Subgroups — #11791678

Order 1152.

450 conjugacy classes of subgroups.

The subgroup lattice has 225 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#63431	C₂	1	1	yes	maximal, characteristic
2	C₉×#384	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×C₉⋊C₈	C₂	1	1	yes	maximal, characteristic
4	#10588206	C₂	1	4	yes	maximal
5	#69582	—	3	1	no	maximal
6	C₂²×Dic₁₈	—	2	1	no	—
7	C₂×C₉×C₄⋊C₄	—	2	1	no	—
8	C₂×C₉×C₄⋊C₄	—	2	1	no	—
9	C₂×C₉×C₄⋊C₄	C₂²	1	2	yes	—
10	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊Dic₉	—	2	1	no	—
12	C₂×C₄⋊Dic₉	C₂²	1	2	yes	—
13	C₂²×C₉⋊C₈	C₂²	1	2	yes	—
14	#63431	—	2	2	no	—
15	#63431	C₄	1	4	yes	complemented
16	C₄×C₉⋊C₈	—	2	2	no	—
17	C₂×#869	—	3	1	no	—
18	C₃×#384	D₃	1	1	yes	characteristic
19	C₂×C₄×C₃⋊C₈	—	3	1	no	—
20	#4529	—	3	4	no	—
21	C₂²×C₄×C₉	—	2	2	no	—
22	C₂²×C₄×C₉	D₄	1	1	yes	characteristic
23	C₂²×C₄×C₉	D₄	1	1	yes	characteristic
24	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
25	C₂²×Dic₉	—	2	2	no	—
26	C₂×Dic₁₈	—	2	4	no	—
27	C₂×Dic₁₈	—	4	2	no	—
28	C₂×Dic₁₈	—	4	2	no	—
29	C₉×C₄⋊C₄	—	2	4	no	—
30	C₉×C₄⋊C₄	—	4	2	no	—
31	C₉×C₄⋊C₄	—	4	2	no	—
32	C₄²×C₉	C₂×C₄	1	2	yes	—
33	C₄²×C₉	D₄	1	2	yes	—
34	C₄⋊Dic₉	—	2	4	no	—
35	C₄⋊Dic₉	—	2	2	no	—
36	C₄⋊Dic₉	C₂×C₄	1	4	yes	—
37	C₂×C₉⋊C₈	—	2	2	no	—
38	C₂×C₉⋊C₈	—	2	2	no	—
39	C₂×C₉⋊C₈	Q₈	1	4	yes	complemented
40	#1990	—	9	1	no	Sylow
41	C₂²×Dic₆	—	6	1	no	—
42	C₂×C₃×C₄⋊C₄	—	2	1	no	—
43	C₂×C₃×C₄⋊C₄	—	2	1	no	—
44	C₂×C₃×C₄⋊C₄	C₂×D₃	1	2	yes	—
45	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
46	C₂×C₄⋊Dic₃	—	3	2	no	—
47	C₂×C₄⋊Dic₃	—	6	1	no	—
48	C₂²×C₃⋊C₈	—	3	2	no	—
49	#869	—	3	4	no	—
50	#869	—	6	2	no	—
51	C₄×C₃⋊C₈	—	6	2	no	—
52	C₂³×C₉	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₉	—	4	2	no	—
54	C₂×C₄×C₉	—	4	2	no	—
55	C₂×C₄×C₉	—	4	2	no	—
56	C₂×C₄×C₉	C₂²⋊C₄	1	2	yes	—
57	C₂×C₄×C₉	C₂²⋊C₄	1	2	yes	—
58	C₂×C₄×C₉	D₈	1	2	yes	—
59	C₂×C₄×C₉	SD₁₆	1	2	yes	—
60	C₂×C₄×C₉	SD₁₆	1	2	yes	—
61	C₂×C₄×C₉	Q₁₆	1	2	yes	—
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×Q₈	1	2	yes	—
65	C₂×C₄×C₉	#41	1	2	yes	—
66	C₂×Dic₉	—	2	4	no	—
67	C₂×Dic₉	—	4	2	no	—
68	C₂×Dic₉	—	4	2	no	—
69	Dic₁₈	—	4	8	no	—
70	C₉⋊C₈	—	2	4	no	—
71	C₂×C₄⋊Q₈	—	9	1	no	—
72	C₂×C₄×C₈	—	9	1	no	—
73	#384	D₉	1	1	yes	characteristic
74	#340	—	9	4	no	—
75	C₂²×C₃×C₄	—	2	2	no	—
76	C₂²×C₃×C₄	#67	1	1	yes	characteristic
77	C₂²×C₃×C₄	#67	1	1	yes	characteristic
78	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
79	C₂²×Dic₃	—	6	2	no	—
80	C₂×Dic₆	—	6	4	no	—
81	C₂×Dic₆	—	12	2	no	—
82	C₂×Dic₆	—	12	2	no	—
