Finite Group Databasebeta

← #85526

Subgroups — #85526

Order 384.

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

normal subgroups only

This group has 300 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×#994C211yesmaximal, characteristic, complemented
2#5923C211yesmaximal, characteristic, complemented
3#5429C212yesmaximal, complemented
4#5428C212yesmaximal, complemented
5#5423C211yesmaximal, characteristic, complemented
6#3019—31nomaximal, Sylow
7#994—21no—
8#994C2212yes—
9C2×#41⋊C3—21no—
10C2×#41⋊C3C2211yescharacteristic
11#986C2211yescharacteristic, complemented
12#985—21no—
13#984C2211yescharacteristic, complemented
14#983—21no—
15#942—41nomaximal
16GU(2,𝔽3)C2212yescomplemented
17C2×H(5,𝔽2)D311yescharacteristic, complemented
18#575—31no—
19#575—31no—
20#535—31no—
21#455—32no—
22#455—32no—
23#453—32no—
24#453—32no—
25#421—31no—
26#421—31no—
27C2×#100—31no—
28C2×C3×Q8—41no—
29#235—41no—
30#41⋊C3—22no—
31#41⋊C3D411yescharacteristic, complemented
32#41⋊C3C2311yescharacteristic
33C2×SL(2,𝔽3)D411yescharacteristic, complemented
34GL(2,𝔽3)—21no—
35#226—21no—
36#216—42no—
37#215—42no—
38#208—41no—
39H(5,𝔽2)—21no—
40H(5,𝔽2)—32no—
41H(5,𝔽2)—61no—
42H(5,𝔽2)C2×D312yes—
43C2×#41—21no—
44C2×#41—31no—
45C2×#41—31no—
46C2×#41C2×D311yescharacteristic
47C22×D4—31no—
48C22×D4—61no—
49#137—31no—
50#137—32no—
51#137—31no—
52#137—61no—
53#137—61no—
54AGL(1,ℤ/8)—31no—
55AGL(1,ℤ/8)—32no—
56AGL(1,ℤ/8)—31no—
57AGL(1,ℤ/8)—61no—
58AGL(1,ℤ/8)—61no—
59#135—61no—
60#135—62no—
61#135—61no—
62C2×M16—31no—
63C2×M16—31no—
64#127—32no—
65#124—32no—
66#121—62no—
67#120—62no—
68#117—31no—
69#104—32no—
70#104—32no—
71#104—32no—
72#104—32no—
73#100—32no—
74#100—32no—
75C3×Q8—42no—
76C3×Q8—81no—
77C2×C3×C4—41no—
78C2×C3×C4—81no—
79#67—81no—
80D12—41no—
81C4×D3—81no—
82Dic6—41no—
83SL(2,𝔽3)C2×D411yescharacteristic
84C3⋊C8—42no—
85C24—32no—
86#41—22no—
87#41—31no—
88#41—31no—
89#41—32no—
90#41—32no—
91#41—61no—
92#41—61no—
93#41—61no—
94#41—61no—
95#41—61no—
96#41—62no—
97#41—61no—
98#41#6711yescharacteristic
99#41C22×D311yescharacteristic
100C2×Q8—31no—
101C2×Q8#6711yescharacteristic
102C2×Q8S411yescharacteristic
103C2×D4—31no—
104C2×D4—32no—
105C2×D4—32no—
106C2×D4—31no—
107C2×D4—31no—
108C2×D4—61no—
109C2×D4—61no—
110C2×D4—61no—
111C2×D4—61no—
112C2×D4—62no—
113C2×D4—62no—
114C2×D4—64no—
115C2×D4—61no—
116C2×D4—64no—
117C22×C4—31no—
118C22×C4—61no—
119C22×C4—61no—
120Q16—61no—
121Q16—61no—
122Q16—62no—
123SD16—61no—
124SD16—61no—
125SD16—62no—
126SD16—61no—
127SD16—61no—
128SD16—62no—
129D8—61no—
130D8—61no—
131D8—62no—
132M16—31no—
133M16—31no—
134M16—32no—
135M16—61no—
136M16—61no—
137C2×C8—61no—
138C2×C8—61no—
139C4⋊C4—61no—
140C22⋊C4—61no—
141C22⋊C4—64no—
142C42—32no—
143C22×C3—41no—
144C2×D3—81no—
145C3×C4—41no—
146C3×C4—41no—
147C3×C4—82no—
148Dic3—81no—
149C23—31no—
150C23—34no—
151C23—62no—
152C23—61no—
153C23—61no—
154C23—242no—
155Q8—21no—
156Q8—31no—
157Q8—61no—
158Q8—61no—
159Q8C2×#6711yescharacteristic
160Q8C2×S412yes—
161D4—31no—
162D4—31no—
163D4—32no—
164D4—62no—
165D4—62no—
166D4—61no—
167D4—62no—
168D4—62no—
169D4—61no—
170D4—61no—
171D4—61no—
172D4—121no—
173D4—121no—
174D4—244no—
175C2×C4—21no—
176C2×C4—31no—
177C2×C4—31no—
178C2×C4—31no—
179C2×C4—31no—
180C2×C4—31no—
181C2×C4—62no—
182C2×C4—62no—
183C2×C4—62no—
184C2×C4—61no—
185C2×C4—121no—
186C2×C4—121no—
187C2×C4C2×S411yescharacteristic
188C8—61no—
189C8—61no—
190C8—62no—
191C2×C3—41no—
192C2×C3—81no—
193D3—161no—
194C22—31no—
195C22—31no—
196C22—62no—
197C22—121no—
198C22—122no—
199C22—124no—
200C22—122no—
201C22—121no—
202C22—242no—
203C22GL(2,ℤ/4)11yescharacteristic
204C4—22no—
205C4—31no—
206C4—31no—
207C4—122no—
208C4—121no—
209C4GL(2,ℤ/4)11yescharacteristic
210C4C22×S411yescharacteristic
211C3—41noSylow
212C2—21no—
213C2—61no—
214C2—61no—
215C2—122no—
216C2—241no—
217C2C2×GL(2,ℤ/4)11yescharacteristic
subgroupG/H
center, socleC2C2×GL(2,ℤ/4)
commutator, cosocle#41⋊C3C23
FrattiniC4C22×S4
FittingC2×H(5,𝔽2)D3
hypercenterC2×Q8S4
hypocenterSL(2,𝔽3)C2×D4
LCSG ▷ #41⋊C3 ▷ SL(2,𝔽3)
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
1. C2×#994C2semidirect
2. #5923C2semidirect
3. #5429C2semidirect
4. #5428C2semidirect
5. #5423C2semidirect
11. #986C22semidirect
13. #984C22semidirect
16. GU(2,𝔽3)C22semidirect
17. C2×H(5,𝔽2)D3semidirect
31. #41⋊C3D4semidirect
33. 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
C2×#9942
#59232
#54292
#54282
#54232
#30193
#9424