Subgroups — C₂×#10596041 #11903679

Order 1152.

3144 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×Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂×Q₈×C₃⋊Dic₃	C₂	1	2	yes	maximal
3	#10596041	C₂	1	16	yes	maximal, direct product factor
4	C₂×#64099	C₂	1	6	yes	maximal
5	C₂×#64078	C₂	1	3	yes	maximal
6	C₂×C₄×#634	C₂	1	3	yes	maximal
7	C₂×#5570	—	3	4	no	maximal
8	C₂²×C₃²×Q₈	C₂²	1	1	yes	characteristic
9	C₂²×#634	C₂²	1	3	yes	—
10	C₃²×C₄×Q₈	C₂²	1	8	yes	—
11	C₂×C₃²×C₄⋊C₄	C₂²	1	3	yes	—
12	C₂×C₃²×C₄²	C₂²	1	3	yes	—
13	Q₈×C₃⋊Dic₃	C₂²	1	16	yes	—
14	C₂×C₄⋊(C₃⋊Dic₃)	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊(C₃⋊Dic₃)	C₂²	1	6	yes	—
16	C₂×C₄⋊(C₃⋊Dic₃)	C₂²	1	6	yes	—
17	C₂×#3750	C₂²	1	6	yes	—
18	C₂×C₄×C₃⋊Dic₃	C₂²	1	6	yes	—
19	#64099	C₂²	1	48	yes	—
20	#64078	C₂²	1	24	yes	—
21	C₄×#634	C₂²	1	24	yes	—
22	C₂×C₃×C₄×Q₈	D₃	1	4	yes	—
23	C₂×Q₈×Dic₃	—	3	8	no	—
24	#5570	—	3	64	no	—
25	C₂×#890	—	3	24	no	—
26	C₂×#869	—	3	12	no	—
27	C₂×C₄×Dic₆	—	3	12	no	—
28	C₂×C₃²×Q₈	Q₈	1	8	yes	complemented
29	C₂×C₃²×Q₈	C₂³	1	4	yes	—
30	C₂²×C₃²×C₄	C₂³	1	3	yes	—
31	C₂²×C₃²×C₄	C₂³	1	3	yes	—
32	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
33	C₂²×C₃⋊Dic₃	C₂³	1	6	yes	—
34	C₂²×C₃⋊Dic₃	C₂³	1	2	yes	—
35	C₂×#634	—	2	12	no	—
36	C₂×#634	C₂³	1	12	yes	—
37	C₃²×C₄⋊C₄	C₂³	1	12	yes	—
38	C₃²×C₄²	C₂³	1	12	yes	—
39	C₄⋊(C₃⋊Dic₃)	C₂³	1	4	yes	—
40	C₄⋊(C₃⋊Dic₃)	C₂³	1	24	yes	—
41	C₄⋊(C₃⋊Dic₃)	C₂³	1	24	yes	—
42	#3750	C₂³	1	24	yes	—
43	C₄×C₃⋊Dic₃	C₂³	1	24	yes	—
44	C₂×#557	—	9	1	no	Sylow
45	C₂²×C₃×Q₈	C₂×D₃	1	4	yes	—
46	C₂²×Dic₆	—	3	12	no	—
47	C₃×C₄×Q₈	C₂×D₃	1	32	yes	—
48	C₂×C₃×C₄⋊C₄	C₂×D₃	1	12	yes	—
49	C₂×C₃×C₄²	C₂×D₃	1	12	yes	—
50	Q₈×Dic₃	—	3	64	no	—
51	C₂×C₄⋊Dic₃	—	3	4	no	—
52	C₂×C₄⋊Dic₃	—	3	24	no	—
53	C₂×C₄⋊Dic₃	—	3	24	no	—
54	C₂×Dic₃⋊C₄	—	3	24	no	—
55	C₂×C₄×Dic₃	—	3	24	no	—
56	#890	—	3	192	no	—
57	#869	—	3	96	no	—
58	C₄×Dic₆	—	3	96	no	—
59	C₂³×C₃²	C₂⁴	1	1	yes	characteristic
60	C₃²×Q₈	C₂×Q₈	1	16	yes	complemented
61	C₂×C₃²×C₄	—	2	6	no	—
62	C₂×C₃²×C₄	C₂×Q₈	1	12	yes	—
63	C₂×C₃²×C₄	#41	1	4	yes	—
64	C₂×C₃²×C₄	C₂⁴	1	6	yes	—
65	C₂×C₃²×C₄	C₂⁴	1	6	yes	—
66	C₂×C₃²×C₄	C₂⁴	1	2	yes	—
67	C₂×C₃⋊Dic₃	—	2	12	no	—
68	C₂×C₃⋊Dic₃	—	2	4	no	—
69	C₂×C₃⋊Dic₃	C₂⁴	1	4	yes	—
70	C₂×C₃⋊Dic₃	C₂⁴	1	12	yes	—
