Subgroups — Q₁₆×#108 #225754

Order 512.

543 conjugacy classes of subgroups.

The subgroup lattice has 287 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×#108	C₂	1	2	yes	maximal
2	M₁₆×Q₁₆	C₂	1	2	yes	maximal
3	C₂×C₈×Q₁₆	C₂	1	1	yes	maximal, characteristic
4	#9624	C₂	1	2	yes	maximal
5	#8907	C₂	1	4	yes	maximal
6	#8823	C₂	1	1	yes	maximal, characteristic
7	#8659	C₂	1	2	yes	maximal
8	C₈×#108	C₂	1	1	yes	maximal, characteristic
9	Q₈×M₁₆	C₂²	1	4	yes	—
10	C₂×C₈×Q₈	C₂²	1	2	yes	—
11	C₂×C₄×Q₁₆	C₂²	1	1	yes	characteristic
12	#1950	—	2	4	no	—
13	#1950	C₂²	1	4	yes	—
14	#1848	—	2	2	no	—
15	#1848	C₂²	1	2	yes	—
16	C₄×#108	—	2	2	no	—
17	C₄×#108	C₂²	1	1	yes	characteristic
18	#1591	—	2	2	no	—
19	#1584	—	2	4	no	—
20	C₈×Q₁₆	—	2	2	no	—
21	C₈×Q₁₆	C₄	1	2	yes	—
22	C₈×Q₁₆	C₄	1	1	yes	characteristic
23	C₈×Q₁₆	C₄	1	2	yes	—
24	C₈×Q₁₆	C₄	1	1	yes	characteristic
25	C₈×Q₁₆	C₄	1	1	yes	characteristic
26	C₈×Q₁₆	C₄	1	1	yes	characteristic
27	#1567	C₂²	1	2	yes	—
28	#1567	C₂²	1	2	yes	—
29	#1564	C₂²	1	1	yes	characteristic
30	C₂×#335	C₂²	1	1	yes	characteristic
31	#1487	C₂²	1	4	yes	—
32	C₂×Q₈⋊C₈	C₂²	1	2	yes	—
33	C₈×M₁₆	C₂²	1	2	yes	—
34	C₂×C₈²	C₂²	1	1	yes	characteristic
35	#1301	C₂²	1	4	yes	—
36	#1278	C₂²	1	2	yes	—
37	C₂²×Q₁₆	Q₈	1	1	yes	characteristic
38	C₂×C₄×Q₈	C₂³	1	2	yes	—
39	#446	—	2	4	no	—
40	#446	—	4	4	no	—
41	C₈×Q₈	—	2	2	no	—
42	C₈×Q₈	—	2	2	no	—
43	C₈×Q₈	—	2	4	no	—
44	C₈×Q₈	C₂×C₄	1	2	yes	—
45	C₈×Q₈	C₂×C₄	1	2	yes	—
46	C₈×Q₈	C₂×C₄	1	2	yes	—
47	C₈×Q₈	C₂×C₄	1	2	yes	—
48	C₄×Q₁₆	C₂×C₄	1	2	yes	—
49	C₄×Q₁₆	C₂×C₄	1	1	yes	characteristic
50	C₄×Q₁₆	C₂×C₄	1	1	yes	characteristic
51	C₄×Q₁₆	D₄	1	1	yes	characteristic
52	C₄×Q₁₆	D₄	1	2	yes	—
53	C₄×Q₁₆	D₄	1	1	yes	characteristic
54	#431	C₂³	1	2	yes	—
55	C₂×#108	—	2	2	no	—
56	C₂×#108	—	2	4	no	—
57	C₂×#108	—	2	2	no	—
58	C₂×#108	D₄	1	1	yes	characteristic
59	C₂×#108	D₄	1	1	yes	characteristic
60	C₂×#107	Q₈	1	1	yes	characteristic
61	#423	—	2	4	no	—
62	#423	C₂³	1	4	yes	—
63	C₂×C₄⋊C₈	—	2	2	no	—
64	C₂×C₄⋊C₈	C₂³	1	2	yes	—
65	C₂×#103	Q₈	1	2	yes	—
66	C₄×M₁₆	—	2	4	no	—
67	C₄×M₁₆	C₂³	1	2	yes	—
68	C₂×C₄×C₈	—	2	2	no	—
69	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
70	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
