Finite Group Databasebeta

← #85522

Subgroups — #85522

Order 384.

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

normal subgroups only

This group has 311 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
1C22×#41⋊C3C211yesmaximal, characteristic, complemented
2C2×#983C211yesmaximal, characteristic, complemented
3#5427C212yesmaximal, complemented
4#5425C212yesmaximal, complemented
5#5420C211yesmaximal, characteristic, complemented
6#3005—31nomaximal, Sylow
7C2×#41⋊C3—22no—
8C2×#41⋊C3C2212yes—
9C22×SL(2,𝔽3)C2211yescharacteristic
10#983—22no—
11C2×GL(2,𝔽3)C2211yescharacteristic, complemented
12C2×#226C2211yescharacteristic, complemented
13#929—41nomaximal
14#859C2212yescomplemented
15C22×#41D311yescharacteristic, complemented
16C2×#137—31no—
17C2×AGL(1,ℤ/8)—31no—
18#525—31no—
19#452—32no—
20#452—32no—
21#449—32no—
22#449—32no—
23#419—31no—
24#418—31no—
25#407—31no—
26C22×C3×C4—41no—
27C2×#67—41no—
28#41⋊C3—22no—
29C2×SL(2,𝔽3)D412yescomplemented
30C2×SL(2,𝔽3)C2311yescharacteristic
31GL(2,𝔽3)—21no—
32#226—21no—
33#217—41no—
34(C2×D3)⋊C4—42no—
35Dic3⋊C4—42no—
36C2×#41—22no—
37C2×#41—32no—
38C2×#41—31no—
39C2×#41—62no—
40C2×#41C2×D312yes—
41C22×Q8C2×D311yescharacteristic
42C22×D4—31no—
43C23×C4—31no—
44#137—62no—
45#137—62no—
46AGL(1,ℤ/8)—64no—
47C2×Q16—31no—
48C2×Q16—31no—
49C2×SD16—32no—
50C2×SD16—31no—
51C2×SD16—31no—
52C2×D8—32no—
53C2×M16—31no—
54C2×M16—31no—
55#123—61no—
56C22⋊Q8—32no—
57#121—32no—
58#120—61no—
59C4×D4—62no—
60#117—31no—
61#103—32no—
62#103—32no—
63#102—32no—
64#102—32no—
65C22⋊C8—32no—
66C22⋊C8—32no—
67C23×C3—41no—
68C22×D3—41no—
69C2×C3×C4—42no—
70C2×C3×C4—82no—
71#67—82no—
72C2×Dic3—42no—
73C2×Dic3—41no—
74SL(2,𝔽3)C2×D411yescharacteristic
75C24—31no—
76#41—22no—
77#41—62no—
78#41—62no—
79#41—64no—
80#41—62no—
81#41—64no—
82C2×Q8—31no—
83C2×Q8—32no—
84C2×Q8—31no—
85C2×Q8#6712yes—
86C2×Q8C22×D311yescharacteristic
87C2×D4—32no—
88C2×D4—34no—
89C2×D4—31no—
90C2×D4—61no—
91C2×D4—62no—
92C2×D4—62no—
93C2×D4—61no—
94C22×C4—31no—
95C22×C4—32no—
96C22×C4—32no—
97C22×C4—62no—
98C22×C4—62no—
99C22×C4—61no—
100C22×C4S411yescharacteristic
101Q16—62no—
102Q16—61no—
103Q16—61no—
104SD16—64no—
105SD16—62no—
106SD16—61no—
107SD16—61no—
108D8—64no—
109M16—62no—
110M16—62no—
111C2×C8—31no—
112C2×C8—31no—
113C2×C8—32no—
114C4⋊C4—32no—
115C4⋊C4—62no—
116C22⋊C4—62no—
117C22⋊C4—61no—
118C22⋊C4—62no—
119C42—61no—
120C22×C3—42no—
121C22×C3—41no—
122C22×C3—82no—
123C2×D3—81no—
124C2×D3—82no—
125C3×C4—82no—
126Dic3—82no—
127Dic3—81no—
128C23—32no—
129C23—62no—
130C23—61no—
131C23—122no—
132C23C2×S411yescharacteristic
133Q8—32no—
134Q8—31no—
135Q8—61no—
136Q8—61no—
137Q8—61no—
138Q8C2×#6711yescharacteristic
139D4—34no—
140D4—62no—
141D4—64no—
142D4—61no—
143D4—61no—
144D4—124no—
145D4—122no—
146C2×C4—22no—
147C2×C4—32no—
148C2×C4—32no—
149C2×C4—31no—
150C2×C4—31no—
151C2×C4—62no—
152C2×C4—62no—
153C2×C4—64no—
154C2×C4—62no—
155C2×C4—61no—
156C2×C4—122no—
157C2×C4C2×S412yes—
158C8—62no—
159C8—61no—
160C8—61no—
161C2×C3—42no—
162C2×C3—41no—
163C2×C3—82no—
164D3—161no—
165C22—22no—
166C22—62no—
167C22—62no—
168C22—62no—
169C22—124no—
170C22—122no—
171C22—121no—
172C22GL(2,ℤ/4)12yes—
173C22C22×S411yescharacteristic
174C4—22no—
175C4—31no—
176C4—31no—
177C4—32no—
178C4—122no—
179C4—121no—
180C3—41noSylow
181C2—22no—
182C2—122no—
183C2—241no—
184C2#592312yes—
185C2C2×GL(2,ℤ/4)11yescharacteristic
subgroupG/H
center, Frattini, socleC22C22×S4
commutator, cosocleC2×SL(2,𝔽3)C23
FittingC22×#41D3
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
1. C22×#41⋊C3C2semidirect
2. C2×#983C2semidirect
3. #5427C2semidirect
4. #5425C2semidirect
5. #5420C2semidirect
11. C2×GL(2,𝔽3)C22semidirect
12. C2×#226C22semidirect
14. #859C22semidirect
15. C22×#41D3semidirect
29. C2×SL(2,𝔽3)D4semidirect

Maximal subgroups

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

subgroupindex
C22×#41⋊C32
C2×#9832
#54272
#54252
#54202
#30053
#9294