Subgroups — C₂×#10588689 #11859265

Order 1152.

828 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	C₂×C₄×D₄×C₉	C₂	1	1	yes	maximal, characteristic
2	C₂²×#3667	C₂	1	2	yes	maximal, complemented
3	C₂×#63389	C₂	1	2	yes	maximal, complemented
4	#10588689	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₄⋊(C₉⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₄×C₉⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#5012	—	3	1	no	maximal
8	C₂²×D₄×C₉	C₄	1	1	yes	characteristic
9	C₂³×C₄×C₉	C₂²	1	2	yes	—
10	C₄×D₄×C₉	C₄	1	4	yes	—
11	C₄×D₄×C₉	C₂²	1	4	yes	—
12	C₂×C₉×C₄⋊C₄	C₄	1	1	yes	characteristic
13	C₂×C₉×C₂²⋊C₄	C₄	1	2	yes	—
14	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
15	C₂×#3667	—	2	8	no	—
16	C₂×#3667	C₂²	1	8	yes	complemented
17	C₂²×C₉⋊C₈	C₂²	1	1	yes	characteristic
18	C₂²×C₉⋊C₈	C₂²	1	2	yes	—
19	C₂²×C₉⋊C₈	C₂²	1	1	yes	characteristic
20	#63389	C₂²	1	8	yes	complemented
21	C₄⋊(C₉⋊C₈)	C₂²	1	4	yes	complemented
22	C₄×C₉⋊C₈	C₂²	1	4	yes	complemented
23	C₂×C₃×C₄×D₄	D₃	1	1	yes	characteristic
24	C₂²×#208	—	3	2	no	—
25	C₂×#830	—	3	2	no	—
26	#5012	—	3	8	no	—
27	C₂×C₄⋊(C₃⋊C₈)	—	3	1	no	—
28	C₂×C₄×C₃⋊C₈	—	3	1	no	—
29	C₂⁴×C₉	C₂×C₄	1	2	yes	—
30	C₂×D₄×C₉	—	2	4	no	—
31	C₂×D₄×C₉	C₂×C₄	1	4	yes	—
32	C₂²×C₄×C₉	—	2	4	no	—
33	C₂²×C₄×C₉	—	2	4	no	—
34	C₂²×C₄×C₉	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₄×C₉	C₂×C₄	1	2	yes	—
36	C₂²×C₄×C₉	C₂×C₄	1	4	yes	—
37	C₂²×C₄×C₉	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₄×C₉	C₂³	1	4	yes	—
39	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
40	C₉×C₄⋊C₄	C₂×C₄	1	4	yes	—
41	C₉×C₂²⋊C₄	C₂×C₄	1	8	yes	—
42	C₄²×C₉	C₂×C₄	1	2	yes	—
43	C₄²×C₉	C₂³	1	2	yes	—
44	#3667	—	2	16	no	—
45	C₂×C₉⋊C₈	—	2	4	no	—
46	C₂×C₉⋊C₈	—	2	2	no	—
47	C₂×C₉⋊C₈	D₄	1	4	yes	complemented
48	C₂×C₉⋊C₈	C₂³	1	2	yes	—
49	C₂×C₉⋊C₈	C₂³	1	4	yes	—
50	C₂×C₉⋊C₈	C₂³	1	2	yes	—
51	C₂×#436	—	9	1	no	Sylow
52	C₂²×C₃×D₄	Dic₃	1	1	yes	characteristic
53	C₂³×C₃×C₄	C₂×D₃	1	2	yes	—
54	C₃×C₄×D₄	Dic₃	1	4	yes	—
55	C₃×C₄×D₄	C₂×D₃	1	4	yes	—
56	C₂×C₃×C₄⋊C₄	Dic₃	1	1	yes	characteristic
57	C₂×C₃×C₂²⋊C₄	Dic₃	1	2	yes	—
58	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
59	C₂×#208	—	3	8	no	—
60	C₂×#208	—	6	8	no	—
61	C₂²×C₃⋊C₈	—	3	1	no	—
62	C₂²×C₃⋊C₈	—	3	1	no	—
63	C₂²×C₃⋊C₈	—	3	2	no	—
64	#830	—	3	8	no	—
65	C₄⋊(C₃⋊C₈)	—	3	4	no	—
66	C₄×C₃⋊C₈	—	3	4	no	—
67	C₂³×C₉	—	2	4	no	—
68	C₂³×C₉	—	2	4	no	—
69	C₂³×C₉	—	2	4	no	—
70	C₂³×C₉	C₂²×C₄	1	4	yes	—
71	C₂³×C₉	C₂²×C₄	1	1	yes	characteristic
72	D₄×C₉	—	2	8	no	—
73	C₂×C₄×C₉	—	2	4	no	—
74	C₂×C₄×C₉	—	2	2	no	—
75	C₂×C₄×C₉	—	2	8	no	—
76	C₂×C₄×C₉	—	2	4	no	—
77	C₂×C₄×C₉	—	2	4	no	—
78	C₂×C₄×C₉	M₁₆	1	4	yes	complemented
79	C₂×C₄×C₉	C₂²×C₄	1	2	yes	—
80	C₂×C₄×C₉	C₂²×C₄	1	4	yes	—
