Subgroups — #11790998

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×C₉×Q₁₆	C₂	1	1	yes	maximal, characteristic
2	#10588909	C₂	1	2	yes	maximal
3	C₂×#63395	C₂	1	2	yes	maximal
4	C₂×#63378	C₂	1	1	yes	maximal, characteristic
5	C₂×C₈×Dic₉	C₂	1	1	yes	maximal, characteristic
6	#72066	—	3	1	no	maximal
7	C₂²×Q₈×C₉	C₂²	1	2	yes	—
8	C₂×C₉×Q₁₆	—	2	2	no	—
9	C₂×C₉×Q₁₆	—	2	2	no	—
10	C₂×C₉×Q₁₆	C₄	1	4	yes	complemented
11	C₂²×C₈×C₉	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊Dic₉	C₂²	1	2	yes	—
13	C₂×C₄×Dic₉	C₂²	1	1	yes	characteristic
14	C₂²×C₉⋊C₈	C₂²	1	1	yes	characteristic
15	#63395	—	2	4	no	—
16	C₉⋊#95	—	2	4	no	—
17	#63378	—	2	2	no	—
18	C₈×Dic₉	—	2	2	no	—
19	C₂²×C₃×Q₁₆	D₃	1	1	yes	characteristic
20	#5229	—	3	2	no	—
21	C₂×#835	—	3	2	no	—
22	C₂×#818	—	3	1	no	—
23	C₂×C₈×Dic₃	—	3	1	no	—
24	C₂×Q₈×C₉	—	2	2	no	—
25	C₂×Q₈×C₉	—	2	2	no	—
26	C₂×Q₈×C₉	—	2	2	no	—
27	C₂×Q₈×C₉	—	2	2	no	—
28	C₂×Q₈×C₉	—	2	2	no	—
29	C₂×Q₈×C₉	C₂×C₄	1	4	yes	—
30	C₂²×C₄×C₉	—	2	2	no	—
31	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
32	C₂²×Dic₉	—	2	2	no	—
33	C₂²×Dic₉	D₄	1	2	yes	—
34	C₉×Q₁₆	—	2	8	no	—
35	C₂×C₈×C₉	C₂×C₄	1	2	yes	—
36	C₂×C₈×C₉	D₄	1	2	yes	—
37	C₂×C₈×C₉	D₄	1	2	yes	—
38	C₄⋊Dic₉	—	2	2	no	—
39	C₄⋊Dic₉	—	2	2	no	—
40	C₄×Dic₉	—	2	1	no	—
41	C₄×Dic₉	—	2	1	no	—
42	C₂×C₉⋊C₈	—	2	1	no	—
43	C₂×C₉⋊C₈	—	2	1	no	—
44	C₂×C₉⋊C₈	D₄	1	2	yes	—
45	#1969	—	9	1	no	Sylow
46	C₂²×C₃×Q₈	C₂×D₃	1	2	yes	—
47	C₂×C₃×Q₁₆	—	2	2	no	—
48	C₂×C₃×Q₁₆	—	2	2	no	—
49	C₂×C₃×Q₁₆	Dic₃	1	4	yes	—
50	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
51	C₂×C₄⋊Dic₃	—	3	2	no	—
52	C₂×C₄×Dic₃	—	3	1	no	—
53	C₂²×C₃⋊C₈	—	3	1	no	—
54	#835	—	6	4	no	—
55	C₃⋊#95	—	6	4	no	—
56	#818	—	6	2	no	—
57	C₈×Dic₃	—	6	2	no	—
58	C₂³×C₉	C₂×D₄	1	1	yes	characteristic
59	Q₈×C₉	—	2	4	no	—
60	Q₈×C₉	—	2	4	no	—
61	Q₈×C₉	—	4	2	no	—
62	Q₈×C₉	—	4	2	no	—
63	C₂×C₄×C₉	—	2	4	no	—
64	C₂×C₄×C₉	—	4	2	no	—
65	C₂×C₄×C₉	—	4	2	no	—
66	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
67	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
68	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
69	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
70	C₂×C₄×C₉	#41	1	1	yes	characteristic
71	C₂×C₄×C₉	#41	1	1	yes	characteristic
72	C₂×Dic₉	—	2	2	no	—
73	C₂×Dic₉	—	2	2	no	—
74	C₂×Dic₉	—	4	2	no	—
75	C₂×Dic₉	—	4	4	no	—
76	C₂×Dic₉	Q₁₆	1	4	yes	complemented
77	C₈×C₉	C₂²⋊C₄	1	4	yes	—
