Subgroups — Q₈×#14519710 #29698850

Order 15616.

838 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₈×#92418	C₂	1	1	yes	maximal, characteristic
2	Q₈×C₄⋊Dic₆₁	C₂	1	1	yes	maximal, characteristic
3	#21446577	C₂	1	3	yes	maximal
4	#21446575	C₂	1	3	yes	maximal
5	#21445978	C₂	1	3	yes	maximal
6	C₄×#14519710	C₂	1	3	yes	maximal
7	Q₈×C₄⋊C₄×C₆₁	C₂	1	1	yes	maximal, characteristic
8	C₂×Q₈×Dic₆₁	—	2	1	no	—
9	Q₈×#92418	C₄	1	8	yes	complemented
10	C₂×#14519710	C₂²	1	3	yes	—
11	C₂×#14519710	C₂²	1	6	yes	—
12	C₂×#14519710	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊#92418	C₂²	1	3	yes	—
14	C₂×C₄×#92418	C₂²	1	3	yes	—
15	#16839661	—	2	3	no	—
16	#16839632	C₂²	1	3	yes	—
17	C₄×C₄⋊Dic₆₁	C₂²	1	3	yes	—
18	#16839524	C₂²	1	3	yes	—
19	#16839524	C₂²	1	3	yes	—
20	C₂×C₄×Q₈×C₆₁	—	2	1	no	—
21	C₂×C₄×Q₈×C₆₁	C₂²	1	1	yes	characteristic
22	C₆₁×#389	—	2	3	no	—
23	C₆₁×#384	C₂²	1	3	yes	—
24	C₄×C₄⋊C₄×C₆₁	C₂²	1	3	yes	—
25	C₂²×#92418	C₂³	1	3	yes	—
26	C₂²×#92418	C₂³	1	1	yes	characteristic
27	Q₈×Dic₆₁	—	4	4	no	—
28	C₂×C₄⋊Dic₆₁	—	2	3	no	—
29	C₂×C₄⋊Dic₆₁	—	2	3	no	—
30	C₂×C₄⋊Dic₆₁	C₂³	1	3	yes	—
31	C₂×C₄⋊Dic₆₁	C₂³	1	1	yes	characteristic
32	C₂×C₄×Dic₆₁	—	2	3	no	—
33	#14519710	—	2	6	no	—
34	#14519710	—	4	6	no	—
35	#14519710	Q₈	1	4	yes	direct product factor
36	C₄⋊#92418	C₂×C₄	1	12	yes	—
37	C₄×#92418	C₂×C₄	1	12	yes	—
38	#14519684	—	2	3	no	—
39	C₂²×Q₈×C₆₁	D₄	1	1	yes	characteristic
40	C₄×Q₈×C₆₁	—	4	4	no	—
41	C₄×Q₈×C₆₁	C₂×C₄	1	4	yes	—
42	C₄×Q₈×C₆₁	Q₈	1	4	yes	—
43	C₂×C₄⋊C₄×C₆₁	—	2	3	no	—
44	C₂×C₄⋊C₄×C₆₁	—	2	3	no	—
45	C₂×C₄⋊C₄×C₆₁	D₄	1	3	yes	—
46	C₂×C₄⋊C₄×C₆₁	C₂³	1	3	yes	—
47	C₂×C₄⋊C₄×C₆₁	C₂³	1	1	yes	characteristic
48	C₂×C₄²×C₆₁	—	2	3	no	—
49	C₂×C₄²×C₆₁	C₂³	1	3	yes	—
50	#95×C₆₁	—	2	3	no	—
51	C₂×Q₈×C₆₁	C₄⋊C₄	1	4	yes	—
52	C₂×Q₈×C₆₁	D₈	1	4	yes	—
53	C₂×Q₈×C₆₁	Q₁₆	1	4	yes	—
54	C₂²×C₄×C₆₁	—	2	1	no	—
55	C₂²×C₄×C₆₁	—	2	3	no	—
56	C₂²×C₄×C₆₁	C₂×D₄	1	3	yes	—
57	C₂²×C₄×C₆₁	C₂×D₄	1	3	yes	—
58	C₂²×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
59	C₂²×Dic₆₁	—	2	1	no	—
60	C₂²×Dic₆₁	—	2	3	no	—
61	C₄⋊C₄×C₆₁	—	2	1	no	—
62	C₄⋊C₄×C₆₁	—	2	3	no	—
63	C₄⋊C₄×C₆₁	—	2	3	no	—
64	C₄⋊C₄×C₆₁	—	4	6	no	—
65	C₄⋊C₄×C₆₁	—	8	3	no	—
66	C₄⋊C₄×C₆₁	C₄⋊C₄	1	12	yes	—
67	C₄⋊C₄×C₆₁	C₂×Q₈	1	2	yes	—
68	C₄²×C₆₁	—	4	6	no	—
69	C₄²×C₆₁	C₂²×C₄	1	6	yes	—
70	C₄²×C₆₁	C₂×Q₈	1	6	yes	—
71	C₄⋊Dic₆₁	—	2	1	no	—
72	C₄⋊Dic₆₁	—	2	3	no	—
