Subgroups — C₃³×C₄²×C₃₂ #72192011

Order 13824.

9912 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃³×C₄²×C₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃³×C₄×C₃₂	C₂	1	6	yes	maximal
3	C₃²×C₄²×C₃₂	C₃	1	13	yes	maximal, direct product factor
4	C₃³×C₄²×C₈	C₄	1	1	yes	characteristic
5	C₂×C₃³×C₄×C₁₆	C₄	1	3	yes	—
6	C₂×C₃³×C₄×C₁₆	C₂²	1	3	yes	—
7	C₂²×C₃³×C₃₂	C₂²	1	4	yes	—
8	C₃³×C₄×C₃₂	C₄	1	24	yes	direct product factor
9	C₃²×C₄²×C₁₆	C₂×C₃	1	13	yes	—
10	C₂×C₃²×C₄×C₃₂	C₂×C₃	1	78	yes	—
11	C₃×C₄²×C₃₂	C₃²	1	13	yes	direct product factor
12	C₂²×C₃³×C₁₆	C₂×C₄	1	3	yes	—
13	C₂²×C₃³×C₁₆	C₂³	1	1	yes	characteristic
14	C₂×C₃³×C₄×C₈	C₈	1	3	yes	—
15	C₂×C₃³×C₄×C₈	C₂×C₄	1	3	yes	—
16	C₃³×C₄³	C₈	1	1	yes	characteristic
17	C₂×C₃³×C₃₂	C₂×C₄	1	24	yes	—
18	C₃³×C₄×C₁₆	C₈	1	12	yes	—
19	C₃³×C₄×C₁₆	C₂×C₄	1	12	yes	—
20	C₂²×C₃²×C₃₂	C₂²×C₃	1	52	yes	—
21	C₂×C₃²×C₄×C₁₆	C₃×C₄	1	39	yes	—
22	C₂×C₃²×C₄×C₁₆	C₂²×C₃	1	39	yes	—
23	C₃²×C₄²×C₈	C₃×C₄	1	13	yes	—
24	C₃²×C₄×C₃₂	C₃×C₄	1	312	yes	direct product factor
25	C₂²×C₃³×C₈	C₂×C₈	1	3	yes	—
26	C₂²×C₃³×C₈	C₂²×C₄	1	1	yes	characteristic
27	C₂×C₃³×C₄²	C₁₆	1	4	yes	—
28	C₂×C₃³×C₄²	C₂×C₈	1	3	yes	—
29	C₂×C₃³×C₁₆	C₄²	1	6	yes	—
30	C₂×C₃³×C₁₆	C₂×C₈	1	12	yes	—
31	C₂×C₃³×C₁₆	C₂²×C₄	1	6	yes	—
32	C₃³×C₄×C₈	C₁₆	1	12	yes	—
33	C₃³×C₄×C₈	C₄²	1	6	yes	—
34	C₃³×C₄×C₈	C₂×C₈	1	6	yes	—
35	C₃³×C₃₂	C₄²	1	16	yes	direct product factor
36	C₂×C₃×C₄×C₃₂	C₂×C₃²	1	78	yes	—
37	C₃×C₄²×C₁₆	C₂×C₃²	1	13	yes	—
38	C₂²×C₃²×C₁₆	C₂×C₃×C₄	1	39	yes	—
39	C₂²×C₃²×C₁₆	C₂³×C₃	1	13	yes	—
40	C₂×C₃²×C₄×C₈	C₃×C₈	1	39	yes	—
41	C₂×C₃²×C₄×C₈	C₂×C₃×C₄	1	39	yes	—
42	C₃²×C₄³	C₃×C₈	1	13	yes	—
43	C₂×C₃²×C₃₂	C₂×C₃×C₄	1	312	yes	—
44	C₃²×C₄×C₁₆	C₃×C₈	1	156	yes	—
45	C₃²×C₄×C₁₆	C₂×C₃×C₄	1	156	yes	—
46	C₄²×C₃₂	C₃³	1	1	yes	characteristic, direct product factor, Sylow
