Subgroups — C₂×#10588234 #11783389

Order 1152.

648 conjugacy classes of subgroups.

The subgroup lattice has 333 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₉×M₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄⋊Dic₉	C₂	1	1	yes	maximal, characteristic
3	C₂²×#3667	C₂	1	1	yes	maximal, characteristic
4	#10588234	C₂	1	4	yes	maximal, direct product factor
5	C₂×#4554	—	3	1	no	maximal
6	C₂³×C₄×C₉	C₂²	1	1	yes	characteristic
7	C₂³×Dic₉	C₄	1	2	yes	—
8	C₂×C₉×M₁₆	—	2	2	no	—
9	C₂×C₉×M₁₆	—	2	1	no	—
10	C₂×C₉×M₁₆	C₄	1	4	yes	complemented
11	C₂×C₉×M₁₆	C₂²	1	2	yes	—
12	C₂²×C₈×C₉	—	2	1	no	—
13	C₂×C₄⋊Dic₉	—	2	1	no	—
14	C₂×C₄⋊Dic₉	—	2	1	no	—
15	C₂×C₄⋊Dic₉	—	2	1	no	—
16	C₂×C₄⋊Dic₉	—	2	1	no	—
17	C₂×C₄⋊Dic₉	C₄	1	2	yes	—
18	C₂×C₄⋊Dic₉	C₂²	1	2	yes	—
19	C₂×#3667	—	2	2	no	—
20	C₂×#3667	—	2	1	no	—
21	C₂×#3667	C₄	1	4	yes	complemented
22	C₂×#3667	C₂²	1	2	yes	—
23	C₂²×C₉⋊C₈	—	2	1	no	—
24	C₂²×C₃×M₁₆	D₃	1	1	yes	characteristic
25	C₂²×C₄⋊Dic₃	—	3	1	no	—
26	C₂²×#208	—	3	1	no	—
27	#4554	—	3	4	no	—
28	C₂⁴×C₉	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₄×C₉	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₄×C₉	C₂×C₄	1	2	yes	—
31	C₂²×C₄×C₉	C₂×C₄	1	2	yes	—
32	C₂²×C₄×C₉	D₄	1	2	yes	—
33	C₂²×C₄×C₉	D₄	1	1	yes	characteristic
34	C₂²×C₄×C₉	D₄	1	1	yes	characteristic
35	C₂²×C₄×C₉	D₄	1	1	yes	characteristic
36	C₂²×C₄×C₉	D₄	1	1	yes	characteristic
37	C₂²×C₄×C₉	Q₈	1	2	yes	—
38	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
39	C₂²×Dic₉	—	2	2	no	—
40	C₂²×Dic₉	—	2	2	no	—
41	C₂²×Dic₉	—	4	4	no	—
42	C₂²×Dic₉	C₂×C₄	1	4	yes	—
43	C₉×M₁₆	—	2	4	no	—
44	C₉×M₁₆	—	2	2	no	—
45	C₉×M₁₆	C₂×C₄	1	4	yes	complemented
46	C₂×C₈×C₉	—	2	4	no	—
47	C₂×C₈×C₉	—	2	2	no	—
48	C₄⋊Dic₉	—	2	2	no	—
49	C₄⋊Dic₉	—	2	2	no	—
50	C₄⋊Dic₉	—	2	2	no	—
51	C₄⋊Dic₉	—	2	2	no	—
52	#3667	—	2	4	no	—
53	#3667	—	2	2	no	—
54	#3667	C₂×C₄	1	4	yes	complemented
55	C₂×C₉⋊C₈	—	2	4	no	—
56	C₂×C₉⋊C₈	—	2	2	no	—
57	C₂×#343	—	9	1	no	Sylow
58	C₂³×C₃×C₄	C₂×D₃	1	1	yes	characteristic
59	C₂³×Dic₃	—	3	2	no	—
60	C₂×C₃×M₁₆	—	2	2	no	—
61	C₂×C₃×M₁₆	—	2	1	no	—
62	C₂×C₃×M₁₆	Dic₃	1	4	yes	—
63	C₂×C₃×M₁₆	C₂×D₃	1	2	yes	—
64	C₂²×C₃×C₈	—	2	1	no	—
65	C₂×C₄⋊Dic₃	—	3	2	no	—
66	C₂×C₄⋊Dic₃	—	3	2	no	—
67	C₂×C₄⋊Dic₃	—	6	1	no	—
68	C₂×C₄⋊Dic₃	—	6	1	no	—
69	C₂×C₄⋊Dic₃	—	6	1	no	—
70	C₂×C₄⋊Dic₃	—	6	1	no	—
