Finite Group Databasebeta

Subgroups — #19731

Order 256.

365 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 365 conjugacy classes of subgroups — too many to draw. Only its 119 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	#3230	C₂	1	2	yes	maximal, complemented
2	#3228	C₂	1	2	yes	maximal
3	#3185	C₂	1	1	yes	maximal, characteristic, complemented
4	#3152	C₂	1	1	yes	maximal, characteristic
5	#1723	C₂	1	2	yes	maximal, complemented
6	#1722	C₂	1	1	yes	maximal, characteristic, complemented
7	#1719	C₂	1	2	yes	maximal, complemented
8	#1716	C₂	1	1	yes	maximal, characteristic, complemented
9	#1713	C₂	1	1	yes	maximal, characteristic, complemented
10	#1708	C₂	1	1	yes	maximal, characteristic, complemented
11	#1453	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#137	—	2	2	no	—
13	#533	C₂²	1	2	yes	—
14	#532	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊Q₈	C₂²	1	1	yes	characteristic
16	#500	C₂²	1	2	yes	—
17	#500	C₂²	1	2	yes	—
18	#500	C₂²	1	2	yes	—
19	#498	C₂²	1	1	yes	characteristic
20	#498	C₂²	1	2	yes	—
21	#497	—	2	2	no	—
22	#494	C₂²	1	1	yes	characteristic
23	#494	C₂²	1	1	yes	characteristic
24	#494	C₂²	1	1	yes	characteristic
25	#492	C₂²	1	1	yes	characteristic
26	#492	C₂²	1	1	yes	characteristic
27	#487	C₂²	1	2	yes	—
28	#487	C₂²	1	2	yes	—
29	#487	C₂²	1	2	yes	—
30	#486	C₂²	1	2	yes	—
31	#486	C₂²	1	2	yes	—
32	#484	—	2	4	no	—
33	#482	—	2	4	no	—
34	#478	—	2	4	no	—
35	#477	—	2	4	no	—
36	#470	—	2	2	no	—
37	#441	—	2	2	no	—
38	#431	C₂²	1	2	yes	—
39	C₄×M₁₆	C₂²	1	1	yes	characteristic
40	#322	C₂²	1	1	yes	characteristic
41	#322	C₂²	1	2	yes	—
42	C₈²	C₂²	1	1	yes	characteristic
43	C₂×#41	—	2	1	no	—
44	C₂²×Q₈	—	2	1	no	—
45	#137	—	4	8	no	—
46	C₂×Q₁₆	—	2	2	no	—
47	C₂×Q₁₆	—	2	2	no	—
48	C₂×Q₁₆	—	2	2	no	—
49	C₂×SD₁₆	—	2	2	no	—
50	C₂×SD₁₆	—	2	2	no	—
51	C₂×M₁₆	D₄	1	2	yes	—
52	C₄⋊Q₈	—	2	2	no	—
53	C₄⋊Q₈	—	2	1	no	—
54	C₄⋊Q₈	C₂³	1	1	yes	characteristic
55	C₄⋊Q₈	C₂³	1	1	yes	characteristic
56	C₄⋊Q₈	C₂³	1	1	yes	characteristic
57	C₄⋊Q₈	C₂³	1	2	yes	—
58	C₄⋊Q₈	C₂³	1	2	yes	—
59	#127	C₂³	1	1	yes	characteristic
60	#125	—	2	2	no	—
61	#124	—	2	1	no	—
62	C₂²⋊Q₈	—	2	2	no	—
63	C₂²⋊Q₈	—	2	2	no	—
64	#121	—	2	2	no	—
65	C₄×Q₈	—	2	2	no	—
66	C₄×Q₈	—	2	2	no	—
67	C₄×D₄	—	2	2	no	—
68	#117	—	2	2	no	—
69	C₂×C₄⋊C₄	—	2	2	no	—
70	C₂×C₄²	C₂³	1	1	yes	characteristic
71	#107	—	2	4	no	—
72	#107	—	2	4	no	—
73	#107	—	2	4	no	—
74	#106	—	2	2	no	—
75	#106	—	2	4	no	—
76	C₄⋊C₈	D₄	1	4	yes	—
77	#103	—	2	4	no	—
78	#103	—	2	4	no	—
79	#103	—	2	4	no	—
80	#102	—	2	4	no	—
81	#102	—	2	4	no	—
82	C₂²⋊C₈	D₄	1	4	yes	—
83	#97	D₄	1	2	yes	—
84	C₄×C₈	C₂³	1	1	yes	characteristic
85	C₄×C₈	C₂³	1	2	yes	—
