Finite Group Databasebeta

Subgroups — #33638

Order 256.

257 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 257 conjugacy classes of subgroups — too many to draw. Only its 117 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	#3163	C₂	1	1	yes	maximal, characteristic, complemented
2	#2957	C₂	1	1	yes	maximal, characteristic
3	#2231	C₂	1	4	yes	maximal, complemented
4	#2230	C₂	1	4	yes	maximal, complemented
5	#2155	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#365	C₂	1	1	yes	maximal, characteristic, complemented
7	#2153	C₂	1	1	yes	maximal, characteristic, complemented
8	#2145	C₂	1	1	yes	maximal, characteristic
9	C₂×#360	C₂	1	1	yes	maximal, characteristic
10	C₂×#135	C₂²	1	1	yes	characteristic
11	#568	—	2	1	no	—
12	#544	—	2	1	no	—
13	C₂×M₃₂	C₂²	1	1	yes	characteristic
14	C₂×M₃₂	C₂²	1	1	yes	characteristic
15	#501	—	2	2	no	—
16	#499	C₂²	1	2	yes	complemented
17	#498	C₂²	1	2	yes	complemented
18	#444	—	2	2	no	—
19	#443	C₂²	1	4	yes	—
20	C₂×#108	C₂²	1	1	yes	characteristic
21	#428	—	2	1	no	—
22	#427	C₂²	1	1	yes	characteristic
23	C₂×#106	C₂²	1	1	yes	characteristic
24	#421	—	2	1	no	—
25	#405	C₂²	1	1	yes	characteristic
26	#365	C₂²	1	4	yes	complemented
27	#365	C₂²	1	2	yes	complemented
28	#365	C₂²	1	2	yes	complemented
29	#364	C₂²	1	4	yes	complemented
30	#360	C₂²	1	4	yes	—
31	#360	C₂²	1	2	yes	—
32	#360	C₂²	1	2	yes	—
33	C₂×#41	—	2	1	no	—
34	#135	—	2	1	no	—
35	#135	—	2	1	no	—
36	#135	Q₈	1	2	yes	complemented
37	#135	C₂³	1	2	yes	—
38	C₂×Q₁₆	Q₈	1	1	yes	characteristic, complemented
39	C₂×SD₁₆	—	2	1	no	—
40	C₂×D₈	Q₈	1	1	yes	characteristic, complemented
41	#131	—	2	4	no	—
42	#131	—	4	2	no	—
43	C₂×M₁₆	—	2	1	no	—
44	C₂×M₁₆	—	4	2	no	—
45	C₂×M₁₆	D₄	1	1	yes	characteristic
46	C₂²×C₈	C₂³	1	1	yes	characteristic
47	C₄⋊Q₈	—	2	2	no	—
48	C₄⋊Q₈	—	4	1	no	—
49	#125	—	2	2	no	—
50	#125	—	4	1	no	—
51	C₂²⋊Q₈	—	4	2	no	—
52	#117	—	2	1	no	—
53	#117	D₄	1	1	yes	characteristic
54	C₂×C₄⋊C₄	—	2	1	no	—
55	M₃₂	—	2	1	no	—
56	M₃₂	—	2	1	no	—
57	M₃₂	C₂³	1	2	yes	—
58	M₃₂	C₂³	1	2	yes	—
59	C₂×C₁₆	—	2	1	no	—
60	C₂×C₁₆	—	2	1	no	—
61	#108	—	2	1	no	—
62	#108	C₂³	1	2	yes	—
63	#107	—	2	1	no	—
64	#107	—	2	2	no	—
65	#106	—	2	1	no	—
66	#106	—	2	2	no	—
67	#106	C₂³	1	2	yes	—
68	#106	C₂³	1	1	yes	characteristic
69	#106	C₂³	1	1	yes	characteristic
70	#104	—	2	4	no	—
71	#97	D₄	1	2	yes	—
72	C₄×C₈	C₂³	1	2	yes	—
73	#41	—	2	2	no	—
74	#41	—	2	2	no	—
75	#41	—	4	1	no	—
76	#41	—	4	1	no	—
77	C₂×Q₈	—	2	1	no	—
78	C₂×Q₈	—	4	2	no	—
79	C₂×D₄	—	2	1	no	—
80	C₂×D₄	—	4	1	no	—
81	C₂²×C₄	—	4	1	no	—
82	C₂²×C₄	—	4	1	no	—
83	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
84	Q₁₆	—	2	1	no	—
85	Q₁₆	C₂×Q₈	1	2	yes	—
86	SD₁₆	—	2	2	no	—
87	SD₁₆	—	2	2	no	—
88	D₈	—	2	1	no	—
89	D₈	C₂×Q₈	1	2	yes	—
90	M₁₆	—	2	2	no	—
91	M₁₆	—	4	1	no	—
92	M₁₆	—	4	4	no	—
