Finite Group Databasebeta

Subgroups — #19898

Order 256.

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

normal subgroups only

This group has 362 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	#3341	C₂	1	1	yes	maximal, characteristic, complemented
2	#3325	C₂	1	2	yes	maximal, complemented
3	D₄×D₈	C₂	1	1	yes	maximal, characteristic, complemented
4	#1706	C₂	1	1	yes	maximal, characteristic, complemented
5	#1697	C₂	1	2	yes	maximal, complemented
6	#1675	C₂	1	1	yes	maximal, characteristic, complemented
7	#1667	C₂	1	1	yes	maximal, characteristic, complemented
8	#1629	C₂	1	2	yes	maximal, complemented
9	#1624	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	C₂²×D₈	—	2	1	no	—
13	#555	C₂²	1	2	yes	complemented
14	D₄²	C₂²	1	2	yes	—
15	#500	C₂²	1	1	yes	characteristic, complemented
16	#493	C₂²	1	1	yes	characteristic
17	#492	C₂²	1	2	yes	complemented
18	#486	C₂²	1	2	yes	complemented
19	#482	—	2	2	no	—
20	#482	—	2	2	no	—
21	#476	C₂²	1	2	yes	—
22	#476	C₂²	1	2	yes	complemented
23	#469	—	2	2	no	—
24	#466	—	2	1	no	—
25	#466	—	2	1	no	—
26	#459	C₂²	1	2	yes	—
27	#459	C₂²	1	2	yes	complemented
28	#447	—	2	2	no	—
29	#447	—	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	complemented
33	C₈×D₄	C₂²	1	1	yes	characteristic
34	C₈×D₄	C₂²	1	1	yes	characteristic, complemented
35	#427	—	2	1	no	—
36	#418	—	2	2	no	—
37	#335	C₂²	1	1	yes	characteristic, complemented
38	#335	C₂²	1	1	yes	characteristic, complemented
39	#333	C₂²	1	2	yes	complemented
40	#331	C₂²	1	2	yes	complemented
41	D₄⋊C₈	C₂²	1	1	yes	characteristic
42	D₄⋊C₈	C₂²	1	1	yes	characteristic, complemented
43	D₄⋊C₈	C₂²	1	2	yes	complemented
44	C₈²	C₂²	1	1	yes	characteristic, complemented
45	C₂²×D₄	—	2	2	no	—
46	C₂²×D₄	—	2	2	no	—
47	C₂×SD₁₆	—	4	2	no	—
48	C₂×D₈	—	2	2	no	—
49	C₂×D₈	—	2	2	no	—
50	C₂×D₈	—	2	1	no	—
51	C₂×D₈	—	2	1	no	—
52	C₂×D₈	—	4	4	no	—
53	C₂×D₈	D₄	1	1	yes	characteristic
54	C₂×D₈	D₄	1	1	yes	characteristic
55	C₂×M₁₆	—	2	2	no	—
56	C₂²×C₈	—	2	1	no	—
57	C₂²×C₈	—	2	1	no	—
58	C₄⋊Q₈	C₂³	1	2	yes	—
59	#127	C₂³	1	2	yes	—
60	#124	—	4	2	no	—
61	#121	—	2	2	no	—
62	#121	—	2	2	no	—
63	#121	—	2	2	no	—
64	#121	—	2	2	no	—
65	#120	—	4	2	no	—
66	C₄×D₄	C₂³	1	1	yes	characteristic
67	C₄×D₄	C₂³	1	2	yes	—
68	C₄×D₄	C₂³	1	1	yes	characteristic
69	#117	—	2	2	no	—
70	#117	—	2	2	no	—
71	#107	—	2	1	no	—
72	#107	—	2	1	no	—
73	#107	D₄	1	1	yes	characteristic
74	#107	D₄	1	1	yes	characteristic
75	#106	—	2	2	no	—
76	#106	—	2	2	no	—
77	C₄⋊C₈	C₂³	1	1	yes	characteristic
78	C₄⋊C₈	C₂³	1	2	yes	—
79	C₄⋊C₈	C₂³	1	1	yes	characteristic
80	#102	—	2	2	no	—
81	#102	—	2	2	no	—
82	#102	—	2	2	no	—
83	#102	—	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	C₂⁴	—	4	2	no	—
96	C₂×Q₈	—	2	2	no	—
97	C₂×D₄	—	2	2	no	—
98	C₂×D₄	—	2	2	no	—
99	C₂×D₄	—	2	2	no	—
100	C₂×D₄	—	4	2	no	—
101	C₂×D₄	—	4	2	no	—
102	C₂×D₄	—	4	4	no	—
103	C₂×D₄	—	4	4	no	—
104	C₂×D₄	—	4	2	no	—
105	C₂×D₄	—	8	2	no	—
106	C₂×D₄	—	8	2	no	—
107	C₂×D₄	C₂×D₄	1	2	yes	—
108	C₂×D₄	C₂×D₄	1	1	yes	characteristic
109	C₂×D₄	C₂×D₄	1	1	yes	characteristic
110	C₂²×C₄	—	2	1	no	—
111	C₂²×C₄	—	2	2	no	—
112	C₂²×C₄	—	2	1	no	—
113	SD₁₆	—	8	4	no	—
114	D₈	—	2	2	no	—
115	D₈	—	2	2	no	—
116	D₈	—	4	4	no	—
117	D₈	—	4	2	no	—
118	D₈	—	8	1	no	—
119	D₈	—	8	2	no	—
