Finite Group Databasebeta

← #85403

Subgroups — #85403

Order 384.

208 conjugacy classes of subgroups; only the 34 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 34 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
1C4⋊(#41⋊C3)C211yesmaximal, characteristic
2#5425C212yesmaximal, complemented
3#5424C212yesmaximal, complemented
4C4×#226C211yesmaximal, characteristic, complemented
5#5386C211yesmaximal, characteristic, complemented
6#3304—31nomaximal, Sylow
7C2×#41⋊C3C2212yes—
8C2×#226C2211yescharacteristic
9#887—41nomaximal
10C4×SL(2,𝔽3)C2211yescharacteristic
11#859C2212yes—
12#859C2211yescharacteristic
13#556—31no—
14#550D311yescharacteristic
15#486—31no—
16#482—32no—
17#475—31no—
18#465—32no—
19#460—31no—
20#452—32no—
21C4×Q16—31no—
22C8×D4—31no—
23#416—32no—
24C2×Dic6—41no—
25#41⋊C3—22no—
26C2×SL(2,𝔽3)C2311yescharacteristic
27#226—21no—
28C3×C4⋊C4—41no—
29Dic3⋊C4—42no—
30C4×Dic3—41no—
31C4×Dic3—42no—
32C2×#41C2×D312yes—
33C2×Q16—31no—
34C2×SD16—32no—
35C22×C8—32no—
36C4⋊Q8—31no—
37#127—31no—
38#126—62no—
39#124—61no—
40C22⋊Q8—32no—
41#121—32no—
42C4×Q8—31no—
43C4×Q8—61no—
44C4×Q8C2×D311yescharacteristic
45C4×D4—31no—
46#117—32no—
47#107—31no—
48#106—32no—
49C4⋊C8—31no—
50#103—31no—
51#103—31no—
52#103—32no—
53#102—32no—
54#102—32no—
55#102—32no—
56C22⋊C8—32no—
57C4×C8—31no—
58C2×C3×C4—41no—
59C2×C3×C4—42no—
60C2×Dic3—41no—
61C2×Dic3—42no—
62C2×Dic3—41no—
63Dic6—44no—
64SL(2,𝔽3)#4111yescharacteristic
65#41—22no—
66#41—62no—
67C2×Q8—31no—
68C2×Q8—61no—
69C2×Q8C22×D311yescharacteristic
70C2×D4—32no—
71C2×D4—31no—
72C2×D4—61no—
73C22×C4—32no—
74Q16—61no—
75Q16—61no—
76SD16—122no—
77C2×C8—31no—
78C2×C8—31no—
79C2×C8—32no—
80C2×C8—122no—
81C4⋊C4—31no—
82C4⋊C4—32no—
83C4⋊C4—31no—
84C4⋊C4—61no—
85C4⋊C4—61no—
86C4⋊C4—62no—
87C4⋊C4S411yescharacteristic
88C22⋊C4—32no—
89C22⋊C4—62no—
90C22⋊C4—62no—
91C42—31no—
92C42—62no—
93C42—61no—
94C22×C3—41no—
95C3×C4—42no—
96C3×C4—82no—
97Dic3—44no—
98Dic3—42no—
99Dic3—81no—
100C23—32no—
101Q8—31no—
102Q8—61no—
103Q8—61no—
104Q8—122no—
105Q8#23711yescharacteristic
106D4—62no—
107D4—62no—
108D4—62no—
109D4—241no—
110C2×C4—31no—
111C2×C4—31no—
112C2×C4—62no—
113C2×C4—62no—
114C2×C4—61no—
115C2×C4—62no—
116C2×C4—61no—
117C2×C4C2×S412yes—
118C2×C4C2×S411yescharacteristic
119C8—61no—
120C8—62no—
121C8—61no—
122C2×C3—41no—
123C2×C3—41no—
124C2×C3—41no—
125C22—62no—
126C22—62no—
127C22—62no—
128C22C22×S411yescharacteristic
129C4—22no—
130C4—31no—
131C4—31no—
132C4—61no—
133C4—121no—
134C4—122no—
135C4—121no—
136C4#98512yes—
137C3—41noSylow
138C2—122no—
139C2#591411yescharacteristic
140C2C2×#98511yescharacteristic
141C2#592411yescharacteristic
subgroupG/H
center, Frattini, socleC22C22×S4
commutator, cosocleC2×SL(2,𝔽3)C23
Fitting#550D3
prime radicalC4⋊(#41⋊C3)C2
hypercenterC4⋊C4S4
hypocenterSL(2,𝔽3)#41
LCSG ▷ C2×SL(2,𝔽3) ▷ SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
2. #5425C2semidirect
3. #5424C2semidirect
4. C4×#226C2semidirect
5. #5386C2semidirect

Maximal subgroups

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

subgroupindex
C4⋊(#41⋊C3)2
#54252
#54242
C4×#2262
#53862
#33043
#8874