Subgroups — C₂×C₃²×#393 #11769434

Order 1152.

3108 conjugacy classes of subgroups.

The subgroup lattice has 186 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, complemented
2	C₂²×C₃²×C₂²⋊C₄	C₂	1	3	yes	maximal
3	C₃²×#393	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₃²×#95	C₂	1	3	yes	maximal, complemented
5	C₂×C₃×#393	C₃	1	4	yes	maximal, direct product factor
6	C₂⁵×C₃²	C₂²	1	1	yes	characteristic
7	C₂²×C₃²×Q₈	—	2	12	no	—
8	C₂²×C₃²×Q₈	C₂²	1	4	yes	complemented
9	C₂³×C₃²×C₄	C₂²	1	3	yes	—
10	C₂³×C₃²×C₄	C₂²	1	3	yes	—
11	C₂×C₃²×C₂²⋊C₄	—	2	12	no	—
12	C₂×C₃²×C₂²⋊C₄	C₂²	1	12	yes	—
13	C₃²×#95	C₂²	1	12	yes	complemented
14	C₂³×C₃×Q₈	C₂×C₃	1	4	yes	complemented
15	C₂²×C₃×C₂²⋊C₄	C₂×C₃	1	12	yes	—
16	C₃×#393	C₂×C₃	1	32	yes	direct product factor
17	C₂×C₃×#95	C₂×C₃	1	12	yes	complemented
18	C₂⁴×C₃²	—	2	6	no	—
19	C₂⁴×C₃²	—	2	6	no	—
20	C₂⁴×C₃²	Q₈	1	4	yes	complemented
21	C₂⁴×C₃²	C₂³	1	2	yes	—
22	C₂⁴×C₃²	C₂³	1	1	yes	characteristic
23	C₂×C₃²×Q₈	—	2	32	no	—
24	C₂×C₃²×Q₈	—	2	24	no	—
25	C₂²×C₃²×C₄	—	2	12	no	—
26	C₂²×C₃²×C₄	—	2	6	no	—
27	C₂²×C₃²×C₄	—	2	12	no	—
28	C₂²×C₃²×C₄	D₄	1	12	yes	—
29	C₂²×C₃²×C₄	C₂³	1	6	yes	—
30	C₂²×C₃²×C₄	C₂³	1	6	yes	—
31	C₃²×C₂²⋊C₄	—	2	24	no	—
32	C₂×#393	C₃²	1	1	yes	characteristic, direct product factor, Sylow
33	C₂⁵×C₃	C₂²×C₃	1	4	yes	—
34	C₂²×C₃×Q₈	—	2	48	no	—
35	C₂²×C₃×Q₈	C₂²×C₃	1	16	yes	complemented
36	C₂³×C₃×C₄	C₂²×C₃	1	12	yes	—
37	C₂³×C₃×C₄	C₂²×C₃	1	12	yes	—
38	C₂×C₃×C₂²⋊C₄	—	2	48	no	—
39	C₂×C₃×C₂²⋊C₄	C₂²×C₃	1	48	yes	—
40	C₃×#95	C₂²×C₃	1	48	yes	complemented
41	C₂³×C₃²	—	2	24	no	—
42	C₂³×C₃²	—	2	6	no	—
43	C₂³×C₃²	—	2	6	no	—
44	C₂³×C₃²	—	4	4	no	—
45	C₂³×C₃²	—	4	12	no	—
46	C₂³×C₃²	C₂×D₄	1	6	yes	—
47	C₂³×C₃²	C₂×Q₈	1	2	yes	—
48	C₂³×C₃²	C₂×Q₈	1	4	yes	complemented
49	C₂³×C₃²	#41	1	6	yes	—
50	C₂³×C₃²	C₂⁴	1	1	yes	characteristic
51	C₃²×Q₈	—	2	32	no	—
52	C₂×C₃²×C₄	—	2	24	no	—
53	C₂×C₃²×C₄	—	2	12	no	—
54	C₂×C₃²×C₄	—	2	6	no	—
55	C₂×C₃²×C₄	—	2	12	no	—
56	C₂×C₃²×C₄	—	4	12	no	—
57	C₂×C₃²×C₄	C₂×D₄	1	12	yes	—
58	C₂³×Q₈	C₂×C₃²	1	1	yes	characteristic, complemented
59	C₂²×C₂²⋊C₄	C₂×C₃²	1	3	yes	—
60	#393	C₂×C₃²	1	8	yes	direct product factor
61	C₂×#95	C₂×C₃²	1	3	yes	complemented
62	C₂⁴×C₃	—	2	24	no	—
63	C₂⁴×C₃	—	2	24	no	—
64	C₂⁴×C₃	C₃×Q₈	1	16	yes	complemented
65	C₂⁴×C₃	C₂³×C₃	1	8	yes	—
66	C₂⁴×C₃	C₂³×C₃	1	4	yes	—
67	C₂×C₃×Q₈	—	2	128	no	—
68	C₂×C₃×Q₈	—	2	96	no	—
69	C₂²×C₃×C₄	—	2	48	no	—
70	C₂²×C₃×C₄	—	2	24	no	—
71	C₂²×C₃×C₄	—	2	48	no	—
72	C₂²×C₃×C₄	C₃×D₄	1	48	yes	—
73	C₂²×C₃×C₄	C₂³×C₃	1	24	yes	—
74	C₂²×C₃×C₄	C₂³×C₃	1	24	yes	—
75	C₃×C₂²⋊C₄	—	2	96	no	—
76	C₂²×C₃²	—	2	12	no	—
77	C₂²×C₃²	—	4	2	no	—
78	C₂²×C₃²	—	4	6	no	—
79	C₂²×C₃²	—	4	16	no	—
80	C₂²×C₃²	#120	1	4	yes	—
81	C₂²×C₃²	C₂²⋊Q₈	1	12	yes	—
82	C₂²×C₃²	#124	1	12	yes	—
83	C₂²×C₃²	C₂²×D₄	1	3	yes	—
84	C₂²×C₃²	C₂²×Q₈	1	1	yes	characteristic
85	C₂²×C₃²	C₂×#41	1	3	yes	—
