Subgroups — #3563

Order 128.

456 conjugacy classes of subgroups.

The subgroup lattice has 188 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#564	C₂	1	2	yes	maximal, complemented
2	#563	C₂	1	2	yes	maximal, complemented
3	#557	C₂	1	2	yes	maximal, complemented
4	#554	C₂	1	2	yes	maximal, complemented
5	#552	C₂	1	4	yes	maximal, complemented
6	#552	C₂	1	2	yes	maximal, complemented
7	#552	C₂	1	2	yes	maximal, complemented
8	#544	C₂	1	2	yes	maximal, complemented
9	#544	C₂	1	2	yes	maximal, complemented
10	#541	C₂	1	2	yes	maximal, complemented
11	#541	C₂	1	4	yes	maximal, complemented
12	#539	C₂	1	2	yes	maximal
13	#533	C₂	1	1	yes	maximal, characteristic, complemented
14	#529	C₂	1	1	yes	maximal, characteristic
15	C₂×#125	C₂	1	1	yes	maximal, characteristic, complemented
16	C₄⋊Q₈	C₂²	1	1	yes	characteristic
17	C₄⋊Q₈	C₂²	1	1	yes	characteristic
18	C₄⋊Q₈	C₂²	1	4	yes	—
19	C₄⋊Q₈	C₂²	1	2	yes	—
20	C₄⋊Q₈	C₂²	1	4	yes	—
21	#126	C₂²	1	2	yes	—
22	#126	C₂²	1	4	yes	—
23	#126	C₂²	1	4	yes	—
24	#126	C₂²	1	4	yes	—
25	#126	C₂²	1	4	yes	—
26	#125	C₂²	1	2	yes	—
27	#125	C₂²	1	2	yes	—
28	#125	C₂²	1	2	yes	—
29	#125	C₂²	1	2	yes	—
30	#125	C₂²	1	1	yes	characteristic
31	#125	C₂²	1	2	yes	—
32	#125	C₂²	1	4	yes	—
33	#125	C₂²	1	4	yes	—
34	#125	C₂²	1	4	yes	—
35	#125	C₂²	1	4	yes	—
36	#125	C₂²	1	4	yes	—
37	#125	C₂²	1	4	yes	—
38	#125	C₂²	1	2	yes	—
39	#125	C₂²	1	4	yes	—
40	#125	C₂²	1	4	yes	—
41	#124	C₂²	1	1	yes	characteristic
42	#123	C₂²	1	2	yes	—
43	#123	C₂²	1	4	yes	—
44	#123	C₂²	1	4	yes	—
45	C₂²⋊Q₈	C₂²	1	1	yes	characteristic
46	C₂²⋊Q₈	C₂²	1	2	yes	—
47	C₂²⋊Q₈	C₂²	1	2	yes	—
48	C₂²⋊Q₈	C₂²	1	4	yes	—
49	C₂²⋊Q₈	C₂²	1	2	yes	—
50	C₂²⋊Q₈	C₂²	1	2	yes	—
51	C₂²⋊Q₈	C₂²	1	4	yes	—
52	C₂²⋊Q₈	C₂²	1	4	yes	—
53	C₂²⋊Q₈	C₂²	1	4	yes	—
54	#121	C₂²	1	1	yes	characteristic
55	C₄×Q₈	C₂²	1	1	yes	characteristic
56	C₄×Q₈	C₂²	1	2	yes	—
57	C₄×Q₈	C₂²	1	2	yes	—
58	C₄×Q₈	C₂²	1	4	yes	—
59	C₄×Q₈	C₂²	1	2	yes	—
60	C₄×Q₈	C₂²	1	2	yes	—
