Subgroups — C₃×#70906 #11777923

Order 1152.

351 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₄×Q₁₆	C₂	1	1	yes	maximal, characteristic
2	C₃×#5024	C₂	1	2	yes	maximal
3	C₃×#4535	C₂	1	2	yes	maximal
4	C₃×#4457	C₂	1	1	yes	maximal, characteristic
5	C₃×#4455	C₂	1	1	yes	maximal, characteristic
6	C₃×#1593	—	3	1	no	maximal
7	#70906	C₃	1	1	yes	maximal, characteristic, direct product factor
8	C₂×C₃²×Q₁₆	C₄	1	1	yes	characteristic
9	C₃²×C₄×Q₈	C₂²	1	2	yes	—
10	C₃²×#107	C₄	1	1	yes	characteristic
11	C₃²×#103	C₄	1	2	yes	—
12	C₃²×C₄×C₈	C₂²	1	1	yes	characteristic
13	C₃×C₄⋊(C₃⋊C₈)	—	2	2	no	—
14	C₃×C₄⋊(C₃⋊C₈)	C₂²	1	2	yes	—
15	C₃×#803	—	2	2	no	—
16	C₃×C₄×C₃⋊C₈	C₂²	1	1	yes	characteristic
17	C₃×C₄×C₃⋊C₈	C₂²	1	1	yes	characteristic
18	C₃×#446	—	3	2	no	—
19	C₃×C₄×Q₁₆	—	2	1	no	—
20	C₃×C₄×Q₁₆	D₃	1	1	yes	characteristic
21	C₃×C₄×Q₁₆	C₂×C₃	1	1	yes	characteristic
22	#5024	C₂×C₃	1	2	yes	—
23	C₃×#334	—	3	1	no	—
24	C₃×Q₈⋊C₈	—	3	2	no	—
25	C₃×#322	—	3	1	no	—
26	#4535	C₂×C₃	1	2	yes	—
27	#4457	C₂×C₃	1	1	yes	characteristic
28	#4455	C₂×C₃	1	1	yes	characteristic
29	C₂×C₃²×Q₈	C₂×C₄	1	2	yes	—
30	C₃²×Q₁₆	—	2	2	no	—
31	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
32	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
33	C₃²×C₄⋊C₄	—	2	2	no	—
34	C₃²×C₄⋊C₄	C₂×C₄	1	2	yes	—
35	C₃²×C₄²	—	2	2	no	—
36	C₃²×C₄²	C₂³	1	1	yes	characteristic
37	C₂×C₃×C₃⋊C₈	—	2	2	no	—
38	C₂×C₃×C₃⋊C₈	—	2	1	no	—
39	C₂×C₃×C₃⋊C₈	D₄	1	2	yes	—
40	#1593	—	3	1	no	Sylow
41	C₂×C₃×Q₁₆	—	2	1	no	—
42	C₂×C₃×Q₁₆	Dic₃	1	1	yes	characteristic
43	C₂×C₃×Q₁₆	C₃×C₄	1	1	yes	characteristic
44	C₃×C₄×Q₈	—	2	2	no	—
45	C₃×C₄×Q₈	C₂×D₃	1	2	yes	—
46	C₃×C₄×Q₈	C₂²×C₃	1	2	yes	—
47	C₃×#107	—	2	1	no	—
48	C₃×#107	Dic₃	1	1	yes	characteristic
49	C₃×#107	C₃×C₄	1	1	yes	characteristic
50	C₃×C₄⋊C₈	—	3	2	no	—
51	C₃×C₄⋊C₈	—	6	2	no	—
52	C₃×#103	—	2	2	no	—
53	C₃×#103	Dic₃	1	2	yes	—
54	C₃×#103	C₃×C₄	1	2	yes	—
55	C₃×#97	—	6	2	no	—
56	C₃×C₄×C₈	—	2	1	no	—
57	C₃×C₄×C₈	—	3	1	no	—
58	C₃×C₄×C₈	—	3	1	no	—
