Finite Group Databasebeta

← #27170321

Subgroups — #27170321

Order 12928.

212 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 53 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#20265912C211yesmaximal, characteristic
2#20265524C211yesmaximal, characteristic
3#20265462C211yesmaximal, characteristic, complemented
4#20265388C211yesmaximal, characteristic, complemented
5#20264920C211yesmaximal, characteristic
6#20264909C211yesmaximal, characteristic, complemented
7Q8⋊C8×C101C211yesmaximal, characteristic, complemented
8C2×Dic404—21no—
9#15692255C2211yescharacteristic
10#15692252C2211yescharacteristic
11#15692244—21no—
12#15692243—21no—
13#15692240—22no—
14C4×Dic202C2211yescharacteristic
15C4×D404—21no—
16#15692217—21no—
17#15692186—21no—
18C4⋊#11717579C2211yescharacteristic
19#15692180—21no—
20#15692179—21no—
21#15692177—22no—
22C4×Q8×C101C2211yescharacteristic
23C4⋊C8×C101C2211yescharacteristic
24C4×C8×C101C2211yescharacteristic
25C2×Dic202—21no—
26C2×Dic202D411yescharacteristic
27C2×D404—21no—
28C2×D404—21no—
29C2×D404D412yes—
30C2×C4×D101—22no—
31Dic404—42no—
32C2×#11717579—21no—
33C4⋊Dic101—21no—
34C4⋊Dic101—21no—
35C4⋊Dic101D411yescharacteristic
36#13108053—21no—
37#13108051—22no—
38#13108051—22no—
39#13108051—22no—
40C4×Dic101—21no—
41C2×Q8×C101D411yescharacteristic
42C2×C8×C101—21no—
43C2×C8×C101—21no—
44C4⋊C4×C101—21no—
45C4⋊C4×C101D411yescharacteristic
46C42×C101—21no—
47C42×C101C2311yescharacteristic
48C22×D101—22no—
49Q8×C101—21no—
50Q8×C101—21no—
51C2×C4×C101—21no—
52C2×C4×C101—21no—
53C2×C4×C101C2×D411yescharacteristic
54C2×C4×C101C2×D411yescharacteristic
55C2×C4×C101C2×D411yescharacteristic
56C2×Dic101—21no—
57C2×Dic101—21no—
58C2×Dic101—41no—
59D404—42no—
60D404—41no—
61D404—41no—
62D404—41no—
63D404—41no—
64C4×D101—82no—
65Dic202—21no—
66Dic202—21no—
67Dic202—41no—
68Dic202—41no—
69C8×C101—41no—
70C8×C101—41no—
71#11717579—41no—
72C22×C101#12011yescharacteristic
73C2×D101—42no—
74C2×D101—42no—
75C2×D101—42no—
76C4×C101—21no—
77C4×C101—41no—
78C4×C101—41no—
79C4×C101#13511yescharacteristic
80C4×C101#13511yescharacteristic
81C4×C101AGL(1,ℤ/8)11yescharacteristic
82C4×C101#13711yescharacteristic
83Dic101—41no—
84Dic101—41no—
85Dic101—81no—
86C2×C101#44911yescharacteristic
87C2×C101#45211yescharacteristic
88C2×C101#45511yescharacteristic
89D101—82no—
90#1640—1011nomaximal, Sylow
91C101#164011yescharacteristic, complemented, Sylow
92#559—1011no—
93#486—1011no—
94#475—1011no—
95#462—1011no—
96#331—1011no—
97Q8⋊C8—1011no—
98Q8⋊C8D10111yescharacteristic, complemented
99C2×Q16—2021no—
100C4⋊Q8—1011no—
101#127—1011no—
102#124—2021no—
103#124—2021no—
104#123—2022no—
105C4×Q8—1011no—
106C4×Q8C2×D10111yescharacteristic
107C4×D4—2021no—
108C4×D4—2021no—
109#106—2021no—
110C4⋊C8—1011no—
111C4⋊C8C2×D10111yescharacteristic
112#103—2021no—
113#103—2021no—
114#102—2022no—
115C4×C8C2×D10111yescharacteristic
116C2×Q8—1011no—
117C2×Q8—2021no—
118C2×Q8D40411yescharacteristic
119C2×D4—1012no—
120C2×D4—2021no—
121C2×D4—2021no—
122C22×C4—2022no—
123Q16—4042no—
124C2×C8—21no—
125C2×C8—21no—
126C2×C8—2021no—
127C4⋊C4—21no—
128C4⋊C4—1011no—
129C4⋊C4—2021no—
130C4⋊C4—2021no—
131C4⋊C4—2021no—
132C4⋊C4D40411yescharacteristic
133C22⋊C4—2022no—
134C22⋊C4—2022no—
135C22⋊C4—2022no—
136C42—21no—
137C42—2021no—
138C42C22×D10111yescharacteristic
139C23—2022no—
140Q8—21no—
141Q8—21no—
142Q8—4041no—
143Q8—4041no—
144Q8—2021no—
145Q8—2021no—
146D4—4042no—
147D4—4041no—
148D4—4041no—
149D4—4041no—
150D4—4041no—
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—8082no—
154C2×C4—4041no—
155C2×C4—2021no—
156C2×C4—2021no—
157C2×C4D4×D10111yescharacteristic
158C2×C4D4×D10111yescharacteristic
159C2×C4C2×D40411yescharacteristic
160C8—41no—
161C8—41no—
162C8—4041no—
163C22—4042no—
164C22—4042no—
165C22—4042no—
166C22#1569222511yescharacteristic
167C4—21no—
168C4—41no—
169C4—41no—
170C4—8081no—
171C4—4041no—
172C4—4041no—
173C4#1569227711yescharacteristic
174C4#1569227811yescharacteristic
175C4#1569228211yescharacteristic
176C4#1569228811yescharacteristic
177C2—8082no—
178C2#2026532511yescharacteristic
179C2#2026534111yescharacteristic
180C2#2026535611yescharacteristic
subgroupG/H
centerC22#15692225
commutator, cosocleC42×C101C23
FrattiniC42C22×D101
FittingQ8⋊C8×C101C2
socleC22×C101#120
prime radical#15692252C22
hypercenterQ8⋊C8D101
hypocenterC101#1640
LCSG ▷ C42×C101 ▷ C22×C101 ▷ C101
UCSC1 ◁ C22 ◁ C42 ◁ Q8⋊C8

Complements

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

NH ≅ G/Ntype
3. #20265462C2semidirect
4. #20265388C2semidirect
6. #20264909C2semidirect
7. Q8⋊C8×C101C2semidirect
91. C101#1640semidirect
98. Q8⋊C8D101semidirect

Maximal subgroups

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

subgroupindex
#202659122
#202655242
#202654622
#202653882
#202649202
#202649092
Q8⋊C8×C1012
#1640101