Subgroups — #109170

Order 512.

636 conjugacy classes of subgroups.

The subgroup lattice has 299 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×#108	C₂	1	1	yes	maximal, characteristic
2	C₂²×#334	C₂	1	1	yes	maximal, characteristic
3	C₂×#1450	C₂	1	1	yes	maximal, characteristic
4	C₂×#1283	C₂	1	2	yes	maximal
5	#7873	C₂	1	2	yes	maximal
6	C₂×#431	C₂²	1	1	yes	characteristic
7	C₂×#431	C₂²	1	1	yes	characteristic
8	C₂²×#108	C₄	1	2	yes	—
9	C₂²×C₄⋊C₈	C₂²	1	2	yes	—
10	C₂×C₄×M₁₆	C₂²	1	2	yes	—
11	C₂²×C₄×C₈	C₂²	1	1	yes	characteristic
12	C₄×#108	—	2	2	no	—
13	C₄×#108	—	2	2	no	—
14	C₂×#341	C₄	1	2	yes	—
15	C₂×#334	—	2	1	no	—
16	C₂×#334	—	2	1	no	—
17	C₂×#334	—	2	2	no	—
18	C₂×#334	C₄	1	4	yes	—
19	#1450	—	2	2	no	—
20	#1450	C₄	1	4	yes	—
21	C₂×#322	—	2	1	no	—
22	C₂×C₈²	—	2	1	no	—
23	#1283	—	2	4	no	—
24	#1277	—	2	4	no	—
25	C₂²×M₁₆	C₂×C₄	1	4	yes	—
26	C₂³×C₈	—	2	2	no	—
27	C₂³×C₈	C₂×C₄	1	1	yes	characteristic
28	C₂³×C₈	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₄²	C₂³	1	1	yes	characteristic
30	#431	—	2	1	no	—
31	#431	—	2	1	no	—
32	#431	—	2	1	no	—
33	#431	—	2	1	no	—
34	#431	C₂×C₄	1	2	yes	—
35	#431	C₂×C₄	1	2	yes	—
36	#431	D₄	1	2	yes	—
37	#431	Q₈	1	2	yes	—
38	C₂×#108	—	2	4	no	—
39	C₂×#108	—	2	4	no	—
40	C₂×#108	C₈	1	8	yes	complemented
41	C₂×C₄⋊C₈	—	2	1	no	—
42	C₂×C₄⋊C₈	—	2	2	no	—
43	C₂×C₄⋊C₈	—	2	2	no	—
44	C₂×C₄⋊C₈	—	2	2	no	—
45	C₂×C₄⋊C₈	—	2	2	no	—
46	C₂×C₄⋊C₈	—	2	2	no	—
47	C₂×C₄⋊C₈	C₂×C₄	1	4	yes	—
48	C₂×C₄⋊C₈	D₄	1	2	yes	—
49	C₂×C₂²⋊C₈	—	2	1	no	—
50	C₂×C₂²⋊C₈	Q₈	1	2	yes	—
51	C₄×M₁₆	—	2	2	no	—
52	C₄×M₁₆	—	2	2	no	—
53	C₄×M₁₆	—	2	2	no	—
54	C₄×M₁₆	—	2	2	no	—
55	C₂×#97	—	2	2	no	—
56	C₂×C₄×C₈	—	2	1	no	—
57	C₂×C₄×C₈	—	2	1	no	—
58	C₂×C₄×C₈	—	2	2	no	—
59	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
60	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
61	C₂×C₄×C₈	D₄	1	1	yes	characteristic
62	C₂×C₄×C₈	D₄	1	1	yes	characteristic
63	C₂×C₄×C₈	D₄	1	2	yes	—
64	C₂×C₄×C₈	D₄	1	2	yes	—
65	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
66	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
67	#341	—	2	4	no	—
68	#334	—	2	4	no	—
69	#334	—	2	4	no	—
70	#322	—	2	4	no	—
71	C₈²	—	2	4	no	—
72	C₂³×C₄	C₂²×C₄	1	1	yes	characteristic
73	C₂³×C₄	C₂×Q₈	1	1	yes	characteristic
74	C₂³×C₄	#41	1	1	yes	characteristic
75	C₂×M₁₆	—	2	4	no	—
76	C₂×M₁₆	—	2	4	no	—
77	C₂×M₁₆	—	2	4	no	—
