Subgroups — #11790995

Order 1152.

468 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Dic₃₆	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₈×C₉	C₂	1	1	yes	maximal, characteristic
3	C₂×#63378	C₂	1	1	yes	maximal, characteristic
4	C₂×#63376	C₂	1	2	yes	maximal
5	#10588609	C₂	1	2	yes	maximal
6	#69775	—	3	1	no	maximal
7	C₂²×Dic₁₈	C₂²	1	2	yes	—
8	C₂²×C₈×C₉	C₂²	1	1	yes	characteristic
9	C₂²×C₈×C₉	C₂²	1	1	yes	characteristic
10	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊Dic₉	C₂²	1	2	yes	—
12	C₂×Dic₃₆	—	2	2	no	—
13	C₂×Dic₃₆	—	2	2	no	—
14	C₂×Dic₃₆	C₄	1	4	yes	complemented
15	C₄×C₈×C₉	—	2	2	no	—
16	C₉⋊#95	—	2	4	no	—
17	#63378	—	2	2	no	—
18	#63376	—	2	4	no	—
19	C₂²×Dic₁₂	—	3	1	no	—
20	C₂×C₃×C₄×C₈	D₃	1	1	yes	characteristic
21	C₂×#818	—	3	1	no	—
22	C₂×#816	—	3	2	no	—
23	#4929	—	3	2	no	—
24	C₂²×C₄×C₉	D₄	1	2	yes	—
25	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
26	C₂²×Dic₉	—	2	2	no	—
27	C₂²×Dic₉	—	2	2	no	—
28	C₂×Dic₁₈	—	2	2	no	—
29	C₂×Dic₁₈	—	2	2	no	—
30	C₂×Dic₁₈	—	2	2	no	—
31	C₂×Dic₁₈	—	2	2	no	—
32	C₂×Dic₁₈	—	2	2	no	—
33	C₂×Dic₁₈	C₂×C₄	1	4	yes	—
34	C₂×C₈×C₉	—	2	1	no	—
35	C₂×C₈×C₉	—	2	1	no	—
36	C₂×C₈×C₉	C₂×C₄	1	2	yes	—
37	C₂×C₈×C₉	D₄	1	2	yes	—
38	C₂×C₈×C₉	D₄	1	2	yes	—
39	C₂×C₈×C₉	D₄	1	2	yes	—
40	C₄²×C₉	—	2	1	no	—
41	C₄²×C₉	—	2	1	no	—
42	C₄⋊Dic₉	—	2	2	no	—
43	C₄⋊Dic₉	—	2	2	no	—
44	Dic₃₆	—	2	8	no	—
45	#1969	—	9	1	no	Sylow
46	C₂²×Dic₆	—	3	2	no	—
47	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
48	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
49	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
50	C₂×C₄⋊Dic₃	—	3	2	no	—
51	C₂×Dic₁₂	—	3	4	no	—
52	C₂×Dic₁₂	—	6	2	no	—
53	C₂×Dic₁₂	—	6	2	no	—
54	C₃×C₄×C₈	—	2	2	no	—
55	C₃⋊#95	—	6	4	no	—
56	#818	—	6	2	no	—
57	#816	—	6	4	no	—
58	C₂³×C₉	C₂×D₄	1	1	yes	characteristic
59	C₂×C₄×C₉	—	2	2	no	—
60	C₂×C₄×C₉	—	2	2	no	—
61	C₂×C₄×C₉	Q₁₆	1	4	yes	complemented
62	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
63	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
64	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
65	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
66	C₂×C₄×C₉	#41	1	1	yes	characteristic
67	C₂×C₄×C₉	#41	1	1	yes	characteristic
68	C₂×Dic₉	—	2	4	no	—
69	C₂×Dic₉	—	4	2	no	—
70	C₂×Dic₉	—	4	2	no	—
71	C₂×Dic₉	—	4	2	no	—
72	C₂×Dic₉	—	4	4	no	—
73	Dic₁₈	—	2	4	no	—
74	Dic₁₈	—	2	4	no	—
75	Dic₁₈	—	4	2	no	—
76	Dic₁₈	—	4	2	no	—
77	C₈×C₉	—	2	2	no	—
78	C₈×C₉	C₂²⋊C₄	1	4	yes	—
79	C₂²×Q₁₆	—	9	1	no	—
80	C₂×#107	—	9	1	no	—
81	C₂×#103	—	9	2	no	—
82	C₂×C₄×C₈	D₉	1	1	yes	characteristic
