Finite Group Databasebeta

Subgroups — #19908

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	#3391	C₂	1	1	yes	maximal, characteristic, complemented
2	#3332	C₂	1	2	yes	maximal, complemented
3	#3283	C₂	1	1	yes	maximal, characteristic, complemented
4	#1709	C₂	1	1	yes	maximal, characteristic, complemented
5	#1701	C₂	1	2	yes	maximal, complemented
6	#1670	C₂	1	1	yes	maximal, characteristic, complemented
7	#1667	C₂	1	1	yes	maximal, characteristic, complemented
8	#1642	C₂	1	2	yes	maximal, complemented
9	#1625	C₂	1	2	yes	maximal, complemented
10	#1589	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×SD₁₆	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#135	—	2	1	no	—
13	#550	C₂²	1	2	yes	—
14	#547	C₂²	1	2	yes	—
15	#494	C₂²	1	1	yes	characteristic
16	#493	—	2	1	no	—
17	#493	C₂²	1	1	yes	characteristic
18	#486	C₂²	1	2	yes	—
19	#486	C₂²	1	2	yes	—
20	#481	—	2	2	no	—
21	#475	C₂²	1	2	yes	—
22	#469	—	2	2	no	—
23	#467	—	2	1	no	—
24	#461	C₂²	1	2	yes	—
25	#459	—	2	2	no	—
26	#459	C₂²	1	2	yes	—
27	#459	C₂²	1	2	yes	—
28	#452	—	2	2	no	—
29	#449	—	2	2	no	—
30	C₈×Q₈	C₂²	1	1	yes	characteristic
31	C₄×SD₁₆	C₂²	1	1	yes	characteristic
32	C₄×D₈	—	2	1	no	—
33	C₄×D₈	C₂²	1	1	yes	characteristic
34	#436	C₂²	1	2	yes	—
35	C₈×D₄	C₂²	1	1	yes	characteristic
36	#417	—	2	2	no	—
37	#335	C₂²	1	1	yes	characteristic
38	#334	C₂²	1	1	yes	characteristic
39	#332	C₂²	1	2	yes	—
40	#331	C₂²	1	2	yes	—
41	Q₈⋊C₈	C₂²	1	1	yes	characteristic
42	D₄⋊C₈	C₂²	1	1	yes	characteristic
43	D₄⋊C₈	C₂²	1	2	yes	—
44	C₈²	C₂²	1	1	yes	characteristic
45	C₂×#41	—	2	2	no	—
46	C₂×#41	—	2	2	no	—
47	#135	—	4	4	no	—
48	C₂×Q₁₆	—	2	1	no	—
49	C₂×Q₁₆	—	2	1	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	2	no	—
54	C₂×D₈	—	2	1	no	—
55	C₂×D₈	—	2	1	no	—
56	C₂×D₈	D₄	1	1	yes	characteristic
57	C₂×M₁₆	—	2	2	no	—
58	C₂²×C₈	—	2	1	no	—
59	C₄⋊Q₈	C₂³	1	2	yes	—
60	#127	—	2	2	no	—
61	#127	C₂³	1	2	yes	—
62	#123	—	4	2	no	—
63	C₂²⋊Q₈	—	2	2	no	—
64	#121	—	2	2	no	—
65	#121	—	2	2	no	—
66	#121	—	4	2	no	—
67	C₄×Q₈	C₂³	1	1	yes	characteristic
68	C₄×D₄	—	2	2	no	—
69	C₄×D₄	C₂³	1	1	yes	characteristic
70	C₄×D₄	C₂³	1	2	yes	—
71	C₂×C₄⋊C₄	—	2	2	no	—
72	#107	D₄	1	1	yes	characteristic
73	#106	—	2	2	no	—
74	#106	D₄	1	1	yes	characteristic
75	C₄⋊C₈	—	2	1	no	—
76	C₄⋊C₈	C₂³	1	2	yes	—
77	C₄⋊C₈	C₂³	1	1	yes	characteristic
78	C₄⋊C₈	C₂³	1	1	yes	characteristic
79	#103	—	2	2	no	—
80	#103	—	2	2	no	—
81	#103	D₄	1	1	yes	characteristic
82	#102	—	2	2	no	—
83	#102	—	2	2	no	—
84	#102	—	2	2	no	—
85	#102	—	2	2	no	—
86	#102	—	4	2	no	—
87	#102	D₄	1	2	yes	—
88	#102	D₄	1	1	yes	characteristic
89	C₂²⋊C₈	—	2	1	no	—
90	C₂²⋊C₈	—	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	—	4	4	no	—
97	#41	—	8	2	no	—
98	#41	—	8	2	no	—
99	C₂×Q₈	—	2	2	no	—
100	C₂×Q₈	C₂×D₄	1	1	yes	characteristic
101	C₂×D₄	—	2	2	no	—
102	C₂×D₄	—	2	2	no	—
103	C₂×D₄	—	2	2	no	—
104	C₂×D₄	—	4	2	no	—
105	C₂×D₄	—	4	2	no	—
106	C₂×D₄	—	4	2	no	—
107	C₂×D₄	C₂×D₄	1	1	yes	characteristic
108	C₂×D₄	C₂×D₄	1	2	yes	—
