Finite Group Databasebeta

Subgroups — #2426

Order 128.

316 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 112 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×#124	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#121	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
5	#396	C₂	1	2	yes	maximal, complemented
6	#394	C₂	1	2	yes	maximal, complemented
7	#390	C₂	1	1	yes	maximal, characteristic, complemented
8	#390	C₂	1	2	yes	maximal, complemented
9	#389	C₂	1	1	yes	maximal, characteristic, complemented
10	#388	C₂	1	2	yes	maximal, complemented
11	C₄×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×Q₈	C₂²	1	1	yes	characteristic
13	C₂²×D₄	C₂²	1	1	yes	characteristic
14	C₂²×D₄	C₂²	1	2	yes	—
15	C₂³×C₄	C₂²	1	1	yes	characteristic
16	#124	—	2	4	no	—
17	C₂²⋊Q₈	—	2	4	no	—
18	#121	—	2	4	no	—
19	C₄×Q₈	—	2	4	no	—
20	C₂×C₄⋊C₄	C₂²	1	2	yes	—
21	C₂×C₄⋊C₄	C₂²	1	2	yes	—
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	2	yes	—
26	C₂×C₂²⋊C₄	C₂²	1	2	yes	—
27	C₂×C₂²⋊C₄	C₂²	1	2	yes	—
28	C₂×C₂²⋊C₄	C₂²	1	2	yes	—
29	C₂×C₂²⋊C₄	C₂²	1	1	yes	characteristic
30	C₂×C₂²⋊C₄	C₂²	1	1	yes	characteristic
31	C₂×C₂²⋊C₄	C₂²	1	2	yes	—
32	C₂×C₂²⋊C₄	C₂²	1	2	yes	—
33	C₂×C₄²	C₂²	1	1	yes	characteristic
34	C₂×C₄²	C₂²	1	2	yes	—
35	C₂×C₄²	C₂²	1	1	yes	characteristic
36	#95	C₂²	1	1	yes	characteristic
37	#95	C₂²	1	1	yes	characteristic
38	#95	C₂²	1	2	yes	—
39	C₂⁴	C₂³	1	2	yes	—
40	C₂⁴	C₂³	1	1	yes	characteristic
41	C₂×Q₈	—	2	2	no	—
42	C₂×Q₈	—	2	2	no	—
43	C₂×Q₈	D₄	1	4	yes	complemented
44	C₂×D₄	—	2	4	no	—
45	C₂×D₄	—	2	4	no	—
46	C₂×D₄	—	2	2	no	—
47	C₂×D₄	—	2	2	no	—
48	C₂×D₄	—	2	2	no	—
49	C₂×D₄	—	4	2	no	—
50	C₂²×C₄	—	2	2	no	—
51	C₂²×C₄	—	2	2	no	—
52	C₂²×C₄	—	2	2	no	—
53	C₂²×C₄	C₂³	1	1	yes	characteristic
54	C₂²×C₄	C₂³	1	1	yes	characteristic
55	C₂²×C₄	C₂³	1	1	yes	characteristic
56	C₂²×C₄	C₂³	1	1	yes	characteristic
57	C₂²×C₄	C₂³	1	2	yes	—
58	C₂²×C₄	C₂³	1	2	yes	—
59	C₂²×C₄	C₂³	1	2	yes	—
60	C₂²×C₄	C₂³	1	1	yes	characteristic
61	C₂²×C₄	C₂³	1	1	yes	characteristic
62	C₄⋊C₄	—	2	4	no	—
63	C₄⋊C₄	—	2	2	no	—
64	C₄⋊C₄	—	2	4	no	—
65	C₄⋊C₄	—	4	2	no	—
66	C₄⋊C₄	D₄	1	4	yes	—
67	C₂²⋊C₄	—	2	2	no	—
68	C₂²⋊C₄	—	2	4	no	—
69	C₂²⋊C₄	—	2	4	no	—
70	C₂²⋊C₄	—	2	4	no	—
71	C₂²⋊C₄	—	2	4	no	—
72	C₂²⋊C₄	—	2	2	no	—
73	C₂²⋊C₄	—	2	4	no	—
74	C₂²⋊C₄	—	4	2	no	—
75	C₄²	—	2	4	no	—
76	C₄²	—	2	2	no	—
77	C₄²	—	2	2	no	—
78	C₂³	—	2	1	no	—
79	C₂³	—	2	1	no	—
80	C₂³	—	2	1	no	—
81	C₂³	—	2	1	no	—
82	C₂³	—	2	1	no	—
