Subgroups — #498263

Order 512.

508 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#33626	C₂	1	1	yes	maximal, characteristic
2	#23170	C₂	1	1	yes	maximal, characteristic, complemented
3	#22688	C₂	1	1	yes	maximal, characteristic
4	#13285	C₂	1	1	yes	maximal, characteristic, complemented
5	#13179	C₂	1	1	yes	maximal, characteristic, complemented
6	#13169	C₂	1	2	yes	maximal, complemented
7	#13113	C₂	1	2	yes	maximal, complemented
8	#12939	C₂	1	2	yes	maximal
9	C₁₆×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
10	#12641	C₂	1	2	yes	maximal
11	C₄×#359	C₂	1	1	yes	maximal, characteristic
12	#3160	C₂²	1	1	yes	characteristic
13	#2962	—	2	1	no	—
14	C₂×#443	C₂²	1	2	yes	—
15	C₄×#135	C₂²	1	1	yes	characteristic
16	#2906	—	2	1	no	—
17	#2229	C₄	1	2	yes	complemented
18	#2228	C₄	1	2	yes	complemented
19	#2227	C₄	1	1	yes	characteristic, complemented
20	#2226	C₄	1	1	yes	characteristic, complemented
21	#2225	C₄	1	1	yes	characteristic, complemented
22	#2224	C₄	1	1	yes	characteristic, complemented
23	C₂×#368	C₂²	1	2	yes	—
24	#2158	C₂²	1	2	yes	—
25	C₂×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
26	C₂×#359	C₂²	1	1	yes	characteristic
27	C₂×#359	C₂²	1	1	yes	characteristic
28	#2138	C₂²	1	2	yes	—
29	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
30	C₂×C₄×C₁₆	C₂²	1	2	yes	—
31	#1986	—	2	1	no	—
32	#1945	C₂²	1	2	yes	—
33	#1945	C₂²	1	2	yes	—
34	#1922	—	2	2	no	—
35	#1866	—	2	2	no	—
36	#1848	C₂²	1	1	yes	characteristic
37	#1806	—	2	1	no	—
38	C₄×#108	C₂²	1	1	yes	characteristic
39	#1772	C₂²	1	2	yes	—
40	C₈×C₄⋊C₄	C₂²	1	1	yes	characteristic
41	#1382	C₂²	1	1	yes	characteristic
42	#1382	C₂²	1	2	yes	—
43	#1382	C₂²	1	1	yes	characteristic
44	#1382	C₂²	1	2	yes	—
45	#1381	C₂²	1	2	yes	—
46	#1374	C₂²	1	2	yes	—
47	C₂×#135	C₂³	1	1	yes	characteristic
48	C₂×#131	—	2	2	no	—
49	#533	—	2	1	no	—
50	C₄×#41	—	2	1	no	—
51	C₂²×C₁₆	C₂³	1	1	yes	characteristic
52	C₂²×C₁₆	C₂³	1	2	yes	—
53	C₂²×C₁₆	C₂³	1	1	yes	characteristic
54	#500	C₂×C₄	1	1	yes	characteristic
55	#500	C₂×C₄	1	1	yes	characteristic
56	#499	—	2	1	no	—