86	C₃²×C₄	—	2	12	no	—
87	C₃²×C₄	—	4	6	no	—
88	C₂⁵	C₂²×C₃²	1	1	yes	characteristic
89	C₂²×Q₈	—	2	12	no	—
90	C₂²×Q₈	C₂²×C₃²	1	4	yes	complemented
91	C₂³×C₄	C₂²×C₃²	1	3	yes	—
92	C₂³×C₄	C₂²×C₃²	1	3	yes	—
93	C₂×C₂²⋊C₄	—	2	12	no	—
94	C₂×C₂²⋊C₄	C₂²×C₃²	1	12	yes	—
95	#95	C₂²×C₃²	1	12	yes	complemented
96	C₂³×C₃	—	2	96	no	—
97	C₂³×C₃	—	2	24	no	—
98	C₂³×C₃	—	2	24	no	—
99	C₂³×C₃	—	4	16	no	—
100	C₂³×C₃	—	4	48	no	—
101	C₂³×C₃	C₂×C₃×D₄	1	24	yes	—
102	C₂³×C₃	C₂×C₃×Q₈	1	8	yes	—
103	C₂³×C₃	C₂×C₃×Q₈	1	16	yes	complemented
104	C₂³×C₃	C₃×#41	1	24	yes	—
105	C₂³×C₃	C₂⁴×C₃	1	4	yes	—
106	C₃×Q₈	—	2	128	no	—
107	C₂×C₃×C₄	—	2	96	no	—
108	C₂×C₃×C₄	—	2	48	no	—
109	C₂×C₃×C₄	—	2	24	no	—
110	C₂×C₃×C₄	—	2	48	no	—
111	C₂×C₃×C₄	—	4	48	no	—
112	C₂×C₃×C₄	C₂×C₃×D₄	1	48	yes	—
113	C₂×C₃²	—	4	4	no	—
114	C₂×C₃²	#393	1	8	yes	direct product factor
115	C₂×C₃²	C₂×#120	1	1	yes	characteristic
116	C₂×C₃²	C₂×C₂²⋊Q₈	1	3	yes	—
117	C₂×C₃²	C₂×#124	1	3	yes	—
118	C₂⁴	—	2	6	no	—
119	C₂⁴	—	2	6	no	—
120	C₂⁴	C₃²×Q₈	1	4	yes	complemented
121	C₂⁴	C₂³×C₃²	1	2	yes	—
122	C₂⁴	C₂³×C₃²	1	1	yes	characteristic
123	C₂×Q₈	—	2	32	no	—
124	C₂×Q₈	—	2	24	no	—
125	C₂²×C₄	—	2	12	no	—
126	C₂²×C₄	—	2	6	no	—
127	C₂²×C₄	—	2	12	no	—
128	C₂²×C₄	C₃²×D₄	1	12	yes	—
129	C₂²×C₄	C₂³×C₃²	1	6	yes	—
130	C₂²×C₄	C₂³×C₃²	1	6	yes	—
131	C₂²⋊C₄	—	2	24	no	—
132	C₂²×C₃	—	2	48	no	—
133	C₂²×C₃	—	4	64	no	—
134	C₂²×C₃	—	4	8	no	—
135	C₂²×C₃	—	4	24	no	—
136	C₂²×C₃	C₃×#120	1	16	yes	—
137	C₂²×C₃	C₃×C₂²⋊Q₈	1	48	yes	—
138	C₂²×C₃	C₃×#124	1	48	yes	—
139	C₂²×C₃	C₂²×C₃×D₄	1	12	yes	—
140	C₂²×C₃	C₂²×C₃×Q₈	1	4	yes	—
141	C₂²×C₃	C₂×C₃×#41	1	12	yes	—
142	C₃×C₄	—	2	48	no	—
143	C₃×C₄	—	4	24	no	—
144	C₃²	C₂×#393	1	1	yes	characteristic, direct product factor, Sylow
145	C₂³	—	2	6	no	—
146	C₂³	—	2	6	no	—
147	C₂³	—	2	24	no	—
148	C₂³	—	4	4	no	—
149	C₂³	—	4	12	no	—
150	C₂³	C₂×C₃²×D₄	1	6	yes	—
151	C₂³	C₂×C₃²×Q₈	1	2	yes	—
152	C₂³	C₂×C₃²×Q₈	1	4	yes	complemented
153	C₂³	C₃²×#41	1	6	yes	—
154	C₂³	C₂⁴×C₃²	1	1	yes	characteristic
155	Q₈	—	2	32	no	—
156	C₂×C₄	—	2	12	no	—
157	C₂×C₄	—	2	24	no	—
158	C₂×C₄	—	2	6	no	—
159	C₂×C₄	—	2	12	no	—
160	C₂×C₄	—	4	12	no	—
161	C₂×C₄	C₂×C₃²×D₄	1	12	yes	—
162	C₂×C₃	—	4	16	no	—
163	C₂×C₃	C₃×#393	1	32	yes	direct product factor
164	C₂×C₃	C₂×C₃×#120	1	4	yes	—
165	C₂×C₃	C₂×C₃×C₂²⋊Q₈	1	12	yes	—
166	C₂×C₃	C₂×C₃×#124	1	12	yes	—
167	C₂²	—	2	12	no	—
168	C₂²	—	4	16	no	—
169	C₂²	—	4	2	no	—
170	C₂²	—	4	6	no	—
171	C₂²	C₃²×#120	1	4	yes	—
172	C₂²	C₃²×C₂²⋊Q₈	1	12	yes	—
173	C₂²	C₃²×#124	1	12	yes	—
174	C₂²	C₂²×C₃²×D₄	1	3	yes	—
175	C₂²	C₂²×C₃²×Q₈	1	1	yes	characteristic
176	C₂²	C₂×C₃²×#41	1	3	yes	—
177	C₄	—	2	12	no	—
178	C₄	—	4	6	no	—
179	C₃	C₂×C₃×#393	1	4	yes	direct product factor
180	C₂	—	4	4	no	—
181	C₂	C₃²×#393	1	8	yes	direct product factor
182	C₂	C₂×C₃²×#120	1	1	yes	characteristic
183	C₂	C₂×C₃²×C₂²⋊Q₈	1	3	yes	—
184	C₂	C₂×C₃²×#124	1	3	yes	—

