Subgroups — C₃×#72004 #11796000

Order 1152.

589 conjugacy classes of subgroups.

The subgroup lattice has 310 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×#131	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×#812	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₃×C₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#4549	C₂	1	4	yes	maximal, complemented
5	C₃×#2113	—	3	1	no	maximal
6	#72004	C₃	1	1	yes	maximal, characteristic, direct product factor
7	C₂×C₃²×#41	C₄	1	1	yes	characteristic
8	C₃²×#131	—	2	2	no	—
9	C₃²×#131	—	2	2	no	—
10	C₂×C₃²×M₁₆	C₄	1	1	yes	characteristic
11	C₂×C₃²×M₁₆	C₄	1	2	yes	—
12	C₂²×C₃²×C₈	C₂²	1	2	yes	—
13	C₂²×C₃²×C₈	C₂²	1	1	yes	characteristic
14	C₃×#812	—	2	2	no	—
15	C₂×C₃×C₃⋊C₁₆	—	2	2	no	—
16	C₂×C₃×C₃⋊C₁₆	C₂²	1	2	yes	—
17	C₂×C₃×C₃⋊C₁₆	C₂²	1	2	yes	—
18	C₂×C₃×#131	—	2	1	no	—
19	C₂×C₃×#131	D₃	1	1	yes	characteristic
20	C₂×C₃×#131	C₂×C₃	1	1	yes	characteristic
21	C₂×C₃×M₃₂	—	3	1	no	—
22	C₂²×C₃×C₁₆	—	3	1	no	—
23	C₂×#812	C₂×C₃	1	1	yes	characteristic, complemented
24	C₂²×C₃⋊C₁₆	C₂×C₃	1	1	yes	characteristic, complemented
25	C₃×C₂²⋊C₁₆	—	3	4	no	—
26	#4549	C₂×C₃	1	4	yes	complemented
27	C₃²×#41	—	2	2	no	—
28	C₃²×#41	—	2	2	no	—
29	C₂×C₃²×Q₈	C₈	1	1	yes	characteristic
30	C₂×C₃²×D₄	C₈	1	2	yes	—
31	C₂×C₃²×D₄	C₈	1	1	yes	characteristic
32	C₂²×C₃²×C₄	C₂×C₄	1	2	yes	—
33	C₂²×C₃²×C₄	C₂×C₄	1	1	yes	characteristic
34	C₃²×M₁₆	—	2	2	no	—
35	C₃²×M₁₆	—	2	4	no	—
36	C₂×C₃²×C₈	—	2	2	no	—
37	C₂×C₃²×C₈	—	2	2	no	—
38	C₂×C₃²×C₈	C₂×C₄	1	2	yes	—
39	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
40	C₂×C₃²×C₈	D₄	1	2	yes	—
41	C₂×C₃²×C₈	D₄	1	2	yes	—
42	C₂×C₃²×C₈	C₂³	1	1	yes	characteristic
43	C₃×C₃⋊C₁₆	—	2	2	no	—
44	C₃×C₃⋊C₁₆	—	2	2	no	—
45	#2113	—	3	1	no	Sylow
46	C₂×C₃×#41	—	2	1	no	—
47	C₂×C₃×#41	Dic₃	1	1	yes	characteristic
48	C₂×C₃×#41	C₃×C₄	1	1	yes	characteristic
49	C₃×#131	—	2	4	no	—
50	C₃×#131	—	2	4	no	—
51	C₃×#131	—	2	2	no	—
52	C₃×#131	—	2	2	no	—
53	C₃×#131	—	2	2	no	—
54	C₃×#131	—	2	2	no	—
55	C₂×C₃×M₁₆	—	2	1	no	—
56	C₂×C₃×M₁₆	—	2	2	no	—
57	C₂×C₃×M₁₆	Dic₃	1	1	yes	characteristic
