Finite Group Databasebeta

← #13178140

Subgroups — #13178140

Order 1728.

1230 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×#64083C211yesmaximal, characteristic, complemented
2C3×#64081C211yesmaximal, characteristic, complemented
3C3×#64079C211yesmaximal, characteristic, complemented
4#11723696C212yesmaximal, complemented
5#11723691C212yesmaximal, complemented
6#11723687C214yesmaximal, complemented
7#11723686C214yesmaximal, complemented
8#10593989—31nomaximal
9#10593950—31nomaximal
10#10593949—31nomaximal
11C2×C3×C4⋊(C3⋊D3)C2212yes—
12C2×C3×C4⋊(C3⋊D3)C2214yes—
13C2×C3×C4×C3⋊D3C2211yescharacteristic
14C2×#6295C2214yes—
15C2×#6294C2214yes—
16C33×C42C2211yescharacteristic
17C3×#3752C2212yes—
18C3×C4⋊(C3⋊Dic3)C2212yes—
19C3×#3750C2214yes—
20C3×#3750C2212yes—
21C3×C4×C3⋊Dic3C2211yescharacteristic
22#88814C2214yes—
23#88814C2214yes—
24#64083D311yescharacteristic, complemented
25#64081D311yescharacteristic, complemented
26#64079D311yescharacteristic, complemented
27C3×#874—31no—
28C3×#874—31no—
29C3×#874—61no—
30C3×#872—31no—
31C3×#872—31no—
32C3×#872—61no—
33C3×#870—31no—
34C3×#870—31no—
35C3×#870—61no—
36#63916—32no—
37#63910—34no—
38#63909—32no—
39#63908—32no—
40#63898—32no—
41#63897—34no—
42#63894—32no—
43Dic3×D12—32no—
44#63886—34no—
45#63885—34no—
46#63884—34no—
47#63884—34no—
48C22×C3×C3⋊D3C2311yescharacteristic
49C22×C3×C3⋊D3C2314yes—
50C2×C33×C4C2312yes—
51C2×C33×C4C2311yescharacteristic
52C2×C3×C3⋊Dic3C2312yes—
53C2×C3×C3⋊Dic3C2314yes—
54C2×C3×C3⋊Dic3C2311yescharacteristic
55C3×C4⋊(C3⋊D3)—24no—
56C3×C4⋊(C3⋊D3)—44no—
57C3×C4×C3⋊D3—22no—
58#6295—44no—
59#6294—44no—
60#5719—91no—
61#5697—91no—
62#5535—91no—
63C2×C4⋊(C3⋊D3)C2×D312yes—
64C2×C4⋊(C3⋊D3)C2×D314yes—
65C2×C4×C3⋊D3C2×D311yescharacteristic
66C2×C3×D12—32no—
67C2×C3×D12—32no—
68C2×C3×D12—34no—
69C2×C3×D12—34no—
70C2×C3×D12—62no—
71C2×C3×D12—64no—
72C2×C3×C4×D3—31no—
73C2×C3×C4×D3—31no—
74C2×C3×C4×D3—61no—
75C2×#626—34no—
76C2×#626—34no—
77C2×#625—34no—
78C2×#624—34no—
79C2×D3×Dic3—34no—
80C2×D3×Dic3—34no—
81C32×C42—21no—
82C32×C42—21no—
83C32×C42—21no—
84C32×C42—41no—
85C32×C42C2×D311yescharacteristic
86C32×C42C2×D311yescharacteristic
87C32×C42C2×D311yescharacteristic
88#3752C2×D312yes—
89C4⋊(C3⋊Dic3)C2×D312yes—
90#3750C2×D314yes—
91#3750C2×D312yes—
92C4×C3⋊Dic3C2×D311yescharacteristic
93C3×(C2×D3)⋊C4—32no—
94C3×(C2×D3)⋊C4—32no—
95C3×(C2×D3)⋊C4—62no—
96C3×C4⋊Dic3—32no—
97C3×C4⋊Dic3—32no—
98C3×C4⋊Dic3—62no—
99C3×Dic3⋊C4—32no—
100C3×Dic3⋊C4—34no—
101C3×Dic3⋊C4—32no—
102C3×Dic3⋊C4—34no—
103C3×Dic3⋊C4—64no—
104C3×Dic3⋊C4—62no—
105C3×C4×Dic3—31no—
106C3×C4×Dic3—31no—
107C3×C4×Dic3—61no—
108#3722—34no—
109#3722—34no—
110(C2×D3)⋊Dic3—34no—
111(C2×D3)⋊Dic3—34no—
112(C2×D3)⋊Dic3—34no—
113(C2×D3)⋊Dic3—34no—
114C22×C33C2411yescharacteristic
115C2×C3×C3⋊D3—21no—