78	C₂×M₁₆	—	2	4	no	—
79	C₂×M₁₆	—	2	4	no	—
80	C₂×M₁₆	C₂×C₈	1	8	yes	—
81	C₂²×C₈	—	2	2	no	—
82	C₂²×C₈	—	2	2	no	—
83	C₂²×C₈	—	2	4	no	—
84	C₂²×C₈	—	2	2	no	—
85	C₂²×C₈	—	2	2	no	—
86	C₂²×C₈	—	2	4	no	—
87	C₂²×C₈	—	4	2	no	—
88	C₂²×C₈	—	4	2	no	—
89	C₂²×C₈	—	8	4	no	—
90	C₂²×C₈	C₄²	1	2	yes	—
91	C₂²×C₈	C₄²	1	2	yes	—
92	C₂²×C₈	C₂²⋊C₄	1	2	yes	—
93	C₂²×C₈	C₂²⋊C₄	1	2	yes	—
94	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
95	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
96	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
97	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
98	C₂²×C₈	C₂×C₈	1	4	yes	—
99	C₂²×C₈	M₁₆	1	4	yes	—
100	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
101	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
102	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
103	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
104	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
105	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
106	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
107	C₂×C₄²	C₂×Q₈	1	1	yes	characteristic
108	C₂×C₄²	C₂×Q₈	1	1	yes	characteristic
109	C₂×C₄²	#41	1	1	yes	characteristic
110	C₂×C₄²	#41	1	1	yes	characteristic
111	C₂×C₄²	#41	1	1	yes	characteristic
112	#108	—	2	16	no	—
113	C₄⋊C₈	—	2	4	no	—
114	C₄⋊C₈	—	2	2	no	—
115	C₄⋊C₈	—	2	4	no	—
116	C₄⋊C₈	—	2	2	no	—
117	C₄⋊C₈	—	2	2	no	—
118	C₄⋊C₈	—	4	2	no	—
119	C₄⋊C₈	—	4	2	no	—
120	C₄⋊C₈	C₄⋊C₄	1	4	yes	—
121	C₂²⋊C₈	—	2	2	no	—
122	C₂²⋊C₈	—	2	2	no	—
123	C₂²⋊C₈	—	2	2	no	—
124	C₂²⋊C₈	C₄⋊C₄	1	4	yes	—
125	#97	—	4	2	no	—
126	#97	—	4	2	no	—
127	C₄×C₈	—	2	2	no	—
128	C₄×C₈	—	2	2	no	—
129	C₄×C₈	—	2	2	no	—
130	C₄×C₈	—	2	2	no	—
131	C₄×C₈	—	4	2	no	—
132	C₄×C₈	—	4	2	no	—
133	C₄×C₈	C₂²⋊C₄	1	2	yes	—
134	C₄×C₈	C₂²⋊C₄	1	2	yes	—
135	C₄×C₈	C₂²⋊C₄	1	2	yes	—
136	C₄×C₈	C₂²⋊C₄	1	2	yes	—
137	C₄×C₈	C₄⋊C₄	1	2	yes	—
138	C₄×C₈	C₄⋊C₄	1	2	yes	—
139	C₄×C₈	C₄⋊C₄	1	2	yes	—
140	C₄×C₈	C₄⋊C₄	1	2	yes	—
141	C₂⁴	C₄×Q₈	1	1	yes	characteristic
142	C₂²×C₄	—	2	4	no	—
143	C₂²×C₄	—	4	2	no	—
144	C₂²×C₄	C₂×C₄²	1	1	yes	characteristic
145	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
146	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
147	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
148	C₂²×C₄	#117	1	1	yes	characteristic
149	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
