Subgroups — #146995

Order 512.

714 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×M₃₂	C₂	1	1	yes	maximal, characteristic
2	#20797	C₂	1	1	yes	maximal, characteristic
3	C₂×#1844	C₂	1	1	yes	maximal, characteristic
4	C₂×#1386	C₂	1	4	yes	maximal
5	C₂²×M₃₂	C₂²	1	2	yes	—
6	C₂×#427	—	2	1	no	—
7	C₂×#427	C₂²	1	2	yes	—
8	C₂²×#106	—	2	1	no	—
9	C₂²×#106	C₂²	1	2	yes	—
10	C₂²×C₄×C₈	C₂²	1	1	yes	characteristic
11	#2285	—	2	1	no	—
12	C₂×#384	—	2	1	no	—
13	C₄×M₃₂	—	2	2	no	—
14	C₄×M₃₂	C₄	1	4	yes	complemented
15	C₂×#346	—	2	1	no	—
16	C₂×C₄×C₁₆	—	2	1	no	—
17	#1846	—	2	1	no	—
18	#1846	C₄	1	2	yes	complemented
19	#1845	C₄	1	2	yes	complemented
20	#1844	—	2	2	no	—
21	#1844	—	2	1	no	—
22	#1844	C₄	1	4	yes	complemented
23	#1386	—	2	8	no	—
24	C₂³×C₈	D₄	1	1	yes	characteristic
25	C₂³×C₈	C₂³	1	1	yes	characteristic
26	C₂×#117	—	2	2	no	—
27	C₂²×C₄⋊C₄	—	2	2	no	—
28	C₂²×C₄⋊C₄	—	4	2	no	—
29	C₂²×C₄²	D₄	1	1	yes	characteristic
30	C₂×M₃₂	—	2	2	no	—
31	C₂×M₃₂	—	2	2	no	—
32	C₂×M₃₂	—	2	2	no	—
33	C₂×M₃₂	C₂×C₄	1	4	yes	—
34	C₂×M₃₂	D₄	1	4	yes	—
35	C₂×M₃₂	Q₈	1	4	yes	—
36	C₂²×C₁₆	—	2	2	no	—
37	#427	—	2	4	no	—
38	#427	—	2	4	no	—
39	#427	—	4	2	no	—
40	#427	—	4	2	no	—
41	C₂×#107	—	2	2	no	—
42	C₂×#107	C₂×C₄	1	4	yes	—
43	C₂×#106	—	2	4	no	—
44	C₂×#106	—	2	4	no	—
45	C₂×#106	—	2	4	no	—
46	C₂×#106	—	2	2	no	—
47	C₂×#106	—	2	2	no	—
48	C₂×#106	—	2	2	no	—
49	C₂×#106	—	4	2	no	—
50	C₂×#106	—	4	2	no	—
51	C₂×#106	C₂×C₄	1	4	yes	—
52	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
53	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
54	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
55	C₂×C₄×C₈	C₂×C₄	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	Q₈	1	2	yes	—
60	#384	—	2	4	no	—
61	#384	—	2	2	no	—
62	#384	—	4	2	no	—
63	#382	—	2	2	no	—
64	#380	—	4	2	no	—
65	C₄×C₄⋊C₄	—	4	2	no	—
66	#346	—	2	4	no	—
67	C₄×C₁₆	—	2	4	no	—
68	C₂³×C₄	—	4	2	no	—
69	C₂³×C₄	D₈	1	2	yes	—
70	C₂³×C₄	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₈	C₂²⋊C₄	1	2	yes	—
72	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
73	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
74	C₂²×C₈	C₄⋊C₄	1	2	yes	—
75	C₂²×C₈	D₈	1	2	yes	—
76	C₂²×C₈	SD₁₆	1	2	yes	—
77	C₂²×C₈	SD₁₆	1	2	yes	—
78	C₂²×C₈	Q₁₆	1	2	yes	—
79	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