120	M₁₆	—	8	2	no	—
121	C₂×C₈	—	2	1	no	—
122	C₂×C₈	—	2	1	no	—
123	C₂×C₈	—	2	2	no	—
124	C₂×C₈	—	2	1	no	—
125	C₂×C₈	—	4	2	no	—
126	C₂×C₈	—	4	2	no	—
127	C₂×C₈	C₂×D₄	1	1	yes	characteristic
128	C₂×C₈	C₂×D₄	1	1	yes	characteristic
129	C₂×C₈	C₂×D₄	1	1	yes	characteristic
130	C₂×C₈	C₂×D₄	1	1	yes	characteristic
131	C₄⋊C₄	—	2	2	no	—
132	C₄⋊C₄	—	2	2	no	—
133	C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
134	C₄⋊C₄	C₂×D₄	1	2	yes	—
135	C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
136	C₂²⋊C₄	—	2	1	no	—
137	C₂²⋊C₄	—	2	2	no	—
138	C₂²⋊C₄	—	2	1	no	—
139	C₂²⋊C₄	—	4	2	no	—
140	C₂²⋊C₄	—	4	2	no	—
141	C₂²⋊C₄	—	4	2	no	—
142	C₄²	—	4	2	no	—
143	C₄²	C₂⁴	1	1	yes	characteristic
144	C₂³	—	2	1	no	—
145	C₂³	—	2	1	no	—
146	C₂³	—	2	2	no	—
147	C₂³	—	4	2	no	—
148	C₂³	—	8	4	no	—
149	C₂³	—	8	4	no	—
150	C₂³	—	16	2	no	—
151	Q₈	—	8	2	no	—
152	D₄	—	2	1	no	—
153	D₄	—	2	2	no	—
154	D₄	—	2	1	no	—
155	D₄	—	2	1	no	—
156	D₄	—	2	2	no	—
157	D₄	—	4	2	no	—
158	D₄	—	4	2	no	—
159	D₄	—	4	2	no	—
160	D₄	—	4	2	no	—
161	D₄	—	8	2	no	—
162	D₄	—	8	4	no	—
163	D₄	—	8	4	no	—
164	D₄	—	16	2	no	—
165	D₄	#135	1	2	yes	—
166	C₂×C₄	—	2	1	no	—
167	C₂×C₄	—	2	2	no	—
168	C₂×C₄	—	2	1	no	—
169	C₂×C₄	—	4	2	no	—
170	C₂×C₄	—	4	1	no	—
171	C₂×C₄	—	4	1	no	—
172	C₂×C₄	—	4	2	no	—
173	C₂×C₄	—	4	1	no	—
174	C₂×C₄	—	4	1	no	—
175	C₂×C₄	—	4	2	no	—
176	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
177	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
178	C₂×C₄	H(5,𝔽₂)	1	1	yes	characteristic
179	C₈	—	2	2	no	—
180	C₈	—	2	1	no	—
181	C₈	—	2	1	no	—
182	C₈	—	2	1	no	—
183	C₈	—	4	1	no	—
184	C₈	—	4	1	no	—
185	C₈	—	4	2	no	—
186	C₈	AGL(1,ℤ/8)	1	2	yes	—
187	C₂²	—	2	2	no	—
188	C₂²	—	4	1	no	—
189	C₂²	—	4	1	no	—
190	C₂²	—	4	1	no	—
191	C₂²	—	4	2	no	—
192	C₂²	—	4	2	no	—
193	C₂²	—	4	1	no	—
194	C₂²	—	4	1	no	—
195	C₂²	—	4	2	no	—
196	C₂²	—	8	2	no	—
197	C₂²	—	8	2	no	—
198	C₂²	—	8	2	no	—
199	C₂²	—	16	8	no	—
200	C₂²	D₄²	1	1	yes	characteristic
201	C₄	—	2	1	no	—
202	C₄	—	4	1	no	—
203	C₄	—	4	1	no	—
204	C₄	—	4	2	no	—
205	C₄	—	8	2	no	—
206	C₄	C₂×#135	1	1	yes	characteristic
207	C₄	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
208	C₄	#576	1	1	yes	characteristic
209	C₄	#577	1	1	yes	characteristic
210	C₂	—	4	2	no	—
211	C₂	—	8	1	no	—
212	C₂	—	8	2	no	—
213	C₂	—	16	2	no	—
214	C₂	#3272	1	1	yes	characteristic
215	C₂	#3280	1	1	yes	characteristic
216	C₂	#3292	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. #3341	C₂	semidirect
2. #3325	C₂	semidirect
3. D₄×D₈	C₂	semidirect
4. #1706	C₂	semidirect
5. #1697	C₂	semidirect
6. #1675	C₂	semidirect
7. #1667	C₂	semidirect
8. #1629	C₂	semidirect
9. #1624	C₂	semidirect
10. #1589	C₂	semidirect
11. C₈×D₈	C₂	semidirect
13. #555	C₂²	semidirect
15. #500	C₂²	semidirect
17. #492	C₂²	semidirect
18. #486	C₂²	semidirect
22. #476	C₂²	semidirect
27. #459	C₂²	semidirect
32. #436	C₂²	semidirect
34. C₈×D₄	C₂²	semidirect
37. #335	C₂²	semidirect
38. #335	C₂²	semidirect
39. #333	C₂²	semidirect
40. #331	C₂²	semidirect
42. D₄⋊C₈	C₂²	semidirect
43. D₄⋊C₈	C₂²	semidirect
44. C₈²	C₂²	semidirect

Maximal subgroups

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

subgroup	index
#3341	2
#3325	2
D₄×D₈	2
#1706	2
#1697	2
#1675	2
#1667	2
#1629	2
#1624	2
#1589	2
C₈×D₈	2