58	C₂×C₃×M₁₆	Dic₃	1	2	yes	—
59	C₂×C₃×M₁₆	C₃×C₄	1	1	yes	characteristic
60	C₂×C₃×M₁₆	C₃×C₄	1	2	yes	—
61	C₂²×C₃×C₈	—	2	2	no	—
62	C₂²×C₃×C₈	—	2	1	no	—
63	C₂²×C₃×C₈	C₂×D₃	1	2	yes	—
64	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
65	C₂²×C₃×C₈	C₂²×C₃	1	2	yes	—
66	C₂²×C₃×C₈	C₂²×C₃	1	1	yes	characteristic
67	C₃×M₃₂	—	6	2	no	—
68	C₂×C₃×C₁₆	—	3	2	no	—
69	C₂×C₃×C₁₆	—	3	2	no	—
70	C₂×C₃×C₁₆	—	6	2	no	—
71	#812	—	2	2	no	—
72	C₂×C₃⋊C₁₆	—	2	2	no	—
73	C₂×C₃⋊C₁₆	C₂²×C₃	1	2	yes	—
74	C₂×C₃⋊C₁₆	C₂²×C₃	1	2	yes	—
75	C₂³×C₃²	C₂×C₈	1	2	yes	—
76	C₂³×C₃²	C₂×C₈	1	1	yes	characteristic
77	C₃²×Q₈	—	2	2	no	—
78	C₃²×D₄	—	2	4	no	—
79	C₃²×D₄	—	2	2	no	—
80	C₂×C₃²×C₄	—	2	2	no	—
81	C₂×C₃²×C₄	—	2	2	no	—
82	C₂×C₃²×C₄	C₂²⋊C₄	1	2	yes	—
83	C₂×C₃²×C₄	C₂×C₈	1	2	yes	—
84	C₂×C₃²×C₄	C₂×C₈	1	1	yes	characteristic
85	C₂×C₃²×C₄	M₁₆	1	2	yes	—
86	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
87	C₃²×C₈	—	2	2	no	—
88	C₃²×C₈	C₂²⋊C₄	1	2	yes	—
89	C₃²×C₈	C₂×D₄	1	1	yes	characteristic
90	C₃²×C₈	C₂×D₄	1	1	yes	characteristic
91	C₂×#131	C₃×D₃	1	1	yes	characteristic
92	C₂×M₃₂	—	3	1	no	—
93	C₂²×C₁₆	—	3	1	no	—
94	C₂²⋊C₁₆	—	3	4	no	—
95	C₃×#41	—	2	4	no	—
96	C₃×#41	—	2	4	no	—
97	C₃×#41	—	2	2	no	—
98	C₃×#41	—	2	2	no	—
99	C₃×#41	—	2	2	no	—
100	C₃×#41	—	2	2	no	—
101	C₂×C₃×Q₈	—	2	1	no	—
102	C₂×C₃×Q₈	C₃⋊C₈	1	1	yes	characteristic
103	C₂×C₃×Q₈	C₃×C₈	1	1	yes	characteristic
104	C₂×C₃×D₄	—	2	2	no	—
105	C₂×C₃×D₄	—	2	1	no	—
106	C₂×C₃×D₄	C₃⋊C₈	1	2	yes	—
107	C₂×C₃×D₄	C₃⋊C₈	1	1	yes	characteristic
108	C₂×C₃×D₄	C₃×C₈	1	2	yes	—
109	C₂×C₃×D₄	C₃×C₈	1	1	yes	characteristic
110	C₂²×C₃×C₄	—	2	2	no	—
111	C₂²×C₃×C₄	—	2	1	no	—
112	C₂²×C₃×C₄	C₂×Dic₃	1	2	yes	—
113	C₂²×C₃×C₄	C₂×Dic₃	1	1	yes	characteristic
114	C₂²×C₃×C₄	C₂×C₃×C₄	1	2	yes	—
115	C₂²×C₃×C₄	C₂×C₃×C₄	1	1	yes	characteristic
116	C₃×M₁₆	—	2	4	no	—
117	C₃×M₁₆	—	2	8	no	—
118	C₃×M₁₆	—	2	2	no	—
119	C₃×M₁₆	—	2	2	no	—
120	C₃×M₁₆	—	2	4	no	—
121	C₃×M₁₆	—	2	4	no	—
122	C₂×C₃×C₈	—	2	4	no	—
123	C₂×C₃×C₈	—	2	4	no	—
124	C₂×C₃×C₈	—	2	2	no	—
125	C₂×C₃×C₈	—	2	2	no	—
126	C₂×C₃×C₈	—	2	2	no	—
127	C₂×C₃×C₈	—	2	2	no	—
128	C₂×C₃×C₈	—	2	2	no	—
