Subgroups — C₃×#71432 #11796003

Order 1152.

501 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#901	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃²×M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₃×C₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×(C₂×D₃)⋊C₁₆	C₂	1	4	yes	maximal, complemented
5	C₃×#2113	—	3	1	no	maximal
6	#71432	C₃	1	1	yes	maximal, characteristic, direct product factor
7	C₂×C₃×#235	C₄	1	1	yes	characteristic
8	C₂²×C₃²×C₈	C₂²	1	1	yes	characteristic
9	C₂×C₃×#208	C₄	1	1	yes	characteristic
10	C₃×#901	—	2	2	no	—
11	C₃×#901	—	2	2	no	—
12	C₂×C₃×ΓL(1,𝔽₂₅)	C₄	1	2	yes	—
13	C₂×C₃×D₃×C₈	C₂²	1	2	yes	—
14	C₃²×M₃₂	—	2	2	no	—
15	C₂×C₃²×C₁₆	C₂²	1	2	yes	—
16	C₂×C₃×C₃⋊C₁₆	—	2	2	no	—
17	C₂×C₃×C₃⋊C₁₆	C₂²	1	2	yes	—
18	C₂×C₃×#131	—	3	1	no	—
19	C₂×#901	C₂×C₃	1	1	yes	characteristic
20	C₂×C₃×M₃₂	—	2	1	no	—
21	C₂×C₃×M₃₂	D₃	1	1	yes	characteristic, complemented
22	C₂×C₃×M₃₂	C₂×C₃	1	1	yes	characteristic, complemented
23	C₂²×C₃×C₁₆	—	3	1	no	—
24	C₂²×C₃⋊C₁₆	C₂×C₃	1	1	yes	characteristic, complemented
25	C₃×C₂²⋊C₁₆	—	3	4	no	—
26	(C₂×D₃)⋊C₁₆	C₂×C₃	1	4	yes	complemented
27	C₂²×C₃²×C₄	C₂×C₄	1	1	yes	characteristic
28	C₂×C₃×#67	C₈	1	2	yes	—
29	C₃×#235	—	2	2	no	—
30	C₃×#235	—	2	2	no	—
31	C₂×C₃×D₁₂	C₈	1	1	yes	characteristic
32	C₂×C₃×C₄×D₃	C₂×C₄	1	2	yes	—
33	C₂×C₃×Dic₆	C₈	1	1	yes	characteristic
34	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
35	C₂×C₃²×C₈	D₄	1	2	yes	—
36	C₂×C₃²×C₈	D₄	1	2	yes	—
37	C₂×C₃²×C₈	C₂³	1	1	yes	characteristic
38	C₃×#208	—	2	2	no	—
39	C₂×C₃×C₃⋊C₈	C₂×C₄	1	2	yes	—
40	C₃×ΓL(1,𝔽₂₅)	—	2	4	no	—
41	C₃×D₃×C₈	—	2	2	no	—
42	C₃×D₃×C₈	—	2	2	no	—
43	C₃²×C₁₆	—	2	2	no	—
44	C₃×C₃⋊C₁₆	—	2	2	no	—
45	#2113	—	3	1	no	Sylow
46	C₂×C₃×#41	—	3	1	no	—
47	C₂×#235	C₃×C₄	1	1	yes	characteristic
48	C₃×#131	—	6	2	no	—
49	C₃×#131	—	6	2	no	—
50	C₂×C₃×M₁₆	—	3	1	no	—
51	C₂×C₃×M₁₆	—	3	2	no	—
52	C₂²×C₃×C₈	—	2	1	no	—
53	C₂²×C₃×C₈	—	3	2	no	—
