Subgroups — #191542

Order 512.

697 conjugacy classes of subgroups.

The subgroup lattice has 287 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20826	C₂	1	1	yes	maximal, characteristic
2	C₂×#1848	C₂	1	1	yes	maximal, characteristic
3	#17817	C₂	1	2	yes	maximal
4	#17796	C₂	1	2	yes	maximal
5	C₂×#1451	C₂	1	1	yes	maximal, characteristic
6	#8828	C₂	1	4	yes	maximal, complemented
7	#8826	C₂	1	2	yes	maximal, complemented
8	#8825	C₂	1	2	yes	maximal, complemented
9	#2920	C₂²	1	2	yes	—
10	C₂×#431	C₂²	1	1	yes	characteristic
11	C₂²×#108	—	2	2	no	—
12	C₂×#423	C₂²	1	2	yes	—
13	C₂×C₄×M₁₆	C₂²	1	1	yes	characteristic
14	C₂²×C₄×C₈	C₂²	1	1	yes	characteristic
15	#1848	—	2	2	no	—
16	#1848	C₂²	1	2	yes	—
17	#1848	C₂²	1	2	yes	—
18	#1848	C₂²	1	1	yes	characteristic
19	#1848	C₂²	1	1	yes	characteristic
20	#1837	C₂²	1	2	yes	—
21	#1810	—	2	2	no	—
22	C₄×#108	—	2	2	no	—
23	#1574	—	2	4	no	—
24	#1573	—	2	4	no	—
25	#1569	—	2	4	no	—
26	#1567	C₂²	1	4	yes	—
27	#1567	C₂²	1	4	yes	—
28	#1566	C₂²	1	4	yes	—
29	#1566	C₂²	1	4	yes	—
30	#1565	—	2	4	no	—
31	#1451	C₄	1	4	yes	—
32	#1451	C₄	1	4	yes	—
33	C₂×#322	C₂²	1	2	yes	—
34	C₂×#322	C₂²	1	2	yes	—
35	C₂²×M₁₆	—	2	2	no	—
36	C₂²×M₁₆	Q₈	1	2	yes	—
37	C₂³×C₈	D₄	1	2	yes	—
38	C₂²×C₄²	C₂³	1	1	yes	characteristic
39	#432	—	2	2	no	—
40	#432	—	2	2	no	—
41	#431	—	2	2	no	—
42	#431	—	2	2	no	—
43	#431	C₂×C₄	1	2	yes	—
44	#431	C₂×C₄	1	2	yes	—
45	#431	D₄	1	2	yes	—
46	#431	D₄	1	2	yes	—
47	C₂×#108	—	2	2	no	—
48	C₂×#108	—	2	2	no	—
49	C₂×#108	—	2	4	no	—
50	C₂×#108	—	2	4	no	—
51	C₂×#108	—	4	8	no	—
52	#423	—	2	2	no	—
53	#423	—	2	2	no	—
54	#423	—	2	2	no	—
55	#423	C₂³	1	2	yes	—
56	#423	C₂³	1	4	yes	—
57	#423	C₂³	1	2	yes	—
58	C₂×C₄⋊C₈	—	2	2	no	—
59	C₂×C₄⋊C₈	—	2	2	no	—
60	C₂×C₄⋊C₈	Q₈	1	2	yes	—
61	#407	—	2	2	no	—
62	C₂×C₂²⋊C₈	Q₈	1	2	yes	—
63	C₄×M₁₆	—	2	2	no	—
64	C₄×M₁₆	C₂×C₄	1	2	yes	—
65	C₄×M₁₆	C₂×C₄	1	2	yes	—
66	C₄×M₁₆	D₄	1	2	yes	—
67	C₄×M₁₆	D₄	1	2	yes	—
68	C₂×#97	Q₈	1	2	yes	—
69	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
70	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
71	C₂×C₄×C₈	D₄	1	2	yes	—
72	C₂×C₄×C₈	Q₈	1	2	yes	—
73	C₂×C₄×C₈	Q₈	1	2	yes	—
74	C₂×C₄×C₈	C₂³	1	2	yes	—
75	C₂×C₄×C₈	C₂³	1	2	yes	—
76	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
77	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
78	#341	—	2	4	no	—
79	#335	—	2	8	no	—
80	#335	—	2	8	no	—
81	#334	—	2	8	no	—
82	#334	—	2	8	no	—
83	#322	C₂×C₄	1	8	yes	—
84	#322	C₂×C₄	1	8	yes	—
85	C₂³×C₄	C₂×D₄	1	1	yes	characteristic
86	C₂³×C₄	C₂×Q₈	1	1	yes	characteristic
87	C₂³×C₄	C₂×Q₈	1	1	yes	characteristic
88	C₂×M₁₆	—	2	4	no	—
89	C₂×M₁₆	—	2	2	no	—
90	C₂×M₁₆	—	2	2	no	—
91	C₂×M₁₆	—	4	4	no	—
92	C₂×M₁₆	—	4	2	no	—
93	C₂×M₁₆	—	4	8	no	—
94	C₂×M₁₆	C₄⋊C₄	1	4	yes	—
95	C₂×M₁₆	C₄⋊C₄	1	4	yes	—
96	C₂²×C₈	—	2	4	no	—
97	C₂²×C₈	—	2	4	no	—
98	C₂²×C₈	—	4	2	no	—
99	C₂²×C₈	C₄⋊C₄	1	4	yes	—
100	C₂²×C₈	C₄⋊C₄	1	4	yes	—
101	C₂²×C₈	C₂×D₄	1	2	yes	—
102	C₂²×C₈	C₂×D₄	1	2	yes	—
103	C₂²×C₈	C₂×Q₈	1	4	yes	—
