Finite Group Databasebeta

← C₂×#2213 #33592

Subgroups — C₂×#2213 #33592

Order 256.

395 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 395 conjugacy classes of subgroups — too many to draw. Only its 119 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×#509	C₂	1	2	yes	maximal, complemented
2	C₂×#493	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#443	C₂	1	1	yes	maximal, characteristic, complemented
4	#2213	C₂	1	8	yes	maximal, direct product factor
5	C₂×#364	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#361	C₂	1	2	yes	maximal, complemented
7	C₂×#135	C₂²	1	1	yes	characteristic
8	C₂²×D₈	—	2	1	no	—
9	C₂²×D₈	C₂²	1	2	yes	—
10	C₂×#131	—	2	1	no	—
11	C₂×#127	—	2	1	no	—
12	#509	—	2	4	no	—
13	#509	C₂²	1	8	yes	complemented
14	C₂×SD₃₂	—	2	2	no	—
15	C₂×D₁₆	—	2	2	no	—
16	C₂×M₃₂	C₂²	1	2	yes	—
17	#493	—	2	2	no	—
18	#493	C₂²	1	4	yes	complemented
19	#443	—	2	2	no	—
20	#443	C₂²	1	4	yes	complemented
21	C₂×#108	C₂²	1	1	yes	characteristic
22	C₂×#104	—	2	1	no	—
23	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
24	#364	C₂²	1	4	yes	complemented
25	#361	C₂²	1	8	yes	complemented
26	C₂×#41	—	2	1	no	—
27	C₂²×D₄	—	2	2	no	—
28	C₂²×D₄	—	4	2	no	—
29	#135	—	2	1	no	—
30	#135	—	2	1	no	—
31	#135	D₄	1	2	yes	complemented
32	#135	C₂³	1	2	yes	—
33	C₂×Q₁₆	D₄	1	1	yes	characteristic, complemented
34	C₂×SD₁₆	—	2	1	no	—
35	C₂×D₈	—	2	2	no	—
36	C₂×D₈	—	2	2	no	—
37	C₂×D₈	—	2	4	no	—
38	C₂×D₈	—	2	2	no	—
39	C₂×D₈	—	2	2	no	—
40	C₂×D₈	—	2	2	no	—
41	C₂×D₈	—	4	1	no	—
42	C₂×D₈	—	4	1	no	—
43	C₂×D₈	—	4	2	no	—
44	C₂×D₈	D₄	1	1	yes	characteristic, complemented
45	C₂×D₈	C₂³	1	4	yes	—
46	#131	—	2	4	no	—
47	#131	—	4	2	no	—
48	C₂×M₁₆	—	2	1	no	—
49	C₂×M₁₆	—	4	1	no	—
50	C₂²×C₈	—	4	1	no	—
51	C₂²×C₈	D₄	1	1	yes	characteristic
52	C₂²×C₈	C₂³	1	1	yes	characteristic
53	#127	—	2	4	no	—
54	#127	—	4	2	no	—
55	C₂×C₄²	D₄	1	1	yes	characteristic
56	SD₃₂	—	2	8	no	—
57	D₁₆	—	2	8	no	—
58	M₃₂	—	2	2	no	—
59	M₃₂	C₂³	1	4	yes	—
60	C₂×C₁₆	—	2	2	no	—
61	#108	—	2	1	no	—
62	#108	C₂³	1	2	yes	—
63	#104	—	2	4	no	—
64	C₄×C₈	D₄	1	2	yes	—
65	C₄×C₈	C₂³	1	2	yes	—
66	C₂⁴	—	4	2	no	—
67	#41	—	2	2	no	—
68	#41	—	2	2	no	—
69	#41	—	4	1	no	—
70	#41	—	4	1	no	—
71	C₂×Q₈	—	2	1	no	—
72	C₂×D₄	—	2	1	no	—
73	C₂×D₄	—	2	4	no	—
74	C₂×D₄	—	4	8	no	—
75	C₂×D₄	—	4	1	no	—
76	C₂×D₄	—	4	2	no	—
77	C₂×D₄	—	4	2	no	—
78	C₂×D₄	—	4	2	no	—
79	C₂×D₄	—	4	2	no	—
80	C₂×D₄	—	4	2	no	—
81	C₂×D₄	—	8	8	no	—
82	C₂²×C₄	—	2	1	no	—
83	C₂²×C₄	—	4	1	no	—
84	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
85	Q₁₆	—	2	1	no	—
86	Q₁₆	C₂×D₄	1	2	yes	complemented
87	SD₁₆	—	2	2	no	—
88	SD₁₆	—	2	2	no	—
89	D₈	—	2	1	no	—
90	D₈	—	2	4	no	—
91	D₈	—	2	4	no	—
92	D₈	—	4	2	no	—
93	D₈	—	4	2	no	—
94	D₈	—	4	4	no	—
95	D₈	—	4	4	no	—
96	D₈	C₂×D₄	1	2	yes	complemented
