Subgroups — Q₈×#16860809 #44563942

Order 32128.

1068 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×Dic₅₀₂	C₂	1	1	yes	maximal, characteristic, complemented
2	#30632412	C₂	1	3	yes	maximal, complemented
3	#30632395	C₂	1	6	yes	maximal, complemented
4	#30632394	C₂	1	3	yes	maximal, complemented
5	Q₈×C₂²⋊D₂₅₁	C₂	1	2	yes	maximal, complemented
6	#30632242	C₂	1	3	yes	maximal, complemented
7	#30632239	C₂	1	6	yes	maximal, complemented
8	C₄×#16860809	C₂	1	3	yes	maximal
9	C₂×Q₈×Dic₂₅₁	C₂	1	2	yes	maximal, complemented
10	C₄×Q₈×D₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
11	D₄×Q₈×C₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#16860809	C₂²	1	1	yes	characteristic
13	C₂×Q₈×D₂₅₁	C₂²	1	1	yes	characteristic
14	#22548715	C₂²	1	3	yes	—
15	#22548714	C₂²	1	6	yes	—
16	#22548713	C₂²	1	6	yes	—
17	#22548706	C₂²	1	6	yes	—
18	#22548705	C₂²	1	12	yes	—
19	#22548695	C₂²	1	6	yes	—
20	#22548694	C₂²	1	6	yes	—
21	#22548692	C₂²	1	12	yes	—
22	#22548691	C₂²	1	6	yes	—
23	Q₈×Dic₂₅₁	C₂²	1	6	yes	—
24	Q₈×Dic₂₅₁	C₂²	1	2	yes	—
25	Q₈×Dic₂₅₁	C₂²	1	6	yes	—
26	Q₈×Dic₂₅₁	C₂²	1	1	yes	characteristic
27	C₄×Dic₅₀₂	C₂²	1	3	yes	—
28	C₄×Dic₅₀₂	C₂²	1	3	yes	—
29	#22548680	C₂²	1	6	yes	—
30	#22548680	C₂²	1	3	yes	—
31	#22548680	C₂²	1	3	yes	—
32	D₄×Dic₂₅₁	C₂²	1	3	yes	—
33	C₄×C₂²⋊D₂₅₁	C₂²	1	6	yes	—
34	(C₂²⋊D₂₅₁)⋊C₄	C₂²	1	6	yes	—
35	#22548673	C₂²	1	3	yes	—
36	#22548671	C₂²	1	6	yes	—
37	C₂×C₄⋊Dic₂₅₁	C₂²	1	6	yes	—
38	C₄⋊C₄×D₂₅₁	C₂²	1	3	yes	—
39	C₂×C₄×Dic₂₅₁	C₂²	1	6	yes	—
40	C₄²×D₂₅₁	C₂²	1	3	yes	—
41	C₂²×Q₈×C₂₅₁	C₂²	1	2	yes	—
42	C₄⋊Q₈×C₂₅₁	C₂²	1	3	yes	—
43	C₂²⋊Q₈×C₂₅₁	C₂²	1	6	yes	—
44	C₄×Q₈×C₂₅₁	C₂²	1	1	yes	characteristic
45	C₄×D₄×C₂₅₁	C₂²	1	3	yes	—
46	#16860809	Q₈	1	8	yes	direct product factor
47	C₂×Dic₅₀₂	C₂³	1	3	yes	—
48	C₂×Dic₅₀₂	C₂³	1	6	yes	—
49	C₂×Dic₅₀₂	C₂³	1	1	yes	characteristic
50	Q₈×D₂₅₁	—	2	4	no	—
51	C₂×C₂²⋊D₂₅₁	C₂³	1	2	yes	—
52	C₂²×Dic₂₅₁	C₂³	1	6	yes	—
53	C₂²×Dic₂₅₁	C₂³	1	2	yes	—
54	C₂×C₄×D₂₅₁	C₂³	1	3	yes	—
55	C₂×C₄×D₂₅₁	C₂³	1	3	yes	—
56	C₂×C₄×D₂₅₁	C₂³	1	1	yes	characteristic
57	C₄⋊Dic₂₅₁	C₂³	1	3	yes	—
58	C₄⋊Dic₂₅₁	C₂³	1	6	yes	—
59	C₄⋊Dic₂₅₁	C₂³	1	12	yes	—
60	C₄⋊Dic₂₅₁	C₂³	1	3	yes	—
61	Dic₂₅₁⋊C₄	C₂³	1	6	yes	—
62	Dic₂₅₁⋊C₄	C₂³	1	6	yes	—
63	Dic₂₅₁⋊C₄	C₂³	1	6	yes	—
64	Dic₂₅₁⋊C₄	C₂³	1	12	yes	—
65	C₂²⋊Dic₂₅₁	C₂³	1	6	yes	—
66	(C₂×D₂₅₁)⋊C₄	C₂³	1	6	yes	—
67	C₄×Dic₂₅₁	C₂³	1	3	yes	—
68	C₄×Dic₂₅₁	C₂³	1	3	yes	—
69	C₄×Dic₂₅₁	C₂³	1	3	yes	—
70	C₄×Dic₂₅₁	C₂³	1	6	yes	—
71	C₄×Dic₂₅₁	C₂³	1	6	yes	—
72	C₄×Dic₂₅₁	C₂³	1	6	yes	—
73	C₂×Q₈×C₂₅₁	—	2	8	no	—
74	C₂×Q₈×C₂₅₁	C₂³	1	6	yes	—
75	C₂×Q₈×C₂₅₁	C₂³	1	1	yes	characteristic
