Finite Group Databasebeta

Subgroups — #3318

Order 128.

231 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 231 conjugacy classes of subgroups — too many to draw. Only its 88 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	#553	C₂	1	1	yes	maximal, characteristic, complemented
2	D₄²	C₂	1	1	yes	maximal, characteristic, complemented
3	#490	C₂	1	1	yes	maximal, characteristic, complemented
4	#482	C₂	1	1	yes	maximal, characteristic, complemented
5	#481	C₂	1	1	yes	maximal, characteristic, complemented
6	#478	C₂	1	1	yes	maximal, characteristic, complemented
7	#469	C₂	1	1	yes	maximal, characteristic, complemented
8	#466	C₂	1	1	yes	maximal, characteristic, complemented
9	#459	C₂	1	1	yes	maximal, characteristic, complemented
10	#447	C₂	1	1	yes	maximal, characteristic, complemented
11	#447	C₂	1	1	yes	maximal, characteristic, complemented
12	#442	C₂	1	1	yes	maximal, characteristic, complemented
13	#435	C₂	1	1	yes	maximal, characteristic, complemented
14	#418	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂×#102	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂²×D₄	—	2	1	no	—
17	C₂²×D₄	C₂²	1	1	yes	characteristic
18	C₂²×D₄	C₂²	1	1	yes	characteristic
19	C₂×D₈	C₂²	1	1	yes	characteristic
20	C₂×D₈	C₂²	1	1	yes	characteristic, complemented
21	C₂×D₈	C₂²	1	1	yes	characteristic, complemented
22	C₂×M₁₆	C₂²	1	1	yes	characteristic, complemented
23	C₂²×C₈	C₂²	1	1	yes	characteristic, complemented
24	#127	C₂²	1	1	yes	characteristic
25	#126	—	2	1	no	—
26	#125	C₂²	1	1	yes	characteristic, complemented
27	#123	—	2	1	no	—
28	#121	—	2	1	no	—
29	#121	—	2	1	no	—
30	#121	C₂²	1	1	yes	characteristic
31	#121	C₂²	1	1	yes	characteristic
32	#121	C₂²	1	1	yes	characteristic, complemented
33	#121	C₂²	1	1	yes	characteristic, complemented
34	#120	—	2	1	no	—
35	#120	—	2	1	no	—
36	C₄×D₄	—	2	1	no	—
37	C₄×D₄	C₂²	1	1	yes	characteristic
38	C₄×D₄	C₂²	1	1	yes	characteristic
39	C₄×D₄	C₂²	1	1	yes	characteristic, complemented
40	#117	C₂²	1	1	yes	characteristic, complemented
41	C₂×C₄⋊C₄	C₂²	1	1	yes	characteristic, complemented
42	#107	C₂²	1	1	yes	characteristic, complemented
43	#106	C₂²	1	1	yes	characteristic, complemented
44	#106	C₂²	1	1	yes	characteristic, complemented
45	C₄⋊C₈	C₂²	1	1	yes	characteristic, complemented
46	#102	C₂²	1	1	yes	characteristic
47	#102	C₂²	1	1	yes	characteristic
48	#102	C₂²	1	1	yes	characteristic
49	#102	C₂²	1	1	yes	characteristic, complemented
50	#102	C₂²	1	1	yes	characteristic, complemented
51	#102	C₂²	1	1	yes	characteristic, complemented
