Subgroups — C₂×#10592635 #11801085

Order 1152.

2136 conjugacy classes of subgroups.

The subgroup lattice has 174 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃²×C₄²	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄⋊(C₃⋊Dic₃)	C₂	1	2	yes	maximal
3	C₂×#63658	C₂	1	4	yes	maximal
4	#10592635	C₂	1	8	yes	maximal, direct product factor
5	C₂×#4936	—	3	4	no	maximal
6	C₂³×C₃²×C₄	C₂²	1	2	yes	—
7	C₂³×C₃²×C₄	C₂²	1	1	yes	characteristic
8	C₂³×C₃⋊Dic₃	C₂²	1	4	yes	—
9	C₂×C₃²×C₄²	C₄	1	8	yes	complemented
10	C₂×C₃²×C₄²	C₂²	1	4	yes	—
11	C₂×C₄⋊(C₃⋊Dic₃)	—	2	8	no	—
12	C₂×C₄⋊(C₃⋊Dic₃)	C₂²	1	8	yes	—
13	#63658	C₂²	1	16	yes	—
14	C₂²×C₃×C₄²	D₃	1	4	yes	—
15	C₂²×C₄⋊Dic₃	—	3	8	no	—
16	C₂×C₃⋊#95	—	3	16	no	—
17	#4936	—	3	32	no	—
18	C₂⁴×C₃²	C₂³	1	1	yes	characteristic
19	C₂²×C₃²×C₄	—	2	8	no	—
20	C₂²×C₃²×C₄	C₂×C₄	1	8	yes	—
21	C₂²×C₃²×C₄	C₂×C₄	1	4	yes	—
22	C₂²×C₃²×C₄	D₄	1	4	yes	—
23	C₂²×C₃²×C₄	Q₈	1	4	yes	—
24	C₂²×C₃²×C₄	C₂³	1	4	yes	—
25	C₂²×C₃²×C₄	C₂³	1	2	yes	—
26	C₂²×C₃⋊Dic₃	—	2	8	no	—
27	C₂²×C₃⋊Dic₃	—	2	16	no	—
28	C₂²×C₃⋊Dic₃	C₂³	1	8	yes	—
29	C₃²×C₄²	C₂×C₄	1	16	yes	complemented
30	C₄⋊(C₃⋊Dic₃)	—	2	16	no	—
31	C₂×#382	—	9	1	no	Sylow
32	C₂³×C₃×C₄	C₂×D₃	1	8	yes	—
33	C₂³×C₃×C₄	C₂×D₃	1	4	yes	—
34	C₂³×Dic₃	—	3	16	no	—
35	C₂×C₃×C₄²	Dic₃	1	32	yes	complemented
36	C₂×C₃×C₄²	C₂×D₃	1	16	yes	—
37	C₂×C₄⋊Dic₃	—	3	32	no	—
38	C₂×C₄⋊Dic₃	—	6	32	no	—
39	C₃⋊#95	—	3	64	no	—
40	C₂³×C₃²	C₂²×C₄	1	2	yes	—
41	C₂³×C₃²	C₂×D₄	1	2	yes	—
42	C₂³×C₃²	C₂×Q₈	1	2	yes	—
43	C₂³×C₃²	#41	1	8	yes	—
44	C₂³×C₃²	C₂⁴	1	1	yes	characteristic
45	C₂×C₃²×C₄	—	2	16	no	—
46	C₂×C₃²×C₄	—	2	8	no	—
47	C₂×C₃²×C₄	C₄⋊C₄	1	16	yes	—
48	C₂×C₃²×C₄	C₂²×C₄	1	8	yes	—
49	C₂×C₃²×C₄	C₂²×C₄	1	4	yes	—
50	C₂×C₃²×C₄	C₂×D₄	1	4	yes	—
51	C₂×C₃²×C₄	C₂×Q₈	1	4	yes	—
52	C₂×C₃⋊Dic₃	—	2	8	no	—
53	C₂×C₃⋊Dic₃	—	2	16	no	—
54	C₂×C₃⋊Dic₃	—	4	16	no	—
55	C₂²×C₄⋊C₄	—	9	2	no	—
56	C₂²×C₄²	C₃⋊D₃	1	1	yes	characteristic
57	#382	—	9	8	no	—
58	C₂×#95	—	9	4	no	—
59	C₂⁴×C₃	C₂²×D₃	1	4	yes	—
60	C₂²×C₃×C₄	—	2	32	no	—
61	C₂²×C₃×C₄	Dic₆	1	16	yes	—
62	C₂²×C₃×C₄	D₁₂	1	16	yes	—
63	C₂²×C₃×C₄	C₂×Dic₃	1	32	yes	—
64	C₂²×C₃×C₄	C₂×Dic₃	1	16	yes	—
65	C₂²×C₃×C₄	C₂²×D₃	1	16	yes	—
66	C₂²×C₃×C₄	C₂²×D₃	1	8	yes	—
67	C₂²×Dic₃	—	3	32	no	—
68	C₂²×Dic₃	—	6	32	no	—
69	C₂²×Dic₃	—	6	64	no	—
70	C₃×C₄²	C₂×Dic₃	1	64	yes	complemented
71	C₄⋊Dic₃	—	6	64	no	—
72	C₂²×C₃²	C₂×C₄⋊C₄	1	4	yes	—
73	C₂²×C₃²	#117	1	8	yes	—
74	C₂²×C₃²	#124	1	8	yes	—
75	C₂²×C₃²	#125	1	8	yes	—
76	C₂²×C₃²	C₂³×C₄	1	1	yes	characteristic
77	C₂²×C₃²	C₂²×D₄	1	1	yes	characteristic
78	C₂²×C₃²	C₂²×Q₈	1	1	yes	characteristic
79	C₂²×C₃²	C₂×#41	1	4	yes	—
80	C₃²×C₄	—	2	8	no	—
81	C₃²×C₄	C₂×C₄⋊C₄	1	8	yes	—
82	C₃⋊Dic₃	—	4	8	no	—
83	C₂³×C₄	—	9	4	no	—
84	C₂³×C₄	C₂×C₃⋊D₃	1	2	yes	—
