Finite Group Databasebeta

Subgroups — #33634

Order 256.

267 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 267 conjugacy classes of subgroups — too many to draw. Only its 119 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	#3164	C₂	1	1	yes	maximal, characteristic, complemented
2	#2951	C₂	1	1	yes	maximal, characteristic
3	#2229	C₂	1	2	yes	maximal, complemented
4	#2228	C₂	1	2	yes	maximal, complemented
5	#2226	C₂	1	2	yes	maximal, complemented
6	#2225	C₂	1	2	yes	maximal, complemented
7	#2159	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#367	C₂	1	1	yes	maximal, characteristic, complemented
9	#2151	C₂	1	1	yes	maximal, characteristic, complemented
10	#2139	C₂	1	1	yes	maximal, characteristic
11	#2138	C₂	1	1	yes	maximal, characteristic
12	C₂×#135	C₂²	1	1	yes	characteristic
13	#544	—	2	1	no	—
14	#520	—	2	1	no	—
15	C₂×M₃₂	C₂²	1	1	yes	characteristic
16	C₂²×C₁₆	C₂²	1	1	yes	characteristic
17	#501	—	2	2	no	—
18	#500	C₂²	1	2	yes	—
19	#499	C₂²	1	2	yes	—
20	#440	—	2	2	no	—
21	C₄×Q₁₆	C₂²	1	2	yes	—
22	C₄×D₈	C₂²	1	2	yes	—
23	#428	—	2	1	no	—
24	#427	C₂²	1	1	yes	characteristic
25	C₂×#107	C₂²	1	1	yes	characteristic
26	C₂×#107	C₂²	1	1	yes	characteristic
27	#417	—	2	1	no	—
28	#405	C₂²	1	1	yes	characteristic
29	#367	C₂²	1	2	yes	—
30	#367	C₂²	1	2	yes	—
31	#367	C₂²	1	2	yes	—
32	#366	C₂²	1	2	yes	—
33	C₄⋊C₁₆	C₂²	1	2	yes	—
34	C₄⋊C₁₆	C₂²	1	2	yes	—
35	#358	C₂²	1	1	yes	characteristic
36	#358	C₂²	1	1	yes	characteristic
37	#358	C₂²	1	2	yes	—
38	#357	C₂²	1	1	yes	characteristic
39	#357	C₂²	1	1	yes	characteristic
40	#357	C₂²	1	2	yes	—
41	C₂×#41	—	2	1	no	—
42	#135	—	2	2	no	—
43	#135	Q₈	1	4	yes	complemented
44	C₂×Q₁₆	C₂³	1	1	yes	characteristic
45	C₂×SD₁₆	—	2	1	no	—
46	C₂×D₈	C₂³	1	1	yes	characteristic
47	C₂×M₁₆	D₄	1	1	yes	characteristic
48	C₂²×C₈	C₂³	1	1	yes	characteristic
49	C₄⋊Q₈	—	2	2	no	—
50	C₄⋊Q₈	—	4	1	no	—
51	#125	—	2	2	no	—
52	#125	—	4	1	no	—
53	C₂²⋊Q₈	—	4	2	no	—
54	C₄×Q₈	—	2	2	no	—
55	C₄×D₄	—	2	2	no	—
56	C₄×D₄	—	4	2	no	—
57	#117	—	2	1	no	—
58	#117	—	4	1	no	—
59	#117	D₄	1	1	yes	characteristic
60	C₂×C₄⋊C₄	—	2	1	no	—
61	C₂×C₄⋊C₄	—	2	1	no	—
62	C₂×C₄⋊C₄	—	4	1	no	—
63	M₃₂	—	2	2	no	—
64	C₂×C₁₆	—	2	2	no	—
65	C₂×C₁₆	C₂³	1	2	yes	—
66	C₂×C₁₆	C₂³	1	1	yes	characteristic
67	C₂×C₁₆	C₂³	1	1	yes	characteristic
68	#107	—	2	1	no	—
69	#107	—	2	1	no	—
70	#107	—	2	2	no	—
71	#107	C₂³	1	1	yes	characteristic
72	#107	C₂³	1	2	yes	—
73	#107	C₂³	1	1	yes	characteristic
74	#107	C₂³	1	2	yes	—
75	#106	—	2	1	no	—
76	#106	—	2	2	no	—
77	#103	—	2	2	no	—
78	#102	—	2	2	no	—
79	#97	D₄	1	2	yes	—
80	C₄×C₈	C₂³	1	2	yes	—
81	#41	—	2	4	no	—
82	#41	—	4	2	no	—
83	C₂×Q₈	—	2	1	no	—
84	C₂×Q₈	—	4	2	no	—