104	C₂²×C₈	C₂×Q₈	1	4	yes	—
105	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
106	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
107	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
108	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
109	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
110	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
111	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
112	C₂×C₄²	C₂×Q₈	1	1	yes	characteristic
113	C₂×C₄²	C₂×Q₈	1	1	yes	characteristic
114	C₂×C₄²	C₂×Q₈	1	1	yes	characteristic
115	C₂×C₄²	C₂×Q₈	1	1	yes	characteristic
116	C₂×C₄²	C₂⁴	1	1	yes	characteristic
117	#108	—	4	8	no	—
118	#108	—	4	4	no	—
119	#108	—	4	8	no	—
120	#108	—	4	4	no	—
121	C₄⋊C₈	—	2	2	no	—
122	C₄⋊C₈	—	2	2	no	—
123	C₄⋊C₈	—	2	4	no	—
124	C₄⋊C₈	—	2	4	no	—
125	C₄⋊C₈	—	2	2	no	—
126	C₄⋊C₈	—	2	2	no	—
127	C₄⋊C₈	—	4	2	no	—
128	C₄⋊C₈	—	4	2	no	—
129	C₄⋊C₈	C₄⋊C₄	1	4	yes	—
130	C₄⋊C₈	C₄⋊C₄	1	4	yes	—
131	C₂²⋊C₈	—	4	2	no	—
132	C₂²⋊C₈	—	4	2	no	—
133	C₂²⋊C₈	C₄⋊C₄	1	4	yes	—
134	C₂²⋊C₈	C₄⋊C₄	1	4	yes	—
135	#97	—	4	2	no	—
136	#97	C₄⋊C₄	1	4	yes	—
137	#97	C₄⋊C₄	1	4	yes	—
138	C₄×C₈	—	4	2	no	—
139	C₄×C₈	C₄⋊C₄	1	4	yes	—
140	C₄×C₈	C₄⋊C₄	1	4	yes	—
141	C₄×C₈	C₂²×C₄	1	2	yes	—
142	C₄×C₈	C₂²×C₄	1	2	yes	—
143	C₄×C₈	C₂²×C₄	1	4	yes	—
144	C₄×C₈	C₂²×C₄	1	4	yes	—
145	C₄×C₈	C₂×D₄	1	4	yes	—
146	C₄×C₈	C₂×D₄	1	4	yes	—
147	C₄×C₈	C₂×Q₈	1	2	yes	—
148	C₄×C₈	C₂×Q₈	1	2	yes	—
149	C₂⁴	C₄⋊Q₈	1	1	yes	characteristic
150	C₂²×C₄	—	2	2	no	—
151	C₂²×C₄	—	2	2	no	—
152	C₂²×C₄	—	2	4	no	—
153	C₂²×C₄	—	4	2	no	—
154	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
155	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
156	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
157	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
158	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
159	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
160	C₂²×C₄	#127	1	1	yes	characteristic
161	C₂²×C₄	#127	1	1	yes	characteristic
162	C₂²×C₄	C₄⋊Q₈	1	1	yes	characteristic
163	C₂²×C₄	C₄⋊Q₈	1	1	yes	characteristic
164	C₂²×C₄	C₄⋊Q₈	1	1	yes	characteristic
165	C₂²×C₄	C₄⋊Q₈	1	1	yes	characteristic
166	C₂²×C₄	C₄⋊Q₈	1	1	yes	characteristic
167	C₂²×C₄	C₄⋊Q₈	1	1	yes	characteristic
168	C₂²×C₄	C₄⋊Q₈	1	1	yes	characteristic
169	C₂²×C₄	C₂²×D₄	1	1	yes	characteristic
170	C₂²×C₄	C₂²×Q₈	1	1	yes	characteristic
171	C₂²×C₄	C₂²×Q₈	1	1	yes	characteristic
172	M₁₆	—	2	8	no	—
173	M₁₆	—	4	4	no	—
174	M₁₆	—	4	8	no	—
175	M₁₆	—	4	4	no	—
176	M₁₆	—	4	4	no	—
177	M₁₆	—	8	4	no	—
178	C₂×C₈	—	2	2	no	—
179	C₂×C₈	—	2	2	no	—
180	C₂×C₈	—	2	8	no	—
181	C₂×C₈	—	2	8	no	—
182	C₂×C₈	—	4	4	no	—
183	C₂×C₈	—	4	4	no	—
184	C₂×C₈	—	4	2	no	—
185	C₂×C₈	—	4	2	no	—
186	C₂×C₈	—	4	4	no	—
187	C₂×C₈	—	8	4	no	—
188	C₂×C₈	C₂×C₄⋊C₄	1	4	yes	—
189	C₂×C₈	C₂×C₄⋊C₄	1	8	yes	—
190	C₂×C₈	C₂×C₄⋊C₄	1	8	yes	—
191	C₂×C₈	C₂×C₄⋊C₄	1	4	yes	—
192	C₂×C₈	AGL(1,ℤ/8)	1	4	yes	—
193	C₂×C₈	#137	1	4	yes	—
194	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