54	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
55	C₂²×C₃×C₈	C₂²×C₃	1	1	yes	characteristic
56	C₂×#208	C₃×C₄	1	1	yes	characteristic
57	#901	—	2	2	no	—
58	#901	—	2	2	no	—
59	C₂×ΓL(1,𝔽₂₅)	C₃×C₄	1	2	yes	—
60	C₂×D₃×C₈	C₂²×C₃	1	2	yes	—
61	C₃×M₃₂	—	2	4	no	—
62	C₃×M₃₂	—	2	2	no	—
63	C₃×M₃₂	—	2	2	no	—
64	C₂×C₃×C₁₆	—	2	2	no	—
65	C₂×C₃×C₁₆	—	3	2	no	—
66	C₂×C₃×C₁₆	—	6	2	no	—
67	C₂×C₃×C₁₆	C₂×D₃	1	2	yes	—
68	C₂×C₃×C₁₆	C₂²×C₃	1	2	yes	—
69	C₂×C₃⋊C₁₆	—	2	2	no	—
70	C₂×C₃⋊C₁₆	C₂²×C₃	1	2	yes	—
71	C₂³×C₃²	C₂×C₈	1	1	yes	characteristic
72	C₂²×C₃×D₃	C₂×C₈	1	2	yes	—
73	C₂×C₃²×C₄	C₂²⋊C₄	1	2	yes	—
74	C₂×C₃²×C₄	C₂×C₈	1	1	yes	characteristic
75	C₂×C₃²×C₄	M₁₆	1	2	yes	—
76	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
77	C₃×#67	—	2	4	no	—
78	C₂×C₃×Dic₃	C₂×C₈	1	2	yes	—
79	C₃×D₁₂	—	2	2	no	—
80	C₃×C₄×D₃	—	2	2	no	—
81	C₃×C₄×D₃	—	2	2	no	—
82	C₃×Dic₆	—	2	2	no	—
83	C₃²×C₈	C₂²⋊C₄	1	2	yes	—
84	C₃²×C₈	C₂×D₄	1	1	yes	characteristic
85	C₃²×C₈	C₂×D₄	1	1	yes	characteristic
86	C₃×C₃⋊C₈	—	2	2	no	—
87	C₂×#131	—	3	1	no	—
88	C₂×M₃₂	C₃×D₃	1	1	yes	characteristic, complemented
89	C₂²×C₁₆	—	3	1	no	—
90	C₂²⋊C₁₆	—	3	4	no	—
91	C₃×#41	—	6	2	no	—
92	C₃×#41	—	6	2	no	—
93	C₂×C₃×Q₈	—	3	1	no	—
94	C₂×C₃×D₄	—	3	2	no	—
95	C₂×C₃×D₄	—	3	1	no	—
96	C₂²×C₃×C₄	—	2	1	no	—
97	C₂²×C₃×C₄	—	3	2	no	—
98	C₂²×C₃×C₄	C₄×D₃	1	1	yes	characteristic
99	C₂²×C₃×C₄	C₂×C₃×C₄	1	1	yes	characteristic
100	C₂×#67	C₃×C₈	1	2	yes	—
101	#235	—	2	2	no	—
102	#235	—	2	2	no	—
103	C₂×D₁₂	C₃×C₈	1	1	yes	characteristic
104	C₂×C₄×D₃	C₂×C₃×C₄	1	2	yes	—
105	C₂×Dic₆	C₃×C₈	1	1	yes	characteristic
106	C₃×M₁₆	—	6	2	no	—
107	C₃×M₁₆	—	6	4	no	—
108	C₂×C₃×C₈	—	2	1	no	—
109	C₂×C₃×C₈	—	2	2	no	—
110	C₂×C₃×C₈	—	2	2	no	—
111	C₂×C₃×C₈	—	2	1	no	—
112	C₂×C₃×C₈	—	3	2	no	—
113	C₂×C₃×C₈	—	6	2	no	—
114	C₂×C₃×C₈	—	6	2	no	—
115	C₂×C₃×C₈	C₄×D₃	1	1	yes	characteristic
116	C₂×C₃×C₈	D₁₂	1	2	yes	—
117	C₂×C₃×C₈	#67	1	2	yes	—
118	C₂×C₃×C₈	C₂×C₃×C₄	1	1	yes	characteristic
