Finite Group Databasebeta

← C3×#10593551 #13167628

Subgroups — C3×#10593551 #13167628

Order 1728.

338 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 338 conjugacy classes of subgroups — too many to draw. Only its 82 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×C4⋊#157C211yesmaximal, characteristic
2C2×C3×#3777C211yesmaximal, characteristic
3C3×#63781C212yesmaximal
4C3×#63747C211yesmaximal, characteristic
5C3×#63747C212yesmaximal
6#10593551C311yesmaximal, characteristic, direct product factor
7C22×C3×#157C2212yes—
8C22×C3×#157C2211yescharacteristic
9C22×C3×#157C2212yes—
10C22×C3×#157C2211yescharacteristic
11C2×C3×C4×C3⋊D3C2211yescharacteristic
12C3×C4⋊#157—22no—
13C3×#3777—22no—
14C2×C4⋊#157C2×C311yescharacteristic
15C2×#3777C2×C311yescharacteristic
16#63781C2×C312yes—
17#63747C2×C312yes—
18#63747C2×C311yescharacteristic
19C22×C3×C3⋊D3C2311yescharacteristic
20C2×C3×#157—22no—
21C2×C3×#157—22no—
22C2×C3×#157—21no—
23C2×C3×#157—22no—
24C2×C3×#157—24no—
25C2×C3×#157—22no—
26C2×C3×#157—22no—
27C2×C3×#157—22no—
28C2×C3×#157D412yes—
29C2×C33×C4Q811yescharacteristic
30C2×C3×C3⋊Dic3Q811yescharacteristic
31C3×C4×C3⋊D3—21no—
32C3×C4×C3⋊D3—21no—
33C3×#400—91nomaximal
34C22×#157C22×C311yescharacteristic
35C22×#157C22×C312yes—
36C22×#157C22×C312yes—
37C22×#157C22×C311yescharacteristic
38C2×C4×C3⋊D3C22×C311yescharacteristic
39C2×C3×C4×D3—121no—
40C4⋊#157—22no—
41#3777—22no—
42C22×C33C2×Q811yescharacteristic
43C2×C3×C3⋊D3C2×D411yescharacteristic
44C2×C3×C3⋊D3#4112yes—
45C2×C3×C3⋊D3#4112yes—
46C2×C3×C3⋊D3#4111yescharacteristic
47C3×#157—22no—
48C3×#157—41no—
49C3×#157—42no—
50C3×#157—42no—
51C33×C4—21no—
52C3×C3⋊Dic3—21no—
53C2×C3×C4⋊C4—91no—
54C2×C3×C4⋊C4—91no—
55C3×#95—91no—
56C3×#95—92no—
57C3×#95—92no—
58C22×C3⋊D3C23×C311yescharacteristic
59C22×C3×D3—121no—
60C2×#157—22no—
61C2×#157—22no—
62C2×#157—21no—
63C2×#157—22no—
64C2×#157—24no—
65C2×#157—22no—
66C2×#157—22no—
67C2×#157—22no—
68C2×#157C3×D412yes—
69C2×C32×C4—41no—
70C2×C32×C4—81no—
71C2×C32×C4C3×Q811yescharacteristic
72C2×C3⋊Dic3C3×Q811yescharacteristic
73C4×C3⋊D3—21no—
74C4×C3⋊D3—21no—
75C2×C3×Dic3—121no—
76C3×C4×D3—122no—
77C3×C4×D3—122no—
78#400—91noSylow
79C2×C33C22⋊Q811yescharacteristic
80C2×C33#12512yes—
81C3×C3⋊D3#12311yescharacteristic
82C3×C3⋊D3#12411yescharacteristic
83C3×C3⋊D3#12612yes—
84C22×C3×C4—91no—
85C22×C3×C4—91no—
86C22×C3×C4—92no—
87C22×C3×C4—91no—
88C22×C3×C4—92no—
89C2×C4×D3—121no—
90C3×C4⋊C4—182no—
91C3×C4⋊C4—182no—
92C22×C32—41no—
93C22×C32—81no—
94C22×C32C2×C3×Q811yescharacteristic
95C2×C3⋊D3C2×C3×D411yescharacteristic
96C2×C3⋊D3C3×#4112yes—
97C2×C3⋊D3C3×#4112yes—
98C2×C3⋊D3C3×#4111yescharacteristic
99C2×C3×D3—121no—
100C2×C3×D3—121no—
101C2×C3×D3—122no—
102C2×C3×D3—122no—
103#157—22no—
104#157—41no—
105#157—42no—
106#157—42no—
