Subgroups — C₂²×#63392 #11857718

Order 1152.

1608 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×D₄×C₉	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄⋊Dic₉	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂³×C₉⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#63392	C₂	1	12	yes	maximal, direct product factor
5	C₂²×#832	—	3	1	no	maximal
6	C₂⁵×C₉	—	2	1	no	—
7	C₂²×D₄×C₉	—	2	3	no	—
8	C₂²×D₄×C₉	—	2	4	no	—
9	C₂²×D₄×C₉	C₄	1	8	yes	complemented
10	C₂²×D₄×C₉	C₂²	1	6	yes	—
11	C₂³×C₄×C₉	C₂²	1	1	yes	characteristic
12	C₂³×Dic₉	—	2	1	no	—
13	C₂×C₄⋊Dic₉	—	2	3	no	—
14	C₂×C₄⋊Dic₉	C₂²	1	6	yes	complemented
15	C₂²×C₉⋊C₈	—	2	4	no	—
16	C₂²×C₉⋊C₈	C₂²	1	6	yes	complemented
17	#63392	C₂²	1	16	yes	direct product factor
18	C₂³×C₃×D₄	D₃	1	1	yes	characteristic
19	C₂²×C₄⋊Dic₃	—	3	1	no	—
20	C₂³×C₃⋊C₈	—	3	1	no	—
21	C₂×#832	—	3	12	no	—
22	C₂⁴×C₉	—	2	8	no	—
23	C₂⁴×C₉	—	2	6	no	—
24	C₂⁴×C₉	—	4	4	no	—
25	C₂⁴×C₉	—	4	4	no	—
26	C₂⁴×C₉	D₄	1	1	yes	characteristic
27	C₂×D₄×C₉	—	2	24	no	—
28	C₂×D₄×C₉	—	2	6	no	—
29	C₂×D₄×C₉	—	2	12	no	—
30	C₂×D₄×C₉	C₂×C₄	1	24	yes	complemented
31	C₂×D₄×C₉	C₂³	1	4	yes	—
32	C₂²×C₄×C₉	C₂×C₄	1	4	yes	—
33	C₂²×C₄×C₉	D₄	1	3	yes	—
34	C₂²×C₄×C₉	D₄	1	4	yes	—
35	C₂²×C₄×C₉	C₂³	1	3	yes	—
36	C₂²×Dic₉	—	2	6	no	—
37	C₂²×Dic₉	—	4	4	no	—
38	C₄⋊Dic₉	—	2	6	no	—
39	C₄⋊Dic₉	C₂³	1	4	yes	complemented
40	C₂×C₉⋊C₈	—	2	12	no	—
41	C₂×C₉⋊C₈	C₂³	1	4	yes	complemented
42	C₂²×#102	—	9	1	no	Sylow
43	C₂⁵×C₃	—	2	1	no	—
44	C₂²×C₃×D₄	—	2	3	no	—
45	C₂²×C₃×D₄	—	2	4	no	—
46	C₂²×C₃×D₄	Dic₃	1	8	yes	—
47	C₂²×C₃×D₄	C₂×D₃	1	6	yes	—
48	C₂³×C₃×C₄	C₂×D₃	1	1	yes	characteristic
49	C₂³×Dic₃	—	6	1	no	—
50	C₂×C₄⋊Dic₃	—	3	6	no	—
51	C₂×C₄⋊Dic₃	—	6	3	no	—
52	C₂²×C₃⋊C₈	—	3	6	no	—
53	C₂²×C₃⋊C₈	—	6	4	no	—
54	#832	—	3	16	no	—
55	C₂³×C₉	—	2	24	no	—
56	C₂³×C₉	—	2	4	no	—
57	C₂³×C₉	—	4	32	no	—
58	C₂³×C₉	—	4	12	no	—
59	C₂³×C₉	—	4	12	no	—
60	C₂³×C₉	C₂²⋊C₄	1	4	yes	—
61	C₂³×C₉	D₈	1	4	yes	—
62	C₂³×C₉	SD₁₆	1	4	yes	—
63	C₂³×C₉	C₂×D₄	1	3	yes	—
64	D₄×C₉	—	2	24	no	—
65	D₄×C₉	—	2	4	no	—
66	D₄×C₉	C₂²×C₄	1	8	yes	complemented
67	C₂×C₄×C₉	C₂²⋊C₄	1	12	yes	—
68	C₂×C₄×C₉	C₂²×C₄	1	6	yes	—
69	C₂×C₄×C₉	C₂×D₄	1	3	yes	—
70	C₂×C₄×C₉	C₂×D₄	1	6	yes	—
71	C₂×C₄×C₉	C₂⁴	1	1	yes	characteristic
72	C₂×Dic₉	—	2	4	no	—
73	C₂×Dic₉	—	4	12	no	—
74	C₉⋊C₈	—	2	4	no	—
75	C₂³×D₄	D₉	1	1	yes	characteristic
