Finite Group Databasebeta

← #72115

Subgroups — #72115

Order 384.

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

normal subgroups only

This group has 264 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×#575C211yesmaximal, characteristic
2#5240C212yesmaximal, complemented
3#5233C212yesmaximal, complemented
4#5119C211yesmaximal, characteristic, complemented
5#4557C211yesmaximal, characteristic, complemented
6#2053—31nomaximal, Sylow
7C2×C3×#41C2212yes—
8C3×#137—22no—
9C3×AGL(1,ℤ/8)—22no—
10C3×#135—22no—
11C2×C3×M16C2211yescharacteristic
12#926C2212yes—
13#926C2211yescharacteristic
14C22×C3⋊C8C2211yescharacteristic
15#835—22no—
16#834—24no—
17#832—22no—
18Dic3⋊C8—22no—
19#575D311yescharacteristic
20#424—31no—
21#416—32no—
22#410—32no—
23H(3,ℤ/4)—31no—
24C3×#41—22no—
25C3×#41—22no—
26C3×#41—22no—
27C3×#41—22no—
28C2×C3×Q8D412yes—
29C2×C3×D4—22no—
30C2×C3×D4D412yes—
31C22×C3×C4—22no—
32C22×C3×C4C2311yescharacteristic
33C3×Q16—22no—
34C3×SD16—24no—
35C3×D8—22no—
36C3×M16—22no—
37C2×C3×C8D412yes—
38#217—22no—
39#217D412yes—
40C4⋊Dic3—21no—
41C4⋊Dic3—22no—
42C4×Dic3—21no—
43C4×Dic3—22no—
44C2×C3⋊C8—21no—
45C2×C3⋊C8—22no—
46C2×#41C2×D312yes—
47#137—22no—
48AGL(1,ℤ/8)—22no—
49#135—22no—
50C2×M16C2×D311yescharacteristic
51C22×C8—31no—
52#117—31no—
53#117—32no—
54C4⋊C8—62no—
55#103—62no—
56#102—62no—
57#99—64no—
58C23×C3—22no—
59C23×C3C2×D411yescharacteristic
60C3×Q8—22no—
61C3×Q8—22no—
62C3×D4—22no—
63C3×D4—22no—
64C3×D4—42no—
65C3×D4—42no—
66C2×C3×C4—22no—
67C2×C3×C4—42no—
68C2×C3×C4—42no—
69C2×C3×C4C2×D412yes—
70C2×C3×C4C2×D411yescharacteristic
71C2×C3×C4#4111yescharacteristic
72C2×C3×C4#4112yes—
73C2×Dic3—22no—
74C2×Dic3—42no—
75C3×C8—22no—
76C3⋊C8—41no—
77#41—22no—
78#41—22no—
79#41—22no—
80#41—22no—
81C2×Q8#6712yes—
82C2×D4—22no—
83C2×D4#6712yes—
84C22×C4—22no—
85C22×C4C22×D311yescharacteristic
86Q16—22no—
87SD16—24no—
88D8—22no—
89M16—22no—
90C2×C8—62no—
91C2×C8—61no—
92C2×C8#6712yes—
93C4⋊C4—61no—
94C4⋊C4—62no—
95C22⋊C4—32no—
96C22⋊C4—62no—
97C42—61no—
98C42—62no—
99C22×C3—41no—
100C22×C3—42no—
101C22×C3—82no—
102C22×C3#12111yescharacteristic
103C22×C3#12112yes—
104C3×C4—42no—
105C3×C4#12011yescharacteristic
106C3×C4#12312yes—
107C3×C4#12411yescharacteristic
108Dic3—42no—
109Dic3—82no—
110C23—22no—
111C23D3×D411yescharacteristic
112Q8—22no—
113Q8—22no—
114D4—22no—
115D4—22no—
116D4—42no—
117D4—42no—
118C2×C4—22no—
119C2×C4—42no—
120C2×C4—42no—
121C2×C4—62no—
122C2×C4—122no—
123C2×C4#23711yescharacteristic
124C2×C4#23712yes—
125C2×C4C2×#6712yes—
126C2×C4C2×#6711yescharacteristic
127C8—22no—
128C8—121no—
129C2×C3—21no—
130C2×C3—22no—
131C2×C3—82no—
132C2×C3#39411yescharacteristic
133C22—41no—
134C22—42no—
135C22—82no—
136C22#93811yescharacteristic
137C22#93912yes—
138C4—42no—
139C4—122no—
140C4—242no—
141C4#93512yes—
142C4#93611yescharacteristic
143C4#95311yescharacteristic
144C3#205311yescharacteristic, complemented, Sylow
145C2—21no—
146C2—22no—
147C2—82no—
148C2#522211yescharacteristic
subgroupG/H
centerC4#936
commutator, cosocleC22×C3×C4C23
FrattiniC22×C4C22×D3
Fitting, prime radicalC3×#575C2
socleC2×C3#394
hypercenter#575D3
hypocenterC3#2053
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C4 ◁ C22×C4 ◁ #575

Complements

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

NH ≅ G/Ntype
2. #5240C2semidirect
3. #5233C2semidirect
4. #5119C2semidirect
5. #4557C2semidirect
144. C3#2053semidirect

Maximal subgroups

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

subgroupindex
C3×#5752
#52402
#52332
#51192
#45572
#20533