Subgroups — D₄×#1204 #11755005

Order 960.

560 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#1204	C₂	1	2	yes	maximal, complemented
2	D₄×Dic₁₅	C₂	1	1	yes	maximal, characteristic
3	C₅×D₄×Dic₃	C₂	1	1	yes	maximal, characteristic
4	C₃×D₄×Dic₅	C₂	1	1	yes	maximal, characteristic
5	#91576	C₂	1	2	yes	maximal, complemented
6	#91575	C₂	1	2	yes	maximal, complemented
7	#91574	C₂	1	2	yes	maximal, complemented
8	#91470	C₂	1	1	yes	maximal, characteristic, complemented
9	#91469	C₂	1	1	yes	maximal, characteristic, complemented
10	#91468	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×#1204	C₂	1	1	yes	maximal, characteristic, complemented
12	D₄×Dic₁₀	—	3	1	no	maximal, Hall
13	C₂×C₃×C₅×D₄	C₂²	1	1	yes	characteristic
14	C₂²×Dic₁₅	C₂²	1	2	yes	—
15	C₂²×C₅×Dic₃	C₂²	1	2	yes	—
16	C₂²×C₃×Dic₅	C₂²	1	2	yes	—
17	C₂×#1204	—	2	8	no	—
18	C₂×#1204	C₂²	1	2	yes	—
19	C₂×#1204	C₂²	1	2	yes	—
20	C₂×#1204	C₂²	1	2	yes	—
21	C₂×#1204	C₂²	1	1	yes	characteristic
22	#6719	C₂²	1	2	yes	—
23	C₄⋊Dic₁₅	C₂²	1	1	yes	characteristic
24	C₄×Dic₁₅	C₂²	1	1	yes	characteristic
25	C₅×#217	C₂²	1	2	yes	—
26	C₅×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
27	C₄×C₅×Dic₃	C₂²	1	1	yes	characteristic
28	C₃×C₂²⋊Dic₅	C₂²	1	2	yes	—
29	C₃×C₄⋊Dic₅	C₂²	1	1	yes	characteristic
30	C₃×C₄×Dic₅	C₂²	1	1	yes	characteristic
31	Dic₃⋊Dic₅	C₂²	1	1	yes	characteristic
32	Dic₃⋊Dic₅	C₂²	1	2	yes	—
33	#6669	C₂²	1	1	yes	characteristic
34	#6669	C₂²	1	2	yes	—
35	#6668	C₂²	1	1	yes	characteristic
36	#6668	C₂²	1	2	yes	—
37	D₄×Dic₆	—	5	1	no	maximal, Hall
38	C₂²×Dic₁₀	—	3	2	no	—
39	C₄×C₅×D₄	—	3	1	no	—
40	D₄×Dic₅	—	3	1	no	—
41	D₄×Dic₅	D₃	1	1	yes	characteristic
42	#4070	—	3	2	no	—
43	#4034	—	3	1	no	—
44	#4034	—	3	1	no	—
45	#4024	—	3	2	no	—
46	#4024	—	3	2	no	—
47	#4015	—	3	1	no	—
48	C₄×Dic₁₀	—	3	1	no	—
49	C₂³×C₃×C₅	C₂³	1	2	yes	—
50	C₃×C₅×D₄	Q₈	1	4	yes	complemented
51	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
52	C₂×Dic₁₅	—	2	4	no	—
53	C₂×Dic₁₅	C₂³	1	1	yes	characteristic
