Subgroups — #3515

Order 128.

507 conjugacy classes of subgroups.

The subgroup lattice has 217 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#557	C₂	1	1	yes	maximal, characteristic, complemented
2	#554	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	4	yes	maximal, complemented
7	#552	C₂	1	2	yes	maximal, complemented
8	#547	C₂	1	1	yes	maximal, characteristic, complemented
9	#547	C₂	1	2	yes	maximal, complemented
10	#541	C₂	1	2	yes	maximal, complemented
11	#539	C₂	1	4	yes	maximal, complemented
12	#533	C₂	1	1	yes	maximal, characteristic, complemented
13	#529	C₂	1	2	yes	maximal, complemented
14	#529	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂×#125	C₂	1	2	yes	maximal, complemented
16	#520	C₂	1	2	yes	maximal, complemented
17	C₄×#41	C₂	1	1	yes	maximal, characteristic, complemented
18	C₂×#41	C₂²	1	1	yes	characteristic
19	C₂×#41	C₂²	1	2	yes	—
20	C₄⋊Q₈	C₂²	1	1	yes	characteristic
21	C₄⋊Q₈	C₂²	1	1	yes	characteristic
22	C₄⋊Q₈	C₂²	1	2	yes	—
23	#126	C₂²	1	4	yes	—
24	#126	C₂²	1	2	yes	—
25	#126	C₂²	1	4	yes	—
26	#126	C₂²	1	4	yes	—
27	#125	C₂²	1	1	yes	characteristic
28	#125	C₂²	1	2	yes	—
29	#125	C₂²	1	4	yes	—
30	#125	C₂²	1	4	yes	—
31	#125	C₂²	1	2	yes	—
32	#125	C₂²	1	2	yes	—
33	#125	C₂²	1	4	yes	—
34	#125	C₂²	1	4	yes	—
35	#125	C₂²	1	2	yes	—
36	#124	C₂²	1	1	yes	characteristic
37	#123	C₂²	1	2	yes	—
38	#123	C₂²	1	4	yes	—
39	#123	C₂²	1	4	yes	—
40	#123	C₂²	1	4	yes	—
41	#123	C₂²	1	4	yes	—
42	C₂²⋊Q₈	C₂²	1	1	yes	characteristic
43	C₂²⋊Q₈	C₂²	1	2	yes	—
44	C₂²⋊Q₈	C₂²	1	2	yes	—
45	C₂²⋊Q₈	C₂²	1	2	yes	—
46	C₂²⋊Q₈	C₂²	1	4	yes	—
47	C₂²⋊Q₈	C₂²	1	4	yes	—
48	C₂²⋊Q₈	C₂²	1	2	yes	—
49	C₂²⋊Q₈	C₂²	1	2	yes	—
50	#121	C₂²	1	2	yes	—
51	#121	C₂²	1	1	yes	characteristic
52	#121	C₂²	1	4	yes	—
53	C₄×Q₈	C₂²	1	1	yes	characteristic
54	C₄×Q₈	C₂²	1	1	yes	characteristic
55	C₄×Q₈	C₂²	1	2	yes	—
56	C₄×Q₈	C₂²	1	2	yes	—
57	C₄×Q₈	C₂²	1	2	yes	—
58	C₄×Q₈	C₂²	1	1	yes	characteristic
59	C₄×Q₈	C₂²	1	2	yes	—
60	C₄×D₄	C₂²	1	1	yes	characteristic
61	C₄×D₄	C₂²	1	2	yes	—
62	C₄×D₄	C₂²	1	1	yes	characteristic
63	C₄×D₄	C₂²	1	2	yes	—
64	C₄×D₄	C₂²	1	4	yes	—
65	C₄×D₄	C₂²	1	2	yes	—
