Finite Group Databasebeta

← #13178221

Subgroups — #13178221

Order 1728.

1016 conjugacy classes of subgroups.

The subgroup lattice has 578 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#64095C211yesmaximal, characteristic
2C3×#64088C212yesmaximal, complemented
3#11723724C211yesmaximal, characteristic
4#11723721C212yesmaximal, complemented
5#11723718C211yesmaximal, characteristic
6C4⋊#6296C212yesmaximal, complemented
7#11723692C212yesmaximal, complemented
8#11723676C212yesmaximal, complemented
9#11723669C212yesmaximal, complemented
10#10594169—32nomaximal
11#10594036—31nomaximal
12C2×C3×#638C2211yescharacteristic
13C22×C3×C3⋊Dic3C2211yescharacteristic
14C2×#6296C2212yes—
15C2×#6296C2211yescharacteristic
16C2×#6294C2211yescharacteristic
17C33×C22⋊C4C2211yescharacteristic
18C3×#3757C2212yes—
19C3×#3757C2212yes—
20C3×#3752C2212yes—
21C3×C4⋊(C3⋊Dic3)C2212yes—
22C3×C4⋊(C3⋊Dic3)C2212yes—
23C3×#3750C2212yes—
24C3×#3750C2212yes—
25C3×C4×C3⋊Dic3C2212yes—
26#88815C2212yes—
27#88815C2212yes—
28#88815C2212yes—
29#88815C2212yes—
30#88814C2211yescharacteristic
31#88814C2212yes—
32#88813C2211yescharacteristic
33#64095D311yescharacteristic
34#64088D312yescomplemented
35C3×#886—32no—
36C3×#886—61no—
37C3×#879—32no—
38C3×#879—32no—
39C3×#879—62no—
40#63982—32no—
41#63975—31no—
42#63969—32no—
43#63964—32no—
44#63964—32no—
45#63959—32no—
46#63953—31no—
47#63919—32no—
48#63918—32no—
49#63918—32no—
50#63899—32no—
51#63895—32no—
52#63893—32no—
53#63868—32no—
54#63862—32no—
55#63859—32no—
56#63847—32no—
57#63846—32no—
58#63845—32no—
59C23×C33C2311yescharacteristic
60C22×C3×C3⋊D3C2311yescharacteristic
61C2×C33×C4C2312yes—
62C3×#638—41no—
63C2×C3×C3⋊Dic3—41no—
64C2×C3×C3⋊Dic3C2312yes—
65C2×C3×C3⋊Dic3C2312yes—
66C2×C3×C3⋊Dic3C2312yes—
67C2×C3×C3⋊Dic3C2312yes—
68C2×C3×C3⋊Dic3C2311yescharacteristic
69C2×C3×C3⋊Dic3C2312yes—
70#6296—42no—
71#6296—41no—
72#6294—41no—
73#5712—92no—
74#5651—91no—
75C2×#638C2×D311yescharacteristic
76C22×C3⋊Dic3C2×D311yescharacteristic
77C2×C3×#67—32no—
78C2×C3×#67—61no—
79C22×C3×Dic3—32no—
80C22×C3×Dic3—61no—
81C2×#627—32no—
82C2×#627—32no—
83C2×#627—32no—
84C2×#627—31no—
85C2×#627—32no—
86C2×#624—31no—
87C2×D3×Dic3—32no—
88C32×C22⋊C4—22no—
89C32×C22⋊C4—21no—
90C32×C22⋊C4—41no—
91C32×C22⋊C4C2×D311yescharacteristic
92C32×C22⋊C4C2×D312yes—
93#3757C2×D312yes—
94#3757C2×D312yes—
