Subgroups — C₂×#10588392 #11860151

Order 1152.

912 conjugacy classes of subgroups.

The subgroup lattice has 231 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×D₇₂	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×D₃₆	C₂	1	2	yes	maximal, complemented
3	C₂×C₉×#97	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#63380	C₂	1	2	yes	maximal, complemented
5	C₂×#63377	C₂	1	1	yes	maximal, characteristic, complemented
6	#10588392	C₂	1	8	yes	maximal, direct product factor
7	C₂×#4711	—	3	1	no	maximal
8	C₂²×D₃₆	C₂²	1	2	yes	—
9	C₂²×C₄×D₉	—	2	2	no	—
10	C₂²×C₈×C₉	C₂²	1	1	yes	characteristic
11	C₂²×C₈×C₉	C₂²	1	1	yes	characteristic
12	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
13	C₂×(C₂×D₉)⋊C₄	—	2	2	no	—
14	C₂×C₄⋊Dic₉	C₂²	1	2	yes	—
15	C₂×D₇₂	C₄	1	8	yes	complemented
16	C₂×D₇₂	C₂²	1	4	yes	—
17	C₄×D₃₆	—	2	4	no	—
18	C₄×D₃₆	C₂²	1	8	yes	complemented
19	C₉×#97	C₂²	1	4	yes	complemented
20	#63380	C₂²	1	8	yes	complemented
21	#63377	C₂²	1	4	yes	complemented
22	C₂²×D₂₄	—	3	1	no	—
23	C₂×C₄×D₁₂	—	3	2	no	—
24	C₂×C₃×#97	D₃	1	1	yes	characteristic
25	C₂×#821	—	3	2	no	—
26	C₂×#817	—	3	1	no	—
27	#4711	—	3	8	no	—
28	C₂³×D₉	—	2	2	no	—
29	C₂²×C₄×C₉	D₄	1	2	yes	—
30	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
31	C₂²×Dic₉	—	2	2	no	—
32	C₂×D₃₆	—	2	2	no	—
33	C₂×D₃₆	—	2	4	no	—
34	C₂×D₃₆	C₂×C₄	1	8	yes	complemented
35	C₂×D₃₆	C₂³	1	4	yes	—
36	C₂×C₄×D₉	—	2	8	no	—
37	C₂×C₄×D₉	—	4	8	no	—
38	C₂×C₈×C₉	—	2	2	no	—
39	C₂×C₈×C₉	C₂×C₄	1	4	yes	complemented
40	C₂×C₈×C₉	C₂³	1	2	yes	—
41	C₂×C₈×C₉	C₂³	1	2	yes	—
42	C₄²×C₉	D₄	1	2	yes	—
43	C₄²×C₉	C₂³	1	2	yes	—
44	(C₂×D₉)⋊C₄	—	2	8	no	—
45	C₄⋊Dic₉	—	2	2	no	—
46	C₄⋊Dic₉	C₂³	1	4	yes	—
47	D₇₂	C₂×C₄	1	16	yes	complemented
48	C₂×#442	—	9	1	no	Sylow
49	C₂²×D₁₂	—	3	2	no	—
50	C₂²×C₄×D₃	—	6	2	no	—
51	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
52	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
53	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
54	C₂×(C₂×D₃)⋊C₄	—	6	2	no	—
