Finite Group Databasebeta

← #13162037

Subgroups — #13162037

Order 1728.

908 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#41×C3⋊D3C211yesmaximal, characteristic
2#11723673C211yesmaximal, characteristic, complemented
3#11723656C211yesmaximal, characteristic, complemented
4C3×#63664C212yesmaximal, complemented
5#11722077C212yesmaximal, complemented
6#10591672—31nomaximal
7#10591670—32nomaximal
8C33×#41C2211yescharacteristic
9C3×#3832C411yescharacteristic, complemented
10C3×Q8×C3⋊D3C411yescharacteristic, complemented
11C3×#3830—21no—
12C3×#3830C411yescharacteristic, complemented
13C3×D4×C3⋊D3—21no—
14C3×D4×C3⋊D3C411yescharacteristic, complemented
15C3×#3828—21no—
16C3×#3828C2211yescharacteristic
17C2×C3×C4×C3⋊D3—21no—
18C2×C3×C4×C3⋊D3C2211yescharacteristic
19C3×C4×C3⋊Dic3C2212yescomplemented
20C3×#3748C2212yescomplemented
21#88815—21no—
22#88814—21no—
23#88809—21no—
24#88808—21no—
25#41×C3⋊D3D311yescharacteristic
26C3×D3×#41—32no—
27C3×D3×#41—61no—
28#63874—31no—
29#63855—32no—
30#63818—31no—
31D3×#208—32no—
32#63664D312yescomplemented
33C3×#837—32no—
34C3×#837—32no—
35C3×#837—62no—
36#63570—32no—
37#63569—32no—
38#63568—32no—
39C22×C3×C3⋊D3—21no—
40C22×C3×C3⋊D3D411yescharacteristic
41C33×Q8C2×C411yescharacteristic
42C33×D4—21no—
43C33×D4C2×C411yescharacteristic
44C2×C33×C4—21no—
45C2×C33×C4C2311yescharacteristic
46C3×#638—21no—
47C3×#638—21no—
48C3×#638—21no—
49C2×C3×C3⋊Dic3—21no—
50C2×C3×C3⋊Dic3—22no—
51C2×C3×C3⋊Dic3D411yescharacteristic
52C3×C4⋊(C3⋊D3)—21no—
53C3×C4⋊(C3⋊D3)C2×C411yescharacteristic
54C3×C4×C3⋊D3—21no—
55C3×C4×C3⋊D3—21no—
56C3×C4×C3⋊D3—21no—
57C3×C4×C3⋊D3C2×C411yescharacteristic
58C3×C4×C3⋊D3C2×C411yescharacteristic
59C3×C4×C3⋊D3D411yescharacteristic
60C3×C4×C3⋊D3D411yescharacteristic
61C3×#634—21no—
62C3×#634C2×C411yescharacteristic
63C3×C3⋊(C3⋊C8)—22no—
64#5236—91no—
65#4821—92no—
66C32×#41—21no—
67C32×#41—22no—
68C32×#41—41no—
69C32×#41C2×D312yes—
70C32×#41C2×D311yescharacteristic
71#3832Dic311yescharacteristic, complemented
72Q8×C3⋊D3Dic311yescharacteristic, complemented
73#3830—21no—
74#3830Dic311yescharacteristic, complemented
75D4×C3⋊D3—21no—
76D4×C3⋊D3Dic311yescharacteristic, complemented
77#3828—21no—
78#3828C2×D311yescharacteristic
79C2×C4×C3⋊D3—21no—
80C2×C4×C3⋊D3C2×D311yescharacteristic
81C3×#239—32no—
82C3×#239—61no—
83C3×D3×Q8—32no—
84C3×D3×Q8—61no—
85C3×#237—32no—
86C3×#237—62no—
87C3×#237—61no—
88C3×#237—62no—
89C3×D3×D4—32no—
90C3×D3×D4—62no—
91C3×D3×D4—61no—
92C3×D3×D4—62no—
93C3×#235—32no—
94C3×#235—62no—
95C3×#235—61no—
96C3×#235—62no—
97C2×C3×C4×D3—32no—
98C2×C3×C4×D3—62no—
99C2×C3×C4×D3—61no—