107C32×C4—21no—
108C32×C4—42no—
109C32×C4—82no—
110C3⋊Dic3—21no—
111C3×Dic3—122no—
112C2×C4⋊C4—91no—
113C2×C4⋊C4—91no—
114#95—91no—
115#95—92no—
116#95—92no—
117C33#40011yescharacteristic, complemented, Sylow
118C23×C3—91no—
119C22×D3—121no—
120C2×C3×C4—41no—
121C2×C3×C4—81no—
122C2×C3×C4—92no—
123C2×C3×C4—91no—
124C2×C3×C4—181no—
125C2×C3×C4—181no—
126C2×C3×C4—182no—
127C2×C3×C4—182no—
128C2×C3×C4—181no—
129C2×C3×C4—182no—
130C2×C3×C4—182no—
131C2×C3×C4—182no—
132C2×C3×C4—184no—
133C2×C3×C4—182no—
134C2×C3×C4PSU(3,𝔽2)11yescharacteristic
135C2×Dic3—121no—
136C4×D3—122no—
137C4×D3—122no—
138C2×C32—42no—
139C2×C32—41no—
140C2×C32—81no—
141C2×C32—82no—
142C2×C32C3×C22⋊Q811yescharacteristic
143C2×C32C3×#12512yes—
144C3⋊D3C3×#12311yescharacteristic
145C3⋊D3C3×#12411yescharacteristic
146C3⋊D3C3×#12612yes—
147C3×D3—121no—
148C3×D3—121no—
149C3×D3—122no—
150C22×C4—91no—
151C22×C4—91no—
152C22×C4—92no—
153C22×C4—91no—
154C22×C4—92no—
155C4⋊C4—182no—
156C4⋊C4—182no—
157C22×C3—41no—
158C22×C3—81no—
159C22×C3—92no—
160C22×C3—92no—
161C22×C3—91no—
162C22×C3—91no—
163C22×C3C2×PSU(3,𝔽2)11yescharacteristic
164C2×D3—121no—
165C2×D3—121no—
166C2×D3—122no—
167C2×D3—122no—
168C3×C4—21no—
169C3×C4—42no—
170C3×C4—82no—
171C3×C4—182no—
172C3×C4—181no—
173C3×C4—361no—
174C3×C4—362no—
175C3×C4—362no—
176Dic3—122no—
177C32—41no—
178C32—81no—
179C32C3×#40011yescharacteristic, complemented
180C23—91no—
181C2×C4—92no—
182C2×C4—91no—
183C2×C4—181no—
184C2×C4—181no—
185C2×C4—182no—
186C2×C4—182no—
187C2×C4—181no—
188C2×C4—182no—
189C2×C4—182no—
190C2×C4—182no—
191C2×C4—184no—
192C2×C4—182no—
193C2×C4C3×PSU(3,𝔽2)11yescharacteristic
194C2×C3—42no—
195C2×C3—41no—
196C2×C3—82no—
197C2×C3—81no—
198C2×C3—91no—
199C2×C3—91no—
200C2×C3—92no—
201C2×C3#6424312yes—
202C2×C3C22⋊PSU(3,𝔽2)11yescharacteristic
203D3—121no—
204D3—121no—
205D3—122no—
206C22—92no—
207C22—92no—
208C22—91no—
209C22—91no—
210C22C2×C3×PSU(3,𝔽2)11yescharacteristic
211C4—21no—
212C4—182no—
213C4—181no—
214C4—361no—
215C4—362no—
216C4—362no—
217C3—41no—
218C3—81no—
219C3#1059355111yescharacteristic, direct product factor
220C2—91no—
221C2—91no—
222C2—92no—
223C2C3×#6424312yes—
224C2C3×C22⋊PSU(3,𝔽2)11yescharacteristic
subgroupG/H
centerC22×C3C2×PSU(3,𝔽2)
commutator, cosocleC22×C3⋊D3C23×C3
FrattiniC22C2×C3×PSU(3,𝔽2)
FittingC2×C33×C4Q8
socleC22×C33C2×Q8
prime radicalC22×C3×C3⋊D3C23
hypercenterC2×C3×C4PSU(3,𝔽2)
hypocenterC32C3×#400
LCSG ▷ C22×C3⋊D3 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4

Complements

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

NH ≅ G/Ntype
6. #10593551C3direct (N × H)
117. C33#400semidirect
179. C32C3×#400semidirect
219. C3#10593551direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×C4⋊#1572
C2×C3×#37772
C3×#637812
C3×#637472
#105935513
C3×#4009