Finite Group Databasebeta

Subgroups — #2933

Order 128.

251 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 251 conjugacy classes of subgroups — too many to draw. Only its 142 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#131	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	#525	C₂	1	1	yes	maximal, characteristic
4	#436	C₂	1	2	yes	maximal, complemented
5	#435	C₂	1	2	yes	maximal, complemented
6	#435	C₂	1	2	yes	maximal, complemented
7	C₈×D₄	C₂	1	2	yes	maximal, complemented
8	#424	C₂	1	1	yes	maximal, characteristic, complemented
9	#408	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₂²⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
11	#405	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#41	C₂²	1	1	yes	characteristic
13	C₂³×C₄	C₂²	1	1	yes	characteristic
14	#131	—	2	4	no	—
15	C₂×M₁₆	—	2	4	no	—
16	C₂×M₁₆	C₂²	1	1	yes	characteristic
17	C₂×M₁₆	C₂²	1	2	yes	—
18	C₂×M₁₆	C₂²	1	1	yes	characteristic
19	C₂×M₁₆	C₂²	1	2	yes	complemented
20	C₂²×C₈	C₂²	1	2	yes	—
21	C₂²×C₈	C₂²	1	1	yes	characteristic
22	C₂²×C₈	C₂²	1	1	yes	characteristic
23	C₂²×C₈	C₂²	1	2	yes	complemented
24	#123	C₄	1	2	yes	—
25	C₂²⋊Q₈	C₄	1	1	yes	characteristic
26	C₂²⋊Q₈	C₄	1	1	yes	characteristic
27	#121	C₄	1	1	yes	characteristic
28	#121	C₄	1	1	yes	characteristic
29	#120	C₄	1	2	yes	—
30	C₄×D₄	C₂²	1	2	yes	—
31	C₄×D₄	C₂²	1	2	yes	—
32	#117	C₂²	1	1	yes	characteristic
33	C₄⋊C₈	C₂²	1	2	yes	complemented
34	C₄⋊C₈	C₂²	1	2	yes	complemented
35	C₂²⋊C₈	C₂²	1	1	yes	characteristic
36	C₂²⋊C₈	C₂²	1	1	yes	characteristic
37	C₂²⋊C₈	C₂²	1	1	yes	characteristic
38	C₂²⋊C₈	C₂²	1	1	yes	characteristic
39	C₂²⋊C₈	C₂²	1	2	yes	complemented
40	C₂²⋊C₈	C₂²	1	2	yes	complemented
41	#97	C₂²	1	2	yes	complemented
42	C₄×C₈	C₂²	1	2	yes	complemented
43	C₂⁴	C₂×C₄	1	1	yes	characteristic
44	#41	—	2	2	no	—
45	#41	—	2	2	no	—
46	C₂×Q₈	C₂×C₄	1	1	yes	characteristic
47	C₂×D₄	C₂×C₄	1	2	yes	—
48	C₂×D₄	C₂×C₄	1	2	yes	—
49	C₂×D₄	C₂×C₄	1	1	yes	characteristic
50	C₂×D₄	C₂×C₄	1	2	yes	—
51	C₂²×C₄	—	2	2	no	—
52	C₂²×C₄	—	2	2	no	—
53	C₂²×C₄	C₂×C₄	1	1	yes	characteristic
54	C₂²×C₄	C₂×C₄	1	1	yes	characteristic
55	C₂²×C₄	C₂×C₄	1	1	yes	characteristic
56	C₂²×C₄	C₂³	1	1	yes	characteristic
57	C₂²×C₄	C₂³	1	1	yes	characteristic
58	C₂²×C₄	C₂³	1	2	yes	—
59	C₂²×C₄	C₂³	1	1	yes	characteristic
60	M₁₆	—	2	2	no	—
61	M₁₆	—	2	4	no	—
62	M₁₆	—	2	4	no	—
63	M₁₆	D₄	1	4	yes	complemented
64	C₂×C₈	—	2	2	no	—
65	C₂×C₈	—	2	2	no	—
66	C₂×C₈	—	2	4	no	—
67	C₂×C₈	—	2	4	no	—
68	C₂×C₈	C₂³	1	1	yes	characteristic
69	C₂×C₈	C₂³	1	1	yes	characteristic
70	C₂×C₈	C₂³	1	1	yes	characteristic
71	C₂×C₈	C₂³	1	1	yes	characteristic
72	C₂×C₈	C₂³	1	2	yes	—
73	C₂×C₈	C₂³	1	2	yes	—
74	C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
