Subgroups — C₂×C₃²×#392 #11769433

Order 1152.

5436 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₃²×D₄	C₂	1	3	yes	maximal, complemented
2	C₂²×C₃²×C₂²⋊C₄	C₂	1	3	yes	maximal, complemented
3	C₃²×#392	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₃²×#95	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₃×#392	C₃	1	4	yes	maximal, direct product factor
6	C₂⁵×C₃²	C₂²	1	3	yes	—
7	C₂²×C₃²×D₄	—	2	12	no	—
8	C₂²×C₃²×D₄	—	2	24	no	—
9	C₂²×C₃²×D₄	C₂²	1	12	yes	complemented
10	C₂³×C₃²×C₄	C₂²	1	1	yes	characteristic
11	C₂³×C₃²×C₄	C₂²	1	3	yes	—
12	C₂×C₃²×C₂²⋊C₄	—	2	12	no	—
13	C₂×C₃²×C₂²⋊C₄	C₂²	1	12	yes	complemented
14	C₃²×#95	C₂²	1	4	yes	complemented
15	C₂³×C₃×D₄	C₂×C₃	1	12	yes	complemented
16	C₂²×C₃×C₂²⋊C₄	C₂×C₃	1	12	yes	complemented
17	C₃×#392	C₂×C₃	1	32	yes	direct product factor
18	C₂×C₃×#95	C₂×C₃	1	4	yes	complemented
19	C₂⁴×C₃²	—	2	12	no	—
20	C₂⁴×C₃²	—	2	12	no	—
21	C₂⁴×C₃²	—	2	6	no	—
22	C₂⁴×C₃²	—	2	6	no	—
23	C₂⁴×C₃²	D₄	1	12	yes	complemented
24	C₂⁴×C₃²	C₂³	1	6	yes	—
25	C₂⁴×C₃²	C₂³	1	1	yes	characteristic
26	C₂×C₃²×D₄	—	2	96	no	—
27	C₂×C₃²×D₄	—	2	48	no	—
28	C₂×C₃²×D₄	—	2	24	no	—
29	C₂²×C₃²×C₄	—	2	12	no	—
30	C₂²×C₃²×C₄	—	2	6	no	—
31	C₂²×C₃²×C₄	D₄	1	12	yes	complemented
32	C₂²×C₃²×C₄	C₂³	1	2	yes	—
33	C₂²×C₃²×C₄	C₂³	1	6	yes	—
34	C₃²×C₂²⋊C₄	—	2	24	no	—
35	C₂×#392	C₃²	1	1	yes	characteristic, direct product factor, Sylow
36	C₂⁵×C₃	C₂²×C₃	1	12	yes	—
37	C₂²×C₃×D₄	—	2	96	no	—
38	C₂²×C₃×D₄	—	2	48	no	—
39	C₂²×C₃×D₄	C₂²×C₃	1	48	yes	complemented
40	C₂³×C₃×C₄	C₂²×C₃	1	4	yes	—
41	C₂³×C₃×C₄	C₂²×C₃	1	12	yes	—
42	C₂×C₃×C₂²⋊C₄	—	2	48	no	—
43	C₂×C₃×C₂²⋊C₄	C₂²×C₃	1	48	yes	complemented
44	C₃×#95	C₂²×C₃	1	16	yes	complemented
45	C₂³×C₃²	—	2	48	no	—
46	C₂³×C₃²	—	2	24	no	—
47	C₂³×C₃²	—	2	6	no	—
48	C₂³×C₃²	—	2	12	no	—
49	C₂³×C₃²	—	2	12	no	—
50	C₂³×C₃²	—	2	6	no	—
51	C₂³×C₃²	—	4	12	no	—
52	C₂³×C₃²	—	4	24	no	—
53	C₂³×C₃²	—	4	12	no	—
54	C₂³×C₃²	C₂×D₄	1	6	yes	—
55	C₂³×C₃²	C₂×D₄	1	6	yes	—
56	C₂³×C₃²	C₂×D₄	1	12	yes	complemented
