Finite Group Databasebeta

← C3×#10591094 #13158122

Subgroups — C3×#10591094 #13158122

Order 1728.

746 conjugacy classes of subgroups.

The subgroup lattice has 520 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C4⋊(C3⋊Dic3)C211yesmaximal, characteristic, complemented
2C32×#930C211yesmaximal, characteristic
3C32×#830C211yesmaximal, characteristic, complemented
4C3×#63575C212yesmaximal, complemented
5C3×#63563C212yesmaximal, complemented
6C3×#5033—31nomaximal
7C3×#4736—31nomaximal
8#10591094C311yesmaximal, characteristic, direct product factor
9C22×C33×C4C2211yescharacteristic
10C22×C3×C3⋊Dic3—21no—
11C2×C32×#67—21no—
12C2×C32×D12C2211yescharacteristic
13C3×C4⋊(C3⋊Dic3)—21no—
14C3×C4⋊(C3⋊Dic3)C2212yes—
15C32×(C2×D3)⋊C4—21no—
16C32×C4⋊Dic3C2211yescharacteristic
17C2×C32×C3⋊C8C2212yes—
18C32×#121—31no—
19C2×C4⋊(C3⋊Dic3)C2×C311yescharacteristic, complemented
20C3×#930—21no—
21C3×#930D311yescharacteristic
22C3×#930C2×C311yescharacteristic
23C2×C3×C4⋊Dic3—31no—
24C2×C3×C4⋊Dic3—31no—
25C2×C3×C4⋊Dic3—61no—
26C32×C22⋊C8—31no—
27C3×#832—32no—
28C3×#830—21no—
29C3×#830D311yescharacteristic, complemented
30C3×#830C2×C311yescharacteristic, complemented
31C3×#821—32no—
32C3×#818—32no—
33C3×#807—32no—
34#63575C2×C312yescomplemented
35#63563C2×C312yescomplemented
36C23×C33D411yescharacteristic
37C22×C32×D3—21no—
38C2×C33×C4—22no—
39C2×C33×C4D411yescharacteristic
40C2×C33×C4C2311yescharacteristic
41C2×C3×C3⋊Dic3—22no—
42C2×C3×C3⋊Dic3—42no—
43C32×#67—42no—
44C2×C32×Dic3—21no—
45C32×D12—22no—
46C32×C3⋊C8—22no—
47C3×#480—91no—
48#5033—31no—
49#4736—31no—
50C2×C32×D4—31no—
51C2×C32×D4—61no—
52C22×C32×C4—21no—
53C22×C32×C4—21no—
54C22×C32×C4—21no—
55C22×C32×C4—41no—
56C22×C32×C4C2×D311yescharacteristic
57C22×C32×C4C2×D311yescharacteristic
58C22×C32×C4C22×C311yescharacteristic
59C22×C3⋊Dic3—21no—
60C2×C3×#67—22no—
61C2×C3×#67—21no—
62C2×C3×#67—21no—
63C22×C3×Dic3—62no—
64C22×C3×Dic3—61no—
65C22×C3×Dic3—61no—
66C2×C3×D12—21no—
67C2×C3×D12C2×D311yescharacteristic
68C2×C3×D12C22×C311yescharacteristic
69C2×C32×C8—32no—
70C32×C4⋊C4—31no—
71C32×C22⋊C4—61no—
72C4⋊(C3⋊Dic3)—21no—
73C4⋊(C3⋊Dic3)C22×C312yes—
74C3×(C2×D3)⋊C4—22no—
75C3×(C2×D3)⋊C4—21no—
76C3×(C2×D3)⋊C4—21no—
77C3×C4⋊Dic3—21no—
78C3×C4⋊Dic3—32no—
79C3×C4⋊Dic3—32no—
80C3×C4⋊Dic3—62no—
81C3×C4⋊Dic3—62no—
82C3×C4⋊Dic3—61no—
83C3×C4⋊Dic3—61no—
84C3×C4⋊Dic3C2×D311yescharacteristic
85C3×C4⋊Dic3C22×C311yescharacteristic
