Subgroups — #3534

Order 128.

583 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#562	C₂	1	2	yes	maximal, complemented
2	#560	C₂	1	2	yes	maximal, complemented
3	#553	C₂	1	2	yes	maximal, complemented
4	#551	C₂	1	2	yes	maximal, complemented
5	#546	C₂	1	2	yes	maximal, complemented
6	#546	C₂	1	2	yes	maximal, complemented
7	#546	C₂	1	2	yes	maximal, complemented
8	D₄²	C₂	1	2	yes	maximal, complemented
9	#540	C₂	1	2	yes	maximal, complemented
10	#539	C₂	1	2	yes	maximal, complemented
11	#539	C₂	1	1	yes	maximal, characteristic, complemented
12	#538	C₂	1	1	yes	maximal, characteristic, complemented
13	#538	C₂	1	2	yes	maximal, complemented
14	#537	C₂	1	1	yes	maximal, characteristic, complemented
15	#534	C₂	1	2	yes	maximal, complemented
16	#534	C₂	1	1	yes	maximal, characteristic, complemented
17	#525	C₂	1	1	yes	maximal, characteristic, complemented
18	C₂×#123	C₂	1	1	yes	maximal, characteristic, complemented
19	C₂×#121	C₂	1	1	yes	maximal, characteristic, complemented
20	C₂×#41	C₂²	1	1	yes	characteristic, complemented
21	C₂×#41	C₂²	1	2	yes	complemented
22	C₂²×D₄	C₂²	1	1	yes	characteristic
23	C₂²×D₄	C₂²	1	1	yes	characteristic, complemented
24	C₂²×D₄	C₂²	1	2	yes	complemented
25	C₂²×D₄	C₂²	1	2	yes	complemented
26	C₂²×D₄	C₂²	1	2	yes	complemented
27	C₂³×C₄	C₂²	1	1	yes	characteristic, complemented
28	#127	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	#125	C₂²	1	2	yes	complemented
34	#125	C₂²	1	2	yes	complemented
35	#124	C₂²	1	2	yes	complemented
36	#124	C₂²	1	2	yes	complemented
37	#124	C₂²	1	2	yes	complemented
38	#123	C₂²	1	2	yes	complemented
39	#123	C₂²	1	2	yes	complemented
40	#123	C₂²	1	2	yes	complemented
41	#123	C₂²	1	2	yes	complemented
42	#123	C₂²	1	2	yes	complemented
43	#123	C₂²	1	2	yes	complemented
44	#123	C₂²	1	2	yes	complemented
45	#123	C₂²	1	2	yes	complemented
46	#123	C₂²	1	2	yes	complemented
47	#123	C₂²	1	2	yes	complemented
48	#123	C₂²	1	2	yes	complemented
49	#123	C₂²	1	2	yes	complemented
50	#123	C₂²	1	2	yes	complemented
51	#123	C₂²	1	2	yes	complemented
52	C₂²⋊Q₈	C₂²	1	1	yes	characteristic, complemented
53	C₂²⋊Q₈	C₂²	1	1	yes	characteristic, complemented
54	C₂²⋊Q₈	C₂²	1	2	yes	complemented
55	C₂²⋊Q₈	C₂²	1	2	yes	complemented
56	C₂²⋊Q₈	C₂²	1	2	yes	complemented
57	C₂²⋊Q₈	C₂²	1	2	yes	complemented
