Finite Group Databasebeta

← C3×#10591153 #13158330

Subgroups — C3×#10591153 #13158330

Order 1728.

950 conjugacy classes of subgroups.

The subgroup lattice has 524 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3828C211yesmaximal, characteristic
2C2×C3×#3712C211yesmaximal, characteristic, complemented
3C2×C3×#3710C211yesmaximal, characteristic, complemented
4C32×#830C212yesmaximal, complemented
5C3×#63557C212yesmaximal, complemented
6C3×#4727—32nomaximal
7#10591153C311yesmaximal, characteristic, direct product factor
8C22×C33×C4C2211yescharacteristic
9C2×C3×#638C411yescharacteristic
10C2×C3×#638C411yescharacteristic
11C3×#3828—22no—
12C3×#3828—22no—
13C2×C3×C4⋊(C3⋊D3)C411yescharacteristic
14C2×C3×C4×C3⋊D3C2211yescharacteristic
15C2×C3×C4×C3⋊D3C2211yescharacteristic
16C2×C3×#634C411yescharacteristic
17C2×C32×C3⋊C8C2212yes—
18C2×C32×C3⋊C8C2212yes—
19C3×#3712—22no—
20C3×#3710—22no—
21C2×#3828C2×C311yescharacteristic
22C2×C3×#235—32no—
23C2×C3×#235—61no—
24C2×C3×ΓL(1,𝔽25)—32no—
25C2×C3×D3×C8—32no—
26C2×#3712C2×C311yescharacteristic, complemented
27C2×#3710C2×C311yescharacteristic, complemented
28C32×C22⋊C8—32no—
29C3×#830—22no—
30C3×#830D312yescomplemented
31C3×#830C2×C312yescomplemented
32C3×(C2×D3)⋊C8—32no—
33C3×(C2×D3)⋊C8—32no—
34#63557C2×C312yescomplemented
35C23×C33C2×C411yescharacteristic
36C22×C3×C3⋊D3C2×C411yescharacteristic
37C22×C3×C3⋊D3C2×C411yescharacteristic
38C2×C33×C4—21no—
39C2×C33×C4—21no—
40C2×C33×C4C2×C411yescharacteristic
41C2×C33×C4C2311yescharacteristic
42C3×#638—22no—
43C3×#638—22no—
44C2×C3×C3⋊Dic3C2×C411yescharacteristic
45C2×C3×C3⋊Dic3C2×C411yescharacteristic
46C3×C4⋊(C3⋊D3)—22no—
47C3×C4×C3⋊D3—21no—
48C3×C4×C3⋊D3—21no—
49C3×C4×C3⋊D3D412yes—
50C3×C4×C3⋊D3D412yes—
51C3×#634—22no—
52C32×C3⋊C8—22no—
53C32×C3⋊C8—22no—
54C3×#408—91no—
55#4727—32no—
56C22×C32×C4—21no—
57C22×C32×C4—22no—
58C22×C32×C4—41no—
59C22×C32×C4C2×D312yes—
60C22×C32×C4C22×C311yescharacteristic
61C2×#638C3×C411yescharacteristic
62C2×#638C3×C411yescharacteristic
63#3828—22no—
64#3828—22no—
65C2×C4⋊(C3⋊D3)C3×C411yescharacteristic
66C2×C4×C3⋊D3C22×C311yescharacteristic
67C2×C4×C3⋊D3C22×C311yescharacteristic
68C2×#634C3×C411yescharacteristic
69C2×C3×#67—32no—
70C2×C3×#67—32no—
71C2×C3×#67—61no—
72C2×C3×#67—61no—
73C3×#235—64no—
74C3×#235—64no—
75C3×#235—64no—
76C3×#235—64no—
77C2×C3×D12—32no—
