Subgroups — C₂×#10592805 #11858424

Order 1152.

1536 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₃²×C₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#64135	C₂	1	2	yes	maximal
3	#10592805	C₂	1	8	yes	maximal, direct product factor
4	C₂×#63645	C₂	1	2	yes	maximal, complemented
5	C₂×#63644	C₂	1	2	yes	maximal, complemented
6	C₂×#5106	—	3	4	no	maximal
7	C₂³×C₃²×C₄	C₂²	1	1	yes	characteristic
8	C₂²×C₃²×C₈	C₄	1	8	yes	complemented
9	C₂²×C₃²×C₈	C₂²	1	1	yes	characteristic
10	C₂²×C₃²×C₈	C₂²	1	1	yes	characteristic
11	C₂²×C₃²×C₈	C₂²	1	4	yes	—
12	C₂×#3757	—	2	2	no	—
13	#64135	—	2	4	no	—
14	#64135	C₂²	1	8	yes	—
15	C₂×C₄⋊(C₃⋊Dic₃)	C₂²	1	4	yes	—
16	C₂×C₄×C₃⋊Dic₃	—	2	2	no	—
17	#63645	C₂²	1	8	yes	complemented
18	#63644	C₂²	1	8	yes	complemented
19	C₂³×C₃×C₈	D₃	1	4	yes	—
20	C₂×#926	—	3	8	no	—
21	#5106	—	3	32	no	—
22	C₂×#818	—	3	8	no	—
23	C₂×#817	—	3	8	no	—
24	C₂⁴×C₃²	D₄	1	1	yes	characteristic
25	C₂²×C₃²×C₄	C₂×C₄	1	4	yes	—
26	C₂²×C₃²×C₄	D₄	1	1	yes	characteristic
27	C₂²×C₃²×C₄	D₄	1	2	yes	—
28	C₂²×C₃²×C₄	Q₈	1	4	yes	—
29	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
30	C₂²×C₃²×C₄	C₂³	1	2	yes	—
31	C₂²×C₃⋊Dic₃	—	2	4	no	—
32	C₂×C₃²×C₈	C₂×C₄	1	4	yes	—
33	C₂×C₃²×C₈	C₂×C₄	1	4	yes	—
34	C₂×C₃²×C₈	C₂×C₄	1	16	yes	complemented
35	C₂×C₃²×C₈	C₂³	1	2	yes	—
36	C₂×C₃²×C₈	C₂³	1	2	yes	—
37	#3757	—	2	8	no	—
38	C₄⋊(C₃⋊Dic₃)	—	2	4	no	—
39	C₄⋊(C₃⋊Dic₃)	C₂³	1	8	yes	—
40	C₄×C₃⋊Dic₃	—	2	8	no	—
41	C₂×#427	—	9	1	no	Sylow
42	C₂³×C₃×C₄	C₂×D₃	1	4	yes	—
43	C₂²×C₃×C₈	Dic₃	1	32	yes	complemented
44	C₂²×C₃×C₈	C₂×D₃	1	4	yes	—
45	C₂²×C₃×C₈	C₂×D₃	1	4	yes	—
46	C₂²×C₃×C₈	C₂×D₃	1	16	yes	—
47	C₂×#217	—	6	8	no	—
48	#926	—	3	32	no	—
49	#926	—	6	16	no	—
50	C₂×C₄⋊Dic₃	—	3	16	no	—
51	C₂×C₄×Dic₃	—	6	8	no	—
52	#818	—	3	32	no	—
53	#817	—	3	32	no	—
54	C₂³×C₃²	—	2	4	no	—
55	C₂³×C₃²	C₄⋊C₄	1	4	yes	—
56	C₂³×C₃²	C₂×D₄	1	2	yes	—
57	C₂³×C₃²	C₂×D₄	1	1	yes	characteristic
58	C₂×C₃²×C₄	C₄⋊C₄	1	8	yes	—
