Finite Group Databasebeta

← #27170229

Subgroups — #27170229

Order 12928.

210 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 210 conjugacy classes of subgroups — too many to draw. Only its 55 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#20265902C211yesmaximal, characteristic, complemented
2#20265525C211yesmaximal, characteristic
3#20265378C211yesmaximal, characteristic
4#20265378C211yesmaximal, characteristic
5#20264907C211yesmaximal, characteristic, complemented
6#20264907C211yesmaximal, characteristic, complemented
7#332×C101C211yesmaximal, characteristic, complemented
8C2×#13108063—21no—
9C2×#13108063—21no—
10#15692253—21no—
11#15692252C2211yescharacteristic
12#15692246—21no—
13#15692245—22no—
14Q8×Dic101—21no—
15C4×Dic202C2211yescharacteristic
16C4×Dic202C2211yescharacteristic
17#15692220—21no—
18#15692177—21no—
19#15692177—21no—
20#15692175—22no—
21C4×#11717579C2211yescharacteristic
22C4⋊Q8×C101C2211yescharacteristic
23C4⋊C8×C101C2211yescharacteristic
24C4⋊C8×C101C2211yescharacteristic
25C2×Dic202D411yescharacteristic
26C2×Dic202D411yescharacteristic
27C2×D404—21no—
28C2×D404—21no—
29C2×D404—21no—
30#13108063—42no—
31#13108063—42no—
32C2×#11717579—21no—
33C4⋊Dic101—22no—
34C4⋊Dic101D411yescharacteristic
35C4⋊Dic101D411yescharacteristic
36#13108053—22no—
37#13108053—22no—
38#13108053—21no—
39#13108053—21no—
40C4×Dic101—22no—
41C4×Dic101—21no—
42C4×Dic101—21no—
43C2×Q8×C101—21no—
44C2×C8×C101—21no—
45C2×C8×C101—21no—
46C4⋊C4×C101—21no—
47C4⋊C4×C101D412yes—
48C42×C101C2311yescharacteristic
49C22×D101—41no—
50Q8×C101—41no—
51Q8×C101—41no—
52C2×C4×C101—22no—
53C2×C4×C101C2×D411yescharacteristic
54C2×C4×C101C2×D411yescharacteristic
55C2×C4×C101C2×D411yescharacteristic
56C2×Dic101—21no—
57C2×Dic101—21no—
58C2×Dic101—21no—
59C2×Dic101—21no—
60D404—42no—
61D404—41no—
62D404—41no—
63D404—41no—
64D404—41no—
65Dic202—21no—
66Dic202—21no—
67Dic202—21no—
68Dic202SD1612yescomplemented
69C8×C101—41no—
70C8×C101—41no—
71#11717579—41no—
72C22×C101#12011yescharacteristic
73C2×D101—81no—
74C2×D101—81no—
75C2×D101—81no—
76C4×C101—21no—
77C4×C101—42no—
78C4×C101C2×SD1611yescharacteristic
79C4×C101#13511yescharacteristic
80C4×C101AGL(1,ℤ/8)11yescharacteristic
81C4×C101#13711yescharacteristic
82Dic101—22no—
83Dic101—41no—
84Dic101—41no—
85Dic101—41no—
86C2×C101#44811yescharacteristic
87C2×C101#44911yescharacteristic
88C2×C101#45411yescharacteristic
89D101—161no—
90#1626—1011nomaximal, Sylow
91C101#162611yescharacteristic, complemented, Sylow
92#557—1011no—
93#486—1011no—
94#460—1011no—
95#460—1011no—
96#332D10111yescharacteristic, complemented
97Q8⋊C8—1011no—
98Q8⋊C8—1011no—
99C2×SD16—2021no—
100C2×SD16—2021no—
101C4⋊Q8—2021no—
102C4⋊Q8C2×D10111yescharacteristic
103#127—1011no—
104#125—2021no—
105#125—2022no—
106C4×Q8—1011no—
107C4×Q8—1011no—
108C4×Q8—2021no—
109C4×Q8—2021no—
110C4⋊C8C2×D10111yescharacteristic
111C4⋊C8C2×D10111yescharacteristic
112#102—2021no—
113#102—2021no—
114#102—2022no—
115C4×C8—1011no—
116C2×Q8—21no—
117C2×Q8—1011no—
118C2×Q8—1011no—
119C2×D4—2021no—
120C2×D4—2021no—
121C2×D4—2021no—
122SD16—4042no—
123SD16—4042no—
124C2×C8—21no—
125C2×C8—21no—
126C2×C8—2021no—
127C4⋊C4—21no—
128C4⋊C4—1011no—
129C4⋊C4—1011no—
130C4⋊C4—2022no—
131C4⋊C4—2022no—
132C4⋊C4—2022no—
133C4⋊C4—2021no—
134C4⋊C4—2021no—
135C4⋊C4D40412yes—
136C42—2022no—
137C42—2021no—
138C42—2021no—
139C42C22×D10111yescharacteristic
140C23—4041no—
141Q8—41no—
142Q8—41no—
143Q8—1012no—
144Q8—2021no—
145Q8—2021no—
146Q8—2021no—
147D4—4042no—
148D4—4041no—
149D4—4041no—
150D4—4041no—
151D4—4041no—
152C2×C4—22no—
153C2×C4—2021no—
154C2×C4—2021no—
155C2×C4—2021no—
156C2×C4—2021no—
157C2×C4D4×D10111yescharacteristic
158C2×C4D4×D10111yescharacteristic
159C2×C4C2×D40411yescharacteristic
160C8—41no—
161C8—41no—
162C8—4041no—
163C22—8081no—
164C22—8081no—
165C22—8081no—
166C22#1569222511yescharacteristic
167C4—21no—
168C4—42no—
169C4—4041no—
170C4—4041no—
171C4—4041no—
172C4—2022no—
173C4SD16×D10111yescharacteristic
174C4#1569227611yescharacteristic
175C4#1569227911yescharacteristic
176C4#1569228711yescharacteristic
177C2—16161no—
178C2#2026531711yescharacteristic
179C2#2026532311yescharacteristic
180C2#2026535211yescharacteristic
subgroupG/H
centerC22#15692225
commutator, cosocleC42×C101C23
FrattiniC42C22×D101
Fitting#332×C101C2
socleC22×C101#120
prime radical#15692252C22
hypercenter#332D101
hypocenterC101#1626
LCSG ▷ C42×C101 ▷ C22×C101 ▷ C101
UCSC1 ◁ C22 ◁ C42 ◁ #332

Complements

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

NH ≅ G/Ntype
1. #20265902C2semidirect
5. #20264907C2semidirect
6. #20264907C2semidirect
7. #332×C101C2semidirect
68. Dic202SD16semidirect
91. C101#1626semidirect
96. #332D101semidirect

Maximal subgroups

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

subgroupindex
#202659022
#202655252
#202653782
#202649072
#332×C1012
#1626101