Finite Group Databasebeta

← #27169449

Subgroups — #27169449

Order 12928.

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

normal subgroups only

This group has 332 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×#15692217C211yesmaximal, characteristic
2C22⋊C8×D101C212yesmaximal, complemented
3#20264903C211yesmaximal, characteristic, complemented
4#20264900C212yesmaximal, complemented
5#324×C101C211yesmaximal, characteristic, complemented
6C22×D404C411yescharacteristic
7C22×C4×D101C411yescharacteristic
8C22×C4×D101C2211yescharacteristic
9C2×C8×D101—22no—
10#15692217—22no—
11#15692217—22no—
12C2×#13108051C411yescharacteristic
13C2×#13108051C2211yescharacteristic
14C2×C4×Dic101C411yescharacteristic
15C22⋊#11717579C2212yes—
16#15692167—22no—
17C2×C4⋊C4×C101C2211yescharacteristic
18C22⋊C8×C101C2212yes—
19C23×D101C2×C411yescharacteristic
20C23×D101C2×C411yescharacteristic
21C2×D404—22no—
22C2×D404—22no—
23C2×D404C814yescomplemented
24C22×Dic101C2×C411yescharacteristic
25C22×Dic101C2×C411yescharacteristic
26C2×C4×D101—21no—
27C2×C4×D101—21no—
28C2×C4×D101—42no—
29C2×C4×D101—42no—
30C2×C4×D101D412yes—
31C2×C4×D101D412yes—
32C2×#11717579—22no—
33C8×D101—44no—
34#13108051—22no—
35#13108051—22no—
36C4×Dic101—21no—
37C4×Dic101—21no—
38C22×C4×C101C2×C411yescharacteristic
39C22×C4×C101C2×C411yescharacteristic
40C22×C4×C101C2311yescharacteristic
41C2×C8×C101—22no—
42C4⋊C4×C101—22no—
43C23×C101C22×C411yescharacteristic
44C22×D101—21no—
45C22×D101—21no—
46C22×D101—41no—
47C22×D101—41no—
48C22×D101—41no—
49C22×D101—41no—
50C22×D101C22⋊C412yes—
51C22×D101C2×C812yes—
52C22×D101C2×C812yescomplemented
53C22×D101M1612yes—
54C2×C4×C101—21no—
55C2×C4×C101—21no—
56C2×C4×C101—21no—
57C2×C4×C101—41no—
58C2×C4×C101—41no—
59C2×C4×C101C2×C812yescomplemented
60C2×C4×C101C2×D412yes—
61C2×Dic101—22no—
62C2×Dic101—21no—
63C2×Dic101—41no—
64C2×Dic101C22⋊C411yescharacteristic
65C2×Dic101C22⋊C411yescharacteristic
66D404—28no—
67C4×D101—24no—
68C4×D101—42no—
69C4×D101—42no—
70C4×D101—42no—
71C8×C101—42no—
72#11717579—42no—
73C22×C101—21no—
74C22×C101—21no—
75C22×C101C2×C22⋊C411yescharacteristic
76C22×C101C22×C811yescharacteristic
77C22×C101C2×M1611yescharacteristic
78C2×D101—22no—
79C2×D101—22no—
80C2×D101—22no—
81C2×D101—42no—
82C2×D101—42no—
83C2×D101—42no—
84C2×D101—41no—
85C2×D101—41no—
86C2×D101—42no—
87C2×D101C22⋊C814yes—
88C4×C101—22no—
89C4×C101—22no—
90C4×C101—41no—
91Dic101—21no—
92Dic101—41no—
93Dic101#9912yes—
94C2×C101—21no—
95C2×C101—21no—
96C2×C101C2×C22⋊C811yescharacteristic
97C2×C101C2×#9911yescharacteristic
98C2×C101#41311yescharacteristic
99D101—24no—
100D101—42no—
101#1459—1011nomaximal, Sylow
102C101#145911yescharacteristic, complemented, Sylow
