Finite Group Databasebeta

← #13166401

Subgroups — #13166401

Order 1728.

332 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 74 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#11723349C211yesmaximal, characteristic
2#11723098C211yesmaximal, characteristic, complemented
3C2×C3×#3708C211yesmaximal, characteristic, complemented
4#11722720C212yesmaximal, complemented
5D3×#3708C212yesmaximal, complemented
6#10593838—31nomaximal
7C32×#235C411yescharacteristic
8#88855C411yescharacteristic
9C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
10#88786C412yes—
11D3×C3⋊(C3⋊C8)C2212yes—
12C3⋊#3708C2212yes—
13C3×#3708C2211yescharacteristic
14C3×#3708C2211yescharacteristic
15#64157—31no—
16#63814—62no—
17#63773—31no—
18#63773—32no—
19C2×#3708—32no—
20C2×#3708D311yescharacteristic, complemented
21C2×C33×C4C2×C411yescharacteristic
22C32×#67C812yes—
23C32×D12C811yescharacteristic
24C32×C4×D3C2×C412yes—
25C32×Dic6C811yescharacteristic
26C3⋊(C3⋊(C3⋊C8))C2×C412yes—
27C3×C3⋊(C3⋊C8)C2×C411yescharacteristic
28C3×C3⋊(C3⋊C8)C2311yescharacteristic
29#4901—91nomaximal
30C32×#41—31no—
31C3×#235—22no—
32#3748—31no—
33#3748—32no—
34#3748—62no—
35#3748—122no—
36C2×C3⋊(C3⋊C8)—32no—
37C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
38C2×C3×C3⋊C8—62no—
39#3711—64no—
40D3×C3⋊C8—64no—
41#3708—32no—
42#3708C2×D311yescharacteristic
43#3708C2×D311yescharacteristic
44C22×C33C2×C811yescharacteristic
45C2×C32×D3C2×C812yes—
46C33×C4C2×C811yescharacteristic
47C33×C4C22×C411yescharacteristic
48C32×Dic3C2×C812yes—
49#948—182no—
50#901—91no—
51C2×C3×C16—91no—
52#812—91no—
53#798—92no—
54D3×C16—92no—
55C32×Q8—31no—
56C32×D4—32no—
57C32×D4—31no—
58C2×C32×C4—22no—
59C2×C32×C4—32no—
60C2×C32×C4—42no—
61C2×C32×C4C4×D311yescharacteristic
62C3×#67—24no—
63C3×D12—22no—
64C3×C4×D3—24no—
65C3×Dic6—22no—
66C3⋊(C3⋊C8)—32no—
67C3⋊(C3⋊C8)—64no—
68C3⋊(C3⋊C8)—124no—
69C3⋊(C3⋊C8)C4×D311yescharacteristic
70C3⋊(C3⋊C8)C22×D311yescharacteristic
71C3×C3⋊C8—62no—
72C3×C3⋊C8—62no—
73#504—271noSylow
74C2×C33—21no—
75C2×C33C22×C811yescharacteristic
76C32×D3—22no—
77C3×#41—62no—
78#235#15711yescharacteristic
79C2×C3×C8—91no—
80#208—91no—
81#208—184no—
82#208—182no—
83#208—362no—
84C2×C3⋊C8—62no—
85C2×C3⋊C8—184no—
86ΓL(1,𝔽25)—92no—
87D3×C8—92no—
88C3×C16—91no—
89C3×C16—91no—
90C3⋊C16—92no—
91C22×C32—22no—
92C22×C32—32no—
93C22×C32—42no—
94C22×C32D3×C811yescharacteristic
95C2×C3×D3—24no—
96C32×C4—22no—
97C32×C4—22no—
98C32×C4—32no—
99C32×C4—42no—
100C32×C4—42no—
101C32×C4D3×C811yescharacteristic
102C32×C4C2×C4×D311yescharacteristic
103C3×Dic3—24no—
104#131—271no—
105M32—272no—
106M32—271no—
107C2×C16—91no—
108C2×C16—272no—
109C33#50411yescharacteristic, complemented, Sylow
110C3×Q8—62no—
111C3×D4—62no—
112C3×D4—64no—
113C2×C3×C4—22no—
114C2×C3×C4—42no—
115C2×C3×C4—64no—
116C2×C3×C4C2×#15711yescharacteristic
117#67#62212yes—
118D12#62211yescharacteristic
119C4×D3C2×#15712yes—
120Dic6#62211yescharacteristic
121C3×C8—91no—
122C3×C8—91no—
123C3⋊C8—62no—
124C3⋊C8—62no—
125C3⋊C8—92no—
126C3⋊C8—184no—
127C3⋊C8—364no—
128C2×C32—22no—
129C2×C32—21no—
130C2×C32—42no—
131C2×C32—42no—
132C2×C32—62no—
133C2×C32—82no—
134C2×C32C2×D3×C811yescharacteristic
135C3×D3—44no—
136#41—31no—
137M16—272no—
138M16—271no—
139C2×C8—91no—
140C2×C8—272no—
141C16—91no—
142C16—91no—
143C16—272no—
144C22×C3—22no—
145C22×C3—42no—
146C22×C3—64no—
147C22×C3C2×#62211yescharacteristic
148C2×D3C2×#62212yes—
149C3×C4—22no—
150C3×C4—22no—
151C3×C4—42no—
152C3×C4—42no—
153C3×C4—64no—
154C3×C4C2×#62211yescharacteristic
155C3×C4C22×#15711yescharacteristic
156Dic3C2×#62212yes—
157C32—22no—
158C32—42no—
159C32#490111yescharacteristic, complemented
160Q8—31no—
161D4—32no—
162D4—31no—
163C2×C4—32no—
164C2×C4D3×#15711yescharacteristic
165C8—91no—
166C8—91no—
167C8—272no—
168C2×C3—21no—
169C2×C3—22no—
170C2×C3—42no—
171C2×C3—42no—
172C2×C3—82no—
173C2×C3—124no—
174C2×C3C22×#62211yescharacteristic
175D3—22no—
176C22—32no—
177C22D3×#62211yescharacteristic
178C4—32no—
179C4D3×#62211yescharacteristic
180C4C2×D3×#15711yescharacteristic
181C3—22no—
182C3—42no—
183C3#1059383811yescharacteristic, complemented
184C2—21no—
185C2—62no—
186C2C2×D3×#62211yescharacteristic
subgroupG/H
center, FrattiniC4C2×D3×#157
commutator, socleC2×C33C22×C8
FittingC2×C33×C4C2×C4
prime radicalC32×#235C4
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. #11722720C2semidirect
5. D3×#3708C2semidirect
20. C2×#3708D3semidirect
109. C33#504semidirect
159. C32#4901semidirect
183. C3#10593838semidirect

Maximal subgroups

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

subgroupindex
#117233492
#117230982
C2×C3×#37082
#117227202
D3×#37082
#105938383
#49019