Subgroups — #191540

Order 512.

717 conjugacy classes of subgroups.

The subgroup lattice has 297 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×C₈×M₁₆	C₂	1	1	yes	maximal, characteristic
6	#8828	C₂	1	4	yes	maximal, complemented
7	#8823	C₂	1	2	yes	maximal, complemented
8	#8822	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	C₈×M₁₆	C₄	1	2	yes	—
32	C₈×M₁₆	C₄	1	4	yes	—
33	C₈×M₁₆	C₄	1	2	yes	—
34	C₂×#322	C₂²	1	2	yes	—
35	C₂×C₈²	C₂²	1	2	yes	—
36	C₂²×M₁₆	—	2	2	no	—
37	C₂²×M₁₆	Q₈	1	2	yes	—
38	C₂³×C₈	D₄	1	2	yes	—
39	C₂²×C₄²	C₂³	1	1	yes	characteristic
40	#432	—	2	2	no	—
41	#432	—	2	2	no	—
42	#431	—	2	2	no	—
43	#431	—	2	2	no	—
44	#431	C₂×C₄	1	2	yes	—
45	#431	C₂×C₄	1	2	yes	—
46	#431	D₄	1	2	yes	—
47	#431	D₄	1	2	yes	—
48	C₂×#108	—	2	2	no	—
49	C₂×#108	—	2	2	no	—
50	C₂×#108	—	2	4	no	—
51	C₂×#108	—	2	4	no	—
52	C₂×#108	—	4	8	no	—
53	#423	—	2	2	no	—
54	#423	—	2	2	no	—
55	#423	—	2	2	no	—
56	#423	C₂³	1	2	yes	—
57	#423	C₂³	1	4	yes	—
58	#423	C₂³	1	2	yes	—
59	C₂×C₄⋊C₈	—	2	2	no	—
60	C₂×C₄⋊C₈	—	2	2	no	—
61	C₂×C₄⋊C₈	Q₈	1	2	yes	—
62	#407	—	2	2	no	—
63	C₂×C₂²⋊C₈	Q₈	1	2	yes	—
64	C₄×M₁₆	—	2	2	no	—
65	C₄×M₁₆	C₂×C₄	1	1	yes	characteristic
66	C₄×M₁₆	C₂×C₄	1	2	yes	—
67	C₄×M₁₆	C₂×C₄	1	1	yes	characteristic
68	C₄×M₁₆	D₄	1	2	yes	—
69	C₄×M₁₆	D₄	1	2	yes	—
70	C₂×#97	Q₈	1	2	yes	—
71	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
72	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
73	C₂×C₄×C₈	D₄	1	2	yes	—
74	C₂×C₄×C₈	Q₈	1	2	yes	—
75	C₂×C₄×C₈	Q₈	1	2	yes	—
76	C₂×C₄×C₈	C₂³	1	2	yes	—
77	C₂×C₄×C₈	C₂³	1	2	yes	—
78	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
79	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
80	#341	—	2	4	no	—
81	#335	—	2	8	no	—
82	#335	—	2	8	no	—
83	#334	—	2	8	no	—
84	#334	—	2	8	no	—
85	#322	C₂×C₄	1	4	yes	—
86	#322	C₂×C₄	1	4	yes	—
87	C₈²	C₂×C₄	1	4	yes	—
88	C₈²	C₂×C₄	1	4	yes	—
89	C₂³×C₄	C₂×D₄	1	1	yes	characteristic
90	C₂³×C₄	C₂×Q₈	1	1	yes	characteristic
91	C₂³×C₄	C₂×Q₈	1	1	yes	characteristic
92	C₂×M₁₆	—	2	8	no	—
93	C₂×M₁₆	—	2	4	no	—
94	C₂×M₁₆	—	2	2	no	—
95	C₂×M₁₆	—	2	2	no	—
96	C₂×M₁₆	—	4	2	no	—
97	C₂×M₁₆	—	4	8	no	—
98	C₂×M₁₆	C₄⋊C₄	1	2	yes	—
99	C₂×M₁₆	C₄⋊C₄	1	4	yes	—
100	C₂×M₁₆	C₄⋊C₄	1	2	yes	—
101	C₂²×C₈	—	2	4	no	—
102	C₂²×C₈	—	2	4	no	—
103	C₂²×C₈	—	4	2	no	—
104	C₂²×C₈	C₄⋊C₄	1	4	yes	—
105	C₂²×C₈	C₄⋊C₄	1	4	yes	—
106	C₂²×C₈	C₂×D₄	1	2	yes	—
107	C₂²×C₈	C₂×D₄	1	2	yes	—
108	C₂²×C₈	C₂×Q₈	1	4	yes	—
109	C₂²×C₈	C₂×Q₈	1	4	yes	—
110	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
111	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
112	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
113	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
114	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
