Finite Group Databasebeta

Subgroups — #33869

Order 256.

273 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 273 conjugacy classes of subgroups — too many to draw. Only its 101 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	#3342	C₂	1	1	yes	maximal, characteristic, complemented
2	#3283	C₂	1	1	yes	maximal, characteristic
3	#2241	C₂	1	1	yes	maximal, characteristic, complemented
4	#2237	C₂	1	2	yes	maximal, complemented
5	#2224	C₂	1	1	yes	maximal, characteristic, complemented
6	#2216	C₂	1	2	yes	maximal, complemented
7	#2209	C₂	1	1	yes	maximal, characteristic, complemented
8	#2189	C₂	1	2	yes	maximal, complemented
9	C₄×SD₃₂	C₂	1	1	yes	maximal, characteristic, complemented
10	D₄×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
11	#2138	C₂	1	2	yes	maximal, complemented
12	C₂×#135	C₂²	1	2	yes	—
13	#556	—	2	1	no	—
14	#547	—	2	1	no	—
15	C₂×Q₃₂	C₂²	1	2	yes	—
16	C₂×SD₃₂	C₂²	1	1	yes	characteristic
17	C₂²×C₁₆	C₂²	1	2	yes	—
18	#500	C₂²	1	1	yes	characteristic
19	#494	C₂²	1	1	yes	characteristic
20	#485	—	2	2	no	—
21	#475	—	2	1	no	—
22	#467	C₂²	1	2	yes	—
23	#467	C₂²	1	2	yes	—
24	#461	—	2	1	no	—
25	#452	—	2	2	no	—
26	C₄×Q₁₆	C₂²	1	1	yes	characteristic
27	C₄×D₈	C₂²	1	1	yes	characteristic
28	C₈×D₄	C₂²	1	1	yes	characteristic
29	#427	C₂²	1	2	yes	—
30	#367	C₂²	1	1	yes	characteristic
31	#366	C₂²	1	2	yes	—
32	C₄⋊C₁₆	C₂²	1	1	yes	characteristic
33	#358	C₂²	1	2	yes	—
34	#358	C₂²	1	2	yes	—
35	#358	C₂²	1	1	yes	characteristic
36	#358	C₂²	1	2	yes	—
37	#357	C₂²	1	1	yes	characteristic
38	#357	C₂²	1	2	yes	—
39	C₂²⋊C₁₆	C₂²	1	2	yes	—
40	C₄×C₁₆	C₂²	1	1	yes	characteristic
41	C₂×#41	—	2	2	no	—
42	#135	—	2	4	no	—
43	#135	—	4	2	no	—
44	C₂×Q₁₆	—	2	1	no	—
45	C₂×Q₁₆	C₂³	1	2	yes	—
46	C₂×Q₁₆	C₂³	1	1	yes	characteristic
47	C₂×SD₁₆	—	2	2	no	—
48	C₂×D₈	C₂³	1	1	yes	characteristic
49	C₂²×C₈	C₂³	1	2	yes	—
50	C₄⋊Q₈	—	2	1	no	—
51	C₄⋊Q₈	—	2	1	no	—
52	#126	—	4	2	no	—
53	#124	—	4	1	no	—
54	#123	—	4	2	no	—
55	C₂²⋊Q₈	—	2	2	no	—
56	C₂²⋊Q₈	—	2	2	no	—
57	#121	—	4	1	no	—
58	C₄×Q₈	—	2	1	no	—
59	C₄×Q₈	—	4	1	no	—
60	C₄×D₄	—	2	1	no	—
61	C₄×D₄	D₄	1	1	yes	characteristic
62	#117	—	2	2	no	—
63	C₂×C₄⋊C₄	—	4	1	no	—
64	Q₃₂	—	4	2	no	—
65	SD₃₂	—	2	2	no	—
66	C₂×C₁₆	—	4	2	no	—
67	C₂×C₁₆	C₂³	1	1	yes	characteristic
68	C₂×C₁₆	C₂³	1	1	yes	characteristic
69	C₂×C₁₆	C₂³	1	2	yes	—
70	#107	—	2	1	no	—
71	#107	C₂³	1	2	yes	—
72	#107	C₂³	1	1	yes	characteristic
73	#107	C₂³	1	1	yes	characteristic
74	#106	—	2	2	no	—
75	C₄⋊C₈	D₄	1	1	yes	characteristic
76	#103	—	2	1	no	—
77	#103	—	2	2	no	—
78	#103	—	2	2	no	—
79	#102	—	2	1	no	—
80	C₂²⋊C₈	D₄	1	2	yes	—
81	C₄×C₈	C₂³	1	1	yes	characteristic
82	#41	—	4	4	no	—
83	#41	—	8	2	no	—
84	C₂×Q₈	—	2	1	no	—
85	C₂×Q₈	—	2	2	no	—
86	C₂×Q₈	—	4	1	no	—
