Finite Group Databasebeta

← #27169443

Subgroups — #27169443

Order 12928.

264 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 79 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
1C2×#15692211C211yesmaximal, characteristic
2#20265227C212yesmaximal, complemented
3#20264902C212yesmaximal, complemented
4#20264901C211yesmaximal, characteristic
5#323×C101C211yesmaximal, characteristic
6C23×Dic101C411yescharacteristic
7#15692211—22no—
8#15692211—22no—
9C2×#13108053C411yescharacteristic
10C2×#13108053C2211yescharacteristic
11C2×C22⋊Dic101C411yescharacteristic
12C2×C4×Dic101C411yescharacteristic
13C2×C4×Dic101C2211yescharacteristic
14#15692184—22no—
15C22⋊#11717579C2212yes—
16C8×Dic101—22no—
17C2×C22⋊C4×C101C2211yescharacteristic
18C22⋊C8×C101C2212yes—
19C22×Dic101—21no—
20C22×Dic101—21no—
21C22×Dic101—22no—
22C22×Dic101C814yescomplemented
23C22×Dic101C2×C411yescharacteristic
24C22×Dic101C2×C411yescharacteristic
25C22×Dic101C2×C411yescharacteristic
26C22×Dic101C2×C411yescharacteristic
27C2×#11717579—22no—
28#13108053—22no—
29#13108053—22no—
30C22⋊Dic101—22no—
31C4×Dic101—21no—
32C4×Dic101—21no—
33C4×Dic101D412yes—
34C4×Dic101D412yes—
35C24×C101C2×C411yescharacteristic
36C22×C4×C101C2×C411yescharacteristic
37C22×C4×C101C2311yescharacteristic
38C2×C8×C101—22no—
39C22⋊C4×C101—22no—
40C23×C101—21no—
41C23×C101—21no—
42C23×C101—41no—
43C23×C101—41no—
44C23×C101C2×C812yes—
45C23×C101C22×C411yescharacteristic
46C2×C4×C101—21no—
47C2×C4×C101—41no—
48C2×C4×C101—41no—
49C2×C4×C101C2×D412yes—
50C2×Dic101—21no—
51C2×Dic101—21no—
52C2×Dic101—24no—
53C2×Dic101—24no—
54C2×Dic101—21no—
55C2×Dic101—41no—
56C2×Dic101—41no—
57C2×Dic101C22⋊C412yes—
58C2×Dic101C22⋊C411yescharacteristic
59C2×Dic101C22⋊C411yescharacteristic
60C2×Dic101C2×C812yes—
61C2×Dic101C2×C812yes—
62C2×Dic101M1612yes—
63C8×C101—42no—
64#11717579—42no—
65C22×C101—21no—
66C22×C101—21no—
67C22×C101—22no—
68C22×C101—42no—
69C22×C101—42no—
70C22×C101—41no—
71C22×C101—41no—
72C22×C101C2×C22⋊C411yescharacteristic
73C22×C101C22×C811yescharacteristic
74C22×C101C2×M1611yescharacteristic
75C4×C101—22no—
76C4×C101—41no—
77Dic101—22no—
78Dic101—22no—
79Dic101—41no—
80Dic101C22⋊C812yes—
81Dic101C22⋊C812yes—
82C2×C101—21no—
83C2×C101—21no—
84C2×C101—42no—
85C2×C101C2×C22⋊C811yescharacteristic
86C2×C101#41011yescharacteristic
87C2×C101#41311yescharacteristic
88#1458—1011nomaximal, Sylow
89C101#145811yescharacteristic, complemented, Sylow
90C2×#117—1011no—
91#431—1012no—
92#324—1012no—
93#323—1011no—
94#323D10111yescharacteristic
95C23×C4—1011no—
96#117—2022no—
97#117—2022no—
98C2×C4⋊C4—1011no—
99C2×C4⋊C4—1011no—
100C2×C22⋊C4—1011no—
101C2×C22⋊C4C2×D10111yescharacteristic
102C2×C42—1011no—
103C2×C42—1011no—
104C4⋊C8—2022no—
105C22⋊C8—1012no—
106C22⋊C8C2×D10112yes—
107C4×C8—2022no—
108C24C4×D10111yescharacteristic
109C22×C4—1011no—
110C22×C4—1011no—
111C22×C4—1014no—
112C22×C4—1011no—
113C22×C4—1011no—
114C22×C4—2021no—
115C22×C4—2021no—
116C22×C4—2022no—
117C22×C4C4×D10111yescharacteristic
118C22×C4C22×D10111yescharacteristic
119C2×C8—22no—
120C2×C8—2022no—
121C4⋊C4—2022no—
122C4⋊C4—2022no—
123C22⋊C4—22no—
124C22⋊C4—2022no—
125C42—1012no—
126C42—1012no—
127C42—2021no—
128C42—2021no—
129C23—21no—
130C23—21no—
131C23—41no—
132C23—41no—
133C23C8×D10112yes—
134C23C2×C4×D10111yescharacteristic
135C2×C4—21no—
136C2×C4—41no—
137C2×C4—41no—
138C2×C4—1011no—
139C2×C4—1011no—
140C2×C4—1012no—
141C2×C4—1012no—
142C2×C4—1012no—
143C2×C4—1012no—
144C2×C4—4041no—
145C2×C4—4041no—
146C2×C4—2024no—
147C2×C4—2024no—
148C2×C4—2021no—
149C2×C4—2021no—
150C2×C4—2021no—
151C2×C4D4×D10112yes—
152C8—42no—
153C8—4042no—
154C22—21no—
155C22—21no—
156C22—22no—
157C22—42no—
158C22—42no—
159C22—41no—
160C22—41no—
161C22C22⋊C4×D10111yescharacteristic
162C22C2×C8×D10111yescharacteristic
163C22M16×D10111yescharacteristic
164C4—22no—
165C4—41no—
166C4—1012no—
167C4—1012no—
168C4—4041no—
169C4—2022no—
170C4—2022no—
171C2—21no—
172C2—21no—
173C2—42no—
174C2C22⋊C8×D10111yescharacteristic
175C2#2026514911yescharacteristic
176C2#2026515811yescharacteristic
subgroupG/H
centerC22C22⋊C4×D101
commutatorC22×C101C22×C8
FrattiniC22×C4C22×D101
Fitting#323×C101C2
socleC22×C101C2×C22⋊C4
prime radicalC24×C101C2×C4
cosocleC22×C4×C101C23
hypercenter#323D101
hypocenterC101#1458
LCSG ▷ C22×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ #323

Complements

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

NH ≅ G/Ntype
2. #20265227C2semidirect
3. #20264902C2semidirect
22. C22×Dic101C8semidirect
89. C101#1458semidirect

Maximal subgroups

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

subgroupindex
C2×#156922112
#202652272
#202649022
#202649012
#323×C1012
#1458101