Subgroups — #231306

Order 512.

558 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23248	C₂	1	2	yes	maximal
2	C₄×#500	C₂	1	1	yes	maximal, characteristic
3	#9868	C₂	1	2	yes	maximal
4	#9594	C₂	1	1	yes	maximal, characteristic
5	#9594	C₂	1	1	yes	maximal, characteristic
6	#9540	C₂	1	2	yes	maximal
7	#9536	C₂	1	2	yes	maximal
8	#8282	C₂	1	1	yes	maximal, characteristic
9	C₈×#107	C₂	1	1	yes	maximal, characteristic
10	#8274	C₂	1	1	yes	maximal, characteristic
11	C₈×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
12	C₂×#500	C₄	1	1	yes	characteristic
13	C₄×C₄⋊Q₈	C₂²	1	2	yes	—
14	#1990	C₄	1	2	yes	—
15	#1985	—	2	2	no	—
16	#1984	—	2	2	no	—
17	#1942	C₄	1	1	yes	characteristic
18	#1942	C₄	1	2	yes	—
19	#1942	C₄	1	1	yes	characteristic
20	#1940	—	2	2	no	—
21	#1939	C₂²	1	2	yes	—
22	#1939	C₂²	1	2	yes	—
23	#1845	C₄	1	1	yes	characteristic
24	C₄×#107	C₂²	1	2	yes	—
25	C₄×#107	C₂²	1	1	yes	characteristic
26	C₄×#107	C₂²	1	1	yes	characteristic
27	#1771	—	2	2	no	—
28	#1770	C₂²	1	2	yes	—
29	C₈×C₄⋊C₄	C₂²	1	2	yes	—
30	#1767	C₂²	1	2	yes	—
31	C₄×C₄⋊C₈	C₂²	1	1	yes	characteristic
32	C₄×C₄⋊C₈	C₂²	1	1	yes	characteristic
33	C₄²×C₈	C₂²	1	1	yes	characteristic
34	C₂×#335	C₂²	1	1	yes	characteristic
35	C₂×#335	C₂²	1	1	yes	characteristic
36	C₂×C₈²	C₂²	1	1	yes	characteristic
37	C₂×C₈²	C₂²	1	1	yes	characteristic
38	#1279	C₂²	1	2	yes	—
39	#1279	C₂²	1	2	yes	—
40	#1279	C₂²	1	2	yes	—
41	#1279	C₂²	1	2	yes	—
42	#1277	C₂²	1	2	yes	—
43	#1273	C₂²	1	1	yes	characteristic
44	#1273	C₂²	1	1	yes	characteristic
45	C₂×C₄⋊Q₈	C₂×C₄	1	2	yes	—
46	C₂×C₄×Q₈	—	2	2	no	—
47	#500	—	2	4	no	—
48	C₂×#107	C₂×C₄	1	2	yes	—
49	C₂×#107	C₂×C₄	1	1	yes	characteristic
50	C₂×#107	C₂×C₄	1	1	yes	characteristic
51	C₂×#107	C₂×C₄	1	1	yes	characteristic
52	C₂×#107	C₂×C₄	1	2	yes	—
53	C₂×#107	C₂×C₄	1	1	yes	characteristic
54	C₂×C₄⋊C₈	—	2	1	no	—
55	C₂×C₄⋊C₈	—	2	2	no	—
56	C₂×C₄⋊C₈	—	2	2	no	—
57	C₂×C₄⋊C₈	D₄	1	1	yes	characteristic
58	C₂×C₄⋊C₈	D₄	1	1	yes	characteristic
59	C₂×C₄⋊C₈	C₂³	1	2	yes	—
60	C₂×C₄⋊C₈	C₂³	1	2	yes	—
61	C₂×#97	—	2	2	no	—
62	C₂×#97	—	2	2	no	—
63	C₂×C₄×C₈	—	2	2	no	—
64	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
65	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
66	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
67	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
68	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
69	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
70	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
71	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
72	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
