Finite Group Databasebeta

← #27186566

Subgroups — #27186566

Order 12928.

220 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 66 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#20266195C211yesmaximal, characteristic
2#20266121C211yesmaximal, characteristic
3#20266031C214yesmaximal
4#20265732C211yesmaximal, characteristic
5C2×C4⋊#87938—21no—
6#15692322—21no—
7#15692312C2212yes—
8C4×#11717581—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—
22C4⋊#87938—42no—
23C2×Dic202—21no—
24C2×C4×D101D411yescharacteristic
25C2×C4×D101D411yescharacteristic
26C2×C4×D101C2311yescharacteristic
27C4⋊Dic101—21no—
28C4⋊Dic101—22no—
29C4⋊Dic101C2×C412yes—
30#13108053—22no—
31#13108051—22no—
32C4×Dic101—22no—
33C4×Dic101C2×C411yescharacteristic
34C4×Dic101D411yescharacteristic
35C4×Dic101D411yescharacteristic
36C2×Q8×C101C2×C412yes—
37C4⋊C4×C101—21no—
38C4⋊C4×C101—21no—
39C42×C101C2×C411yescharacteristic
40C22×D101C2×D411yescharacteristic
41C2×#87938—42no—
42C2×#87938—42no—
43C2×#87938—42no—
44Q8×C101—42no—
45C2×C4×C101—22no—
46C2×C4×C101C22⋊C411yescharacteristic
47C2×C4×C101C22⋊C411yescharacteristic
48C2×C4×C101C22×C411yescharacteristic
49C2×Dic101—22no—
50C2×Dic101C22⋊C411yescharacteristic
51C2×Dic101C22⋊C411yescharacteristic
52C2×Dic101C2×D411yescharacteristic
53C4×D101—41no—
54C4×D101—41no—
55C4×D101SD1612yes—
56C4×D101Q1612yes—
57Dic202—42no—
58#11717581—42no—
59C22×C101C2×C22⋊C411yescharacteristic
60C2×D101—21no—
61C2×D101—21no—
62C2×D101C2×SD1611yescharacteristic
63C2×D101C2×Q1611yescharacteristic
64#87938—82no—
65C4×C101—21no—
66C4×C101—21no—
67C4×C101—42no—
68C4×C101#10312yes—
69Dic101—21no—
70Dic101—21no—
71Dic101—42no—
72Dic101#10312yes—
73C2×C101#41011yescharacteristic
74C2×C101C2×#10311yescharacteristic
75C2×C101#42111yescharacteristic
76D101—21no—
77D101—21no—
78#1518—1011nomaximal, Sylow
79C101#151811yescharacteristic, complemented, Sylow
80#533—1011no—
81#431—1011no—
82#382—1011no—
83#328—1014no—
84C4⋊Q8—1011no—
85C4⋊Q8#8793811yescharacteristic, complemented
86#125—2021no—
87C22⋊Q8—1012no—
88C22⋊Q8—2021no—
89C4×Q8—1012no—
90C4×Q8—2021no—
91#117—2021no—
92C2×C4⋊C4—2021no—
93C2×C42—1011no—
94C4⋊C8—2021no—
95C22⋊C8—1012no—
96C4×C8—2021no—
97#95—1012no—
98#95—2021no—
99C2×Q8—2021no—
100C2×Q8C2×#8793812yes—
101C22×C4—1011no—
102C22×C4—1011no—
103C22×C4—1011no—
104C22×C4—2022no—
105C2×C8—2022no—
106C4⋊C4—21no—
107C4⋊C4—21no—
108C4⋊C4—1012no—
109C4⋊C4—4042no—
110C4⋊C4—2022no—
111C4⋊C4—2022no—
112C4⋊C4—2021no—
113C22⋊C4—2022no—
114C42—1011no—
115C42—1011no—
116C42—1011no—
117C42—2022no—
118C42C2×#8793811yescharacteristic
119C23—1011no—
120Q8—42no—
121Q8—4042no—
122C2×C4—22no—
123C2×C4—1011no—
124C2×C4—1012no—
125C2×C4—1012no—
126C2×C4—1011no—
127C2×C4—1011no—
128C2×C4—4041no—
129C2×C4—4041no—
130C2×C4—4042no—
131C2×C4—4042no—
132C2×C4—4042no—
133C2×C4—2022no—
134C2×C4C22⋊#8793811yescharacteristic
135C2×C4C22⋊#8793811yescharacteristic
136C2×C4C22×#8793811yescharacteristic
137C8—4042no—
138C22—1011no—
139C22—1011no—
140C22—2021no—
141C22—2021no—
142C22C2×C22⋊#8793811yescharacteristic
143C4—21no—
144C4—21no—
145C4—42no—
146C4—8082no—
147C4—1012no—
148C4—4042no—
149C4—2021no—
150C4—2021no—
151C4#1569231812yes—
152C2—2021no—
153C2—2021no—
154C2#2026615711yescharacteristic
155C2C2×#1569231811yescharacteristic
156C2#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#1518
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
79. C101#1518semidirect
85. C4⋊Q8#87938semidirect

Maximal subgroups

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

subgroupindex
#202661952
#202661212
#202660312
#202657322
#1518101