Subgroups — #109428

Order 512.

686 conjugacy classes of subgroups.

The subgroup lattice has 298 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20797	C₂	1	1	yes	maximal, characteristic
2	C₂×#1846	C₂	1	1	yes	maximal, characteristic
3	C₂×#1450	C₂	1	1	yes	maximal, characteristic
4	#8026	C₂	1	4	yes	maximal
5	C₂×#431	C₂²	1	2	yes	—
6	C₂×#427	—	2	1	no	—
7	C₂×#427	—	2	1	no	—
8	C₂²×#107	—	2	1	no	—
9	C₂²×#106	—	2	1	no	—
10	C₂²×C₄×C₈	C₂²	1	1	yes	characteristic
11	#2285	C₂²	1	2	yes	—
12	C₂×#382	C₂²	1	2	yes	—
13	#1846	—	2	2	no	—
14	#1846	—	2	1	no	—
15	#1846	—	2	1	no	—
16	#1846	C₄	1	4	yes	complemented
17	#1845	C₄	1	2	yes	complemented
18	#1844	C₄	1	2	yes	complemented
19	#1450	—	2	2	no	—
20	#1450	C₄	1	4	yes	complemented
21	C₂×#322	—	2	1	no	—
22	C₂×C₈²	—	2	1	no	—
23	#1294	—	2	8	no	—
24	C₂³×C₈	D₄	1	1	yes	characteristic
25	C₂³×C₈	D₄	1	1	yes	characteristic
26	C₂×#117	—	2	2	no	—
27	C₂²×C₄⋊C₄	—	2	2	no	—
28	C₂²×C₄²	C₂³	1	1	yes	characteristic
29	#431	—	2	2	no	—
30	#431	—	2	2	no	—
31	#431	C₂×C₄	1	4	yes	—
32	#431	Q₈	1	4	yes	—
33	#427	—	4	2	no	—
34	#427	—	4	2	no	—
35	#427	—	4	2	no	—
36	#427	—	4	2	no	—
37	C₂×#107	—	2	2	no	—
38	C₂×#107	—	2	2	no	—
39	C₂×#107	—	2	4	no	—
40	C₂×#107	—	4	2	no	—
41	C₂×#107	—	4	2	no	—
42	C₂×#106	—	2	2	no	—
43	C₂×#106	—	2	2	no	—
44	C₂×#106	—	2	4	no	—
45	C₂×#106	—	4	2	no	—
46	C₂×#106	—	4	2	no	—
47	C₂×C₄⋊C₈	—	2	2	no	—
48	C₂×C₂²⋊C₈	D₄	1	4	yes	—
49	C₂×C₄×C₈	—	2	2	no	—
50	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
51	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
53	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
54	C₂×C₄×C₈	D₄	1	1	yes	characteristic
55	C₂×C₄×C₈	D₄	1	1	yes	characteristic
56	C₂×C₄×C₈	D₄	1	2	yes	—
57	C₂×C₄×C₈	Q₈	1	2	yes	—
58	#384	C₂×C₄	1	4	yes	—
59	#382	—	2	4	no	—
60	#382	—	2	2	no	—
61	#382	—	2	2	no	—
62	#382	C₂×C₄	1	4	yes	—
63	#380	—	2	4	no	—
64	C₄×C₄⋊C₄	—	2	4	no	—
65	C₂×#95	—	2	4	no	—
66	#322	—	2	4	no	—
67	C₈²	—	2	4	no	—
68	C₂³×C₄	—	4	2	no	—
69	C₂³×C₄	C₂×D₄	1	2	yes	—
70	C₂³×C₄	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₈	—	2	4	no	—
72	C₂²×C₈	C₂²⋊C₄	1	2	yes	—
73	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
74	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
75	C₂²×C₈	C₂²⋊C₄	1	2	yes	—