81	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₄×C₉	C₂²×C₄	1	2	yes	—
83	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
84	C₂×C₄×C₉	#41	1	2	yes	—
85	C₂×C₄×C₉	C₂⁴	1	1	yes	characteristic
86	C₉⋊C₈	—	2	4	no	—
87	C₉⋊C₈	—	2	2	no	—
88	C₉⋊C₈	C₂×D₄	1	4	yes	complemented
89	C₂²×M₁₆	—	9	2	no	—
90	C₂×C₄×D₄	D₉	1	1	yes	characteristic
91	#436	—	9	8	no	—
92	C₂×C₄⋊C₈	—	9	1	no	—
93	C₂×C₂²⋊C₈	—	9	2	no	—
94	C₂×C₄×C₈	—	9	1	no	—
95	C₂⁴×C₃	C₂×Dic₃	1	2	yes	—
96	C₂×C₃×D₄	—	2	4	no	—
97	C₂×C₃×D₄	C₂×Dic₃	1	4	yes	—
98	C₂²×C₃×C₄	—	2	4	no	—
99	C₂²×C₃×C₄	—	2	4	no	—
100	C₂²×C₃×C₄	C₂×Dic₃	1	1	yes	characteristic
101	C₂²×C₃×C₄	C₂×Dic₃	1	2	yes	—
102	C₂²×C₃×C₄	C₂×Dic₃	1	4	yes	—
103	C₂²×C₃×C₄	C₂×Dic₃	1	1	yes	characteristic
104	C₂²×C₃×C₄	C₂²×D₃	1	4	yes	—
105	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
106	C₃×C₄⋊C₄	C₂×Dic₃	1	4	yes	—
107	C₃×C₂²⋊C₄	C₂×Dic₃	1	8	yes	—
108	C₃×C₄²	C₂×Dic₃	1	2	yes	—
109	C₃×C₄²	C₂²×D₃	1	2	yes	—
110	#208	—	6	16	no	—
111	C₂×C₃⋊C₈	—	3	4	no	—
112	C₂×C₃⋊C₈	—	3	2	no	—
113	C₂×C₃⋊C₈	—	3	2	no	—
114	C₂×C₃⋊C₈	—	3	4	no	—
115	C₂×C₃⋊C₈	—	6	4	no	—
116	C₂×C₃⋊C₈	—	6	2	no	—
117	C₂²×C₉	—	2	4	no	—
118	C₂²×C₉	—	2	4	no	—
119	C₂²×C₉	—	2	4	no	—
120	C₂²×C₉	—	4	8	no	—
121	C₂²×C₉	C₄×D₄	1	2	yes	—
122	C₂²×C₉	C₂×M₁₆	1	2	yes	—
123	C₂²×C₉	#131	1	2	yes	—
124	C₂²×C₉	C₂³×C₄	1	1	yes	characteristic
125	C₄×C₉	—	2	4	no	—
126	C₄×C₉	—	2	2	no	—
127	C₄×C₉	C₄×D₄	1	2	yes	—
128	C₄×C₉	C₂×M₁₆	1	4	yes	complemented
129	C₄×C₉	C₂²×D₄	1	1	yes	characteristic
130	C₄×C₉	C₂×#41	1	1	yes	characteristic
131	C₂²×D₄	Dic₉	1	1	yes	characteristic
132	C₂³×C₄	C₂×D₉	1	2	yes	—
133	C₂×M₁₆	—	9	8	no	—
134	C₂×M₁₆	—	18	8	no	—
135	C₂²×C₈	—	9	1	no	—
136	C₂²×C₈	—	9	2	no	—
137	C₂²×C₈	—	9	1	no	—
138	C₄×D₄	Dic₉	1	4	yes	—
139	C₄×D₄	C₂×D₉	1	4	yes	—
140	C₂×C₄⋊C₄	Dic₉	1	1	yes	characteristic
141	C₂×C₂²⋊C₄	Dic₉	1	2	yes	—
142	C₂×C₄²	C₂×D₉	1	1	yes	characteristic
143	C₄⋊C₈	—	9	4	no	—
144	C₂²⋊C₈	—	9	8	no	—
145	C₄×C₈	—	9	4	no	—
146	C₂³×C₃	—	2	4	no	—
147	C₂³×C₃	—	2	4	no	—
148	C₂³×C₃	—	2	4	no	—
149	C₂³×C₃	C₂²×Dic₃	1	4	yes	—
150	C₂³×C₃	C₂²×Dic₃	1	1	yes	characteristic
151	C₃×D₄	—	2	8	no	—
152	C₂×C₃×C₄	—	2	4	no	—
153	C₂×C₃×C₄	—	2	2	no	—
154	C₂×C₃×C₄	—	2	8	no	—
155	C₂×C₃×C₄	—	2	4	no	—
156	C₂×C₃×C₄	—	2	4	no	—
157	C₂×C₃×C₄	#208	1	4	yes	—
158	C₂×C₃×C₄	D₃×D₄	1	2	yes	—
159	C₂×C₃×C₄	#237	1	2	yes	—
160	C₂×C₃×C₄	C₂²×Dic₃	1	2	yes	—
161	C₂×C₃×C₄	C₂²×Dic₃	1	4	yes	—
162	C₂×C₃×C₄	C₂²×Dic₃	1	2	yes	—
163	C₂×C₃×C₄	C₂²×Dic₃	1	1	yes	characteristic
164	C₂×C₃×C₄	C₂³×D₃	1	1	yes	characteristic
165	C₃⋊C₈	—	3	4	no	—
166	C₃⋊C₈	—	6	4	no	—
167	C₃⋊C₈	—	6	2	no	—
168	C₂×C₉	—	4	4	no	—
169	C₂×C₉	#436	1	4	yes	complemented
170	C₂×C₉	C₂×C₄×D₄	1	1	yes	characteristic
171	C₂×C₉	C₂²×M₁₆	1	1	yes	characteristic
172	C₂×C₉	C₂×#131	1	1	yes	characteristic
173	C₂⁴	C₂×Dic₉	1	2	yes	—
