Subgroups — #147047

Order 512.

674 conjugacy classes of subgroups.

The subgroup lattice has 338 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	C₂×#1845	C₂	1	1	yes	maximal, characteristic
3	C₂×#1772	C₂	1	1	yes	maximal, characteristic
4	C₂×#1387	C₂	1	2	yes	maximal
5	#12557	C₂	1	2	yes	maximal
6	C₂²×M₃₂	C₂²	1	2	yes	—
7	C₂²×#108	C₂²	1	1	yes	characteristic
8	C₂×#427	C₂²	1	1	yes	characteristic
9	C₂²×#107	—	2	1	no	—
10	C₂²×#107	C₂²	1	2	yes	—
11	C₂×#405	—	2	1	no	—
12	C₂²×C₄×C₈	C₂²	1	1	yes	characteristic
13	C₂×#384	—	2	1	no	—
14	C₂×#364	—	2	2	no	—
15	C₂×#364	—	2	2	no	—
16	C₂×#364	C₄	1	4	yes	complemented
17	#1845	—	2	2	no	—
18	#1845	C₄	1	4	yes	complemented
19	#1772	—	2	2	no	—
20	#1772	C₄	1	4	yes	complemented
21	C₂×#339	—	2	1	no	—
22	#1387	—	2	4	no	—
23	#1380	—	2	4	no	—
24	C₂²×M₁₆	—	2	1	no	—
25	C₂³×C₈	D₄	1	1	yes	characteristic
26	C₂³×C₈	C₂³	1	1	yes	characteristic
27	C₂×#117	—	2	1	no	—
28	C₂²×C₄⋊C₄	—	2	2	no	—
29	C₂²×C₄⋊C₄	—	4	2	no	—
30	C₂²×C₄²	D₄	1	1	yes	characteristic
31	C₂×M₃₂	—	2	2	no	—
32	C₂×M₃₂	—	2	2	no	—
33	C₂×M₃₂	—	2	2	no	—
34	C₂×M₃₂	—	2	2	no	—
35	C₂×M₃₂	—	2	2	no	—
36	C₂×M₃₂	C₂×C₄	1	4	yes	—
37	C₂²×C₁₆	—	2	2	no	—
38	C₂×#108	—	2	1	no	—
39	C₂×#108	—	2	1	no	—
40	C₂×#108	—	2	1	no	—
41	C₂×#108	C₂×C₄	1	2	yes	—
42	C₂×#108	D₄	1	2	yes	—
43	C₂×#108	Q₈	1	2	yes	—
44	#427	—	2	1	no	—
45	#427	—	2	1	no	—
46	#427	C₂×C₄	1	2	yes	—
47	#427	Q₈	1	2	yes	—
48	C₂×#107	—	2	2	no	—
49	C₂×#107	—	2	2	no	—
50	C₂×#107	—	2	2	no	—
51	C₂×#107	—	2	2	no	—
52	C₂×#107	—	2	4	no	—
53	C₂×#107	—	2	2	no	—
54	C₂×#107	—	4	1	no	—
55	C₂×#107	—	4	1	no	—
56	C₂×#107	—	4	2	no	—
57	C₂×#107	C₂×C₄	1	4	yes	—
58	C₂×#107	D₄	1	2	yes	—
59	C₂×#106	—	2	1	no	—
60	#405	—	2	4	no	—
61	#405	—	4	2	no	—
62	C₂×#97	—	4	1	no	—
63	C₂×C₄×C₈	—	4	1	no	—
64	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
65	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
66	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
67	C₂×C₄×C₈	D₄	1	2	yes	—
68	C₂×C₄×C₈	D₄	1	2	yes	—
69	C₂×C₄×C₈	Q₈	1	2	yes	—
70	#384	—	2	4	no	—
71	#384	—	4	2	no	—
72	#364	—	2	8	no	—
73	#339	—	2	4	no	—
74	C₂³×C₄	—	2	1	no	—
75	C₂³×C₄	—	4	2	no	—
76	C₂³×C₄	C₂×D₄	1	1	yes	characteristic
77	C₂×M₁₆	—	2	2	no	—
78	C₂×M₁₆	—	2	2	no	—
79	C₂×M₁₆	—	2	2	no	—
80	C₂×M₁₆	—	4	1	no	—
81	C₂×M₁₆	—	4	1	no	—
82	C₂×M₁₆	—	4	1	no	—
83	C₂²×C₈	—	4	1	no	—
84	C₂²×C₈	C₂²⋊C₄	1	2	yes	—
85	C₂²×C₈	C₂²⋊C₄	1	2	yes	—
86	C₂²×C₈	C₄⋊C₄	1	2	yes	—
87	C₂²×C₈	D₈	1	1	yes	characteristic
88	C₂²×C₈	D₈	1	1	yes	characteristic
89	C₂²×C₈	SD₁₆	1	2	yes	—
90	C₂²×C₈	SD₁₆	1	2	yes	—
91	C₂²×C₈	Q₁₆	1	2	yes	—
92	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
93	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
94	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
