Finite Group Databasebeta

Subgroups — #21296

Order 256.

365 conjugacy classes of subgroups; only the 163 normal subgroups are shown.

normal subgroups only

This group has 365 conjugacy classes of subgroups — too many to draw. Only its 163 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	#2943	C₂	1	1	yes	maximal, characteristic
2	#2922	C₂	1	2	yes	maximal, complemented
3	#1975	C₂	1	1	yes	maximal, characteristic
4	#1974	C₂	1	1	yes	maximal, characteristic
5	#1946	C₂	1	1	yes	maximal, characteristic, complemented
6	#1945	C₂	1	1	yes	maximal, characteristic, complemented
7	#1938	C₂	1	2	yes	maximal, complemented
8	#1937	C₂	1	2	yes	maximal, complemented
9	#1778	C₂	1	1	yes	maximal, characteristic, complemented
10	#1759	C₂	1	2	yes	maximal, complemented
11	#1726	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#135	C₂²	1	1	yes	characteristic
13	C₄×#41	C₂²	1	2	yes	—
14	#442	C₄	1	2	yes	complemented
15	#441	C₄	1	2	yes	complemented
16	#440	C₄	1	4	yes	complemented
17	#433	—	2	4	no	—
18	#431	—	2	4	no	—
19	C₂×#108	C₂²	1	1	yes	characteristic
20	C₂×#108	C₂²	1	1	yes	characteristic
21	#427	C₂²	1	1	yes	characteristic
22	#424	C₂²	1	2	yes	—
23	#424	C₂²	1	2	yes	—
24	C₂×#104	C₂²	1	2	yes	—
25	C₂×#104	C₂²	1	2	yes	—
26	#416	C₂²	1	2	yes	—
27	#408	C₂²	1	2	yes	—
28	#408	C₂²	1	2	yes	—
29	C₄×M₁₆	C₂²	1	2	yes	—
30	C₂×#97	C₂²	1	1	yes	characteristic
31	C₂×#97	C₂²	1	1	yes	characteristic
32	C₂×#97	C₂²	1	1	yes	characteristic
33	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
34	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
35	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
36	#376	C₂²	1	1	yes	characteristic
37	#341	C₂²	1	2	yes	—
38	#339	C₂²	1	2	yes	—
39	#339	C₂²	1	2	yes	—
40	C₂×#41	C₂³	1	2	yes	—
41	#135	—	2	2	no	—
42	#135	—	2	2	no	—
43	C₂×Q₁₆	C₂×C₄	1	1	yes	characteristic
44	C₂×SD₁₆	C₂×C₄	1	2	yes	—
45	C₂×D₈	C₂×C₄	1	1	yes	characteristic
46	C₂×M₁₆	C₂³	1	2	yes	—
47	C₂×M₁₆	C₂³	1	2	yes	—
48	C₂²×C₈	C₂³	1	1	yes	characteristic
49	C₂²×C₈	C₂³	1	1	yes	characteristic
50	C₂²×C₈	C₂³	1	1	yes	characteristic
51	C₂²×C₈	C₂³	1	1	yes	characteristic
52	C₄×Q₈	C₂×C₄	1	4	yes	—
53	C₄×D₄	—	2	4	no	—
54	C₄×D₄	C₂×C₄	1	4	yes	—
55	#117	—	2	2	no	—
56	#117	C₂³	1	2	yes	—
57	C₂×C₄²	—	2	2	no	—
58	C₂×C₄²	C₂³	1	1	yes	characteristic
59	C₂×C₄²	C₂³	1	1	yes	characteristic
60	C₂×C₄²	C₂³	1	1	yes	characteristic
61	#108	—	2	2	no	—
62	#108	—	2	2	no	—
63	#107	C₂×C₄	1	2	yes	—
64	#106	C₂×C₄	1	2	yes	—
65	C₄⋊C₈	—	2	4	no	—
66	C₄⋊C₈	—	2	4	no	—
67	#104	—	2	4	no	—
68	#104	—	2	4	no	—
69	#103	C₂×C₄	1	4	yes	—
70	#102	C₂×C₄	1	4	yes	—
71	C₂²⋊C₈	—	2	4	no	—
72	C₂²⋊C₈	—	2	4	no	—
73	#97	—	2	2	no	—
74	#97	—	2	2	no	—
75	#97	—	2	2	no	—
76	#97	C₂×C₄	1	2	yes	—
77	#97	C₂×C₄	1	2	yes	—
78	C₄×C₈	—	2	2	no	—
79	C₄×C₈	—	2	2	no	—
80	C₄×C₈	—	2	2	no	—
81	C₄×C₈	D₄	1	4	yes	—
82	#95	—	2	2	no	—
83	#41	—	2	2	no	—
84	#41	—	2	2	no	—
85	#41	—	2	2	no	—
86	#41	—	2	2	no	—
87	C₂×Q₈	C₂²×C₄	1	2	yes	—
88	C₂×D₄	—	2	2	no	—
