Finite Group Databasebeta

← #13178737

Subgroups — #13178737

Order 1728.

1526 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3832C211yesmaximal, characteristic, complemented
2C2×#89042C211yesmaximal, characteristic, complemented
3C3×#64155C212yesmaximal, complemented
4#11723696C212yesmaximal, complemented
5#11723695C212yesmaximal, complemented
6#11723689C211yesmaximal, characteristic, complemented
7#11723687C214yesmaximal, complemented
8#11723671C212yesmaximal, complemented
9#10595404—31nomaximal
10#10595398—32nomaximal
11C2×C33×Q8C2211yescharacteristic
12C3×#3832—24no—
13C2×C3×C4⋊(C3⋊D3)C2212yes—
14C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
15C2×C3×C4×C3⋊D3C2212yes—
16C2×C3×C4×C3⋊D3C2212yes—
17C2×C3×C4×C3⋊D3C2211yescharacteristic
18C2×#6296C2211yescharacteristic
19C2×#6295C2212yes—
20C2×#6295C2211yescharacteristic
21C2×#6295C2214yes—
22C2×#6294C2212yes—
23#89042—24no—
24C3×#3752C2214yes—
25C3×C4⋊(C3⋊Dic3)C2212yes—
26C3×#3750C2214yes—
27#88815C2212yes—
28#88814C2214yes—
29C2×#3832D311yescharacteristic, complemented
30C2×C3×#239—32no—
31C2×C3×#239—61no—
32C2×#3798—32no—
33C2×#3798—31no—
34#64155D312yescomplemented
35C3×#946—32no—
36C3×#946—32no—
37C3×#946—62no—
38#63916—32no—
39#63912—32no—
40#63910—34no—
41#63909—32no—
42#63908—32no—
43#63906—34no—
44#63900—31no—
45D3×C4⋊Dic3—32no—
46#63886—34no—
47#63885—34no—
48#63865—32no—
49#63851—34no—
50C22×C3×C3⋊D3C2312yes—
51C22×C3×C3⋊D3C2312yes—
52C22×C3×C3⋊D3C2311yescharacteristic
53C33×Q8—22no—
54C2×C33×C4C2312yes—
55C2×C33×C4C2311yescharacteristic
56C2×C3×C3⋊Dic3C2312yes—
57C2×C3×C3⋊Dic3C2314yes—
58C2×C3×C3⋊Dic3C2311yescharacteristic
59C3×C4⋊(C3⋊D3)—24no—
60C3×C4⋊(C3⋊D3)—22no—
61C3×C4×C3⋊D3—24no—
62C3×C4×C3⋊D3—24no—
63C3×C4×C3⋊D3D414yescomplemented
64#6296—22no—
65#6295—24no—
66#6295—22no—
67#6295—44no—
68#6294—42no—
69#5833—91no—
70#5629—92no—
71C2×C32×Q8—21no—
72C2×C32×Q8—22no—
73C2×C32×Q8—41no—
74C2×C32×Q8C2×D311yescharacteristic
75C2×C32×Q8C2×D312yes—
76#3832—24no—
77C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
78C2×C4⋊(C3⋊D3)C2×D312yes—
79C2×C4×C3⋊D3C2×D312yes—
80C2×C4×C3⋊D3C2×D312yes—
81C2×C4×C3⋊D3C2×D311yescharacteristic
82C3×#239—68no—
83C3×#239—68no—
84C2×C3×D12—34no—
85C2×C3×D12—32no—
86C2×C3×D12—61no—
87C2×C3×D12—62no—
88C2×C3×C4×D3—32no—
89C2×C3×C4×D3—32no—
90C2×C3×C4×D3—32no—
91C2×C3×C4×D3—34no—
92C2×C3×C4×D3—61no—
93C2×C3×C4×D3—62no—
94C2×C3×C4×D3—62no—
95C2×#627—32no—
96C2×#627—31no—
