Finite Group Databasebeta

Subgroups — #19360

Order 256.

285 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 285 conjugacy classes of subgroups — too many to draw. Only its 109 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	#3252	C₂	1	2	yes	maximal
2	#3160	C₂	1	1	yes	maximal, characteristic, complemented
3	#2966	C₂	1	1	yes	maximal, characteristic
4	#1696	C₂	1	1	yes	maximal, characteristic, complemented
5	#1695	C₂	1	1	yes	maximal, characteristic, complemented
6	#1694	C₂	1	1	yes	maximal, characteristic, complemented
7	#1693	C₂	1	1	yes	maximal, characteristic, complemented
8	#1672	C₂	1	2	yes	maximal, complemented
9	#1671	C₂	1	2	yes	maximal, complemented
10	#1566	C₂	1	1	yes	maximal, characteristic, complemented
11	#1559	C₂	1	2	yes	maximal, complemented
12	#533	C₂²	1	2	yes	—
13	#529	C₂²	1	1	yes	characteristic
14	#500	C₂²	1	1	yes	characteristic
15	#500	C₂²	1	1	yes	characteristic
16	#499	—	2	1	no	—
17	#499	—	2	1	no	—
18	#498	C₂²	1	2	yes	—
19	#479	—	2	2	no	—
20	#477	C₂²	1	2	yes	—
21	#476	C₂²	1	2	yes	—
22	#470	—	2	2	no	—
23	#468	—	2	2	no	—
24	#464	—	2	2	no	—
25	#462	C₂²	1	2	yes	—
26	#461	C₂²	1	2	yes	—
27	#446	C₂²	1	1	yes	characteristic
28	C₈×Q₈	C₂²	1	1	yes	characteristic
29	#436	C₂²	1	1	yes	characteristic
30	#435	—	2	1	no	—
31	C₈×D₄	—	2	1	no	—
32	C₈×D₄	C₂²	1	1	yes	characteristic
33	#433	—	2	1	no	—
34	#432	C₂²	1	1	yes	characteristic
35	#427	—	2	1	no	—
36	#427	—	2	1	no	—
37	#423	C₂²	1	2	yes	—
38	#405	—	2	1	no	—
39	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
40	#334	C₂²	1	2	yes	—
41	#334	C₂²	1	1	yes	characteristic
42	#334	C₂²	1	1	yes	characteristic
43	#333	C₂²	1	2	yes	—
44	#333	C₂²	1	2	yes	—
45	#332	C₂²	1	2	yes	—
46	#332	C₂²	1	2	yes	—
47	C₂×Q₁₆	—	2	2	no	—
48	C₂×SD₁₆	—	2	2	no	—
49	C₂×M₁₆	—	2	2	no	—
50	C₂×M₁₆	—	2	1	no	—
51	C₂²×C₈	—	2	1	no	—
52	C₂²×C₈	D₄	1	1	yes	characteristic
53	C₂²×C₈	D₄	1	1	yes	characteristic
54	C₄⋊Q₈	C₂³	1	2	yes	—
55	C₄⋊Q₈	C₂³	1	2	yes	—
56	#126	—	2	1	no	—
57	#125	—	2	2	no	—
58	#125	D₄	1	1	yes	characteristic
59	#124	D₄	1	1	yes	characteristic
60	#123	—	2	1	no	—
61	C₂²⋊Q₈	—	2	2	no	—
62	C₂²⋊Q₈	—	2	2	no	—
63	C₂²⋊Q₈	D₄	1	1	yes	characteristic
64	#121	D₄	1	1	yes	characteristic
65	C₄×Q₈	—	2	2	no	—
66	C₄×Q₈	—	2	2	no	—
67	C₄×Q₈	C₂³	1	1	yes	characteristic
68	C₄×D₄	—	2	1	no	—
69	C₄×D₄	C₂³	1	1	yes	characteristic
70	#117	—	2	1	no	—
71	#117	—	2	2	no	—
72	C₂×C₄²	C₂³	1	1	yes	characteristic
73	#107	—	2	1	no	—
74	#107	—	2	1	no	—
75	#107	—	2	1	no	—
76	#107	—	2	1	no	—
77	#107	—	2	2	no	—
78	#106	—	2	2	no	—
79	#106	—	2	2	no	—
80	#106	—	2	2	no	—
81	C₄⋊C₈	—	2	1	no	—
82	C₄⋊C₈	—	2	1	no	—
83	C₄⋊C₈	—	2	2	no	—
84	C₄⋊C₈	C₂³	1	1	yes	characteristic
85	C₄⋊C₈	C₂³	1	1	yes	characteristic
86	C₄⋊C₈	C₂³	1	2	yes	—
87	C₄⋊C₈	C₂³	1	2	yes	—
88	#103	—	2	2	no	—
89	#103	—	2	2	no	—
90	#103	—	2	2	no	—
91	#102	—	2	2	no	—
92	C₂²⋊C₈	—	2	1	no	—
93	C₂²⋊C₈	—	2	1	no	—
94	C₂²⋊C₈	—	2	1	no	—
95	#97	—	2	1	no	—
96	#97	—	2	1	no	—
97	C₄×C₈	—	2	1	no	—
98	C₄×C₈	D₄	1	1	yes	characteristic
99	C₄×C₈	D₄	1	1	yes	characteristic
100	C₄×C₈	C₂³	1	1	yes	characteristic
101	C₄×C₈	C₂³	1	1	yes	characteristic
102	C₂×Q₈	—	2	2	no	—
103	C₂×Q₈	—	2	2	no	—
104	C₂×Q₈	C₂×D₄	1	1	yes	characteristic
