Subgroups — #29699015

Order 15616.

642 conjugacy classes of subgroups.

The subgroup lattice has 248 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	1	yes	maximal, characteristic
2	C₂×Q₈×#92418	C₂	1	1	yes	maximal, characteristic
3	#21446594	C₂	1	3	yes	maximal
4	#21446592	C₂	1	3	yes	maximal
5	#21445956	C₂	1	3	yes	maximal
6	C₄×#14519712	C₂	1	3	yes	maximal
7	Q₈×M₁₆×C₆₁	C₂	1	1	yes	maximal, characteristic
8	#16839893	—	2	3	no	—
9	Q₈×#92418	—	2	1	no	—
10	Q₈×#92418	C₄	1	6	yes	—
11	Q₈×#92418	C₄	1	2	yes	—
12	C₂×#14519712	C₂²	1	3	yes	—
13	C₂×#14519712	C₂²	1	3	yes	—
14	#16839786	C₂²	1	3	yes	—
15	C₂×C₄⋊#92418	C₂²	1	3	yes	—
16	C₄×#11758194	C₂²	1	3	yes	—
17	C₂×C₄×#92418	C₂²	1	3	yes	—
18	#16839526	C₂²	1	3	yes	—
19	#16839526	C₂²	1	6	yes	—
20	#16839526	C₂²	1	1	yes	characteristic
21	C₂×C₄×Q₈×C₆₁	C₂²	1	1	yes	characteristic
22	C₆₁×#446	—	2	3	no	—
23	C₈×Q₈×C₆₁	—	2	1	no	—
24	C₆₁×#423	C₂²	1	3	yes	—
25	C₄×M₁₆×C₆₁	C₂²	1	3	yes	—
26	C₂×#11758194	C₂³	1	3	yes	—
27	C₂×#11758194	C₂³	1	1	yes	characteristic
28	C₂²×#92418	C₂³	1	1	yes	characteristic
29	C₂²×#92418	C₂³	1	3	yes	—
30	#14519712	—	2	6	no	—
31	#14519712	—	2	6	no	—
32	C₄⋊#92418	—	2	3	no	—
33	C₄⋊#92418	—	2	3	no	—
34	C₄⋊#92418	C₂×C₄	1	6	yes	—
35	C₄⋊#92418	C₂×C₄	1	6	yes	—
36	#14519688	—	2	3	no	—
37	C₄×#92418	—	2	3	no	—
38	C₄×#92418	C₂×C₄	1	6	yes	—
39	C₄×#92418	C₂×C₄	1	6	yes	—
40	C₂²×Q₈×C₆₁	Q₈	1	1	yes	characteristic
41	C₂×M₁₆×C₆₁	C₂³	1	3	yes	—
42	C₂×M₁₆×C₆₁	C₂³	1	1	yes	characteristic
43	C₄×Q₈×C₆₁	C₂×C₄	1	3	yes	—
44	C₄×Q₈×C₆₁	C₂×C₄	1	1	yes	characteristic
45	C₄×Q₈×C₆₁	D₄	1	1	yes	characteristic
46	C₄×Q₈×C₆₁	D₄	1	3	yes	—
47	C₂×C₄⋊C₄×C₆₁	Q₈	1	3	yes	—
48	C₂×C₄²×C₆₁	C₂³	1	3	yes	—
49	C₄⋊C₈×C₆₁	—	2	3	no	—
50	C₄⋊C₈×C₆₁	—	2	3	no	—
51	#97×C₆₁	—	2	3	no	—
52	C₄×C₈×C₆₁	—	2	3	no	—
53	C₂×Q₈×C₆₁	—	2	4	no	—
54	C₂×Q₈×C₆₁	C₄⋊C₄	1	3	yes	—
55	C₂×Q₈×C₆₁	C₄⋊C₄	1	1	yes	characteristic
56	C₂²×C₄×C₆₁	C₂×Q₈	1	3	yes	—
57	C₂²×C₄×C₆₁	C₂×Q₈	1	3	yes	—
58	C₂²×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
59	M₁₆×C₆₁	—	2	1	no	—
60	M₁₆×C₆₁	—	2	3	no	—
61	M₁₆×C₆₁	—	2	3	no	—
