Subgroups — #147048

Order 512.

616 conjugacy classes of subgroups.

The subgroup lattice has 359 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#364	C₂	1	1	yes	maximal, characteristic
2	#20797	C₂	1	1	yes	maximal, characteristic
3	C₂×#1772	C₂	1	1	yes	maximal, characteristic
4	#12567	C₂	1	2	yes	maximal
5	#12561	C₂	1	2	yes	maximal
6	C₂²×M₃₂	C₂²	1	2	yes	—
7	C₂²×#108	C₂²	1	1	yes	characteristic
8	C₂×#427	—	2	1	no	—
9	C₂×#427	C₂²	1	2	yes	—
10	C₂×#427	C₂²	1	1	yes	characteristic
11	C₂×#405	—	2	1	no	—
12	C₂²×C₄×C₈	C₂²	1	1	yes	characteristic
13	#2285	—	2	1	no	—
14	C₂×#364	—	2	1	no	—
15	C₂×#364	—	2	1	no	—
16	C₂×#364	—	2	2	no	—
17	C₂×#364	C₄	1	4	yes	complemented
18	#1846	—	2	1	no	—
19	#1846	C₄	1	2	yes	complemented
20	#1845	C₄	1	2	yes	complemented
21	#1844	—	2	1	no	—
22	#1772	—	2	2	no	—
23	#1772	C₄	1	4	yes	complemented
24	C₂×#339	—	2	1	no	—
25	#1386	—	2	4	no	—
26	#1382	—	2	4	no	—
27	C₂²×M₁₆	—	2	1	no	—
28	C₂³×C₈	D₄	1	1	yes	characteristic
29	C₂³×C₈	C₂³	1	1	yes	characteristic
30	C₂×#117	—	2	1	no	—
31	C₂×#117	—	2	2	no	—
32	C₂²×C₄²	D₄	1	1	yes	characteristic
33	C₂×M₃₂	—	2	2	no	—
34	C₂×M₃₂	—	2	2	no	—
35	C₂×M₃₂	—	2	2	no	—
36	C₂×M₃₂	—	2	2	no	—
37	C₂×M₃₂	—	2	2	no	—
38	C₂×M₃₂	C₂×C₄	1	4	yes	—
39	C₂²×C₁₆	—	2	2	no	—
40	C₂×#108	—	2	1	no	—
41	C₂×#108	—	2	1	no	—
42	C₂×#108	—	2	1	no	—
43	C₂×#108	C₂×C₄	1	2	yes	—
44	C₂×#108	D₄	1	2	yes	—
45	C₂×#108	Q₈	1	2	yes	—
46	#427	—	2	1	no	—
47	#427	—	2	1	no	—
48	#427	—	2	2	no	—
49	#427	—	2	2	no	—
50	#427	—	2	2	no	—
51	#427	—	2	2	no	—
52	#427	—	4	1	no	—
53	#427	—	4	1	no	—
54	#427	—	4	1	no	—
55	#427	—	4	1	no	—
56	#427	C₂×C₄	1	2	yes	—
57	#427	Q₈	1	2	yes	—
58	C₂×#107	—	2	1	no	—
59	C₂×#107	—	2	2	no	—
60	C₂×#107	C₂×C₄	1	4	yes	—
61	C₂×#106	—	2	2	no	—
62	C₂×#106	—	2	2	no	—
63	C₂×#106	D₄	1	2	yes	—
64	#405	—	2	4	no	—
65	#405	—	4	2	no	—
66	C₂×#97	—	4	1	no	—
67	C₂×C₄×C₈	—	4	1	no	—
68	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
69	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
70	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
71	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
72	C₂×C₄×C₈	D₄	1	1	yes	characteristic
73	C₂×C₄×C₈	D₄	1	1	yes	characteristic
74	C₂×C₄×C₈	D₄	1	2	yes	—
75	C₂×C₄×C₈	Q₈	1	2	yes	—
76	#384	—	2	2	no	—
77	#382	—	2	2	no	—
78	#380	—	4	2	no	—
79	C₄×C₄⋊C₄	—	4	2	no	—
80	#364	—	2	4	no	—
81	#364	—	2	4	no	—
82	#339	—	2	4	no	—
83	C₂³×C₄	—	2	1	no	—
84	C₂³×C₄	C₂×D₄	1	1	yes	characteristic
85	C₂×M₁₆	—	2	2	no	—
86	C₂×M₁₆	—	2	2	no	—
87	C₂×M₁₆	—	2	2	no	—
88	C₂×M₁₆	—	4	1	no	—
89	C₂×M₁₆	—	4	1	no	—
90	C₂×M₁₆	—	4	1	no	—
91	C₂²×C₈	—	4	1	no	—
92	C₂²×C₈	C₂²⋊C₄	1	2	yes	—
93	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
94	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
95	C₂²×C₈	C₄⋊C₄	1	2	yes	—
96	C₂²×C₈	D₈	1	1	yes	characteristic
97	C₂²×C₈	D₈	1	1	yes	characteristic
98	C₂²×C₈	SD₁₆	1	2	yes	—
99	C₂²×C₈	SD₁₆	1	1	yes	characteristic
100	C₂²×C₈	SD₁₆	1	1	yes	characteristic