95#3752C2×D312yes—
96C4⋊(C3⋊Dic3)C2×D312yes—
97C4⋊(C3⋊Dic3)C2×D312yes—
98#3750C2×D312yes—
99#3750C2×D312yes—
100C4×C3⋊Dic3C2×D312yes—
101C3×#217—32no—
102C3×#217—32no—
103C3×#217—32no—
104C3×#217—32no—
105C3×#217—62no—
106C3×#217—62no—
107C3×(C2×D3)⋊C4—32no—
108C3×(C2×D3)⋊C4—32no—
109C3×(C2×D3)⋊C4—62no—
110C3×C4⋊Dic3—32no—
111C3×C4⋊Dic3—32no—
112C3×C4⋊Dic3—32no—
113C3×C4⋊Dic3—32no—
114C3×C4⋊Dic3—62no—
115C3×C4⋊Dic3—62no—
116C3×Dic3⋊C4—32no—
117C3×Dic3⋊C4—32no—
118C3×Dic3⋊C4—32no—
119C3×Dic3⋊C4—32no—
120C3×Dic3⋊C4—62no—
121C3×Dic3⋊C4—62no—
122C3×C4×Dic3—32no—
123C3×C4×Dic3—32no—
124C3×C4×Dic3—62no—
125#3724—32no—
126#3724—32no—
127#3724—32no—
128#3724—32no—
129Dic3⋊Dic3—32no—
130Dic3⋊Dic3—32no—
131Dic3⋊Dic3—32no—
132Dic3⋊Dic3—32no—
133Dic3⋊Dic3—32no—
134Dic3⋊Dic3—32no—
135Dic3⋊Dic3—32no—
136Dic3⋊Dic3—32no—
137#3722—31no—
138#3722—32no—
139(C2×D3)⋊Dic3—32no—
140(C2×D3)⋊Dic3—32no—
141(C2×D3)⋊Dic3—32no—
142Dic32—31no—
143Dic32—32no—
144C22×C33—21no—
145C22×C33—22no—
146C22×C33C2411yescharacteristic
147C2×C3×C3⋊D3—21no—
148C2×C3×C3⋊D3—22no—
149C33×C4—22no—
150C3×C3⋊Dic3—22no—
151C3×C3⋊Dic3—22no—
152C3×C3⋊Dic3—22no—
153C3×C3⋊Dic3—21no—
154C3×C3⋊Dic3—22no—
155C3×C3⋊Dic3—22no—
156C3×#126—92no—
157C3×#123—91no—
158#936—91no—
159#935—91no—
160#924—92no—
161#898—92no—
162#897—92no—
163#896—92no—
164#890—92no—
165#889—92no—
166#889—92no—
167#888—92no—
168#888—92no—
169#886—32no—
170#886—61no—
171#885—92no—
172#883—92no—
173#879—32no—
174#879—32no—
175#879—62no—
176#879—92no—
177#878—92no—
178#878—92no—
179#878—92no—
180#876—92no—
181#869—92no—
182C23×C32—21no—
183C23×C32—22no—
184C23×C32—41no—
185C23×C32C22×D311yescharacteristic
186C23×C32C22×D312yes—
187C22×C3⋊D3C22×D311yescharacteristic
188C22×C3×D3—32no—
189C22×C3×D3—61no—
190C2×C32×C4—22no—
191C2×C32×C4—22no—
192C2×C32×C4—22no—
193C2×C32×C4—42no—
194C2×C32×C4C22×D312yes—
195C2×C32×C4C22×D312yes—
196C2×C32×C4C22×D312yes—
197#638—41no—
198C2×C3⋊Dic3—41no—
199C2×C3⋊Dic3C22×D312yes—
200C2×C3⋊Dic3C22×D311yescharacteristic
201C2×C3⋊Dic3C22×D312yes—
202C2×C3⋊Dic3C22×D312yes—
203C2×C3⋊Dic3C22×D312yes—
204C2×C3⋊Dic3C22×D312yes—
205C3×#67—122no—
206C3×#67—122no—
207C2×C3×Dic3—32no—
208C2×C3×Dic3—32no—
209C2×C3×Dic3—32no—
210C2×C3×Dic3—32no—
211C2×C3×Dic3—32no—
212C2×C3×Dic3—32no—
213C2×C3×Dic3—32no—
214C2×C3×Dic3—32no—
215C2×C3×Dic3—32no—
216C2×C3×Dic3—32no—
