Subgroups — #226402

Order 512.

632 conjugacy classes of subgroups.

The subgroup lattice has 295 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22722	C₂	1	2	yes	maximal
2	C₄×C₈×Q₈	C₂	1	1	yes	maximal, characteristic
3	#9605	C₂	1	4	yes	maximal
4	#9602	C₂	1	2	yes	maximal
5	#8805	C₂	1	2	yes	maximal
6	#8805	C₂	1	1	yes	maximal, characteristic
7	#8273	C₂	1	2	yes	maximal
8	C₄×#334	C₂	1	1	yes	maximal, characteristic
9	C₂×#446	—	2	2	no	—
10	C₂×C₈×Q₈	C₄	1	1	yes	characteristic
11	C₂×C₈×Q₈	C₄	1	1	yes	characteristic
12	C₄²×Q₈	C₂²	1	1	yes	characteristic
13	#1940	—	2	4	no	—
14	#1939	—	2	2	no	—
15	#1842	C₂²	1	2	yes	—
16	#1842	C₂²	1	4	yes	—
17	C₈×C₄⋊C₄	C₄	1	2	yes	—
18	C₈×C₄⋊C₄	C₄	1	1	yes	characteristic
19	C₈×C₄⋊C₄	C₄	1	2	yes	—
20	C₈×C₄⋊C₄	C₄	1	1	yes	characteristic
21	#1767	C₂²	1	4	yes	—
22	C₄×C₄⋊C₈	C₂²	1	2	yes	—
23	C₄×C₄⋊C₈	C₂²	1	1	yes	characteristic
24	C₄×C₄⋊C₈	C₂²	1	2	yes	—
25	C₄²×C₈	C₂²	1	1	yes	characteristic
26	C₄²×C₈	C₂²	1	2	yes	—
27	C₂×#334	C₂²	1	1	yes	characteristic
28	C₂×#334	C₂²	1	2	yes	—
29	C₂×#334	C₂²	1	4	yes	—
30	C₂×#334	C₂²	1	2	yes	—
31	C₂×#334	C₂²	1	1	yes	characteristic
32	#1277	C₂²	1	2	yes	—
33	#1277	C₂²	1	4	yes	—
34	C₂×C₄×Q₈	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄×Q₈	D₄	1	1	yes	characteristic
36	C₂×C₄×Q₈	Q₈	1	1	yes	characteristic
37	#446	—	4	8	no	—
38	C₈×Q₈	—	2	4	no	—
39	C₈×Q₈	—	2	4	no	—
40	C₂×C₄⋊C₈	—	2	2	no	—
41	C₂×C₄⋊C₈	—	2	4	no	—
42	C₂×C₄⋊C₈	—	2	4	no	—
43	C₂×C₄⋊C₈	—	2	2	no	—
44	C₂×C₄⋊C₈	C₂×C₄	1	2	yes	—
45	C₂×C₄⋊C₈	C₂×C₄	1	2	yes	—
46	C₂×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄⋊C₈	C₂³	1	2	yes	—
49	C₂×C₄⋊C₈	C₂³	1	4	yes	—
50	C₂×C₄⋊C₈	C₂³	1	2	yes	—
51	C₂×#97	—	2	4	no	—
52	C₂×C₄×C₈	—	2	2	no	—
53	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
54	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
55	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
56	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
57	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
58	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
59	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
60	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
61	C₂×C₄×C₈	C₂³	1	2	yes	—
