Finite Group Databasebeta

← #13167458

Subgroups — #13167458

Order 1728.

388 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 388 conjugacy classes of subgroups — too many to draw. Only its 83 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
1C2×C3×#3787C211yesmaximal, characteristic
2#11723656C212yesmaximal
3#11722787C214yesmaximal, complemented
4#10593331—31nomaximal
5C2×C3×C4×C3⋊D3C2211yescharacteristic
6C2×C3×#622C412yes—
7C3×#3787C412yes—
8C3×#3787C2212yes—
9C3×#3748—22no—
10#88809C2214yes—
11#88808—22no—
12C2×#3787D311yescharacteristic
13#63818—32no—
14D3×#208—62no—
15#63733—34no—
16C22×C3×C3⋊D3Q811yescharacteristic
17C2×C33×C4D411yescharacteristic
18C2×C3×C3⋊Dic3C2×C411yescharacteristic
19C3×C4×C3⋊D3C2×C411yescharacteristic
20C3×C4×C3⋊D3D411yescharacteristic
21C3×C4×C3⋊D3Q811yescharacteristic
22C3×C4×C3⋊D3C2311yescharacteristic
23C3×#622C2×C414yes—
24C3×C3⋊(C3⋊C8)—24no—
25C2×#819—91nomaximal
26C2×C4×C3⋊D3C2×D311yescharacteristic
27C2×C3×C4×D3—62no—
28C2×#622Dic312yes—
29#3787Dic312yes—
30#3787C2×D312yes—
31#3748—22no—
32C3×#208—62no—
33C3×#208—62no—
34C3×#208—122no—
35#3712—34no—
36#3711—64no—
37#3710—62no—
38D3×C3⋊C8—64no—
39C22×C33C4⋊C411yescharacteristic
40C2×C3×C3⋊D3—22no—
41C2×C3×C3⋊D3C4⋊C411yescharacteristic
42C2×C3×C3⋊D3C2×Q811yescharacteristic
43C33×C4C4⋊C411yescharacteristic
44C33×C4C2×D411yescharacteristic
45C3×C3⋊Dic3C4⋊C411yescharacteristic
46C3×C3⋊Dic3C22×C411yescharacteristic
47C22×C3×C8—91no—
48C2×#208—92no—
49D3×M16—182no—
50#819—94no—
51C22×C3⋊D3Dic611yescharacteristic
52C22×C3×D3—62no—
53C2×C32×C4—22no—
54C2×C32×C4—42no—
55C2×C32×C4D1211yescharacteristic
56C2×C3⋊Dic3C2×Dic311yescharacteristic
57C4×C3⋊D3Dic611yescharacteristic
58C4×C3⋊D3D1211yescharacteristic
59C4×C3⋊D3C2×Dic311yescharacteristic
60C4×C3⋊D3C22×D311yescharacteristic
61C2×C3×Dic3—62no—
62C3×C4×D3—62no—
63C3×C4×D3—62no—
64C3×C4×D3—62no—
65C3×C4×D3—62no—
66#622C2×Dic314yes—
67C3⋊(C3⋊C8)—24no—
68C3×C3⋊C8—64no—
69C3×C3⋊C8—64no—
70C3×C3⋊C8—124no—
71C2×#108—271noSylow
72C2×C33—21no—
73C2×C33C2×C4⋊C411yescharacteristic
74C3×C3⋊D3—21no—
75C3×C3⋊D3#10812yescomplemented
76C22×C3×C4—91no—
77C2×C4×D3—62no—
78C3×M16—182no—
79C2×C3×C8—92no—
80C2×C3×C8—92no—
81C2×C3×C8—92no—
82#208—62no—
83#208—62no—
84#208—94no—
85#208—122no—
86C2×C3⋊C8—182no—
87ΓL(1,𝔽25)—184no—
88D3×C8—184no—
89C22×C32—22no—
90C22×C32—42no—
91C22×C32C4⋊Dic311yescharacteristic
92C2×C3⋊D3—22no—
93C2×C3⋊D3C4⋊Dic311yescharacteristic
94C2×C3⋊D3C2×Dic611yescharacteristic
95C2×C3×D3—62no—
96C2×C3×D3—62no—
97C2×C3×D3—124no—
98C32×C4—22no—
99C32×C4—22no—