150	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
151	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
152	C₂²×C₄	C₄×Q₈	1	1	yes	characteristic
153	C₂²×C₄	C₄×Q₈	1	1	yes	characteristic
154	C₂²×C₄	#121	1	1	yes	characteristic
155	C₂²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
156	C₂²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
157	C₂²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
158	C₂²×C₄	#124	1	1	yes	characteristic
159	C₂²×C₄	#125	1	1	yes	characteristic
160	C₂²×C₄	C₄⋊Q₈	1	1	yes	characteristic
161	C₂²×C₄	C₄⋊Q₈	1	1	yes	characteristic
162	C₂²×C₄	C₂²×C₈	1	2	yes	—
163	C₂²×C₄	C₂×M₁₆	1	2	yes	—
164	C₂²×C₄	#131	1	2	yes	—
165	C₂²×C₄	#131	1	2	yes	—
166	M₁₆	—	2	8	no	—
167	M₁₆	—	2	8	no	—
168	M₁₆	—	4	4	no	—
169	M₁₆	—	4	4	no	—
170	C₂×C₈	—	2	2	no	—
171	C₂×C₈	—	2	2	no	—
172	C₂×C₈	—	2	4	no	—
173	C₂×C₈	—	2	2	no	—
174	C₂×C₈	—	2	2	no	—
175	C₂×C₈	—	2	4	no	—
176	C₂×C₈	—	2	4	no	—
177	C₂×C₈	—	2	8	no	—
178	C₂×C₈	—	4	4	no	—
179	C₂×C₈	—	4	4	no	—
180	C₂×C₈	—	4	4	no	—
181	C₂×C₈	—	4	4	no	—
182	C₂×C₈	—	4	4	no	—
183	C₂×C₈	—	8	4	no	—
184	C₂×C₈	—	8	4	no	—
185	C₂×C₈	—	8	4	no	—
186	C₂×C₈	#95	1	2	yes	—
187	C₂×C₈	#95	1	2	yes	—
188	C₂×C₈	#95	1	2	yes	—
189	C₂×C₈	#95	1	2	yes	—
190	C₂×C₈	C₄×C₈	1	4	yes	—
191	C₂×C₈	#97	1	4	yes	—
192	C₂×C₈	C₂²⋊C₈	1	8	yes	—
193	C₂×C₈	C₄⋊C₈	1	4	yes	—
194	C₂×C₈	C₄⋊C₈	1	4	yes	—
195	C₄²	C₂×C₂²⋊C₄	1	1	yes	characteristic
196	C₄²	C₂×C₂²⋊C₄	1	1	yes	characteristic
197	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
198	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
199	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
200	C₄²	#117	1	1	yes	characteristic
201	C₄²	C₄×D₄	1	1	yes	characteristic
202	C₄²	C₄×Q₈	1	1	yes	characteristic
203	C₄²	#121	1	1	yes	characteristic
204	C₄²	#121	1	1	yes	characteristic
205	C₄²	#121	1	1	yes	characteristic
206	C₄²	C₂²⋊Q₈	1	1	yes	characteristic
207	C₄²	#124	1	1	yes	characteristic
208	C₄²	#125	1	1	yes	characteristic
209	C₄²	#127	1	1	yes	characteristic
210	C₄²	C₄⋊Q₈	1	1	yes	characteristic
211	C₂³	—	2	2	no	—
212	C₂³	—	2	2	no	—
213	C₂³	C₄×C₄⋊C₄	1	1	yes	characteristic
214	C₂³	#389	1	1	yes	characteristic
215	C₂³	#391	1	1	yes	characteristic
216	C₂³	C₈×Q₈	1	2	yes	—
217	C₂³	#446	1	2	yes	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	4	1	no	—
226	C₂×C₄	—	4	1	no	—
