Finite Group Databasebeta

Subgroups — #13005

Order 256.

246 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 73 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	#2912	C₂	1	1	yes	maximal, characteristic, complemented
3	#2145	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#360	C₂	1	1	yes	maximal, characteristic, complemented
5	#2117	C₂	1	1	yes	maximal, characteristic, complemented
6	#1389	C₂	1	1	yes	maximal, characteristic, complemented
7	#1383	C₂	1	1	yes	maximal, characteristic, complemented
8	#575	—	2	1	no	—
9	C₂×#135	C₂²	1	1	yes	characteristic
10	C₂×#131	C₂²	1	1	yes	characteristic
11	#520	—	2	1	no	—
12	C₂×M₃₂	C₂²	1	1	yes	characteristic
13	C₂×M₃₂	C₂²	1	1	yes	characteristic
14	#473	C₄	1	2	yes	complemented
15	#472	C₄	1	2	yes	complemented
16	#471	—	2	2	no	—
17	C₂×#108	C₂²	1	1	yes	characteristic
18	#428	—	2	1	no	—
19	#428	—	2	2	no	—
20	#427	—	2	1	no	—
21	#427	C₂²	1	1	yes	characteristic
22	C₂×#106	—	2	1	no	—
23	C₂×#106	C₂²	1	1	yes	characteristic
24	#413	—	2	1	no	—
25	#360	—	2	2	no	—
26	#360	—	2	1	no	—
27	#360	—	2	1	no	—
28	#350	—	2	2	no	—
29	C₂×#41	—	2	1	no	—
30	C₂×#41	D₄	1	1	yes	characteristic
31	#137	—	2	2	no	—
32	#137	—	4	1	no	—
33	AGL(1,ℤ/8)	—	2	2	no	—
34	AGL(1,ℤ/8)	—	4	1	no	—
35	#135	—	2	1	no	—
36	#135	—	2	1	no	—
37	#135	—	2	1	no	—
38	#135	—	2	1	no	—
39	#135	—	4	2	no	—
40	C₂×Q₁₆	C₂×C₄	1	1	yes	characteristic, complemented
41	C₂×SD₁₆	—	2	1	no	—
42	C₂×D₈	C₂×C₄	1	1	yes	characteristic, complemented
43	#131	—	2	1	no	—
44	#131	—	2	1	no	—
45	#131	C₂×C₄	1	2	yes	—
46	#131	D₄	1	2	yes	—
47	C₂×M₁₆	—	2	1	no	—
48	C₂×M₁₆	—	2	1	no	—
49	C₂×M₁₆	D₄	1	1	yes	characteristic
50	C₂²×C₈	—	2	1	no	—
51	C₂²×C₈	C₂³	1	1	yes	characteristic
52	C₄×Q₈	—	4	1	no	—
53	C₄×D₄	—	4	2	no	—
54	C₄×D₄	—	4	1	no	—
55	#117	—	2	2	no	—
56	#117	—	2	1	no	—
57	C₂×C₄⋊C₄	—	2	2	no	—
58	C₂×C₄⋊C₄	—	2	1	no	—
59	M₃₂	—	2	1	no	—
60	M₃₂	—	2	1	no	—
61	M₃₂	—	2	1	no	—
62	M₃₂	—	2	1	no	—
63	C₂×C₁₆	—	2	1	no	—
64	C₂×C₁₆	—	2	1	no	—
65	#108	—	2	1	no	—
66	#108	C₂×C₄	1	2	yes	complemented
67	#107	—	2	2	no	—
68	#107	—	2	1	no	—
69	#106	—	2	2	no	—
70	#106	—	2	1	no	—
71	#106	D₄	1	1	yes	characteristic
72	#106	D₄	1	2	yes	—
73	#106	D₄	1	1	yes	characteristic
74	#101	—	2	2	no	—
75	#100	—	2	2	no	—
76	#41	—	2	1	no	—
77	#41	—	2	1	no	—
78	#41	—	4	1	no	—
79	#41	—	4	1	no	—
80	#41	—	4	1	no	—
81	#41	—	4	1	no	—
82	#41	C₂²⋊C₄	1	2	yes	—
83	#41	SD₁₆	1	2	yes	—
84	C₂×Q₈	—	2	1	no	—
85	C₂×Q₈	D₈	1	1	yes	characteristic
86	C₂×D₄	—	2	1	no	—
87	C₂×D₄	—	2	1	no	—
88	C₂×D₄	—	4	1	no	—
89	C₂×D₄	D₈	1	1	yes	characteristic
90	C₂²×C₄	—	2	1	no	—
91	C₂²×C₄	—	4	2	no	—
92	C₂²×C₄	—	4	1	no	—
93	C₂²×C₄	—	4	1	no	—
94	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
95	Q₁₆	—	2	1	no	—
96	Q₁₆	—	2	1	no	—
97	Q₁₆	—	4	2	no	—
98	SD₁₆	—	4	2	no	—
99	SD₁₆	—	4	2	no	—
100	SD₁₆	—	4	1	no	—
101	SD₁₆	—	4	1	no	—
102	D₈	—	2	1	no	—
103	D₈	—	2	1	no	—
104	D₈	—	4	2	no	—
105	M₁₆	—	2	1	no	—
