Finite Group Databasebeta

← C3×#10591226 #13158074

Subgroups — C3×#10591226 #13158074

Order 1728.

850 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C4⋊(C3⋊Dic3)C211yesmaximal, characteristic
2C2×C32×Dic3⋊C4C211yesmaximal, characteristic
3C32×C3⋊#95C211yesmaximal, characteristic
4C3×#63579C212yesmaximal
5C3×#63579C212yesmaximal
6C3×#4986—31nomaximal
7C3×#4659—31nomaximal
8#10591226C311yesmaximal, characteristic, direct product factor
9C22×C33×C4C2211yescharacteristic
10C22×C3×C3⋊Dic3C2212yes—
11C22×C32×Dic3C2211yescharacteristic
12C22×C32×Dic3C2212yes—
13C22×C32×Dic3C2211yescharacteristic
14C3×C4⋊(C3⋊Dic3)—22no—
15C32×Dic3⋊C4—22no—
16C2×C32×C4⋊C4—31no—
17C2×C4⋊(C3⋊Dic3)C2×C311yescharacteristic
18C2×C3×C4⋊Dic3—31no—
19C2×C3×C4⋊Dic3—31no—
20C2×C3×C4⋊Dic3—61no—
21C2×C3×Dic3⋊C4—21no—
22C2×C3×Dic3⋊C4D311yescharacteristic
23C2×C3×Dic3⋊C4C2×C311yescharacteristic
24C32×#95—31no—
25C3×C3⋊#95—21no—
26C3×C3⋊#95—32no—
27C3×C3⋊#95—32no—
28C3×C3⋊#95—32no—
29C3×C3⋊#95—32no—
30C3×C3⋊#95D311yescharacteristic
31C3×C3⋊#95C2×C311yescharacteristic
32#63579C2×C312yes—
33#63579C2×C312yes—
34C23×C33C2311yescharacteristic
35C2×C33×C4—22no—
36C2×C33×C4D412yes—
37C2×C3×C3⋊Dic3—22no—
38C2×C3×C3⋊Dic3—24no—
39C2×C32×Dic3—21no—
40C2×C32×Dic3—21no—
41C2×C32×Dic3—21no—
42C2×C32×Dic3—22no—
43C2×C32×Dic3—22no—
44C2×C32×Dic3—22no—
45C2×C32×Dic3—22no—
46C2×C32×Dic3Q812yes—
47C3×#400—91no—
48#4986—31no—
49#4659—31no—
50C22×C32×C4—21no—
51C22×C32×C4—21no—
52C22×C32×C4—21no—
53C22×C32×C4—31no—
54C22×C32×C4—31no—
55C22×C32×C4—32no—
56C22×C32×C4—41no—
57C22×C32×C4C2×D311yescharacteristic
58C22×C32×C4C2×D311yescharacteristic
59C22×C32×C4C22×C311yescharacteristic
60C22×C3⋊Dic3C22×C312yes—
61C22×C3×Dic3—21no—
62C22×C3×Dic3—21no—
63C22×C3×Dic3—22no—
64C22×C3×Dic3—32no—
65C22×C3×Dic3—32no—
66C22×C3×Dic3—62no—
67C22×C3×Dic3C2×D311yescharacteristic
68C22×C3×Dic3C2×D312yes—
69C22×C3×Dic3C2×D311yescharacteristic
70C22×C3×Dic3C22×C311yescharacteristic
71C22×C3×Dic3C22×C312yes—
72C22×C3×Dic3C22×C311yescharacteristic
73C32×C4⋊C4—62no—
74C4⋊(C3⋊Dic3)—22no—
75C3×C4⋊Dic3—64no—
76C3×C4⋊Dic3—62no—
77C3×C4⋊Dic3—62no—
78C3×Dic3⋊C4—24no—
79C3×Dic3⋊C4—22no—
80C3×Dic3⋊C4—22no—
81C22×C33C2×D411yescharacteristic
82C22×C33C2×Q811yescharacteristic
83C22×C33#4111yescharacteristic
84C22×C33#4112yes—
85C22×C33#4112yes—
86C33×C4—22no—
87C3×C3⋊Dic3—42no—
