Finite Group Databasebeta

← C3×#10593356 #13166697

Subgroups — C3×#10593356 #13166697

Order 1728.

464 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 464 conjugacy classes of subgroups — too many to draw. Only its 62 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×#63914C211yesmaximal, characteristic
2C3×#63870C211yesmaximal, characteristic, complemented
3C3×#63777C211yesmaximal, characteristic, complemented
4C3×#63739C212yesmaximal, complemented
5C3×#63739C212yesmaximal, complemented
6#10593356C311yesmaximal, characteristic, direct product factor
7C2×C32×D12—21no—
8C2×C3×#625—21no—
9C2×C3×#625C2212yes—
10C2×C3×#622C2212yes—
11C3×C4×C3⋊Dic3C2211yescharacteristic
12C32×Dic3⋊C4—21no—
13C3×#3724C2211yescharacteristic
14C3×#3724C2211yescharacteristic
15C3×Dic3⋊Dic3—21no—
16C3×#940—61no—
17C3×#889—61no—
18#63914C2×C311yescharacteristic
19#63870C2×C311yescharacteristic, complemented
20#63777C2×C311yescharacteristic, complemented
21#63739C2×C312yescomplemented
22#63739C2×C312yescomplemented
23C22×C32×D3—22no—
24C2×C33×C4D411yescharacteristic
25C2×C3×C3⋊Dic3D411yescharacteristic
26C2×C3×C3⋊Dic3C2311yescharacteristic
27C2×C32×Dic3—21no—
28C2×C32×Dic3—21no—
29C32×D12—42no—
30C3×#625—24no—
31C3×#625—42no—
32C3×#622—22no—
33C3×#490—91nomaximal
34C2×C32×D4—61no—
35C2×C3×#67—62no—
36C2×C3×#67—62no—
37C2×C3×D12—21no—
38C2×C3×D12—21no—
39C2×C3×D12—41no—
40C2×#625—21no—
41C2×#625C22×C312yes—
42C2×#622C22×C312yes—
43C32×C4⋊C4—61no—
44C4×C3⋊Dic3C22×C311yescharacteristic
45C3×C4⋊Dic3—61no—
46C3×Dic3⋊C4—21no—
47C3×Dic3⋊C4—21no—
48C3×Dic3⋊C4—41no—
49C3×Dic3⋊C4—61no—
50C3×Dic3⋊C4—61no—
51C3×Dic3⋊C4—61no—
52C3×C4×Dic3—61no—
53C3×C4×Dic3—61no—
54#3724C22×C311yescharacteristic
55#3724C22×C311yescharacteristic
56Dic3⋊Dic3—21no—
57C22×C33C2×D411yescharacteristic
58C2×C32×D3—42no—
59C2×C32×D3—42no—
60C2×C32×D3—42no—
61C33×C4—21no—
62C3×C3⋊Dic3—21no—
63C3×C3⋊Dic3#4112yes—
64C32×Dic3—41no—
65C32×Dic3—41no—
66C3×#127—91no—
67C3×#125—91no—
68#940—61no—
69#889—61no—
70C3×#102—92no—
71C3×#102—92no—
72C3×#97—91no—
73C23×C32—62no—
74C22×C3×D3—22no—
75C22×C3×D3—22no—
76C22×C3×D3—42no—
77C32×D4—122no—
78C2×C32×C4—21no—
79C2×C32×C4—21no—
80C2×C32×C4—41no—
81C2×C32×C4—41no—
82C2×C32×C4—61no—
83C2×C32×C4—61no—
84C2×C32×C4C3×D411yescharacteristic
85C2×C3⋊Dic3C3×D411yescharacteristic
86C2×C3⋊Dic3C23×C311yescharacteristic
87C3×#67—124no—
88C3×#67—124no—
89C2×C3×Dic3—21no—
90C2×C3×Dic3—21no—
91C2×C3×Dic3—21no—
92C2×C3×Dic3—21no—
93C2×C3×Dic3—41no—
94C2×C3×Dic3—41no—
95C2×C3×Dic3—61no—
96C2×C3×Dic3—61no—
97C2×C3×Dic3—61no—
98C2×C3×Dic3—61no—
99C3×D12—42no—
100C3×D12—42no—
101C3×D12—44no—
102#625—24no—
103#625—42no—
104#622—22no—
105#490—91noSylow
106C2×C33#12411yescharacteristic
107C2×C33AGL(1,ℤ/8)11yescharacteristic
108C2×C33AGL(1,ℤ/8)11yescharacteristic
109C32×D3—82no—
110C2×C3×D4—61no—
111C2×C3×D4—61no—
112C2×C3×D4—92no—
113C2×C3×D4—121no—
114C2×C3×D4—181no—
115C2×#67—62no—
116C2×#67—62no—
117C2×D12—21no—
118C2×C3×C8—92no—
119C3×C4⋊C4—61no—
120C3×C4⋊C4—61no—
121C3×C4⋊C4—91no—
122C3×C4⋊C4—91no—
123C3×C4⋊C4—121no—
124C3×C4⋊C4—181no—
125C3×C42—91no—
126C4⋊Dic3—61no—
127Dic3⋊C4—21no—
128Dic3⋊C4—61no—
129Dic3⋊C4—61no—
130Dic3⋊C4—61no—
131C4×Dic3—61no—
132C4×Dic3—61no—
133C22×C32—21no—
134C22×C32—21no—
135C22×C32—41no—
136C22×C32—41no—
137C22×C32—122no—
138C22×C32—122no—
139C22×C32—122no—
140C22×C32C2×C3×D411yescharacteristic
141C2×C3×D3—42no—
142C2×C3×D3—42no—
143C2×C3×D3—42no—
144C2×C3×D3—44no—
145C2×C3×D3—42no—
146C2×C3×D3—42no—
147C2×C3×D3—42no—
148C2×C3×D3—82no—
149C2×C3×D3—82no—
150C32×C4—22no—
151C32×C4—21no—
152C32×C4—41no—
153C32×C4—42no—
