Subgroups — #3535

Order 128.

519 conjugacy classes of subgroups.

The subgroup lattice has 295 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#563	C₂	1	2	yes	maximal, complemented
2	#562	C₂	1	2	yes	maximal, complemented
3	#554	C₂	1	2	yes	maximal, complemented
4	#552	C₂	1	2	yes	maximal, complemented
5	#551	C₂	1	2	yes	maximal, complemented
6	#551	C₂	1	2	yes	maximal, complemented
7	#547	C₂	1	2	yes	maximal, complemented
8	#546	C₂	1	2	yes	maximal, complemented
9	#544	C₂	1	1	yes	maximal, characteristic, complemented
10	#543	C₂	1	1	yes	maximal, characteristic, complemented
11	#542	C₂	1	2	yes	maximal, complemented
12	#539	C₂	1	1	yes	maximal, characteristic, complemented
13	#539	C₂	1	2	yes	maximal, complemented
14	#539	C₂	1	2	yes	maximal, complemented
15	#538	C₂	1	1	yes	maximal, characteristic, complemented
16	#536	C₂	1	2	yes	maximal, complemented
17	#525	C₂	1	1	yes	maximal, characteristic, complemented
18	C₂×#123	C₂	1	1	yes	maximal, characteristic, complemented
19	C₂×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
20	C₂×#41	C₂²	1	2	yes	—
21	C₂×#41	C₂²	1	1	yes	characteristic, complemented
22	C₂²×Q₈	C₂²	1	1	yes	characteristic, complemented
23	C₂²×D₄	C₂²	1	1	yes	characteristic
24	C₂³×C₄	C₂²	1	1	yes	characteristic
25	C₄⋊Q₈	C₂²	1	2	yes	complemented
26	C₄⋊Q₈	C₂²	1	2	yes	complemented
27	C₄⋊Q₈	C₂²	1	2	yes	complemented
28	#126	C₂²	1	2	yes	complemented
29	#126	C₂²	1	2	yes	complemented
30	#126	C₂²	1	2	yes	complemented
31	#126	C₂²	1	2	yes	complemented
32	#126	C₂²	1	2	yes	complemented
33	#126	C₂²	1	2	yes	complemented
34	#125	C₂²	1	2	yes	complemented
35	#125	C₂²	1	2	yes	complemented
36	#125	C₂²	1	2	yes	complemented
37	#125	C₂²	1	2	yes	complemented
38	#125	C₂²	1	2	yes	complemented
39	#124	C₂²	1	2	yes	—
40	#124	C₂²	1	2	yes	complemented
41	#124	C₂²	1	2	yes	complemented
42	#124	C₂²	1	2	yes	complemented
43	#123	C₂²	1	2	yes	—
44	#123	C₂²	1	2	yes	—
45	#123	C₂²	1	2	yes	—
46	#123	C₂²	1	2	yes	—
47	#123	C₂²	1	2	yes	—
48	#123	C₂²	1	2	yes	—
49	#123	C₂²	1	2	yes	—
50	#123	C₂²	1	2	yes	complemented
51	#123	C₂²	1	2	yes	complemented
52	#123	C₂²	1	2	yes	complemented
53	#123	C₂²	1	2	yes	complemented