78C2×C3×D12—61no—
79C2×C3×C4×D3—32no—
80C2×C3×C4×D3—32no—
81C2×C3×C4×D3—61no—
82C2×C3×C4×D3—61no—
83C2×C3×Dic6—32no—
84C2×C3×Dic6—61no—
85C2×C32×C8—32no—
86C2×C32×C8—32no—
87C2×C3×C3⋊C8—22no—
88C2×C3×C3⋊C8—22no—
89C2×C3×C3⋊C8C2×D312yes—
90C2×C3×C3⋊C8C2×D312yes—
91C2×C3×C3⋊C8C22×C312yes—
92C2×C3×C3⋊C8C22×C312yes—
93C3×ΓL(1,𝔽25)—64no—
94C3×D3×C8—64no—
95#3712—22no—
96#3710—22no—
97C22×C33—21no—
98C22×C33—22no—
99C22×C33C22×C411yescharacteristic
100C2×C3×C3⋊D3—22no—
101C2×C3×C3⋊D3—21no—
102C2×C3×C3⋊D3—22no—
103C2×C3×C3⋊D3C22⋊C412yes—
104C33×C4—21no—
105C33×C4C2×D411yescharacteristic
106C33×C4C2×D411yescharacteristic
107C3×C3⋊Dic3—21no—
108C3×C3⋊Dic3C22⋊C412yes—
109C2×C3×#41—91no—
110C2×#235—32no—
111C2×#235—61no—
112C2×C3×M16—91no—
113C22×C3×C8—91no—
114C2×ΓL(1,𝔽25)—32no—
115C2×D3×C8—32no—
116C3×C22⋊C8—32no—
117C3×C22⋊C8—32no—
118C3×C22⋊C8—62no—
119C3×C22⋊C8—92no—
120#830C3×D312yescomplemented
121(C2×D3)⋊C8—32no—
122(C2×D3)⋊C8—32no—
123C23×C32—21no—
124C23×C32—22no—
125C23×C32—41no—
126C23×C32C4×D312yes—
127C23×C32C2×C3×C411yescharacteristic
128C22×C3⋊D3C2×C3×C411yescharacteristic
129C22×C3⋊D3C2×C3×C411yescharacteristic
130C22×C3×D3—32no—
131C22×C3×D3—32no—
132C22×C3×D3—61no—
133C22×C3×D3—61no—
134C2×C32×C4—21no—
135C2×C32×C4—21no—
136C2×C32×C4—22no—
137C2×C32×C4—22no—
138C2×C32×C4—22no—
139C2×C32×C4—22no—
140C2×C32×C4—21no—
141C2×C32×C4—22no—
142C2×C32×C4—21no—
143C2×C32×C4—22no—
144C2×C32×C4—41no—
145C2×C32×C4—41no—
146C2×C32×C4—42no—
147C2×C32×C4—42no—
148C2×C32×C4—42no—
149C2×C32×C4—42no—
150C2×C32×C4C4×D312yes—
151C2×C32×C4C2×C3×C411yescharacteristic
152C2×C32×C4C22×D312yes—
153C2×C32×C4C23×C311yescharacteristic
154#638—22no—
155#638—22no—
156C2×C3⋊Dic3C2×C3×C411yescharacteristic
157C2×C3⋊Dic3C2×C3×C411yescharacteristic
158C4⋊(C3⋊D3)—22no—
159C4×C3⋊D3—21no—
160C4×C3⋊D3—21no—
161C4×C3⋊D3C3×D412yes—
162C4×C3⋊D3C3×D412yes—
163#634—22no—
164C3×#67—64no—
165C3×#67—64no—
166C3×#67—64no—
167C3×#67—64no—
168C2×C3×Dic3—32no—
169C2×C3×Dic3—32no—
170C2×C3×Dic3—61no—
171C2×C3×Dic3—61no—
172C3×D12—64no—
173C3×D12—64no—
174C3×C4×D3—34no—
175C3×C4×D3—34no—
176C3×C4×D3—62no—
177C3×C4×D3—62no—
178C3×C4×D3—62no—
179C3×C4×D3—62no—
180C3×C4×D3—62no—
181C3×C4×D3—62no—
182C3×Dic6—64no—
183C3×Dic6—64no—
184C32×C8—62no—
185C32×C8—62no—
186C3×C3⋊C8—22no—
187C3×C3⋊C8—22no—
188C3×C3⋊C8—22no—
