Subgroups — M₁₆×Dic₁₈ #11861501

Order 1152.

582 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Dic₁₈	C₂	1	1	yes	maximal, characteristic
2	C₄×C₉×M₁₆	C₂	1	1	yes	maximal, characteristic
3	#10588802	C₂	1	2	yes	maximal
4	M₁₆×Dic₉	C₂	1	2	yes	maximal
5	#10588604	C₂	1	1	yes	maximal, characteristic
6	#10588381	C₂	1	4	yes	maximal, complemented
7	#10588359	C₂	1	2	yes	maximal, complemented
8	C₈×Dic₁₈	C₂	1	2	yes	maximal, complemented
9	M₁₆×Dic₆	—	3	1	no	maximal
10	C₂²×Dic₁₈	C₄	1	1	yes	characteristic
11	C₂×C₉×M₁₆	C₂²	1	1	yes	characteristic
12	C₂×C₉×M₁₆	C₂²	1	1	yes	characteristic
13	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊Dic₉	C₄	1	1	yes	characteristic
15	C₂×Dic₉⋊C₄	C₄	1	2	yes	—
16	C₂×C₄×Dic₉	C₂²	1	2	yes	—
17	C₂×#3667	C₂²	1	2	yes	—
18	C₄×Dic₁₈	C₄	1	1	yes	characteristic
19	C₄×Dic₁₈	C₄	1	2	yes	—
20	C₄×Dic₁₈	C₄	1	1	yes	characteristic
21	C₄×Dic₁₈	C₂²	1	1	yes	characteristic
22	C₄×Dic₁₈	C₂²	1	1	yes	characteristic
23	C₄×Dic₁₈	C₂²	1	2	yes	—
24	C₉×#97	C₂²	1	2	yes	—
25	C₄×C₈×C₉	C₂²	1	2	yes	—
26	#63375	C₂²	1	4	yes	—
27	Dic₉⋊C₈	C₂²	1	4	yes	—
28	Dic₉⋊C₈	C₂²	1	4	yes	—
29	C₈×Dic₉	C₂²	1	4	yes	—
30	C₄⋊(C₉⋊C₈)	C₂²	1	2	yes	—
31	C₄⋊(C₉⋊C₈)	C₂²	1	2	yes	—
32	C₂×C₄×Dic₆	—	3	1	no	—
33	C₃×C₄×M₁₆	D₃	1	1	yes	characteristic
34	#5118	—	3	2	no	—
35	Dic₃×M₁₆	—	3	2	no	—
36	#4924	—	3	1	no	—
37	#4701	—	3	4	no	—
38	#4679	—	3	2	no	—
39	C₈×Dic₆	—	3	2	no	—
40	C₂²×C₄×C₉	C₂×C₄	1	1	yes	characteristic
41	C₂²×C₄×C₉	C₂×C₄	1	1	yes	characteristic
42	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
43	C₂²×Dic₉	C₂×C₄	1	2	yes	—
44	C₂²×Dic₉	C₂×C₄	1	2	yes	—
45	C₂×Dic₁₈	—	2	4	no	—
46	C₂×Dic₁₈	C₂×C₄	1	1	yes	characteristic
47	C₂×Dic₁₈	C₂×C₄	1	2	yes	—
48	C₂×Dic₁₈	C₂×C₄	1	1	yes	characteristic
49	C₉×M₁₆	—	2	2	no	—
50	C₉×M₁₆	Q₈	1	4	yes	complemented
51	C₂×C₈×C₉	C₂³	1	2	yes	—
52	C₂×C₈×C₉	C₂³	1	2	yes	—
53	C₄²×C₉	C₂×C₄	1	1	yes	characteristic
54	C₄²×C₉	C₂×C₄	1	1	yes	characteristic
55	C₄²×C₉	C₂³	1	1	yes	characteristic
56	C₄²×C₉	C₂³	1	1	yes	characteristic
57	C₄⋊Dic₉	C₂×C₄	1	2	yes	—
58	C₄⋊Dic₉	C₂×C₄	1	1	yes	characteristic