83	C₂³	—	2	1	no	—
84	C₂³	—	2	2	no	—
85	C₂³	—	2	2	no	—
86	C₂³	—	2	2	no	—
87	C₂³	—	2	2	no	—
88	C₂³	—	2	2	no	—
89	C₂³	—	2	2	no	—
90	C₂³	—	2	2	no	—
91	C₂³	#41	1	2	yes	—
92	C₂³	C₂⁴	1	1	yes	characteristic
93	Q₈	—	2	8	no	—
94	D₄	—	4	8	no	—
95	D₄	—	4	4	no	—
96	C₂×C₄	—	2	1	no	—
97	C₂×C₄	—	2	2	no	—
98	C₂×C₄	—	2	2	no	—
99	C₂×C₄	—	2	2	no	—
100	C₂×C₄	—	2	2	no	—
101	C₂×C₄	—	2	1	no	—
102	C₂×C₄	—	2	1	no	—
103	C₂×C₄	—	2	1	no	—
104	C₂×C₄	—	2	1	no	—
105	C₂×C₄	—	2	1	no	—
106	C₂×C₄	—	2	2	no	—
107	C₂×C₄	—	2	1	no	—
108	C₂×C₄	—	2	1	no	—
109	C₂×C₄	—	2	1	no	—
110	C₂×C₄	—	2	1	no	—
111	C₂×C₄	—	2	2	no	—
112	C₂×C₄	—	2	1	no	—
113	C₂×C₄	—	2	1	no	—
114	C₂×C₄	—	2	2	no	—
115	C₂×C₄	—	2	1	no	—
116	C₂×C₄	—	2	1	no	—
117	C₂×C₄	—	4	4	no	—
118	C₂×C₄	C₂×D₄	1	4	yes	—
119	C₂×C₄	C₂×D₄	1	4	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	C₂×C₄	#41	1	2	yes	—
124	C₂²	—	2	2	no	—
125	C₂²	—	2	2	no	—
126	C₂²	—	2	2	no	—
127	C₂²	—	4	1	no	—
128	C₂²	—	4	1	no	—
129	C₂²	—	4	1	no	—
130	C₂²	—	4	1	no	—
131	C₂²	—	4	2	no	—
132	C₂²	—	4	2	no	—
133	C₂²	—	4	2	no	—
134	C₂²	—	4	2	no	—
135	C₂²	—	4	2	no	—
136	C₂²	—	4	2	no	—
137	C₂²	—	4	2	no	—
138	C₂²	C₂²×D₄	1	1	yes	characteristic
139	C₂²	C₂²×D₄	1	1	yes	characteristic
140	C₂²	C₂×#41	1	1	yes	characteristic
141	C₂²	C₂×#41	1	1	yes	characteristic
142	C₂²	C₂×#41	1	1	yes	characteristic
143	C₂²	H(5,𝔽₂)	1	1	yes	characteristic
144	C₂²	2₋¹⁺⁴	1	1	yes	characteristic
145	C₄	—	2	4	no	—
146	C₄	—	2	4	no	—
147	C₄	—	2	2	no	—
148	C₄	—	4	1	no	—
149	C₄	—	4	1	no	—
150	C₄	—	4	1	no	—
151	C₄	—	4	1	no	—
152	C₄	—	4	2	no	—
153	C₄	#121	1	4	yes	—
154	C₂	—	4	2	no	—
155	C₂	—	8	2	no	—
156	C₂	C₂×#121	1	1	yes	characteristic
157	C₂	#529	1	1	yes	characteristic
158	C₂	#542	1	1	yes	characteristic
159	C₂	#546	1	1	yes	characteristic
160	C₂	#548	1	1	yes	characteristic
161	C₂	#548	1	1	yes	characteristic
162	C₂	#550	1	1	yes	characteristic

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁴
prime radical	C₂×#121	C₂
LCS	G ▷ C₂³ ▷ 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
1. C₂×#124	C₂	semidirect
2. C₂×C₂²⋊Q₈	C₂	semidirect
4. C₂×C₄×Q₈	C₂	semidirect
5. #396	C₂	semidirect
6. #394	C₂	semidirect
7. #390	C₂	semidirect
8. #390	C₂	semidirect
9. #389	C₂	semidirect
10. #388	C₂	semidirect
11. C₄×C₂²⋊C₄	C₂	semidirect
43. C₂×Q₈	D₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×#124	2
C₂×C₂²⋊Q₈	2
C₂×#121	2
C₂×C₄×Q₈	2
#396	2
#394	2
#390	2
#389	2
#388	2
C₄×C₂²⋊C₄	2