Subgroups — #231523

Order 512.

559 conjugacy classes of subgroups.

The subgroup lattice has 266 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23170	C₂	1	2	yes	maximal
2	#19604	C₂	1	1	yes	maximal, characteristic
3	#18068	C₂	1	2	yes	maximal
4	#9749	C₂	1	2	yes	maximal
5	#9718	C₂	1	2	yes	maximal
6	#9056	C₂	1	4	yes	maximal
7	C₈×#108	C₂	1	1	yes	maximal, characteristic
8	C₈×#107	C₂	1	1	yes	maximal, characteristic
9	#3170	C₂²	1	2	yes	—
10	#3170	C₂²	1	2	yes	—
11	#3160	C₂²	1	2	yes	—
12	#2962	C₂²	1	4	yes	—
13	#1986	C₂²	1	4	yes	—
14	#1945	—	2	2	no	—
15	#1922	—	2	4	no	—
16	C₄×#108	C₂²	1	1	yes	characteristic
17	C₄×#107	C₂²	1	1	yes	characteristic
18	#1772	—	2	2	no	—
19	C₈×C₄⋊C₄	C₂²	1	2	yes	—
20	#1713	C₄	1	1	yes	characteristic, complemented
21	#1713	C₄	1	1	yes	characteristic, complemented
22	#1710	C₄	1	2	yes	complemented
23	#1707	C₄	1	2	yes	complemented
24	#1707	C₄	1	2	yes	complemented
25	#1591	—	2	2	no	—
26	#1584	—	2	4	no	—
27	C₈×Q₁₆	—	2	2	no	—
28	#1564	C₂²	1	1	yes	characteristic
29	C₂×#335	C₂²	1	1	yes	characteristic
30	#1503	C₂²	1	4	yes	—
31	#1481	C₂²	1	4	yes	—
32	C₈×M₁₆	C₂²	1	2	yes	—
33	C₂×C₈²	C₂²	1	1	yes	characteristic
34	#1279	C₂²	1	2	yes	—
35	#1278	C₂²	1	2	yes	—
36	#533	C₂³	1	4	yes	—
37	#500	C₂×C₄	1	2	yes	—
38	#500	C₂×C₄	1	2	yes	—
39	#500	C₂×C₄	1	2	yes	—
40	#500	C₂×C₄	1	2	yes	—
41	#499	—	2	2	no	—
42	#499	—	2	2	no	—
43	#498	C₂×C₄	1	4	yes	—
44	#494	C₂×C₄	1	2	yes	—
45	#494	C₂×C₄	1	2	yes	—
46	#487	C₂×C₄	1	4	yes	—
47	#487	C₂×C₄	1	4	yes	—
48	#467	—	2	2	no	—
49	#467	—	2	2	no	—
50	#467	—	4	2	no	—
51	#446	—	2	4	no	—
52	#446	—	4	4	no	—
53	C₈×Q₈	—	2	4	no	—
54	C₄×Q₁₆	—	2	2	no	—
55	C₄×Q₁₆	—	2	2	no	—
56	C₄×Q₁₆	—	4	2	no	—
57	#431	C₂³	1	2	yes	—
58	C₂×#108	D₄	1	1	yes	characteristic
59	C₂×#108	D₄	1	1	yes	characteristic
60	#427	—	2	2	no	—
61	#427	—	2	2	no	—
62	C₂×#107	—	2	1	no	—
63	#424	—	2	4	no	—
64	C₂×C₄⋊C₈	C₂³	1	2	yes	—
65	#405	—	2	4	no	—
66	C₄×M₁₆	C₂³	1	2	yes	—
67	C₂×C₄×C₈	—	2	2	no	—
68	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
69	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
70	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
71	C₄×C₄⋊C₄	C₂³	1	2	yes	—
72	#341	D₄	1	2	yes	—
73	#340	—	2	1	no	—
74	#339	—	2	4	no	—
75	#339	—	2	4	no	—
76	#336	—	2	2	no	—
77	#335	—	2	1	no	—
78	#335	D₄	1	1	yes	characteristic
79	#335	D₄	1	2	yes	—
80	#335	D₄	1	1	yes	characteristic
81	#334	—	2	1	no	—
82	#328	—	4	4	no	—
83	Q₈⋊C₈	—	2	4	no	—
84	Q₈⋊C₈	—	2	4	no	—
85	#322	—	2	2	no	—
86	C₈²	—	2	2	no	—
87	C₈²	C₂×C₄	1	1	yes	characteristic
88	C₈²	C₂×C₄	1	1	yes	characteristic
89	C₈²	C₂×C₄	1	1	yes	characteristic
90	C₈²	C₂×C₄	1	1	yes	characteristic
91	C₂×Q₁₆	—	2	2	no	—
