Subgroups — C₃²×#2551 #11769600

Order 1152.

1668 conjugacy classes of subgroups.

The subgroup lattice has 189 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×C₄⋊Q₈	C₂	1	4	yes	maximal
2	C₃²×#395	C₂	1	4	yes	maximal
3	C₃²×#389	C₂	1	4	yes	maximal
4	C₃²×#384	C₂	1	1	yes	maximal, characteristic
5	C₃²×C₄×C₄⋊C₄	C₂	1	2	yes	maximal
6	C₃×#2551	C₃	1	4	yes	maximal, direct product factor
7	C₂²×C₃²×Q₈	C₂²	1	4	yes	—
8	C₃²×C₄⋊Q₈	—	2	16	no	—
9	C₂×C₃²×C₄⋊C₄	C₂²	1	4	yes	—
10	C₂×C₃²×C₄⋊C₄	C₂²	1	8	yes	—
11	C₂×C₃²×C₄⋊C₄	C₂²	1	2	yes	—
12	C₂×C₃²×C₄⋊C₄	C₂²	1	2	yes	—
13	C₂×C₃²×C₄⋊C₄	C₂²	1	2	yes	—
14	C₂×C₃²×C₄²	C₂²	1	4	yes	—
15	C₂×C₃²×C₄²	C₂²	1	1	yes	characteristic
16	C₃²×#95	C₂²	1	4	yes	—
17	C₃²×#95	C₂²	1	4	yes	—
18	C₂×C₃×C₄⋊Q₈	C₂×C₃	1	16	yes	—
19	C₃×#395	C₂×C₃	1	16	yes	—
20	C₃×#389	C₂×C₃	1	16	yes	—
21	C₃×#384	C₂×C₃	1	4	yes	—
22	C₃×C₄×C₄⋊C₄	C₂×C₃	1	8	yes	—
23	C₂×C₃²×Q₈	—	2	16	no	—
24	C₂×C₃²×Q₈	—	4	4	no	—
25	C₂²×C₃²×C₄	C₂³	1	4	yes	—
26	C₂²×C₃²×C₄	C₂³	1	4	yes	—
27	C₂²×C₃²×C₄	C₂³	1	4	yes	—
28	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
29	C₂²×C₃²×C₄	C₂³	1	2	yes	—
30	C₃²×C₄⋊C₄	—	2	16	no	—
31	C₃²×C₄⋊C₄	—	2	8	no	—
32	C₃²×C₄⋊C₄	—	4	2	no	—
33	C₃²×C₄⋊C₄	D₄	1	8	yes	—
34	C₃²×C₄⋊C₄	Q₈	1	8	yes	complemented
35	C₃²×C₄²	—	2	8	no	—
36	C₃²×C₄²	Q₈	1	4	yes	complemented
37	#2551	C₃²	1	1	yes	characteristic, direct product factor, Sylow
38	C₂²×C₃×Q₈	C₂²×C₃	1	16	yes	—
39	C₃×C₄⋊Q₈	—	2	64	no	—
40	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	16	yes	—
41	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	32	yes	—
42	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	8	yes	—
43	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	8	yes	—
44	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	8	yes	—
45	C₂×C₃×C₄²	C₂²×C₃	1	16	yes	—
46	C₂×C₃×C₄²	C₂²×C₃	1	4	yes	—
47	C₃×#95	C₂²×C₃	1	16	yes	—
48	C₃×#95	C₂²×C₃	1	16	yes	—
49	C₂³×C₃²	C₂⁴	1	1	yes	characteristic
50	C₃²×Q₈	—	4	16	no	—
51	C₂×C₃²×C₄	—	2	4	no	—
52	C₂×C₃²×C₄	—	2	4	no	—
53	C₂×C₃²×C₄	—	2	1	no	—
54	C₂×C₃²×C₄	—	2	4	no	—
55	C₂×C₃²×C₄	—	2	1	no	—
56	C₂×C₃²×C₄	—	2	4	no	—
57	C₂×C₃²×C₄	—	2	8	no	—
58	C₂×C₃²×C₄	—	2	4	no	—
59	C₂×C₃²×C₄	C₂×D₄	1	8	yes	—
60	C₂×C₃²×C₄	C₂×D₄	1	4	yes	—
61	C₂×C₃²×C₄	C₂×Q₈	1	2	yes	—
