Subgroups — #498637

Order 512.

492 conjugacy classes of subgroups.

The subgroup lattice has 298 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	#13264	C₂	1	2	yes	maximal
3	#13263	C₂	1	2	yes	maximal
4	#13160	C₂	1	2	yes	maximal
5	#13159	C₂	1	1	yes	maximal, characteristic
6	#13159	C₂	1	1	yes	maximal, characteristic
7	#12663	C₂	1	2	yes	maximal
8	#12662	C₂	1	1	yes	maximal, characteristic
9	C₁₆×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
10	#12653	C₂	1	1	yes	maximal, characteristic
11	C₄×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
12	C₂×C₈×Q₈	C₂²	1	2	yes	—
13	C₄×C₄⋊Q₈	C₄	1	1	yes	characteristic
14	C₂×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊C₁₆	C₂²	1	2	yes	—
17	C₂×C₄⋊C₁₆	C₂²	1	2	yes	—
18	C₂×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
20	C₂×#346	C₂²	1	2	yes	—
21	C₂×#346	C₂²	1	2	yes	—
22	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
24	C₂×C₄×C₁₆	C₂²	1	2	yes	—
25	#1984	C₄	1	2	yes	—
26	#1939	C₄	1	2	yes	—
27	#1939	C₄	1	1	yes	characteristic
28	#1939	C₄	1	1	yes	characteristic
29	#1842	C₄	1	1	yes	characteristic
30	C₈×C₄⋊C₄	C₂²	1	1	yes	characteristic
31	C₈×C₄⋊C₄	C₂²	1	1	yes	characteristic
32	C₈×C₄⋊C₄	C₂²	1	2	yes	—
33	C₄²×C₈	C₂²	1	1	yes	characteristic
34	#1374	C₂²	1	2	yes	—
35	#1374	C₂²	1	2	yes	—
36	#1374	C₂²	1	2	yes	—
37	#1374	C₂²	1	2	yes	—
38	#1374	C₂²	1	1	yes	characteristic
39	#1374	C₂²	1	1	yes	characteristic
40	#1374	C₂²	1	2	yes	—
41	C₂×C₄⋊Q₈	C₈	1	1	yes	characteristic
42	C₂×C₄×Q₈	C₂×C₄	1	2	yes	—
43	C₂²×C₁₆	C₂³	1	1	yes	characteristic
44	C₂²×C₁₆	C₂³	1	1	yes	characteristic
45	C₂²×C₁₆	C₂³	1	2	yes	—
46	C₂²×C₁₆	C₂³	1	1	yes	characteristic
47	C₂²×C₁₆	C₂³	1	1	yes	characteristic
48	C₂²×C₁₆	C₂³	1	2	yes	—
49	C₈×Q₈	—	4	2	no	—
50	C₈×Q₈	—	4	2	no	—
51	C₂×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
53	C₂×C₄⋊C₈	C₂×C₄	1	2	yes	—
54	C₂×C₄⋊C₈	C₂×C₄	1	2	yes	—
55	C₂×C₄⋊C₈	C₂×C₄	1	2	yes	—
56	C₂×C₄⋊C₈	C₂×C₄	1	2	yes	—
57	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
58	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
59	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
60	C₂×C₄×C₈	C₂³	1	2	yes	—
61	C₂×C₄×C₈	C₂³	1	2	yes	—
62	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
63	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
64	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
65	#391	C₈	1	2	yes	—
66	#389	C₈	1	1	yes	characteristic
67	#389	C₈	1	1	yes	characteristic
68	#389	C₈	1	2	yes	—
69	#384	C₈	1	1	yes	characteristic
70	C₄×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
71	C₄×C₄⋊C₄	C₂×C₄	1	2	yes	—
72	C₄×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
73	C₄³	C₂×C₄	1	1	yes	characteristic
74	C₄⋊C₁₆	—	2	2	no	—
75	C₄⋊C₁₆	—	2	2	no	—
76	C₄⋊C₁₆	—	4	2	no	—
77	C₄⋊C₁₆	—	4	2	no	—
78	C₄⋊C₁₆	Q₈	1	4	yes	complemented
79	C₄⋊C₁₆	Q₈	1	4	yes	complemented
80	#346	—	2	4	no	—
81	#346	—	4	2	no	—
82	C₄×C₁₆	—	2	2	no	—
83	C₄×C₁₆	—	2	2	no	—
