Subgroups — C₃×#69809 #11784191

Order 1152.

657 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×#97	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×Dic₃⋊C₈	C₂	1	2	yes	maximal
3	C₃×C₄²×Dic₃	C₂	1	1	yes	maximal, characteristic
4	C₂×C₃×#803	C₂	1	1	yes	maximal, characteristic
5	C₃×#4550	C₂	1	2	yes	maximal
6	C₃×#1767	—	3	1	no	maximal
7	#69809	C₃	1	1	yes	maximal, characteristic, direct product factor
8	C₂²×C₃²×C₈	C₂²	1	2	yes	—
9	C₂×C₃²×C₄²	C₂²	1	1	yes	characteristic
10	C₂×C₃×C₄×Dic₃	C₄	1	2	yes	—
11	C₂×C₃×C₄×Dic₃	C₄	1	2	yes	—
12	C₂×C₃×C₄×Dic₃	C₂²	1	1	yes	characteristic
13	C₂×C₃×C₄×Dic₃	C₂²	1	1	yes	characteristic
14	C₂²×C₃×C₃⋊C₈	C₂²	1	2	yes	—
15	C₃²×#97	—	2	2	no	—
16	C₃×Dic₃⋊C₈	C₄	1	8	yes	complemented
17	C₃×#803	—	2	2	no	—
18	C₂×C₃×C₄⋊C₈	—	3	2	no	—
19	C₂×C₃×#97	—	2	1	no	—
20	C₂×C₃×#97	—	3	1	no	—
21	C₂×C₃×#97	D₃	1	1	yes	characteristic
22	C₂×C₃×#97	C₂×C₃	1	1	yes	characteristic
23	C₃×C₄³	—	3	1	no	—
24	C₂×Dic₃⋊C₈	C₂×C₃	1	2	yes	—
25	C₄²×Dic₃	C₂×C₃	1	1	yes	characteristic
26	C₂×#803	C₂×C₃	1	1	yes	characteristic
27	C₃×#336	—	3	2	no	—
28	#4550	C₂×C₃	1	2	yes	—
29	C₂²×C₃²×C₄	C₂×C₄	1	2	yes	—
30	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
31	C₂²×C₃×Dic₃	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₃×Dic₃	C₂×C₄	1	2	yes	—
33	C₂²×C₃×Dic₃	C₂×C₄	1	1	yes	characteristic
34	C₂×C₃²×C₈	—	2	2	no	—
35	C₂×C₃²×C₈	—	2	2	no	—
36	C₂×C₃²×C₈	C₂×C₄	1	4	yes	—
37	C₃²×C₄²	D₄	1	2	yes	—
38	C₃²×C₄²	Q₈	1	2	yes	—
39	C₃×C₄×Dic₃	—	2	2	no	—
40	C₃×C₄×Dic₃	—	2	4	no	—
41	C₃×C₄×Dic₃	—	2	4	no	—
42	C₃×C₄×Dic₃	C₂×C₄	1	4	yes	—
43	C₂×C₃×C₃⋊C₈	—	2	2	no	—
44	C₂×C₃×C₃⋊C₈	—	2	2	no	—
45	C₂×C₃×C₃⋊C₈	C₂×C₄	1	4	yes	—
46	#1767	—	3	1	no	Sylow
47	C₂²×C₃×C₈	—	2	2	no	—
48	C₂²×C₃×C₈	—	3	2	no	—
49	C₂²×C₃×C₈	C₂×D₃	1	2	yes	—
50	C₂²×C₃×C₈	C₂²×C₃	1	2	yes	—
51	C₂×C₃×C₄²	—	2	1	no	—
52	C₂×C₃×C₄²	—	3	1	no	—
53	C₂×C₃×C₄²	—	3	2	no	—
54	C₂×C₃×C₄²	—	3	2	no	—
55	C₂×C₃×C₄²	—	3	1	no	—
56	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
57	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
