Subgroups — #498609

Order 512.

526 conjugacy classes of subgroups.

The subgroup lattice has 351 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#33697	C₂	1	1	yes	maximal, characteristic
2	#23179	C₂	1	1	yes	maximal, characteristic
3	#23170	C₂	1	1	yes	maximal, characteristic, complemented
4	#13274	C₂	1	1	yes	maximal, characteristic, complemented
5	#13251	C₂	1	1	yes	maximal, characteristic
6	#13151	C₂	1	2	yes	maximal, complemented
7	#13134	C₂	1	2	yes	maximal, complemented
8	#12939	C₂	1	2	yes	maximal, complemented
9	C₄×#368	C₂	1	1	yes	maximal, characteristic, complemented
10	C₁₆×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
11	#12641	C₂	1	2	yes	maximal, complemented
12	#3160	C₂²	1	1	yes	characteristic
13	#3147	C₂²	1	1	yes	characteristic
14	#2962	—	2	1	no	—
15	C₂×#443	C₂²	1	2	yes	—
16	#2935	—	2	1	no	—
17	#2242	C₄	1	2	yes	complemented
18	#2242	C₄	1	2	yes	complemented
19	#2241	C₄	1	1	yes	characteristic, complemented
20	#2241	C₄	1	1	yes	characteristic, complemented
21	#2240	C₄	1	1	yes	characteristic, complemented
22	#2240	C₄	1	1	yes	characteristic, complemented
23	C₂×#368	C₂²	1	1	yes	characteristic
24	C₂×#368	C₂²	1	1	yes	characteristic
25	C₂×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
26	C₂×#359	C₂²	1	2	yes	—
27	C₂×#359	C₂²	1	2	yes	—
28	#2138	C₂²	1	2	yes	—
29	#2138	C₂²	1	2	yes	—
30	C₂×C₄×C₁₆	C₂²	1	2	yes	—
31	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
32	#1986	—	2	1	no	—
33	#1980	—	2	1	no	—
34	#1974	C₂²	1	2	yes	—
35	#1945	C₂²	1	2	yes	—
36	#1922	—	2	2	no	—
37	#1876	—	2	2	no	—
38	C₄×#108	C₂²	1	1	yes	characteristic
39	C₄×#108	C₂²	1	1	yes	characteristic
40	#1772	C₂²	1	2	yes	—
41	C₈×C₄⋊C₄	C₂²	1	1	yes	characteristic
42	#1382	C₂²	1	2	yes	—
43	#1382	C₂²	1	2	yes	—
44	#1381	C₂²	1	1	yes	characteristic
45	#1381	C₂²	1	1	yes	characteristic
46	#1374	C₂²	1	2	yes	—
47	C₂×#135	—	2	1	no	—
48	C₂×#135	C₂³	1	2	yes	—
49	C₂×#131	—	2	2	no	—
50	#533	—	2	1	no	—
51	#532	—	2	1	no	—
52	C₂²×C₁₆	C₂³	1	2	yes	—
53	C₂²×C₁₆	C₂³	1	1	yes	characteristic
54	C₂²×C₁₆	C₂³	1	1	yes	characteristic
55	#500	C₂×C₄	1	1	yes	characteristic
56	#500	C₂×C₄	1	1	yes	characteristic
57	#499	—	2	1	no	—
58	#499	C₂×C₄	1	2	yes	—
59	#498	—	2	1	no	—
60	#495	—	2	1	no	—
61	#495	C₂×C₄	1	2	yes	—
62	#494	C₂×C₄	1	1	yes	characteristic
63	#493	C₂×C₄	1	1	yes	characteristic
64	#492	—	2	1	no	—
65	#446	—	4	2	no	—
66	#446	—	4	1	no	—
67	C₈×Q₈	—	4	1	no	—
68	#443	—	2	4	no	—
69	#443	—	4	2	no	—
70	#436	—	4	1	no	—
71	#435	—	4	2	no	—
