Subgroups — C₂×#10588887 #11857353

Order 1152.

1512 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×C₄×C₉	C₂	1	1	yes	maximal, characteristic
2	C₂²×#3667	C₂	1	2	yes	maximal, complemented
3	#10588887	C₂	1	8	yes	maximal, direct product factor
4	C₂×#63389	C₂	1	4	yes	maximal, complemented
5	C₂×#5207	—	3	1	no	maximal
6	C₂⁵×C₉	C₄	1	1	yes	characteristic
7	C₂³×C₄×C₉	—	2	8	no	—
8	C₂³×C₄×C₉	C₄	1	1	yes	characteristic
9	C₂³×C₄×C₉	C₄	1	4	yes	—
10	C₂³×C₄×C₉	C₄	1	2	yes	—
11	C₂³×C₄×C₉	C₂²	1	2	yes	—
12	C₂³×C₄×C₉	C₂²	1	4	yes	—
13	C₂³×C₄×C₉	C₂²	1	1	yes	characteristic
14	C₂×#3667	—	2	8	no	—
15	C₂×#3667	C₂²	1	8	yes	complemented
16	C₂²×C₉⋊C₈	C₂²	1	4	yes	complemented
17	#63389	C₂²	1	16	yes	complemented
18	C₂⁴×C₃×C₄	D₃	1	1	yes	characteristic
19	C₂²×#208	—	3	2	no	—
20	#5207	—	3	8	no	—
21	C₂×#830	—	3	4	no	—
22	C₂⁴×C₉	—	2	4	no	—
23	C₂⁴×C₉	—	2	8	no	—
24	C₂⁴×C₉	C₂×C₄	1	4	yes	—
25	C₂⁴×C₉	C₂×C₄	1	1	yes	characteristic
26	C₂⁴×C₉	C₂×C₄	1	2	yes	—
27	C₂²×C₄×C₉	—	2	4	no	—
28	C₂²×C₄×C₉	—	2	16	no	—
29	C₂²×C₄×C₉	—	2	8	no	—
30	C₂²×C₄×C₉	—	2	8	no	—
31	C₂²×C₄×C₉	—	2	16	no	—
32	C₂²×C₄×C₉	C₂×C₄	1	4	yes	—
33	C₂²×C₄×C₉	C₂×C₄	1	2	yes	—
34	C₂²×C₄×C₉	C₂×C₄	1	4	yes	—
35	C₂²×C₄×C₉	C₂×C₄	1	8	yes	—
36	C₂²×C₄×C₉	C₂×C₄	1	2	yes	—
37	C₂²×C₄×C₉	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₄×C₉	D₄	1	8	yes	—
39	C₂²×C₄×C₉	C₂³	1	4	yes	—
40	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
41	C₂²×C₄×C₉	C₂³	1	2	yes	—
42	#3667	—	2	16	no	—
43	C₂×C₉⋊C₈	—	2	8	no	—
44	C₂×C₉⋊C₈	C₂³	1	8	yes	complemented
45	C₂×#407	—	9	1	no	Sylow
46	C₂⁵×C₃	Dic₃	1	1	yes	characteristic
47	C₂³×C₃×C₄	—	2	8	no	—
48	C₂³×C₃×C₄	Dic₃	1	1	yes	characteristic
49	C₂³×C₃×C₄	Dic₃	1	4	yes	—
50	C₂³×C₃×C₄	Dic₃	1	2	yes	—
51	C₂³×C₃×C₄	C₂×D₃	1	2	yes	—
52	C₂³×C₃×C₄	C₂×D₃	1	4	yes	—
53	C₂³×C₃×C₄	C₂×D₃	1	1	yes	characteristic
54	C₂×#208	—	3	8	no	—
55	C₂×#208	—	6	8	no	—
56	C₂²×C₃⋊C₈	—	3	4	no	—
57	#830	—	3	16	no	—
58	C₂³×C₉	—	2	32	no	—
59	C₂³×C₉	—	2	8	no	—
60	C₂³×C₉	—	2	4	no	—
61	C₂³×C₉	—	2	4	no	—
62	C₂³×C₉	—	2	4	no	—
63	C₂³×C₉	—	2	16	no	—
64	C₂³×C₉	C₂²⋊C₄	1	4	yes	—
65	C₂³×C₉	M₁₆	1	8	yes	complemented
66	C₂³×C₉	C₂²×C₄	1	2	yes	—