71	C₂×#208	—	3	4	no	—
72	C₂×#208	—	3	2	no	—
73	C₂×#208	—	6	1	no	—
74	C₂×#208	—	6	2	no	—
75	C₂²×C₃⋊C₈	—	6	1	no	—
76	C₂³×C₉	—	2	2	no	—
77	C₂³×C₉	—	2	2	no	—
78	C₂³×C₉	C₄²	1	2	yes	—
79	C₂³×C₉	C₂²⋊C₄	1	1	yes	characteristic
80	C₂³×C₉	C₂²⋊C₄	1	1	yes	characteristic
81	C₂³×C₉	C₄⋊C₄	1	2	yes	—
82	C₂³×C₉	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₄×C₉	C₄²	1	2	yes	—
84	C₂×C₄×C₉	C₂²⋊C₄	1	1	yes	characteristic
85	C₂×C₄×C₉	C₂²⋊C₄	1	1	yes	characteristic
86	C₂×C₄×C₉	C₂²⋊C₄	1	2	yes	—
87	C₂×C₄×C₉	C₂²⋊C₄	1	2	yes	—
88	C₂×C₄×C₉	C₂²⋊C₄	1	2	yes	—
89	C₂×C₄×C₉	C₂²⋊C₄	1	2	yes	—
90	C₂×C₄×C₉	C₄⋊C₄	1	2	yes	—
91	C₂×C₄×C₉	C₄⋊C₄	1	2	yes	—
92	C₂×C₄×C₉	C₄⋊C₄	1	2	yes	—
93	C₂×C₄×C₉	C₄⋊C₄	1	2	yes	—
94	C₂×C₄×C₉	C₄⋊C₄	1	2	yes	—
95	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
96	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
97	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
98	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
99	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
100	C₂×C₄×C₉	C₂×Q₈	1	1	yes	characteristic
101	C₂×Dic₉	—	2	4	no	—
102	C₂×Dic₉	—	4	4	no	—
103	C₂×Dic₉	—	4	4	no	—
104	C₂×Dic₉	—	4	4	no	—
105	C₈×C₉	—	2	4	no	—
106	C₉⋊C₈	—	2	4	no	—
107	C₂²×M₁₆	—	9	1	no	—
108	C₂²×M₁₆	D₉	1	1	yes	characteristic
109	C₂²×C₄⋊C₄	—	9	1	no	—
110	#343	—	9	4	no	—
111	C₂⁴×C₃	C₄×D₃	1	1	yes	characteristic
112	C₂²×C₃×C₄	Dic₆	1	2	yes	—
113	C₂²×C₃×C₄	C₄×D₃	1	1	yes	characteristic
114	C₂²×C₃×C₄	C₄×D₃	1	2	yes	—
115	C₂²×C₃×C₄	D₁₂	1	1	yes	characteristic
116	C₂²×C₃×C₄	D₁₂	1	1	yes	characteristic
117	C₂²×C₃×C₄	C₂×Dic₃	1	2	yes	—
118	C₂²×C₃×C₄	#67	1	1	yes	characteristic
119	C₂²×C₃×C₄	#67	1	1	yes	characteristic
120	C₂²×C₃×C₄	#67	1	2	yes	—
121	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
122	C₂²×Dic₃	—	3	4	no	—
123	C₂²×Dic₃	—	6	2	no	—
124	C₂²×Dic₃	—	6	2	no	—
125	C₂²×Dic₃	—	12	4	no	—
126	C₃×M₁₆	—	2	4	no	—
127	C₃×M₁₆	—	2	2	no	—
128	C₃×M₁₆	C₂×Dic₃	1	4	yes	—
129	C₂×C₃×C₈	—	2	4	no	—
130	C₂×C₃×C₈	—	2	2	no	—
131	C₄⋊Dic₃	—	6	2	no	—
132	C₄⋊Dic₃	—	6	2	no	—
133	C₄⋊Dic₃	—	6	2	no	—
134	C₄⋊Dic₃	—	6	2	no	—
135	#208	—	3	4	no	—
136	#208	—	6	4	no	—
137	#208	—	6	2	no	—
138	C₂×C₃⋊C₈	—	6	4	no	—
139	C₂×C₃⋊C₈	—	6	2	no	—
140	C₂²×C₉	—	2	4	no	—
141	C₂²×C₉	—	2	4	no	—
142	C₂²×C₉	—	2	1	no	—
143	C₂²×C₉	—	2	1	no	—
144	C₂²×C₉	—	2	1	no	—
145	C₂²×C₉	—	2	1	no	—
