Subgroups — C₂²×C₃×#399 #10653147

Order 768.

5838 conjugacy classes of subgroups.

The subgroup lattice has 180 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×#399	C₂	1	24	yes	maximal, direct product factor
4	C₂²×C₃×#95	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×#399	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×#399	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×#399	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	24	no	—
25	C₂³×C₃×C₄	—	2	12	no	—
26	C₂³×C₃×C₄	—	2	24	no	—
27	C₂³×C₃×C₄	D₄	1	24	yes	—
28	C₂³×C₃×C₄	C₂³	1	18	yes	—
29	C₂³×C₃×C₄	C₂³	1	18	yes	—
30	C₂×C₃×C₄⋊C₄	—	2	144	no	—
31	C₂×C₃×C₄⋊C₄	C₂³	1	48	yes	complemented
32	C₂×C₃×C₂²⋊C₄	—	2	144	no	—
33	C₂×C₃×C₂²⋊C₄	C₂³	1	48	yes	—
34	C₃×#95	C₂³	1	16	yes	complemented
35	C₂⁶	C₂²×C₃	1	1	yes	characteristic
36	C₂⁴×C₄	C₂²×C₃	1	3	yes	—
37	C₂⁴×C₄	C₂²×C₃	1	3	yes	—
38	C₂²×C₄⋊C₄	—	2	24	no	—
39	C₂²×C₄⋊C₄	C₂²×C₃	1	36	yes	complemented
40	C₂²×C₂²⋊C₄	—	2	24	no	—
41	C₂²×C₂²⋊C₄	C₂²×C₃	1	36	yes	—
42	#399	C₂²×C₃	1	64	yes	direct product factor
43	C₂×#95	C₂²×C₃	1	12	yes	complemented
44	C₂⁴×C₃	—	2	96	no	—
45	C₂⁴×C₃	—	2	36	no	—
46	C₂⁴×C₃	—	2	36	no	—
47	C₂⁴×C₃	—	4	48	no	—
48	C₂⁴×C₃	—	4	16	no	—
49	C₂⁴×C₃	C₂×D₄	1	12	yes	—
50	C₂⁴×C₃	C₂×Q₈	1	24	yes	—
51	C₂⁴×C₃	C₂×Q₈	1	4	yes	—
52	C₂⁴×C₃	#41	1	12	yes	—
53	C₂⁴×C₃	C₂⁴	1	4	yes	—
54	C₂⁴×C₃	C₂⁴	1	3	yes	—
55	C₂²×C₃×C₄	—	2	96	no	—
56	C₂²×C₃×C₄	—	2	72	no	—
57	C₂²×C₃×C₄	—	2	36	no	—
58	C₂²×C₃×C₄	—	2	72	no	—
59	C₂²×C₃×C₄	—	4	48	no	—
60	C₂²×C₃×C₄	C₂×D₄	1	72	yes	—
61	C₂²×C₃×C₄	C₂⁴	1	12	yes	—
62	C₂²×C₃×C₄	C₂⁴	1	12	yes	—
63	C₃×C₄⋊C₄	—	2	96	no	—
64	C₃×C₂²⋊C₄	—	2	96	no	—
65	C₂⁵	—	2	12	no	—
66	C₂⁵	—	2	12	no	—
67	C₂⁵	C₃×Q₈	1	8	yes	—
68	C₂⁵	C₂³×C₃	1	6	yes	—
69	C₂⁵	C₂³×C₃	1	1	yes	characteristic
70	C₂³×C₄	—	2	24	no	—
71	C₂³×C₄	—	2	12	no	—
72	C₂³×C₄	—	2	24	no	—
73	C₂³×C₄	C₃×D₄	1	24	yes	—
74	C₂³×C₄	C₂³×C₃	1	18	yes	—