80	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
81	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
82	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₈	C₂×D₄	1	2	yes	—
85	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
86	C₂²×C₈	C₂×Q₈	1	2	yes	—
87	C₂²×C₈	#41	1	2	yes	—
88	C₂²×C₈	#41	1	2	yes	—
89	#117	—	4	2	no	—
90	#117	—	4	2	no	—
91	#117	—	4	2	no	—
92	#117	—	4	2	no	—
93	C₂×C₄⋊C₄	—	2	4	no	—
94	C₂×C₄⋊C₄	—	2	4	no	—
95	C₂×C₄⋊C₄	—	4	2	no	—
96	C₂×C₄⋊C₄	—	4	2	no	—
97	C₂×C₄⋊C₄	—	4	2	no	—
98	C₂×C₄⋊C₄	—	4	2	no	—
99	C₂×C₄⋊C₄	—	4	8	no	—
100	C₂×C₄⋊C₄	—	4	4	no	—
101	C₂×C₄⋊C₄	—	4	2	no	—
102	C₂×C₄⋊C₄	—	8	8	no	—
103	C₂×C₂²⋊C₄	—	4	2	no	—
104	C₂×C₄²	—	4	2	no	—
105	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
106	C₂×C₄²	C₂²⋊C₄	1	2	yes	—
107	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
108	C₂×C₄²	C₄⋊C₄	1	2	yes	—
109	C₂×C₄²	SD₁₆	1	2	yes	—
110	C₂×C₄²	SD₁₆	1	2	yes	—
111	C₂×C₄²	Q₁₆	1	2	yes	—
112	M₃₂	—	2	4	no	—
113	M₃₂	—	2	4	no	—
114	M₃₂	—	2	4	no	—
115	M₃₂	C₄⋊C₄	1	8	yes	complemented
116	C₂×C₁₆	—	2	4	no	—
117	C₂×C₁₆	—	2	4	no	—
118	C₂×C₁₆	—	2	4	no	—
119	#107	—	2	8	no	—
120	#107	—	4	4	no	—
121	#106	—	2	8	no	—
122	#106	—	4	8	no	—
123	#106	—	4	4	no	—
124	#106	—	4	4	no	—
125	#106	—	4	4	no	—
126	C₄×C₈	C₄²	1	2	yes	—
127	C₄×C₈	C₄²	1	2	yes	—
128	C₄×C₈	C₂²⋊C₄	1	2	yes	—
129	C₄×C₈	C₂²⋊C₄	1	2	yes	—
130	C₄×C₈	C₄⋊C₄	1	2	yes	—
131	C₄×C₈	C₄⋊C₄	1	2	yes	—
132	C₄×C₈	C₄⋊C₄	1	2	yes	—
133	C₄×C₈	C₄⋊C₄	1	2	yes	—
134	#95	—	4	4	no	—
135	C₂⁴	C₂×D₈	1	1	yes	characteristic
136	C₂²×C₄	—	2	8	no	—
137	C₂²×C₄	—	4	4	no	—
138	C₂²×C₄	—	4	4	no	—
139	C₂²×C₄	—	8	8	no	—
140	C₂²×C₄	—	8	2	no	—
141	C₂²×C₄	—	8	2	no	—
142	C₂²×C₄	#102	1	4	yes	—
143	C₂²×C₄	#102	1	2	yes	—
144	C₂²×C₄	#102	1	2	yes	—
145	C₂²×C₄	#107	1	4	yes	—
146	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
147	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
148	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
149	C₂²×C₄	#124	1	1	yes	characteristic
150	C₂²×C₄	#124	1	1	yes	characteristic
151	C₂²×C₄	#127	1	1	yes	characteristic
152	C₂²×C₄	C₄⋊Q₈	1	1	yes	characteristic
153	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
154	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
155	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
