Finite Group Databasebeta

Subgroups — #13324

Order 256.

257 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 257 conjugacy classes of subgroups — too many to draw. Only its 57 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	C₂×#510	C₂	1	1	yes	maximal, characteristic, complemented
2	#3067	C₂	1	1	yes	maximal, characteristic
3	C₂×#473	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#362	C₂	1	1	yes	maximal, characteristic, complemented
5	#2143	C₂	1	1	yes	maximal, characteristic, complemented
6	#2115	C₂	1	1	yes	maximal, characteristic, complemented
7	#1388	C₂	1	1	yes	maximal, characteristic, complemented
8	#575	—	2	1	no	—
9	C₂×#137	—	2	1	no	—
10	C₂×#135	C₂²	1	1	yes	characteristic
11	C₂²×Q₁₆	C₂²	1	1	yes	characteristic
12	C₂×#131	C₂²	1	1	yes	characteristic
13	#510	—	2	2	no	—
14	#510	—	2	2	no	—
15	C₂×Q₃₂	—	2	1	no	—
16	C₂×SD₃₂	—	2	1	no	—
17	C₂×M₃₂	C₂²	1	1	yes	characteristic
18	C₂×M₃₂	C₂²	1	1	yes	characteristic
19	#473	—	2	2	no	—
20	#473	—	2	1	no	—
21	#473	—	2	2	no	—
22	#472	—	2	1	no	—
23	#471	—	2	2	no	—
24	C₂×#108	C₂²	1	1	yes	characteristic
25	C₂×#108	C₂²	1	1	yes	characteristic
26	#413	—	2	1	no	—
27	C₂×#101	—	2	1	no	—
28	#362	—	2	2	no	—
29	#359	—	2	2	no	—
30	#349	—	2	2	no	—
31	C₂×#41	—	2	1	no	—
32	C₂×#41	D₄	1	1	yes	characteristic
33	C₂²×Q₈	—	2	1	no	—
34	#137	—	2	4	no	—
35	#137	—	2	2	no	—
36	#137	—	4	2	no	—
37	#137	—	4	1	no	—
38	AGL(1,ℤ/8)	—	2	2	no	—
39	AGL(1,ℤ/8)	—	4	1	no	—
40	#135	—	2	1	no	—
41	#135	—	2	1	no	—
42	#135	—	2	1	no	—
43	#135	—	2	1	no	—
44	#135	—	4	2	no	—
45	C₂×Q₁₆	—	2	1	no	—
46	C₂×Q₁₆	—	2	1	no	—
47	C₂×Q₁₆	—	2	1	no	—
48	C₂×Q₁₆	—	2	1	no	—
49	C₂×Q₁₆	—	2	1	no	—
50	C₂×Q₁₆	—	4	1	no	—
51	C₂×Q₁₆	D₄	1	2	yes	—
52	C₂×Q₁₆	D₄	1	1	yes	characteristic
53	C₂×SD₁₆	—	2	1	no	—
54	C₂×SD₁₆	—	4	1	no	—
55	C₂×D₈	D₄	1	1	yes	characteristic
56	#131	—	2	1	no	—
57	#131	—	2	1	no	—
58	#131	—	2	1	no	—
59	#131	—	2	1	no	—
60	C₂×M₁₆	—	2	1	no	—
61	C₂×M₁₆	—	2	1	no	—
62	C₂×M₁₆	—	2	1	no	—
63	C₂×M₁₆	D₄	1	1	yes	characteristic
64	C₂²×C₈	—	2	1	no	—
65	C₂²×C₈	C₂³	1	1	yes	characteristic
66	Q₃₂	—	2	4	no	—
67	SD₃₂	—	2	4	no	—
68	M₃₂	—	2	1	no	—
69	M₃₂	—	2	1	no	—
70	M₃₂	—	2	1	no	—
71	M₃₂	D₄	1	2	yes	—
72	C₂×C₁₆	—	2	1	no	—
73	C₂×C₁₆	—	2	1	no	—
74	#108	—	2	1	no	—
75	#108	—	2	1	no	—
76	#108	D₄	1	2	yes	—
77	#108	D₄	1	2	yes	—
78	#101	—	2	4	no	—
79	#101	—	2	2	no	—
80	#100	—	2	2	no	—
81	#41	—	2	1	no	—
82	#41	—	2	1	no	—
83	#41	—	2	1	no	—
84	#41	—	2	1	no	—
85	#41	—	4	1	no	—
86	#41	—	4	1	no	—
87	#41	—	4	1	no	—
88	#41	—	4	1	no	—
89	C₂×Q₈	—	2	1	no	—
90	C₂×Q₈	—	2	2	no	—
91	C₂×Q₈	—	4	1	no	—
92	C₂×Q₈	—	4	1	no	—
93	C₂×Q₈	—	4	1	no	—
94	C₂×Q₈	—	4	1	no	—
95	C₂×Q₈	—	4	1	no	—
96	C₂×Q₈	D₈	1	1	yes	characteristic
97	C₂×D₄	—	2	1	no	—
98	C₂×D₄	—	2	1	no	—
99	C₂×D₄	—	4	1	no	—
100	C₂×D₄	D₈	1	1	yes	characteristic
101	C₂²×C₄	—	2	1	no	—
102	C₂²×C₄	—	4	1	no	—
103	C₂²×C₄	—	4	1	no	—
104	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