66	C₄×D₄	C₂²	1	4	yes	—
67	C₄×D₄	C₂²	1	2	yes	—
68	C₄×D₄	C₂²	1	1	yes	characteristic
69	C₄×D₄	C₂²	1	2	yes	—
70	#117	C₂²	1	1	yes	characteristic
71	#117	C₂²	1	2	yes	—
72	#117	C₂²	1	2	yes	—
73	#117	C₂²	1	4	yes	—
74	#117	C₂²	1	4	yes	—
75	#117	C₂²	1	2	yes	—
76	C₂×C₄⋊C₄	C₂²	1	4	yes	—
77	C₂×C₄⋊C₄	C₂²	1	2	yes	—
78	C₂×C₄⋊C₄	C₂²	1	2	yes	—
79	C₂×C₄⋊C₄	C₂²	1	4	yes	—
80	C₂×C₄⋊C₄	C₂²	1	2	yes	—
81	C₂×C₄²	C₂²	1	2	yes	—
82	C₂×C₄²	C₂²	1	1	yes	characteristic
83	#41	—	2	4	no	—
84	#41	—	2	8	no	—
85	C₂×Q₈	C₂³	1	1	yes	characteristic
86	C₂×Q₈	C₂³	1	1	yes	characteristic
87	C₂×Q₈	C₂³	1	2	yes	—
88	C₂×Q₈	C₂³	1	2	yes	—
89	C₂×D₄	C₂³	1	1	yes	characteristic
90	C₂×D₄	C₂³	1	2	yes	—
91	C₂×D₄	C₂³	1	4	yes	—
92	C₂×D₄	C₂³	1	2	yes	—
93	C₂×D₄	C₂³	1	1	yes	characteristic
94	C₂²×C₄	C₂³	1	2	yes	—
95	C₂²×C₄	C₂³	1	2	yes	—
96	C₂²×C₄	C₂³	1	1	yes	characteristic
97	C₂²×C₄	C₂³	1	1	yes	characteristic
98	C₂²×C₄	C₂³	1	4	yes	—
99	C₂²×C₄	C₂³	1	2	yes	—
100	C₂²×C₄	C₂³	1	2	yes	—
101	C₂²×C₄	C₂³	1	1	yes	characteristic
102	C₂²×C₄	C₂³	1	4	yes	—
103	C₂²×C₄	C₂³	1	4	yes	—
104	C₄⋊C₄	C₂³	1	1	yes	characteristic
105	C₄⋊C₄	C₂³	1	4	yes	—
106	C₄⋊C₄	C₂³	1	2	yes	—
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	2	yes	—
112	C₄⋊C₄	C₂³	1	4	yes	—
113	C₄⋊C₄	C₂³	1	2	yes	—
114	C₄⋊C₄	C₂³	1	4	yes	—
115	C₄⋊C₄	C₂³	1	4	yes	—
116	C₄⋊C₄	C₂³	1	4	yes	—
117	C₄⋊C₄	C₂³	1	4	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	1	yes	characteristic
123	C₄⋊C₄	C₂³	1	2	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	C₂²⋊C₄	C₂³	1	2	yes	—
129	C₂²⋊C₄	C₂³	1	2	yes	—
130	C₂²⋊C₄	C₂³	1	2	yes	—
131	C₂²⋊C₄	C₂³	1	4	yes	—
132	C₂²⋊C₄	C₂³	1	2	yes	—
133	C₂²⋊C₄	C₂³	1	2	yes	—
134	C₂²⋊C₄	C₂³	1	2	yes	—
135	C₂²⋊C₄	C₂³	1	4	yes	—
136	C₂²⋊C₄	C₂³	1	4	yes	—
137	C₂²⋊C₄	C₂³	1	4	yes	—
138	C₂²⋊C₄	C₂³	1	2	yes	—
139	C₂²⋊C₄	C₂³	1	2	yes	—
140	C₄²	C₂³	1	1	yes	characteristic
141	C₄²	C₂³	1	2	yes	—
142	C₄²	C₂³	1	1	yes	characteristic
143	C₄²	C₂³	1	1	yes	characteristic
144	C₄²	C₂³	1	2	yes	—
145	C₄²	C₂³	1	1	yes	characteristic
146	C₄²	C₂³	1	2	yes	—
147	C₄²	C₂³	1	4	yes	—
148	C₄²	C₂³	1	4	yes	—
149	C₄²	C₂³	1	2	yes	—
150	C₂³	C₂⁴	1	1	yes	characteristic
