Finite Group Databasebeta

Subgroups — #9169

Order 256.

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

normal subgroups only

This group has 239 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	#3079	C₂	1	1	yes	maximal, characteristic
2	#2906	C₂	1	1	yes	maximal, characteristic
3	#1549	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#333	C₂	1	1	yes	maximal, characteristic, complemented
5	#1477	C₂	1	1	yes	maximal, characteristic
6	#1301	C₂	1	1	yes	maximal, characteristic, complemented
7	#1292	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#131	—	2	1	no	—
9	C₂×C₄⋊Q₈	C₂²	1	1	yes	characteristic
10	C₄×#41	C₂²	1	1	yes	characteristic
11	#477	—	2	1	no	—
12	#476	—	2	1	no	—
13	#475	C₄	1	2	yes	—
14	#474	C₄	1	2	yes	—
15	#431	C₂²	1	1	yes	characteristic
16	#428	—	2	1	no	—
17	#424	—	2	1	no	—
18	#424	—	2	2	no	—
19	#423	—	2	1	no	—
20	#423	C₂²	1	1	yes	characteristic
21	#423	C₂²	1	1	yes	characteristic
22	C₂×C₄⋊C₈	—	2	1	no	—
23	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
24	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
25	#417	—	2	1	no	—
26	#333	—	2	1	no	—
27	#333	—	2	1	no	—
28	#332	—	2	2	no	—
29	Q₈⋊C₈	—	2	1	no	—
30	D₄⋊C₈	—	2	1	no	—
31	C₂×#41	D₄	1	1	yes	characteristic
32	C₂²×Q₈	—	2	1	no	—
33	#131	—	4	2	no	—
34	#131	—	4	2	no	—
35	C₂×M₁₆	—	2	2	no	—
36	C₂×M₁₆	—	2	1	no	—
37	C₂×M₁₆	—	2	1	no	—
38	C₂²×C₈	—	2	2	no	—
39	C₂²×C₈	—	2	1	no	—
40	C₂²×C₈	—	2	1	no	—
41	C₄⋊Q₈	—	2	1	no	—
42	C₄⋊Q₈	—	2	1	no	—
43	C₄⋊Q₈	—	2	1	no	—
44	C₄⋊Q₈	C₂×C₄	1	2	yes	—
45	C₄×Q₈	C₂×C₄	1	1	yes	characteristic
46	C₄×Q₈	D₄	1	1	yes	characteristic
47	C₄×D₄	—	2	1	no	—
48	C₄×D₄	—	2	1	no	—
49	C₄×D₄	C₂×C₄	1	1	yes	characteristic
50	C₄×D₄	D₄	1	1	yes	characteristic
51	#117	—	2	1	no	—
52	#117	D₄	1	1	yes	characteristic
53	C₂×C₄⋊C₄	—	2	1	no	—
54	C₂×C₄⋊C₄	—	2	1	no	—
55	C₂×C₄²	—	2	1	no	—
56	C₂×C₄²	C₂³	1	1	yes	characteristic
57	#107	—	2	2	no	—
58	#106	—	2	2	no	—
59	C₄⋊C₈	—	2	2	no	—
60	C₄⋊C₈	—	2	2	no	—
61	C₄⋊C₈	—	2	1	no	—
62	C₄⋊C₈	—	2	1	no	—
63	C₄⋊C₈	—	2	1	no	—
64	C₄⋊C₈	—	2	1	no	—
65	C₄⋊C₈	—	2	1	no	—
66	C₄⋊C₈	—	2	1	no	—
67	C₄⋊C₈	C₂×C₄	1	2	yes	—
68	C₄⋊C₈	D₄	1	2	yes	—
69	C₄⋊C₈	D₄	1	1	yes	characteristic
70	C₄⋊C₈	D₄	1	1	yes	characteristic
71	#103	—	2	2	no	—
72	#102	—	2	2	no	—
73	C₂²⋊C₈	—	2	1	no	—
74	C₄×C₈	—	2	1	no	—
75	#41	—	2	1	no	—
76	#41	—	2	1	no	—
77	#41	SD₁₆	1	2	yes	—
78	#41	Q₁₆	1	2	yes	—
79	C₂×Q₈	—	2	2	no	—
80	C₂×Q₈	—	4	1	no	—
81	C₂×Q₈	—	8	2	no	—
82	C₂×Q₈	C₂²⋊C₄	1	1	yes	characteristic
83	C₂×D₄	—	2	1	no	—
84	C₂×D₄	C₂²⋊C₄	1	1	yes	characteristic
85	C₂²×C₄	—	2	1	no	—
86	C₂²×C₄	—	2	1	no	—
87	C₂²×C₄	—	2	1	no	—
88	C₂²×C₄	—	4	1	no	—
89	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
90	C₂²×C₄	#41	1	1	yes	characteristic
91	C₂²×C₄	#41	1	1	yes	characteristic
92	M₁₆	—	4	4	no	—
93	M₁₆	—	4	1	no	—
94	M₁₆	—	4	1	no	—
95	M₁₆	—	4	1	no	—
96	M₁₆	—	4	1	no	—
97	C₂×C₈	—	2	2	no	—
98	C₂×C₈	—	2	1	no	—
99	C₂×C₈	—	2	2	no	—
