Finite Group Databasebeta

← #85359

Subgroups — #85359

Order 384.

259 conjugacy classes of subgroups; only the 38 normal subgroups are shown.

normal subgroups only

This group has 259 conjugacy classes of subgroups — too many to draw. Only its 38 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×#983C211yesmaximal, characteristic
2C22×#226C211yesmaximal, characteristic, complemented
3#519⋊C3C211yesmaximal, characteristic, complemented
4#5395C212yesmaximal, complemented
5C4⋊#226C212yesmaximal, complemented
6#3044—31nomaximal, Sylow
7C22×SL(2,𝔽3)C2211yescharacteristic
8#983—22no—
9C2×GL(2,𝔽3)C2211yescharacteristic
10C2×#226—22no—
11C2×#226C2211yescharacteristic
12C2×#226C2212yes—
13#886—41nomaximal
14C4×SL(2,𝔽3)C2212yes—
15C2×#137—31no—
16C22×Q16—31no—
17#536—31no—
18#519D311yescharacteristic, complemented
19#464—32no—
20#464—32no—
21#462—32no—
22#462—32no—
23#424—31no—
24#419—31no—
25#416—31no—
26C2×#67—41no—
27C22×Dic3—41no—
28C2×SL(2,𝔽3)D412yes—
29C2×SL(2,𝔽3)C2311yescharacteristic
30GL(2,𝔽3)—21no—
31#226—22no—
32#226—21no—
33C3×C22⋊C4—41no—
34(C2×D3)⋊C4—42no—
35C4⋊Dic3—42no—
36C2×#41—31no—
37C22×Q8—31no—
38C22×Q8C2×D311yescharacteristic
39#137—62no—
40#137—62no—
41C2×Q16—32no—
42C2×Q16—31no—
43C2×Q16—32no—
44C2×Q16—31no—
45C2×Q16—62no—
46C2×Q16—62no—
47C2×SD16—31no—
48C2×SD16—31no—
49C2×M16—31no—
50C22×C8—31no—
51C4⋊Q8—32no—
52#124—32no—
53#123—61no—
54#123—61no—
55C22⋊Q8—62no—
56C4×Q8—32no—
57C4×Q8C2×D312yes—
58#117—31no—
59#117—31no—
60C4⋊C8—32no—
61C4⋊C8—32no—
62#103—32no—
63#103—32no—
64#103—32no—
65#102—32no—
66C23×C3—41no—
67C22×D3—41no—
68C2×C3×C4—42no—
69#67—82no—
70C2×Dic3—41no—
71C2×Dic3—42no—
72C2×Dic3—82no—
73SL(2,𝔽3)C2×D411yescharacteristic
74#41—62no—
75#41—62no—
76C2×Q8—31no—
77C2×Q8—32no—
78C2×Q8—32no—
79C2×Q8—31no—
80C2×Q8—61no—
81C2×Q8—62no—
82C2×Q8—62no—
83C2×Q8D1212yes—
84C2×Q8C22×D311yescharacteristic
85C2×D4—31no—
86C2×D4—61no—
87C22×C4—31no—
88C22×C4—61no—
89C22×C4—61no—
90Q16—64no—
91Q16—62no—
92Q16—62no—
93Q16—61no—
94Q16—62no—
95Q16—61no—
96SD16—62no—
97SD16—61no—
98SD16—61no—
99M16—62no—
100C2×C8—32no—
101C2×C8—31no—
102C2×C8—31no—
103C2×C8—62no—
104C4⋊C4—32no—
105C4⋊C4—32no—
106C4⋊C4—62no—
107C4⋊C4—62no—
108C22⋊C4—31no—
109C22⋊C4—62no—
110C22⋊C4—62no—
111C22⋊C4S411yescharacteristic
112C42—32no—
113C42—61no—
114C22×C3—42no—
115C22×C3—41no—
116C22×C3—82no—
117C2×D3—81no—
118C2×D3—82no—
119C3×C4—82no—
120Dic3—82no—
121Dic3—81no—
122C23—61no—
123C23C2×S411yescharacteristic
124Q8—32no—
125Q8—31no—
126Q8—61no—
127Q8—62no—
128Q8—61no—
129Q8—62no—
130Q8—61no—
131Q8—122no—
132Q8C2×D1211yescharacteristic
133D4—61no—
134D4—61no—
135D4—122no—
136C2×C4—32no—
137C2×C4—32no—
138C2×C4—31no—
139C2×C4—31no—
140C2×C4—32no—
141C2×C4—62no—
142C2×C4—61no—
143C2×C4—122no—
144C2×C4—122no—
145C2×C4C2×S412yes—
146C8—62no—
147C8—61no—
148C8—61no—
149C2×C3—41no—
150C2×C3—42no—
151C2×C3—82no—
152D3—161no—
153C22—22no—
154C22—121no—
155C22—122no—
156C22C4⋊S412yes—
157C22C22×S411yescharacteristic
158C4—22no—
159C4—31no—
160C4—32no—
161C4—31no—
162C4—62no—
163C4—121no—
164C4—122no—
165C3—41noSylow
166C2—22no—
167C2—241no—
168C2C2×C4⋊S411yescharacteristic
169C2#592712yes—
subgroupG/H
center, Frattini, socleC22C22×S4
commutator, cosocleC2×SL(2,𝔽3)C23
Fitting#519D3
prime radicalC2×#983C2
hypercenterC22⋊C4S4
hypocenterSL(2,𝔽3)C2×D4
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
2. C22×#226C2semidirect
3. #519⋊C3C2semidirect
4. #5395C2semidirect
5. C4⋊#226C2semidirect
18. #519D3semidirect

Maximal subgroups

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

subgroupindex
C2×#9832
C22×#2262
#519⋊C32
#53952
C4⋊#2262
#30443
#8864