71	#634	—	2	24	no	—
72	#557	—	9	16	no	—
73	C₂×C₄⋊Q₈	—	9	3	no	—
74	C₂×#125	—	9	6	no	—
75	C₂×C₄×Q₈	—	9	2	no	—
76	C₂×C₄×Q₈	—	9	3	no	—
77	C₂×C₄×Q₈	C₃⋊D₃	1	1	yes	characteristic
78	C₂×C₃×Q₈	Dic₆	1	32	yes	complemented
79	C₂×C₃×Q₈	C₂²×D₃	1	16	yes	—
80	C₂²×C₃×C₄	C₂²×D₃	1	12	yes	—
81	C₂²×C₃×C₄	C₂²×D₃	1	12	yes	—
82	C₂²×C₃×C₄	C₂²×D₃	1	4	yes	—
83	C₂²×Dic₃	—	3	8	no	—
84	C₂²×Dic₃	—	3	24	no	—
85	C₂×Dic₆	—	3	48	no	—
86	C₂×Dic₆	—	6	48	no	—
87	C₃×C₄⋊C₄	C₂²×D₃	1	48	yes	—
88	C₃×C₄²	C₂²×D₃	1	48	yes	—
89	C₄⋊Dic₃	—	3	16	no	—
90	C₄⋊Dic₃	—	3	96	no	—
91	C₄⋊Dic₃	—	3	96	no	—
92	Dic₃⋊C₄	—	3	96	no	—
93	C₄×Dic₃	—	3	96	no	—
94	C₂²×C₃²	C₂²×Q₈	1	2	yes	—
95	C₂²×C₃²	C₂×#41	1	2	yes	—
96	C₂²×C₃²	2₋¹⁺⁴	1	2	yes	—
97	C₂²×C₃²	C₂⁵	1	1	yes	characteristic
98	C₃²×C₄	—	2	6	no	—
99	C₃²×C₄	C₂²×Q₈	1	12	yes	—
100	C₃²×C₄	C₂×#41	1	4	yes	—
101	C₃⋊Dic₃	—	2	12	no	—
102	C₃⋊Dic₃	—	2	4	no	—
103	C₂²×Q₈	—	9	3	no	—
104	C₂²×Q₈	C₂×C₃⋊D₃	1	1	yes	characteristic
105	C₄⋊Q₈	—	9	24	no	—
106	#125	—	9	48	no	—
107	C₄×Q₈	—	9	16	no	—
108	C₄×Q₈	—	9	24	no	—
109	C₄×Q₈	C₂×C₃⋊D₃	1	8	yes	—
110	C₂×C₄⋊C₄	—	9	1	no	—
111	C₂×C₄⋊C₄	—	9	6	no	—
112	C₂×C₄⋊C₄	—	9	6	no	—
113	C₂×C₄⋊C₄	—	9	6	no	—
114	C₂×C₄⋊C₄	C₂×C₃⋊D₃	1	3	yes	—
115	C₂×C₄²	—	9	6	no	—
116	C₂×C₄²	C₂×C₃⋊D₃	1	3	yes	—
117	C₂³×C₃	C₂³×D₃	1	4	yes	—
118	C₃×Q₈	C₂×Dic₆	1	64	yes	complemented
119	C₂×C₃×C₄	—	2	24	no	—
120	C₂×C₃×C₄	C₂×Dic₆	1	48	yes	—
121	C₂×C₃×C₄	#239	1	16	yes	—
122	C₂×C₃×C₄	C₂³×D₃	1	24	yes	—
123	C₂×C₃×C₄	C₂³×D₃	1	24	yes	—
124	C₂×C₃×C₄	C₂³×D₃	1	8	yes	—
125	C₂×Dic₃	—	3	16	no	—
126	C₂×Dic₃	—	3	48	no	—
127	C₂×Dic₃	—	6	48	no	—
128	C₂×Dic₃	—	6	16	no	—
129	Dic₆	—	6	96	no	—
130	C₂×C₃²	#557	1	4	yes	complemented
131	C₂×C₃²	C₂³×Q₈	1	1	yes	characteristic
132	C₂×C₃²	C₂²×#41	1	1	yes	characteristic
133	C₂×C₃²	C₂×2₋¹⁺⁴	1	1	yes	characteristic
134	C₂×Q₈	—	9	12	no	—
135	C₂×Q₈	—	18	12	no	—
136	C₂×Q₈	#634	1	8	yes	complemented
137	C₂×Q₈	C₂²×C₃⋊D₃	1	4	yes	—
138	C₂²×C₄	—	9	2	no	—
139	C₂²×C₄	—	9	6	no	—
140	C₂²×C₄	C₂²×C₃⋊D₃	1	3	yes	—
141	C₂²×C₄	C₂²×C₃⋊D₃	1	3	yes	—
142	C₂²×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
143	C₄⋊C₄	—	9	4	no	—
144	C₄⋊C₄	—	9	24	no	—
145	C₄⋊C₄	—	9	24	no	—
146	C₄⋊C₄	—	9	24	no	—
147	C₄⋊C₄	C₂²×C₃⋊D₃	1	12	yes	—
148	C₄²	—	9	24	no	—
