Finite Group Databasebeta

Subgroups — #33632

Order 256.

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

normal subgroups only

This group has 260 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
2	#2950	C₂	1	1	yes	maximal, characteristic
3	#2229	C₂	1	2	yes	maximal, complemented
4	#2229	C₂	1	2	yes	maximal, complemented
5	#2227	C₂	1	2	yes	maximal, complemented
6	#2225	C₂	1	2	yes	maximal, complemented
7	#2159	C₂	1	1	yes	maximal, characteristic
8	#2158	C₂	1	1	yes	maximal, characteristic
9	#2151	C₂	1	1	yes	maximal, characteristic
10	#2141	C₂	1	1	yes	maximal, characteristic
11	C₂×#358	C₂	1	1	yes	maximal, characteristic
12	C₂²×Q₁₆	C₂²	1	1	yes	characteristic
13	#544	—	2	1	no	—
14	#519	—	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	#441	—	2	2	no	—
21	C₄×Q₁₆	C₂²	1	2	yes	—
22	C₄×Q₁₆	C₂²	1	2	yes	—
23	#428	—	2	1	no	—
24	#427	C₂²	1	1	yes	characteristic
25	#427	C₂²	1	1	yes	characteristic
26	C₂×#107	C₂²	1	1	yes	characteristic
27	#419	—	2	1	no	—
28	#405	C₂²	1	1	yes	characteristic
29	#367	C₂²	1	1	yes	characteristic
30	#367	C₂²	1	1	yes	characteristic
31	#367	C₂²	1	2	yes	—
32	#366	C₂²	1	1	yes	characteristic
33	#366	C₂²	1	1	yes	characteristic
34	#366	C₂²	1	2	yes	—
35	C₄⋊C₁₆	C₂²	1	2	yes	—
36	C₄⋊C₁₆	C₂²	1	2	yes	—
37	#358	C₂²	1	2	yes	—
38	#358	C₂²	1	2	yes	—
39	#358	C₂²	1	2	yes	—
40	#358	C₂²	1	2	yes	—
41	C₂²×Q₈	—	2	1	no	—
42	C₂×Q₁₆	—	2	1	no	—
43	C₂×Q₁₆	—	2	2	no	—
44	C₂×Q₁₆	Q₈	1	4	yes	complemented
45	C₂×Q₁₆	C₂³	1	2	yes	—
46	C₂×M₁₆	D₄	1	1	yes	characteristic
47	C₂²×C₈	C₂³	1	1	yes	characteristic
48	C₄⋊Q₈	—	2	2	no	—
49	C₄⋊Q₈	—	4	1	no	—
50	#125	—	2	2	no	—
51	#125	—	4	1	no	—
52	C₂²⋊Q₈	—	4	2	no	—
53	C₄×Q₈	—	2	2	no	—
54	C₄×Q₈	—	2	2	no	—
55	C₄×Q₈	—	4	1	no	—
56	C₄×Q₈	—	4	1	no	—
57	#117	—	2	1	no	—
58	#117	—	2	1	no	—
59	#117	—	4	2	no	—
60	#117	D₄	1	1	yes	characteristic
61	C₂×C₄⋊C₄	—	2	1	no	—
62	M₃₂	—	2	2	no	—
63	C₂×C₁₆	—	2	2	no	—
64	C₂×C₁₆	C₂³	1	1	yes	characteristic
65	C₂×C₁₆	C₂³	1	2	yes	—
66	C₂×C₁₆	C₂³	1	1	yes	characteristic
67	#107	—	2	1	no	—
68	#107	—	2	2	no	—
69	#107	C₂³	1	2	yes	—
70	#107	C₂³	1	1	yes	characteristic
71	#107	C₂³	1	1	yes	characteristic
72	#107	C₂³	1	1	yes	characteristic
73	#107	C₂³	1	1	yes	characteristic
74	#106	—	2	1	no	—
75	#106	—	2	1	no	—
76	#106	—	2	2	no	—
77	#103	—	2	2	no	—
78	#103	—	2	2	no	—
79	#97	D₄	1	2	yes	—
80	C₄×C₈	C₂³	1	2	yes	—
81	C₂×Q₈	—	2	4	no	—
82	C₂×Q₈	—	2	2	no	—