129	C₂×C₃×C₈	—	2	1	no	—
130	C₂×C₃×C₈	—	2	2	no	—
131	C₂×C₃×C₈	—	2	2	no	—
132	C₂×C₃×C₈	—	2	1	no	—
133	C₂×C₃×C₈	C₂×Dic₃	1	2	yes	—
134	C₂×C₃×C₈	C₂×Dic₃	1	1	yes	characteristic
135	C₂×C₃×C₈	#67	1	2	yes	—
136	C₂×C₃×C₈	#67	1	2	yes	—
137	C₂×C₃×C₈	C₂×C₃×C₄	1	2	yes	—
138	C₂×C₃×C₈	C₂×C₃×C₄	1	1	yes	characteristic
139	C₂×C₃×C₈	C₃×D₄	1	2	yes	—
140	C₂×C₃×C₈	C₃×D₄	1	2	yes	—
141	C₂×C₃×C₈	C₂²×D₃	1	1	yes	characteristic
142	C₂×C₃×C₈	C₂³×C₃	1	1	yes	characteristic
143	C₃×C₁₆	—	6	2	no	—
144	C₃×C₁₆	—	6	2	no	—
145	C₃⋊C₁₆	—	2	2	no	—
146	C₃⋊C₁₆	—	2	2	no	—
147	C₂²×C₃²	—	2	2	no	—
148	C₂²×C₃²	—	2	2	no	—
149	C₂²×C₃²	—	2	4	no	—
150	C₂²×C₃²	C₂²⋊C₈	1	2	yes	—
151	C₂²×C₃²	C₂²×C₈	1	1	yes	characteristic
152	C₃²×C₄	—	2	2	no	—
153	C₃²×C₄	C₂²⋊C₈	1	2	yes	—
154	C₃²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
155	C₃²×C₄	C₂×M₁₆	1	1	yes	characteristic
156	C₂×#41	C₃×Dic₃	1	1	yes	characteristic
157	#131	—	2	2	no	—
158	#131	—	2	2	no	—
159	C₂×M₁₆	C₃×Dic₃	1	1	yes	characteristic
160	C₂×M₁₆	C₃×Dic₃	1	2	yes	—
161	C₂²×C₈	C₂×C₃×D₃	1	2	yes	—
162	C₂²×C₈	C₂×C₃×D₃	1	1	yes	characteristic
163	M₃₂	—	6	2	no	—
164	C₂×C₁₆	—	3	2	no	—
165	C₂×C₁₆	—	3	2	no	—
166	C₂×C₁₆	—	6	2	no	—
167	C₂³×C₃	—	2	2	no	—
168	C₂³×C₃	—	2	1	no	—
169	C₂³×C₃	C₂×C₃⋊C₈	1	2	yes	—
170	C₂³×C₃	C₂×C₃⋊C₈	1	1	yes	characteristic
171	C₂³×C₃	C₂×C₃×C₈	1	2	yes	—
172	C₂³×C₃	C₂×C₃×C₈	1	1	yes	characteristic
173	C₃×Q₈	—	2	4	no	—
174	C₃×Q₈	—	2	2	no	—
175	C₃×Q₈	—	2	2	no	—
176	C₃×D₄	—	2	8	no	—
177	C₃×D₄	—	2	4	no	—
178	C₃×D₄	—	2	4	no	—
179	C₃×D₄	—	2	4	no	—
180	C₃×D₄	—	2	2	no	—
181	C₃×D₄	—	2	2	no	—
182	C₂×C₃×C₄	—	2	4	no	—
183	C₂×C₃×C₄	—	2	4	no	—
184	C₂×C₃×C₄	—	2	2	no	—
185	C₂×C₃×C₄	—	2	2	no	—
186	C₂×C₃×C₄	—	2	2	no	—
187	C₂×C₃×C₄	—	2	2	no	—
188	C₂×C₃×C₄	—	2	2	no	—
189	C₂×C₃×C₄	—	2	1	no	—
190	C₂×C₃×C₄	—	2	2	no	—
191	C₂×C₃×C₄	—	2	2	no	—
192	C₂×C₃×C₄	—	2	1	no	—
193	C₂×C₃×C₄	C₂×C₃⋊C₈	1	2	yes	—
194	C₂×C₃×C₄	C₂×C₃⋊C₈	1	1	yes	characteristic
195	C₂×C₃×C₄	#208	1	2	yes	—
196	C₂×C₃×C₄	#217	1	2	yes	—
197	C₂×C₃×C₄	C₃×C₂²⋊C₄	1	2	yes	—
198	C₂×C₃×C₄	C₂×C₃×C₈	1	2	yes	—
199	C₂×C₃×C₄	C₂×C₃×C₈	1	1	yes	characteristic
