Subgroups — #109040

Order 512.

616 conjugacy classes of subgroups.

The subgroup lattice has 303 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	#10848	C₂	1	2	yes	maximal
5	#7957	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	C₄	1	4	yes	complemented
19	C₂×#335	—	2	2	no	—
20	C₈×M₁₆	—	2	2	no	—
21	C₈×M₁₆	—	2	2	no	—
22	C₂×#322	C₄	1	2	yes	complemented
23	C₂×C₈²	C₄	1	2	yes	complemented
24	#1286	—	2	4	no	—
25	C₂²×M₁₆	—	2	2	no	—
26	C₂²×M₁₆	D₄	1	2	yes	—
27	C₂³×C₈	C₂×C₄	1	2	yes	—
28	C₂×#117	—	2	2	no	—
29	C₂²×C₄²	C₂³	1	1	yes	characteristic
30	#431	—	2	1	no	—
31	#431	—	2	1	no	—
32	#431	D₄	1	2	yes	—
33	#431	Q₈	1	2	yes	—
34	C₂×#107	—	2	4	no	—
35	#423	—	2	2	no	—
36	#423	—	2	2	no	—
37	#423	—	2	2	no	—
38	#423	C₂×C₄	1	4	yes	—
39	C₂×C₄⋊C₈	—	2	1	no	—
40	C₂×C₄⋊C₈	—	2	2	no	—
41	C₂×C₄⋊C₈	—	2	2	no	—
42	C₂×C₂²⋊C₈	—	2	1	no	—
43	C₄×M₁₆	—	2	1	no	—
44	C₄×M₁₆	—	2	1	no	—
45	C₄×M₁₆	—	2	1	no	—
46	C₄×M₁₆	—	2	1	no	—
47	C₂×#97	Q₈	1	2	yes	—
48	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
49	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
50	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
51	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
53	C₂×C₄×C₈	D₄	1	2	yes	—
54	C₂×C₄×C₈	D₄	1	1	yes	characteristic
55	C₂×C₄×C₈	D₄	1	1	yes	characteristic
56	C₂×C₄×C₈	D₄	1	2	yes	—
57	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
58	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
59	#384	—	2	2	no	—
60	#382	—	2	2	no	—
61	#380	—	2	2	no	—
62	#380	C₂×C₄	1	4	yes	—
63	C₄×C₄⋊C₄	—	2	2	no	—
64	C₄×C₄⋊C₄	—	2	2	no	—
65	#344	—	4	8	no	—
66	#340	—	2	4	no	—
67	#335	—	2	4	no	—
68	#335	—	2	4	no	—
69	#322	—	2	4	no	—
70	C₈²	—	2	4	no	—
71	C₂³×C₄	C₂²×C₄	1	1	yes	characteristic
72	C₂³×C₄	C₂×D₄	1	1	yes	characteristic
73	C₂³×C₄	#41	1	1	yes	characteristic
74	C₂×M₁₆	—	2	2	no	—
75	C₂×M₁₆	—	2	2	no	—
76	C₂×M₁₆	—	2	2	no	—
77	C₂×M₁₆	—	2	2	no	—
78	C₂×M₁₆	—	2	4	no	—
79	C₂×M₁₆	—	4	2	no	—
80	C₂×M₁₆	—	4	4	no	—
81	C₂×M₁₆	—	4	4	no	—
82	C₂×M₁₆	D₈	1	4	yes	—
83	C₂×M₁₆	Q₁₆	1	4	yes	—
84	C₂²×C₈	—	2	2	no	—
85	C₂²×C₈	—	4	4	no	—
86	C₂²×C₈	—	4	2	no	—
87	C₂²×C₈	C₄²	1	2	yes	—
88	C₂²×C₈	C₄²	1	2	yes	—
89	C₂²×C₈	C₂²⋊C₄	1	2	yes	—
90	C₂²×C₈	C₂²⋊C₄	1	2	yes	—
91	C₂²×C₈	C₄⋊C₄	1	2	yes	—
92	C₂²×C₈	C₄⋊C₄	1	4	yes	—
93	C₂²×C₈	C₄⋊C₄	1	2	yes	—
94	#117	—	4	4	no	—
95	#117	—	4	4	no	—
96	C₂×C₄⋊C₄	—	2	4	no	—
97	C₂×C₄⋊C₄	—	4	2	no	—
98	C₂×C₄⋊C₄	—	4	2	no	—
99	C₂×C₂²⋊C₄	—	2	4	no	—
100	C₂×C₄²	—	4	2	no	—
101	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
102	C₂×C₄²	C₂²×C₄	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	C₂×Q₈	1	1	yes	characteristic