85	C₂³×C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
86	C₂×C₄⋊C₄	—	9	8	no	—
87	C₂×C₄⋊C₄	—	18	8	no	—
88	C₂×C₄²	C₃⋊Dic₃	1	8	yes	complemented
89	C₂×C₄²	C₂×C₃⋊D₃	1	4	yes	—
90	#95	—	9	16	no	—
91	C₂³×C₃	C₂×Dic₆	1	8	yes	—
92	C₂³×C₃	C₂×D₁₂	1	8	yes	—
93	C₂³×C₃	#235	1	32	yes	—
94	C₂³×C₃	C₂²×Dic₃	1	8	yes	—
95	C₂³×C₃	C₂³×D₃	1	4	yes	—
96	C₂×C₃×C₄	—	2	64	no	—
97	C₂×C₃×C₄	—	2	32	no	—
98	C₂×C₃×C₄	C₄⋊Dic₃	1	64	yes	—
99	C₂×C₃×C₄	C₂×Dic₆	1	16	yes	—
100	C₂×C₃×C₄	C₂×D₁₂	1	16	yes	—
101	C₂×C₃×C₄	C₂²×Dic₃	1	32	yes	—
102	C₂×C₃×C₄	C₂²×Dic₃	1	16	yes	—
103	C₂×Dic₃	—	6	32	no	—
104	C₂×Dic₃	—	6	64	no	—
105	C₂×Dic₃	—	12	64	no	—
106	C₂×C₃²	#382	1	8	yes	complemented
107	C₂×C₃²	C₂²×C₄⋊C₄	1	1	yes	characteristic
108	C₂×C₃²	C₂×#117	1	2	yes	—
109	C₂×C₃²	C₂×#124	1	2	yes	—
110	C₂×C₃²	C₂×#125	1	2	yes	—
111	C₂⁴	C₂²×C₃⋊D₃	1	1	yes	characteristic
112	C₂²×C₄	—	2	8	no	—
113	C₂²×C₄	—	9	8	no	—
114	C₂²×C₄	—	18	8	no	—
115	C₂²×C₄	—	18	16	no	—
116	C₂²×C₄	#634	1	4	yes	—
117	C₂²×C₄	C₄⋊(C₃⋊D₃)	1	4	yes	—
118	C₂²×C₄	C₂×C₃⋊Dic₃	1	8	yes	—
119	C₂²×C₄	C₂×C₃⋊Dic₃	1	4	yes	—
120	C₂²×C₄	C₂²×C₃⋊D₃	1	4	yes	—
121	C₂²×C₄	C₂²×C₃⋊D₃	1	2	yes	—
122	C₄⋊C₄	—	18	16	no	—
123	C₄²	C₂×C₃⋊Dic₃	1	16	yes	complemented
124	C₂²×C₃	#870	1	32	yes	—
125	C₂²×C₃	#875	1	32	yes	—
126	C₂²×C₃	C₂×C₄⋊Dic₃	1	16	yes	—
127	C₂²×C₃	#926	1	32	yes	—
128	C₂²×C₃	C₂²×Dic₆	1	4	yes	—
129	C₂²×C₃	C₂²×D₁₂	1	4	yes	—
130	C₂²×C₃	C₂×#235	1	16	yes	—
131	C₂²×C₃	C₂³×Dic₃	1	4	yes	—
132	C₃×C₄	—	2	32	no	—
133	C₃×C₄	C₂×C₄⋊Dic₃	1	32	yes	—
134	Dic₃	—	12	32	no	—
135	C₃²	C₂×#382	1	1	yes	characteristic, complemented, Sylow
136	C₂³	C₂×#634	1	2	yes	—
137	C₂³	C₂×C₄⋊(C₃⋊D₃)	1	2	yes	—
138	C₂³	#3828	1	8	yes	—
139	C₂³	C₂²×C₃⋊Dic₃	1	2	yes	—
140	C₂³	C₂³×C₃⋊D₃	1	1	yes	characteristic
141	C₂×C₄	—	2	8	no	—
142	C₂×C₄	—	2	16	no	—
143	C₂×C₄	—	18	8	no	—
144	C₂×C₄	—	18	16	no	—
145	C₂×C₄	—	36	16	no	—
146	C₂×C₄	C₄⋊(C₃⋊Dic₃)	1	16	yes	—
147	C₂×C₄	C₂×#634	1	4	yes	—
148	C₂×C₄	C₂×C₄⋊(C₃⋊D₃)	1	4	yes	—
149	C₂×C₄	C₂²×C₃⋊Dic₃	1	8	yes	—
150	C₂×C₄	C₂²×C₃⋊Dic₃	1	4	yes	—
151	C₂×C₃	#4936	1	32	yes	complemented
152	C₂×C₃	C₂×#870	1	8	yes	—
153	C₂×C₃	C₂×#875	1	8	yes	—
154	C₂×C₃	C₂²×C₄⋊Dic₃	1	4	yes	—
155	C₂×C₃	C₂×#926	1	8	yes	—
156	C₂²	#64079	1	8	yes	—
157	C₂²	#64084	1	8	yes	—
158	C₂²	C₂×C₄⋊(C₃⋊Dic₃)	1	4	yes	—
159	C₂²	#64135	1	8	yes	—
160	C₂²	C₂²×#634	1	1	yes	characteristic
161	C₂²	C₂²×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
162	C₂²	C₂×#3828	1	4	yes	—
163	C₂²	C₂³×C₃⋊Dic₃	1	1	yes	characteristic
164	C₄	—	2	8	no	—
165	C₄	—	36	8	no	—
166	C₄	C₂×C₄⋊(C₃⋊Dic₃)	1	8	yes	—
167	C₃	C₂×#4936	1	4	yes	complemented
168	C₂	#10592635	1	8	yes	direct product factor
169	C₂	C₂×#64079	1	2	yes	—
170	C₂	C₂×#64084	1	2	yes	—
171	C₂	C₂²×C₄⋊(C₃⋊Dic₃)	1	1	yes	characteristic
172	C₂	C₂×#64135	1	2	yes	—

