Subgroups — #11756488

Order 960.

412 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	#91835	C₂	1	2	yes	maximal
3	#91833	C₂	1	2	yes	maximal, complemented
4	#91785	C₂	1	2	yes	maximal
5	#91781	C₂	1	1	yes	maximal, characteristic
6	C₄⋊C₄×D₁₅	C₂	1	1	yes	maximal, characteristic
7	#91779	C₂	1	1	yes	maximal, characteristic, complemented
8	#91778	C₂	1	2	yes	maximal, complemented
9	#91777	C₂	1	1	yes	maximal, characteristic, complemented
10	#91761	C₂	1	1	yes	maximal, characteristic
11	#91759	C₂	1	1	yes	maximal, characteristic, complemented
12	#66063	—	3	1	no	maximal, Hall
13	C₂×C₃×C₅×Q₈	C₂²	1	2	yes	—
14	C₂×C₄×D₁₅	C₂²	1	2	yes	—
15	C₂×C₄×D₁₅	C₂²	1	1	yes	characteristic
16	C₂×Dic₃₀	C₂²	1	2	yes	—
17	C₃×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
18	C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
19	C₃×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
20	C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
21	#6714	C₂²	1	1	yes	characteristic
22	#6714	C₂²	1	1	yes	characteristic
23	#6714	C₂²	1	2	yes	—
24	C₄⋊Dic₁₅	C₂²	1	1	yes	characteristic
25	C₄⋊Dic₁₅	C₂²	1	1	yes	characteristic
26	C₄⋊Dic₁₅	C₂²	1	2	yes	—
27	#6712	C₂²	1	2	yes	—
28	#6712	C₂²	1	2	yes	—
29	#6712	C₂²	1	1	yes	characteristic
30	#6712	C₂²	1	1	yes	characteristic
31	#6712	C₂²	1	2	yes	—
32	#6712	C₂²	1	2	yes	—
33	#6712	C₂²	1	2	yes	—
34	C₄×Dic₁₅	C₂²	1	1	yes	characteristic
35	C₄×Dic₁₅	C₂²	1	2	yes	—
36	#5729	—	5	1	no	maximal, Hall
37	C₅×C₄⋊Q₈	D₃	1	1	yes	characteristic, complemented
38	#4092	—	3	2	no	—
39	#4090	—	3	2	no	—
40	#4042	—	3	2	no	—
41	#4038	—	3	1	no	—
42	D₅×C₄⋊C₄	—	3	1	no	—
43	#4036	—	3	1	no	—
44	#4035	—	3	2	no	—
45	#4034	—	3	1	no	—
46	#4018	—	3	1	no	—
47	#4016	—	3	1	no	—
48	C₂²×D₁₅	C₂³	1	1	yes	characteristic
49	C₃×C₅×Q₈	—	4	2	no	—
50	C₂×C₃×C₄×C₅	C₂³	1	2	yes	—
51	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
52	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
53	C₂×C₃×C₄×C₅	C₂³	1	2	yes	—
54	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
55	C₂×Dic₁₅	C₂³	1	1	yes	characteristic
56	C₂×Dic₁₅	C₂³	1	2	yes	—
57	C₂×Dic₁₅	C₂³	1	1	yes	characteristic
58	C₂×Dic₁₅	C₂³	1	2	yes	—
59	C₂×Dic₁₅	C₂³	1	1	yes	characteristic
60	C₄×D₁₅	—	4	2	no	—