54	C₂×Dic₁₅	C₂³	1	2	yes	—
55	C₂×Dic₁₅	C₂³	1	1	yes	characteristic
56	C₂×C₅×Dic₃	—	2	4	no	—
57	C₂×C₅×Dic₃	C₂³	1	1	yes	characteristic
58	C₂×C₅×Dic₃	C₂³	1	2	yes	—
59	C₂×C₅×Dic₃	C₂³	1	1	yes	characteristic
60	C₂×C₃×Dic₅	—	2	4	no	—
61	C₂×C₃×Dic₅	C₂³	1	1	yes	characteristic
62	C₂×C₃×Dic₅	C₂³	1	2	yes	—
63	C₂×C₃×Dic₅	C₂³	1	1	yes	characteristic
64	#1204	—	2	4	no	—
65	#1204	—	2	4	no	—
66	#1204	—	2	4	no	—
67	#1204	D₄	1	4	yes	direct product factor
68	C₂²×Dic₆	—	5	2	no	—
69	C₃×C₄×D₄	—	5	1	no	—
70	D₄×Dic₃	—	5	1	no	—
71	D₄×Dic₃	D₅	1	1	yes	characteristic
72	#924	—	5	2	no	—
73	#888	—	5	1	no	—
74	#888	—	5	1	no	—
75	#878	—	5	2	no	—
76	#878	—	5	2	no	—
77	#869	—	5	1	no	—
78	C₄×Dic₆	—	5	1	no	—
79	C₂×C₅×D₄	C₂×D₃	1	1	yes	characteristic
80	C₂²×C₄×C₅	—	3	2	no	—
81	C₂²×Dic₅	—	3	2	no	—
82	C₂²×Dic₅	C₂×D₃	1	2	yes	—
83	C₂×Dic₁₀	—	3	1	no	—
84	C₂×Dic₁₀	—	3	2	no	—
85	C₂×Dic₁₀	—	3	2	no	—
86	C₂×Dic₁₀	—	3	2	no	—
87	C₂×Dic₁₀	—	6	8	no	—
88	C₅×C₄⋊C₄	—	3	1	no	—
89	C₅×C₂²⋊C₄	—	3	2	no	—
90	C₄²×C₅	—	3	1	no	—
91	C₂²⋊Dic₅	—	3	2	no	—
92	C₂²⋊Dic₅	C₂×D₃	1	2	yes	—
93	C₄⋊Dic₅	—	3	1	no	—
94	C₄⋊Dic₅	—	3	1	no	—
95	C₄⋊Dic₅	—	3	2	no	—
96	C₄⋊Dic₅	C₂×D₃	1	1	yes	characteristic
97	#683	—	3	1	no	—
98	#683	—	3	2	no	—
99	#683	—	3	1	no	—
100	#683	—	3	2	no	—
101	C₄×Dic₅	—	3	1	no	—
102	C₄×Dic₅	C₂×D₃	1	1	yes	characteristic
103	D₄×Q₈	—	15	1	no	Sylow
104	C₂²×C₃×C₅	—	2	2	no	—
105	C₂²×C₃×C₅	—	2	2	no	—
106	C₂²×C₃×C₅	C₂×Q₈	1	4	yes	complemented
107	C₂²×C₃×C₅	C₂⁴	1	1	yes	characteristic
108	C₃×C₄×C₅	C₂×Q₈	1	2	yes	complemented
109	Dic₁₅	—	2	2	no	—
110	Dic₁₅	—	2	1	no	—
111	Dic₁₅	C₂×D₄	1	2	yes	—
112	C₃×Dic₅	—	2	2	no	—
113	C₃×Dic₅	—	2	1	no	—
114	C₃×Dic₅	C₂×D₄	1	2	yes	—
115	C₅×Dic₃	—	2	2	no	—
116	C₅×Dic₃	—	2	1	no	—
117	C₅×Dic₃	C₂×D₄	1	2	yes	—
118	C₂×C₃×D₄	C₂×D₅	1	1	yes	characteristic
119	C₂²×C₃×C₄	—	5	2	no	—
120	C₂²×Dic₃	—	5	2	no	—
121	C₂²×Dic₃	C₂×D₅	1	2	yes	—
122	C₂×Dic₆	—	5	1	no	—
123	C₂×Dic₆	—	5	2	no	—
124	C₂×Dic₆	—	5	2	no	—
125	C₂×Dic₆	—	5	2	no	—
