Subgroups — #191543

Order 512.

653 conjugacy classes of subgroups.

The subgroup lattice has 275 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×#1837	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	#8827	C₂	1	2	yes	maximal, complemented
8	#8827	C₂	1	2	yes	maximal, complemented
9	#2920	C₂²	1	2	yes	—
10	C₂×#431	C₂²	1	1	yes	characteristic
11	C₂×#423	C₂²	1	2	yes	—
12	C₂×C₄×M₁₆	C₂²	1	1	yes	characteristic
13	C₂²×C₄×C₈	C₂²	1	1	yes	characteristic
14	#1848	C₂²	1	2	yes	—
15	#1837	—	2	2	no	—
16	#1837	C₂²	1	2	yes	—
17	#1837	C₂²	1	2	yes	—
18	#1837	C₂²	1	1	yes	characteristic
19	#1837	C₂²	1	1	yes	characteristic
20	#1810	—	2	2	no	—
21	C₄×#108	—	2	2	no	—
22	C₂×#341	—	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	4	no	—
48	#423	—	2	2	no	—
49	#423	—	2	2	no	—
50	#423	—	2	2	no	—
51	#423	C₂³	1	4	yes	—
52	#423	C₂³	1	2	yes	—
53	#423	C₂³	1	2	yes	—
54	C₂×C₄⋊C₈	—	2	2	no	—
55	C₂×C₄⋊C₈	—	2	2	no	—
56	C₂×C₄⋊C₈	Q₈	1	2	yes	—
57	#407	—	2	2	no	—
58	C₂×C₂²⋊C₈	Q₈	1	2	yes	—
59	C₄×M₁₆	—	2	2	no	—
60	C₄×M₁₆	C₂×C₄	1	2	yes	—
61	C₄×M₁₆	C₂×C₄	1	2	yes	—
62	C₄×M₁₆	D₄	1	2	yes	—
63	C₄×M₁₆	D₄	1	2	yes	—
64	C₂×#97	Q₈	1	2	yes	—
65	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
66	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
67	C₂×C₄×C₈	D₄	1	2	yes	—
68	C₂×C₄×C₈	Q₈	1	2	yes	—
69	C₂×C₄×C₈	Q₈	1	2	yes	—
70	C₂×C₄×C₈	C₂³	1	2	yes	—
71	C₂×C₄×C₈	C₂³	1	2	yes	—
72	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
73	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
74	#341	—	2	2	no	—
75	#341	—	2	2	no	—
76	#341	—	2	4	no	—
77	#341	—	2	4	no	—
78	#335	—	2	8	no	—
79	#335	—	2	8	no	—
80	#334	—	2	8	no	—
81	#334	—	2	8	no	—
82	#322	C₂×C₄	1	8	yes	—
83	#322	C₂×C₄	1	8	yes	—
84	C₂³×C₄	C₂×D₄	1	1	yes	characteristic
85	C₂³×C₄	C₂×Q₈	1	1	yes	characteristic
86	C₂³×C₄	C₂×Q₈	1	1	yes	characteristic
87	C₂×M₁₆	—	2	4	no	—
88	C₂×M₁₆	—	2	2	no	—
89	C₂×M₁₆	—	2	2	no	—
90	C₂×M₁₆	—	4	4	no	—
91	C₂×M₁₆	—	4	2	no	—
92	C₂×M₁₆	—	4	8	no	—
93	C₂×M₁₆	C₄⋊C₄	1	4	yes	—
94	C₂×M₁₆	C₄⋊C₄	1	4	yes	—
95	C₂²×C₈	—	4	4	no	—
96	C₂²×C₈	—	4	2	no	—
97	C₂²×C₈	C₄⋊C₄	1	4	yes	—
98	C₂²×C₈	C₄⋊C₄	1	4	yes	—
99	C₂²×C₈	C₂×D₄	1	2	yes	—
100	C₂²×C₈	C₂×D₄	1	2	yes	—
101	C₂²×C₈	C₂×Q₈	1	4	yes	—
102	C₂²×C₈	C₂×Q₈	1	4	yes	—
103	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
104	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
105	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
