Subgroups — C₃×#68929 #11786106

Order 1152.

405 conjugacy classes of subgroups.

The subgroup lattice has 222 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#5118	C₂	1	2	yes	maximal
2	C₂×C₃×#815	C₂	1	1	yes	maximal, characteristic
3	C₃²×#341	C₂	1	1	yes	maximal, characteristic
4	C₃×#4558	C₂	1	2	yes	maximal
5	C₃×#4552	C₂	1	1	yes	maximal, characteristic
6	C₃×#1838	—	3	1	no	maximal
7	#68929	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	C₃×#815	C₄	1	4	yes	—
14	C₃×Dic₃⋊C₈	—	2	4	no	—
15	C₃×#423	—	3	2	no	—
16	C₂×C₃×#97	—	3	1	no	—
17	#5118	C₂×C₃	1	2	yes	—
18	C₂×#815	C₂×C₃	1	1	yes	characteristic
19	C₃×#341	—	2	1	no	—
20	C₃×#341	—	3	2	no	—
21	C₃×#341	—	3	1	no	—
22	C₃×#341	D₃	1	1	yes	characteristic
23	C₃×#341	C₂×C₃	1	1	yes	characteristic
24	#4558	C₂×C₃	1	2	yes	—
25	#4552	C₂×C₃	1	1	yes	characteristic
26	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
27	C₂²×C₃×Dic₃	Q₈	1	2	yes	—
28	C₃²×M₁₆	—	2	2	no	—
29	C₃²×M₁₆	—	2	2	no	—
30	C₂×C₃²×C₈	—	2	2	no	—
31	C₂×C₃²×C₈	—	2	2	no	—
32	C₂×C₃²×C₈	C₂×C₄	1	2	yes	—
33	C₃×C₄×Dic₃	C₂×C₄	1	2	yes	—
34	C₃×C₄×Dic₃	D₄	1	2	yes	—
35	C₃×#208	—	2	2	no	—
36	C₃×#208	—	2	2	no	—
37	C₂×C₃×C₃⋊C₈	—	2	2	no	—
38	C₂×C₃×C₃⋊C₈	—	2	2	no	—
39	C₂×C₃×C₃⋊C₈	C₂×C₄	1	2	yes	—
40	#1838	—	3	1	no	Sylow
41	C₂×C₃×M₁₆	—	2	2	no	—
42	C₂×C₃×M₁₆	—	3	2	no	—
43	C₂×C₃×M₁₆	C₂×D₃	1	2	yes	—
44	C₂×C₃×M₁₆	C₂²×C₃	1	2	yes	—
45	C₂²×C₃×C₈	—	2	1	no	—
46	C₂²×C₃×C₈	—	3	1	no	—
47	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
48	C₂²×C₃×C₈	C₂²×C₃	1	1	yes	characteristic
49	C₂×C₃×C₄²	—	3	1	no	—
50	C₂×C₄×Dic₃	C₂²×C₃	1	1	yes	characteristic
51	C₂×#208	C₂²×C₃	1	2	yes	—
52	C₂²×C₃⋊C₈	C₂²×C₃	1	1	yes	characteristic
53	C₃×C₄⋊C₈	—	6	4	no	—
54	C₃×#97	—	3	4	no	—
55	#815	C₃×C₄	1	4	yes	—
56	Dic₃⋊C₈	—	2	4	no	—
57	C₂³×C₃²	C₂×Q₈	1	1	yes	characteristic
58	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
60	C₂×C₃²×C₄	#41	1	2	yes	—
61	C₂×C₃²×C₄	#41	1	2	yes	—
62	C₂×C₃×Dic₃	—	4	2	no	—
63	C₂×C₃×Dic₃	C₄⋊C₄	1	4	yes	—
64	C₃²×C₈	—	2	2	no	—
65	C₃²×C₈	—	4	2	no	—
66	C₃×C₃⋊C₈	—	2	2	no	—
67	C₃×C₃⋊C₈	—	4	2	no	—
68	#423	—	3	2	no	—
69	C₂×#97	—	3	1	no	—
70	#341	—	3	2	no	—
71	#341	—	3	1	no	—
72	#341	C₃×D₃	1	1	yes	characteristic
73	C₂²×C₃×C₄	—	2	1	no	—
74	C₂²×C₃×C₄	—	3	2	no	—
75	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
76	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
77	C₂²×Dic₃	C₃×Q₈	1	2	yes	—
78	C₃×M₁₆	—	2	4	no	—
79	C₃×M₁₆	—	2	2	no	—