58	C₂²⋊Q₈	C₂²	1	2	yes	complemented
59	#121	C₂²	1	1	yes	characteristic
60	#121	C₂²	1	2	yes	—
61	#121	C₂²	1	1	yes	characteristic, complemented
62	#121	C₂²	1	1	yes	characteristic, complemented
63	#121	C₂²	1	1	yes	characteristic, complemented
64	#121	C₂²	1	2	yes	complemented
65	#121	C₂²	1	2	yes	complemented
66	#121	C₂²	1	1	yes	characteristic, complemented
67	#121	C₂²	1	1	yes	characteristic, complemented
68	#121	C₂²	1	2	yes	complemented
69	#121	C₂²	1	2	yes	complemented
70	#121	C₂²	1	2	yes	complemented
71	#121	C₂²	1	2	yes	complemented
72	#121	C₂²	1	2	yes	complemented
73	#121	C₂²	1	2	yes	complemented
74	#121	C₂²	1	2	yes	complemented
75	#121	C₂²	1	2	yes	complemented
76	#121	C₂²	1	2	yes	complemented
77	#121	C₂²	1	2	yes	complemented
78	#121	C₂²	1	2	yes	complemented
79	#121	C₂²	1	2	yes	complemented
80	#120	C₂²	1	2	yes	—
81	#120	C₂²	1	2	yes	complemented
82	#120	C₂²	1	2	yes	complemented
83	#120	C₂²	1	2	yes	complemented
84	#120	C₂²	1	2	yes	complemented
85	#120	C₂²	1	2	yes	complemented
86	#120	C₂²	1	2	yes	complemented
87	#120	C₂²	1	2	yes	complemented
88	C₄×D₄	C₂²	1	2	yes	complemented
89	C₄×D₄	C₂²	1	2	yes	complemented
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	C₄×D₄	C₂²	1	2	yes	complemented
94	C₄×D₄	C₂²	1	2	yes	complemented
95	C₄×D₄	C₂²	1	2	yes	complemented
96	#117	C₂²	1	1	yes	characteristic, complemented
97	#117	C₂²	1	2	yes	complemented
98	C₂×C₄⋊C₄	C₂²	1	1	yes	characteristic, complemented
99	C₂×C₄⋊C₄	C₂²	1	1	yes	characteristic, complemented
100	C₂×C₄⋊C₄	C₂²	1	1	yes	characteristic, complemented
101	C₂×C₂²⋊C₄	C₂²	1	2	yes	complemented
102	C₂×C₂²⋊C₄	C₂²	1	1	yes	characteristic, complemented
103	C₂×C₂²⋊C₄	C₂²	1	1	yes	characteristic, complemented
104	C₂×C₂²⋊C₄	C₂²	1	1	yes	characteristic, complemented
105	C₂×C₂²⋊C₄	C₂²	1	2	yes	complemented
106	C₂×C₂²⋊C₄	C₂²	1	2	yes	complemented
107	C₂⁴	C₂³	1	1	yes	characteristic
108	C₂⁴	C₂³	1	2	yes	—
109	C₂⁴	C₂³	1	1	yes	characteristic
110	C₂⁴	C₂³	1	2	yes	—
111	#41	—	4	2	no	—
112	#41	—	4	4	no	—
113	C₂×Q₈	C₂³	1	1	yes	characteristic
114	C₂×Q₈	C₂³	1	2	yes	—
115	C₂×D₄	—	4	2	no	—
116	C₂×D₄	—	4	2	no	—
117	C₂×D₄	—	4	4	no	—
118	C₂×D₄	—	4	4	no	—
119	C₂×D₄	—	4	4	no	—
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	2	yes	—
127	C₂×D₄	C₂³	1	2	yes	—