93	M₁₆	SD₁₆	1	2	yes	—
94	M₁₆	SD₁₆	1	2	yes	—
95	C₂×C₈	—	4	1	no	—
96	C₂×C₈	—	4	4	no	—
97	C₂×C₈	C₂×D₄	1	1	yes	characteristic
98	C₂×C₈	C₂×D₄	1	2	yes	—
99	C₂×C₈	C₂×Q₈	1	1	yes	characteristic
100	C₂×C₈	C₂×Q₈	1	1	yes	characteristic
101	C₂×C₈	#41	1	1	yes	characteristic
102	C₂×C₈	#41	1	1	yes	characteristic
103	C₂×C₈	C₂⁴	1	1	yes	characteristic
104	C₄⋊C₄	—	2	1	no	—
105	C₄⋊C₄	—	2	2	no	—
106	C₄⋊C₄	—	2	1	no	—
107	C₄⋊C₄	—	2	1	no	—
108	C₄⋊C₄	—	4	2	no	—
109	C₄⋊C₄	—	4	2	no	—
110	C₄⋊C₄	—	4	2	no	—
111	C₄⋊C₄	—	4	1	no	—
112	C₂²⋊C₄	—	2	1	no	—
113	C₂²⋊C₄	—	4	1	no	—
114	C₄²	—	4	1	no	—
115	C₄²	C₂×D₄	1	2	yes	—
116	C₁₆	—	2	2	no	—
117	C₁₆	—	2	2	no	—
118	C₂³	—	4	1	no	—
119	C₂³	AGL(1,ℤ/8)	1	1	yes	characteristic
120	Q₈	—	2	2	no	—
121	Q₈	—	4	1	no	—
122	Q₈	—	16	2	no	—
123	D₄	—	2	2	no	—
124	D₄	—	4	2	no	—
125	D₄	—	4	2	no	—
126	D₄	—	4	1	no	—
127	C₂×C₄	—	2	2	no	—
128	C₂×C₄	—	4	1	no	—
129	C₂×C₄	—	4	2	no	—
130	C₂×C₄	—	4	2	no	—
131	C₂×C₄	—	4	1	no	—
132	C₂×C₄	—	4	1	no	—
133	C₂×C₄	—	4	2	no	—
134	C₂×C₄	—	16	1	no	—
135	C₂×C₄	C₂²⋊Q₈	1	1	yes	characteristic
136	C₂×C₄	C₂²⋊Q₈	1	1	yes	characteristic
137	C₂×C₄	C₂×SD₁₆	1	1	yes	characteristic
138	C₂×C₄	C₂×SD₁₆	1	1	yes	characteristic
139	C₂×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
140	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
141	C₈	—	4	2	no	—
142	C₈	C₂²⋊Q₈	1	1	yes	characteristic
143	C₈	C₂²⋊Q₈	1	1	yes	characteristic
144	C₈	C₂×SD₁₆	1	4	yes	—
145	C₈	C₂²×Q₈	1	1	yes	characteristic
146	C₈	C₂×#41	1	1	yes	characteristic
147	C₂²	—	4	1	no	—
148	C₂²	—	4	2	no	—
149	C₂²	—	16	1	no	—
150	C₂²	#476	1	1	yes	characteristic
151	C₂²	#476	1	1	yes	characteristic
152	C₂²	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
153	C₄	—	4	2	no	—
154	C₄	—	4	2	no	—
155	C₄	—	8	1	no	—
156	C₄	—	8	2	no	—
157	C₄	—	8	1	no	—
158	C₄	#476	1	1	yes	characteristic
159	C₄	#476	1	1	yes	characteristic
160	C₄	C₂×C₂²⋊Q₈	1	1	yes	characteristic
161	C₄	C₂²×SD₁₆	1	1	yes	characteristic
162	C₂	—	2	1	no	—
163	C₂	—	2	1	no	—
164	C₂	—	2	1	no	—
165	C₂	—	8	2	no	—
166	C₂	C₂×#476	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	C₂×#476
commutator, Frattini, cosocle	C₂×C₈	C₂⁴
prime radical	C₂×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #405 ◁ 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. #3163	C₂	semidirect
3. #2231	C₂	semidirect
4. #2230	C₂	semidirect
5. #2155	C₂	semidirect
6. C₂×#365	C₂	semidirect
7. #2153	C₂	semidirect
16. #499	C₂²	semidirect
17. #498	C₂²	semidirect
26. #365	C₂²	semidirect
27. #365	C₂²	semidirect
28. #365	C₂²	semidirect
29. #364	C₂²	semidirect
36. #135	Q₈	semidirect
38. C₂×Q₁₆	Q₈	semidirect
40. C₂×D₈	Q₈	semidirect

Maximal subgroups

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

subgroup	index
#3163	2
#2957	2
#2231	2
#2230	2
#2155	2
C₂×#365	2
#2153	2
#2145	2
C₂×#360	2