Finite Group Databasebeta

← #10682491

Subgroups — #10682491

Order 768.

456 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 456 conjugacy classes of subgroups — too many to draw. Only its 67 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
1C3×#1782C211yesmaximal, characteristic
2#72137C212yesmaximal
3#7319—31nomaximal, Sylow
4C22×C3×#41C2211yescharacteristic
5C2×C3×#99—22no—
6C3×#381C412yes—
7C2×#836—22no—
8C2×#833—22no—
9#5207C414yescomplemented
10#1785—32no—
11#1782D311yescharacteristic
12C2×C3×#41—42no—
13C2×C3×#41—44no—
14C22×C3×Q8D411yescharacteristic
15C22×C3×D4D412yes—
16C22×C3×D4Q811yescharacteristic
17C23×C3×C4C2×C412yes—
18C23×C3×C4C2×C411yescharacteristic
19C2×C3×C22⋊C4—22no—
20C2×#208—24no—
21C3×#99—44no—
22C3×#95—42no—
23#836—44no—
24#833—44no—
25#830—44no—
26C22×#41C2×D311yescharacteristic
27C2×#101—62no—
28C2×#100—62no—
29C2×#99—22no—
30#407—34no—
31#381Dic312yes—
32C24×C3C22⋊C411yescharacteristic
33C24×C3C4⋊C412yes—
34C3×#41—416no—
35C2×C3×Q8—24no—
36C2×C3×Q8—41no—
37C2×C3×D4—24no—
38C2×C3×D4—24no—
39C2×C3×D4—42no—
40C2×C3×D4—41no—
41C2×C3×D4—42no—
42C22×C3×C4—22no—
43C22×C3×C4—24no—
44C22×C3×C4—42no—
45C22×C3×C4—42no—
46C22×C3×C4—42no—
47C22×C3×C4—42no—
48C22×C3×C4C4211yescharacteristic
49C22×C3×C4C22⋊C412yes—
50C22×C3×C4C4⋊C411yescharacteristic
51C3×C22⋊C4—44no—
52#208—44no—
53#208—44no—
54C2×C3⋊C8—44no—
55C2×#41—42no—
56C2×#41—44no—
57C22×Q8D1211yescharacteristic
58C22×D4Dic611yescharacteristic
59C22×D4#6712yes—
60C23×C4C4×D312yes—
61C23×C4C2×Dic311yescharacteristic
62C2×M16—64no—
63C2×C22⋊C4—22no—
64#101—124no—
65#100—124no—
66#99—44no—
67C22⋊C8—124no—
68#95—42no—
69C23×C3—22no—
70C23×C3—22no—
71C23×C3—22no—
72C23×C3—22no—
73C23×C3—81no—
74C23×C3—82no—
75C23×C3#9511yescharacteristic
76C23×C3#9912yes—
77C3×Q8—44no—
78C3×D4—48no—
79C3×D4—44no—
80C2×C3×C4—22no—
81C2×C3×C4—22no—
82C2×C3×C4—21no—
83C2×C3×C4—22no—
84C2×C3×C4—21no—
85C2×C3×C4—22no—
86C2×C3×C4—48no—
87C2×C3×C4—44no—
88C2×C3×C4—82no—
89C2×C3×C4—84no—
90C2×C3×C4#9914yes—
91C3⋊C8—84no—
92C24Dic3⋊C412yes—
93C24#21711yescharacteristic
94#41—416no—
95C2×Q8—24no—
96C2×Q8—41no—
97C2×D4—24no—
98C2×D4—24no—
99C2×D4—41no—
100C2×D4—42no—
101C2×D4—42no—
102C22×C4—22no—
103C22×C4—24no—
104C22×C4—42no—
105C22×C4—42no—
106C22×C4—42no—
107C22×C4—42no—
108C22×C4C4×Dic311yescharacteristic
109C22×C4C4⋊Dic311yescharacteristic
110C22×C4(C2×D3)⋊C412yes—
111M16—124no—
112M16—124no—
113C2×C8—124no—
114C22⋊C4—44no—
115C22×C3—22no—
116C22×C3—41no—
117C22×C3—42no—
118C22×C3—44no—
119C22×C3—42no—
120C22×C3—82no—
121C22×C3—82no—
122C22×C3—82no—
123C22×C3#34211yescharacteristic
124C22×C3#34212yes—
125C3×C4—24no—
126C3×C4—41no—
127C3×C4—41no—
128C3×C4—162no—
129C23—22no—
130C23—22no—
131C23—22no—
132C23—22no—
133C23—82no—
134C23—81no—
135C23C3⋊#9511yescharacteristic
136C23#83412yes—
137Q8—44no—
138D4—48no—
139D4—44no—
140C2×C4—22no—
141C2×C4—22no—
142C2×C4—21no—
143C2×C4—21no—
144C2×C4—22no—
145C2×C4—22no—
146C2×C4—48no—
147C2×C4—44no—
148C2×C4—82no—
149C2×C4—84no—
150C2×C4#80614yes—
151C8—244no—
152C2×C3—21no—
153C2×C3—22no—
154C2×C3—42no—
155C2×C3—82no—
156C2×C3#130411yescharacteristic
157C22—22no—
158C22—41no—
159C22—42no—
160C22—44no—
161C22—42no—
162C22—82no—
163C22—82no—
164C22—82no—
165C22#452611yescharacteristic
166C22#452712yes—
167C4—24no—
168C4—41no—
169C4—41no—
170C4—162no—
171C3#731911yescharacteristic, complemented, Sylow
172C2—21no—
173C2—22no—
174C2—42no—
175C2—82no—
176C2C3⋊#130411yescharacteristic
subgroupG/H
centerC2C3⋊#1304
commutatorC22×C3×C4C42
FrattiniC22×#41C2×D3
FittingC3×#1782C2
socleC2×C3#1304
prime radical, cosocleC22×C3×#41C22
hypercenter#1782D3
hypocenterC3#7319
LCSG ▷ C22×C3×C4 ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ C23 ◁ C22×#41 ◁ #1782

Complements

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

NH ≅ G/Ntype
9. #5207C4semidirect
171. C3#7319semidirect

Maximal subgroups

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

subgroupindex
C3×#17822
#721372
#73193