Subgroups — #497951

Order 512.

466 conjugacy classes of subgroups.

The subgroup lattice has 305 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×M₃₂	C₂	1	1	yes	maximal, characteristic
2	Q₈×#108	C₂	1	1	yes	maximal, characteristic
3	#22725	C₂	1	1	yes	maximal, characteristic
4	#13185	C₂	1	1	yes	maximal, characteristic
5	#13185	C₂	1	2	yes	maximal
6	#13184	C₂	1	2	yes	maximal
7	#13183	C₂	1	1	yes	maximal, characteristic
8	#12887	C₂	1	2	yes	maximal
9	#12886	C₂	1	1	yes	maximal, characteristic
10	C₄×#365	C₂	1	1	yes	maximal, characteristic
11	C₄×#365	C₂	1	2	yes	maximal
12	Q₈×M₁₆	—	2	1	no	—
13	C₂×C₈×Q₈	C₂²	1	1	yes	characteristic
14	#2355	—	2	1	no	—
15	#2183	C₄	1	2	yes	complemented
16	#2183	C₄	1	4	yes	complemented
17	Q₈×C₁₆	C₄	1	2	yes	complemented
18	C₂×#365	C₂²	1	2	yes	—
19	C₂×#365	C₂²	1	1	yes	characteristic
20	C₂×#365	C₂²	1	2	yes	—
21	C₂×#365	C₂²	1	1	yes	characteristic
22	#2150	C₂²	1	1	yes	characteristic
23	#2150	C₂²	1	2	yes	—
24	C₄×M₃₂	C₂²	1	1	yes	characteristic
25	C₄×M₃₂	C₂²	1	2	yes	—
26	#1950	—	2	1	no	—
27	#1950	—	2	2	no	—
28	#1949	—	2	2	no	—
29	#1947	—	2	1	no	—
30	#1848	C₂²	1	1	yes	characteristic
31	#1848	C₂²	1	2	yes	—
32	#1846	C₂²	1	2	yes	—
33	#1844	C₂²	1	1	yes	characteristic
34	C₄×#108	C₂²	1	1	yes	characteristic
35	C₄×#108	C₂²	1	2	yes	—
36	C₄×#106	C₂²	1	2	yes	—
37	C₄×#106	C₂²	1	1	yes	characteristic
38	#1383	C₂²	1	2	yes	—
39	#1383	C₂²	1	2	yes	—
40	#1383	C₂²	1	1	yes	characteristic
41	#1383	C₂²	1	2	yes	—
42	#1383	C₂²	1	2	yes	—
43	#1383	C₂²	1	1	yes	characteristic
44	C₂×C₄×Q₈	D₄	1	1	yes	characteristic
45	C₂×M₃₂	C₂³	1	1	yes	characteristic
46	C₂×M₃₂	C₂³	1	1	yes	characteristic
47	C₂×M₃₂	C₂³	1	2	yes	—
48	#446	—	4	1	no	—
49	#446	—	4	2	no	—
50	C₈×Q₈	—	4	1	no	—
51	C₈×Q₈	C₂×C₄	1	1	yes	characteristic
52	C₈×Q₈	C₂×C₄	1	1	yes	characteristic
53	C₈×Q₈	C₂×C₄	1	2	yes	—
54	C₈×Q₈	Q₈	1	1	yes	characteristic
55	C₈×Q₈	Q₈	1	2	yes	—
56	C₈×Q₈	Q₈	1	1	yes	characteristic
57	C₂×#108	—	2	1	no	—
58	C₂×#108	—	2	2	no	—
59	C₂×#108	C₂³	1	2	yes	—
60	C₂×#108	C₂³	1	1	yes	characteristic
61	C₂×#108	C₂³	1	1	yes	characteristic
62	C₂×#107	—	2	2	no	—
63	C₂×#106	—	2	1	no	—
64	C₂×#106	C₂³	1	2	yes	—
65	C₂×#106	C₂³	1	1	yes	characteristic
66	C₂×#106	C₂³	1	1	yes	characteristic
67	#423	—	2	2	no	—
68	#423	—	2	1	no	—
69	C₂×C₄⋊C₈	D₄	1	1	yes	characteristic
70	C₂×C₄⋊C₈	D₄	1	2	yes	—
