Finite Group Databasebeta

← #27186556

Subgroups — #27186556

Order 12928.

208 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 56 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#20266200C211yesmaximal, characteristic
2#20266122C211yesmaximal, characteristic
3#20266031C214yesmaximal
4#20265732C211yesmaximal, characteristic
5#15692322—21no—
6#15692312C2212yes—
7#15692310—21no—
8(C2×#87938)⋊C4—21no—
9(C2×#87938)⋊C4—21no—
10(C2×#87938)⋊C4C2212yes—
11#15692255C411yescharacteristic, complemented
12#15692246—21no—
13#15692235—21no—
14#15692231C2212yes—
15Q8×Dic101C412yescomplemented
16#15692220—21no—
17#15692213—21no—
18C42×D101C2211yescharacteristic
19C4⋊Q8×C101C411yescharacteristic, complemented
20C22×#87938—22no—
21C2×#11717581—22no—
22C2×Dic202—21no—
23C2×C4×D101D411yescharacteristic
24C2×C4×D101D411yescharacteristic
25C2×C4×D101C2311yescharacteristic
26C4⋊Dic101—21no—
27C4⋊Dic101—22no—
28C4⋊Dic101C2×C412yes—
29#13108053—22no—
30#13108051—22no—
31C4×Dic101—22no—
32C4×Dic101C2×C411yescharacteristic
33C4×Dic101D411yescharacteristic
34C4×Dic101D411yescharacteristic
35C2×Q8×C101C2×C412yes—
36C4⋊C4×C101—21no—
37C4⋊C4×C101—21no—
38C42×C101C2×C411yescharacteristic
39C22×D101C2×D411yescharacteristic
40C2×#87938—42no—
41C2×#87938—42no—
42C2×#87938—42no—
43Q8×C101—42no—
44C2×C4×C101—22no—
45C2×C4×C101C22⋊C411yescharacteristic
46C2×C4×C101C22⋊C411yescharacteristic
47C2×C4×C101C22×C411yescharacteristic
48C2×Dic101—22no—
49C2×Dic101C22⋊C411yescharacteristic
50C2×Dic101C22⋊C411yescharacteristic
51C2×Dic101C2×D411yescharacteristic
52C4×D101—22no—
53C4×D101—41no—
54C4×D101—41no—
55Dic202—42no—
56#11717581—42no—
57C22×C101C2×C22⋊C411yescharacteristic
58C2×D101—21no—
59C2×D101—21no—
60C2×D101#13712yes—
61#87938—82no—
62C4×C101—21no—
63C4×C101—21no—
64C4×C101—21no—
65C4×C101—42no—
66Dic101—21no—
67Dic101—21no—
68Dic101—21no—
69Dic101—42no—
70C2×C101#41011yescharacteristic
71C2×C101#41911yescharacteristic
72C2×C101#42111yescharacteristic
73D101—22no—
74#1515—1011nomaximal, Sylow
75C101#151511yescharacteristic, complemented, Sylow
76#533—1011no—
77#432—1011no—
78#383—1011no—
79#328—1014no—
80C4⋊Q8—1011no—
81C4⋊Q8#8793811yescharacteristic, complemented
82#125—2021no—
83C22⋊Q8—1012no—
84C22⋊Q8—2021no—
85C4×Q8—1012no—
86C4×Q8—2021no—
87#117—2021no—
88C2×C42—1011no—
89C4⋊C8—2021no—
90C22⋊C8—1012no—
91#97—2021no—
92#95—1012no—
93#95—2021no—
94#95—2021no—
95C2×Q8—2021no—
96C2×Q8C2×#8793812yes—
97C22×C4—1011no—
98C22×C4—1011no—
99C22×C4—1011no—
100C22×C4—2022no—
101C2×C8—2022no—
102C4⋊C4—21no—
103C4⋊C4—21no—
104C4⋊C4—1012no—
105C4⋊C4—2022no—
106C4⋊C4—2022no—
107C4⋊C4—2021no—
108C22⋊C4—2022no—
109C42—1011no—
110C42—1011no—
111C42—1011no—
112C42—2022no—
113C42C2×#8793811yescharacteristic
114C23—1011no—
115Q8—42no—
116Q8—4042no—
117C2×C4—22no—
118C2×C4—1011no—
119C2×C4—1011no—
120C2×C4—1011no—
121C2×C4—4041no—
122C2×C4—4041no—
123C2×C4—4042no—
124C2×C4—4042no—
125C2×C4—4042no—
126C2×C4—2022no—
127C2×C4—2022no—
128C2×C4C22⋊#8793811yescharacteristic
129C2×C4C22⋊#8793811yescharacteristic
130C2×C4C22×#8793811yescharacteristic
131C8—4042no—
132C22—1012no—
133C22—2021no—
134C22—2021no—
135C22C2×C22⋊#8793811yescharacteristic
136C4—21no—
137C4—21no—
138C4—21no—
139C4—42no—
140C4—8082no—
141C4—4042no—
142C4—2021no—
143C4—2021no—
144C4—2021no—
145C2—2022no—
146C2#2026615711yescharacteristic
147C2#2026617011yescharacteristic
148C2#2026617611yescharacteristic
subgroupG/H
centerC22C2×C22⋊#87938
commutatorC2×C4×C101C22×C4
FrattiniC2×C4C22×#87938
FittingC4⋊Q8×C101C4
socleC22×C101C2×C22⋊C4
prime radicalC22×D101C2×D4
cosocleC2×C4×D101C23
hypercenterC4⋊Q8#87938
hypocenterC101#1515
LCSG ▷ C2×C4×C101 ▷ C22×C101 ▷ C101
UCSC1 ◁ C22 ◁ C42 ◁ C4⋊Q8

Complements

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

NH ≅ G/Ntype
11. #15692255C4semidirect
15. Q8×Dic101C4semidirect
19. C4⋊Q8×C101C4semidirect
75. C101#1515semidirect
81. C4⋊Q8#87938semidirect

Maximal subgroups

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

subgroupindex
#202662002
#202661222
#202660312
#202657322
#1515101