Finite Group Databasebeta

← #27170403

Subgroups — #27170403

Order 12928.

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

normal subgroups only

This group has 200 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#20265901C211yesmaximal, characteristic
2#20265573C211yesmaximal, characteristic, complemented
3#20265458C211yesmaximal, characteristic
4#20265388C211yesmaximal, characteristic, complemented
5#20264907C211yesmaximal, characteristic, complemented
6#20264904C211yesmaximal, characteristic
7#332×C101C211yesmaximal, characteristic, complemented
8C2×Dic404—21no—
9#15692255C2211yescharacteristic
10#15692250—22no—
11#15692244—21no—
12#15692235—21no—
13Q8×Dic101—21no—
14C4×Dic202C2211yescharacteristic
15C4×D404C2211yescharacteristic
16#15692213—21no—
17#15692189—21no—
18#15692186—22no—
19#15692180—21no—
20#15692177—21no—
21C4×#11717579C2211yescharacteristic
22C4⋊Q8×C101C2211yescharacteristic
23C4⋊C8×C101C2211yescharacteristic
24C4⋊C8×C101C2211yescharacteristic
25C2×Dic202—21no—
26C2×Dic202D411yescharacteristic
27C2×D404D411yescharacteristic
28C2×C4×D101—21no—
29Dic404—42no—
30C2×#11717579—21no—
31C4⋊Dic101—22no—
32C4⋊Dic101—21no—
33C4⋊Dic101—21no—
34C4⋊Dic101D411yescharacteristic
35C4⋊Dic101D411yescharacteristic
36#13108053—21no—
37#13108051—22no—
38#13108051—22no—
39#13108051—21no—
40C4×Dic101—22no—
41C4×Dic101—21no—
42C2×Q8×C101—21no—
43C2×C8×C101—21no—
44C2×C8×C101—21no—
45C4⋊C4×C101—21no—
46C4⋊C4×C101D412yes—
47C42×C101C2311yescharacteristic
48C22×D101—21no—
49Q8×C101—41no—
50Q8×C101—41no—
51C2×C4×C101—22no—
52C2×C4×C101C2×D411yescharacteristic
53C2×C4×C101C2×D411yescharacteristic
54C2×C4×C101C2×D411yescharacteristic
55C2×Dic101—21no—
56C2×Dic101—21no—
57C2×Dic101—21no—
58C2×Dic101—41no—
59D404—21no—
60D404—21no—
61C4×D101—41no—
62C4×D101—41no—
63Dic202—21no—
64Dic202—21no—
65Dic202—41no—
66Dic202—41no—
67C8×C101—41no—
68C8×C101—41no—
69#11717579—22no—
70C22×C101#12011yescharacteristic
71C2×D101—41no—
72C2×D101—41no—
73C2×D101—41no—
74C4×C101—21no—
75C4×C101—42no—
76C4×C101#13511yescharacteristic
77C4×C101#13511yescharacteristic
78C4×C101AGL(1,ℤ/8)11yescharacteristic
79C4×C101#13711yescharacteristic
80Dic101—41no—
81Dic101—41no—
82Dic101—41no—
83Dic101—81no—
84C2×C101#44911yescharacteristic
85C2×C101#45211yescharacteristic
86C2×C101#45611yescharacteristic
87D101—81no—
88#1652—1011nomaximal, Sylow
89C101#165211yescharacteristic, complemented, Sylow
90#556—1011no—
91#498—1011no—
92#474—1011no—
93#462—1011no—
94#332D10111yescharacteristic, complemented
95Q8⋊C8—1011no—
96D4⋊C8—1011no—
97C2×Q16—2021no—
98C4⋊Q8—1011no—
99C4⋊Q8C2×D10111yescharacteristic
100#126—2022no—
101#124—2021no—
102C22⋊Q8—2021no—
103C4×Q8—1011no—
104C4×Q8—2021no—
105C4×D4—1011no—
106#117—2021no—
107#107—2021no—
108#106—2022no—
109C4⋊C8C2×D10111yescharacteristic
110C4⋊C8C2×D10111yescharacteristic
111#103—2021no—
112#102—2021no—
113C4×C8—1011no—
114C2×Q8—21no—
115C2×Q8—1011no—
116C2×Q8—2021no—
117C2×D4—1011no—
118C22×C4—2021no—
119Q16—4042no—
120C2×C8—21no—
121C2×C8—21no—
122C2×C8—2021no—
123C4⋊C4—21no—
124C4⋊C4—1011no—
125C4⋊C4—1011no—
126C4⋊C4—2022no—
127C4⋊C4—2021no—
128C4⋊C4—2021no—
129C4⋊C4—2021no—
130C4⋊C4D40412yes—
131C22⋊C4—2022no—
132C22⋊C4—2022no—
133C22⋊C4—2021no—
134C42—2022no—
135C42—2021no—
136C42C22×D10111yescharacteristic
137C23—2021no—
138Q8—41no—
139Q8—41no—
140Q8—4041no—
141Q8—4041no—
142Q8—2021no—
143Q8—2021no—
144D4—2021no—
145D4—2021no—
146C2×C4—22no—
147C2×C4—4041no—
148C2×C4—4041no—
149C2×C4—4041no—
150C2×C4—2021no—
151C2×C4—2021no—
152C2×C4—2021no—
153C2×C4D4×D10111yescharacteristic
154C2×C4D4×D10111yescharacteristic
155C2×C4C2×D40411yescharacteristic
156C8—41no—
157C8—41no—
158C8—2022no—
159C22—4041no—
160C22—4041no—
161C22—4041no—
162C22#1569222511yescharacteristic
163C4—21no—
164C4—42no—
165C4—8081no—
166C4—4041no—
167C4—4041no—
168C4—4041no—
169C4#1569227411yescharacteristic
170C4#1569227811yescharacteristic
171C4#1569228211yescharacteristic
172C4#1569228411yescharacteristic
173C2—8081no—
174C2#2026532511yescharacteristic
175C2#2026534011yescharacteristic
176C2#2026536211yescharacteristic
subgroupG/H
centerC22#15692225
commutator, cosocleC42×C101C23
FrattiniC42C22×D101
Fitting#332×C101C2
socleC22×C101#120
prime radicalC2×D404D4
hypercenter#332D101
hypocenterC101#1652
LCSG ▷ C42×C101 ▷ C22×C101 ▷ C101
UCSC1 ◁ C22 ◁ C42 ◁ #332

Complements

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

NH ≅ G/Ntype
2. #20265573C2semidirect
4. #20265388C2semidirect
5. #20264907C2semidirect
7. #332×C101C2semidirect
89. C101#1652semidirect
94. #332D101semidirect

Maximal subgroups

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

subgroupindex
#202659012
#202655732
#202654582
#202653882
#202649072
#202649042
#332×C1012
#1652101