Finite Group Databasebeta

← #85358

Subgroups — #85358

Order 384.

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

normal subgroups only

This group has 222 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#519⋊C3C211yesmaximal, characteristic, complemented
2#5421C211yesmaximal, characteristic, complemented
3#5421C211yesmaximal, characteristic, complemented
4#5394C211yesmaximal, characteristic, complemented
5C4⋊GL(2,𝔽3)C211yesmaximal, characteristic, complemented
6C4×GL(2,𝔽3)C211yesmaximal, characteristic, complemented
7#5389C211yesmaximal, characteristic, complemented
8#3126—31nomaximal, Sylow
9C22×SL(2,𝔽3)C2211yescharacteristic
10C2×GL(2,𝔽3)C2211yescharacteristic
11C2×GL(2,𝔽3)C2211yescharacteristic
12#884—41nomaximal
13C4×SL(2,𝔽3)C2211yescharacteristic
14C4×SL(2,𝔽3)C2211yescharacteristic
15#859C2211yescharacteristic
16#859C2211yescharacteristic
17#540—31no—
18#519D311yescharacteristic, complemented
19#482—31no—
20#482—31no—
21#479—31no—
22#479—31no—
23#460—31no—
24#460—31no—
25#448—31no—
26#448—31no—
27#440—31no—
28#440—31no—
29C4×SD16—31no—
30C4×SD16—31no—
31#433—31no—
32C2×#67—41no—
33C2×#67—41no—
34C2×D12—41no—
35C2×C4×D3—41no—
36C2×SL(2,𝔽3)—21no—
37C2×SL(2,𝔽3)C2311yescharacteristic
38GL(2,𝔽3)—21no—
39GL(2,𝔽3)—21no—
40C3×C22⋊C4—41no—
41(C2×D3)⋊C4—41no—
42Dic3⋊C4—41no—
43C22×Q8C2×D311yescharacteristic
44C2×SD16—31no—
45C2×SD16—31no—
46C2×SD16—31no—
47C2×SD16—31no—
48#127—31no—
49#125—31no—
50#123—61no—
51#123—61no—
52#121—31no—
53#121—31no—
54#121—61no—
55#121—61no—
56C4×Q8—31no—
57C4×Q8—31no—
58C4×Q8C2×D311yescharacteristic
59C4×Q8C2×D311yescharacteristic
60C4×D4—31no—
61C4×D4—31no—
62#117—31no—
63#117—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#102—31no—
75#102—31no—
76#102—31no—
77#102—31no—
78C22⋊C8—31no—
79C22⋊C8—31no—
80#97—31no—
81C4×C8—31no—
82C23×C3—41no—
83C22×D3—41no—
84C22×D3—41no—
85C2×C3×C4—41no—
86C2×C3×C4—41no—
87#67—82no—
88#67—82no—
89C2×Dic3—41no—
90C2×Dic3—41no—
91D12—82no—
92C4×D3—82no—
93SL(2,𝔽3)#4111yescharacteristic
94C2×Q8—21no—
95C2×Q8—31no—
96C2×Q8—61no—
97C2×Q8C22×D311yescharacteristic
98C2×D4—31no—
99C2×D4—31no—
100C2×D4—61no—
101C2×D4—61no—
102C2×D4—61no—
103C2×D4—61no—
104C22×C4—31no—
105C22×C4—61no—
106C22×C4—61no—
107SD16—61no—
108SD16—61no—
109SD16—62no—
110SD16—62no—
111SD16—61no—
112SD16—61no—
113C2×C8—31no—
114C2×C8—31no—
115C2×C8—31no—
116C2×C8—31no—
117C4⋊C4—31no—
118C4⋊C4—31no—
119C4⋊C4—31no—
120C4⋊C4—31no—
121C4⋊C4—31no—
122C4⋊C4—31no—
123C4⋊C4—61no—
124C4⋊C4—61no—
125C22⋊C4—31no—
126C22⋊C4—61no—
127C22⋊C4—61no—
128C22⋊C4—61no—
129C22⋊C4—61no—
130C22⋊C4S411yescharacteristic
131C42—31no—
132C42—31no—
133C42—61no—
134C22×C3—41no—
135C22×C3—81no—
136C22×C3—81no—
137C22×C3—81no—
138C2×D3—42no—
139C2×D3—81no—
140C2×D3—81no—
141C2×D3—81no—
142C2×D3—81no—
143C2×D3—81no—
144C3×C4—81no—
145C3×C4—81no—
146Dic3—42no—
147Dic3—81no—
148C23—61no—
149C23—61no—
150C23C2×S411yescharacteristic
151Q8—31no—
152Q8—32no—
153Q8—61no—
154Q8#23511yescharacteristic
155D4—61no—
156D4—61no—
157D4—61no—
158D4—61no—
159D4—241no—
160D4—241no—
161D4—241no—
162D4—241no—
163C2×C4—31no—
164C2×C4—31no—
165C2×C4—31no—
166C2×C4—31no—
167C2×C4—61no—
168C2×C4—61no—
169C2×C4—61no—
170C2×C4—61no—
171C2×C4—241no—
172C2×C4—241no—
173C2×C4C2×S411yescharacteristic
174C2×C4C2×S411yescharacteristic
175C8—61no—
176C8—61no—
177C8—61no—
178C8—61no—
179C2×C3—41no—
180C2×C3—41no—
181C2×C3—41no—
182C2×C3—161no—
183D3—82no—
184D3—161no—
185C22—21no—
186C22—21no—
187C22—21no—
188C22—121no—
189C22—121no—
190C22—121no—
191C22—121no—
192C22—121no—
193C22—121no—
194C22C22×S411yescharacteristic
195C4—21no—
196C4—21no—
197C4—31no—
198C4—31no—
199C4—61no—
200C4—61no—
201C4—61no—
202C4—121no—
203C4—121no—
204C3—41noSylow
205C2—41no—
206C2—241no—
207C2—241no—
208C2#591211yescharacteristic
209C2#592611yescharacteristic
210C2#592611yescharacteristic
subgroupG/H
center, Frattini, socleC22C22×S4
commutator, cosocleC2×SL(2,𝔽3)C23
Fitting#519D3
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. #519⋊C3C2semidirect
2. #5421C2semidirect
3. #5421C2semidirect
4. #5394C2semidirect
5. C4⋊GL(2,𝔽3)C2semidirect
6. C4×GL(2,𝔽3)C2semidirect
7. #5389C2semidirect
18. #519D3semidirect

Maximal subgroups

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

subgroupindex
#519⋊C32
#54212
#53942
C4⋊GL(2,𝔽3)2
C4×GL(2,𝔽3)2
#53892
#31263
#8844