Subgroups — C₂²×C₃×#395 #10653141

Order 768.

6046 conjugacy classes of subgroups.

The subgroup lattice has 162 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×C₃×Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂³×C₃×C₄⋊C₄	C₂	1	3	yes	maximal
3	C₂×C₃×#395	C₂	1	24	yes	maximal, direct product factor
4	C₂²×C₃×#95	C₂	1	3	yes	maximal
5	C₂²×#395	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
6	C₂³×C₃×Q₈	—	2	24	no	—
7	C₂³×C₃×Q₈	C₂²	1	12	yes	—
8	C₂⁴×C₃×C₄	C₂²	1	1	yes	characteristic
9	C₂⁴×C₃×C₄	C₂²	1	3	yes	—
10	C₂⁴×C₃×C₄	C₂²	1	3	yes	—
11	C₂²×C₃×C₄⋊C₄	—	2	24	no	—
12	C₂²×C₃×C₄⋊C₄	C₂²	1	36	yes	—
13	C₃×#395	C₂²	1	64	yes	direct product factor
14	C₂×C₃×#95	C₂²	1	36	yes	—
15	C₂⁴×Q₈	C₂×C₃	1	1	yes	characteristic
16	C₂³×C₄⋊C₄	C₂×C₃	1	3	yes	—
17	C₂×#395	C₂×C₃	1	24	yes	direct product factor
18	C₂²×#95	C₂×C₃	1	3	yes	—
19	C₂⁵×C₃	C₂³	1	1	yes	characteristic
20	C₂²×C₃×Q₈	—	2	128	no	—
21	C₂²×C₃×Q₈	—	2	144	no	—
22	C₂²×C₃×Q₈	C₂³	1	16	yes	—
23	C₂³×C₃×C₄	—	2	24	no	—
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	D₄	1	24	yes	—
28	C₂³×C₃×C₄	Q₈	1	24	yes	complemented
29	C₂³×C₃×C₄	C₂³	1	6	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	—
34	C₃×#95	C₂³	1	48	yes	—
35	C₂³×Q₈	—	2	24	no	—
36	C₂³×Q₈	C₂²×C₃	1	12	yes	—
37	C₂⁴×C₄	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	—
42	#395	C₂²×C₃	1	64	yes	direct product factor
43	C₂×#95	C₂²×C₃	1	36	yes	—
44	C₂⁴×C₃	C₂×D₄	1	12	yes	—
45	C₂⁴×C₃	C₂×Q₈	1	12	yes	—
46	C₂⁴×C₃	#41	1	4	yes	—
47	C₂⁴×C₃	C₂⁴	1	3	yes	—
48	C₂×C₃×Q₈	—	2	384	no	—
49	C₂×C₃×Q₈	—	2	96	no	—
50	C₂²×C₃×C₄	—	2	96	no	—
51	C₂²×C₃×C₄	—	2	96	no	—
52	C₂²×C₃×C₄	—	2	36	no	—
53	C₂²×C₃×C₄	—	2	72	no	—
54	C₂²×C₃×C₄	—	2	36	no	—
55	C₂²×C₃×C₄	—	2	36	no	—
56	C₂²×C₃×C₄	—	4	16	no	—
57	C₂²×C₃×C₄	C₂×D₄	1	72	yes	—
58	C₂²×C₃×C₄	C₂×Q₈	1	72	yes	complemented
59	C₂²×C₃×C₄	C₂⁴	1	4	yes	—
60	C₂²×C₃×C₄	C₂⁴	1	12	yes	—
61	C₂²×C₃×C₄	C₂⁴	1	12	yes	—
62	C₃×C₄⋊C₄	—	2	96	no	—
63	C₂⁵	C₂³×C₃	1	1	yes	characteristic
64	C₂²×Q₈	—	2	128	no	—
65	C₂²×Q₈	—	2	144	no	—
66	C₂²×Q₈	C₂³×C₃	1	16	yes	—
67	C₂³×C₄	—	2	12	no	—
68	C₂³×C₄	—	2	24	no	—
69	C₂³×C₄	—	2	12	no	—
70	C₂³×C₄	—	2	12	no	—
71	C₂³×C₄	C₃×D₄	1	24	yes	—
72	C₂³×C₄	C₃×Q₈	1	24	yes	complemented
73	C₂³×C₄	C₂³×C₃	1	6	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	—
78	#95	C₂³×C₃	1	48	yes	—
79	C₂³×C₃	#120	1	16	yes	—
80	C₂³×C₃	C₂²⋊Q₈	1	48	yes	—
81	C₂³×C₃	C₄⋊Q₈	1	48	yes	—
82	C₂³×C₃	C₂²×D₄	1	18	yes	—
83	C₂³×C₃	C₂²×Q₈	1	18	yes	—
84	C₂³×C₃	C₂×#41	1	6	yes	—
85	C₂³×C₃	C₂⁵	1	1	yes	characteristic
86	C₃×Q₈	—	2	128	no	—
87	C₂×C₃×C₄	—	2	144	no	—
88	C₂×C₃×C₄	—	2	144	no	—
89	C₂×C₃×C₄	—	2	12	no	—
90	C₂×C₃×C₄	—	2	24	no	—
91	C₂×C₃×C₄	—	2	12	no	—
92	C₂×C₃×C₄	—	2	12	no	—
93	C₂×C₃×C₄	—	4	24	no	—
94	C₂×C₃×C₄	C₂²×D₄	1	24	yes	—
95	C₂×C₃×C₄	C₂²×Q₈	1	24	yes	complemented
96	C₂⁴	C₂×C₃×D₄	1	12	yes	—
97	C₂⁴	C₂×C₃×Q₈	1	12	yes	—
98	C₂⁴	C₃×#41	1	4	yes	—
99	C₂⁴	C₂⁴×C₃	1	3	yes	—
100	C₂×Q₈	—	2	384	no	—
101	C₂×Q₈	—	2	96	no	—
102	C₂²×C₄	—	2	96	no	—
103	C₂²×C₄	—	2	96	no	—
104	C₂²×C₄	—	2	36	no	—
105	C₂²×C₄	—	2	72	no	—
106	C₂²×C₄	—	2	36	no	—
107	C₂²×C₄	—	2	36	no	—
108	C₂²×C₄	—	4	16	no	—
109	C₂²×C₄	C₂×C₃×D₄	1	72	yes	—
110	C₂²×C₄	C₂×C₃×Q₈	1	72	yes	complemented
111	C₂²×C₄	C₂⁴×C₃	1	4	yes	—
112	C₂²×C₄	C₂⁴×C₃	1	12	yes	—
113	C₂²×C₄	C₂⁴×C₃	1	12	yes	—
114	C₄⋊C₄	—	2	96	no	—
115	C₂²×C₃	#395	1	64	yes	direct product factor
116	C₂²×C₃	C₂×#120	1	12	yes	—
117	C₂²×C₃	C₂×C₂²⋊Q₈	1	36	yes	—
118	C₂²×C₃	C₂×C₄⋊Q₈	1	36	yes	—
119	C₂²×C₃	C₂³×D₄	1	3	yes	—
120	C₂²×C₃	C₂³×Q₈	1	3	yes	—
121	C₂²×C₃	C₂²×#41	1	1	yes	characteristic
122	C₃×C₄	—	2	24	no	—
123	C₃×C₄	—	2	24	no	—
124	C₃×C₄	—	4	4	no	—
125	C₂³	C₃×#120	1	16	yes	—
126	C₂³	C₃×C₂²⋊Q₈	1	48	yes	—
127	C₂³	C₃×C₄⋊Q₈	1	48	yes	—
128	C₂³	C₂²×C₃×D₄	1	18	yes	—
129	C₂³	C₂²×C₃×Q₈	1	18	yes	—
130	C₂³	C₂×C₃×#41	1	6	yes	—
131	C₂³	C₂⁵×C₃	1	1	yes	characteristic
132	Q₈	—	2	128	no	—
133	C₂×C₄	—	2	12	no	—
134	C₂×C₄	—	2	24	no	—
135	C₂×C₄	—	2	12	no	—
136	C₂×C₄	—	2	144	no	—
137	C₂×C₄	—	2	144	no	—
138	C₂×C₄	—	2	12	no	—
139	C₂×C₄	—	4	24	no	—
140	C₂×C₄	C₂²×C₃×D₄	1	24	yes	—
141	C₂×C₄	C₂²×C₃×Q₈	1	24	yes	complemented
142	C₂×C₃	C₂×#395	1	24	yes	direct product factor
143	C₂×C₃	C₂²×#120	1	1	yes	characteristic
144	C₂×C₃	C₂²×C₂²⋊Q₈	1	3	yes	—
145	C₂×C₃	C₂²×C₄⋊Q₈	1	3	yes	—
146	C₂²	C₃×#395	1	64	yes	direct product factor
147	C₂²	C₂×C₃×#120	1	12	yes	—
148	C₂²	C₂×C₃×C₂²⋊Q₈	1	36	yes	—
149	C₂²	C₂×C₃×C₄⋊Q₈	1	36	yes	—
150	C₂²	C₂³×C₃×D₄	1	3	yes	—
151	C₂²	C₂³×C₃×Q₈	1	3	yes	—
152	C₂²	C₂²×C₃×#41	1	1	yes	characteristic
153	C₄	—	2	24	no	—
154	C₄	—	2	24	no	—
155	C₄	—	4	4	no	—
156	C₃	C₂²×#395	1	1	yes	characteristic, direct product factor, Sylow
157	C₂	C₂×C₃×#395	1	24	yes	direct product factor
158	C₂	C₂²×C₃×#120	1	1	yes	characteristic
159	C₂	C₂²×C₃×C₂²⋊Q₈	1	3	yes	—
160	C₂	C₂²×C₃×C₄⋊Q₈	1	3	yes	—

