19127638C2semidirect
6. #19127637C2semidirect
7. #19127636C2semidirect
9. #19127570C2semidirect
10. #19127552C2semidirect
11. #399×C83C2semidirect
45. C2×C22⋊D83Q8semidirect
61. Dic83⋊C4D4semidirect
137. Dic83C22⋊Q8semidirect
147. C83#2542semidirect
155. #399D83semidirect

Maximal subgroups

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

subgroupindex
C2×#145817912
C2×#145817702
C2×#145817692
C2×(C22⋊D83)⋊C42
#191276382
#191276372
#191276362
#191276012
#191275702
#191275522
#399×C832
#254283