71	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
72	#341	—	2	4	no	—
73	#341	D₄	1	2	yes	—
74	#335	—	2	2	no	—
75	#335	—	2	2	no	—
76	#335	—	2	1	no	—
77	#335	—	4	2	no	—
78	#335	C₂×C₄	1	2	yes	—
79	#335	C₂×C₄	1	1	yes	characteristic
80	#335	C₂×C₄	1	1	yes	characteristic
81	#334	—	2	1	no	—
82	Q₈⋊C₈	—	2	4	no	—
83	Q₈⋊C₈	—	2	4	no	—
84	Q₈⋊C₈	C₂×C₄	1	2	yes	—
85	Q₈⋊C₈	C₂×C₄	1	2	yes	—
86	Q₈⋊C₈	C₂×C₄	1	2	yes	—
87	Q₈⋊C₈	C₂×C₄	1	2	yes	—
88	#322	—	2	2	no	—
89	C₈²	—	2	2	no	—
90	C₈²	C₂×C₄	1	1	yes	characteristic
91	C₈²	C₂×C₄	1	1	yes	characteristic
92	C₈²	C₂×C₄	1	1	yes	characteristic
93	C₈²	C₂×C₄	1	1	yes	characteristic
94	C₂²×Q₈	C₂×Q₈	1	2	yes	—
95	C₂×Q₁₆	—	2	4	no	—
96	C₂×Q₁₆	C₄⋊C₄	1	2	yes	—
97	C₂×Q₁₆	C₄⋊C₄	1	1	yes	characteristic
98	C₂×Q₁₆	C₄⋊C₄	1	1	yes	characteristic
99	C₂×M₁₆	—	2	4	no	—
100	C₂×M₁₆	C₂×D₄	1	2	yes	—
101	C₂×M₁₆	C₂×D₄	1	2	yes	—
102	C₂²×C₈	—	2	2	no	—
103	C₂²×C₈	—	2	1	no	—
104	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
105	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
106	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
107	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
108	C₄×Q₈	—	2	2	no	—
109	C₄×Q₈	—	2	2	no	—
110	C₄×Q₈	C₂²×C₄	1	2	yes	—
111	C₄×Q₈	C₂²×C₄	1	2	yes	—
112	C₄×Q₈	C₂×D₄	1	2	yes	—
113	C₄×Q₈	C₂×D₄	1	2	yes	—
114	C₂×C₄⋊C₄	—	2	2	no	—
115	C₂×C₄⋊C₄	C₂×Q₈	1	2	yes	—
116	C₂×C₄²	—	2	2	no	—
117	C₂×C₄²	C₂⁴	1	1	yes	characteristic
118	#108	—	2	2	no	—
119	#108	—	4	4	no	—
120	#108	—	8	2	no	—
121	#108	—	8	4	no	—
122	#108	Q₁₆	1	4	yes	direct product factor
123	#107	C₄⋊C₄	1	2	yes	—
124	#107	C₄⋊C₄	1	1	yes	characteristic
125	#107	C₄⋊C₄	1	1	yes	characteristic
126	C₄⋊C₈	—	2	2	no	—
127	C₄⋊C₈	—	2	2	no	—
128	C₄⋊C₈	—	2	2	no	—
129	C₄⋊C₈	—	2	2	no	—
130	C₄⋊C₈	—	2	4	no	—
131	C₄⋊C₈	—	2	2	no	—
132	C₄⋊C₈	—	2	2	no	—
133	C₄⋊C₈	—	2	4	no	—
134	C₄⋊C₈	—	4	4	no	—
135	C₄⋊C₈	—	4	4	no	—
136	C₄⋊C₈	C₂²×C₄	1	2	yes	—
137	C₄⋊C₈	C₂²×C₄	1	2	yes	—
138	#103	C₄⋊C₄	1	2	yes	—
139	#103	C₄⋊C₄	1	2	yes	—
140	#103	C₄⋊C₄	1	2	yes	—
141	#103	C₄⋊C₄	1	2	yes	—
142	C₂²⋊C₈	—	2	2	no	—
143	#97	—	2	2	no	—
144	#97	—	4	4	no	—
145	C₄×C₈	—	2	2	no	—
146	C₄×C₈	—	2	2	no	—
147	C₄×C₈	—	2	2	no	—
148	C₄×C₈	—	2	2	no	—
149	C₄×C₈	—	2	2	no	—
