Finite Group Databasebeta

← #45663493

Subgroups — #45663493

Order 32896.

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

normal subgroups only

This group has 194 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#30657461C211yesmaximal, characteristic
2#30657089C211yesmaximal, characteristic, complemented
3#30657033C211yesmaximal, characteristic, complemented
4#30657028C211yesmaximal, characteristic
5#30656467C211yesmaximal, characteristic, complemented
6#30656464C211yesmaximal, characteristic
7#332×C257C211yesmaximal, characteristic, complemented
8#22551592C2211yescharacteristic
9#22551587—22no—
10#22551581—21no—
11#22551572—21no—
12Q8×Dic257—21no—
13C4×Dic514C2211yescharacteristic
14C4×D1028C2211yescharacteristic
15#22551550—21no—
16#22551526—21no—
17#22551524—21no—
18#22551517—21no—
19#22551515—22no—
20#22551514—21no—
21C4×#14523339C2211yescharacteristic
22C4⋊Q8×C257C2211yescharacteristic
23C4⋊C8×C257C2211yescharacteristic
24C4⋊C8×C257C2211yescharacteristic
25C2×Dic514—21no—
26C2×Dic514D411yescharacteristic
27C2×D1028D411yescharacteristic
28C2×C4×D257—21no—
29C2×#14523339—21no—
30C4⋊Dic257—22no—
31C4⋊Dic257—21no—
32C4⋊Dic257—21no—
33C4⋊Dic257D411yescharacteristic
34C4⋊Dic257D411yescharacteristic
35#16861597—21no—
36#16861595—22no—
37#16861595—22no—
38#16861595—21no—
39C4×Dic257—22no—
40C4×Dic257—21no—
41C2×Q8×C257—21no—
42C2×C8×C257—21no—
43C2×C8×C257—21no—
44C4⋊C4×C257—21no—
45C4⋊C4×C257D412yes—
46C42×C257C2311yescharacteristic
47C22×D257—21no—
48Dic514—21no—
49Dic514—21no—
50Dic514—42no—
51D1028—21no—
52D1028—21no—
53C2×Dic257—21no—
54C2×Dic257—21no—
55C2×Dic257—21no—
56C2×Dic257—41no—
57C4×D257—41no—
58C4×D257—41no—
59#14523339—41no—
60Q8×C257—41no—
61Q8×C257—41no—
62C2×C4×C257—22no—
63C2×C4×C257C2×D411yescharacteristic
64C2×C4×C257C2×D411yescharacteristic
65C2×C4×C257C2×D411yescharacteristic
66C8×C257—41no—
67C8×C257—41no—
68C22×C257#12011yescharacteristic
69C2×D257—41no—
70C2×D257—41no—
71C2×D257—41no—
72C4×C257—21no—
73C4×C257—42no—
74C4×C257#13511yescharacteristic
75C4×C257#13511yescharacteristic
76C4×C257#13711yescharacteristic
77C4×C257#13711yescharacteristic
78Dic257—41no—
79Dic257—41no—
80Dic257—41no—
81Dic257—81no—
82C2×C257#45211yescharacteristic
83C2×C257#45211yescharacteristic
84C2×C257#45411yescharacteristic
85D257—81no—
86C257#163811yescharacteristic, complemented, Sylow
87#1638—2571nomaximal, Sylow
88#556—2571no—
89#487—2571no—
90#477—2571no—
91#476—2571no—
92#332D25711yescharacteristic, complemented
93Q8⋊C8—2571no—
94D4⋊C8—2571no—
95C4⋊Q8—2571no—
96C4⋊Q8C2×D25711yescharacteristic
97#126—5142no—
98#124—5141no—
99C22⋊Q8—5141no—
100C4×Q8—2571no—
101C4×Q8—5141no—
102C4×D4—2571no—
103#117—5141no—
104#107—5141no—
105#106—5141no—
106C4⋊C8C2×D25711yescharacteristic
107C4⋊C8C2×D25711yescharacteristic
108#103—5141no—
109#103—5142no—
110#102—5141no—
111C4×C8—2571no—
112C2×Q8—2571no—
113C2×Q8—21no—
114C2×Q8—5141no—
115C2×D4—2571no—
116C22×C4—5141no—
117C2×C8—21no—
118C2×C8—21no—
119C2×C8—5141no—
120C4⋊C4—2571no—
121C4⋊C4—2571no—
122C4⋊C4—21no—
123C4⋊C4—5142no—
124C4⋊C4—5141no—
125C4⋊C4—5141no—
126C4⋊C4—5141no—
127C4⋊C4D102812yes—
128C22⋊C4—5142no—
129C22⋊C4—5142no—
130C22⋊C4—5141no—
131C42—5142no—
132C42—5141no—
133C42C22×D25711yescharacteristic
134C23—5141no—
135Q8—5141no—
136Q8—5141no—
137Q8—41no—
138Q8—41no—
139Q8—10282no—
140D4—5141no—
141D4—5141no—
142C2×C4—22no—
143C2×C4—5141no—
144C2×C4—5141no—
145C2×C4—5141no—
146C2×C4—10281no—
147C2×C4—10281no—
148C2×C4—10281no—
149C2×C4D4×D25711yescharacteristic
150C2×C4D4×D25711yescharacteristic
151C2×C4C2×D102811yescharacteristic
152C8—41no—
153C8—41no—
154C8—10281no—
155C22—10281no—
156C22—10281no—
157C22—10281no—
158C22#2255156211yescharacteristic
159C4—21no—
160C4—42no—
161C4—10281no—
162C4—10281no—
163C4—10281no—
164C4—20561no—
165C4#2255161111yescharacteristic
166C4#2255161311yescharacteristic
167C4#2255162111yescharacteristic
168C4#2255162411yescharacteristic
169C2—20561no—
170C2#3065690011yescharacteristic
171C2#3065690211yescharacteristic
172C2#3065691211yescharacteristic
subgroupG/H
centerC22#22551562
commutator, cosocleC42×C257C23
FrattiniC42C22×D257
Fitting#332×C257C2
socleC22×C257#120
prime radicalC2×D1028D4
hypercenter#332D257
hypocenterC257#1638
LCSG ▷ C42×C257 ▷ C22×C257 ▷ C257
UCSC1 ◁ C22 ◁ C42 ◁ #332

Complements

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

NH ≅ G/Ntype
2. #30657089C2semidirect
3. #30657033C2semidirect
5. #30656467C2semidirect
7. #332×C257C2semidirect
86. C257#1638semidirect
92. #332D257semidirect

Maximal subgroups

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

subgroupindex
#306574612
#306570892
#306570332
#306570282
#306564672
#306564642
#332×C2572
#1638257