Finite Group Databasebeta

Subgroups — #24985

Order 256.

323 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 323 conjugacy classes of subgroups — too many to draw. Only its 105 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	#3067	C₂	1	1	yes	maximal, characteristic
2	C₄×#135	C₂	1	1	yes	maximal, characteristic, complemented
3	#2080	C₂	1	2	yes	maximal, complemented
4	#2053	C₂	1	2	yes	maximal, complemented
5	#1976	C₂	1	1	yes	maximal, characteristic, complemented
6	#1945	C₂	1	1	yes	maximal, characteristic, complemented
7	#1906	C₂	1	2	yes	maximal, complemented
8	#1833	C₂	1	1	yes	maximal, characteristic, complemented
9	#1827	C₂	1	2	yes	maximal, complemented
10	#1812	C₂	1	1	yes	maximal, characteristic, complemented
11	#1764	C₂	1	1	yes	maximal, characteristic, complemented
12	#575	C₂²	1	2	yes	—
13	C₂×#135	C₂²	1	1	yes	characteristic
14	C₂×#131	C₂²	1	1	yes	characteristic
15	C₄×#41	C₂²	1	2	yes	—
16	#473	—	2	1	no	—
17	#472	—	2	1	no	—
18	#471	—	2	2	no	—
19	C₄×Q₁₆	—	2	1	no	—
20	C₄×SD₁₆	—	2	2	no	—
21	C₄×D₈	—	2	1	no	—
22	C₂×#108	C₂²	1	1	yes	characteristic
23	#427	C₂²	1	1	yes	characteristic
24	#427	C₂²	1	1	yes	characteristic
25	#424	C₂²	1	2	yes	—
26	#421	C₂²	1	2	yes	—
27	C₂×#104	C₂²	1	1	yes	characteristic
28	#416	C₂²	1	1	yes	characteristic
29	#416	C₂²	1	2	yes	—
30	#413	C₂²	1	2	yes	—
31	#410	C₂²	1	2	yes	—
32	#410	C₂²	1	2	yes	—
33	#405	C₂²	1	1	yes	characteristic
34	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
35	#344	C₂²	1	2	yes	—
36	#341	C₂²	1	1	yes	characteristic
37	#339	C₂²	1	1	yes	characteristic
38	#339	C₂²	1	2	yes	—
39	H(3,ℤ/4)	C₂²	1	2	yes	—
40	H(3,ℤ/4)	C₂²	1	2	yes	—
41	C₂×#41	C₂³	1	2	yes	—
42	C₂×#41	C₂³	1	1	yes	characteristic
43	#137	—	2	2	no	—
44	#137	—	2	2	no	—
45	AGL(1,ℤ/8)	—	2	2	no	—
46	AGL(1,ℤ/8)	—	2	2	no	—
47	#135	—	2	1	no	—
48	#135	—	2	1	no	—
49	#135	—	2	1	no	—
50	#135	—	2	1	no	—
51	#135	—	4	2	no	—
52	C₂×Q₁₆	D₄	1	1	yes	characteristic
53	C₂×SD₁₆	D₄	1	2	yes	—
54	C₂×D₈	D₄	1	1	yes	characteristic
55	#131	—	2	1	no	—
56	#131	—	2	1	no	—
57	#131	—	2	1	no	—
58	#131	—	2	1	no	—
59	C₂×M₁₆	—	2	1	no	—
60	C₂×M₁₆	C₂³	1	1	yes	characteristic
61	C₂×M₁₆	C₂³	1	2	yes	—
62	C₂×M₁₆	C₂³	1	1	yes	characteristic
63	C₂²×C₈	—	2	1	no	—
64	C₂²×C₈	C₂³	1	1	yes	characteristic