76	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
77	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
78	C₂²×C₈	C₄⋊C₄	1	2	yes	—
79	C₂²×C₈	C₄⋊C₄	1	2	yes	—
80	C₂²×C₈	D₈	1	2	yes	—
81	C₂²×C₈	D₈	1	2	yes	—
82	C₂²×C₈	SD₁₆	1	2	yes	—
83	C₂²×C₈	SD₁₆	1	2	yes	—
84	C₂²×C₈	SD₁₆	1	2	yes	—
85	C₂²×C₈	SD₁₆	1	2	yes	—
86	C₂²×C₈	Q₁₆	1	2	yes	—
87	C₂²×C₈	Q₁₆	1	2	yes	—
88	#117	—	4	2	no	—
89	#117	—	4	2	no	—
90	#117	—	4	2	no	—
91	#117	—	4	2	no	—
92	C₂×C₄⋊C₄	—	2	4	no	—
93	C₂×C₄⋊C₄	—	2	4	no	—
94	C₂×C₄⋊C₄	—	2	8	no	—
95	C₂×C₄⋊C₄	—	4	2	no	—
96	C₂×C₄⋊C₄	—	4	2	no	—
97	C₂×C₄⋊C₄	—	4	2	no	—
98	C₂×C₄⋊C₄	—	4	2	no	—
99	C₂×C₄⋊C₄	—	4	2	no	—
100	C₂×C₂²⋊C₄	—	4	2	no	—
101	C₂×C₄²	—	4	2	no	—
102	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
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×Q₈	1	1	yes	characteristic
108	C₂×C₄²	C₂×Q₈	1	2	yes	—
109	C₂×C₄²	#41	1	2	yes	—
110	C₂×C₄²	#41	1	2	yes	—
111	#107	—	4	8	no	—
112	#107	—	4	4	no	—
113	#107	—	4	4	no	—
114	#106	—	4	8	no	—
115	#106	—	4	4	no	—
116	#106	—	4	4	no	—
117	C₄⋊C₈	—	2	4	no	—
118	C₄⋊C₈	—	2	4	no	—
119	C₂²⋊C₈	—	2	4	no	—
120	C₂²⋊C₈	C₄⋊C₄	1	8	yes	—
121	C₄×C₈	—	2	4	no	—
122	C₄×C₈	—	2	4	no	—
123	C₄×C₈	C₄²	1	2	yes	—
124	C₄×C₈	C₄²	1	2	yes	—
125	C₄×C₈	C₂²⋊C₄	1	2	yes	—
126	C₄×C₈	C₂²⋊C₄	1	2	yes	—
127	C₄×C₈	C₄⋊C₄	1	2	yes	—
128	C₄×C₈	C₄⋊C₄	1	2	yes	—
129	C₄×C₈	C₄⋊C₄	1	2	yes	—
130	C₄×C₈	C₄⋊C₄	1	2	yes	—
131	#95	—	2	8	no	—
132	#95	—	4	4	no	—
133	#95	—	4	4	no	—
134	C₂⁴	#127	1	1	yes	characteristic
135	C₂²×C₄	—	4	4	no	—
136	C₂²×C₄	—	4	4	no	—
137	C₂²×C₄	—	4	4	no	—
138	C₂²×C₄	—	8	8	no	—
139	C₂²×C₄	—	8	2	no	—
140	C₂²×C₄	—	8	2	no	—
141	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
142	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
143	C₂²×C₄	C₂×C₄⋊C₄	1	2	yes	—
144	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
145	C₂²×C₄	C₄×D₄	1	2	yes	—
146	C₂²×C₄	C₄×D₄	1	2	yes	—
147	C₂²×C₄	#121	1	2	yes	—
148	C₂²×C₄	#121	1	2	yes	—
149	C₂²×C₄	#124	1	1	yes	characteristic
150	C₂²×C₄	#124	1	1	yes	characteristic
151	C₂²×C₄	C₄⋊Q₈	1	2	yes	—
152	C₂²×C₄	C₄⋊Q₈	1	1	yes	characteristic
153	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
154	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
155	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
156	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
