Finite Group Databasebeta

Subgroups — #19512

Order 256.

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

normal subgroups only

This group has 294 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	#3229	C₂	1	1	yes	maximal, characteristic, complemented
3	#3206	C₂	1	1	yes	maximal, characteristic
4	#2981	C₂	1	1	yes	maximal, characteristic
5	#1699	C₂	1	2	yes	maximal, complemented
6	#1691	C₂	1	2	yes	maximal, complemented
7	#1679	C₂	1	2	yes	maximal, complemented
8	#1671	C₂	1	2	yes	maximal, complemented
9	#1572	C₂	1	1	yes	maximal, characteristic, complemented
10	#1556	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	#498	C₂²	1	1	yes	characteristic
17	#498	C₂²	1	1	yes	characteristic
18	#492	—	2	1	no	—
19	#490	—	2	1	no	—
20	#487	—	2	1	no	—
21	#487	C₂²	1	1	yes	characteristic
22	#487	C₂²	1	1	yes	characteristic
23	#483	—	2	2	no	—
24	#482	—	2	2	no	—
25	#476	C₂²	1	2	yes	—
26	#475	—	2	2	no	—
27	#474	C₂²	1	2	yes	—
28	#461	C₂²	1	2	yes	—
29	#461	C₂²	1	2	yes	—
30	#448	—	2	2	no	—
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	#332	C₂²	1	2	yes	—
43	#332	C₂²	1	2	yes	—
44	#332	C₂²	1	2	yes	—
45	C₂×#41	—	2	1	no	—
46	C₂²×Q₈	—	2	1	no	—
47	C₂×SD₁₆	—	2	2	no	—
48	C₂×SD₁₆	—	2	2	no	—
49	C₂×M₁₆	—	2	2	no	—
50	C₂²×C₈	—	2	2	no	—
51	C₄⋊Q₈	—	2	1	no	—
52	C₄⋊Q₈	—	2	1	no	—
53	C₄⋊Q₈	—	2	1	no	—
54	C₄⋊Q₈	D₄	1	1	yes	characteristic
55	C₄⋊Q₈	C₂³	1	1	yes	characteristic
56	C₄⋊Q₈	C₂³	1	1	yes	characteristic
57	C₄⋊Q₈	C₂³	1	1	yes	characteristic
58	C₄⋊Q₈	C₂³	1	1	yes	characteristic
59	#127	D₄	1	1	yes	characteristic
60	#125	—	2	1	no	—
61	#124	—	2	1	no	—
62	C₂²⋊Q₈	—	2	1	no	—
63	C₂²⋊Q₈	—	2	1	no	—
64	#121	—	2	1	no	—
65	#121	D₄	1	2	yes	—
66	C₄×Q₈	—	2	1	no	—
67	C₄×Q₈	—	2	1	no	—
68	C₄×D₄	—	2	1	no	—
69	C₄×D₄	C₂³	1	2	yes	—
70	#117	—	2	1	no	—
71	C₂×C₄⋊C₄	—	2	1	no	—
72	C₂×C₄⋊C₄	—	2	1	no	—
73	C₂×C₄²	C₂³	1	1	yes	characteristic
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	#103	—	2	2	no	—
85	#103	—	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	SD₁₆	—	8	2	no	—
117	SD₁₆	—	8	2	no	—
118	M₁₆	—	8	2	no	—
119	C₂×C₈	—	2	2	no	—
120	C₂×C₈	—	2	2	no	—
121	C₂×C₈	—	2	2	no	—
122	C₂×C₈	—	8	2	no	—
123	C₂×C₈	C₂×D₄	1	2	yes	—
124	C₂×C₈	C₂×D₄	1	2	yes	—
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	—	2	1	no	—
133	C₄⋊C₄	—	2	1	no	—
134	C₄⋊C₄	—	4	1	no	—
135	C₄⋊C₄	—	4	1	no	—
136	C₄⋊C₄	—	4	1	no	—
137	C₄⋊C₄	—	4	1	no	—
138	C₄⋊C₄	C₂×D₄	1	2	yes	—
139	C₂²⋊C₄	—	2	2	no	—
140	C₂²⋊C₄	—	2	2	no	—
141	C₂²⋊C₄	—	4	1	no	—
142	C₄²	—	4	1	no	—
143	C₄²	C₂×D₄	1	1	yes	characteristic
144	C₄²	C₂×D₄	1	1	yes	characteristic
145	C₄²	#41	1	1	yes	characteristic
146	C₄²	C₂⁴	1	1	yes	characteristic
147	C₂³	—	2	2	no	—
148	C₂³	#121	1	1	yes	characteristic
149	Q₈	—	4	2	no	—
150	Q₈	—	4	2	no	—
151	Q₈	—	4	2	no	—
152	Q₈	—	8	1	no	—
153	Q₈	—	8	1	no	—
154	Q₈	—	8	2	no	—
155	D₄	—	2	2	no	—
156	D₄	—	2	2	no	—
157	D₄	—	8	1	no	—
158	D₄	—	8	1	no	—
159	D₄	—	8	2	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	2	2	no	—
162	C₂×C₄	—	2	2	no	—
163	C₂×C₄	—	4	1	no	—
164	C₂×C₄	—	4	2	no	—
165	C₂×C₄	—	4	2	no	—
166	C₂×C₄	—	4	1	no	—
167	C₂×C₄	—	4	1	no	—
168	C₂×C₄	—	4	1	no	—
169	C₂×C₄	—	4	1	no	—
170	C₂×C₄	—	8	2	no	—
171	C₂×C₄	#121	1	1	yes	characteristic
172	C₂×C₄	#121	1	1	yes	characteristic
173	C₂×C₄	#121	1	1	yes	characteristic
174	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
175	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
176	C₂×C₄	C₂×#41	1	1	yes	characteristic
177	C₈	—	2	2	no	—
178	C₈	—	2	2	no	—
179	C₈	—	4	2	no	—
180	C₈	—	4	2	no	—
181	C₈	—	4	2	no	—
182	C₂²	—	2	1	no	—
183	C₂²	—	2	1	no	—
184	C₂²	—	2	1	no	—
185	C₂²	—	4	2	no	—
186	C₂²	—	4	2	no	—
187	C₂²	—	4	2	no	—
188	C₂²	C₂×#121	1	1	yes	characteristic
189	C₄	—	2	1	no	—
190	C₄	—	2	1	no	—
191	C₄	—	2	1	no	—
192	C₄	—	2	1	no	—
193	C₄	—	4	2	no	—
194	C₄	—	8	1	no	—
195	C₄	—	8	1	no	—
196	C₄	—	8	1	no	—
197	C₄	—	8	1	no	—
198	C₄	#577	1	2	yes	—
199	C₄	#577	1	2	yes	—
200	C₂	—	4	1	no	—
201	C₂	—	8	2	no	—
202	C₂	#3042	1	1	yes	characteristic
203	C₂	#3057	1	1	yes	characteristic
204	C₂	#3069	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. #3229	C₂	semidirect
5. #1699	C₂	semidirect
6. #1691	C₂	semidirect
7. #1679	C₂	semidirect
8. #1671	C₂	semidirect
9. #1572	C₂	semidirect
10. #1556	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
#3229	2
#3206	2
#2981	2
#1699	2
#1691	2
#1679	2
#1671	2
#1572	2
#1556	2
#1554	2