Subgroups — C₃×#69755 #11784217

Order 1152.

601 conjugacy classes of subgroups.

The subgroup lattice has 255 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×C₄×C₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×Dic₃⋊C₈	C₂	1	2	yes	maximal
3	C₃×#4932	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×#1768	—	3	1	no	maximal
7	#69755	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	C₂²	1	1	yes	characteristic
11	C₂×C₃×C₄×Dic₃	C₂²	1	1	yes	characteristic
12	C₂²×C₃×C₃⋊C₈	C₂²	1	2	yes	—
13	C₃²×C₄×C₈	—	2	2	no	—
14	C₃×C₃⋊#95	C₄	1	4	yes	—
15	C₃×Dic₃⋊C₈	C₄	1	8	yes	complemented
16	C₃×#803	—	2	2	no	—
17	C₂×C₃×C₄⋊C₈	—	3	2	no	—
18	C₂×C₃×#97	—	3	1	no	—
19	C₂×C₃×C₄×C₈	—	2	1	no	—
20	C₂×C₃×C₄×C₈	D₃	1	1	yes	characteristic
21	C₂×C₃×C₄×C₈	C₂×C₃	1	1	yes	characteristic
22	C₃×#376	—	3	1	no	—
23	C₂×Dic₃⋊C₈	C₂×C₃	1	2	yes	—
24	#4932	C₂×C₃	1	1	yes	characteristic
25	C₂×#803	C₂×C₃	1	1	yes	characteristic
26	C₃×#336	—	3	2	no	—
27	#4550	C₂×C₃	1	2	yes	—
28	C₂²×C₃²×C₄	C₂×C₄	1	2	yes	—
29	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
30	C₂²×C₃×Dic₃	C₂×C₄	1	2	yes	—
31	C₂²×C₃×Dic₃	C₂×C₄	1	2	yes	—
32	C₂×C₃²×C₈	—	2	2	no	—
33	C₂×C₃²×C₈	—	2	2	no	—
34	C₂×C₃²×C₈	C₂×C₄	1	4	yes	—
35	C₃²×C₄²	D₄	1	2	yes	—
36	C₃²×C₄²	Q₈	1	2	yes	—
37	C₃×C₄×Dic₃	—	2	2	no	—
38	C₃×C₄×Dic₃	C₂×C₄	1	4	yes	—
39	C₂×C₃×C₃⋊C₈	—	2	2	no	—
40	C₂×C₃×C₃⋊C₈	—	2	2	no	—
41	C₂×C₃×C₃⋊C₈	C₂×C₄	1	4	yes	—
42	#1768	—	3	1	no	Sylow
43	C₂²×C₃×C₈	—	2	2	no	—
44	C₂²×C₃×C₈	—	3	2	no	—
45	C₂²×C₃×C₈	C₂×D₃	1	2	yes	—
46	C₂²×C₃×C₈	C₂²×C₃	1	2	yes	—
47	C₂×C₃×C₄²	—	2	1	no	—
48	C₂×C₃×C₄²	—	3	1	no	—
49	C₂×C₃×C₄²	—	3	1	no	—
50	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
51	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
52	C₂×C₄×Dic₃	C₂²×C₃	1	1	yes	characteristic
53	C₂×C₄×Dic₃	C₂²×C₃	1	1	yes	characteristic
54	C₂²×C₃⋊C₈	C₂²×C₃	1	2	yes	—
55	C₃×C₄⋊C₈	—	3	8	no	—
56	C₃×#97	—	6	2	no	—
57	C₃×C₄×C₈	—	2	4	no	—
58	C₃×C₄×C₈	—	2	2	no	—
59	C₃×C₄×C₈	—	2	2	no	—
60	C₃×#95	—	3	4	no	—
61	C₃⋊#95	C₃×C₄	1	4	yes	—
62	Dic₃⋊C₈	C₃×C₄	1	8	yes	complemented
63	#803	—	2	2	no	—
64	C₂³×C₃²	C₂²×C₄	1	1	yes	characteristic
