Finite Group Databasebeta

Subgroups — #19358

Order 256.

304 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 304 conjugacy classes of subgroups — too many to draw. Only its 111 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	#3218	C₂	1	2	yes	maximal
2	C₂×#500	C₂	1	1	yes	maximal, characteristic, complemented
3	#2966	C₂	1	1	yes	maximal, characteristic
4	#1694	C₂	1	2	yes	maximal, complemented
5	#1693	C₂	1	2	yes	maximal, complemented
6	#1674	C₂	1	2	yes	maximal, complemented
7	#1673	C₂	1	2	yes	maximal, complemented
8	#1568	C₂	1	1	yes	maximal, characteristic, complemented
9	#1556	C₂	1	2	yes	maximal, complemented
10	C₂×C₄⋊Q₈	C₂²	1	2	yes	—
11	#529	C₂²	1	1	yes	characteristic
12	#500	—	2	1	no	—
13	#500	—	2	1	no	—
14	#500	C₂²	1	2	yes	—
15	#500	C₂²	1	2	yes	—
16	#497	—	2	2	no	—
17	#491	—	2	2	no	—
18	#462	C₂²	1	4	yes	—
19	#461	C₂²	1	4	yes	—
20	#451	—	2	2	no	—
21	#448	—	2	2	no	—
22	#446	C₂²	1	1	yes	characteristic
23	C₈×Q₈	C₂²	1	1	yes	characteristic
24	#436	C₂²	1	1	yes	characteristic
25	#435	—	2	1	no	—
26	C₈×D₄	—	2	1	no	—
27	C₈×D₄	C₂²	1	1	yes	characteristic
28	#433	—	2	1	no	—
29	#432	C₂²	1	1	yes	characteristic
30	#432	C₂²	1	2	yes	—
31	C₂×#107	—	2	1	no	—
32	C₂×#107	—	2	1	no	—
33	#405	—	2	1	no	—
34	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
35	#335	C₂²	1	2	yes	—
36	#334	C₂²	1	2	yes	—
37	#332	C₂²	1	4	yes	—
38	#332	C₂²	1	4	yes	—
39	C₂²×Q₈	—	2	2	no	—
40	C₂×Q₁₆	—	2	4	no	—
41	C₂×SD₁₆	—	2	4	no	—
42	C₂×M₁₆	—	2	1	no	—
43	C₂²×C₈	—	2	1	no	—
44	C₂²×C₈	D₄	1	1	yes	characteristic
45	C₂²×C₈	D₄	1	1	yes	characteristic
46	C₄⋊Q₈	—	2	2	no	—
47	C₄⋊Q₈	—	2	2	no	—
48	C₄⋊Q₈	—	2	2	no	—
49	C₄⋊Q₈	C₂³	1	4	yes	—
50	#126	—	2	1	no	—
51	#125	D₄	1	1	yes	characteristic
52	#124	D₄	1	1	yes	characteristic
53	#123	—	2	1	no	—
54	C₂²⋊Q₈	D₄	1	1	yes	characteristic
55	#121	D₄	1	1	yes	characteristic
56	C₄×Q₈	C₂³	1	1	yes	characteristic
57	C₄×D₄	—	2	1	no	—
58	C₄×D₄	C₂³	1	1	yes	characteristic
59	#117	—	2	1	no	—
60	C₂×C₄⋊C₄	—	2	2	no	—
61	C₂×C₄⋊C₄	—	2	2	no	—
62	C₂×C₄²	C₂³	1	1	yes	characteristic
63	#107	—	2	2	no	—
64	#107	—	2	2	no	—
65	#107	—	2	2	no	—
66	#107	—	2	2	no	—
67	C₄⋊C₈	—	2	1	no	—
68	C₄⋊C₈	—	2	1	no	—
69	C₄⋊C₈	C₂³	1	1	yes	characteristic
70	C₄⋊C₈	C₂³	1	1	yes	characteristic
71	C₄⋊C₈	C₂³	1	4	yes	—
72	#103	—	2	4	no	—
73	#103	—	2	4	no	—
74	C₂²⋊C₈	—	2	1	no	—
75	C₂²⋊C₈	—	2	2	no	—
76	C₂²⋊C₈	—	2	1	no	—
77	C₂²⋊C₈	—	2	1	no	—
78	#97	—	2	1	no	—
79	#97	—	2	1	no	—
80	#97	—	2	2	no	—
81	C₄×C₈	—	2	1	no	—
82	C₄×C₈	D₄	1	1	yes	characteristic
83	C₄×C₈	D₄	1	1	yes	characteristic
84	C₄×C₈	C₂³	1	1	yes	characteristic
85	C₄×C₈	C₂³	1	1	yes	characteristic
86	C₂×Q₈	—	2	4	no	—
87	C₂×Q₈	—	4	2	no	—
88	C₂×Q₈	—	8	4	no	—
89	C₂×Q₈	C₂×D₄	1	1	yes	characteristic
90	C₂×D₄	—	2	1	no	—
91	C₂×D₄	C₂×D₄	1	1	yes	characteristic
92	C₂²×C₄	—	2	1	no	—
93	C₂²×C₄	—	4	2	no	—
94	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