76	C₂×D₄×C₂₅₁	C₂³	1	1	yes	characteristic
77	C₂²×C₄×C₂₅₁	C₂³	1	6	yes	—
78	C₄⋊C₄×C₂₅₁	C₂³	1	3	yes	—
79	C₄⋊C₄×C₂₅₁	C₂³	1	3	yes	—
80	C₄⋊C₄×C₂₅₁	C₂³	1	6	yes	—
81	C₂²⋊C₄×C₂₅₁	C₂³	1	6	yes	—
82	C₄²×C₂₅₁	C₂³	1	3	yes	—
83	C₂²×D₂₅₁	C₂⁴	1	1	yes	characteristic
84	Dic₅₀₂	—	2	6	no	—
85	Dic₅₀₂	—	4	6	no	—
86	Dic₅₀₂	C₂×Q₈	1	4	yes	complemented
87	C₂²⋊D₂₅₁	C₂×Q₈	1	8	yes	complemented
88	C₂×Dic₂₅₁	—	4	6	no	—
89	C₂×Dic₂₅₁	C₂×Q₈	1	8	yes	complemented
90	C₂×Dic₂₅₁	C₂⁴	1	3	yes	—
91	C₂×Dic₂₅₁	C₂⁴	1	6	yes	—
92	C₂×Dic₂₅₁	C₂⁴	1	3	yes	—
93	C₂×Dic₂₅₁	C₂⁴	1	1	yes	characteristic
94	C₂×Dic₂₅₁	C₂⁴	1	2	yes	—
95	C₄×D₂₅₁	—	2	6	no	—
96	C₄×D₂₅₁	—	4	3	no	—
97	C₄×D₂₅₁	C₂×Q₈	1	4	yes	complemented
98	C₂³×C₂₅₁	C₂⁴	1	2	yes	—
99	Q₈×C₂₅₁	—	2	12	no	—
100	Q₈×C₂₅₁	#41	1	4	yes	complemented
101	D₄×C₂₅₁	C₂×Q₈	1	4	yes	complemented
102	C₂×C₄×C₂₅₁	—	2	12	no	—
103	C₂×C₄×C₂₅₁	C₂⁴	1	3	yes	—
104	C₂×C₄×C₂₅₁	C₂⁴	1	6	yes	—
105	C₂×C₄×C₂₅₁	C₂⁴	1	1	yes	characteristic
106	C₂×C₄×C₂₅₁	C₂⁴	1	3	yes	—
107	C₂²×C₂₅₁	—	2	2	no	—
108	C₂²×C₂₅₁	—	2	2	no	—
109	C₂²×C₂₅₁	C₂²×Q₈	1	4	yes	—
110	C₂²×C₂₅₁	C₂⁵	1	1	yes	characteristic
111	C₂×D₂₅₁	—	2	1	no	—
112	C₂×D₂₅₁	—	2	1	no	—
113	C₂×D₂₅₁	C₂²×Q₈	1	2	yes	—
114	C₄×C₂₅₁	—	2	6	no	—
115	C₄×C₂₅₁	—	2	3	no	—
116	C₄×C₂₅₁	C₂²×Q₈	1	2	yes	—
117	C₄×C₂₅₁	C₂×#41	1	6	yes	—
118	Dic₂₅₁	—	2	3	no	—
119	Dic₂₅₁	—	2	3	no	—
120	Dic₂₅₁	—	2	6	no	—
121	Dic₂₅₁	C₂²×Q₈	1	2	yes	—
122	Dic₂₅₁	C₂²×Q₈	1	4	yes	—
123	C₂×C₂₅₁	—	2	4	no	—
124	C₂×C₂₅₁	C₂³×Q₈	1	1	yes	characteristic
125	C₂×C₂₅₁	C₂²×#41	1	1	yes	characteristic
126	C₂×C₂₅₁	#585	1	1	yes	characteristic
127	D₂₅₁	—	2	2	no	—
128	C₂₅₁	Q₈×#41	1	1	yes	characteristic, complemented, Sylow
129	Q₈×#41	—	251	1	no	maximal, Sylow
130	Q₈²	—	251	1	no	—
131	#557	—	251	3	no	—
132	#554	—	251	6	no	—
133	#554	—	251	3	no	—
134	D₄×Q₈	—	251	2	no	—
135	D₄×Q₈	D₂₅₁	1	1	yes	characteristic, complemented
136	#533	—	251	3	no	—
137	#533	—	251	6	no	—
138	C₄×#41	—	251	3	no	—
139	C₂×C₄×Q₈	—	251	2	no	—
140	C₂×C₄×Q₈	—	251	1	no	—
141	C₂×#41	—	251	1	no	—
142	C₂²×Q₈	—	251	1	no	—
143	C₂²×Q₈	C₂×D₂₅₁	1	2	yes	—
144	C₄⋊Q₈	—	251	6	no	—
145	C₄⋊Q₈	—	251	6	no	—
146	C₄⋊Q₈	—	251	3	no	—
147	C₄⋊Q₈	C₂×D₂₅₁	1	3	yes	—
148	#125	—	251	12	no	—
149	#125	—	251	6	no	—
150	C₂²⋊Q₈	—	251	6	no	—
151	C₂²⋊Q₈	—	251	6	no	—
152	C₂²⋊Q₈	—	251	6	no	—
153	C₂²⋊Q₈	—	251	12	no	—
154	C₂²⋊Q₈	C₂×D₂₅₁	1	6	yes	—
155	C₄×Q₈	—	251	1	no	—
156	C₄×Q₈	—	251	6	no	—
157	C₄×Q₈	—	251	2	no	—
158	C₄×Q₈	—	251	6	no	—
159	C₄×Q₈	—	251	3	no	—
160	C₄×Q₈	—	251	3	no	—
161	C₄×Q₈	—	251	6	no	—
162	C₄×Q₈	—	251	3	no	—
163	C₄×Q₈	—	251	3	no	—
164	C₄×Q₈	C₂×D₂₅₁	1	1	yes	characteristic
165	C₄×D₄	—	251	3	no	—
