Finite Group Databasebeta

Subgroups — #10116

Order 256.

246 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 53 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	#3254	C₂	1	1	yes	maximal, characteristic
2	#3227	C₂	1	1	yes	maximal, characteristic
3	#3165	C₂	1	1	yes	maximal, characteristic, complemented
4	#1966	C₂	1	1	yes	maximal, characteristic, complemented
5	#1525	C₂	1	1	yes	maximal, characteristic
6	#1485	C₂	1	1	yes	maximal, characteristic, complemented
7	#1299	C₂	1	1	yes	maximal, characteristic, complemented
8	#533	C₂²	1	1	yes	characteristic
9	#532	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊Q₈	C₂²	1	1	yes	characteristic
11	#501	—	2	2	no	—
12	#500	—	2	1	no	—
13	#498	—	2	1	no	—
14	#498	—	2	1	no	—
15	#486	—	2	1	no	—
16	#481	—	2	2	no	—
17	#477	—	2	1	no	—
18	#476	—	2	2	no	—
19	#474	—	2	1	no	—
20	#468	—	2	2	no	—
21	#461	—	2	1	no	—
22	#460	—	2	1	no	—
23	#431	C₂²	1	1	yes	characteristic
24	#428	—	2	2	no	—
25	#423	C₂²	1	1	yes	characteristic
26	C₂×#101	—	2	2	no	—
27	C₄×M₁₆	C₂²	1	1	yes	characteristic
28	C₄×M₁₆	C₂²	1	1	yes	characteristic
29	#329	—	2	2	no	—
30	Q₈⋊C₈	—	2	2	no	—
31	C₂×#41	—	2	1	no	—
32	C₂²×Q₈	—	2	1	no	—
33	C₂×SD₁₆	—	2	2	no	—
34	C₂×M₁₆	—	2	1	no	—
35	C₂×M₁₆	—	2	1	no	—
36	C₂×M₁₆	D₄	1	2	yes	—
37	C₄⋊Q₈	—	2	1	no	—
38	C₄⋊Q₈	—	2	1	no	—
39	C₄⋊Q₈	—	2	1	no	—
40	C₄⋊Q₈	—	2	1	no	—
41	C₄⋊Q₈	D₄	1	1	yes	characteristic
42	C₄⋊Q₈	D₄	1	1	yes	characteristic
43	C₄⋊Q₈	D₄	1	1	yes	characteristic
44	#127	D₄	1	1	yes	characteristic
45	#125	—	2	1	no	—
46	#124	—	2	1	no	—
47	C₂²⋊Q₈	—	2	1	no	—
48	C₂²⋊Q₈	—	2	1	no	—
49	#121	—	2	1	no	—
50	#121	—	2	1	no	—
51	C₄×Q₈	—	2	1	no	—
52	C₄×Q₈	—	2	1	no	—
53	C₄×D₄	—	2	1	no	—
54	C₄×D₄	—	2	1	no	—
55	#117	—	2	1	no	—
56	C₂×C₄⋊C₄	—	2	1	no	—
57	C₂×C₄⋊C₄	D₄	1	2	yes	—
58	C₂×C₄²	C₂³	1	1	yes	characteristic
59	#107	—	2	4	no	—
60	#107	—	2	2	no	—
61	#106	—	2	4	no	—
62	#106	—	2	4	no	—
63	C₄⋊C₈	—	2	1	no	—
64	C₄⋊C₈	—	2	1	no	—
65	C₄⋊C₈	D₄	1	2	yes	—
66	#103	—	2	2	no	—
67	#102	—	2	2	no	—
68	#102	—	2	4	no	—
69	#101	—	4	4	no	—
70	C₂²⋊C₈	—	2	1	no	—
71	#97	—	2	1	no	—
72	#97	D₄	1	2	yes	—
73	C₄×C₈	—	2	1	no	—
74	C₄×C₈	—	2	1	no	—
75	C₄×C₈	—	2	1	no	—
76	#41	—	8	2	no	—
77	C₂×Q₈	—	2	1	no	—
78	C₂×Q₈	—	2	1	no	—
79	C₂×Q₈	—	2	1	no	—
80	C₂×Q₈	—	4	2	no	—
81	C₂×Q₈	—	4	2	no	—
82	C₂×Q₈	—	4	1	no	—
83	C₂×Q₈	—	4	1	no	—
84	C₂×D₄	—	2	1	no	—
85	C₂×D₄	—	2	1	no	—
86	C₂×D₄	—	2	1	no	—
87	C₂×D₄	—	4	1	no	—
88	C₂²×C₄	—	2	2	no	—
89	C₂²×C₄	—	4	1	no	—
90	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
91	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