200	C₂×C₃×C₄	C₃×M₁₆	1	2	yes	—
201	C₂×C₃×C₄	C₂²×Dic₃	1	1	yes	characteristic
202	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
203	C₃×C₈	—	2	4	no	—
204	C₃×C₈	—	2	2	no	—
205	C₃×C₈	—	2	2	no	—
206	C₃×C₈	—	2	2	no	—
207	C₃×C₈	—	2	1	no	—
208	C₃×C₈	—	2	1	no	—
209	C₃×C₈	#217	1	2	yes	—
210	C₃×C₈	C₃×C₂²⋊C₄	1	2	yes	—
211	C₃×C₈	C₂×#67	1	1	yes	characteristic
212	C₃×C₈	C₂×#67	1	1	yes	characteristic
213	C₃×C₈	C₂×C₃×D₄	1	1	yes	characteristic
214	C₃×C₈	C₂×C₃×D₄	1	1	yes	characteristic
215	C₂×C₃²	—	2	2	no	—
216	C₂×C₃²	—	4	2	no	—
217	C₂×C₃²	C₂×C₂²⋊C₈	1	1	yes	characteristic
218	C₂×C₃²	#504	1	2	yes	—
219	#41	—	2	2	no	—
220	#41	—	2	2	no	—
221	C₂×Q₈	C₃×C₃⋊C₈	1	1	yes	characteristic
222	C₂×D₄	C₃×C₃⋊C₈	1	2	yes	—
223	C₂×D₄	C₃×C₃⋊C₈	1	1	yes	characteristic
224	C₂²×C₄	C₂×C₃×Dic₃	1	2	yes	—
225	C₂²×C₄	C₂×C₃×Dic₃	1	1	yes	characteristic
226	M₁₆	—	2	2	no	—
227	M₁₆	—	2	4	no	—
228	C₂×C₈	—	2	2	no	—
229	C₂×C₈	—	2	2	no	—
230	C₂×C₈	C₂×C₃×Dic₃	1	2	yes	—
231	C₂×C₈	C₂×C₃×Dic₃	1	1	yes	characteristic
232	C₂×C₈	C₃×#67	1	2	yes	—
233	C₂×C₈	C₃×#67	1	2	yes	—
234	C₂×C₈	C₂²×C₃×D₃	1	1	yes	characteristic
235	C₁₆	—	6	2	no	—
236	C₁₆	—	6	2	no	—
237	C₂²×C₃	—	2	2	no	—
238	C₂²×C₃	—	2	2	no	—
239	C₂²×C₃	—	2	4	no	—
240	C₂²×C₃	—	2	4	no	—
241	C₂²×C₃	—	2	4	no	—
242	C₂²×C₃	—	2	2	no	—
243	C₂²×C₃	—	2	2	no	—
244	C₂²×C₃	—	2	1	no	—
245	C₂²×C₃	—	2	2	no	—
246	C₂²×C₃	—	4	2	no	—
247	C₂²×C₃	—	4	4	no	—
248	C₂²×C₃	#830	1	2	yes	—
249	C₂²×C₃	C₃×C₂²⋊C₈	1	2	yes	—
250	C₂²×C₃	C₂²×C₃⋊C₈	1	1	yes	characteristic
251	C₂²×C₃	C₂²×C₃×C₈	1	1	yes	characteristic
252	C₃×C₄	—	2	4	no	—
253	C₃×C₄	—	2	2	no	—
254	C₃×C₄	—	2	2	no	—
255	C₃×C₄	—	2	1	no	—
256	C₃×C₄	—	2	1	no	—
257	C₃×C₄	—	2	2	no	—
258	C₃×C₄	#830	1	2	yes	—
259	C₃×C₄	C₃×C₂²⋊C₈	1	2	yes	—
260	C₃×C₄	C₂×#208	1	1	yes	characteristic
261	C₃×C₄	C₂×#217	1	1	yes	characteristic
262	C₃×C₄	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
263	C₃×C₄	C₂×C₃×M₁₆	1	1	yes	characteristic
264	C₃²	#2113	1	1	yes	characteristic, complemented, Sylow
265	C₂³	C₂×C₃×C₃⋊C₈	1	2	yes	—
266	C₂³	C₂×C₃×C₃⋊C₈	1	1	yes	characteristic
267	Q₈	—	2	2	no	—
