Subgroups — C₂³×#552 #10585927

Order 512.

38378 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#552	C₂	1	112	yes	maximal, direct product factor
2	C₂³×#126	C₂	1	2	yes	maximal, complemented
3	C₂³×#125	C₂	1	2	yes	maximal, complemented
4	C₂³×#125	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂³×#123	C₂	1	2	yes	maximal
6	C₂³×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂³×C₄×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂³×C₄×D₄	C₂	1	1	yes	maximal, characteristic
10	C₂³×#117	C₂	1	2	yes	maximal, complemented
11	C₂³×#117	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂⁴×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂⁴×Q₈	C₂²	1	1	yes	characteristic
14	C₂⁴×D₄	C₂²	1	1	yes	characteristic
15	C₂⁵×C₄	C₂²	1	2	yes	—
16	C₂⁵×C₄	C₂²	1	1	yes	characteristic
17	C₂⁵×C₄	C₂²	1	1	yes	characteristic
18	C₂×#552	C₂²	1	1792	yes	direct product factor
19	C₂²×#126	C₂²	1	112	yes	complemented
20	C₂²×#125	C₂²	1	56	yes	complemented
21	C₂²×#125	C₂²	1	112	yes	complemented
22	C₂²×#123	C₂²	1	112	yes	—
23	C₂²×C₂²⋊Q₈	C₂²	1	56	yes	complemented
24	C₂²×C₂²⋊Q₈	C₂²	1	56	yes	complemented
25	C₂²×C₄×Q₈	C₂²	1	56	yes	complemented
26	C₂²×C₄×D₄	C₂²	1	56	yes	—
27	C₂²×#117	C₂²	1	112	yes	complemented
28	C₂²×#117	C₂²	1	56	yes	complemented
29	C₂³×C₄⋊C₄	C₂²	1	1	yes	characteristic
30	C₂³×C₄⋊C₄	C₂²	1	1	yes	characteristic
31	C₂³×C₄⋊C₄	C₂²	1	2	yes	—
32	C₂³×C₄⋊C₄	C₂²	1	2	yes	—
33	C₂³×C₄⋊C₄	C₂²	1	1	yes	characteristic
34	C₂³×C₄⋊C₄	C₂²	1	2	yes	—
35	C₂³×C₄⋊C₄	C₂²	1	1	yes	characteristic
36	C₂³×C₄⋊C₄	C₂²	1	2	yes	—
37	C₂³×C₄⋊C₄	C₂²	1	2	yes	—
38	C₂³×C₄⋊C₄	C₂²	1	2	yes	—
39	C₂³×C₄⋊C₄	C₂²	1	56	yes	complemented
40	C₂³×C₂²⋊C₄	C₂²	1	1	yes	characteristic
41	C₂³×C₂²⋊C₄	C₂²	1	2	yes	—
42	C₂³×C₂²⋊C₄	C₂²	1	2	yes	—
43	C₂³×C₂²⋊C₄	C₂²	1	1	yes	characteristic
44	C₂³×C₂²⋊C₄	C₂²	1	2	yes	—
45	C₂³×C₄²	C₂²	1	2	yes	—
46	C₂³×C₄²	C₂²	1	2	yes	—
47	C₂³×C₄²	C₂²	1	1	yes	characteristic
48	C₂⁶	C₂³	1	1	yes	characteristic
49	C₂⁶	C₂³	1	1	yes	characteristic
50	C₂³×Q₈	—	2	16	no	—
51	C₂³×Q₈	C₂³	1	28	yes	—
