Subgroups — #3564

Order 128.

468 conjugacy classes of subgroups.

The subgroup lattice has 271 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	#563	C₂	1	2	yes	maximal, complemented
3	#556	C₂	1	2	yes	maximal, complemented
4	#554	C₂	1	2	yes	maximal, complemented
5	#552	C₂	1	2	yes	maximal, complemented
6	#552	C₂	1	2	yes	maximal, complemented
7	#551	C₂	1	2	yes	maximal, complemented
8	#551	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	#541	C₂	1	2	yes	maximal, complemented
13	#541	C₂	1	2	yes	maximal, complemented
14	#539	C₂	1	1	yes	maximal, characteristic, complemented
15	#539	C₂	1	2	yes	maximal, complemented
16	#538	C₂	1	1	yes	maximal, characteristic
17	#533	C₂	1	1	yes	maximal, characteristic, complemented
18	#529	C₂	1	1	yes	maximal, characteristic, complemented
19	C₂×#126	C₂	1	1	yes	maximal, characteristic, complemented
20	C₄⋊Q₈	C₂²	1	1	yes	characteristic, complemented
21	C₄⋊Q₈	C₂²	1	1	yes	characteristic, complemented
22	C₄⋊Q₈	C₂²	1	2	yes	complemented
23	C₄⋊Q₈	C₂²	1	2	yes	complemented
24	C₄⋊Q₈	C₂²	1	2	yes	complemented
25	#126	C₂²	1	2	yes	—
26	#126	C₂²	1	2	yes	—
27	#126	C₂²	1	2	yes	—
28	#126	C₂²	1	2	yes	—
29	#126	C₂²	1	2	yes	—
30	#126	C₂²	1	2	yes	—
31	#126	C₂²	1	2	yes	—
32	#126	C₂²	1	2	yes	—
33	#126	C₂²	1	2	yes	—
34	#126	C₂²	1	2	yes	complemented
35	#126	C₂²	1	2	yes	complemented
36	#126	C₂²	1	2	yes	complemented
37	#126	C₂²	1	2	yes	complemented
38	#125	C₂²	1	1	yes	characteristic, complemented
39	#125	C₂²	1	2	yes	complemented
40	#125	C₂²	1	2	yes	complemented
41	#125	C₂²	1	2	yes	complemented
42	#125	C₂²	1	2	yes	complemented
43	#125	C₂²	1	2	yes	complemented
44	#125	C₂²	1	2	yes	complemented
45	#125	C₂²	1	2	yes	complemented
46	#125	C₂²	1	2	yes	complemented
47	#125	C₂²	1	2	yes	complemented
48	#124	C₂²	1	2	yes	—
49	#124	C₂²	1	2	yes	—
50	#124	C₂²	1	2	yes	—
51	#124	C₂²	1	1	yes	characteristic, complemented
52	#123	C₂²	1	2	yes	—
53	#123	C₂²	1	2	yes	—
54	#123	C₂²	1	2	yes	—
55	#123	C₂²	1	2	yes	—
56	#123	C₂²	1	2	yes	—
57	#123	C₂²	1	2	yes	—
58	#123	C₂²	1	2	yes	—
59	#123	C₂²	1	2	yes	complemented
60	C₂²⋊Q₈	C₂²	1	1	yes	characteristic
