Subgroups — #11861334

Order 1152.

1092 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	#10595941	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃²×C₄×C₈	C₂	1	1	yes	maximal, characteristic
3	#10592801	C₂	1	2	yes	maximal
4	#10592800	C₂	1	2	yes	maximal
5	#10592623	C₂	1	1	yes	maximal, characteristic
6	#10592378	C₂	1	4	yes	maximal, complemented
7	C₈×#634	C₂	1	4	yes	maximal, complemented
8	#78445	—	3	4	no	maximal
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	#64135	C₂²	1	2	yes	—
13	#64133	C₄	1	4	yes	—
14	C₂×#3748	C₂²	1	2	yes	—
15	#64079	C₄	1	2	yes	—
16	#64078	C₄	1	2	yes	—
17	C₄×#634	C₂²	1	4	yes	—
18	C₃²×C₄×C₈	C₂²	1	2	yes	—
19	C₃²×C₄×C₈	C₂²	1	2	yes	—
20	#63642	C₂²	1	4	yes	—
21	#63641	C₂²	1	8	yes	—
22	C₈×C₃⋊Dic₃	C₂²	1	4	yes	—
23	C₄⋊(C₃⋊(C₃⋊C₈))	C₂²	1	4	yes	—
24	#5470	—	3	4	no	—
25	C₂×C₃×C₄×C₈	D₃	1	4	yes	—
26	#5102	—	3	8	no	—
27	#5101	—	3	8	no	—
28	#4924	—	3	4	no	—
29	#4679	—	3	16	no	—
30	C₈×Dic₆	—	3	16	no	—
31	C₂²×C₃²×C₄	C₂×C₄	1	2	yes	—
32	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
33	C₂×#634	C₂×C₄	1	4	yes	—
34	C₂×C₃²×C₈	—	2	2	no	—
35	C₂×C₃²×C₈	Q₈	1	4	yes	complemented
36	C₂×C₃²×C₈	C₂³	1	2	yes	—
37	C₂×C₃²×C₈	C₂³	1	1	yes	characteristic
38	C₂×C₃²×C₈	C₂³	1	1	yes	characteristic
39	C₃²×C₄²	C₂×C₄	1	1	yes	characteristic
40	C₃²×C₄²	C₂×C₄	1	1	yes	characteristic
41	C₃²×C₄²	C₂³	1	2	yes	—
42	#3757	C₂×C₄	1	4	yes	—
43	C₄⋊(C₃⋊Dic₃)	C₂×C₄	1	4	yes	—
44	C₄⋊(C₃⋊Dic₃)	C₂×C₄	1	4	yes	—
45	#3750	C₂×C₄	1	8	yes	—
46	C₄×C₃⋊Dic₃	C₂³	1	4	yes	—
47	#3748	—	2	4	no	—
48	C₂×C₃⋊(C₃⋊C₈)	C₂³	1	4	yes	—
49	#2960	—	9	1	no	Sylow
50	C₂²×C₃×C₈	C₂×D₃	1	4	yes	—
51	C₂²×C₃×C₈	C₂×D₃	1	4	yes	—
52	C₂×C₃×C₄²	C₂×D₃	1	4	yes	—
53	#926	—	3	8	no	—
54	#924	—	3	16	no	—
55	C₂×#208	—	3	8	no	—
56	#870	—	3	8	no	—
57	#869	—	3	8	no	—
58	C₄×Dic₆	—	3	16	no	—
59	C₃×C₄×C₈	C₂×D₃	1	8	yes	—
60	C₃×C₄×C₈	C₂×D₃	1	8	yes	—
61	#815	—	3	16	no	—
62	Dic₃⋊C₈	—	3	32	no	—
63	C₈×Dic₃	—	3	16	no	—
64	C₄⋊(C₃⋊C₈)	—	3	16	no	—
65	C₂³×C₃²	C₂²×C₄	1	1	yes	characteristic
66	C₂×C₃²×C₄	—	2	4	no	—
67	C₂×C₃²×C₄	C₂²×C₄	1	2	yes	—
68	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₃²×C₄	C₂²×C₄	1	2	yes	—
70	C₂×C₃²×C₄	C₂×Q₈	1	2	yes	—
71	C₂×C₃²×C₄	#41	1	2	yes	—
72	C₂×C₃²×C₄	C₂⁴	1	1	yes	characteristic
