Finite Group Databasebeta

Subgroups — #27993

Order 256.

295 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 295 conjugacy classes of subgroups — too many to draw. Only its 127 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	#2509	C₂	1	2	yes	maximal
2	#2101	C₂	1	2	yes	maximal
3	#2088	C₂	1	2	yes	maximal
4	#1942	C₂	1	1	yes	maximal, characteristic
5	#1942	C₂	1	1	yes	maximal, characteristic
6	#1942	C₂	1	1	yes	maximal, characteristic
7	#1942	C₂	1	1	yes	maximal, characteristic
8	#1919	C₂	1	2	yes	maximal
9	#1846	C₂	1	1	yes	maximal, characteristic
10	C₄×#107	C₂	1	1	yes	maximal, characteristic
11	C₈×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
12	C₂×#107	C₂²	1	1	yes	characteristic
13	C₂×#107	C₂²	1	1	yes	characteristic
14	C₂×#107	C₂²	1	1	yes	characteristic
15	C₂×#107	C₂²	1	1	yes	characteristic
16	C₂×#106	C₂²	1	2	yes	—
17	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
21	#400	—	2	2	no	—
22	#400	—	2	2	no	—
23	#389	—	2	2	no	—
24	#389	C₂²	1	2	yes	—
25	#389	C₂²	1	2	yes	—
26	#389	C₂²	1	2	yes	—
27	#389	C₂²	1	2	yes	—
28	#387	—	2	2	no	—
29	#382	C₂²	1	2	yes	—
30	C₄×C₄⋊C₄	C₂²	1	2	yes	—
31	C₄×C₄⋊C₄	C₂²	1	1	yes	characteristic
32	#340	C₂²	1	2	yes	—
33	#340	C₂²	1	2	yes	—
34	#340	C₂²	1	2	yes	—
35	#340	C₂²	1	2	yes	—
36	#340	C₂²	1	2	yes	—
37	#336	C₂²	1	1	yes	characteristic
38	#336	C₂²	1	1	yes	characteristic
39	C₂²×C₈	C₂³	1	1	yes	characteristic
40	C₂²×C₈	C₂³	1	1	yes	characteristic
41	C₂²×C₈	C₂³	1	1	yes	characteristic
42	C₂²×C₈	C₂³	1	1	yes	characteristic
43	C₂×C₄⋊C₄	—	2	2	no	—
44	C₂×C₄⋊C₄	—	2	2	no	—
45	C₂×C₄⋊C₄	—	2	2	no	—
46	C₂×C₄⋊C₄	—	2	2	no	—
47	C₂×C₄⋊C₄	D₄	1	1	yes	characteristic
48	C₂×C₄⋊C₄	D₄	1	1	yes	characteristic
49	C₂×C₄⋊C₄	C₂³	1	2	yes	—
50	C₂×C₄⋊C₄	C₂³	1	2	yes	—
51	C₂×C₄⋊C₄	C₂³	1	2	yes	—
52	C₂×C₄⋊C₄	C₂³	1	2	yes	—
53	C₂×C₄²	—	2	2	no	—
54	C₂×C₄²	C₂³	1	1	yes	characteristic
55	C₂×C₄²	C₂³	1	1	yes	characteristic
56	C₂×C₄²	C₂³	1	1	yes	characteristic
57	#107	—	2	2	no	—
58	#107	—	2	2	no	—
59	#107	—	2	2	no	—
60	#107	Q₈	1	4	yes	—
61	#106	—	2	4	no	—
62	C₄⋊C₈	—	2	2	no	—
63	C₄×C₈	—	2	2	no	—
64	C₄×C₈	—	2	2	no	—
65	C₄×C₈	—	2	2	no	—
66	#95	—	2	2	no	—
67	#95	—	2	2	no	—
68	#95	—	2	2	no	—
69	#95	—	2	2	no	—
70	#95	—	2	2	no	—
71	#95	—	2	2	no	—
72	#95	—	2	2	no	—
73	#95	D₄	1	1	yes	characteristic
74	#95	D₄	1	1	yes	characteristic
75	C₂²×C₄	—	2	2	no	—
76	C₂²×C₄	—	2	2	no	—
77	C₂²×C₄	—	2	2	no	—
78	C₂²×C₄	—	2	2	no	—
79	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
85	C₂²×C₄	C₂⁴	1	1	yes	characteristic
86	C₂×C₈	—	2	1	no	—
87	C₂×C₈	—	2	1	no	—
88	C₂×C₈	—	2	1	no	—
89	C₂×C₈	—	2	1	no	—
