Finite Group Databasebeta

← C3×#10590872 #13157834

Subgroups — C3×#10590872 #13157834

Order 1728.

542 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#63869C211yesmaximal, characteristic, complemented
2C3×#63864C211yesmaximal, characteristic
3C3×#63648C211yesmaximal, characteristic
4C32×#816C211yesmaximal, characteristic, complemented
5C32×C8×Dic3C211yesmaximal, characteristic, complemented
6C3×#63566C211yesmaximal, characteristic, complemented
7C3×#63564C211yesmaximal, characteristic
8C3×#4800—31nomaximal
9C3×#4697—31nomaximal
10#10590872C311yesmaximal, characteristic, direct product factor
11C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
12C2×C32×Dic6C2211yescharacteristic
13C2×C33×C8C2211yescharacteristic
14C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
15C32×C4⋊Dic3C2211yescharacteristic
16C32×C4×Dic3C2211yescharacteristic
17C2×C32×C3⋊C8C2211yescharacteristic
18C3×#3724—21no—
19C3×Dic3⋊Dic3—21no—
20C3×#3722—21no—
21C3×#3722—21no—
22C3×#947—31no—
23C3×#890—31no—
24C3×#875—31no—
25C3×#870—31no—
26#63869C2×C311yescharacteristic, complemented
27#63864C2×C311yescharacteristic
28C32×#103—31no—
29C32×C4×C8—31no—
30#63648C2×C311yescharacteristic
31C3×#835—31no—
32C3×#821—31no—
33C3×#821—31no—
34C3×#821—31no—
35C3×#821—61no—
36C3×#816—21no—
37C3×#816—31no—
38C3×#816D311yescharacteristic, complemented
39C3×#816C2×C311yescharacteristic, complemented
40C3×C8×Dic3—21no—
41C3×C8×Dic3D311yescharacteristic, complemented
42C3×C8×Dic3C2×C311yescharacteristic, complemented
43C3×#809—31no—
44#63566C2×C311yescharacteristic, complemented
45#63564C2×C311yescharacteristic
46C22×C3×C3⋊D3—21no—
47C2×C33×C4C2311yescharacteristic
48C2×C3×C3⋊Dic3—21no—
49C3×C4⋊(C3⋊D3)—21no—
50C3×C4⋊(C3⋊D3)—21no—
51C2×C32×Dic3—21no—
52C2×C32×Dic3—21no—
53C2×C32×Dic3D411yescharacteristic
54C2×C32×Dic3D411yescharacteristic
55C32×Dic6—21no—
56C32×Dic6—21no—
57C33×C8—21no—
58C32×C3⋊C8—21no—
59C3×#488—91no—
60#4800—31no—
61#4697—31no—
62C2×C32×Q8—31no—
63C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
64C2×C3×D12—31no—
65C2×C3×D12—31no—
66C2×C3×D12—61no—
67C2×C3×Dic6—21no—
68C2×C3×Dic6C2×D311yescharacteristic
69C2×C3×Dic6C22×C311yescharacteristic
70C2×C32×C8—21no—
71C2×C32×C8—21no—
72C2×C32×C8—21no—
73C2×C32×C8—31no—
74C2×C32×C8—41no—
75C2×C32×C8C2×D311yescharacteristic
76C2×C32×C8C2×D311yescharacteristic
77C2×C32×C8C22×C311yescharacteristic
78C32×C4⋊C4—31no—
79C32×C42—31no—
80C4⋊(C3⋊Dic3)C22×C311yescharacteristic
81C3×(C2×D3)⋊C4—61no—
82C3×(C2×D3)⋊C4—61no—
