Finite Group Databasebeta

← #24920350

Subgroups — #24920350

Order 10624.

324 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 133 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#19128429C211yesmaximal, characteristic
2#19128104C211yesmaximal, characteristic
3#19128100C212yesmaximal
4#19128008C212yesmaximal
5#19128005C212yesmaximal
6#19127986C211yesmaximal, characteristic
7#19127983C211yesmaximal, characteristic
8Q8×C83⋊C8C211yesmaximal, characteristic
9C4×#13050580C211yesmaximal, characteristic
10C4×#13050580C212yesmaximal
11#557×C83C211yesmaximal, characteristic
12C2×#13050580C2211yescharacteristic
13#14581792C2211yescharacteristic
14#14581790—21no—
15#14581784C2212yes—
16#14581782—22no—
17C4×Dic166C2211yescharacteristic
18C4×Dic166C2212yes—
19#14581757—21no—
20#14581725C2211yescharacteristic
21#14581725C2212yes—
22#14581725C2212yes—
23#14581723C2212yes—
24#14581723C2212yes—
25C4⋊(C83⋊C8)C2211yescharacteristic
26C4⋊(C83⋊C8)C2212yes—
27#14581716C2212yes—
28#14581716C2211yescharacteristic
29#14581715C2212yes—
30#14581715C2211yescharacteristic
31C4×C83⋊C8C2212yes—
32C4×C83⋊C8C2211yescharacteristic
33C4⋊Q8×C83—21no—
34C4⋊Q8×C83C2211yescharacteristic
35#125×C83—22no—
36#125×C83C2212yes—
37C4×Q8×C83—21no—
38C4×Q8×C83C2211yescharacteristic
39C4×Q8×C83C2212yes—
40C4×Q8×C83C2211yescharacteristic
41C2×Dic166—21no—
42C2×Dic166C2311yescharacteristic
43#13050580Q814yescomplemented
44C2×C83⋊C8C2312yes—
45C2×C83⋊C8C2311yescharacteristic
46C2×C83⋊C8C2311yescharacteristic
47C4⋊Dic83—21no—
48C4⋊Dic83C2311yescharacteristic
49C4⋊Dic83C2312yes—
50Dic83⋊C4—21no—
51Dic83⋊C4—22no—
52Dic83⋊C4—22no—
53Dic83⋊C4—22no—
54C4×Dic83—22no—
55C4×Dic83—21no—
56C2×Q8×C83—21no—
57C2×Q8×C83D411yescharacteristic
58C2×Q8×C83C2311yescharacteristic
59C4⋊C4×C83—22no—
60C4⋊C4×C83—21no—
61C4⋊C4×C83—22no—
62C4⋊C4×C83—21no—
63C4⋊C4×C83—22no—
64C4⋊C4×C83D412yes—
65C4⋊C4×C83D411yescharacteristic
66C4⋊C4×C83C2311yescharacteristic
67C4⋊C4×C83C2312yes—
68C42×C83—22no—
69C42×C83—21no—
70C42×C83C2312yes—
71C42×C83C2311yescharacteristic
72Q8×C83—22no—
73Q8×C83—21no—
74Q8×C83—42no—
75Q8×C83C2×Q812yes—
76C2×C4×C83—21no—
77C2×C4×C83—21no—
78C2×C4×C83—22no—
79C2×C4×C83C2×D412yes—
80C2×C4×C83C2×D411yescharacteristic
81C2×C4×C83C2×D412yes—
82C2×C4×C83C2×D411yescharacteristic
83C2×C4×C83C2411yescharacteristic
84C2×Dic83—21no—
85C2×Dic83—21no—
86C2×Dic83—22no—
87Dic166—21no—
88Dic166—42no—
89Dic166C2×Q812yes—
90C83⋊C8—22no—
91C83⋊C8—21no—
92C83⋊C8C2×Q812yes—
93C22×C83C22×D411yescharacteristic
94C4×C83—22no—
95C4×C83—22no—
96C4×C83—22no—
97C4×C83—21no—
98C4×C83—42no—
99C4×C83—41no—
100C4×C83#13512yes—
101C4×C83C22×Q811yescharacteristic
102C4×C832−1+411yescharacteristic
103Dic83—22no—
104Dic83—42no—
105Dic83—41no—
106C2×C83D4×Q811yescharacteristic