57	#499	C₂×C₄	1	2	yes	—
58	#498	—	2	1	no	—
59	#446	—	4	2	no	—
60	#446	—	4	1	no	—
61	C₈×Q₈	—	4	1	no	—
62	#443	—	2	4	no	—
63	#443	—	4	2	no	—
64	C₄×Q₁₆	C₂×C₄	1	1	yes	characteristic
65	C₄×Q₁₆	C₂×C₄	1	1	yes	characteristic
66	C₄×SD₁₆	—	2	1	no	—
67	C₄×SD₁₆	—	2	1	no	—
68	C₄×D₈	C₂×C₄	1	1	yes	characteristic
69	C₄×D₈	C₂×C₄	1	1	yes	characteristic
70	C₂×#108	—	2	1	no	—
71	C₂×#108	C₂³	1	2	yes	—
72	C₂×#108	C₂³	1	1	yes	characteristic
73	C₂×#108	C₂³	1	1	yes	characteristic
74	#427	—	2	1	no	—
75	#427	C₂³	1	2	yes	—
76	#427	C₂³	1	1	yes	characteristic
77	#424	—	2	2	no	—
78	#424	—	2	2	no	—
79	#424	—	4	2	no	—
80	#423	—	2	1	no	—
81	#423	—	4	1	no	—
82	C₂×C₄⋊C₈	—	4	1	no	—
83	C₂×C₄⋊C₈	D₄	1	1	yes	characteristic
84	C₂×#104	—	2	2	no	—
85	#416	—	2	1	no	—
86	#405	—	2	2	no	—
87	C₄×M₁₆	—	2	1	no	—
88	C₂×C₄×C₈	C₂³	1	2	yes	—
89	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
90	C₄×C₄⋊C₄	D₄	1	1	yes	characteristic
91	#368	—	4	2	no	—
92	#367	C₂×C₄	1	2	yes	—
93	#367	C₂×C₄	1	2	yes	—
94	#366	C₂×C₄	1	2	yes	—
95	#366	C₂×C₄	1	2	yes	—
96	C₄⋊C₁₆	C₂×C₄	1	1	yes	characteristic
97	C₄⋊C₁₆	C₂×C₄	1	1	yes	characteristic
98	C₄⋊C₁₆	C₂×C₄	1	2	yes	—
99	#359	—	2	2	no	—
100	#359	Q₈	1	4	yes	complemented
101	#358	C₂×C₄	1	2	yes	—
102	#358	C₂×C₄	1	2	yes	—
103	#357	C₂×C₄	1	2	yes	—
104	#357	C₂×C₄	1	2	yes	—
105	C₄×C₁₆	—	2	1	no	—
106	C₄×C₁₆	—	2	1	no	—
107	C₄×C₁₆	—	4	2	no	—
108	#341	—	2	1	no	—
109	#339	—	2	2	no	—
110	#339	—	2	2	no	—
111	#339	—	2	2	no	—
112	#336	D₄	1	2	yes	—
113	C₂×#41	—	2	1	no	—
114	#135	—	2	1	no	—
115	#135	—	2	1	no	—
116	#135	C₂×Q₈	1	2	yes	—
117	#135	#41	1	2	yes	—
118	C₂×Q₁₆	C₂²×C₄	1	1	yes	characteristic
119	C₂×SD₁₆	—	2	1	no	—
120	C₂×D₈	C₂²×C₄	1	1	yes	characteristic
121	#131	—	4	4	no	—
122	#131	—	8	2	no	—
123	C₂×M₁₆	—	2	1	no	—
124	C₂×M₁₆	—	2	2	no	—
125	C₂×M₁₆	—	2	1	no	—
126	C₂×M₁₆	—	4	2	no	—
127	C₂²×C₈	—	4	2	no	—
128	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
129	C₂²×C₈	C₂×D₄	1	2	yes	—
130	C₂²×C₈	C₂⁴	1	1	yes	characteristic
131	C₄⋊Q₈	—	2	1	no	—
132	C₄⋊Q₈	—	2	1	no	—
133	#125	—	2	2	no	—
134	C₂²⋊Q₈	—	4	1	no	—