59	C₂×C₃²×C₄	C₄⋊C₄	1	4	yes	—
60	C₂×C₃²×C₄	C₂²×C₄	1	2	yes	—
61	C₂×C₃²×C₄	C₂²×C₄	1	4	yes	—
62	C₂×C₃²×C₄	C₂×D₄	1	2	yes	—
63	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
64	C₂×C₃²×C₄	C₂×Q₈	1	4	yes	—
65	C₂×C₃²×C₄	C₂×Q₈	1	2	yes	—
66	C₂×C₃²×C₄	C₂⁴	1	1	yes	characteristic
67	C₂×C₃⋊Dic₃	—	2	8	no	—
68	C₂×C₃⋊Dic₃	—	4	8	no	—
69	C₃²×C₈	C₂²×C₄	1	4	yes	—
70	C₃²×C₈	C₂²×C₄	1	4	yes	—
71	C₂³×C₈	C₃⋊D₃	1	1	yes	characteristic
72	C₂×#117	—	9	2	no	—
73	#427	—	9	8	no	—
74	C₂×#107	—	9	2	no	—
75	C₂×#106	—	9	2	no	—
76	C₂⁴×C₃	D₁₂	1	4	yes	—
77	C₂²×C₃×C₄	Dic₆	1	16	yes	—
78	C₂²×C₃×C₄	D₁₂	1	8	yes	—
79	C₂²×C₃×C₄	D₁₂	1	4	yes	—
80	C₂²×C₃×C₄	C₂×Dic₃	1	16	yes	—
81	C₂²×C₃×C₄	C₂²×D₃	1	4	yes	—
82	C₂²×C₃×C₄	C₂²×D₃	1	8	yes	—
83	C₂²×Dic₃	—	6	16	no	—
84	C₂×C₃×C₈	C₂×Dic₃	1	16	yes	—
85	C₂×C₃×C₈	C₂×Dic₃	1	16	yes	—
86	C₂×C₃×C₈	C₂×Dic₃	1	64	yes	complemented
87	C₂×C₃×C₈	C₂²×D₃	1	8	yes	—
88	C₂×C₃×C₈	C₂²×D₃	1	8	yes	—
89	#217	—	6	32	no	—
90	C₄⋊Dic₃	—	3	32	no	—
91	C₄⋊Dic₃	—	6	16	no	—
92	C₄×Dic₃	—	6	32	no	—
93	C₂²×C₃²	—	2	4	no	—
94	C₂²×C₃²	—	2	8	no	—
95	C₂²×C₃²	C₂×C₄⋊C₄	1	4	yes	—
96	C₂²×C₃²	C₂×C₄⋊C₄	1	2	yes	—
97	C₂²×C₃²	#135	1	4	yes	—
98	C₂²×C₃²	C₂²×D₄	1	1	yes	characteristic
99	C₃²×C₄	C₂×C₄⋊C₄	1	2	yes	—
100	C₃²×C₄	C₂×C₄⋊C₄	1	4	yes	—
101	C₃²×C₄	C₂³×C₄	1	1	yes	characteristic
102	C₃²×C₄	C₂²×Q₈	1	1	yes	characteristic
103	C₃⋊Dic₃	—	4	8	no	—
104	C₂³×C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
105	C₂²×C₈	C₃⋊Dic₃	1	8	yes	complemented
106	C₂²×C₈	C₂×C₃⋊D₃	1	1	yes	characteristic
107	C₂²×C₈	C₂×C₃⋊D₃	1	1	yes	characteristic
108	C₂²×C₈	C₂×C₃⋊D₃	1	4	yes	—
109	#117	—	9	8	no	—
110	#117	—	18	4	no	—
111	C₂×C₄⋊C₄	—	9	4	no	—
112	C₂×C₂²⋊C₄	—	18	2	no	—
113	C₂×C₄²	—	18	2	no	—
114	#107	—	9	8	no	—
115	#106	—	9	8	no	—
116	C₂³×C₃	—	2	16	no	—
117	C₂³×C₃	C₄⋊Dic₃	1	16	yes	—
118	C₂³×C₃	C₂×D₁₂	1	8	yes	—
119	C₂³×C₃	C₂×D₁₂	1	4	yes	—
120	C₂×C₃×C₄	C₄⋊Dic₃	1	32	yes	—
121	C₂×C₃×C₄	C₄⋊Dic₃	1	16	yes	—