58	C₂×C₄×Dic₃	C₃×C₄	1	2	yes	—
59	C₂×C₄×Dic₃	C₃×C₄	1	2	yes	—
60	C₂×C₄×Dic₃	C₂²×C₃	1	1	yes	characteristic
61	C₂×C₄×Dic₃	C₂²×C₃	1	1	yes	characteristic
62	C₂²×C₃⋊C₈	C₂²×C₃	1	2	yes	—
63	C₃×C₄⋊C₈	—	3	8	no	—
64	C₃×#97	—	2	4	no	—
65	C₃×#97	—	2	2	no	—
66	C₃×#97	—	2	2	no	—
67	C₃×#97	—	6	2	no	—
68	Dic₃⋊C₈	C₃×C₄	1	8	yes	complemented
69	#803	—	2	2	no	—
70	C₂³×C₃²	C₂²×C₄	1	1	yes	characteristic
71	C₂×C₃²×C₄	—	2	2	no	—
72	C₂×C₃²×C₄	—	2	2	no	—
73	C₂×C₃²×C₄	C₄⋊C₄	1	4	yes	—
74	C₂×C₃²×C₄	C₂²×C₄	1	2	yes	—
75	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
76	C₂×C₃²×C₄	C₂×Q₈	1	1	yes	characteristic
77	C₂×C₃²×C₄	#41	1	2	yes	—
78	C₂×C₃×Dic₃	—	2	2	no	—
79	C₂×C₃×Dic₃	—	2	2	no	—
80	C₂×C₃×Dic₃	—	2	2	no	—
81	C₂×C₃×Dic₃	C₄²	1	2	yes	—
82	C₂×C₃×Dic₃	C₄²	1	2	yes	—
83	C₂×C₃×Dic₃	M₁₆	1	4	yes	—
84	C₂×C₃×Dic₃	M₁₆	1	4	yes	—
85	C₃²×C₈	—	2	4	no	—
86	C₃×C₃⋊C₈	—	2	4	no	—
87	C₂×C₄⋊C₈	—	3	2	no	—
88	C₂×#97	—	3	1	no	—
89	C₂×#97	C₃×D₃	1	1	yes	characteristic
90	C₄³	—	3	1	no	—
91	#336	—	3	2	no	—
92	C₂²×C₃×C₄	—	2	2	no	—
93	C₂²×C₃×C₄	—	2	1	no	—
94	C₂²×C₃×C₄	—	3	1	no	—
95	C₂²×C₃×C₄	—	3	1	no	—
96	C₂²×C₃×C₄	—	3	2	no	—
97	C₂²×C₃×C₄	C₄×D₃	1	2	yes	—
98	C₂²×C₃×C₄	C₂×C₃×C₄	1	2	yes	—
99	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
100	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
101	C₂²×Dic₃	C₂×C₃×C₄	1	1	yes	characteristic
102	C₂²×Dic₃	C₂×C₃×C₄	1	2	yes	—
103	C₂²×Dic₃	C₂×C₃×C₄	1	1	yes	characteristic
104	C₂×C₃×C₈	—	2	4	no	—
105	C₂×C₃×C₈	—	2	4	no	—
106	C₂×C₃×C₈	—	2	2	no	—
107	C₂×C₃×C₈	—	2	2	no	—
108	C₂×C₃×C₈	—	2	2	no	—
109	C₂×C₃×C₈	—	2	2	no	—
110	C₂×C₃×C₈	—	2	4	no	—
111	C₂×C₃×C₈	—	3	4	no	—
112	C₂×C₃×C₈	—	6	2	no	—
113	C₂×C₃×C₈	—	6	2	no	—
114	C₂×C₃×C₈	C₄×D₃	1	4	yes	—
115	C₂×C₃×C₈	C₂×C₃×C₄	1	4	yes	—
116	C₃×C₄²	—	2	2	no	—
117	C₃×C₄²	—	2	2	no	—
118	C₃×C₄²	—	3	4	no	—
119	C₃×C₄²	—	6	4	no	—
120	C₃×C₄²	—	6	4	no	—
121	C₃×C₄²	—	6	2	no	—
122	C₃×C₄²	Dic₆	1	2	yes	—
123	C₃×C₄²	#67	1	2	yes	—
124	C₃×C₄²	C₃×D₄	1	2	yes	—
125	C₃×C₄²	C₃×Q₈	1	2	yes	—
