Finite Group Databasebeta

← C3×#10590292 #13157325

Subgroups — C3×#10590292 #13157325

Order 1728.

554 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
1C3×C4×#634C211yesmaximal, characteristic
2C32×#870C211yesmaximal, characteristic
3C32×C4⋊(C3⋊C8)C211yesmaximal, characteristic
4C3×#63575C211yesmaximal, characteristic
5C3×#63573C211yesmaximal, characteristic
6C3×#63567C211yesmaximal, characteristic
7C3×#63567C211yesmaximal, characteristic
8C3×#5063—31nomaximal
9C3×#4829—31nomaximal
10#10590292C311yesmaximal, characteristic, direct product factor
11C2×C3×#634C2211yescharacteristic
12C33×C42C2211yescharacteristic
13C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
14C3×#3750—21no—
15C3×C4×C3⋊Dic3—21no—
16C32×C4⋊Dic3C2211yescharacteristic
17C32×C4⋊Dic3C2211yescharacteristic
18C32×Dic3⋊C4—21no—
19C32×Dic3⋊C4—21no—
20C2×C32×C3⋊C8C2211yescharacteristic
21C2×C32×C3⋊C8C2211yescharacteristic
22C32×#125—31no—
23C4×#634C2×C311yescharacteristic
24C3×#870—21no—
25C3×#870D311yescharacteristic
26C3×#870C2×C311yescharacteristic
27C3×C4×Dic6—31no—
28C3×C4×Dic6—31no—
29C3×C4×Dic6—61no—
30C32×C4⋊C8—31no—
31C3×#818—31no—
32C3×#817—31no—
33C3×#816—31no—
34C3×#816—31no—
35C3×#810—31no—
36C3×#810—31no—
37C3×#808—31no—
38C3×#807—31no—
39C3×C4⋊(C3⋊C8)—21no—
40C3×C4⋊(C3⋊C8)D311yescharacteristic
41C3×C4⋊(C3⋊C8)C2×C311yescharacteristic
42#63575C2×C311yescharacteristic
43#63573C2×C311yescharacteristic
44#63567C2×C311yescharacteristic
45#63567C2×C311yescharacteristic
46C2×C33×C4D411yescharacteristic
47C2×C33×C4D411yescharacteristic
48C2×C33×C4C2311yescharacteristic
49C2×C3×C3⋊Dic3—21no—
50C2×C3×C3⋊Dic3—21no—
51C3×#634—21no—
52C3×#634Q812yes—
53C2×C32×Dic3—21no—
54C2×C32×Dic3—21no—
55C32×C3⋊C8—21no—
56C32×C3⋊C8—21no—
57C3×#479—91no—
58#5063—31no—
59#4829—31no—
60C2×#634C22×C311yescharacteristic
61C2×C3×Dic6—31no—
62C2×C3×Dic6—31no—
63C2×C3×Dic6—61no—
64C2×C32×C8—31no—
65C2×C32×C8—31no—
66C32×C4⋊C4—31no—
67C32×C4⋊C4—31no—
68C32×C4⋊C4—61no—
69C32×C4⋊C4—61no—
70C32×C42—21no—
71C32×C42—21no—
72C32×C42—21no—
73C32×C42—41no—
74C32×C42C2×D311yescharacteristic
75C32×C42C2×D311yescharacteristic
76C32×C42C22×C311yescharacteristic
77C4⋊(C3⋊Dic3)C22×C311yescharacteristic
78#3750—21no—
79C4×C3⋊Dic3—21no—
80C3×C4⋊Dic3—21no—
81C3×C4⋊Dic3—21no—
82C3×C4⋊Dic3—31no—
83C3×C4⋊Dic3—31no—
84C3×C4⋊Dic3—61no—
85C3×C4⋊Dic3C2×D311yescharacteristic