52	C₂³×D₄	—	2	16	no	—
53	C₂³×D₄	C₂³	1	28	yes	—
54	C₂⁴×C₄	—	2	32	no	—
55	C₂⁴×C₄	—	2	16	no	—
56	C₂⁴×C₄	—	2	16	no	—
57	C₂⁴×C₄	C₂³	1	56	yes	—
58	C₂⁴×C₄	C₂³	1	28	yes	—
59	C₂⁴×C₄	C₂³	1	28	yes	—
60	C₂⁴×C₄	C₂³	1	2	yes	—
61	C₂⁴×C₄	C₂³	1	1	yes	characteristic
62	C₂⁴×C₄	C₂³	1	2	yes	—
63	C₂⁴×C₄	C₂³	1	1	yes	characteristic
64	C₂⁴×C₄	C₂³	1	1	yes	characteristic
65	C₂⁴×C₄	C₂³	1	2	yes	—
66	C₂⁴×C₄	C₂³	1	1	yes	characteristic
67	C₂⁴×C₄	C₂³	1	2	yes	—
68	C₂⁴×C₄	C₂³	1	1	yes	characteristic
69	#552	C₂³	1	4096	yes	direct product factor
70	C₂×#126	C₂³	1	896	yes	complemented
71	C₂×#125	C₂³	1	896	yes	complemented
72	C₂×#125	C₂³	1	448	yes	complemented
73	C₂×#123	C₂³	1	896	yes	—
74	C₂×C₂²⋊Q₈	C₂³	1	448	yes	complemented
75	C₂×C₂²⋊Q₈	C₂³	1	448	yes	complemented
76	C₂×C₄×Q₈	C₂³	1	448	yes	complemented
77	C₂×C₄×D₄	C₂³	1	448	yes	—
78	C₂×#117	C₂³	1	448	yes	complemented
79	C₂×#117	C₂³	1	896	yes	complemented
80	C₂²×C₄⋊C₄	C₂³	1	28	yes	—
81	C₂²×C₄⋊C₄	C₂³	1	28	yes	—
82	C₂²×C₄⋊C₄	C₂³	1	56	yes	—
83	C₂²×C₄⋊C₄	C₂³	1	28	yes	—
84	C₂²×C₄⋊C₄	C₂³	1	56	yes	—
85	C₂²×C₄⋊C₄	C₂³	1	56	yes	—
86	C₂²×C₄⋊C₄	C₂³	1	28	yes	—
87	C₂²×C₄⋊C₄	C₂³	1	56	yes	—
88	C₂²×C₄⋊C₄	C₂³	1	56	yes	—
89	C₂²×C₄⋊C₄	C₂³	1	56	yes	—
90	C₂²×C₄⋊C₄	C₂³	1	448	yes	complemented
91	C₂²×C₂²⋊C₄	C₂³	1	28	yes	—
92	C₂²×C₂²⋊C₄	C₂³	1	56	yes	—
93	C₂²×C₂²⋊C₄	C₂³	1	56	yes	—
94	C₂²×C₂²⋊C₄	C₂³	1	28	yes	—
95	C₂²×C₂²⋊C₄	C₂³	1	56	yes	—
96	C₂²×C₄²	C₂³	1	56	yes	—
97	C₂²×C₄²	C₂³	1	56	yes	—
98	C₂²×C₄²	C₂³	1	28	yes	—
99	C₂⁵	—	2	8	no	—
100	C₂⁵	—	2	8	no	—
101	C₂⁵	—	2	8	no	—
102	C₂⁵	—	2	8	no	—
103	C₂⁵	—	2	8	no	—
104	C₂⁵	#41	1	16	yes	—
105	C₂⁵	C₂⁴	1	14	yes	—
106	C₂⁵	C₂⁴	1	14	yes	—
107	C₂⁵	C₂⁴	1	1	yes	characteristic
108	C₂²×Q₈	—	2	224	no	—
109	C₂²×Q₈	C₂⁴	1	112	yes	—
110	C₂²×D₄	—	2	224	no	—
111	C₂²×D₄	C₂⁴	1	112	yes	—
112	C₂³×C₄	—	2	448	no	—
113	C₂³×C₄	—	2	16	no	—
114	C₂³×C₄	—	2	16	no	—
115	C₂³×C₄	—	2	8	no	—
116	C₂³×C₄	—	2	8	no	—
117	C₂³×C₄	—	2	8	no	—
118	C₂³×C₄	—	2	16	no	—
119	C₂³×C₄	—	2	8	no	—
120	C₂³×C₄	—	2	16	no	—
121	C₂³×C₄	—	2	224	no	—
122	C₂³×C₄	—	2	224	no	—