86C2×C3×C3⋊C8—22no—
87C2×C3×C3⋊C8C2×D312yes—
88C2×C3×C3⋊C8C22×C312yes—
89C22×C33—21no—
90C22×C33—21no—
91C22×C33D812yes—
92C22×C33C2×D411yescharacteristic
93C2×C32×D3—41no—
94C2×C32×D3—41no—
95C2×C32×D3—41no—
96C33×C4—21no—
97C33×C4#4112yes—
98C3×C3⋊Dic3—42no—
99C32×Dic3—41no—
100C3×#121—31no—
101C3×#121—31no—
102C3×#121—61no—
103C2×C3×C4⋊C4—91no—
104#930C3×D311yescharacteristic
105C2×C4⋊Dic3—31no—
106C2×C4⋊Dic3—31no—
107C2×C4⋊Dic3—61no—
108C3×#107—92no—
109C3×#102—92no—
110C3×C22⋊C8—31no—
111C3×C22⋊C8—31no—
112C3×C22⋊C8—61no—
113#832—32no—
114#830C3×D311yescharacteristic, complemented
115#821—32no—
116#818—32no—
117#807—32no—
118C23×C32—21no—
119C23×C32—21no—
120C23×C32—21no—
121C23×C32—41no—
122C23×C32—61no—
123C23×C32D1211yescharacteristic
124C23×C32#6711yescharacteristic
125C23×C32C3×D411yescharacteristic
126C22×C3×D3—22no—
127C22×C3×D3—21no—
128C22×C3×D3—21no—
129C32×D4—62no—
130C32×D4—122no—
131C2×C32×C4—22no—
132C2×C32×C4—24no—
133C2×C32×C4—22no—
134C2×C32×C4—22no—
135C2×C32×C4—21no—
136C2×C32×C4—21no—
137C2×C32×C4—21no—
138C2×C32×C4—21no—
139C2×C32×C4—21no—
140C2×C32×C4—21no—
141C2×C32×C4—41no—
142C2×C32×C4—41no—
143C2×C32×C4—42no—
144C2×C32×C4—42no—
145C2×C32×C4—44no—
146C2×C32×C4—61no—
147C2×C32×C4D1211yescharacteristic
148C2×C32×C4#6711yescharacteristic
149C2×C32×C4C3×D411yescharacteristic
150C2×C32×C4C22×D311yescharacteristic
151C2×C32×C4C22×D311yescharacteristic
152C2×C32×C4C23×C311yescharacteristic
153C2×C3⋊Dic3—22no—
154C2×C3⋊Dic3—42no—
155C3×#67—44no—
156C3×#67—42no—
157C3×#67—42no—
158C2×C3×Dic3—22no—
159C2×C3×Dic3—21no—
160C2×C3×Dic3—21no—
161C2×C3×Dic3—64no—
162C2×C3×Dic3—62no—
163C2×C3×Dic3—62no—
164C2×C3×Dic3—124no—
165C2×C3×Dic3—122no—
166C2×C3×Dic3—122no—
167C3×D12—22no—
168C3×D12—22no—
169C3×D12—42no—
170C32×C8—62no—
171C3×C3⋊C8—22no—
172C3×C3⋊C8—22no—
173C3×C3⋊C8—42no—
174#480—91noSylow
175C2×C33—22no—
176C2×C33#12311yescharacteristic
177C2×C33C2×D811yescharacteristic
178C2×C33#13711yescharacteristic
179C32×D3—81no—
180C2×C3×D4—31no—
181C2×C3×D4—31no—
182C2×C3×D4—61no—
183C2×C3×D4—62no—
184C2×C3×D4—61no—
185C2×C3×D4—61no—
186C22×C3×C4—21no—
187C22×C3×C4—21no—
188C22×C3×C4—21no—
189C22×C3×C4—41no—
190C22×C3×C4—181no—
191C22×C3×C4D3211yescharacteristic
192C22×C3×C4C2×C3×D311yescharacteristic
193C22×C3×C4C2×C3×D311yescharacteristic
194C2×#67—21no—
195C22×Dic3—62no—
196C22×Dic3—61no—
197C22×Dic3—61no—
198C2×D12C2×C3×D311yescharacteristic
