Finite Group Databasebeta

← C3×#73886 #11807797

Subgroups — C3×#73886 #11807797

Order 1152.

1045 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#924C211yesmaximal, characteristic
2C32×#380C211yesmaximal, characteristic
3C3×#5211C211yesmaximal, characteristic
4C3×#4978C212yesmaximal, complemented
5C3×#4976C212yesmaximal, complemented
6C3×#4945C212yesmaximal
7C3×#4944C212yesmaximal
8C3×#4654C212yesmaximal, complemented
9C3×#4646C212yesmaximal, complemented
10C3×#2378—31nomaximal
11#73886C311yesmaximal, characteristic, direct product factor
12C23×C32×C4C2211yescharacteristic
13C22×C3×Dic6C2211yescharacteristic
14C2×C32×C4⋊C4C2212yes—
15C2×C32×C22⋊C4C2211yescharacteristic
16C2×C32×C22⋊C4C2211yescharacteristic
17C2×C3×#217C2212yes—
18C2×C3×#217C2212yes—
19C2×C3×C4⋊Dic3C2211yescharacteristic
20C3×#924C418yescomplemented
21C2×C3×Dic3⋊C4C2212yes—
22C2×C3×Dic3⋊C4C2212yes—
23C2×C3×Dic3⋊C4C2212yes—
24C2×C3×C4×Dic3C2212yes—
25C2×C3×C4×Dic3C2212yes—
26C32×#95C2212yes—
27C3×C3⋊#95C2212yes—
28C3×C3⋊#95C2212yes—
29C3×C3⋊#95C2212yes—
30C3×C3⋊#95C2212yes—
31C3×C3⋊#95C2212yes—
32C3×C3⋊#95C2211yescharacteristic
33C3×C3⋊#95C2211yescharacteristic
34C2×C3×C22⋊Q8—31no—
35C2×#924C2×C311yescharacteristic
36C3×#391—32no—
37C3×#389—32no—
38C3×#388—32no—
39C3×#388—32no—
40C3×#387—32no—
41C3×#387—32no—
42C3×#381—31no—
43C3×#380—21no—
44C3×#380D311yescharacteristic
45C3×#380C2×C311yescharacteristic
46#5211C2×C311yescharacteristic
47#4978C2×C312yescomplemented
48#4976C2×C312yescomplemented
49#4945C2×C312yes—
50#4944C2×C312yes—
51#4654C2×C312yescomplemented
52#4646C2×C312yescomplemented
53C24×C32C2311yescharacteristic
54C22×C32×C4—42no—
55C22×C32×C4C2×C414yes—
56C22×C32×C4C2312yes—
57C22×C32×C4C2312yes—
58C22×C32×C4C2311yescharacteristic
59C22×C32×C4C2311yescharacteristic
60C22×C3×Dic3C2312yes—
61C22×C3×Dic3C2312yes—
62C22×C3×Dic3C2312yes—
63C22×C3×Dic3C2312yes—
64C2×C3×Dic6—42no—
65C2×C3×Dic6C2×C414yes—
66C32×C4⋊C4—42no—
67C32×C22⋊C4—41no—
68C32×C22⋊C4—41no—
69C3×#217—42no—
70C3×#217C2×C418yes—
71C3×C4⋊Dic3C2×C414yes—
72C3×Dic3⋊C4—42no—
73C3×Dic3⋊C4—42no—
74C3×Dic3⋊C4C2×C418yes—
75C3×C4×Dic3—42no—
76C3×C4×Dic3—42no—
77#2378—31noSylow
78C22×C3×Q8—31no—
79C23×C3×C4—21no—
80C23×C3×C4C2×D311yescharacteristic
81C23×C3×C4C22×C311yescharacteristic
82C22×Dic6C22×C311yescharacteristic
