Subgroups — C₂×#10588800 #11858637

Order 1152.

690 conjugacy classes of subgroups.

The subgroup lattice has 219 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₉×M₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂²×#3667	C₂	1	2	yes	maximal
3	#10588800	C₂	1	8	yes	maximal, direct product factor
4	C₂×#63372	C₂	1	4	yes	maximal, complemented
5	C₂×#5125	—	3	1	no	maximal
6	C₂³×C₄×C₉	C₂²	1	1	yes	characteristic
7	C₂×C₉×M₁₆	C₄	1	4	yes	—
8	C₂×C₉×M₁₆	C₄	1	4	yes	—
9	C₂×C₉×M₁₆	C₂²	1	4	yes	—
10	C₂²×C₈×C₉	C₂²	1	2	yes	—
11	C₂×#3667	—	2	2	no	—
12	C₂×#3667	—	2	4	no	—
13	C₂×#3667	C₂²	1	4	yes	—
14	C₂×#3667	C₂²	1	8	yes	—
15	C₂²×C₉⋊C₈	—	2	2	no	—
16	#63372	C₂²	1	16	yes	complemented
17	C₂²×C₃×M₁₆	D₃	1	1	yes	characteristic
18	C₂²×#208	—	3	2	no	—
19	#5125	—	3	8	no	—
20	C₂×#819	—	3	4	no	—
21	C₂⁴×C₉	Q₈	1	1	yes	characteristic
22	C₂²×C₄×C₉	C₂×C₄	1	2	yes	—
23	C₂²×C₄×C₉	C₂×C₄	1	2	yes	—
24	C₂²×C₄×C₉	D₄	1	2	yes	—
25	C₂²×C₄×C₉	D₄	1	2	yes	—
26	C₂²×C₄×C₉	Q₈	1	1	yes	characteristic
27	C₂²×C₄×C₉	Q₈	1	2	yes	—
28	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
29	C₂²×C₄×C₉	C₂³	1	2	yes	—
30	C₉×M₁₆	C₂×C₄	1	16	yes	—
31	C₂×C₈×C₉	C₂×C₄	1	4	yes	—
32	C₂×C₈×C₉	C₂×C₄	1	4	yes	—
33	C₂×C₈×C₉	C₂³	1	4	yes	—
34	#3667	—	2	8	no	—
35	#3667	—	2	4	no	—
36	#3667	C₂³	1	8	yes	—
37	C₂×C₉⋊C₈	—	2	4	no	—
38	C₂×C₉⋊C₈	—	2	8	no	—
39	C₂×#430	—	9	1	no	Sylow
40	C₂³×C₃×C₄	C₂×D₃	1	1	yes	characteristic
41	C₂×C₃×M₁₆	Dic₃	1	4	yes	—
42	C₂×C₃×M₁₆	Dic₃	1	4	yes	—
43	C₂×C₃×M₁₆	C₂×D₃	1	4	yes	—
44	C₂²×C₃×C₈	C₂×D₃	1	2	yes	—
45	C₂×#208	—	3	8	no	—
46	C₂×#208	—	3	4	no	—
47	C₂×#208	—	6	2	no	—
48	C₂×#208	—	6	4	no	—
49	C₂²×C₃⋊C₈	—	6	2	no	—
50	#819	—	3	16	no	—
51	C₂³×C₉	—	4	2	no	—
52	C₂³×C₉	C₄⋊C₄	1	2	yes	—
53	C₂³×C₉	C₄⋊C₄	1	2	yes	—
54	C₂³×C₉	C₂×Q₈	1	1	yes	characteristic
55	C₂³×C₉	C₂×Q₈	1	2	yes	—
56	C₂×C₄×C₉	C₄⋊C₄	1	4	yes	—
57	C₂×C₄×C₉	C₄⋊C₄	1	4	yes	—
58	C₂×C₄×C₉	C₄⋊C₄	1	2	yes	—
59	C₂×C₄×C₉	C₄⋊C₄	1	2	yes	—
