Finite Group Databasebeta

Subgroups — #19214

Order 256.

329 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 329 conjugacy classes of subgroups — too many to draw. Only its 115 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	#3209	C₂	1	1	yes	maximal, characteristic, complemented
2	#3208	C₂	1	1	yes	maximal, characteristic
3	C₂×#486	C₂	1	1	yes	maximal, characteristic
4	#2960	C₂	1	1	yes	maximal, characteristic, complemented
5	#1702	C₂	1	4	yes	maximal, complemented
6	#1682	C₂	1	4	yes	maximal, complemented
7	#1564	C₂	1	1	yes	maximal, characteristic, complemented
8	#1549	C₂	1	1	yes	maximal, characteristic, complemented
9	#1545	C₂	1	1	yes	maximal, characteristic
10	#533	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊Q₈	C₂²	1	1	yes	characteristic
12	C₂×#127	C₂²	1	1	yes	characteristic
13	#494	—	2	1	no	—
14	#492	—	2	1	no	—
15	#487	—	2	1	no	—
16	#486	—	2	1	no	—
17	#486	—	2	1	no	—
18	#486	—	2	1	no	—
19	#486	C₂²	1	2	yes	—
20	#486	C₂²	1	2	yes	—
21	#462	C₂²	1	4	yes	complemented
22	#460	C₂²	1	4	yes	—
23	#451	—	2	2	no	—
24	#450	—	2	2	no	—
25	#446	—	2	1	no	—
26	#446	C₂²	1	2	yes	complemented
27	C₈×Q₈	—	2	1	no	—
28	C₈×Q₈	C₂²	1	2	yes	complemented
29	#431	C₂²	1	1	yes	characteristic
30	#431	C₂²	1	1	yes	characteristic
31	#424	—	2	1	no	—
32	#423	C₂²	1	1	yes	characteristic
33	C₂×#102	—	2	1	no	—
34	C₂×#102	—	2	1	no	—
35	#405	—	2	1	no	—
36	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
37	#335	C₂²	1	2	yes	complemented
38	#334	C₂²	1	2	yes	complemented
39	#332	C₂²	1	4	yes	complemented
40	#331	C₂²	1	4	yes	—
41	C₂²×Q₈	—	2	1	no	—
42	C₂²×D₄	—	2	1	no	—
43	C₂²×D₄	—	2	1	no	—
44	C₂²×D₄	—	2	1	no	—
45	C₂×Q₁₆	—	2	4	no	—
46	C₂×SD₁₆	—	2	4	no	—
47	C₂×M₁₆	—	2	1	no	—
48	C₂²×C₈	D₄	1	1	yes	characteristic
49	C₂²×C₈	D₄	1	1	yes	characteristic
50	C₄⋊Q₈	—	2	1	no	—
51	C₄⋊Q₈	—	2	1	no	—
52	C₄⋊Q₈	—	2	1	no	—
53	C₄⋊Q₈	D₄	1	1	yes	characteristic
54	C₄⋊Q₈	D₄	1	1	yes	characteristic
55	C₄⋊Q₈	C₂³	1	2	yes	—
56	#127	—	2	1	no	—
57	#127	—	2	1	no	—
58	#127	—	2	1	no	—
59	#127	C₂³	1	2	yes	—
60	#125	—	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	C₂³	1	2	yes	—
65	#117	—	2	1	no	—
66	C₂×C₄⋊C₄	—	2	1	no	—
67	C₂×C₄⋊C₄	—	2	1	no	—
68	C₂×C₄²	C₂³	1	1	yes	characteristic
69	C₄⋊C₈	—	2	2	no	—
70	C₄⋊C₈	—	2	2	no	—
71	C₄⋊C₈	—	2	1	no	—
72	C₄⋊C₈	C₂³	1	2	yes	—
73	C₄⋊C₈	C₂³	1	2	yes	—
74	C₄⋊C₈	C₂³	1	2	yes	—
75	#103	—	2	4	no	—
76	#102	—	2	2	no	—
77	#102	—	2	2	no	—
78	#102	—	2	2	no	—
79	#102	—	2	2	no	—
80	#102	—	2	4	no	—
81	C₂²⋊C₈	—	2	1	no	—
82	C₂²⋊C₈	—	2	1	no	—
83	#97	—	2	2	no	—
84	C₄×C₈	—	2	2	no	—
85	C₄×C₈	—	2	1	no	—
86	C₄×C₈	—	2	1	no	—
87	C₄×C₈	D₄	1	1	yes	characteristic
88	C₄×C₈	D₄	1	1	yes	characteristic
89	C₄×C₈	C₂³	1	1	yes	characteristic
90	C₄×C₈	C₂³	1	1	yes	characteristic
91	C₂⁴	—	4	1	no	—
92	C₂×Q₈	—	2	1	no	—
93	C₂×Q₈	—	2	2	no	—
94	C₂×Q₈	—	4	1	no	—
95	C₂×Q₈	—	8	2	no	—
96	C₂×Q₈	C₂×D₄	1	2	yes	—
97	C₂×D₄	—	2	2	no	—
98	C₂×D₄	—	2	2	no	—
99	C₂×D₄	—	2	2	no	—
100	C₂×D₄	—	4	1	no	—
