Subgroups — C₂×C₃²×#395 #11769436

Order 1152.

2724 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	C₂³×C₃²×Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₃²×C₄⋊C₄	C₂	1	3	yes	maximal
3	C₃²×#395	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₃²×#95	C₂	1	3	yes	maximal
5	C₂×C₃×#395	C₃	1	4	yes	maximal, direct product factor
6	C₂²×C₃²×Q₈	—	2	12	no	—
7	C₂²×C₃²×Q₈	C₂²	1	4	yes	—
8	C₂³×C₃²×C₄	C₂²	1	1	yes	characteristic
9	C₂³×C₃²×C₄	C₂²	1	3	yes	—
10	C₂³×C₃²×C₄	C₂²	1	3	yes	—
11	C₂×C₃²×C₄⋊C₄	—	2	12	no	—
12	C₂×C₃²×C₄⋊C₄	C₂²	1	12	yes	—
13	C₃²×#95	C₂²	1	12	yes	—
14	C₂³×C₃×Q₈	C₂×C₃	1	4	yes	—
15	C₂²×C₃×C₄⋊C₄	C₂×C₃	1	12	yes	—
16	C₃×#395	C₂×C₃	1	32	yes	direct product factor
17	C₂×C₃×#95	C₂×C₃	1	12	yes	—
18	C₂⁴×C₃²	C₂³	1	1	yes	characteristic
19	C₂×C₃²×Q₈	—	2	32	no	—
20	C₂×C₃²×Q₈	—	2	24	no	—
21	C₂²×C₃²×C₄	—	2	12	no	—
22	C₂²×C₃²×C₄	—	2	6	no	—
23	C₂²×C₃²×C₄	—	2	6	no	—
24	C₂²×C₃²×C₄	—	2	6	no	—
25	C₂²×C₃²×C₄	D₄	1	12	yes	—
26	C₂²×C₃²×C₄	Q₈	1	12	yes	complemented
27	C₂²×C₃²×C₄	C₂³	1	2	yes	—
28	C₂²×C₃²×C₄	C₂³	1	6	yes	—
29	C₂²×C₃²×C₄	C₂³	1	6	yes	—
30	C₃²×C₄⋊C₄	—	2	24	no	—
31	C₂×#395	C₃²	1	1	yes	characteristic, direct product factor, Sylow
32	C₂²×C₃×Q₈	—	2	48	no	—
33	C₂²×C₃×Q₈	C₂²×C₃	1	16	yes	—
34	C₂³×C₃×C₄	C₂²×C₃	1	4	yes	—
35	C₂³×C₃×C₄	C₂²×C₃	1	12	yes	—
36	C₂³×C₃×C₄	C₂²×C₃	1	12	yes	—
37	C₂×C₃×C₄⋊C₄	—	2	48	no	—
38	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	48	yes	—
39	C₃×#95	C₂²×C₃	1	48	yes	—
40	C₂³×C₃²	C₂×D₄	1	6	yes	—
41	C₂³×C₃²	C₂×Q₈	1	6	yes	—
42	C₂³×C₃²	#41	1	2	yes	—
43	C₂³×C₃²	C₂⁴	1	1	yes	characteristic
44	C₃²×Q₈	—	2	32	no	—
45	C₂×C₃²×C₄	—	2	24	no	—
46	C₂×C₃²×C₄	—	2	24	no	—
47	C₂×C₃²×C₄	—	2	6	no	—
48	C₂×C₃²×C₄	—	2	12	no	—
49	C₂×C₃²×C₄	—	2	6	no	—
50	C₂×C₃²×C₄	—	2	6	no	—
51	C₂×C₃²×C₄	—	4	4	no	—
52	C₂×C₃²×C₄	C₂×D₄	1	12	yes	—
53	C₂×C₃²×C₄	C₂×Q₈	1	12	yes	complemented
54	C₂³×Q₈	C₂×C₃²	1	1	yes	characteristic
55	C₂²×C₄⋊C₄	C₂×C₃²	1	3	yes	—
56	#395	C₂×C₃²	1	8	yes	direct product factor
57	C₂×#95	C₂×C₃²	1	3	yes	—
58	C₂⁴×C₃	C₂³×C₃	1	4	yes	—
59	C₂×C₃×Q₈	—	2	128	no	—
60	C₂×C₃×Q₈	—	2	96	no	—
61	C₂²×C₃×C₄	—	2	24	no	—
62	C₂²×C₃×C₄	—	2	48	no	—
63	C₂²×C₃×C₄	—	2	24	no	—
64	C₂²×C₃×C₄	—	2	24	no	—
65	C₂²×C₃×C₄	C₃×D₄	1	48	yes	—
66	C₂²×C₃×C₄	C₃×Q₈	1	48	yes	complemented
67	C₂²×C₃×C₄	C₂³×C₃	1	8	yes	—
68	C₂²×C₃×C₄	C₂³×C₃	1	24	yes	—
69	C₂²×C₃×C₄	C₂³×C₃	1	24	yes	—
70	C₃×C₄⋊C₄	—	2	96	no	—
71	C₂²×C₃²	#120	1	4	yes	—
72	C₂²×C₃²	C₂²⋊Q₈	1	12	yes	—
73	C₂²×C₃²	C₄⋊Q₈	1	12	yes	—
74	C₂²×C₃²	C₂²×D₄	1	3	yes	—
75	C₂²×C₃²	C₂²×Q₈	1	3	yes	—
76	C₂²×C₃²	C₂×#41	1	1	yes	characteristic
