Finite Group Databasebeta

Subgroups — #3012

Order 128.

382 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 382 conjugacy classes of subgroups — too many to draw. Only its 108 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×H(5,𝔽₂)	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×AGL(1,ℤ/8)	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#135	C₂	1	1	yes	maximal, characteristic, complemented
4	#537	C₂	1	1	yes	maximal, characteristic, complemented
5	#452	C₂	1	2	yes	maximal, complemented
6	#450	C₂	1	2	yes	maximal, complemented
7	#449	C₂	1	2	yes	maximal, complemented
8	#447	C₂	1	2	yes	maximal, complemented
9	#417	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#102	C₂	1	1	yes	maximal, characteristic, complemented
11	#408	C₂	1	1	yes	maximal, characteristic, complemented
12	H(5,𝔽₂)	—	2	4	no	—
13	H(5,𝔽₂)	—	2	4	no	—
14	C₂×#41	—	2	1	no	—
15	C₂×#41	C₂²	1	2	yes	—
16	C₂×#41	C₂²	1	1	yes	characteristic
17	C₂×#41	C₂²	1	1	yes	characteristic
18	C₂×#41	C₂²	1	1	yes	characteristic, complemented
19	C₂²×D₄	—	2	2	no	—
20	C₂²×D₄	—	2	1	no	—
21	C₂²×D₄	C₂²	1	2	yes	—
22	C₂²×D₄	C₂²	1	1	yes	characteristic
23	AGL(1,ℤ/8)	—	2	4	no	—
24	#135	—	2	4	no	—
25	C₂×Q₁₆	C₂²	1	1	yes	characteristic, complemented
26	C₂×SD₁₆	C₂²	1	1	yes	characteristic, complemented
27	C₂×SD₁₆	C₂²	1	1	yes	characteristic, complemented
28	C₂×SD₁₆	C₂²	1	2	yes	complemented
29	C₂×D₈	C₂²	1	1	yes	characteristic, complemented
30	C₂×D₈	C₂²	1	2	yes	complemented
31	C₂×M₁₆	C₂²	1	1	yes	characteristic, complemented
32	C₂²×C₈	C₂²	1	1	yes	characteristic, complemented
33	C₂²⋊Q₈	—	2	1	no	—
34	C₂²⋊Q₈	C₂²	1	2	yes	complemented
35	#121	—	2	1	no	—
36	#121	—	2	2	no	—
37	#121	C₂²	1	2	yes	complemented
38	C₂×C₄⋊C₄	C₂²	1	1	yes	characteristic, complemented
39	#103	C₂²	1	2	yes	complemented
40	#102	C₂²	1	2	yes	complemented
41	#102	C₂²	1	2	yes	complemented
42	#102	C₂²	1	2	yes	complemented
43	C₂²⋊C₈	C₂²	1	2	yes	complemented
44	C₂²⋊C₈	C₂²	1	2	yes	complemented
45	C₂⁴	—	2	2	no	—
46	C₂⁴	—	2	1	no	—
47	#41	—	2	8	no	—
48	#41	—	2	4	no	—
49	#41	—	2	4	no	—
50	#41	—	2	2	no	—
51	#41	—	2	2	no	—
52	#41	—	2	2	no	—
53	#41	—	2	2	no	—
54	#41	—	2	2	no	—
55	#41	D₄	1	4	yes	complemented
56	C₂×Q₈	D₄	1	1	yes	characteristic
57	C₂×Q₈	C₂³	1	1	yes	characteristic
58	C₂×Q₈	C₂³	1	1	yes	characteristic
59	C₂×D₄	—	2	16	no	—
60	C₂×D₄	—	2	8	no	—
61	C₂×D₄	—	2	2	no	—
62	C₂×D₄	—	2	1	no	—
63	C₂×D₄	—	2	4	no	—
64	C₂×D₄	—	2	4	no	—
65	C₂×D₄	—	2	1	no	—
66	C₂×D₄	—	2	2	no	—
67	C₂×D₄	—	2	2	no	—
68	C₂×D₄	—	2	1	no	—
69	C₂×D₄	—	2	2	no	—
70	C₂×D₄	D₄	1	2	yes	—
71	C₂×D₄	D₄	1	1	yes	characteristic
72	C₂×D₄	D₄	1	4	yes	complemented
73	C₂×D₄	C₂³	1	1	yes	characteristic
74	C₂×D₄	C₂³	1	1	yes	characteristic
75	C₂×D₄	C₂³	1	2	yes	—
76	C₂²×C₄	—	2	2	no	—
77	C₂²×C₄	—	2	1	no	—
78	C₂²×C₄	—	2	1	no	—
79	C₂²×C₄	—	2	1	no	—
80	C₂²×C₄	C₂³	1	2	yes	—
81	C₂²×C₄	C₂³	1	1	yes	characteristic
82	Q₁₆	—	2	2	no	—
83	SD₁₆	—	2	2	no	—
84	SD₁₆	—	2	2	no	—
85	SD₁₆	—	2	4	no	—
86	D₈	—	2	2	no	—
87	D₈	—	2	4	no	—
88	M₁₆	—	2	2	no	—
89	C₂×C₈	—	2	2	no	—
