Finite Group Databasebeta

Subgroups — #19902

Order 256.

336 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 103 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	#3334	C₂	1	1	yes	maximal, characteristic, complemented
2	#3330	C₂	1	2	yes	maximal, complemented
3	#3283	C₂	1	1	yes	maximal, characteristic, complemented
4	#1707	C₂	1	1	yes	maximal, characteristic, complemented
5	#1699	C₂	1	2	yes	maximal, complemented
6	#1673	C₂	1	1	yes	maximal, characteristic, complemented
7	#1669	C₂	1	1	yes	maximal, characteristic, complemented
8	#1654	C₂	1	2	yes	maximal, complemented
9	#1654	C₂	1	2	yes	maximal, complemented
10	#1589	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×D₈	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#135	—	2	1	no	—
13	#547	C₂²	1	2	yes	—
14	#547	C₂²	1	2	yes	—
15	#500	C₂²	1	1	yes	characteristic
16	#498	C₂²	1	2	yes	—
17	#494	C₂²	1	1	yes	characteristic
18	#487	C₂²	1	2	yes	—
19	#484	—	2	2	no	—
20	#480	—	2	2	no	—
21	#474	C₂²	1	2	yes	—
22	#474	C₂²	1	2	yes	—
23	#468	—	2	2	no	—
24	#467	—	2	1	no	—
25	#466	—	2	1	no	—
26	#461	C₂²	1	2	yes	—
27	#461	C₂²	1	2	yes	—
28	#452	—	2	2	no	—
29	#449	—	2	2	no	—
30	C₄×D₈	C₂²	1	1	yes	characteristic
31	C₄×D₈	C₂²	1	1	yes	characteristic
32	#436	C₂²	1	2	yes	—
33	C₈×D₄	C₂²	1	1	yes	characteristic
34	C₈×D₄	C₂²	1	1	yes	characteristic
35	C₂×#107	—	2	1	no	—
36	#417	—	2	2	no	—
37	#335	C₂²	1	1	yes	characteristic
38	#335	C₂²	1	1	yes	characteristic
39	#332	C₂²	1	2	yes	—
40	#332	C₂²	1	2	yes	—
41	D₄⋊C₈	C₂²	1	1	yes	characteristic
42	D₄⋊C₈	C₂²	1	1	yes	characteristic
43	D₄⋊C₈	C₂²	1	2	yes	—
44	C₈²	C₂²	1	1	yes	characteristic
45	C₂×#41	—	2	2	no	—
46	C₂×#41	—	2	2	no	—
47	#135	—	4	4	no	—
48	C₂×Q₁₆	—	2	1	no	—
49	C₂×Q₁₆	—	2	1	no	—
50	C₂×SD₁₆	—	2	2	no	—
51	C₂×SD₁₆	—	2	2	no	—
52	C₂×D₈	D₄	1	1	yes	characteristic
53	C₂×D₈	D₄	1	1	yes	characteristic
54	C₂×M₁₆	—	2	2	no	—
55	C₂²×C₈	—	2	1	no	—
56	C₂²×C₈	—	2	1	no	—
57	C₄⋊Q₈	C₂³	1	2	yes	—
58	C₄⋊Q₈	C₂³	1	2	yes	—
59	#123	—	4	2	no	—
60	#123	—	4	2	no	—
61	C₂²⋊Q₈	—	2	2	no	—
62	C₂²⋊Q₈	—	2	2	no	—
63	#121	—	2	2	no	—
64	#121	—	2	2	no	—
65	C₄×D₄	C₂³	1	1	yes	characteristic
66	C₄×D₄	C₂³	1	1	yes	characteristic
67	C₄×D₄	C₂³	1	2	yes	—
68	C₂×C₄⋊C₄	—	2	2	no	—
69	C₂×C₄⋊C₄	—	2	2	no	—
70	#107	—	2	1	no	—
71	#107	—	2	1	no	—
72	#107	—	2	2	no	—
73	#107	—	2	2	no	—
74	#107	D₄	1	1	yes	characteristic
75	#107	D₄	1	1	yes	characteristic
76	#106	—	4	2	no	—
77	C₄⋊C₈	C₂³	1	1	yes	characteristic
78	C₄⋊C₈	C₂³	1	1	yes	characteristic
79	C₄⋊C₈	C₂³	1	2	yes	—
80	#103	—	2	2	no	—
81	#103	—	2	2	no	—
82	#103	—	2	2	no	—
83	#103	—	2	2	no	—
84	#102	—	2	2	no	—
85	#102	—	2	2	no	—
86	#102	D₄	1	1	yes	characteristic
87	#102	D₄	1	2	yes	—
88	#102	D₄	1	1	yes	characteristic
89	C₂²⋊C₈	—	2	1	no	—
90	C₂²⋊C₈	—	2	1	no	—
91	C₂²⋊C₈	—	2	2	no	—
92	C₄×C₈	C₂³	1	1	yes	characteristic
93	C₄×C₈	C₂³	1	1	yes	characteristic
94	C₄×C₈	C₂³	1	1	yes	characteristic
95	#41	—	4	4	no	—
96	#41	—	4	4	no	—
97	#41	—	8	2	no	—
98	#41	—	8	2	no	—
99	C₂×Q₈	—	2	2	no	—
100	C₂×Q₈	—	2	2	no	—
101	C₂×D₄	—	4	2	no	—
102	C₂×D₄	—	4	2	no	—
103	C₂×D₄	C₂×D₄	1	2	yes	—
104	C₂×D₄	C₂×D₄	1	1	yes	characteristic
105	C₂×D₄	C₂×D₄	1	1	yes	characteristic
