Subgroups — #109273

Order 512.

616 conjugacy classes of subgroups.

The subgroup lattice has 307 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20817	C₂	1	1	yes	maximal, characteristic
2	C₂×#1567	C₂	1	1	yes	maximal, characteristic
3	C₂×C₈×M₁₆	C₂	1	1	yes	maximal, characteristic
4	#10855	C₂	1	2	yes	maximal
5	#8095	C₂	1	2	yes	maximal
6	C₂×#431	C₂²	1	1	yes	characteristic
7	C₂×#423	C₂²	1	2	yes	—
8	C₂×C₄×M₁₆	C₂²	1	1	yes	characteristic
9	C₂²×C₄×C₈	C₂²	1	1	yes	characteristic
10	#2285	C₂²	1	2	yes	—
11	#1845	—	2	1	no	—
12	#1834	—	2	1	no	—
13	#1808	C₄	1	4	yes	—
14	C₄×#107	—	2	2	no	—
15	C₂×#344	—	2	4	no	—
16	#1567	—	2	1	no	—
17	#1567	—	2	1	no	—
18	#1567	—	2	2	no	—
19	C₂×#335	C₄	1	4	yes	complemented
20	C₈×M₁₆	—	2	1	no	—
21	C₈×M₁₆	—	2	1	no	—
22	C₈×M₁₆	C₄	1	2	yes	complemented
23	C₈×M₁₆	C₄	1	2	yes	complemented
24	C₂×#322	—	2	1	no	—
25	C₂×C₈²	—	2	1	no	—
26	#1303	—	2	4	no	—
27	C₂²×M₁₆	—	2	2	no	—
28	C₂²×M₁₆	D₄	1	2	yes	—
29	C₂³×C₈	C₂×C₄	1	2	yes	—
30	C₂×#117	—	2	2	no	—
31	C₂²×C₄²	C₂³	1	1	yes	characteristic
32	#431	—	2	1	no	—
33	#431	—	2	1	no	—
34	#431	C₂×C₄	1	2	yes	—
35	#431	D₄	1	2	yes	—
36	C₂×#107	—	2	4	no	—
37	#423	—	2	2	no	—
38	#423	—	2	2	no	—
39	#423	—	2	2	no	—
40	#423	—	2	2	no	—
41	C₂×C₄⋊C₈	—	2	2	no	—
42	C₂×C₄⋊C₈	C₂×C₄	1	4	yes	—
43	C₂×C₄⋊C₈	Q₈	1	2	yes	—
44	C₂×C₂²⋊C₈	—	2	1	no	—
45	C₄×M₁₆	—	2	1	no	—
46	C₄×M₁₆	—	2	1	no	—
47	C₄×M₁₆	C₂×C₄	1	2	yes	—
48	C₄×M₁₆	Q₈	1	2	yes	—
49	C₂×#97	—	2	1	no	—
50	C₂×C₄×C₈	—	2	1	no	—
51	C₂×C₄×C₈	—	2	1	no	—
52	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
53	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
54	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
55	C₂×C₄×C₈	D₄	1	1	yes	characteristic
56	C₂×C₄×C₈	D₄	1	1	yes	characteristic
57	C₂×C₄×C₈	D₄	1	2	yes	—
58	C₂×C₄×C₈	D₄	1	1	yes	characteristic
59	C₂×C₄×C₈	D₄	1	1	yes	characteristic
60	C₂×C₄×C₈	Q₈	1	2	yes	—
61	#384	—	2	2	no	—
62	#382	—	2	2	no	—
63	#380	—	2	2	no	—
64	#380	C₂×C₄	1	4	yes	—
65	C₄×C₄⋊C₄	—	2	2	no	—
66	C₄×C₄⋊C₄	—	2	2	no	—
67	#344	—	4	8	no	—
68	#340	—	2	4	no	—
69	#335	—	2	4	no	—
70	#335	—	2	4	no	—
71	#322	—	2	2	no	—
72	#322	—	2	2	no	—
73	C₈²	—	2	2	no	—
74	C₈²	—	2	2	no	—
75	C₂³×C₄	C₂²×C₄	1	1	yes	characteristic
76	C₂³×C₄	C₂×D₄	1	1	yes	characteristic
77	C₂³×C₄	#41	1	1	yes	characteristic
78	C₂×M₁₆	—	2	2	no	—
79	C₂×M₁₆	—	2	2	no	—
80	C₂×M₁₆	—	2	2	no	—
81	C₂×M₁₆	—	4	2	no	—