77	C₃²×C₄	—	2	12	no	—
78	C₃²×C₄	—	2	12	no	—
79	C₃²×C₄	—	4	2	no	—
80	C₂²×Q₈	—	2	12	no	—
81	C₂²×Q₈	C₂²×C₃²	1	4	yes	—
82	C₂³×C₄	C₂²×C₃²	1	1	yes	characteristic
83	C₂³×C₄	C₂²×C₃²	1	3	yes	—
84	C₂³×C₄	C₂²×C₃²	1	3	yes	—
85	C₂×C₄⋊C₄	—	2	12	no	—
86	C₂×C₄⋊C₄	C₂²×C₃²	1	12	yes	—
87	#95	C₂²×C₃²	1	12	yes	—
88	C₂³×C₃	C₂×C₃×D₄	1	24	yes	—
89	C₂³×C₃	C₂×C₃×Q₈	1	24	yes	—
90	C₂³×C₃	C₃×#41	1	8	yes	—
91	C₂³×C₃	C₂⁴×C₃	1	4	yes	—
92	C₃×Q₈	—	2	128	no	—
93	C₂×C₃×C₄	—	2	96	no	—
94	C₂×C₃×C₄	—	2	96	no	—
95	C₂×C₃×C₄	—	2	24	no	—
96	C₂×C₃×C₄	—	2	48	no	—
97	C₂×C₃×C₄	—	2	24	no	—
98	C₂×C₃×C₄	—	2	24	no	—
99	C₂×C₃×C₄	—	4	16	no	—
100	C₂×C₃×C₄	C₂×C₃×D₄	1	48	yes	—
101	C₂×C₃×C₄	C₂×C₃×Q₈	1	48	yes	complemented
102	C₂×C₃²	#395	1	8	yes	direct product factor
103	C₂×C₃²	C₂×#120	1	1	yes	characteristic
104	C₂×C₃²	C₂×C₂²⋊Q₈	1	3	yes	—
105	C₂×C₃²	C₂×C₄⋊Q₈	1	3	yes	—
106	C₂⁴	C₂³×C₃²	1	1	yes	characteristic
107	C₂×Q₈	—	2	32	no	—
108	C₂×Q₈	—	2	24	no	—
109	C₂²×C₄	—	2	6	no	—
110	C₂²×C₄	—	2	12	no	—
111	C₂²×C₄	—	2	6	no	—
112	C₂²×C₄	—	2	6	no	—
113	C₂²×C₄	C₃²×D₄	1	12	yes	—
114	C₂²×C₄	C₃²×Q₈	1	12	yes	complemented
115	C₂²×C₄	C₂³×C₃²	1	2	yes	—
116	C₂²×C₄	C₂³×C₃²	1	6	yes	—
117	C₂²×C₄	C₂³×C₃²	1	6	yes	—
118	C₄⋊C₄	—	2	24	no	—
119	C₂²×C₃	C₃×#120	1	16	yes	—
120	C₂²×C₃	C₃×C₂²⋊Q₈	1	48	yes	—
121	C₂²×C₃	C₃×C₄⋊Q₈	1	48	yes	—
122	C₂²×C₃	C₂²×C₃×D₄	1	12	yes	—
123	C₂²×C₃	C₂²×C₃×Q₈	1	12	yes	—
124	C₂²×C₃	C₂×C₃×#41	1	4	yes	—
125	C₃×C₄	—	2	48	no	—
126	C₃×C₄	—	2	48	no	—
127	C₃×C₄	—	4	8	no	—
128	C₃²	C₂×#395	1	1	yes	characteristic, direct product factor, Sylow
129	C₂³	C₂×C₃²×D₄	1	6	yes	—
130	C₂³	C₂×C₃²×Q₈	1	6	yes	—
131	C₂³	C₃²×#41	1	2	yes	—
132	C₂³	C₂⁴×C₃²	1	1	yes	characteristic
133	Q₈	—	2	32	no	—
134	C₂×C₄	—	2	6	no	—
135	C₂×C₄	—	2	12	no	—
136	C₂×C₄	—	2	6	no	—
137	C₂×C₄	—	2	24	no	—
138	C₂×C₄	—	2	24	no	—
139	C₂×C₄	—	2	6	no	—
140	C₂×C₄	—	4	4	no	—
141	C₂×C₄	C₂×C₃²×D₄	1	12	yes	—
142	C₂×C₄	C₂×C₃²×Q₈	1	12	yes	complemented
143	C₂×C₃	C₃×#395	1	32	yes	direct product factor
144	C₂×C₃	C₂×C₃×#120	1	4	yes	—
145	C₂×C₃	C₂×C₃×C₂²⋊Q₈	1	12	yes	—
146	C₂×C₃	C₂×C₃×C₄⋊Q₈	1	12	yes	—
147	C₂²	C₃²×#120	1	4	yes	—
148	C₂²	C₃²×C₂²⋊Q₈	1	12	yes	—
149	C₂²	C₃²×C₄⋊Q₈	1	12	yes	—
150	C₂²	C₂²×C₃²×D₄	1	3	yes	—
151	C₂²	C₂²×C₃²×Q₈	1	3	yes	—
152	C₂²	C₂×C₃²×#41	1	1	yes	characteristic
153	C₄	—	2	12	no	—
154	C₄	—	2	12	no	—
155	C₄	—	4	2	no	—
156	C₃	C₂×C₃×#395	1	4	yes	direct product factor
157	C₂	C₃²×#395	1	8	yes	direct product factor
158	C₂	C₂×C₃²×#120	1	1	yes	characteristic
159	C₂	C₂×C₃²×C₂²⋊Q₈	1	3	yes	—
160	C₂	C₂×C₃²×C₄⋊Q₈	1	3	yes	—