62	M₁₆×C₆₁	#41	1	2	yes	—
63	C₂×C₈×C₆₁	—	2	1	no	—
64	C₂×C₈×C₆₁	—	2	3	no	—
65	C₄⋊C₄×C₆₁	C₄⋊C₄	1	6	yes	—
66	C₄⋊C₄×C₆₁	C₄⋊C₄	1	3	yes	—
67	C₄⋊C₄×C₆₁	C₄⋊C₄	1	3	yes	—
68	C₄²×C₆₁	C₂²×C₄	1	3	yes	—
69	C₄²×C₆₁	C₂²×C₄	1	3	yes	—
70	C₄²×C₆₁	C₂×D₄	1	3	yes	—
71	C₄²×C₆₁	C₂×D₄	1	3	yes	—
72	#11758194	—	2	1	no	—
73	#11758194	—	2	3	no	—
74	#11758194	—	2	3	no	—
75	#11758194	#41	1	2	yes	—
76	C₂×#92418	—	2	1	no	—
77	C₂×#92418	—	2	3	no	—
78	C₂×#92418	—	2	3	no	—
79	C₂×#92418	—	2	1	no	—
80	C₂×#92418	—	2	1	no	—
81	C₂×#92418	—	2	3	no	—
82	C₂×#92418	C₂²×C₄	1	2	yes	—
83	C₂×#92418	C₂²×C₄	1	6	yes	—
84	C₂³×C₆₁	C₂²×Q₈	1	1	yes	characteristic
85	Q₈×C₆₁	—	2	6	no	—
86	Q₈×C₆₁	#108	1	4	yes	complemented
87	C₂×C₄×C₆₁	—	2	6	no	—
88	C₂×C₄×C₆₁	—	4	3	no	—
89	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	3	yes	—
90	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	3	yes	—
91	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	3	yes	—
92	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	3	yes	—
93	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
94	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
95	C₂×C₄×C₆₁	C₂×#41	1	1	yes	characteristic
96	C₂×C₄×C₆₁	C₂×#41	1	1	yes	characteristic
97	C₂×C₄×C₆₁	2₋¹⁺⁴	1	1	yes	characteristic
98	C₂×C₄×C₆₁	2₋¹⁺⁴	1	1	yes	characteristic
99	C₈×C₆₁	—	2	2	no	—
100	C₈×C₆₁	—	4	3	no	—
101	#92418	—	2	6	no	—
102	#92418	—	2	2	no	—
103	#92418	—	2	2	no	—
104	#92418	—	4	3	no	—
105	#22746	—	61	1	no	maximal, Sylow
106	C₂²×C₆₁	—	2	1	no	—
107	C₂²×C₆₁	—	2	1	no	—
108	C₂²×C₆₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
109	C₂²×C₆₁	#557	1	1	yes	characteristic
110	C₂²×C₆₁	#557	1	1	yes	characteristic
111	C₄×C₆₁	—	2	3	no	—
112	C₄×C₆₁	—	2	3	no	—
113	C₄×C₆₁	—	2	3	no	—
114	C₄×C₆₁	C₂×#108	1	6	yes	—
115	C₄×C₆₁	C₄×#41	1	1	yes	characteristic
116	C₄×C₆₁	#519	1	1	yes	characteristic
117	C₄×C₆₁	#548	1	1	yes	characteristic
118	C₄×C₆₁	#548	1	1	yes	characteristic
119	Q₈×M₁₆	—	61	1	no	—
120	Q₈×M₁₆	D₆₁	1	1	yes	characteristic
121	C₂×C₈×Q₈	—	61	1	no	—
122	#1950	—	61	3	no	—
123	#1950	—	61	3	no	—
124	#1837	—	61	3	no	—
