Subgroups — C₃×#70924 #11797593

Order 1152.

341 conjugacy classes of subgroups.

The subgroup lattice has 237 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×#4461	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#2171	—	3	1	no	maximal
7	#70924	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×#346	—	3	1	no	—
26	#4561	C₂×C₃	1	2	yes	complemented
27	#4463	C₂×C₃	1	1	yes	characteristic, complemented
28	#4461	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	2	yes	—
42	#2171	—	3	1	no	Sylow
43	C₃×#135	—	2	1	no	—
44	C₃×#135	Dic₃	1	1	yes	characteristic
45	C₃×#135	C₃×C₄	1	1	yes	characteristic
46	C₃×#131	—	2	2	no	—
47	C₃×#131	C₂×D₃	1	2	yes	—
48	C₃×#131	C₂²×C₃	1	2	yes	—
49	C₃×M₃₂	—	3	2	no	—
50	C₃×M₃₂	—	6	2	no	—
51	C₂×C₃×C₁₆	—	3	1	no	—
52	C₂×C₃×C₁₆	—	3	1	no	—
53	C₂×C₃×C₁₆	—	6	2	no	—
54	C₃×#108	—	2	1	no	—
55	C₃×#108	Dic₃	1	1	yes	characteristic
56	C₃×#108	C₃×C₄	1	1	yes	characteristic
57	C₃×#104	—	2	2	no	—
58	C₃×#104	Dic₃	1	2	yes	—
59	C₃×#104	C₃×C₄	1	2	yes	—
60	C₃×C₄×C₈	—	2	1	no	—
61	C₃×C₄×C₈	C₂×D₃	1	1	yes	characteristic
62	C₃×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
63	#812	—	2	2	no	—
64	#812	C₂²×C₃	1	2	yes	—
65	C₂×C₃⋊C₁₆	—	2	2	no	—
66	C₂×C₃⋊C₁₆	C₂²×C₃	1	1	yes	characteristic
67	C₂×C₃⋊C₁₆	C₂²×C₃	1	1	yes	characteristic
68	C₃²×Q₈	C₂×C₈	1	2	yes	—
69	C₃²×D₄	—	2	2	no	—
70	C₃²×D₄	C₂×C₈	1	2	yes	—
71	C₂×C₃²×C₄	—	2	2	no	—
72	C₂×C₃²×C₄	—	2	1	no	—
73	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
74	C₃²×C₈	—	2	2	no	—
75	C₃²×C₈	C₂×C₈	1	1	yes	characteristic
76	C₃²×C₈	C₂×C₈	1	1	yes	characteristic
77	C₃²×C₈	C₂×D₄	1	1	yes	characteristic
78	C₃²×C₈	#41	1	1	yes	characteristic
79	#504	—	3	2	no	—
80	#443	C₃×D₃	1	1	yes	characteristic
81	#364	—	3	1	no	—
82	#350	—	3	2	no	—
83	#346	—	3	1	no	—
84	C₃×#41	—	2	2	no	—
85	C₃×#41	C₂×Dic₃	1	2	yes	—
86	C₃×#41	C₂×C₃×C₄	1	2	yes	—
87	C₃×Q₁₆	—	2	1	no	—
88	C₃×Q₁₆	C₃⋊C₈	1	1	yes	characteristic
89	C₃×Q₁₆	C₃×C₈	1	1	yes	characteristic
90	C₃×SD₁₆	—	2	2	no	—
91	C₃×SD₁₆	C₃⋊C₈	1	2	yes	—
92	C₃×SD₁₆	C₃×C₈	1	2	yes	—
93	C₃×D₈	—	2	1	no	—
94	C₃×D₈	C₃⋊C₈	1	1	yes	characteristic
95	C₃×D₈	C₃×C₈	1	1	yes	characteristic
96	C₃×M₁₆	—	2	2	no	—
97	C₃×M₁₆	—	2	2	no	—
98	C₃×M₁₆	—	2	2	no	—
99	C₃×M₁₆	—	4	2	no	—
100	C₃×M₁₆	C₂×Dic₃	1	2	yes	—
101	C₃×M₁₆	C₂×C₃×C₄	1	2	yes	—
102	C₂×C₃×C₈	—	2	2	no	—
