Finite Group Databasebeta

Subgroups — #19957

Order 256.

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

normal subgroups only

This group has 292 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	#3380	C₂	1	1	yes	maximal, characteristic, complemented
2	#3357	C₂	1	2	yes	maximal, complemented
3	#3341	C₂	1	1	yes	maximal, characteristic
4	#1713	C₂	1	1	yes	maximal, characteristic, complemented
5	#1695	C₂	1	2	yes	maximal, complemented
6	#1676	C₂	1	1	yes	maximal, characteristic, complemented
7	#1675	C₂	1	1	yes	maximal, characteristic, complemented
8	#1634	C₂	1	2	yes	maximal, complemented
9	#1632	C₂	1	2	yes	maximal, complemented
10	#1589	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×SD₁₆	C₂	1	1	yes	maximal, characteristic, complemented
12	#555	C₂²	1	2	yes	—
13	#554	C₂²	1	2	yes	—
14	#500	C₂²	1	1	yes	characteristic
15	#500	C₂²	1	2	yes	—
16	#500	C₂²	1	1	yes	characteristic
17	#499	—	2	1	no	—
18	#494	C₂²	1	2	yes	—
19	#485	—	2	2	no	—
20	#482	—	2	2	no	—
21	#480	—	2	2	no	—
22	#478	—	2	2	no	—
23	#477	C₂²	1	2	yes	—
24	#476	C₂²	1	2	yes	—
25	#476	C₂²	1	2	yes	—
26	#476	C₂²	1	2	yes	—
27	#468	—	2	2	no	—
28	#466	—	2	1	no	—
29	C₈×Q₈	C₂²	1	1	yes	characteristic
30	C₄×SD₁₆	C₂²	1	1	yes	characteristic
31	C₄×D₈	—	2	1	no	—
32	C₄×D₈	C₂²	1	1	yes	characteristic
33	#436	C₂²	1	2	yes	—
34	C₈×D₄	C₂²	1	1	yes	characteristic
35	#428	—	2	2	no	—
36	#427	—	2	1	no	—
37	#335	C₂²	1	1	yes	characteristic
38	#334	C₂²	1	1	yes	characteristic
39	#333	C₂²	1	2	yes	—
40	#333	C₂²	1	2	yes	—
41	Q₈⋊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×Q₁₆	—	4	2	no	—
46	C₂×SD₁₆	D₄	1	1	yes	characteristic
47	C₂×D₈	D₄	1	1	yes	characteristic
48	C₂×M₁₆	—	2	2	no	—
49	C₂²×C₈	—	2	1	no	—
50	C₄⋊Q₈	C₂³	1	2	yes	—
51	C₄⋊Q₈	C₂³	1	2	yes	—
52	#125	—	2	2	no	—
53	#124	—	4	2	no	—
54	C₂²⋊Q₈	—	2	2	no	—
55	C₂²⋊Q₈	—	4	2	no	—
56	#121	—	2	2	no	—
57	#121	—	2	2	no	—
58	C₄×Q₈	C₂³	1	1	yes	characteristic
59	C₄×D₄	—	2	2	no	—
60	C₄×D₄	C₂³	1	2	yes	—
61	C₄×D₄	C₂³	1	1	yes	characteristic
62	#117	—	2	2	no	—
63	#117	—	2	2	no	—
64	C₂×C₄⋊C₄	—	2	2	no	—
65	#107	—	2	1	no	—
66	#107	—	2	2	no	—
67	#107	—	2	1	no	—
68	#107	—	2	1	no	—
69	#107	—	2	2	no	—
70	#107	—	2	1	no	—
71	#107	—	2	2	no	—
72	#107	D₄	1	1	yes	characteristic
73	#106	—	2	2	no	—
74	#106	—	2	2	no	—
75	#106	—	2	2	no	—
76	#106	D₄	1	1	yes	characteristic
77	C₄⋊C₈	—	2	1	no	—
78	C₄⋊C₈	C₂³	1	1	yes	characteristic
79	C₄⋊C₈	C₂³	1	2	yes	—
80	C₄⋊C₈	C₂³	1	1	yes	characteristic
81	#103	—	2	2	no	—
82	#103	D₄	1	1	yes	characteristic
83	#102	—	2	2	no	—
84	#102	—	2	2	no	—
85	#102	—	2	2	no	—
86	#102	D₄	1	2	yes	—
87	#102	D₄	1	1	yes	characteristic
88	C₂²⋊C₈	—	2	1	no	—
89	C₂²⋊C₈	—	2	2	no	—
90	C₄×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	C₂×Q₈	—	2	2	no	—
95	C₂×Q₈	—	2	2	no	—
96	C₂×Q₈	C₂×D₄	1	1	yes	characteristic
97	C₂×D₄	—	4	2	no	—
98	C₂×D₄	C₂×D₄	1	2	yes	—
