Subgroups — C₂³×C₃³×C₈² #72192044

Order 13824.

116536 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	C₂³×C₃³×C₄×C₈	C₂	1	3	yes	maximal
2	C₂²×C₃³×C₈²	C₂	1	28	yes	maximal, direct product factor
3	C₂³×C₃²×C₈²	C₃	1	13	yes	maximal, direct product factor
4	C₂³×C₃³×C₄²	C₂²	1	1	yes	characteristic
5	C₂⁴×C₃³×C₈	C₄	1	6	yes	—
6	C₂²×C₃³×C₄×C₈	C₄	1	42	yes	—
7	C₂²×C₃³×C₄×C₈	C₂²	1	42	yes	—
8	C₂×C₃³×C₈²	C₂²	1	112	yes	direct product factor
9	C₂³×C₃²×C₄×C₈	C₂×C₃	1	39	yes	—
10	C₂²×C₃²×C₈²	C₂×C₃	1	364	yes	direct product factor
11	C₂³×C₃×C₈²	C₃²	1	13	yes	direct product factor
12	C₂⁴×C₃³×C₄	C₂×C₄	1	3	yes	—
13	C₂³×C₃³×C₈	C₈	1	96	yes	direct product factor
14	C₂³×C₃³×C₈	C₂×C₄	1	84	yes	—
15	C₂²×C₃³×C₄²	C₂×C₄	1	21	yes	—
16	C₂²×C₃³×C₄²	C₂³	1	7	yes	—
17	C₂×C₃³×C₄×C₈	C₂×C₄	1	252	yes	—
18	C₂×C₃³×C₄×C₈	C₂³	1	84	yes	—
19	C₃³×C₈²	C₂³	1	64	yes	direct product factor
20	C₂⁴×C₃²×C₈	C₃×C₄	1	78	yes	—
21	C₂³×C₃²×C₄²	C₂²×C₃	1	13	yes	—
22	C₂²×C₃²×C₄×C₈	C₃×C₄	1	546	yes	—
23	C₂²×C₃²×C₄×C₈	C₂²×C₃	1	546	yes	—
24	C₂×C₃²×C₈²	C₂²×C₃	1	1456	yes	direct product factor
25	C₂⁵×C₃³	C₄²	1	1	yes	characteristic
26	C₂³×C₃³×C₄	C₄²	1	21	yes	—
27	C₂³×C₃³×C₄	C₂×C₈	1	48	yes	—
28	C₂³×C₃³×C₄	C₂²×C₄	1	21	yes	—
29	C₂²×C₃³×C₈	C₂×C₈	1	672	yes	direct product factor
30	C₂²×C₃³×C₈	C₂²×C₄	1	168	yes	—
31	C₂×C₃³×C₄²	C₄²	1	42	yes	—
32	C₂×C₃³×C₄²	C₂²×C₄	1	63	yes	—
33	C₂×C₃³×C₄²	C₂⁴	1	7	yes	—
34	C₃³×C₄×C₈	C₂²×C₄	1	168	yes	—
35	C₃³×C₄×C₈	C₂⁴	1	24	yes	—
36	C₂³×C₃×C₄×C₈	C₂×C₃²	1	39	yes	—
37	C₂²×C₃×C₈²	C₂×C₃²	1	364	yes	direct product factor
38	C₂⁴×C₃²×C₄	C₂×C₃×C₄	1	39	yes	—
39	C₂³×C₃²×C₈	C₃×C₈	1	1248	yes	direct product factor
40	C₂³×C₃²×C₈	C₂×C₃×C₄	1	1092	yes	—
41	C₂²×C₃²×C₄²	C₂×C₃×C₄	1	273	yes	—
42	C₂²×C₃²×C₄²	C₂³×C₃	1	91	yes	—
43	C₂×C₃²×C₄×C₈	C₂×C₃×C₄	1	3276	yes	—
44	C₂×C₃²×C₄×C₈	C₂³×C₃	1	1092	yes	—
45	C₃²×C₈²	C₂³×C₃	1	832	yes	direct product factor
46	C₂³×C₈²	C₃³	1	1	yes	characteristic, direct product factor, Sylow
47	C₂⁴×C₃³	C₄×C₈	1	24	yes	—
48	C₂⁴×C₃³	C₂×C₄²	1	7	yes	—
49	C₂²×C₃³×C₄	C₄×C₈	1	168	yes	—
50	C₂²×C₃³×C₄	C₂×C₄²	1	63	yes	—
51	C₂²×C₃³×C₄	C₂²×C₈	1	168	yes	—
