Finite Group Databasebeta

← #11923077

Subgroups — #11923077

Order 1152.

750 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 750 conjugacy classes of subgroups — too many to draw. Only its 145 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
1C2×C3×#994C211yesmaximal, characteristic
2#10593660C213yesmaximal, complemented
3#10593658C213yesmaximal, complemented
4#85528—33nomaximal
5#84099—31nomaximal
6C3×#994C414yescomplemented
7C2×C3×#41⋊C3C2213yes—
8Q8×C3⋊Dic3—41nomaximal
9C3⋊#859C2211yescharacteristic, complemented
10C3⋊#859C2213yescomplemented
11C2×C3×H(5,𝔽2)D311yescharacteristic
12C2×#994D313yes—
13#5827—31no—
14C2×#948—31no—
15#5426—39no—
16#5424—39no—
17#5233—33no—
18#5232—33no—
19#5215—33no—
20C2×#832—33no—
21C2×C32×Q8—41no—
22C3×#41⋊C3C2×C416yescomplemented
23C2×C3×SL(2,𝔽3)C2311yescharacteristic
24C4⋊(C3⋊Dic3)—43no—
25C4×C3⋊Dic3—43no—
26#2897—91noSylow
27C3×H(5,𝔽2)—34no—
28C3×H(5,𝔽2)Dic314yescomplemented
29C2×C3×#41—31no—
30C2×C3×#41C2×D313yes—
31C22×C3×D4—33no—
32#994Dic3112yescomplemented
33C2×#41⋊C3C2×D319yes—
34#948—64no—
35Q8×Dic3—61no—
36Q8×Dic3—123no—
37D4×Dic3—63no—
38#926—33no—
39C2×C4⋊Dic3—33no—
40C2×#208—33no—
41C22×C3⋊C8—33no—
42#859—33no—
43#859—39no—
44#835—33no—
45#835—31no—
46#832—33no—
47#832—36no—
48#832—33no—
49C32×Q8—44no—
50C2×C32×C4—43no—
51C2×C3⋊Dic3—41no—
52C2×C3⋊Dic3—43no—
53C3×SL(2,𝔽3)C22×C411yescharacteristic
54C2×H(5,𝔽2)C3⋊D311yescharacteristic
55C2×#131—91no—
56#520—91no—
57#418—93no—
58#417—93no—
59#416—93no—
60C2×#102—93no—
61C24×C3—61no—
62C3×#41—34no—
63C3×#41—34no—
64C3×#41—36no—
65C3×#41C2×Dic316yes—
66C2×C3×Q8—43no—
67C2×C3×Q8S411yescharacteristic
68C2×C3×Q8C22×D311yescharacteristic
69C2×C3×D4—312no—
70C2×C3×D4—33no—
71C2×C3×D4—33no—
72C2×C3×D4—66no—
73C22×C3×C4—33no—
74C22×Dic3—63no—
75#41⋊C3C2×Dic3118yescomplemented
76C2×SL(2,𝔽3)C22×D313yes—
77#217—63no—
78C4⋊Dic3—33no—
79C4⋊Dic3—31no—
80C4⋊Dic3—63no—
81C4⋊Dic3—129no—
82C4×Dic3—63no—
83C4×Dic3—129no—
84#208—66no—
85C2×C3⋊C8—31no—
86C2×C3⋊C8—33no—
87C2×C3⋊C8—66no—
88C22×C32—41no—
89C32×C4—46no—
90C3⋊Dic3—42no—
91C3⋊Dic3—83no—
92H(5,𝔽2)—34no—
93H(5,𝔽2)C3⋊Dic314yescomplemented
94C2×#41—31no—
95C2×#41C2×C3⋊D313yes—
96C22×D4—33no—
97#131—184no—
98C2×M16—93no—
99C22×C8—93no—
100C4×Q8—181no—
101C4×D4—183no—
102#117—93no—
103C2×C4⋊C4—93no—
104#103—91no—
105#103—93no—
106#102—93no—
107#102—96no—
108#102—93no—
109C23×C3—33no—
110C23×C3—64no—
111C23×C3—241no—
112C23×C3—241no—
113C3×Q8—31no—
114C3×Q8—412no—
115C3×Q8SL(2,ℤ/4)14yes—
