Finite Group Databasebeta

Subgroups — #8902

Order 256.

219 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 219 conjugacy classes of subgroups — too many to draw. Only its 77 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	#2906	C₂	1	1	yes	maximal, characteristic
2	#1806	C₂	1	1	yes	maximal, characteristic
3	#1571	C₂	1	1	yes	maximal, characteristic, complemented
4	#1567	C₂	1	1	yes	maximal, characteristic, complemented
5	#1476	C₂	1	1	yes	maximal, characteristic
6	#1303	C₂	1	1	yes	maximal, characteristic, complemented
7	#1279	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#131	—	2	1	no	—
9	C₄×#41	C₂²	1	1	yes	characteristic
10	#431	C₂²	1	1	yes	characteristic
11	#424	—	2	1	no	—
12	#424	—	2	2	no	—
13	#423	—	2	1	no	—
14	#423	C₂²	1	1	yes	characteristic
15	#423	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊C₈	—	2	1	no	—
17	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
18	C₂×#104	—	2	1	no	—
19	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
20	C₄×C₄⋊C₄	C₂²	1	1	yes	characteristic
21	#339	—	2	1	no	—
22	#335	—	2	2	no	—
23	#335	—	2	1	no	—
24	#335	—	2	1	no	—
25	Q₈⋊C₈	C₄	1	1	yes	characteristic, complemented
26	Q₈⋊C₈	C₄	1	1	yes	characteristic, complemented
27	D₄⋊C₈	C₄	1	1	yes	characteristic, complemented
28	D₄⋊C₈	C₄	1	1	yes	characteristic, complemented
29	C₂×#41	D₄	1	1	yes	characteristic
30	#131	—	4	2	no	—
31	#131	—	4	2	no	—
32	C₂×M₁₆	—	2	2	no	—
33	C₂×M₁₆	—	2	1	no	—
34	C₂×M₁₆	—	2	1	no	—
35	C₂²×C₈	—	2	2	no	—
36	C₂²×C₈	—	2	1	no	—
37	C₂²×C₈	—	2	1	no	—
38	C₄×Q₈	C₂×C₄	1	1	yes	characteristic
39	C₄×Q₈	Q₈	1	1	yes	characteristic
40	C₄×D₄	—	2	1	no	—
41	C₄×D₄	—	2	1	no	—
42	C₄×D₄	C₂×C₄	1	1	yes	characteristic
43	C₄×D₄	Q₈	1	1	yes	characteristic
44	#117	—	2	1	no	—
45	#117	D₄	1	1	yes	characteristic
46	C₂×C₄⋊C₄	—	2	1	no	—
47	C₂×C₄²	—	2	1	no	—
48	C₂×C₄²	—	2	1	no	—
49	C₂×C₄²	C₂³	1	1	yes	characteristic
50	C₄⋊C₈	—	2	2	no	—
51	C₄⋊C₈	—	2	2	no	—
52	C₄⋊C₈	—	2	1	no	—
53	C₄⋊C₈	—	2	1	no	—
54	C₄⋊C₈	—	2	1	no	—
55	C₄⋊C₈	—	2	1	no	—
56	C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
57	C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
58	C₄⋊C₈	D₄	1	1	yes	characteristic
59	C₄⋊C₈	D₄	1	1	yes	characteristic
60	C₄⋊C₈	D₄	1	2	yes	—
61	#104	—	4	2	no	—
62	C₂²⋊C₈	—	2	1	no	—
63	C₄×C₈	—	2	1	no	—
64	C₄×C₈	—	2	1	no	—
65	C₄×C₈	C₂×C₄	1	1	yes	characteristic
66	C₄×C₈	C₂×C₄	1	1	yes	characteristic
67	#95	—	2	1	no	—
68	#41	—	2	1	no	—
69	#41	—	2	1	no	—
70	#41	D₈	1	2	yes	—
71	#41	Q₁₆	1	2	yes	—
72	C₂×Q₈	C₄⋊C₄	1	1	yes	characteristic
73	C₂×D₄	—	2	1	no	—
74	C₂×D₄	C₄⋊C₄	1	1	yes	characteristic
75	C₂²×C₄	—	2	1	no	—
76	C₂²×C₄	—	2	1	no	—
77	C₂²×C₄	—	2	1	no	—
78	C₂²×C₄	—	4	1	no	—
79	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄	#41	1	1	yes	characteristic
81	C₂²×C₄	#41	1	1	yes	characteristic
82	M₁₆	—	4	4	no	—
83	M₁₆	—	4	1	no	—
84	M₁₆	—	4	1	no	—
85	M₁₆	—	4	1	no	—
86	M₁₆	—	4	1	no	—
87	C₂×C₈	—	2	1	no	—
88	C₂×C₈	—	2	1	no	—
89	C₂×C₈	—	2	1	no	—
90	C₂×C₈	—	2	1	no	—
