Finite Group Databasebeta

Subgroups — #3537

Order 128.

635 conjugacy classes of subgroups; only the 384 normal subgroups are shown.

normal subgroups only

This group has 635 conjugacy classes of subgroups — too many to draw. Only its 384 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	#560	C₂	1	4	yes	maximal, complemented
2	#553	C₂	1	4	yes	maximal, complemented
3	D₄²	C₂	1	4	yes	maximal, complemented
4	D₄²	C₂	1	4	yes	maximal, complemented
5	#540	C₂	1	2	yes	maximal, complemented
6	#538	C₂	1	4	yes	maximal, complemented
7	#535	C₂	1	2	yes	maximal, complemented
8	#534	C₂	1	4	yes	maximal, complemented
9	#525	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#121	C₂	1	2	yes	maximal, complemented
11	C₂×#41	C₂²	1	1	yes	characteristic, complemented
12	C₂²×D₄	C₂²	1	2	yes	complemented
13	C₂²×D₄	C₂²	1	4	yes	complemented
14	C₂²×D₄	C₂²	1	8	yes	complemented
15	C₂²×D₄	C₂²	1	2	yes	complemented
16	C₂³×C₄	C₂²	1	1	yes	characteristic, complemented
17	#127	C₂²	1	4	yes	complemented
18	#127	C₂²	1	2	yes	complemented
19	#126	C₂²	1	8	yes	complemented
20	#125	C₂²	1	2	yes	complemented
21	#124	C₂²	1	4	yes	complemented
22	#123	C₂²	1	2	yes	complemented
23	#123	C₂²	1	8	yes	complemented
24	#123	C₂²	1	8	yes	complemented
25	C₂²⋊Q₈	C₂²	1	2	yes	complemented
26	#121	C₂²	1	4	yes	complemented
27	#121	C₂²	1	4	yes	complemented
28	#121	C₂²	1	4	yes	complemented
29	#121	C₂²	1	4	yes	complemented
30	#121	C₂²	1	2	yes	complemented
31	#121	C₂²	1	8	yes	complemented
32	#121	C₂²	1	4	yes	complemented
33	#121	C₂²	1	8	yes	complemented
34	#121	C₂²	1	8	yes	complemented
35	#120	C₂²	1	2	yes	complemented
36	#120	C₂²	1	4	yes	complemented
37	#120	C₂²	1	8	yes	complemented
38	#120	C₂²	1	4	yes	complemented
39	#120	C₂²	1	8	yes	complemented
40	C₄×D₄	C₂²	1	4	yes	complemented
41	C₄×D₄	C₂²	1	8	yes	complemented
42	C₄×D₄	C₂²	1	4	yes	complemented
43	#117	C₂²	1	1	yes	characteristic, complemented
44	#117	C₂²	1	2	yes	complemented
45	C₂×C₄⋊C₄	C₂²	1	2	yes	complemented
46	C₂×C₂²⋊C₄	C₂²	1	4	yes	complemented
47	C₂⁴	C₂³	1	1	yes	characteristic
48	C₂⁴	C₂³	1	4	yes	—
49	C₂⁴	C₂³	1	4	yes	—
50	#41	—	4	2	no	—
51	C₂×Q₈	C₂³	1	1	yes	characteristic
52	C₂×D₄	—	4	4	no	—
53	C₂×D₄	—	4	4	no	—
54	C₂×D₄	—	4	8	no	—
55	C₂×D₄	—	4	16	no	—
56	C₂×D₄	C₂³	1	4	yes	—
57	C₂×D₄	C₂³	1	8	yes	—
58	C₂×D₄	C₂³	1	4	yes	—
59	C₂×D₄	C₂³	1	1	yes	characteristic
60	C₂×D₄	C₂³	1	2	yes	—
61	C₂×D₄	C₂³	1	4	yes	—
62	C₂×D₄	C₂³	1	8	yes	—
63	C₂×D₄	C₂³	1	8	yes	—
64	C₂×D₄	C₂³	1	4	yes	—
65	C₂×D₄	C₂³	1	8	yes	—
66	C₂²×C₄	—	4	2	no	—
67	C₂²×C₄	C₂³	1	2	yes	—
68	C₂²×C₄	C₂³	1	2	yes	—
69	C₂²×C₄	C₂³	1	1	yes	characteristic
70	C₂²×C₄	C₂³	1	2	yes	—
71	C₂²×C₄	C₂³	1	4	yes	—
72	C₂²×C₄	C₂³	1	8	yes	—
73	C₂²×C₄	C₂³	1	1	yes	characteristic
74	C₄⋊C₄	C₂³	1	4	yes	—
75	C₄⋊C₄	C₂³	1	4	yes	—
76	C₄⋊C₄	C₂³	1	2	yes	—
77	C₄⋊C₄	C₂³	1	4	yes	—
78	C₄⋊C₄	C₂³	1	8	yes	—
79	C₂²⋊C₄	C₂³	1	8	yes	—
80	C₂²⋊C₄	C₂³	1	8	yes	—
81	C₂²⋊C₄	C₂³	1	4	yes	—
82	C₂²⋊C₄	C₂³	1	2	yes	—
83	C₂²⋊C₄	C₂³	1	4	yes	—
84	C₂²⋊C₄	C₂³	1	8	yes	—
85	C₂²⋊C₄	C₂³	1	8	yes	—
86	C₂²⋊C₄	C₂³	1	4	yes	—
87	C₄²	C₂³	1	2	yes	—
88	C₄²	C₂³	1	4	yes	—
89	C₂³	—	2	4	no	—
90	C₂³	—	2	8	no	—
91	C₂³	—	4	1	no	—
92	C₂³	—	4	4	no	—
93	C₂³	—	4	4	no	—
94	C₂³	—	4	4	no	—
95	C₂³	—	4	8	no	—
96	C₂³	C₂⁴	1	2	yes	—
97	C₂³	C₂⁴	1	8	yes	—
98	C₂³	C₂⁴	1	1	yes	characteristic
99	C₂³	C₂⁴	1	2	yes	—
100	Q₈	—	4	1	no	—
101	D₄	—	2	16	no	—
102	D₄	—	2	8	no	—
103	D₄	—	2	8	no	—
104	D₄	—	4	4	no	—
105	D₄	—	4	4	no	—
106	D₄	—	4	8	no	—
107	D₄	—	4	8	no	—
108	D₄	—	4	8	no	—
109	D₄	—	4	1	no	—
110	D₄	—	4	2	no	—
111	C₂×C₄	—	2	4	no	—
112	C₂×C₄	—	2	2	no	—
113	C₂×C₄	—	2	4	no	—
114	C₂×C₄	—	4	1	no	—
115	C₂×C₄	—	4	2	no	—
116	C₂×C₄	—	4	4	no	—
117	C₂×C₄	—	4	8	no	—
118	C₂×C₄	C₂⁴	1	2	yes	—
119	C₂×C₄	C₂⁴	1	4	yes	—
120	C₂×C₄	C₂⁴	1	8	yes	—
121	C₂×C₄	C₂⁴	1	1	yes	characteristic
122	C₂×C₄	C₂⁴	1	2	yes	—
123	C₂×C₄	C₂⁴	1	1	yes	characteristic
124	C₂²	—	2	2	no	—
125	C₂²	—	2	2	no	—
126	C₂²	—	2	1	no	—
127	C₂²	—	2	2	no	—
128	C₂²	—	2	8	no	—
129	C₂²	—	2	4	no	—
130	C₂²	—	2	8	no	—
131	C₂²	—	2	8	no	—
132	C₂²	—	2	2	no	—
133	C₂²	—	4	4	no	—
134	C₂²	—	8	8	no	—
135	C₂²	—	8	8	no	—
136	C₂²	H(5,𝔽₂)	1	4	yes	—
137	C₂²	C₂⁵	1	1	yes	characteristic
138	C₄	—	2	2	no	—
139	C₄	—	2	1	no	—
140	C₄	—	2	2	no	—
141	C₄	—	2	8	no	—
142	C₄	—	2	4	no	—
143	C₄	H(5,𝔽₂)	1	2	yes	—
144	C₂	—	2	4	no	—
145	C₂	—	4	1	no	—
146	C₂	—	4	2	no	—
147	C₂	—	4	8	no	—
148	C₂	C₂×H(5,𝔽₂)	1	2	yes	—
149	C₂	C₂×H(5,𝔽₂)	1	1	yes	characteristic

