Finite Group Databasebeta

Subgroups — #19712

Order 256.

352 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 352 conjugacy classes of subgroups — too many to draw. Only its 115 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	#3228	C₂	1	2	yes	maximal, complemented
2	#3210	C₂	1	2	yes	maximal, complemented
3	#3170	C₂	1	1	yes	maximal, characteristic, complemented
4	#3169	C₂	1	1	yes	maximal, characteristic
5	#1723	C₂	1	2	yes	maximal, complemented
6	#1722	C₂	1	2	yes	maximal, complemented
7	#1720	C₂	1	2	yes	maximal, complemented
8	#1717	C₂	1	2	yes	maximal, complemented
9	#1452	C₂	1	1	yes	maximal, characteristic, complemented
10	#533	C₂²	1	2	yes	—
11	#532	C₂²	1	2	yes	—
12	#500	C₂²	1	1	yes	characteristic
13	#500	C₂²	1	1	yes	characteristic
14	#500	C₂²	1	2	yes	—
15	#498	C₂²	1	2	yes	—
16	#494	C₂²	1	1	yes	characteristic
17	#494	C₂²	1	2	yes	—
18	#493	C₂²	1	1	yes	characteristic
19	#492	C₂²	1	2	yes	—
20	#487	C₂²	1	2	yes	—
21	#487	C₂²	1	2	yes	—
22	#486	C₂²	1	2	yes	—
23	#486	C₂²	1	2	yes	—
24	#486	C₂²	1	2	yes	—
25	#486	C₂²	1	2	yes	—
26	#482	—	2	4	no	—
27	#478	—	2	4	no	—
28	#467	—	2	2	no	—
29	#466	—	2	2	no	—
30	#464	—	2	4	no	—
31	#452	—	2	4	no	—
32	C₄×Q₁₆	—	2	2	no	—
33	C₄×D₈	—	2	2	no	—
34	#431	C₂²	1	1	yes	characteristic
35	#431	C₂²	1	1	yes	characteristic
36	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
37	#322	C₂²	1	2	yes	—
38	#322	C₂²	1	2	yes	—
39	C₂×#41	—	2	2	no	—
40	C₂×Q₁₆	—	2	2	no	—
41	C₂×Q₁₆	—	2	4	no	—
42	C₂×SD₁₆	—	2	4	no	—
43	C₂×D₈	—	2	2	no	—
44	C₂²×C₈	D₄	1	2	yes	—
45	C₄⋊Q₈	C₂³	1	2	yes	—
46	C₄⋊Q₈	C₂³	1	2	yes	—
47	C₄⋊Q₈	C₂³	1	2	yes	—
48	#127	C₂³	1	2	yes	—
49	#125	—	2	2	no	—
50	#124	—	2	2	no	—
51	C₂²⋊Q₈	—	2	2	no	—
52	C₂²⋊Q₈	—	2	2	no	—
53	#121	—	2	2	no	—
54	#121	—	2	2	no	—
55	C₄×Q₈	—	2	2	no	—
56	C₄×Q₈	—	2	2	no	—
57	C₄×D₄	—	2	2	no	—
58	C₄×D₄	—	2	2	no	—
59	#117	—	2	2	no	—
60	C₂×C₄²	C₂³	1	1	yes	characteristic
61	#107	—	2	2	no	—
62	#107	—	2	2	no	—
63	#107	—	2	4	no	—
64	#106	—	2	4	no	—
65	C₄⋊C₈	D₄	1	2	yes	—
66	C₄⋊C₈	D₄	1	2	yes	—
67	#103	—	2	4	no	—
68	#103	—	2	4	no	—
69	#102	—	2	4	no	—
70	#102	—	2	4	no	—
71	#102	—	2	4	no	—
72	#102	—	2	4	no	—
73	C₂²⋊C₈	D₄	1	2	yes	—
74	C₂²⋊C₈	D₄	1	2	yes	—
75	C₄×C₈	D₄	1	2	yes	—
76	C₄×C₈	C₂³	1	1	yes	characteristic
77	C₄×C₈	C₂³	1	1	yes	characteristic
78	C₄×C₈	C₂³	1	2	yes	—
79	C₄×C₈	C₂³	1	2	yes	—
80	#41	—	8	4	no	—
81	C₂×Q₈	—	2	2	no	—
82	C₂×Q₈	—	2	2	no	—
83	C₂×Q₈	—	2	2	no	—
84	C₂×D₄	—	2	2	no	—
85	C₂×D₄	—	2	2	no	—
86	C₂×D₄	—	2	2	no	—
87	C₂×D₄	—	4	2	no	—