59	C₄⋊Dic₉	C₂×C₄	1	1	yes	characteristic
60	Dic₉⋊C₄	C₂×C₄	1	2	yes	—
61	Dic₉⋊C₄	C₂×C₄	1	2	yes	—
62	Dic₉⋊C₄	C₂×C₄	1	2	yes	—
63	Dic₉⋊C₄	C₂×C₄	1	2	yes	—
64	C₄×Dic₉	C₂×C₄	1	2	yes	—
65	C₄×Dic₉	C₂×C₄	1	2	yes	—
66	C₄×Dic₉	C₂³	1	2	yes	—
67	C₄×Dic₉	C₂³	1	2	yes	—
68	#3667	—	2	4	no	—
69	C₂×C₉⋊C₈	C₂³	1	4	yes	—
70	Q₈×M₁₆	—	9	1	no	Sylow
71	C₂²×Dic₆	—	3	1	no	—
72	C₂×C₃×M₁₆	C₂×D₃	1	1	yes	characteristic
73	C₂×C₃×M₁₆	C₂×D₃	1	1	yes	characteristic
74	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
75	C₂×C₄⋊Dic₃	—	3	1	no	—
76	C₂×Dic₃⋊C₄	—	3	2	no	—
77	C₂×C₄×Dic₃	—	3	2	no	—
78	C₂×#208	—	3	2	no	—
79	C₄×Dic₆	—	3	1	no	—
80	C₄×Dic₆	—	3	2	no	—
81	C₄×Dic₆	—	3	1	no	—
82	C₄×Dic₆	—	3	1	no	—
83	C₄×Dic₆	—	3	1	no	—
84	C₄×Dic₆	—	3	2	no	—
85	C₃×#97	C₂×D₃	1	2	yes	—
86	C₃×C₄×C₈	C₂×D₃	1	2	yes	—
87	#815	—	3	4	no	—
88	Dic₃⋊C₈	—	3	4	no	—
89	Dic₃⋊C₈	—	3	4	no	—
90	C₈×Dic₃	—	3	4	no	—
91	C₄⋊(C₃⋊C₈)	—	3	2	no	—
92	C₄⋊(C₃⋊C₈)	—	3	2	no	—
93	C₂³×C₉	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₄×C₉	—	2	2	no	—
95	C₂×C₄×C₉	—	4	1	no	—
96	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
97	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
98	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
99	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
100	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
101	C₂×C₄×C₉	C₂×Q₈	1	2	yes	—
102	C₂×C₄×C₉	#41	1	2	yes	—
103	C₂×C₄×C₉	C₂⁴	1	1	yes	characteristic
104	C₂×Dic₉	—	2	4	no	—
105	C₂×Dic₉	—	4	2	no	—
106	C₂×Dic₉	C₂²×C₄	1	2	yes	—
107	C₂×Dic₉	C₂²×C₄	1	2	yes	—
108	C₂×Dic₉	C₂²×C₄	1	2	yes	—
109	C₂×Dic₉	C₂²×C₄	1	2	yes	—
110	Dic₁₈	—	2	2	no	—
111	Dic₁₈	—	2	4	no	—
112	Dic₁₈	M₁₆	1	4	yes	direct product factor
113	C₈×C₉	—	2	2	no	—
114	C₈×C₉	C₂×Q₈	1	4	yes	complemented
115	C₉⋊C₈	—	2	4	no	—
116	C₂×C₄×Q₈	—	9	1	no	—
117	#446	—	9	4	no	—
118	#446	—	9	2	no	—
119	C₈×Q₈	—	9	2	no	—
120	#423	—	9	2	no	—
121	#423	—	9	1	no	—
122	C₄×M₁₆	—	9	2	no	—
123	C₄×M₁₆	D₉	1	1	yes	characteristic
124	C₂²×C₃×C₄	C₄×D₃	1	1	yes	characteristic
125	C₂²×C₃×C₄	C₄×D₃	1	1	yes	characteristic
126	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
