Finite Group Databasebeta

Subgroups — #19959

Order 256.

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

normal subgroups only

This group has 285 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	#3393	C₂	1	1	yes	maximal, characteristic
2	#3380	C₂	1	1	yes	maximal, characteristic
3	#3356	C₂	1	2	yes	maximal, complemented
4	#1707	C₂	1	1	yes	maximal, characteristic, complemented
5	#1691	C₂	1	2	yes	maximal, complemented
6	#1676	C₂	1	1	yes	maximal, characteristic, complemented
7	#1670	C₂	1	1	yes	maximal, characteristic, complemented
8	#1643	C₂	1	2	yes	maximal, complemented
9	#1638	C₂	1	2	yes	maximal, complemented
10	#1589	C₂	1	1	yes	maximal, characteristic
11	C₈×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
12	#556	C₂²	1	2	yes	—
13	#554	C₂²	1	2	yes	—
14	#500	C₂²	1	1	yes	characteristic
15	#499	—	2	1	no	—
16	#498	C₂²	1	2	yes	—
17	#495	—	2	1	no	—
18	#494	C₂²	1	1	yes	characteristic
19	#487	C₂²	1	2	yes	—
20	#485	—	2	2	no	—
21	#483	—	2	2	no	—
22	#478	—	2	2	no	—
23	#477	C₂²	1	2	yes	—
24	#476	C₂²	1	2	yes	—
25	#475	C₂²	1	2	yes	—
26	#470	—	2	2	no	—
27	#464	—	2	2	no	—
28	#461	C₂²	1	2	yes	—
29	C₈×Q₈	C₂²	1	1	yes	characteristic
30	C₈×Q₈	C₂²	1	1	yes	characteristic
31	C₄×Q₁₆	—	2	1	no	—
32	C₄×Q₁₆	C₂²	1	1	yes	characteristic
33	C₄×D₈	—	2	1	no	—
34	C₄×D₈	C₂²	1	1	yes	characteristic
35	#436	C₂²	1	2	yes	—
36	#428	—	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	Q₈⋊C₈	C₂²	1	1	yes	characteristic
42	Q₈⋊C₈	C₂²	1	1	yes	characteristic
43	D₄⋊C₈	C₂²	1	2	yes	—
44	C₈²	C₂²	1	1	yes	characteristic
45	C₂×Q₁₆	—	2	1	no	—
46	C₂×Q₁₆	—	2	1	no	—
47	C₂×Q₁₆	D₄	1	1	yes	characteristic
48	C₂×SD₁₆	—	2	2	no	—
49	C₂×D₈	D₄	1	1	yes	characteristic
50	C₂×M₁₆	—	2	2	no	—
51	C₄⋊Q₈	C₂³	1	2	yes	—
52	C₄⋊Q₈	C₂³	1	2	yes	—
53	#126	—	4	2	no	—
54	#125	—	2	2	no	—
55	#124	—	2	2	no	—
56	C₂²⋊Q₈	—	2	2	no	—
57	C₂²⋊Q₈	—	2	2	no	—
58	C₂²⋊Q₈	—	4	2	no	—
59	C₄×Q₈	—	2	2	no	—
60	C₄×Q₈	C₂³	1	1	yes	characteristic
61	C₄×Q₈	C₂³	1	1	yes	characteristic
62	C₄×D₄	—	2	2	no	—
63	C₄×D₄	C₂³	1	2	yes	—
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	2	no	—
69	#107	D₄	1	1	yes	characteristic
70	#107	D₄	1	1	yes	characteristic
71	#106	—	2	2	no	—
72	#106	—	2	2	no	—
73	#106	—	2	2	no	—
74	#106	—	2	2	no	—
75	C₄⋊C₈	—	2	1	no	—
76	C₄⋊C₈	—	2	1	no	—
77	C₄⋊C₈	C₂³	1	2	yes	—
78	C₄⋊C₈	C₂³	1	1	yes	characteristic
79	C₄⋊C₈	C₂³	1	1	yes	characteristic
80	#103	—	2	2	no	—
81	#103	—	2	2	no	—
82	#103	—	2	2	no	—
83	#103	—	4	2	no	—
84	#103	D₄	1	1	yes	characteristic
85	#103	D₄	1	1	yes	characteristic
86	#102	—	2	2	no	—
87	#102	D₄	1	2	yes	—
88	C₂²⋊C₈	—	2	2	no	—