61	C₄×D₁₅	Q₈	1	4	yes	complemented
62	Dic₃₀	—	4	2	no	—
63	C₃×C₄⋊Q₈	D₅	1	1	yes	characteristic, complemented
64	#946	—	5	2	no	—
65	#944	—	5	2	no	—
66	#896	—	5	2	no	—
67	#892	—	5	1	no	—
68	D₃×C₄⋊C₄	—	5	1	no	—
69	#890	—	5	1	no	—
70	#889	—	5	2	no	—
71	#888	—	5	1	no	—
72	#872	—	5	1	no	—
73	#870	—	5	1	no	—
74	C₂×C₅×Q₈	C₂×D₃	1	2	yes	—
75	C₂×C₄×D₅	—	3	1	no	—
76	C₂×C₄×D₅	—	3	2	no	—
77	C₂×Dic₁₀	—	3	2	no	—
78	C₅×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
79	C₅×C₄⋊C₄	C₂×D₃	1	2	yes	—
80	C₅×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
81	C₄²×C₅	C₂×D₃	1	1	yes	characteristic
82	#685	—	3	1	no	—
83	#685	—	3	1	no	—
84	#685	—	3	2	no	—
85	C₄⋊Dic₅	—	3	1	no	—
86	C₄⋊Dic₅	—	3	1	no	—
87	C₄⋊Dic₅	—	3	2	no	—
88	#683	—	3	2	no	—
89	#683	—	3	2	no	—
90	#683	—	3	1	no	—
91	#683	—	3	1	no	—
92	#683	—	3	2	no	—
93	#683	—	3	2	no	—
94	#683	—	3	2	no	—
95	C₄×Dic₅	—	3	1	no	—
96	C₄×Dic₅	—	3	2	no	—
97	#544	—	15	1	no	Sylow
98	C₂²×C₃×C₅	C₂⁴	1	1	yes	characteristic
99	C₂×D₁₅	—	2	1	no	—
100	C₂×D₁₅	—	2	1	no	—
101	C₂×D₁₅	C₂×Q₈	1	2	yes	—
102	C₃×C₄×C₅	—	2	2	no	—
103	C₃×C₄×C₅	—	2	2	no	—
104	C₃×C₄×C₅	—	2	1	no	—
105	C₃×C₄×C₅	—	2	1	no	—
106	C₃×C₄×C₅	C₂×Q₈	1	2	yes	—
107	Dic₁₅	—	2	1	no	—
108	Dic₁₅	—	2	2	no	—
109	Dic₁₅	—	2	1	no	—
110	Dic₁₅	—	2	2	no	—
111	Dic₁₅	C₂×Q₈	1	2	yes	—
112	C₂×C₃×Q₈	C₂×D₅	1	2	yes	—
113	C₂×C₄×D₃	—	5	1	no	—
114	C₂×C₄×D₃	—	5	2	no	—
115	C₂×Dic₆	—	5	2	no	—
116	C₃×C₄⋊C₄	C₂×D₅	1	1	yes	characteristic
117	C₃×C₄⋊C₄	C₂×D₅	1	2	yes	—
118	C₃×C₄⋊C₄	C₂×D₅	1	1	yes	characteristic
119	C₃×C₄²	C₂×D₅	1	1	yes	characteristic
120	(C₂×D₃)⋊C₄	—	5	1	no	—
121	(C₂×D₃)⋊C₄	—	5	1	no	—
122	(C₂×D₃)⋊C₄	—	5	2	no	—
123	C₄⋊Dic₃	—	5	1	no	—
124	C₄⋊Dic₃	—	5	1	no	—
125	C₄⋊Dic₃	—	5	2	no	—
126	Dic₃⋊C₄	—	5	2	no	—
127	Dic₃⋊C₄	—	5	2	no	—
128	Dic₃⋊C₄	—	5	1	no	—
129	Dic₃⋊C₄	—	5	1	no	—
130	Dic₃⋊C₄	—	5	2	no	—
131	Dic₃⋊C₄	—	5	2	no	—
132	Dic₃⋊C₄	—	5	2	no	—
133	C₄×Dic₃	—	5	1	no	—
134	C₄×Dic₃	—	5	2	no	—
135	C₂²×D₅	—	3	1	no	—
136	C₅×Q₈	—	4	2	no	—
137	C₂×C₄×C₅	C₂²×D₃	1	2	yes	—
138	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
139	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
140	C₂×C₄×C₅	C₂²×D₃	1	2	yes	—
141	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