115	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
116	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
117	C₂×C₄²	C₂×Q₈	1	1	yes	characteristic
118	C₂×C₄²	C₂×Q₈	1	1	yes	characteristic
119	C₂×C₄²	C₂×Q₈	1	1	yes	characteristic
120	C₂×C₄²	C₂×Q₈	1	1	yes	characteristic
121	C₂×C₄²	C₂⁴	1	1	yes	characteristic
122	#108	—	4	8	no	—
123	#108	—	4	4	no	—
124	#108	—	4	8	no	—
125	#108	—	4	4	no	—
126	C₄⋊C₈	—	2	2	no	—
127	C₄⋊C₈	—	2	2	no	—
128	C₄⋊C₈	—	2	4	no	—
129	C₄⋊C₈	—	2	4	no	—
130	C₄⋊C₈	—	2	2	no	—
131	C₄⋊C₈	—	2	2	no	—
132	C₄⋊C₈	—	4	2	no	—
133	C₄⋊C₈	—	4	2	no	—
134	C₄⋊C₈	C₄⋊C₄	1	4	yes	—
135	C₄⋊C₈	C₄⋊C₄	1	4	yes	—
136	C₂²⋊C₈	—	4	2	no	—
137	C₂²⋊C₈	—	4	2	no	—
138	C₂²⋊C₈	C₄⋊C₄	1	4	yes	—
139	C₂²⋊C₈	C₄⋊C₄	1	4	yes	—
140	#97	—	4	2	no	—
141	#97	C₄⋊C₄	1	2	yes	—
142	#97	C₄⋊C₄	1	4	yes	—
143	#97	C₄⋊C₄	1	2	yes	—
144	C₄×C₈	—	4	2	no	—
145	C₄×C₈	C₄⋊C₄	1	4	yes	—
146	C₄×C₈	C₄⋊C₄	1	4	yes	—
147	C₄×C₈	C₂²×C₄	1	2	yes	—
148	C₄×C₈	C₂²×C₄	1	2	yes	—
149	C₄×C₈	C₂²×C₄	1	2	yes	—
150	C₄×C₈	C₂²×C₄	1	4	yes	—
151	C₄×C₈	C₂²×C₄	1	2	yes	—
152	C₄×C₈	C₂×D₄	1	4	yes	—
153	C₄×C₈	C₂×D₄	1	4	yes	—
154	C₄×C₈	C₂×Q₈	1	2	yes	—
155	C₄×C₈	C₂×Q₈	1	2	yes	—
156	C₂⁴	C₄⋊Q₈	1	1	yes	characteristic
157	C₂²×C₄	—	2	2	no	—
158	C₂²×C₄	—	2	2	no	—
159	C₂²×C₄	—	2	4	no	—
160	C₂²×C₄	—	4	2	no	—
161	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
162	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
163	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
164	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
165	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
166	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
167	C₂²×C₄	#127	1	1	yes	characteristic
168	C₂²×C₄	#127	1	1	yes	characteristic
169	C₂²×C₄	C₄⋊Q₈	1	1	yes	characteristic
170	C₂²×C₄	C₄⋊Q₈	1	1	yes	characteristic
171	C₂²×C₄	C₄⋊Q₈	1	1	yes	characteristic
172	C₂²×C₄	C₄⋊Q₈	1	1	yes	characteristic
173	C₂²×C₄	C₄⋊Q₈	1	1	yes	characteristic
174	C₂²×C₄	C₄⋊Q₈	1	1	yes	characteristic
175	C₂²×C₄	C₄⋊Q₈	1	1	yes	characteristic
176	C₂²×C₄	C₂²×D₄	1	1	yes	characteristic
177	C₂²×C₄	C₂²×Q₈	1	1	yes	characteristic
178	C₂²×C₄	C₂²×Q₈	1	1	yes	characteristic
179	M₁₆	—	2	4	no	—
180	M₁₆	—	2	8	no	—
181	M₁₆	—	4	8	no	—
182	M₁₆	—	4	4	no	—
183	M₁₆	—	4	4	no	—
184	M₁₆	—	8	4	no	—
185	M₁₆	#108	1	8	yes	complemented
186	C₂×C₈	—	2	2	no	—
187	C₂×C₈	—	2	2	no	—
188	C₂×C₈	—	2	8	no	—
189	C₂×C₈	—	2	8	no	—
190	C₂×C₈	—	2	8	no	—
191	C₂×C₈	—	4	4	no	—
192	C₂×C₈	—	4	2	no	—
193	C₂×C₈	—	4	2	no	—
194	C₂×C₈	—	4	4	no	—
195	C₂×C₈	—	8	4	no	—
196	C₂×C₈	C₂×C₄⋊C₄	1	4	yes	—
197	C₂×C₈	C₂×C₄⋊C₄	1	4	yes	—
198	C₂×C₈	C₂×C₄⋊C₄	1	8	yes	—
199	C₂×C₈	C₂×C₄⋊C₄	1	4	yes	—
200	C₂×C₈	C₂×C₄⋊C₄	1	4	yes	—
201	C₂×C₈	AGL(1,ℤ/8)	1	4	yes	—
202	C₂×C₈	#137	1	4	yes	—
