Subgroups — #30619467

Order 15616.

714 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21461988	C₂	1	4	yes	maximal
2	#21461916	C₂	1	2	yes	maximal
3	#21461872	C₂	1	4	yes	maximal
4	#21461734	C₂	1	1	yes	maximal, characteristic
5	C₄⋊C₄×#92420	C₂	1	2	yes	maximal
6	#21461533	C₂	1	1	yes	maximal, characteristic
7	C₄⋊Q₈×Dic₆₁	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄⋊#92420	C₂²	1	2	yes	—
9	C₂×C₄⋊#92420	C₂²	1	4	yes	—
10	C₂×C₄×#92420	C₂²	1	1	yes	characteristic
11	C₂×C₄×#92420	C₂²	1	4	yes	—
12	C₂×C₄×#92420	C₂²	1	1	yes	characteristic
13	#16840628	C₂²	1	2	yes	—
14	#16840628	C₂²	1	2	yes	—
15	#16840628	C₂²	1	4	yes	—
16	#16840628	C₂²	1	2	yes	—
17	#16840628	C₂²	1	4	yes	—
18	#16840628	C₂²	1	2	yes	—
19	C₂×Q₈×Dic₆₁	C₂²	1	2	yes	—
20	#16839669	C₄	1	2	yes	—
21	#16839661	C₄	1	2	yes	—
22	#16839661	C₄	1	2	yes	—
23	#16839632	C₄	1	1	yes	characteristic
24	C₄⋊C₄×Dic₆₁	C₂²	1	2	yes	—
25	C₄⋊C₄×Dic₆₁	C₂²	1	2	yes	—
26	C₄²×Dic₆₁	C₂²	1	1	yes	characteristic
27	C₂×C₄⋊Q₈×C₆₁	C₄	1	1	yes	characteristic
28	C₂²×#92420	C₂³	1	2	yes	—
29	C₂²×#92420	C₂³	1	4	yes	—
30	C₂²×#92420	C₂³	1	2	yes	—
31	C₄⋊#92420	—	2	4	no	—
32	C₄⋊#92420	—	4	4	no	—
33	C₄×#92420	—	2	2	no	—
34	C₄×#92420	—	4	4	no	—
35	C₄×#92420	Q₈	1	4	yes	complemented
36	Q₈×Dic₆₁	—	4	4	no	—
37	C₂×C₄⋊Dic₆₁	C₂×C₄	1	2	yes	—
38	C₂×C₄⋊Dic₆₁	C₂×C₄	1	2	yes	—
39	C₂×C₄⋊Dic₆₁	C₂×C₄	1	4	yes	—
40	C₂×C₄⋊Dic₆₁	C₂×C₄	1	2	yes	—
41	C₂×C₄×Dic₆₁	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄×Dic₆₁	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄×Dic₆₁	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄×Dic₆₁	C₂³	1	4	yes	—
45	C₂×C₄×Dic₆₁	C₂³	1	1	yes	characteristic
46	C₂×C₄×Dic₆₁	C₂³	1	1	yes	characteristic
47	C₂×C₄×Dic₆₁	C₂³	1	1	yes	characteristic
48	#14519684	C₂×C₄	1	4	yes	—
49	#14519684	C₂×C₄	1	4	yes	—
50	C₂²×Q₈×C₆₁	C₂×C₄	1	2	yes	—
51	C₄⋊Q₈×C₆₁	C₈	1	8	yes	complemented
52	C₂×C₄⋊C₄×C₆₁	C₂×C₄	1	2	yes	—
53	C₂×C₄⋊C₄×C₆₁	C₂×C₄	1	2	yes	—
54	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
55	C₂×Q₈×C₆₁	—	4	2	no	—
56	C₂×Q₈×C₆₁	—	4	2	no	—
57	C₂×Q₈×C₆₁	C₂×C₈	1	8	yes	—
58	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
59	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
60	C₂²×C₄×C₆₁	C₂²×C₄	1	4	yes	—
61	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
62	C₂²×Dic₆₁	C₂²×C₄	1	1	yes	characteristic
63	C₂²×Dic₆₁	C₂²×C₄	1	4	yes	—
