Finite Group Databasebeta

Subgroups — #19515

Order 256.

288 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 288 conjugacy classes of subgroups — too many to draw. Only its 105 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	#3235	C₂	1	1	yes	maximal, characteristic
2	#3230	C₂	1	1	yes	maximal, characteristic, complemented
3	#3227	C₂	1	1	yes	maximal, characteristic
4	#2981	C₂	1	1	yes	maximal, characteristic
5	#1703	C₂	1	2	yes	maximal, complemented
6	#1695	C₂	1	2	yes	maximal, complemented
7	#1683	C₂	1	2	yes	maximal, complemented
8	#1675	C₂	1	2	yes	maximal, complemented
9	#1572	C₂	1	1	yes	maximal, characteristic, complemented
10	#1558	C₂	1	1	yes	maximal, characteristic, complemented
11	#1554	C₂	1	1	yes	maximal, characteristic, complemented
12	#533	C₂²	1	1	yes	characteristic
13	#532	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊Q₈	C₂²	1	1	yes	characteristic
15	#501	—	2	1	no	—
16	#500	C₂²	1	1	yes	characteristic
17	#500	C₂²	1	1	yes	characteristic
18	#498	—	2	1	no	—
19	#490	—	2	1	no	—
20	#487	C₂²	1	1	yes	characteristic
21	#487	C₂²	1	1	yes	characteristic
22	#486	—	2	1	no	—
23	#484	—	2	2	no	—
24	#482	—	2	2	no	—
25	#481	—	2	2	no	—
26	#477	—	2	2	no	—
27	#476	C₂²	1	2	yes	—
28	#476	C₂²	1	2	yes	—
29	#476	C₂²	1	2	yes	—
30	#474	C₂²	1	2	yes	—
31	#436	C₂²	1	2	yes	—
32	#435	—	2	2	no	—
33	C₈×D₄	C₂²	1	2	yes	—
34	#432	C₂²	1	1	yes	characteristic
35	#432	C₂²	1	1	yes	characteristic
36	#431	C₂²	1	1	yes	characteristic
37	#431	C₂²	1	1	yes	characteristic
38	#408	—	2	2	no	—
39	#335	C₂²	1	2	yes	—
40	#334	C₂²	1	2	yes	—
41	#333	C₂²	1	2	yes	—
42	#333	C₂²	1	2	yes	—
43	#333	C₂²	1	2	yes	—
44	#332	C₂²	1	2	yes	—
45	C₂×#41	—	2	1	no	—
46	C₂²×Q₈	—	2	1	no	—
47	C₂×M₁₆	—	2	2	no	—
48	C₂²×C₈	—	2	2	no	—
49	C₄⋊Q₈	—	2	1	no	—
50	C₄⋊Q₈	—	2	1	no	—
51	C₄⋊Q₈	—	2	1	no	—
52	C₄⋊Q₈	D₄	1	1	yes	characteristic
53	C₄⋊Q₈	C₂³	1	1	yes	characteristic
54	C₄⋊Q₈	C₂³	1	1	yes	characteristic
55	C₄⋊Q₈	C₂³	1	1	yes	characteristic
56	C₄⋊Q₈	C₂³	1	1	yes	characteristic
57	#127	D₄	1	1	yes	characteristic
58	#125	—	2	1	no	—
59	#124	—	2	1	no	—
60	C₂²⋊Q₈	—	2	1	no	—
61	C₂²⋊Q₈	—	2	1	no	—
62	#121	—	2	1	no	—
63	#121	D₄	1	2	yes	—
64	C₄×Q₈	—	2	1	no	—
65	C₄×Q₈	—	2	1	no	—
66	C₄×D₄	—	2	1	no	—
67	C₄×D₄	C₂³	1	2	yes	—
68	#117	—	2	1	no	—
69	C₂×C₄⋊C₄	—	2	1	no	—
70	C₂×C₄⋊C₄	—	2	1	no	—
71	C₂×C₄²	C₂³	1	1	yes	characteristic
72	#107	—	2	2	no	—
73	#107	—	2	2	no	—
74	#107	—	2	2	no	—
75	#106	—	2	2	no	—
76	#106	—	2	2	no	—
77	#106	—	2	2	no	—
78	C₄⋊C₈	D₄	1	2	yes	—
79	C₄⋊C₈	C₂³	1	2	yes	—
80	C₄⋊C₈	C₂³	1	2	yes	—
81	C₄⋊C₈	C₂³	1	2	yes	—
82	#103	—	2	2	no	—
83	#103	—	2	2	no	—
84	#102	—	2	2	no	—
85	#102	—	2	2	no	—
86	#102	—	2	2	no	—
87	#102	—	2	2	no	—
88	C₂²⋊C₈	—	2	1	no	—
89	C₂²⋊C₈	—	2	1	no	—
90	C₂²⋊C₈	—	2	1	no	—
91	C₂²⋊C₈	—	2	2	no	—
92	C₂²⋊C₈	—	2	2	no	—
93	C₂²⋊C₈	D₄	1	2	yes	—
94	#97	—	2	1	no	—
95	#97	—	2	1	no	—
96	C₄×C₈	—	2	1	no	—
97	C₄×C₈	C₂³	1	1	yes	characteristic
98	C₄×C₈	C₂³	1	1	yes	characteristic