67	C₂³×C₉	C₂²×C₄	1	4	yes	—
68	C₂³×C₉	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₄×C₉	—	2	8	no	—
70	C₂×C₄×C₉	—	2	4	no	—
71	C₂×C₄×C₉	—	2	8	no	—
72	C₂×C₄×C₉	—	2	16	no	—
73	C₂×C₄×C₉	—	2	8	no	—
74	C₂×C₄×C₉	C₂²⋊C₄	1	8	yes	—
75	C₂×C₄×C₉	C₂²⋊C₄	1	4	yes	—
76	C₂×C₄×C₉	C₂²×C₄	1	4	yes	—
77	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₄×C₉	C₂²×C₄	1	2	yes	—
79	C₂×C₄×C₉	C₂×D₄	1	4	yes	—
80	C₂×C₄×C₉	C₂×D₄	1	8	yes	—
81	C₂×C₄×C₉	C₂⁴	1	1	yes	characteristic
82	C₉⋊C₈	—	2	8	no	—
83	C₂⁴×C₄	D₉	1	1	yes	characteristic
84	C₂²×M₁₆	—	9	2	no	—
85	#407	—	9	8	no	—
86	C₂×C₂²⋊C₈	—	9	4	no	—
87	C₂⁴×C₃	—	2	8	no	—
88	C₂⁴×C₃	—	2	4	no	—
89	C₂⁴×C₃	C₂×Dic₃	1	4	yes	—
90	C₂⁴×C₃	C₂×Dic₃	1	2	yes	—
91	C₂⁴×C₃	C₂×Dic₃	1	1	yes	characteristic
92	C₂²×C₃×C₄	—	2	4	no	—
93	C₂²×C₃×C₄	—	2	16	no	—
94	C₂²×C₃×C₄	—	2	8	no	—
95	C₂²×C₃×C₄	—	2	8	no	—
96	C₂²×C₃×C₄	—	2	16	no	—
97	C₂²×C₃×C₄	C₂×Dic₃	1	4	yes	—
98	C₂²×C₃×C₄	C₂×Dic₃	1	2	yes	—
99	C₂²×C₃×C₄	C₂×Dic₃	1	4	yes	—
100	C₂²×C₃×C₄	C₂×Dic₃	1	8	yes	—
101	C₂²×C₃×C₄	C₂×Dic₃	1	2	yes	—
102	C₂²×C₃×C₄	C₂×Dic₃	1	1	yes	characteristic
103	C₂²×C₃×C₄	#67	1	8	yes	—
104	C₂²×C₃×C₄	C₂²×D₃	1	4	yes	—
105	C₂²×C₃×C₄	C₂²×D₃	1	2	yes	—
106	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
107	#208	—	6	16	no	—
108	C₂×C₃⋊C₈	—	3	8	no	—
109	C₂×C₃⋊C₈	—	6	8	no	—
110	C₂²×C₉	—	2	32	no	—
111	C₂²×C₉	—	2	4	no	—
112	C₂²×C₉	—	2	4	no	—
113	C₂²×C₉	—	2	4	no	—
114	C₂²×C₉	—	2	8	no	—
115	C₂²×C₉	—	2	16	no	—
116	C₂²×C₉	C₂×C₂²⋊C₄	1	4	yes	—
117	C₂²×C₉	C₂×C₂²⋊C₄	1	2	yes	—
118	C₂²×C₉	C₂×M₁₆	1	4	yes	—
119	C₂²×C₉	C₂×M₁₆	1	8	yes	complemented
120	C₂²×C₉	C₂³×C₄	1	1	yes	characteristic
121	C₄×C₉	—	2	4	no	—
122	C₄×C₉	C₂×C₂²⋊C₄	1	2	yes	—
123	C₄×C₉	C₂×C₂²⋊C₄	1	4	yes	—
124	C₄×C₉	C₂²×D₄	1	2	yes	—
125	C₂⁵	Dic₉	1	1	yes	characteristic
126	C₂³×C₄	—	2	8	no	—
127	C₂³×C₄	Dic₉	1	1	yes	characteristic
128	C₂³×C₄	Dic₉	1	4	yes	—
129	C₂³×C₄	Dic₉	1	2	yes	—
130	C₂³×C₄	C₂×D₉	1	4	yes	—
131	C₂³×C₄	C₂×D₉	1	2	yes	—
132	C₂³×C₄	C₂×D₉	1	1	yes	characteristic
133	C₂×M₁₆	—	9	8	no	—
134	C₂×M₁₆	—	18	8	no	—
135	C₂²×C₈	—	9	4	no	—
136	C₂²⋊C₈	—	9	16	no	—
137	C₂³×C₃	—	2	32	no	—
138	C₂³×C₃	—	2	4	no	—
139	C₂³×C₃	—	2	4	no	—
140	C₂³×C₃	—	2	16	no	—
