Subgroups — C₂×#10588235 #11783173

Order 1152.

732 conjugacy classes of subgroups.

The subgroup lattice has 321 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	C₂×#63488	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₄×Dic₉	C₂	1	1	yes	maximal, characteristic
4	#10588235	C₂	1	4	yes	maximal, direct product factor
5	C₂×#4555	—	3	1	no	maximal
6	C₂³×C₄×C₉	C₂²	1	1	yes	characteristic
7	C₂³×Dic₉	—	2	1	no	—
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×#3676	—	2	1	no	—
14	#63488	—	2	1	no	—
15	#63488	—	2	1	no	—
16	#63488	C₄	1	2	yes	complemented
17	#63488	C₂²	1	2	yes	—
18	C₂×C₄⋊Dic₉	C₄	1	2	yes	complemented
19	C₂×C₄×Dic₉	—	2	1	no	—
20	C₂×C₄×Dic₉	—	2	2	no	—
21	C₂×C₄×Dic₉	—	2	1	no	—
22	C₂×C₄×Dic₉	C₄	1	4	yes	complemented
23	C₂×C₄×Dic₉	C₂²	1	2	yes	—
24	C₂²×C₃×M₁₆	D₃	1	1	yes	characteristic
25	C₂×#926	—	3	1	no	—
26	C₂²×C₄×Dic₃	—	3	1	no	—
27	#4555	—	3	4	no	—
28	C₂⁴×C₉	D₄	1	1	yes	characteristic
29	C₂²×C₄×C₉	C₂×C₄	1	2	yes	—
30	C₂²×C₄×C₉	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₄×C₉	C₂×C₄	1	2	yes	—
32	C₂²×C₄×C₉	C₂×C₄	1	1	yes	characteristic
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	2	yes	—
37	C₂²×C₄×C₉	Q₈	1	2	yes	—
38	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
39	C₂²×Dic₉	—	2	8	no	—
40	C₂²×Dic₉	—	2	4	no	—
41	C₂²×Dic₉	—	2	2	no	—
42	C₂²×Dic₉	—	2	2	no	—
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	#3676	—	2	2	no	—
49	#3676	—	2	2	no	—
50	C₄⋊Dic₉	—	2	2	no	—
51	C₄⋊Dic₉	C₂×C₄	1	4	yes	complemented
52	C₄×Dic₉	—	2	2	no	—
53	C₄×Dic₉	—	2	4	no	—
54	C₄×Dic₉	—	2	2	no	—
55	C₄×Dic₉	—	2	2	no	—
56	C₄×Dic₉	C₂×C₄	1	4	yes	complemented
57	C₂×#339	—	9	1	no	Sylow
58	C₂³×C₃×C₄	C₂×D₃	1	1	yes	characteristic
59	C₂³×Dic₃	—	6	1	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×#217	—	6	1	no	—
66	#926	—	3	2	no	—
67	#926	—	3	2	no	—
68	#926	—	6	1	no	—
69	#926	—	6	1	no	—
70	C₂×C₄⋊Dic₃	—	3	2	no	—
71	C₂×C₄×Dic₃	—	3	4	no	—
72	C₂×C₄×Dic₃	—	3	2	no	—
73	C₂×C₄×Dic₃	—	6	1	no	—
74	C₂×C₄×Dic₃	—	6	2	no	—
75	C₂×C₄×Dic₃	—	6	1	no	—
76	C₂³×C₉	—	2	4	no	—
77	C₂³×C₉	C₂²⋊C₄	1	1	yes	characteristic
