Finite Group Databasebeta

← #3517⋊C3 #85617

Subgroups — #3517⋊C3 #85617

Order 384.

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

normal subgroups only

This group has 267 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
1#557⋊C3C211yesmaximal, characteristic, complemented
2C4⋊(#41⋊C3)C211yesmaximal, characteristic, complemented
3C4⋊(#41⋊C3)C212yesmaximal, complemented
4#520⋊C3C212yesmaximal, complemented
5C4×#41⋊C3C211yesmaximal, characteristic, complemented
6#3517C311yesmaximal, characteristic, complemented, Sylow
7C2×#41⋊C3C2211yescharacteristic
8C2×#41⋊C3C2212yes—
9C3×#125—41nomaximal
10C4×SL(2,𝔽3)C2211yescharacteristic
11C4×SL(2,𝔽3)C2211yescharacteristic
12C4×SL(2,𝔽3)C2212yes—
13#557C2×C311yescharacteristic
14#553—32no—
15#550—31no—
16#550C2×C311yescharacteristic
17#550C2×C312yes—
18#547—31no—
19#540—32no—
20#532—31no—
21#529—31no—
22#520C2×C312yes—
23C4×#41C2×C311yescharacteristic
24#41⋊C3—21no—
25#41⋊C3—22no—
26C2×SL(2,𝔽3)C2311yescharacteristic
27C3×C4⋊C4—42no—
28C3×C4⋊C4—41no—
29C3×C4⋊C4—42no—
30C3×C4⋊C4—41no—
31C3×C42—41no—
32C2×#41—32no—
33C2×#41C22×C311yescharacteristic
34C2×#41C22×C312yes—
35C4⋊Q8—31no—
36#127—31no—
37#127—32no—
38#126—32no—
39#125—32no—
40#125A411yescharacteristic
41#124—31no—
42#123—34no—
43#123—32no—
44C22⋊Q8—31no—
45#121—32no—
46#121—34no—
47#121—31no—
48#121—32no—
49#121—32no—
50#121—32no—
51C4×Q8—31no—
52C4×Q8C22×C311yescharacteristic
53C4×Q8C22×C311yescharacteristic
54C4×Q8C22×C312yes—
55C4×D4—32no—
56C4×D4—31no—
57C4×D4—31no—
58C4×D4—31no—
59C4×D4—32no—
60C4×D4—32no—
61C4×D4—32no—
62#117—32no—
63#117—31no—
64C2×C4⋊C4—32no—
65C2×C42—31no—
66C2×C3×C4—41no—
67C2×C3×C4—41no—
68C2×C3×C4—42no—
69C2×C3×C4—41no—
70C2×C3×C4—42no—
71SL(2,𝔽3)#4111yescharacteristic, complemented
72#41—21no—
73#41—22no—
74#41—61no—
75#41—62no—
76#41—124no—
77C2×Q8—31no—
78C2×Q8C23×C311yescharacteristic
79C2×D4—31no—
80C2×D4—32no—
81C2×D4—32no—
82C2×D4—34no—
83C2×D4—31no—
84C2×D4—32no—
85C22×C4—34no—
86C22×C4—31no—
87C22×C4—31no—
88C22×C4—32no—
89C22×C4—31no—
90C4⋊C4—31no—
91C4⋊C4—32no—
92C4⋊C4—31no—
93C4⋊C4—32no—
94C4⋊C4—32no—
95C4⋊C4—32no—
96C4⋊C4C2×A412yes—
97C4⋊C4C2×A411yescharacteristic
98C4⋊C4C2×A412yes—
99C4⋊C4C2×A411yescharacteristic
100C22⋊C4—32no—
101C22⋊C4—32no—
102C22⋊C4—32no—
103C22⋊C4—32no—
104C22⋊C4—32no—
105C22⋊C4—32no—
106C42—31no—
107C42—31no—
108C42—31no—
109C42—32no—
110C42C2×A411yescharacteristic
111C22×C3—41no—
112C3×C4—42no—
113C3×C4—81no—
114C3×C4—82no—
115C3×C4—81no—
116C3×C4—82no—
117C23—31no—
118C23—32no—
119Q8—31no—
120Q8—62no—
121Q8C3×#4111yescharacteristic
122D4—61no—
123D4—62no—
124D4—61no—
125D4—62no—
126D4—62no—
127D4—122no—
128D4—122no—
129D4—124no—
130C2×C4—31no—
131C2×C4—31no—
132C2×C4—31no—
133C2×C4—32no—
134C2×C4—61no—
135C2×C4—62no—
136C2×C4—61no—
137C2×C4—62no—
138C2×C4—62no—
139C2×C4—121no—
140C2×C4—124no—
141C2×C4C22×A411yescharacteristic
142C2×C4C22×A411yescharacteristic
143C2×C4C22×A412yes—
144C2×C4C22×A412yes—
145C2×C4C22×A411yescharacteristic
146C2×C3—41no—
147C2×C3—41no—
148C2×C3—41no—
149C22—61no—
150C22—62no—
151C22—61no—
152C22—62no—
153C22—61no—
154C22—62no—
155C22C23×A411yescharacteristic
156C4—21no—
157C4—22no—
158C4—21no—
159C4—22no—
160C4—31no—
161C4—31no—
162C4—61no—
163C4—62no—
164C4—61no—
165C4#99412yes—
166C3—41noSylow
167C2—121no—
168C2—122no—
169C2A4×#4111yescharacteristic
170C2C2×#99411yescharacteristic
171C2#594511yescharacteristic
subgroupG/H
center, Frattini, socleC22C23×A4
commutator, cosocleC2×Q8C23×C3
Fitting#3517C3
hypercenter#125A4
hypocenterQ8C3×#41
LCSG ▷ C2×Q8 ▷ Q8
UCSC1 ◁ C22 ◁ #125

Complements

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

NH ≅ G/Ntype
1. #557⋊C3C2semidirect
2. C4⋊(#41⋊C3)C2semidirect
3. C4⋊(#41⋊C3)C2semidirect
4. #520⋊C3C2semidirect
5. C4×#41⋊C3C2semidirect
6. #3517C3semidirect
71. SL(2,𝔽3)#41semidirect

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
#520⋊C32
C4×#41⋊C32
#35173
C3×#1254