Subgroups — C₃²×#1581 #11768630

Order 1152.

546 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₄×D₈	C₂	1	1	yes	maximal, characteristic
2	C₃²×#436	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃²×C₈×D₄	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃²×#335	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃²×D₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃²×D₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃²×#322	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#1581	C₃	1	4	yes	maximal, direct product factor
9	C₂×C₃²×D₈	C₄	1	1	yes	characteristic
10	C₂×C₃²×M₁₆	—	2	1	no	—
11	C₂²×C₃²×C₈	—	2	1	no	—
12	C₃²×C₄×D₄	C₂²	1	1	yes	characteristic
13	C₃²×C₄×D₄	C₂²	1	1	yes	characteristic
14	C₃²×#107	C₄	1	1	yes	characteristic
15	C₃²×C₄⋊C₈	C₂²	1	1	yes	characteristic
16	C₃²×C₄⋊C₈	C₂²	1	1	yes	characteristic
17	C₃²×#102	C₄	1	1	yes	characteristic
18	C₃²×#102	C₄	1	1	yes	characteristic
19	C₃²×C₂²⋊C₈	—	2	1	no	—
20	C₃²×C₂²⋊C₈	—	2	1	no	—
21	C₃²×C₄×C₈	C₂²	1	1	yes	characteristic
22	C₃²×C₄×C₈	C₂²	1	1	yes	characteristic
23	C₃²×C₄×C₈	C₂²	1	1	yes	characteristic
24	C₃×C₄×D₈	C₂×C₃	1	4	yes	—
25	C₃×#436	C₂×C₃	1	4	yes	complemented
26	C₃×C₈×D₄	C₂×C₃	1	4	yes	complemented
27	C₃×#335	C₂×C₃	1	4	yes	complemented
28	C₃×D₄⋊C₈	C₂×C₃	1	4	yes	complemented
29	C₃×D₄⋊C₈	C₂×C₃	1	4	yes	complemented
30	C₃×#322	C₂×C₃	1	4	yes	complemented
31	C₂×C₃²×D₄	C₂×C₄	1	1	yes	characteristic
32	C₂×C₃²×D₄	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₃²×C₄	—	2	1	no	—
34	C₂²×C₃²×C₄	—	2	1	no	—
35	C₃²×D₈	—	2	2	no	—
36	C₃²×M₁₆	—	4	2	no	—
37	C₂×C₃²×C₈	—	2	1	no	—
38	C₂×C₃²×C₈	—	2	1	no	—
39	C₂×C₃²×C₈	—	2	1	no	—
40	C₂×C₃²×C₈	—	4	2	no	—
41	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
42	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
43	C₂×C₃²×C₈	D₄	1	1	yes	characteristic
44	C₂×C₃²×C₈	D₄	1	1	yes	characteristic
45	C₃²×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
46	C₃²×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
47	C₃²×C₂²⋊C₄	—	2	1	no	—
48	C₃²×C₂²⋊C₄	—	2	1	no	—
49	C₃²×C₄²	C₂³	1	1	yes	characteristic
50	#1581	C₃²	1	1	yes	characteristic, direct product factor, Sylow
51	C₂×C₃×D₈	C₃×C₄	1	4	yes	—
52	C₂×C₃×M₁₆	—	2	4	no	—
53	C₂²×C₃×C₈	—	2	4	no	—
54	C₃×C₄×D₄	C₂²×C₃	1	4	yes	—
55	C₃×C₄×D₄	C₂²×C₃	1	4	yes	—
56	C₃×#107	C₃×C₄	1	4	yes	—
