Finite Group Databasebeta

← C2×C3×#5518 #11901552

Subgroups — C2×C3×#5518 #11901552

Order 1152.

2173 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C32×#117C211yesmaximal, characteristic
2C2×C3×#924C212yesmaximal
3C3×#5518C2116yesmaximal, direct product factor
4C2×C3×#889C214yesmaximal, complemented
5C2×C3×#879C214yesmaximal
6C2×C3×#870C211yesmaximal, characteristic, complemented
7C2×C3×#869C211yesmaximal, characteristic, complemented
8C2×C3×C4×Dic6C212yesmaximal, complemented
9C2×C3×#541—31nomaximal
10C2×#5518C311yesmaximal, characteristic, direct product factor
11C23×C32×C4C2211yescharacteristic
12C22×C3×Dic6C2212yes—
13C32×#117C2218yes—
14C2×C32×C4⋊C4C2212yes—
15C2×C32×C22⋊C4C2212yes—
16C2×C32×C42C2211yescharacteristic
17C2×C32×C42C2211yescharacteristic
18C2×C3×#217C2214yes—
19C2×C3×C4⋊Dic3C2214yes—
20C2×C3×C4⋊Dic3C2212yes—
21C3×#924C22116yes—
22C2×C3×Dic3⋊C4C2214yes—
23C2×C3×Dic3⋊C4C2214yes—
24C2×C3×Dic3⋊C4C2214yes—
25C2×C3×C4×Dic3C2214yes—
26C3×#889C22132yescomplemented
27C3×#879C22132yes—
28C3×#870C2218yescomplemented
29C3×#869C2218yescomplemented
30C3×C4×Dic6C22116yescomplemented
31C3×#541—316no—
32C2×C3×C4⋊Q8—31no—
33C2×C3×#126—34no—
34C2×C3×#125—34no—
35C2×C3×#125—31no—
36C2×C3×C22⋊Q8—32no—
37C2×C3×C4×Q8—32no—
38C2×C3×#117—21no—
39C2×C3×#117D311yescharacteristic
40C2×C3×#117C2×C311yescharacteristic
41C2×#924C2×C312yes—
42#5518C2×C3116yesdirect product factor
43C2×#889C2×C314yescomplemented
44C2×#879C2×C314yes—
45C2×#870C2×C311yescharacteristic, complemented
46C2×#869C2×C311yescharacteristic, complemented
47C2×C4×Dic6C2×C312yescomplemented
48C24×C32C2311yescharacteristic
49C22×C32×C4—24no—
50C22×C32×C4C2312yes—
51C22×C32×C4C2311yescharacteristic
52C22×C32×C4C2312yes—
53C22×C32×C4C2314yes—
54C22×C32×C4C2311yescharacteristic
55C22×C3×Dic3C2314yes—
56C22×C3×Dic3C2314yes—
57C2×C3×Dic6—28no—
58C2×C3×Dic6C2318yes—
59C32×C4⋊C4C2318yes—
60C32×C22⋊C4C2318yes—
61C32×C42C2314yes—
62C32×C42C2314yes—
63C3×#217C23116yes—
64C3×C4⋊Dic3C23116yes—
65C3×C4⋊Dic3C2318yes—
66C3×Dic3⋊C4C23116yes—
67C3×Dic3⋊C4C23116yes—
68C3×Dic3⋊C4C23116yes—
69C3×C4×Dic3C23116yes—
70C2×#541—31noSylow
71C22×C3×Q8—32no—
72C23×C3×C4—21no—
73C23×C3×C4C2×D311yescharacteristic
74C23×C3×C4C22×C311yescharacteristic
75C22×Dic6C22×C312yes—
76C3×C4⋊Q8—38no—
77C3×#126—332no—
78C3×#125—332no—
79C3×#125—38no—
80C3×C22⋊Q8—316no—
81C3×C4×Q8—316no—
82C3×#117—28no—
83C3×#117C2×D318yes—
84C3×#117C22×C318yes—
85C2×C3×C4⋊C4—22no—
86C2×C3×C4⋊C4—34no—
87C2×C3×C4⋊C4—34no—
88C2×C3×C4⋊C4—34no—
89C2×C3×C4⋊C4—32no—
90C2×C3×C4⋊C4—34no—
91C2×C3×C4⋊C4C2×D312yes—
92C2×C3×C4⋊C4C22×C312yes—
93C2×C3×C22⋊C4—22no—
94C2×C3×C22⋊C4—34no—
95C2×C3×C22⋊C4C2×D312yes—
96C2×C3×C22⋊C4C22×C312yes—
97C2×C3×C42—21no—
98C2×C3×C42—21no—
99C2×C3×C42—34no—
100C2×C3×C42C2×D311yescharacteristic
101C2×C3×C42C2×D311yescharacteristic
102C2×C3×C42C22×C311yescharacteristic
103C2×C3×C42C22×C311yescharacteristic
104C2×#217C22×C314yes—
105C2×C4⋊Dic3C22×C312yes—
106C2×C4⋊Dic3C22×C314yes—
107#924C22×C3116yes—
108C2×Dic3⋊C4C22×C314yes—
109C2×Dic3⋊C4C22×C314yes—
110C2×Dic3⋊C4C22×C314yes—
