Finite Group Databasebeta

Subgroups — #33270

Order 256.

327 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 327 conjugacy classes of subgroups — too many to draw. Only its 123 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	#2863	C₂	1	2	yes	maximal
2	#1971	C₂	1	1	yes	maximal, characteristic
3	#1971	C₂	1	1	yes	maximal, characteristic
4	#1969	C₂	1	1	yes	maximal, characteristic
5	#1944	C₂	1	1	yes	maximal, characteristic
6	#1943	C₂	1	1	yes	maximal, characteristic
7	#1941	C₂	1	1	yes	maximal, characteristic
8	#1935	C₂	1	2	yes	maximal
9	#1935	C₂	1	2	yes	maximal
10	#1933	C₂	1	2	yes	maximal
11	C₄×#97	C₂	1	1	yes	maximal, characteristic
12	C₂²×Q₁₆	C₂²	1	1	yes	characteristic
13	C₂×#107	C₂²	1	1	yes	characteristic
14	C₂×#106	C₂²	1	1	yes	characteristic
15	C₂×#106	C₂²	1	1	yes	characteristic
16	C₂×#103	C₂²	1	2	yes	—
17	C₂×#103	C₂²	1	2	yes	—
18	C₂×#103	C₂²	1	2	yes	—
19	C₂×#97	C₂²	1	1	yes	characteristic
20	C₂×#97	C₂²	1	1	yes	characteristic
21	C₂×#97	C₂²	1	1	yes	characteristic
22	C₂×#97	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
24	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
25	#391	C₂²	1	2	yes	—
26	#391	C₂²	1	2	yes	—
27	#391	C₂²	1	2	yes	—
28	#389	C₂²	1	2	yes	—
29	#389	C₂²	1	2	yes	—
30	#389	C₂²	1	2	yes	—
31	#387	—	2	2	no	—
32	#387	—	2	2	no	—
33	#387	—	2	2	no	—
34	#383	—	2	2	no	—
35	C₄³	C₂²	1	1	yes	characteristic
36	#340	C₂²	1	2	yes	—
37	#340	C₂²	1	2	yes	—
38	#340	C₂²	1	2	yes	—
39	C₂²×Q₈	C₂³	1	2	yes	—
40	C₂×Q₁₆	—	2	2	no	—
41	C₂×Q₁₆	—	2	2	no	—
42	C₂×Q₁₆	—	2	2	no	—
43	C₂²×C₈	C₂³	1	1	yes	characteristic
44	C₂²×C₈	C₂³	1	1	yes	characteristic
45	C₂²×C₈	C₂³	1	1	yes	characteristic
46	C₂²×C₈	C₂³	1	1	yes	characteristic
47	C₂×C₄⋊C₄	—	2	2	no	—
48	C₂×C₄⋊C₄	—	2	2	no	—
49	C₂×C₄⋊C₄	—	2	2	no	—
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	D₄	1	2	yes	—
54	C₂×C₄²	D₄	1	2	yes	—
55	C₂×C₄²	C₂³	1	1	yes	characteristic
56	C₂×C₄²	C₂³	1	1	yes	characteristic
57	C₂×C₄²	C₂³	1	1	yes	characteristic
58	#107	—	2	2	no	—
59	#106	—	2	2	no	—
60	#106	—	2	2	no	—
61	#103	—	2	4	no	—
62	#103	—	2	4	no	—
63	#103	—	2	4	no	—
64	#97	—	2	2	no	—
65	#97	—	2	2	no	—
66	#97	—	2	2	no	—
67	#97	—	2	2	no	—
68	C₄×C₈	—	2	2	no	—
69	C₄×C₈	—	2	2	no	—
70	#95	—	2	2	no	—
71	#95	—	2	2	no	—
72	#95	—	2	2	no	—
73	#95	—	2	2	no	—
74	#95	—	2	2	no	—
75	#95	—	2	2	no	—
76	#95	—	2	2	no	—
77	#95	—	2	2	no	—
78	#95	—	2	2	no	—
79	C₂×Q₈	—	2	2	no	—
80	C₂×Q₈	—	2	2	no	—