78	C₂³×C₉	C₂²⋊C₄	1	2	yes	—
79	C₂³×C₉	C₂²⋊C₄	1	1	yes	characteristic
80	C₂³×C₉	C₄⋊C₄	1	2	yes	—
81	C₂³×C₉	C₂×D₄	1	1	yes	characteristic
82	C₂×C₄×C₉	C₄²	1	2	yes	—
83	C₂×C₄×C₉	C₄²	1	2	yes	—
84	C₂×C₄×C₉	C₂²⋊C₄	1	2	yes	—
85	C₂×C₄×C₉	C₂²⋊C₄	1	1	yes	characteristic
86	C₂×C₄×C₉	C₂²⋊C₄	1	1	yes	characteristic
87	C₂×C₄×C₉	C₂²⋊C₄	1	2	yes	—
88	C₂×C₄×C₉	C₂²⋊C₄	1	2	yes	—
89	C₂×C₄×C₉	C₄⋊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	C₂²×C₄	1	1	yes	characteristic
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×Q₈	1	1	yes	characteristic
100	C₂×Dic₉	—	2	16	no	—
101	C₂×Dic₉	—	2	8	no	—
102	C₂×Dic₉	—	2	4	no	—
103	C₂×Dic₉	—	2	4	no	—
104	C₂×Dic₉	—	4	4	no	—
105	C₈×C₉	—	2	4	no	—
106	C₂²×M₁₆	D₉	1	1	yes	characteristic
107	C₂×#117	—	9	1	no	—
108	C₂²×C₄²	—	9	1	no	—
109	#339	—	9	4	no	—
110	C₂⁴×C₃	D₁₂	1	1	yes	characteristic
111	C₂²×C₃×C₄	Dic₆	1	2	yes	—
112	C₂²×C₃×C₄	C₄×D₃	1	2	yes	—
113	C₂²×C₃×C₄	C₄×D₃	1	1	yes	characteristic
114	C₂²×C₃×C₄	C₄×D₃	1	1	yes	characteristic
115	C₂²×C₃×C₄	D₁₂	1	1	yes	characteristic
116	C₂²×C₃×C₄	C₂×Dic₃	1	2	yes	—
117	C₂²×C₃×C₄	#67	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	C₂²×D₃	1	1	yes	characteristic
121	C₂²×Dic₃	—	6	8	no	—
122	C₂²×Dic₃	—	6	4	no	—
123	C₂²×Dic₃	—	6	2	no	—
124	C₂²×Dic₃	—	6	2	no	—
125	C₃×M₁₆	—	2	4	no	—
126	C₃×M₁₆	—	2	2	no	—
127	C₃×M₁₆	C₂×Dic₃	1	4	yes	—
128	C₂×C₃×C₈	—	2	4	no	—
129	C₂×C₃×C₈	—	2	2	no	—
130	#217	—	6	2	no	—
131	#217	—	6	2	no	—
132	C₄⋊Dic₃	—	3	4	no	—
133	C₄⋊Dic₃	—	6	2	no	—
134	C₄×Dic₃	—	3	4	no	—
135	C₄×Dic₃	—	6	4	no	—
136	C₄×Dic₃	—	6	2	no	—
137	C₄×Dic₃	—	6	2	no	—
138	C₄×Dic₃	—	6	2	no	—
139	C₂²×C₉	—	2	4	no	—
140	C₂²×C₉	—	2	4	no	—
141	C₂²×C₉	—	2	2	no	—
142	C₂²×C₉	—	2	2	no	—
143	C₂²×C₉	#95	1	2	yes	—
144	C₂²×C₉	#95	1	2	yes	—
145	C₂²×C₉	#104	1	4	yes	—
146	C₂²×C₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
147	C₂²×C₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
148	C₂²×C₉	C₂×C₄⋊C₄	1	1	yes	characteristic
149	C₄×C₉	#95	1	2	yes	—
150	C₄×C₉	#95	1	2	yes	—
151	C₄×C₉	C₂×C₄²	1	1	yes	characteristic
152	C₄×C₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