73	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
74	#391	—	2	2	no	—
75	#389	—	2	2	no	—
76	#389	C₂×C₄	1	2	yes	—
77	#389	C₂×C₄	1	2	yes	—
78	#384	C₂×C₄	1	2	yes	—
79	C₄×C₄⋊C₄	—	2	2	no	—
80	C₄×C₄⋊C₄	C₂³	1	2	yes	—
81	C₄×C₄⋊C₄	C₂³	1	2	yes	—
82	C₄³	C₂³	1	1	yes	characteristic
83	#340	C₂×C₄	1	2	yes	—
84	#340	C₂×C₄	1	2	yes	—
85	#340	C₂×C₄	1	2	yes	—
86	#340	C₂×C₄	1	2	yes	—
87	#336	—	2	2	no	—
88	#336	—	2	2	no	—
89	#336	—	2	2	no	—
90	#336	—	2	2	no	—
91	#336	—	2	2	no	—
92	#336	—	2	1	no	—
93	#336	D₄	1	1	yes	characteristic
94	#336	D₄	1	1	yes	characteristic
95	#335	—	2	2	no	—
96	#335	Q₈	1	4	yes	complemented
97	C₈²	—	2	2	no	—
98	C₈²	—	2	2	no	—
99	C₂²×Q₈	—	2	2	no	—
100	C₂²×C₈	—	2	2	no	—
101	C₂²×C₈	—	2	1	no	—
102	C₂²×C₈	—	2	1	no	—
103	C₂²×C₈	—	2	2	no	—
104	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
105	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
106	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
107	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
108	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
109	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
110	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
111	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
112	C₄⋊Q₈	—	2	4	no	—
113	C₄⋊Q₈	—	4	2	no	—
114	C₄×Q₈	—	8	2	no	—
115	C₄×Q₈	—	8	2	no	—
116	C₂×C₄⋊C₄	—	2	2	no	—
117	C₂×C₄⋊C₄	—	2	2	no	—
118	C₂×C₄⋊C₄	—	2	2	no	—
119	C₂×C₄⋊C₄	—	2	2	no	—
120	C₂×C₄⋊C₄	—	2	2	no	—
121	C₂×C₄⋊C₄	C₂²×C₄	1	2	yes	—
122	C₂×C₄⋊C₄	C₂²×C₄	1	2	yes	—
123	C₂×C₄⋊C₄	C₂²×C₄	1	2	yes	—
124	C₂×C₄⋊C₄	C₂²×C₄	1	2	yes	—
125	C₂×C₄²	—	2	2	no	—
126	C₂×C₄²	—	2	2	no	—
127	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
128	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
129	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
130	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
131	C₂×C₄²	#41	1	1	yes	characteristic
132	C₂×C₄²	#41	1	1	yes	characteristic
133	C₂×C₄²	C₂⁴	1	1	yes	characteristic
134	#107	—	2	2	no	—
135	#107	—	2	4	no	—
136	#107	—	2	2	no	—
137	#107	—	2	2	no	—
138	#107	—	2	4	no	—
139	#107	M₁₆	1	4	yes	—
140	C₄⋊C₈	—	2	4	no	—
141	C₄⋊C₈	—	2	2	no	—
142	C₄⋊C₈	—	2	2	no	—
143	C₄⋊C₈	—	2	2	no	—
144	C₄⋊C₈	—	2	2	no	—
145	C₄⋊C₈	—	8	2	no	—
146	C₄⋊C₈	—	8	2	no	—
147	C₄⋊C₈	Q₁₆	1	4	yes	—
148	C₄⋊C₈	C₂×Q₈	1	4	yes	—
149	#97	—	4	4	no	—
150	#97	—	8	2	no	—
151	C₄×C₈	—	2	2	no	—
152	C₄×C₈	—	2	2	no	—
153	C₄×C₈	—	2	2	no	—
154	C₄×C₈	—	2	1	no	—
155	C₄×C₈	—	2	1	no	—
156	C₄×C₈	—	2	1	no	—
157	C₄×C₈	—	2	1	no	—
158	C₄×C₈	—	2	1	no	—
159	C₄×C₈	—	2	1	no	—
160	C₄×C₈	—	2	1	no	—