128	C₂×D₄	C₂³	1	2	yes	—
129	C₂×D₄	C₂³	1	1	yes	characteristic
130	C₂×D₄	C₂³	1	2	yes	—
131	C₂×D₄	C₂³	1	1	yes	characteristic
132	C₂×D₄	C₂³	1	2	yes	—
133	C₂×D₄	C₂³	1	2	yes	—
134	C₂×D₄	C₂³	1	2	yes	—
135	C₂×D₄	C₂³	1	2	yes	—
136	C₂×D₄	C₂³	1	2	yes	—
137	C₂×D₄	C₂³	1	1	yes	characteristic
138	C₂×D₄	C₂³	1	2	yes	—
139	C₂×D₄	C₂³	1	2	yes	—
140	C₂²×C₄	—	4	2	no	—
141	C₂²×C₄	C₂³	1	1	yes	characteristic
142	C₂²×C₄	C₂³	1	1	yes	characteristic
143	C₂²×C₄	C₂³	1	1	yes	characteristic
144	C₂²×C₄	C₂³	1	1	yes	characteristic
145	C₂²×C₄	C₂³	1	2	yes	—
146	C₂²×C₄	C₂³	1	1	yes	characteristic
147	C₂²×C₄	C₂³	1	1	yes	characteristic
148	C₂²×C₄	C₂³	1	2	yes	—
149	C₂²×C₄	C₂³	1	2	yes	—
150	C₂²×C₄	C₂³	1	2	yes	—
151	C₂²×C₄	C₂³	1	2	yes	—
152	C₂²×C₄	C₂³	1	1	yes	characteristic
153	C₂²×C₄	C₂³	1	1	yes	characteristic
154	C₂²×C₄	C₂³	1	2	yes	—
155	C₂²×C₄	C₂³	1	2	yes	—
156	C₂²×C₄	C₂³	1	1	yes	characteristic
157	C₄⋊C₄	C₂³	1	2	yes	—
158	C₄⋊C₄	C₂³	1	1	yes	characteristic
159	C₄⋊C₄	C₂³	1	1	yes	characteristic
160	C₄⋊C₄	C₂³	1	2	yes	—
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	1	yes	characteristic
165	C₄⋊C₄	C₂³	1	2	yes	—
166	C₄⋊C₄	C₂³	1	1	yes	characteristic
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	C₄⋊C₄	C₂³	1	2	yes	—
172	C₄⋊C₄	C₂³	1	2	yes	—
173	C₄⋊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	C₂²⋊C₄	C₂³	1	2	yes	—
195	C₂²⋊C₄	C₂³	1	2	yes	—
196	C₂²⋊C₄	C₂³	1	2	yes	—
197	C₂²⋊C₄	C₂³	1	2	yes	—
198	C₂²⋊C₄	C₂³	1	2	yes	—
199	C₄²	C₂³	1	2	yes	—
200	C₄²	C₂³	1	2	yes	—
201	C₄²	C₂³	1	2	yes	—
202	C₂³	—	2	2	no	—
203	C₂³	—	2	2	no	—
204	C₂³	—	2	2	no	—
205	C₂³	—	2	4	no	—
206	C₂³	—	4	1	no	—
207	C₂³	—	4	1	no	—
208	C₂³	—	4	2	no	—
209	C₂³	—	4	2	no	—
210	C₂³	—	4	2	no	—
211	C₂³	—	4	1	no	—
212	C₂³	—	4	2	no	—
213	C₂³	—	4	2	no	—
214	C₂³	C₂⁴	1	2	yes	—
215	C₂³	C₂⁴	1	2	yes	—
216	C₂³	C₂⁴	1	1	yes	characteristic
217	C₂³	C₂⁴	1	2	yes	—
218	C₂³	C₂⁴	1	2	yes	—
219	C₂³	C₂⁴	1	1	yes	characteristic
220	C₂³	C₂⁴	1	1	yes	characteristic
221	Q₈	—	4	1	no	—
222	Q₈	—	4	2	no	—
223	D₄	—	2	4	no	—
224	D₄	—	2	4	no	—
225	D₄	—	2	4	no	—
226	D₄	—	2	4	no	—
227	D₄	—	2	4	no	—
228	D₄	—	4	1	no	—
229	D₄	—	4	2	no	—
230	D₄	—	4	1	no	—
231	D₄	—	4	2	no	—
232	D₄	—	4	2	no	—
233	D₄	—	4	2	no	—
234	D₄	—	4	2	no	—
235	D₄	—	4	2	no	—
236	D₄	—	4	2	no	—
237	D₄	—	4	2	no	—
238	D₄	—	4	2	no	—
