Finite Group Databasebeta

← #24913836

Subgroups — #24913836

Order 10624.

592 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 592 conjugacy classes of subgroups — too many to draw. Only its 157 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×#14581771C212yesmaximal, complemented
2C2×#14581769C212yesmaximal, complemented
3#19127624C212yesmaximal, complemented
4#19127624C212yesmaximal, complemented
5#19127622C212yesmaximal, complemented
6#19127604C211yesmaximal, characteristic
7#19127598C212yesmaximal, complemented
8#19127581C211yesmaximal, characteristic
9#386×C83C211yesmaximal, characteristic
10C22×Dic166C2212yes—
11C23×Dic83C2211yescharacteristic
12#14581771—28no—
13#14581769—28no—
14C2×C4⋊Dic83C2212yes—
15C2×C4⋊Dic83C2212yes—
16C2×C4⋊Dic83C2211yescharacteristic
17C2×Dic83⋊C4C2212yes—
18C2×Dic83⋊C4C2212yes—
19C2×Dic83⋊C4C2212yes—
20C2×Dic83⋊C4C2212yes—
21C2×C22⋊Dic83C2212yes—
22C2×C22⋊Dic83C2212yes—
23C2×C22⋊Dic83C2212yes—
24C2×C22⋊Dic83C2211yescharacteristic
25C2×C4×Dic83C2211yescharacteristic
26C83⋊#95C2212yes—
27C83⋊#95C2212yes—
28C22×D4×C83C2211yescharacteristic
29C23×C4×C83C2211yescharacteristic
30C2×C22⋊C4×C83C2212yes—
31C2×C22⋊C4×C83C2211yescharacteristic
32#95×C83C2212yes—
33C2×Dic166—24no—
34C2×Dic166—24no—
35C2×Dic166—42no—
36C22×Dic83—24no—
37C22×Dic83—41no—
38C22×Dic83C2312yes—
39C22×Dic83C2312yes—
40C22×Dic83C2312yes—
41C22×Dic83C2312yes—
42C4⋊Dic83—24no—
43C4⋊Dic83—24no—
44C4⋊Dic83—41no—
45Dic83⋊C4—24no—
46Dic83⋊C4—24no—
47Dic83⋊C4—42no—
48Dic83⋊C4D418yescomplemented
49C22⋊Dic83—24no—
50C22⋊Dic83—24no—
51C22⋊Dic83—24no—
52C22⋊Dic83—41no—
53C4×Dic83—41no—
54C24×C83C2311yescharacteristic
55C24×C83C2311yescharacteristic
56C2×D4×C83—41no—
57C2×D4×C83—41no—
58C2×D4×C83Q814yescomplemented
59C22×C4×C83—24no—
60C22×C4×C83—41no—
61C22×C4×C83C2312yes—
62C22×C4×C83C2312yes—
63C22×C4×C83C2311yescharacteristic
64C22⋊C4×C83—24no—
65C22⋊C4×C83—41no—
66C23×C83—22no—
67C23×C83—22no—
68C23×C83—21no—
69C23×C83—21no—
70C23×C83—21no—
71C23×C83—21no—
72C23×C83—22no—
73C23×C83—22no—
74C23×C83C2×Q812yes—
75C23×C83C2×Q812yes—
76C23×C83C2411yescharacteristic
77D4×C83—44no—
78C2×C4×C83—22no—
79C2×C4×C83—22no—
80C2×C4×C83—21no—
81C2×C4×C83—21no—
82C2×C4×C83—22no—
83C2×C4×C83—22no—
84C2×C4×C83—22no—
85C2×C4×C83—44no—
86C2×C4×C83C2×D414yes—
87C2×C4×C83C2×Q812yes—
88C2×Dic83—22no—
89C2×Dic83—22no—
90C2×Dic83—22no—
91C2×Dic83—22no—
92C2×Dic83—22no—
93C2×Dic83—22no—
94C2×Dic83—22no—
95C2×Dic83—22no—
96C2×Dic83—22no—
97C2×Dic83—22no—
98C2×Dic83—44no—
99C2×Dic83C2×D414yes—
100C2×Dic83C2×D414yes—
101Dic166—48no—
102C22×C83—22no—
103C22×C83—22no—
104C22×C83—24no—
105C22×C83—24no—
106C22×C83—42no—
107C22×C83—42no—
108C22×C83—41no—
109C22×C83—41no—
110C22×C83—41no—
111C22×C83—41no—
112C22×C83C22×D412yes—
113C22×C83C22×Q811yescharacteristic
114C22×C83H(5,𝔽2)11yescharacteristic
115C22×C83H(5,𝔽2)11yescharacteristic
116C22×C832−1+412yes—
117C4×C83—22no—
118C4×C83—24no—
119C4×C83—42no—
120Dic83—24no—
121Dic83—24no—
122Dic83—42no—
123Dic83—42no—
124C2×C83—42no—
125C2×C83—42no—
126C2×C83#53712yes—
127C2×C83#54311yescharacteristic
128C2×C83D4211yescharacteristic
