Subgroups — C₃×#68923 #11795299

Order 1152.

397 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#5121	C₂	1	2	yes	maximal
2	C₂×C₃×#818	C₂	1	1	yes	maximal, characteristic
3	C₃×#5102	C₂	1	1	yes	maximal, characteristic
4	C₃²×#341	C₂	1	1	yes	maximal, characteristic
5	C₃×#4554	C₂	1	2	yes	maximal
6	C₃×#2089	—	3	1	no	maximal
7	#68923	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	C₃×#926	C₂²	1	1	yes	characteristic
11	C₂×C₃×C₄⋊Dic₃	C₂²	1	2	yes	—
12	C₂×C₃×#208	C₂²	1	1	yes	characteristic
13	C₃×#818	—	2	2	no	—
14	C₃×#818	—	2	2	no	—
15	C₃×#817	—	2	2	no	—
16	C₃×Dic₃⋊C₈	—	2	2	no	—
17	C₃×#428	—	3	2	no	—
18	C₂×C₃×#107	—	3	1	no	—
19	C₃×#424	—	3	1	no	—
20	#5121	C₂×C₃	1	2	yes	—
21	C₂×#818	C₂×C₃	1	1	yes	characteristic
22	#5102	C₂×C₃	1	1	yes	characteristic
23	C₃×#343	—	3	2	no	—
24	C₃×#341	—	2	1	no	—
25	C₃×#341	D₃	1	1	yes	characteristic
26	C₃×#341	C₂×C₃	1	1	yes	characteristic
27	#4554	C₂×C₃	1	2	yes	—
28	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
29	C₂²×C₃×Dic₃	—	2	2	no	—
30	C₃²×M₁₆	—	2	2	no	—
31	C₃²×M₁₆	Q₈	1	4	yes	—
32	C₂×C₃²×C₈	—	2	2	no	—
33	C₂×C₃²×C₈	—	2	2	no	—
34	C₂×C₃²×C₈	D₄	1	2	yes	—
35	C₃×#217	—	2	1	no	—
36	C₃×C₄⋊Dic₃	—	2	2	no	—
37	C₃×C₄⋊Dic₃	—	2	2	no	—
38	C₃×C₄⋊Dic₃	D₄	1	1	yes	characteristic
39	C₃×C₄⋊Dic₃	D₄	1	1	yes	characteristic
40	C₃×C₄×Dic₃	—	2	1	no	—
41	C₃×#208	—	2	2	no	—
42	C₂×C₃×C₃⋊C₈	—	2	1	no	—
43	#2089	—	3	1	no	Sylow
44	C₂×C₃×M₁₆	—	2	2	no	—
45	C₂×C₃×M₁₆	—	3	1	no	—
46	C₂×C₃×M₁₆	C₂×D₃	1	2	yes	—
47	C₂×C₃×M₁₆	C₂²×C₃	1	2	yes	—
48	C₂²×C₃×C₈	—	2	1	no	—
49	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
50	C₂²×C₃×C₈	C₂²×C₃	1	1	yes	characteristic
51	C₃×#117	—	3	1	no	—
52	C₂×C₃×C₄⋊C₄	—	3	2	no	—
53	#926	C₂²×C₃	1	1	yes	characteristic
54	C₂×C₄⋊Dic₃	C₂²×C₃	1	2	yes	—
55	C₂×#208	C₂²×C₃	1	1	yes	characteristic
56	C₃×#107	—	6	2	no	—
57	C₃×#107	—	6	2	no	—
58	C₃×#106	—	6	2	no	—
59	C₃×C₄⋊C₈	—	6	2	no	—
60	#818	—	2	2	no	—
61	#818	—	2	2	no	—
62	#817	—	2	2	no	—
63	Dic₃⋊C₈	—	2	2	no	—
64	C₂³×C₃²	#41	1	1	yes	characteristic
65	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