	subgroup	G/H
center, socle, prime radical	C₂⁴×C₃²	C₂³
commutator, Frattini, cosocle	C₂³	C₂⁴×C₃²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₃² ◁ G

Complements

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

N	H ≅ G/N	type
3. C₃²×#395	C₂	direct (N × H)
5. C₂×C₃×#395	C₃	direct (N × H)
16. C₃×#395	C₂×C₃	direct (N × H)
26. C₂²×C₃²×C₄	Q₈	semidirect
31. C₂×#395	C₃²	direct (N × H)
53. C₂×C₃²×C₄	C₂×Q₈	semidirect
56. #395	C₂×C₃²	direct (N × H)
66. C₂²×C₃×C₄	C₃×Q₈	semidirect
101. C₂×C₃×C₄	C₂×C₃×Q₈	semidirect
102. C₂×C₃²	#395	direct (N × H)
114. C₂²×C₄	C₃²×Q₈	semidirect
128. C₃²	C₂×#395	direct (N × H)
142. C₂×C₄	C₂×C₃²×Q₈	semidirect
143. C₂×C₃	C₃×#395	direct (N × H)
156. C₃	C₂×C₃×#395	direct (N × H)
157. C₂	C₃²×#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
C₂³×C₃²×Q₈	2
C₂²×C₃²×C₄⋊C₄	2
C₃²×#395	2
C₂×C₃²×#95	2
C₂×C₃×#395	3

Subgroups — C2×C32×#395 #11769436

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃²×#395 #11769436