Subgroups — C₃×#69822 #11790728

Order 1152.

689 conjugacy classes of subgroups.

The subgroup lattice has 267 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#869	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃²×#97	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×#816	C₂	1	4	yes	maximal
4	C₃×#4936	C₂	1	1	yes	maximal, characteristic
5	C₃×#1960	—	3	1	no	maximal
6	#69822	C₃	1	1	yes	maximal, characteristic, direct product factor
7	C₂²×C₃×Dic₆	C₂²	1	2	yes	—
8	C₂²×C₃²×C₈	C₂²	1	2	yes	—
9	C₂×C₃²×C₄²	C₂²	1	1	yes	characteristic
10	C₂×C₃×C₄⋊Dic₃	—	2	2	no	—
11	C₂×C₃×C₄⋊Dic₃	C₂²	1	2	yes	—
12	C₃×#869	—	2	2	no	—
13	C₃×#869	C₄	1	4	yes	complemented
14	C₃²×#97	—	2	2	no	—
15	C₃×C₃⋊#95	—	2	2	no	—
16	C₃×#816	—	2	8	no	—
17	C₂×C₃×C₄⋊Q₈	—	3	1	no	—
18	C₂×#869	C₂×C₃	1	1	yes	characteristic
19	C₂×C₃×#103	—	3	4	no	—
20	C₂×C₃×#97	—	2	1	no	—
21	C₂×C₃×#97	D₃	1	1	yes	characteristic
22	C₂×C₃×#97	C₂×C₃	1	1	yes	characteristic
23	C₃×#382	—	3	1	no	—
24	C₂×#816	C₂×C₃	1	4	yes	—
25	#4936	C₂×C₃	1	1	yes	characteristic
26	C₂²×C₃²×C₄	D₄	1	2	yes	—
27	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
28	C₂²×C₃×Dic₃	—	2	2	no	—
29	C₂²×C₃×Dic₃	—	2	2	no	—
30	C₂×C₃×Dic₆	—	2	4	no	—
31	C₂×C₃×Dic₆	—	2	4	no	—
32	C₂×C₃×Dic₆	—	2	2	no	—
33	C₂×C₃×Dic₆	C₂×C₄	1	4	yes	—
34	C₂×C₃²×C₈	—	2	2	no	—
35	C₂×C₃²×C₈	—	2	2	no	—
36	C₂×C₃²×C₈	D₄	1	4	yes	—
37	C₃²×C₄²	C₂×C₄	1	2	yes	—
38	C₃²×C₄²	D₄	1	2	yes	—
39	C₃×C₄⋊Dic₃	—	2	8	no	—
40	C₃×C₄⋊Dic₃	—	2	4	no	—
41	#1960	—	3	1	no	Sylow
42	C₂²×C₃×Q₈	—	3	2	no	—
43	C₂²×Dic₆	C₂²×C₃	1	2	yes	—
44	C₂²×C₃×C₈	—	2	2	no	—
45	C₂²×C₃×C₈	C₂×D₃	1	2	yes	—
46	C₂²×C₃×C₈	C₂²×C₃	1	2	yes	—
47	C₃×C₄⋊Q₈	—	3	4	no	—
48	C₃×C₄⋊Q₈	—	6	2	no	—
49	C₂×C₃×C₄⋊C₄	—	3	2	no	—
50	C₂×C₃×C₄⋊C₄	—	6	2	no	—
51	C₂×C₃×C₄²	—	2	1	no	—
52	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
53	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
54	C₂×C₄⋊Dic₃	—	2	2	no	—
55	C₂×C₄⋊Dic₃	C₂²×C₃	1	2	yes	—
56	#869	—	2	2	no	—
57	#869	C₃×C₄	1	4	yes	complemented
58	C₃×#103	—	6	8	no	—
59	C₃×#97	—	2	4	no	—
60	C₃×#97	—	2	2	no	—
61	C₃×#97	—	2	2	no	—
62	C₃×#95	—	6	2	no	—
63	C₃⋊#95	—	2	2	no	—
64	#816	—	2	8	no	—
65	C₂³×C₃²	C₂×D₄	1	1	yes	characteristic
