Subgroups — #134359

Order 512.

434 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄²×Q₁₆	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	C₂×#1332	C₂	1	2	yes	maximal
4	#8806	C₂	1	1	yes	maximal, characteristic
5	#7171	C₂	1	2	yes	maximal
6	C₂×C₄×Q₁₆	C₄	1	1	yes	characteristic
7	C₂×C₄×Q₁₆	C₄	1	1	yes	characteristic
8	C₂×C₄×Q₁₆	C₂²	1	1	yes	characteristic
9	C₄²×Q₈	—	2	1	no	—
10	C₂×C₄⋊C₁₆	—	2	1	no	—
11	C₂×C₄×C₁₆	C₂²	1	2	yes	—
12	#1842	—	2	1	no	—
13	C₄×#107	C₄	1	1	yes	characteristic
14	C₄×#107	C₄	1	1	yes	characteristic
15	C₄×#107	C₂²	1	1	yes	characteristic
16	C₄×#103	—	2	1	no	—
17	C₄×#103	—	2	1	no	—
18	C₄×#103	—	2	1	no	—
19	C₄²×C₈	C₂²	1	1	yes	characteristic
20	C₂×#335	—	2	1	no	—
21	C₂×#335	C₂²	1	2	yes	—
22	#1374	—	2	1	no	—
23	#1332	—	2	4	no	—
24	C₂²×Q₁₆	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄×Q₈	—	2	1	no	—
26	C₂×C₄×Q₈	—	2	1	no	—
27	C₂×C₄×Q₈	—	2	1	no	—
28	C₂²×C₁₆	—	2	2	no	—
29	C₄×Q₁₆	—	2	2	no	—
30	C₄×Q₁₆	—	2	2	no	—
31	C₄×Q₁₆	—	2	2	no	—
32	C₄×Q₁₆	—	2	1	no	—
33	C₄×Q₁₆	—	2	1	no	—
34	C₄×Q₁₆	C₈	1	4	yes	complemented
35	C₄×Q₁₆	D₄	1	2	yes	—
36	C₄×Q₁₆	Q₈	1	2	yes	—
37	C₂×#107	C₂×C₄	1	1	yes	characteristic
38	C₂×#107	C₂×C₄	1	1	yes	characteristic
39	C₂×#107	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄⋊C₈	—	2	2	no	—
41	C₂×C₄⋊C₈	—	4	1	no	—
42	C₂×C₄⋊C₈	—	4	1	no	—
43	C₂×#103	—	2	1	no	—
44	C₂×#103	—	2	1	no	—
45	C₂×#103	—	2	1	no	—
46	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄×C₈	D₄	1	1	yes	characteristic
49	C₂×C₄×C₈	D₄	1	1	yes	characteristic
50	C₂×C₄×C₈	D₄	1	1	yes	characteristic
51	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
52	C₄×C₄⋊C₄	—	2	1	no	—
53	C₄×C₄⋊C₄	—	2	1	no	—
54	C₄×C₄⋊C₄	—	2	1	no	—
55	C₄×C₄⋊C₄	—	4	1	no	—
56	C₄×C₄⋊C₄	—	4	1	no	—
57	C₄×C₄⋊C₄	—	4	1	no	—
58	C₄³	—	4	1	no	—
59	C₄³	D₄	1	1	yes	characteristic
60	C₄⋊C₁₆	—	2	4	no	—
61	C₄×C₁₆	—	2	2	no	—
62	C₄×C₁₆	—	2	2	no	—
63	#340	—	2	1	no	—
64	#340	—	2	1	no	—
65	#340	—	2	1	no	—
66	#335	—	2	2	no	—
67	#335	—	2	2	no	—
68	#335	—	4	2	no	—
69	C₂²×Q₈	—	2	1	no	—
70	C₂×Q₁₆	—	2	1	no	—
71	C₂×Q₁₆	—	2	1	no	—
72	C₂×Q₁₆	—	2	1	no	—
73	C₂×Q₁₆	C₄⋊C₄	1	2	yes	—
74	C₂×Q₁₆	C₂×C₈	1	2	yes	—
75	C₂×Q₁₆	M₁₆	1	2	yes	—
76	C₂²×C₈	—	4	2	no	—
