Finite Group Databasebeta

← #85330

Subgroups — #85330

Order 384.

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

normal subgroups only

This group has 214 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
1C2×C4×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
2#5421C211yesmaximal, characteristic
3#5419C211yesmaximal, characteristic, complemented
4#5395C211yesmaximal, characteristic, complemented
5#5394C211yesmaximal, characteristic, complemented
6#5389C211yesmaximal, characteristic, complemented
7#5388C211yesmaximal, characteristic, complemented
8#3117—31nomaximal, Sylow
9C22×SL(2,𝔽3)C2211yescharacteristic
10C2×GL(2,𝔽3)C2211yescharacteristic
11C2×#226C2211yescharacteristic
12#929—41nomaximal
13C4×SL(2,𝔽3)C2211yescharacteristic
14C4×SL(2,𝔽3)C2211yescharacteristic
15#859C2211yescharacteristic
16#859C2211yescharacteristic
17#529—31no—
18C2×C4×Q8D311yescharacteristic, complemented
19#484—31no—
20#481—31no—
21#479—31no—
22#479—31no—
23#464—31no—
24#464—31no—
25#451—31no—
26#448—31no—
27#441—31no—
28#441—31no—
29#440—31no—
30#440—31no—
31#432—31no—
32C22×C3×C4—41no—
33C2×#67—41no—
34C2×SL(2,𝔽3)—21no—
35C2×SL(2,𝔽3)C2311yescharacteristic
36GL(2,𝔽3)—21no—
37#226—21no—
38#217—41no—
39(C2×D3)⋊C4—41no—
40(C2×D3)⋊C4—41no—
41Dic3⋊C4—41no—
42Dic3⋊C4—41no—
43C22×Q8C2×D311yescharacteristic
44C2×Q16—31no—
45C2×Q16—31no—
46C2×SD16—31no—
47C2×SD16—31no—
48#126—61no—
49#125—31no—
50#124—31no—
51#123—61no—
52C22⋊Q8—31no—
53#121—31no—
54C4×Q8—31no—
55C4×Q8—31no—
56C4×Q8—31no—
57C4×Q8C2×D311yescharacteristic
58C4×Q8C2×D311yescharacteristic
59C4×D4—31no—
60C4×D4—61no—
61#117—61no—
62C2×C4⋊C4—31no—
63C2×C42—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—
81#97—31no—
82C23×C3—41no—
83C22×D3—41no—
84C2×C3×C4—41no—
85C2×C3×C4—41no—
86C2×C3×C4—82no—
87#67—82no—
88C2×Dic3—41no—
89C2×Dic3—41no—
90C2×Dic3—41no—
91SL(2,𝔽3)#4111yescharacteristic
92C2×Q8—21no—
93C2×Q8—31no—
94C2×Q8—31no—
95C2×Q8—61no—
96C2×Q8C22×D311yescharacteristic
97C2×D4—31no—
98C2×D4—61no—
99C22×C4—31no—
100C22×C4—31no—
101C22×C4—61no—
102C22×C4S411yescharacteristic
103Q16—61no—
104Q16—62no—
105Q16—61no—
106SD16—61no—
107SD16—62no—
108SD16—61no—
109C2×C8—31no—
110C2×C8—31no—
111C2×C8—31no—
112C2×C8—31no—
113C4⋊C4—31no—
114C4⋊C4—31no—
115C4⋊C4—31no—
116C4⋊C4—31no—
117C4⋊C4—61no—
118C4⋊C4—61no—
119C4⋊C4—61no—
120C4⋊C4—61no—
121C4⋊C4—61no—
122C22⋊C4—61no—
123C22⋊C4—61no—
124C22⋊C4—61no—
125C22⋊C4—61no—
126C22⋊C4—61no—
127C42—31no—
128C42—31no—
129C42—31no—
130C42—31no—
131C42—61no—
132C22×C3—42no—
133C22×C3—41no—
134C22×C3—81no—
135C22×C3—81no—
136C2×D3—81no—
137C2×D3—81no—
138C2×D3—81no—
139C3×C4—81no—
140C3×C4—81no—
141Dic3—81no—
142Dic3—81no—
143Dic3—81no—
144C23—61no—
145C23C2×S411yescharacteristic
146Q8—31no—
147Q8—32no—
148Q8—61no—
149Q8—61no—
150Q8—61no—
151Q8#23511yescharacteristic
152D4—61no—
153D4—61no—
154D4—122no—
155C2×C4—22no—
156C2×C4—31no—
157C2×C4—31no—
158C2×C4—31no—
159C2×C4—31no—
160C2×C4—61no—
161C2×C4—61no—
162C2×C4—61no—
163C2×C4—61no—
164C2×C4—61no—
165C2×C4—121no—
166C2×C4—122no—
167C2×C4C2×S411yescharacteristic
168C2×C4C2×S411yescharacteristic
169C8—61no—
170C8—61no—
171C8—61no—
172C8—61no—
173C2×C3—41no—
174C2×C3—41no—
175C2×C3—41no—
176C2×C3—82no—
177D3—161no—
178C22—21no—
179C22—21no—
180C22—121no—
181C22—121no—
182C22—121no—
183C22#98312yes—
184C22C22×S411yescharacteristic
185C4—21no—
186C4—21no—
187C4—31no—
188C4—31no—
189C4—61no—
190C4—61no—
191C4—61no—
192C4—121no—
193C4—121no—
194C4—121no—
195C3—41noSylow
196C2—22no—
197C2—241no—
198C2#591211yescharacteristic
199C2C2×#98311yescharacteristic
200C2#592311yescharacteristic
subgroupG/H
center, Frattini, socleC22C22×S4
commutator, cosocleC2×SL(2,𝔽3)C23
FittingC2×C4×Q8D3
prime radical#5421C2
hypercenterC22×C4S4
hypocenterSL(2,𝔽3)#41
LCSG ▷ C2×SL(2,𝔽3) ▷ SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C2×C4×SL(2,𝔽3)C2semidirect
3. #5419C2semidirect
4. #5395C2semidirect
5. #5394C2semidirect
6. #5389C2semidirect
7. #5388C2semidirect
18. C2×C4×Q8D3semidirect

Maximal subgroups

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

subgroupindex
C2×C4×SL(2,𝔽3)2
#54212
#54192
#53952
#53942
#53892
#53882
#31173
#9294