103	C₂×C₃×C₈	—	2	2	no	—
104	C₂×C₃×C₈	—	2	1	no	—
105	C₂×C₃×C₈	—	2	1	no	—
106	C₂×C₃×C₈	—	4	2	no	—
107	C₂×C₃×C₈	C₂×Dic₃	1	1	yes	characteristic
108	C₂×C₃×C₈	C₂×C₃×C₄	1	1	yes	characteristic
109	C₂×C₃×C₈	C₂²×D₃	1	1	yes	characteristic
110	C₂×C₃×C₈	C₂³×C₃	1	1	yes	characteristic
111	C₃×C₄²	—	2	1	no	—
112	C₃×C₄²	C₂×Dic₃	1	1	yes	characteristic
113	C₃×C₄²	C₂×C₃×C₄	1	1	yes	characteristic
114	C₃×C₁₆	—	3	2	no	—
115	C₃×C₁₆	—	6	2	no	—
116	C₃×C₁₆	—	6	1	no	—
117	C₃⋊C₁₆	—	2	2	no	—
118	C₃⋊C₁₆	—	2	1	no	—
119	C₃⋊C₁₆	C₃×D₄	1	2	yes	—
120	C₂²×C₃²	—	2	2	no	—
121	C₂²×C₃²	#131	1	1	yes	characteristic
122	C₃²×C₄	—	2	2	no	—
123	C₃²×C₄	—	4	1	no	—
124	C₃²×C₄	C₄×D₄	1	1	yes	characteristic
125	C₃²×C₄	C₂²×C₈	1	1	yes	characteristic
126	#135	C₃×Dic₃	1	1	yes	characteristic
127	#131	C₂×C₃×D₃	1	2	yes	—
128	M₃₂	—	3	2	no	—
129	M₃₂	—	6	2	no	—
130	C₂×C₁₆	—	3	1	no	—
131	C₂×C₁₆	—	3	1	no	—
132	C₂×C₁₆	—	6	2	no	—
133	#108	C₃×Dic₃	1	1	yes	characteristic
134	#104	C₃×Dic₃	1	2	yes	—
135	C₄×C₈	C₂×C₃×D₃	1	1	yes	characteristic
136	C₃×Q₈	—	2	2	no	—
137	C₃×Q₈	C₂×C₃⋊C₈	1	2	yes	—
138	C₃×Q₈	C₂×C₃×C₈	1	2	yes	—
139	C₃×D₄	—	2	2	no	—
140	C₃×D₄	—	2	2	no	—
141	C₃×D₄	—	2	2	no	—
142	C₃×D₄	—	4	2	no	—
143	C₃×D₄	C₂×C₃⋊C₈	1	2	yes	—
144	C₃×D₄	C₂×C₃×C₈	1	2	yes	—
145	C₂×C₃×C₄	—	2	2	no	—
146	C₂×C₃×C₄	—	2	2	no	—
147	C₂×C₃×C₄	—	2	1	no	—
148	C₂×C₃×C₄	—	2	1	no	—
149	C₂×C₃×C₄	—	2	1	no	—
150	C₂×C₃×C₄	—	4	2	no	—
151	C₂×C₃×C₄	—	4	1	no	—
152	C₂×C₃×C₄	C₂²×Dic₃	1	1	yes	characteristic
153	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
154	C₃×C₈	—	2	2	no	—
155	C₃×C₈	—	2	2	no	—
156	C₃×C₈	—	2	1	no	—
157	C₃×C₈	—	2	1	no	—
158	C₃×C₈	—	2	1	no	—
159	C₃×C₈	—	2	1	no	—
160	C₃×C₈	—	4	2	no	—
161	C₃×C₈	C₂×C₃⋊C₈	1	1	yes	characteristic
162	C₃×C₈	C₂×C₃⋊C₈	1	1	yes	characteristic
163	C₃×C₈	C₂×C₃×C₈	1	1	yes	characteristic
164	C₃×C₈	C₂×C₃×C₈	1	1	yes	characteristic
165	C₃×C₈	D₃×D₄	1	1	yes	characteristic
166	C₃×C₈	#237	1	1	yes	characteristic
167	C₃×C₈	C₂×C₃×D₄	1	1	yes	characteristic
168	C₃×C₈	C₃×#41	1	1	yes	characteristic
169	C₂×C₃²	—	2	1	no	—
170	C₂×C₃²	—	4	2	no	—
171	C₂×C₃²	C₈×D₄	1	1	yes	characteristic
172	#41	C₂×C₃×Dic₃	1	2	yes	—
173	Q₁₆	C₃×C₃⋊C₈	1	1	yes	characteristic
174	SD₁₆	C₃×C₃⋊C₈	1	2	yes	—
175	D₈	C₃×C₃⋊C₈	1	1	yes	characteristic
176	M₁₆	—	2	2	no	—
177	M₁₆	C₂×C₃×Dic₃	1	2	yes	—
