Finite Group Databasebeta

Subgroups — #33627

Order 256.

299 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 299 conjugacy classes of subgroups — too many to draw. Only its 121 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	#3160	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₄×D₈	C₂	1	1	yes	maximal, characteristic
3	#2228	C₂	1	4	yes	maximal, complemented
4	#2226	C₂	1	2	yes	maximal, complemented
5	#2224	C₂	1	2	yes	maximal, complemented
6	#2159	C₂	1	2	yes	maximal, complemented
7	#2150	C₂	1	1	yes	maximal, characteristic, complemented
8	#2140	C₂	1	2	yes	maximal
9	C₂²×D₈	C₂²	1	1	yes	characteristic
10	#533	—	2	1	no	—
11	C₂×C₄×D₄	—	2	1	no	—
12	C₂×M₃₂	C₂²	1	2	yes	—
13	#500	C₂²	1	1	yes	characteristic, complemented
14	#500	C₂²	1	1	yes	characteristic, complemented
15	#499	—	2	1	no	—
16	#499	C₂²	1	2	yes	complemented
17	#498	—	2	1	no	—
18	C₄×D₈	—	2	1	no	—
19	C₄×D₈	—	2	1	no	—
20	C₄×D₈	C₂²	1	2	yes	—
21	C₄×D₈	C₂²	1	2	yes	—
22	#427	—	2	1	no	—
23	#427	C₂²	1	2	yes	—
24	C₂×#107	C₂²	1	1	yes	characteristic
25	C₂×#102	—	2	1	no	—
26	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
27	#367	C₂²	1	4	yes	complemented
28	#366	C₂²	1	4	yes	complemented
29	C₄⋊C₁₆	C₂²	1	2	yes	complemented
30	C₄⋊C₁₆	C₂²	1	2	yes	complemented
31	#357	C₂²	1	4	yes	—
32	#357	C₂²	1	4	yes	—
33	C₂²×D₄	—	2	1	no	—
34	C₂³×C₄	—	4	1	no	—
35	C₂×D₈	—	2	1	no	—
36	C₂×D₈	—	2	2	no	—
37	C₂×D₈	Q₈	1	4	yes	complemented
38	C₂×D₈	C₂³	1	2	yes	—
39	C₂²×C₈	D₄	1	1	yes	characteristic
40	C₂²×C₈	C₂³	1	1	yes	characteristic
41	C₄⋊Q₈	—	2	1	no	—
42	C₄⋊Q₈	—	2	1	no	—
43	#125	—	2	2	no	—
44	C₂²⋊Q₈	—	4	1	no	—
45	C₂²⋊Q₈	—	4	1	no	—
46	C₄×Q₈	—	4	1	no	—
47	C₄×Q₈	—	4	1	no	—
48	C₄×D₄	—	2	2	no	—
49	C₄×D₄	—	2	2	no	—
50	C₄×D₄	—	4	1	no	—
51	C₄×D₄	—	4	1	no	—
52	#117	—	2	2	no	—
53	C₂×C₄⋊C₄	—	2	1	no	—
54	C₂×C₂²⋊C₄	—	4	1	no	—
55	C₂×C₄²	D₄	1	1	yes	characteristic
56	M₃₂	—	2	4	no	—
57	C₂×C₁₆	C₂³	1	4	yes	—
58	#107	—	2	1	no	—
59	#107	—	2	1	no	—
60	#107	—	2	1	no	—
61	#107	C₂³	1	2	yes	—
62	#107	C₂³	1	2	yes	—
63	#107	C₂³	1	2	yes	—
64	#106	—	2	2	no	—
65	#106	—	2	2	no	—
66	#102	—	2	2	no	—
67	#102	—	2	2	no	—
68	C₄×C₈	D₄	1	1	yes	characteristic
69	C₄×C₈	D₄	1	1	yes	characteristic
70	C₄×C₈	C₂³	1	1	yes	characteristic
71	C₄×C₈	C₂³	1	1	yes	characteristic
72	C₂⁴	—	4	1	no	—
73	C₂×Q₈	—	4	1	no	—
74	C₂×Q₈	—	4	1	no	—
75	C₂×D₄	—	2	4	no	—
76	C₂×D₄	—	2	2	no	—
77	C₂×D₄	—	4	1	no	—
78	C₂×D₄	—	4	2	no	—
79	C₂²×C₄	—	4	1	no	—
80	C₂²×C₄	—	4	2	no	—
81	C₂²×C₄	—	4	2	no	—
82	C₂²×C₄	—	8	4	no	—
83	C₂²×C₄	D₈	1	1	yes	characteristic
84	C₂²×C₄	D₈	1	1	yes	characteristic
85	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
86	D₈	—	2	4	no	—
87	D₈	—	2	2	no	—
88	D₈	C₂×Q₈	1	4	yes	—
