Finite Group Databasebeta

Subgroups — #19264

Order 256.

325 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 325 conjugacy classes of subgroups — too many to draw. Only its 113 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	#3169	C₂	1	1	yes	maximal, characteristic
2	#3112	C₂	1	2	yes	maximal, complemented
3	C₂×C₈×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
4	#1678	C₂	1	4	yes	maximal, complemented
5	#1676	C₂	1	2	yes	maximal, complemented
6	#1668	C₂	1	2	yes	maximal, complemented
7	#1568	C₂	1	1	yes	maximal, characteristic, complemented
8	#1551	C₂	1	2	yes	maximal, complemented
9	#532	C₂²	1	2	yes	—
10	C₂×C₄×Q₈	C₂²	1	1	yes	characteristic
11	#500	C₂²	1	1	yes	characteristic
12	#493	C₂²	1	1	yes	characteristic
13	#486	C₂²	1	2	yes	—
14	#480	—	2	2	no	—
15	#477	C₂²	1	4	yes	—
16	#466	—	2	1	no	—
17	#466	—	2	1	no	—
18	#460	C₂²	1	4	yes	—
19	#448	—	2	2	no	—
20	C₈×Q₈	—	2	1	no	—
21	C₈×Q₈	—	2	1	no	—
22	C₈×Q₈	C₂²	1	2	yes	—
23	C₈×Q₈	C₂²	1	2	yes	—
24	#440	—	2	4	no	—
25	C₄×D₈	—	2	1	no	—
26	C₄×D₈	—	2	1	no	—
27	#432	C₂²	1	2	yes	—
28	C₂×C₄⋊C₈	—	2	1	no	—
29	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
30	C₂×C₄×C₈	—	2	1	no	—
31	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
32	#335	C₂²	1	2	yes	—
33	#334	C₂²	1	2	yes	—
34	#333	C₂²	1	4	yes	—
35	#331	C₂²	1	4	yes	—
36	C₂×#41	—	2	2	no	—
37	C₂²×Q₈	D₄	1	1	yes	characteristic
38	C₂×SD₁₆	—	2	4	no	—
39	C₂×D₈	—	2	1	no	—
40	C₂×D₈	—	2	1	no	—
41	C₂²×C₈	—	2	1	no	—
42	C₂²×C₈	D₄	1	1	yes	characteristic
43	C₂²×C₈	D₄	1	1	yes	characteristic
44	C₄⋊Q₈	C₂³	1	2	yes	—
45	#127	C₂³	1	2	yes	—
46	#124	—	2	2	no	—
47	#121	—	2	2	no	—
48	#121	—	2	2	no	—
49	C₄×Q₈	—	2	1	no	—
50	C₄×Q₈	—	2	1	no	—
51	C₄×Q₈	D₄	1	2	yes	—
52	C₄×Q₈	C₂³	1	2	yes	—
53	C₄×D₄	—	2	2	no	—
54	C₄×D₄	—	2	2	no	—
55	C₂×C₄⋊C₄	—	2	1	no	—
56	C₂×C₄⋊C₄	D₄	1	1	yes	characteristic
57	C₂×C₄²	—	2	1	no	—
58	C₂×C₄²	C₂³	1	1	yes	characteristic
59	#107	—	2	1	no	—
60	#107	—	2	1	no	—
61	#107	—	2	4	no	—
62	C₄⋊C₈	—	2	2	no	—
63	C₄⋊C₈	—	2	2	no	—
64	C₄⋊C₈	—	2	1	no	—
65	C₄⋊C₈	C₂³	1	2	yes	—
66	C₄⋊C₈	C₂³	1	4	yes	—
67	#103	—	2	4	no	—
68	#102	—	2	2	no	—
69	#102	—	2	2	no	—
70	#102	—	2	4	no	—
71	C₂²⋊C₈	—	2	2	no	—
72	#97	—	2	2	no	—
73	C₄×C₈	—	2	2	no	—
74	C₄×C₈	—	2	2	no	—
75	C₄×C₈	D₄	1	1	yes	characteristic
76	C₄×C₈	D₄	1	1	yes	characteristic
77	C₄×C₈	C₂³	1	1	yes	characteristic
78	C₄×C₈	C₂³	1	1	yes	characteristic
79	#41	—	8	4	no	—
80	C₂×Q₈	—	2	4	no	—
81	C₂×Q₈	—	2	1	no	—
82	C₂×Q₈	—	2	2	no	—
83	C₂×Q₈	C₂×D₄	1	2	yes	—
84	C₂×D₄	—	2	2	no	—
85	C₂×D₄	—	2	2	no	—
86	C₂×D₄	—	2	2	no	—
87	C₂×D₄	—	4	2	no	—
88	C₂²×C₄	—	2	1	no	—