62	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
63	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
64	C₄×C₄⋊C₄	C₂×C₄	1	2	yes	—
65	C₄×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
66	C₄×C₄⋊C₄	D₄	1	1	yes	characteristic
67	C₄×C₄⋊C₄	D₄	1	2	yes	—
68	C₄×C₄⋊C₄	Q₈	1	1	yes	characteristic
69	C₄×C₄⋊C₄	Q₈	1	2	yes	—
70	C₄³	C₂³	1	1	yes	characteristic
71	C₄³	C₂³	1	2	yes	—
72	#336	—	2	4	no	—
73	#336	—	2	2	no	—
74	#336	C₂×C₄	1	2	yes	—
75	#336	C₂×C₄	1	2	yes	—
76	#336	C₂×C₄	1	2	yes	—
77	#334	—	2	2	no	—
78	#334	—	2	4	no	—
79	#334	—	4	4	no	—
80	#334	—	4	2	no	—
81	#334	Q₈	1	4	yes	complemented
82	C₂²×Q₈	C₄⋊C₄	1	1	yes	characteristic
83	C₂²×C₈	—	2	2	no	—
84	C₂²×C₈	—	2	2	no	—
85	C₂²×C₈	—	2	4	no	—
86	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
87	C₂²×C₈	C₂²×C₄	1	2	yes	—
88	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
89	C₂²×C₈	C₂²×C₄	1	2	yes	—
90	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
91	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
92	C₄×Q₈	—	2	4	no	—
93	C₄×Q₈	—	2	2	no	—
94	C₄×Q₈	—	2	2	no	—
95	C₄×Q₈	SD₁₆	1	8	yes	—
96	C₂×C₄⋊C₄	C₄⋊C₄	1	1	yes	characteristic
97	C₂×C₄⋊C₄	C₄⋊C₄	1	2	yes	—
98	C₂×C₄⋊C₄	C₄⋊C₄	1	2	yes	—
99	C₂×C₄⋊C₄	C₄⋊C₄	1	2	yes	—
100	C₂×C₄⋊C₄	C₄⋊C₄	1	1	yes	characteristic
101	C₂×C₄⋊C₄	C₄⋊C₄	1	1	yes	characteristic
102	C₂×C₄²	C₂²×C₄	1	2	yes	—
103	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
104	C₂×C₄²	C₂²×C₄	1	2	yes	—
105	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
106	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
107	C₂×C₄²	C₂×D₄	1	2	yes	—
108	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
109	C₂×C₄²	C₂×D₄	1	2	yes	—
110	C₂×C₄²	C₂×Q₈	1	1	yes	characteristic
111	C₂×C₄²	C₂×Q₈	1	2	yes	—
112	C₂×C₄²	C₂×Q₈	1	2	yes	—
113	C₂×C₄²	C₂×Q₈	1	1	yes	characteristic
114	C₂×C₄²	C₂⁴	1	1	yes	characteristic
115	C₄⋊C₈	—	2	4	no	—
116	C₄⋊C₈	—	2	4	no	—
117	C₄⋊C₈	—	2	2	no	—
118	C₄⋊C₈	—	2	2	no	—
119	C₄⋊C₈	—	2	4	no	—
120	C₄⋊C₈	—	2	2	no	—
121	C₄⋊C₈	—	2	4	no	—
122	C₄⋊C₈	—	2	2	no	—
123	C₄⋊C₈	—	4	4	no	—
124	C₄⋊C₈	—	4	8	no	—
125	C₄⋊C₈	—	8	4	no	—
126	C₄⋊C₈	—	8	2	no	—
127	C₄⋊C₈	M₁₆	1	4	yes	—
128	C₄⋊C₈	C₂×Q₈	1	4	yes	—
129	#97	—	4	8	no	—