125	C₄×#108	—	61	3	no	—
126	C₂×C₆₁	—	2	1	no	—
127	C₂×C₆₁	—	2	1	no	—
128	C₂×C₆₁	#2351	1	1	yes	characteristic
129	C₂×C₆₁	C₂²×#108	1	1	yes	characteristic
130	C₂×C₆₁	#2916	1	1	yes	characteristic
131	C₂×C₄×Q₈	C₂×D₆₁	1	1	yes	characteristic
132	#446	—	2	3	no	—
133	#446	—	122	3	no	—
134	C₈×Q₈	—	2	1	no	—
135	C₈×Q₈	—	61	6	no	—
136	C₈×Q₈	—	61	2	no	—
137	C₈×Q₈	—	122	1	no	—
138	C₂×#108	—	61	3	no	—
139	C₂×#108	—	61	3	no	—
140	#423	—	61	3	no	—
141	#423	C₂×D₆₁	1	3	yes	—
142	C₂×C₄⋊C₈	—	61	3	no	—
143	C₄×M₁₆	—	61	3	no	—
144	C₄×M₁₆	C₂×D₆₁	1	3	yes	—
145	C₂×C₄×C₈	—	61	3	no	—
146	#341	—	61	3	no	—
147	#341	—	61	6	no	—
148	#341	—	61	1	no	—
149	C₆₁	#22746	1	1	yes	characteristic, complemented, Sylow
150	C₂²×Q₈	Dic₁₂₂	1	1	yes	characteristic
151	C₂×M₁₆	—	61	1	no	—
152	C₂×M₁₆	—	61	3	no	—
153	C₂×M₁₆	C₂²×D₆₁	1	3	yes	—
154	C₂×M₁₆	C₂²×D₆₁	1	1	yes	characteristic
155	C₂²×C₈	—	61	1	no	—
156	C₂²×C₈	—	61	3	no	—
157	C₄×Q₈	C₄×D₆₁	1	3	yes	—
158	C₄×Q₈	C₄×D₆₁	1	1	yes	characteristic
159	C₄×Q₈	C₂²⋊D₆₁	1	1	yes	characteristic
160	C₄×Q₈	C₂²⋊D₆₁	1	3	yes	—
161	C₂×C₄⋊C₄	Dic₁₂₂	1	3	yes	—
162	C₂×C₄²	C₂²×D₆₁	1	3	yes	—
163	#108	—	122	6	no	—
164	#108	—	122	6	no	—
165	C₄⋊C₈	—	2	3	no	—
166	C₄⋊C₈	—	2	3	no	—
167	C₄⋊C₈	—	61	6	no	—
168	C₄⋊C₈	—	61	6	no	—
169	C₄⋊C₈	—	122	3	no	—
170	C₄⋊C₈	—	122	3	no	—
171	#97	—	2	3	no	—
172	#97	—	122	3	no	—
173	C₄×C₈	—	2	3	no	—
174	C₄×C₈	—	61	6	no	—
175	C₄×C₈	—	61	6	no	—
176	C₄×C₈	—	122	3	no	—
177	C₂×Q₈	—	2	4	no	—
178	C₂×Q₈	#11758196	1	3	yes	—
179	C₂×Q₈	#11758196	1	1	yes	characteristic
180	C₂²×C₄	C₂×Dic₁₂₂	1	3	yes	—
181	C₂²×C₄	C₂×Dic₁₂₂	1	3	yes	—
182	C₂²×C₄	C₂³×D₆₁	1	1	yes	characteristic
183	M₁₆	—	2	1	no	—
184	M₁₆	—	2	3	no	—
185	M₁₆	—	2	3	no	—
186	M₁₆	—	61	2	no	—
187	M₁₆	—	122	1	no	—
188	M₁₆	—	122	3	no	—
189	M₁₆	—	122	3	no	—
190	M₁₆	#11758226	1	2	yes	—
191	C₂×C₈	—	2	1	no	—
192	C₂×C₈	—	2	3	no	—
193	C₂×C₈	—	61	2	no	—
194	C₂×C₈	—	61	6	no	—
195	C₂×C₈	—	122	1	no	—
196	C₂×C₈	—	122	3	no	—
197	C₂×C₈	—	122	3	no	—
198	C₂×C₈	—	122	1	no	—
199	C₂×C₈	—	122	1	no	—
200	C₂×C₈	—	122	3	no	—
201	C₄⋊C₄	#11758196	1	6	yes	—
