Finite Group Databasebeta

← C3×#10593328 #13166732

Subgroups — C3×#10593328 #13166732

Order 1728.

428 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 428 conjugacy classes of subgroups — too many to draw. Only its 102 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
2C3×#63818C212yesmaximal
3C3×#63732C214yesmaximal, complemented
4#10593328C311yesmaximal, characteristic, direct product factor
5C22×C3×#157C412yes—
6C2×C3×C4×C3⋊D3C2211yescharacteristic
7C3×C4×#157C411yescharacteristic
8C3×C4×#157C411yescharacteristic
9C3×C4×#157C2212yes—
10C32×#208—22no—
11C3×#3712—22no—
12C3×#3710C2214yes—
13C2×C4×#157C2×C311yescharacteristic
14C3×D3×M16—62no—
15#63818C2×C312yes—
16#63732C2×C314yescomplemented
17C22×C3×C3⋊D3C2×C411yescharacteristic
18C2×C3×#157—24no—
19C2×C3×#157C2×C412yes—
20C2×C3×#157C2×C412yes—
21C2×C33×C4D411yescharacteristic
22C2×C3×C3⋊Dic3Q811yescharacteristic
23C3×C4×C3⋊D3C2×C411yescharacteristic
24C3×C4×C3⋊D3D411yescharacteristic
25C3×C4×C3⋊D3Q811yescharacteristic
26C3×C4×C3⋊D3C2311yescharacteristic
27C32×C3⋊C8—24no—
28C3×#423—91nomaximal
29C22×#157C3×C412yes—
30C2×C4×C3⋊D3C22×C311yescharacteristic
31C2×C3×C4×D3—62no—
32C4×#157C3×C411yescharacteristic
33C4×#157C3×C411yescharacteristic
34C4×#157C22×C312yes—
35C32×M16—62no—
36C3×#208—22no—
37C3×#208—22no—
38C3×#208—42no—
39C3×ΓL(1,𝔽25)—64no—
40C3×D3×C8—64no—
41#3712—22no—
42#3710C22×C314yes—
43C22×C33C4⋊C411yescharacteristic
44C2×C3×C3⋊D3—22no—
45C2×C3×C3⋊D3C4⋊C411yescharacteristic
46C2×C3×C3⋊D3C22×C411yescharacteristic
47C3×#157—22no—
48C3×#157M1614yescomplemented
49C33×C4C4⋊C411yescharacteristic
50C33×C4C2×D411yescharacteristic
51C3×C3⋊Dic3C4⋊C411yescharacteristic
52C3×C3⋊Dic3C2×Q811yescharacteristic
53C2×C3×M16—92no—
54C2×C3×C42—91no—
55D3×M16—62no—
56C3×C4⋊C8—94no—
57C22×C3⋊D3C2×C3×C411yescharacteristic
58C22×C3×D3—62no—
59C2×#157—24no—
60C2×#157C2×C3×C412yes—
61C2×#157C2×C3×C412yes—
62C2×C32×C4—22no—
63C2×C32×C4—42no—
64C2×C32×C4C3×D411yescharacteristic
65C2×C3⋊Dic3C3×Q811yescharacteristic
66C4×C3⋊D3C2×C3×C411yescharacteristic
67C4×C3⋊D3C3×D411yescharacteristic
68C4×C3⋊D3C3×Q811yescharacteristic
69C4×C3⋊D3C23×C311yescharacteristic
70C2×C3×Dic3—62no—
71C3×C4×D3—62no—
72C3×C4×D3—62no—
73C3×C4×D3—62no—
74C3×C4×D3—62no—
75C32×C8—64no—
76C3×C3⋊C8—24no—
77C3×C3⋊C8—24no—
78C3×C3⋊C8—44no—
79#423—91noSylow
80C2×C33—21no—
81C2×C33C2×C4⋊C411yescharacteristic
82C3×C3⋊D3—21no—
83C3×C3⋊D3C2×M1612yes—
84C22×C3×C4—91no—
85C22×C3×C4—92no—
86C2×C4×D3—62no—
87C3×M16—62no—
88C3×M16—62no—
89C3×M16—122no—
90C3×M16—182no—
91C2×C3×C8—94no—
92C3×C42—92no—
93C3×C42—91no—
94C3×C42—91no—
95#208—22no—
96ΓL(1,𝔽25)—64no—
97D3×C8—64no—
98C22×C32—22no—
99C22×C32—42no—
100C22×C32C3×C4⋊C411yescharacteristic
101C2×C3⋊D3—22no—
102C2×C3⋊D3C3×C4⋊C411yescharacteristic
103C2×C3⋊D3C22×C3×C411yescharacteristic
104C2×C3×D3—62no—
105C2×C3×D3—62no—
106C2×C3×D3—124no—
107#157—22no—
