Subgroups — #30602150

Order 15616.

490 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21461734	C₂	1	1	yes	maximal, characteristic
2	#21461533	C₂	1	1	yes	maximal, characteristic
3	#21460655	C₂	1	4	yes	maximal
4	C₄⋊Q₈×Dic₆₁	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊#92420	C₂²	1	2	yes	—
6	C₂×C₄×#92420	C₂²	1	2	yes	—
7	#16840628	—	2	1	no	—
8	#16840628	—	2	1	no	—
9	#16840628	—	2	1	no	—
10	#16840628	—	2	1	no	—
11	C₂×Q₈×Dic₆₁	—	2	1	no	—
12	#16839669	—	2	1	no	—
13	#16839661	—	2	1	no	—
14	#16839661	C₄	1	2	yes	—
15	#16839632	C₄	1	1	yes	characteristic
16	C₄⋊C₄×Dic₆₁	—	2	1	no	—
17	C₄⋊C₄×Dic₆₁	C₂²	1	2	yes	—
18	C₄²×Dic₆₁	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊Q₈×C₆₁	C₄	1	1	yes	characteristic
20	C₂²×#92420	—	2	2	no	—
21	C₂²×#92420	—	2	2	no	—
22	C₄⋊#92420	—	2	4	no	—
23	C₄×#92420	—	2	2	no	—
24	C₄×#92420	Q₈	1	4	yes	complemented
25	Q₈×Dic₆₁	—	4	4	no	—
26	C₂×C₄⋊Dic₆₁	—	2	1	no	—
27	C₂×C₄⋊Dic₆₁	—	2	1	no	—
28	C₂×C₄⋊Dic₆₁	—	2	2	no	—
29	C₂×C₄⋊Dic₆₁	C₂×C₄	1	2	yes	—
30	C₂×C₄×Dic₆₁	—	2	2	no	—
31	C₂×C₄×Dic₆₁	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄×Dic₆₁	D₄	1	1	yes	characteristic
33	C₂×C₄×Dic₆₁	D₄	1	1	yes	characteristic
34	C₂×C₄×Dic₆₁	D₄	1	1	yes	characteristic
35	C₂×C₄×Dic₆₁	D₄	1	1	yes	characteristic
36	C₂×C₄×Dic₆₁	C₂³	1	1	yes	characteristic
37	#14519684	—	2	2	no	—
38	#14519684	—	2	2	no	—
39	C₂²×Q₈×C₆₁	—	2	1	no	—
40	C₄⋊Q₈×C₆₁	—	2	2	no	—
41	C₄⋊Q₈×C₆₁	C₈	1	4	yes	complemented
42	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
43	C₂×C₄⋊C₄×C₆₁	C₂×C₄	1	2	yes	—
44	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
45	C₂×Q₈×C₆₁	—	2	4	no	—
46	C₂×Q₈×C₆₁	—	4	2	no	—
47	C₂×Q₈×C₆₁	—	4	2	no	—
48	C₂²×C₄×C₆₁	—	2	2	no	—
49	C₂²×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
50	C₂²×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
51	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
52	C₂²×Dic₆₁	—	2	2	no	—
53	C₂²×Dic₆₁	C₂²⋊C₄	1	1	yes	characteristic
54	C₂²×Dic₆₁	C₂²⋊C₄	1	1	yes	characteristic
55	C₂²×Dic₆₁	C₂×D₄	1	1	yes	characteristic
56	C₂²×Dic₆₁	C₂×D₄	1	1	yes	characteristic
57	C₂×#92420	—	2	4	no	—
58	C₂×#92420	—	4	2	no	—
59	C₂×#92420	—	4	2	no	—
60	C₂×#92420	—	4	2	no	—
61	C₂×#92420	—	4	2	no	—
