Finite Group Databasebeta

← #45663009

Subgroups — #45663009

Order 32896.

216 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 73 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#30657292C211yesmaximal, characteristic
2#30656759C211yesmaximal, characteristic
3C4⋊C8×D257C211yesmaximal, characteristic
4#30656485C211yesmaximal, characteristic, complemented
5#30656485C211yesmaximal, characteristic, complemented
6#30656483C211yesmaximal, characteristic, complemented
7#332×C257C211yesmaximal, characteristic, complemented
8C2×#16861601—21no—
9C2×C8×D257—21no—
10#22551592C2211yescharacteristic
11#22551583—21no—
12#22551572C412yes—
13#22551568—21no—
14Q8×Dic257—21no—
15#22551557C412yes—
16#22551550—21no—
17C42×D257C2211yescharacteristic
18C4⋊#14523339C2211yescharacteristic
19C4⋊#14523339C2211yescharacteristic
20#22551521—21no—
21#22551521—21no—
22C4⋊Q8×C257C2211yescharacteristic
23C4⋊C8×C257C2211yescharacteristic
24C4⋊C8×C257C2211yescharacteristic
25C2×Dic514C2×C412yes—
26C2×C4×D257C2×C411yescharacteristic
27C2×C4×D257D411yescharacteristic
28C2×C4×D257D411yescharacteristic
29#16861601—42no—
30C2×#14523339—21no—
31C2×#14523339—21no—
32C8×D257—42no—
33C4⋊Dic257—22no—
34C4⋊Dic257—21no—
35C4⋊Dic257—21no—
36#16861597—22no—
37#16861595—22no—
38C4×Dic257—22no—
39C4×Dic257C2×C411yescharacteristic
40C4×Dic257D411yescharacteristic
41C4×Dic257D411yescharacteristic
42C2×Q8×C257—21no—
43C2×C8×C257—21no—
44C2×C8×C257—21no—
45C4⋊C4×C257—21no—
46C4⋊C4×C257C2×C412yes—
47C42×C257C2311yescharacteristic
48C22×D257C22⋊C411yescharacteristic
49Dic514—42no—
50C2×Dic257—22no—
51C2×Dic257C22⋊C411yescharacteristic
52C2×Dic257C22⋊C411yescharacteristic
53C2×Dic257C22⋊C411yescharacteristic
54C4×D257—21no—
55C4×D257—21no—
56C4×D257—41no—
57C4×D257SD1612yes—
58C4×D257Q1612yes—
59#14523339—41no—
60#14523339—41no—
61Q8×C257—41no—
62Q8×C257—41no—
63C2×C4×C257—22no—
64C2×C4×C257C22×C411yescharacteristic
65C2×C4×C257C2×D411yescharacteristic
66C2×C4×C257C2×D411yescharacteristic
67C8×C257—41no—
68C8×C257—41no—
69C22×C257C2×C22⋊C411yescharacteristic
70C2×D257—21no—
71C2×D257—21no—
72C2×D257#10312yes—
73C4×C257—21no—
74C4×C257—42no—
75C4×C257C2×SD1611yescharacteristic
76C4×C257C2×Q1611yescharacteristic
77C4×C257AGL(1,ℤ/8)11yescharacteristic
78C4×C257#13711yescharacteristic
79Dic257—21no—
80Dic257—21no—
81Dic257—42no—
82Dic257#10312yes—
83C2×C257#41311yescharacteristic
84C2×C257C2×#10311yescharacteristic
85C2×C257#41711yescharacteristic
86D257—22no—
87C257#155011yescharacteristic, complemented, Sylow
88#1550—2571nomaximal, Sylow
89#533—2571no—
90#423—2571no—
91C2×C4⋊C8—2571no—
92#333—2571no—
93#333—2571no—
94#332—2571no—
95#332D25711yescharacteristic, complemented
96C2×M16—5141no—
97C22×C8—5141no—
98C4⋊Q8—2571no—
99C4⋊Q8C2×D25711yescharacteristic
100#125—5141no—
101C22⋊Q8—2572no—
102C22⋊Q8—5141no—
103C4×Q8—2572no—
104C4×Q8—5141no—
105#117—5141no—
106C2×C42—2571no—
107C4⋊C8—2571no—
108C4⋊C8—2571no—
109C4⋊C8—5141no—
110C4⋊C8—5141no—
111C4⋊C8C2×D25711yescharacteristic
112C4⋊C8C2×D25711yescharacteristic
113C2×Q8—2572no—
114C2×Q8—21no—
115C22×C4—2571no—
116C22×C4—2571no—
117C22×C4—2571no—
118M16—10282no—
119C2×C8—21no—
120C2×C8—21no—
121C2×C8—5141no—
122C2×C8—5141no—
123C2×C8—10282no—
124C4⋊C4—21no—
125C4⋊C4—5142no—
126C4⋊C4—5142no—
127C4⋊C4—5141no—
128C4⋊C4—5141no—
129C4⋊C4C4×D25712yes—
130C22⋊C4—5142no—
131C42—2571no—
132C42—2571no—
133C42—2571no—
134C42—5142no—
135C42C22×D25711yescharacteristic
136C23—2571no—
137Q8—41no—
138Q8—41no—
139Q8—10282no—
140C2×C4—2571no—
141C2×C4—2572no—
142C2×C4—2572no—
143C2×C4—2571no—
144C2×C4—2571no—
145C2×C4—22no—
146C2×C4—5141no—
147C2×C4—5141no—
148C2×C4—5142no—
149C2×C4—10281no—
150C2×C4C2×C4×D25711yescharacteristic
151C2×C4D4×D25711yescharacteristic
152C2×C4D4×D25711yescharacteristic
153C8—41no—
154C8—41no—
155C8—10281no—
156C8—10281no—
157C22—2572no—
158C22—5141no—
159C22—5141no—
160C22C22⋊C4×D25711yescharacteristic
161C4—2572no—
162C4—21no—
163C4—5141no—
164C4—5141no—
165C4—42no—
166C4—10282no—
167C4SD16×D25711yescharacteristic
168C4Q16×D25711yescharacteristic
169C4#2255161611yescharacteristic
170C4#2255162111yescharacteristic
171C2—5142no—
172C2#3065672011yescharacteristic
173C2#103×D25711yescharacteristic
174C2#3065673511yescharacteristic
subgroupG/H
centerC22C22⋊C4×D257
commutatorC2×C4×C257C22×C4
FrattiniC42C22×D257
Fitting#332×C257C2
socleC22×C257C2×C22⋊C4
prime radicalC22×D257C22⋊C4
cosocleC42×C257C23
hypercenter#332D257
hypocenterC257#1550
LCSG ▷ C2×C4×C257 ▷ C22×C257 ▷ C257
UCSC1 ◁ C22 ◁ C42 ◁ #332

Complements

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

NH ≅ G/Ntype
4. #30656485C2semidirect
5. #30656485C2semidirect
6. #30656483C2semidirect
7. #332×C257C2semidirect
87. C257#1550semidirect
95. #332D257semidirect

Maximal subgroups

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

subgroupindex
#306572922
#306567592
C4⋊C8×D2572
#306564852
#306564832
#332×C2572
#1550257