Finite Group Databasebeta

← #11723579

Subgroups — #11723579

Order 864.

432 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 432 conjugacy classes of subgroups — too many to draw. Only its 92 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
1C32×D3×Q8C211yesmaximal, characteristic, complemented
2#89021C211yesmaximal, characteristic
3#88863C211yesmaximal, characteristic, complemented
4C3×#3755C211yesmaximal, characteristic, complemented
5#88799C211yesmaximal, characteristic, complemented
6#88793C211yesmaximal, characteristic, complemented
7#88786C211yesmaximal, characteristic, complemented
8#64151—31nomaximal
9#63946—34nomaximal
10C33×Q8C2211yescharacteristic
11#6308C2211yescharacteristic
12C3×C4⋊(C3⋊D3)C2211yescharacteristic
13C32×C4×D3—21no—
14C32×C4×D3C2211yescharacteristic
15C32×Dic6—21no—
16C32×Dic6C2211yescharacteristic
17#6290—21no—
18Dic3×C3⋊D3—21no—
19C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
20C3×C3⋊(C3⋊C8)C2211yescharacteristic
21C2×C32×Q8—31no—
22#3828—31no—
23C3×D3×Q8D314yescomplemented
24#3795—34no—
25#3756—31no—
26#3756—34no—
27#3756—31no—
28#3756—64no—
29#3755—31no—
30#3755D311yescharacteristic, complemented
31#3748—31no—
32C3×#215—34no—
33#3719—34no—
34#3715—34no—
35#3711—34no—
36C2×C3×C3⋊D3—21no—
37C2×C32×D3D411yescharacteristic
38C33×C4—21no—
39C33×C4C2311yescharacteristic
40C3⋊(C3⋊Dic3)—21no—
41C32×Dic3—21no—
42C32×Dic3D411yescharacteristic
43#942—94no—
44#915—91no—
45C32×Q8—24no—
46C32×Q8—31no—
47C32×Q8—61no—
48C32×Q8C2×D314yes—
49C32×Q8C2×D311yescharacteristic
50C2×C32×C4—31no—
51C2×C32×C4—61no—
52#638—61no—
53C4⋊(C3⋊D3)C2×D311yescharacteristic
54C4×C3⋊D3—61no—
55#634—31no—
56#634—34no—
57#634—64no—
58C3×D12—34no—
59C3×C4×D3—24no—
60C3×C4×D3C2×D314yes—
61C3×Dic6—24no—
62C3×Dic6C2×D314yes—
63#625—64no—
64D3×Dic3—64no—
65C3⋊(C3⋊C8)—31no—
66C3⋊(C3⋊C8)—34no—
67C3⋊(C3⋊C8)—64no—
68C3⋊(C3⋊C8)C2×D311yescharacteristic
69C3×C3⋊C8—34no—
70C2×C33C2×D411yescharacteristic
71C3×C3⋊D3—41no—
72C32×D3—21no—
73C2×C3×Q8—34no—
74D3×Q8C3⋊D311yescharacteristic, complemented
75#237—91no—
76#235—94no—
77C3×SD16—91no—
78#216—94no—
79#216—94no—
80#216—91no—
81#216—184no—
82#215—34no—
83#215—94no—
84#214—91no—
85#208—94no—
86Dic12—91no—
87ΓL(1,𝔽25)—91no—
88C22×C32—31no—
89C2×C3⋊D3—21no—
90C2×C3×D3—64no—
91C2×C3×D3#6714yes—
92C32×C4—21no—
93C32×C4—24no—
94C32×C4—24no—
95C32×C4—31no—
96C32×C4—44no—
97C32×C4—61no—
98C32×C4C22×D314yes—
99C32×C4C22×D311yescharacteristic
100C3⋊Dic3—61no—
101C3⋊Dic3—64no—
102C3⋊Dic3—124no—
103C3×Dic3—24no—
104C3×Dic3#6714yes—
105#137—271noSylow
106C33#13711yescharacteristic, complemented, Sylow
107C3×Q8—24no—
108C3×Q8—34no—
109C3×Q8—64no—
110C3×Q8D3214yes—
111C3×Q8C2×C3⋊D311yescharacteristic
112C3×D4—91no—
113C2×C3×C4—34no—
114C2×C3×C4—64no—
115#67—184no—
116#67—181no—
117C2×Dic3—181no—
118D12—34no—
119C4×D3—21no—
120C4×D3—184no—
121C4×D3C2×C3⋊D311yescharacteristic
122Dic6—21no—
123Dic6—94no—
124Dic6—91no—
125Dic6—184no—
126Dic6C2×C3⋊D311yescharacteristic
127C3×C8—91no—
128C3⋊C8—34no—
129C3⋊C8—94no—
130C3⋊C8—91no—
131C3⋊C8—184no—
132C2×C32—24no—
133C2×C32—61no—
134C2×C32D3×D411yescharacteristic
135C2×C32C2×#6714yes—
136C3⋊D3—41no—
137C3×D3—24no—
138C3×D3—124no—
139#41—271no—
140C2×Q8—31no—
141Q16—271no—
142Q16—271no—
143SD16—91no—
144SD16—271no—
145M16—271no—
146C22×C3—34no—
147C22×C3—181no—
148C2×D3—64no—
149C2×D3#63811yescharacteristic
150C3×C4—24no—
151C3×C4—21no—
152C3×C4—24no—
153C3×C4—34no—
154C3×C4—44no—
155C3×C4—64no—
156C3×C4C2×D3214yes—
157C3×C4C22×C3⋊D311yescharacteristic
158Dic3—21no—
159Dic3—184no—
160Dic3—181no—
161Dic3—364no—
162Dic3#63811yescharacteristic
163C32—24no—
164C32#91511yescharacteristic, complemented
165C32#94214yescomplemented
166Q8—31no—
167Q8—61no—
168Q8—271no—
169Q8D3×C3⋊D311yescharacteristic
170D4—91no—
171D4—541no—
172C2×C4—31no—
173C2×C4—61no—
174C2×C4—541no—
175C8—91no—
176C8—271no—
177C2×C3—24no—
178C2×C3—64no—
179C2×C3—361no—
180C2×C3D3×#6714yes—
181C2×C3C2×#63811yescharacteristic
182D3—21no—
183D3—124no—
184C22—31no—
185C22—181no—
186C4—21no—
187C4—31no—
188C4—61no—
189C4—541no—
190C4C2×D3×C3⋊D311yescharacteristic
191C3—24no—
192C3#6394614yescomplemented
193C3#6415111yescharacteristic, complemented
194C2—61no—
195C2—361no—
196C2D3×#63811yescharacteristic
subgroupG/H
centerC2D3×#638
commutator, cosocleC33×C4C23
FrattiniC4C2×D3×C3⋊D3
FittingC33×Q8C22
socleC2×C33C2×D4
prime radical#89021C2
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. C32×D3×Q8C2semidirect
3. #88863C2semidirect
4. C3×#3755C2semidirect
5. #88799C2semidirect
6. #88793C2semidirect
7. #88786C2semidirect
23. C3×D3×Q8D3semidirect
30. #3755D3semidirect
74. D3×Q8C3⋊D3semidirect
106. C33#137semidirect
164. C32#915semidirect
165. C32#942semidirect
192. C3#63946semidirect
193. C3#64151semidirect

Maximal subgroups

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

subgroupindex
C32×D3×Q82
#890212
#888632
C3×#37552
#887992
#887932
#887862
#641513
#639463