62	C₂×C₃²×C₄	C₂×Q₈	1	8	yes	—
63	C₂×C₃²×C₄	C₂×Q₈	1	4	yes	—
64	C₂×C₃²×C₄	C₂×Q₈	1	4	yes	—
65	C₂×C₄⋊Q₈	C₂×C₃²	1	4	yes	—
66	#395	C₂×C₃²	1	4	yes	—
67	#389	C₂×C₃²	1	4	yes	—
68	#384	C₂×C₃²	1	1	yes	characteristic
69	C₄×C₄⋊C₄	C₂×C₃²	1	2	yes	—
70	C₂×C₃×Q₈	—	2	64	no	—
71	C₂×C₃×Q₈	—	4	16	no	—
72	C₂²×C₃×C₄	C₂³×C₃	1	16	yes	—
73	C₂²×C₃×C₄	C₂³×C₃	1	16	yes	—
74	C₂²×C₃×C₄	C₂³×C₃	1	4	yes	—
75	C₂²×C₃×C₄	C₂³×C₃	1	16	yes	—
76	C₂²×C₃×C₄	C₂³×C₃	1	8	yes	—
77	C₃×C₄⋊C₄	—	2	64	no	—
78	C₃×C₄⋊C₄	—	2	32	no	—
79	C₃×C₄⋊C₄	—	4	8	no	—
80	C₃×C₄⋊C₄	C₃×D₄	1	32	yes	—
81	C₃×C₄⋊C₄	C₃×Q₈	1	32	yes	complemented
82	C₃×C₄²	—	2	32	no	—
83	C₃×C₄²	C₃×Q₈	1	16	yes	complemented
84	C₂²×C₃²	C₂²×D₄	1	2	yes	—
85	C₂²×C₃²	C₂²×Q₈	1	1	yes	characteristic
86	C₂²×C₃²	C₂²×Q₈	1	2	yes	—
87	C₂²×C₃²	H(5,𝔽₂)	1	1	yes	characteristic
88	C₂²×C₃²	2₋¹⁺⁴	1	1	yes	characteristic
89	C₃²×C₄	—	2	2	no	—
90	C₃²×C₄	—	2	8	no	—
91	C₃²×C₄	—	2	8	no	—
92	C₃²×C₄	—	4	4	no	—
93	C₃²×C₄	C₄⋊Q₈	1	8	yes	complemented
94	C₂²×Q₈	C₂²×C₃²	1	4	yes	—
95	C₄⋊Q₈	—	2	16	no	—
96	C₂×C₄⋊C₄	C₂²×C₃²	1	4	yes	—
97	C₂×C₄⋊C₄	C₂²×C₃²	1	8	yes	—
98	C₂×C₄⋊C₄	C₂²×C₃²	1	2	yes	—
99	C₂×C₄⋊C₄	C₂²×C₃²	1	2	yes	—
100	C₂×C₄⋊C₄	C₂²×C₃²	1	2	yes	—
101	C₂×C₄²	C₂²×C₃²	1	4	yes	—
102	C₂×C₄²	C₂²×C₃²	1	1	yes	characteristic
103	#95	C₂²×C₃²	1	4	yes	—
104	#95	C₂²×C₃²	1	4	yes	—
105	C₂³×C₃	C₂⁴×C₃	1	4	yes	—
106	C₃×Q₈	—	4	64	no	—
107	C₂×C₃×C₄	—	2	16	no	—
108	C₂×C₃×C₄	—	2	16	no	—
109	C₂×C₃×C₄	—	2	4	no	—
110	C₂×C₃×C₄	—	2	16	no	—
111	C₂×C₃×C₄	—	2	4	no	—
112	C₂×C₃×C₄	—	2	16	no	—
113	C₂×C₃×C₄	—	2	32	no	—
114	C₂×C₃×C₄	—	2	16	no	—
115	C₂×C₃×C₄	C₂×C₃×D₄	1	32	yes	—
116	C₂×C₃×C₄	C₂×C₃×D₄	1	16	yes	—
117	C₂×C₃×C₄	C₂×C₃×Q₈	1	8	yes	—
118	C₂×C₃×C₄	C₂×C₃×Q₈	1	32	yes	—
119	C₂×C₃×C₄	C₂×C₃×Q₈	1	16	yes	—
120	C₂×C₃×C₄	C₂×C₃×Q₈	1	16	yes	—
121	C₂×C₃²	C₂×C₄⋊Q₈	1	2	yes	—
122	C₂×C₃²	#544	1	1	yes	characteristic
123	C₂×C₃²	D₄²	1	1	yes	characteristic
124	C₂×C₃²	D₄×Q₈	1	2	yes	—
125	C₂×C₃²	Q₈²	1	1	yes	characteristic
126	C₂×Q₈	—	2	16	no	—
127	C₂×Q₈	—	4	4	no	—
128	C₂²×C₄	C₂³×C₃²	1	4	yes	—
129	C₂²×C₄	C₂³×C₃²	1	4	yes	—
130	C₂²×C₄	C₂³×C₃²	1	1	yes	characteristic
131	C₂²×C₄	C₂³×C₃²	1	4	yes	—
132	C₂²×C₄	C₂³×C₃²	1	2	yes	—
133	C₄⋊C₄	—	2	16	no	—
134	C₄⋊C₄	—	2	8	no	—
135	C₄⋊C₄	—	4	2	no	—
