Subgroups — C₃×#70923 #11797557

Order 1152.

345 conjugacy classes of subgroups.

The subgroup lattice has 241 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×#443	C₂	1	1	yes	maximal, characteristic
2	C₃×#5140	C₂	1	2	yes	maximal, complemented
3	C₃×#4561	C₂	1	2	yes	maximal, complemented
4	C₃×#4463	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×C₄×C₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#2170	—	3	1	no	maximal
7	#70923	C₃	1	1	yes	maximal, characteristic, direct product factor
8	C₃²×#135	C₄	1	1	yes	characteristic
9	C₃²×#131	C₂²	1	2	yes	—
10	C₃²×#108	C₄	1	1	yes	characteristic
11	C₃²×#104	C₄	1	2	yes	—
12	C₃²×C₄×C₈	C₂²	1	1	yes	characteristic
13	C₃×#812	—	2	2	no	—
14	C₃×#812	C₂²	1	2	yes	—
15	C₂×C₃×C₃⋊C₁₆	—	2	2	no	—
16	C₂×C₃×C₃⋊C₁₆	C₂²	1	1	yes	characteristic
17	C₂×C₃×C₃⋊C₁₆	C₂²	1	1	yes	characteristic
18	C₃×#504	—	3	2	no	—
19	C₃×#443	—	2	1	no	—
20	C₃×#443	D₃	1	1	yes	characteristic
21	C₃×#443	C₂×C₃	1	1	yes	characteristic
22	#5140	C₂×C₃	1	2	yes	complemented
23	C₃×#364	—	3	1	no	—
24	C₃×#350	—	3	2	no	—
25	C₃×C₄×C₁₆	—	3	1	no	—
26	#4561	C₂×C₃	1	2	yes	complemented
27	#4463	C₂×C₃	1	1	yes	characteristic, complemented
28	C₄×C₃⋊C₁₆	C₂×C₃	1	1	yes	characteristic, complemented
29	C₃²×#41	C₂×C₄	1	2	yes	—
30	C₃²×Q₁₆	C₈	1	1	yes	characteristic
31	C₃²×SD₁₆	C₈	1	2	yes	—
32	C₃²×D₈	C₈	1	1	yes	characteristic
33	C₃²×M₁₆	—	2	2	no	—
34	C₃²×M₁₆	C₂×C₄	1	2	yes	—
35	C₂×C₃²×C₈	—	2	2	no	—
36	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
37	C₂×C₃²×C₈	C₂³	1	1	yes	characteristic
38	C₃²×C₄²	C₂×C₄	1	1	yes	characteristic
39	C₃×C₃⋊C₁₆	—	2	2	no	—
40	C₃×C₃⋊C₁₆	—	2	1	no	—
41	C₃×C₃⋊C₁₆	D₄	1	1	yes	characteristic
42	C₃×C₃⋊C₁₆	D₄	1	1	yes	characteristic
43	#2170	—	3	1	no	Sylow
44	C₃×#135	—	2	1	no	—
45	C₃×#135	Dic₃	1	1	yes	characteristic
46	C₃×#135	C₃×C₄	1	1	yes	characteristic
47	C₃×#131	—	2	2	no	—
48	C₃×#131	C₂×D₃	1	2	yes	—
49	C₃×#131	C₂²×C₃	1	2	yes	—
50	C₃×M₃₂	—	3	2	no	—
51	C₃×M₃₂	—	6	2	no	—
52	C₂×C₃×C₁₆	—	3	1	no	—
53	C₂×C₃×C₁₆	—	3	1	no	—
54	C₂×C₃×C₁₆	—	6	2	no	—
55	C₃×#108	—	2	1	no	—
56	C₃×#108	Dic₃	1	1	yes	characteristic
57	C₃×#108	C₃×C₄	1	1	yes	characteristic
58	C₃×#104	—	2	2	no	—
59	C₃×#104	Dic₃	1	2	yes	—
60	C₃×#104	C₃×C₄	1	2	yes	—
61	C₃×C₄×C₈	—	2	1	no	—
62	C₃×C₄×C₈	C₂×D₃	1	1	yes	characteristic
63	C₃×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
64	#812	—	2	2	no	—
65	#812	C₂²×C₃	1	2	yes	—
66	C₂×C₃⋊C₁₆	—	2	2	no	—
67	C₂×C₃⋊C₁₆	C₂²×C₃	1	1	yes	characteristic
68	C₂×C₃⋊C₁₆	C₂²×C₃	1	1	yes	characteristic
69	C₃²×Q₈	C₂×C₈	1	2	yes	—
70	C₃²×D₄	—	2	2	no	—
71	C₃²×D₄	C₂×C₈	1	2	yes	—
