Finite Group Databasebeta

← #24914481

Subgroups — #24914481

Order 10624.

528 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 149 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×#14581768C212yesmaximal, complemented
2#19127638C212yesmaximal, complemented
3#19127633C212yesmaximal, complemented
4#19127623C212yesmaximal, complemented
5#19127603C211yesmaximal, characteristic
6#19127599C211yesmaximal, characteristic, complemented
7#19127579C212yesmaximal
8#19127575C212yesmaximal, complemented
9#391×C83C211yesmaximal, characteristic, complemented
10C22×D332C2211yescharacteristic
11C22×C4×D83C2212yes—
12#14581768—28no—
13C2×C4⋊Dic83C2211yescharacteristic
14C2×C4⋊Dic83C2212yes—
15C2×C4⋊Dic83C2211yescharacteristic
16C2×C4⋊Dic83C2211yescharacteristic
17C2×C4⋊Dic83C2212yes—
18C2×Dic83⋊C4C2212yes—
19C2×Dic83⋊C4C2212yes—
20C2×(C2×D83)⋊C4C2212yes—
21C2×(C2×D83)⋊C4C2212yes—
22C2×(C2×D83)⋊C4C2212yes—
23C2×(C2×D83)⋊C4C2212yes—
24C2×(C2×D83)⋊C4C2212yes—
25C83⋊#95C2212yes—
26C83⋊#95C2212yes—
27C22×Q8×C83C2211yescharacteristic
28C2×C4⋊C4×C83C2211yescharacteristic
29C2×C42×C83C2211yescharacteristic
30#95×C83C2212yes—
31#95×C83C2212yes—
32C23×D83C2312yes—
33C2×D332—41no—
34C2×D332—41no—
35C2×D332Q814yescomplemented
36C22×Dic83C2312yes—
37C22×Dic83C2312yes—
38C22×Dic83C2312yes—
39C2×C4×D83—24no—
40C2×C4×D83—42no—
41C2×C4×D83—42no—
42C4⋊Dic83—24no—
43C4⋊Dic83—22no—
44C4⋊Dic83—42no—
45C4⋊Dic83—41no—
46C4⋊Dic83—41no—
47Dic83⋊C4—24no—
48Dic83⋊C4—24no—
49(C2×D83)⋊C4—24no—
50(C2×D83)⋊C4—24no—
51(C2×D83)⋊C4—24no—
52(C2×D83)⋊C4—42no—
53(C2×D83)⋊C4—42no—
54C2×Q8×C83—41no—
55C2×Q8×C83—41no—
56C2×Q8×C83D414yescomplemented
57C22×C4×C83C2312yes—
58C22×C4×C83C2312yes—
59C22×C4×C83C2311yescharacteristic
60C22×C4×C83C2311yescharacteristic
61C22×C4×C83C2311yescharacteristic
62C4⋊C4×C83—41no—
63C42×C83—41no—
64C23×C83C2411yescharacteristic
65C22×D83—22no—
66C22×D83—22no—
67C22×D83—22no—
68C22×D83—22no—
69C22×D83—22no—
70C22×D83—22no—
71C22×D83C2×Q814yes—
72Q8×C83—44no—
73C2×C4×C83—21no—
74C2×C4×C83—21no—
75C2×C4×C83—21no—
76C2×C4×C83—21no—
77C2×C4×C83—22no—
78C2×C4×C83—22no—
79C2×C4×C83—21no—
80C2×C4×C83—21no—
81C2×C4×C83—22no—
82C2×C4×C83—22no—
83C2×C4×C83—22no—
84C2×C4×C83C2×D412yes—
85C2×C4×C83C2×D414yes—
86C2×C4×C83C2×Q812yes—
87C2×C4×C83#4112yes—
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—
97D332—44no—
98C4×D83—48no—
99C22×C83C22×D411yescharacteristic
100C22×C83C22×Q811yescharacteristic
101C22×C83C2×#4111yescharacteristic
102C22×C83H(5,𝔽2)11yescharacteristic
103C22×C83H(5,𝔽2)11yescharacteristic
104C22×C83H(5,𝔽2)11yescharacteristic
105C22×C832−1+411yescharacteristic
106C2×D83—24no—
107C2×D83—24no—
108C2×D83—24no—
109C2×D83—42no—
110C2×D83—42no—
111C2×D83—42no—
112C2×D83—42no—
113C4×C83—24no—
114C4×C83—22no—
115C4×C83—22no—
116C4×C83—22no—
117C4×C83—42no—
118Dic83—42no—
119Dic83—42no—
120Dic83—42no—
121C2×C83#53411yescharacteristic
122C2×C83#53911yescharacteristic
123C2×C83#54311yescharacteristic
124C2×C83D4×Q811yescharacteristic
125C2×C83#55011yescharacteristic
126C2×C83#55411yescharacteristic
127C2×C83#56211yescharacteristic
128D83—44no—
129#2732—831nomaximal, Sylow
130C83#273211yescharacteristic, complemented, Sylow
