Subgroups — C₂²×#63450 #11900714

Order 1152.

2436 conjugacy classes of subgroups.

The subgroup lattice has 198 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×Dic₁₈	C₂	1	2	yes	maximal
2	C₂²×C₉×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₄⋊Dic₉	C₂	1	1	yes	maximal, characteristic
4	C₂²×Dic₉⋊C₄	C₂	1	2	yes	maximal
5	C₂²×C₄×Dic₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#63450	C₂	1	24	yes	maximal, direct product factor
7	C₂²×#888	—	3	1	no	maximal
8	C₂³×C₄×C₉	C₂²	1	2	yes	—
9	C₂³×C₄×C₉	C₂²	1	1	yes	characteristic
10	C₂³×Dic₉	C₂²	1	2	yes	—
11	C₂³×Dic₉	C₂²	1	1	yes	characteristic
12	C₂³×Dic₉	C₂²	1	1	yes	characteristic
13	C₂²×Dic₁₈	—	2	16	no	—
14	C₂²×Dic₁₈	C₂²	1	24	yes	—
15	C₂×C₉×C₄⋊C₄	C₂²	1	12	yes	—
16	C₂×C₄⋊Dic₉	C₂²	1	12	yes	—
17	C₂×Dic₉⋊C₄	C₂²	1	24	yes	—
18	C₂×C₄×Dic₉	C₂²	1	12	yes	—
19	#63450	C₂²	1	64	yes	direct product factor
20	C₂³×Dic₆	—	3	2	no	—
21	C₂²×C₃×C₄⋊C₄	D₃	1	1	yes	characteristic
22	C₂²×C₄⋊Dic₃	—	3	1	no	—
23	C₂²×Dic₃⋊C₄	—	3	2	no	—
24	C₂²×C₄×Dic₃	—	3	1	no	—
25	C₂×#888	—	3	24	no	—
26	C₂⁴×C₉	C₂³	1	1	yes	characteristic
27	C₂²×C₄×C₉	—	2	8	no	—
28	C₂²×C₄×C₉	Q₈	1	8	yes	complemented
29	C₂²×C₄×C₉	C₂³	1	12	yes	—
30	C₂²×C₄×C₉	C₂³	1	6	yes	—
31	C₂²×Dic₉	—	2	8	no	—
32	C₂²×Dic₉	D₄	1	8	yes	—
33	C₂²×Dic₉	Q₈	1	8	yes	complemented
34	C₂²×Dic₉	C₂³	1	12	yes	—
35	C₂²×Dic₉	C₂³	1	6	yes	—
36	C₂²×Dic₉	C₂³	1	6	yes	—
37	C₂×Dic₁₈	—	2	96	no	—
38	C₂×Dic₁₈	C₂³	1	32	yes	—
39	C₉×C₄⋊C₄	C₂³	1	16	yes	—
40	C₄⋊Dic₉	C₂³	1	16	yes	—
41	Dic₉⋊C₄	C₂³	1	32	yes	—
42	C₄×Dic₉	C₂³	1	16	yes	—
43	C₂²×C₄⋊Q₈	—	9	1	no	Sylow
44	C₂³×C₃×C₄	C₂×D₃	1	2	yes	—
45	C₂³×C₃×C₄	C₂×D₃	1	1	yes	characteristic
46	C₂³×Dic₃	—	3	1	no	—
47	C₂³×Dic₃	—	3	1	no	—
48	C₂³×Dic₃	—	3	2	no	—
49	C₂²×Dic₆	—	3	24	no	—
50	C₂²×Dic₆	—	6	16	no	—
51	C₂×C₃×C₄⋊C₄	C₂×D₃	1	12	yes	—
52	C₂×C₄⋊Dic₃	—	3	12	no	—
53	C₂×Dic₃⋊C₄	—	3	24	no	—
54	C₂×C₄×Dic₃	—	3	12	no	—
55	#888	—	3	64	no	—
56	C₂³×C₉	C₂×D₄	1	4	yes	—
57	C₂³×C₉	C₂×Q₈	1	4	yes	—
58	C₂³×C₉	C₂×Q₈	1	4	yes	—
59	C₂³×C₉	C₂⁴	1	3	yes	—
60	C₂×C₄×C₉	—	2	24	no	—
61	C₂×C₄×C₉	C₂×Q₈	1	24	yes	complemented
62	C₂×C₄×C₉	C₂⁴	1	8	yes	—
63	C₂×C₄×C₉	C₂⁴	1	4	yes	—
64	C₂×Dic₉	—	2	24	no	—
65	C₂×Dic₉	C₂×D₄	1	24	yes	—
66	C₂×Dic₉	C₂×Q₈	1	24	yes	complemented
67	C₂×Dic₉	C₂⁴	1	8	yes	—
68	C₂×Dic₉	C₂⁴	1	4	yes	—
69	C₂×Dic₉	C₂⁴	1	4	yes	—
70	Dic₁₈	—	2	64	no	—