87	C₂×D₄	—	2	1	no	—
88	C₂×D₄	—	4	2	no	—
89	C₂×D₄	D₈	1	1	yes	characteristic
90	C₂²×C₄	—	4	1	no	—
91	C₂²×C₄	—	4	2	no	—
92	C₂²×C₄	C₂×D₄	1	2	yes	—
93	Q₁₆	—	2	1	no	—
94	Q₁₆	—	2	2	no	—
95	Q₁₆	—	2	1	no	—
96	Q₁₆	—	2	2	no	—
97	Q₁₆	—	8	1	no	—
98	SD₁₆	—	4	4	no	—
99	D₈	—	2	1	no	—
100	D₈	#41	1	2	yes	—
101	C₂×C₈	—	2	2	no	—
102	C₂×C₈	—	2	2	no	—
103	C₂×C₈	C₂×D₄	1	1	yes	characteristic
104	C₂×C₈	C₂×D₄	1	2	yes	—
105	C₂×C₈	C₂⁴	1	1	yes	characteristic
106	C₄⋊C₄	—	2	1	no	—
107	C₄⋊C₄	—	2	1	no	—
108	C₄⋊C₄	—	2	2	no	—
109	C₄⋊C₄	—	4	1	no	—
110	C₄⋊C₄	—	4	2	no	—
111	C₄⋊C₄	—	4	1	no	—
112	C₄⋊C₄	—	4	2	no	—
113	C₄⋊C₄	—	4	1	no	—
114	C₄⋊C₄	—	4	1	no	—
115	C₄⋊C₄	D₈	1	1	yes	characteristic
116	C₂²⋊C₄	—	4	2	no	—
117	C₂²⋊C₄	—	4	2	no	—
118	C₂²⋊C₄	—	4	2	no	—
119	C₂²⋊C₄	—	4	1	no	—
120	C₂²⋊C₄	D₈	1	2	yes	—
121	C₄²	—	4	2	no	—
122	C₄²	—	4	1	no	—
123	C₄²	C₂×D₄	1	1	yes	characteristic
124	C₁₆	—	2	1	no	—
125	C₁₆	—	2	1	no	—
126	C₁₆	—	2	2	no	—
127	C₂³	—	4	1	no	—
128	C₂³	C₂×D₈	1	2	yes	—
129	Q₈	—	4	2	no	—
130	Q₈	—	4	1	no	—
131	Q₈	—	4	2	no	—
132	Q₈	—	4	1	no	—
133	Q₈	—	8	2	no	—
134	D₄	—	2	2	no	—
135	D₄	—	2	2	no	—
136	D₄	—	4	1	no	—
137	D₄	—	8	4	no	—
138	C₂×C₄	—	2	2	no	—
139	C₂×C₄	—	2	2	no	—
140	C₂×C₄	—	4	1	no	—
141	C₂×C₄	—	4	2	no	—
142	C₂×C₄	—	4	1	no	—
143	C₂×C₄	—	4	1	no	—
144	C₂×C₄	—	4	2	no	—
145	C₂×C₄	—	8	4	no	—
146	C₂×C₄	—	8	2	no	—
147	C₂×C₄	C₂×D₈	1	2	yes	—
148	C₂×C₄	C₂×D₈	1	1	yes	characteristic
149	C₂×C₄	C₂×D₈	1	1	yes	characteristic
150	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
151	C₈	—	2	2	no	—
152	C₈	—	2	1	no	—
153	C₈	C₂×#41	1	1	yes	characteristic
154	C₈	H(5,𝔽₂)	1	1	yes	characteristic
155	C₂²	—	2	2	no	—
156	C₂²	—	2	2	no	—
157	C₂²	—	2	2	no	—
158	C₂²	—	4	2	no	—
159	C₂²	—	8	1	no	—
160	C₂²	—	8	1	no	—
161	C₂²	C₂²×D₈	1	1	yes	characteristic
162	C₄	—	2	1	no	—
163	C₄	—	2	2	no	—
164	C₄	—	8	1	no	—
165	C₄	—	8	2	no	—
166	C₄	—	8	1	no	—
167	C₄	—	8	2	no	—
168	C₄	—	8	1	no	—
169	C₄	#508	1	2	yes	—
170	C₄	#546	1	1	yes	characteristic
171	C₄	#577	1	1	yes	characteristic
172	C₂	—	4	2	no	—
173	C₂	—	8	2	no	—
174	C₂	#3294	1	1	yes	characteristic
175	C₂	C₂×#508	1	1	yes	characteristic
176	C₂	#3417	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴
prime radical	#3283	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ C₈×D₄ ◁ 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. #3342	C₂	semidirect
3. #2241	C₂	semidirect
4. #2237	C₂	semidirect
5. #2224	C₂	semidirect
6. #2216	C₂	semidirect
7. #2209	C₂	semidirect
8. #2189	C₂	semidirect
9. C₄×SD₃₂	C₂	semidirect
10. D₄×C₁₆	C₂	semidirect
11. #2138	C₂	semidirect

Maximal subgroups

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

subgroup	index
#3342	2
#3283	2
#2241	2
#2237	2
#2224	2
#2216	2
#2209	2
#2189	2
C₄×SD₃₂	2
D₄×C₁₆	2
#2138	2