97C2×#626—32no—
98C2×#626—32no—
99C2×#626—32no—
100C2×#626—34no—
101C2×#626—34no—
102C2×#625—32no—
103C2×#625—31no—
104C2×#625—34no—
105C2×#624—32no—
106C2×D3×Dic3—34no—
107#3798—68no—
108#3798—64no—
109#3752C2×D314yes—
110C4⋊(C3⋊Dic3)C2×D312yes—
111#3750C2×D314yes—
112C3×(C2×D3)⋊C4—34no—
113C3×(C2×D3)⋊C4—34no—
114C3×(C2×D3)⋊C4—64no—
115C3×C4⋊Dic3—32no—
116C3×C4⋊Dic3—32no—
117C3×C4⋊Dic3—62no—
118C3×Dic3⋊C4—34no—
119C3×Dic3⋊C4—34no—
120C3×Dic3⋊C4—64no—
121#3724—32no—
122Dic3⋊Dic3—34no—
123#3722—34no—
124(C2×D3)⋊Dic3—34no—
125(C2×D3)⋊Dic3—34no—
126C22×C33C2411yescharacteristic
127C2×C3×C3⋊D3—21no—
128C2×C3×C3⋊D3—21no—
129C2×C3×C3⋊D3—22no—
130C2×C3×C3⋊D3—22no—
131C2×C3×C3⋊D3—22no—
132C2×C3×C3⋊D3—22no—
133C2×C3×C3⋊D3—22no—
134C2×C3×C3⋊D3—22no—
135C2×C3×C3⋊D3C2×D412yes—
136C33×C4—22no—
137C33×C4C2×D412yes—
138C3×C3⋊Dic3—22no—
139C3×C3⋊Dic3—24no—
140C3×C3⋊Dic3C2×D412yes—
141C2×C3×#41—91no—
142C2×#239—32no—
143C2×#239—61no—
144C2×#235—91no—
145C2×#235—92no—
146C3×C22⋊Q8—92no—
147#946—32no—
148#946—32no—
149#946—62no—
150#939—94no—
151#938—92no—
152#930—92no—
153#930—92no—
154C2×C4⋊Dic3—91no—
155#924—92no—
156#896—94no—
157#895—94no—
158#895—92no—
159D3×C4⋊C4—92no—
160#884—94no—
161#884—94no—
162C22×C3⋊D3C22×D312yes—
163C22×C3⋊D3C22×D312yes—
164C22×C3⋊D3C22×D311yescharacteristic
165C22×C3×D3—32no—
166C22×C3×D3—34no—
167C22×C3×D3—32no—
168C22×C3×D3—32no—
169C22×C3×D3—62no—
170C22×C3×D3—61no—
171C22×C3×D3—62no—
172C32×Q8—22no—
173C32×Q8—24no—
174C32×Q8—24no—
175C32×Q8—44no—
176C32×Q8—44no—
177C2×C32×C4—22no—
178C2×C32×C4—22no—
179C2×C32×C4—21no—
180C2×C32×C4—24no—
181C2×C32×C4—41no—
182C2×C32×C4—42no—
183C2×C32×C4C22×D312yes—
184C2×C32×C4C22×D314yes—
185C2×C32×C4C22×D312yes—
186C2×C32×C4C22×D311yescharacteristic
187C2×C3⋊Dic3C22×D312yes—
188C2×C3⋊Dic3C22×D314yes—
189C2×C3⋊Dic3C22×D311yescharacteristic
190C4⋊(C3⋊D3)—24no—
191C4⋊(C3⋊D3)—22no—
192C4×C3⋊D3—24no—
193C4×C3⋊D3—24no—
194C4×C3⋊D3#6714yescomplemented
195C2×C3×Dic3—32no—
196C2×C3×Dic3—32no—
197C2×C3×Dic3—34no—
198C2×C3×Dic3—32no—
199C2×C3×Dic3—34no—
200C2×C3×Dic3—61no—
201C2×C3×Dic3—62no—
202C2×C3×Dic3—64no—
203C3×D12—68no—
204C3×D12—68no—
205C3×D12—64no—
206C3×D12—64no—
207C3×C4×D3—38no—
208C3×C4×D3—64no—
209C3×C4×D3—68no—
210C3×C4×D3—68no—
211C3×C4×D3—64no—
212C3×C4×D3—64no—
213C3×C4×D3—124no—
214#627—64no—
215#627—62no—
216#626—64no—
217#626—64no—
218#626—64no—
219#626—124no—
220#626—124no—
221#625—64no—
222#625—62no—