55	C₂×C₄⋊Dic₃	—	3	2	no	—
56	C₂×D₂₄	—	3	8	no	—
57	C₂×D₂₄	—	3	4	no	—
58	C₄×D₁₂	—	3	8	no	—
59	C₄×D₁₂	—	6	4	no	—
60	C₃×#97	C₂×D₃	1	4	yes	—
61	#821	—	3	8	no	—
62	#817	—	3	4	no	—
63	C₂³×C₉	C₂×D₄	1	1	yes	characteristic
64	C₂²×D₉	—	2	8	no	—
65	C₂²×D₉	—	2	4	no	—
66	C₂²×D₉	—	4	4	no	—
67	C₂²×D₉	—	4	4	no	—
68	C₂×C₄×C₉	—	2	4	no	—
69	C₂×C₄×C₉	C₂²×C₄	1	2	yes	—
70	C₂×C₄×C₉	C₂×D₄	1	4	yes	—
71	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
72	C₂×C₄×C₉	#41	1	2	yes	—
73	C₂×C₄×C₉	C₂⁴	1	1	yes	characteristic
74	C₂×Dic₉	—	2	4	no	—
75	C₂×Dic₉	—	4	4	no	—
76	D₃₆	—	2	8	no	—
77	D₃₆	—	2	4	no	—
78	D₃₆	C₂²×C₄	1	8	yes	complemented
79	C₄×D₉	—	4	16	no	—
80	C₈×C₉	—	2	2	no	—
81	C₈×C₉	C₂²×C₄	1	4	yes	complemented
82	C₂²×D₈	—	9	1	no	—
83	C₂×C₄×D₄	—	9	2	no	—
84	#442	—	9	8	no	—
85	C₂×#106	—	9	1	no	—
86	C₂×#102	—	9	2	no	—
87	C₂×#97	D₉	1	1	yes	characteristic, complemented
88	C₂³×D₃	—	6	2	no	—
89	C₂²×C₃×C₄	D₁₂	1	2	yes	—
90	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
91	C₂²×Dic₃	—	6	2	no	—
92	C₂×D₁₂	—	3	8	no	—
93	C₂×D₁₂	—	3	4	no	—
94	C₂×D₁₂	—	6	2	no	—
95	C₂×D₁₂	—	6	4	no	—
96	C₂×C₄×D₃	—	6	8	no	—
97	C₂×C₄×D₃	—	12	8	no	—
98	C₂×C₃×C₈	—	2	2	no	—
99	C₂×C₃×C₈	C₄×D₃	1	4	yes	—
100	C₂×C₃×C₈	C₂²×D₃	1	2	yes	—
101	C₂×C₃×C₈	C₂²×D₃	1	2	yes	—
102	C₃×C₄²	D₁₂	1	2	yes	—
103	C₃×C₄²	C₂²×D₃	1	2	yes	—
104	(C₂×D₃)⋊C₄	—	6	8	no	—
105	C₄⋊Dic₃	—	3	4	no	—
106	C₄⋊Dic₃	—	6	2	no	—
107	D₂₄	—	3	16	no	—
108	C₂²×C₉	C₄×D₄	1	2	yes	—
109	C₂²×C₉	AGL(1,ℤ/8)	1	4	yes	—
110	C₂²×C₉	C₂²×D₄	1	1	yes	characteristic
111	C₂×D₉	—	2	8	no	—
112	C₂×D₉	—	4	4	no	—
113	C₂×D₉	—	4	4	no	—
114	C₂×D₉	—	4	16	no	—
115	C₄×C₉	—	2	4	no	—
116	C₄×C₉	C₄×D₄	1	2	yes	—
117	C₄×C₉	C₂³×C₄	1	1	yes	characteristic
118	C₄×C₉	C₂×#41	1	1	yes	characteristic
119	Dic₉	—	4	4	no	—
120	C₂²×D₄	—	9	2	no	—
121	C₂³×C₄	—	18	2	no	—
122	C₂×D₈	—	9	8	no	—
123	C₂×D₈	—	9	4	no	—
124	C₂²×C₈	C₂×D₉	1	1	yes	characteristic
125	C₂²×C₈	C₂×D₉	1	1	yes	characteristic
126	C₄×D₄	—	9	8	no	—
127	C₄×D₄	—	18	4	no	—
128	C₂×C₄⋊C₄	—	9	2	no	—
129	C₂×C₂²⋊C₄	—	18	2	no	—
