Finite Group Databasebeta

← #13178666

Subgroups — #13178666

Order 1728.

1582 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3830C211yesmaximal, characteristic, complemented
2C2×#89042C211yesmaximal, characteristic, complemented
3C3×#64147C212yesmaximal, complemented
4#11723725C214yesmaximal, complemented
5#11723721C212yesmaximal, complemented
6#11723717C212yesmaximal, complemented
7#11723689C211yesmaximal, characteristic, complemented
8#11723674C212yesmaximal, complemented
9#10595240—31nomaximal
10#10595232—32nomaximal
11C2×C33×D4C2211yescharacteristic
12C2×C3×#638C2212yes—
13C2×C3×#638C2214yes—
14C22×C3×C3⋊Dic3C2212yes—
15C3×#3830—24no—
16C2×C3×C4×C3⋊D3C2212yes—
17C2×C3×C4×C3⋊D3C2211yescharacteristic
18C2×C3×#634C2211yescharacteristic
19C2×#6296C2211yescharacteristic
20C2×#6295C2211yescharacteristic
21C2×#6295C2212yes—
22C2×#6294C2212yes—
23#89042—24no—
24C3×#3757C2214yes—
25C3×C4⋊(C3⋊Dic3)C2212yes—
26#88815C2214yes—
27#88815C2212yes—
28#88814C2214yes—
29C2×#3830D311yescharacteristic, complemented
30C2×C3×#237—32no—
31C2×C3×#237—61no—
32C2×#3798—32no—
33C2×#3798—31no—
34#64147D312yescomplemented
35C3×#938—32no—
36C3×#938—32no—
37C3×#938—62no—
38#63982—32no—
39#63980—32no—
40#63978—34no—
41#63971—34no—
42#63970—34no—
43#63964—34no—
44#63958—34no—
45#63900—31no—
46D3×C4⋊Dic3—32no—
47#63871—32no—
48#63857—32no—
49#63856—32no—
50C23×C33C2312yes—
51C22×C3×C3⋊D3C2312yes—
52C22×C3×C3⋊D3C2311yescharacteristic
53C33×D4—22no—
54C2×C33×C4C2311yescharacteristic
55C3×#638—24no—
56C3×#638—44no—
57C2×C3×C3⋊Dic3—24no—
58C2×C3×C3⋊Dic3C2314yes—
59C2×C3×C3⋊Dic3C2312yes—
60C2×C3×C3⋊Dic3C2312yes—
61C2×C3×C3⋊Dic3C2311yescharacteristic
62C3×C4×C3⋊D3—24no—
63C3×C4×C3⋊D3D414yescomplemented
64C3×#634—22no—
65#6296—22no—
66#6295—24no—
67#6295—22no—
68#6294—42no—
69#5833—91no—
70#5608—92no—
71C2×C32×D4—21no—
72C2×C32×D4—22no—
73C2×C32×D4—41no—
74C2×C32×D4C2×D311yescharacteristic
75C2×C32×D4C2×D312yes—
76C2×#638C2×D312yes—
77C2×#638C2×D314yes—
78C22×C3⋊Dic3C2×D312yes—
79#3830—24no—
80C2×C4×C3⋊D3C2×D312yes—
81C2×C4×C3⋊D3C2×D311yescharacteristic
82C2×#634C2×D311yescharacteristic
83C2×C3×#67—34no—
84C2×C3×#67—34no—
85C2×C3×#67—34no—
86C2×C3×#67—62no—
87C2×C3×#67—64no—
88C22×C3×Dic3—34no—
89C22×C3×Dic3—62no—
90C3×#237—68no—
91C3×#237—68no—
92C2×C3×C4×D3—32no—
93C2×C3×C4×D3—32no—
94C2×C3×C4×D3—32no—
95C2×C3×C4×D3—61no—
96C2×C3×C4×D3—62no—
97C2×C3×Dic6—32no—
98C2×C3×Dic6—61no—
99C2×#627—32no—
100C2×#627—31no—