88	C₂²×C₄	—	4	2	no	—
89	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
90	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
91	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
92	Q₁₆	—	4	4	no	—
93	Q₁₆	—	8	4	no	—
94	SD₁₆	—	8	4	no	—
95	D₈	—	4	4	no	—
96	C₂×C₈	—	4	2	no	—
97	C₂×C₈	C₂×D₄	1	2	yes	—
98	C₂×C₈	C₂×D₄	1	2	yes	—
99	C₂×C₈	C₂×D₄	1	4	yes	—
100	C₂×C₈	C₂×D₄	1	4	yes	—
101	C₄⋊C₄	—	2	2	no	—
102	C₄⋊C₄	—	2	2	no	—
103	C₄⋊C₄	—	2	2	no	—
104	C₄⋊C₄	—	2	2	no	—
105	C₄⋊C₄	—	2	2	no	—
106	C₄⋊C₄	—	2	2	no	—
107	C₄⋊C₄	—	4	2	no	—
108	C₄⋊C₄	—	4	2	no	—
109	C₂²⋊C₄	—	4	2	no	—
110	C₂²⋊C₄	—	4	2	no	—
111	C₂²⋊C₄	—	4	2	no	—
112	C₄²	—	4	2	no	—
113	C₄²	C₂×D₄	1	1	yes	characteristic
114	C₄²	C₂×D₄	1	1	yes	characteristic
115	C₄²	C₂×D₄	1	1	yes	characteristic
116	C₄²	C₂⁴	1	1	yes	characteristic
117	C₂³	—	4	2	no	—
118	C₂³	#127	1	1	yes	characteristic
119	Q₈	—	4	4	no	—
120	Q₈	—	4	4	no	—
121	Q₈	—	4	4	no	—
122	D₄	—	4	4	no	—
123	D₄	—	4	4	no	—
124	D₄	—	4	4	no	—
125	D₄	—	8	4	no	—
126	C₂×C₄	—	2	2	no	—
127	C₂×C₄	—	2	2	no	—
128	C₂×C₄	—	2	2	no	—
129	C₂×C₄	—	4	2	no	—
130	C₂×C₄	—	4	2	no	—
131	C₂×C₄	—	4	2	no	—
132	C₂×C₄	—	8	4	no	—
133	C₂×C₄	#127	1	1	yes	characteristic
134	C₂×C₄	#127	1	1	yes	characteristic
135	C₂×C₄	#127	1	1	yes	characteristic
136	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
137	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
138	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
139	C₈	—	2	2	no	—
140	C₈	—	2	2	no	—
141	C₈	—	2	4	no	—
142	C₈	—	2	4	no	—
143	C₂²	—	2	1	no	—
144	C₂²	—	2	1	no	—
145	C₂²	—	2	1	no	—
146	C₂²	—	8	2	no	—
147	C₂²	—	8	2	no	—
148	C₂²	—	8	2	no	—
149	C₂²	C₂×#127	1	1	yes	characteristic
150	C₄	—	2	1	no	—
151	C₄	—	2	1	no	—
152	C₄	—	2	1	no	—
153	C₄	—	8	2	no	—
154	C₄	—	8	2	no	—
155	C₄	—	8	2	no	—
156	C₄	#575	1	2	yes	—
157	C₄	#576	1	2	yes	—
158	C₄	#578	1	2	yes	—
159	C₂	—	4	1	no	—
160	C₂	—	16	2	no	—
161	C₂	#3150	1	1	yes	characteristic
162	C₂	#3155	1	1	yes	characteristic
163	C₂	#3156	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂×#127
commutator, Frattini, cosocle	C₄²	C₂⁴
prime radical	#3169	C₂
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. #3228	C₂	semidirect
2. #3210	C₂	semidirect
3. #3170	C₂	semidirect
5. #1723	C₂	semidirect
6. #1722	C₂	semidirect
7. #1720	C₂	semidirect
8. #1717	C₂	semidirect
9. #1452	C₂	semidirect

Maximal subgroups

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

subgroup	index
#3228	2
#3210	2
#3170	2
#3169	2
#1723	2
#1722	2
#1720	2
#1717	2
#1452	2