Subgroups — #120878

Order 512.

464 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20769	C₂	1	1	yes	maximal, characteristic
2	#10858	C₂	1	1	yes	maximal, characteristic
3	#8079	C₂	1	1	yes	maximal, characteristic
4	#8001	C₂	1	1	yes	maximal, characteristic
5	C₂×#1286	C₂	1	1	yes	maximal, characteristic
6	C₂×#1286	C₂	1	1	yes	maximal, characteristic
7	#7882	C₂	1	1	yes	maximal, characteristic
8	C₂×#431	C₂²	1	1	yes	characteristic
9	C₂×#431	C₂²	1	1	yes	characteristic
10	C₂×#423	C₂²	1	1	yes	characteristic
11	C₂×#423	C₂²	1	1	yes	characteristic
12	C₂×#423	C₂²	1	1	yes	characteristic
13	C₂×C₄×M₁₆	C₂²	1	1	yes	characteristic
14	C₂×#382	C₂²	1	1	yes	characteristic
15	#1838	—	2	2	no	—
16	#1837	—	2	2	no	—
17	C₂×#343	—	2	1	no	—
18	C₂×#343	—	2	1	no	—
19	#1738	—	2	1	no	—
20	#1738	—	2	1	no	—
21	#1301	—	2	2	no	—
22	#1292	—	2	2	no	—
23	#1286	—	2	2	no	—
24	#1286	—	2	2	no	—
25	#1278	C₄	1	4	yes	complemented
26	C₂²×M₁₆	—	2	1	no	—
27	C₂²×M₁₆	—	2	1	no	—
28	C₂²×M₁₆	—	2	1	no	—
29	C₂²×M₁₆	D₄	1	1	yes	characteristic
30	C₂²×M₁₆	D₄	1	1	yes	characteristic
31	C₂²×C₄⋊C₄	—	2	1	no	—
32	C₂²×C₄²	C₂³	1	1	yes	characteristic
33	#431	—	2	1	no	—
34	#431	—	2	1	no	—
35	#431	—	2	1	no	—
36	#431	—	2	1	no	—
37	#431	—	2	1	no	—
38	#431	—	2	1	no	—
39	#431	—	2	1	no	—
40	#431	C₂×C₄	1	2	yes	—
41	#423	—	2	1	no	—
42	#423	—	2	1	no	—
43	#423	—	2	1	no	—
44	#423	—	2	1	no	—
45	#423	—	2	1	no	—
46	#423	—	2	1	no	—
47	#423	—	2	1	no	—
48	#423	—	2	1	no	—
49	#423	—	2	1	no	—
50	#423	D₄	1	2	yes	—
51	#423	D₄	1	2	yes	—
52	#423	D₄	1	2	yes	—
53	C₂×C₄⋊C₈	—	2	1	no	—
54	C₂×C₄⋊C₈	—	2	1	no	—
55	C₂×C₄⋊C₈	—	2	1	no	—
56	C₂×C₄⋊C₈	—	2	1	no	—
57	C₂×C₄⋊C₈	—	2	1	no	—
58	C₂×C₄⋊C₈	—	2	1	no	—
59	C₂×C₄⋊C₈	—	2	1	no	—
60	C₂×C₄⋊C₈	—	2	1	no	—
61	C₂×C₂²⋊C₈	—	2	1	no	—
62	C₂×C₂²⋊C₈	—	2	1	no	—
63	C₄×M₁₆	—	2	1	no	—
64	C₄×M₁₆	—	2	1	no	—
65	C₄×M₁₆	C₂×C₄	1	2	yes	—
66	C₄×M₁₆	Q₈	1	2	yes	—
67	C₂×#97	—	2	1	no	—
68	C₂×C₄×C₈	—	2	1	no	—
69	C₂×C₄×C₈	—	2	1	no	—
70	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
71	#382	—	2	1	no	—
72	#382	—	2	1	no	—
73	#382	—	2	1	no	—
74	#382	—	2	1	no	—
75	C₂×#95	—	2	1	no	—
76	C₂×#95	Q₈	1	2	yes	—
77	#343	—	4	2	no	—
78	#343	—	4	2	no	—
79	#341	—	2	4	no	—
80	#341	—	2	4	no	—