100C2×C3×C4×D3—62no—
101C4×C3⋊Dic3C2×D312yescomplemented
102#3748C2×D312yescomplemented
103C3×C4×Dic3—32no—
104C3×C4×Dic3—32no—
105C3×C4×Dic3—62no—
106C3×#208—32no—
107C3×#208—32no—
108C3×#208—62no—
109#3724—61no—
110Dic3⋊Dic3—62no—
111#3722—61no—
112(C2×D3)⋊Dic3—62no—
113#3712—61no—
114#3711—62no—
115#3710—61no—
116D3×C3⋊C8—62no—
117C22×C33—21no—
118C22×C33C2×D411yescharacteristic
119C2×C3×C3⋊D3—21no—
120C2×C3×C3⋊D3—41no—
121C2×C3×C3⋊D3—42no—
122C2×C3×C3⋊D3C22⋊C411yescharacteristic
123C2×C3×C3⋊D3C22⋊C411yescharacteristic
124C33×C4—21no—
125C33×C4C22×C411yescharacteristic
126C33×C4C2×D411yescharacteristic
127C3×C3⋊Dic3—21no—
128C3×C3⋊Dic3—42no—
129C3×C3⋊Dic3C22⋊C411yescharacteristic
130C3×C3⋊Dic3C22⋊C411yescharacteristic
131C2×C3×#41—91no—
132D3×#41—32no—
133D3×#41—61no—
134#926—91no—
135C2×#208—91no—
136D3×M16—92no—
137#872—92no—
138C3×#104—92no—
139#837—32no—
140#837—32no—
141#837—62no—
142#837—92no—
143#825—92no—
144#805—92no—
145C22×C3⋊D3—21no—
146C22×C3⋊D3#6711yescharacteristic
147C22×C3×D3—32no—
148C22×C3×D3—61no—
149C22×C3×D3—62no—
150C22×C3×D3—62no—
151C32×Q8—22no—
152C32×Q8—21no—
153C32×Q8—41no—
154C32×Q8C4×D312yes—
155C32×Q8C2×Dic311yescharacteristic
156C32×D4—21no—
157C32×D4—22no—
158C32×D4—22no—
159C32×D4—22no—
160C32×D4—21no—
161C32×D4—41no—
162C32×D4—42no—
163C32×D4—42no—
164C32×D4C4×D312yes—
165C32×D4C2×Dic311yescharacteristic
166C2×C32×C4—21no—
167C2×C32×C4—22no—
168C2×C32×C4—22no—
169C2×C32×C4—22no—
170C2×C32×C4—21no—
171C2×C32×C4—41no—
172C2×C32×C4—42no—
173C2×C32×C4—42no—
174C2×C32×C4C22×D312yes—
175C2×C32×C4C22×D311yescharacteristic
176#638—21no—
177#638—21no—
178#638—21no—
179C2×C3⋊Dic3—21no—
180C2×C3⋊Dic3—22no—
181C2×C3⋊Dic3#6711yescharacteristic
182C4⋊(C3⋊D3)—21no—
183C4⋊(C3⋊D3)C2×Dic311yescharacteristic
184C4×C3⋊D3—21no—
185C4×C3⋊D3—21no—
186C4×C3⋊D3—21no—
187C4×C3⋊D3C2×Dic311yescharacteristic
188C4×C3⋊D3C2×Dic311yescharacteristic
189C4×C3⋊D3#6711yescharacteristic
190C4×C3⋊D3#6711yescharacteristic
191#634—21no—
192#634C2×Dic311yescharacteristic
193C3×#67—62no—
194C3×#67—62no—
195C3×#67—62no—
196C3×#67—62no—
197C3×#67—62no—
198C3×#67—62no—
199C2×C3×Dic3—32no—
200C2×C3×Dic3—61no—
201C2×C3×Dic3—62no—
202C2×C3×Dic3—62no—
203C2×C3×Dic3—62no—
204C2×C3×Dic3—62no—
205C2×C3×Dic3—122no—
206C3×D12—32no—
207C3×D12—61no—
208C3×D12—62no—
209C3×D12—62no—
210C3×C4×D3—32no—
211C3×C4×D3—32no—
212C3×C4×D3—32no—
213C3×C4×D3—32no—
214C3×C4×D3—61no—
215C3×C4×D3—61no—
216C3×C4×D3—61no—
217C3×C4×D3—61no—
218C3×C4×D3—62no—
219C3×C4×D3—62no—
220C3×C4×D3—62no—
221C3×C4×D3—62no—
222C3×C4×D3—62no—
223C3×C4×D3—62no—
224C3×Dic6—32no—