60	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₄×C₉	C₂²×C₄	1	4	yes	—
63	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
64	C₂×C₄×C₉	C₂×D₄	1	4	yes	—
65	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
66	C₂×C₄×C₉	C₂×Q₈	1	1	yes	characteristic
67	C₂×C₄×C₉	C₂×Q₈	1	2	yes	—
68	C₂×C₄×C₉	C₂⁴	1	1	yes	characteristic
69	C₈×C₉	C₂²×C₄	1	8	yes	—
70	C₉⋊C₈	—	2	8	no	—
71	C₂²×M₁₆	—	9	2	no	—
72	C₂²×M₁₆	D₉	1	1	yes	characteristic
73	#430	—	9	8	no	—
74	C₂×#108	—	9	4	no	—
75	C₂⁴×C₃	Dic₆	1	1	yes	characteristic
76	C₂²×C₃×C₄	Dic₆	1	1	yes	characteristic
77	C₂²×C₃×C₄	Dic₆	1	2	yes	—
78	C₂²×C₃×C₄	D₁₂	1	2	yes	—
79	C₂²×C₃×C₄	D₁₂	1	2	yes	—
80	C₂²×C₃×C₄	C₂×Dic₃	1	2	yes	—
81	C₂²×C₃×C₄	C₂×Dic₃	1	2	yes	—
82	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
83	C₂²×C₃×C₄	C₂²×D₃	1	2	yes	—
84	C₃×M₁₆	C₂×Dic₃	1	16	yes	—
85	C₂×C₃×C₈	C₂×Dic₃	1	4	yes	—
86	C₂×C₃×C₈	C₂×Dic₃	1	4	yes	—
87	C₂×C₃×C₈	C₂²×D₃	1	4	yes	—
88	#208	—	3	8	no	—
89	#208	—	6	8	no	—
90	#208	—	6	4	no	—
91	C₂×C₃⋊C₈	—	6	8	no	—
92	C₂×C₃⋊C₈	—	6	4	no	—
93	C₂²×C₉	—	2	4	no	—
94	C₂²×C₉	—	2	2	no	—
95	C₂²×C₉	—	4	2	no	—
96	C₂²×C₉	—	4	2	no	—
97	C₂²×C₉	C₂×C₄⋊C₄	1	1	yes	characteristic
98	C₂²×C₉	C₂×C₄⋊C₄	1	4	yes	—
99	C₂²×C₉	C₂×C₄⋊C₄	1	1	yes	characteristic
100	C₂²×C₉	C₂²×Q₈	1	1	yes	characteristic
101	C₄×C₉	C₂×C₄⋊C₄	1	1	yes	characteristic
102	C₄×C₉	C₂×C₄⋊C₄	1	1	yes	characteristic
103	C₄×C₉	C₂×C₄⋊C₄	1	4	yes	—
104	C₄×C₉	C₂³×C₄	1	1	yes	characteristic
105	C₄×C₉	C₂²×D₄	1	1	yes	characteristic
106	C₂³×C₄	C₂×D₉	1	1	yes	characteristic
107	C₂×M₁₆	—	9	4	no	—
108	C₂×M₁₆	—	9	8	no	—
109	C₂×M₁₆	—	18	4	no	—
110	C₂×M₁₆	—	18	2	no	—
111	C₂×M₁₆	Dic₉	1	4	yes	—
112	C₂×M₁₆	Dic₉	1	4	yes	—
113	C₂×M₁₆	C₂×D₉	1	4	yes	—
114	C₂²×C₈	—	18	2	no	—
115	C₂²×C₈	C₂×D₉	1	2	yes	—
116	#108	—	9	16	no	—
117	C₂³×C₃	—	4	2	no	—
118	C₂³×C₃	C₄⋊Dic₃	1	2	yes	—
119	C₂³×C₃	C₄⋊Dic₃	1	2	yes	—
120	C₂³×C₃	C₂×Dic₆	1	1	yes	characteristic
121	C₂³×C₃	C₂×Dic₆	1	2	yes	—
122	C₂×C₃×C₄	C₄⋊Dic₃	1	4	yes	—