110	C₂×C₄²	#41	1	1	yes	characteristic
111	C₂×C₄²	#41	1	1	yes	characteristic
112	C₂×C₄²	#41	1	1	yes	characteristic
113	#107	—	4	8	no	—
114	C₄⋊C₈	—	2	4	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	C₂²⋊C₈	—	2	2	no	—
121	C₂²⋊C₈	—	2	2	no	—
122	#97	—	2	2	no	—
123	#97	—	2	2	no	—
124	C₄×C₈	—	2	2	no	—
125	C₄×C₈	—	2	2	no	—
126	C₄×C₈	—	2	2	no	—
127	C₄×C₈	C₂²⋊C₄	1	2	yes	—
128	C₄×C₈	C₂²⋊C₄	1	2	yes	—
129	C₄×C₈	C₂²⋊C₄	1	2	yes	—
130	C₄×C₈	C₂²⋊C₄	1	2	yes	—
131	C₄×C₈	C₄⋊C₄	1	4	yes	—
132	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
133	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
134	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
135	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
136	C₄×C₈	C₄⋊C₄	1	2	yes	—
137	C₄×C₈	C₄⋊C₄	1	2	yes	—
138	#95	—	2	4	no	—
139	#95	—	4	2	no	—
140	C₂⁴	C₄×D₄	1	1	yes	characteristic
141	C₂²×C₄	—	4	2	no	—
142	C₂²×C₄	—	4	2	no	—
143	C₂²×C₄	—	4	4	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×Q₈	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	#121	1	1	yes	characteristic
156	C₂²×C₄	#121	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	#127	1	1	yes	characteristic
161	C₂²×C₄	C₄⋊Q₈	1	1	yes	characteristic
162	C₂²×C₄	C₂×D₈	1	2	yes	—
163	C₂²×C₄	C₂×Q₁₆	1	2	yes	—
164	C₂²×C₄	#135	1	2	yes	—
165	C₂²×C₄	#135	1	2	yes	—
166	M₁₆	—	2	4	no	—
167	M₁₆	—	2	4	no	—
168	M₁₆	—	2	4	no	—
169	M₁₆	—	2	4	no	—
170	M₁₆	—	4	8	no	—
171	M₁₆	—	8	2	no	—
172	M₁₆	—	8	2	no	—
173	C₂×C₈	—	2	4	no	—
174	C₂×C₈	—	2	4	no	—
175	C₂×C₈	—	2	4	no	—
176	C₂×C₈	—	2	4	no	—
177	C₂×C₈	—	2	4	no	—
178	C₂×C₈	—	2	4	no	—
179	C₂×C₈	—	4	2	no	—
180	C₂×C₈	—	4	2	no	—
181	C₂×C₈	—	4	2	no	—
182	C₂×C₈	—	4	4	no	—
183	C₂×C₈	—	4	4	no	—
184	C₂×C₈	—	8	4	no	—
185	C₂×C₈	#95	1	2	yes	—
186	C₂×C₈	#95	1	2	yes	—
187	C₂×C₈	#95	1	2	yes	—
188	C₂×C₈	#95	1	2	yes	—
189	C₂×C₈	#107	1	8	yes	complemented
190	C₄⋊C₄	—	4	8	no	—
191	C₄⋊C₄	—	8	4	no	—
192	C₄⋊C₄	—	8	2	no	—
193	C₄⋊C₄	—	8	2	no	—
194	C₂²⋊C₄	—	4	8	no	—
195	C₄²	—	8	4	no	—
196	C₄²	C₂×C₂²⋊C₄	1	1	yes	characteristic
197	C₄²	C₂×C₂²⋊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	C₂×C₄⋊C₄	1	1	yes	characteristic
201	C₄²	#117	1	1	yes	characteristic
202	C₄²	C₄×D₄	1	1	yes	characteristic
203	C₄²	C₄×Q₈	1	1	yes	characteristic
204	C₄²	#121	1	1	yes	characteristic
205	C₄²	C₂²⋊Q₈	1	1	yes	characteristic
206	C₄²	C₂²⋊Q₈	1	1	yes	characteristic
207	C₄²	C₂²⋊Q₈	1	1	yes	characteristic
208	C₄²	#124	1	1	yes	characteristic
209	C₄²	#125	1	1	yes	characteristic
210	C₄²	C₄⋊Q₈	1	1	yes	characteristic
211	C₄²	C₄⋊Q₈	1	1	yes	characteristic