	subgroup	G/H
center, socle, prime radical	C₂⁵×C₃	C₂³
commutator, Frattini, cosocle	C₂³	C₂⁵×C₃
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
3. C₂×C₃×#395	C₂	direct (N × H)
5. C₂²×#395	C₃	direct (N × H)
13. C₃×#395	C₂²	direct (N × H)
17. C₂×#395	C₂×C₃	direct (N × H)
28. C₂³×C₃×C₄	Q₈	semidirect
42. #395	C₂²×C₃	direct (N × H)
58. C₂²×C₃×C₄	C₂×Q₈	semidirect
72. C₂³×C₄	C₃×Q₈	semidirect
95. C₂×C₃×C₄	C₂²×Q₈	semidirect
110. C₂²×C₄	C₂×C₃×Q₈	semidirect
115. C₂²×C₃	#395	direct (N × H)
141. C₂×C₄	C₂²×C₃×Q₈	semidirect
142. C₂×C₃	C₂×#395	direct (N × H)
146. C₂²	C₃×#395	direct (N × H)
156. C₃	C₂²×#395	direct (N × H)
157. C₂	C₂×C₃×#395	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×C₃×Q₈	2
C₂³×C₃×C₄⋊C₄	2
C₂×C₃×#395	2
C₂²×C₃×#95	2
C₂²×#395	3

Subgroups — C22×C3×#395 #10653141

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃×#395 #10653141