195	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
196	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
197	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
198	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
199	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
200	C₄²	#127	1	1	yes	characteristic
201	C₄²	#127	1	1	yes	characteristic
202	C₄²	C₄⋊Q₈	1	1	yes	characteristic
203	C₄²	C₄⋊Q₈	1	1	yes	characteristic
204	C₄²	C₄⋊Q₈	1	1	yes	characteristic
205	C₄²	C₄⋊Q₈	1	1	yes	characteristic
206	C₄²	C₂³×C₄	1	1	yes	characteristic
207	C₄²	C₂²×D₄	1	1	yes	characteristic
208	C₄²	C₂²×D₄	1	1	yes	characteristic
209	C₄²	C₂²×Q₈	1	1	yes	characteristic
210	C₂³	—	2	2	no	—
211	C₂³	—	2	2	no	—
212	C₂³	—	4	1	no	—
213	C₂³	#384	1	1	yes	characteristic
214	C₂³	#384	1	1	yes	characteristic
215	C₂³	C₂×C₄⋊Q₈	1	1	yes	characteristic
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	2	2	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	2	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	4	1	no	—
232	C₂×C₄	—	4	1	no	—
233	C₂×C₄	—	4	1	no	—
234	C₂×C₄	#384	1	1	yes	characteristic
235	C₂×C₄	#384	1	1	yes	characteristic
236	C₂×C₄	#384	1	1	yes	characteristic
237	C₂×C₄	#384	1	1	yes	characteristic
238	C₂×C₄	#384	1	1	yes	characteristic
239	C₂×C₄	#384	1	1	yes	characteristic
240	C₂×C₄	#430	1	4	yes	—
241	C₂×C₄	C₂²×C₄⋊C₄	1	1	yes	characteristic
242	C₂×C₄	C₂²×C₄⋊C₄	1	1	yes	characteristic
243	C₂×C₄	C₂²×C₄⋊C₄	1	1	yes	characteristic
244	C₂×C₄	C₂×#127	1	1	yes	characteristic
245	C₂×C₄	C₂×C₄⋊Q₈	1	1	yes	characteristic
246	C₂×C₄	C₂×C₄⋊Q₈	1	1	yes	characteristic
247	C₂×C₄	C₂×AGL(1,ℤ/8)	1	2	yes	—
248	C₂×C₄	C₂×#137	1	2	yes	—
249	C₈	—	2	4	no	—
250	C₈	—	2	8	no	—
251	C₈	—	2	8	no	—
252	C₈	—	8	4	no	—
253	C₈	—	8	2	no	—
254	C₈	—	8	2	no	—
255	C₈	#428	1	8	yes	—
256	C₂²	—	2	1	no	—
257	C₂²	—	2	1	no	—
258	C₂²	—	2	1	no	—
259	C₂²	—	2	1	no	—
260	C₂²	—	2	1	no	—
261	C₂²	—	2	1	no	—
262	C₂²	—	2	1	no	—
263	C₂²	—	4	4	no	—
264	C₂²	#1849	1	2	yes	—
265	C₂²	C₂×#384	1	1	yes	characteristic
266	C₂²	#3165	1	2	yes	—
267	C₄	—	2	1	no	—
268	C₄	—	2	1	no	—
269	C₄	—	2	1	no	—
270	C₄	—	2	1	no	—
271	C₄	—	2	1	no	—
272	C₄	—	2	1	no	—
273	C₄	—	2	1	no	—
274	C₄	#1849	1	2	yes	—
275	C₄	C₂×#428	1	2	yes	—
276	C₄	C₂×#430	1	2	yes	—
277	C₄	#2916	1	2	yes	—
278	C₄	#3151	1	1	yes	characteristic
279	C₄	#3152	1	1	yes	characteristic
280	C₂	—	2	2	no	—
281	C₂	—	2	2	no	—
282	C₂	—	4	1	no	—
283	C₂	#20800	1	1	yes	characteristic
284	C₂	C₂×#1849	1	1	yes	characteristic
285	C₂	#20807	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×#127
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#384
prime radical	C₂⁴	C₄⋊Q₈
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
6. #8828	C₂	semidirect
7. #8826	C₂	semidirect
8. #8825	C₂	semidirect

Maximal subgroups

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

subgroup	index
#20826	2
C₂×#1848	2
#17817	2
#17796	2
C₂×#1451	2
#8828	2
#8826	2
#8825	2