111C2×C4×Dic3C22×C314yes—
112#889C22×C3132yescomplemented
113#879C22×C3132yes—
114#870C22×C318yescomplemented
115#869C22×C318yescomplemented
116C4×Dic6C22×C3116yescomplemented
117C23×C32—22no—
118C23×C32—24no—
119C23×C32C2412yes—
120C23×C32C2411yescharacteristic
121C2×C32×C4—22no—
122C2×C32×C4—28no—
123C2×C32×C4—24no—
124C2×C32×C4—24no—
125C2×C32×C4#4114yes—
126C2×C32×C4C2414yes—
127C2×C32×C4C2412yes—
128C2×C32×C4C2414yes—
129C2×C32×C4C2412yes—
130C2×C3×Dic3—28no—
131C2×C3×Dic3—28no—
132C2×C3×Dic3C2418yes—
133C2×C3×Dic3C2418yes—
134C3×Dic6—216no—
135#541—316no—
136C2×C4⋊Q8—31no—
137C2×#126—34no—
138C2×#125—34no—
139C2×#125—31no—
140C2×C22⋊Q8—32no—
141C2×C4×Q8—32no—
142C2×#117C3×D311yescharacteristic
143C24×C3—21no—
144C24×C3C22×D311yescharacteristic
145C24×C3C23×C311yescharacteristic
146C2×C3×Q8—38no—
147C2×C3×Q8—68no—
148C22×C3×C4—28no—
149C22×C3×C4—24no—
150C22×C3×C4—24no—
151C22×C3×C4—21no—
152C22×C3×C4—21no—
153C22×C3×C4—24no—
154C22×C3×C4—22no—
155C22×C3×C4—22no—
156C22×C3×C4—34no—
157C22×C3×C4—34no—
158C22×C3×C4C22×D314yes—
159C22×C3×C4C22×D312yes—
160C22×C3×C4C22×D312yes—
161C22×C3×C4C22×D311yescharacteristic
162C22×C3×C4C22×D311yescharacteristic
163C22×C3×C4C23×C312yes—
164C22×C3×C4C23×C311yescharacteristic
165C22×C3×C4C23×C312yes—
166C22×C3×C4C23×C314yes—
167C22×C3×C4C23×C311yescharacteristic
168C22×Dic3C23×C314yes—
169C22×Dic3C23×C314yes—
170C2×Dic6—28no—
171C2×Dic6C23×C318yes—
172C3×C4⋊C4—28no—
173C3×C4⋊C4—316no—
174C3×C4⋊C4—316no—
175C3×C4⋊C4—316no—
176C3×C4⋊C4—38no—
177C3×C4⋊C4—316no—
178C3×C4⋊C4C22×D318yes—
179C3×C4⋊C4C23×C318yes—
180C3×C22⋊C4—28no—
181C3×C22⋊C4—316no—
182C3×C22⋊C4C22×D318yes—
183C3×C22⋊C4C23×C318yes—
184C3×C42—24no—
185C3×C42—24no—
186C3×C42—316no—
187C3×C42C22×D314yes—
188C3×C42C22×D314yes—
189C3×C42C23×C314yes—
190C3×C42C23×C314yes—
191#217C23×C3116yes—
192C4⋊Dic3C23×C318yes—
193C4⋊Dic3C23×C3116yes—
194Dic3⋊C4C23×C3116yes—
195Dic3⋊C4C23×C3116yes—
196Dic3⋊C4C23×C3116yes—
197C4×Dic3C23×C3116yes—
198C22×C32—22no—
199C22×C32—24no—
200C22×C32—44no—
201C22×C32C2×#4112yes—
202C22×C322−1+414yes—
203C22×C32C2511yescharacteristic
204C32×C4—22no—
205C32×C4—24no—
206C32×C4—24no—
207C32×C4C2×#4114yes—
208C3×Dic3—28no—
209C3×Dic3—28no—
210C22×Q8—32no—
211C23×C4C2×C3×D311yescharacteristic
212C4⋊Q8—38no—
213#126—332no—
214#125—332no—
215#125—38no—
216C22⋊Q8—316no—
217C4×Q8—316no—
218#117C2×C3×D318yes—
219C2×C4⋊C4—32no—
220C2×C4⋊C4—34no—
221C2×C4⋊C4—34no—
222C2×C4⋊C4—34no—
223C2×C4⋊C4—34no—
224C2×C4⋊C4C2×C3×D312yes—
225C2×C22⋊C4—34no—
226C2×C22⋊C4C2×C3×D312yes—
227C2×C42—34no—
228C2×C42C2×C3×D311yescharacteristic
229C2×C42C2×C3×D311yescharacteristic
230C23×C3—24no—
231C23×C3—22no—
232C23×C3—24no—
233C23×C3—22no—
234C23×C3—24no—
235C23×C3—21no—
236C23×C3—22no—
237C23×C3—44no—
238C23×C3C23×D312yes—
239C23×C3C23×D311yescharacteristic
240C23×C3C24×C312yes—
241C23×C3C24×C311yescharacteristic
242C3×Q8—616no—
243C2×C3×C4—24no—
244C2×C3×C4—216no—
245C2×C3×C4—22no—
246C2×C3×C4—24no—
247C2×C3×C4—24no—
248C2×C3×C4—22no—
249C2×C3×C4—24no—
250C2×C3×C4—24no—
251C2×C3×C4—28no—
252C2×C3×C4—28no—
253C2×C3×C4—22no—