217C2×C3×Dic3—32no—
218C2×C3×Dic3—62no—
219C2×C3×Dic3—62no—
220C2×C3×Dic3—62no—
221C2×C3×Dic3—62no—
222C2×C3×Dic3—61no—
223C2×C3×Dic3—62no—
224C2×C3×Dic3—122no—
225C2×C3×Dic3—122no—
226#627—122no—
227#627—121no—
228#627—122no—
229#627—122no—
230#627—122no—
231#624—121no—
232D3×Dic3—122no—
233#563—271noSylow
234C2×C33—41no—
235C2×C33H(5,𝔽2)11yescharacteristic
236C2×C332−1+412yes—
237C3×C3⋊D3—41no—
238C2×C3×D4—91no—
239C22×C3×C4—91no—
240C2×#67—32no—
241C2×#67—61no—
242C22×Dic3—32no—
243C22×Dic3—61no—
244C22×Dic3—91no—
245C2×C4×D3—92no—
246C2×Dic6—92no—
247C2×Dic6—92no—
248C2×Dic6—92no—
249C2×Dic6—92no—
250C2×Dic6—91no—
251C3×C4⋊C4—92no—
252C3×C4⋊C4—92no—
253C3×C4⋊C4—92no—
254C3×C4⋊C4—92no—
255C3×C22⋊C4—22no—
256C3×C22⋊C4—21no—
257C3×C22⋊C4—41no—
258C3×C22⋊C4—92no—
259C3×C22⋊C4—92no—
260C3×C22⋊C4—92no—
261C3×C22⋊C4D3211yescharacteristic
262C3×C22⋊C4D3212yes—
263C3×C42—92no—
264#217—32no—
265#217—32no—
266#217—32no—
267#217—32no—
268#217—62no—
269#217—62no—
270#217—92no—
271#217—91no—
272(C2×D3)⋊C4—32no—
273(C2×D3)⋊C4—32no—
274(C2×D3)⋊C4—62no—
275(C2×D3)⋊C4—92no—
276(C2×D3)⋊C4—92no—
277(C2×D3)⋊C4—92no—
278C4⋊Dic3—32no—
279C4⋊Dic3—32no—
280C4⋊Dic3—32no—
281C4⋊Dic3—32no—
282C4⋊Dic3—62no—
283C4⋊Dic3—62no—
284C4⋊Dic3—92no—
285C4⋊Dic3—92no—
286C4⋊Dic3—92no—
287C4⋊Dic3—92no—
288Dic3⋊C4—32no—
289Dic3⋊C4—32no—
290Dic3⋊C4—32no—
291Dic3⋊C4—32no—
292Dic3⋊C4—62no—
293Dic3⋊C4—62no—
294Dic3⋊C4—92no—
295Dic3⋊C4—92no—
296Dic3⋊C4—92no—
297Dic3⋊C4—92no—
298Dic3⋊C4—92no—
299Dic3⋊C4—92no—
300Dic3⋊C4—92no—
301Dic3⋊C4—92no—
302C4×Dic3—32no—
303C4×Dic3—32no—
304C4×Dic3—62no—
305C4×Dic3—92no—
306C4×Dic3—91no—
307C22×C32—21no—
308C22×C32—22no—
309C22×C32—22no—
310C22×C32—22no—
311C22×C32—22no—
312C22×C32—22no—
313C22×C32—22no—
314C22×C32—21no—
315C22×C32—41no—
316C22×C32—42no—
317C22×C32—42no—
318C22×C32—42no—
319C22×C32—42no—
320C22×C32—42no—
321C22×C32—82no—
322C22×C32C23×D311yescharacteristic
323C22×C32C23×D312yes—
324C2×C3⋊D3—21no—
325C2×C3⋊D3—22no—
326C2×C3×D3—62no—
327C2×C3×D3—62no—
328C2×C3×D3—62no—
329C2×C3×D3—121no—
330C2×C3×D3—122no—
331C32×C4—22no—
332C32×C4—22no—
333C32×C4—22no—
334C32×C4—42no—
335C32×C4—42no—
336C32×C4—42no—
337C32×C4—82no—
338C3⋊Dic3—22no—
339C3⋊Dic3—21no—
340C3⋊Dic3—22no—
341C3⋊Dic3—22no—
342C3⋊Dic3—22no—
343C3⋊Dic3—22no—
344C3×Dic3—62no—
345C3×Dic3—62no—
346C3×Dic3—62no—
347C3×Dic3—62no—