54	#123	C₂²	1	2	yes	complemented
55	#123	C₂²	1	2	yes	complemented
56	#123	C₂²	1	2	yes	complemented
57	C₂²⋊Q₈	C₂²	1	1	yes	characteristic
58	C₂²⋊Q₈	C₂²	1	1	yes	characteristic
59	C₂²⋊Q₈	C₂²	1	2	yes	—
60	C₂²⋊Q₈	C₂²	1	1	yes	characteristic
61	C₂²⋊Q₈	C₂²	1	2	yes	—
62	C₂²⋊Q₈	C₂²	1	2	yes	—
63	C₂²⋊Q₈	C₂²	1	1	yes	characteristic, complemented
64	C₂²⋊Q₈	C₂²	1	1	yes	characteristic, complemented
65	C₂²⋊Q₈	C₂²	1	2	yes	complemented
66	C₂²⋊Q₈	C₂²	1	1	yes	characteristic, complemented
67	C₂²⋊Q₈	C₂²	1	2	yes	complemented
68	C₂²⋊Q₈	C₂²	1	2	yes	complemented
69	C₂²⋊Q₈	C₂²	1	2	yes	complemented
70	C₂²⋊Q₈	C₂²	1	2	yes	complemented
71	C₂²⋊Q₈	C₂²	1	2	yes	complemented
72	C₂²⋊Q₈	C₂²	1	2	yes	complemented
73	C₂²⋊Q₈	C₂²	1	2	yes	complemented
74	C₂²⋊Q₈	C₂²	1	2	yes	complemented
75	C₂²⋊Q₈	C₂²	1	2	yes	complemented
76	C₂²⋊Q₈	C₂²	1	2	yes	complemented
77	#121	C₂²	1	1	yes	characteristic
78	#121	C₂²	1	2	yes	—
79	#121	C₂²	1	2	yes	—
80	#121	C₂²	1	2	yes	—
81	#121	C₂²	1	1	yes	characteristic, complemented
82	#121	C₂²	1	2	yes	complemented
83	#120	C₂²	1	2	yes	—
84	#120	C₂²	1	2	yes	—
85	C₄×Q₈	C₂²	1	2	yes	complemented
86	C₄×D₄	C₂²	1	2	yes	—
87	C₄×D₄	C₂²	1	2	yes	—
88	C₄×D₄	C₂²	1	2	yes	—
89	C₄×D₄	C₂²	1	2	yes	—
90	C₄×D₄	C₂²	1	2	yes	complemented
91	C₄×D₄	C₂²	1	2	yes	complemented
92	C₄×D₄	C₂²	1	2	yes	complemented
93	#117	C₂²	1	2	yes	—
94	#117	C₂²	1	1	yes	characteristic, complemented
95	#117	C₂²	1	2	yes	complemented
96	#117	C₂²	1	2	yes	complemented
97	C₂×C₄⋊C₄	C₂²	1	1	yes	characteristic
98	C₂×C₄⋊C₄	C₂²	1	1	yes	characteristic
99	C₂×C₄⋊C₄	C₂²	1	1	yes	characteristic, complemented
100	C₂×C₄⋊C₄	C₂²	1	2	yes	complemented
101	C₂×C₄⋊C₄	C₂²	1	2	yes	complemented
102	C₂×C₂²⋊C₄	C₂²	1	2	yes	—
103	C₂×C₂²⋊C₄	C₂²	1	1	yes	characteristic
104	C₂×C₂²⋊C₄	C₂²	1	1	yes	characteristic
105	C₂×C₂²⋊C₄	C₂²	1	1	yes	characteristic
106	C₂×C₂²⋊C₄	C₂²	1	2	yes	—
107	C₂⁴	C₂³	1	1	yes	characteristic
108	C₂⁴	C₂³	1	1	yes	characteristic
109	#41	—	4	4	no	—
110	#41	—	4	2	no	—
111	C₂×Q₈	—	4	2	no	—
112	C₂×Q₈	C₂³	1	2	yes	—
113	C₂×Q₈	C₂³	1	1	yes	characteristic
114	C₂×Q₈	C₂³	1	2	yes	—
115	C₂×Q₈	C₂³	1	1	yes	characteristic
116	C₂×Q₈	C₂³	1	2	yes	—
117	C₂×Q₈	C₂³	1	1	yes	characteristic
