Finite Group Databasebeta

Subgroups — #3055

Order 128.

274 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 274 conjugacy classes of subgroups — too many to draw. Only its 102 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×AGL(1,ℤ/8)	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, complemented
4	C₂×#121	C₂	1	1	yes	maximal, characteristic, complemented
5	#469	C₂	1	2	yes	maximal, complemented
6	#468	C₂	1	2	yes	maximal, complemented
7	#466	C₂	1	2	yes	maximal, complemented
8	#465	C₂	1	2	yes	maximal, complemented
9	#428	C₂	1	1	yes	maximal, characteristic, complemented
10	#418	C₂	1	1	yes	maximal, characteristic, complemented
11	#417	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#41	C₂²	1	1	yes	characteristic
13	C₂²×D₄	—	2	1	no	—
14	C₂²×D₄	C₂²	1	1	yes	characteristic
15	C₂³×C₄	C₂²	1	1	yes	characteristic
16	AGL(1,ℤ/8)	—	2	4	no	—
17	C₂×SD₁₆	C₂²	1	2	yes	complemented
18	C₂×D₈	C₂²	1	2	yes	complemented
19	C₂×M₁₆	—	2	4	no	—
20	C₂×M₁₆	C₂²	1	1	yes	characteristic
21	C₂×M₁₆	C₂²	1	1	yes	characteristic
22	C₂×M₁₆	C₂²	1	2	yes	complemented
23	C₂²×C₈	C₂²	1	2	yes	complemented
24	#123	—	2	1	no	—
25	C₂²⋊Q₈	C₂²	1	1	yes	characteristic, complemented
26	C₂²⋊Q₈	C₂²	1	1	yes	characteristic, complemented
27	#121	—	2	1	no	—
28	#121	—	2	1	no	—
29	#121	C₂²	1	2	yes	—
30	#121	C₂²	1	2	yes	—
31	#121	C₂²	1	1	yes	characteristic, complemented
32	#121	C₂²	1	1	yes	characteristic, complemented
33	#120	—	2	1	no	—
34	C₄×D₄	—	2	1	no	—
35	C₄×D₄	—	2	1	no	—
36	#117	C₂²	1	1	yes	characteristic
37	C₂×C₄⋊C₄	C₂²	1	1	yes	characteristic
38	C₂×C₂²⋊C₄	—	2	1	no	—
39	#107	C₂²	1	2	yes	complemented
40	#106	C₂²	1	2	yes	complemented
41	#103	C₂²	1	2	yes	complemented
42	#102	C₂²	1	2	yes	complemented
43	#102	C₂²	1	2	yes	complemented
44	#102	C₂²	1	2	yes	complemented
45	C₂⁴	—	2	1	no	—
46	C₂⁴	D₄	1	1	yes	characteristic
47	#41	—	2	2	no	—
48	#41	—	2	2	no	—
49	C₂×Q₈	C₂³	1	1	yes	characteristic
50	C₂×D₄	—	2	1	no	—
51	C₂×D₄	—	2	1	no	—
52	C₂×D₄	—	2	1	no	—
53	C₂×D₄	—	2	2	no	—
54	C₂×D₄	—	2	2	no	—
55	C₂×D₄	—	2	2	no	—
56	C₂×D₄	—	2	1	no	—
57	C₂×D₄	—	2	2	no	—
58	C₂×D₄	—	4	4	no	—
59	C₂×D₄	C₂³	1	2	yes	—
60	C₂×D₄	C₂³	1	1	yes	characteristic
61	C₂²×C₄	—	2	2	no	—
62	C₂²×C₄	—	2	2	no	—
63	C₂²×C₄	—	2	1	no	—
64	C₂²×C₄	—	2	1	no	—
65	C₂²×C₄	D₄	1	1	yes	characteristic
66	C₂²×C₄	D₄	1	1	yes	characteristic
67	C₂²×C₄	D₄	1	1	yes	characteristic
68	C₂²×C₄	C₂³	1	1	yes	characteristic
69	C₂²×C₄	C₂³	1	1	yes	characteristic
70	C₂²×C₄	C₂³	1	1	yes	characteristic
71	SD₁₆	—	2	4	no	—
72	D₈	—	2	4	no	—
73	M₁₆	—	2	2	no	—
74	M₁₆	—	2	4	no	—
75	M₁₆	D₄	1	4	yes	complemented
76	C₂×C₈	—	2	4	no	—
77	C₂×C₈	C₂³	1	2	yes	—
78	C₂×C₈	C₂³	1	2	yes	—
79	C₄⋊C₄	—	2	1	no	—
80	C₄⋊C₄	—	2	1	no	—
81	C₄⋊C₄	—	2	1	no	—
82	C₄⋊C₄	C₂³	1	1	yes	characteristic
83	C₄⋊C₄	C₂³	1	1	yes	characteristic
84	C₄⋊C₄	C₂³	1	2	yes	—
85	C₂²⋊C₄	—	2	1	no	—
86	C₂²⋊C₄	—	2	1	no	—
