Subgroups — C₂²×C₃×#396 #10653143

Order 768.

5838 conjugacy classes of subgroups.

The subgroup lattice has 192 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₃×C₄⋊C₄	C₂	1	3	yes	maximal, complemented
2	C₂³×C₃×C₂²⋊C₄	C₂	1	3	yes	maximal
3	C₂×C₃×#396	C₂	1	24	yes	maximal, direct product factor
4	C₂²×C₃×#95	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×#396	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
6	C₂⁶×C₃	C₂²	1	1	yes	characteristic
7	C₂⁴×C₃×C₄	C₂²	1	3	yes	—
8	C₂⁴×C₃×C₄	C₂²	1	3	yes	—
9	C₂²×C₃×C₄⋊C₄	—	2	24	no	—
10	C₂²×C₃×C₄⋊C₄	C₂²	1	36	yes	complemented
11	C₂²×C₃×C₂²⋊C₄	—	2	24	no	—
12	C₂²×C₃×C₂²⋊C₄	C₂²	1	36	yes	—
13	C₃×#396	C₂²	1	64	yes	direct product factor
14	C₂×C₃×#95	C₂²	1	12	yes	complemented
15	C₂³×C₄⋊C₄	C₂×C₃	1	3	yes	complemented
16	C₂³×C₂²⋊C₄	C₂×C₃	1	3	yes	—
17	C₂×#396	C₂×C₃	1	24	yes	direct product factor
18	C₂²×#95	C₂×C₃	1	1	yes	characteristic, complemented
19	C₂⁵×C₃	—	2	12	no	—
20	C₂⁵×C₃	—	2	12	no	—
21	C₂⁵×C₃	Q₈	1	8	yes	—
22	C₂⁵×C₃	C₂³	1	6	yes	—
23	C₂⁵×C₃	C₂³	1	1	yes	characteristic
24	C₂³×C₃×C₄	—	2	12	no	—
25	C₂³×C₃×C₄	—	2	12	no	—
26	C₂³×C₃×C₄	—	2	12	no	—
27	C₂³×C₃×C₄	—	2	12	no	—
28	C₂³×C₃×C₄	—	2	12	no	—
29	C₂³×C₃×C₄	D₄	1	24	yes	—
30	C₂³×C₃×C₄	C₂³	1	18	yes	—
31	C₂³×C₃×C₄	C₂³	1	18	yes	—
32	C₂×C₃×C₄⋊C₄	—	2	144	no	—
33	C₂×C₃×C₄⋊C₄	C₂³	1	48	yes	complemented
34	C₂×C₃×C₂²⋊C₄	—	2	144	no	—
35	C₂×C₃×C₂²⋊C₄	C₂³	1	48	yes	—
36	C₃×#95	C₂³	1	16	yes	complemented
37	C₂⁶	C₂²×C₃	1	1	yes	characteristic
38	C₂⁴×C₄	C₂²×C₃	1	3	yes	—
39	C₂⁴×C₄	C₂²×C₃	1	3	yes	—
40	C₂²×C₄⋊C₄	—	2	24	no	—
41	C₂²×C₄⋊C₄	C₂²×C₃	1	36	yes	complemented
42	C₂²×C₂²⋊C₄	—	2	24	no	—
43	C₂²×C₂²⋊C₄	C₂²×C₃	1	36	yes	—
44	#396	C₂²×C₃	1	64	yes	direct product factor
45	C₂×#95	C₂²×C₃	1	12	yes	complemented
46	C₂⁴×C₃	—	2	96	no	—
47	C₂⁴×C₃	—	2	36	no	—
48	C₂⁴×C₃	—	2	36	no	—
49	C₂⁴×C₃	—	4	48	no	—
50	C₂⁴×C₃	—	4	16	no	—
51	C₂⁴×C₃	C₂×D₄	1	12	yes	—
52	C₂⁴×C₃	C₂×Q₈	1	24	yes	—
53	C₂⁴×C₃	C₂×Q₈	1	4	yes	—
54	C₂⁴×C₃	#41	1	12	yes	—
55	C₂⁴×C₃	C₂⁴	1	4	yes	—
56	C₂⁴×C₃	C₂⁴	1	3	yes	—
57	C₂²×C₃×C₄	—	2	96	no	—
58	C₂²×C₃×C₄	—	2	36	no	—
59	C₂²×C₃×C₄	—	2	36	no	—
60	C₂²×C₃×C₄	—	2	36	no	—
61	C₂²×C₃×C₄	—	2	36	no	—
62	C₂²×C₃×C₄	—	2	36	no	—
63	C₂²×C₃×C₄	—	4	48	no	—
64	C₂²×C₃×C₄	C₂×D₄	1	72	yes	—
65	C₂²×C₃×C₄	C₂⁴	1	12	yes	—
66	C₂²×C₃×C₄	C₂⁴	1	12	yes	—
