Finite Group Databasebeta

Subgroups — #22606

Order 256.

352 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 352 conjugacy classes of subgroups — too many to draw. Only its 161 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	#2924	C₂	1	2	yes	maximal
2	C₂×#427	C₂	1	1	yes	maximal, characteristic
3	#1945	C₂	1	2	yes	maximal, complemented
4	#1922	C₂	1	4	yes	maximal, complemented
5	#1826	C₂	1	2	yes	maximal
6	#1810	C₂	1	1	yes	maximal, characteristic
7	#1772	C₂	1	2	yes	maximal, complemented
8	C₈×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic
9	C₂³×C₈	C₂²	1	1	yes	characteristic
10	C₂×#117	C₂²	1	2	yes	—
11	#433	—	2	4	no	—
12	#432	—	2	2	no	—
13	#431	—	2	2	no	—
14	C₂×#108	C₂²	1	2	yes	—
15	#427	C₄	1	1	yes	characteristic, complemented
16	#427	C₄	1	1	yes	characteristic, complemented
17	#427	C₄	1	2	yes	complemented
18	#427	C₂²	1	2	yes	—
19	#427	C₂²	1	1	yes	characteristic
20	#427	C₂²	1	1	yes	characteristic
21	C₂×#107	C₄	1	1	yes	characteristic, complemented
22	C₂×#107	C₄	1	1	yes	characteristic, complemented
23	C₂×#106	C₄	1	2	yes	complemented
24	#424	C₂²	1	4	yes	—
25	#407	C₂²	1	2	yes	—
26	C₂×C₂²⋊C₈	C₂²	1	1	yes	characteristic
27	#405	C₂²	1	4	yes	—
28	C₂×C₄×C₈	C₂²	1	2	yes	—
29	C₄×C₂²⋊C₄	C₂²	1	1	yes	characteristic
30	#341	C₂²	1	2	yes	—
31	#339	C₂²	1	4	yes	—
32	#339	C₂²	1	4	yes	—
33	#336	C₂²	1	2	yes	—
34	C₂³×C₄	C₂³	1	1	yes	characteristic
35	C₂×M₁₆	C₂³	1	4	yes	—
36	C₂²×C₈	—	2	4	no	—
37	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
41	C₂²×C₈	C₂³	1	1	yes	characteristic
42	C₂²×C₈	C₂³	1	2	yes	—
43	C₂²×C₈	C₂³	1	1	yes	characteristic
44	#117	—	2	2	no	—
45	#117	—	2	2	no	—
46	#117	C₂×C₄	1	2	yes	—
47	#117	C₂×C₄	1	2	yes	—
48	#117	C₂³	1	2	yes	—
49	#117	C₂³	1	2	yes	—
50	C₂×C₄⋊C₄	C₂×C₄	1	2	yes	—
51	C₂×C₄⋊C₄	C₂×C₄	1	2	yes	—
52	C₂×C₂²⋊C₄	—	2	2	no	—
53	C₂×C₂²⋊C₄	—	2	1	no	—
54	C₂×C₄²	—	2	2	no	—
55	C₂×C₄²	C₂³	1	2	yes	—
56	#108	—	2	4	no	—
57	#107	C₂×C₄	1	2	yes	—
58	#107	C₂×C₄	1	2	yes	—
59	#107	C₂×C₄	1	1	yes	characteristic
60	#107	C₂×C₄	1	1	yes	characteristic
61	#107	C₂×C₄	1	1	yes	characteristic
62	#107	C₂×C₄	1	1	yes	characteristic
63	#106	C₂×C₄	1	2	yes	—
64	#106	C₂×C₄	1	2	yes	—
65	#106	C₂×C₄	1	2	yes	—
66	#106	C₂×C₄	1	2	yes	—
67	C₄⋊C₈	—	2	4	no	—
68	C₄⋊C₈	—	2	4	no	—
69	C₂²⋊C₈	—	2	2	no	—
70	C₂²⋊C₈	—	2	2	no	—
71	C₂²⋊C₈	D₄	1	1	yes	characteristic
72	C₂²⋊C₈	D₄	1	1	yes	characteristic
73	C₂²⋊C₈	D₄	1	2	yes	—
74	#97	—	2	4	no	—
75	C₄×C₈	—	2	4	no	—
76	C₄×C₈	—	2	4	no	—
77	#95	—	2	1	no	—
78	C₂⁴	#41	1	1	yes	characteristic
79	C₂²×C₄	—	2	2	no	—
80	C₂²×C₄	—	2	2	no	—
81	C₂²×C₄	—	2	2	no	—
82	C₂²×C₄	—	2	2	no	—
83	C₂²×C₄	—	2	2	no	—
84	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
85	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
86	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
87	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
88	C₂²×C₄	#41	1	1	yes	characteristic
89	C₂²×C₄	C₂⁴	1	1	yes	characteristic
90	M₁₆	—	2	4	no	—
91	M₁₆	—	2	4	no	—
92	C₂×C₈	—	2	2	no	—