239	D₄	—	4	2	no	—
240	D₄	—	4	2	no	—
241	D₄	—	4	1	no	—
242	D₄	—	4	2	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	4	1	no	—
247	C₂×C₄	—	4	1	no	—
248	C₂×C₄	—	4	1	no	—
249	C₂×C₄	—	4	1	no	—
250	C₂×C₄	—	4	1	no	—
251	C₂×C₄	—	4	2	no	—
252	C₂×C₄	—	4	1	no	—
253	C₂×C₄	—	4	2	no	—
254	C₂×C₄	—	4	2	no	—
255	C₂×C₄	—	4	2	no	—
256	C₂×C₄	—	4	2	no	—
257	C₂×C₄	—	4	2	no	—
258	C₂×C₄	—	4	2	no	—
259	C₂×C₄	C₂⁴	1	2	yes	—
260	C₂×C₄	C₂⁴	1	1	yes	characteristic
261	C₂×C₄	C₂⁴	1	2	yes	—
262	C₂×C₄	C₂⁴	1	2	yes	—
263	C₂×C₄	C₂⁴	1	2	yes	—
264	C₂×C₄	C₂⁴	1	2	yes	—
265	C₂×C₄	C₂⁴	1	2	yes	—
266	C₂×C₄	C₂⁴	1	2	yes	—
267	C₂×C₄	C₂⁴	1	2	yes	—
268	C₂×C₄	C₂⁴	1	1	yes	characteristic
269	C₂×C₄	C₂⁴	1	1	yes	characteristic
270	C₂×C₄	C₂⁴	1	1	yes	characteristic
271	C₂²	—	2	1	no	—
272	C₂²	—	2	1	no	—
273	C₂²	—	2	1	no	—
274	C₂²	—	2	1	no	—
275	C₂²	—	2	1	no	—
276	C₂²	—	2	2	no	—
277	C₂²	—	2	2	no	—
278	C₂²	—	2	2	no	—
279	C₂²	—	2	2	no	—
280	C₂²	—	2	2	no	—
281	C₂²	—	2	2	no	—
282	C₂²	—	2	2	no	—
283	C₂²	—	2	1	no	—
284	C₂²	—	2	2	no	—
285	C₂²	—	2	2	no	—
286	C₂²	—	2	1	no	—
287	C₂²	—	2	2	no	—
288	C₂²	—	2	2	no	—
289	C₂²	—	2	2	no	—
290	C₂²	—	4	4	no	—
291	C₂²	—	8	4	no	—
292	C₂²	—	8	2	no	—
293	C₂²	—	8	4	no	—
294	C₂²	H(5,𝔽₂)	1	2	yes	—
295	C₂²	H(5,𝔽₂)	1	2	yes	—
296	C₂²	C₂⁵	1	1	yes	characteristic
297	C₄	—	2	1	no	—
298	C₄	—	2	1	no	—
299	C₄	—	2	1	no	—
300	C₄	—	2	1	no	—
301	C₄	—	2	2	no	—
302	C₄	—	2	2	no	—
303	C₄	—	2	2	no	—
304	C₄	—	2	2	no	—
305	C₄	—	2	2	no	—
306	C₄	—	2	2	no	—
307	C₄	—	2	2	no	—
308	C₄	—	2	2	no	—
309	C₂	—	2	2	no	—
310	C₂	—	2	2	no	—
311	C₂	—	4	1	no	—
312	C₂	—	4	2	no	—
313	C₂	—	4	2	no	—
314	C₂	—	4	2	no	—
315	C₂	—	4	2	no	—
316	C₂	C₂×H(5,𝔽₂)	1	1	yes	characteristic
317	C₂	C₂×H(5,𝔽₂)	1	1	yes	characteristic
318	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. #562	C₂	semidirect
2. #560	C₂	semidirect
3. #553	C₂	semidirect
4. #551	C₂	semidirect
5. #546	C₂	semidirect
6. #546	C₂	semidirect
7. #546	C₂	semidirect
8. D₄²	C₂	semidirect
9. #540	C₂	semidirect
10. #539	C₂	semidirect
11. #539	C₂	semidirect
12. #538	C₂	semidirect
13. #538	C₂	semidirect
14. #537	C₂	semidirect
15. #534	C₂	semidirect
16. #534	C₂	semidirect
17. #525	C₂	semidirect
18. C₂×#123	C₂	semidirect
19. C₂×#121	C₂	semidirect
20. C₂×#41	C₂²	semidirect
21. C₂×#41	C₂²	semidirect