52	#102	C₂²	1	1	yes	characteristic, complemented
53	#102	C₂²	1	1	yes	characteristic, complemented
54	#102	C₂²	1	1	yes	characteristic, complemented
55	#102	C₂²	1	1	yes	characteristic, complemented
56	C₂²⋊C₈	C₂²	1	1	yes	characteristic, complemented
57	C₂²⋊C₈	C₂²	1	1	yes	characteristic, complemented
58	#97	C₂²	1	1	yes	characteristic, complemented
59	C₂⁴	—	2	1	no	—
60	C₂⁴	—	2	1	no	—
61	C₂×D₄	—	2	2	no	—
62	C₂×D₄	—	2	2	no	—
63	C₂×D₄	—	2	1	no	—
64	C₂×D₄	—	2	1	no	—
65	C₂×D₄	—	2	1	no	—
66	C₂×D₄	—	2	1	no	—
67	C₂×D₄	—	2	1	no	—
68	C₂×D₄	—	2	1	no	—
69	C₂×D₄	—	2	1	no	—
70	C₂×D₄	—	2	1	no	—
71	C₂×D₄	—	4	1	no	—
72	C₂×D₄	—	4	1	no	—
73	C₂×D₄	—	4	4	no	—
74	C₂×D₄	D₄	1	1	yes	characteristic
75	C₂×D₄	C₂³	1	1	yes	characteristic
76	C₂×D₄	C₂³	1	1	yes	characteristic
77	C₂×D₄	C₂³	1	1	yes	characteristic
78	C₂×D₄	C₂³	1	1	yes	characteristic
79	C₂²×C₄	—	2	1	no	—
80	C₂²×C₄	—	2	1	no	—
81	C₂²×C₄	—	2	1	no	—
82	C₂²×C₄	C₂³	1	1	yes	characteristic
83	C₂²×C₄	C₂³	1	1	yes	characteristic
84	D₈	—	2	2	no	—
85	D₈	—	4	1	no	—
86	D₈	—	4	1	no	—
87	M₁₆	—	4	1	no	—
88	C₂×C₈	—	4	1	no	—
89	C₂×C₈	C₂³	1	1	yes	characteristic
90	C₂×C₈	C₂³	1	1	yes	characteristic
91	C₂×C₈	C₂³	1	1	yes	characteristic
92	C₂×C₈	C₂³	1	1	yes	characteristic
93	C₄⋊C₄	—	2	1	no	—
94	C₄⋊C₄	—	2	1	no	—
95	C₄⋊C₄	—	2	1	no	—
96	C₄⋊C₄	D₄	1	1	yes	characteristic
97	C₄⋊C₄	C₂³	1	1	yes	characteristic
98	C₄⋊C₄	C₂³	1	1	yes	characteristic
99	C₄⋊C₄	C₂³	1	1	yes	characteristic
100	C₄⋊C₄	C₂³	1	1	yes	characteristic
101	C₂²⋊C₄	—	2	1	no	—
102	C₂²⋊C₄	—	2	1	no	—
103	C₂²⋊C₄	—	2	1	no	—
104	C₂²⋊C₄	—	2	1	no	—
105	C₂²⋊C₄	—	2	1	no	—
106	C₂²⋊C₄	—	2	1	no	—
107	C₂²⋊C₄	—	2	1	no	—
108	C₂²⋊C₄	D₄	1	1	yes	characteristic
109	C₂²⋊C₄	D₄	1	1	yes	characteristic
110	C₄²	—	2	1	no	—
111	C₄²	C₂³	1	1	yes	characteristic
112	C₂³	—	2	1	no	—
113	C₂³	—	2	1	no	—
114	C₂³	—	2	1	no	—
115	C₂³	—	2	1	no	—
116	C₂³	—	4	2	no	—
117	C₂³	—	4	2	no	—
118	C₂³	—	4	2	no	—
119	C₂³	—	4	2	no	—
120	C₂³	—	8	1	no	—
121	C₂³	—	8	1	no	—
122	C₂³	C₂×D₄	1	1	yes	characteristic
123	C₂³	C₂×D₄	1	1	yes	characteristic
124	D₄	—	2	2	no	—
125	D₄	—	2	1	no	—
126	D₄	—	2	1	no	—
127	D₄	—	2	1	no	—
128	D₄	—	2	1	no	—
129	D₄	—	2	1	no	—
130	D₄	—	2	1	no	—
131	D₄	—	2	1	no	—
132	D₄	—	4	2	no	—
133	D₄	—	4	2	no	—
134	D₄	—	4	2	no	—
135	D₄	—	4	2	no	—
136	D₄	—	4	2	no	—
137	D₄	—	4	2	no	—
138	D₄	—	8	1	no	—
139	D₄	—	8	1	no	—
140	D₄	#41	1	2	yes	—
141	C₂×C₄	—	2	1	no	—
142	C₂×C₄	—	2	1	no	—
143	C₂×C₄	—	2	1	no	—
144	C₂×C₄	—	2	1	no	—
145	C₂×C₄	—	2	1	no	—
146	C₂×C₄	—	2	1	no	—
147	C₂×C₄	—	2	1	no	—
