Subgroups — C₃²×#2168 #11769217

Order 1152.

492 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₃²×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃²×C₈×D₄	C₂	1	1	yes	maximal, characteristic
4	C₃²×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃²×C₂²⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃²×C₂²⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃²×#346	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#2168	C₃	1	4	yes	maximal, direct product factor
9	C₂²×C₃²×C₈	C₂²	1	1	yes	characteristic
10	C₂²×C₃²×C₈	C₂²	1	1	yes	characteristic
11	C₃²×C₄×D₄	C₄	1	1	yes	characteristic
12	C₃²×M₃₂	—	2	2	no	—
13	C₂×C₃²×C₁₆	—	2	2	no	—
14	C₂×C₃²×C₁₆	C₂²	1	1	yes	characteristic
15	C₂×C₃²×C₁₆	C₂²	1	1	yes	characteristic
16	C₂×C₃²×C₁₆	C₂²	1	1	yes	characteristic
17	C₂×C₃²×C₁₆	C₂²	1	1	yes	characteristic
18	C₃²×C₄⋊C₈	C₄	1	1	yes	characteristic
19	C₃²×C₂²⋊C₈	C₄	1	1	yes	characteristic
20	C₃²×C₂²⋊C₈	C₄	1	1	yes	characteristic
21	C₃²×C₄×C₈	C₂²	1	1	yes	characteristic
22	C₂×C₃×M₃₂	C₂×C₃	1	4	yes	complemented
23	C₂²×C₃×C₁₆	C₂×C₃	1	4	yes	complemented
24	C₃×C₈×D₄	C₂×C₃	1	4	yes	—
25	C₃×C₄⋊C₁₆	C₂×C₃	1	4	yes	complemented
26	C₃×C₂²⋊C₁₆	C₂×C₃	1	4	yes	complemented
27	C₃×C₂²⋊C₁₆	C₂×C₃	1	4	yes	complemented
28	C₃×#346	C₂×C₃	1	4	yes	complemented
29	C₂×C₃²×D₄	C₈	1	1	yes	characteristic
30	C₂²×C₃²×C₄	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₃²×C₄	C₂×C₄	1	1	yes	characteristic
32	C₂×C₃²×C₈	—	2	1	no	—
33	C₂×C₃²×C₈	—	2	1	no	—
34	C₂×C₃²×C₈	—	2	1	no	—
35	C₂×C₃²×C₈	—	2	1	no	—
36	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
37	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
38	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
39	C₂×C₃²×C₈	C₂³	1	1	yes	characteristic
40	C₃²×C₄⋊C₄	C₈	1	1	yes	characteristic
41	C₃²×C₂²⋊C₄	C₈	1	1	yes	characteristic
42	C₃²×C₂²⋊C₄	C₈	1	1	yes	characteristic
43	C₃²×C₄²	C₂×C₄	1	1	yes	characteristic
44	C₃²×C₁₆	—	2	1	no	—
45	C₃²×C₁₆	—	2	1	no	—
46	C₃²×C₁₆	—	2	1	no	—
47	C₃²×C₁₆	D₄	1	2	yes	complemented
48	#2168	C₃²	1	1	yes	characteristic, direct product factor, Sylow
49	C₂²×C₃×C₈	C₂²×C₃	1	4	yes	—
50	C₂²×C₃×C₈	C₂²×C₃	1	4	yes	—
51	C₃×C₄×D₄	C₃×C₄	1	4	yes	—
52	C₃×M₃₂	—	2	8	no	—
53	C₂×C₃×C₁₆	—	2	8	no	—
