Finite Group Databasebeta

← #27169451

Subgroups — #27169451

Order 12928.

312 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 81 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×#15692215C211yesmaximal, characteristic
2#20265227C212yesmaximal, complemented
3#20264901C211yesmaximal, characteristic, complemented
4#20264900C212yesmaximal, complemented
5#323×C101C211yesmaximal, characteristic, complemented
6C22×C22⋊D101C411yescharacteristic
7C23×Dic101C411yescharacteristic
8C22×C4×D101C411yescharacteristic
9#15692215—22no—
10#15692215—22no—
11C2×#13108053C411yescharacteristic
12C2×#13108051C2211yescharacteristic
13C2×C4×Dic101C2211yescharacteristic
14#15692184—22no—
15C22⋊#11717579C2212yes—
16C8×Dic101—22no—
17C2×C22⋊C4×C101C2211yescharacteristic
18C22⋊C8×C101C2212yes—
19C23×D101C2×C411yescharacteristic
20C2×C22⋊D101—22no—
21C2×C22⋊D101—22no—
22C2×C22⋊D101C814yescomplemented
23C22×Dic101—21no—
24C22×Dic101—21no—
25C22×Dic101—42no—
26C22×Dic101C2×C411yescharacteristic
27C22×Dic101C2×C411yescharacteristic
28C22×Dic101C2×C411yescharacteristic
29C2×C4×D101—21no—
30C2×C4×D101—21no—
31C2×C4×D101—42no—
32C2×#11717579—22no—
33#13108053—22no—
34#13108051—22no—
35C4×Dic101D412yes—
36C4×Dic101D412yes—
37C24×C101C2×C411yescharacteristic
38C22×C4×C101C2×C411yescharacteristic
39C22×C4×C101C2311yescharacteristic
40C2×C8×C101—22no—
41C22⋊C4×C101—22no—
42C23×C101—21no—
43C23×C101—21no—
44C23×C101—41no—
45C23×C101—41no—
46C23×C101C2×C812yes—
47C23×C101C22×C411yescharacteristic
48C22×D101—21no—
49C22×D101—21no—
50C22×D101—41no—
51C22×D101—41no—
52C22×D101C2×C812yes—
53C2×C4×C101—21no—
54C2×C4×C101—41no—
55C2×C4×C101—41no—
56C2×C4×C101C2×D412yes—
57C22⋊D101—28no—
58C2×Dic101—22no—
59C2×Dic101—21no—
60C2×Dic101—42no—
61C2×Dic101—42no—
62C2×Dic101—41no—
63C2×Dic101C22⋊C411yescharacteristic
64C2×Dic101C22⋊C412yes—
65C2×Dic101C22⋊C411yescharacteristic
66C2×Dic101C2×C812yes—
67C2×Dic101M1612yes—
68C4×D101—42no—
69C4×D101—42no—
70C8×C101—42no—
71#11717579—42no—
72C22×C101—21no—
73C22×C101—21no—
74C22×C101—22no—
75C22×C101—42no—
76C22×C101—42no—
77C22×C101—41no—
78C22×C101—41no—
79C22×C101C2×C22⋊C411yescharacteristic
80C22×C101C22×C811yescharacteristic
81C22×C101C2×M1611yescharacteristic
82C2×D101—22no—
83C2×D101—42no—
84C2×D101—41no—
85C2×D101—41no—
86C2×D101—42no—
87C4×C101—22no—
88C4×C101—41no—
89Dic101—21no—
90Dic101—41no—
91Dic101C22⋊C814yes—
92Dic101#10012yes—
93C2×C101—21no—
94C2×C101—21no—
95C2×C101—42no—
96C2×C101C2×C22⋊C811yescharacteristic
97C2×C101#41011yescharacteristic
98C2×C101C2×#10011yescharacteristic
99D101—42no—
100#1460—1011nomaximal, Sylow
101C101#146011yescharacteristic, complemented, Sylow
102C2×C4×D4—1011no—
103#431—1012no—
104#323—1011no—
105#323—1012no—
106#323D10111yescharacteristic, complemented
107C22×D4—1011no—
108C23×C4—1011no—
109C23×C4—1011no—
110C4×D4—2022no—
111C4×D4—2022no—
112C2×C4⋊C4—1011no—
113C2×C22⋊C4—1011no—
114C2×C22⋊C4C2×D10111yescharacteristic
115C2×C42—1011no—
116C4⋊C8—2022no—
117C22⋊C8—1012no—
118C22⋊C8C2×D10112yes—
119C4×C8—2022no—
120C24—1011no—
121C24C4×D10111yescharacteristic
122C2×D4—1014no—
123C2×D4—2022no—
124C2×D4—2022no—
125C22×C4—1011no—
126C22×C4—1011no—
127C22×C4—1011no—
128C22×C4—4042no—
129C22×C4—4042no—
130C22×C4—2021no—
131C22×C4—2021no—
132C22×C4—2021no—
133C22×C4—2021no—
134C22×C4C4×D10111yescharacteristic
135C22×C4C22×D10111yescharacteristic
136C2×C8—22no—
137C2×C8—2022no—
138C4⋊C4—2022no—
139C22⋊C4—22no—
140C22⋊C4—2022no—
141C42—1012no—
142C42—1012no—
143C23—21no—
144C23—21no—
145C23—41no—
146C23—41no—
147C23—1012no—
148C23—4041no—
149C23—4041no—
150C23—2021no—
151C23—2021no—
152C23C8×D10112yes—
153C23C2×C4×D10111yescharacteristic
154D4—2028no—
155C2×C4—21no—
156C2×C4—41no—
157C2×C4—41no—
158C2×C4—1012no—
159C2×C4—1011no—
160C2×C4—1012no—
161C2×C4—1012no—
162C2×C4—1011no—
163C2×C4—4042no—
164C2×C4—4042no—
165C2×C4—4042no—
166C2×C4—4042no—
167C2×C4—4041no—
168C2×C4—2022no—
169C2×C4—2021no—
170C2×C4D4×D10112yes—
171C8—42no—
172C8—4042no—
173C22—21no—
174C22—21no—
175C22—22no—
176C22—42no—
177C22—42no—
178C22—41no—
179C22—41no—
180C22—4042no—
181C22—4042no—
182C22—4041no—
183C22—4041no—
184C22—2022no—
185C22C22⋊C4×D10111yescharacteristic
186C22C2×C8×D10111yescharacteristic
187C22M16×D10111yescharacteristic
188C4—22no—
189C4—41no—
190C4—1014no—
191C4—1012no—
192C4—4041no—
193C4—2021no—
194C2—21no—
195C2—21no—
196C2—42no—
197C2—4042no—
198C2C22⋊C8×D10111yescharacteristic
199C2#2026514911yescharacteristic
200C2#100×D10111yescharacteristic
subgroupG/H
centerC22C22⋊C4×D101
commutatorC22×C101C22×C8
FrattiniC22×C4C22×D101
Fitting#323×C101C2
socleC22×C101C2×C22⋊C4
prime radicalC22×C22⋊D101C4
cosocleC22×C4×C101C23
hypercenter#323D101
hypocenterC101#1460
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. #20264901C2semidirect
4. #20264900C2semidirect
5. #323×C101C2semidirect
22. C2×C22⋊D101C8semidirect
101. C101#1460semidirect
106. #323D101semidirect

Maximal subgroups

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

subgroupindex
C2×#156922152
#202652272
#202649012
#202649002
#323×C1012
#1460101