116C2×C3×C3⋊D3—24no—
117C2×C3×C3⋊D3—24no—
118C2×C3×C3⋊D3—22no—
119C2×C3×C3⋊D3—24no—
120C33×C4—22no—
121C33×C4#4112yes—
122C3×C3⋊Dic3—21no—
123C3×C3⋊Dic3—22no—
124C3×C3⋊Dic3—24no—
125C3×#127—91no—
126C3×#125—91no—
127C3×#117—91no—
128#939—94no—
129#938—92no—
130#935—94no—
131D4×Dic3—92no—
132#930—92no—
133#895—94no—
134#895—92no—
135#894—94no—
136#894—94no—
137#893—92no—
138#884—94no—
139#874—31no—
140#874—31no—
141#874—61no—
142C4×D12—92no—
143#872—31no—
144#872—31no—
145#872—61no—
146#870—31no—
147#870—31no—
148#870—61no—
149C22×C3⋊D3C22×D314yes—
150C22×C3⋊D3C22×D311yescharacteristic
151C22×C3×D3—34no—
152C22×C3×D3—34no—
153C22×C3×D3—31no—
154C22×C3×D3—31no—
155C22×C3×D3—61no—
156C22×C3×D3—64no—
157C2×C32×C4—21no—
158C2×C32×C4—21no—
159C2×C32×C4—22no—
160C2×C32×C4—21no—
161C2×C32×C4—22no—
162C2×C32×C4—22no—
163C2×C32×C4—41no—
164C2×C32×C4—42no—
165C2×C32×C4C22×D312yes—
166C2×C32×C4C22×D312yes—
167C2×C32×C4C22×D312yes—
168C2×C32×C4C22×D311yescharacteristic
169C2×C32×C4C22×D311yescharacteristic
170C2×C32×C4C22×D311yescharacteristic
171C2×C3⋊Dic3C22×D311yescharacteristic
172C2×C3⋊Dic3C22×D312yes—
173C2×C3⋊Dic3C22×D314yes—
174C4⋊(C3⋊D3)—24no—
175C4⋊(C3⋊D3)—44no—
176C4×C3⋊D3—22no—
177C2×C3×Dic3—31no—
178C2×C3×Dic3—32no—
179C2×C3×Dic3—34no—
180C2×C3×Dic3—31no—
181C2×C3×Dic3—32no—
182C2×C3×Dic3—34no—
183C2×C3×Dic3—61no—
184C2×C3×Dic3—62no—
185C2×C3×Dic3—64no—
186C3×D12—64no—
187C3×D12—64no—
188C3×D12—124no—
189C3×D12—124no—
190C3×D12—124no—
191C3×D12—128no—
192C3×C4×D3—64no—
193C3×C4×D3—62no—
194C3×C4×D3—62no—
195#626—124no—
196#626—124no—
197#625—124no—
198#624—124no—
199D3×Dic3—124no—
200D3×Dic3—124no—
201#540—271noSylow
202C2×C33C2×#4111yescharacteristic
203C2×C33H(5,𝔽2)12yes—
204C3×C3⋊D3—41no—
205C3×C3⋊D3—44no—
206C2×C3×D4—92no—
207C2×C3×D4—94no—
208C22×C3×C4—91no—
209C2×#67—94no—
210C2×#67—94no—
211C22×Dic3—94no—
212C2×D12—32no—
213C2×D12—32no—
214C2×D12—34no—
215C2×D12—34no—
216C2×D12—62no—
217C2×D12—64no—
218C2×D12—94no—
219C2×C4×D3—31no—
220C2×C4×D3—31no—
221C2×C4×D3—61no—
222C2×C4×D3—94no—
223C2×C4×D3—94no—
224C3×C4⋊C4—92no—
225C3×C4⋊C4—92no—
226C3×C4⋊C4—94no—
227C3×C22⋊C4—92no—
228C3×C42—21no—
229C3×C42—21no—
230C3×C42—21no—
231C3×C42—41no—
232C3×C42—91no—
233C3×C42D3211yescharacteristic
234C3×C42D3211yescharacteristic
235C3×C42D3211yescharacteristic
236#217—94no—
237#217—94no—
238(C2×D3)⋊C4—32no—
239(C2×D3)⋊C4—32no—
240(C2×D3)⋊C4—62no—
241(C2×D3)⋊C4—94no—
242(C2×D3)⋊C4—94no—
243(C2×D3)⋊C4—94no—
244(C2×D3)⋊C4—94no—
245C4⋊Dic3—32no—
246C4⋊Dic3—32no—
247C4⋊Dic3—62no—
248Dic3⋊C4—34no—
249Dic3⋊C4—34no—
250Dic3⋊C4—32no—
251Dic3⋊C4—32no—
252Dic3⋊C4—64no—
253Dic3⋊C4—62no—
254C4×Dic3—31no—
255C4×Dic3—31no—
256C4×Dic3—61no—
257C22×C32—21no—
258C22×C32—21no—
259C22×C32—21no—
260C22×C32—41no—
261C22×C32C23×D311yescharacteristic