59	C₃×C₄×C₈	C₂×D₃	1	1	yes	characteristic
60	C₃×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
61	C₄⋊(C₃⋊C₈)	—	2	2	no	—
62	C₄⋊(C₃⋊C₈)	C₂²×C₃	1	2	yes	—
63	#803	—	2	2	no	—
64	C₄×C₃⋊C₈	C₂²×C₃	1	1	yes	characteristic
65	C₄×C₃⋊C₈	C₂²×C₃	1	1	yes	characteristic
66	C₃²×Q₈	—	2	2	no	—
67	C₃²×Q₈	—	2	2	no	—
68	C₂×C₃²×C₄	—	2	2	no	—
69	C₂×C₃²×C₄	—	2	2	no	—
70	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
71	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
72	C₂×C₃²×C₄	#41	1	1	yes	characteristic
73	C₃²×C₈	—	2	1	no	—
74	C₃²×C₈	M₁₆	1	2	yes	—
75	C₃×C₃⋊C₈	—	2	2	no	—
76	C₃×C₃⋊C₈	—	4	1	no	—
77	C₃×C₃⋊C₈	—	4	2	no	—
78	#446	—	3	2	no	—
79	C₄×Q₁₆	C₃×D₃	1	1	yes	characteristic
80	#334	—	3	1	no	—
81	Q₈⋊C₈	—	3	2	no	—
82	#322	—	3	1	no	—
83	C₂×C₃×Q₈	—	2	2	no	—
84	C₂×C₃×Q₈	C₂×Dic₃	1	2	yes	—
85	C₂×C₃×Q₈	C₂×C₃×C₄	1	2	yes	—
86	C₃×Q₁₆	—	2	4	no	—
87	C₃×Q₁₆	—	2	2	no	—
88	C₃×Q₁₆	—	2	2	no	—
89	C₂×C₃×C₈	—	2	1	no	—
90	C₂×C₃×C₈	—	2	1	no	—
91	C₂×C₃×C₈	—	3	2	no	—
92	C₂×C₃×C₈	—	6	2	no	—
93	C₂×C₃×C₈	—	6	1	no	—
94	C₂×C₃×C₈	C₂×Dic₃	1	1	yes	characteristic
95	C₂×C₃×C₈	C₂×Dic₃	1	1	yes	characteristic
96	C₂×C₃×C₈	C₂×C₃×C₄	1	1	yes	characteristic
97	C₂×C₃×C₈	C₂×C₃×C₄	1	1	yes	characteristic
98	C₃×C₄⋊C₄	—	2	2	no	—
99	C₃×C₄⋊C₄	—	2	2	no	—
100	C₃×C₄⋊C₄	—	2	2	no	—
101	C₃×C₄⋊C₄	—	4	2	no	—
102	C₃×C₄⋊C₄	C₂×Dic₃	1	2	yes	—
103	C₃×C₄⋊C₄	C₂×C₃×C₄	1	2	yes	—
104	C₃×C₄²	—	2	2	no	—
105	C₃×C₄²	—	2	2	no	—
106	C₃×C₄²	—	2	1	no	—
107	C₃×C₄²	—	4	2	no	—
108	C₃×C₄²	C₂²×D₃	1	1	yes	characteristic
109	C₃×C₄²	C₂³×C₃	1	1	yes	characteristic
110	C₂×C₃⋊C₈	—	2	2	no	—
111	C₂×C₃⋊C₈	—	2	1	no	—
112	C₂×C₃⋊C₈	C₃×D₄	1	2	yes	—
113	C₂²×C₃²	C₄×D₄	1	1	yes	characteristic
114	C₃²×C₄	—	2	1	no	—
115	C₃²×C₄	—	4	2	no	—
116	C₃²×C₄	—	4	2	no	—
117	C₃²×C₄	C₂×M₁₆	1	1	yes	characteristic
118	C₃²×C₄	#131	1	1	yes	characteristic
119	C₃²×C₄	#137	1	2	yes	—
120	C₂×Q₁₆	C₃×Dic₃	1	1	yes	characteristic
121	C₄×Q₈	C₂×C₃×D₃	1	2	yes	—
122	#107	C₃×Dic₃	1	1	yes	characteristic
123	C₄⋊C₈	—	3	2	no	—
124	C₄⋊C₈	—	6	2	no	—
125	#103	C₃×Dic₃	1	2	yes	—
