Subgroups — C₃×#68724 #11785512

Order 1152.

547 conjugacy classes of subgroups.

The subgroup lattice has 298 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#907	C₂	1	1	yes	maximal, characteristic
2	C₃×#5132	C₂	1	2	yes	maximal
3	C₂×C₃×#822	C₂	1	2	yes	maximal, complemented
4	C₃²×#338	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×C₃⋊#338	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#1813	—	3	1	no	maximal
7	#68724	C₃	1	1	yes	maximal, characteristic, direct product factor
8	C₂×C₃×#235	C₂²	1	1	yes	characteristic
9	C₂×C₃²×M₁₆	C₂²	1	2	yes	—
10	C₂×C₃²×M₁₆	C₂²	1	1	yes	characteristic
11	C₂×C₃×#208	C₂²	1	2	yes	—
12	C₂²×C₃×C₃⋊C₈	C₂²	1	1	yes	characteristic
13	C₃×#907	—	2	4	no	—
14	C₂×C₃×ΓL(1,𝔽₂₅)	C₄	1	2	yes	—
15	C₂×C₃×D₃×C₈	C₄	1	2	yes	—
16	C₃×#824	—	2	2	no	—
17	C₃×#823	—	2	2	no	—
18	C₃×#822	—	2	4	no	—
19	C₂×C₃×#131	—	3	1	no	—
20	C₂×#907	C₂×C₃	1	1	yes	characteristic
21	C₂×C₃×#108	—	3	2	no	—
22	C₃×#413	—	3	2	no	—
23	#5132	C₂×C₃	1	2	yes	—
24	C₂×#822	C₂×C₃	1	2	yes	complemented
25	C₃×#338	—	2	1	no	—
26	C₃×#338	—	3	1	no	—
27	C₃×#338	D₃	1	1	yes	characteristic, complemented
28	C₃×#338	C₂×C₃	1	1	yes	characteristic, complemented
29	C₃⋊#338	C₂×C₃	1	1	yes	characteristic, complemented
30	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
31	C₂×C₃×#67	Q₈	1	2	yes	—
32	C₃×#235	—	2	2	no	—
33	C₃×#235	—	2	2	no	—
34	C₂×C₃×D₁₂	D₄	1	1	yes	characteristic
35	C₂×C₃×C₄×D₃	C₂×C₄	1	2	yes	—
36	C₂×C₃×Dic₆	D₄	1	1	yes	characteristic
37	C₃²×M₁₆	—	2	2	no	—
38	C₃²×M₁₆	—	2	2	no	—
39	C₃²×M₁₆	—	2	2	no	—
40	C₂×C₃²×C₈	—	2	2	no	—
41	C₂×C₃²×C₈	C₂×C₄	1	2	yes	—
42	C₃×#208	—	2	2	no	—
43	C₃×#208	—	2	2	no	—
44	C₂×C₃×C₃⋊C₈	—	2	2	no	—
45	C₂×C₃×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
46	C₂×C₃×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
47	C₂×C₃×C₃⋊C₈	D₄	1	4	yes	—
48	C₃×ΓL(1,𝔽₂₅)	—	2	4	no	—
49	C₃×D₃×C₈	—	2	4	no	—
50	#1813	—	3	1	no	Sylow
51	C₂×C₃×#41	—	3	1	no	—
52	C₂×#235	C₂²×C₃	1	1	yes	characteristic
53	C₃×#131	—	6	4	no	—
54	C₂×C₃×M₁₆	—	2	2	no	—
55	C₂×C₃×M₁₆	—	2	1	no	—
56	C₂×C₃×M₁₆	—	3	2	no	—
57	C₂×C₃×M₁₆	—	3	2	no	—
58	C₂×C₃×M₁₆	C₂×D₃	1	2	yes	—
59	C₂×C₃×M₁₆	C₂×D₃	1	1	yes	characteristic
60	C₂×C₃×M₁₆	C₂²×C₃	1	2	yes	—
61	C₂×C₃×M₁₆	C₂²×C₃	1	1	yes	characteristic
62	C₂²×C₃×C₈	—	3	1	no	—
63	C₂²×C₃×C₈	—	3	2	no	—
64	C₂×#208	C₂²×C₃	1	2	yes	—
65	C₂²×C₃⋊C₈	C₂²×C₃	1	1	yes	characteristic
66	#907	—	2	4	no	—
67	C₂×ΓL(1,𝔽₂₅)	C₃×C₄	1	2	yes	—
68	C₂×D₃×C₈	C₃×C₄	1	2	yes	—
69	C₃×#108	—	6	4	no	—
70	C₃×#101	—	6	2	no	—
71	C₃×#100	—	6	2	no	—
72	#824	—	2	2	no	—
73	#823	—	2	2	no	—
74	#822	—	2	4	no	—
75	C₂³×C₃²	C₂×Q₈	1	1	yes	characteristic
76	C₂²×C₃×D₃	C₄⋊C₄	1	2	yes	—
77	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₃²×C₄	C₂×D₄	1	2	yes	—
79	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
80	C₂×C₃²×C₄	#41	1	2	yes	—
81	C₃×#67	—	2	4	no	—
82	C₂×C₃×Dic₃	C₄⋊C₄	1	2	yes	—
83	C₃×D₁₂	—	2	2	no	—
84	C₃×C₄×D₃	—	2	2	no	—
85	C₃×C₄×D₃	—	2	2	no	—