174	C₂×D₄	—	2	4	no	—
175	C₂×D₄	C₂×Dic₉	1	4	yes	—
176	C₂²×C₄	—	2	4	no	—
177	C₂²×C₄	—	2	4	no	—
178	C₂²×C₄	C₂×Dic₉	1	1	yes	characteristic
179	C₂²×C₄	C₂×Dic₉	1	2	yes	—
180	C₂²×C₄	C₂×Dic₉	1	4	yes	—
181	C₂²×C₄	C₂×Dic₉	1	1	yes	characteristic
182	C₂²×C₄	C₂²×D₉	1	4	yes	—
183	C₂²×C₄	C₂²×D₉	1	1	yes	characteristic
184	M₁₆	—	18	16	no	—
185	C₂×C₈	—	9	4	no	—
186	C₂×C₈	—	9	2	no	—
187	C₂×C₈	—	9	2	no	—
188	C₂×C₈	—	9	4	no	—
189	C₂×C₈	—	18	4	no	—
190	C₂×C₈	—	18	2	no	—
191	C₄⋊C₄	C₂×Dic₉	1	4	yes	—
192	C₂²⋊C₄	C₂×Dic₉	1	8	yes	—
193	C₄²	C₂×Dic₉	1	2	yes	—
194	C₄²	C₂²×D₉	1	2	yes	—
195	C₂²×C₃	—	2	4	no	—
196	C₂²×C₃	—	2	4	no	—
197	C₂²×C₃	—	2	4	no	—
198	C₂²×C₃	—	4	8	no	—
199	C₂²×C₃	C₂×#208	1	2	yes	—
200	C₂²×C₃	D₄×Dic₃	1	2	yes	—
201	C₂²×C₃	#948	1	2	yes	—
202	C₂²×C₃	C₂³×Dic₃	1	1	yes	characteristic
203	C₃×C₄	—	2	4	no	—
204	C₃×C₄	—	2	2	no	—
205	C₃×C₄	C₂×#208	1	4	yes	—
206	C₃×C₄	D₄×Dic₃	1	2	yes	—
207	C₃×C₄	C₂×D₃×D₄	1	1	yes	characteristic
208	C₃×C₄	C₂×#237	1	1	yes	characteristic
209	C₉	C₂×#436	1	1	yes	characteristic, complemented, Sylow
210	C₂³	—	2	4	no	—
211	C₂³	—	2	4	no	—
212	C₂³	—	2	4	no	—
213	C₂³	C₂²×Dic₉	1	4	yes	—
214	C₂³	C₂²×Dic₉	1	1	yes	characteristic
215	D₄	—	2	8	no	—
216	C₂×C₄	—	2	4	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	8	no	—
219	C₂×C₄	—	2	4	no	—
220	C₂×C₄	—	2	4	no	—
221	C₂×C₄	#3667	1	4	yes	complemented
222	C₂×C₄	D₄×D₉	1	2	yes	—
223	C₂×C₄	#3699	1	2	yes	—
224	C₂×C₄	C₂²×Dic₉	1	2	yes	—
225	C₂×C₄	C₂²×Dic₉	1	4	yes	—
226	C₂×C₄	C₂²×Dic₉	1	1	yes	characteristic
227	C₂×C₄	C₂²×Dic₉	1	2	yes	—
228	C₂×C₄	C₂³×D₉	1	1	yes	characteristic
229	C₈	—	9	4	no	—
230	C₈	—	18	4	no	—
231	C₈	—	18	2	no	—
232	C₂×C₃	—	4	4	no	—
233	C₂×C₃	#5012	1	4	yes	—
234	C₂×C₃	C₂²×#208	1	1	yes	characteristic
235	C₂×C₃	C₂×D₄×Dic₃	1	1	yes	characteristic
236	C₂×C₃	C₂×#948	1	1	yes	characteristic
237	C₂²	—	2	4	no	—
238	C₂²	—	2	4	no	—
239	C₂²	—	2	4	no	—
240	C₂²	—	4	8	no	—
241	C₂²	C₂×#3667	1	2	yes	—
242	C₂²	D₄×Dic₉	1	2	yes	—
243	C₂²	#63510	1	2	yes	—
244	C₂²	C₂³×Dic₉	1	1	yes	characteristic
245	C₄	—	2	4	no	—
246	C₄	—	2	2	no	—
247	C₄	C₂×#3667	1	4	yes	complemented
248	C₄	D₄×Dic₉	1	2	yes	—
249	C₄	C₂×D₄×D₉	1	1	yes	characteristic
250	C₄	C₂×#3699	1	1	yes	characteristic
251	C₃	C₂×#5012	1	1	yes	characteristic
252	C₂	—	4	4	no	—
253	C₂	#10588689	1	4	yes	direct product factor
254	C₂	C₂²×#3667	1	1	yes	characteristic
255	C₂	C₂×D₄×Dic₉	1	1	yes	characteristic
256	C₂	C₂×#63510	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²×D₉
commutator	C₂²×C₉	C₂³×C₄
Frattini	C₂×C₃×C₄	C₂³×D₃
Fitting	C₂×C₄×D₄×C₉	C₂
socle	C₂³×C₃	C₂²×Dic₃
prime radical	C₂²×C₃×D₄	Dic₃
cosocle	C₂×C₄×C₉	C₂⁴
hypercenter	C₂×C₄×D₄	D₉
hypocenter	C₉	C₂×#436
LCS	G ▷ C₂²×C₉ ▷ C₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×D₄

