Finite Group Databasebeta

← #24918851

Subgroups — #24918851

Order 10624.

386 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 125 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
1#19128258C211yesmaximal, characteristic, complemented
2#19128251C211yesmaximal, characteristic
3#19127997C212yesmaximal, complemented
4#19127982C212yesmaximal, complemented
5#19127947C214yesmaximal, complemented
6#19127907C212yesmaximal, complemented
7#19127902C212yesmaximal, complemented
8#423×C83C211yesmaximal, characteristic, complemented
9C2×#13050590—21no—
10C2×#13050579C2214yes—
11#14581792C2211yescharacteristic
12#14581792C2211yescharacteristic
13#14581792C2211yescharacteristic
14#14581789C2211yescharacteristic
15#14581784—21no—
16#14581780—21no—
17#14581771—21no—
18#14581771C2212yes—
19#14581766—21no—
20#14581766C2212yes—
21C4×Dic166—21no—
22C4×Dic166C2212yes—
23C4×D332—21no—
24C4×D332C2212yes—
25#14581751—21no—
26#14581726C2214yes—
27#14581717C2214yes—
28#14581714C2214yes—
29C2×M16×C83C2212yes—
30C2×C42×C83C2211yescharacteristic
31C4⋊C8×C83C2212yes—
32C4⋊C8×C83C2212yes—
33#13050590—44no—
34C2×Dic166—21no—
35C2×Dic166—21no—
36C2×Dic166C2312yes—
37C2×D332—21no—
38C2×D332—21no—
39C2×D332C2312yes—
40C2×C22⋊D83—22no—
41C2×C4×D83—22no—
42#13050579—44no—
43C4⋊Dic83—21no—
44C4⋊Dic83—21no—
45C4⋊Dic83—21no—
46C4⋊Dic83—21no—
47C4⋊Dic83C2312yes—
48C4⋊Dic83C2312yes—
49Dic83⋊C4—22no—
50Dic83⋊C4—22no—
51C22⋊Dic83—22no—
52(C2×D83)⋊C4—22no—
53(C2×D83)⋊C4—22no—
54C4×Dic83—22no—
55C22×C4×C83D411yescharacteristic
56C22×C4×C83D411yescharacteristic
57C22×C4×C83C2311yescharacteristic
58M16×C83—42no—
59C2×C8×C83C2314yes—
60C42×C83D411yescharacteristic
61C42×C83D411yescharacteristic
62C42×C83C2311yescharacteristic
63C42×C83C2311yescharacteristic
64C23×C83C2×D411yescharacteristic
65C22×D83—22no—
66C2×C4×C83—22no—
67C2×C4×C83—22no—
68C2×C4×C83—21no—
69C2×C4×C83—21no—
70C2×C4×C83C2×D411yescharacteristic
71C2×C4×C83C2×D411yescharacteristic
72C2×C4×C83C2×D411yescharacteristic
73C2×C4×C83C2×D411yescharacteristic
74C2×C4×C83C2×D411yescharacteristic
75C2×C4×C83C2411yescharacteristic
76C22⋊D83—44no—
77C2×Dic83—22no—
78C2×Dic83—22no—
79C2×Dic83—22no—
80D332—22no—
81D332—22no—
82D332—42no—
83D332—42no—
84C4×D83—44no—
85Dic166—22no—
86Dic166—22no—
87Dic166—42no—
88Dic166—42no—
89C8×C83—24no—
90C22×C83—21no—
91C22×C83—21no—
92C22×C83—21no—
93C22×C83C22×D411yescharacteristic
94C2×D83—42no—
95C2×D83—42no—
96C2×D83—42no—
97C4×C83—21no—
98C4×C83—21no—
99C4×C83—21no—
100C4×C83AGL(1,ℤ/8)12yes—
101C4×C83#13712yes—
102C4×C83H(5,𝔽2)11yescharacteristic
103C4×C832−1+411yescharacteristic
104Dic83—42no—
105Dic83—42no—
106Dic83—42no—
107C2×C83—41no—
108C2×C83#53711yescharacteristic
109C2×C83C2×AGL(1,ℤ/8)11yescharacteristic
110C2×C83C2×#13711yescharacteristic
111D83—82no—
112#3254—831nomaximal, Sylow
113C83#325411yescharacteristic, complemented, Sylow
114#533—831no—