81	C₂³×C₄	—	2	2	no	—
82	C₂³×C₄	C₂×D₄	1	1	yes	characteristic
83	C₂³×C₄	C₂×Q₈	1	1	yes	characteristic
84	C₂³×C₄	#41	1	1	yes	characteristic
85	C₂×M₁₆	—	2	1	no	—
86	C₂×M₁₆	—	2	1	no	—
87	C₂×M₁₆	—	2	1	no	—
88	C₂×M₁₆	—	2	1	no	—
89	C₂×M₁₆	—	2	1	no	—
90	C₂×M₁₆	—	2	1	no	—
91	C₂×M₁₆	—	2	1	no	—
92	C₂×M₁₆	—	2	1	no	—
93	C₂×M₁₆	—	2	1	no	—
94	C₂×M₁₆	—	2	1	no	—
95	C₂×M₁₆	—	2	2	no	—
96	C₂×M₁₆	—	2	2	no	—
97	C₂×M₁₆	—	2	2	no	—
98	C₂×M₁₆	—	4	1	no	—
99	C₂×M₁₆	—	4	1	no	—
100	C₂×M₁₆	—	4	1	no	—
101	C₂×M₁₆	—	4	1	no	—
102	C₂×M₁₆	—	4	1	no	—
103	C₂×M₁₆	—	4	1	no	—
104	C₂×M₁₆	—	4	1	no	—
105	C₂×M₁₆	—	4	1	no	—
106	C₂×M₁₆	—	4	1	no	—
107	C₂×M₁₆	—	4	1	no	—
108	C₂×M₁₆	—	4	1	no	—
109	C₂×M₁₆	—	4	1	no	—
110	C₂×M₁₆	—	4	1	no	—
111	C₂×M₁₆	—	4	1	no	—
112	C₂×M₁₆	—	4	1	no	—
113	C₂×M₁₆	C₄⋊C₄	1	2	yes	—
114	C₂×M₁₆	C₄⋊C₄	1	2	yes	—
115	C₂²×C₈	—	2	1	no	—
116	C₂²×C₈	—	2	1	no	—
117	C₂²×C₈	—	2	2	no	—
118	C₂²×C₈	—	4	1	no	—
119	C₂²×C₈	—	4	1	no	—
120	C₂²×C₈	—	4	1	no	—
121	C₂²×C₈	—	4	1	no	—
122	C₂×C₄⋊C₄	—	4	1	no	—
123	C₂×C₄⋊C₄	—	4	1	no	—
124	C₂×C₄⋊C₄	—	4	1	no	—
125	C₂×C₄⋊C₄	—	4	1	no	—
126	C₂×C₄⋊C₄	—	4	1	no	—
127	C₂×C₄⋊C₄	—	4	1	no	—
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	C₂×D₄	1	1	yes	characteristic
132	C₂×C₄²	C₂×Q₈	1	1	yes	characteristic
133	C₂×C₄²	#41	1	1	yes	characteristic
134	C₂×C₄²	#41	1	1	yes	characteristic
135	C₂×C₄²	#41	1	1	yes	characteristic
136	C₂×C₄²	#41	1	1	yes	characteristic
137	C₂×C₄²	#41	1	1	yes	characteristic
138	C₂×C₄²	#41	1	1	yes	characteristic
139	C₂×C₄²	#41	1	1	yes	characteristic
140	C₄⋊C₈	—	2	2	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	—	2	2	no	—
146	C₄⋊C₈	—	2	2	no	—
147	C₄⋊C₈	—	4	1	no	—
148	C₄⋊C₈	—	4	1	no	—
149	C₄⋊C₈	—	4	1	no	—
150	C₄⋊C₈	—	4	1	no	—
151	C₄⋊C₈	—	4	1	no	—
152	C₄⋊C₈	—	4	1	no	—
153	C₄⋊C₈	—	4	1	no	—
154	C₄⋊C₈	—	4	1	no	—
155	C₄⋊C₈	—	4	1	no	—
156	C₂²⋊C₈	—	2	2	no	—
157	C₂²⋊C₈	—	4	1	no	—
158	C₂²⋊C₈	—	4	1	no	—
159	C₂²⋊C₈	—	4	1	no	—
160	#97	—	2	2	no	—
161	#97	—	2	2	no	—
162	C₄×C₈	—	2	2	no	—
163	C₄×C₈	—	2	2	no	—
164	C₄×C₈	—	2	2	no	—
165	C₄×C₈	—	2	2	no	—
166	C₄×C₈	—	2	2	no	—
167	C₄×C₈	—	4	1	no	—
168	#95	—	2	2	no	—
169	#95	—	2	2	no	—
170	#95	—	4	1	no	—
171	#95	—	4	1	no	—
172	C₂⁴	C₂²⋊Q₈	1	1	yes	characteristic
