Finite Group Databasebeta

Subgroups — #34524

Order 256.

1244 conjugacy classes of subgroups; only the 701 normal subgroups are shown.

normal subgroups only

This group has 1244 conjugacy classes of subgroups — too many to draw. Only its 701 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(3,𝔽₄)	C₂	1	1	yes	maximal, characteristic
2	#2379	C₂	1	3	yes	maximal, complemented
3	#2372	C₂	1	6	yes	maximal, complemented
4	#2358	C₂	1	6	yes	maximal, complemented
5	#2326	C₂	1	6	yes	maximal, complemented
6	#2321	C₂	1	6	yes	maximal, complemented
7	C₄×#124	C₂	1	3	yes	maximal, complemented
8	H(3,𝔽₄)	C₄	1	16	yes	complemented
9	C₂×#124	C₂²	1	6	yes	—
10	C₂×#124	C₂²	1	3	yes	—
11	C₂×#120	C₂²	1	6	yes	—
12	C₂×C₄×Q₈	C₂²	1	3	yes	complemented
13	C₂×C₄×D₄	C₂²	1	3	yes	—
14	C₂×C₄×D₄	C₂²	1	6	yes	—
15	C₂²×C₂²⋊C₄	C₂²	1	6	yes	complemented
16	#391	C₂²	1	6	yes	complemented
17	#391	C₂²	1	3	yes	complemented
18	#390	C₂²	1	6	yes	—
19	#390	C₂²	1	3	yes	—
20	#388	C₂²	1	12	yes	complemented
21	#388	C₂²	1	12	yes	complemented
22	#388	C₂²	1	12	yes	complemented
23	#386	C₂²	1	12	yes	—
24	#386	C₂²	1	6	yes	—
25	#385	C₂²	1	6	yes	complemented
26	#385	C₂²	1	12	yes	complemented
27	#384	C₂²	1	2	yes	complemented
28	#382	C₂²	1	3	yes	complemented
29	#379	C₂²	1	2	yes	complemented
30	C₄×C₂²⋊C₄	C₂²	1	6	yes	complemented
31	C₄×C₂²⋊C₄	C₂²	1	12	yes	complemented
32	#376	C₂²	1	6	yes	complemented
33	C₄³	C₂²	1	1	yes	characteristic, complemented
34	C₂⁵	C₂³	1	2	yes	—
35	C₂²×Q₈	C₂³	1	3	yes	—
36	C₂²×D₄	C₂³	1	3	yes	—
37	C₂²×D₄	C₂³	1	6	yes	—
38	C₂³×C₄	C₂³	1	6	yes	—
39	C₂³×C₄	C₂³	1	6	yes	—
40	#124	C₂×C₄	1	48	yes	complemented
41	#124	C₂×C₄	1	24	yes	complemented
42	#120	C₂×C₄	1	48	yes	complemented
43	C₄×Q₈	—	4	6	no	—
44	C₄×D₄	—	4	12	no	—
45	C₄×D₄	—	4	6	no	—
46	C₂×C₄⋊C₄	C₂³	1	3	yes	—
47	C₂×C₄⋊C₄	C₂³	1	6	yes	—
48	C₂×C₄⋊C₄	C₂³	1	3	yes	—
49	C₂×C₄⋊C₄	C₂³	1	6	yes	—
50	C₂×C₂²⋊C₄	—	4	12	no	—
51	C₂×C₂²⋊C₄	C₂³	1	12	yes	—
52	C₂×C₂²⋊C₄	C₂³	1	12	yes	—
53	C₂×C₂²⋊C₄	C₂³	1	6	yes	—
54	C₂×C₂²⋊C₄	C₂³	1	6	yes	—
55	C₂×C₂²⋊C₄	C₂³	1	6	yes	—
56	C₂×C₂²⋊C₄	C₂³	1	12	yes	—
57	C₂×C₄²	C₂³	1	3	yes	—
58	C₂×C₄²	C₂³	1	6	yes	—
59	C₂×C₄²	C₂³	1	3	yes	—
60	C₂×C₄²	C₂³	1	6	yes	—
61	C₂×C₄²	C₂³	1	1	yes	characteristic
62	C₂×C₄²	C₂³	1	2	yes	—
63	#95	C₂³	1	12	yes	—
64	#95	C₂³	1	12	yes	—
65	#95	C₂³	1	12	yes	—
66	C₂⁴	—	4	6	no	—
67	C₂⁴	—	4	6	no	—
68	C₂⁴	C₂²×C₄	1	8	yes	—
69	C₂⁴	C₂⁴	1	6	yes	—
70	C₂×Q₈	—	4	3	no	—
71	C₂×Q₈	—	4	3	no	—
72	C₂×Q₈	C₂²×C₄	1	12	yes	—
73	C₂×D₄	—	4	6	no	—
74	C₂×D₄	—	4	3	no	—
75	C₂×D₄	—	4	3	no	—
76	C₂×D₄	—	4	6	no	—
77	C₂×D₄	C₂²×C₄	1	24	yes	—
78	C₂×D₄	C₂²×C₄	1	12	yes	—