203	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
204	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
205	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
206	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
207	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
208	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
209	C₄²	#127	1	1	yes	characteristic
210	C₄²	#127	1	1	yes	characteristic
211	C₄²	C₄⋊Q₈	1	1	yes	characteristic
212	C₄²	C₄⋊Q₈	1	1	yes	characteristic
213	C₄²	C₄⋊Q₈	1	1	yes	characteristic
214	C₄²	C₄⋊Q₈	1	1	yes	characteristic
215	C₄²	C₂³×C₄	1	1	yes	characteristic
216	C₄²	C₂²×D₄	1	1	yes	characteristic
217	C₄²	C₂²×D₄	1	1	yes	characteristic
218	C₄²	C₂²×Q₈	1	1	yes	characteristic
219	C₂³	—	2	2	no	—
220	C₂³	—	2	2	no	—
221	C₂³	—	4	1	no	—
222	C₂³	#384	1	1	yes	characteristic
223	C₂³	#384	1	1	yes	characteristic
224	C₂³	C₂×C₄⋊Q₈	1	1	yes	characteristic
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	1	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	4	1	no	—
241	C₂×C₄	—	4	1	no	—
242	C₂×C₄	—	4	1	no	—
243	C₂×C₄	#384	1	1	yes	characteristic
244	C₂×C₄	#384	1	1	yes	characteristic
245	C₂×C₄	#384	1	1	yes	characteristic
246	C₂×C₄	#384	1	1	yes	characteristic
247	C₂×C₄	#384	1	1	yes	characteristic
248	C₂×C₄	#384	1	1	yes	characteristic
249	C₂×C₄	C₂×#108	1	4	yes	—
250	C₂×C₄	C₂²×C₄⋊C₄	1	1	yes	characteristic
251	C₂×C₄	C₂²×C₄⋊C₄	1	1	yes	characteristic
252	C₂×C₄	C₂²×C₄⋊C₄	1	1	yes	characteristic
253	C₂×C₄	C₂×#127	1	1	yes	characteristic
254	C₂×C₄	C₂×C₄⋊Q₈	1	1	yes	characteristic
255	C₂×C₄	C₂×C₄⋊Q₈	1	1	yes	characteristic
256	C₂×C₄	C₂×AGL(1,ℤ/8)	1	2	yes	—
257	C₂×C₄	C₂×#137	1	2	yes	—
258	C₈	—	2	4	no	—
259	C₈	—	2	4	no	—
260	C₈	—	2	8	no	—
261	C₈	—	8	4	no	—
262	C₈	—	8	2	no	—
263	C₈	—	8	2	no	—
264	C₈	#428	1	8	yes	—
265	C₈	C₂×#108	1	8	yes	complemented
266	C₂²	—	2	1	no	—
267	C₂²	—	2	1	no	—
268	C₂²	—	2	1	no	—
269	C₂²	—	2	1	no	—
270	C₂²	—	2	1	no	—
271	C₂²	—	2	1	no	—
272	C₂²	—	2	1	no	—
273	C₂²	—	4	4	no	—
274	C₂²	#1848	1	2	yes	—
275	C₂²	C₂×#384	1	1	yes	characteristic
276	C₂²	#3165	1	2	yes	—
277	C₄	—	2	1	no	—
278	C₄	—	2	1	no	—
279	C₄	—	2	1	no	—
280	C₄	—	2	1	no	—
281	C₄	—	2	1	no	—
282	C₄	—	2	1	no	—
283	C₄	—	2	1	no	—
284	C₄	#1848	1	2	yes	—
285	C₄	C₂×#428	1	2	yes	—
286	C₄	C₂²×#108	1	2	yes	—
287	C₄	#2916	1	2	yes	—
288	C₄	#3151	1	1	yes	characteristic
289	C₄	#3152	1	1	yes	characteristic
290	C₂	—	2	2	no	—
291	C₂	—	2	2	no	—
292	C₂	—	4	1	no	—
293	C₂	#20800	1	1	yes	characteristic
294	C₂	C₂×#1848	1	1	yes	characteristic
295	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. #8823	C₂	semidirect
8. #8822	C₂	semidirect
185. M₁₆	#108	semidirect
265. C₈	C₂×#108	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×C₈×M₁₆	2
#8828	2
#8823	2
#8822	2