Finite Group Databasebeta

Subgroups — #19989

Order 256.

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

normal subgroups only

This group has 310 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	#3334	C₂	1	1	yes	maximal, characteristic, complemented
3	#3303	C₂	1	2	yes	maximal, complemented
4	#1710	C₂	1	1	yes	maximal, characteristic, complemented
5	#1683	C₂	1	2	yes	maximal, complemented
6	#1675	C₂	1	1	yes	maximal, characteristic, complemented
7	#1673	C₂	1	1	yes	maximal, characteristic, complemented
8	#1646	C₂	1	2	yes	maximal, complemented
9	#1635	C₂	1	2	yes	maximal, complemented
10	C₈×D₈	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×D₈	C₂	1	1	yes	maximal, characteristic, complemented
12	#555	C₂²	1	2	yes	—
13	#547	C₂²	1	2	yes	—
14	#500	C₂²	1	1	yes	characteristic
15	#500	C₂²	1	1	yes	characteristic
16	#487	C₂²	1	2	yes	—
17	#487	C₂²	1	2	yes	—
18	#485	—	2	2	no	—
19	#482	—	2	2	no	—
20	#480	—	2	2	no	—
21	#476	C₂²	1	2	yes	—
22	#476	C₂²	1	2	yes	—
23	#474	C₂²	1	2	yes	—
24	#466	—	2	1	no	—
25	#466	—	2	1	no	—
26	#465	—	2	2	no	—
27	#461	C₂²	1	2	yes	—
28	#449	—	2	2	no	—
29	C₄×D₈	C₂²	1	1	yes	characteristic
30	C₄×D₈	C₂²	1	1	yes	characteristic
31	C₈×D₄	C₂²	1	1	yes	characteristic
32	C₈×D₄	C₂²	1	1	yes	characteristic
33	C₈×D₄	C₂²	1	2	yes	—
34	#427	—	2	1	no	—
35	C₂×#107	—	2	1	no	—
36	#416	—	2	2	no	—
37	#335	C₂²	1	1	yes	characteristic
38	#335	C₂²	1	1	yes	characteristic
39	#333	C₂²	1	2	yes	—
40	#332	C₂²	1	2	yes	—
41	D₄⋊C₈	C₂²	1	1	yes	characteristic
42	D₄⋊C₈	C₂²	1	2	yes	—
43	D₄⋊C₈	C₂²	1	1	yes	characteristic
44	C₈²	C₂²	1	1	yes	characteristic
45	C₂×#41	—	2	2	no	—
46	C₂×SD₁₆	—	2	2	no	—
47	C₂×D₈	D₄	1	1	yes	characteristic
48	C₂×D₈	D₄	1	1	yes	characteristic
49	C₂²×C₈	—	2	1	no	—
50	C₂²×C₈	—	2	1	no	—
51	C₂²×C₈	—	2	2	no	—
52	C₄⋊Q₈	C₂³	1	2	yes	—
53	C₄⋊Q₈	C₂³	1	2	yes	—
54	#124	—	4	2	no	—
55	#123	—	4	2	no	—
56	C₂²⋊Q₈	—	2	2	no	—
57	#121	—	2	2	no	—
58	#121	—	2	2	no	—
59	#121	—	2	2	no	—
60	C₄×D₄	C₂³	1	1	yes	characteristic
61	C₄×D₄	C₂³	1	2	yes	—
62	C₄×D₄	C₂³	1	1	yes	characteristic
63	#117	—	2	2	no	—
64	#117	—	2	2	no	—
65	C₂×C₄⋊C₄	—	2	2	no	—
66	#107	—	2	1	no	—
67	#107	—	2	1	no	—
68	#107	—	2	1	no	—
69	#107	—	2	1	no	—
70	#107	—	2	2	no	—
71	#107	D₄	1	1	yes	characteristic
72	#107	D₄	1	1	yes	characteristic
73	#106	—	2	2	no	—
74	#106	—	2	2	no	—
75	C₄⋊C₈	C₂³	1	1	yes	characteristic
76	C₄⋊C₈	C₂³	1	2	yes	—
77	C₄⋊C₈	C₂³	1	1	yes	characteristic
78	#103	—	2	2	no	—
79	#103	—	2	2	no	—
80	#103	—	2	2	no	—
81	#103	—	4	2	no	—
82	#102	—	2	2	no	—
83	#102	—	2	2	no	—
84	#102	—	2	2	no	—
85	#102	D₄	1	1	yes	characteristic
86	#102	D₄	1	1	yes	characteristic
87	#102	D₄	1	2	yes	—
88	C₂²⋊C₈	—	2	2	no	—
89	C₂²⋊C₈	—	2	1	no	—
90	C₂²⋊C₈	—	2	1	no	—
91	C₄×C₈	C₂³	1	1	yes	characteristic
92	C₄×C₈	C₂³	1	1	yes	characteristic
93	C₄×C₈	C₂³	1	1	yes	characteristic
94	#41	—	4	4	no	—
95	#41	—	8	2	no	—
96	C₂×Q₈	—	2	2	no	—
97	C₂×Q₈	—	2	2	no	—
98	C₂×D₄	—	4	2	no	—
99	C₂×D₄	—	4	2	no	—
