Finite Group Databasebeta

Subgroups — #19994

Order 256.

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

normal subgroups only

This group has 326 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, complemented
2	#3341	C₂	1	1	yes	maximal, characteristic, complemented
3	#3305	C₂	1	2	yes	maximal, complemented
4	#1706	C₂	1	1	yes	maximal, characteristic, complemented
5	#1677	C₂	1	2	yes	maximal, complemented
6	#1676	C₂	1	1	yes	maximal, characteristic, complemented
7	#1667	C₂	1	1	yes	maximal, characteristic, complemented
8	#1631	C₂	1	2	yes	maximal, complemented
9	#1624	C₂	1	2	yes	maximal, complemented
10	C₈×SD₁₆	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×D₈	C₂	1	1	yes	maximal, characteristic, complemented
12	#555	C₂²	1	2	yes	—
13	#550	C₂²	1	2	yes	—
14	#500	C₂²	1	1	yes	characteristic
15	#493	—	2	1	no	—
16	#493	C₂²	1	1	yes	characteristic
17	#492	C₂²	1	2	yes	—
18	#486	C₂²	1	2	yes	—
19	#482	—	2	2	no	—
20	#482	—	2	2	no	—
21	#477	C₂²	1	2	yes	—
22	#476	C₂²	1	2	yes	—
23	#466	—	2	1	no	—
24	#466	—	2	2	no	—
25	#459	—	2	2	no	—
26	#459	C₂²	1	2	yes	—
27	#459	C₂²	1	2	yes	—
28	#449	—	2	2	no	—
29	C₈×Q₈	C₂²	1	1	yes	characteristic
30	C₄×SD₁₆	C₂²	1	1	yes	characteristic
31	C₄×D₈	—	2	1	no	—
32	C₄×D₈	C₂²	1	1	yes	characteristic
33	C₈×D₄	C₂²	1	1	yes	characteristic
34	C₈×D₄	C₂²	1	2	yes	—
35	#427	—	2	1	no	—
36	#416	—	2	2	no	—
37	#335	C₂²	1	1	yes	characteristic
38	#334	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	D₄⋊C₈	C₂²	1	2	yes	—
43	D₄⋊C₈	C₂²	1	1	yes	characteristic
44	C₈²	C₂²	1	1	yes	characteristic
45	C₂×#41	—	2	2	no	—
46	C₂×SD₁₆	—	4	2	no	—
47	C₂×SD₁₆	D₄	1	1	yes	characteristic
48	C₂×D₈	—	2	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	C₂²×C₈	—	2	1	no	—
54	C₂²×C₈	—	2	2	no	—
55	C₄⋊Q₈	C₂³	1	2	yes	—
56	#127	—	2	2	no	—
57	#127	C₂³	1	2	yes	—
58	#124	—	4	2	no	—
59	#121	—	2	2	no	—
60	#121	—	2	2	no	—
61	#121	—	2	2	no	—
62	#121	—	4	2	no	—
63	C₄×Q₈	C₂³	1	1	yes	characteristic
64	C₄×D₄	—	2	2	no	—
65	C₄×D₄	C₂³	1	2	yes	—
66	C₄×D₄	C₂³	1	1	yes	characteristic
67	#117	—	2	2	no	—
68	#117	—	2	2	no	—
69	#107	—	2	1	no	—
70	#107	—	2	1	no	—
71	#107	—	2	2	no	—
72	#107	D₄	1	1	yes	characteristic
73	#106	—	2	2	no	—
74	#106	D₄	1	1	yes	characteristic
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	#102	—	2	2	no	—
82	#102	—	2	2	no	—
83	#102	—	2	2	no	—
84	#102	—	2	2	no	—
85	#102	—	2	2	no	—
86	#102	D₄	1	2	yes	—
87	#102	D₄	1	1	yes	characteristic
88	C₂²⋊C₈	—	2	2	no	—
89	C₂²⋊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×D₄	—	2	2	no	—
99	C₂×D₄	—	2	2	no	—
100	C₂×D₄	—	2	2	no	—
101	C₂×D₄	—	4	2	no	—
102	C₂×D₄	—	4	2	no	—
103	C₂×D₄	—	4	2	no	—
104	C₂×D₄	C₂×D₄	1	2	yes	—
105	C₂×D₄	C₂×D₄	1	1	yes	characteristic
106	C₂²×C₄	—	2	1	no	—
107	C₂²×C₄	—	2	2	no	—
108	C₂²×C₄	—	4	2	no	—