86C3×C4⋊Dic3C2×D311yescharacteristic
87C3×C4⋊Dic3C22×C311yescharacteristic
88C3×C4⋊Dic3C22×C311yescharacteristic
89C3×Dic3⋊C4—22no—
90C3×Dic3⋊C4—22no—
91C3×Dic3⋊C4—21no—
92C3×Dic3⋊C4—21no—
93C3×Dic3⋊C4—21no—
94C3×Dic3⋊C4—21no—
95C3×Dic3⋊C4—62no—
96C3×Dic3⋊C4—61no—
97C3×Dic3⋊C4—61no—
98C3×C4×Dic3—62no—
99C3×C4×Dic3—61no—
100C3×C4×Dic3—61no—
101C2×C3×C3⋊C8—21no—
102C2×C3×C3⋊C8—21no—
103C2×C3×C3⋊C8C2×D311yescharacteristic
104C2×C3×C3⋊C8C2×D311yescharacteristic
105C2×C3×C3⋊C8C22×C311yescharacteristic
106C2×C3×C3⋊C8C22×C311yescharacteristic
107C22×C33C2×D411yescharacteristic
108C33×C4—21no—
109C33×C4—21no—
110C33×C4C2×Q811yescharacteristic
111C33×C4#4111yescharacteristic
112C3×C3⋊Dic3—22no—
113C3×C3⋊Dic3—41no—
114C32×Dic3—41no—
115C32×Dic3—41no—
116C3×#125—31no—
117C3×#125—31no—
118C3×#125—61no—
119C3×C4×Q8—91no—
120#870C3×D311yescharacteristic
121C4×Dic6—31no—
122C4×Dic6—31no—
123C4×Dic6—61no—
124C3×#107—91no—
125C3×#106—91no—
126C3×C4⋊C8—31no—
127C3×C4⋊C8—31no—
128C3×C4⋊C8—61no—
129C3×#103—91no—
130C3×#103—91no—
131#818—31no—
132#817—31no—
133#816—31no—
134#816—31no—
135#810—31no—
136#810—31no—
137#808—31no—
138#807—31no—
139C4⋊(C3⋊C8)C3×D311yescharacteristic
140C2×C32×C4—21no—
141C2×C32×C4—21no—
142C2×C32×C4—21no—
143C2×C32×C4—21no—
144C2×C32×C4—21no—
145C2×C32×C4—21no—
146C2×C32×C4—21no—
147C2×C32×C4—21no—
148C2×C32×C4—21no—
149C2×C32×C4—41no—
150C2×C32×C4—41no—
151C2×C32×C4—41no—
152C2×C32×C4—61no—
153C2×C32×C4—61no—
154C2×C32×C4D1211yescharacteristic
155C2×C32×C4D1211yescharacteristic
156C2×C32×C4#6711yescharacteristic
157C2×C32×C4#6711yescharacteristic
158C2×C32×C4C3×D411yescharacteristic
159C2×C32×C4C3×D411yescharacteristic
160C2×C32×C4C22×D311yescharacteristic
161C2×C32×C4C22×D311yescharacteristic
162C2×C32×C4C23×C311yescharacteristic
163C2×C3⋊Dic3—21no—
164C2×C3⋊Dic3—21no—
165#634—21no—
166#634C3×Q812yes—
167C2×C3×Dic3—22no—
168C2×C3×Dic3—22no—
169C2×C3×Dic3—21no—
170C2×C3×Dic3—21no—
171C2×C3×Dic3—21no—
172C2×C3×Dic3—21no—
173C2×C3×Dic3—62no—
174C2×C3×Dic3—61no—
175C2×C3×Dic3—61no—
176C2×C3×Dic3—62no—
177C2×C3×Dic3—61no—
178C2×C3×Dic3—61no—
179C3×Dic6—32no—
180C3×Dic6—32no—
181C3×Dic6—62no—
182C3×Dic6—62no—
183C3×Dic6—61no—
184C3×Dic6—61no—
185C32×C8—61no—
186C32×C8—61no—
187C3×C3⋊C8—21no—
188C3×C3⋊C8—21no—
189C3×C3⋊C8—21no—
190C3×C3⋊C8—21no—
191C3×C3⋊C8—41no—
192C3×C3⋊C8—41no—
193#479—91noSylow
