Subgroups — #498034

Order 512.

558 conjugacy classes of subgroups.

The subgroup lattice has 310 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#33423	C₂	1	2	yes	maximal, complemented
2	C₄×#443	C₂	1	1	yes	maximal, characteristic
3	#13195	C₂	1	2	yes	maximal
4	#13122	C₂	1	4	yes	maximal, complemented
5	#12900	C₂	1	1	yes	maximal, characteristic, complemented
6	#12787	C₂	1	2	yes	maximal, complemented
7	#12664	C₂	1	2	yes	maximal, complemented
8	C₄×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#443	C₄	1	1	yes	characteristic
10	C₂×#443	C₂²	1	1	yes	characteristic
11	C₄×#135	C₄	1	1	yes	characteristic
12	C₄×#131	C₂²	1	2	yes	—
13	#2164	—	2	4	no	—
14	#2163	—	2	2	no	—
15	#2162	—	2	2	no	—
16	C₂×#364	C₂²	1	2	yes	—
17	C₂×#364	C₂²	1	2	yes	—
18	#2150	C₂²	1	2	yes	—
19	C₂×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
21	C₂×#350	C₂²	1	4	yes	—
22	#2113	C₂²	1	4	yes	—
23	#2108	C₂²	1	4	yes	—
24	C₂×C₄×C₁₆	C₂²	1	2	yes	—
25	C₄×#108	C₄	1	1	yes	characteristic
26	#1772	C₄	1	1	yes	characteristic
27	#1772	C₂²	1	1	yes	characteristic
28	#1764	C₄	1	2	yes	—
29	#1764	C₂²	1	2	yes	—
30	C₄×#104	C₄	1	2	yes	—
31	C₄²×C₈	C₂²	1	1	yes	characteristic
32	#1379	C₂²	1	4	yes	—
33	#1374	C₂²	1	2	yes	—
34	C₂×#135	C₂×C₄	1	1	yes	characteristic
35	C₂×#131	C₂×C₄	1	2	yes	—
36	C₂×#131	C₂³	1	2	yes	—
37	C₄×#41	C₂×C₄	1	2	yes	—
38	C₂×M₃₂	C₂³	1	4	yes	—
39	C₂²×C₁₆	C₂³	1	2	yes	—
40	C₂²×C₁₆	C₂³	1	2	yes	—
41	#443	—	2	4	no	—
42	#443	—	2	2	no	—
43	#443	—	2	2	no	—
44	C₄×Q₁₆	C₈	1	1	yes	characteristic
45	C₄×Q₁₆	C₈	1	1	yes	characteristic
46	C₄×SD₁₆	C₈	1	2	yes	—
47	C₄×SD₁₆	C₈	1	2	yes	—
48	C₄×D₈	C₈	1	1	yes	characteristic
49	C₄×D₈	C₈	1	1	yes	characteristic
50	C₂×#108	C₂×C₄	1	1	yes	characteristic
51	C₂×#108	C₂×C₄	1	1	yes	characteristic
52	#427	C₂×C₄	1	1	yes	characteristic
53	C₂×#104	C₂×C₄	1	2	yes	—
54	C₂×#104	C₂×C₄	1	2	yes	—
55	#416	C₂×C₄	1	2	yes	—
56	#405	—	2	2	no	—
57	#405	—	2	2	no	—
58	#405	C₂×C₄	1	2	yes	—
59	#405	C₂³	1	2	yes	—
60	C₄×M₁₆	—	2	2	no	—
61	C₄×M₁₆	C₂×C₄	1	2	yes	—
62	C₂×C₄×C₈	—	2	2	no	—
63	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
64	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
65	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
66	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
67	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