75	C₂³×C₄	C₂³×C₃	1	18	yes	—
76	C₂×C₄⋊C₄	—	2	144	no	—
77	C₂×C₄⋊C₄	C₂³×C₃	1	48	yes	complemented
78	C₂×C₂²⋊C₄	—	2	144	no	—
79	C₂×C₂²⋊C₄	C₂³×C₃	1	48	yes	—
80	#95	C₂³×C₃	1	16	yes	complemented
81	C₂³×C₃	—	2	144	no	—
82	C₂³×C₃	—	2	12	no	—
83	C₂³×C₃	—	2	12	no	—
84	C₂³×C₃	—	4	72	no	—
85	C₂³×C₃	—	4	24	no	—
86	C₂³×C₃	—	4	128	no	—
87	C₂³×C₃	C₂²⋊Q₈	1	48	yes	—
88	C₂³×C₃	#123	1	48	yes	—
89	C₂³×C₃	#127	1	16	yes	—
90	C₂³×C₃	C₂²×D₄	1	18	yes	—
91	C₂³×C₃	C₂²×Q₈	1	8	yes	—
92	C₂³×C₃	C₂²×Q₈	1	6	yes	—
93	C₂³×C₃	C₂×#41	1	18	yes	—
94	C₂³×C₃	C₂⁵	1	1	yes	characteristic
95	C₂×C₃×C₄	—	2	144	no	—
96	C₂×C₃×C₄	—	2	24	no	—
97	C₂×C₃×C₄	—	2	12	no	—
98	C₂×C₃×C₄	—	2	24	no	—
99	C₂×C₃×C₄	—	4	72	no	—
100	C₂×C₃×C₄	C₂²×D₄	1	24	yes	—
101	C₂⁴	—	2	96	no	—
102	C₂⁴	—	2	36	no	—
103	C₂⁴	—	2	36	no	—
104	C₂⁴	—	4	48	no	—
105	C₂⁴	—	4	16	no	—
106	C₂⁴	C₂×C₃×D₄	1	12	yes	—
107	C₂⁴	C₂×C₃×Q₈	1	24	yes	—
108	C₂⁴	C₂×C₃×Q₈	1	4	yes	—
109	C₂⁴	C₃×#41	1	12	yes	—
110	C₂⁴	C₂⁴×C₃	1	4	yes	—
111	C₂⁴	C₂⁴×C₃	1	3	yes	—
112	C₂²×C₄	—	2	96	no	—
113	C₂²×C₄	—	2	72	no	—
114	C₂²×C₄	—	2	36	no	—
115	C₂²×C₄	—	2	72	no	—
116	C₂²×C₄	—	4	48	no	—
117	C₂²×C₄	C₂×C₃×D₄	1	72	yes	—
118	C₂²×C₄	C₂⁴×C₃	1	12	yes	—
119	C₂²×C₄	C₂⁴×C₃	1	12	yes	—
120	C₄⋊C₄	—	2	96	no	—
121	C₂²⋊C₄	—	2	96	no	—
122	C₂²×C₃	—	2	24	no	—
123	C₂²×C₃	—	4	96	no	—
124	C₂²×C₃	—	4	12	no	—
125	C₂²×C₃	—	4	4	no	—
126	C₂²×C₃	#399	1	64	yes	direct product factor
127	C₂²×C₃	C₂×C₂²⋊Q₈	1	36	yes	—
128	C₂²×C₃	C₂×#123	1	36	yes	—
129	C₂²×C₃	C₂×#127	1	12	yes	—
130	C₂²×C₃	C₂³×D₄	1	3	yes	—
131	C₂²×C₃	C₂³×Q₈	1	1	yes	characteristic
132	C₂²×C₃	C₂²×#41	1	3	yes	—
133	C₃×C₄	—	2	24	no	—
134	C₃×C₄	—	4	12	no	—
135	C₂³	—	2	12	no	—
136	C₂³	—	2	12	no	—
137	C₂³	—	2	144	no	—
138	C₂³	—	4	128	no	—
139	C₂³	—	4	72	no	—
140	C₂³	—	4	24	no	—
141	C₂³	C₃×C₂²⋊Q₈	1	48	yes	—
142	C₂³	C₃×#123	1	48	yes	—
143	C₂³	C₃×#127	1	16	yes	—
144	C₂³	C₂²×C₃×D₄	1	18	yes	—
145	C₂³	C₂²×C₃×Q₈	1	8	yes	—