130	C₄×C₈	—	2	4	no	—
131	C₄×C₈	—	2	4	no	—
132	C₄×C₈	—	2	4	no	—
133	C₄×C₈	—	2	2	no	—
134	C₄×C₈	—	2	2	no	—
135	C₄×C₈	—	2	1	no	—
136	C₄×C₈	—	2	2	no	—
137	C₄×C₈	—	2	1	no	—
138	C₄×C₈	—	2	2	no	—
139	C₄×C₈	—	4	1	no	—
140	C₄×C₈	—	4	2	no	—
141	C₄×C₈	—	4	4	no	—
142	C₄×C₈	M₁₆	1	4	yes	—
143	C₄×C₈	C₂×Q₈	1	2	yes	—
144	C₄×C₈	#41	1	2	yes	—
145	#95	C₄⋊C₄	1	2	yes	—
146	#95	C₄⋊C₄	1	2	yes	—
147	#95	C₄⋊C₄	1	2	yes	—
148	C₂×Q₈	—	2	2	no	—
149	C₂×Q₈	—	2	2	no	—
150	C₂×Q₈	#106	1	4	yes	—
151	C₂²×C₄	C₂×C₄⋊C₄	1	2	yes	—
152	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
153	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
154	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
155	C₂²×C₄	C₂×C₄⋊C₄	1	2	yes	—
156	C₂²×C₄	C₂×C₄⋊C₄	1	2	yes	—
157	C₂²×C₄	C₂×C₄⋊C₄	1	2	yes	—
158	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
159	C₂²×C₄	C₂³×C₄	1	1	yes	characteristic
160	C₂²×C₄	C₂²×D₄	1	1	yes	characteristic
161	C₂²×C₄	C₂²×Q₈	1	1	yes	characteristic
162	C₂×C₈	—	2	1	no	—
163	C₂×C₈	—	2	2	no	—
164	C₂×C₈	—	2	1	no	—
165	C₂×C₈	—	2	2	no	—
166	C₂×C₈	—	2	1	no	—
167	C₂×C₈	—	2	1	no	—
168	C₂×C₈	—	2	2	no	—
169	C₂×C₈	—	2	1	no	—
170	C₂×C₈	—	2	4	no	—
171	C₂×C₈	—	2	2	no	—
172	C₂×C₈	—	2	1	no	—
173	C₂×C₈	—	2	1	no	—
174	C₂×C₈	—	2	2	no	—
175	C₂×C₈	—	4	2	no	—
176	C₂×C₈	—	4	2	no	—
177	C₂×C₈	—	4	2	no	—
178	C₂×C₈	—	4	4	no	—
179	C₂×C₈	—	4	2	no	—
180	C₂×C₈	—	4	4	no	—
181	C₂×C₈	—	4	2	no	—
182	C₂×C₈	—	4	4	no	—
183	C₂×C₈	C₄×Q₈	1	2	yes	—
184	C₂×C₈	C₄×Q₈	1	2	yes	—
185	C₂×C₈	C₂×M₁₆	1	4	yes	—
186	C₂×C₈	C₂×M₁₆	1	2	yes	—
187	C₂×C₈	C₂×M₁₆	1	2	yes	—
188	C₂×C₈	#131	1	2	yes	—
189	C₄⋊C₄	—	2	4	no	—
190	C₄⋊C₄	—	2	2	no	—
191	C₄⋊C₄	—	2	4	no	—
192	C₄⋊C₄	#106	1	8	yes	—
193	C₄⋊C₄	#106	1	4	yes	—
194	C₄⋊C₄	#108	1	4	yes	—
195	C₄²	—	2	2	no	—
196	C₄²	—	2	2	no	—
197	C₄²	—	2	2	no	—
198	C₄²	—	2	1	no	—
199	C₄²	—	2	2	no	—
200	C₄²	—	2	1	no	—
201	C₄²	—	2	2	no	—
202	C₄²	—	2	4	no	—
203	C₄²	—	2	4	no	—
204	C₄²	—	2	2	no	—
205	C₄²	—	4	1	no	—
206	C₄²	—	4	2	no	—
207	C₄²	C₂×M₁₆	1	2	yes	—
208	C₄²	C₂×M₁₆	1	2	yes	—
209	C₄²	C₂×SD₁₆	1	4	yes	—
210	C₄²	C₂×SD₁₆	1	8	yes	—
211	C₄²	C₂²×Q₈	1	1	yes	characteristic
212	C₄²	C₂×#41	1	1	yes	characteristic
213	C₄²	H(5,𝔽₂)	1	1	yes	characteristic