81	C₂×Q₈	—	2	2	no	—
82	C₂×Q₈	—	2	2	no	—
83	C₂×Q₈	—	2	2	no	—
84	C₂×Q₈	—	2	2	no	—
85	C₂²×C₄	—	2	2	no	—
86	C₂²×C₄	—	2	2	no	—
87	C₂²×C₄	—	2	2	no	—
88	C₂²×C₄	—	2	2	no	—
89	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
90	C₂²×C₄	C₂×D₄	1	2	yes	—
91	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
92	C₂²×C₄	C₂×D₄	1	2	yes	—
93	C₂²×C₄	C₂⁴	1	1	yes	characteristic
94	Q₁₆	—	4	4	no	—
95	C₂×C₈	—	2	1	no	—
96	C₂×C₈	—	2	1	no	—
97	C₂×C₈	—	2	1	no	—
98	C₂×C₈	—	2	1	no	—
99	C₂×C₈	—	2	1	no	—
100	C₂×C₈	—	2	1	no	—
101	C₂×C₈	#41	1	2	yes	—
102	C₂×C₈	#41	1	2	yes	—
103	C₂×C₈	#41	1	2	yes	—
104	C₂×C₈	#41	1	2	yes	—
105	C₂×C₈	#41	1	2	yes	—
106	C₂×C₈	#41	1	2	yes	—
107	C₄⋊C₄	—	2	2	no	—
108	C₄⋊C₄	—	2	2	no	—
109	C₄⋊C₄	—	2	2	no	—
110	C₄⋊C₄	—	2	2	no	—
111	C₄⋊C₄	—	2	2	no	—
112	C₄⋊C₄	—	2	2	no	—
113	C₄⋊C₄	—	4	4	no	—
114	C₄⋊C₄	—	4	4	no	—
115	C₄⋊C₄	—	4	4	no	—
116	C₄²	—	2	4	no	—
117	C₄²	—	2	1	no	—
118	C₄²	—	2	1	no	—
119	C₄²	—	2	1	no	—
120	C₄²	—	2	1	no	—
121	C₄²	—	2	1	no	—
122	C₄²	—	2	1	no	—
123	C₄²	—	4	2	no	—
124	C₂³	C₂²×D₄	1	1	yes	characteristic
125	Q₈	—	4	2	no	—
126	Q₈	—	4	2	no	—
127	Q₈	—	4	2	no	—
128	Q₈	—	4	2	no	—
129	C₂×C₄	—	2	2	no	—
130	C₂×C₄	—	2	2	no	—
131	C₂×C₄	—	2	2	no	—
132	C₂×C₄	—	2	1	no	—
133	C₂×C₄	—	2	1	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	—	2	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	—	4	2	no	—
149	C₂×C₄	—	4	2	no	—
150	C₂×C₄	#137	1	4	yes	—
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	C₂×C₄	C₂×#41	1	1	yes	characteristic
156	C₂×C₄	C₂×#41	1	1	yes	characteristic
157	C₈	—	2	2	no	—
158	C₈	—	2	2	no	—
159	C₈	—	2	2	no	—
160	C₈	#126	1	4	yes	—
161	C₂²	#532	1	1	yes	characteristic
162	C₂²	#532	1	1	yes	characteristic
163	C₂²	#532	1	1	yes	characteristic
164	C₂²	C₂×#137	1	2	yes	—
165	C₂²	#575	1	2	yes	—
166	C₄	—	2	4	no	—
167	C₄	—	4	2	no	—
168	C₄	—	4	1	no	—
169	C₄	—	4	1	no	—
170	C₄	—	8	2	no	—
171	C₄	—	8	2	no	—
172	C₄	—	8	2	no	—
173	C₄	—	8	2	no	—
174	C₄	C₂×#126	1	1	yes	characteristic
175	C₄	#529	1	1	yes	characteristic
176	C₄	#529	1	1	yes	characteristic
177	C₄	#529	1	1	yes	characteristic
178	C₂	#2860	1	1	yes	characteristic
179	C₂	#3172	1	2	yes	—
180	C₂	#3181	1	2	yes	—
181	C₂	#3183	1	1	yes	characteristic
182	C₂	#3185	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 ◁ C₄³ ◁ 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
#2863	2
#1971	2
#1969	2
#1944	2
#1943	2
#1941	2
#1935	2
#1933	2
C₄×#97	2