126	C₄×Dic₃	—	2	2	no	—
127	C₄×Dic₃	—	2	4	no	—
128	C₄×Dic₃	—	2	4	no	—
129	C₄×Dic₃	C₂×C₃×C₄	1	4	yes	—
130	C₂×C₃⋊C₈	—	2	2	no	—
131	C₂×C₃⋊C₈	—	2	2	no	—
132	C₂×C₃⋊C₈	C₂×C₃×C₄	1	4	yes	—
133	C₂²×C₃²	C₂×C₄²	1	1	yes	characteristic
134	C₂²×C₃²	C₂×C₄⋊C₄	1	1	yes	characteristic
135	C₂²×C₃²	#117	1	1	yes	characteristic
136	C₂²×C₃²	C₂×M₁₆	1	2	yes	—
137	C₂²×C₃²	C₂×M₁₆	1	2	yes	—
138	C₃²×C₄	—	2	4	no	—
139	C₃²×C₄	C₄×D₄	1	2	yes	—
140	C₃²×C₄	C₄×Q₈	1	2	yes	—
141	C₃×Dic₃	—	2	2	no	—
142	C₃×Dic₃	—	2	4	no	—
143	C₃×Dic₃	#97	1	4	yes	complemented
144	C₂²×C₈	—	3	2	no	—
145	C₂²×C₈	C₂×C₃×D₃	1	2	yes	—
146	C₂×C₄²	—	3	1	no	—
147	C₂×C₄²	—	3	2	no	—
148	C₂×C₄²	—	3	2	no	—
149	C₂×C₄²	—	3	1	no	—
150	C₂×C₄²	C₂×C₃×D₃	1	1	yes	characteristic
151	C₄⋊C₈	—	3	8	no	—
152	#97	—	2	2	no	—
153	#97	—	6	2	no	—
154	C₂³×C₃	—	2	1	no	—
155	C₂³×C₃	C₂×C₄×D₃	1	1	yes	characteristic
156	C₂³×C₃	C₂²×C₃×C₄	1	1	yes	characteristic
157	C₂×C₃×C₄	—	2	2	no	—
158	C₂×C₃×C₄	—	2	2	no	—
159	C₂×C₃×C₄	—	2	2	no	—
160	C₂×C₃×C₄	—	2	2	no	—
161	C₂×C₃×C₄	—	2	4	no	—
162	C₂×C₃×C₄	—	2	1	no	—
163	C₂×C₃×C₄	—	2	2	no	—
164	C₂×C₃×C₄	—	2	2	no	—
165	C₂×C₃×C₄	—	2	1	no	—
166	C₂×C₃×C₄	—	3	2	no	—
167	C₂×C₃×C₄	—	3	4	no	—
168	C₂×C₃×C₄	—	3	2	no	—
169	C₂×C₃×C₄	—	3	4	no	—
170	C₂×C₃×C₄	—	4	2	no	—
171	C₂×C₃×C₄	—	4	2	no	—
172	C₂×C₃×C₄	—	6	2	no	—
173	C₂×C₃×C₄	—	6	2	no	—
174	C₂×C₃×C₄	—	6	2	no	—
175	C₂×C₃×C₄	Dic₃⋊C₄	1	4	yes	—
176	C₂×C₃×C₄	C₃×C₄⋊C₄	1	4	yes	—
177	C₂×C₃×C₄	C₂×Dic₆	1	1	yes	characteristic
178	C₂×C₃×C₄	C₂×C₄×D₃	1	2	yes	—
179	C₂×C₃×C₄	#235	1	2	yes	—
180	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
181	C₂×C₃×C₄	C₂²×C₃×C₄	1	2	yes	—
182	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
183	C₂×C₃×C₄	C₂×C₃×Q₈	1	1	yes	characteristic
184	C₂×C₃×C₄	C₃×#41	1	2	yes	—
185	C₂×Dic₃	—	2	2	no	—
186	C₂×Dic₃	—	2	2	no	—
187	C₂×Dic₃	—	2	2	no	—
188	C₂×Dic₃	C₃×C₄²	1	2	yes	—
189	C₂×Dic₃	C₃×C₄²	1	2	yes	—
190	C₂×Dic₃	C₃×M₁₆	1	4	yes	—
191	C₂×Dic₃	C₃×M₁₆	1	4	yes	—
192	C₃×C₈	—	2	8	no	—
193	C₃×C₈	—	2	4	no	—
194	C₃×C₈	—	2	4	no	—
195	C₃×C₈	—	6	4	no	—
