Finite Group Databasebeta

← C2×C3×#5570 #11903680

Subgroups — C2×C3×#5570 #11903680

Order 1152.

2245 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C32×C4×Q8C211yesmaximal, characteristic
2C2×C3×Q8×Dic3C212yesmaximal
3C3×#5570C2116yesmaximal, direct product factor
4C2×C3×#890C216yesmaximal
5C2×C3×#869C213yesmaximal
6C2×C3×C4×Dic6C213yesmaximal
7C2×C3×#557—31nomaximal
8C2×#5570C311yesmaximal, characteristic, direct product factor
9C22×C32×Q8C2211yescharacteristic
10C22×C3×Dic6C2213yes—
11C32×C4×Q8C2218yes—
12C2×C32×C4⋊C4C2213yes—
13C2×C32×C42C2213yes—
14C3×Q8×Dic3C22116yes—
15C2×C3×C4⋊Dic3C2216yes—
16C2×C3×C4⋊Dic3C2216yes—
17C2×C3×C4⋊Dic3C2211yescharacteristic
18C2×C3×Dic3⋊C4C2216yes—
19C2×C3×C4×Dic3C2216yes—
20C3×#890C22148yes—
21C3×#869C22124yes—
22C3×C4×Dic6C22124yes—
23C3×#557—316no—
24C2×C3×C4⋊Q8—33no—
25C2×C3×#125—36no—
26C2×C3×C4×Q8—21no—
27C2×C3×C4×Q8—32no—
28C2×C3×C4×Q8—33no—
29C2×C3×C4×Q8D311yescharacteristic
30C2×C3×C4×Q8C2×C311yescharacteristic
31C2×Q8×Dic3C2×C312yes—
32#5570C2×C3116yesdirect product factor
33C2×#890C2×C316yes—
34C2×#869C2×C313yes—
35C2×C4×Dic6C2×C313yes—
36C2×C32×Q8Q818yescomplemented
37C2×C32×Q8C2314yes—
38C22×C32×C4C2313yes—
39C22×C32×C4C2313yes—
40C22×C32×C4C2311yescharacteristic
41C22×C3×Dic3C2312yes—
42C22×C3×Dic3C2316yes—
43C2×C3×Dic6—212no—
44C2×C3×Dic6C23112yes—
45C32×C4⋊C4C23112yes—
46C32×C42C23112yes—
47C3×C4⋊Dic3C23124yes—
48C3×C4⋊Dic3C23124yes—
49C3×C4⋊Dic3C2314yes—
50C3×Dic3⋊C4C23124yes—
51C3×C4×Dic3C23124yes—
52C2×#557—31noSylow
53C22×C3×Q8—21no—
54C22×C3×Q8—33no—
55C22×C3×Q8C2×D311yescharacteristic
56C22×C3×Q8C22×C311yescharacteristic
57C22×Dic6C22×C313yes—
58C3×C4⋊Q8—324no—
59C3×#125—348no—
60C3×C4×Q8—28no—
61C3×C4×Q8—316no—
62C3×C4×Q8—324no—
63C3×C4×Q8C2×D318yes—
64C3×C4×Q8C22×C318yes—
65C2×C3×C4⋊C4—23no—
66C2×C3×C4⋊C4—36no—
67C2×C3×C4⋊C4—31no—
68C2×C3×C4⋊C4—36no—
69C2×C3×C4⋊C4—36no—
70C2×C3×C4⋊C4C2×D313yes—
71C2×C3×C4⋊C4C22×C313yes—
72C2×C3×C42—23no—
73C2×C3×C42—36no—
74C2×C3×C42C2×D313yes—
75C2×C3×C42C22×C313yes—
76Q8×Dic3C22×C3116yes—
77C2×C4⋊Dic3C22×C311yescharacteristic
78C2×C4⋊Dic3C22×C316yes—
79C2×C4⋊Dic3C22×C316yes—
80C2×Dic3⋊C4C22×C316yes—
81C2×C4×Dic3C22×C316yes—
82#890C22×C3148yes—
83#869C22×C3124yes—
84C4×Dic6C22×C3124yes—
85C23×C32C2411yescharacteristic
86C32×Q8C2×Q8116yescomplemented
87C2×C32×C4—26no—
88C2×C32×C4C2×Q8112yes—
89C2×C32×C4#4114yes—
90C2×C32×C4C2416yes—
91C2×C32×C4C2416yes—
92C2×C32×C4C2412yes—
93C2×C3×Dic3—212no—
94C2×C3×Dic3—24no—
95C2×C3×Dic3C2414yes—
96C2×C3×Dic3C24112yes—
97C3×Dic6—224no—
98#557—316no—
99C2×C4⋊Q8—33no—
100C2×#125—36no—
101C2×C4×Q8—32no—
102C2×C4×Q8—33no—
