Finite Group Databasebeta

← #27170316

Subgroups — #27170316

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#20265525C211yesmaximal, characteristic
3#20265459C211yesmaximal, characteristic, complemented
4#20264920C211yesmaximal, characteristic
5Q8⋊#11717579C211yesmaximal, characteristic, complemented
6#20264907C211yesmaximal, characteristic, complemented
7#462×C101C211yesmaximal, characteristic, complemented
8#15692252C2211yescharacteristic
9#15692244—21no—
10#15692243—21no—
11#15692240—22no—
12C4×Dic202C2211yescharacteristic
13C4×D404—21no—
14#15692217—21no—
15#15692186—21no—
16C4⋊#11717579C2211yescharacteristic
17#15692178—21no—
18#15692175—22no—
19C4×#11717579C2211yescharacteristic
20C2×Q16×C101—21no—
21C4⋊Q8×C101C2211yescharacteristic
22C4×Q8×C101C2211yescharacteristic
23C4⋊C8×C101C2211yescharacteristic
24#103×C101—21no—
25C2×Dic202D411yescharacteristic
26C2×D404—21no—
27C2×D404—21no—
28C2×D404D412yes—
29C2×C4×D101—22no—
30C2×#11717579—21no—
31C2×#11717579—21no—
32C4⋊Dic101D411yescharacteristic
33#13108053—21no—
34#13108051—22no—
35#13108051—22no—
36#13108051—22no—
37C4×Dic101—21no—
38C2×Q8×C101—21no—
39C2×Q8×C101D411yescharacteristic
40Q16×C101—42no—
41C2×C8×C101—21no—
42C4⋊C4×C101—21no—
43C4⋊C4×C101—21no—
44C4⋊C4×C101—21no—
45C4⋊C4×C101D411yescharacteristic
46C42×C101—21no—
47C42×C101C2311yescharacteristic
48C22×D101—22no—
49Q8×C101—21no—
50Q8×C101—21no—
51Q8×C101—41no—
52Q8×C101—41no—
53C2×C4×C101—21no—
54C2×C4×C101—21no—
55C2×C4×C101—41no—
56C2×C4×C101C2×D411yescharacteristic
57C2×C4×C101C2×D411yescharacteristic
58C2×C4×C101C2×D411yescharacteristic
59C2×Dic101—21no—
60C2×Dic101—21no—
61D404—42no—
62D404—41no—
63D404—41no—
64D404—41no—
65D404—41no—
66C4×D101—82no—
67Dic202—21no—
68Dic202—21no—
69C8×C101—41no—
70#11717579—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—81no—
80C4×C101#13511yescharacteristic
81C4×C101#13511yescharacteristic
82C4×C101AGL(1,ℤ/8)11yescharacteristic
83C4×C101#13711yescharacteristic
84Dic101—41no—
85Dic101—41no—
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#462D10111yescharacteristic, complemented
96#331—1011no—
97Q8⋊C8—1011no—
98Q8⋊C8—1011no—
99C2×Q16—21no—
100C4⋊Q8C2×D10111yescharacteristic
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—21no—
113#103—2021no—
114#102—2022no—
115C4×C8—1011no—
116C2×Q8—21no—
117C2×Q8—1011no—
118C2×Q8C22⋊D10111yescharacteristic
119C2×D4—1012no—
120C2×D4—2021no—
121C2×D4—2021no—
122C22×C4—2022no—
123Q16—42no—
124C2×C8—21no—
125C2×C8—2021no—
126C2×C8—2021no—
127C4⋊C4—21no—
128C4⋊C4—21no—
129C4⋊C4—21no—
130C4⋊C4—1011no—
131C4⋊C4—2021no—
132C4⋊C4C22⋊D10111yescharacteristic
133C22⋊C4—2022no—
134C22⋊C4—2022no—
135C22⋊C4—2022no—
136C42—21no—
137C42—2021no—
138C42C22×D10111yescharacteristic
139C23—2022no—
140Q8—21no—
141Q8—21no—
142Q8—41no—
143Q8—41no—
144Q8—2021no—
145Q8—2021no—
146D4—4042no—
147D4—4041no—
148D4—4041no—
149D4—4041no—
150D4—4041no—
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—41no—
154C2×C4—8082no—
155C2×C4—2021no—
156C2×C4—2021no—
157C2×C4D4×D10111yescharacteristic
158C2×C4D4×D10111yescharacteristic
159C2×C4C2×C22⋊D10111yescharacteristic
160C8—41no—
161C8—4041no—
162C8—4041no—
163C22—4042no—
164C22—4042no—
165C22—4042no—
166C22#1569222311yescharacteristic
167C4—21no—
168C4—41no—
169C4—41no—
170C4—81no—
171C4—4041no—
172C4—4041no—
173C4#1569227511yescharacteristic
174C4#1569227811yescharacteristic
175C4#1569228111yescharacteristic
176C4#1569228711yescharacteristic
177C2—8082no—
178C2#2026532011yescharacteristic
179C2#2026533811yescharacteristic
180C2#2026535311yescharacteristic
subgroupG/H
centerC22#15692223
commutator, cosocleC42×C101C23
FrattiniC42C22×D101
Fitting#462×C101C2
socleC22×C101#120
prime radical#15692252C22
hypercenter#462D101
hypocenterC101#1640
LCSG ▷ C42×C101 ▷ C22×C101 ▷ C101
UCSC1 ◁ C22 ◁ C42 ◁ #462

Complements

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

NH ≅ G/Ntype
3. #20265459C2semidirect
5. Q8⋊#11717579C2semidirect
6. #20264907C2semidirect
7. #462×C101C2semidirect
91. C101#1640semidirect
95. #462D101semidirect

Maximal subgroups

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

subgroupindex
#202659122
#202655252
#202654592
#202649202
Q8⋊#117175792
#202649072
#462×C1012
#1640101