73	C₂×C₃⋊Dic₃	C₂²×C₄	1	8	yes	—
74	#634	—	2	8	no	—
75	C₃²×C₈	—	2	2	no	—
76	C₃²×C₈	C₂×Q₈	1	2	yes	—
77	C₃²×C₈	C₂×Q₈	1	2	yes	—
78	C₃⋊(C₃⋊C₈)	—	2	4	no	—
79	#533	—	9	1	no	—
80	#446	—	9	4	no	—
81	C₈×Q₈	—	9	4	no	—
82	#424	—	9	2	no	—
83	#423	—	9	1	no	—
84	#405	—	9	2	no	—
85	C₂×C₄×C₈	C₃⋊D₃	1	1	yes	characteristic
86	C₂²×C₃×C₄	C₄×D₃	1	8	yes	—
87	C₂²×C₃×C₄	C₂²×D₃	1	4	yes	—
88	C₂×Dic₆	—	3	16	no	—
89	C₂×C₃×C₈	—	2	8	no	—
90	C₂×C₃×C₈	Dic₆	1	16	yes	complemented
91	C₂×C₃×C₈	C₂²×D₃	1	8	yes	—
92	C₂×C₃×C₈	C₂²×D₃	1	4	yes	—
93	C₂×C₃×C₈	C₂²×D₃	1	4	yes	—
94	C₃×C₄²	C₄×D₃	1	4	yes	—
95	C₃×C₄²	C₄×D₃	1	4	yes	—
96	C₃×C₄²	C₂²×D₃	1	8	yes	—
97	#217	—	3	16	no	—
98	C₄⋊Dic₃	—	3	16	no	—
99	C₄⋊Dic₃	—	3	16	no	—
100	Dic₃⋊C₄	—	3	32	no	—
101	C₄×Dic₃	—	3	16	no	—
102	#208	—	6	16	no	—
103	C₂×C₃⋊C₈	—	3	16	no	—
104	C₂²×C₃²	—	2	2	no	—
105	C₂²×C₃²	C₄×Q₈	1	2	yes	—
106	C₂²×C₃²	C₂³×C₄	1	1	yes	characteristic
107	C₃²×C₄	—	2	2	no	—
108	C₃²×C₄	C₄×Q₈	1	2	yes	—
109	C₃²×C₄	#131	1	4	yes	—
110	C₃²×C₄	C₂²×Q₈	1	1	yes	characteristic
111	C₃²×C₄	C₂×#41	1	1	yes	characteristic
112	C₃⋊Dic₃	—	2	8	no	—
113	C₂×M₁₆	—	9	2	no	—
114	C₂²×C₈	C₂×C₃⋊D₃	1	1	yes	characteristic
115	C₂²×C₈	C₂×C₃⋊D₃	1	1	yes	characteristic
116	C₄⋊Q₈	—	9	2	no	—
117	#125	—	9	2	no	—
118	C₂²⋊Q₈	—	9	4	no	—
119	C₄×Q₈	—	9	4	no	—
120	#117	—	9	2	no	—
121	C₂×C₄²	C₂×C₃⋊D₃	1	1	yes	characteristic
122	C₄⋊C₈	—	9	8	no	—
123	C₄⋊C₈	—	9	4	no	—
124	#97	—	9	4	no	—
125	C₄×C₈	—	9	4	no	—
126	C₄×C₈	C₂×C₃⋊D₃	1	2	yes	—
127	C₄×C₈	C₂×C₃⋊D₃	1	2	yes	—
128	C₂³×C₃	C₂×C₄×D₃	1	4	yes	—
129	C₂×C₃×C₄	—	2	16	no	—
130	C₂×C₃×C₄	C₂×Dic₆	1	8	yes	—
131	C₂×C₃×C₄	C₂×C₄×D₃	1	8	yes	—
132	C₂×C₃×C₄	C₂×C₄×D₃	1	4	yes	—
133	C₂×C₃×C₄	C₂×C₄×D₃	1	8	yes	—
134	C₂×C₃×C₄	#235	1	8	yes	—
135	C₂×C₃×C₄	C₂³×D₃	1	4	yes	—
136	C₂×Dic₃	—	3	32	no	—
137	Dic₆	—	6	32	no	—
138	C₃×C₈	—	2	8	no	—
139	C₃×C₈	C₂×Dic₆	1	8	yes	—
140	C₃×C₈	C₂×Dic₆	1	8	yes	—
141	C₃⋊C₈	—	6	16	no	—
142	C₂×C₃²	—	2	2	no	—
143	C₂×C₃²	C₂×C₄×Q₈	1	1	yes	characteristic
144	C₂×C₃²	C₂×#131	1	2	yes	—
145	C₂×Q₈	—	9	4	no	—
146	C₂²×C₄	C₄×C₃⋊D₃	1	2	yes	—
147	C₂²×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
148	M₁₆	—	18	4	no	—
149	C₂×C₈	—	2	2	no	—
150	C₂×C₈	—	9	4	no	—
151	C₂×C₈	#634	1	4	yes	complemented
