Finite Group Databasebeta

← #45666013

Subgroups — #45666013

Order 32896.

230 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 230 conjugacy classes of subgroups — too many to draw. Only its 55 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#30657312C211yesmaximal, characteristic
2#30656720C212yesmaximal
3#30656536C211yesmaximal, characteristic, complemented
4#30656535C212yesmaximal, complemented
5#356×C257C211yesmaximal, characteristic, complemented
6C2×#16861619C2211yescharacteristic
7C2×Q8×D257C411yescharacteristic
8M16×D257—22no—
9#22551590C2211yescharacteristic
10#22551581C411yescharacteristic
11#22551580C411yescharacteristic
12#22551577—22no—
13#22551570—22no—
14#22551549C411yescharacteristic
15#22551511C2212yes—
16#22551508—22no—
17C4⋊Q8×C257C2211yescharacteristic
18#101×C257C2212yes—
19#16861619—42no—
20C2×Dic514—21no—
21C2×Dic514C2×C411yescharacteristic
22Q8×D257—42no—
23C2×D1028C2×C411yescharacteristic
24C2×D1028D412yes—
25C2×C4×D257—21no—
26C2×C4×D257C2×C411yescharacteristic
27C2×C4×D257D412yes—
28#16861602—22no—
29#16861601—42no—
30C8×D257—42no—
31#16861597—21no—
32#16861597—22no—
33#16861595—22no—
34#16861595—22no—
35#16861595—21no—
36C4×Dic257C2×C411yescharacteristic
37C2×Q8×C257C2×C411yescharacteristic
38C2×Q8×C257C2311yescharacteristic
39M16×C257—22no—
40C4⋊C4×C257—22no—
41C42×C257C2×C411yescharacteristic
42C22×D257C22⋊C412yes—
43C22×D257C22⋊C411yescharacteristic
44Dic514—42no—
45Dic514—41no—
46D1028—41no—
47D1028—42no—
48C2×Dic257—21no—
49C2×Dic257—21no—
50C2×Dic257C22⋊C411yescharacteristic
51C4×D257—22no—
52C4×D257—22no—
53C4×D257—22no—
54C4×D257—42no—
55#14523339—42no—
56Q8×C257—41no—
57Q8×C257—41no—
58C2×C4×C257—21no—
59C2×C4×C257—21no—
60C2×C4×C257C22×C411yescharacteristic
61C2×C4×C257C2×D412yes—
62C8×C257—42no—
63C22×C257C2×C22⋊C411yescharacteristic
64C2×D257—21no—
65C2×D257—22no—
66C2×D257—42no—
67C2×D257—41no—
68C4×C257—22no—
69C4×C257—21no—
70C4×C257—41no—
71C4×C257—41no—
72Dic257—41no—
73Dic257—41no—
74Dic257#9912yes—
75C2×C257—21no—
76C2×C257C2×#9911yescharacteristic
77D257—41no—
78D257—42no—
79C257#213511yescharacteristic, complemented, Sylow
80#2135—2571nomaximal, Sylow
81#536—2571no—
82#413—2572no—
83#356—2571no—
84#356—2572no—
85#356D25711yescharacteristic, complemented
86C2×#41—2571no—
87C22×Q8—2571no—
88C2×M16—5142no—
89C4⋊Q8—2571no—
90C4⋊Q8C2×D25711yescharacteristic
91#124—2571no—
92#124—2571no—
93#123—5142no—
94C22⋊Q8—5142no—
95#117—2571no—
96#101—2572no—
97#101C2×D25712yes—
98#100—5142no—
99#41—10282no—
100C2×Q8—2571no—
101C2×Q8—5141no—
102C2×Q8—10282no—
103C2×Q8C4×D25711yescharacteristic
104C2×Q8C22×D25711yescharacteristic
105C2×D4—2571no—
106C2×D4—2572no—
107C22×C4—2571no—
108C22×C4—2572no—
109C22×C4—5141no—
110M16—22no—
111M16—5142no—
112M16—10282no—
113C2×C8—10282no—
114C4⋊C4—22no—
115C4⋊C4—5141no—
116C4⋊C4—5142no—
117C22⋊C4—5141no—
118C22⋊C4—5142no—
119C22⋊C4—5142no—
120C42—2571no—
121C42C4×D25711yescharacteristic
122C23—2572no—
123C23—2571no—
124Q8—41no—
125Q8—41no—
126Q8—10282no—
127Q8—10281no—
128D4—10282no—
129D4—10281no—
130C2×C4—2571no—
131C2×C4—21no—
132C2×C4—21no—
133C2×C4—5142no—
134C2×C4—5142no—
135C2×C4—5142no—
136C2×C4—5141no—
137C2×C4—5141no—
138C2×C4—10282no—
139C2×C4C2×C4×D25711yescharacteristic
140C2×C4D4×D25712yes—
141C8—42no—
142C8—10282no—
143C22—5141no—
144C22—5142no—
145C22—10282no—
146C22—10281no—
147C22C22⋊C4×D25711yescharacteristic
148C4—2572no—
149C4—22no—
150C4—21no—
151C4—41no—
152C4—41no—
153C4—10281no—
154C4—10281no—
155C2—21no—
156C2—10281no—
157C2—10282no—
158C2#99×D25711yescharacteristic
subgroupG/H
centerC2#99×D257
commutatorC2×C4×C257C22×C4
FrattiniC2×Q8C22×D257
Fitting#356×C257C2
socleC2×C257C2×#99
prime radicalC2×#16861619C22
cosocleC2×Q8×C257C23
hypercenter#356D257
hypocenterC257#2135
LCSG ▷ C2×C4×C257 ▷ C22×C257 ▷ C2×C257 ▷ C257
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ #356

Complements

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

NH ≅ G/Ntype
3. #30656536C2semidirect
4. #30656535C2semidirect
5. #356×C257C2semidirect
79. C257#2135semidirect
85. #356D257semidirect

Maximal subgroups

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

subgroupindex
#306573122
#306567202
#306565362
#306565352
#356×C2572
#2135257