83	C₃×C₄⋊C₄	—	2	4	no	—
84	C₃×C₄⋊C₄	—	4	2	no	—
85	C₃×C₄⋊C₄	—	4	2	no	—
86	C₃×C₄²	C₄×D₃	1	2	yes	—
87	C₃×C₄²	D₁₂	1	2	yes	—
88	C₄⋊Dic₃	—	3	4	no	—
89	C₄⋊Dic₃	—	6	4	no	—
90	C₄⋊Dic₃	—	6	2	no	—
91	C₂×C₃⋊C₈	—	3	4	no	—
92	C₂×C₃⋊C₈	—	6	2	no	—
93	C₂×C₃⋊C₈	—	6	2	no	—
94	C₂²×C₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
95	C₂²×C₉	#124	1	1	yes	characteristic
96	C₂²×C₉	C₄⋊Q₈	1	1	yes	characteristic
97	C₂²×C₉	C₂×D₈	1	1	yes	characteristic
98	C₂²×C₉	C₂×SD₁₆	1	1	yes	characteristic
99	C₂²×C₉	C₂×SD₁₆	1	1	yes	characteristic
100	C₂²×C₉	C₂×Q₁₆	1	1	yes	characteristic
101	C₄×C₉	—	8	2	no	—
102	C₄×C₉	#102	1	4	yes	—
103	C₄×C₉	#103	1	4	yes	—
104	C₄×C₉	C₄×Q₈	1	2	yes	—
105	C₄×C₉	C₂²⋊Q₈	1	2	yes	—
106	Dic₉	—	4	4	no	—
107	C₂²×Q₈	—	18	1	no	—
108	C₂²×C₈	—	9	2	no	—
109	C₄⋊Q₈	—	9	4	no	—
110	C₄⋊Q₈	—	18	2	no	—
111	C₂×C₄⋊C₄	—	2	1	no	—
112	C₂×C₄⋊C₄	—	2	1	no	—
113	C₂×C₄⋊C₄	—	9	2	no	—
114	C₂×C₄⋊C₄	—	18	1	no	—
115	C₂×C₄⋊C₄	C₂×D₉	1	2	yes	—
116	C₂×C₄²	C₂×D₉	1	1	yes	characteristic
117	C₄×C₈	—	18	2	no	—
118	C₂³×C₃	C₂×#67	1	1	yes	characteristic
119	C₂×C₃×C₄	—	4	2	no	—
120	C₂×C₃×C₄	—	4	2	no	—
121	C₂×C₃×C₄	—	4	2	no	—
122	C₂×C₃×C₄	(C₂×D₃)⋊C₄	1	2	yes	—
123	C₂×C₃×C₄	(C₂×D₃)⋊C₄	1	2	yes	—
124	C₂×C₃×C₄	#213	1	2	yes	—
125	C₂×C₃×C₄	#214	1	2	yes	—
126	C₂×C₃×C₄	#215	1	2	yes	—
127	C₂×C₃×C₄	#216	1	2	yes	—
128	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
129	C₂×C₃×C₄	C₂×D₁₂	1	1	yes	characteristic
130	C₂×C₃×C₄	#237	1	2	yes	—
131	C₂×C₃×C₄	D₃×Q₈	1	2	yes	—
132	C₂×Dic₃	—	6	4	no	—
133	C₂×Dic₃	—	12	2	no	—
134	C₂×Dic₃	—	12	2	no	—
135	Dic₆	—	12	8	no	—
136	C₃⋊C₈	—	6	4	no	—
137	C₂×C₉	#391	1	1	yes	characteristic
138	C₂×C₉	C₂×#102	1	1	yes	characteristic
139	C₂×C₉	C₂×#103	1	1	yes	characteristic
140	C₂×C₉	#486	1	1	yes	characteristic
141	C₂×C₉	#487	1	1	yes	characteristic
142	C₂×C₉	#498	1	1	yes	characteristic
143	C₂×C₉	#500	1	1	yes	characteristic
144	C₂×Q₈	—	18	4	no	—
145	C₂×Q₈	—	36	2	no	—
146	C₂×Q₈	—	36	2	no	—
147	C₂²×C₄	—	2	2	no	—
148	C₂²×C₄	—	18	2	no	—
149	C₂²×C₄	#611	1	1	yes	characteristic
150	C₂²×C₄	#611	1	1	yes	characteristic
151	C₂²×C₄	C₂²×D₉	1	1	yes	characteristic
152	C₂×C₈	—	9	4	no	—
153	C₂×C₈	—	18	2	no	—
154	C₂×C₈	—	18	2	no	—
155	C₄⋊C₄	—	2	4	no	—
156	C₄⋊C₄	—	4	2	no	—
157	C₄⋊C₄	—	4	2	no	—
158	C₄⋊C₄	—	9	4	no	—
159	C₄⋊C₄	—	18	4	no	—
160	C₄⋊C₄	—	18	2	no	—
161	C₄²	C₄×D₉	1	2	yes	—
162	C₄²	D₃₆	1	2	yes	—
163	C₂²×C₃	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
164	C₂²×C₃	C₂×#213	1	1	yes	characteristic
165	C₂²×C₃	C₂×#214	1	1	yes	characteristic
