Finite Group Databasebeta

← C3×#10590032 #13154882

Subgroups — C3×#10590032 #13154882

Order 1728.

300 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 300 conjugacy classes of subgroups — too many to draw. Only its 58 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×C4×#627C211yesmaximal, characteristic
2C3×C4×#622C211yesmaximal, characteristic
3C3×#63572C211yesmaximal, characteristic
4#10590032C311yesmaximal, characteristic, direct product factor
5C2×C3×#627C411yescharacteristic
6C2×C3×#622—21no—
7C3×C4×C3⋊Dic3C2211yescharacteristic
8C32×C4×Dic3—21no—
9C2×C32×C3⋊C8—21no—
10C3×#3724C411yescharacteristic
11C3×Dic3⋊Dic3—21no—
12C3×C4×Dic6—61no—
13C4×#627C2×C311yescharacteristic
14C4×#622C2×C311yescharacteristic
15C3×Dic3⋊C8—61no—
16#63572C2×C311yescharacteristic
17C2×C33×C4D411yescharacteristic
18C2×C3×C3⋊Dic3C2×C411yescharacteristic
19C2×C3×C3⋊Dic3D411yescharacteristic
20C2×C32×Dic3—21no—
21C2×C32×Dic3—21no—
22C3×#627—21no—
23C3×#627C812yescomplemented
24C3×#622—22no—
25C32×C3⋊C8—41no—
26C3×Q8⋊C8—91nomaximal
27C2×C3×Dic6—61no—
28C2×#627C3×C411yescharacteristic
29C2×#622—21no—
30C2×C32×C8—61no—
31C32×C42—61no—
32C4×C3⋊Dic3C22×C311yescharacteristic
33C3×C4⋊Dic3—61no—
34C3×Dic3⋊C4—61no—
35C3×Dic3⋊C4—61no—
36C3×C4×Dic3—21no—
37C3×C4×Dic3—21no—
38C3×C4×Dic3—41no—
39C3×C4×Dic3—61no—
40C3×C4×Dic3—61no—
41C2×C3×C3⋊C8—21no—
42C2×C3×C3⋊C8—21no—
43C2×C3×C3⋊C8—41no—
44#3724C3×C411yescharacteristic
45Dic3⋊Dic3—21no—
46C22×C33C22⋊C411yescharacteristic
47C33×C4SD1611yescharacteristic
48C33×C4Q1611yescharacteristic
49C3×C3⋊Dic3—21no—
50C3×C3⋊Dic3C2×C811yescharacteristic
51C3×C3⋊Dic3M1611yescharacteristic
52C32×Dic3—22no—
53C32×Dic3—41no—
54C3×C4×Q8—91no—
55C4×Dic6—61no—
56C3×C4⋊C8—91no—
57C3×C4×C8—91no—
58Dic3⋊C8—61no—
59C2×C32×C4—21no—
60C2×C32×C4—21no—
61C2×C32×C4—41no—
62C2×C32×C4—41no—
63C2×C32×C4—61no—
64C2×C32×C4—61no—
65C2×C32×C4C3×D411yescharacteristic
66C2×C3⋊Dic3C2×C3×C411yescharacteristic
67C2×C3⋊Dic3C3×D411yescharacteristic
68C2×C3×Dic3—21no—
69C2×C3×Dic3—21no—
70C2×C3×Dic3—21no—
71C2×C3×Dic3—21no—
72C2×C3×Dic3—41no—
73C2×C3×Dic3—41no—
74C2×C3×Dic3—61no—
75C2×C3×Dic3—61no—
76C2×C3×Dic3—61no—
77C2×C3×Dic3—61no—
78C3×Dic6—62no—
79C3×Dic6—62no—
80#627—21no—
81#627C3×C812yescomplemented
82#622—22no—
83C32×C8—121no—
84C3×C3⋊C8—41no—
85C3×C3⋊C8—42no—
86C3×C3⋊C8—41no—
87Q8⋊C8—91noSylow
88C2×C33C22⋊C811yescharacteristic
89C2×C33#10311yescharacteristic
90C2×C33#10411yescharacteristic
91C2×C3×Q8—91no—
92C2×Dic6—61no—
93C2×C3×C8—61no—
94C2×C3×C8—61no—
95C2×C3×C8—121no—
96C2×C3×C8—181no—
97C3×C4⋊C4—91no—
98C3×C4⋊C4—181no—
99C3×C42—61no—
100C3×C42—61no—
101C3×C42—91no—
102C3×C42—121no—
103C4⋊Dic3—61no—
104Dic3⋊C4—61no—
105Dic3⋊C4—61no—
106C4×Dic3—21no—
107C4×Dic3—61no—
108C4×Dic3—61no—
109C2×C3⋊C8—21no—
110C22×C32—21no—
111C22×C32—21no—
112C22×C32—41no—
113C22×C32—41no—
114C22×C32C3×C22⋊C411yescharacteristic
115C32×C4—21no—
116C32×C4—21no—
117C32×C4—21no—
118C32×C4—21no—
119C32×C4—41no—
120C32×C4—41no—
121C32×C4—41no—
122C32×C4—41no—
123C32×C4—62no—
124C32×C4—121no—
125C32×C4C3×SD1611yescharacteristic
126C32×C4C3×Q1611yescharacteristic
127C3⋊Dic3—21no—
128C3⋊Dic3C2×C3×C811yescharacteristic
129C3⋊Dic3C3×M1611yescharacteristic
130C3×Dic3—22no—
131C3×Dic3—22no—
132C3×Dic3—42no—
133C3×Dic3—41no—