73	C₄⋊Dic₆₁	—	2	3	no	—
74	C₄⋊Dic₆₁	—	4	6	no	—
75	C₄⋊Dic₆₁	—	8	3	no	—
76	C₄⋊Dic₆₁	C₂×Q₈	1	2	yes	—
77	C₄×Dic₆₁	—	4	6	no	—
78	C₂×#92418	—	2	3	no	—
79	C₂×#92418	—	2	3	no	—
80	C₂×#92418	C₂²×C₄	1	6	yes	—
81	C₂×#92418	C₂²×C₄	1	2	yes	—
82	C₂×#92418	C₂×Q₈	1	2	yes	—
83	C₂×#92418	#41	1	2	yes	—
84	C₂³×C₆₁	C₂²×D₄	1	1	yes	characteristic
85	Q₈×C₆₁	#107	1	16	yes	complemented
86	C₂×C₄×C₆₁	—	2	3	no	—
87	C₂×C₄×C₆₁	—	2	3	no	—
88	C₂×C₄×C₆₁	—	2	2	no	—
89	C₂×C₄×C₆₁	—	4	1	no	—
90	C₂×C₄×C₆₁	—	4	1	no	—
91	C₂×C₄×C₆₁	—	4	3	no	—
92	C₂×C₄×C₆₁	—	4	3	no	—
93	C₂×C₄×C₆₁	—	4	3	no	—
94	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	6	yes	—
95	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	6	yes	—
96	C₂×C₄×C₆₁	C₂×D₈	1	6	yes	—
97	C₂×C₄×C₆₁	C₂×Q₁₆	1	6	yes	—
98	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
99	C₂×C₄×C₆₁	C₂²×Q₈	1	1	yes	characteristic
100	C₂×C₄×C₆₁	C₂²×Q₈	1	1	yes	characteristic
101	C₂×C₄×C₆₁	C₂×#41	1	1	yes	characteristic
102	C₂×C₄×C₆₁	H(5,𝔽₂)	1	1	yes	characteristic
103	C₂×C₄×C₆₁	2₋¹⁺⁴	1	1	yes	characteristic
104	C₂×Dic₆₁	—	2	2	no	—
105	C₂×Dic₆₁	—	4	1	no	—
106	C₂×Dic₆₁	—	4	1	no	—
107	C₂×Dic₆₁	—	4	3	no	—
108	C₂×Dic₆₁	—	4	3	no	—
109	C₂×Dic₆₁	—	4	3	no	—
110	#92418	—	2	6	no	—
111	#92418	C₄×Q₈	1	4	yes	complemented
112	Q₈×#107	—	61	1	no	maximal, Sylow
113	C₂²×C₆₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
114	C₂²×C₆₁	D₄×Q₈	1	1	yes	characteristic
115	C₂²×C₆₁	#550	1	1	yes	characteristic
116	C₂²×C₆₁	C₂²×D₈	1	1	yes	characteristic
117	C₂²×C₆₁	C₂²×Q₁₆	1	1	yes	characteristic
118	C₂²×C₆₁	#576	1	1	yes	characteristic
119	C₂²×C₆₁	#578	1	1	yes	characteristic
120	C₄×C₆₁	—	2	6	no	—
121	C₄×C₆₁	—	4	2	no	—
122	C₄×C₆₁	—	8	3	no	—
123	C₄×C₆₁	C₂×#107	1	12	yes	—
124	C₄×C₆₁	C₂×C₄×Q₈	1	1	yes	characteristic
125	C₄×C₆₁	#520	1	1	yes	characteristic
126	C₄×C₆₁	#557	1	1	yes	characteristic
127	C₄×C₆₁	Q₈²	1	1	yes	characteristic
128	Dic₆₁	—	4	2	no	—
129	Dic₆₁	—	8	3	no	—
130	C₂×C₈×Q₈	—	61	1	no	—
131	Q₈×C₄⋊C₄	—	61	1	no	—
132	Q₈×C₄⋊C₄	D₆₁	1	1	yes	characteristic
133	#1947	—	61	3	no	—
134	#1947	—	61	3	no	—
135	#1845	—	61	3	no	—
136	C₄×#107	—	61	3	no	—
137	C₂×C₆₁	Q₈×C₄⋊C₄	1	1	yes	characteristic
138	C₂×C₆₁	C₂²×#107	1	1	yes	characteristic
139	C₂×C₆₁	#2913	1	1	yes	characteristic
140	C₂×C₆₁	Q₈×D₈	1	1	yes	characteristic
141	C₂×C₆₁	Q₈×Q₁₆	1	1	yes	characteristic
