Finite Group Databasebeta

← #45663643

Subgroups — #45663643

Order 32896.

196 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 196 conjugacy classes of subgroups — too many to draw. Only its 49 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#30657107C211yesmaximal, characteristic, complemented
2#30657096C212yesmaximal, complemented
3#30656479C211yesmaximal, characteristic, complemented
4#30656477C212yesmaximal, complemented
5#562×C257C211yesmaximal, characteristic
6#22551592C2211yescharacteristic
7#22551516—22no—
8#22551515—22no—
9#22551513—22no—
10#22551503C2212yes—
11#22551503C2211yescharacteristic
12#125×C257C2211yescharacteristic
13#124×C257C2212yes—
14#123×C257—22no—
15C22⋊Q8×C257—22no—
16#121×C257—22no—
17C2×Dic514—21no—
18C2×#14523339—22no—
19C2×#14523339—21no—
20C4⋊Dic257—22no—
21C2×Q8×C257D412yes—
22C2×D4×C257—22no—
23C2×D4×C257D412yes—
24C22×C4×C257—22no—
25C4⋊C4×C257—22no—
26C4⋊C4×C257—22no—
27C4⋊C4×C257D412yes—
28C22⋊C4×C257—22no—
29C22⋊C4×C257—22no—
30C22⋊C4×C257—22no—
31C42×C257C2311yescharacteristic
32Dic514—42no—
33C2×Dic257—41no—
34#14523339—41no—
35#14523339—42no—
36C23×C257—22no—
37Q8×C257—42no—
38D4×C257—42no—
39D4×C257—82no—
40C2×C4×C257—22no—
41C2×C4×C257—22no—
42C2×C4×C257—82no—
43C2×C4×C257C2×D411yescharacteristic
44C2×C4×C257C2×D412yes—
45C22×C257—42no—
46C22×C257—42no—
47C22×C257—42no—
48C22×C257#12011yescharacteristic
49C4×C257—21no—
50C4×C257—22no—
51C4×C257—42no—
52C4×C257—42no—
53Dic257—81no—
54C2×C257—82no—
55C2×C257#45411yescharacteristic
56C2×C257#45512yes—
57C257#166211yescharacteristic, complemented, Sylow
58#1662—2571nomaximal, Sylow
59#562D25711yescharacteristic
60#491—2571no—
61#489—2572no—
62#330—2571no—
63#329—2572no—
64C4⋊Q8—2571no—
65#125C2×D25711yescharacteristic
66#124C2×D25712yes—
67#123—22no—
68C22⋊Q8—22no—
69#121—22no—
70#103—5142no—
71#103—5142no—
72#102—5142no—
73#97—2571no—
74#97—2572no—
75C2×Q8—5141no—
76C2×Q8C22⋊D25712yes—
77C2×D4—22no—
78C2×D4C22⋊D25712yes—
79C22×C4—22no—
80C2×C8—5142no—
81C2×C8—5141no—
82C4⋊C4—22no—
83C4⋊C4—22no—
84C4⋊C4—5142no—
85C4⋊C4C22⋊D25712yes—
86C22⋊C4—22no—
87C22⋊C4—22no—
88C22⋊C4—22no—
89C42C22×D25711yescharacteristic
90C23—22no—
91Q8—42no—
92Q8—10282no—
93D4—42no—
94D4—82no—
95C2×C4—22no—
96C2×C4—22no—
97C2×C4—10281no—
98C2×C4—82no—
99C2×C4C2×C22⋊D25711yescharacteristic
100C2×C4C2×C22⋊D25712yes—
101C8—10281no—
102C8—10282no—
103C22—42no—
104C22—42no—
105C22—42no—
106C22#2255156111yescharacteristic
107C4—21no—
108C4—22no—
109C4—42no—
110C4—42no—
111C4—20561no—
112C2—82no—
113C2#3065691011yescharacteristic
114C2#3065691512yes—
subgroupG/H
centerC22#22551561
commutator, cosocleC42×C257C23
FrattiniC42C22×D257
Fitting, prime radical#562×C257C2
socleC22×C257#120
hypercenter#562D257
hypocenterC257#1662
LCSG ▷ C42×C257 ▷ C22×C257 ▷ C257
UCSC1 ◁ C22 ◁ C42 ◁ #562

Complements

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

NH ≅ G/Ntype
1. #30657107C2semidirect
2. #30657096C2semidirect
3. #30656479C2semidirect
4. #30656477C2semidirect
57. C257#1662semidirect

Maximal subgroups

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

subgroupindex
#306571072
#306570962
#306564792
#306564772
#562×C2572
#1662257