57	C₃×C₄⋊C₈	C₂²×C₃	1	4	yes	—
58	C₃×C₄⋊C₈	C₂²×C₃	1	4	yes	—
59	C₃×#102	C₃×C₄	1	4	yes	—
60	C₃×#102	C₃×C₄	1	4	yes	—
61	C₃×C₂²⋊C₈	—	2	4	no	—
62	C₃×C₂²⋊C₈	—	2	4	no	—
63	C₃×C₄×C₈	C₂²×C₃	1	4	yes	—
64	C₃×C₄×C₈	C₂²×C₃	1	4	yes	—
65	C₃×C₄×C₈	C₂²×C₃	1	4	yes	—
66	C₂³×C₃²	—	2	1	no	—
67	C₂³×C₃²	—	2	1	no	—
68	C₃²×D₄	—	2	1	no	—
69	C₃²×D₄	—	2	1	no	—
70	C₃²×D₄	—	2	1	no	—
71	C₃²×D₄	M₁₆	1	2	yes	complemented
72	C₂×C₃²×C₄	—	2	1	no	—
73	C₂×C₃²×C₄	—	2	1	no	—
74	C₂×C₃²×C₄	—	4	1	no	—
75	C₂×C₃²×C₄	—	4	1	no	—
76	C₂×C₃²×C₄	—	4	1	no	—
77	C₂×C₃²×C₄	—	4	1	no	—
78	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
80	C₂×C₃²×C₄	#41	1	1	yes	characteristic
81	C₃²×C₈	—	2	1	no	—
82	C₃²×C₈	—	2	1	no	—
83	C₃²×C₈	—	2	1	no	—
84	C₃²×C₈	—	4	1	no	—
85	C₃²×C₈	—	4	1	no	—
86	C₃²×C₈	—	4	1	no	—
87	C₃²×C₈	D₈	1	2	yes	complemented
88	C₄×D₈	C₂×C₃²	1	1	yes	characteristic
89	#436	C₂×C₃²	1	1	yes	characteristic, complemented
90	C₈×D₄	C₂×C₃²	1	1	yes	characteristic, complemented
91	#335	C₂×C₃²	1	1	yes	characteristic, complemented
92	D₄⋊C₈	C₂×C₃²	1	1	yes	characteristic, complemented
93	D₄⋊C₈	C₂×C₃²	1	1	yes	characteristic, complemented
94	#322	C₂×C₃²	1	1	yes	characteristic, complemented
95	C₂×C₃×D₄	C₂×C₃×C₄	1	4	yes	—
96	C₂×C₃×D₄	C₂×C₃×C₄	1	4	yes	—
97	C₂²×C₃×C₄	—	2	4	no	—
98	C₂²×C₃×C₄	—	2	4	no	—
99	C₃×D₈	—	2	8	no	—
100	C₃×M₁₆	—	4	8	no	—
101	C₂×C₃×C₈	—	2	4	no	—
102	C₂×C₃×C₈	—	2	4	no	—
103	C₂×C₃×C₈	—	2	4	no	—
104	C₂×C₃×C₈	—	4	8	no	—
105	C₂×C₃×C₈	C₂×C₃×C₄	1	4	yes	—
106	C₂×C₃×C₈	C₂×C₃×C₄	1	4	yes	—
107	C₂×C₃×C₈	C₃×D₄	1	4	yes	—
108	C₂×C₃×C₈	C₃×D₄	1	4	yes	—
109	C₃×C₄⋊C₄	C₂×C₃×C₄	1	4	yes	—
110	C₃×C₄⋊C₄	C₂×C₃×C₄	1	4	yes	—
111	C₃×C₂²⋊C₄	—	2	4	no	—
112	C₃×C₂²⋊C₄	—	2	4	no	—
113	C₃×C₄²	C₂³×C₃	1	4	yes	—
114	C₂²×C₃²	—	2	2	no	—
115	C₂²×C₃²	—	4	1	no	—
116	C₂²×C₃²	—	4	1	no	—
117	C₂²×C₃²	—	4	1	no	—
118	C₂²×C₃²	—	4	1	no	—
119	C₂²×C₃²	—	4	1	no	—
120	C₂²×C₃²	C₄×D₄	1	1	yes	characteristic
121	C₃²×C₄	—	2	1	no	—
122	C₃²×C₄	—	4	1	no	—
123	C₃²×C₄	—	4	1	no	—
124	C₃²×C₄	C₂×M₁₆	1	1	yes	characteristic
125	C₃²×C₄	#131	1	1	yes	characteristic
126	C₃²×C₄	C₂×D₈	1	1	yes	characteristic
127	C₃²×C₄	#135	1	1	yes	characteristic
128	C₂×D₈	C₃²×C₄	1	1	yes	characteristic
129	C₂×M₁₆	—	2	1	no	—
130	C₂²×C₈	—	2	1	no	—
131	C₄×D₄	C₂²×C₃²	1	1	yes	characteristic
132	C₄×D₄	C₂²×C₃²	1	1	yes	characteristic
133	#107	C₃²×C₄	1	1	yes	characteristic