106	C₂×C₄²	C₂×D₄	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×Q₈	1	1	yes	characteristic
111	C₂×C₄²	C₂×Q₈	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	C₂⁴	1	1	yes	characteristic
115	#108	—	4	8	no	—
116	C₄⋊C₈	—	2	2	no	—
117	C₄⋊C₈	—	2	2	no	—
118	C₄⋊C₈	—	2	4	no	—
119	C₄⋊C₈	—	2	4	no	—
120	C₄⋊C₈	—	2	2	no	—
121	C₄⋊C₈	—	2	2	no	—
122	C₄⋊C₈	—	4	2	no	—
123	C₄⋊C₈	—	4	2	no	—
124	C₄⋊C₈	C₄⋊C₄	1	4	yes	—
125	C₄⋊C₈	C₄⋊C₄	1	4	yes	—
126	C₂²⋊C₈	—	4	2	no	—
127	C₂²⋊C₈	—	4	2	no	—
128	C₂²⋊C₈	C₄⋊C₄	1	4	yes	—
129	C₂²⋊C₈	C₄⋊C₄	1	4	yes	—
130	#97	—	4	2	no	—
131	#97	C₄⋊C₄	1	4	yes	—
132	#97	C₄⋊C₄	1	4	yes	—
133	C₄×C₈	—	4	2	no	—
134	C₄×C₈	C₄⋊C₄	1	4	yes	—
135	C₄×C₈	C₄⋊C₄	1	4	yes	—
136	C₄×C₈	C₂²×C₄	1	2	yes	—
137	C₄×C₈	C₂²×C₄	1	2	yes	—
138	C₄×C₈	C₂²×C₄	1	4	yes	—
139	C₄×C₈	C₂²×C₄	1	4	yes	—
140	C₄×C₈	C₂×D₄	1	4	yes	—
141	C₄×C₈	C₂×D₄	1	4	yes	—
142	C₄×C₈	C₂×Q₈	1	2	yes	—
143	C₄×C₈	C₂×Q₈	1	2	yes	—
144	C₂⁴	C₄⋊Q₈	1	1	yes	characteristic
145	C₂²×C₄	—	2	4	no	—
146	C₂²×C₄	—	2	4	no	—
147	C₂²×C₄	—	4	2	no	—
148	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
149	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
150	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
151	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
152	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
153	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
154	C₂²×C₄	#127	1	1	yes	characteristic
155	C₂²×C₄	#127	1	1	yes	characteristic
156	C₂²×C₄	C₄⋊Q₈	1	1	yes	characteristic
157	C₂²×C₄	C₄⋊Q₈	1	1	yes	characteristic
158	C₂²×C₄	C₄⋊Q₈	1	1	yes	characteristic
159	C₂²×C₄	C₄⋊Q₈	1	1	yes	characteristic
160	C₂²×C₄	C₄⋊Q₈	1	1	yes	characteristic
161	C₂²×C₄	C₄⋊Q₈	1	1	yes	characteristic
162	C₂²×C₄	C₄⋊Q₈	1	1	yes	characteristic
163	C₂²×C₄	C₂²×D₄	1	1	yes	characteristic
164	C₂²×C₄	C₂²×Q₈	1	1	yes	characteristic
165	C₂²×C₄	C₂²×Q₈	1	1	yes	characteristic
166	M₁₆	—	2	8	no	—
167	M₁₆	—	4	4	no	—
168	M₁₆	—	4	4	no	—
169	M₁₆	—	4	4	no	—
170	M₁₆	—	4	8	no	—
171	M₁₆	—	8	4	no	—
172	C₂×C₈	—	2	4	no	—
173	C₂×C₈	—	2	4	no	—
174	C₂×C₈	—	4	4	no	—
175	C₂×C₈	—	4	4	no	—
176	C₂×C₈	—	4	4	no	—
177	C₂×C₈	—	4	4	no	—
178	C₂×C₈	—	4	2	no	—
179	C₂×C₈	—	4	2	no	—
180	C₂×C₈	—	4	4	no	—
181	C₂×C₈	—	8	4	no	—
182	C₂×C₈	C₂×C₄⋊C₄	1	4	yes	—
183	C₂×C₈	C₂×C₄⋊C₄	1	4	yes	—
184	C₂×C₈	C₂×C₄⋊C₄	1	8	yes	—
185	C₂×C₈	C₂×C₄⋊C₄	1	8	yes	—
186	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
187	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
188	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