115#532—831no—
116#477—832no—
117#474—832no—
118#468—834no—
119#461—832no—
120#460—832no—
121#423D8311yescharacteristic, complemented
122C2×#41—1661no—
123C2×SD16—834no—
124C2×M16C2×D8312yes—
125C4⋊Q8—831no—
126C4⋊Q8—831no—
127C4⋊Q8—831no—
128#127—831no—
129#125—1661no—
130#124—1661no—
131C22⋊Q8—832no—
132C22⋊Q8—1661no—
133#121—832no—
134#121—1661no—
135C4×Q8—832no—
136C4×Q8—1661no—
137C4×D4—832no—
138C4×D4—1661no—
139#117—1661no—
140C2×C42C2×D8311yescharacteristic
141#107—834no—
142C4⋊C8C2×D8312yes—
143C4⋊C8C2×D8312yes—
144#103—834no—
145#102—834no—
146#41—3324no—
147C2×Q8—832no—
148C2×Q8—1661no—
149C2×Q8—1661no—
150C2×D4—832no—
151C2×D4—1662no—
152C2×D4—1661no—
153C2×D4—1661no—
154C22×C4—1662no—
155C22×C4D33211yescharacteristic
156C22×C4D33211yescharacteristic
157C22×C4C22×D8311yescharacteristic
158SD16—3324no—
159M16—42no—
160C2×C8C22×D8314yes—
161C4⋊C4—832no—
162C4⋊C4—832no—
163C4⋊C4—1661no—
164C4⋊C4—1661no—
165C4⋊C4—1661no—
166C4⋊C4—1662no—
167C4⋊C4—1662no—
168C4⋊C4—1661no—
169C22⋊C4—1662no—
170C22⋊C4—1662no—
171C22⋊C4—1662no—
172C42—1662no—
173C42D33211yescharacteristic
174C42D33211yescharacteristic
175C42C22×D8311yescharacteristic
176C42C22×D8311yescharacteristic
177C23—1662no—
178C23C2×D33211yescharacteristic
179Q8—3322no—
180Q8—3322no—
181Q8—1662no—
182Q8—1662no—
183D4—3322no—
184D4—3324no—
185D4—3322no—
186D4—1662no—
187D4—1662no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—3324no—
193C2×C4—1662no—
194C2×C4—1662no—
195C2×C4—1662no—
196C2×C4C2×D33211yescharacteristic
197C2×C4C2×D33211yescharacteristic
198C2×C4C2×D33211yescharacteristic
199C2×C4C2×D33211yescharacteristic
200C2×C4C2×D33211yescharacteristic
201C2×C4C23×D8311yescharacteristic
202C8—24no—
203C22—21no—
204C22—21no—
205C22—21no—
206C22—3322no—
207C22—3322no—
208C22—3322no—
209C22C22×D33211yescharacteristic
210C4—21no—
211C4—21no—
212C4—21no—
213C4—3322no—
214C4—3322no—
215C4—3322no—
216C4#1458182012yes—
217C4#1458182512yes—
218C4#1458183711yescharacteristic
219C4#1458184011yescharacteristic
220C2—41no—
221C2—6642no—
222C2#1912828211yescharacteristic
223C2C2×#1458182011yescharacteristic
224C2C2×#1458182511yescharacteristic
subgroupG/H
centerC22C22×D332
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#423×C83C2
socleC22×C83C22×D4
prime radical#19128251C2
hypercenter#423D83
hypocenterC83#3254
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C2×C42 ◁ #423

Complements

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

NH ≅ G/Ntype
1. #19128258C2semidirect
3. #19127997C2semidirect
4. #19127982C2semidirect
5. #19127947C2semidirect
6. #19127907C2semidirect
7. #19127902C2semidirect
8. #423×C83C2semidirect
113. C83#3254semidirect
121. #423D83semidirect

Maximal subgroups

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

subgroupindex
#191282582
#191282512
#191279972
#191279822
#191279472
#191279072
#191279022
#423×C832
#325483