103C2×C4×Q8C3×D311yescharacteristic
104C2×C3×Q8—28no—
105C2×C3×Q8—24no—
106C2×C3×Q8—312no—
107C2×C3×Q8—612no—
108C2×C3×Q8Dic618yescomplemented
109C2×C3×Q8C3×Q818yescomplemented
110C2×C3×Q8C22×D314yes—
111C2×C3×Q8C23×C314yes—
112C22×C3×C4—21no—
113C22×C3×C4—23no—
114C22×C3×C4—23no—
115C22×C3×C4—32no—
116C22×C3×C4—36no—
117C22×C3×C4C22×D313yes—
118C22×C3×C4C22×D313yes—
119C22×C3×C4C22×D311yescharacteristic
120C22×C3×C4C23×C313yes—
121C22×C3×C4C23×C313yes—
122C22×C3×C4C23×C311yescharacteristic
123C22×Dic3C23×C312yes—
124C22×Dic3C23×C316yes—
125C2×Dic6—212no—
126C2×Dic6C23×C3112yes—
127C3×C4⋊C4—212no—
128C3×C4⋊C4—324no—
129C3×C4⋊C4—34no—
130C3×C4⋊C4—324no—
131C3×C4⋊C4—324no—
132C3×C4⋊C4C22×D3112yes—
133C3×C4⋊C4C23×C3112yes—
134C3×C42—212no—
135C3×C42—324no—
136C3×C42C22×D3112yes—
137C3×C42C23×C3112yes—
138C4⋊Dic3C23×C314yes—
139C4⋊Dic3C23×C3124yes—
140C4⋊Dic3C23×C3124yes—
141Dic3⋊C4C23×C3124yes—
142C4×Dic3C23×C3124yes—
143C22×C32C22×Q812yes—
144C22×C32C2×#4112yes—
145C22×C322−1+412yes—
146C22×C32C2511yescharacteristic
147C32×C4—26no—
148C32×C4C22×Q8112yes—
149C32×C4C2×#4114yes—
150C3×Dic3—212no—
151C3×Dic3—24no—
152C22×Q8—33no—
153C22×Q8C2×C3×D311yescharacteristic
154C4⋊Q8—324no—
155#125—348no—
156C4×Q8—316no—
157C4×Q8—324no—
158C4×Q8C2×C3×D318yes—
159C2×C4⋊C4—31no—
160C2×C4⋊C4—36no—
161C2×C4⋊C4—36no—
162C2×C4⋊C4—36no—
163C2×C4⋊C4C2×C3×D313yes—
164C2×C42—36no—
165C2×C42C2×C3×D313yes—
166C23×C3—21no—
167C23×C3C23×D311yescharacteristic
168C23×C3C24×C311yescharacteristic
169C3×Q8—216no—
170C3×Q8—624no—
171C3×Q8C2×Dic6116yescomplemented
172C3×Q8C2×C3×Q8116yescomplemented
173C2×C3×C4—26no—
174C2×C3×C4—26no—
175C2×C3×C4—22no—
176C2×C3×C4—26no—
177C2×C3×C4—26no—
178C2×C3×C4—212no—
179C2×C3×C4—24no—
180C2×C3×C4—34no—
181C2×C3×C4—312no—
182C2×C3×C4—46no—
183C2×C3×C4—612no—
184C2×C3×C4—64no—
185C2×C3×C4C2×Dic6112yes—
186C2×C3×C4#23914yes—
187C2×C3×C4C2×C3×Q8112yes—
188C2×C3×C4C3×#4114yes—
189C2×C3×C4C23×D316yes—
190C2×C3×C4C23×D316yes—
191C2×C3×C4C23×D312yes—
192C2×C3×C4C24×C316yes—
193C2×C3×C4C24×C316yes—
194C2×C3×C4C24×C312yes—
195C2×Dic3—212no—
196C2×Dic3—24no—
197C2×Dic3C24×C314yes—
198C2×Dic3C24×C3112yes—
199Dic6—224no—
200C2×C32#55714yescomplemented
201C2×C32C23×Q811yescharacteristic
202C2×C32C22×#4111yescharacteristic
203C2×C32C2×2−1+411yescharacteristic
204C2×Q8—312no—
205C2×Q8—612no—
206C2×Q8C3×Dic618yescomplemented
207C2×Q8C22×C3×D314yes—
208C22×C4—32no—
209C22×C4—36no—
210C22×C4C22×C3×D313yes—
211C22×C4C22×C3×D313yes—
212C22×C4C22×C3×D311yescharacteristic
213C4⋊C4—34no—
214C4⋊C4—324no—
215C4⋊C4—324no—
216C4⋊C4—324no—
217C4⋊C4C22×C3×D3112yes—
218C42—324no—
219C42C22×C3×D3112yes—
220C22×C3—22no—