173	C₂²×C₄	—	4	2	no	—
174	C₂²×C₄	—	4	2	no	—
175	C₂²×C₄	—	4	2	no	—
176	C₂²×C₄	—	8	1	no	—
177	C₂²×C₄	—	8	1	no	—
178	C₂²×C₄	—	8	2	no	—
179	C₂²×C₄	—	8	2	no	—
180	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
181	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
182	C₂²×C₄	C₄×Q₈	1	1	yes	characteristic
183	C₂²×C₄	#120	1	1	yes	characteristic
184	C₂²×C₄	#121	1	1	yes	characteristic
185	C₂²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
186	C₂²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
187	C₂²×C₄	#123	1	1	yes	characteristic
188	C₂²×C₄	#124	1	1	yes	characteristic
189	C₂²×C₄	#124	1	1	yes	characteristic
190	C₂²×C₄	#124	1	1	yes	characteristic
191	C₂²×C₄	#124	1	1	yes	characteristic
192	C₂²×C₄	#125	1	1	yes	characteristic
193	C₂²×C₄	#125	1	1	yes	characteristic
194	C₂²×C₄	#125	1	1	yes	characteristic
195	C₂²×C₄	#126	1	1	yes	characteristic
196	C₂²×C₄	#126	1	1	yes	characteristic
197	C₂²×C₄	C₄⋊Q₈	1	1	yes	characteristic
198	C₂²×C₄	#135	1	1	yes	characteristic
199	C₂²×C₄	#135	1	1	yes	characteristic
200	C₂²×C₄	#135	1	1	yes	characteristic
201	C₂²×C₄	#135	1	1	yes	characteristic
202	C₂²×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
203	C₂²×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
204	C₂²×C₄	#137	1	1	yes	characteristic
205	C₂²×C₄	#137	1	1	yes	characteristic
206	M₁₆	—	2	2	no	—
207	M₁₆	—	2	2	no	—
208	M₁₆	—	2	2	no	—
209	M₁₆	—	2	2	no	—
210	M₁₆	—	4	1	no	—
211	M₁₆	—	4	1	no	—
212	M₁₆	—	4	1	no	—
213	M₁₆	—	4	1	no	—
214	M₁₆	—	4	2	no	—
215	M₁₆	—	4	2	no	—
216	M₁₆	—	4	2	no	—
217	M₁₆	—	4	2	no	—
218	M₁₆	—	4	2	no	—
219	M₁₆	—	4	2	no	—
220	M₁₆	—	8	1	no	—
221	M₁₆	—	8	1	no	—
222	M₁₆	—	8	1	no	—
223	M₁₆	—	8	1	no	—
224	M₁₆	—	8	1	no	—
225	M₁₆	—	8	1	no	—
226	C₂×C₈	—	2	2	no	—
227	C₂×C₈	—	2	2	no	—
228	C₂×C₈	—	4	1	no	—
229	C₂×C₈	—	4	1	no	—
230	C₂×C₈	—	4	1	no	—
231	C₂×C₈	—	4	1	no	—
232	C₂×C₈	—	4	2	no	—
233	C₂×C₈	—	4	2	no	—
234	C₂×C₈	—	4	2	no	—
235	C₂×C₈	—	4	2	no	—
236	C₂×C₈	—	4	2	no	—
237	C₂×C₈	—	8	2	no	—
238	C₂×C₈	—	8	1	no	—
239	C₂×C₈	—	8	1	no	—
240	C₂×C₈	—	8	1	no	—
241	C₂×C₈	—	8	1	no	—
242	C₂×C₈	—	8	1	no	—
243	C₂×C₈	—	8	1	no	—
244	C₂×C₈	—	8	2	no	—
245	C₄⋊C₄	—	8	1	no	—
246	C₄⋊C₄	—	8	1	no	—