83C3×(C2×D3)⋊C4—61no—
84C3×(C2×D3)⋊C4—61no—
85C3×C4⋊Dic3—21no—
86C3×C4⋊Dic3—31no—
87C3×C4⋊Dic3—31no—
88C3×C4⋊Dic3—61no—
89C3×C4⋊Dic3—61no—
90C3×C4⋊Dic3C2×D311yescharacteristic
91C3×C4⋊Dic3C22×C311yescharacteristic
92C3×Dic3⋊C4—61no—
93C3×Dic3⋊C4—61no—
94C3×Dic3⋊C4—61no—
95C3×C4×Dic3—21no—
96C3×C4×Dic3C2×D311yescharacteristic
97C3×C4×Dic3C22×C311yescharacteristic
98C2×C3×C3⋊C8—21no—
99C2×C3×C3⋊C8C2×D311yescharacteristic
100C2×C3×C3⋊C8C22×C311yescharacteristic
101#3724—21no—
102Dic3⋊Dic3—21no—
103#3722—21no—
104#3722—21no—
105C22×C33C2×D411yescharacteristic
106C2×C3×C3⋊D3—41no—
107C2×C3×C3⋊D3—41no—
108C2×C3×C3⋊D3—41no—
109C33×C4#4111yescharacteristic
110C33×C4#4111yescharacteristic
111C3×C3⋊Dic3—41no—
112C32×Dic3—21no—
113C32×Dic3—21no—
114C32×Dic3—41no—
115C32×Dic3—41no—
116C3×#125—91no—
117C3×#124—91no—
118#947—31no—
119#890—31no—
120#875—31no—
121#870—31no—
122C3×#103—31no—
123C3×#103—31no—
124C3×#103—61no—
125C3×#103—91no—
126C3×#102—91no—
127C3×#102—91no—
128C3×C4×C8—31no—
129C3×C4×C8—31no—
130C3×C4×C8—61no—
131#835—31no—
132#821—31no—
133#821—31no—
134#821—31no—
135#821—61no—
136#816—31no—
137#816C3×D311yescharacteristic, complemented
138C8×Dic3C3×D311yescharacteristic, complemented
139#809—31no—
140C22×C3⋊D3—21no—
141C22×C3×D3—61no—
142C22×C3×D3—61no—
143C22×C3×D3—121no—
144C32×Q8—61no—
145C32×Q8—61no—
146C2×C32×C4—21no—
147C2×C32×C4—21no—
148C2×C32×C4—21no—
149C2×C32×C4—31no—
150C2×C32×C4—31no—
151C2×C32×C4—41no—
152C2×C32×C4—61no—
153C2×C32×C4—61no—
154C2×C32×C4C22×D311yescharacteristic
155C2×C32×C4C22×D311yescharacteristic
156C2×C32×C4C23×C311yescharacteristic
157C2×C3⋊Dic3—21no—
158C4⋊(C3⋊D3)—21no—
159C4⋊(C3⋊D3)—21no—
160C2×C3×Dic3—21no—
161C2×C3×Dic3—21no—
162C2×C3×Dic3—21no—
163C2×C3×Dic3—21no—
164C2×C3×Dic3—21no—
165C2×C3×Dic3—21no—
166C2×C3×Dic3—41no—
167C2×C3×Dic3—41no—
168C2×C3×Dic3—61no—
169C2×C3×Dic3—61no—
170C2×C3×Dic3—121no—
171C2×C3×Dic3D1211yescharacteristic
172C2×C3×Dic3D1211yescharacteristic
173C2×C3×Dic3C3×D411yescharacteristic
174C2×C3×Dic3C3×D411yescharacteristic
175C3×D12—62no—
176C3×D12—61no—
177C3×D12—61no—
178C3×D12—61no—
179C3×D12—61no—
180C3×D12—121no—
181C3×Dic6—22no—
182C3×Dic6—21no—
183C3×Dic6—21no—
184C3×Dic6—21no—
185C3×Dic6—21no—
186C3×Dic6—41no—
187C32×C8—21no—
188C32×C8—22no—
189C32×C8—21no—
190C32×C8—21no—
191C32×C8—41no—
192C32×C8—41no—
193C32×C8—42no—
194C32×C8—61no—
195C3×C3⋊C8—22no—
196C3×C3⋊C8—21no—
197C3×C3⋊C8—21no—
198#488—91noSylow
199C2×C33#12411yescharacteristic
200C2×C33#13511yescharacteristic
201C2×C33#13511yescharacteristic