83C3×C22⋊Q8—38no—
84C2×C3×C4⋊C4—22no—
85C2×C3×C4⋊C4—32no—
86C2×C3×C4⋊C4—31no—
87C2×C3×C4⋊C4—32no—
88C2×C3×C4⋊C4—32no—
89C2×C3×C4⋊C4C2×D312yes—
90C2×C3×C4⋊C4C22×C312yes—
91C2×C3×C22⋊C4—21no—
92C2×C3×C22⋊C4—21no—
93C2×C3×C22⋊C4—32no—
94C2×C3×C22⋊C4—32no—
95C2×C3×C22⋊C4C2×D311yescharacteristic
96C2×C3×C22⋊C4C2×D311yescharacteristic
97C2×C3×C22⋊C4C22×C311yescharacteristic
98C2×C3×C22⋊C4C22×C311yescharacteristic
99C2×C3×C42—32no—
100C2×C3×C42—32no—
101C2×#217C22×C312yes—
102C2×#217C22×C312yes—
103C2×C4⋊Dic3C22×C311yescharacteristic
104#924C3×C418yescomplemented
105C2×Dic3⋊C4C22×C312yes—
106C2×Dic3⋊C4C22×C312yes—
107C2×Dic3⋊C4C22×C312yes—
108C2×C4×Dic3C22×C312yes—
109C2×C4×Dic3C22×C312yes—
110C3×#95—22no—
111C3×#95—31no—
112C3×#95—32no—
113C3×#95—31no—
114C3×#95—32no—
115C3×#95—32no—
116C3×#95—32no—
117C3×#95—32no—
118C3×#95C2×D312yes—
119C3×#95C22×C312yes—
120C3⋊#95C22×C312yes—
121C3⋊#95C22×C312yes—
122C3⋊#95C22×C312yes—
123C3⋊#95C22×C312yes—
124C3⋊#95C22×C312yes—
125C3⋊#95C22×C311yescharacteristic
126C3⋊#95C22×C311yescharacteristic
127C23×C32—21no—
128C23×C32—21no—
129C23×C32—22no—
130C23×C32—22no—
131C23×C32C22×C412yes—
132C23×C32C2411yescharacteristic
133C2×C32×C4—22no—
134C2×C32×C4—22no—
135C2×C32×C4—22no—
136C2×C32×C4—22no—
137C2×C32×C4—22no—
138C2×C32×C4—22no—
139C2×C32×C4—22no—
140C2×C32×C4—22no—
141C2×C32×C4—44no—
142C2×C32×C4C22×C412yes—
143C2×C32×C4C22×C412yes—
144C2×C3×Dic3—22no—
145C2×C3×Dic3—22no—
146C2×C3×Dic3—22no—
147C2×C3×Dic3—22no—
148C2×C3×Dic3—22no—
149C2×C3×Dic3—22no—
150C2×C3×Dic3—22no—
151C2×C3×Dic3—22no—
152C2×C3×Dic3—22no—
153C2×C3×Dic3—22no—
154C2×C3×Dic3C22×C414yes—
155C2×C3×Dic3C22×C414yes—
156C3×Dic6—44no—
157C2×C22⋊Q8—31no—
158#391—32no—
159#389—32no—
160#388—32no—
161#388—32no—
162#387—32no—
163#387—32no—
164#381—31no—
165#380C3×D311yescharacteristic
166C24×C3—21no—
167C24×C3C22×D311yescharacteristic
168C24×C3C23×C311yescharacteristic
169C2×C3×Q8—34no—
170C2×C3×Q8—122no—
171C22×C3×C4—24no—
172C22×C3×C4—21no—
173C22×C3×C4—21no—
174C22×C3×C4—22no—
175C22×C3×C4—22no—
176C22×C3×C4—32no—
177C22×C3×C4—32no—
178C22×C3×C4—32no—
179C22×C3×C4—32no—
180C22×C3×C4—42no—
181C22×C3×C4—42no—
182C22×C3×C4—44no—
183C22×C3×C4C4×D314yes—
184C22×C3×C4C2×C3×C414yes—
185C22×C3×C4C22×D312yes—
186C22×C3×C4C22×D312yes—
187C22×C3×C4C22×D311yescharacteristic