146	C₂²×C₉	#95	1	2	yes	—
147	C₂²×C₉	#95	1	2	yes	—
148	C₂²×C₉	#100	1	2	yes	—
149	C₂²×C₉	#101	1	2	yes	—
150	C₂²×C₉	C₂×C₄²	1	1	yes	characteristic
151	C₂²×C₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
152	C₂²×C₉	C₂×C₄⋊C₄	1	1	yes	characteristic
153	C₄×C₉	#95	1	2	yes	—
154	C₄×C₉	#95	1	2	yes	—
155	C₄×C₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
156	C₄×C₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
157	C₄×C₉	C₂×C₄⋊C₄	1	1	yes	characteristic
158	C₄×C₉	C₂×C₄⋊C₄	1	1	yes	characteristic
159	Dic₉	—	4	4	no	—
160	C₂³×C₄	—	9	2	no	—
161	C₂³×C₄	C₂×D₉	1	1	yes	characteristic
162	C₂×M₁₆	—	2	2	no	—
163	C₂×M₁₆	—	2	1	no	—
164	C₂×M₁₆	—	9	4	no	—
165	C₂×M₁₆	—	9	2	no	—
166	C₂×M₁₆	—	18	1	no	—
167	C₂×M₁₆	—	18	2	no	—
168	C₂×M₁₆	Dic₉	1	4	yes	complemented
169	C₂×M₁₆	C₂×D₉	1	2	yes	—
170	C₂²×C₈	—	2	1	no	—
171	C₂²×C₈	—	18	1	no	—
172	C₂×C₄⋊C₄	—	9	2	no	—
173	C₂×C₄⋊C₄	—	9	2	no	—
174	C₂×C₄⋊C₄	—	18	1	no	—
175	C₂×C₄⋊C₄	—	18	1	no	—
176	C₂×C₄⋊C₄	—	18	1	no	—
177	C₂×C₄⋊C₄	—	18	1	no	—
178	C₂³×C₃	—	2	2	no	—
179	C₂³×C₃	—	2	2	no	—
180	C₂³×C₃	C₄×Dic₃	1	2	yes	—
181	C₂³×C₃	Dic₃⋊C₄	1	2	yes	—
182	C₂³×C₃	(C₂×D₃)⋊C₄	1	1	yes	characteristic
183	C₂³×C₃	(C₂×D₃)⋊C₄	1	1	yes	characteristic
184	C₂³×C₃	C₂×C₄×D₃	1	1	yes	characteristic
185	C₂×C₃×C₄	C₄×Dic₃	1	2	yes	—
186	C₂×C₃×C₄	Dic₃⋊C₄	1	2	yes	—
187	C₂×C₃×C₄	Dic₃⋊C₄	1	2	yes	—
188	C₂×C₃×C₄	Dic₃⋊C₄	1	2	yes	—
189	C₂×C₃×C₄	C₄⋊Dic₃	1	2	yes	—
190	C₂×C₃×C₄	C₄⋊Dic₃	1	2	yes	—
191	C₂×C₃×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
192	C₂×C₃×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
193	C₂×C₃×C₄	(C₂×D₃)⋊C₄	1	2	yes	—
194	C₂×C₃×C₄	(C₂×D₃)⋊C₄	1	2	yes	—
195	C₂×C₃×C₄	#217	1	2	yes	—
196	C₂×C₃×C₄	#217	1	2	yes	—
197	C₂×C₃×C₄	C₂×Dic₆	1	1	yes	characteristic
198	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
199	C₂×C₃×C₄	C₂×D₁₂	1	1	yes	characteristic
200	C₂×C₃×C₄	C₂²×Dic₃	1	1	yes	characteristic
201	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
202	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
203	C₂×Dic₃	—	6	4	no	—
204	C₂×Dic₃	—	12	4	no	—
205	C₂×Dic₃	—	12	4	no	—
206	C₂×Dic₃	—	12	4	no	—
207	C₃×C₈	—	2	4	no	—
208	C₃⋊C₈	—	6	4	no	—
209	C₂×C₉	—	2	2	no	—
210	C₂×C₉	—	2	1	no	—
211	C₂×C₉	—	2	1	no	—
212	C₂×C₉	#343	1	4	yes	complemented
213	C₂×C₉	C₂×#95	1	1	yes	characteristic