116C3×Q8C22×Dic311yescharacteristic
117C3×D4—36no—
118C3×D4—312no—
119C3×D4—63no—
120C2×C3×C4—31no—
121C2×C3×C4—36no—
122C2×C3×C4—36no—
123C2×C3×C4—49no—
124C2×C3×C4C2×S413yes—
125C2×Dic3—61no—
126C2×Dic3—63no—
127C2×Dic3—123no—
128C2×Dic3—129no—
129C2×Dic3—243no—
130SL(2,𝔽3)C22×Dic313yes—
131C3⋊C8—61no—
132C3⋊C8—63no—
133C2×C32—41no—
134C2×C32—41no—
135C2×C32—41no—
136C24—61no—
137#41—34no—
138#41—34no—
139#41—36no—
140#41C2×C3⋊Dic316yes—
141C2×Q8C3⋊S411yescharacteristic
142C2×Q8C22×C3⋊D311yescharacteristic
143C2×D4—312no—
144C2×D4—33no—
145C2×D4—33no—
146C2×D4—66no—
147C22×C4—33no—
148C22×C4—183no—
149M16—186no—
150C2×C8—91no—
151C2×C8—93no—
152C2×C8—186no—
153C4⋊C4—93no—
154C4⋊C4—91no—
155C4⋊C4—183no—
156C22⋊C4—183no—
157C42—183no—
158C22×C3—36no—
159C22×C3—43no—
160C22×C3—63no—
161C22×C3—63no—
162C22×C3—244no—
163C22×C3C22×S411yescharacteristic
164C3×C4—31no—
165C3×C4—31no—
166C3×C4—418no—
167C3×C4C2×SL(2,ℤ/4)16yes—
168Dic3—126no—
169Dic3—121no—
170Dic3—123no—
171Dic3—249no—
172C32—41noSylow
173C23—33no—
174C23—64no—
175C23—241no—
176C23—241no—
177Q8—31no—
178Q8C3⋊SL(2,ℤ/4)14yes—
179Q8C22×C3⋊Dic311yescharacteristic
180D4—36no—
181D4—312no—
182D4—63no—
183C2×C4—31no—
184C2×C4—36no—
185C2×C4—36no—
186C2×C4—181no—
187C2×C4—183no—
188C2×C4—723no—
189C2×C4C2×C3⋊S413yes—
190C8—181no—
191C8—183no—
192C2×C3—43no—
193C2×C3—43no—
194C2×C3—43no—
195C2×C3—66no—
196C2×C3#592411yescharacteristic
197C2×C3#592511yescharacteristic
198C2×C3C22×SL(2,ℤ/4)11yescharacteristic
199C22—36no—
200C22—63no—
201C22—63no—
202C22—244no—
203C22C22×C3⋊S411yescharacteristic
204C4—31no—
205C4—31no—
206C4—361no—
207C4—363no—
208C4C2×C3⋊SL(2,ℤ/4)16yes—
209C3—43no—
210C3#8552811yescharacteristic, complemented
211C2—66no—
212C2#1059658711yescharacteristic
213C2#1059658811yescharacteristic
214C2C22×C3⋊SL(2,ℤ/4)11yescharacteristic
subgroupG/H
center, FrattiniC22C22×C3⋊S4
commutator, hypocenterC3×SL(2,𝔽3)C22×C4
FittingC2×C3×H(5,𝔽2)D3
socleC22×C3C22×S4
prime radicalC2×C3×#994C2
cosocleC2×C3×SL(2,𝔽3)C23
hypercenterC2×Q8C3⋊S4
LCSG ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
2. #10593660C2semidirect
3. #10593658C2semidirect
6. C3×#994C4semidirect
9. C3⋊#859C22semidirect
10. C3⋊#859C22semidirect
22. C3×#41⋊C3C2×C4semidirect
28. C3×H(5,𝔽2)Dic3semidirect
32. #994Dic3semidirect
75. #41⋊C3C2×Dic3semidirect
93. H(5,𝔽2)C3⋊Dic3semidirect
210. C3#85528semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#9942
#105936602
#105936582
#855283
#840993
Q8×C3⋊Dic34