101C2×#626—32no—
102C2×#626—32no—
103C2×#626—32no—
104C2×#625—32no—
105C2×#625—31no—
106C2×#624—32no—
107C2×D3×Dic3—34no—
108#3798—68no—
109#3798—64no—
110#3757C2×D314yes—
111C4⋊(C3⋊Dic3)C2×D312yes—
112C3×#217—34no—
113C3×#217—34no—
114C3×#217—64no—
115C3×C4⋊Dic3—32no—
116C3×C4⋊Dic3—32no—
117C3×C4⋊Dic3—62no—
118#3724—34no—
119#3724—32no—
120Dic3⋊Dic3—34no—
121Dic3⋊Dic3—34no—
122Dic3⋊Dic3—34no—
123#3722—34no—
124(C2×D3)⋊Dic3—34no—
125(C2×D3)⋊Dic3—34no—
126C22×C33—22no—
127C22×C33—22no—
128C22×C33—22no—
129C22×C33C2411yescharacteristic
130C2×C3×C3⋊D3—21no—
131C2×C3×C3⋊D3—21no—
132C2×C3×C3⋊D3—22no—
133C2×C3×C3⋊D3—22no—
134C2×C3×C3⋊D3—22no—
135C2×C3×C3⋊D3C2×D412yes—
136C33×C4C2×D412yes—
137C3×C3⋊Dic3—22no—
138C3×C3⋊Dic3—22no—
139C3×C3⋊Dic3—24no—
140C3×C3⋊Dic3C2×D412yes—
141C2×C3×#41—91no—
142C2×#237—32no—
143C2×#237—61no—
144C2×#235—91no—
145C2×#235—92no—
146C3×#121—92no—
147#946—92no—
148#939—94no—
149#939—94no—
150#938—32no—
151#938—32no—
152#938—62no—
153#938—94no—
154#930—92no—
155C2×C4⋊Dic3—91no—
156#924—92no—
157#924—92no—
158#897—92no—
159D3×C4⋊C4—92no—
160#884—94no—
161#878—94no—
162C23×C32—22no—
163C23×C32—24no—
164C23×C32—42no—
165C23×C32C22×D312yes—
166C23×C32C22×D314yes—
167C22×C3⋊D3C22×D312yes—
168C22×C3⋊D3C22×D311yescharacteristic
169C22×C3×D3—32no—
170C22×C3×D3—32no—
171C22×C3×D3—32no—
172C22×C3×D3—61no—
173C22×C3×D3—62no—
174C32×D4—22no—
175C32×D4—24no—
176C32×D4—24no—
177C32×D4—44no—
178C32×D4—44no—
179C2×C32×C4—22no—
180C2×C32×C4—21no—
181C2×C32×C4—41no—
182C2×C32×C4C22×D312yes—
183C2×C32×C4C22×D311yescharacteristic
184#638—24no—
185#638—44no—
186C2×C3⋊Dic3—24no—
187C2×C3⋊Dic3C22×D312yes—
188C2×C3⋊Dic3C22×D314yes—
189C2×C3⋊Dic3C22×D312yes—
190C2×C3⋊Dic3C22×D311yescharacteristic
191C4×C3⋊D3—24no—
192C4×C3⋊D3#6714yescomplemented
193#634—22no—
194C3×#67—68no—
195C3×#67—68no—
196C3×#67—124no—
197C3×#67—124no—
198C3×#67—128no—
199C2×C3×Dic3—32no—
200C2×C3×Dic3—34no—
201C2×C3×Dic3—34no—
202C2×C3×Dic3—34no—
203C2×C3×Dic3—32no—
204C2×C3×Dic3—32no—
205C2×C3×Dic3—62no—
206C2×C3×Dic3—61no—
207C2×C3×Dic3—64no—
208C2×C3×Dic3—62no—
209C2×C3×Dic3—68no—
210C2×C3×Dic3—68no—
211C3×C4×D3—38no—
212C3×C4×D3—64no—
213C3×C4×D3—64no—
214C3×C4×D3—64no—
215C3×C4×D3—124no—
216C3×Dic6—64no—
217C3×Dic6—64no—
218#627—64no—
219#627—62no—
220#626—64no—
221#626—64no—
222#626—64no—
223#625—64no—
224#625—62no—
225#624—122no—
226D3×Dic3—124no—
227#537—271noSylow
228C2×C33—42no—
229C2×C33C22×D411yescharacteristic