189C3×C3⋊C8—22no—
190C3×C3⋊C8—42no—
191C3×C3⋊C8—42no—
192#408—91noSylow
193C2×C33—41no—
194C2×C33C2×C22⋊C411yescharacteristic
195C2×C33#13112yes—
196C3×C3⋊D3—22no—
197C3×C3⋊D3—41no—
198C3×#41—182no—
199C3×#41—182no—
200C2×C3×Q8—91no—
201C2×C3×D4—91no—
202C2×C3×D4—91no—
203C2×C3×D4—91no—
204C22×C3×C4—21no—
205C22×C3×C4—22no—
206C22×C3×C4—41no—
207C22×C3×C4—91no—
208C22×C3×C4—91no—
209C22×C3×C4D3211yescharacteristic
210C22×C3×C4C2×C3×D312yes—
211C2×#67—32no—
212C2×#67—32no—
213C2×#67—61no—
214C2×#67—61no—
215#235—64no—
216#235—64no—
217#235—64no—
218#235—64no—
219C2×D12—32no—
220C2×D12—61no—
221C2×C4×D3—32no—
222C2×C4×D3—32no—
223C2×C4×D3—61no—
224C2×C4×D3—61no—
225C2×Dic6—32no—
226C2×Dic6—61no—
227C3×M16—182no—
228C2×C3×C8—32no—
229C2×C3×C8—32no—
230C2×C3×C8—32no—
231C2×C3×C8—32no—
232C2×C3×C8—62no—
233C2×C3×C8—62no—
234C2×C3×C8—182no—
235C2×C3⋊C8C2×C3×D312yes—
236C2×C3⋊C8C2×C3×D312yes—
237ΓL(1,𝔽25)—64no—
238D3×C8—64no—
239C22×C32—22no—
240C22×C32—21no—
241C22×C32—22no—
242C22×C32—24no—
243C22×C32—22no—
244C22×C32—22no—
245C22×C32—22no—
246C22×C32—21no—
247C22×C32—22no—
248C22×C32—41no—
249C22×C32—42no—
250C22×C32—42no—
251C22×C32—42no—
252C22×C32—42no—
253C22×C32—44no—
254C22×C32C2×C4×D312yes—
255C22×C32C22×C3×C411yescharacteristic
256C2×C3⋊D3—21no—
257C2×C3⋊D3—22no—
258C2×C3⋊D3—22no—
259C2×C3⋊D3C3×C22⋊C412yes—
260C2×C3×D3—34no—
261C2×C3×D3—62no—
262C2×C3×D3—62no—
263C2×C3×D3—62no—
264C2×C3×D3—62no—
265C2×C3×D3—62no—
266C2×C3×D3—62no—
267C2×C3×D3—62no—
268C2×C3×D3—122no—
269C2×C3×D3—122no—
270C32×C4—21no—
271C32×C4—22no—
272C32×C4—22no—
273C32×C4—21no—
274C32×C4—21no—
275C32×C4—22no—
276C32×C4—22no—
277C32×C4—41no—
278C32×C4—41no—
279C32×C4—42no—
280C32×C4—42no—
281C32×C4D3×D412yes—
282C32×C4D3×D412yes—
283C32×C4C2×C3×D411yescharacteristic
284C32×C4C2×C3×D411yescharacteristic
285C3⋊Dic3—21no—
286C3⋊Dic3C3×C22⋊C412yes—
287C3×Dic3—34no—
288C3×Dic3—62no—
289C3×Dic3—62no—
290C3×Dic3—62no—
291C2×#41—91no—
292C2×M16—91no—
293C22×C8—91no—
294C22⋊C8—32no—
295C22⋊C8—92no—
296C33#40811yescharacteristic, complemented, Sylow
297C23×C3—22no—
298C23×C3—21no—
299C23×C3—41no—
300C23×C3—91no—
301C23×C3—91no—
302C23×C3#62411yescharacteristic
303C23×C3C3×C4×D312yes—
304C22×D3—32no—
305C22×D3—32no—
306C22×D3—61no—
307C22×D3—61no—
308C3×Q8—182no—
309C3×D4—182no—
310C3×D4—182no—
311C3×D4—182no—
312C2×C3×C4—22no—
