Finite Group Databasebeta

Subgroups — #72227046

Order 4096.

438 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 438 conjugacy classes of subgroups — too many to draw. Only its 129 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	#72225402	C₂	1	2	yes	maximal
2	#72225398	—	2	2	no	—
3	#72225397	C₄	1	4	yes	complemented
4	#72225395	—	2	2	no	—
5	#72216379	C₂	1	1	yes	maximal, characteristic
6	#72216298	—	2	2	no	—
7	#72216239	C₄	1	1	yes	characteristic
8	C₁₆×M₆₄	C₂²	1	1	yes	characteristic
9	#72216197	C₄	1	1	yes	characteristic
10	#153582	—	4	2	no	—
11	#152407	—	4	2	no	—
12	#152407	—	4	2	no	—
13	C₈×M₆₄	D₄	1	2	yes	—
14	#135796	—	4	1	no	—
15	#135781	—	2	1	no	—
16	#135778	—	2	1	no	—
17	#135778	—	4	1	no	—
18	C₂×C₁₆²	C₂×C₄	1	1	yes	characteristic
19	#94748	—	2	4	no	—
20	#94746	—	4	2	no	—
21	#93664	D₄	1	1	yes	characteristic
22	#93664	Q₈	1	1	yes	characteristic
23	C₁₆×C₃₂	C₂×C₄	1	2	yes	—
24	C₂×#1426	—	8	4	no	—
25	C₂×#1426	—	8	2	no	—
26	C₂×#1426	—	8	2	no	—
27	C₄×M₆₄	D₈	1	2	yes	—
28	C₄×M₆₄	SD₁₆	1	2	yes	—
29	#12909	—	4	2	no	—
30	#12906	—	4	2	no	—
31	#12906	—	8	2	no	—
32	#12023	—	8	2	no	—
33	C₂×C₈×C₁₆	—	2	1	no	—
34	C₂×C₈×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
35	#8823	—	4	1	no	—
36	#8818	—	4	1	no	—
37	#7544	—	4	4	no	—
38	#7513	—	8	2	no	—
39	#7511	—	8	2	no	—
40	#7329	SD₁₆	1	2	yes	—
41	#7329	Q₁₆	1	2	yes	—
42	C₈×C₃₂	C₂²⋊C₄	1	2	yes	—
43	C₈×C₃₂	C₄⋊C₄	1	2	yes	—
44	C₁₆²	—	2	1	no	—
45	C₁₆²	C₄²	1	1	yes	characteristic
46	C₁₆²	C₄⋊C₄	1	1	yes	characteristic
47	C₂×M₆₄	—	2	2	no	—
48	C₂×M₆₄	D₁₆	1	2	yes	—
49	C₂×M₆₄	SD₃₂	1	2	yes	—
50	C₂×#368	—	8	2	no	—
51	C₂×#368	—	8	2	no	—
52	C₂×#367	—	8	2	no	—
53	C₂×#367	—	8	2	no	—
54	C₂×#367	—	16	4	no	—
55	C₂×C₄×C₁₆	—	2	2	no	—
56	C₂×C₄×C₁₆	#102	1	2	yes	—
57	#1848	—	8	1	no	—
58	#1845	—	8	2	no	—
59	#1845	—	8	1	no	—
60	#1567	—	8	2	no	—
61	C₂×C₈²	#100	1	1	yes	characteristic
62	M₁₂₈	—	8	4	no	—
63	#1426	—	16	8	no	—
64	#1426	—	16	4	no	—
65	#1426	—	16	4	no	—
66	#1397	—	2	2	no	—
67	#1397	SD₃₂	1	2	yes	—
68	#1397	Q₃₂	1	2	yes	—
69	C₄×C₃₂	#102	1	2	yes	—
70	C₄×C₃₂	#103	1	2	yes	—
71	C₄×C₃₂	#106	1	2	yes	—
72	C₄×C₃₂	#107	1	2	yes	—
73	#1372	—	16	4	no	—
74	C₈×C₁₆	—	2	2	no	—
75	C₈×C₁₆	—	2	1	no	—
76	C₈×C₁₆	—	2	1	no	—
77	C₈×C₁₆	#95	1	1	yes	characteristic
78	C₈×C₁₆	#101	1	1	yes	characteristic
79	C₈×C₁₆	#103	1	2	yes	—
80	C₂²×C₁₆	—	2	4	no	—
81	C₂²×C₁₆	#357	1	2	yes	—
82	C₂²×C₁₆	#361	1	2	yes	—
83	C₂×#108	—	16	2	no	—
84	C₂×#107	—	16	4	no	—
85	C₂×#107	—	16	2	no	—
86	#423	—	16	1	no	—
87	C₂×C₄×C₈	—	2	1	no	—
88	C₂×C₄×C₈	#331	1	1	yes	characteristic
89	#384	—	16	1	no	—
90	M₆₄	—	2	8	no	—
91	M₆₄	—	2	4	no	—
92	M₆₄	—	2	4	no	—
93	C₂×C₃₂	—	2	2	no	—
94	C₂×C₃₂	—	2	2	no	—
95	C₂×C₃₂	#357	1	2	yes	—
96	C₂×C₃₂	#358	1	2	yes	—
97	C₂×C₃₂	#366	1	2	yes	—
98	C₂×C₃₂	#367	1	2	yes	—
99	#368	—	16	4	no	—
100	#368	—	16	2	no	—
101	#368	—	16	2	no	—
102	#367	—	16	4	no	—
103	#367	—	16	2	no	—