150	C₄×C₈	—	2	2	no	—
151	C₄×C₈	—	2	1	no	—
152	C₄×C₈	—	4	4	no	—
153	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
154	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
155	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
156	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
157	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
158	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
159	C₄×C₈	C₂×D₄	1	1	yes	characteristic
160	C₄×C₈	C₂×D₄	1	1	yes	characteristic
161	C₄×C₈	#41	1	1	yes	characteristic
162	C₄×C₈	#41	1	1	yes	characteristic
163	C₂×Q₈	—	2	2	no	—
164	C₂×Q₈	—	2	4	no	—
165	C₂×Q₈	—	4	2	no	—
166	C₂×Q₈	C₂×C₄⋊C₄	1	2	yes	—
167	C₂×Q₈	C₂×C₄⋊C₄	1	2	yes	—
168	C₂²×C₄	—	2	2	no	—
169	C₂²×C₄	—	2	2	no	—
170	C₂²×C₄	C₂²×D₄	1	1	yes	characteristic
171	C₂²×C₄	C₂²×Q₈	1	1	yes	characteristic
172	C₂²×C₄	2₋¹⁺⁴	1	1	yes	characteristic
173	Q₁₆	—	2	4	no	—
174	Q₁₆	—	2	2	no	—
175	Q₁₆	#108	1	4	yes	direct product factor
176	M₁₆	—	2	2	no	—
177	M₁₆	—	2	2	no	—
178	M₁₆	—	2	2	no	—
179	M₁₆	—	4	4	no	—
180	M₁₆	—	4	4	no	—
181	M₁₆	C₂×Q₁₆	1	4	yes	—
182	C₂×C₈	—	2	1	no	—
183	C₂×C₈	—	2	2	no	—
184	C₂×C₈	—	2	1	no	—
185	C₂×C₈	—	2	1	no	—
186	C₂×C₈	—	2	2	no	—
187	C₂×C₈	—	2	2	no	—
188	C₂×C₈	—	2	2	no	—
189	C₂×C₈	—	2	1	no	—
190	C₂×C₈	—	2	2	no	—
191	C₂×C₈	—	4	1	no	—
192	C₂×C₈	—	4	2	no	—
193	C₂×C₈	—	4	2	no	—
194	C₂×C₈	—	4	2	no	—
195	C₂×C₈	—	4	4	no	—
196	C₂×C₈	—	4	2	no	—
197	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
198	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
199	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
200	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
201	C₂×C₈	C₄×D₄	1	1	yes	characteristic
202	C₂×C₈	C₄×D₄	1	1	yes	characteristic
203	C₂×C₈	C₄×D₄	1	1	yes	characteristic
204	C₂×C₈	C₄×D₄	1	1	yes	characteristic
205	C₂×C₈	C₂×Q₁₆	1	2	yes	—
206	C₂×C₈	#135	1	2	yes	—
207	C₄⋊C₄	—	2	2	no	—
208	C₄⋊C₄	—	2	2	no	—
209	C₄⋊C₄	—	2	2	no	—
210	C₄⋊C₄	—	2	2	no	—
211	C₄⋊C₄	—	2	2	no	—
212	C₄⋊C₄	C₂×C₄⋊C₄	1	2	yes	—
213	C₄⋊C₄	C₂×C₄⋊C₄	1	2	yes	—
214	C₄²	—	2	2	no	—
215	C₄²	—	2	2	no	—
216	C₄²	—	2	2	no	—
217	C₄²	—	2	2	no	—
218	C₄²	C₂³×C₄	1	1	yes	characteristic
219	C₄²	C₂²×D₄	1	1	yes	characteristic
220	C₄²	C₂×#41	1	1	yes	characteristic
