Finite Group Databasebeta

Subgroups — #9354

Order 256.

235 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 235 conjugacy classes of subgroups — too many to draw. Only its 65 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	#3039	C₂	1	1	yes	maximal, characteristic
2	#2991	C₂	1	1	yes	maximal, characteristic
3	#1997	C₂	1	1	yes	maximal, characteristic, complemented
4	#1548	C₂	1	1	yes	maximal, characteristic, complemented
5	#1489	C₂	1	1	yes	maximal, characteristic
6	#1309	C₂	1	1	yes	maximal, characteristic, complemented
7	#1299	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#137	C₄	1	2	yes	—
9	C₂×C₄⋊Q₈	C₂²	1	1	yes	characteristic
10	C₄×#41	C₂²	1	1	yes	characteristic
11	#462	—	2	1	no	—
12	#462	—	2	1	no	—
13	#461	—	2	1	no	—
14	#461	—	2	1	no	—
15	C₈×Q₈	—	2	1	no	—
16	#436	—	2	2	no	—
17	#436	—	2	1	no	—
18	#431	—	2	1	no	—
19	#431	C₂²	1	1	yes	characteristic
20	#423	—	2	1	no	—
21	#423	C₂²	1	1	yes	characteristic
22	#423	C₂²	1	1	yes	characteristic
23	#423	C₂²	1	1	yes	characteristic
24	#419	C₄	1	2	yes	—
25	C₄×M₁₆	C₂²	1	1	yes	characteristic
26	#343	—	2	2	no	—
27	#333	—	2	1	no	—
28	#333	—	2	1	no	—
29	Q₈⋊C₈	—	2	1	no	—
30	D₄⋊C₈	—	2	1	no	—
31	C₂×#41	C₂×C₄	1	1	yes	characteristic
32	C₂²×Q₈	C₂×C₄	1	2	yes	—
33	#137	—	4	4	no	—
34	C₂×Q₁₆	—	2	2	no	—
35	C₂×SD₁₆	—	2	2	no	—
36	C₂×M₁₆	—	2	2	no	—
37	C₂×M₁₆	—	2	1	no	—
38	C₂×M₁₆	—	2	1	no	—
39	C₂×M₁₆	C₂×C₄	1	2	yes	—
40	C₂×M₁₆	D₄	1	2	yes	—
41	C₄⋊Q₈	—	2	1	no	—
42	C₄⋊Q₈	—	2	1	no	—
43	C₄⋊Q₈	—	2	1	no	—
44	C₄⋊Q₈	—	2	1	no	—
45	C₄×Q₈	D₄	1	1	yes	characteristic
46	C₄×Q₈	D₄	1	1	yes	characteristic
47	C₄×D₄	—	2	1	no	—
48	C₄×D₄	—	2	1	no	—
49	C₄×D₄	D₄	1	1	yes	characteristic
50	C₄×D₄	D₄	1	1	yes	characteristic
51	#117	—	2	1	no	—
52	#117	C₂×C₄	1	1	yes	characteristic
53	C₂×C₄⋊C₄	—	2	1	no	—
54	C₂×C₄⋊C₄	—	2	1	no	—
55	C₂×C₄²	—	2	1	no	—
56	C₂×C₄²	C₂³	1	1	yes	characteristic
57	C₄⋊C₈	—	2	2	no	—
58	C₄⋊C₈	—	2	1	no	—
59	C₄⋊C₈	—	2	1	no	—
60	C₄⋊C₈	—	2	1	no	—
61	C₄⋊C₈	—	2	1	no	—
62	C₄⋊C₈	—	2	1	no	—
63	C₄⋊C₈	—	2	1	no	—
64	C₄⋊C₈	D₄	1	1	yes	characteristic
65	C₄⋊C₈	D₄	1	1	yes	characteristic
66	#103	—	2	2	no	—
67	#103	—	2	2	no	—
68	C₂²⋊C₈	—	2	2	no	—
69	C₂²⋊C₈	—	2	1	no	—
70	#97	—	2	1	no	—
71	C₄×C₈	—	2	2	no	—
72	C₄×C₈	—	2	1	no	—
73	C₄×C₈	—	2	1	no	—
74	#41	—	2	2	no	—
75	#41	—	4	1	no	—
76	C₂×Q₈	—	2	2	no	—
77	C₂×Q₈	—	2	2	no	—
78	C₂×Q₈	—	4	2	no	—
79	C₂×Q₈	—	4	2	no	—
80	C₂×Q₈	C₂²⋊C₄	1	1	yes	characteristic
81	C₂×D₄	—	2	1	no	—
82	C₂×D₄	C₂²⋊C₄	1	1	yes	characteristic
83	C₂²×C₄	—	2	1	no	—
84	C₂²×C₄	—	2	1	no	—
85	C₂²×C₄	—	2	1	no	—
86	C₂²×C₄	—	2	2	no	—
87	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
88	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