166	C₄×D₄	—	251	6	no	—
167	C₄×D₄	—	251	6	no	—
168	C₄×D₄	C₂×D₂₅₁	1	3	yes	—
169	#117	—	251	3	no	—
170	#117	—	251	6	no	—
171	C₂×C₄⋊C₄	—	251	6	no	—
172	C₂×C₄⋊C₄	—	251	3	no	—
173	C₂×C₄²	—	251	6	no	—
174	C₂×C₄²	—	251	3	no	—
175	#41	—	251	8	no	—
176	C₂×Q₈	—	2	8	no	—
177	C₂×Q₈	—	502	4	no	—
178	C₂×Q₈	—	251	6	no	—
179	C₂×Q₈	—	251	3	no	—
180	C₂×Q₈	—	251	1	no	—
181	C₂×Q₈	C₂²×D₂₅₁	1	6	yes	—
182	C₂×Q₈	C₂²×D₂₅₁	1	1	yes	characteristic
183	C₂×D₄	—	251	2	no	—
184	C₂×D₄	C₂²×D₂₅₁	1	1	yes	characteristic
185	C₂²×C₄	—	251	6	no	—
186	C₂²×C₄	—	251	3	no	—
187	C₂²×C₄	—	251	3	no	—
188	C₂²×C₄	—	251	2	no	—
189	C₂²×C₄	—	251	1	no	—
190	C₂²×C₄	C₂²×D₂₅₁	1	6	yes	—
191	C₄⋊C₄	—	251	3	no	—
192	C₄⋊C₄	—	251	6	no	—
193	C₄⋊C₄	—	251	12	no	—
194	C₄⋊C₄	—	251	6	no	—
195	C₄⋊C₄	—	251	6	no	—
196	C₄⋊C₄	—	251	6	no	—
197	C₄⋊C₄	—	251	3	no	—
198	C₄⋊C₄	—	251	12	no	—
199	C₄⋊C₄	C₂²×D₂₅₁	1	3	yes	—
200	C₄⋊C₄	C₂²×D₂₅₁	1	3	yes	—
201	C₄⋊C₄	C₂²×D₂₅₁	1	6	yes	—
202	C₂²⋊C₄	—	251	6	no	—
203	C₂²⋊C₄	—	251	6	no	—
204	C₂²⋊C₄	C₂²×D₂₅₁	1	6	yes	—
205	C₄²	—	251	3	no	—
206	C₄²	—	251	6	no	—
207	C₄²	—	251	3	no	—
208	C₄²	—	251	3	no	—
209	C₄²	—	251	6	no	—
210	C₄²	—	251	6	no	—
211	C₄²	C₂²×D₂₅₁	1	3	yes	—
212	C₂³	—	251	1	no	—
213	C₂³	C₂³×D₂₅₁	1	2	yes	—
214	Q₈	—	2	12	no	—
215	Q₈	—	1004	6	no	—
216	Q₈	—	502	6	no	—
217	Q₈	—	251	4	no	—
218	Q₈	#16860809	1	4	yes	direct product factor
219	D₄	—	251	8	no	—
220	D₄	Q₈×D₂₅₁	1	4	yes	complemented
221	C₂×C₄	—	2	12	no	—
222	C₂×C₄	—	1004	3	no	—
223	C₂×C₄	—	1004	6	no	—
224	C₂×C₄	—	502	6	no	—
225	C₂×C₄	—	251	3	no	—
226	C₂×C₄	—	251	3	no	—
227	C₂×C₄	—	251	6	no	—
228	C₂×C₄	—	251	8	no	—
229	C₂×C₄	—	251	1	no	—
230	C₂×C₄	—	251	2	no	—
231	C₂×C₄	—	251	4	no	—
232	C₂×C₄	C₂³×D₂₅₁	1	6	yes	—
233	C₂×C₄	C₂³×D₂₅₁	1	3	yes	—
234	C₂×C₄	C₂³×D₂₅₁	1	1	yes	characteristic
235	C₂×C₄	C₂³×D₂₅₁	1	3	yes	—
236	C₂²	—	2	2	no	—
237	C₂²	—	2	2	no	—
238	C₂²	—	502	1	no	—
239	C₂²	—	502	1	no	—
240	C₂²	—	251	2	no	—
241	C₂²	C₂×Q₈×D₂₅₁	1	4	yes	—
242	C₂²	C₂⁴×D₂₅₁	1	1	yes	characteristic
243	C₄	—	2	6	no	—
244	C₄	—	2	3	no	—
245	C₄	—	502	3	no	—
246	C₄	—	502	3	no	—
247	C₄	—	502	6	no	—
248	C₄	—	251	2	no	—
249	C₄	—	251	4	no	—
250	C₄	C₂×Q₈×D₂₅₁	1	2	yes	—
251	C₄	C₂×#16860809	1	6	yes	—
252	C₂	—	2	4	no	—
253	C₂	—	502	2	no	—
254	C₂	C₂²×Q₈×D₂₅₁	1	1	yes	characteristic
255	C₂	C₂²×#16860809	1	1	yes	characteristic
256	C₂	#30632524	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₂₅₁
commutator, socle, cosocle	C₂²×C₂₅₁	C₂⁵
Fitting	D₄×Q₈×C₂₅₁	C₂
prime radical	C₂×#16860809	C₂²
hypercenter	D₄×Q₈	D₂₅₁
hypocenter	C₂₅₁	Q₈×#41
LCS	G ▷ C₂²×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₂² ◁ D₄×Q₈