95	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
96	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
97	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
98	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
99	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
100	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
101	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
102	C₂²×C₈	#41	1	1	yes	characteristic
103	C₂²×C₈	#41	1	1	yes	characteristic
104	C₂²×C₈	#41	1	1	yes	characteristic
105	C₂²×C₈	#41	1	1	yes	characteristic
106	#117	—	2	2	no	—
107	#117	—	2	2	no	—
108	#117	—	4	1	no	—
109	#117	—	4	1	no	—
110	C₂×C₄⋊C₄	—	2	2	no	—
111	C₂×C₄⋊C₄	—	2	4	no	—
112	C₂×C₄⋊C₄	—	4	2	no	—
113	C₂×C₄⋊C₄	—	4	2	no	—
114	C₂×C₄⋊C₄	—	4	2	no	—
115	C₂×C₄⋊C₄	—	4	2	no	—
116	C₂×C₄⋊C₄	—	4	8	no	—
117	C₂×C₄⋊C₄	—	4	2	no	—
118	C₂×C₄⋊C₄	—	8	8	no	—
119	C₂×C₂²⋊C₄	—	4	1	no	—
120	C₂×C₄²	—	2	1	no	—
121	C₂×C₄²	—	2	1	no	—
122	C₂×C₄²	—	2	1	no	—
123	C₂×C₄²	—	4	1	no	—
124	C₂×C₄²	C₂²⋊C₄	1	2	yes	—
125	C₂×C₄²	C₂²⋊C₄	1	2	yes	—
126	C₂×C₄²	C₄⋊C₄	1	2	yes	—
127	M₃₂	—	2	4	no	—
128	M₃₂	—	2	4	no	—
129	M₃₂	—	4	2	no	—
130	M₃₂	—	4	2	no	—
131	C₂×C₁₆	—	2	4	no	—
132	C₂×C₁₆	—	4	2	no	—
133	C₂×C₁₆	—	4	2	no	—
134	#108	—	2	2	no	—
135	#108	—	2	2	no	—
136	#108	—	2	2	no	—
137	#108	C₄⋊C₄	1	4	yes	complemented
138	#107	—	2	2	no	—
139	#107	—	2	4	no	—
140	#107	—	2	4	no	—
141	#107	—	4	4	no	—
142	#107	—	4	4	no	—
143	#107	—	4	2	no	—
144	#107	—	4	2	no	—
145	#107	C₄⋊C₄	1	4	yes	complemented
146	#106	—	2	2	no	—
147	#106	—	2	2	no	—
148	#97	—	4	4	no	—
149	C₄×C₈	—	4	4	no	—
150	C₄×C₈	C₄²	1	4	yes	—
151	C₄×C₈	C₂²⋊C₄	1	4	yes	—
152	C₄×C₈	C₄⋊C₄	1	4	yes	—
153	C₄×C₈	C₄⋊C₄	1	4	yes	—
154	C₂⁴	AGL(1,ℤ/8)	1	1	yes	characteristic
155	C₂²×C₄	—	2	4	no	—
156	C₂²×C₄	—	2	4	no	—
157	C₂²×C₄	—	2	2	no	—
158	C₂²×C₄	—	2	2	no	—
159	C₂²×C₄	—	2	2	no	—
160	C₂²×C₄	—	4	2	no	—
161	C₂²×C₄	—	4	4	no	—
162	C₂²×C₄	—	8	8	no	—
163	C₂²×C₄	—	8	2	no	—
164	C₂²×C₄	—	8	2	no	—
165	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
166	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
167	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
168	C₂²×C₄	#121	1	1	yes	characteristic
169	C₂²×C₄	#121	1	1	yes	characteristic
170	C₂²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
171	C₂²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
172	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
173	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
174	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
175	C₂²×C₄	C₂×Q₁₆	1	1	yes	characteristic
176	C₂²×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
177	C₂²×C₄	#137	1	1	yes	characteristic
178	C₂²×C₄	#137	1	1	yes	characteristic
179	M₁₆	—	2	4	no	—
180	M₁₆	—	4	2	no	—
181	M₁₆	—	4	2	no	—
182	M₁₆	—	4	2	no	—