106	M₁₆	—	2	2	no	—
107	M₁₆	—	2	2	no	—
108	M₁₆	—	2	2	no	—
109	M₁₆	—	4	1	no	—
110	M₁₆	C₂²⋊C₄	1	2	yes	—
111	C₂×C₈	—	2	1	no	—
112	C₂×C₈	—	2	2	no	—
113	C₂×C₈	—	2	2	no	—
114	C₂×C₈	—	4	1	no	—
115	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
116	C₂×C₈	C₂×D₄	1	1	yes	characteristic
117	C₂×C₈	C₂×D₄	1	1	yes	characteristic
118	C₂×C₈	C₂×D₄	1	1	yes	characteristic
119	C₂×C₈	#41	1	1	yes	characteristic
120	C₂×C₈	#41	1	1	yes	characteristic
121	C₄⋊C₄	—	2	1	no	—
122	C₄⋊C₄	—	2	2	no	—
123	C₄⋊C₄	—	2	1	no	—
124	C₄⋊C₄	—	4	2	no	—
125	C₄⋊C₄	—	4	1	no	—
126	C₂²⋊C₄	—	4	2	no	—
127	C₂²⋊C₄	—	4	1	no	—
128	C₄²	—	4	2	no	—
129	C₄²	—	4	1	no	—
130	C₁₆	—	4	1	no	—
131	C₁₆	—	4	1	no	—
132	C₂³	—	2	1	no	—
133	C₂³	—	4	1	no	—
134	C₂³	AGL(1,ℤ/8)	1	1	yes	characteristic
135	Q₈	—	2	1	no	—
136	Q₈	—	4	1	no	—
137	Q₈	—	4	1	no	—
138	Q₈	#102	1	2	yes	—
139	D₄	—	2	2	no	—
140	D₄	—	2	2	no	—
141	D₄	—	2	1	no	—
142	D₄	—	4	1	no	—
143	D₄	—	4	1	no	—
144	D₄	—	8	1	no	—
145	D₄	—	8	1	no	—
146	D₄	#102	1	2	yes	—
147	C₂×C₄	—	2	1	no	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	—	2	2	no	—
150	C₂×C₄	—	4	1	no	—
151	C₂×C₄	—	4	1	no	—
152	C₂×C₄	—	4	2	no	—
153	C₂×C₄	—	4	1	no	—
154	C₂×C₄	—	8	1	no	—
155	C₂×C₄	—	8	1	no	—
156	C₂×C₄	—	16	1	no	—
157	C₂×C₄	—	16	2	no	—
158	C₂×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
159	C₂×C₄	#120	1	1	yes	characteristic
160	C₂×C₄	#123	1	1	yes	characteristic
161	C₂×C₄	C₂×D₈	1	1	yes	characteristic
162	C₂×C₄	C₂×SD₁₆	1	1	yes	characteristic
163	C₂×C₄	#137	1	1	yes	characteristic
164	C₈	—	2	2	no	—
165	C₈	—	4	2	no	—
166	C₈	C₄×D₄	1	1	yes	characteristic
167	C₈	C₄×D₄	1	1	yes	characteristic
168	C₈	#121	1	1	yes	characteristic
169	C₈	#121	1	1	yes	characteristic
170	C₂²	—	2	2	no	—
171	C₂²	—	4	1	no	—
172	C₂²	—	4	2	no	—
173	C₂²	—	8	1	no	—
174	C₂²	—	16	1	no	—
175	C₂²	#417	1	1	yes	characteristic
176	C₂²	#447	1	1	yes	characteristic
177	C₂²	#481	1	1	yes	characteristic
178	C₄	—	2	2	no	—
179	C₄	—	8	1	no	—
180	C₄	—	8	2	no	—
181	C₄	—	8	1	no	—
182	C₄	—	8	1	no	—
183	C₄	#386	1	1	yes	characteristic
184	C₄	C₂×#102	1	1	yes	characteristic
185	C₄	#448	1	1	yes	characteristic
186	C₄	#480	1	1	yes	characteristic
187	C₂	—	2	1	no	—
188	C₂	—	2	1	no	—
189	C₂	—	2	1	no	—
190	C₂	—	4	2	no	—
191	C₂	—	16	1	no	—
192	C₂	#1874	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	#1874
commutator	C₂×C₈	C₂²×C₄
Frattini, cosocle	C₂²×C₈	C₂³
prime radical	#3067	C₂
LCS	G ▷ C₂×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
2. #2912	C₂	semidirect
3. #2145	C₂	semidirect
4. C₂×#360	C₂	semidirect
5. #2117	C₂	semidirect
6. #1389	C₂	semidirect
7. #1383	C₂	semidirect
14. #473	C₄	semidirect
15. #472	C₄	semidirect
40. C₂×Q₁₆	C₂×C₄	semidirect
42. C₂×D₈	C₂×C₄	semidirect
66. #108	C₂×C₄	semidirect

Maximal subgroups

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

subgroup	index
#3067	2
#2912	2
#2145	2
C₂×#360	2
#2117	2
#1389	2
#1383	2