66	C₂×C₃²×C₄	—	2	4	no	—
67	C₂×C₃²×C₄	C₂²⋊C₄	1	4	yes	—
68	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₃²×C₄	C₂×D₄	1	2	yes	—
70	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
71	C₂×C₃²×C₄	#41	1	2	yes	—
72	C₂×C₃×Dic₃	—	2	4	no	—
73	C₂×C₃×Dic₃	—	4	2	no	—
74	C₂×C₃×Dic₃	—	4	2	no	—
75	C₂×C₃×Dic₃	—	4	2	no	—
76	C₂×C₃×Dic₃	—	4	4	no	—
77	C₃×Dic₆	—	2	8	no	—
78	C₃×Dic₆	—	4	4	no	—
79	C₃²×C₈	—	2	4	no	—
80	C₂×C₄⋊Q₈	—	3	1	no	—
81	C₂×#103	—	3	4	no	—
82	C₂×#97	C₃×D₃	1	1	yes	characteristic
83	#382	—	3	1	no	—
84	C₂×C₃×Q₈	—	3	4	no	—
85	C₂×C₃×Q₈	—	6	2	no	—
86	C₂×C₃×Q₈	—	6	4	no	—
87	C₂×C₃×Q₈	—	6	4	no	—
88	C₂²×C₃×C₄	—	2	2	no	—
89	C₂²×C₃×C₄	—	2	1	no	—
90	C₂²×C₃×C₄	—	6	2	no	—
91	C₂²×C₃×C₄	—	6	2	no	—
92	C₂²×C₃×C₄	D₁₂	1	2	yes	—
93	C₂²×C₃×C₄	C₃×D₄	1	2	yes	—
94	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
95	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
96	C₂²×Dic₃	—	2	2	no	—
97	C₂²×Dic₃	—	2	2	no	—
98	C₂×Dic₆	—	2	4	no	—
99	C₂×Dic₆	—	2	4	no	—
100	C₂×Dic₆	—	2	2	no	—
101	C₂×Dic₆	C₂×C₃×C₄	1	4	yes	—
102	C₂×C₃×C₈	—	2	4	no	—
103	C₂×C₃×C₈	—	2	4	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	—	2	4	no	—
109	C₂×C₃×C₈	D₁₂	1	4	yes	—
110	C₂×C₃×C₈	C₃×D₄	1	4	yes	—
111	C₃×C₄⋊C₄	—	6	8	no	—
112	C₃×C₄⋊C₄	—	6	4	no	—
113	C₃×C₄²	—	2	2	no	—
114	C₃×C₄²	—	2	2	no	—
115	C₃×C₄²	C₄×D₃	1	2	yes	—
116	C₃×C₄²	#67	1	2	yes	—
117	C₃×C₄²	C₂×C₃×C₄	1	2	yes	—
118	C₃×C₄²	C₃×D₄	1	2	yes	—
119	C₄⋊Dic₃	—	2	8	no	—
120	C₄⋊Dic₃	—	2	4	no	—
121	C₂²×C₃²	C₂×C₂²⋊C₄	1	1	yes	characteristic
122	C₂²×C₃²	#124	1	1	yes	characteristic
123	C₂²×C₃²	#127	1	1	yes	characteristic
124	C₂²×C₃²	#137	1	4	yes	—
125	C₃²×C₄	—	2	4	no	—
126	C₃²×C₄	C₄×D₄	1	2	yes	—
127	C₃²×C₄	#121	1	2	yes	—
128	C₃×Dic₃	—	4	4	no	—
129	C₃×Dic₃	—	8	2	no	—
130	C₂²×Q₈	—	3	2	no	—
131	C₂²×C₈	C₂×C₃×D₃	1	2	yes	—
132	C₄⋊Q₈	—	3	4	no	—
133	C₄⋊Q₈	—	6	2	no	—
134	C₂×C₄⋊C₄	—	3	2	no	—
135	C₂×C₄⋊C₄	—	6	2	no	—
136	C₂×C₄²	C₂×C₃×D₃	1	1	yes	characteristic
137	#103	—	6	8	no	—
138	#97	—	2	2	no	—
139	#95	—	6	2	no	—
140	C₂³×C₃	—	2	1	no	—
141	C₂³×C₃	C₂×D₁₂	1	1	yes	characteristic
142	C₂³×C₃	C₂×C₃×D₄	1	1	yes	characteristic
143	C₃×Q₈	—	6	8	no	—
144	C₃×Q₈	—	12	4	no	—
145	C₂×C₃×C₄	—	2	4	no	—