122	C₂×C₃×C₄	C₂×Dic₆	1	16	yes	—
123	C₂×C₃×C₄	C₂×Dic₆	1	8	yes	—
124	C₂×C₃×C₄	C₂×D₁₂	1	8	yes	—
125	C₂×C₃×C₄	C₂×D₁₂	1	4	yes	—
126	C₂×C₃×C₄	C₂²×Dic₃	1	8	yes	—
127	C₂×C₃×C₄	C₂²×Dic₃	1	16	yes	—
128	C₂×C₃×C₄	C₂³×D₃	1	4	yes	—
129	C₂×Dic₃	—	6	32	no	—
130	C₂×Dic₃	—	12	32	no	—
131	C₃×C₈	C₂²×Dic₃	1	16	yes	—
132	C₃×C₈	C₂²×Dic₃	1	16	yes	—
133	C₂×C₃²	—	2	4	no	—
134	C₂×C₃²	#427	1	4	yes	complemented
135	C₂×C₃²	C₂²×C₄⋊C₄	1	1	yes	characteristic
136	C₂×C₃²	C₂×#135	1	2	yes	—
137	C₂⁴	C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
138	C₂²×C₄	—	18	4	no	—
139	C₂²×C₄	#634	1	4	yes	—
140	C₂²×C₄	C₄⋊(C₃⋊D₃)	1	2	yes	—
141	C₂²×C₄	C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
142	C₂²×C₄	C₂×C₃⋊Dic₃	1	4	yes	—
143	C₂²×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
144	C₂²×C₄	C₂²×C₃⋊D₃	1	2	yes	—
145	C₂×C₈	C₂×C₃⋊Dic₃	1	4	yes	—
146	C₂×C₈	C₂×C₃⋊Dic₃	1	4	yes	—
147	C₂×C₈	C₂×C₃⋊Dic₃	1	16	yes	complemented
148	C₂×C₈	C₂²×C₃⋊D₃	1	2	yes	—
149	C₂×C₈	C₂²×C₃⋊D₃	1	2	yes	—
150	C₄⋊C₄	—	9	8	no	—
151	C₄⋊C₄	—	18	4	no	—
152	C₂²⋊C₄	—	18	8	no	—
153	C₄²	—	18	8	no	—
154	C₂²×C₃	—	2	16	no	—
155	C₂²×C₃	—	2	32	no	—
156	C₂²×C₃	#904	1	16	yes	—
157	C₂²×C₃	C₂×C₄⋊Dic₃	1	16	yes	—
158	C₂²×C₃	C₂×C₄⋊Dic₃	1	8	yes	—
159	C₂²×C₃	C₂²×D₁₂	1	4	yes	—
160	C₃×C₄	C₂×C₄⋊Dic₃	1	16	yes	—
161	C₃×C₄	C₂×C₄⋊Dic₃	1	8	yes	—
162	C₃×C₄	C₂²×Dic₆	1	4	yes	—
163	C₃×C₄	C₂³×Dic₃	1	4	yes	—
164	Dic₃	—	12	32	no	—
165	C₃²	C₂×#427	1	1	yes	characteristic, complemented, Sylow
166	C₂³	—	2	4	no	—
167	C₂³	C₄⋊(C₃⋊Dic₃)	1	4	yes	—
168	C₂³	C₂×C₄⋊(C₃⋊D₃)	1	2	yes	—
169	C₂³	C₂×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
170	C₂×C₄	—	18	8	no	—
171	C₂×C₄	—	36	8	no	—
172	C₂×C₄	C₄⋊(C₃⋊Dic₃)	1	8	yes	—
173	C₂×C₄	C₄⋊(C₃⋊Dic₃)	1	4	yes	—
174	C₂×C₄	C₂×#634	1	4	yes	—
175	C₂×C₄	C₂×#634	1	2	yes	—
176	C₂×C₄	C₂×C₄⋊(C₃⋊D₃)	1	2	yes	—
177	C₂×C₄	C₂×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
178	C₂×C₄	C₂²×C₃⋊Dic₃	1	2	yes	—