71	C₂³×Q₈	—	9	2	no	—
72	C₂×C₄⋊Q₈	—	9	24	no	—
73	C₂²×C₄⋊C₄	—	9	2	no	—
74	C₂²×C₄⋊C₄	—	9	1	no	—
75	C₂²×C₄⋊C₄	D₉	1	1	yes	characteristic
76	C₂²×C₄²	—	9	1	no	—
77	C₂⁴×C₃	C₂²×D₃	1	1	yes	characteristic
78	C₂²×C₃×C₄	—	2	8	no	—
79	C₂²×C₃×C₄	Dic₆	1	8	yes	—
80	C₂²×C₃×C₄	C₂²×D₃	1	12	yes	—
81	C₂²×C₃×C₄	C₂²×D₃	1	6	yes	—
82	C₂²×Dic₃	—	3	8	no	—
83	C₂²×Dic₃	—	3	8	no	—
84	C₂²×Dic₃	—	3	6	no	—
85	C₂²×Dic₃	—	3	6	no	—
86	C₂²×Dic₃	—	3	12	no	—
87	C₂²×Dic₃	—	6	8	no	—
88	C₂×Dic₆	—	3	32	no	—
89	C₂×Dic₆	—	6	96	no	—
90	C₃×C₄⋊C₄	C₂²×D₃	1	16	yes	—
91	C₄⋊Dic₃	—	3	16	no	—
92	Dic₃⋊C₄	—	3	32	no	—
93	C₄×Dic₃	—	3	16	no	—
94	C₂²×C₉	C₄⋊Q₈	1	16	yes	complemented
95	C₂²×C₉	C₂²×D₄	1	6	yes	—
96	C₂²×C₉	C₂²×Q₈	1	6	yes	—
97	C₂²×C₉	C₂²×Q₈	1	6	yes	—
98	C₂²×C₉	C₂⁵	1	1	yes	characteristic
99	C₄×C₉	—	2	8	no	—
100	C₄×C₉	C₂²×Q₈	1	8	yes	complemented
101	Dic₉	—	2	8	no	—
102	Dic₉	C₂²×D₄	1	8	yes	—
103	Dic₉	C₂²×Q₈	1	8	yes	complemented
104	C₂²×Q₈	—	9	24	no	—
105	C₂²×Q₈	—	18	16	no	—
106	C₂³×C₄	—	9	1	no	—
107	C₂³×C₄	—	9	1	no	—
108	C₂³×C₄	—	9	2	no	—
109	C₂³×C₄	C₂×D₉	1	2	yes	—
110	C₂³×C₄	C₂×D₉	1	1	yes	characteristic
111	C₄⋊Q₈	—	9	64	no	—
112	C₂×C₄⋊C₄	—	9	24	no	—
113	C₂×C₄⋊C₄	—	9	12	no	—
114	C₂×C₄⋊C₄	C₂×D₉	1	12	yes	—
115	C₂×C₄²	—	9	12	no	—
116	C₂³×C₃	C₂×Dic₆	1	4	yes	—
117	C₂³×C₃	D₃×D₄	1	4	yes	—
118	C₂³×C₃	D₃×Q₈	1	4	yes	—
119	C₂³×C₃	C₂³×D₃	1	3	yes	—
120	C₂×C₃×C₄	—	2	24	no	—
121	C₂×C₃×C₄	C₂×Dic₆	1	24	yes	—
122	C₂×C₃×C₄	C₂³×D₃	1	8	yes	—
123	C₂×C₃×C₄	C₂³×D₃	1	4	yes	—
124	C₂×Dic₃	—	3	24	no	—
125	C₂×Dic₃	—	3	24	no	—
126	C₂×Dic₃	—	3	4	no	—
127	C₂×Dic₃	—	3	4	no	—
128	C₂×Dic₃	—	3	8	no	—
129	C₂×Dic₃	—	6	24	no	—
130	Dic₆	—	6	64	no	—
131	C₂×C₉	C₂×C₄⋊Q₈	1	12	yes	complemented
132	C₂×C₉	C₂³×D₄	1	1	yes	characteristic
133	C₂×C₉	C₂³×Q₈	1	1	yes	characteristic
134	C₂×C₉	C₂³×Q₈	1	1	yes	characteristic
135	C₂⁴	C₂²×D₉	1	1	yes	characteristic
136	C₂×Q₈	—	9	32	no	—
137	C₂×Q₈	—	18	96	no	—
138	C₂²×C₄	—	2	8	no	—
139	C₂²×C₄	—	9	8	no	—
140	C₂²×C₄	—	9	8	no	—
141	C₂²×C₄	—	9	6	no	—
142	C₂²×C₄	—	9	6	no	—
143	C₂²×C₄	—	9	12	no	—
144	C₂²×C₄	—	18	8	no	—
145	C₂²×C₄	Dic₁₈	1	8	yes	complemented
146	C₂²×C₄	C₂²×D₉	1	12	yes	—
147	C₂²×C₄	C₂²×D₉	1	6	yes	—
148	C₄⋊C₄	—	9	32	no	—
149	C₄⋊C₄	—	9	16	no	—
150	C₄⋊C₄	C₂²×D₉	1	16	yes	—
151	C₄²	—	9	16	no	—
152	C₂²×C₃	#888	1	16	yes	—
153	C₂²×C₃	C₂²×Dic₆	1	6	yes	—