148	C₂×C₄	—	2	1	no	—
149	C₂×C₄	—	4	2	no	—
150	C₂×C₄	—	8	1	no	—
151	C₂×C₄	—	8	1	no	—
152	C₂×C₄	C₂×D₄	1	1	yes	characteristic
153	C₂×C₄	C₂×D₄	1	1	yes	characteristic
154	C₂×C₄	C₂×D₄	1	1	yes	characteristic
155	C₂×C₄	C₂×D₄	1	1	yes	characteristic
156	C₂×C₄	C₂⁴	1	1	yes	characteristic
157	C₈	—	2	1	no	—
158	C₈	—	2	1	no	—
159	C₈	—	2	1	no	—
160	C₈	—	2	1	no	—
161	C₂²	—	2	1	no	—
162	C₂²	—	2	1	no	—
163	C₂²	—	2	1	no	—
164	C₂²	—	2	2	no	—
165	C₂²	—	2	1	no	—
166	C₂²	—	2	1	no	—
167	C₂²	—	4	1	no	—
168	C₂²	—	4	1	no	—
169	C₂²	—	4	1	no	—
170	C₂²	—	4	1	no	—
171	C₂²	—	4	1	no	—
172	C₂²	—	4	1	no	—
173	C₂²	—	4	1	no	—
174	C₂²	—	4	1	no	—
175	C₂²	—	4	1	no	—
176	C₂²	—	4	1	no	—
177	C₂²	—	4	1	no	—
178	C₂²	—	8	4	no	—
179	C₂²	—	8	4	no	—
180	C₂²	AGL(1,ℤ/8)	1	2	yes	—
181	C₂²	C₂²×D₄	1	1	yes	characteristic
182	C₄	—	2	1	no	—
183	C₄	—	2	1	no	—
184	C₄	—	2	1	no	—
185	C₄	—	2	1	no	—
186	C₄	—	4	1	no	—
187	C₄	—	4	1	no	—
188	C₄	—	4	1	no	—
189	C₄	—	4	1	no	—
190	C₄	C₂×#41	1	1	yes	characteristic
191	C₄	H(5,𝔽₂)	1	1	yes	characteristic
192	C₂	—	2	2	no	—
193	C₂	—	4	1	no	—
194	C₂	—	4	2	no	—
195	C₂	—	8	1	no	—
196	C₂	—	8	1	no	—
197	C₂	—	8	1	no	—
198	C₂	#546	1	1	yes	characteristic
199	C₂	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
200	C₂	#576	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 ◁ C₄×D₄ ◁ G
LECS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #553	C₂	semidirect
2. D₄²	C₂	semidirect
3. #490	C₂	semidirect
4. #482	C₂	semidirect
5. #481	C₂	semidirect
6. #478	C₂	semidirect
7. #469	C₂	semidirect
8. #466	C₂	semidirect
9. #459	C₂	semidirect
10. #447	C₂	semidirect
11. #447	C₂	semidirect
12. #442	C₂	semidirect
13. #435	C₂	semidirect
14. #418	C₂	semidirect
15. C₂×#102	C₂	semidirect
20. C₂×D₈	C₂²	semidirect
21. C₂×D₈	C₂²	semidirect
22. C₂×M₁₆	C₂²	semidirect
23. C₂²×C₈	C₂²	semidirect
26. #125	C₂²	semidirect
32. #121	C₂²	semidirect
33. #121	C₂²	semidirect
39. C₄×D₄	C₂²	semidirect
40. #117	C₂²	semidirect
41. C₂×C₄⋊C₄	C₂²	semidirect
42. #107	C₂²	semidirect
43. #106	C₂²	semidirect
44. #106	C₂²	semidirect
45. C₄⋊C₈	C₂²	semidirect
49. #102	C₂²	semidirect
50. #102	C₂²	semidirect
51. #102	C₂²	semidirect
52. #102	C₂²	semidirect
53. #102	C₂²	semidirect
54. #102	C₂²	semidirect
55. #102	C₂²	semidirect
56. C₂²⋊C₈	C₂²	semidirect
57. C₂²⋊C₈	C₂²	semidirect
58. #97	C₂²	semidirect

Maximal subgroups

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

subgroup	index
#553	2
D₄²	2
#490	2
#482	2
#481	2
#478	2
#469	2
#466	2
#459	2
#447	2
#442	2
#435	2
#418	2
C₂×#102	2