52	C₂²×C₃³×C₄	C₂³×C₄	1	21	yes	—
53	C₂×C₃³×C₈	C₂²×C₈	1	672	yes	direct product factor
54	C₂×C₃³×C₈	C₂³×C₄	1	48	yes	—
55	C₃³×C₄²	C₂×C₄²	1	42	yes	—
56	C₃³×C₄²	C₂³×C₄	1	21	yes	—
57	C₃³×C₄²	C₂⁵	1	1	yes	characteristic
58	C₂⁴×C₃×C₈	C₃²×C₄	1	78	yes	—
59	C₂³×C₃×C₄²	C₂²×C₃²	1	13	yes	—
60	C₂²×C₃×C₄×C₈	C₃²×C₄	1	546	yes	—
61	C₂²×C₃×C₄×C₈	C₂²×C₃²	1	546	yes	—
62	C₂×C₃×C₈²	C₂²×C₃²	1	1456	yes	direct product factor
63	C₂⁵×C₃²	C₃×C₄²	1	13	yes	—
64	C₂³×C₃²×C₄	C₃×C₄²	1	273	yes	—
65	C₂³×C₃²×C₄	C₂×C₃×C₈	1	624	yes	—
66	C₂³×C₃²×C₄	C₂²×C₃×C₄	1	273	yes	—
67	C₂²×C₃²×C₈	C₂×C₃×C₈	1	8736	yes	direct product factor
68	C₂²×C₃²×C₈	C₂²×C₃×C₄	1	2184	yes	—
69	C₂×C₃²×C₄²	C₃×C₄²	1	546	yes	—
70	C₂×C₃²×C₄²	C₂²×C₃×C₄	1	819	yes	—
71	C₂×C₃²×C₄²	C₂⁴×C₃	1	91	yes	—
72	C₃²×C₄×C₈	C₂²×C₃×C₄	1	2184	yes	—
73	C₃²×C₄×C₈	C₂⁴×C₃	1	312	yes	—
74	C₂³×C₄×C₈	C₂×C₃³	1	3	yes	—
75	C₂²×C₈²	C₂×C₃³	1	28	yes	direct product factor
76	C₂³×C₃³	C₈²	1	64	yes	direct product factor
77	C₂³×C₃³	C₂×C₄×C₈	1	84	yes	—
78	C₂³×C₃³	C₂²×C₄²	1	7	yes	—
79	C₂×C₃³×C₄	C₂×C₄×C₈	1	252	yes	—
80	C₂×C₃³×C₄	C₂²×C₄²	1	21	yes	—
81	C₂×C₃³×C₄	C₂³×C₈	1	84	yes	—
82	C₂×C₃³×C₄	C₂⁴×C₄	1	3	yes	—
83	C₃³×C₈	C₂³×C₈	1	96	yes	direct product factor
84	C₂⁴×C₃×C₄	C₂×C₃²×C₄	1	39	yes	—
85	C₂³×C₃×C₈	C₃²×C₈	1	1248	yes	direct product factor
86	C₂³×C₃×C₈	C₂×C₃²×C₄	1	1092	yes	—
87	C₂²×C₃×C₄²	C₂×C₃²×C₄	1	273	yes	—
88	C₂²×C₃×C₄²	C₂³×C₃²	1	91	yes	—
89	C₂×C₃×C₄×C₈	C₂×C₃²×C₄	1	3276	yes	—
90	C₂×C₃×C₄×C₈	C₂³×C₃²	1	1092	yes	—
91	C₃×C₈²	C₂³×C₃²	1	832	yes	direct product factor
92	C₂⁴×C₃²	C₃×C₄×C₈	1	312	yes	—
93	C₂⁴×C₃²	C₂×C₃×C₄²	1	91	yes	—
94	C₂²×C₃²×C₄	C₃×C₄×C₈	1	2184	yes	—
95	C₂²×C₃²×C₄	C₂×C₃×C₄²	1	819	yes	—
96	C₂²×C₃²×C₄	C₂²×C₃×C₈	1	2184	yes	—
97	C₂²×C₃²×C₄	C₂³×C₃×C₄	1	273	yes	—
98	C₂×C₃²×C₈	C₂²×C₃×C₈	1	8736	yes	direct product factor
99	C₂×C₃²×C₈	C₂³×C₃×C₄	1	624	yes	—
100	C₃²×C₄²	C₂×C₃×C₄²	1	546	yes	—
101	C₃²×C₄²	C₂³×C₃×C₄	1	273	yes	—
102	C₃²×C₄²	C₂⁵×C₃	1	13	yes	—
103	C₂⁴×C₈	C₃³×C₄	1	6	yes	—
104	C₂³×C₄²	C₂²×C₃³	1	1	yes	characteristic
105	C₂²×C₄×C₈	C₃³×C₄	1	42	yes	—
106	C₂²×C₄×C₈	C₂²×C₃³	1	42	yes	—
