Finite Group Databasebeta

Subgroups — #19191

Order 256.

331 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 331 conjugacy classes of subgroups — too many to draw. Only its 123 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	#3227	C₂	1	1	yes	maximal, characteristic
2	#3206	C₂	1	1	yes	maximal, characteristic
3	C₂×#498	C₂	1	1	yes	maximal, characteristic, complemented
4	#2931	C₂	1	1	yes	maximal, characteristic
5	#1691	C₂	1	4	yes	maximal, complemented
6	#1671	C₂	1	4	yes	maximal, complemented
7	#1564	C₂	1	1	yes	maximal, characteristic, complemented
8	#1549	C₂	1	1	yes	maximal, characteristic, complemented
9	#1548	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#131	—	2	1	no	—
11	#532	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊Q₈	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊Q₈	C₂²	1	1	yes	characteristic
14	#498	—	2	1	no	—
15	#498	—	2	1	no	—
16	#498	—	2	1	no	—
17	#498	C₂²	1	2	yes	—
18	#498	C₂²	1	2	yes	—
19	#492	—	2	1	no	—
20	#487	—	2	1	no	—
21	#486	—	2	1	no	—
22	#481	—	2	2	no	—
23	#476	C₂²	1	4	yes	—
24	#461	C₂²	1	4	yes	—
25	#448	—	2	2	no	—
26	#436	—	2	1	no	—
27	#436	C₂²	1	2	yes	—
28	C₈×D₄	—	2	1	no	—
29	C₈×D₄	C₂²	1	2	yes	—
30	#431	C₂²	1	1	yes	characteristic
31	#431	C₂²	1	1	yes	characteristic
32	C₂×#106	—	2	1	no	—
33	C₂×#106	—	2	1	no	—
34	#423	C₂²	1	1	yes	characteristic
35	#408	—	2	1	no	—
36	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
37	#335	C₂²	1	2	yes	—
38	#334	C₂²	1	2	yes	—
39	#333	C₂²	1	4	yes	—
40	#332	C₂²	1	4	yes	—
41	C₂×#41	—	2	1	no	—
42	C₂²×Q₈	—	2	1	no	—
43	C₂²×Q₈	—	2	1	no	—
44	C₂×SD₁₆	—	2	4	no	—
45	#131	—	4	4	no	—
46	C₂×M₁₆	—	2	1	no	—
47	C₂×M₁₆	—	2	2	no	—
48	C₂²×C₈	—	2	2	no	—
49	C₂²×C₈	D₄	1	1	yes	characteristic
50	C₂²×C₈	D₄	1	1	yes	characteristic
51	C₄⋊Q₈	—	2	1	no	—
52	C₄⋊Q₈	—	2	1	no	—
53	C₄⋊Q₈	—	2	1	no	—
54	C₄⋊Q₈	—	2	1	no	—
55	C₄⋊Q₈	—	2	1	no	—
56	C₄⋊Q₈	—	2	1	no	—
57	C₄⋊Q₈	D₄	1	1	yes	characteristic
58	C₄⋊Q₈	C₂³	1	2	yes	—
59	C₄⋊Q₈	C₂³	1	2	yes	—
60	#127	D₄	1	1	yes	characteristic
61	#124	—	2	1	no	—
62	#121	—	2	1	no	—
63	#121	D₄	1	2	yes	—
64	C₄×D₄	—	2	1	no	—
65	C₄×D₄	C₂³	1	2	yes	—
66	C₂×C₄⋊C₄	—	2	1	no	—
67	C₂×C₄⋊C₄	—	2	1	no	—
68	C₂×C₄⋊C₄	—	2	1	no	—
69	C₂×C₄⋊C₄	—	2	1	no	—
70	C₂×C₄²	C₂³	1	1	yes	characteristic
71	#106	—	2	2	no	—
72	#106	—	2	2	no	—
73	#106	—	2	2	no	—
74	#106	—	2	2	no	—
75	#106	—	2	4	no	—
76	C₄⋊C₈	—	2	1	no	—
77	C₄⋊C₈	C₂³	1	2	yes	—
78	C₄⋊C₈	C₂³	1	2	yes	—
79	C₄⋊C₈	C₂³	1	2	yes	—
80	#103	—	2	4	no	—
81	#102	—	2	4	no	—
82	C₂²⋊C₈	—	2	1	no	—
83	C₂²⋊C₈	—	2	1	no	—
84	C₂²⋊C₈	—	2	2	no	—
85	C₂²⋊C₈	—	2	2	no	—
86	C₄×C₈	—	2	1	no	—
87	C₄×C₈	—	2	1	no	—
88	C₄×C₈	D₄	1	1	yes	characteristic
89	C₄×C₈	D₄	1	1	yes	characteristic
90	C₄×C₈	C₂³	1	1	yes	characteristic
91	C₄×C₈	C₂³	1	1	yes	characteristic
92	#41	—	4	2	no	—
93	#41	—	4	2	no	—
94	C₂×Q₈	—	2	1	no	—
95	C₂×Q₈	—	2	2	no	—
96	C₂×Q₈	—	2	2	no	—
97	C₂×Q₈	—	4	1	no	—
98	C₂×Q₈	—	4	1	no	—
99	C₂×Q₈	—	8	2	no	—
100	C₂×Q₈	—	8	2	no	—
101	C₂×D₄	—	2	2	no	—
102	C₂×D₄	—	2	1	no	—
