Subgroups — C₂×C₈×#634 #11861250

Order 1152.

1320 conjugacy classes of subgroups.

The subgroup lattice has 219 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×#634	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃²×C₄×C₈	C₂	1	1	yes	maximal, characteristic
3	C₂×#63641	C₂	1	2	yes	maximal
4	C₂×C₈×C₃⋊Dic₃	C₂	1	2	yes	maximal
5	C₂×C₄⋊(C₃⋊(C₃⋊C₈))	C₂	1	1	yes	maximal, characteristic
6	C₈×#634	C₂	1	8	yes	maximal, direct product factor
7	C₂×C₈×Dic₆	—	3	4	no	maximal
8	C₂²×#634	C₄	1	1	yes	characteristic
9	C₂²×C₃²×C₈	C₂²	1	1	yes	characteristic
10	C₂²×C₃²×C₈	C₂²	1	1	yes	characteristic
11	C₂×C₃²×C₄²	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊(C₃⋊Dic₃)	C₄	1	1	yes	characteristic
13	C₂×#3750	C₄	1	2	yes	—
14	C₂×C₄×C₃⋊Dic₃	C₂²	1	2	yes	—
15	C₂²×C₃⋊(C₃⋊C₈)	C₂²	1	2	yes	—
16	C₄×#634	C₄	1	4	yes	—
17	C₄×#634	C₂²	1	4	yes	—
18	C₃²×C₄×C₈	C₂²	1	4	yes	—
19	#63641	C₂²	1	8	yes	—
20	C₈×C₃⋊Dic₃	C₂²	1	8	yes	—
21	C₄⋊(C₃⋊(C₃⋊C₈))	C₂²	1	4	yes	—
22	C₂×C₄×Dic₆	—	3	4	no	—
23	C₂×C₃×C₄×C₈	D₃	1	4	yes	—
24	C₂×Dic₃⋊C₈	—	3	8	no	—
25	C₂×C₈×Dic₃	—	3	8	no	—
26	C₂×C₄⋊(C₃⋊C₈)	—	3	4	no	—
27	C₈×Dic₆	—	3	32	no	—
28	C₂²×C₃²×C₄	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₃²×C₄	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
31	C₂²×C₃⋊Dic₃	C₂×C₄	1	2	yes	—
32	C₂²×C₃⋊Dic₃	C₂×C₄	1	2	yes	—
33	C₂×#634	C₈	1	8	yes	direct product factor
34	C₂×#634	C₂×C₄	1	4	yes	—
35	C₂×C₃²×C₈	—	2	2	no	—
36	C₂×C₃²×C₈	Q₈	1	4	yes	complemented
37	C₂×C₃²×C₈	C₂³	1	2	yes	—
38	C₂×C₃²×C₈	C₂³	1	2	yes	—
39	C₃²×C₄²	C₂×C₄	1	2	yes	—
40	C₃²×C₄²	C₂³	1	2	yes	—
41	C₄⋊(C₃⋊Dic₃)	C₂×C₄	1	4	yes	—
42	#3750	C₂×C₄	1	8	yes	—
43	C₄×C₃⋊Dic₃	C₂×C₄	1	4	yes	—
44	C₄×C₃⋊Dic₃	C₂³	1	4	yes	—
45	C₂×C₃⋊(C₃⋊C₈)	—	2	4	no	—
46	C₂×C₃⋊(C₃⋊C₈)	C₂³	1	4	yes	—
47	C₂×C₈×Q₈	—	9	1	no	Sylow
48	C₂²×Dic₆	—	3	4	no	—
49	C₂²×C₃×C₈	C₂×D₃	1	4	yes	—
50	C₂²×C₃×C₈	C₂×D₃	1	4	yes	—
51	C₂×C₃×C₄²	C₂×D₃	1	4	yes	—
52	C₂×C₄⋊Dic₃	—	3	4	no	—
53	C₂×Dic₃⋊C₄	—	3	8	no	—
