Finite Group Databasebeta

← #24920338

Subgroups — #24920338

Order 10624.

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

normal subgroups only

This group has 338 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, complemented
2#19128100C212yesmaximal, complemented
3#19128097C211yesmaximal, characteristic, complemented
4#19128005C212yesmaximal, complemented
5#19128001C212yesmaximal
6#19127993C211yesmaximal, characteristic, complemented
7#19127981C211yesmaximal, characteristic
8Q8×C83⋊C8C211yesmaximal, characteristic, complemented
9C4×#13050578C211yesmaximal, characteristic
10C4×#13050578C212yesmaximal
11#554×C83C211yesmaximal, characteristic
12C2×#13050578C2211yescharacteristic
13#14581792C2211yescharacteristic
14#14581790—21no—
15#14581784C2212yes—
16#14581782—22no—
17C4×Dic166C2211yescharacteristic
18C4×Dic166C2212yes—
19#14581757—21no—
20#14581725C2212yes—
21#14581725C2212yes—
22#14581723C2212yes—
23#14581723C2212yes—
24#14581723C2211yescharacteristic
25C4⋊(C83⋊C8)C2211yescharacteristic
26C4⋊(C83⋊C8)C2212yes—
27#14581715C2212yes—
28#14581715C2211yescharacteristic
29#14581713C2212yes—
30#14581713C2211yescharacteristic
31C4×C83⋊C8C2211yescharacteristic
32C4×C83⋊C8C2212yes—
33C4⋊Q8×C83C2211yescharacteristic
34#125×C83C2212yes—
35C22⋊Q8×C83—21no—
36C22⋊Q8×C83—22no—
37C4×Q8×C83C2211yescharacteristic
38C4×D4×C83C2211yescharacteristic
39C4×D4×C83C2212yes—
40C2×C4⋊C4×C83—21no—
41C2×Dic166—21no—
42C2×Dic166C2311yescharacteristic
43#13050578Q814yescomplemented
44C2×C83⋊C8C2311yescharacteristic
45C2×C83⋊C8C2312yes—
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×D4×C83C2311yescharacteristic
59C22×C4×C83—21no—
60C22×C4×C83—22no—
61C4⋊C4×C83—22no—
62C4⋊C4×C83—21no—
63C4⋊C4×C83—22no—
64C4⋊C4×C83D412yes—
65C4⋊C4×C83D411yescharacteristic
66C4⋊C4×C83C2311yescharacteristic
67C4⋊C4×C83C2312yes—
68C22⋊C4×C83—21no—
69C22⋊C4×C83—22no—
70C42×C83C2312yes—
71C42×C83C2311yescharacteristic
72C23×C83—21no—
73Q8×C83—22no—
74Q8×C83—81no—
75D4×C83—21no—
76D4×C83C2×Q812yes—
77C2×C4×C83—21no—
78C2×C4×C83—22no—
79C2×C4×C83—42no—
80C2×C4×C83—82no—
81C2×C4×C83C2×D412yes—
82C2×C4×C83C2×D411yescharacteristic
83C2×C4×C83C2×D412yes—
84C2×C4×C83C2×D411yescharacteristic
85C2×C4×C83C2411yescharacteristic
86C2×Dic83—21no—
87C2×Dic83—21no—
88C2×Dic83—22no—
89Dic166—21no—
90Dic166—42no—
91Dic166C2×Q812yes—
92C83⋊C8—21no—
93C83⋊C8—22no—
94C83⋊C8C2×Q812yes—
95C22×C83—22no—
96C22×C83—41no—
97C22×C83—41no—
98C22×C83C22×D411yescharacteristic
99C4×C83—22no—
100C4×C83—22no—
101C4×C83—21no—
102C4×C83—41no—
103C4×C83—42no—
104C4×C83#13512yes—
105C4×C83C22×Q811yescharacteristic
106C4×C832−1+411yescharacteristic
107Dic83—22no—
108Dic83—42no—
109Dic83—41no—
110C2×C83—42no—
111C2×C83D4×Q811yescharacteristic
112C2×C83C2×#13511yescharacteristic
113C2×C83#57711yescharacteristic
114#3381—831nomaximal, Sylow
115C83#338111yescharacteristic, complemented, Sylow
