Finite Group Databasebeta

Subgroups — #19648

Order 256.

332 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 123 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	#3231	C₂	1	2	yes	maximal
2	#3185	C₂	1	2	yes	maximal
3	#3170	C₂	1	1	yes	maximal, characteristic
4	#3160	C₂	1	1	yes	maximal, characteristic
5	#1718	C₂	1	2	yes	maximal, complemented
6	#1716	C₂	1	2	yes	maximal, complemented
7	#1713	C₂	1	1	yes	maximal, characteristic, complemented
8	#1710	C₂	1	1	yes	maximal, characteristic, complemented
9	#1707	C₂	1	1	yes	maximal, characteristic, complemented
10	#1707	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×M₁₆	C₂	1	1	yes	maximal, characteristic
12	#533	C₂²	1	2	yes	—
13	#533	C₂²	1	2	yes	—
14	#500	C₂²	1	1	yes	characteristic
15	#500	C₂²	1	1	yes	characteristic
16	#500	C₂²	1	1	yes	characteristic
17	#500	C₂²	1	2	yes	—
18	#499	—	2	2	no	—
19	#498	C₂²	1	2	yes	—
20	#498	C₂²	1	2	yes	—
21	#498	C₂²	1	2	yes	—
22	#494	C₂²	1	1	yes	characteristic
23	#494	C₂²	1	2	yes	—
24	#487	C₂²	1	2	yes	—
25	#487	C₂²	1	2	yes	—
26	#487	C₂²	1	2	yes	—
27	#487	C₂²	1	2	yes	—
28	#487	C₂²	1	2	yes	—
29	#485	—	2	4	no	—
30	#479	—	2	4	no	—
31	#470	—	2	4	no	—
32	#467	—	2	2	no	—
33	#441	—	2	4	no	—
34	C₄×Q₁₆	—	2	2	no	—
35	#431	C₂²	1	1	yes	characteristic
36	#427	—	2	2	no	—
37	C₄×M₁₆	C₂²	1	1	yes	characteristic
38	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
39	#322	C₂²	1	1	yes	characteristic
40	#322	C₂²	1	1	yes	characteristic
41	C₈²	C₂²	1	1	yes	characteristic
42	C₈²	C₂²	1	1	yes	characteristic
43	C₂×Q₁₆	—	2	2	no	—
44	C₂×Q₁₆	—	2	4	no	—
45	C₂×M₁₆	D₄	1	2	yes	—
46	C₂²×C₈	D₄	1	2	yes	—
47	C₄⋊Q₈	C₂³	1	2	yes	—
48	C₄⋊Q₈	C₂³	1	2	yes	—
49	C₄⋊Q₈	C₂³	1	2	yes	—
50	C₄⋊Q₈	C₂³	1	2	yes	—
51	#125	—	2	2	no	—
52	#125	—	2	2	no	—
53	C₂²⋊Q₈	—	2	2	no	—
54	C₂²⋊Q₈	—	2	2	no	—
55	C₂²⋊Q₈	—	2	2	no	—
56	C₂²⋊Q₈	—	2	2	no	—
57	C₄×Q₈	—	2	2	no	—
58	C₄×Q₈	—	2	2	no	—
59	C₄×Q₈	—	2	2	no	—
60	C₄×Q₈	—	2	2	no	—
61	#117	—	2	2	no	—
62	#117	—	2	2	no	—
63	C₂×C₄²	C₂³	1	1	yes	characteristic
64	#107	—	2	2	no	—
65	#107	—	2	2	no	—
66	#107	—	2	2	no	—
67	#107	—	2	4	no	—
68	#106	—	2	4	no	—
69	#106	—	2	4	no	—
70	#106	—	2	4	no	—
71	C₄⋊C₈	D₄	1	2	yes	—
72	#103	—	2	4	no	—
73	#103	—	2	4	no	—
74	#103	—	2	4	no	—
75	#103	—	2	4	no	—
76	#103	—	2	4	no	—
77	C₂²⋊C₈	D₄	1	2	yes	—
78	#97	D₄	1	2	yes	—
79	C₄×C₈	D₄	1	2	yes	—
80	C₄×C₈	C₂³	1	1	yes	characteristic
81	C₄×C₈	C₂³	1	1	yes	characteristic
82	C₄×C₈	C₂³	1	1	yes	characteristic
83	C₄×C₈	C₂³	1	1	yes	characteristic
84	C₄×C₈	C₂³	1	1	yes	characteristic
85	C₄×C₈	C₂³	1	1	yes	characteristic