Complements

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

N	H ≅ G/N	type
1. Q₈×Dic₅₀₂	C₂	semidirect
2. #30632412	C₂	semidirect
3. #30632395	C₂	semidirect
4. #30632394	C₂	semidirect
5. Q₈×C₂²⋊D₂₅₁	C₂	semidirect
6. #30632242	C₂	semidirect
7. #30632239	C₂	semidirect
9. C₂×Q₈×Dic₂₅₁	C₂	semidirect
10. C₄×Q₈×D₂₅₁	C₂	semidirect
11. D₄×Q₈×C₂₅₁	C₂	semidirect
46. #16860809	Q₈	direct (N × H)
86. Dic₅₀₂	C₂×Q₈	semidirect
87. C₂²⋊D₂₅₁	C₂×Q₈	semidirect
89. C₂×Dic₂₅₁	C₂×Q₈	semidirect
97. C₄×D₂₅₁	C₂×Q₈	semidirect
100. Q₈×C₂₅₁	#41	semidirect
101. D₄×C₂₅₁	C₂×Q₈	semidirect
128. C₂₅₁	Q₈×#41	semidirect
135. D₄×Q₈	D₂₅₁	semidirect
218. Q₈	#16860809	direct (N × H)
220. D₄	Q₈×D₂₅₁	semidirect

Maximal subgroups

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

subgroup	index
Q₈×Dic₅₀₂	2
#30632412	2
#30632395	2
#30632394	2
Q₈×C₂²⋊D₂₅₁	2
#30632242	2
#30632239	2
C₄×#16860809	2
C₂×Q₈×Dic₂₅₁	2
C₄×Q₈×D₂₅₁	2
D₄×Q₈×C₂₅₁	2
Q₈×#41	251

Subgroups — Q8×#16860809 #44563942

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×#16860809 #44563942