Finite Group Databasebeta

Subgroups — #33866

Order 256.

274 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 274 conjugacy classes of subgroups — too many to draw. Only its 109 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	#3338	C₂	1	1	yes	maximal, characteristic
2	D₄×Q₁₆	C₂	1	1	yes	maximal, characteristic
3	#2241	C₂	1	1	yes	maximal, characteristic, complemented
4	#2236	C₂	1	2	yes	maximal, complemented
5	#2227	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	#2188	C₂	1	2	yes	maximal, complemented
9	C₄×Q₃₂	C₂	1	1	yes	maximal, characteristic, complemented
10	D₄×C₁₆	C₂	1	1	yes	maximal, characteristic
11	C₂×#358	C₂	1	2	yes	maximal, complemented
12	C₂²×Q₁₆	C₂²	1	2	yes	—
13	#554	—	2	1	no	—
14	D₄×Q₈	—	2	1	no	—
15	C₂×Q₃₂	C₂²	1	1	yes	characteristic
16	C₂×Q₃₂	C₂²	1	2	yes	—
17	C₂²×C₁₆	C₂²	1	2	yes	—
18	#500	C₂²	1	1	yes	characteristic
19	#494	C₂²	1	1	yes	characteristic
20	#484	—	2	2	no	—
21	#477	—	2	1	no	—
22	#467	C₂²	1	2	yes	—
23	#467	C₂²	1	2	yes	—
24	#462	—	2	1	no	—
25	#451	—	2	2	no	—
26	C₄×Q₁₆	C₂²	1	1	yes	characteristic
27	C₄×Q₁₆	C₂²	1	1	yes	characteristic
28	C₈×D₄	C₂²	1	1	yes	characteristic
29	C₂×#107	C₂²	1	2	yes	—
30	#366	C₂²	1	1	yes	characteristic
31	#366	C₂²	1	2	yes	—
32	C₄⋊C₁₆	C₂²	1	1	yes	characteristic
33	#358	C₂²	1	1	yes	characteristic
34	#358	C₂²	1	2	yes	—
35	#358	C₂²	1	2	yes	—
36	#358	C₂²	1	2	yes	—
37	#358	C₂²	1	1	yes	characteristic
38	#358	C₂²	1	2	yes	—
39	C₂²⋊C₁₆	C₂²	1	2	yes	—
40	C₄×C₁₆	C₂²	1	1	yes	characteristic
41	C₂²×Q₈	—	2	2	no	—
42	C₂×Q₁₆	—	2	4	no	—
43	C₂×Q₁₆	—	2	2	no	—
44	C₂×Q₁₆	—	2	1	no	—
45	C₂×Q₁₆	—	4	2	no	—
46	C₂×Q₁₆	C₂³	1	2	yes	—
47	C₂×Q₁₆	C₂³	1	1	yes	characteristic
48	C₂×Q₁₆	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	C₄⋊Q₈	—	4	1	no	—
53	#125	—	4	1	no	—
54	C₂²⋊Q₈	—	2	2	no	—
55	C₂²⋊Q₈	—	2	2	no	—
56	C₂²⋊Q₈	—	4	2	no	—
57	C₂²⋊Q₈	—	4	2	no	—
58	C₄×Q₈	—	2	1	no	—
59	C₄×Q₈	—	2	1	no	—
60	C₄×D₄	—	4	1	no	—
61	C₄×D₄	—	4	1	no	—
62	C₄×D₄	D₄	1	1	yes	characteristic
63	C₂×C₄⋊C₄	—	2	2	no	—
64	Q₃₂	—	2	2	no	—
65	Q₃₂	—	4	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	2	no	—
71	#107	—	2	1	no	—
72	#107	C₂³	1	1	yes	characteristic
73	#107	C₂³	1	2	yes	—
74	#107	C₂³	1	1	yes	characteristic
75	C₄⋊C₈	D₄	1	1	yes	characteristic
76	#103	—	2	1	no	—
77	#103	—	2	1	no	—
