Finite Group Databasebeta

Subgroups — #19962

Order 256.

346 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 346 conjugacy classes of subgroups — too many to draw. Only its 103 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	#3399	C₂	1	2	yes	maximal, complemented
2	#3396	C₂	1	1	yes	maximal, characteristic, complemented
3	#3287	C₂	1	1	yes	maximal, characteristic, complemented
4	#1712	C₂	1	1	yes	maximal, characteristic, complemented
5	#1688	C₂	1	2	yes	maximal, complemented
6	#1668	C₂	1	1	yes	maximal, characteristic, complemented
7	#1667	C₂	1	1	yes	maximal, characteristic, complemented
8	#1621	C₂	1	2	yes	maximal, complemented
9	#1620	C₂	1	2	yes	maximal, complemented
10	#1591	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×SD₁₆	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#135	—	2	1	no	—
13	#559	C₂²	1	2	yes	—
14	#550	C₂²	1	2	yes	—
15	#496	—	2	2	no	—
16	#495	—	2	1	no	—
17	#493	C₂²	1	1	yes	characteristic
18	#493	C₂²	1	2	yes	—
19	#493	C₂²	1	1	yes	characteristic
20	#493	C₂²	1	2	yes	—
21	#466	—	2	1	no	—
22	#463	—	2	2	no	—
23	#463	—	2	2	no	—
24	#460	C₂²	1	2	yes	—
25	#460	C₂²	1	2	yes	—
26	#460	C₂²	1	2	yes	—
27	#459	—	2	2	no	—
28	#459	C₂²	1	2	yes	—
29	#449	—	2	2	no	—
30	#446	C₂²	1	2	yes	—
31	C₈×Q₈	C₂²	1	1	yes	characteristic
32	#442	—	2	2	no	—
33	C₄×Q₁₆	C₂²	1	1	yes	characteristic
34	C₄×SD₁₆	C₂²	1	1	yes	characteristic
35	C₄×D₈	—	2	1	no	—
36	C₈×D₄	C₂²	1	1	yes	characteristic
37	#335	C₂²	1	1	yes	characteristic
38	#334	C₂²	1	1	yes	characteristic
39	#331	C₂²	1	2	yes	—
40	#331	C₂²	1	2	yes	—
41	Q₈⋊C₈	C₂²	1	1	yes	characteristic
42	Q₈⋊C₈	C₂²	1	2	yes	—
43	D₄⋊C₈	C₂²	1	1	yes	characteristic
44	C₈²	C₂²	1	1	yes	characteristic
45	C₂×#41	—	2	2	no	—
46	#135	—	4	4	no	—
47	C₂×Q₁₆	D₄	1	1	yes	characteristic
48	C₂×SD₁₆	—	2	2	no	—
49	C₂×SD₁₆	—	2	2	no	—
50	C₂×SD₁₆	—	2	2	no	—
51	C₂×SD₁₆	D₄	1	1	yes	characteristic
52	C₂×D₈	—	2	2	no	—
53	C₂×D₈	—	2	1	no	—
54	C₂×D₈	—	2	1	no	—
55	C₂×D₈	—	2	2	no	—
56	C₂×D₈	—	2	1	no	—
57	C₂×D₈	—	2	2	no	—
58	C₂×D₈	—	2	1	no	—
59	C₂×D₈	—	4	2	no	—
60	C₂²×C₈	—	2	1	no	—
61	#127	—	2	2	no	—
62	#127	C₂³	1	2	yes	—
63	#127	C₂³	1	2	yes	—
64	#124	—	2	2	no	—
65	#124	—	2	2	no	—
66	#123	—	4	2	no	—
67	#121	—	2	2	no	—
68	#121	—	4	2	no	—
69	C₄×Q₈	C₂³	1	2	yes	—
70	C₄×Q₈	C₂³	1	1	yes	characteristic
71	C₄×D₄	—	2	2	no	—
72	C₄×D₄	—	2	2	no	—
73	C₄×D₄	—	2	2	no	—
74	C₄×D₄	C₂³	1	1	yes	characteristic
75	#107	D₄	1	1	yes	characteristic
76	#106	D₄	1	1	yes	characteristic
77	C₄⋊C₈	—	2	1	no	—
78	C₄⋊C₈	—	2	2	no	—
79	C₄⋊C₈	C₂³	1	1	yes	characteristic
80	C₄⋊C₈	C₂³	1	1	yes	characteristic
81	C₄⋊C₈	C₂³	1	2	yes	—
82	#103	D₄	1	1	yes	characteristic
83	#103	D₄	1	2	yes	—
84	#102	—	2	2	no	—
85	#102	—	2	2	no	—
86	#102	—	2	2	no	—
87	#102	—	2	2	no	—
88	#102	D₄	1	1	yes	characteristic
89	C₂²⋊C₈	—	2	1	no	—
90	#97	—	2	2	no	—
91	C₄×C₈	—	2	1	no	—
92	C₄×C₈	C₂³	1	1	yes	characteristic
93	C₄×C₈	C₂³	1	1	yes	characteristic
94	C₄×C₈	C₂³	1	1	yes	characteristic
95	#41	—	4	4	no	—
96	#41	—	8	2	no	—
97	C₂×Q₈	C₂×D₄	1	1	yes	characteristic
98	C₂×Q₈	C₂×D₄	1	2	yes	—
99	C₂×D₄	—	2	2	no	—
100	C₂×D₄	—	2	2	no	—
101	C₂×D₄	—	2	2	no	—
102	C₂×D₄	—	2	2	no	—
103	C₂×D₄	—	2	2	no	—
104	C₂×D₄	—	2	2	no	—
105	C₂×D₄	—	4	2	no	—
106	C₂×D₄	—	4	2	no	—