87	C₂²⋊C₄	—	2	1	no	—
88	C₂²⋊C₄	—	2	2	no	—
89	C₂²⋊C₄	—	2	2	no	—
90	C₂²⋊C₄	—	2	1	no	—
91	C₂²⋊C₄	—	2	1	no	—
92	C₄²	—	2	1	no	—
93	C₂³	—	2	2	no	—
94	C₂³	—	2	2	no	—
95	C₂³	—	2	1	no	—
96	C₂³	—	2	2	no	—
97	C₂³	—	4	1	no	—
98	C₂³	—	4	2	no	—
99	C₂³	—	4	2	no	—
100	C₂³	—	4	2	no	—
101	C₂³	C₂×D₄	1	1	yes	characteristic
102	C₂³	C₂×D₄	1	1	yes	characteristic
103	C₂³	C₂×D₄	1	1	yes	characteristic
104	Q₈	—	2	1	no	—
105	Q₈	—	2	1	no	—
106	D₄	—	2	1	no	—
107	D₄	—	2	2	no	—
108	D₄	—	2	2	no	—
109	D₄	—	2	1	no	—
110	D₄	—	4	2	no	—
111	D₄	—	4	4	no	—
112	D₄	—	4	2	no	—
113	D₄	—	4	2	no	—
114	D₄	—	4	4	no	—
115	D₄	—	4	2	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	—	2	2	no	—
118	C₂×C₄	—	2	2	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	2	no	—
127	C₂×C₄	—	4	2	no	—
128	C₂×C₄	—	4	2	no	—
129	C₂×C₄	C₂×D₄	1	1	yes	characteristic
130	C₂×C₄	C₂×D₄	1	1	yes	characteristic
131	C₂×C₄	C₂×D₄	1	1	yes	characteristic
132	C₂×C₄	C₂×D₄	1	2	yes	—
133	C₂×C₄	#41	1	2	yes	—
134	C₂×C₄	C₂⁴	1	1	yes	characteristic
135	C₈	—	2	2	no	—
136	C₈	C₂×D₄	1	4	yes	complemented
137	C₂²	—	2	1	no	—
138	C₂²	—	2	1	no	—
139	C₂²	—	2	1	no	—
140	C₂²	—	2	1	no	—
141	C₂²	—	2	1	no	—
142	C₂²	—	2	1	no	—
143	C₂²	—	2	1	no	—
144	C₂²	—	4	4	no	—
145	C₂²	—	4	1	no	—
146	C₂²	—	4	2	no	—
147	C₂²	—	4	2	no	—
148	C₂²	—	4	1	no	—
149	C₂²	—	4	1	no	—
150	C₂²	—	4	2	no	—
151	C₂²	—	8	4	no	—
152	C₂²	#121	1	2	yes	—
153	C₂²	AGL(1,ℤ/8)	1	2	yes	complemented
154	C₂²	C₂²×D₄	1	1	yes	characteristic
155	C₄	—	2	1	no	—
156	C₄	—	2	1	no	—
157	C₄	—	4	1	no	—
158	C₄	—	4	2	no	—
159	C₄	—	4	1	no	—
160	C₄	—	4	1	no	—
161	C₄	#121	1	2	yes	—
162	C₄	C₂²×D₄	1	1	yes	characteristic
163	C₄	C₂×#41	1	1	yes	characteristic
164	C₂	—	2	2	no	—
165	C₂	—	2	2	no	—
166	C₂	—	4	1	no	—
167	C₂	—	8	1	no	—
168	C₂	—	8	2	no	—
169	C₂	C₂×#121	1	1	yes	characteristic
170	C₂	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
171	C₂	#575	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×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×AGL(1,ℤ/8)	C₂	semidirect
2. C₂²×M₁₆	C₂	semidirect
3. #525	C₂	semidirect
4. C₂×#121	C₂	semidirect
5. #469	C₂	semidirect
6. #468	C₂	semidirect
7. #466	C₂	semidirect
8. #465	C₂	semidirect
9. #428	C₂	semidirect
10. #418	C₂	semidirect
11. #417	C₂	semidirect
17. C₂×SD₁₆	C₂²	semidirect
18. C₂×D₈	C₂²	semidirect
22. C₂×M₁₆	C₂²	semidirect
23. C₂²×C₈	C₂²	semidirect
25. C₂²⋊Q₈	C₂²	semidirect
26. C₂²⋊Q₈	C₂²	semidirect
31. #121	C₂²	semidirect
32. #121	C₂²	semidirect
39. #107	C₂²	semidirect
40. #106	C₂²	semidirect
41. #103	C₂²	semidirect
42. #102	C₂²	semidirect
43. #102	C₂²	semidirect
44. #102	C₂²	semidirect
75. M₁₆	D₄	semidirect
136. C₈	C₂×D₄	semidirect
153. C₂²	AGL(1,ℤ/8)	semidirect

Maximal subgroups

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

subgroup	index
C₂×AGL(1,ℤ/8)	2
C₂²×M₁₆	2
#525	2
C₂×#121	2
#469	2
#468	2
#466	2
#465	2
#428	2
#418	2
#417	2