Finite Group Databasebeta

← #72255419

Subgroups — #72255419

Order 2304.

612 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 612 conjugacy classes of subgroups — too many to draw. Only its 161 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#11797877C211yesmaximal, characteristic
2#11797589C212yesmaximal, complemented
3#11771655C212yesmaximal, complemented
4#11771319C211yesmaximal, characteristic, complemented
5C32×#1369C211yesmaximal, characteristic, complemented
6#10707076—34nomaximal
7#10592606—22no—
8#10592603C411yescharacteristic
9#10592399C2212yes—
10C32×#364C2212yes—
11C32×#346C2211yescharacteristic
12#10589381—22no—
13#10589376C412yes—
14#10589372C411yescharacteristic
15#10589327C2211yescharacteristic
16#10589321C2211yescharacteristic
17#69301—38no—
18#69085—34no—
19C3×#1369D314yescomplemented
20#67569—38no—
21#67511—34no—
22#64113C2×C412yes—
23#64110—22no—
24C32×M32—22no—
25C2×C32×C16C2×C411yescharacteristic
26C2×C32×C16C2×C411yescharacteristic
27C32×C4×C8C2311yescharacteristic
28#63646C2×C412yes—
29C2×C3⋊(C3⋊C16)D412yes—
30#63632—22no—
31C4×C3⋊(C3⋊C8)—21no—
32#63628C811yescharacteristic, complemented
33#63627C812yescomplemented
34#63626C811yescharacteristic, complemented
35#63625—42no—
36C16×C3⋊D3—42no—
37#12068—91noSylow
38#4907—68no—
39#4904—34no—
40#4700—38no—
41C3×#364C2×D318yes—
42C3×#346C2×D314yes—
43#4515—68no—
44#4510—38no—
45#4506—34no—
46#4461—34no—
47#4455—34no—
48#3828—22no—
49C2×C32×C8C22×C411yescharacteristic
50C2×C32×C8C2×D411yescharacteristic
51C32×C42#4111yescharacteristic
52#3748—22no—
53C2×C3⋊(C3⋊C8)—41no—
54#3746C2×C812yes—
55#3745C2×C812yes—
56#3744—22no—
57#3743—42no—
58C8×C3⋊D3—42no—
59C32×C16—21no—
60C32×C16—42no—
61C32×C16C2×C811yescharacteristic
62C32×C16C2×C811yescharacteristic
63C3⋊(C3⋊C16)—22no—
64#2179—91no—
65#2171—92no—
66#1369C3⋊D311yescharacteristic, complemented
67#1336—92no—
68#1313—91no—
69#904—38no—
70#901—68no—
71C3×M32—28no—
72C2×C3×C16C4×D314yes—
73C2×C3×C16C4×D314yes—
74C3×C4×C8C22×D314yes—
75#819—38no—
76C2×C3⋊C16—38no—
77#805—68no—
78C4×C3⋊C8—64no—
79Dic24—34no—
80#800—38no—
81D48—34no—
82#798—128no—
83D3×C16—128no—
84C2×C32×C4—21no—
85C2×C32×C4C4×D411yescharacteristic
86#638—42no—
87C4⋊(C3⋊D3)—22no—
88C4×C3⋊D3—42no—
89#634—22no—
90C32×C8C22×C811yescharacteristic
91C32×C8#13111yescharacteristic
92C32×C8AGL(1,ℤ/8)12yes—
93C3⋊(C3⋊C8)—42no—
94C3⋊(C3⋊C8)—81no—
95#508—91no—
96#504—182no—
97#443—92no—
98#368—91no—
99#364C2×C3⋊D312yes—
100#359—92no—
101#350—182no—
102#346—91no—
103#346C2×C3⋊D311yescharacteristic
104#322—91no—
105#235—68no—
106C2×C3×C8C2×C4×D314yes—
107C2×C3×C8D3×D414yes—
108C3×C42#23914yes—
109#208—68no—
110C2×C3⋊C8—124no—
