Finite Group Databasebeta

← #85543

Subgroups — #85543

Order 384.

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

normal subgroups only

This group has 306 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
1#5945C211yesmaximal, characteristic, complemented
2C2×#986C211yesmaximal, characteristic, complemented
3#5431C212yesmaximal, complemented
4#5429C212yesmaximal, complemented
5#5423C211yesmaximal, characteristic, complemented
6#3021—31nomaximal, Sylow
72−1+4⋊C3—21no—
82−1+4⋊C3C2211yescharacteristic, complemented
9#994C2211yescharacteristic, complemented
10C2×#41⋊C3—21no—
11C2×#41⋊C3C2211yescharacteristic
12#986—21no—
13#986C2212yescomplemented
14C2×GL(2,𝔽3)—21no—
15#949—41nomaximal
16GU(2,𝔽3)C2212yescomplemented
17#585D311yescharacteristic, complemented
18C2×AGL(1,ℤ/8)—31no—
19C2×AGL(1,ℤ/8)—31no—
20#535—31no—
21#454—32no—
22#454—32no—
23#453—32no—
24#453—32no—
25#421—31no—
26#421—31no—
27#413—31no—
28C3×#41—41no—
29C2×D12—41no—
30#41⋊C3—21no—
31#41⋊C3D411yescharacteristic, complemented
32#41⋊C3C2311yescharacteristic
33C2×SL(2,𝔽3)—21no—
34C2×SL(2,𝔽3)D411yescharacteristic, complemented
35GL(2,𝔽3)—22no—
36#215—42no—
37#213—42no—
38#208—41no—
392−1+4—21no—
402−1+4—31no—
412−1+4C2×D311yescharacteristic, complemented
42H(5,𝔽2)—31no—
43H(5,𝔽2)—61no—
44H(5,𝔽2)C2×D311yescharacteristic, complemented
45C2×#41—21no—
46C2×#41—31no—
47C2×#41—31no—
48C2×#41—61no—
49C2×#41C2×D311yescharacteristic
50C22×D4—31no—
51AGL(1,ℤ/8)—32no—
52AGL(1,ℤ/8)—32no—
53AGL(1,ℤ/8)—32no—
54AGL(1,ℤ/8)—32no—
55AGL(1,ℤ/8)—61no—
56AGL(1,ℤ/8)—61no—
57AGL(1,ℤ/8)—61no—
58AGL(1,ℤ/8)—61no—
59C2×SD16—61no—
60C2×SD16—61no—
61C2×D8—61no—
62C2×D8—61no—
63C2×M16—31no—
64C2×M16—31no—
65#127—32no—
66#124—32no—
67#121—62no—
68#120—62no—
69#117—31no—
70#104—32no—
71#104—32no—
72#104—32no—
73#104—32no—
74#101—32no—
75#100—32no—
76C22×D3—81no—
77C3×Q8—41no—
78C3×D4—41no—
79C3×D4—81no—
80C2×C3×C4—41no—
81C2×C3×C4—81no—
82D12—42no—
83D12—81no—
84SL(2,𝔽3)C2×D411yescharacteristic
85C3⋊C8—42no—
86C24—61no—
87#41—21no—
88#41—31no—
89#41—31no—
90#41—31no—
91#41—31no—
92#41—31no—
93#41—61no—
94#41—61no—
95#41—61no—
96#41—61no—
97#41—62no—
98#41—61no—
99#41—61no—
100#41—61no—
101#41—61no—
102#41#6711yescharacteristic
103#41S411yescharacteristic
104#41C22×D311yescharacteristic
105C2×Q8—21no—
106C2×Q8—31no—
107C2×Q8—31no—
108C2×Q8—61no—
109C2×Q8#6711yescharacteristic
110C2×D4—31no—
111C2×D4—31no—
112C2×D4—31no—
113C2×D4—32no—
114C2×D4—32no—
115C2×D4—61no—