65	C₂×C₃²×C₄	—	2	4	no	—
66	C₂×C₃²×C₄	C₄⋊C₄	1	4	yes	—
67	C₂×C₃²×C₄	C₂²×C₄	1	2	yes	—
68	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
69	C₂×C₃²×C₄	C₂×Q₈	1	1	yes	characteristic
70	C₂×C₃²×C₄	#41	1	2	yes	—
71	C₂×C₃×Dic₃	—	2	4	no	—
72	C₂×C₃×Dic₃	—	2	2	no	—
73	C₂×C₃×Dic₃	—	2	4	no	—
74	C₂×C₃×Dic₃	C₄²	1	4	yes	—
75	C₃²×C₈	—	2	4	no	—
76	C₃×C₃⋊C₈	—	2	4	no	—
77	C₂×C₄⋊C₈	—	3	2	no	—
78	C₂×#97	—	3	1	no	—
79	C₂×C₄×C₈	C₃×D₃	1	1	yes	characteristic
80	#376	—	3	1	no	—
81	#336	—	3	2	no	—
82	C₂²×C₃×C₄	—	2	2	no	—
83	C₂²×C₃×C₄	—	2	1	no	—
84	C₂²×C₃×C₄	—	3	2	no	—
85	C₂²×C₃×C₄	—	3	2	no	—
86	C₂²×C₃×C₄	C₄×D₃	1	2	yes	—
87	C₂²×C₃×C₄	C₂×C₃×C₄	1	2	yes	—
88	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
89	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
90	C₂²×Dic₃	C₂×C₃×C₄	1	2	yes	—
91	C₂²×Dic₃	C₂×C₃×C₄	1	2	yes	—
92	C₂×C₃×C₈	—	2	4	no	—
93	C₂×C₃×C₈	—	2	4	no	—
94	C₂×C₃×C₈	—	2	2	no	—
95	C₂×C₃×C₈	—	2	2	no	—
96	C₂×C₃×C₈	—	2	2	no	—
97	C₂×C₃×C₈	—	2	2	no	—
98	C₂×C₃×C₈	—	2	4	no	—
99	C₂×C₃×C₈	—	3	4	no	—
100	C₂×C₃×C₈	—	6	2	no	—
101	C₂×C₃×C₈	—	6	2	no	—
102	C₂×C₃×C₈	C₄×D₃	1	4	yes	—
103	C₂×C₃×C₈	C₂×C₃×C₄	1	4	yes	—
104	C₃×C₄²	—	2	2	no	—
105	C₃×C₄²	—	2	2	no	—
106	C₃×C₄²	—	3	4	no	—
107	C₃×C₄²	—	6	2	no	—
108	C₃×C₄²	Dic₆	1	2	yes	—
109	C₃×C₄²	#67	1	2	yes	—
110	C₃×C₄²	C₃×D₄	1	2	yes	—
111	C₃×C₄²	C₃×Q₈	1	2	yes	—
112	C₄×Dic₃	—	2	2	no	—
113	C₄×Dic₃	C₂×C₃×C₄	1	4	yes	—
114	C₂×C₃⋊C₈	—	2	2	no	—
115	C₂×C₃⋊C₈	—	2	2	no	—
116	C₂×C₃⋊C₈	C₂×C₃×C₄	1	4	yes	—
117	C₂²×C₃²	C₂×C₄²	1	1	yes	characteristic
118	C₂²×C₃²	C₂×C₄⋊C₄	1	1	yes	characteristic
119	C₂²×C₃²	#117	1	1	yes	characteristic
120	C₂²×C₃²	#131	1	4	yes	—
121	C₃²×C₄	—	2	4	no	—
122	C₃²×C₄	C₄×D₄	1	2	yes	—
123	C₃²×C₄	C₄×Q₈	1	2	yes	—
124	C₃×Dic₃	—	2	4	no	—
125	C₃×Dic₃	—	4	2	no	—
126	C₂²×C₈	—	3	2	no	—
127	C₂²×C₈	C₂×C₃×D₃	1	2	yes	—
128	C₂×C₄²	—	3	1	no	—
129	C₂×C₄²	—	3	1	no	—
130	C₂×C₄²	C₂×C₃×D₃	1	1	yes	characteristic
131	C₄⋊C₈	—	3	8	no	—
132	#97	—	6	2	no	—
133	C₄×C₈	—	2	2	no	—
134	#95	—	3	4	no	—
135	C₂³×C₃	—	2	1	no	—
136	C₂³×C₃	C₂×C₄×D₃	1	1	yes	characteristic
137	C₂³×C₃	C₂²×C₃×C₄	1	1	yes	characteristic