157	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
158	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
159	C₂²×C₄	C₂×Q₁₆	1	1	yes	characteristic
160	C₂²×C₄	C₂×Q₁₆	1	1	yes	characteristic
161	C₂×C₈	—	2	8	no	—
162	C₂×C₈	—	4	8	no	—
163	C₂×C₈	#95	1	2	yes	—
164	C₂×C₈	#95	1	2	yes	—
165	C₂×C₈	#95	1	2	yes	—
166	C₂×C₈	#95	1	2	yes	—
167	C₂×C₈	#102	1	4	yes	—
168	C₂×C₈	#102	1	2	yes	—
169	C₂×C₈	#102	1	2	yes	—
170	C₂×C₈	#102	1	4	yes	—
171	C₂×C₈	#102	1	2	yes	—
172	C₂×C₈	#102	1	2	yes	—
173	C₂×C₈	#103	1	2	yes	—
174	C₂×C₈	#103	1	2	yes	—
175	C₂×C₈	#103	1	4	yes	—
176	C₂×C₈	#103	1	2	yes	—
177	C₂×C₈	#103	1	2	yes	—
178	C₂×C₈	#103	1	4	yes	—
179	C₂×C₈	#106	1	4	yes	—
180	C₂×C₈	#106	1	4	yes	—
181	C₂×C₈	#107	1	4	yes	—
182	C₂×C₈	#107	1	4	yes	—
183	C₄⋊C₄	—	4	4	no	—
184	C₄⋊C₄	—	4	4	no	—
185	C₄⋊C₄	—	4	4	no	—
186	C₄⋊C₄	—	4	4	no	—
187	C₄⋊C₄	—	8	8	no	—
188	C₄⋊C₄	—	8	2	no	—
189	C₄⋊C₄	—	8	2	no	—
190	C₂²⋊C₄	—	8	2	no	—
191	C₂²⋊C₄	—	8	2	no	—
192	C₄²	—	8	2	no	—
193	C₄²	—	8	2	no	—
194	C₄²	C₂×C₄²	1	1	yes	characteristic
195	C₄²	C₂×C₂²⋊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	#117	1	2	yes	—
199	C₄²	C₄×Q₈	1	2	yes	—
200	C₄²	C₄×Q₈	1	2	yes	—
201	C₄²	C₂²⋊Q₈	1	2	yes	—
202	C₄²	C₂²⋊Q₈	1	2	yes	—
203	C₄²	#125	1	2	yes	—
204	C₂³	—	2	4	no	—
205	C₂³	#384	1	1	yes	characteristic
206	C₂³	#390	1	1	yes	characteristic
207	C₂³	#390	1	1	yes	characteristic
208	C₂³	#492	1	1	yes	characteristic
209	C₂³	#492	1	1	yes	characteristic
210	C₂³	#493	1	1	yes	characteristic
211	C₂³	#494	1	1	yes	characteristic
212	C₂×C₄	—	2	4	no	—
213	C₂×C₄	—	2	4	no	—
214	C₂×C₄	—	4	2	no	—
215	C₂×C₄	—	4	2	no	—
216	C₂×C₄	—	4	2	no	—
217	C₂×C₄	—	4	2	no	—
218	C₂×C₄	—	8	8	no	—
219	C₂×C₄	—	8	4	no	—
220	C₂×C₄	—	8	4	no	—
221	C₂×C₄	—	8	8	no	—
222	C₂×C₄	—	16	4	no	—
223	C₂×C₄	—	16	4	no	—
224	C₂×C₄	C₂×#95	1	1	yes	characteristic
225	C₂×C₄	C₄×C₄⋊C₄	1	2	yes	—
226	C₂×C₄	#380	1	2	yes	—
227	C₂×C₄	#382	1	1	yes	characteristic
228	C₂×C₄	#389	1	2	yes	—
229	C₂×C₄	#389	1	2	yes	—
230	C₂×C₄	#391	1	1	yes	characteristic
231	C₂×C₄	#391	1	1	yes	characteristic
232	C₂×C₄	C₂×#102	1	1	yes	characteristic
233	C₂×C₄	C₂×#102	1	1	yes	characteristic
234	C₂×C₄	C₂×#102	1	1	yes	characteristic
235	C₂×C₄	C₂×#102	1	1	yes	characteristic
236	C₂×C₄	C₂×#103	1	1	yes	characteristic