88C32×Dic3—22no—
89C32×Dic3—41no—
90C32×Dic3—42no—
91C2×C3×C4⋊C4—31no—
92C2×C3×C4⋊C4—31no—
93C2×C3×C4⋊C4—61no—
94C2×C3×C4⋊C4—91no—
95C2×C4⋊Dic3—31no—
96C2×C4⋊Dic3—31no—
97C2×C4⋊Dic3—61no—
98C2×Dic3⋊C4C3×D311yescharacteristic
99C3×#95—31no—
100C3×#95—31no—
101C3×#95—61no—
102C3×#95—92no—
103C3×#95—92no—
104C3⋊#95—32no—
105C3⋊#95—32no—
106C3⋊#95—32no—
107C3⋊#95—32no—
108C3⋊#95C3×D311yescharacteristic
109C23×C32—21no—
110C23×C32—21no—
111C23×C32—21no—
112C23×C32—41no—
113C23×C32C22×D311yescharacteristic
114C23×C32C22×D311yescharacteristic
115C23×C32C23×C311yescharacteristic
116C2×C32×C4—22no—
117C2×C32×C4—24no—
118C2×C32×C4—22no—
119C2×C32×C4—22no—
120C2×C32×C4—22no—
121C2×C32×C4—22no—
122C2×C32×C4—22no—
123C2×C32×C4—32no—
124C2×C32×C4—42no—
125C2×C32×C4—42no—
126C2×C32×C4—42no—
127C2×C32×C4—44no—
128C2×C32×C4—61no—
129C2×C32×C4—61no—
130C2×C32×C4—61no—
131C2×C32×C4—62no—
132C2×C32×C4—62no—
133C2×C32×C4—62no—
134C2×C32×C4—62no—
135C2×C32×C4D1212yes—
136C2×C32×C4#6712yes—
137C2×C32×C4C3×D412yes—
138C2×C3⋊Dic3—22no—
139C2×C3⋊Dic3—24no—
140C2×C3×Dic3—22no—
141C2×C3×Dic3—22no—
142C2×C3×Dic3—21no—
143C2×C3×Dic3—21no—
144C2×C3×Dic3—21no—
145C2×C3×Dic3—21no—
146C2×C3×Dic3—22no—
147C2×C3×Dic3—24no—
148C2×C3×Dic3—22no—
149C2×C3×Dic3—22no—
150C2×C3×Dic3—22no—
151C2×C3×Dic3—22no—
152C2×C3×Dic3—21no—
153C2×C3×Dic3—21no—
154C2×C3×Dic3—22no—
155C2×C3×Dic3—22no—
156C2×C3×Dic3—22no—
157C2×C3×Dic3—22no—
158C2×C3×Dic3—22no—
159C2×C3×Dic3—42no—
160C2×C3×Dic3—42no—
161C2×C3×Dic3—42no—
162C2×C3×Dic3—64no—
163C2×C3×Dic3—62no—
164C2×C3×Dic3—64no—
165C2×C3×Dic3—64no—
166C2×C3×Dic3—62no—
167C2×C3×Dic3—124no—
168C2×C3×Dic3Dic612yes—
169C2×C3×Dic3C3×Q812yes—
170#400—91noSylow
171C2×C33C22⋊Q811yescharacteristic
172C2×C33#12311yescharacteristic
173C2×C33#12411yescharacteristic
174C2×C33#12512yes—
175C2×C33#12612yes—
176C22×C3×C4—21no—
177C22×C3×C4—21no—
178C22×C3×C4—21no—
179C22×C3×C4—31no—
180C22×C3×C4—31no—
181C22×C3×C4—32no—
182C22×C3×C4—31no—
183C22×C3×C4—32no—
184C22×C3×C4—31no—
185C22×C3×C4—41no—
186C22×C3×C4—61no—
187C22×C3×C4—62no—
188C22×C3×C4—61no—
189C22×C3×C4—92no—
190C22×C3×C4D3211yescharacteristic
191C22×C3×C4C2×C3×D311yescharacteristic
192C22×C3×C4C2×C3×D311yescharacteristic
193C22×Dic3—32no—
194C22×Dic3—32no—
195C22×Dic3—62no—
196C22×Dic3C2×C3×D312yes—
197C22×Dic3C2×C3×D311yescharacteristic
