Finite Group Databasebeta

← #13166268

Subgroups — #13166268

Order 1728.

372 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 372 conjugacy classes of subgroups — too many to draw. Only its 58 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#11724250C212yesmaximal, complemented
2#11723339C211yesmaximal, characteristic
3#11723096C211yesmaximal, characteristic, complemented
4C3×C8×#157C211yesmaximal, characteristic, complemented
5#11722711C212yesmaximal, complemented
6#10593779—31nomaximal
7#89014C2212yes—
8#88989C2212yes—
9C3×C4×#157C2211yescharacteristic
10C3×#3787C2211yescharacteristic
11#88925—22no—
12D3×#622—22no—
13#88853C411yescharacteristic, complemented
14C3×C8×C3⋊D3C2211yescharacteristic
15C32×D24C411yescharacteristic, complemented
16#88794C412yescomplemented
17#64287—32no—
18#64121—31no—
19#63807—62no—
20#63783—32no—
21#63771—31no—
22C8×#157D311yescharacteristic, complemented
23C2×C3×#157—21no—
24#6308C2×C412yes—
25C3×C4×C3⋊D3C2311yescharacteristic
26C32×D12C2×C412yes—
27#6290—22no—
28D3×C3⋊Dic3—22no—
29C3⋊#622—22no—
30C3×#622D411yescharacteristic
31C3×#622D411yescharacteristic
32C33×C8C2×C411yescharacteristic
33C3×C3⋊(C3⋊C8)C2×C411yescharacteristic
34#4700—91nomaximal
35#3830—32no—
36#3795—64no—
37#3792—62no—
38C2×#622—62no—
39C4×#157C2×D311yescharacteristic
40#3788—32no—
41#3787C2×D311yescharacteristic
42C32×D8—31no—
43#3754—32no—
44#3746—31no—
45#3746—62no—
46#3746—122no—
47C8×C3⋊D3C2×D311yescharacteristic
48C3×D24—22no—
49C3×D3×C8—62no—
50#3716—64no—
51C2×C3×C3⋊D3#4111yescharacteristic
52C2×C32×D3—22no—
53C3×#157—22no—
54C33×C4C22×C411yescharacteristic
55C3⋊(C3⋊Dic3)—22no—
56C3×C3⋊Dic3C2×D411yescharacteristic
57#912—182no—
58#904—91no—
59#901—92no—
60C3×C4×C8—91no—
61#819—91no—
62#805—92no—
63C2×#157—21no—
64C32×D4—32no—
65#638—62no—
66C2×C3⋊Dic3—62no—
67C4×C3⋊D3C22×D311yescharacteristic
68#634—32no—
69#634—64no—
70#634—124no—
71C3×D12—24no—
72C3×C4×D3—62no—
73#625—124no—
74D3×Dic3—124no—
75#622—62no—
76#622D1211yescharacteristic
77#622D1211yescharacteristic
78C32×C8—22no—
79C32×C8—42no—
80C32×C8C4×D311yescharacteristic
81C3⋊(C3⋊C8)C4×D311yescharacteristic
82C3×C3⋊C8—62no—
83#443—271noSylow
84C2×C33C4×D411yescharacteristic
85C3×C3⋊D3—21no—
86C32×D3—42no—
87#237—184no—
88#235—92no—
89C3×D8—62no—
90C2×C3×C8—91no—
91C2×C3×C8—91no—
92C3×C42—91no—
93#214—184no—
94#208—92no—
95Dic12—91no—
96Dic12—182no—
97Dic12—362no—
98D24#15711yescharacteristic, complemented
99#204—92no—
100ΓL(1,𝔽25)—182no—
101D3×C8—62no—
102D3×C8—182no—
103C22×C32—62no—
104C2×C3⋊D3#23511yescharacteristic
105C2×C3×D3—44no—
106C2×C3×D3—62no—
107#157—22no—
108C32×C4—22no—
109C32×C4—42no—
110C32×C4C2×C4×D311yescharacteristic
111C3⋊Dic3—62no—
112C3⋊Dic3—124no—
113C3⋊Dic3—244no—
114C3⋊Dic3C2×D1211yescharacteristic
115C3×Dic3—62no—
116#135—271no—
117#131—272no—
118#108—271no—
119#104—272no—
120C4×C8—91no—
121C33#44311yescharacteristic, complemented, Sylow
122C3×D4—64no—
123C2×C3×C4—91no—
124C2×C3×C4—181no—
125#67—182no—
126#67—364no—
127C2×Dic3—364no—
128D12C2×#15712yes—
129C4×D3—62no—
130C4×D3—182no—
131Dic6—92no—
132Dic6—184no—
133Dic6—364no—
134C3×C8—22no—
135C3×C8—42no—
136C3×C8—91no—
137C3×C8—91no—
138C3×C8—91no—
139C3×C8C2×#15711yescharacteristic
140C3⋊C8—62no—
141C3⋊C8—182no—
142C2×C32—22no—
143C2×C32—42no—
144C2×C32—122no—
145C2×C32C4×D1211yescharacteristic
146C3⋊D3—21no—
147C3×D3—84no—
148C3×D3—122no—
149#41—272no—
150Q16—271no—
151SD16—272no—
152D8—31no—
153M16—272no—
154M16—542no—
155C2×C8—91no—
156C2×C8—91no—
157C2×C8—542no—
158C42—91no—
159C22×C3—91no—
160C22×C3—124no—
161C2×D3—22no—
162C2×D3—62no—
163C3×C4—22no—
164C3×C4—42no—
165C3×C4—91no—
166C3×C4—182no—
167C3×C4C22×#15711yescharacteristic
168Dic3—62no—
169Dic3—182no—
170Dic3—364no—
171Dic3—724no—
172C32—22no—
173C32—42no—
174C32#470011yescharacteristic, complemented
175Q8—272no—
176D4—32no—
177D4—542no—
178C2×C4—91no—
179C2×C4—181no—
180C2×C4—542no—
181C8—91no—
182C8—91no—
183C8—91no—
184C8—542no—
185C8D3×#15711yescharacteristic
186C2×C3—22no—
187C2×C3—42no—
188C2×C3—181no—
189C2×C3—244no—
190C2×C3D4×#15711yescharacteristic
191D3—42no—
192D3—122no—
193C22—62no—
194C22—91no—
195C4—91no—
196C4—182no—
197C4—542no—
198C4C2×D3×#15711yescharacteristic
199C3—22no—
200C3—42no—
201C3#1059377911yescharacteristic, complemented
202C2—122no—
203C2—181no—
204C2D12×#15711yescharacteristic
subgroupG/H
centerC2D12×#157
commutatorC33×C4C22×C4
FrattiniC4C2×D3×#157
FittingC33×C8C2×C4
socleC2×C33C4×D4
prime radical#11723339C2
cosocleC3×C4×C3⋊D3C23
hypercenterC8D3×#157
hypocenterC33#443
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C4 ◁ C8

Complements

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

NH ≅ G/Ntype
1. #11724250C2semidirect
3. #11723096C2semidirect
4. C3×C8×#157C2semidirect
5. #11722711C2semidirect
13. #88853C4semidirect
15. C32×D24C4semidirect
16. #88794C4semidirect
22. C8×#157D3semidirect
98. D24#157semidirect
121. C33#443semidirect
174. C32#4700semidirect
201. C3#10593779semidirect

Maximal subgroups

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

subgroupindex
#117242502
#117233392
#117230962
C3×C8×#1572
#117227112
#105937793
#47009