Subgroups — C₃×#69829 #11784398

Order 1152.

521 conjugacy classes of subgroups.

The subgroup lattice has 294 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×#819	C₂	1	1	yes	maximal, characteristic
3	C₃×#5106	C₂	1	1	yes	maximal, characteristic
4	C₃×#5101	C₂	1	2	yes	maximal
5	C₃×#4523	C₂	1	2	yes	maximal
6	C₃×#1773	—	3	1	no	maximal
7	#69829	C₃	1	1	yes	maximal, characteristic, direct product factor
8	C₂²×C₃²×C₈	C₂²	1	1	yes	characteristic
9	C₂²×C₃²×C₈	C₂²	1	1	yes	characteristic
10	C₂×C₃²×C₄²	C₂²	1	1	yes	characteristic
11	C₃×#926	C₂²	1	2	yes	—
12	C₂×C₃×#208	C₂²	1	2	yes	—
13	C₃²×#97	C₄	1	4	yes	complemented
14	C₃×#819	C₄	1	4	yes	complemented
15	C₃×#818	C₄	1	2	yes	complemented
16	C₃×#817	C₄	1	2	yes	complemented
17	C₃×#815	—	2	2	no	—
18	C₃×C₈×Dic₃	—	2	2	no	—
19	C₂×C₃×#108	—	3	1	no	—
20	C₃×#427	—	3	1	no	—
21	C₃×#405	—	3	2	no	—
22	C₂×C₃×#97	—	2	1	no	—
23	C₂×C₃×#97	D₃	1	1	yes	characteristic
24	C₂×C₃×#97	C₂×C₃	1	1	yes	characteristic
25	C₂×#819	C₂×C₃	1	1	yes	characteristic
26	#5106	C₂×C₃	1	1	yes	characteristic
27	#5101	C₂×C₃	1	2	yes	—
28	C₃×#339	—	3	2	no	—
29	#4523	C₂×C₃	1	2	yes	—
30	C₂²×C₃²×C₄	—	2	1	no	—
31	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
32	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
33	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
34	C₂×C₃²×C₈	C₂×C₄	1	2	yes	—
35	C₂×C₃²×C₈	C₂×C₄	1	2	yes	—
36	C₂×C₃²×C₈	C₂×C₄	1	2	yes	—
37	C₂×C₃²×C₈	D₄	1	2	yes	—
38	C₂×C₃²×C₈	Q₈	1	2	yes	—
39	C₃²×C₄²	—	2	1	no	—
40	C₃²×C₄²	C₂×C₄	1	2	yes	—
41	C₃×#217	—	2	2	no	—
42	C₃×C₄⋊Dic₃	C₂×C₄	1	4	yes	—
43	C₃×C₄×Dic₃	—	2	2	no	—
44	C₃×#208	—	2	2	no	—
45	C₃×#208	C₂×C₄	1	4	yes	—
46	C₂×C₃×C₃⋊C₈	—	2	2	no	—
47	#1773	—	3	1	no	Sylow
48	C₂×C₃×M₁₆	—	3	2	no	—
49	C₂²×C₃×C₈	—	2	1	no	—
50	C₂²×C₃×C₈	—	2	1	no	—
51	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
52	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
53	C₂²×C₃×C₈	C₂²×C₃	1	1	yes	characteristic
54	C₂²×C₃×C₈	C₂²×C₃	1	1	yes	characteristic
55	C₃×#117	—	3	2	no	—
56	C₂×C₃×C₄²	—	2	1	no	—
57	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
58	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
59	#926	C₂²×C₃	1	2	yes	—
60	C₂×#208	C₂²×C₃	1	2	yes	—
61	C₃×#108	—	3	4	no	—
62	C₃×#107	—	3	2	no	—
63	C₃×#106	—	3	2	no	—
64	C₃×#97	—	2	4	no	—
65	C₃×#97	—	6	2	no	—
66	C₃×#97	Dic₃	1	4	yes	complemented
67	C₃×#97	C₃×C₄	1	4	yes	complemented
68	C₃×C₄×C₈	—	6	2	no	—
69	#819	C₃×C₄	1	4	yes	complemented
70	#818	C₃×C₄	1	2	yes	complemented
71	#817	C₃×C₄	1	2	yes	complemented
72	#815	—	2	2	no	—
73	C₈×Dic₃	—	2	2	no	—
74	C₂³×C₃²	#41	1	1	yes	characteristic
