Finite Group Databasebeta

← #85367

Subgroups — #85367

Order 384.

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

normal subgroups only

This group has 215 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, complemented
2#5427C211yesmaximal, characteristic, complemented
3#5427C211yesmaximal, 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#3119—31nomaximal, Sylow
9C2×#41⋊C3C2211yescharacteristic
10C2×GL(2,𝔽3)C2211yescharacteristic
11C2×GL(2,𝔽3)C2211yescharacteristic
12#894—41nomaximal
13C4×SL(2,𝔽3)C2211yescharacteristic
14C4×SL(2,𝔽3)C2211yescharacteristic
15#859C2211yescharacteristic
16#859C2211yescharacteristic
17#540—31no—
18#520D311yescharacteristic, complemented
19#481—31no—
20#480—31no—
21#479—31no—
22#478—31no—
23#460—31no—
24#459—31no—
25#449—31no—
26#449—31no—
27#442—31no—
28#440—31no—
29C4×SD16—31no—
30C4×D8—31no—
31#433—31no—
32C2×D12—41no—
33C2×C4×D3—41no—
34#41⋊C3—21no—
35C2×SL(2,𝔽3)C2311yescharacteristic
36GL(2,𝔽3)—21no—
37GL(2,𝔽3)—21no—
38C3×C4⋊C4—41no—
39(C2×D3)⋊C4—41no—
40(C2×D3)⋊C4—41no—
41(C2×D3)⋊C4—41no—
42Dic3⋊C4—41no—
43C2×#41C2×D311yescharacteristic
44C2×SD16—31no—
45C2×SD16—31no—
46C2×D8—31no—
47C2×D8—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×Q8C2×D311yescharacteristic
57C4×Q8C2×D311yescharacteristic
58C4×D4—31no—
59C4×D4—31no—
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#102—31no—
73#102—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—
82C22×D3—41no—
83C22×D3—41no—
84C2×C3×C4—41no—
85C2×C3×C4—41no—
86C2×C3×C4—41no—
87C2×Dic3—41no—
88C2×Dic3—41no—
89D12—82no—
90C4×D3—82no—
91SL(2,𝔽3)#4111yescharacteristic
92#41—21no—
93#41—61no—
94C2×Q8C22×D311yescharacteristic
95C2×D4—31no—
96C2×D4—31no—
97C2×D4—31no—
98C2×D4—61no—
99C2×D4—61no—
100C2×D4—61no—
101C2×D4—61no—
102C22×C4—31no—
103C22×C4—61no—
104C22×C4—61no—
105C22×C4—61no—
106SD16—61no—
107SD16—61no—
108SD16—61no—
109SD16—61no—
110D8—62no—
111D8—62no—
112C2×C8—31no—
113C2×C8—31no—
114C2×C8—31no—
115C2×C8—31no—
116C4⋊C4—31no—
117C4⋊C4—31no—
118C4⋊C4—31no—
119C4⋊C4—31no—
120C4⋊C4—31no—
121C4⋊C4—61no—
122C4⋊C4—61no—
123C4⋊C4S411yescharacteristic
124C22⋊C4—31no—
125C22⋊C4—31no—
126C22⋊C4—61no—
127C22⋊C4—61no—
128C22⋊C4—61no—
129C22⋊C4—61no—
130C42—31no—
131C42—31no—
132C22×C3—41no—
133C2×D3—42no—
134C2×D3—81no—
135C2×D3—81no—
136C2×D3—81no—
137C2×D3—81no—
138C2×D3—81no—
139C3×C4—81no—
140C3×C4—81no—
141C3×C4—81no—
142Dic3—81no—
143Dic3—81no—
144C23—31no—
145C23—61no—
146C23—61no—
147Q8—31no—
148Q8#23511yescharacteristic
149D4—32no—
150D4—61no—
151D4—61no—
152D4—61no—
153D4—61no—
154D4—61no—
155D4—241no—
156D4—241no—
157D4—241no—
158D4—241no—
159C2×C4—31no—
160C2×C4—31no—
161C2×C4—31no—
162C2×C4—61no—
163C2×C4—61no—
164C2×C4—61no—
165C2×C4—61no—
166C2×C4—241no—
167C2×C4—241no—
168C2×C4—241no—
169C2×C4C2×S411yescharacteristic
170C2×C4C2×S411yescharacteristic
171C2×C4C2×S411yescharacteristic
172C8—61no—
173C8—61no—
174C8—61no—
175C8—61no—
176C2×C3—41no—
177C2×C3—41no—
178C2×C3—41no—
179D3—82no—
180D3—161no—
181C22—61no—
182C22—61no—
183C22—61no—
184C22—121no—
185C22—121no—
186C22—121no—
187C22—121no—
188C22—121no—
189C22—121no—
190C22C22×S411yescharacteristic
191C4—21no—
192C4—21no—
193C4—21no—
194C4—31no—
195C4—31no—
196C4—61no—
197C4—61no—
198C4—121no—
199C4—121no—
200C3—41noSylow
201C2—121no—
202C2—241no—
203C2—241no—
204C2#591211yescharacteristic
205C2#592511yescharacteristic
206C2#592611yescharacteristic
subgroupG/H
center, Frattini, socleC22C22×S4
commutator, cosocleC2×SL(2,𝔽3)C23
Fitting#520D3
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
1. #520⋊C3C2semidirect
2. #5427C2semidirect
3. #5427C2semidirect
4. #5394C2semidirect
5. C4⋊GL(2,𝔽3)C2semidirect
6. C4×GL(2,𝔽3)C2semidirect
7. #5389C2semidirect
18. #520D3semidirect

Maximal subgroups

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

subgroupindex
#520⋊C32
#54272
#53942
C4⋊GL(2,𝔽3)2
C4×GL(2,𝔽3)2
#53892
#31193
#8944