86	C₄×C₈	C₂³	1	1	yes	characteristic
87	C₄×C₈	C₂³	1	2	yes	—
88	#41	—	4	4	no	—
89	C₂×Q₈	—	2	2	no	—
90	C₂×Q₈	—	2	2	no	—
91	C₂×Q₈	—	2	1	no	—
92	C₂×Q₈	—	2	1	no	—
93	C₂×Q₈	—	2	1	no	—
94	C₂×Q₈	—	4	4	no	—
95	C₂×Q₈	—	4	1	no	—
96	C₂×D₄	—	2	2	no	—
97	C₂×D₄	—	2	1	no	—
98	C₂×D₄	—	4	1	no	—
99	C₂²×C₄	—	4	1	no	—
100	C₂²×C₄	—	4	1	no	—
101	C₂²×C₄	C₂×D₄	1	2	yes	—
102	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
103	Q₁₆	—	4	4	no	—
104	Q₁₆	—	4	4	no	—
105	Q₁₆	—	4	4	no	—
106	SD₁₆	—	4	4	no	—
107	SD₁₆	—	4	4	no	—
108	M₁₆	—	2	4	no	—
109	C₂×C₈	C₂×D₄	1	2	yes	—
110	C₂×C₈	C₂×D₄	1	4	yes	—
111	C₂×C₈	C₂×D₄	1	4	yes	—
112	C₂×C₈	C₂×D₄	1	2	yes	—
113	C₄⋊C₄	—	2	2	no	—
114	C₄⋊C₄	—	2	2	no	—
115	C₄⋊C₄	—	2	2	no	—
116	C₄⋊C₄	—	2	2	no	—
117	C₄⋊C₄	—	2	2	no	—
118	C₄⋊C₄	—	2	2	no	—
119	C₄⋊C₄	—	2	2	no	—
120	C₄⋊C₄	—	4	2	no	—
121	C₄⋊C₄	—	4	2	no	—
122	C₄⋊C₄	—	4	2	no	—
123	C₂²⋊C₄	—	4	2	no	—
124	C₂²⋊C₄	—	4	2	no	—
125	C₄²	—	4	2	no	—
126	C₄²	C₂×D₄	1	1	yes	characteristic
127	C₄²	C₂×D₄	1	2	yes	—
128	C₄²	C₂⁴	1	1	yes	characteristic
129	C₂³	—	4	1	no	—
130	C₂³	#127	1	1	yes	characteristic
131	Q₈	—	4	2	no	—
132	Q₈	—	4	2	no	—
133	Q₈	—	4	2	no	—
134	Q₈	—	4	4	no	—
135	Q₈	—	4	4	no	—
136	Q₈	—	8	2	no	—
137	D₄	—	4	2	no	—
138	D₄	—	4	4	no	—
139	D₄	—	8	2	no	—
140	C₂×C₄	—	2	2	no	—
141	C₂×C₄	—	2	4	no	—
142	C₂×C₄	—	4	2	no	—
143	C₂×C₄	—	4	2	no	—
144	C₂×C₄	—	4	1	no	—
145	C₂×C₄	—	4	1	no	—
146	C₂×C₄	—	4	1	no	—
147	C₂×C₄	—	8	2	no	—
148	C₂×C₄	—	8	2	no	—
149	C₂×C₄	#127	1	2	yes	—
150	C₂×C₄	#127	1	1	yes	characteristic
151	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
152	C₂×C₄	C₂²×D₄	1	2	yes	—
153	C₈	—	2	2	no	—
154	C₈	—	2	4	no	—
155	C₈	—	2	4	no	—
156	C₈	AGL(1,ℤ/8)	1	4	yes	—
157	C₂²	—	2	1	no	—
158	C₂²	—	2	2	no	—
159	C₂²	—	8	1	no	—
160	C₂²	—	8	2	no	—
161	C₂²	C₂×#127	1	1	yes	characteristic
162	C₄	—	2	1	no	—
163	C₄	—	2	2	no	—
164	C₄	—	8	1	no	—
165	C₄	—	8	1	no	—
166	C₄	—	8	1	no	—
167	C₄	—	8	2	no	—
168	C₄	—	8	2	no	—
169	C₄	C₂×AGL(1,ℤ/8)	1	2	yes	—
170	C₄	#578	1	4	yes	—
171	C₂	—	4	1	no	—
172	C₂	—	16	1	no	—
173	C₂	#3151	1	1	yes	characteristic
174	C₂	#3156	1	2	yes	—

	subgroup	G/H
center, socle	C₂²	C₂×#127
commutator, Frattini, cosocle	C₄²	C₂⁴
prime radical	#532	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. #3230	C₂	semidirect
3. #3185	C₂	semidirect
5. #1723	C₂	semidirect
6. #1722	C₂	semidirect
7. #1719	C₂	semidirect
8. #1716	C₂	semidirect
9. #1713	C₂	semidirect
10. #1708	C₂	semidirect
11. #1453	C₂	semidirect

Maximal subgroups

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

subgroup	index
#3230	2
#3228	2
#3185	2
#3152	2
#1723	2
#1722	2
#1719	2
#1716	2
#1713	2
#1708	2
#1453	2