247	C₄⋊C₄	—	8	1	no	—
248	C₄⋊C₄	—	8	1	no	—
249	C₄²	#117	1	1	yes	characteristic
250	C₄²	#117	1	1	yes	characteristic
251	C₄²	C₄×D₄	1	1	yes	characteristic
252	C₄²	C₄×D₄	1	1	yes	characteristic
253	C₄²	#121	1	1	yes	characteristic
254	C₄²	#121	1	1	yes	characteristic
255	C₄²	C₂²⋊Q₈	1	1	yes	characteristic
256	C₄²	C₂²⋊Q₈	1	1	yes	characteristic
257	C₄²	#123	1	1	yes	characteristic
258	C₄²	#123	1	1	yes	characteristic
259	C₄²	#123	1	1	yes	characteristic
260	C₄²	#123	1	1	yes	characteristic
261	C₄²	#125	1	1	yes	characteristic
262	C₄²	#125	1	1	yes	characteristic
263	C₄²	#126	1	1	yes	characteristic
264	C₄²	#126	1	1	yes	characteristic
265	C₂³	—	4	1	no	—
266	C₂³	—	4	1	no	—
267	C₂³	#389	1	1	yes	characteristic
268	C₂³	#393	1	1	yes	characteristic
269	C₂³	#400	1	1	yes	characteristic
270	C₂³	#478	1	1	yes	characteristic
271	C₂³	#478	1	1	yes	characteristic
272	C₂³	#479	1	1	yes	characteristic
273	C₂³	#479	1	1	yes	characteristic
274	C₂×C₄	—	4	1	no	—
275	C₂×C₄	—	4	1	no	—
276	C₂×C₄	—	4	1	no	—
277	C₂×C₄	—	4	1	no	—
278	C₂×C₄	—	4	1	no	—
279	C₂×C₄	—	4	1	no	—
280	C₂×C₄	—	4	1	no	—
281	C₂×C₄	—	4	1	no	—
282	C₂×C₄	—	4	1	no	—
283	C₂×C₄	—	4	1	no	—
284	C₂×C₄	—	4	1	no	—
285	C₂×C₄	—	4	1	no	—
286	C₂×C₄	—	8	2	no	—
287	C₂×C₄	—	8	2	no	—
288	C₂×C₄	—	8	2	no	—
289	C₂×C₄	—	16	2	no	—
290	C₂×C₄	—	16	2	no	—
291	C₂×C₄	#382	1	1	yes	characteristic
292	C₂×C₄	#385	1	1	yes	characteristic
293	C₂×C₄	#387	1	1	yes	characteristic
294	C₂×C₄	#387	1	1	yes	characteristic
295	C₂×C₄	#388	1	1	yes	characteristic
296	C₂×C₄	#388	1	1	yes	characteristic
297	C₂×C₄	#394	1	1	yes	characteristic
298	C₂×C₄	#397	1	1	yes	characteristic
299	C₂×C₄	#398	1	1	yes	characteristic
300	C₂×C₄	#398	1	1	yes	characteristic
301	C₂×C₄	#400	1	1	yes	characteristic
302	C₂×C₄	#400	1	1	yes	characteristic
303	C₂×C₄	#427	1	1	yes	characteristic
304	C₂×C₄	#427	1	1	yes	characteristic
305	C₂×C₄	#428	1	1	yes	characteristic
306	C₂×C₄	#428	1	1	yes	characteristic
307	C₂×C₄	#449	1	1	yes	characteristic
308	C₂×C₄	#449	1	1	yes	characteristic
309	C₂×C₄	#452	1	1	yes	characteristic
310	C₂×C₄	#452	1	1	yes	characteristic
311	C₂×C₄	#463	1	1	yes	characteristic
312	C₂×C₄	#463	1	1	yes	characteristic