107	C₂×C₈²	C₂²×C₃³	1	112	yes	direct product factor
108	C₂²×C₃³	C₂×C₈²	1	112	yes	direct product factor
109	C₂²×C₃³	C₂²×C₄×C₈	1	42	yes	—
110	C₂²×C₃³	C₂³×C₄²	1	1	yes	characteristic
111	C₃³×C₄	C₂²×C₄×C₈	1	42	yes	—
112	C₃³×C₄	C₂⁴×C₈	1	6	yes	—
113	C₂⁵×C₃	C₃²×C₄²	1	13	yes	—
114	C₂³×C₃×C₄	C₃²×C₄²	1	273	yes	—
115	C₂³×C₃×C₄	C₂×C₃²×C₈	1	624	yes	—
116	C₂³×C₃×C₄	C₂²×C₃²×C₄	1	273	yes	—
117	C₂²×C₃×C₈	C₂×C₃²×C₈	1	8736	yes	direct product factor
118	C₂²×C₃×C₈	C₂²×C₃²×C₄	1	2184	yes	—
119	C₂×C₃×C₄²	C₃²×C₄²	1	546	yes	—
120	C₂×C₃×C₄²	C₂²×C₃²×C₄	1	819	yes	—
121	C₂×C₃×C₄²	C₂⁴×C₃²	1	91	yes	—
122	C₃×C₄×C₈	C₂²×C₃²×C₄	1	2184	yes	—
123	C₃×C₄×C₈	C₂⁴×C₃²	1	312	yes	—
124	C₂³×C₃²	C₃×C₈²	1	832	yes	direct product factor
125	C₂³×C₃²	C₂×C₃×C₄×C₈	1	1092	yes	—
126	C₂³×C₃²	C₂²×C₃×C₄²	1	91	yes	—
127	C₂×C₃²×C₄	C₂×C₃×C₄×C₈	1	3276	yes	—
128	C₂×C₃²×C₄	C₂²×C₃×C₄²	1	273	yes	—
129	C₂×C₃²×C₄	C₂³×C₃×C₈	1	1092	yes	—
130	C₂×C₃²×C₄	C₂⁴×C₃×C₄	1	39	yes	—
131	C₃²×C₈	C₂³×C₃×C₈	1	1248	yes	direct product factor
132	C₂⁴×C₄	C₂×C₃³×C₄	1	3	yes	—
133	C₂³×C₈	C₃³×C₈	1	96	yes	direct product factor
134	C₂³×C₈	C₂×C₃³×C₄	1	84	yes	—
135	C₂²×C₄²	C₂×C₃³×C₄	1	21	yes	—
136	C₂²×C₄²	C₂³×C₃³	1	7	yes	—
137	C₂×C₄×C₈	C₂×C₃³×C₄	1	252	yes	—
138	C₂×C₄×C₈	C₂³×C₃³	1	84	yes	—
139	C₈²	C₂³×C₃³	1	64	yes	direct product factor
140	C₂×C₃³	C₂²×C₈²	1	28	yes	direct product factor
141	C₂×C₃³	C₂³×C₄×C₈	1	3	yes	—
142	C₂⁴×C₃	C₃²×C₄×C₈	1	312	yes	—
143	C₂⁴×C₃	C₂×C₃²×C₄²	1	91	yes	—
144	C₂²×C₃×C₄	C₃²×C₄×C₈	1	2184	yes	—
145	C₂²×C₃×C₄	C₂×C₃²×C₄²	1	819	yes	—
146	C₂²×C₃×C₄	C₂²×C₃²×C₈	1	2184	yes	—
147	C₂²×C₃×C₄	C₂³×C₃²×C₄	1	273	yes	—
148	C₂×C₃×C₈	C₂²×C₃²×C₈	1	8736	yes	direct product factor
149	C₂×C₃×C₈	C₂³×C₃²×C₄	1	624	yes	—
150	C₃×C₄²	C₂×C₃²×C₄²	1	546	yes	—
151	C₃×C₄²	C₂³×C₃²×C₄	1	273	yes	—
152	C₃×C₄²	C₂⁵×C₃²	1	13	yes	—
153	C₂²×C₃²	C₂×C₃×C₈²	1	1456	yes	direct product factor
154	C₂²×C₃²	C₂²×C₃×C₄×C₈	1	546	yes	—
155	C₂²×C₃²	C₂³×C₃×C₄²	1	13	yes	—
156	C₃²×C₄	C₂²×C₃×C₄×C₈	1	546	yes	—
157	C₃²×C₄	C₂⁴×C₃×C₈	1	78	yes	—
158	C₂⁵	C₃³×C₄²	1	1	yes	characteristic
159	C₂³×C₄	C₃³×C₄²	1	21	yes	—
160	C₂³×C₄	C₂×C₃³×C₈	1	48	yes	—