141	C₂³×C₃	—	2	8	no	—
142	C₂³×C₃	—	2	4	no	—
143	C₂³×C₃	#208	1	8	yes	—
144	C₂³×C₃	#217	1	4	yes	—
145	C₂³×C₃	C₂²×Dic₃	1	4	yes	—
146	C₂³×C₃	C₂²×Dic₃	1	2	yes	—
147	C₂³×C₃	C₂²×Dic₃	1	1	yes	characteristic
148	C₂×C₃×C₄	—	2	8	no	—
149	C₂×C₃×C₄	—	2	4	no	—
150	C₂×C₃×C₄	—	2	8	no	—
151	C₂×C₃×C₄	—	2	16	no	—
152	C₂×C₃×C₄	—	2	8	no	—
153	C₂×C₃×C₄	#217	1	8	yes	—
154	C₂×C₃×C₄	#217	1	4	yes	—
155	C₂×C₃×C₄	C₂²×Dic₃	1	4	yes	—
156	C₂×C₃×C₄	C₂²×Dic₃	1	2	yes	—
157	C₂×C₃×C₄	C₂²×Dic₃	1	1	yes	characteristic
158	C₂×C₃×C₄	C₂×#67	1	4	yes	—
159	C₂×C₃×C₄	C₂×#67	1	8	yes	—
160	C₂×C₃×C₄	C₂³×D₃	1	1	yes	characteristic
161	C₃⋊C₈	—	6	8	no	—
162	C₂×C₉	—	2	4	no	—
163	C₂×C₉	—	2	8	no	—
164	C₂×C₉	#407	1	4	yes	complemented
165	C₂×C₉	C₂²×C₂²⋊C₄	1	1	yes	characteristic
166	C₂×C₉	C₂²×M₁₆	1	2	yes	—
167	C₂⁴	—	2	8	no	—
168	C₂⁴	—	2	4	no	—
169	C₂⁴	C₂×Dic₉	1	4	yes	—
170	C₂⁴	C₂×Dic₉	1	2	yes	—
171	C₂⁴	C₂×Dic₉	1	1	yes	characteristic
172	C₂²×C₄	—	2	4	no	—
173	C₂²×C₄	—	2	16	no	—
174	C₂²×C₄	—	2	8	no	—
175	C₂²×C₄	—	2	8	no	—
176	C₂²×C₄	—	2	16	no	—
177	C₂²×C₄	C₂×Dic₉	1	4	yes	—
178	C₂²×C₄	C₂×Dic₉	1	2	yes	—
179	C₂²×C₄	C₂×Dic₉	1	8	yes	—
180	C₂²×C₄	C₂×Dic₉	1	4	yes	—
181	C₂²×C₄	C₂×Dic₉	1	1	yes	characteristic
182	C₂²×C₄	C₂×Dic₉	1	2	yes	—
183	C₂²×C₄	#611	1	8	yes	—
184	C₂²×C₄	C₂²×D₉	1	4	yes	—
185	C₂²×C₄	C₂²×D₉	1	2	yes	—
186	C₂²×C₄	C₂²×D₉	1	1	yes	characteristic
187	M₁₆	—	18	16	no	—
188	C₂×C₈	—	9	8	no	—
189	C₂×C₈	—	18	8	no	—
190	C₂²×C₃	—	2	32	no	—
191	C₂²×C₃	—	2	16	no	—
192	C₂²×C₃	—	2	8	no	—
193	C₂²×C₃	—	2	4	no	—
194	C₂²×C₃	—	2	4	no	—
195	C₂²×C₃	—	2	4	no	—
196	C₂²×C₃	C₂×#208	1	4	yes	—
197	C₂²×C₃	C₂×#208	1	8	yes	—
198	C₂²×C₃	C₂×#217	1	4	yes	—
199	C₂²×C₃	C₂×#217	1	2	yes	—
200	C₂²×C₃	C₂³×Dic₃	1	1	yes	characteristic
201	C₃×C₄	—	2	4	no	—
202	C₃×C₄	C₂×#217	1	2	yes	—
203	C₃×C₄	C₂×#217	1	4	yes	—
204	C₃×C₄	C₂²×#67	1	2	yes	—
205	C₉	C₂×#407	1	1	yes	characteristic, complemented, Sylow
206	C₂³	—	2	32	no	—
207	C₂³	—	2	8	no	—
208	C₂³	—	2	4	no	—
209	C₂³	—	2	4	no	—
210	C₂³	—	2	4	no	—
211	C₂³	—	2	16	no	—
212	C₂³	#3667	1	8	yes	complemented
213	C₂³	#3676	1	4	yes	—
214	C₂³	C₂²×Dic₉	1	2	yes	—
215	C₂³	C₂²×Dic₉	1	4	yes	—