119	C₂×C₃×C₈	C₃×D₄	1	2	yes	—
120	C₂×C₃×C₈	C₃×D₄	1	2	yes	—
121	C₂×C₃×C₈	C₂²×D₃	1	1	yes	characteristic
122	C₂×C₃×C₈	C₂³×C₃	1	1	yes	characteristic
123	#208	—	2	2	no	—
124	C₂×C₃⋊C₈	C₂×C₃×C₄	1	2	yes	—
125	ΓL(1,𝔽₂₅)	—	2	4	no	—
126	D₃×C₈	—	2	2	no	—
127	D₃×C₈	—	2	2	no	—
128	C₃×C₁₆	—	2	4	no	—
129	C₃×C₁₆	—	2	2	no	—
130	C₃×C₁₆	—	2	2	no	—
131	C₃×C₁₆	—	6	2	no	—
132	C₃⋊C₁₆	—	2	2	no	—
133	C₂²×C₃²	—	2	2	no	—
134	C₂²×C₃²	C₂²⋊C₈	1	2	yes	—
135	C₂²×C₃²	C₂²×C₈	1	1	yes	characteristic
136	C₂×C₃×D₃	—	2	2	no	—
137	C₂×C₃×D₃	—	2	4	no	—
138	C₃²×C₄	C₂²⋊C₈	1	2	yes	—
139	C₃²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
140	C₃²×C₄	C₂×M₁₆	1	1	yes	characteristic
141	C₃×Dic₃	—	2	2	no	—
142	C₂×#41	—	3	1	no	—
143	#131	—	6	2	no	—
144	#131	—	6	2	no	—
145	C₂×M₁₆	—	3	1	no	—
146	C₂×M₁₆	—	3	2	no	—
147	C₂²×C₈	—	3	2	no	—
148	C₂²×C₈	C₂×C₃×D₃	1	1	yes	characteristic
149	M₃₂	—	2	2	no	—
150	C₂×C₁₆	—	3	2	no	—
151	C₂×C₁₆	—	6	2	no	—
152	C₂×C₁₆	C₂×C₃×D₃	1	2	yes	—
153	C₂³×C₃	—	2	1	no	—
154	C₂³×C₃	—	3	2	no	—
155	C₂³×C₃	D₃×C₈	1	1	yes	characteristic
156	C₂³×C₃	C₂×C₃×C₈	1	1	yes	characteristic
157	C₂²×D₃	C₂×C₃×C₈	1	2	yes	—
158	C₃×Q₈	—	6	2	no	—
159	C₃×D₄	—	6	4	no	—
160	C₃×D₄	—	6	2	no	—
161	C₂×C₃×C₄	—	2	1	no	—
162	C₂×C₃×C₄	—	2	2	no	—
163	C₂×C₃×C₄	—	2	2	no	—
164	C₂×C₃×C₄	—	2	1	no	—
165	C₂×C₃×C₄	—	3	2	no	—
166	C₂×C₃×C₄	—	6	2	no	—
167	C₂×C₃×C₄	—	6	2	no	—
168	C₂×C₃×C₄	D₃×C₈	1	1	yes	characteristic
169	C₂×C₃×C₄	ΓL(1,𝔽₂₅)	1	2	yes	—
170	C₂×C₃×C₄	(C₂×D₃)⋊C₄	1	2	yes	—
171	C₂×C₃×C₄	C₃×C₂²⋊C₄	1	2	yes	—
172	C₂×C₃×C₄	C₂×C₃×C₈	1	1	yes	characteristic
173	C₂×C₃×C₄	C₃×M₁₆	1	2	yes	—
174	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
175	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
176	#67	—	2	4	no	—
177	C₂×Dic₃	C₂×C₃×C₈	1	2	yes	—
178	D₁₂	—	2	2	no	—
179	C₄×D₃	—	2	2	no	—
180	C₄×D₃	—	2	2	no	—
181	Dic₆	—	2	2	no	—
182	C₃×C₈	—	2	2	no	—
183	C₃×C₈	—	2	1	no	—
184	C₃×C₈	—	2	1	no	—
185	C₃×C₈	—	6	2	no	—
186	C₃×C₈	(C₂×D₃)⋊C₄	1	2	yes	—
