Finite Group Databasebeta

Subgroups — #9281

Order 256.

228 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 59 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	#3207	C₂	1	1	yes	maximal, characteristic
2	#2981	C₂	1	1	yes	maximal, characteristic, complemented
3	#1505	C₂	1	1	yes	maximal, characteristic, complemented
4	#1505	C₂	1	1	yes	maximal, characteristic, complemented
5	#1496	C₂	1	1	yes	maximal, characteristic, complemented
6	#1287	C₂	1	1	yes	maximal, characteristic, complemented
7	#1274	C₂	1	1	yes	maximal, characteristic, complemented
8	#532	C₂²	1	1	yes	characteristic
9	#532	C₂²	1	1	yes	characteristic
10	#493	C₄	1	1	yes	characteristic, complemented
11	#492	C₄	1	1	yes	characteristic, complemented
12	#487	C₄	1	1	yes	characteristic, complemented
13	#486	C₄	1	1	yes	characteristic, complemented
14	#463	—	2	2	no	—
15	#449	—	2	2	no	—
16	#436	—	2	1	no	—
17	#435	—	2	2	no	—
18	C₈×D₄	—	2	1	no	—
19	#432	C₂²	1	1	yes	characteristic
20	#432	C₂²	1	1	yes	characteristic
21	#431	C₂²	1	1	yes	characteristic
22	#431	C₂²	1	1	yes	characteristic
23	#408	—	2	2	no	—
24	#376	C₂²	1	1	yes	characteristic
25	#327	—	2	2	no	—
26	#327	—	2	2	no	—
27	D₄⋊C₈	—	2	1	no	—
28	D₄⋊C₈	—	2	1	no	—
29	C₂×#41	—	2	1	no	—
30	C₂×#41	—	2	1	no	—
31	C₂×SD₁₆	—	2	2	no	—
32	C₂×D₈	—	2	2	no	—
33	C₂×M₁₆	—	2	2	no	—
34	C₂²×C₈	—	2	2	no	—
35	C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
36	C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
37	#127	C₂×C₄	1	1	yes	characteristic
38	#127	C₂×C₄	1	1	yes	characteristic
39	#124	—	2	1	no	—
40	#124	—	2	1	no	—
41	#121	—	2	1	no	—
42	#121	—	2	1	no	—
43	#121	—	2	1	no	—
44	#121	D₄	1	2	yes	—
45	C₄×D₄	—	2	1	no	—
46	C₄×D₄	—	2	1	no	—
47	C₄×D₄	—	2	1	no	—
48	C₄×D₄	D₄	1	2	yes	—
49	C₂×C₄²	—	2	1	no	—
50	C₂×C₄²	C₂³	1	1	yes	characteristic
51	C₄⋊C₈	—	2	1	no	—
52	C₄⋊C₈	—	2	1	no	—
53	C₄⋊C₈	—	2	1	no	—
54	C₄⋊C₈	—	2	1	no	—
55	#103	—	2	2	no	—
56	#102	—	2	2	no	—
57	C₂²⋊C₈	—	2	2	no	—
58	C₂²⋊C₈	—	2	2	no	—
59	C₂²⋊C₈	—	2	1	no	—
60	C₂²⋊C₈	—	2	1	no	—
61	C₂²⋊C₈	—	2	1	no	—
62	C₂²⋊C₈	D₄	1	2	yes	—
63	#97	—	2	1	no	—
64	#97	—	2	1	no	—
65	C₄×C₈	C₂×C₄	1	1	yes	characteristic
66	C₄×C₈	C₂×C₄	1	1	yes	characteristic
67	C₄×C₈	D₄	1	1	yes	characteristic
68	C₄×C₈	D₄	1	1	yes	characteristic
69	#95	—	2	1	no	—
70	#95	—	2	1	no	—
71	#41	—	8	2	no	—
72	#41	—	8	2	no	—
73	C₂×Q₈	—	2	1	no	—
74	C₂×Q₈	—	2	1	no	—
75	C₂×D₄	—	2	2	no	—
76	C₂×D₄	—	2	1	no	—
77	C₂×D₄	—	2	1	no	—
78	C₂×D₄	—	2	1	no	—
79	C₂×D₄	—	2	1	no	—
80	C₂×D₄	—	2	1	no	—
81	C₂×D₄	—	4	1	no	—
82	C₂×D₄	C₂²⋊C₄	1	2	yes	—
83	C₂²×C₄	—	2	2	no	—
84	C₂²×C₄	—	4	1	no	—
85	C₂²×C₄	—	4	1	no	—
86	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
87	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
