Finite Group Databasebeta

← #13167446

Subgroups — #13167446

Order 1728.

336 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 59 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#11723679C212yesmaximal
2C3×C4×#622C211yesmaximal, characteristic
3#11722797C211yesmaximal, characteristic
4#11722797C211yesmaximal, characteristic
5#11722796C212yesmaximal
6#10593390—31nomaximal
7C2×C3×#622C2211yescharacteristic
8C2×C3×#622C2211yescharacteristic
9C3×#3750—22no—
10C3×C4×C3⋊Dic3C2211yescharacteristic
11#88815—22no—
12#88815C2212yes—
13#88815C2212yes—
14#63870—32no—
15#63867—62no—
16C4×#622D311yescharacteristic
17#63743—31no—
18#63743—31no—
19#63742—32no—
20C2×C33×C4D411yescharacteristic
21C2×C3×C3⋊Dic3—22no—
22C2×C3×C3⋊Dic3—22no—
23C2×C3×C3⋊Dic3D411yescharacteristic
24C2×C3×C3⋊Dic3C2311yescharacteristic
25C3×#622Q812yes—
26C3×#622Q812yes—
27#4683—91nomaximal
28C2×#622C2×D311yescharacteristic
29C2×#622C2×D311yescharacteristic
30#3750—22no—
31C4×C3⋊Dic3C2×D311yescharacteristic
32C3×Dic3⋊C4—62no—
33C3×Dic3⋊C4—62no—
34C3×Dic3⋊C4—122no—
35C3×C4×Dic3—62no—
36#3724—32no—
37#3724—32no—
38#3724—62no—
39Dic3⋊Dic3—62no—
40Dic3⋊Dic3—62no—
41Dic3⋊Dic3—62no—
42Dic3⋊Dic3—62no—
43C22×C33C2×D411yescharacteristic
44C33×C4—21no—
45C3×C3⋊Dic3—21no—
46C3×C3⋊Dic3—42no—
47C3×C3⋊Dic3—42no—
48C3×C3⋊Dic3C2×Q811yescharacteristic
49C3×C3⋊Dic3C2×Q811yescharacteristic
50#889—182no—
51#870—92no—
52C3×C4×C8—91no—
53#818—91no—
54#818—91no—
55#817—92no—
56C2×C32×C4—22no—
57C2×C32×C4—42no—
58C2×C32×C4D1211yescharacteristic
59C2×C3⋊Dic3—22no—
60C2×C3⋊Dic3—22no—
61C2×C3⋊Dic3D1211yescharacteristic
62C2×C3⋊Dic3C22×D311yescharacteristic
63C2×C3×Dic3—62no—
64C2×C3×Dic3—62no—
65C2×C3×Dic3—62no—
66C2×C3×Dic3—62no—
67C2×C3×Dic3—62no—
68C2×C3×Dic3—62no—
69C2×C3×Dic3—122no—
70C2×C3×Dic3—122no—
71#622Dic612yes—
72#622Dic612yes—
73#499—271noSylow
74C2×C33C4⋊Q811yescharacteristic
75C2×C33#13511yescharacteristic
76C2×C33#13511yescharacteristic
77C2×C3×C8—91no—
78C2×C3×C8—91no—
79C3×C4⋊C4—182no—
80C3×C42—91no—
81C4⋊Dic3—92no—
82C4⋊Dic3—92no—
83C4⋊Dic3—182no—
84Dic3⋊C4—62no—
85Dic3⋊C4—62no—
86Dic3⋊C4—122no—
87Dic3⋊C4—182no—
88Dic3⋊C4—182no—
89Dic3⋊C4—182no—
90Dic3⋊C4—182no—
91C4×Dic3—62no—
92C22×C32—22no—
93C22×C32—42no—
94C22×C32C2×D1211yescharacteristic
95C32×C4—21no—
96C32×C4—42no—
97C32×C4—82no—
98C3⋊Dic3—21no—
99C3⋊Dic3—42no—
100C3⋊Dic3—42no—
101C3⋊Dic3C2×Dic611yescharacteristic