89	C₂×D₄	C₂²×C₄	1	2	yes	—
90	C₂²×C₄	—	2	1	no	—
91	C₂²×C₄	—	2	1	no	—
92	C₂²×C₄	—	2	4	no	—
93	C₂²×C₄	—	2	2	no	—
94	C₂²×C₄	#41	1	2	yes	—
95	C₂²×C₄	C₂⁴	1	1	yes	characteristic
96	Q₁₆	—	2	2	no	—
97	SD₁₆	—	2	4	no	—
98	D₈	—	2	2	no	—
99	M₁₆	—	2	2	no	—
100	M₁₆	—	2	2	no	—
101	M₁₆	—	2	2	no	—
102	M₁₆	—	2	2	no	—
103	C₂×C₈	—	2	1	no	—
104	C₂×C₈	—	2	1	no	—
105	C₂×C₈	—	2	1	no	—
106	C₂×C₈	—	2	1	no	—
107	C₂×C₈	—	2	1	no	—
108	C₂×C₈	—	2	1	no	—
109	C₂×C₈	—	2	2	no	—
110	C₂×C₈	—	2	2	no	—
111	C₂×C₈	C₂²×C₄	1	2	yes	—
112	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
113	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
114	C₂×C₈	C₂×D₄	1	2	yes	—
115	C₂×C₈	C₂×D₄	1	2	yes	—
116	C₂×C₈	#41	1	2	yes	—
117	C₂×C₈	#41	1	2	yes	—
118	C₄⋊C₄	—	2	4	no	—
119	C₄⋊C₄	C₂²×C₄	1	4	yes	—
120	C₂²⋊C₄	—	2	2	no	—
121	C₂²⋊C₄	—	2	4	no	—
122	C₄²	—	2	4	no	—
123	C₄²	—	2	2	no	—
124	C₄²	—	2	1	no	—
125	C₄²	—	2	1	no	—
126	C₄²	C₂²×C₄	1	2	yes	—
127	C₄²	C₂×D₄	1	2	yes	—
128	C₄²	#41	1	2	yes	—
129	C₄²	#41	1	2	yes	—
130	C₂³	—	2	2	no	—
131	C₂³	C₂×#41	1	1	yes	characteristic
132	Q₈	—	2	2	no	—
133	Q₈	—	2	2	no	—
134	D₄	—	2	2	no	—
135	D₄	—	2	2	no	—
136	D₄	—	4	2	no	—
137	D₄	—	4	2	no	—
138	C₂×C₄	—	2	4	no	—
139	C₂×C₄	—	2	2	no	—
140	C₂×C₄	—	2	2	no	—
141	C₂×C₄	—	2	2	no	—
142	C₂×C₄	—	4	2	no	—
143	C₂×C₄	—	4	8	no	—
144	C₂×C₄	—	4	2	no	—
145	C₂×C₄	—	4	2	no	—
146	C₂×C₄	—	4	2	no	—
147	C₂×C₄	—	4	2	no	—
148	C₂×C₄	C₄×D₄	1	4	yes	—
149	C₂×C₄	C₂³×C₄	1	1	yes	characteristic
150	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
151	C₂×C₄	C₂×#41	1	1	yes	characteristic
152	C₂×C₄	C₂×#41	1	1	yes	characteristic
153	C₂×C₄	C₂×#41	1	1	yes	characteristic
154	C₂×C₄	C₂×#41	1	1	yes	characteristic
155	C₈	—	2	2	no	—
156	C₈	—	2	2	no	—
157	C₈	—	2	2	no	—
158	C₈	—	4	2	no	—
159	C₈	—	4	2	no	—
160	C₈	#117	1	2	yes	—
161	C₈	#117	1	2	yes	—
162	C₂²	—	2	2	no	—
163	C₂²	—	4	4	no	—
164	C₂²	—	4	2	no	—
165	C₂²	C₂×C₄×D₄	1	1	yes	characteristic
166	C₂²	#529	1	1	yes	characteristic
167	C₂²	#529	1	1	yes	characteristic
168	C₄	—	2	4	no	—
169	C₄	—	4	2	no	—
170	C₄	—	4	2	no	—
171	C₄	—	4	4	no	—
172	C₄	—	4	2	no	—
173	C₄	C₂×#117	1	1	yes	characteristic
174	C₄	C₄×#41	1	1	yes	characteristic
175	C₄	#532	1	1	yes	characteristic
176	C₄	#532	1	1	yes	characteristic
177	C₂	—	2	1	no	—
178	C₂	—	2	1	no	—
179	C₂	—	8	2	no	—
180	C₂	#2293	1	1	yes	characteristic
181	C₂	#2955	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄
Frattini, cosocle	C₂²×C₄	C₂⁴
socle	C₂²	C₂×C₄×D₄
prime radical	C₂×#135	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. #2922	C₂	semidirect
5. #1946	C₂	semidirect
6. #1945	C₂	semidirect
7. #1938	C₂	semidirect
8. #1937	C₂	semidirect
9. #1778	C₂	semidirect
10. #1759	C₂	semidirect
11. #1726	C₂	semidirect
14. #442	C₄	semidirect
15. #441	C₄	semidirect
16. #440	C₄	semidirect

Maximal subgroups

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

subgroup	index
#2943	2
#2922	2
#1975	2
#1974	2
#1946	2
#1945	2
#1938	2
#1937	2
#1778	2
#1759	2
#1726	2