198C22×Dic3C2×C3×D311yescharacteristic
199C3×C4⋊C4—64no—
200C3×C4⋊C4—62no—
201C3×C4⋊C4—62no—
202C3×C4⋊C4—182no—
203C4⋊Dic3—64no—
204C4⋊Dic3—62no—
205C4⋊Dic3—62no—
206Dic3⋊C4—22no—
207C22×C32—21no—
208C22×C32—21no—
209C22×C32—22no—
210C22×C32—22no—
211C22×C32—21no—
212C22×C32—21no—
213C22×C32—21no—
214C22×C32—21no—
215C22×C32—22no—
216C22×C32—21no—
217C22×C32—22no—
218C22×C32—22no—
219C22×C32—21no—
220C22×C32—22no—
221C22×C32—21no—
222C22×C32—41no—
223C22×C32—41no—
224C22×C32—42no—
225C22×C32—41no—
226C22×C32—42no—
227C22×C32C2×Dic611yescharacteristic
228C22×C32C2×D1211yescharacteristic
229C22×C32#23511yescharacteristic
230C22×C32#23512yes—
231C22×C32#23711yescharacteristic
232C22×C32#23712yes—
233C22×C32#23712yes—
234C22×C32D3×Q811yescharacteristic
235C22×C32#23912yes—
236C22×C32C2×#6711yescharacteristic
237C22×C32C2×C3×D411yescharacteristic
238C22×C32C2×C3×Q811yescharacteristic
239C22×C32C3×#4111yescharacteristic
240C22×C32C3×#4112yes—
241C22×C32C3×#4112yes—
242C32×C4—22no—
243C32×C4—24no—
244C32×C4—22no—
245C32×C4—22no—
246C32×C4—42no—
247C32×C4—42no—
248C32×C4—44no—
249C32×C4—62no—
250C32×C4—121no—
251C32×C4—122no—
252C3⋊Dic3—42no—
253C3×Dic3—24no—
254C3×Dic3—22no—
255C3×Dic3—22no—
256C3×Dic3—44no—
257C3×Dic3—42no—
258C3×Dic3—42no—
259C3×Dic3—41no—
260C3×Dic3—41no—
261C3×Dic3—42no—
262C3×Dic3—124no—
263C3×Dic3—122no—
264C3×Dic3—122no—
265C2×C4⋊C4—31no—
266C2×C4⋊C4—91no—
267#95—31no—
268#95—92no—
269#95—92no—
270C33#40011yescharacteristic, complemented, Sylow
271C23×C3—21no—
272C23×C3—21no—
273C23×C3—21no—
274C23×C3—41no—
275C23×C3C2×D3211yescharacteristic
276C23×C3C22×C3×D311yescharacteristic
277C23×C3C22×C3×D311yescharacteristic
278C2×C3×C4—24no—
279C2×C3×C4—22no—
280C2×C3×C4—22no—
281C2×C3×C4—22no—
282C2×C3×C4—22no—
283C2×C3×C4—22no—
284C2×C3×C4—22no—
285C2×C3×C4—32no—
286C2×C3×C4—32no—
287C2×C3×C4—42no—
288C2×C3×C4—42no—
289C2×C3×C4—44no—
290C2×C3×C4—42no—
291C2×C3×C4—62no—
292C2×C3×C4—62no—
293C2×C3×C4—62no—
294C2×C3×C4—61no—
295C2×C3×C4—61no—
296C2×C3×C4—61no—
297C2×C3×C4—61no—
298C2×C3×C4—62no—
299C2×C3×C4—64no—
300C2×C3×C4—62no—
301C2×C3×C4—62no—
302C2×C3×C4—62no—
303C2×C3×C4—62no—
304C2×C3×C4—61no—
305C2×C3×C4—61no—
306C2×C3×C4—62no—
307C2×C3×C4—62no—
308C2×C3×C4—62no—
309C2×C3×C4—62no—
310C2×C3×C4—122no—
311C2×C3×C4—122no—
312C2×C3×C4—122no—
313C2×C3×C4—184no—
314C2×C3×C4—182no—
315C2×C3×C4#62612yes—