101	C₂²×C₈	Q₁₆	1	2	yes	—
102	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
103	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
104	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
105	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
106	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
107	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
108	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
109	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
110	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
111	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
112	C₂²×C₈	#41	1	1	yes	characteristic
113	C₂²×C₈	#41	1	1	yes	characteristic
114	C₂²×C₈	#41	1	1	yes	characteristic
115	C₂²×C₈	#41	1	1	yes	characteristic
116	#117	—	2	2	no	—
117	#117	—	2	2	no	—
118	#117	—	4	1	no	—
119	#117	—	4	1	no	—
120	#117	—	4	2	no	—
121	#117	—	4	2	no	—
122	#117	—	4	2	no	—
123	#117	—	4	2	no	—
124	C₂×C₄⋊C₄	—	2	2	no	—
125	C₂×C₄⋊C₄	—	2	4	no	—
126	C₂×C₄⋊C₄	—	4	4	no	—
127	C₂×C₂²⋊C₄	—	4	1	no	—
128	C₂×C₂²⋊C₄	—	4	2	no	—
129	C₂×C₄²	—	2	1	no	—
130	C₂×C₄²	—	2	1	no	—
131	C₂×C₄²	—	2	1	no	—
132	C₂×C₄²	—	4	1	no	—
133	C₂×C₄²	—	4	2	no	—
134	C₂×C₄²	C₂²⋊C₄	1	2	yes	—
135	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
136	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
137	C₂×C₄²	C₄⋊C₄	1	2	yes	—
138	M₃₂	—	2	4	no	—
139	M₃₂	—	2	4	no	—
140	M₃₂	—	4	2	no	—
141	M₃₂	—	4	2	no	—
142	C₂×C₁₆	—	2	4	no	—
143	C₂×C₁₆	—	4	2	no	—
144	C₂×C₁₆	—	4	2	no	—
145	#108	—	2	2	no	—
146	#108	—	2	2	no	—
147	#108	—	2	2	no	—
148	#108	C₄⋊C₄	1	4	yes	complemented
149	#107	—	2	2	no	—
150	#107	—	2	2	no	—
151	#107	—	2	4	no	—
152	#107	—	2	4	no	—
153	#107	—	4	2	no	—
154	#107	—	4	2	no	—
155	#106	—	2	2	no	—
156	#106	—	4	2	no	—
157	#106	—	4	2	no	—
158	#106	—	4	2	no	—
159	#106	—	4	2	no	—
160	#106	C₄⋊C₄	1	4	yes	complemented
161	#97	—	4	4	no	—
162	C₄×C₈	—	4	4	no	—
163	C₄×C₈	C₄²	1	2	yes	—
164	C₄×C₈	C₄²	1	2	yes	—
165	C₄×C₈	C₂²⋊C₄	1	2	yes	—
166	C₄×C₈	C₂²⋊C₄	1	2	yes	—
167	C₄×C₈	C₄⋊C₄	1	2	yes	—
168	C₄×C₈	C₄⋊C₄	1	2	yes	—
169	C₄×C₈	C₄⋊C₄	1	2	yes	—
170	C₄×C₈	C₄⋊C₄	1	2	yes	—
171	#95	—	4	4	no	—
172	C₂⁴	AGL(1,ℤ/8)	1	1	yes	characteristic
173	C₂²×C₄	—	2	1	no	—
174	C₂²×C₄	—	2	2	no	—
175	C₂²×C₄	—	2	1	no	—
176	C₂²×C₄	—	2	2	no	—
177	C₂²×C₄	—	4	4	no	—
178	C₂²×C₄	—	4	2	no	—
179	C₂²×C₄	—	4	4	no	—
180	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
181	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
182	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
183	C₂²×C₄	#121	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	C₂×D₈	1	1	yes	characteristic
188	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
189	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
190	C₂²×C₄	C₂×Q₁₆	1	1	yes	characteristic
191	C₂²×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
192	C₂²×C₄	#137	1	1	yes	characteristic