202C3×C3⋊D3—81no—
203C2×C3×Q8—31no—
204C2×C3×Q8—31no—
205C2×C3×Q8—61no—
206C2×C3×D4—91no—
207C2×D12—31no—
208C2×D12—31no—
209C2×D12—61no—
210C2×Dic6C2×C3×D311yescharacteristic
211C2×C3×C8—21no—
212C2×C3×C8—21no—
213C2×C3×C8—21no—
214C2×C3×C8—31no—
215C2×C3×C8—31no—
216C2×C3×C8—41no—
217C2×C3×C8—61no—
218C2×C3×C8D3211yescharacteristic
219C2×C3×C8C2×C3×D311yescharacteristic
220C2×C3×C8C2×C3×D311yescharacteristic
221C3×C4⋊C4—31no—
222C3×C4⋊C4—31no—
223C3×C4⋊C4—61no—
224C3×C4⋊C4—91no—
225C3×C4⋊C4—181no—
226C3×C4⋊C4—181no—
227C3×C22⋊C4—181no—
228C3×C22⋊C4—181no—
229C3×C42—31no—
230C3×C42—31no—
231C3×C42—61no—
232(C2×D3)⋊C4—61no—
233(C2×D3)⋊C4—61no—
234(C2×D3)⋊C4—61no—
235(C2×D3)⋊C4—61no—
236C4⋊Dic3—31no—
237C4⋊Dic3—31no—
238C4⋊Dic3—61no—
239C4⋊Dic3—61no—
240C4⋊Dic3C2×C3×D311yescharacteristic
241Dic3⋊C4—61no—
242Dic3⋊C4—61no—
243Dic3⋊C4—61no—
244C4×Dic3C2×C3×D311yescharacteristic
245C2×C3⋊C8C2×C3×D311yescharacteristic
246C22×C32—21no—
247C22×C32—21no—
248C22×C32—21no—
249C22×C32—41no—
250C22×C32C2×D1211yescharacteristic
251C22×C32D3×D411yescharacteristic
252C22×C32C2×C3×D411yescharacteristic
253C2×C3⋊D3—41no—
254C2×C3⋊D3—41no—
255C2×C3⋊D3—41no—
256C2×C3×D3—122no—
257C2×C3×D3—121no—
258C2×C3×D3—121no—
259C2×C3×D3—121no—
260C2×C3×D3—121no—
261C2×C3×D3—121no—
262C2×C3×D3—121no—
263C2×C3×D3—241no—
264C2×C3×D3—241no—
265C32×C4—21no—
266C32×C4—21no—
267C32×C4—21no—
268C32×C4—21no—
269C32×C4—21no—
270C32×C4—21no—
271C32×C4—41no—
272C32×C4—41no—
273C32×C4—61no—
274C32×C4—61no—
275C32×C4—121no—
276C32×C4—121no—
277C32×C4#23511yescharacteristic
278C32×C4#23511yescharacteristic
279C32×C4#23511yescharacteristic
280C32×C4#23711yescharacteristic
281C32×C4C3×#4111yescharacteristic
282C32×C4C3×#4111yescharacteristic
283C3⋊Dic3—41no—
284C3×Dic3—22no—
285C3×Dic3—22no—
286C3×Dic3—21no—
287C3×Dic3—21no—
288C3×Dic3—21no—
289C3×Dic3—21no—
290C3×Dic3—42no—
291C3×Dic3—41no—
292C3×Dic3—41no—
293C3×Dic3—41no—
294C3×Dic3—41no—
295C3×Dic3—81no—
296C3×Dic3—121no—
297C3×Dic3—121no—
298C3×Dic3—241no—
299#125—91no—
300#124—91no—
301#103—31no—
302#103—91no—
303#102—91no—
304#102—91no—
305C4×C8—31no—
306C33#48811yescharacteristic, complemented, Sylow
307C23×C3—181no—
308C22×D3—61no—
309C22×D3—61no—
310C22×D3—121no—
311C3×Q8—62no—
312C3×Q8—61no—
313C3×Q8—61no—
314C3×Q8—61no—
315C3×Q8—61no—
316C3×Q8—121no—
317C3×D4—181no—
318C3×D4—181no—
319C2×C3×C4—21no—
320C2×C3×C4—21no—
321C2×C3×C4—21no—
322C2×C3×C4—31no—
323C2×C3×C4—31no—
324C2×C3×C4—31no—
325C2×C3×C4—31no—
326C2×C3×C4—41no—