68	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
69	C₄³	C₂×C₄	1	1	yes	characteristic
70	#364	—	2	4	no	—
71	#364	—	4	2	no	—
72	C₄⋊C₁₆	—	2	2	no	—
73	C₄⋊C₁₆	—	2	2	no	—
74	C₄⋊C₁₆	—	2	1	no	—
75	C₄⋊C₁₆	—	2	1	no	—
76	C₄⋊C₁₆	D₄	1	1	yes	characteristic
77	C₄⋊C₁₆	D₄	1	1	yes	characteristic
78	C₄⋊C₁₆	D₄	1	2	yes	—
79	#350	—	4	4	no	—
80	C₂²⋊C₁₆	—	2	4	no	—
81	C₂²⋊C₁₆	—	2	4	no	—
82	#346	—	2	4	no	—
83	C₄×C₁₆	—	2	4	no	—
84	C₄×C₁₆	—	2	4	no	—
85	#341	C₂×C₄	1	2	yes	—
86	#339	C₂×C₄	1	2	yes	—
87	#339	C₂×C₄	1	2	yes	—
88	C₂×#41	C₂²×C₄	1	2	yes	—
89	#135	—	2	2	no	—
90	#135	—	2	2	no	—
91	C₂×Q₁₆	C₂×C₈	1	1	yes	characteristic
92	C₂×SD₁₆	C₂×C₈	1	2	yes	—
93	C₂×D₈	C₂×C₈	1	1	yes	characteristic
94	#131	—	2	4	no	—
95	#131	—	2	2	no	—
96	#131	—	2	2	no	—
97	#131	—	2	2	no	—
98	#131	—	2	2	no	—
99	#131	—	4	2	no	—
100	C₂×M₁₆	—	2	2	no	—
101	C₂×M₁₆	—	2	2	no	—
102	C₂×M₁₆	C₂²×C₄	1	2	yes	—
103	C₂×M₁₆	C₂²×C₄	1	2	yes	—
104	C₂²×C₈	—	2	2	no	—
105	C₂²×C₈	—	2	2	no	—
106	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
107	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
108	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
109	C₂²×C₈	C₂⁴	1	1	yes	characteristic
110	C₄×Q₈	C₂×C₈	1	2	yes	—
111	C₄×Q₈	C₂×C₈	1	2	yes	—
112	C₄×D₄	—	2	2	no	—
113	C₄×D₄	—	2	2	no	—
114	C₄×D₄	C₂×C₈	1	2	yes	—
115	C₄×D₄	C₂×C₈	1	2	yes	—
116	#117	—	2	2	no	—
117	#117	C₂²×C₄	1	2	yes	—
118	C₂×C₄²	—	2	1	no	—
119	C₂×C₄²	—	2	2	no	—
120	C₂×C₄²	—	2	1	no	—
121	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
122	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
123	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
124	M₃₂	—	2	4	no	—
125	M₃₂	—	2	4	no	—
126	C₂×C₁₆	—	2	2	no	—
127	C₂×C₁₆	—	2	2	no	—
128	C₂×C₁₆	—	2	4	no	—
129	C₂×C₁₆	—	2	2	no	—
130	C₂×C₁₆	—	2	2	no	—
131	C₂×C₁₆	—	2	2	no	—
132	C₂×C₁₆	C₂×D₄	1	2	yes	—
133	C₂×C₁₆	C₂×D₄	1	2	yes	—
134	#108	—	2	2	no	—
135	#108	M₁₆	1	4	yes	—
136	#107	C₂×C₈	1	1	yes	characteristic
137	#107	C₂×C₈	1	1	yes	characteristic
138	#106	C₂×C₈	1	2	yes	—
139	#104	—	2	4	no	—
140	#104	—	2	4	no	—
141	#103	C₂×C₈	1	2	yes	—
142	#103	C₂×C₈	1	2	yes	—
143	#102	C₂×C₈	1	2	yes	—
144	#102	C₂×C₈	1	2	yes	—
145	#97	—	2	2	no	—
146	#97	—	2	2	no	—
147	#97	—	2	2	no	—
148	C₄×C₈	—	2	2	no	—
149	C₄×C₈	—	2	2	no	—
150	C₄×C₈	—	2	2	no	—
151	C₄×C₈	—	2	1	no	—
152	C₄×C₈	—	2	1	no	—