65	C₂²×C₈	C₂³	1	1	yes	characteristic
66	C₄×Q₈	—	2	2	no	—
67	C₄×D₄	—	2	2	no	—
68	C₄×D₄	—	4	2	no	—
69	#117	—	2	2	no	—
70	#117	C₂³	1	2	yes	—
71	#117	C₂³	1	1	yes	characteristic
72	#117	C₂³	1	2	yes	—
73	C₂×C₄²	—	2	2	no	—
74	C₂×C₄²	C₂³	1	1	yes	characteristic
75	#108	—	2	1	no	—
76	#108	—	2	1	no	—
77	#107	—	2	1	no	—
78	#107	—	2	1	no	—
79	#106	—	2	1	no	—
80	#106	—	2	1	no	—
81	C₄⋊C₈	—	4	2	no	—
82	#104	—	2	1	no	—
83	#104	—	2	1	no	—
84	#104	—	4	2	no	—
85	#103	—	2	2	no	—
86	#103	—	2	1	no	—
87	#102	—	2	2	no	—
88	#102	—	2	1	no	—
89	#101	—	2	2	no	—
90	#100	—	2	2	no	—
91	#99	—	4	2	no	—
92	#99	—	4	2	no	—
93	#97	—	2	1	no	—
94	C₄×C₈	—	2	1	no	—
95	C₄×C₈	—	2	1	no	—
96	C₄×C₈	—	2	1	no	—
97	#41	—	2	1	no	—
98	#41	—	2	1	no	—
99	#41	—	2	1	no	—
100	#41	—	2	2	no	—
101	#41	—	2	2	no	—
102	#41	—	2	2	no	—
103	#41	—	2	2	no	—
104	#41	—	2	1	no	—
105	C₂×Q₈	C₂×D₄	1	2	yes	—
106	C₂×Q₈	#41	1	1	yes	characteristic
107	C₂×D₄	—	2	1	no	—
108	C₂×D₄	—	2	2	no	—
109	C₂×D₄	C₂×D₄	1	2	yes	—
110	C₂×D₄	#41	1	1	yes	characteristic
111	C₂²×C₄	—	2	1	no	—
112	C₂²×C₄	—	2	2	no	—
113	C₂²×C₄	—	2	1	no	—
114	C₂²×C₄	—	4	2	no	—
115	C₂²×C₄	C₂⁴	1	1	yes	characteristic
116	Q₁₆	—	2	1	no	—
117	Q₁₆	—	2	1	no	—
118	Q₁₆	—	4	2	no	—
119	SD₁₆	—	2	2	no	—
120	SD₁₆	—	2	2	no	—
121	SD₁₆	—	4	2	no	—
122	SD₁₆	—	4	2	no	—
123	D₈	—	2	1	no	—
124	D₈	—	2	1	no	—
125	D₈	—	4	2	no	—
126	M₁₆	—	2	1	no	—
127	M₁₆	—	2	2	no	—
128	M₁₆	—	2	1	no	—
129	M₁₆	—	2	1	no	—
130	M₁₆	—	2	2	no	—
131	M₁₆	—	2	1	no	—
132	M₁₆	—	4	1	no	—
133	M₁₆	—	4	1	no	—
134	C₂×C₈	—	2	1	no	—
135	C₂×C₈	—	2	1	no	—
136	C₂×C₈	—	2	1	no	—
137	C₂×C₈	—	2	1	no	—
138	C₂×C₈	—	2	1	no	—
139	C₂×C₈	—	2	2	no	—
140	C₂×C₈	—	4	1	no	—
141	C₂×C₈	—	4	1	no	—
142	C₂×C₈	C₂×D₄	1	1	yes	characteristic
143	C₂×C₈	C₂×D₄	1	1	yes	characteristic
144	C₂×C₈	#41	1	1	yes	characteristic
145	C₂×C₈	#41	1	1	yes	characteristic
146	C₂×C₈	#41	1	1	yes	characteristic
147	C₂×C₈	#41	1	1	yes	characteristic
148	C₄⋊C₄	—	2	2	no	—
149	C₄⋊C₄	—	2	1	no	—
150	C₄⋊C₄	—	2	2	no	—
151	C₄⋊C₄	—	4	2	no	—
152	C₂²⋊C₄	—	2	2	no	—
153	C₂²⋊C₄	—	2	1	no	—
154	C₂²⋊C₄	—	2	2	no	—
155	C₂²⋊C₄	—	4	2	no	—
156	C₄²	—	2	1	no	—
157	C₄²	—	2	1	no	—
158	C₄²	—	2	2	no	—
159	C₄²	—	2	2	no	—
160	C₄²	—	4	2	no	—
161	C₄²	#41	1	1	yes	characteristic
162	C₄²	#41	1	1	yes	characteristic
163	C₂³	—	2	2	no	—
164	C₂³	—	2	1	no	—
