Finite Group Databasebeta

← C3×#85528 #11922812

Subgroups — C3×#85528 #11922812

Order 1152.

661 conjugacy classes of subgroups; only the 124 normal subgroups are shown.

normal subgroups only

This group has 661 conjugacy classes of subgroups — too many to draw. Only its 124 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
2C3×#5426C213yesmaximal, complemented
3C3×#5424C213yesmaximal, complemented
4#85528C311yesmaximal, characteristic, direct product factor
5C3×#2897—31nomaximal
6C3×#994C414yescomplemented
7C2×C3×#41⋊C3C2213yes—
8C3×Q8×Dic3—41nomaximal
9C3×#859C2211yescharacteristic, complemented
10C3×#859C2213yescomplemented
11C2×C3×H(5,𝔽2)D311yescharacteristic
12C2×#994—21no—
13C2×#994C2×C311yescharacteristic
14C2×C3×#131—31no—
15C3×#520—31no—
16#5426C2×C313yescomplemented
17#5424C2×C313yescomplemented
18C3×#418—33no—
19C3×#417—33no—
20C3×#416—33no—
21C2×C3×#102—33no—
22C2×C32×Q8—41no—
23C3×#41⋊C3C2×C416yescomplemented
24C2×C3×SL(2,𝔽3)C2311yescharacteristic
25C3×C4⋊Dic3—43no—
26C3×C4×Dic3—43no—
27#2897—31noSylow
28C3×H(5,𝔽2)—34no—
29C3×H(5,𝔽2)Dic314yescomplemented
30C2×C3×#41—31no—
31C2×C3×#41C2×D313yes—
32C22×C3×D4—33no—
33#994—24no—
34#994C3×C414yescomplemented
35C2×#41⋊C3—23no—
36C2×#41⋊C3C22×C313yes—
37C3×#131—64no—
38C2×C3×M16—33no—
39C22×C3×C8—33no—
40C3×C4×Q8—61no—
41C3×C4×D4—63no—
42C3×#117—33no—
43C2×C3×C4⋊C4—33no—
44Q8×Dic3—41no—
45#859C22×C311yescharacteristic, complemented
46#859C22×C313yescomplemented
47C3×#103—31no—
48C3×#103—33no—
49C3×#102—33no—
50C3×#102—36no—
51C3×#102—33no—
52C32×Q8—44no—
53C2×C32×C4—43no—
54C2×C3×Dic3—41no—
55C2×C3×Dic3—43no—
56C3×SL(2,𝔽3)C22×C411yescharacteristic
57C2×H(5,𝔽2)C3×D311yescharacteristic
58C2×#131—31no—
59#520—31no—
60#418—33no—
61#417—33no—
62#416—33no—
63C2×#102—33no—
64C24×C3—61no—
65C3×#41—34no—
66C3×#41—34no—
67C3×#41—36no—
68C3×#41C2×Dic316yes—
69C2×C3×Q8—41no—
70C2×C3×Q8—81no—
71C2×C3×Q8S411yescharacteristic
72C2×C3×Q8C22×D311yescharacteristic
73C2×C3×D4—312no—
74C2×C3×D4—33no—
75C2×C3×D4—33no—
76C2×C3×D4—66no—
77C22×C3×C4—33no—
78C22×C3×C4—63no—
79#41⋊C3—26no—
80#41⋊C3C2×C3×C416yescomplemented
81C2×SL(2,𝔽3)—21no—
82C2×SL(2,𝔽3)C23×C311yescharacteristic
83C3×M16—66no—
84C2×C3×C8—31no—
85C2×C3×C8—33no—
86C2×C3×C8—66no—
87C3×C4⋊C4—33no—
88C3×C4⋊C4—31no—
89C3×C4⋊C4—63no—
90C3×C22⋊C4—63no—
91C3×C42—63no—
92C4⋊Dic3—43no—
93C4×Dic3—43no—
94C22×C32—41no—
95C32×C4—46no—
96C3×Dic3—42no—
97C3×Dic3—83no—
98H(5,𝔽2)—34no—
99H(5,𝔽2)C3×Dic314yescomplemented
100C2×#41—31no—
101C2×#41C2×C3×D313yes—
102C22×D4—33no—
103#131—64no—
104C2×M16—33no—
105C22×C8—33no—
106C4×Q8—61no—
107C4×D4—63no—
108#117—33no—
109C2×C4⋊C4—33no—
110#103—31no—
111#103—33no—
112#102—33no—
113#102—36no—
114#102—33no—
115C23×C3—33no—
116C23×C3—64no—
117C23×C3—241no—
118C23×C3—241no—
119C3×Q8—31no—
120C3×Q8—44no—
121C3×Q8—84no—