89	C₂²×C₄	#41	1	1	yes	characteristic
90	Q₁₆	—	4	4	no	—
91	SD₁₆	—	4	4	no	—
92	M₁₆	—	2	2	no	—
93	M₁₆	—	4	1	no	—
94	M₁₆	—	4	2	no	—
95	M₁₆	—	4	2	no	—
96	M₁₆	—	8	2	no	—
97	M₁₆	—	8	1	no	—
98	C₂×C₈	—	2	2	no	—
99	C₂×C₈	—	2	1	no	—
100	C₂×C₈	—	2	2	no	—
101	C₂×C₈	—	2	1	no	—
102	C₂×C₈	—	4	1	no	—
103	C₂×C₈	—	4	1	no	—
104	C₄⋊C₄	—	2	1	no	—
105	C₄⋊C₄	—	2	1	no	—
106	C₄⋊C₄	—	4	1	no	—
107	C₄⋊C₄	—	4	1	no	—
108	C₄⋊C₄	—	4	1	no	—
109	C₄⋊C₄	—	4	1	no	—
110	C₄⋊C₄	C₂²⋊C₄	1	1	yes	characteristic
111	C₄⋊C₄	C₂²⋊C₄	1	1	yes	characteristic
112	C₂²⋊C₄	—	2	1	no	—
113	C₂²⋊C₄	—	2	1	no	—
114	C₂²⋊C₄	—	2	1	no	—
115	C₄²	—	2	1	no	—
116	C₄²	—	2	1	no	—
117	C₄²	—	2	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	#41	1	1	yes	characteristic
122	C₂³	—	2	1	no	—
123	C₂³	C₄×D₄	1	1	yes	characteristic
124	Q₈	—	2	1	no	—
125	Q₈	—	4	2	no	—
126	Q₈	—	4	2	no	—
127	Q₈	—	8	2	no	—
128	Q₈	#100	1	2	yes	—
129	D₄	—	2	1	no	—
130	D₄	—	2	1	no	—
131	D₄	—	4	2	no	—
132	C₂×C₄	—	2	1	no	—
133	C₂×C₄	—	2	1	no	—
134	C₂×C₄	—	2	1	no	—
135	C₂×C₄	—	2	1	no	—
136	C₂×C₄	—	2	1	no	—
137	C₂×C₄	—	2	1	no	—
138	C₂×C₄	—	2	1	no	—
139	C₂×C₄	—	2	1	no	—
140	C₂×C₄	—	2	1	no	—
141	C₂×C₄	—	4	2	no	—
142	C₂×C₄	—	4	1	no	—
143	C₂×C₄	—	4	1	no	—
144	C₂×C₄	—	4	1	no	—
145	C₂×C₄	—	4	1	no	—
146	C₂×C₄	—	4	2	no	—
147	C₂×C₄	—	8	2	no	—
148	C₂×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
149	C₂×C₄	C₄×D₄	1	1	yes	characteristic
150	C₂×C₄	#120	1	1	yes	characteristic
151	C₂×C₄	#121	1	1	yes	characteristic
152	C₂×C₄	#121	1	1	yes	characteristic
153	C₂×C₄	#123	1	1	yes	characteristic
154	C₈	—	4	2	no	—
155	C₈	—	4	2	no	—
156	C₈	—	4	2	no	—
157	C₈	—	4	1	no	—
158	C₈	—	4	1	no	—
159	C₈	—	8	1	no	—
160	C₂²	—	2	1	no	—
161	C₂²	—	2	1	no	—
162	C₂²	—	2	1	no	—
163	C₂²	—	4	1	no	—
164	C₂²	—	4	1	no	—
165	C₂²	—	4	1	no	—
166	C₂²	#386	1	1	yes	characteristic
167	C₄	—	2	1	no	—
168	C₄	—	2	1	no	—
169	C₄	—	2	2	no	—
170	C₄	—	2	1	no	—
171	C₄	—	4	1	no	—
172	C₄	—	4	1	no	—
173	C₄	—	8	2	no	—
174	C₄	C₂×#100	1	1	yes	characteristic
175	C₄	#413	1	1	yes	characteristic
176	C₄	#455	1	1	yes	characteristic
177	C₄	#456	1	1	yes	characteristic
178	C₄	#471	1	1	yes	characteristic
179	C₄	#473	1	1	yes	characteristic
180	C₂	—	4	1	no	—
181	C₂	—	8	1	no	—
182	C₂	#1882	1	1	yes	characteristic
183	C₂	#1900	1	1	yes	characteristic
184	C₂	#1908	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	#386
commutator	C₂²×C₄	C₂²×C₄
Frattini, cosocle	C₂×C₄²	C₂³
prime radical	C₂×#41	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ 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. #1997	C₂	semidirect
4. #1548	C₂	semidirect
6. #1309	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
#3039	2
#2991	2
#1997	2
#1548	2
#1489	2
#1309	2
#1299	2