Subgroups — C₃×#70694 #11786326

Order 1152.

477 conjugacy classes of subgroups.

The subgroup lattice has 269 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×#108	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×#822	C₂	1	2	yes	maximal
3	C₃×#5118	C₂	1	2	yes	maximal
4	C₂×C₃×#819	C₂	1	1	yes	maximal, characteristic
5	C₂×C₃×#815	C₂	1	1	yes	maximal, characteristic
6	C₃×#1849	—	3	1	no	maximal
7	#70694	C₃	1	1	yes	maximal, characteristic, direct product factor
8	C₂×C₃²×M₁₆	C₂²	1	2	yes	—
9	C₂²×C₃²×C₈	C₂²	1	1	yes	characteristic
10	C₂×C₃×C₄×Dic₃	C₂²	1	1	yes	characteristic
11	C₂×C₃×#208	C₂²	1	2	yes	—
12	C₂²×C₃×C₃⋊C₈	C₂²	1	1	yes	characteristic
13	C₃²×#108	—	2	2	no	—
14	C₃×#822	—	2	4	no	—
15	C₃×#819	—	2	2	no	—
16	C₃×#815	C₄	1	4	yes	—
17	C₃×Dic₃⋊C₈	—	2	4	no	—
18	C₂×C₃×#108	—	2	1	no	—
19	C₂×C₃×#108	—	3	2	no	—
20	C₂×C₃×#108	—	3	1	no	—
21	C₂×C₃×#108	D₃	1	1	yes	characteristic
22	C₂×C₃×#108	C₂×C₃	1	1	yes	characteristic
23	C₃×#423	—	3	2	no	—
24	C₂×C₃×#97	—	3	1	no	—
25	C₂×#822	C₂×C₃	1	2	yes	—
26	#5118	C₂×C₃	1	2	yes	—
27	C₂×#819	C₂×C₃	1	1	yes	characteristic
28	C₂×#815	C₂×C₃	1	1	yes	characteristic
29	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
30	C₂²×C₃×Dic₃	Q₈	1	2	yes	—
31	C₃²×M₁₆	—	2	2	no	—
32	C₃²×M₁₆	—	2	2	no	—
33	C₂×C₃²×C₈	—	2	2	no	—
34	C₂×C₃²×C₈	C₂×C₄	1	2	yes	—
35	C₂×C₃²×C₈	D₄	1	2	yes	—
36	C₂×C₃²×C₈	Q₈	1	2	yes	—
37	C₃×C₄×Dic₃	C₂×C₄	1	2	yes	—
38	C₃×C₄×Dic₃	D₄	1	2	yes	—
39	C₃×#208	—	2	2	no	—
40	C₃×#208	—	2	2	no	—
41	C₂×C₃×C₃⋊C₈	—	2	2	no	—
42	C₂×C₃×C₃⋊C₈	C₂×C₄	1	2	yes	—
43	C₂×C₃×C₃⋊C₈	D₄	1	2	yes	—
44	C₂×C₃×C₃⋊C₈	Q₈	1	2	yes	—
45	#1849	—	3	1	no	Sylow
46	C₂×C₃×M₁₆	—	2	2	no	—
47	C₂×C₃×M₁₆	—	3	2	no	—
48	C₂×C₃×M₁₆	C₂×D₃	1	2	yes	—
49	C₂×C₃×M₁₆	C₂²×C₃	1	2	yes	—
50	C₂²×C₃×C₈	—	2	1	no	—
51	C₂²×C₃×C₈	—	3	1	no	—
52	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
53	C₂²×C₃×C₈	C₂²×C₃	1	1	yes	characteristic
54	C₂×C₃×C₄²	—	3	1	no	—
55	C₂×C₄×Dic₃	C₂²×C₃	1	1	yes	characteristic
56	C₂×#208	C₂²×C₃	1	2	yes	—
57	C₂²×C₃⋊C₈	C₂²×C₃	1	1	yes	characteristic
58	C₃×#108	—	2	4	no	—
59	C₃×#108	—	2	2	no	—
60	C₃×#108	—	2	2	no	—
61	C₃×#108	—	6	4	no	—
62	C₃×#108	—	6	2	no	—
63	C₃×C₄⋊C₈	—	6	4	no	—
64	C₃×#97	—	3	4	no	—
65	#822	—	2	4	no	—
66	#819	—	2	2	no	—
67	#815	C₃×C₄	1	4	yes	—
68	Dic₃⋊C₈	—	2	4	no	—
69	C₂³×C₃²	C₂×Q₈	1	1	yes	characteristic
70	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