105	C₂×D₄	—	2	1	no	—
106	C₂×D₄	C₂×D₄	1	1	yes	characteristic
107	C₂²×C₄	—	2	1	no	—
108	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
109	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
110	C₂²×C₄	#41	1	1	yes	characteristic
111	Q₁₆	—	8	2	no	—
112	SD₁₆	—	8	2	no	—
113	M₁₆	—	4	2	no	—
114	M₁₆	—	8	2	no	—
115	C₂×C₈	—	2	2	no	—
116	C₂×C₈	—	2	2	no	—
117	C₂×C₈	—	2	1	no	—
118	C₂×C₈	—	2	1	no	—
119	C₂×C₈	—	2	2	no	—
120	C₂×C₈	—	2	2	no	—
121	C₂×C₈	—	4	2	no	—
122	C₂×C₈	C₂×D₄	1	1	yes	characteristic
123	C₂×C₈	C₂×D₄	1	1	yes	characteristic
124	C₂×C₈	C₂×D₄	1	1	yes	characteristic
125	C₂×C₈	C₂×D₄	1	1	yes	characteristic
126	C₄⋊C₄	—	2	1	no	—
127	C₄⋊C₄	—	2	1	no	—
128	C₄⋊C₄	—	2	1	no	—
129	C₄⋊C₄	—	2	1	no	—
130	C₄⋊C₄	—	2	2	no	—
131	C₄⋊C₄	—	2	2	no	—
132	C₄⋊C₄	—	2	2	no	—
133	C₄⋊C₄	—	2	2	no	—
134	C₄⋊C₄	—	4	2	no	—
135	C₄⋊C₄	—	4	2	no	—
136	C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
137	C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
138	C₂²⋊C₄	—	2	1	no	—
139	C₂²⋊C₄	—	2	1	no	—
140	C₂²⋊C₄	—	2	1	no	—
141	C₂²⋊C₄	—	2	1	no	—
142	C₂²⋊C₄	—	2	1	no	—
143	C₂²⋊C₄	—	4	2	no	—
144	C₄²	—	2	1	no	—
145	C₄²	—	4	2	no	—
146	C₄²	C₂×D₄	1	1	yes	characteristic
147	C₄²	C₂×D₄	1	1	yes	characteristic
148	C₄²	#41	1	1	yes	characteristic
149	C₄²	C₂⁴	1	1	yes	characteristic
150	C₂³	—	2	1	no	—
151	C₂³	#121	1	1	yes	characteristic
152	Q₈	—	2	2	no	—
153	Q₈	—	4	2	no	—
154	Q₈	—	4	2	no	—
155	Q₈	—	8	2	no	—
156	D₄	—	2	2	no	—
157	D₄	—	4	2	no	—
158	C₂×C₄	—	2	1	no	—
159	C₂×C₄	—	2	1	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	2	1	no	—
162	C₂×C₄	—	2	1	no	—
163	C₂×C₄	—	2	1	no	—
164	C₂×C₄	—	4	1	no	—
165	C₂×C₄	—	4	1	no	—
166	C₂×C₄	—	4	1	no	—
167	C₂×C₄	—	4	2	no	—
168	C₂×C₄	—	4	2	no	—
169	C₂×C₄	#121	1	1	yes	characteristic
170	C₂×C₄	#121	1	1	yes	characteristic
171	C₂×C₄	#121	1	1	yes	characteristic
172	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
173	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
174	C₂×C₄	C₂×#41	1	1	yes	characteristic
175	C₈	—	2	1	no	—
176	C₈	—	2	1	no	—
177	C₈	—	4	1	no	—
178	C₈	—	4	1	no	—
179	C₈	—	4	2	no	—
180	C₈	—	4	2	no	—
181	C₈	#137	1	2	yes	—
182	C₈	#137	1	2	yes	—
183	C₂²	—	2	1	no	—
184	C₂²	—	2	1	no	—
185	C₂²	—	2	1	no	—
186	C₂²	—	4	1	no	—
187	C₂²	—	4	1	no	—
188	C₂²	—	4	1	no	—
189	C₂²	C₂×#121	1	1	yes	characteristic
190	C₄	—	2	1	no	—
191	C₄	—	2	1	no	—
192	C₄	—	2	1	no	—
193	C₄	—	2	1	no	—
194	C₄	—	4	1	no	—
195	C₄	—	4	1	no	—
196	C₄	—	4	1	no	—
197	C₄	—	8	2	no	—
198	C₄	—	8	2	no	—
199	C₄	C₂×#137	1	1	yes	characteristic
200	C₄	C₂×#137	1	1	yes	characteristic
201	C₄	#577	1	2	yes	—
202	C₂	—	4	1	no	—
203	C₂	—	8	1	no	—
204	C₂	#3039	1	1	yes	characteristic
205	C₂	#3060	1	1	yes	characteristic
206	C₂	#3067	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. #3160	C₂	semidirect
4. #1696	C₂	semidirect
5. #1695	C₂	semidirect
6. #1694	C₂	semidirect
7. #1693	C₂	semidirect
8. #1672	C₂	semidirect
9. #1671	C₂	semidirect
10. #1566	C₂	semidirect
11. #1559	C₂	semidirect

Maximal subgroups

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

subgroup	index
#3252	2
#3160	2
#2966	2
#1696	2
#1695	2
#1694	2
#1693	2
#1672	2
#1671	2
#1566	2
#1559	2