Finite Group Databasebeta

Subgroups — #8905

Order 256.

196 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 196 conjugacy classes of subgroups — too many to draw. Only its 69 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	#2925	C₂	1	1	yes	maximal, characteristic
2	#1806	C₂	1	1	yes	maximal, characteristic
3	#1571	C₂	1	1	yes	maximal, characteristic, complemented
4	#1566	C₂	1	1	yes	maximal, characteristic, complemented
5	#1476	C₂	1	1	yes	maximal, characteristic
6	#1303	C₂	1	1	yes	maximal, characteristic, complemented
7	#1279	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×#41	C₂²	1	1	yes	characteristic
9	#433	—	2	2	no	—
10	#432	—	2	1	no	—
11	#431	—	2	1	no	—
12	#431	C₂²	1	1	yes	characteristic
13	#423	C₂²	1	1	yes	characteristic
14	#423	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
16	C₂×#104	—	2	1	no	—
17	#408	—	2	1	no	—
18	#405	—	2	1	no	—
19	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
20	C₄×C₄⋊C₄	C₂²	1	1	yes	characteristic
21	#339	—	2	1	no	—
22	#335	—	2	1	no	—
23	#335	—	2	1	no	—
24	#334	—	2	2	no	—
25	Q₈⋊C₈	C₄	1	1	yes	characteristic, complemented
26	Q₈⋊C₈	C₄	1	1	yes	characteristic, complemented
27	D₄⋊C₈	C₄	1	1	yes	characteristic, complemented
28	D₄⋊C₈	C₄	1	1	yes	characteristic, complemented
29	C₂×#41	D₄	1	1	yes	characteristic
30	C₂×M₁₆	—	2	1	no	—
31	C₂×M₁₆	—	2	1	no	—
32	C₂²×C₈	—	2	1	no	—
33	C₂²×C₈	—	2	1	no	—
34	C₄×Q₈	C₂×C₄	1	1	yes	characteristic
35	C₄×Q₈	Q₈	1	1	yes	characteristic
36	C₄×D₄	—	2	1	no	—
37	C₄×D₄	—	2	1	no	—
38	C₄×D₄	C₂×C₄	1	1	yes	characteristic
39	C₄×D₄	Q₈	1	1	yes	characteristic
40	#117	—	2	1	no	—
41	#117	D₄	1	1	yes	characteristic
42	C₂×C₄⋊C₄	—	2	1	no	—
43	C₂×C₄²	—	2	1	no	—
44	C₂×C₄²	—	2	1	no	—
45	C₂×C₄²	C₂³	1	1	yes	characteristic
46	C₄⋊C₈	—	2	1	no	—
47	C₄⋊C₈	—	2	1	no	—
48	C₄⋊C₈	—	2	1	no	—
49	C₄⋊C₈	—	2	1	no	—
50	C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
51	C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
52	C₄⋊C₈	D₄	1	2	yes	—
53	C₄⋊C₈	D₄	1	1	yes	characteristic
54	C₄⋊C₈	D₄	1	1	yes	characteristic
55	#104	—	4	2	no	—
56	C₂²⋊C₈	—	2	2	no	—
57	C₂²⋊C₈	—	2	2	no	—
58	C₂²⋊C₈	—	2	1	no	—
59	#97	—	2	2	no	—
60	C₄×C₈	—	2	2	no	—
61	C₄×C₈	—	2	1	no	—
62	C₄×C₈	—	2	1	no	—
63	C₄×C₈	C₂×C₄	1	1	yes	characteristic
64	C₄×C₈	C₂×C₄	1	1	yes	characteristic
65	#95	—	2	1	no	—
66	#41	—	2	1	no	—
67	#41	—	2	1	no	—
68	#41	—	2	1	no	—
69	#41	—	2	1	no	—
70	C₂×Q₈	C₄⋊C₄	1	1	yes	characteristic
71	C₂×D₄	—	2	1	no	—
72	C₂×D₄	C₄⋊C₄	1	1	yes	characteristic
73	C₂²×C₄	—	2	1	no	—
74	C₂²×C₄	—	2	1	no	—
75	C₂²×C₄	—	2	1	no	—
76	C₂²×C₄	—	4	1	no	—
77	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄	#41	1	1	yes	characteristic
79	C₂²×C₄	#41	1	1	yes	characteristic
80	M₁₆	—	4	1	no	—
81	M₁₆	—	4	1	no	—
82	M₁₆	—	4	1	no	—
83	M₁₆	—	4	1	no	—
84	C₂×C₈	—	2	1	no	—
85	C₂×C₈	—	2	1	no	—
86	C₂×C₈	—	2	1	no	—
87	C₂×C₈	—	2	1	no	—
88	C₂×C₈	—	2	1	no	—
89	C₂×C₈	—	2	2	no	—
90	C₂×C₈	—	2	1	no	—