316C2×C3×C4C3×D1212yes—
317C2×C3×C4C3×#6712yes—
318C2×Dic3—21no—
319C2×Dic3—21no—
320C2×Dic3—22no—
321C2×Dic3—22no—
322C2×Dic3—22no—
323C2×Dic3—22no—
324C2×Dic3—21no—
325C2×Dic3—64no—
326C2×Dic3—62no—
327C2×Dic3—64no—
328C2×Dic3—64no—
329C2×Dic3—62no—
330C2×Dic3—124no—
331C2×Dic3C3×Dic612yes—
332C2×C32—22no—
333C2×C32—21no—
334C2×C32—22no—
335C2×C32—22no—
336C2×C32—21no—
337C2×C32—22no—
338C2×C32—21no—
339C2×C32—21no—
340C2×C32—21no—
341C2×C32—22no—
342C2×C32—21no—
343C2×C32—22no—
344C2×C32—21no—
345C2×C32—21no—
346C2×C32—21no—
347C2×C32—41no—
348C2×C32—41no—
349C2×C32—41no—
350C2×C32—42no—
351C2×C32—42no—
352C2×C32#87511yescharacteristic
353C2×C32#87912yes—
354C2×C32#88611yescharacteristic
355C2×C32#88912yes—
356C2×C32#89012yes—
357C2×C32#89711yescharacteristic
358C2×C32#89812yes—
359C2×C32#92411yescharacteristic
360C2×C32#93511yescharacteristic
361C2×C32#94711yescharacteristic
362C2×C32C3×C22⋊Q811yescharacteristic
363C2×C32C3×#12311yescharacteristic
364C2×C32C3×#12411yescharacteristic
365C2×C32C3×#12512yes—
366C2×C32C3×#12612yes—
367C22×C4—31no—
368C22×C4—32no—
369C22×C4—31no—
370C22×C4—92no—
371C22×C4C3×D3211yescharacteristic
372C4⋊C4—62no—
373C4⋊C4—182no—
374C22×C3—21no—
375C22×C3—21no—
376C22×C3—21no—
377C22×C3—21no—
378C22×C3—21no—
379C22×C3—21no—
380C22×C3—22no—
381C22×C3—22no—
382C22×C3—22no—
383C22×C3—22no—
384C22×C3—22no—
385C22×C3—22no—
386C22×C3—21no—
387C22×C3—21no—
388C22×C3—21no—
389C22×C3—41no—
390C22×C3—41no—
391C22×C3—42no—
392C22×C3—42no—
393C22×C3—41no—
394C22×C3D3×Dic611yescharacteristic
395C22×C3#379511yescharacteristic
396C22×C3#379912yes—
397C22×C3#380512yes—
398C22×C3C2×#62611yescharacteristic
399C22×C3C2×C3×Dic611yescharacteristic
400C22×C3C2×C3×D1211yescharacteristic
401C22×C3C3×#23511yescharacteristic
402C22×C3C3×#23512yes—
403C22×C3C3×#23711yescharacteristic
404C22×C3C3×#23712yes—
405C22×C3C3×#23712yes—
406C22×C3C3×D3×Q811yescharacteristic
407C22×C3C3×#23912yes—
408C22×C3C2×C3×#6711yescharacteristic
409C3×C4—24no—
410C3×C4—22no—
411C3×C4—22no—
412C3×C4—22no—
413C3×C4—42no—
414C3×C4—42no—
415C3×C4—44no—
416C3×C4—64no—
417C3×C4—62no—
418C3×C4—62no—
419C3×C4—122no—
420C3×C4—124no—
421C3×C4—121no—
422C3×C4—122no—
423C3×C4—122no—
424C3×C4—121no—
425C3×C4—362no—
426Dic3—22no—
427Dic3—42no—
428Dic3—41no—
429Dic3—124no—
430Dic3—122no—
431Dic3—122no—
432C32—21no—
433C32—21no—
