Finite Group Databasebeta

← #13166405

Subgroups — #13166405

Order 1728.

372 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 372 conjugacy classes of subgroups — too many to draw. Only its 66 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#11723357C211yesmaximal, characteristic
2#11723098C211yesmaximal, characteristic, complemented
3C2×C3×#3708C211yesmaximal, characteristic, complemented
4#11722721C212yesmaximal, complemented
5#11722719C212yesmaximal, complemented
6#10593772—31nomaximal
7#89076C411yescharacteristic
8C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
9C32×#208C411yescharacteristic
10#88788C412yes—
11#88785C2212yes—
12C3⋊#3708C2212yes—
13C3×#3708C2211yescharacteristic
14C3×#3708C2211yescharacteristic
15#64116—31no—
16#63817—62no—
17#63773—31no—
18C2×#3708D311yescharacteristic, complemented
19#63765—32no—
20#63764—32no—
21C2×C33×C4C2×C411yescharacteristic
22#6312C812yes—
23C4⋊(C3⋊(C3⋊D3))C811yescharacteristic
24C4×C3⋊(C3⋊D3)C2×C412yes—
25#6308C811yescharacteristic
26C3×C3⋊(C3⋊C8)C2×C411yescharacteristic
27C3×C3⋊(C3⋊C8)C2311yescharacteristic
28C32×C3⋊C8C2×C412yes—
29#4901—91nomaximal
30#3828—31no—
31#3828—62no—
32#3828—122no—
33C32×M16—31no—
34C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
35#3743—32no—
36C8×C3⋊D3—32no—
37C3×#208—22no—
38C2×C3×C3⋊C8—62no—
39#3712—64no—
40#3710—64no—
41#3708—32no—
42#3708C2×D311yescharacteristic
43#3708C2×D311yescharacteristic
44C22×C33C2×C811yescharacteristic
45C2×C3⋊(C3⋊D3)C2×C812yes—
46C33×C4C2×C811yescharacteristic
47C33×C4C22×C411yescharacteristic
48C3⋊(C3⋊Dic3)C2×C812yes—
49#907—182no—
50#901—91no—
51C2×C3×C16—91no—
52#812—91no—
53#798—92no—
54D3×C16—92no—
55C2×C32×C4—22no—
56C2×C32×C4—42no—
57C2×C32×C4C4×D311yescharacteristic
58#638—32no—
59#638—64no—
60#638—124no—
61C4⋊(C3⋊D3)—31no—
62C4⋊(C3⋊D3)—62no—
63C4⋊(C3⋊D3)—122no—
64C4×C3⋊D3—32no—
65C4×C3⋊D3—64no—
66C4×C3⋊D3—124no—
67#634—31no—
68#634—62no—
69#634—122no—
70C32×C8—32no—
71C3⋊(C3⋊C8)C4×D311yescharacteristic
72C3⋊(C3⋊C8)C22×D311yescharacteristic
73C3×C3⋊C8—24no—
74C3×C3⋊C8—62no—
75C3×C3⋊C8—62no—
76#504—271noSylow
77C2×C33—21no—
78C2×C33C22×C811yescharacteristic
79C3⋊(C3⋊D3)—22no—
80#235—91no—
81#235—182no—
82#235—362no—
83C3×M16—62no—
84C2×C3×C8—91no—
85#208#15711yescharacteristic
86C2×C3⋊C8—62no—
87ΓL(1,𝔽25)—92no—
88ΓL(1,𝔽25)—184no—
89D3×C8—92no—
90D3×C8—184no—
91C3×C16—91no—
92C3×C16—91no—
93C3⋊C16—92no—
94C22×C32—22no—
95C22×C32—42no—
96C22×C32D3×C811yescharacteristic
97C2×C3⋊D3—32no—
98C2×C3⋊D3—64no—
99C2×C3⋊D3—124no—
100C32×C4—22no—
101C32×C4—22no—
102C32×C4—42no—
103C32×C4—42no—
104C32×C4D3×C811yescharacteristic
105C32×C4C2×C4×D311yescharacteristic
106C3⋊Dic3—32no—
107C3⋊Dic3—64no—
108C3⋊Dic3—124no—
109#131—271no—
110M32—272no—
111M32—271no—
112C2×C16—91no—
113C2×C16—272no—
114C33#50411yescharacteristic, complemented, Sylow
115C2×C3×C4—22no—
116C2×C3×C4—42no—
117C2×C3×C4C2×#15711yescharacteristic
118#67—92no—
119#67—184no—
120#67—364no—
121D12—91no—
122D12—182no—
123D12—362no—
124C4×D3—92no—
125C4×D3—184no—
126C4×D3—364no—
127Dic6—91no—
128Dic6—182no—
129Dic6—362no—
130C3×C8—64no—
131C3×C8—91no—
132C3×C8—91no—
133C3⋊C8—62no—
134C3⋊C8—62no—
135C3⋊C8C2×#15712yes—
136C2×C32—22no—
137C2×C32—21no—
138C2×C32—42no—
139C2×C32—42no—
140C2×C32—82no—
141C2×C32C2×D3×C811yescharacteristic
142C3⋊D3—62no—
143C3⋊D3—124no—
144C3⋊D3—244no—
145#41—271no—
146M16—31no—
147M16—272no—
148C2×C8—91no—
149C2×C8—272no—
150C16—91no—
151C16—91no—
152C16—272no—
153C22×C3—22no—
154C22×C3—42no—
155C22×C3#378711yescharacteristic
156C2×D3—92no—
157C2×D3—184no—
158C2×D3—364no—
159C3×C4—22no—
160C3×C4—22no—
161C3×C4—42no—
162C3×C4—42no—
163C3×C4#378711yescharacteristic
164C3×C4C22×#15711yescharacteristic
165Dic3—92no—
166Dic3—184no—
167Dic3—364no—
168C32—22no—
169C32—42no—
170C32#490111yescharacteristic, complemented
171Q8—271no—
172D4—272no—
173D4—271no—
174C2×C4—272no—
175C2×C4D3×#15711yescharacteristic
176C8—32no—
177C8—91no—
178C8—91no—
179C2×C3—21no—
180C2×C3—22no—
181C2×C3—42no—
182C2×C3—42no—
183C2×C3—82no—
184C2×C3C2×#378711yescharacteristic
185D3—182no—
186D3—364no—
187D3—724no—
188C22—272no—
189C22#8892411yescharacteristic
190C4—272no—
191C4#8892411yescharacteristic
192C4C2×D3×#15711yescharacteristic
193C3—22no—
194C3—42no—
195C3#1059377211yescharacteristic, complemented
196C2—21no—
197C2—542no—
198C2C2×#8892411yescharacteristic
subgroupG/H
center, FrattiniC4C2×D3×#157
commutator, socleC2×C33C22×C8
FittingC2×C33×C4C2×C4
prime radical#89076C4
cosocleC3×C3⋊(C3⋊C8)C23
hypercenterC2×C4D3×#157
hypocenterC33#504
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C4 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. #11723098C2semidirect
3. C2×C3×#3708C2semidirect
4. #11722721C2semidirect
5. #11722719C2semidirect
18. C2×#3708D3semidirect
114. C33#504semidirect
170. C32#4901semidirect
195. C3#10593772semidirect

Maximal subgroups

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

subgroupindex
#117233572
#117230982
C2×C3×#37082
#117227212
#117227192
#105937723
#49019