138	C₂×C₃×C₄	—	2	8	no	—
139	C₂×C₃×C₄	—	2	4	no	—
140	C₂×C₃×C₄	—	2	4	no	—
141	C₂×C₃×C₄	—	2	4	no	—
142	C₂×C₃×C₄	—	2	1	no	—
143	C₂×C₃×C₄	—	2	2	no	—
144	C₂×C₃×C₄	—	2	2	no	—
145	C₂×C₃×C₄	—	2	1	no	—
146	C₂×C₃×C₄	—	3	4	no	—
147	C₂×C₃×C₄	—	6	4	no	—
148	C₂×C₃×C₄	—	6	2	no	—
149	C₂×C₃×C₄	—	6	4	no	—
150	C₂×C₃×C₄	Dic₃⋊C₄	1	4	yes	—
151	C₂×C₃×C₄	C₃×C₄⋊C₄	1	4	yes	—
152	C₂×C₃×C₄	C₂×Dic₆	1	1	yes	characteristic
153	C₂×C₃×C₄	C₂×C₄×D₃	1	2	yes	—
154	C₂×C₃×C₄	#235	1	2	yes	—
155	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
156	C₂×C₃×C₄	C₂²×C₃×C₄	1	2	yes	—
157	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
158	C₂×C₃×C₄	C₂×C₃×Q₈	1	1	yes	characteristic
159	C₂×C₃×C₄	C₃×#41	1	2	yes	—
160	C₂×Dic₃	—	2	4	no	—
161	C₂×Dic₃	—	2	2	no	—
162	C₂×Dic₃	—	2	4	no	—
163	C₂×Dic₃	C₃×C₄²	1	4	yes	—
164	C₃×C₈	—	2	8	no	—
165	C₃×C₈	—	2	4	no	—
166	C₃×C₈	—	2	4	no	—
167	C₃×C₈	—	6	4	no	—
168	C₃⋊C₈	—	2	4	no	—
169	C₂×C₃²	C₄×C₄⋊C₄	1	1	yes	characteristic
170	C₂×C₃²	#405	1	2	yes	—
171	C₂×C₃²	#424	1	2	yes	—
172	C₂×C₃²	#433	1	2	yes	—
173	C₂²×C₄	—	3	2	no	—
174	C₂²×C₄	—	3	2	no	—
175	C₂²×C₄	C₃×C₄×D₃	1	2	yes	—
176	C₂²×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
177	C₂×C₈	—	2	2	no	—
178	C₂×C₈	—	2	2	no	—
179	C₂×C₈	—	3	4	no	—
180	C₂×C₈	—	6	2	no	—
181	C₂×C₈	—	6	2	no	—
182	C₂×C₈	C₃×C₄×D₃	1	4	yes	—
183	C₄²	—	3	4	no	—
184	C₄²	—	6	2	no	—
185	C₄²	C₃×Dic₆	1	2	yes	—
186	C₄²	C₃×#67	1	2	yes	—
187	C₂²×C₃	—	2	4	no	—
188	C₂²×C₃	—	2	1	no	—
189	C₂²×C₃	—	2	1	no	—
190	C₂²×C₃	—	2	1	no	—
191	C₂²×C₃	C₄²×D₃	1	1	yes	characteristic
192	C₂²×C₃	#872	1	1	yes	characteristic
193	C₂²×C₃	#901	1	4	yes	—
194	C₂²×C₃	C₂×Dic₃⋊C₄	1	1	yes	characteristic
195	C₂²×C₃	C₂×C₃×C₄²	1	1	yes	characteristic
196	C₂²×C₃	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
197	C₂²×C₃	C₃×#117	1	1	yes	characteristic
198	C₂²×C₃	C₃×#131	1	4	yes	—
199	C₃×C₄	—	2	8	no	—
200	C₃×C₄	—	2	4	no	—
201	C₃×C₄	—	2	4	no	—
202	C₃×C₄	—	2	2	no	—
203	C₃×C₄	—	2	2	no	—
204	C₃×C₄	—	6	4	no	—
205	C₃×C₄	—	12	2	no	—
206	C₃×C₄	C₄×Dic₆	1	2	yes	—
207	C₃×C₄	C₄×#67	1	2	yes	—
208	C₃×C₄	C₃×C₄×D₄	1	2	yes	—
209	C₃×C₄	C₃×C₄×Q₈	1	2	yes	—