82	C₂×M₁₆	—	4	2	no	—
83	C₂×M₁₆	—	4	4	no	—
84	C₂×M₁₆	—	4	4	no	—
85	C₂×M₁₆	C₄⋊C₄	1	4	yes	—
86	C₂×M₁₆	D₈	1	4	yes	—
87	C₂×M₁₆	Q₁₆	1	4	yes	—
88	C₂²×C₈	—	2	2	no	—
89	C₂²×C₈	—	2	2	no	—
90	C₂²×C₈	—	2	4	no	—
91	C₂²×C₈	—	4	4	no	—
92	C₂²×C₈	C₄²	1	2	yes	—
93	C₂²×C₈	C₄²	1	2	yes	—
94	C₂²×C₈	C₂²⋊C₄	1	2	yes	—
95	C₂²×C₈	C₂²⋊C₄	1	2	yes	—
96	C₂²×C₈	C₄⋊C₄	1	2	yes	—
97	C₂²×C₈	C₄⋊C₄	1	2	yes	—
98	#117	—	4	4	no	—
99	#117	—	4	4	no	—
100	C₂×C₄⋊C₄	—	2	4	no	—
101	C₂×C₄⋊C₄	—	4	2	no	—
102	C₂×C₄⋊C₄	—	4	2	no	—
103	C₂×C₂²⋊C₄	—	2	4	no	—
104	C₂×C₄²	—	4	2	no	—
105	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
106	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
107	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
108	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
109	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
110	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
111	C₂×C₄²	C₂×Q₈	1	1	yes	characteristic
112	C₂×C₄²	C₂×Q₈	1	1	yes	characteristic
113	C₂×C₄²	C₂×Q₈	1	1	yes	characteristic
114	C₂×C₄²	#41	1	1	yes	characteristic
115	C₂×C₄²	#41	1	1	yes	characteristic
116	C₂×C₄²	#41	1	1	yes	characteristic
117	#107	—	4	8	no	—
118	C₄⋊C₈	—	2	4	no	—
119	C₄⋊C₈	—	2	4	no	—
120	C₄⋊C₈	—	2	2	no	—
121	C₄⋊C₈	—	4	2	no	—
122	C₄⋊C₈	—	4	2	no	—
123	C₄⋊C₈	C₄⋊C₄	1	4	yes	—
124	C₂²⋊C₈	—	2	2	no	—
125	C₂²⋊C₈	—	2	2	no	—
126	#97	—	2	2	no	—
127	#97	—	2	2	no	—
128	C₄×C₈	—	2	2	no	—
129	C₄×C₈	—	2	2	no	—
130	C₄×C₈	—	2	2	no	—
131	C₄×C₈	—	2	2	no	—
132	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
133	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
134	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
135	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
136	C₄×C₈	C₂²⋊C₄	1	2	yes	—
137	C₄×C₈	C₂²⋊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	C₄×C₈	C₄⋊C₄	1	2	yes	—
142	#95	—	2	4	no	—
143	#95	—	4	2	no	—
144	C₂⁴	C₄×D₄	1	1	yes	characteristic
145	C₂²×C₄	—	4	2	no	—
146	C₂²×C₄	—	4	2	no	—
147	C₂²×C₄	—	4	4	no	—
148	C₂²×C₄	C₂×C₄²	1	1	yes	characteristic
149	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
150	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
151	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
152	C₂²×C₄	#117	1	1	yes	characteristic
153	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
154	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
155	C₂²×C₄	C₄×Q₈	1	1	yes	characteristic