146	C₂³	C₂²×C₃×Q₈	1	6	yes	—
147	C₂³	C₂×C₃×#41	1	18	yes	—
148	C₂³	C₂⁵×C₃	1	1	yes	characteristic
149	C₂×C₄	—	2	24	no	—
150	C₂×C₄	—	2	144	no	—
151	C₂×C₄	—	2	12	no	—
152	C₂×C₄	—	2	24	no	—
153	C₂×C₄	—	4	72	no	—
154	C₂×C₄	C₂²×C₃×D₄	1	24	yes	—
155	C₂×C₃	—	4	8	no	—
156	C₂×C₃	C₂×#399	1	24	yes	direct product factor
157	C₂×C₃	C₂²×C₂²⋊Q₈	1	3	yes	—
158	C₂×C₃	C₂²×#123	1	3	yes	—
159	C₂×C₃	C₂²×#127	1	1	yes	characteristic
160	C₂²	—	2	24	no	—
161	C₂²	—	4	12	no	—
162	C₂²	—	4	4	no	—
163	C₂²	—	4	96	no	—
164	C₂²	C₃×#399	1	64	yes	direct product factor
165	C₂²	C₂×C₃×C₂²⋊Q₈	1	36	yes	—
166	C₂²	C₂×C₃×#123	1	36	yes	—
167	C₂²	C₂×C₃×#127	1	12	yes	—
168	C₂²	C₂³×C₃×D₄	1	3	yes	—
169	C₂²	C₂³×C₃×Q₈	1	1	yes	characteristic
170	C₂²	C₂²×C₃×#41	1	3	yes	—
171	C₄	—	2	24	no	—
172	C₄	—	4	12	no	—
173	C₃	C₂²×#399	1	1	yes	characteristic, direct product factor, Sylow
174	C₂	—	4	8	no	—
175	C₂	C₂×C₃×#399	1	24	yes	direct product factor
176	C₂	C₂²×C₃×C₂²⋊Q₈	1	3	yes	—
177	C₂	C₂²×C₃×#123	1	3	yes	—
178	C₂	C₂²×C₃×#127	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×#399	C₂	direct (N × H)
4. C₂²×C₃×#95	C₂	semidirect
5. C₂²×#399	C₃	direct (N × H)
10. C₂²×C₃×C₄⋊C₄	C₂²	semidirect
13. C₃×#399	C₂²	direct (N × H)
14. C₂×C₃×#95	C₂²	semidirect
15. C₂³×C₄⋊C₄	C₂×C₃	semidirect
17. C₂×#399	C₂×C₃	direct (N × H)
18. C₂²×#95	C₂×C₃	semidirect
31. C₂×C₃×C₄⋊C₄	C₂³	semidirect
34. C₃×#95	C₂³	semidirect
39. C₂²×C₄⋊C₄	C₂²×C₃	semidirect
42. #399	C₂²×C₃	direct (N × H)
43. C₂×#95	C₂²×C₃	semidirect
77. C₂×C₄⋊C₄	C₂³×C₃	semidirect
80. #95	C₂³×C₃	semidirect
126. C₂²×C₃	#399	direct (N × H)
156. C₂×C₃	C₂×#399	direct (N × H)
164. C₂²	C₃×#399	direct (N × H)
173. C₃	C₂²×#399	direct (N × H)
175. C₂	C₂×C₃×#399	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×#399	2
C₂²×C₃×#95	2
C₂²×#399	3

Subgroups — C22×C3×#399 #10653147

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃×#399 #10653147