Finite Group Databasebeta

Subgroups — #19261

Order 256.

355 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 355 conjugacy classes of subgroups — too many to draw. Only its 123 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	#3169	C₂	1	1	yes	maximal, characteristic
2	#3037	C₂	1	2	yes	maximal, complemented
3	C₂×C₈×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
4	#1682	C₂	1	4	yes	maximal, complemented
5	#1674	C₂	1	2	yes	maximal, complemented
6	#1668	C₂	1	2	yes	maximal, complemented
7	#1567	C₂	1	1	yes	maximal, characteristic, complemented
8	#1547	C₂	1	2	yes	maximal, complemented
9	C₂×#137	—	2	2	no	—
10	#532	C₂²	1	2	yes	—
11	C₂×C₄×Q₈	C₂²	1	1	yes	characteristic
12	#500	C₂²	1	1	yes	characteristic
13	#493	C₂²	1	1	yes	characteristic
14	#486	C₂²	1	2	yes	—
15	#466	—	2	1	no	—
16	#466	—	2	1	no	—
17	#464	—	2	4	no	—
18	#462	C₂²	1	4	yes	—
19	#460	C₂²	1	4	yes	—
20	C₈×Q₈	—	2	1	no	—
21	C₈×Q₈	—	2	1	no	—
22	C₈×Q₈	C₂²	1	2	yes	—
23	C₈×Q₈	C₂²	1	2	yes	—
24	C₄×D₈	—	2	1	no	—
25	C₄×D₈	—	2	1	no	—
26	#423	C₂²	1	2	yes	—
27	C₂×C₄⋊C₈	—	2	1	no	—
28	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
29	#417	—	2	2	no	—
30	C₂×C₄×C₈	—	2	1	no	—
31	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
32	#335	C₂²	1	2	yes	—
33	#335	C₂²	1	2	yes	—
34	#332	C₂²	1	4	yes	—
35	#331	C₂²	1	4	yes	—
36	C₂×#41	—	2	2	no	—
37	C₂²×Q₈	D₄	1	1	yes	characteristic
38	#137	—	4	8	no	—
39	C₂×Q₁₆	—	2	4	no	—
40	C₂×SD₁₆	—	2	4	no	—
41	C₂×D₈	—	2	1	no	—
42	C₂×D₈	—	2	1	no	—
43	C₂×M₁₆	—	2	2	no	—
44	C₂²×C₈	—	2	1	no	—
45	C₂²×C₈	D₄	1	1	yes	characteristic
46	C₂²×C₈	D₄	1	1	yes	characteristic
47	C₄⋊Q₈	C₂³	1	2	yes	—
48	#127	C₂³	1	2	yes	—
49	#124	—	2	2	no	—
50	#121	—	2	2	no	—
51	#121	—	2	2	no	—
52	C₄×Q₈	—	2	1	no	—
53	C₄×Q₈	—	2	1	no	—
54	C₄×Q₈	D₄	1	2	yes	—
55	C₄×Q₈	C₂³	1	2	yes	—
56	C₄×D₄	—	2	2	no	—
57	C₄×D₄	—	2	2	no	—
58	C₂×C₄⋊C₄	—	2	1	no	—
59	C₂×C₄⋊C₄	D₄	1	1	yes	characteristic
60	C₂×C₄²	—	2	1	no	—
61	C₂×C₄²	C₂³	1	1	yes	characteristic
62	#107	—	2	1	no	—
63	#107	—	2	1	no	—
64	C₄⋊C₈	—	2	2	no	—
65	C₄⋊C₈	—	2	2	no	—
66	C₄⋊C₈	—	2	1	no	—
67	C₄⋊C₈	—	2	2	no	—
68	C₄⋊C₈	C₂³	1	2	yes	—
69	C₄⋊C₈	C₂³	1	4	yes	—
70	#103	—	2	4	no	—
71	#102	—	2	2	no	—
72	#102	—	2	2	no	—
73	#102	—	2	4	no	—
74	C₄×C₈	—	2	2	no	—
75	C₄×C₈	—	2	2	no	—
76	C₄×C₈	D₄	1	1	yes	characteristic
77	C₄×C₈	D₄	1	1	yes	characteristic
78	C₄×C₈	C₂³	1	1	yes	characteristic
79	C₄×C₈	C₂³	1	1	yes	characteristic
80	#41	—	4	4	no	—
81	#41	—	4	4	no	—
82	C₂×Q₈	—	2	1	no	—
83	C₂×Q₈	—	2	2	no	—
84	C₂×Q₈	—	2	2	no	—
85	C₂×Q₈	D₈	1	4	yes	complemented
86	C₂×Q₈	C₂×D₄	1	2	yes	—
87	C₂×D₄	—	2	2	no	—
88	C₂×D₄	—	2	2	no	—
89	C₂×D₄	—	2	2	no	—
90	C₂×D₄	—	4	2	no	—
91	C₂²×C₄	—	2	1	no	—
92	C₂²×C₄	—	2	1	no	—
93	C₂²×C₄	—	4	2	no	—
94	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
