Subgroups — #3475

Order 128.

566 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#556	C₂	1	1	yes	maximal, characteristic, complemented
2	#555	C₂	1	1	yes	maximal, characteristic, complemented
3	#554	C₂	1	1	yes	maximal, characteristic, complemented
4	#553	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	4	yes	maximal, complemented
8	#547	C₂	1	1	yes	maximal, characteristic, complemented
9	#546	C₂	1	2	yes	maximal, complemented
10	#533	C₂	1	1	yes	maximal, characteristic, complemented
11	#529	C₂	1	4	yes	maximal, complemented
12	#529	C₂	1	2	yes	maximal, complemented
13	#525	C₂	1	2	yes	maximal, complemented
14	C₄×#41	C₂	1	1	yes	maximal, characteristic, complemented
15	C₄×#41	C₂	1	2	yes	maximal, complemented
16	C₂×C₄×D₄	C₂	1	1	yes	maximal, characteristic, complemented
17	C₂×#117	C₂	1	2	yes	maximal, complemented
18	C₂×#41	C₂²	1	1	yes	characteristic
19	C₂×#41	C₂²	1	2	yes	—
20	C₂²×D₄	C₂²	1	1	yes	characteristic
21	C₂³×C₄	C₂²	1	2	yes	—
22	C₄⋊Q₈	C₂²	1	1	yes	characteristic, complemented
23	C₄⋊Q₈	C₂²	1	1	yes	characteristic, complemented
24	#126	C₂²	1	4	yes	complemented
25	#126	C₂²	1	4	yes	complemented
26	#126	C₂²	1	4	yes	complemented
27	#125	C₂²	1	4	yes	complemented
28	#125	C₂²	1	2	yes	complemented
29	#125	C₂²	1	2	yes	complemented
30	#124	C₂²	1	2	yes	—
31	#124	C₂²	1	4	yes	complemented
32	#123	C₂²	1	4	yes	—
33	#123	C₂²	1	4	yes	—
34	#123	C₂²	1	4	yes	complemented
35	#123	C₂²	1	4	yes	complemented
36	C₂²⋊Q₈	C₂²	1	2	yes	—
37	C₂²⋊Q₈	C₂²	1	2	yes	—
38	C₂²⋊Q₈	C₂²	1	2	yes	complemented
39	C₂²⋊Q₈	C₂²	1	2	yes	complemented
40	C₂²⋊Q₈	C₂²	1	2	yes	complemented
41	C₂²⋊Q₈	C₂²	1	4	yes	complemented
42	#121	C₂²	1	4	yes	—
43	#121	C₂²	1	2	yes	—
44	#121	C₂²	1	2	yes	complemented
45	#121	C₂²	1	2	yes	complemented
46	#120	C₂²	1	4	yes	—
47	C₄×Q₈	C₂²	1	2	yes	complemented
48	C₄×Q₈	C₂²	1	1	yes	characteristic, complemented
49	C₄×Q₈	C₂²	1	2	yes	complemented
50	C₄×Q₈	C₂²	1	1	yes	characteristic, complemented
51	C₄×Q₈	C₂²	1	2	yes	complemented
52	C₄×D₄	C₂²	1	1	yes	characteristic
53	C₄×D₄	C₂²	1	1	yes	characteristic
54	C₄×D₄	C₂²	1	2	yes	—
55	C₄×D₄	C₂²	1	1	yes	characteristic
