Finite Group Databasebeta

← C3×#10590081 #13154931

Subgroups — C3×#10590081 #13154931

Order 1728.

336 conjugacy classes of subgroups; only the 34 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 34 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×#63953C211yesmaximal, characteristic
2C3×#63786C211yesmaximal, characteristic
3C3×#63581C211yesmaximal, characteristic
4#10590081C311yesmaximal, characteristic, direct product factor
5C2×C3×#638C2211yescharacteristic
6C2×C3×#627C411yescharacteristic, complemented
7C3×C22⋊#157—21no—
8C32×#217—21no—
9C32×#217—21no—
10C3×#3722—21no—
11C3×Dic32C411yescharacteristic, complemented
12C3×#885—61no—
13#63953C2×C311yescharacteristic
14#63786C2×C311yescharacteristic
15C3×#806—61no—
16#63581C2×C311yescharacteristic
17C23×C33D411yescharacteristic
18C22×C3×C3⋊D3D411yescharacteristic
19C2×C3×#157—41no—
20C3×#638—41no—
21C2×C3×C3⋊Dic3C2×C411yescharacteristic
22C2×C32×Dic3—21no—
23C2×C32×Dic3—21no—
24C2×C32×Dic3—41no—
25C3×#627—41no—
26C3×#354—91nomaximal
27C2×#638C22×C311yescharacteristic
28C2×C3×#67—61no—
29C2×C3×#67—61no—
30C2×C3×Dic6—61no—
31C2×#627C3×C411yescharacteristic, complemented
32C22⋊#157—21no—
33C32×C22⋊C4—61no—
34C32×C22⋊C4—61no—
35C3×#217—21no—
36C3×#217—21no—
37C3×#217—21no—
38C3×#217—21no—
39C3×#217—41no—
40C3×#217—41no—
41C3×(C2×D3)⋊C4—61no—
42C3×(C2×D3)⋊C4—61no—
43C3×C4×Dic3—61no—
44#3722—21no—
45Dic32C3×C411yescharacteristic, complemented
46C22×C33—21no—
47C22×C33—41no—
48C22×C33C22⋊C411yescharacteristic
49C2×C3×C3⋊D3—21no—
50C2×C3×C3⋊D3—41no—
51C3×#157—81no—
52C3×C3⋊Dic3—21no—
53C32×Dic3—41no—
54C32×Dic3—41no—
55C32×Dic3—81no—
56C3×#124—91no—
57#885—61no—
58C3×#99—91no—
59C3×#99—91no—
60#806—61no—
61C23×C32—21no—
62C23×C32—21no—
63C23×C32—41no—
64C23×C32—41no—
65C23×C32C3×D411yescharacteristic
66C22×C3⋊D3C3×D411yescharacteristic
67C22×C3×D3—61no—
68C22×C3×D3—61no—
69C2×#157—41no—
70C2×C32×C4—61no—
71C2×C32×C4—61no—
72C2×C32×C4—121no—
73#638—41no—
74C2×C3⋊Dic3C2×C3×C411yescharacteristic
75C3×#67—122no—
76C3×#67—121no—
77C3×#67—121no—
78C2×C3×Dic3—21no—
79C2×C3×Dic3—21no—
80C2×C3×Dic3—21no—
81C2×C3×Dic3—21no—
82C2×C3×Dic3—41no—
83C2×C3×Dic3—41no—
84C2×C3×Dic3—41no—
85C2×C3×Dic3—41no—
86C2×C3×Dic3—42no—
87C2×C3×Dic3—61no—
88C2×C3×Dic3—61no—
89C3×Dic6—122no—
90#627—41no—
91#354—91noSylow
92C2×C33—21no—
93C2×C33—41no—
94C2×C33#9911yescharacteristic
95C3×C3⋊D3—41no—
96C2×C3×Q8—91no—
97C2×C3×D4—91no—
98C2×#67—61no—
99C2×#67—61no—
100C2×Dic6—61no—
101C3×C22⋊C4—61no—
102C3×C22⋊C4—61no—
103C3×C22⋊C4—61no—
104C3×C22⋊C4—61no—
105C3×C22⋊C4—121no—
106C3×C22⋊C4—121no—
107C3×C22⋊C4—181no—
108C3×C22⋊C4—181no—
109C3×C42—91no—
110#217—21no—
111#217—21no—
112(C2×D3)⋊C4—61no—
113(C2×D3)⋊C4—61no—
114C4×Dic3—61no—
115C22×C32—21no—
116C22×C32—21no—
117C22×C32—21no—
118C22×C32—21no—
119C22×C32—21no—
120C22×C32—41no—
121C22×C32—41no—
122C22×C32—41no—
123C22×C32—41no—
124C22×C32—41no—
125C22×C32—41no—
126C22×C32—41no—
127C22×C32—41no—
128C22×C32—42no—
129C22×C32—81no—
130C22×C32—81no—
131C22×C32—81no—
132C22×C32—82no—
133C22×C32C3×C22⋊C411yescharacteristic
134C2×C3⋊D3—21no—
135C2×C3⋊D3—41no—
136C2×C3×D3—121no—
137C2×C3×D3—121no—
138C2×C3×D3—121no—
139C2×C3×D3—121no—
140C2×C3×D3—241no—
141#157—81no—
142C32×C4—121no—
143C32×C4—121no—
144C32×C4—241no—
145C3⋊Dic3—21no—
146C3×Dic3—41no—
147C3×Dic3—41no—
148C3×Dic3—41no—
149C3×Dic3—41no—
150C3×Dic3—62no—
151C3×Dic3—81no—
