Finite Group Databasebeta

← #27186542

Subgroups — #27186542

Order 12928.

274 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 274 conjugacy classes of subgroups — too many to draw. Only its 60 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#20266149C211yesmaximal, characteristic
2#20266120C211yesmaximal, characteristic
3#20266031C214yesmaximal, complemented
4C2×#15692231C211yesmaximal, characteristic
5C2×#13108083—21no—
6C2×C22⋊#87938—21no—
7#15692312—21no—
8#15692312C2212yes—
9(C2×#87938)⋊C4—21no—
10(C2×#87938)⋊C4C2212yes—
11C22×C4×D101C2211yescharacteristic
12#15692231—21no—
13#15692231—21no—
14#15692231C412yescomplemented
15#15692231C2212yes—
16C2×C4⋊Dic101C411yescharacteristic, complemented
17C2×#13108053—21no—
18C2×#13108051—21no—
19C22×Q8×C101C411yescharacteristic, complemented
20C22×#87938—22no—
21#13108083—42no—
22C2×#11717581—22no—
23C22⋊#87938—42no—
24C23×D101D411yescharacteristic
25C22×Dic101—21no—
26C22×Dic101D411yescharacteristic
27C2×C4×D101—42no—
28C2×C4×D101C2×C411yescharacteristic
29C2×C4×D101D411yescharacteristic
30C2×C4×D101D411yescharacteristic
31C2×C4×D101C2311yescharacteristic
32C4⋊Dic101—21no—
33C4⋊Dic101C2×C412yes—
34#13108053—22no—
35#13108053—22no—
36#13108051—22no—
37#13108051—22no—
38C2×Q8×C101—21no—
39C2×Q8×C101—42no—
40C2×Q8×C101C2×C412yes—
41C22×C4×C101—21no—
42C22×C4×C101C2×C411yescharacteristic
43C23×C101C22⋊C411yescharacteristic
44C22×D101—22no—
45C22×D101—22no—
46C22×D101—41no—
47C22×D101C22⋊C411yescharacteristic
48C22×D101C2×D411yescharacteristic
49C2×#87938—42no—
50C2×#87938—42no—
51C2×#87938—42no—
52Q8×C101—42no—
53Q8×C101—42no—
54C2×C4×C101—22no—
55C2×C4×C101—22no—
56C2×C4×C101—42no—
57C2×C4×C101C22⋊C411yescharacteristic
58C2×C4×C101C22×C411yescharacteristic
59C2×Dic101—22no—
60C2×Dic101—22no—
61C2×Dic101—42no—
62C2×Dic101C22⋊C411yescharacteristic
63C2×Dic101C2×D411yescharacteristic
64C4×D101—41no—
65C4×D101—41no—
66C4×D101—41no—
67C4×D101—41no—
68#11717581—42no—
69C22×C101—21no—
70C22×C101—21no—
71C22×C101#9912yes—
72C22×C101C2×C22⋊C411yescharacteristic
73C2×D101—21no—
74C2×D101—21no—
75C2×D101—21no—
76C2×D101—21no—
77C2×D101—21no—
78C2×D101—44no—
79C2×D101#13712yes—
80#87938—82no—
81C4×C101—21no—
82C4×C101—21no—
83C4×C101—42no—
84Dic101—21no—
85Dic101—21no—
86Dic101—42no—
87C2×C101—22no—
88C2×C101C2×#9911yescharacteristic
89C2×C101#41911yescharacteristic
90C2×C101#42111yescharacteristic
91D101—22no—
92D101—41no—
93#1511—1011nomaximal, Sylow
94C101#151111yescharacteristic, complemented, Sylow
95C2×C22⋊Q8—1011no—
96#407—1011no—
97#381—1011no—
98#328—1014no—
99C22×Q8#8793811yescharacteristic, complemented
100C23×C4—1011no—
101C2×M16—2021no—
102C22⋊Q8—1012no—
103C22⋊Q8—1012no—
104C22⋊Q8—2021no—
105C22⋊Q8—2021no—
106C2×C4⋊C4—1011no—
107C2×C4⋊C4—2021no—
108C2×C22⋊C4—2021no—
109C2×C22⋊C4—2021no—
110C22⋊C8—1012no—
111C22⋊C8—2021no—
112#95—1012no—
113#95—2021no—
114C24—1011no—
115C2×Q8—21no—
116C2×Q8—42no—
117C2×Q8C2×#8793812yes—
118C22×C4—21no—
119C22×C4—1011no—
120C22×C4—1011no—
121C22×C4—1011no—
122C22×C4—1011no—
123C22×C4—1011no—
124C22×C4—4042no—
125C22×C4—2022no—
126C22×C4—2021no—
127C22×C4C2×#8793811yescharacteristic
128M16—4042no—
129C2×C8—2022no—
130C4⋊C4—1012no—
131C4⋊C4—2022no—
132C4⋊C4—2022no—
133C4⋊C4—2021no—
134C22⋊C4—4042no—
135C22⋊C4—2022no—
136C22⋊C4—2022no—
137C23—1011no—
138C23—1011no—
139C23—4041no—
140C23—2022no—
141C23—2022no—
142C23C22⋊#8793811yescharacteristic
143Q8—42no—
144Q8—42no—
145C2×C4—22no—
146C2×C4—22no—
147C2×C4—42no—
148C2×C4—1011no—
149C2×C4—1011no—
150C2×C4—4041no—
151C2×C4—4041no—
152C2×C4—4041no—
153C2×C4—4041no—
154C2×C4—4042no—
155C2×C4—4042no—
156C2×C4—4042no—
157C2×C4—4042no—
158C2×C4—2022no—
159C2×C4—2022no—
160C2×C4C22⋊#8793811yescharacteristic
161C2×C4C22×#8793811yescharacteristic
162C8—4042no—
163C22—21no—
164C22—21no—
165C22—1012no—
166C22—4044no—
167C22—2021no—
168C22—2021no—
169C22—2021no—
170C22—2021no—
171C22—2021no—
172C22#1569231312yes—
173C22C2×C22⋊#8793811yescharacteristic
174C4—21no—
175C4—21no—
176C4—42no—
177C4—8082no—
178C4—4042no—
179C4—2021no—
180C4—2021no—
181C2—22no—
182C2—4041no—
183C2—2022no—
184C2C2×#1569231311yescharacteristic
185C2#2026617011yescharacteristic
186C2#2026617611yescharacteristic
subgroupG/H
centerC22C2×C22⋊#87938
commutatorC2×C4×C101C22×C4
FrattiniC2×C4C22×#87938
FittingC22×Q8×C101C4
socleC22×C101C2×C22⋊C4
prime radicalC23×D101D4
cosocleC2×C4×D101C23
hypercenterC22×Q8#87938
hypocenterC101#1511
LCSG ▷ C2×C4×C101 ▷ C22×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×Q8

Complements

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

NH ≅ G/Ntype
3. #20266031C2semidirect
14. #15692231C4semidirect
16. C2×C4⋊Dic101C4semidirect
19. C22×Q8×C101C4semidirect
94. C101#1511semidirect
99. C22×Q8#87938semidirect

Maximal subgroups

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

subgroupindex
#202661492
#202661202
#202660312
C2×#156922312
#1511101