202	C₄⋊C₄	#11758196	1	3	yes	—
203	C₄⋊C₄	#11758196	1	3	yes	—
204	C₄²	C₂×C₄×D₆₁	1	3	yes	—
205	C₄²	C₂×C₄×D₆₁	1	3	yes	—
206	C₄²	C₂×C₂²⋊D₆₁	1	3	yes	—
207	C₄²	C₂×C₂²⋊D₆₁	1	3	yes	—
208	C₂³	C₂²×Dic₁₂₂	1	1	yes	characteristic
209	Q₈	—	2	6	no	—
210	Q₈	#14519712	1	4	yes	complemented
211	C₂×C₄	—	2	6	no	—
212	C₂×C₄	—	4	3	no	—
213	C₂×C₄	C₂×#11758196	1	3	yes	—
214	C₂×C₄	C₂×#11758196	1	3	yes	—
215	C₂×C₄	C₂×#11758196	1	3	yes	—
216	C₂×C₄	C₂×#11758196	1	3	yes	—
217	C₂×C₄	C₂²×C₄×D₆₁	1	1	yes	characteristic
218	C₂×C₄	C₂²×C₂²⋊D₆₁	1	1	yes	characteristic
219	C₂×C₄	#41×D₆₁	1	1	yes	characteristic
220	C₂×C₄	C₂×#11758226	1	1	yes	characteristic
221	C₂×C₄	#14519824	1	1	yes	characteristic
222	C₂×C₄	#14519825	1	1	yes	characteristic
223	C₈	—	2	2	no	—
224	C₈	—	4	3	no	—
225	C₈	—	122	6	no	—
226	C₈	—	122	2	no	—
227	C₈	—	122	2	no	—
228	C₈	—	244	3	no	—
229	C₂²	—	2	1	no	—
230	C₂²	—	2	1	no	—
231	C₂²	C₂²×#11758196	1	1	yes	characteristic
232	C₂²	#16840488	1	1	yes	characteristic
233	C₂²	#16840491	1	1	yes	characteristic
234	C₄	—	2	3	no	—
235	C₄	—	2	3	no	—
236	C₄	—	2	3	no	—
237	C₄	C₂×#14519712	1	6	yes	—
238	C₄	C₄×#11758226	1	1	yes	characteristic
239	C₄	#16840238	1	1	yes	characteristic
240	C₄	#16840419	1	1	yes	characteristic
241	C₄	#16840423	1	1	yes	characteristic
242	C₂	—	2	1	no	—
243	C₂	—	2	1	no	—
244	C₂	#21448737	1	1	yes	characteristic
245	C₂	C₂²×#14519712	1	1	yes	characteristic
246	C₂	#21454428	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊D₆₁
commutator	C₂×C₄×C₆₁	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₁
Fitting	Q₈×M₁₆×C₆₁	C₂
socle	C₂²×C₆₁	C₂²×C₄⋊C₄
prime radical	C₂³×C₆₁	C₂²×Q₈
cosocle	C₂²×C₄×C₆₁	C₂⁴
hypercenter	Q₈×M₁₆	D₆₁
hypocenter	C₆₁	#22746
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×Q₈ ◁ Q₈×M₁₆

Complements

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

N	H ≅ G/N	type
86. Q₈×C₆₁	#108	semidirect
149. C₆₁	#22746	semidirect
210. Q₈	#14519712	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
C₂×Q₈×#92418	2
#21446594	2
#21446592	2
#21445956	2
C₄×#14519712	2
Q₈×M₁₆×C₆₁	2
#22746	61