225C3×Dic6—61no—
226C3×Dic6—62no—
227C3×Dic6—62no—
228C3⋊(C3⋊C8)—22no—
229C3×C3⋊C8—62no—
230C3×C3⋊C8—62no—
231C3×C3⋊C8—122no—
232#421—271noSylow
233C2×C33—21no—
234C2×C33—41no—
235C2×C33C2×C22⋊C411yescharacteristic
236C3×C3⋊D3—21no—
237C3×C3⋊D3—21no—
238C3×C3⋊D3—41no—
239C3×#41—21no—
240C3×#41—22no—
241C3×#41—41no—
242C3×#41—91no—
243C3×#41—91no—
244C3×#41—91no—
245C3×#41—181no—
246C3×#41—181no—
247C3×#41D3212yes—
248C3×#41D3211yescharacteristic
249C2×C3×Q8—91no—
250C2×C3×D4—91no—
251C2×C3×D4—181no—
252C22×C3×C4—91no—
253C22×C3×C4—181no—
254#239—32no—
255#239—61no—
256D3×Q8—32no—
257D3×Q8—61no—
258#237—32no—
259#237—61no—
260#237—62no—
261#237—62no—
262D3×D4—32no—
263D3×D4—61no—
264D3×D4—62no—
265D3×D4—62no—
266#235—32no—
267#235—61no—
268#235—62no—
269#235—62no—
270C2×C4×D3—32no—
271C2×C4×D3—61no—
272C2×C4×D3—62no—
273C2×C4×D3—62no—
274C3×M16—92no—
275C3×C42—92no—
276#217—181no—
277(C2×D3)⋊C4—182no—
278C4⋊Dic3—181no—
279Dic3⋊C4—182no—
280C4×Dic3—32no—
281C4×Dic3—32no—
282C4×Dic3—62no—
283#208—32no—
284#208—32no—
285#208—62no—
286#208—181no—
287C2×C3⋊C8—181no—
288ΓL(1,𝔽25)—182no—
289D3×C8—182no—
290C22×C32—21no—
291C22×C32—22no—
292C22×C32—22no—
293C22×C32—22no—
294C22×C32—21no—
295C22×C32—41no—
296C22×C32—42no—
297C22×C32—42no—
298C22×C32D3×D412yes—
299C22×C32C2×#6711yescharacteristic
300C2×C3⋊D3—21no—
301C2×C3⋊D3—41no—
302C2×C3⋊D3—42no—
303C2×C3⋊D3#21711yescharacteristic
304C2×C3⋊D3#21711yescharacteristic
305C2×C3×D3—32no—
306C2×C3×D3—32no—
307C2×C3×D3—61no—
308C2×C3×D3—61no—
309C2×C3×D3—62no—
310C2×C3×D3—62no—
311C2×C3×D3—122no—
312C2×C3×D3—122no—
313C2×C3×D3—124no—
314C2×C3×D3—124no—
315C32×C4—21no—
316C32×C4—22no—
317C32×C4—22no—
318C32×C4—22no—
319C32×C4—21no—
320C32×C4—22no—
321C32×C4—21no—
322C32×C4—41no—
323C32×C4—41no—
324C32×C4—42no—
325C32×C4—42no—
326C32×C4C2×C4×D312yes—
327C32×C4D3×D412yes—
328C32×C4C22×Dic311yescharacteristic
329C32×C4C2×#6711yescharacteristic
330C3⋊Dic3—21no—
331C3⋊Dic3—42no—
332C3⋊Dic3#21711yescharacteristic
333C3⋊Dic3#21711yescharacteristic
334C3×Dic3—32no—
335C3×Dic3—32no—
336C3×Dic3—61no—
337C3×Dic3—61no—
338C3×Dic3—62no—
339C3×Dic3—62no—
340C3×Dic3—122no—
341C3×Dic3—122no—
342C3×Dic3—124no—
343C2×#41—91no—
344C2×M16—271no—
345#117—271no—
346#104—92no—
347#104—272no—
348C33#42111yescharacteristic, complemented, Sylow
349C23×C3—91no—
350C23×C3—181no—
351C22×D3—32no—
352C22×D3—61no—
353C22×D3—62no—
354C22×D3—62no—
355C3×Q8—21no—
356C3×Q8—22no—
357C3×Q8—41no—
358C3×Q8—91no—
359C3×Q8—181no—
360C3×Q8D3×Dic312yes—
361C3×Q8#62411yescharacteristic
362C3×D4—22no—
363C3×D4—22no—
364C3×D4—21no—
365C3×D4—21no—
