Finite Group Databasebeta

← #85394

Subgroups — #85394

Order 384.

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

normal subgroups only

This group has 212 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
1D4×SL(2,𝔽3)C211yesmaximal, characteristic
2#5420C211yesmaximal, characteristic, complemented
3#5419C211yesmaximal, characteristic, complemented
4#5419C211yesmaximal, characteristic, complemented
5C2×#859C211yesmaximal, characteristic, complemented
6#5389C211yesmaximal, characteristic, complemented
7#5388C211yesmaximal, characteristic, complemented
8#3312—31nomaximal, Sylow
9C22×SL(2,𝔽3)C2211yescharacteristic
10C22×SL(2,𝔽3)C2211yescharacteristic
11C2×#226C2211yescharacteristic
12#935—41nomaximal
13C4×SL(2,𝔽3)C2211yescharacteristic
14#859C2211yescharacteristic
15#859C2211yescharacteristic
16#859C2211yescharacteristic
17#552—31no—
18D4×Q8D311yescharacteristic
19#491—31no—
20#485—31no—
21#483—31no—
22#479—31no—
23#470—31no—
24#467—31no—
25#462—31no—
26#451—31no—
27#451—31no—
28#441—31no—
29#435—31no—
30#419—31no—
31C2×#103—31no—
32C2×C3×D4—41no—
33C22×Dic3—41no—
34C2×SL(2,𝔽3)—21no—
35C2×SL(2,𝔽3)—21no—
36C2×SL(2,𝔽3)C2311yescharacteristic
37#226—21no—
38#217—41no—
39#217—41no—
40#217—41no—
41Dic3⋊C4—41no—
42Dic3⋊C4—41no—
43C22×Q8C2×D311yescharacteristic
44C22×Q8C2×D311yescharacteristic
45C2×Q16—31no—
46C2×Q16—31no—
47C2×Q16—31no—
48C2×M16—31no—
49C22×C8—31no—
50C4⋊Q8—31no—
51#126—61no—
52#125—31no—
53#125—61no—
54#123—61no—
55C22⋊Q8—31no—
56C22⋊Q8—31no—
57C22⋊Q8—31no—
58C22⋊Q8—31no—
59C4×Q8—31no—
60C4×Q8C2×D311yescharacteristic
61C4×D4—31no—
62#117—31no—
63#117—61no—
64C2×C4⋊C4—31no—
65#107—31no—
66#106—31no—
67#106—31no—
68C4⋊C8—31no—
69#103—31no—
70#103—31no—
71#103—31no—
72#103—31no—
73#103—31no—
74#103—31no—
75#103—31no—
76#103—31no—
77#103—31no—
78#103—31no—
79C22⋊C8—31no—
80C22⋊C8—31no—
81#97—31no—
82C23×C3—41no—
83C23×C3—41no—
84C3×D4—82no—
85C2×C3×C4—41no—
86C2×Dic3—41no—
87C2×Dic3—41no—
88C2×Dic3—41no—
89C2×Dic3—41no—
90C2×Dic3—82no—
91SL(2,𝔽3)#4111yescharacteristic
92C2×Q8—21no—
93C2×Q8—21no—
94C2×Q8—31no—
95C2×Q8—31no—
96C2×Q8—31no—
97C2×Q8—61no—
98C2×Q8—61no—
99C2×Q8C22×D311yescharacteristic
100C2×D4S411yescharacteristic
101C22×C4—31no—
102C22×C4—31no—
103C22×C4—61no—
104Q16—61no—
105Q16—61no—
106Q16—121no—
107Q16—121no—
108M16—121no—
109C2×C8—31no—
110C2×C8—31no—
111C2×C8—31no—
112C2×C8—31no—
113C2×C8—121no—
114C4⋊C4—31no—
115C4⋊C4—31no—
116C4⋊C4—31no—
117C4⋊C4—31no—
118C4⋊C4—31no—
119C4⋊C4—61no—
120C4⋊C4—61no—
121C4⋊C4—61no—
122C4⋊C4—61no—
123C4⋊C4—61no—
124C4⋊C4—61no—
125C4⋊C4—61no—
126C22⋊C4—31no—
127C22⋊C4—31no—
128C22⋊C4—61no—
129C22⋊C4—61no—
130C22⋊C4—61no—
131C42—31no—
132C42—61no—
133C42—61no—
134C22×C3—42no—
135C22×C3—41no—
136C22×C3—81no—
137C22×C3—81no—
138C22×C3—81no—
139C22×C3—81no—
140C22×C3—81no—
141C3×C4—81no—
142Dic3—81no—
143Dic3—81no—
144Dic3—81no—
145Dic3—81no—
146C23C2×S411yescharacteristic
147C23C2×S411yescharacteristic
148Q8—31no—
149Q8—61no—
150Q8—61no—
151Q8—61no—
152Q8—61no—
153Q8—61no—
154Q8—61no—
155Q8#23711yescharacteristic
156D4—22no—
157C2×C4—31no—
158C2×C4—31no—
159C2×C4—31no—
160C2×C4—31no—
161C2×C4—61no—
162C2×C4—61no—
163C2×C4—61no—
164C2×C4—61no—
165C2×C4—61no—
166C2×C4—61no—
167C2×C4—61no—
168C2×C4—61no—
169C2×C4—122no—
170C2×C4C2×S411yescharacteristic
171C8—61no—
172C8—61no—
173C8—61no—
174C8—61no—
175C2×C3—41no—
176C2×C3—41no—
177C2×C3—41no—
178C2×C3—82no—
179C2×C3—161no—
180C22—21no—
181C22—21no—
182C22—21no—
183C22—21no—
184C22—21no—
185C22#98312yes—
186C22C22×S411yescharacteristic
187C4—21no—
188C4—31no—
189C4—31no—
190C4—61no—
191C4—61no—
192C4—61no—
193C4—121no—
194C4—121no—
195C4—121no—
196C4—121no—
197C3—41noSylow
198C2—22no—
199C2—41no—
200C2#591411yescharacteristic
201C2C2×#98311yescharacteristic
202C2#592711yescharacteristic
subgroupG/H
center, Frattini, socleC22C22×S4
commutator, cosocleC2×SL(2,𝔽3)C23
FittingD4×Q8D3
prime radicalD4×SL(2,𝔽3)C2
hypercenterC2×D4S4
hypocenterSL(2,𝔽3)#41
LCSG ▷ C2×SL(2,𝔽3) ▷ SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
2. #5420C2semidirect
3. #5419C2semidirect
4. #5419C2semidirect
5. C2×#859C2semidirect
6. #5389C2semidirect
7. #5388C2semidirect

Maximal subgroups

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

subgroupindex
D4×SL(2,𝔽3)2
#54202
#54192
C2×#8592
#53892
#53882
#33123
#9354