Subgroups — #29698912

Order 15616.

638 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×#11758194	C₂	1	2	yes	maximal
2	#21446595	C₂	1	2	yes	maximal
3	#21446595	C₂	1	4	yes	maximal
4	#21445994	C₂	1	2	yes	maximal
5	#21445991	C₂	1	1	yes	maximal, characteristic
6	#21445725	C₂	1	2	yes	maximal
7	C₄×#14519713	C₂	1	1	yes	maximal, characteristic
8	C₂×C₆₁×#446	C₂	1	1	yes	maximal, characteristic
9	#16839893	—	2	2	no	—
10	#16839893	—	2	4	no	—
11	Q₈×#92418	—	2	2	no	—
12	C₂×#14519713	C₂²	1	1	yes	characteristic
13	C₂×#14519713	C₂²	1	2	yes	—
14	C₂×#14519713	C₂²	1	2	yes	—
15	C₂×#14519713	C₂²	1	4	yes	—
16	C₂×#14519713	C₂²	1	1	yes	characteristic
17	#16839786	C₂²	1	2	yes	—
18	#16839786	C₂²	1	4	yes	—
19	C₄×#11758194	C₂²	1	2	yes	—
20	C₄×#11758194	C₂²	1	4	yes	—
21	#16839527	C₂²	1	2	yes	—
22	#16839527	C₂²	1	4	yes	—
23	C₂×C₄×Q₈×C₆₁	C₂²	1	1	yes	characteristic
24	C₆₁×#446	C₄	1	4	yes	—
25	C₆₁×#446	C₄	1	4	yes	—
26	C₂×C₄⋊C₈×C₆₁	C₂²	1	1	yes	characteristic
27	C₂×C₄⋊C₈×C₆₁	C₂²	1	2	yes	—
28	C₂×#97×C₆₁	C₂²	1	2	yes	—
29	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
30	C₂×#11758194	C₂³	1	2	yes	—
31	C₂×#11758194	C₂³	1	4	yes	—
32	C₂×#11758194	C₂³	1	2	yes	—
33	#14519713	—	2	2	no	—
34	#14519713	—	2	4	no	—
35	#14519713	—	4	2	no	—
36	#14519713	—	4	4	no	—
37	#14519713	Q₈	1	4	yes	complemented
38	C₄⋊#92418	—	2	2	no	—
39	C₄⋊#92418	—	2	4	no	—
40	C₄⋊#92418	—	2	4	no	—
41	C₄⋊#92418	—	2	2	no	—
42	#14519688	—	2	4	no	—
43	#14519688	—	2	2	no	—
44	C₄×#92418	—	2	2	no	—
45	C₄×#92418	—	2	4	no	—
46	C₂²×Q₈×C₆₁	Q₈	1	1	yes	characteristic
47	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
48	C₂²×C₈×C₆₁	C₂³	1	2	yes	—
49	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
50	C₄×Q₈×C₆₁	C₂×C₄	1	2	yes	—
51	C₄×Q₈×C₆₁	C₂×C₄	1	2	yes	—
52	C₄×Q₈×C₆₁	D₄	1	2	yes	—
53	C₄×Q₈×C₆₁	D₄	1	2	yes	—
54	C₂×C₄⋊C₄×C₆₁	Q₈	1	1	yes	characteristic
55	C₂×C₄⋊C₄×C₆₁	Q₈	1	2	yes	—
56	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
57	C₂×C₄²×C₆₁	C₂³	1	2	yes	—
58	C₄⋊C₈×C₆₁	C₂×C₄	1	2	yes	—
59	C₄⋊C₈×C₆₁	C₂×C₄	1	2	yes	—
60	C₄⋊C₈×C₆₁	C₂×C₄	1	8	yes	—
61	#97×C₆₁	C₂×C₄	1	8	yes	—
62	C₄×C₈×C₆₁	C₂×C₄	1	2	yes	—
63	C₄×C₈×C₆₁	C₂×C₄	1	2	yes	—
64	C₂×Q₈×C₆₁	—	4	2	no	—
65	C₂×Q₈×C₆₁	C₄⋊C₄	1	2	yes	—
66	C₂×Q₈×C₆₁	C₄⋊C₄	1	2	yes	—
67	C₂²×C₄×C₆₁	C₂×Q₈	1	1	yes	characteristic
68	C₂²×C₄×C₆₁	C₂×Q₈	1	1	yes	characteristic
69	C₂²×C₄×C₆₁	C₂×Q₈	1	2	yes	—
70	C₂²×C₄×C₆₁	C₂×Q₈	1	2	yes	—
71	C₂²×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
72	C₂×C₈×C₆₁	—	2	1	no	—
73	C₂×C₈×C₆₁	—	2	2	no	—
74	C₂×C₈×C₆₁	—	2	1	no	—
