Finite Group Databasebeta

← #27187296

Subgroups — #27187296

Order 12928.

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

normal subgroups only

This group has 336 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
1C2×#15692319C214yesmaximal, complemented
2C4×#13108084C211yesmaximal, characteristic, complemented
3C4×C4⋊#87938C211yesmaximal, characteristic, complemented
4#20265725C211yesmaximal, characteristic
5C2×#13108084C2212yes—
6C2×C4⋊#87938—21no—
7C2×C4×#87938C2212yes—
8#15692319—28no—
9#15692310—21no—
10C4×#11717581—21no—
11(C2×#87938)⋊C4—21no—
12C2×#13108073C2212yes—
13#15692255C411yescharacteristic, complemented
14#15692242C412yescomplemented
15#15692226—21no—
16#15692226—21no—
17D4×Dic101—21no—
18D4×Dic101—21no—
19C42×D101C2211yescharacteristic
20#127×C101C411yescharacteristic, complemented
21C22×#87938—22no—
22#13108084—22no—
23#13108084D414yescomplemented
24C2×#11717581—22no—
25C4⋊#87938—24no—
26C4×#87938—24no—
27#13108073—24no—
28#13108073—24no—
29C2×Dic202C2×C412yes—
30C2×C22⋊D101—22no—
31C22×Dic101—22no—
32C2×C4×D101D411yescharacteristic
33C2×C4×D101D411yescharacteristic
34C2×C4×D101C2311yescharacteristic
35C4⋊Dic101—21no—
36C4⋊Dic101—21no—
37C22⋊Dic101—22no—
38C22⋊Dic101—22no—
39C4×Dic101C2×C411yescharacteristic
40C4×Dic101D411yescharacteristic
41C4×Dic101D411yescharacteristic
42C2×D4×C101—21no—
43C2×D4×C101—21no—
44C2×D4×C101C2×C412yes—
45C42×C101C2×C411yescharacteristic
46C23×C101—22no—
47C22×D101C2×D411yescharacteristic
48C2×#87938—24no—
49C2×#87938—42no—
50C2×#87938—42no—
51D4×C101—24no—
52D4×C101—42no—
53D4×C101—42no—
54C2×C4×C101C22⋊C411yescharacteristic
55C2×C4×C101C22⋊C411yescharacteristic
56C2×C4×C101C22×C411yescharacteristic
57C22⋊D101—42no—
58C22⋊D101—42no—
59C2×Dic101—22no—
60C2×Dic101—44no—
61C2×Dic101C22⋊C411yescharacteristic
62C2×Dic101C22⋊C411yescharacteristic
63C2×Dic101C2×D411yescharacteristic
64C4×D101—22no—
65C4×D101—41no—
66C4×D101C2×D412yes—
67C4×D101#4112yes—
68Dic202—24no—
69#11717581—24no—
70C22×C101—42no—
71C22×C101—42no—
72C22×C101—42no—
73C22×C101C2×C22⋊C411yescharacteristic
74C2×D101—21no—
75C2×D101—21no—
76C2×D101#12411yescharacteristic
77C2×D101#12711yescharacteristic
78#87938—44no—
79C4×C101—21no—
80C4×C101#10412yescomplemented
81C4×C101C4×D412yes—
82Dic101—21no—
83Dic101—21no—
84Dic101—42no—
85Dic101#12112yes—
86C2×C101—82no—
87C2×C101#39011yescharacteristic
88C2×C101C2×#10411yescharacteristic
89C2×C101#42111yescharacteristic
90D101—21no—
91D101—21no—
92#1980—1011nomaximal, Sylow
93C101#198011yescharacteristic, complemented, Sylow
94#532—1011no—
95C2×#104—1014no—
96C4×M16—1011no—
97C4×C4⋊C4—1011no—
98C2×#41—1012no—
99C2×M16—1012no—
100C4⋊Q8—1011no—
101#127#8793811yescharacteristic, complemented
102#124—1012no—
103#121—2021no—
104#121—2021no—
105C4×D4—2021no—
106C4×D4—2021no—
107C2×C4⋊C4—2021no—
108C2×C42—1011no—
109C2×C42—1012no—
110#104—2028no—
111#97—2021no—
112C4×C8—2021no—
113#95—2021no—
114#41—2024no—
115#41—2024no—
116C2×Q8—1012no—
117C2×D4—21no—
118C2×D4—21no—
119C2×D4—2022no—
120C2×D4C2×#8793812yes—
121C22×C4—1011no—
122C22×C4—1011no—
123C22×C4—1011no—
124C22×C4—2022no—
125C22×C4—2022no—
126M16—1014no—
127M16—2022no—
128C2×C8—2022no—
129C4⋊C4—2024no—
130C4⋊C4—2021no—
131C4⋊C4—2021no—
132C22⋊C4—2022no—
133C22⋊C4—2022no—
134C42—1011no—
135C42—1011no—
136C42—1011no—
137C42—2024no—
138C42C2×#8793811yescharacteristic
139C23—22no—
140C23—1011no—
141Q8—2024no—
142D4—24no—
143D4—42no—
144D4—42no—
145D4—4042no—
146D4—4042no—
147C2×C4—1011no—
148C2×C4—1012no—
149C2×C4—1012no—
150C2×C4—1011no—
151C2×C4—1011no—
152C2×C4—4041no—
153C2×C4—4044no—
154C2×C4—4042no—
155C2×C4—4042no—
156C2×C4—2024no—
157C2×C4—2022no—
158C2×C4—2022no—
159C2×C4C22⋊#8793811yescharacteristic
160C2×C4C22⋊#8793811yescharacteristic
161C2×C4C22×#8793811yescharacteristic
162C8—2024no—
163C22—42no—
164C22—42no—
165C22—42no—
166C22—1011no—
167C22—1011no—
168C22—2021no—
169C22—2021no—
170C22C2×C22⋊#8793811yescharacteristic
171C4—21no—
172C4—1012no—
173C4—4044no—
174C4—4042no—
175C4—2021no—
176C4—2021no—
177C4#1569231912yescomplemented
178C4D4×#8793812yes—
179C2—82no—
180C2—2021no—
181C2—2021no—
182C2#2026612911yescharacteristic
183C2C2×#1569231911yescharacteristic
184C2#2026617511yescharacteristic
subgroupG/H
centerC22C2×C22⋊#87938
commutatorC2×C4×C101C22×C4
FrattiniC2×C4C22×#87938
Fitting#127×C101C4
socleC22×C101C2×C22⋊C4
prime radical#20265725C2
cosocleC2×C4×D101C23
hypercenter#127#87938
hypocenterC101#1980
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C42 ◁ #127

Complements

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

NH ≅ G/Ntype
1. C2×#15692319C2semidirect
2. C4×#13108084C2semidirect
3. C4×C4⋊#87938C2semidirect
13. #15692255C4semidirect
14. #15692242C4semidirect
20. #127×C101C4semidirect
23. #13108084D4semidirect
80. C4×C101#104semidirect
93. C101#1980semidirect
101. #127#87938semidirect
177. C4#15692319semidirect

Maximal subgroups

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

subgroupindex
C2×#156923192
C4×#131080842
C4×C4⋊#879382
#202657252
#1980101