54	C₂×C₃×C₁₆	C₂²×C₃	1	4	yes	—
55	C₂×C₃×C₁₆	C₂²×C₃	1	4	yes	—
56	C₂×C₃×C₁₆	C₂²×C₃	1	4	yes	—
57	C₂×C₃×C₁₆	C₂²×C₃	1	4	yes	—
58	C₃×C₄⋊C₈	C₃×C₄	1	4	yes	—
59	C₃×C₂²⋊C₈	C₃×C₄	1	4	yes	—
60	C₃×C₂²⋊C₈	C₃×C₄	1	4	yes	—
61	C₃×C₄×C₈	C₂²×C₃	1	4	yes	—
62	C₂³×C₃²	C₂×C₈	1	1	yes	characteristic
63	C₂³×C₃²	C₂×C₈	1	1	yes	characteristic
64	C₃²×D₄	—	2	2	no	—
65	C₂×C₃²×C₄	—	2	1	no	—
66	C₂×C₃²×C₄	—	2	1	no	—
67	C₂×C₃²×C₄	—	2	1	no	—
68	C₂×C₃²×C₄	—	2	1	no	—
69	C₂×C₃²×C₄	C₂×C₈	1	1	yes	characteristic
70	C₂×C₃²×C₄	C₂×C₈	1	1	yes	characteristic
71	C₂×C₃²×C₄	C₂×C₈	1	1	yes	characteristic
72	C₂×C₃²×C₄	C₂×C₈	1	1	yes	characteristic
73	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
74	C₃²×C₈	—	2	1	no	—
75	C₃²×C₈	—	2	1	no	—
76	C₃²×C₈	—	2	1	no	—
77	C₃²×C₈	C₂×D₄	1	1	yes	characteristic
78	C₃²×C₈	#41	1	1	yes	characteristic
79	C₂×M₃₂	C₂×C₃²	1	1	yes	characteristic, complemented
80	C₂²×C₁₆	C₂×C₃²	1	1	yes	characteristic, complemented
81	C₈×D₄	C₂×C₃²	1	1	yes	characteristic
82	C₄⋊C₁₆	C₂×C₃²	1	1	yes	characteristic, complemented
83	C₂²⋊C₁₆	C₂×C₃²	1	1	yes	characteristic, complemented
84	C₂²⋊C₁₆	C₂×C₃²	1	1	yes	characteristic, complemented
85	#346	C₂×C₃²	1	1	yes	characteristic, complemented
86	C₂×C₃×D₄	C₃×C₈	1	4	yes	—
87	C₂²×C₃×C₄	C₂×C₃×C₄	1	4	yes	—
88	C₂²×C₃×C₄	C₂×C₃×C₄	1	4	yes	—
89	C₂×C₃×C₈	—	2	4	no	—
90	C₂×C₃×C₈	—	2	4	no	—
91	C₂×C₃×C₈	—	2	4	no	—
92	C₂×C₃×C₈	—	2	4	no	—
93	C₂×C₃×C₈	C₂×C₃×C₄	1	4	yes	—
94	C₂×C₃×C₈	C₂×C₃×C₄	1	4	yes	—
95	C₂×C₃×C₈	C₂×C₃×C₄	1	4	yes	—
96	C₂×C₃×C₈	C₂³×C₃	1	4	yes	—
97	C₃×C₄⋊C₄	C₃×C₈	1	4	yes	—
98	C₃×C₂²⋊C₄	C₃×C₈	1	4	yes	—
99	C₃×C₂²⋊C₄	C₃×C₈	1	4	yes	—
100	C₃×C₄²	C₂×C₃×C₄	1	4	yes	—
101	C₃×C₁₆	—	2	4	no	—
102	C₃×C₁₆	—	2	4	no	—
103	C₃×C₁₆	—	2	4	no	—
104	C₃×C₁₆	C₃×D₄	1	8	yes	complemented
105	C₂²×C₃²	—	2	1	no	—
106	C₂²×C₃²	—	2	1	no	—
107	C₂²×C₃²	—	2	1	no	—
108	C₂²×C₃²	—	2	1	no	—
109	C₂²×C₃²	—	2	1	no	—
110	C₂²×C₃²	M₃₂	1	2	yes	complemented
111	C₂²×C₃²	C₂²×C₈	1	1	yes	characteristic
112	C₃²×C₄	—	2	1	no	—
113	C₃²×C₄	—	2	1	no	—
114	C₃²×C₄	—	2	1	no	—
115	C₃²×C₄	—	2	1	no	—
116	C₃²×C₄	C₄×D₄	1	1	yes	characteristic
117	C₃²×C₄	#131	1	1	yes	characteristic
118	C₂²×C₈	C₂²×C₃²	1	1	yes	characteristic
119	C₂²×C₈	C₂²×C₃²	1	1	yes	characteristic