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂²	C₂⁵
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. #560	C₂	semidirect
2. #553	C₂	semidirect
3. D₄²	C₂	semidirect
4. D₄²	C₂	semidirect
5. #540	C₂	semidirect
6. #538	C₂	semidirect
7. #535	C₂	semidirect
8. #534	C₂	semidirect
9. #525	C₂	semidirect
10. C₂×#121	C₂	semidirect
11. C₂×#41	C₂²	semidirect
12. C₂²×D₄	C₂²	semidirect
13. C₂²×D₄	C₂²	semidirect
14. C₂²×D₄	C₂²	semidirect
15. C₂²×D₄	C₂²	semidirect
16. C₂³×C₄	C₂²	semidirect
17. #127	C₂²	semidirect
18. #127	C₂²	semidirect
19. #126	C₂²	semidirect
20. #125	C₂²	semidirect
21. #124	C₂²	semidirect
22. #123	C₂²	semidirect
23. #123	C₂²	semidirect
24. #123	C₂²	semidirect
25. C₂²⋊Q₈	C₂²	semidirect
26. #121	C₂²	semidirect
27. #121	C₂²	semidirect
28. #121	C₂²	semidirect
29. #121	C₂²	semidirect
30. #121	C₂²	semidirect
31. #121	C₂²	semidirect
32. #121	C₂²	semidirect
33. #121	C₂²	semidirect
34. #121	C₂²	semidirect
35. #120	C₂²	semidirect
36. #120	C₂²	semidirect
37. #120	C₂²	semidirect
38. #120	C₂²	semidirect
39. #120	C₂²	semidirect
40. C₄×D₄	C₂²	semidirect
41. C₄×D₄	C₂²	semidirect
42. C₄×D₄	C₂²	semidirect
43. #117	C₂²	semidirect
44. #117	C₂²	semidirect
45. C₂×C₄⋊C₄	C₂²	semidirect
46. C₂×C₂²⋊C₄	C₂²	semidirect

Maximal subgroups

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

subgroup	index
#560	2
#553	2
D₄²	2
#540	2
#538	2
#535	2
#534	2
#525	2
C₂×#121	2