126	#97	—	6	2	no	—
127	C₄×C₈	—	3	1	no	—
128	C₄×C₈	—	3	1	no	—
129	C₄×C₈	C₂×C₃×D₃	1	1	yes	characteristic
130	C₃×Q₈	—	2	4	no	—
131	C₃×Q₈	—	2	2	no	—
132	C₃×Q₈	—	2	2	no	—
133	C₃×Q₈	—	2	2	no	—
134	C₃×Q₈	—	2	2	no	—
135	C₃×Q₈	—	4	2	no	—
136	C₂×C₃×C₄	—	2	2	no	—
137	C₂×C₃×C₄	—	2	2	no	—
138	C₂×C₃×C₄	—	2	2	no	—
139	C₂×C₃×C₄	—	2	2	no	—
140	C₂×C₃×C₄	—	2	1	no	—
141	C₂×C₃×C₄	—	2	1	no	—
142	C₂×C₃×C₄	—	2	1	no	—
143	C₂×C₃×C₄	—	4	2	no	—
144	C₂×C₃×C₄	—	4	2	no	—
145	C₂×C₃×C₄	D₃×D₄	1	1	yes	characteristic
146	C₂×C₃×C₄	#237	1	1	yes	characteristic
147	C₂×C₃×C₄	C₂²×Dic₃	1	1	yes	characteristic
148	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
149	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
150	C₂×C₃×C₄	C₃×#41	1	1	yes	characteristic
151	C₃×C₈	—	2	1	no	—
152	C₃×C₈	—	2	1	no	—
153	C₃×C₈	—	2	2	no	—
154	C₃×C₈	—	4	1	no	—
155	C₃×C₈	—	6	2	no	—
156	C₃×C₈	—	12	1	no	—
157	C₃×C₈	—	12	2	no	—
158	C₃×C₈	#208	1	2	yes	—
159	C₃×C₈	C₃×M₁₆	1	2	yes	—
160	C₃⋊C₈	—	2	2	no	—
161	C₃⋊C₈	—	4	1	no	—
162	C₃⋊C₈	—	4	2	no	—
163	C₂×C₃²	#436	1	1	yes	characteristic
164	C₂×C₃²	#441	1	1	yes	characteristic
165	C₂×C₃²	#444	1	1	yes	characteristic
166	C₂×Q₈	C₂×C₃×Dic₃	1	2	yes	—
167	Q₁₆	—	2	2	no	—
168	C₂×C₈	—	3	2	no	—
169	C₂×C₈	—	6	2	no	—
170	C₂×C₈	—	6	1	no	—
171	C₂×C₈	C₂×C₃×Dic₃	1	1	yes	characteristic
172	C₂×C₈	C₂×C₃×Dic₃	1	1	yes	characteristic
173	C₄⋊C₄	—	2	2	no	—
174	C₄⋊C₄	C₂×C₃×Dic₃	1	2	yes	—
175	C₄²	—	2	2	no	—
176	C₄²	C₂²×C₃×D₃	1	1	yes	characteristic
177	C₂²×C₃	—	2	1	no	—
178	C₂²×C₃	D₄×Dic₃	1	1	yes	characteristic
179	C₂²×C₃	C₃×C₄×D₄	1	1	yes	characteristic
180	C₃×C₄	—	2	1	no	—
181	C₃×C₄	—	2	1	no	—
182	C₃×C₄	—	2	2	no	—
183	C₃×C₄	—	2	1	no	—
184	C₃×C₄	—	2	1	no	—
185	C₃×C₄	—	4	1	no	—
186	C₃×C₄	—	4	4	no	—
187	C₃×C₄	—	4	2	no	—
188	C₃×C₄	—	4	2	no	—
189	C₃×C₄	—	4	2	no	—
190	C₃×C₄	—	4	2	no	—
191	C₃×C₄	—	8	2	no	—
192	C₃×C₄	#918	1	2	yes	—
193	C₃×C₄	C₂×#208	1	1	yes	characteristic
194	C₃×C₄	#948	1	1	yes	characteristic
195	C₃×C₄	C₂×C₃×M₁₆	1	1	yes	characteristic