123	C₂×C₃×C₄	C₄⋊Dic₃	1	2	yes	—
124	C₂×C₃×C₄	C₄⋊Dic₃	1	4	yes	—
125	C₂×C₃×C₄	C₄⋊Dic₃	1	2	yes	—
126	C₂×C₃×C₄	C₂×Dic₆	1	1	yes	characteristic
127	C₂×C₃×C₄	C₂×Dic₆	1	2	yes	—
128	C₂×C₃×C₄	C₂×D₁₂	1	4	yes	—
129	C₂×C₃×C₄	C₂×D₁₂	1	1	yes	characteristic
130	C₂×C₃×C₄	C₂×D₁₂	1	1	yes	characteristic
131	C₂×C₃×C₄	C₂²×Dic₃	1	1	yes	characteristic
132	C₂×C₃×C₄	C₂²×Dic₃	1	1	yes	characteristic
133	C₂×C₃×C₄	C₂²×Dic₃	1	4	yes	—
134	C₂×C₃×C₄	C₂³×D₃	1	1	yes	characteristic
135	C₃×C₈	C₂²×Dic₃	1	8	yes	—
136	C₃⋊C₈	—	6	8	no	—
137	C₂×C₉	—	2	4	no	—
138	C₂×C₉	—	2	1	no	—
139	C₂×C₉	—	2	1	no	—
140	C₂×C₉	#430	1	2	yes	complemented
141	C₂×C₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
142	C₂⁴	Dic₁₈	1	1	yes	characteristic
143	C₂²×C₄	Dic₁₈	1	1	yes	characteristic
144	C₂²×C₄	Dic₁₈	1	2	yes	—
145	C₂²×C₄	D₃₆	1	2	yes	—
146	C₂²×C₄	D₃₆	1	2	yes	—
147	C₂²×C₄	C₂×Dic₉	1	2	yes	—
148	C₂²×C₄	C₂×Dic₉	1	2	yes	—
149	C₂²×C₄	C₂²×D₉	1	2	yes	—
150	C₂²×C₄	C₂²×D₉	1	1	yes	characteristic
151	M₁₆	—	9	8	no	—
152	M₁₆	—	18	8	no	—
153	M₁₆	—	18	4	no	—
154	M₁₆	C₂×Dic₉	1	16	yes	—
155	C₂×C₈	—	18	8	no	—
156	C₂×C₈	—	18	4	no	—
157	C₂×C₈	C₂×Dic₉	1	4	yes	—
158	C₂×C₈	C₂×Dic₉	1	4	yes	—
159	C₂×C₈	C₂²×D₉	1	4	yes	—
160	C₂²×C₃	—	2	4	no	—
161	C₂²×C₃	—	2	2	no	—
162	C₂²×C₃	—	4	2	no	—
163	C₂²×C₃	—	4	2	no	—
164	C₂²×C₃	C₂×C₄⋊Dic₃	1	4	yes	—
165	C₂²×C₃	C₂×C₄⋊Dic₃	1	1	yes	characteristic
166	C₂²×C₃	C₂×C₄⋊Dic₃	1	1	yes	characteristic
167	C₂²×C₃	C₂²×Dic₆	1	1	yes	characteristic
168	C₃×C₄	C₂×C₄⋊Dic₃	1	4	yes	—
169	C₃×C₄	C₂×C₄⋊Dic₃	1	1	yes	characteristic
170	C₃×C₄	C₂×C₄⋊Dic₃	1	1	yes	characteristic
171	C₃×C₄	C₂²×D₁₂	1	1	yes	characteristic
172	C₃×C₄	C₂³×Dic₃	1	1	yes	characteristic
173	C₉	C₂×#430	1	1	yes	characteristic, complemented, Sylow
174	C₂³	—	4	2	no	—
175	C₂³	C₄⋊Dic₉	1	2	yes	—
176	C₂³	C₄⋊Dic₉	1	2	yes	—
177	C₂³	C₂×Dic₁₈	1	1	yes	characteristic
178	C₂³	C₂×Dic₁₈	1	2	yes	—
179	C₂×C₄	C₄⋊Dic₉	1	2	yes	—
180	C₂×C₄	C₄⋊Dic₉	1	4	yes	—
181	C₂×C₄	C₄⋊Dic₉	1	2	yes	—