227	C₂×C₄	—	4	1	no	—
228	C₂×C₄	—	4	1	no	—
229	C₂×C₄	C₂×#95	1	1	yes	characteristic
230	C₂×C₄	C₄×C₄⋊C₄	1	1	yes	characteristic
231	C₂×C₄	#380	1	1	yes	characteristic
232	C₂×C₄	#380	1	1	yes	characteristic
233	C₂×C₄	#382	1	1	yes	characteristic
234	C₂×C₄	#384	1	1	yes	characteristic
235	C₂×C₄	#389	1	1	yes	characteristic
236	C₂×C₄	#389	1	1	yes	characteristic
237	C₂×C₄	#389	1	1	yes	characteristic
238	C₂×C₄	#390	1	1	yes	characteristic
239	C₂×C₄	#390	1	1	yes	characteristic
240	C₂×C₄	#391	1	1	yes	characteristic
241	C₂×C₄	C₂×C₄×C₈	1	1	yes	characteristic
242	C₂×C₄	C₂×#97	1	1	yes	characteristic
243	C₂×C₄	#405	1	1	yes	characteristic
244	C₂×C₄	#405	1	1	yes	characteristic
245	C₂×C₄	C₂×C₂²⋊C₈	1	2	yes	—
246	C₂×C₄	#408	1	2	yes	—
247	C₂×C₄	C₂×C₄⋊C₈	1	2	yes	—
248	C₂×C₄	#424	1	2	yes	—
249	C₂×C₄	C₈×D₄	1	2	yes	—
250	C₂×C₄	C₈×D₄	1	2	yes	—
251	C₂×C₄	#436	1	2	yes	—
252	C₂×C₄	#436	1	2	yes	—
253	C₂×C₄	C₈×Q₈	1	2	yes	—
254	C₂×C₄	#446	1	2	yes	—
255	C₂×C₄	#471	1	4	yes	—
256	C₈	—	2	2	no	—
257	C₈	—	2	2	no	—
258	C₈	—	4	4	no	—
259	C₈	—	4	4	no	—
260	C₈	—	4	8	no	—
261	C₈	—	8	4	no	—
262	C₈	#336	1	4	yes	—
263	C₈	#336	1	4	yes	—
264	C₂²	—	2	1	no	—
265	C₂²	—	2	1	no	—
266	C₂²	—	2	1	no	—
267	C₂²	—	2	1	no	—
268	C₂²	—	2	4	no	—
269	C₂²	—	2	4	no	—
270	C₂²	C₄⋊#95	1	1	yes	characteristic
271	C₂²	C₈×C₄⋊C₄	1	1	yes	characteristic
272	C₂²	#1770	1	1	yes	characteristic
273	C₂²	#1939	1	2	yes	—
274	C₂²	#1951	1	2	yes	—
275	C₂²	#1953	1	2	yes	—
276	C₂²	#1984	1	2	yes	—
277	C₄	—	2	1	no	—
278	C₄	—	2	1	no	—
279	C₄	—	2	1	no	—
280	C₄	—	2	1	no	—
281	C₄	—	2	1	no	—
282	C₄	—	2	1	no	—
283	C₄	C₂×#336	1	1	yes	characteristic
284	C₄	#1729	1	1	yes	characteristic
285	C₄	C₈×C₄⋊C₄	1	1	yes	characteristic
286	C₄	#1770	1	1	yes	characteristic
287	C₄	#1815	1	2	yes	—
288	C₄	#1939	1	2	yes	—
289	C₄	#1954	1	2	yes	—
290	C₄	#1977	1	2	yes	—
291	C₂	—	2	2	no	—
292	C₂	—	2	1	no	—
293	C₂	—	2	1	no	—
294	C₂	#9607	1	4	yes	—
295	C₂	#10834	1	1	yes	characteristic
296	C₂	#10869	1	1	yes	characteristic
297	C₂	#10873	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	#121
commutator	C₂×C₄	C₂×C₄×C₈
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂³	#389
prime radical	C₂⁴	C₄×Q₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
40. C₂×#108	C₈	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×#108	2
C₂²×#334	2
C₂×#1450	2
C₂×#1283	2
#7873	2