126	C₂×Dic₆	—	10	8	no	—
127	C₃×C₄⋊C₄	—	5	1	no	—
128	C₃×C₂²⋊C₄	—	5	2	no	—
129	C₃×C₄²	—	5	1	no	—
130	#217	—	5	2	no	—
131	#217	C₂×D₅	1	2	yes	—
132	C₄⋊Dic₃	—	5	1	no	—
133	C₄⋊Dic₃	—	5	1	no	—
134	C₄⋊Dic₃	—	5	2	no	—
135	C₄⋊Dic₃	C₂×D₅	1	1	yes	characteristic
136	Dic₃⋊C₄	—	5	1	no	—
137	Dic₃⋊C₄	—	5	2	no	—
138	Dic₃⋊C₄	—	5	1	no	—
139	Dic₃⋊C₄	—	5	2	no	—
140	C₄×Dic₃	—	5	1	no	—
141	C₄×Dic₃	C₂×D₅	1	1	yes	characteristic
142	C₂³×C₅	C₂²×D₃	1	2	yes	—
143	C₅×D₄	Dic₆	1	4	yes	complemented
144	C₂×C₄×C₅	—	3	1	no	—
145	C₂×C₄×C₅	—	3	1	no	—
146	C₂×C₄×C₅	—	3	2	no	—
147	C₂×C₄×C₅	—	6	4	no	—
148	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
149	C₂×Dic₅	—	2	4	no	—
150	C₂×Dic₅	—	3	1	no	—
151	C₂×Dic₅	—	3	1	no	—
152	C₂×Dic₅	—	3	2	no	—
153	C₂×Dic₅	—	6	4	no	—
154	C₂×Dic₅	C₂²×D₃	1	1	yes	characteristic
155	C₂×Dic₅	C₂²×D₃	1	2	yes	—
156	C₂×Dic₅	C₂²×D₃	1	1	yes	characteristic
157	Dic₁₀	—	3	4	no	—
158	Dic₁₀	—	6	4	no	—
159	Dic₁₀	—	6	4	no	—
160	Dic₁₀	—	6	4	no	—
161	C₂²×Q₈	—	15	2	no	—
162	C₄⋊Q₈	—	15	1	no	—
163	C₄⋊Q₈	—	15	1	no	—
164	C₄⋊Q₈	—	15	1	no	—
165	C₂²⋊Q₈	—	15	2	no	—
166	C₂²⋊Q₈	—	15	2	no	—
167	C₂²⋊Q₈	—	15	2	no	—
168	C₄×Q₈	—	15	1	no	—
169	C₄×D₄	—	3	1	no	—
170	C₄×D₄	—	5	1	no	—
171	C₄×D₄	—	15	1	no	—
172	C₂×C₃×C₅	—	2	4	no	—
173	C₂×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
174	C₂×C₃×C₅	C₂²×Q₈	1	1	yes	characteristic
175	C₂×C₃×C₅	2₋¹⁺⁴	1	1	yes	characteristic
176	C₂³×C₃	C₂²×D₅	1	2	yes	—
177	C₃×D₄	Dic₁₀	1	4	yes	complemented
178	C₂×C₃×C₄	—	5	1	no	—
179	C₂×C₃×C₄	—	5	1	no	—
180	C₂×C₃×C₄	—	5	2	no	—
181	C₂×C₃×C₄	—	10	4	no	—
182	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
183	C₂×Dic₃	—	2	4	no	—
184	C₂×Dic₃	—	5	1	no	—
185	C₂×Dic₃	—	5	1	no	—
186	C₂×Dic₃	—	5	2	no	—
187	C₂×Dic₃	—	10	4	no	—
188	C₂×Dic₃	C₂²×D₅	1	1	yes	characteristic
189	C₂×Dic₃	C₂²×D₅	1	2	yes	—
190	C₂×Dic₃	C₂²×D₅	1	1	yes	characteristic
191	Dic₆	—	5	4	no	—
192	Dic₆	—	10	4	no	—
193	Dic₆	—	10	4	no	—
194	Dic₆	—	10	4	no	—
195	C₂²×C₅	—	2	2	no	—
196	C₂²×C₅	—	2	2	no	—