85	C₂×D₄	—	2	1	no	—
86	C₂×D₄	—	4	1	no	—
87	C₂²×C₄	—	4	1	no	—
88	C₂²×C₄	—	4	1	no	—
89	C₂²×C₄	—	4	1	no	—
90	C₂²×C₄	—	4	2	no	—
91	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
92	Q₁₆	—	2	1	no	—
93	Q₁₆	C₂×Q₈	1	2	yes	—
94	SD₁₆	—	2	4	no	—
95	D₈	—	2	1	no	—
96	D₈	C₂×Q₈	1	2	yes	—
97	M₁₆	—	2	2	no	—
98	C₂×C₈	C₂×D₄	1	2	yes	—
99	C₂×C₈	C₂×D₄	1	1	yes	characteristic
100	C₂×C₈	C₂×Q₈	1	2	yes	—
101	C₂×C₈	#41	1	2	yes	—
102	C₂×C₈	C₂⁴	1	1	yes	characteristic
103	C₄⋊C₄	—	2	2	no	—
104	C₄⋊C₄	—	2	1	no	—
105	C₄⋊C₄	—	2	2	no	—
106	C₄⋊C₄	—	2	1	no	—
107	C₄⋊C₄	—	4	1	no	—
108	C₄⋊C₄	—	4	1	no	—
109	C₄⋊C₄	—	4	2	no	—
110	C₄⋊C₄	—	4	2	no	—
111	C₄⋊C₄	—	4	2	no	—
112	C₄⋊C₄	—	4	2	no	—
113	C₄⋊C₄	D₈	1	1	yes	characteristic
114	C₄⋊C₄	D₈	1	1	yes	characteristic
115	C₂²⋊C₄	—	4	1	no	—
116	C₂²⋊C₄	—	4	2	no	—
117	C₂²⋊C₄	D₈	1	2	yes	—
118	C₄²	—	4	1	no	—
119	C₄²	—	4	2	no	—
120	C₄²	C₂×D₄	1	2	yes	—
121	C₁₆	—	2	1	no	—
122	C₁₆	—	2	1	no	—
123	C₁₆	—	2	2	no	—
124	C₂³	—	4	1	no	—
125	C₂³	C₂×D₈	1	1	yes	characteristic
126	Q₈	—	2	2	no	—
127	Q₈	—	4	1	no	—
128	Q₈	—	16	2	no	—
129	D₄	—	2	2	no	—
130	D₄	—	4	4	no	—
131	D₄	—	4	1	no	—
132	C₂×C₄	—	4	1	no	—
133	C₂×C₄	—	4	4	no	—
134	C₂×C₄	—	4	1	no	—
135	C₂×C₄	—	4	2	no	—
136	C₂×C₄	—	4	1	no	—
137	C₂×C₄	—	4	2	no	—
138	C₂×C₄	—	16	2	no	—
139	C₂×C₄	—	16	1	no	—
140	C₂×C₄	—	16	1	no	—
141	C₂×C₄	C₂²⋊Q₈	1	2	yes	—
142	C₂×C₄	C₂×D₈	1	2	yes	—
143	C₂×C₄	C₂×D₈	1	2	yes	—
144	C₂×C₄	C₂×D₈	1	1	yes	characteristic
145	C₂×C₄	#137	1	2	yes	—
146	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
147	C₈	—	2	2	no	—
148	C₈	C₂²⋊Q₈	1	2	yes	—
149	C₈	C₂²×Q₈	1	1	yes	characteristic
150	C₈	C₂×#41	1	1	yes	characteristic
151	C₂²	—	2	1	no	—
152	C₂²	—	2	1	no	—
153	C₂²	—	4	2	no	—
154	C₂²	—	8	1	no	—
155	C₂²	—	8	1	no	—
156	C₂²	#474	1	2	yes	—
157	C₂²	C₂²×D₈	1	1	yes	characteristic
158	C₄	—	2	2	no	—
159	C₄	—	2	2	no	—
160	C₄	—	4	2	no	—
161	C₄	—	8	1	no	—
162	C₄	—	8	2	no	—
163	C₄	—	8	2	no	—
164	C₄	—	8	1	no	—
165	C₄	#474	1	2	yes	—
166	C₄	C₂×C₂²⋊Q₈	1	1	yes	characteristic
167	C₄	C₂×#137	1	1	yes	characteristic
168	C₂	—	2	2	no	—
169	C₂	—	8	2	no	—
170	C₂	C₂×#474	1	1	yes	characteristic
171	C₂	#3416	1	1	yes	characteristic
172	C₂	#3416	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴
prime radical	C₂×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. #3164	C₂	semidirect
3. #2229	C₂	semidirect
4. #2228	C₂	semidirect
5. #2226	C₂	semidirect
6. #2225	C₂	semidirect
7. #2159	C₂	semidirect
8. C₂×#367	C₂	semidirect
9. #2151	C₂	semidirect
43. #135	Q₈	semidirect

Maximal subgroups

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

subgroup	index
#3164	2
#2951	2
#2229	2
#2228	2
#2226	2
#2225	2
#2159	2
C₂×#367	2
#2151	2
#2139	2
#2138	2