130	C₂×C₄²	C₂×D₉	1	1	yes	characteristic
131	#106	—	9	4	no	—
132	#102	—	9	8	no	—
133	#97	C₂×D₉	1	4	yes	complemented
134	C₂³×C₃	C₂×D₁₂	1	1	yes	characteristic
135	C₂²×D₃	—	6	8	no	—
136	C₂²×D₃	—	6	4	no	—
137	C₂²×D₃	—	12	4	no	—
138	C₂²×D₃	—	12	4	no	—
139	C₂×C₃×C₄	—	2	4	no	—
140	C₂×C₃×C₄	C₂×C₄×D₃	1	2	yes	—
141	C₂×C₃×C₄	C₂×D₁₂	1	4	yes	—
142	C₂×C₃×C₄	C₂×D₁₂	1	1	yes	characteristic
143	C₂×C₃×C₄	#235	1	2	yes	—
144	C₂×C₃×C₄	C₂³×D₃	1	1	yes	characteristic
145	C₂×Dic₃	—	6	4	no	—
146	C₂×Dic₃	—	12	4	no	—
147	D₁₂	—	3	8	no	—
148	D₁₂	—	6	8	no	—
149	D₁₂	—	6	4	no	—
150	C₄×D₃	—	12	16	no	—
151	C₃×C₈	—	2	2	no	—
152	C₃×C₈	C₂×C₄×D₃	1	4	yes	—
153	C₂×C₉	#442	1	4	yes	complemented
154	C₂×C₉	C₂×C₄×D₄	1	1	yes	characteristic
155	C₂×C₉	C₂×AGL(1,ℤ/8)	1	2	yes	—
156	D₉	—	4	8	no	—
157	C₂⁴	—	18	2	no	—
158	C₂×D₄	—	9	8	no	—
159	C₂×D₄	—	9	4	no	—
160	C₂×D₄	—	18	2	no	—
161	C₂×D₄	—	18	4	no	—
162	C₂²×C₄	—	18	2	no	—
163	C₂²×C₄	—	18	8	no	—
164	C₂²×C₄	—	36	8	no	—
165	C₂²×C₄	D₃₆	1	2	yes	—
166	C₂²×C₄	C₂²×D₉	1	1	yes	characteristic
167	D₈	—	9	16	no	—
168	C₂×C₈	—	2	2	no	—
169	C₂×C₈	C₄×D₉	1	4	yes	complemented
170	C₂×C₈	C₂²×D₉	1	2	yes	—
171	C₂×C₈	C₂²×D₉	1	2	yes	—
172	C₄⋊C₄	—	9	4	no	—
173	C₄⋊C₄	—	18	2	no	—
174	C₂²⋊C₄	—	18	8	no	—
175	C₄²	D₃₆	1	2	yes	—
176	C₄²	C₂²×D₉	1	2	yes	—
177	C₂²×C₃	C₄×D₁₂	1	2	yes	—
178	C₂²×C₃	#908	1	4	yes	—
179	C₂²×C₃	C₂²×D₁₂	1	1	yes	characteristic
180	C₂×D₃	—	6	8	no	—
181	C₂×D₃	—	12	16	no	—
182	C₂×D₃	—	12	4	no	—
183	C₂×D₃	—	12	4	no	—
184	C₃×C₄	—	2	4	no	—
185	C₃×C₄	C₄×D₁₂	1	2	yes	—
186	C₃×C₄	C₂²×C₄×D₃	1	1	yes	characteristic
187	C₃×C₄	C₂×#235	1	1	yes	characteristic
188	Dic₃	—	12	4	no	—
189	C₉	C₂×#442	1	1	yes	characteristic, complemented, Sylow
190	C₂³	—	18	8	no	—
191	C₂³	—	18	4	no	—
192	C₂³	—	36	4	no	—
193	C₂³	—	36	4	no	—
194	C₂³	C₂×D₃₆	1	1	yes	characteristic
195	D₄	—	9	8	no	—
196	D₄	—	18	8	no	—
197	D₄	—	18	4	no	—
198	C₂×C₄	—	2	4	no	—
199	C₂×C₄	—	18	4	no	—
200	C₂×C₄	—	36	16	no	—
201	C₂×C₄	—	36	4	no	—
202	C₂×C₄	C₂×C₄×D₉	1	2	yes	—