196	C₃⋊C₈	—	2	4	no	—
197	C₂×C₃²	C₄×C₄⋊C₄	1	1	yes	characteristic
198	C₂×C₃²	C₂×#97	1	1	yes	characteristic
199	C₂×C₃²	C₄×M₁₆	1	1	yes	characteristic
200	C₂×C₃²	#423	1	2	yes	—
201	C₂×C₃²	#432	1	2	yes	—
202	C₂²×C₄	—	3	1	no	—
203	C₂²×C₄	—	3	2	no	—
204	C₂²×C₄	—	3	1	no	—
205	C₂²×C₄	C₃×C₄×D₃	1	2	yes	—
206	C₂²×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
207	C₂×C₈	—	2	2	no	—
208	C₂×C₈	—	2	2	no	—
209	C₂×C₈	—	3	4	no	—
210	C₂×C₈	—	6	2	no	—
211	C₂×C₈	—	6	2	no	—
212	C₂×C₈	C₃×C₄×D₃	1	4	yes	—
213	C₄²	—	3	4	no	—
214	C₄²	—	6	4	no	—
215	C₄²	—	6	4	no	—
216	C₄²	—	6	2	no	—
217	C₄²	C₃×Dic₆	1	2	yes	—
218	C₄²	C₃×#67	1	2	yes	—
219	C₂²×C₃	—	2	2	no	—
220	C₂²×C₃	—	2	2	no	—
221	C₂²×C₃	—	2	1	no	—
222	C₂²×C₃	—	2	1	no	—
223	C₂²×C₃	—	2	1	no	—
224	C₂²×C₃	C₄²×D₃	1	1	yes	characteristic
225	C₂²×C₃	#872	1	1	yes	characteristic
226	C₂²×C₃	D₃×M₁₆	1	2	yes	—
227	C₂²×C₃	D₃×M₁₆	1	2	yes	—
228	C₂²×C₃	C₂×Dic₃⋊C₄	1	1	yes	characteristic
229	C₂²×C₃	C₂×C₃×C₄²	1	1	yes	characteristic
230	C₂²×C₃	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
231	C₂²×C₃	C₃×#117	1	1	yes	characteristic
232	C₂²×C₃	C₂×C₃×M₁₆	1	2	yes	—
233	C₂²×C₃	C₂×C₃×M₁₆	1	2	yes	—
234	C₃×C₄	—	2	4	no	—
235	C₃×C₄	—	2	4	no	—
236	C₃×C₄	—	2	2	no	—
237	C₃×C₄	—	2	2	no	—
238	C₃×C₄	—	3	4	no	—
239	C₃×C₄	—	4	4	no	—
240	C₃×C₄	—	6	4	no	—
241	C₃×C₄	—	6	2	no	—
242	C₃×C₄	C₄×Dic₆	1	2	yes	—
243	C₃×C₄	C₄×#67	1	2	yes	—
244	C₃×C₄	C₃×C₄×D₄	1	2	yes	—
245	C₃×C₄	C₃×C₄×Q₈	1	2	yes	—
246	Dic₃	—	2	2	no	—
247	Dic₃	—	2	4	no	—
248	Dic₃	C₃×#97	1	4	yes	complemented
249	C₃²	#1767	1	1	yes	characteristic, complemented, Sylow
250	C₂³	C₂×C₃×C₄×D₃	1	1	yes	characteristic
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	3	4	no	—
254	C₂×C₄	—	3	2	no	—
255	C₂×C₄	—	3	4	no	—
256	C₂×C₄	—	3	2	no	—
257	C₂×C₄	—	6	2	no	—
258	C₂×C₄	—	6	2	no	—
259	C₂×C₄	—	6	2	no	—
260	C₂×C₄	C₃×Dic₃⋊C₄	1	4	yes	—
261	C₂×C₄	C₂×C₃×Dic₆	1	1	yes	characteristic
262	C₂×C₄	C₂×C₃×C₄×D₃	1	2	yes	—
263	C₂×C₄	C₃×#235	1	2	yes	—
264	C₂×C₄	C₂×C₃×#67	1	1	yes	characteristic
