Subgroups — D₄×C₉×D₈ #11768000

Order 1152.

930 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₉×D₈	C₂	1	2	yes	maximal, complemented
2	D₄²×C₉	C₂	1	2	yes	maximal, complemented
3	C₉×#493	C₂	1	1	yes	maximal, characteristic, complemented
4	C₉×#466	C₂	1	2	yes	maximal, complemented
5	C₉×#459	C₂	1	2	yes	maximal, complemented
6	C₉×#447	C₂	1	4	yes	maximal, complemented
7	C₄×C₉×D₈	C₂	1	1	yes	maximal, characteristic, complemented
8	C₈×D₄×C₉	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃×D₄×D₈	C₃	1	1	yes	maximal, characteristic
10	C₂²×D₄×C₉	—	2	2	no	—
11	C₂²×D₄×C₉	C₂²	1	4	yes	complemented
12	C₂×C₉×D₈	—	2	8	no	—
13	C₂×C₉×D₈	C₂²	1	1	yes	characteristic
14	C₂×C₉×D₈	C₂²	1	4	yes	complemented
15	C₂×C₉×D₈	C₂²	1	2	yes	complemented
16	C₂×C₉×D₈	C₂²	1	2	yes	complemented
17	C₂²×C₈×C₉	C₂²	1	2	yes	complemented
18	C₉×#127	C₂²	1	2	yes	complemented
19	C₉×#121	—	2	2	no	—
20	C₉×#121	C₂²	1	4	yes	complemented
21	C₉×#120	—	2	4	no	—
22	C₄×D₄×C₉	C₂²	1	1	yes	characteristic
23	C₄×D₄×C₉	C₂²	1	2	yes	complemented
24	C₉×#107	C₂²	1	1	yes	characteristic, complemented
25	C₉×C₄⋊C₈	C₂²	1	1	yes	characteristic, complemented
26	C₉×#102	C₂²	1	2	yes	complemented
27	C₉×#102	C₂²	1	4	yes	complemented
28	C₉×C₂²⋊C₈	C₂²	1	2	yes	complemented
29	C₄×C₈×C₉	C₂²	1	1	yes	characteristic, complemented
30	C₂²×C₃×D₈	C₂×C₃	1	2	yes	—
31	C₃×D₄²	C₂×C₃	1	2	yes	—
32	C₃×#493	C₂×C₃	1	1	yes	characteristic
33	C₃×#466	C₂×C₃	1	2	yes	—
34	C₃×#459	C₂×C₃	1	2	yes	—
35	C₃×#447	C₂×C₃	1	4	yes	—
36	C₃×C₄×D₈	C₂×C₃	1	1	yes	characteristic
37	C₃×C₈×D₄	C₂×C₃	1	1	yes	characteristic
38	C₂⁴×C₉	—	2	4	no	—
39	C₂×D₄×C₉	—	2	8	no	—
40	C₂×D₄×C₉	—	2	4	no	—
41	C₂×D₄×C₉	—	2	4	no	—
42	C₂×D₄×C₉	—	2	2	no	—
43	C₂×D₄×C₉	—	2	2	no	—
44	C₂×D₄×C₉	—	4	4	no	—
45	C₂×D₄×C₉	—	4	8	no	—
46	C₂×D₄×C₉	D₄	1	1	yes	characteristic
47	C₂×D₄×C₉	C₂³	1	4	yes	—
48	C₂×D₄×C₉	C₂³	1	2	yes	—
49	C₂²×C₄×C₉	—	2	2	no	—
50	C₂²×C₄×C₉	C₂³	1	2	yes	—
51	C₉×D₈	—	2	8	no	—
52	C₉×D₈	—	2	4	no	—
53	C₉×D₈	—	4	2	no	—
54	C₉×D₈	D₄	1	4	yes	direct product factor
