Finite Group Databasebeta

Subgroups — #19721

Order 256.

399 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 399 conjugacy classes of subgroups — too many to draw. Only its 119 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	#3227	C₂	1	2	yes	maximal, complemented
2	#3207	C₂	1	2	yes	maximal, complemented
3	#3184	C₂	1	1	yes	maximal, characteristic, complemented
4	#3152	C₂	1	1	yes	maximal, characteristic, complemented
5	#1721	C₂	1	1	yes	maximal, characteristic, complemented
6	#1720	C₂	1	2	yes	maximal, complemented
7	#1716	C₂	1	1	yes	maximal, characteristic, complemented
8	#1715	C₂	1	2	yes	maximal, complemented
9	#1709	C₂	1	1	yes	maximal, characteristic, complemented
10	#1705	C₂	1	1	yes	maximal, characteristic, complemented
11	#1453	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#137	—	2	2	no	—
13	#532	C₂²	1	1	yes	characteristic
14	#532	C₂²	1	2	yes	—
15	C₂×C₄⋊Q₈	C₂²	1	1	yes	characteristic
16	#498	C₂²	1	1	yes	characteristic
17	#498	C₂²	1	2	yes	—
18	#498	C₂²	1	2	yes	—
19	#497	—	2	2	no	—
20	#494	C₂²	1	1	yes	characteristic
21	#494	C₂²	1	1	yes	characteristic
22	#493	C₂²	1	1	yes	characteristic
23	#493	C₂²	1	2	yes	—
24	#492	C₂²	1	1	yes	characteristic
25	#492	C₂²	1	1	yes	characteristic
26	#492	C₂²	1	2	yes	—
27	#487	C₂²	1	2	yes	—
28	#486	C₂²	1	2	yes	—
29	#486	C₂²	1	2	yes	—
30	#486	C₂²	1	2	yes	—
31	#486	C₂²	1	2	yes	—
32	#481	—	2	4	no	—
33	#476	—	2	4	no	—
34	#469	—	2	2	no	—
35	#463	—	2	4	no	—
36	#449	—	2	4	no	—
37	#442	—	2	2	no	—
38	#431	C₂²	1	2	yes	—
39	C₄×M₁₆	C₂²	1	1	yes	characteristic
40	#322	C₂²	1	2	yes	—
41	#322	C₂²	1	1	yes	characteristic
42	C₈²	C₂²	1	1	yes	characteristic
43	C₂×#41	—	2	1	no	—
44	C₂×#41	—	2	2	no	—
45	C₂²×Q₈	—	2	1	no	—
46	#137	—	4	8	no	—
47	C₂×Q₁₆	—	2	2	no	—
48	C₂×Q₁₆	—	2	2	no	—
49	C₂×SD₁₆	—	2	2	no	—
50	C₂×SD₁₆	—	2	2	no	—
51	C₂×SD₁₆	—	2	4	no	—
52	C₂×D₈	—	2	2	no	—
53	C₂×D₈	—	2	4	no	—
54	C₂×M₁₆	D₄	1	2	yes	—
55	C₄⋊Q₈	—	2	2	no	—
56	C₄⋊Q₈	—	2	1	no	—
57	C₄⋊Q₈	C₂³	1	1	yes	characteristic
58	C₄⋊Q₈	C₂³	1	1	yes	characteristic
59	C₄⋊Q₈	C₂³	1	1	yes	characteristic
60	C₄⋊Q₈	C₂³	1	2	yes	—
61	#127	C₂³	1	2	yes	—
62	#127	C₂³	1	1	yes	characteristic
63	#124	—	2	2	no	—
64	#124	—	2	1	no	—
65	#121	—	2	2	no	—
66	#121	—	2	2	no	—
67	#121	—	2	2	no	—
68	C₄×D₄	—	2	2	no	—
69	C₄×D₄	—	2	2	no	—
70	C₄×D₄	—	2	2	no	—
71	C₂×C₄⋊C₄	—	2	2	no	—
72	C₂×C₄²	C₂³	1	1	yes	characteristic
73	#106	—	2	2	no	—
74	#106	—	2	4	no	—
75	#106	—	2	4	no	—
76	C₄⋊C₈	D₄	1	4	yes	—
77	#103	—	2	4	no	—
78	#102	—	2	4	no	—
79	#102	—	2	4	no	—
80	#102	—	2	4	no	—
81	#102	—	2	4	no	—
82	C₂²⋊C₈	D₄	1	4	yes	—
83	#97	D₄	1	2	yes	—
84	C₄×C₈	C₂³	1	1	yes	characteristic
85	C₄×C₈	C₂³	1	2	yes	—
86	C₄×C₈	C₂³	1	1	yes	characteristic
87	C₄×C₈	C₂³	1	2	yes	—
88	#41	—	4	4	no	—
89	#41	—	8	4	no	—
90	C₂×Q₈	—	2	2	no	—
