Finite Group Databasebeta

← C3×#86571 #11901932

Subgroups — C3×#86571 #11901932

Order 1152.

2113 conjugacy classes of subgroups.

The subgroup lattice has 271 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×#543C211yesmaximal, characteristic
2C3×#5623C2112yesmaximal, complemented
3C3×#5622C2112yesmaximal, complemented
4C3×#5483C213yesmaximal
5C2×C3×#877C213yesmaximal
6C3×#3458—31nomaximal
7#86571C311yesmaximal, characteristic, direct product factor
8C23×C3×Dic3C2211yescharacteristic
9C32×C22⋊Q8C22112yes—
10C2×C32×C22⋊C4C2213yes—
11C2×C3×#217C2213yes—
12C3×Q8×Dic3C2214yescomplemented
13C3×#944C22112yescomplemented
14C3×#924C22112yes—
15C2×C3×Dic3⋊C4C2216yes—
16C2×C3×C4×Dic3C2216yes—
17C3×#889C22124yescomplemented
18C3×#888C22112yescomplemented
19C3×#887C22112yescomplemented
20C3×#878C22124yes—
21C3×#877C22112yes—
22C3×#877C22112yes—
23C3×#544—312no—
24C3×#543—21no—
25C3×#543—33no—
26C3×#543D311yescharacteristic
27C3×#543C2×C311yescharacteristic
28C3×#533—312no—
29C2×C3×#117—33no—
30#5623C2×C3112yescomplemented
31#5622C2×C3112yescomplemented
32#5483C2×C313yes—
33C2×#877C2×C313yes—
34C24×C32C2311yescharacteristic
35C2×C32×Q8C2314yes—
36C22×C32×C4C2316yes—
37C22×C3×Dic3Q818yescomplemented
38C22×C3×Dic3C2316yes—
39C22×C3×Dic3C2313yes—
40C22×C3×Dic3C2313yes—
41C2×C3×Dic6C23112yes—
42C32×C4⋊C4C23112yes—
43C32×C22⋊C4C23112yes—
44C3×#217C23112yes—
45C3×C4⋊Dic3C23112yes—
46C3×Dic3⋊C4C23124yes—
47C3×Dic3⋊C4C23124yes—
48C3×C4×Dic3C23112yes—
49C3×C4×Dic3C23112yes—
50#3458—31noSylow
51C23×C3×C4—31no—
52C23×Dic3C22×C311yescharacteristic
53C3×C4⋊Q8—312no—
54C3×C4⋊Q8—312no—
55C3×#125—324no—
56C3×C22⋊Q8—212no—
57C3×C22⋊Q8—312no—
58C3×C22⋊Q8—324no—
59C3×C22⋊Q8C2×D3112yes—
60C3×C22⋊Q8C22×C3112yes—
61C3×C4×Q8—34no—
62C3×C4×Q8—312no—
63C3×#117—312no—
64C3×#117—312no—
65C2×C3×C4⋊C4—36no—
66C2×C3×C22⋊C4—23no—
67C2×C3×C22⋊C4—33no—
68C2×C3×C22⋊C4C2×D313yes—
69C2×C3×C22⋊C4C22×C313yes—
70C2×C3×C42—36no—
71C2×#217C22×C313yes—
72Q8×Dic3C22×C314yescomplemented
73#944C22×C3112yescomplemented
74#924C22×C3112yes—
75C2×Dic3⋊C4C22×C316yes—
76C2×C4×Dic3C22×C316yes—
77#889C22×C3124yescomplemented
78#888C22×C3112yescomplemented
79#887C22×C3112yescomplemented
80#878C22×C3124yes—
81#877C22×C3112yes—
82#877C22×C3112yes—
83C23×C32—42no—
84C23×C32C2×Q814yes—
85C23×C32C2413yes—
86C32×Q8—44no—
87C2×C32×C4—46no—
88C2×C32×C4C24112yes—
89C2×C3×Dic3—46no—