71	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
72	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
73	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
74	C₂×C₃²×C₄	C₂×Q₈	1	1	yes	characteristic
75	C₂×C₃²×C₄	C₂×Q₈	1	1	yes	characteristic
76	C₂×C₃×Dic₃	—	4	2	no	—
77	C₂×C₃×Dic₃	C₄⋊C₄	1	4	yes	—
78	C₃²×C₈	—	4	2	no	—
79	C₃²×C₈	C₄⋊C₄	1	4	yes	—
80	C₃×C₃⋊C₈	—	4	2	no	—
81	C₃×C₃⋊C₈	C₄⋊C₄	1	4	yes	—
82	C₂×#108	—	3	2	no	—
83	C₂×#108	—	3	1	no	—
84	C₂×#108	C₃×D₃	1	1	yes	characteristic
85	#423	—	3	2	no	—
86	C₂×#97	—	3	1	no	—
87	C₂²×C₃×C₄	—	2	1	no	—
88	C₂²×C₃×C₄	—	3	2	no	—
89	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
90	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
91	C₂²×Dic₃	C₃×Q₈	1	2	yes	—
92	C₃×M₁₆	—	2	4	no	—
93	C₃×M₁₆	—	2	2	no	—
94	C₃×M₁₆	—	2	2	no	—
95	C₃×M₁₆	—	2	2	no	—
96	C₃×M₁₆	—	2	2	no	—
97	C₃×M₁₆	—	4	2	no	—
98	C₃×M₁₆	—	6	2	no	—
99	C₃×M₁₆	—	6	2	no	—
100	C₂×C₃×C₈	—	2	2	no	—
101	C₂×C₃×C₈	—	2	2	no	—
102	C₂×C₃×C₈	—	2	2	no	—
103	C₂×C₃×C₈	—	2	2	no	—
104	C₂×C₃×C₈	—	2	2	no	—
105	C₂×C₃×C₈	—	3	2	no	—
106	C₂×C₃×C₈	—	3	2	no	—
107	C₂×C₃×C₈	—	3	2	no	—
108	C₂×C₃×C₈	—	4	2	no	—
109	C₂×C₃×C₈	—	6	2	no	—
110	C₂×C₃×C₈	Dic₆	1	2	yes	—
111	C₂×C₃×C₈	C₄×D₃	1	2	yes	—
112	C₂×C₃×C₈	#67	1	2	yes	—
113	C₂×C₃×C₈	C₂×C₃×C₄	1	2	yes	—
114	C₂×C₃×C₈	C₃×D₄	1	2	yes	—
115	C₂×C₃×C₈	C₃×Q₈	1	2	yes	—
116	C₃×C₄²	—	3	2	no	—
117	C₃×C₄²	—	3	2	no	—
118	C₄×Dic₃	C₂×C₃×C₄	1	2	yes	—
119	C₄×Dic₃	C₃×D₄	1	2	yes	—
120	#208	—	2	2	no	—
121	#208	—	2	2	no	—
122	C₂×C₃⋊C₈	—	2	2	no	—
123	C₂×C₃⋊C₈	C₂×C₃×C₄	1	2	yes	—
124	C₂×C₃⋊C₈	C₃×D₄	1	2	yes	—
125	C₂×C₃⋊C₈	C₃×Q₈	1	2	yes	—
126	C₂²×C₃²	—	2	2	no	—
127	C₂²×C₃²	C₂×C₄⋊C₄	1	1	yes	characteristic
128	C₂²×C₃²	C₄⋊Q₈	1	1	yes	characteristic
129	C₂²×C₃²	C₄⋊Q₈	1	1	yes	characteristic
130	C₃²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
131	C₃²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
132	C₃²×C₄	#127	1	1	yes	characteristic
133	C₃²×C₄	C₄⋊Q₈	1	1	yes	characteristic
134	C₃×Dic₃	—	2	4	no	—
135	C₂×M₁₆	—	3	2	no	—
136	C₂×M₁₆	C₂×C₃×D₃	1	2	yes	—
137	C₂²×C₈	—	3	1	no	—
138	C₂²×C₈	C₂×C₃×D₃	1	1	yes	characteristic
139	C₂×C₄²	—	3	1	no	—
140	#108	—	2	2	no	—
141	#108	—	6	4	no	—
142	#108	—	6	2	no	—
143	C₄⋊C₈	—	6	4	no	—
144	#97	—	3	4	no	—
145	C₂³×C₃	—	2	1	no	—
146	C₂³×C₃	D₃×Q₈	1	1	yes	characteristic
147	C₂³×C₃	C₂×C₃×Q₈	1	1	yes	characteristic