153	C₄×C₉	C₂×C₄⋊C₄	1	1	yes	characteristic
154	C₄×C₉	C₂×C₄⋊C₄	1	1	yes	characteristic
155	Dic₉	—	2	8	no	—
156	Dic₉	—	4	4	no	—
157	C₂³×C₄	—	18	1	no	—
158	C₂³×C₄	C₂×D₉	1	1	yes	characteristic
159	C₂×M₁₆	—	2	2	no	—
160	C₂×M₁₆	—	2	1	no	—
161	C₂×M₁₆	Dic₉	1	4	yes	complemented
162	C₂×M₁₆	C₂×D₉	1	2	yes	—
163	C₂²×C₈	—	2	1	no	—
164	#117	—	9	2	no	—
165	#117	—	9	2	no	—
166	#117	—	18	1	no	—
167	#117	—	18	1	no	—
168	C₂×C₄⋊C₄	—	9	2	no	—
169	C₂×C₂²⋊C₄	—	18	1	no	—
170	C₂×C₄²	—	9	4	no	—
171	C₂×C₄²	—	9	2	no	—
172	C₂×C₄²	—	18	1	no	—
173	C₂×C₄²	—	18	2	no	—
174	C₂×C₄²	—	18	1	no	—
175	C₂³×C₃	—	2	4	no	—
176	C₂³×C₃	C₄⋊Dic₃	1	2	yes	—
177	C₂³×C₃	(C₂×D₃)⋊C₄	1	1	yes	characteristic
178	C₂³×C₃	(C₂×D₃)⋊C₄	1	2	yes	—
179	C₂³×C₃	(C₂×D₃)⋊C₄	1	1	yes	characteristic
180	C₂³×C₃	C₂×D₁₂	1	1	yes	characteristic
181	C₂×C₃×C₄	C₄×Dic₃	1	2	yes	—
182	C₂×C₃×C₄	C₄×Dic₃	1	2	yes	—
183	C₂×C₃×C₄	Dic₃⋊C₄	1	2	yes	—
184	C₂×C₃×C₄	Dic₃⋊C₄	1	2	yes	—
185	C₂×C₃×C₄	Dic₃⋊C₄	1	2	yes	—
186	C₂×C₃×C₄	Dic₃⋊C₄	1	2	yes	—
187	C₂×C₃×C₄	C₄⋊Dic₃	1	2	yes	—
188	C₂×C₃×C₄	(C₂×D₃)⋊C₄	1	2	yes	—
189	C₂×C₃×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
190	C₂×C₃×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
191	C₂×C₃×C₄	#217	1	2	yes	—
192	C₂×C₃×C₄	#217	1	2	yes	—
193	C₂×C₃×C₄	C₂×Dic₆	1	1	yes	characteristic
194	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
195	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
196	C₂×C₃×C₄	C₂²×Dic₃	1	1	yes	characteristic
197	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
198	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
199	C₂×Dic₃	—	6	8	no	—
200	C₂×Dic₃	—	6	16	no	—
201	C₂×Dic₃	—	6	4	no	—
202	C₂×Dic₃	—	6	4	no	—
203	C₂×Dic₃	—	12	4	no	—
204	C₃×C₈	—	2	4	no	—
205	C₂×C₉	—	2	1	no	—
206	C₂×C₉	—	2	2	no	—
207	C₂×C₉	—	2	1	no	—
208	C₂×C₉	#339	1	4	yes	complemented
209	C₂×C₉	C₂×#95	1	1	yes	characteristic
210	C₂×C₉	C₂×#104	1	2	yes	—
211	C₂⁴	D₃₆	1	1	yes	characteristic
212	C₂²×C₄	—	18	8	no	—
213	C₂²×C₄	—	18	4	no	—
214	C₂²×C₄	—	18	2	no	—
215	C₂²×C₄	—	18	2	no	—
216	C₂²×C₄	Dic₁₈	1	2	yes	—
217	C₂²×C₄	C₄×D₉	1	2	yes	—