153	C₄×C₈	—	2	2	no	—
154	C₄×C₈	—	4	1	no	—
155	C₄×C₈	C₂×C₈	1	1	yes	characteristic
156	C₄×C₈	C₂×C₈	1	1	yes	characteristic
157	C₄×C₈	C₂×C₈	1	1	yes	characteristic
158	C₄×C₈	C₂×C₈	1	1	yes	characteristic
159	C₄×C₈	C₂×D₄	1	1	yes	characteristic
160	C₄×C₈	C₂×D₄	1	1	yes	characteristic
161	C₄×C₈	#41	1	2	yes	—
162	C₄×C₈	#41	1	2	yes	—
163	C₄×C₈	#41	1	1	yes	characteristic
164	C₄×C₈	#41	1	1	yes	characteristic
165	#41	—	2	2	no	—
166	#41	—	2	2	no	—
167	#41	—	2	2	no	—
168	#41	—	2	2	no	—
169	C₂×Q₈	C₂²×C₈	1	2	yes	—
170	C₂×D₄	—	2	2	no	—
171	C₂×D₄	C₂²×C₈	1	2	yes	—
172	C₂²×C₄	—	2	1	no	—
173	C₂²×C₄	—	2	1	no	—
174	C₂²×C₄	—	2	2	no	—
175	C₂²×C₄	—	2	2	no	—
176	C₂²×C₄	—	2	2	no	—
177	C₂²×C₄	#131	1	1	yes	characteristic
178	C₂²×C₄	#131	1	1	yes	characteristic
179	C₂²×C₄	C₂³×C₄	1	1	yes	characteristic
180	Q₁₆	—	2	2	no	—
181	SD₁₆	—	2	4	no	—
182	D₈	—	2	2	no	—
183	M₁₆	—	2	2	no	—
184	M₁₆	—	2	2	no	—
185	M₁₆	—	2	2	no	—
186	M₁₆	—	4	4	no	—
187	M₁₆	—	4	2	no	—
188	M₁₆	—	4	2	no	—
189	M₁₆	C₂×M₁₆	1	4	yes	—
190	C₂×C₈	—	2	2	no	—
191	C₂×C₈	—	2	1	no	—
192	C₂×C₈	—	2	1	no	—
193	C₂×C₈	—	2	1	no	—
194	C₂×C₈	—	2	1	no	—
195	C₂×C₈	—	2	2	no	—
196	C₂×C₈	—	4	4	no	—
197	C₂×C₈	—	4	2	no	—
198	C₂×C₈	—	4	2	no	—
199	C₂×C₈	#117	1	2	yes	—
200	C₂×C₈	C₄×D₄	1	1	yes	characteristic
201	C₂×C₈	C₄×D₄	1	1	yes	characteristic
202	C₂×C₈	C₂²×C₈	1	1	yes	characteristic
203	C₂×C₈	C₂²×C₈	1	1	yes	characteristic
204	C₂×C₈	C₂²×C₈	1	1	yes	characteristic
205	C₂×C₈	C₂²×C₈	1	1	yes	characteristic
206	C₂×C₈	C₂×M₁₆	1	2	yes	—
207	C₂×C₈	C₂²×D₄	1	1	yes	characteristic
208	C₂×C₈	C₂×#41	1	1	yes	characteristic
209	C₂×C₈	C₂×#41	1	1	yes	characteristic
210	C₂×C₈	C₂×#41	1	1	yes	characteristic
211	C₂×C₈	H(5,𝔽₂)	1	1	yes	characteristic
212	C₂×C₈	2₋¹⁺⁴	1	1	yes	characteristic
213	C₄⋊C₄	—	2	2	no	—
214	C₄⋊C₄	—	2	2	no	—
215	C₄⋊C₄	C₂²×C₈	1	2	yes	—
216	C₄⋊C₄	C₂²×C₈	1	2	yes	—
217	C₂²⋊C₄	—	2	2	no	—
218	C₂²⋊C₄	—	2	2	no	—
219	C₂²⋊C₄	—	2	2	no	—
220	C₄²	—	2	4	no	—
221	C₄²	—	2	2	no	—
222	C₄²	—	2	1	no	—
223	C₄²	—	2	2	no	—
224	C₄²	—	2	2	no	—
225	C₄²	—	2	1	no	—
226	C₄²	—	2	1	no	—
227	C₄²	—	4	2	no	—
228	C₄²	#117	1	2	yes	—
229	C₄²	C₄×D₄	1	1	yes	characteristic
230	C₄²	C₄×D₄	1	1	yes	characteristic
231	C₄²	C₂²×C₈	1	1	yes	characteristic
232	C₄²	C₂²×C₈	1	1	yes	characteristic
233	C₁₆	—	2	2	no	—
234	C₁₆	—	2	2	no	—
235	C₁₆	—	4	4	no	—
236	C₁₆	—	4	2	no	—