90C2×C3×Dic3C2×Q8112yes—
91C2×C3×Dic3C2×Q8112yes—
92C2×C3×Dic3C24112yes—
93C2×C3×Dic3C2411yescharacteristic
94C2×C3×Dic3C2413yes—
95C3×Dic6—412no—
96#544—312no—
97#543—33no—
98#543C3×D311yescharacteristic
99#533—312no—
100C2×#117—33no—
101C24×C3—21no—
102C24×C3C22×D311yescharacteristic
103C24×C3C23×C311yescharacteristic
104C2×C3×Q8—24no—
105C2×C3×Q8—312no—
106C2×C3×Q8C22×D314yes—
107C2×C3×Q8C23×C314yes—
108C22×C3×C4—26no—
109C22×C3×C4—38no—
110C22×C3×C4—33no—
111C22×C3×C4—33no—
112C22×C3×C4—36no—
113C22×C3×C4C22×D316yes—
114C22×C3×C4C23×C316yes—
115C22×Dic3C3×Q818yescomplemented
116C22×Dic3C23×C316yes—
117C22×Dic3C23×C313yes—
118C22×Dic3C23×C313yes—
119C2×Dic6C23×C3112yes—
120C3×C4⋊C4—212no—
121C3×C4⋊C4—324no—
122C3×C4⋊C4—324no—
123C3×C4⋊C4—312no—
124C3×C4⋊C4C22×D3112yes—
125C3×C4⋊C4C23×C3112yes—
126C3×C22⋊C4—212no—
127C3×C22⋊C4—312no—
128C3×C22⋊C4C22×D3112yes—
129C3×C22⋊C4C23×C3112yes—
130C3×C42—312no—
131C3×C42—312no—
132#217C23×C3112yes—
133C4⋊Dic3C23×C3112yes—
134Dic3⋊C4C23×C3124yes—
135Dic3⋊C4C23×C3124yes—
136C4×Dic3C23×C3112yes—
137C4×Dic3C23×C3112yes—
138C22×C32—26no—
139C22×C32—44no—
140C22×C32C22×Q816yes—
141C22×C32C2511yescharacteristic
142C32×C4—212no—
143C3×Dic3—212no—
144C3×Dic3C22×Q812yes—
145C3×Dic3C22×Q816yes—
146C23×C4—31no—
147C4⋊Q8—312no—
148C4⋊Q8—312no—
149#125—324no—
150C22⋊Q8—312no—
151C22⋊Q8—324no—
152C22⋊Q8C2×C3×D3112yes—
153C4×Q8—34no—
154C4×Q8—312no—
155#117—312no—
156#117—312no—
157C2×C4⋊C4—36no—
158C2×C22⋊C4—33no—
159C2×C22⋊C4C2×C3×D313yes—
160C2×C42—36no—
161C23×C3—23no—
162C23×C3—24no—
163C23×C3—42no—
164C23×C3—42no—
165C23×C3—82no—
166C23×C3D3×Q814yes—
167C23×C3C2×C3×Q814yes—
168C23×C3C23×D313yes—
169C23×C3C24×C313yes—
170C3×Q8—44no—
171C3×Q8—44no—
172C3×Q8—84no—
173C3×Q8—1212no—
174C2×C3×C4—212no—
175C2×C3×C4—312no—
176C2×C3×C4—312no—
177C2×C3×C4—31no—
178C2×C3×C4—33no—
179C2×C3×C4—312no—
180C2×C3×C4—46no—
181C2×C3×C4—46no—
182C2×C3×C4—86no—
183C2×C3×C4—126no—
184C2×C3×C4C23×D3112yes—
185C2×C3×C4C24×C3112yes—
186C2×Dic3—46no—
187C2×Dic3C2×C3×Q8112yes—
188C2×Dic3C2×C3×Q8112yes—
189C2×Dic3C24×C3112yes—
190C2×Dic3C24×C311yescharacteristic
191C2×Dic3C24×C313yes—
192Dic6—412no—
193C2×C32—26no—
194C2×C32C23×Q811yescharacteristic
195C2×C32#58512yes—
196C24C22×C3×D311yescharacteristic
197C2×Q8—312no—
198C2×Q8C22×C3×D314yes—
199C22×C4—33no—
200C22×C4—33no—
201C22×C4—38no—
202C22×C4—36no—
