Subgroups — C₂×#10588226 #11795954

Order 1152.

408 conjugacy classes of subgroups.

The subgroup lattice has 186 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₈×C₉	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₉⋊C₁₆	C₂	1	2	yes	maximal, complemented
3	#10588226	C₂	1	4	yes	maximal, direct product factor
4	C₂×#4549	—	3	1	no	maximal
5	C₂³×C₄×C₉	C₄	1	1	yes	characteristic
6	C₂²×C₈×C₉	—	2	4	no	—
7	C₂²×C₈×C₉	C₄	1	1	yes	characteristic
8	C₂²×C₈×C₉	C₄	1	2	yes	—
9	C₂²×C₈×C₉	C₂²	1	2	yes	—
10	C₂²×C₈×C₉	C₂²	1	1	yes	characteristic
11	C₂×C₉⋊C₁₆	—	2	4	no	—
12	C₂×C₉⋊C₁₆	C₂²	1	4	yes	complemented
13	C₂³×C₃×C₈	D₃	1	1	yes	characteristic
14	C₂²×C₃⋊C₁₆	—	3	2	no	—
15	#4549	—	3	4	no	—
16	C₂⁴×C₉	C₈	1	1	yes	characteristic
17	C₂²×C₄×C₉	—	2	2	no	—
18	C₂²×C₄×C₉	—	2	2	no	—
19	C₂²×C₄×C₉	C₈	1	1	yes	characteristic
20	C₂²×C₄×C₉	C₈	1	2	yes	—
21	C₂²×C₄×C₉	C₂×C₄	1	2	yes	—
22	C₂²×C₄×C₉	C₂×C₄	1	1	yes	characteristic
23	C₂×C₈×C₉	—	2	2	no	—
24	C₂×C₈×C₉	—	2	4	no	—
25	C₂×C₈×C₉	—	2	4	no	—
26	C₂×C₈×C₉	C₂×C₄	1	2	yes	—
27	C₂×C₈×C₉	C₂×C₄	1	1	yes	characteristic
28	C₂×C₈×C₉	D₄	1	4	yes	—
29	C₂×C₈×C₉	C₂³	1	1	yes	characteristic
30	C₉⋊C₁₆	—	2	4	no	—
31	C₂×C₂²⋊C₁₆	—	9	1	no	Sylow
32	C₂³×C₃×C₄	Dic₃	1	1	yes	characteristic
33	C₂²×C₃×C₈	—	2	4	no	—
34	C₂²×C₃×C₈	Dic₃	1	1	yes	characteristic
35	C₂²×C₃×C₈	Dic₃	1	2	yes	—
36	C₂²×C₃×C₈	C₂×D₃	1	2	yes	—
37	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
38	C₂×C₃⋊C₁₆	—	3	4	no	—
39	C₂×C₃⋊C₁₆	—	6	4	no	—
40	C₂³×C₉	—	2	2	no	—
41	C₂³×C₉	—	2	2	no	—
42	C₂³×C₉	C₁₆	1	4	yes	complemented
43	C₂³×C₉	C₂×C₈	1	2	yes	—
44	C₂³×C₉	C₂×C₈	1	1	yes	characteristic
45	C₂×C₄×C₉	—	2	2	no	—
46	C₂×C₄×C₉	—	2	4	no	—
47	C₂×C₄×C₉	—	2	2	no	—
48	C₂×C₄×C₉	—	2	2	no	—
49	C₂×C₄×C₉	C₂²⋊C₄	1	2	yes	—
50	C₂×C₄×C₉	C₂×C₈	1	2	yes	—
51	C₂×C₄×C₉	C₂×C₈	1	1	yes	characteristic
52	C₂×C₄×C₉	M₁₆	1	2	yes	—
53	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
54	C₈×C₉	—	2	2	no	—
55	C₈×C₉	C₂²⋊C₄	1	2	yes	—
56	C₈×C₉	C₂×D₄	1	2	yes	—
57	C₂³×C₈	D₉	1	1	yes	characteristic
58	C₂²×C₁₆	—	9	2	no	—
59	C₂²⋊C₁₆	—	9	4	no	—
60	C₂⁴×C₃	C₃⋊C₈	1	1	yes	characteristic
61	C₂²×C₃×C₄	—	2	2	no	—
62	C₂²×C₃×C₄	—	2	2	no	—
63	C₂²×C₃×C₄	C₃⋊C₈	1	1	yes	characteristic
64	C₂²×C₃×C₄	C₃⋊C₈	1	2	yes	—
65	C₂²×C₃×C₄	C₂×Dic₃	1	2	yes	—
66	C₂²×C₃×C₄	C₂×Dic₃	1	1	yes	characteristic
