Finite Group Databasebeta

← #13165777

Subgroups — #13165777

Order 1728.

230 conjugacy classes of subgroups; only the 48 normal subgroups are shown.

normal subgroups only

This group has 230 conjugacy classes of subgroups — too many to draw. Only its 48 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
1C3×#64243C211yesmaximal, characteristic
2Dic6×#157C211yesmaximal, characteristic
3#11722807C211yesmaximal, characteristic
4#11722805C211yesmaximal, characteristic
5#11722804C211yesmaximal, characteristic
6#11722708C211yesmaximal, characteristic
7#11722708C211yesmaximal, characteristic
8#10593560—31nomaximal
9C3⋊D3×Dic6C2211yescharacteristic
10C4⋊(C3⋊#157)C2211yescharacteristic
11#88988C2211yescharacteristic
12#88988C2211yescharacteristic
13C3×C4⋊#157C2211yescharacteristic
14C3×C4⋊#157C2211yescharacteristic
15C3×C4×#157C2211yescharacteristic
16C3×#3777—21no—
17C3×#3777—21no—
18Dic3⋊#157—21no—
19Dic3×#157—21no—
20Q8×#157—31no—
21#64243D311yescharacteristic
22#63784—31no—
23#63784—31no—
24#63746—31no—
25#63745—31no—
26#63744—31no—
27C2×C3⋊#157—21no—
28C2×C3×#157—21no—
29C2×C3×#157—21no—
30C2×C3×#157D411yescharacteristic
31C2×C3×#157D411yescharacteristic
32#6308Q811yescharacteristic
33C3×C4×C3⋊D3C2311yescharacteristic
34C32×Dic6Q811yescharacteristic
35#6293—21no—
36Dic3×C3⋊D3—21no—
37C3⋊#622—21no—
38C3⋊#622—21no—
39#5063—91nomaximal
40Q8×C3⋊D3—31no—
41D3×Dic6—121no—
42C4⋊#157—31no—
43C4⋊#157—61no—
44C4⋊#157C2×D311yescharacteristic
45C4⋊#157C2×D311yescharacteristic
46C4×#157—61no—
47C4×#157C2×D311yescharacteristic
48#3787—31no—
49#3787—31no—
50#3777—21no—
51#3777—21no—
52C2×C3×C3⋊D3C2×D411yescharacteristic
53C3⋊#157—41no—
54C3×#157—21no—
55C3×#157—21no—
56C3×#157—41no—
57C3×#157—41no—
58C33×C4C2×Q811yescharacteristic
59C3⋊(C3⋊Dic3)—21no—
60C3×C3⋊Dic3#4111yescharacteristic
61C32×Dic3—21no—
62C3×#125—91no—
63C4×Dic6—91no—
64#810—91no—
65#810—91no—
66#808—91no—
67#807—91no—
68C4⋊(C3⋊C8)—91no—
69C2×#157—21no—
70C2×#157—21no—
71C2×#157—61no—
72C2×#157#6711yescharacteristic
73C2×#157#6711yescharacteristic
74C32×Q8—31no—
75C4×C3⋊D3—61no—
76C4×C3⋊D3C22×D311yescharacteristic
77#634—31no—
78#634—61no—
79#634—121no—
80#634—241no—
81C3×C4×D3—121no—
82C3×Dic6—41no—
83#627—122no—
84D3×Dic3—122no—
85#622—61no—
86#622—61no—
87#479—271noSylow
88C2×C33C22⋊Q811yescharacteristic
89C3×C3⋊D3#13511yescharacteristic
90C3×C3⋊D3#13711yescharacteristic
91D3×Q8—361no—
92C2×Dic6—91no—
93C3×C4⋊C4—91no—
94C3×C4⋊C4—91no—
95C3×C4⋊C4—181no—
96C3×C4⋊C4—181no—
97C3×C42—91no—
98C4⋊Dic3—91no—
99Dic3⋊C4—181no—
100C4×Dic3—181no—
101C2×C3⋊C8—91no—
102C2×C3⋊C8—91no—
103C2×C3⋊D3C2×#6711yescharacteristic