92	C₂×Q₁₆	—	2	2	no	—
93	C₂×Q₁₆	—	4	2	no	—
94	C₂×M₁₆	C₂×D₄	1	2	yes	—
95	C₂×M₁₆	C₂×D₄	1	2	yes	—
96	C₂²×C₈	—	2	2	no	—
97	C₂²×C₈	—	2	1	no	—
98	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
99	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
100	C₂²×C₈	#41	1	1	yes	characteristic
101	C₂²×C₈	#41	1	1	yes	characteristic
102	C₄⋊Q₈	C₂²×C₄	1	4	yes	—
103	C₄⋊Q₈	C₂²×C₄	1	4	yes	—
104	#125	—	2	4	no	—
105	C₂²⋊Q₈	—	2	4	no	—
106	C₂²⋊Q₈	—	2	4	no	—
107	C₄×Q₈	—	2	4	no	—
108	C₄×Q₈	—	2	4	no	—
109	#117	—	2	4	no	—
110	C₂×C₄⋊C₄	—	2	2	no	—
111	C₂×C₄²	—	2	2	no	—
112	C₂×C₄²	C₂⁴	1	1	yes	characteristic
113	#108	—	2	2	no	—
114	#108	Q₁₆	1	4	yes	complemented
115	#107	—	2	4	no	—
116	#107	—	2	2	no	—
117	#107	—	2	2	no	—
118	#107	—	2	2	no	—
119	#107	—	2	2	no	—
120	#107	—	2	2	no	—
121	#107	—	2	2	no	—
122	#107	—	4	2	no	—
123	#106	—	2	4	no	—
124	#106	—	2	4	no	—
125	C₄⋊C₈	—	2	2	no	—
126	C₄⋊C₈	—	2	2	no	—
127	C₄⋊C₈	—	4	4	no	—
128	C₄⋊C₈	—	4	4	no	—
129	C₄⋊C₈	C₂×D₄	1	2	yes	—
130	C₄⋊C₈	C₂×D₄	1	2	yes	—
131	#103	—	2	4	no	—
132	#103	—	2	4	no	—
133	#103	—	4	4	no	—
134	C₂²⋊C₈	—	2	2	no	—
135	#97	—	2	2	no	—
136	#97	—	4	4	no	—
137	C₄×C₈	—	2	2	no	—
138	C₄×C₈	—	2	2	no	—
139	C₄×C₈	—	2	1	no	—
140	C₄×C₈	—	4	4	no	—
141	C₄×C₈	—	4	4	no	—
142	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
143	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
144	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
145	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
146	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
147	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
148	C₄×C₈	C₂×D₄	1	1	yes	characteristic
149	C₄×C₈	C₂×D₄	1	1	yes	characteristic
150	C₄×C₈	#41	1	1	yes	characteristic
151	C₄×C₈	#41	1	1	yes	characteristic
152	#95	—	2	2	no	—
153	C₂×Q₈	—	2	4	no	—
154	C₂×Q₈	—	2	4	no	—
155	C₂²×C₄	—	4	2	no	—
156	C₂²×C₄	C₂²×D₄	1	1	yes	characteristic
157	C₂²×C₄	C₂×#41	1	1	yes	characteristic
158	C₂²×C₄	H(5,𝔽₂)	1	1	yes	characteristic
159	Q₁₆	—	4	4	no	—
160	Q₁₆	—	8	2	no	—
161	Q₁₆	—	8	4	no	—
162	M₁₆	—	2	2	no	—
163	M₁₆	—	2	2	no	—
164	M₁₆	—	2	2	no	—
165	M₁₆	C₂×Q₁₆	1	4	yes	—
166	C₂×C₈	—	2	1	no	—
167	C₂×C₈	—	2	2	no	—
168	C₂×C₈	—	2	1	no	—
169	C₂×C₈	—	2	1	no	—
170	C₂×C₈	—	2	2	no	—
171	C₂×C₈	—	2	1	no	—
172	C₂×C₈	—	2	2	no	—
173	C₂×C₈	—	2	2	no	—
174	C₂×C₈	—	2	2	no	—
175	C₂×C₈	—	4	1	no	—
176	C₂×C₈	—	4	2	no	—
177	C₂×C₈	—	4	2	no	—
178	C₂×C₈	—	4	2	no	—
179	C₂×C₈	—	4	2	no	—
180	C₂×C₈	C₄×D₄	1	1	yes	characteristic
181	C₂×C₈	C₄×D₄	1	1	yes	characteristic
182	C₂×C₈	C₄×D₄	1	1	yes	characteristic
183	C₂×C₈	C₄×D₄	1	1	yes	characteristic
184	C₂×C₈	C₄×D₄	1	1	yes	characteristic
185	C₂×C₈	C₄×D₄	1	1	yes	characteristic