111Dic12—38no—
112D24—38no—
113#204—68no—
114ΓL(1,𝔽25)—128no—
115D3×C8—128no—
116C3×C16—24no—
117C3×C16—48no—
118C3×C16D3×C814yes—
119C3×C16D3×C814yes—
120C3⋊C16—68no—
121C22×C32#44411yescharacteristic
122C2×C3⋊D3—42no—
123C32×C4—41no—
124C32×C4C8×D411yescharacteristic
125C32×C4#44211yescharacteristic
126C3⋊Dic3—42no—
127#135—92no—
128#131—182no—
129Q32—91no—
130SD32—92no—
131D16—91no—
132M32—22no—
133M32—362no—
134C2×C16—92no—
135C2×C16—362no—
136C2×C16C4×C3⋊D311yescharacteristic
137C2×C16C4×C3⋊D311yescharacteristic
138#108—92no—
139#104—182no—
140C4×C8—181no—
141C4×C8C22×C3⋊D311yescharacteristic
142C2×C3×C4—24no—
143C2×C3×C4#89314yes—
144#67—128no—
145D12—68no—
146C4×D3—128no—
147Dic6—68no—
148C3×C8C2×D3×C814yes—
149C3×C8#90714yes—
150C3×C8#91418yes—
151C3⋊C8—128no—
152C3⋊C8—244no—
153C2×C32—21no—
154C2×C32#158011yescharacteristic
155C3⋊D3—82no—
156#41—182no—
157Q16—92no—
158SD16—182no—
159D8—92no—
160M16—182no—
161M16—362no—
162C2×C8—362no—
163C2×C8—361no—
164C2×C8C2×C4×C3⋊D311yescharacteristic
165C2×C8D4×C3⋊D311yescharacteristic
166C42#383211yescharacteristic
167C16—21no—
168C16—42no—
169C16—182no—
170C16C8×C3⋊D311yescharacteristic
171C16C8×C3⋊D311yescharacteristic
172C22×C3#489414yes—
173C2×D3—128no—
174C3×C4—44no—
175C3×C4#483414yes—
176C3×C4#486914yes—
177Dic3—128no—
178C32#1206811yescharacteristic, complemented, Sylow
179Q8—182no—
180D4—182no—
181D4—362no—
182C2×C4—21no—
183C2×C4—362no—
184C2×C4#6410211yescharacteristic
185C8—362no—
186C8—721no—
187C8C2×C8×C3⋊D311yescharacteristic
188C8#6411611yescharacteristic
189C8#6412312yes—
190C2×C3—24no—
191C2×C3#6851814yes—
192D3—248no—
193C22—362no—
194C22#1059259311yescharacteristic
195C4—41no—
196C4—362no—
197C4#1059253311yescharacteristic
198C4#1059256811yescharacteristic
199C3#1070707614yescomplemented
200C2—21no—
201C2—722no—
202C2#1177733711yescharacteristic
subgroupG/H
centerC4#10592568
commutatorC32×C8C22×C8
FrattiniC4×C8C22×C3⋊D3
FittingC32×#1369C2
socleC2×C32#1580
prime radical#10592603C4
cosocleC32×C4×C8C23
hypercenter#1369C3⋊D3
hypocenterC32#12068
LCSG ▷ C32×C8 ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C4 ◁ C2×C8 ◁ C4×C8 ◁ #1369

Complements

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

NH ≅ G/Ntype
2. #11797589C2semidirect
3. #11771655C2semidirect
4. #11771319C2semidirect
5. C32×#1369C2semidirect
19. C3×#1369D3semidirect
32. #63628C8semidirect
33. #63627C8semidirect
34. #63626C8semidirect
66. #1369C3⋊D3semidirect
178. C32#12068semidirect
199. C3#10707076semidirect

Maximal subgroups

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

subgroupindex
#117978772
#117975892
#117716552
#117713192
C32×#13692
#107070763