91	C₂×C₈	—	4	2	no	—
92	C₂×C₈	—	4	1	no	—
93	C₂×C₈	—	4	1	no	—
94	C₂×C₈	—	4	1	no	—
95	C₄⋊C₄	—	2	1	no	—
96	C₄⋊C₄	—	2	1	no	—
97	C₄⋊C₄	—	2	2	no	—
98	C₄⋊C₄	—	4	1	no	—
99	C₄⋊C₄	C₄⋊C₄	1	1	yes	characteristic
100	C₄⋊C₄	C₄⋊C₄	1	1	yes	characteristic
101	C₂²⋊C₄	—	2	1	no	—
102	C₂²⋊C₄	—	2	1	no	—
103	C₂²⋊C₄	—	2	1	no	—
104	C₄²	—	2	1	no	—
105	C₄²	—	2	1	no	—
106	C₄²	—	2	1	no	—
107	C₄²	—	4	1	no	—
108	C₄²	—	4	1	no	—
109	C₄²	C₂²×C₄	1	1	yes	characteristic
110	C₄²	C₂×D₄	1	1	yes	characteristic
111	C₄²	C₂×D₄	1	1	yes	characteristic
112	C₄²	C₂×Q₈	1	1	yes	characteristic
113	C₂³	—	2	1	no	—
114	C₂³	#123	1	1	yes	characteristic
115	Q₈	—	2	1	no	—
116	Q₈	—	2	1	no	—
117	D₄	—	2	1	no	—
118	D₄	—	2	1	no	—
119	D₄	—	4	1	no	—
120	D₄	—	4	1	no	—
121	C₂×C₄	—	2	1	no	—
122	C₂×C₄	—	2	1	no	—
123	C₂×C₄	—	2	1	no	—
124	C₂×C₄	—	2	1	no	—
125	C₂×C₄	—	2	1	no	—
126	C₂×C₄	—	4	1	no	—
127	C₂×C₄	—	4	1	no	—
128	C₂×C₄	—	4	1	no	—
129	C₂×C₄	—	4	1	no	—
130	C₂×C₄	—	4	1	no	—
131	C₂×C₄	—	4	2	no	—
132	C₂×C₄	—	8	1	no	—
133	C₂×C₄	—	8	1	no	—
134	C₂×C₄	—	8	1	no	—
135	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
136	C₂×C₄	C₄×D₄	1	1	yes	characteristic
137	C₂×C₄	C₄×D₄	1	1	yes	characteristic
138	C₂×C₄	#120	1	1	yes	characteristic
139	C₂×C₄	C₂²⋊Q₈	1	1	yes	characteristic
140	C₂×C₄	C₂²⋊Q₈	1	1	yes	characteristic
141	C₂×C₄	#135	1	1	yes	characteristic
142	C₂×C₄	#135	1	1	yes	characteristic
143	C₂×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
144	C₂×C₄	#137	1	1	yes	characteristic
145	C₈	—	2	2	no	—
146	C₈	—	4	1	no	—
147	C₈	—	4	1	no	—
148	C₈	—	4	1	no	—
149	C₈	—	4	1	no	—
150	C₈	—	4	1	no	—
151	C₈	—	4	2	no	—
152	C₈	—	8	1	no	—
153	C₂²	—	2	1	no	—
154	C₂²	—	2	1	no	—
155	C₂²	—	4	1	no	—
156	C₂²	—	4	1	no	—
157	C₂²	—	4	1	no	—
158	C₂²	#385	1	1	yes	characteristic
159	C₂²	#482	1	1	yes	characteristic
160	C₂²	#485	1	1	yes	characteristic
161	C₄	—	2	1	no	—
162	C₄	—	2	1	no	—
163	C₄	—	2	1	no	—
164	C₄	—	4	1	no	—
165	C₄	—	4	1	no	—
166	C₄	—	4	1	no	—
167	C₄	—	8	2	no	—
168	C₄	#427	1	1	yes	characteristic
169	C₄	#428	1	1	yes	characteristic
170	C₄	#443	1	1	yes	characteristic
171	C₄	#444	1	1	yes	characteristic
172	C₄	#449	1	1	yes	characteristic
173	C₄	#452	1	1	yes	characteristic
174	C₂	—	2	1	no	—
175	C₂	—	2	1	no	—
176	C₂	—	8	1	no	—
177	C₂	#1854	1	1	yes	characteristic
178	C₂	#1861	1	1	yes	characteristic
179	C₂	#1866	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#120
commutator	C₄²	C₂²×C₄
Frattini, cosocle	C₂×C₄²	C₂³
socle	C₂²	#385
prime radical	C₂×#41	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ 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. #1571	C₂	semidirect
4. #1566	C₂	semidirect
6. #1303	C₂	semidirect
7. #1279	C₂	semidirect
25. Q₈⋊C₈	C₄	semidirect
26. Q₈⋊C₈	C₄	semidirect
27. D₄⋊C₈	C₄	semidirect
28. D₄⋊C₈	C₄	semidirect

Maximal subgroups

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

subgroup	index
#2925	2
#1806	2
#1571	2
#1566	2
#1476	2
#1303	2
#1279	2