67	C₂×C₃×C₈	—	2	2	no	—
68	C₂×C₃×C₈	—	2	4	no	—
69	C₂×C₃×C₈	—	2	4	no	—
70	C₂×C₃×C₈	C₂×Dic₃	1	2	yes	—
71	C₂×C₃×C₈	C₂×Dic₃	1	1	yes	characteristic
72	C₂×C₃×C₈	#67	1	4	yes	—
73	C₂×C₃×C₈	C₂²×D₃	1	1	yes	characteristic
74	C₃⋊C₁₆	—	6	4	no	—
75	C₂²×C₉	—	2	8	no	—
76	C₂²×C₉	—	2	2	no	—
77	C₂²×C₉	—	2	2	no	—
78	C₂²×C₉	C₂²⋊C₈	1	2	yes	—
79	C₂²×C₉	C₂×C₁₆	1	2	yes	—
80	C₂²×C₉	C₂×C₁₆	1	4	yes	complemented
81	C₂²×C₉	M₃₂	1	2	yes	—
82	C₂²×C₉	C₂²×C₈	1	1	yes	characteristic
83	C₄×C₉	—	2	2	no	—
84	C₄×C₉	C₂²⋊C₈	1	2	yes	—
85	C₄×C₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
86	C₄×C₉	C₂×M₁₆	1	1	yes	characteristic
87	C₂³×C₄	Dic₉	1	1	yes	characteristic
88	C₂²×C₈	—	2	4	no	—
89	C₂²×C₈	Dic₉	1	1	yes	characteristic
90	C₂²×C₈	Dic₉	1	2	yes	—
91	C₂²×C₈	C₂×D₉	1	2	yes	—
92	C₂²×C₈	C₂×D₉	1	1	yes	characteristic
93	C₂×C₁₆	—	9	4	no	—
94	C₂×C₁₆	—	18	4	no	—
95	C₂³×C₃	—	2	2	no	—
96	C₂³×C₃	—	2	2	no	—
97	C₂³×C₃	C₃⋊C₁₆	1	4	yes	—
98	C₂³×C₃	C₂×C₃⋊C₈	1	2	yes	—
99	C₂³×C₃	C₂×C₃⋊C₈	1	1	yes	characteristic
100	C₂×C₃×C₄	—	2	2	no	—
101	C₂×C₃×C₄	—	2	4	no	—
102	C₂×C₃×C₄	—	2	2	no	—
103	C₂×C₃×C₄	—	2	2	no	—
104	C₂×C₃×C₄	C₂×C₃⋊C₈	1	2	yes	—
105	C₂×C₃×C₄	C₂×C₃⋊C₈	1	1	yes	characteristic
106	C₂×C₃×C₄	#208	1	2	yes	—
107	C₂×C₃×C₄	#217	1	2	yes	—
108	C₂×C₃×C₄	C₂²×Dic₃	1	1	yes	characteristic
109	C₃×C₈	—	2	2	no	—
110	C₃×C₈	#217	1	2	yes	—
111	C₃×C₈	C₂×#67	1	2	yes	—
112	C₂×C₉	—	2	4	no	—
113	C₂×C₉	C₂²⋊C₁₆	1	4	yes	complemented
114	C₂×C₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
115	C₂×C₉	C₂²×C₁₆	1	1	yes	characteristic
116	C₂×C₉	C₂×M₃₂	1	1	yes	characteristic
117	C₂⁴	C₉⋊C₈	1	1	yes	characteristic
118	C₂²×C₄	—	2	2	no	—
119	C₂²×C₄	—	2	2	no	—
120	C₂²×C₄	C₉⋊C₈	1	1	yes	characteristic
121	C₂²×C₄	C₉⋊C₈	1	2	yes	—
122	C₂²×C₄	C₂×Dic₉	1	2	yes	—
123	C₂²×C₄	C₂×Dic₉	1	1	yes	characteristic
124	C₂×C₈	—	2	2	no	—
125	C₂×C₈	—	2	4	no	—
126	C₂×C₈	—	2	4	no	—
127	C₂×C₈	C₂×Dic₉	1	2	yes	—
128	C₂×C₈	C₂×Dic₉	1	1	yes	characteristic
129	C₂×C₈	#611	1	4	yes	—
130	C₂×C₈	C₂²×D₉	1	1	yes	characteristic
131	C₁₆	—	18	4	no	—
132	C₂²×C₃	—	2	2	no	—
133	C₂²×C₃	—	2	8	no	—
134	C₂²×C₃	—	2	2	no	—
135	C₂²×C₃	C₂×C₃⋊C₁₆	1	2	yes	—
136	C₂²×C₃	C₂×C₃⋊C₁₆	1	4	yes	—
137	C₂²×C₃	#812	1	2	yes	—
138	C₂²×C₃	#830	1	2	yes	—
139	C₂²×C₃	C₂²×C₃⋊C₈	1	1	yes	characteristic
140	C₃×C₄	—	2	2	no	—
141	C₃×C₄	#830	1	2	yes	—
