Finite Group Databasebeta

Subgroups — #9171

Order 256.

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

normal subgroups only

This group has 255 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	#3039	C₂	1	1	yes	maximal, characteristic
2	#2906	C₂	1	1	yes	maximal, characteristic
3	#1546	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#332	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×#137	—	2	1	no	—
9	C₂×#131	—	2	1	no	—
10	C₂×C₄⋊Q₈	C₂²	1	1	yes	characteristic
11	C₄×#41	C₂²	1	1	yes	characteristic
12	#462	—	2	1	no	—
13	#462	C₄	1	2	yes	—
14	#461	—	2	1	no	—
15	#461	C₄	1	2	yes	—
16	#431	C₂²	1	1	yes	characteristic
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	#419	—	2	1	no	—
26	#332	—	2	1	no	—
27	#332	—	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	#137	—	4	2	no	—
34	#137	—	4	2	no	—
35	C₂×Q₁₆	—	2	2	no	—
36	C₂×SD₁₆	—	2	2	no	—
37	#131	—	4	2	no	—
38	#131	—	4	2	no	—
39	C₂×M₁₆	—	2	2	no	—
40	C₂×M₁₆	—	2	1	no	—
41	C₂×M₁₆	—	2	1	no	—
42	C₂²×C₈	—	2	2	no	—
43	C₂²×C₈	—	2	1	no	—
44	C₂²×C₈	—	2	1	no	—
45	C₄⋊Q₈	—	2	1	no	—
46	C₄⋊Q₈	—	2	1	no	—
47	C₄⋊Q₈	—	2	1	no	—
48	C₄⋊Q₈	C₂×C₄	1	2	yes	—
49	C₄×Q₈	C₂×C₄	1	1	yes	characteristic
50	C₄×Q₈	D₄	1	1	yes	characteristic
51	C₄×D₄	—	2	1	no	—
52	C₄×D₄	—	2	1	no	—
53	C₄×D₄	C₂×C₄	1	1	yes	characteristic
54	C₄×D₄	D₄	1	1	yes	characteristic
55	#117	—	2	1	no	—
56	#117	D₄	1	1	yes	characteristic
57	C₂×C₄⋊C₄	—	2	1	no	—
58	C₂×C₄⋊C₄	—	2	1	no	—
59	C₂×C₄²	—	2	1	no	—
60	C₂×C₄²	C₂³	1	1	yes	characteristic
61	C₄⋊C₈	—	2	2	no	—
62	C₄⋊C₈	—	2	2	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	—	2	1	no	—
68	C₄⋊C₈	—	2	1	no	—
69	C₄⋊C₈	C₂×C₄	1	2	yes	—
70	C₄⋊C₈	D₄	1	1	yes	characteristic
71	C₄⋊C₈	D₄	1	1	yes	characteristic
72	C₄⋊C₈	D₄	1	2	yes	—
73	#103	—	2	2	no	—
74	#103	—	2	2	no	—
75	C₂²⋊C₈	—	2	1	no	—
76	C₄×C₈	—	2	1	no	—
77	#41	—	2	1	no	—
78	#41	—	2	1	no	—
79	#41	D₈	1	2	yes	—
80	#41	SD₁₆	1	2	yes	—
81	C₂×Q₈	—	2	2	no	—
82	C₂×Q₈	—	4	1	no	—
83	C₂×Q₈	—	4	1	no	—
84	C₂×Q₈	—	4	1	no	—
85	C₂×Q₈	—	4	1	no	—
86	C₂×Q₈	—	4	1	no	—
87	C₂×Q₈	C₂²⋊C₄	1	1	yes	characteristic
88	C₂×D₄	—	2	1	no	—
89	C₂×D₄	C₂²⋊C₄	1	1	yes	characteristic
90	C₂²×C₄	—	2	1	no	—
91	C₂²×C₄	—	2	1	no	—
92	C₂²×C₄	—	2	1	no	—
93	C₂²×C₄	—	4	1	no	—
94	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
95	C₂²×C₄	#41	1	1	yes	characteristic
96	C₂²×C₄	#41	1	1	yes	characteristic
97	Q₁₆	—	4	4	no	—
98	SD₁₆	—	4	4	no	—
99	M₁₆	—	4	4	no	—
100	M₁₆	—	4	1	no	—
101	M₁₆	—	4	1	no	—
102	M₁₆	—	4	1	no	—
103	M₁₆	—	4	1	no	—
104	C₂×C₈	—	2	2	no	—
105	C₂×C₈	—	2	1	no	—
106	C₂×C₈	—	2	2	no	—
107	C₂×C₈	—	2	1	no	—
108	C₂×C₈	—	4	4	no	—