214	C₄²	2₋¹⁺⁴	1	1	yes	characteristic
215	C₂³	C₂²×C₄⋊C₄	1	1	yes	characteristic
216	Q₈	—	2	8	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	#423	1	2	yes	—
236	C₂×C₄	#423	1	2	yes	—
237	C₂×C₄	C₂×#106	1	2	yes	—
238	C₂×C₄	C₂×#106	1	4	yes	—
239	C₂×C₄	C₂×#106	1	4	yes	—
240	C₂×C₄	C₂×#106	1	2	yes	—
241	C₂×C₄	C₂×#108	1	4	yes	—
242	C₂×C₄	C₂×#108	1	2	yes	—
243	C₂×C₄	C₂×C₄×Q₈	1	1	yes	characteristic
244	C₂×C₄	#520	1	1	yes	characteristic
245	C₂×C₄	D₄×Q₈	1	1	yes	characteristic
246	C₂×C₄	#550	1	1	yes	characteristic
247	C₂×C₄	#557	1	1	yes	characteristic
248	C₂×C₄	Q₈²	1	1	yes	characteristic
249	C₂×C₄	C₂²×M₁₆	1	1	yes	characteristic
250	C₂×C₄	C₂²×M₁₆	1	1	yes	characteristic
251	C₂×C₄	C₂×#131	1	1	yes	characteristic
252	C₂×C₄	#568	1	1	yes	characteristic
253	C₂×C₄	C₂²×SD₁₆	1	2	yes	—
254	C₂×C₄	#577	1	2	yes	—
255	C₈	—	2	2	no	—
256	C₈	—	2	2	no	—
257	C₈	—	2	4	no	—
258	C₈	—	4	4	no	—
259	C₈	—	4	2	no	—
260	C₈	—	4	1	no	—
261	C₈	—	8	2	no	—
262	C₈	—	8	4	no	—
263	C₈	#446	1	4	yes	complemented
264	C₂²	Q₈×C₄⋊C₄	1	1	yes	characteristic
265	C₂²	C₂×#423	1	1	yes	characteristic
266	C₂²	#2906	1	1	yes	characteristic
267	C₂²	C₂²×#106	1	1	yes	characteristic
268	C₂²	#2912	1	1	yes	characteristic
269	C₂²	C₂²×#108	1	1	yes	characteristic
270	C₂²	#2916	1	1	yes	characteristic
271	C₄	—	2	4	no	—
272	C₄	—	2	2	no	—
273	C₄	—	2	2	no	—
274	C₄	—	2	4	no	—
275	C₄	—	2	4	no	—
276	C₄	—	2	1	no	—
277	C₄	—	2	1	no	—
278	C₄	—	4	2	no	—
279	C₄	—	4	1	no	—
280	C₄	#1566	1	4	yes	—
281	C₄	C₂×#446	1	1	yes	characteristic
282	C₄	Q₈×M₁₆	1	1	yes	characteristic
283	C₄	#2989	1	1	yes	characteristic
284	C₄	#2990	1	1	yes	characteristic
285	C₄	Q₈×SD₁₆	1	2	yes	—
286	C₄	#3392	1	2	yes	—
287	C₂	C₂×#1566	1	1	yes	characteristic
288	C₂	#17798	1	1	yes	characteristic
289	C₂	#17826	1	1	yes	characteristic
290	C₂	#17834	1	1	yes	characteristic
291	C₂	#22722	1	1	yes	characteristic
292	C₂	Q₈×#106	1	1	yes	characteristic
293	C₂	Q₈×#108	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
81. #334	Q₈	semidirect
263. C₈	#446	semidirect

Maximal subgroups

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

subgroup	index
#22722	2
C₄×C₈×Q₈	2
#9605	2
#9602	2
#8805	2
#8273	2
C₄×#334	2