102C3⋊Dic3C2×Dic611yescharacteristic
103C3×Dic3—62no—
104C3×Dic3—62no—
105C3×Dic3—122no—
106C3×Dic3—122no—
107C3×Dic3—122no—
108C3×Dic3—124no—
109C3×Dic3—122no—
110C3×Dic3—122no—
111C3×Dic3—242no—
112#125—272no—
113#107—271no—
114#107—271no—
115#106—272no—
116C4×C8—91no—
117C33#49911yescharacteristic, complemented, Sylow
118C2×C3×C4—22no—
119C2×C3×C4—42no—
120C2×C3×C4—91no—
121C2×C3×C4—91no—
122C2×C3×C4—182no—
123C2×C3×C4—182no—
124C2×C3×C4GO+(4,𝔽2)11yescharacteristic
125C2×Dic3—62no—
126C2×Dic3—62no—
127C2×Dic3—62no—
128C2×Dic3—62no—
129C2×Dic3—62no—
130C2×Dic3—62no—
131C2×Dic3—122no—
132C2×Dic3—122no—
133C3×C8—92no—
134C3×C8—92no—
135C2×C32—22no—
136C2×C32—22no—
137C2×C32—22no—
138C2×C32—42no—
139C2×C32—42no—
140C2×C32—42no—
141C2×C32#86911yescharacteristic
142C2×C32#90411yescharacteristic
143C2×C32#90411yescharacteristic
144C2×C8—91no—
145C2×C8—91no—
146C4⋊C4—182no—
147C4⋊C4—272no—
148C4⋊C4—272no—
149C4⋊C4—542no—
150C42—91no—
151C22×C3—22no—
152C22×C3—42no—
153C22×C3C2×GO+(4,𝔽2)11yescharacteristic
154C3×C4—21no—
155C3×C4—42no—
156C3×C4—82no—
157C3×C4—91no—
158C3×C4—91no—
159C3×C4—181no—
160C3×C4—362no—
161C3×C4—362no—
162Dic3—62no—
163Dic3—62no—
164Dic3—122no—
165Dic3—122no—
166Dic3—122no—
167Dic3—124no—
168Dic3—122no—
169Dic3—122no—
170Dic3—242no—
171C32—22no—
172C32—42no—
173C32#468311yescharacteristic, complemented
174C2×C4—91no—
175C2×C4—91no—
176C2×C4—182no—
177C2×C4—182no—
178C2×C4#157⋊D311yescharacteristic
179C8—92no—
180C8—92no—
181C2×C3—22no—
182C2×C3—22no—
183C2×C3—22no—
184C2×C3—42no—
185C2×C3—42no—
186C2×C3—42no—
187C2×C3#6422211yescharacteristic
188C2×C3#6422311yescharacteristic
189C2×C3#6423711yescharacteristic
190C22C2×#157⋊D311yescharacteristic
191C4—21no—
192C4—91no—
193C4—91no—
194C4—181no—
195C4—362no—
196C4—362no—
197C3—22no—
198C3—42no—
199C3#1059339011yescharacteristic, complemented
200C2#1172433511yescharacteristic
201C2#1172433711yescharacteristic
202C2#1172435111yescharacteristic
subgroupG/H
center, FrattiniC22C2×#157⋊D3
commutator, cosocleC2×C3×C3⋊Dic3C23
FittingC2×C33×C4D4
socle, prime radicalC22×C33C2×D4
hypercenterC2×C4#157⋊D3
hypocenterC33#499
LCSG ▷ C2×C3×C3⋊Dic3 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
117. C33#499semidirect
173. C32#4683semidirect
199. C3#10593390semidirect

Maximal subgroups

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

subgroupindex
#117236792
C3×C4×#6222
#117227972
#117227962
#105933903
#46839