186	C₂×C₈	C₄×D₄	1	1	yes	characteristic
187	C₂×C₈	C₄×D₄	1	1	yes	characteristic
188	C₂×C₈	C₂×Q₁₆	1	2	yes	—
189	C₂×C₈	#135	1	2	yes	—
190	C₄⋊C₄	—	2	4	no	—
191	C₄⋊C₄	—	2	4	no	—
192	C₄⋊C₄	—	2	4	no	—
193	C₄⋊C₄	—	2	4	no	—
194	C₄⋊C₄	—	2	4	no	—
195	C₄⋊C₄	—	4	2	no	—
196	C₄⋊C₄	—	4	4	no	—
197	C₄⋊C₄	—	4	4	no	—
198	C₂²⋊C₄	—	4	4	no	—
199	C₄²	—	4	4	no	—
200	C₄²	—	4	2	no	—
201	C₄²	—	4	2	no	—
202	C₄²	C₂³×C₄	1	1	yes	characteristic
203	C₄²	C₂²×D₄	1	1	yes	characteristic
204	C₄²	C₂×#41	1	1	yes	characteristic
205	C₄²	H(5,𝔽₂)	1	1	yes	characteristic
206	C₂³	#550	1	1	yes	characteristic
207	Q₈	—	4	4	no	—
208	Q₈	—	4	4	no	—
209	Q₈	—	8	4	no	—
210	C₂×C₄	—	2	1	no	—
211	C₂×C₄	—	2	1	no	—
212	C₂×C₄	—	4	1	no	—
213	C₂×C₄	—	4	4	no	—
214	C₂×C₄	—	4	4	no	—
215	C₂×C₄	—	4	4	no	—
216	C₂×C₄	—	8	2	no	—
217	C₂×C₄	—	8	2	no	—
218	C₂×C₄	C₂×C₄×D₄	1	1	yes	characteristic
219	C₂×C₄	C₂×C₄×D₄	1	1	yes	characteristic
220	C₂×C₄	#518	1	1	yes	characteristic
221	C₂×C₄	D₄²	1	1	yes	characteristic
222	C₂×C₄	#550	1	1	yes	characteristic
223	C₂×C₄	#559	1	1	yes	characteristic
224	C₂×C₄	C₂²×Q₁₆	1	1	yes	characteristic
225	C₂×C₄	C₂×#135	1	1	yes	characteristic
226	C₂×C₄	#578	1	1	yes	characteristic
227	C₂×C₄	#578	1	1	yes	characteristic
228	C₈	—	2	2	no	—
229	C₈	—	2	1	no	—
230	C₈	—	2	1	no	—
231	C₈	—	2	1	no	—
232	C₈	—	2	1	no	—
233	C₈	—	2	1	no	—
234	C₈	—	2	4	no	—
235	C₈	—	4	1	no	—
236	C₈	—	4	2	no	—
237	C₈	—	4	2	no	—
238	C₈	—	4	2	no	—
239	C₈	—	8	2	no	—
240	C₈	C₄×Q₁₆	1	2	yes	—
241	C₈	C₄×Q₁₆	1	2	yes	—
242	C₈	#443	1	2	yes	—
243	C₂²	—	2	1	no	—
244	C₂²	—	2	1	no	—
245	C₂²	#2366	1	1	yes	characteristic
246	C₂²	#3393	1	1	yes	characteristic
247	C₂²	#3395	1	1	yes	characteristic
248	C₄	—	2	1	no	—
249	C₄	—	2	1	no	—
250	C₄	—	2	1	no	—
251	C₄	—	8	4	no	—
252	C₄	—	8	4	no	—
253	C₄	—	8	4	no	—
254	C₄	C₂×C₄×Q₁₆	1	1	yes	characteristic
255	C₄	#2950	1	1	yes	characteristic
256	C₄	C₂×#443	1	1	yes	characteristic
257	C₄	#2957	1	1	yes	characteristic
258	C₄	#3283	1	1	yes	characteristic
259	C₄	D₄×Q₁₆	1	1	yes	characteristic
260	C₂	—	2	1	no	—
261	C₂	—	2	1	no	—
262	C₂	#23179	1	1	yes	characteristic
263	C₂	#23184	1	1	yes	characteristic
264	C₂	#23196	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₄²
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	#2366
prime radical	C₂³	#550
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
20. #1713	C₄	semidirect
21. #1713	C₄	semidirect
22. #1710	C₄	semidirect
23. #1707	C₄	semidirect
24. #1707	C₄	semidirect
114. #108	Q₁₆	semidirect

Maximal subgroups

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

subgroup	index
#23170	2
#19604	2
#18068	2
#9749	2
#9718	2
#9056	2
C₈×#108	2
C₈×#107	2