199C2×C3×C8—32no—
200C2×C3×C8—32no—
201C2×C3×C8—62no—
202C3×C4⋊C4—31no—
203C3×C4⋊C4—31no—
204C3×C4⋊C4—61no—
205C3×C4⋊C4—92no—
206C3×C4⋊C4—181no—
207C3×C22⋊C4—62no—
208C3×C22⋊C4—61no—
209C3×C22⋊C4—61no—
210(C2×D3)⋊C4—21no—
211C4⋊Dic3—32no—
212C4⋊Dic3—32no—
213C4⋊Dic3—62no—
214C4⋊Dic3—62no—
215C4⋊Dic3—61no—
216C4⋊Dic3—61no—
217C4⋊Dic3C2×C3×D311yescharacteristic
218C2×C3⋊C8C2×C3×D312yes—
219C22×C32—21no—
220C22×C32—21no—
221C22×C32—22no—
222C22×C32—22no—
223C22×C32—21no—
224C22×C32—21no—
225C22×C32—21no—
226C22×C32—21no—
227C22×C32—22no—
228C22×C32—22no—
229C22×C32—22no—
230C22×C32—21no—
231C22×C32—21no—
232C22×C32—21no—
233C22×C32—42no—
234C22×C32—41no—
235C22×C32—41no—
236C22×C32—41no—
237C22×C32—41no—
238C22×C32—41no—
239C22×C32—42no—
240C22×C32—42no—
241C22×C32—121no—
242C22×C32—121no—
243C22×C32—121no—
244C22×C32D2412yes—
245C22×C32#21312yes—
246C22×C32C3×D812yes—
247C22×C32C2×D1211yescharacteristic
248C22×C32C2×#6711yescharacteristic
249C22×C32C2×C3×D411yescharacteristic
250C2×C3×D3—42no—
251C2×C3×D3—42no—
252C2×C3×D3—42no—
253C2×C3×D3—41no—
254C2×C3×D3—41no—
255C2×C3×D3—41no—
256C2×C3×D3—41no—
257C2×C3×D3—41no—
258C2×C3×D3—41no—
259C32×C4—21no—
260C32×C4—22no—
261C32×C4—21no—
262C32×C4—21no—
263C32×C4—22no—
264C32×C4—22no—
265C32×C4—22no—
266C32×C4—42no—
267C32×C4—41no—
268C32×C4—41no—
269C32×C4—42no—
270C32×C4—121no—
271C32×C4#23712yes—
272C32×C4#23712yes—
273C32×C4C3×#4112yes—
274C3⋊Dic3—42no—
275C3×Dic3—42no—
276C3×Dic3—41no—
277C3×Dic3—41no—
278C3×Dic3—124no—
279C3×Dic3—122no—
280C3×Dic3—122no—
281#121—31no—
282C2×C4⋊C4—91no—
283#107—92no—
284#102—92no—
285C22⋊C8—31no—
286C33#48011yescharacteristic, complemented, Sylow
287C23×C3—21no—
288C23×C3—21no—
289C23×C3—21no—
290C23×C3—41no—
291C23×C3—62no—
292C23×C3—61no—
293C23×C3—61no—
294C23×C3#62611yescharacteristic
295C23×C3C3×D1211yescharacteristic
296C23×C3C3×#6711yescharacteristic
297C22×D3—21no—
298C3×D4—62no—
299C3×D4—62no—
300C3×D4—122no—
301C3×D4—124no—
302C3×D4—122no—
303C3×D4—122no—
304C2×C3×C4—24no—
305C2×C3×C4—22no—
306C2×C3×C4—22no—
307C2×C3×C4—22no—
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—42no—
315C2×C3×C4—42no—
316C2×C3×C4—44no—
317C2×C3×C4—41no—
318C2×C3×C4—41no—
319C2×C3×C4—62no—
320C2×C3×C4—61no—
321C2×C3×C4—61no—
322C2×C3×C4—182no—
323C2×C3×C4—362no—
324C2×C3×C4#62611yescharacteristic
325C2×C3×C4C3×D1211yescharacteristic
326C2×C3×C4C3×#6711yescharacteristic