104C2×C3×D3—121no—
105#157—21no—
106#157—21no—
107#157—41no—
108#157—41no—
109#157—121no—
110C32×C4—41no—
111C32×C4—61no—
112C32×C4—81no—
113C32×C4D3×Q811yescharacteristic
114C3⋊Dic3—61no—
115C3⋊Dic3—122no—
116C3⋊Dic3—242no—
117C3⋊Dic3#23911yescharacteristic
118C3×Dic3—42no—
119C3×Dic3—121no—
120#125—91no—
121C4×Q8—271no—
122#107—271no—
123#106—271no—
124C4⋊C8—271no—
125#103—271no—
126#103—271no—
127C33#47911yescharacteristic, complemented, Sylow
128C3×Q8—121no—
129C2×C3×C4—91no—
130C2×C3×C4—91no—
131C2×C3×C4—91no—
132C2×C3×C4—181no—
133C2×C3×C4—181no—
134C2×Dic3—181no—
135C2×Dic3—181no—
136C4×D3—121no—
137C4×D3—362no—
138Dic6—91no—
139Dic6—181no—
140Dic6—361no—
141Dic6—362no—
142Dic6—721no—
143Dic6PSU(3,𝔽2)11yescharacteristic
144C3⋊C8—181no—
145C3⋊C8—181no—
146C2×C32—41no—
147C2×C32—81no—
148C2×C32#94611yescharacteristic
149C3⋊D3#95011yescharacteristic
150C3⋊D3#95111yescharacteristic
151C3×D3—121no—
152C3×D3—121no—
153C2×Q8—271no—
154C2×C8—271no—
155C2×C8—271no—
156C4⋊C4—91no—
157C4⋊C4—91no—
158C4⋊C4—181no—
159C4⋊C4—181no—
160C4⋊C4—271no—
161C4⋊C4—541no—
162C42—91no—
163C42—541no—
164C22×C3—91no—
165C2×D3—121no—
166C3×C4—41no—
167C3×C4—81no—
168C3×C4—91no—
169C3×C4—122no—
170C3×C4—181no—
171C3×C4—181no—
172C3×C4—361no—
173C3×C4—361no—
174C3×C4C2×PSU(3,𝔽2)11yescharacteristic
175Dic3—21no—
176Dic3—121no—
177Dic3—181no—
178Dic3—362no—
179Dic3—361no—
180Dic3—722no—
181C32—41no—
182C32—81no—
183C32#506311yescharacteristic, complemented
184Q8—31no—
185Q8—271no—
186Q8—541no—
187C2×C4—91no—
188C2×C4—91no—
189C2×C4—91no—
190C2×C4—181no—
191C2×C4—181no—
192C2×C4—541no—
193C2×C4—541no—
194C8—541no—
195C8—541no—
196C2×C3—41no—
197C2×C3—81no—
198C2×C3—91no—
199C2×C3—91no—
200C2×C3C22⋊PSU(3,𝔽2)11yescharacteristic
201D3—121no—
202D3—121no—
203C22—91no—
204C4—61no—
205C4—91no—
206C4—181no—
207C4—181no—
208C4—361no—
209C4—361no—
210C4—541no—
211C4—1081no—
212C4D3×PSU(3,𝔽2)11yescharacteristic
213C3—41no—
214C3—81no—
215C3#1059356011yescharacteristic, complemented
216C2—91no—
217C2—91no—
218C2#1172414911yescharacteristic
subgroupG/H
centerC2#11724149
commutator, cosocleC3×C4×C3⋊D3C23
Frattini, hypercenterC4D3×PSU(3,𝔽2)
FittingC33×C4C2×Q8
socleC2×C33C22⋊Q8
prime radicalC2×C3×C3⋊D3C2×D4
hypocenterC33#479
LCSG ▷ C3×C4×C3⋊D3 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C4

Complements

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

NH ≅ G/Ntype
127. C33#479semidirect
183. C32#5063semidirect
215. C3#10593560semidirect

Maximal subgroups

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

subgroupindex
C3×#642432
Dic6×#1572
#117228072
#117228052
#117228042
#117227082
#105935603
#50639