47	C₂²×C₃³×C₄	C₂×C₁₆	1	6	yes	—
48	C₂²×C₃³×C₄	C₂²×C₈	1	1	yes	characteristic
49	C₂×C₃³×C₈	C₄×C₈	1	6	yes	—
50	C₂×C₃³×C₈	C₂×C₁₆	1	12	yes	—
51	C₂×C₃³×C₈	C₂×C₄²	1	3	yes	—
52	C₂×C₃³×C₈	C₂²×C₈	1	3	yes	—
53	C₃³×C₄²	C₃₂	1	16	yes	direct product factor
54	C₃³×C₄²	C₄×C₈	1	6	yes	—
55	C₃³×C₄²	C₂×C₁₆	1	6	yes	—
56	C₃³×C₁₆	C₄×C₈	1	12	yes	—
57	C₃³×C₁₆	C₂×C₄²	1	4	yes	—
58	C₂²×C₃×C₃₂	C₂²×C₃²	1	52	yes	—
59	C₂×C₃×C₄×C₁₆	C₃²×C₄	1	39	yes	—
60	C₂×C₃×C₄×C₁₆	C₂²×C₃²	1	39	yes	—
61	C₃×C₄²×C₈	C₃²×C₄	1	13	yes	—
62	C₃×C₄×C₃₂	C₃²×C₄	1	312	yes	direct product factor
63	C₂²×C₃²×C₈	C₂×C₃×C₈	1	39	yes	—
64	C₂²×C₃²×C₈	C₂²×C₃×C₄	1	13	yes	—
65	C₂×C₃²×C₄²	C₃×C₁₆	1	52	yes	—
66	C₂×C₃²×C₄²	C₂×C₃×C₈	1	39	yes	—
67	C₂×C₃²×C₁₆	C₃×C₄²	1	78	yes	—
68	C₂×C₃²×C₁₆	C₂×C₃×C₈	1	156	yes	—
69	C₂×C₃²×C₁₆	C₂²×C₃×C₄	1	78	yes	—
70	C₃²×C₄×C₈	C₃×C₁₆	1	156	yes	—
71	C₃²×C₄×C₈	C₃×C₄²	1	78	yes	—
72	C₃²×C₄×C₈	C₂×C₃×C₈	1	78	yes	—
73	C₃²×C₃₂	C₃×C₄²	1	208	yes	direct product factor
74	C₂×C₄×C₃₂	C₂×C₃³	1	6	yes	—
75	C₄²×C₁₆	C₂×C₃³	1	1	yes	characteristic
76	C₂³×C₃³	C₂²×C₁₆	1	1	yes	characteristic
77	C₂×C₃³×C₄	C₄×C₁₆	1	12	yes	—
78	C₂×C₃³×C₄	C₂×C₃₂	1	24	yes	—
79	C₂×C₃³×C₄	C₂×C₄×C₈	1	3	yes	—
80	C₂×C₃³×C₄	C₂²×C₁₆	1	3	yes	—
81	C₃³×C₈	C₄×C₁₆	1	12	yes	—
82	C₃³×C₈	C₄³	1	1	yes	characteristic
83	C₃³×C₈	C₂×C₄×C₈	1	3	yes	—
84	C₂²×C₃×C₁₆	C₂×C₃²×C₄	1	39	yes	—
85	C₂²×C₃×C₁₆	C₂³×C₃²	1	13	yes	—
86	C₂×C₃×C₄×C₈	C₃²×C₈	1	39	yes	—
87	C₂×C₃×C₄×C₈	C₂×C₃²×C₄	1	39	yes	—
88	C₃×C₄³	C₃²×C₈	1	13	yes	—
89	C₂×C₃×C₃₂	C₂×C₃²×C₄	1	312	yes	—
90	C₃×C₄×C₁₆	C₃²×C₈	1	156	yes	—
91	C₃×C₄×C₁₆	C₂×C₃²×C₄	1	156	yes	—
92	C₂²×C₃²×C₄	C₂×C₃×C₁₆	1	78	yes	—
93	C₂²×C₃²×C₄	C₂²×C₃×C₈	1	13	yes	—
94	C₂×C₃²×C₈	C₃×C₄×C₈	1	78	yes	—
95	C₂×C₃²×C₈	C₂×C₃×C₁₆	1	156	yes	—
96	C₂×C₃²×C₈	C₂×C₃×C₄²	1	39	yes	—
97	C₂×C₃²×C₈	C₂²×C₃×C₈	1	39	yes	—