142	C₃×C₄	C₂×#208	1	1	yes	characteristic
143	C₃×C₄	C₂×#217	1	1	yes	characteristic
144	C₉	C₂×C₂²⋊C₁₆	1	1	yes	characteristic, complemented, Sylow
145	C₂³	—	2	2	no	—
146	C₂³	—	2	2	no	—
147	C₂³	C₉⋊C₁₆	1	4	yes	complemented
148	C₂³	C₂×C₉⋊C₈	1	2	yes	—
149	C₂³	C₂×C₉⋊C₈	1	1	yes	characteristic
150	C₂×C₄	—	2	2	no	—
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	2	4	no	—
153	C₂×C₄	—	2	2	no	—
154	C₂×C₄	C₂×C₉⋊C₈	1	2	yes	—
155	C₂×C₄	C₂×C₉⋊C₈	1	1	yes	characteristic
156	C₂×C₄	#3667	1	2	yes	—
157	C₂×C₄	#3676	1	2	yes	—
158	C₂×C₄	C₂²×Dic₉	1	1	yes	characteristic
159	C₈	—	2	2	no	—
160	C₈	#3676	1	2	yes	—
161	C₈	C₂×#611	1	2	yes	—
162	C₂×C₃	—	2	4	no	—
163	C₂×C₃	#4549	1	4	yes	—
164	C₂×C₃	C₂²×C₃⋊C₁₆	1	1	yes	characteristic
165	C₂×C₃	C₂×#812	1	1	yes	characteristic
166	C₂×C₃	C₂×#830	1	1	yes	characteristic
167	C₂²	—	2	8	no	—
168	C₂²	—	2	2	no	—
169	C₂²	—	2	2	no	—
170	C₂²	C₂×C₉⋊C₁₆	1	2	yes	—
171	C₂²	C₂×C₉⋊C₁₆	1	4	yes	complemented
172	C₂²	#63371	1	2	yes	—
173	C₂²	#63389	1	2	yes	—
174	C₂²	C₂²×C₉⋊C₈	1	1	yes	characteristic
175	C₄	—	2	2	no	—
176	C₄	#63389	1	2	yes	—
177	C₄	C₂×#3667	1	1	yes	characteristic
178	C₄	C₂×#3676	1	1	yes	characteristic
179	C₃	C₂×#4549	1	1	yes	characteristic
180	C₂	—	2	4	no	—
181	C₂	#10588226	1	4	yes	direct product factor
182	C₂	C₂²×C₉⋊C₁₆	1	1	yes	characteristic
183	C₂	C₂×#63371	1	1	yes	characteristic
184	C₂	C₂×#63389	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₈	C₂×D₉
commutator	C₂×C₉	C₂²×C₁₆
Frattini	C₂×C₃×C₈	C₂²×D₃
Fitting	C₂³×C₈×C₉	C₂
socle	C₂³×C₃	C₂×C₃⋊C₈
prime radical	C₂⁴×C₃	C₃⋊C₈
cosocle	C₂×C₈×C₉	C₂³
hypercenter	C₂³×C₈	D₉
hypocenter	C₉	C₂×C₂²⋊C₁₆
LCS	G ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂²×C₈ ◁ C₂³×C₈

Complements

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

N	H ≅ G/N	type
2. C₂²×C₉⋊C₁₆	C₂	semidirect
3. #10588226	C₂	direct (N × H)
12. C₂×C₉⋊C₁₆	C₂²	semidirect
42. C₂³×C₉	C₁₆	semidirect
80. C₂²×C₉	C₂×C₁₆	semidirect
113. C₂×C₉	C₂²⋊C₁₆	semidirect
144. C₉	C₂×C₂²⋊C₁₆	semidirect
147. C₂³	C₉⋊C₁₆	semidirect
171. C₂²	C₂×C₉⋊C₁₆	semidirect
181. C₂	#10588226	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₈×C₉	2
C₂²×C₉⋊C₁₆	2
#10588226	2
C₂×#4549	3

Subgroups — C2×#10588226 #11795954

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#10588226 #11795954