193	C₂²×C₄	#137	1	1	yes	characteristic
194	M₁₆	—	2	4	no	—
195	M₁₆	—	4	2	no	—
196	M₁₆	—	4	2	no	—
197	M₁₆	—	4	2	no	—
198	C₂×C₈	—	4	2	no	—
199	C₂×C₈	—	4	2	no	—
200	C₂×C₈	—	4	2	no	—
201	C₂×C₈	#95	1	2	yes	—
202	C₂×C₈	#95	1	2	yes	—
203	C₂×C₈	#102	1	2	yes	—
204	C₂×C₈	#102	1	2	yes	—
205	C₂×C₈	#102	1	1	yes	characteristic
206	C₂×C₈	#102	1	1	yes	characteristic
207	C₂×C₈	#102	1	1	yes	characteristic
208	C₂×C₈	#102	1	1	yes	characteristic
209	C₂×C₈	#103	1	2	yes	—
210	C₂×C₈	#103	1	2	yes	—
211	C₂×C₈	#103	1	2	yes	—
212	C₂×C₈	#103	1	2	yes	—
213	C₂×C₈	#106	1	2	yes	—
214	C₂×C₈	#106	1	2	yes	—
215	C₂×C₈	#107	1	2	yes	—
216	C₂×C₈	#107	1	2	yes	—
217	C₂×C₈	C₂×C₄²	1	1	yes	characteristic
218	C₂×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
219	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
220	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
221	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
222	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
223	C₂×C₈	#117	1	1	yes	characteristic
224	C₂×C₈	#117	1	1	yes	characteristic
225	C₂×C₈	C₄×D₄	1	1	yes	characteristic
226	C₂×C₈	C₄×D₄	1	1	yes	characteristic
227	C₂×C₈	C₄×D₄	1	1	yes	characteristic
228	C₂×C₈	C₄×D₄	1	1	yes	characteristic
229	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
230	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
231	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
232	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
233	C₂×C₈	#121	1	1	yes	characteristic
234	C₂×C₈	#121	1	1	yes	characteristic
235	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
236	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
237	C₂×C₈	#124	1	1	yes	characteristic
238	C₂×C₈	#124	1	1	yes	characteristic
239	C₂×C₈	#125	1	1	yes	characteristic
240	C₂×C₈	#125	1	1	yes	characteristic
241	C₂×C₈	#127	1	1	yes	characteristic
242	C₂×C₈	C₄⋊Q₈	1	1	yes	characteristic
243	C₂×C₈	C₄⋊Q₈	1	1	yes	characteristic
244	C₂×C₈	C₄⋊Q₈	1	1	yes	characteristic
245	C₄⋊C₄	—	2	4	no	—
246	C₄⋊C₄	—	4	2	no	—
247	C₄⋊C₄	—	4	4	no	—
248	C₄⋊C₄	—	4	4	no	—
249	C₄⋊C₄	—	8	8	no	—
250	C₂²⋊C₄	—	4	2	no	—
251	C₂²⋊C₄	—	4	2	no	—
252	C₂²⋊C₄	—	8	2	no	—
253	C₂²⋊C₄	—	8	2	no	—
254	C₄²	—	2	2	no	—
255	C₄²	—	2	1	no	—
256	C₄²	—	2	1	no	—
257	C₄²	—	2	2	no	—
258	C₄²	—	4	2	no	—
259	C₄²	—	4	2	no	—
260	C₄²	—	8	2	no	—
261	C₄²	—	8	2	no	—
262	C₄²	#95	1	2	yes	—
263	C₄²	#95	1	2	yes	—
264	C₁₆	—	4	4	no	—
265	C₂³	—	4	2	no	—
266	C₂³	#417	1	1	yes	characteristic
267	C₂³	#418	1	1	yes	characteristic
268	C₂³	#428	1	1	yes	characteristic
269	C₂³	#459	1	1	yes	characteristic
270	C₂³	#460	1	1	yes	characteristic
271	C₂³	#476	1	1	yes	characteristic
272	C₂³	#477	1	1	yes	characteristic
273	C₂×C₄	—	2	4	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	4	4	no	—
277	C₂×C₄	—	4	4	no	—
278	C₂×C₄	—	4	2	no	—
279	C₂×C₄	—	4	4	no	—
280	C₂×C₄	—	8	4	no	—
281	C₂×C₄	—	8	8	no	—
282	C₂×C₄	—	8	4	no	—
283	C₂×C₄	C₂×#95	1	1	yes	characteristic