72	C₈×D₄	—	4	1	no	—
73	C₂×#108	C₂³	1	1	yes	characteristic
74	C₂×#108	C₂³	1	1	yes	characteristic
75	C₂×#108	C₂³	1	1	yes	characteristic
76	C₂×#108	C₂³	1	1	yes	characteristic
77	#427	—	2	1	no	—
78	#427	C₂³	1	2	yes	—
79	#424	—	2	2	no	—
80	C₂×C₄⋊C₈	D₄	1	1	yes	characteristic
81	C₂×#104	—	2	2	no	—
82	C₂×#104	—	2	2	no	—
83	#408	—	2	2	no	—
84	#405	—	2	2	no	—
85	C₄×M₁₆	—	2	1	no	—
86	C₄×M₁₆	—	2	1	no	—
87	C₂×C₄×C₈	C₂³	1	2	yes	—
88	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
89	C₄×C₄⋊C₄	D₄	1	1	yes	characteristic
90	#368	—	2	2	no	—
91	#368	—	2	2	no	—
92	C₄⋊C₁₆	—	2	1	no	—
93	C₄⋊C₁₆	—	2	1	no	—
94	#359	—	2	4	no	—
95	#359	—	4	2	no	—
96	#358	C₂×C₄	1	2	yes	—
97	#358	C₂×C₄	1	2	yes	—
98	#358	C₂×C₄	1	2	yes	—
99	#358	C₂×C₄	1	2	yes	—
100	#357	C₂×C₄	1	2	yes	—
101	#357	C₂×C₄	1	2	yes	—
102	#357	C₂×C₄	1	2	yes	—
103	#357	C₂×C₄	1	2	yes	—
104	C₄×C₁₆	—	4	2	no	—
105	C₄×C₁₆	C₂×C₄	1	1	yes	characteristic
106	C₄×C₁₆	C₂×C₄	1	1	yes	characteristic
107	C₄×C₁₆	C₂×C₄	1	1	yes	characteristic
108	C₄×C₁₆	C₂×C₄	1	1	yes	characteristic
109	#341	—	2	1	no	—
110	#341	—	2	1	no	—
111	#339	—	2	2	no	—
112	#339	—	2	2	no	—
113	#336	D₄	1	2	yes	—
114	C₂×#41	—	2	2	no	—
115	#135	—	2	2	no	—
116	#135	—	2	2	no	—
117	#135	—	2	2	no	—
118	#135	—	2	2	no	—
119	#135	—	8	1	no	—
120	#135	—	8	1	no	—
121	C₂×Q₁₆	—	2	1	no	—
122	C₂×Q₁₆	C₂²×C₄	1	2	yes	—
123	C₂×SD₁₆	—	2	2	no	—
124	C₂×SD₁₆	—	2	2	no	—
125	C₂×D₈	—	2	1	no	—
126	C₂×D₈	C₂²×C₄	1	2	yes	—
127	#131	—	4	4	no	—
128	#131	—	8	2	no	—
129	C₂×M₁₆	—	2	1	no	—
130	C₂×M₁₆	—	2	1	no	—
131	C₂×M₁₆	—	2	1	no	—
132	C₂×M₁₆	—	2	1	no	—
133	C₂×M₁₆	—	4	2	no	—
134	C₂²×C₈	—	4	2	no	—
135	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
136	C₂²×C₈	C₂×D₄	1	2	yes	—
137	C₂²×C₈	C₂⁴	1	1	yes	characteristic
138	C₄⋊Q₈	—	2	1	no	—
139	C₄⋊Q₈	—	2	1	no	—
140	C₄⋊Q₈	—	2	1	no	—
141	#127	—	2	1	no	—
142	#125	—	2	2	no	—
143	#124	—	2	2	no	—
144	C₂²⋊Q₈	—	4	1	no	—
145	C₂²⋊Q₈	—	4	1	no	—
146	#121	—	4	1	no	—
147	#121	—	4	1	no	—
148	C₄×Q₈	—	4	1	no	—
149	C₄×Q₈	—	4	1	no	—
150	C₄×D₄	—	4	1	no	—
151	C₄×D₄	—	4	1	no	—
152	#117	—	2	2	no	—
153	C₂×C₄⋊C₄	—	2	1	no	—
154	C₂×C₄²	C₂×D₄	1	2	yes	—
155	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
156	C₂×C₁₆	—	2	1	no	—
157	C₂×C₁₆	—	2	1	no	—
158	C₂×C₁₆	—	2	1	no	—
159	C₂×C₁₆	—	2	1	no	—
160	C₂×C₁₆	—	2	2	no	—
161	C₂×C₁₆	—	2	2	no	—
162	C₂×C₁₆	—	2	2	no	—
163	C₂×C₁₆	C₂²×C₄	1	1	yes	characteristic
164	C₂×C₁₆	C₂²×C₄	1	1	yes	characteristic
165	C₂×C₁₆	C₂²×C₄	1	1	yes	characteristic
166	C₂×C₁₆	C₂²×C₄	1	1	yes	characteristic