197	C₂²×C₅	C₂×Dic₆	1	4	yes	complemented
198	C₂²×C₅	C₂³×D₃	1	1	yes	characteristic
199	C₄×C₅	—	3	2	no	—
200	C₄×C₅	—	6	2	no	—
201	C₄×C₅	—	6	1	no	—
202	C₄×C₅	C₂×Dic₆	1	2	yes	complemented
203	Dic₅	—	2	2	no	—
204	Dic₅	—	2	1	no	—
205	Dic₅	—	3	2	no	—
206	Dic₅	—	6	2	no	—
207	Dic₅	—	6	1	no	—
208	Dic₅	D₃×D₄	1	2	yes	—
209	C₂×Q₈	—	15	1	no	—
210	C₂×Q₈	—	15	2	no	—
211	C₂×Q₈	—	15	2	no	—
212	C₂×Q₈	—	15	2	no	—
213	C₂×Q₈	—	30	8	no	—
214	C₂×D₄	D₃×D₅	1	1	yes	characteristic
215	C₂²×C₄	—	3	2	no	—
216	C₂²×C₄	—	5	2	no	—
217	C₂²×C₄	—	15	2	no	—
218	C₄⋊C₄	—	3	1	no	—
219	C₄⋊C₄	—	5	1	no	—
220	C₄⋊C₄	—	15	1	no	—
221	C₄⋊C₄	—	15	1	no	—
222	C₄⋊C₄	—	15	1	no	—
223	C₄⋊C₄	—	15	2	no	—
224	C₄⋊C₄	—	15	2	no	—
225	C₄⋊C₄	—	15	2	no	—
226	C₄⋊C₄	—	15	1	no	—
227	C₂²⋊C₄	—	3	2	no	—
228	C₂²⋊C₄	—	5	2	no	—
229	C₂²⋊C₄	—	15	2	no	—
230	C₄²	—	3	1	no	—
231	C₄²	—	5	1	no	—
232	C₄²	—	15	1	no	—
233	C₃×C₅	D₄×Q₈	1	1	yes	characteristic, complemented, Hall
234	C₂²×C₃	—	2	2	no	—
235	C₂²×C₃	—	2	2	no	—
236	C₂²×C₃	C₂×Dic₁₀	1	4	yes	complemented
237	C₂²×C₃	C₂³×D₅	1	1	yes	characteristic
238	C₃×C₄	—	5	2	no	—
239	C₃×C₄	—	10	2	no	—
240	C₃×C₄	—	10	1	no	—
241	C₃×C₄	C₂×Dic₁₀	1	2	yes	complemented
242	Dic₃	—	2	2	no	—
243	Dic₃	—	2	1	no	—
244	Dic₃	—	5	2	no	—
245	Dic₃	—	10	2	no	—
246	Dic₃	—	10	1	no	—
247	Dic₃	D₄×D₅	1	2	yes	—
248	C₂×C₅	—	2	4	no	—
249	C₂×C₅	C₂²×Dic₆	1	1	yes	characteristic
250	C₂×C₅	C₂×D₃×D₄	1	1	yes	characteristic
251	C₂×C₅	#1010	1	1	yes	characteristic
252	C₂³	C₂×D₃×D₅	1	2	yes	—
253	Q₈	—	15	4	no	—
254	Q₈	—	30	4	no	—
255	Q₈	—	30	4	no	—
256	Q₈	—	30	4	no	—
257	D₄	#1204	1	4	yes	direct product factor
258	C₂×C₄	—	3	1	no	—
259	C₂×C₄	—	3	1	no	—
260	C₂×C₄	—	3	2	no	—
261	C₂×C₄	—	5	1	no	—
262	C₂×C₄	—	5	1	no	—
263	C₂×C₄	—	5	2	no	—
264	C₂×C₄	—	6	4	no	—
265	C₂×C₄	—	10	4	no	—
266	C₂×C₄	—	15	1	no	—
267	C₂×C₄	—	15	1	no	—
268	C₂×C₄	—	15	2	no	—
269	C₂×C₄	—	30	4	no	—
270	C₂×C₄	C₂×D₃×D₅	1	1	yes	characteristic
271	C₂×C₃	—	2	4	no	—
272	C₂×C₃	C₂²×Dic₁₀	1	1	yes	characteristic