230C2×C33H(5,𝔽2)11yescharacteristic
231C2×C332−1+411yescharacteristic
232C3×C3⋊D3—22no—
233C3×C3⋊D3—42no—
234C3×#41—184no—
235C2×C3×Q8—91no—
236C2×C3×D4—21no—
237C2×C3×D4—22no—
238C2×C3×D4—41no—
239C2×C3×D4—92no—
240C2×C3×D4—94no—
241C2×C3×D4D3211yescharacteristic
242C2×C3×D4D3212yes—
243C22×C3×C4—91no—
244C22×C3×C4—92no—
245C22×C3×C4—92no—
246C2×#67—34no—
247C2×#67—34no—
248C2×#67—34no—
249C2×#67—62no—
250C2×#67—64no—
251C2×#67—92no—
252C2×#67—92no—
253C2×#67—92no—
254C22×Dic3—34no—
255C22×Dic3—62no—
256C22×Dic3—92no—
257#237—68no—
258#237—68no—
259#235—188no—
260#235—184no—
261C2×D12—92no—
262C2×D12—91no—
263C2×C4×D3—32no—
264C2×C4×D3—32no—
265C2×C4×D3—32no—
266C2×C4×D3—61no—
267C2×C4×D3—62no—
268C2×C4×D3—94no—
269C2×Dic6—32no—
270C2×Dic6—61no—
271C2×Dic6—92no—
272C2×Dic6—91no—
273C3×C4⋊C4—92no—
274C3×C22⋊C4—94no—
275#217—34no—
276#217—34no—
277#217—64no—
278#217—94no—
279(C2×D3)⋊C4—94no—
280(C2×D3)⋊C4—94no—
281C4⋊Dic3—32no—
282C4⋊Dic3—32no—
283C4⋊Dic3—62no—
284C4⋊Dic3—94no—
285C4⋊Dic3—92no—
286Dic3⋊C4—94no—
287Dic3⋊C4—94no—
288Dic3⋊C4—94no—
289C22×C32—22no—
290C22×C32—24no—
291C22×C32—24no—
292C22×C32—24no—
293C22×C32—22no—
294C22×C32—22no—
295C22×C32—24no—
296C22×C32—21no—
297C22×C32—22no—
298C22×C32—41no—
299C22×C32—44no—
300C22×C32—44no—
301C22×C32—42no—
302C22×C32—42no—
303C22×C32—44no—
304C22×C32—44no—
305C22×C32—82no—
306C22×C32—82no—
307C22×C32C23×D312yes—
308C22×C32C23×D311yescharacteristic
309C2×C3⋊D3—22no—
310C2×C3⋊D3—21no—
311C2×C3⋊D3—21no—
312C2×C3⋊D3—22no—
313C2×C3⋊D3—22no—
314C2×C3⋊D3C2×#6712yes—
315C2×C3×D3—34no—
316C2×C3×D3—62no—
317C2×C3×D3—62no—
318C2×C3×D3—62no—
319C2×C3×D3—64no—
320C2×C3×D3—62no—
321C2×C3×D3—62no—
322C2×C3×D3—62no—
323C2×C3×D3—62no—
324C2×C3×D3—62no—
325C2×C3×D3—62no—
326C2×C3×D3—122no—
327C2×C3×D3—121no—
328C2×C3×D3—121no—
329C2×C3×D3—122no—
330C32×C4—24no—
331C32×C4—22no—
332C32×C4—42no—
333C32×C4D3×D414yes—
334C32×C4C2×#6712yes—
335C3⋊Dic3—22no—
336C3⋊Dic3—22no—
337C3⋊Dic3—24no—
338C3⋊Dic3C2×#6712yes—
339C3×Dic3—34no—
340C3×Dic3—62no—
341C3×Dic3—64no—
342C3×Dic3—62no—
343C3×Dic3—64no—
344C3×Dic3—64no—
345C3×Dic3—64no—
346C3×Dic3—62no—
347C3×Dic3—122no—
348C3×Dic3—124no—
349C2×#41—91no—
350C2×#41—271no—
351C22⋊Q8—272no—
352C22⋊Q8—272no—
353#121—92no—
354#121—274no—
355#121—272no—
356C2×C4⋊C4—271no—
357C33#53711yescharacteristic, complemented, Sylow
358C23×C3—22no—
359C23×C3—24no—
360C23×C3—42no—
361C23×C3—91no—
362C23×C3—92no—
363C23×C3C2×D3212yes—
364C23×C3C2×D3214yes—