187	C₃×C₈	C₃×C₂²⋊C₄	1	2	yes	—
188	C₃×C₈	C₂×D₁₂	1	1	yes	characteristic
189	C₃×C₈	C₂×#67	1	1	yes	characteristic
190	C₃×C₈	C₂×C₃×D₄	1	1	yes	characteristic
191	C₃×C₈	C₂×C₃×D₄	1	1	yes	characteristic
192	C₃⋊C₈	—	2	2	no	—
193	C₂×C₃²	—	2	2	no	—
194	C₂×C₃²	C₂×C₂²⋊C₈	1	1	yes	characteristic
195	C₂×C₃²	#504	1	2	yes	—
196	C₃×D₃	—	4	2	no	—
197	#41	—	6	2	no	—
198	#41	—	6	2	no	—
199	C₂×Q₈	—	3	1	no	—
200	C₂×D₄	—	3	2	no	—
201	C₂×D₄	—	3	1	no	—
202	C₂²×C₄	—	3	2	no	—
203	C₂²×C₄	C₃×C₄×D₃	1	1	yes	characteristic
204	M₁₆	—	6	2	no	—
205	M₁₆	—	6	4	no	—
206	C₂×C₈	—	3	2	no	—
207	C₂×C₈	—	6	2	no	—
208	C₂×C₈	—	6	2	no	—
209	C₂×C₈	C₃×C₄×D₃	1	1	yes	characteristic
210	C₂×C₈	C₃×D₁₂	1	2	yes	—
211	C₂×C₈	C₃×#67	1	2	yes	—
212	C₂×C₈	C₂²×C₃×D₃	1	1	yes	characteristic
213	C₁₆	—	2	2	no	—
214	C₁₆	—	6	2	no	—
215	C₂²×C₃	—	2	2	no	—
216	C₂²×C₃	—	2	2	no	—
217	C₂²×C₃	—	2	1	no	—
218	C₂²×C₃	—	2	2	no	—
219	C₂²×C₃	—	4	2	no	—
220	C₂²×C₃	—	6	4	no	—
221	C₂²×C₃	—	6	2	no	—
222	C₂²×C₃	(C₂×D₃)⋊C₈	1	2	yes	—
223	C₂²×C₃	C₃×C₂²⋊C₈	1	2	yes	—
224	C₂²×C₃	C₂×D₃×C₈	1	1	yes	characteristic
225	C₂²×C₃	C₂²×C₃×C₈	1	1	yes	characteristic
226	C₂×D₃	—	2	4	no	—
227	C₂×D₃	—	2	2	no	—
228	C₃×C₄	—	2	1	no	—
229	C₃×C₄	—	2	1	no	—
230	C₃×C₄	—	2	2	no	—
231	C₃×C₄	—	6	2	no	—
232	C₃×C₄	(C₂×D₃)⋊C₈	1	2	yes	—
233	C₃×C₄	C₃×C₂²⋊C₈	1	2	yes	—
234	C₃×C₄	C₂×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
235	C₃×C₄	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
236	C₃×C₄	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
237	C₃×C₄	C₂×C₃×M₁₆	1	1	yes	characteristic
238	Dic₃	—	2	2	no	—
239	C₃²	#2113	1	1	yes	characteristic, complemented, Sylow
240	C₂³	—	3	2	no	—
241	C₂³	C₃×D₃×C₈	1	1	yes	characteristic
242	Q₈	—	6	2	no	—
243	D₄	—	6	4	no	—
244	D₄	—	6	2	no	—
245	C₂×C₄	—	3	2	no	—
246	C₂×C₄	—	6	2	no	—
247	C₂×C₄	—	6	2	no	—
248	C₂×C₄	C₃×D₃×C₈	1	1	yes	characteristic
249	C₂×C₄	C₃×ΓL(1,𝔽₂₅)	1	2	yes	—
250	C₂×C₄	C₃×(C₂×D₃)⋊C₄	1	2	yes	—
