Finite Group Databasebeta

← C3×#10593442 #13166809

Subgroups — C3×#10593442 #13166809

Order 1728.

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

normal subgroups only

This group has 540 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×#3775C214yesmaximal, complemented
2C3×#63919C211yesmaximal, characteristic, complemented
3C3×#63914C211yesmaximal, characteristic
4C3×C4×#622C211yesmaximal, characteristic, complemented
5#10593442C311yesmaximal, characteristic, direct product factor
6C2×C32×D12—21no—
7C2×C3×#627C2212yes—
8C2×C3×#625—21no—
9C2×C3×#625C2212yes—
10C2×C3×#622C2212yes—
11C3×#3775—28no—
12C3×C4×C3⋊Dic3C2211yescharacteristic
13C32×C4⋊Dic3—21no—
14C3×#3724—21no—
15C2×#3775C2×C314yescomplemented
16C3×#940—61no—
17C3×#888—61no—
18#63919C2×C311yescharacteristic, complemented
19#63914C2×C311yescharacteristic
20C4×#622C2×C311yescharacteristic, complemented
21C22×C32×D3—22no—
22C2×C33×C4D411yescharacteristic
23C2×C3×C3⋊Dic3D411yescharacteristic
24C2×C3×C3⋊Dic3C2311yescharacteristic
25C2×C32×Dic3—22no—
26C32×D12—42no—
27C3×#627—24no—
28C3×#625—24no—
29C3×#625—42no—
30C3×#622D414yescomplemented
31C3×#492—91nomaximal
32C2×C32×D4—61no—
33C2×C3×#67—62no—
34C2×C3×#67—62no—
35C2×C3×D12—21no—
36C2×C3×D12—21no—
37C2×C3×D12—41no—
38C2×C3×Dic6—62no—
39C2×#627C22×C312yes—
40C2×#625—21no—
41C2×#625C22×C312yes—
42C2×#622C22×C312yes—
43#3775—28no—
44C32×C4⋊C4—61no—
45C4×C3⋊Dic3C22×C311yescharacteristic
46C3×C4⋊Dic3—21no—
47C3×C4⋊Dic3—21no—
48C3×C4⋊Dic3—41no—
49C3×Dic3⋊C4—62no—
50C3×C4×Dic3—61no—
51C3×C4×Dic3—61no—
52#3724—21no—
53C22×C33C2×D411yescharacteristic
54C2×C32×D3—42no—
55C2×C32×D3—42no—
56C2×C32×D3—42no—
57C33×C4SD1612yescomplemented
58C3×C3⋊Dic3SD1612yescomplemented
59C3×C3⋊Dic3C2×D412yes—
60C32×Dic3—42no—
61C2×C3×SD16—94no—
62C3×C4⋊Q8—91no—
63C3×#127—91no—
64#940—61no—
65#888—61no—
66C3×C4×C8—91no—
67C23×C32—62no—
68C22×C3×D3—22no—
69C22×C3×D3—22no—
70C22×C3×D3—42no—
71C32×D4—122no—
72C2×C32×C4—21no—
73C2×C32×C4—21no—
74C2×C32×C4—41no—
75C2×C32×C4—41no—
76C2×C32×C4—62no—
77C2×C32×C4C3×D411yescharacteristic
78C2×C3⋊Dic3C3×D411yescharacteristic
79C2×C3⋊Dic3C23×C311yescharacteristic
80C3×#67—124no—
81C3×#67—124no—
82C2×C3×Dic3—22no—
83C2×C3×Dic3—22no—
84C2×C3×Dic3—42no—
85C2×C3×Dic3—61no—
86C2×C3×Dic3—61no—
87C2×C3×Dic3—61no—
88C2×C3×Dic3—61no—
89C3×D12—42no—
90C3×D12—44no—
91C3×D12—42no—
92C3×Dic6—124no—
93#627—24no—
94#625—24no—