194C2×C33C22⋊Q811yescharacteristic
195C2×C33#13511yescharacteristic
196C2×C33#13711yescharacteristic
197C2×C3×Q8—91no—
198C2×Dic6—31no—
199C2×Dic6—31no—
200C2×Dic6—61no—
201C2×C3×C8—31no—
202C2×C3×C8—31no—
203C2×C3×C8—31no—
204C2×C3×C8—31no—
205C2×C3×C8—61no—
206C2×C3×C8—61no—
207C3×C4⋊C4—31no—
208C3×C4⋊C4—31no—
209C3×C4⋊C4—31no—
210C3×C4⋊C4—31no—
211C3×C4⋊C4—61no—
212C3×C4⋊C4—61no—
213C3×C4⋊C4—62no—
214C3×C4⋊C4—62no—
215C3×C4⋊C4—61no—
216C3×C4⋊C4—61no—
217C3×C4⋊C4—61no—
218C3×C4⋊C4—61no—
219C3×C4⋊C4—91no—
220C3×C4⋊C4—181no—
221C3×C42—21no—
222C3×C42—21no—
223C3×C42—21no—
224C3×C42—41no—
225C3×C42—181no—
226C3×C42D3211yescharacteristic
227C3×C42C2×C3×D311yescharacteristic
228C3×C42C2×C3×D311yescharacteristic
229C4⋊Dic3—31no—
230C4⋊Dic3—31no—
231C4⋊Dic3—61no—
232C4⋊Dic3C2×C3×D311yescharacteristic
233C4⋊Dic3C2×C3×D311yescharacteristic
234Dic3⋊C4—21no—
235Dic3⋊C4—21no—
236Dic3⋊C4—62no—
237Dic3⋊C4—61no—
238Dic3⋊C4—61no—
239C4×Dic3—62no—
240C4×Dic3—61no—
241C4×Dic3—61no—
242C2×C3⋊C8C2×C3×D311yescharacteristic
243C2×C3⋊C8C2×C3×D311yescharacteristic
244C22×C32—21no—
245C22×C32—21no—
246C22×C32—21no—
247C22×C32—41no—
248C22×C32C2×D1211yescharacteristic
249C22×C32C2×#6711yescharacteristic
250C22×C32C2×C3×D411yescharacteristic
251C32×C4—21no—
252C32×C4—21no—
253C32×C4—22no—
254C32×C4—22no—
255C32×C4—21no—
256C32×C4—21no—
257C32×C4—21no—
258C32×C4—21no—
259C32×C4—21no—
260C32×C4—21no—
261C32×C4—21no—
262C32×C4—21no—
263C32×C4—21no—
264C32×C4—21no—
265C32×C4—41no—
266C32×C4—41no—
267C32×C4—41no—
268C32×C4—41no—
269C32×C4—41no—
270C32×C4—41no—
271C32×C4—42no—
272C32×C4—42no—
273C32×C4—121no—
274C32×C4—121no—
275C32×C4#23711yescharacteristic
276C32×C4D3×Q811yescharacteristic
277C32×C4D3×Q811yescharacteristic
278C32×C4#23911yescharacteristic
279C32×C4C2×C3×Q811yescharacteristic
280C32×C4C3×#4111yescharacteristic
281C3⋊Dic3—22no—
282C3⋊Dic3—41no—
283C3×Dic3—42no—
284C3×Dic3—42no—
285C3×Dic3—41no—
286C3×Dic3—41no—
287C3×Dic3—41no—
288C3×Dic3—41no—
289C3×Dic3—64no—
290C3×Dic3—62no—
291C3×Dic3—62no—
292C3×Dic3—122no—
293C3×Dic3—121no—
294C3×Dic3—121no—
295#125—31no—
296C4×Q8—91no—
297#107—91no—
298#106—91no—
299C4⋊C8—31no—
300#103—91no—
301#103—91no—
302C33#47911yescharacteristic, complemented, Sylow
303C3×Q8—92no—
304C3×Q8—181no—
305C2×C3×C4—21no—
306C2×C3×C4—21no—
307C2×C3×C4—21no—
308C2×C3×C4—21no—
309C2×C3×C4—21no—