212	C₂³	—	4	1	no	—
213	C₂³	—	4	1	no	—
214	C₂³	C₄×C₄⋊C₄	1	1	yes	characteristic
215	C₂³	#389	1	1	yes	characteristic
216	C₂³	#390	1	1	yes	characteristic
217	C₂³	C₄×D₈	1	2	yes	—
218	C₂³	C₄×Q₁₆	1	2	yes	—
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	—	4	1	no	—
230	C₂×C₄	—	4	1	no	—
231	C₂×C₄	—	8	4	no	—
232	C₂×C₄	—	8	4	no	—
233	C₂×C₄	—	8	4	no	—
234	C₂×C₄	C₂×#95	1	1	yes	characteristic
235	C₂×C₄	C₄×C₄⋊C₄	1	1	yes	characteristic
236	C₂×C₄	#380	1	1	yes	characteristic
237	C₂×C₄	#380	1	1	yes	characteristic
238	C₂×C₄	#382	1	1	yes	characteristic
239	C₂×C₄	#384	1	1	yes	characteristic
240	C₂×C₄	#389	1	1	yes	characteristic
241	C₂×C₄	#389	1	1	yes	characteristic
242	C₂×C₄	#389	1	1	yes	characteristic
243	C₂×C₄	#390	1	1	yes	characteristic
244	C₂×C₄	#391	1	1	yes	characteristic
245	C₂×C₄	#391	1	1	yes	characteristic
246	C₂×C₄	C₂×#107	1	2	yes	—
247	C₂×C₄	#427	1	2	yes	—
248	C₂×C₄	C₄×D₈	1	2	yes	—
249	C₂×C₄	C₄×Q₁₆	1	2	yes	—
250	C₂×C₄	#466	1	2	yes	—
251	C₂×C₄	#466	1	2	yes	—
252	C₂×C₄	#467	1	2	yes	—
253	C₂×C₄	#467	1	2	yes	—
254	C₂×C₄	#493	1	1	yes	characteristic
255	C₂×C₄	#494	1	1	yes	characteristic
256	C₂×C₄	#495	1	1	yes	characteristic
257	C₂×C₄	#495	1	1	yes	characteristic
258	C₂×C₄	#499	1	2	yes	—
259	C₂×C₄	#500	1	2	yes	—
260	C₈	—	2	8	no	—
261	C₈	—	2	2	no	—
262	C₈	—	2	2	no	—
263	C₈	—	2	2	no	—
264	C₈	—	4	4	no	—
265	C₈	—	8	4	no	—
266	C₈	#344	1	4	yes	complemented
267	C₂²	—	2	1	no	—
268	C₂²	—	2	1	no	—
269	C₂²	—	2	1	no	—
270	C₂²	—	2	1	no	—
271	C₂²	—	2	1	no	—
272	C₂²	—	2	1	no	—
273	C₂²	—	4	2	no	—
274	C₂²	—	4	2	no	—
275	C₂²	C₄⋊#95	1	1	yes	characteristic
276	C₂²	C₄×#107	1	2	yes	—
277	C₂²	#1942	1	2	yes	—
278	C₂²	#1969	1	1	yes	characteristic
279	C₂²	#1970	1	1	yes	characteristic
280	C₄	—	2	1	no	—
281	C₄	—	2	1	no	—
282	C₄	—	2	1	no	—
283	C₄	—	2	1	no	—
284	C₄	—	2	1	no	—
285	C₄	—	2	1	no	—
286	C₄	—	16	4	no	—
287	C₄	C₂×#344	1	2	yes	—
288	C₄	#1808	1	2	yes	—
289	C₄	#1833	1	2	yes	—
290	C₄	#1845	1	1	yes	characteristic
291	C₄	#1846	1	1	yes	characteristic
292	C₄	#1942	1	2	yes	—
293	C₄	#1969	1	1	yes	characteristic
294	C₄	#1970	1	1	yes	characteristic
295	C₂	—	2	1	no	—
296	C₂	—	2	2	no	—
297	C₂	—	2	1	no	—
298	C₂	—	2	2	no	—
299	C₂	#10848	1	1	yes	characteristic
300	C₂	#10862	1	1	yes	characteristic
301	C₂	#10868	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#380
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
18. #1567	C₄	semidirect
22. C₂×#322	C₄	semidirect
23. C₂×C₈²	C₄	semidirect
189. C₂×C₈	#107	semidirect
266. 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
#10848	2
#7957	2