91	C₂×Q₈	—	2	1	no	—
92	C₂×Q₈	—	2	1	no	—
93	C₂×Q₈	—	2	1	no	—
94	C₂×Q₈	—	4	4	no	—
95	C₂×Q₈	—	4	1	no	—
96	C₂×D₄	—	2	2	no	—
97	C₂×D₄	—	2	2	no	—
98	C₂×D₄	—	2	2	no	—
99	C₂×D₄	—	2	2	no	—
100	C₂×D₄	—	2	1	no	—
101	C₂×D₄	—	4	1	no	—
102	C₂×D₄	—	4	2	no	—
103	C₂²×C₄	—	4	1	no	—
104	C₂²×C₄	—	4	1	no	—
105	C₂²×C₄	—	4	2	no	—
106	C₂²×C₄	C₂×D₄	1	2	yes	—
107	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
108	Q₁₆	—	4	4	no	—
109	Q₁₆	—	4	4	no	—
110	SD₁₆	—	4	4	no	—
111	SD₁₆	—	4	4	no	—
112	SD₁₆	—	8	4	no	—
113	D₈	—	4	4	no	—
114	D₈	—	8	4	no	—
115	M₁₆	—	2	4	no	—
116	C₂×C₈	C₂×D₄	1	2	yes	—
117	C₂×C₈	C₂×D₄	1	4	yes	—
118	C₂×C₈	C₂×D₄	1	4	yes	—
119	C₂×C₈	C₂×D₄	1	2	yes	—
120	C₄⋊C₄	—	2	2	no	—
121	C₄⋊C₄	—	2	2	no	—
122	C₄⋊C₄	—	2	2	no	—
123	C₄⋊C₄	—	2	2	no	—
124	C₄⋊C₄	—	2	2	no	—
125	C₄⋊C₄	—	4	2	no	—
126	C₂²⋊C₄	—	4	2	no	—
127	C₂²⋊C₄	—	4	2	no	—
128	C₂²⋊C₄	—	4	2	no	—
129	C₄²	C₂×D₄	1	1	yes	characteristic
130	C₄²	C₂×D₄	1	2	yes	—
131	C₄²	C₂⁴	1	1	yes	characteristic
132	C₂³	—	4	1	no	—
133	C₂³	—	4	2	no	—
134	C₂³	#127	1	1	yes	characteristic
135	Q₈	—	4	2	no	—
136	Q₈	—	4	2	no	—
137	Q₈	—	4	2	no	—
138	Q₈	—	4	4	no	—
139	Q₈	—	8	2	no	—
140	D₄	—	4	4	no	—
141	D₄	—	4	2	no	—
142	D₄	—	4	4	no	—
143	D₄	—	4	4	no	—
144	D₄	—	4	4	no	—
145	D₄	—	8	2	no	—
146	D₄	—	8	4	no	—
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	—	2	4	no	—
149	C₂×C₄	—	4	1	no	—
150	C₂×C₄	—	4	2	no	—
151	C₂×C₄	—	4	1	no	—
152	C₂×C₄	—	4	1	no	—
153	C₂×C₄	—	8	2	no	—
154	C₂×C₄	—	8	4	no	—
155	C₂×C₄	—	8	2	no	—
156	C₂×C₄	#127	1	2	yes	—
157	C₂×C₄	#127	1	1	yes	characteristic
158	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
159	C₂×C₄	C₂²×D₄	1	2	yes	—
160	C₈	—	2	2	no	—
161	C₈	—	2	4	no	—
162	C₈	—	2	4	no	—
163	C₈	AGL(1,ℤ/8)	1	4	yes	—
164	C₂²	—	2	1	no	—
165	C₂²	—	2	2	no	—
166	C₂²	—	8	1	no	—
167	C₂²	—	8	2	no	—
168	C₂²	—	8	2	no	—
169	C₂²	—	8	2	no	—
170	C₂²	—	8	2	no	—
171	C₂²	C₂×#127	1	1	yes	characteristic
172	C₄	—	2	1	no	—
173	C₄	—	2	2	no	—
174	C₄	—	8	1	no	—
175	C₄	—	8	1	no	—
176	C₄	—	8	1	no	—
177	C₄	—	8	2	no	—
178	C₄	C₂×AGL(1,ℤ/8)	1	2	yes	—
179	C₄	#577	1	4	yes	—
180	C₂	—	4	1	no	—
181	C₂	—	16	1	no	—
182	C₂	—	16	2	no	—
183	C₂	#3151	1	1	yes	characteristic
184	C₂	#3154	1	2	yes	—

	subgroup	G/H
center, socle	C₂²	C₂×#127
commutator, Frattini, cosocle	C₄²	C₂⁴
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. #3227	C₂	semidirect
2. #3207	C₂	semidirect
3. #3184	C₂	semidirect
4. #3152	C₂	semidirect
5. #1721	C₂	semidirect
6. #1720	C₂	semidirect
7. #1716	C₂	semidirect
8. #1715	C₂	semidirect
9. #1709	C₂	semidirect
10. #1705	C₂	semidirect
11. #1453	C₂	semidirect

Maximal subgroups

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

subgroup	index
#3227	2
#3207	2
#3184	2
#3152	2
#1721	2
#1720	2
#1716	2
#1715	2
#1709	2
#1705	2
#1453	2