Finite Group Databasebeta

Subgroups — #19520

Order 256.

306 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 105 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	#3237	C₂	1	1	yes	maximal, characteristic, complemented
2	#3208	C₂	1	1	yes	maximal, characteristic
3	#3208	C₂	1	1	yes	maximal, characteristic
4	#2985	C₂	1	1	yes	maximal, characteristic, complemented
5	#1698	C₂	1	2	yes	maximal, complemented
6	#1686	C₂	1	2	yes	maximal, complemented
7	#1678	C₂	1	2	yes	maximal, complemented
8	#1666	C₂	1	2	yes	maximal, complemented
9	#1572	C₂	1	1	yes	maximal, characteristic, complemented
10	#1555	C₂	1	1	yes	maximal, characteristic
11	#1554	C₂	1	1	yes	maximal, characteristic, complemented
12	#533	C₂²	1	1	yes	characteristic
13	#533	C₂²	1	1	yes	characteristic
14	C₂×#127	C₂²	1	1	yes	characteristic
15	#501	—	2	1	no	—
16	#492	—	2	1	no	—
17	#492	C₂²	1	1	yes	characteristic
18	#492	C₂²	1	1	yes	characteristic
19	#491	—	2	1	no	—
20	#486	—	2	1	no	—
21	#486	C₂²	1	1	yes	characteristic
22	#486	C₂²	1	1	yes	characteristic
23	#485	—	2	2	no	—
24	#477	C₂²	1	2	yes	complemented
25	#475	C₂²	1	2	yes	complemented
26	#460	—	2	2	no	—
27	#460	C₂²	1	2	yes	—
28	#460	C₂²	1	2	yes	—
29	#450	—	2	2	no	—
30	#450	—	2	2	no	—
31	#446	—	2	2	no	—
32	#446	C₂²	1	2	yes	complemented
33	C₈×Q₈	C₂²	1	2	yes	complemented
34	#433	—	2	2	no	—
35	#432	C₂²	1	1	yes	characteristic
36	#432	C₂²	1	1	yes	characteristic
37	#431	C₂²	1	1	yes	characteristic
38	#431	C₂²	1	1	yes	characteristic
39	#335	C₂²	1	2	yes	complemented
40	#334	C₂²	1	2	yes	complemented
41	#333	C₂²	1	2	yes	complemented
42	#332	C₂²	1	2	yes	complemented
43	#331	C₂²	1	2	yes	—
44	#331	C₂²	1	2	yes	—
45	C₂²×D₄	—	2	1	no	—
46	C₂²×D₄	—	2	1	no	—
47	C₂²×D₄	—	2	1	no	—
48	C₂×SD₁₆	—	2	2	no	—
49	C₂×SD₁₆	—	2	2	no	—
50	C₂×SD₁₆	—	2	2	no	—
51	C₂×SD₁₆	—	2	2	no	—
52	C₄⋊Q₈	D₄	1	1	yes	characteristic
53	C₄⋊Q₈	D₄	1	1	yes	characteristic
54	C₄⋊Q₈	C₂³	1	1	yes	characteristic
55	C₄⋊Q₈	C₂³	1	1	yes	characteristic
56	#127	—	2	1	no	—
57	#127	—	2	1	no	—
58	#127	—	2	1	no	—
59	#127	C₂³	1	1	yes	characteristic
60	#127	C₂³	1	1	yes	characteristic
61	#125	—	2	1	no	—
62	#125	—	2	1	no	—
63	C₂²⋊Q₈	—	2	1	no	—
64	C₂²⋊Q₈	—	2	1	no	—
65	C₂²⋊Q₈	—	2	1	no	—
66	C₂²⋊Q₈	D₄	1	2	yes	—
67	C₄×Q₈	—	2	1	no	—
68	C₄×Q₈	—	2	1	no	—
69	C₄×Q₈	—	2	1	no	—
70	C₄×Q₈	C₂³	1	2	yes	—
71	#117	—	2	1	no	—
72	#117	—	2	1	no	—
73	C₂×C₄²	C₂³	1	1	yes	characteristic
74	#107	—	2	2	no	—
75	#106	—	2	2	no	—
76	C₄⋊C₈	—	2	2	no	—
77	C₄⋊C₈	—	2	2	no	—
78	C₄⋊C₈	D₄	1	2	yes	—
79	C₄⋊C₈	C₂³	1	2	yes	—
80	C₄⋊C₈	C₂³	1	2	yes	—
81	C₄⋊C₈	C₂³	1	2	yes	—
82	#103	—	2	2	no	—
83	#103	—	2	2	no	—
84	#102	—	2	2	no	—
85	#102	—	2	2	no	—
86	#102	—	2	2	no	—
87	#102	—	2	2	no	—
88	C₂²⋊C₈	—	2	1	no	—
89	C₂²⋊C₈	—	2	1	no	—
90	C₂²⋊C₈	—	2	1	no	—
91	C₂²⋊C₈	D₄	1	2	yes	—
92	#97	—	2	2	no	—
93	#97	—	2	1	no	—
94	#97	—	2	1	no	—
95	C₄×C₈	—	2	2	no	—
96	C₄×C₈	—	2	1	no	—
97	C₄×C₈	C₂³	1	1	yes	characteristic
98	C₄×C₈	C₂³	1	1	yes	characteristic
99	C₂⁴	—	4	1	no	—
100	C₂×Q₈	—	2	1	no	—
101	C₂×Q₈	—	2	1	no	—
102	C₂×Q₈	—	2	1	no	—
103	C₂×Q₈	C₂×D₄	1	2	yes	—
104	C₂×D₄	—	2	1	no	—
105	C₂×D₄	—	2	1	no	—
106	C₂×D₄	—	2	1	no	—
107	C₂×D₄	—	2	1	no	—
108	C₂×D₄	—	2	1	no	—
109	C₂×D₄	—	2	1	no	—
110	C₂×D₄	—	4	1	no	—