161	C₂³×C₄	C₂²×C₃³×C₄	1	21	yes	—
162	C₂²×C₈	C₂×C₃³×C₈	1	672	yes	direct product factor
163	C₂²×C₈	C₂²×C₃³×C₄	1	168	yes	—
164	C₂×C₄²	C₃³×C₄²	1	42	yes	—
165	C₂×C₄²	C₂²×C₃³×C₄	1	63	yes	—
166	C₂×C₄²	C₂⁴×C₃³	1	7	yes	—
167	C₄×C₈	C₂²×C₃³×C₄	1	168	yes	—
168	C₄×C₈	C₂⁴×C₃³	1	24	yes	—
169	C₃³	C₂³×C₈²	1	1	yes	characteristic, direct product factor, Sylow
170	C₂³×C₃	C₃²×C₈²	1	832	yes	direct product factor
171	C₂³×C₃	C₂×C₃²×C₄×C₈	1	1092	yes	—
172	C₂³×C₃	C₂²×C₃²×C₄²	1	91	yes	—
173	C₂×C₃×C₄	C₂×C₃²×C₄×C₈	1	3276	yes	—
174	C₂×C₃×C₄	C₂²×C₃²×C₄²	1	273	yes	—
175	C₂×C₃×C₄	C₂³×C₃²×C₈	1	1092	yes	—
176	C₂×C₃×C₄	C₂⁴×C₃²×C₄	1	39	yes	—
177	C₃×C₈	C₂³×C₃²×C₈	1	1248	yes	direct product factor
178	C₂×C₃²	C₂²×C₃×C₈²	1	364	yes	direct product factor
179	C₂×C₃²	C₂³×C₃×C₄×C₈	1	39	yes	—
180	C₂⁴	C₃³×C₄×C₈	1	24	yes	—
181	C₂⁴	C₂×C₃³×C₄²	1	7	yes	—
182	C₂²×C₄	C₃³×C₄×C₈	1	168	yes	—
183	C₂²×C₄	C₂×C₃³×C₄²	1	63	yes	—
184	C₂²×C₄	C₂²×C₃³×C₈	1	168	yes	—
185	C₂²×C₄	C₂³×C₃³×C₄	1	21	yes	—
186	C₂×C₈	C₂²×C₃³×C₈	1	672	yes	direct product factor
187	C₂×C₈	C₂³×C₃³×C₄	1	48	yes	—
188	C₄²	C₂×C₃³×C₄²	1	42	yes	—
189	C₄²	C₂³×C₃³×C₄	1	21	yes	—
190	C₄²	C₂⁵×C₃³	1	1	yes	characteristic
191	C₂²×C₃	C₂×C₃²×C₈²	1	1456	yes	direct product factor
192	C₂²×C₃	C₂²×C₃²×C₄×C₈	1	546	yes	—
193	C₂²×C₃	C₂³×C₃²×C₄²	1	13	yes	—
194	C₃×C₄	C₂²×C₃²×C₄×C₈	1	546	yes	—
195	C₃×C₄	C₂⁴×C₃²×C₈	1	78	yes	—
196	C₃²	C₂³×C₃×C₈²	1	13	yes	direct product factor
197	C₂³	C₃³×C₈²	1	64	yes	direct product factor
198	C₂³	C₂×C₃³×C₄×C₈	1	84	yes	—
199	C₂³	C₂²×C₃³×C₄²	1	7	yes	—
200	C₂×C₄	C₂×C₃³×C₄×C₈	1	252	yes	—
201	C₂×C₄	C₂²×C₃³×C₄²	1	21	yes	—
202	C₂×C₄	C₂³×C₃³×C₈	1	84	yes	—
203	C₂×C₄	C₂⁴×C₃³×C₄	1	3	yes	—
204	C₈	C₂³×C₃³×C₈	1	96	yes	direct product factor
205	C₂×C₃	C₂²×C₃²×C₈²	1	364	yes	direct product factor
206	C₂×C₃	C₂³×C₃²×C₄×C₈	1	39	yes	—
207	C₂²	C₂×C₃³×C₈²	1	112	yes	direct product factor
208	C₂²	C₂²×C₃³×C₄×C₈	1	42	yes	—
209	C₂²	C₂³×C₃³×C₄²	1	1	yes	characteristic
210	C₄	C₂²×C₃³×C₄×C₈	1	42	yes	—
211	C₄	C₂⁴×C₃³×C₈	1	6	yes	—
212	C₃	C₂³×C₃²×C₈²	1	13	yes	direct product factor