284	C₂×C₄	#380	1	1	yes	characteristic
285	C₂×C₄	#380	1	1	yes	characteristic
286	C₂×C₄	#389	1	1	yes	characteristic
287	C₂×C₄	#389	1	1	yes	characteristic
288	C₂×C₄	#390	1	1	yes	characteristic
289	C₂×C₄	#391	1	1	yes	characteristic
290	C₂×C₄	C₂×#102	1	1	yes	characteristic
291	C₂×C₄	C₂×#102	1	1	yes	characteristic
292	C₂×C₄	C₂×#103	1	1	yes	characteristic
293	C₂×C₄	C₂×#103	1	1	yes	characteristic
294	C₂×C₄	#417	1	1	yes	characteristic
295	C₂×C₄	#419	1	1	yes	characteristic
296	C₂×C₄	C₂×#106	1	1	yes	characteristic
297	C₂×C₄	C₂×#107	1	1	yes	characteristic
298	C₂×C₄	#428	1	1	yes	characteristic
299	C₂×C₄	#459	1	1	yes	characteristic
300	C₂×C₄	#460	1	1	yes	characteristic
301	C₂×C₄	#461	1	1	yes	characteristic
302	C₂×C₄	#461	1	1	yes	characteristic
303	C₂×C₄	#462	1	1	yes	characteristic
304	C₂×C₄	#462	1	1	yes	characteristic
305	C₂×C₄	#474	1	1	yes	characteristic
306	C₂×C₄	#474	1	1	yes	characteristic
307	C₂×C₄	#475	1	1	yes	characteristic
308	C₂×C₄	#475	1	1	yes	characteristic
309	C₂×C₄	#476	1	1	yes	characteristic
310	C₂×C₄	#477	1	1	yes	characteristic
311	C₈	—	4	4	no	—
312	C₈	#340	1	2	yes	—
313	C₈	#340	1	2	yes	—
314	C₈	#340	1	2	yes	—
315	C₈	#340	1	2	yes	—
316	C₈	C₄×C₄⋊C₄	1	1	yes	characteristic
317	C₈	C₄×C₄⋊C₄	1	1	yes	characteristic
318	C₈	#382	1	1	yes	characteristic
319	C₈	#384	1	1	yes	characteristic
320	C₈	#389	1	1	yes	characteristic
321	C₈	#389	1	1	yes	characteristic
322	C₈	#390	1	1	yes	characteristic
323	C₈	#391	1	1	yes	characteristic
324	C₂²	—	2	2	no	—
325	C₂²	—	2	2	no	—
326	C₂²	—	2	2	no	—
327	C₂²	—	4	1	no	—
328	C₂²	—	4	1	no	—
329	C₂²	—	4	1	no	—
330	C₂²	—	4	1	no	—
331	C₂²	#1736	1	1	yes	characteristic
332	C₂²	#1802	1	1	yes	characteristic
333	C₂²	#1803	1	1	yes	characteristic
334	C₂²	#1947	1	1	yes	characteristic
335	C₂²	#1948	1	1	yes	characteristic
336	C₂²	#1982	1	1	yes	characteristic
337	C₂²	#1987	1	1	yes	characteristic
338	C₄	—	2	4	no	—
339	C₄	—	4	2	no	—
340	C₄	—	8	4	no	—
341	C₄	—	16	4	no	—
342	C₄	C₄⋊#95	1	1	yes	characteristic
343	C₄	C₂×#340	1	1	yes	characteristic
344	C₄	#1802	1	1	yes	characteristic
345	C₄	#1803	1	1	yes	characteristic
346	C₄	#1947	1	1	yes	characteristic
347	C₄	#1948	1	1	yes	characteristic
348	C₄	#1981	1	1	yes	characteristic
349	C₄	#1988	1	1	yes	characteristic
350	C₂	—	2	1	no	—
351	C₂	—	2	1	no	—
352	C₂	—	2	1	no	—
353	C₂	—	2	1	no	—
354	C₂	—	2	1	no	—
355	C₂	—	2	1	no	—
356	C₂	#10849	1	1	yes	characteristic
357	C₂	#13259	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#459
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1982
prime radical	C₂⁴	AGL(1,ℤ/8)
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ 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
17. C₂×#364	C₄	semidirect
19. #1846	C₄	semidirect
20. #1845	C₄	semidirect
23. #1772	C₄	semidirect
148. #108	C₄⋊C₄	semidirect
160. #106	C₄⋊C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#364	2
#20797	2
C₂×#1772	2
#12567	2
#12561	2