131C2×C22⋊Q8—832no—
132#399—832no—
133#398—832no—
134#396—832no—
135#391D8311yescharacteristic, complemented
136#390—831no—
137#389—831no—
138#386—832no—
139#385—832no—
140C22×Q8C2×D8311yescharacteristic
141C22×D4—831no—
142C23×C4—832no—
143C22⋊Q8—1668no—
144C2×C4⋊C4—831no—
145C2×C4⋊C4—831no—
146C2×C4⋊C4—832no—
147C2×C4⋊C4—832no—
148C2×C4⋊C4—832no—
149C2×C4⋊C4—832no—
150C2×C4⋊C4—831no—
151C2×C4⋊C4C2×D8311yescharacteristic
152C2×C22⋊C4—832no—
153C2×C22⋊C4—832no—
154C2×C22⋊C4—832no—
155C2×C22⋊C4—832no—
156C2×C22⋊C4—832no—
157C2×C42C2×D8311yescharacteristic
158#95—832no—
159#95—832no—
160#95C2×D8312yes—
161#95C2×D8312yes—
162C24—832no—
163C2×Q8—41no—
164C2×Q8—41no—
165C2×Q8D33214yescomplemented
166C2×D4—3321no—
167C2×D4—3321no—
168C2×D4—834no—
169C22×C4—3322no—
170C22×C4—3322no—
171C22×C4—832no—
172C22×C4—832no—
173C22×C4—832no—
174C22×C4—1664no—
175C22×C4C22×D8312yes—
176C22×C4C22×D8312yes—
177C22×C4C22×D8311yescharacteristic
178C22×C4C22×D8311yescharacteristic
179C22×C4C22×D8311yescharacteristic
180C4⋊C4—41no—
181C4⋊C4—3321no—
182C4⋊C4—3321no—
183C4⋊C4—3322no—
184C4⋊C4—1664no—
185C4⋊C4—1664no—
186C4⋊C4—1664no—
187C4⋊C4—1662no—
188C22⋊C4—3322no—
189C22⋊C4—3322no—
190C22⋊C4—1664no—
191C22⋊C4—1664no—
192C22⋊C4—1664no—
193C42—41no—
194C23—834no—
195C23—1662no—
196C23—1662no—
197C23—1662no—
198C23—1662no—
199C23—1662no—
200C23—1662no—
201C23C23×D8311yescharacteristic
202Q8—44no—
203D4—3324no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—3328no—
216C2×C4—1662no—
217C2×C4—1662no—
218C2×C4—1662no—
219C2×C4—1662no—
220C2×C4—1662no—
221C2×C4—1662no—
222C2×C4—1662no—
223C2×C4—1662no—
224C2×C4—1662no—
225C2×C4C2×D33212yes—
226C2×C4C2×D33214yes—
227C2×C4Q8×D8312yes—
228C2×C4#1305059112yes—
229C22—3322no—
230C22—3322no—
231C22—3322no—
232C22—3322no—
233C22—1664no—
234C22—1664no—
235C22—1664no—
236C22C22×D33211yescharacteristic
237C22C2×Q8×D8311yescharacteristic
238C22C2×#1305059111yescharacteristic
239C22#1458183711yescharacteristic
240C22#1458183711yescharacteristic
241C22#1458183811yescharacteristic
242C22#1458183911yescharacteristic
243C4—22no—
244C4—24no—
245C4—22no—
246C4—22no—
247C4—42no—
248C4—3322no—
249C4—3322no—
250C4—3322no—
251C2—3324no—
252C2#1912826311yescharacteristic
253C2#1912830011yescharacteristic
254C2#1912832611yescharacteristic
255C2Q8×D33211yescharacteristic
256C2#1912837311yescharacteristic
257C2#1912841211yescharacteristic
258C2#1912844511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
Fitting#391×C83C2
prime radicalC22×D332C22
hypercenter#391D83
hypocenterC83#2732
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ #391

Complements

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

NH ≅ G/Ntype
1. C2×#14581768C2semidirect
2. #19127638C2semidirect
3. #19127633C2semidirect
4. #19127623C2semidirect
6. #19127599C2semidirect
8. #19127575C2semidirect
9. #391×C83C2semidirect
35. C2×D332Q8semidirect
56. C2×Q8×C83D4semidirect
130. C83#2732semidirect
135. #391D83semidirect
165. C2×Q8D332semidirect

Maximal subgroups

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

subgroupindex
C2×#145817682
#191276382
#191276332
#191276232
#191276032
#191275992
#191275792
#191275752
#391×C832
#273283