61	C₂²⋊Q₈	C₂²	1	1	yes	characteristic
62	C₂²⋊Q₈	C₂²	1	1	yes	characteristic
63	C₂²⋊Q₈	C₂²	1	2	yes	—
64	C₂²⋊Q₈	C₂²	1	2	yes	—
65	C₂²⋊Q₈	C₂²	1	2	yes	—
66	C₂²⋊Q₈	C₂²	1	2	yes	—
67	C₂²⋊Q₈	C₂²	1	2	yes	—
68	C₂²⋊Q₈	C₂²	1	2	yes	—
69	C₂²⋊Q₈	C₂²	1	2	yes	—
70	C₂²⋊Q₈	C₂²	1	2	yes	—
71	C₂²⋊Q₈	C₂²	1	2	yes	—
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	#120	C₂²	1	2	yes	—
80	C₄×Q₈	C₂²	1	1	yes	characteristic, complemented
81	C₄×Q₈	C₂²	1	2	yes	complemented
82	C₄×Q₈	C₂²	1	1	yes	characteristic, complemented
83	C₄×Q₈	C₂²	1	1	yes	characteristic, complemented
84	C₄×Q₈	C₂²	1	2	yes	complemented
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	—
91	C₄×D₄	C₂²	1	1	yes	characteristic, complemented
92	#117	C₂²	1	2	yes	—
93	#117	C₂²	1	2	yes	—
94	#117	C₂²	1	2	yes	—
95	#117	C₂²	1	2	yes	—
96	#117	C₂²	1	2	yes	complemented
97	#117	C₂²	1	2	yes	complemented
98	C₂×C₄⋊C₄	C₂²	1	1	yes	characteristic
99	C₂×C₄⋊C₄	C₂²	1	1	yes	characteristic
100	C₂×C₄⋊C₄	C₂²	1	1	yes	characteristic
101	C₂×C₄⋊C₄	C₂²	1	2	yes	complemented
102	C₂×C₂²⋊C₄	C₂²	1	1	yes	characteristic
103	C₂×C₂²⋊C₄	C₂²	1	1	yes	characteristic
104	C₂×C₂²⋊C₄	C₂²	1	1	yes	characteristic
105	C₂×C₂²⋊C₄	C₂²	1	2	yes	—
106	C₂×C₄²	C₂²	1	1	yes	characteristic
107	C₂⁴	C₂³	1	1	yes	characteristic
108	C₂×Q₈	C₂³	1	1	yes	characteristic
109	C₂×Q₈	C₂³	1	2	yes	—
110	C₂×Q₈	C₂³	1	1	yes	characteristic
111	C₂×Q₈	C₂³	1	2	yes	—
112	C₂×Q₈	C₂³	1	2	yes	—
113	C₂×Q₈	C₂³	1	1	yes	characteristic
114	C₂×D₄	C₂³	1	2	yes	—
115	C₂×D₄	C₂³	1	2	yes	—
116	C₂×D₄	C₂³	1	2	yes	—
117	C₂×D₄	C₂³	1	1	yes	characteristic
118	C₂²×C₄	C₂³	1	1	yes	characteristic
119	C₂²×C₄	C₂³	1	1	yes	characteristic
120	C₂²×C₄	C₂³	1	1	yes	characteristic
121	C₂²×C₄	C₂³	1	1	yes	characteristic
122	C₂²×C₄	C₂³	1	1	yes	characteristic
123	C₂²×C₄	C₂³	1	1	yes	characteristic
124	C₂²×C₄	C₂³	1	2	yes	—
125	C₂²×C₄	C₂³	1	2	yes	—
126	C₂²×C₄	C₂³	1	2	yes	—
127	C₂²×C₄	C₂³	1	2	yes	—
128	C₂²×C₄	C₂³	1	2	yes	—
129	C₄⋊C₄	C₂³	1	2	yes	—
130	C₄⋊C₄	C₂³	1	2	yes	—
131	C₄⋊C₄	C₂³	1	2	yes	—
132	C₄⋊C₄	C₂³	1	2	yes	—
133	C₄⋊C₄	C₂³	1	2	yes	—
134	C₄⋊C₄	C₂³	1	2	yes	—
135	C₄⋊C₄	C₂³	1	2	yes	—
136	C₄⋊C₄	C₂³	1	2	yes	—
137	C₄⋊C₄	C₂³	1	2	yes	—