203C22×C4C22×C3×D316yes—
204C4⋊C4—324no—
205C4⋊C4—324no—
206C4⋊C4—312no—
207C4⋊C4C22×C3×D3112yes—
208C22⋊C4—312no—
209C22⋊C4C22×C3×D3112yes—
210C42—312no—
211C42—312no—
212C22×C3—26no—
213C22×C3—26no—
214C22×C3—26no—
215C22×C3—21no—
216C22×C3—46no—
217C22×C3—44no—
218C22×C3—44no—
219C22×C3—84no—
220C22×C3C2×D3×Q816yes—
221C22×C3C22×C3×Q816yes—
222C22×C3C24×D311yescharacteristic
223C22×C3C25×C311yescharacteristic
224C3×C4—212no—
225C3×C4—212no—
226C3×C4—32no—
227C3×C4—36no—
228C3×C4—412no—
229C3×C4—612no—
230Dic3—212no—
231Dic3C22×C3×Q812yes—
232Dic3C22×C3×Q816yes—
233C32#345811yescharacteristic, complemented, Sylow
234C23—42no—
235C23C3×D3×Q814yes—
236C23C23×C3×D313yes—
237Q8—44no—
238Q8—1212no—
239C2×C4—312no—
240C2×C4—312no—
241C2×C4—31no—
242C2×C4—33no—
243C2×C4—312no—
244C2×C4—46no—
245C2×C4—126no—
246C2×C4C23×C3×D3112yes—
247C2×C3—26no—
248C2×C3—26no—
249C2×C3—21no—
250C2×C3—22no—
251C2×C3—46no—
252C2×C3C22×D3×Q811yescharacteristic
253C2×C3#596612yes—
254C2×C3C23×C3×Q811yescharacteristic
255C2×C3C3×#58512yes—
256C22—26no—
257C22—44no—
258C22C2×C3×D3×Q816yes—
259C22C24×C3×D311yescharacteristic
260C4—212no—
261C4—32no—
262C4—36no—
263C4—612no—
264C3—21no—
265C3#8657111yescharacteristic, direct product factor
266C3C3×#345811yescharacteristic, complemented
267C2—26no—
268C2C22×C3×D3×Q811yescharacteristic
269C2C3×#596612yes—
subgroupG/H
centerC22×C3C24×D3
commutator, cosocleC22×C3C25×C3
FrattiniC22C24×C3×D3
FittingC32×#543C2
socleC22×C32C25
prime radicalC24×C32C23
hypercenterC3×#543D3
hypocenterC3C3×#3458
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C3×#543

Complements

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

NH ≅ G/Ntype
2. C3×#5623C2semidirect
3. C3×#5622C2semidirect
7. #86571C3direct (N × H)
12. C3×Q8×Dic3C22semidirect
13. C3×#944C22semidirect
17. C3×#889C22semidirect
18. C3×#888C22semidirect
19. C3×#887C22semidirect
30. #5623C2×C3semidirect
31. #5622C2×C3semidirect
37. C22×C3×Dic3Q8semidirect
72. Q8×Dic3C22×C3semidirect
73. #944C22×C3semidirect
77. #889C22×C3semidirect
78. #888C22×C3semidirect
79. #887C22×C3semidirect
115. C22×Dic3C3×Q8semidirect
233. C32#3458semidirect
265. C3#86571direct (N × H)
266. C3C3×#3458semidirect

Maximal subgroups

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

subgroupindex
C32×#5432
C3×#56232
C3×#56222
C3×#54832
C2×C3×#8772
C3×#34583
#865713