76	C₂³×C₈	—	9	1	no	—
77	C₂²×C₄⋊C₄	—	9	1	no	—
78	C₂×#102	—	9	12	no	—
79	C₂⁴×C₃	—	2	8	no	—
80	C₂⁴×C₃	—	2	6	no	—
81	C₂⁴×C₃	—	4	4	no	—
82	C₂⁴×C₃	—	4	4	no	—
83	C₂⁴×C₃	#67	1	1	yes	characteristic
84	C₂×C₃×D₄	—	2	24	no	—
85	C₂×C₃×D₄	—	2	6	no	—
86	C₂×C₃×D₄	—	2	12	no	—
87	C₂×C₃×D₄	C₂×Dic₃	1	24	yes	—
88	C₂×C₃×D₄	C₂²×D₃	1	4	yes	—
89	C₂²×C₃×C₄	C₂×Dic₃	1	4	yes	—
90	C₂²×C₃×C₄	#67	1	3	yes	—
91	C₂²×C₃×C₄	#67	1	4	yes	—
92	C₂²×C₃×C₄	C₂²×D₃	1	3	yes	—
93	C₂²×Dic₃	—	6	6	no	—
94	C₂²×Dic₃	—	12	4	no	—
95	C₄⋊Dic₃	—	3	4	no	—
96	C₄⋊Dic₃	—	6	6	no	—
97	C₂×C₃⋊C₈	—	3	4	no	—
98	C₂×C₃⋊C₈	—	6	12	no	—
99	C₂²×C₉	—	2	8	no	—
100	C₂²×C₉	—	4	4	no	—
101	C₂²×C₉	—	4	4	no	—
102	C₂²×C₉	—	4	48	no	—
103	C₂²×C₉	#102	1	16	yes	complemented
104	C₂²×C₉	C₂×C₂²⋊C₄	1	6	yes	—
105	C₂²×C₉	C₂×D₈	1	6	yes	—
106	C₂²×C₉	C₂×SD₁₆	1	6	yes	—
107	C₂²×C₉	C₂²×D₄	1	1	yes	characteristic
108	C₄×C₉	C₂×C₂²⋊C₄	1	6	yes	—
109	C₄×C₉	C₂³×C₄	1	1	yes	characteristic
110	C₄×C₉	C₂²×D₄	1	1	yes	characteristic
111	Dic₉	—	4	4	no	—
112	C₂⁵	—	2	1	no	—
113	C₂²×D₄	—	2	3	no	—
114	C₂²×D₄	—	2	4	no	—
115	C₂²×D₄	Dic₉	1	8	yes	complemented
116	C₂²×D₄	C₂×D₉	1	6	yes	—
117	C₂³×C₄	—	18	1	no	—
118	C₂³×C₄	C₂×D₉	1	1	yes	characteristic
119	C₂²×C₈	—	9	6	no	—
120	C₂²×C₈	—	18	4	no	—
121	C₂×C₄⋊C₄	—	9	6	no	—
122	C₂×C₄⋊C₄	—	18	3	no	—
123	#102	—	9	16	no	—
124	C₂³×C₃	—	2	24	no	—
125	C₂³×C₃	—	2	4	no	—
126	C₂³×C₃	—	4	32	no	—
127	C₂³×C₃	—	4	12	no	—
128	C₂³×C₃	—	4	12	no	—
129	C₂³×C₃	#213	1	4	yes	—
130	C₂³×C₃	#214	1	4	yes	—
131	C₂³×C₃	#217	1	4	yes	—
132	C₂³×C₃	C₂×#67	1	3	yes	—
133	C₃×D₄	—	2	24	no	—
134	C₃×D₄	—	2	4	no	—
135	C₃×D₄	C₂²×Dic₃	1	8	yes	—
136	C₂×C₃×C₄	#217	1	12	yes	—
137	C₂×C₃×C₄	C₂²×Dic₃	1	6	yes	—
138	C₂×C₃×C₄	C₂×#67	1	3	yes	—
139	C₂×C₃×C₄	C₂×#67	1	6	yes	—
140	C₂×C₃×C₄	C₂³×D₃	1	1	yes	characteristic
141	C₂×Dic₃	—	6	4	no	—
142	C₂×Dic₃	—	12	12	no	—
143	C₃⋊C₈	—	6	4	no	—
144	C₂×C₉	—	4	8	no	—
145	C₂×C₉	C₂×#102	1	12	yes	complemented
146	C₂×C₉	C₂²×C₂²⋊C₄	1	1	yes	characteristic
147	C₂×C₉	C₂²×D₈	1	1	yes	characteristic
148	C₂×C₉	C₂²×SD₁₆	1	1	yes	characteristic
149	C₂⁴	—	2	8	no	—
150	C₂⁴	—	2	6	no	—
151	C₂⁴	—	4	4	no	—
152	C₂⁴	—	4	4	no	—
153	C₂⁴	#611	1	1	yes	characteristic
154	C₂×D₄	—	2	24	no	—
155	C₂×D₄	—	2	6	no	—
156	C₂×D₄	—	2	12	no	—
157	C₂×D₄	C₂×Dic₉	1	24	yes	complemented
158	C₂×D₄	C₂²×D₉	1	4	yes	—