Complements

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

N	H ≅ G/N	type
2. C₂²×#3667	C₂	semidirect
3. C₂×#63389	C₂	semidirect
4. #10588689	C₂	direct (N × H)
5. C₂×C₄⋊(C₉⋊C₈)	C₂	semidirect
6. C₂×C₄×C₉⋊C₈	C₂	semidirect
16. C₂×#3667	C₂²	semidirect
20. #63389	C₂²	semidirect
21. C₄⋊(C₉⋊C₈)	C₂²	semidirect
22. C₄×C₉⋊C₈	C₂²	semidirect
47. C₂×C₉⋊C₈	D₄	semidirect
78. C₂×C₄×C₉	M₁₆	semidirect
88. C₉⋊C₈	C₂×D₄	semidirect
128. C₄×C₉	C₂×M₁₆	semidirect
169. C₂×C₉	#436	semidirect
209. C₉	C₂×#436	semidirect
221. C₂×C₄	#3667	semidirect
247. C₄	C₂×#3667	semidirect
253. C₂	#10588689	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₄×D₄×C₉	2
C₂²×#3667	2
C₂×#63389	2
#10588689	2
C₂×C₄⋊(C₉⋊C₈)	2
C₂×C₄×C₉⋊C₈	2
C₂×#5012	3

Subgroups — C2×#10588689 #11859265

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#10588689 #11859265