120	C₄×D₄	C₃²×C₄	1	1	yes	characteristic
121	M₃₂	—	2	2	no	—
122	C₂×C₁₆	—	2	2	no	—
123	C₂×C₁₆	C₂²×C₃²	1	1	yes	characteristic
124	C₂×C₁₆	C₂²×C₃²	1	1	yes	characteristic
125	C₂×C₁₆	C₂²×C₃²	1	1	yes	characteristic
126	C₂×C₁₆	C₂²×C₃²	1	1	yes	characteristic
127	C₄⋊C₈	C₃²×C₄	1	1	yes	characteristic
128	C₂²⋊C₈	C₃²×C₄	1	1	yes	characteristic
129	C₂²⋊C₈	C₃²×C₄	1	1	yes	characteristic
130	C₄×C₈	C₂²×C₃²	1	1	yes	characteristic
131	C₂³×C₃	C₂×C₃×C₈	1	4	yes	—
132	C₂³×C₃	C₂×C₃×C₈	1	4	yes	—
133	C₃×D₄	—	2	8	no	—
134	C₂×C₃×C₄	—	2	4	no	—
135	C₂×C₃×C₄	—	2	4	no	—
136	C₂×C₃×C₄	—	2	4	no	—
137	C₂×C₃×C₄	—	2	4	no	—
138	C₂×C₃×C₄	C₂×C₃×C₈	1	4	yes	—
139	C₂×C₃×C₄	C₂×C₃×C₈	1	4	yes	—
140	C₂×C₃×C₄	C₂×C₃×C₈	1	4	yes	—
141	C₂×C₃×C₄	C₂×C₃×C₈	1	4	yes	—
142	C₂×C₃×C₄	C₂²×C₃×C₄	1	4	yes	—
143	C₃×C₈	—	2	4	no	—
144	C₃×C₈	—	2	4	no	—
145	C₃×C₈	—	2	4	no	—
146	C₃×C₈	C₂×C₃×D₄	1	4	yes	—
147	C₃×C₈	C₃×#41	1	4	yes	—
148	C₂×C₃²	—	2	2	no	—
149	C₂×C₃²	—	4	1	no	—
150	C₂×C₃²	C₈×D₄	1	1	yes	characteristic
151	C₂×C₃²	C₂×M₃₂	1	1	yes	characteristic
152	C₂×C₃²	#504	1	1	yes	characteristic
153	C₂×D₄	C₃²×C₈	1	1	yes	characteristic
154	C₂²×C₄	C₂×C₃²×C₄	1	1	yes	characteristic
155	C₂²×C₄	C₂×C₃²×C₄	1	1	yes	characteristic
156	C₂×C₈	—	2	1	no	—
157	C₂×C₈	—	2	1	no	—
158	C₂×C₈	—	2	1	no	—
159	C₂×C₈	—	2	1	no	—
160	C₂×C₈	C₂×C₃²×C₄	1	1	yes	characteristic
161	C₂×C₈	C₂×C₃²×C₄	1	1	yes	characteristic
162	C₂×C₈	C₂×C₃²×C₄	1	1	yes	characteristic
163	C₂×C₈	C₂³×C₃²	1	1	yes	characteristic
164	C₄⋊C₄	C₃²×C₈	1	1	yes	characteristic
165	C₂²⋊C₄	C₃²×C₈	1	1	yes	characteristic
166	C₂²⋊C₄	C₃²×C₈	1	1	yes	characteristic
167	C₄²	C₂×C₃²×C₄	1	1	yes	characteristic
168	C₁₆	—	2	1	no	—
169	C₁₆	—	2	1	no	—
170	C₁₆	—	2	1	no	—
171	C₁₆	C₃²×D₄	1	2	yes	complemented
172	C₂²×C₃	—	2	4	no	—
173	C₂²×C₃	—	2	4	no	—
174	C₂²×C₃	—	2	4	no	—
175	C₂²×C₃	—	2	4	no	—
176	C₂²×C₃	—	2	4	no	—
177	C₂²×C₃	C₃×M₃₂	1	8	yes	complemented
178	C₂²×C₃	C₂²×C₃×C₈	1	4	yes	—
179	C₃×C₄	—	2	4	no	—
180	C₃×C₄	—	2	4	no	—
181	C₃×C₄	—	2	4	no	—
182	C₃×C₄	—	2	4	no	—
183	C₃×C₄	C₃×C₄×D₄	1	4	yes	—
184	C₃×C₄	C₃×#131	1	4	yes	—
185	C₃²	#2168	1	1	yes	characteristic, direct product factor, Sylow
186	C₂³	C₂×C₃²×C₈	1	1	yes	characteristic
187	C₂³	C₂×C₃²×C₈	1	1	yes	characteristic
188	D₄	—	2	2	no	—
189	C₂×C₄	—	2	1	no	—