57	C₂³×C₃²	#41	1	2	yes	—
58	C₂³×C₃²	C₂⁴	1	1	yes	characteristic
59	C₃²×D₄	—	2	96	no	—
60	C₂×C₃²×C₄	—	2	24	no	—
61	C₂×C₃²×C₄	—	2	12	no	—
62	C₂×C₃²×C₄	—	2	6	no	—
63	C₂×C₃²×C₄	—	4	4	no	—
64	C₂×C₃²×C₄	C₂×D₄	1	12	yes	complemented
65	C₂³×D₄	C₂×C₃²	1	3	yes	complemented
66	C₂²×C₂²⋊C₄	C₂×C₃²	1	3	yes	complemented
67	#392	C₂×C₃²	1	8	yes	direct product factor
68	C₂×#95	C₂×C₃²	1	1	yes	characteristic, complemented
69	C₂⁴×C₃	—	2	24	no	—
70	C₂⁴×C₃	—	2	48	no	—
71	C₂⁴×C₃	—	2	48	no	—
72	C₂⁴×C₃	—	2	24	no	—
73	C₂⁴×C₃	C₃×D₄	1	48	yes	complemented
74	C₂⁴×C₃	C₂³×C₃	1	24	yes	—
75	C₂⁴×C₃	C₂³×C₃	1	4	yes	—
76	C₂×C₃×D₄	—	2	384	no	—
77	C₂×C₃×D₄	—	2	192	no	—
78	C₂×C₃×D₄	—	2	96	no	—
79	C₂²×C₃×C₄	—	2	48	no	—
80	C₂²×C₃×C₄	—	2	24	no	—
81	C₂²×C₃×C₄	C₃×D₄	1	48	yes	complemented
82	C₂²×C₃×C₄	C₂³×C₃	1	8	yes	—
83	C₂²×C₃×C₄	C₂³×C₃	1	24	yes	—
84	C₃×C₂²⋊C₄	—	2	96	no	—
85	C₂²×C₃²	—	2	24	no	—
86	C₂²×C₃²	—	2	12	no	—
87	C₂²×C₃²	—	4	6	no	—
88	C₂²×C₃²	—	4	12	no	—
89	C₂²×C₃²	—	4	6	no	—
90	C₂²×C₃²	—	4	48	no	—
91	C₂²×C₃²	#120	1	12	yes	—
92	C₂²×C₃²	#121	1	12	yes	—
93	C₂²×C₃²	#127	1	4	yes	—
94	C₂²×C₃²	C₂²×D₄	1	3	yes	—
95	C₂²×C₃²	C₂²×D₄	1	3	yes	—
96	C₂²×C₃²	C₂×#41	1	1	yes	characteristic
97	C₃²×C₄	—	2	12	no	—
98	C₃²×C₄	—	4	2	no	—
99	C₂⁵	C₂²×C₃²	1	3	yes	—
100	C₂²×D₄	—	2	24	no	—
101	C₂²×D₄	—	2	12	no	—
102	C₂²×D₄	C₂²×C₃²	1	12	yes	complemented
103	C₂³×C₄	C₂²×C₃²	1	1	yes	characteristic
104	C₂³×C₄	C₂²×C₃²	1	3	yes	—
105	C₂×C₂²⋊C₄	—	2	12	no	—
106	C₂×C₂²⋊C₄	C₂²×C₃²	1	12	yes	complemented
107	#95	C₂²×C₃²	1	4	yes	complemented
108	C₂³×C₃	—	2	192	no	—
109	C₂³×C₃	—	2	96	no	—
110	C₂³×C₃	—	2	24	no	—
111	C₂³×C₃	—	2	48	no	—
112	C₂³×C₃	—	2	48	no	—
113	C₂³×C₃	—	2	24	no	—
114	C₂³×C₃	—	4	48	no	—
115	C₂³×C₃	—	4	96	no	—
116	C₂³×C₃	—	4	48	no	—
117	C₂³×C₃	C₂×C₃×D₄	1	24	yes	—
118	C₂³×C₃	C₂×C₃×D₄	1	24	yes	—
119	C₂³×C₃	C₂×C₃×D₄	1	48	yes	complemented
120	C₂³×C₃	C₃×#41	1	8	yes	—
121	C₂³×C₃	C₂⁴×C₃	1	4	yes	—
122	C₃×D₄	—	2	384	no	—
123	C₂×C₃×C₄	—	2	96	no	—
124	C₂×C₃×C₄	—	2	48	no	—