327C2×C3×C4C2×D3211yescharacteristic
328C2×C3×C4C22×C3×D311yescharacteristic
329C2×C3×C4C22×C3×D311yescharacteristic
330#67—42no—
331C2×Dic3—21no—
332C2×Dic3—64no—
333C2×Dic3—62no—
334C2×Dic3—62no—
335C2×Dic3—124no—
336C2×Dic3—122no—
337C2×Dic3—122no—
338D12—22no—
339C3×C8—62no—
340C3×C8—62no—
341C3×C8—122no—
342C3⋊C8—22no—
343C2×C32—22no—
344C2×C32—24no—
345C2×C32—22no—
346C2×C32—22no—
347C2×C32—21no—
348C2×C32—21no—
349C2×C32—21no—
350C2×C32—21no—
351C2×C32—21no—
352C2×C32—21no—
353C2×C32—21no—
354C2×C32—21no—
355C2×C32—21no—
356C2×C32—41no—
357C2×C32—41no—
358C2×C32—41no—
359C2×C32—42no—
360C2×C32—42no—
361C2×C32—44no—
362C2×C32—241no—
363C2×C32#88611yescharacteristic
364C2×C32C2×D2411yescharacteristic
365C2×C32#90911yescharacteristic
366C2×C32C2×#21311yescharacteristic
367C2×C32#93511yescharacteristic
368C2×C32#95111yescharacteristic
369C2×C32C3×#12311yescharacteristic
370C2×C32C2×C3×D811yescharacteristic
371C2×C32C3×#13711yescharacteristic
372C3×D3—82no—
373C3×D3—81no—
374C3×D3—81no—
375C2×D4—31no—
376C2×D4—61no—
377C22×C4—181no—
378C22×C4C3×D3211yescharacteristic
379C2×C8—32no—
380C4⋊C4—31no—
381C4⋊C4—92no—
382C4⋊C4—181no—
383C22⋊C4—61no—
384C22×C3—22no—
385C22×C3—22no—
386C22×C3—21no—
387C22×C3—21no—
388C22×C3—21no—
389C22×C3—21no—
390C22×C3—21no—
391C22×C3—21no—
392C22×C3—21no—
393C22×C3—21no—
394C22×C3—21no—
395C22×C3—22no—
396C22×C3—22no—
397C22×C3—22no—
398C22×C3—41no—
399C22×C3—41no—
400C22×C3—41no—
401C22×C3—41no—
402C22×C3—42no—
403C22×C3—42no—
404C22×C3—41no—
405C22×C3—42no—
406C22×C3—122no—
407C22×C3—122no—
408C22×C3—122no—
409C22×C3—121no—
410C22×C3—121no—
411C22×C3—121no—
412C22×C3—121no—
413C22×C3—121no—
414C22×C3—121no—
415C22×C3#371412yes—
416C22×C3C3×D2412yes—
417C22×C3C3×#21312yes—
418C22×C3C2×#62611yescharacteristic
419C22×C3C2×C3×D1211yescharacteristic
420C22×C3C2×C3×#6711yescharacteristic
421C2×D3—41no—
422C2×D3—41no—
423C2×D3—41no—
424C3×C4—22no—
425C3×C4—21no—
426C3×C4—21no—
427C3×C4—21no—
428C3×C4—22no—
429C3×C4—22no—
430C3×C4—22no—
431C3×C4—41no—
432C3×C4—41no—
433C3×C4—42no—
434C3×C4—42no—
435C3×C4—122no—
436C3×C4—121no—
437C3×C4—121no—
438C3×C4—362no—
439C3×C4#380512yes—
440C3×C4C3×#23712yes—
441C3×C4C3×#23712yes—
442Dic3—41no—
443Dic3—124no—
444Dic3—122no—
445Dic3—122no—
446C32—21no—
447C32—21no—
448C32—21no—
449C32—41no—
450C32#473611yescharacteristic, complemented
451C32#503311yescharacteristic, complemented
