Subgroups — C₂×#10589346 #11776752

Order 1152.

600 conjugacy classes of subgroups.

The subgroup lattice has 192 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊(C₃⋊(C₃⋊C₈))	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×C₃⋊(C₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
4	#10589346	C₂	1	4	yes	maximal, direct product factor
5	C₂×#4480	—	3	4	no	maximal
6	C₂²×C₃²×C₈	—	2	1	no	—
7	C₂×C₃²×C₄²	C₂²	1	1	yes	characteristic
8	C₂²×C₃⋊(C₃⋊C₈)	—	2	1	no	—
9	C₂²×C₃⋊(C₃⋊C₈)	C₄	1	1	yes	characteristic
10	C₂²×C₃⋊(C₃⋊C₈)	C₄	1	1	yes	characteristic
11	C₃²×C₄⋊C₈	—	2	1	no	—
12	C₃²×C₄⋊C₈	C₂²	1	2	yes	—
13	C₄⋊(C₃⋊(C₃⋊C₈))	—	2	1	no	—
14	C₄⋊(C₃⋊(C₃⋊C₈))	C₂²	1	2	yes	—
15	C₄×C₃⋊(C₃⋊C₈)	C₄	1	2	yes	—
16	C₄×C₃⋊(C₃⋊C₈)	C₂²	1	2	yes	—
17	C₂×C₃×C₄⋊C₈	D₃	1	4	yes	—
18	C₂×C₄⋊(C₃⋊C₈)	—	3	4	no	—
19	C₂×C₄×C₃⋊C₈	—	3	4	no	—
20	#4480	—	3	16	no	—
21	C₂²×C₃²×C₄	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₃²×C₄	D₄	1	1	yes	characteristic
23	C₂²×C₃²×C₄	Q₈	1	1	yes	characteristic
24	C₂×C₃²×C₈	—	2	2	no	—
25	C₂×C₃²×C₈	—	4	2	no	—
26	C₃²×C₄²	C₂×C₄	1	1	yes	characteristic
27	C₃²×C₄²	D₄	1	1	yes	characteristic
28	C₃²×C₄²	Q₈	1	1	yes	characteristic
29	C₃²×C₄²	C₂³	1	1	yes	characteristic
30	C₂×C₃⋊(C₃⋊C₈)	—	2	2	no	—
31	C₂×C₃⋊(C₃⋊C₈)	—	2	2	no	—
32	C₂×C₃⋊(C₃⋊C₈)	—	4	2	no	—
33	C₂×C₃⋊(C₃⋊C₈)	C₈	1	4	yes	complemented
34	C₂×C₃⋊(C₃⋊C₈)	C₂×C₄	1	2	yes	—
35	C₂×C₃⋊(C₃⋊C₈)	C₂×C₄	1	2	yes	—
36	C₂×#334	—	9	1	no	Sylow
37	C₂²×C₃×C₈	—	2	4	no	—
38	C₂×C₃×C₄²	C₂×D₃	1	4	yes	—
39	C₂²×C₃⋊C₈	—	3	4	no	—
40	C₂²×C₃⋊C₈	—	3	4	no	—
41	C₂²×C₃⋊C₈	—	6	4	no	—
42	C₃×C₄⋊C₈	—	2	4	no	—
43	C₃×C₄⋊C₈	C₂×D₃	1	8	yes	—
44	C₄⋊(C₃⋊C₈)	—	3	8	no	—
45	C₄⋊(C₃⋊C₈)	—	6	4	no	—
46	C₄×C₃⋊C₈	—	3	8	no	—
47	C₄×C₃⋊C₈	—	3	8	no	—
48	C₂³×C₃²	C₄⋊C₄	1	1	yes	characteristic
49	C₂×C₃²×C₄	—	2	2	no	—
50	C₂×C₃²×C₄	C₄⋊C₄	1	1	yes	characteristic
51	C₂×C₃²×C₄	C₄⋊C₄	1	1	yes	characteristic
52	C₂×C₃²×C₄	C₄⋊C₄	1	1	yes	characteristic
53	C₂×C₃²×C₄	C₂×C₈	1	2	yes	—
54	C₂×C₃²×C₄	M₁₆	1	2	yes	—
55	C₂×C₃²×C₄	SD₁₆	1	2	yes	—
56	C₂×C₃²×C₄	SD₁₆	1	2	yes	—
57	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
59	C₂×C₃²×C₄	C₂×Q₈	1	1	yes	characteristic
60	C₃²×C₈	—	4	2	no	—
61	C₃⋊(C₃⋊C₈)	—	2	2	no	—
62	C₃⋊(C₃⋊C₈)	—	4	2	no	—
63	C₃⋊(C₃⋊C₈)	C₂×C₈	1	4	yes	complemented
64	C₂×C₄⋊C₈	—	9	1	no	—