72	C₂×C₃²×C₄	—	2	1	no	—
73	C₂×C₃²×C₄	—	2	2	no	—
74	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
75	C₃²×C₈	—	2	2	no	—
76	C₃²×C₈	C₂×C₈	1	1	yes	characteristic
77	C₃²×C₈	C₂×C₈	1	1	yes	characteristic
78	C₃²×C₈	C₂×D₄	1	1	yes	characteristic
79	C₃²×C₈	#41	1	1	yes	characteristic
80	#504	—	3	2	no	—
81	#443	C₃×D₃	1	1	yes	characteristic
82	#364	—	3	1	no	—
83	#350	—	3	2	no	—
84	C₄×C₁₆	—	3	1	no	—
85	C₃×#41	—	2	2	no	—
86	C₃×#41	C₂×Dic₃	1	2	yes	—
87	C₃×#41	C₂×C₃×C₄	1	2	yes	—
88	C₃×Q₁₆	—	2	1	no	—
89	C₃×Q₁₆	C₃⋊C₈	1	1	yes	characteristic
90	C₃×Q₁₆	C₃×C₈	1	1	yes	characteristic
91	C₃×SD₁₆	—	2	2	no	—
92	C₃×SD₁₆	C₃⋊C₈	1	2	yes	—
93	C₃×SD₁₆	C₃×C₈	1	2	yes	—
94	C₃×D₈	—	2	1	no	—
95	C₃×D₈	C₃⋊C₈	1	1	yes	characteristic
96	C₃×D₈	C₃×C₈	1	1	yes	characteristic
97	C₃×M₁₆	—	2	2	no	—
98	C₃×M₁₆	—	2	2	no	—
99	C₃×M₁₆	—	2	2	no	—
100	C₃×M₁₆	—	4	2	no	—
101	C₃×M₁₆	C₂×Dic₃	1	2	yes	—
102	C₃×M₁₆	C₂×C₃×C₄	1	2	yes	—
103	C₂×C₃×C₈	—	2	2	no	—
104	C₂×C₃×C₈	—	2	2	no	—
105	C₂×C₃×C₈	—	2	1	no	—
106	C₂×C₃×C₈	—	2	1	no	—
107	C₂×C₃×C₈	—	4	2	no	—
108	C₂×C₃×C₈	C₂×Dic₃	1	1	yes	characteristic
109	C₂×C₃×C₈	C₂×C₃×C₄	1	1	yes	characteristic
110	C₂×C₃×C₈	C₂²×D₃	1	1	yes	characteristic
111	C₂×C₃×C₈	C₂³×C₃	1	1	yes	characteristic
112	C₃×C₄²	—	2	1	no	—
113	C₃×C₄²	C₂×Dic₃	1	1	yes	characteristic
114	C₃×C₄²	C₂×C₃×C₄	1	1	yes	characteristic
115	C₃×C₁₆	—	3	1	no	—
116	C₃×C₁₆	—	3	1	no	—
117	C₃×C₁₆	—	6	2	no	—
118	C₃×C₁₆	—	6	1	no	—
119	C₃⋊C₁₆	—	2	2	no	—
120	C₃⋊C₁₆	—	2	1	no	—
121	C₃⋊C₁₆	C₃×D₄	1	1	yes	characteristic
122	C₃⋊C₁₆	C₃×D₄	1	1	yes	characteristic
123	C₂²×C₃²	—	2	2	no	—
124	C₂²×C₃²	#131	1	1	yes	characteristic
125	C₃²×C₄	—	2	2	no	—
126	C₃²×C₄	—	2	2	no	—
127	C₃²×C₄	C₄×D₄	1	1	yes	characteristic
128	C₃²×C₄	C₂²×C₈	1	1	yes	characteristic
129	#135	C₃×Dic₃	1	1	yes	characteristic
130	#131	C₂×C₃×D₃	1	2	yes	—
131	M₃₂	—	3	2	no	—
132	M₃₂	—	6	2	no	—
133	C₂×C₁₆	—	3	1	no	—
134	C₂×C₁₆	—	3	1	no	—
135	C₂×C₁₆	—	6	2	no	—
136	#108	C₃×Dic₃	1	1	yes	characteristic
137	#104	C₃×Dic₃	1	2	yes	—
138	C₄×C₈	C₂×C₃×D₃	1	1	yes	characteristic
139	C₃×Q₈	—	2	2	no	—
140	C₃×Q₈	C₂×C₃⋊C₈	1	2	yes	—
141	C₃×Q₈	C₂×C₃×C₈	1	2	yes	—
142	C₃×D₄	—	2	2	no	—
143	C₃×D₄	—	2	2	no	—
144	C₃×D₄	—	2	2	no	—
145	C₃×D₄	—	4	2	no	—
146	C₃×D₄	C₂×C₃⋊C₈	1	2	yes	—
147	C₃×D₄	C₂×C₃×C₈	1	2	yes	—
148	C₂×C₃×C₄	—	2	1	no	—
149	C₂×C₃×C₄	—	2	1	no	—
150	C₂×C₃×C₄	—	2	2	no	—
151	C₂×C₃×C₄	—	2	2	no	—
152	C₂×C₃×C₄	—	2	1	no	—
153	C₂×C₃×C₄	—	4	2	no	—
154	C₂×C₃×C₄	—	4	1	no	—
155	C₂×C₃×C₄	C₂²×Dic₃	1	1	yes	characteristic
156	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
157	C₃×C₈	—	2	2	no	—