75	C₂×C₈×C₆₁	—	2	2	no	—
76	C₂×C₈×C₆₁	C₂²×C₄	1	2	yes	—
77	C₂×C₈×C₆₁	C₂²×C₄	1	4	yes	—
78	C₂×C₈×C₆₁	C₂²×C₄	1	2	yes	—
79	C₂×C₈×C₆₁	C₂×Q₈	1	2	yes	—
80	C₂×C₈×C₆₁	#41	1	2	yes	—
81	C₄⋊C₄×C₆₁	C₄⋊C₄	1	4	yes	—
82	C₄⋊C₄×C₆₁	C₄⋊C₄	1	4	yes	—
83	C₄⋊C₄×C₆₁	C₄⋊C₄	1	2	yes	—
84	C₄⋊C₄×C₆₁	C₄⋊C₄	1	2	yes	—
85	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
86	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
87	C₄²×C₆₁	C₂²×C₄	1	4	yes	—
88	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
89	C₄²×C₆₁	C₂×D₄	1	4	yes	—
90	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
91	#11758194	—	2	2	no	—
92	#11758194	—	2	2	no	—
93	#11758194	—	2	4	no	—
94	#11758194	—	2	4	no	—
95	#11758194	—	2	2	no	—
96	#11758194	C₂×Q₈	1	4	yes	—
97	C₂×#92418	—	2	2	no	—
98	C₂×#92418	—	2	2	no	—
99	C₂×#92418	—	2	4	no	—
100	C₂³×C₆₁	C₂²×Q₈	1	1	yes	characteristic
101	Q₈×C₆₁	—	2	4	no	—
102	Q₈×C₆₁	—	4	2	no	—
103	C₂×C₄×C₆₁	—	2	2	no	—
104	C₂×C₄×C₆₁	—	4	1	no	—
105	C₂×C₄×C₆₁	—	4	2	no	—
106	C₂×C₄×C₆₁	—	4	2	no	—
107	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
108	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
109	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	4	yes	—
110	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
111	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	4	yes	—
112	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
113	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
114	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
115	C₂×C₄×C₆₁	C₂²×Q₈	1	1	yes	characteristic
116	C₂×C₄×C₆₁	C₂×#41	1	1	yes	characteristic
117	C₂×C₄×C₆₁	H(5,𝔽₂)	1	1	yes	characteristic
118	C₂×C₄×C₆₁	2₋¹⁺⁴	1	1	yes	characteristic
119	C₈×C₆₁	—	2	2	no	—
120	C₈×C₆₁	—	2	4	no	—
121	C₈×C₆₁	C₄×Q₈	1	4	yes	—
122	#92418	—	2	4	no	—
123	#92418	—	4	2	no	—
124	#92418	—	4	4	no	—
125	#22733	—	61	1	no	maximal, Sylow
126	C₂²×C₆₁	—	2	1	no	—
127	C₂²×C₆₁	—	2	1	no	—
128	C₂²×C₆₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
129	C₂²×C₆₁	#557	1	1	yes	characteristic
130	C₂²×C₆₁	Q₈²	1	1	yes	characteristic
131	C₄×C₆₁	—	2	1	no	—
132	C₄×C₆₁	—	2	4	no	—
133	C₄×C₆₁	—	2	1	no	—
134	C₄×C₆₁	—	2	1	no	—
135	C₄×C₆₁	—	2	4	no	—
136	C₄×C₆₁	#430	1	2	yes	—
137	C₄×C₆₁	C₂×C₄×Q₈	1	1	yes	characteristic
138	C₄×C₆₁	#520	1	1	yes	characteristic
139	C₄×C₆₁	D₄×Q₈	1	1	yes	characteristic
140	C₄×C₆₁	#550	1	1	yes	characteristic
141	Q₈×M₁₆	—	61	2	no	—
142	C₂×#446	D₆₁	1	1	yes	characteristic
143	#1950	—	61	4	no	—
144	#1950	—	61	2	no	—
145	#1849	—	61	2	no	—
146	#1848	—	61	1	no	—