166	C₂²×C₃	#936	1	1	yes	characteristic
167	C₂²×C₃	C₂×#215	1	1	yes	characteristic
168	C₂²×C₃	C₂×#216	1	1	yes	characteristic
169	C₂²×C₃	#944	1	1	yes	characteristic
170	C₃×C₄	—	8	2	no	—
171	C₃×C₄	#809	1	4	yes	—
172	C₃×C₄	#810	1	4	yes	—
173	C₃×C₄	#887	1	2	yes	—
174	C₃×C₄	#897	1	2	yes	—
175	Dic₃	—	12	4	no	—
176	C₉	#1990	1	1	yes	characteristic, complemented, Sylow
177	C₂³	C₂×#611	1	1	yes	characteristic
178	Q₈	—	36	8	no	—
179	C₂×C₄	—	4	2	no	—
180	C₂×C₄	—	4	2	no	—
181	C₂×C₄	—	4	2	no	—
182	C₂×C₄	—	18	4	no	—
183	C₂×C₄	—	36	2	no	—
184	C₂×C₄	—	36	2	no	—
185	C₂×C₄	(C₂×D₉)⋊C₄	1	2	yes	—
186	C₂×C₄	(C₂×D₉)⋊C₄	1	2	yes	—
187	C₂×C₄	#3672	1	2	yes	—
188	C₂×C₄	#3673	1	2	yes	—
189	C₂×C₄	#3674	1	2	yes	—
190	C₂×C₄	#3675	1	2	yes	—
191	C₂×C₄	C₂×C₄×D₉	1	1	yes	characteristic
192	C₂×C₄	C₂×D₃₆	1	1	yes	characteristic
193	C₂×C₄	#3699	1	2	yes	—
194	C₂×C₄	Q₈×D₉	1	2	yes	—
195	C₈	—	18	4	no	—
196	C₂×C₃	C₂×#809	1	1	yes	characteristic
197	C₂×C₃	C₂×#810	1	1	yes	characteristic
198	C₂×C₃	#4973	1	1	yes	characteristic
199	C₂×C₃	#5070	1	1	yes	characteristic
200	C₂×C₃	#5078	1	1	yes	characteristic
201	C₂×C₃	#5079	1	1	yes	characteristic
202	C₂×C₃	#5080	1	1	yes	characteristic
203	C₂²	C₂×(C₂×D₉)⋊C₄	1	1	yes	characteristic
204	C₂²	C₂×#3672	1	1	yes	characteristic
205	C₂²	C₂×#3673	1	1	yes	characteristic
206	C₂²	#63498	1	1	yes	characteristic
207	C₂²	C₂×#3674	1	1	yes	characteristic
208	C₂²	C₂×#3675	1	1	yes	characteristic
209	C₂²	#63506	1	1	yes	characteristic
210	C₄	—	8	2	no	—
211	C₄	—	36	4	no	—
212	C₄	#63368	1	4	yes	—
213	C₄	#63369	1	4	yes	—
214	C₄	#63449	1	2	yes	—
215	C₄	#63459	1	2	yes	—
216	C₃	#69582	1	1	yes	characteristic
217	C₂	C₂×#63368	1	1	yes	characteristic
218	C₂	C₂×#63369	1	1	yes	characteristic
219	C₂	#10588653	1	1	yes	characteristic
220	C₂	#10588750	1	1	yes	characteristic
221	C₂	#10588758	1	1	yes	characteristic
222	C₂	#10588759	1	1	yes	characteristic
223	C₂	#10588760	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#611
commutator	C₂×C₄×C₉	C₂²×C₄
Frattini	C₂²×C₃×C₄	C₂²×D₃
Fitting	C₉×#384	C₂
socle, prime radical	C₂³×C₃	C₂×#67
cosocle	C₂²×C₄×C₉	C₂³
hypercenter	#384	D₉
hypocenter	C₉	#1990
LCS	G ▷ C₂×C₄×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

Complements

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

N	H ≅ G/N	type
15. #63431	C₄	semidirect
39. C₂×C₉⋊C₈	Q₈	semidirect
176. C₉	#1990	semidirect

Maximal subgroups

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

subgroup	index
C₂×#63431	2
C₉×#384	2
C₂×C₄×C₉⋊C₈	2
#10588206	2
#69582	3