Finite Group Databasebeta

Subgroups — #19706

Order 256.

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

normal subgroups only

This group has 335 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	#3230	C₂	1	1	yes	maximal, characteristic
2	#3229	C₂	1	1	yes	maximal, characteristic
3	#3229	C₂	1	1	yes	maximal, characteristic
4	#3209	C₂	1	1	yes	maximal, characteristic
5	#3160	C₂	1	1	yes	maximal, characteristic
6	C₂×#487	C₂	1	1	yes	maximal, characteristic
7	#1723	C₂	1	2	yes	maximal, complemented
8	#1718	C₂	1	2	yes	maximal, complemented
9	#1718	C₂	1	2	yes	maximal, complemented
10	#1716	C₂	1	2	yes	maximal, complemented
11	#1452	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	C₂×C₄⋊Q₈	C₂²	1	1	yes	characteristic
16	#500	C₂²	1	1	yes	characteristic
17	#500	C₂²	1	1	yes	characteristic
18	#500	C₂²	1	2	yes	—
19	#499	—	2	2	no	—
20	#498	C₂²	1	1	yes	characteristic
21	#498	C₂²	1	1	yes	characteristic
22	#498	C₂²	1	2	yes	—
23	#498	C₂²	1	2	yes	—
24	#494	C₂²	1	2	yes	—
25	#487	—	2	2	no	—
26	#487	C₂²	1	2	yes	—
27	#487	C₂²	1	2	yes	—
28	#487	C₂²	1	2	yes	—
29	#487	C₂²	1	2	yes	—
30	#487	C₂²	1	2	yes	—
31	#487	C₂²	1	2	yes	—
32	#484	—	2	2	no	—
33	#483	—	2	2	no	—
34	#483	—	2	2	no	—
35	#477	—	2	2	no	—
36	#475	—	2	2	no	—
37	#475	—	2	2	no	—
38	#462	—	2	2	no	—
39	#451	—	2	2	no	—
40	#431	C₂²	1	1	yes	characteristic
41	#431	C₂²	1	1	yes	characteristic
42	#427	—	2	2	no	—
43	C₂×#103	—	2	2	no	—
44	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
45	#322	C₂²	1	2	yes	—
46	#322	C₂²	1	2	yes	—
47	C₂²×Q₈	—	2	1	no	—
48	C₂²×Q₈	—	2	1	no	—
49	C₂×Q₁₆	—	2	4	no	—
50	C₂²×C₈	D₄	1	2	yes	—
51	C₄⋊Q₈	—	2	1	no	—
52	C₄⋊Q₈	—	2	1	no	—
53	C₄⋊Q₈	—	2	1	no	—
54	C₄⋊Q₈	—	2	1	no	—
55	C₄⋊Q₈	—	2	1	no	—
56	C₄⋊Q₈	—	2	1	no	—
57	C₄⋊Q₈	C₂³	1	2	yes	—
58	C₄⋊Q₈	C₂³	1	2	yes	—
59	C₄⋊Q₈	C₂³	1	1	yes	characteristic
60	C₄⋊Q₈	C₂³	1	1	yes	characteristic
61	C₄⋊Q₈	C₂³	1	1	yes	characteristic
62	C₄⋊Q₈	C₂³	1	1	yes	characteristic
63	#125	—	2	1	no	—
64	#125	—	2	1	no	—
65	C₂²⋊Q₈	—	2	1	no	—
66	C₂²⋊Q₈	—	2	1	no	—
67	C₂²⋊Q₈	—	2	1	no	—
68	C₂²⋊Q₈	—	2	1	no	—
69	C₄×Q₈	—	2	1	no	—
70	C₄×Q₈	—	2	1	no	—
71	C₄×Q₈	—	2	1	no	—
72	C₄×Q₈	—	2	1	no	—
73	#117	—	2	1	no	—
74	#117	—	2	1	no	—
75	C₂×C₄⋊C₄	—	2	1	no	—
76	C₂×C₄⋊C₄	—	2	1	no	—
77	C₂×C₄⋊C₄	—	2	1	no	—
78	C₂×C₄⋊C₄	—	2	1	no	—
79	C₂×C₄²	C₂³	1	1	yes	characteristic
80	#107	—	2	2	no	—
81	#107	—	2	2	no	—
82	#107	—	2	4	no	—
83	#106	—	2	2	no	—
84	#106	—	2	2	no	—
85	#106	—	2	4	no	—
86	#106	—	2	4	no	—
87	C₄⋊C₈	D₄	1	2	yes	—
88	C₄⋊C₈	D₄	1	2	yes	—
89	#103	—	2	4	no	—
90	#103	—	2	4	no	—
91	#103	—	2	4	no	—
92	#103	—	2	4	no	—
93	#103	—	2	4	no	—
94	#103	—	2	4	no	—
95	C₂²⋊C₈	D₄	1	2	yes	—
96	C₂²⋊C₈	D₄	1	2	yes	—
97	C₄×C₈	D₄	1	2	yes	—
98	C₄×C₈	C₂³	1	1	yes	characteristic
99	C₄×C₈	C₂³	1	1	yes	characteristic
100	C₄×C₈	C₂³	1	2	yes	—
101	C₄×C₈	C₂³	1	2	yes	—
102	C₂×Q₈	—	2	1	no	—
103	C₂×Q₈	—	2	1	no	—
104	C₂×Q₈	—	2	1	no	—