106	C₂²×C₄	—	2	1	no	—
107	C₂²×C₄	—	2	2	no	—
108	C₂²×C₄	—	2	1	no	—
109	C₂²×C₄	—	4	2	no	—
110	C₂²×C₄	—	4	2	no	—
111	Q₁₆	—	4	2	no	—
112	Q₁₆	—	8	1	no	—
113	SD₁₆	—	4	4	no	—
114	SD₁₆	—	8	2	no	—
115	D₈	—	2	2	no	—
116	D₈	—	2	2	no	—
117	M₁₆	—	8	2	no	—
118	C₂×C₈	—	2	1	no	—
119	C₂×C₈	—	2	1	no	—
120	C₂×C₈	—	2	2	no	—
121	C₂×C₈	—	2	1	no	—
122	C₂×C₈	—	4	2	no	—
123	C₂×C₈	—	4	2	no	—
124	C₂×C₈	C₂×D₄	1	1	yes	characteristic
125	C₂×C₈	C₂×D₄	1	1	yes	characteristic
126	C₂×C₈	C₂×D₄	1	1	yes	characteristic
127	C₂×C₈	C₂×D₄	1	1	yes	characteristic
128	C₄⋊C₄	—	2	2	no	—
129	C₄⋊C₄	—	2	2	no	—
130	C₄⋊C₄	—	2	2	no	—
131	C₄⋊C₄	—	2	2	no	—
132	C₄⋊C₄	—	4	2	no	—
133	C₄⋊C₄	—	4	2	no	—
134	C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
135	C₄⋊C₄	C₂×D₄	1	2	yes	—
136	C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
137	C₂²⋊C₄	—	2	1	no	—
138	C₂²⋊C₄	—	2	2	no	—
139	C₂²⋊C₄	—	2	1	no	—
140	C₂²⋊C₄	—	4	2	no	—
141	C₂²⋊C₄	—	4	2	no	—
142	C₄²	C₂⁴	1	1	yes	characteristic
143	C₂³	—	2	1	no	—
144	C₂³	—	2	1	no	—
145	C₂³	—	2	2	no	—
146	Q₈	—	4	2	no	—
147	Q₈	—	4	2	no	—
148	Q₈	—	4	2	no	—
149	Q₈	—	4	2	no	—
150	D₄	—	2	1	no	—
151	D₄	—	2	2	no	—
152	D₄	—	2	1	no	—
153	D₄	—	2	1	no	—
154	D₄	—	2	2	no	—
155	D₄	—	8	4	no	—
156	D₄	—	8	4	no	—
157	D₄	#135	1	2	yes	complemented
158	C₂×C₄	—	2	1	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	2	1	no	—
161	C₂×C₄	—	4	2	no	—
162	C₂×C₄	—	4	1	no	—
163	C₂×C₄	—	4	1	no	—
164	C₂×C₄	—	4	2	no	—
165	C₂×C₄	—	4	1	no	—
166	C₂×C₄	—	4	1	no	—
167	C₂×C₄	—	4	2	no	—
168	C₂×C₄	—	4	2	no	—
169	C₂×C₄	—	8	4	no	—
170	C₂×C₄	—	8	4	no	—
171	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
172	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
173	C₂×C₄	H(5,𝔽₂)	1	1	yes	characteristic
174	C₈	—	2	2	no	—
175	C₈	—	2	1	no	—
176	C₈	—	2	1	no	—
177	C₈	—	2	1	no	—
178	C₈	—	4	1	no	—
179	C₈	—	4	1	no	—
180	C₈	—	4	2	no	—
181	C₈	AGL(1,ℤ/8)	1	2	yes	—
182	C₂²	—	2	2	no	—
183	C₂²	—	4	1	no	—
184	C₂²	—	4	1	no	—
185	C₂²	—	4	1	no	—
186	C₂²	—	4	2	no	—
187	C₂²	—	4	2	no	—
188	C₂²	—	4	1	no	—
189	C₂²	—	4	1	no	—
190	C₂²	—	4	2	no	—
191	C₂²	D₄²	1	1	yes	characteristic
192	C₄	—	2	1	no	—
193	C₄	—	4	1	no	—
194	C₄	—	4	1	no	—
195	C₄	—	4	2	no	—
196	C₄	—	8	2	no	—
197	C₄	—	8	2	no	—
198	C₄	C₂×#135	1	1	yes	characteristic
199	C₄	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
200	C₄	#576	1	1	yes	characteristic
201	C₄	#577	1	1	yes	characteristic
202	C₂	—	4	2	no	—
203	C₂	—	8	1	no	—
204	C₂	—	8	2	no	—
205	C₂	#3272	1	1	yes	characteristic
206	C₂	#3280	1	1	yes	characteristic
207	C₂	#3293	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	D₄²
commutator, Frattini, cosocle	C₄²	C₂⁴
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #3334	C₂	semidirect
2. #3330	C₂	semidirect
3. #3283	C₂	semidirect
4. #1707	C₂	semidirect
5. #1699	C₂	semidirect
6. #1673	C₂	semidirect
7. #1669	C₂	semidirect
8. #1654	C₂	semidirect
9. #1654	C₂	semidirect
10. #1589	C₂	semidirect
11. C₈×D₈	C₂	semidirect
157. D₄	#135	semidirect

Maximal subgroups

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

subgroup	index
#3334	2
#3330	2
#3283	2
#1707	2
#1699	2
#1673	2
#1669	2
#1654	2
#1589	2
C₈×D₈	2