90	C₂×C₈	—	2	1	no	—
91	C₂×C₈	—	2	1	no	—
92	C₂×C₈	—	2	1	no	—
93	C₂×C₈	—	2	1	no	—
94	C₂×C₈	C₂×Q₈	1	2	yes	—
95	C₂×C₈	#41	1	2	yes	—
96	C₂×C₈	#41	1	2	yes	—
97	C₂×C₈	#41	1	2	yes	—
98	C₄⋊C₄	—	2	2	no	—
99	C₄⋊C₄	—	2	2	no	—
100	C₄⋊C₄	—	2	2	no	—
101	C₄⋊C₄	—	2	2	no	—
102	C₄⋊C₄	—	2	2	no	—
103	C₄⋊C₄	—	2	2	no	—
104	C₄⋊C₄	—	2	2	no	—
105	C₄⋊C₄	—	2	2	no	—
106	C₄⋊C₄	—	2	2	no	—
107	C₄⋊C₄	—	4	4	no	—
108	C₄⋊C₄	—	4	4	no	—
109	C₄⋊C₄	—	8	2	no	—
110	C₄⋊C₄	—	8	2	no	—
111	C₄⋊C₄	C₂×Q₈	1	4	yes	—
112	C₄²	—	2	1	no	—
113	C₄²	—	2	1	no	—
114	C₄²	—	2	1	no	—
115	C₄²	—	2	1	no	—
116	C₄²	—	2	1	no	—
117	C₄²	—	2	1	no	—
118	C₄²	—	8	2	no	—
119	C₂³	C₂²×D₄	1	1	yes	characteristic
120	C₂×C₄	—	2	1	no	—
121	C₂×C₄	—	2	1	no	—
122	C₂×C₄	—	2	1	no	—
123	C₂×C₄	—	2	1	no	—
124	C₂×C₄	—	2	1	no	—
125	C₂×C₄	—	2	1	no	—
126	C₂×C₄	—	2	1	no	—
127	C₂×C₄	—	2	1	no	—
128	C₂×C₄	—	2	4	no	—
129	C₂×C₄	—	2	1	no	—
130	C₂×C₄	—	2	1	no	—
131	C₂×C₄	—	2	1	no	—
132	C₂×C₄	—	2	1	no	—
133	C₂×C₄	—	2	1	no	—
134	C₂×C₄	—	2	1	no	—
135	C₂×C₄	—	2	1	no	—
136	C₂×C₄	—	2	1	no	—
137	C₂×C₄	—	4	2	no	—
138	C₂×C₄	—	4	2	no	—
139	C₂×C₄	—	4	2	no	—
140	C₂×C₄	—	4	2	no	—
141	C₂×C₄	—	4	2	no	—
142	C₂×C₄	—	4	2	no	—
143	C₂×C₄	—	4	2	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	#135	1	2	yes	—
149	C₂×C₄	#135	1	2	yes	—
150	C₂×C₄	C₂²×Q₈	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	2₋¹⁺⁴	1	1	yes	characteristic
155	C₂×C₄	2₋¹⁺⁴	1	1	yes	characteristic
156	C₈	—	2	2	no	—
157	C₈	—	4	1	no	—
158	C₈	—	4	1	no	—
159	C₈	#125	1	4	yes	—
160	C₂²	#532	1	1	yes	characteristic
161	C₂²	#547	1	1	yes	characteristic
162	C₂²	D₄×Q₈	1	1	yes	characteristic
163	C₂²	C₂×#135	1	1	yes	characteristic
164	C₂²	C₂×#135	1	1	yes	characteristic
165	C₂²	#576	1	1	yes	characteristic
166	C₂²	#578	1	1	yes	characteristic
167	C₄	—	2	2	no	—
168	C₄	—	2	2	no	—
169	C₄	—	4	1	no	—
170	C₄	—	4	1	no	—
171	C₄	—	4	1	no	—
172	C₄	—	4	4	no	—
173	C₄	—	4	1	no	—
174	C₄	—	8	2	no	—
175	C₄	—	8	2	no	—
176	C₄	—	8	2	no	—
177	C₄	C₂×#125	1	1	yes	characteristic
178	C₄	#541	1	1	yes	characteristic
179	C₄	#552	1	1	yes	characteristic
180	C₄	#557	1	1	yes	characteristic
181	C₂	#2506	1	1	yes	characteristic
182	C₂	#3168	1	1	yes	characteristic
183	C₂	#3175	1	1	yes	characteristic
184	C₂	#3339	1	1	yes	characteristic
185	C₂	#3342	1	1	yes	characteristic
186	C₂	#3380	1	1	yes	characteristic
187	C₂	#3383	1	1	yes	characteristic

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

Maximal subgroups

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

subgroup	index
#2509	2
#2101	2
#2088	2
#1942	2
#1919	2
#1846	2
C₄×#107	2
C₈×C₄⋊C₄	2