138	C₄⋊C₄	C₂³	1	2	yes	—
139	C₄⋊C₄	C₂³	1	2	yes	—
140	C₄⋊C₄	C₂³	1	2	yes	—
141	C₄⋊C₄	C₂³	1	2	yes	—
142	C₄⋊C₄	C₂³	1	2	yes	—
143	C₄⋊C₄	C₂³	1	2	yes	—
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	1	yes	characteristic
153	C₄⋊C₄	C₂³	1	2	yes	—
154	C₄⋊C₄	C₂³	1	1	yes	characteristic
155	C₄⋊C₄	C₂³	1	2	yes	—
156	C₄⋊C₄	C₂³	1	2	yes	—
157	C₄⋊C₄	C₂³	1	1	yes	characteristic
158	C₄⋊C₄	C₂³	1	2	yes	—
159	C₄⋊C₄	C₂³	1	1	yes	characteristic
160	C₄⋊C₄	C₂³	1	2	yes	—
161	C₄⋊C₄	C₂³	1	2	yes	—
162	C₂²⋊C₄	C₂³	1	2	yes	—
163	C₂²⋊C₄	C₂³	1	2	yes	—
164	C₂²⋊C₄	C₂³	1	2	yes	—
165	C₂²⋊C₄	C₂³	1	2	yes	—
166	C₂²⋊C₄	C₂³	1	2	yes	—
167	C₂²⋊C₄	C₂³	1	2	yes	—
168	C₂²⋊C₄	C₂³	1	2	yes	—
169	C₂²⋊C₄	C₂³	1	2	yes	—
170	C₂²⋊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	C₄²	C₂³	1	1	yes	characteristic
184	C₄²	C₂³	1	1	yes	characteristic
185	C₄²	C₂³	1	1	yes	characteristic
186	C₄²	C₂³	1	1	yes	characteristic
187	C₄²	C₂³	1	2	yes	—
188	C₄²	C₂³	1	2	yes	—
189	C₄²	C₂³	1	2	yes	—
190	C₄²	C₂³	1	2	yes	—
191	C₄²	C₂³	1	2	yes	—
192	C₄²	C₂³	1	2	yes	—
193	C₄²	C₂³	1	2	yes	—
194	C₂³	—	2	2	no	—
195	C₂³	—	4	1	no	—
196	C₂³	—	4	1	no	—
197	C₂³	C₂⁴	1	2	yes	—
198	C₂³	C₂⁴	1	2	yes	—
199	C₂³	C₂⁴	1	1	yes	characteristic
200	Q₈	—	4	1	no	—
201	Q₈	—	4	2	no	—
202	Q₈	—	4	2	no	—
203	Q₈	—	4	1	no	—
204	Q₈	—	4	2	no	—
205	Q₈	—	4	1	no	—
206	D₄	—	2	4	no	—
207	D₄	—	4	2	no	—
208	D₄	—	4	1	no	—
209	D₄	—	4	2	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	4	2	no	—
213	C₂×C₄	—	4	1	no	—
214	C₂×C₄	—	4	1	no	—
215	C₂×C₄	—	4	1	no	—
216	C₂×C₄	—	4	2	no	—
217	C₂×C₄	—	4	2	no	—
218	C₂×C₄	—	4	2	no	—
219	C₂×C₄	—	4	2	no	—
220	C₂×C₄	—	4	1	no	—
221	C₂×C₄	C₂⁴	1	1	yes	characteristic
222	C₂×C₄	C₂⁴	1	1	yes	characteristic
223	C₂×C₄	C₂⁴	1	2	yes	—
224	C₂×C₄	C₂⁴	1	2	yes	—
225	C₂×C₄	C₂⁴	1	1	yes	characteristic
226	C₂×C₄	C₂⁴	1	2	yes	—
227	C₂×C₄	C₂⁴	1	1	yes	characteristic
228	C₂×C₄	C₂⁴	1	1	yes	characteristic
229	C₂×C₄	C₂⁴	1	2	yes	—
230	C₂×C₄	C₂⁴	1	2	yes	—
231	C₂×C₄	C₂⁴	1	1	yes	characteristic
232	C₂×C₄	C₂⁴	1	2	yes	—
233	C₂×C₄	C₂⁴	1	2	yes	—
234	C₂×C₄	C₂⁴	1	2	yes	—
235	C₂×C₄	C₂⁴	1	2	yes	—
236	C₂×C₄	C₂⁴	1	2	yes	—
237	C₂²	—	2	1	no	—
238	C₂²	—	2	1	no	—