64	C₂²×Dic₆₁	C₂²×C₄	1	1	yes	characteristic
65	C₂²×Dic₆₁	C₂²×C₄	1	1	yes	characteristic
66	C₂²×Dic₆₁	C₂⁴	1	1	yes	characteristic
67	C₂×#92420	—	2	2	no	—
68	C₂×#92420	—	2	2	no	—
69	C₂×#92420	—	2	2	no	—
70	C₂×#92420	—	2	4	no	—
71	C₂×#92420	—	2	4	no	—
72	C₂×#92420	—	2	2	no	—
73	C₂×#92420	—	2	2	no	—
74	C₂×#92420	—	2	4	no	—
75	C₂×#92420	C₂×Q₈	1	4	yes	—
76	C₄⋊C₄×C₆₁	C₂×C₈	1	8	yes	—
77	C₄⋊C₄×C₆₁	C₂×C₈	1	8	yes	—
78	C₄²×C₆₁	C₂×C₈	1	4	yes	—
79	C₄⋊Dic₆₁	—	2	4	no	—
80	C₄⋊Dic₆₁	—	4	2	no	—
81	C₄⋊Dic₆₁	—	4	4	no	—
82	C₄⋊Dic₆₁	—	4	2	no	—
83	C₄×Dic₆₁	—	2	2	no	—
84	C₄×Dic₆₁	—	2	4	no	—
85	C₄×Dic₆₁	—	2	4	no	—
86	C₄×Dic₆₁	—	2	2	no	—
87	C₄×Dic₆₁	—	2	2	no	—
88	C₄×Dic₆₁	—	4	1	no	—
89	C₄×Dic₆₁	—	4	1	no	—
90	C₄×Dic₆₁	C₂×Q₈	1	2	yes	—
91	C₄×Dic₆₁	#41	1	2	yes	—
92	C₂³×C₆₁	C₂³×C₄	1	1	yes	characteristic
93	Q₈×C₆₁	—	4	8	no	—
94	C₂×C₄×C₆₁	—	2	1	no	—
95	C₂×C₄×C₆₁	—	2	4	no	—
96	C₂×C₄×C₆₁	—	2	1	no	—
97	C₂×C₄×C₆₁	—	2	1	no	—
98	C₂×C₄×C₆₁	—	2	1	no	—
99	C₂×C₄×C₆₁	—	2	4	no	—
100	C₂×C₄×C₆₁	C₄×Q₈	1	2	yes	—
101	C₂×C₄×C₆₁	C₂²×C₈	1	2	yes	—
102	C₂×C₄×C₆₁	C₂²×C₈	1	2	yes	—
103	C₂×C₄×C₆₁	C₂²×C₈	1	8	yes	—
104	C₂×C₄×C₆₁	C₂²×C₈	1	2	yes	—
105	C₂×C₄×C₆₁	#131	1	2	yes	—
106	C₂×Dic₆₁	—	2	1	no	—
107	C₂×Dic₆₁	—	2	1	no	—
108	C₂×Dic₆₁	—	2	1	no	—
109	C₂×Dic₆₁	—	2	4	no	—
110	C₂×Dic₆₁	—	2	1	no	—
111	C₂×Dic₆₁	—	2	1	no	—
112	C₂×Dic₆₁	—	2	1	no	—
113	C₂×Dic₆₁	—	2	4	no	—
114	C₂×Dic₆₁	—	2	1	no	—
115	C₂×Dic₆₁	—	2	1	no	—
116	C₂×Dic₆₁	—	2	4	no	—
117	C₂×Dic₆₁	C₄×Q₈	1	2	yes	—
118	C₂×Dic₆₁	C₂²×Q₈	1	1	yes	characteristic
119	C₂×Dic₆₁	C₂×#41	1	1	yes	characteristic
120	C₂×Dic₆₁	H(5,𝔽₂)	1	2	yes	—
121	C₂×Dic₆₁	2₋¹⁺⁴	1	2	yes	—
122	#92420	—	2	4	no	—
123	#92420	—	4	2	no	—
124	#92420	—	4	4	no	—
125	#22307	—	61	1	no	maximal, Sylow
126	C₂²×C₆₁	C₂×C₄×Q₈	1	1	yes	characteristic
127	C₂²×C₆₁	#518	1	1	yes	characteristic
128	C₂²×C₆₁	#519	1	1	yes	characteristic
129	C₂²×C₆₁	C₂³×C₈	1	1	yes	characteristic
130	C₂²×C₆₁	C₂×#131	1	1	yes	characteristic
131	C₂²×C₆₁	#568	1	1	yes	characteristic
132	C₂²×C₆₁	#568	1	1	yes	characteristic
133	C₄×C₆₁	—	2	8	no	—
134	C₄×C₆₁	—	2	2	no	—
135	C₄×C₆₁	—	2	2	no	—
136	C₄×C₆₁	C₈×Q₈	1	4	yes	—
137	Dic₆₁	—	2	2	no	—
138	Dic₆₁	—	4	1	no	—
139	Dic₆₁	—	4	4	no	—
140	Dic₆₁	—	4	1	no	—
141	Dic₆₁	#542	1	2	yes	—
142	Dic₆₁	#544	1	2	yes	—
143	C₄×C₄⋊Q₈	—	61	1	no	—
144	#1984	—	61	4	no	—
145	#1939	—	61	2	no	—
146	#1916	—	61	4	no	—