104	#367	—	16	2	no	—
105	#367	—	32	8	no	—
106	C₄×C₁₆	—	2	2	no	—
107	C₄×C₁₆	—	2	2	no	—
108	C₄×C₁₆	—	2	4	no	—
109	C₄×C₁₆	#332	1	2	yes	—
110	C₄×C₁₆	#340	1	2	yes	—
111	C₄×C₁₆	#358	1	2	yes	—
112	C₄×C₁₆	#362	1	2	yes	—
113	#335	—	16	4	no	—
114	C₈²	—	2	1	no	—
115	C₈²	#333	1	1	yes	characteristic
116	C₈²	#343	1	1	yes	characteristic
117	C₆₄	—	16	4	no	—
118	C₂×M₁₆	—	32	1	no	—
119	C₂²×C₈	—	2	2	no	—
120	C₂²×C₈	#1361	1	2	yes	—
121	C₂×C₄⋊C₄	—	32	2	no	—
122	C₂×C₄⋊C₄	—	32	1	no	—
123	C₂×C₄²	#1357	1	1	yes	characteristic
124	C₂×C₁₆	—	2	4	no	—
125	C₂×C₁₆	—	2	4	no	—
126	C₂×C₁₆	—	2	2	no	—
127	C₂×C₁₆	—	2	2	no	—
128	C₂×C₁₆	—	2	4	no	—
129	C₂×C₁₆	—	4	8	no	—
130	C₂×C₁₆	#1364	1	2	yes	—
131	C₂×C₁₆	#1365	1	2	yes	—
132	C₂×C₁₆	#1380	1	2	yes	—
133	C₂×C₁₆	#1387	1	2	yes	—
134	#108	—	32	2	no	—
135	#108	—	32	2	no	—
136	#107	—	32	2	no	—
137	#107	—	32	2	no	—
138	#107	—	32	8	no	—
139	C₄⋊C₈	—	32	2	no	—
140	C₄×C₈	—	2	2	no	—
141	C₄×C₈	—	2	1	no	—
142	C₄×C₈	—	2	1	no	—
143	C₄×C₈	#1296	1	1	yes	characteristic
144	C₄×C₈	#1358	1	1	yes	characteristic
145	C₄×C₈	#1362	1	2	yes	—
146	C₃₂	—	2	4	no	—
147	C₃₂	—	2	4	no	—
148	C₃₂	—	2	2	no	—
149	C₃₂	—	2	2	no	—
150	C₃₂	#1418	1	4	yes	—
151	C₃₂	#1426	1	4	yes	—
152	C₂²×C₄	—	2	1	no	—
153	C₂²×C₄	—	64	1	no	—
154	C₂²×C₄	#7096	1	1	yes	characteristic
155	M₁₆	—	64	2	no	—
156	C₂×C₈	—	2	2	no	—
157	C₂×C₈	—	2	2	no	—
158	C₂×C₈	—	2	2	no	—
159	C₂×C₈	—	2	2	no	—
160	C₂×C₈	—	4	4	no	—
161	C₂×C₈	—	64	1	no	—
162	C₂×C₈	#7097	1	2	yes	—
163	C₂×C₈	#7274	1	2	yes	—
164	C₄⋊C₄	—	64	2	no	—
165	C₄⋊C₄	—	64	4	no	—
166	C₄²	—	2	1	no	—
167	C₄²	#7098	1	1	yes	characteristic
168	C₄²	#7269	1	1	yes	characteristic
169	C₁₆	—	2	8	no	—
170	C₁₆	—	2	2	no	—
171	C₁₆	—	2	2	no	—
172	C₁₆	—	4	4	no	—
173	C₁₆	#7453	1	4	yes	—
174	C₁₆	#7474	1	2	yes	—
175	C₁₆	#7477	1	2	yes	—
176	C₂³	#93868	1	1	yes	characteristic
177	C₂×C₄	—	2	1	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	2	1	no	—
180	C₂×C₄	—	4	2	no	—
181	C₂×C₄	—	128	1	no	—
182	C₂×C₄	—	128	2	no	—
183	C₂×C₄	#93064	1	1	yes	characteristic
184	C₂×C₄	#93868	1	1	yes	characteristic
185	C₈	—	2	4	no	—
186	C₈	—	2	2	no	—
187	C₈	—	4	2	no	—
188	C₈	—	128	1	no	—
189	C₈	#94355	1	2	yes	—
190	C₈	#94364	1	2	yes	—
191	C₂²	—	2	1	no	—
192	C₂²	—	4	1	no	—
193	C₂²	#72216560	1	1	yes	characteristic
194	C₄	—	256	1	no	—
195	C₄	—	2	2	no	—
196	C₄	—	2	1	no	—
197	C₄	—	4	1	no	—
198	C₄	#72227044	1	2	yes	—
199	C₂	—	2	1	no	—
200	C₂	—	4	1	no	—
201	C₂	#72227045	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	#72227045
commutator	C₁₆²	C₄²
Frattini, cosocle	C₁₆×M₆₄	C₂²
prime radical	C₂³	#93868
LCS	G ▷ C₁₆² ▷ C₈×C₁₆ ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ C₂×C₈×C₁₆ ◁ C₁₆×M₆₄ ◁ G
LECS	G ▷ C₁₆×M₆₄ ▷ C₈×C₁₆ ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
3. #72225397	C₄	semidirect

Maximal subgroups

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

subgroup	index
#72225402	2
#72216379	2