66	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
67	C₂×C₃²×C₄	C₂×Q₈	1	2	yes	—
68	C₂×C₃²×C₄	#41	1	2	yes	—
69	C₂×C₃×Dic₃	—	2	1	no	—
70	C₂×C₃×Dic₃	—	2	1	no	—
71	C₂×C₃×Dic₃	—	4	2	no	—
72	C₂×C₃×Dic₃	—	8	2	no	—
73	C₃²×C₈	—	2	4	no	—
74	C₃²×C₈	—	2	2	no	—
75	C₃×C₃⋊C₈	—	4	1	no	—
76	#428	—	3	2	no	—
77	C₂×#107	—	3	1	no	—
78	#424	—	3	1	no	—
79	#343	—	3	2	no	—
80	#341	C₃×D₃	1	1	yes	characteristic
81	C₂²×C₃×C₄	—	2	1	no	—
82	C₂²×C₃×C₄	—	6	2	no	—
83	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
84	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
85	C₂²×Dic₃	—	2	2	no	—
86	C₃×M₁₆	—	2	2	no	—
87	C₃×M₁₆	—	2	2	no	—
88	C₃×M₁₆	—	2	4	no	—
89	C₃×M₁₆	—	4	2	no	—
90	C₃×M₁₆	—	6	2	no	—
91	C₃×M₁₆	Dic₆	1	4	yes	—
92	C₃×M₁₆	C₃×Q₈	1	4	yes	—
93	C₂×C₃×C₈	—	2	2	no	—
94	C₂×C₃×C₈	—	2	2	no	—
95	C₂×C₃×C₈	—	2	2	no	—
96	C₂×C₃×C₈	—	2	2	no	—
97	C₂×C₃×C₈	—	2	2	no	—
98	C₂×C₃×C₈	—	4	2	no	—
99	C₂×C₃×C₈	—	4	2	no	—
100	C₂×C₃×C₈	—	6	1	no	—
101	C₂×C₃×C₈	D₁₂	1	2	yes	—
102	C₂×C₃×C₈	C₃×D₄	1	2	yes	—
103	C₃×C₄⋊C₄	—	3	1	no	—
104	C₃×C₄⋊C₄	—	3	1	no	—
105	C₃×C₄⋊C₄	—	6	2	no	—
106	C₃×C₄⋊C₄	—	6	2	no	—
107	C₃×C₂²⋊C₄	—	6	1	no	—
108	C₃×C₄²	—	6	1	no	—
109	#217	—	2	1	no	—
110	C₄⋊Dic₃	—	2	2	no	—
111	C₄⋊Dic₃	—	2	2	no	—
112	C₄⋊Dic₃	C₃×D₄	1	1	yes	characteristic
113	C₄⋊Dic₃	C₃×D₄	1	1	yes	characteristic
114	C₄×Dic₃	—	2	1	no	—
115	#208	—	2	2	no	—
116	C₂×C₃⋊C₈	—	2	1	no	—
117	C₂²×C₃²	—	2	1	no	—
118	C₂²×C₃²	—	2	1	no	—
119	C₂²×C₃²	#121	1	1	yes	characteristic
120	C₂²×C₃²	#125	1	2	yes	—
121	C₃²×C₄	C₂²⋊Q₈	1	2	yes	—
122	C₃²×C₄	#123	1	1	yes	characteristic
123	C₃²×C₄	C₄⋊Q₈	1	1	yes	characteristic
124	C₃×Dic₃	—	4	1	no	—
125	C₃×Dic₃	—	4	1	no	—
126	C₃×Dic₃	—	8	2	no	—
127	C₂×M₁₆	—	3	1	no	—
128	C₂×M₁₆	C₂×C₃×D₃	1	2	yes	—
129	C₂²×C₈	C₂×C₃×D₃	1	1	yes	characteristic
130	#117	—	3	1	no	—
131	C₂×C₄⋊C₄	—	3	2	no	—
132	#107	—	6	2	no	—
133	#107	—	6	2	no	—
134	#106	—	6	2	no	—
135	C₄⋊C₈	—	6	2	no	—
136	C₂³×C₃	—	2	1	no	—
137	C₂³×C₃	#239	1	1	yes	characteristic
138	C₂³×C₃	C₃×#41	1	1	yes	characteristic