55	C₂×C₈×C₉	—	2	4	no	—
56	C₂×C₈×C₉	C₂³	1	2	yes	—
57	C₂×C₈×C₉	C₂³	1	1	yes	characteristic
58	C₂×C₈×C₉	C₂³	1	1	yes	characteristic
59	C₉×C₄⋊C₄	D₄	1	1	yes	characteristic
60	C₉×C₄⋊C₄	C₂³	1	2	yes	—
61	C₉×C₂²⋊C₄	—	2	2	no	—
62	C₉×C₂²⋊C₄	—	2	4	no	—
63	C₉×C₂²⋊C₄	D₄	1	2	yes	—
64	C₄²×C₉	C₂³	1	1	yes	characteristic
65	D₄×D₈	C₉	1	1	yes	characteristic, direct product factor, Sylow
66	C₂²×C₃×D₄	—	2	2	no	—
67	C₂²×C₃×D₄	C₂²×C₃	1	4	yes	—
68	C₂×C₃×D₈	—	2	8	no	—
69	C₂×C₃×D₈	C₂²×C₃	1	1	yes	characteristic
70	C₂×C₃×D₈	C₂²×C₃	1	4	yes	—
71	C₂×C₃×D₈	C₂²×C₃	1	2	yes	—
72	C₂×C₃×D₈	C₂²×C₃	1	2	yes	—
73	C₂²×C₃×C₈	C₂²×C₃	1	2	yes	—
74	C₃×#127	C₂²×C₃	1	2	yes	—
75	C₃×#121	—	2	2	no	—
76	C₃×#121	C₂²×C₃	1	4	yes	—
77	C₃×#120	—	2	4	no	—
78	C₃×C₄×D₄	C₂²×C₃	1	1	yes	characteristic
79	C₃×C₄×D₄	C₂²×C₃	1	2	yes	—
80	C₃×#107	C₂²×C₃	1	1	yes	characteristic
81	C₃×C₄⋊C₈	C₂²×C₃	1	1	yes	characteristic
82	C₃×#102	C₂²×C₃	1	2	yes	—
83	C₃×#102	C₂²×C₃	1	4	yes	—
84	C₃×C₂²⋊C₈	C₂²×C₃	1	2	yes	—
85	C₃×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
86	C₂³×C₉	—	2	2	no	—
87	C₂³×C₉	—	2	4	no	—
88	C₂³×C₉	—	4	8	no	—
89	C₂³×C₉	—	4	8	no	—
90	C₂³×C₉	—	8	4	no	—
91	C₂³×C₉	C₂×D₄	1	2	yes	—
92	D₄×C₉	—	2	4	no	—
93	D₄×C₉	—	2	2	no	—
94	D₄×C₉	—	2	4	no	—
95	D₄×C₉	—	4	8	no	—
96	D₄×C₉	—	4	8	no	—
97	D₄×C₉	—	4	4	no	—
98	D₄×C₉	—	8	2	no	—
99	D₄×C₉	D₈	1	4	yes	direct product factor
100	D₄×C₉	C₂×D₄	1	4	yes	complemented
101	C₂×C₄×C₉	—	2	2	no	—
102	C₂×C₄×C₉	—	2	2	no	—
103	C₂×C₄×C₉	—	2	2	no	—
104	C₂×C₄×C₉	—	4	4	no	—
105	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
106	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
107	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
108	C₂×C₄×C₉	C₂⁴	1	1	yes	characteristic
109	C₈×C₉	—	2	2	no	—
110	C₈×C₉	—	2	1	no	—
111	C₈×C₉	C₂×D₄	1	2	yes	complemented
112	C₂²×D₈	C₂×C₉	1	2	yes	complemented
113	D₄²	C₂×C₉	1	2	yes	complemented
114	#493	C₂×C₉	1	1	yes	characteristic, complemented
115	#466	C₂×C₉	1	2	yes	complemented
116	#459	C₂×C₉	1	2	yes	complemented
117	#447	C₂×C₉	1	4	yes	complemented