75	C₂×C₃²×C₄	—	2	2	no	—
76	C₂×C₃²×C₄	—	4	2	no	—
77	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
81	C₂×C₃²×C₄	C₂×Q₈	1	1	yes	characteristic
82	C₂×C₃²×C₄	#41	1	1	yes	characteristic
83	C₂×C₃×Dic₃	—	2	4	no	—
84	C₃²×C₈	C₄²	1	2	yes	—
85	C₃²×C₈	C₄²	1	2	yes	—
86	C₃²×C₈	C₄⋊C₄	1	4	yes	—
87	C₃×C₃⋊C₈	—	2	4	no	—
88	C₂×#108	—	3	1	no	—
89	#427	—	3	1	no	—
90	#405	—	3	2	no	—
91	C₂×#97	C₃×D₃	1	1	yes	characteristic
92	#339	—	3	2	no	—
93	C₂²×C₃×C₄	—	2	2	no	—
94	C₂²×C₃×C₄	—	2	1	no	—
95	C₂²×C₃×C₄	—	2	1	no	—
96	C₂²×C₃×C₄	—	2	1	no	—
97	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
98	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
99	C₃×M₁₆	—	3	4	no	—
100	C₃×M₁₆	—	6	2	no	—
101	C₂×C₃×C₈	—	2	2	no	—
102	C₂×C₃×C₈	—	2	1	no	—
103	C₂×C₃×C₈	—	2	1	no	—
104	C₂×C₃×C₈	—	2	2	no	—
105	C₂×C₃×C₈	—	2	2	no	—
106	C₂×C₃×C₈	—	2	2	no	—
107	C₂×C₃×C₈	—	2	2	no	—
108	C₂×C₃×C₈	—	6	2	no	—
109	C₂×C₃×C₈	Dic₆	1	2	yes	—
110	C₂×C₃×C₈	C₄×D₃	1	1	yes	characteristic
111	C₂×C₃×C₈	C₄×D₃	1	1	yes	characteristic
112	C₂×C₃×C₈	C₄×D₃	1	2	yes	—
113	C₂×C₃×C₈	D₁₂	1	2	yes	—
114	C₂×C₃×C₈	C₂×Dic₃	1	2	yes	—
115	C₂×C₃×C₈	C₂×Dic₃	1	2	yes	—
116	C₂×C₃×C₈	C₂×C₃×C₄	1	1	yes	characteristic
117	C₂×C₃×C₈	C₂×C₃×C₄	1	1	yes	characteristic
118	C₂×C₃×C₈	C₂×C₃×C₄	1	2	yes	—
119	C₂×C₃×C₈	C₂×C₃×C₄	1	2	yes	—
120	C₂×C₃×C₈	C₂×C₃×C₄	1	2	yes	—
121	C₂×C₃×C₈	C₃×D₄	1	2	yes	—
122	C₂×C₃×C₈	C₃×Q₈	1	2	yes	—
123	C₃×C₄⋊C₄	—	3	4	no	—
124	C₃×C₂²⋊C₄	—	6	2	no	—
125	C₃×C₄²	—	2	2	no	—
126	C₃×C₄²	—	2	1	no	—
127	C₃×C₄²	—	2	1	no	—
128	C₃×C₄²	—	2	2	no	—
129	C₃×C₄²	—	6	2	no	—
130	C₃×C₄²	C₂×Dic₃	1	2	yes	—
131	C₃×C₄²	C₂×C₃×C₄	1	2	yes	—
132	#217	—	2	2	no	—
133	C₄⋊Dic₃	C₂×C₃×C₄	1	4	yes	—
134	C₄×Dic₃	—	2	2	no	—
135	#208	—	2	2	no	—
136	#208	C₂×C₃×C₄	1	4	yes	—
137	C₂×C₃⋊C₈	—	2	2	no	—
138	C₂²×C₃²	—	2	2	no	—
139	C₂²×C₃²	#117	1	1	yes	characteristic
140	C₂²×C₃²	C₄×D₄	1	1	yes	characteristic
141	C₂²×C₃²	C₄×Q₈	1	1	yes	characteristic
142	C₃²×C₄	—	4	2	no	—
143	C₃²×C₄	C₂×C₄²	1	1	yes	characteristic
144	C₃²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
145	C₃²×C₄	C₄×D₄	1	1	yes	characteristic
146	C₃²×C₄	C₄×Q₈	1	1	yes	characteristic
147	C₃×Dic₃	—	4	4	no	—
148	C₂×M₁₆	—	3	2	no	—
149	C₂²×C₈	C₂×C₃×D₃	1	1	yes	characteristic
150	C₂²×C₈	C₂×C₃×D₃	1	1	yes	characteristic
151	#117	—	3	2	no	—
152	C₂×C₄²	C₂×C₃×D₃	1	1	yes	characteristic
153	#108	—	3	4	no	—
154	#107	—	3	2	no	—
155	#106	—	3	2	no	—
156	#97	—	6	2	no	—
157	#97	C₃×Dic₃	1	4	yes	complemented
158	C₄×C₈	—	6	2	no	—
159	C₂³×C₃	—	2	1	no	—
160	C₂³×C₃	#235	1	1	yes	characteristic
161	C₂³×C₃	C₃×#41	1	1	yes	characteristic
162	C₂×C₃×C₄	—	2	4	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	—	2	1	no	—