71	C₄×M₁₆	—	2	2	no	—
72	C₄×M₁₆	—	2	1	no	—
73	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
74	C₂×C₄×C₈	C₂³	1	2	yes	—
75	#391	—	4	1	no	—
76	#387	—	4	2	no	—
77	#384	—	2	1	no	—
78	#382	—	2	2	no	—
79	C₄×C₄⋊C₄	—	2	1	no	—
80	C₄×C₄⋊C₄	—	2	2	no	—
81	C₄×C₄⋊C₄	—	4	1	no	—
82	#365	—	2	4	no	—
83	#365	—	2	2	no	—
84	#365	—	2	4	no	—
85	#365	—	2	2	no	—
86	C₄⋊C₁₆	C₂×C₄	1	2	yes	—
87	C₄⋊C₁₆	C₂×C₄	1	4	yes	—
88	C₄⋊C₁₆	C₂×C₄	1	2	yes	—
89	C₄⋊C₁₆	C₂×C₄	1	4	yes	—
90	#346	C₂×C₄	1	2	yes	—
91	#346	C₂×C₄	1	4	yes	—
92	C₄×C₁₆	C₂×C₄	1	2	yes	—
93	C₄×C₁₆	C₂×C₄	1	4	yes	—
94	#341	—	2	1	no	—
95	#341	—	2	2	no	—
96	#340	—	2	2	no	—
97	#340	—	2	1	no	—
98	C₂²×Q₈	SD₁₆	1	1	yes	characteristic
99	C₂×M₁₆	—	2	1	no	—
100	C₂×M₁₆	—	2	1	no	—
101	C₂×M₁₆	—	2	2	no	—
102	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
103	C₂²×C₈	C₂×D₄	1	2	yes	—
104	C₂²×C₈	C₂⁴	1	1	yes	characteristic
105	C₄×Q₈	C₄⋊C₄	1	1	yes	characteristic
106	C₄×Q₈	C₄⋊C₄	1	1	yes	characteristic
107	C₄×Q₈	C₄⋊C₄	1	2	yes	—
108	C₄×Q₈	SD₁₆	1	1	yes	characteristic
109	C₄×Q₈	SD₁₆	1	1	yes	characteristic
110	C₄×Q₈	SD₁₆	1	2	yes	—
111	C₂×C₄⋊C₄	—	2	1	no	—
112	C₂×C₄⋊C₄	—	2	1	no	—
113	C₂×C₄⋊C₄	—	2	2	no	—
114	C₂×C₄⋊C₄	—	4	1	no	—
115	C₂×C₄⋊C₄	—	4	1	no	—
116	C₂×C₄⋊C₄	SD₁₆	1	1	yes	characteristic
117	C₂×C₄⋊C₄	SD₁₆	1	2	yes	—
118	C₂×C₄²	—	4	1	no	—
119	C₂×C₄²	—	4	2	no	—
120	C₂×C₄²	C₂×D₄	1	2	yes	—
121	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
122	M₃₂	—	2	2	no	—
123	M₃₂	—	2	1	no	—
124	M₃₂	—	2	1	no	—
125	M₃₂	—	2	2	no	—
126	M₃₂	—	2	1	no	—
127	M₃₂	—	2	1	no	—
128	C₂×C₁₆	C₂²×C₄	1	2	yes	—
129	C₂×C₁₆	C₂²×C₄	1	2	yes	—
130	C₂×C₁₆	C₂²×C₄	1	4	yes	—
131	#108	—	2	2	no	—
132	#108	—	2	1	no	—
133	#108	—	2	1	no	—
134	#108	—	2	2	no	—
135	#108	—	2	1	no	—
136	#108	—	8	1	no	—
137	#108	—	8	2	no	—
138	#108	#41	1	2	yes	—
139	#107	—	8	2	no	—
140	#106	—	2	1	no	—
141	#106	—	2	2	no	—
142	#106	—	2	1	no	—
143	#106	—	2	2	no	—
144	#106	—	2	1	no	—
145	#106	—	8	1	no	—
146	#106	#41	1	2	yes	—
147	C₄⋊C₈	—	4	2	no	—
148	C₄⋊C₈	—	4	1	no	—
149	C₄⋊C₈	—	4	2	no	—
150	C₄⋊C₈	—	4	1	no	—
151	C₄⋊C₈	C₄⋊C₄	1	1	yes	characteristic
152	C₄⋊C₈	C₄⋊C₄	1	2	yes	—
153	C₄⋊C₈	C₄⋊C₄	1	1	yes	characteristic
154	C₄⋊C₈	C₄⋊C₄	1	2	yes	—
155	C₄⋊C₈	C₄⋊C₄	1	2	yes	—
156	C₄⋊C₈	C₄⋊C₄	1	2	yes	—
157	C₄⋊C₈	C₄⋊C₄	1	2	yes	—
158	#97	—	4	2	no	—
159	#97	—	4	1	no	—
160	C₄×C₈	—	4	2	no	—
161	C₄×C₈	—	4	1	no	—