127	C₂²×Dic₃	—	3	2	no	—
128	C₂²×Dic₃	—	3	2	no	—
129	C₂×Dic₆	—	3	1	no	—
130	C₂×Dic₆	—	3	2	no	—
131	C₂×Dic₆	—	3	1	no	—
132	C₂×Dic₆	—	6	4	no	—
133	C₃×M₁₆	—	2	2	no	—
134	C₃×M₁₆	Dic₆	1	4	yes	—
135	C₂×C₃×C₈	C₂²×D₃	1	2	yes	—
136	C₂×C₃×C₈	C₂²×D₃	1	2	yes	—
137	C₃×C₄²	C₄×D₃	1	1	yes	characteristic
138	C₃×C₄²	C₄×D₃	1	1	yes	characteristic
139	C₃×C₄²	C₂²×D₃	1	1	yes	characteristic
140	C₃×C₄²	C₂²×D₃	1	1	yes	characteristic
141	C₄⋊Dic₃	—	3	2	no	—
142	C₄⋊Dic₃	—	3	1	no	—
143	C₄⋊Dic₃	—	3	1	no	—
144	Dic₃⋊C₄	—	3	2	no	—
145	Dic₃⋊C₄	—	3	2	no	—
146	Dic₃⋊C₄	—	3	2	no	—
147	Dic₃⋊C₄	—	3	2	no	—
148	C₄×Dic₃	—	3	2	no	—
149	C₄×Dic₃	—	3	2	no	—
150	C₄×Dic₃	—	3	2	no	—
151	C₄×Dic₃	—	3	2	no	—
152	#208	—	6	4	no	—
153	C₂×C₃⋊C₈	—	3	4	no	—
154	C₂²×C₉	—	2	1	no	—
155	C₂²×C₉	—	2	1	no	—
156	C₂²×C₉	C₄×Q₈	1	2	yes	—
157	C₂²×C₉	C₂³×C₄	1	1	yes	characteristic
158	C₄×C₉	—	2	1	no	—
159	C₄×C₉	—	2	1	no	—
160	C₄×C₉	—	2	1	no	—
161	C₄×C₉	C₄×Q₈	1	2	yes	—
162	C₄×C₉	C₂×M₁₆	1	2	yes	—
163	C₄×C₉	C₂²×Q₈	1	1	yes	characteristic
164	C₄×C₉	C₂×#41	1	1	yes	characteristic
165	Dic₉	—	2	2	no	—
166	Dic₉	—	2	2	no	—
167	Dic₉	—	2	2	no	—
168	Dic₉	C₂×M₁₆	1	4	yes	—
169	C₂²×Q₈	—	9	1	no	—
170	C₂×M₁₆	—	9	2	no	—
171	C₂×M₁₆	C₂×D₉	1	1	yes	characteristic
172	C₂×M₁₆	C₂×D₉	1	1	yes	characteristic
173	C₄×Q₈	—	9	1	no	—
174	C₄×Q₈	—	9	2	no	—
175	C₄×Q₈	—	9	1	no	—
176	C₄×Q₈	—	9	1	no	—
177	C₄×Q₈	—	9	2	no	—
178	C₄×Q₈	—	9	1	no	—
179	C₂×C₄⋊C₄	—	9	2	no	—
180	C₂×C₄⋊C₄	—	9	1	no	—
181	C₂×C₄²	—	9	2	no	—
182	C₂×C₄²	C₂×D₉	1	1	yes	characteristic
183	C₄⋊C₈	—	9	4	no	—
184	C₄⋊C₈	—	9	4	no	—
185	C₄⋊C₈	—	9	2	no	—
186	C₄⋊C₈	—	9	2	no	—
187	#97	—	9	4	no	—
188	#97	C₂×D₉	1	2	yes	—
189	C₄×C₈	—	9	4	no	—
190	C₄×C₈	C₂×D₉	1	2	yes	—
191	C₂³×C₃	C₂×C₄×D₃	1	1	yes	characteristic
192	C₂×C₃×C₄	—	2	2	no	—
193	C₂×C₃×C₄	—	4	1	no	—
194	C₂×C₃×C₄	C₂×Dic₆	1	2	yes	—
195	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
196	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
197	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
198	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
