Finite Group Databasebeta

Subgroups — #2324

Order 128.

334 conjugacy classes of subgroups; only the 148 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 148 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×#121	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×D₄	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#117	C₂	1	1	yes	maximal, characteristic, complemented
4	#390	C₂	1	1	yes	maximal, characteristic, complemented
5	#390	C₂	1	2	yes	maximal, complemented
6	#389	C₂	1	1	yes	maximal, characteristic, complemented
7	#388	C₂	1	2	yes	maximal, complemented
8	#386	C₂	1	2	yes	maximal, complemented
9	#385	C₂	1	2	yes	maximal, complemented
10	#380	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×D₄	C₂²	1	1	yes	characteristic
13	C₂²×D₄	C₂²	1	2	yes	—
14	C₂³×C₄	C₂²	1	1	yes	characteristic
15	C₂³×C₄	C₂²	1	2	yes	—
16	#121	C₄	1	8	yes	complemented
17	C₄×D₄	—	2	4	no	—
18	#117	—	2	4	no	—
19	C₂×C₄⋊C₄	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊C₄	C₂²	1	1	yes	characteristic
21	C₂×C₄⋊C₄	C₂²	1	1	yes	characteristic
22	C₂×C₄⋊C₄	C₂²	1	1	yes	characteristic
23	C₂×C₄⋊C₄	C₂²	1	2	yes	—
24	C₂×C₄⋊C₄	C₂²	1	1	yes	characteristic
25	C₂×C₂²⋊C₄	C₂²	1	1	yes	characteristic
26	C₂×C₂²⋊C₄	C₂²	1	1	yes	characteristic
27	C₂×C₂²⋊C₄	C₂²	1	2	yes	—
28	C₂×C₂²⋊C₄	C₂²	1	2	yes	—
29	C₂×C₂²⋊C₄	C₂²	1	2	yes	—
30	C₂×C₂²⋊C₄	C₂²	1	2	yes	—
31	C₂×C₂²⋊C₄	C₂²	1	2	yes	—
32	C₂×C₄²	C₂²	1	1	yes	characteristic
33	C₂×C₄²	C₂²	1	2	yes	—
34	C₂×C₄²	C₂²	1	1	yes	characteristic
35	#95	C₂²	1	2	yes	—
36	#95	C₂²	1	2	yes	—
37	#95	C₂²	1	2	yes	—
38	C₂⁴	C₂³	1	1	yes	characteristic
39	C₂⁴	C₂³	1	2	yes	—
40	C₂×D₄	—	2	2	no	—
41	C₂×D₄	—	2	2	no	—
42	C₂×D₄	—	4	2	no	—
43	C₂×D₄	—	4	2	no	—
44	C₂×D₄	C₂×C₄	1	4	yes	—
45	C₂×D₄	C₂×C₄	1	8	yes	complemented
46	C₂²×C₄	—	2	2	no	—
47	C₂²×C₄	—	2	2	no	—
48	C₂²×C₄	—	2	4	no	—
49	C₂²×C₄	—	4	2	no	—
50	C₂²×C₄	—	4	2	no	—
51	C₂²×C₄	C₂×C₄	1	4	yes	complemented
52	C₂²×C₄	C₂³	1	1	yes	characteristic
53	C₂²×C₄	C₂³	1	2	yes	—
54	C₂²×C₄	C₂³	1	2	yes	—
55	C₂²×C₄	C₂³	1	1	yes	characteristic
56	C₂²×C₄	C₂³	1	2	yes	—
57	C₂²×C₄	C₂³	1	1	yes	characteristic
58	C₂²×C₄	C₂³	1	1	yes	characteristic
59	C₂²×C₄	C₂³	1	1	yes	characteristic
60	C₂²×C₄	C₂³	1	1	yes	characteristic
61	C₄⋊C₄	—	2	2	no	—
62	C₄⋊C₄	—	2	2	no	—
63	C₄⋊C₄	—	2	2	no	—
64	C₄⋊C₄	—	2	2	no	—
65	C₄⋊C₄	—	4	2	no	—
66	C₄⋊C₄	C₂×C₄	1	4	yes	complemented
67	C₂²⋊C₄	—	2	2	no	—
68	C₂²⋊C₄	—	2	2	no	—
69	C₂²⋊C₄	—	2	4	no	—
70	C₂²⋊C₄	—	4	2	no	—
71	C₂²⋊C₄	—	4	2	no	—
72	C₂²⋊C₄	—	4	2	no	—
73	C₂²⋊C₄	C₂×C₄	1	8	yes	complemented
74	C₄²	—	2	2	no	—
75	C₄²	—	4	2	no	—
76	C₄²	D₄	1	4	yes	complemented
77	C₂³	—	2	1	no	—
78	C₂³	—	2	2	no	—