77	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
78	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
79	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
80	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
81	C₄×Q₈	—	2	4	no	—
82	C₄×Q₈	—	2	4	no	—
83	C₄×Q₈	—	2	2	no	—
84	C₄×Q₈	—	2	2	no	—
85	C₄×Q₈	—	4	2	no	—
86	C₄×Q₈	—	4	2	no	—
87	C₄×Q₈	—	4	1	no	—
88	C₄×Q₈	—	4	1	no	—
89	C₂×C₄⋊C₄	—	2	1	no	—
90	C₂×C₄⋊C₄	—	2	1	no	—
91	C₂×C₄⋊C₄	—	2	1	no	—
92	C₂×C₄⋊C₄	—	4	1	no	—
93	C₂×C₄⋊C₄	—	4	1	no	—
94	C₂×C₄⋊C₄	—	4	1	no	—
95	C₂×C₄²	—	4	1	no	—
96	C₂×C₄²	—	4	1	no	—
97	C₂×C₄²	—	4	1	no	—
98	C₂×C₄²	—	4	1	no	—
99	C₂×C₄²	—	4	1	no	—
100	C₂×C₄²	—	4	1	no	—
101	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
102	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
103	C₂×C₄²	D₈	1	1	yes	characteristic
104	C₂×C₄²	D₈	1	1	yes	characteristic
105	C₂×C₄²	SD₁₆	1	1	yes	characteristic
106	C₂×C₄²	SD₁₆	1	1	yes	characteristic
107	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
108	C₂×C₁₆	—	2	4	no	—
109	C₂×C₁₆	—	4	2	no	—
110	C₂×C₁₆	—	4	2	no	—
111	#107	—	2	1	no	—
112	#107	—	2	1	no	—
113	#107	—	2	1	no	—
114	#107	C₄⋊C₄	1	2	yes	—
115	#107	C₂×C₈	1	2	yes	—
116	#107	M₁₆	1	2	yes	—
117	C₄⋊C₈	—	4	2	no	—
118	C₄⋊C₈	—	4	2	no	—
119	C₄⋊C₈	—	8	2	no	—
120	C₄⋊C₈	—	8	2	no	—
121	#103	—	2	4	no	—
122	#103	—	2	4	no	—
123	#103	—	2	2	no	—
124	#103	—	2	2	no	—
125	C₄×C₈	—	2	1	no	—
126	C₄×C₈	—	2	1	no	—
127	C₄×C₈	—	2	2	no	—
128	C₄×C₈	—	2	1	no	—
129	C₄×C₈	—	2	1	no	—
130	C₄×C₈	—	4	1	no	—
131	C₄×C₈	C₂×C₈	1	2	yes	—
132	C₄×C₈	M₁₆	1	2	yes	—
133	C₄×C₈	C₂×D₄	1	1	yes	characteristic
134	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
135	C₄×C₈	#41	1	1	yes	characteristic
136	C₄×C₈	#41	1	1	yes	characteristic
137	#95	—	4	1	no	—
138	#95	—	4	1	no	—
139	#95	—	4	1	no	—
140	C₂×Q₈	—	2	2	no	—
141	C₂×Q₈	—	2	2	no	—
142	C₂×Q₈	—	2	2	no	—
143	C₂×Q₈	—	4	1	no	—
144	C₂×Q₈	—	4	1	no	—
145	C₂×Q₈	—	4	1	no	—
146	C₂²×C₄	—	4	1	no	—
147	C₂²×C₄	—	4	1	no	—
148	C₂²×C₄	—	4	1	no	—
149	C₂²×C₄	—	4	1	no	—
150	C₂²×C₄	#102	1	1	yes	characteristic
151	C₂²×C₄	#102	1	1	yes	characteristic
152	C₂²×C₄	#102	1	1	yes	characteristic
153	C₂²×C₄	#102	1	1	yes	characteristic
154	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