167	C₂×C₃×C₄	—	2	1	no	—
168	C₂×C₃×C₄	—	2	1	no	—
169	C₂×C₃×C₄	—	2	1	no	—
170	C₂×C₃×C₄	—	2	1	no	—
171	C₂×C₃×C₄	—	4	4	no	—
172	C₂×C₃×C₄	—	4	2	no	—
173	C₂×C₃×C₄	—	4	2	no	—
174	C₂×C₃×C₄	—	6	4	no	—
175	C₂×C₃×C₄	C₂×Dic₆	1	1	yes	characteristic
176	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
177	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
178	C₂×C₃×C₄	C₂×D₁₂	1	1	yes	characteristic
179	C₂×C₃×C₄	#235	1	1	yes	characteristic
180	C₂×C₃×C₄	C₂²×Dic₃	1	1	yes	characteristic
181	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
182	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
183	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
184	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
185	C₂×C₃×C₄	C₂×C₃×Q₈	1	1	yes	characteristic
186	C₂×C₃×C₄	C₃×#41	1	1	yes	characteristic
187	C₂×Dic₃	—	2	4	no	—
188	C₃×C₈	—	2	2	no	—
189	C₃×C₈	—	2	2	no	—
190	C₃×C₈	—	2	4	no	—
191	C₃×C₈	—	6	4	no	—
192	C₃×C₈	C₄×Dic₃	1	2	yes	—
193	C₃×C₈	C₄×Dic₃	1	2	yes	—
194	C₃×C₈	C₄⋊Dic₃	1	4	yes	—
195	C₃×C₈	C₃×C₄²	1	2	yes	—
196	C₃×C₈	C₃×C₄²	1	2	yes	—
197	C₃×C₈	C₃×C₄⋊C₄	1	4	yes	—
198	C₃⋊C₈	—	2	4	no	—
199	C₂×C₃²	—	2	1	no	—
200	C₂×C₃²	—	2	1	no	—
201	C₂×C₃²	C₄×C₄⋊C₄	1	1	yes	characteristic
202	C₂×C₃²	#444	1	2	yes	—
203	C₂²×C₄	—	2	1	no	—
204	C₂²×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
205	M₁₆	—	3	4	no	—
206	M₁₆	—	6	2	no	—
207	C₂×C₈	—	6	2	no	—
208	C₂×C₈	C₃×Dic₆	1	2	yes	—
209	C₂×C₈	C₃×C₄×D₃	1	1	yes	characteristic
210	C₂×C₈	C₃×C₄×D₃	1	1	yes	characteristic
211	C₂×C₈	C₃×C₄×D₃	1	2	yes	—
212	C₂×C₈	C₃×D₁₂	1	2	yes	—
213	C₂×C₈	C₂×C₃×Dic₃	1	2	yes	—
214	C₂×C₈	C₂×C₃×Dic₃	1	2	yes	—
215	C₄⋊C₄	—	3	4	no	—
216	C₂²⋊C₄	—	6	2	no	—
217	C₄²	—	2	1	no	—
218	C₄²	—	6	2	no	—
219	C₄²	C₂×C₃×Dic₃	1	2	yes	—
220	C₂²×C₃	—	2	4	no	—
221	C₂²×C₃	—	2	2	no	—
222	C₂²×C₃	—	2	2	no	—
223	C₂²×C₃	—	2	1	no	—
224	C₂²×C₃	—	2	1	no	—
225	C₂²×C₃	—	2	1	no	—
226	C₂²×C₃	C₄×Dic₆	1	1	yes	characteristic
227	C₂²×C₃	C₄×D₁₂	1	1	yes	characteristic
228	C₂²×C₃	#926	1	1	yes	characteristic
229	C₂²×C₃	C₃×#117	1	1	yes	characteristic
230	C₂²×C₃	C₃×C₄×D₄	1	1	yes	characteristic
231	C₂²×C₃	C₃×C₄×Q₈	1	1	yes	characteristic
232	C₃×C₄	—	2	1	no	—
233	C₃×C₄	—	2	1	no	—
234	C₃×C₄	—	2	1	no	—
235	C₃×C₄	—	2	1	no	—
236	C₃×C₄	—	4	4	no	—
237	C₃×C₄	—	4	2	no	—
238	C₃×C₄	—	4	2	no	—
239	C₃×C₄	—	12	4	no	—
240	C₃×C₄	C₄×Dic₆	1	1	yes	characteristic
241	C₃×C₄	C₄×D₁₂	1	1	yes	characteristic
242	C₃×C₄	C₂×C₄×Dic₃	1	1	yes	characteristic
243	C₃×C₄	C₂×C₄⋊Dic₃	1	1	yes	characteristic
244	C₃×C₄	C₂×C₃×C₄²	1	1	yes	characteristic
245	C₃×C₄	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