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	2	no	—
143	C₂×C₃×C₄	—	6	1	no	—
144	C₂×C₃×C₄	—	6	1	no	—
145	C₂×C₃×C₄	—	12	2	no	—
146	C₂×C₃×C₄	—	24	2	no	—
147	C₂×C₃×C₄	C₂×Dic₆	1	2	yes	—
148	C₂×C₃×C₄	C₂×D₁₂	1	1	yes	characteristic
149	C₂×C₃×C₄	D₃×D₄	1	1	yes	characteristic
150	C₂×C₃×C₄	#237	1	2	yes	—
151	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
152	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
153	C₂×C₃×C₄	C₂×C₃×Q₈	1	2	yes	—
154	C₂×C₃×C₄	C₃×#41	1	2	yes	—
155	C₂×Dic₃	—	2	1	no	—
156	C₂×Dic₃	—	2	1	no	—
157	C₂×Dic₃	—	4	2	no	—
158	C₂×Dic₃	—	8	2	no	—
159	C₃×C₈	—	2	4	no	—
160	C₃×C₈	—	2	2	no	—
161	C₃×C₈	—	2	4	no	—
162	C₃×C₈	—	2	2	no	—
163	C₃×C₈	—	4	2	no	—
164	C₃×C₈	—	4	4	no	—
165	C₃×C₈	—	12	1	no	—
166	C₃⋊C₈	—	4	1	no	—
167	C₂×C₃²	—	2	2	no	—
168	C₂×C₃²	#398	1	1	yes	characteristic
169	C₂×C₃²	#472	1	1	yes	characteristic
170	C₂×C₃²	#473	1	1	yes	characteristic
171	C₂²×C₄	—	6	2	no	—
172	C₂²×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
173	M₁₆	—	2	2	no	—
174	M₁₆	—	6	2	no	—
175	M₁₆	C₃×Dic₆	1	4	yes	—
176	C₂×C₈	—	2	2	no	—
177	C₂×C₈	—	2	2	no	—
178	C₂×C₈	—	6	1	no	—
179	C₂×C₈	C₃×D₁₂	1	2	yes	—
180	C₄⋊C₄	—	3	1	no	—
181	C₄⋊C₄	—	3	1	no	—
182	C₄⋊C₄	—	6	2	no	—
183	C₄⋊C₄	—	6	2	no	—
184	C₂²⋊C₄	—	6	1	no	—
185	C₄²	—	6	1	no	—
186	C₂²×C₃	—	2	1	no	—
187	C₂²×C₃	—	2	1	no	—
188	C₂²×C₃	—	2	1	no	—
189	C₂²×C₃	—	2	1	no	—
190	C₂²×C₃	—	2	2	no	—
191	C₂²×C₃	—	2	1	no	—
192	C₂²×C₃	—	4	1	no	—
193	C₂²×C₃	—	4	1	no	—
194	C₂²×C₃	#890	1	2	yes	—
195	C₂²×C₃	#895	1	1	yes	characteristic
196	C₂²×C₃	C₃×#121	1	1	yes	characteristic
197	C₂²×C₃	C₃×#125	1	2	yes	—
198	C₃×C₄	—	2	2	no	—
199	C₃×C₄	—	2	1	no	—
200	C₃×C₄	—	2	1	no	—
201	C₃×C₄	—	12	1	no	—
202	C₃×C₄	—	12	1	no	—
203	C₃×C₄	—	24	2	no	—
204	C₃×C₄	#869	1	1	yes	characteristic
205	C₃×C₄	#878	1	2	yes	—
206	C₃×C₄	#886	1	1	yes	characteristic
207	C₃×C₄	C₃×C₂²⋊Q₈	1	2	yes	—
208	C₃×C₄	C₃×#123	1	1	yes	characteristic
209	C₃×C₄	C₃×C₄⋊Q₈	1	1	yes	characteristic
210	Dic₃	—	4	1	no	—
211	Dic₃	—	4	1	no	—
212	Dic₃	—	8	2	no	—
213	C₃²	#2089	1	1	yes	characteristic, complemented, Sylow
