Finite Group Databasebeta

← #85362

Subgroups — #85362

Order 384.

192 conjugacy classes of subgroups; only the 32 normal subgroups are shown.

normal subgroups only

This group has 192 conjugacy classes of subgroups — too many to draw. Only its 32 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#520⋊C3C211yesmaximal, characteristic
2#5425C211yesmaximal, characteristic
3#5425C211yesmaximal, characteristic
4C4⋊#226C211yesmaximal, characteristic, complemented
5#5389C211yesmaximal, characteristic, complemented
6#5388C211yesmaximal, characteristic, complemented
7C4×#226C211yesmaximal, characteristic, complemented
8#3120—31nomaximal, Sylow
9C2×#41⋊C3C2211yescharacteristic
10C2×#226C2211yescharacteristic
11C2×#226C2211yescharacteristic
12#889—41nomaximal
13C4×SL(2,𝔽3)C2211yescharacteristic
14C4×SL(2,𝔽3)C2211yescharacteristic
15#859C2211yescharacteristic
16#859C2211yescharacteristic
17#541—31no—
18#520D311yescharacteristic
19#484—31no—
20#483—31no—
21#479—31no—
22#478—31no—
23#462—31no—
24#461—31no—
25#452—31no—
26#452—31no—
27#441—31no—
28#440—31no—
29C4×Q16—31no—
30C4×SD16—31no—
31#433—31no—
32#41⋊C3—21no—
33C2×SL(2,𝔽3)C2311yescharacteristic
34#226—21no—
35#226—21no—
36C3×C4⋊C4—41no—
37C4⋊Dic3—41no—
38Dic3⋊C4—41no—
39Dic3⋊C4—41no—
40Dic3⋊C4—41no—
41Dic3⋊C4—41no—
42C4×Dic3—41no—
43C2×#41C2×D311yescharacteristic
44C2×Q16—31no—
45C2×Q16—31no—
46C2×SD16—31no—
47C2×SD16—31no—
48C4⋊Q8—31no—
49#126—61no—
50#126—61no—
51#125—31no—
52#125—61no—
53#125—61no—
54C22⋊Q8—31no—
55C22⋊Q8—31no—
56C4×Q8—31no—
57C4×Q8—31no—
58C4×Q8C2×D311yescharacteristic
59C4×Q8C2×D311yescharacteristic
60C4×D4—31no—
61C4×D4—31no—
62#117—31no—
63C2×C4⋊C4—31no—
64#107—31no—
65#107—31no—
66#106—31no—
67#106—31no—
68C4⋊C8—31no—
69C4⋊C8—31no—
70#103—31no—
71#103—31no—
72#103—31no—
73#103—31no—
74#103—31no—
75#103—31no—
76#102—31no—
77#102—31no—
78C22⋊C8—31no—
79C22⋊C8—31no—
80#97—31no—
81C4×C8—31no—
82C2×C3×C4—41no—
83C2×C3×C4—41no—
84C2×C3×C4—41no—
85C2×Dic3—41no—
86C2×Dic3—41no—
87C2×Dic3—41no—
88C2×Dic3—41no—
89SL(2,𝔽3)#4111yescharacteristic
90#41—21no—
91#41—61no—
92C2×Q8—31no—
93C2×Q8—31no—
94C2×Q8C22×D311yescharacteristic
95C2×D4—31no—
96C22×C4—31no—
97C22×C4—61no—
98Q16—61no—
99Q16—61no—
100Q16—61no—
101Q16—61no—
102SD16—62no—
103SD16—62no—
104C2×C8—31no—
105C2×C8—31no—
106C2×C8—31no—
107C2×C8—31no—
108C4⋊C4—31no—
109C4⋊C4—31no—
110C4⋊C4—31no—
111C4⋊C4—31no—
112C4⋊C4—31no—
113C4⋊C4—61no—
114C4⋊C4—61no—
115C4⋊C4—61no—
116C4⋊C4—61no—
117C4⋊C4—61no—
118C4⋊C4—61no—
119C4⋊C4—61no—
120C4⋊C4—61no—
121C4⋊C4S411yescharacteristic
122C22⋊C4—31no—
123C22⋊C4—31no—
124C22⋊C4—61no—
125C22⋊C4—61no—
126C42—31no—
127C42—31no—
128C42—61no—
129C42—61no—
130C22×C3—41no—
131C3×C4—81no—
132C3×C4—81no—
133C3×C4—81no—
134Dic3—42no—
135Dic3—81no—
136Dic3—81no—
137Dic3—81no—
138C23—31no—
139Q8—31no—
140Q8—61no—
141Q8—61no—
142Q8—61no—
143Q8—61no—
144Q8#23511yescharacteristic
145D4—32no—
146D4—61no—
147C2×C4—31no—
148C2×C4—31no—
149C2×C4—31no—
150C2×C4—61no—
151C2×C4—61no—
152C2×C4—61no—
153C2×C4—61no—
154C2×C4—61no—
155C2×C4—61no—
156C2×C4—241no—
157C2×C4C2×S411yescharacteristic
158C2×C4C2×S411yescharacteristic
159C2×C4C2×S411yescharacteristic
160C8—61no—
161C8—61no—
162C8—61no—
163C8—61no—
164C2×C3—41no—
165C2×C3—41no—
166C2×C3—41no—
167C22—61no—
168C22—61no—
169C22—61no—
170C22C22×S411yescharacteristic
171C4—21no—
172C4—21no—
173C4—21no—
174C4—31no—
175C4—31no—
176C4—61no—
177C4—61no—
178C4—121no—
179C4—121no—
180C4—121no—
181C4—121no—
182C3—41noSylow
183C2—121no—
184C2#591211yescharacteristic
185C2#592411yescharacteristic
186C2#592711yescharacteristic
subgroupG/H
center, Frattini, socleC22C22×S4
commutator, cosocleC2×SL(2,𝔽3)C23
Fitting#520D3
prime radicalC2×#41⋊C3C22
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
4. C4⋊#226C2semidirect
5. #5389C2semidirect
6. #5388C2semidirect
7. C4×#226C2semidirect

Maximal subgroups

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

subgroupindex
#520⋊C32
#54252
C4⋊#2262
#53892
#53882
C4×#2262
#31203
#8894