95#625—42no—
96#622C3×D414yescomplemented
97#492—91noSylow
98C2×C33#12711yescharacteristic
99C2×C33C2×SD1611yescharacteristic
100C2×C33C2×SD1611yescharacteristic
101C32×D3—82no—
102C2×C3×Q8—92no—
103C2×C3×D4—61no—
104C2×C3×D4—61no—
105C2×C3×D4—92no—
106C2×C3×D4—121no—
107C2×C3×D4—181no—
108C2×#67—62no—
109C2×#67—62no—
110C2×D12—21no—
111C2×Dic6—62no—
112C3×SD16—188no—
113C2×C3×C8—92no—
114C3×C4⋊C4—61no—
115C3×C4⋊C4—61no—
116C3×C4⋊C4—121no—
117C3×C4⋊C4—181no—
118C3×C42—91no—
119C4⋊Dic3—21no—
120Dic3⋊C4—62no—
121C4×Dic3—61no—
122C4×Dic3—61no—
123C22×C32—21no—
124C22×C32—21no—
125C22×C32—41no—
126C22×C32—41no—
127C22×C32—122no—
128C22×C32—122no—
129C22×C32—122no—
130C22×C32C2×C3×D411yescharacteristic
131C2×C3×D3—42no—
132C2×C3×D3—42no—
133C2×C3×D3—42no—
134C2×C3×D3—44no—
135C2×C3×D3—42no—
136C2×C3×D3—42no—
137C2×C3×D3—42no—
138C2×C3×D3—82no—
139C2×C3×D3—82no—
140C32×C4—22no—
141C32×C4—22no—
142C32×C4—42no—
143C32×C4—42no—
144C32×C4—122no—
145C32×C4C3×SD1612yescomplemented
146C3⋊Dic3C3×SD1612yescomplemented
147C3⋊Dic3C2×C3×D412yes—
148C3×Dic3—42no—
149C3×Dic3—42no—
150C3×Dic3—62no—
151C3×Dic3—62no—
152C3×Dic3—62no—
153C3×Dic3—62no—
154C3×Dic3—82no—
155C2×SD16—94no—
156C4⋊Q8—91no—
157#127—91no—
158C4×C8—91no—
159C33#49211yescharacteristic, complemented, Sylow
160C23×C3—62no—
161C23×C3—62no—
162C23×C3—122no—
163C22×D3—22no—
164C3×Q8—184no—
165C3×D4—122no—
166C3×D4—124no—
167C3×D4—122no—
168C3×D4—184no—
169C3×D4—362no—
170C2×C3×C4—21no—
171C2×C3×C4—21no—
172C2×C3×C4—41no—
173C2×C3×C4—41no—
174C2×C3×C4—62no—
175C2×C3×C4—62no—
176C2×C3×C4—91no—
177C2×C3×C4—91no—
178C2×C3×C4—122no—
179C2×C3×C4GO+(4,𝔽2)11yescharacteristic
180#67—124no—
181#67—124no—
182C2×Dic3—22no—
183C2×Dic3—61no—
184C2×Dic3—61no—
185C2×Dic3—61no—
186C2×Dic3—61no—
187D12—42no—
188Dic6—124no—
189C3×C8—94no—
190C2×C32—21no—
191C2×C32—21no—
192C2×C32—21no—
193C2×C32—21no—
194C2×C32—21no—
195C2×C32—21no—
196C2×C32—41no—
197C2×C32—41no—
198C2×C32—41no—
199C2×C32—41no—
200C2×C32—41no—
201C2×C32—41no—
202C2×C32—242no—
203C2×C32C3×#12711yescharacteristic
204C2×C32C2×C3×SD1611yescharacteristic
205C2×C32C2×C3×SD1611yescharacteristic
206C3×D3—82no—
207C3×D3—84no—
208C3×D3—82no—
209C2×Q8—92no—
210C2×D4—61no—
211C2×D4—92no—
212C2×D4—181no—
213SD16—188no—
214C2×C8—92no—
215C4⋊C4—61no—
216C4⋊C4—181no—
217C42—91no—
218C22×C3—21no—
219C22×C3—21no—
220C22×C3—41no—
221C22×C3—41no—
222C22×C3—122no—
