Finite Group Databasebeta

← #27186563

Subgroups — #27186563

Order 12928.

264 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 66 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#20266195C211yesmaximal, characteristic, complemented
2#20266121C211yesmaximal, characteristic, complemented
3#20266029C214yesmaximal, complemented
4#20265728C211yesmaximal, characteristic
5C2×C4⋊#87938—21no—
6#15692322—21no—
7#15692312C2212yes—
8C4×#11717581—21no—
9(C2×#87938)⋊C4—21no—
10(C2×#87938)⋊C4C2212yes—
11C2×#13108075—21no—
12#15692252C411yescharacteristic, complemented
13#15692244—21no—
14#15692230—21no—
15#15692230C2212yes—
16#15692217—21no—
17#15692217C412yescomplemented
18C42×D101C2211yescharacteristic
19C4⋊Q8×C101C411yescharacteristic, complemented
20C22×#87938—22no—
21C2×#11717581—22no—
22C4⋊#87938—42no—
23#13108075—42no—
24#13108075—42no—
25C2×D404—22no—
26C2×D404—21no—
27C2×D404—21no—
28C2×D404C2×C412yes—
29C2×C4×D101—22no—
30C2×C4×D101D411yescharacteristic
31C2×C4×D101D411yescharacteristic
32C2×C4×D101C2311yescharacteristic
33#13108051—22no—
34#13108051—22no—
35C4×Dic101C2×C411yescharacteristic
36C4×Dic101D411yescharacteristic
37C4×Dic101D411yescharacteristic
38C2×Q8×C101—21no—
39C4⋊C4×C101—21no—
40C4⋊C4×C101C2×C412yes—
41C42×C101C2×C411yescharacteristic
42C22×D101—22no—
43C22×D101C2×D411yescharacteristic
44C2×#87938—42no—
45C2×#87938—42no—
46C2×#87938—42no—
47Q8×C101—42no—
48C2×C4×C101—22no—
49C2×C4×C101C22⋊C411yescharacteristic
50C2×C4×C101C22⋊C411yescharacteristic
51C2×C4×C101C22×C411yescharacteristic
52C2×Dic101C22⋊C411yescharacteristic
53C2×Dic101C22⋊C411yescharacteristic
54C2×Dic101C2×D411yescharacteristic
55D404—42no—
56D404—42no—
57D404—42no—
58D404—42no—
59D404—42no—
60C4×D101—41no—
61C4×D101—41no—
62C4×D101—44no—
63C4×D101D812yes—
64C4×D101SD1612yes—
65#11717581—42no—
66C22×C101C2×C22⋊C411yescharacteristic
67C2×D101—21no—
68C2×D101—21no—
69C2×D101—42no—
70C2×D101—42no—
71C2×D101—42no—
72C2×D101C2×D811yescharacteristic
73C2×D101C2×SD1611yescharacteristic
74#87938—82no—
75C4×C101—21no—
76C4×C101—21no—
77C4×C101—42no—
78C4×C101#10212yes—
79Dic101—21no—
80Dic101—21no—
81Dic101#10212yes—
82C2×C101#41011yescharacteristic
83C2×C101C2×#10211yescharacteristic
84C2×C101#42111yescharacteristic
85D101—21no—
86D101—21no—
87D101—82no—
88#1517—1011nomaximal, Sylow
89C101#151711yescharacteristic, complemented, Sylow
90#532—1011no—
91#431—1011no—
92#382—1011no—
93#327—1014no—
94C2×#41—2021no—
95C4⋊Q8#8793811yescharacteristic, complemented
96#127—1011no—
97#124—2021no—
98#121—1012no—
99#121—2021no—
100C4×D4—1012no—
101C4×D4—2021no—
102C2×C4⋊C4—2021no—
103C2×C42—1011no—
104C4⋊C8—2021no—
105C22⋊C8—1012no—
106C4×C8—2021no—
107#95—1012no—
108#95—2021no—
109#41—4042no—
110#41—4042no—
111C2×Q8—21no—
112C2×D4—1012no—
113C2×D4—2022no—
114C2×D4—2021no—
115C2×D4—2021no—
116C22×C4—1011no—
117C22×C4—1011no—
118C22×C4—1011no—
119C22×C4—2022no—
120C22×C4—2022no—
121C2×C8—2022no—
122C4⋊C4—21no—
123C4⋊C4—4042no—
124C4⋊C4C2×#8793812yes—
125C22⋊C4—2022no—
126C22⋊C4—2022no—
127C42—1011no—
128C42—1011no—
129C42—1011no—
130C42C2×#8793811yescharacteristic
131C23—1011no—
132C23—2022no—
133Q8—42no—
134D4—4042no—
135D4—4042no—
136D4—4042no—
137D4—4042no—
138D4—4042no—
139C2×C4—22no—
140C2×C4—1011no—
141C2×C4—1012no—
142C2×C4—1012no—
143C2×C4—1011no—
144C2×C4—1011no—
145C2×C4—4041no—
146C2×C4—4041no—
147C2×C4—4044no—
148C2×C4—4042no—
149C2×C4—4042no—
150C2×C4—4042no—
151C2×C4C22⋊#8793811yescharacteristic
152C2×C4C22⋊#8793811yescharacteristic
153C2×C4C22×#8793811yescharacteristic
154C8—4042no—
155C22—1011no—
156C22—1011no—
157C22—4042no—
158C22—4042no—
159C22—4042no—
160C22—2021no—
161C22—2021no—
162C22C2×C22⋊#8793811yescharacteristic
163C4—21no—
164C4—21no—
165C4—42no—
166C4—8082no—
167C4—1012no—
168C4—2021no—
169C4—2021no—
170C4#1569231712yes—
171C2—8082no—
172C2—2021no—
173C2—2021no—
174C2#2026615811yescharacteristic
175C2C2×#1569231711yescharacteristic
176C2#2026617611yescharacteristic
subgroupG/H
centerC22C2×C22⋊#87938
commutatorC2×C4×C101C22×C4
FrattiniC2×C4C22×#87938
FittingC4⋊Q8×C101C4
socleC22×C101C2×C22⋊C4
prime radical#20265728C2
cosocleC2×C4×D101C23
hypercenterC4⋊Q8#87938
hypocenterC101#1517
LCSG ▷ C2×C4×C101 ▷ C22×C101 ▷ C101
UCSC1 ◁ C22 ◁ C42 ◁ C4⋊Q8

Complements

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

NH ≅ G/Ntype
1. #20266195C2semidirect
2. #20266121C2semidirect
3. #20266029C2semidirect
12. #15692252C4semidirect
17. #15692217C4semidirect
19. C4⋊Q8×C101C4semidirect
89. C101#1517semidirect
95. C4⋊Q8#87938semidirect

Maximal subgroups

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

subgroupindex
#202661952
#202661212
#202660292
#202657282
#1517101