89	C₂²×C₄	—	2	1	no	—
90	C₂²×C₄	—	4	2	no	—
91	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
92	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
93	C₂²×C₄	#41	1	1	yes	characteristic
94	SD₁₆	—	4	8	no	—
95	D₈	—	4	2	no	—
96	D₈	—	4	2	no	—
97	C₂×C₈	—	2	2	no	—
98	C₂×C₈	—	2	2	no	—
99	C₂×C₈	—	2	4	no	—
100	C₂×C₈	—	4	1	no	—
101	C₂×C₈	—	4	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	1	no	—
108	C₄⋊C₄	—	2	2	no	—
109	C₄⋊C₄	—	2	2	no	—
110	C₄⋊C₄	—	2	2	no	—
111	C₄⋊C₄	C₂×D₄	1	2	yes	—
112	C₂²⋊C₄	—	4	2	no	—
113	C₂²⋊C₄	—	4	2	no	—
114	C₄²	—	2	2	no	—
115	C₄²	—	2	2	no	—
116	C₄²	C₂×D₄	1	1	yes	characteristic
117	C₄²	C₂×D₄	1	1	yes	characteristic
118	C₄²	#41	1	1	yes	characteristic
119	C₄²	C₂⁴	1	1	yes	characteristic
120	C₂³	—	4	2	no	—
121	C₂³	#121	1	1	yes	characteristic
122	Q₈	—	2	4	no	—
123	Q₈	—	2	2	no	—
124	Q₈	—	8	2	no	—
125	Q₈	#135	1	4	yes	—
126	D₄	—	4	4	no	—
127	D₄	—	4	2	no	—
128	D₄	—	4	2	no	—
129	D₄	—	8	2	no	—
130	D₄	—	8	4	no	—
131	C₂×C₄	—	2	4	no	—
132	C₂×C₄	—	2	2	no	—
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	2	1	no	—
136	C₂×C₄	—	2	1	no	—
137	C₂×C₄	—	2	2	no	—
138	C₂×C₄	—	4	2	no	—
139	C₂×C₄	—	4	2	no	—
140	C₂×C₄	—	8	4	no	—
141	C₂×C₄	#121	1	1	yes	characteristic
142	C₂×C₄	#121	1	1	yes	characteristic
143	C₂×C₄	#121	1	1	yes	characteristic
144	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
145	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
146	C₂×C₄	C₂×#41	1	1	yes	characteristic
147	C₈	—	2	1	no	—
148	C₈	—	2	1	no	—
149	C₈	—	2	1	no	—
150	C₈	—	4	4	no	—
151	C₈	—	4	2	no	—
152	C₈	#137	1	2	yes	—
153	C₂²	—	2	1	no	—
154	C₂²	—	2	1	no	—
155	C₂²	—	8	2	no	—
156	C₂²	—	8	2	no	—
157	C₂²	—	8	2	no	—
158	C₂²	#472	1	2	yes	—
159	C₂²	C₂×#121	1	1	yes	characteristic
160	C₄	—	2	1	no	—
161	C₄	—	2	4	no	—
162	C₄	—	2	1	no	—
163	C₄	—	2	1	no	—
164	C₄	—	2	1	no	—
165	C₄	—	4	2	no	—
166	C₄	—	8	2	no	—
167	C₄	C₂×#135	1	2	yes	—
168	C₄	C₂×#137	1	1	yes	characteristic
169	C₄	#575	1	1	yes	characteristic
170	C₂	—	2	2	no	—
171	C₂	—	16	2	no	—
172	C₂	#3037	1	1	yes	characteristic
173	C₂	#3050	1	1	yes	characteristic
174	C₂	C₂×#472	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂×#121
commutator, Frattini, cosocle	C₄²	C₂⁴
prime radical	#3169	C₂
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
2. #3112	C₂	semidirect
3. C₂×C₈×Q₈	C₂	semidirect
4. #1678	C₂	semidirect
5. #1676	C₂	semidirect
6. #1668	C₂	semidirect
7. #1568	C₂	semidirect
8. #1551	C₂	semidirect

Maximal subgroups

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

subgroup	index
#3169	2
#3112	2
C₂×C₈×Q₈	2
#1678	2
#1676	2
#1668	2
#1568	2
#1551	2