Subgroups — #3555

Order 128.

455 conjugacy classes of subgroups.

The subgroup lattice has 207 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#563	C₂	1	4	yes	maximal, complemented
2	Q₈²	C₂	1	1	yes	maximal, characteristic, complemented
3	#557	C₂	1	1	yes	maximal, characteristic, complemented
4	#557	C₂	1	2	yes	maximal, complemented
5	#556	C₂	1	2	yes	maximal, complemented
6	#556	C₂	1	2	yes	maximal, complemented
7	#556	C₂	1	1	yes	maximal, characteristic, complemented
8	#554	C₂	1	1	yes	maximal, characteristic, complemented
9	#552	C₂	1	2	yes	maximal, complemented
10	#544	C₂	1	2	yes	maximal, complemented
11	#542	C₂	1	2	yes	maximal
12	#541	C₂	1	2	yes	maximal, complemented
13	#541	C₂	1	2	yes	maximal, complemented
14	#541	C₂	1	4	yes	maximal, complemented
15	#533	C₂	1	1	yes	maximal, characteristic, complemented
16	#533	C₂	1	1	yes	maximal, characteristic, complemented
17	#529	C₂	1	1	yes	maximal, characteristic
18	C₄⋊Q₈	C₂²	1	1	yes	characteristic
19	C₄⋊Q₈	C₂²	1	1	yes	characteristic
20	C₄⋊Q₈	C₂²	1	1	yes	characteristic
21	C₄⋊Q₈	C₂²	1	1	yes	characteristic
22	C₄⋊Q₈	C₂²	1	2	yes	—
23	C₄⋊Q₈	C₂²	1	4	yes	—
24	C₄⋊Q₈	C₂²	1	2	yes	—
25	C₄⋊Q₈	C₂²	1	2	yes	—
26	C₄⋊Q₈	C₂²	1	4	yes	—
27	#126	C₂²	1	2	yes	—
28	#126	C₂²	1	4	yes	—
29	#126	C₂²	1	4	yes	—
30	#126	C₂²	1	4	yes	—
31	#126	C₂²	1	4	yes	—
32	#126	C₂²	1	4	yes	—
33	#126	C₂²	1	4	yes	—
34	#125	C₂²	1	1	yes	characteristic
35	#125	C₂²	1	2	yes	—
36	#125	C₂²	1	2	yes	—
37	#125	C₂²	1	4	yes	—
38	#125	C₂²	1	4	yes	—
39	#125	C₂²	1	4	yes	—
40	#125	C₂²	1	2	yes	—
41	#125	C₂²	1	4	yes	—
42	#125	C₂²	1	4	yes	—
43	#125	C₂²	1	2	yes	—
44	#124	C₂²	1	1	yes	characteristic
45	#124	C₂²	1	2	yes	—
46	#124	C₂²	1	4	yes	—
47	#123	C₂²	1	2	yes	—
48	#123	C₂²	1	4	yes	—
49	C₂²⋊Q₈	C₂²	1	1	yes	characteristic
50	C₂²⋊Q₈	C₂²	1	2	yes	—
51	C₂²⋊Q₈	C₂²	1	1	yes	characteristic
52	C₂²⋊Q₈	C₂²	1	1	yes	characteristic
53	C₂²⋊Q₈	C₂²	1	2	yes	—
54	C₂²⋊Q₈	C₂²	1	2	yes	—
55	C₂²⋊Q₈	C₂²	1	4	yes	—
56	C₂²⋊Q₈	C₂²	1	2	yes	—
57	C₂²⋊Q₈	C₂²	1	2	yes	—
58	C₂²⋊Q₈	C₂²	1	4	yes	—
59	C₂²⋊Q₈	C₂²	1	4	yes	—
60	#121	C₂²	1	1	yes	characteristic