107	C₂×D₄	C₂×D₄	1	1	yes	characteristic
108	C₂²×C₄	—	2	1	no	—
109	C₂²×C₄	—	4	2	no	—
110	C₂²×C₄	—	4	2	no	—
111	Q₁₆	—	2	2	no	—
112	SD₁₆	—	2	2	no	—
113	SD₁₆	—	4	4	no	—
114	SD₁₆	—	8	2	no	—
115	SD₁₆	—	8	2	no	—
116	D₈	—	4	4	no	—
117	D₈	—	4	2	no	—
118	D₈	—	4	2	no	—
119	D₈	—	8	2	no	—
120	D₈	—	8	1	no	—
121	D₈	—	8	1	no	—
122	D₈	—	8	4	no	—
123	D₈	—	8	2	no	—
124	C₂×C₈	—	2	1	no	—
125	C₂×C₈	—	2	1	no	—
126	C₂×C₈	—	2	2	no	—
127	C₂×C₈	—	2	1	no	—
128	C₂×C₈	—	4	2	no	—
129	C₂×C₈	C₂×D₄	1	1	yes	characteristic
130	C₂×C₈	C₂×D₄	1	1	yes	characteristic
131	C₂×C₈	C₂×D₄	1	1	yes	characteristic
132	C₂×C₈	C₂×D₄	1	1	yes	characteristic
133	C₄⋊C₄	—	2	2	no	—
134	C₄⋊C₄	—	2	1	no	—
135	C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
136	C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
137	C₄⋊C₄	C₂×D₄	1	2	yes	—
138	C₂²⋊C₄	—	2	1	no	—
139	C₂²⋊C₄	—	4	2	no	—
140	C₂²⋊C₄	—	4	2	no	—
141	C₂²⋊C₄	—	4	2	no	—
142	C₂²⋊C₄	—	4	2	no	—
143	C₄²	—	2	1	no	—
144	C₄²	—	2	2	no	—
145	C₄²	C₂⁴	1	1	yes	characteristic
146	C₂³	—	2	1	no	—
147	C₂³	—	4	2	no	—
148	C₂³	—	4	2	no	—
149	Q₈	—	2	1	no	—
150	Q₈	—	2	2	no	—
151	Q₈	—	2	1	no	—
152	Q₈	—	2	2	no	—
153	D₄	—	2	1	no	—
154	D₄	—	2	1	no	—
155	D₄	—	4	2	no	—
156	D₄	—	4	2	no	—
157	D₄	—	4	2	no	—
158	D₄	—	4	2	no	—
159	D₄	—	4	2	no	—
160	D₄	—	4	2	no	—
161	D₄	—	4	2	no	—
162	D₄	—	4	2	no	—
163	D₄	—	8	2	no	—
164	D₄	—	8	2	no	—
165	D₄	—	8	4	no	—
166	D₄	—	16	2	no	—
167	C₂×C₄	—	2	1	no	—
168	C₂×C₄	—	2	1	no	—
169	C₂×C₄	—	2	2	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	1	no	—
172	C₂×C₄	—	4	1	no	—
173	C₂×C₄	—	4	1	no	—
174	C₂×C₄	—	8	4	no	—
175	C₂×C₄	—	16	2	no	—
176	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
177	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
178	C₂×C₄	H(5,𝔽₂)	1	1	yes	characteristic
179	C₈	—	2	2	no	—
180	C₈	—	2	1	no	—
181	C₈	—	2	1	no	—
182	C₈	—	4	1	no	—
183	C₈	—	4	2	no	—
184	C₈	—	4	1	no	—
185	C₈	#135	1	2	yes	complemented
186	C₈	AGL(1,ℤ/8)	1	2	yes	—
187	C₂²	—	4	1	no	—
188	C₂²	—	4	1	no	—
189	C₂²	—	4	1	no	—
190	C₂²	—	8	2	no	—
191	C₂²	—	8	2	no	—
192	C₂²	—	8	2	no	—
193	C₂²	—	8	2	no	—
194	C₂²	—	8	2	no	—
195	C₂²	—	8	2	no	—
196	C₂²	D₄²	1	1	yes	characteristic
197	C₄	—	2	1	no	—
198	C₄	—	4	1	no	—
199	C₄	—	4	2	no	—
200	C₄	—	4	1	no	—
201	C₄	—	4	1	no	—
202	C₄	—	4	2	no	—
203	C₄	C₂×#135	1	1	yes	characteristic
204	C₄	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
205	C₄	#576	1	1	yes	characteristic
206	C₄	#576	1	1	yes	characteristic
207	C₂	—	8	1	no	—
208	C₂	—	16	2	no	—
209	C₂	—	16	2	no	—
210	C₂	#3274	1	1	yes	characteristic
211	C₂	#3287	1	1	yes	characteristic
212	C₂	#3292	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	D₄²
commutator, Frattini, cosocle	C₄²	C₂⁴
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #3399	C₂	semidirect
2. #3396	C₂	semidirect
3. #3287	C₂	semidirect
4. #1712	C₂	semidirect
5. #1688	C₂	semidirect
6. #1668	C₂	semidirect
7. #1667	C₂	semidirect
8. #1621	C₂	semidirect
9. #1620	C₂	semidirect
10. #1591	C₂	semidirect
11. C₈×SD₁₆	C₂	semidirect
185. C₈	#135	semidirect

Maximal subgroups

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

subgroup	index
#3399	2
#3396	2
#3287	2
#1712	2
#1688	2
#1668	2
#1667	2
#1621	2
#1620	2
#1591	2
C₈×SD₁₆	2