148	C₂×C₃×C₄	—	2	1	no	—
149	C₂×C₃×C₄	—	2	1	no	—
150	C₂×C₃×C₄	—	2	1	no	—
151	C₂×C₃×C₄	—	2	1	no	—
152	C₂×C₃×C₄	—	2	1	no	—
153	C₂×C₃×C₄	—	2	1	no	—
154	C₂×C₃×C₄	—	3	4	no	—
155	C₂×C₃×C₄	—	12	2	no	—
156	C₂×C₃×C₄	C₂×Dic₆	1	1	yes	characteristic
157	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
158	C₂×C₃×C₄	D₃×D₄	1	1	yes	characteristic
159	C₂×C₃×C₄	D₃×D₄	1	1	yes	characteristic
160	C₂×C₃×C₄	D₃×Q₈	1	1	yes	characteristic
161	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
162	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
163	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
164	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
165	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
166	C₂×C₃×C₄	C₂×C₃×Q₈	1	1	yes	characteristic
167	C₂×C₃×C₄	C₂×C₃×Q₈	1	1	yes	characteristic
168	C₂×Dic₃	—	4	2	no	—
169	C₂×Dic₃	C₃×C₄⋊C₄	1	4	yes	—
170	C₃×C₈	—	2	4	no	—
171	C₃×C₈	—	3	4	no	—
172	C₃×C₈	—	4	4	no	—
173	C₃×C₈	—	4	2	no	—
174	C₃×C₈	—	4	2	no	—
175	C₃×C₈	—	12	2	no	—
176	C₃×C₈	Dic₃⋊C₄	1	4	yes	—
177	C₃×C₈	C₃×C₄⋊C₄	1	4	yes	—
178	C₃⋊C₈	—	4	2	no	—
179	C₃⋊C₈	C₃×C₄⋊C₄	1	4	yes	—
180	C₂×C₃²	—	2	1	no	—
181	C₂×C₃²	—	2	1	no	—
182	C₂×C₃²	#384	1	1	yes	characteristic
183	C₂×C₃²	#430	1	2	yes	—
184	C₂²×C₄	—	3	2	no	—
185	C₂²×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
186	M₁₆	—	2	2	no	—
187	M₁₆	—	2	2	no	—
188	M₁₆	—	6	2	no	—
189	M₁₆	—	6	2	no	—
190	C₂×C₈	—	2	2	no	—
191	C₂×C₈	—	3	2	no	—
192	C₂×C₈	—	3	2	no	—
193	C₂×C₈	—	3	2	no	—
194	C₂×C₈	—	6	2	no	—
195	C₂×C₈	C₃×Dic₆	1	2	yes	—
196	C₂×C₈	C₃×C₄×D₃	1	2	yes	—
197	C₂×C₈	C₃×#67	1	2	yes	—
198	C₄²	—	3	2	no	—
199	C₄²	—	3	2	no	—
200	C₂²×C₃	—	2	2	no	—
201	C₂²×C₃	—	2	2	no	—
202	C₂²×C₃	—	2	1	no	—
203	C₂²×C₃	—	2	1	no	—
204	C₂²×C₃	—	2	1	no	—
205	C₂²×C₃	—	4	2	no	—
206	C₂²×C₃	#888	1	1	yes	characteristic
207	C₂²×C₃	D₃×C₄⋊C₄	1	1	yes	characteristic
208	C₂²×C₃	#944	1	1	yes	characteristic
209	C₂²×C₃	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
210	C₂²×C₃	C₃×C₄⋊Q₈	1	1	yes	characteristic
211	C₂²×C₃	C₃×C₄⋊Q₈	1	1	yes	characteristic
212	C₃×C₄	—	2	1	no	—
213	C₃×C₄	—	2	1	no	—
214	C₃×C₄	—	2	1	no	—
215	C₃×C₄	—	2	1	no	—
216	C₃×C₄	—	6	4	no	—
217	C₃×C₄	#888	1	1	yes	characteristic
218	C₃×C₄	D₃×C₄⋊C₄	1	1	yes	characteristic
219	C₃×C₄	C₂×Dic₃⋊C₄	1	1	yes	characteristic
220	C₃×C₄	#940	1	1	yes	characteristic