237	C₂×C₄	C₂×#103	1	1	yes	characteristic
238	C₂×C₄	C₂×#103	1	1	yes	characteristic
239	C₂×C₄	C₂×#103	1	1	yes	characteristic
240	C₂×C₄	C₂×#106	1	1	yes	characteristic
241	C₂×C₄	C₂×#106	1	1	yes	characteristic
242	C₂×C₄	C₂×#107	1	1	yes	characteristic
243	C₂×C₄	C₂×#107	1	1	yes	characteristic
244	C₂×C₄	#486	1	1	yes	characteristic
245	C₂×C₄	#486	1	1	yes	characteristic
246	C₂×C₄	#486	1	1	yes	characteristic
247	C₂×C₄	#486	1	1	yes	characteristic
248	C₂×C₄	#487	1	1	yes	characteristic
249	C₂×C₄	#487	1	1	yes	characteristic
250	C₂×C₄	#487	1	1	yes	characteristic
251	C₂×C₄	#487	1	1	yes	characteristic
252	C₂×C₄	#498	1	1	yes	characteristic
253	C₂×C₄	#498	1	1	yes	characteristic
254	C₂×C₄	#500	1	1	yes	characteristic
255	C₂×C₄	#500	1	1	yes	characteristic
256	C₈	—	4	8	no	—
257	C₈	#340	1	4	yes	—
258	C₈	#340	1	4	yes	—
259	C₈	#340	1	4	yes	—
260	C₈	#340	1	4	yes	—
261	C₂²	—	2	2	no	—
262	C₂²	—	2	2	no	—
263	C₂²	—	2	2	no	—
264	C₂²	—	2	2	no	—
265	C₂²	—	2	2	no	—
266	C₂²	—	2	2	no	—
267	C₂²	C₄⋊#95	1	1	yes	characteristic
268	C₂²	#1844	1	1	yes	characteristic
269	C₂²	#1845	1	1	yes	characteristic
270	C₂²	#1956	1	1	yes	characteristic
271	C₂²	#1956	1	1	yes	characteristic
272	C₂²	#1957	1	1	yes	characteristic
273	C₂²	#1957	1	1	yes	characteristic
274	C₂²	#1976	1	4	yes	—
275	C₄	—	2	2	no	—
276	C₄	—	2	2	no	—
277	C₄	—	2	2	no	—
278	C₄	—	2	2	no	—
279	C₄	—	16	4	no	—
280	C₄	—	16	4	no	—
281	C₄	C₂×#340	1	1	yes	characteristic
282	C₄	C₂×#340	1	1	yes	characteristic
283	C₄	#1834	1	1	yes	characteristic
284	C₄	#1834	1	1	yes	characteristic
285	C₄	#1990	1	1	yes	characteristic
286	C₄	#1990	1	1	yes	characteristic
287	C₄	#1990	1	1	yes	characteristic
288	C₄	#1990	1	1	yes	characteristic
289	C₂	—	2	1	no	—
290	C₂	—	2	1	no	—
291	C₂	—	2	1	no	—
292	C₂	—	2	1	no	—
293	C₂	#9755	1	2	yes	—
294	C₂	#9756	1	2	yes	—
295	C₂	#10841	1	1	yes	characteristic
296	C₂	#10871	1	2	yes	—

	subgroup	G/H
center, socle	C₂³	#390
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#127
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
16. #1846	C₄	semidirect
17. #1845	C₄	semidirect
18. #1844	C₄	semidirect
20. #1450	C₄	semidirect

Maximal subgroups

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

subgroup	index
#20797	2
C₂×#1846	2
C₂×#1450	2
#8026	2