223#625—124no—
224#624—122no—
225D3×Dic3—124no—
226#537—271noSylow
227C2×C33C22×D411yescharacteristic
228C2×C33H(5,𝔽2)11yescharacteristic
229C2×C332−1+411yescharacteristic
230C3×C3⋊D3—22no—
231C3×C3⋊D3—42no—
232C3×C3⋊D3—42no—
233C3×#41—184no—
234C2×C3×Q8—21no—
235C2×C3×Q8—22no—
236C2×C3×Q8—41no—
237C2×C3×Q8D3211yescharacteristic
238C2×C3×Q8D3212yes—
239C2×C3×D4—91no—
240C2×C3×D4—92no—
241C22×C3×C4—91no—
242C22×C3×C4—92no—
243C22×C3×C4—92no—
244C2×#67—92no—
245C2×#67—92no—
246C2×#67—92no—
247C2×#67—94no—
248C2×#67—94no—
249C22×Dic3—92no—
250#239—68no—
251#239—68no—
252#235—188no—
253#235—184no—
254C2×D12—34no—
255C2×D12—32no—
256C2×D12—61no—
257C2×D12—62no—
258C2×D12—92no—
259C2×D12—91no—
260C2×D12—94no—
261C2×C4×D3—32no—
262C2×C4×D3—34no—
263C2×C4×D3—32no—
264C2×C4×D3—32no—
265C2×C4×D3—61no—
266C2×C4×D3—62no—
267C2×C4×D3—62no—
268C2×C4×D3—94no—
269C2×Dic6—92no—
270C2×Dic6—91no—
271C3×C4⋊C4—92no—
272C3×C4⋊C4—94no—
273C3×C22⋊C4—94no—
274#217—94no—
275(C2×D3)⋊C4—34no—
276(C2×D3)⋊C4—34no—
277(C2×D3)⋊C4—64no—
278(C2×D3)⋊C4—94no—
279(C2×D3)⋊C4—94no—
280C4⋊Dic3—32no—
281C4⋊Dic3—32no—
282C4⋊Dic3—62no—
283C4⋊Dic3—92no—
284Dic3⋊C4—34no—
285Dic3⋊C4—34no—
286Dic3⋊C4—64no—
287Dic3⋊C4—94no—
288C22×C32—22no—
289C22×C32—21no—
290C22×C32—41no—
291C22×C32C23×D312yes—
292C22×C32C23×D311yescharacteristic
293C2×C3⋊D3—22no—
294C2×C3⋊D3—22no—
295C2×C3⋊D3—21no—
296C2×C3⋊D3—21no—
297C2×C3⋊D3—22no—
298C2×C3⋊D3—22no—
299C2×C3⋊D3—22no—
300C2×C3⋊D3—22no—
301C2×C3⋊D3C2×#6712yes—
302C2×C3×D3—34no—
303C2×C3×D3—62no—
304C2×C3×D3—62no—
305C2×C3×D3—62no—
306C2×C3×D3—64no—
307C2×C3×D3—64no—
308C2×C3×D3—62no—
309C2×C3×D3—64no—
310C2×C3×D3—64no—
311C2×C3×D3—62no—
312C2×C3×D3—62no—
313C2×C3×D3—64no—
314C2×C3×D3—62no—
315C2×C3×D3—62no—
316C2×C3×D3—62no—
317C2×C3×D3—122no—
318C2×C3×D3—121no—
319C2×C3×D3—121no—
320C2×C3×D3—122no—
321C2×C3×D3—122no—
322C2×C3×D3—122no—
323C32×C4—22no—
324C32×C4—24no—
325C32×C4—24no—
326C32×C4—24no—
327C32×C4—22no—
328C32×C4—42no—
329C32×C4—44no—
330C32×C4—44no—
331C32×C4D3×D414yes—
332C32×C4C2×#6712yes—
333C3⋊Dic3—22no—
334C3⋊Dic3—24no—
335C3⋊Dic3C2×#6712yes—
336C3×Dic3—34no—
337C3×Dic3—62no—
338C3×Dic3—62no—
339C3×Dic3—64no—
340C3×Dic3—64no—
341C3×Dic3—62no—
342C3×Dic3—122no—
343C3×Dic3—124no—
344C2×#41—91no—
345C2×#41—271no—
346C22⋊Q8—92no—
347C22⋊Q8—272no—
348#121—272no—
349#121—274no—
350#121—272no—
351C2×C4⋊C4—271no—
352C33#53711yescharacteristic, complemented, Sylow
353C23×C3—91no—