84	C₄×C₁₆	—	4	2	no	—
85	#336	C₂×C₄	1	2	yes	—
86	#336	C₂×C₄	1	2	yes	—
87	#336	C₂×C₄	1	2	yes	—
88	#336	C₂×C₄	1	2	yes	—
89	C₂²×Q₈	C₂×C₈	1	2	yes	—
90	C₂²×C₈	C₂²×C₄	1	2	yes	—
91	C₂²×C₈	C₂²×C₄	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	C₂⁴	1	1	yes	characteristic
96	C₄⋊Q₈	—	2	4	no	—
97	C₄×Q₈	—	4	2	no	—
98	C₄×Q₈	—	4	2	no	—
99	C₂×C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
100	C₂×C₄⋊C₄	C₂×C₈	1	2	yes	—
101	C₂×C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
102	C₂×C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
103	C₂×C₄⋊C₄	C₂×C₈	1	2	yes	—
104	C₂×C₄⋊C₄	C₂×C₈	1	2	yes	—
105	C₂×C₄⋊C₄	C₂×C₈	1	2	yes	—
106	C₂×C₄⋊C₄	C₂×C₈	1	2	yes	—
107	C₂×C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
108	C₂×C₄²	C₂×C₈	1	1	yes	characteristic
109	C₂×C₄²	C₂×C₈	1	1	yes	characteristic
110	C₂×C₄²	C₂×C₈	1	1	yes	characteristic
111	C₂×C₄²	C₂×C₈	1	1	yes	characteristic
112	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
113	C₂×C₄²	C₂²×C₄	1	2	yes	—
114	C₂×C₄²	C₂²×C₄	1	2	yes	—
115	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
116	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
117	C₂×C₁₆	—	2	1	no	—
118	C₂×C₁₆	—	2	1	no	—
119	C₂×C₁₆	—	2	2	no	—
120	C₂×C₁₆	—	2	1	no	—
121	C₂×C₁₆	—	2	2	no	—
122	C₂×C₁₆	—	2	1	no	—
123	C₂×C₁₆	—	2	1	no	—
124	C₂×C₁₆	—	2	1	no	—
125	C₂×C₁₆	—	2	2	no	—
126	C₂×C₁₆	—	2	1	no	—
127	C₂×C₁₆	—	2	1	no	—
128	C₂×C₁₆	—	2	2	no	—
129	C₂×C₁₆	—	2	1	no	—
130	C₂×C₁₆	—	2	1	no	—
131	C₂×C₁₆	—	2	2	no	—
132	C₂×C₁₆	C₂×Q₈	1	2	yes	—
133	C₂×C₁₆	C₂×Q₈	1	2	yes	—
134	C₂×C₁₆	C₂×Q₈	1	4	yes	—
135	C₄⋊C₈	—	2	4	no	—
136	C₄⋊C₈	—	2	2	no	—
137	C₄⋊C₈	—	2	2	no	—
138	C₄⋊C₈	—	4	2	no	—
139	C₄⋊C₈	—	4	2	no	—
140	C₄⋊C₈	—	4	2	no	—
141	C₄×C₈	—	2	2	no	—
142	C₄×C₈	—	2	2	no	—
143	C₄×C₈	—	2	2	no	—
144	C₄×C₈	—	2	2	no	—
145	C₄×C₈	—	2	1	no	—
146	C₄×C₈	—	2	1	no	—
147	C₄×C₈	—	2	2	no	—
148	C₄×C₈	—	2	2	no	—
149	C₄×C₈	—	4	1	no	—
150	C₄×C₈	C₂×Q₈	1	2	yes	—
151	C₄×C₈	C₂×Q₈	1	2	yes	—
152	C₄×C₈	#41	1	2	yes	—
153	C₄×C₈	#41	1	2	yes	—
154	#95	C₂×C₈	1	2	yes	—
155	#95	C₂×C₈	1	2	yes	—
156	#95	C₂×C₈	1	2	yes	—
157	#95	C₂×C₈	1	2	yes	—
158	C₂×Q₈	—	2	4	no	—
159	C₂×Q₈	—	2	4	no	—
160	C₂×Q₈	—	4	2	no	—
161	C₂²×C₄	C₂²×C₈	1	2	yes	—
162	C₂²×C₄	C₂²×C₈	1	2	yes	—
163	C₂²×C₄	C₂²×C₈	1	2	yes	—
164	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
165	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
166	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
167	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
168	C₂²×C₄	C₂²×C₈	1	2	yes	—
169	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
170	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
171	C₂²×C₄	C₂³×C₄	1	1	yes	characteristic
172	C₂×C₈	—	2	1	no	—
173	C₂×C₈	—	2	1	no	—
174	C₂×C₈	—	2	1	no	—
175	C₂×C₈	—	2	1	no	—
176	C₂×C₈	—	2	1	no	—
177	C₂×C₈	—	2	2	no	—
178	C₂×C₈	—	2	2	no	—
179	C₂×C₈	—	2	1	no	—
180	C₂×C₈	—	2	1	no	—
181	C₂×C₈	—	2	2	no	—
182	C₂×C₈	—	2	2	no	—
