Finite Group Databasebeta

Subgroups — #19523

Order 256.

274 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 274 conjugacy classes of subgroups — too many to draw. Only its 105 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	#3237	C₂	1	1	yes	maximal, characteristic
2	#3229	C₂	1	1	yes	maximal, characteristic
3	#3209	C₂	1	1	yes	maximal, characteristic
4	#2985	C₂	1	1	yes	maximal, characteristic
5	#1704	C₂	1	2	yes	maximal, complemented
6	#1690	C₂	1	2	yes	maximal, complemented
7	#1684	C₂	1	2	yes	maximal, complemented
8	#1670	C₂	1	2	yes	maximal, complemented
9	#1572	C₂	1	1	yes	maximal, characteristic
10	#1556	C₂	1	1	yes	maximal, characteristic
11	#1554	C₂	1	1	yes	maximal, characteristic
12	#533	C₂²	1	1	yes	characteristic
13	#533	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊Q₈	C₂²	1	1	yes	characteristic
15	#501	—	2	1	no	—
16	#498	—	2	1	no	—
17	#494	C₂²	1	1	yes	characteristic
18	#494	C₂²	1	1	yes	characteristic
19	#491	—	2	1	no	—
20	#487	—	2	1	no	—
21	#487	C₂²	1	1	yes	characteristic
22	#487	C₂²	1	1	yes	characteristic
23	#485	—	2	2	no	—
24	#483	—	2	2	no	—
25	#477	C₂²	1	2	yes	—
26	#475	C₂²	1	2	yes	—
27	#475	C₂²	1	2	yes	—
28	#475	C₂²	1	2	yes	—
29	#462	—	2	2	no	—
30	#451	—	2	2	no	—
31	#446	—	2	2	no	—
32	#446	C₂²	1	2	yes	—
33	C₈×Q₈	C₂²	1	2	yes	—
34	#433	—	2	2	no	—
35	#432	C₂²	1	1	yes	characteristic
36	#432	C₂²	1	1	yes	characteristic
37	#431	C₂²	1	1	yes	characteristic
38	#431	C₂²	1	1	yes	characteristic
39	#335	C₂²	1	2	yes	—
40	#334	C₂²	1	2	yes	—
41	#333	C₂²	1	2	yes	—
42	#332	C₂²	1	2	yes	—
43	#332	C₂²	1	2	yes	—
44	#332	C₂²	1	2	yes	—
45	C₂²×Q₈	—	2	1	no	—
46	C₂×Q₁₆	—	2	2	no	—
47	C₂×Q₁₆	—	2	2	no	—
48	C₄⋊Q₈	—	2	1	no	—
49	C₄⋊Q₈	—	2	1	no	—
50	C₄⋊Q₈	—	2	1	no	—
51	C₄⋊Q₈	D₄	1	1	yes	characteristic
52	C₄⋊Q₈	D₄	1	1	yes	characteristic
53	C₄⋊Q₈	C₂³	1	1	yes	characteristic
54	C₄⋊Q₈	C₂³	1	1	yes	characteristic
55	C₄⋊Q₈	C₂³	1	1	yes	characteristic
56	C₄⋊Q₈	C₂³	1	1	yes	characteristic
57	#125	—	2	1	no	—
58	#125	—	2	1	no	—
59	C₂²⋊Q₈	—	2	1	no	—
60	C₂²⋊Q₈	—	2	1	no	—
61	C₂²⋊Q₈	—	2	1	no	—
62	C₂²⋊Q₈	D₄	1	2	yes	—
63	C₄×Q₈	—	2	1	no	—
64	C₄×Q₈	—	2	1	no	—
65	C₄×Q₈	—	2	1	no	—
66	C₄×Q₈	C₂³	1	2	yes	—
67	#117	—	2	1	no	—
68	#117	—	2	1	no	—
69	C₂×C₄⋊C₄	—	2	1	no	—
70	C₂×C₄⋊C₄	—	2	1	no	—
71	C₂×C₄²	C₂³	1	1	yes	characteristic
72	#107	—	2	2	no	—
73	#106	—	2	2	no	—
74	#106	—	2	2	no	—
75	#106	—	2	2	no	—
76	C₄⋊C₈	—	2	2	no	—
77	C₄⋊C₈	—	2	2	no	—
78	C₄⋊C₈	D₄	1	2	yes	—
79	C₄⋊C₈	C₂³	1	2	yes	—
80	C₄⋊C₈	C₂³	1	2	yes	—
81	C₄⋊C₈	C₂³	1	2	yes	—
82	#103	—	2	2	no	—
83	#103	—	2	2	no	—
84	#103	—	2	2	no	—
85	#103	—	2	2	no	—
86	#103	—	2	2	no	—
87	#103	—	2	2	no	—
88	C₂²⋊C₈	—	2	1	no	—
89	C₂²⋊C₈	—	2	1	no	—
90	C₂²⋊C₈	—	2	1	no	—
91	C₂²⋊C₈	D₄	1	2	yes	—
92	#97	—	2	2	no	—
93	#97	—	2	1	no	—
94	#97	—	2	1	no	—
95	C₄×C₈	—	2	2	no	—
96	C₄×C₈	—	2	1	no	—
97	C₄×C₈	C₂³	1	1	yes	characteristic
