Subgroups — #134365

Order 512.

418 conjugacy classes of subgroups.

The subgroup lattice has 291 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	#12653	C₂	1	1	yes	maximal, characteristic
3	C₂×#1334	C₂	1	2	yes	maximal
4	#8807	C₂	1	1	yes	maximal, characteristic
5	#7178	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×#346	C₂²	1	2	yes	—
12	#1831	—	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×#334	C₂²	1	2	yes	—
22	#1374	—	2	1	no	—
23	#1334	—	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×#103	—	2	1	no	—
42	C₂×#103	—	2	1	no	—
43	C₂×#103	—	2	1	no	—
44	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄×C₈	D₄	1	1	yes	characteristic
47	C₂×C₄×C₈	D₄	1	1	yes	characteristic
48	C₂×C₄×C₈	D₄	1	1	yes	characteristic
49	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
50	C₄×C₄⋊C₄	—	2	1	no	—
51	C₄×C₄⋊C₄	—	2	1	no	—
52	C₄×C₄⋊C₄	—	2	1	no	—
53	C₄×C₄⋊C₄	—	4	1	no	—
54	C₄×C₄⋊C₄	—	4	1	no	—
55	C₄×C₄⋊C₄	—	4	1	no	—
56	C₄³	—	4	1	no	—
57	C₄³	D₄	1	1	yes	characteristic
58	C₄⋊C₁₆	—	2	4	no	—
59	#346	—	2	2	no	—
60	#346	—	2	2	no	—
61	#340	—	2	1	no	—
62	#340	—	2	1	no	—
63	#340	—	2	1	no	—
64	#336	—	4	1	no	—
65	#336	—	4	1	no	—
66	#335	—	4	2	no	—
67	#334	—	2	2	no	—
68	#334	—	2	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	#103	—	2	4	no	—
120	#103	—	2	4	no	—
121	#103	—	2	2	no	—
122	#103	—	2	2	no	—
123	C₄×C₈	—	2	1	no	—
124	C₄×C₈	—	2	1	no	—
125	C₄×C₈	—	2	2	no	—
126	C₄×C₈	—	2	1	no	—
127	C₄×C₈	—	2	1	no	—
128	C₄×C₈	—	4	1	no	—
129	C₄×C₈	C₂×C₈	1	2	yes	—
130	C₄×C₈	M₁₆	1	2	yes	—
131	C₄×C₈	C₂×D₄	1	1	yes	characteristic
132	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
133	C₄×C₈	#41	1	1	yes	characteristic
134	C₄×C₈	#41	1	1	yes	characteristic
135	#95	—	4	1	no	—
136	#95	—	4	1	no	—
137	#95	—	4	1	no	—
138	C₂×Q₈	—	2	2	no	—
139	C₂×Q₈	—	2	2	no	—
140	C₂×Q₈	—	2	2	no	—
141	C₂×Q₈	—	4	1	no	—
142	C₂×Q₈	—	4	1	no	—
143	C₂×Q₈	—	4	1	no	—
144	C₂²×C₄	—	4	1	no	—
145	C₂²×C₄	—	4	1	no	—
146	C₂²×C₄	—	4	1	no	—
147	C₂²×C₄	—	4	1	no	—
148	C₂²×C₄	#102	1	1	yes	characteristic
149	C₂²×C₄	#102	1	1	yes	characteristic
150	C₂²×C₄	#102	1	1	yes	characteristic
151	C₂²×C₄	#102	1	1	yes	characteristic
152	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
153	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
154	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
155	Q₁₆	—	2	2	no	—
156	Q₁₆	—	2	2	no	—
157	Q₁₆	—	2	2	no	—
158	Q₁₆	C₄⋊C₈	1	4	yes	—
159	C₂×C₈	—	2	1	no	—
160	C₂×C₈	—	2	1	no	—
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	—	8	2	no	—
167	C₂×C₈	—	8	2	no	—
168	C₂×C₈	—	8	2	no	—
169	C₂×C₈	C₂²⋊C₈	1	2	yes	—
170	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
171	C₂×C₈	#117	1	1	yes	characteristic
172	C₂×C₈	#121	1	1	yes	characteristic
173	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
174	C₂×C₈	C₂²×C₈	1	1	yes	characteristic
175	C₂×C₈	C₂×M₁₆	1	1	yes	characteristic
176	C₂×C₈	C₂×M₁₆	1	1	yes	characteristic