310C2×C3×C4—21no—
311C2×C3×C4—21no—
312C2×C3×C4—21no—
313C2×C3×C4—21no—
314C2×C3×C4—41no—
315C2×C3×C4—41no—
316C2×C3×C4—41no—
317C2×C3×C4—62no—
318C2×C3×C4—62no—
319C2×C3×C4—61no—
320C2×C3×C4—61no—
321C2×C3×C4—61no—
322C2×C3×C4—61no—
323C2×C3×C4—181no—
324C2×C3×C4—181no—
325C2×C3×C4#62611yescharacteristic
326C2×C3×C4#62611yescharacteristic
327C2×C3×C4C3×D1211yescharacteristic
328C2×C3×C4C3×D1211yescharacteristic
329C2×C3×C4C3×#6711yescharacteristic
330C2×C3×C4C3×#6711yescharacteristic
331C2×C3×C4C2×D3211yescharacteristic
332C2×C3×C4C22×C3×D311yescharacteristic
333C2×C3×C4C22×C3×D311yescharacteristic
334C2×Dic3—21no—
335C2×Dic3—21no—
336C2×Dic3—62no—
337C2×Dic3—61no—
338C2×Dic3—61no—
339C2×Dic3—62no—
340C2×Dic3—61no—
341C2×Dic3—61no—
342Dic6—32no—
343Dic6—32no—
344Dic6—62no—
345Dic6—62no—
346Dic6—61no—
347Dic6—61no—
348C3×C8—61no—
349C3×C8—61no—
350C3×C8—61no—
351C3×C8—61no—
352C3×C8—121no—
353C3×C8—121no—
354C3⋊C8—21no—
355C3⋊C8—21no—
356C2×C32—21no—
357C2×C32—21no—
358C2×C32—21no—
359C2×C32—21no—
360C2×C32—21no—
361C2×C32—21no—
362C2×C32—21no—
363C2×C32—21no—
364C2×C32—21no—
365C2×C32—41no—
366C2×C32—41no—
367C2×C32—41no—
368C2×C32#89711yescharacteristic
369C2×C32#90411yescharacteristic
370C2×C32#90911yescharacteristic
371C2×C32#94611yescharacteristic
372C2×C32#95011yescharacteristic
373C2×C32#95111yescharacteristic
374C2×C32C3×C22⋊Q811yescharacteristic
375C2×C32C3×#13511yescharacteristic
376C2×C32C3×#13711yescharacteristic
377C2×Q8—91no—
378C2×C8—31no—
379C2×C8—31no—
380C4⋊C4—31no—
381C4⋊C4—31no—
382C4⋊C4—61no—
383C4⋊C4—61no—
384C4⋊C4—91no—
385C4⋊C4—181no—
386C42—181no—
387C42C3×D3211yescharacteristic
388C22×C3—21no—
389C22×C3—21no—
390C22×C3—21no—
391C22×C3—41no—
392C22×C3C2×#62611yescharacteristic
393C22×C3C2×C3×D1211yescharacteristic
394C22×C3C2×C3×#6711yescharacteristic
395C3×C4—22no—
396C3×C4—22no—
397C3×C4—21no—
398C3×C4—21no—
399C3×C4—21no—
400C3×C4—21no—
401C3×C4—21no—
402C3×C4—21no—
403C3×C4—21no—
404C3×C4—21no—
405C3×C4—21no—
406C3×C4—21no—
407C3×C4—21no—
408C3×C4—21no—
409C3×C4—41no—
410C3×C4—41no—
411C3×C4—41no—
412C3×C4—41no—
413C3×C4—42no—
414C3×C4—42no—
415C3×C4—41no—
416C3×C4—41no—
417C3×C4—122no—
418C3×C4—122no—
419C3×C4—121no—
420C3×C4—121no—
421C3×C4—121no—
422C3×C4—121no—
423C3×C4—182no—
424C3×C4—361no—
425C3×C4#379611yescharacteristic
426C3×C4#379711yescharacteristic
427C3×C4C3×#23711yescharacteristic