154C32×C4—81no—
155C32×C4—121no—
156C32×C4—121no—
157C3⋊Dic3—21no—
158C3⋊Dic3C3×#4112yes—
159C3×Dic3—41no—
160C3×Dic3—41no—
161C3×Dic3—42no—
162C3×Dic3—41no—
163C3×Dic3—41no—
164C3×Dic3—62no—
165C3×Dic3—62no—
166C3×Dic3—62no—
167C3×Dic3—81no—
168C3×Dic3—121no—
169#127—91no—
170#125—91no—
171#102—92no—
172#102—92no—
173#97—91no—
174C33#49011yescharacteristic, complemented, Sylow
175C23×C3—62no—
176C23×C3—62no—
177C23×C3—122no—
178C22×D3—22no—
179C3×D4—122no—
180C3×D4—122no—
181C3×D4—124no—
182C3×D4—184no—
183C3×D4—362no—
184C2×C3×C4—21no—
185C2×C3×C4—21no—
186C2×C3×C4—41no—
187C2×C3×C4—41no—
188C2×C3×C4—61no—
189C2×C3×C4—61no—
190C2×C3×C4—61no—
191C2×C3×C4—61no—
192C2×C3×C4—91no—
193C2×C3×C4—91no—
194C2×C3×C4—121no—
195C2×C3×C4—121no—
196C2×C3×C4GO+(4,𝔽2)11yescharacteristic
197#67—124no—
198#67—124no—
199C2×Dic3—21no—
200C2×Dic3—21no—
201C2×Dic3—61no—
202C2×Dic3—61no—
203C2×Dic3—61no—
204C2×Dic3—61no—
205D12—42no—
206C3×C8—182no—
207C2×C32—21no—
208C2×C32—21no—
209C2×C32—21no—
210C2×C32—21no—
211C2×C32—21no—
212C2×C32—21no—
213C2×C32—41no—
214C2×C32—41no—
215C2×C32—41no—
216C2×C32—41no—
217C2×C32—41no—
218C2×C32—41no—
219C2×C32—242no—
220C2×C32C3×#12411yescharacteristic
221C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
222C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
223C3×D3—82no—
224C3×D3—84no—
225C3×D3—82no—
226C2×D4—61no—
227C2×D4—92no—
228C2×D4—181no—
229C2×C8—92no—
230C4⋊C4—61no—
231C4⋊C4—91no—
232C4⋊C4—91no—
233C4⋊C4—181no—
234C42—91no—
235C22×C3—21no—
236C22×C3—21no—
237C22×C3—41no—
238C22×C3—41no—
239C22×C3—122no—
240C22×C3—122no—
241C22×C3—122no—
242C22×C3—124no—
243C22×C3—122no—
244C22×C3—122no—
245C22×C3—122no—
246C22×C3—242no—
247C22×C3—242no—
248C22×C3C2×GO+(4,𝔽2)11yescharacteristic
249C2×D3—42no—
250C2×D3—42no—
251C2×D3—42no—
252C3×C4—22no—
253C3×C4—21no—
254C3×C4—41no—
255C3×C4—42no—
256C3×C4—81no—
257C3×C4—92no—
258C3×C4—121no—
259C3×C4—121no—
260C3×C4—122no—
261C3×C4—121no—
262C3×C4—121no—
263C3×C4—181no—
264C3×C4—241no—
265Dic3—41no—
266Dic3—41no—
267Dic3—62no—
268Dic3—62no—
269Dic3—62no—
270Dic3—121no—
271C32—21no—
272C32—21no—
273C32—41no—
274C32—41no—
275C32C3×#49011yescharacteristic, complemented
276C23—62no—
277D4—122no—
278D4—184no—
279D4—362no—
280C2×C4—61no—
281C2×C4—61no—
282C2×C4—91no—
283C2×C4—91no—
284C2×C4C3×GO+(4,𝔽2)11yescharacteristic
285C8—182no—
286C2×C3—21no—
287C2×C3—21no—
288C2×C3—21no—
289C2×C3—21no—
290C2×C3—21no—
291C2×C3—21no—
292C2×C3—41no—
293C2×C3—41no—
294C2×C3—41no—
295C2×C3—41no—
296C2×C3—41no—
297C2×C3—41no—
298C2×C3—242no—
299C2×C3—244no—
300C2×C3—242no—
301C2×C3#6422111yescharacteristic
302C2×C3#6422411yescharacteristic
303C2×C3#6423611yescharacteristic
304D3—82no—
305C22—122no—
306C22—122no—
307C22—122no—
308C22C2×C3×GO+(4,𝔽2)11yescharacteristic
309C4—21no—
310C4—92no—
311C4—121no—
312C4—121no—
313C4—181no—
314C3—21no—
315C3—21no—
316C3—41no—
317C3—41no—
318C3#1059335611yescharacteristic, direct product factor
319C2—242no—
320C2C3×#6422111yescharacteristic
321C2C3×#6422411yescharacteristic
322C2C3×#6423611yescharacteristic
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×#490
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
2. C3×#63870C2semidirect
3. C3×#63777C2semidirect
4. C3×#63739C2semidirect
5. C3×#63739C2semidirect
6. #10593356C3direct (N × H)
19. #63870C2×C3semidirect
20. #63777C2×C3semidirect
21. #63739C2×C3semidirect
22. #63739C2×C3semidirect
174. C33#490semidirect
275. C32C3×#490semidirect
318. C3#10593356direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#639142
C3×#638702
C3×#637772
C3×#637392
#105933563
C3×#4909