223C22×C3—122no—
224C22×C3—122no—
225C22×C3—124no—
226C22×C3—122no—
227C22×C3—122no—
228C22×C3—122no—
229C22×C3—242no—
230C22×C3—242no—
231C22×C3C2×GO+(4,𝔽2)11yescharacteristic
232C2×D3—42no—
233C2×D3—42no—
234C2×D3—42no—
235C3×C4—22no—
236C3×C4—22no—
237C3×C4—42no—
238C3×C4—42no—
239C3×C4—92no—
240C3×C4—92no—
241C3×C4—122no—
242C3×C4—122no—
243C3×C4—242no—
244C3×C4#377512yescomplemented
245Dic3—42no—
246Dic3—62no—
247Dic3—62no—
248Dic3—62no—
249Dic3—62no—
250C32—21no—
251C32—21no—
252C32—41no—
253C32—41no—
254C32C3×#49211yescharacteristic, complemented
255C23—62no—
256Q8—184no—
257D4—122no—
258D4—184no—
259D4—362no—
260C2×C4—62no—
261C2×C4—91no—
262C2×C4—91no—
263C2×C4C3×GO+(4,𝔽2)11yescharacteristic
264C8—94no—
265C2×C3—21no—
266C2×C3—21no—
267C2×C3—21no—
268C2×C3—21no—
269C2×C3—21no—
270C2×C3—21no—
271C2×C3—41no—
272C2×C3—41no—
273C2×C3—41no—
274C2×C3—41no—
275C2×C3—41no—
276C2×C3—41no—
277C2×C3—242no—
278C2×C3—244no—
279C2×C3—242no—
280C2×C3#6422711yescharacteristic
281C2×C3#6423111yescharacteristic
282C2×C3C2×#377511yescharacteristic
283D3—82no—
284C22—122no—
285C22—122no—
286C22—122no—
287C22C2×C3×GO+(4,𝔽2)11yescharacteristic
288C4—92no—
289C4—92no—
290C4—122no—
291C4C3×#377512yescomplemented
292C3—21no—
293C3—21no—
294C3—41no—
295C3—41no—
296C3#1059344211yescharacteristic, direct product factor
297C2—242no—
298C2C3×#6422711yescharacteristic
299C2C3×#6423111yescharacteristic
300C2C2×C3×#377511yescharacteristic
subgroupG/H
centerC22×C3C2×GO+(4,𝔽2)
commutator, cosocleC2×C3⋊Dic3C23×C3
FrattiniC22C2×C3×GO+(4,𝔽2)
FittingC2×C33×C4D4
socleC22×C33C2×D4
prime radicalC3×#63914C2
hypercenterC2×C3×C4GO+(4,𝔽2)
hypocenterC32C3×#492
LCSG ▷ C2×C3⋊Dic3 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4

Complements

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

NH ≅ G/Ntype
1. C2×C3×#3775C2semidirect
2. C3×#63919C2semidirect
4. C3×C4×#622C2semidirect
5. #10593442C3direct (N × H)
15. C2×#3775C2×C3semidirect
18. #63919C2×C3semidirect
20. C4×#622C2×C3semidirect
30. C3×#622D4semidirect
57. C33×C4SD16semidirect
58. C3×C3⋊Dic3SD16semidirect
96. #622C3×D4semidirect
145. C32×C4C3×SD16semidirect
146. C3⋊Dic3C3×SD16semidirect
159. C33#492semidirect
244. C3×C4#3775semidirect
254. C32C3×#492semidirect
291. C4C3×#3775semidirect
296. C3#10593442direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#37752
C3×#639192
C3×#639142
C3×C4×#6222
#105934423
C3×#4929