155	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
156	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
157	Q₁₆	—	2	2	no	—
158	Q₁₆	—	2	2	no	—
159	Q₁₆	—	2	2	no	—
160	Q₁₆	C₄⋊C₈	1	4	yes	complemented
161	C₂×C₈	—	2	1	no	—
162	C₂×C₈	—	2	1	no	—
163	C₂×C₈	—	2	1	no	—
164	C₂×C₈	—	2	1	no	—
165	C₂×C₈	—	2	1	no	—
166	C₂×C₈	—	2	1	no	—
167	C₂×C₈	—	2	1	no	—
168	C₂×C₈	—	8	2	no	—
169	C₂×C₈	—	8	2	no	—
170	C₂×C₈	—	8	2	no	—
171	C₂×C₈	C₂²⋊C₈	1	2	yes	—
172	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
173	C₂×C₈	#117	1	1	yes	characteristic
174	C₂×C₈	#121	1	1	yes	characteristic
175	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
176	C₂×C₈	C₂²×C₈	1	1	yes	characteristic
177	C₂×C₈	C₂×M₁₆	1	1	yes	characteristic
178	C₂×C₈	C₂×M₁₆	1	1	yes	characteristic
179	C₂×C₈	C₂×M₁₆	1	1	yes	characteristic
180	C₄⋊C₄	—	2	2	no	—
181	C₄⋊C₄	—	2	2	no	—
182	C₄⋊C₄	—	2	2	no	—
183	C₄⋊C₄	—	4	4	no	—
184	C₄⋊C₄	—	4	4	no	—
185	C₄⋊C₄	—	4	1	no	—
186	C₄⋊C₄	—	4	1	no	—
187	C₄⋊C₄	—	4	1	no	—
188	C₄⋊C₄	—	4	2	no	—
189	C₄⋊C₄	—	4	2	no	—
190	C₄²	—	2	2	no	—
191	C₄²	—	2	1	no	—
192	C₄²	—	2	1	no	—
193	C₄²	—	2	1	no	—
194	C₄²	—	2	1	no	—
195	C₄²	—	4	1	no	—
196	C₄²	—	4	4	no	—
197	C₄²	—	4	4	no	—
198	C₄²	—	4	2	no	—
199	C₄²	—	4	2	no	—
200	C₄²	—	8	2	no	—
201	C₄²	—	8	2	no	—
202	C₄²	—	8	1	no	—
203	C₄²	—	8	1	no	—
204	C₄²	C₂²⋊C₈	1	2	yes	—
205	C₄²	#104	1	2	yes	—
206	C₄²	SD₃₂	1	2	yes	—
207	C₄²	Q₃₂	1	2	yes	—
208	C₄²	#121	1	1	yes	characteristic
209	C₄²	C₂²⋊Q₈	1	1	yes	characteristic
210	C₄²	AGL(1,ℤ/8)	1	1	yes	characteristic
211	C₄²	#137	1	1	yes	characteristic
212	C₁₆	—	4	4	no	—
213	C₂³	C₂×#102	1	1	yes	characteristic
214	Q₈	—	2	4	no	—
215	Q₈	—	4	2	no	—
216	Q₈	—	4	2	no	—
217	Q₈	—	4	2	no	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	2	1	no	—
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	4	2	no	—
228	C₂×C₄	—	4	2	no	—
229	C₂×C₄	—	4	2	no	—
230	C₂×C₄	—	4	2	no	—
231	C₂×C₄	—	4	2	no	—
232	C₂×C₄	—	4	2	no	—
233	C₂×C₄	—	8	1	no	—
234	C₂×C₄	—	8	1	no	—
235	C₂×C₄	—	8	1	no	—
236	C₂×C₄	—	8	1	no	—
237	C₂×C₄	—	8	1	no	—
238	C₂×C₄	—	8	1	no	—
239	C₂×C₄	D₄⋊C₈	1	2	yes	—
240	C₂×C₄	D₄⋊C₈	1	2	yes	—
241	C₂×C₄	#358	1	2	yes	—
242	C₂×C₄	#358	1	2	yes	—
243	C₂×C₄	#359	1	2	yes	—
244	C₂×C₄	#380	1	1	yes	characteristic