179	C₂×C₄	C₂²×C₃⋊Dic₃	1	4	yes	—
180	C₂×C₄	C₂³×C₃⋊D₃	1	1	yes	characteristic
181	C₈	C₂²×C₃⋊Dic₃	1	4	yes	—
182	C₈	C₂²×C₃⋊Dic₃	1	4	yes	—
183	C₂×C₃	—	2	16	no	—
184	C₂×C₃	#5106	1	16	yes	complemented
185	C₂×C₃	C₂×#904	1	8	yes	—
186	C₂×C₃	C₂²×C₄⋊Dic₃	1	4	yes	—
187	C₂²	—	2	8	no	—
188	C₂²	—	2	4	no	—
189	C₂²	#64113	1	4	yes	—
190	C₂²	C₂×C₄⋊(C₃⋊Dic₃)	1	4	yes	—
191	C₂²	C₂×C₄⋊(C₃⋊Dic₃)	1	2	yes	—
192	C₂²	C₂²×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
193	C₄	—	36	8	no	—
194	C₄	C₂×C₄⋊(C₃⋊Dic₃)	1	2	yes	—
195	C₄	C₂×C₄⋊(C₃⋊Dic₃)	1	4	yes	—
196	C₄	C₂²×#634	1	1	yes	characteristic
197	C₄	C₂³×C₃⋊Dic₃	1	1	yes	characteristic
198	C₃	C₂×#5106	1	4	yes	complemented
199	C₂	—	2	4	no	—
200	C₂	#10592805	1	4	yes	direct product factor
201	C₂	C₂×#64113	1	2	yes	—
202	C₂	C₂²×C₄⋊(C₃⋊Dic₃)	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₄⋊(C₃⋊D₃)
commutator	C₃²×C₄	C₂³×C₄
Frattini	C₂×C₄	C₂³×C₃⋊D₃
Fitting	C₂³×C₃²×C₈	C₂
socle	C₂³×C₃²	C₂×D₄
prime radical	C₂⁴×C₃²	D₄
cosocle	C₂×C₃²×C₄	C₂⁴
hypercenter	C₂³×C₈	C₃⋊D₃
hypocenter	C₃²	C₂×#427
LCS	G ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂³×C₈

Complements

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

N	H ≅ G/N	type
3. #10592805	C₂	direct (N × H)
4. C₂×#63645	C₂	semidirect
5. C₂×#63644	C₂	semidirect
8. C₂²×C₃²×C₈	C₄	semidirect
17. #63645	C₂²	semidirect
18. #63644	C₂²	semidirect
34. C₂×C₃²×C₈	C₂×C₄	semidirect
43. C₂²×C₃×C₈	Dic₃	semidirect
86. C₂×C₃×C₈	C₂×Dic₃	semidirect
105. C₂²×C₈	C₃⋊Dic₃	semidirect
134. C₂×C₃²	#427	semidirect
147. C₂×C₈	C₂×C₃⋊Dic₃	semidirect
165. C₃²	C₂×#427	semidirect
184. C₂×C₃	#5106	semidirect
198. C₃	C₂×#5106	semidirect
200. C₂	#10592805	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×C₈	2
C₂×#64135	2
#10592805	2
C₂×#63645	2
C₂×#63644	2
C₂×#5106	3

Subgroups — C2×#10592805 #11858424

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#10592805 #11858424