105	Q₁₆	—	2	2	no	—
106	Q₁₆	—	2	1	no	—
107	Q₁₆	—	2	2	no	—
108	Q₁₆	—	2	1	no	—
109	Q₁₆	—	4	4	no	—
110	Q₁₆	—	4	2	no	—
111	Q₁₆	—	4	1	no	—
112	Q₁₆	—	4	1	no	—
113	SD₁₆	—	4	4	no	—
114	SD₁₆	—	4	2	no	—
115	SD₁₆	—	4	2	no	—
116	SD₁₆	—	4	1	no	—
117	SD₁₆	—	4	1	no	—
118	D₈	—	2	1	no	—
119	D₈	—	2	1	no	—
120	D₈	—	4	2	no	—
121	M₁₆	—	2	1	no	—
122	M₁₆	—	2	2	no	—
123	M₁₆	—	2	1	no	—
124	M₁₆	—	2	2	no	—
125	M₁₆	—	4	1	no	—
126	M₁₆	—	4	1	no	—
127	M₁₆	—	4	1	no	—
128	M₁₆	—	4	1	no	—
129	C₂×C₈	—	2	1	no	—
130	C₂×C₈	—	4	1	no	—
131	C₂×C₈	—	4	1	no	—
132	C₂×C₈	—	4	1	no	—
133	C₂×C₈	—	4	1	no	—
134	C₂×C₈	C₂×D₄	1	1	yes	characteristic
135	C₂×C₈	C₂×D₄	1	1	yes	characteristic
136	C₂×C₈	C₂×D₄	1	1	yes	characteristic
137	C₂×C₈	C₂×D₄	1	1	yes	characteristic
138	C₂×C₈	C₂×D₄	1	1	yes	characteristic
139	C₂×C₈	#41	1	1	yes	characteristic
140	C₁₆	—	2	2	no	—
141	C₁₆	—	4	1	no	—
142	C₂³	—	2	1	no	—
143	C₂³	—	4	1	no	—
144	C₂³	#135	1	1	yes	characteristic
145	Q₈	—	2	1	no	—
146	Q₈	—	2	1	no	—
147	Q₈	—	4	1	no	—
148	Q₈	—	4	2	no	—
149	Q₈	—	4	2	no	—
150	Q₈	—	4	1	no	—
151	Q₈	—	8	1	no	—
152	Q₈	—	8	1	no	—
153	D₄	—	2	1	no	—
154	D₄	—	2	1	no	—
155	D₄	—	4	1	no	—
156	D₄	—	4	1	no	—
157	D₄	—	4	1	no	—
158	D₄	—	4	1	no	—
159	D₄	—	8	1	no	—
160	D₄	—	8	1	no	—
161	C₂×C₄	—	2	1	no	—
162	C₂×C₄	—	4	1	no	—
163	C₂×C₄	—	4	1	no	—
164	C₂×C₄	—	4	1	no	—
165	C₂×C₄	—	4	2	no	—
166	C₂×C₄	—	8	1	no	—
167	C₂×C₄	—	8	1	no	—
168	C₂×C₄	—	8	1	no	—
169	C₂×C₄	—	8	1	no	—
170	C₂×C₄	#120	1	1	yes	characteristic
171	C₂×C₄	#120	1	1	yes	characteristic
172	C₂×C₄	#121	1	1	yes	characteristic
173	C₂×C₄	C₂×D₈	1	1	yes	characteristic
174	C₂×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
175	C₂×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
176	C₈	—	4	1	no	—
177	C₈	—	4	2	no	—
178	C₈	—	4	2	no	—
179	C₈	#120	1	1	yes	characteristic
180	C₈	#121	1	1	yes	characteristic
181	C₈	#121	1	1	yes	characteristic
182	C₈	#127	1	1	yes	characteristic
183	C₂²	—	4	1	no	—
184	C₂²	—	4	1	no	—
185	C₂²	—	8	1	no	—
186	C₂²	—	8	1	no	—
187	C₂²	—	16	1	no	—
188	C₂²	#449	1	1	yes	characteristic
189	C₂²	#449	1	1	yes	characteristic
190	C₂²	#466	1	1	yes	characteristic
191	C₄	—	4	1	no	—
192	C₄	—	8	1	no	—
193	C₄	—	8	2	no	—
194	C₄	#392	1	1	yes	characteristic
195	C₄	#447	1	1	yes	characteristic
196	C₄	#447	1	1	yes	characteristic
197	C₄	#469	1	1	yes	characteristic
198	C₂	—	2	1	no	—
199	C₂	—	2	1	no	—
200	C₂	—	2	1	no	—
201	C₂	—	8	1	no	—
202	C₂	—	16	1	no	—
203	C₂	#1999	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	#1999
commutator, Frattini, cosocle	C₂²×C₈	C₂³
prime radical	#3067	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. C₂×#510	C₂	semidirect
3. C₂×#473	C₂	semidirect
4. C₂×#362	C₂	semidirect
5. #2143	C₂	semidirect
6. #2115	C₂	semidirect
7. #1388	C₂	semidirect

Maximal subgroups

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

subgroup	index
C₂×#510	2
#3067	2
C₂×#473	2
C₂×#362	2
#2143	2
#2115	2
#1388	2