183	C₂×C₈	—	4	2	no	—
184	C₂×C₈	—	4	2	no	—
185	C₂×C₈	—	4	2	no	—
186	C₂×C₈	#95	1	4	yes	—
187	C₂×C₈	#102	1	2	yes	—
188	C₂×C₈	#102	1	2	yes	—
189	C₂×C₈	#102	1	2	yes	—
190	C₂×C₈	#102	1	2	yes	—
191	C₂×C₈	#103	1	4	yes	—
192	C₂×C₈	#103	1	4	yes	—
193	C₂×C₈	#106	1	4	yes	—
194	C₂×C₈	#107	1	2	yes	—
195	C₂×C₈	#107	1	2	yes	—
196	C₂×C₈	C₂×C₄²	1	1	yes	characteristic
197	C₂×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
198	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
199	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
200	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
201	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
202	C₂×C₈	#117	1	1	yes	characteristic
203	C₂×C₈	#117	1	1	yes	characteristic
204	C₂×C₈	C₄×D₄	1	1	yes	characteristic
205	C₂×C₈	C₄×D₄	1	1	yes	characteristic
206	C₂×C₈	C₄×D₄	1	1	yes	characteristic
207	C₂×C₈	C₄×D₄	1	1	yes	characteristic
208	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
209	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
210	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
211	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
212	C₂×C₈	#121	1	1	yes	characteristic
213	C₂×C₈	#121	1	1	yes	characteristic
214	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
215	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
216	C₂×C₈	#124	1	1	yes	characteristic
217	C₂×C₈	#124	1	1	yes	characteristic
218	C₂×C₈	#125	1	1	yes	characteristic
219	C₂×C₈	#125	1	1	yes	characteristic
220	C₂×C₈	#127	1	1	yes	characteristic
221	C₂×C₈	C₄⋊Q₈	1	1	yes	characteristic
222	C₂×C₈	C₄⋊Q₈	1	1	yes	characteristic
223	C₂×C₈	C₄⋊Q₈	1	1	yes	characteristic
224	C₄⋊C₄	—	2	4	no	—
225	C₄⋊C₄	—	4	2	no	—
226	C₄⋊C₄	—	4	4	no	—
227	C₄⋊C₄	—	4	4	no	—
228	C₄⋊C₄	—	8	16	no	—
229	C₄⋊C₄	—	8	2	no	—
230	C₄⋊C₄	—	8	2	no	—
231	C₂²⋊C₄	—	4	2	no	—
232	C₂²⋊C₄	—	4	2	no	—
233	C₄²	—	2	2	no	—
234	C₄²	—	2	2	no	—
235	C₄²	—	2	2	no	—
236	C₄²	—	4	2	no	—
237	C₄²	—	4	2	no	—
238	C₄²	#95	1	4	yes	—
239	C₁₆	—	4	4	no	—
240	C₂³	—	2	2	no	—
241	C₂³	—	2	2	no	—
242	C₂³	#417	1	1	yes	characteristic
243	C₂³	#418	1	1	yes	characteristic
244	C₂³	#428	1	1	yes	characteristic
245	C₂³	#459	1	1	yes	characteristic
246	C₂³	#460	1	1	yes	characteristic
247	C₂³	#476	1	1	yes	characteristic
248	C₂³	#477	1	1	yes	characteristic
249	C₂×C₄	—	2	4	no	—
250	C₂×C₄	—	2	8	no	—
251	C₂×C₄	—	2	8	no	—
252	C₂×C₄	—	2	4	no	—
253	C₂×C₄	—	2	4	no	—
254	C₂×C₄	—	4	4	no	—
255	C₂×C₄	—	8	2	no	—
256	C₂×C₄	—	8	2	no	—
257	C₂×C₄	—	8	8	no	—
258	C₂×C₄	—	8	4	no	—
259	C₂×C₄	—	16	4	no	—
260	C₂×C₄	—	16	4	no	—
261	C₂×C₄	C₂×#95	1	1	yes	characteristic
262	C₂×C₄	#380	1	1	yes	characteristic
263	C₂×C₄	#380	1	1	yes	characteristic
264	C₂×C₄	#389	1	1	yes	characteristic
265	C₂×C₄	#389	1	1	yes	characteristic