142	C₂×C₆₁	#3391	1	1	yes	characteristic
143	C₂×C₆₁	#3395	1	1	yes	characteristic
144	C₂×C₄×Q₈	—	2	1	no	—
145	C₂×C₄×Q₈	—	122	1	no	—
146	C₂×C₄×Q₈	C₂×D₆₁	1	1	yes	characteristic
147	C₈×Q₈	—	61	8	no	—
148	C₂×#107	—	61	1	no	—
149	C₂×#107	—	61	3	no	—
150	C₂×#107	—	61	6	no	—
151	C₂×C₄⋊C₈	—	61	3	no	—
152	C₂×C₄×C₈	—	61	3	no	—
153	#389	—	2	3	no	—
154	#389	—	122	3	no	—
155	#384	—	61	3	no	—
156	#384	C₂×D₆₁	1	3	yes	—
157	C₄×C₄⋊C₄	—	61	3	no	—
158	C₄×C₄⋊C₄	C₂×D₆₁	1	3	yes	—
159	#340	—	61	3	no	—
160	#340	—	61	3	no	—
161	C₆₁	Q₈×#107	1	1	yes	characteristic, complemented, Sylow
162	C₂²×Q₈	C₂²⋊D₆₁	1	1	yes	characteristic
163	C₂²×C₈	—	61	1	no	—
164	C₂²×C₈	—	61	3	no	—
165	C₄×Q₈	—	4	4	no	—
166	C₄×Q₈	—	244	4	no	—
167	C₄×Q₈	Dic₁₂₂	1	4	yes	—
168	C₄×Q₈	C₄×D₆₁	1	4	yes	—
169	C₂×C₄⋊C₄	—	2	3	no	—
170	C₂×C₄⋊C₄	—	2	3	no	—
171	C₂×C₄⋊C₄	—	61	1	no	—
172	C₂×C₄⋊C₄	—	61	3	no	—
173	C₂×C₄⋊C₄	—	122	3	no	—
174	C₂×C₄⋊C₄	—	122	3	no	—
175	C₂×C₄⋊C₄	C₂²⋊D₆₁	1	3	yes	—
176	C₂×C₄⋊C₄	C₂²×D₆₁	1	3	yes	—
177	C₂×C₄⋊C₄	C₂²×D₆₁	1	1	yes	characteristic
178	C₂×C₄²	—	2	3	no	—
179	C₂×C₄²	—	122	3	no	—
180	C₂×C₄²	C₂²×D₆₁	1	3	yes	—
181	#107	—	61	4	no	—
182	#107	—	122	6	no	—
183	#107	—	244	6	no	—
184	C₄⋊C₈	—	61	12	no	—
185	C₄×C₈	—	61	12	no	—
186	#95	—	2	3	no	—
187	#95	—	122	3	no	—
188	C₂×Q₈	#11758196	1	4	yes	—
189	C₂×Q₈	#11758199	1	4	yes	—
190	C₂×Q₈	#11758202	1	4	yes	—
191	C₂²×C₄	—	2	1	no	—
192	C₂²×C₄	—	2	3	no	—
193	C₂²×C₄	—	122	1	no	—
194	C₂²×C₄	—	122	3	no	—
195	C₂²×C₄	C₂×C₂²⋊D₆₁	1	3	yes	—
196	C₂²×C₄	C₂×C₂²⋊D₆₁	1	3	yes	—
197	C₂²×C₄	C₂³×D₆₁	1	1	yes	characteristic
198	C₂×C₈	—	61	2	no	—
199	C₂×C₈	—	61	2	no	—
200	C₂×C₈	—	61	2	no	—
201	C₂×C₈	—	61	6	no	—
202	C₂×C₈	—	122	3	no	—
203	C₂×C₈	—	122	3	no	—
204	C₄⋊C₄	—	2	1	no	—
205	C₄⋊C₄	—	2	3	no	—
206	C₄⋊C₄	—	2	3	no	—
207	C₄⋊C₄	—	4	6	no	—
208	C₄⋊C₄	—	8	3	no	—
209	C₄⋊C₄	—	61	2	no	—
210	C₄⋊C₄	—	122	1	no	—
211	C₄⋊C₄	—	122	3	no	—
212	C₄⋊C₄	—	122	3	no	—
213	C₄⋊C₄	—	488	3	no	—
214	C₄⋊C₄	—	244	6	no	—
215	C₄⋊C₄	#11758196	1	12	yes	—
216	C₄⋊C₄	Q₈×D₆₁	1	2	yes	—
217	C₄²	—	4	6	no	—
218	C₄²	—	244	6	no	—
219	C₄²	C₂×Dic₁₂₂	1	6	yes	—
220	C₄²	C₂×C₄×D₆₁	1	6	yes	—
221	C₂³	C₂²×C₂²⋊D₆₁	1	1	yes	characteristic
222	Q₈	#14519710	1	16	yes	direct product factor
223	C₂×C₄	—	2	3	no	—
224	C₂×C₄	—	2	3	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	4	1	no	—
227	C₂×C₄	—	4	1	no	—
228	C₂×C₄	—	4	3	no	—