156	C₂²×C₄	C₄×Q₈	1	1	yes	characteristic
157	C₂²×C₄	C₄×Q₈	1	1	yes	characteristic
158	C₂²×C₄	#121	1	1	yes	characteristic
159	C₂²×C₄	#121	1	1	yes	characteristic
160	C₂²×C₄	#121	1	1	yes	characteristic
161	C₂²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
162	C₂²×C₄	#124	1	1	yes	characteristic
163	C₂²×C₄	#125	1	1	yes	characteristic
164	C₂²×C₄	#127	1	1	yes	characteristic
165	C₂²×C₄	C₄⋊Q₈	1	1	yes	characteristic
166	C₂²×C₄	C₂×D₈	1	2	yes	—
167	C₂²×C₄	C₂×Q₁₆	1	2	yes	—
168	C₂²×C₄	#135	1	2	yes	—
169	C₂²×C₄	#135	1	2	yes	—
170	M₁₆	—	2	4	no	—
171	M₁₆	—	2	4	no	—
172	M₁₆	—	2	4	no	—
173	M₁₆	—	8	4	no	—
174	M₁₆	—	8	2	no	—
175	M₁₆	—	8	2	no	—
176	M₁₆	#107	1	8	yes	complemented
177	C₂×C₈	—	2	4	no	—
178	C₂×C₈	—	2	4	no	—
179	C₂×C₈	—	2	4	no	—
180	C₂×C₈	—	2	4	no	—
181	C₂×C₈	—	2	4	no	—
182	C₂×C₈	—	2	4	no	—
183	C₂×C₈	—	2	4	no	—
184	C₂×C₈	—	4	2	no	—
185	C₂×C₈	—	4	2	no	—
186	C₂×C₈	—	4	2	no	—
187	C₂×C₈	—	4	4	no	—
188	C₂×C₈	—	4	4	no	—
189	C₂×C₈	—	4	8	no	—
190	C₂×C₈	#95	1	2	yes	—
191	C₂×C₈	#95	1	2	yes	—
192	C₂×C₈	#95	1	2	yes	—
193	C₂×C₈	#95	1	2	yes	—
194	C₄⋊C₄	—	4	8	no	—
195	C₄⋊C₄	—	8	4	no	—
196	C₄⋊C₄	—	8	4	no	—
197	C₂²⋊C₄	—	4	8	no	—
198	C₄²	—	8	2	no	—
199	C₄²	—	8	2	no	—
200	C₄²	C₂×C₂²⋊C₄	1	1	yes	characteristic
201	C₄²	C₂×C₂²⋊C₄	1	1	yes	characteristic
202	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
203	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
204	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
205	C₄²	#117	1	1	yes	characteristic
206	C₄²	C₄×D₄	1	1	yes	characteristic
207	C₄²	C₄×Q₈	1	1	yes	characteristic
208	C₄²	#121	1	1	yes	characteristic
209	C₄²	C₂²⋊Q₈	1	1	yes	characteristic
210	C₄²	C₂²⋊Q₈	1	1	yes	characteristic
211	C₄²	C₂²⋊Q₈	1	1	yes	characteristic
212	C₄²	#124	1	1	yes	characteristic
213	C₄²	#125	1	1	yes	characteristic
214	C₄²	C₄⋊Q₈	1	1	yes	characteristic
215	C₄²	C₄⋊Q₈	1	1	yes	characteristic
216	C₂³	—	4	1	no	—
217	C₂³	—	4	1	no	—
218	C₂³	C₄×C₄⋊C₄	1	1	yes	characteristic
219	C₂³	#389	1	1	yes	characteristic
220	C₂³	#390	1	1	yes	characteristic
221	C₂³	C₄×D₈	1	2	yes	—
222	C₂³	C₄×Q₁₆	1	2	yes	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	4	1	no	—
230	C₂×C₄	—	4	1	no	—
231	C₂×C₄	—	4	1	no	—
232	C₂×C₄	—	4	1	no	—
233	C₂×C₄	—	4	1	no	—
234	C₂×C₄	—	4	1	no	—
235	C₂×C₄	—	8	4	no	—
236	C₂×C₄	—	8	4	no	—
237	C₂×C₄	—	8	4	no	—
238	C₂×C₄	C₂×#95	1	1	yes	characteristic