123	C₂³×C₄	#41	1	16	yes	—
124	C₂³×C₄	C₂⁴	1	28	yes	—
125	C₂³×C₄	C₂⁴	1	28	yes	—
126	C₂³×C₄	C₂⁴	1	14	yes	—
127	C₂³×C₄	C₂⁴	1	14	yes	—
128	C₂³×C₄	C₂⁴	1	28	yes	—
129	C₂³×C₄	C₂⁴	1	14	yes	—
130	C₂³×C₄	C₂⁴	1	28	yes	—
131	C₂³×C₄	C₂⁴	1	14	yes	—
132	C₂³×C₄	C₂⁴	1	224	yes	—
133	C₂³×C₄	C₂⁴	1	112	yes	—
134	C₂³×C₄	C₂⁴	1	112	yes	—
135	C₂³×C₄	C₂⁴	1	14	yes	—
136	#126	C₂⁴	1	1024	yes	complemented
137	#125	C₂⁴	1	1024	yes	complemented
138	#125	C₂⁴	1	512	yes	complemented
139	#123	C₂⁴	1	1024	yes	—
140	C₂²⋊Q₈	C₂⁴	1	512	yes	complemented
141	C₂²⋊Q₈	C₂⁴	1	512	yes	complemented
142	C₄×Q₈	C₂⁴	1	512	yes	complemented
143	C₄×D₄	C₂⁴	1	512	yes	—
144	#117	C₂⁴	1	512	yes	complemented
145	#117	C₂⁴	1	1024	yes	complemented
146	C₂×C₄⋊C₄	C₂⁴	1	112	yes	—
147	C₂×C₄⋊C₄	C₂⁴	1	112	yes	—
148	C₂×C₄⋊C₄	C₂⁴	1	224	yes	—
149	C₂×C₄⋊C₄	C₂⁴	1	224	yes	—
150	C₂×C₄⋊C₄	C₂⁴	1	112	yes	—
151	C₂×C₄⋊C₄	C₂⁴	1	224	yes	—
152	C₂×C₄⋊C₄	C₂⁴	1	112	yes	—
153	C₂×C₄⋊C₄	C₂⁴	1	224	yes	—
154	C₂×C₄⋊C₄	C₂⁴	1	224	yes	—
155	C₂×C₄⋊C₄	C₂⁴	1	224	yes	—
156	C₂×C₄⋊C₄	C₂⁴	1	512	yes	complemented
157	C₂×C₂²⋊C₄	C₂⁴	1	112	yes	—
158	C₂×C₂²⋊C₄	C₂⁴	1	224	yes	—
159	C₂×C₂²⋊C₄	C₂⁴	1	224	yes	—
160	C₂×C₂²⋊C₄	C₂⁴	1	112	yes	—
161	C₂×C₂²⋊C₄	C₂⁴	1	224	yes	—
162	C₂×C₄²	C₂⁴	1	224	yes	—
163	C₂×C₄²	C₂⁴	1	224	yes	—
164	C₂×C₄²	C₂⁴	1	112	yes	—
165	C₂⁴	—	2	128	no	—
166	C₂⁴	—	2	56	no	—
167	C₂⁴	—	2	56	no	—
168	C₂⁴	—	2	56	no	—
169	C₂⁴	—	2	56	no	—
170	C₂⁴	—	2	56	no	—
171	C₂⁴	—	4	64	no	—
172	C₂⁴	C₂×#41	1	112	yes	—
173	C₂⁴	C₂×#41	1	8	yes	—
174	C₂⁴	C₂×#41	1	8	yes	—
175	C₂⁴	2₋¹⁺⁴	1	8	yes	—
176	C₂⁴	C₂⁵	1	28	yes	—
177	C₂⁴	C₂⁵	1	28	yes	—
178	C₂⁴	C₂⁵	1	7	yes	—
179	C₂×Q₈	—	2	448	no	—
180	C₂×Q₈	C₂⁵	1	64	yes	—
181	C₂×D₄	—	2	448	no	—
182	C₂×D₄	C₂⁵	1	64	yes	—
183	C₂²×C₄	—	2	896	no	—
184	C₂²×C₄	—	2	112	no	—
185	C₂²×C₄	—	2	112	no	—
186	C₂²×C₄	—	2	56	no	—
187	C₂²×C₄	—	2	56	no	—
188	C₂²×C₄	—	2	56	no	—
189	C₂²×C₄	—	2	112	no	—
190	C₂²×C₄	—	2	112	no	—
191	C₂²×C₄	—	2	56	no	—
192	C₂²×C₄	—	2	448	no	—
193	C₂²×C₄	—	2	448	no	—
194	C₂²×C₄	C₂×#41	1	112	yes	—
