Finite Group Databasebeta

Subgroups — #9728

Order 256.

207 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 207 conjugacy classes of subgroups — too many to draw. Only its 65 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	#3167	C₂	1	1	yes	maximal, characteristic
2	C₄×#108	C₂	1	1	yes	maximal, characteristic, complemented
3	#1526	C₂	1	1	yes	maximal, characteristic, complemented
4	#1525	C₂	1	1	yes	maximal, characteristic
5	#1485	C₂	1	1	yes	maximal, characteristic, complemented
6	#1484	C₂	1	1	yes	maximal, characteristic
7	#1451	C₂	1	1	yes	maximal, characteristic, complemented
8	#533	C₂²	1	1	yes	characteristic
9	#532	C₂²	1	1	yes	characteristic
10	#498	C₄	1	1	yes	characteristic
11	#492	C₄	1	1	yes	characteristic
12	#487	C₄	1	1	yes	characteristic
13	#486	C₄	1	1	yes	characteristic
14	#465	—	2	1	no	—
15	#465	—	2	1	no	—
16	C₄×SD₁₆	—	2	1	no	—
17	C₄×SD₁₆	—	2	1	no	—
18	#431	C₂²	1	1	yes	characteristic
19	C₂×#108	—	2	1	no	—
20	C₄×M₁₆	C₂²	1	1	yes	characteristic
21	C₄×M₁₆	C₂²	1	1	yes	characteristic
22	C₄×M₁₆	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
24	#341	—	2	1	no	—
25	#330	—	2	2	no	—
26	#329	—	2	2	no	—
27	Q₈⋊C₈	—	2	2	no	—
28	D₄⋊C₈	—	2	2	no	—
29	#322	—	2	1	no	—
30	#322	—	2	1	no	—
31	C₂×#41	—	2	1	no	—
32	C₂×SD₁₆	—	2	1	no	—
33	C₂×SD₁₆	—	2	1	no	—
34	C₂×M₁₆	—	2	1	no	—
35	C₂×M₁₆	—	2	1	no	—
36	C₂×M₁₆	—	2	1	no	—
37	C₂²×C₈	D₄	1	1	yes	characteristic
38	C₂²×C₈	D₄	1	1	yes	characteristic
39	C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
40	C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
41	C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
42	#127	C₂×C₄	1	1	yes	characteristic
43	#125	—	2	1	no	—
44	#124	—	2	1	no	—
45	C₂²⋊Q₈	—	2	1	no	—
46	C₂²⋊Q₈	—	2	1	no	—
47	#121	—	2	1	no	—
48	#121	—	2	1	no	—
49	C₄×Q₈	—	2	1	no	—
50	C₄×Q₈	—	2	1	no	—
51	C₄×D₄	—	2	1	no	—
52	C₄×D₄	—	2	1	no	—
53	#117	—	2	1	no	—
54	C₂×C₄²	C₂³	1	1	yes	characteristic
55	#108	—	2	4	no	—
56	#106	—	2	1	no	—
57	#106	—	2	1	no	—
58	C₄⋊C₈	D₄	1	2	yes	—
59	#103	—	2	1	no	—
60	#103	—	2	1	no	—
61	#102	—	2	1	no	—
62	#102	—	2	1	no	—
63	C₂²⋊C₈	—	2	1	no	—
64	#97	—	2	1	no	—
65	#97	—	2	1	no	—
66	#97	D₄	1	2	yes	—
67	C₄×C₈	—	2	1	no	—
68	C₄×C₈	—	2	1	no	—
69	C₄×C₈	—	2	1	no	—
70	C₄×C₈	—	2	1	no	—
71	C₄×C₈	C₂×C₄	1	1	yes	characteristic
72	C₄×C₈	C₂×C₄	1	1	yes	characteristic
73	C₄×C₈	D₄	1	1	yes	characteristic
74	C₄×C₈	D₄	1	1	yes	characteristic
75	#41	—	8	2	no	—
76	C₂×Q₈	—	2	1	no	—
77	C₂×Q₈	—	2	1	no	—
78	C₂×Q₈	—	2	1	no	—
79	C₂×D₄	—	2	1	no	—
80	C₂×D₄	—	2	1	no	—
81	C₂×D₄	—	2	1	no	—
82	C₂×D₄	—	4	1	no	—
83	C₂²×C₄	—	4	1	no	—
84	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