61	C₄×D₄	C₂²	1	1	yes	characteristic
62	C₄×D₄	C₂²	1	2	yes	—
63	C₄×D₄	C₂²	1	4	yes	—
64	#117	C₂²	1	2	yes	—
65	#117	C₂²	1	2	yes	—
66	#117	C₂²	1	4	yes	—
67	#117	C₂²	1	2	yes	—
68	#117	C₂²	1	4	yes	—
69	C₂×C₄⋊C₄	C₂²	1	2	yes	—
70	C₂×C₄⋊C₄	C₂²	1	1	yes	characteristic
71	C₂×C₄⋊C₄	C₂²	1	2	yes	—
72	C₂×C₄⋊C₄	C₂²	1	1	yes	characteristic
73	C₂×C₄⋊C₄	C₂²	1	2	yes	—
74	C₂×C₄²	C₂²	1	1	yes	characteristic
75	C₂×Q₈	C₂³	1	2	yes	—
76	C₂×Q₈	C₂³	1	4	yes	—
77	C₂×Q₈	C₂³	1	2	yes	—
78	C₂×Q₈	C₂³	1	1	yes	characteristic
79	C₂×D₄	C₂³	1	2	yes	—
80	C₂×D₄	C₂³	1	1	yes	characteristic
81	C₂²×C₄	C₂³	1	1	yes	characteristic
82	C₂²×C₄	C₂³	1	1	yes	characteristic
83	C₂²×C₄	C₂³	1	2	yes	—
84	C₂²×C₄	C₂³	1	2	yes	—
85	C₂²×C₄	C₂³	1	1	yes	characteristic
86	C₂²×C₄	C₂³	1	2	yes	—
87	C₂²×C₄	C₂³	1	4	yes	—
88	C₄⋊C₄	C₂³	1	2	yes	—
89	C₄⋊C₄	C₂³	1	2	yes	—
90	C₄⋊C₄	C₂³	1	2	yes	—
91	C₄⋊C₄	C₂³	1	4	yes	—
92	C₄⋊C₄	C₂³	1	2	yes	—
93	C₄⋊C₄	C₂³	1	4	yes	—
94	C₄⋊C₄	C₂³	1	4	yes	—
95	C₄⋊C₄	C₂³	1	4	yes	—
96	C₄⋊C₄	C₂³	1	4	yes	—
97	C₄⋊C₄	C₂³	1	2	yes	—
98	C₄⋊C₄	C₂³	1	4	yes	—
99	C₄⋊C₄	C₂³	1	2	yes	—
100	C₄⋊C₄	C₂³	1	4	yes	—
101	C₄⋊C₄	C₂³	1	2	yes	—
102	C₄⋊C₄	C₂³	1	4	yes	—
103	C₄⋊C₄	C₂³	1	2	yes	—
104	C₄⋊C₄	C₂³	1	2	yes	—
105	C₄⋊C₄	C₂³	1	1	yes	characteristic
106	C₄⋊C₄	C₂³	1	4	yes	—
107	C₄⋊C₄	C₂³	1	2	yes	—
108	C₄⋊C₄	C₂³	1	2	yes	—
109	C₄⋊C₄	C₂³	1	1	yes	characteristic
110	C₄⋊C₄	C₂³	1	4	yes	—
111	C₄⋊C₄	C₂³	1	4	yes	—
112	C₄⋊C₄	C₂³	1	4	yes	—
113	C₄⋊C₄	C₂³	1	4	yes	—
114	C₄⋊C₄	C₂³	1	2	yes	—
115	C₄⋊C₄	C₂³	1	4	yes	—
116	C₂²⋊C₄	C₂³	1	4	yes	—
117	C₂²⋊C₄	C₂³	1	2	yes	—
118	C₂²⋊C₄	C₂³	1	2	yes	—
119	C₂²⋊C₄	C₂³	1	2	yes	—
120	C₂²⋊C₄	C₂³	1	4	yes	—
121	C₂²⋊C₄	C₂³	1	2	yes	—
122	C₂²⋊C₄	C₂³	1	4	yes	—
123	C₂²⋊C₄	C₂³	1	4	yes	—
124	C₂²⋊C₄	C₂³	1	2	yes	—
125	C₄²	C₂³	1	1	yes	characteristic
126	C₄²	C₂³	1	1	yes	characteristic
127	C₄²	C₂³	1	1	yes	characteristic
128	C₄²	C₂³	1	1	yes	characteristic
129	C₄²	C₂³	1	4	yes	—
130	C₄²	C₂³	1	4	yes	—
131	C₄²	C₂³	1	2	yes	—