327C2×C3×C4—61no—
328C2×C3×C4—61no—
329C2×C3×C4—61no—
330C2×C3×C4—61no—
331C2×C3×C4—61no—
332C2×C3×C4—61no—
333C2×C3×C4—121no—
334C2×C3×C4—121no—
335C2×C3×C4—181no—
336C2×C3×C4C2×D3211yescharacteristic
337C2×C3×C4C22×C3×D311yescharacteristic
338C2×C3×C4C22×C3×D311yescharacteristic
339C2×Dic3—21no—
340C2×Dic3—21no—
341C2×Dic3—61no—
342C2×Dic3—61no—
343C2×Dic3—121no—
344C2×Dic3C3×D1211yescharacteristic
345C2×Dic3C3×D1211yescharacteristic
346D12—62no—
347D12—61no—
348D12—61no—
349D12—61no—
350D12—61no—
351D12—121no—
352Dic6—21no—
353Dic6—21no—
354C3×C8—22no—
355C3×C8—21no—
356C3×C8—21no—
357C3×C8—21no—
358C3×C8—41no—
359C3×C8—41no—
360C3×C8—42no—
361C3×C8—62no—
362C3×C8—61no—
363C3×C8—61no—
364C3⋊C8—21no—
365C2×C32—21no—
366C2×C32—21no—
367C2×C32—21no—
368C2×C32—21no—
369C2×C32—21no—
370C2×C32—21no—
371C2×C32—21no—
372C2×C32—21no—
373C2×C32—21no—
374C2×C32—41no—
375C2×C32—41no—
376C2×C32—41no—
377C2×C32#87511yescharacteristic
378C2×C32#88511yescharacteristic
379C2×C32#90411yescharacteristic
380C2×C32#90411yescharacteristic
381C2×C32#91611yescharacteristic
382C2×C32#91911yescharacteristic
383C2×C32C3×#12411yescharacteristic
384C2×C32C3×#13511yescharacteristic
385C2×C32C3×#13511yescharacteristic
386C3⋊D3—81no—
387C3×D3—242no—
388C3×D3—241no—
389C3×D3—241no—
390C2×Q8—31no—
391C2×D4—91no—
392C2×C8—31no—
393C2×C8C3×D3211yescharacteristic
394C4⋊C4—31no—
395C4⋊C4—91no—
396C4⋊C4—181no—
397C4⋊C4—181no—
398C22⋊C4—181no—
399C22⋊C4—181no—
400C42—31no—
401C22×C3—21no—
402C22×C3—21no—
403C22×C3—21no—
404C22×C3—41no—
405C22×C3—361no—
406C22×C3—361no—
407C22×C3—361no—
408C22×C3D3×D1211yescharacteristic
409C22×C3C2×C3×D1211yescharacteristic
410C22×C3C3×D3×D411yescharacteristic
411C2×D3—122no—
412C2×D3—121no—
413C2×D3—121no—
414C2×D3—121no—
415C2×D3—121no—
416C2×D3—121no—
417C2×D3—121no—
418C2×D3—241no—
419C2×D3—241no—
420C3×C4—21no—
421C3×C4—21no—
422C3×C4—21no—
423C3×C4—21no—
424C3×C4—21no—
425C3×C4—21no—
426C3×C4—41no—
427C3×C4—41no—
428C3×C4—62no—
429C3×C4—62no—
430C3×C4—61no—
431C3×C4—61no—
432C3×C4—61no—
433C3×C4—61no—
434C3×C4—122no—
435C3×C4—121no—
436C3×C4—121no—
437C3×C4—121no—
438C3×C4—121no—
439C3×C4—241no—
440C3×C4—361no—
441C3×C4#379811yescharacteristic
442C3×C4#380411yescharacteristic
443C3×C4C3×#23511yescharacteristic
444C3×C4C3×#23511yescharacteristic
445C3×C4C3×#23511yescharacteristic
446C3×C4C3×#23711yescharacteristic
447Dic3—21no—
448Dic3—21no—
449Dic3—41no—
450Dic3—41no—
451Dic3—121no—
452Dic3—121no—
453Dic3—241no—
454C32—21no—
455C32—21no—
456C32—21no—