109	C₂²×C₄	—	2	2	no	—
110	C₂²×C₄	—	2	1	no	—
111	C₂²×C₄	—	4	2	no	—
112	C₂²×C₄	—	4	2	no	—
113	Q₁₆	—	4	2	no	—
114	Q₁₆	—	8	1	no	—
115	SD₁₆	—	2	2	no	—
116	SD₁₆	—	4	4	no	—
117	D₈	—	2	2	no	—
118	D₈	—	4	2	no	—
119	D₈	—	8	1	no	—
120	D₈	—	8	2	no	—
121	D₈	—	8	2	no	—
122	M₁₆	—	8	2	no	—
123	C₂×C₈	—	2	1	no	—
124	C₂×C₈	—	2	2	no	—
125	C₂×C₈	—	2	1	no	—
126	C₂×C₈	—	2	1	no	—
127	C₂×C₈	—	4	2	no	—
128	C₂×C₈	C₂×D₄	1	1	yes	characteristic
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	C₄⋊C₄	—	2	1	no	—
133	C₄⋊C₄	—	2	2	no	—
134	C₄⋊C₄	—	2	2	no	—
135	C₄⋊C₄	—	4	2	no	—
136	C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
137	C₄⋊C₄	C₂×D₄	1	2	yes	—
138	C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
139	C₂²⋊C₄	—	2	2	no	—
140	C₂²⋊C₄	—	2	1	no	—
141	C₂²⋊C₄	—	4	2	no	—
142	C₂²⋊C₄	—	4	2	no	—
143	C₄²	—	2	1	no	—
144	C₄²	C₂⁴	1	1	yes	characteristic
145	C₂³	—	2	1	no	—
146	C₂³	—	2	2	no	—
147	C₂³	—	4	2	no	—
148	Q₈	—	2	1	no	—
149	Q₈	—	4	2	no	—
150	Q₈	—	4	2	no	—
151	Q₈	#135	1	2	yes	complemented
152	D₄	—	2	1	no	—
153	D₄	—	2	2	no	—
154	D₄	—	2	1	no	—
155	D₄	—	2	2	no	—
156	D₄	—	4	2	no	—
157	D₄	—	4	2	no	—
158	D₄	—	4	2	no	—
159	D₄	—	4	2	no	—
160	D₄	—	8	4	no	—
161	D₄	—	8	2	no	—
162	D₄	—	8	4	no	—
163	D₄	—	16	2	no	—
164	C₂×C₄	—	2	1	no	—
165	C₂×C₄	—	2	1	no	—
166	C₂×C₄	—	2	2	no	—
167	C₂×C₄	—	2	1	no	—
168	C₂×C₄	—	4	2	no	—
169	C₂×C₄	—	4	1	no	—
170	C₂×C₄	—	4	2	no	—
171	C₂×C₄	—	4	1	no	—
172	C₂×C₄	—	4	2	no	—
173	C₂×C₄	—	8	4	no	—
174	C₂×C₄	—	8	4	no	—
175	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
176	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
177	C₂×C₄	H(5,𝔽₂)	1	1	yes	characteristic
178	C₈	—	2	1	no	—
179	C₈	—	2	1	no	—
180	C₈	—	2	1	no	—
181	C₈	—	4	1	no	—
182	C₈	—	4	1	no	—
183	C₈	—	4	2	no	—
184	C₈	—	4	1	no	—
185	C₈	AGL(1,ℤ/8)	1	2	yes	—
186	C₂²	—	4	1	no	—
187	C₂²	—	4	2	no	—
188	C₂²	—	4	2	no	—
189	C₂²	—	4	1	no	—
190	C₂²	—	4	2	no	—
191	C₂²	—	4	1	no	—
192	C₂²	—	8	2	no	—
193	C₂²	—	8	2	no	—
194	C₂²	—	8	2	no	—
195	C₂²	D₄²	1	1	yes	characteristic
196	C₄	—	2	2	no	—
197	C₄	—	2	1	no	—
198	C₄	—	4	1	no	—
199	C₄	—	4	2	no	—
200	C₄	—	4	1	no	—
201	C₄	—	8	2	no	—
202	C₄	C₂×#135	1	1	yes	characteristic
203	C₄	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
204	C₄	#576	1	1	yes	characteristic
205	C₄	#576	1	1	yes	characteristic
206	C₂	—	8	1	no	—
207	C₂	—	8	2	no	—
208	C₂	—	16	2	no	—
209	C₂	#3274	1	1	yes	characteristic
210	C₂	#3280	1	1	yes	characteristic
211	C₂	#3290	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. #3391	C₂	semidirect
2. #3332	C₂	semidirect
3. #3283	C₂	semidirect
4. #1709	C₂	semidirect
5. #1701	C₂	semidirect
6. #1670	C₂	semidirect
7. #1667	C₂	semidirect
8. #1642	C₂	semidirect
9. #1625	C₂	semidirect
10. #1589	C₂	semidirect
11. C₈×SD₁₆	C₂	semidirect
151. Q₈	#135	semidirect

Maximal subgroups

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

subgroup	index
#3391	2
#3332	2
#3283	2
#1709	2
#1701	2
#1670	2
#1667	2
#1642	2
#1625	2
#1589	2
C₈×SD₁₆	2