118	C₂×D₄	—	4	2	no	—
119	C₂×D₄	C₂³	1	2	yes	—
120	C₂×D₄	C₂³	1	2	yes	—
121	C₂×D₄	C₂³	1	2	yes	—
122	C₂×D₄	C₂³	1	2	yes	—
123	C₂×D₄	C₂³	1	2	yes	—
124	C₂×D₄	C₂³	1	2	yes	—
125	C₂×D₄	C₂³	1	2	yes	—
126	C₂×D₄	C₂³	1	1	yes	characteristic
127	C₂²×C₄	—	4	2	no	—
128	C₂²×C₄	C₂³	1	1	yes	characteristic
129	C₂²×C₄	C₂³	1	2	yes	—
130	C₂²×C₄	C₂³	1	1	yes	characteristic
131	C₂²×C₄	C₂³	1	1	yes	characteristic
132	C₂²×C₄	C₂³	1	1	yes	characteristic
133	C₂²×C₄	C₂³	1	1	yes	characteristic
134	C₂²×C₄	C₂³	1	1	yes	characteristic
135	C₂²×C₄	C₂³	1	2	yes	—
136	C₂²×C₄	C₂³	1	2	yes	—
137	C₂²×C₄	C₂³	1	2	yes	—
138	C₂²×C₄	C₂³	1	1	yes	characteristic
139	C₂²×C₄	C₂³	1	2	yes	—
140	C₂²×C₄	C₂³	1	2	yes	—
141	C₂²×C₄	C₂³	1	2	yes	—
142	C₂²×C₄	C₂³	1	1	yes	characteristic
143	C₂²×C₄	C₂³	1	1	yes	characteristic
144	C₄⋊C₄	C₂³	1	2	yes	—
145	C₄⋊C₄	C₂³	1	2	yes	—
146	C₄⋊C₄	C₂³	1	2	yes	—
147	C₄⋊C₄	C₂³	1	2	yes	—
148	C₄⋊C₄	C₂³	1	2	yes	—
149	C₄⋊C₄	C₂³	1	2	yes	—
150	C₄⋊C₄	C₂³	1	2	yes	—
151	C₄⋊C₄	C₂³	1	2	yes	—
152	C₄⋊C₄	C₂³	1	2	yes	—
153	C₄⋊C₄	C₂³	1	1	yes	characteristic
154	C₄⋊C₄	C₂³	1	2	yes	—
155	C₄⋊C₄	C₂³	1	2	yes	—
156	C₄⋊C₄	C₂³	1	1	yes	characteristic
157	C₄⋊C₄	C₂³	1	2	yes	—
158	C₄⋊C₄	C₂³	1	2	yes	—
159	C₄⋊C₄	C₂³	1	1	yes	characteristic
160	C₄⋊C₄	C₂³	1	1	yes	characteristic
161	C₄⋊C₄	C₂³	1	2	yes	—
162	C₄⋊C₄	C₂³	1	2	yes	—
163	C₄⋊C₄	C₂³	1	2	yes	—
164	C₄⋊C₄	C₂³	1	2	yes	—
165	C₄⋊C₄	C₂³	1	2	yes	—
166	C₄⋊C₄	C₂³	1	2	yes	—
167	C₄⋊C₄	C₂³	1	2	yes	—
168	C₄⋊C₄	C₂³	1	2	yes	—
169	C₄⋊C₄	C₂³	1	2	yes	—
170	C₄⋊C₄	C₂³	1	2	yes	—
171	C₂²⋊C₄	C₂³	1	2	yes	—
172	C₂²⋊C₄	C₂³	1	2	yes	—
173	C₂²⋊C₄	C₂³	1	2	yes	—
174	C₂²⋊C₄	C₂³	1	2	yes	—
175	C₂²⋊C₄	C₂³	1	2	yes	—
176	C₂²⋊C₄	C₂³	1	2	yes	—
177	C₂²⋊C₄	C₂³	1	2	yes	—
178	C₂²⋊C₄	C₂³	1	2	yes	—
179	C₂²⋊C₄	C₂³	1	2	yes	—
180	C₂²⋊C₄	C₂³	1	2	yes	—
181	C₂²⋊C₄	C₂³	1	2	yes	—
182	C₂²⋊C₄	C₂³	1	2	yes	—
183	C₂²⋊C₄	C₂³	1	2	yes	—
184	C₂²⋊C₄	C₂³	1	2	yes	—
185	C₂²⋊C₄	C₂³	1	2	yes	—
186	C₂²⋊C₄	C₂³	1	2	yes	—
187	C₂²⋊C₄	C₂³	1	2	yes	—
188	C₂²⋊C₄	C₂³	1	2	yes	—
189	C₂²⋊C₄	C₂³	1	2	yes	—
190	C₂²⋊C₄	C₂³	1	2	yes	—
191	C₂²⋊C₄	C₂³	1	2	yes	—
192	C₂²⋊C₄	C₂³	1	2	yes	—
193	C₂²⋊C₄	C₂³	1	2	yes	—
194	C₄²	C₂³	1	2	yes	—
195	C₄²	C₂³	1	2	yes	—
196	C₄²	C₂³	1	2	yes	—
197	C₄²	C₂³	1	2	yes	—