65	C₂×C₄⋊C₈	C₃⋊D₃	1	1	yes	characteristic
66	C₂×C₄×C₈	—	9	1	no	—
67	#334	—	9	4	no	—
68	C₂²×C₃×C₄	Dic₆	1	4	yes	—
69	C₂²×C₃×C₄	C₄×D₃	1	4	yes	—
70	C₂²×C₃×C₄	#67	1	4	yes	—
71	C₂×C₃×C₈	—	2	8	no	—
72	C₂×C₃×C₈	—	4	8	no	—
73	C₃×C₄²	Dic₆	1	4	yes	—
74	C₃×C₄²	C₄×D₃	1	4	yes	—
75	C₃×C₄²	#67	1	4	yes	—
76	C₃×C₄²	C₂²×D₃	1	4	yes	—
77	C₂×C₃⋊C₈	—	3	16	no	—
78	C₂×C₃⋊C₈	—	3	8	no	—
79	C₂×C₃⋊C₈	—	3	8	no	—
80	C₂×C₃⋊C₈	—	6	8	no	—
81	C₂×C₃⋊C₈	—	6	8	no	—
82	C₂×C₃⋊C₈	—	12	8	no	—
83	C₂²×C₃²	C₄⋊C₈	1	2	yes	—
84	C₂²×C₃²	#106	1	2	yes	—
85	C₂²×C₃²	#108	1	2	yes	—
86	C₂²×C₃²	C₂×C₄⋊C₄	1	1	yes	characteristic
87	C₃²×C₄	—	2	1	no	—
88	C₃²×C₄	—	2	1	no	—
89	C₃²×C₄	C₄⋊C₈	1	2	yes	—
90	C₃²×C₄	#106	1	2	yes	—
91	C₃²×C₄	C₂²×C₈	1	1	yes	characteristic
92	C₃²×C₄	C₂×M₁₆	1	1	yes	characteristic
93	C₃²×C₄	C₂×SD₁₆	1	1	yes	characteristic
94	C₃²×C₄	C₂×SD₁₆	1	1	yes	characteristic
95	C₂²×C₈	—	2	1	no	—
96	C₂²×C₈	—	9	1	no	—
97	C₂²×C₈	—	9	1	no	—
98	C₂²×C₈	—	18	1	no	—
99	C₂×C₄²	C₂×C₃⋊D₃	1	1	yes	characteristic
100	C₄⋊C₈	—	2	1	no	—
101	C₄⋊C₈	—	9	2	no	—
102	C₄⋊C₈	—	18	1	no	—
103	C₄⋊C₈	C₂×C₃⋊D₃	1	2	yes	—
104	C₄×C₈	—	9	2	no	—
105	C₄×C₈	—	9	2	no	—
106	C₂³×C₃	Dic₃⋊C₄	1	4	yes	—
107	C₂×C₃×C₄	—	2	8	no	—
108	C₂×C₃×C₄	D₃×C₈	1	8	yes	—
109	C₂×C₃×C₄	ΓL(1,𝔽₂₅)	1	8	yes	—
110	C₂×C₃×C₄	Dic₃⋊C₄	1	4	yes	—
111	C₂×C₃×C₄	Dic₃⋊C₄	1	4	yes	—
112	C₂×C₃×C₄	Dic₃⋊C₄	1	4	yes	—
113	C₂×C₃×C₄	#214	1	8	yes	—
114	C₂×C₃×C₄	#215	1	8	yes	—
115	C₂×C₃×C₄	C₂×Dic₆	1	4	yes	—
116	C₂×C₃×C₄	C₂×C₄×D₃	1	4	yes	—
117	C₂×C₃×C₄	C₂×#67	1	4	yes	—
118	C₃×C₈	—	4	8	no	—
119	C₃⋊C₈	—	3	16	no	—
120	C₃⋊C₈	—	6	8	no	—
121	C₃⋊C₈	—	12	8	no	—
122	C₂×C₃²	#334	1	4	yes	complemented
123	C₂×C₃²	C₂×C₄⋊C₈	1	1	yes	characteristic
124	C₂×C₃²	C₂×#106	1	1	yes	characteristic
125	C₂×C₃²	C₂×#108	1	1	yes	characteristic
126	C₂²×C₄	#634	1	1	yes	characteristic
127	C₂²×C₄	C₄×C₃⋊D₃	1	1	yes	characteristic
128	C₂²×C₄	#638	1	1	yes	characteristic
129	C₂×C₈	—	2	2	no	—
130	C₂×C₈	—	4	2	no	—
131	C₂×C₈	—	9	4	no	—
132	C₂×C₈	—	9	2	no	—
133	C₂×C₈	—	9	2	no	—
134	C₂×C₈	—	18	2	no	—
135	C₂×C₈	—	18	2	no	—
136	C₂×C₈	—	36	2	no	—
137	C₄²	#634	1	1	yes	characteristic
138	C₄²	C₄×C₃⋊D₃	1	1	yes	characteristic
139	C₄²	#638	1	1	yes	characteristic
140	C₄²	C₂²×C₃⋊D₃	1	1	yes	characteristic
141	C₂²×C₃	#808	1	8	yes	—
142	C₂²×C₃	Dic₃⋊C₈	1	8	yes	—
143	C₂²×C₃	#822	1	8	yes	—
