Finite Group Databasebeta

← #11723584

Subgroups — #11723584

Order 864.

432 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 432 conjugacy classes of subgroups — too many to draw. Only its 84 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
1C3×Q8×C3⋊D3C211yesmaximal, characteristic, complemented
2#89014C211yesmaximal, characteristic
3#88863C211yesmaximal, characteristic, complemented
4C32×#215C211yesmaximal, characteristic, complemented
5#88800C211yesmaximal, characteristic, complemented
6#88792C211yesmaximal, characteristic, complemented
7#88787C211yesmaximal, characteristic, complemented
8#64124—31nomaximal
9#63946—34nomaximal
10C33×Q8C2211yescharacteristic
11#6308C2211yescharacteristic
12C3×C4×C3⋊D3—21no—
13C3×C4×C3⋊D3C2211yescharacteristic
14C3×#634—21no—
15C3×#634C2211yescharacteristic
16C32×D12C2211yescharacteristic
17#6290—21no—
18D3×C3⋊Dic3—21no—
19C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
20C32×C3⋊C8C2211yescharacteristic
21Q8×C3⋊D3D311yescharacteristic, complemented
22#3830—31no—
23C3×D3×Q8—34no—
24#3795—34no—
25C32×SD16—31no—
26#3756—31no—
27#3756—34no—
28#3756—64no—
29#3754—31no—
30#3746—31no—
31#3743—31no—
32C3×#215D314yescomplemented
33#3719—34no—
34#3715—34no—
35#3711—34no—
36C2×C3×C3⋊D3D411yescharacteristic
37C2×C32×D3—21no—
38C33×C4—21no—
39C33×C4C2311yescharacteristic
40C3⋊(C3⋊Dic3)—21no—
41C3×C3⋊Dic3—21no—
42C3×C3⋊Dic3D411yescharacteristic
43#942—91no—
44#915—94no—
45C32×Q8—24no—
46C32×Q8C2×D314yes—
47C32×Q8C2×D311yescharacteristic
48C32×D4—31no—
49#638—61no—
50C2×C3⋊Dic3—61no—
51C4×C3⋊D3—21no—
52C4×C3⋊D3C2×D311yescharacteristic
53#634—21no—
54#634—31no—
55#634—34no—
56#634—64no—
57#634C2×D311yescharacteristic
58C3×D12C2×D314yes—
59C3×C4×D3—34no—
60C3×C4×D3—64no—
61C3×Dic6—34no—
62C3×Dic6—64no—
63#625—64no—
64D3×Dic3—64no—
65C32×C8—31no—
66C3⋊(C3⋊C8)—31no—
67C3⋊(C3⋊C8)—34no—
68C3⋊(C3⋊C8)—64no—
69C3×C3⋊C8C2×D314yes—
70C2×C33C2×D411yescharacteristic
71C3×C3⋊D3—21no—
72C32×D3—41no—
73C2×C3×Q8—91no—
74D3×Q8—34no—
75#237—94no—
76#235—91no—
77C3×SD16—34no—
78#216—94no—
79#216—91no—
80#216—91no—
81#216—184no—
82#215—91no—
83#215C3⋊D311yescharacteristic, complemented
84#214—94no—
85#208—91no—
86Dic12—94no—
87ΓL(1,𝔽25)—94no—
88C22×C32—61no—
89C2×C3⋊D3#6711yescharacteristic
90C2×C3×D3—24no—
91C2×C3×D3—34no—
92C32×C4—21no—
93C32×C4—24no—
94C32×C4—24no—
95C32×C4—44no—
96C32×C4C22×D314yes—
97C32×C4C22×D311yescharacteristic
98C3⋊Dic3—21no—
99C3⋊Dic3—61no—
100C3⋊Dic3—64no—
101C3⋊Dic3—124no—
102C3⋊Dic3#6711yescharacteristic
103C3×Dic3—34no—
104C3×Dic3—64no—
105#137—271noSylow
106C33#13711yescharacteristic, complemented, Sylow
107C3×Q8—24no—
108C3×Q8—91no—
109C3×Q8—181no—
110C3×Q8D3214yes—
111C3×Q8C2×C3⋊D311yescharacteristic
112C3×D4—34no—
113C2×C3×C4—91no—
114C2×C3×C4—181no—
115#67—181no—
116#67—184no—
117C2×Dic3—184no—
118D12C2×C3⋊D311yescharacteristic
119C4×D3—34no—
120C4×D3—64no—
121C4×D3—181no—
122Dic6—34no—
123Dic6—64no—
124Dic6—94no—
125Dic6—91no—
126Dic6—184no—
127C3×C8—34no—
128C3⋊C8—94no—
129C3⋊C8—91no—
130C3⋊C8—184no—
131C3⋊C8C2×C3⋊D311yescharacteristic
132C2×C32—24no—
133C2×C32—121no—
134C2×C32D3×D414yes—
135C2×C32C2×#6711yescharacteristic
136C3⋊D3—21no—
137C3×D3—44no—
138C3×D3—64no—
139#41—271no—
140C2×Q8—91no—
141Q16—271no—
142Q16—271no—
143SD16—31no—
144SD16—271no—
145M16—271no—
146C22×C3—64no—
147C22×C3—91no—
148C2×D3—21no—
149C2×D3—34no—
150C3×C4—24no—
151C3×C4—21no—
152C3×C4—24no—
153C3×C4—44no—
154C3×C4—91no—
155C3×C4—181no—
156C3×C4C2×D3214yes—
157C3×C4C22×C3⋊D311yescharacteristic
158Dic3—34no—
159Dic3—64no—
160Dic3—184no—
161Dic3—181no—
162Dic3—364no—
163C32—24no—
164C32#91514yescomplemented
165C32#94211yescharacteristic, complemented
166Q8—91no—
167Q8—181no—
168Q8—271no—
169Q8D3×C3⋊D311yescharacteristic
170D4—31no—
171D4—541no—
172C2×C4—91no—
173C2×C4—181no—
174C2×C4—541no—
175C8—31no—
176C8—271no—
177C2×C3—24no—
178C2×C3—124no—
179C2×C3—181no—
180C2×C3D3×#6714yes—
181C2×C3D4×C3⋊D311yescharacteristic
182D3—41no—
183D3—64no—
184C22—61no—
185C22—91no—
186C4—21no—
187C4—91no—
188C4—181no—
189C4—541no—
190C4C2×D3×C3⋊D311yescharacteristic
191C3—24no—
192C3#6394614yescomplemented
193C3#6412411yescharacteristic, complemented
194C2—121no—
195C2—181no—
196C2C3⋊D3×#6711yescharacteristic
subgroupG/H
centerC2C3⋊D3×#67
commutator, cosocleC33×C4C23
FrattiniC4C2×D3×C3⋊D3
FittingC33×Q8C22
socleC2×C33C2×D4
prime radical#89014C2
hypercenterQ8D3×C3⋊D3
hypocenterC33#137
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C4 ◁ Q8

Complements

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

NH ≅ G/Ntype
1. C3×Q8×C3⋊D3C2semidirect
3. #88863C2semidirect
4. C32×#215C2semidirect
5. #88800C2semidirect
6. #88792C2semidirect
7. #88787C2semidirect
21. Q8×C3⋊D3D3semidirect
32. C3×#215D3semidirect
83. #215C3⋊D3semidirect
106. C33#137semidirect
164. C32#915semidirect
165. C32#942semidirect
192. C3#63946semidirect
193. C3#64124semidirect

Maximal subgroups

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

subgroupindex
C3×Q8×C3⋊D32
#890142
#888632
C32×#2152
#888002
#887922
#887872
#641243
#639463