Finite Group Databasebeta

← #27186651

Subgroups — #27186651

Order 12928.

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

normal subgroups only

This group has 252 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, complemented
2C2×#15692322C211yesmaximal, characteristic, complemented
3#20266037C212yesmaximal, complemented
4#20266035C212yesmaximal, complemented
5#20265723C211yesmaximal, characteristic
6C22×#11717581—21no—
7#15692322—21no—
8#15692322C2212yes—
9#15692322C2212yes—
10C22⋊#11717581—21no—
11#15692310—21no—
12C2×#13108074C411yescharacteristic
13#15692253C2211yescharacteristic
14#15692251C2211yescharacteristic
15#15692244C411yescharacteristic
16#15692242C411yescharacteristic
17#15692227—21no—
18#15692227—21no—
19C4×C22⋊D101—21no—
20C4×C22⋊D101—21no—
21C2×C4×Dic101C2211yescharacteristic
22C2×#41×C101C411yescharacteristic
23C2×#11717581—22no—
24C2×#11717581—22no—
25C2×#11717581—42no—
26#13108074—42no—
27C2×Dic202C2×C411yescharacteristic
28C2×D404C2×C411yescharacteristic
29C2×C22⋊D101—21no—
30C2×C22⋊D101—21no—
31C2×C22⋊D101—21no—
32C22×Dic101D411yescharacteristic
33C22×Dic101D411yescharacteristic
34C2×C4×D101—21no—
35#13108053—21no—
36#13108053—21no—
37C22⋊Dic101—21no—
38C22⋊Dic101—21no—
39#13108051—21no—
40#13108051—21no—
41C4×Dic101C2×C411yescharacteristic
42C4×Dic101D411yescharacteristic
43C4×Dic101D411yescharacteristic
44C4×Dic101C2311yescharacteristic
45#41×C101—42no—
46C2×Q8×C101C2×C411yescharacteristic
47C2×D4×C101—21no—
48C2×D4×C101C2×C411yescharacteristic
49C22×C4×C101—21no—
50C22×C4×C101C2×C411yescharacteristic
51C23×C101—21no—
52C23×C101C22⋊C411yescharacteristic
53C22×D101—21no—
54Q8×C101—41no—
55D4×C101—41no—
56D4×C101—42no—
57C2×C4×C101—22no—
58C2×C4×C101—21no—
59C2×C4×C101—42no—
60C2×C4×C101C22⋊C411yescharacteristic
61C2×C4×C101C22×C411yescharacteristic
62C22⋊D101—42no—
63C22⋊D101—42no—
64C22⋊D101—41no—
65C22⋊D101—41no—
66C2×Dic101—22no—
67C2×Dic101—21no—
68C2×Dic101—21no—
69C2×Dic101—21no—
70C2×Dic101C22⋊C411yescharacteristic
71C2×Dic101C22⋊C411yescharacteristic
72C2×Dic101C2×D411yescharacteristic
73C2×Dic101C2×D411yescharacteristic
74D404—41no—
75C4×D101—42no—
76Dic202—41no—
77#11717581—42no—
78#11717581—42no—
79C22×C101—21no—
80C22×C101—21no—
81C22×C101—41no—
82C22×C101—41no—
83C22×C101—41no—
84C22×C101#10012yes—
85C22×C101C2×C22⋊C411yescharacteristic
86C2×D101—41no—
87C2×D101—41no—
88C2×D101—41no—
89C4×C101—21no—
90C4×C101—21no—
91C4×C101—41no—
92Dic101—21no—
93Dic101—21no—
94Dic101—41no—
95Dic101#13512yes—
96Dic101AGL(1,ℤ/8)11yescharacteristic
97Dic101#13711yescharacteristic
98C2×C101—22no—
99C2×C101—81no—
100C2×C101C2×#10011yescharacteristic
101C2×C101#41611yescharacteristic