156	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
157	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
158	C₂²×C₄	C₂×Q₁₆	1	1	yes	characteristic
159	C₂²×C₄	C₂×Q₁₆	1	1	yes	characteristic
160	C₂×C₈	#95	1	2	yes	—
161	C₂×C₈	#95	1	2	yes	—
162	C₂×C₈	#102	1	4	yes	—
163	C₂×C₈	#102	1	2	yes	—
164	C₂×C₈	#102	1	2	yes	—
165	C₂×C₈	#103	1	2	yes	—
166	C₂×C₈	#103	1	4	yes	—
167	C₂×C₈	#103	1	2	yes	—
168	C₂×C₈	#106	1	4	yes	—
169	C₂×C₈	#107	1	4	yes	—
170	C₂×C₈	C₂×C₄²	1	1	yes	characteristic
171	C₂×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
172	C₂×C₈	C₂×C₄⋊C₄	1	2	yes	—
173	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
174	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
175	C₂×C₈	#117	1	2	yes	—
176	C₂×C₈	C₄×D₄	1	2	yes	—
177	C₂×C₈	C₄×D₄	1	2	yes	—
178	C₂×C₈	C₄×Q₈	1	2	yes	—
179	C₂×C₈	C₄×Q₈	1	2	yes	—
180	C₂×C₈	#121	1	2	yes	—
181	C₂×C₈	#121	1	2	yes	—
182	C₂×C₈	C₂²⋊Q₈	1	2	yes	—
183	C₂×C₈	C₂²⋊Q₈	1	2	yes	—
184	C₂×C₈	#125	1	2	yes	—
185	C₂×C₈	C₄⋊Q₈	1	2	yes	—
186	C₄⋊C₄	—	4	4	no	—
187	C₄⋊C₄	—	4	4	no	—
188	C₄⋊C₄	—	4	4	no	—
189	C₄⋊C₄	—	4	4	no	—
190	C₄⋊C₄	—	8	16	no	—
191	C₄⋊C₄	—	8	8	no	—
192	C₄⋊C₄	—	8	2	no	—
193	C₄⋊C₄	—	8	2	no	—
194	C₂²⋊C₄	—	8	2	no	—
195	C₂²⋊C₄	—	8	2	no	—
196	C₄²	—	8	2	no	—
197	C₄²	—	8	2	no	—
198	C₄²	#95	1	2	yes	—
199	C₄²	#95	1	2	yes	—
200	C₄²	#103	1	2	yes	—
201	C₄²	#103	1	4	yes	—
202	C₄²	#103	1	2	yes	—
203	C₄²	#106	1	4	yes	—
204	C₁₆	—	2	8	no	—
205	C₂³	—	2	4	no	—
206	C₂³	C₂×#102	1	1	yes	characteristic
207	C₂³	C₂×#102	1	1	yes	characteristic
208	C₂³	C₂×#107	1	1	yes	characteristic
209	C₂³	#486	1	1	yes	characteristic
210	C₂³	#486	1	1	yes	characteristic
211	C₂³	#493	1	1	yes	characteristic
212	C₂³	#500	1	1	yes	characteristic
213	C₂×C₄	—	2	4	no	—
214	C₂×C₄	—	2	8	no	—
215	C₂×C₄	—	2	8	no	—
216	C₂×C₄	—	8	8	no	—
217	C₂×C₄	—	8	8	no	—
218	C₂×C₄	—	8	4	no	—
219	C₂×C₄	—	8	4	no	—
220	C₂×C₄	—	16	4	no	—
221	C₂×C₄	—	16	4	no	—
222	C₂×C₄	#340	1	4	yes	—
223	C₂×C₄	#340	1	4	yes	—
224	C₂×C₄	#360	1	8	yes	—
225	C₂×C₄	C₂×#95	1	1	yes	characteristic
226	C₂×C₄	#382	1	1	yes	characteristic
227	C₂×C₄	#384	1	1	yes	characteristic
228	C₂×C₄	#390	1	1	yes	characteristic
229	C₂×C₄	#390	1	1	yes	characteristic
230	C₂×C₄	#391	1	1	yes	characteristic
231	C₂×C₄	#391	1	1	yes	characteristic
232	C₂×C₄	C₂×#102	1	1	yes	characteristic
233	C₂×C₄	C₂×#102	1	1	yes	characteristic