142	C₂×Dic₅	—	3	1	no	—
143	C₂×Dic₅	—	3	1	no	—
144	C₂×Dic₅	—	3	2	no	—
145	C₂×Dic₅	—	3	1	no	—
146	C₂×Dic₅	—	3	2	no	—
147	C₄×D₅	—	3	4	no	—
148	C₄×D₅	—	12	2	no	—
149	Dic₁₀	—	12	2	no	—
150	C₄⋊Q₈	—	15	2	no	—
151	C₄⋊Q₈	—	15	1	no	—
152	C₄⋊Q₈	D₁₅	1	1	yes	characteristic, complemented
153	#125	—	15	2	no	—
154	#125	—	15	1	no	—
155	#125	—	15	1	no	—
156	C₂²⋊Q₈	—	15	2	no	—
157	C₂²⋊Q₈	—	15	2	no	—
158	#117	—	15	1	no	—
159	#117	—	15	1	no	—
160	C₂×C₄⋊C₄	—	15	1	no	—
161	C₂×C₃×C₅	C₂²×Q₈	1	1	yes	characteristic
162	C₂×C₃×C₅	H(5,𝔽₂)	1	1	yes	characteristic
163	C₂×C₃×C₅	2₋¹⁺⁴	1	1	yes	characteristic
164	D₁₅	—	2	2	no	—
165	C₂²×D₃	—	5	1	no	—
166	C₃×Q₈	—	4	2	no	—
167	C₂×C₃×C₄	C₂²×D₅	1	2	yes	—
168	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
169	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
170	C₂×C₃×C₄	C₂²×D₅	1	2	yes	—
171	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
172	C₂×Dic₃	—	5	1	no	—
173	C₂×Dic₃	—	5	1	no	—
174	C₂×Dic₃	—	5	2	no	—
175	C₂×Dic₃	—	5	1	no	—
176	C₂×Dic₃	—	5	2	no	—
177	C₄×D₃	—	5	4	no	—
178	C₄×D₃	—	20	2	no	—
179	Dic₆	—	20	2	no	—
180	C₂²×C₅	C₂³×D₃	1	1	yes	characteristic
181	C₂×D₅	—	3	2	no	—
182	C₂×D₅	—	6	1	no	—
183	C₂×D₅	—	6	1	no	—
184	C₄×C₅	—	2	2	no	—
185	C₄×C₅	—	2	2	no	—
186	C₄×C₅	—	2	1	no	—
187	C₄×C₅	—	2	1	no	—
188	C₄×C₅	D₃×Q₈	1	2	yes	—
189	Dic₅	—	3	2	no	—
190	Dic₅	—	6	1	no	—
191	Dic₅	—	6	2	no	—
192	Dic₅	—	6	1	no	—
193	Dic₅	—	6	2	no	—
194	C₂×Q₈	—	15	2	no	—
195	C₂×Q₈	C₂×D₁₅	1	2	yes	—
196	C₂²×C₄	—	15	1	no	—
197	C₂²×C₄	—	15	2	no	—
198	C₄⋊C₄	—	15	2	no	—
199	C₄⋊C₄	—	15	1	no	—
200	C₄⋊C₄	—	15	1	no	—
201	C₄⋊C₄	—	15	1	no	—
202	C₄⋊C₄	—	15	1	no	—
203	C₄⋊C₄	—	15	2	no	—
204	C₄⋊C₄	—	15	2	no	—
205	C₄⋊C₄	—	15	2	no	—
206	C₄⋊C₄	—	15	2	no	—
207	C₄⋊C₄	—	15	2	no	—
208	C₄⋊C₄	C₂×D₁₅	1	1	yes	characteristic
209	C₄⋊C₄	C₂×D₁₅	1	1	yes	characteristic
210	C₄⋊C₄	C₂×D₁₅	1	2	yes	—
211	C₂²⋊C₄	—	15	2	no	—
212	C₂²⋊C₄	—	15	1	no	—
213	C₂²⋊C₄	—	15	1	no	—
214	C₄²	—	15	2	no	—
215	C₄²	—	15	1	no	—
216	C₄²	C₂×D₁₅	1	1	yes	characteristic
217	C₃×C₅	#544	1	1	yes	characteristic, complemented, Hall
218	C₂²×C₃	C₂³×D₅	1	1	yes	characteristic
219	C₂×D₃	—	5	2	no	—
220	C₂×D₃	—	10	1	no	—
221	C₂×D₃	—	10	1	no	—
222	C₃×C₄	—	2	2	no	—
223	C₃×C₄	—	2	2	no	—
224	C₃×C₄	—	2	1	no	—