54	C₂×C₄×Dic₃	—	3	8	no	—
55	C₂²×C₃⋊C₈	—	3	8	no	—
56	C₄×Dic₆	—	3	16	no	—
57	C₄×Dic₆	—	3	16	no	—
58	C₃×C₄×C₈	C₂×D₃	1	16	yes	—
59	Dic₃⋊C₈	—	3	32	no	—
60	C₈×Dic₃	—	3	32	no	—
61	C₄⋊(C₃⋊C₈)	—	3	16	no	—
62	C₂³×C₃²	C₂²×C₄	1	1	yes	characteristic
63	C₂×C₃²×C₄	—	2	2	no	—
64	C₂×C₃²×C₄	C₂×C₈	1	4	yes	—
65	C₂×C₃²×C₄	C₂²×C₄	1	2	yes	—
66	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
67	C₂×C₃²×C₄	C₂²×C₄	1	2	yes	—
68	C₂×C₃²×C₄	C₂×Q₈	1	2	yes	—
69	C₂×C₃²×C₄	#41	1	2	yes	—
70	C₂×C₃²×C₄	C₂⁴	1	1	yes	characteristic
71	C₂×C₃⋊Dic₃	—	2	4	no	—
72	C₂×C₃⋊Dic₃	C₂×C₈	1	8	yes	—
73	C₂×C₃⋊Dic₃	C₂²×C₄	1	4	yes	—
74	C₂×C₃⋊Dic₃	C₂²×C₄	1	4	yes	—
75	#634	C₂×C₈	1	16	yes	direct product factor
76	C₃²×C₈	—	2	2	no	—
77	C₃²×C₈	C₂×Q₈	1	4	yes	complemented
78	C₃⋊(C₃⋊C₈)	—	2	4	no	—
79	C₂×C₄×Q₈	—	9	1	no	—
80	C₈×Q₈	—	9	8	no	—
81	C₂×C₄⋊C₈	—	9	2	no	—
82	C₂×C₄⋊C₈	—	9	1	no	—
83	C₂×C₄×C₈	—	9	2	no	—
84	C₂×C₄×C₈	C₃⋊D₃	1	1	yes	characteristic
85	C₂²×C₃×C₄	C₄×D₃	1	4	yes	—
86	C₂²×C₃×C₄	C₄×D₃	1	4	yes	—
87	C₂²×C₃×C₄	C₂²×D₃	1	4	yes	—
88	C₂²×Dic₃	—	3	8	no	—
89	C₂²×Dic₃	—	3	8	no	—
90	C₂×Dic₆	—	3	32	no	—
91	C₂×Dic₆	—	3	16	no	—
92	C₂×C₃×C₈	—	2	8	no	—
93	C₂×C₃×C₈	Dic₆	1	16	yes	complemented
94	C₂×C₃×C₈	C₂²×D₃	1	8	yes	—
95	C₂×C₃×C₈	C₂²×D₃	1	8	yes	—
96	C₃×C₄²	C₄×D₃	1	8	yes	—
97	C₃×C₄²	C₂²×D₃	1	8	yes	—
98	C₄⋊Dic₃	—	3	16	no	—
99	Dic₃⋊C₄	—	3	32	no	—
100	C₄×Dic₃	—	3	16	no	—
101	C₄×Dic₃	—	3	16	no	—
102	C₂×C₃⋊C₈	—	3	16	no	—
103	C₂×C₃⋊C₈	—	6	16	no	—
104	C₂²×C₃²	C₄×Q₈	1	2	yes	—
105	C₂²×C₃²	C₂²×C₈	1	2	yes	—
106	C₂²×C₃²	#131	1	2	yes	—
107	C₂²×C₃²	C₂³×C₄	1	1	yes	characteristic
108	C₃²×C₄	—	2	2	no	—
109	C₃²×C₄	C₄×Q₈	1	2	yes	—
110	C₃²×C₄	C₂²×C₈	1	4	yes	—
111	C₃²×C₄	C₂²×Q₈	1	1	yes	characteristic
112	C₃²×C₄	C₂×#41	1	1	yes	characteristic
113	C₃⋊Dic₃	—	2	4	no	—
114	C₃⋊Dic₃	C₂²×C₈	1	8	yes	—
115	C₂²×Q₈	—	9	1	no	—
116	C₂²×C₈	—	9	2	no	—
117	C₂²×C₈	C₂×C₃⋊D₃	1	1	yes	characteristic
118	C₂²×C₈	C₂×C₃⋊D₃	1	1	yes	characteristic
119	C₄×Q₈	—	9	4	no	—
120	C₄×Q₈	—	9	4	no	—