428C3×C4C3×D3×Q811yescharacteristic
429C3×C4C3×D3×Q811yescharacteristic
430C3×C4C3×#23911yescharacteristic
431Dic3—41no—
432Dic3—41no—
433Dic3—64no—
434Dic3—62no—
435Dic3—62no—
436Dic3—122no—
437Dic3—121no—
438Dic3—121no—
439C32—21no—
440C32—21no—
441C32—21no—
442C32—41no—
443C32#482911yescharacteristic, complemented
444C32#506311yescharacteristic, complemented
445C32C3×#47911yescharacteristic, complemented
446Q8—92no—
447Q8—181no—
448C2×C4—61no—
449C2×C4—61no—
450C2×C4—181no—
451C2×C4—181no—
452C2×C4C3×#62611yescharacteristic
453C2×C4C3×#62611yescharacteristic
454C2×C4C2×C3×D3211yescharacteristic
455C8—61no—
456C8—61no—
457C2×C3—21no—
458C2×C3—21no—
459C2×C3—21no—
460C2×C3—21no—
461C2×C3—21no—
462C2×C3—21no—
463C2×C3—21no—
464C2×C3—21no—
465C2×C3—21no—
466C2×C3—41no—
467C2×C3—41no—
468C2×C3—41no—
469C2×C3#6382911yescharacteristic
470C2×C3#6383111yescharacteristic
471C2×C3#6387111yescharacteristic
472C2×C3C3×#89711yescharacteristic
473C2×C3C3×#90411yescharacteristic
474C2×C3C3×#90911yescharacteristic
475C2×C3C3×#94611yescharacteristic
476C2×C3C3×#95011yescharacteristic
477C2×C3C3×#95111yescharacteristic
478C22C2×C3×#62611yescharacteristic
479C4—21no—
480C4—21no—
481C4—121no—
482C4—121no—
483C4—182no—
484C4—361no—
485C4C3×#379611yescharacteristic
486C4C3×#379711yescharacteristic
487C3—21no—
488C3—21no—
489C3—21no—
490C3—41no—
491C3#1059029211yescharacteristic, direct product factor
492C3C3×#482911yescharacteristic, complemented
493C3C3×#506311yescharacteristic, complemented
494C2C3×#6382911yescharacteristic
495C2C3×#6383111yescharacteristic
496C2C3×#6387111yescharacteristic
subgroupG/H
centerC22×C3C2×#626
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C2×C3×D32
FittingC33×C42C22
socle, prime radicalC22×C33C2×D4
hypercenterC3×C42D32
hypocenterC32C3×#479
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C3×C42

Complements

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

NH ≅ G/Ntype
10. #10590292C3direct (N × H)
302. C33#479semidirect
443. C32#4829semidirect
444. C32#5063semidirect
445. C32C3×#479semidirect
491. C3#10590292direct (N × H)
492. C3C3×#4829semidirect
493. C3C3×#5063semidirect

Maximal subgroups

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

subgroupindex
C3×C4×#6342
C32×#8702
C32×C4⋊(C3⋊C8)2
C3×#635752
C3×#635732
C3×#635672
C3×#50633
C3×#48293
#105902923