78	C₉⋊C₈	—	2	2	no	—
79	C₂²×Q₁₆	D₉	1	1	yes	characteristic
80	C₂×#107	—	9	1	no	—
81	C₂×#103	—	9	2	no	—
82	C₂×C₄×C₈	—	9	1	no	—
83	#391	—	9	2	no	—
84	C₂×C₃×Q₈	—	2	2	no	—
85	C₂×C₃×Q₈	—	2	2	no	—
86	C₂×C₃×Q₈	—	2	2	no	—
87	C₂×C₃×Q₈	—	2	2	no	—
88	C₂×C₃×Q₈	—	2	2	no	—
89	C₂×C₃×Q₈	C₂×Dic₃	1	4	yes	—
90	C₂²×C₃×C₄	—	2	2	no	—
91	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
92	C₂²×Dic₃	—	3	2	no	—
93	C₂²×Dic₃	—	6	2	no	—
94	C₃×Q₁₆	—	2	8	no	—
95	C₂×C₃×C₈	C₂×Dic₃	1	2	yes	—
96	C₂×C₃×C₈	#67	1	2	yes	—
97	C₂×C₃×C₈	#67	1	2	yes	—
98	C₄⋊Dic₃	—	6	2	no	—
99	C₄⋊Dic₃	—	6	2	no	—
100	C₄×Dic₃	—	6	1	no	—
101	C₄×Dic₃	—	6	1	no	—
102	C₂×C₃⋊C₈	—	3	2	no	—
103	C₂×C₃⋊C₈	—	6	1	no	—
104	C₂×C₃⋊C₈	—	6	1	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	—	4	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	—	2	4	no	—
116	Dic₉	—	8	2	no	—
117	C₂²×Q₈	C₂×D₉	1	2	yes	—
118	C₂×Q₁₆	—	2	2	no	—
119	C₂×Q₁₆	—	2	2	no	—
120	C₂×Q₁₆	Dic₉	1	4	yes	complemented
121	C₂²×C₈	—	9	1	no	—
122	C₂²×C₈	C₂×D₉	1	1	yes	characteristic
123	C₂×C₄⋊C₄	—	9	2	no	—
124	C₂×C₄²	—	9	1	no	—
125	#107	—	18	2	no	—
126	#103	—	18	4	no	—
127	C₄×C₈	—	18	2	no	—
128	#95	—	18	4	no	—
129	C₂³×C₃	D₃×D₄	1	1	yes	characteristic
130	C₃×Q₈	—	2	4	no	—
131	C₃×Q₈	—	2	4	no	—
132	C₃×Q₈	—	4	2	no	—
133	C₃×Q₈	—	4	2	no	—
134	C₂×C₃×C₄	—	2	4	no	—
135	C₂×C₃×C₄	—	4	2	no	—
136	C₂×C₃×C₄	—	4	2	no	—
137	C₂×C₃×C₄	D₃×D₄	1	1	yes	characteristic
138	C₂×C₃×C₄	#237	1	1	yes	characteristic
139	C₂×C₃×C₄	#237	1	1	yes	characteristic
140	C₂×C₃×C₄	C₂²×Dic₃	1	1	yes	characteristic
141	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
142	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
143	C₂×Dic₃	—	3	4	no	—
144	C₂×Dic₃	—	6	2	no	—
145	C₂×Dic₃	—	6	2	no	—
146	C₂×Dic₃	—	12	4	no	—
147	C₂×Dic₃	—	12	2	no	—
148	C₃×C₈	#217	1	4	yes	—
149	C₃⋊C₈	—	6	2	no	—
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	—	2	2	no	—
156	C₂×Q₈	—	2	2	no	—
157	C₂×Q₈	—	2	2	no	—
158	C₂×Q₈	—	2	2	no	—
159	C₂×Q₈	—	2	2	no	—
160	C₂×Q₈	C₂×Dic₉	1	4	yes	—
161	C₂²×C₄	—	2	2	no	—
162	C₂²×C₄	—	9	2	no	—
163	C₂²×C₄	—	18	2	no	—
164	C₂²×C₄	C₂²×D₉	1	1	yes	characteristic
165	Q₁₆	—	2	8	no	—
166	C₂×C₈	—	9	2	no	—
167	C₂×C₈	—	18	1	no	—
168	C₂×C₈	—	18	1	no	—
169	C₂×C₈	C₂×Dic₉	1	2	yes	—
170	C₂×C₈	#611	1	2	yes	—
171	C₂×C₈	#611	1	2	yes	—
172	C₄⋊C₄	—	18	2	no	—
173	C₄⋊C₄	—	18	2	no	—
174	C₄²	—	18	1	no	—
175	C₄²	—	18	1	no	—
176	C₂²×C₃	D₃×Q₁₆	1	2	yes	—