161	C₄×C₈	—	2	1	no	—
162	C₄×C₈	—	2	1	no	—
163	C₄×C₈	—	2	1	no	—
164	C₄×C₈	—	4	1	no	—
165	C₄×C₈	—	8	2	no	—
166	C₄×C₈	C₂×Q₈	1	2	yes	—
167	C₄×C₈	#41	1	2	yes	—
168	#95	—	2	2	no	—
169	#95	—	2	2	no	—
170	#95	—	2	2	no	—
171	#95	—	2	2	no	—
172	C₂×Q₈	—	4	4	no	—
173	C₂×Q₈	—	4	4	no	—
174	C₂×Q₈	—	8	2	no	—
175	C₂²×C₄	—	2	2	no	—
176	C₂²×C₄	—	2	2	no	—
177	C₂²×C₄	—	2	2	no	—
178	C₂²×C₄	—	2	2	no	—
179	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
180	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
181	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
182	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
183	C₂²×C₄	C₂³×C₄	1	1	yes	characteristic
184	C₂²×C₄	C₂²×D₄	1	1	yes	characteristic
185	C₂²×C₄	C₂×#41	1	1	yes	characteristic
186	C₂×C₈	—	2	1	no	—
187	C₂×C₈	—	2	1	no	—
188	C₂×C₈	—	2	1	no	—
189	C₂×C₈	—	2	1	no	—
190	C₂×C₈	—	2	1	no	—
191	C₂×C₈	—	2	1	no	—
192	C₂×C₈	—	2	1	no	—
193	C₂×C₈	—	2	1	no	—
194	C₂×C₈	—	2	2	no	—
195	C₂×C₈	—	2	2	no	—
196	C₂×C₈	—	2	1	no	—
197	C₂×C₈	—	2	1	no	—
198	C₂×C₈	—	2	1	no	—
199	C₂×C₈	—	2	4	no	—
200	C₂×C₈	—	2	1	no	—
201	C₂×C₈	—	2	1	no	—
202	C₂×C₈	—	2	1	no	—
203	C₂×C₈	—	2	1	no	—
204	C₂×C₈	—	2	1	no	—
205	C₂×C₈	—	2	1	no	—
206	C₂×C₈	—	4	1	no	—
207	C₂×C₈	—	4	1	no	—
208	C₂×C₈	—	4	1	no	—
209	C₂×C₈	—	4	1	no	—
210	C₂×C₈	—	4	2	no	—
211	C₂×C₈	—	4	2	no	—
212	C₂×C₈	—	4	2	no	—
213	C₂×C₈	—	4	2	no	—
214	C₂×C₈	—	4	2	no	—
215	C₂×C₈	C₄×Q₈	1	2	yes	—
216	C₂×C₈	C₄×Q₈	1	2	yes	—
217	C₂×C₈	C₂×M₁₆	1	2	yes	—
218	C₂×C₈	#131	1	2	yes	—
219	C₂×C₈	#131	1	2	yes	—
220	C₂×C₈	C₂×Q₁₆	1	2	yes	—
221	C₂×C₈	C₂×Q₁₆	1	2	yes	—
222	C₄⋊C₄	—	2	2	no	—
223	C₄⋊C₄	—	2	2	no	—
224	C₄⋊C₄	—	2	2	no	—
225	C₄⋊C₄	—	2	2	no	—
226	C₄⋊C₄	—	2	2	no	—
227	C₄⋊C₄	—	2	2	no	—
228	C₄⋊C₄	—	2	2	no	—
229	C₄⋊C₄	—	4	4	no	—
230	C₄⋊C₄	—	8	2	no	—
231	C₄⋊C₄	—	8	2	no	—
232	C₄⋊C₄	—	8	2	no	—
233	C₄⋊C₄	—	8	2	no	—
234	C₄⋊C₄	C₂×M₁₆	1	4	yes	—
235	C₄²	—	2	1	no	—
236	C₄²	—	2	1	no	—
237	C₄²	—	2	1	no	—
238	C₄²	—	2	1	no	—
239	C₄²	—	2	1	no	—
240	C₄²	—	2	1	no	—
241	C₄²	—	2	2	no	—
242	C₄²	—	4	1	no	—
243	C₄²	—	4	2	no	—
244	C₄²	—	4	2	no	—
245	C₄²	—	4	2	no	—
246	C₄²	—	4	2	no	—
247	C₄²	C₂×Q₁₆	1	2	yes	—
248	C₄²	#135	1	2	yes	—
249	C₄²	C₂²×Q₈	1	1	yes	characteristic
250	C₄²	C₂×#41	1	1	yes	characteristic
251	C₄²	2₋¹⁺⁴	1	1	yes	characteristic
252	C₄²	2₋¹⁺⁴	1	1	yes	characteristic
253	C₂³	C₂×C₄×D₄	1	1	yes	characteristic
254	Q₈	—	8	8	no	—
255	C₂×C₄	—	2	1	no	—
256	C₂×C₄	—	2	1	no	—
257	C₂×C₄	—	2	1	no	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	2	1	no	—
260	C₂×C₄	—	2	1	no	—
261	C₂×C₄	—	2	1	no	—
262	C₂×C₄	—	2	1	no	—