67	C₃×C₄⋊C₄	—	2	96	no	—
68	C₃×C₂²⋊C₄	—	2	96	no	—
69	C₂⁵	—	2	12	no	—
70	C₂⁵	—	2	12	no	—
71	C₂⁵	C₃×Q₈	1	8	yes	—
72	C₂⁵	C₂³×C₃	1	6	yes	—
73	C₂⁵	C₂³×C₃	1	1	yes	characteristic
74	C₂³×C₄	—	2	12	no	—
75	C₂³×C₄	—	2	12	no	—
76	C₂³×C₄	—	2	12	no	—
77	C₂³×C₄	—	2	12	no	—
78	C₂³×C₄	—	2	12	no	—
79	C₂³×C₄	C₃×D₄	1	24	yes	—
80	C₂³×C₄	C₂³×C₃	1	18	yes	—
81	C₂³×C₄	C₂³×C₃	1	18	yes	—
82	C₂×C₄⋊C₄	—	2	144	no	—
83	C₂×C₄⋊C₄	C₂³×C₃	1	48	yes	complemented
84	C₂×C₂²⋊C₄	—	2	144	no	—
85	C₂×C₂²⋊C₄	C₂³×C₃	1	48	yes	—
86	#95	C₂³×C₃	1	16	yes	complemented
87	C₂³×C₃	—	2	144	no	—
88	C₂³×C₃	—	2	12	no	—
89	C₂³×C₃	—	2	12	no	—
90	C₂³×C₃	—	4	72	no	—
91	C₂³×C₃	—	4	24	no	—
92	C₂³×C₃	—	4	128	no	—
93	C₂³×C₃	#121	1	48	yes	—
94	C₂³×C₃	C₂²⋊Q₈	1	48	yes	—
95	C₂³×C₃	#126	1	16	yes	—
96	C₂³×C₃	C₂²×D₄	1	18	yes	—
97	C₂³×C₃	C₂²×Q₈	1	8	yes	—
98	C₂³×C₃	C₂²×Q₈	1	6	yes	—
99	C₂³×C₃	C₂×#41	1	18	yes	—
100	C₂³×C₃	C₂⁵	1	1	yes	characteristic
101	C₂×C₃×C₄	—	2	144	no	—
102	C₂×C₃×C₄	—	2	12	no	—
103	C₂×C₃×C₄	—	2	12	no	—
104	C₂×C₃×C₄	—	2	12	no	—
105	C₂×C₃×C₄	—	2	12	no	—
106	C₂×C₃×C₄	—	2	12	no	—
107	C₂×C₃×C₄	—	4	72	no	—
108	C₂×C₃×C₄	C₂²×D₄	1	24	yes	—
109	C₂⁴	—	2	96	no	—
110	C₂⁴	—	2	36	no	—
111	C₂⁴	—	2	36	no	—
112	C₂⁴	—	4	48	no	—
113	C₂⁴	—	4	16	no	—
114	C₂⁴	C₂×C₃×D₄	1	12	yes	—
115	C₂⁴	C₂×C₃×Q₈	1	24	yes	—
116	C₂⁴	C₂×C₃×Q₈	1	4	yes	—
117	C₂⁴	C₃×#41	1	12	yes	—
118	C₂⁴	C₂⁴×C₃	1	4	yes	—
119	C₂⁴	C₂⁴×C₃	1	3	yes	—
120	C₂²×C₄	—	2	96	no	—
121	C₂²×C₄	—	2	36	no	—
122	C₂²×C₄	—	2	36	no	—
123	C₂²×C₄	—	2	36	no	—
124	C₂²×C₄	—	2	36	no	—
125	C₂²×C₄	—	2	36	no	—
126	C₂²×C₄	—	4	48	no	—
127	C₂²×C₄	C₂×C₃×D₄	1	72	yes	—
128	C₂²×C₄	C₂⁴×C₃	1	12	yes	—
129	C₂²×C₄	C₂⁴×C₃	1	12	yes	—
130	C₄⋊C₄	—	2	96	no	—
131	C₂²⋊C₄	—	2	96	no	—
132	C₂²×C₃	—	2	24	no	—
133	C₂²×C₃	—	4	96	no	—
134	C₂²×C₃	—	4	12	no	—
135	C₂²×C₃	—	4	4	no	—
136	C₂²×C₃	#396	1	64	yes	direct product factor
137	C₂²×C₃	C₂×#121	1	36	yes	—
138	C₂²×C₃	C₂×C₂²⋊Q₈	1	36	yes	—
139	C₂²×C₃	C₂×#126	1	12	yes	—
140	C₂²×C₃	C₂³×D₄	1	3	yes	—
141	C₂²×C₃	C₂³×Q₈	1	1	yes	characteristic
142	C₂²×C₃	C₂²×#41	1	3	yes	—
143	C₃×C₄	—	2	24	no	—
144	C₃×C₄	—	4	12	no	—
145	C₂³	—	2	12	no	—
146	C₂³	—	2	12	no	—
147	C₂³	—	2	144	no	—
148	C₂³	—	4	128	no	—
149	C₂³	—	4	72	no	—
150	C₂³	—	4	24	no	—
151	C₂³	C₃×#121	1	48	yes	—
152	C₂³	C₃×C₂²⋊Q₈	1	48	yes	—