98	C₃²×C₄²	C₃×C₃₂	1	208	yes	direct product factor
99	C₃²×C₄²	C₃×C₄×C₈	1	78	yes	—
100	C₃²×C₄²	C₂×C₃×C₁₆	1	78	yes	—
101	C₃²×C₁₆	C₃×C₄×C₈	1	156	yes	—
102	C₃²×C₁₆	C₂×C₃×C₄²	1	52	yes	—
103	C₂²×C₃₂	C₂²×C₃³	1	4	yes	—
104	C₂×C₄×C₁₆	C₃³×C₄	1	3	yes	—
105	C₂×C₄×C₁₆	C₂²×C₃³	1	3	yes	—
106	C₄²×C₈	C₃³×C₄	1	1	yes	characteristic
107	C₄×C₃₂	C₃³×C₄	1	24	yes	direct product factor
108	C₂²×C₃³	C₂×C₄×C₁₆	1	3	yes	—
109	C₂²×C₃³	C₂²×C₃₂	1	4	yes	—
110	C₃³×C₄	C₄×C₃₂	1	24	yes	direct product factor
111	C₃³×C₄	C₄²×C₈	1	1	yes	characteristic
112	C₃³×C₄	C₂×C₄×C₁₆	1	3	yes	—
113	C₂²×C₃×C₈	C₂×C₃²×C₈	1	39	yes	—
114	C₂²×C₃×C₈	C₂²×C₃²×C₄	1	13	yes	—
115	C₂×C₃×C₄²	C₃²×C₁₆	1	52	yes	—
116	C₂×C₃×C₄²	C₂×C₃²×C₈	1	39	yes	—
117	C₂×C₃×C₁₆	C₃²×C₄²	1	78	yes	—
118	C₂×C₃×C₁₆	C₂×C₃²×C₈	1	156	yes	—
119	C₂×C₃×C₁₆	C₂²×C₃²×C₄	1	78	yes	—
120	C₃×C₄×C₈	C₃²×C₁₆	1	156	yes	—
121	C₃×C₄×C₈	C₃²×C₄²	1	78	yes	—
122	C₃×C₄×C₈	C₂×C₃²×C₈	1	78	yes	—
123	C₃×C₃₂	C₃²×C₄²	1	208	yes	direct product factor
124	C₂³×C₃²	C₂²×C₃×C₁₆	1	13	yes	—
125	C₂×C₃²×C₄	C₃×C₄×C₁₆	1	156	yes	—
126	C₂×C₃²×C₄	C₂×C₃×C₃₂	1	312	yes	—
127	C₂×C₃²×C₄	C₂×C₃×C₄×C₈	1	39	yes	—
128	C₂×C₃²×C₄	C₂²×C₃×C₁₆	1	39	yes	—
129	C₃²×C₈	C₃×C₄×C₁₆	1	156	yes	—
130	C₃²×C₈	C₃×C₄³	1	13	yes	—
131	C₃²×C₈	C₂×C₃×C₄×C₈	1	39	yes	—
132	C₂²×C₁₆	C₂×C₃³×C₄	1	3	yes	—
133	C₂²×C₁₆	C₂³×C₃³	1	1	yes	characteristic
134	C₂×C₄×C₈	C₃³×C₈	1	3	yes	—
135	C₂×C₄×C₈	C₂×C₃³×C₄	1	3	yes	—
136	C₄³	C₃³×C₈	1	1	yes	characteristic
137	C₂×C₃₂	C₂×C₃³×C₄	1	24	yes	—
138	C₄×C₁₆	C₃³×C₈	1	12	yes	—
139	C₄×C₁₆	C₂×C₃³×C₄	1	12	yes	—
140	C₂×C₃³	C₄²×C₁₆	1	1	yes	characteristic
141	C₂×C₃³	C₂×C₄×C₃₂	1	6	yes	—
142	C₂²×C₃×C₄	C₂×C₃²×C₁₆	1	78	yes	—
143	C₂²×C₃×C₄	C₂²×C₃²×C₈	1	13	yes	—
144	C₂×C₃×C₈	C₃²×C₄×C₈	1	78	yes	—
145	C₂×C₃×C₈	C₂×C₃²×C₁₆	1	156	yes	—
146	C₂×C₃×C₈	C₂×C₃²×C₄²	1	39	yes	—