245	C₂×C₄	C₂×C₂²⋊C₈	1	1	yes	characteristic
246	C₂×C₄	#407	1	1	yes	characteristic
247	C₂×C₄	#417	1	1	yes	characteristic
248	C₂×C₄	C₂×#104	1	1	yes	characteristic
249	C₂×C₄	#421	1	1	yes	characteristic
250	C₂×C₄	#459	1	1	yes	characteristic
251	C₂×C₄	#461	1	1	yes	characteristic
252	C₂×C₄	#474	1	1	yes	characteristic
253	C₂×C₄	#476	1	1	yes	characteristic
254	C₂×C₄	C₂×SD₃₂	1	1	yes	characteristic
255	C₂×C₄	C₂×Q₃₂	1	1	yes	characteristic
256	C₂×C₄	#509	1	1	yes	characteristic
257	C₂×C₄	#510	1	1	yes	characteristic
258	C₈	—	2	1	no	—
259	C₈	—	2	1	no	—
260	C₈	—	2	2	no	—
261	C₈	—	4	1	no	—
262	C₈	—	16	2	no	—
263	C₈	C₂×C₄⋊C₈	1	1	yes	characteristic
264	C₈	#423	1	1	yes	characteristic
265	C₈	#431	1	1	yes	characteristic
266	C₈	#432	1	1	yes	characteristic
267	C₂²	C₂×D₄⋊C₈	1	1	yes	characteristic
268	C₂²	#1486	1	1	yes	characteristic
269	C₂²	#1802	1	1	yes	characteristic
270	C₂²	C₂×#358	1	1	yes	characteristic
271	C₂²	#2139	1	1	yes	characteristic
272	C₂²	C₂×#359	1	1	yes	characteristic
273	C₂²	#2143	1	1	yes	characteristic
274	C₄	—	2	1	no	—
275	C₄	—	2	1	no	—
276	C₄	—	2	2	no	—
277	C₄	—	2	2	no	—
278	C₄	—	4	4	no	—
279	C₄	—	4	1	no	—
280	C₄	—	8	2	no	—
281	C₄	—	8	2	no	—
282	C₄	—	8	2	no	—
283	C₄	#1332	1	4	yes	complemented
284	C₄	#1477	1	1	yes	characteristic
285	C₄	#1489	1	1	yes	characteristic
286	C₄	#1797	1	1	yes	characteristic
287	C₄	#1806	1	1	yes	characteristic
288	C₄	#2207	1	1	yes	characteristic
289	C₄	#2209	1	1	yes	characteristic
290	C₄	#2225	1	1	yes	characteristic
291	C₄	#2227	1	1	yes	characteristic
292	C₂	#8563	1	1	yes	characteristic
293	C₂	C₂×#1332	1	1	yes	characteristic
294	C₂	#11592	1	1	yes	characteristic
295	C₂	#11658	1	1	yes	characteristic
296	C₂	#11660	1	1	yes	characteristic
297	C₂	#12812	1	1	yes	characteristic
298	C₂	#12815	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₈
commutator	C₂×C₈	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle, prime radical	C₂³	C₂×#102
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
34. C₄×Q₁₆	C₈	semidirect
160. Q₁₆	C₄⋊C₈	semidirect
283. C₄	#1332	semidirect

Maximal subgroups

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

subgroup	index
C₄²×Q₁₆	2
C₄×C₄⋊C₁₆	2
C₂×#1332	2
#8806	2
#7171	2