56	C₄×D₄	C₂²	1	2	yes	—
57	C₄×D₄	C₂²	1	1	yes	characteristic
58	C₄×D₄	C₂²	1	2	yes	—
59	C₄×D₄	C₂²	1	2	yes	—
60	C₄×D₄	C₂²	1	2	yes	—
61	C₄×D₄	C₂²	1	4	yes	complemented
62	C₄×D₄	C₂²	1	1	yes	characteristic, complemented
63	C₄×D₄	C₂²	1	4	yes	complemented
64	C₄×D₄	C₂²	1	1	yes	characteristic, complemented
65	C₄×D₄	C₂²	1	4	yes	complemented
66	#117	C₂²	1	1	yes	characteristic
67	#117	C₂²	1	2	yes	—
68	#117	C₂²	1	4	yes	—
69	#117	C₂²	1	4	yes	complemented
70	#117	C₂²	1	2	yes	complemented
71	#117	C₂²	1	2	yes	complemented
72	#117	C₂²	1	2	yes	complemented
73	#117	C₂²	1	4	yes	complemented
74	#117	C₂²	1	2	yes	complemented
75	C₂×C₄⋊C₄	C₂²	1	1	yes	characteristic
76	C₂×C₄⋊C₄	C₂²	1	2	yes	complemented
77	C₂×C₄⋊C₄	C₂²	1	2	yes	complemented
78	C₂×C₂²⋊C₄	C₂²	1	4	yes	—
79	C₂×C₂²⋊C₄	C₂²	1	2	yes	—
80	C₂×C₄²	C₂²	1	1	yes	characteristic
81	C₂×C₄²	C₂²	1	4	yes	complemented
82	C₂×C₄²	C₂²	1	2	yes	complemented
83	C₂⁴	C₂³	1	2	yes	—
84	#41	—	2	8	no	—
85	#41	—	2	4	no	—
86	C₂×Q₈	C₂³	1	2	yes	—
87	C₂×Q₈	C₂³	1	1	yes	characteristic
88	C₂×Q₈	C₂³	1	2	yes	—
89	C₂×D₄	—	2	4	no	—
90	C₂×D₄	C₂³	1	2	yes	—
91	C₂×D₄	C₂³	1	2	yes	—
92	C₂×D₄	C₂³	1	1	yes	characteristic
93	C₂×D₄	C₂³	1	4	yes	—
94	C₂×D₄	C₂³	1	1	yes	characteristic
95	C₂×D₄	C₂³	1	4	yes	—
96	C₂×D₄	C₂³	1	1	yes	characteristic
97	C₂²×C₄	—	2	8	no	—
98	C₂²×C₄	C₂³	1	1	yes	characteristic
99	C₂²×C₄	C₂³	1	4	yes	—
100	C₂²×C₄	C₂³	1	2	yes	—
101	C₂²×C₄	C₂³	1	2	yes	—
102	C₂²×C₄	C₂³	1	2	yes	—
103	C₂²×C₄	C₂³	1	2	yes	—
104	C₂²×C₄	C₂³	1	2	yes	—
105	C₂²×C₄	C₂³	1	2	yes	—
106	C₂²×C₄	C₂³	1	2	yes	—
107	C₂²×C₄	C₂³	1	1	yes	characteristic
108	C₂²×C₄	C₂³	1	4	yes	—
109	C₂²×C₄	C₂³	1	2	yes	—
110	C₂²×C₄	C₂³	1	1	yes	characteristic
111	C₄⋊C₄	C₂³	1	1	yes	characteristic
112	C₄⋊C₄	C₂³	1	1	yes	characteristic
113	C₄⋊C₄	C₂³	1	2	yes	—
114	C₄⋊C₄	C₂³	1	1	yes	characteristic
115	C₄⋊C₄	C₂³	1	1	yes	characteristic
116	C₄⋊C₄	C₂³	1	2	yes	—
117	C₄⋊C₄	C₂³	1	2	yes	—
118	C₄⋊C₄	C₂³	1	4	yes	—
119	C₄⋊C₄	C₂³	1	4	yes	—
120	C₄⋊C₄	C₂³	1	2	yes	—
121	C₄⋊C₄	C₂³	1	2	yes	—
122	C₄⋊C₄	C₂³	1	2	yes	—
123	C₄⋊C₄	C₂³	1	4	yes	—
124	C₄⋊C₄	C₂³	1	4	yes	—
125	C₄⋊C₄	C₂³	1	4	yes	—
126	C₄⋊C₄	C₂³	1	4	yes	—
127	C₄⋊C₄	C₂³	1	2	yes	—
