Finite Group Databasebeta

← #85404

Subgroups — #85404

Order 384.

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

normal subgroups only

This group has 257 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×#984C212yesmaximal, complemented
2C4⋊(#41⋊C3)C211yesmaximal, characteristic
3#5425C212yesmaximal, complemented
4C4⋊#226C211yesmaximal, characteristic, complemented
5C4×#226C211yesmaximal, characteristic, complemented
6#3278—31nomaximal, Sylow
7C2×#41⋊C3C2212yes—
8#984—24no—
9C2×#226C2212yes—
10C2×#226C2211yescharacteristic
11#888—41nomaximal
12C4×SL(2,𝔽3)C2211yescharacteristic
13#859C2211yescharacteristic
14C2×#137—32no—
15#550D311yescharacteristic
16D4×Q8—31no—
17#492—31no—
18#468—32no—
19#462—31no—
20#460—31no—
21#452—32no—
22#448—32no—
23C4×Q16—31no—
24#436—31no—
25C2×Dic6—42no—
26#41⋊C3—22no—
27C2×SL(2,𝔽3)C2311yescharacteristic
28#226—22no—
29#226D412yescomplemented
30C3×C4⋊C4—41no—
31C4⋊Dic3—41no—
32Dic3⋊C4—42no—
33C4×Dic3—41no—
34C2×#41C2×D312yes—
35C22×Q8—32no—
36#137—64no—
37#137—64no—
38C2×Q16—31no—
39C2×Q16—32no—
40C2×SD16—32no—
41C2×SD16—32no—
42C2×SD16—32no—
43C2×M16—32no—
44C4⋊Q8—31no—
45C4⋊Q8—61no—
46#127—31no—
47C22⋊Q8—32no—
48C22⋊Q8—62no—
49#121—32no—
50C4×Q8—31no—
51C4×Q8C2×D311yescharacteristic
52C4×D4—31no—
53C4×D4—61no—
54#107—31no—
55C4⋊C8—31no—
56#103—31no—
57#103—32no—
58#103—31no—
59#102—32no—
60C22⋊C8—32no—
61C4×C8—31no—
62C2×C3×C4—41no—
63C2×C3×C4—42no—
64C2×Dic3—41no—
65C2×Dic3—41no—
66C2×Dic3—42no—
67Dic6—84no—
68SL(2,𝔽3)C2×D411yescharacteristic
69#41—22no—
70#41—62no—
71C2×Q8—31no—
72C2×Q8—32no—
73C2×Q8—62no—
74C2×Q8—64no—
75C2×Q8—122no—
76C2×Q8C22×D311yescharacteristic
77C2×D4—32no—
78C2×D4—31no—
79C2×D4—61no—
80C22×C4—32no—
81C22×C4—62no—
82Q16—32no—
83Q16—32no—
84Q16—62no—
85Q16—62no—
86SD16—64no—
87SD16—64no—
88SD16—122no—
89M16—64no—
90C2×C8—31no—
91C2×C8—31no—
92C2×C8—32no—
93C4⋊C4—31no—
94C4⋊C4—31no—
95C4⋊C4—61no—
96C4⋊C4—61no—
97C4⋊C4—61no—
98C4⋊C4—62no—
99C4⋊C4S411yescharacteristic
100C22⋊C4—32no—
101C22⋊C4—62no—
102C42—31no—
103C42—61no—
104C22×C3—41no—
105C3×C4—42no—
106C3×C4—82no—
107Dic3—42no—
108Dic3—42no—
109Dic3—82no—
110C23—32no—
111Q8—31no—
112Q8—32no—
113Q8—61no—
114Q8—62no—
115Q8—62no—
116Q8—124no—
117Q8D3×D411yescharacteristic
118D4—62no—
119D4—62no—
120D4—62no—
121D4—241no—
122C2×C4—31no—
123C2×C4—31no—
124C2×C4—62no—
125C2×C4—62no—
126C2×C4—61no—
127C2×C4—61no—
128C2×C4—62no—
129C2×C4—124no—
130C2×C4C2×S412yes—
131C2×C4C2×S411yescharacteristic
132C8—32no—
133C8—62no—
134C8—61no—
135C2×C3—41no—
136C2×C3—41no—
137C2×C3—41no—
138C22—62no—
139C22—62no—
140C22—62no—
141C22C22×S411yescharacteristic
142C4—22no—
143C4—31no—
144C4—31no—
145C4—62no—
146C4—61no—
147C4—122no—
148C4—121no—
149C4#98412yescomplemented
150C3—41noSylow
151C2—122no—
152C2D4×S411yescharacteristic
153C2C2×#98411yescharacteristic
154C2#592411yescharacteristic
subgroupG/H
center, Frattini, socleC22C22×S4
commutator, cosocleC2×SL(2,𝔽3)C23
Fitting#550D3
prime radicalC4⋊(#41⋊C3)C2
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×#984C2semidirect
3. #5425C2semidirect
4. C4⋊#226C2semidirect
5. C4×#226C2semidirect
29. #226D4semidirect
149. C4#984semidirect

Maximal subgroups

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

subgroupindex
C2×#9842
C4⋊(#41⋊C3)2
#54252
C4⋊#2262
C4×#2262
#32783
#8884