Finite Group Databasebeta

Subgroups — #33951

Order 256.

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

normal subgroups only

This group has 249 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	#3380	C₂	1	1	yes	maximal, characteristic, complemented
2	Q₈×D₈	C₂	1	1	yes	maximal, characteristic, complemented
3	#2255	C₂	1	2	yes	maximal, complemented
4	#2254	C₂	1	1	yes	maximal, characteristic, complemented
5	#2228	C₂	1	2	yes	maximal, complemented
6	#2226	C₂	1	2	yes	maximal, complemented
7	#2226	C₂	1	1	yes	maximal, characteristic, complemented
8	#2224	C₂	1	1	yes	maximal, characteristic, complemented
9	#2183	C₂	1	1	yes	maximal, characteristic, complemented
10	#2177	C₂	1	2	yes	maximal
11	C₄×D₁₆	C₂	1	1	yes	maximal, characteristic
12	#554	—	2	1	no	—
13	D₄×Q₈	—	2	1	no	—
14	C₂×D₁₆	C₂²	1	1	yes	characteristic
15	#500	C₂²	1	1	yes	characteristic
16	#500	C₂²	1	2	yes	—
17	#500	C₂²	1	1	yes	characteristic
18	#499	C₂²	1	2	yes	—
19	#478	—	2	2	no	—
20	#476	—	2	1	no	—
21	#474	—	2	1	no	—
22	#474	—	2	2	no	—
23	C₈×Q₈	C₂²	1	1	yes	characteristic
24	C₄×D₈	C₂²	1	1	yes	characteristic
25	C₄×D₈	C₂²	1	2	yes	—
26	C₄×D₈	C₂²	1	1	yes	characteristic
27	C₄×D₈	C₂²	1	2	yes	—
28	#367	C₂²	1	2	yes	—
29	#367	C₂²	1	2	yes	—
30	#367	C₂²	1	2	yes	—
31	#366	C₂²	1	1	yes	characteristic
32	#366	C₂²	1	2	yes	—
33	C₄⋊C₁₆	C₂²	1	1	yes	characteristic
34	C₄⋊C₁₆	C₂²	1	2	yes	—
35	#357	C₂²	1	1	yes	characteristic
36	#357	C₂²	1	1	yes	characteristic
37	#357	C₂²	1	2	yes	—
38	#357	C₂²	1	2	yes	—
39	#346	C₂²	1	2	yes	—
40	C₄×C₁₆	C₂²	1	1	yes	characteristic
41	C₂²×Q₈	—	4	1	no	—
42	C₂×D₈	C₂³	1	1	yes	characteristic
43	C₂×D₈	C₂³	1	1	yes	characteristic
44	C₄⋊Q₈	—	2	2	no	—
45	C₄⋊Q₈	—	2	1	no	—
46	C₄⋊Q₈	—	2	1	no	—
47	#125	—	2	2	no	—
48	C₂²⋊Q₈	—	4	1	no	—
49	C₂²⋊Q₈	—	4	1	no	—
50	C₂²⋊Q₈	—	4	2	no	—
51	C₂²⋊Q₈	—	4	2	no	—
52	C₄×Q₈	D₄	1	1	yes	characteristic
53	C₄×D₄	—	2	2	no	—
54	C₄×D₄	—	2	1	no	—
55	C₄×D₄	—	2	1	no	—
56	C₄×D₄	—	2	2	no	—
57	C₂×C₄⋊C₄	—	4	1	no	—
58	D₁₆	Q₈	1	4	yes	complemented
59	C₂×C₁₆	C₂³	1	1	yes	characteristic
60	C₂×C₁₆	C₂³	1	2	yes	—
61	C₂×C₁₆	C₂³	1	1	yes	characteristic
62	#107	—	2	2	no	—
63	#107	—	2	1	no	—
64	#107	—	2	1	no	—
65	#107	C₂³	1	1	yes	characteristic
66	#107	C₂³	1	1	yes	characteristic
67	#107	C₂³	1	2	yes	—
68	#107	C₂³	1	2	yes	—
69	#106	—	2	2	no	—
70	C₄⋊C₈	D₄	1	1	yes	characteristic
71	C₄⋊C₈	D₄	1	2	yes	—
72	#102	—	2	2	no	—
73	#102	—	2	1	no	—
74	#102	—	2	2	no	—
75	#102	—	2	1	no	—
76	C₄×C₈	C₂³	1	1	yes	characteristic
77	C₄×C₈	C₂³	1	2	yes	—
78	C₂×Q₈	—	4	2	no	—
79	C₂×Q₈	—	4	1	no	—
80	C₂×Q₈	—	4	1	no	—
81	C₂×Q₈	—	8	4	no	—
82	C₂×Q₈	D₈	1	1	yes	characteristic