86	C₃×Dic₆	—	2	2	no	—
87	C₃²×C₈	—	2	2	no	—
88	C₃²×C₈	—	4	2	no	—
89	C₃×C₃⋊C₈	—	4	2	no	—
90	C₃×C₃⋊C₈	C₂²⋊C₄	1	2	yes	—
91	C₃×C₃⋊C₈	C₂²⋊C₄	1	2	yes	—
92	C₂×#131	—	3	1	no	—
93	C₂×#108	—	3	2	no	—
94	#413	—	3	2	no	—
95	#338	—	3	1	no	—
96	#338	C₃×D₃	1	1	yes	characteristic, complemented
97	C₃×#41	—	6	2	no	—
98	C₃×#41	—	6	2	no	—
99	C₂×C₃×Q₈	—	3	1	no	—
100	C₂×C₃×D₄	—	3	2	no	—
101	C₂×C₃×D₄	—	3	1	no	—
102	C₂²×C₃×C₄	—	2	1	no	—
103	C₂²×C₃×C₄	—	3	2	no	—
104	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
105	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
106	C₂×#67	C₃×Q₈	1	2	yes	—
107	#235	—	2	2	no	—
108	#235	—	2	2	no	—
109	C₂×D₁₂	C₃×D₄	1	1	yes	characteristic
110	C₂×C₄×D₃	C₂×C₃×C₄	1	2	yes	—
111	C₂×Dic₆	C₃×D₄	1	1	yes	characteristic
112	C₃×M₁₆	—	2	2	no	—
113	C₃×M₁₆	—	2	2	no	—
114	C₃×M₁₆	—	2	2	no	—
115	C₃×M₁₆	—	2	2	no	—
116	C₃×M₁₆	—	2	2	no	—
117	C₃×M₁₆	—	2	2	no	—
118	C₃×M₁₆	—	4	2	no	—
119	C₃×M₁₆	—	4	2	no	—
120	C₃×M₁₆	—	4	2	no	—
121	C₃×M₁₆	—	6	4	no	—
122	C₃×M₁₆	—	6	2	no	—
123	C₃×M₁₆	—	6	2	no	—
124	C₂×C₃×C₈	—	2	4	no	—
125	C₂×C₃×C₈	—	2	2	no	—
126	C₂×C₃×C₈	—	2	2	no	—
127	C₂×C₃×C₈	—	2	2	no	—
128	C₂×C₃×C₈	—	3	1	no	—
129	C₂×C₃×C₈	—	3	1	no	—
130	C₂×C₃×C₈	—	3	4	no	—
131	C₂×C₃×C₈	—	6	4	no	—
132	C₂×C₃×C₈	—	6	2	no	—
133	C₂×C₃×C₈	C₄×D₃	1	2	yes	—
134	C₂×C₃×C₈	C₂×C₃×C₄	1	2	yes	—
135	#208	—	2	2	no	—
136	#208	—	2	2	no	—
137	C₂×C₃⋊C₈	—	2	2	no	—
138	C₂×C₃⋊C₈	C₂×C₃×C₄	1	1	yes	characteristic
139	C₂×C₃⋊C₈	C₂×C₃×C₄	1	1	yes	characteristic
140	C₂×C₃⋊C₈	C₃×D₄	1	4	yes	—
141	ΓL(1,𝔽₂₅)	—	2	4	no	—
142	D₃×C₈	—	2	4	no	—
143	C₂²×C₃²	—	4	1	no	—
144	C₂²×C₃²	C₂×C₄⋊C₄	1	1	yes	characteristic
145	C₂²×C₃²	C₂²⋊Q₈	1	2	yes	—
146	C₂×C₃×D₃	—	2	2	no	—
147	C₂×C₃×D₃	—	4	2	no	—
148	C₃²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
149	C₃²×C₄	#117	1	1	yes	characteristic
150	C₃²×C₄	#121	1	2	yes	—
151	C₃×Dic₃	—	2	2	no	—
152	C₂×#41	—	3	1	no	—
153	#131	—	6	4	no	—
154	C₂×M₁₆	—	3	2	no	—
155	C₂×M₁₆	—	3	2	no	—
156	C₂×M₁₆	C₂×C₃×D₃	1	2	yes	—
157	C₂×M₁₆	C₂×C₃×D₃	1	1	yes	characteristic
158	C₂²×C₈	—	3	1	no	—
159	C₂²×C₈	—	3	2	no	—
160	#108	—	6	4	no	—
161	#101	—	6	2	no	—
162	#100	—	6	2	no	—
163	C₂³×C₃	—	2	1	no	—
164	C₂³×C₃	—	3	2	no	—
165	C₂³×C₃	D₃×Q₈	1	1	yes	characteristic
166	C₂³×C₃	C₂×C₃×Q₈	1	1	yes	characteristic
167	C₂²×D₃	C₃×C₄⋊C₄	1	2	yes	—
168	C₃×Q₈	—	6	2	no	—
169	C₃×D₄	—	6	4	no	—
170	C₃×D₄	—	6	2	no	—
171	C₂×C₃×C₄	—	2	1	no	—
172	C₂×C₃×C₄	—	2	2	no	—
173	C₂×C₃×C₄	—	2	2	no	—
174	C₂×C₃×C₄	—	2	1	no	—
175	C₂×C₃×C₄	—	3	2	no	—
176	C₂×C₃×C₄	—	6	2	no	—
177	C₂×C₃×C₄	—	6	2	no	—
178	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
179	C₂×C₃×C₄	#235	1	2	yes	—
180	C₂×C₃×C₄	D₃×D₄	1	2	yes	—
181	C₂×C₃×C₄	D₃×D₄	1	1	yes	characteristic
182	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
183	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
184	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
185	C₂×C₃×C₄	C₃×#41	1	2	yes	—
186	#67	—	2	4	no	—