135	C₂²⋊Q₈	—	4	1	no	—
136	C₄×Q₈	—	2	1	no	—
137	C₄×Q₈	—	2	1	no	—
138	C₄×Q₈	—	4	1	no	—
139	C₄×Q₈	—	4	1	no	—
140	C₄×D₄	—	2	1	no	—
141	C₄×D₄	—	2	1	no	—
142	C₄×D₄	—	4	1	no	—
143	C₄×D₄	—	4	1	no	—
144	#117	—	2	1	no	—
145	#117	—	2	2	no	—
146	#117	—	4	1	no	—
147	C₂×C₄⋊C₄	—	2	1	no	—
148	C₂×C₄²	—	4	1	no	—
149	C₂×C₄²	C₂×D₄	1	2	yes	—
150	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
151	C₂×C₁₆	—	2	1	no	—
152	C₂×C₁₆	—	2	1	no	—
153	C₂×C₁₆	—	2	2	no	—
154	C₂×C₁₆	—	2	2	no	—
155	C₂×C₁₆	—	2	1	no	—
156	C₂×C₁₆	—	2	1	no	—
157	C₂×C₁₆	—	2	1	no	—
158	C₂×C₁₆	C₂²×C₄	1	2	yes	—
159	C₂×C₁₆	C₂²×C₄	1	2	yes	—
160	C₂×C₁₆	C₂×Q₈	1	2	yes	—
161	#108	—	2	1	no	—
162	#108	—	2	2	no	—
163	#108	—	2	1	no	—
164	#108	—	2	2	no	—
165	#108	—	2	1	no	—
166	#108	—	8	1	no	—
167	#108	C₂×Q₈	1	2	yes	—
168	#107	—	2	1	no	—
169	#107	—	2	1	no	—
170	#107	C₂²×C₄	1	2	yes	—
171	#107	C₂²×C₄	1	2	yes	—
172	#107	C₂²×C₄	1	1	yes	characteristic
173	#107	C₂²×C₄	1	1	yes	characteristic
174	#106	—	2	2	no	—
175	#106	—	2	1	no	—
176	#106	—	2	2	no	—
177	C₄⋊C₈	—	4	2	no	—
178	C₄⋊C₈	—	4	2	no	—
179	C₄⋊C₈	—	4	2	no	—
180	C₄⋊C₈	—	4	2	no	—
181	C₄⋊C₈	—	4	1	no	—
182	C₄⋊C₈	—	4	1	no	—
183	C₄⋊C₈	D₈	1	1	yes	characteristic
184	C₄⋊C₈	D₈	1	1	yes	characteristic
185	C₄⋊C₈	D₈	1	2	yes	—
186	#104	—	4	4	no	—
187	#103	—	2	1	no	—
188	#103	—	2	1	no	—
189	#102	—	2	1	no	—
190	#102	—	2	1	no	—
191	#97	—	4	2	no	—
192	#97	—	4	1	no	—
193	C₄×C₈	—	2	2	no	—
194	C₄×C₈	—	2	2	no	—
195	C₄×C₈	—	4	2	no	—
196	C₄×C₈	—	4	1	no	—
197	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
198	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
199	C₄×C₈	#41	1	1	yes	characteristic
200	C₄×C₈	#41	1	1	yes	characteristic
201	#95	—	2	1	no	—
202	#41	—	2	2	no	—
203	#41	—	2	2	no	—
204	#41	—	4	1	no	—
205	#41	—	4	1	no	—
206	C₂×Q₈	—	2	1	no	—
207	C₂×Q₈	—	4	1	no	—
208	C₂×Q₈	—	4	1	no	—
209	C₂×D₄	—	2	1	no	—
210	C₂×D₄	—	4	1	no	—
211	C₂²×C₄	—	2	2	no	—
212	C₂²×C₄	—	4	1	no	—
213	C₂²×C₄	—	4	1	no	—
214	C₂²×C₄	—	4	1	no	—
215	C₂²×C₄	#135	1	1	yes	characteristic
216	C₂²×C₄	#135	1	1	yes	characteristic
217	C₂²×C₄	C₂²×D₄	1	1	yes	characteristic