103	C₂×D₄	—	2	1	no	—
104	C₂×D₄	C₂×D₄	1	2	yes	—
105	C₂²×C₄	—	2	2	no	—
106	C₂²×C₄	—	4	1	no	—
107	C₂²×C₄	—	4	1	no	—
108	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
109	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
110	C₂²×C₄	#41	1	1	yes	characteristic
111	SD₁₆	—	8	4	no	—
112	M₁₆	—	4	2	no	—
113	M₁₆	—	4	4	no	—
114	C₂×C₈	—	2	2	no	—
115	C₂×C₈	—	2	2	no	—
116	C₂×C₈	—	2	2	no	—
117	C₂×C₈	—	2	2	no	—
118	C₂×C₈	—	4	4	no	—
119	C₂×C₈	SD₁₆	1	4	yes	complemented
120	C₂×C₈	C₂×D₄	1	1	yes	characteristic
121	C₂×C₈	C₂×D₄	1	1	yes	characteristic
122	C₂×C₈	C₂×D₄	1	1	yes	characteristic
123	C₂×C₈	C₂×D₄	1	1	yes	characteristic
124	C₄⋊C₄	—	2	1	no	—
125	C₄⋊C₄	—	2	2	no	—
126	C₄⋊C₄	—	2	2	no	—
127	C₄⋊C₄	—	2	2	no	—
128	C₄⋊C₄	—	2	2	no	—
129	C₄⋊C₄	—	4	1	no	—
130	C₄⋊C₄	—	4	1	no	—
131	C₄⋊C₄	—	4	1	no	—
132	C₄⋊C₄	—	4	1	no	—
133	C₄⋊C₄	C₂×D₄	1	2	yes	—
134	C₂²⋊C₄	—	2	2	no	—
135	C₂²⋊C₄	—	2	2	no	—
136	C₄²	C₂×D₄	1	1	yes	characteristic
137	C₄²	C₂×D₄	1	1	yes	characteristic
138	C₄²	#41	1	1	yes	characteristic
139	C₄²	C₂⁴	1	1	yes	characteristic
140	C₂³	—	2	2	no	—
141	C₂³	#121	1	1	yes	characteristic
142	Q₈	—	4	2	no	—
143	Q₈	—	4	4	no	—
144	Q₈	—	8	2	no	—
145	Q₈	—	8	2	no	—
146	Q₈	—	8	2	no	—
147	D₄	—	2	4	no	—
148	D₄	—	4	4	no	—
149	D₄	—	4	2	no	—
150	D₄	—	4	2	no	—
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	—	2	2	no	—
154	C₂×C₄	—	4	2	no	—
155	C₂×C₄	—	4	2	no	—
156	C₂×C₄	—	4	2	no	—
157	C₂×C₄	—	4	2	no	—
158	C₂×C₄	—	8	2	no	—
159	C₂×C₄	—	8	2	no	—
160	C₂×C₄	#121	1	1	yes	characteristic
161	C₂×C₄	#121	1	1	yes	characteristic
162	C₂×C₄	#121	1	1	yes	characteristic
163	C₂×C₄	C₂×SD₁₆	1	2	yes	—
164	C₂×C₄	#137	1	2	yes	—
165	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
166	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
167	C₂×C₄	C₂×#41	1	1	yes	characteristic
168	C₈	—	2	1	no	—
169	C₈	—	2	1	no	—
170	C₈	—	4	2	no	—
171	C₈	—	4	2	no	—
172	C₈	—	4	2	no	—
173	C₈	C₂×SD₁₆	1	2	yes	—
174	C₈	C₂×SD₁₆	1	2	yes	—
175	C₂²	—	2	1	no	—
176	C₂²	—	2	1	no	—
177	C₂²	—	4	2	no	—
178	C₂²	—	4	2	no	—
179	C₂²	—	4	2	no	—
180	C₂²	#461	1	2	yes	—
181	C₂²	C₂×#121	1	1	yes	characteristic
182	C₄	—	2	1	no	—
183	C₄	—	2	1	no	—
184	C₄	—	4	2	no	—
185	C₄	—	8	2	no	—
186	C₄	—	8	2	no	—
187	C₄	#461	1	2	yes	—
188	C₄	#471	1	2	yes	—
189	C₄	C₂²×SD₁₆	1	1	yes	characteristic
190	C₄	C₂×#137	1	1	yes	characteristic
191	C₄	#577	1	2	yes	—
192	C₂	—	2	2	no	—
193	C₂	—	8	2	no	—
194	C₂	C₂×#461	1	1	yes	characteristic
195	C₂	#3057	1	1	yes	characteristic
196	C₂	C₂×#471	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂×#121
commutator, Frattini, cosocle	C₄²	C₂⁴
prime radical	#532	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
3. C₂×#498	C₂	semidirect
5. #1691	C₂	semidirect
6. #1671	C₂	semidirect
7. #1564	C₂	semidirect
8. #1549	C₂	semidirect
9. #1548	C₂	semidirect
119. C₂×C₈	SD₁₆	semidirect

Maximal subgroups

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

subgroup	index
#3227	2
#3206	2
C₂×#498	2
#2931	2
#1691	2
#1671	2
#1564	2
#1549	2
#1548	2