Finite Group Databasebeta

Subgroups — #19909

Order 256.

319 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 319 conjugacy classes of subgroups — too many to draw. Only its 103 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	#3391	C₂	1	1	yes	maximal, characteristic
2	#3380	C₂	1	1	yes	maximal, characteristic, complemented
3	#3333	C₂	1	2	yes	maximal, complemented
4	#1706	C₂	1	1	yes	maximal, characteristic, complemented
5	#1697	C₂	1	2	yes	maximal, complemented
6	#1676	C₂	1	1	yes	maximal, characteristic, complemented
7	#1668	C₂	1	1	yes	maximal, characteristic, complemented
8	#1631	C₂	1	2	yes	maximal, complemented
9	#1622	C₂	1	2	yes	maximal, complemented
10	#1589	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
12	#554	C₂²	1	2	yes	—
13	#550	C₂²	1	2	yes	—
14	#500	C₂²	1	1	yes	characteristic
15	#499	—	2	1	no	—
16	#493	—	2	1	no	—
17	#493	C₂²	1	1	yes	characteristic
18	#492	C₂²	1	2	yes	—
19	#486	C₂²	1	2	yes	—
20	#480	—	2	2	no	—
21	#478	—	2	2	no	—
22	#477	C₂²	1	2	yes	—
23	#476	C₂²	1	2	yes	—
24	#468	—	2	2	no	—
25	#460	C₂²	1	2	yes	—
26	#459	—	2	2	no	—
27	#459	C₂²	1	2	yes	—
28	#452	—	2	2	no	—
29	C₈×Q₈	C₂²	1	1	yes	characteristic
30	C₈×Q₈	C₂²	1	1	yes	characteristic
31	C₄×Q₁₆	C₂²	1	1	yes	characteristic
32	C₄×D₈	—	2	1	no	—
33	C₄×D₈	—	2	1	no	—
34	C₄×D₈	C₂²	1	1	yes	characteristic
35	#436	C₂²	1	2	yes	—
36	#417	—	2	2	no	—
37	#335	C₂²	1	1	yes	characteristic
38	#335	C₂²	1	1	yes	characteristic
39	#333	C₂²	1	2	yes	—
40	#331	C₂²	1	2	yes	—
41	Q₈⋊C₈	C₂²	1	1	yes	characteristic
42	Q₈⋊C₈	C₂²	1	1	yes	characteristic
43	D₄⋊C₈	C₂²	1	2	yes	—
44	C₈²	C₂²	1	1	yes	characteristic
45	C₂×#41	—	2	2	no	—
46	C₂×Q₁₆	D₄	1	1	yes	characteristic
47	C₂×SD₁₆	—	2	2	no	—
48	C₂×SD₁₆	—	4	2	no	—
49	C₂×D₈	—	2	2	no	—
50	C₂×D₈	—	2	1	no	—
51	C₂×D₈	—	2	1	no	—
52	C₂×D₈	D₄	1	1	yes	characteristic
53	C₂×M₁₆	—	2	2	no	—
54	C₄⋊Q₈	C₂³	1	2	yes	—
55	#127	—	2	2	no	—
56	#127	C₂³	1	2	yes	—
57	#125	—	2	2	no	—
58	C₂²⋊Q₈	—	2	2	no	—
59	C₂²⋊Q₈	—	4	2	no	—
60	#121	—	2	2	no	—
61	#121	—	4	2	no	—
62	C₄×Q₈	C₂³	1	1	yes	characteristic
63	C₄×Q₈	C₂³	1	1	yes	characteristic
64	C₄×D₄	—	2	2	no	—
65	C₄×D₄	—	2	2	no	—
66	C₄×D₄	C₂³	1	2	yes	—
67	C₂×C₄⋊C₄	—	2	2	no	—
68	#107	—	2	1	no	—
69	#107	—	2	1	no	—
70	#107	—	2	2	no	—
71	#107	D₄	1	1	yes	characteristic
72	#107	D₄	1	1	yes	characteristic
73	#106	—	2	2	no	—
74	C₄⋊C₈	—	2	1	no	—
75	C₄⋊C₈	—	2	1	no	—
76	C₄⋊C₈	C₂³	1	2	yes	—
77	C₄⋊C₈	C₂³	1	1	yes	characteristic
78	C₄⋊C₈	C₂³	1	1	yes	characteristic
79	#103	—	2	2	no	—
80	#103	D₄	1	1	yes	characteristic
81	#103	D₄	1	1	yes	characteristic
82	#102	—	2	2	no	—
83	#102	—	2	2	no	—
84	#102	—	2	2	no	—
85	#102	—	2	2	no	—
86	#102	—	2	2	no	—
87	#102	D₄	1	2	yes	—
88	C₂²⋊C₈	—	2	2	no	—
89	C₄×C₈	—	2	1	no	—
90	C₄×C₈	—	2	1	no	—
91	C₄×C₈	C₂³	1	1	yes	characteristic
92	C₄×C₈	C₂³	1	1	yes	characteristic
93	C₄×C₈	C₂³	1	1	yes	characteristic
94	#41	—	4	4	no	—
95	#41	—	8	2	no	—
96	C₂×Q₈	—	2	2	no	—
97	C₂×Q₈	C₂×D₄	1	1	yes	characteristic
