Finite Group Databasebeta

← #27187415

Subgroups — #27187415

Order 12928.

238 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 52 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#20266158C212yesmaximal
2#20266084C212yesmaximal, complemented
3#20266083C212yesmaximal, complemented
4#20265752C211yesmaximal, characteristic
5#15692334—22no—
6#15692314—22no—
7#15692314C2214yes—
8C2×#13108075C2211yescharacteristic
9C2×Q8×D101C411yescharacteristic, complemented
10#15692253C411yescharacteristic, complemented
11#15692244C2211yescharacteristic
12#15692243C2211yescharacteristic
13#15692240—22no—
14#15692233—22no—
15#15692212C411yescharacteristic, complemented
16C4⋊Q8×C101C411yescharacteristic, complemented
17C4⋊#87938—42no—
18C22⋊#87938—24no—
19C4×#87938—42no—
20#13108075—42no—
21C2×Dic202—21no—
22C2×Dic202C2×C411yescharacteristic
23Q8×D101—42no—
24C2×D404D412yes—
25C2×D404C2311yescharacteristic
26C2×C4×D101—21no—
27C2×C4×D101C2×C411yescharacteristic
28C2×C4×D101D412yes—
29#13108053—22no—
30#13108053—21no—
31#13108051—21no—
32#13108051—22no—
33#13108051—22no—
34C4×Dic101C2×C411yescharacteristic
35C2×Q8×C101C2×C411yescharacteristic
36C2×Q8×C101C2×C411yescharacteristic
37C4⋊C4×C101—22no—
38C42×C101C2×C411yescharacteristic
39C22×D101C22⋊C411yescharacteristic
40C22×D101C2×D412yes—
41C2×#87938—44no—
42Q8×C101—41no—
43Q8×C101—41no—
44C2×C4×C101—21no—
45C2×C4×C101—21no—
46C2×C4×C101C22⋊C412yes—
47C2×C4×C101C22×C411yescharacteristic
48C2×Dic101—21no—
49C2×Dic101—21no—
50C2×Dic101C22⋊C411yescharacteristic
51D404—42no—
52D404—41no—
53C4×D101—22no—
54C4×D101—22no—
55C4×D101—22no—
56C4×D101—42no—
57Dic202—42no—
58Dic202—41no—
59C22×C101C2×C22⋊C411yescharacteristic
60C2×D101—21no—
61C2×D101—22no—
62C2×D101—41no—
63C2×D101—42no—
64#87938—84no—
65C4×C101—21no—
66C4×C101—22no—
67C4×C101—41no—
68C4×C101—41no—
69Dic101—41no—
70Dic101—41no—
71Dic101#9912yes—
72C2×C101—21no—
73C2×C101C2×#9911yescharacteristic
74D101—42no—
75D101—41no—
76#2129—1011nomaximal, Sylow
77C101#212911yescharacteristic, complemented, Sylow
78#536—1011no—
79#410—1012no—
80#354—1012no—
81#354—1012no—
82C2×#41—1011no—
83C22×Q8—1011no—
84C4⋊Q8—1011no—
85C4⋊Q8#8793811yescharacteristic, complemented
86#124—1011no—
87#124—1011no—
88#123—2022no—
89C22⋊Q8—2022no—
90#117—1011no—
91#117—2022no—
92#99—1014no—
93#99—2022no—
94#41—4042no—
95C2×Q8—1011no—
96C2×Q8—4042no—
97C2×Q8—2021no—
98C2×Q8C2×#8793811yescharacteristic
99C2×Q8C2×#8793811yescharacteristic
100C2×D4—1012no—
101C2×D4—1011no—
102C22×C4—1012no—
103C22×C4—1011no—
104C22×C4—2021no—
105C4⋊C4—22no—
106C4⋊C4—4042no—
107C4⋊C4—2021no—
108C4⋊C4—2022no—
109C22⋊C4—2021no—
110C22⋊C4—2022no—
111C22⋊C4—2022no—
112C22⋊C4—2024no—
113C42—1011no—
114C42—4042no—
115C42C2×#8793811yescharacteristic
116C23—1012no—
117C23—1011no—
118Q8—41no—
119Q8—41no—
120Q8—4041no—
121Q8—4042no—
122D4—4042no—
123D4—4041no—
124C2×C4—21no—
125C2×C4—21no—
126C2×C4—1011no—
127C2×C4—4042no—
128C2×C4—4044no—
129C2×C4—2022no—
130C2×C4—2022no—
131C2×C4—2022no—
132C2×C4—2021no—
133C2×C4—2021no—
134C2×C4C22⋊#8793812yes—
135C2×C4C22×#8793811yescharacteristic
136C22—4042no—
137C22—4041no—
138C22—2022no—
139C22—2021no—
140C22C2×C22⋊#8793811yescharacteristic
141C4—22no—
142C4—21no—
143C4—41no—
144C4—41no—
145C4—8084no—
146C4—1012no—
147C4—4041no—
148C4—4041no—
149C2—21no—
150C2—4042no—
151C2—4041no—
152C2D101⋊#9911yescharacteristic
subgroupG/H
centerC2D101⋊#99
commutatorC2×C4×C101C22×C4
FrattiniC2×C4C22×#87938
FittingC4⋊Q8×C101C4
socleC2×C101C2×#99
prime radicalC2×#13108075C22
cosocleC2×D404C23
hypercenterC4⋊Q8#87938
hypocenterC101#2129
LCSG ▷ C2×C4×C101 ▷ C22×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ C4⋊Q8

Complements

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

NH ≅ G/Ntype
2. #20266084C2semidirect
3. #20266083C2semidirect
9. C2×Q8×D101C4semidirect
10. #15692253C4semidirect
15. #15692212C4semidirect
16. C4⋊Q8×C101C4semidirect
77. C101#2129semidirect
85. C4⋊Q8#87938semidirect

Maximal subgroups

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

subgroupindex
#202661582
#202660842
#202660832
#202657522
#2129101