210	Dic₃	—	2	4	no	—
211	Dic₃	—	4	2	no	—
212	C₃²	#1768	1	1	yes	characteristic, complemented, Sylow
213	C₂³	C₂×C₃×C₄×D₃	1	1	yes	characteristic
214	C₂×C₄	—	2	4	no	—
215	C₂×C₄	—	3	4	no	—
216	C₂×C₄	—	6	4	no	—
217	C₂×C₄	—	6	4	no	—
218	C₂×C₄	—	6	2	no	—
219	C₂×C₄	C₃×Dic₃⋊C₄	1	4	yes	—
220	C₂×C₄	C₂×C₃×Dic₆	1	1	yes	characteristic
221	C₂×C₄	C₂×C₃×C₄×D₃	1	2	yes	—
222	C₂×C₄	C₃×#235	1	2	yes	—
223	C₂×C₄	C₂×C₃×#67	1	1	yes	characteristic
224	C₈	—	2	4	no	—
225	C₈	—	6	4	no	—
226	C₂×C₃	—	2	1	no	—
227	C₂×C₃	—	2	2	no	—
228	C₂×C₃	—	2	2	no	—
229	C₂×C₃	—	2	2	no	—
230	C₂×C₃	#4689	1	2	yes	—
231	C₂×C₃	#4690	1	2	yes	—
232	C₂×C₃	C₄×Dic₃⋊C₄	1	1	yes	characteristic
233	C₂×C₃	#5102	1	2	yes	—
234	C₂×C₃	C₃×C₄×C₄⋊C₄	1	1	yes	characteristic
235	C₂×C₃	C₃×#405	1	2	yes	—
236	C₂×C₃	C₃×#424	1	2	yes	—
237	C₂×C₃	C₃×#433	1	2	yes	—
238	C₂²	C₃×C₄²×D₃	1	1	yes	characteristic
239	C₂²	C₃×#872	1	1	yes	characteristic
240	C₂²	C₃×#901	1	4	yes	—
241	C₂²	C₂×C₃×Dic₃⋊C₄	1	1	yes	characteristic
242	C₄	—	2	4	no	—
243	C₄	—	6	4	no	—
244	C₄	—	12	2	no	—
245	C₄	C₃×C₄×Dic₆	1	2	yes	—
246	C₄	C₃×C₄×#67	1	2	yes	—
247	C₃	—	2	1	no	—
248	C₃	#69755	1	1	yes	characteristic, direct product factor
249	C₃	C₃×#1768	1	1	yes	characteristic, complemented
250	C₂	C₃×#4689	1	2	yes	—
251	C₂	C₃×#4690	1	2	yes	—
252	C₂	C₃×C₄×Dic₃⋊C₄	1	1	yes	characteristic
253	C₂	C₃×#5102	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×C₄×C₈	C₂
socle, prime radical	C₂³×C₃²	C₂²×C₄
cosocle	C₂²×C₃×C₄	C₂³×C₃
hypercenter	C₂×C₃×C₄×C₈	D₃
hypocenter	C₃	C₃×#1768
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₃×C₄ ◁ C₂×C₃×C₄×C₈

Complements

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

N	H ≅ G/N	type
7. #69755	C₃	direct (N × H)
15. C₃×Dic₃⋊C₈	C₄	semidirect
62. Dic₃⋊C₈	C₃×C₄	semidirect
212. C₃²	#1768	semidirect
248. C₃	#69755	direct (N × H)
249. C₃	C₃×#1768	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃²×C₄×C₈	2
C₂×C₃×Dic₃⋊C₈	2
C₃×#4932	2
C₂×C₃×#803	2
C₃×#4550	2
C₃×#1768	3
#69755	3

Subgroups — C3×#69755 #11784217

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#69755 #11784217