147	#1831	—	61	1	no	—
148	C₈×C₄⋊C₄	—	61	2	no	—
149	#1744	—	61	1	no	—
150	C₂×C₆₁	#2331	1	1	yes	characteristic
151	C₂×C₆₁	C₂×C₈×Q₈	1	1	yes	characteristic
152	C₂×C₆₁	#2961	1	1	yes	characteristic
153	C₂×C₆₁	#2972	1	1	yes	characteristic
154	C₂×C₆₁	#2981	1	1	yes	characteristic
155	C₂×C₆₁	#2982	1	1	yes	characteristic
156	C₂×C₆₁	#2985	1	1	yes	characteristic
157	C₂×C₄⋊Q₈	#6923	1	1	yes	characteristic
158	C₂×C₄×Q₈	—	61	2	no	—
159	C₂×C₄⋊C₈	—	61	2	no	—
160	C₂×C₄⋊C₈	—	61	4	no	—
161	C₂×C₄×C₈	—	61	1	no	—
162	C₂×C₄×C₈	—	61	1	no	—
163	C₂×C₄×C₈	—	61	4	no	—
164	#391	—	61	2	no	—
165	#389	—	61	2	no	—
166	#389	—	61	2	no	—
167	#384	—	61	1	no	—
168	C₄×C₄⋊C₄	—	61	2	no	—
169	C₄×C₄⋊C₄	—	61	2	no	—
170	C₄³	—	61	1	no	—
171	#336	—	61	2	no	—
172	#336	—	61	2	no	—
173	#336	—	61	2	no	—
174	#336	—	61	4	no	—
175	#336	—	61	4	no	—
176	#336	—	61	2	no	—
177	C₆₁	#22307	1	1	yes	characteristic, complemented, Sylow
178	C₂²×Q₈	C₂×#6923	1	2	yes	—
179	C₂²×C₈	—	61	2	no	—
180	C₂²×C₈	—	61	2	no	—
181	C₂²×C₈	—	61	4	no	—
182	C₄⋊Q₈	#92420	1	8	yes	complemented
183	C₄×Q₈	—	244	4	no	—
184	C₂×C₄⋊C₄	—	61	2	no	—
185	C₂×C₄⋊C₄	—	61	2	no	—
186	C₂×C₄⋊C₄	—	61	4	no	—
187	C₂×C₄⋊C₄	—	61	2	no	—
188	C₂×C₄⋊C₄	C₂×#6923	1	2	yes	—
189	C₂×C₄⋊C₄	C₂×#6923	1	2	yes	—
190	C₂×C₄²	—	61	1	no	—
191	C₂×C₄²	—	61	1	no	—
192	C₂×C₄²	—	61	4	no	—
193	C₂×C₄²	—	61	1	no	—
194	C₂×C₄²	—	61	1	no	—
195	C₂×C₄²	—	61	1	no	—
196	C₂×C₄²	—	61	1	no	—
197	C₂×C₄²	C₂×#6923	1	1	yes	characteristic
198	C₄⋊C₈	—	122	4	no	—
199	C₄⋊C₈	—	244	4	no	—
200	C₄×C₈	—	61	4	no	—
201	C₄×C₈	—	122	2	no	—
202	C₄×C₈	—	244	4	no	—
203	#95	—	61	4	no	—
204	#95	—	61	4	no	—
205	C₂×Q₈	—	4	2	no	—
206	C₂×Q₈	—	4	2	no	—
207	C₂×Q₈	C₂×#92420	1	8	yes	—
208	C₂²×C₄	—	61	1	no	—
209	C₂²×C₄	—	61	1	no	—
210	C₂²×C₄	—	61	1	no	—
211	C₂²×C₄	—	61	4	no	—
212	C₂²×C₄	—	61	1	no	—
213	C₂²×C₄	C₂²×#6923	1	4	yes	—
214	C₂²×C₄	C₂²×#6923	1	1	yes	characteristic
215	C₂²×C₄	C₂²×#6923	1	1	yes	characteristic
216	C₂²×C₄	C₂²×#6923	1	1	yes	characteristic
217	C₂×C₈	—	61	4	no	—
218	C₂×C₈	—	122	2	no	—
219	C₂×C₈	—	122	2	no	—
220	C₂×C₈	—	122	2	no	—
221	C₂×C₈	—	122	4	no	—
222	C₂×C₈	—	122	4	no	—
223	C₂×C₈	—	122	2	no	—
224	C₂×C₈	—	122	2	no	—
225	C₂×C₈	—	122	4	no	—
226	C₄⋊C₄	—	122	4	no	—
227	C₄⋊C₄	—	244	2	no	—
228	C₄⋊C₄	—	244	2	no	—
229	C₄⋊C₄	—	244	4	no	—
230	C₄⋊C₄	C₂×#92420	1	8	yes	—
231	C₄⋊C₄	C₂×#92420	1	8	yes	—
232	C₄²	—	61	2	no	—