146	C₂×C₃×C₄	—	2	4	no	—
147	C₂×C₃×C₄	—	2	4	no	—
148	C₂×C₃×C₄	—	2	1	no	—
149	C₂×C₃×C₄	—	2	2	no	—
150	C₂×C₃×C₄	—	2	2	no	—
151	C₂×C₃×C₄	—	2	1	no	—
152	C₂×C₃×C₄	—	4	4	no	—
153	C₂×C₃×C₄	—	6	4	no	—
154	C₂×C₃×C₄	—	12	2	no	—
155	C₂×C₃×C₄	—	12	2	no	—
156	C₂×C₃×C₄	—	12	2	no	—
157	C₂×C₃×C₄	—	12	4	no	—
158	C₂×C₃×C₄	(C₂×D₃)⋊C₄	1	4	yes	—
159	C₂×C₃×C₄	C₃×C₂²⋊C₄	1	4	yes	—
160	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
161	C₂×C₃×C₄	C₂×D₁₂	1	2	yes	—
162	C₂×C₃×C₄	#235	1	2	yes	—
163	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
164	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
165	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
166	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
167	C₂×C₃×C₄	C₃×#41	1	2	yes	—
168	C₂×Dic₃	—	2	4	no	—
169	C₂×Dic₃	—	4	2	no	—
170	C₂×Dic₃	—	4	2	no	—
171	C₂×Dic₃	—	4	2	no	—
172	C₂×Dic₃	—	4	4	no	—
173	Dic₆	—	2	8	no	—
174	Dic₆	—	4	4	no	—
175	C₃×C₈	—	2	8	no	—
176	C₃×C₈	—	2	4	no	—
177	C₃×C₈	—	2	4	no	—
178	C₂×C₃²	#390	1	1	yes	characteristic
179	C₂×C₃²	#419	1	2	yes	—
180	C₂×C₃²	#491	1	2	yes	—
181	C₂×C₃²	#497	1	2	yes	—
182	C₂×Q₈	—	3	4	no	—
183	C₂×Q₈	—	6	2	no	—
184	C₂×Q₈	—	6	4	no	—
185	C₂×Q₈	—	6	4	no	—
186	C₂²×C₄	—	6	2	no	—
187	C₂²×C₄	—	6	2	no	—
188	C₂²×C₄	C₃×D₁₂	1	2	yes	—
189	C₂²×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
190	C₂×C₈	—	2	2	no	—
191	C₂×C₈	—	2	2	no	—
192	C₂×C₈	C₃×D₁₂	1	4	yes	—
193	C₄⋊C₄	—	6	8	no	—
194	C₄⋊C₄	—	6	4	no	—
195	C₄²	C₃×C₄×D₃	1	2	yes	—
196	C₄²	C₃×#67	1	2	yes	—
197	C₂²×C₃	—	2	4	no	—
198	C₂²×C₃	—	2	1	no	—
199	C₂²×C₃	—	2	1	no	—
200	C₂²×C₃	—	2	1	no	—
201	C₂²×C₃	#874	1	1	yes	characteristic
202	C₂²×C₃	#875	1	1	yes	characteristic
203	C₂²×C₃	#909	1	4	yes	—
204	C₂²×C₃	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
205	C₂²×C₃	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
206	C₂²×C₃	C₃×#124	1	1	yes	characteristic
207	C₂²×C₃	C₃×#127	1	1	yes	characteristic
208	C₂²×C₃	C₃×#137	1	4	yes	—
209	C₃×C₄	—	2	4	no	—
210	C₃×C₄	—	2	4	no	—
211	C₃×C₄	—	2	2	no	—
212	C₃×C₄	—	2	2	no	—
213	C₃×C₄	—	4	4	no	—
214	C₃×C₄	—	12	4	no	—
215	C₃×C₄	—	24	2	no	—
216	C₃×C₄	C₄×D₁₂	1	2	yes	—
217	C₃×C₄	#930	1	2	yes	—
218	C₃×C₄	C₃×C₄×D₄	1	2	yes	—