239	C₂²	—	2	2	no	—
240	C₂²	—	2	2	no	—
241	C₂²	—	2	2	no	—
242	C₂²	—	2	2	no	—
243	C₂²	—	2	2	no	—
244	C₂²	—	2	2	no	—
245	C₂²	—	8	2	no	—
246	C₂²	2₋¹⁺⁴	1	2	yes	—
247	C₂²	C₂⁵	1	1	yes	characteristic
248	C₄	—	2	1	no	—
249	C₄	—	2	1	no	—
250	C₄	—	2	2	no	—
251	C₄	—	2	2	no	—
252	C₄	—	2	1	no	—
253	C₄	—	2	2	no	—
254	C₄	—	2	2	no	—
255	C₄	—	2	2	no	—
256	C₄	—	2	2	no	—
257	C₄	—	2	2	no	—
258	C₄	—	2	1	no	—
259	C₄	—	2	1	no	—
260	C₄	—	2	2	no	—
261	C₄	—	2	2	no	—
262	C₄	—	2	1	no	—
263	C₄	—	2	2	no	—
264	C₂	—	2	2	no	—
265	C₂	—	4	2	no	—
266	C₂	—	4	2	no	—
267	C₂	C₂×2₋¹⁺⁴	1	1	yes	characteristic
268	C₂	#585	1	1	yes	characteristic
269	C₂	#585	1	1	yes	characteristic

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂²	C₂⁵
prime radical	#538	C₂
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. #563	C₂	semidirect
3. #556	C₂	semidirect
4. #554	C₂	semidirect
5. #552	C₂	semidirect
6. #552	C₂	semidirect
7. #551	C₂	semidirect
8. #551	C₂	semidirect
9. #544	C₂	semidirect
10. #543	C₂	semidirect
11. #542	C₂	semidirect
12. #541	C₂	semidirect
13. #541	C₂	semidirect
14. #539	C₂	semidirect
15. #539	C₂	semidirect
17. #533	C₂	semidirect
18. #529	C₂	semidirect
19. C₂×#126	C₂	semidirect
20. C₄⋊Q₈	C₂²	semidirect
21. C₄⋊Q₈	C₂²	semidirect
22. C₄⋊Q₈	C₂²	semidirect
23. C₄⋊Q₈	C₂²	semidirect
24. C₄⋊Q₈	C₂²	semidirect
34. #126	C₂²	semidirect
35. #126	C₂²	semidirect
36. #126	C₂²	semidirect
37. #126	C₂²	semidirect
38. #125	C₂²	semidirect
39. #125	C₂²	semidirect
40. #125	C₂²	semidirect
41. #125	C₂²	semidirect
42. #125	C₂²	semidirect
43. #125	C₂²	semidirect
44. #125	C₂²	semidirect
45. #125	C₂²	semidirect
46. #125	C₂²	semidirect
47. #125	C₂²	semidirect
51. #124	C₂²	semidirect
59. #123	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
80. C₄×Q₈	C₂²	semidirect
81. C₄×Q₈	C₂²	semidirect
82. C₄×Q₈	C₂²	semidirect
83. C₄×Q₈	C₂²	semidirect
84. C₄×Q₈	C₂²	semidirect
85. C₄×Q₈	C₂²	semidirect
91. C₄×D₄	C₂²	semidirect
96. #117	C₂²	semidirect
97. #117	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
#556	2
#554	2
#552	2
#551	2
#544	2
#543	2
#542	2
#541	2
#539	2
#538	2
#533	2
#529	2
C₂×#126	2