97	M₁₆	—	2	2	no	—
98	M₁₆	—	4	1	no	—
99	M₁₆	—	4	4	no	—
100	C₂×C₈	—	4	1	no	—
101	C₂×C₈	—	4	4	no	—
102	C₂×C₈	D₈	1	1	yes	characteristic
103	C₂×C₈	D₈	1	1	yes	characteristic
104	C₂×C₈	D₈	1	2	yes	—
105	C₂×C₈	C₂×D₄	1	1	yes	characteristic
106	C₂×C₈	C₂×D₄	1	1	yes	characteristic
107	C₂×C₈	C₂×D₄	1	1	yes	characteristic
108	C₂×C₈	C₂×D₄	1	2	yes	—
109	C₂×C₈	#41	1	1	yes	characteristic
110	C₂×C₈	#41	1	1	yes	characteristic
111	C₂×C₈	C₂⁴	1	1	yes	characteristic
112	C₄²	—	2	1	no	—
113	C₄²	C₂×D₄	1	2	yes	—
114	C₁₆	—	2	4	no	—
115	C₂³	—	4	1	no	—
116	C₂³	—	4	4	no	—
117	C₂³	—	8	8	no	—
118	C₂³	—	8	2	no	—
119	C₂³	—	8	2	no	—
120	C₂³	AGL(1,ℤ/8)	1	1	yes	characteristic
121	Q₈	—	2	2	no	—
122	Q₈	—	4	1	no	—
123	D₄	—	2	2	no	—
124	D₄	—	4	2	no	—
125	D₄	—	4	2	no	—
126	D₄	—	4	4	no	—
127	D₄	—	4	4	no	—
128	D₄	—	4	1	no	—
129	D₄	—	8	16	no	—
130	D₄	—	8	2	no	—
131	D₄	—	8	2	no	—
132	C₂×C₄	—	2	4	no	—
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	—	4	1	no	—
135	C₂×C₄	—	4	2	no	—
136	C₂×C₄	—	4	2	no	—
137	C₂×C₄	#121	1	1	yes	characteristic
138	C₂×C₄	#121	1	1	yes	characteristic
139	C₂×C₄	C₂×D₈	1	1	yes	characteristic
140	C₂×C₄	C₂×D₈	1	1	yes	characteristic
141	C₂×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
142	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
143	C₈	—	4	2	no	—
144	C₈	#121	1	1	yes	characteristic
145	C₈	#121	1	1	yes	characteristic
146	C₈	C₂×D₈	1	2	yes	—
147	C₈	C₂×D₈	1	2	yes	—
148	C₈	C₂²×D₄	1	1	yes	characteristic
149	C₈	C₂×#41	1	1	yes	characteristic
150	C₂²	—	2	2	no	—
151	C₂²	—	4	2	no	—
152	C₂²	—	8	8	no	—
153	C₂²	—	8	2	no	—
154	C₂²	—	8	4	no	—
155	C₂²	—	16	4	no	—
156	C₂²	—	16	4	no	—
157	C₂²	#459	1	1	yes	characteristic
158	C₂²	#459	1	1	yes	characteristic
159	C₂²	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
160	C₄	—	2	4	no	—
161	C₄	—	4	2	no	—
162	C₄	#459	1	1	yes	characteristic
163	C₄	#459	1	1	yes	characteristic
164	C₄	C₂×#121	1	1	yes	characteristic
165	C₄	C₂²×D₈	1	1	yes	characteristic
166	C₂	—	2	1	no	—
167	C₂	—	2	1	no	—
168	C₂	—	8	2	no	—
169	C₂	—	16	4	no	—
170	C₂	#2213	1	2	yes	direct product factor
171	C₂	C₂×#459	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	#459
commutator, Frattini, cosocle	C₂×C₈	C₂⁴
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. C₂×#509	C₂	semidirect
2. C₂×#493	C₂	semidirect
3. C₂×#443	C₂	semidirect
4. #2213	C₂	direct (N × H)
5. C₂×#364	C₂	semidirect
6. C₂×#361	C₂	semidirect
13. #509	C₂²	semidirect
18. #493	C₂²	semidirect
20. #443	C₂²	semidirect
24. #364	C₂²	semidirect
25. #361	C₂²	semidirect
31. #135	D₄	semidirect
33. C₂×Q₁₆	D₄	semidirect
44. C₂×D₈	D₄	semidirect
86. Q₁₆	C₂×D₄	semidirect
96. D₈	C₂×D₄	semidirect
170. C₂	#2213	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#509	2
C₂×#493	2
C₂×#443	2
#2213	2
C₂×#364	2
C₂×#361	2