162	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
163	C₄×C₈	C₂²×C₄	1	2	yes	—
164	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
165	C₄×C₈	C₂²×C₄	1	2	yes	—
166	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
167	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
168	C₄×C₈	C₂×Q₈	1	2	yes	—
169	C₄×C₈	C₂×Q₈	1	2	yes	—
170	#95	—	4	1	no	—
171	#95	—	4	2	no	—
172	#95	—	4	2	no	—
173	#95	—	4	2	no	—
174	C₂×Q₈	—	4	2	no	—
175	C₂×Q₈	#106	1	1	yes	characteristic
176	C₂×Q₈	#106	1	1	yes	characteristic
177	C₂×Q₈	#106	1	2	yes	—
178	C₂²×C₄	—	4	1	no	—
179	C₂²×C₄	—	4	1	no	—
180	C₂²×C₄	—	4	2	no	—
181	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
182	C₂²×C₄	C₂×SD₁₆	1	2	yes	—
183	C₂²×C₄	C₂×SD₁₆	1	2	yes	—
184	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
185	C₂²×C₄	C₂²×D₄	1	1	yes	characteristic
186	M₁₆	—	2	2	no	—
187	M₁₆	—	4	1	no	—
188	M₁₆	—	4	2	no	—
189	M₁₆	—	4	1	no	—
190	M₁₆	—	4	2	no	—
191	M₁₆	—	4	1	no	—
192	C₂×C₈	—	2	1	no	—
193	C₂×C₈	—	2	2	no	—
194	C₂×C₈	—	2	1	no	—
195	C₂×C₈	—	2	2	no	—
196	C₂×C₈	—	4	1	no	—
197	C₂×C₈	—	4	2	no	—
198	C₂×C₈	—	4	1	no	—
199	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
200	C₂×C₈	C₂×C₄⋊C₄	1	2	yes	—
201	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
202	C₂×C₈	C₂×C₄⋊C₄	1	2	yes	—
203	C₂×C₈	C₂³×C₄	1	1	yes	characteristic
204	C₂×C₈	C₂²×Q₈	1	1	yes	characteristic
205	C₂×C₈	C₂×#41	1	1	yes	characteristic
206	C₂×C₈	C₂×#41	1	1	yes	characteristic
207	C₂×C₈	2₋¹⁺⁴	1	1	yes	characteristic
208	C₂×C₈	2₋¹⁺⁴	1	1	yes	characteristic
209	C₄⋊C₄	—	2	2	no	—
210	C₄⋊C₄	—	4	1	no	—
211	C₄⋊C₄	—	4	1	no	—
212	C₄⋊C₄	—	4	2	no	—
213	C₄⋊C₄	—	4	2	no	—
214	C₄⋊C₄	—	4	1	no	—
215	C₄⋊C₄	—	8	2	no	—
216	C₄⋊C₄	—	16	1	no	—
217	C₄⋊C₄	#106	1	2	yes	—
218	C₄⋊C₄	#106	1	2	yes	—
219	C₄⋊C₄	#106	1	2	yes	—
220	C₄⋊C₄	#106	1	2	yes	—
221	C₄⋊C₄	#106	1	2	yes	—
222	C₄⋊C₄	#106	1	1	yes	characteristic
223	C₄⋊C₄	#106	1	1	yes	characteristic
224	C₄²	—	8	2	no	—
225	C₄²	—	16	2	no	—
226	C₄²	C₂×C₄⋊C₄	1	2	yes	—
227	C₄²	C₂×C₄⋊C₄	1	2	yes	—
228	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
229	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
230	C₄²	C₂×SD₁₆	1	2	yes	—
231	C₄²	C₂×SD₁₆	1	1	yes	characteristic
232	C₄²	C₂×SD₁₆	1	2	yes	—
233	C₄²	C₂×SD₁₆	1	1	yes	characteristic
234	C₁₆	—	2	2	no	—
235	C₁₆	—	2	4	no	—
236	C₁₆	—	2	2	no	—
