Finite Group Databasebeta

← #24908271

Subgroups — #24908271

Order 10624.

336 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 99 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×#14581781C211yesmaximal, characteristic
2C2×#14581716C211yesmaximal, characteristic, complemented
3C2×#14581714C211yesmaximal, characteristic
4C2×#14581712C211yesmaximal, characteristic
5C2×C8×Dic83C211yesmaximal, characteristic, complemented
6#19127566C211yesmaximal, characteristic, complemented
7C2×#103×C83C211yesmaximal, characteristic, complemented
8C22×D332C2211yescharacteristic
9C22×C83⋊C8C2211yescharacteristic
10#14581781—22no—
11#14581781C414yescomplemented
12C2×C4⋊Dic83C2211yescharacteristic
13C2×(C2×D83)⋊C4—22no—
14C2×C4×Dic83C2211yescharacteristic
15#14581716—22no—
16#14581714—22no—
17#14581712—22no—
18C8×Dic83—22no—
19C83⋊#95—22no—
20C22×Q8×C83C2211yescharacteristic
21C22×C8×C83C2211yescharacteristic
22C2×C4⋊C4×C83C2211yescharacteristic
23#103×C83—22no—
24C23×D83—21no—
25C2×D332—21no—
26C2×D332—21no—
27C2×D332—21no—
28C2×D332—21no—
29C2×D332—21no—
30C2×D332C2×C412yes—
31C22×Dic83—21no—
32C22×Dic83D412yes—
33C2×C83⋊C8—21no—
34C2×C83⋊C8—21no—
35C2×C83⋊C8D412yes—
36C4⋊Dic83—21no—
37C4⋊Dic83—21no—
38(C2×D83)⋊C4—28no—
39C4×Dic83C2×C412yes—
40C4×Dic83D412yes—
41C2×Q8×C83—21no—
42C2×Q8×C83—21no—
43C2×Q8×C83—21no—
44C2×Q8×C83—21no—
45C2×Q8×C83—21no—
46C2×Q8×C83C2×C412yes—
47C22×C4×C83—21no—
48C22×C4×C83—21no—
49C22×C4×C83C2311yescharacteristic
50C2×C8×C83—21no—
51C2×C8×C83—21no—
52C2×C8×C83D412yes—
53C4⋊C4×C83—21no—
54C4⋊C4×C83—21no—
55C23×C83C2×D411yescharacteristic
56C22×D83—22no—
57C22×D83—41no—
58C22×D83—41no—
59C22×D83—42no—
60C22×D83—42no—
61Q8×C83—22no—
62Q8×C83—22no—
63Q8×C83—41no—
64Q8×C83—41no—
65C2×C4×C83—22no—
66C2×C4×C83—41no—
67C2×C4×C83—41no—
68C2×C4×C83—41no—
69C2×C4×C83—42no—
70C2×C4×C83C22×C411yescharacteristic
71C2×C4×C83C2×D411yescharacteristic
72C2×C4×C83C2×D411yescharacteristic
73C2×C4×C83C2×D411yescharacteristic
74C2×C4×C83#4111yescharacteristic
75C2×C4×C83#4111yescharacteristic
76C2×Dic83—22no—
77C2×Dic83—22no—
78C2×Dic83—41no—
79C2×Dic83—42no—
80C2×Dic83C22⋊C414yes—
81D332—22no—
82D332—22no—
83D332—41no—
84D332—41no—
85C8×C83—22no—
86C83⋊C8—22no—
87C22×C83C2×C22⋊C411yescharacteristic
88C22×C83#12411yescharacteristic
89C22×C83#12711yescharacteristic
90C22×C83#13512yes—
91C22×C83#13512yes—
92C2×D83—42no—
93C2×D83—44no—
94C2×D83—82no—
95C2×D83—81no—
96C2×D83—81no—
97C4×C83—42no—
98C4×C83—81no—
99C4×C83C4×D411yescharacteristic
100C4×C83C4×D411yescharacteristic
101C4×C83#12111yescharacteristic
102C4×C83#12111yescharacteristic
103Dic83—24no—
104Dic83—81no—
105C2×C83#39011yescharacteristic
106C2×C83#41612yes—
107C2×C83#48812yes—
108C2×C83#49511yescharacteristic
109C2×C83#49511yescharacteristic
110D83—82no—
111#1958—831nomaximal, Sylow
112C83#195811yescharacteristic, complemented, Sylow
113C2×#124—831no—
114C2×#103—831no—