218	C₂²×C₄	C₄×D₉	1	1	yes	characteristic
219	C₂²×C₄	C₄×D₉	1	1	yes	characteristic
220	C₂²×C₄	D₃₆	1	1	yes	characteristic
221	C₂²×C₄	C₂×Dic₉	1	2	yes	—
222	C₂²×C₄	#611	1	2	yes	—
223	C₂²×C₄	#611	1	1	yes	characteristic
224	C₂²×C₄	#611	1	1	yes	characteristic
225	C₂²×C₄	C₂²×D₉	1	1	yes	characteristic
226	M₁₆	—	2	4	no	—
227	M₁₆	—	2	2	no	—
228	M₁₆	C₂×Dic₉	1	4	yes	complemented
229	C₂×C₈	—	2	4	no	—
230	C₂×C₈	—	2	2	no	—
231	C₄⋊C₄	—	9	4	no	—
232	C₄⋊C₄	—	18	2	no	—
233	C₂²⋊C₄	—	18	2	no	—
234	C₂²⋊C₄	—	18	2	no	—
235	C₄²	—	9	4	no	—
236	C₄²	—	18	4	no	—
237	C₄²	—	18	2	no	—
238	C₄²	—	18	2	no	—
239	C₄²	—	18	2	no	—
240	C₂²×C₃	—	2	2	no	—
241	C₂²×C₃	—	2	2	no	—
242	C₂²×C₃	—	2	4	no	—
243	C₂²×C₃	—	2	4	no	—
244	C₂²×C₃	#825	1	4	yes	—
245	C₂²×C₃	C₃⋊#95	1	2	yes	—
246	C₂²×C₃	C₃⋊#95	1	2	yes	—
247	C₂²×C₃	C₂×C₄⋊Dic₃	1	1	yes	characteristic
248	C₂²×C₃	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
249	C₂²×C₃	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
250	C₃×C₄	C₃⋊#95	1	2	yes	—
251	C₃×C₄	C₃⋊#95	1	2	yes	—
252	C₃×C₄	C₂×C₄×Dic₃	1	1	yes	characteristic
253	C₃×C₄	C₂×Dic₃⋊C₄	1	1	yes	characteristic
254	C₃×C₄	C₂×Dic₃⋊C₄	1	1	yes	characteristic
255	C₃×C₄	C₂×#217	1	1	yes	characteristic
256	Dic₃	—	6	8	no	—
257	Dic₃	—	12	4	no	—
258	C₉	C₂×#339	1	1	yes	characteristic, complemented, Sylow
259	C₂³	—	2	4	no	—
260	C₂³	C₄⋊Dic₉	1	2	yes	—
261	C₂³	(C₂×D₉)⋊C₄	1	1	yes	characteristic
262	C₂³	(C₂×D₉)⋊C₄	1	2	yes	—
263	C₂³	(C₂×D₉)⋊C₄	1	1	yes	characteristic
264	C₂³	C₂×D₃₆	1	1	yes	characteristic
265	C₂×C₄	—	18	16	no	—
266	C₂×C₄	—	18	8	no	—
267	C₂×C₄	—	18	4	no	—
268	C₂×C₄	—	18	4	no	—
269	C₂×C₄	—	36	4	no	—
270	C₂×C₄	C₄×Dic₉	1	2	yes	—
271	C₂×C₄	C₄×Dic₉	1	2	yes	—
272	C₂×C₄	Dic₉⋊C₄	1	2	yes	—
273	C₂×C₄	Dic₉⋊C₄	1	2	yes	—
274	C₂×C₄	Dic₉⋊C₄	1	2	yes	—
275	C₂×C₄	Dic₉⋊C₄	1	2	yes	—
276	C₂×C₄	C₄⋊Dic₉	1	2	yes	—
277	C₂×C₄	(C₂×D₉)⋊C₄	1	2	yes	—
278	C₂×C₄	(C₂×D₉)⋊C₄	1	1	yes	characteristic
279	C₂×C₄	(C₂×D₉)⋊C₄	1	1	yes	characteristic
280	C₂×C₄	#3676	1	2	yes	—
281	C₂×C₄	#3676	1	2	yes	—
282	C₂×C₄	C₂×Dic₁₈	1	1	yes	characteristic
283	C₂×C₄	C₂×C₄×D₉	1	1	yes	characteristic