102C2×C101#41711yescharacteristic
103D101—81no—
104#1551—1011nomaximal, Sylow
105C101#155111yescharacteristic, complemented, Sylow
106#532—1011no—
107#432—1011no—
108C2×C4⋊C8—1011no—
109#333—1012no—
110#331—1012no—
111C2×#41—1011no—
112C2×#41#8793811yescharacteristic
113C22×C8—2021no—
114C4⋊Q8—1011no—
115#127—1011no—
116#124—1011no—
117#124—1011no—
118#121—2021no—
119#121—2021no—
120C4×D4—2021no—
121C4×D4—2021no—
122C2×C42—1011no—
123C4⋊C8—1012no—
124C4⋊C8—1012no—
125C4⋊C8—2021no—
126C22⋊C8—2021no—
127#97—2021no—
128#41—42no—
129#41—4042no—
130C2×Q8—1011no—
131C2×Q8C2×#8793811yescharacteristic
132C2×D4—21no—
133C2×D4—1011no—
134C2×D4—2021no—
135C2×D4—2021no—
136C2×D4—2021no—
137C2×D4C2×#8793811yescharacteristic
138C22×C4—21no—
139C22×C4—1011no—
140C22×C4—1011no—
141C22×C4—2021no—
142C22×C4C2×#8793811yescharacteristic
143C2×C8—4042no—
144C2×C8—2022no—
145C2×C8—2022no—
146C4⋊C4—2021no—
147C4⋊C4—2021no—
148C22⋊C4—2021no—
149C22⋊C4—2021no—
150C22⋊C4—2021no—
151C22⋊C4—2021no—
152C42—1011no—
153C42—1011no—
154C42—1011no—
155C42—1011no—
156C23—21no—
157C23—2021no—
158C23C22⋊#8793811yescharacteristic
159Q8—41no—
160Q8—4041no—
161D4—41no—
162D4—42no—
163D4—4041no—
164D4—4042no—
165D4—4041no—
166D4—4041no—
167D4—4042no—
168C2×C4—22no—
169C2×C4—21no—
170C2×C4—42no—
171C2×C4—1011no—
172C2×C4—1011no—
173C2×C4—1011no—
174C2×C4—1011no—
175C2×C4—4042no—
176C2×C4—2021no—
177C2×C4—2022no—
178C2×C4—2021no—
179C2×C4—2021no—
180C2×C4C22⋊#8793811yescharacteristic
181C2×C4C22×#8793811yescharacteristic
182C8—4042no—
183C8—4042no—
184C22—21no—
185C22—21no—
186C22—41no—
187C22—41no—
188C22—41no—
189C22—4041no—
190C22—4041no—
191C22—4041no—
192C22#1569231512yes—
193C22C2×C22⋊#8793811yescharacteristic
194C4—21no—
195C4—21no—
196C4—41no—
197C4—1012no—
198C4—1011no—
199C4—1011no—
200C4—4041no—
201C4—2021no—
202C4—2021no—
203C2—22no—
204C2—81no—
205C2—8081no—
206C2C2×#1569231511yescharacteristic
207C2#2026616711yescharacteristic
208C2#2026616811yescharacteristic
subgroupG/H
centerC22C2×C22⋊#87938
commutatorC2×C4×C101C22×C4
FrattiniC2×C4C22×#87938
FittingC2×#41×C101C4
socleC22×C101C2×C22⋊C4
prime radical#20265723C2
cosocleC4×Dic101C23
hypercenterC2×#41#87938
hypocenterC101#1551
LCSG ▷ C2×C4×C101 ▷ C22×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#41

Complements

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

NH ≅ G/Ntype
1. #20266197C2semidirect
2. C2×#15692322C2semidirect
3. #20266037C2semidirect
4. #20266035C2semidirect
105. C101#1551semidirect

Maximal subgroups

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

subgroupindex
#202661972
C2×#156923222
#202660372
#202660352
#202657232
#1551101