118	C₄×D₈	C₂×C₉	1	1	yes	characteristic, complemented
119	C₈×D₄	C₂×C₉	1	1	yes	characteristic, complemented
120	C₂⁴×C₃	—	2	4	no	—
121	C₂×C₃×D₄	—	2	8	no	—
122	C₂×C₃×D₄	—	2	4	no	—
123	C₂×C₃×D₄	—	2	4	no	—
124	C₂×C₃×D₄	—	2	2	no	—
125	C₂×C₃×D₄	—	2	2	no	—
126	C₂×C₃×D₄	—	4	4	no	—
127	C₂×C₃×D₄	—	4	8	no	—
128	C₂×C₃×D₄	C₃×D₄	1	1	yes	characteristic
129	C₂×C₃×D₄	C₂³×C₃	1	4	yes	—
130	C₂×C₃×D₄	C₂³×C₃	1	2	yes	—
131	C₂²×C₃×C₄	—	2	2	no	—
132	C₂²×C₃×C₄	C₂³×C₃	1	2	yes	—
133	C₃×D₈	—	2	8	no	—
134	C₃×D₈	—	2	4	no	—
135	C₃×D₈	—	4	2	no	—
136	C₃×D₈	C₃×D₄	1	4	yes	—
137	C₂×C₃×C₈	—	2	4	no	—
138	C₂×C₃×C₈	C₂³×C₃	1	1	yes	characteristic
139	C₂×C₃×C₈	C₂³×C₃	1	2	yes	—
140	C₂×C₃×C₈	C₂³×C₃	1	1	yes	characteristic
141	C₃×C₄⋊C₄	C₃×D₄	1	1	yes	characteristic
142	C₃×C₄⋊C₄	C₂³×C₃	1	2	yes	—
143	C₃×C₂²⋊C₄	—	2	2	no	—
144	C₃×C₂²⋊C₄	—	2	4	no	—
145	C₃×C₂²⋊C₄	C₃×D₄	1	2	yes	—
146	C₃×C₄²	C₂³×C₃	1	1	yes	characteristic
147	C₂²×C₉	—	2	2	no	—
148	C₂²×C₉	—	2	2	no	—
149	C₂²×C₉	—	2	4	no	—
150	C₂²×C₉	—	4	2	no	—
151	C₂²×C₉	—	4	2	no	—
152	C₂²×C₉	—	4	4	no	—
153	C₂²×C₉	—	4	4	no	—
154	C₂²×C₉	—	4	4	no	—
155	C₂²×C₉	—	8	16	no	—
156	C₂²×C₉	C₂×D₈	1	4	yes	complemented
157	C₂²×C₉	C₂²×D₄	1	1	yes	characteristic
158	C₄×C₉	—	2	2	no	—
159	C₄×C₉	—	2	1	no	—
160	C₄×C₉	—	4	2	no	—
161	C₄×C₉	C₂×D₈	1	2	yes	complemented
162	C₄×C₉	C₂²×D₄	1	1	yes	characteristic
163	C₄×C₉	H(5,𝔽₂)	1	1	yes	characteristic
164	C₂²×D₄	—	2	2	no	—
165	C₂²×D₄	C₂²×C₉	1	4	yes	complemented
166	C₂×D₈	—	2	8	no	—
167	C₂×D₈	C₂²×C₉	1	1	yes	characteristic
168	C₂×D₈	C₂²×C₉	1	4	yes	complemented
169	C₂×D₈	C₂²×C₉	1	2	yes	complemented
170	C₂×D₈	C₂²×C₉	1	2	yes	complemented
171	C₂²×C₈	C₂²×C₉	1	2	yes	complemented
172	#127	C₂²×C₉	1	2	yes	complemented
173	#121	—	2	2	no	—
174	#121	C₂²×C₉	1	4	yes	complemented
175	#120	—	2	4	no	—
176	C₄×D₄	C₂²×C₉	1	1	yes	characteristic
177	C₄×D₄	C₂²×C₉	1	2	yes	complemented
178	#107	C₂²×C₉	1	1	yes	characteristic, complemented
179	C₄⋊C₈	C₂²×C₉	1	1	yes	characteristic, complemented
180	#102	C₂²×C₉	1	2	yes	complemented
