Finite Group Databasebeta

Subgroups — #19954

Order 256.

309 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 309 conjugacy classes of subgroups — too many to draw. Only its 101 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	#3354	C₂	1	2	yes	maximal, complemented
2	#3303	C₂	1	2	yes	maximal, complemented
3	#1707	C₂	1	1	yes	maximal, characteristic, complemented
4	#1691	C₂	1	2	yes	maximal, complemented
5	#1683	C₂	1	2	yes	maximal, complemented
6	#1647	C₂	1	2	yes	maximal, complemented
7	#1634	C₂	1	2	yes	maximal, complemented
8	#1589	C₂	1	1	yes	maximal, characteristic, complemented
9	C₈×D₈	C₂	1	1	yes	maximal, characteristic, complemented
10	#555	C₂²	1	2	yes	—
11	#547	C₂²	1	2	yes	—
12	#500	C₂²	1	1	yes	characteristic
13	#498	C₂²	1	2	yes	—
14	#494	C₂²	1	1	yes	characteristic
15	#487	C₂²	1	2	yes	—
16	#485	—	2	2	no	—
17	#482	—	2	2	no	—
18	#481	—	2	2	no	—
19	#476	C₂²	1	2	yes	—
20	#476	C₂²	1	2	yes	—
21	#474	C₂²	1	2	yes	—
22	#469	—	2	2	no	—
23	#465	—	2	2	no	—
24	#461	C₂²	1	2	yes	—
25	#449	—	2	2	no	—
26	C₄×D₈	C₂²	1	1	yes	characteristic
27	C₄×D₈	C₂²	1	1	yes	characteristic
28	#436	C₂²	1	2	yes	—
29	C₈×D₄	C₂²	1	2	yes	—
30	#428	—	2	2	no	—
31	#416	—	2	2	no	—
32	#335	C₂²	1	1	yes	characteristic
33	#335	C₂²	1	1	yes	characteristic
34	#333	C₂²	1	2	yes	—
35	#332	C₂²	1	2	yes	—
36	D₄⋊C₈	C₂²	1	2	yes	—
37	D₄⋊C₈	C₂²	1	2	yes	—
38	C₈²	C₂²	1	1	yes	characteristic
39	C₂×#41	—	2	2	no	—
40	C₂×Q₁₆	—	4	1	no	—
41	C₂×SD₁₆	—	2	2	no	—
42	C₂×D₈	D₄	1	1	yes	characteristic
43	C₂×D₈	D₄	1	1	yes	characteristic
44	C₂×M₁₆	—	2	2	no	—
45	C₂²×C₈	—	2	2	no	—
46	C₄⋊Q₈	C₂³	1	2	yes	—
47	C₄⋊Q₈	C₂³	1	2	yes	—
48	#124	—	4	2	no	—
49	#123	—	4	2	no	—
50	C₂²⋊Q₈	—	2	2	no	—
51	#121	—	2	2	no	—
52	#121	—	2	2	no	—
53	#121	—	2	2	no	—
54	C₄×D₄	C₂³	1	2	yes	—
55	C₄×D₄	C₂³	1	2	yes	—
56	#117	—	2	2	no	—
57	#117	—	2	2	no	—
58	C₂×C₄⋊C₄	—	2	2	no	—
59	#107	—	2	2	no	—
60	#107	—	4	1	no	—
61	#107	D₄	1	1	yes	characteristic
62	#107	D₄	1	1	yes	characteristic
63	#106	—	2	2	no	—
64	#106	—	2	2	no	—
65	#106	—	2	2	no	—
66	#106	—	2	2	no	—
67	C₄⋊C₈	C₂³	1	2	yes	—
68	C₄⋊C₈	C₂³	1	2	yes	—
69	#103	—	2	2	no	—
70	#103	—	2	2	no	—
71	#103	—	2	2	no	—
72	#102	—	2	2	no	—
73	#102	—	2	2	no	—
74	#102	—	2	2	no	—
75	#102	D₄	1	2	yes	—
76	#102	D₄	1	2	yes	—
77	C₂²⋊C₈	—	2	2	no	—
78	C₂²⋊C₈	—	2	2	no	—
79	C₄×C₈	C₂³	1	1	yes	characteristic
80	C₄×C₈	C₂³	1	1	yes	characteristic
81	C₄×C₈	C₂³	1	1	yes	characteristic
82	#41	—	4	4	no	—
83	#41	—	8	2	no	—