83	C₂×D₄	—	2	1	no	—
84	C₂×D₄	—	2	1	no	—
85	C₂²×C₄	—	4	1	no	—
86	C₂²×C₄	—	4	2	no	—
87	C₂²×C₄	—	4	1	no	—
88	C₂²×C₄	—	4	2	no	—
89	D₈	—	2	1	no	—
90	D₈	—	2	1	no	—
91	D₈	C₂×Q₈	1	2	yes	—
92	D₈	C₂×Q₈	1	2	yes	complemented
93	C₂×C₈	C₂×D₄	1	2	yes	—
94	C₂×C₈	C₂×D₄	1	1	yes	characteristic
95	C₂×C₈	C₂⁴	1	1	yes	characteristic
96	C₄⋊C₄	—	2	1	no	—
97	C₄⋊C₄	—	2	2	no	—
98	C₄⋊C₄	—	2	2	no	—
99	C₄⋊C₄	—	2	1	no	—
100	C₄⋊C₄	—	4	2	no	—
101	C₄⋊C₄	—	4	1	no	—
102	C₄⋊C₄	—	4	2	no	—
103	C₄⋊C₄	—	4	1	no	—
104	C₄⋊C₄	—	4	2	no	—
105	C₄⋊C₄	D₈	1	1	yes	characteristic
106	C₄⋊C₄	D₈	1	2	yes	—
107	C₂²⋊C₄	—	4	1	no	—
108	C₂²⋊C₄	—	4	2	no	—
109	C₂²⋊C₄	—	4	1	no	—
110	C₂²⋊C₄	—	4	2	no	—
111	C₄²	C₂×D₄	1	2	yes	—
112	C₄²	C₂×D₄	1	1	yes	characteristic
113	C₁₆	—	2	1	no	—
114	C₁₆	—	2	2	no	—
115	C₁₆	C₂×Q₈	1	2	yes	complemented
116	C₂³	—	4	1	no	—
117	C₂³	—	4	1	no	—
118	Q₈	—	2	2	no	—
119	Q₈	—	8	4	no	—
120	Q₈	—	8	2	no	—
121	Q₈	—	16	1	no	—
122	D₄	—	2	2	no	—
123	D₄	—	2	2	no	—
124	D₄	—	4	1	no	—
125	D₄	—	4	1	no	—
126	C₂×C₄	—	4	1	no	—
127	C₂×C₄	—	4	2	no	—
128	C₂×C₄	—	4	2	no	—
129	C₂×C₄	—	4	1	no	—
130	C₂×C₄	—	8	4	no	—
131	C₂×C₄	—	8	2	no	—
132	C₂×C₄	—	8	2	no	—
133	C₂×C₄	—	16	2	no	—
134	C₂×C₄	C₂×D₈	1	1	yes	characteristic
135	C₂×C₄	C₂×D₈	1	2	yes	—
136	C₂×C₄	C₂×D₈	1	2	yes	—
137	C₂×C₄	C₂×D₈	1	1	yes	characteristic
138	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
139	C₈	—	2	2	no	—
140	C₈	—	2	1	no	—
141	C₈	C₂²×Q₈	1	1	yes	characteristic
142	C₈	2₋¹⁺⁴	1	1	yes	characteristic
143	C₂²	—	4	2	no	—
144	C₂²	—	4	2	no	—
145	C₂²	—	8	1	no	—
146	C₂²	—	8	1	no	—
147	C₂²	—	8	1	no	—
148	C₂²	—	8	1	no	—
149	C₂²	C₂²×D₈	1	1	yes	characteristic
150	C₄	—	2	2	no	—
151	C₄	—	2	2	no	—
152	C₄	—	2	1	no	—
153	C₄	—	8	1	no	—
154	C₄	—	8	2	no	—
155	C₄	—	8	1	no	—
156	C₄	—	8	2	no	—
157	C₄	#509	1	2	yes	—
158	C₄	D₄×Q₈	1	1	yes	characteristic
159	C₄	#578	1	1	yes	characteristic
160	C₂	—	8	2	no	—
161	C₂	—	8	2	no	—
162	C₂	Q₈×D₈	1	1	yes	characteristic
163	C₂	C₂×#509	1	1	yes	characteristic
164	C₂	#3416	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴
prime radical	C₂×D₁₆	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈ ◁ 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. #3380	C₂	semidirect
2. Q₈×D₈	C₂	semidirect
3. #2255	C₂	semidirect
4. #2254	C₂	semidirect
5. #2228	C₂	semidirect
6. #2226	C₂	semidirect
7. #2226	C₂	semidirect
8. #2224	C₂	semidirect
9. #2183	C₂	semidirect
58. D₁₆	Q₈	semidirect
92. D₈	C₂×Q₈	semidirect
115. C₁₆	C₂×Q₈	semidirect

Maximal subgroups

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

subgroup	index
#3380	2
Q₈×D₈	2
#2255	2
#2254	2
#2228	2
#2226	2
#2224	2
#2183	2
#2177	2
C₄×D₁₆	2