Finite Group Databasebeta

Subgroups — #29154

Order 256.

306 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 117 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×#443	C₂	1	1	yes	maximal, characteristic
3	#2031	C₂	1	4	yes	maximal, complemented
4	#1976	C₂	1	2	yes	maximal
5	#1833	C₂	1	2	yes	maximal, complemented
6	#1814	C₂	1	2	yes	maximal, complemented
7	#1812	C₂	1	2	yes	maximal, complemented
8	#1772	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#135	C₂²	1	1	yes	characteristic
10	C₂×#131	C₂²	1	2	yes	—
11	#533	C₂²	1	2	yes	—
12	#500	—	2	1	no	—
13	#499	—	2	1	no	—
14	#499	—	2	1	no	—
15	#498	—	2	1	no	—
16	#443	—	2	1	no	—
17	#443	—	2	1	no	—
18	#443	—	2	1	no	—
19	#443	—	2	1	no	—
20	C₂×#108	C₂²	1	1	yes	characteristic
21	C₂×#108	C₂²	1	1	yes	characteristic
22	#427	C₂²	1	1	yes	characteristic
23	#427	C₂²	1	2	yes	—
24	#427	C₂²	1	2	yes	—
25	#421	C₂²	1	4	yes	—
26	C₂×#104	C₂²	1	2	yes	—
27	#410	C₂²	1	4	yes	—
28	#405	C₂²	1	2	yes	—
29	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
30	#344	C₂²	1	4	yes	—
31	#339	C₂²	1	2	yes	—
32	H(3,ℤ/4)	C₂²	1	4	yes	—
33	C₂×#41	C₂³	1	2	yes	—
34	#135	—	2	1	no	—
35	#135	—	2	1	no	—
36	#135	—	2	1	no	—
37	#135	—	2	1	no	—
38	C₂×Q₁₆	Q₈	1	1	yes	characteristic, complemented
39	C₂×SD₁₆	Q₈	1	2	yes	complemented
40	C₂×D₈	Q₈	1	1	yes	characteristic, complemented
41	#131	—	2	2	no	—
42	#131	—	2	2	no	—
43	#131	—	2	2	no	—
44	#131	—	2	2	no	—
45	C₂×M₁₆	—	2	2	no	—
46	C₂×M₁₆	C₂³	1	2	yes	—
47	C₂×M₁₆	C₂³	1	2	yes	—
48	C₂²×C₈	—	2	2	no	—
49	C₂²×C₈	C₂³	1	1	yes	characteristic
50	C₂²×C₈	C₂³	1	1	yes	characteristic
51	C₄⋊Q₈	—	2	2	no	—
52	#125	—	2	2	no	—
53	C₂²⋊Q₈	—	4	2	no	—
54	C₄×Q₈	—	4	2	no	—
55	#117	C₂³	1	4	yes	—
56	#117	C₂³	1	2	yes	—
57	C₂×C₄²	C₂³	1	1	yes	characteristic
58	#108	—	2	1	no	—
59	#108	—	2	1	no	—
60	#108	—	2	1	no	—
61	#108	—	2	1	no	—
62	#107	—	2	2	no	—
63	#107	—	2	2	no	—
64	#107	—	2	1	no	—
65	#106	—	2	2	no	—
66	#106	—	2	2	no	—
67	#106	—	2	1	no	—
68	#104	—	2	2	no	—
69	#104	—	2	2	no	—
70	#104	—	4	4	no	—
71	#99	—	4	4	no	—
72	#97	—	2	2	no	—
73	C₄×C₈	—	2	2	no	—
74	C₄×C₈	D₄	1	1	yes	characteristic
75	C₄×C₈	D₄	1	1	yes	characteristic
76	C₄×C₈	D₄	1	1	yes	characteristic
77	C₄×C₈	D₄	1	1	yes	characteristic
78	#41	—	2	2	no	—
79	#41	—	2	2	no	—
80	#41	—	2	2	no	—
81	#41	—	2	2	no	—
82	C₂×Q₈	—	4	2	no	—
83	C₂×Q₈	C₂×Q₈	1	2	yes	—
84	C₂×D₄	—	2	2	no	—
85	C₂×D₄	C₂×Q₈	1	2	yes	—
86	C₂²×C₄	—	2	1	no	—
87	C₂²×C₄	—	2	2	no	—
88	C₂²×C₄	C₂⁴	1	1	yes	characteristic
89	Q₁₆	—	2	1	no	—
90	Q₁₆	—	2	1	no	—
91	SD₁₆	—	2	2	no	—
92	SD₁₆	—	2	2	no	—
93	D₈	—	2	1	no	—
94	D₈	—	2	1	no	—
95	M₁₆	—	2	2	no	—
96	M₁₆	—	2	2	no	—
97	M₁₆	—	2	2	no	—
98	M₁₆	—	2	2	no	—
99	M₁₆	—	4	2	no	—
100	M₁₆	—	4	2	no	—
101	C₂×C₈	—	2	2	no	—
102	C₂×C₈	—	2	2	no	—
103	C₂×C₈	—	4	2	no	—