86	C₂×Q₈	—	2	2	no	—
87	C₂×Q₈	—	2	2	no	—
88	C₂×Q₈	—	2	2	no	—
89	C₂×Q₈	—	2	2	no	—
90	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
91	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
92	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
93	Q₁₆	—	4	8	no	—
94	Q₁₆	—	4	4	no	—
95	M₁₆	—	2	4	no	—
96	C₂×C₈	—	2	4	no	—
97	C₂×C₈	C₂×D₄	1	2	yes	—
98	C₂×C₈	C₂×D₄	1	2	yes	—
99	C₂×C₈	C₂×D₄	1	2	yes	—
100	C₂×C₈	C₂×D₄	1	2	yes	—
101	C₂×C₈	C₂×D₄	1	2	yes	—
102	C₂×C₈	C₂×D₄	1	2	yes	—
103	C₄⋊C₄	—	2	2	no	—
104	C₄⋊C₄	—	2	2	no	—
105	C₄⋊C₄	—	2	2	no	—
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	—	2	2	no	—
111	C₄⋊C₄	—	4	2	no	—
112	C₄⋊C₄	—	4	2	no	—
113	C₄⋊C₄	—	4	2	no	—
114	C₄⋊C₄	—	4	2	no	—
115	C₂²⋊C₄	—	4	2	no	—
116	C₂²⋊C₄	—	4	2	no	—
117	C₄²	—	4	2	no	—
118	C₄²	—	4	2	no	—
119	C₄²	C₂×D₄	1	1	yes	characteristic
120	C₄²	C₂×D₄	1	1	yes	characteristic
121	C₄²	C₂×D₄	1	1	yes	characteristic
122	C₄²	C₂⁴	1	1	yes	characteristic
123	C₂³	#127	1	1	yes	characteristic
124	Q₈	—	4	4	no	—
125	Q₈	—	4	4	no	—
126	Q₈	—	4	4	no	—
127	Q₈	—	4	4	no	—
128	C₂×C₄	—	2	2	no	—
129	C₂×C₄	—	2	2	no	—
130	C₂×C₄	—	2	2	no	—
131	C₂×C₄	—	4	2	no	—
132	C₂×C₄	—	4	2	no	—
133	C₂×C₄	—	4	2	no	—
134	C₂×C₄	—	4	2	no	—
135	C₂×C₄	#127	1	1	yes	characteristic
136	C₂×C₄	#127	1	1	yes	characteristic
137	C₂×C₄	#127	1	1	yes	characteristic
138	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
139	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
140	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
141	C₈	—	2	2	no	—
142	C₈	—	2	2	no	—
143	C₈	—	2	2	no	—
144	C₈	—	2	2	no	—
145	C₈	#135	1	4	yes	—
146	C₈	#137	1	4	yes	—
147	C₂²	—	2	1	no	—
148	C₂²	—	2	1	no	—
149	C₂²	—	2	1	no	—
150	C₂²	C₂×#127	1	1	yes	characteristic
151	C₄	—	2	1	no	—
152	C₄	—	2	1	no	—
153	C₄	—	2	1	no	—
154	C₄	—	8	2	no	—
155	C₄	—	8	2	no	—
156	C₄	—	8	2	no	—
157	C₄	—	8	2	no	—
158	C₄	C₂×#135	1	2	yes	—
159	C₄	C₂×#137	1	2	yes	—
160	C₄	#578	1	2	yes	—
161	C₂	—	4	1	no	—
162	C₂	#3147	1	1	yes	characteristic
163	C₂	#3152	1	1	yes	characteristic
164	C₂	#3156	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂×#127
commutator, Frattini, cosocle	C₄²	C₂⁴
prime radical	C₂³	#127
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
5. #1718	C₂	semidirect
6. #1716	C₂	semidirect
7. #1713	C₂	semidirect
8. #1710	C₂	semidirect
9. #1707	C₂	semidirect
10. #1707	C₂	semidirect

Maximal subgroups

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

subgroup	index
#3231	2
#3185	2
#3170	2
#3160	2
#1718	2
#1716	2
#1713	2
#1710	2
#1707	2
C₈×M₁₆	2