98	C₄×C₈	C₂³	1	1	yes	characteristic
99	C₂×Q₈	—	2	1	no	—
100	C₂×Q₈	—	2	1	no	—
101	C₂×Q₈	—	2	1	no	—
102	C₂×Q₈	—	2	1	no	—
103	C₂×Q₈	—	2	1	no	—
104	C₂×Q₈	—	4	1	no	—
105	C₂×Q₈	—	8	2	no	—
106	C₂×Q₈	C₂×D₄	1	2	yes	—
107	C₂²×C₄	—	4	1	no	—
108	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
109	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
110	C₂²×C₄	#41	1	1	yes	characteristic
111	Q₁₆	—	8	2	no	—
112	Q₁₆	—	8	2	no	—
113	C₂×C₈	—	2	2	no	—
114	C₂×C₈	—	2	2	no	—
115	C₂×C₈	—	2	2	no	—
116	C₂×C₈	C₂×D₄	1	2	yes	—
117	C₂×C₈	C₂×D₄	1	2	yes	—
118	C₄⋊C₄	—	2	1	no	—
119	C₄⋊C₄	—	2	1	no	—
120	C₄⋊C₄	—	2	1	no	—
121	C₄⋊C₄	—	2	2	no	—
122	C₄⋊C₄	—	2	2	no	—
123	C₄⋊C₄	—	2	1	no	—
124	C₄⋊C₄	—	2	1	no	—
125	C₄⋊C₄	—	2	1	no	—
126	C₄⋊C₄	—	2	1	no	—
127	C₄⋊C₄	—	2	1	no	—
128	C₄⋊C₄	—	2	1	no	—
129	C₄⋊C₄	—	2	1	no	—
130	C₄⋊C₄	—	2	1	no	—
131	C₄⋊C₄	—	4	1	no	—
132	C₄⋊C₄	—	4	1	no	—
133	C₄⋊C₄	—	4	1	no	—
134	C₄⋊C₄	—	4	1	no	—
135	C₄⋊C₄	C₂×D₄	1	2	yes	—
136	C₂²⋊C₄	—	2	2	no	—
137	C₂²⋊C₄	—	4	1	no	—
138	C₄²	—	2	2	no	—
139	C₄²	—	4	1	no	—
140	C₄²	C₂×D₄	1	1	yes	characteristic
141	C₄²	C₂×D₄	1	1	yes	characteristic
142	C₄²	#41	1	1	yes	characteristic
143	C₄²	C₂⁴	1	1	yes	characteristic
144	C₂³	#121	1	1	yes	characteristic
145	Q₈	—	2	4	no	—
146	Q₈	—	4	2	no	—
147	Q₈	—	4	2	no	—
148	Q₈	—	4	2	no	—
149	Q₈	—	8	1	no	—
150	Q₈	—	8	1	no	—
151	Q₈	—	8	2	no	—
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	—	2	2	no	—
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	—	4	1	no	—
157	C₂×C₄	—	4	1	no	—
158	C₂×C₄	—	4	1	no	—
159	C₂×C₄	—	4	1	no	—
160	C₂×C₄	—	4	1	no	—
161	C₂×C₄	—	8	2	no	—
162	C₂×C₄	#121	1	1	yes	characteristic
163	C₂×C₄	#121	1	1	yes	characteristic
164	C₂×C₄	#121	1	1	yes	characteristic
165	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
166	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
167	C₂×C₄	C₂×#41	1	1	yes	characteristic
168	C₈	—	2	2	no	—
169	C₈	—	2	2	no	—
170	C₈	—	4	2	no	—
171	C₈	—	4	2	no	—
172	C₈	—	4	2	no	—
173	C₂²	—	2	1	no	—
174	C₂²	—	2	1	no	—
175	C₂²	—	2	1	no	—
176	C₂²	C₂×#121	1	1	yes	characteristic
177	C₄	—	2	1	no	—
178	C₄	—	2	1	no	—
179	C₄	—	2	1	no	—
180	C₄	—	2	1	no	—
181	C₄	—	4	2	no	—
182	C₄	—	4	2	no	—
183	C₄	—	8	1	no	—
184	C₄	—	8	1	no	—
185	C₄	—	8	1	no	—
186	C₄	—	8	1	no	—
187	C₄	#577	1	2	yes	—
188	C₄	#578	1	2	yes	—
189	C₂	—	4	1	no	—
190	C₂	#3043	1	1	yes	characteristic
191	C₂	#3059	1	1	yes	characteristic
192	C₂	#3069	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂×#121
commutator, Frattini, cosocle	C₄²	C₂⁴
prime radical	C₂³	#121
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
5. #1704	C₂	semidirect
6. #1690	C₂	semidirect
7. #1684	C₂	semidirect
8. #1670	C₂	semidirect

Maximal subgroups

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

subgroup	index
#3237	2
#3229	2
#3209	2
#2985	2
#1704	2
#1690	2
#1684	2
#1670	2
#1572	2
#1556	2
#1554	2