61	C₄×Q₈	C₂²	1	1	yes	characteristic
62	C₄×Q₈	C₂²	1	2	yes	—
63	C₄×Q₈	C₂²	1	1	yes	characteristic
64	C₄×Q₈	C₂²	1	2	yes	—
65	C₄×Q₈	C₂²	1	2	yes	—
66	C₄×Q₈	C₂²	1	1	yes	characteristic
67	C₄×Q₈	C₂²	1	2	yes	—
68	C₄×Q₈	C₂²	1	4	yes	—
69	C₄×Q₈	C₂²	1	2	yes	—
70	C₄×Q₈	C₂²	1	2	yes	—
71	C₄×Q₈	C₂²	1	2	yes	—
72	C₄×Q₈	C₂²	1	2	yes	—
73	C₄×D₄	C₂²	1	2	yes	—
74	C₄×D₄	C₂²	1	1	yes	characteristic
75	C₄×D₄	C₂²	1	2	yes	—
76	#117	C₂²	1	2	yes	—
77	#117	C₂²	1	2	yes	—
78	#117	C₂²	1	2	yes	—
79	#117	C₂²	1	4	yes	—
80	#117	C₂²	1	2	yes	—
81	C₂×C₄⋊C₄	C₂²	1	2	yes	—
82	C₂×C₄²	C₂²	1	1	yes	characteristic
83	C₂×Q₈	C₂³	1	1	yes	characteristic
84	C₂×Q₈	C₂³	1	1	yes	characteristic
85	C₂×Q₈	C₂³	1	1	yes	characteristic
86	C₂×Q₈	C₂³	1	2	yes	—
87	C₂×Q₈	C₂³	1	2	yes	—
88	C₂×Q₈	C₂³	1	2	yes	—
89	C₂×Q₈	C₂³	1	4	yes	—
90	C₂×D₄	C₂³	1	1	yes	characteristic
91	C₂×D₄	C₂³	1	2	yes	—
92	C₂²×C₄	C₂³	1	1	yes	characteristic
93	C₂²×C₄	C₂³	1	1	yes	characteristic
94	C₂²×C₄	C₂³	1	2	yes	—
95	C₂²×C₄	C₂³	1	1	yes	characteristic
96	C₂²×C₄	C₂³	1	4	yes	—
97	C₄⋊C₄	C₂³	1	1	yes	characteristic
98	C₄⋊C₄	C₂³	1	2	yes	—
99	C₄⋊C₄	C₂³	1	4	yes	—
100	C₄⋊C₄	C₂³	1	2	yes	—
101	C₄⋊C₄	C₂³	1	2	yes	—
102	C₄⋊C₄	C₂³	1	1	yes	characteristic
103	C₄⋊C₄	C₂³	1	2	yes	—
104	C₄⋊C₄	C₂³	1	2	yes	—
105	C₄⋊C₄	C₂³	1	4	yes	—
106	C₄⋊C₄	C₂³	1	2	yes	—
107	C₄⋊C₄	C₂³	1	2	yes	—
108	C₄⋊C₄	C₂³	1	1	yes	characteristic
109	C₄⋊C₄	C₂³	1	2	yes	—
110	C₄⋊C₄	C₂³	1	2	yes	—
111	C₄⋊C₄	C₂³	1	1	yes	characteristic
112	C₄⋊C₄	C₂³	1	2	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	2	yes	—
120	C₄⋊C₄	C₂³	1	2	yes	—
121	C₄⋊C₄	C₂³	1	2	yes	—
122	C₄⋊C₄	C₂³	1	4	yes	—
123	C₄⋊C₄	C₂³	1	2	yes	—
124	C₄⋊C₄	C₂³	1	4	yes	—
125	C₄⋊C₄	C₂³	1	4	yes	—
126	C₂²⋊C₄	C₂³	1	2	yes	—
127	C₂²⋊C₄	C₂³	1	2	yes	—
128	C₂²⋊C₄	C₂³	1	2	yes	—
129	C₂²⋊C₄	C₂³	1	2	yes	—
130	C₂²⋊C₄	C₂³	1	4	yes	—
131	C₂²⋊C₄	C₂³	1	2	yes	—
132	C₂²⋊C₄	C₂³	1	4	yes	—
133	C₂²⋊C₄	C₂³	1	4	yes	—
134	C₂²⋊C₄	C₂³	1	2	yes	—
135	C₂²⋊C₄	C₂³	1	4	yes	—
136	C₂²⋊C₄	C₂³	1	2	yes	—
137	C₄²	C₂³	1	1	yes	characteristic
138	C₄²	C₂³	1	1	yes	characteristic
139	C₄²	C₂³	1	1	yes	characteristic
140	C₄²	C₂³	1	1	yes	characteristic