84	C₂×Q₈	—	2	2	no	—
85	C₂×Q₈	—	2	2	no	—
86	C₂×D₄	—	4	2	no	—
87	C₂×D₄	—	4	2	no	—
88	C₂×D₄	C₂×D₄	1	2	yes	—
89	C₂×D₄	C₂×D₄	1	2	yes	—
90	C₂²×C₄	—	2	2	no	—
91	C₂²×C₄	—	2	2	no	—
92	C₂²×C₄	—	4	2	no	—
93	Q₁₆	—	8	2	no	—
94	SD₁₆	—	8	2	no	—
95	D₈	—	2	2	no	—
96	D₈	—	4	1	no	—
97	M₁₆	—	4	4	no	—
98	C₂×C₈	—	2	1	no	—
99	C₂×C₈	—	2	2	no	—
100	C₂×C₈	—	2	2	no	—
101	C₂×C₈	—	8	2	no	—
102	C₂×C₈	C₂×D₄	1	1	yes	characteristic
103	C₂×C₈	C₂×D₄	1	1	yes	characteristic
104	C₂×C₈	C₂×D₄	1	1	yes	characteristic
105	C₂×C₈	C₂×D₄	1	1	yes	characteristic
106	C₄⋊C₄	—	2	2	no	—
107	C₄⋊C₄	—	2	2	no	—
108	C₄⋊C₄	—	2	2	no	—
109	C₄⋊C₄	—	2	2	no	—
110	C₄⋊C₄	—	4	2	no	—
111	C₄⋊C₄	C₂×D₄	1	2	yes	—
112	C₄⋊C₄	C₂×D₄	1	2	yes	—
113	C₂²⋊C₄	—	2	2	no	—
114	C₂²⋊C₄	—	2	2	no	—
115	C₂²⋊C₄	—	4	2	no	—
116	C₂²⋊C₄	—	4	2	no	—
117	C₂²⋊C₄	—	4	2	no	—
118	C₄²	—	4	2	no	—
119	C₄²	C₂⁴	1	1	yes	characteristic
120	C₂³	—	2	2	no	—
121	C₂³	—	2	2	no	—
122	Q₈	—	4	2	no	—
123	Q₈	—	4	2	no	—
124	Q₈	—	8	2	no	—
125	D₄	—	2	2	no	—
126	D₄	—	2	2	no	—
127	D₄	—	2	2	no	—
128	D₄	—	2	2	no	—
129	D₄	—	8	4	no	—
130	D₄	—	16	2	no	—
131	C₂×C₄	—	2	2	no	—
132	C₂×C₄	—	2	2	no	—
133	C₂×C₄	—	4	2	no	—
134	C₂×C₄	—	4	2	no	—
135	C₂×C₄	—	4	2	no	—
136	C₂×C₄	—	4	2	no	—
137	C₂×C₄	—	4	2	no	—
138	C₂×C₄	—	4	2	no	—
139	C₂×C₄	—	8	4	no	—
140	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
141	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
142	C₂×C₄	H(5,𝔽₂)	1	1	yes	characteristic
143	C₈	—	2	2	no	—
144	C₈	—	2	1	no	—
145	C₈	—	2	1	no	—
146	C₈	—	2	1	no	—
147	C₈	—	4	2	no	—
148	C₈	—	4	2	no	—
149	C₈	#135	1	2	yes	—
150	C₂²	—	4	2	no	—
151	C₂²	—	4	2	no	—
152	C₂²	—	4	2	no	—
153	C₂²	—	4	2	no	—
154	C₂²	—	4	2	no	—
155	C₂²	—	4	2	no	—
156	C₂²	D₄²	1	1	yes	characteristic
157	C₄	—	2	1	no	—
158	C₄	—	4	2	no	—
159	C₄	—	4	2	no	—
160	C₄	—	8	2	no	—
161	C₄	—	8	2	no	—
162	C₄	C₂×#135	1	1	yes	characteristic
163	C₄	#575	1	1	yes	characteristic
164	C₄	#576	1	1	yes	characteristic
165	C₄	#578	1	1	yes	characteristic
166	C₂	—	8	2	no	—
167	C₂	—	8	2	no	—
168	C₂	#3273	1	1	yes	characteristic
169	C₂	#3283	1	1	yes	characteristic
170	C₂	#3291	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. #3354	C₂	semidirect
2. #3303	C₂	semidirect
3. #1707	C₂	semidirect
4. #1691	C₂	semidirect
5. #1683	C₂	semidirect
6. #1647	C₂	semidirect
7. #1634	C₂	semidirect
8. #1589	C₂	semidirect
9. 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
#3354	2
#3303	2
#1707	2
#1691	2
#1683	2
#1647	2
#1634	2
#1589	2
C₈×D₈	2