181	#102	C₂²×C₉	1	4	yes	complemented
182	C₂²⋊C₈	C₂²×C₉	1	2	yes	complemented
183	C₄×C₈	C₂²×C₉	1	1	yes	characteristic, complemented
184	C₂³×C₃	—	2	2	no	—
185	C₂³×C₃	—	2	4	no	—
186	C₂³×C₃	—	4	8	no	—
187	C₂³×C₃	—	4	8	no	—
188	C₂³×C₃	—	8	4	no	—
189	C₂³×C₃	C₂×C₃×D₄	1	2	yes	—
190	C₃×D₄	—	2	4	no	—
191	C₃×D₄	—	2	2	no	—
192	C₃×D₄	—	2	4	no	—
193	C₃×D₄	—	4	8	no	—
194	C₃×D₄	—	4	8	no	—
195	C₃×D₄	—	4	4	no	—
196	C₃×D₄	—	8	2	no	—
197	C₃×D₄	C₃×D₈	1	4	yes	—
198	C₃×D₄	C₂×C₃×D₄	1	4	yes	—
199	C₂×C₃×C₄	—	2	2	no	—
200	C₂×C₃×C₄	—	2	2	no	—
201	C₂×C₃×C₄	—	2	2	no	—
202	C₂×C₃×C₄	—	4	4	no	—
203	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
204	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
205	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
206	C₂×C₃×C₄	C₂⁴×C₃	1	1	yes	characteristic
207	C₃×C₈	—	2	2	no	—
208	C₃×C₈	—	2	1	no	—
209	C₃×C₈	C₂×C₃×D₄	1	2	yes	—
210	C₂×C₉	—	2	4	no	—
211	C₂×C₉	—	4	4	no	—
212	C₂×C₉	—	8	4	no	—
213	C₂×C₉	D₄²	1	1	yes	characteristic
214	C₂×C₉	C₂²×D₈	1	1	yes	characteristic
215	C₂×C₉	#576	1	1	yes	characteristic
216	C₂⁴	—	2	4	no	—
217	C₂×D₄	—	2	8	no	—
218	C₂×D₄	—	2	4	no	—
219	C₂×D₄	—	2	4	no	—
220	C₂×D₄	—	2	2	no	—
221	C₂×D₄	—	2	2	no	—
222	C₂×D₄	—	4	4	no	—
223	C₂×D₄	—	4	8	no	—
224	C₂×D₄	D₄×C₉	1	1	yes	characteristic
225	C₂×D₄	C₂³×C₉	1	4	yes	—
226	C₂×D₄	C₂³×C₉	1	2	yes	—
227	C₂²×C₄	—	2	2	no	—
228	C₂²×C₄	C₂³×C₉	1	2	yes	—
229	D₈	—	2	8	no	—
230	D₈	—	2	4	no	—
231	D₈	—	4	2	no	—
232	D₈	D₄×C₉	1	4	yes	direct product factor
233	C₂×C₈	—	2	4	no	—
234	C₂×C₈	C₂³×C₉	1	1	yes	characteristic
235	C₂×C₈	C₂³×C₉	1	2	yes	—
236	C₂×C₈	C₂³×C₉	1	1	yes	characteristic
237	C₄⋊C₄	D₄×C₉	1	1	yes	characteristic
238	C₄⋊C₄	C₂³×C₉	1	2	yes	—
239	C₂²⋊C₄	—	2	2	no	—
240	C₂²⋊C₄	—	2	4	no	—
241	C₂²⋊C₄	D₄×C₉	1	2	yes	—
242	C₄²	C₂³×C₉	1	1	yes	characteristic
243	C₂²×C₃	—	2	2	no	—
244	C₂²×C₃	—	2	2	no	—
245	C₂²×C₃	—	2	4	no	—
246	C₂²×C₃	—	4	2	no	—
247	C₂²×C₃	—	4	2	no	—
248	C₂²×C₃	—	4	4	no	—
249	C₂²×C₃	—	4	4	no	—
250	C₂²×C₃	—	4	4	no	—
251	C₂²×C₃	—	8	16	no	—
252	C₂²×C₃	C₂×C₃×D₈	1	4	yes	—
