Finite Group Databasebeta

← #27187396

Subgroups — #27187396

Order 12928.

234 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 54 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#20266163C211yesmaximal, characteristic
2#20266156C211yesmaximal, characteristic
3#20266080C214yesmaximal, complemented
4#20265753C211yesmaximal, characteristic
5D101⋊M16—21no—
6#15692334—21no—
7#15692316—21no—
8#15692315C2212yes—
9#15692314—21no—
10#15692313C2212yes—
11C2×#13108074C2211yescharacteristic
12C22×Dic202C411yescharacteristic, complemented
13#15692255—21no—
14#15692243—21no—
15#15692239C412yescomplemented
16#15692239C2212yes—
17#15692235—21no—
18#15692235—21no—
19#117×C101C411yescharacteristic, complemented
20#13108084—41no—
21#13108083—22no—
22D101⋊C8—41no—
23C4⋊#87938—41no—
24C22⋊#87938—22no—
25C4×#87938—41no—
26#13108074—41no—
27#13108074—41no—
28C2×Dic202—21no—
29C2×Dic202—21no—
30C2×Dic202—41no—
31C2×Dic202—41no—
32C2×Dic202D411yescharacteristic
33C2×D404D411yescharacteristic
34C2×C22⋊D101C2×C411yescharacteristic
35C2×C22⋊D101C2311yescharacteristic
36C22×Dic101C2×C412yes—
37C2×C4×D101D411yescharacteristic
38C2×C4×D101D411yescharacteristic
39C4⋊Dic101—22no—
40C4⋊Dic101—22no—
41#13108051—22no—
42#13108051—22no—
43C22×C4×C101C2×C411yescharacteristic
44C4⋊C4×C101—21no—
45C22⋊C4×C101C2×C412yes—
46C42×C101—21no—
47C23×C101C22×C411yescharacteristic
48C22×D101C22⋊C411yescharacteristic
49C22×D101C2×D411yescharacteristic
50C2×#87938—42no—
51C2×C4×C101—21no—
52C2×C4×C101—21no—
53C2×C4×C101—22no—
54C2×C4×C101C22⋊C411yescharacteristic
55C2×C4×C101C22⋊C411yescharacteristic
56C22⋊D101—41no—
57C22⋊D101—41no—
58C2×Dic101—22no—
59C2×Dic101—42no—
60C2×Dic101C22⋊C411yescharacteristic
61C2×Dic101C2×D411yescharacteristic
62D404—41no—
63C4×D101—21no—
64C4×D101—21no—
65C4×D101—21no—
66C4×D101—21no—
67Dic202—41no—
68Dic202—42no—
69Dic202—41no—
70Dic202—41no—
71#11717581—42no—
72C22×C101—21no—
73C22×C101—41no—
74C22×C101C2×C22⋊C411yescharacteristic
75C2×D101—21no—
76C2×D101—21no—
77C2×D101—41no—
78C2×D101—41no—
79#87938—82no—
80C4×C101—21no—
81C4×C101—42no—
82C4×C101#9911yescharacteristic
83C4×C101#9911yescharacteristic
84Dic101—21no—
85Dic101—21no—
86Dic101—42no—
87C2×C101—21no—
88C2×C101—41no—
89C2×C101C2×#9911yescharacteristic
90D101—41no—
91D101—41no—
92#2123—1011nomaximal, Sylow
93C101#212311yescharacteristic, complemented, Sylow
94#536—1011no—
95#413—1011no—
96#410—1011no—
97#352—1014no—
98C2×#41—1011no—
99C22×Q8—1011no—
100C2×M16—2021no—
101C4⋊Q8—2021no—
102#124—2021no—
103#123—1012no—
104#123—1012no—
105C22⋊Q8—2021no—
106C22⋊Q8—2021no—
107#117—2021no—
108#117#8793811yescharacteristic, complemented
109#101—2021no—
110#100—1012no—
111#99—1012no—
112#99—2021no—
113#41—4041no—
114#41—4041no—
115C2×Q8—1011no—
116C2×Q8—4041no—
117C2×Q8—4041no—
118C2×Q8—2021no—
119C2×Q8—2021no—
120C2×D4—1011no—
121C2×D4—1011no—
122C2×D4—1011no—
123C22×C4—1011no—
124C22×C4—1011no—
125C22×C4—1012no—
126C22×C4C2×#8793811yescharacteristic
127M16—4041no—
128M16—2022no—
129C2×C8—4041no—
130C4⋊C4—21no—
131C4⋊C4—4041no—
132C4⋊C4—2022no—
133C4⋊C4—2022no—
134C22⋊C4—2022no—
135C22⋊C4—2022no—
136C22⋊C4—2022no—
137C22⋊C4C2×#8793812yes—
138C42—21no—
139C42—4041no—
140C23—1011no—
141C23—1011no—
142C23C22×#8793811yescharacteristic
143Q8—4042no—
144Q8—4041no—
145Q8—4041no—
146Q8—4041no—
147D4—4041no—
148D4—4041no—
149D4—4041no—
150C2×C4—21no—
151C2×C4—21no—
152C2×C4—22no—
153C2×C4—1011no—
154C2×C4—1011no—
155C2×C4—4042no—
156C2×C4—4042no—
157C2×C4—2022no—
158C2×C4—2021no—
159C2×C4—2021no—
160C2×C4—2021no—
161C2×C4—2021no—
162C2×C4C22⋊#8793811yescharacteristic
163C2×C4C22⋊#8793811yescharacteristic
164C8—4042no—
165C22—21no—
166C22—41no—
167C22—4041no—
168C22—4041no—
169C22—2021no—
170C22—2021no—
171C22C2×C22⋊#8793811yescharacteristic
172C4—21no—
173C4—42no—
174C4—8082no—
175C4—4042no—
176C4—2021no—
177C4—2021no—
178C4#1569231411yescharacteristic
179C4#1569231411yescharacteristic
180C2—21no—
181C2—41no—
182C2—4041no—
183C2—4041no—
184C2C2×#1569231411yescharacteristic
subgroupG/H
centerC2C2×#15692314
commutatorC23×C101C22×C4
FrattiniC23C22×#87938
Fitting#117×C101C4
socleC2×C101C2×#99
prime radicalC2×#13108074C22
cosocleC2×C22⋊D101C23
hypercenter#117#87938
hypocenterC101#2123
LCSG ▷ C23×C101 ▷ C22×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C2 ◁ C2×C4 ◁ C22×C4 ◁ #117

Complements

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

NH ≅ G/Ntype
3. #20266080C2semidirect
12. C22×Dic202C4semidirect
15. #15692239C4semidirect
19. #117×C101C4semidirect
93. C101#2123semidirect
108. #117#87938semidirect

Maximal subgroups

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

subgroupindex
#202661632
#202661562
#202660802
#202657532
#2123101