Finite Group Databasebeta

← #27187321

Subgroups — #27187321

Order 12928.

324 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 74 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#20266119C211yesmaximal, characteristic
2C4×C22⋊#87938C211yesmaximal, characteristic
3#20266055C214yesmaximal
4C2×#15692231C211yesmaximal, characteristic
5C2×C4⋊#87938—21no—
6C2×C22⋊#87938C2212yes—
7C2×C22⋊#87938C2212yes—
8C2×C4×#87938—21no—
9(C2×#87938)⋊C4—21no—
10(C2×#87938)⋊C4—21no—
11C22×C4×D101C2211yescharacteristic
12#15692231—21no—
13#15692231—21no—
14#15692231—21no—
15#15692231—21no—
16C2×C4⋊Dic101C411yescharacteristic, complemented
17C2×#13108053C412yescomplemented
18C2×#13108051C2212yes—
19C22×Q8×C101C411yescharacteristic, complemented
20C22×#87938—22no—
21C22×#87938—22no—
22C4⋊#87938—42no—
23C22⋊#87938—22no—
24C22⋊#87938—24no—
25C22⋊#87938Q814yescomplemented
26C4×#87938—42no—
27C23×D101C2311yescharacteristic
28C22×Dic101C2×C412yes—
29C22×Dic101C2×C411yescharacteristic
30C2×C4×D101—42no—
31C2×C4×D101D411yescharacteristic
32C2×C4×D101D411yescharacteristic
33C2×C4×D101D411yescharacteristic
34C2×C4×D101D411yescharacteristic
35C4⋊Dic101—21no—
36C4⋊Dic101—21no—
37#13108053—22no—
38#13108053—22no—
39#13108051—22no—
40#13108051—22no—
41C2×Q8×C101—21no—
42C2×Q8×C101—21no—
43C2×Q8×C101—42no—
44C22×C4×C101C2×C412yes—
45C22×C4×C101C2×C411yescharacteristic
46C23×C101C22×C411yescharacteristic
47C22×D101—41no—
48C22×D101—41no—
49C22×D101C2×D411yescharacteristic
50C22×D101C2×D411yescharacteristic
51C22×D101C2×Q812yes—
52C22×D101#4112yes—
53C2×#87938—24no—
54C2×#87938—42no—
55C2×#87938—42no—
56C2×#87938—42no—
57C2×#87938—42no—
58C2×#87938—42no—
59Q8×C101—42no—
60Q8×C101—42no—
61C2×C4×C101—22no—
62C2×C4×C101—22no—
63C2×C4×C101—42no—
64C2×C4×C101C22⋊C411yescharacteristic
65C2×C4×C101C22⋊C411yescharacteristic
66C2×Dic101—22no—
67C2×Dic101—41no—
68C2×Dic101—42no—
69C2×Dic101C22⋊C411yescharacteristic
70C2×Dic101C22⋊C411yescharacteristic
71C4×D101—22no—
72C4×D101—22no—
73C4×D101—41no—
74C4×D101—41no—
75C22×C101—21no—
76C22×C101—21no—
77C22×C101C2×C22⋊C411yescharacteristic
78C22×C101C4×Q812yes—
79C2×D101—21no—
80C2×D101—21no—
81C2×D101—21no—
82C2×D101—21no—
83C2×D101—42no—
84C2×D101—42no—
85C2×D101C22⋊Q812yes—
86C2×D101#12411yescharacteristic
87C2×D101C4⋊Q811yescharacteristic
88#87938—44no—
89#87938—82no—
90C4×C101—21no—
91C4×C101—42no—
92C4×C101#9912yes—
93Dic101—21no—
94Dic101—21no—
95Dic101—42no—
96C2×C101—22no—
97C2×C101#39111yescharacteristic
98C2×C101C2×#9911yescharacteristic
99C2×C101#41011yescharacteristic
100D101—22no—
101D101—21no—
102D101—21no—
103#1996—1011nomaximal, Sylow