83	#391	—	9	2	no	—
84	C₂²×C₃×C₄	D₁₂	1	2	yes	—
85	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
86	C₂²×Dic₃	—	6	2	no	—
87	C₂²×Dic₃	—	6	2	no	—
88	C₂×Dic₆	—	3	4	no	—
89	C₂×Dic₆	—	6	2	no	—
90	C₂×Dic₆	—	6	2	no	—
91	C₂×Dic₆	—	6	2	no	—
92	C₂×Dic₆	—	6	2	no	—
93	C₂×Dic₆	—	6	2	no	—
94	C₂×C₃×C₈	—	2	1	no	—
95	C₂×C₃×C₈	—	2	1	no	—
96	C₂×C₃×C₈	C₄×D₃	1	2	yes	—
97	C₂×C₃×C₈	D₁₂	1	2	yes	—
98	C₂×C₃×C₈	D₁₂	1	2	yes	—
99	C₂×C₃×C₈	#67	1	2	yes	—
100	C₃×C₄²	—	2	1	no	—
101	C₃×C₄²	—	2	1	no	—
102	C₄⋊Dic₃	—	6	2	no	—
103	C₄⋊Dic₃	—	6	2	no	—
104	Dic₁₂	—	6	8	no	—
105	C₂²×C₉	C₄×D₄	1	1	yes	characteristic
106	C₂²×C₉	#121	1	1	yes	characteristic
107	C₂²×C₉	#127	1	1	yes	characteristic
108	C₂²×C₉	C₂×Q₁₆	1	2	yes	—
109	C₂²×C₉	#135	1	2	yes	—
110	C₄×C₉	—	2	4	no	—
111	C₄×C₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
112	C₄×C₉	C₄×D₄	1	1	yes	characteristic
113	C₄×C₉	#121	1	1	yes	characteristic
114	C₄×C₉	#124	1	1	yes	characteristic
115	Dic₉	—	4	4	no	—
116	Dic₉	—	8	2	no	—
117	C₂²×Q₈	—	9	2	no	—
118	C₂×Q₁₆	—	9	4	no	—
119	C₂×Q₁₆	—	18	2	no	—
120	C₂×Q₁₆	—	18	2	no	—
121	C₂²×C₈	C₂×D₉	1	1	yes	characteristic
122	C₂²×C₈	C₂×D₉	1	1	yes	characteristic
123	C₂×C₄⋊C₄	—	9	2	no	—
124	C₂×C₄²	C₂×D₉	1	1	yes	characteristic
125	#107	—	18	2	no	—
126	#103	—	18	4	no	—
127	C₄×C₈	—	2	2	no	—
128	#95	—	18	4	no	—
129	C₂³×C₃	C₂×D₁₂	1	1	yes	characteristic
130	C₂×C₃×C₄	—	2	2	no	—
131	C₂×C₃×C₄	—	2	2	no	—
132	C₂×C₃×C₄	Dic₁₂	1	4	yes	—
133	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
134	C₂×C₃×C₄	C₂×D₁₂	1	1	yes	characteristic
135	C₂×C₃×C₄	C₂×D₁₂	1	1	yes	characteristic
136	C₂×C₃×C₄	#235	1	1	yes	characteristic
137	C₂×C₃×C₄	#235	1	1	yes	characteristic
138	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
139	C₂×Dic₃	—	6	4	no	—
140	C₂×Dic₃	—	12	2	no	—
141	C₂×Dic₃	—	12	2	no	—
142	C₂×Dic₃	—	12	2	no	—
143	C₂×Dic₃	—	12	4	no	—
144	Dic₆	—	6	4	no	—
145	Dic₆	—	6	4	no	—
146	Dic₆	—	12	2	no	—
147	Dic₆	—	12	2	no	—
148	C₃×C₈	—	2	2	no	—
149	C₃×C₈	(C₂×D₃)⋊C₄	1	4	yes	—
150	C₂×C₉	#390	1	1	yes	characteristic
151	C₂×C₉	C₄×Q₁₆	1	2	yes	—
152	C₂×C₉	#467	1	2	yes	—
153	C₂×C₉	#494	1	1	yes	characteristic
154	C₂×C₉	#495	1	1	yes	characteristic
155	C₂×Q₈	—	9	4	no	—
156	C₂×Q₈	—	18	2	no	—
157	C₂×Q₈	—	18	2	no	—
158	C₂×Q₈	—	18	2	no	—
159	C₂×Q₈	—	18	2	no	—
160	C₂×Q₈	—	18	2	no	—
161	C₂²×C₄	—	18	2	no	—
162	C₂²×C₄	—	18	2	no	—
163	C₂²×C₄	D₃₆	1	2	yes	—
164	C₂²×C₄	C₂²×D₉	1	1	yes	characteristic
165	Q₁₆	—	18	8	no	—
166	C₂×C₈	—	2	1	no	—
167	C₂×C₈	—	2	1	no	—
168	C₂×C₈	C₄×D₉	1	2	yes	—
169	C₂×C₈	D₃₆	1	2	yes	—
170	C₂×C₈	D₃₆	1	2	yes	—