100	C₂×C₈	—	2	1	no	—
101	C₂×C₈	—	4	4	no	—
102	C₂×C₈	—	4	1	no	—
103	C₂×C₈	—	4	1	no	—
104	C₂×C₈	—	4	1	no	—
105	C₂×C₈	—	4	1	no	—
106	C₂×C₈	—	8	1	no	—
107	C₄⋊C₄	—	2	1	no	—
108	C₄⋊C₄	—	2	1	no	—
109	C₄⋊C₄	—	2	2	no	—
110	C₄⋊C₄	—	2	2	no	—
111	C₄⋊C₄	—	4	1	no	—
112	C₄⋊C₄	—	4	1	no	—
113	C₄⋊C₄	C₂²⋊C₄	1	1	yes	characteristic
114	C₄⋊C₄	C₂²⋊C₄	1	1	yes	characteristic
115	C₂²⋊C₄	—	2	1	no	—
116	C₂²⋊C₄	—	2	1	no	—
117	C₂²⋊C₄	—	2	1	no	—
118	C₄²	—	2	1	no	—
119	C₄²	—	2	1	no	—
120	C₄²	—	2	1	no	—
121	C₄²	C₂²×C₄	1	1	yes	characteristic
122	C₄²	C₂×D₄	1	1	yes	characteristic
123	C₄²	C₂×D₄	1	1	yes	characteristic
124	C₄²	C₂×D₄	1	1	yes	characteristic
125	C₂³	—	2	1	no	—
126	C₂³	#123	1	1	yes	characteristic
127	Q₈	—	2	1	no	—
128	Q₈	—	8	2	no	—
129	Q₈	—	8	2	no	—
130	Q₈	#103	1	2	yes	—
131	D₄	—	2	2	no	—
132	D₄	—	2	2	no	—
133	D₄	—	2	1	no	—
134	D₄	#103	1	2	yes	—
135	C₂×C₄	—	2	1	no	—
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	—	2	2	no	—
138	C₂×C₄	—	2	1	no	—
139	C₂×C₄	—	2	1	no	—
140	C₂×C₄	—	2	1	no	—
141	C₂×C₄	—	2	1	no	—
142	C₂×C₄	—	4	1	no	—
143	C₂×C₄	—	4	1	no	—
144	C₂×C₄	—	4	1	no	—
145	C₂×C₄	—	4	2	no	—
146	C₂×C₄	—	8	1	no	—
147	C₂×C₄	—	8	2	no	—
148	C₂×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
149	C₂×C₄	C₄×D₄	1	1	yes	characteristic
150	C₂×C₄	C₄×D₄	1	1	yes	characteristic
151	C₂×C₄	#120	1	1	yes	characteristic
152	C₂×C₄	#121	1	1	yes	characteristic
153	C₂×C₄	#121	1	1	yes	characteristic
154	C₂×C₄	C₂×SD₁₆	1	1	yes	characteristic
155	C₂×C₄	C₂×Q₁₆	1	1	yes	characteristic
156	C₂×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
157	C₂×C₄	#137	1	1	yes	characteristic
158	C₈	—	4	2	no	—
159	C₈	—	4	1	no	—
160	C₈	—	4	2	no	—
161	C₈	—	4	1	no	—
162	C₈	—	8	1	no	—
163	C₈	—	8	1	no	—
164	C₂²	—	2	1	no	—
165	C₂²	—	2	1	no	—
166	C₂²	—	2	2	no	—
167	C₂²	—	4	1	no	—
168	C₂²	—	4	1	no	—
169	C₂²	#386	1	1	yes	characteristic
170	C₂²	#483	1	1	yes	characteristic
171	C₂²	#484	1	1	yes	characteristic
172	C₄	—	2	2	no	—
173	C₄	—	2	1	no	—
174	C₄	—	2	1	no	—
175	C₄	—	2	1	no	—
176	C₄	—	4	1	no	—
177	C₄	—	4	1	no	—
178	C₄	—	8	2	no	—
179	C₄	C₂×#103	1	1	yes	characteristic
180	C₄	#417	1	1	yes	characteristic
181	C₄	#450	1	1	yes	characteristic
182	C₄	#451	1	1	yes	characteristic
183	C₄	#471	1	1	yes	characteristic
184	C₄	#473	1	1	yes	characteristic
185	C₂	—	2	1	no	—
186	C₂	—	2	1	no	—
187	C₂	—	4	2	no	—
188	C₂	#1872	1	1	yes	characteristic
189	C₂	#1903	1	1	yes	characteristic
190	C₂	#1908	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	#386
commutator	C₄²	C₂²×C₄
Frattini, cosocle	C₂×C₄²	C₂³
prime radical	C₂×#41	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ 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. #1549	C₂	semidirect
4. C₂×#333	C₂	semidirect
6. #1301	C₂	semidirect
7. #1292	C₂	semidirect

Maximal subgroups

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

subgroup	index
#3079	2
#2906	2
#1549	2
C₂×#333	2
#1477	2
#1301	2
#1292	2