221	C₄²	H(5,𝔽₂)	1	1	yes	characteristic
222	C₂³	D₄×Q₈	1	1	yes	characteristic
223	Q₈	—	2	2	no	—
224	Q₈	—	2	2	no	—
225	Q₈	—	2	2	no	—
226	Q₈	—	4	4	no	—
227	Q₈	C₂×#108	1	4	yes	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	4	1	no	—
235	C₂×C₄	—	4	4	no	—
236	C₂×C₄	—	8	2	no	—
237	C₂×C₄	C₂²×C₄⋊C₄	1	1	yes	characteristic
238	C₂×C₄	C₂×C₄×D₄	1	1	yes	characteristic
239	C₂×C₄	#520	1	1	yes	characteristic
240	C₂×C₄	D₄²	1	1	yes	characteristic
241	C₂×C₄	#547	1	1	yes	characteristic
242	C₂×C₄	#554	1	1	yes	characteristic
243	C₂×C₄	C₂²×Q₁₆	1	1	yes	characteristic
244	C₂×C₄	C₂×#135	1	1	yes	characteristic
245	C₂×C₄	#576	1	1	yes	characteristic
246	C₂×C₄	#578	1	1	yes	characteristic
247	C₈	—	2	2	no	—
248	C₈	—	2	1	no	—
249	C₈	—	2	1	no	—
250	C₈	—	2	1	no	—
251	C₈	—	2	1	no	—
252	C₈	—	2	1	no	—
253	C₈	—	2	4	no	—
254	C₈	—	4	1	no	—
255	C₈	—	4	2	no	—
256	C₈	—	4	2	no	—
257	C₈	—	4	2	no	—
258	C₈	—	8	2	no	—
259	C₈	—	8	4	no	—
260	C₈	C₂×#108	1	2	yes	—
261	C₈	C₄×Q₁₆	1	2	yes	—
262	C₈	C₄×Q₁₆	1	2	yes	—
263	C₂²	—	2	1	no	—
264	C₂²	—	2	1	no	—
265	C₂²	D₄×C₄⋊C₄	1	1	yes	characteristic
266	C₂²	Q₈×Q₁₆	1	1	yes	characteristic
267	C₂²	#3383	1	1	yes	characteristic
268	C₄	—	2	1	no	—
269	C₄	—	2	4	no	—
270	C₄	—	2	1	no	—
271	C₄	—	2	1	no	—
272	C₄	—	4	2	no	—
273	C₄	—	4	2	no	—
274	C₄	—	4	2	no	—
275	C₄	C₂²×#108	1	1	yes	characteristic
276	C₄	#2916	1	1	yes	characteristic
277	C₄	C₂×C₄×Q₁₆	1	1	yes	characteristic
278	C₄	#2951	1	1	yes	characteristic
279	C₄	D₄×Q₁₆	1	1	yes	characteristic
280	C₄	#3287	1	1	yes	characteristic
281	C₂	—	2	1	no	—
282	C₂	—	2	1	no	—
283	C₂	D₄×#108	1	1	yes	characteristic
284	C₂	C₄⋊C₄×Q₁₆	1	1	yes	characteristic
285	C₂	#22694	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₄²
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	D₄×C₄⋊C₄
prime radical	C₂³	D₄×Q₈
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
122. #108	Q₁₆	direct (N × H)
175. Q₁₆	#108	direct (N × H)

Maximal subgroups

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

subgroup	index
Q₈×#108	2
M₁₆×Q₁₆	2
C₂×C₈×Q₁₆	2
#9624	2
#8907	2
#8823	2
#8659	2
C₈×#108	2

Subgroups — Q16×#108 #225754

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₁₆×#108 #225754