	subgroup	G/H
center, socle	C₂⁴×C₃²	C₂³
commutator, Frattini, cosocle	C₂³	C₂⁴×C₃²
prime radical	C₂⁵×C₃²	C₂²
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
1. C₂³×C₃²×Q₈	C₂	semidirect
3. C₃²×#393	C₂	direct (N × H)
4. C₂×C₃²×#95	C₂	semidirect
5. C₂×C₃×#393	C₃	direct (N × H)
8. C₂²×C₃²×Q₈	C₂²	semidirect
13. C₃²×#95	C₂²	semidirect
14. C₂³×C₃×Q₈	C₂×C₃	semidirect
16. C₃×#393	C₂×C₃	direct (N × H)
17. C₂×C₃×#95	C₂×C₃	semidirect
20. C₂⁴×C₃²	Q₈	semidirect
32. C₂×#393	C₃²	direct (N × H)
35. C₂²×C₃×Q₈	C₂²×C₃	semidirect
40. C₃×#95	C₂²×C₃	semidirect
48. C₂³×C₃²	C₂×Q₈	semidirect
58. C₂³×Q₈	C₂×C₃²	semidirect
60. #393	C₂×C₃²	direct (N × H)
61. C₂×#95	C₂×C₃²	semidirect
64. C₂⁴×C₃	C₃×Q₈	semidirect
90. C₂²×Q₈	C₂²×C₃²	semidirect
95. #95	C₂²×C₃²	semidirect
103. C₂³×C₃	C₂×C₃×Q₈	semidirect
114. C₂×C₃²	#393	direct (N × H)
120. C₂⁴	C₃²×Q₈	semidirect
144. C₃²	C₂×#393	direct (N × H)
152. C₂³	C₂×C₃²×Q₈	semidirect
163. C₂×C₃	C₃×#393	direct (N × H)
179. C₃	C₂×C₃×#393	direct (N × H)
181. C₂	C₃²×#393	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×C₂²⋊C₄	2
C₃²×#393	2
C₂×C₃²×#95	2
C₂×C₃×#393	3

Subgroups — C2×C32×#393 #11769434

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃²×#393 #11769434