195	C₂²×C₄	C₂⁵	1	28	yes	—
196	C₂²×C₄	C₂⁵	1	28	yes	—
197	C₂²×C₄	C₂⁵	1	56	yes	—
198	C₂²×C₄	C₂⁵	1	56	yes	—
199	C₂²×C₄	C₂⁵	1	56	yes	—
200	C₂²×C₄	C₂⁵	1	28	yes	—
201	C₂²×C₄	C₂⁵	1	28	yes	—
202	C₂²×C₄	C₂⁵	1	56	yes	—
203	C₂²×C₄	C₂⁵	1	128	yes	—
204	C₂²×C₄	C₂⁵	1	64	yes	—
205	C₂²×C₄	C₂⁵	1	64	yes	—
206	C₂²×C₄	C₂⁵	1	28	yes	—
207	C₄⋊C₄	C₂⁵	1	64	yes	—
208	C₄⋊C₄	C₂⁵	1	64	yes	—
209	C₄⋊C₄	C₂⁵	1	128	yes	—
210	C₄⋊C₄	C₂⁵	1	128	yes	—
211	C₄⋊C₄	C₂⁵	1	64	yes	—
212	C₄⋊C₄	C₂⁵	1	128	yes	—
213	C₄⋊C₄	C₂⁵	1	64	yes	—
214	C₄⋊C₄	C₂⁵	1	128	yes	—
215	C₄⋊C₄	C₂⁵	1	128	yes	—
216	C₄⋊C₄	C₂⁵	1	128	yes	—
217	C₂²⋊C₄	C₂⁵	1	64	yes	—
218	C₂²⋊C₄	C₂⁵	1	128	yes	—
219	C₂²⋊C₄	C₂⁵	1	128	yes	—
220	C₂²⋊C₄	C₂⁵	1	64	yes	—
221	C₂²⋊C₄	C₂⁵	1	128	yes	—
222	C₄²	C₂⁵	1	128	yes	—
223	C₄²	C₂⁵	1	128	yes	—
224	C₄²	C₂⁵	1	64	yes	—
225	C₂³	—	2	56	no	—
226	C₂³	—	2	56	no	—
227	C₂³	—	2	448	no	—
228	C₂³	—	2	56	no	—
229	C₂³	—	2	56	no	—
230	C₂³	—	2	56	no	—
231	C₂³	—	4	224	no	—
232	C₂³	#552	1	64	yes	direct product factor
233	C₂³	C₂²×#41	1	112	yes	—
234	C₂³	C₂²×#41	1	28	yes	—
235	C₂³	C₂²×#41	1	28	yes	—
236	C₂³	C₂×2₋¹⁺⁴	1	28	yes	—
237	C₂³	C₂⁶	1	8	yes	—
238	C₂³	C₂⁶	1	8	yes	—
239	C₂³	C₂⁶	1	7	yes	—
240	Q₈	—	2	128	no	—
241	D₄	—	2	128	no	—
242	C₂×C₄	—	2	256	no	—
243	C₂×C₄	—	2	112	no	—
244	C₂×C₄	—	2	112	no	—
245	C₂×C₄	—	2	56	no	—
246	C₂×C₄	—	2	56	no	—
247	C₂×C₄	—	2	56	no	—
248	C₂×C₄	—	2	112	no	—
249	C₂×C₄	—	2	56	no	—
250	C₂×C₄	—	2	112	no	—
251	C₂×C₄	—	2	128	no	—
252	C₂×C₄	—	2	128	no	—
253	C₂×C₄	C₂²×#41	1	112	yes	—
254	C₂×C₄	C₂⁶	1	8	yes	—
255	C₂×C₄	C₂⁶	1	8	yes	—
256	C₂×C₄	C₂⁶	1	16	yes	—
257	C₂×C₄	C₂⁶	1	16	yes	—
258	C₂×C₄	C₂⁶	1	16	yes	—
259	C₂×C₄	C₂⁶	1	8	yes	—
260	C₂×C₄	C₂⁶	1	8	yes	—
261	C₂×C₄	C₂⁶	1	16	yes	—
262	C₂×C₄	C₂⁶	1	8	yes	—
263	C₂²	—	2	8	no	—
264	C₂²	—	2	8	no	—
265	C₂²	—	2	224	no	—
266	C₂²	—	2	8	no	—
267	C₂²	—	2	8	no	—
268	C₂²	—	2	8	no	—
269	C₂²	—	4	112	no	—
270	C₂²	C₂×#552	1	112	yes	direct product factor
271	C₂²	C₂³×#41	1	16	yes	—
272	C₂²	C₂³×#41	1	14	yes	—
273	C₂²	C₂³×#41	1	14	yes	—
274	C₂²	C₂²×2₋¹⁺⁴	1	14	yes	—