	subgroup	G/H
center	C₂⁴	C₂²×C₃⋊D₃
commutator	C₂²×C₃²	C₂³×C₄
Frattini	C₂³	C₂³×C₃⋊D₃
Fitting	C₂²×C₃²×C₄²	C₂
socle, prime radical	C₂⁴×C₃²	C₂³
cosocle	C₂³×C₃²	C₂⁴
hypercenter	C₂²×C₄²	C₃⋊D₃
hypocenter	C₃²	C₂×#382
LCS	G ▷ C₂²×C₃² ▷ C₃²
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄²

Complements

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

N	H ≅ G/N	type
4. #10592635	C₂	direct (N × H)
9. C₂×C₃²×C₄²	C₄	semidirect
29. C₃²×C₄²	C₂×C₄	semidirect
35. C₂×C₃×C₄²	Dic₃	semidirect
70. C₃×C₄²	C₂×Dic₃	semidirect
88. C₂×C₄²	C₃⋊Dic₃	semidirect
106. C₂×C₃²	#382	semidirect
123. C₄²	C₂×C₃⋊Dic₃	semidirect
135. C₃²	C₂×#382	semidirect
151. C₂×C₃	#4936	semidirect
167. C₃	C₂×#4936	semidirect
168. C₂	#10592635	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₃²×C₄²	2
C₂²×C₄⋊(C₃⋊Dic₃)	2
C₂×#63658	2
#10592635	2
C₂×#4936	3

Subgroups — C2×#10592635 #11801085

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#10592635 #11801085