203	C₂×C₄	C₂×D₃₆	1	4	yes	—
204	C₂×C₄	C₂×D₃₆	1	1	yes	characteristic
205	C₂×C₄	#3697	1	2	yes	—
206	C₂×C₄	C₂³×D₉	1	1	yes	characteristic
207	C₈	—	2	2	no	—
208	C₈	C₂×C₄×D₉	1	4	yes	complemented
209	C₂×C₃	#4711	1	4	yes	—
210	C₂×C₃	C₂×C₄×D₁₂	1	1	yes	characteristic
211	C₂×C₃	C₂×#908	1	2	yes	—
212	D₃	—	12	8	no	—
213	C₂²	—	18	8	no	—
214	C₂²	—	36	4	no	—
215	C₂²	—	36	4	no	—
216	C₂²	—	36	16	no	—
217	C₂²	C₄×D₃₆	1	2	yes	—
218	C₂²	#63470	1	4	yes	—
219	C₂²	C₂²×D₃₆	1	1	yes	characteristic
220	C₄	—	2	4	no	—
221	C₄	—	36	4	no	—
222	C₄	C₄×D₃₆	1	2	yes	—
223	C₄	C₂²×C₄×D₉	1	1	yes	characteristic
224	C₄	C₂×#3697	1	1	yes	characteristic
225	C₃	C₂×#4711	1	1	yes	characteristic
226	C₂	—	36	8	no	—
227	C₂	#10588392	1	4	yes	direct product factor
228	C₂	C₂×C₄×D₃₆	1	1	yes	characteristic
229	C₂	C₂×#63470	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₃₆
commutator	C₄×C₉	C₂³×C₄
Frattini	C₂×C₃×C₄	C₂³×D₃
Fitting	C₂×C₉×#97	C₂
socle	C₂³×C₃	C₂×D₁₂
prime radical	C₂²×D₇₂	C₂
cosocle	C₂×C₄×C₉	C₂⁴
hypercenter	C₂×#97	D₉
hypocenter	C₉	C₂×#442
LCS	G ▷ C₄×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#97

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×D₃₆	C₂	semidirect
3. C₂×C₉×#97	C₂	semidirect
4. C₂×#63380	C₂	semidirect
5. C₂×#63377	C₂	semidirect
6. #10588392	C₂	direct (N × H)
15. C₂×D₇₂	C₄	semidirect
18. C₄×D₃₆	C₂²	semidirect
19. C₉×#97	C₂²	semidirect
20. #63380	C₂²	semidirect
21. #63377	C₂²	semidirect
34. C₂×D₃₆	C₂×C₄	semidirect
39. C₂×C₈×C₉	C₂×C₄	semidirect
47. D₇₂	C₂×C₄	semidirect
78. D₃₆	C₂²×C₄	semidirect
81. C₈×C₉	C₂²×C₄	semidirect
87. C₂×#97	D₉	semidirect
133. #97	C₂×D₉	semidirect
153. C₂×C₉	#442	semidirect
169. C₂×C₈	C₄×D₉	semidirect
189. C₉	C₂×#442	semidirect
208. C₈	C₂×C₄×D₉	semidirect
227. C₂	#10588392	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×D₇₂	2
C₂×C₄×D₃₆	2
C₂×C₉×#97	2
C₂×#63380	2
C₂×#63377	2
#10588392	2
C₂×#4711	3

Subgroups — C2×#10588392 #11860151

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#10588392 #11860151