158	C₃×C₈	—	2	2	no	—
159	C₃×C₈	—	2	1	no	—
160	C₃×C₈	—	2	1	no	—
161	C₃×C₈	—	2	1	no	—
162	C₃×C₈	—	2	1	no	—
163	C₃×C₈	—	4	2	no	—
164	C₃×C₈	C₂×C₃⋊C₈	1	1	yes	characteristic
165	C₃×C₈	C₂×C₃⋊C₈	1	1	yes	characteristic
166	C₃×C₈	C₂×C₃×C₈	1	1	yes	characteristic
167	C₃×C₈	C₂×C₃×C₈	1	1	yes	characteristic
168	C₃×C₈	D₃×D₄	1	1	yes	characteristic
169	C₃×C₈	#237	1	1	yes	characteristic
170	C₃×C₈	C₂×C₃×D₄	1	1	yes	characteristic
171	C₃×C₈	C₃×#41	1	1	yes	characteristic
172	C₂×C₃²	—	2	1	no	—
173	C₂×C₃²	—	4	2	no	—
174	C₂×C₃²	C₈×D₄	1	1	yes	characteristic
175	#41	C₂×C₃×Dic₃	1	2	yes	—
176	Q₁₆	C₃×C₃⋊C₈	1	1	yes	characteristic
177	SD₁₆	C₃×C₃⋊C₈	1	2	yes	—
178	D₈	C₃×C₃⋊C₈	1	1	yes	characteristic
179	M₁₆	—	2	2	no	—
180	M₁₆	C₂×C₃×Dic₃	1	2	yes	—
181	C₂×C₈	—	2	2	no	—
182	C₂×C₈	C₂×C₃×Dic₃	1	1	yes	characteristic
183	C₂×C₈	C₂²×C₃×D₃	1	1	yes	characteristic
184	C₄²	C₂×C₃×Dic₃	1	1	yes	characteristic
185	C₁₆	—	3	1	no	—
186	C₁₆	—	3	1	no	—
187	C₁₆	—	6	2	no	—
188	C₁₆	—	6	1	no	—
189	C₂²×C₃	—	2	2	no	—
190	C₂²×C₃	—	2	2	no	—
191	C₂²×C₃	—	2	1	no	—
192	C₂²×C₃	—	4	2	no	—
193	C₂²×C₃	#948	1	1	yes	characteristic
194	C₂²×C₃	C₃×#131	1	1	yes	characteristic
195	C₃×C₄	—	2	2	no	—
196	C₃×C₄	—	2	2	no	—
197	C₃×C₄	—	2	2	no	—
198	C₃×C₄	—	2	2	no	—
199	C₃×C₄	—	2	1	no	—
200	C₃×C₄	—	2	1	no	—
201	C₃×C₄	—	4	2	no	—
202	C₃×C₄	—	4	2	no	—
203	C₃×C₄	C₂²×C₃⋊C₈	1	1	yes	characteristic
204	C₃×C₄	D₄×Dic₃	1	1	yes	characteristic
205	C₃×C₄	C₃×C₄×D₄	1	1	yes	characteristic
206	C₃×C₄	C₂²×C₃×C₈	1	1	yes	characteristic
207	C₃²	#2170	1	1	yes	characteristic, complemented, Sylow
208	Q₈	C₂×C₃×C₃⋊C₈	1	2	yes	—
209	D₄	—	2	2	no	—
210	D₄	C₂×C₃×C₃⋊C₈	1	2	yes	—
211	C₂×C₄	—	2	1	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	C₂²×C₃×Dic₃	1	1	yes	characteristic
214	C₈	—	2	2	no	—
215	C₈	C₂×C₃×C₃⋊C₈	1	1	yes	characteristic
216	C₈	C₂×C₃×C₃⋊C₈	1	1	yes	characteristic
217	C₈	C₃×D₃×D₄	1	1	yes	characteristic
218	C₈	C₃×#237	1	1	yes	characteristic
219	C₂×C₃	—	2	1	no	—
220	C₂×C₃	—	2	1	no	—
221	C₂×C₃	—	2	1	no	—
222	C₂×C₃	—	4	1	no	—
223	C₂×C₃	—	4	2	no	—
224	C₂×C₃	—	4	2	no	—
225	C₂×C₃	—	8	2	no	—
226	C₂×C₃	D₄×C₃⋊C₈	1	1	yes	characteristic
227	C₂×C₃	C₃×C₈×D₄	1	1	yes	characteristic
228	C₂²	—	2	2	no	—
229	C₂²	C₃×#948	1	1	yes	characteristic
230	C₄	—	2	2	no	—
231	C₄	—	2	2	no	—
232	C₄	C₂²×C₃×C₃⋊C₈	1	1	yes	characteristic
233	C₄	C₃×D₄×Dic₃	1	1	yes	characteristic
234	C₃	—	2	1	no	—
235	C₃	#70923	1	1	yes	characteristic, direct product factor
236	C₃	C₃×#2170	1	1	yes	characteristic, complemented
237	C₂	—	2	1	no	—
238	C₂	—	4	2	no	—
239	C₂	C₃×D₄×C₃⋊C₈	1	1	yes	characteristic