104	C₂×C₈	—	4	2	no	—
105	C₂×C₈	C₂×D₄	1	1	yes	characteristic
106	C₂×C₈	C₂×D₄	1	1	yes	characteristic
107	C₂×C₈	C₂×D₄	1	1	yes	characteristic
108	C₂×C₈	C₂×D₄	1	1	yes	characteristic
109	C₂×C₈	C₂×Q₈	1	1	yes	characteristic
110	C₂×C₈	C₂×Q₈	1	1	yes	characteristic
111	C₂×C₈	#41	1	1	yes	characteristic
112	C₂×C₈	#41	1	1	yes	characteristic
113	C₂×C₈	#41	1	1	yes	characteristic
114	C₂×C₈	#41	1	1	yes	characteristic
115	C₂×C₈	#41	1	1	yes	characteristic
116	C₂×C₈	#41	1	1	yes	characteristic
117	C₄⋊C₄	—	2	4	no	—
118	C₄⋊C₄	—	2	2	no	—
119	C₄⋊C₄	—	4	4	no	—
120	C₄⋊C₄	—	4	2	no	—
121	C₂²⋊C₄	—	2	2	no	—
122	C₂²⋊C₄	—	2	4	no	—
123	C₄²	—	2	1	no	—
124	C₄²	—	2	2	no	—
125	C₄²	—	2	4	no	—
126	C₄²	C₂×D₄	1	1	yes	characteristic
127	C₄²	C₂×D₄	1	1	yes	characteristic
128	C₂³	—	2	2	no	—
129	C₂³	H(5,𝔽₂)	1	1	yes	characteristic
130	Q₈	—	2	2	no	—
131	Q₈	—	2	2	no	—
132	Q₈	—	8	4	no	—
133	D₄	—	2	2	no	—
134	D₄	—	2	2	no	—
135	D₄	—	4	2	no	—
136	D₄	—	4	2	no	—
137	C₂×C₄	—	2	1	no	—
138	C₂×C₄	—	2	1	no	—
139	C₂×C₄	—	2	2	no	—
140	C₂×C₄	—	4	2	no	—
141	C₂×C₄	—	4	2	no	—
142	C₂×C₄	—	4	2	no	—
143	C₂×C₄	—	4	4	no	—
144	C₂×C₄	—	4	2	no	—
145	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
146	C₂×C₄	C₂²×Q₈	1	1	yes	characteristic
147	C₂×C₄	C₂×#41	1	1	yes	characteristic
148	C₂×C₄	C₂×#41	1	1	yes	characteristic
149	C₂×C₄	C₂×#41	1	1	yes	characteristic
150	C₂×C₄	H(5,𝔽₂)	1	1	yes	characteristic
151	C₈	—	4	2	no	—
152	C₈	—	4	2	no	—
153	C₈	C₂²⋊Q₈	1	1	yes	characteristic
154	C₈	C₂²⋊Q₈	1	1	yes	characteristic
155	C₈	C₂²⋊Q₈	1	1	yes	characteristic
156	C₈	C₂²⋊Q₈	1	1	yes	characteristic
157	C₈	#124	1	1	yes	characteristic
158	C₈	#124	1	1	yes	characteristic
159	C₈	#124	1	1	yes	characteristic
160	C₈	#124	1	1	yes	characteristic
161	C₂²	—	4	1	no	—
162	C₂²	—	4	2	no	—
163	C₂²	—	8	2	no	—
164	C₂²	#535	1	1	yes	characteristic
165	C₂²	#554	1	1	yes	characteristic
166	C₂²	#555	1	1	yes	characteristic
167	C₄	—	2	2	no	—
168	C₄	—	4	1	no	—
169	C₄	—	4	2	no	—
170	C₄	—	8	2	no	—
171	C₄	—	8	4	no	—
172	C₄	C₂×C₂²⋊Q₈	1	1	yes	characteristic
173	C₄	C₂×#124	1	1	yes	characteristic
174	C₄	#554	1	1	yes	characteristic
175	C₄	#555	1	1	yes	characteristic
176	C₂	—	2	1	no	—
177	C₂	—	2	1	no	—
178	C₂	—	2	1	no	—
179	C₂	—	8	2	no	—
180	C₂	#2563	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#555
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴
socle	C₂	#2563
prime radical	C₂×#135	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #3160	C₂	semidirect
3. #2031	C₂	semidirect
5. #1833	C₂	semidirect
6. #1814	C₂	semidirect
7. #1812	C₂	semidirect
8. #1772	C₂	semidirect
38. C₂×Q₁₆	Q₈	semidirect
39. C₂×SD₁₆	Q₈	semidirect
40. 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×#443	2
#2031	2
#1976	2
#1833	2
#1814	2
#1812	2
#1772	2