354C23×C3—92no—
355C23×C3—92no—
356C22×D3—32no—
357C22×D3—32no—
358C22×D3—32no—
359C22×D3—34no—
360C22×D3—62no—
361C22×D3—61no—
362C22×D3—62no—
363C3×Q8—24no—
364C3×Q8—24no—
365C3×Q8—22no—
366C3×Q8—44no—
367C3×Q8—44no—
368C3×D4—184no—
369C3×D4—182no—
370C2×C3×C4—22no—
371C2×C3×C4—22no—
372C2×C3×C4—21no—
373C2×C3×C4—24no—
374C2×C3×C4—42no—
375C2×C3×C4—41no—
376C2×C3×C4—91no—
377C2×C3×C4—94no—
378C2×C3×C4—94no—
379C2×C3×C4—92no—
380C2×C3×C4—184no—
381C2×C3×C4—184no—
382C2×C3×C4C2×D3212yes—
383C2×C3×C4C2×D3214yes—
384C2×C3×C4C2×D3212yes—
385C2×C3×C4C2×D3211yescharacteristic
386#67—184no—
387#67—184no—
388#67—184no—
389#67—364no—
390#67—364no—
391C2×Dic3—32no—
392C2×Dic3—32no—
393C2×Dic3—32no—
394C2×Dic3—34no—
395C2×Dic3—34no—
396C2×Dic3—61no—
397C2×Dic3—62no—
398C2×Dic3—64no—
399C2×Dic3—362no—
400D12—68no—
401D12—68no—
402D12—64no—
403D12—64no—
404D12—182no—
405D12—184no—
406D12—364no—
407C4×D3—38no—
408C4×D3—64no—
409C4×D3—68no—
410C4×D3—68no—
411C4×D3—64no—
412C4×D3—64no—
413C4×D3—124no—
414C4×D3—364no—
415Dic6—184no—
416Dic6—182no—
417C2×C32—22no—
418C2×C32—21no—
419C2×C32—22no—
420C2×C32—21no—
421C2×C32—22no—
422C2×C32—21no—
423C2×C32—41no—
424C2×C32—41no—
425C2×C32—41no—
426C2×C32C2×D3×D412yes—
427C2×C32#100712yes—
428C2×C32#100911yescharacteristic
429C2×C32#100912yes—
430C2×C32#101011yescharacteristic
431C2×C32C22×#6711yescharacteristic
432C3⋊D3—22no—
433C3⋊D3—42no—
434C3⋊D3—42no—
435C3×D3—64no—
436C3×D3—122no—
437C3×D3—124no—
438C3×D3—124no—
439C3×D3—122no—
440C3×D3—124no—
441C3×D3—122no—
442#41—184no—
443#41—544no—
444C2×Q8—271no—
445C2×Q8#111311yescharacteristic
446C2×D4—92no—
447C2×D4—91no—
448C2×D4—271no—
449C2×D4—272no—
450C2×D4—274no—
451C22×C4—91no—
452C22×C4—92no—
453C22×C4—92no—
454C22×C4—272no—
455C4⋊C4—92no—
456C4⋊C4—94no—
457C4⋊C4—272no—
458C22⋊C4—94no—
459C22⋊C4—274no—
460C22×C3—22no—
461C22×C3—21no—
462C22×C3—41no—
463C22×C3—92no—
464C22×C3—181no—
465C22×C3—181no—
466C22×C3—182no—
467C22×C3—182no—
468C22×C3—182no—
469C22×C3—182no—
470C22×C3—182no—
471C22×C3—182no—
472C22×C3C22×D3212yes—
473C22×C3C22×D3211yescharacteristic
474C2×D3—34no—
475C2×D3—62no—
476C2×D3—62no—
477C2×D3—62no—
478C2×D3—64no—
479C2×D3—64no—
480C2×D3—62no—
481C2×D3—64no—
482C2×D3—62no—
483C2×D3—62no—
484C2×D3—64no—
485C2×D3—62no—
486C2×D3—64no—
487C2×D3—62no—
488C2×D3—62no—
489C2×D3—122no—
490C2×D3—121no—
491C2×D3—121no—
492C2×D3—122no—
493C2×D3—122no—
494C2×D3—122no—
495C3×C4—24no—