183	C₂×C₈	—	2	2	no	—
184	C₂×C₈	—	2	2	no	—
185	C₂×C₈	C₄×Q₈	1	2	yes	—
186	C₂×C₈	C₄×Q₈	1	2	yes	—
187	C₂×C₈	C₂²×Q₈	1	1	yes	characteristic
188	C₂×C₈	C₂²×Q₈	1	1	yes	characteristic
189	C₂×C₈	C₂×#41	1	1	yes	characteristic
190	C₂×C₈	C₂×#41	1	1	yes	characteristic
191	C₂×C₈	H(5,𝔽₂)	1	1	yes	characteristic
192	C₂×C₈	H(5,𝔽₂)	1	1	yes	characteristic
193	C₄⋊C₄	—	2	4	no	—
194	C₄⋊C₄	—	2	2	no	—
195	C₄⋊C₄	—	2	2	no	—
196	C₄⋊C₄	—	2	2	no	—
197	C₄⋊C₄	—	4	2	no	—
198	C₄⋊C₄	—	4	2	no	—
199	C₄⋊C₄	—	4	2	no	—
200	C₄⋊C₄	—	4	2	no	—
201	C₄⋊C₄	M₃₂	1	4	yes	—
202	C₄²	—	2	1	no	—
203	C₄²	—	2	2	no	—
204	C₄²	—	2	2	no	—
205	C₄²	—	2	1	no	—
206	C₄²	—	2	2	no	—
207	C₄²	—	2	2	no	—
208	C₄²	—	2	2	no	—
209	C₄²	—	2	2	no	—
210	C₄²	—	2	2	no	—
211	C₄²	—	4	1	no	—
212	C₄²	C₄×Q₈	1	2	yes	—
213	C₄²	C₄×Q₈	1	2	yes	—
214	C₄²	#131	1	2	yes	—
215	C₄²	#131	1	2	yes	—
216	C₁₆	—	2	2	no	—
217	C₁₆	—	2	2	no	—
218	C₁₆	—	2	4	no	—
219	C₁₆	—	4	1	no	—
220	C₁₆	—	4	1	no	—
221	C₁₆	—	4	2	no	—
222	C₂³	C₂³×C₈	1	1	yes	characteristic
223	Q₈	—	4	8	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	C₈×Q₈	1	2	yes	—
247	C₂×C₄	C₈×Q₈	1	2	yes	—
248	C₂×C₄	C₈×Q₈	1	2	yes	—
249	C₂×C₄	C₈×Q₈	1	2	yes	—
250	C₂×C₄	C₂×M₃₂	1	4	yes	—
251	C₂×C₄	C₂×M₃₂	1	2	yes	—
252	C₂×C₄	#504	1	2	yes	—
253	C₂×C₄	#504	1	2	yes	—
254	C₂×C₄	C₂×C₄×Q₈	1	1	yes	characteristic
255	C₂×C₄	C₂×C₄×Q₈	1	1	yes	characteristic
256	C₂×C₄	#518	1	1	yes	characteristic
257	C₂×C₄	C₂×#131	1	1	yes	characteristic
258	C₂×C₄	C₂×#131	1	1	yes	characteristic
259	C₂×C₄	#568	1	1	yes	characteristic
260	C₈	—	2	2	no	—
261	C₈	—	2	2	no	—
262	C₈	—	4	1	no	—
263	C₈	—	4	2	no	—
264	C₈	—	4	2	no	—
265	C₈	#554	1	1	yes	characteristic
266	C₈	#554	1	1	yes	characteristic
267	C₈	#555	1	1	yes	characteristic
268	C₈	Q₈²	1	1	yes	characteristic
269	C₂²	C₂×C₈×Q₈	1	1	yes	characteristic
270	C₂²	C₂×C₈×Q₈	1	1	yes	characteristic
271	C₂²	#2972	1	1	yes	characteristic
272	C₂²	C₂²×M₃₂	1	1	yes	characteristic
273	C₂²	C₂×#504	1	1	yes	characteristic
274	C₂²	C₂×#504	1	1	yes	characteristic
275	C₂²	#3407	1	1	yes	characteristic
276	C₄	—	2	2	no	—
277	C₄	—	2	2	no	—
278	C₄	—	2	2	no	—
279	C₄	—	2	4	no	—
280	C₄	—	2	2	no	—
281	C₄	—	4	2	no	—
282	C₄	—	4	1	no	—
283	C₄	—	4	2	no	—
284	C₄	—	4	2	no	—
285	C₄	#2183	1	4	yes	—
286	C₄	#2369	1	1	yes	characteristic
287	C₄	#2962	1	1	yes	characteristic
288	C₄	#2962	1	1	yes	characteristic
289	C₄	#2979	1	1	yes	characteristic
290	C₂	#23247	1	1	yes	characteristic
291	C₂	C₂×#2183	1	1	yes	characteristic
292	C₂	#33474	1	1	yes	characteristic
293	C₂	#33476	1	1	yes	characteristic
294	C₂	Q₈×M₃₂	1	1	yes	characteristic
295	C₂	#33488	1	1	yes	characteristic
296	C₂	#33495	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
78. C₄⋊C₁₆	Q₈	semidirect
79. C₄⋊C₁₆	Q₈	semidirect

Maximal subgroups

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

subgroup	index
C₈×C₄⋊Q₈	2
#13264	2
#13263	2
#13160	2
#13159	2
#12663	2
#12662	2
C₁₆×C₄⋊C₄	2
#12653	2
C₄×C₄⋊C₁₆	2