229	C₂×C₄	—	4	3	no	—
230	C₂×C₄	—	4	3	no	—
231	C₂×C₄	—	122	2	no	—
232	C₂×C₄	—	244	3	no	—
233	C₂×C₄	—	244	3	no	—
234	C₂×C₄	—	244	3	no	—
235	C₂×C₄	—	244	1	no	—
236	C₂×C₄	—	244	1	no	—
237	C₂×C₄	C₂×#11758196	1	6	yes	—
238	C₂×C₄	C₂×#11758196	1	6	yes	—
239	C₂×C₄	C₂×#11758199	1	6	yes	—
240	C₂×C₄	C₂×#11758202	1	6	yes	—
241	C₂×C₄	C₂²×C₄×D₆₁	1	1	yes	characteristic
242	C₂×C₄	C₂×Q₈×D₆₁	1	1	yes	characteristic
243	C₂×C₄	C₂²×Dic₁₂₂	1	1	yes	characteristic
244	C₂×C₄	#41×D₆₁	1	1	yes	characteristic
245	C₂×C₄	#14519823	1	1	yes	characteristic
246	C₂×C₄	#14519825	1	1	yes	characteristic
247	C₈	—	61	4	no	—
248	C₈	—	122	6	no	—
249	C₂²	C₂²×#11758196	1	1	yes	characteristic
250	C₂²	Q₈×C₂²⋊D₆₁	1	1	yes	characteristic
251	C₂²	#16840431	1	1	yes	characteristic
252	C₂²	C₂²×#11758199	1	1	yes	characteristic
253	C₂²	C₂²×#11758202	1	1	yes	characteristic
254	C₂²	#16840565	1	1	yes	characteristic
255	C₂²	#16840577	1	1	yes	characteristic
256	C₄	—	2	6	no	—
257	C₄	—	4	2	no	—
258	C₄	—	8	3	no	—
259	C₄	—	488	3	no	—
260	C₄	—	244	2	no	—
261	C₄	C₂×#14519710	1	12	yes	—
262	C₄	C₄×Q₈×D₆₁	1	1	yes	characteristic
263	C₄	#16840244	1	1	yes	characteristic
264	C₄	#16840488	1	1	yes	characteristic
265	C₄	Q₈×Dic₁₂₂	1	1	yes	characteristic
266	C₂	Q₈×#11758196	1	1	yes	characteristic
267	C₂	C₂²×#14519710	1	1	yes	characteristic
268	C₂	#21454417	1	1	yes	characteristic
269	C₂	Q₈×#11758199	1	1	yes	characteristic
270	C₂	Q₈×#11758202	1	1	yes	characteristic
271	C₂	#21458791	1	1	yes	characteristic
272	C₂	#21458822	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²×C₂²⋊D₆₁
commutator	C₂×C₄×C₆₁	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₁
Fitting	Q₈×C₄⋊C₄×C₆₁	C₂
socle, prime radical	C₂³×C₆₁	C₂²×D₄
cosocle	C₂²×C₄×C₆₁	C₂⁴
hypercenter	Q₈×C₄⋊C₄	D₆₁
hypocenter	C₆₁	Q₈×#107
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×Q₈ ◁ Q₈×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
9. Q₈×#92418	C₄	semidirect
35. #14519710	Q₈	direct (N × H)
85. Q₈×C₆₁	#107	semidirect
111. #92418	C₄×Q₈	semidirect
161. C₆₁	Q₈×#107	semidirect
222. Q₈	#14519710	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×Q₈×#92418	2
Q₈×C₄⋊Dic₆₁	2
#21446577	2
#21446575	2
#21445978	2
C₄×#14519710	2
Q₈×C₄⋊C₄×C₆₁	2
Q₈×#107	61

Subgroups — Q8×#14519710 #29698850

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×#14519710 #29698850