496C3×C4—24no—
497C3×C4—22no—
498C3×C4—22no—
499C3×C4—24no—
500C3×C4—44no—
501C3×C4—44no—
502C3×C4—42no—
503C3×C4—92no—
504C3×C4—184no—
505C3×C4—182no—
506C3×C4D3×#6714yes—
507C3×C4#381112yes—
508Dic3—34no—
509Dic3—62no—
510Dic3—64no—
511Dic3—64no—
512Dic3—62no—
513Dic3—62no—
514Dic3—122no—
515Dic3—124no—
516C32—22no—
517C32—21no—
518C32—41no—
519C32#562912yescomplemented
520C32#583311yescharacteristic, complemented
521C23—91no—
522C23—92no—
523C23—92no—
524Q8—22no—
525Q8—542no—
526D4—184no—
527D4—182no—
528D4—542no—
529D4—544no—
530D4—1084no—
531C2×C4—91no—
532C2×C4—94no—
533C2×C4—94no—
534C2×C4—92no—
535C2×C4—184no—
536C2×C4—184no—
537C2×C4—1082no—
538C2×C4C2×#111312yes—
539C2×C4C2×#111311yescharacteristic
540C2×C3—22no—
541C2×C3—21no—
542C2×C3—22no—
543C2×C3—21no—
544C2×C3—21no—
545C2×C3—22no—
546C2×C3—41no—
547C2×C3—41no—
548C2×C3—41no—
549C2×C3—182no—
550C2×C3—362no—
551C2×C3—362no—
552C2×C3#6431512yes—
553C2×C3#6431611yescharacteristic
554C2×C3#6432012yes—
555C2×C3#6432011yescharacteristic
556C2×C3C2×D3×#6712yes—
557C2×C3C2×#381111yescharacteristic
558D3—64no—
559D3—122no—
560D3—124no—
561D3—124no—
562D3—122no—
563D3—124no—
564D3—122no—
565C22—92no—
566C22—181no—
567C22—181no—
568C22—182no—
569C22—182no—
570C22—182no—
571C22—182no—
572C22—182no—
573C22—182no—
574C22C22×#111311yescharacteristic
575C4—22no—
576C4—92no—
577C4—184no—
578C4—182no—
579C4#8904912yes—
580C3—21no—
581C3—22no—
582C3—41no—
583C3#1059539812yescomplemented
584C3#1059540411yescharacteristic, complemented
585C2—182no—
586C2—362no—
587C2—362no—
588C2#1172462411yescharacteristic
589C2#1172462711yescharacteristic
590C2C2×#8904911yescharacteristic
subgroupG/H
center, FrattiniC22C22×#1113
commutator, socle, cosocleC22×C33C24
FittingC2×C33×Q8C22
hypercenterC2×Q8#1113
hypocenterC33#537
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
1. C2×C3×#3832C2semidirect
2. C2×#89042C2semidirect
3. C3×#64155C2semidirect
4. #11723696C2semidirect
5. #11723695C2semidirect
6. #11723689C2semidirect
7. #11723687C2semidirect
8. #11723671C2semidirect
29. C2×#3832D3semidirect
34. #64155D3semidirect
63. C3×C4×C3⋊D3D4semidirect
194. C4×C3⋊D3#67semidirect
352. C33#537semidirect
519. C32#5629semidirect
520. C32#5833semidirect
583. C3#10595398semidirect
584. C3#10595404semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#38322
C2×#890422
C3×#641552
#117236962
#117236952
#117236892
#117236872
#117236712
#105954043
#105953983