109	C₂×C₈	—	4	1	no	—
110	C₂×C₈	—	4	1	no	—
111	C₂×C₈	—	4	1	no	—
112	C₂×C₈	—	4	1	no	—
113	C₂×C₈	—	8	1	no	—
114	C₄⋊C₄	—	2	1	no	—
115	C₄⋊C₄	—	2	1	no	—
116	C₄⋊C₄	—	2	2	no	—
117	C₄⋊C₄	—	2	2	no	—
118	C₄⋊C₄	—	4	1	no	—
119	C₄⋊C₄	—	4	1	no	—
120	C₄⋊C₄	C₂²⋊C₄	1	1	yes	characteristic
121	C₄⋊C₄	C₂²⋊C₄	1	1	yes	characteristic
122	C₂²⋊C₄	—	2	1	no	—
123	C₂²⋊C₄	—	2	1	no	—
124	C₂²⋊C₄	—	2	1	no	—
125	C₄²	—	2	1	no	—
126	C₄²	—	2	1	no	—
127	C₄²	—	2	1	no	—
128	C₄²	C₂²×C₄	1	1	yes	characteristic
129	C₄²	C₂×D₄	1	1	yes	characteristic
130	C₄²	C₂×D₄	1	1	yes	characteristic
131	C₄²	C₂×D₄	1	1	yes	characteristic
132	C₂³	—	2	1	no	—
133	C₂³	#123	1	1	yes	characteristic
134	Q₈	—	2	1	no	—
135	Q₈	—	4	2	no	—
136	Q₈	—	4	2	no	—
137	Q₈	—	8	1	no	—
138	Q₈	—	8	1	no	—
139	Q₈	#102	1	2	yes	—
140	D₄	—	2	2	no	—
141	D₄	—	2	2	no	—
142	D₄	—	2	1	no	—
143	D₄	#102	1	2	yes	—
144	C₂×C₄	—	2	1	no	—
145	C₂×C₄	—	2	2	no	—
146	C₂×C₄	—	2	2	no	—
147	C₂×C₄	—	2	1	no	—
148	C₂×C₄	—	2	1	no	—
149	C₂×C₄	—	2	1	no	—
150	C₂×C₄	—	2	1	no	—
151	C₂×C₄	—	4	1	no	—
152	C₂×C₄	—	4	1	no	—
153	C₂×C₄	—	4	1	no	—
154	C₂×C₄	—	4	2	no	—
155	C₂×C₄	—	8	1	no	—
156	C₂×C₄	—	8	1	no	—
157	C₂×C₄	—	8	1	no	—
158	C₂×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
159	C₂×C₄	C₄×D₄	1	1	yes	characteristic
160	C₂×C₄	C₄×D₄	1	1	yes	characteristic
161	C₂×C₄	#120	1	1	yes	characteristic
162	C₂×C₄	#121	1	1	yes	characteristic
163	C₂×C₄	#121	1	1	yes	characteristic
164	C₂×C₄	C₂×D₈	1	1	yes	characteristic
165	C₂×C₄	C₂×SD₁₆	1	1	yes	characteristic
166	C₂×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
167	C₂×C₄	#137	1	1	yes	characteristic
168	C₈	—	4	2	no	—
169	C₈	—	4	1	no	—
170	C₈	—	4	2	no	—
171	C₈	—	4	1	no	—
172	C₈	—	8	1	no	—
173	C₈	—	8	1	no	—
174	C₂²	—	2	1	no	—
175	C₂²	—	2	1	no	—
176	C₂²	—	2	2	no	—
177	C₂²	—	4	1	no	—
178	C₂²	—	4	1	no	—
179	C₂²	#386	1	1	yes	characteristic
180	C₂²	#480	1	1	yes	characteristic
181	C₂²	#481	1	1	yes	characteristic
182	C₄	—	2	2	no	—
183	C₄	—	2	1	no	—
184	C₄	—	2	1	no	—
185	C₄	—	2	1	no	—
186	C₄	—	4	1	no	—
187	C₄	—	4	1	no	—
188	C₄	—	8	2	no	—
189	C₄	C₂×#102	1	1	yes	characteristic
190	C₄	#417	1	1	yes	characteristic
191	C₄	#447	1	1	yes	characteristic
192	C₄	#448	1	1	yes	characteristic
193	C₄	#471	1	1	yes	characteristic
194	C₄	#473	1	1	yes	characteristic
195	C₂	—	2	1	no	—
196	C₂	—	2	1	no	—
197	C₂	—	4	2	no	—
198	C₂	#1874	1	1	yes	characteristic
199	C₂	#1902	1	1	yes	characteristic
200	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. #1546	C₂	semidirect
4. C₂×#332	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
#3039	2
#2906	2
#1546	2
C₂×#332	2
#1477	2
#1301	2
#1292	2