188C22×C3×C4C22×D311yescharacteristic
189C22×C3×C4C23×C312yes—
190C22×C3×C4C23×C312yes—
191C22×C3×C4C23×C311yescharacteristic
192C22×C3×C4C23×C311yescharacteristic
193C22×Dic3C23×C312yes—
194C22×Dic3C23×C312yes—
195C22×Dic3C23×C312yes—
196C22×Dic3C23×C312yes—
197C2×Dic6—42no—
198C2×Dic6C2×C3×C414yes—
199C3×C4⋊C4—38no—
200C3×C4⋊C4—34no—
201C3×C4⋊C4—42no—
202C3×C4⋊C4—42no—
203C3×C4⋊C4—44no—
204C3×C4⋊C4—122no—
205C3×C4⋊C4—122no—
206C3×C22⋊C4—38no—
207C3×C22⋊C4—41no—
208C3×C22⋊C4—41no—
209C3×C22⋊C4—41no—
210C3×C22⋊C4—41no—
211C3×C22⋊C4—42no—
212C3×C22⋊C4—42no—
213C3×C22⋊C4—122no—
214C3×C42—122no—
215C3×C42—122no—
216#217—42no—
217#217C2×C3×C418yes—
218C4⋊Dic3C2×C3×C414yes—
219Dic3⋊C4—42no—
220Dic3⋊C4—42no—
221Dic3⋊C4C2×C3×C418yes—
222C4×Dic3—42no—
223C4×Dic3—42no—
224C22×C32—22no—
225C22×C32—22no—
226C22×C32—22no—
227C22×C32—42no—
228C22×C32—42no—
229C22×C32C23×C411yescharacteristic
230C22×C32H(5,𝔽2)12yes—
231C22×C32H(5,𝔽2)11yescharacteristic
232C22×C322−1+411yescharacteristic
233C22×C322−1+412yes—
234C32×C4—22no—
235C32×C4—22no—
236C32×C4—42no—
237C32×C4—42no—
238C3×Dic3—24no—
239C3×Dic3—24no—
240C3×Dic3—42no—
241C3×Dic3—42no—
242C22×Q8—31no—
243C23×C4C2×C3×D311yescharacteristic
244C22⋊Q8—38no—
245C2×C4⋊C4—31no—
246C2×C4⋊C4—32no—
247C2×C4⋊C4—32no—
248C2×C4⋊C4—32no—
249C2×C4⋊C4C2×C3×D312yes—
250C2×C22⋊C4—32no—
251C2×C22⋊C4—32no—
252C2×C22⋊C4C2×C3×D311yescharacteristic
253C2×C22⋊C4C2×C3×D311yescharacteristic
254C2×C42—32no—
255C2×C42—32no—
256#95—31no—
257#95—32no—
258#95—31no—
259#95—32no—
260#95—32no—
261#95—32no—
262#95—32no—
263#95C2×C3×D312yes—
264C23×C3—22no—
265C23×C3—22no—
266C23×C3—21no—
267C23×C3—22no—
268C23×C3—21no—
269C23×C3—22no—
270C23×C3—22no—
271C23×C3—21no—
272C23×C3—21no—
273C23×C3—22no—
274C23×C3—21no—
275C23×C3—22no—
276C23×C3—42no—
277C23×C3—42no—
278C23×C3C2×C4×D312yes—
279C23×C3C22×C3×C412yes—
280C23×C3C23×D311yescharacteristic
281C23×C3C24×C311yescharacteristic
282C3×Q8—124no—
283C2×C3×C4—24no—
284C2×C3×C4—22no—
285C2×C3×C4—22no—
286C2×C3×C4—22no—
287C2×C3×C4—22no—
288C2×C3×C4—22no—
289C2×C3×C4—22no—
290C2×C3×C4—22no—
291C2×C3×C4—22no—
292C2×C3×C4—22no—
293C2×C3×C4—22no—
294C2×C3×C4—22no—
295C2×C3×C4—22no—
296C2×C3×C4—22no—
297C2×C3×C4—22no—
298C2×C3×C4—22no—
299C2×C3×C4—22no—
300C2×C3×C4—22no—
301C2×C3×C4—22no—