116#557—831no—
117#554D8311yescharacteristic
118#499—832no—
119#498—831no—
120#479—832no—
121#478—832no—
122#477—831no—
123#474—831no—
124C8×Q8—831no—
125C4×SD16—831no—
126C4×SD16—832no—
127C2×SD16—831no—
128C4⋊Q8—831no—
129C4⋊Q8—1661no—
130C4⋊Q8C2×D8311yescharacteristic
131#125—832no—
132#125—1662no—
133#125C2×D8312yes—
134C22⋊Q8—21no—
135C22⋊Q8—22no—
136C4×Q8—831no—
137C4×Q8—832no—
138C4×Q8—1661no—
139C4×Q8C2×D8311yescharacteristic
140C4×D4C2×D8311yescharacteristic
141C4×D4C2×D8312yes—
142C2×C4⋊C4—21no—
143#107—832no—
144#107—832no—
145#106—832no—
146#106—831no—
147#106—832no—
148C4⋊C8—831no—
149C4⋊C8—832no—
150#103—832no—
151#103—831no—
152#102—832no—
153#102—831no—
154C4×C8—831no—
155C4×C8—832no—
156C2×Q8—21no—
157C2×Q8—831no—
158C2×Q8—1661no—
159C2×Q8C22⋊D8311yescharacteristic
160C2×D4C22×D8311yescharacteristic
161C22×C4—21no—
162C22×C4—22no—
163SD16—834no—
164C2×C8—831no—
165C2×C8—831no—
166C2×C8—832no—
167C4⋊C4—22no—
168C4⋊C4—21no—
169C4⋊C4—22no—
170C4⋊C4—832no—
171C4⋊C4—831no—
172C4⋊C4—1661no—
173C4⋊C4—1662no—
174C4⋊C4—1662no—
175C4⋊C4—1662no—
176C4⋊C4—1661no—
177C4⋊C4C22⋊D8311yescharacteristic
178C4⋊C4C22⋊D8312yes—
179C4⋊C4C22×D8312yes—
180C4⋊C4C22×D8311yescharacteristic
181C22⋊C4—21no—
182C22⋊C4—22no—
183C42—1662no—
184C42—1661no—
185C42C22×D8312yes—
186C42C22×D8311yescharacteristic
187C23—21no—
188Q8—22no—
189Q8—81no—
190Q8—3322no—
191Q8—832no—
192Q8—1661no—
193D4—21no—
194D4Q8×D8312yes—
195C2×C4—21no—
196C2×C4—22no—
197C2×C4—42no—
198C2×C4—82no—
199C2×C4—1661no—
200C2×C4—1662no—
201C2×C4—1661no—
202C2×C4C2×C22⋊D8312yes—
203C2×C4C2×C22⋊D8312yes—
204C2×C4C2×C22⋊D8311yescharacteristic
205C2×C4C2×C22⋊D8311yescharacteristic
206C2×C4C23×D8311yescharacteristic
207C8—832no—
208C8—1661no—
209C8—1662no—
210C22—22no—
211C22—41no—
212C22—41no—
213C22C22×C22⋊D8311yescharacteristic
214C4—22no—
215C4—22no—
216C4—21no—
217C4—41no—
218C4—42no—
219C4—3322no—
220C4—3321no—
221C4—1662no—
222C4#1458181212yes—
223C4C2×Q8×D8311yescharacteristic
224C4#1458184011yescharacteristic
225C2—42no—
226C2Q8×C22⋊D8311yescharacteristic
227C2C2×#1458181211yescharacteristic
228C2#1912851211yescharacteristic
subgroupG/H
centerC22C22×C22⋊D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#554×C83C2
socleC22×C83C22×D4
prime radicalC2×D4×C83C23
hypercenter#554D83
hypocenterC83#3381
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C4×Q8 ◁ #554

Complements

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

NH ≅ G/Ntype
1. #19128429C2semidirect
2. #19128100C2semidirect
3. #19128097C2semidirect
4. #19128005C2semidirect
6. #19127993C2semidirect
8. Q8×C83⋊C8C2semidirect
43. #13050578Q8semidirect
115. C83#3381semidirect

Maximal subgroups

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

subgroupindex
#191284292
#191281002
#191280972
#191280052
#191280012
#191279932
#191279812
Q8×C83⋊C82
C4×#130505782
#554×C832
#338183