Finite Group Databasebeta

← #72224662

Subgroups — #72224662

Order 2048.

218 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 218 conjugacy classes of subgroups — too many to draw. Only its 74 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#72224659C211yesmaximal, characteristic, complemented
2#72224654C211yesmaximal, characteristic, complemented
3#72224652C211yesmaximal, characteristic
4#72224648C211yesmaximal, characteristic, complemented
5#72224647C211yesmaximal, characteristic, complemented
6#72224644C211yesmaximal, characteristic, complemented
7#72224639C211yesmaximal, characteristic, complemented
8#153598C411yescharacteristic
9#151103—21no—
10#151086C2211yescharacteristic
11#151085C2211yescharacteristic
12#122963—21no—
13#94754C411yescharacteristic
14#94614C411yescharacteristic
15#94613C411yescharacteristic
16#94509C2211yescharacteristic
17#94506C2211yescharacteristic
18#93878C2211yescharacteristic
19#93877C2211yescharacteristic
20#93759—21no—
21#93757—21no—
22#93668C2211yescharacteristic
23#13742C811yescharacteristic
24#13656C2×C411yescharacteristic
25#13654C2×C411yescharacteristic
26#13648—41no—
27#12062—41no—
28#12047—21no—
29#12045—21no—
30#7548C811yescharacteristic
31#7541C811yescharacteristic
32#7540C811yescharacteristic
33#7511—21no—
34#7511C2×C411yescharacteristic
35#7511C2×C411yescharacteristic
36#7464C2×C411yescharacteristic
37#7463C2×C411yescharacteristic
38#7456—21no—
39#7388—21no—
40#7385—21no—
41#7382—41no—
42#7332D411yescharacteristic
43#7330C2311yescharacteristic
44#7099—21no—
45#7070—41no—
46#7053D411yescharacteristic
47#2262C2×C811yescharacteristic
48C2×D32C2×C811yescharacteristic
49#2258—81no—
50#2178—21no—
51C4×D16—21no—
52#2170—41no—
53D4×C16—81no—
54C8×D8—41no—
55SD128C1612yescomplemented
56D64C1612yescomplemented
57M128—21no—
58M128—41no—
59M128C2×C811yescharacteristic
60M128C2×C811yescharacteristic
61#1425C2×C811yescharacteristic
62#1424C2×C811yescharacteristic
63#1422—41no—
64#1417—21no—
65#1415—21no—
66#1401—81no—
67#1397—21no—
68#1397C22×C411yescharacteristic
69#1397#4111yescharacteristic
70#1373—21no—
71#1372—21no—
72#1367—41no—
73#1335—41no—
74#1331—41no—
75D4⋊C16—81no—
76#1312—21no—
77#1311C2×D411yescharacteristic
78#508—21no—
79C2×D16—21no—
80#504—81no—
81C22×C16—161no—
82#443—41no—
83C4×D8—41no—
84C8×D4—81no—
85Q64C2×C1611yescharacteristic, complemented
86SD64—21no—
87D32—21no—
88D32C2×C1612yes—
89D32C2×C1611yescharacteristic, complemented
90M64—81no—
91M64—161no—
92C2×C32—21no—
93C2×C32—41no—
94C2×C32—161no—
95C2×C32C22×C811yescharacteristic
96C2×C32#13111yescharacteristic
97#368—21no—
98#366—21no—
99#364—41no—
100C4⋊C16—81no—
101#359—41no—
102#357—41no—
103#350—81no—
104C22⋊C16—161no—
105C4×C16—41no—
106C4×C16C4×D411yescharacteristic
107C4×C16AGL(1,ℤ/8)11yescharacteristic
108#335—41no—
109D4⋊C8—81no—
110C82AGL(1,ℤ/8)11yescharacteristic
111C64—21no—
112C64—41no—
113C64—81no—
114C64C2×C1612yescomplemented
115#135—41no—
116C2×D8—41no—
117#131—81no—
118C22×C8—161no—
119C4×D4—81no—
120Q32—21no—
121SD32—41no—
122D16—22no—
123D16—21no—
124D16—41no—
125M32—81no—
126M32—161no—
127C2×C16—21no—
128C2×C16—81no—
129C2×C16—161no—
130C2×C16—161no—
131C2×C16—322no—
132C2×C16C8×D411yescharacteristic
133C2×C16#44411yescharacteristic
134#108—41no—
135#107—41no—
136C4⋊C8—81no—
137#104—81no—
138#102—81no—
139C22⋊C8—161no—
140C4×C8—21no—
141C4×C8#44211yescharacteristic
142C32—21no—
143C32—41no—
144C32—81no—
145C32—161no—
146C32C22×C1611yescharacteristic
147C32#50411yescharacteristic
148#41—81no—
149C2×D4—81no—
150C22×C4—161no—
151Q16—41no—
152SD16—81no—
153D8—42no—
154D8—41no—
155D8—81no—
156M16—81no—
157M16—161no—
158C2×C8—41no—
159C2×C8—161no—
160C2×C8—161no—
161C2×C8—321no—
162C2×C8—321no—
163C2×C8#158011yescharacteristic
164C2×C8AGL(1,ℤ/16)11yescharacteristic
165C4⋊C4—81no—
166C22⋊C4—161no—
167C42AGL(1,ℤ/16)11yescharacteristic
168C16—21no—
169C16—41no—
170C16—161no—
171C16—161no—
172C16—321no—
173C16D4×C1611yescharacteristic
174C16#217111yescharacteristic
175C23—161no—
176Q8—81no—
177D4—82no—
178D4—81no—
179D4—161no—
180D4—161no—
181C2×C4—21no—
182C2×C4—161no—
183C2×C4—161no—
184C2×C4—321no—
185C2×C4—321no—
186C2×C4#1206711yescharacteristic
187C8—21no—
188C8—81no—
189C8—161no—
190C8—321no—
191C8#1206211yescharacteristic
192C8#1206811yescharacteristic
193C22—161no—
194C22—162no—
195C22—321no—
196C22—321no—
197C22AGL(1,ℤ/32)11yescharacteristic
198C4—41no—
199C4—161no—
200C4—321no—
201C4#12296311yescharacteristic
202C4AGL(1,ℤ/32)11yescharacteristic
203C2—21no—
204C2—321no—
205C2—322no—
206C2#7222463911yescharacteristic
subgroupG/H
center, socleC2#72224639
commutatorC32C22×C16
Frattini, cosocle#7330C23
prime radical#13742C8
LCSG ▷ C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×C8 ◁ #1311 ◁ #93668 ◁ G
LECSG ▷ #7330 ▷ C4×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #72224659C2semidirect
2. #72224654C2semidirect
4. #72224648C2semidirect
5. #72224647C2semidirect
6. #72224644C2semidirect
7. #72224639C2semidirect
55. SD128C16semidirect
56. D64C16semidirect
85. Q64C2×C16semidirect
89. D32C2×C16semidirect
114. C64C2×C16semidirect

Maximal subgroups

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

subgroupindex
#722246592
#722246542
#722246522
#722246482
#722246472
#722246442
#722246392