Subgroups — #3495

Order 128.

511 conjugacy classes of subgroups.

The subgroup lattice has 226 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#556	C₂	1	2	yes	maximal, complemented
2	#554	C₂	1	2	yes	maximal, complemented
3	#552	C₂	1	2	yes	maximal, complemented
4	#552	C₂	1	4	yes	maximal, complemented
5	#552	C₂	1	2	yes	maximal, complemented
6	#551	C₂	1	4	yes	maximal, complemented
7	#544	C₂	1	1	yes	maximal, characteristic, complemented
8	#542	C₂	1	2	yes	maximal, complemented
9	#541	C₂	1	2	yes	maximal, complemented
10	#536	C₂	1	1	yes	maximal, characteristic, complemented
11	#533	C₂	1	2	yes	maximal, complemented
12	#529	C₂	1	2	yes	maximal, complemented
13	#525	C₂	1	1	yes	maximal, characteristic
14	C₂×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
15	#520	C₂	1	1	yes	maximal, characteristic, complemented
16	#519	C₂	1	1	yes	maximal, characteristic, complemented
17	C₂×#117	C₂	1	1	yes	maximal, characteristic, complemented
18	C₂×#41	C₂²	1	1	yes	characteristic
19	C₂²×Q₈	C₂²	1	1	yes	characteristic
20	C₂³×C₄	C₂²	1	1	yes	characteristic
21	C₄⋊Q₈	C₂²	1	2	yes	complemented
22	C₄⋊Q₈	C₂²	1	2	yes	complemented
23	C₄⋊Q₈	C₂²	1	2	yes	complemented
24	#126	C₂²	1	4	yes	—
25	#126	C₂²	1	4	yes	—
26	#126	C₂²	1	4	yes	complemented
27	#126	C₂²	1	4	yes	complemented
28	#125	C₂²	1	2	yes	complemented
29	#125	C₂²	1	4	yes	complemented
30	#125	C₂²	1	4	yes	complemented
31	#125	C₂²	1	4	yes	complemented
32	#125	C₂²	1	2	yes	complemented
33	#124	C₂²	1	4	yes	—
34	#124	C₂²	1	2	yes	complemented
35	#123	C₂²	1	2	yes	—
36	#123	C₂²	1	4	yes	—
37	#123	C₂²	1	4	yes	—
38	#123	C₂²	1	4	yes	—
39	C₂²⋊Q₈	C₂²	1	2	yes	—
40	C₂²⋊Q₈	C₂²	1	2	yes	—
41	C₂²⋊Q₈	C₂²	1	2	yes	—
42	C₂²⋊Q₈	C₂²	1	2	yes	—
43	C₂²⋊Q₈	C₂²	1	1	yes	characteristic
44	C₂²⋊Q₈	C₂²	1	1	yes	characteristic
45	C₂²⋊Q₈	C₂²	1	4	yes	—
46	C₂²⋊Q₈	C₂²	1	4	yes	—
47	C₂²⋊Q₈	C₂²	1	2	yes	complemented
48	C₂²⋊Q₈	C₂²	1	2	yes	complemented
49	C₂²⋊Q₈	C₂²	1	4	yes	complemented
50	#121	C₂²	1	1	yes	characteristic
51	#121	C₂²	1	1	yes	characteristic
52	#120	C₂²	1	2	yes	—
53	C₄×Q₈	C₂²	1	2	yes	complemented
54	C₄×Q₈	C₂²	1	2	yes	complemented
55	C₄×Q₈	C₂²	1	2	yes	complemented
56	C₄×Q₈	C₂²	1	2	yes	complemented
57	C₄×Q₈	C₂²	1	2	yes	complemented
58	C₄×Q₈	C₂²	1	4	yes	complemented
59	C₄×D₄	C₂²	1	4	yes	—
60	C₄×D₄	C₂²	1	2	yes	—
61	C₄×D₄	C₂²	1	2	yes	—
62	C₄×D₄	C₂²	1	4	yes	—
63	C₄×D₄	C₂²	1	2	yes	complemented
64	#117	C₂²	1	1	yes	characteristic
65	#117	C₂²	1	1	yes	characteristic
66	#117	C₂²	1	2	yes	—
67	#117	C₂²	1	1	yes	characteristic
68	#117	C₂²	1	2	yes	—
69	#117	C₂²	1	1	yes	characteristic