103C2×C4×D4—1011no—
104C2×C22⋊C8—1012no—
105#324—1011no—
106#324D10111yescharacteristic, complemented
107#323—1012no—
108C22×D4—1011no—
109C23×C4—1011no—
110C23×C4—1011no—
111C22×C8—2022no—
112C4×D4—2022no—
113C4×D4—2022no—
114C2×C4⋊C4C2×D10111yescharacteristic
115C2×C22⋊C4—1011no—
116C2×C22⋊C4—1011no—
117C2×C42—1011no—
118C22⋊C8—1012no—
119C22⋊C8—2022no—
120C22⋊C8C2×D10112yes—
121C24—1011no—
122C24—1011no—
123C2×D4—1014no—
124C2×D4—2022no—
125C2×D4—2022no—
126C22×C4—1011no—
127C22×C4—1012no—
128C22×C4—1012no—
129C22×C4—1011no—
130C22×C4—4042no—
131C22×C4—4042no—
132C22×C4—2021no—
133C22×C4—2021no—
134C22×C4C4×D10111yescharacteristic
135C22×C4C4×D10111yescharacteristic
136C22×C4C22×D10111yescharacteristic
137C2×C8—22no—
138C2×C8—4044no—
139C2×C8—2022no—
140C4⋊C4—22no—
141C22⋊C4—2022no—
142C22⋊C4—2022no—
143C42—2021no—
144C42—2021no—
145C23—1012no—
146C23—1012no—
147C23—1012no—
148C23—1012no—
149C23—4041no—
150C23—4041no—
151C23—4041no—
152C23—4041no—
153C23—2021no—
154C23—2021no—
155C23C2×C4×D10111yescharacteristic
156D4—2028no—
157C2×C4—21no—
158C2×C4—21no—
159C2×C4—21no—
160C2×C4—41no—
161C2×C4—41no—
162C2×C4—1011no—
163C2×C4—1011no—
164C2×C4—4042no—
165C2×C4—4042no—
166C2×C4—4042no—
167C2×C4—4041no—
168C2×C4—2024no—
169C2×C4—2022no—
170C2×C4—2021no—
171C2×C4C8×D10112yescomplemented
172C2×C4D4×D10112yes—
173C8—42no—
174C8—4042no—
175C22—21no—
176C22—21no—
177C22—1014no—
178C22—4042no—
179C22—4042no—
180C22—4042no—
181C22—4042no—
182C22—4041no—
183C22—4041no—
184C22—2022no—
185C22—2022no—
186C22—2022no—
187C22C22⋊C4×D10111yescharacteristic
188C22C2×C8×D10111yescharacteristic
189C22M16×D10111yescharacteristic
190C4—22no—
191C4—22no—
192C4—41no—
193C4—1012no—
194C4—4041no—
195C4—2021no—
196C2—21no—
197C2—21no—
198C2—4042no—
199C2—2024no—
200C2C22⋊C8×D10111yescharacteristic
201C2#99×D10111yescharacteristic
202C2#2026516011yescharacteristic
subgroupG/H
centerC22C22⋊C4×D101
commutatorC22×C101C22×C8
FrattiniC22×C4C22×D101
Fitting#324×C101C2
socleC22×C101C2×C22⋊C4
prime radicalC22×D404C4
cosocleC22×C4×C101C23
hypercenter#324D101
hypocenterC101#1459
LCSG ▷ C22×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ #324

Complements

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

NH ≅ G/Ntype
2. C22⋊C8×D101C2semidirect
3. #20264903C2semidirect
4. #20264900C2semidirect
5. #324×C101C2semidirect
23. C2×D404C8semidirect
52. C22×D101C2×C8semidirect
59. C2×C4×C101C2×C8semidirect
102. C101#1459semidirect
106. #324D101semidirect
171. C2×C4C8×D101semidirect

Maximal subgroups

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

subgroupindex
C2×#156922172
C22⋊C8×D1012
#202649032
#202649002
#324×C1012
#1459101