182	C₂×C₄	C₄⋊Dic₉	1	4	yes	—
183	C₂×C₄	C₂×Dic₁₈	1	2	yes	—
184	C₂×C₄	C₂×Dic₁₈	1	1	yes	characteristic
185	C₂×C₄	C₂×D₃₆	1	4	yes	—
186	C₂×C₄	C₂×D₃₆	1	1	yes	characteristic
187	C₂×C₄	C₂×D₃₆	1	1	yes	characteristic
188	C₂×C₄	C₂²×Dic₉	1	1	yes	characteristic
189	C₂×C₄	C₂²×Dic₉	1	1	yes	characteristic
190	C₂×C₄	C₂²×Dic₉	1	4	yes	—
191	C₂×C₄	C₂³×D₉	1	1	yes	characteristic
192	C₈	—	18	8	no	—
193	C₈	C₂²×Dic₉	1	8	yes	—
194	C₂×C₃	—	2	4	no	—
195	C₂×C₃	—	2	1	no	—
196	C₂×C₃	—	2	1	no	—
197	C₂×C₃	#5125	1	2	yes	—
198	C₂×C₃	C₂²×C₄⋊Dic₃	1	1	yes	characteristic
199	C₂²	—	2	4	no	—
200	C₂²	—	2	2	no	—
201	C₂²	—	4	2	no	—
202	C₂²	—	4	2	no	—
203	C₂²	C₂×C₄⋊Dic₉	1	4	yes	—
204	C₂²	C₂×C₄⋊Dic₉	1	1	yes	characteristic
205	C₂²	C₂×C₄⋊Dic₉	1	1	yes	characteristic
206	C₂²	C₂²×Dic₁₈	1	1	yes	characteristic
207	C₄	C₂×C₄⋊Dic₉	1	4	yes	—
208	C₄	C₂×C₄⋊Dic₉	1	1	yes	characteristic
209	C₄	C₂×C₄⋊Dic₉	1	1	yes	characteristic
210	C₄	C₂²×D₃₆	1	1	yes	characteristic
211	C₄	C₂³×Dic₉	1	1	yes	characteristic
212	C₃	C₂×#5125	1	1	yes	characteristic
213	C₂	—	2	1	no	—
214	C₂	—	2	4	no	—
215	C₂	—	2	1	no	—
216	C₂	#10588800	1	2	yes	direct product factor
217	C₂	C₂²×C₄⋊Dic₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₃₆
commutator	C₄×C₉	C₂³×C₄
Frattini	C₂×C₃×C₄	C₂³×D₃
Fitting	C₂²×C₉×M₁₆	C₂
socle	C₂²×C₃	C₂×C₄⋊Dic₃
prime radical	C₂⁴×C₃	Dic₆
cosocle	C₂×C₄×C₉	C₂⁴
hypercenter	C₂²×M₁₆	D₉
hypocenter	C₉	C₂×#430
LCS	G ▷ C₄×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×M₁₆

Complements

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

N	H ≅ G/N	type
3. #10588800	C₂	direct (N × H)
4. C₂×#63372	C₂	semidirect
16. #63372	C₂²	semidirect
140. C₂×C₉	#430	semidirect
173. C₉	C₂×#430	semidirect
216. C₂	#10588800	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×C₉×M₁₆	2
C₂²×#3667	2
#10588800	2
C₂×#63372	2
C₂×#5125	3

Subgroups — C2×#10588800 #11858637

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#10588800 #11858637