101	C₂×D₄	—	4	1	no	—
102	C₂×D₄	—	4	1	no	—
103	C₂×D₄	—	4	2	no	—
104	C₂×D₄	—	4	2	no	—
105	C₂×D₄	—	8	2	no	—
106	C₂×D₄	—	8	2	no	—
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	4	no	—
112	SD₁₆	—	8	4	no	—
113	M₁₆	—	4	2	no	—
114	C₂×C₈	—	2	2	no	—
115	C₂×C₈	—	2	2	no	—
116	C₂×C₈	—	2	2	no	—
117	C₂×C₈	—	2	2	no	—
118	C₂×C₈	—	2	2	no	—
119	C₂×C₈	C₂×D₄	1	1	yes	characteristic
120	C₂×C₈	C₂×D₄	1	1	yes	characteristic
121	C₂×C₈	C₂×D₄	1	1	yes	characteristic
122	C₂×C₈	C₂×D₄	1	1	yes	characteristic
123	C₄⋊C₄	—	2	1	no	—
124	C₄⋊C₄	—	2	1	no	—
125	C₄⋊C₄	—	2	1	no	—
126	C₄⋊C₄	—	2	2	no	—
127	C₄⋊C₄	—	2	2	no	—
128	C₄⋊C₄	—	2	2	no	—
129	C₄⋊C₄	—	2	2	no	—
130	C₄⋊C₄	—	4	1	no	—
131	C₄⋊C₄	—	4	1	no	—
132	C₄⋊C₄	C₂×D₄	1	2	yes	—
133	C₂²⋊C₄	—	2	2	no	—
134	C₄²	—	2	2	no	—
135	C₄²	C₂×D₄	1	1	yes	characteristic
136	C₄²	C₂×D₄	1	1	yes	characteristic
137	C₄²	#41	1	1	yes	characteristic
138	C₄²	C₂⁴	1	1	yes	characteristic
139	C₂³	—	4	2	no	—
140	C₂³	—	8	2	no	—
141	C₂³	—	8	2	no	—
142	C₂³	—	8	2	no	—
143	C₂³	#121	1	1	yes	characteristic
144	Q₈	—	2	4	no	—
145	Q₈	—	4	2	no	—
146	Q₈	—	4	4	no	—
147	Q₈	—	8	2	no	—
148	D₄	—	4	2	no	—
149	D₄	—	4	2	no	—
150	D₄	—	4	4	no	—
151	D₄	—	8	2	no	—
152	D₄	—	8	2	no	—
153	D₄	—	8	2	no	—
154	D₄	—	8	2	no	—
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	4	2	no	—
160	C₂×C₄	—	8	2	no	—
161	C₂×C₄	#121	1	1	yes	characteristic
162	C₂×C₄	#121	1	1	yes	characteristic
163	C₂×C₄	#121	1	1	yes	characteristic
164	C₂×C₄	#135	1	2	yes	—
165	C₂×C₄	#137	1	2	yes	—
166	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
167	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
168	C₂×C₄	C₂×#41	1	1	yes	characteristic
169	C₈	—	2	1	no	—
170	C₈	—	2	1	no	—
171	C₈	—	2	1	no	—
172	C₈	—	2	1	no	—
173	C₈	—	4	2	no	—
174	C₈	—	4	2	no	—
175	C₈	—	4	2	no	—
176	C₂²	—	2	1	no	—
177	C₂²	—	2	1	no	—
178	C₂²	—	8	2	no	—
179	C₂²	—	8	2	no	—
180	C₂²	—	8	2	no	—
181	C₂²	—	16	4	no	—
182	C₂²	#464	1	2	yes	—
183	C₂²	C₂×#121	1	1	yes	characteristic
184	C₄	—	2	1	no	—
185	C₄	—	2	1	no	—
186	C₄	—	4	2	no	—
187	C₄	—	4	2	no	—
188	C₄	—	8	2	no	—
189	C₄	#464	1	2	yes	—
190	C₄	#471	1	2	yes	—
191	C₄	C₂×#135	1	1	yes	characteristic
192	C₄	C₂×#137	1	1	yes	characteristic
193	C₄	#576	1	2	yes	—
194	C₂	—	2	2	no	—
195	C₂	—	16	2	no	—
196	C₂	C₂×#464	1	1	yes	characteristic
197	C₂	#3058	1	1	yes	characteristic
198	C₂	C₂×#471	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂×#121
commutator, Frattini, cosocle	C₄²	C₂⁴
prime radical	C₂×#127	C₂²
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
1. #3209	C₂	semidirect
4. #2960	C₂	semidirect
5. #1702	C₂	semidirect
6. #1682	C₂	semidirect
7. #1564	C₂	semidirect
8. #1549	C₂	semidirect
21. #462	C₂²	semidirect
26. #446	C₂²	semidirect
28. C₈×Q₈	C₂²	semidirect
37. #335	C₂²	semidirect
38. #334	C₂²	semidirect
39. #332	C₂²	semidirect

Maximal subgroups

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

subgroup	index
#3209	2
#3208	2
C₂×#486	2
#2960	2
#1702	2
#1682	2
#1564	2
#1549	2
#1545	2