262C22×C32C23×D311yescharacteristic
263C22×C32C23×D311yescharacteristic
264C2×C3⋊D3—21no—
265C2×C3⋊D3—24no—
266C2×C3⋊D3—24no—
267C2×C3⋊D3—24no—
268C2×C3⋊D3—22no—
269C2×C3×D3—62no—
270C2×C3×D3—64no—
271C2×C3×D3—61no—
272C2×C3×D3—62no—
273C2×C3×D3—61no—
274C2×C3×D3—64no—
275C2×C3×D3—64no—
276C2×C3×D3—64no—
277C2×C3×D3—62no—
278C2×C3×D3—64no—
279C2×C3×D3—64no—
280C2×C3×D3—122no—
281C2×C3×D3—124no—
282C2×C3×D3—124no—
283C2×C3×D3—124no—
284C32×C4—22no—
285C32×C4—24no—
286C32×C4—22no—
287C32×C4—22no—
288C32×C4—22no—
289C32×C4—22no—
290C32×C4—22no—
291C32×C4—42no—
292C32×C4—42no—
293C32×C4—42no—
294C32×C4—44no—
295C32×C4#23512yes—
296C32×C4#23712yes—
297C32×C4#23912yes—
298C3⋊Dic3—21no—
299C3⋊Dic3—22no—
300C3⋊Dic3—24no—
301C3×Dic3—62no—
302C3×Dic3—61no—
303C3×Dic3—62no—
304C3×Dic3—64no—
305C3×Dic3—61no—
306C3×Dic3—62no—
307C3×Dic3—64no—
308C3×Dic3—122no—
309C3×Dic3—124no—
310#127—91no—
311#125—91no—
312#123—274no—
313#121—274no—
314#121—272no—
315C4×D4—272no—
316#117—91no—
317C33#54011yescharacteristic, complemented, Sylow
318C23×C3—94no—
319C23×C3—91no—
320C22×D3—31no—
321C22×D3—31no—
322C22×D3—34no—
323C22×D3—34no—
324C22×D3—61no—
325C22×D3—64no—
326C3×D4—184no—
327C3×D4—364no—
328C2×C3×C4—21no—
329C2×C3×C4—21no—
330C2×C3×C4—22no—
331C2×C3×C4—21no—
332C2×C3×C4—22no—
333C2×C3×C4—22no—
334C2×C3×C4—42no—
335C2×C3×C4—41no—
336C2×C3×C4—91no—
337C2×C3×C4—92no—
338C2×C3×C4—94no—
339C2×C3×C4—182no—
340C2×C3×C4C2×D3212yes—
341C2×C3×C4C2×D3212yes—
342C2×C3×C4C2×D3212yes—
343C2×C3×C4C2×D3211yescharacteristic
344C2×C3×C4C2×D3211yescharacteristic
345C2×C3×C4C2×D3211yescharacteristic
346#67—364no—
347#67—364no—
348C2×Dic3—32no—
349C2×Dic3—32no—
350C2×Dic3—34no—
351C2×Dic3—34no—
352C2×Dic3—31no—
353C2×Dic3—31no—
354C2×Dic3—61no—
355C2×Dic3—62no—
356C2×Dic3—64no—
357C2×Dic3—364no—
358D12—64no—
359D12—64no—
360D12—124no—
361D12—124no—
362D12—124no—
363D12—128no—
364D12—364no—
365C4×D3—64no—
366C4×D3—62no—
367C4×D3—62no—
368C4×D3—364no—
369C4×D3—364no—
370C2×C32—21no—
371C2×C32—21no—
372C2×C32—21no—
373C2×C32—22no—
374C2×C32—22no—
375C2×C32—22no—
376C2×C32—41no—
377C2×C32—42no—
378C2×C32C2×#23511yescharacteristic
379C2×C32C2×#23711yescharacteristic
380C2×C32#100412yes—
381C2×C32C2×#23911yescharacteristic
382C2×C32#100912yes—
383C2×C32#100912yes—
384C3⋊D3—41no—
385C3⋊D3—44no—
386C3×D3—122no—
387C3×D3—124no—
388C3×D3—121no—
389C3×D3—121no—
390C3×D3—124no—
391C3×D3—244no—
392C2×D4—92no—
393C2×D4—94no—
394C2×D4—274no—
395C22×C4—91no—
396C22×C4—274no—
397C4⋊C4—92no—
398C4⋊C4—92no—
399C4⋊C4—94no—
400C22⋊C4—92no—
401C22⋊C4—274no—
402C22⋊C4—274no—
403C42—91no—
404C42#111311yescharacteristic
405C22×C3—21no—
406C22×C3—21no—
407C22×C3—21no—
408C22×C3—41no—
409C22×C3—181no—
410C22×C3—184no—
411C22×C3—182no—
412C22×C3—184no—
413C22×C3—184no—