348C3×Dic3—62no—
349C3×Dic3—62no—
350C3×Dic3—62no—
351C3×Dic3—62no—
352C3×Dic3—62no—
353C3×Dic3—62no—
354C3×Dic3—62no—
355C3×Dic3—122no—
356C3×Dic3—122no—
357C3×Dic3—122no—
358C3×Dic3—122no—
359C3×Dic3—122no—
360C3×Dic3—121no—
361C4⋊Q8—272no—
362#126—92no—
363#126—272no—
364#125—272no—
365#124—271no—
366#123—91no—
367#123—271no—
368C22⋊Q8—272no—
369C22⋊Q8—272no—
370C33#56311yescharacteristic, complemented, Sylow
371C23×C3—21no—
372C23×C3—22no—
373C23×C3—41no—
374C23×C3—91no—
375C23×C3C2×D3211yescharacteristic
376C23×C3C2×D3212yes—
377C22×D3—32no—
378C22×D3—61no—
379C3×D4—361no—
380C2×C3×C4—22no—
381C2×C3×C4—22no—
382C2×C3×C4—22no—
383C2×C3×C4—42no—
384C2×C3×C4—91no—
385C2×C3×C4—92no—
386C2×C3×C4—92no—
387C2×C3×C4—92no—
388C2×C3×C4—92no—
389C2×C3×C4—92no—
390C2×C3×C4—361no—
391C2×C3×C4C2×D3212yes—
392C2×C3×C4C2×D3212yes—
393C2×C3×C4C2×D3212yes—
394#67—122no—
395#67—122no—
396C2×Dic3—32no—
397C2×Dic3—32no—
398C2×Dic3—32no—
399C2×Dic3—32no—
400C2×Dic3—32no—
401C2×Dic3—32no—
402C2×Dic3—32no—
403C2×Dic3—32no—
404C2×Dic3—32no—
405C2×Dic3—32no—
406C2×Dic3—32no—
407C2×Dic3—62no—
408C2×Dic3—62no—
409C2×Dic3—62no—
410C2×Dic3—62no—
411C2×Dic3—61no—
412C2×Dic3—62no—
413C2×Dic3—122no—
414C2×Dic3—122no—
415C2×Dic3—361no—
416C4×D3—362no—
417Dic6—362no—
418Dic6—362no—
419Dic6—362no—
420Dic6—361no—
421Dic6—362no—
422C2×C32—22no—
423C2×C32—21no—
424C2×C32—22no—
425C2×C32—22no—
426C2×C32—22no—
427C2×C32—41no—
428C2×C32—42no—
429C2×C32—41no—
430C2×C32—42no—
431C2×C32—42no—
432C2×C32—82no—
433C2×C32—82no—
434C2×C32#100411yescharacteristic
435C2×C32#100912yes—
436C2×C32#101012yes—
437C2×C32#101012yes—
438C2×C32#101012yes—
439C3⋊D3—41no—
440C3×D3—122no—
441C3×D3—241no—
442C2×Q8—272no—
443C2×Q8—271no—
444C2×D4—91no—
445C22×C4—91no—
446C22×C4—271no—
447C4⋊C4—92no—
448C4⋊C4—92no—
449C4⋊C4—92no—
450C4⋊C4—92no—
451C4⋊C4—272no—
452C4⋊C4—272no—
453C4⋊C4—272no—
454C4⋊C4—272no—
455C22⋊C4—92no—
456C22⋊C4—92no—
457C22⋊C4—92no—
458C22⋊C4—272no—
459C22⋊C4—271no—
460C22⋊C4#111311yescharacteristic
461C42—92no—
462C42—271no—
463C22×C3—22no—
464C22×C3—22no—
465C22×C3—21no—
466C22×C3—22no—
467C22×C3—22no—
468C22×C3—22no—
469C22×C3—22no—
470C22×C3—21no—
471C22×C3—42no—
472C22×C3—42no—
473C22×C3—42no—
474C22×C3—42no—
475C22×C3—42no—
476C22×C3—41no—
477C22×C3—82no—
478C22×C3—181no—
479C22×C3—182no—
480C22×C3C22×D3212yes—