92	C₂²×C₄	#41	1	1	yes	characteristic
93	SD₁₆	—	8	2	no	—
94	M₁₆	—	2	4	no	—
95	M₁₆	—	2	2	no	—
96	M₁₆	—	4	1	no	—
97	M₁₆	—	4	2	no	—
98	C₂×C₈	—	2	2	no	—
99	C₂×C₈	—	2	2	no	—
100	C₂×C₈	—	2	2	no	—
101	C₂×C₈	—	4	1	no	—
102	C₄⋊C₄	—	2	2	no	—
103	C₄⋊C₄	—	2	1	no	—
104	C₄⋊C₄	—	2	1	no	—
105	C₄⋊C₄	—	2	1	no	—
106	C₄⋊C₄	—	2	1	no	—
107	C₄⋊C₄	—	2	2	no	—
108	C₄⋊C₄	—	2	1	no	—
109	C₄⋊C₄	—	2	1	no	—
110	C₄⋊C₄	—	4	1	no	—
111	C₄⋊C₄	—	4	1	no	—
112	C₄⋊C₄	—	4	1	no	—
113	C₄⋊C₄	—	4	1	no	—
114	C₂²⋊C₄	—	4	1	no	—
115	C₂²⋊C₄	—	4	1	no	—
116	C₂²⋊C₄	—	4	1	no	—
117	C₄²	—	4	1	no	—
118	C₄²	C₂×D₄	1	1	yes	characteristic
119	C₄²	C₂×D₄	1	1	yes	characteristic
120	C₄²	C₂×D₄	1	1	yes	characteristic
121	C₄²	C₂×D₄	1	1	yes	characteristic
122	C₂³	—	4	1	no	—
123	C₂³	#121	1	1	yes	characteristic
124	Q₈	—	4	1	no	—
125	Q₈	—	4	1	no	—
126	Q₈	—	4	2	no	—
127	Q₈	—	8	1	no	—
128	Q₈	—	8	2	no	—
129	Q₈	—	8	1	no	—
130	Q₈	—	8	1	no	—
131	D₄	—	4	1	no	—
132	D₄	—	4	1	no	—
133	D₄	—	4	2	no	—
134	D₄	—	8	1	no	—
135	D₄	—	8	2	no	—
136	C₂×C₄	—	2	1	no	—
137	C₂×C₄	—	2	1	no	—
138	C₂×C₄	—	2	2	no	—
139	C₂×C₄	—	2	2	no	—
140	C₂×C₄	—	4	4	no	—
141	C₂×C₄	—	4	1	no	—
142	C₂×C₄	—	4	1	no	—
143	C₂×C₄	—	4	1	no	—
144	C₂×C₄	—	4	2	no	—
145	C₂×C₄	—	8	2	no	—
146	C₂×C₄	#120	1	1	yes	characteristic
147	C₂×C₄	#120	1	1	yes	characteristic
148	C₂×C₄	#120	1	1	yes	characteristic
149	C₂×C₄	#121	1	1	yes	characteristic
150	C₂×C₄	#121	1	1	yes	characteristic
151	C₂×C₄	#127	1	1	yes	characteristic
152	C₈	—	2	4	no	—
153	C₈	—	4	2	no	—
154	C₈	—	4	2	no	—
155	C₈	—	4	2	no	—
156	C₂²	—	2	1	no	—
157	C₂²	—	2	1	no	—
158	C₂²	—	2	1	no	—
159	C₂²	—	8	1	no	—
160	C₂²	—	8	1	no	—
161	C₂²	—	8	1	no	—
162	C₂²	#392	1	1	yes	characteristic
163	C₄	—	2	1	no	—
164	C₄	—	2	1	no	—
165	C₄	—	2	1	no	—
166	C₄	—	8	2	no	—
167	C₄	—	8	1	no	—
168	C₄	—	8	1	no	—
169	C₄	—	8	1	no	—
170	C₄	#453	1	1	yes	characteristic
171	C₄	#454	1	1	yes	characteristic
172	C₄	#456	1	2	yes	—
173	C₄	#471	1	2	yes	—
174	C₂	—	4	1	no	—
175	C₂	—	16	1	no	—
176	C₂	#2005	1	1	yes	characteristic
177	C₂	#2006	1	1	yes	characteristic
178	C₂	#2019	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	#392
commutator, Frattini, cosocle	C₂×C₄²	C₂³
prime radical	#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ 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. #3165	C₂	semidirect
4. #1966	C₂	semidirect
6. #1485	C₂	semidirect
7. #1299	C₂	semidirect

Maximal subgroups

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

subgroup	index
#3254	2
#3227	2
#3165	2
#1966	2
#1525	2
#1485	2
#1299	2