302C2×C3×C4—34no—
303C2×C3×C4—34no—
304C2×C3×C4—42no—
305C2×C3×C4—42no—
306C2×C3×C4—42no—
307C2×C3×C4—42no—
308C2×C3×C4—42no—
309C2×C3×C4—42no—
310C2×C3×C4—42no—
311C2×C3×C4—44no—
312C2×C3×C4—44no—
313C2×C3×C4—48no—
314C2×C3×C4—62no—
315C2×C3×C4—62no—
316C2×C3×C4—62no—
317C2×C3×C4—62no—
318C2×C3×C4—62no—
319C2×C3×C4—62no—
320C2×C3×C4—62no—
321C2×C3×C4—62no—
322C2×C3×C4—62no—
323C2×C3×C4—62no—
324C2×C3×C4C2×C4×D312yes—
325C2×C3×C4C2×C4×D312yes—
326C2×C3×C4C22×C3×C412yes—
327C2×C3×C4C22×C3×C412yes—
328C2×Dic3—22no—
329C2×Dic3—22no—
330C2×Dic3—22no—
331C2×Dic3—22no—
332C2×Dic3—22no—
333C2×Dic3—22no—
334C2×Dic3—22no—
335C2×Dic3—22no—
336C2×Dic3—22no—
337C2×Dic3—22no—
338C2×Dic3C22×C3×C414yes—
339C2×Dic3C22×C3×C414yes—
340Dic6—44no—
341C2×C32—42no—
342C2×C32#51811yescharacteristic
343C2×C32#51911yescharacteristic
344C2×C32#52011yescharacteristic
345C2×C32#56212yes—
346C2×C32#56312yes—
347C24C22×C3×D311yescharacteristic
348C2×Q8—34no—
349C2×Q8—122no—
350C22×C4—32no—
351C22×C4—32no—
352C22×C4—32no—
353C22×C4—32no—
354C22×C4—42no—
355C22×C4C3×C4×D314yes—
356C22×C4C22×C3×D312yes—
357C22×C4C22×C3×D312yes—
358C22×C4C22×C3×D311yescharacteristic
359C22×C4C22×C3×D311yescharacteristic
360C4⋊C4—38no—
361C4⋊C4—34no—
362C4⋊C4—42no—
363C4⋊C4—122no—
364C4⋊C4—122no—
365C22⋊C4—38no—
366C22⋊C4—41no—
367C22⋊C4—41no—
368C22⋊C4—122no—
369C42—122no—
370C42—122no—
371C22×C3—24no—
372C22×C3—22no—
373C22×C3—22no—
374C22×C3—22no—
375C22×C3—22no—
376C22×C3—22no—
377C22×C3—22no—
378C22×C3—21no—
379C22×C3—22no—
380C22×C3—21no—
381C22×C3—21no—
382C22×C3—22no—
383C22×C3—42no—
384C22×C3—42no—
385C22×C3—44no—
386C22×C3—44no—
387C22×C3—42no—
388C22×C3—42no—
389C22×C3—42no—
390C22×C3—42no—
391C22×C3C22×C4×D311yescharacteristic
392C22×C3#100412yes—
393C22×C3#100411yescharacteristic
394C22×C3#100711yescharacteristic
395C22×C3#101012yes—
396C22×C3C23×C3×C411yescharacteristic
397C22×C3C3×H(5,𝔽2)11yescharacteristic
398C22×C3C3×H(5,𝔽2)12yes—
399C22×C3C3×2−1+411yescharacteristic
400C22×C3C3×2−1+412yes—
401C3×C4—24no—
402C3×C4—22no—
403C3×C4—22no—
404C3×C4—22no—
405C3×C4—22no—
406C3×C4—42no—
407C3×C4—42no—
408C3×C4—42no—
409C3×C4—42no—
410C3×C4—42no—
411C3×C4—64no—
412C3×C4—64no—
413C3×C4—82no—
414C3×C4—82no—
415C3×C4—122no—
416C3×C4—122no—
417Dic3—24no—
418Dic3—24no—
419Dic3—42no—
420Dic3—42no—