125	C₂×C₃×C₄	—	2	24	no	—
126	C₂×C₃×C₄	—	4	16	no	—
127	C₂×C₃×C₄	C₂×C₃×D₄	1	48	yes	complemented
128	C₂×C₃²	—	4	12	no	—
129	C₂×C₃²	#392	1	8	yes	direct product factor
130	C₂×C₃²	C₂×#120	1	3	yes	—
131	C₂×C₃²	C₂×#121	1	3	yes	—
132	C₂×C₃²	C₂×#127	1	1	yes	characteristic
133	C₂⁴	—	2	6	no	—
134	C₂⁴	—	2	12	no	—
135	C₂⁴	—	2	12	no	—
136	C₂⁴	—	2	6	no	—
137	C₂⁴	C₃²×D₄	1	12	yes	complemented
138	C₂⁴	C₂³×C₃²	1	6	yes	—
139	C₂⁴	C₂³×C₃²	1	1	yes	characteristic
140	C₂×D₄	—	2	96	no	—
141	C₂×D₄	—	2	48	no	—
142	C₂×D₄	—	2	24	no	—
143	C₂²×C₄	—	2	12	no	—
144	C₂²×C₄	—	2	6	no	—
145	C₂²×C₄	C₃²×D₄	1	12	yes	complemented
146	C₂²×C₄	C₂³×C₃²	1	2	yes	—
147	C₂²×C₄	C₂³×C₃²	1	6	yes	—
148	C₂²⋊C₄	—	2	24	no	—
149	C₂²×C₃	—	2	96	no	—
150	C₂²×C₃	—	2	48	no	—
151	C₂²×C₃	—	4	192	no	—
152	C₂²×C₃	—	4	24	no	—
153	C₂²×C₃	—	4	48	no	—
154	C₂²×C₃	—	4	24	no	—
155	C₂²×C₃	C₃×#120	1	48	yes	—
156	C₂²×C₃	C₃×#121	1	48	yes	—
157	C₂²×C₃	C₃×#127	1	16	yes	—
158	C₂²×C₃	C₂²×C₃×D₄	1	12	yes	—
159	C₂²×C₃	C₂²×C₃×D₄	1	12	yes	—
160	C₂²×C₃	C₂×C₃×#41	1	4	yes	—
161	C₃×C₄	—	2	48	no	—
162	C₃×C₄	—	4	8	no	—
163	C₃²	C₂×#392	1	1	yes	characteristic, direct product factor, Sylow
164	C₂³	—	2	6	no	—
165	C₂³	—	2	12	no	—
166	C₂³	—	2	12	no	—
167	C₂³	—	2	6	no	—
168	C₂³	—	2	48	no	—
169	C₂³	—	2	24	no	—
170	C₂³	—	4	12	no	—
171	C₂³	—	4	24	no	—
172	C₂³	—	4	12	no	—
173	C₂³	C₂×C₃²×D₄	1	6	yes	—
174	C₂³	C₂×C₃²×D₄	1	6	yes	—
175	C₂³	C₂×C₃²×D₄	1	12	yes	complemented
176	C₂³	C₃²×#41	1	2	yes	—
177	C₂³	C₂⁴×C₃²	1	1	yes	characteristic
178	D₄	—	2	96	no	—
179	C₂×C₄	—	2	12	no	—
180	C₂×C₄	—	2	24	no	—
181	C₂×C₄	—	2	6	no	—
182	C₂×C₄	—	4	4	no	—
183	C₂×C₄	C₂×C₃²×D₄	1	12	yes	complemented
184	C₂×C₃	—	4	48	no	—
185	C₂×C₃	C₃×#392	1	32	yes	direct product factor
186	C₂×C₃	C₂×C₃×#120	1	12	yes	—
187	C₂×C₃	C₂×C₃×#121	1	12	yes	—
188	C₂×C₃	C₂×C₃×#127	1	4	yes	—
189	C₂²	—	2	24	no	—
190	C₂²	—	2	12	no	—
191	C₂²	—	4	48	no	—
192	C₂²	—	4	6	no	—
193	C₂²	—	4	12	no	—
194	C₂²	—	4	6	no	—
195	C₂²	C₃²×#120	1	12	yes	—