62	C₂×#92420	—	4	2	no	—
63	C₄⋊C₄×C₆₁	—	2	4	no	—
64	C₄⋊C₄×C₆₁	—	2	2	no	—
65	C₄⋊C₄×C₆₁	C₂×C₈	1	4	yes	—
66	C₄²×C₆₁	C₂×C₈	1	2	yes	—
67	C₄²×C₆₁	M₁₆	1	2	yes	—
68	C₄⋊Dic₆₁	—	2	4	no	—
69	C₄⋊Dic₆₁	—	4	4	no	—
70	C₄⋊Dic₆₁	—	4	2	no	—
71	C₄⋊Dic₆₁	—	4	2	no	—
72	C₄×Dic₆₁	—	2	2	no	—
73	C₄×Dic₆₁	—	2	2	no	—
74	C₄×Dic₆₁	—	2	2	no	—
75	C₄×Dic₆₁	—	4	1	no	—
76	C₄×Dic₆₁	—	4	1	no	—
77	C₄×Dic₆₁	—	4	4	no	—
78	C₄×Dic₆₁	C₂×Q₈	1	2	yes	—
79	C₄×Dic₆₁	#41	1	2	yes	—
80	C₂³×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
81	Q₈×C₆₁	—	4	8	no	—
82	C₂×C₄×C₆₁	—	2	1	no	—
83	C₂×C₄×C₆₁	—	2	1	no	—
84	C₂×C₄×C₆₁	—	2	1	no	—
85	C₂×C₄×C₆₁	—	2	1	no	—
86	C₂×C₄×C₆₁	—	2	4	no	—
87	C₂×C₄×C₆₁	—	4	2	no	—
88	C₂×C₄×C₆₁	—	4	2	no	—
89	C₂×C₄×C₆₁	C₂²⋊C₈	1	2	yes	—
90	C₂×C₄×C₆₁	C₂²⋊C₈	1	2	yes	—
91	C₂×C₄×C₆₁	C₄×Q₈	1	2	yes	—
92	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
93	C₂×C₄×C₆₁	C₂×M₁₆	1	1	yes	characteristic
94	C₂×C₄×C₆₁	#131	1	2	yes	—
95	C₂×Dic₆₁	—	2	1	no	—
96	C₂×Dic₆₁	—	2	1	no	—
97	C₂×Dic₆₁	—	2	1	no	—
98	C₂×Dic₆₁	—	2	1	no	—
99	C₂×Dic₆₁	—	2	1	no	—
100	C₂×Dic₆₁	—	2	1	no	—
101	C₂×Dic₆₁	—	2	1	no	—
102	C₂×Dic₆₁	—	2	1	no	—
103	C₂×Dic₆₁	—	4	2	no	—
104	C₂×Dic₆₁	—	4	2	no	—
105	C₂×Dic₆₁	—	4	2	no	—
106	C₂×Dic₆₁	C₂²⋊Q₈	1	2	yes	—
107	C₂×Dic₆₁	#124	1	1	yes	characteristic
108	C₂×Dic₆₁	C₄⋊Q₈	1	1	yes	characteristic
109	C₂×Dic₆₁	AGL(1,ℤ/8)	1	2	yes	—
110	C₂×Dic₆₁	#137	1	2	yes	—
111	#92420	—	4	4	no	—
112	#92420	—	8	2	no	—
113	#9873	—	61	1	no	maximal, Sylow
114	C₂²×C₆₁	#391	1	1	yes	characteristic
115	C₂²×C₆₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
116	C₂²×C₆₁	#408	1	1	yes	characteristic
117	C₂²×C₆₁	#418	1	1	yes	characteristic
118	C₂²×C₆₁	#419	1	1	yes	characteristic
119	C₂²×C₆₁	#421	1	1	yes	characteristic
120	C₂²×C₆₁	#421	1	1	yes	characteristic
121	C₄×C₆₁	—	2	2	no	—
122	C₄×C₆₁	—	2	2	no	—
123	C₄×C₆₁	—	4	4	no	—
124	C₄×C₆₁	C₈×Q₈	1	2	yes	—
125	C₄×C₆₁	#446	1	2	yes	—
126	Dic₆₁	—	2	2	no	—
127	Dic₆₁	—	4	1	no	—
128	Dic₆₁	—	4	1	no	—
129	Dic₆₁	—	8	2	no	—
130	Dic₆₁	#489	1	2	yes	—
131	Dic₆₁	#501	1	2	yes	—
132	C₄×C₄⋊Q₈	—	61	1	no	—
133	#1831	—	61	1	no	—
134	#1744	—	61	1	no	—
135	#1279	—	61	4	no	—
136	C₂×C₆₁	#1478	1	1	yes	characteristic
137	C₂×C₆₁	#1479	1	1	yes	characteristic