216	C₂³	C₂²×Dic₉	1	1	yes	characteristic
217	C₂×C₄	—	2	8	no	—
218	C₂×C₄	—	2	4	no	—
219	C₂×C₄	—	2	16	no	—
220	C₂×C₄	—	2	8	no	—
221	C₂×C₄	—	2	8	no	—
222	C₂×C₄	#3676	1	8	yes	—
223	C₂×C₄	#3676	1	4	yes	—
224	C₂×C₄	C₂²×Dic₉	1	4	yes	—
225	C₂×C₄	C₂²×Dic₉	1	2	yes	—
226	C₂×C₄	C₂²×Dic₉	1	1	yes	characteristic
227	C₂×C₄	C₂×#611	1	4	yes	—
228	C₂×C₄	C₂×#611	1	8	yes	—
229	C₂×C₄	C₂³×D₉	1	1	yes	characteristic
230	C₈	—	18	8	no	—
231	C₂×C₃	—	2	8	no	—
232	C₂×C₃	—	2	4	no	—
233	C₂×C₃	#5207	1	4	yes	—
234	C₂×C₃	C₂²×#208	1	2	yes	—
235	C₂×C₃	C₂²×#217	1	1	yes	characteristic
236	C₂²	—	2	32	no	—
237	C₂²	—	2	4	no	—
238	C₂²	—	2	4	no	—
239	C₂²	—	2	16	no	—
240	C₂²	—	2	8	no	—
241	C₂²	—	2	4	no	—
242	C₂²	C₂×#3667	1	4	yes	—
243	C₂²	C₂×#3667	1	8	yes	complemented
244	C₂²	C₂×#3676	1	4	yes	—
245	C₂²	C₂×#3676	1	2	yes	—
246	C₂²	C₂³×Dic₉	1	1	yes	characteristic
247	C₄	—	2	4	no	—
248	C₄	C₂×#3676	1	4	yes	—
249	C₄	C₂×#3676	1	2	yes	—
250	C₄	C₂²×#611	1	2	yes	—
251	C₃	C₂×#5207	1	1	yes	characteristic
252	C₂	—	2	4	no	—
253	C₂	—	2	8	no	—
254	C₂	#10588887	1	4	yes	direct product factor
255	C₂	C₂²×#3667	1	2	yes	—
256	C₂	C₂²×#3676	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²×D₉
commutator	C₂²×C₉	C₂³×C₄
Frattini	C₂×C₃×C₄	C₂³×D₃
Fitting	C₂⁴×C₄×C₉	C₂
socle	C₂³×C₃	C₂²×Dic₃
prime radical	C₂⁵×C₃	Dic₃
cosocle	C₂×C₄×C₉	C₂⁴
hypercenter	C₂⁴×C₄	D₉
hypocenter	C₉	C₂×#407
LCS	G ▷ C₂²×C₉ ▷ C₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂⁴×C₄

Complements

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

N	H ≅ G/N	type
2. C₂²×#3667	C₂	semidirect
3. #10588887	C₂	direct (N × H)
4. C₂×#63389	C₂	semidirect
15. C₂×#3667	C₂²	semidirect
16. C₂²×C₉⋊C₈	C₂²	semidirect
17. #63389	C₂²	semidirect
44. C₂×C₉⋊C₈	C₂³	semidirect
65. C₂³×C₉	M₁₆	semidirect
119. C₂²×C₉	C₂×M₁₆	semidirect
164. C₂×C₉	#407	semidirect
205. C₉	C₂×#407	semidirect
212. C₂³	#3667	semidirect
243. C₂²	C₂×#3667	semidirect
254. C₂	#10588887	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×C₄×C₉	2
C₂²×#3667	2
#10588887	2
C₂×#63389	2
C₂×#5207	3

Subgroups — C2×#10588887 #11857353

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#10588887 #11857353