196	C₂²	C₃²×#121	1	12	yes	—
197	C₂²	C₃²×#127	1	4	yes	—
198	C₂²	C₂²×C₃²×D₄	1	3	yes	—
199	C₂²	C₂²×C₃²×D₄	1	3	yes	—
200	C₂²	C₂×C₃²×#41	1	1	yes	characteristic
201	C₄	—	2	12	no	—
202	C₄	—	4	2	no	—
203	C₃	C₂×C₃×#392	1	4	yes	direct product factor
204	C₂	—	4	12	no	—
205	C₂	C₃²×#392	1	8	yes	direct product factor
206	C₂	C₂×C₃²×#120	1	3	yes	—
207	C₂	C₂×C₃²×#121	1	3	yes	—
208	C₂	C₂×C₃²×#127	1	1	yes	characteristic

	subgroup	G/H
center, socle	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
1. C₂³×C₃²×D₄	C₂	semidirect
2. C₂²×C₃²×C₂²⋊C₄	C₂	semidirect
3. C₃²×#392	C₂	direct (N × H)
4. C₂×C₃²×#95	C₂	semidirect
5. C₂×C₃×#392	C₃	direct (N × H)
9. C₂²×C₃²×D₄	C₂²	semidirect
13. C₂×C₃²×C₂²⋊C₄	C₂²	semidirect
14. C₃²×#95	C₂²	semidirect
15. C₂³×C₃×D₄	C₂×C₃	semidirect
16. C₂²×C₃×C₂²⋊C₄	C₂×C₃	semidirect
17. C₃×#392	C₂×C₃	direct (N × H)
18. C₂×C₃×#95	C₂×C₃	semidirect
23. C₂⁴×C₃²	D₄	semidirect
31. C₂²×C₃²×C₄	D₄	semidirect
35. C₂×#392	C₃²	direct (N × H)
39. C₂²×C₃×D₄	C₂²×C₃	semidirect
43. C₂×C₃×C₂²⋊C₄	C₂²×C₃	semidirect
44. C₃×#95	C₂²×C₃	semidirect
56. C₂³×C₃²	C₂×D₄	semidirect
64. C₂×C₃²×C₄	C₂×D₄	semidirect
65. C₂³×D₄	C₂×C₃²	semidirect
66. C₂²×C₂²⋊C₄	C₂×C₃²	semidirect
67. #392	C₂×C₃²	direct (N × H)
68. C₂×#95	C₂×C₃²	semidirect
73. C₂⁴×C₃	C₃×D₄	semidirect
81. C₂²×C₃×C₄	C₃×D₄	semidirect
102. C₂²×D₄	C₂²×C₃²	semidirect
106. C₂×C₂²⋊C₄	C₂²×C₃²	semidirect
107. #95	C₂²×C₃²	semidirect
119. C₂³×C₃	C₂×C₃×D₄	semidirect
127. C₂×C₃×C₄	C₂×C₃×D₄	semidirect
129. C₂×C₃²	#392	direct (N × H)
137. C₂⁴	C₃²×D₄	semidirect
145. C₂²×C₄	C₃²×D₄	semidirect
163. C₃²	C₂×#392	direct (N × H)
175. C₂³	C₂×C₃²×D₄	semidirect
183. C₂×C₄	C₂×C₃²×D₄	semidirect
185. C₂×C₃	C₃×#392	direct (N × H)
203. C₃	C₂×C₃×#392	direct (N × H)
205. C₂	C₃²×#392	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×D₄	2
C₂²×C₃²×C₂²⋊C₄	2
C₃²×#392	2
C₂×C₃²×#95	2
C₂×C₃×#392	3

Subgroups — C2×C32×#392 #11769433

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃²×#392 #11769433