121	C₂×C₄⋊C₄	—	9	2	no	—
122	C₂×C₄⋊C₄	—	9	1	no	—
123	C₂×C₄²	—	9	2	no	—
124	C₂×C₄²	C₂×C₃⋊D₃	1	1	yes	characteristic
125	C₄⋊C₈	—	9	8	no	—
126	C₄⋊C₈	—	9	4	no	—
127	C₄×C₈	—	9	8	no	—
128	C₄×C₈	C₂×C₃⋊D₃	1	4	yes	—
129	C₂³×C₃	C₂×C₄×D₃	1	4	yes	—
130	C₂×C₃×C₄	—	2	8	no	—
131	C₂×C₃×C₄	D₃×C₈	1	16	yes	—
132	C₂×C₃×C₄	C₂×Dic₆	1	8	yes	—
133	C₂×C₃×C₄	C₂×C₄×D₃	1	8	yes	—
134	C₂×C₃×C₄	C₂×C₄×D₃	1	4	yes	—
135	C₂×C₃×C₄	C₂×C₄×D₃	1	8	yes	—
136	C₂×C₃×C₄	#235	1	8	yes	—
137	C₂×C₃×C₄	C₂³×D₃	1	4	yes	—
138	C₂×Dic₃	—	3	32	no	—
139	C₂×Dic₃	—	3	16	no	—
140	C₂×Dic₃	—	3	16	no	—
141	C₂×Dic₃	—	6	16	no	—
142	Dic₆	—	3	64	no	—
143	C₃×C₈	—	2	8	no	—
144	C₃×C₈	C₂×Dic₆	1	16	yes	complemented
145	C₃⋊C₈	—	6	16	no	—
146	C₂×C₃²	C₈×Q₈	1	4	yes	complemented
147	C₂×C₃²	C₂×C₄×Q₈	1	1	yes	characteristic
148	C₂×C₃²	C₂³×C₈	1	1	yes	characteristic
149	C₂×C₃²	C₂×#131	1	1	yes	characteristic
150	C₂×Q₈	—	9	8	no	—
151	C₂×Q₈	—	9	4	no	—
152	C₂²×C₄	—	9	2	no	—
153	C₂²×C₄	—	9	2	no	—
154	C₂²×C₄	C₄×C₃⋊D₃	1	1	yes	characteristic
155	C₂²×C₄	C₄×C₃⋊D₃	1	1	yes	characteristic
156	C₂²×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
157	C₂×C₈	—	2	2	no	—
158	C₂×C₈	—	9	4	no	—
159	C₂×C₈	—	18	4	no	—
160	C₂×C₈	#634	1	4	yes	direct product factor
161	C₂×C₈	C₂²×C₃⋊D₃	1	2	yes	—
162	C₂×C₈	C₂²×C₃⋊D₃	1	2	yes	—
163	C₄⋊C₄	—	9	8	no	—
164	C₄⋊C₄	—	9	4	no	—
165	C₄²	—	9	4	no	—
166	C₄²	—	9	4	no	—
167	C₄²	C₄×C₃⋊D₃	1	2	yes	—
168	C₄²	C₂²×C₃⋊D₃	1	2	yes	—
169	C₂²×C₃	C₄×Dic₆	1	8	yes	—
170	C₂²×C₃	C₂×D₃×C₈	1	8	yes	—
171	C₂²×C₃	#901	1	8	yes	—
172	C₂²×C₃	C₂²×C₄×D₃	1	4	yes	—
173	C₃×C₄	—	2	8	no	—
174	C₃×C₄	C₄×Dic₆	1	8	yes	—
175	C₃×C₄	C₂×D₃×C₈	1	16	yes	—
176	C₃×C₄	C₂²×Dic₆	1	4	yes	—
177	C₃×C₄	C₂×#235	1	4	yes	—
178	Dic₃	—	3	32	no	—
179	Dic₃	—	6	16	no	—
180	C₃²	C₂×C₈×Q₈	1	1	yes	characteristic, complemented, Sylow
181	C₂³	C₂×C₄×C₃⋊D₃	1	1	yes	characteristic
182	Q₈	—	9	16	no	—
183	C₂×C₄	—	2	2	no	—
184	C₂×C₄	—	9	4	no	—
185	C₂×C₄	—	9	8	no	—
186	C₂×C₄	—	9	4	no	—
187	C₂×C₄	—	18	4	no	—