457C32—41no—
458C32#469711yescharacteristic, complemented
459C32#480011yescharacteristic, complemented
460C32C3×#48811yescharacteristic, complemented
461C23—181no—
462Q8—61no—
463Q8—61no—
464D4—181no—
465D4—181no—
466C2×C4—31no—
467C2×C4—31no—
468C2×C4—61no—
469C2×C4—61no—
470C2×C4—181no—
471C2×C4C2×C3×D3211yescharacteristic
472C8—21no—
473C8—61no—
474C2×C3—21no—
475C2×C3—21no—
476C2×C3—21no—
477C2×C3—21no—
478C2×C3—21no—
479C2×C3—21no—
480C2×C3—21no—
481C2×C3—21no—
482C2×C3—21no—
483C2×C3—41no—
484C2×C3—41no—
485C2×C3—41no—
486C2×C3—721no—
487C2×C3#6380611yescharacteristic
488C2×C3#6380811yescharacteristic
489C2×C3#6385211yescharacteristic
490C2×C3C3×#87511yescharacteristic
491C2×C3C3×#88511yescharacteristic
492C2×C3C3×#90411yescharacteristic
493C2×C3C3×#90411yescharacteristic
494C2×C3C3×#91611yescharacteristic
495C2×C3C3×#91911yescharacteristic
496D3—242no—
497D3—241no—
498D3—241no—
499C22—361no—
500C22—361no—
501C22—361no—
502C22C3×D3×D1211yescharacteristic
503C4—61no—
504C4—61no—
505C4—121no—
506C4—121no—
507C4—361no—
508C4C3×#379811yescharacteristic
509C4C3×#380411yescharacteristic
510C3—21no—
511C3—21no—
512C3—21no—
513C3—41no—
514C3#1059087211yescharacteristic, direct product factor
515C3C3×#469711yescharacteristic, complemented
516C3C3×#480011yescharacteristic, complemented
517C2—721no—
518C2C3×#6380611yescharacteristic
519C2C3×#6380811yescharacteristic
520C2C3×#6385211yescharacteristic
subgroupG/H
centerC22×C3D3×D12
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C2×C3×D32
FittingC2×C33×C8C22
socleC22×C33C2×D4
prime radicalC2×C3×C4⋊(C3⋊D3)C22
hypercenterC2×C3×C8D32
hypocenterC32C3×#488
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C2×C3×C8

Complements

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

NH ≅ G/Ntype
1. C3×#63869C2semidirect
4. C32×#816C2semidirect
5. C32×C8×Dic3C2semidirect
6. C3×#63566C2semidirect
10. #10590872C3direct (N × H)
26. #63869C2×C3semidirect
38. C3×#816D3semidirect
39. C3×#816C2×C3semidirect
41. C3×C8×Dic3D3semidirect
42. C3×C8×Dic3C2×C3semidirect
44. #63566C2×C3semidirect
137. #816C3×D3semidirect
138. C8×Dic3C3×D3semidirect
306. C33#488semidirect
458. C32#4697semidirect
459. C32#4800semidirect
460. C32C3×#488semidirect
514. C3#10590872direct (N × H)
515. C3C3×#4697semidirect
516. C3C3×#4800semidirect

Maximal subgroups

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

subgroupindex
C3×#638692
C3×#638642
C3×#636482
C32×#8162
C32×C8×Dic32
C3×#635662
C3×#635642
C3×#48003
C3×#46973
#105908723