218	Q₁₆	—	2	1	no	—
219	Q₁₆	C₄×Q₈	1	2	yes	—
220	SD₁₆	—	2	2	no	—
221	SD₁₆	—	2	2	no	—
222	D₈	—	2	1	no	—
223	D₈	C₄×Q₈	1	2	yes	—
224	M₁₆	—	2	2	no	—
225	M₁₆	—	4	1	no	—
226	M₁₆	—	4	2	no	—
227	M₁₆	—	4	1	no	—
228	M₁₆	—	4	2	no	—
229	M₁₆	—	4	1	no	—
230	M₁₆	—	8	4	no	—
231	C₂×C₈	—	2	2	no	—
232	C₂×C₈	—	2	1	no	—
233	C₂×C₈	—	2	2	no	—
234	C₂×C₈	—	2	1	no	—
235	C₂×C₈	—	4	1	no	—
236	C₂×C₈	—	4	1	no	—
237	C₂×C₈	—	4	2	no	—
238	C₂×C₈	—	8	4	no	—
239	C₂×C₈	C₄×D₄	1	1	yes	characteristic
240	C₂×C₈	C₄×D₄	1	1	yes	characteristic
241	C₂×C₈	C₂×D₈	1	2	yes	—
242	C₂×C₈	C₂×D₈	1	2	yes	—
243	C₂×C₈	C₂³×C₄	1	1	yes	characteristic
244	C₂×C₈	C₂²×Q₈	1	1	yes	characteristic
245	C₂×C₈	C₂×#41	1	1	yes	characteristic
246	C₂×C₈	C₂×#41	1	1	yes	characteristic
247	C₂×C₈	H(5,𝔽₂)	1	1	yes	characteristic
248	C₂×C₈	2₋¹⁺⁴	1	1	yes	characteristic
249	C₄⋊C₄	—	2	4	no	—
250	C₄⋊C₄	—	2	1	no	—
251	C₄⋊C₄	—	2	1	no	—
252	C₄⋊C₄	—	2	2	no	—
253	C₄⋊C₄	—	2	2	no	—
254	C₄⋊C₄	—	4	1	no	—
255	C₄⋊C₄	—	4	1	no	—
256	C₄⋊C₄	—	4	2	no	—
257	C₄⋊C₄	—	4	2	no	—
258	C₄⋊C₄	—	4	1	no	—
259	C₄⋊C₄	—	4	1	no	—
260	C₂²⋊C₄	—	4	1	no	—
261	C₂²⋊C₄	—	4	2	no	—
262	C₂²⋊C₄	—	4	1	no	—
263	C₂²⋊C₄	—	4	1	no	—
264	C₄²	—	2	2	no	—
265	C₄²	—	2	2	no	—
266	C₄²	—	4	2	no	—
267	C₄²	—	4	1	no	—
268	C₄²	—	4	1	no	—
269	C₄²	—	4	1	no	—
270	C₄²	C₄×D₄	1	1	yes	characteristic
271	C₄²	C₄×D₄	1	1	yes	characteristic
272	C₄²	C₂×D₈	1	1	yes	characteristic
273	C₄²	C₂×D₈	1	1	yes	characteristic
274	C₁₆	—	2	2	no	—
275	C₁₆	—	2	2	no	—
276	C₁₆	—	2	2	no	—
277	C₁₆	—	4	1	no	—
278	C₂³	—	4	1	no	—
279	C₂³	C₂×#135	1	1	yes	characteristic
280	Q₈	—	2	2	no	—
281	Q₈	—	4	1	no	—
282	Q₈	—	8	2	no	—
283	Q₈	—	16	1	no	—
284	D₄	—	2	2	no	—
285	D₄	—	4	2	no	—
286	D₄	—	4	2	no	—
287	D₄	—	4	1	no	—
288	C₂×C₄	—	2	2	no	—
289	C₂×C₄	—	2	4	no	—
290	C₂×C₄	—	4	1	no	—
291	C₂×C₄	—	4	2	no	—
292	C₂×C₄	—	4	1	no	—
293	C₂×C₄	—	4	2	no	—
294	C₂×C₄	—	4	2	no	—
295	C₂×C₄	—	4	2	no	—
296	C₂×C₄	—	4	1	no	—
297	C₂×C₄	—	4	1	no	—
298	C₂×C₄	—	4	2	no	—
299	C₂×C₄	—	4	2	no	—
300	C₂×C₄	—	8	2	no	—
301	C₂×C₄	—	16	1	no	—
302	C₂×C₄	C₄×D₈	1	1	yes	characteristic