93	C₂×C₈	—	2	2	no	—
94	C₂×C₈	—	2	2	no	—
95	C₂×C₈	—	2	2	no	—
96	C₂×C₈	—	2	2	no	—
97	C₂×C₈	—	2	2	no	—
98	C₂×C₈	—	2	2	no	—
99	C₂×C₈	—	2	4	no	—
100	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
101	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
102	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
103	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
104	C₂×C₈	C₂×D₄	1	2	yes	—
105	C₂×C₈	C₂×D₄	1	2	yes	—
106	C₂×C₈	#41	1	4	yes	—
107	C₄⋊C₄	—	2	2	no	—
108	C₄⋊C₄	—	2	2	no	—
109	C₄⋊C₄	C₂²×C₄	1	2	yes	—
110	C₄⋊C₄	C₂²×C₄	1	2	yes	—
111	C₄⋊C₄	C₂²×C₄	1	2	yes	—
112	C₄⋊C₄	C₂²×C₄	1	2	yes	—
113	C₂²⋊C₄	—	2	4	no	—
114	C₂²⋊C₄	—	2	2	no	—
115	C₂²⋊C₄	—	2	2	no	—
116	C₂²⋊C₄	—	2	2	no	—
117	C₂²⋊C₄	—	2	2	no	—
118	C₄²	—	2	2	no	—
119	C₄²	—	2	2	no	—
120	C₄²	—	2	2	no	—
121	C₄²	—	2	2	no	—
122	C₄²	—	2	2	no	—
123	C₄²	—	2	2	no	—
124	C₂³	—	2	2	no	—
125	C₂³	—	2	2	no	—
126	C₂³	—	4	1	no	—
127	C₂³	C₄×D₄	1	1	yes	characteristic
128	C₂³	C₄×D₄	1	1	yes	characteristic
129	C₂³	C₂×#41	1	1	yes	characteristic
130	C₂×C₄	—	2	2	no	—
131	C₂×C₄	—	2	1	no	—
132	C₂×C₄	—	2	1	no	—
133	C₂×C₄	—	2	1	no	—
134	C₂×C₄	—	2	1	no	—
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	—	2	4	no	—
138	C₂×C₄	—	2	2	no	—
139	C₂×C₄	—	2	2	no	—
140	C₂×C₄	—	2	2	no	—
141	C₂×C₄	—	2	2	no	—
142	C₂×C₄	—	4	2	no	—
143	C₂×C₄	—	4	2	no	—
144	C₂×C₄	—	4	4	no	—
145	C₂×C₄	—	8	2	no	—
146	C₂×C₄	C₄×D₄	1	1	yes	characteristic
147	C₂×C₄	C₄×D₄	1	1	yes	characteristic
148	C₂×C₄	C₂³×C₄	1	1	yes	characteristic
149	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
150	C₂×C₄	C₂×#41	1	2	yes	—
151	C₂×C₄	2₋¹⁺⁴	1	2	yes	—
152	C₈	—	2	1	no	—
153	C₈	—	2	1	no	—
154	C₈	—	2	2	no	—
155	C₈	—	2	2	no	—
156	C₈	—	4	4	no	—
157	C₈	#117	1	2	yes	—
158	C₈	#117	1	2	yes	—
159	C₂²	—	2	1	no	—
160	C₂²	—	2	1	no	—
161	C₂²	—	2	1	no	—
162	C₂²	—	2	1	no	—
163	C₂²	—	2	1	no	—
164	C₂²	—	2	1	no	—
165	C₂²	—	2	1	no	—
166	C₂²	—	4	4	no	—
167	C₂²	#443	1	2	yes	—
168	C₂²	C₂×C₄×D₄	1	1	yes	characteristic
169	C₂²	#552	1	2	yes	—
170	C₄	—	2	1	no	—
171	C₄	—	2	1	no	—
172	C₄	—	4	4	no	—
173	C₄	—	4	2	no	—
174	C₄	—	4	2	no	—
175	C₄	—	4	2	no	—
176	C₄	—	4	2	no	—
177	C₄	C₂×#117	1	1	yes	characteristic
178	C₄	#519	1	1	yes	characteristic
179	C₄	#547	1	2	yes	—
180	C₂	—	2	2	no	—
181	C₂	—	2	2	no	—
182	C₂	—	4	1	no	—
183	C₂	#2344	1	1	yes	characteristic
184	C₂	C₂×#443	1	1	yes	characteristic
185	C₂	#2957	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄
Frattini, cosocle	C₂²×C₄	C₂⁴
socle	C₂²	C₂×C₄×D₄
prime radical	C₂⁴	#41
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₂²⋊C₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
3. #1945	C₂	semidirect
4. #1922	C₂	semidirect
7. #1772	C₂	semidirect
15. #427	C₄	semidirect
16. #427	C₄	semidirect
17. #427	C₄	semidirect
21. C₂×#107	C₄	semidirect
22. C₂×#107	C₄	semidirect
23. C₂×#106	C₄	semidirect

Maximal subgroups

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

subgroup	index
#2924	2
C₂×#427	2
#1945	2
#1922	2
#1826	2
#1810	2
#1772	2
C₈×C₂²⋊C₄	2