253	C₂²×C₃	C₂²×C₃×D₄	1	1	yes	characteristic
254	C₃×C₄	—	2	1	no	—
255	C₃×C₄	—	2	2	no	—
256	C₃×C₄	—	4	2	no	—
257	C₃×C₄	C₂×C₃×D₈	1	2	yes	—
258	C₃×C₄	C₂²×C₃×D₄	1	1	yes	characteristic
259	C₃×C₄	C₃×H(5,𝔽₂)	1	1	yes	characteristic
260	C₉	D₄×D₈	1	1	yes	characteristic, direct product factor, Sylow
261	C₂³	—	2	2	no	—
262	C₂³	—	2	4	no	—
263	C₂³	—	4	8	no	—
264	C₂³	—	4	8	no	—
265	C₂³	—	8	4	no	—
266	C₂³	C₂×D₄×C₉	1	2	yes	—
267	D₄	—	2	4	no	—
268	D₄	—	2	2	no	—
269	D₄	—	2	4	no	—
270	D₄	—	4	8	no	—
271	D₄	—	4	8	no	—
272	D₄	—	4	4	no	—
273	D₄	—	8	2	no	—
274	D₄	C₉×D₈	1	4	yes	direct product factor
275	D₄	C₂×D₄×C₉	1	4	yes	complemented
276	C₂×C₄	—	2	2	no	—
277	C₂×C₄	—	2	2	no	—
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	4	4	no	—
280	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
281	C₂×C₄	C₂×D₄×C₉	1	2	yes	—
282	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
283	C₂×C₄	C₂⁴×C₉	1	1	yes	characteristic
284	C₈	—	2	2	no	—
285	C₈	—	2	1	no	—
286	C₈	C₂×D₄×C₉	1	2	yes	complemented
287	C₂×C₃	—	2	4	no	—
288	C₂×C₃	—	4	4	no	—
289	C₂×C₃	—	8	4	no	—
290	C₂×C₃	C₃×D₄²	1	1	yes	characteristic
291	C₂×C₃	C₂²×C₃×D₈	1	1	yes	characteristic
292	C₂×C₃	C₃×#576	1	1	yes	characteristic
293	C₂²	—	2	2	no	—
294	C₂²	—	2	2	no	—
295	C₂²	—	2	4	no	—
296	C₂²	—	4	2	no	—
297	C₂²	—	4	2	no	—
298	C₂²	—	4	4	no	—
299	C₂²	—	4	4	no	—
300	C₂²	—	4	4	no	—
301	C₂²	—	8	16	no	—
302	C₂²	C₂×C₉×D₈	1	4	yes	complemented
303	C₂²	C₂²×D₄×C₉	1	1	yes	characteristic
304	C₄	—	2	2	no	—
305	C₄	—	2	1	no	—
306	C₄	—	4	2	no	—
307	C₄	C₂×C₉×D₈	1	2	yes	complemented
308	C₄	C₂²×D₄×C₉	1	1	yes	characteristic
309	C₄	C₉×H(5,𝔽₂)	1	1	yes	characteristic
310	C₃	C₃×D₄×D₈	1	1	yes	characteristic
311	C₂	—	2	4	no	—
312	C₂	—	4	4	no	—
313	C₂	—	8	4	no	—
314	C₂	D₄²×C₉	1	1	yes	characteristic
315	C₂	C₂²×C₉×D₈	1	1	yes	characteristic
316	C₂	C₉×#576	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₉	C₂²×D₄
commutator	C₂×C₄	C₂⁴×C₉
Frattini, cosocle	C₂×C₃×C₄	C₂⁴×C₃
socle	C₂²×C₃	C₂²×C₃×D₄
prime radical	C₃×D₄×D₈	C₃