75	C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
76	C₄⋊C₄	C₂×C₄	1	2	yes	—
77	C₄⋊C₄	C₂×C₄	1	2	yes	—
78	C₂²⋊C₄	C₂×C₄	1	2	yes	—
79	C₂²⋊C₄	C₂×C₄	1	2	yes	—
80	C₂²⋊C₄	C₂×C₄	1	2	yes	—
81	C₂²⋊C₄	C₂×C₄	1	2	yes	—
82	C₂²⋊C₄	C₂×C₄	1	2	yes	—
83	C₄²	C₂³	1	2	yes	—
84	C₂³	—	2	2	no	—
85	C₂³	—	2	2	no	—
86	C₂³	—	4	1	no	—
87	C₂³	C₂²×C₄	1	1	yes	characteristic
88	C₂³	C₂²×C₄	1	2	yes	—
89	C₂³	C₂²×C₄	1	1	yes	characteristic
90	C₂³	C₂²×C₄	1	1	yes	characteristic
91	Q₈	—	2	2	no	—
92	D₄	—	2	4	no	—
93	D₄	—	2	4	no	—
94	D₄	—	2	2	no	—
95	D₄	—	4	2	no	—
96	C₂×C₄	—	2	2	no	—
97	C₂×C₄	—	2	2	no	—
98	C₂×C₄	—	2	2	no	—
99	C₂×C₄	—	2	1	no	—
100	C₂×C₄	—	2	2	no	—
101	C₂×C₄	—	2	1	no	—
102	C₂×C₄	—	2	1	no	—
103	C₂×C₄	—	2	1	no	—
104	C₂×C₄	—	2	2	no	—
105	C₂×C₄	C₂²×C₄	1	2	yes	—
106	C₂×C₄	C₂²×C₄	1	2	yes	—
107	C₂×C₄	C₂²×C₄	1	1	yes	characteristic
108	C₂×C₄	C₂²×C₄	1	1	yes	characteristic
109	C₂×C₄	C₂²×C₄	1	2	yes	—
110	C₂×C₄	C₂²×C₄	1	1	yes	characteristic
111	C₂×C₄	C₂×D₄	1	2	yes	—
112	C₂×C₄	#41	1	2	yes	—
113	C₂×C₄	C₂⁴	1	1	yes	characteristic
114	C₈	—	2	1	no	—
115	C₈	—	2	1	no	—
116	C₈	—	2	2	no	—
117	C₈	—	2	1	no	—
118	C₈	—	2	1	no	—
119	C₈	C₂×D₄	1	4	yes	—
120	C₂²	—	2	1	no	—
121	C₂²	—	2	1	no	—
122	C₂²	—	2	1	no	—
123	C₂²	—	2	1	no	—
124	C₂²	—	2	1	no	—
125	C₂²	—	2	2	no	—
126	C₂²	—	2	2	no	—
127	C₂²	—	2	1	no	—
128	C₂²	—	2	1	no	—
129	C₂²	—	2	2	no	—
130	C₂²	—	4	4	no	—
131	C₂²	C₄×D₄	1	2	yes	—
132	C₂²	#131	1	2	yes	complemented
133	C₂²	C₂³×C₄	1	1	yes	characteristic
134	C₄	—	2	1	no	—
135	C₄	—	2	1	no	—
136	C₄	—	2	2	no	—
137	C₄	—	2	2	no	—
138	C₄	—	2	2	no	—
139	C₄	C₄×D₄	1	2	yes	—
140	C₄	C₂²×D₄	1	1	yes	characteristic
141	C₄	C₂×#41	1	1	yes	characteristic
142	C₂	—	2	2	no	—
143	C₂	—	2	2	no	—
144	C₂	—	4	2	no	—
145	C₂	—	4	1	no	—
146	C₂	C₂×C₄×D₄	1	1	yes	characteristic
147	C₂	C₂×#131	1	1	yes	characteristic
148	C₂	#568	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, cosocle	C₂×C₄	C₂⁴
commutator, socle	C₂²	C₂³×C₄
prime radical	#525	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ G
LECS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. C₂×#131	C₂	semidirect
2. C₂²×M₁₆	C₂	semidirect
4. #436	C₂	semidirect
5. #435	C₂	semidirect
6. #435	C₂	semidirect
7. C₈×D₄	C₂	semidirect
8. #424	C₂	semidirect
9. #408	C₂	semidirect
10. C₂×C₂²⋊C₈	C₂	semidirect
11. #405	C₂	semidirect
19. C₂×M₁₆	C₂²	semidirect
23. C₂²×C₈	C₂²	semidirect
33. C₄⋊C₈	C₂²	semidirect
34. C₄⋊C₈	C₂²	semidirect
39. C₂²⋊C₈	C₂²	semidirect
40. C₂²⋊C₈	C₂²	semidirect
41. #97	C₂²	semidirect
42. C₄×C₈	C₂²	semidirect
63. M₁₆	D₄	semidirect
132. C₂²	#131	semidirect

Maximal subgroups

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

subgroup	index
C₂×#131	2
C₂²×M₁₆	2
#525	2
#436	2
#435	2
C₈×D₄	2
#424	2
#408	2
C₂×C₂²⋊C₈	2
#405	2