Finite Group Databasebeta

← #85365

Subgroups — #85365

Order 384.

265 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 265 conjugacy classes of subgroups — too many to draw. Only its 46 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×#985C211yesmaximal, characteristic
2#520⋊C3C211yesmaximal, characteristic, complemented
3#5426C211yesmaximal, characteristic, complemented
4#5394C212yesmaximal, complemented
5C4×#226C212yesmaximal, complemented
6#2951—31nomaximal, Sylow
7C2×#41⋊C3C2211yescharacteristic
8#985C414yescomplemented
9C2×GL(2,𝔽3)C2211yescharacteristic
10C2×#226C2211yescharacteristic
11#892—41nomaximal
12C4×SL(2,𝔽3)C2212yes—
13#859C2212yes—
14C2×#135—31no—
15#520—31no—
16#520D311yescharacteristic, complemented
17#440—32no—
18#440—32no—
19C4×Q16—32no—
20C4×D8—32no—
21C2×#107—31no—
22#417—31no—
23#417—31no—
24#405—31no—
25C2×C4×D3—41no—
26#41⋊C3C2×C412yescomplemented
27C2×SL(2,𝔽3)C2311yescharacteristic
28GL(2,𝔽3)C2×C412yescomplemented
29#226C2×C412yescomplemented
30C3×C4⋊C4—41no—
31(C2×D3)⋊C4—42no—
32C4⋊Dic3—41no—
33C4×Dic3—42no—
34C2×#41—31no—
35C2×#41C2×D311yescharacteristic
36#135—34no—
37#135—34no—
38C2×Q16—31no—
39C2×SD16—31no—
40C2×SD16—31no—
41C2×D8—31no—
42C2×M16—31no—
43C22×C8—31no—
44C4×Q8—32no—
45C4×Q8C2×D312yes—
46C4×D4—32no—
47C4×D4—32no—
48C4×D4—62no—
49#117—31no—
50#117—61no—
51C2×C4⋊C4—31no—
52C2×C4⋊C4—31no—
53C2×C4⋊C4—61no—
54#107—32no—
55#107—32no—
56#103—32no—
57#103—32no—
58#102—32no—
59#102—32no—
60#97—32no—
61C4×C8—32no—
62C22×D3—41no—
63C2×C3×C4—41no—
64C2×C3×C4—42no—
65C2×Dic3—41no—
66C2×Dic3—42no—
67C4×D3—44no—
68SL(2,𝔽3)C22×C411yescharacteristic
69#41—32no—
70#41—34no—
71#41—62no—
72#41C4×D312yes—
73C2×Q8—31no—
74C2×Q8C22×D311yescharacteristic
75C2×D4—31no—
76C2×D4—31no—
77C2×D4—61no—
78C22×C4—31no—
79C22×C4—61no—
80C22×C4—62no—
81C22×C4—61no—
82C22×C4—61no—
83Q16—32no—
84Q16—32no—
85SD16—34no—
86SD16—32no—
87SD16—32no—
88D8—34no—
89M16—62no—
90C2×C8—31no—
91C2×C8—31no—
92C2×C8—34no—
93C2×C8—32no—
94C4⋊C4—32no—
95C4⋊C4—31no—
96C4⋊C4—32no—
97C4⋊C4—62no—
98C4⋊C4—61no—
99C4⋊C4S411yescharacteristic
100C22⋊C4—32no—
101C22⋊C4—62no—
102C42—32no—
103C42—62no—
104C22×C3—41no—
105C2×D3—42no—
106C2×D3—81no—
107C2×D3—81no—
108C3×C4—42no—
109C3×C4—82no—
110Dic3—42no—
111Dic3—82no—
112C23—31no—
113C23—61no—
114Q8—31no—
115Q8—32no—
116Q8—61no—
117Q8C2×C4×D311yescharacteristic
118D4—32no—
119D4—32no—
120D4—61no—
121D4—61no—
122D4—64no—
123C2×C4—31no—
124C2×C4—32no—
125C2×C4—32no—
126C2×C4—32no—
127C2×C4—61no—
128C2×C4—64no—
129C2×C4—62no—
130C2×C4—241no—
131C2×C4—242no—
132C2×C4—241no—
133C2×C4C2×S412yes—
134C2×C4C2×S411yescharacteristic
135C8—32no—
136C8—32no—
137C8—62no—
138C2×C3—41no—
139C2×C3—41no—
140C2×C3—41no—
141D3—82no—
142C22—32no—
143C22—61no—
144C22—61no—
145C22—62no—
146C22—121no—
147C22—121no—
148C22C22×S411yescharacteristic
149C4—22no—
150C4—31no—
151C4—31no—
152C4—62no—
153C4—62no—
154C4—122no—
155C4C4×S412yes—
156C3—41noSylow
157C2—62no—
158C2—122no—
159C2C2×C4×S411yescharacteristic
160C2#592511yescharacteristic
161C2#592711yescharacteristic
subgroupG/H
center, Frattini, socleC22C22×S4
commutator, hypocenterSL(2,𝔽3)C22×C4
Fitting#520D3
prime radicalC2×#985C2
cosocleC2×SL(2,𝔽3)C23
hypercenterC4⋊C4S4
LCSG ▷ SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
2. #520⋊C3C2semidirect
3. #5426C2semidirect
4. #5394C2semidirect
5. C4×#226C2semidirect
8. #985C4semidirect
16. #520D3semidirect
26. #41⋊C3C2×C4semidirect
28. GL(2,𝔽3)C2×C4semidirect
29. #226C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#9852
#520⋊C32
#54262
#53942
C4×#2262
#29513
#8924