144	C₂²×C₃	C₂×Dic₃⋊C₄	1	4	yes	—
145	C₃×C₄	—	2	4	no	—
146	C₃×C₄	—	2	4	no	—
147	C₃×C₄	#808	1	8	yes	—
148	C₃×C₄	Dic₃⋊C₈	1	8	yes	—
149	C₃×C₄	C₂×D₃×C₈	1	4	yes	—
150	C₃×C₄	C₂×ΓL(1,𝔽₂₅)	1	4	yes	—
151	C₃×C₄	C₂×#214	1	4	yes	—
152	C₃×C₄	C₂×#215	1	4	yes	—
153	C₃²	C₂×#334	1	1	yes	characteristic, complemented, Sylow
154	C₂³	#3750	1	1	yes	characteristic
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	C₈×C₃⋊D₃	1	2	yes	—
157	C₂×C₄	#3743	1	2	yes	—
158	C₂×C₄	#3750	1	1	yes	characteristic
159	C₂×C₄	#3750	1	1	yes	characteristic
160	C₂×C₄	#3750	1	1	yes	characteristic
161	C₂×C₄	#3754	1	2	yes	—
162	C₂×C₄	#3755	1	2	yes	—
163	C₂×C₄	C₂×#634	1	1	yes	characteristic
164	C₂×C₄	C₂×C₄×C₃⋊D₃	1	1	yes	characteristic
165	C₂×C₄	C₂×#638	1	1	yes	characteristic
166	C₈	—	4	2	no	—
167	C₈	—	9	4	no	—
168	C₈	—	18	2	no	—
169	C₈	—	36	2	no	—
170	C₂×C₃	#4480	1	16	yes	complemented
171	C₂×C₃	C₂×#808	1	4	yes	—
172	C₂×C₃	C₂×Dic₃⋊C₈	1	4	yes	—
173	C₂×C₃	C₂×#822	1	4	yes	—
174	C₂²	#63635	1	2	yes	—
175	C₂²	#63641	1	2	yes	—
176	C₂²	#63649	1	2	yes	—
177	C₂²	C₂×#3750	1	1	yes	characteristic
178	C₄	—	2	1	no	—
179	C₄	—	2	1	no	—
180	C₄	#63635	1	2	yes	—
181	C₄	#63641	1	2	yes	—
182	C₄	C₂×C₈×C₃⋊D₃	1	1	yes	characteristic
183	C₄	C₂×#3743	1	1	yes	characteristic
184	C₄	C₂×#3754	1	1	yes	characteristic
185	C₄	C₂×#3755	1	1	yes	characteristic
186	C₃	C₂×#4480	1	4	yes	complemented
187	C₂	#10589346	1	4	yes	direct product factor
188	C₂	C₂×#63635	1	1	yes	characteristic
189	C₂	C₂×#63641	1	1	yes	characteristic
190	C₂	C₂×#63649	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	#638
commutator	C₃²×C₄	C₂²×C₈
Frattini	C₄²	C₂²×C₃⋊D₃
Fitting	C₂×C₃²×C₄⋊C₈	C₂
socle, prime radical	C₂³×C₃²	C₄⋊C₄
cosocle	C₃²×C₄²	C₂³
hypercenter	C₂×C₄⋊C₈	C₃⋊D₃
hypocenter	C₃²	C₂×#334
LCS	G ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
4. #10589346	C₂	direct (N × H)
33. C₂×C₃⋊(C₃⋊C₈)	C₈	semidirect
63. C₃⋊(C₃⋊C₈)	C₂×C₈	semidirect
122. C₂×C₃²	#334	semidirect
153. C₃²	C₂×#334	semidirect
170. C₂×C₃	#4480	semidirect
186. C₃	C₂×#4480	semidirect
187. C₂	#10589346	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⋊C₈	2
C₂×C₄⋊(C₃⋊(C₃⋊C₈))	2
C₂×C₄×C₃⋊(C₃⋊C₈)	2
#10589346	2
C₂×#4480	3

Subgroups — C2×#10589346 #11776752

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#10589346 #11776752