Finite Group Databasebeta

← C₂×#2217 #33599

Subgroups — C₂×#2217 #33599

Order 256.

377 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 377 conjugacy classes of subgroups — too many to draw. Only its 135 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	C₂²×SD₃₂	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂³×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#467	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#466	C₂	1	1	yes	maximal, characteristic
5	#2217	C₂	1	8	yes	maximal, direct product factor
6	C₂×#367	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#358	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#357	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×Q₁₆	C₂²	1	1	yes	characteristic
10	C₂²×D₈	C₂²	1	1	yes	characteristic
11	C₂³×C₈	C₂²	1	1	yes	characteristic
12	C₂×C₂²⋊Q₈	—	2	1	no	—
13	C₂×#121	—	2	1	no	—
14	C₂×SD₃₂	—	2	4	no	—
15	C₂×SD₃₂	C₂²	1	4	yes	complemented
16	C₂²×C₁₆	—	2	4	no	—
17	C₂²×C₁₆	C₂²	1	1	yes	characteristic
18	C₂²×C₁₆	C₂²	1	1	yes	characteristic
19	C₂²×C₁₆	C₂²	1	4	yes	complemented
20	#467	—	2	2	no	—
21	#467	C₂²	1	4	yes	complemented
22	#466	—	2	2	no	—
23	#466	C₂²	1	4	yes	—
24	C₂×#107	C₂²	1	1	yes	characteristic
25	C₂×#107	C₂²	1	1	yes	characteristic
26	C₂×#103	—	2	1	no	—
27	C₂×#102	—	2	1	no	—
28	#367	C₂²	1	4	yes	complemented
29	#358	C₂²	1	4	yes	complemented
30	#357	C₂²	1	4	yes	complemented
31	C₂²×Q₈	—	2	1	no	—
32	C₂²×D₄	—	2	1	no	—
33	C₂²×D₄	—	4	1	no	—
34	C₂³×C₄	D₄	1	1	yes	characteristic
35	C₂×Q₁₆	—	2	2	no	—
36	C₂×Q₁₆	—	2	1	no	—
37	C₂×Q₁₆	—	2	2	no	—
38	C₂×Q₁₆	C₂³	1	2	yes	—
39	C₂×D₈	—	2	2	no	—
40	C₂×D₈	—	2	1	no	—
41	C₂×D₈	—	2	2	no	—
42	C₂×D₈	C₂³	1	2	yes	—
43	C₂²×C₈	—	2	2	no	—
44	C₂²×C₈	—	2	2	no	—
45	C₂²×C₈	D₄	1	1	yes	characteristic
46	C₂²×C₈	D₄	1	2	yes	—
47	C₂²×C₈	C₂³	1	2	yes	—
48	C₂²×C₈	C₂³	1	1	yes	characteristic
49	C₂²⋊Q₈	—	2	4	no	—
50	C₂²⋊Q₈	—	4	2	no	—
51	#121	—	2	4	no	—
52	#121	—	4	2	no	—
53	C₂×C₄⋊C₄	—	2	1	no	—
54	C₂×C₄⋊C₄	—	2	1	no	—
55	C₂×C₄⋊C₄	—	4	1	no	—
56	C₂×C₂²⋊C₄	—	4	1	no	—
57	C₂×C₂²⋊C₄	—	4	1	no	—
58	SD₃₂	—	2	8	no	—
59	C₂×C₁₆	—	2	2	no	—
60	C₂×C₁₆	—	2	8	no	—
61	C₂×C₁₆	D₄	1	4	yes	complemented
62	C₂×C₁₆	C₂³	1	2	yes	—
63	C₂×C₁₆	C₂³	1	2	yes	—
64	#107	—	2	1	no	—
65	#107	—	2	1	no	—
66	#107	C₂³	1	2	yes	—
67	#107	C₂³	1	2	yes	—
68	#103	—	2	4	no	—
69	#102	—	2	4	no	—
70	C₂⁴	—	4	1	no	—
71	C₂⁴	D₈	1	1	yes	characteristic
72	C₂×Q₈	—	2	2	no	—
73	C₂×Q₈	—	4	1	no	—
74	C₂×Q₈	—	4	2	no	—
75	C₂×Q₈	—	4	2	no	—
76	C₂×D₄	—	2	2	no	—
77	C₂×D₄	—	4	4	no	—
78	C₂×D₄	—	4	2	no	—
79	C₂×D₄	—	4	1	no	—
80	C₂×D₄	—	4	2	no	—
81	C₂×D₄	—	8	4	no	—
82	C₂²×C₄	—	2	2	no	—
83	C₂²×C₄	—	2	2	no	—
84	C₂²×C₄	—	4	1	no	—
85	C₂²×C₄	—	4	1	no	—
86	C₂²×C₄	—	4	1	no	—
87	C₂²×C₄	D₈	1	2	yes	—
88	C₂²×C₄	D₈	1	1	yes	characteristic
89	C₂²×C₄	C₂×D₄	1	2	yes	—
90	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