233	C₄²	—	61	2	no	—
234	C₄²	—	122	4	no	—
235	C₄²	—	122	4	no	—
236	C₄²	—	122	2	no	—
237	C₄²	—	122	2	no	—
238	C₄²	—	122	2	no	—
239	C₄²	—	244	1	no	—
240	C₄²	—	244	1	no	—
241	C₄²	C₂×#92420	1	4	yes	—
242	C₂³	C₂³×#6923	1	1	yes	characteristic
243	Q₈	—	4	8	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	2	4	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	2	1	no	—
248	C₂×C₄	—	2	4	no	—
249	C₂×C₄	—	2	1	no	—
250	C₂×C₄	—	61	1	no	—
251	C₂×C₄	—	61	1	no	—
252	C₂×C₄	—	61	2	no	—
253	C₂×C₄	—	61	2	no	—
254	C₂×C₄	—	61	2	no	—
255	C₂×C₄	—	122	1	no	—
256	C₂×C₄	—	122	1	no	—
257	C₂×C₄	—	122	1	no	—
258	C₂×C₄	—	122	4	no	—
259	C₂×C₄	—	122	1	no	—
260	C₂×C₄	—	122	1	no	—
261	C₂×C₄	—	122	1	no	—
262	C₂×C₄	—	122	4	no	—
263	C₂×C₄	—	122	1	no	—
264	C₂×C₄	—	122	1	no	—
265	C₂×C₄	—	122	4	no	—
266	C₂×C₄	Q₈×#6923	1	2	yes	—
267	C₂×C₄	C₂²×#92420	1	2	yes	—
268	C₂×C₄	C₂²×#92420	1	2	yes	—
269	C₂×C₄	C₂²×#92420	1	8	yes	—
270	C₂×C₄	C₂²×#92420	1	2	yes	—
271	C₂×C₄	#14519865	1	2	yes	—
272	C₈	—	122	4	no	—
273	C₈	—	244	2	no	—
274	C₈	—	244	4	no	—
275	C₂²	C₂×Q₈×#6923	1	1	yes	characteristic
276	C₂²	#16840829	1	1	yes	characteristic
277	C₂²	#16840830	1	1	yes	characteristic
278	C₂²	C₂³×#92420	1	1	yes	characteristic
279	C₂²	C₂×#14519865	1	1	yes	characteristic
280	C₂²	#16840840	1	1	yes	characteristic
281	C₂²	#16840842	1	1	yes	characteristic
282	C₄	—	2	8	no	—
283	C₄	—	2	2	no	—
284	C₄	—	2	2	no	—
285	C₄	—	61	2	no	—
286	C₄	—	61	2	no	—
287	C₄	—	122	2	no	—
288	C₄	—	244	1	no	—
289	C₄	—	244	4	no	—
290	C₄	—	244	1	no	—
291	C₄	Q₈×#92420	1	4	yes	—
292	C₂	#21462314	1	1	yes	characteristic
293	C₂	C₂×Q₈×#92420	1	1	yes	characteristic
294	C₂	#21462612	1	1	yes	characteristic
295	C₂	#21462666	1	1	yes	characteristic
296	C₂	#21462711	1	1	yes	characteristic
297	C₂	#21462712	1	1	yes	characteristic
298	C₂	#21462727	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×#6923
commutator	C₂²×C₆₁	C₂³×C₈
Fitting	C₂×C₄⋊Q₈×C₆₁	C₄
socle, prime radical	C₂³×C₆₁	C₂³×C₄
cosocle	C₂²×Dic₆₁	C₂⁴
hypercenter	C₂×C₄⋊Q₈	#6923
hypocenter	C₆₁	#22307
LCS	G ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
35. C₄×#92420	Q₈	semidirect
51. C₄⋊Q₈×C₆₁	C₈	semidirect
177. C₆₁	#22307	semidirect
182. 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
#21461988	2
#21461916	2
#21461872	2
#21461734	2
C₄⋊C₄×#92420	2
#21461533	2
C₄⋊Q₈×Dic₆₁	2
#22307	61