159	C₂²×C₄	—	18	6	no	—
160	C₂²×C₄	—	36	4	no	—
161	C₂²×C₄	C₂×Dic₉	1	4	yes	—
162	C₂²×C₄	#611	1	3	yes	—
163	C₂²×C₄	#611	1	4	yes	—
164	C₂²×C₄	C₂²×D₉	1	3	yes	—
165	C₂×C₈	—	9	4	no	—
166	C₂×C₈	—	18	12	no	—
167	C₄⋊C₄	—	9	4	no	—
168	C₄⋊C₄	—	18	6	no	—
169	C₂²×C₃	—	2	8	no	—
170	C₂²×C₃	—	4	48	no	—
171	C₂²×C₃	—	4	4	no	—
172	C₂²×C₃	—	4	4	no	—
173	C₂²×C₃	#832	1	16	yes	—
174	C₂²×C₃	C₂×#213	1	6	yes	—
175	C₂²×C₃	C₂×#214	1	6	yes	—
176	C₂²×C₃	C₂×#217	1	6	yes	—
177	C₂²×C₃	C₂²×#67	1	1	yes	characteristic
178	C₃×C₄	C₂×#217	1	6	yes	—
179	C₃×C₄	C₂³×Dic₃	1	1	yes	characteristic
180	C₃×C₄	C₂²×#67	1	1	yes	characteristic
181	Dic₃	—	12	4	no	—
182	C₉	C₂²×#102	1	1	yes	characteristic, complemented, Sylow
183	C₂³	—	2	24	no	—
184	C₂³	—	2	4	no	—
185	C₂³	—	4	12	no	—
186	C₂³	—	4	12	no	—
187	C₂³	—	4	32	no	—
188	C₂³	#3672	1	4	yes	—
189	C₂³	#3673	1	4	yes	—
190	C₂³	#3676	1	4	yes	—
191	C₂³	C₂×#611	1	3	yes	—
192	D₄	—	2	24	no	—
193	D₄	—	2	4	no	—
194	D₄	C₂²×Dic₉	1	8	yes	complemented
195	C₂×C₄	—	18	4	no	—
196	C₂×C₄	—	36	12	no	—
197	C₂×C₄	#3676	1	12	yes	—
198	C₂×C₄	C₂²×Dic₉	1	6	yes	—
199	C₂×C₄	C₂×#611	1	3	yes	—
200	C₂×C₄	C₂×#611	1	6	yes	—
201	C₂×C₄	C₂³×D₉	1	1	yes	characteristic
202	C₈	—	18	4	no	—
203	C₂×C₃	—	4	8	no	—
204	C₂×C₃	C₂×#832	1	12	yes	—
205	C₂×C₃	C₂²×#213	1	1	yes	characteristic
206	C₂×C₃	C₂²×#214	1	1	yes	characteristic
207	C₂×C₃	C₂²×#217	1	1	yes	characteristic
208	C₂²	—	2	8	no	—
209	C₂²	—	4	4	no	—
210	C₂²	—	4	4	no	—
211	C₂²	—	4	48	no	—
212	C₂²	#63392	1	16	yes	direct product factor
213	C₂²	C₂×#3672	1	6	yes	—
214	C₂²	C₂×#3673	1	6	yes	—
215	C₂²	C₂×#3676	1	6	yes	—
216	C₂²	C₂²×#611	1	1	yes	characteristic
217	C₄	—	36	4	no	—
218	C₄	C₂×#3676	1	6	yes	—
219	C₄	C₂³×Dic₉	1	1	yes	characteristic
220	C₄	C₂²×#611	1	1	yes	characteristic
221	C₃	C₂²×#832	1	1	yes	characteristic
222	C₂	—	4	8	no	—
223	C₂	C₂×#63392	1	12	yes	direct product factor
224	C₂	C₂²×#3672	1	1	yes	characteristic
225	C₂	C₂²×#3673	1	1	yes	characteristic
226	C₂	C₂²×#3676	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	#611
commutator	C₄×C₉	C₂³×C₄
Frattini	C₂×C₃×C₄	C₂³×D₃
Fitting	C₂³×D₄×C₉	C₂
socle	C₂⁴×C₃	#67
prime radical	C₂³×C₃×D₄	D₃
cosocle	C₂×C₄×C₉	C₂⁴
hypercenter	C₂³×D₄	D₉
hypocenter	C₉	C₂²×#102
LCS	G ▷ C₄×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄ ◁ C₂³×D₄