134	C₄⋊C₈	C₂²×C₃²	1	1	yes	characteristic
135	C₄⋊C₈	C₂²×C₃²	1	1	yes	characteristic
136	#102	C₃²×C₄	1	1	yes	characteristic
137	#102	C₃²×C₄	1	1	yes	characteristic
138	C₂²⋊C₈	—	2	1	no	—
139	C₂²⋊C₈	—	2	1	no	—
140	C₄×C₈	C₂²×C₃²	1	1	yes	characteristic
141	C₄×C₈	C₂²×C₃²	1	1	yes	characteristic
142	C₄×C₈	C₂²×C₃²	1	1	yes	characteristic
143	C₂³×C₃	—	2	4	no	—
144	C₂³×C₃	—	2	4	no	—
145	C₃×D₄	—	2	4	no	—
146	C₃×D₄	—	2	4	no	—
147	C₃×D₄	—	2	4	no	—
148	C₃×D₄	C₃×M₁₆	1	8	yes	complemented
149	C₂×C₃×C₄	—	2	4	no	—
150	C₂×C₃×C₄	—	2	4	no	—
151	C₂×C₃×C₄	—	4	4	no	—
152	C₂×C₃×C₄	—	4	4	no	—
153	C₂×C₃×C₄	—	4	4	no	—
154	C₂×C₃×C₄	—	4	4	no	—
155	C₂×C₃×C₄	C₂²×C₃×C₄	1	4	yes	—
156	C₂×C₃×C₄	C₂×C₃×D₄	1	4	yes	—
157	C₂×C₃×C₄	C₃×#41	1	4	yes	—
158	C₃×C₈	—	2	4	no	—
159	C₃×C₈	—	2	4	no	—
160	C₃×C₈	—	2	4	no	—
161	C₃×C₈	—	4	4	no	—
162	C₃×C₈	—	4	4	no	—
163	C₃×C₈	—	4	4	no	—
164	C₃×C₈	C₃×D₈	1	8	yes	complemented
165	C₂×C₃²	—	4	2	no	—
166	C₂×C₃²	—	8	1	no	—
167	C₂×C₃²	#435	1	1	yes	characteristic
168	C₂×C₃²	C₄×D₈	1	1	yes	characteristic
169	C₂×C₃²	#444	1	1	yes	characteristic
170	C₂×D₄	C₂×C₃²×C₄	1	1	yes	characteristic
171	C₂×D₄	C₂×C₃²×C₄	1	1	yes	characteristic
172	C₂²×C₄	—	2	1	no	—
173	C₂²×C₄	—	2	1	no	—
174	D₈	—	2	2	no	—
175	M₁₆	—	4	2	no	—
176	C₂×C₈	—	2	1	no	—
177	C₂×C₈	—	2	1	no	—
178	C₂×C₈	—	2	1	no	—
179	C₂×C₈	—	4	2	no	—
180	C₂×C₈	C₂×C₃²×C₄	1	1	yes	characteristic
181	C₂×C₈	C₂×C₃²×C₄	1	1	yes	characteristic
182	C₂×C₈	C₃²×D₄	1	1	yes	characteristic
183	C₂×C₈	C₃²×D₄	1	1	yes	characteristic
184	C₄⋊C₄	C₂×C₃²×C₄	1	1	yes	characteristic
185	C₄⋊C₄	C₂×C₃²×C₄	1	1	yes	characteristic
186	C₂²⋊C₄	—	2	1	no	—
187	C₂²⋊C₄	—	2	1	no	—
188	C₄²	C₂³×C₃²	1	1	yes	characteristic
189	C₂²×C₃	—	2	8	no	—
190	C₂²×C₃	—	4	4	no	—
191	C₂²×C₃	—	4	4	no	—
192	C₂²×C₃	—	4	4	no	—
193	C₂²×C₃	—	4	4	no	—
194	C₂²×C₃	—	4	4	no	—
195	C₂²×C₃	C₃×C₄×D₄	1	4	yes	—
196	C₃×C₄	—	2	4	no	—
197	C₃×C₄	—	4	4	no	—
198	C₃×C₄	—	4	4	no	—
199	C₃×C₄	C₂×C₃×M₁₆	1	4	yes	—
200	C₃×C₄	C₃×#131	1	4	yes	—
201	C₃×C₄	C₂×C₃×D₈	1	4	yes	—
202	C₃×C₄	C₃×#135	1	4	yes	—
203	C₃²	#1581	1	1	yes	characteristic, direct product factor, Sylow
204	C₂³	—	2	1	no	—
205	C₂³	—	2	1	no	—
206	D₄	—	2	1	no	—
207	D₄	—	2	1	no	—
208	D₄	—	2	1	no	—
209	D₄	C₃²×M₁₆	1	2	yes	complemented
210	C₂×C₄	—	2	1	no	—
211	C₂×C₄	—	2	1	no	—
212	C₂×C₄	—	4	1	no	—
213	C₂×C₄	—	4	1	no	—
214	C₂×C₄	—	4	1	no	—