365C22×D3—32no—
366C22×D3—32no—
367C22×D3—32no—
368C22×D3—61no—
369C22×D3—62no—
370C3×Q8—182no—
371C3×D4—24no—
372C3×D4—24no—
373C3×D4—22no—
374C3×D4—44no—
375C3×D4—44no—
376C3×D4—184no—
377C3×D4—364no—
378C2×C3×C4—22no—
379C2×C3×C4—21no—
380C2×C3×C4—41no—
381C2×C3×C4—91no—
382C2×C3×C4—94no—
383C2×C3×C4—92no—
384C2×C3×C4—92no—
385C2×C3×C4—94no—
386C2×C3×C4—184no—
387C2×C3×C4—184no—
388C2×C3×C4C2×D3212yes—
389C2×C3×C4C2×D3211yescharacteristic
390#67—68no—
391#67—68no—
392#67—124no—
393#67—124no—
394#67—128no—
395#67—184no—
396#67—184no—
397#67—184no—
398C2×Dic3—32no—
399C2×Dic3—34no—
400C2×Dic3—32no—
401C2×Dic3—34no—
402C2×Dic3—34no—
403C2×Dic3—32no—
404C2×Dic3—62no—
405C2×Dic3—61no—
406C2×Dic3—64no—
407C2×Dic3—62no—
408C2×Dic3—68no—
409C2×Dic3—68no—
410C2×Dic3—362no—
411D12—182no—
412D12—184no—
413C4×D3—38no—
414C4×D3—64no—
415C4×D3—64no—
416C4×D3—64no—
417C4×D3—124no—
418C4×D3—364no—
419Dic6—64no—
420Dic6—64no—
421Dic6—184no—
422Dic6—182no—
423C2×C32—22no—
424C2×C32—21no—
425C2×C32—21no—
426C2×C32—22no—
427C2×C32—22no—
428C2×C32—21no—
429C2×C32—41no—
430C2×C32—41no—
431C2×C32—41no—
432C2×C32—42no—
433C2×C32—44no—
434C2×C32—44no—
435C2×C32—84no—
436C2×C32—84no—
437C2×C32C2×D3×D412yes—
438C2×C32#100412yes—
439C2×C32#100911yescharacteristic
440C2×C32#101011yescharacteristic
441C2×C32#101012yes—
442C2×C32C22×#6711yescharacteristic
443C3⋊D3—22no—
444C3⋊D3—42no—
445C3×D3—64no—
446C3×D3—122no—
447C3×D3—124no—
448C3×D3—122no—
449C3×D3—122no—
450#41—184no—
451#41—544no—
452C2×Q8—91no—
453C2×Q8—271no—
454C2×D4—92no—
455C2×D4—94no—
456C2×D4—271no—
457C2×D4—272no—
458C2×D4#111311yescharacteristic
459C22×C4—91no—
460C22×C4—92no—
461C22×C4—92no—
462C22×C4—272no—
463C4⋊C4—92no—
464C4⋊C4—272no—
465C4⋊C4—274no—
466C22⋊C4—94no—
467C22⋊C4—274no—
468C22×C3—24no—
469C22×C3—24no—
470C22×C3—22no—
471C22×C3—22no—
472C22×C3—24no—
473C22×C3—24no—
474C22×C3—22no—
475C22×C3—22no—
476C22×C3—21no—
477C22×C3—42no—
478C22×C3—42no—
479C22×C3—44no—
480C22×C3—44no—
481C22×C3—44no—
482C22×C3—44no—
483C22×C3—41no—
484C22×C3—82no—
485C22×C3—82no—
486C22×C3—92no—
487C22×C3—181no—
488C22×C3—181no—
489C22×C3—182no—
490C22×C3—182no—
491C22×C3—182no—
492C22×C3C22×D3211yescharacteristic
493C22×C3C22×D3212yes—
494C2×D3—34no—
495C2×D3—62no—
496C2×D3—62no—
497C2×D3—62no—
498C2×D3—64no—
499C2×D3—62no—
500C2×D3—62no—
501C2×D3—62no—
502C2×D3—62no—
503C2×D3—62no—
504C2×D3—62no—
505C2×D3—122no—
506C2×D3—121no—
507C2×D3—121no—