138	C₂×C₆₁	#1496	1	1	yes	characteristic
139	C₂×C₆₁	#1497	1	1	yes	characteristic
140	C₂×C₆₁	#1984	1	1	yes	characteristic
141	C₂×C₆₁	#1993	1	1	yes	characteristic
142	C₂×C₆₁	#1995	1	1	yes	characteristic
143	C₂×C₄⋊Q₈	#6923	1	1	yes	characteristic
144	C₂×C₄×Q₈	—	122	1	no	—
145	C₂×C₄⋊C₈	—	61	2	no	—
146	C₂×C₄×C₈	—	61	2	no	—
147	#391	—	122	1	no	—
148	#389	—	61	2	no	—
149	#389	—	122	1	no	—
150	#384	—	61	1	no	—
151	C₄×C₄⋊C₄	—	61	2	no	—
152	C₄×C₄⋊C₄	—	122	1	no	—
153	C₄³	—	61	1	no	—
154	#336	—	122	1	no	—
155	#336	—	122	1	no	—
156	#336	—	122	1	no	—
157	#336	—	122	1	no	—
158	C₆₁	#9873	1	1	yes	characteristic, complemented, Sylow
159	C₂²×Q₈	—	2	1	no	—
160	C₂²×C₈	—	122	2	no	—
161	C₂²×C₈	—	122	2	no	—
162	C₄⋊Q₈	—	2	2	no	—
163	C₄⋊Q₈	#92420	1	4	yes	complemented
164	C₄×Q₈	—	244	4	no	—
165	C₂×C₄⋊C₄	—	2	1	no	—
166	C₂×C₄⋊C₄	—	61	2	no	—
167	C₂×C₄⋊C₄	—	122	2	no	—
168	C₂×C₄⋊C₄	—	122	1	no	—
169	C₂×C₄⋊C₄	—	122	1	no	—
170	C₂×C₄⋊C₄	C₂×#6923	1	2	yes	—
171	C₂×C₄²	—	61	1	no	—
172	C₂×C₄²	—	61	1	no	—
173	C₂×C₄²	—	61	1	no	—
174	C₂×C₄²	—	61	1	no	—
175	C₂×C₄²	—	61	1	no	—
176	C₂×C₄²	—	61	1	no	—
177	C₂×C₄²	—	122	2	no	—
178	C₂×C₄²	C₂×#6923	1	1	yes	characteristic
179	C₄⋊C₈	—	122	4	no	—
180	C₄×C₈	—	61	4	no	—
181	C₄×C₈	—	122	2	no	—
182	#95	—	122	2	no	—
183	#95	—	122	2	no	—
184	C₂×Q₈	—	2	4	no	—
185	C₂×Q₈	—	4	2	no	—
186	C₂×Q₈	—	4	2	no	—
187	C₂²×C₄	—	2	2	no	—
188	C₂²×C₄	—	61	1	no	—
189	C₂²×C₄	—	61	1	no	—
190	C₂²×C₄	—	61	1	no	—
191	C₂²×C₄	—	61	1	no	—
192	C₂²×C₄	—	122	2	no	—
193	C₂²×C₄	C₂²⋊#6923	1	1	yes	characteristic
194	C₂²×C₄	C₂²⋊#6923	1	1	yes	characteristic
195	C₂²×C₄	C₂²×#6923	1	1	yes	characteristic
196	C₂×C₈	—	122	4	no	—
197	C₂×C₈	—	244	2	no	—
198	C₂×C₈	—	244	2	no	—
199	C₂×C₈	—	244	2	no	—
200	C₂×C₈	—	244	2	no	—
201	C₂×C₈	—	244	2	no	—
202	C₄⋊C₄	—	2	4	no	—
203	C₄⋊C₄	—	2	2	no	—
204	C₄⋊C₄	—	122	4	no	—
205	C₄⋊C₄	—	244	2	no	—
206	C₄⋊C₄	—	244	2	no	—
207	C₄⋊C₄	—	244	4	no	—
208	C₄⋊C₄	C₂×#92420	1	4	yes	—
209	C₄²	—	61	2	no	—
210	C₄²	—	61	2	no	—
211	C₄²	—	122	2	no	—
212	C₄²	—	122	2	no	—
213	C₄²	—	122	2	no	—
214	C₄²	—	244	1	no	—
215	C₄²	—	244	1	no	—
216	C₄²	—	244	4	no	—
217	C₄²	C₂×#92420	1	2	yes	—
218	C₄²	#11758217	1	2	yes	—
219	C₂³	C₂×C₂²⋊#6923	1	1	yes	characteristic