284	C₂×C₄	C₂×C₄×D₉	1	1	yes	characteristic
285	C₂×C₄	C₂²×Dic₉	1	1	yes	characteristic
286	C₂×C₄	C₂×#611	1	1	yes	characteristic
287	C₂×C₄	C₂×#611	1	1	yes	characteristic
288	C₈	—	2	4	no	—
289	C₂×C₃	—	2	1	no	—
290	C₂×C₃	—	2	2	no	—
291	C₂×C₃	—	2	1	no	—
292	C₂×C₃	#4555	1	4	yes	—
293	C₂×C₃	C₂×#825	1	2	yes	—
294	C₂×C₃	C₂×C₃⋊#95	1	1	yes	characteristic
295	C₂²	—	2	4	no	—
296	C₂²	—	2	4	no	—
297	C₂²	—	2	2	no	—
298	C₂²	—	2	2	no	—
299	C₂²	#63384	1	4	yes	—
300	C₂²	C₉⋊#95	1	2	yes	—
301	C₂²	C₉⋊#95	1	2	yes	—
302	C₂²	C₂×C₄⋊Dic₉	1	1	yes	characteristic
303	C₂²	C₂×(C₂×D₉)⋊C₄	1	1	yes	characteristic
304	C₂²	C₂×(C₂×D₉)⋊C₄	1	1	yes	characteristic
305	C₄	—	18	8	no	—
306	C₄	—	36	4	no	—
307	C₄	C₉⋊#95	1	2	yes	—
308	C₄	C₉⋊#95	1	2	yes	—
309	C₄	C₂×C₄×Dic₉	1	1	yes	characteristic
310	C₄	C₂×Dic₉⋊C₄	1	1	yes	characteristic
311	C₄	C₂×Dic₉⋊C₄	1	1	yes	characteristic
312	C₄	C₂×#3676	1	1	yes	characteristic
313	C₃	C₂×#4555	1	1	yes	characteristic
314	C₂	—	2	1	no	—
315	C₂	—	2	2	no	—
316	C₂	—	2	1	no	—
317	C₂	#10588235	1	4	yes	direct product factor
318	C₂	C₂×#63384	1	2	yes	—
319	C₂	C₂×C₉⋊#95	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	#611
commutator	C₄×C₉	C₂×C₄²
Frattini	C₂²×C₃×C₄	C₂²×D₃
Fitting	C₂²×C₉×M₁₆	C₂
socle	C₂³×C₃	(C₂×D₃)⋊C₄
prime radical	C₂⁴×C₃	D₁₂
cosocle	C₂²×C₄×C₉	C₂³
hypercenter	C₂²×M₁₆	D₉
hypocenter	C₉	C₂×#339
LCS	G ▷ C₄×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂²×M₁₆

Complements

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

N	H ≅ G/N	type
4. #10588235	C₂	direct (N × H)
10. C₂×C₉×M₁₆	C₄	semidirect
16. #63488	C₄	semidirect
18. C₂×C₄⋊Dic₉	C₄	semidirect
22. C₂×C₄×Dic₉	C₄	semidirect
45. C₉×M₁₆	C₂×C₄	semidirect
51. C₄⋊Dic₉	C₂×C₄	semidirect
56. C₄×Dic₉	C₂×C₄	semidirect
161. C₂×M₁₆	Dic₉	semidirect
208. C₂×C₉	#339	semidirect
228. M₁₆	C₂×Dic₉	semidirect
258. C₉	C₂×#339	semidirect
317. C₂	#10588235	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
C₂×#63488	2
C₂²×C₄×Dic₉	2
#10588235	2
C₂×#4555	3

Subgroups — C2×#10588235 #11783173

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#10588235 #11783173