70	#117	C₂²	1	1	yes	characteristic
71	#117	C₂²	1	4	yes	—
72	#117	C₂²	1	4	yes	—
73	#117	C₂²	1	2	yes	complemented
74	#117	C₂²	1	4	yes	complemented
75	C₂×C₄⋊C₄	C₂²	1	1	yes	characteristic
76	C₂×C₄⋊C₄	C₂²	1	1	yes	characteristic
77	C₂×C₄⋊C₄	C₂²	1	1	yes	characteristic
78	C₂×C₄⋊C₄	C₂²	1	2	yes	—
79	C₂×C₄⋊C₄	C₂²	1	2	yes	complemented
80	C₂×C₂²⋊C₄	C₂²	1	2	yes	—
81	C₂×C₂²⋊C₄	C₂²	1	2	yes	—
82	C₂×C₄²	C₂²	1	2	yes	—
83	C₂⁴	C₂³	1	1	yes	characteristic
84	#41	—	2	4	no	—
85	C₂×Q₈	—	2	4	no	—
86	C₂×Q₈	C₂³	1	2	yes	—
87	C₂×Q₈	C₂³	1	2	yes	—
88	C₂×Q₈	C₂³	1	1	yes	characteristic
89	C₂×Q₈	C₂³	1	4	yes	—
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	2	yes	—
94	C₂²×C₄	—	2	4	no	—
95	C₂²×C₄	C₂³	1	1	yes	characteristic
96	C₂²×C₄	C₂³	1	2	yes	—
97	C₂²×C₄	C₂³	1	2	yes	—
98	C₂²×C₄	C₂³	1	1	yes	characteristic
99	C₂²×C₄	C₂³	1	1	yes	characteristic
100	C₂²×C₄	C₂³	1	2	yes	—
101	C₂²×C₄	C₂³	1	1	yes	characteristic
102	C₂²×C₄	C₂³	1	1	yes	characteristic
103	C₂²×C₄	C₂³	1	2	yes	—
104	C₂²×C₄	C₂³	1	2	yes	—
105	C₂²×C₄	C₂³	1	4	yes	—
106	C₂²×C₄	C₂³	1	1	yes	characteristic
107	C₄⋊C₄	C₂³	1	1	yes	characteristic
108	C₄⋊C₄	C₂³	1	2	yes	—
109	C₄⋊C₄	C₂³	1	1	yes	characteristic
110	C₄⋊C₄	C₂³	1	2	yes	—
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	4	yes	—
115	C₄⋊C₄	C₂³	1	2	yes	—
116	C₄⋊C₄	C₂³	1	4	yes	—
117	C₄⋊C₄	C₂³	1	4	yes	—
118	C₄⋊C₄	C₂³	1	1	yes	characteristic
119	C₄⋊C₄	C₂³	1	1	yes	characteristic
120	C₄⋊C₄	C₂³	1	4	yes	—
121	C₄⋊C₄	C₂³	1	4	yes	—
122	C₄⋊C₄	C₂³	1	4	yes	—
123	C₄⋊C₄	C₂³	1	4	yes	—
124	C₄⋊C₄	C₂³	1	4	yes	—
125	C₄⋊C₄	C₂³	1	2	yes	—
126	C₄⋊C₄	C₂³	1	2	yes	—
127	C₄⋊C₄	C₂³	1	4	yes	—
128	C₄⋊C₄	C₂³	1	4	yes	—
129	C₂²⋊C₄	C₂³	1	2	yes	—
130	C₂²⋊C₄	C₂³	1	2	yes	—
131	C₂²⋊C₄	C₂³	1	2	yes	—
132	C₂²⋊C₄	C₂³	1	2	yes	—
133	C₂²⋊C₄	C₂³	1	4	yes	—
134	C₂²⋊C₄	C₂³	1	4	yes	—
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	4	yes	—
139	C₂²⋊C₄	C₂³	1	4	yes	—
140	C₂²⋊C₄	C₂³	1	2	yes	—
141	C₂²⋊C₄	C₂³	1	2	yes	—
142	C₂²⋊C₄	C₂³	1	4	yes	—
143	C₄²	C₂³	1	2	yes	—
144	C₄²	C₂³	1	2	yes	—
145	C₄²	C₂³	1	2	yes	—
146	C₄²	C₂³	1	2	yes	—
147	C₄²	C₂³	1	4	yes	—
148	C₄²	C₂³	1	4	yes	—
149	C₄²	C₂³	1	4	yes	—
150	C₄²	C₂³	1	2	yes	—
151	C₂³	—	2	2	no	—
152	C₂³	—	2	2	no	—
153	C₂³	—	4	1	no	—
154	C₂³	C₂⁴	1	1	yes	characteristic
155	C₂³	C₂⁴	1	2	yes	—
156	C₂³	C₂⁴	1	1	yes	characteristic
157	C₂³	C₂⁴	1	1	yes	characteristic
158	Q₈	—	2	4	no	—
159	Q₈	—	2	4	no	—
160	Q₈	—	2	2	no	—
161	Q₈	—	4	4	no	—
162	D₄	—	2	4	no	—
163	D₄	—	2	4	no	—
164	D₄	—	2	2	no	—
165	D₄	—	4	2	no	—
166	C₂×C₄	—	2	2	no	—
167	C₂×C₄	—	2	4	no	—
168	C₂×C₄	—	2	4	no	—
