Finite Group Databasebeta

← #3482⋊C3 #85603

Subgroups — #3482⋊C3 #85603

Order 384.

294 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 294 conjugacy classes of subgroups — too many to draw. Only its 66 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#557⋊C3C211yesmaximal, characteristic, complemented
2D4×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
3#548⋊C3C212yesmaximal, complemented
4C4×#41⋊C3C211yesmaximal, characteristic, complemented
5C2×C4×SL(2,𝔽3)C212yesmaximal, complemented
6#3482C311yesmaximal, characteristic, complemented, Sylow
7C2×#41⋊C3C2211yescharacteristic
8C22×SL(2,𝔽3)C2212yes—
9C3×C4×D4—41nomaximal
10C4×SL(2,𝔽3)C2211yescharacteristic
11C4×SL(2,𝔽3)C2212yes—
12C4×SL(2,𝔽3)C2211yescharacteristic
13#557C2×C311yescharacteristic, complemented
14#556—31no—
15#552—32no—
16D4×Q8C2×C311yescharacteristic
17#548C2×C312yescomplemented
18#547—31no—
19#533—31no—
20#529—32no—
21C4×#41—31no—
22C4×#41C2×C311yescharacteristic, complemented
23C2×C4×Q8C2×C312yescomplemented
24C2×C3×D4—41no—
25C22×C3×C4—42no—
26#41⋊C3—21no—
27C2×SL(2,𝔽3)—22no—
28C2×SL(2,𝔽3)C2311yescharacteristic
29C3×C4⋊C4—41no—
30C3×C22⋊C4—42no—
31C3×C42—41no—
32C2×#41—31no—
33C2×#41C22×C311yescharacteristic
34C22×Q8C22×C312yes—
35C4⋊Q8—31no—
36C4⋊Q8—31no—
37#126—34no—
38#125—32no—
39#125—32no—
40#124—32no—
41#123—34no—
42C22⋊Q8—32no—
43C22⋊Q8—32no—
44C22⋊Q8—32no—
45#121—32no—
46C4×Q8—31no—
47C4×Q8—31no—
48C4×Q8—32no—
49C4×Q8—32no—
50C4×Q8C22×C311yescharacteristic
51C4×Q8C22×C312yes—
52C4×Q8C22×C311yescharacteristic
53C4×D4—31no—
54C4×D4—32no—
55C4×D4—31no—
56C4×D4—31no—
57C4×D4—32no—
58C4×D4A411yescharacteristic
59#117—32no—
60#117—31no—
61#117—32no—
62#117—31no—
63C2×C4⋊C4—32no—
64C2×C42—32no—
65C2×C42—31no—
66C23×C3—42no—
67C3×D4—44no—
68C2×C3×C4—41no—
69C2×C3×C4—42no—
70C2×C3×C4—41no—
71C2×C3×C4—41no—
72C2×C3×C4—84no—
73SL(2,𝔽3)#4111yescharacteristic, complemented
74#41—21no—
75#41—64no—
76#41—61no—
77C2×Q8—22no—
78C2×Q8—31no—
79C2×Q8—32no—
80C2×Q8—62no—
81C2×Q8C23×C311yescharacteristic
82C2×D4—31no—
83C2×D4—32no—
84C2×D4C2×A411yescharacteristic
85C22×C4—32no—
86C22×C4—32no—
87C22×C4—31no—
88C22×C4—31no—
89C22×C4—31no—
90C22×C4C2×A412yes—
91C4⋊C4—31no—
92C4⋊C4—32no—
93C4⋊C4—31no—
94C4⋊C4—34no—
95C4⋊C4—32no—
96C4⋊C4—32no—
97C4⋊C4—31no—
98C4⋊C4—32no—
99C4⋊C4—32no—
100C4⋊C4C2×A411yescharacteristic
101C22⋊C4—32no—
102C22⋊C4—34no—
103C22⋊C4—32no—
104C22⋊C4—32no—
105C22⋊C4C2×A412yes—
106C42—31no—
107C42—32no—
108C42—31no—
109C42—32no—
110C42—31no—
111C42—32no—
112C42C2×A411yescharacteristic
113C22×C3—44no—
114C22×C3—41no—
115C22×C3—82no—
116C22×C3—82no—
117C3×C4—42no—
118C3×C4—42no—
119C3×C4—82no—
120C3×C4—81no—
121C23—31no—
122C23C22×A412yes—
123Q8—31no—
124Q8—62no—
125Q8—62no—
126Q8—62no—
127Q8C3×#4111yescharacteristic
128D4—61no—
129D4—61no—
130D4—64no—
131D4#41⋊C314yescomplemented
132C2×C4—24no—
133C2×C4—31no—
134C2×C4—31no—
135C2×C4—31no—
136C2×C4—32no—
137C2×C4—32no—
138C2×C4—61no—
139C2×C4—61no—
140C2×C4—62no—
141C2×C4—62no—
142C2×C4—64no—
143C2×C4—62no—
144C2×C4—62no—
145C2×C4C22×A412yes—
146C2×C4C22×A411yescharacteristic
147C2×C4C22×A411yescharacteristic
148C2×C4C22×A411yescharacteristic
149C2×C3—41no—
150C2×C3—41no—
151C2×C3—41no—
152C2×C3—84no—
153C22—22no—
154C22—22no—
155C22—61no—
156C22—61no—
157C22—61no—
158C22C2×#41⋊C314yes—
159C22C23×A411yescharacteristic
160C4—21no—
161C4—22no—
162C4—31no—
163C4—31no—
164C4—61no—
165C4—62no—
166C4—62no—
167C4—61no—
168C4C2×#41⋊C312yes—
169C42−1+4⋊C312yes—
170C3—41noSylow
171C2—24no—
172C2—121no—
173C2C22×#41⋊C311yescharacteristic
174C2A4×#4111yescharacteristic
175C2C2×2−1+4⋊C311yescharacteristic
subgroupG/H
centerC2×C4C22×A4
commutator, cosocleC2×Q8C23×C3
Frattini, socleC22C23×A4
Fitting#3482C3
hypercenterC4×D4A4
hypocenterQ8C3×#41
LCSG ▷ C2×Q8 ▷ Q8
UCSC1 ◁ C2×C4 ◁ C4×D4

Complements

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

NH ≅ G/Ntype
1. #557⋊C3C2semidirect
2. D4×SL(2,𝔽3)C2semidirect
3. #548⋊C3C2semidirect
4. C4×#41⋊C3C2semidirect
5. C2×C4×SL(2,𝔽3)C2semidirect
6. #3482C3semidirect
13. #557C2×C3semidirect
17. #548C2×C3semidirect
22. C4×#41C2×C3semidirect
23. C2×C4×Q8C2×C3semidirect
73. SL(2,𝔽3)#41semidirect
131. D4#41⋊C3semidirect

Maximal subgroups

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

subgroupindex
#557⋊C32
D4×SL(2,𝔽3)2
#548⋊C32
C4×#41⋊C32
C2×C4×SL(2,𝔽3)2
#34823
C3×C4×D44