149	C₄²	C₂²×C₃⋊D₃	1	12	yes	—
150	C₂²×C₃	C₂²×Dic₆	1	8	yes	—
151	C₂²×C₃	C₂×#239	1	8	yes	—
152	C₂²×C₃	#1010	1	8	yes	—
153	C₂²×C₃	C₂⁴×D₃	1	4	yes	—
154	C₃×C₄	—	2	24	no	—
155	C₃×C₄	C₂²×Dic₆	1	48	yes	—
156	C₃×C₄	C₂×#239	1	16	yes	—
157	Dic₃	—	6	48	no	—
158	Dic₃	—	6	16	no	—
159	C₃²	C₂×#557	1	1	yes	characteristic, complemented, Sylow
160	C₂³	C₂³×C₃⋊D₃	1	1	yes	characteristic
161	Q₈	—	18	24	no	—
162	Q₈	C₂×#634	1	16	yes	complemented
163	C₂×C₄	—	2	6	no	—
164	C₂×C₄	—	9	4	no	—
165	C₂×C₄	—	9	12	no	—
166	C₂×C₄	—	18	12	no	—
167	C₂×C₄	—	18	4	no	—
168	C₂×C₄	C₂×#634	1	12	yes	—
169	C₂×C₄	#3832	1	4	yes	—
170	C₂×C₄	C₂³×C₃⋊D₃	1	6	yes	—
171	C₂×C₄	C₂³×C₃⋊D₃	1	6	yes	—
172	C₂×C₄	C₂³×C₃⋊D₃	1	2	yes	—
173	C₂×C₃	#5570	1	16	yes	complemented
174	C₂×C₃	C₂³×Dic₆	1	4	yes	—
175	C₂×C₃	C₂²×#239	1	4	yes	—
176	C₂×C₃	C₂×#1010	1	4	yes	—
177	C₂²	C₂²×#634	1	2	yes	—
178	C₂²	C₂×#3832	1	2	yes	—
179	C₂²	#64367	1	2	yes	—
180	C₂²	C₂⁴×C₃⋊D₃	1	1	yes	characteristic
181	C₄	—	2	6	no	—
182	C₄	—	18	12	no	—
183	C₄	—	18	4	no	—
184	C₄	C₂²×#634	1	12	yes	—
185	C₄	C₂×#3832	1	4	yes	—
186	C₃	C₂×#5570	1	4	yes	complemented
187	C₂	#10596041	1	4	yes	direct product factor
188	C₂	C₂³×#634	1	1	yes	characteristic
189	C₂	C₂²×#3832	1	1	yes	characteristic
190	C₂	C₂×#64367	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂³×C₃⋊D₃
commutator, cosocle	C₂²×C₃²	C₂⁵
Frattini	C₂²	C₂⁴×C₃⋊D₃
Fitting	C₂×C₃²×C₄×Q₈	C₂
socle, prime radical	C₂³×C₃²	C₂⁴
hypercenter	C₂×C₄×Q₈	C₃⋊D₃
hypocenter	C₃²	C₂×#557
LCS	G ▷ C₂²×C₃² ▷ C₃²
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×Q₈

Complements

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

N	H ≅ G/N	type
3. #10596041	C₂	direct (N × H)
28. C₂×C₃²×Q₈	Q₈	semidirect
60. C₃²×Q₈	C₂×Q₈	semidirect
78. C₂×C₃×Q₈	Dic₆	semidirect
118. C₃×Q₈	C₂×Dic₆	semidirect
130. C₂×C₃²	#557	semidirect
136. C₂×Q₈	#634	semidirect
159. C₃²	C₂×#557	semidirect
162. Q₈	C₂×#634	semidirect
173. C₂×C₃	#5570	semidirect
186. C₃	C₂×#5570	semidirect
187. C₂	#10596041	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
C₂×Q₈×C₃⋊Dic₃	2
#10596041	2
C₂×#64099	2
C₂×#64078	2
C₂×C₄×#634	2
C₂×#5570	3

Subgroups — C2×#10596041 #11903679

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#10596041 #11903679