219	C₃×C₄	C₃×#121	1	2	yes	—
220	Dic₃	—	4	4	no	—
221	Dic₃	—	8	2	no	—
222	C₃²	#1960	1	1	yes	characteristic, complemented, Sylow
223	C₂³	C₂×C₃×D₁₂	1	1	yes	characteristic
224	Q₈	—	6	8	no	—
225	Q₈	—	12	4	no	—
226	C₂×C₄	—	2	4	no	—
227	C₂×C₄	—	6	4	no	—
228	C₂×C₄	—	12	2	no	—
229	C₂×C₄	—	12	2	no	—
230	C₂×C₄	—	12	4	no	—
231	C₂×C₄	—	12	2	no	—
232	C₂×C₄	C₃×(C₂×D₃)⋊C₄	1	4	yes	—
233	C₂×C₄	C₂×C₃×C₄×D₃	1	1	yes	characteristic
234	C₂×C₄	C₂×C₃×D₁₂	1	2	yes	—
235	C₂×C₄	C₃×#235	1	2	yes	—
236	C₂×C₄	C₂×C₃×#67	1	1	yes	characteristic
237	C₈	—	2	4	no	—
238	C₂×C₃	—	2	1	no	—
239	C₂×C₃	—	2	2	no	—
240	C₂×C₃	—	2	2	no	—
241	C₂×C₃	—	2	2	no	—
242	C₂×C₃	#4703	1	2	yes	—
243	C₂×C₃	#4715	1	2	yes	—
244	C₂×C₃	#4939	1	1	yes	characteristic
245	C₂×C₃	#5120	1	2	yes	—
246	C₂×C₃	C₃×#390	1	1	yes	characteristic
247	C₂×C₃	C₃×#419	1	2	yes	—
248	C₂×C₃	C₃×#491	1	2	yes	—
249	C₂×C₃	C₃×#497	1	2	yes	—
250	C₂²	C₃×#874	1	1	yes	characteristic
251	C₂²	C₃×#875	1	1	yes	characteristic
252	C₂²	C₃×#909	1	4	yes	—
253	C₂²	C₂×C₃×(C₂×D₃)⋊C₄	1	1	yes	characteristic
254	C₄	—	2	4	no	—
255	C₄	—	12	4	no	—
256	C₄	—	24	2	no	—
257	C₄	C₃×C₄×D₁₂	1	2	yes	—
258	C₄	C₃×#930	1	2	yes	—
259	C₃	—	2	1	no	—
260	C₃	#69822	1	1	yes	characteristic, direct product factor
261	C₃	C₃×#1960	1	1	yes	characteristic, complemented
262	C₂	C₃×#4703	1	2	yes	—
263	C₂	C₃×#4715	1	2	yes	—
264	C₂	C₃×#4939	1	1	yes	characteristic
265	C₂	C₃×#5120	1	2	yes	—

	subgroup	G/H
center	C₂³×C₃	C₂×D₁₂
commutator	C₂×C₃×C₄	C₂²×C₃×C₄
Frattini	C₂²×C₄	C₂²×C₃×D₃
Fitting	C₂×C₃²×#97	C₂
socle, prime radical	C₂³×C₃²	C₂×D₄
cosocle	C₂²×C₃×C₄	C₂³×C₃
hypercenter	C₂×C₃×#97	D₃
hypocenter	C₃	C₃×#1960
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂³×C₃ ◁ C₂×C₃×C₄² ◁ C₂×C₃×#97

Complements

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

N	H ≅ G/N	type
6. #69822	C₃	direct (N × H)
13. C₃×#869	C₄	semidirect
57. #869	C₃×C₄	semidirect
222. C₃²	#1960	semidirect
260. C₃	#69822	direct (N × H)
261. C₃	C₃×#1960	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×#869	2
C₂×C₃²×#97	2
C₂×C₃×#816	2
C₃×#4936	2
C₃×#1960	3
#69822	3

Subgroups — C3×#69822 #11790728

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#69822 #11790728