199	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
200	C₂×C₃×C₄	#235	1	2	yes	—
201	C₂×C₃×C₄	C₂³×D₃	1	1	yes	characteristic
202	C₂×Dic₃	—	3	2	no	—
203	C₂×Dic₃	—	3	2	no	—
204	C₂×Dic₃	—	3	2	no	—
205	C₂×Dic₃	—	3	2	no	—
206	C₂×Dic₃	—	6	4	no	—
207	C₂×Dic₃	—	12	2	no	—
208	Dic₆	—	3	4	no	—
209	Dic₆	—	6	4	no	—
210	Dic₆	—	6	2	no	—
211	C₃×C₈	—	2	2	no	—
212	C₃×C₈	C₂×Dic₆	1	4	yes	—
213	C₃⋊C₈	—	6	4	no	—
214	C₂×C₉	—	2	2	no	—
215	C₂×C₉	C₂×C₄×Q₈	1	1	yes	characteristic
216	C₂×C₉	C₂²×M₁₆	1	1	yes	characteristic
217	C₂×C₉	#568	1	1	yes	characteristic
218	C₂×Q₈	—	9	1	no	—
219	C₂×Q₈	—	9	2	no	—
220	C₂×Q₈	—	9	1	no	—
221	C₂×Q₈	—	18	4	no	—
222	C₂²×C₄	—	9	2	no	—
223	C₂²×C₄	—	9	2	no	—
224	C₂²×C₄	C₄×D₉	1	1	yes	characteristic
225	C₂²×C₄	C₄×D₉	1	1	yes	characteristic
226	C₂²×C₄	C₂²×D₉	1	1	yes	characteristic
227	M₁₆	—	2	2	no	—
228	M₁₆	—	18	4	no	—
229	M₁₆	Dic₁₈	1	4	yes	direct product factor
230	C₂×C₈	—	9	4	no	—
231	C₂×C₈	C₂²×D₉	1	2	yes	—
232	C₂×C₈	C₂²×D₉	1	2	yes	—
233	C₄⋊C₄	—	9	2	no	—
234	C₄⋊C₄	—	9	2	no	—
235	C₄⋊C₄	—	9	2	no	—
236	C₄⋊C₄	—	9	2	no	—
237	C₄⋊C₄	—	9	2	no	—
238	C₄⋊C₄	—	9	1	no	—
239	C₄⋊C₄	—	9	1	no	—
240	C₄²	—	9	2	no	—
241	C₄²	—	9	2	no	—
242	C₄²	—	9	2	no	—
243	C₄²	—	9	2	no	—
244	C₄²	C₄×D₉	1	1	yes	characteristic
245	C₄²	C₄×D₉	1	1	yes	characteristic
246	C₄²	C₂²×D₉	1	1	yes	characteristic
247	C₄²	C₂²×D₉	1	1	yes	characteristic
248	C₂²×C₃	—	2	1	no	—
249	C₂²×C₃	—	2	1	no	—
250	C₂²×C₃	C₄×Dic₆	1	2	yes	—
251	C₂²×C₃	C₂²×C₄×D₃	1	1	yes	characteristic
252	C₃×C₄	—	2	1	no	—
253	C₃×C₄	—	2	1	no	—
254	C₃×C₄	—	2	1	no	—
255	C₃×C₄	C₄×Dic₆	1	2	yes	—
256	C₃×C₄	D₃×M₁₆	1	2	yes	—
257	C₃×C₄	C₂²×Dic₆	1	1	yes	characteristic
258	C₃×C₄	C₂×#235	1	1	yes	characteristic
259	Dic₃	—	3	4	no	—
260	Dic₃	—	6	2	no	—
261	Dic₃	—	6	2	no	—
262	Dic₃	—	6	2	no	—
263	C₉	Q₈×M₁₆	1	1	yes	characteristic, complemented, Sylow
264	C₂³	C₂×C₄×D₉	1	1	yes	characteristic
265	Q₈	—	9	4	no	—
266	Q₈	—	18	4	no	—
267	Q₈	—	18	2	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	4	1	no	—
270	C₂×C₄	—	9	2	no	—
271	C₂×C₄	—	9	2	no	—
272	C₂×C₄	—	9	2	no	—