220	Q₈	—	4	8	no	—
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	2	1	no	—
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	4	no	—
226	C₂×C₄	—	4	2	no	—
227	C₂×C₄	—	4	2	no	—
228	C₂×C₄	—	61	1	no	—
229	C₂×C₄	—	61	1	no	—
230	C₂×C₄	—	61	2	no	—
231	C₂×C₄	—	61	2	no	—
232	C₂×C₄	—	61	2	no	—
233	C₂×C₄	—	122	1	no	—
234	C₂×C₄	—	122	1	no	—
235	C₂×C₄	—	122	1	no	—
236	C₂×C₄	—	122	1	no	—
237	C₂×C₄	—	122	1	no	—
238	C₂×C₄	—	122	1	no	—
239	C₂×C₄	—	122	1	no	—
240	C₂×C₄	—	122	1	no	—
241	C₂×C₄	—	244	2	no	—
242	C₂×C₄	—	244	2	no	—
243	C₂×C₄	—	244	2	no	—
244	C₂×C₄	C₂²⋊#92420	1	2	yes	—
245	C₂×C₄	C₂²⋊#92420	1	2	yes	—
246	C₂×C₄	Q₈×#6923	1	2	yes	—
247	C₂×C₄	C₂²×#92420	1	1	yes	characteristic
248	C₂×C₄	C₂×#11758217	1	1	yes	characteristic
249	C₂×C₄	#14519865	1	2	yes	—
250	C₈	—	488	2	no	—
251	C₈	—	244	4	no	—
252	C₂²	(C₂×#6923)⋊Q₈	1	1	yes	characteristic
253	C₂²	C₂×C₂²⋊#92420	1	1	yes	characteristic
254	C₂²	#16840737	1	1	yes	characteristic
255	C₂²	#16840754	1	1	yes	characteristic
256	C₂²	#16840755	1	1	yes	characteristic
257	C₂²	#16840760	1	1	yes	characteristic
258	C₂²	#16840761	1	1	yes	characteristic
259	C₄	—	2	2	no	—
260	C₄	—	2	2	no	—
261	C₄	—	4	4	no	—
262	C₄	—	61	2	no	—
263	C₄	—	61	2	no	—
264	C₄	—	122	2	no	—
265	C₄	—	488	2	no	—
266	C₄	—	244	1	no	—
267	C₄	—	244	1	no	—
268	C₄	Q₈×#92420	1	2	yes	—
269	C₄	#16840813	1	2	yes	—
270	C₂	#21461118	1	1	yes	characteristic
271	C₂	#21461121	1	1	yes	characteristic
272	C₂	#21461180	1	1	yes	characteristic
273	C₂	#21461183	1	1	yes	characteristic
274	C₂	#21461988	1	1	yes	characteristic
275	C₂	#21462003	1	1	yes	characteristic
276	C₂	#21462007	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊#6923
commutator	C₂×C₄×C₆₁	C₂²×C₈
Frattini	C₂²×C₄	C₂²×#6923
Fitting	C₂×C₄⋊Q₈×C₆₁	C₄
socle, prime radical	C₂³×C₆₁	C₂×C₂²⋊C₄
cosocle	C₂×C₄×Dic₆₁	C₂³
hypercenter	C₂×C₄⋊Q₈	#6923
hypocenter	C₆₁	#9873
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
24. C₄×#92420	Q₈	semidirect
41. C₄⋊Q₈×C₆₁	C₈	semidirect
158. C₆₁	#9873	semidirect
163. C₄⋊Q₈	#92420	semidirect

Maximal subgroups

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

subgroup	index
#21461734	2
#21461533	2
#21460655	2
C₄⋊Q₈×Dic₆₁	2
#9873	61