	subgroup	G/H
center	C₃×C₈	D₃×D₄
commutator	C₃×C₄	C₂²×C₃×C₈
Frattini	C₂×C₈	C₂²×C₃×D₃
Fitting	C₃²×#443	C₂
socle	C₂×C₃²	C₈×D₄
prime radical	C₃²×#135	C₄
cosocle	C₂×C₃×C₈	C₂³×C₃
hypercenter	C₃×#443	D₃
hypocenter	C₃	C₃×#2170
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₃×C₈ ◁ C₂×C₃×C₈ ◁ C₃×#443

Complements

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

N	H ≅ G/N	type
2. C₃×#5140	C₂	semidirect
3. C₃×#4561	C₂	semidirect
4. C₃×#4463	C₂	semidirect
5. C₃×C₄×C₃⋊C₁₆	C₂	semidirect
7. #70923	C₃	direct (N × H)
22. #5140	C₂×C₃	semidirect
26. #4561	C₂×C₃	semidirect
27. #4463	C₂×C₃	semidirect
28. C₄×C₃⋊C₁₆	C₂×C₃	semidirect
207. C₃²	#2170	semidirect
235. C₃	#70923	direct (N × H)
236. C₃	C₃×#2170	semidirect

Maximal subgroups

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

subgroup	index
C₃²×#443	2
C₃×#5140	2
C₃×#4561	2
C₃×#4463	2
C₃×C₄×C₃⋊C₁₆	2
C₃×#2170	3
#70923	3

Subgroups — C3×#70923 #11797557

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#70923 #11797557