79	C₂³	—	2	1	no	—
80	C₂³	—	2	2	no	—
81	C₂³	—	2	1	no	—
82	C₂³	—	2	2	no	—
83	C₂³	—	2	2	no	—
84	C₂³	—	2	2	no	—
85	C₂³	—	2	1	no	—
86	C₂³	—	2	1	no	—
87	C₂³	—	2	1	no	—
88	C₂³	—	2	2	no	—
89	C₂³	C₂²×C₄	1	4	yes	—
90	C₂³	C₂²×C₄	1	2	yes	—
91	C₂³	C₂⁴	1	1	yes	characteristic
92	D₄	—	2	8	no	—
93	D₄	—	4	8	no	—
94	C₂×C₄	—	2	1	no	—
95	C₂×C₄	—	2	1	no	—
96	C₂×C₄	—	2	1	no	—
97	C₂×C₄	—	2	2	no	—
98	C₂×C₄	—	2	2	no	—
99	C₂×C₄	—	2	1	no	—
100	C₂×C₄	—	2	1	no	—
101	C₂×C₄	—	2	1	no	—
102	C₂×C₄	—	2	8	no	—
103	C₂×C₄	—	2	1	no	—
104	C₂×C₄	—	2	1	no	—
105	C₂×C₄	—	2	2	no	—
106	C₂×C₄	—	2	2	no	—
107	C₂×C₄	—	2	2	no	—
108	C₂×C₄	—	2	1	no	—
109	C₂×C₄	—	2	1	no	—
110	C₂×C₄	—	2	2	no	—
111	C₂×C₄	—	2	2	no	—
112	C₂×C₄	—	2	1	no	—
113	C₂×C₄	—	2	2	no	—
114	C₂×C₄	—	2	1	no	—
115	C₂×C₄	—	4	8	no	—
116	C₂×C₄	C₂²×C₄	1	2	yes	—
117	C₂×C₄	C₂²×C₄	1	4	yes	—
118	C₂×C₄	C₂²×C₄	1	2	yes	—
119	C₂×C₄	C₂×D₄	1	2	yes	—
120	C₂×C₄	C₂×D₄	1	2	yes	—
121	C₂×C₄	C₂×D₄	1	2	yes	—
122	C₂×C₄	#41	1	2	yes	—
123	C₂²	—	2	2	no	—
124	C₂²	—	2	4	no	—
125	C₂²	—	2	2	no	—
126	C₂²	—	2	4	no	—
127	C₂²	—	2	2	no	—
128	C₂²	—	2	4	no	—
129	C₂²	—	4	2	no	—
130	C₂²	—	4	1	no	—
131	C₂²	—	4	1	no	—
132	C₂²	—	4	2	no	—
133	C₂²	—	4	1	no	—
134	C₂²	—	4	2	no	—
135	C₂²	—	4	1	no	—
136	C₂²	—	4	2	no	—
137	C₂²	C₂³×C₄	1	1	yes	characteristic
138	C₂²	C₂²×D₄	1	1	yes	characteristic
139	C₂²	C₂×#41	1	1	yes	characteristic
140	C₂²	H(5,𝔽₂)	1	1	yes	characteristic
141	C₂²	H(5,𝔽₂)	1	1	yes	characteristic
142	C₂²	H(5,𝔽₂)	1	1	yes	characteristic
143	C₂²	2₋¹⁺⁴	1	1	yes	characteristic
144	C₄	—	2	2	no	—
145	C₄	—	2	4	no	—
146	C₄	—	2	2	no	—
147	C₄	—	2	2	no	—
148	C₄	—	4	1	no	—
149	C₄	—	4	2	no	—
150	C₄	—	4	1	no	—
151	C₄	—	4	2	no	—
152	C₄	C₄×D₄	1	4	yes	—
153	C₂	—	4	2	no	—
154	C₂	—	4	4	no	—
155	C₂	C₂×C₄×D₄	1	1	yes	characteristic
156	C₂	#518	1	1	yes	characteristic
157	C₂	#520	1	1	yes	characteristic
158	C₂	#535	1	1	yes	characteristic
159	C₂	#537	1	1	yes	characteristic
160	C₂	#540	1	1	yes	characteristic
161	C₂	#542	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, socle, cosocle	C₂³	C₂⁴
commutator	C₂²	C₂³×C₄
prime radical	C₂×#121	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×D₄	C₂	semidirect
3. C₂×#117	C₂	semidirect
4. #390	C₂	semidirect
5. #390	C₂	semidirect
6. #389	C₂	semidirect
7. #388	C₂	semidirect
8. #386	C₂	semidirect
9. #385	C₂	semidirect
10. #380	C₂	semidirect
11. C₄×C₄⋊C₄	C₂	semidirect
16. #121	C₄	semidirect
45. C₂×D₄	C₂×C₄	semidirect
51. C₂²×C₄	C₂×C₄	semidirect
66. C₄⋊C₄	C₂×C₄	semidirect
73. C₂²⋊C₄	C₂×C₄	semidirect
76. C₄²	D₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×#121	2
C₂×C₄×D₄	2
C₂×#117	2
#390	2
#389	2
#388	2
#386	2
#385	2
#380	2
C₄×C₄⋊C₄	2