Finite Group Databasebeta

← #27187137

Subgroups — #27187137

Order 12928.

360 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 68 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#20266244C211yesmaximal, characteristic
2#20266170C212yesmaximal, complemented
3D101⋊#100C211yesmaximal, characteristic
4#20266048C212yesmaximal, complemented
5#124×D101C211yesmaximal, characteristic
6D101⋊M16C2212yes—
7D4×#87938—24no—
8#15692334C2212yes—
9C2×C22⋊#87938—22no—
10#15692318—24no—
11#15692315—22no—
12C2×Q8×D101C2211yescharacteristic
13C2×D4×D101C2211yescharacteristic
14#15692244C411yescharacteristic, complemented
15#15692243C411yescharacteristic, complemented
16#15692242C411yescharacteristic, complemented
17#15692241—22no—
18C22⋊C4×D101—22no—
19C42×D101C2211yescharacteristic
20#124×C101C411yescharacteristic, complemented
21C22×#87938—24no—
22#13108084—22no—
23#13108083—22no—
24D101⋊C8—22no—
25C4⋊#87938D414yes—
26C22⋊#87938—24no—
27C22⋊#87938—22no—
28C4×#87938—22no—
29C23×D101D412yes—
30C2×Dic202—21no—
31C2×Dic202C2×C411yescharacteristic, complemented
32Q8×D101—22no—
33Q8×D101—22no—
34C2×D404C2×C411yescharacteristic
35C2×C22⋊D101D412yes—
36D4×D101—22no—
37D4×D101—22no—
38C2×C4×D101—21no—
39C2×C4×D101—21no—
40C2×C4×D101C2311yescharacteristic
41C22⋊Dic101—22no—
42#13108051—22no—
43#13108051—22no—
44C4×Dic101—21no—
45C4×Dic101C2×C411yescharacteristic, complemented
46C2×Q8×C101C2×C411yescharacteristic, complemented
47C2×D4×C101C2×C411yescharacteristic
48C22⋊C4×C101—22no—
49C42×C101C2×C411yescharacteristic, complemented
50C23×C101C22⋊C412yes—
51C22×D101—22no—
52C22×D101—22no—
53C22×D101—44no—
54C22×D101C22⋊C412yes—
55C22×D101C2×D411yescharacteristic
56C2×#87938—24no—
57C2×#87938—84no—
58Q8×C101—22no—
59D4×C101—22no—
60C2×C4×C101—21no—
61C2×C4×C101—21no—
62C2×C4×C101C22×C411yescharacteristic
63C22⋊D101—24no—
64C2×Dic101—21no—
65C2×Dic101—21no—
66C2×Dic101C2×D411yescharacteristic
67D404—22no—
68C4×D101—42no—
69C4×D101—41no—
70C4×D101—41no—
71C4×D101C2×D412yes—
72C4×D101#4112yes—
73Dic202—22no—
74Dic202—42no—
75#11717581—42no—
76C22×C101—22no—
77C22×C101—42no—
78C22×C101C2×C22⋊C411yescharacteristic
79C2×D101—22no—
80C2×D101—22no—
81C2×D101—44no—
82C2×D101—42no—
83C2×D101—82no—
84C2×D101#12011yescharacteristic
85C2×D101#12311yescharacteristic
86#87938—44no—
87C4×C101—41no—
88C4×C101—41no—
89C4×C101C4×D412yes—
90Dic101—41no—
91Dic101—41no—
92Dic101#12112yes—
93C2×C101—21no—
94C2×C101—42no—
95C2×C101#38611yescharacteristic
96D101—21no—
97D101—42no—
98D101#45512yes—
99#1885—1011nomaximal, Sylow
100C101#188511yescharacteristic, complemented, Sylow
101C2×#124—1011no—
102#518—1011no—
103#419—1012no—
104C2×#100—1011no—
105#339—1012no—
106C22×Q8—1011no—
107C22×D4—1011no—
108C2×M16—1012no—
109#124—1011no—
110#124—1011no—
111#124—1011no—
112#124—2022no—
113#124#8793811yescharacteristic, complemented
114C4×D4—2024no—
115#117—1012no—
116C2×C22⋊C4—2022no—
117C2×C22⋊C4—2022no—
118C2×C42—1011no—
119#103—2024no—
120#100—2022no—
121C24—1012no—
122C2×Q8—1011no—
123C2×Q8—2021no—
124C2×Q8—2022no—
125C2×Q8—2022no—
126C2×Q8C2×#8793811yescharacteristic, complemented
127C2×D4—1012no—
128C2×D4—1011no—
129C2×D4—2022no—
130C2×D4—2022no—
131C2×D4C2×#8793811yescharacteristic
132C22×C4—1011no—
133C22×C4—2021no—
134C22×C4—2021no—
135C22×C4—2024no—
136M16—2022no—
137M16—2022no—
138C2×C8—2022no—
139C4⋊C4—1014no—
140C22⋊C4—22no—
141C22⋊C4—2022no—
142C22⋊C4—2022no—
143C22⋊C4—2022no—
144C22⋊C4—2024no—
145C22⋊C4—2022no—
146C42—1011no—
147C42—2021no—
148C42—2022no—
149C42C2×#8793811yescharacteristic, complemented
150C23—1011no—
151C23—1012no—
152C23—4044no—
153C23—2022no—
154C23—2022no—
155C23C22⋊#8793812yes—
156Q8—22no—
157Q8—4042no—
158Q8—2022no—
159D4—22no—
160D4—2024no—
161D4—2022no—
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—8084no—
165C2×C4—1011no—
166C2×C4—1012no—
167C2×C4—1012no—
168C2×C4—4041no—
169C2×C4—4041no—
170C2×C4—4042no—
171C2×C4—2024no—
172C2×C4—2021no—
173C2×C4—2021no—
174C2×C4C22×#8793811yescharacteristic
175C8—4042no—
176C22—22no—
177C22—42no—
178C22—8082no—
179C22—1011no—
180C22—1011no—
181C22—4042no—
182C22—4044no—
183C22—2022no—
184C22—2022no—
185C22C2×C22⋊#8793811yescharacteristic
186C4—41no—
187C4—41no—
188C4—1012no—
189C4—4041no—
190C4—4044no—
191C4—4041no—
192C4D4×#8793812yes—
193C2—21no—
194C2—42no—
195C2—1012no—
196C2—4042no—
197C2—2021no—
198C2#2026612511yescharacteristic
subgroupG/H
centerC2#20266125
commutatorC2×C4×C101C22×C4
FrattiniC2×C4C22×#87938
Fitting#124×C101C4
socleC2×C101#386
prime radicalC2×D4×D101C22
cosocleC2×C4×D101C23
hypercenter#124#87938
hypocenterC101#1885
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C2 ◁ C2×C4 ◁ #124

Complements

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

NH ≅ G/Ntype
2. #20266170C2semidirect
4. #20266048C2semidirect
14. #15692244C4semidirect
15. #15692243C4semidirect
16. #15692242C4semidirect
20. #124×C101C4semidirect
31. C2×Dic202C2×C4semidirect
45. C4×Dic101C2×C4semidirect
46. C2×Q8×C101C2×C4semidirect
49. C42×C101C2×C4semidirect
100. C101#1885semidirect
113. #124#87938semidirect
126. C2×Q8C2×#87938semidirect
149. C42C2×#87938semidirect

Maximal subgroups

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

subgroupindex
#202662442
#202661702
D101⋊#1002
#202660482
#124×D1012
#1885101