136	C₄⋊C₄	C₃²×D₄	1	8	yes	—
137	C₄⋊C₄	C₃²×Q₈	1	8	yes	complemented
138	C₄²	—	2	8	no	—
139	C₄²	C₃²×Q₈	1	4	yes	complemented
140	C₂²×C₃	C₂²×C₃×D₄	1	8	yes	—
141	C₂²×C₃	C₂²×C₃×Q₈	1	8	yes	—
142	C₂²×C₃	C₂²×C₃×Q₈	1	4	yes	—
143	C₂²×C₃	C₃×H(5,𝔽₂)	1	4	yes	—
144	C₂²×C₃	C₃×2₋¹⁺⁴	1	4	yes	—
145	C₃×C₄	—	2	8	no	—
146	C₃×C₄	—	2	32	no	—
147	C₃×C₄	—	2	32	no	—
148	C₃×C₄	—	4	16	no	—
149	C₃×C₄	C₃×C₄⋊Q₈	1	32	yes	complemented
150	C₃²	#2551	1	1	yes	characteristic, direct product factor, Sylow
151	C₂³	C₂⁴×C₃²	1	1	yes	characteristic
152	Q₈	—	4	16	no	—
153	C₂×C₄	—	2	4	no	—
154	C₂×C₄	—	2	4	no	—
155	C₂×C₄	—	2	1	no	—
156	C₂×C₄	—	2	4	no	—
157	C₂×C₄	—	2	1	no	—
158	C₂×C₄	—	2	4	no	—
159	C₂×C₄	—	2	8	no	—
160	C₂×C₄	—	2	4	no	—
161	C₂×C₄	C₂×C₃²×D₄	1	8	yes	—
162	C₂×C₄	C₂×C₃²×D₄	1	4	yes	—
163	C₂×C₄	C₂×C₃²×Q₈	1	2	yes	—
164	C₂×C₄	C₂×C₃²×Q₈	1	8	yes	—
165	C₂×C₄	C₂×C₃²×Q₈	1	4	yes	—
166	C₂×C₄	C₂×C₃²×Q₈	1	4	yes	—
167	C₂×C₃	C₂×C₃×C₄⋊Q₈	1	8	yes	—
168	C₂×C₃	C₃×#544	1	4	yes	—
169	C₂×C₃	C₃×D₄²	1	4	yes	—
170	C₂×C₃	C₃×D₄×Q₈	1	8	yes	—
171	C₂×C₃	C₃×Q₈²	1	4	yes	—
172	C₂²	C₂²×C₃²×D₄	1	2	yes	—
173	C₂²	C₂²×C₃²×Q₈	1	2	yes	—
174	C₂²	C₂²×C₃²×Q₈	1	1	yes	characteristic
175	C₂²	C₃²×H(5,𝔽₂)	1	1	yes	characteristic
176	C₂²	C₃²×2₋¹⁺⁴	1	1	yes	characteristic
177	C₄	—	2	2	no	—
178	C₄	—	2	8	no	—
179	C₄	—	2	8	no	—
180	C₄	—	4	4	no	—
181	C₄	C₃²×C₄⋊Q₈	1	8	yes	complemented
182	C₃	C₃×#2551	1	4	yes	direct product factor
183	C₂	C₂×C₃²×C₄⋊Q₈	1	2	yes	—
184	C₂	C₃²×#544	1	1	yes	characteristic
185	C₂	C₃²×D₄²	1	1	yes	characteristic
186	C₂	C₃²×D₄×Q₈	1	2	yes	—
187	C₂	C₃²×Q₈²	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
6. C₃×#2551	C₃	direct (N × H)
34. C₃²×C₄⋊C₄	Q₈	semidirect
36. C₃²×C₄²	Q₈	semidirect
37. #2551	C₃²	direct (N × H)
81. C₃×C₄⋊C₄	C₃×Q₈	semidirect
83. C₃×C₄²	C₃×Q₈	semidirect
93. C₃²×C₄	C₄⋊Q₈	semidirect
137. C₄⋊C₄	C₃²×Q₈	semidirect
139. C₄²	C₃²×Q₈	semidirect
149. C₃×C₄	C₃×C₄⋊Q₈	semidirect
150. C₃²	#2551	direct (N × H)
181. C₄	C₃²×C₄⋊Q₈	semidirect
182. C₃	C₃×#2551	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃²×C₄⋊Q₈	2
C₃²×#395	2
C₃²×#389	2
C₃²×#384	2
C₃²×C₄×C₄⋊C₄	2
C₃×#2551	3

Subgroups — C32×#2551 #11769600

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×#2551 #11769600