Finite Group Databasebeta

← #85533

Subgroups — #85533

Order 384.

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

normal subgroups only

This group has 292 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
1C2×2−1+4⋊C3C211yesmaximal, characteristic, complemented
2#5923C211yesmaximal, characteristic, complemented
3#5431C212yesmaximal, complemented
4#5430C212yesmaximal, complemented
5#5423C211yesmaximal, characteristic, complemented
6#3020—31nomaximal, Sylow
72−1+4⋊C3—21no—
82−1+4⋊C3C2212yes—
9C2×#41⋊C3C2211yescharacteristic
10C22×SL(2,𝔽3)—21no—
11#986C2211yescharacteristic, complemented
12#985—21no—
13#984C2211yescharacteristic, complemented
14#983—21no—
15#932—41nomaximal
16GU(2,𝔽3)C2212yescomplemented
17C2×2−1+4D311yescharacteristic, complemented
18#575—31no—
19#575—31no—
20#536—31no—
21#456—32no—
22#456—32no—
23#454—32no—
24#454—32no—
25#421—31no—
26#421—31no—
27C2×#101—31no—
28C2×C3×D4—41no—
29#235—41no—
30#41⋊C3D411yescharacteristic
31#41⋊C3C2311yescharacteristic
32C2×SL(2,𝔽3)—22no—
33C2×SL(2,𝔽3)D411yescharacteristic
34GL(2,𝔽3)—21no—
35#226—21no—
36#214—42no—
37#213—42no—
38#208—41no—
392−1+4—21no—
402−1+4—32no—
412−1+4—61no—
422−1+4C2×D312yes—
43C2×#41—31no—
44C2×#41—31no—
45C2×#41—61no—
46C2×#41C2×D311yescharacteristic
47C22×Q8—21no—
48C22×Q8—31no—
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—
64C4⋊Q8—32no—
65#124—32no—
66#123—62no—
67C22⋊Q8—62no—
68#117—31no—
69#104—32no—
70#104—32no—
71#104—32no—
72#104—32no—
73#101—32no—
74#101—32no—
75C23×C3—81no—
76C3×D4—42no—
77C3×D4—81no—
78C2×C3×C4—41no—
79#67—81no—
80D12—41no—
81C4×D3—81no—
82Dic6—41no—
83SL(2,𝔽3)C2×D411yescharacteristic
84C3⋊C8—42no—
85#41—31no—
86#41—31no—
87#41—32no—
88#41—32no—
89#41—62no—
90#41—61no—
91#41—61no—
92#41—64no—
93#41—61no—
94#41—61no—
95#41—61no—
96#41—62no—
97#41—61no—
98#41#6711yescharacteristic
99#41C22×D311yescharacteristic
100C2×Q8—22no—
101C2×Q8—32no—
102C2×Q8—32no—
103C2×Q8—31no—
104C2×Q8—31no—
105C2×Q8—61no—
106C2×Q8—61no—
107C2×Q8—61no—
108C2×Q8—62no—
109C2×Q8#6711yescharacteristic
110C2×D4—31no—
111C2×D4—31no—
112C2×D4—61no—
113C2×D4—61no—
114C2×D4S411yescharacteristic
115C22×C4—31no—
116C22×C4—32no—
117C22×C4—61no—
118C22×C4—61no—
119Q16—61no—
120Q16—62no—
121Q16—61no—
122SD16—61no—
123SD16—61no—
124SD16—61no—
125SD16—62no—
126SD16—61no—
127SD16—62no—
128D8—61no—
129D8—61no—
130D8—62no—
131M16—31no—
132M16—31no—
133M16—32no—
134M16—61no—
135M16—61no—
136C2×C8—61no—
137C2×C8—61no—
138C4⋊C4—61no—
139C4⋊C4—64no—
140C22⋊C4—64no—
141C22⋊C4—61no—
142C42—32no—
143C22×C3—41no—
144C22×C3—82no—
145C22×C3—162no—
146C2×D3—81no—
147C3×C4—41no—
148C3×C4—41no—
149Dic3—81no—
150C23—21no—
151C23—31no—
152C23—61no—
153Q8—31no—
154Q8—32no—
155Q8—62no—
156Q8—62no—
157Q8—61no—
158Q8—61no—
159Q8—61no—
160Q8C2×#6711yescharacteristic
161D4—21no—
162D4—31no—
163D4—31no—
164D4—61no—
165D4—61no—
166D4—62no—
167D4—62no—
168D4—61no—
169D4—121no—
170D4—121no—
171D4C2×S412yes—
172C2×C4—31no—
173C2×C4—31no—
174C2×C4—31no—
175C2×C4—31no—
176C2×C4—31no—
177C2×C4—34no—
178C2×C4—62no—
179C2×C4—62no—
180C2×C4—62no—
181C2×C4—61no—
182C2×C4—62no—
183C2×C4—61no—
184C2×C4—121no—
185C2×C4—121no—
186C2×C4C2×S411yescharacteristic
187C8—61no—
188C8—61no—
189C8—62no—
190C2×C3—41no—
191C2×C3—81no—
192C2×C3—162no—
193D3—161no—
194C22—22no—
195C22—31no—
196C22—31no—
197C22—42no—
198C22—121no—
199C22—122no—
200C22—121no—
201C22GL(2,ℤ/4)11yescharacteristic
202C4—31no—
203C4—31no—
204C4—62no—
205C4—122no—
206C4—121no—
207C4GL(2,ℤ/4)11yescharacteristic
208C4C22×S411yescharacteristic
209C3—41noSylow
210C2—21no—
211C2—42no—
212C2—61no—
213C2—61no—
214C2—241no—
215C2C2×GL(2,ℤ/4)11yescharacteristic
subgroupG/H
center, socleC2C2×GL(2,ℤ/4)
commutator, cosocle#41⋊C3C23
FrattiniC4C22×S4
FittingC2×2−1+4D3
hypercenterC2×D4S4
hypocenterSL(2,𝔽3)C2×D4
LCSG ▷ #41⋊C3 ▷ SL(2,𝔽3)
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C2×2−1+4⋊C3C2semidirect
2. #5923C2semidirect
3. #5431C2semidirect
4. #5430C2semidirect
5. #5423C2semidirect
11. #986C22semidirect
13. #984C22semidirect
16. GU(2,𝔽3)C22semidirect
17. C2×2−1+4D3semidirect

Maximal subgroups

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

subgroupindex
C2×2−1+4⋊C32
#59232
#54312
#54302
#54232
#30203
#9324