151	C₂³	C₂⁴	1	2	yes	—
152	C₂³	C₂⁴	1	2	yes	—
153	Q₈	—	2	2	no	—
154	Q₈	—	2	4	no	—
155	Q₈	—	2	2	no	—
156	Q₈	—	4	2	no	—
157	D₄	—	2	4	no	—
158	D₄	—	2	8	no	—
159	D₄	—	2	2	no	—
160	D₄	—	4	2	no	—
161	D₄	#41	1	4	yes	—
162	C₂×C₄	—	2	4	no	—
163	C₂×C₄	—	2	8	no	—
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	2	4	no	—
166	C₂×C₄	—	2	4	no	—
167	C₂×C₄	—	2	2	no	—
168	C₂×C₄	—	4	2	no	—
169	C₂×C₄	—	4	4	no	—
170	C₂×C₄	—	4	1	no	—
171	C₂×C₄	—	4	4	no	—
172	C₂×C₄	C₂⁴	1	1	yes	characteristic
173	C₂×C₄	C₂⁴	1	1	yes	characteristic
174	C₂×C₄	C₂⁴	1	1	yes	characteristic
175	C₂×C₄	C₂⁴	1	2	yes	—
176	C₂×C₄	C₂⁴	1	2	yes	—
177	C₂×C₄	C₂⁴	1	2	yes	—
178	C₂×C₄	C₂⁴	1	1	yes	characteristic
179	C₂×C₄	C₂⁴	1	2	yes	—
180	C₂×C₄	C₂⁴	1	4	yes	—
181	C₂×C₄	C₂⁴	1	4	yes	—
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	1	yes	characteristic
186	C₂²	—	2	2	no	—
187	C₂²	—	2	2	no	—
188	C₂²	—	2	1	no	—
189	C₂²	—	2	2	no	—
190	C₂²	—	2	1	no	—
191	C₂²	—	2	2	no	—
192	C₂²	—	2	2	no	—
193	C₂²	—	2	1	no	—
194	C₂²	C₂×#41	1	4	yes	—
195	C₂²	C₂⁵	1	1	yes	characteristic
196	C₄	—	2	1	no	—
197	C₄	—	2	2	no	—
198	C₄	—	2	2	no	—
199	C₄	—	2	4	no	—
200	C₄	—	2	1	no	—
201	C₄	—	2	2	no	—
202	C₄	—	2	2	no	—
203	C₄	—	2	1	no	—
204	C₄	—	2	2	no	—
205	C₄	—	2	2	no	—
206	C₄	—	2	4	no	—
207	C₄	—	2	1	no	—
208	C₄	C₂×#41	1	2	yes	—
209	C₄	2₋¹⁺⁴	1	2	yes	—
210	C₂	—	2	4	no	—
211	C₂	—	4	1	no	—
212	C₂	—	4	2	no	—
213	C₂	C₂²×#41	1	1	yes	characteristic
214	C₂	C₂×2₋¹⁺⁴	1	1	yes	characteristic
215	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. #557	C₂	semidirect
2. #554	C₂	semidirect
3. #554	C₂	semidirect
4. #553	C₂	semidirect
5. #552	C₂	semidirect
6. #552	C₂	semidirect
7. #552	C₂	semidirect
8. #547	C₂	semidirect
9. #547	C₂	semidirect
10. #541	C₂	semidirect
11. #539	C₂	semidirect
12. #533	C₂	semidirect
13. #529	C₂	semidirect
14. #529	C₂	semidirect
15. C₂×#125	C₂	semidirect
16. #520	C₂	semidirect
17. C₄×#41	C₂	semidirect

Maximal subgroups

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

subgroup	index
#557	2
#554	2
#553	2
#552	2
#547	2
#541	2
#539	2
#533	2
#529	2
C₂×#125	2
#520	2
C₄×#41	2