95	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
96	C₂²×C₄	#41	1	1	yes	characteristic
97	Q₁₆	—	8	4	no	—
98	SD₁₆	—	8	4	no	—
99	M₁₆	—	4	2	no	—
100	C₂×C₈	—	2	2	no	—
101	C₂×C₈	—	2	2	no	—
102	C₂×C₈	—	2	1	no	—
103	C₂×C₈	—	2	1	no	—
104	C₂×C₈	—	2	4	no	—
105	C₂×C₈	—	4	2	no	—
106	C₂×C₈	C₂×D₄	1	1	yes	characteristic
107	C₂×C₈	C₂×D₄	1	1	yes	characteristic
108	C₂×C₈	C₂×D₄	1	1	yes	characteristic
109	C₂×C₈	C₂×D₄	1	1	yes	characteristic
110	C₄⋊C₄	—	2	1	no	—
111	C₄⋊C₄	—	2	1	no	—
112	C₄⋊C₄	—	2	1	no	—
113	C₄⋊C₄	—	2	1	no	—
114	C₄⋊C₄	—	2	4	no	—
115	C₄⋊C₄	—	2	4	no	—
116	C₄⋊C₄	—	4	2	no	—
117	C₄⋊C₄	—	4	2	no	—
118	C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
119	C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
120	C₂²⋊C₄	—	2	1	no	—
121	C₂²⋊C₄	—	2	1	no	—
122	C₂²⋊C₄	—	2	1	no	—
123	C₂²⋊C₄	—	2	1	no	—
124	C₂²⋊C₄	—	2	1	no	—
125	C₄²	—	2	1	no	—
126	C₄²	C₂×D₄	1	1	yes	characteristic
127	C₄²	C₂×D₄	1	1	yes	characteristic
128	C₄²	#41	1	1	yes	characteristic
129	C₄²	C₂⁴	1	1	yes	characteristic
130	C₂³	—	2	1	no	—
131	C₂³	#121	1	1	yes	characteristic
132	Q₈	—	2	2	no	—
133	Q₈	—	4	4	no	—
134	Q₈	—	4	4	no	—
135	Q₈	—	8	4	no	—
136	D₄	—	2	2	no	—
137	D₄	—	4	2	no	—
138	C₂×C₄	—	2	2	no	—
139	C₂×C₄	—	2	1	no	—
140	C₂×C₄	—	2	1	no	—
141	C₂×C₄	—	2	2	no	—
142	C₂×C₄	—	2	1	no	—
143	C₂×C₄	—	2	1	no	—
144	C₂×C₄	—	2	1	no	—
145	C₂×C₄	—	4	1	no	—
146	C₂×C₄	—	4	1	no	—
147	C₂×C₄	—	4	4	no	—
148	C₂×C₄	—	8	4	no	—
149	C₂×C₄	#121	1	1	yes	characteristic
150	C₂×C₄	#121	1	1	yes	characteristic
151	C₂×C₄	#121	1	1	yes	characteristic
152	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
153	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
154	C₂×C₄	C₂×#41	1	1	yes	characteristic
155	C₈	—	2	1	no	—
156	C₈	—	2	1	no	—
157	C₈	—	4	1	no	—
158	C₈	—	4	1	no	—
159	C₈	—	4	4	no	—
160	C₈	#137	1	2	yes	—
161	C₈	#137	1	2	yes	—
162	C₂²	—	2	1	no	—
163	C₂²	—	2	1	no	—
164	C₂²	—	4	1	no	—
165	C₂²	—	4	1	no	—
166	C₂²	—	4	1	no	—
167	C₂²	#473	1	2	yes	—
168	C₂²	C₂×#121	1	1	yes	characteristic
169	C₄	—	2	1	no	—
170	C₄	—	2	1	no	—
171	C₄	—	2	1	no	—
172	C₄	—	2	1	no	—
173	C₄	—	4	1	no	—
174	C₄	—	4	1	no	—
175	C₄	—	4	1	no	—
176	C₄	—	8	4	no	—
177	C₄	C₂×#137	1	1	yes	characteristic
178	C₄	C₂×#137	1	1	yes	characteristic
179	C₄	#576	1	2	yes	—
180	C₂	—	2	2	no	—
181	C₂	—	8	1	no	—
182	C₂	#3039	1	1	yes	characteristic
183	C₂	#3058	1	1	yes	characteristic
184	C₂	C₂×#473	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂×#121
commutator, Frattini, cosocle	C₄²	C₂⁴
prime radical	#121	D₄
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. C₂×#500	C₂	semidirect
4. #1694	C₂	semidirect
5. #1693	C₂	semidirect
6. #1674	C₂	semidirect
7. #1673	C₂	semidirect
8. #1568	C₂	semidirect
9. #1556	C₂	semidirect

Maximal subgroups

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

subgroup	index
#3218	2
C₂×#500	2
#2966	2
#1694	2
#1693	2
#1674	2
#1673	2
#1568	2
#1556	2