251	C₂×C₄	C₂×C₃×C₄×D₃	1	1	yes	characteristic
252	C₈	—	6	2	no	—
253	C₈	C₃×(C₂×D₃)⋊C₄	1	2	yes	—
254	C₈	C₂×C₃×D₁₂	1	1	yes	characteristic
255	C₈	C₂×C₃×#67	1	1	yes	characteristic
256	C₂×C₃	—	2	2	no	—
257	C₂×C₃	—	2	2	no	—
258	C₂×C₃	—	2	2	no	—
259	C₂×C₃	—	2	1	no	—
260	C₂×C₃	—	4	2	no	—
261	C₂×C₃	—	12	2	no	—
262	C₂×C₃	#4907	1	2	yes	—
263	C₂×C₃	C₂×(C₂×D₃)⋊C₈	1	1	yes	characteristic
264	C₂×C₃	C₂×C₃×C₂²⋊C₈	1	1	yes	characteristic
265	C₂×C₃	C₃×#504	1	2	yes	—
266	D₃	—	4	2	no	—
267	C₂²	—	2	2	no	—
268	C₂²	—	6	4	no	—
269	C₂²	—	6	2	no	—
270	C₂²	C₃×(C₂×D₃)⋊C₈	1	2	yes	—
271	C₂²	C₂×C₃×D₃×C₈	1	1	yes	characteristic
272	C₄	—	6	2	no	—
273	C₄	C₃×(C₂×D₃)⋊C₈	1	2	yes	—
274	C₄	C₂×C₃×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
275	C₄	C₂×C₃×(C₂×D₃)⋊C₄	1	1	yes	characteristic
276	C₃	—	2	1	no	—
277	C₃	#71432	1	1	yes	characteristic, direct product factor
278	C₃	C₃×#2113	1	1	yes	characteristic, complemented
279	C₂	—	2	2	no	—
280	C₂	—	12	2	no	—
281	C₂	C₃×#4907	1	2	yes	—
282	C₂	C₂×C₃×(C₂×D₃)⋊C₈	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×M₃₂	C₂
socle	C₂²×C₃²	C₂²×C₈
prime radical	C₂×C₃×#235	C₄
cosocle	C₂×C₃×C₈	C₂³×C₃
hypercenter	C₂×C₃×M₃₂	D₃
hypocenter	C₃	C₃×#2113
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₃×C₈ ◁ C₂×C₃×M₃₂

Complements

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

N	H ≅ G/N	type
2. C₂×C₃²×M₃₂	C₂	semidirect
3. C₂²×C₃×C₃⋊C₁₆	C₂	semidirect
4. C₃×(C₂×D₃)⋊C₁₆	C₂	semidirect
6. #71432	C₃	direct (N × H)
21. C₂×C₃×M₃₂	D₃	semidirect
22. C₂×C₃×M₃₂	C₂×C₃	semidirect
24. C₂²×C₃⋊C₁₆	C₂×C₃	semidirect
26. (C₂×D₃)⋊C₁₆	C₂×C₃	semidirect
88. C₂×M₃₂	C₃×D₃	semidirect
239. C₃²	#2113	semidirect
277. C₃	#71432	direct (N × H)
278. 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×C₃×#901	2
C₂×C₃²×M₃₂	2
C₂²×C₃×C₃⋊C₁₆	2
C₃×(C₂×D₃)⋊C₁₆	2
C₃×#2113	3
#71432	3

Subgroups — C3×#71432 #11796003

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#71432 #11796003