128	C₂²⋊C₄	C₂³	1	2	yes	—
129	C₂²⋊C₄	C₂³	1	4	yes	—
130	C₂²⋊C₄	C₂³	1	2	yes	—
131	C₂²⋊C₄	C₂³	1	4	yes	—
132	C₂²⋊C₄	C₂³	1	4	yes	—
133	C₂²⋊C₄	C₂³	1	2	yes	—
134	C₂²⋊C₄	C₂³	1	2	yes	—
135	C₂²⋊C₄	C₂³	1	2	yes	—
136	C₂²⋊C₄	C₂³	1	4	yes	—
137	C₂²⋊C₄	C₂³	1	2	yes	—
138	C₂²⋊C₄	C₂³	1	2	yes	—
139	C₂²⋊C₄	C₂³	1	4	yes	—
140	C₂²⋊C₄	C₂³	1	4	yes	—
141	C₂²⋊C₄	C₂³	1	4	yes	—
142	C₄²	C₂³	1	1	yes	characteristic
143	C₄²	C₂³	1	1	yes	characteristic
144	C₄²	C₂³	1	4	yes	—
145	C₄²	C₂³	1	4	yes	—
146	C₄²	C₂³	1	2	yes	—
147	C₄²	C₂³	1	2	yes	—
148	C₄²	C₂³	1	1	yes	characteristic
149	C₄²	C₂³	1	1	yes	characteristic
150	C₄²	C₂³	1	2	yes	—
151	C₄²	C₂³	1	4	yes	—
152	C₂³	—	2	4	no	—
153	C₂³	—	2	4	no	—
154	C₂³	—	4	2	no	—
155	C₂³	C₂⁴	1	2	yes	—
156	C₂³	C₂⁴	1	2	yes	—
157	C₂³	C₂⁴	1	1	yes	characteristic
158	C₂³	C₂⁴	1	2	yes	—
159	Q₈	—	2	2	no	—
160	Q₈	—	2	4	no	—
161	Q₈	—	4	2	no	—
162	D₄	—	2	4	no	—
163	D₄	—	2	4	no	—
164	D₄	—	2	8	no	—
165	D₄	—	2	2	no	—
166	D₄	—	2	2	no	—
167	D₄	—	4	4	no	—
168	D₄	#41	1	4	yes	complemented
169	C₂×C₄	—	2	4	no	—
170	C₂×C₄	—	2	8	no	—
171	C₂×C₄	—	2	4	no	—
172	C₂×C₄	—	2	4	no	—
173	C₂×C₄	—	2	4	no	—
174	C₂×C₄	—	2	4	no	—
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	2	4	no	—
177	C₂×C₄	—	2	4	no	—
178	C₂×C₄	—	4	2	no	—
179	C₂×C₄	—	4	4	no	—
180	C₂×C₄	—	4	1	no	—
181	C₂×C₄	#41	1	4	yes	complemented
182	C₂×C₄	C₂⁴	1	2	yes	—
183	C₂×C₄	C₂⁴	1	2	yes	—
184	C₂×C₄	C₂⁴	1	1	yes	characteristic
185	C₂×C₄	C₂⁴	1	2	yes	—
186	C₂×C₄	C₂⁴	1	1	yes	characteristic
187	C₂×C₄	C₂⁴	1	4	yes	—
188	C₂×C₄	C₂⁴	1	2	yes	—
189	C₂×C₄	C₂⁴	1	4	yes	—
190	C₂×C₄	C₂⁴	1	1	yes	characteristic
191	C₂×C₄	C₂⁴	1	2	yes	—
192	C₂×C₄	C₂⁴	1	1	yes	characteristic
193	C₂×C₄	C₂⁴	1	1	yes	characteristic
194	C₂×C₄	C₂⁴	1	1	yes	characteristic
195	C₂²	—	2	2	no	—
196	C₂²	—	2	1	no	—
197	C₂²	—	2	2	no	—
198	C₂²	—	2	1	no	—
199	C₂²	—	2	2	no	—
200	C₂²	—	2	2	no	—
201	C₂²	—	2	2	no	—
202	C₂²	—	2	2	no	—
203	C₂²	—	2	2	no	—
204	C₂²	—	2	2	no	—
205	C₂²	—	4	8	no	—
206	C₂²	C₂×#41	1	2	yes	—
207	C₂²	C₂×#41	1	4	yes	—
208	C₂²	C₂⁵	1	1	yes	characteristic
209	C₄	—	2	2	no	—
210	C₄	—	2	2	no	—
211	C₄	—	2	2	no	—
212	C₄	—	2	1	no	—
213	C₄	—	2	4	no	—
214	C₄	—	2	2	no	—