313	C₂×C₄	#464	1	1	yes	characteristic
314	C₂×C₄	#464	1	1	yes	characteristic
315	C₂×C₄	#482	1	1	yes	characteristic
316	C₂×C₄	#482	1	1	yes	characteristic
317	C₂×C₄	#485	1	1	yes	characteristic
318	C₂×C₄	#485	1	1	yes	characteristic
319	C₂×C₄	#488	1	1	yes	characteristic
320	C₂×C₄	#488	1	1	yes	characteristic
321	C₂×C₄	#488	1	1	yes	characteristic
322	C₂×C₄	#488	1	1	yes	characteristic
323	C₂×C₄	#489	1	1	yes	characteristic
324	C₂×C₄	#489	1	1	yes	characteristic
325	C₂×C₄	#490	1	1	yes	characteristic
326	C₂×C₄	#491	1	1	yes	characteristic
327	C₈	—	4	2	no	—
328	C₈	—	4	2	no	—
329	C₈	—	8	2	no	—
330	C₈	—	8	2	no	—
331	C₈	—	8	2	no	—
332	C₈	—	8	2	no	—
333	C₈	—	8	2	no	—
334	C₂²	—	2	1	no	—
335	C₂²	—	2	1	no	—
336	C₂²	—	2	1	no	—
337	C₂²	—	2	1	no	—
338	C₂²	—	2	1	no	—
339	C₂²	—	2	1	no	—
340	C₂²	—	4	1	no	—
341	C₂²	—	4	1	no	—
342	C₂²	—	4	1	no	—
343	C₂²	—	4	1	no	—
344	C₂²	#1442	1	1	yes	characteristic
345	C₂²	#1949	1	1	yes	characteristic
346	C₂²	#1949	1	1	yes	characteristic
347	C₂²	#2025	1	1	yes	characteristic
348	C₂²	#2026	1	1	yes	characteristic
349	C₂²	#2101	1	1	yes	characteristic
350	C₂²	#2101	1	1	yes	characteristic
351	C₄	—	2	1	no	—
352	C₄	—	2	1	no	—
353	C₄	—	2	1	no	—
354	C₄	—	2	1	no	—
355	C₄	—	2	1	no	—
356	C₄	—	2	1	no	—
357	C₄	—	2	1	no	—
358	C₄	—	2	1	no	—
359	C₄	—	16	2	no	—
360	C₄	#1835	1	1	yes	characteristic
361	C₄	#1836	1	1	yes	characteristic
362	C₄	#1854	1	1	yes	characteristic
363	C₄	#1854	1	1	yes	characteristic
364	C₄	#2042	1	1	yes	characteristic
365	C₄	#2042	1	1	yes	characteristic
366	C₄	#2073	1	1	yes	characteristic
367	C₄	#2074	1	1	yes	characteristic
368	C₂	—	2	1	no	—
369	C₂	—	2	1	no	—
370	C₂	—	2	1	no	—
371	C₂	—	2	1	no	—
372	C₂	—	2	1	no	—
373	C₂	—	2	1	no	—
374	C₂	#11266	1	1	yes	characteristic
375	C₂	#11268	1	1	yes	characteristic
376	C₂	#11278	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	#1442
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₂²⋊Q₈
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ 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
25. #1278	C₄	semidirect

Maximal subgroups

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

subgroup	index
#20769	2
#10858	2
#8079	2
#8001	2
C₂×#1286	2
#7882	2