115C2×#103D8311yescharacteristic, complemented
116C2×#102—831no—
117C2×#102—831no—
118C2×C4×C8—831no—
119#382—831no—
120C22×Q8C2×D8311yescharacteristic
121C22×D4—831no—
122C22×C8—831no—
123C22×C8C2×D8311yescharacteristic
124#124—834no—
125#124—1662no—
126C2×C4⋊C4—831no—
127C2×C4⋊C4C2×D8311yescharacteristic
128C2×C22⋊C4—1662no—
129C2×C42—831no—
130#103—22no—
131#103—1662no—
132#102—1662no—
133#102—1662no—
134C4×C8—1662no—
135#95—1662no—
136C24—1661no—
137C2×Q8—21no—
138C2×Q8—21no—
139C2×Q8—21no—
140C2×Q8—21no—
141C2×Q8—21no—
142C2×Q8C4×D8312yes—
143C2×D4—832no—
144C2×D4—1661no—
145C2×D4—1661no—
146C2×D4—1661no—
147C2×D4—1661no—
148C2×D4—1661no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—832no—
152C22×C4—1661no—
153C22×C4C22×D8311yescharacteristic
154C2×C8—21no—
155C2×C8—21no—
156C2×C8—832no—
157C2×C8—1661no—
158C2×C8—1661no—
159C2×C8C22⋊D8312yes—
160C4⋊C4—21no—
161C4⋊C4—21no—
162C4⋊C4—1661no—
163C4⋊C4—1661no—
164C22⋊C4—1668no—
165C42—832no—
166C42—832no—
167C23—3321no—
168C23—3321no—
169C23—3322no—
170C23—3322no—
171C23—1662no—
172C23D4×D8311yescharacteristic
173Q8—22no—
174Q8—22no—
175Q8—41no—
176Q8—41no—
177D4—3321no—
178D4—3321no—
179D4—1662no—
180D4—1662no—
181C2×C4—22no—
182C2×C4—41no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—42no—
186C2×C4—3321no—
187C2×C4—3322no—
188C2×C4—834no—
189C2×C4—1662no—
190C2×C4—1662no—
191C2×C4C2×C4×D8311yescharacteristic
192C2×C4D4×D8311yescharacteristic
193C2×C4D4×D8311yescharacteristic
194C2×C4C2×C22⋊D8311yescharacteristic
195C2×C4#1305058911yescharacteristic
196C2×C4#1305059011yescharacteristic
197C8—22no—
198C8—1662no—
199C22—3322no—
200C22—3324no—
201C22—6641no—
202C22—6641no—
203C22—6642no—
204C22C22⋊C4×D8311yescharacteristic
205C22#1458177811yescharacteristic
206C22#1458178811yescharacteristic
207C22#1458181312yes—
208C22#1458181512yes—
209C4—42no—
210C4—81no—
211C4—6641no—
212C4—1664no—
213C4(C22⋊D83)⋊C411yescharacteristic
214C4C4×C22⋊D8311yescharacteristic
215C4#1458176411yescharacteristic
216C4#1458176511yescharacteristic
217C2—6642no—
218C2#1912760111yescharacteristic
219C2#1912769612yes—
220C2#1912805712yes—
221C2#1912808211yescharacteristic
222C2#1912808611yescharacteristic
subgroupG/H
centerC23D4×D83
commutatorC2×C4×C83C22×C4
FrattiniC22×C4C22×D83
FittingC2×#103×C83C2
socleC23×C83C2×D4
prime radicalC22×D332C22
cosocleC22×C4×C83C23
hypercenterC2×#103D83
hypocenterC83#1958
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C22×C4 ◁ C2×#103

Complements

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

NH ≅ G/Ntype
2. C2×#14581716C2semidirect
5. C2×C8×Dic83C2semidirect
6. #19127566C2semidirect
7. C2×#103×C83C2semidirect
11. #14581781C4semidirect
112. C83#1958semidirect
115. C2×#103D83semidirect

Maximal subgroups

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

subgroupindex
C2×#145817812
C2×#145817162
C2×#145817142
C2×#145817122
C2×C8×Dic832
#191275662
C2×#103×C832
#195883