167	#108	—	2	1	no	—
168	#108	—	2	1	no	—
169	#108	—	2	1	no	—
170	#108	—	2	1	no	—
171	#108	—	2	1	no	—
172	#108	—	2	1	no	—
173	#108	#41	1	2	yes	—
174	#108	#41	1	2	yes	—
175	#107	—	2	1	no	—
176	#107	—	2	1	no	—
177	#107	C₂²×C₄	1	2	yes	—
178	#107	C₂²×C₄	1	2	yes	—
179	#106	—	2	2	no	—
180	#106	—	2	2	no	—
181	C₄⋊C₈	—	4	2	no	—
182	C₄⋊C₈	—	4	2	no	—
183	C₄⋊C₈	D₈	1	1	yes	characteristic
184	C₄⋊C₈	D₈	1	2	yes	—
185	C₄⋊C₈	D₈	1	1	yes	characteristic
186	#104	—	4	4	no	—
187	#104	—	8	2	no	—
188	C₂²⋊C₈	—	4	2	no	—
189	C₂²⋊C₈	—	4	2	no	—
190	#97	—	4	2	no	—
191	#97	—	4	1	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	—	4	1	no	—
198	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
199	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
200	C₄×C₈	#41	1	1	yes	characteristic
201	C₄×C₈	#41	1	1	yes	characteristic
202	#95	—	2	1	no	—
203	#41	—	4	2	no	—
204	#41	—	4	2	no	—
205	#41	—	4	2	no	—
206	#41	—	4	2	no	—
207	C₂×Q₈	—	2	2	no	—
208	C₂×Q₈	—	4	1	no	—
209	C₂×Q₈	—	4	1	no	—
210	C₂×D₄	—	2	2	no	—
211	C₂×D₄	—	4	2	no	—
212	C₂×D₄	—	4	1	no	—
213	C₂×D₄	—	4	1	no	—
214	C₂²×C₄	—	2	2	no	—
215	C₂²×C₄	—	4	2	no	—
216	C₂²×C₄	#135	1	1	yes	characteristic
217	C₂²×C₄	#135	1	1	yes	characteristic
218	C₂²×C₄	C₂²×D₄	1	1	yes	characteristic
219	Q₁₆	—	2	2	no	—
220	Q₁₆	—	2	2	no	—
221	Q₁₆	—	8	1	no	—
222	SD₁₆	—	4	2	no	—
223	SD₁₆	—	4	2	no	—
224	SD₁₆	—	8	2	no	—
225	D₈	—	2	2	no	—
226	D₈	—	2	2	no	—
227	D₈	—	8	1	no	—
228	M₁₆	—	2	2	no	—
229	M₁₆	—	2	2	no	—
230	M₁₆	—	4	1	no	—
231	M₁₆	—	4	1	no	—
232	M₁₆	—	4	1	no	—
233	M₁₆	—	4	1	no	—
234	M₁₆	—	4	1	no	—
235	M₁₆	—	4	1	no	—
236	M₁₆	—	8	4	no	—
237	C₂×C₈	—	2	2	no	—
238	C₂×C₈	—	2	2	no	—
239	C₂×C₈	—	2	1	no	—
240	C₂×C₈	—	2	1	no	—
241	C₂×C₈	—	4	1	no	—
242	C₂×C₈	—	4	1	no	—
243	C₂×C₈	—	4	1	no	—
244	C₂×C₈	—	4	1	no	—
245	C₂×C₈	—	8	4	no	—
246	C₂×C₈	C₄×D₄	1	1	yes	characteristic
247	C₂×C₈	C₄×D₄	1	1	yes	characteristic
248	C₂×C₈	C₂×D₈	1	2	yes	—
249	C₂×C₈	C₂×D₈	1	2	yes	—
250	C₂×C₈	C₂³×C₄	1	1	yes	characteristic
251	C₂×C₈	C₂×#41	1	1	yes	characteristic
252	C₂×C₈	C₂×#41	1	1	yes	characteristic
253	C₂×C₈	C₂×#41	1	1	yes	characteristic
254	C₂×C₈	H(5,𝔽₂)	1	1	yes	characteristic
255	C₂×C₈	2₋¹⁺⁴	1	1	yes	characteristic
256	C₄⋊C₄	—	2	4	no	—
257	C₄⋊C₄	—	2	2	no	—
258	C₄⋊C₄	—	2	2	no	—
259	C₄⋊C₄	—	4	1	no	—
260	C₄⋊C₄	—	4	1	no	—
261	C₄⋊C₄	—	4	2	no	—
262	C₄⋊C₄	—	4	2	no	—
263	C₄⋊C₄	—	4	1	no	—
264	C₄⋊C₄	—	4	1	no	—
265	C₂²⋊C₄	—	4	2	no	—
266	C₂²⋊C₄	—	4	2	no	—
267	C₂²⋊C₄	—	4	2	no	—
268	C₄²	—	2	2	no	—
269	C₄²	—	2	2	no	—
270	C₄²	—	4	2	no	—