421C32#237811yescharacteristic, complemented, Sylow
422C23—21no—
423C23—22no—
424C23—21no—
425C23—22no—
426C23C2×C3×C4×D312yes—
427C23C23×C3×D311yescharacteristic
428Q8—124no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—34no—
438C2×C4—34no—
439C2×C4—44no—
440C2×C4—62no—
441C2×C4—62no—
442C2×C4—62no—
443C2×C4—62no—
444C2×C4—62no—
445C2×C4—62no—
446C2×C4—62no—
447C2×C4—62no—
448C2×C4—62no—
449C2×C4—62no—
450C2×C4C2×C3×C4×D312yes—
451C2×C4C2×C3×C4×D312yes—
452C2×C3—21no—
453C2×C3—21no—
454C2×C3—22no—
455C2×C3—22no—
456C2×C3—21no—
457C2×C3—44no—
458C2×C3—42no—
459C2×C3—42no—
460C2×C3#548611yescharacteristic
461C2×C3#550411yescharacteristic
462C2×C3#551611yescharacteristic
463C2×C3#562112yes—
464C2×C3#564612yes—
465C2×C3C3×#51811yescharacteristic
466C2×C3C3×#51911yescharacteristic
467C2×C3C3×#52011yescharacteristic
468C2×C3C3×#56212yes—
469C2×C3C3×#56312yes—
470C22—22no—
471C22—22no—
472C22—22no—
473C22—42no—
474C22—42no—
475C22C22×C3×C4×D311yescharacteristic
476C22C3×#100411yescharacteristic
477C22C3×#100412yes—
478C22C3×#100711yescharacteristic
479C22C3×#101012yes—
480C4—22no—
481C4—22no—
482C4—42no—
483C4—42no—
484C4—64no—
485C4—64no—
486C4—122no—
487C4—122no—
488C3—21no—
489C3#7388611yescharacteristic, direct product factor
490C3C3×#237811yescharacteristic, complemented
491C2—42no—
492C2C3×#548611yescharacteristic
493C2C3×#550411yescharacteristic
494C2C3×#551611yescharacteristic
495C2C3×#562112yes—
496C2C3×#564612yes—
subgroupG/H
centerC23×C3C23×D3
commutatorC22×C3C23×C3×C4
FrattiniC23C23×C3×D3
FittingC32×#380C2
socleC23×C32C24
prime radicalC24×C32C23
cosocleC23×C3C24×C3
hypercenterC3×#380D3
hypocenterC3C3×#2378
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C23×C3 ◁ C3×#380

Complements

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

NH ≅ G/Ntype
4. C3×#4978C2semidirect
5. C3×#4976C2semidirect
8. C3×#4654C2semidirect
9. C3×#4646C2semidirect
11. #73886C3direct (N × H)
20. C3×#924C4semidirect
47. #4978C2×C3semidirect
48. #4976C2×C3semidirect
51. #4654C2×C3semidirect
52. #4646C2×C3semidirect
104. #924C3×C4semidirect
421. C32#2378semidirect
489. C3#73886direct (N × H)
490. C3C3×#2378semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#9242
C32×#3802
C3×#52112
C3×#49782
C3×#49762
C3×#49452
C3×#49442
C3×#46542
C3×#46462
C3×#23783
#738863