147	C₂×C₃×C₈	C₂²×C₃²×C₈	1	39	yes	—
148	C₃×C₄²	C₃²×C₃₂	1	208	yes	direct product factor
149	C₃×C₄²	C₃²×C₄×C₈	1	78	yes	—
150	C₃×C₄²	C₂×C₃²×C₁₆	1	78	yes	—
151	C₃×C₁₆	C₃²×C₄×C₈	1	156	yes	—
152	C₃×C₁₆	C₂×C₃²×C₄²	1	52	yes	—
153	C₂²×C₃²	C₂×C₃×C₄×C₁₆	1	39	yes	—
154	C₂²×C₃²	C₂²×C₃×C₃₂	1	52	yes	—
155	C₃²×C₄	C₃×C₄×C₃₂	1	312	yes	direct product factor
156	C₃²×C₄	C₃×C₄²×C₈	1	13	yes	—
157	C₃²×C₄	C₂×C₃×C₄×C₁₆	1	39	yes	—
158	C₂²×C₈	C₂×C₃³×C₈	1	3	yes	—
159	C₂²×C₈	C₂²×C₃³×C₄	1	1	yes	characteristic
160	C₂×C₄²	C₃³×C₁₆	1	4	yes	—
161	C₂×C₄²	C₂×C₃³×C₈	1	3	yes	—
162	C₂×C₁₆	C₃³×C₄²	1	6	yes	—
163	C₂×C₁₆	C₂×C₃³×C₈	1	12	yes	—
164	C₂×C₁₆	C₂²×C₃³×C₄	1	6	yes	—
165	C₄×C₈	C₃³×C₁₆	1	12	yes	—
166	C₄×C₈	C₃³×C₄²	1	6	yes	—
167	C₄×C₈	C₂×C₃³×C₈	1	6	yes	—
168	C₃₂	C₃³×C₄²	1	16	yes	direct product factor
169	C₃³	C₄²×C₃₂	1	1	yes	characteristic, direct product factor, Sylow
170	C₂³×C₃	C₂²×C₃²×C₁₆	1	13	yes	—
171	C₂×C₃×C₄	C₃²×C₄×C₁₆	1	156	yes	—
172	C₂×C₃×C₄	C₂×C₃²×C₃₂	1	312	yes	—
173	C₂×C₃×C₄	C₂×C₃²×C₄×C₈	1	39	yes	—
174	C₂×C₃×C₄	C₂²×C₃²×C₁₆	1	39	yes	—
175	C₃×C₈	C₃²×C₄×C₁₆	1	156	yes	—
176	C₃×C₈	C₃²×C₄³	1	13	yes	—
177	C₃×C₈	C₂×C₃²×C₄×C₈	1	39	yes	—
178	C₂×C₃²	C₃×C₄²×C₁₆	1	13	yes	—
179	C₂×C₃²	C₂×C₃×C₄×C₃₂	1	78	yes	—
180	C₂²×C₄	C₂×C₃³×C₁₆	1	6	yes	—
181	C₂²×C₄	C₂²×C₃³×C₈	1	1	yes	characteristic
182	C₂×C₈	C₃³×C₄×C₈	1	6	yes	—
183	C₂×C₈	C₂×C₃³×C₁₆	1	12	yes	—
184	C₂×C₈	C₂×C₃³×C₄²	1	3	yes	—
185	C₂×C₈	C₂²×C₃³×C₈	1	3	yes	—
186	C₄²	C₃³×C₃₂	1	16	yes	direct product factor
187	C₄²	C₃³×C₄×C₈	1	6	yes	—
188	C₄²	C₂×C₃³×C₁₆	1	6	yes	—
189	C₁₆	C₃³×C₄×C₈	1	12	yes	—
190	C₁₆	C₂×C₃³×C₄²	1	4	yes	—
191	C₂²×C₃	C₂×C₃²×C₄×C₁₆	1	39	yes	—
192	C₂²×C₃	C₂²×C₃²×C₃₂	1	52	yes	—
193	C₃×C₄	C₃²×C₄×C₃₂	1	312	yes	direct product factor
194	C₃×C₄	C₃²×C₄²×C₈	1	13	yes	—
195	C₃×C₄	C₂×C₃²×C₄×C₁₆	1	39	yes	—
