Finite Group Databasebeta

← #13166271

Subgroups — #13166271

Order 1728.

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

normal subgroups only

This group has 380 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#11724252C212yesmaximal, complemented
2#11723341C211yesmaximal, characteristic
3#11723096C211yesmaximal, characteristic, complemented
4C3×C8×#157C211yesmaximal, characteristic, complemented
5#11722713C212yesmaximal, complemented
6#10593787—31nomaximal
7#89018C2212yes—
8#88989C2212yes—
9C3×C4×#157C2211yescharacteristic
10C3×#3787C2211yescharacteristic
11#88926—22no—
12#88924—22no—
13#88852C411yescharacteristic, complemented
14C3×C8×C3⋊D3C2211yescharacteristic
15C32×Dic12C411yescharacteristic, complemented
16#88795C412yescomplemented
17#64289—32no—
18#64128—31no—
19#63808—62no—
20#63785—32no—
21#63771—31no—
22C8×#157D311yescharacteristic, complemented
23C2×C3×#157—21no—
24C4⋊(C3⋊(C3⋊D3))C2×C412yes—
25C3×C4×C3⋊D3C2311yescharacteristic
26C32×Dic6C2×C412yes—
27#6292—22no—
28#6289—22no—
29C3⋊#622—22no—
30C3×#622D411yescharacteristic
31C3×#622D411yescharacteristic
32C33×C8C2×C411yescharacteristic
33C3×C3⋊(C3⋊C8)C2×C411yescharacteristic
34#4700—91nomaximal
35#3832—32no—
36#3799—64no—
37C4×#157C2×D311yescharacteristic
38#3788—32no—
39#3788—62no—
40#3787—62no—
41#3787C2×D311yescharacteristic
42C32×Q16—31no—
43#3755—32no—
44#3745—31no—
45#3745—62no—
46#3745—122no—
47C8×C3⋊D3C2×D311yescharacteristic
48C3×Dic12—22no—
49C3×D3×C8—62no—
50#3717—64no—
51C2×C3⋊(C3⋊D3)—22no—
52C2×C3×C3⋊D3#4111yescharacteristic
53C3×#157—22no—
54C33×C4C22×C411yescharacteristic
55C3×C3⋊Dic3C2×D411yescharacteristic
56C32×Dic3—22no—
57#919—182no—
58#904—91no—
59#901—92no—
60C3×C4×C8—91no—
61#819—91no—
62#805—92no—
63C2×#157—21no—
64C32×Q8—32no—
65C4⋊(C3⋊D3)—32no—
66C4⋊(C3⋊D3)—64no—
67C4⋊(C3⋊D3)—62no—
68C4⋊(C3⋊D3)—124no—
69C4×C3⋊D3—62no—
70C4×C3⋊D3C22×D311yescharacteristic
71C3×C4×D3—62no—
72C3×Dic6—24no—
73#626—124no—
74#624—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)—42no—
86C3×C3⋊D3—21no—
87#239—184no—
88#235—92no—
89C3×Q16—62no—
90C2×C3×C8—91no—
91C2×C3×C8—91no—
92C3×C42—91no—
93#215—184no—
94#208—92no—
95Dic12#15711yescharacteristic, complemented
96D24—91no—
97D24—182no—
98D24—362no—
99#204—92no—
100ΓL(1,𝔽25)—182no—
101D3×C8—62no—
102D3×C8—182no—
103C2×C3⋊D3—62no—
104C2×C3⋊D3—124no—
105C2×C3⋊D3—244no—
106C2×C3⋊D3#23511yescharacteristic
107C2×C3×D3—62no—
108#157—22no—
109C32×C4—22no—
110C32×C4—42no—
111C32×C4—62no—
112C32×C4C2×C4×D311yescharacteristic
113C3⋊Dic3C2×D1211yescharacteristic
114C3×Dic3—44no—
115C3×Dic3—62no—
116#135—271no—
117#131—272no—
118#108—271no—
119#104—272no—
120C4×C8—91no—
121C33#44311yescharacteristic, complemented, Sylow
122C3×Q8—64no—
123C2×C3×C4—91no—
124C2×C3×C4—181no—
125#67—182no—
126D12—92no—
127D12—184no—
128D12—364no—
129D12—364no—
130C4×D3—62no—
131C4×D3—182no—
132C4×D3—364no—
133Dic6C2×#15712yes—
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×C32C4×D1211yescharacteristic
145C3⋊D3—21no—
146C3⋊D3—122no—
147C3⋊D3—244no—
148C3⋊D3—484no—
149C3×D3—122no—
150#41—272no—
151Q16—31no—
152SD16—272no—
153D8—271no—
154M16—272no—
155M16—542no—
156C2×C8—91no—
157C2×C8—91no—
158C2×C8—542no—
159C42—91no—
160C22×C3—91no—
161C2×D3—62no—
162C2×D3—182no—
163C2×D3—364no—
164C2×D3—724no—
165C3×C4—22no—
166C3×C4—42no—
167C3×C4—91no—
168C3×C4—124no—
169C3×C4—182no—
170C3×C4C22×#15711yescharacteristic
171Dic3—22no—
172Dic3—62no—
173C32—22no—
174C32—42no—
175C32#470011yescharacteristic, complemented
176Q8—32no—
177D4—272no—
178D4—542no—
179C2×C4—91no—
180C2×C4—181no—
181C2×C4—542no—
182C8—91no—
183C8—91no—
184C8—91no—
185C8—542no—
186C8D3×#15711yescharacteristic
187C2×C3—22no—
188C2×C3—42no—
189C2×C3—181no—
190C2×C3D4×#15711yescharacteristic
191D3—122no—
192D3—362no—
193D3—724no—
194D3—1444no—
195C22—91no—
196C22—542no—
197C4—62no—
198C4—91no—
199C4—182no—
200C4C2×D3×#15711yescharacteristic
201C3—22no—
202C3—42no—
203C3#1059378711yescharacteristic, complemented
204C2—181no—
205C2—1082no—
206C2D12×#15711yescharacteristic
subgroupG/H
centerC2D12×#157
commutatorC33×C4C22×C4
FrattiniC4C2×D3×#157
FittingC33×C8C2×C4
socleC2×C33C4×D4
prime radical#11723341C2
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. #11724252C2semidirect
3. #11723096C2semidirect
4. C3×C8×#157C2semidirect
5. #11722713C2semidirect
13. #88852C4semidirect
15. C32×Dic12C4semidirect
16. #88795C4semidirect
22. C8×#157D3semidirect
95. Dic12#157semidirect
121. C33#443semidirect
175. C32#4700semidirect
203. C3#10593787semidirect

Maximal subgroups

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

subgroupindex
#117242522
#117233412
#117230962
C3×C8×#1572
#117227132
#105937873
#47009