132	C₄²	C₂³	1	4	yes	—
133	C₄²	C₂³	1	4	yes	—
134	C₂³	C₂⁴	1	2	yes	—
135	C₂³	C₂⁴	1	1	yes	characteristic
136	Q₈	—	2	4	no	—
137	Q₈	—	4	4	no	—
138	Q₈	—	4	1	no	—
139	Q₈	—	4	2	no	—
140	D₄	—	2	4	no	—
141	D₄	—	4	1	no	—
142	C₂×C₄	—	2	2	no	—
143	C₂×C₄	—	2	4	no	—
144	C₂×C₄	—	2	2	no	—
145	C₂×C₄	—	2	4	no	—
146	C₂×C₄	—	4	2	no	—
147	C₂×C₄	—	4	4	no	—
148	C₂×C₄	—	4	1	no	—
149	C₂×C₄	C₂⁴	1	1	yes	characteristic
150	C₂×C₄	C₂⁴	1	4	yes	—
151	C₂×C₄	C₂⁴	1	2	yes	—
152	C₂×C₄	C₂⁴	1	2	yes	—
153	C₂×C₄	C₂⁴	1	1	yes	characteristic
154	C₂×C₄	C₂⁴	1	1	yes	characteristic
155	C₂×C₄	C₂⁴	1	2	yes	—
156	C₂×C₄	C₂⁴	1	4	yes	—
157	C₂×C₄	C₂⁴	1	1	yes	characteristic
158	C₂×C₄	C₂⁴	1	4	yes	—
159	C₂×C₄	C₂⁴	1	1	yes	characteristic
160	C₂×C₄	C₂⁴	1	4	yes	—
161	C₂×C₄	C₂⁴	1	1	yes	characteristic
162	C₂²	—	2	1	no	—
163	C₂²	—	2	1	no	—
164	C₂²	—	2	2	no	—
165	C₂²	—	2	2	no	—
166	C₂²	—	2	2	no	—
167	C₂²	2₋¹⁺⁴	1	2	yes	—
168	C₂²	C₂⁵	1	1	yes	characteristic
169	C₄	—	2	1	no	—
170	C₄	—	2	1	no	—
171	C₄	—	2	4	no	—
172	C₄	—	2	1	no	—
173	C₄	—	2	2	no	—
174	C₄	—	2	2	no	—
175	C₄	—	2	4	no	—
176	C₄	—	2	4	no	—
177	C₄	—	2	1	no	—
178	C₄	—	2	4	no	—
179	C₄	—	2	1	no	—
180	C₄	—	2	2	no	—
181	C₄	2₋¹⁺⁴	1	2	yes	—
182	C₂	—	2	2	no	—
183	C₂	—	4	2	no	—
184	C₂	C₂×2₋¹⁺⁴	1	1	yes	characteristic
185	C₂	C₂×2₋¹⁺⁴	1	1	yes	characteristic
186	C₂	#585	1	1	yes	characteristic

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂²	C₂⁵
prime radical	#121	C₂²
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. #564	C₂	semidirect
2. #563	C₂	semidirect
3. #557	C₂	semidirect
4. #554	C₂	semidirect
5. #552	C₂	semidirect
6. #552	C₂	semidirect
7. #552	C₂	semidirect
8. #544	C₂	semidirect
9. #544	C₂	semidirect
10. #541	C₂	semidirect
11. #541	C₂	semidirect
13. #533	C₂	semidirect
15. C₂×#125	C₂	semidirect

Maximal subgroups

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

subgroup	index
#564	2
#563	2
#557	2
#554	2
#552	2
#544	2
#541	2
#539	2
#533	2
#529	2
C₂×#125	2