198	C₄²	C₂³	1	2	yes	—
199	C₂³	—	2	2	no	—
200	C₂³	—	2	2	no	—
201	C₂³	—	2	2	no	—
202	C₂³	—	4	1	no	—
203	C₂³	—	4	1	no	—
204	C₂³	—	4	1	no	—
205	C₂³	C₂⁴	1	1	yes	characteristic
206	C₂³	C₂⁴	1	2	yes	—
207	C₂³	C₂⁴	1	2	yes	—
208	C₂³	C₂⁴	1	1	yes	characteristic
209	C₂³	C₂⁴	1	1	yes	characteristic
210	Q₈	—	4	1	no	—
211	Q₈	—	4	2	no	—
212	Q₈	—	4	1	no	—
213	Q₈	—	4	2	no	—
214	Q₈	—	4	1	no	—
215	Q₈	—	4	2	no	—
216	D₄	—	2	4	no	—
217	D₄	—	2	4	no	—
218	D₄	—	2	4	no	—
219	D₄	—	4	2	no	—
220	D₄	—	4	2	no	—
221	D₄	—	4	2	no	—
222	D₄	—	4	1	no	—
223	D₄	—	4	2	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	4	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	4	1	no	—
229	C₂×C₄	—	4	1	no	—
230	C₂×C₄	—	4	1	no	—
231	C₂×C₄	—	4	1	no	—
232	C₂×C₄	—	4	2	no	—
233	C₂×C₄	—	4	2	no	—
234	C₂×C₄	—	4	1	no	—
235	C₂×C₄	—	4	2	no	—
236	C₂×C₄	—	4	1	no	—
237	C₂×C₄	—	4	2	no	—
238	C₂×C₄	—	4	2	no	—
239	C₂×C₄	—	4	2	no	—
240	C₂×C₄	C₂⁴	1	2	yes	—
241	C₂×C₄	C₂⁴	1	2	yes	—
242	C₂×C₄	C₂⁴	1	2	yes	—
243	C₂×C₄	C₂⁴	1	2	yes	—
244	C₂×C₄	C₂⁴	1	1	yes	characteristic
245	C₂×C₄	C₂⁴	1	2	yes	—
246	C₂×C₄	C₂⁴	1	2	yes	—
247	C₂×C₄	C₂⁴	1	2	yes	—
248	C₂×C₄	C₂⁴	1	2	yes	—
249	C₂×C₄	C₂⁴	1	2	yes	—
250	C₂×C₄	C₂⁴	1	2	yes	—
251	C₂×C₄	C₂⁴	1	1	yes	characteristic
252	C₂×C₄	C₂⁴	1	1	yes	characteristic
253	C₂×C₄	C₂⁴	1	1	yes	characteristic
254	C₂²	—	2	1	no	—
255	C₂²	—	2	1	no	—
256	C₂²	—	2	1	no	—
257	C₂²	—	2	1	no	—
258	C₂²	—	2	2	no	—
259	C₂²	—	2	1	no	—
260	C₂²	—	2	2	no	—
261	C₂²	—	2	2	no	—
262	C₂²	—	2	1	no	—
263	C₂²	—	2	2	no	—
264	C₂²	—	2	2	no	—
265	C₂²	—	2	2	no	—
266	C₂²	—	2	1	no	—
267	C₂²	—	4	4	no	—
268	C₂²	—	8	2	no	—
269	C₂²	H(5,𝔽₂)	1	2	yes	—
270	C₂²	2₋¹⁺⁴	1	2	yes	—
271	C₂²	C₂⁵	1	1	yes	characteristic
272	C₄	—	2	1	no	—
273	C₄	—	2	1	no	—
274	C₄	—	2	1	no	—
275	C₄	—	2	1	no	—
276	C₄	—	2	2	no	—
277	C₄	—	2	2	no	—
278	C₄	—	2	2	no	—
279	C₄	—	2	2	no	—
280	C₄	—	2	2	no	—
281	C₄	—	2	2	no	—
282	C₄	—	2	2	no	—
283	C₄	—	2	2	no	—
284	C₄	—	2	2	no	—
285	C₄	—	2	2	no	—
286	C₂	—	2	2	no	—
287	C₂	—	2	2	no	—
288	C₂	—	4	2	no	—
289	C₂	—	4	1	no	—
290	C₂	—	4	2	no	—
291	C₂	C₂×H(5,𝔽₂)	1	1	yes	characteristic
292	C₂	C₂×2₋¹⁺⁴	1	1	yes	characteristic
293	C₂	#585	1	1	yes	characteristic

