Finite Group Databasebeta

← #69642

Subgroups — #69642

Order 384.

324 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 89 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#5798C211yesmaximal, characteristic
2C3×#390C211yesmaximal, characteristic
3C2×#833C211yesmaximal, characteristic
4#4964C212yesmaximal, complemented
5#4523C212yesmaximal, complemented
6#1856—31nomaximal, Sylow
7C22×C3×D4C2211yescharacteristic
8C2×C3×C4⋊C4C2211yescharacteristic
9C2×C3×C22⋊C4—22no—
10C2×C3×C42C2211yescharacteristic
11C2×#217—22no—
12D4×Dic3—22no—
13D4×Dic3—22no—
14#926C2212yes—
15C2×#208C2212yes—
16#833C414yescomplemented
17#808—22no—
18#807—22no—
19#518—31no—
20#428—32no—
21C2×#100—31no—
22#390D311yescharacteristic
23#339—32no—
24C24×C3D412yes—
25C2×C3×D4—22no—
26C2×C3×D4—22no—
27C2×C3×D4C2×C412yes—
28C2×C3×D4Q812yes—
29C22×C3×C4—21no—
30C22×C3×C4—21no—
31C22×C3×C4C2311yescharacteristic
32C22×Dic3—22no—
33C22×Dic3—22no—
34C3×C4⋊C4—22no—
35C3×C22⋊C4—44no—
36C3×C42—22no—
37#217—24no—
38#217—22no—
39C4⋊Dic3D412yes—
40C4⋊Dic3D412yes—
41C4×Dic3—22no—
42#208—22no—
43#208C2×C414yescomplemented
44C2×C3⋊C8—22no—
45C22×D4C2×D311yescharacteristic
46C2×M16—32no—
47C4×D4—62no—
48C4×D4—62no—
49#117—32no—
50C2×C4⋊C4C2×D311yescharacteristic
51C2×C22⋊C4—22no—
52C2×C22⋊C4—62no—
53C2×C42C2×D311yescharacteristic
54#107—62no—
55#106—62no—
56#100—34no—
57C23×C3—22no—
58C23×C3—22no—
59C23×C3—42no—
60C23×C3—42no—
61C23×C3C4⋊C414yes—
62C23×C3C2×D411yescharacteristic
63C3×D4—24no—
64C3×D4—24no—
65C2×C3×C4—22no—
66C2×C3×C4—41no—
67C2×C3×C4—41no—
68C2×C3×C4—42no—
69C2×C3×C4—41no—
70C2×C3×C4C22×C411yescharacteristic
71C2×C3×C4C2×D412yes—
72C2×C3×C4C2×Q811yescharacteristic
73C2×C3×C4#4112yes—
74C2×Dic3—22no—
75C2×Dic3—22no—
76C2×Dic3—82no—
77C2×Dic3—82no—
78C3⋊C8—24no—
79C24#6712yes—
80C2×D4—22no—
81C2×D4—22no—
82C2×D4Dic612yes—
83C2×D4C4×D312yes—
84C22×C4—21no—
85C22×C4—21no—
86C22×C4—62no—
87C22×C4—62no—
88C22×C4C22×D311yescharacteristic
89M16—34no—
90M16—62no—
91C2×C8—62no—
92C4⋊C4—22no—
93C4⋊C4—32no—
94C4⋊C4—32no—
95C22⋊C4—44no—
96C22⋊C4—64no—
97C22⋊C4—62no—
98C42—22no—
99C42—62no—
100C22×C3—21no—
101C22×C3—21no—
102C22×C3—24no—
103C22×C3—44no—
104C22×C3—42no—
105C22×C3—42no—
106C22×C3—82no—
107C22×C3C2×C4⋊C411yescharacteristic
108C22×C3#12011yescharacteristic
109C22×C3#12311yescharacteristic
110C3×C4—42no—
111C3×C4—81no—
112C3×C4C4×D412yes—
113C3×C4C22⋊Q812yes—
114Dic3—42no—
115Dic3—42no—
116C23—22no—
117C23—22no—
118C23—42no—
119C23—42no—
120C23Dic3⋊C414yes—
121C23C2×#6711yescharacteristic
122D4—24no—
123D4—24no—
124C2×C4—22no—
125C2×C4—41no—
126C2×C4—41no—
127C2×C4—42no—
128C2×C4—41no—
129C2×C4—62no—
130C2×C4—62no—
131C2×C4—242no—
132C2×C4—242no—
133C2×C4C2×Dic611yescharacteristic
134C2×C4C2×C4×D311yescharacteristic
135C2×C4D3×D412yes—
136C2×C4#23712yes—
137C8—64no—
138C2×C3—21no—
139C2×C3—21no—
140C2×C3—44no—
141C2×C3#38511yescharacteristic
142C2×C3#45311yescharacteristic
143C2×C3#45511yescharacteristic
144C22—21no—
145C22—21no—
146C22—24no—
147C22—44no—
148C22—42no—
149C22—42no—
150C22—82no—
151C22C2×Dic3⋊C411yescharacteristic
152C22#93511yescharacteristic
153C22#93711yescharacteristic
154C4—42no—
155C4—81no—
156C4—122no—
157C4—122no—
158C4#87812yes—
159C4#67⋊C412yes—
160C3#185611yescharacteristic, complemented, Sylow
161C2—21no—
162C2—21no—
163C2—44no—
164C2#494611yescharacteristic
165C2#506111yescharacteristic
166C2#507711yescharacteristic
subgroupG/H
centerC22#937
commutatorC2×C3×C4C22×C4
FrattiniC22×C4C22×D3
FittingC3×#390C2
socleC22×C3#120
prime radicalC22×C3×D4C22
cosocleC22×C3×C4C23
hypercenter#390D3
hypocenterC3#1856
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #390

Complements

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

NH ≅ G/Ntype
4. #4964C2semidirect
5. #4523C2semidirect
16. #833C4semidirect
43. #208C2×C4semidirect
160. C3#1856semidirect

Maximal subgroups

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

subgroupindex
#57982
C3×#3902
C2×#8332
#49642
#45232
#18563