116C2×D4—61no—
117C2×D4—61no—
118C2×D4—61no—
119C2×D4—61no—
120C2×D4—61no—
121C2×D4—61no—
122C2×D4—62no—
123C2×D4—64no—
124C2×D4—61no—
125C22×C4—31no—
126C22×C4—32no—
127C22×C4—61no—
128SD16—62no—
129SD16—62no—
130SD16—62no—
131SD16—62no—
132D8—62no—
133D8—62no—
134D8—62no—
135D8—62no—
136M16—32no—
137M16—32no—
138M16—61no—
139M16—61no—
140C2×C8—61no—
141C2×C8—61no—
142C4⋊C4—61no—
143C22⋊C4—64no—
144C22⋊C4—61no—
145C42—32no—
146C22×C3—41no—
147C22×C3—81no—
148C2×D3—82no—
149C2×D3—162no—
150C3×C4—41no—
151C3×C4—41no—
152C3×C4—81no—
153C23—31no—
154C23—32no—
155C23—61no—
156C23—62no—
157C23—121no—
158C23—121no—
159C23—242no—
160Q8—31no—
161Q8—31no—
162Q8—61no—
163Q8—61no—
164Q8C2×#6711yescharacteristic
165Q8C2×S411yescharacteristic
166D4—21no—
167D4—31no—
168D4—31no—
169D4—31no—
170D4—61no—
171D4—61no—
172D4—61no—
173D4—61no—
174D4—62no—
175D4—61no—
176D4—62no—
177D4—61no—
178D4—62no—
179D4—121no—
180D4—121no—
181D4—244no—
182D4C2×S411yescharacteristic
183C2×C4—21no—
184C2×C4—31no—
185C2×C4—31no—
186C2×C4—31no—
187C2×C4—31no—
188C2×C4—31no—
189C2×C4—32no—
190C2×C4—61no—
191C2×C4—61no—
192C2×C4—61no—
193C2×C4—62no—
194C2×C4—61no—
195C2×C4—62no—
196C2×C4C2×S411yescharacteristic
197C8—62no—
198C8—62no—
199C2×C3—41no—
200C2×C3—81no—
201C2×C3—161no—
202D3—162no—
203C22—21no—
204C22—31no—
205C22—31no—
206C22—61no—
207C22—121no—
208C22—122no—
209C22—124no—
210C22—122no—
211C22—242no—
212C22—241no—
213C22—242no—
214C22GL(2,ℤ/4)11yescharacteristic
215C4—21no—
216C4—31no—
217C4—31no—
218C4—61no—
219C4—122no—
220C4GL(2,ℤ/4)11yescharacteristic
221C4C22×S411yescharacteristic
222C3—41noSylow
223C2—21no—
224C2—41no—
225C2—61no—
226C2—61no—
227C2—121no—
228C2—242no—
229C2C2×GL(2,ℤ/4)11yescharacteristic
subgroupG/H
center, socleC2C2×GL(2,ℤ/4)
commutator, cosocle#41⋊C3C23
FrattiniC4C22×S4
Fitting#585D3
hypercenter#41S4
hypocenterSL(2,𝔽3)C2×D4
LCSG ▷ #41⋊C3 ▷ SL(2,𝔽3)
UCSC1 ◁ C2 ◁ C2×C4 ◁ #41

Complements

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

NH ≅ G/Ntype
1. #5945C2semidirect
2. C2×#986C2semidirect
3. #5431C2semidirect
4. #5429C2semidirect
5. #5423C2semidirect
8. 2−1+4⋊C3C22semidirect
9. #994C22semidirect
13. #986C22semidirect
16. GU(2,𝔽3)C22semidirect
17. #585D3semidirect
31. #41⋊C3D4semidirect
34. C2×SL(2,𝔽3)D4semidirect
41. 2−1+4C2×D3semidirect
44. H(5,𝔽2)C2×D3semidirect

Maximal subgroups

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

subgroupindex
#59452
C2×#9862
#54312
#54292
#54232
#30213
#9494