100C32×C4—42no—
101C32×C4—42no—
102C32×C4C4⋊Dic311yescharacteristic
103C32×C4C2×D1211yescharacteristic
104C3⋊Dic3C4⋊Dic311yescharacteristic
105C3⋊Dic3C22×Dic311yescharacteristic
106C3×Dic3—62no—
107C3×Dic3—62no—
108C2×M16—272no—
109C22×C8—91no—
110#108—274no—
111C33C2×#10811yescharacteristic, complemented, Sylow
112C23×C3—91no—
113C22×D3—62no—
114C2×C3×C4—22no—
115C2×C3×C4—42no—
116C2×C3×C4—91no—
117C2×C3×C4—91no—
118C2×C3×C4—91no—
119C2×C3×C4—91no—
120C2×C3×C4—91no—
121C2×C3×C4GO+(4,𝔽2)11yescharacteristic
122C2×Dic3—62no—
123C4×D3—62no—
124C4×D3—62no—
125C4×D3—62no—
126C4×D3—62no—
127C3×C8—94no—
128C3×C8—184no—
129C3⋊C8—64no—
130C3⋊C8—64no—
131C3⋊C8—124no—
132C2×C32—22no—
133C2×C32—21no—
134C2×C32—42no—
135C2×C32—42no—
136C2×C32—82no—
137C2×C32C2×C4⋊Dic311yescharacteristic
138C3⋊D3—21no—
139C3⋊D3#81912yescomplemented
140C3×D3—64no—
141C3×D3—122no—
142C22×C4—91no—
143M16—182no—
144M16—274no—
145C2×C8—92no—
146C2×C8—92no—
147C2×C8—92no—
148C2×C8—542no—
149C22×C3—22no—
150C22×C3—42no—
151C22×C3—91no—
152C22×C3—91no—
153C22×C3—182no—
154C22×C3#377311yescharacteristic
155C2×D3—62no—
156C2×D3—62no—
157C2×D3—124no—
158C3×C4—22no—
159C3×C4—22no—
160C3×C4—42no—
161C3×C4—42no—
162C3×C4—91no—
163C3×C4—91no—
164C3×C4#377311yescharacteristic
165C3×C4C2×GO+(4,𝔽2)11yescharacteristic
166Dic3—62no—
167Dic3—62no—
168C32—22no—
169C32—42no—
170C32C2×#81911yescharacteristic, complemented
171C23—91no—
172C2×C4—91no—
173C2×C4—91no—
174C2×C4—91no—
175C2×C4—91no—
176C2×C4—91no—
177C2×C4#157⋊D311yescharacteristic
178C8—94no—
179C8—184no—
180C2×C3—21no—
181C2×C3—22no—
182C2×C3—42no—
183C2×C3—42no—
184C2×C3—82no—
185C2×C3—92no—
186C2×C3—181no—
187C2×C3C2×#377311yescharacteristic
188D3—64no—
189D3—122no—
190C22—91no—
191C22—91no—
192C22—182no—
193C22#8894011yescharacteristic
194C4—91no—
195C4—91no—
196C4#8894011yescharacteristic
197C4C2×#157⋊D311yescharacteristic
198C3—22no—
199C3—42no—
200C3#1059333111yescharacteristic, complemented
201C2—21no—
202C2—92no—
203C2—181no—
204C2C2×#8894011yescharacteristic
subgroupG/H
center, FrattiniC4C2×#157⋊D3
commutatorC3×C3⋊Dic3C22×C4
FittingC2×C33×C4D4
socleC2×C33C2×C4⋊C4
prime radicalC22×C3×C3⋊D3Q8
cosocleC3×C4×C3⋊D3C23
hypercenterC2×C4#157⋊D3
hypocenterC33C2×#108
LCSG ▷ C3×C3⋊Dic3 ▷ C2×C33 ▷ C33
UCSC1 ◁ C4 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
3. #11722787C2semidirect
75. C3×C3⋊D3#108semidirect
111. C33C2×#108semidirect
139. C3⋊D3#819semidirect
170. C32C2×#819semidirect
200. C3#10593331semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#37872
#117236562
#117227872
#105933313
C2×#8199