196	C₃×C₄	C₃×#131	1	1	yes	characteristic
197	C₃×C₄	C₃×#137	1	2	yes	—
198	C₃²	#1593	1	1	yes	characteristic, complemented, Sylow
199	Q₈	—	2	2	no	—
200	Q₈	—	2	2	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	C₃×D₃×D₄	1	1	yes	characteristic
204	C₂×C₄	C₃×#237	1	1	yes	characteristic
205	C₂×C₄	C₂²×C₃×Dic₃	1	1	yes	characteristic
206	C₈	—	2	1	no	—
207	C₈	—	6	2	no	—
208	C₈	—	12	1	no	—
209	C₈	—	12	2	no	—
210	C₈	C₃×#208	1	2	yes	—
211	C₂×C₃	—	2	1	no	—
212	C₂×C₃	—	2	1	no	—
213	C₂×C₃	—	2	1	no	—
214	C₂×C₃	#5012	1	1	yes	characteristic
215	C₂×C₃	#5184	1	1	yes	characteristic
216	C₂×C₃	#5197	1	1	yes	characteristic
217	C₂×C₃	C₃×#436	1	1	yes	characteristic
218	C₂×C₃	C₃×#441	1	1	yes	characteristic
219	C₂×C₃	C₃×#444	1	1	yes	characteristic
220	C₂²	C₃×D₄×Dic₃	1	1	yes	characteristic
221	C₄	—	2	1	no	—
222	C₄	—	4	2	no	—
223	C₄	—	4	2	no	—
224	C₄	C₃×#918	1	2	yes	—
225	C₄	C₂×C₃×#208	1	1	yes	characteristic
226	C₄	C₃×#948	1	1	yes	characteristic
227	C₃	—	2	1	no	—
228	C₃	#70906	1	1	yes	characteristic, direct product factor
229	C₃	C₃×#1593	1	1	yes	characteristic, complemented
230	C₂	C₃×#5012	1	1	yes	characteristic
231	C₂	C₃×#5184	1	1	yes	characteristic
232	C₂	C₃×#5197	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₄	D₃×D₄
commutator	C₂×C₃×C₄	C₂²×C₃×C₄
Frattini	C₄²	C₂²×C₃×D₃
Fitting	C₃²×C₄×Q₁₆	C₂
socle, prime radical	C₂²×C₃²	C₄×D₄
cosocle	C₃×C₄²	C₂³×C₃
hypercenter	C₃×C₄×Q₁₆	D₃
hypocenter	C₃	C₃×#1593
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₃×C₄² ◁ C₃×C₄×Q₁₆

Complements

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

N	H ≅ G/N	type
7. #70906	C₃	direct (N × H)
198. C₃²	#1593	semidirect
228. C₃	#70906	direct (N × H)
229. C₃	C₃×#1593	semidirect

Maximal subgroups

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

subgroup	index
C₃²×C₄×Q₁₆	2
C₃×#5024	2
C₃×#4535	2
C₃×#4457	2
C₃×#4455	2
C₃×#1593	3
#70906	3

Subgroups — C3×#70906 #11777923

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#70906 #11777923