213	C₂	C₂²×C₃³×C₈²	1	28	yes	direct product factor
214	C₂	C₂³×C₃³×C₄×C₈	1	3	yes	—

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₄²	C₂⁵×C₃³
socle, prime radical	C₂⁵×C₃³	C₄²
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×C₃³×C₈²	C₂	direct (N × H)
3. C₂³×C₃²×C₈²	C₃	direct (N × H)
8. C₂×C₃³×C₈²	C₂²	direct (N × H)
10. C₂²×C₃²×C₈²	C₂×C₃	direct (N × H)
11. C₂³×C₃×C₈²	C₃²	direct (N × H)
13. C₂³×C₃³×C₈	C₈	direct (N × H)
19. C₃³×C₈²	C₂³	direct (N × H)
24. C₂×C₃²×C₈²	C₂²×C₃	direct (N × H)
29. C₂²×C₃³×C₈	C₂×C₈	direct (N × H)
37. C₂²×C₃×C₈²	C₂×C₃²	direct (N × H)
39. C₂³×C₃²×C₈	C₃×C₈	direct (N × H)
45. C₃²×C₈²	C₂³×C₃	direct (N × H)
46. C₂³×C₈²	C₃³	direct (N × H)
53. C₂×C₃³×C₈	C₂²×C₈	direct (N × H)
62. C₂×C₃×C₈²	C₂²×C₃²	direct (N × H)
67. C₂²×C₃²×C₈	C₂×C₃×C₈	direct (N × H)
75. C₂²×C₈²	C₂×C₃³	direct (N × H)
76. C₂³×C₃³	C₈²	direct (N × H)
83. C₃³×C₈	C₂³×C₈	direct (N × H)
85. C₂³×C₃×C₈	C₃²×C₈	direct (N × H)
91. C₃×C₈²	C₂³×C₃²	direct (N × H)
98. C₂×C₃²×C₈	C₂²×C₃×C₈	direct (N × H)
107. C₂×C₈²	C₂²×C₃³	direct (N × H)
108. C₂²×C₃³	C₂×C₈²	direct (N × H)
117. C₂²×C₃×C₈	C₂×C₃²×C₈	direct (N × H)
124. C₂³×C₃²	C₃×C₈²	direct (N × H)
131. C₃²×C₈	C₂³×C₃×C₈	direct (N × H)
133. C₂³×C₈	C₃³×C₈	direct (N × H)
139. C₈²	C₂³×C₃³	direct (N × H)
140. C₂×C₃³	C₂²×C₈²	direct (N × H)
148. C₂×C₃×C₈	C₂²×C₃²×C₈	direct (N × H)
153. C₂²×C₃²	C₂×C₃×C₈²	direct (N × H)
162. C₂²×C₈	C₂×C₃³×C₈	direct (N × H)
169. C₃³	C₂³×C₈²	direct (N × H)
170. C₂³×C₃	C₃²×C₈²	direct (N × H)
177. C₃×C₈	C₂³×C₃²×C₈	direct (N × H)
178. C₂×C₃²	C₂²×C₃×C₈²	direct (N × H)
186. C₂×C₈	C₂²×C₃³×C₈	direct (N × H)
191. C₂²×C₃	C₂×C₃²×C₈²	direct (N × H)
196. C₃²	C₂³×C₃×C₈²	direct (N × H)
197. C₂³	C₃³×C₈²	direct (N × H)
204. C₈	C₂³×C₃³×C₈	direct (N × H)
205. C₂×C₃	C₂²×C₃²×C₈²	direct (N × H)
207. C₂²	C₂×C₃³×C₈²	direct (N × H)
212. C₃	C₂³×C₃²×C₈²	direct (N × H)
213. C₂	C₂²×C₃³×C₈²	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₃³×C₄×C₈	2
C₂²×C₃³×C₈²	2
C₂³×C₃²×C₈²	3

Subgroups — C23×C33×C82 #72192044

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×C₃³×C₈² #72192044