196	C₃²	C₃×C₄²×C₃₂	1	13	yes	direct product factor
197	C₂³	C₂²×C₃³×C₁₆	1	1	yes	characteristic
198	C₂×C₄	C₃³×C₄×C₁₆	1	12	yes	—
199	C₂×C₄	C₂×C₃³×C₃₂	1	24	yes	—
200	C₂×C₄	C₂×C₃³×C₄×C₈	1	3	yes	—
201	C₂×C₄	C₂²×C₃³×C₁₆	1	3	yes	—
202	C₈	C₃³×C₄×C₁₆	1	12	yes	—
203	C₈	C₃³×C₄³	1	1	yes	characteristic
204	C₈	C₂×C₃³×C₄×C₈	1	3	yes	—
205	C₂×C₃	C₂×C₃²×C₄×C₃₂	1	78	yes	—
206	C₂×C₃	C₃²×C₄²×C₁₆	1	13	yes	—
207	C₂²	C₂²×C₃³×C₃₂	1	4	yes	—
208	C₂²	C₂×C₃³×C₄×C₁₆	1	3	yes	—
209	C₄	C₃³×C₄×C₃₂	1	24	yes	direct product factor
210	C₄	C₂×C₃³×C₄×C₁₆	1	3	yes	—
211	C₄	C₃³×C₄²×C₈	1	1	yes	characteristic
212	C₃	C₃²×C₄²×C₃₂	1	13	yes	direct product factor
213	C₂	C₂×C₃³×C₄×C₃₂	1	6	yes	—
214	C₂	C₃³×C₄²×C₁₆	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂²×C₁₆	C₂³×C₃³
socle, prime radical	C₂³×C₃³	C₂²×C₁₆
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₃²×C₄²×C₃₂	C₃	direct (N × H)
8. C₃³×C₄×C₃₂	C₄	direct (N × H)
11. C₃×C₄²×C₃₂	C₃²	direct (N × H)
24. C₃²×C₄×C₃₂	C₃×C₄	direct (N × H)
35. C₃³×C₃₂	C₄²	direct (N × H)
46. C₄²×C₃₂	C₃³	direct (N × H)
53. C₃³×C₄²	C₃₂	direct (N × H)
62. C₃×C₄×C₃₂	C₃²×C₄	direct (N × H)
73. C₃²×C₃₂	C₃×C₄²	direct (N × H)
98. C₃²×C₄²	C₃×C₃₂	direct (N × H)
107. C₄×C₃₂	C₃³×C₄	direct (N × H)
110. C₃³×C₄	C₄×C₃₂	direct (N × H)
123. C₃×C₃₂	C₃²×C₄²	direct (N × H)
148. C₃×C₄²	C₃²×C₃₂	direct (N × H)
155. C₃²×C₄	C₃×C₄×C₃₂	direct (N × H)
168. C₃₂	C₃³×C₄²	direct (N × H)
169. C₃³	C₄²×C₃₂	direct (N × H)
186. C₄²	C₃³×C₃₂	direct (N × H)
193. C₃×C₄	C₃²×C₄×C₃₂	direct (N × H)
196. C₃²	C₃×C₄²×C₃₂	direct (N × H)
209. C₄	C₃³×C₄×C₃₂	direct (N × H)
212. C₃	C₃²×C₄²×C₃₂	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃³×C₄²×C₁₆	2
C₂×C₃³×C₄×C₃₂	2
C₃²×C₄²×C₃₂	3

Subgroups — C33×C42×C32 #72192011

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃³×C₄²×C₃₂ #72192011