99	#41	—	8	2	no	—
100	C₂×Q₈	—	2	1	no	—
101	C₂×Q₈	—	2	1	no	—
102	C₂×Q₈	—	2	1	no	—
103	C₂×Q₈	—	2	1	no	—
104	C₂×Q₈	—	2	1	no	—
105	C₂×Q₈	—	4	1	no	—
106	C₂×Q₈	—	8	2	no	—
107	C₂×D₄	—	2	2	no	—
108	C₂×D₄	—	2	1	no	—
109	C₂×D₄	—	2	1	no	—
110	C₂×D₄	C₂×D₄	1	2	yes	—
111	C₂²×C₄	—	2	2	no	—
112	C₂²×C₄	—	4	1	no	—
113	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
114	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
115	C₂²×C₄	#41	1	1	yes	characteristic
116	M₁₆	—	8	2	no	—
117	C₂×C₈	—	2	2	no	—
118	C₂×C₈	—	2	2	no	—
119	C₂×C₈	—	2	2	no	—
120	C₂×C₈	—	8	2	no	—
121	C₂×C₈	C₂×D₄	1	2	yes	—
122	C₂×C₈	C₂×D₄	1	2	yes	—
123	C₄⋊C₄	—	2	1	no	—
124	C₄⋊C₄	—	2	1	no	—
125	C₄⋊C₄	—	2	1	no	—
126	C₄⋊C₄	—	2	1	no	—
127	C₄⋊C₄	—	2	1	no	—
128	C₄⋊C₄	—	2	1	no	—
129	C₄⋊C₄	—	2	1	no	—
130	C₄⋊C₄	—	2	1	no	—
131	C₄⋊C₄	—	2	1	no	—
132	C₄⋊C₄	—	4	1	no	—
133	C₄⋊C₄	—	4	1	no	—
134	C₄⋊C₄	—	4	1	no	—
135	C₄⋊C₄	—	4	1	no	—
136	C₄⋊C₄	C₂×D₄	1	2	yes	—
137	C₂²⋊C₄	—	2	2	no	—
138	C₂²⋊C₄	—	2	2	no	—
139	C₂²⋊C₄	—	4	1	no	—
140	C₄²	—	4	1	no	—
141	C₄²	C₂×D₄	1	1	yes	characteristic
142	C₄²	C₂×D₄	1	1	yes	characteristic
143	C₄²	#41	1	1	yes	characteristic
144	C₄²	C₂⁴	1	1	yes	characteristic
145	C₂³	—	2	2	no	—
146	C₂³	#121	1	1	yes	characteristic
147	Q₈	—	4	2	no	—
148	Q₈	—	8	1	no	—
149	Q₈	—	8	1	no	—
150	Q₈	—	8	1	no	—
151	Q₈	—	8	1	no	—
152	Q₈	—	8	2	no	—
153	D₄	—	2	4	no	—
154	D₄	—	8	1	no	—
155	D₄	—	8	1	no	—
156	D₄	—	8	2	no	—
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	4	1	no	—
161	C₂×C₄	—	4	2	no	—
162	C₂×C₄	—	4	2	no	—
163	C₂×C₄	—	4	1	no	—
164	C₂×C₄	—	4	1	no	—
165	C₂×C₄	—	4	1	no	—
166	C₂×C₄	—	4	1	no	—
167	C₂×C₄	—	8	2	no	—
168	C₂×C₄	#121	1	1	yes	characteristic
169	C₂×C₄	#121	1	1	yes	characteristic
170	C₂×C₄	#121	1	1	yes	characteristic
171	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
172	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
173	C₂×C₄	C₂×#41	1	1	yes	characteristic
174	C₈	—	2	2	no	—
175	C₈	—	2	2	no	—
176	C₈	—	4	2	no	—
177	C₈	—	4	2	no	—
178	C₈	—	4	2	no	—
179	C₂²	—	2	1	no	—
180	C₂²	—	2	1	no	—
181	C₂²	—	2	1	no	—
182	C₂²	—	4	2	no	—
183	C₂²	—	4	2	no	—
184	C₂²	—	4	2	no	—
185	C₂²	C₂×#121	1	1	yes	characteristic
186	C₄	—	2	1	no	—
187	C₄	—	2	1	no	—
188	C₄	—	2	1	no	—
189	C₄	—	2	1	no	—
190	C₄	—	4	2	no	—
191	C₄	—	8	1	no	—
192	C₄	—	8	1	no	—
193	C₄	—	8	1	no	—
194	C₄	—	8	1	no	—
195	C₄	#577	1	2	yes	—
196	C₄	#578	1	2	yes	—
197	C₂	—	4	1	no	—
198	C₂	—	8	2	no	—
199	C₂	#3042	1	1	yes	characteristic
200	C₂	#3059	1	1	yes	characteristic
201	C₂	#3068	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
2. #3230	C₂	semidirect
5. #1703	C₂	semidirect
6. #1695	C₂	semidirect
7. #1683	C₂	semidirect
8. #1675	C₂	semidirect
9. #1572	C₂	semidirect
10. #1558	C₂	semidirect
11. #1554	C₂	semidirect

Maximal subgroups

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

subgroup	index
#3235	2
#3230	2
#3227	2
#2981	2
#1703	2
#1695	2
#1683	2
#1675	2
#1572	2
#1558	2
#1554	2