Finite Group Databasebeta

Subgroups — #19426

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	#3229	C₂	1	1	yes	maximal, characteristic, complemented
2	#3077	C₂	1	1	yes	maximal, characteristic
3	C₂×#461	C₂	1	1	yes	maximal, characteristic
4	#2975	C₂	1	1	yes	maximal, characteristic
5	#1699	C₂	1	4	yes	maximal, complemented
6	#1691	C₂	1	4	yes	maximal, complemented
7	#1571	C₂	1	1	yes	maximal, characteristic, complemented
8	#1546	C₂	1	1	yes	maximal, characteristic, complemented
9	#1542	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×SD₁₆	—	2	1	no	—
11	#533	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊Q₈	C₂²	1	1	yes	characteristic
13	C₂×C₄×D₄	C₂²	1	1	yes	characteristic
14	#498	C₂²	1	2	yes	complemented
15	#487	C₂²	1	2	yes	complemented
16	#483	—	2	2	no	—
17	#478	—	2	2	no	—
18	#476	C₂²	1	2	yes	—
19	#475	—	2	2	no	—
20	#474	C₂²	1	2	yes	—
21	#461	—	2	2	no	—
22	#461	C₂²	1	4	yes	—
23	#436	—	2	2	no	—
24	#436	C₂²	1	4	yes	—
25	#431	C₂²	1	1	yes	characteristic
26	#431	C₂²	1	1	yes	characteristic
27	#428	—	2	1	no	—
28	#423	C₂²	1	1	yes	characteristic
29	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
30	#418	—	2	1	no	—
31	C₂×#103	—	2	1	no	—
32	#407	—	2	2	no	—
33	#335	C₂²	1	4	yes	complemented
34	#333	C₂²	1	2	yes	complemented
35	#332	C₂²	1	2	yes	complemented
36	#332	C₂²	1	4	yes	complemented
37	C₂²×Q₈	—	2	1	no	—
38	C₂²×D₄	D₄	1	1	yes	characteristic
39	C₂³×C₄	—	2	1	no	—
40	C₂×SD₁₆	—	2	4	no	—
41	C₂×SD₁₆	—	4	4	no	—
42	C₂×M₁₆	—	2	1	no	—
43	C₂×M₁₆	—	2	4	no	—
44	C₂²×C₈	—	2	1	no	—
45	C₄⋊Q₈	—	2	1	no	—
46	C₄⋊Q₈	—	2	1	no	—
47	C₄⋊Q₈	—	2	1	no	—
48	C₄⋊Q₈	C₂³	1	2	yes	—
49	C₄⋊Q₈	C₂³	1	1	yes	characteristic
50	C₄⋊Q₈	C₂³	1	1	yes	characteristic
51	#125	—	2	1	no	—
52	C₂²⋊Q₈	—	2	1	no	—
53	C₂²⋊Q₈	—	2	1	no	—
54	C₄×Q₈	—	2	1	no	—
55	C₄×Q₈	—	2	1	no	—
56	C₄×D₄	—	2	1	no	—
57	C₄×D₄	—	2	1	no	—
58	C₄×D₄	D₄	1	2	yes	—
59	C₄×D₄	C₂³	1	2	yes	—
60	#117	—	2	1	no	—
61	C₂×C₄⋊C₄	—	2	1	no	—
62	C₂×C₄⋊C₄	—	2	1	no	—
63	C₂×C₄⋊C₄	D₄	1	1	yes	characteristic
64	C₂×C₂²⋊C₄	—	2	1	no	—
65	C₂×C₄²	C₂³	1	1	yes	characteristic
66	#107	—	2	2	no	—
67	#106	—	2	4	no	—
68	#106	—	2	2	no	—
69	C₄⋊C₈	—	2	1	no	—
70	C₄⋊C₈	—	2	1	no	—
71	C₄⋊C₈	D₄	1	2	yes	—
72	C₄⋊C₈	C₂³	1	2	yes	—
73	C₄⋊C₈	C₂³	1	2	yes	—
74	C₄⋊C₈	C₂³	1	2	yes	—
75	#103	—	2	4	no	—
76	#103	—	2	4	no	—
77	#102	—	2	2	no	—
78	#102	—	2	2	no	—
79	C₂²⋊C₈	—	2	1	no	—
80	C₂²⋊C₈	—	2	4	no	—
81	C₂²⋊C₈	D₄	1	2	yes	—
82	C₄×C₈	—	2	1	no	—
83	C₄×C₈	C₂³	1	2	yes	—
84	C₂⁴	—	2	1	no	—
85	C₂×Q₈	—	2	1	no	—
86	C₂×Q₈	—	2	2	no	—
87	C₂×Q₈	—	2	1	no	—
88	C₂×Q₈	—	4	1	no	—
89	C₂×Q₈	—	4	2	no	—
90	C₂×Q₈	—	4	2	no	—
91	C₂×D₄	—	2	1	no	—
92	C₂×D₄	—	2	2	no	—
93	C₂×D₄	—	2	2	no	—
94	C₂×D₄	C₂×D₄	1	2	yes	—
95	C₂²×C₄	—	2	1	no	—
96	C₂²×C₄	—	2	2	no	—
97	C₂²×C₄	—	2	2	no	—
98	C₂²×C₄	—	4	1	no	—
99	C₂²×C₄	—	8	2	no	—
100	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