23. C₂²×D₄	C₂²	semidirect
24. C₂²×D₄	C₂²	semidirect
25. C₂²×D₄	C₂²	semidirect
26. C₂²×D₄	C₂²	semidirect
27. C₂³×C₄	C₂²	semidirect
28. #127	C₂²	semidirect
29. #126	C₂²	semidirect
30. #126	C₂²	semidirect
31. #126	C₂²	semidirect
32. #126	C₂²	semidirect
33. #125	C₂²	semidirect
34. #125	C₂²	semidirect
35. #124	C₂²	semidirect
36. #124	C₂²	semidirect
37. #124	C₂²	semidirect
38. #123	C₂²	semidirect
39. #123	C₂²	semidirect
40. #123	C₂²	semidirect
41. #123	C₂²	semidirect
42. #123	C₂²	semidirect
43. #123	C₂²	semidirect
44. #123	C₂²	semidirect
45. #123	C₂²	semidirect
46. #123	C₂²	semidirect
47. #123	C₂²	semidirect
48. #123	C₂²	semidirect
49. #123	C₂²	semidirect
50. #123	C₂²	semidirect
51. #123	C₂²	semidirect
52. C₂²⋊Q₈	C₂²	semidirect
53. C₂²⋊Q₈	C₂²	semidirect
54. C₂²⋊Q₈	C₂²	semidirect
55. C₂²⋊Q₈	C₂²	semidirect
56. C₂²⋊Q₈	C₂²	semidirect
57. C₂²⋊Q₈	C₂²	semidirect
58. C₂²⋊Q₈	C₂²	semidirect
61. #121	C₂²	semidirect
62. #121	C₂²	semidirect
63. #121	C₂²	semidirect
64. #121	C₂²	semidirect
65. #121	C₂²	semidirect
66. #121	C₂²	semidirect
67. #121	C₂²	semidirect
68. #121	C₂²	semidirect
69. #121	C₂²	semidirect
70. #121	C₂²	semidirect
71. #121	C₂²	semidirect
72. #121	C₂²	semidirect
73. #121	C₂²	semidirect
74. #121	C₂²	semidirect
75. #121	C₂²	semidirect
76. #121	C₂²	semidirect
77. #121	C₂²	semidirect
78. #121	C₂²	semidirect
79. #121	C₂²	semidirect
81. #120	C₂²	semidirect
82. #120	C₂²	semidirect
83. #120	C₂²	semidirect
84. #120	C₂²	semidirect
85. #120	C₂²	semidirect
86. #120	C₂²	semidirect
87. #120	C₂²	semidirect
88. C₄×D₄	C₂²	semidirect
89. C₄×D₄	C₂²	semidirect
90. C₄×D₄	C₂²	semidirect
91. C₄×D₄	C₂²	semidirect
92. C₄×D₄	C₂²	semidirect
93. C₄×D₄	C₂²	semidirect
94. C₄×D₄	C₂²	semidirect
95. C₄×D₄	C₂²	semidirect
96. #117	C₂²	semidirect
97. #117	C₂²	semidirect
98. C₂×C₄⋊C₄	C₂²	semidirect
99. C₂×C₄⋊C₄	C₂²	semidirect
100. C₂×C₄⋊C₄	C₂²	semidirect
101. C₂×C₂²⋊C₄	C₂²	semidirect
102. C₂×C₂²⋊C₄	C₂²	semidirect
103. C₂×C₂²⋊C₄	C₂²	semidirect
104. C₂×C₂²⋊C₄	C₂²	semidirect
105. C₂×C₂²⋊C₄	C₂²	semidirect
106. C₂×C₂²⋊C₄	C₂²	semidirect

Maximal subgroups

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

subgroup	index
#562	2
#560	2
#553	2
#551	2
#546	2
D₄²	2
#540	2
#539	2
#538	2
#537	2
#534	2
#525	2
C₂×#123	2
C₂×#121	2