100	C₂×D₄	C₂×D₄	1	2	yes	—
101	C₂×D₄	C₂×D₄	1	1	yes	characteristic
102	C₂×D₄	C₂×D₄	1	1	yes	characteristic
103	C₂²×C₄	—	2	1	no	—
104	C₂²×C₄	—	2	1	no	—
105	C₂²×C₄	—	2	2	no	—
106	C₂²×C₄	—	4	2	no	—
107	SD₁₆	—	8	2	no	—
108	D₈	—	2	2	no	—
109	D₈	—	2	2	no	—
110	C₂×C₈	—	2	1	no	—
111	C₂×C₈	—	2	1	no	—
112	C₂×C₈	—	2	2	no	—
113	C₂×C₈	—	2	1	no	—
114	C₂×C₈	—	4	2	no	—
115	C₂×C₈	—	4	2	no	—
116	C₂×C₈	—	8	2	no	—
117	C₂×C₈	C₂×D₄	1	1	yes	characteristic
118	C₂×C₈	C₂×D₄	1	1	yes	characteristic
119	C₂×C₈	C₂×D₄	1	1	yes	characteristic
120	C₂×C₈	C₂×D₄	1	1	yes	characteristic
121	C₄⋊C₄	—	2	2	no	—
122	C₄⋊C₄	—	2	2	no	—
123	C₄⋊C₄	—	2	2	no	—
124	C₄⋊C₄	—	2	2	no	—
125	C₄⋊C₄	—	4	2	no	—
126	C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
127	C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
128	C₄⋊C₄	C₂×D₄	1	2	yes	—
129	C₂²⋊C₄	—	2	1	no	—
130	C₂²⋊C₄	—	2	2	no	—
131	C₂²⋊C₄	—	2	1	no	—
132	C₂²⋊C₄	—	4	2	no	—
133	C₂²⋊C₄	—	4	2	no	—
134	C₂²⋊C₄	—	4	2	no	—
135	C₄²	—	4	2	no	—
136	C₄²	C₂⁴	1	1	yes	characteristic
137	C₂³	—	2	1	no	—
138	C₂³	—	2	2	no	—
139	C₂³	—	2	1	no	—
140	Q₈	—	4	2	no	—
141	Q₈	—	4	2	no	—
142	Q₈	—	8	2	no	—
143	D₄	—	2	1	no	—
144	D₄	—	2	2	no	—
145	D₄	—	2	1	no	—
146	D₄	—	2	1	no	—
147	D₄	—	2	2	no	—
148	D₄	—	8	4	no	—
149	D₄	—	16	2	no	—
150	D₄	#135	1	2	yes	—
151	C₂×C₄	—	2	1	no	—
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	—	2	1	no	—
154	C₂×C₄	—	4	1	no	—
155	C₂×C₄	—	4	1	no	—
156	C₂×C₄	—	4	2	no	—
157	C₂×C₄	—	4	2	no	—
158	C₂×C₄	—	4	1	no	—
159	C₂×C₄	—	4	1	no	—
160	C₂×C₄	—	4	2	no	—
161	C₂×C₄	—	4	2	no	—
162	C₂×C₄	—	8	4	no	—
163	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
164	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
165	C₂×C₄	H(5,𝔽₂)	1	1	yes	characteristic
166	C₈	—	2	1	no	—
167	C₈	—	2	1	no	—
168	C₈	—	2	1	no	—
169	C₈	—	4	1	no	—
170	C₈	—	4	1	no	—
171	C₈	—	4	1	no	—
172	C₈	—	4	2	no	—
173	C₈	#135	1	2	yes	—
174	C₂²	—	2	2	no	—
175	C₂²	—	4	1	no	—
176	C₂²	—	4	1	no	—
177	C₂²	—	4	2	no	—
178	C₂²	—	4	1	no	—
179	C₂²	—	4	1	no	—
180	C₂²	—	4	2	no	—
181	C₂²	—	4	1	no	—
182	C₂²	—	4	2	no	—
183	C₂²	D₄²	1	1	yes	characteristic
184	C₄	—	2	1	no	—
185	C₄	—	4	2	no	—
186	C₄	—	4	1	no	—
187	C₄	—	4	1	no	—
188	C₄	—	8	2	no	—
189	C₄	—	8	2	no	—
190	C₄	C₂×#135	1	1	yes	characteristic
191	C₄	C₂×#135	1	1	yes	characteristic
192	C₄	#576	1	1	yes	characteristic
193	C₄	#578	1	1	yes	characteristic
194	C₂	—	4	2	no	—
195	C₂	—	8	2	no	—
196	C₂	—	8	1	no	—
197	C₂	#3280	1	1	yes	characteristic
198	C₂	#3283	1	1	yes	characteristic
199	C₂	#3290	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. #3334	C₂	semidirect
3. #3303	C₂	semidirect
4. #1710	C₂	semidirect
5. #1683	C₂	semidirect
6. #1675	C₂	semidirect
7. #1673	C₂	semidirect
8. #1646	C₂	semidirect
9. #1635	C₂	semidirect
10. C₈×D₈	C₂	semidirect
11. C₈×D₈	C₂	semidirect

Maximal subgroups

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

subgroup	index
#3341	2
#3334	2
#3303	2
#1710	2
#1683	2
#1675	2
#1673	2
#1646	2
#1635	2
C₈×D₈	2