147	#1778	—	61	2	no	—
148	C₄×#108	—	61	1	no	—
149	C₂×C₆₁	—	2	1	no	—
150	C₂×C₆₁	—	2	1	no	—
151	C₂×C₆₁	Q₈×C₄⋊C₄	1	1	yes	characteristic
152	C₂×C₆₁	C₂×#430	1	1	yes	characteristic
153	C₂×C₆₁	#2916	1	1	yes	characteristic
154	C₂×C₄×Q₈	C₂×D₆₁	1	1	yes	characteristic
155	#446	—	122	2	no	—
156	#446	—	122	4	no	—
157	#446	Dic₆₁	1	4	yes	—
158	#446	Dic₆₁	1	4	yes	—
159	C₈×Q₈	—	122	2	no	—
160	C₂×#108	—	61	1	no	—
161	C₂×#108	—	61	2	no	—
162	C₂×#108	—	61	1	no	—
163	C₂×#108	—	61	2	no	—
164	C₂×#108	—	61	4	no	—
165	#423	—	61	2	no	—
166	#423	—	61	4	no	—
167	C₂×C₄⋊C₈	C₂×D₆₁	1	1	yes	characteristic
168	C₂×C₄⋊C₈	C₂×D₆₁	1	2	yes	—
169	C₄×M₁₆	—	61	2	no	—
170	C₄×M₁₆	—	61	4	no	—
171	C₂×#97	C₂×D₆₁	1	2	yes	—
172	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
173	#341	—	61	2	no	—
174	#341	—	61	4	no	—
175	C₆₁	#22733	1	1	yes	characteristic, complemented, Sylow
176	C₂²×Q₈	Dic₁₂₂	1	1	yes	characteristic
177	C₂×M₁₆	—	61	2	no	—
178	C₂×M₁₆	—	61	2	no	—
179	C₂×M₁₆	—	61	4	no	—
180	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
181	C₂²×C₈	C₂²×D₆₁	1	2	yes	—
182	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
183	C₄×Q₈	D₂₄₄	1	2	yes	—
184	C₄×Q₈	D₂₄₄	1	2	yes	—
185	C₄×Q₈	C₂×Dic₆₁	1	2	yes	—
186	C₄×Q₈	C₂×Dic₆₁	1	2	yes	—
187	C₂×C₄⋊C₄	Dic₁₂₂	1	1	yes	characteristic
188	C₂×C₄⋊C₄	Dic₁₂₂	1	2	yes	—
189	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
190	C₂×C₄²	C₂²×D₆₁	1	2	yes	—
191	#108	—	61	4	no	—
192	#108	—	122	4	no	—
193	#108	—	122	2	no	—
194	#108	—	244	2	no	—
195	#108	—	244	4	no	—
196	C₄⋊C₈	—	122	2	no	—
197	C₄⋊C₈	—	122	4	no	—
198	C₄⋊C₈	—	122	4	no	—
199	C₄⋊C₈	—	122	2	no	—
200	C₄⋊C₈	C₂×Dic₆₁	1	2	yes	—
201	C₄⋊C₈	C₂×Dic₆₁	1	2	yes	—
202	C₄⋊C₈	C₂×Dic₆₁	1	8	yes	—
203	#97	—	122	4	no	—
204	#97	—	122	2	no	—
205	#97	C₂×Dic₆₁	1	8	yes	—
206	C₄×C₈	—	122	2	no	—
207	C₄×C₈	—	122	4	no	—
208	C₄×C₈	C₂×Dic₆₁	1	2	yes	—
209	C₄×C₈	C₂×Dic₆₁	1	2	yes	—
210	C₂×Q₈	—	4	2	no	—
211	C₂×Q₈	C₄⋊Dic₆₁	1	2	yes	—
212	C₂×Q₈	C₄⋊Dic₆₁	1	2	yes	—
213	C₂²×C₄	C₂×Dic₁₂₂	1	1	yes	characteristic
214	C₂²×C₄	C₂×Dic₁₂₂	1	1	yes	characteristic
215	C₂²×C₄	C₂×Dic₁₂₂	1	2	yes	—
216	C₂²×C₄	C₂×Dic₁₂₂	1	2	yes	—
217	C₂²×C₄	C₂³×D₆₁	1	1	yes	characteristic
218	M₁₆	—	61	4	no	—
219	M₁₆	—	122	2	no	—
220	M₁₆	—	122	4	no	—
221	M₁₆	—	122	2	no	—
222	M₁₆	—	122	4	no	—
223	M₁₆	—	122	2	no	—
224	C₂×C₈	—	2	1	no	—
225	C₂×C₈	—	2	2	no	—
226	C₂×C₈	—	2	1	no	—
227	C₂×C₈	—	2	2	no	—
228	C₂×C₈	—	122	2	no	—
229	C₂×C₈	—	122	2	no	—
230	C₂×C₈	—	122	4	no	—
231	C₂×C₈	Q₈×D₆₁	1	2	yes	—
232	C₂×C₈	#11758226	1	2	yes	—
233	C₂×C₈	C₂²×Dic₆₁	1	2	yes	—