254C2×C3×C4—22no—
255C2×C3×C4—24no—
256C2×C3×C4—24no—
257C2×C3×C4—24no—
258C2×C3×C4—38no—
259C2×C3×C4—38no—
260C2×C3×C4—44no—
261C2×C3×C4—44no—
262C2×C3×C4—68no—
263C2×C3×C4—68no—
264C2×C3×C4#23514yes—
265C2×C3×C4C3×#4114yes—
266C2×C3×C4C23×D314yes—
267C2×C3×C4C23×D314yes—
268C2×C3×C4C23×D312yes—
269C2×C3×C4C23×D312yes—
270C2×C3×C4C24×C314yes—
271C2×C3×C4C24×C312yes—
272C2×C3×C4C24×C314yes—
273C2×C3×C4C24×C312yes—
274C2×Dic3—28no—
275C2×Dic3—28no—
276C2×Dic3C24×C318yes—
277C2×Dic3C24×C318yes—
278Dic6—216no—
279C2×C32—42no—
280C2×C32#54114yescomplemented
281C2×C32C22×#4111yescharacteristic
282C2×C32C2×2−1+412yes—
283C24C22×C3×D311yescharacteristic
284C2×Q8—38no—
285C2×Q8—68no—
286C22×C4—24no—
287C22×C4—34no—
288C22×C4—34no—
289C22×C4C22×C3×D312yes—
290C22×C4C22×C3×D312yes—
291C22×C4C22×C3×D311yescharacteristic
292C22×C4C22×C3×D314yes—
293C22×C4C22×C3×D311yescharacteristic
294C4⋊C4—38no—
295C4⋊C4—316no—
296C4⋊C4—316no—
297C4⋊C4—316no—
298C4⋊C4—316no—
299C4⋊C4C22×C3×D318yes—
300C22⋊C4—316no—
301C22⋊C4C22×C3×D318yes—
302C42—316no—
303C42C22×C3×D314yes—
304C42C22×C3×D314yes—
305C22×C3—24no—
306C22×C3—22no—
307C22×C3—24no—
308C22×C3—22no—
309C22×C3—24no—
310C22×C3—22no—
311C22×C3—24no—
312C22×C3—21no—
313C22×C3—44no—
314C22×C3—48no—
315C22×C3—44no—
316C22×C3—44no—
317C22×C3C2×#23512yes—
318C22×C3#101014yes—
319C22×C3C2×C3×#4112yes—
320C22×C3C3×2−1+414yes—
321C22×C3C24×D311yescharacteristic
322C22×C3C25×C311yescharacteristic
323C3×C4—24no—
324C3×C4—22no—
325C3×C4—24no—
326C3×C4—24no—
327C3×C4—22no—
328C3×C4—24no—
329C3×C4—24no—
330C3×C4—24no—
331C3×C4—44no—
332C3×C4—44no—
333C3×C4—68no—
334C3×C4—68no—
335C3×C4C2×#23514yes—
336C3×C4C2×C3×#4114yes—
337Dic3—28no—
338Dic3—28no—
339C32C2×#54111yescharacteristic, complemented, Sylow
340C23—22no—
341C23—24no—
342C23C23×C3×D312yes—
343C23C23×C3×D311yescharacteristic
344Q8—616no—
345C2×C4—22no—
346C2×C4—24no—
347C2×C4—24no—
348C2×C4—28no—
349C2×C4—38no—
350C2×C4—38no—
351C2×C4—68no—
352C2×C4—68no—
353C2×C4C3×#23514yes—
354C2×C4C23×C3×D314yes—
355C2×C4C23×C3×D314yes—
356C2×C4C23×C3×D312yes—
357C2×C4C23×C3×D312yes—
358C2×C3—24no—
359C2×C3—21no—
360C2×C3—22no—
361C2×C3—44no—
362C2×C3—42no—
363C2×C3—42no—
364C2×C3#551814yesdirect product factor
365C2×C3C3×#54114yescomplemented
366C2×C3C22×#23511yescharacteristic
367C2×C3C2×#101012yes—
368C2×C3C22×C3×#4111yescharacteristic
369C2×C3C2×C3×2−1+412yes—
370C22—22no—
371C22—24no—
372C22—44no—
373C22C2×C3×#23512yes—
374C22C3×#101014yes—
375C22C24×C3×D311yescharacteristic
376C4—22no—
377C4—24no—
378C4—24no—
379C4—68no—
380C4—68no—
381C4C2×C3×#23514yes—
382C3—21no—
383C3C2×#551811yescharacteristic, direct product factor
384C3C2×C3×#54111yescharacteristic, complemented
385C2—42no—
386C2C3×#551814yesdirect product factor
387C2C22×C3×#23511yescharacteristic
388C2C2×C3×#101012yes—
subgroupG/H
centerC23×C3C23×D3
commutator, cosocleC22×C3C25×C3
FrattiniC22C24×C3×D3
FittingC2×C32×#117C2
socleC23×C32C24
prime radicalC24×C32C23
hypercenterC2×C3×#117D3
hypocenterC3C2×C3×#541
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C23×C3 ◁ C2×C3×#117