80	C₃×M₁₆	—	2	2	no	—
81	C₃×M₁₆	—	2	2	no	—
82	C₃×M₁₆	—	2	2	no	—
83	C₃×M₁₆	—	4	2	no	—
84	C₃×M₁₆	—	6	2	no	—
85	C₃×M₁₆	—	6	2	no	—
86	C₂×C₃×C₈	—	2	2	no	—
87	C₂×C₃×C₈	—	2	2	no	—
88	C₂×C₃×C₈	—	2	2	no	—
89	C₂×C₃×C₈	—	2	2	no	—
90	C₂×C₃×C₈	—	2	2	no	—
91	C₂×C₃×C₈	—	3	2	no	—
92	C₂×C₃×C₈	—	4	2	no	—
93	C₂×C₃×C₈	—	4	2	no	—
94	C₂×C₃×C₈	—	6	2	no	—
95	C₂×C₃×C₈	—	6	2	no	—
96	C₂×C₃×C₈	C₄×D₃	1	2	yes	—
97	C₂×C₃×C₈	C₂×C₃×C₄	1	2	yes	—
98	C₃×C₄²	—	3	2	no	—
99	C₃×C₄²	—	3	2	no	—
100	C₄×Dic₃	C₂×C₃×C₄	1	2	yes	—
101	C₄×Dic₃	C₃×D₄	1	2	yes	—
102	#208	—	2	2	no	—
103	#208	—	2	2	no	—
104	C₂×C₃⋊C₈	—	2	2	no	—
105	C₂×C₃⋊C₈	—	2	2	no	—
106	C₂×C₃⋊C₈	C₂×C₃×C₄	1	2	yes	—
107	C₂²×C₃²	—	2	2	no	—
108	C₂²×C₃²	C₂×C₄⋊C₄	1	1	yes	characteristic
109	C₂²×C₃²	#125	1	2	yes	—
110	C₃²×C₄	#117	1	1	yes	characteristic
111	C₃²×C₄	#117	1	1	yes	characteristic
112	C₃²×C₄	#124	1	2	yes	—
113	C₃×Dic₃	—	2	4	no	—
114	C₂×M₁₆	—	3	2	no	—
115	C₂×M₁₆	C₂×C₃×D₃	1	2	yes	—
116	C₂²×C₈	—	3	1	no	—
117	C₂²×C₈	C₂×C₃×D₃	1	1	yes	characteristic
118	C₂×C₄²	—	3	1	no	—
119	C₄⋊C₈	—	6	4	no	—
120	#97	—	3	4	no	—
121	C₂³×C₃	—	2	1	no	—
122	C₂³×C₃	D₃×Q₈	1	1	yes	characteristic
123	C₂³×C₃	C₂×C₃×Q₈	1	1	yes	characteristic
124	C₂×C₃×C₄	—	2	1	no	—
125	C₂×C₃×C₄	—	2	2	no	—
126	C₂×C₃×C₄	—	2	1	no	—
127	C₂×C₃×C₄	—	2	2	no	—
128	C₂×C₃×C₄	—	3	4	no	—
129	C₂×C₃×C₄	—	12	2	no	—
130	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
131	C₂×C₃×C₄	#235	1	2	yes	—
132	C₂×C₃×C₄	D₃×D₄	1	1	yes	characteristic
133	C₂×C₃×C₄	#237	1	2	yes	—
134	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
135	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
136	C₂×C₃×C₄	C₃×#41	1	2	yes	—
137	C₂×C₃×C₄	C₃×#41	1	2	yes	—
138	C₂×Dic₃	—	4	2	no	—
139	C₂×Dic₃	C₃×C₄⋊C₄	1	4	yes	—
140	C₃×C₈	—	2	2	no	—
141	C₃×C₈	—	2	2	no	—
142	C₃×C₈	—	4	2	no	—
143	C₃×C₈	—	4	4	no	—
144	C₃×C₈	—	4	2	no	—
145	C₃×C₈	—	4	2	no	—
146	C₃×C₈	—	6	2	no	—
147	C₃×C₈	—	12	2	no	—
148	C₃⋊C₈	—	2	2	no	—
149	C₃⋊C₈	—	4	2	no	—
150	C₂×C₃²	—	2	2	no	—
151	C₂×C₃²	#382	1	1	yes	characteristic
152	C₂×C₃²	#430	1	2	yes	—
153	C₂²×C₄	—	3	2	no	—
154	C₂²×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
155	M₁₆	—	2	2	no	—
156	M₁₆	—	2	2	no	—
157	M₁₆	—	6	2	no	—
158	M₁₆	—	6	2	no	—
159	C₂×C₈	—	2	2	no	—
160	C₂×C₈	—	2	2	no	—
161	C₂×C₈	—	3	2	no	—
162	C₂×C₈	—	6	2	no	—
163	C₂×C₈	—	6	2	no	—
164	C₂×C₈	C₃×C₄×D₃	1	2	yes	—
165	C₄²	—	3	2	no	—
166	C₄²	—	3	2	no	—
167	C₂²×C₃	—	2	2	no	—
168	C₂²×C₃	—	2	2	no	—