275	C₂²	C₂⁷	1	1	yes	characteristic
276	C₄	—	2	16	no	—
277	C₄	—	2	16	no	—
278	C₄	—	2	8	no	—
279	C₄	—	2	8	no	—
280	C₄	—	2	8	no	—
281	C₄	—	2	16	no	—
282	C₄	—	2	8	no	—
283	C₄	—	2	16	no	—
284	C₄	C₂³×#41	1	16	yes	—
285	C₂	—	2	16	no	—
286	C₂	—	4	8	no	—
287	C₂	C₂²×#552	1	28	yes	direct product factor
288	C₂	C₂⁴×#41	1	1	yes	characteristic
289	C₂	C₂⁴×#41	1	1	yes	characteristic
290	C₂	C₂³×2₋¹⁺⁴	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂⁵	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁷
prime radical	C₂⁴×D₄	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
1. C₂²×#552	C₂	direct (N × H)
2. C₂³×#126	C₂	semidirect
3. C₂³×#125	C₂	semidirect
4. C₂³×#125	C₂	semidirect
6. C₂³×C₂²⋊Q₈	C₂	semidirect
7. C₂³×C₂²⋊Q₈	C₂	semidirect
8. C₂³×C₄×Q₈	C₂	semidirect
10. C₂³×#117	C₂	semidirect
11. C₂³×#117	C₂	semidirect
12. C₂⁴×C₄⋊C₄	C₂	semidirect
18. C₂×#552	C₂²	direct (N × H)
19. C₂²×#126	C₂²	semidirect
20. C₂²×#125	C₂²	semidirect
21. C₂²×#125	C₂²	semidirect
23. C₂²×C₂²⋊Q₈	C₂²	semidirect
24. C₂²×C₂²⋊Q₈	C₂²	semidirect
25. C₂²×C₄×Q₈	C₂²	semidirect
27. C₂²×#117	C₂²	semidirect
28. C₂²×#117	C₂²	semidirect
39. C₂³×C₄⋊C₄	C₂²	semidirect
69. #552	C₂³	direct (N × H)
70. C₂×#126	C₂³	semidirect
71. C₂×#125	C₂³	semidirect
72. C₂×#125	C₂³	semidirect
74. C₂×C₂²⋊Q₈	C₂³	semidirect
75. C₂×C₂²⋊Q₈	C₂³	semidirect
76. C₂×C₄×Q₈	C₂³	semidirect
78. C₂×#117	C₂³	semidirect
79. C₂×#117	C₂³	semidirect
90. C₂²×C₄⋊C₄	C₂³	semidirect
136. #126	C₂⁴	semidirect
137. #125	C₂⁴	semidirect
138. #125	C₂⁴	semidirect
140. C₂²⋊Q₈	C₂⁴	semidirect
141. C₂²⋊Q₈	C₂⁴	semidirect
142. C₄×Q₈	C₂⁴	semidirect
144. #117	C₂⁴	semidirect
145. #117	C₂⁴	semidirect
156. C₂×C₄⋊C₄	C₂⁴	semidirect
232. C₂³	#552	direct (N × H)
270. C₂²	C₂×#552	direct (N × H)
287. C₂	C₂²×#552	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#552	2
C₂³×#126	2
C₂³×#125	2
C₂³×#123	2
C₂³×C₂²⋊Q₈	2
C₂³×C₄×Q₈	2
C₂³×C₄×D₄	2
C₂³×#117	2
C₂⁴×C₄⋊C₄	2

Subgroups — C23×#552 #10585927

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#552 #10585927