88	C₂²×C₄	#41	1	1	yes	characteristic
89	SD₁₆	—	8	2	no	—
90	D₈	—	8	2	no	—
91	M₁₆	—	8	2	no	—
92	C₂×C₈	—	2	1	no	—
93	C₂×C₈	—	2	1	no	—
94	C₂×C₈	—	2	1	no	—
95	C₂×C₈	—	2	1	no	—
96	C₂×C₈	—	2	2	no	—
97	C₂×C₈	—	4	1	no	—
98	C₂×C₈	—	8	2	no	—
99	C₄⋊C₄	—	2	1	no	—
100	C₄⋊C₄	—	2	1	no	—
101	C₄⋊C₄	—	2	1	no	—
102	C₄⋊C₄	C₂²⋊C₄	1	2	yes	—
103	C₂²⋊C₄	—	2	2	no	—
104	C₂²⋊C₄	—	2	2	no	—
105	C₂²⋊C₄	—	4	1	no	—
106	C₂²⋊C₄	—	4	1	no	—
107	C₄²	—	2	2	no	—
108	C₄²	—	4	1	no	—
109	C₄²	C₂²×C₄	1	1	yes	characteristic
110	C₄²	C₂×D₄	1	1	yes	characteristic
111	C₄²	C₂×D₄	1	1	yes	characteristic
112	C₄²	#41	1	1	yes	characteristic
113	C₂³	—	2	2	no	—
114	C₂³	—	4	1	no	—
115	C₂³	#121	1	1	yes	characteristic
116	Q₈	—	4	2	no	—
117	Q₈	—	8	1	no	—
118	D₄	—	4	2	no	—
119	D₄	—	4	2	no	—
120	D₄	—	4	1	no	—
121	D₄	—	4	1	no	—
122	D₄	—	4	1	no	—
123	D₄	—	4	1	no	—
124	D₄	—	8	1	no	—
125	D₄	—	8	1	no	—
126	D₄	—	8	2	no	—
127	D₄	—	8	2	no	—
128	C₂×C₄	—	2	1	no	—
129	C₂×C₄	—	2	1	no	—
130	C₂×C₄	—	2	2	no	—
131	C₂×C₄	—	4	1	no	—
132	C₂×C₄	—	4	1	no	—
133	C₂×C₄	—	4	2	no	—
134	C₂×C₄	—	4	2	no	—
135	C₂×C₄	—	4	2	no	—
136	C₂×C₄	—	4	1	no	—
137	C₂×C₄	—	8	2	no	—
138	C₂×C₄	—	8	1	no	—
139	C₂×C₄	—	8	1	no	—
140	C₂×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
141	C₂×C₄	C₄×D₄	1	1	yes	characteristic
142	C₂×C₄	C₄×D₄	1	1	yes	characteristic
143	C₂×C₄	#120	1	1	yes	characteristic
144	C₂×C₄	#121	1	1	yes	characteristic
145	C₂×C₄	#123	1	1	yes	characteristic
146	C₈	—	4	1	no	—
147	C₈	—	4	2	no	—
148	C₈	—	4	1	no	—
149	C₈	—	4	1	no	—
150	C₈	—	4	1	no	—
151	C₈	—	8	1	no	—
152	C₂²	—	2	1	no	—
153	C₂²	—	2	1	no	—
154	C₂²	—	2	1	no	—
155	C₂²	—	4	2	no	—
156	C₂²	—	4	2	no	—
157	C₂²	—	4	2	no	—
158	C₂²	—	8	1	no	—
159	C₂²	—	8	1	no	—
160	C₂²	—	8	1	no	—
161	C₂²	#386	1	1	yes	characteristic
162	C₄	—	2	1	no	—
163	C₄	—	2	1	no	—
164	C₄	—	2	1	no	—
165	C₄	—	2	1	no	—
166	C₄	—	2	1	no	—
167	C₄	—	4	2	no	—
168	C₄	—	8	2	no	—
169	C₄	—	8	1	no	—
170	C₄	#443	1	1	yes	characteristic
171	C₄	#444	1	1	yes	characteristic
172	C₂	—	4	1	no	—
173	C₂	—	8	2	no	—
174	C₂	—	16	1	no	—
175	C₂	#1876	1	1	yes	characteristic
176	C₂	#1904	1	1	yes	characteristic
177	C₂	#1910	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	#386
commutator	C₄²	C₂²×C₄
Frattini, cosocle	C₂×C₄²	C₂³
prime radical	#3207	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
2. #2981	C₂	semidirect
3. #1505	C₂	semidirect
4. #1505	C₂	semidirect
5. #1496	C₂	semidirect
6. #1287	C₂	semidirect
7. #1274	C₂	semidirect
10. #493	C₄	semidirect
11. #492	C₄	semidirect
12. #487	C₄	semidirect
13. #486	C₄	semidirect

Maximal subgroups

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

subgroup	index
#3207	2
#2981	2
#1505	2
#1496	2
#1287	2
#1274	2