Subgroups — #2373

Order 128.

248 conjugacy classes of subgroups.

The subgroup lattice has 180 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#126	C₂	1	1	yes	maximal, characteristic
2	#389	C₂	1	1	yes	maximal, characteristic, complemented
3	#388	C₂	1	1	yes	maximal, characteristic
4	#388	C₂	1	1	yes	maximal, characteristic
5	#388	C₂	1	1	yes	maximal, characteristic
6	#387	C₂	1	1	yes	maximal, characteristic, complemented
7	#387	C₂	1	1	yes	maximal, characteristic, complemented
8	#387	C₂	1	1	yes	maximal, characteristic, complemented
9	#385	C₂	1	1	yes	maximal, characteristic
10	C₄×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
12	C₄×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
13	C₄×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic
14	C₄×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic
15	#376	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂³×C₄	C₂²	1	1	yes	characteristic
17	#126	C₄	1	8	yes	complemented
18	C₂×C₄⋊C₄	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊C₄	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊C₄	C₂²	1	1	yes	characteristic
21	C₂×C₄⋊C₄	C₂²	1	1	yes	characteristic
22	C₂×C₄⋊C₄	C₂²	1	1	yes	characteristic
23	C₂×C₄⋊C₄	C₂²	1	1	yes	characteristic
24	C₂×C₄⋊C₄	C₂²	1	1	yes	characteristic
25	C₂×C₄⋊C₄	C₂²	1	1	yes	characteristic
26	C₂×C₂²⋊C₄	C₂²	1	1	yes	characteristic
27	C₂×C₂²⋊C₄	C₂²	1	1	yes	characteristic
28	C₂×C₂²⋊C₄	C₂²	1	1	yes	characteristic
29	C₂×C₂²⋊C₄	C₂²	1	1	yes	characteristic
30	C₂×C₂²⋊C₄	C₂²	1	1	yes	characteristic
31	C₂×C₂²⋊C₄	C₂²	1	1	yes	characteristic
32	C₂×C₄²	C₂²	1	1	yes	characteristic
33	C₂×C₄²	C₂²	1	1	yes	characteristic
34	C₂×C₄²	C₂²	1	1	yes	characteristic
35	C₂×C₄²	C₂²	1	1	yes	characteristic
36	C₂×C₄²	C₂²	1	1	yes	characteristic
37	C₂×C₄²	C₂²	1	1	yes	characteristic
38	C₂×C₄²	C₂²	1	1	yes	characteristic
39	C₂×C₄²	C₂²	1	1	yes	characteristic
40	#95	C₂²	1	1	yes	characteristic
41	#95	C₂²	1	1	yes	characteristic
42	#95	C₂²	1	1	yes	characteristic
43	#95	C₂²	1	1	yes	characteristic
44	#95	C₂²	1	1	yes	characteristic
45	#95	C₂²	1	1	yes	characteristic
46	#95	C₂²	1	1	yes	characteristic
47	#95	C₂²	1	1	yes	characteristic
48	#95	C₂²	1	1	yes	characteristic
49	#95	C₂²	1	1	yes	characteristic
50	#95	C₂²	1	1	yes	characteristic
51	#95	C₂²	1	1	yes	characteristic
52	C₂⁴	C₂³	1	1	yes	characteristic
53	C₂²×C₄	—	2	2	no	—
54	C₂²×C₄	—	2	2	no	—
55	C₂²×C₄	—	4	1	no	—
56	C₂²×C₄	C₂³	1	1	yes	characteristic
57	C₂²×C₄	C₂³	1	1	yes	characteristic
58	C₂²×C₄	C₂³	1	1	yes	characteristic
59	C₂²×C₄	C₂³	1	1	yes	characteristic
60	C₂²×C₄	C₂³	1	1	yes	characteristic
61	C₂²×C₄	C₂³	1	1	yes	characteristic
62	C₂²×C₄	C₂³	1	1	yes	characteristic
63	C₂²×C₄	C₂³	1	1	yes	characteristic
64	C₂²×C₄	C₂³	1	1	yes	characteristic
65	C₂²×C₄	C₂³	1	1	yes	characteristic
66	C₂²×C₄	C₂³	1	1	yes	characteristic
67	C₂²×C₄	C₂³	1	1	yes	characteristic
68	C₂²×C₄	C₂³	1	1	yes	characteristic
69	C₂²×C₄	C₂³	1	1	yes	characteristic
70	C₄⋊C₄	—	2	2	no	—
71	C₄⋊C₄	—	2	2	no	—
72	C₄⋊C₄	—	2	2	no	—
73	C₄⋊C₄	—	2	2	no	—
74	C₄⋊C₄	—	4	1	no	—
75	C₄⋊C₄	C₂×C₄	1	4	yes	complemented