105	C₂×Q₈	—	2	2	no	—
106	C₂×Q₈	—	2	2	no	—
107	C₂×Q₈	—	2	1	no	—
108	C₂×Q₈	—	4	1	no	—
109	C₂×Q₈	—	4	1	no	—
110	C₂×Q₈	—	4	4	no	—
111	C₂×Q₈	—	8	2	no	—
112	C₂²×C₄	—	4	1	no	—
113	C₂²×C₄	—	4	1	no	—
114	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
115	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
116	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
117	Q₁₆	—	8	4	no	—
118	C₂×C₈	—	2	4	no	—
119	C₂×C₈	C₂×D₄	1	2	yes	—
120	C₂×C₈	C₂×D₄	1	2	yes	—
121	C₂×C₈	C₂×D₄	1	4	yes	—
122	C₂×C₈	C₂×D₄	1	4	yes	—
123	C₄⋊C₄	—	2	2	no	—
124	C₄⋊C₄	—	2	1	no	—
125	C₄⋊C₄	—	2	2	no	—
126	C₄⋊C₄	—	2	2	no	—
127	C₄⋊C₄	—	2	2	no	—
128	C₄⋊C₄	—	2	1	no	—
129	C₄⋊C₄	—	2	1	no	—
130	C₄⋊C₄	—	2	1	no	—
131	C₄⋊C₄	—	2	1	no	—
132	C₄⋊C₄	—	2	1	no	—
133	C₄⋊C₄	—	2	1	no	—
134	C₄⋊C₄	—	2	1	no	—
135	C₄⋊C₄	—	4	1	no	—
136	C₄⋊C₄	—	4	1	no	—
137	C₄⋊C₄	—	4	1	no	—
138	C₄⋊C₄	—	4	1	no	—
139	C₄⋊C₄	—	4	1	no	—
140	C₄⋊C₄	—	4	1	no	—
141	C₄⋊C₄	—	4	1	no	—
142	C₄⋊C₄	—	4	1	no	—
143	C₂²⋊C₄	—	4	1	no	—
144	C₂²⋊C₄	—	4	1	no	—
145	C₄²	—	4	1	no	—
146	C₄²	—	4	1	no	—
147	C₄²	C₂×D₄	1	1	yes	characteristic
148	C₄²	C₂×D₄	1	1	yes	characteristic
149	C₄²	C₂×D₄	1	1	yes	characteristic
150	C₄²	C₂⁴	1	1	yes	characteristic
151	C₂³	#127	1	1	yes	characteristic
152	Q₈	—	4	4	no	—
153	Q₈	—	4	2	no	—
154	Q₈	—	4	4	no	—
155	Q₈	—	4	2	no	—
156	Q₈	—	4	2	no	—
157	Q₈	—	4	2	no	—
158	Q₈	—	8	2	no	—
159	Q₈	—	8	2	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	2	2	no	—
162	C₂×C₄	—	4	1	no	—
163	C₂×C₄	—	4	1	no	—
164	C₂×C₄	—	4	1	no	—
165	C₂×C₄	—	4	1	no	—
166	C₂×C₄	—	4	2	no	—
167	C₂×C₄	—	4	2	no	—
168	C₂×C₄	—	8	2	no	—
169	C₂×C₄	—	8	2	no	—
170	C₂×C₄	#127	1	1	yes	characteristic
171	C₂×C₄	#127	1	1	yes	characteristic
172	C₂×C₄	#127	1	1	yes	characteristic
173	C₂×C₄	#137	1	4	yes	—
174	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
175	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
176	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
177	C₈	—	2	2	no	—
178	C₈	—	2	2	no	—
179	C₈	—	2	4	no	—
180	C₈	—	2	4	no	—
181	C₂²	—	2	1	no	—
182	C₂²	—	2	1	no	—
183	C₂²	#497	1	2	yes	—
184	C₂²	C₂×#127	1	1	yes	characteristic
185	C₄	—	2	1	no	—
186	C₄	—	2	1	no	—
187	C₄	—	8	2	no	—
188	C₄	—	8	2	no	—
189	C₄	—	8	1	no	—
190	C₄	—	8	1	no	—
191	C₄	—	8	1	no	—
192	C₄	—	8	1	no	—
193	C₄	#497	1	2	yes	—
194	C₄	C₂×#137	1	2	yes	—
195	C₄	#577	1	2	yes	—
196	C₄	#578	1	2	yes	—
197	C₂	—	2	2	no	—
198	C₂	C₂×#497	1	1	yes	characteristic
199	C₂	#3154	1	1	yes	characteristic
200	C₂	#3156	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂×#127
commutator, Frattini, cosocle	C₄²	C₂⁴
prime radical	C₂³	#127
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
7. #1723	C₂	semidirect
8. #1718	C₂	semidirect
9. #1718	C₂	semidirect
10. #1716	C₂	semidirect

Maximal subgroups

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

subgroup	index
#3230	2
#3229	2
#3209	2
#3160	2
C₂×#487	2
#1723	2
#1718	2
#1716	2
#1452	2