271	C₄²	C₄×D₄	1	1	yes	characteristic
272	C₄²	C₄×D₄	1	1	yes	characteristic
273	C₄²	C₂×D₈	1	1	yes	characteristic
274	C₄²	C₂×D₈	1	1	yes	characteristic
275	C₁₆	—	2	1	no	—
276	C₁₆	—	2	1	no	—
277	C₁₆	—	2	1	no	—
278	C₁₆	—	2	1	no	—
279	C₁₆	—	4	2	no	—
280	C₂³	—	4	2	no	—
281	C₂³	C₂×#135	1	1	yes	characteristic
282	Q₈	—	4	2	no	—
283	Q₈	—	4	2	no	—
284	Q₈	—	8	2	no	—
285	Q₈	—	16	1	no	—
286	D₄	—	4	2	no	—
287	D₄	—	4	2	no	—
288	D₄	—	8	2	no	—
289	D₄	—	8	2	no	—
290	D₄	—	8	2	no	—
291	D₄	—	16	1	no	—
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	—	2	4	no	—
294	C₂×C₄	—	4	1	no	—
295	C₂×C₄	—	4	2	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	—	4	2	no	—
299	C₂×C₄	—	4	2	no	—
300	C₂×C₄	—	8	2	no	—
301	C₂×C₄	—	8	2	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	#548	1	1	yes	characteristic
309	C₂×C₄	C₂²×D₈	1	1	yes	characteristic
310	C₂×C₄	#577	1	1	yes	characteristic
311	C₂×C₄	#577	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	—	4	2	no	—
318	C₈	—	8	1	no	—
319	C₈	—	8	1	no	—
320	C₈	C₄×#41	1	1	yes	characteristic
321	C₈	#520	1	1	yes	characteristic
322	C₈	#556	1	1	yes	characteristic
323	C₈	#559	1	1	yes	characteristic
324	C₂²	—	2	1	no	—
325	C₂²	—	2	1	no	—
326	C₂²	—	8	2	no	—
327	C₂²	—	8	2	no	—
328	C₂²	—	8	2	no	—
329	C₂²	C₂×C₄×D₈	1	1	yes	characteristic
330	C₂²	#3304	1	1	yes	characteristic
331	C₂²	#3365	1	1	yes	characteristic
332	C₄	—	2	1	no	—
333	C₄	—	2	1	no	—
334	C₄	—	2	1	no	—
335	C₄	—	4	4	no	—
336	C₄	—	8	2	no	—
337	C₄	—	8	2	no	—
338	C₄	—	8	2	no	—
339	C₄	#2178	1	2	yes	—
340	C₄	#2367	1	1	yes	characteristic
341	C₄	#2949	1	1	yes	characteristic
342	C₄	#3294	1	1	yes	characteristic
343	C₄	#3358	1	1	yes	characteristic
344	C₂	—	2	1	no	—
345	C₂	—	2	1	no	—
346	C₂	—	16	2	no	—
347	C₂	#23216	1	1	yes	characteristic
348	C₂	C₂×#2178	1	1	yes	characteristic
349	C₂	#33465	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	#3147	C₂²
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
3. #23170	C₂	semidirect
4. #13274	C₂	semidirect
6. #13151	C₂	semidirect
7. #13134	C₂	semidirect
8. #12939	C₂	semidirect
9. C₄×#368	C₂	semidirect
10. C₁₆×C₄⋊C₄	C₂	semidirect
11. #12641	C₂	semidirect
17. #2242	C₄	semidirect
18. #2242	C₄	semidirect
19. #2241	C₄	semidirect
20. #2241	C₄	semidirect
21. #2240	C₄	semidirect
22. #2240	C₄	semidirect

Maximal subgroups

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

subgroup	index
#33697	2
#23179	2
#23170	2
#13274	2
#13251	2
#13151	2
#13134	2
#12939	2
C₄×#368	2
C₁₆×C₄⋊C₄	2
#12641	2