178	C₂×C₈	—	2	2	no	—
179	C₂×C₈	C₂×C₃×Dic₃	1	1	yes	characteristic
180	C₂×C₈	C₂²×C₃×D₃	1	1	yes	characteristic
181	C₄²	C₂×C₃×Dic₃	1	1	yes	characteristic
182	C₁₆	—	3	2	no	—
183	C₁₆	—	6	2	no	—
184	C₁₆	—	6	1	no	—
185	C₂²×C₃	—	2	2	no	—
186	C₂²×C₃	—	2	2	no	—
187	C₂²×C₃	—	2	1	no	—
188	C₂²×C₃	—	4	2	no	—
189	C₂²×C₃	#948	1	1	yes	characteristic
190	C₂²×C₃	C₃×#131	1	1	yes	characteristic
191	C₃×C₄	—	2	2	no	—
192	C₃×C₄	—	2	2	no	—
193	C₃×C₄	—	2	1	no	—
194	C₃×C₄	—	2	1	no	—
195	C₃×C₄	—	4	2	no	—
196	C₃×C₄	—	4	1	no	—
197	C₃×C₄	—	4	1	no	—
198	C₃×C₄	—	4	2	no	—
199	C₃×C₄	C₂²×C₃⋊C₈	1	1	yes	characteristic
200	C₃×C₄	D₄×Dic₃	1	1	yes	characteristic
201	C₃×C₄	C₃×C₄×D₄	1	1	yes	characteristic
202	C₃×C₄	C₂²×C₃×C₈	1	1	yes	characteristic
203	C₃²	#2171	1	1	yes	characteristic, complemented, Sylow
204	Q₈	C₂×C₃×C₃⋊C₈	1	2	yes	—
205	D₄	—	2	2	no	—
206	D₄	C₂×C₃×C₃⋊C₈	1	2	yes	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	1	no	—
209	C₂×C₄	C₂²×C₃×Dic₃	1	1	yes	characteristic
210	C₈	—	2	2	no	—
211	C₈	C₂×C₃×C₃⋊C₈	1	1	yes	characteristic
212	C₈	C₂×C₃×C₃⋊C₈	1	1	yes	characteristic
213	C₈	C₃×D₃×D₄	1	1	yes	characteristic
214	C₈	C₃×#237	1	1	yes	characteristic
215	C₂×C₃	—	2	1	no	—
216	C₂×C₃	—	2	1	no	—
217	C₂×C₃	—	2	1	no	—
218	C₂×C₃	—	4	1	no	—
219	C₂×C₃	—	4	2	no	—
220	C₂×C₃	—	4	2	no	—
221	C₂×C₃	—	8	2	no	—
222	C₂×C₃	D₄×C₃⋊C₈	1	1	yes	characteristic
223	C₂×C₃	C₃×C₈×D₄	1	1	yes	characteristic
224	C₂²	—	2	2	no	—
225	C₂²	C₃×#948	1	1	yes	characteristic
226	C₄	—	2	2	no	—
227	C₄	—	4	1	no	—
228	C₄	C₂²×C₃×C₃⋊C₈	1	1	yes	characteristic
229	C₄	C₃×D₄×Dic₃	1	1	yes	characteristic
230	C₃	—	2	1	no	—
231	C₃	#70924	1	1	yes	characteristic, direct product factor
232	C₃	C₃×#2171	1	1	yes	characteristic, complemented
233	C₂	—	2	1	no	—
234	C₂	—	4	2	no	—
235	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×#2171
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×#4461	C₂	semidirect
7. #70924	C₃	direct (N × H)
22. #5140	C₂×C₃	semidirect
26. #4561	C₂×C₃	semidirect
27. #4463	C₂×C₃	semidirect
28. #4461	C₂×C₃	semidirect
203. C₃²	#2171	semidirect
231. C₃	#70924	direct (N × H)
232. C₃	C₃×#2171	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×#4461	2
C₃×#2171	3
#70924	3

Subgroups — C3×#70924 #11797593

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#70924 #11797593