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₉ ◁ C₄×D₄×C₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×C₉×D₈	C₂	semidirect
2. D₄²×C₉	C₂	semidirect
3. C₉×#493	C₂	semidirect
4. C₉×#466	C₂	semidirect
5. C₉×#459	C₂	semidirect
6. C₉×#447	C₂	semidirect
7. C₄×C₉×D₈	C₂	semidirect
8. C₈×D₄×C₉	C₂	semidirect
11. C₂²×D₄×C₉	C₂²	semidirect
14. C₂×C₉×D₈	C₂²	semidirect
15. C₂×C₉×D₈	C₂²	semidirect
16. C₂×C₉×D₈	C₂²	semidirect
17. C₂²×C₈×C₉	C₂²	semidirect
18. C₉×#127	C₂²	semidirect
20. C₉×#121	C₂²	semidirect
23. C₄×D₄×C₉	C₂²	semidirect
24. C₉×#107	C₂²	semidirect
25. C₉×C₄⋊C₈	C₂²	semidirect
26. C₉×#102	C₂²	semidirect
27. C₉×#102	C₂²	semidirect
28. C₉×C₂²⋊C₈	C₂²	semidirect
29. C₄×C₈×C₉	C₂²	semidirect
54. C₉×D₈	D₄	direct (N × H)
65. D₄×D₈	C₉	direct (N × H)
99. D₄×C₉	D₈	direct (N × H)
100. D₄×C₉	C₂×D₄	semidirect
111. C₈×C₉	C₂×D₄	semidirect
112. C₂²×D₈	C₂×C₉	semidirect
113. D₄²	C₂×C₉	semidirect
114. #493	C₂×C₉	semidirect
115. #466	C₂×C₉	semidirect
116. #459	C₂×C₉	semidirect
117. #447	C₂×C₉	semidirect
118. C₄×D₈	C₂×C₉	semidirect
119. C₈×D₄	C₂×C₉	semidirect
156. C₂²×C₉	C₂×D₈	semidirect
161. C₄×C₉	C₂×D₈	semidirect
165. C₂²×D₄	C₂²×C₉	semidirect
168. C₂×D₈	C₂²×C₉	semidirect
169. C₂×D₈	C₂²×C₉	semidirect
170. C₂×D₈	C₂²×C₉	semidirect
171. C₂²×C₈	C₂²×C₉	semidirect
172. #127	C₂²×C₉	semidirect
174. #121	C₂²×C₉	semidirect
177. C₄×D₄	C₂²×C₉	semidirect
178. #107	C₂²×C₉	semidirect
179. C₄⋊C₈	C₂²×C₉	semidirect
180. #102	C₂²×C₉	semidirect
181. #102	C₂²×C₉	semidirect
182. C₂²⋊C₈	C₂²×C₉	semidirect
183. C₄×C₈	C₂²×C₉	semidirect
232. D₈	D₄×C₉	direct (N × H)
260. C₉	D₄×D₈	direct (N × H)
274. D₄	C₉×D₈	direct (N × H)
275. D₄	C₂×D₄×C₉	semidirect
286. C₈	C₂×D₄×C₉	semidirect
302. C₂²	C₂×C₉×D₈	semidirect
307. C₄	C₂×C₉×D₈	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₉×D₈	2
D₄²×C₉	2
C₉×#493	2
C₉×#466	2
C₉×#459	2
C₉×#447	2
C₄×C₉×D₈	2
C₈×D₄×C₉	2
C₃×D₄×D₈	3

Subgroups — D4×C9×D8 #11768000

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₄×C₉×D₈ #11768000