98	C₂×Q₈	C₂×D₄	1	1	yes	characteristic
99	C₂×D₄	—	2	2	no	—
100	C₂×D₄	—	2	2	no	—
101	C₂×D₄	—	2	2	no	—
102	C₂×D₄	—	4	2	no	—
103	C₂×D₄	—	4	2	no	—
104	C₂×D₄	C₂×D₄	1	2	yes	—
105	C₂²×C₄	—	2	2	no	—
106	C₂²×C₄	—	4	2	no	—
107	C₂²×C₄	—	4	2	no	—
108	Q₁₆	—	2	2	no	—
109	SD₁₆	—	8	4	no	—
110	SD₁₆	—	8	2	no	—
111	D₈	—	2	2	no	—
112	D₈	—	4	2	no	—
113	D₈	—	8	1	no	—
114	D₈	—	8	2	no	—
115	M₁₆	—	8	2	no	—
116	C₂×C₈	—	2	1	no	—
117	C₂×C₈	—	2	1	no	—
118	C₂×C₈	—	2	1	no	—
119	C₂×C₈	—	2	2	no	—
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	1	no	—
126	C₄⋊C₄	—	2	2	no	—
127	C₄⋊C₄	—	2	2	no	—
128	C₄⋊C₄	—	4	2	no	—
129	C₄⋊C₄	—	4	2	no	—
130	C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
131	C₄⋊C₄	C₂×D₄	1	2	yes	—
132	C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
133	C₂²⋊C₄	—	2	2	no	—
134	C₂²⋊C₄	—	4	2	no	—
135	C₂²⋊C₄	—	4	2	no	—
136	C₄²	—	2	1	no	—
137	C₄²	—	2	1	no	—
138	C₄²	C₂⁴	1	1	yes	characteristic
139	C₂³	—	2	2	no	—
140	C₂³	—	4	2	no	—
141	Q₈	—	2	1	no	—
142	Q₈	—	2	1	no	—
143	Q₈	—	2	1	no	—
144	Q₈	—	8	2	no	—
145	Q₈	#135	1	2	yes	—
146	D₄	—	2	2	no	—
147	D₄	—	2	2	no	—
148	D₄	—	4	2	no	—
149	D₄	—	4	2	no	—
150	D₄	—	4	2	no	—
151	D₄	—	4	2	no	—
152	D₄	—	8	4	no	—
153	D₄	—	8	2	no	—
154	D₄	—	16	2	no	—
155	C₂×C₄	—	2	1	no	—
156	C₂×C₄	—	2	1	no	—
157	C₂×C₄	—	2	1	no	—
158	C₂×C₄	—	2	1	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	4	2	no	—
161	C₂×C₄	—	4	2	no	—
162	C₂×C₄	—	4	2	no	—
163	C₂×C₄	—	8	4	no	—
164	C₂×C₄	—	16	2	no	—
165	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
166	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
167	C₂×C₄	H(5,𝔽₂)	1	1	yes	characteristic
168	C₈	—	2	2	no	—
169	C₈	—	2	1	no	—
170	C₈	—	2	1	no	—
171	C₈	—	2	1	no	—
172	C₈	—	4	2	no	—
173	C₈	—	4	1	no	—
174	C₈	—	4	1	no	—
175	C₈	#137	1	2	yes	—
176	C₂²	—	4	2	no	—
177	C₂²	—	4	2	no	—
178	C₂²	—	4	2	no	—
179	C₂²	—	8	2	no	—
180	C₂²	—	8	2	no	—
181	C₂²	—	8	2	no	—
182	C₂²	D₄²	1	1	yes	characteristic
183	C₄	—	2	2	no	—
184	C₄	—	2	1	no	—
185	C₄	—	4	2	no	—
186	C₄	—	4	1	no	—
187	C₄	—	4	1	no	—
188	C₄	—	4	1	no	—
189	C₄	—	8	2	no	—
190	C₄	C₂×#135	1	1	yes	characteristic
191	C₄	C₂×#137	1	1	yes	characteristic
192	C₄	#576	1	1	yes	characteristic
193	C₄	#577	1	1	yes	characteristic
194	C₂	—	8	2	no	—
195	C₂	—	16	2	no	—
196	C₂	#3278	1	1	yes	characteristic
197	C₂	#3280	1	1	yes	characteristic
198	C₂	#3292	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	D₄²
commutator, Frattini, cosocle	C₄²	C₂⁴
prime radical	#3391	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
2. #3380	C₂	semidirect
3. #3333	C₂	semidirect
4. #1706	C₂	semidirect
5. #1697	C₂	semidirect
6. #1676	C₂	semidirect
7. #1668	C₂	semidirect
8. #1631	C₂	semidirect
9. #1622	C₂	semidirect
10. #1589	C₂	semidirect
11. C₈×Q₁₆	C₂	semidirect

Maximal subgroups

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

subgroup	index
#3391	2
#3380	2
#3333	2
#1706	2
#1697	2
#1676	2
#1668	2
#1631	2
#1622	2
#1589	2
C₈×Q₁₆	2