Finite Group Databasebeta

Subgroups — #12646

Order 256.

214 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 214 conjugacy classes of subgroups — too many to draw. Only its 93 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×#443	C₂	1	1	yes	maximal, characteristic
2	#2148	C₂	1	1	yes	maximal, characteristic
3	#2145	C₂	1	1	yes	maximal, characteristic
4	C₄×M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
5	#1772	C₂	1	1	yes	maximal, characteristic, complemented
6	#1389	C₂	1	1	yes	maximal, characteristic, complemented
7	#1388	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#135	C₂²	1	1	yes	characteristic
9	C₂×#131	—	2	1	no	—
10	C₂×M₃₂	C₂²	1	1	yes	characteristic
11	C₂×M₃₂	C₂²	1	1	yes	characteristic
12	#443	—	2	1	no	—
13	#443	—	2	1	no	—
14	#443	C₄	1	2	yes	complemented
15	#443	C₄	1	2	yes	complemented
16	C₂×#108	C₂²	1	1	yes	characteristic
17	C₂×#108	C₂²	1	1	yes	characteristic
18	#427	C₂²	1	1	yes	characteristic
19	C₂×#104	—	2	1	no	—
20	#405	—	2	1	no	—
21	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
22	#362	C₄	1	2	yes	complemented
23	#361	C₄	1	2	yes	complemented
24	#360	C₄	1	4	yes	complemented
25	#346	—	2	1	no	—
26	C₄×C₁₆	—	2	1	no	—
27	#339	—	2	1	no	—
28	C₂×#41	—	2	1	no	—
29	#135	—	2	1	no	—
30	#135	—	2	1	no	—
31	#135	C₂×C₄	1	2	yes	—
32	#135	C₂×C₄	1	2	yes	—
33	C₂×Q₁₆	C₂×C₄	1	1	yes	characteristic
34	C₂×SD₁₆	—	2	1	no	—
35	C₂×D₈	C₂×C₄	1	1	yes	characteristic
36	#131	—	2	2	no	—
37	#131	—	2	2	no	—
38	#131	—	4	1	no	—
39	#131	—	4	1	no	—
40	C₂×M₁₆	—	2	1	no	—
41	C₂×M₁₆	—	2	1	no	—
42	C₂×M₁₆	—	4	1	no	—
43	C₂²×C₈	—	4	1	no	—
44	C₂²×C₈	D₄	1	1	yes	characteristic
45	C₂²×C₈	C₂³	1	1	yes	characteristic
46	#117	—	2	1	no	—
47	C₂×C₄²	D₄	1	1	yes	characteristic
48	M₃₂	—	2	1	no	—
49	M₃₂	—	2	1	no	—
50	M₃₂	C₂×C₄	1	2	yes	—
51	M₃₂	C₂×C₄	1	2	yes	—
52	C₂×C₁₆	—	2	1	no	—
53	C₂×C₁₆	—	2	1	no	—
54	#108	—	2	1	no	—
55	#108	—	2	1	no	—
56	#108	C₂×C₄	1	2	yes	—
57	#108	C₂×C₄	1	2	yes	—
58	#107	—	2	1	no	—
59	#106	C₂×C₄	1	2	yes	—
60	#104	—	2	2	no	—
61	#104	—	2	2	no	—
62	#97	—	4	1	no	—
63	C₄×C₈	—	4	1	no	—
64	C₄×C₈	C₂×C₄	1	1	yes	characteristic
65	C₄×C₈	C₂×C₄	1	1	yes	characteristic
66	C₄×C₈	D₄	1	1	yes	characteristic
67	C₄×C₈	D₄	1	1	yes	characteristic
68	#41	—	2	2	no	—
69	#41	—	2	2	no	—
70	#41	—	4	1	no	—
71	#41	—	4	1	no	—
72	C₂×Q₈	—	2	1	no	—
73	C₂×D₄	—	2	1	no	—
74	C₂×D₄	—	4	1	no	—
75	C₂²×C₄	—	2	1	no	—
76	C₂²×C₄	—	4	1	no	—
77	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
78	Q₁₆	—	2	1	no	—
79	Q₁₆	C₄²	1	2	yes	—
80	SD₁₆	—	2	2	no	—
81	SD₁₆	—	2	2	no	—
82	D₈	—	2	1	no	—
83	D₈	C₄²	1	2	yes	—
84	M₁₆	—	2	2	no	—
85	M₁₆	—	2	2	no	—
86	M₁₆	—	4	1	no	—
87	M₁₆	—	4	1	no	—
88	M₁₆	—	4	2	no	—
89	M₁₆	—	4	2	no	—
90	C₂×C₈	—	4	1	no	—