111	C₂×D₄	—	4	1	no	—
112	C₂×D₄	—	4	1	no	—
113	C₂×D₄	—	8	2	no	—
114	C₂×D₄	—	8	2	no	—
115	C₂×D₄	—	8	2	no	—
116	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
117	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
118	C₂²×C₄	#41	1	1	yes	characteristic
119	SD₁₆	—	8	2	no	—
120	SD₁₆	—	8	2	no	—
121	SD₁₆	—	8	2	no	—
122	SD₁₆	—	8	2	no	—
123	C₂×C₈	—	2	2	no	—
124	C₂×C₈	—	2	2	no	—
125	C₂×C₈	—	2	2	no	—
126	C₂×C₈	C₂×D₄	1	2	yes	—
127	C₂×C₈	C₂×D₄	1	2	yes	—
128	C₄⋊C₄	—	2	1	no	—
129	C₄⋊C₄	—	2	1	no	—
130	C₄⋊C₄	—	2	1	no	—
131	C₄⋊C₄	—	2	2	no	—
132	C₄⋊C₄	—	2	2	no	—
133	C₄⋊C₄	—	2	1	no	—
134	C₄⋊C₄	—	2	1	no	—
135	C₄⋊C₄	—	2	1	no	—
136	C₄⋊C₄	—	2	1	no	—
137	C₄⋊C₄	—	4	1	no	—
138	C₄⋊C₄	—	4	1	no	—
139	C₄⋊C₄	C₂×D₄	1	2	yes	—
140	C₂²⋊C₄	—	2	2	no	—
141	C₂²⋊C₄	—	4	1	no	—
142	C₄²	—	2	2	no	—
143	C₄²	—	4	1	no	—
144	C₄²	C₂×D₄	1	1	yes	characteristic
145	C₄²	C₂×D₄	1	1	yes	characteristic
146	C₄²	#41	1	1	yes	characteristic
147	C₄²	C₂⁴	1	1	yes	characteristic
148	C₂³	—	4	1	no	—
149	C₂³	—	4	1	no	—
150	C₂³	—	8	2	no	—
151	C₂³	—	8	2	no	—
152	C₂³	—	8	2	no	—
153	C₂³	#121	1	1	yes	characteristic
154	Q₈	—	2	2	no	—
155	Q₈	—	2	2	no	—
156	Q₈	—	4	2	no	—
157	Q₈	—	8	1	no	—
158	Q₈	—	8	1	no	—
159	D₄	—	4	2	no	—
160	D₄	—	4	2	no	—
161	D₄	—	4	2	no	—
162	D₄	—	4	2	no	—
163	D₄	—	8	1	no	—
164	D₄	—	8	1	no	—
165	D₄	—	8	2	no	—
166	D₄	—	8	2	no	—
167	D₄	—	8	2	no	—
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	2	2	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	4	1	no	—
173	C₂×C₄	—	4	1	no	—
174	C₂×C₄	—	4	1	no	—
175	C₂×C₄	#121	1	1	yes	characteristic
176	C₂×C₄	#121	1	1	yes	characteristic
177	C₂×C₄	#121	1	1	yes	characteristic
178	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
179	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
180	C₂×C₄	C₂×#41	1	1	yes	characteristic
181	C₈	—	2	2	no	—
182	C₈	—	2	2	no	—
183	C₈	—	4	2	no	—
184	C₈	—	4	2	no	—
185	C₈	—	4	2	no	—
186	C₂²	—	2	1	no	—
187	C₂²	—	2	1	no	—
188	C₂²	—	2	1	no	—
189	C₂²	—	8	1	no	—
190	C₂²	—	8	1	no	—
191	C₂²	—	8	1	no	—
192	C₂²	—	8	1	no	—
193	C₂²	—	8	1	no	—
194	C₂²	—	8	1	no	—
195	C₂²	—	16	4	no	—
196	C₂²	C₂×#121	1	1	yes	characteristic
197	C₄	—	2	1	no	—
198	C₄	—	2	1	no	—
199	C₄	—	2	1	no	—
200	C₄	—	2	1	no	—
201	C₄	—	4	2	no	—
202	C₄	—	4	2	no	—
203	C₄	—	8	1	no	—
204	C₄	—	8	1	no	—
205	C₄	#577	1	2	yes	—
206	C₄	#577	1	2	yes	—
207	C₂	—	4	1	no	—
208	C₂	—	16	1	no	—
209	C₂	—	16	1	no	—
210	C₂	#3043	1	1	yes	characteristic
211	C₂	#3057	1	1	yes	characteristic
212	C₂	#3068	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂×#121
commutator, Frattini, cosocle	C₄²	C₂⁴
prime radical	C₂×#127	C₂²
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
1. #3237	C₂	semidirect
4. #2985	C₂	semidirect
5. #1698	C₂	semidirect
6. #1686	C₂	semidirect
7. #1678	C₂	semidirect
8. #1666	C₂	semidirect
9. #1572	C₂	semidirect
11. #1554	C₂	semidirect
24. #477	C₂²	semidirect
25. #475	C₂²	semidirect
32. #446	C₂²	semidirect
33. C₈×Q₈	C₂²	semidirect
39. #335	C₂²	semidirect
40. #334	C₂²	semidirect
41. #333	C₂²	semidirect
42. #332	C₂²	semidirect

Maximal subgroups

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

subgroup	index
#3237	2
#3208	2
#2985	2
#1698	2
#1686	2
#1678	2
#1666	2
#1572	2
#1555	2
#1554	2