129C2×C83D4×Q812yes—
130C2×C83#56311yescharacteristic
131#2683—831nomaximal, Sylow
132C83#268311yescharacteristic, complemented, Sylow
133C2×C22⋊Q8—832no—
134C2×C22⋊Q8—832no—
135#396—832no—
136#396—832no—
137#395—832no—
138#390—831no—
139#389—832no—
140#386—831no—
141#386D8311yescharacteristic
142C22×Q8—832no—
143C22×D4C2×D8311yescharacteristic
144C23×C4—831no—
145C23×C4C2×D8311yescharacteristic
146C22⋊Q8—1668no—
147C22⋊Q8—1668no—
148C2×C4⋊C4—832no—
149C2×C4⋊C4—831no—
150C2×C4⋊C4—832no—
151C2×C4⋊C4—832no—
152C2×C4⋊C4—832no—
153C2×C4⋊C4—832no—
154C2×C4⋊C4—832no—
155C2×C22⋊C4—831no—
156C2×C22⋊C4—832no—
157C2×C22⋊C4—832no—
158C2×C22⋊C4—832no—
159C2×C22⋊C4C2×D8312yes—
160C2×C22⋊C4C2×D8311yescharacteristic
161C2×C42—831no—
162#95—832no—
163#95—832no—
164#95C2×D8312yes—
165C24C22×D8311yescharacteristic
166C24C22×D8311yescharacteristic
167C2×Q8—3322no—
168C2×Q8—1664no—
169C2×Q8—1664no—
170C2×D4—41no—
171C2×D4—41no—
172C2×D4Dic16614yescomplemented
173C22×C4—24no—
174C22×C4—41no—
175C22×C4—3321no—
176C22×C4—832no—
177C22×C4—832no—
178C22×C4—832no—
179C22×C4—832no—
180C22×C4—1664no—
181C22×C4C22×D8312yes—
182C22×C4C22×D8312yes—
183C22×C4C22×D8311yescharacteristic
184C4⋊C4—3322no—
185C4⋊C4—3321no—
186C4⋊C4—838no—
187C4⋊C4—1664no—
188C4⋊C4—1664no—
189C4⋊C4—1664no—
190C4⋊C4—1664no—
191C22⋊C4—24no—
192C22⋊C4—41no—
193C22⋊C4—3321no—
194C22⋊C4—1664no—
195C22⋊C4—1664no—
196C22⋊C4—1664no—
197C42—3321no—
198C23—22no—
199C23—22no—
200C23—21no—
201C23—21no—
202C23—21no—
203C23—21no—
204C23—22no—
205C23—22no—
206C23C2×Dic16612yes—
207C23C2×Dic16612yes—
208C23C23×D8311yescharacteristic
209Q8—3328no—
210D4—44no—
211C2×C4—22no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—44no—
219C2×C4—3324no—
220C2×C4—834no—
221C2×C4—834no—
222C2×C4—1662no—
223C2×C4—1662no—
224C2×C4—1662no—
225C2×C4—1662no—
226C2×C4—1662no—
227C2×C4—1662no—
228C2×C4—1662no—
229C2×C4—1662no—
230C2×C4—1662no—
231C2×C4—1662no—
232C2×C4D4×D8314yes—
233C2×C4C2×Dic16612yes—
234C22—22no—
235C22—22no—
236C22—24no—
237C22—24no—
238C22—41no—
239C22—41no—
240C22—41no—
241C22—41no—
242C22—42no—
243C22—42no—
244C22C2×D4×D8312yes—
245C22C22×Dic16611yescharacteristic
246C22#1458183711yescharacteristic
247C22#1458183711yescharacteristic
248C22#1458183912yes—
249C4—22no—
250C4—24no—
251C4—42no—
252C4—3322no—
253C4—3322no—
254C4—1664no—
255C4—1664no—
256C2—42no—
257C2—42no—
258C2#1912828012yes—
259C2#1912832711yescharacteristic
260C2#1912833311yescharacteristic
261C2D4×Dic16612yes—
262C2#1912845411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
Fitting#386×C83C2
prime radicalC22×D4×C83C22
hypercenter#386D83
hypocenterC83#2683
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
1. C2×#14581771C2semidirect
2. C2×#14581769C2semidirect
3. #19127624C2semidirect
4. #19127624C2semidirect
5. #19127622C2semidirect
7. #19127598C2semidirect
48. Dic83⋊C4D4semidirect
58. C2×D4×C83Q8semidirect
132. C83#2683semidirect
172. C2×D4Dic166semidirect

Maximal subgroups

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

subgroupindex
C2×#145817712
C2×#145817692
#191276242
#191276222
#191276042
#191275982
#191275812
#386×C832
#268383