101	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
102	C₂²×C₄	#41	1	1	yes	characteristic
103	SD₁₆	—	4	8	no	—
104	M₁₆	—	4	2	no	—
105	M₁₆	—	8	4	no	—
106	C₂×C₈	—	2	2	no	—
107	C₂×C₈	—	2	2	no	—
108	C₂×C₈	—	2	2	no	—
109	C₂×C₈	—	4	2	no	—
110	C₂×C₈	C₂×D₄	1	4	yes	—
111	C₄⋊C₄	—	2	1	no	—
112	C₄⋊C₄	—	2	1	no	—
113	C₄⋊C₄	—	2	1	no	—
114	C₄⋊C₄	—	2	2	no	—
115	C₄⋊C₄	—	2	1	no	—
116	C₄⋊C₄	—	2	2	no	—
117	C₄⋊C₄	—	2	1	no	—
118	C₄⋊C₄	—	4	1	no	—
119	C₄⋊C₄	—	4	1	no	—
120	C₄⋊C₄	—	4	1	no	—
121	C₄⋊C₄	—	4	1	no	—
122	C₄⋊C₄	C₂×D₄	1	2	yes	—
123	C₂²⋊C₄	—	2	2	no	—
124	C₂²⋊C₄	—	2	2	no	—
125	C₂²⋊C₄	—	4	1	no	—
126	C₄²	—	4	1	no	—
127	C₄²	C₂×D₄	1	1	yes	characteristic
128	C₄²	C₂×D₄	1	1	yes	characteristic
129	C₄²	#41	1	1	yes	characteristic
130	C₄²	C₂⁴	1	1	yes	characteristic
131	C₂³	—	2	2	no	—
132	C₂³	—	4	2	no	—
133	C₂³	—	8	1	no	—
134	C₂³	—	8	1	no	—
135	C₂³	#121	1	1	yes	characteristic
136	Q₈	—	4	2	no	—
137	Q₈	—	4	2	no	—
138	Q₈	—	4	2	no	—
139	Q₈	—	8	1	no	—
140	Q₈	—	8	2	no	—
141	D₄	—	2	2	no	—
142	D₄	—	2	2	no	—
143	D₄	—	4	2	no	—
144	C₂×C₄	—	2	2	no	—
145	C₂×C₄	—	2	2	no	—
146	C₂×C₄	—	4	1	no	—
147	C₂×C₄	—	4	4	no	—
148	C₂×C₄	—	4	1	no	—
149	C₂×C₄	—	4	1	no	—
150	C₂×C₄	—	4	2	no	—
151	C₂×C₄	—	8	2	no	—
152	C₂×C₄	—	8	1	no	—
153	C₂×C₄	—	8	2	no	—
154	C₂×C₄	#121	1	1	yes	characteristic
155	C₂×C₄	#121	1	1	yes	characteristic
156	C₂×C₄	#121	1	1	yes	characteristic
157	C₂×C₄	AGL(1,ℤ/8)	1	2	yes	—
158	C₂×C₄	#137	1	2	yes	—
159	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
160	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
161	C₂×C₄	C₂×#41	1	1	yes	characteristic
162	C₈	—	2	4	no	—
163	C₈	—	4	2	no	—
164	C₈	—	4	2	no	—
165	C₈	—	4	2	no	—
166	C₂²	—	2	1	no	—
167	C₂²	—	2	1	no	—
168	C₂²	—	4	2	no	—
169	C₂²	—	4	2	no	—
170	C₂²	—	4	2	no	—
171	C₂²	—	8	4	no	—
172	C₂²	#468	1	2	yes	—
173	C₂²	C₂×#121	1	1	yes	characteristic
174	C₄	—	2	1	no	—
175	C₄	—	2	1	no	—
176	C₄	—	2	1	no	—
177	C₄	—	4	2	no	—
178	C₄	—	8	1	no	—
179	C₄	—	8	2	no	—
180	C₄	—	8	1	no	—
181	C₄	#468	1	2	yes	—
182	C₄	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
183	C₄	C₂×#137	1	1	yes	characteristic
184	C₄	#577	1	2	yes	—
185	C₂	—	2	2	no	—
186	C₂	—	8	2	no	—
187	C₂	#3042	1	1	yes	characteristic
188	C₂	C₂×#468	1	1	yes	characteristic
189	C₂	#3069	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂×#121
commutator, Frattini, cosocle	C₄²	C₂⁴
prime radical	C₂²×D₄	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
1. #3229	C₂	semidirect
5. #1699	C₂	semidirect
6. #1691	C₂	semidirect
7. #1571	C₂	semidirect
8. #1546	C₂	semidirect
9. #1542	C₂	semidirect
14. #498	C₂²	semidirect
15. #487	C₂²	semidirect
33. #335	C₂²	semidirect
34. #333	C₂²	semidirect
35. #332	C₂²	semidirect
36. #332	C₂²	semidirect

Maximal subgroups

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

subgroup	index
#3229	2
#3077	2
C₂×#461	2
#2975	2
#1699	2
#1691	2
#1571	2
#1546	2
#1542	2