239	C₂×C₄	C₄×C₄⋊C₄	1	1	yes	characteristic
240	C₂×C₄	#380	1	1	yes	characteristic
241	C₂×C₄	#380	1	1	yes	characteristic
242	C₂×C₄	#382	1	1	yes	characteristic
243	C₂×C₄	#384	1	1	yes	characteristic
244	C₂×C₄	#389	1	1	yes	characteristic
245	C₂×C₄	#389	1	1	yes	characteristic
246	C₂×C₄	#389	1	1	yes	characteristic
247	C₂×C₄	#390	1	1	yes	characteristic
248	C₂×C₄	#391	1	1	yes	characteristic
249	C₂×C₄	#391	1	1	yes	characteristic
250	C₂×C₄	C₂×#107	1	2	yes	—
251	C₂×C₄	#427	1	2	yes	—
252	C₂×C₄	C₄×D₈	1	2	yes	—
253	C₂×C₄	C₄×Q₁₆	1	2	yes	—
254	C₂×C₄	#466	1	2	yes	—
255	C₂×C₄	#466	1	2	yes	—
256	C₂×C₄	#467	1	2	yes	—
257	C₂×C₄	#467	1	2	yes	—
258	C₂×C₄	#493	1	1	yes	characteristic
259	C₂×C₄	#494	1	1	yes	characteristic
260	C₂×C₄	#495	1	1	yes	characteristic
261	C₂×C₄	#495	1	1	yes	characteristic
262	C₂×C₄	#499	1	2	yes	—
263	C₂×C₄	#500	1	2	yes	—
264	C₈	—	2	8	no	—
265	C₈	—	2	2	no	—
266	C₈	—	2	2	no	—
267	C₈	—	2	2	no	—
268	C₈	—	4	4	no	—
269	C₈	—	8	4	no	—
270	C₈	#344	1	4	yes	complemented
271	C₂²	—	2	1	no	—
272	C₂²	—	2	1	no	—
273	C₂²	—	2	1	no	—
274	C₂²	—	2	1	no	—
275	C₂²	—	2	1	no	—
276	C₂²	—	2	1	no	—
277	C₂²	—	4	2	no	—
278	C₂²	—	4	2	no	—
279	C₂²	C₄⋊#95	1	1	yes	characteristic
280	C₂²	C₄×#107	1	2	yes	—
281	C₂²	#1942	1	2	yes	—
282	C₂²	#1969	1	1	yes	characteristic
283	C₂²	#1970	1	1	yes	characteristic
284	C₄	—	2	1	no	—
285	C₄	—	2	1	no	—
286	C₄	—	2	1	no	—
287	C₄	—	2	1	no	—
288	C₄	—	2	1	no	—
289	C₄	—	2	1	no	—
290	C₄	—	16	4	no	—
291	C₄	C₂×#344	1	2	yes	—
292	C₄	#1808	1	2	yes	—
293	C₄	#1814	1	2	yes	—
294	C₄	#1845	1	1	yes	characteristic
295	C₄	#1846	1	1	yes	characteristic
296	C₄	#1942	1	2	yes	—
297	C₄	#1969	1	1	yes	characteristic
298	C₄	#1970	1	1	yes	characteristic
299	C₂	—	2	1	no	—
300	C₂	—	2	2	no	—
301	C₂	—	2	1	no	—
302	C₂	—	2	2	no	—
303	C₂	#10855	1	1	yes	characteristic
304	C₂	#10862	1	1	yes	characteristic
305	C₂	#10872	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#390
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	C₄⋊#95
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×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
19. C₂×#335	C₄	semidirect
22. C₈×M₁₆	C₄	semidirect
23. C₈×M₁₆	C₄	semidirect
176. M₁₆	#107	semidirect
270. C₈	#344	semidirect

Maximal subgroups

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

subgroup	index
#20817	2
C₂×#1567	2
C₂×C₈×M₁₆	2
#10855	2
#8095	2