Finite Group Databasebeta

← #27186475

Subgroups — #27186475

Order 12928.

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

normal subgroups only

This group has 252 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#20266199C211yesmaximal, characteristic
2C4×#13108084C211yesmaximal, characteristic
3#20266027C212yesmaximal, complemented
4Q8⋊#11717581C212yesmaximal, complemented
5C4×Q8×D101C211yesmaximal, characteristic
6C2×#13108084—21no—
7C4⋊#11717581C2212yes—
8#15692312—21no—
9#15692310—21no—
10#15692310—21no—
11C4×#11717581C2212yes—
12C2×Q8×D101C411yescharacteristic
13Q8×Dic101C2211yescharacteristic
14C4×Dic202—21no—
15C4×Dic202C411yescharacteristic
16#15692220—21no—
17#15692220C2211yescharacteristic
18C4⋊C4×D101—21no—
19C4⋊C4×D101C411yescharacteristic
20C42×D101—21no—
21C42×D101C2211yescharacteristic
22C4×Q8×C101C411yescharacteristic
23#13108084—24no—
24C2×#11717581—22no—
25C2×#11717581—22no—
26C2×Dic202—21no—
27C2×Dic202C2×C411yescharacteristic
28Q8×D101—22no—
29Q8×D101—22no—
30C2×C4×D101—21no—
31C2×C4×D101—21no—
32C2×C4×D101C2×C411yescharacteristic
33C2×C4×D101D411yescharacteristic
34C2×C4×D101D411yescharacteristic
35C4⋊Dic101—21no—
36C4⋊Dic101C2×C411yescharacteristic
37#13108053—21no—
38#13108053—21no—
39#13108053—21no—
40C4×Dic101—21no—
41C4×Dic101—21no—
42C4×Dic101—21no—
43C4×Dic101D411yescharacteristic
44C4×Dic101D411yescharacteristic
45C4×Dic101C2311yescharacteristic
46C2×Q8×C101C2×C411yescharacteristic
47C4⋊C4×C101—21no—
48C4⋊C4×C101C2×C411yescharacteristic
49C42×C101—21no—
50C42×C101C2×C411yescharacteristic
51C22×D101C22⋊C411yescharacteristic
52Q8×C101—21no—
53Q8×C101M1612yescomplemented
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×Dic101C2×D411yescharacteristic
63C2×Dic101C2×D411yescharacteristic
64C4×D101—24no—
65C4×D101—22no—
66C4×D101—21no—
67C4×D101—21no—
68C4×D101—22no—
69C4×D101—42no—
70Dic202—24no—
71Dic202—21no—
72Dic202M1612yescomplemented
73#11717581—24no—
74#11717581—42no—
75C22×C101C2×C22⋊C411yescharacteristic
76C2×D101—21no—
77C2×D101—21no—
78C2×D101#10412yes—
79C4×C101—22no—
80C4×C101—21no—
81C4×C101—21no—
82C4×C101—41no—
83C4×C101C2×M1611yescharacteristic
84C4×C101C2×M1611yescharacteristic
85Dic101—21no—
86Dic101—22no—
87Dic101—21no—
88Dic101—41no—
89Dic101#13712yes—
90C2×C101#40711yescharacteristic
91C2×C101#41911yescharacteristic
92C2×C101C2×#10411yescharacteristic
93D101—22no—
94#1491—1011nomaximal, Sylow
95C101#149111yescharacteristic, complemented, Sylow
96C2×C4×Q8—1011no—
97#432—1011no—
98C4×M16—1011no—
99Q8⋊C8—1012no—
100Q8⋊C8—1012no—
101C22×Q8—1011no—
102C2×M16—2021no—
103C4×Q8—1011no—
104C4×Q8—1011no—
105C4×Q8—1011no—
106C4×Q8—2021no—
107C4×Q8—2021no—
108C4×Q8#8793811yescharacteristic
109C2×C4⋊C4—1011no—
110C2×C4⋊C4—2021no—
111C2×C42—1011no—
112C2×C42—2021no—
113C4⋊C8—1012no—
114C22⋊C8—2021no—
115#97—2021no—
116#97—2021no—
117C4×C8—1012no—
118C2×Q8—1011no—
119C2×Q8—2021no—
120C2×Q8—2022no—
121C2×Q8—2022no—
122C2×Q8C2×#8793811yescharacteristic
123C22×C4—1011no—
124C22×C4—1011no—
125C22×C4—1011no—
126C22×C4—2021no—
127C22×C4—2021no—
128M16—2024no—
129C2×C8—2022no—
130C2×C8—2022no—
131C4⋊C4—21no—
132C4⋊C4—1011no—
133C4⋊C4—2021no—
134C4⋊C4—2021no—
135C4⋊C4—2021no—
136C4⋊C4—2021no—
137C4⋊C4C2×#8793811yescharacteristic
138C42—21no—
139C42—1011no—
140C42—1011no—
141C42—1011no—
142C42—2021no—
143C42—2021no—
144C42—2021no—
145C42C2×#8793811yescharacteristic
146C23—1011no—
147Q8—21no—
148Q8—1012no—
149Q8—2024no—
150Q8—2021no—
151Q8#1310808412yescomplemented
152C2×C4—21no—
153C2×C4—21no—
154C2×C4—1011no—
155C2×C4—1011no—
156C2×C4—1011no—
157C2×C4—4042no—
158C2×C4—2022no—
159C2×C4—2022no—
160C2×C4—2024no—
161C2×C4—2021no—
162C2×C4—2021no—
163C2×C4—2021no—
164C2×C4—2021no—
165C2×C4C22⋊#8793811yescharacteristic
166C2×C4C22⋊#8793811yescharacteristic
167C2×C4C22×#8793811yescharacteristic
168C8—4042no—
169C8—2024no—
170C22—1012no—
171C22—2021no—
172C22—2021no—
173C22C2×C22⋊#8793811yescharacteristic
174C4—22no—
175C4—21no—
176C4—21no—
177C4—41no—
178C4—1012no—
179C4—4041no—
180C4—2021no—
181C4—2022no—
182C4—2021no—
183C4D101⋊M1611yescharacteristic
184C4C2×#1310808411yescharacteristic
185C2—2022no—
186C2#2026614811yescharacteristic
187C2#2026617011yescharacteristic
188C2D101⋊#10411yescharacteristic
subgroupG/H
centerC22C2×C22⋊#87938
commutatorC2×C4×C101C22×C4
FrattiniC2×C4C22×#87938
FittingC4×Q8×C101C4
socleC22×C101C2×C22⋊C4
prime radicalC22×D101C22⋊C4
cosocleC4×Dic101C23
hypercenterC4×Q8#87938
hypocenterC101#1491
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C42 ◁ C4×Q8

Complements

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

NH ≅ G/Ntype
3. #20266027C2semidirect
4. Q8⋊#11717581C2semidirect
53. Q8×C101M16semidirect
72. Dic202M16semidirect
95. C101#1491semidirect
151. Q8#13108084semidirect

Maximal subgroups

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

subgroupindex
#202661992
C4×#131080842
#202660272
Q8⋊#117175812
C4×Q8×D1012
#1491101