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂²	C₂⁵
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ G
LECS	G ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #563	C₂	semidirect
2. #562	C₂	semidirect
3. #554	C₂	semidirect
4. #552	C₂	semidirect
5. #551	C₂	semidirect
6. #551	C₂	semidirect
7. #547	C₂	semidirect
8. #546	C₂	semidirect
9. #544	C₂	semidirect
10. #543	C₂	semidirect
11. #542	C₂	semidirect
12. #539	C₂	semidirect
13. #539	C₂	semidirect
14. #539	C₂	semidirect
15. #538	C₂	semidirect
16. #536	C₂	semidirect
17. #525	C₂	semidirect
18. C₂×#123	C₂	semidirect
19. C₂×C₂²⋊Q₈	C₂	semidirect
21. C₂×#41	C₂²	semidirect
22. C₂²×Q₈	C₂²	semidirect
25. C₄⋊Q₈	C₂²	semidirect
26. C₄⋊Q₈	C₂²	semidirect
27. C₄⋊Q₈	C₂²	semidirect
28. #126	C₂²	semidirect
29. #126	C₂²	semidirect
30. #126	C₂²	semidirect
31. #126	C₂²	semidirect
32. #126	C₂²	semidirect
33. #126	C₂²	semidirect
34. #125	C₂²	semidirect
35. #125	C₂²	semidirect
36. #125	C₂²	semidirect
37. #125	C₂²	semidirect
38. #125	C₂²	semidirect
40. #124	C₂²	semidirect
41. #124	C₂²	semidirect
42. #124	C₂²	semidirect
50. #123	C₂²	semidirect
51. #123	C₂²	semidirect
52. #123	C₂²	semidirect
53. #123	C₂²	semidirect
54. #123	C₂²	semidirect
55. #123	C₂²	semidirect
56. #123	C₂²	semidirect
63. C₂²⋊Q₈	C₂²	semidirect
64. C₂²⋊Q₈	C₂²	semidirect
65. C₂²⋊Q₈	C₂²	semidirect
66. C₂²⋊Q₈	C₂²	semidirect
67. C₂²⋊Q₈	C₂²	semidirect
68. C₂²⋊Q₈	C₂²	semidirect
69. C₂²⋊Q₈	C₂²	semidirect
70. C₂²⋊Q₈	C₂²	semidirect
71. C₂²⋊Q₈	C₂²	semidirect
72. C₂²⋊Q₈	C₂²	semidirect
73. C₂²⋊Q₈	C₂²	semidirect
74. C₂²⋊Q₈	C₂²	semidirect
75. C₂²⋊Q₈	C₂²	semidirect
76. C₂²⋊Q₈	C₂²	semidirect
81. #121	C₂²	semidirect
82. #121	C₂²	semidirect
85. C₄×Q₈	C₂²	semidirect
90. C₄×D₄	C₂²	semidirect
91. C₄×D₄	C₂²	semidirect
92. C₄×D₄	C₂²	semidirect
94. #117	C₂²	semidirect
95. #117	C₂²	semidirect
96. #117	C₂²	semidirect
99. C₂×C₄⋊C₄	C₂²	semidirect
100. C₂×C₄⋊C₄	C₂²	semidirect
101. C₂×C₄⋊C₄	C₂²	semidirect

Maximal subgroups

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

subgroup	index
#563	2
#562	2
#554	2
#552	2
#551	2
#547	2
#546	2
#544	2
#543	2
#542	2
#539	2
#538	2
#536	2
#525	2
C₂×#123	2
C₂×C₂²⋊Q₈	2