187	C₂×Dic₃	C₃×C₄⋊C₄	1	2	yes	—
188	D₁₂	—	2	2	no	—
189	C₄×D₃	—	2	2	no	—
190	C₄×D₃	—	2	2	no	—
191	Dic₆	—	2	2	no	—
192	C₃×C₈	—	2	2	no	—
193	C₃×C₈	—	2	2	no	—
194	C₃×C₈	—	3	2	no	—
195	C₃×C₈	—	3	2	no	—
196	C₃×C₈	—	4	2	no	—
197	C₃×C₈	—	4	4	no	—
198	C₃×C₈	—	4	2	no	—
199	C₃×C₈	—	4	2	no	—
200	C₃×C₈	—	12	2	no	—
201	C₃⋊C₈	—	4	2	no	—
202	C₃⋊C₈	C₃×C₂²⋊C₄	1	2	yes	—
203	C₃⋊C₈	C₃×C₂²⋊C₄	1	2	yes	—
204	C₂×C₃²	—	2	1	no	—
205	C₂×C₃²	—	2	2	no	—
206	C₂×C₃²	#380	1	1	yes	characteristic
207	C₃×D₃	—	4	2	no	—
208	#41	—	6	2	no	—
209	#41	—	6	2	no	—
210	C₂×Q₈	—	3	1	no	—
211	C₂×D₄	—	3	2	no	—
212	C₂×D₄	—	3	1	no	—
213	C₂²×C₄	—	3	2	no	—
214	C₂²×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
215	M₁₆	—	2	2	no	—
216	M₁₆	—	2	2	no	—
217	M₁₆	—	2	2	no	—
218	M₁₆	—	6	4	no	—
219	M₁₆	—	6	2	no	—
220	M₁₆	—	6	2	no	—
221	C₂×C₈	—	2	2	no	—
222	C₂×C₈	—	3	1	no	—
223	C₂×C₈	—	3	1	no	—
224	C₂×C₈	—	3	4	no	—
225	C₂×C₈	—	6	4	no	—
226	C₂×C₈	—	6	2	no	—
227	C₂×C₈	C₃×C₄×D₃	1	2	yes	—
228	C₂²×C₃	—	2	1	no	—
229	C₂²×C₃	—	2	2	no	—
230	C₂²×C₃	—	4	1	no	—
231	C₂²×C₃	—	4	1	no	—
232	C₂²×C₃	—	6	2	no	—
233	C₂²×C₃	—	8	1	no	—
234	C₂²×C₃	—	12	2	no	—
235	C₂²×C₃	D₃×C₄⋊C₄	1	1	yes	characteristic
236	C₂²×C₃	#896	1	2	yes	—
237	C₂²×C₃	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
238	C₂²×C₃	C₃×C₂²⋊Q₈	1	2	yes	—
239	C₂×D₃	—	2	2	no	—
240	C₂×D₃	—	4	2	no	—
241	C₃×C₄	—	2	1	no	—
242	C₃×C₄	—	2	2	no	—
243	C₃×C₄	—	2	1	no	—
244	C₃×C₄	—	6	2	no	—
245	C₃×C₄	#872	1	1	yes	characteristic
246	C₃×C₄	D₃×C₂²⋊C₄	1	1	yes	characteristic
247	C₃×C₄	#884	1	2	yes	—
248	C₃×C₄	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
249	C₃×C₄	C₃×#117	1	1	yes	characteristic
250	C₃×C₄	C₃×#121	1	2	yes	—
251	Dic₃	—	2	2	no	—
252	C₃²	#1813	1	1	yes	characteristic, complemented, Sylow
253	C₂³	—	3	2	no	—
254	C₂³	C₃×D₃×Q₈	1	1	yes	characteristic
255	Q₈	—	6	2	no	—
256	D₄	—	6	4	no	—
257	D₄	—	6	2	no	—
258	C₂×C₄	—	3	2	no	—
259	C₂×C₄	—	6	2	no	—
260	C₂×C₄	—	6	2	no	—
261	C₂×C₄	C₂×C₃×C₄×D₃	1	1	yes	characteristic
262	C₂×C₄	C₃×#235	1	2	yes	—
263	C₂×C₄	C₃×D₃×D₄	1	2	yes	—
264	C₂×C₄	C₃×D₃×D₄	1	1	yes	characteristic
265	C₈	—	2	2	no	—
266	C₈	—	3	2	no	—
267	C₈	—	3	2	no	—
268	C₈	—	4	2	no	—
269	C₈	—	12	2	no	—
270	C₂×C₃	—	2	1	no	—
271	C₂×C₃	—	2	2	no	—
272	C₂×C₃	—	2	1	no	—
273	C₂×C₃	—	2	2	no	—
274	C₂×C₃	—	2	1	no	—
275	C₂×C₃	—	4	1	no	—
276	C₂×C₃	—	4	2	no	—
277	C₂×C₃	—	12	2	no	—
278	C₂×C₃	#4667	1	1	yes	characteristic
279	C₂×C₃	C₃×#380	1	1	yes	characteristic
280	D₃	—	4	2	no	—
281	C₂²	—	4	1	no	—
282	C₂²	—	6	2	no	—
283	C₂²	—	12	2	no	—
284	C₂²	C₃×D₃×C₄⋊C₄	1	1	yes	characteristic
285	C₂²	C₃×#896	1	2	yes	—
286	C₄	—	6	2	no	—
287	C₄	C₃×#872	1	1	yes	characteristic
288	C₄	C₃×D₃×C₂²⋊C₄	1	1	yes	characteristic
289	C₄	C₃×#884	1	2	yes	—
290	C₃	—	2	1	no	—
291	C₃	#68724	1	1	yes	characteristic, direct product factor
292	C₃	C₃×#1813	1	1	yes	characteristic, complemented
293	C₂	—	2	1	no	—
294	C₂	—	2	2	no	—
295	C₂	—	12	2	no	—
296	C₂	C₃×#4667	1	1	yes	characteristic