225	C₃×C₄	—	2	1	no	—
226	C₃×C₄	Q₈×D₅	1	2	yes	—
227	Dic₃	—	5	2	no	—
228	Dic₃	—	10	1	no	—
229	Dic₃	—	10	2	no	—
230	Dic₃	—	10	1	no	—
231	Dic₃	—	10	2	no	—
232	C₂×C₅	#1004	1	1	yes	characteristic
233	C₂×C₅	C₂×D₃×Q₈	1	1	yes	characteristic
234	C₂×C₅	#1007	1	1	yes	characteristic
235	D₅	—	6	2	no	—
236	C₂³	—	15	1	no	—
237	Q₈	—	4	2	no	—
238	Q₈	—	60	2	no	—
239	C₂×C₄	—	15	1	no	—
240	C₂×C₄	—	15	4	no	—
241	C₂×C₄	—	15	1	no	—
242	C₂×C₄	—	15	2	no	—
243	C₂×C₄	—	15	1	no	—
244	C₂×C₄	—	15	2	no	—
245	C₂×C₄	—	60	2	no	—
246	C₂×C₄	C₂²×D₁₅	1	2	yes	—
247	C₂×C₄	C₂²×D₁₅	1	1	yes	characteristic
248	C₂×C₄	C₂²×D₁₅	1	1	yes	characteristic
249	C₂×C₄	C₂²×D₁₅	1	2	yes	—
250	C₂×C₄	C₂²×D₁₅	1	1	yes	characteristic
251	C₂×C₃	#4144	1	1	yes	characteristic
252	C₂×C₃	C₂×Q₈×D₅	1	1	yes	characteristic
253	C₂×C₃	#4147	1	1	yes	characteristic
254	D₃	—	10	2	no	—
255	C₅	#5729	1	1	yes	characteristic, complemented, Sylow
256	C₂²	—	15	2	no	—
257	C₂²	—	30	1	no	—
258	C₂²	—	30	1	no	—
259	C₂²	C₂³×D₁₅	1	1	yes	characteristic
260	C₄	—	2	2	no	—
261	C₄	—	2	1	no	—
262	C₄	—	2	2	no	—
263	C₄	—	2	1	no	—
264	C₄	—	15	2	no	—
265	C₄	—	30	2	no	—
266	C₄	—	30	1	no	—
267	C₄	—	30	1	no	—
268	C₄	—	30	2	no	—
269	C₄	Q₈×D₁₅	1	2	yes	—
270	C₃	#66063	1	1	yes	characteristic, complemented, Sylow
271	C₂	—	30	2	no	—
272	C₂	#92095	1	1	yes	characteristic
273	C₂	C₂×Q₈×D₁₅	1	1	yes	characteristic
274	C₂	#92098	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₁₅
commutator, socle, cosocle	C₂²×C₃×C₅	C₂⁴
Fitting	C₃×C₅×C₄⋊Q₈	C₂
prime radical	C₂²×D₁₅	C₂³
hypercenter	C₄⋊Q₈	D₁₅
hypocenter	C₃×C₅	#544
LCS	G ▷ C₂²×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂² ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
1. C₃×C₅×C₄⋊Q₈	C₂	semidirect
3. #91833	C₂	semidirect
7. #91779	C₂	semidirect
8. #91778	C₂	semidirect
9. #91777	C₂	semidirect
11. #91759	C₂	semidirect
37. C₅×C₄⋊Q₈	D₃	semidirect
61. C₄×D₁₅	Q₈	semidirect
63. C₃×C₄⋊Q₈	D₅	semidirect
152. C₄⋊Q₈	D₁₅	semidirect
217. C₃×C₅	#544	semidirect
255. C₅	#5729	semidirect
270. C₃	#66063	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₅×C₄⋊Q₈	2
#91835	2
#91833	2
#91785	2
#91781	2
C₄⋊C₄×D₁₅	2
#91779	2
#91778	2
#91777	2
#91761	2
#91759	2
#66063	3
#5729	5