188	C₂×C₄	C₈×C₃⋊D₃	1	4	yes	—
189	C₂×C₄	C₂×#634	1	2	yes	—
190	C₂×C₄	C₂×C₄×C₃⋊D₃	1	2	yes	—
191	C₂×C₄	C₂×C₄×C₃⋊D₃	1	1	yes	characteristic
192	C₂×C₄	C₂×C₄×C₃⋊D₃	1	2	yes	—
193	C₂×C₄	#3828	1	2	yes	—
194	C₂×C₄	C₂³×C₃⋊D₃	1	1	yes	characteristic
195	C₈	—	2	2	no	—
196	C₈	—	18	4	no	—
197	C₈	C₂×#634	1	4	yes	direct product factor
198	C₂×C₃	C₈×Dic₆	1	16	yes	complemented
199	C₂×C₃	C₂×C₄×Dic₆	1	4	yes	—
200	C₂×C₃	C₂²×D₃×C₈	1	4	yes	—
201	C₂×C₃	C₂×#901	1	4	yes	—
202	C₂²	C₄×#634	1	2	yes	—
203	C₂²	C₂×C₈×C₃⋊D₃	1	2	yes	—
204	C₂²	#64110	1	2	yes	—
205	C₂²	C₂²×C₄×C₃⋊D₃	1	1	yes	characteristic
206	C₄	—	2	2	no	—
207	C₄	—	9	8	no	—
208	C₄	—	18	4	no	—
209	C₄	C₄×#634	1	2	yes	—
210	C₄	C₂×C₈×C₃⋊D₃	1	4	yes	—
211	C₄	C₂²×#634	1	1	yes	characteristic
212	C₄	C₂×#3828	1	1	yes	characteristic
213	C₃	C₂×C₈×Dic₆	1	4	yes	complemented
214	C₂	C₈×#634	1	4	yes	direct product factor
215	C₂	C₂×C₄×#634	1	1	yes	characteristic
216	C₂	C₂²×C₈×C₃⋊D₃	1	1	yes	characteristic
217	C₂	C₂×#64110	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₈	C₂×C₃⋊D₃
commutator	C₂×C₃²	C₂³×C₈
Frattini	C₂×C₄	C₂³×C₃⋊D₃
Fitting	C₂×C₃²×C₄×C₈	C₂
socle, prime radical	C₂³×C₃²	C₂²×C₄
cosocle	C₂×C₃²×C₄	C₂⁴
hypercenter	C₂×C₄×C₈	C₃⋊D₃
hypocenter	C₃²	C₂×C₈×Q₈
LCS	G ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
6. C₈×#634	C₂	direct (N × H)
33. C₂×#634	C₈	direct (N × H)
36. C₂×C₃²×C₈	Q₈	semidirect
75. #634	C₂×C₈	direct (N × H)
77. C₃²×C₈	C₂×Q₈	semidirect
93. C₂×C₃×C₈	Dic₆	semidirect
144. C₃×C₈	C₂×Dic₆	semidirect
146. C₂×C₃²	C₈×Q₈	semidirect
160. C₂×C₈	#634	direct (N × H)
180. C₃²	C₂×C₈×Q₈	semidirect
197. C₈	C₂×#634	direct (N × H)
198. C₂×C₃	C₈×Dic₆	semidirect
213. C₃	C₂×C₈×Dic₆	semidirect
214. C₂	C₈×#634	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×C₄×#634	2
C₂×C₃²×C₄×C₈	2
C₂×#63641	2
C₂×C₈×C₃⋊Dic₃	2
C₂×C₄⋊(C₃⋊(C₃⋊C₈))	2
C₈×#634	2
C₂×C₈×Dic₆	3

Subgroups — C2×C8×#634 #11861250

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₈×#634 #11861250