Finite Group Databasebeta

Subgroups — #10415

Order 256.

242 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 242 conjugacy classes of subgroups — too many to draw. Only its 75 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	#2567	C₂	1	1	yes	maximal, characteristic
2	#1956	C₂	1	1	yes	maximal, characteristic
3	#1948	C₂	1	1	yes	maximal, characteristic
4	#1803	C₂	1	1	yes	maximal, characteristic
5	C₂×#333	C₂	1	1	yes	maximal, characteristic
6	C₂×Q₈⋊C₈	C₂	1	1	yes	maximal, characteristic
7	#1279	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄⋊Q₈	C₂²	1	1	yes	characteristic
9	C₂×C₄×Q₈	C₂²	1	1	yes	characteristic
10	C₂×#106	—	2	1	no	—
11	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
13	C₂×#103	—	2	1	no	—
14	C₂×#103	—	2	2	no	—
15	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
16	#395	—	2	2	no	—
17	#391	—	2	1	no	—
18	#389	—	2	1	no	—
19	#389	—	2	1	no	—
20	#384	—	2	1	no	—
21	#384	C₂²	1	1	yes	characteristic
22	C₄×C₄⋊C₄	C₂²	1	1	yes	characteristic
23	#340	—	2	1	no	—
24	#340	—	2	1	no	—
25	#333	—	2	2	no	—
26	Q₈⋊C₈	—	2	2	no	—
27	C₂²×Q₈	D₄	1	1	yes	characteristic
28	C₂²×Q₈	D₄	1	2	yes	—
29	C₂²×C₈	—	2	1	no	—
30	C₂²×C₈	—	2	1	no	—
31	C₂²×C₈	—	2	1	no	—
32	C₄⋊Q₈	—	2	1	no	—
33	C₄⋊Q₈	—	2	1	no	—
34	C₄⋊Q₈	—	2	1	no	—
35	C₄⋊Q₈	—	2	1	no	—
36	C₄×Q₈	—	2	1	no	—
37	C₄×Q₈	—	2	1	no	—
38	C₄×Q₈	—	2	1	no	—
39	C₄×Q₈	—	2	1	no	—
40	C₂×C₄⋊C₄	—	2	1	no	—
41	C₂×C₄⋊C₄	—	2	1	no	—
42	C₂×C₄⋊C₄	—	2	1	no	—
43	C₂×C₄⋊C₄	—	2	1	no	—
44	C₂×C₄⋊C₄	—	2	1	no	—
45	C₂×C₄⋊C₄	—	2	1	no	—
46	C₂×C₄⋊C₄	—	2	2	no	—
47	C₂×C₄⋊C₄	—	2	2	no	—
48	C₂×C₄⋊C₄	D₄	1	1	yes	characteristic
49	C₂×C₄⋊C₄	D₄	1	1	yes	characteristic
50	C₂×C₄⋊C₄	D₄	1	1	yes	characteristic
51	C₂×C₄²	—	2	1	no	—
52	C₂×C₄²	—	2	1	no	—
53	C₂×C₄²	C₂³	1	1	yes	characteristic
54	#106	—	4	2	no	—
55	C₄⋊C₈	—	2	1	no	—
56	C₄⋊C₈	—	2	1	no	—
57	C₄⋊C₈	—	2	1	no	—
58	C₄⋊C₈	D₄	1	2	yes	—
59	#103	—	4	4	no	—
60	#103	—	4	2	no	—
61	C₄×C₈	—	2	1	no	—
62	C₄×C₈	—	2	1	no	—
63	#95	—	2	2	no	—
64	#95	—	2	2	no	—
65	#95	—	2	1	no	—
66	C₂×Q₈	—	2	1	no	—
67	C₂×Q₈	—	2	1	no	—
68	C₂×Q₈	—	2	2	no	—
69	C₂×Q₈	—	2	1	no	—
70	C₂×Q₈	—	2	2	no	—
71	C₂×Q₈	—	2	1	no	—
72	C₂×Q₈	—	2	1	no	—
73	C₂×Q₈	—	4	2	no	—
74	C₂×Q₈	—	4	2	no	—
75	C₂×Q₈	SD₁₆	1	2	yes	—
76	C₂²×C₄	—	2	1	no	—
77	C₂²×C₄	—	2	1	no	—
78	C₂²×C₄	—	2	1	no	—
79	C₂²×C₄	—	2	2	no	—
80	C₂²×C₄	—	2	1	no	—
81	C₂²×C₄	—	4	1	no	—
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	C₂×C₈	—	2	2	no	—
86	C₂×C₈	—	2	2	no	—
87	C₂×C₈	—	4	1	no	—
88	C₂×C₈	—	4	1	no	—
89	C₂×C₈	—	4	1	no	—
90	C₂×C₈	—	4	1	no	—
91	C₂×C₈	—	4	1	no	—
92	C₂×C₈	—	4	1	no	—
93	C₂×C₈	—	4	1	no	—
94	C₄⋊C₄	—	2	1	no	—
95	C₄⋊C₄	—	2	1	no	—
96	C₄⋊C₄	—	2	1	no	—
97	C₄⋊C₄	—	2	1	no	—
98	C₄⋊C₄	—	4	2	no	—
99	C₄⋊C₄	—	4	1	no	—
100	C₄⋊C₄	—	4	1	no	—
101	C₄⋊C₄	—	4	1	no	—
102	C₄⋊C₄	—	4	1	no	—
103	C₄⋊C₄	—	4	1	no	—
104	C₄⋊C₄	—	4	1	no	—
105	C₄⋊C₄	—	4	1	no	—