234	C₂×C₈	C₂²×Dic₆₁	1	4	yes	—
235	C₂×C₈	C₂²×Dic₆₁	1	2	yes	—
236	C₄⋊C₄	C₄⋊Dic₆₁	1	4	yes	—
237	C₄⋊C₄	C₄⋊Dic₆₁	1	2	yes	—
238	C₄⋊C₄	C₄⋊Dic₆₁	1	4	yes	—
239	C₄⋊C₄	C₄⋊Dic₆₁	1	2	yes	—
240	C₄²	C₂×D₂₄₄	1	1	yes	characteristic
241	C₄²	C₂×D₂₄₄	1	4	yes	—
242	C₄²	C₂×D₂₄₄	1	1	yes	characteristic
243	C₄²	C₂²×Dic₆₁	1	1	yes	characteristic
244	C₄²	C₂²×Dic₆₁	1	1	yes	characteristic
245	C₄²	C₂²×Dic₆₁	1	4	yes	—
246	C₂³	C₂²×Dic₁₂₂	1	1	yes	characteristic
247	Q₈	—	2	4	no	—
248	Q₈	—	4	2	no	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	4	1	no	—
251	C₂×C₄	—	4	2	no	—
252	C₂×C₄	—	4	2	no	—
253	C₂×C₄	C₂×C₄⋊Dic₆₁	1	1	yes	characteristic
254	C₂×C₄	C₂×C₄⋊Dic₆₁	1	4	yes	—
255	C₂×C₄	C₂×C₄⋊Dic₆₁	1	1	yes	characteristic
256	C₂×C₄	C₂×C₄⋊Dic₆₁	1	1	yes	characteristic
257	C₂×C₄	C₂×C₄⋊Dic₆₁	1	4	yes	—
258	C₂×C₄	C₂×C₄⋊Dic₆₁	1	1	yes	characteristic
259	C₂×C₄	C₂³×Dic₆₁	1	1	yes	characteristic
260	C₂×C₄	C₂²×D₂₄₄	1	1	yes	characteristic
261	C₂×C₄	C₂×Q₈×D₆₁	1	1	yes	characteristic
262	C₂×C₄	C₂×#11758226	1	1	yes	characteristic
263	C₂×C₄	#14519823	1	1	yes	characteristic
264	C₂×C₄	#14519824	1	1	yes	characteristic
265	C₈	—	2	2	no	—
266	C₈	—	2	4	no	—
267	C₈	—	122	4	no	—
268	C₈	—	244	2	no	—
269	C₈	—	244	4	no	—
270	C₈	Q₈×Dic₆₁	1	4	yes	—
271	C₂²	—	2	1	no	—
272	C₂²	—	2	1	no	—
273	C₂²	C₂²×C₄⋊Dic₆₁	1	1	yes	characteristic
274	C₂²	#16840491	1	1	yes	characteristic
275	C₂²	Q₈×Dic₁₂₂	1	1	yes	characteristic
276	C₄	—	2	1	no	—
277	C₄	—	2	4	no	—
278	C₄	—	2	1	no	—
279	C₄	—	2	4	no	—
280	C₄	—	2	1	no	—
281	C₄	#16839811	1	2	yes	—
282	C₄	C₂×Q₈×Dic₆₁	1	1	yes	characteristic
283	C₄	#16840245	1	1	yes	characteristic
284	C₄	Q₈×D₂₄₄	1	1	yes	characteristic
285	C₄	#16840434	1	1	yes	characteristic
286	C₂	—	2	1	no	—
287	C₂	—	2	1	no	—
288	C₂	Q₈×C₄⋊Dic₆₁	1	1	yes	characteristic
289	C₂	C₂×#16839811	1	1	yes	characteristic
290	C₂	#21454430	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₂₄₄
commutator	C₂×C₄×C₆₁	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₁
Fitting	C₂×C₆₁×#446	C₂
socle	C₂²×C₆₁	C₂²×C₄⋊C₄
prime radical	C₂³×C₆₁	C₂²×Q₈
cosocle	C₂²×C₄×C₆₁	C₂⁴
hypercenter	C₂×#446	D₆₁
hypocenter	C₆₁	#22733
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×Q₈ ◁ C₂×#446

Complements

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

N	H ≅ G/N	type
37. #14519713	Q₈	semidirect
175. C₆₁	#22733	semidirect

Maximal subgroups

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

subgroup	index
Q₈×#11758194	2
#21446595	2
#21445994	2
#21445991	2
#21445725	2
C₄×#14519713	2
C₂×C₆₁×#446	2
#22733	61