215	C₂×C₄	—	4	1	no	—
216	C₂×C₄	C₂²×C₃²×C₄	1	1	yes	characteristic
217	C₂×C₄	C₂×C₃²×D₄	1	1	yes	characteristic
218	C₂×C₄	C₃²×#41	1	1	yes	characteristic
219	C₈	—	2	1	no	—
220	C₈	—	2	1	no	—
221	C₈	—	2	1	no	—
222	C₈	—	4	1	no	—
223	C₈	—	4	1	no	—
224	C₈	—	4	1	no	—
225	C₈	C₃²×D₈	1	2	yes	complemented
226	C₂×C₃	—	4	8	no	—
227	C₂×C₃	—	8	4	no	—
228	C₂×C₃	C₃×#435	1	4	yes	—
229	C₂×C₃	C₃×C₄×D₈	1	4	yes	—
230	C₂×C₃	C₃×#444	1	4	yes	—
231	C₂²	—	2	2	no	—
232	C₂²	—	4	1	no	—
233	C₂²	—	4	1	no	—
234	C₂²	—	4	1	no	—
235	C₂²	—	4	1	no	—
236	C₂²	—	4	1	no	—
237	C₂²	C₃²×C₄×D₄	1	1	yes	characteristic
238	C₄	—	2	1	no	—
239	C₄	—	4	1	no	—
240	C₄	—	4	1	no	—
241	C₄	C₂×C₃²×M₁₆	1	1	yes	characteristic
242	C₄	C₃²×#131	1	1	yes	characteristic
243	C₄	C₂×C₃²×D₈	1	1	yes	characteristic
244	C₄	C₃²×#135	1	1	yes	characteristic
245	C₃	C₃×#1581	1	4	yes	direct product factor
246	C₂	—	4	2	no	—
247	C₂	—	8	1	no	—
248	C₂	C₃²×#435	1	1	yes	characteristic
249	C₂	C₃²×C₄×D₈	1	1	yes	characteristic
250	C₂	C₃²×#444	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃²×C₄	C₂×D₄
commutator	C₂×C₄	C₂²×C₃²×C₄
Frattini, cosocle	C₄²	C₂³×C₃²
socle	C₂²×C₃²	C₄×D₄
prime radical	C₂×C₃²×D₈	C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃²×C₄ ◁ C₃²×C₄×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃²×#436	C₂	semidirect
3. C₃²×C₈×D₄	C₂	semidirect
4. C₃²×#335	C₂	semidirect
5. C₃²×D₄⋊C₈	C₂	semidirect
6. C₃²×D₄⋊C₈	C₂	semidirect
7. C₃²×#322	C₂	semidirect
8. C₃×#1581	C₃	direct (N × H)
25. C₃×#436	C₂×C₃	semidirect
26. C₃×C₈×D₄	C₂×C₃	semidirect
27. C₃×#335	C₂×C₃	semidirect
28. C₃×D₄⋊C₈	C₂×C₃	semidirect
29. C₃×D₄⋊C₈	C₂×C₃	semidirect
30. C₃×#322	C₂×C₃	semidirect
50. #1581	C₃²	direct (N × H)
71. C₃²×D₄	M₁₆	semidirect
87. C₃²×C₈	D₈	semidirect
89. #436	C₂×C₃²	semidirect
90. C₈×D₄	C₂×C₃²	semidirect
91. #335	C₂×C₃²	semidirect
92. D₄⋊C₈	C₂×C₃²	semidirect
93. D₄⋊C₈	C₂×C₃²	semidirect
94. #322	C₂×C₃²	semidirect
148. C₃×D₄	C₃×M₁₆	semidirect
164. C₃×C₈	C₃×D₈	semidirect
203. C₃²	#1581	direct (N × H)
209. D₄	C₃²×M₁₆	semidirect
225. C₈	C₃²×D₈	semidirect
245. C₃	C₃×#1581	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃²×C₄×D₈	2
C₃²×#436	2
C₃²×C₈×D₄	2
C₃²×#335	2
C₃²×D₄⋊C₈	2
C₃²×#322	2
C₃×#1581	3

Subgroups — C32×#1581 #11768630

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×#1581 #11768630