Finite Group Databasebeta

← #3502⋊C3 #85613

Subgroups — #3502⋊C3 #85613

Order 384.

254 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 254 conjugacy classes of subgroups — too many to draw. Only its 62 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
2#557⋊C3C212yesmaximal, complemented
3#557⋊C3C211yesmaximal, characteristic, complemented
4#519⋊C3C212yesmaximal
5C2×C4×SL(2,𝔽3)C211yesmaximal, characteristic
6#3502C311yesmaximal, characteristic, complemented, Sylow
7C22×SL(2,𝔽3)C2211yescharacteristic
8C3×C22⋊Q8—41nomaximal
9C4×SL(2,𝔽3)C2211yescharacteristic
10C4×SL(2,𝔽3)C2211yescharacteristic
11C4×SL(2,𝔽3)C2212yes—
12C4×SL(2,𝔽3)C2212yes—
13Q82—31no—
14Q82C2×C311yescharacteristic, complemented
15#557—31no—
16#557—32no—
17#557C2×C312yescomplemented
18#557C2×C311yescharacteristic, complemented
19#544—32no—
20#533—31no—
21#533—31no—
22#519C2×C312yes—
23C2×C4×Q8C2×C311yescharacteristic
24C2×C3×Q8—41no—
25C22×C3×C4—41no—
26C2×SL(2,𝔽3)Q812yescomplemented
27C2×SL(2,𝔽3)C2311yescharacteristic
28C3×C4⋊C4—42no—
29C3×C4⋊C4—41no—
30C3×C22⋊C4—42no—
31C22×Q8C22×C311yescharacteristic
32C4⋊Q8—31no—
33C4⋊Q8—32no—
34C4⋊Q8—34no—
35C4⋊Q8—31no—
36C4⋊Q8—31no—
37C4⋊Q8—31no—
38C4⋊Q8—32no—
39#125—32no—
40#125—32no—
41#125—32no—
42#125—34no—
43#125—32no—
44C22⋊Q8—32no—
45C22⋊Q8—31no—
46C22⋊Q8—32no—
47C22⋊Q8A411yescharacteristic
48C4×Q8—32no—
49C4×Q8—32no—
50C4×Q8—31no—
51C4×Q8—32no—
52C4×Q8—31no—
53C4×Q8—32no—
54C4×Q8—32no—
55C4×Q8—31no—
56C4×Q8—31no—
57C4×Q8C22×C311yescharacteristic
58C4×Q8C22×C311yescharacteristic
59C4×Q8C22×C312yes—
60C4×Q8C22×C312yes—
61#117—32no—
62#117—32no—
63C2×C4⋊C4—31no—
64C2×C42—31no—
65C23×C3—41no—
66C3×Q8—82no—
67C2×C3×C4—41no—
68C2×C3×C4—42no—
69C2×C3×C4—42no—
70C2×C3×C4—41no—
71C2×C3×C4—82no—
72SL(2,𝔽3)C2×Q811yescharacteristic
73C2×Q8—32no—
74C2×Q8—31no—
75C2×Q8—32no—
76C2×Q8—31no—
77C2×Q8—32no—
78C2×Q8C3×Q812yescomplemented
79C2×Q8C2×A411yescharacteristic
80C2×Q8C23×C311yescharacteristic
81C22×C4—31no—
82C22×C4—31no—
83C22×C4C2×A411yescharacteristic
84C4⋊C4—31no—
85C4⋊C4—31no—
86C4⋊C4—32no—
87C4⋊C4—32no—
88C4⋊C4—32no—
89C4⋊C4—32no—
90C4⋊C4—34no—
91C4⋊C4—32no—
92C4⋊C4—32no—
93C4⋊C4—32no—
94C4⋊C4—31no—
95C4⋊C4—32no—
96C4⋊C4—32no—
97C4⋊C4—32no—
98C4⋊C4C2×A412yes—
99C4⋊C4C2×A411yescharacteristic
100C22⋊C4—32no—
101C22⋊C4C2×A412yes—
102C42—34no—
103C42—31no—
104C42—31no—
105C42—31no—
106C42—31no—
107C42—32no—
108C42—32no—
109C22×C3—42no—
110C22×C3—41no—
111C22×C3—81no—
112C22×C3—81no—
113C3×C4—42no—
114C3×C4—82no—
115C3×C4—81no—
116C3×C4—82no—
117C23C22×A411yescharacteristic
118Q8—22no—
119Q8—32no—
120Q8—31no—
121Q8—32no—
122Q8—62no—
123Q8—122no—
124Q8—122no—
125Q8C2×C3×Q811yescharacteristic
126C2×C4—22no—
127C2×C4—32no—
128C2×C4—32no—
129C2×C4—31no—
130C2×C4—31no—
131C2×C4—32no—
132C2×C4—32no—
133C2×C4—31no—
134C2×C4—31no—
135C2×C4—121no—
136C2×C4C22×A411yescharacteristic
137C2×C4C22×A412yes—
138C2×C4C22×A412yes—
139C2×C4C22×A411yescharacteristic
140C2×C3—41no—
141C2×C3—41no—
142C2×C3—41no—
143C2×C3—82no—
144C22—21no—
145C22—21no—
146C22Q8×A412yes—
147C22C23×A411yescharacteristic
148C4—21no—
149C4—22no—
150C4—22no—
151C4—31no—
152C4—32no—
153C4—31no—
154C4—62no—
155C4—61no—
156C4—61no—
157C4—62no—
158C42−1+4⋊C312yes—
159C3—41noSylow
160C2—22no—
161C2C2×Q8×A411yescharacteristic
162C2C2×2−1+4⋊C311yescharacteristic
163C2#594511yescharacteristic
subgroupG/H
center, Frattini, socleC22C23×A4
commutator, cosocleC2×Q8C23×C3
Fitting#3502C3
prime radicalC22×SL(2,𝔽3)C22
hypercenterC22⋊Q8A4
hypocenterQ8C2×C3×Q8
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. #557⋊C3C2semidirect
3. #557⋊C3C2semidirect
6. #3502C3semidirect
14. Q82C2×C3semidirect
17. #557C2×C3semidirect
18. #557C2×C3semidirect
26. C2×SL(2,𝔽3)Q8semidirect
78. C2×Q8C3×Q8semidirect

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
#557⋊C32
#519⋊C32
C2×C4×SL(2,𝔽3)2
#35023
C3×C22⋊Q84