Finite Group Databasebeta

← #27170252

Subgroups — #27170252

Order 12928.

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

normal subgroups only

This group has 218 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#20265551C211yesmaximal, characteristic
3#20265469C211yesmaximal, characteristic, complemented
4#20264920C211yesmaximal, characteristic
5Q8⋊#11717579C211yesmaximal, characteristic, complemented
6#20264907C211yesmaximal, characteristic, complemented
7#477×C101C211yesmaximal, characteristic, complemented
8C2×#13108061—22no—
9#15692252C2211yescharacteristic
10#15692244—21no—
11#15692243—21no—
12#15692240—22no—
13C4×Dic202C2211yescharacteristic
14C4×D404—21no—
15#15692217—21no—
16#15692188—21no—
17C4⋊#11717579C2211yescharacteristic
18#15692178—21no—
19C4×#11717579C2211yescharacteristic
20C4⋊Q8×C101C2211yescharacteristic
21C4×Q8×C101C2211yescharacteristic
22#107×C101—21no—
23C4⋊C8×C101C2211yescharacteristic
24#103×C101—21no—
25C2×Dic202D411yescharacteristic
26C2×D404—21no—
27C2×D404—21no—
28C2×D404D412yes—
29C2×C4×D101—22no—
30#13108061—44no—
31C2×#11717579—21no—
32C2×#11717579—21no—
33C4⋊Dic101D411yescharacteristic
34#13108053—21no—
35#13108051—22no—
36#13108051—22no—
37#13108051—22no—
38C4×Dic101—21no—
39C2×Q8×C101—21no—
40C2×Q8×C101D411yescharacteristic
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—42no—
52C2×C4×C101—21no—
53C2×C4×C101—21no—
54C2×C4×C101—41no—
55C2×C4×C101C2×D411yescharacteristic
56C2×C4×C101C2×D411yescharacteristic
57C2×C4×C101C2×D411yescharacteristic
58C2×Dic101—21no—
59C2×Dic101—21no—
60D404—42no—
61D404—41no—
62D404—41no—
63D404—41no—
64D404—41no—
65C4×D101—82no—
66Dic202—21no—
67Dic202—21no—
68C8×C101—41no—
69#11717579—22no—
70#11717579—41no—
71C22×C101#12011yescharacteristic
72C2×D101—42no—
73C2×D101—42no—
74C2×D101—42no—
75C4×C101—21no—
76C4×C101—41no—
77C4×C101—41no—
78C4×C101—81no—
79C4×C101#13511yescharacteristic
80C4×C101#13511yescharacteristic
81C4×C101#13711yescharacteristic
82C4×C101#13711yescharacteristic
83Dic101—41no—
84Dic101—41no—
85C2×C101#45211yescharacteristic
86C2×C101#45211yescharacteristic
87C2×C101#45311yescharacteristic
88D101—82no—
89#1630—1011nomaximal, Sylow
90C101#163011yescharacteristic, complemented, Sylow
91#559—1011no—
92#492—1011no—
93#477—1011no—
94#477D10111yescharacteristic, complemented
95#331—1011no—
96Q8⋊C8—1011no—
97Q8⋊C8—1011no—
98C2×SD16—2022no—
99C4⋊Q8C2×D10111yescharacteristic
100#127—1011no—
101#124—2021no—
102#124—2021no—
103#123—2022no—
104C4×Q8—1011no—
105C4×Q8C2×D10111yescharacteristic
106C4×D4—2021no—
107C4×D4—2021no—
108#107—21no—
109#107—2021no—
110C4⋊C8—1011no—
111C4⋊C8C2×D10111yescharacteristic
112#103—21no—
113#103—2021no—
114C4×C8—1011no—
115C2×Q8—21no—
116C2×Q8—1011no—
117C2×Q8C22⋊D10111yescharacteristic
118C2×D4—1012no—
119C2×D4—2021no—
120C2×D4—2021no—
121C22×C4—2022no—
122SD16—4044no—
123C2×C8—21no—
124C2×C8—2021no—
125C2×C8—2021no—
126C4⋊C4—21no—
127C4⋊C4—21no—
128C4⋊C4—21no—
129C4⋊C4—1011no—
130C4⋊C4—2021no—
131C4⋊C4C22⋊D10111yescharacteristic
132C22⋊C4—2022no—
133C22⋊C4—2022no—
134C22⋊C4—2022no—
135C42—21no—
136C42—2021no—
137C42C22×D10111yescharacteristic
138C23—2022no—
139Q8—21no—
140Q8—21no—
141Q8—42no—
142Q8—2021no—
143Q8—2021no—
144D4—4042no—
145D4—4041no—
146D4—4041no—
147D4—4041no—
148D4—4041no—
149C2×C4—21no—
150C2×C4—21no—
151C2×C4—41no—
152C2×C4—8082no—
153C2×C4—2021no—
154C2×C4—2021no—
155C2×C4D4×D10111yescharacteristic
156C2×C4D4×D10111yescharacteristic
157C2×C4C2×C22⋊D10111yescharacteristic
158C8—41no—
159C8—4041no—
160C8—2022no—
161C22—4042no—
162C22—4042no—
163C22—4042no—
164C22#1569222311yescharacteristic
165C4—21no—
166C4—41no—
167C4—41no—
168C4—81no—
169C4—4041no—
170C4—4041no—
171C4#1569227511yescharacteristic
172C4#1569227811yescharacteristic
173C4#1569228611yescharacteristic
174C4#1569228711yescharacteristic
175C2—8082no—
176C2#2026533711yescharacteristic
177C2#2026533811yescharacteristic
178C2#2026534311yescharacteristic
subgroupG/H
centerC22#15692223
commutator, cosocleC42×C101C23
FrattiniC42C22×D101
Fitting#477×C101C2
socleC22×C101#120
prime radical#15692252C22
hypercenter#477D101
hypocenterC101#1630
LCSG ▷ C42×C101 ▷ C22×C101 ▷ C101
UCSC1 ◁ C22 ◁ C42 ◁ #477

Complements

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

NH ≅ G/Ntype
3. #20265469C2semidirect
5. Q8⋊#11717579C2semidirect
6. #20264907C2semidirect
7. #477×C101C2semidirect
90. C101#1630semidirect
94. #477D101semidirect

Maximal subgroups

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

subgroupindex
#202659122
#202655512
#202654692
#202649202
Q8⋊#117175792
#202649072
#477×C1012
#1630101