221C22×C3—22no—
222C22×C3—22no—
223C22×C3—21no—
224C22×C3C22×Dic612yes—
225C22×C3C2×#23912yes—
226C22×C3#101012yes—
227C22×C3C22×C3×Q812yes—
228C22×C3C2×C3×#4112yes—
229C22×C3C3×2−1+412yes—
230C22×C3C24×D311yescharacteristic
231C22×C3C25×C311yescharacteristic
232C3×C4—26no—
233C3×C4—26no—
234C3×C4—212no—
235C3×C4—24no—
236C3×C4—46no—
237C3×C4—612no—
238C3×C4—64no—
239C3×C4C22×Dic6112yes—
240C3×C4C2×#23914yes—
241C3×C4C22×C3×Q8112yes—
242C3×C4C2×C3×#4114yes—
243Dic3—212no—
244Dic3—24no—
245C32C2×#55711yescharacteristic, complemented, Sylow
246C23C23×C3×D311yescharacteristic
247Q8—624no—
248Q8C2×C3×Dic6116yescomplemented
249C2×C4—26no—
250C2×C4—34no—
251C2×C4—312no—
252C2×C4—612no—
253C2×C4—64no—
254C2×C4C2×C3×Dic6112yes—
255C2×C4C3×#23914yes—
256C2×C4C23×C3×D316yes—
257C2×C4C23×C3×D316yes—
258C2×C4C23×C3×D312yes—
259C2×C3—24no—
260C2×C3—21no—
261C2×C3—21no—
262C2×C3—21no—
263C2×C3#557014yesdirect product factor
264C2×C3C3×#55714yescomplemented
265C2×C3C23×Dic611yescharacteristic
266C2×C3C22×#23911yescharacteristic
267C2×C3C2×#101011yescharacteristic
268C2×C3C23×C3×Q811yescharacteristic
269C2×C3C22×C3×#4111yescharacteristic
270C2×C3C2×C3×2−1+411yescharacteristic
271C22C22×C3×Dic612yes—
272C22C2×C3×#23912yes—
273C22C3×#101012yes—
274C22C24×C3×D311yescharacteristic
275C4—26no—
276C4—612no—
277C4—64no—
278C4C22×C3×Dic6112yes—
279C4C2×C3×#23914yes—
280C3—21no—
281C3C2×#557011yescharacteristic, direct product factor
282C3C2×C3×#55711yescharacteristic, complemented
283C2C3×#557014yesdirect product factor
284C2C23×C3×Dic611yescharacteristic
285C2C22×C3×#23911yescharacteristic
286C2C2×C3×#101011yescharacteristic
subgroupG/H
centerC23×C3C23×D3
commutator, cosocleC22×C3C25×C3
FrattiniC22C24×C3×D3
FittingC2×C32×C4×Q8C2
socle, prime radicalC23×C32C24
hypercenterC2×C3×C4×Q8D3
hypocenterC3C2×C3×#557
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C23×C3 ◁ C2×C3×C4×Q8

Complements

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

NH ≅ G/Ntype
3. C3×#5570C2direct (N × H)
8. C2×#5570C3direct (N × H)
32. #5570C2×C3direct (N × H)
36. C2×C32×Q8Q8semidirect
86. C32×Q8C2×Q8semidirect
108. C2×C3×Q8Dic6semidirect
109. C2×C3×Q8C3×Q8semidirect
171. C3×Q8C2×Dic6semidirect
172. C3×Q8C2×C3×Q8semidirect
200. C2×C32#557semidirect
206. C2×Q8C3×Dic6semidirect
245. C32C2×#557semidirect
248. Q8C2×C3×Dic6semidirect
263. C2×C3#5570direct (N × H)
264. C2×C3C3×#557semidirect
281. C3C2×#5570direct (N × H)
282. C3C2×C3×#557semidirect
283. C2C3×#5570direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C32×C4×Q82
C2×C3×Q8×Dic32
C3×#55702
C2×C3×#8902
C2×C3×#8692
C2×C3×C4×Dic62
C2×C3×#5573
C2×#55703