Finite Group Databasebeta

← #3518⋊C3 #85618

Subgroups — #3518⋊C3 #85618

Order 384.

265 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 265 conjugacy classes of subgroups — too many to draw. Only its 56 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
2D4×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
3#548⋊C3C212yesmaximal, complemented
4#519⋊C3C212yesmaximal, complemented
5C4×#41⋊C3C211yesmaximal, characteristic, complemented
6#3518C311yesmaximal, characteristic, complemented, Sylow
7C2×#41⋊C3C2211yescharacteristic
8C22×SL(2,𝔽3)C2212yes—
9C3×#124—41nomaximal
10C4×SL(2,𝔽3)C2211yescharacteristic
11C4×SL(2,𝔽3)C2211yescharacteristic
12C4×SL(2,𝔽3)C2212yes—
13Q82C2×C311yescharacteristic, complemented
14#556—32no—
15#556—31no—
16#555—31no—
17D4×Q8C2×C311yescharacteristic
18#548C2×C312yescomplemented
19#542—32no—
20#533—31no—
21#529—31no—
22#519C2×C312yescomplemented
23C4×#41C2×C311yescharacteristic, complemented
24C2×C3×Q8—41no—
25C2×C3×D4—41no—
26#41⋊C3—21no—
27C2×SL(2,𝔽3)—22no—
28C2×SL(2,𝔽3)C2311yescharacteristic
29C3×C22⋊C4—42no—
30C3×C22⋊C4—42no—
31C3×C42—41no—
32C2×#41C22×C311yescharacteristic
33C22×Q8C22×C312yes—
34C4⋊Q8—31no—
35C4⋊Q8—31no—
36C4⋊Q8—32no—
37#126—34no—
38#126—32no—
39#125—31no—
40#124—34no—
41#124—32no—
42#124A411yescharacteristic
43#123—32no—
44C22⋊Q8—32no—
45C22⋊Q8—31no—
46C22⋊Q8—32no—
47C22⋊Q8—32no—
48#121—32no—
49#121—31no—
50C4×Q8—32no—
51C4×Q8—31no—
52C4×Q8—32no—
53C4×Q8—32no—
54C4×Q8C22×C311yescharacteristic
55C4×Q8C22×C311yescharacteristic
56C4×Q8C22×C312yes—
57C4×D4—31no—
58C4×D4—31no—
59C4×D4—31no—
60C4×D4—32no—
61#117—32no—
62#117—32no—
63#117—32no—
64#117—31no—
65C2×C42—31no—
66C23×C3—42no—
67C3×Q8—82no—
68C3×D4—82no—
69C2×C3×C4—41no—
70C2×C3×C4—41no—
71C2×C3×C4—42no—
72C2×C3×C4—41no—
73SL(2,𝔽3)#4111yescharacteristic
74#41—21no—
75#41—61no—
76C2×Q8—22no—
77C2×Q8—32no—
78C2×Q8—32no—
79C2×Q8—62no—
80C2×Q8C2×A411yescharacteristic
81C2×Q8C23×C311yescharacteristic
82C2×D4—31no—
83C2×D4—32no—
84C2×D4C2×A411yescharacteristic
85C22×C4—32no—
86C22×C4—31no—
87C22×C4—31no—
88C22×C4—31no—
89C4⋊C4—31no—
90C4⋊C4—31no—
91C4⋊C4—32no—
92C4⋊C4—32no—
93C4⋊C4—31no—
94C4⋊C4—31no—
95C4⋊C4—32no—
96C4⋊C4—32no—
97C4⋊C4—32no—
98C4⋊C4—32no—
99C22⋊C4—32no—
100C22⋊C4—34no—
101C22⋊C4—32no—
102C22⋊C4—32no—
103C22⋊C4—32no—
104C22⋊C4C2×A412yes—
105C22⋊C4C2×A412yes—
106C42—34no—
107C42—31no—
108C42—31no—
109C42—31no—
110C42—32no—
111C42C2×A411yescharacteristic
112C22×C3—41no—
113C22×C3—82no—
114C22×C3—82no—
115C22×C3—82no—
116C3×C4—42no—
117C3×C4—81no—
118C3×C4—82no—
119C3×C4—81no—
120C23—31no—
121C23C22×A412yes—
122Q8—22no—
123Q8—31no—
124Q8—62no—
125Q8—62no—
126Q8—122no—
127Q8C3×#4111yescharacteristic
128D4—22no—
129D4—61no—
130D4—61no—
131D4—122no—
132C2×C4—31no—
133C2×C4—31no—
134C2×C4—32no—
135C2×C4—32no—
136C2×C4—31no—
137C2×C4—61no—
138C2×C4—61no—
139C2×C4—62no—
140C2×C4—62no—
141C2×C4—62no—
142C2×C4—121no—
143C2×C4C22×A411yescharacteristic
144C2×C4C22×A411yescharacteristic
145C2×C4C22×A412yes—
146C2×C4C22×A411yescharacteristic
147C2×C3—41no—
148C2×C3—41no—
149C2×C3—41no—
150C2×C3—162no—
151C22—22no—
152C22—22no—
153C22—22no—
154C22—61no—
155C22—61no—
156C22—61no—
157C22C23×A411yescharacteristic
158C4—21no—
159C4—22no—
160C4—21no—
161C4—31no—
162C4—31no—
163C4—62no—
164C4—61no—
165C4—61no—
166C4—62no—
167C42−1+4⋊C312yes—
168C3—41noSylow
169C2—42no—
170C2—121no—
171C2A4×#4111yescharacteristic
172C2C2×2−1+4⋊C311yescharacteristic
173C2#594511yescharacteristic
subgroupG/H
center, Frattini, socleC22C23×A4
commutator, cosocleC2×Q8C23×C3
Fitting#3518C3
hypercenter#124A4
hypocenterQ8C3×#41
LCSG ▷ C2×Q8 ▷ Q8
UCSC1 ◁ C22 ◁ #124

Complements

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

NH ≅ G/Ntype
1. Q8×SL(2,𝔽3)C2semidirect
2. D4×SL(2,𝔽3)C2semidirect
3. #548⋊C3C2semidirect
4. #519⋊C3C2semidirect
5. C4×#41⋊C3C2semidirect
6. #3518C3semidirect
13. Q82C2×C3semidirect
18. #548C2×C3semidirect
22. #519C2×C3semidirect
23. C4×#41C2×C3semidirect

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
D4×SL(2,𝔽3)2
#548⋊C32
#519⋊C32
C4×#41⋊C32
#35183
C3×#1244