221	C₃×C₄	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
222	C₃×C₄	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
223	C₃×C₄	C₃×#127	1	1	yes	characteristic
224	C₃×C₄	C₃×C₄⋊Q₈	1	1	yes	characteristic
225	Dic₃	—	2	4	no	—
226	C₃²	#1849	1	1	yes	characteristic, complemented, Sylow
227	C₂³	C₃×D₃×Q₈	1	1	yes	characteristic
228	C₂×C₄	—	3	4	no	—
229	C₂×C₄	—	12	2	no	—
230	C₂×C₄	C₂×C₃×Dic₆	1	1	yes	characteristic
231	C₂×C₄	C₂×C₃×C₄×D₃	1	1	yes	characteristic
232	C₂×C₄	C₃×D₃×D₄	1	1	yes	characteristic
233	C₂×C₄	C₃×D₃×D₄	1	1	yes	characteristic
234	C₂×C₄	C₃×D₃×Q₈	1	1	yes	characteristic
235	C₂×C₄	C₂×C₃×#67	1	1	yes	characteristic
236	C₈	—	3	4	no	—
237	C₈	—	4	2	no	—
238	C₈	—	12	2	no	—
239	C₈	C₃×Dic₃⋊C₄	1	4	yes	—
240	C₂×C₃	—	2	1	no	—
241	C₂×C₃	—	2	1	no	—
242	C₂×C₃	—	2	1	no	—
243	C₂×C₃	—	2	1	no	—
244	C₂×C₃	—	2	1	no	—
245	C₂×C₃	—	2	2	no	—
246	C₂×C₃	—	4	1	no	—
247	C₂×C₃	—	4	1	no	—
248	C₂×C₃	#4893	1	2	yes	—
249	C₂×C₃	#4975	1	1	yes	characteristic
250	C₂×C₃	C₃×#384	1	1	yes	characteristic
251	C₂×C₃	C₃×#430	1	2	yes	—
252	C₂²	—	2	2	no	—
253	C₂²	C₃×#888	1	1	yes	characteristic
254	C₂²	C₃×D₃×C₄⋊C₄	1	1	yes	characteristic
255	C₂²	C₃×#944	1	1	yes	characteristic
256	C₄	—	6	4	no	—
257	C₄	C₃×#888	1	1	yes	characteristic
258	C₄	C₃×D₃×C₄⋊C₄	1	1	yes	characteristic
259	C₄	C₂×C₃×Dic₃⋊C₄	1	1	yes	characteristic
260	C₄	C₃×#940	1	1	yes	characteristic
261	C₃	—	2	1	no	—
262	C₃	#70694	1	1	yes	characteristic, direct product factor
263	C₃	C₃×#1849	1	1	yes	characteristic, complemented
264	C₂	—	2	1	no	—
265	C₂	—	2	1	no	—
266	C₂	C₃×#4893	1	2	yes	—
267	C₂	C₃×#4975	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₄	D₃×D₄
commutator	C₂×C₃×C₄	C₂²×C₃×C₄
Frattini	C₂²×C₄	C₂²×C₃×D₃
Fitting	C₂×C₃²×#108	C₂
socle	C₂²×C₃²	C₂×C₄⋊C₄
prime radical	C₂³×C₃²	C₂×Q₈
cosocle	C₂²×C₃×C₄	C₂³×C₃
hypercenter	C₂×C₃×#108	D₃
hypocenter	C₃	C₃×#1849
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₂²×C₃×C₄ ◁ C₂×C₃×#108

Complements

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

N	H ≅ G/N	type
7. #70694	C₃	direct (N × H)
226. C₃²	#1849	semidirect
262. C₃	#70694	direct (N × H)
263. C₃	C₃×#1849	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃²×#108	2
C₂×C₃×#822	2
C₃×#5118	2
C₂×C₃×#819	2
C₂×C₃×#815	2
C₃×#1849	3
#70694	3

Subgroups — C3×#70694 #11786326

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#70694 #11786326