90	C₂×C₈	C₂³	1	1	yes	characteristic
91	C₂×C₈	C₂³	1	1	yes	characteristic
92	C₂×C₈	C₂³	1	2	yes	—
93	C₄⋊C₄	—	2	1	no	—
94	C₄⋊C₄	—	2	2	no	—
95	C₄⋊C₄	C₂³	1	2	yes	—
96	C₂²⋊C₄	—	2	2	no	—
97	C₂²⋊C₄	—	2	2	no	—
98	C₂³	—	2	8	no	—
99	C₂³	—	2	2	no	—
100	C₂³	—	2	1	no	—
101	C₂³	—	2	4	no	—
102	C₂³	—	2	1	no	—
103	C₂³	—	8	1	no	—
104	C₂³	—	8	1	no	—
105	C₂³	—	8	2	no	—
106	C₂³	—	8	2	no	—
107	C₂³	C₂×D₄	1	2	yes	—
108	C₂³	C₂×D₄	1	1	yes	characteristic
109	Q₈	—	2	2	no	—
110	Q₈	—	2	1	no	—
111	Q₈	—	2	1	no	—
112	Q₈	—	2	1	no	—
113	Q₈	C₂×D₄	1	2	yes	—
114	D₄	—	2	8	no	—
115	D₄	—	2	8	no	—
116	D₄	—	2	4	no	—
117	D₄	—	2	4	no	—
118	D₄	—	2	2	no	—
119	D₄	—	2	4	no	—
120	D₄	—	2	2	no	—
121	D₄	—	2	1	no	—
122	D₄	—	2	1	no	—
123	D₄	—	2	1	no	—
124	D₄	—	4	2	no	—
125	D₄	—	8	2	no	—
126	D₄	C₂×D₄	1	2	yes	—
127	D₄	C₂×D₄	1	4	yes	—
128	C₂×C₄	—	2	8	no	—
129	C₂×C₄	—	2	1	no	—
130	C₂×C₄	—	2	4	no	—
131	C₂×C₄	—	2	2	no	—
132	C₂×C₄	—	2	2	no	—
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	—	2	1	no	—
135	C₂×C₄	—	4	2	no	—
136	C₂×C₄	—	8	1	no	—
137	C₂×C₄	C₂×D₄	1	2	yes	—
138	C₂×C₄	C₂×D₄	1	1	yes	characteristic
139	C₂×C₄	C₂×D₄	1	2	yes	—
140	C₂×C₄	C₂×D₄	1	2	yes	—
141	C₂×C₄	C₂⁴	1	1	yes	characteristic
142	C₈	—	2	1	no	—
143	C₈	—	2	1	no	—
144	C₈	—	2	2	no	—
145	C₂²	—	2	1	no	—
146	C₂²	—	2	1	no	—
147	C₂²	—	2	2	no	—
148	C₂²	—	2	4	no	—
149	C₂²	—	2	2	no	—
150	C₂²	—	2	2	no	—
151	C₂²	—	2	2	no	—
152	C₂²	—	4	2	no	—
153	C₂²	—	4	1	no	—
154	C₂²	—	4	2	no	—
155	C₂²	—	4	1	no	—
156	C₂²	—	4	1	no	—
157	C₂²	—	4	1	no	—
158	C₂²	—	4	1	no	—
159	C₂²	—	8	4	no	—
160	C₂²	—	8	8	no	—
161	C₂²	#120	1	2	yes	—
162	C₂²	C₂²×D₄	1	1	yes	characteristic
163	C₄	—	2	2	no	—
164	C₄	—	2	2	no	—
165	C₄	—	4	1	no	—
166	C₄	—	4	2	no	—
167	C₄	#120	1	2	yes	—
168	C₄	C₂²×D₄	1	1	yes	characteristic
169	C₄	C₂²×D₄	1	1	yes	characteristic
170	C₂	—	2	2	no	—
171	C₂	—	4	2	no	—
172	C₂	—	4	4	no	—
173	C₂	—	4	2	no	—
174	C₂	—	8	1	no	—
175	C₂	C₂×#120	1	1	yes	characteristic
176	C₂	#576	1	1	yes	characteristic
177	C₂	#577	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 ◁ C₂×#41 ◁ 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×H(5,𝔽₂)	C₂	semidirect
2. C₂×AGL(1,ℤ/8)	C₂	semidirect
3. C₂×#135	C₂	semidirect
4. #537	C₂	semidirect
5. #452	C₂	semidirect
6. #450	C₂	semidirect
7. #449	C₂	semidirect
8. #447	C₂	semidirect
9. #417	C₂	semidirect
10. C₂×#102	C₂	semidirect
11. #408	C₂	semidirect
18. C₂×#41	C₂²	semidirect
25. C₂×Q₁₆	C₂²	semidirect
26. C₂×SD₁₆	C₂²	semidirect
27. C₂×SD₁₆	C₂²	semidirect
28. C₂×SD₁₆	C₂²	semidirect
29. C₂×D₈	C₂²	semidirect
30. C₂×D₈	C₂²	semidirect
31. C₂×M₁₆	C₂²	semidirect
32. C₂²×C₈	C₂²	semidirect
34. C₂²⋊Q₈	C₂²	semidirect
37. #121	C₂²	semidirect
38. C₂×C₄⋊C₄	C₂²	semidirect
39. #103	C₂²	semidirect
40. #102	C₂²	semidirect
41. #102	C₂²	semidirect
42. #102	C₂²	semidirect
43. C₂²⋊C₈	C₂²	semidirect
44. C₂²⋊C₈	C₂²	semidirect
55. #41	D₄	semidirect
72. C₂×D₄	D₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×H(5,𝔽₂)	2
C₂×AGL(1,ℤ/8)	2
C₂×#135	2
#537	2
#452	2
#450	2
#449	2
#447	2
#417	2
C₂×#102	2
#408	2