134C3×Dic3—42no—
135C3×Dic3—41no—
136C3×Dic3—61no—
137C3×Dic3—61no—
138C3×Dic3—61no—
139C3×Dic3—61no—
140C3×Dic3—121no—
141C3×Dic3—121no—
142C4×Q8—91no—
143C4⋊C8—91no—
144C4×C8—91no—
145C33Q8⋊C811yescharacteristic, complemented, Sylow
146C3×Q8—92no—
147C3×Q8—181no—
148C2×C3×C4—21no—
149C2×C3×C4—21no—
150C2×C3×C4—41no—
151C2×C3×C4—41no—
152C2×C3×C4—61no—
153C2×C3×C4—61no—
154C2×C3×C4—61no—
155C2×C3×C4—61no—
156C2×C3×C4—91no—
157C2×C3×C4—91no—
158C2×C3×C4—121no—
159C2×C3×C4—121no—
160C2×C3×C4GO+(4,𝔽2)11yescharacteristic
161C2×Dic3—21no—
162C2×Dic3—21no—
163C2×Dic3—61no—
164C2×Dic3—61no—
165C2×Dic3—61no—
166C2×Dic3—61no—
167Dic6—62no—
168Dic6—62no—
169C3×C8—121no—
170C3×C8—122no—
171C3×C8—121no—
172C3×C8—182no—
173C3⋊C8—41no—
174C2×C32—21no—
175C2×C32—21no—
176C2×C32—21no—
177C2×C32—21no—
178C2×C32—21no—
179C2×C32—21no—
180C2×C32—41no—
181C2×C32—41no—
182C2×C32—41no—
183C2×C32—41no—
184C2×C32—41no—
185C2×C32—41no—
186C2×C32C3×C22⋊C811yescharacteristic
187C2×C32C3×#10311yescharacteristic
188C2×C32C3×#10411yescharacteristic
189C2×Q8—91no—
190C2×C8—61no—
191C2×C8—181no—
192C4⋊C4—91no—
193C4⋊C4—181no—
194C42—61no—
195C42—91no—
196C22×C3—21no—
197C22×C3—21no—
198C22×C3—41no—
199C22×C3—41no—
200C22×C3D32⋊C411yescharacteristic
201C3×C4—21no—
202C3×C4—21no—
203C3×C4—21no—
204C3×C4—21no—
205C3×C4—41no—
206C3×C4—41no—
207C3×C4—41no—
208C3×C4—41no—
209C3×C4—62no—
210C3×C4—62no—
211C3×C4—91no—
212C3×C4—91no—
213C3×C4—122no—
214C3×C4—121no—
215C3×C4—122no—
216C3×C4—121no—
217C3×C4—181no—
218C3×C4#377511yescharacteristic
219C3×C4#377611yescharacteristic
220Dic3—22no—
221Dic3—41no—
222Dic3—61no—
223Dic3—61no—
224Dic3—61no—
225Dic3—61no—
226Dic3—121no—
227Dic3—121no—
228C32—21no—
229C32—21no—
230C32—41no—
231C32—41no—
232C32C3×Q8⋊C811yescharacteristic, complemented
233Q8—92no—
234Q8—181no—
235C2×C4—61no—
236C2×C4—61no—
237C2×C4—91no—
238C2×C4—91no—
239C2×C4C3×GO+(4,𝔽2)11yescharacteristic
240C8—121no—
241C8—182no—
242C2×C3—21no—
243C2×C3—21no—
244C2×C3—21no—
245C2×C3—21no—
246C2×C3—21no—
247C2×C3—21no—
248C2×C3—41no—
249C2×C3—41no—
250C2×C3—41no—
251C2×C3—41no—
252C2×C3—41no—
253C2×C3—41no—
254C2×C3D32⋊C811yescharacteristic
255C2×C3#6373111yescharacteristic
256C2×C3#6374011yescharacteristic
257C22C3×D32⋊C411yescharacteristic
258C4—62no—
259C4—91no—
260C4—91no—
261C4—121no—
262C4—181no—
263C4C3×#377511yescharacteristic
264C4C3×#377611yescharacteristic
265C3—21no—
266C3—21no—
267C3—41no—
268C3—41no—
269C3#1059003211yescharacteristic, direct product factor
270C2C3×D32⋊C811yescharacteristic
271C2C3×#6373111yescharacteristic
272C2C3×#6374011yescharacteristic
subgroupG/H
center, hypercenterC2×C3×C4GO+(4,𝔽2)
commutatorC3⋊Dic3C2×C3×C8
FrattiniC2×C4C3×GO+(4,𝔽2)
FittingC2×C33×C4D4
socle, prime radicalC22×C33C22⋊C4
cosocleC4×C3⋊Dic3C22×C3
hypocenterC32C3×Q8⋊C8
LCSG ▷ C3⋊Dic3 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C3×C4

Complements

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

NH ≅ G/Ntype
4. #10590032C3direct (N × H)
23. C3×#627C8semidirect
81. #627C3×C8semidirect
145. C33Q8⋊C8semidirect
232. C32C3×Q8⋊C8semidirect
269. C3#10590032direct (N × H)

Maximal subgroups

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

subgroupindex
C3×C4×#6272
C3×C4×#6222
C3×#635722
#105900323
C3×Q8⋊C89