366C3×D4—22no—
367C3×D4—42no—
368C3×D4—42no—
369C3×D4—41no—
370C3×D4—91no—
371C3×D4—181no—
372C3×D4—181no—
373C3×D4—181no—
374C3×D4—181no—
375C3×D4D3×Dic312yes—
376C3×D4#62411yescharacteristic
377C2×C3×C4—22no—
378C2×C3×C4—22no—
379C2×C3×C4—21no—
380C2×C3×C4—22no—
381C2×C3×C4—21no—
382C2×C3×C4—42no—
383C2×C3×C4—42no—
384C2×C3×C4—41no—
385C2×C3×C4—91no—
386C2×C3×C4—91no—
387C2×C3×C4—91no—
388C2×C3×C4—91no—
389C2×C3×C4—91no—
390C2×C3×C4—181no—
391C2×C3×C4—181no—
392C2×C3×C4—181no—
393C2×C3×C4—181no—
394C2×C3×C4—182no—
395C2×C3×C4C2×D3212yes—
396C2×C3×C4C2×D3211yescharacteristic
397#67—62no—
398#67—62no—
399#67—62no—
400#67—62no—
401#67—62no—
402#67—62no—
403C2×Dic3—32no—
404C2×Dic3—61no—
405C2×Dic3—62no—
406C2×Dic3—62no—
407C2×Dic3—62no—
408C2×Dic3—62no—
409C2×Dic3—122no—
410D12—32no—
411D12—61no—
412D12—62no—
413D12—62no—
414C4×D3—32no—
415C4×D3—32no—
416C4×D3—32no—
417C4×D3—32no—
418C4×D3—61no—
419C4×D3—61no—
420C4×D3—61no—
421C4×D3—61no—
422C4×D3—62no—
423C4×D3—62no—
424C4×D3—62no—
425C4×D3—62no—
426C4×D3—62no—
427C4×D3—62no—
428Dic6—32no—
429Dic6—61no—
430Dic6—62no—
431Dic6—62no—
432C3×C8—182no—
433C3⋊C8—62no—
434C3⋊C8—62no—
435C3⋊C8—122no—
436C2×C32—21no—
437C2×C32—22no—
438C2×C32—22no—
439C2×C32—21no—
440C2×C32—41no—
441C2×C32—42no—
442C2×C32—41no—
443C2×C32—41no—
444C2×C32—42no—
445C2×C32—42no—
446C2×C32—81no—
447C2×C32—82no—
448C2×C32—82no—
449C2×C32D3×C22⋊C412yes—
450C2×C32C2×#21711yescharacteristic
451C3⋊D3—21no—
452C3⋊D3—21no—
453C3⋊D3—41no—
454C3×D3—62no—
455C3×D3—62no—
456C3×D3—62no—
457C3×D3—121no—
458C3×D3—122no—
459C3×D3—122no—
460#41—91no—
461#41—91no—
462#41—91no—
463#41—181no—
464#41—181no—
465#41#111311yescharacteristic
466C2×Q8—91no—
467C2×D4—91no—
468C2×D4—181no—
469C22×C4—91no—
470C22×C4—181no—
471M16—92no—
472M16—541no—
473C2×C8—541no—
474C4⋊C4—541no—
475C22⋊C4—541no—
476C42—92no—
477C22×C3—22no—
478C22×C3—22no—
479C22×C3—21no—
480C22×C3—21no—
481C22×C3—22no—
482C22×C3—42no—
483C22×C3—42no—
484C22×C3—41no—
485C22×C3—91no—
486C22×C3—91no—
487C22×C3—181no—
488C22×C3—361no—
489C22×C3—362no—
490C22×C3D3×#6712yes—
491C22×C3#381111yescharacteristic
492C2×D3—32no—
493C2×D3—32no—
494C2×D3—61no—
495C2×D3—61no—
496C2×D3—62no—
497C2×D3—62no—
498C2×D3—122no—
499C2×D3—122no—
500C2×D3—124no—
501C2×D3—124no—
502C3×C4—22no—
503C3×C4—22no—
504C3×C4—21no—
505C3×C4—21no—
506C3×C4—22no—
507C3×C4—22no—
508C3×C4—21no—
509C3×C4—42no—
510C3×C4—42no—
511C3×C4—41no—
512C3×C4—41no—
513C3×C4—91no—
514C3×C4—91no—
515C3×C4—181no—
516C3×C4—362no—
517C3×C4C2×D3×Dic312yes—
518C3×C4C2×#62411yescharacteristic
519C3×C4D3×#6712yes—
