Finite Group Databasebeta

← #3522⋊C3 #85619

Subgroups — #3522⋊C3 #85619

Order 384.

279 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 279 conjugacy classes of subgroups — too many to draw. Only its 58 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
1Q8×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
2C4⋊(#41⋊C3)C211yesmaximal, characteristic, complemented
3#548⋊C3C212yesmaximal, complemented
4#520⋊C3C212yesmaximal, complemented
5C2×C4×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
6#3522C311yesmaximal, characteristic, complemented, Sylow
7C2×#41⋊C3C2212yes—
8C22×SL(2,𝔽3)C2211yescharacteristic
9C3×C22⋊Q8—41nomaximal
10C4×SL(2,𝔽3)C2211yescharacteristic
11C4×SL(2,𝔽3)C2211yescharacteristic
12C4×SL(2,𝔽3)C2212yes—
13#559—31no—
14Q82C2×C311yescharacteristic, complemented
15#554—31no—
16#550C2×C311yescharacteristic, complemented
17#548C2×C312yes—
18#547—32no—
19#542—32no—
20#532—31no—
21#529—31no—
22#520C2×C312yescomplemented
23C2×C4×Q8C2×C311yescharacteristic
24C2×C3×Q8—41no—
25C22×C3×C4—41no—
26#41⋊C3—22no—
27C2×SL(2,𝔽3)—21no—
28C2×SL(2,𝔽3)C2311yescharacteristic
29C3×C4⋊C4—42no—
30C3×C4⋊C4—41no—
31C3×C22⋊C4—42no—
32C2×#41—32no—
33C2×#41C22×C312yes—
34C22×Q8C22×C311yescharacteristic
35C4⋊Q8—31no—
36C4⋊Q8—32no—
37#127—31no—
38#126—32no—
39#125—31no—
40#124—32no—
41#124—31no—
42#124—32no—
43#123—34no—
44#123—32no—
45C22⋊Q8—34no—
46C22⋊Q8—32no—
47C22⋊Q8A411yescharacteristic
48#121—32no—
49#121—31no—
50#121—32no—
51C4×Q8—31no—
52C4×Q8—31no—
53C4×Q8—31no—
54C4×Q8C22×C311yescharacteristic
55C4×Q8C22×C311yescharacteristic
56C4×Q8C22×C312yes—
57C4×D4—32no—
58C4×D4—31no—
59C4×D4—32no—
60C4×D4—32no—
61C4×D4—32no—
62#117—32no—
63C2×C4⋊C4—31no—
64C2×C4⋊C4—32no—
65C2×C42—31no—
66C23×C3—41no—
67C3×Q8—82no—
68C2×C3×C4—41no—
69C2×C3×C4—42no—
70C2×C3×C4—41no—
71C2×C3×C4—42no—
72C2×C3×C4—82no—
73SL(2,𝔽3)#4111yescharacteristic, complemented
74#41—22no—
75#41—62no—
76#41—124no—
77C2×Q8—21no—
78C2×Q8—32no—
79C2×Q8—31no—
80C2×Q8—61no—
81C2×Q8C2×A411yescharacteristic
82C2×Q8C23×C311yescharacteristic
83C2×D4—32no—
84C2×D4—32no—
85C2×D4—32no—
86C2×D4—31no—
87C22×C4—31no—
88C22×C4—31no—
89C22×C4—34no—
90C22×C4—32no—
91C22×C4C2×A411yescharacteristic
92C4⋊C4—31no—
93C4⋊C4—32no—
94C4⋊C4—31no—
95C4⋊C4—32no—
96C4⋊C4—32no—
97C4⋊C4—31no—
98C4⋊C4—32no—
99C4⋊C4—32no—
100C4⋊C4C2×A412yes—
101C4⋊C4C2×A411yescharacteristic
102C22⋊C4—34no—
103C22⋊C4—32no—
104C22⋊C4—32no—
105C22⋊C4—32no—
106C22⋊C4—32no—
107C22⋊C4C2×A412yes—
108C42—31no—
109C42—31no—
110C42—31no—
111C42—31no—
112C42—32no—
113C22×C3—42no—
114C22×C3—41no—
115C22×C3—81no—
116C22×C3—81no—
117C3×C4—42no—
118C3×C4—81no—
119C3×C4—82no—
120C3×C4—82no—
121C23—32no—
122C23C22×A411yescharacteristic
123Q8—22no—
124Q8—31no—
125Q8—61no—
126Q8—61no—
127Q8—122no—
128Q8C3×#4111yescharacteristic
129D4—62no—
130D4—62no—
131D4—64no—
132D4—62no—
133D4—122no—
134C2×C4—22no—
135C2×C4—31no—
136C2×C4—31no—
137C2×C4—31no—
138C2×C4—31no—
139C2×C4—32no—
140C2×C4—62no—
141C2×C4—62no—
142C2×C4—61no—
143C2×C4—61no—
144C2×C4—62no—
145C2×C4—124no—
146C2×C4C22×A411yescharacteristic
147C2×C4C22×A412yes—
148C2×C4C22×A412yes—
149C2×C4C22×A411yescharacteristic
150C2×C3—41no—
151C2×C3—41no—
152C2×C3—41no—
153C2×C3—82no—
154C22—21no—
155C22—21no—
156C22—62no—
157C22—62no—
158C22—62no—
159C22#99412yes—
160C22C23×A411yescharacteristic
161C4—22no—
162C4—21no—
163C4—22no—
164C4—31no—
165C4—31no—
166C4—61no—
167C4—61no—
168C4—61no—
169C4—62no—
170C42−1+4⋊C312yes—
171C3—41noSylow
172C2—22no—
173C2—122no—
174C2A4×#4111yescharacteristic
175C2C2×#99411yescharacteristic
176C2C2×2−1+4⋊C311yescharacteristic
subgroupG/H
center, Frattini, socleC22C23×A4
commutator, cosocleC2×Q8C23×C3
Fitting#3522C3
hypercenterC22⋊Q8A4
hypocenterQ8C3×#41
LCSG ▷ C2×Q8 ▷ Q8
UCSC1 ◁ C22 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. Q8×SL(2,𝔽3)C2semidirect
2. C4⋊(#41⋊C3)C2semidirect
3. #548⋊C3C2semidirect
4. #520⋊C3C2semidirect
5. C2×C4×SL(2,𝔽3)C2semidirect
6. #3522C3semidirect
14. Q82C2×C3semidirect
16. #550C2×C3semidirect
22. #520C2×C3semidirect
73. SL(2,𝔽3)#41semidirect

Maximal subgroups

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

subgroupindex
Q8×SL(2,𝔽3)2
C4⋊(#41⋊C3)2
#548⋊C32
#520⋊C32
C2×C4×SL(2,𝔽3)2
#35223
C3×C22⋊Q84