99	C₂×D₄	C₂×D₄	1	1	yes	characteristic
100	C₂²×C₄	—	2	2	no	—
101	C₂²×C₄	—	2	1	no	—
102	C₂²×C₄	—	4	2	no	—
103	Q₁₆	—	8	4	no	—
104	SD₁₆	—	2	2	no	—
105	D₈	—	2	2	no	—
106	M₁₆	—	4	4	no	—
107	C₂×C₈	—	2	1	no	—
108	C₂×C₈	—	2	1	no	—
109	C₂×C₈	—	2	1	no	—
110	C₂×C₈	—	2	2	no	—
111	C₂×C₈	—	4	2	no	—
112	C₂×C₈	C₂×D₄	1	1	yes	characteristic
113	C₂×C₈	C₂×D₄	1	1	yes	characteristic
114	C₂×C₈	C₂×D₄	1	1	yes	characteristic
115	C₂×C₈	C₂×D₄	1	1	yes	characteristic
116	C₄⋊C₄	—	2	1	no	—
117	C₄⋊C₄	—	2	2	no	—
118	C₄⋊C₄	—	2	2	no	—
119	C₄⋊C₄	—	2	2	no	—
120	C₄⋊C₄	—	2	2	no	—
121	C₄⋊C₄	—	4	2	no	—
122	C₄⋊C₄	—	4	2	no	—
123	C₄⋊C₄	C₂×D₄	1	2	yes	—
124	C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
125	C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
126	C₂²⋊C₄	—	2	2	no	—
127	C₂²⋊C₄	—	2	1	no	—
128	C₂²⋊C₄	—	4	2	no	—
129	C₂²⋊C₄	—	4	2	no	—
130	C₂²⋊C₄	—	4	2	no	—
131	C₄²	—	2	1	no	—
132	C₄²	—	4	2	no	—
133	C₄²	C₂⁴	1	1	yes	characteristic
134	C₂³	—	2	2	no	—
135	C₂³	—	2	1	no	—
136	Q₈	—	2	1	no	—
137	Q₈	—	2	1	no	—
138	Q₈	—	8	2	no	—
139	Q₈	—	8	2	no	—
140	D₄	—	2	1	no	—
141	D₄	—	2	2	no	—
142	D₄	—	2	2	no	—
143	D₄	—	2	1	no	—
144	D₄	—	16	2	no	—
145	C₂×C₄	—	2	1	no	—
146	C₂×C₄	—	2	1	no	—
147	C₂×C₄	—	2	1	no	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	—	4	2	no	—
150	C₂×C₄	—	4	1	no	—
151	C₂×C₄	—	4	1	no	—
152	C₂×C₄	—	4	2	no	—
153	C₂×C₄	—	4	2	no	—
154	C₂×C₄	—	4	2	no	—
155	C₂×C₄	—	16	2	no	—
156	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
157	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
158	C₂×C₄	H(5,𝔽₂)	1	1	yes	characteristic
159	C₈	—	2	2	no	—
160	C₈	—	2	1	no	—
161	C₈	—	2	1	no	—
162	C₈	—	4	2	no	—
163	C₈	—	4	1	no	—
164	C₈	—	4	1	no	—
165	C₈	#135	1	2	yes	—
166	C₈	#137	1	2	yes	—
167	C₂²	—	4	1	no	—
168	C₂²	—	4	2	no	—
169	C₂²	—	4	2	no	—
170	C₂²	—	4	1	no	—
171	C₂²	—	4	2	no	—
172	C₂²	—	4	1	no	—
173	C₂²	D₄²	1	1	yes	characteristic
174	C₄	—	2	1	no	—
175	C₄	—	4	1	no	—
176	C₄	—	4	2	no	—
177	C₄	—	4	1	no	—
178	C₄	—	4	1	no	—
179	C₄	—	8	2	no	—
180	C₄	—	8	2	no	—
181	C₄	C₂×#135	1	1	yes	characteristic
182	C₄	C₂×#137	1	1	yes	characteristic
183	C₄	#578	1	1	yes	characteristic
184	C₄	#578	1	1	yes	characteristic
185	C₂	—	8	1	no	—
186	C₂	—	8	2	no	—
187	C₂	#3276	1	1	yes	characteristic
188	C₂	#3283	1	1	yes	characteristic
189	C₂	#3292	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	D₄²
commutator, Frattini, cosocle	C₄²	C₂⁴
prime radical	#3341	C₂
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. #3380	C₂	semidirect
2. #3357	C₂	semidirect
4. #1713	C₂	semidirect
5. #1695	C₂	semidirect
6. #1676	C₂	semidirect
7. #1675	C₂	semidirect
8. #1634	C₂	semidirect
9. #1632	C₂	semidirect
10. #1589	C₂	semidirect
11. C₈×SD₁₆	C₂	semidirect

Maximal subgroups

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

subgroup	index
#3380	2
#3357	2
#3341	2
#1713	2
#1695	2
#1676	2
#1675	2
#1634	2
#1632	2
#1589	2
C₈×SD₁₆	2