Finite Group Databasebeta

← #27186525

Subgroups — #27186525

Order 12928.

228 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 58 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#20266200C211yesmaximal, characteristic
2#20266113C211yesmaximal, characteristic
3#20266031C212yesmaximal
4#20266030C212yesmaximal
5#20265731C211yesmaximal, characteristic
6C2×C4×#87938—21no—
7#15692322—21no—
8#15692312C2212yes—
9#15692310—21no—
10(C2×#87938)⋊C4—21no—
11(C2×#87938)⋊C4C2212yes—
12#15692254C411yescharacteristic, complemented
13#15692253C411yescharacteristic, complemented
14#15692245—21no—
15#15692235C2211yescharacteristic
16#15692233—21no—
17#15692231C2211yescharacteristic
18C4×Dic202—21no—
19C4×Dic202C411yescharacteristic, complemented
20#15692212—21no—
21C42×D101C2211yescharacteristic
22C4×Q8×C101C411yescharacteristic, complemented
23C22×#87938—22no—
24C2×#11717581—22no—
25C4×#87938—24no—
26C2×Dic202—21no—
27C2×Dic202C2×C411yescharacteristic
28C2×C4×D101D411yescharacteristic
29C2×C4×D101D411yescharacteristic
30C2×C4×D101C2311yescharacteristic
31C4⋊Dic101—21no—
32C4⋊Dic101C2×C411yescharacteristic
33#13108053—21no—
34#13108053—21no—
35#13108053—21no—
36#13108053—21no—
37#13108053—21no—
38#13108051—21no—
39#13108051—21no—
40C4×Dic101—21no—
41C4×Dic101C2×C411yescharacteristic
42C4×Dic101D411yescharacteristic
43C4×Dic101D411yescharacteristic
44C2×Q8×C101C2×C411yescharacteristic
45C4⋊C4×C101—21no—
46C4⋊C4×C101C2×C411yescharacteristic
47C42×C101—21no—
48C42×C101C2×C411yescharacteristic
49C22×D101C2×D411yescharacteristic
50C2×#87938—24no—
51C2×#87938—42no—
52C2×#87938—42no—
53Q8×C101—41no—
54C2×C4×C101—21no—
55C2×C4×C101—21no—
56C2×C4×C101C22⋊C411yescharacteristic
57C2×C4×C101C22⋊C411yescharacteristic
58C2×C4×C101C22×C411yescharacteristic
59C2×Dic101—21no—
60C2×Dic101—21no—
61C2×Dic101C22⋊C411yescharacteristic
62C2×Dic101C22⋊C411yescharacteristic
63C2×Dic101C2×D411yescharacteristic
64C4×D101—22no—
65C4×D101—22no—
66C4×D101—41no—
67Dic202—42no—
68Dic202—41no—
69#11717581—42no—
70C22×C101C2×C22⋊C411yescharacteristic
71C2×D101—21no—
72C2×D101—21no—
73C2×D101#13712yes—
74#87938—44no—
75C4×C101—21no—
76C4×C101—21no—
77C4×C101—21no—
78C4×C101—41no—
79C4×C101—41no—
80Dic101—21no—
81Dic101—21no—
82Dic101—41no—
83Dic101—41no—
84Dic101#10412yes—
85C2×C101#41011yescharacteristic
86C2×C101#41911yescharacteristic
87C2×C101C2×#10411yescharacteristic
88D101—22no—
89#1506—1011nomaximal, Sylow
90C101#150611yescharacteristic, complemented, Sylow
91#533—1011no—
92#432—1011no—
93#376—1011no—
94#328—1012no—
95#328—1012no—
96C4⋊Q8—1011no—
97C4⋊Q8—1011no—
98#125—2021no—
99C22⋊Q8—1011no—
100C22⋊Q8—1011no—
101C22⋊Q8—2021no—
102C4×Q8—1011no—
103C4×Q8—2021no—
104C4×Q8#8793811yescharacteristic, complemented
105#117—2021no—
106C2×C42—1011no—
107C2×C42—2021no—
108C4⋊C8—2021no—
109C22⋊C8—1012no—
110#97—2021no—
111#95—1012no—
112#95—2021no—
113C2×Q8—1011no—
114C2×Q8—2021no—
115C2×Q8C2×#8793811yescharacteristic
116C22×C4—1011no—
117C22×C4—1011no—
118C22×C4—1011no—
119C22×C4—2022no—
120C2×C8—2022no—
121C4⋊C4—21no—
122C4⋊C4—1011no—
123C4⋊C4—2021no—
124C4⋊C4—2021no—
125C4⋊C4—2021no—
126C4⋊C4—2021no—
127C4⋊C4—2021no—
128C4⋊C4—2021no—
129C4⋊C4C2×#8793811yescharacteristic
130C22⋊C4—2021no—
131C22⋊C4—2021no—
132C42—21no—
133C42—1011no—
134C42—1011no—
135C42—1011no—
136C42—2024no—
137C42—2021no—
138C42C2×#8793811yescharacteristic
139C23—1011no—
140Q8—41no—
141Q8—4041no—
142Q8—4042no—
143C2×C4—21no—
144C2×C4—21no—
145C2×C4—1011no—
146C2×C4—1011no—
147C2×C4—1011no—
148C2×C4—4041no—
149C2×C4—4042no—
150C2×C4—4042no—
151C2×C4—2022no—
152C2×C4—2022no—
153C2×C4—2024no—
154C2×C4—2021no—
155C2×C4—2021no—
156C2×C4C22⋊#8793811yescharacteristic
157C2×C4C22⋊#8793811yescharacteristic
158C2×C4C22×#8793811yescharacteristic
159C8—4042no—
160C22—1012no—
161C22—2021no—
162C22—2021no—
163C22C2×C22⋊#8793811yescharacteristic
164C4—21no—
165C4—21no—
166C4—21no—
167C4—41no—
168C4—41no—
169C4—1012no—
170C4—4044no—
171C4—4041no—
172C4—4041no—
173C4—2021no—
174C4—2021no—
175C2—2022no—
176C2#2026615611yescharacteristic
177C2#2026617011yescharacteristic
178C2D101⋊#10411yescharacteristic
subgroupG/H
centerC22C2×C22⋊#87938
commutatorC2×C4×C101C22×C4
FrattiniC2×C4C22×#87938
FittingC4×Q8×C101C4
socleC22×C101C2×C22⋊C4
prime radicalC22×D101C2×D4
cosocleC2×C4×D101C23
hypercenterC4×Q8#87938
hypocenterC101#1506
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
12. #15692254C4semidirect
13. #15692253C4semidirect
19. C4×Dic202C4semidirect
22. C4×Q8×C101C4semidirect
90. C101#1506semidirect
104. C4×Q8#87938semidirect

Maximal subgroups

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

subgroupindex
#202662002
#202661132
#202660312
#202660302
#202657312
#1506101