Finite Group Databasebeta

← #27186653

Subgroups — #27186653

Order 12928.

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

normal subgroups only

This group has 208 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#20266197C211yesmaximal, characteristic
2C2×#15692322C211yesmaximal, characteristic
3#20266036C212yesmaximal, complemented
4#20266036C212yesmaximal, complemented
5#20265730C211yesmaximal, characteristic
6C22×#11717581—21no—
7#15692322—21no—
8#15692322C2212yes—
9#15692322C2212yes—
10C22⋊#11717581—21no—
11#15692310—21no—
12#15692254C2211yescharacteristic
13#15692254C2211yescharacteristic
14#15692246C411yescharacteristic
15#15692246C411yescharacteristic
16#15692232—21no—
17#15692232—21no—
18C4×Dic202—21no—
19C4×Dic202—21no—
20#15692214C411yescharacteristic
21C2×C4×Dic101C2211yescharacteristic
22#117×C101C411yescharacteristic
23C2×#11717581—22no—
24C2×#11717581—22no—
25C2×#11717581—42no—
26C2×Dic202—21no—
27C2×Dic202—21no—
28C22×Dic101D411yescharacteristic
29C22×Dic101D411yescharacteristic
30C4⋊Dic101—21no—
31C4⋊Dic101—21no—
32C4⋊Dic101C2×C411yescharacteristic
33C4⋊Dic101C2×C411yescharacteristic
34#13108053—21no—
35#13108053—21no—
36#13108053—21no—
37#13108053—21no—
38C22⋊Dic101—21no—
39C4×Dic101—21no—
40C4×Dic101C2×C411yescharacteristic
41C4×Dic101D411yescharacteristic
42C4×Dic101D411yescharacteristic
43C4×Dic101C2311yescharacteristic
44C22×C4×C101C2×C411yescharacteristic
45C4⋊C4×C101C2×C411yescharacteristic
46C4⋊C4×C101C2×C411yescharacteristic
47C22⋊C4×C101—21no—
48C42×C101—21no—
49C23×C101C22⋊C411yescharacteristic
50C2×C4×C101—22no—
51C2×C4×C101—21no—
52C2×C4×C101—21no—
53C2×C4×C101C22⋊C411yescharacteristic
54C2×C4×C101C22×C411yescharacteristic
55C2×Dic101—22no—
56C2×Dic101—21no—
57C2×Dic101—21no—
58C2×Dic101—21no—
59C2×Dic101—21no—
60C2×Dic101C22⋊C411yescharacteristic
61C2×Dic101C22⋊C411yescharacteristic
62C2×Dic101C2×D411yescharacteristic
63C2×Dic101C2×D411yescharacteristic
64Dic202—42no—
65Dic202—41no—
66Dic202—41no—
67#11717581—42no—
68#11717581—42no—
69C22×C101—21no—
70C22×C101—21no—
71C22×C101#10112yes—
72C22×C101C2×C22⋊C411yescharacteristic
73C4×C101—21no—
74C4×C101—21no—
75C4×C101—41no—
76C4×C101—41no—
77Dic101—21no—
78Dic101—21no—
79Dic101—41no—
80Dic101—41no—
81Dic101#13512yes—
82Dic101AGL(1,ℤ/8)11yescharacteristic
83Dic101#13711yescharacteristic
84C2×C101—22no—
85C2×C101C2×#10111yescharacteristic
86C2×C101#41611yescharacteristic
87C2×C101#41711yescharacteristic
88#1552—1011nomaximal, Sylow
89C101#155211yescharacteristic, complemented, Sylow
90#533—1011no—
91#432—1011no—
92C2×C4⋊C8—1011no—
93#332—1012no—
94#332—1012no—
95C22×C8—2021no—
96C4⋊Q8—1011no—
97C4⋊Q8—1011no—
98#125—1011no—
99#125—1011no—
100C22⋊Q8—2021no—
101C22⋊Q8—2021no—
102C4×Q8—2021no—
103C4×Q8—2021no—
104#117—1011no—
105#117#8793811yescharacteristic
106C2×C42—1011no—
107C4⋊C8—1012no—
108C4⋊C8—1012no—
109C4⋊C8—2021no—
110C22⋊C8—2021no—
111#97—2021no—
112C2×Q8—2021no—
113C2×Q8—2021no—
114C22×C4—1011no—
115C22×C4—1011no—
116C22×C4C2×#8793811yescharacteristic
117C2×C8—4042no—
118C2×C8—2022no—
119C2×C8—2022no—
120C4⋊C4—1011no—
121C4⋊C4—1011no—
122C4⋊C4—2021no—
123C4⋊C4—2021no—
124C4⋊C4—2021no—
125C4⋊C4—2021no—
126C4⋊C4—2021no—
127C4⋊C4—2021no—
128C4⋊C4C2×#8793811yescharacteristic
129C4⋊C4C2×#8793811yescharacteristic
130C22⋊C4—21no—
131C22⋊C4—2021no—
132C42—21no—
133C42—1011no—
134C42—1011no—
135C42—1011no—
136C42—1011no—
137C42—2021no—
138C23C22⋊#8793811yescharacteristic
139Q8—4042no—
140Q8—4041no—
141Q8—4041no—
142C2×C4—22no—
143C2×C4—21no—
144C2×C4—21no—
145C2×C4—1011no—
146C2×C4—1011no—
147C2×C4—1011no—
148C2×C4—1011no—
149C2×C4—2021no—
150C2×C4—2022no—
151C2×C4—2021no—
152C2×C4—2021no—
153C2×C4—2021no—
154C2×C4C22⋊#8793811yescharacteristic
155C2×C4C22×#8793811yescharacteristic
156C8—4042no—
157C8—4042no—
158C22—21no—
159C22—21no—
160C22#1569231612yes—
161C22C2×C22⋊#8793811yescharacteristic
162C4—21no—
163C4—21no—
164C4—41no—
165C4—41no—
166C4—1012no—
167C4—1011no—
168C4—1011no—
169C4—4041no—
170C4—4041no—
171C4—2021no—
172C4—2021no—
173C2—22no—
174C2C2×#1569231611yescharacteristic
175C2#2026616711yescharacteristic
176C2#2026616811yescharacteristic
subgroupG/H
centerC22C2×C22⋊#87938
commutatorC2×C4×C101C22×C4
FrattiniC2×C4C22×#87938
Fitting#117×C101C4
socleC22×C101C2×C22⋊C4
prime radicalC23×C101C22⋊C4
cosocleC4×Dic101C23
hypercenter#117#87938
hypocenterC101#1552
LCSG ▷ C2×C4×C101 ▷ C22×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ #117

Complements

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

NH ≅ G/Ntype
3. #20266036C2semidirect
4. #20266036C2semidirect
89. C101#1552semidirect

Maximal subgroups

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

subgroupindex
#202661972
C2×#156923222
#202660362
#202657302
#1552101