79	C₂²×C₄	—	2	12	no	—
80	C₂²×C₄	—	4	6	no	—
81	C₂²×C₄	—	4	12	no	—
82	C₂²×C₄	—	4	12	no	—
83	C₂²×C₄	C₂⁴	1	6	yes	—
84	C₂²×C₄	C₂⁴	1	6	yes	—
85	C₂²×C₄	C₂⁴	1	1	yes	characteristic
86	C₂²×C₄	C₂⁴	1	6	yes	—
87	C₂²×C₄	C₂⁴	1	3	yes	—
88	C₂²×C₄	C₂⁴	1	3	yes	—
89	C₄⋊C₄	—	4	6	no	—
90	C₄⋊C₄	—	4	3	no	—
91	C₄⋊C₄	—	4	6	no	—
92	C₄⋊C₄	—	4	3	no	—
93	C₂²⋊C₄	—	2	24	no	—
94	C₂²⋊C₄	—	4	12	no	—
95	C₂²⋊C₄	—	4	6	no	—
96	C₂²⋊C₄	—	4	6	no	—
97	C₂²⋊C₄	C₂²×C₄	1	24	yes	—
98	C₂²⋊C₄	C₂²×C₄	1	48	yes	—
99	C₄²	—	2	6	no	—
100	C₄²	—	2	12	no	—
101	C₄²	—	4	3	no	—
102	C₄²	—	4	6	no	—
103	C₄²	C₂²×C₄	1	4	yes	—
104	C₄²	C₂²×C₄	1	8	yes	—
105	C₂³	—	2	12	no	—
106	C₂³	—	2	6	no	—
107	C₂³	—	2	6	no	—
108	C₂³	—	2	12	no	—
109	C₂³	—	4	24	no	—
110	C₂³	—	8	12	no	—
111	C₂³	—	8	4	no	—
112	C₂³	C₂³×C₄	1	12	yes	—
113	C₂³	C₂⁵	1	1	yes	characteristic
114	Q₈	—	4	12	no	—
115	D₄	—	4	24	no	—
116	D₄	—	4	12	no	—
117	C₂×C₄	—	2	3	no	—
118	C₂×C₄	—	2	6	no	—
119	C₂×C₄	—	2	6	no	—
120	C₂×C₄	—	2	3	no	—
121	C₂×C₄	—	2	6	no	—
122	C₂×C₄	—	2	3	no	—
123	C₂×C₄	—	2	6	no	—
124	C₂×C₄	—	2	12	no	—
125	C₂×C₄	—	2	6	no	—
126	C₂×C₄	—	2	12	no	—
127	C₂×C₄	—	8	12	no	—
128	C₂×C₄	—	8	12	no	—
129	C₂×C₄	C₂³×C₄	1	6	yes	—
130	C₂×C₄	C₂³×C₄	1	12	yes	—
131	C₂×C₄	H(5,𝔽₂)	1	6	yes	—
132	C₂²	—	2	24	no	—
133	C₂²	—	2	12	no	—
134	C₂²	—	4	6	no	—
135	C₂²	—	4	12	no	—
136	C₂²	—	4	6	no	—
137	C₂²	—	8	16	no	—
138	C₂²	C₂⁴×C₄	1	1	yes	characteristic
139	C₂²	C₂×H(5,𝔽₂)	1	3	yes	—
140	C₂²	#585	1	3	yes	—
141	C₄	—	2	6	no	—
142	C₄	—	2	12	no	—
143	C₄	—	2	6	no	—
144	C₄	—	4	6	no	—
145	C₄	—	4	1	no	—
146	C₄	—	4	6	no	—
147	C₂	—	4	12	no	—
148	C₂	C₄×H(5,𝔽₂)	1	3	yes	—
149	C₂	#3545	1	3	yes	—
150	C₂	#3549	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, socle, cosocle	C₂³	C₂⁵
commutator	C₂²	C₂⁴×C₄
prime radical	C₂×H(3,𝔽₄)	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. #2379	C₂	semidirect
3. #2372	C₂	semidirect
4. #2358	C₂	semidirect
5. #2326	C₂	semidirect
6. #2321	C₂	semidirect
7. C₄×#124	C₂	semidirect
8. H(3,𝔽₄)	C₄	semidirect
12. C₂×C₄×Q₈	C₂²	semidirect
15. C₂²×C₂²⋊C₄	C₂²	semidirect
16. #391	C₂²	semidirect
17. #391	C₂²	semidirect
20. #388	C₂²	semidirect
21. #388	C₂²	semidirect
22. #388	C₂²	semidirect
25. #385	C₂²	semidirect
26. #385	C₂²	semidirect
27. #384	C₂²	semidirect
28. #382	C₂²	semidirect
29. #379	C₂²	semidirect
30. C₄×C₂²⋊C₄	C₂²	semidirect
31. C₄×C₂²⋊C₄	C₂²	semidirect
32. #376	C₂²	semidirect
33. C₄³	C₂²	semidirect
40. #124	C₂×C₄	semidirect
41. #124	C₂×C₄	semidirect
42. #120	C₂×C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×H(3,𝔽₄)	2
#2379	2
#2372	2
#2358	2
#2326	2
#2321	2
C₄×#124	2