246	C₃×C₄	C₃×C₄×D₄	1	1	yes	characteristic
247	C₃×C₄	C₃×C₄×Q₈	1	1	yes	characteristic
248	Dic₃	—	4	4	no	—
249	C₃²	#1773	1	1	yes	characteristic, complemented, Sylow
250	C₂³	C₃×#235	1	1	yes	characteristic
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	4	2	no	—
253	C₂×C₄	—	6	4	no	—
254	C₂×C₄	C₂×C₃×Dic₆	1	1	yes	characteristic
255	C₂×C₄	C₂×C₃×C₄×D₃	1	1	yes	characteristic
256	C₂×C₄	C₂×C₃×C₄×D₃	1	1	yes	characteristic
257	C₂×C₄	C₂×C₃×D₁₂	1	1	yes	characteristic
258	C₂×C₄	C₃×#235	1	1	yes	characteristic
259	C₂×C₄	C₂²×C₃×Dic₃	1	1	yes	characteristic
260	C₈	—	6	4	no	—
261	C₈	C₃×C₄×Dic₃	1	2	yes	—
262	C₈	C₃×C₄×Dic₃	1	2	yes	—
263	C₈	C₃×C₄⋊Dic₃	1	4	yes	—
264	C₂×C₃	—	2	2	no	—
265	C₂×C₃	—	2	2	no	—
266	C₂×C₃	—	2	1	no	—
267	C₂×C₃	—	2	1	no	—
268	C₂×C₃	—	2	1	no	—
269	C₂×C₃	—	2	1	no	—
270	C₂×C₃	—	2	1	no	—
271	C₂×C₃	—	2	2	no	—
272	C₂×C₃	#4717	1	2	yes	—
273	C₂×C₃	C₄×C₄⋊Dic₃	1	1	yes	characteristic
274	C₂×C₃	C₃×C₄×C₄⋊C₄	1	1	yes	characteristic
275	C₂×C₃	C₃×#444	1	2	yes	—
276	C₂²	—	2	2	no	—
277	C₂²	C₃×C₄×Dic₆	1	1	yes	characteristic
278	C₂²	C₃×C₄×D₁₂	1	1	yes	characteristic
279	C₂²	C₃×#926	1	1	yes	characteristic
280	C₄	—	4	2	no	—
281	C₄	—	12	4	no	—
282	C₄	C₃×C₄×Dic₆	1	1	yes	characteristic
283	C₄	C₃×C₄×D₁₂	1	1	yes	characteristic
284	C₄	C₂×C₃×C₄×Dic₃	1	1	yes	characteristic
285	C₄	C₂×C₃×C₄⋊Dic₃	1	1	yes	characteristic
286	C₃	—	2	1	no	—
287	C₃	#69829	1	1	yes	characteristic, direct product factor
288	C₃	C₃×#1773	1	1	yes	characteristic, complemented
289	C₂	—	2	1	no	—
290	C₂	—	2	1	no	—
291	C₂	C₃×#4717	1	2	yes	—
292	C₂	C₃×C₄×C₄⋊Dic₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₄	C₂×D₁₂
commutator	C₃×C₄	C₂×C₃×C₄²
Frattini	C₂²×C₄	C₂²×C₃×D₃
Fitting	C₂×C₃²×#97	C₂
socle	C₂²×C₃²	C₄×D₄
prime radical	C₂³×C₃²	#41
cosocle	C₂²×C₃×C₄	C₂³×C₃
hypercenter	C₂×C₃×#97	D₃
hypocenter	C₃	C₃×#1773
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₂²×C₃×C₈ ◁ C₂×C₃×#97

Complements

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

N	H ≅ G/N	type
7. #69829	C₃	direct (N × H)
13. C₃²×#97	C₄	semidirect
14. C₃×#819	C₄	semidirect
15. C₃×#818	C₄	semidirect
16. C₃×#817	C₄	semidirect
66. C₃×#97	Dic₃	semidirect
67. C₃×#97	C₃×C₄	semidirect
69. #819	C₃×C₄	semidirect
70. #818	C₃×C₄	semidirect
71. #817	C₃×C₄	semidirect
157. #97	C₃×Dic₃	semidirect
249. C₃²	#1773	semidirect
287. C₃	#69829	direct (N × H)
288. C₃	C₃×#1773	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×#819	2
C₃×#5106	2
C₃×#5101	2
C₃×#4523	2
C₃×#1773	3
#69829	3

Subgroups — C3×#69829 #11784398

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#69829 #11784398