189	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
190	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
191	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
192	C₄²	#127	1	1	yes	characteristic
193	C₄²	#127	1	1	yes	characteristic
194	C₄²	C₄⋊Q₈	1	1	yes	characteristic
195	C₄²	C₄⋊Q₈	1	1	yes	characteristic
196	C₄²	C₄⋊Q₈	1	1	yes	characteristic
197	C₄²	C₄⋊Q₈	1	1	yes	characteristic
198	C₄²	C₂³×C₄	1	1	yes	characteristic
199	C₄²	C₂²×D₄	1	1	yes	characteristic
200	C₄²	C₂²×D₄	1	1	yes	characteristic
201	C₄²	C₂²×Q₈	1	1	yes	characteristic
202	C₂³	—	2	2	no	—
203	C₂³	—	2	2	no	—
204	C₂³	—	4	1	no	—
205	C₂³	#384	1	1	yes	characteristic
206	C₂³	#384	1	1	yes	characteristic
207	C₂³	C₂×C₄⋊Q₈	1	1	yes	characteristic
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	2	2	no	—
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	2	no	—
221	C₂×C₄	—	4	1	no	—
222	C₂×C₄	—	4	1	no	—
223	C₂×C₄	—	4	1	no	—
224	C₂×C₄	#384	1	1	yes	characteristic
225	C₂×C₄	#384	1	1	yes	characteristic
226	C₂×C₄	#384	1	1	yes	characteristic
227	C₂×C₄	#384	1	1	yes	characteristic
228	C₂×C₄	#384	1	1	yes	characteristic
229	C₂×C₄	#384	1	1	yes	characteristic
230	C₂×C₄	#430	1	4	yes	—
231	C₂×C₄	C₂²×C₄⋊C₄	1	1	yes	characteristic
232	C₂×C₄	C₂²×C₄⋊C₄	1	1	yes	characteristic
233	C₂×C₄	C₂²×C₄⋊C₄	1	1	yes	characteristic
234	C₂×C₄	C₂×#127	1	1	yes	characteristic
235	C₂×C₄	C₂×C₄⋊Q₈	1	1	yes	characteristic
236	C₂×C₄	C₂×C₄⋊Q₈	1	1	yes	characteristic
237	C₂×C₄	#575	1	4	yes	—
238	C₈	—	2	4	no	—
239	C₈	—	2	4	no	—
240	C₈	—	2	8	no	—
241	C₈	—	2	8	no	—
242	C₈	—	8	4	no	—
243	C₈	—	8	2	no	—
244	C₈	—	8	2	no	—
245	C₂²	—	2	1	no	—
246	C₂²	—	2	1	no	—
247	C₂²	—	2	1	no	—
248	C₂²	—	2	1	no	—
249	C₂²	—	2	1	no	—
250	C₂²	—	2	1	no	—
251	C₂²	—	2	1	no	—
252	C₂²	—	4	4	no	—
253	C₂²	#1849	1	2	yes	—
254	C₂²	C₂×#384	1	1	yes	characteristic
255	C₂²	#3166	1	2	yes	—
256	C₄	—	2	1	no	—
257	C₄	—	2	1	no	—
258	C₄	—	2	1	no	—
259	C₄	—	2	1	no	—
260	C₄	—	2	1	no	—
261	C₄	—	2	1	no	—
262	C₄	—	2	1	no	—
263	C₄	#1849	1	2	yes	—
264	C₄	#2911	1	2	yes	—
265	C₄	C₂×#430	1	2	yes	—
266	C₄	#2916	1	2	yes	—
267	C₄	#3153	1	2	yes	—
268	C₂	—	2	2	no	—
269	C₂	—	2	2	no	—
270	C₂	—	4	1	no	—
271	C₂	#20801	1	1	yes	characteristic
272	C₂	C₂×#1849	1	1	yes	characteristic
273	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. #8827	C₂	semidirect
8. #8827	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×#1837	2
#17817	2
#17796	2
C₂×#1451	2
#8828	2
#8827	2