85	C₂²×C₄	#41	1	1	yes	characteristic
86	C₂²×C₄	#41	1	1	yes	characteristic
87	SD₁₆	—	4	2	no	—
88	SD₁₆	—	4	2	no	—
89	M₁₆	—	2	2	no	—
90	M₁₆	—	2	2	no	—
91	M₁₆	—	4	2	no	—
92	M₁₆	—	4	1	no	—
93	M₁₆	—	4	1	no	—
94	C₂×C₈	—	2	2	no	—
95	C₂×C₈	—	2	2	no	—
96	C₂×C₈	—	2	2	no	—
97	C₂×C₈	—	4	1	no	—
98	C₂×C₈	—	4	1	no	—
99	C₂×C₈	—	4	1	no	—
100	C₂×C₈	C₂²⋊C₄	1	1	yes	characteristic
101	C₂×C₈	C₂²⋊C₄	1	1	yes	characteristic
102	C₂×C₈	C₂²⋊C₄	1	1	yes	characteristic
103	C₂×C₈	C₂²⋊C₄	1	1	yes	characteristic
104	C₄⋊C₄	—	2	1	no	—
105	C₄⋊C₄	—	2	1	no	—
106	C₄⋊C₄	—	2	1	no	—
107	C₄⋊C₄	—	2	1	no	—
108	C₄⋊C₄	—	2	1	no	—
109	C₄⋊C₄	—	2	1	no	—
110	C₄⋊C₄	—	4	1	no	—
111	C₄⋊C₄	—	4	1	no	—
112	C₂²⋊C₄	—	4	1	no	—
113	C₂²⋊C₄	—	4	1	no	—
114	C₂²⋊C₄	—	4	1	no	—
115	C₄²	—	4	1	no	—
116	C₄²	C₂²×C₄	1	1	yes	characteristic
117	C₄²	C₂×D₄	1	1	yes	characteristic
118	C₄²	C₂×D₄	1	1	yes	characteristic
119	C₄²	C₂×D₄	1	1	yes	characteristic
120	C₂³	—	4	1	no	—
121	C₂³	#124	1	1	yes	characteristic
122	Q₈	—	4	1	no	—
123	Q₈	—	4	1	no	—
124	Q₈	—	4	2	no	—
125	Q₈	—	8	1	no	—
126	D₄	—	4	1	no	—
127	D₄	—	4	1	no	—
128	D₄	—	4	2	no	—
129	D₄	—	8	1	no	—
130	D₄	—	8	2	no	—
131	C₂×C₄	—	2	1	no	—
132	C₂×C₄	—	2	1	no	—
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	4	1	no	—
136	C₂×C₄	—	4	1	no	—
137	C₂×C₄	—	4	1	no	—
138	C₂×C₄	—	8	2	no	—
139	C₂×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
140	C₂×C₄	C₄×D₄	1	1	yes	characteristic
141	C₂×C₄	C₄×D₄	1	1	yes	characteristic
142	C₂×C₄	#121	1	1	yes	characteristic
143	C₂×C₄	#121	1	1	yes	characteristic
144	C₂×C₄	#127	1	1	yes	characteristic
145	C₈	—	2	1	no	—
146	C₈	—	2	1	no	—
147	C₈	—	2	1	no	—
148	C₈	—	4	2	no	—
149	C₈	—	4	2	no	—
150	C₈	—	4	2	no	—
151	C₈	—	4	2	no	—
152	C₈	#104	1	2	yes	—
153	C₂²	—	2	1	no	—
154	C₂²	—	2	1	no	—
155	C₂²	—	2	1	no	—
156	C₂²	—	8	1	no	—
157	C₂²	—	8	1	no	—
158	C₂²	—	8	1	no	—
159	C₂²	#390	1	1	yes	characteristic
160	C₄	—	2	1	no	—
161	C₄	—	2	1	no	—
162	C₄	—	2	1	no	—
163	C₄	—	8	1	no	—
164	C₄	—	8	1	no	—
165	C₄	—	8	1	no	—
166	C₄	C₂×#104	1	1	yes	characteristic
167	C₄	#421	1	1	yes	characteristic
168	C₄	#444	1	2	yes	—
169	C₄	#471	1	2	yes	—
170	C₂	—	4	1	no	—
171	C₂	—	16	1	no	—
172	C₂	#1975	1	1	yes	characteristic
173	C₂	#1977	1	1	yes	characteristic
174	C₂	#1980	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#121
commutator	C₄²	C₂²×C₄
Frattini, cosocle	C₂×C₄²	C₂³
socle	C₂²	#390
prime radical	#532	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
2. C₄×#108	C₂	semidirect
3. #1526	C₂	semidirect
5. #1485	C₂	semidirect
7. #1451	C₂	semidirect

Maximal subgroups

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

subgroup	index
#3167	2
C₄×#108	2
#1526	2
#1525	2
#1485	2
#1484	2
#1451	2