520C3×C4#381111yescharacteristic
521Dic3—32no—
522Dic3—32no—
523Dic3—61no—
524Dic3—61no—
525Dic3—62no—
526Dic3—62no—
527Dic3—122no—
528Dic3—122no—
529Dic3—124no—
530C32—22no—
531C32—21no—
532C32—41no—
533C32#482112yescomplemented
534C32#523611yescharacteristic, complemented
535C23—91no—
536C23—181no—
537Q8—91no—
538Q8—181no—
539Q8#629411yescharacteristic
540D4—21no—
541D4—91no—
542D4—181no—
543D4—181no—
544D4—181no—
545D4—181no—
546D4#629411yescharacteristic
547C2×C4—21no—
548C2×C4—91no—
549C2×C4—91no—
550C2×C4—91no—
551C2×C4—91no—
552C2×C4—91no—
553C2×C4—181no—
554C2×C4—181no—
555C2×C4—181no—
556C2×C4—181no—
557C2×C4—182no—
558C2×C4C2×#111311yescharacteristic
559C8—182no—
560C2×C3—22no—
561C2×C3—21no—
562C2×C3—21no—
563C2×C3—22no—
564C2×C3—42no—
565C2×C3—41no—
566C2×C3—42no—
567C2×C3—42no—
568C2×C3—41no—
569C2×C3—41no—
570C2×C3—81no—
571C2×C3—82no—
572C2×C3—82no—
573C2×C3—181no—
574C2×C3—181no—
575C2×C3—361no—
576C2×C3D3×#21712yes—
577C2×C3#6397411yescharacteristic
578D3—62no—
579D3—62no—
580D3—62no—
581D3—121no—
582D3—122no—
583D3—122no—
584C22—21no—
585C22—91no—
586C22—91no—
587C22—181no—
588C22—361no—
589C22—362no—
590C22#8904911yescharacteristic
591C4—21no—
592C4—91no—
593C4—91no—
594C4—181no—
595C4—362no—
596C4C2×#629411yescharacteristic
597C4#8904911yescharacteristic
598C3—21no—
599C3—22no—
600C3—41no—
601C3#1059167012yescomplemented
602C3#1059167211yescharacteristic, complemented
603C2—21no—
604C2—41no—
605C2—181no—
606C2—181no—
607C2—361no—
608C2#1172372311yescharacteristic
subgroupG/H
centerC4#89049
commutatorC33×C4C22×C4
FrattiniC2×C4C2×#1113
FittingC33×#41C22
socleC2×C33C2×C22⋊C4
prime radicalC3×#41×C3⋊D3C2
cosocleC2×C33×C4C23
hypercenter#41#1113
hypocenterC33#421
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C4 ◁ C2×C4 ◁ #41

Complements

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

NH ≅ G/Ntype
2. #11723673C2semidirect
3. #11723656C2semidirect
4. C3×#63664C2semidirect
5. #11722077C2semidirect
9. C3×#3832C4semidirect
10. C3×Q8×C3⋊D3C4semidirect
12. C3×#3830C4semidirect
14. C3×D4×C3⋊D3C4semidirect
19. C3×C4×C3⋊Dic3C22semidirect
20. C3×#3748C22semidirect
32. #63664D3semidirect
71. #3832Dic3semidirect
72. Q8×C3⋊D3Dic3semidirect
74. #3830Dic3semidirect
76. D4×C3⋊D3Dic3semidirect
101. C4×C3⋊Dic3C2×D3semidirect
102. #3748C2×D3semidirect
348. C33#421semidirect
533. C32#4821semidirect
534. C32#5236semidirect
601. C3#10591670semidirect
602. C3#10591672semidirect

Maximal subgroups

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

subgroupindex
C3×#41×C3⋊D32
#117236732
#117236562
C3×#636642
#117220772
#105916723
#105916703