165	C₂³	H(5,𝔽₂)	1	1	yes	characteristic
166	Q₈	—	2	2	no	—
167	Q₈	—	2	1	no	—
168	Q₈	—	2	1	no	—
169	Q₈	—	2	2	no	—
170	D₄	—	2	2	no	—
171	D₄	—	2	1	no	—
172	D₄	—	2	1	no	—
173	D₄	—	2	2	no	—
174	D₄	—	4	1	no	—
175	D₄	—	4	2	no	—
176	D₄	—	4	1	no	—
177	D₄	—	4	2	no	—
178	C₂×C₄	—	2	1	no	—
179	C₂×C₄	—	2	1	no	—
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	2	1	no	—
182	C₂×C₄	—	4	2	no	—
183	C₂×C₄	—	4	1	no	—
184	C₂×C₄	—	4	2	no	—
185	C₂×C₄	—	4	2	no	—
186	C₂×C₄	—	4	1	no	—
187	C₂×C₄	—	4	2	no	—
188	C₂×C₄	—	4	2	no	—
189	C₂×C₄	—	4	1	no	—
190	C₂×C₄	—	8	4	no	—
191	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
192	C₂×C₄	C₂×#41	1	1	yes	characteristic
193	C₂×C₄	C₂×#41	1	1	yes	characteristic
194	C₂×C₄	C₂×#41	1	1	yes	characteristic
195	C₂×C₄	C₂×#41	1	1	yes	characteristic
196	C₂×C₄	H(5,𝔽₂)	1	1	yes	characteristic
197	C₈	—	4	1	no	—
198	C₈	—	4	1	no	—
199	C₈	—	4	2	no	—
200	C₈	—	4	1	no	—
201	C₈	#123	1	1	yes	characteristic
202	C₈	#123	1	1	yes	characteristic
203	C₈	#123	1	1	yes	characteristic
204	C₈	#123	1	1	yes	characteristic
205	C₂²	—	4	1	no	—
206	C₂²	—	4	1	no	—
207	C₂²	—	4	2	no	—
208	C₂²	—	8	2	no	—
209	C₂²	—	8	1	no	—
210	C₂²	#540	1	1	yes	characteristic
211	C₂²	#546	1	1	yes	characteristic
212	C₂²	#553	1	1	yes	characteristic
213	C₄	—	2	2	no	—
214	C₄	—	4	1	no	—
215	C₄	—	4	2	no	—
216	C₄	—	4	1	no	—
217	C₄	—	8	2	no	—
218	C₄	—	8	1	no	—
219	C₄	—	8	2	no	—
220	C₄	C₂×#123	1	1	yes	characteristic
221	C₄	#529	1	1	yes	characteristic
222	C₄	#546	1	1	yes	characteristic
223	C₄	#555	1	1	yes	characteristic
224	C₂	—	2	1	no	—
225	C₂	—	2	1	no	—
226	C₂	—	2	1	no	—
227	C₂	—	8	2	no	—
228	C₂	—	8	1	no	—
229	C₂	#2420	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#555
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴
socle	C₂	#2420
prime radical	#3067	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. C₄×#135	C₂	semidirect
3. #2080	C₂	semidirect
4. #2053	C₂	semidirect
5. #1976	C₂	semidirect
6. #1945	C₂	semidirect
7. #1906	C₂	semidirect
8. #1833	C₂	semidirect
9. #1827	C₂	semidirect
10. #1812	C₂	semidirect
11. #1764	C₂	semidirect

Maximal subgroups

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

subgroup	index
#3067	2
C₄×#135	2
#2080	2
#2053	2
#1976	2
#1945	2
#1906	2
#1833	2
#1827	2
#1812	2
#1764	2