265	C₈	—	2	4	no	—
266	C₈	—	6	4	no	—
267	C₂×C₃	—	2	1	no	—
268	C₂×C₃	—	2	1	no	—
269	C₂×C₃	—	2	2	no	—
270	C₂×C₃	—	2	1	no	—
271	C₂×C₃	—	2	2	no	—
272	C₂×C₃	D₃×#97	1	1	yes	characteristic
273	C₂×C₃	#4705	1	2	yes	—
274	C₂×C₃	#4707	1	1	yes	characteristic
275	C₂×C₃	C₄×Dic₃⋊C₄	1	1	yes	characteristic
276	C₂×C₃	#5118	1	2	yes	—
277	C₂×C₃	C₃×C₄×C₄⋊C₄	1	1	yes	characteristic
278	C₂×C₃	C₂×C₃×#97	1	1	yes	characteristic
279	C₂×C₃	C₃×C₄×M₁₆	1	1	yes	characteristic
280	C₂×C₃	C₃×#423	1	2	yes	—
281	C₂×C₃	C₃×#432	1	2	yes	—
282	C₂²	C₃×C₄²×D₃	1	1	yes	characteristic
283	C₂²	C₃×#872	1	1	yes	characteristic
284	C₂²	C₃×D₃×M₁₆	1	2	yes	—
285	C₂²	C₃×D₃×M₁₆	1	2	yes	—
286	C₂²	C₂×C₃×Dic₃⋊C₄	1	1	yes	characteristic
287	C₄	—	2	4	no	—
288	C₄	—	3	4	no	—
289	C₄	—	6	4	no	—
290	C₄	—	6	2	no	—
291	C₄	C₃×C₄×Dic₆	1	2	yes	—
292	C₄	C₃×C₄×#67	1	2	yes	—
293	C₃	—	2	1	no	—
294	C₃	#69809	1	1	yes	characteristic, direct product factor
295	C₃	C₃×#1767	1	1	yes	characteristic, complemented
296	C₂	C₃×D₃×#97	1	1	yes	characteristic
297	C₂	C₃×#4705	1	2	yes	—
298	C₂	C₃×#4707	1	1	yes	characteristic
299	C₂	C₃×C₄×Dic₃⋊C₄	1	1	yes	characteristic
300	C₂	C₃×#5118	1	2	yes	—

	subgroup	G/H
center	C₂²×C₃×C₄	C₂²×D₃
commutator	C₂²×C₃	C₂×C₃×C₄²
Frattini	C₂²×C₄	C₂²×C₃×D₃
Fitting	C₂×C₃²×#97	C₂
socle, prime radical	C₂³×C₃²	C₂²×C₄
cosocle	C₂²×C₃×C₄	C₂³×C₃
hypercenter	C₂×C₃×#97	D₃
hypocenter	C₃	C₃×#1767
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₃×C₄ ◁ C₂×C₃×#97

Complements

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

N	H ≅ G/N	type
7. #69809	C₃	direct (N × H)
16. C₃×Dic₃⋊C₈	C₄	semidirect
68. Dic₃⋊C₈	C₃×C₄	semidirect
143. C₃×Dic₃	#97	semidirect
248. Dic₃	C₃×#97	semidirect
249. C₃²	#1767	semidirect
294. C₃	#69809	direct (N × H)
295. C₃	C₃×#1767	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃²×#97	2
C₂×C₃×Dic₃⋊C₈	2
C₃×C₄²×Dic₃	2
C₂×C₃×#803	2
C₃×#4550	2
C₃×#1767	3
#69809	3

Subgroups — C3×#69809 #11784191

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#69809 #11784191