89	C₄×C₈	—	2	1	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×Q₈	C₂×D₄	1	1	yes	characteristic
98	C₂×D₄	C₂×D₄	1	2	yes	—
99	C₂²×C₄	—	2	2	no	—
100	C₂²×C₄	—	4	2	no	—
101	Q₁₆	—	2	2	no	—
102	Q₁₆	—	4	2	no	—
103	Q₁₆	—	8	1	no	—
104	SD₁₆	—	8	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	2	no	—
110	C₂×C₈	—	2	1	no	—
111	C₂×C₈	C₂×D₄	1	1	yes	characteristic
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	C₄⋊C₄	—	2	1	no	—
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	—	4	2	no	—
124	C₄⋊C₄	C₂×D₄	1	2	yes	—
125	C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
126	C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
127	C₂²⋊C₄	—	2	2	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	—	2	1	no	—
133	C₄²	—	4	2	no	—
134	C₄²	C₂⁴	1	1	yes	characteristic
135	C₂³	—	2	2	no	—
136	Q₈	—	2	1	no	—
137	Q₈	—	2	1	no	—
138	Q₈	—	2	1	no	—
139	Q₈	—	2	1	no	—
140	Q₈	—	4	2	no	—
141	Q₈	—	4	2	no	—
142	Q₈	—	8	2	no	—
143	D₄	—	2	2	no	—
144	D₄	—	2	2	no	—
145	C₂×C₄	—	2	1	no	—
146	C₂×C₄	—	2	1	no	—
147	C₂×C₄	—	2	1	no	—
148	C₂×C₄	—	2	1	no	—
149	C₂×C₄	—	2	2	no	—
150	C₂×C₄	—	4	2	no	—
151	C₂×C₄	—	4	2	no	—
152	C₂×C₄	—	4	2	no	—
153	C₂×C₄	—	4	2	no	—
154	C₂×C₄	—	16	2	no	—
155	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
156	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
157	C₂×C₄	H(5,𝔽₂)	1	1	yes	characteristic
158	C₈	—	2	1	no	—
159	C₈	—	2	1	no	—
160	C₈	—	4	1	no	—
161	C₈	—	4	2	no	—
162	C₈	—	4	1	no	—
163	C₈	—	4	1	no	—
164	C₈	#135	1	2	yes	—
165	C₈	#137	1	2	yes	—
166	C₂²	—	4	2	no	—
167	C₂²	—	4	2	no	—
168	C₂²	—	4	2	no	—
169	C₂²	D₄²	1	1	yes	characteristic
170	C₄	—	2	1	no	—
171	C₄	—	4	2	no	—
172	C₄	—	4	1	no	—
173	C₄	—	4	1	no	—
174	C₄	—	4	1	no	—
175	C₄	—	4	1	no	—
176	C₄	—	8	2	no	—
177	C₄	—	8	2	no	—
178	C₄	C₂×#135	1	1	yes	characteristic
179	C₄	C₂×#137	1	1	yes	characteristic
180	C₄	#577	1	1	yes	characteristic
181	C₄	#578	1	1	yes	characteristic
182	C₂	—	8	2	no	—
183	C₂	#3278	1	1	yes	characteristic
184	C₂	#3283	1	1	yes	characteristic
185	C₂	#3290	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	D₄²
commutator, Frattini, cosocle	C₄²	C₂⁴
prime radical	C₂×D₈	D₄
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
3. #3356	C₂	semidirect
4. #1707	C₂	semidirect
5. #1691	C₂	semidirect
6. #1676	C₂	semidirect
7. #1670	C₂	semidirect
8. #1643	C₂	semidirect
9. #1638	C₂	semidirect
11. C₈×Q₁₆	C₂	semidirect

Maximal subgroups

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

subgroup	index
#3393	2
#3380	2
#3356	2
#1707	2
#1691	2
#1676	2
#1670	2
#1643	2
#1638	2
#1589	2
C₈×Q₁₆	2