Complements

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

NH ≅ G/Ntype
3. C3×#5518C2direct (N × H)
4. C2×C3×#889C2semidirect
6. C2×C3×#870C2semidirect
7. C2×C3×#869C2semidirect
8. C2×C3×C4×Dic6C2semidirect
10. C2×#5518C3direct (N × H)
26. C3×#889C22semidirect
28. C3×#870C22semidirect
29. C3×#869C22semidirect
30. C3×C4×Dic6C22semidirect
42. #5518C2×C3direct (N × H)
43. C2×#889C2×C3semidirect
45. C2×#870C2×C3semidirect
46. C2×#869C2×C3semidirect
47. C2×C4×Dic6C2×C3semidirect
112. #889C22×C3semidirect
114. #870C22×C3semidirect
115. #869C22×C3semidirect
116. C4×Dic6C22×C3semidirect
280. C2×C32#541semidirect
339. C32C2×#541semidirect
364. C2×C3#5518direct (N × H)
365. C2×C3C3×#541semidirect
383. C3C2×#5518direct (N × H)
384. C3C2×C3×#541semidirect
386. C2C3×#5518direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C32×#1172
C2×C3×#9242
C3×#55182
C2×C3×#8892
C2×C3×#8792
C2×C3×#8702
C2×C3×#8692
C2×C3×C4×Dic62
C2×C3×#5413
C2×#55183