452C32C3×#48011yescharacteristic, complemented
453C23—61no—
454C23C3×#62611yescharacteristic
455D4—62no—
456D4—122no—
457C2×C4—22no—
458C2×C4—61no—
459C2×C4—182no—
460C2×C4—362no—
461C2×C4C3×#62611yescharacteristic
462C2×C4C2×C3×D3211yescharacteristic
463C8—62no—
464C2×C3—24no—
465C2×C3—22no—
466C2×C3—22no—
467C2×C3—22no—
468C2×C3—21no—
469C2×C3—21no—
470C2×C3—21no—
471C2×C3—21no—
472C2×C3—21no—
473C2×C3—21no—
474C2×C3—21no—
475C2×C3—21no—
476C2×C3—21no—
477C2×C3—42no—
478C2×C3—42no—
479C2×C3—44no—
480C2×C3—41no—
481C2×C3—41no—
482C2×C3—41no—
483C2×C3—242no—
484C2×C3—241no—
485C2×C3—241no—
486C2×C3C2×#371411yescharacteristic
487C2×C3#6383111yescharacteristic
488C2×C3#6396211yescharacteristic
489C2×C3C3×#88611yescharacteristic
490C2×C3C2×C3×D2411yescharacteristic
491C2×C3C3×#90911yescharacteristic
492C2×C3C2×C3×#21311yescharacteristic
493C2×C3C3×#93511yescharacteristic
494C2×C3C3×#95111yescharacteristic
495D3—81no—
496C22—21no—
497C22—21no—
498C22—121no—
499C22—121no—
500C22—121no—
501C22C3×#371412yes—
502C22C2×C3×#62611yescharacteristic
503C4—21no—
504C4—121no—
505C4—362no—
506C4C3×#380512yes—
507C3—21no—
508C3—21no—
509C3—21no—
510C3—41no—
511C3#1059109411yescharacteristic, direct product factor
512C3C3×#473611yescharacteristic, complemented
513C3C3×#503311yescharacteristic, complemented
514C2—22no—
515C2—241no—
516C2C2×C3×#371411yescharacteristic
517C2C3×#6383111yescharacteristic
518C2C3×#6396211yescharacteristic
subgroupG/H
centerC22×C3C2×#626
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C2×C3×D32
FittingC22×C33×C4C22
socleC22×C33C2×D4
prime radicalC32×#930C2
hypercenterC22×C3×C4D32
hypocenterC32C3×#480
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C22×C3×C4

Complements

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

NH ≅ G/Ntype
1. C2×C3×C4⋊(C3⋊Dic3)C2semidirect
3. C32×#830C2semidirect
4. C3×#63575C2semidirect
5. C3×#63563C2semidirect
8. #10591094C3direct (N × H)
19. C2×C4⋊(C3⋊Dic3)C2×C3semidirect
29. C3×#830D3semidirect
30. C3×#830C2×C3semidirect
34. #63575C2×C3semidirect
35. #63563C2×C3semidirect
114. #830C3×D3semidirect
286. C33#480semidirect
450. C32#4736semidirect
451. C32#5033semidirect
452. C32C3×#480semidirect
511. C3#10591094direct (N × H)
512. C3C3×#4736semidirect
513. C3C3×#5033semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C4⋊(C3⋊Dic3)2
C32×#9302
C32×#8302
C3×#635752
C3×#635632
C3×#50333
C3×#47363
#105910943