169	C₂²×C₃	—	2	2	no	—
170	C₂²×C₃	—	2	1	no	—
171	C₂²×C₃	—	4	2	no	—
172	C₂²×C₃	#889	1	2	yes	—
173	C₂²×C₃	D₃×C₄⋊C₄	1	1	yes	characteristic
174	C₂²×C₃	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
175	C₂²×C₃	C₃×#125	1	2	yes	—
176	C₃×C₄	—	2	1	no	—
177	C₃×C₄	—	2	2	no	—
178	C₃×C₄	—	2	1	no	—
179	C₃×C₄	—	6	4	no	—
180	C₃×C₄	#872	1	1	yes	characteristic
181	C₃×C₄	#877	1	1	yes	characteristic
182	C₃×C₄	#885	1	2	yes	—
183	C₃×C₄	C₃×#117	1	1	yes	characteristic
184	C₃×C₄	C₃×#117	1	1	yes	characteristic
185	C₃×C₄	C₃×#124	1	2	yes	—
186	Dic₃	—	2	4	no	—
187	C₃²	#1838	1	1	yes	characteristic, complemented, Sylow
188	C₂³	C₃×D₃×Q₈	1	1	yes	characteristic
189	C₂×C₄	—	3	4	no	—
190	C₂×C₄	—	12	2	no	—
191	C₂×C₄	C₂×C₃×C₄×D₃	1	1	yes	characteristic
192	C₂×C₄	C₃×#235	1	2	yes	—
193	C₂×C₄	C₃×D₃×D₄	1	1	yes	characteristic
194	C₂×C₄	C₃×#237	1	2	yes	—
195	C₈	—	2	2	no	—
196	C₈	—	4	2	no	—
197	C₈	—	6	2	no	—
198	C₈	—	12	2	no	—
199	C₂×C₃	—	2	2	no	—
200	C₂×C₃	—	2	2	no	—
201	C₂×C₃	—	2	1	no	—
202	C₂×C₃	—	2	2	no	—
203	C₂×C₃	—	4	2	no	—
204	C₂×C₃	#4650	1	1	yes	characteristic
205	C₂×C₃	#4893	1	2	yes	—
206	C₂×C₃	C₃×#382	1	1	yes	characteristic
207	C₂×C₃	C₃×#430	1	2	yes	—
208	C₂²	—	2	2	no	—
209	C₂²	C₃×#889	1	2	yes	—
210	C₂²	C₃×D₃×C₄⋊C₄	1	1	yes	characteristic
211	C₄	—	6	4	no	—
212	C₄	C₃×#872	1	1	yes	characteristic
213	C₄	C₃×#877	1	1	yes	characteristic
214	C₄	C₃×#885	1	2	yes	—
215	C₃	—	2	1	no	—
216	C₃	#68929	1	1	yes	characteristic, direct product factor
217	C₃	C₃×#1838	1	1	yes	characteristic, complemented
218	C₂	—	2	2	no	—
219	C₂	C₃×#4650	1	1	yes	characteristic
220	C₂	C₃×#4893	1	2	yes	—

	subgroup	G/H
center	C₂×C₃×C₄	D₃×D₄
commutator	C₂×C₃×C₄	C₂²×C₃×C₄
Frattini	C₂²×C₄	C₂²×C₃×D₃
Fitting	C₃²×#341	C₂
socle	C₂²×C₃²	C₂×C₄⋊C₄
prime radical	C₂³×C₃²	C₂×Q₈
cosocle	C₂²×C₃×C₄	C₂³×C₃
hypercenter	C₃×#341	D₃
hypocenter	C₃	C₃×#1838
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₂²×C₃×C₄ ◁ C₃×#341

Complements

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

N	H ≅ G/N	type
7. #68929	C₃	direct (N × H)
187. C₃²	#1838	semidirect
216. C₃	#68929	direct (N × H)
217. C₃	C₃×#1838	semidirect

Maximal subgroups

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

subgroup	index
C₃×#5118	2
C₂×C₃×#815	2
C₃²×#341	2
C₃×#4558	2
C₃×#4552	2
C₃×#1838	3
#68929	3

Subgroups — C3×#68929 #11786106

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#68929 #11786106