95	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
96	C₂²×C₄	#41	1	1	yes	characteristic
97	Q₁₆	—	4	8	no	—
98	SD₁₆	—	4	8	no	—
99	D₈	—	4	2	no	—
100	D₈	—	4	2	no	—
101	M₁₆	—	4	4	no	—
102	C₂×C₈	—	2	2	no	—
103	C₂×C₈	—	2	2	no	—
104	C₂×C₈	—	2	4	no	—
105	C₂×C₈	—	4	1	no	—
106	C₂×C₈	—	4	2	no	—
107	C₂×C₈	C₂×D₄	1	1	yes	characteristic
108	C₂×C₈	C₂×D₄	1	1	yes	characteristic
109	C₂×C₈	C₂×D₄	1	1	yes	characteristic
110	C₂×C₈	C₂×D₄	1	1	yes	characteristic
111	C₄⋊C₄	—	2	2	no	—
112	C₄⋊C₄	—	2	1	no	—
113	C₄⋊C₄	—	2	2	no	—
114	C₄⋊C₄	—	2	2	no	—
115	C₄⋊C₄	—	2	2	no	—
116	C₄⋊C₄	C₂×D₄	1	2	yes	—
117	C₂²⋊C₄	—	4	2	no	—
118	C₂²⋊C₄	—	4	2	no	—
119	C₄²	—	2	2	no	—
120	C₄²	—	2	2	no	—
121	C₄²	C₂×D₄	1	1	yes	characteristic
122	C₄²	C₂×D₄	1	1	yes	characteristic
123	C₄²	#41	1	1	yes	characteristic
124	C₄²	C₂⁴	1	1	yes	characteristic
125	C₂³	—	4	2	no	—
126	C₂³	#121	1	1	yes	characteristic
127	Q₈	—	2	2	no	—
128	Q₈	—	2	4	no	—
129	Q₈	—	4	2	no	—
130	Q₈	—	4	2	no	—
131	Q₈	C₂×D₈	1	4	yes	—
132	D₄	—	4	2	no	—
133	D₄	—	4	2	no	—
134	D₄	—	4	2	no	—
135	D₄	—	4	2	no	—
136	D₄	—	8	2	no	—
137	D₄	—	8	4	no	—
138	C₂×C₄	—	2	2	no	—
139	C₂×C₄	—	2	4	no	—
140	C₂×C₄	—	2	1	no	—
141	C₂×C₄	—	2	1	no	—
142	C₂×C₄	—	2	2	no	—
143	C₂×C₄	—	4	1	no	—
144	C₂×C₄	—	4	2	no	—
145	C₂×C₄	—	4	2	no	—
146	C₂×C₄	—	8	4	no	—
147	C₂×C₄	#121	1	1	yes	characteristic
148	C₂×C₄	#121	1	1	yes	characteristic
149	C₂×C₄	#121	1	1	yes	characteristic
150	C₂×C₄	C₂×D₈	1	2	yes	—
151	C₂×C₄	#135	1	2	yes	—
152	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
153	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
154	C₂×C₄	C₂×#41	1	1	yes	characteristic
155	C₈	—	2	1	no	—
156	C₈	—	2	1	no	—
157	C₈	—	2	1	no	—
158	C₈	—	4	4	no	—
159	C₈	—	4	2	no	—
160	C₈	#137	1	2	yes	complemented
161	C₂²	—	2	1	no	—
162	C₂²	—	2	1	no	—
163	C₂²	—	8	2	no	—
164	C₂²	—	8	2	no	—
165	C₂²	—	8	2	no	—
166	C₂²	#466	1	2	yes	—
167	C₂²	C₂×#121	1	1	yes	characteristic
168	C₄	—	2	1	no	—
169	C₄	—	2	4	no	—
170	C₄	—	2	1	no	—
171	C₄	—	2	1	no	—
172	C₄	—	4	2	no	—
173	C₄	—	8	2	no	—
174	C₄	#466	1	2	yes	—
175	C₄	C₂²×D₈	1	1	yes	characteristic
176	C₄	C₂×#135	1	1	yes	characteristic
177	C₄	C₂×#137	1	1	yes	characteristic
178	C₄	#575	1	1	yes	characteristic
179	C₂	—	2	2	no	—
180	C₂	—	16	2	no	—
181	C₂	#3037	1	1	yes	characteristic
182	C₂	C₂×#466	1	1	yes	characteristic
183	C₂	#3067	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂×#121
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
2. #3037	C₂	semidirect
3. C₂×C₈×Q₈	C₂	semidirect
4. #1682	C₂	semidirect
5. #1674	C₂	semidirect
6. #1668	C₂	semidirect
7. #1567	C₂	semidirect
8. #1547	C₂	semidirect
85. C₂×Q₈	D₈	semidirect
160. C₈	#137	semidirect

Maximal subgroups

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

subgroup	index
#3169	2
#3037	2
C₂×C₈×Q₈	2
#1682	2
#1674	2
#1668	2
#1567	2
#1547	2