153	C₂³	C₃×#126	1	16	yes	—
154	C₂³	C₂²×C₃×D₄	1	18	yes	—
155	C₂³	C₂²×C₃×Q₈	1	8	yes	—
156	C₂³	C₂²×C₃×Q₈	1	6	yes	—
157	C₂³	C₂×C₃×#41	1	18	yes	—
158	C₂³	C₂⁵×C₃	1	1	yes	characteristic
159	C₂×C₄	—	2	12	no	—
160	C₂×C₄	—	2	12	no	—
161	C₂×C₄	—	2	144	no	—
162	C₂×C₄	—	2	12	no	—
163	C₂×C₄	—	2	12	no	—
164	C₂×C₄	—	2	12	no	—
165	C₂×C₄	—	4	72	no	—
166	C₂×C₄	C₂²×C₃×D₄	1	24	yes	—
167	C₂×C₃	—	4	8	no	—
168	C₂×C₃	C₂×#396	1	24	yes	direct product factor
169	C₂×C₃	C₂²×#121	1	3	yes	—
170	C₂×C₃	C₂²×C₂²⋊Q₈	1	3	yes	—
171	C₂×C₃	C₂²×#126	1	1	yes	characteristic
172	C₂²	—	2	24	no	—
173	C₂²	—	4	12	no	—
174	C₂²	—	4	4	no	—
175	C₂²	—	4	96	no	—
176	C₂²	C₃×#396	1	64	yes	direct product factor
177	C₂²	C₂×C₃×#121	1	36	yes	—
178	C₂²	C₂×C₃×C₂²⋊Q₈	1	36	yes	—
179	C₂²	C₂×C₃×#126	1	12	yes	—
180	C₂²	C₂³×C₃×D₄	1	3	yes	—
181	C₂²	C₂³×C₃×Q₈	1	1	yes	characteristic
182	C₂²	C₂²×C₃×#41	1	3	yes	—
183	C₄	—	2	24	no	—
184	C₄	—	4	12	no	—
185	C₃	C₂²×#396	1	1	yes	characteristic, direct product factor, Sylow
186	C₂	—	4	8	no	—
187	C₂	C₂×C₃×#396	1	24	yes	direct product factor
188	C₂	C₂²×C₃×#121	1	3	yes	—
189	C₂	C₂²×C₃×C₂²⋊Q₈	1	3	yes	—
190	C₂	C₂²×C₃×#126	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂⁵×C₃	C₂³
commutator, Frattini, cosocle	C₂³	C₂⁵×C₃
prime radical	C₂⁶×C₃	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁵×C₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂³×C₃×C₄⋊C₄	C₂	semidirect
3. C₂×C₃×#396	C₂	direct (N × H)
4. C₂²×C₃×#95	C₂	semidirect
5. C₂²×#396	C₃	direct (N × H)
10. C₂²×C₃×C₄⋊C₄	C₂²	semidirect
13. C₃×#396	C₂²	direct (N × H)
14. C₂×C₃×#95	C₂²	semidirect
15. C₂³×C₄⋊C₄	C₂×C₃	semidirect
17. C₂×#396	C₂×C₃	direct (N × H)
18. C₂²×#95	C₂×C₃	semidirect
33. C₂×C₃×C₄⋊C₄	C₂³	semidirect
36. C₃×#95	C₂³	semidirect
41. C₂²×C₄⋊C₄	C₂²×C₃	semidirect
44. #396	C₂²×C₃	direct (N × H)
45. C₂×#95	C₂²×C₃	semidirect
83. C₂×C₄⋊C₄	C₂³×C₃	semidirect
86. #95	C₂³×C₃	semidirect
136. C₂²×C₃	#396	direct (N × H)
168. C₂×C₃	C₂×#396	direct (N × H)
176. C₂²	C₃×#396	direct (N × H)
185. C₃	C₂²×#396	direct (N × H)
187. C₂	C₂×C₃×#396	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×C₃×C₄⋊C₄	2
C₂³×C₃×C₂²⋊C₄	2
C₂×C₃×#396	2
C₂²×C₃×#95	2
C₂²×#396	3

Subgroups — C22×C3×#396 #10653143

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃×#396 #10653143