Finite Group Databasebeta

← #45663494

Subgroups — #45663494

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#30657457C211yesmaximal, characteristic
2#30657088C211yesmaximal, characteristic, complemented
3#30657033C211yesmaximal, characteristic, complemented
4#30657027C211yesmaximal, characteristic
5#30656482C211yesmaximal, characteristic, complemented
6#30656469C211yesmaximal, characteristic, complemented
7D4⋊C8×C257C211yesmaximal, characteristic
8#22551592C2211yescharacteristic
9#22551592C2211yescharacteristic
10#22551585—22no—
11#22551579—21no—
12#22551571—21no—
13C4×Dic514C2211yescharacteristic
14#22551557—21no—
15#22551551—21no—
16#22551526—21no—
17#22551523—21no—
18C4⋊#14523339C2211yescharacteristic
19#22551517—21no—
20#22551517—22no—
21#22551513—21no—
22C4×D4×C257C2211yescharacteristic
23C4⋊C8×C257C2211yescharacteristic
24C4×C8×C257C2211yescharacteristic
25C2×Dic514—21no—
26C2×Dic514—21no—
27C2×Dic514D411yescharacteristic
28C2×#14523339—21no—
29C4⋊Dic257—21no—
30C4⋊Dic257—21no—
31C4⋊Dic257—21no—
32C4⋊Dic257D411yescharacteristic
33C4⋊Dic257D412yes—
34#16861597—22no—
35#16861597—21no—
36C22⋊Dic257—22no—
37C22⋊Dic257—22no—
38C4×Dic257—22no—
39C4×Dic257—21no—
40C2×D4×C257D411yescharacteristic
41C22×C4×C257—21no—
42C2×C8×C257—21no—
43C2×C8×C257—21no—
44C4⋊C4×C257D411yescharacteristic
45C22⋊C4×C257—21no—
46C42×C257C2311yescharacteristic
47Dic514—21no—
48Dic514—21no—
49Dic514—41no—
50Dic514—41no—
51Dic514—42no—
52C2×Dic257—21no—
53C2×Dic257—21no—
54C2×Dic257—22no—
55C2×Dic257—41no—
56#14523339—41no—
57C23×C257—21no—
58D4×C257—21no—
59D4×C257—21no—
60C2×C4×C257—21no—
61C2×C4×C257—41no—
62C2×C4×C257—41no—
63C2×C4×C257C2×D411yescharacteristic
64C2×C4×C257C2×D411yescharacteristic
65C2×C4×C257C2×D411yescharacteristic
66C8×C257—41no—
67C8×C257—41no—
68C22×C257—41no—
69C22×C257—41no—
70C22×C257—41no—
71C22×C257#12011yescharacteristic
72C4×C257—21no—
73C4×C257—41no—
74C4×C257#13511yescharacteristic
75C4×C257#13511yescharacteristic
76C4×C257#13711yescharacteristic
77C4×C257#13711yescharacteristic
78Dic257—41no—
79Dic257—42no—
80Dic257—41no—
81Dic257—81no—
82C2×C257—81no—
83C2×C257#45211yescharacteristic
84C2×C257#45211yescharacteristic
85C2×C257#45411yescharacteristic
86C257#163811yescharacteristic, complemented, Sylow
87#1638—2571nomaximal, Sylow
88#556—2571no—
89#487—2571no—
90#477—2571no—
91#476—2571no—
92#332—2571no—
93Q8⋊C8—2571no—
94D4⋊C8D25711yescharacteristic
95C4⋊Q8—2571no—
96C4⋊Q8—2571no—
97#126—5142no—
98#124—5141no—
99C22⋊Q8—5141no—
100C4×Q8—2571no—
101C4×Q8—5141no—
102C4×D4C2×D25711yescharacteristic
103#117—5141no—
104#107—5141no—
105#106—5141no—
106C4⋊C8—2571no—
107C4⋊C8C2×D25711yescharacteristic
108#103—5142no—
109#103—5141no—
110#102—5141no—
111C4×C8C2×D25711yescharacteristic
112C2×Q8—2571no—
113C2×Q8—5141no—
114C2×Q8—5141no—
115C2×D4D102811yescharacteristic
116C22×C4—21no—
117C2×C8—21no—
118C2×C8—21no—
119C2×C8—5141no—
120C4⋊C4—2572no—
121C4⋊C4—2571no—
122C4⋊C4—5142no—
123C4⋊C4—5141no—
124C4⋊C4—5141no—
125C4⋊C4—5141no—
126C4⋊C4—5141no—
127C4⋊C4D102811yescharacteristic
128C22⋊C4—21no—
129C22⋊C4—5142no—
130C22⋊C4—5142no—
131C42—5142no—
132C42—5141no—
133C42C22×D25711yescharacteristic
134C23—21no—
135Q8—5141no—
136Q8—5141no—
137Q8—10282no—
138Q8—10281no—
139Q8—10281no—
140D4—21no—
141D4—21no—
142C2×C4—21no—
143C2×C4—5141no—
144C2×C4—5142no—
145C2×C4—5141no—
146C2×C4—41no—
147C2×C4—41no—
148C2×C4—10281no—
149C2×C4D4×D25711yescharacteristic
150C2×C4D4×D25711yescharacteristic
151C2×C4C2×D102811yescharacteristic
152C8—41no—
153C8—41no—
154C8—10281no—
155C22—41no—
156C22—41no—
157C22—41no—
158C22#2255156211yescharacteristic
159C4—21no—
160C4—41no—
161C4—10281no—
162C4—10282no—
163C4—10281no—
164C4—20561no—
165C4#2255161111yescharacteristic
166C4#2255161411yescharacteristic
167C4#2255162111yescharacteristic
168C4#2255162511yescharacteristic
169C2—81no—
170C2#3065690011yescharacteristic
171C2#3065690111yescharacteristic
172C2#3065691111yescharacteristic
subgroupG/H
centerC22#22551562
commutator, cosocleC42×C257C23
FrattiniC42C22×D257
FittingD4⋊C8×C257C2
socleC22×C257#120
prime radicalC2×D4×C257D4
hypercenterD4⋊C8D257
hypocenterC257#1638
LCSG ▷ C42×C257 ▷ C22×C257 ▷ C257
UCSC1 ◁ C22 ◁ C42 ◁ D4⋊C8

Complements

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

NH ≅ G/Ntype
2. #30657088C2semidirect
3. #30657033C2semidirect
5. #30656482C2semidirect
6. #30656469C2semidirect
86. C257#1638semidirect

Maximal subgroups

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

subgroupindex
#306574572
#306570882
#306570332
#306570272
#306564822
#306564692
D4⋊C8×C2572
#1638257