214	C₂×C₉	C₂×#100	1	1	yes	characteristic
215	C₂×C₉	C₂×#101	1	1	yes	characteristic
216	C₂⁴	C₄×D₉	1	1	yes	characteristic
217	C₂²×C₄	—	9	4	no	—
218	C₂²×C₄	—	18	2	no	—
219	C₂²×C₄	—	18	2	no	—
220	C₂²×C₄	—	36	4	no	—
221	C₂²×C₄	Dic₁₈	1	2	yes	—
222	C₂²×C₄	C₄×D₉	1	1	yes	characteristic
223	C₂²×C₄	C₄×D₉	1	2	yes	—
224	C₂²×C₄	D₃₆	1	1	yes	characteristic
225	C₂²×C₄	D₃₆	1	1	yes	characteristic
226	C₂²×C₄	C₂×Dic₉	1	2	yes	—
227	C₂²×C₄	#611	1	1	yes	characteristic
228	C₂²×C₄	#611	1	1	yes	characteristic
229	C₂²×C₄	#611	1	2	yes	—
230	C₂²×C₄	C₂²×D₉	1	1	yes	characteristic
231	M₁₆	—	2	4	no	—
232	M₁₆	—	2	2	no	—
233	M₁₆	—	9	4	no	—
234	M₁₆	—	18	4	no	—
235	M₁₆	—	18	2	no	—
236	M₁₆	C₂×Dic₉	1	4	yes	complemented
237	C₂×C₈	—	2	4	no	—
238	C₂×C₈	—	2	2	no	—
239	C₂×C₈	—	18	4	no	—
240	C₂×C₈	—	18	2	no	—
241	C₄⋊C₄	—	18	2	no	—
242	C₄⋊C₄	—	18	2	no	—
243	C₄⋊C₄	—	18	2	no	—
244	C₄⋊C₄	—	18	2	no	—
245	C₂²×C₃	—	2	1	no	—
246	C₂²×C₃	—	2	1	no	—
247	C₂²×C₃	—	2	1	no	—
248	C₂²×C₃	—	2	1	no	—
249	C₂²×C₃	—	2	4	no	—
250	C₂²×C₃	—	2	4	no	—
251	C₂²×C₃	#823	1	2	yes	—
252	C₂²×C₃	#824	1	2	yes	—
253	C₂²×C₃	C₃⋊#95	1	2	yes	—
254	C₂²×C₃	C₃⋊#95	1	2	yes	—
255	C₂²×C₃	C₂×C₄×Dic₃	1	1	yes	characteristic
256	C₂²×C₃	C₂×Dic₃⋊C₄	1	1	yes	characteristic
257	C₂²×C₃	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
258	C₃×C₄	C₃⋊#95	1	2	yes	—
259	C₃×C₄	C₃⋊#95	1	2	yes	—
260	C₃×C₄	C₂×Dic₃⋊C₄	1	1	yes	characteristic
261	C₃×C₄	C₂×C₄⋊Dic₃	1	1	yes	characteristic
262	C₃×C₄	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
263	C₃×C₄	C₂×#217	1	1	yes	characteristic
264	Dic₃	—	12	4	no	—
265	C₉	C₂×#343	1	1	yes	characteristic, complemented, Sylow
266	C₂³	—	2	2	no	—
267	C₂³	—	2	2	no	—
268	C₂³	C₄×Dic₉	1	2	yes	—
269	C₂³	Dic₉⋊C₄	1	2	yes	—
270	C₂³	(C₂×D₉)⋊C₄	1	1	yes	characteristic
271	C₂³	(C₂×D₉)⋊C₄	1	1	yes	characteristic
272	C₂³	C₂×C₄×D₉	1	1	yes	characteristic
273	C₂×C₄	—	18	4	no	—
274	C₂×C₄	—	36	4	no	—
275	C₂×C₄	—	36	4	no	—
276	C₂×C₄	—	36	4	no	—
277	C₂×C₄	C₄×Dic₉	1	2	yes	—
278	C₂×C₄	Dic₉⋊C₄	1	2	yes	—
279	C₂×C₄	Dic₉⋊C₄	1	2	yes	—
280	C₂×C₄	Dic₉⋊C₄	1	2	yes	—
281	C₂×C₄	C₄⋊Dic₉	1	2	yes	—
282	C₂×C₄	C₄⋊Dic₉	1	2	yes	—
283	C₂×C₄	(C₂×D₉)⋊C₄	1	2	yes	—
284	C₂×C₄	(C₂×D₉)⋊C₄	1	2	yes	—
285	C₂×C₄	(C₂×D₉)⋊C₄	1	1	yes	characteristic
286	C₂×C₄	(C₂×D₉)⋊C₄	1	1	yes	characteristic
287	C₂×C₄	#3676	1	2	yes	—