76	C₄⋊C₄	C₂×C₄	1	4	yes	complemented
77	C₄⋊C₄	C₂×C₄	1	4	yes	complemented
78	C₂²⋊C₄	—	2	2	no	—
79	C₂²⋊C₄	—	2	2	no	—
80	C₂²⋊C₄	—	4	1	no	—
81	C₂²⋊C₄	C₂×C₄	1	4	yes	—
82	C₂²⋊C₄	C₂×C₄	1	4	yes	—
83	C₂²⋊C₄	C₂×C₄	1	4	yes	—
84	C₄²	—	2	2	no	—
85	C₄²	—	2	2	no	—
86	C₄²	—	2	2	no	—
87	C₄²	—	2	2	no	—
88	C₄²	—	4	1	no	—
89	C₄²	—	4	1	no	—
90	C₄²	—	4	1	no	—
91	C₄²	C₂×C₄	1	4	yes	complemented
92	C₂³	—	2	1	no	—
93	C₂³	—	2	1	no	—
94	C₂³	—	2	1	no	—
95	C₂³	—	2	1	no	—
96	C₂³	—	2	1	no	—
97	C₂³	—	2	1	no	—
98	C₂³	C₂²×C₄	1	2	yes	—
99	C₂³	C₂⁴	1	1	yes	characteristic
100	C₂×C₄	—	2	1	no	—
101	C₂×C₄	—	2	1	no	—
102	C₂×C₄	—	2	1	no	—
103	C₂×C₄	—	2	1	no	—
104	C₂×C₄	—	2	1	no	—
105	C₂×C₄	—	2	1	no	—
106	C₂×C₄	—	2	1	no	—
107	C₂×C₄	—	2	1	no	—
108	C₂×C₄	—	2	1	no	—
109	C₂×C₄	—	2	1	no	—
110	C₂×C₄	—	2	1	no	—
111	C₂×C₄	—	2	1	no	—
112	C₂×C₄	—	2	1	no	—
113	C₂×C₄	—	2	1	no	—
114	C₂×C₄	—	2	1	no	—
115	C₂×C₄	—	2	1	no	—
116	C₂×C₄	—	2	1	no	—
117	C₂×C₄	—	2	1	no	—
118	C₂×C₄	—	2	1	no	—
119	C₂×C₄	—	2	1	no	—
120	C₂×C₄	—	2	1	no	—
121	C₂×C₄	—	2	1	no	—
122	C₂×C₄	—	2	1	no	—
123	C₂×C₄	—	2	1	no	—
124	C₂×C₄	—	2	1	no	—
125	C₂×C₄	—	2	1	no	—
126	C₂×C₄	—	2	1	no	—
127	C₂×C₄	—	2	1	no	—
128	C₂×C₄	—	2	1	no	—
129	C₂×C₄	—	2	1	no	—
130	C₂×C₄	—	2	1	no	—
131	C₂×C₄	—	2	1	no	—
132	C₂×C₄	—	4	4	no	—
133	C₂×C₄	C₂²×C₄	1	2	yes	—
134	C₂×C₄	C₂²×C₄	1	2	yes	—
135	C₂×C₄	C₂²×C₄	1	2	yes	—
136	C₂×C₄	C₂²×C₄	1	2	yes	—
137	C₂×C₄	C₂²×C₄	1	2	yes	—
138	C₂×C₄	C₂²×C₄	1	2	yes	—
139	C₂×C₄	#41	1	2	yes	—
140	C₂×C₄	#41	1	2	yes	—
141	C₂×C₄	#41	1	2	yes	—
142	C₂×C₄	#41	1	2	yes	—
143	C₂²	—	2	2	no	—
144	C₂²	—	2	2	no	—
145	C₂²	—	2	2	no	—
146	C₂²	—	4	1	no	—
147	C₂²	—	4	1	no	—
148	C₂²	—	4	1	no	—
149	C₂²	—	4	1	no	—
150	C₂²	C₂³×C₄	1	1	yes	characteristic
151	C₂²	C₂×#41	1	1	yes	characteristic
152	C₂²	C₂×#41	1	1	yes	characteristic
153	C₂²	C₂×#41	1	1	yes	characteristic
154	C₂²	C₂×#41	1	1	yes	characteristic
155	C₂²	H(5,𝔽₂)	1	1	yes	characteristic
156	C₂²	2₋¹⁺⁴	1	1	yes	characteristic
157	C₄	—	2	2	no	—
158	C₄	—	2	2	no	—
159	C₄	—	2	2	no	—
160	C₄	—	2	2	no	—
161	C₄	—	2	2	no	—
162	C₄	—	2	2	no	—
163	C₄	—	2	2	no	—
164	C₄	—	2	2	no	—
165	C₄	—	2	2	no	—
166	C₄	—	2	2	no	—
167	C₄	—	4	1	no	—
168	C₄	—	4	1	no	—
169	C₄	—	4	1	no	—
170	C₄	—	4	1	no	—
171	C₂	—	4	2	no	—
172	C₂	C₄×#41	1	1	yes	characteristic
173	C₂	C₄×#41	1	1	yes	characteristic
174	C₂	#520	1	1	yes	characteristic
175	C₂	#551	1	1	yes	characteristic
176	C₂	#552	1	1	yes	characteristic
177	C₂	#553	1	1	yes	characteristic
178	C₂	#556	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, socle, cosocle	C₂³	C₂⁴
commutator	C₂²	C₂³×C₄
prime radical	C₂⁴	C₂³
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. #389	C₂	semidirect
6. #387	C₂	semidirect
7. #387	C₂	semidirect
8. #387	C₂	semidirect
10. C₄×C₄⋊C₄	C₂	semidirect
11. C₄×C₄⋊C₄	C₂	semidirect
12. C₄×C₄⋊C₄	C₂	semidirect
15. #376	C₂	semidirect
17. #126	C₄	semidirect
75. C₄⋊C₄	C₂×C₄	semidirect
76. C₄⋊C₄	C₂×C₄	semidirect
77. C₄⋊C₄	C₂×C₄	semidirect
91. C₄²	C₂×C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×#126	2
#389	2
#388	2
#387	2
#385	2
C₄×C₄⋊C₄	2
C₄×C₂²⋊C₄	2
#376	2