152	C₂×C₈	C₂²×C₃⋊D₃	1	2	yes	—
153	C₂×C₈	C₂²×C₃⋊D₃	1	1	yes	characteristic
154	C₂×C₈	C₂²×C₃⋊D₃	1	1	yes	characteristic
155	C₄⋊C₄	—	9	4	no	—
156	C₄⋊C₄	—	9	8	no	—
157	C₄⋊C₄	—	9	4	no	—
158	C₂²⋊C₄	—	9	4	no	—
159	C₄²	—	9	4	no	—
160	C₄²	C₄×C₃⋊D₃	1	1	yes	characteristic
161	C₄²	C₄×C₃⋊D₃	1	1	yes	characteristic
162	C₄²	C₂²×C₃⋊D₃	1	2	yes	—
163	C₂²×C₃	—	2	8	no	—
164	C₂²×C₃	C₄×Dic₆	1	8	yes	—
165	C₂²×C₃	C₂²×C₄×D₃	1	4	yes	—
166	C₃×C₄	—	2	8	no	—
167	C₃×C₄	C₄×Dic₆	1	8	yes	—
168	C₃×C₄	#901	1	16	yes	—
169	C₃×C₄	C₂²×Dic₆	1	4	yes	—
170	C₃×C₄	C₂×#235	1	4	yes	—
171	Dic₃	—	6	32	no	—
172	C₃²	#2960	1	1	yes	characteristic, complemented, Sylow
173	C₂³	C₂×C₄×C₃⋊D₃	1	1	yes	characteristic
174	Q₈	—	18	8	no	—
175	C₂×C₄	—	2	4	no	—
176	C₂×C₄	—	9	8	no	—
177	C₂×C₄	C₂×#634	1	2	yes	—
178	C₂×C₄	C₂×C₄×C₃⋊D₃	1	2	yes	—
179	C₂×C₄	C₂×C₄×C₃⋊D₃	1	1	yes	characteristic
180	C₂×C₄	C₂×C₄×C₃⋊D₃	1	2	yes	—
181	C₂×C₄	#3828	1	2	yes	—
182	C₂×C₄	C₂³×C₃⋊D₃	1	1	yes	characteristic
183	C₈	—	2	2	no	—
184	C₈	—	18	4	no	—
185	C₈	C₂×#634	1	2	yes	—
186	C₈	C₂×#634	1	2	yes	—
187	C₂×C₃	—	2	8	no	—
188	C₂×C₃	C₂×C₄×Dic₆	1	4	yes	—
189	C₂×C₃	C₂×#901	1	8	yes	—
190	C₂²	—	2	2	no	—
191	C₂²	C₄×#634	1	2	yes	—
192	C₂²	C₂²×C₄×C₃⋊D₃	1	1	yes	characteristic
193	C₄	—	2	2	no	—
194	C₄	—	18	8	no	—
195	C₄	C₄×#634	1	2	yes	—
196	C₄	#64110	1	4	yes	—
197	C₄	C₂²×#634	1	1	yes	characteristic
198	C₄	C₂×#3828	1	1	yes	characteristic
199	C₃	#78445	1	4	yes	complemented
200	C₂	—	2	2	no	—
201	C₂	C₂×C₄×#634	1	1	yes	characteristic
202	C₂	C₂×#64110	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	C₂²×C₃⋊D₃
commutator, socle	C₂²×C₃²	C₂³×C₄
Frattini	C₂×C₄	C₂³×C₃⋊D₃
Fitting	C₂×C₃²×C₄×C₈	C₂
prime radical	C₂³×C₃²	C₂²×C₄
cosocle	C₂×C₃²×C₄	C₂⁴
hypercenter	C₂×C₄×C₈	C₃⋊D₃
hypocenter	C₃²	#2960
LCS	G ▷ C₂²×C₃² ▷ C₃²
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
6. #10592378	C₂	semidirect
7. C₈×#634	C₂	semidirect
35. C₂×C₃²×C₈	Q₈	semidirect
90. C₂×C₃×C₈	Dic₆	semidirect
151. C₂×C₈	#634	semidirect
172. C₃²	#2960	semidirect
199. C₃	#78445	semidirect

Maximal subgroups

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

subgroup	index
#10595941	2
C₂×C₃²×C₄×C₈	2
#10592801	2
#10592800	2
#10592623	2
#10592378	2
C₈×#634	2
#78445	3