237	C₂³	C₂²×SD₁₆	1	1	yes	characteristic
238	Q₈	—	2	2	no	—
239	Q₈	—	2	2	no	—
240	Q₈	—	4	2	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	4	2	no	—
243	C₂×C₄	—	4	1	no	—
244	C₂×C₄	—	4	2	no	—
245	C₂×C₄	—	4	2	no	—
246	C₂×C₄	—	8	1	no	—
247	C₂×C₄	—	8	1	no	—
248	C₂×C₄	—	8	1	no	—
249	C₂×C₄	—	8	2	no	—
250	C₂×C₄	—	8	2	no	—
251	C₂×C₄	—	8	1	no	—
252	C₂×C₄	—	8	2	no	—
253	C₂×C₄	—	8	1	no	—
254	C₂×C₄	C₂×#106	1	2	yes	—
255	C₂×C₄	C₂×#106	1	1	yes	characteristic
256	C₂×C₄	C₂×#106	1	2	yes	—
257	C₂×C₄	C₂×#106	1	1	yes	characteristic
258	C₂×C₄	C₂×#106	1	1	yes	characteristic
259	C₂×C₄	C₂×#106	1	2	yes	—
260	C₂×C₄	C₂×#106	1	2	yes	—
261	C₂×C₄	C₂×#106	1	1	yes	characteristic
262	C₂×C₄	C₂²×C₄⋊C₄	1	1	yes	characteristic
263	C₂×C₄	#548	1	1	yes	characteristic
264	C₂×C₄	#548	1	1	yes	characteristic
265	C₂×C₄	C₂²×SD₁₆	1	1	yes	characteristic
266	C₂×C₄	#576	1	1	yes	characteristic
267	C₂×C₄	#578	1	1	yes	characteristic
268	C₈	—	2	1	no	—
269	C₈	—	2	1	no	—
270	C₈	—	2	2	no	—
271	C₈	—	2	2	no	—
272	C₈	—	4	2	no	—
273	C₈	—	8	2	no	—
274	C₈	—	8	1	no	—
275	C₈	C₄×#41	1	1	yes	characteristic
276	C₈	#519	1	1	yes	characteristic
277	C₈	#557	1	1	yes	characteristic
278	C₈	#557	1	1	yes	characteristic
279	C₂²	—	2	1	no	—
280	C₂²	—	2	1	no	—
281	C₂²	C₂²×#106	1	1	yes	characteristic
282	C₂²	#3359	1	1	yes	characteristic
283	C₂²	#3362	1	1	yes	characteristic
284	C₄	—	2	1	no	—
285	C₄	—	2	2	no	—
286	C₄	—	2	2	no	—
287	C₄	—	2	2	no	—
288	C₄	—	2	2	no	—
289	C₄	—	2	1	no	—
290	C₄	—	2	1	no	—
291	C₄	—	8	2	no	—
292	C₄	—	16	1	no	—
293	C₄	—	16	2	no	—
294	C₄	#2155	1	2	yes	—
295	C₄	#2351	1	1	yes	characteristic
296	C₄	#2913	1	1	yes	characteristic
297	C₄	#3359	1	1	yes	characteristic
298	C₄	#3362	1	1	yes	characteristic
299	C₂	—	2	1	no	—
300	C₂	—	2	1	no	—
301	C₂	#22738	1	1	yes	characteristic
302	C₂	C₂×#2155	1	1	yes	characteristic
303	C₂	#33404	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂²×#106
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂³	C₂²×SD₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×Q₈ ◁ C₂×C₈×Q₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
15. #2183	C₄	semidirect
16. #2183	C₄	semidirect
17. Q₈×C₁₆	C₄	semidirect

Maximal subgroups

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

subgroup	index
Q₈×M₃₂	2
Q₈×#108	2
#22725	2
#13185	2
#13184	2
#13183	2
#12887	2
#12886	2
C₄×#365	2