109	SD₁₆	—	2	2	no	—
110	SD₁₆	—	8	4	no	—
111	D₈	—	2	2	no	—
112	D₈	—	4	2	no	—
113	D₈	—	8	1	no	—
114	D₈	—	8	2	no	—
115	D₈	—	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	—	4	2	no	—
121	C₂×C₈	—	8	2	no	—
122	C₂×C₈	C₂×D₄	1	1	yes	characteristic
123	C₂×C₈	C₂×D₄	1	1	yes	characteristic
124	C₂×C₈	C₂×D₄	1	1	yes	characteristic
125	C₂×C₈	C₂×D₄	1	1	yes	characteristic
126	C₄⋊C₄	—	2	1	no	—
127	C₄⋊C₄	—	2	2	no	—
128	C₄⋊C₄	—	2	2	no	—
129	C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
130	C₄⋊C₄	C₂×D₄	1	2	yes	—
131	C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
132	C₂²⋊C₄	—	2	2	no	—
133	C₂²⋊C₄	—	2	1	no	—
134	C₂²⋊C₄	—	4	2	no	—
135	C₂²⋊C₄	—	4	2	no	—
136	C₂²⋊C₄	—	4	2	no	—
137	C₄²	—	2	1	no	—
138	C₄²	—	4	2	no	—
139	C₄²	C₂⁴	1	1	yes	characteristic
140	C₂³	—	2	1	no	—
141	C₂³	—	2	2	no	—
142	C₂³	—	4	2	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	1	no	—
148	D₄	—	2	1	no	—
149	D₄	—	2	2	no	—
150	D₄	—	4	2	no	—
151	D₄	—	4	2	no	—
152	D₄	—	4	2	no	—
153	D₄	—	4	2	no	—
154	D₄	—	8	4	no	—
155	D₄	—	8	2	no	—
156	D₄	—	16	2	no	—
157	D₄	—	16	2	no	—
158	C₂×C₄	—	2	1	no	—
159	C₂×C₄	—	2	1	no	—
160	C₂×C₄	—	2	1	no	—
161	C₂×C₄	—	2	2	no	—
162	C₂×C₄	—	4	1	no	—
163	C₂×C₄	—	4	2	no	—
164	C₂×C₄	—	4	2	no	—
165	C₂×C₄	—	4	1	no	—
166	C₂×C₄	—	4	2	no	—
167	C₂×C₄	—	8	4	no	—
168	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
169	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
170	C₂×C₄	H(5,𝔽₂)	1	1	yes	characteristic
171	C₈	—	2	2	no	—
172	C₈	—	2	1	no	—
173	C₈	—	2	1	no	—
174	C₈	—	2	1	no	—
175	C₈	—	4	1	no	—
176	C₈	—	4	2	no	—
177	C₈	—	4	1	no	—
178	C₈	#135	1	2	yes	—
179	C₂²	—	4	2	no	—
180	C₂²	—	4	1	no	—
181	C₂²	—	4	1	no	—
182	C₂²	—	4	2	no	—
183	C₂²	—	4	2	no	—
184	C₂²	—	4	1	no	—
185	C₂²	—	8	2	no	—
186	C₂²	—	8	2	no	—
187	C₂²	—	8	2	no	—
188	C₂²	D₄²	1	1	yes	characteristic
189	C₄	—	2	2	no	—
190	C₄	—	2	1	no	—
191	C₄	—	4	2	no	—
192	C₄	—	4	1	no	—
193	C₄	—	4	1	no	—
194	C₄	—	8	2	no	—
195	C₄	C₂×#135	1	1	yes	characteristic
196	C₄	C₂×#135	1	1	yes	characteristic
197	C₄	#576	1	1	yes	characteristic
198	C₄	#577	1	1	yes	characteristic
199	C₂	—	8	2	no	—
200	C₂	—	8	1	no	—
201	C₂	—	16	2	no	—
202	C₂	#3280	1	1	yes	characteristic
203	C₂	#3284	1	1	yes	characteristic
204	C₂	#3292	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	D₄²
commutator, Frattini, cosocle	C₄²	C₂⁴
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
1. #3391	C₂	semidirect
2. #3341	C₂	semidirect
3. #3305	C₂	semidirect
4. #1706	C₂	semidirect
5. #1677	C₂	semidirect
6. #1676	C₂	semidirect
7. #1667	C₂	semidirect
8. #1631	C₂	semidirect
9. #1624	C₂	semidirect
10. C₈×SD₁₆	C₂	semidirect
11. C₈×D₈	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
#3341	2
#3305	2
#1706	2
#1677	2
#1676	2
#1667	2
#1631	2
#1624	2
C₈×SD₁₆	2
C₈×D₈	2