481C22×C3C22×D3211yescharacteristic
482C2×D3—62no—
483C2×D3—62no—
484C2×D3—62no—
485C2×D3—121no—
486C2×D3—122no—
487C3×C4—22no—
488C3×C4—22no—
489C3×C4—22no—
490C3×C4—42no—
491C3×C4—42no—
492C3×C4—42no—
493C3×C4—82no—
494C3×C4—182no—
495C3×C4—182no—
496C3×C4—182no—
497C3×C4—181no—
498C3×C4—182no—
499C3×C4—182no—
500Dic3—62no—
501Dic3—62no—
502Dic3—62no—
503Dic3—62no—
504Dic3—62no—
505Dic3—62no—
506Dic3—62no—
507Dic3—62no—
508Dic3—62no—
509Dic3—62no—
510Dic3—62no—
511Dic3—122no—
512Dic3—121no—
513Dic3—122no—
514Dic3—122no—
515Dic3—122no—
516Dic3—122no—
517C32—22no—
518C32—21no—
519C32—41no—
520C32#565111yescharacteristic, complemented
521C32#571212yescomplemented
522C23—91no—
523C23C2×#111311yescharacteristic
524Q8—1082no—
525Q8—1081no—
526D4—361no—
527C2×C4—92no—
528C2×C4—92no—
529C2×C4—92no—
530C2×C4—91no—
531C2×C4—92no—
532C2×C4—92no—
533C2×C4—361no—
534C2×C4—1081no—
535C2×C4C2×#111312yes—
536C2×C3—21no—
537C2×C3—22no—
538C2×C3—22no—
539C2×C3—22no—
540C2×C3—22no—
541C2×C3—42no—
542C2×C3—42no—
543C2×C3—41no—
544C2×C3—41no—
545C2×C3—42no—
546C2×C3—82no—
547C2×C3—82no—
548C2×C3—361no—
549C2×C3#6430712yes—
550C2×C3#6430811yescharacteristic
551C2×C3#6430912yes—
552C2×C3#6430912yes—
553C2×C3#6430912yes—
554D3—122no—
555D3—241no—
556C22—21no—
557C22—22no—
558C22—181no—
559C22—182no—
560C22C22×#111311yescharacteristic
561C4—22no—
562C4—182no—
563C4—182no—
564C4—181no—
565C4—182no—
566C4—182no—
567C4—182no—
568C3—21no—
569C3—22no—
570C3—41no—
571C3#1059403611yescharacteristic, complemented
572C3#1059416912yescomplemented
573C2—41no—
574C2—361no—
575C2#1172461911yescharacteristic
576C2#1172462012yes—
subgroupG/H
center, FrattiniC22C22×#1113
commutator, socle, cosocleC22×C33C24
FittingC33×C22⋊C4C22
prime radicalC2×C3×#638C22
hypercenterC22⋊C4#1113
hypocenterC33#563
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C3×#64088C2semidirect
4. #11723721C2semidirect
6. C4⋊#6296C2semidirect
7. #11723692C2semidirect
8. #11723676C2semidirect
9. #11723669C2semidirect
34. #64088D3semidirect
370. C33#563semidirect
520. C32#5651semidirect
521. C32#5712semidirect
571. C3#10594036semidirect
572. C3#10594169semidirect

Maximal subgroups

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

subgroupindex
C3×#640952
C3×#640882
#117237242
#117237212
#117237182
C4⋊#62962
#117236922
#117236762
#117236692
#105941693
#105940363