Finite Group Databasebeta

← #45665481

Subgroups — #45665481

Order 32896.

246 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 69 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#30657368C211yesmaximal, characteristic
2#30656746C212yesmaximal
3#30656506C211yesmaximal, characteristic
4#30656497C212yesmaximal
5#391×C257C211yesmaximal, characteristic
6C2×#16861602C2212yes—
7#22551591—22no—
8#22551583—22no—
9#22551571—22no—
10#22551551C2212yes—
11#22551516—22no—
12#22551516—22no—
13C22×Q8×C257C2211yescharacteristic
14C2×C4⋊C4×C257C2211yescharacteristic
15C2×C42×C257C2211yescharacteristic
16#95×C257—22no—
17C2×Dic514—22no—
18#16861602—22no—
19#16861602—22no—
20C2×#14523339—22no—
21C4⋊Dic257—22no—
22C4⋊Dic257D412yes—
23C4⋊Dic257D412yes—
24#16861597—24no—
25C22⋊Dic257—22no—
26C4×Dic257—22no—
27C2×Q8×C257—21no—
28C2×Q8×C257—22no—
29C2×Q8×C257—22no—
30C2×Q8×C257Q812yescomplemented
31C22×C4×C257—21no—
32C22×C4×C257—21no—
33C22×C4×C257D412yes—
34C22×C4×C257C2311yescharacteristic
35C4⋊C4×C257—22no—
36C42×C257—22no—
37Dic514—82no—
38C2×Dic257—22no—
39C2×Dic257—22no—
40#14523339—42no—
41C23×C257C2×D411yescharacteristic
42Q8×C257—24no—
43Q8×C257—42no—
44C2×C4×C257—22no—
45C2×C4×C257—22no—
46C2×C4×C257—22no—
47C2×C4×C257—41no—
48C2×C4×C257—41no—
49C2×C4×C257—41no—
50C2×C4×C257—41no—
51C2×C4×C257—42no—
52C2×C4×C257C2×D412yes—
53C2×C4×C257C2×Q811yescharacteristic
54C2×C4×C257#4112yes—
55C2×C4×C257#4111yescharacteristic
56C22×C257—21no—
57C22×C257—21no—
58C22×C257#12011yescharacteristic
59C22×C257C22⋊Q811yescharacteristic
60C22×C257#12411yescharacteristic
61C4×C257—42no—
62C4×C257—42no—
63C4×C257—42no—
64C4×C257C22⋊Q812yes—
65C4×C257#12412yes—
66Dic257—42no—
67Dic257—42no—
68C2×C257—21no—
69C2×C257—21no—
70C2×C257#39311yescharacteristic
71C2×C257#45511yescharacteristic
72C2×C257#45611yescharacteristic
73C257#202811yescharacteristic, complemented, Sylow
74#2028—2571nomaximal, Sylow
75#544—2571no—
76#419—2572no—
77#391D25711yescharacteristic
78#343—2571no—
79#339—2572no—
80C22×Q8C2×D25711yescharacteristic
81C2×M16—2572no—
82C4⋊Q8—5142no—
83#125—5142no—
84C22⋊Q8—5142no—
85#117—2572no—
86C2×C4⋊C4C2×D25711yescharacteristic
87C2×C42C2×D25711yescharacteristic
88#103—5142no—
89#103—5142no—
90#95—22no—
91C2×Q8—21no—
92C2×Q8—22no—
93C2×Q8—22no—
94C2×Q8—5142no—
95C2×Q8Dic51412yescomplemented
96C22×C4—21no—
97C22×C4—21no—
98C22×C4C22⋊D25712yes—
99C22×C4C22×D25711yescharacteristic
100M16—5142no—
101M16—5142no—
102C2×C8—5142no—
103C4⋊C4—2572no—
104C4⋊C4—2572no—
105C4⋊C4—22no—
106C4⋊C4—5142no—
107C4⋊C4—5144no—
108C22⋊C4—5142no—
109C42—22no—
110C42—5142no—
111C23C2×C22⋊D25711yescharacteristic
112Q8—24no—
113Q8—42no—
114Q8—20562no—
115C2×C4—22no—
116C2×C4—22no—
117C2×C4—22no—
118C2×C4—5142no—
119C2×C4—5142no—
120C2×C4—41no—
121C2×C4—41no—
122C2×C4—41no—
123C2×C4—41no—
124C2×C4—42no—
125C2×C4D4×D25712yes—
126C2×C4C2×Dic51411yescharacteristic
127C2×C4#1686161712yes—
128C2×C4#1686161811yescharacteristic
129C8—10282no—
130C22—21no—
131C22—21no—
132C22#2255156011yescharacteristic
133C22#2255157111yescharacteristic
134C22#2255157911yescharacteristic
135C4—42no—
136C4—42no—
137C4—42no—
138C4—10282no—
139C4—10282no—
140C4#2255156912yes—
141C4#2255157812yes—
142C2—21no—
143C2—21no—
144C2#3065664911yescharacteristic
145C2#3065691411yescharacteristic
146C2#3065691911yescharacteristic
subgroupG/H
centerC22#22551560
commutator, cosocleC22×C4×C257C23
FrattiniC22×C4C22×D257
Fitting#391×C257C2
socleC22×C257#120
prime radicalC23×C257C2×D4
hypercenter#391D257
hypocenterC257#2028
LCSG ▷ C22×C4×C257 ▷ C2×C257 ▷ C257
UCSC1 ◁ C22 ◁ C22×C4 ◁ #391

Complements

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

NH ≅ G/Ntype
30. C2×Q8×C257Q8semidirect
73. C257#2028semidirect
95. C2×Q8Dic514semidirect

Maximal subgroups

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

subgroupindex
#306573682
#306567462
#306565062
#306564972
#391×C2572
#2028257