Finite Group Databasebeta

← #85416

Subgroups — #85416

Order 384.

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

normal subgroups only

This group has 180 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
1#557⋊C3C211yesmaximal, characteristic
2#5391C211yesmaximal, characteristic
3#5389C211yesmaximal, characteristic
4#5389C211yesmaximal, characteristic
5#5388C211yesmaximal, characteristic
6#5388C211yesmaximal, characteristic
7C4×#226C211yesmaximal, characteristic
8#3390—31nomaximal, Sylow
9C2×#226C2211yescharacteristic
10#889—41nomaximal
11C4×SL(2,𝔽3)C2211yescharacteristic
12C4×SL(2,𝔽3)C2211yescharacteristic
13C4×SL(2,𝔽3)C2211yescharacteristic
14#859C2211yescharacteristic
15#859C2211yescharacteristic
16#859C2211yescharacteristic
17#557—31no—
18#557D311yescharacteristic
19#501—31no—
20#501—31no—
21#499—31no—
22#479—31no—
23#479—31no—
24#479—31no—
25#479—31no—
26#477—31no—
27#477—31no—
28#446—31no—
29#441—31no—
30#441—31no—
31C4×Q16—31no—
32C2×SL(2,𝔽3)C2311yescharacteristic
33#226Q812yescomplemented
34C3×C4⋊C4—41no—
35C4⋊Dic3—41no—
36Dic3⋊C4—41no—
37Dic3⋊C4—41no—
38Dic3⋊C4—41no—
39Dic3⋊C4—41no—
40C4×Dic3—41no—
41C2×Q16—31no—
42C4⋊Q8—31no—
43C4⋊Q8—31no—
44C4⋊Q8—61no—
45#125—31no—
46#125—31no—
47#125—31no—
48#125—31no—
49#125—61no—
50#125—61no—
51C4×Q8—31no—
52C4×Q8—31no—
53C4×Q8—31no—
54C4×Q8—31no—
55C4×Q8—61no—
56C4×Q8C2×D311yescharacteristic
57C4×Q8C2×D311yescharacteristic
58C4×Q8C2×D311yescharacteristic
59#107—31no—
60#107—31no—
61#107—31no—
62#107—31no—
63#107—31no—
64#106—31no—
65#106—31no—
66#106—31no—
67#106—31no—
68C4⋊C8—31no—
69C4⋊C8—31no—
70C4⋊C8—31no—
71#103—31no—
72#103—31no—
73#103—31no—
74#103—31no—
75#103—31no—
76#103—31no—
77#97—31no—
78#97—31no—
79C4×C8—31no—
80C2×C3×C4—41no—
81C2×C3×C4—41no—
82C2×C3×C4—41no—
83C2×Dic3—41no—
84C2×Dic3—41no—
85C2×Dic3—41no—
86C2×Dic3—41no—
87SL(2,𝔽3)C2×Q811yescharacteristic
88C2×Q8—31no—
89C2×Q8—31no—
90C2×Q8—61no—
91C2×Q8C22×D311yescharacteristic
92Q16—32no—
93Q16—32no—
94C2×C8—31no—
95C2×C8—31no—
96C2×C8—31no—
97C2×C8—31no—
98C4⋊C4—31no—
99C4⋊C4—31no—
100C4⋊C4—31no—
101C4⋊C4—31no—
102C4⋊C4—31no—
103C4⋊C4—31no—
104C4⋊C4—31no—
105C4⋊C4—31no—
106C4⋊C4—61no—
107C4⋊C4—61no—
108C4⋊C4—61no—
109C4⋊C4—61no—
110C4⋊C4—61no—
111C4⋊C4—61no—
112C4⋊C4—61no—
113C4⋊C4—61no—
114C4⋊C4—61no—
115C4⋊C4S411yescharacteristic
116C42—31no—
117C42—31no—
118C42—31no—
119C42—61no—
120C42—61no—
121C42—61no—
122C22×C3—41no—
123C3×C4—81no—
124C3×C4—81no—
125C3×C4—81no—
126Dic3—42no—
127Dic3—81no—
128Dic3—81no—
129Dic3—81no—
130Q8—31no—
131Q8—32no—
132Q8—61no—
133Q8—121no—
134Q8—122no—
135Q8D3×Q811yescharacteristic
136C2×C4—31no—
137C2×C4—31no—
138C2×C4—31no—
139C2×C4—31no—
140C2×C4—61no—
141C2×C4—61no—
142C2×C4—61no—
143C2×C4—61no—
144C2×C4C2×S411yescharacteristic
145C2×C4C2×S411yescharacteristic
146C2×C4C2×S411yescharacteristic
147C8—32no—
148C8—61no—
149C8—61no—
150C8—61no—
151C2×C3—41no—
152C2×C3—41no—
153C2×C3—41no—
154C22C22×S411yescharacteristic
155C4—21no—
156C4—21no—
157C4—21no—
158C4—31no—
159C4—31no—
160C4—62no—
161C4—61no—
162C4—61no—
163C4—61no—
164C4—121no—
165C4—121no—
166C4—121no—
167C3—41noSylow
168C2Q8×S411yescharacteristic
169C2#592311yescharacteristic
170C2#592711yescharacteristic
subgroupG/H
center, Frattini, socleC22C22×S4
commutator, prime radical, cosocleC2×SL(2,𝔽3)C23
Fitting#557D3
hypercenterC4⋊C4S4
hypocenterSL(2,𝔽3)C2×Q8
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
33. #226Q8semidirect

Maximal subgroups

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

subgroupindex
#557⋊C32
#53912
#53892
#53882
C4×#2262
#33903
#8894