268	D₄	—	2	4	no	—
269	D₄	—	2	2	no	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	C₂×C₃×C₃⋊C₈	1	2	yes	—
273	C₂×C₄	C₂×C₃×C₃⋊C₈	1	1	yes	characteristic
274	C₂×C₄	C₃×#208	1	2	yes	—
275	C₂×C₄	C₃×#217	1	2	yes	—
276	C₂×C₄	C₂²×C₃×Dic₃	1	1	yes	characteristic
277	C₈	—	2	2	no	—
278	C₈	C₃×#217	1	2	yes	—
279	C₈	C₂×C₃×#67	1	1	yes	characteristic
280	C₈	C₂×C₃×#67	1	1	yes	characteristic
281	C₂×C₃	—	2	2	no	—
282	C₂×C₃	—	2	2	no	—
283	C₂×C₃	—	2	1	no	—
284	C₂×C₃	—	2	2	no	—
285	C₂×C₃	—	4	2	no	—
286	C₂×C₃	—	4	4	no	—
287	C₂×C₃	—	4	2	no	—
288	C₂×C₃	—	4	2	no	—
289	C₂×C₃	#5140	1	2	yes	—
290	C₂×C₃	C₂×#830	1	1	yes	characteristic
291	C₂×C₃	C₂×C₃×C₂²⋊C₈	1	1	yes	characteristic
292	C₂×C₃	C₃×#504	1	2	yes	—
293	C₂²	—	2	2	no	—
294	C₂²	—	2	2	no	—
295	C₂²	—	2	4	no	—
296	C₂²	C₃×#830	1	2	yes	—
297	C₂²	C₂²×C₃×C₃⋊C₈	1	1	yes	characteristic
298	C₄	—	2	2	no	—
299	C₄	C₃×#830	1	2	yes	—
300	C₄	C₂×C₃×#208	1	1	yes	characteristic
301	C₄	C₂×C₃×#217	1	1	yes	characteristic
302	C₃	—	2	1	no	—
303	C₃	#72004	1	1	yes	characteristic, direct product factor
304	C₃	C₃×#2113	1	1	yes	characteristic, complemented
305	C₂	—	2	2	no	—
306	C₂	—	4	2	no	—
307	C₂	C₃×#5140	1	2	yes	—
308	C₂	C₂×C₃×#830	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₈	C₂²×D₃
commutator	C₂²×C₃	C₂²×C₃×C₈
Frattini	C₂×C₈	C₂²×C₃×D₃
Fitting	C₂×C₃²×#131	C₂
socle	C₂²×C₃²	C₂²×C₈
prime radical	C₂×C₃²×#41	C₄
cosocle	C₂×C₃×C₈	C₂³×C₃
hypercenter	C₂×C₃×#131	D₃
hypocenter	C₃	C₃×#2113
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₃×C₈ ◁ C₂×C₃×#131

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×#812	C₂	semidirect
3. C₂²×C₃×C₃⋊C₁₆	C₂	semidirect
4. C₃×#4549	C₂	semidirect
6. #72004	C₃	direct (N × H)
23. C₂×#812	C₂×C₃	semidirect
24. C₂²×C₃⋊C₁₆	C₂×C₃	semidirect
26. #4549	C₂×C₃	semidirect
264. C₃²	#2113	semidirect
303. C₃	#72004	direct (N × H)
304. C₃	C₃×#2113	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃²×#131	2
C₂×C₃×#812	2
C₂²×C₃×C₃⋊C₁₆	2
C₃×#4549	2
C₃×#2113	3
#72004	3

Subgroups — C3×#72004 #11796000

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#72004 #11796000