108#157C3×M1614yescomplemented
109C32×C4—22no—
110C32×C4—22no—
111C32×C4—42no—
112C32×C4—42no—
113C32×C4C3×C4⋊C411yescharacteristic
114C32×C4C2×C3×D411yescharacteristic
115C3⋊Dic3C3×C4⋊C411yescharacteristic
116C3⋊Dic3C2×C3×Q811yescharacteristic
117C3×Dic3—62no—
118C3×Dic3—62no—
119C2×M16—92no—
120C2×C42—91no—
121C4⋊C8—94no—
122C33#42311yescharacteristic, complemented, Sylow
123C23×C3—91no—
124C22×D3—62no—
125C2×C3×C4—22no—
126C2×C3×C4—42no—
127C2×C3×C4—92no—
128C2×C3×C4—91no—
129C2×C3×C4—91no—
130C2×C3×C4—91no—
131C2×C3×C4—91no—
132C2×C3×C4—91no—
133C2×C3×C4—92no—
134C2×C3×C4—184no—
135C2×C3×C4GO+(4,𝔽2)11yescharacteristic
136C2×Dic3—62no—
137C4×D3—62no—
138C4×D3—62no—
139C4×D3—62no—
140C4×D3—62no—
141C3×C8—64no—
142C3×C8—64no—
143C3×C8—124no—
144C3⋊C8—24no—
145C2×C32—22no—
146C2×C32—21no—
147C2×C32—42no—
148C2×C32—42no—
149C2×C32—82no—
150C2×C32C2×C3×C4⋊C411yescharacteristic
151C3⋊D3—21no—
152C3⋊D3C2×C3×M1612yes—
153C3×D3—64no—
154C3×D3—122no—
155C22×C4—91no—
156C22×C4—92no—
157M16—62no—
158M16—182no—
159C2×C8—94no—
160C42—92no—
161C42—91no—
162C42—91no—
163C22×C3—22no—
164C22×C3—42no—
165C22×C3—91no—
166C22×C3—91no—
167C22×C3—182no—
168C22×C3#377311yescharacteristic
169C2×D3—62no—
170C2×D3—62no—
171C2×D3—124no—
172C3×C4—22no—
173C3×C4—22no—
174C3×C4—42no—
175C3×C4—42no—
176C3×C4—94no—
177C3×C4—91no—
178C3×C4—91no—
179C3×C4—182no—
180C3×C4#377311yescharacteristic
181C3×C4C2×GO+(4,𝔽2)11yescharacteristic
182Dic3—62no—
183Dic3—62no—
184C32—22no—
185C32—42no—
186C32C3×#42311yescharacteristic, complemented
187C23—91no—
188C2×C4—92no—
189C2×C4—91no—
190C2×C4—91no—
191C2×C4—91no—
192C2×C4—91no—
193C2×C4—91no—
194C2×C4—92no—
195C2×C4—184no—
196C2×C4C3×GO+(4,𝔽2)11yescharacteristic
197C8—64no—
198C2×C3—21no—
199C2×C3—22no—
200C2×C3—42no—
201C2×C3—42no—
202C2×C3—82no—
203C2×C3—92no—
204C2×C3—181no—
205C2×C3C2×#377311yescharacteristic
206D3—64no—
207D3—122no—
208C22—91no—
209C22—91no—
210C22—182no—
211C22C3×#377311yescharacteristic
212C4—94no—
213C4—91no—
214C4—91no—
215C4—182no—
216C4C3×#377311yescharacteristic
217C4C2×C3×GO+(4,𝔽2)11yescharacteristic
218C3—22no—
219C3—42no—
220C3#1059332811yescharacteristic, direct product factor
221C2—21no—
222C2—92no—
223C2—181no—
224C2C2×C3×#377311yescharacteristic
subgroupG/H
centerC3×C4C2×GO+(4,𝔽2)
commutatorC2×C3⋊D3C22×C3×C4
FrattiniC4C2×C3×GO+(4,𝔽2)
FittingC2×C33×C4D4
socleC2×C33C2×C4⋊C4
prime radicalC22×C3×C3⋊D3C2×C4
cosocleC4×C3⋊D3C23×C3
hypercenterC2×C3×C4GO+(4,𝔽2)
hypocenterC32C3×#423
LCSG ▷ C2×C3⋊D3 ▷ C32
UCSC1 ◁ C3×C4 ◁ C2×C3×C4

Complements

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

NH ≅ G/Ntype
3. C3×#63732C2semidirect
4. #10593328C3direct (N × H)
16. #63732C2×C3semidirect
48. C3×#157M16semidirect
108. #157C3×M16semidirect
122. C33#423semidirect
186. C32C3×#423semidirect
220. C3#10593328direct (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
C3×#638182
C3×#637322
#105933283
C3×#4239