234	C₂×C₄	C₂×#103	1	1	yes	characteristic
235	C₂×C₄	C₂×#103	1	1	yes	characteristic
236	C₂×C₄	C₂×#103	1	1	yes	characteristic
237	C₂×C₄	C₂×#103	1	1	yes	characteristic
238	C₂×C₄	C₂×#106	1	1	yes	characteristic
239	C₂×C₄	C₂×#106	1	1	yes	characteristic
240	C₂×C₄	C₂×#107	1	1	yes	characteristic
241	C₂×C₄	#486	1	1	yes	characteristic
242	C₂×C₄	#486	1	1	yes	characteristic
243	C₂×C₄	#487	1	1	yes	characteristic
244	C₂×C₄	#487	1	1	yes	characteristic
245	C₂×C₄	#487	1	1	yes	characteristic
246	C₂×C₄	#487	1	1	yes	characteristic
247	C₂×C₄	#492	1	1	yes	characteristic
248	C₂×C₄	#492	1	1	yes	characteristic
249	C₂×C₄	#494	1	1	yes	characteristic
250	C₂×C₄	#498	1	1	yes	characteristic
251	C₂×C₄	#498	1	1	yes	characteristic
252	C₂×C₄	#500	1	1	yes	characteristic
253	C₈	#340	1	4	yes	—
254	C₈	#340	1	4	yes	—
255	C₈	C₄×C₄⋊C₄	1	2	yes	—
256	C₈	#380	1	2	yes	—
257	C₈	#389	1	2	yes	—
258	C₈	#389	1	2	yes	—
259	C₂²	—	2	2	no	—
260	C₂²	—	2	2	no	—
261	C₂²	—	2	2	no	—
262	C₂²	—	2	2	no	—
263	C₂²	—	2	2	no	—
264	C₂²	—	2	2	no	—
265	C₂²	C₂×#340	1	1	yes	characteristic
266	C₂²	#1834	1	1	yes	characteristic
267	C₂²	#1845	1	1	yes	characteristic
268	C₂²	#1957	1	1	yes	characteristic
269	C₂²	#1957	1	1	yes	characteristic
270	C₂²	#1990	1	1	yes	characteristic
271	C₂²	#1990	1	1	yes	characteristic
272	C₂²	C₂×#360	1	4	yes	—
273	C₄	—	2	4	no	—
274	C₄	—	16	4	no	—
275	C₄	—	16	4	no	—
276	C₄	#1386	1	8	yes	complemented
277	C₄	C₄⋊#95	1	1	yes	characteristic
278	C₄	C₂×#340	1	1	yes	characteristic
279	C₄	#1834	1	1	yes	characteristic
280	C₄	#1844	1	1	yes	characteristic
281	C₄	#1956	1	1	yes	characteristic
282	C₄	#1956	1	1	yes	characteristic
283	C₄	#1990	1	1	yes	characteristic
284	C₄	#1990	1	1	yes	characteristic
285	C₂	—	2	1	no	—
286	C₂	—	2	1	no	—
287	C₂	—	2	1	no	—
288	C₂	—	2	1	no	—
289	C₂	#10841	1	1	yes	characteristic
290	C₂	C₂×#1386	1	2	yes	—
291	C₂	#13210	1	2	yes	—
292	C₂	#13287	1	2	yes	—

	subgroup	G/H
center, socle	C₂³	C₂×#102
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂⁴	C₂×D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
14. C₄×M₃₂	C₄	semidirect
18. #1846	C₄	semidirect
19. #1845	C₄	semidirect
22. #1844	C₄	semidirect
115. M₃₂	C₄⋊C₄	semidirect
276. C₄	#1386	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×M₃₂	2
#20797	2
C₂×#1844	2
C₂×#1386	2