313C2×C3×C4—22no—
314C2×C3×C4—21no—
315C2×C3×C4—21no—
316C2×C3×C4—22no—
317C2×C3×C4—22no—
318C2×C3×C4—21no—
319C2×C3×C4—22no—
320C2×C3×C4—22no—
321C2×C3×C4—21no—
322C2×C3×C4—42no—
323C2×C3×C4—42no—
324C2×C3×C4—42no—
325C2×C3×C4—42no—
326C2×C3×C4—41no—
327C2×C3×C4—41no—
328C2×C3×C4—91no—
329C2×C3×C4—92no—
330C2×C3×C4—92no—
331C2×C3×C4—91no—
332C2×C3×C4—181no—
333C2×C3×C4—181no—
334C2×C3×C4#62411yescharacteristic
335C2×C3×C4C3×C4×D312yes—
336C2×C3×C4C2×D3211yescharacteristic
337C2×C3×C4C22×C3×D312yes—
338#67—64no—
339#67—64no—
340#67—64no—
341#67—64no—
342C2×Dic3—32no—
343C2×Dic3—32no—
344C2×Dic3—61no—
345C2×Dic3—61no—
346D12—64no—
347D12—64no—
348C4×D3—34no—
349C4×D3—34no—
350C4×D3—62no—
351C4×D3—62no—
352C4×D3—62no—
353C4×D3—62no—
354C4×D3—62no—
355C4×D3—62no—
356Dic6—64no—
357Dic6—64no—
358C3×C8—62no—
359C3×C8—62no—
360C3×C8—62no—
361C3×C8—62no—
362C3×C8—122no—
363C3×C8—122no—
364C3⋊C8—22no—
365C3⋊C8—22no—
366C2×C32—22no—
367C2×C32—22no—
368C2×C32—22no—
369C2×C32—22no—
370C2×C32—21no—
371C2×C32—41no—
372C2×C32—42no—
373C2×C32—44no—
374C2×C32—42no—
375C2×C32—44no—
376C2×C32—42no—
377C2×C32—41no—
378C2×C32D3×C22⋊C412yes—
379C2×C32#90112yes—
380C2×C32#90712yes—
381C2×C32C2×C3×C22⋊C411yescharacteristic
382C2×C32C3×#13112yes—
383C3⋊D3—22no—
384C3⋊D3—41no—
385C3×D3—64no—
386C3×D3—122no—
387C3×D3—122no—
388C3×D3—122no—
389#41—182no—
390#41—182no—
391C2×Q8—91no—
392C2×D4—91no—
393C2×D4—91no—
394C2×D4—91no—
395C22×C4—91no—
396C22×C4—91no—
397C22×C4C3×D3211yescharacteristic
398M16—182no—
399C2×C8—32no—
400C2×C8—32no—
401C2×C8—182no—
402C22×C3—22no—
403C22×C3—24no—
404C22×C3—22no—
405C22×C3—22no—
406C22×C3—22no—
407C22×C3—22no—
408C22×C3—21no—
409C22×C3—21no—
410C22×C3—22no—
411C22×C3—42no—
412C22×C3—42no—
413C22×C3—42no—
414C22×C3—42no—
415C22×C3—44no—
416C22×C3—41no—
417C22×C3—92no—
418C22×C3—182no—
419C22×C3—182no—
420C22×C3—181no—
421C22×C3C2×#62411yescharacteristic
422C22×C3C2×C3×C4×D312yes—
423C2×D3—34no—
424C2×D3—62no—
425C2×D3—62no—
426C2×D3—62no—
427C2×D3—62no—
428C2×D3—62no—
429C2×D3—62no—
430C2×D3—62no—
431C2×D3—122no—
432C2×D3—122no—
433C3×C4—22no—
434C3×C4—22no—
435C3×C4—21no—
436C3×C4—22no—
437C3×C4—22no—
438C3×C4—21no—
439C3×C4—21no—
440C3×C4—42no—
441C3×C4—42no—
442C3×C4—41no—
443C3×C4—41no—
444C3×C4—92no—
445C3×C4—181no—
446C3×C4#381111yescharacteristic