434C32—21no—
435C32—41no—
436C32#465911yescharacteristic, complemented
437C32#498611yescharacteristic, complemented
438C32C3×#40011yescharacteristic, complemented
439C23C2×C3×D3211yescharacteristic
440C2×C4—22no—
441C2×C4—32no—
442C2×C4—61no—
443C2×C4—61no—
444C2×C4—62no—
445C2×C4—62no—
446C2×C4—62no—
447C2×C4—62no—
448C2×C4—61no—
449C2×C4—184no—
450C2×C4—182no—
451C2×C4C3×#62612yes—
452C2×C3—21no—
453C2×C3—21no—
454C2×C3—21no—
455C2×C3—22no—
456C2×C3—22no—
457C2×C3—22no—
458C2×C3—21no—
459C2×C3—21no—
460C2×C3—21no—
461C2×C3—22no—
462C2×C3—22no—
463C2×C3—22no—
464C2×C3—21no—
465C2×C3—21no—
466C2×C3—21no—
467C2×C3—41no—
468C2×C3—42no—
469C2×C3—41no—
470C2×C3—42no—
471C2×C3—41no—
472C2×C3#6384512yes—
473C2×C3#6385711yescharacteristic
474C2×C3#6386112yes—
475C2×C3#6386411yescharacteristic
476C2×C3#6396211yescharacteristic
477C2×C3C3×#87511yescharacteristic
478C2×C3C3×#87912yes—
479C2×C3C3×#88611yescharacteristic
480C2×C3C3×#88912yes—
481C2×C3C3×#89012yes—
482C2×C3C3×#89711yescharacteristic
483C2×C3C3×#89812yes—
484C2×C3C3×#92411yescharacteristic
485C2×C3C3×#93511yescharacteristic
486C2×C3C3×#94711yescharacteristic
487C22C3×D3×Dic611yescharacteristic
488C22C3×#379511yescharacteristic
489C22C3×#379912yes—
490C22C3×#380512yes—
491C22C2×C3×#62611yescharacteristic
492C4—22no—
493C4—62no—
494C4—121no—
495C4—122no—
496C4—362no—
497C3—21no—
498C3—21no—
499C3—21no—
500C3—41no—
501C3#1059122611yescharacteristic, direct product factor
502C3C3×#465911yescharacteristic, complemented
503C3C3×#498611yescharacteristic, complemented
504C2C3×#6384512yes—
505C2C3×#6385711yescharacteristic
506C2C3×#6386112yes—
507C2C3×#6386411yescharacteristic
508C2C3×#6396211yescharacteristic
subgroupG/H
centerC23×C3C2×D32
commutator, cosocleC23×C32C23×C3
FrattiniC23C2×C3×D32
FittingC22×C33×C4C22
socle, prime radicalC23×C33C23
hypercenterC22×C3×C4D32
hypocenterC32C3×#400
LCSG ▷ C23×C32 ▷ C32
UCSC1 ◁ C23×C3 ◁ C22×C3×C4

Complements

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

NH ≅ G/Ntype
8. #10591226C3direct (N × H)
270. C33#400semidirect
436. C32#4659semidirect
437. C32#4986semidirect
438. C32C3×#400semidirect
501. C3#10591226direct (N × H)
502. C3C3×#4659semidirect
503. C3C3×#4986semidirect

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
C2×C32×Dic3⋊C42
C32×C3⋊#952
C3×#635792
C3×#49863
C3×#46593
#105912263