Finite Group Databasebeta

← #85402

Subgroups — #85402

Order 384.

281 conjugacy classes of subgroups; only the 36 normal subgroups are shown.

normal subgroups only

This group has 281 conjugacy classes of subgroups — too many to draw. Only its 36 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×#985C212yesmaximal, complemented
2C4⋊(#41⋊C3)C211yesmaximal, characteristic, complemented
3#5427C212yesmaximal, complemented
4C4⋊GL(2,𝔽3)C211yesmaximal, characteristic, complemented
5C4×#226C211yesmaximal, characteristic, complemented
6#3287—31nomaximal, Sylow
7C2×#41⋊C3C2212yes—
8#985—24no—
9C2×GL(2,𝔽3)C2212yes—
10C2×#226C2211yescharacteristic
11#893—41nomaximal
12C4×SL(2,𝔽3)C2211yescharacteristic
13#859C2211yescharacteristic
14C2×#135—32no—
15#550—31no—
16#550D311yescharacteristic, complemented
17#493—31no—
18#466—32no—
19#460—31no—
20#460—31no—
21#449—32no—
22#449—32no—
23C4×Q16—31no—
24C8×D4—31no—
25C2×D12—41no—
26C2×C4×D3—42no—
27#41⋊C3—22no—
28C2×SL(2,𝔽3)C2311yescharacteristic
29GL(2,𝔽3)—22no—
30#226D412yescomplemented
31C3×C4⋊C4—41no—
32(C2×D3)⋊C4—42no—
33C4×Dic3—41no—
34C2×#41—32no—
35C2×#41C2×D312yes—
36#135—64no—
37#135—64no—
38C2×Q16—31no—
39C2×SD16—32no—
40C2×SD16—32no—
41C2×D8—32no—
42C2×D8—32no—
43C22×C8—32no—
44#127—31no—
45#127—31no—
46#127—61no—
47#121—32no—
48#121—32no—
49#121—62no—
50C4×Q8—31no—
51C4×Q8C2×D311yescharacteristic
52C4×D4—31no—
53C4×D4—61no—
54#107—31no—
55C4⋊C8—31no—
56#103—31no—
57#103—31no—
58#102—32no—
59#102—32no—
60C22⋊C8—32no—
61C4×C8—31no—
62C22×D3—42no—
63C2×C3×C4—41no—
64C2×C3×C4—42no—
65C2×Dic3—41no—
66C2×Dic3—41no—
67D12—44no—
68C4×D3—84no—
69SL(2,𝔽3)C2×D411yescharacteristic
70#41—22no—
71#41—62no—
72#41—64no—
73#41—122no—
74C2×Q8—31no—
75C2×Q8C22×D311yescharacteristic
76C2×D4—32no—
77C2×D4—32no—
78C2×D4—31no—
79C2×D4—61no—
80C2×D4—61no—
81C2×D4—62no—
82C2×D4—62no—
83C2×D4—61no—
84C22×C4—32no—
85C22×C4—62no—
86Q16—32no—
87Q16—32no—
88SD16—64no—
89SD16—62no—
90SD16—62no—
91D8—64no—
92D8—122no—
93C2×C8—31no—
94C2×C8—31no—
95C2×C8—32no—
96C2×C8—64no—
97C4⋊C4—31no—
98C4⋊C4—31no—
99C4⋊C4—61no—
100C4⋊C4S411yescharacteristic
101C22⋊C4—32no—
102C22⋊C4—62no—
103C42—31no—
104C42—61no—
105C22×C3—41no—
106C2×D3—44no—
107C2×D3—82no—
108C2×D3—82no—
109C3×C4—42no—
110C3×C4—82no—
111Dic3—42no—
112Dic3—81no—
113C23—32no—
114C23—62no—
115Q8—31no—
116Q8—32no—
117Q8—61no—
118Q8D3×D411yescharacteristic
119D4—62no—
120D4—62no—
121D4—62no—
122D4—62no—
123D4—62no—
124D4—124no—
125D4—122no—
126D4—241no—
127D4—241no—
128D4—242no—
129C2×C4—31no—
130C2×C4—31no—
131C2×C4—62no—
132C2×C4—62no—
133C2×C4—61no—
134C2×C4—61no—
135C2×C4—124no—
136C2×C4C2×S412yes—
137C2×C4C2×S411yescharacteristic
138C8—32no—
139C8—62no—
140C8—61no—
141C2×C3—41no—
142C2×C3—41no—
143C2×C3—41no—
144D3—84no—
145C22—62no—
146C22—62no—
147C22—62no—
148C22—122no—
149C22—122no—
150C22—122no—
151C22C22×S411yescharacteristic
152C4—22no—
153C4—31no—
154C4—31no—
155C4—62no—
156C4—61no—
157C4—121no—
158C4#98512yescomplemented
159C3—41noSylow
160C2—122no—
161C2—242no—
162C2D4×S411yescharacteristic
163C2C2×#98511yescharacteristic
164C2#592511yescharacteristic
subgroupG/H
center, Frattini, socleC22C22×S4
commutator, cosocleC2×SL(2,𝔽3)C23
Fitting#550D3
hypercenterC4⋊C4S4
hypocenterSL(2,𝔽3)C2×D4
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. C2×#985C2semidirect
2. C4⋊(#41⋊C3)C2semidirect
3. #5427C2semidirect
4. C4⋊GL(2,𝔽3)C2semidirect
5. C4×#226C2semidirect
16. #550D3semidirect
30. #226D4semidirect
158. C4#985semidirect

Maximal subgroups

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

subgroupindex
C2×#9852
C4⋊(#41⋊C3)2
#54272
C4⋊GL(2,𝔽3)2
C4×#2262
#32873
#8934