273	C₂×C₃	C₂×D₄×D₅	1	1	yes	characteristic
274	C₂×C₃	#4150	1	1	yes	characteristic
275	C₅	D₄×Dic₆	1	1	yes	characteristic, complemented, Sylow
276	C₂²	—	2	2	no	—
277	C₂²	—	2	2	no	—
278	C₂²	C₂×#1204	1	4	yes	complemented
279	C₂²	C₂²×D₃×D₅	1	1	yes	characteristic
280	C₄	—	3	2	no	—
281	C₄	—	5	2	no	—
282	C₄	—	6	1	no	—
283	C₄	—	6	2	no	—
284	C₄	—	10	1	no	—
285	C₄	—	10	2	no	—
286	C₄	—	15	2	no	—
287	C₄	—	30	1	no	—
288	C₄	—	30	2	no	—
289	C₄	C₂×#1204	1	2	yes	complemented
290	C₃	D₄×Dic₁₀	1	1	yes	characteristic, complemented, Sylow
291	C₂	—	2	4	no	—
292	C₂	#92020	1	1	yes	characteristic
293	C₂	D₃×D₄×D₅	1	1	yes	characteristic
294	C₂	C₂²×#1204	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×D₃×D₅
commutator, socle, cosocle	C₂²×C₃×C₅	C₂⁴
Fitting, prime radical	C₂×C₃×C₅×D₄	C₂²
hypercenter	C₂×D₄	D₃×D₅
hypocenter	C₃×C₅	D₄×Q₈
LCS	G ▷ C₂²×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂² ◁ C₂×D₄

Complements

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

N	H ≅ G/N	type
1. C₂²×#1204	C₂	semidirect
5. #91576	C₂	semidirect
6. #91575	C₂	semidirect
7. #91574	C₂	semidirect
8. #91470	C₂	semidirect
9. #91469	C₂	semidirect
10. #91468	C₂	semidirect
11. C₄×#1204	C₂	semidirect
50. C₃×C₅×D₄	Q₈	semidirect
67. #1204	D₄	direct (N × H)
106. C₂²×C₃×C₅	C₂×Q₈	semidirect
108. C₃×C₄×C₅	C₂×Q₈	semidirect
143. C₅×D₄	Dic₆	semidirect
177. C₃×D₄	Dic₁₀	semidirect
197. C₂²×C₅	C₂×Dic₆	semidirect
202. C₄×C₅	C₂×Dic₆	semidirect
233. C₃×C₅	D₄×Q₈	semidirect
236. C₂²×C₃	C₂×Dic₁₀	semidirect
241. C₃×C₄	C₂×Dic₁₀	semidirect
257. D₄	#1204	direct (N × H)
275. C₅	D₄×Dic₆	semidirect
278. C₂²	C₂×#1204	semidirect
289. C₄	C₂×#1204	semidirect
290. C₃	D₄×Dic₁₀	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#1204	2
D₄×Dic₁₅	2
C₅×D₄×Dic₃	2
C₃×D₄×Dic₅	2
#91576	2
#91575	2
#91574	2
#91470	2
#91469	2
#91468	2
C₄×#1204	2
D₄×Dic₁₀	3
D₄×Dic₆	5

Subgroups — D4×#1204 #11755005

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₄×#1204 #11755005