83	C₂×Q₈	—	4	1	no	—
84	C₂×Q₈	—	4	2	no	—
85	C₂×Q₈	—	4	2	no	—
86	C₂²×C₄	—	4	1	no	—
87	C₂²×C₄	—	4	1	no	—
88	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
89	Q₁₆	—	2	4	no	—
90	Q₁₆	—	2	2	no	—
91	Q₁₆	C₂×Q₈	1	4	yes	—
92	M₁₆	—	2	2	no	—
93	C₂×C₈	C₂×D₄	1	2	yes	—
94	C₂×C₈	C₂×D₄	1	1	yes	characteristic
95	C₂×C₈	C₂×Q₈	1	2	yes	—
96	C₂×C₈	#41	1	2	yes	—
97	C₂×C₈	C₂⁴	1	1	yes	characteristic
98	C₄⋊C₄	—	2	1	no	—
99	C₄⋊C₄	—	2	1	no	—
100	C₄⋊C₄	—	2	1	no	—
101	C₄⋊C₄	—	2	2	no	—
102	C₄⋊C₄	—	2	1	no	—
103	C₄⋊C₄	—	4	1	no	—
104	C₄⋊C₄	—	4	2	no	—
105	C₄⋊C₄	—	4	2	no	—
106	C₄⋊C₄	—	4	2	no	—
107	C₄⋊C₄	—	4	2	no	—
108	C₄⋊C₄	—	4	2	no	—
109	C₄⋊C₄	D₈	1	1	yes	characteristic
110	C₄⋊C₄	D₈	1	1	yes	characteristic
111	C₂²⋊C₄	—	4	1	no	—
112	C₂²⋊C₄	—	4	1	no	—
113	C₂²⋊C₄	D₈	1	2	yes	—
114	C₄²	—	4	1	no	—
115	C₄²	—	4	1	no	—
116	C₄²	—	4	2	no	—
117	C₄²	—	4	2	no	—
118	C₄²	C₂×D₄	1	2	yes	—
119	C₁₆	—	2	1	no	—
120	C₁₆	—	2	1	no	—
121	C₁₆	—	2	2	no	—
122	C₂³	C₂×D₈	1	1	yes	characteristic
123	Q₈	—	2	4	no	—
124	Q₈	—	4	2	no	—
125	Q₈	—	4	4	no	—
126	Q₈	—	16	2	no	—
127	C₂×C₄	—	4	2	no	—
128	C₂×C₄	—	4	4	no	—
129	C₂×C₄	—	4	1	no	—
130	C₂×C₄	—	4	1	no	—
131	C₂×C₄	—	4	1	no	—
132	C₂×C₄	—	4	1	no	—
133	C₂×C₄	—	4	2	no	—
134	C₂×C₄	—	16	1	no	—
135	C₂×C₄	C₂²⋊Q₈	1	2	yes	—
136	C₂×C₄	C₂×D₈	1	2	yes	—
137	C₂×C₄	C₂×D₈	1	2	yes	—
138	C₂×C₄	C₂×D₈	1	1	yes	characteristic
139	C₂×C₄	#137	1	2	yes	—
140	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
141	C₈	—	2	2	no	—
142	C₈	C₂²⋊Q₈	1	2	yes	—
143	C₈	C₂²×Q₈	1	1	yes	characteristic
144	C₈	C₂×#41	1	1	yes	characteristic
145	C₂²	—	2	1	no	—
146	C₂²	—	2	1	no	—
147	C₂²	#474	1	2	yes	—
148	C₂²	C₂²×D₈	1	1	yes	characteristic
149	C₄	—	2	2	no	—
150	C₄	—	2	2	no	—
151	C₄	—	4	4	no	—
152	C₄	—	8	1	no	—
153	C₄	—	8	2	no	—
154	C₄	—	8	1	no	—
155	C₄	—	8	1	no	—
156	C₄	—	8	1	no	—
157	C₄	#474	1	2	yes	—
158	C₄	C₂×C₂²⋊Q₈	1	1	yes	characteristic
159	C₄	C₂×#137	1	1	yes	characteristic
160	C₂	—	2	2	no	—
161	C₂	C₂×#474	1	1	yes	characteristic
162	C₂	#3416	1	1	yes	characteristic
163	C₂	#3417	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴
prime radical	C₂³	C₂×D₈
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
3. #2229	C₂	semidirect
4. #2229	C₂	semidirect
5. #2227	C₂	semidirect
6. #2225	C₂	semidirect
44. C₂×Q₁₆	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
#2950	2
#2229	2
#2227	2
#2225	2
#2159	2
#2158	2
#2151	2
#2141	2
C₂×#358	2