273	C₂×C₄	—	9	2	no	—
274	C₂×C₄	—	18	4	no	—
275	C₂×C₄	—	36	2	no	—
276	C₂×C₄	C₂×Dic₁₈	1	2	yes	—
277	C₂×C₄	C₂×C₄×D₉	1	1	yes	characteristic
278	C₂×C₄	C₂×C₄×D₉	1	1	yes	characteristic
279	C₂×C₄	C₂×C₄×D₉	1	1	yes	characteristic
280	C₂×C₄	C₂×C₄×D₉	1	1	yes	characteristic
281	C₂×C₄	C₂×C₄×D₉	1	1	yes	characteristic
282	C₂×C₄	#3697	1	2	yes	—
283	C₂×C₄	C₂³×D₉	1	1	yes	characteristic
284	C₈	—	2	2	no	—
285	C₈	—	18	4	no	—
286	C₈	C₂×Dic₁₈	1	4	yes	complemented
287	C₂×C₃	—	2	2	no	—
288	C₂×C₃	C₂×C₄×Dic₆	1	1	yes	characteristic
289	C₂×C₃	C₂×D₃×M₁₆	1	1	yes	characteristic
290	C₂×C₃	#5745	1	1	yes	characteristic
291	C₂²	—	2	1	no	—
292	C₂²	—	2	1	no	—
293	C₂²	C₄×Dic₁₈	1	2	yes	—
294	C₂²	C₂²×C₄×D₉	1	1	yes	characteristic
295	C₄	—	2	1	no	—
296	C₄	—	2	1	no	—
297	C₄	—	2	1	no	—
298	C₄	—	9	4	no	—
299	C₄	—	18	2	no	—
300	C₄	—	18	2	no	—
301	C₄	—	18	2	no	—
302	C₄	C₄×Dic₁₈	1	2	yes	—
303	C₄	M₁₆×D₉	1	2	yes	—
304	C₄	C₂²×Dic₁₈	1	1	yes	characteristic
305	C₄	C₂×#3697	1	1	yes	characteristic
306	C₃	M₁₆×Dic₆	1	1	yes	characteristic
307	C₂	—	2	2	no	—
308	C₂	C₂×C₄×Dic₁₈	1	1	yes	characteristic
309	C₂	C₂×M₁₆×D₉	1	1	yes	characteristic
310	C₂	#10589847	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂³×D₉
commutator	C₂²×C₉	C₂³×C₄
Frattini	C₂×C₃×C₄	C₂³×D₃
Fitting	C₄×C₉×M₁₆	C₂
socle	C₂²×C₃	C₂²×C₄×D₃
prime radical	C₂³×C₃	C₂×C₄×D₃
cosocle	C₂×C₄×C₉	C₂⁴
hypercenter	C₄×M₁₆	D₉
hypocenter	C₉	Q₈×M₁₆
LCS	G ▷ C₂²×C₉ ▷ C₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄×M₁₆

Complements

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

N	H ≅ G/N	type
6. #10588381	C₂	semidirect
7. #10588359	C₂	semidirect
8. C₈×Dic₁₈	C₂	semidirect
50. C₉×M₁₆	Q₈	semidirect
112. Dic₁₈	M₁₆	direct (N × H)
114. C₈×C₉	C₂×Q₈	semidirect
229. M₁₆	Dic₁₈	direct (N × H)
263. C₉	Q₈×M₁₆	semidirect
286. C₈	C₂×Dic₁₈	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×Dic₁₈	2
C₄×C₉×M₁₆	2
#10588802	2
M₁₆×Dic₉	2
#10588604	2
#10588381	2
#10588359	2
C₈×Dic₁₈	2
M₁₆×Dic₆	3

Subgroups — M16×Dic18 #11861501

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — M₁₆×Dic₁₈ #11861501