107C2×C83C2×#13511yescharacteristic
108C2×C83#57811yescharacteristic
109#3383—831nomaximal, Sylow
110C83#338311yescharacteristic, complemented, Sylow
111#557—831no—
112#557D8311yescharacteristic
113#500—831no—
114#499—832no—
115#479—832no—
116#479—832no—
117#475—831no—
118#475—831no—
119C8×Q8—831no—
120C4×Q16—832no—
121C4×Q16—831no—
122C2×Q16—831no—
123C4⋊Q8—21no—
124C4⋊Q8—831no—
125C4⋊Q8—1661no—
126C4⋊Q8C2×D8311yescharacteristic
127#125—22no—
128#125—832no—
129#125—1662no—
130#125C2×D8312yes—
131C4×Q8—21no—
132C4×Q8—831no—
133C4×Q8—832no—
134C4×Q8—1661no—
135C4×Q8C2×D8311yescharacteristic
136C4×Q8C2×D8312yes—
137C4×Q8C2×D8311yescharacteristic
138#107—831no—
139#107—832no—
140#107—832no—
141#106—832no—
142#106—832no—
143C4⋊C8—831no—
144C4⋊C8—832no—
145#103—832no—
146#103—831no—
147#103—832no—
148#103—831no—
149C4×C8—832no—
150C4×C8—831no—
151C2×Q8—21no—
152C2×Q8—831no—
153C2×Q8—1661no—
154C2×Q8C22⋊D8311yescharacteristic
155C2×Q8C22×D8311yescharacteristic
156Q16—834no—
157C2×C8—831no—
158C2×C8—832no—
159C2×C8—831no—
160C4⋊C4—22no—
161C4⋊C4—21no—
162C4⋊C4—22no—
163C4⋊C4—21no—
164C4⋊C4—22no—
165C4⋊C4—831no—
166C4⋊C4—832no—
167C4⋊C4—1661no—
168C4⋊C4—1662no—
169C4⋊C4—1662no—
170C4⋊C4—1662no—
171C4⋊C4—1661no—
172C4⋊C4C22⋊D8311yescharacteristic
173C4⋊C4C22⋊D8312yes—
174C4⋊C4C22×D8311yescharacteristic
175C4⋊C4C22×D8312yes—
176C42—22no—
177C42—21no—
178C42—1662no—
179C42—1661no—
180C42C22×D8312yes—
181C42C22×D8311yescharacteristic
182Q8—22no—
183Q8—21no—
184Q8—42no—
185Q8—3322no—
186Q8—832no—
187Q8—1661no—
188Q8Q8×D8312yes—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—22no—
192C2×C4—1661no—
193C2×C4—1662no—
194C2×C4—1661no—
195C2×C4C2×C22⋊D8312yes—
196C2×C4C2×C22⋊D8311yescharacteristic
197C2×C4C2×C22⋊D8312yes—
198C2×C4C2×C22⋊D8311yescharacteristic
199C2×C4C23×D8311yescharacteristic
200C8—832no—
201C8—1662no—
202C8—1661no—
203C22C22×C22⋊D8311yescharacteristic
204C4—22no—
205C4—22no—
206C4—22no—
207C4—21no—
208C4—42no—
209C4—41no—
210C4—3322no—
211C4—3321no—
212C4—1662no—
213C4#1458181212yes—
214C4C2×Q8×D8311yescharacteristic
215C4#1458184011yescharacteristic
216C2Q8×C22⋊D8311yescharacteristic
217C2C2×#1458181211yescharacteristic
218C2#1912851611yescharacteristic
subgroupG/H
centerC22C22×C22⋊D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#557×C83C2
socle, prime radicalC22×C83C22×D4
hypercenter#557D83
hypocenterC83#3383
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C4×Q8 ◁ #557

Complements

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

NH ≅ G/Ntype
43. #13050580Q8semidirect
110. C83#3383semidirect

Maximal subgroups

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

subgroupindex
#191284292
#191281042
#191281002
#191280082
#191280052
#191279862
#191279832
Q8×C83⋊C82
C4×#130505802
#557×C832
#338383