Complements

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

N	H ≅ G/N	type
2. C₂²×C₄⋊Dic₉	C₂	semidirect
3. C₂³×C₉⋊C₈	C₂	semidirect
4. C₂×#63392	C₂	direct (N × H)
9. C₂²×D₄×C₉	C₄	semidirect
14. C₂×C₄⋊Dic₉	C₂²	semidirect
16. C₂²×C₉⋊C₈	C₂²	semidirect
17. #63392	C₂²	direct (N × H)
30. C₂×D₄×C₉	C₂×C₄	semidirect
39. C₄⋊Dic₉	C₂³	semidirect
41. C₂×C₉⋊C₈	C₂³	semidirect
66. D₄×C₉	C₂²×C₄	semidirect
103. C₂²×C₉	#102	semidirect
115. C₂²×D₄	Dic₉	semidirect
145. C₂×C₉	C₂×#102	semidirect
157. C₂×D₄	C₂×Dic₉	semidirect
182. C₉	C₂²×#102	semidirect
194. D₄	C₂²×Dic₉	semidirect
212. C₂²	#63392	direct (N × H)
223. C₂	C₂×#63392	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×D₄×C₉	2
C₂²×C₄⋊Dic₉	2
C₂³×C₉⋊C₈	2
C₂×#63392	2
C₂²×#832	3

Subgroups — C22×#63392 #11857718

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#63392 #11857718