288	C₂×C₄	#3676	1	2	yes	—
289	C₂×C₄	C₂×Dic₁₈	1	1	yes	characteristic
290	C₂×C₄	C₂×C₄×D₉	1	1	yes	characteristic
291	C₂×C₄	C₂×D₃₆	1	1	yes	characteristic
292	C₂×C₄	C₂²×Dic₉	1	1	yes	characteristic
293	C₂×C₄	C₂×#611	1	1	yes	characteristic
294	C₂×C₄	C₂×#611	1	1	yes	characteristic
295	C₈	—	2	4	no	—
296	C₈	—	18	4	no	—
297	C₂×C₃	—	2	2	no	—
298	C₂×C₃	—	2	1	no	—
299	C₂×C₃	—	2	1	no	—
300	C₂×C₃	#4554	1	4	yes	—
301	C₂×C₃	C₂×#823	1	1	yes	characteristic
302	C₂×C₃	C₂×#824	1	1	yes	characteristic
303	C₂×C₃	C₂×C₃⋊#95	1	1	yes	characteristic
304	C₂²	—	2	4	no	—
305	C₂²	—	2	4	no	—
306	C₂²	—	2	1	no	—
307	C₂²	—	2	1	no	—
308	C₂²	—	2	1	no	—
309	C₂²	—	2	1	no	—
310	C₂²	#63382	1	2	yes	—
311	C₂²	#63383	1	2	yes	—
312	C₂²	C₉⋊#95	1	2	yes	—
313	C₂²	C₉⋊#95	1	2	yes	—
314	C₂²	C₂×C₄×Dic₉	1	1	yes	characteristic
315	C₂²	C₂×Dic₉⋊C₄	1	1	yes	characteristic
316	C₂²	C₂×(C₂×D₉)⋊C₄	1	1	yes	characteristic
317	C₄	—	36	4	no	—
318	C₄	C₉⋊#95	1	2	yes	—
319	C₄	C₉⋊#95	1	2	yes	—
320	C₄	C₂×Dic₉⋊C₄	1	1	yes	characteristic
321	C₄	C₂×C₄⋊Dic₉	1	1	yes	characteristic
322	C₄	C₂×(C₂×D₉)⋊C₄	1	1	yes	characteristic
323	C₄	C₂×#3676	1	1	yes	characteristic
324	C₃	C₂×#4554	1	1	yes	characteristic
325	C₂	—	2	2	no	—
326	C₂	—	2	1	no	—
327	C₂	—	2	1	no	—
328	C₂	#10588234	1	4	yes	direct product factor
329	C₂	C₂×#63382	1	1	yes	characteristic
330	C₂	C₂×#63383	1	1	yes	characteristic
331	C₂	C₂×C₉⋊#95	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	(C₂×D₉)⋊C₄
commutator	C₂²×C₉	C₂×C₄²
Frattini	C₂²×C₃×C₄	C₂²×D₃
Fitting	C₂²×C₉×M₁₆	C₂
socle	C₂³×C₃	(C₂×D₃)⋊C₄
prime radical	C₂⁴×C₃	C₄×D₃
cosocle	C₂²×C₄×C₉	C₂³
hypercenter	C₂²×M₁₆	D₉
hypocenter	C₉	C₂×#343
LCS	G ▷ C₂²×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂²×M₁₆

Complements

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

N	H ≅ G/N	type
4. #10588234	C₂	direct (N × H)
10. C₂×C₉×M₁₆	C₄	semidirect
21. C₂×#3667	C₄	semidirect
45. C₉×M₁₆	C₂×C₄	semidirect
54. #3667	C₂×C₄	semidirect
168. C₂×M₁₆	Dic₉	semidirect
212. C₂×C₉	#343	semidirect
236. M₁₆	C₂×Dic₉	semidirect
265. C₉	C₂×#343	semidirect
328. C₂	#10588234	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×C₉×M₁₆	2
C₂²×C₄⋊Dic₉	2
C₂²×#3667	2
#10588234	2
C₂×#4554	3

Subgroups — C2×#10588234 #11783389

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#10588234 #11783389