303	C₂×C₄	C₄×D₈	1	1	yes	characteristic
304	C₂×C₄	C₄×D₈	1	1	yes	characteristic
305	C₂×C₄	C₄×D₈	1	1	yes	characteristic
306	C₂×C₄	C₂×C₄×D₄	1	1	yes	characteristic
307	C₂×C₄	#546	1	1	yes	characteristic
308	C₂×C₄	D₄×Q₈	1	1	yes	characteristic
309	C₂×C₄	C₂²×D₈	1	1	yes	characteristic
310	C₂×C₄	#577	1	1	yes	characteristic
311	C₂×C₄	#578	1	1	yes	characteristic
312	C₈	—	2	1	no	—
313	C₈	—	2	1	no	—
314	C₈	—	2	2	no	—
315	C₈	—	2	2	no	—
316	C₈	—	4	2	no	—
317	C₈	—	8	2	no	—
318	C₈	—	8	1	no	—
319	C₈	C₂×C₄×Q₈	1	1	yes	characteristic
320	C₈	#520	1	1	yes	characteristic
321	C₈	#552	1	1	yes	characteristic
322	C₈	#554	1	1	yes	characteristic
323	C₂²	—	2	1	no	—
324	C₂²	—	2	1	no	—
325	C₂²	—	4	2	no	—
326	C₂²	—	8	1	no	—
327	C₂²	—	8	1	no	—
328	C₂²	C₂×C₄×D₈	1	1	yes	characteristic
329	C₂²	#3302	1	1	yes	characteristic
330	C₂²	#3381	1	1	yes	characteristic
331	C₄	—	2	1	no	—
332	C₄	—	2	1	no	—
333	C₄	—	2	1	no	—
334	C₄	—	4	4	no	—
335	C₄	—	4	2	no	—
336	C₄	—	8	1	no	—
337	C₄	—	8	2	no	—
338	C₄	—	8	1	no	—
339	C₄	—	8	2	no	—
340	C₄	#2178	1	2	yes	—
341	C₄	#2360	1	1	yes	characteristic
342	C₄	#2947	1	1	yes	characteristic
343	C₄	#3294	1	1	yes	characteristic
344	C₄	Q₈×D₈	1	1	yes	characteristic
345	C₂	—	2	1	no	—
346	C₂	—	2	1	no	—
347	C₂	—	8	2	no	—
348	C₂	#22962	1	1	yes	characteristic
349	C₂	C₂×#2178	1	1	yes	characteristic
350	C₂	#33464	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×C₄×D₈
prime radical	C₂×#135	C₂³
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄⋊C₈ ◁ C₈×C₄⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. #23170	C₂	semidirect
4. #13285	C₂	semidirect
5. #13179	C₂	semidirect
6. #13169	C₂	semidirect
7. #13113	C₂	semidirect
9. C₁₆×C₄⋊C₄	C₂	semidirect
17. #2229	C₄	semidirect
18. #2228	C₄	semidirect
19. #2227	C₄	semidirect
20. #2226	C₄	semidirect
21. #2225	C₄	semidirect
22. #2224	C₄	semidirect
100. #359	Q₈	semidirect

Maximal subgroups

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

subgroup	index
#33626	2
#23170	2
#22688	2
#13285	2
#13179	2
#13169	2
#13113	2
#12939	2
C₁₆×C₄⋊C₄	2
#12641	2
C₄×#359	2