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	2	4	no	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	2	1	no	—
267	C₂×C₄	—	4	2	no	—
268	C₂×C₄	—	4	2	no	—
269	C₂×C₄	—	4	2	no	—
270	C₂×C₄	—	4	2	no	—
271	C₂×C₄	—	4	2	no	—
272	C₂×C₄	—	4	2	no	—
273	C₂×C₄	—	4	2	no	—
274	C₂×C₄	—	4	2	no	—
275	C₂×C₄	—	4	2	no	—
276	C₂×C₄	—	4	2	no	—
277	C₂×C₄	—	4	2	no	—
278	C₂×C₄	C₄×Q₁₆	1	2	yes	—
279	C₂×C₄	C₄×Q₁₆	1	2	yes	—
280	C₂×C₄	#443	1	2	yes	—
281	C₂×C₄	C₂×C₄×Q₈	1	1	yes	characteristic
282	C₂×C₄	#519	1	1	yes	characteristic
283	C₂×C₄	#547	1	1	yes	characteristic
284	C₂×C₄	D₄×Q₈	1	1	yes	characteristic
285	C₂×C₄	#556	1	1	yes	characteristic
286	C₂×C₄	#557	1	1	yes	characteristic
287	C₂×C₄	C₂²×M₁₆	1	1	yes	characteristic
288	C₂×C₄	C₂×#131	1	1	yes	characteristic
289	C₂×C₄	C₂×#131	1	1	yes	characteristic
290	C₂×C₄	#568	1	1	yes	characteristic
291	C₂×C₄	C₂²×Q₁₆	1	1	yes	characteristic
292	C₂×C₄	C₂×#135	1	1	yes	characteristic
293	C₂×C₄	#576	1	1	yes	characteristic
294	C₂×C₄	#576	1	1	yes	characteristic
295	C₈	—	2	2	no	—
296	C₈	—	2	2	no	—
297	C₈	—	2	2	no	—
298	C₈	—	2	2	no	—
299	C₈	—	4	2	no	—
300	C₈	—	4	2	no	—
301	C₈	—	4	1	no	—
302	C₈	—	4	1	no	—
303	C₈	—	4	4	no	—
304	C₈	—	4	1	no	—
305	C₈	—	8	2	no	—
306	C₈	#446	1	4	yes	—
307	C₂²	#2365	1	1	yes	characteristic
308	C₂²	#2931	1	1	yes	characteristic
309	C₂²	D₄×M₁₆	1	1	yes	characteristic
310	C₂²	C₂×C₄×Q₁₆	1	1	yes	characteristic
311	C₂²	#2948	1	1	yes	characteristic
312	C₂²	C₂×#443	1	1	yes	characteristic
313	C₂²	#2957	1	1	yes	characteristic
314	C₄	—	2	2	no	—
315	C₄	—	2	2	no	—
316	C₄	—	2	1	no	—
317	C₄	—	2	1	no	—
318	C₄	—	2	2	no	—
319	C₄	—	4	1	no	—
320	C₄	—	4	4	no	—
321	C₄	—	8	2	no	—
322	C₄	—	8	2	no	—
323	C₄	—	8	2	no	—
324	C₄	C₂×#446	1	1	yes	characteristic
325	C₄	#2962	1	1	yes	characteristic
326	C₄	#2984	1	1	yes	characteristic
327	C₄	#2985	1	1	yes	characteristic
328	C₄	#3338	1	1	yes	characteristic
329	C₄	#3341	1	1	yes	characteristic
330	C₄	#3380	1	1	yes	characteristic
331	C₄	Q₈×Q₁₆	1	1	yes	characteristic
332	C₂	#17958	1	1	yes	characteristic
333	C₂	#17970	1	1	yes	characteristic
334	C₂	M₁₆×Q₁₆	1	1	yes	characteristic
335	C₂	#18054	1	1	yes	characteristic
336	C₂	#23151	1	1	yes	characteristic
337	C₂	#23158	1	1	yes	characteristic
338	C₂	#23170	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	C₂×C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×C₄⋊C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
96. #335	Q₈	semidirect

Maximal subgroups

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

subgroup	index
#23248	2
C₄×#500	2
#9868	2
#9594	2
#9540	2
#9536	2
#8282	2
C₈×#107	2
#8274	2
C₈×C₄⋊C₈	2