447C3×C4#381111yescharacteristic
448C3×C4C3×D3×D412yes—
449C3×C4C3×D3×D412yes—
450Dic3—34no—
451Dic3—62no—
452Dic3—62no—
453Dic3—62no—
454C32—22no—
455C32—21no—
456C32—41no—
457C32#472712yescomplemented
458C32C3×#40811yescharacteristic, complemented
459C23—91no—
460C23—91no—
461C23C3×#62411yescharacteristic
462Q8—182no—
463D4—182no—
464D4—182no—
465D4—182no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—91no—
469C2×C4—92no—
470C2×C4—92no—
471C2×C4—91no—
472C2×C4—181no—
473C2×C4—181no—
474C2×C4C3×#62411yescharacteristic
475C2×C4C2×C3×D3211yescharacteristic
476C8—62no—
477C8—62no—
478C2×C3—21no—
479C2×C3—22no—
480C2×C3—22no—
481C2×C3—22no—
482C2×C3—22no—
483C2×C3—42no—
484C2×C3—44no—
485C2×C3—44no—
486C2×C3—42no—
487C2×C3—41no—
488C2×C3—41no—
489C2×C3—42no—
490C2×C3—182no—
491C2×C3—361no—
492C2×C3#6381712yes—
493C2×C3#6397411yescharacteristic
494C2×C3C3×D3×C22⋊C412yes—
495C2×C3C3×#90112yes—
496C2×C3C3×#90712yes—
497D3—64no—
498D3—122no—
499D3—122no—
500D3—122no—
501C22—21no—
502C22—22no—
503C22—92no—
504C22—182no—
505C22—181no—
506C22—182no—
507C22C2×C3×#62411yescharacteristic
508C4—21no—
509C4—92no—
510C4—181no—
511C4C3×#381111yescharacteristic
512C4C3×#381111yescharacteristic
513C3—21no—
514C3—22no—
515C3—41no—
516C3#1059115311yescharacteristic, direct product factor
517C3C3×#472712yescomplemented
518C2—41no—
519C2—182no—
520C2—361no—
521C2C3×#6381712yes—
522C2C3×#6397411yescharacteristic
subgroupG/H
centerC2×C3×C4C2×D32
commutatorC22×C32C22×C3×C4
FrattiniC2×C4C2×C3×D32
FittingC22×C33×C4C22
socleC22×C33C22×C4
prime radicalC2×C3×#3828C2
cosocleC2×C32×C4C23×C3
hypercenterC22×C3×C4D32
hypocenterC32C3×#408
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C2×C3×C4 ◁ C22×C3×C4

Complements

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

NH ≅ G/Ntype
2. C2×C3×#3712C2semidirect
3. C2×C3×#3710C2semidirect
4. C32×#830C2semidirect
5. C3×#63557C2semidirect
7. #10591153C3direct (N × H)
26. C2×#3712C2×C3semidirect
27. C2×#3710C2×C3semidirect
30. C3×#830D3semidirect
31. C3×#830C2×C3semidirect
34. #63557C2×C3semidirect
120. #830C3×D3semidirect
296. C33#408semidirect
457. C32#4727semidirect
458. C32C3×#408semidirect
516. C3#10591153direct (N × H)
517. C3C3×#4727semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#38282
C2×C3×#37122
C2×C3×#37102
C32×#8302
C3×#635572
C3×#47273
#105911533