Finite Group Databasebeta

← #27186451

Subgroups — #27186451

Order 12928.

262 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 62 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#20266193C211yesmaximal, characteristic, complemented
2C4×#13108083C211yesmaximal, characteristic, complemented
3#20266025C212yesmaximal, complemented
4D4⋊#11717581C212yesmaximal, complemented
5#20265722C211yesmaximal, characteristic
6C2×#13108083—21no—
7C4⋊#11717581C2212yes—
8C22⋊#11717581—21no—
9#15692310—21no—
10C4×#11717581—21no—
11C4×#11717581C2212yes—
12C2×#13108073—21no—
13#15692254C411yescharacteristic
14#15692251C411yescharacteristic
15#15692241—21no—
16#15692229C411yescharacteristic
17#15692227—21no—
18D4×Dic101C2211yescharacteristic
19#15692217C2211yescharacteristic
20#15692215—21no—
21C2×C4×Dic101C2211yescharacteristic
22#121×C101C411yescharacteristic
23#13108083—24no—
24C2×#11717581—22no—
25C2×#11717581—22no—
26#13108073—44no—
27C2×Dic202—21no—
28C2×D404C2×C411yescharacteristic
29C2×C22⋊D101—21no—
30C2×C22⋊D101—21no—
31C2×C22⋊D101—21no—
32C22×Dic101—21no—
33C22×Dic101D411yescharacteristic
34C22×Dic101D411yescharacteristic
35C2×C4×D101—21no—
36C4⋊Dic101C2×C411yescharacteristic
37#13108053—21no—
38C22⋊Dic101—21no—
39#13108051—21no—
40#13108051—21no—
41C4×Dic101C2×C411yescharacteristic
42C4×Dic101D411yescharacteristic
43C4×Dic101D411yescharacteristic
44C4×Dic101C2311yescharacteristic
45C2×D4×C101—21no—
46C2×D4×C101C2×C411yescharacteristic
47C22×C4×C101C2×C411yescharacteristic
48C4⋊C4×C101C2×C411yescharacteristic
49C22⋊C4×C101—21no—
50C23×C101—21no—
51C23×C101C22⋊C411yescharacteristic
52C22×D101—21no—
53D4×C101—22no—
54D4×C101—42no—
55C2×C4×C101—22no—
56C2×C4×C101—21no—
57C2×C4×C101C22⋊C411yescharacteristic
58C2×C4×C101C22×C411yescharacteristic
59C22⋊D101—42no—
60C22⋊D101—42no—
61C22⋊D101—42no—
62C2×Dic101—22no—
63C2×Dic101—22no—
64C2×Dic101—21no—
65C2×Dic101—42no—
66C2×Dic101C22⋊C411yescharacteristic
67C2×Dic101C22⋊C411yescharacteristic
68C2×Dic101C2×D411yescharacteristic
69C2×Dic101C2×D411yescharacteristic
70D404—22no—
71C4×D101—42no—
72Dic202—42no—
73#11717581—24no—
74#11717581—42no—
75C22×C101—21no—
76C22×C101—21no—
77C22×C101—21no—
78C22×C101—41no—
79C22×C101—41no—
80C22×C101—41no—
81C22×C101C2×C22⋊C411yescharacteristic
82C2×D101—41no—
83C2×D101—41no—
84C2×D101—41no—
85C4×C101—21no—
86C4×C101—41no—
87C4×C101#13112yes—
88Dic101—21no—
89Dic101—21no—
90Dic101—41no—
91Dic101#10412yes—
92Dic101AGL(1,ℤ/8)12yes—
93C2×C101—41no—
94C2×C101—81no—
95C2×C101#40811yescharacteristic
96C2×C101#41811yescharacteristic
97C2×C101C2×#10411yescharacteristic
98D101—81no—
99#1484—1011nomaximal, Sylow
100C101#148411yescharacteristic, complemented, Sylow
101#532—1011no—
102#431—1011no—
103C4×M16—1011no—
104D4⋊C8—1012no—
105D4⋊C8—1012no—
106C2×#41—2021no—
107C2×M16—2021no—
108C4⋊Q8—1011no—
109#127—1011no—
110#124—2021no—
111#121—1011no—
112#121—2021no—
113#121#8793811yescharacteristic
114C4×D4—1011no—
115C4×D4—1011no—
116C4×D4—2021no—
117C2×C42—1011no—
118C4⋊C8—1012no—
119C22⋊C8—2021no—
120#97—2021no—
121C4×C8—1012no—
122C4×C8—2021no—
123#41—4044no—
124C2×Q8—2021no—
125C2×D4—21no—
126C2×D4—1011no—
127C2×D4—2021no—
128C2×D4—2021no—
129C2×D4—2021no—
130C2×D4C2×#8793811yescharacteristic
131C22×C4—1011no—
132C22×C4—1011no—
133C22×C4—2021no—
134C22×C4—2021no—
135C22×C4C2×#8793811yescharacteristic
136M16—2024no—
137C2×C8—2022no—
138C2×C8—2022no—
139C4⋊C4—1011no—
140C4⋊C4—2021no—
141C4⋊C4C2×#8793811yescharacteristic
142C22⋊C4—21no—
143C22⋊C4—2021no—
144C22⋊C4—2021no—
145C22⋊C4—2021no—
146C42—1011no—
147C42—1011no—
148C42—1011no—
149C42—1011no—
150C23—21no—
151C23—2021no—
152C23C22⋊#8793811yescharacteristic
153Q8—4042no—
154D4—22no—
155D4—42no—
156D4—4042no—
157D4—4042no—
158D4—4042no—
159D4—2022no—
160C2×C4—22no—
161C2×C4—21no—
162C2×C4—1011no—
163C2×C4—1011no—
164C2×C4—1011no—
165C2×C4—1011no—
166C2×C4—4042no—
167C2×C4—4042no—
168C2×C4—2022no—
169C2×C4—2022no—
170C2×C4—2021no—
171C2×C4C22⋊#8793811yescharacteristic
172C2×C4C22×#8793811yescharacteristic
173C8—4042no—
174C8—2024no—
175C22—21no—
176C22—21no—
177C22—21no—
178C22—41no—
179C22—41no—
180C22—41no—
181C22—4041no—
182C22—4041no—
183C22—4041no—
184C22C2×C22⋊#8793811yescharacteristic
185C4—21no—
186C4—41no—
187C4—1012no—
188C4—1012no—
189C4—4041no—
190C4—2021no—
191C4—2021no—
192C4#1569234212yes—
193C2—41no—
194C2—81no—
195C2—8081no—
196C2#2026615011yescharacteristic
197C2#2026616911yescharacteristic
198C2D101⋊#10411yescharacteristic
subgroupG/H
centerC22C2×C22⋊#87938
commutatorC2×C4×C101C22×C4
FrattiniC2×C4C22×#87938
Fitting#121×C101C4
socleC22×C101C2×C22⋊C4
prime radical#20265722C2
cosocleC4×Dic101C23
hypercenter#121#87938
hypocenterC101#1484
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ #121

Complements

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

NH ≅ G/Ntype
1. #20266193C2semidirect
2. C4×#13108083C2semidirect
3. #20266025C2semidirect
4. D4⋊#11717581C2semidirect
100. C101#1484semidirect

Maximal subgroups

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

subgroupindex
#202661932
C4×#131080832
#202660252
D4⋊#117175812
#202657222
#1484101