	subgroup	G/H
center	C₃×C₄	#872
commutator	C₂×C₃×C₄	C₂²×C₃×C₄
Frattini	C₂²×C₄	C₂²×C₃×D₃
Fitting	C₃²×#338	C₂
socle	C₂×C₃²	#380
prime radical	C₂×C₃×#235	C₂²
cosocle	C₂²×C₃×C₄	C₂³×C₃
hypercenter	C₃×#338	D₃
hypocenter	C₃	C₃×#1813
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₃×C₄ ◁ C₂×C₃×M₁₆ ◁ C₃×#338

Complements

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

N	H ≅ G/N	type
3. C₂×C₃×#822	C₂	semidirect
4. C₃²×#338	C₂	semidirect
5. C₃×C₃⋊#338	C₂	semidirect
7. #68724	C₃	direct (N × H)
24. C₂×#822	C₂×C₃	semidirect
27. C₃×#338	D₃	semidirect
28. C₃×#338	C₂×C₃	semidirect
29. C₃⋊#338	C₂×C₃	semidirect
96. #338	C₃×D₃	semidirect
252. C₃²	#1813	semidirect
291. C₃	#68724	direct (N × H)
292. C₃	C₃×#1813	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×#907	2
C₃×#5132	2
C₂×C₃×#822	2
C₃²×#338	2
C₃×C₃⋊#338	2
C₃×#1813	3
#68724	3

Subgroups — C3×#68724 #11785512

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#68724 #11785512