89	C₂×C₈	—	2	2	no	—
90	C₂×C₈	C₂×D₄	1	1	yes	characteristic
91	C₂×C₈	C₂×D₄	1	1	yes	characteristic
92	C₂×C₈	C₂×D₄	1	1	yes	characteristic
93	C₂×C₈	C₂×Q₈	1	2	yes	—
94	C₂×C₈	#41	1	2	yes	—
95	C₂×C₈	C₂⁴	1	1	yes	characteristic
96	C₄⋊C₄	—	2	2	no	—
97	C₄⋊C₄	—	2	2	no	—
98	C₄⋊C₄	—	2	2	no	—
99	C₄⋊C₄	—	4	1	no	—
100	C₄⋊C₄	—	4	2	no	—
101	C₄⋊C₄	—	4	2	no	—
102	C₄⋊C₄	—	4	1	no	—
103	C₄⋊C₄	—	4	1	no	—
104	C₂²⋊C₄	—	4	2	no	—
105	C₂²⋊C₄	—	4	2	no	—
106	C₂²⋊C₄	—	4	2	no	—
107	C₄²	—	4	2	no	—
108	C₄²	D₈	1	1	yes	characteristic
109	C₄²	D₈	1	1	yes	characteristic
110	C₄²	C₂×D₄	1	1	yes	characteristic
111	C₄²	C₂×D₄	1	1	yes	characteristic
112	C₁₆	—	2	4	no	—
113	C₂³	—	4	2	no	—
114	C₂³	—	4	4	no	—
115	C₂³	—	8	2	no	—
116	C₂³	—	8	2	no	—
117	C₂³	C₂×D₈	1	1	yes	characteristic
118	Q₈	—	8	2	no	—
119	Q₈	—	8	2	no	—
120	D₄	—	2	4	no	—
121	D₄	—	4	2	no	—
122	D₄	—	4	4	no	—
123	C₂×C₄	—	2	2	no	—
124	C₂×C₄	—	4	1	no	—
125	C₂×C₄	—	4	2	no	—
126	C₂×C₄	—	4	2	no	—
127	C₂×C₄	—	4	2	no	—
128	C₂×C₄	—	8	4	no	—
129	C₂×C₄	—	8	2	no	—
130	C₂×C₄	—	8	4	no	—
131	C₂×C₄	C₂²⋊Q₈	1	2	yes	—
132	C₂×C₄	C₂×D₈	1	1	yes	characteristic
133	C₂×C₄	C₂×D₈	1	1	yes	characteristic
134	C₂×C₄	C₂×D₈	1	1	yes	characteristic
135	C₂×C₄	C₂×D₈	1	1	yes	characteristic
136	C₂×C₄	C₂×D₈	1	1	yes	characteristic
137	C₂×C₄	#137	1	2	yes	—
138	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
139	C₈	—	2	1	no	—
140	C₈	—	2	1	no	—
141	C₈	C₂²⋊Q₈	1	2	yes	—
142	C₈	C₂²×Q₈	1	1	yes	characteristic
143	C₈	C₂×#41	1	1	yes	characteristic
144	C₂²	—	2	1	no	—
145	C₂²	—	2	1	no	—
146	C₂²	—	4	4	no	—
147	C₂²	—	8	8	no	—
148	C₂²	—	8	2	no	—
149	C₂²	—	8	2	no	—
150	C₂²	#474	1	2	yes	—
151	C₂²	C₂²×D₈	1	1	yes	characteristic
152	C₄	—	2	1	no	—
153	C₄	—	2	1	no	—
154	C₄	—	2	1	no	—
155	C₄	—	8	2	no	—
156	C₄	—	8	2	no	—
157	C₄	—	8	2	no	—
158	C₄	#474	1	2	yes	—
159	C₄	#509	1	2	yes	—
160	C₄	C₂×C₂²⋊Q₈	1	1	yes	characteristic
161	C₄	C₂×#137	1	1	yes	characteristic
162	C₂	—	2	2	no	—
163	C₂	—	8	4	no	—
164	C₂	C₂×#474	1	1	yes	characteristic
165	C₂	C₂×#509	1	1	yes	characteristic
166	C₂	#3414	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴
prime radical	C₂²×D₈	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #3160	C₂	semidirect
3. #2228	C₂	semidirect
4. #2226	C₂	semidirect
5. #2224	C₂	semidirect
6. #2159	C₂	semidirect
7. #2150	C₂	semidirect
13. #500	C₂²	semidirect
14. #500	C₂²	semidirect
16. #499	C₂²	semidirect
27. #367	C₂²	semidirect
28. #366	C₂²	semidirect
29. C₄⋊C₁₆	C₂²	semidirect
30. C₄⋊C₁₆	C₂²	semidirect
37. C₂×D₈	Q₈	semidirect

Maximal subgroups

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

subgroup	index
#3160	2
C₂×C₄×D₈	2
#2228	2
#2226	2
#2224	2
#2159	2
#2150	2
#2140	2