78	#103	—	2	2	no	—
79	#103	—	2	2	no	—
80	C₂²⋊C₈	D₄	1	2	yes	—
81	C₄×C₈	C₂³	1	1	yes	characteristic
82	C₂×Q₈	—	2	1	no	—
83	C₂×Q₈	—	2	1	no	—
84	C₂×Q₈	—	2	2	no	—
85	C₂×Q₈	—	4	4	no	—
86	C₂×Q₈	—	4	2	no	—
87	C₂×Q₈	—	4	1	no	—
88	C₂×Q₈	—	8	2	no	—
89	C₂×D₄	D₈	1	1	yes	characteristic
90	C₂²×C₄	—	4	2	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	1	no	—
97	Q₁₆	—	2	2	no	—
98	Q₁₆	—	4	4	no	—
99	Q₁₆	—	8	1	no	—
100	Q₁₆	#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	2	no	—
110	C₄⋊C₄	—	4	1	no	—
111	C₄⋊C₄	—	4	1	no	—
112	C₄⋊C₄	—	4	1	no	—
113	C₄⋊C₄	—	4	1	no	—
114	C₄⋊C₄	—	4	2	no	—
115	C₄⋊C₄	—	4	2	no	—
116	C₄⋊C₄	—	4	1	no	—
117	C₄⋊C₄	D₈	1	1	yes	characteristic
118	C₂²⋊C₄	—	4	2	no	—
119	C₂²⋊C₄	—	4	2	no	—
120	C₂²⋊C₄	D₈	1	2	yes	—
121	C₄²	—	4	1	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	C₂×D₈	1	2	yes	—
128	Q₈	—	2	2	no	—
129	Q₈	—	4	1	no	—
130	Q₈	—	4	2	no	—
131	Q₈	—	4	1	no	—
132	Q₈	—	4	2	no	—
133	Q₈	—	4	1	no	—
134	Q₈	—	8	4	no	—
135	Q₈	—	16	1	no	—
136	D₄	Q₃₂	1	4	yes	complemented
137	C₂×C₄	—	2	2	no	—
138	C₂×C₄	—	2	2	no	—
139	C₂×C₄	—	4	1	no	—
140	C₂×C₄	—	4	1	no	—
141	C₂×C₄	—	4	1	no	—
142	C₂×C₄	—	4	1	no	—
143	C₂×C₄	—	4	2	no	—
144	C₂×C₄	—	4	2	no	—
145	C₂×C₄	—	8	4	no	—
146	C₂×C₄	—	16	2	no	—
147	C₂×C₄	C₂×D₈	1	1	yes	characteristic
148	C₂×C₄	C₂×D₈	1	2	yes	—
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	C₂×Q₃₂	1	4	yes	—
158	C₂²	C₂²×D₈	1	1	yes	characteristic
159	C₄	—	2	1	no	—
160	C₄	—	2	2	no	—
161	C₄	—	4	2	no	—
162	C₄	—	8	1	no	—
163	C₄	—	8	1	no	—
164	C₄	—	8	2	no	—
165	C₄	—	8	2	no	—
166	C₄	—	8	1	no	—
167	C₄	C₂×Q₃₂	1	2	yes	—
168	C₄	#546	1	1	yes	characteristic
169	C₄	#577	1	1	yes	characteristic
170	C₂	—	2	4	no	—
171	C₂	#3294	1	1	yes	characteristic
172	C₂	C₂²×Q₃₂	1	1	yes	characteristic
173	C₂	#3416	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴
prime radical	C₂×D₄	D₈
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
3. #2241	C₂	semidirect
4. #2236	C₂	semidirect
5. #2227	C₂	semidirect
6. #2216	C₂	semidirect
7. #2209	C₂	semidirect
8. #2188	C₂	semidirect
9. C₄×Q₃₂	C₂	semidirect
11. C₂×#358	C₂	semidirect
136. D₄	Q₃₂	semidirect

Maximal subgroups

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

subgroup	index
#3338	2
D₄×Q₁₆	2
#2241	2
#2236	2
#2227	2
#2216	2
#2209	2
#2188	2
C₄×Q₃₂	2
D₄×C₁₆	2
C₂×#358	2