508C2×D3—122no—
509C3×C4—22no—
510C3×C4—24no—
511C3×C4—42no—
512C3×C4—92no—
513C3×C4—184no—
514C3×C4—182no—
515C3×C4—182no—
516C3×C4D3×#6714yes—
517C3×C4#381112yes—
518Dic3—34no—
519Dic3—62no—
520Dic3—64no—
521Dic3—64no—
522Dic3—64no—
523Dic3—64no—
524Dic3—62no—
525Dic3—62no—
526Dic3—122no—
527Dic3—124no—
528C32—22no—
529C32—21no—
530C32—41no—
531C32#560812yescomplemented
532C32#583311yescharacteristic, complemented
533C23—91no—
534C23—92no—
535C23C2×#111312yes—
536Q8—182no—
537Q8—542no—
538D4—22no—
539D4—184no—
540D4—364no—
541D4—542no—
542D4—544no—
543C2×C4—91no—
544C2×C4—94no—
545C2×C4—92no—
546C2×C4—94no—
547C2×C4—92no—
548C2×C4—184no—
549C2×C4—184no—
550C2×C4—1082no—
551C2×C4C2×#111311yescharacteristic
552C2×C3—21no—
553C2×C3—22no—
554C2×C3—22no—
555C2×C3—21no—
556C2×C3—21no—
557C2×C3—22no—
558C2×C3—44no—
559C2×C3—44no—
560C2×C3—42no—
561C2×C3—41no—
562C2×C3—41no—
563C2×C3—41no—
564C2×C3—84no—
565C2×C3—84no—
566C2×C3—182no—
567C2×C3—362no—
568C2×C3#6430912yes—
569C2×C3#6430911yescharacteristic
570C2×C3#6431311yescharacteristic
571C2×C3#6431412yes—
572C2×C3C2×D3×#6712yes—
573C2×C3C2×#381111yescharacteristic
574D3—64no—
575D3—122no—
576D3—124no—
577D3—122no—
578D3—122no—
579C22—22no—
580C22—22no—
581C22—22no—
582C22—92no—
583C22—181no—
584C22—181no—
585C22—182no—
586C22—182no—
587C22—182no—
588C22C22×#111311yescharacteristic
589C4—92no—
590C4—184no—
591C4—182no—
592C4—182no—
593C4#8904912yes—
594C3—21no—
595C3—22no—
596C3—41no—
597C3#1059523212yescomplemented
598C3#1059524011yescharacteristic, complemented
599C2—42no—
600C2—182no—
601C2—362no—
602C2#1172462011yescharacteristic
603C2#1172462311yescharacteristic
604C2C2×#8904911yescharacteristic
subgroupG/H
center, FrattiniC22C22×#1113
commutator, socle, cosocleC22×C33C24
FittingC2×C33×D4C22
hypercenterC2×D4#1113
hypocenterC33#537
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C2×C3×#3830C2semidirect
2. C2×#89042C2semidirect
3. C3×#64147C2semidirect
4. #11723725C2semidirect
5. #11723721C2semidirect
6. #11723717C2semidirect
7. #11723689C2semidirect
8. #11723674C2semidirect
29. C2×#3830D3semidirect
34. #64147D3semidirect
63. C3×C4×C3⋊D3D4semidirect
192. C4×C3⋊D3#67semidirect
357. C33#537semidirect
531. C32#5608semidirect
532. C32#5833semidirect
597. C3#10595232semidirect
598. C3#10595240semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#38302
C2×#890422
C3×#641472
#117237252
#117237212
#117237172
#117236892
#117236742
#105952403
#105952323