152C3×Dic3—81no—
153C3×Dic3—81no—
154C3×Dic3—82no—
155C3×Dic3—81no—
156C3×Dic3—121no—
157#124—91no—
158#99—91no—
159#99—91no—
160C33#35411yescharacteristic, complemented, Sylow
161C23×C3—21no—
162C23×C3—21no—
163C23×C3—41no—
164C23×C3—41no—
165C23×C3—91no—
166C23×C3GO+(4,𝔽2)11yescharacteristic
167C22×D3—61no—
168C22×D3—61no—
169C3×Q8—361no—
170C3×D4—361no—
171C2×C3×C4—61no—
172C2×C3×C4—61no—
173C2×C3×C4—61no—
174C2×C3×C4—61no—
175C2×C3×C4—91no—
176C2×C3×C4—121no—
177C2×C3×C4—121no—
178C2×C3×C4—121no—
179C2×C3×C4—121no—
180C2×C3×C4—122no—
181C2×C3×C4—361no—
182#67—122no—
183#67—121no—
184#67—121no—
185C2×Dic3—21no—
186C2×Dic3—21no—
187C2×Dic3—41no—
188C2×Dic3—61no—
189C2×Dic3—61no—
190Dic6—122no—
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—41no—
203C2×C32—42no—
204C2×C32—41no—
205C2×C32—81no—
206C2×C32—81no—
207C2×C32—81no—
208C2×C32—82no—
209C2×C32C3×#9911yescharacteristic
210C3⋊D3—41no—
211C3×D3—241no—
212C3×D3—241no—
213C2×Q8—91no—
214C2×D4—91no—
215C22⋊C4—61no—
216C22⋊C4—61no—
217C22⋊C4—181no—
218C22⋊C4—181no—
219C42—91no—
220C22×C3—21no—
221C22×C3—21no—
222C22×C3—21no—
223C22×C3—21no—
224C22×C3—21no—
225C22×C3—41no—
226C22×C3—41no—
227C22×C3—41no—
228C22×C3—41no—
229C22×C3—41no—
230C22×C3—41no—
231C22×C3—42no—
232C22×C3—41no—
233C22×C3—41no—
234C22×C3—81no—
235C22×C3—81no—
236C22×C3—81no—
237C22×C3—82no—
238C22×C3—181no—
239C22×C3—361no—
240C22×C3D32⋊C411yescharacteristic
241C2×D3—121no—
242C2×D3—121no—
243C2×D3—121no—
244C2×D3—121no—
245C2×D3—241no—
246C3×C4—121no—
247C3×C4—121no—
248C3×C4—121no—
249C3×C4—121no—
250C3×C4—181no—
251C3×C4—241no—
252C3×C4—241no—
253C3×C4—241no—
254C3×C4—242no—
255C3×C4—241no—
256C3×C4—721no—
257Dic3—41no—
258Dic3—41no—
259Dic3—62no—
260Dic3—81no—
261Dic3—121no—
262C32—21no—
263C32—21no—
264C32—41no—
265C32—41no—
266C32C3×#35411yescharacteristic, complemented
267C23—91no—
268C23C3×GO+(4,𝔽2)11yescharacteristic
269Q8—361no—
270D4—361no—
271C2×C4—61no—
272C2×C4—61no—
273C2×C4—91no—
274C2×C4—121no—
275C2×C4—361no—
276C2×C3—21no—
277C2×C3—21no—
278C2×C3—21no—
279C2×C3—21no—
280C2×C3—21no—
281C2×C3—41no—
282C2×C3—41no—
283C2×C3—41no—
284C2×C3—41no—
285C2×C3—41no—
286C2×C3—42no—
287C2×C3—41no—
288C2×C3—41no—
289C2×C3—41no—
290C2×C3—81no—
291C2×C3—81no—
292C2×C3—81no—
293C2×C3—82no—
294C2×C3—361no—
295C2×C3#6373811yescharacteristic
296D3—241no—
297D3—241no—
298C22—21no—
299C22—41no—
300C22—181no—
301C22—361no—
302C22C3×D32⋊C411yescharacteristic
303C4—121no—
304C4—121no—
305C4—181no—
306C4—241no—
307C4—721no—
308C3—21no—
309C3—21no—
310C3—41no—
311C3—41no—
312C3#1059008111yescharacteristic, direct product factor
313C2—21no—
314C2—41no—
315C2—361no—
316C2C3×#6373811yescharacteristic
subgroupG/H
centerC2×C3#63738
commutatorC2×C3⋊Dic3C2×C3×C4
FrattiniC23C3×GO+(4,𝔽2)
FittingC23×C33D4
socleC2×C33#99
prime radicalC2×C3×#638C22
cosocleC2×#638C22×C3
hypercenterC23×C3GO+(4,𝔽2)
hypocenterC32C3×#354
LCSG ▷ C2×C3⋊Dic3 ▷ C22×C32 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C3 ◁ C22×C3 ◁ C23×C3

Complements

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

NH ≅ G/Ntype
4. #10590081C3direct (N × H)
6. C2×C3×#627C4semidirect
11. C3×Dic32C4semidirect
31. C2×#627C3×C4semidirect
45. Dic32C3×C4semidirect
160. C33#354semidirect
266. C32C3×#354semidirect
312. C3#10590081direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#639532
C3×#637862
C3×#635812
#105900813
C3×#3549