154	C₂²×C₃	C₂×D₃×D₄	1	6	yes	—
155	C₂²×C₃	C₂×D₃×Q₈	1	6	yes	—
156	C₂²×C₃	C₂⁴×D₃	1	1	yes	characteristic
157	C₃×C₄	—	2	8	no	—
158	C₃×C₄	C₂²×Dic₆	1	8	yes	—
159	Dic₃	—	3	8	no	—
160	Dic₃	—	3	8	no	—
161	Dic₃	—	6	8	no	—
162	C₉	C₂²×C₄⋊Q₈	1	1	yes	characteristic, complemented, Sylow
163	C₂³	C₂×Dic₁₈	1	4	yes	—
164	C₂³	D₄×D₉	1	4	yes	—
165	C₂³	Q₈×D₉	1	4	yes	—
166	C₂³	C₂³×D₉	1	3	yes	—
167	Q₈	—	18	64	no	—
168	C₂×C₄	—	2	24	no	—
169	C₂×C₄	—	9	4	no	—
170	C₂×C₄	—	9	4	no	—
171	C₂×C₄	—	9	24	no	—
172	C₂×C₄	—	9	24	no	—
173	C₂×C₄	—	9	8	no	—
174	C₂×C₄	—	18	24	no	—
175	C₂×C₄	C₂×Dic₁₈	1	24	yes	complemented
176	C₂×C₄	C₂³×D₉	1	8	yes	—
177	C₂×C₄	C₂³×D₉	1	4	yes	—
178	C₂×C₃	C₂×#888	1	12	yes	—
179	C₂×C₃	C₂³×Dic₆	1	1	yes	characteristic
180	C₂×C₃	C₂²×D₃×D₄	1	1	yes	characteristic
181	C₂×C₃	C₂²×D₃×Q₈	1	1	yes	characteristic
182	C₂²	#63450	1	16	yes	direct product factor
183	C₂²	C₂²×Dic₁₈	1	6	yes	—
184	C₂²	C₂×D₄×D₉	1	6	yes	—
185	C₂²	C₂×Q₈×D₉	1	6	yes	—
186	C₂²	C₂⁴×D₉	1	1	yes	characteristic
187	C₄	—	2	8	no	—
188	C₄	—	9	8	no	—
189	C₄	—	9	8	no	—
190	C₄	—	18	8	no	—
191	C₄	C₂²×Dic₁₈	1	8	yes	complemented
192	C₃	C₂²×#888	1	1	yes	characteristic
193	C₂	C₂×#63450	1	12	yes	direct product factor
194	C₂	C₂³×Dic₁₈	1	1	yes	characteristic
195	C₂	C₂²×D₄×D₉	1	1	yes	characteristic
196	C₂	C₂²×Q₈×D₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂²×D₉
commutator, cosocle	C₂²×C₉	C₂⁵
Frattini	C₂²×C₃	C₂⁴×D₃
Fitting	C₂²×C₉×C₄⋊C₄	C₂
socle, prime radical	C₂⁴×C₃	C₂²×D₃
hypercenter	C₂²×C₄⋊C₄	D₉
hypocenter	C₉	C₂²×C₄⋊Q₈
LCS	G ▷ C₂²×C₉ ▷ C₉
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
6. C₂×#63450	C₂	direct (N × H)
19. #63450	C₂²	direct (N × H)
28. C₂²×C₄×C₉	Q₈	semidirect
33. C₂²×Dic₉	Q₈	semidirect
61. C₂×C₄×C₉	C₂×Q₈	semidirect
66. C₂×Dic₉	C₂×Q₈	semidirect
94. C₂²×C₉	C₄⋊Q₈	semidirect
100. C₄×C₉	C₂²×Q₈	semidirect
103. Dic₉	C₂²×Q₈	semidirect
131. C₂×C₉	C₂×C₄⋊Q₈	semidirect
145. C₂²×C₄	Dic₁₈	semidirect
162. C₉	C₂²×C₄⋊Q₈	semidirect
175. C₂×C₄	C₂×Dic₁₈	semidirect
182. C₂²	#63450	direct (N × H)
191. C₄	C₂²×Dic₁₈	semidirect
193. C₂	C₂×#63450	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×Dic₁₈	2
C₂²×C₉×C₄⋊C₄	2
C₂²×C₄⋊Dic₉	2
C₂²×Dic₉⋊C₄	2
C₂²×C₄×Dic₉	2
C₂×#63450	2
C₂²×#888	3

Subgroups — C22×#63450 #11900714

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#63450 #11900714