106	C₄⋊C₄	—	4	1	no	—
107	C₄⋊C₄	—	4	1	no	—
108	C₄⋊C₄	—	4	1	no	—
109	C₄⋊C₄	—	8	2	no	—
110	C₄⋊C₄	—	8	2	no	—
111	C₄⋊C₄	SD₁₆	1	2	yes	—
112	C₄⋊C₄	Q₁₆	1	2	yes	—
113	C₄²	—	4	1	no	—
114	C₄²	—	4	1	no	—
115	C₄²	—	4	1	no	—
116	C₄²	—	4	1	no	—
117	C₄²	C₂×D₄	1	1	yes	characteristic
118	C₄²	#41	1	1	yes	characteristic
119	C₄²	#41	1	1	yes	characteristic
120	C₄²	#41	1	1	yes	characteristic
121	C₂³	#120	1	1	yes	characteristic
122	Q₈	—	2	2	no	—
123	Q₈	—	2	2	no	—
124	Q₈	—	4	4	no	—
125	Q₈	—	4	1	no	—
126	Q₈	—	4	1	no	—
127	Q₈	—	8	2	no	—
128	C₂×C₄	—	2	1	no	—
129	C₂×C₄	—	2	1	no	—
130	C₂×C₄	—	2	2	no	—
131	C₂×C₄	—	2	2	no	—
132	C₂×C₄	—	2	2	no	—
133	C₂×C₄	—	4	1	no	—
134	C₂×C₄	—	4	1	no	—
135	C₂×C₄	—	4	1	no	—
136	C₂×C₄	—	4	1	no	—
137	C₂×C₄	—	4	1	no	—
138	C₂×C₄	—	4	1	no	—
139	C₂×C₄	—	4	1	no	—
140	C₂×C₄	—	4	2	no	—
141	C₂×C₄	—	4	1	no	—
142	C₂×C₄	—	4	2	no	—
143	C₂×C₄	—	4	1	no	—
144	C₂×C₄	—	4	2	no	—
145	C₂×C₄	—	8	1	no	—
146	C₂×C₄	—	8	1	no	—
147	C₂×C₄	—	8	1	no	—
148	C₂×C₄	#121	1	1	yes	characteristic
149	C₂×C₄	#121	1	1	yes	characteristic
150	C₂×C₄	#121	1	1	yes	characteristic
151	C₂×C₄	#123	1	1	yes	characteristic
152	C₂×C₄	#123	1	1	yes	characteristic
153	C₂×C₄	#124	1	1	yes	characteristic
154	C₂×C₄	C₂×SD₁₆	1	1	yes	characteristic
155	C₂×C₄	C₂×SD₁₆	1	1	yes	characteristic
156	C₂×C₄	C₂×Q₁₆	1	1	yes	characteristic
157	C₂×C₄	#135	1	1	yes	characteristic
158	C₂×C₄	#137	1	1	yes	characteristic
159	C₂×C₄	#137	1	1	yes	characteristic
160	C₂×C₄	#137	1	1	yes	characteristic
161	C₂×C₄	#137	1	1	yes	characteristic
162	C₈	—	4	2	no	—
163	C₈	—	4	2	no	—
164	C₈	—	8	1	no	—
165	C₂²	#394	1	1	yes	characteristic
166	C₂²	#448	1	1	yes	characteristic
167	C₂²	#448	1	1	yes	characteristic
168	C₂²	#451	1	1	yes	characteristic
169	C₂²	#452	1	1	yes	characteristic
170	C₂²	#453	1	1	yes	characteristic
171	C₂²	#455	1	1	yes	characteristic
172	C₄	—	2	1	no	—
173	C₄	—	2	1	no	—
174	C₄	—	4	2	no	—
175	C₄	—	4	2	no	—
176	C₄	—	4	2	no	—
177	C₄	—	8	1	no	—
178	C₄	—	8	2	no	—
179	C₄	—	16	1	no	—
180	C₄	#461	1	1	yes	characteristic
181	C₄	#462	1	1	yes	characteristic
182	C₄	#465	1	1	yes	characteristic
183	C₄	#470	1	1	yes	characteristic
184	C₄	#483	1	1	yes	characteristic
185	C₄	#485	1	1	yes	characteristic
186	C₄	#487	1	1	yes	characteristic
187	C₄	#491	1	1	yes	characteristic
188	C₂	#1623	1	1	yes	characteristic
189	C₂	#1632	1	1	yes	characteristic
190	C₂	#1643	1	1	yes	characteristic
191	C₂	#1645	1	1	yes	characteristic
192	C₂	#2035	1	1	yes	characteristic
193	C₂	#2037	1	1	yes	characteristic
194	C₂	#2043	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³	#120
commutator, Frattini, cosocle	C₂×C₄²	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

Maximal subgroups

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

subgroup	index
#2567	2
#1956	2
#1948	2
#1803	2
C₂×#333	2
C₂×Q₈⋊C₈	2
#1279	2