177	C₂²×C₃	#919	1	2	yes	—
178	C₂²×C₃	D₄×Dic₃	1	1	yes	characteristic
179	C₂²×C₃	#938	1	1	yes	characteristic
180	C₂²×C₃	#940	1	1	yes	characteristic
181	C₃×C₄	—	4	4	no	—
182	C₃×C₄	D₄×Dic₃	1	1	yes	characteristic
183	C₃×C₄	#936	1	1	yes	characteristic
184	C₃×C₄	#938	1	1	yes	characteristic
185	C₃×C₄	C₂×#217	1	1	yes	characteristic
186	Dic₃	—	6	4	no	—
187	Dic₃	—	24	2	no	—
188	C₉	#1969	1	1	yes	characteristic, complemented, Sylow
189	C₂³	D₄×D₉	1	1	yes	characteristic
190	Q₈	—	2	4	no	—
191	Q₈	—	2	4	no	—
192	Q₈	—	4	2	no	—
193	Q₈	—	4	2	no	—
194	C₂×C₄	—	2	4	no	—
195	C₂×C₄	—	4	2	no	—
196	C₂×C₄	—	4	2	no	—
197	C₂×C₄	—	9	4	no	—
198	C₂×C₄	—	18	2	no	—
199	C₂×C₄	—	18	2	no	—
200	C₂×C₄	—	36	2	no	—
201	C₂×C₄	—	36	4	no	—
202	C₂×C₄	D₄×D₉	1	1	yes	characteristic
203	C₂×C₄	#3699	1	1	yes	characteristic
204	C₂×C₄	#3699	1	1	yes	characteristic
205	C₂×C₄	C₂²×Dic₉	1	1	yes	characteristic
206	C₂×C₄	C₂×#611	1	1	yes	characteristic
207	C₂×C₄	C₂×#611	1	1	yes	characteristic
208	C₈	—	18	2	no	—
209	C₈	#3676	1	4	yes	—
210	C₂×C₃	Dic₃×Q₁₆	1	2	yes	—
211	C₂×C₃	#5183	1	1	yes	characteristic
212	C₂×C₃	#5188	1	2	yes	—
213	C₂×C₃	#5191	1	1	yes	characteristic
214	C₂×C₃	#5221	1	1	yes	characteristic
215	C₂²	Q₁₆×D₉	1	2	yes	—
216	C₂²	#63481	1	2	yes	—
217	C₂²	D₄×Dic₉	1	1	yes	characteristic
218	C₂²	#63500	1	1	yes	characteristic
219	C₂²	#63502	1	1	yes	characteristic
220	C₄	—	4	4	no	—
221	C₄	—	18	4	no	—
222	C₄	—	72	2	no	—
223	C₄	D₄×Dic₉	1	1	yes	characteristic
224	C₄	#63498	1	1	yes	characteristic
225	C₄	#63500	1	1	yes	characteristic
226	C₄	C₂×#3676	1	1	yes	characteristic
227	C₃	#72066	1	1	yes	characteristic
228	C₂	Q₁₆×Dic₉	1	2	yes	—
229	C₂	#10588863	1	1	yes	characteristic
230	C₂	#10588868	1	2	yes	—
231	C₂	#10588871	1	1	yes	characteristic
232	C₂	#10588901	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	D₄×D₉
commutator	C₂×C₄×C₉	C₂²×C₄
Frattini	C₂²×C₃×C₄	C₂²×D₃
Fitting	C₂²×C₉×Q₁₆	C₂
socle, prime radical	C₂³×C₃	D₃×D₄
cosocle	C₂²×C₄×C₉	C₂³
hypercenter	C₂²×Q₁₆	D₉
hypocenter	C₉	#1969
LCS	G ▷ C₂×C₄×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×Q₁₆

Complements

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

N	H ≅ G/N	type
10. C₂×C₉×Q₁₆	C₄	semidirect
76. C₂×Dic₉	Q₁₆	semidirect
120. C₂×Q₁₆	Dic₉	semidirect
188. C₉	#1969	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₉×Q₁₆	2
#10588909	2
C₂×#63395	2
C₂×#63378	2
C₂×C₈×Dic₉	2
#72066	3