91	Q₁₆	—	2	2	no	—
92	Q₁₆	—	2	2	no	—
93	Q₁₆	—	4	2	no	—
94	D₈	—	2	2	no	—
95	D₈	—	2	2	no	—
96	D₈	—	4	2	no	—
97	C₂×C₈	—	2	2	no	—
98	C₂×C₈	—	2	4	no	—
99	C₂×C₈	—	2	2	no	—
100	C₂×C₈	—	2	2	no	—
101	C₂×C₈	C₂×D₄	1	2	yes	—
102	C₂×C₈	C₂×D₄	1	1	yes	characteristic
103	C₂×C₈	C₂×D₄	1	2	yes	—
104	C₂×C₈	#41	1	2	yes	—
105	C₂×C₈	C₂⁴	1	1	yes	characteristic
106	C₄⋊C₄	—	2	2	no	—
107	C₄⋊C₄	—	2	2	no	—
108	C₄⋊C₄	—	4	1	no	—
109	C₄⋊C₄	—	4	1	no	—
110	C₄⋊C₄	—	4	4	no	—
111	C₂²⋊C₄	—	4	4	no	—
112	C₂²⋊C₄	—	4	4	no	—
113	C₁₆	—	2	2	no	—
114	C₁₆	C₂×D₄	1	4	yes	complemented
115	C₂³	—	2	2	no	—
116	C₂³	—	2	2	no	—
117	C₂³	—	4	2	no	—
118	C₂³	—	8	2	no	—
119	C₂³	—	8	2	no	—
120	C₂³	—	8	2	no	—
121	C₂³	SD₃₂	1	4	yes	complemented
122	C₂³	C₂×D₈	1	2	yes	—
123	C₂³	C₂×D₈	1	1	yes	characteristic
124	Q₈	—	4	2	no	—
125	Q₈	—	4	2	no	—
126	Q₈	—	8	2	no	—
127	D₄	—	4	2	no	—
128	D₄	—	4	2	no	—
129	D₄	—	8	8	no	—
130	D₄	—	8	2	no	—
131	C₂×C₄	—	2	2	no	—
132	C₂×C₄	—	2	2	no	—
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	—	2	4	no	—
135	C₂×C₄	—	4	2	no	—
136	C₂×C₄	—	4	2	no	—
137	C₂×C₄	—	4	2	no	—
138	C₂×C₄	—	8	2	no	—
139	C₂×C₄	—	8	2	no	—
140	C₂×C₄	—	8	2	no	—
141	C₂×C₄	#121	1	2	yes	—
142	C₂×C₄	C₂×D₈	1	2	yes	—
143	C₂×C₄	C₂×D₈	1	1	yes	characteristic
144	C₂×C₄	#135	1	2	yes	—
145	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
146	C₈	—	2	2	no	—
147	C₈	#121	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	—	2	8	no	—
152	C₂²	—	2	2	no	—
153	C₂²	—	8	2	no	—
154	C₂²	—	8	2	no	—
155	C₂²	—	8	2	no	—
156	C₂²	—	16	4	no	—
157	C₂²	#466	1	2	yes	—
158	C₂²	C₂×SD₃₂	1	2	yes	—
159	C₂²	C₂×SD₃₂	1	4	yes	complemented
160	C₂²	#508	1	2	yes	—
161	C₂²	C₂²×D₈	1	1	yes	characteristic
162	C₄	—	2	2	no	—
163	C₄	—	8	2	no	—
164	C₄	—	8	2	no	—
165	C₄	—	8	2	no	—
166	C₄	#466	1	2	yes	—
167	C₄	C₂×#121	1	1	yes	characteristic
168	C₄	C₂×#135	1	1	yes	characteristic
169	C₂	—	2	4	no	—
170	C₂	—	16	2	no	—
171	C₂	#2217	1	4	yes	direct product factor
172	C₂	C₂×#466	1	1	yes	characteristic
173	C₂	C₂²×SD₃₂	1	1	yes	characteristic
174	C₂	C₂×#508	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴
prime radical	C₂×#466	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×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. C₂²×SD₃₂	C₂	semidirect
2. C₂³×C₁₆	C₂	semidirect
3. C₂×#467	C₂	semidirect
5. #2217	C₂	direct (N × H)
6. C₂×#367	C₂	semidirect
7. C₂×#358	C₂	semidirect
8. C₂×#357	C₂	semidirect
15. C₂×SD₃₂	C₂²	semidirect
19. C₂²×C₁₆	C₂²	semidirect
21. #467	C₂²	semidirect
28. #367	C₂²	semidirect
29. #358	C₂²	semidirect
30. #357	C₂²	semidirect
61. C₂×C₁₆	D₄	semidirect
114. C₁₆	C₂×D₄	semidirect
121. C₂³	SD₃₂	semidirect
159. C₂²	C₂×SD₃₂	semidirect
171. C₂	#2217	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×SD₃₂	2
C₂³×C₁₆	2
C₂×#467	2
C₂×#466	2
#2217	2
C₂×#367	2
C₂×#358	2
C₂×#357	2