Finite Group Databasebeta

← #3480⋊C3 #85602

Subgroups — #3480⋊C3 #85602

Order 384.

282 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 282 conjugacy classes of subgroups — too many to draw. Only its 64 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
2C4⋊(#41⋊C3)C211yesmaximal, characteristic, complemented
3#548⋊C3C212yesmaximal, complemented
4C4×#41⋊C3C212yesmaximal, complemented
5C2×C4×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
6#3480C311yesmaximal, characteristic, complemented, Sylow
7C2×#41⋊C3C2212yes—
8C22×SL(2,𝔽3)C2211yescharacteristic
9C3×#117—41nomaximal
10C4×SL(2,𝔽3)C2211yescharacteristic
11C4×SL(2,𝔽3)C2211yescharacteristic
12C4×SL(2,𝔽3)C2212yes—
13#559—31no—
14#557C2×C311yescharacteristic, complemented
15#556—31no—
16#553—32no—
17#550C2×C311yescharacteristic, complemented
18#548C2×C312yes—
19#532—31no—
20#529—32no—
21C4×#41—31no—
22C4×#41C2×C312yescomplemented
23C2×C4×Q8C2×C311yescharacteristic
24C22×C3×C4—41no—
25#41⋊C3—22no—
26C2×SL(2,𝔽3)—21no—
27C2×SL(2,𝔽3)C2311yescharacteristic
28C3×C4⋊C4—41no—
29C3×C4⋊C4—41no—
30C3×C22⋊C4—42no—
31C3×C42—42no—
32C2×#41—31no—
33C2×#41C22×C312yes—
34C22×Q8C22×C311yescharacteristic
35C4⋊Q8—31no—
36#127—31no—
37#126—34no—
38#125—32no—
39#124—32no—
40#124—32no—
41#123—34no—
42C22⋊Q8—32no—
43#121—32no—
44#121—32no—
45#121—32no—
46C4×Q8—31no—
47C4×Q8—31no—
48C4×Q8—31no—
49C4×Q8—31no—
50C4×Q8C22×C311yescharacteristic
51C4×Q8C22×C311yescharacteristic
52C4×Q8C22×C312yes—
53C4×D4—32no—
54C4×D4—32no—
55C4×D4—32no—
56C4×D4—31no—
57C4×D4—31no—
58C4×D4—32no—
59C4×D4—32no—
60#117—32no—
61#117—32no—
62#117A411yescharacteristic
63C2×C4⋊C4—31no—
64C2×C42—31no—
65C2×C42—32no—
66C23×C3—41no—
67C2×C3×C4—44no—
68C2×C3×C4—41no—
69C2×C3×C4—42no—
70C2×C3×C4—42no—
71C2×C3×C4—41no—
72SL(2,𝔽3)#4111yescharacteristic, complemented
73#41—22no—
74#41—64no—
75#41—62no—
76C2×Q8—21no—
77C2×Q8—31no—
78C2×Q8—31no—
79C2×Q8—61no—
80C2×Q8C23×C311yescharacteristic
81C2×D4—32no—
82C2×D4—32no—
83C2×D4—32no—
84C2×D4—31no—
85C22×C4—31no—
86C22×C4—31no—
87C22×C4—32no—
88C22×C4—32no—
89C22×C4—32no—
90C22×C4C2×A411yescharacteristic
91C4⋊C4—31no—
92C4⋊C4—31no—
93C4⋊C4—32no—
94C4⋊C4—32no—
95C4⋊C4—32no—
96C4⋊C4—32no—
97C4⋊C4—32no—
98C4⋊C4C2×A411yescharacteristic
99C4⋊C4C2×A411yescharacteristic
100C22⋊C4—34no—
101C22⋊C4—34no—
102C22⋊C4—32no—
103C22⋊C4—32no—
104C22⋊C4C2×A412yes—
105C42—32no—
106C42—32no—
107C42—31no—
108C42—31no—
109C42—31no—
110C42—31no—
111C42C2×A412yes—
112C22×C3—42no—
113C22×C3—41no—
114C22×C3—81no—
115C22×C3—81no—
116C3×C4—42no—
117C3×C4—42no—
118C3×C4—82no—
119C3×C4—82no—
120C23—32no—
121C23C22×A411yescharacteristic
122Q8—31no—
123Q8—61no—
124Q8—61no—
125Q8—62no—
126Q8C3×#4111yescharacteristic
127D4—62no—
128D4—62no—
129D4—64no—
130D4—62no—
131D4—122no—
132C2×C4—31no—
133C2×C4—31no—
134C2×C4—31no—
135C2×C4—32no—
136C2×C4—31no—
137C2×C4—62no—
138C2×C4—62no—
139C2×C4—61no—
140C2×C4—61no—
141C2×C4—64no—
142C2×C4—122no—
143C2×C4—121no—
144C2×C4#41⋊C314yescomplemented
145C2×C4C22×A412yes—
146C2×C4C22×A412yes—
147C2×C4C22×A411yescharacteristic
148C2×C4C22×A411yescharacteristic
149C2×C3—41no—
150C2×C3—41no—
151C2×C3—41no—
152C2×C3—82no—
153C22—21no—
154C22—21no—
155C22—62no—
156C22—62no—
157C22—62no—
158C22C2×#41⋊C312yes—
159C22C23×A411yescharacteristic
160C4—22no—
161C4—22no—
162C4—31no—
163C4—31no—
164C4—62no—
165C4—61no—
166C4—61no—
167C4—61no—
168C4C2×#41⋊C312yes—
169C4C2×#41⋊C312yes—
170C3—41noSylow
171C2—22no—
172C2—122no—
173C2C22×#41⋊C311yescharacteristic
174C2A4×#4111yescharacteristic
175C2#594511yescharacteristic
subgroupG/H
centerC2×C4C22×A4
commutator, cosocleC2×Q8C23×C3
Frattini, socleC22C23×A4
Fitting#3480C3
hypercenter#117A4
hypocenterQ8C3×#41
LCSG ▷ C2×Q8 ▷ Q8
UCSC1 ◁ C2×C4 ◁ #117

Complements

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

NH ≅ G/Ntype
1. #557⋊C3C2semidirect
2. C4⋊(#41⋊C3)C2semidirect
3. #548⋊C3C2semidirect
4. C4×#41⋊C3C2semidirect
5. C2×C4×SL(2,𝔽3)C2semidirect
6. #3480C3semidirect
14. #557C2×C3semidirect
17. #550C2×C3semidirect
22. C4×#41C2×C3semidirect
72. SL(2,𝔽3)#41semidirect
144. C2×C4#41⋊C3semidirect

Maximal subgroups

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

subgroupindex
#557⋊C32
C4⋊(#41⋊C3)2
#548⋊C32
C4×#41⋊C32
C2×C4×SL(2,𝔽3)2
#34803
C3×#1174