414C22×C3C22×D3211yescharacteristic
415C22×C3C22×D3211yescharacteristic
416C22×C3C22×D3211yescharacteristic
417C2×D3—62no—
418C2×D3—61no—
419C2×D3—64no—
420C2×D3—64no—
421C2×D3—61no—
422C2×D3—64no—
423C2×D3—64no—
424C2×D3—64no—
425C2×D3—64no—
426C2×D3—62no—
427C2×D3—62no—
428C2×D3—122no—
429C2×D3—124no—
430C2×D3—124no—
431C2×D3—124no—
432C3×C4—24no—
433C3×C4—22no—
434C3×C4—22no—
435C3×C4—22no—
436C3×C4—22no—
437C3×C4—22no—
438C3×C4—22no—
439C3×C4—42no—
440C3×C4—42no—
441C3×C4—44no—
442C3×C4—42no—
443C3×C4—181no—
444C3×C4—182no—
445C3×C4—184no—
446C3×C4#379512yes—
447C3×C4#379612yes—
448C3×C4#379912yes—
449Dic3—62no—
450Dic3—61no—
451Dic3—61no—
452Dic3—62no—
453Dic3—64no—
454Dic3—62no—
455Dic3—64no—
456Dic3—122no—
457Dic3—124no—
458C32—21no—
459C32—21no—
460C32—21no—
461C32—41no—
462C32#553511yescharacteristic, complemented
463C32#569711yescharacteristic, complemented
464C32#571911yescharacteristic, complemented
465C23—91no—
466C23—94no—
467D4—184no—
468D4—364no—
469D4—1084no—
470C2×C4—92no—
471C2×C4—91no—
472C2×C4—94no—
473C2×C4—182no—
474C2×C4—1084no—
475C2×C4C2×#111312yes—
476C2×C4C2×#111311yescharacteristic
477C2×C3—21no—
478C2×C3—21no—
479C2×C3—21no—
480C2×C3—22no—
481C2×C3—22no—
482C2×C3—22no—
483C2×C3—41no—
484C2×C3—42no—
485C2×C3—361no—
486C2×C3—364no—
487C2×C3C2×#379511yescharacteristic
488C2×C3C2×#379611yescharacteristic
489C2×C3C2×#379911yescharacteristic
490C2×C3#6430712yes—
491C2×C3#6430712yes—
492C2×C3#6430812yes—
493D3—122no—
494D3—121no—
495D3—124no—
496D3—124no—
497D3—121no—
498D3—244no—
499C22—181no—
500C22—184no—
501C22—184no—
502C22—184no—
503C22—182no—
504C22C22×#111311yescharacteristic
505C4—22no—
506C4—181no—
507C4—182no—
508C4—184no—
509C4#8904112yes—
510C3—21no—
511C3—21no—
512C3—21no—
513C3—41no—
514C3#1059394911yescharacteristic, complemented
515C3#1059395011yescharacteristic, complemented
516C3#1059398911yescharacteristic, complemented
517C2—361no—
518C2—364no—
519C2C2×#8904111yescharacteristic
520C2#1172461912yes—
subgroupG/H
center, FrattiniC22C22×#1113
commutator, socle, cosocleC22×C33C24
FittingC33×C42C22
hypercenterC42#1113
hypocenterC33#540
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
1. C3×#64083C2semidirect
2. C3×#64081C2semidirect
3. C3×#64079C2semidirect
4. #11723696C2semidirect
5. #11723691C2semidirect
6. #11723687C2semidirect
7. #11723686C2semidirect
24. #64083D3semidirect
25. #64081D3semidirect
26. #64079D3semidirect
317. C33#540semidirect
462. C32#5535semidirect
463. C32#5697semidirect
464. C32#5719semidirect
514. C3#10593949semidirect
515. C3#10593950semidirect
516. C3#10593989semidirect

Maximal subgroups

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

subgroupindex
C3×#640832
C3×#640812
C3×#640792
#117236962
#117236912
#117236872
#117236862
#105939893
#105939503
#105939493