91	C₂×C₈	—	2	1	no	—
92	C₂×C₈	—	2	2	no	—
93	C₂×C₈	—	2	1	no	—
94	C₂×C₈	—	4	2	no	—
95	C₂×C₈	—	4	4	no	—
96	C₂×C₈	—	4	1	no	—
97	C₂×C₈	—	4	1	no	—
98	C₂×C₈	—	4	1	no	—
99	C₄⋊C₄	—	2	1	no	—
100	C₄⋊C₄	—	2	1	no	—
101	C₄⋊C₄	—	2	2	no	—
102	C₄⋊C₄	—	4	1	no	—
103	C₄⋊C₄	C₄⋊C₄	1	1	yes	characteristic
104	C₄⋊C₄	C₄⋊C₄	1	1	yes	characteristic
105	C₂²⋊C₄	—	2	1	no	—
106	C₂²⋊C₄	—	2	1	no	—
107	C₂²⋊C₄	—	2	1	no	—
108	C₄²	—	2	1	no	—
109	C₄²	—	2	1	no	—
110	C₄²	—	2	1	no	—
111	C₄²	—	4	1	no	—
112	C₄²	—	4	1	no	—
113	C₄²	C₂²×C₄	1	1	yes	characteristic
114	C₄²	C₂×D₄	1	1	yes	characteristic
115	C₄²	C₂×D₄	1	1	yes	characteristic
116	C₄²	C₂×Q₈	1	1	yes	characteristic
117	C₂³	—	2	1	no	—
118	C₂³	#123	1	1	yes	characteristic
119	Q₈	—	2	1	no	—
120	Q₈	#107	1	2	yes	—
121	D₄	—	2	2	no	—
122	D₄	—	2	2	no	—
123	D₄	—	2	1	no	—
124	D₄	#107	1	2	yes	—
125	C₂×C₄	—	2	1	no	—
126	C₂×C₄	—	2	1	no	—
127	C₂×C₄	—	2	1	no	—
128	C₂×C₄	—	2	2	no	—
129	C₂×C₄	—	2	2	no	—
130	C₂×C₄	—	2	1	no	—
131	C₂×C₄	—	2	1	no	—
132	C₂×C₄	—	4	1	no	—
133	C₂×C₄	—	4	1	no	—
134	C₂×C₄	—	4	1	no	—
135	C₂×C₄	—	4	2	no	—
136	C₂×C₄	—	8	1	no	—
137	C₂×C₄	—	8	1	no	—
138	C₂×C₄	—	8	1	no	—
139	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
140	C₂×C₄	C₄×D₄	1	1	yes	characteristic
141	C₂×C₄	C₄×D₄	1	1	yes	characteristic
142	C₂×C₄	#120	1	1	yes	characteristic
143	C₂×C₄	C₂²⋊Q₈	1	1	yes	characteristic
144	C₂×C₄	C₂²⋊Q₈	1	1	yes	characteristic
145	C₂×C₄	C₂×D₈	1	1	yes	characteristic
146	C₂×C₄	C₂×Q₁₆	1	1	yes	characteristic
147	C₂×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
148	C₂×C₄	#137	1	1	yes	characteristic
149	C₈	—	2	2	no	—
150	C₈	—	4	1	no	—
151	C₈	—	4	1	no	—
152	C₈	—	4	1	no	—
153	C₈	—	4	1	no	—
154	C₈	—	4	2	no	—
155	C₈	—	4	1	no	—
156	C₈	—	8	1	no	—
157	C₂²	—	2	1	no	—
158	C₂²	—	2	1	no	—
159	C₂²	—	2	2	no	—
160	C₂²	—	4	1	no	—
161	C₂²	—	4	1	no	—
162	C₂²	#385	1	1	yes	characteristic
163	C₂²	#480	1	1	yes	characteristic
164	C₂²	#484	1	1	yes	characteristic
165	C₄	—	2	2	no	—
166	C₄	—	2	1	no	—
167	C₄	—	2	1	no	—
168	C₄	—	2	1	no	—
169	C₄	—	4	1	no	—
170	C₄	—	4	1	no	—
171	C₄	—	8	2	no	—
172	C₄	C₂×#107	1	1	yes	characteristic
173	C₄	#428	1	1	yes	characteristic
174	C₄	#443	1	1	yes	characteristic
175	C₄	#444	1	1	yes	characteristic
176	C₄	#447	1	1	yes	characteristic
177	C₄	#451	1	1	yes	characteristic
178	C₂	—	2	1	no	—
179	C₂	—	2	1	no	—
180	C₂	—	4	2	no	—
181	C₂	#1852	1	1	yes	characteristic
182	C₂	#1861	1	1	yes	characteristic
183	C₂	#1866	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#120
commutator	C₄²	C₂²×C₄
Frattini, cosocle	C₂×C₄²	C₂³
socle	C₂²	#385
prime radical	C₂×#41	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
3. #1571	C₂	semidirect
4. #1567	C₂	semidirect
6. #1303	C₂	semidirect
7. #1279	C₂	semidirect
25. Q₈⋊C₈	C₄	semidirect
26. Q₈⋊C₈	C₄	semidirect
27. D₄⋊C₈	C₄	semidirect
28. D₄⋊C₈	C₄	semidirect

Maximal subgroups

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

subgroup	index
#2906	2
#1806	2
#1571	2
#1567	2
#1476	2
#1303	2
#1279	2