177	C₂×C₈	C₂×M₁₆	1	1	yes	characteristic
178	C₄⋊C₄	—	2	2	no	—
179	C₄⋊C₄	—	2	2	no	—
180	C₄⋊C₄	—	2	2	no	—
181	C₄⋊C₄	—	4	4	no	—
182	C₄⋊C₄	—	4	4	no	—
183	C₄⋊C₄	—	4	1	no	—
184	C₄⋊C₄	—	4	1	no	—
185	C₄⋊C₄	—	4	1	no	—
186	C₄⋊C₄	—	4	2	no	—
187	C₄⋊C₄	—	4	2	no	—
188	C₄²	—	2	2	no	—
189	C₄²	—	2	1	no	—
190	C₄²	—	2	1	no	—
191	C₄²	—	4	1	no	—
192	C₄²	—	4	1	no	—
193	C₄²	—	4	1	no	—
194	C₄²	—	4	4	no	—
195	C₄²	—	4	4	no	—
196	C₄²	—	4	2	no	—
197	C₄²	—	4	2	no	—
198	C₄²	—	8	2	no	—
199	C₄²	—	8	2	no	—
200	C₄²	—	8	2	no	—
201	C₄²	C₂²⋊C₈	1	2	yes	—
202	C₄²	#104	1	2	yes	—
203	C₄²	#121	1	1	yes	characteristic
204	C₄²	C₂²⋊Q₈	1	1	yes	characteristic
205	C₄²	AGL(1,ℤ/8)	1	1	yes	characteristic
206	C₄²	#137	1	1	yes	characteristic
207	C₁₆	—	4	4	no	—
208	C₂³	C₂×#102	1	1	yes	characteristic
209	Q₈	—	2	4	no	—
210	Q₈	—	4	2	no	—
211	Q₈	—	4	2	no	—
212	Q₈	—	4	2	no	—
213	C₂×C₄	—	2	1	no	—
214	C₂×C₄	—	2	1	no	—
215	C₂×C₄	—	2	1	no	—
216	C₂×C₄	—	2	1	no	—
217	C₂×C₄	—	2	1	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	—	4	2	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	—	8	1	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	D₄⋊C₈	1	2	yes	—
239	C₂×C₄	D₄⋊C₈	1	2	yes	—
240	C₂×C₄	#362	1	2	yes	—
241	C₂×C₄	#380	1	1	yes	characteristic
242	C₂×C₄	C₂×C₂²⋊C₈	1	1	yes	characteristic
243	C₂×C₄	#407	1	1	yes	characteristic
244	C₂×C₄	#417	1	1	yes	characteristic
245	C₂×C₄	C₂×#104	1	1	yes	characteristic
246	C₂×C₄	#421	1	1	yes	characteristic
247	C₂×C₄	#459	1	1	yes	characteristic
248	C₂×C₄	#461	1	1	yes	characteristic
249	C₂×C₄	#474	1	1	yes	characteristic
250	C₂×C₄	#476	1	1	yes	characteristic
251	C₈	—	2	1	no	—
252	C₈	—	2	1	no	—
253	C₈	—	2	2	no	—
254	C₈	—	4	1	no	—
255	C₈	—	16	2	no	—
256	C₈	C₂×C₄⋊C₈	1	1	yes	characteristic
257	C₈	#423	1	1	yes	characteristic
258	C₈	#431	1	1	yes	characteristic
259	C₈	#432	1	1	yes	characteristic
260	C₂²	C₂×D₄⋊C₈	1	1	yes	characteristic
261	C₂²	#1486	1	1	yes	characteristic
262	C₂²	#1802	1	1	yes	characteristic
263	C₂²	#2145	1	1	yes	characteristic
264	C₂²	#2145	1	1	yes	characteristic
265	C₂²	C₂×#362	1	1	yes	characteristic
266	C₂²	#2148	1	1	yes	characteristic
267	C₄	—	2	1	no	—
268	C₄	—	2	1	no	—
269	C₄	—	2	1	no	—
270	C₄	—	2	1	no	—
271	C₄	—	2	2	no	—
272	C₄	—	2	2	no	—
273	C₄	—	2	2	no	—
274	C₄	—	4	4	no	—
275	C₄	—	4	1	no	—
276	C₄	—	8	2	no	—
277	C₄	—	8	2	no	—
278	C₄	—	8	2	no	—
279	C₄	#1477	1	1	yes	characteristic
280	C₄	#1489	1	1	yes	characteristic
281	C₄	#1797	1	1	yes	characteristic
282	C₄	#1806	1	1	yes	characteristic
283	C₂	#8563	1	1	yes	characteristic
284	C₂	#11597	1	1	yes	characteristic
285	C₂	#11600	1	1	yes	characteristic
286	C₂	#11665	1	1	yes	characteristic
287	C₂	#11668	1	1	yes	characteristic
288	C₂	#12817	1	1	yes	characteristic
289	C₂	#12819	1	1	yes	characteristic

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

Maximal subgroups

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

subgroup	index
C₄²×Q₁₆	2
#12653	2
C₂×#1334	2
#8807	2
#7178	2