169	C₂×C₄	—	2	2	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	—	2	4	no	—
174	C₂×C₄	—	4	4	no	—
175	C₂×C₄	—	4	2	no	—
176	C₂×C₄	—	4	2	no	—
177	C₂×C₄	#41	1	4	yes	—
178	C₂×C₄	C₂⁴	1	2	yes	—
179	C₂×C₄	C₂⁴	1	1	yes	characteristic
180	C₂×C₄	C₂⁴	1	2	yes	—
181	C₂×C₄	C₂⁴	1	2	yes	—
182	C₂×C₄	C₂⁴	1	2	yes	—
183	C₂×C₄	C₂⁴	1	2	yes	—
184	C₂×C₄	C₂⁴	1	4	yes	—
185	C₂×C₄	C₂⁴	1	2	yes	—
186	C₂×C₄	C₂⁴	1	2	yes	—
187	C₂×C₄	C₂⁴	1	4	yes	—
188	C₂×C₄	C₂⁴	1	1	yes	characteristic
189	C₂×C₄	C₂⁴	1	1	yes	characteristic
190	C₂×C₄	C₂⁴	1	1	yes	characteristic
191	C₂²	—	2	1	no	—
192	C₂²	—	2	1	no	—
193	C₂²	—	2	1	no	—
194	C₂²	—	2	1	no	—
195	C₂²	—	2	1	no	—
196	C₂²	—	2	2	no	—
197	C₂²	—	2	2	no	—
198	C₂²	—	2	1	no	—
199	C₂²	—	2	2	no	—
200	C₂²	—	2	1	no	—
201	C₂²	—	4	4	no	—
202	C₂²	C₂×#41	1	2	yes	—
203	C₂²	2₋¹⁺⁴	1	2	yes	—
204	C₂²	C₂⁵	1	1	yes	characteristic
205	C₄	—	2	1	no	—
206	C₄	—	2	1	no	—
207	C₄	—	2	2	no	—
208	C₄	—	2	4	no	—
209	C₄	—	2	2	no	—
210	C₄	—	2	2	no	—
211	C₄	—	2	2	no	—
212	C₄	—	2	2	no	—
213	C₄	—	2	4	no	—
214	C₄	—	2	2	no	—
215	C₄	—	2	2	no	—
216	C₄	C₂×#41	1	2	yes	—
217	C₄	C₂×#41	1	2	yes	—
218	C₂	—	2	2	no	—
219	C₂	—	2	2	no	—
220	C₂	—	4	2	no	—
221	C₂	—	4	1	no	—
222	C₂	C₂²×#41	1	1	yes	characteristic
223	C₂	C₂×2₋¹⁺⁴	1	1	yes	characteristic
224	C₂	#585	1	1	yes	characteristic

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂²	C₂⁵
prime radical	#525	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. #556	C₂	semidirect
2. #554	C₂	semidirect
3. #552	C₂	semidirect
4. #552	C₂	semidirect
5. #552	C₂	semidirect
6. #551	C₂	semidirect
7. #544	C₂	semidirect
8. #542	C₂	semidirect
9. #541	C₂	semidirect
10. #536	C₂	semidirect
11. #533	C₂	semidirect
12. #529	C₂	semidirect
14. C₂×C₂²⋊Q₈	C₂	semidirect
15. #520	C₂	semidirect
16. #519	C₂	semidirect
17. C₂×#117	C₂	semidirect
21. C₄⋊Q₈	C₂²	semidirect
22. C₄⋊Q₈	C₂²	semidirect
23. C₄⋊Q₈	C₂²	semidirect
26. #126	C₂²	semidirect
27. #126	C₂²	semidirect
28. #125	C₂²	semidirect
29. #125	C₂²	semidirect
30. #125	C₂²	semidirect
31. #125	C₂²	semidirect
32. #125	C₂²	semidirect
34. #124	C₂²	semidirect
47. C₂²⋊Q₈	C₂²	semidirect
48. C₂²⋊Q₈	C₂²	semidirect
49. C₂²⋊Q₈	C₂²	semidirect
53. C₄×Q₈	C₂²	semidirect
54. C₄×Q₈	C₂²	semidirect
55. C₄×Q₈	C₂²	semidirect
56. C₄×Q₈	C₂²	semidirect
57. C₄×Q₈	C₂²	semidirect
58. C₄×Q₈	C₂²	semidirect
63. C₄×D₄	C₂²	semidirect
73. #117	C₂²	semidirect
74. #117	C₂²	semidirect
79. 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
#556	2
#554	2
#552	2
#551	2
#544	2
#542	2
#541	2
#536	2
#533	2
#529	2
#525	2
C₂×C₂²⋊Q₈	2
#520	2
#519	2
C₂×#117	2