141	C₄²	C₂³	1	4	yes	—
142	C₄²	C₂³	1	2	yes	—
143	C₄²	C₂³	1	2	yes	—
144	C₄²	C₂³	1	2	yes	—
145	C₄²	C₂³	1	4	yes	—
146	C₄²	C₂³	1	4	yes	—
147	C₄²	C₂³	1	4	yes	—
148	C₂³	C₂⁴	1	1	yes	characteristic
149	C₂³	C₂⁴	1	2	yes	—
150	Q₈	—	2	2	no	—
151	Q₈	—	2	2	no	—
152	Q₈	—	2	2	no	—
153	Q₈	—	2	4	no	—
154	Q₈	—	4	2	no	—
155	Q₈	—	4	2	no	—
156	Q₈	—	4	4	no	—
157	D₄	—	2	2	no	—
158	D₄	—	4	2	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	2	4	no	—
161	C₂×C₄	—	2	2	no	—
162	C₂×C₄	—	4	4	no	—
163	C₂×C₄	—	4	1	no	—
164	C₂×C₄	C₂⁴	1	1	yes	characteristic
165	C₂×C₄	C₂⁴	1	4	yes	—
166	C₂×C₄	C₂⁴	1	2	yes	—
167	C₂×C₄	C₂⁴	1	2	yes	—
168	C₂×C₄	C₂⁴	1	2	yes	—
169	C₂×C₄	C₂⁴	1	2	yes	—
170	C₂×C₄	C₂⁴	1	2	yes	—
171	C₂×C₄	C₂⁴	1	1	yes	characteristic
172	C₂×C₄	C₂⁴	1	2	yes	—
173	C₂×C₄	C₂⁴	1	1	yes	characteristic
174	C₂×C₄	C₂⁴	1	1	yes	characteristic
175	C₂×C₄	C₂⁴	1	4	yes	—
176	C₂×C₄	C₂⁴	1	2	yes	—
177	C₂×C₄	C₂⁴	1	1	yes	characteristic
178	C₂×C₄	C₂⁴	1	1	yes	characteristic
179	C₂²	—	2	1	no	—
180	C₂²	—	2	1	no	—
181	C₂²	—	2	2	no	—
182	C₂²	—	2	2	no	—
183	C₂²	—	2	2	no	—
184	C₂²	—	2	1	no	—
185	C₂²	C₂⁵	1	1	yes	characteristic
186	C₄	—	2	1	no	—
187	C₄	—	2	2	no	—
188	C₄	—	2	1	no	—
189	C₄	—	2	2	no	—
190	C₄	—	2	2	no	—
191	C₄	—	2	4	no	—
192	C₄	—	2	2	no	—
193	C₄	—	2	2	no	—
194	C₄	—	2	4	no	—
195	C₄	—	2	1	no	—
196	C₄	—	2	1	no	—
197	C₄	—	2	2	no	—
198	C₄	—	2	2	no	—
199	C₄	2₋¹⁺⁴	1	2	yes	—
200	C₄	2₋¹⁺⁴	1	2	yes	—
201	C₂	—	4	1	no	—
202	C₂	—	4	2	no	—
203	C₂	C₂×2₋¹⁺⁴	1	1	yes	characteristic
204	C₂	C₂×2₋¹⁺⁴	1	1	yes	characteristic
205	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. #563	C₂	semidirect
2. Q₈²	C₂	semidirect
3. #557	C₂	semidirect
4. #557	C₂	semidirect
5. #556	C₂	semidirect
6. #556	C₂	semidirect
7. #556	C₂	semidirect
8. #554	C₂	semidirect
9. #552	C₂	semidirect
10. #544	C₂	semidirect
12. #541	C₂	semidirect
13. #541	C₂	semidirect
14. #541	C₂	semidirect
15. #533	C₂	semidirect
16. #533	C₂	semidirect

Maximal subgroups

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

subgroup	index
#563	2
Q₈²	2
#557	2
#556	2
#554	2
#552	2
#544	2
#542	2
#541	2
#533	2
#529	2