214	C₂³	C₃×#239	1	1	yes	characteristic
215	C₂×C₄	—	6	1	no	—
216	C₂×C₄	—	6	1	no	—
217	C₂×C₄	—	12	2	no	—
218	C₂×C₄	—	24	2	no	—
219	C₂×C₄	C₂×C₃×Dic₆	1	2	yes	—
220	C₂×C₄	C₂×C₃×D₁₂	1	1	yes	characteristic
221	C₂×C₄	C₃×D₃×D₄	1	1	yes	characteristic
222	C₂×C₄	C₃×#237	1	2	yes	—
223	C₈	—	2	4	no	—
224	C₈	—	2	2	no	—
225	C₈	—	12	1	no	—
226	C₂×C₃	—	2	2	no	—
227	C₂×C₃	—	2	2	no	—
228	C₂×C₃	—	2	1	no	—
229	C₂×C₃	—	2	1	no	—
230	C₂×C₃	—	2	1	no	—
231	C₂×C₃	—	4	2	no	—
232	C₂×C₃	#4661	1	1	yes	characteristic
233	C₂×C₃	#4896	1	1	yes	characteristic
234	C₂×C₃	#4897	1	1	yes	characteristic
235	C₂×C₃	C₃×#398	1	1	yes	characteristic
236	C₂×C₃	C₃×#472	1	1	yes	characteristic
237	C₂×C₃	C₃×#473	1	1	yes	characteristic
238	C₂²	—	2	1	no	—
239	C₂²	—	2	1	no	—
240	C₂²	C₃×#890	1	2	yes	—
241	C₂²	C₃×#895	1	1	yes	characteristic
242	C₄	—	12	1	no	—
243	C₄	—	12	1	no	—
244	C₄	—	24	2	no	—
245	C₄	C₃×#869	1	1	yes	characteristic
246	C₄	C₃×#878	1	2	yes	—
247	C₄	C₃×#886	1	1	yes	characteristic
248	C₃	—	2	1	no	—
249	C₃	#68923	1	1	yes	characteristic, direct product factor
250	C₃	C₃×#2089	1	1	yes	characteristic, complemented
251	C₂	—	2	2	no	—
252	C₂	C₃×#4661	1	1	yes	characteristic
253	C₂	C₃×#4896	1	1	yes	characteristic
254	C₂	C₃×#4897	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃	#895
commutator, cosocle	C₂²×C₃×C₄	C₂³×C₃
Frattini	C₂²×C₄	C₂²×C₃×D₃
Fitting	C₃²×#341	C₂
socle	C₂²×C₃²	#121
prime radical	C₂³×C₃²	#41
hypercenter	C₃×#341	D₃
hypocenter	C₃	C₃×#2089
LCS	G ▷ C₂²×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₃ ◁ C₂²×C₃×C₄ ◁ C₃×#341

Complements

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

N	H ≅ G/N	type
7. #68923	C₃	direct (N × H)
213. C₃²	#2089	semidirect
249. C₃	#68923	direct (N × H)
250. C₃	C₃×#2089	semidirect

Maximal subgroups

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

subgroup	index
C₃×#5121	2
C₂×C₃×#818	2
C₃×#5102	2
C₃²×#341	2
C₃×#4554	2
C₃×#2089	3
#68923	3

Subgroups — C3×#68923 #11795299

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#68923 #11795299