215	C₄	—	2	1	no	—
216	C₄	—	2	2	no	—
217	C₄	—	2	4	no	—
218	C₄	—	2	1	no	—
219	C₄	C₂×#41	1	2	yes	—
220	C₄	C₂×#41	1	2	yes	—
221	C₄	C₂×#41	1	2	yes	—
222	C₂	—	2	2	no	—
223	C₂	—	2	4	no	—
224	C₂	—	4	2	no	—
225	C₂	—	4	2	no	—
226	C₂	C₂²×#41	1	1	yes	characteristic
227	C₂	C₂²×#41	1	1	yes	characteristic
228	C₂	#585	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂⁴
commutator, Frattini, socle, cosocle	C₂²	C₂⁵
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ G
LECS	G ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #556	C₂	semidirect
2. #555	C₂	semidirect
3. #554	C₂	semidirect
4. #553	C₂	semidirect
5. #552	C₂	semidirect
6. #552	C₂	semidirect
7. #551	C₂	semidirect
8. #547	C₂	semidirect
9. #546	C₂	semidirect
10. #533	C₂	semidirect
11. #529	C₂	semidirect
12. #529	C₂	semidirect
13. #525	C₂	semidirect
14. C₄×#41	C₂	semidirect
15. C₄×#41	C₂	semidirect
16. C₂×C₄×D₄	C₂	semidirect
17. C₂×#117	C₂	semidirect
22. C₄⋊Q₈	C₂²	semidirect
23. C₄⋊Q₈	C₂²	semidirect
24. #126	C₂²	semidirect
25. #126	C₂²	semidirect
26. #126	C₂²	semidirect
27. #125	C₂²	semidirect
28. #125	C₂²	semidirect
29. #125	C₂²	semidirect
31. #124	C₂²	semidirect
34. #123	C₂²	semidirect
35. #123	C₂²	semidirect
38. C₂²⋊Q₈	C₂²	semidirect
39. C₂²⋊Q₈	C₂²	semidirect
40. C₂²⋊Q₈	C₂²	semidirect
41. C₂²⋊Q₈	C₂²	semidirect
44. #121	C₂²	semidirect
45. #121	C₂²	semidirect
47. C₄×Q₈	C₂²	semidirect
48. C₄×Q₈	C₂²	semidirect
49. C₄×Q₈	C₂²	semidirect
50. C₄×Q₈	C₂²	semidirect
51. C₄×Q₈	C₂²	semidirect
61. C₄×D₄	C₂²	semidirect
62. C₄×D₄	C₂²	semidirect
63. C₄×D₄	C₂²	semidirect
64. C₄×D₄	C₂²	semidirect
65. C₄×D₄	C₂²	semidirect
69. #117	C₂²	semidirect
70. #117	C₂²	semidirect
71. #117	C₂²	semidirect
72. #117	C₂²	semidirect
73. #117	C₂²	semidirect
74. #117	C₂²	semidirect
76. C₂×C₄⋊C₄	C₂²	semidirect
77. C₂×C₄⋊C₄	C₂²	semidirect
81. C₂×C₄²	C₂²	semidirect
82. C₂×C₄²	C₂²	semidirect
168. D₄	#41	semidirect
181. C₂×C₄	#41	semidirect

Maximal subgroups

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

subgroup	index
#556	2
#555	2
#554	2
#553	2
#552	2
#551	2
#547	2
#546	2
#533	2
#529	2
#525	2
C₄×#41	2
C₂×C₄×D₄	2
C₂×#117	2