190	C₂×C₄	—	2	1	no	—
191	C₂×C₄	—	2	1	no	—
192	C₂×C₄	—	2	1	no	—
193	C₂×C₄	C₂×C₃²×C₈	1	1	yes	characteristic
194	C₂×C₄	C₂×C₃²×C₈	1	1	yes	characteristic
195	C₂×C₄	C₂×C₃²×C₈	1	1	yes	characteristic
196	C₂×C₄	C₂×C₃²×C₈	1	1	yes	characteristic
197	C₂×C₄	C₂²×C₃²×C₄	1	1	yes	characteristic
198	C₈	—	2	1	no	—
199	C₈	—	2	1	no	—
200	C₈	—	2	1	no	—
201	C₈	C₂×C₃²×D₄	1	1	yes	characteristic
202	C₈	C₃²×#41	1	1	yes	characteristic
203	C₂×C₃	—	2	8	no	—
204	C₂×C₃	—	4	4	no	—
205	C₂×C₃	C₃×C₈×D₄	1	4	yes	—
206	C₂×C₃	C₂×C₃×M₃₂	1	4	yes	—
207	C₂×C₃	C₃×#504	1	4	yes	—
208	C₂²	—	2	1	no	—
209	C₂²	—	2	1	no	—
210	C₂²	—	2	1	no	—
211	C₂²	—	2	1	no	—
212	C₂²	—	2	1	no	—
213	C₂²	C₃²×M₃₂	1	2	yes	complemented
214	C₂²	C₂²×C₃²×C₈	1	1	yes	characteristic
215	C₄	—	2	1	no	—
216	C₄	—	2	1	no	—
217	C₄	—	2	1	no	—
218	C₄	—	2	1	no	—
219	C₄	C₃²×C₄×D₄	1	1	yes	characteristic
220	C₄	C₃²×#131	1	1	yes	characteristic
221	C₃	C₃×#2168	1	4	yes	direct product factor
222	C₂	—	2	2	no	—
223	C₂	—	4	1	no	—
224	C₂	C₃²×C₈×D₄	1	1	yes	characteristic
225	C₂	C₂×C₃²×M₃₂	1	1	yes	characteristic
226	C₂	C₃²×#504	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
1. C₂×C₃²×M₃₂	C₂	semidirect
2. C₂²×C₃²×C₁₆	C₂	semidirect
4. C₃²×C₄⋊C₁₆	C₂	semidirect
5. C₃²×C₂²⋊C₁₆	C₂	semidirect
6. C₃²×C₂²⋊C₁₆	C₂	semidirect
7. C₃²×#346	C₂	semidirect
8. C₃×#2168	C₃	direct (N × H)
22. C₂×C₃×M₃₂	C₂×C₃	semidirect
23. C₂²×C₃×C₁₆	C₂×C₃	semidirect
25. C₃×C₄⋊C₁₆	C₂×C₃	semidirect
26. C₃×C₂²⋊C₁₆	C₂×C₃	semidirect
27. C₃×C₂²⋊C₁₆	C₂×C₃	semidirect
28. C₃×#346	C₂×C₃	semidirect
47. C₃²×C₁₆	D₄	semidirect
48. #2168	C₃²	direct (N × H)
79. C₂×M₃₂	C₂×C₃²	semidirect
80. C₂²×C₁₆	C₂×C₃²	semidirect
82. C₄⋊C₁₆	C₂×C₃²	semidirect
83. C₂²⋊C₁₆	C₂×C₃²	semidirect
84. C₂²⋊C₁₆	C₂×C₃²	semidirect
85. #346	C₂×C₃²	semidirect
104. C₃×C₁₆	C₃×D₄	semidirect
110. C₂²×C₃²	M₃₂	semidirect
171. C₁₆	C₃²×D₄	semidirect
177. C₂²×C₃	C₃×M₃₂	semidirect
185. C₃²	#2168	direct (N × H)
213. C₂²	C₃²×M₃₂	semidirect
221. C₃	C₃×#2168	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃²×M₃₂	2
C₂²×C₃²×C₁₆	2
C₃²×C₈×D₄	2
C₃²×C₄⋊C₁₆	2
C₃²×C₂²⋊C₁₆	2
C₃²×#346	2
C₃×#2168	3

Subgroups — C32×#2168 #11769217

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×#2168 #11769217