237	C₂³	—	2	2	no	—
238	C₂³	C₂×#131	1	1	yes	characteristic
239	Q₈	—	2	2	no	—
240	Q₈	—	2	2	no	—
241	D₄	—	2	2	no	—
242	D₄	—	2	2	no	—
243	D₄	—	4	2	no	—
244	D₄	—	4	2	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	4	1	no	—
254	C₂×C₄	—	4	1	no	—
255	C₂×C₄	—	4	1	no	—
256	C₂×C₄	—	4	2	no	—
257	C₂×C₄	—	4	2	no	—
258	C₂×C₄	—	4	4	no	—
259	C₂×C₄	—	8	2	no	—
260	C₂×C₄	C₈×D₄	1	1	yes	characteristic
261	C₂×C₄	C₈×D₄	1	1	yes	characteristic
262	C₂×C₄	C₈×D₄	1	1	yes	characteristic
263	C₂×C₄	C₈×D₄	1	1	yes	characteristic
264	C₂×C₄	C₂×#117	1	1	yes	characteristic
265	C₂×C₄	C₂×C₄×D₄	1	1	yes	characteristic
266	C₂×C₄	#520	1	1	yes	characteristic
267	C₂×C₄	C₂³×C₈	1	1	yes	characteristic
268	C₂×C₄	C₂²×M₁₆	1	1	yes	characteristic
269	C₂×C₄	#568	1	1	yes	characteristic
270	C₈	—	2	1	no	—
271	C₈	—	2	1	no	—
272	C₈	—	2	4	no	—
273	C₈	—	2	1	no	—
274	C₈	—	2	1	no	—
275	C₈	—	4	2	no	—
276	C₈	#431	1	2	yes	—
277	C₈	#431	1	2	yes	—
278	C₈	#547	1	1	yes	characteristic
279	C₈	#550	1	1	yes	characteristic
280	C₈	#555	1	1	yes	characteristic
281	C₈	#556	1	1	yes	characteristic
282	C₂²	—	2	1	no	—
283	C₂²	—	2	1	no	—
284	C₂²	—	4	2	no	—
285	C₂²	—	4	2	no	—
286	C₂²	—	4	2	no	—
287	C₂²	#2925	1	1	yes	characteristic
288	C₂²	C₂×C₈×D₄	1	1	yes	characteristic
289	C₂²	#2991	1	1	yes	characteristic
290	C₄	—	2	4	no	—
291	C₄	—	2	1	no	—
292	C₄	—	2	1	no	—
293	C₄	—	2	1	no	—
294	C₄	—	4	2	no	—
295	C₄	—	4	2	no	—
296	C₄	—	4	2	no	—
297	C₄	—	4	2	no	—
298	C₄	#2170	1	2	yes	—
299	C₄	#2354	1	1	yes	characteristic
300	C₄	C₂×#431	1	1	yes	characteristic
301	C₄	D₄×M₁₆	1	1	yes	characteristic
302	C₄	#2988	1	1	yes	characteristic
303	C₂	—	2	1	no	—
304	C₂	—	2	1	no	—
305	C₂	—	8	2	no	—
306	C₂	#22795	1	1	yes	characteristic
307	C₂	C₂×#2170	1	1	yes	characteristic
308	C₂	#33443	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂²×D₄
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×C₈×D₄
prime radical	C₂×#135	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄⋊C₁₆ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #33423	C₂	semidirect
4. #13122	C₂	semidirect
5. #12900	C₂	semidirect
6. #12787	C₂	semidirect
7. #12664	C₂	semidirect
8. C₄×C₄⋊C₁₆	C₂	semidirect

Maximal subgroups

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

subgroup	index
#33423	2
C₄×#443	2
#13195	2
#13122	2
#12900	2
#12787	2
#12664	2
C₄×C₄⋊C₁₆	2