171	C₂×C₈	#611	1	2	yes	—
172	C₄⋊C₄	—	18	2	no	—
173	C₄⋊C₄	—	18	2	no	—
174	C₄²	—	2	1	no	—
175	C₄²	—	2	1	no	—
176	C₂²×C₃	C₄×D₁₂	1	1	yes	characteristic
177	C₂²×C₃	#874	1	1	yes	characteristic
178	C₂²×C₃	#904	1	2	yes	—
179	C₂²×C₃	C₂×Dic₁₂	1	2	yes	—
180	C₂²×C₃	#930	1	1	yes	characteristic
181	C₃×C₄	—	2	4	no	—
182	C₃×C₄	C₄×D₁₂	1	1	yes	characteristic
183	C₃×C₄	#875	1	1	yes	characteristic
184	C₃×C₄	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
185	C₃×C₄	#930	1	1	yes	characteristic
186	Dic₃	—	12	4	no	—
187	Dic₃	—	24	2	no	—
188	C₉	#1969	1	1	yes	characteristic, complemented, Sylow
189	C₂³	C₂×D₃₆	1	1	yes	characteristic
190	Q₈	—	18	4	no	—
191	Q₈	—	18	4	no	—
192	Q₈	—	36	2	no	—
193	Q₈	—	36	2	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	18	4	no	—
197	C₂×C₄	—	36	2	no	—
198	C₂×C₄	—	36	2	no	—
199	C₂×C₄	—	36	2	no	—
200	C₂×C₄	—	36	4	no	—
201	C₂×C₄	Dic₃₆	1	4	yes	complemented
202	C₂×C₄	C₂×C₄×D₉	1	1	yes	characteristic
203	C₂×C₄	C₂×D₃₆	1	1	yes	characteristic
204	C₂×C₄	C₂×D₃₆	1	1	yes	characteristic
205	C₂×C₄	#3697	1	1	yes	characteristic
206	C₂×C₄	#3697	1	1	yes	characteristic
207	C₂×C₄	C₂×#611	1	1	yes	characteristic
208	C₈	—	2	2	no	—
209	C₈	(C₂×D₉)⋊C₄	1	4	yes	—
210	C₂×C₃	#4696	1	1	yes	characteristic
211	C₂×C₃	C₄×Dic₁₂	1	2	yes	—
212	C₂×C₃	#4699	1	1	yes	characteristic
213	C₂×C₃	#4939	1	1	yes	characteristic
214	C₂×C₃	#5116	1	2	yes	—
215	C₂²	C₄×D₃₆	1	1	yes	characteristic
216	C₂²	#63436	1	1	yes	characteristic
217	C₂²	C₂×Dic₃₆	1	2	yes	—
218	C₂²	#63467	1	2	yes	—
219	C₂²	#63492	1	1	yes	characteristic
220	C₄	—	2	4	no	—
221	C₄	—	36	4	no	—
222	C₄	—	72	2	no	—
223	C₄	C₄×D₃₆	1	1	yes	characteristic
224	C₄	#63437	1	1	yes	characteristic
225	C₄	C₂×(C₂×D₉)⋊C₄	1	1	yes	characteristic
226	C₄	#63492	1	1	yes	characteristic
227	C₃	#69775	1	1	yes	characteristic
228	C₂	C₄×Dic₃₆	1	2	yes	—
229	C₂	#10588365	1	1	yes	characteristic
230	C₂	#10588378	1	1	yes	characteristic
231	C₂	#10588619	1	1	yes	characteristic
232	C₂	#10588788	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₃₆
commutator	C₂×C₄×C₉	C₂²×C₄
Frattini	C₂²×C₃×C₄	C₂²×D₃
Fitting	C₂×C₄×C₈×C₉	C₂
socle, prime radical	C₂³×C₃	C₂×D₁₂
cosocle	C₂²×C₄×C₉	C₂³
hypercenter	C₂×C₄×C₈	D₉
hypocenter	C₉	#1969
LCS	G ▷ C₂×C₄×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
14. C₂×Dic₃₆	C₄	semidirect
61. C₂×C₄×C₉	Q₁₆	semidirect
188. C₉	#1969	semidirect
201. C₂×C₄	Dic₃₆	semidirect

Maximal subgroups

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

subgroup	index
C₂²×Dic₃₆	2
C₂×C₄×C₈×C₉	2
C₂×#63378	2
C₂×#63376	2
#10588609	2
#69775	3