Finite Group Databasebeta

← #11723552

Subgroups — #11723552

Order 864.

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

normal subgroups only

This group has 440 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×#3830C211yesmaximal, characteristic
2C3⋊D3×Dic6C211yesmaximal, characteristic, complemented
3#88861C211yesmaximal, characteristic, complemented
4C32×#214C211yesmaximal, characteristic, complemented
5#88800C211yesmaximal, characteristic, complemented
6#88798C211yesmaximal, characteristic, complemented
7#88787C211yesmaximal, characteristic, complemented
8#64124—31nomaximal
9#63929—34nomaximal
10C33×D4C2211yescharacteristic
11#6308C2211yescharacteristic
12C3×#638—21no—
13C2×C3×C3⋊Dic3—21no—
14C3×C4×C3⋊D3C2211yescharacteristic
15C3×#634C2211yescharacteristic
16C32×Dic6C2211yescharacteristic
17#6293—21no—
18Dic3×C3⋊D3—21no—
19C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
20C32×C3⋊C8C2211yescharacteristic
21Q8×C3⋊D3—31no—
22#3830D311yescharacteristic
23C3×#237—34no—
24D3×Dic6—34no—
25C32×SD16—31no—
26#3756—31no—
27#3754—31no—
28#3754—34no—
29#3754—64no—
30#3746—31no—
31#3743—31no—
32C3×#214D314yescomplemented
33#3719—34no—
34#3718—34no—
35#3711—34no—
36C22×C33—21no—
37C2×C3×C3⋊D3D411yescharacteristic
38C33×C4C2311yescharacteristic
39C3⋊(C3⋊Dic3)—21no—
40C3×C3⋊Dic3—21no—
41C3×C3⋊Dic3D411yescharacteristic
42C32×Dic3—21no—
43#951—91no—
44#915—94no—
45C32×Q8—31no—
46C32×D4—24no—
47C32×D4C2×D314yes—
48C32×D4C2×D311yescharacteristic
49#638—21no—
50C2×C3⋊Dic3—21no—
51C4×C3⋊D3—61no—
52C4×C3⋊D3C2×D311yescharacteristic
53#634—31no—
54#634—34no—
55#634—64no—
56#634—61no—
57#634C2×D311yescharacteristic
58C3×#67—64no—
59C2×C3×Dic3—64no—
60C3×C4×D3—34no—
61C3×Dic6—34no—
62C3×Dic6C2×D314yes—
63#627—64no—
64D3×Dic3—64no—
65C32×C8—31no—
66C3⋊(C3⋊C8)—31no—
67C3⋊(C3⋊C8)—34no—
68C3⋊(C3⋊C8)—64no—
69C3×C3⋊C8C2×D314yes—
70C2×C33—41no—
71C2×C33C2×D411yescharacteristic
72C3×C3⋊D3—21no—
73C3×#41—91no—
74D3×Q8—94no—
75#237—34no—
76C2×Dic6—91no—
77C3×SD16—34no—
78#216—94no—
79#216—91no—
80#216—91no—
81#214—94no—
82#214—91no—
83#214—184no—
84#214C3⋊D311yescharacteristic, complemented
85#208—91no—
86Dic12—94no—
87ΓL(1,𝔽25)—94no—
88C22×C32—21no—
89C22×C32—24no—
90C22×C32—44no—
91C2×C3⋊D3#6711yescharacteristic
92C2×C3×D3—34no—
93C32×C4—24no—
94C32×C4—61no—
95C32×C4C22×D314yes—
96C32×C4C22×D311yescharacteristic
97C3⋊Dic3—21no—
98C3⋊Dic3—61no—
99C3⋊Dic3—64no—
100C3⋊Dic3—124no—
101C3⋊Dic3#6711yescharacteristic
102C3×Dic3—24no—
103C3×Dic3—34no—
104C3×Dic3—64no—
105#137—271noSylow
106C33#13711yescharacteristic, complemented, Sylow
107C3×Q8—34no—
108C3×Q8—91no—
109C3×D4—24no—
110C3×D4—181no—
111C3×D4D3214yes—
112C3×D4C2×C3⋊D311yescharacteristic
113C2×C3×C4—91no—
114C2×C3×C4—181no—
115#67—64no—
116C2×Dic3—64no—
117C2×Dic3—181no—
118C4×D3—34no—
119C4×D3—184no—
120Dic6—34no—
121Dic6—94no—
122Dic6—91no—
123Dic6—184no—
124Dic6—181no—
125Dic6—184no—
126Dic6C2×C3⋊D311yescharacteristic
127C3×C8—34no—
128C3⋊C8—94no—
129C3⋊C8—91no—
130C3⋊C8—184no—
131C3⋊C8C2×C3⋊D311yescharacteristic
132C2×C32—24no—
133C2×C32—41no—
134C2×C32—44no—
135C2×C32—84no—
136C2×C32D3×D414yes—
137C2×C32C2×#6711yescharacteristic
138C3⋊D3—21no—
139C3×D3—64no—
140#41—91no—
141C2×Q8—271no—
142Q16—271no—
143Q16—271no—
144SD16—31no—
145SD16—271no—
146M16—271no—
147C22×C3—24no—
148C22×C3—21no—
149C22×C3—44no—
150C22×C3—91no—
151C2×D3—34no—
152C3×C4—24no—
153C3×C4—64no—
154C3×C4—91no—
155C3×C4—181no—
156C3×C4C2×D3214yes—
157C3×C4C22×C3⋊D311yescharacteristic
158Dic3—21no—
159Dic3—34no—
160Dic3—64no—
161Dic3—184no—
162Dic3—181no—
163Dic3—364no—
164C32—24no—
165C32#91514yescomplemented
166C32#95111yescharacteristic, complemented
167Q8—31no—
168Q8—91no—
169Q8—271no—
170Q8—541no—
171D4—181no—
172D4D3×C3⋊D311yescharacteristic
173C2×C4—91no—
174C2×C4—181no—
175C2×C4—541no—
176C8—31no—
177C8—271no—
178C2×C3—24no—
179C2×C3—41no—
180C2×C3—44no—
181C2×C3—84no—
182C2×C3—181no—
183C2×C3D3×#6714yes—
184C2×C3D4×C3⋊D311yescharacteristic
185D3—64no—
186C22—21no—
187C22—91no—
188C4—61no—
189C4—91no—
190C4—181no—
191C4—541no—
192C4C2×D3×C3⋊D311yescharacteristic
193C3—24no—
194C3#6392914yescomplemented
195C3#6412411yescharacteristic, complemented
196C2—41no—
197C2—181no—
198C2C3⋊D3×#6711yescharacteristic
subgroupG/H
centerC2C3⋊D3×#67
commutator, cosocleC33×C4C23
FrattiniC4C2×D3×C3⋊D3
FittingC33×D4C22
socleC2×C33C2×D4
prime radicalC3×#3830C2
hypercenterD4D3×C3⋊D3
hypocenterC33#137
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C4 ◁ D4

Complements

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

NH ≅ G/Ntype
2. C3⋊D3×Dic6C2semidirect
3. #88861C2semidirect
4. C32×#214C2semidirect
5. #88800C2semidirect
6. #88798C2semidirect
7. #88787C2semidirect
32. C3×#214D3semidirect
84. #214C3⋊D3semidirect
106. C33#137semidirect
165. C32#915semidirect
166. C32#951semidirect
194. C3#63929semidirect
195. C3#64124semidirect

Maximal subgroups

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

subgroupindex
C3×#38302
C3⋊D3×Dic62
#888612
C32×#2142
#888002
#887982
#887872
#641243
#639293