122C3×Q8SL(2,ℤ/4)14yes—
123C3×Q8C22×Dic311yescharacteristic
124C3×D4—36no—
125C3×D4—312no—
126C3×D4—63no—
127C2×C3×C4—31no—
128C2×C3×C4—36no—
129C2×C3×C4—36no—
130C2×C3×C4—43no—
131C2×C3×C4—61no—
132C2×C3×C4—63no—
133C2×C3×C4—83no—
134C2×C3×C4—243no—
135C2×C3×C4C2×S413yes—
136C2×Dic3—41no—
137C2×Dic3—43no—
138SL(2,𝔽3)—21no—
139SL(2,𝔽3)C22×C3×C411yescharacteristic
140C3×C8—61no—
141C3×C8—63no—
142C2×C32—41no—
143C2×C32—41no—
144C2×C32—41no—
145C24—61no—
146#41—34no—
147#41—34no—
148#41—36no—
149#41C2×C3×Dic316yes—
150C2×Q8C3×S411yescharacteristic
151C2×Q8C22×C3×D311yescharacteristic
152C2×D4—312no—
153C2×D4—33no—
154C2×D4—33no—
155C2×D4—66no—
156C22×C4—33no—
157C22×C4—63no—
158M16—66no—
159C2×C8—31no—
160C2×C8—33no—
161C2×C8—66no—
162C4⋊C4—33no—
163C4⋊C4—31no—
164C4⋊C4—63no—
165C22⋊C4—63no—
166C42—63no—
167C22×C3—36no—
168C22×C3—41no—
169C22×C3—63no—
170C22×C3—63no—
171C22×C3—81no—
172C22×C3—244no—
173C22×C3C22×S411yescharacteristic
174C3×C4—31no—
175C3×C4—31no—
176C3×C4—46no—
177C3×C4—86no—
178C3×C4—121no—
179C3×C4—123no—
180C3×C4C2×SL(2,ℤ/4)16yes—
181Dic3—42no—
182Dic3—83no—
183C32—41noSylow
184C23—33no—
185C23—64no—
186C23—241no—
187C23—241no—
188Q8—31no—
189Q8C3×SL(2,ℤ/4)14yes—
190Q8C22×C3×Dic311yescharacteristic
191D4—36no—
192D4—312no—
193D4—63no—
194C2×C4—31no—
195C2×C4—36no—
196C2×C4—36no—
197C2×C4—61no—
198C2×C4—63no—
199C2×C4—243no—
200C2×C4C2×C3×S413yes—
201C8—61no—
202C8—63no—
203C2×C3—41no—
204C2×C3—41no—
205C2×C3—41no—
206C2×C3—66no—
207C2×C3—81no—
208C2×C3—81no—
209C2×C3—81no—
210C2×C3#592411yescharacteristic
211C2×C3#592511yescharacteristic
212C2×C3C22×SL(2,ℤ/4)11yescharacteristic
213C22—36no—
214C22—63no—
215C22—63no—
216C22—244no—
217C22C22×C3×S411yescharacteristic
218C4—31no—
219C4—31no—
220C4—121no—
221C4—123no—
222C4C2×C3×SL(2,ℤ/4)16yes—
223C3—41no—
224C3—81no—
225C3#8552811yescharacteristic, direct product factor
226C2—66no—
227C2C3×#592411yescharacteristic
228C2C3×#592511yescharacteristic
229C2C22×C3×SL(2,ℤ/4)11yescharacteristic
subgroupG/H
center, socleC22×C3C22×S4
commutator, hypocenterSL(2,𝔽3)C22×C3×C4
FrattiniC22C22×C3×S4
FittingC2×C3×H(5,𝔽2)D3
prime radicalC2×C3×#994C2
cosocleC2×SL(2,𝔽3)C23×C3
hypercenterC2×C3×Q8S4
LCSG ▷ SL(2,𝔽3)
UCSC1 ◁ C22×C3 ◁ C2×C3×Q8

Complements

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

NH ≅ G/Ntype
2. C3×#5426C2semidirect
3. C3×#5424C2semidirect
4. #85528C3direct (N × H)
6. C3×#994C4semidirect
9. C3×#859C22semidirect
10. C3×#859C22semidirect
16. #5426C2×C3semidirect
17. #5424C2×C3semidirect
23. C3×#41⋊C3C2×C4semidirect
29. C3×H(5,𝔽2)Dic3semidirect
34. #994C3×C4semidirect
45. #859C22×C3semidirect
46. #859C22×C3semidirect
80. #41⋊C3C2×C3×C4semidirect
99. H(5,𝔽2)C3×Dic3semidirect
225. C3#85528direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#9942
C3×#54262
C3×#54242
#855283
C3×#28973
C3×Q8×Dic34