266	C₂×C₄	#390	1	1	yes	characteristic
267	C₂×C₄	#391	1	1	yes	characteristic
268	C₂×C₄	C₂×#102	1	1	yes	characteristic
269	C₂×C₄	C₂×#102	1	1	yes	characteristic
270	C₂×C₄	C₂×#103	1	1	yes	characteristic
271	C₂×C₄	C₂×#103	1	1	yes	characteristic
272	C₂×C₄	#417	1	1	yes	characteristic
273	C₂×C₄	#419	1	1	yes	characteristic
274	C₂×C₄	C₂×#106	1	1	yes	characteristic
275	C₂×C₄	C₂×#107	1	1	yes	characteristic
276	C₂×C₄	#428	1	1	yes	characteristic
277	C₂×C₄	#459	1	1	yes	characteristic
278	C₂×C₄	#460	1	1	yes	characteristic
279	C₂×C₄	#461	1	1	yes	characteristic
280	C₂×C₄	#461	1	1	yes	characteristic
281	C₂×C₄	#462	1	1	yes	characteristic
282	C₂×C₄	#462	1	1	yes	characteristic
283	C₂×C₄	#474	1	1	yes	characteristic
284	C₂×C₄	#474	1	1	yes	characteristic
285	C₂×C₄	#475	1	1	yes	characteristic
286	C₂×C₄	#475	1	1	yes	characteristic
287	C₂×C₄	#476	1	1	yes	characteristic
288	C₂×C₄	#477	1	1	yes	characteristic
289	C₈	—	4	4	no	—
290	C₈	#340	1	4	yes	—
291	C₈	#340	1	4	yes	—
292	C₈	C₄×C₄⋊C₄	1	1	yes	characteristic
293	C₈	C₄×C₄⋊C₄	1	1	yes	characteristic
294	C₈	#382	1	1	yes	characteristic
295	C₈	#384	1	1	yes	characteristic
296	C₈	#389	1	1	yes	characteristic
297	C₈	#389	1	1	yes	characteristic
298	C₈	#390	1	1	yes	characteristic
299	C₈	#391	1	1	yes	characteristic
300	C₂²	—	2	2	no	—
301	C₂²	—	2	2	no	—
302	C₂²	—	2	2	no	—
303	C₂²	—	2	2	no	—
304	C₂²	—	2	1	no	—
305	C₂²	—	2	1	no	—
306	C₂²	—	2	1	no	—
307	C₂²	—	2	1	no	—
308	C₂²	#1736	1	1	yes	characteristic
309	C₂²	#1802	1	1	yes	characteristic
310	C₂²	#1803	1	1	yes	characteristic
311	C₂²	#1947	1	1	yes	characteristic
312	C₂²	#1948	1	1	yes	characteristic
313	C₂²	#1982	1	1	yes	characteristic
314	C₂²	#1987	1	1	yes	characteristic
315	C₂²	#2213	1	2	yes	—
316	C₂²	#2214	1	2	yes	—
317	C₄	—	2	8	no	—
318	C₄	—	8	4	no	—
319	C₄	—	16	4	no	—
320	C₄	C₄⋊#95	1	1	yes	characteristic
321	C₄	C₂×#340	1	1	yes	characteristic
322	C₄	#1802	1	1	yes	characteristic
323	C₄	#1803	1	1	yes	characteristic
324	C₄	#1947	1	1	yes	characteristic
325	C₄	#1948	1	1	yes	characteristic
326	C₄	#1981	1	1	yes	characteristic
327	C₄	#1988	1	1	yes	characteristic
328	C₂	—	2	1	no	—
329	C₂	—	2	1	no	—
330	C₂	—	2	1	no	—
331	C₂	—	2	1	no	—
332	C₂	#10849	1	1	yes	characteristic
333	C₂	#12821	1	2	yes	—
334	C₂	#13181	1	2	yes	—
335	C₂	#13261	1	1	yes	characteristic
336	C₂	#13262	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³	#459
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂⁴	AGL(1,ℤ/8)
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ 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
16. C₂×#364	C₄	semidirect
18. #1845	C₄	semidirect
20. #1772	C₄	semidirect
137. #108	C₄⋊C₄	semidirect
145. #107	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
C₂×#1845	2
C₂×#1772	2
C₂×#1387	2
#12557	2