104C101#199611yescharacteristic, complemented, Sylow
105C2×C22⋊Q8—1011no—
106#380—1011no—
107C4×C22⋊C4—1011no—
108#342—1014no—
109C22×Q8#8793811yescharacteristic, complemented
110C23×C4—1011no—
111C22⋊Q8—2021no—
112C22⋊Q8—2021no—
113C22⋊Q8—2021no—
114C22⋊Q8—2021no—
115C2×C4⋊C4—1012no—
116C2×C4⋊C4—1011no—
117C2×C4⋊C4—2021no—
118C2×C22⋊C4—1012no—
119C2×C22⋊C4—1012no—
120C2×C22⋊C4—1012no—
121C2×C42—2021no—
122#95—2021no—
123#95—2021no—
124C24—1011no—
125C2×Q8—21no—
126C2×Q8—21no—
127C2×Q8—42no—
128C22×C4—1011no—
129C22×C4—1011no—
130C22×C4—1011no—
131C22×C4—1011no—
132C22×C4—1011no—
133C22×C4—1012no—
134C22×C4—4042no—
135C22×C4—2022no—
136C22×C4—2022no—
137C22×C4C2×#8793812yes—
138C22×C4C2×#8793811yescharacteristic
139C4⋊C4—4042no—
140C4⋊C4—2022no—
141C4⋊C4—2022no—
142C4⋊C4—2021no—
143C4⋊C4—2021no—
144C22⋊C4—1014no—
145C22⋊C4—2022no—
146C22⋊C4—2022no—
147C22⋊C4—2024no—
148C22⋊C4—2022no—
149C42—4042no—
150C23—1012no—
151C23—1012no—
152C23—1011no—
153C23—1011no—
154C23—4041no—
155C23—4041no—
156C23C22×#8793811yescharacteristic
157Q8—42no—
158Q8—42no—
159C2×C4—22no—
160C2×C4—22no—
161C2×C4—42no—
162C2×C4—1011no—
163C2×C4—1011no—
164C2×C4—4041no—
165C2×C4—4041no—
166C2×C4—4042no—
167C2×C4—4042no—
168C2×C4—4042no—
169C2×C4—4042no—
170C2×C4—4042no—
171C2×C4—4042no—
172C2×C4—4041no—
173C2×C4—2024no—
174C2×C4—2022no—
175C2×C4—2022no—
176C2×C4—2022no—
177C2×C4C22⋊#8793811yescharacteristic
178C2×C4C22⋊#8793811yescharacteristic
179C22—21no—
180C22—21no—
181C22—1011no—
182C22—1012no—
183C22—1011no—
184C22—4042no—
185C22—4042no—
186C22—2021no—
187C22—2021no—
188C22—2021no—
189C22—2021no—
190C22C2×C22⋊#8793811yescharacteristic
191C22Q8×#8793812yes—
192C4—21no—
193C4—42no—
194C4—8082no—
195C4—4044no—
196C4—4042no—
197C4—2021no—
198C4—2021no—
199C4#1569231312yes—
200C2—22no—
201C2—2021no—
202C2—2022no—
203C2—2021no—
204C2(C2×#87938)⋊Q811yescharacteristic
205C2C2×#1569231311yescharacteristic
206C2#2026615811yescharacteristic
subgroupG/H
centerC22C2×C22⋊#87938
commutatorC23×C101C22×C4
FrattiniC23C22×#87938
FittingC22×Q8×C101C4
socleC22×C101C2×C22⋊C4
prime radical, cosocleC23×D101C23
hypercenterC22×Q8#87938
hypocenterC101#1996
LCSG ▷ C23×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×Q8

Complements

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

NH ≅ G/Ntype
16. C2×C4⋊Dic101C4semidirect
17. C2×#13108053C4semidirect
19. C22×Q8×C101C4semidirect
25. C22⋊#87938Q8semidirect
104. C101#1996semidirect
109. C22×Q8#87938semidirect

Maximal subgroups

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

subgroupindex
#202661192
C4×C22⋊#879382
#202660552
C2×#156922312
#1996101