91	C₂×C₈	—	4	1	no	—
92	C₂×C₈	—	4	2	no	—
93	C₂×C₈	—	4	2	no	—
94	C₂×C₈	C₂²⋊C₄	1	1	yes	characteristic
95	C₂×C₈	C₂²⋊C₄	1	1	yes	characteristic
96	C₂×C₈	D₈	1	1	yes	characteristic
97	C₂×C₈	D₈	1	1	yes	characteristic
98	C₂×C₈	SD₁₆	1	1	yes	characteristic
99	C₂×C₈	SD₁₆	1	1	yes	characteristic
100	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
101	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
102	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
103	C₂×C₈	C₂×D₄	1	1	yes	characteristic
104	C₂×C₈	#41	1	1	yes	characteristic
105	C₂×C₈	#41	1	1	yes	characteristic
106	C₄⋊C₄	—	2	2	no	—
107	C₂²⋊C₄	—	4	1	no	—
108	C₄²	—	2	1	no	—
109	C₄²	—	4	1	no	—
110	C₄²	C₂²⋊C₄	1	1	yes	characteristic
111	C₄²	C₂²⋊C₄	1	1	yes	characteristic
112	C₁₆	—	2	2	no	—
113	C₁₆	—	2	2	no	—
114	C₂³	—	4	1	no	—
115	C₂³	#135	1	1	yes	characteristic
116	Q₈	—	2	2	no	—
117	Q₈	—	4	1	no	—
118	D₄	—	2	2	no	—
119	D₄	—	4	2	no	—
120	D₄	—	4	2	no	—
121	D₄	—	4	1	no	—
122	C₂×C₄	—	2	1	no	—
123	C₂×C₄	—	2	1	no	—
124	C₂×C₄	—	4	2	no	—
125	C₂×C₄	—	4	1	no	—
126	C₂×C₄	—	4	2	no	—
127	C₂×C₄	—	4	2	no	—
128	C₂×C₄	—	4	2	no	—
129	C₂×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
130	C₂×C₄	C₄×D₄	1	1	yes	characteristic
131	C₂×C₄	C₄×D₄	1	1	yes	characteristic
132	C₂×C₄	C₂×D₈	1	1	yes	characteristic
133	C₂×C₄	C₂×SD₁₆	1	1	yes	characteristic
134	C₂×C₄	#135	1	1	yes	characteristic
135	C₈	—	4	2	no	—
136	C₈	—	4	2	no	—
137	C₈	#102	1	1	yes	characteristic
138	C₈	#102	1	1	yes	characteristic
139	C₈	#102	1	1	yes	characteristic
140	C₈	#102	1	1	yes	characteristic
141	C₈	C₂×C₄²	1	1	yes	characteristic
142	C₈	#117	1	1	yes	characteristic
143	C₈	C₄×D₄	1	1	yes	characteristic
144	C₈	C₄×D₄	1	1	yes	characteristic
145	C₂²	—	4	1	no	—
146	C₂²	—	4	2	no	—
147	C₂²	—	8	2	no	—
148	C₂²	#416	1	1	yes	characteristic
149	C₂²	C₄×D₈	1	1	yes	characteristic
150	C₂²	C₄×SD₁₆	1	1	yes	characteristic
151	C₄	—	2	2	no	—
152	C₄	—	4	1	no	—
153	C₄	—	4	2	no	—
154	C₄	—	8	2	no	—
155	C₄	C₄×C₂²⋊C₄	1	1	yes	characteristic
156	C₄	C₂×#102	1	1	yes	characteristic
157	C₄	C₄×D₈	1	1	yes	characteristic
158	C₄	C₄×SD₁₆	1	1	yes	characteristic
159	C₂	—	2	1	no	—
160	C₂	—	2	1	no	—
161	C₂	—	2	1	no	—
162	C₂	—	8	2	no	—
163	C₂	C₄×#102	1	1	yes	characteristic

	subgroup	G/H
center	C₈	#102
commutator	C₈	C₂×C₄²
Frattini, cosocle	C₂²×C₈	C₂³
socle	C₂	C₄×#102
prime radical	C₂×#135	C₂²
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
4. C₄×M₃₂	C₂	semidirect
5. #1772	C₂	semidirect
6. #1389	C₂	semidirect
7. #1388	C₂	semidirect
14. #443	C₄	semidirect
15. #443	C₄	semidirect
22. #362	C₄	semidirect
23. #361	C₄	semidirect
24. #360	C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×#443	2
#2148	2
#2145	2
C₄×M₃₂	2
#1772	2
#1389	2
#1388	2