Finite Group Databasebeta

← #13162036

Subgroups — #13162036

Order 1728.

972 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#41×C3⋊D3C211yesmaximal, characteristic
2C4×#6294C211yesmaximal, characteristic, complemented
3#11723656C211yesmaximal, characteristic, complemented
4C3×#63664C212yesmaximal, complemented
5#11722077C212yesmaximal, complemented
6#10591671—31nomaximal
7D3×#837—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
19C2×#6294—21no—
20C3×C4×C3⋊Dic3C2212yescomplemented
21C3×#3748C2212yescomplemented
22#88813—21no—
23#88809—21no—
24#88808—21no—
25#41×C3⋊D3D311yescharacteristic
26C3×D3×#41—32no—
27C3×D3×#41—61no—
28C4×#624—31no—
29C4×D3×Dic3—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, complemented
42C33×D4—21no—
43C33×D4C2×C411yescharacteristic, complemented
44C2×C33×C4—21no—
45C2×C33×C4C2311yescharacteristic
46C3×#638—21no—
47C3×#638—21no—
48C3×#638—21no—
49C2×C3×C3⋊Dic3—22no—
50C2×C3×C3⋊Dic3—21no—
51C2×C3×C3⋊Dic3D411yescharacteristic
52C3×C4⋊(C3⋊D3)—21no—
53C3×C4⋊(C3⋊D3)C2×C411yescharacteristic, complemented
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, complemented
63#6294—24no—
64C3×C3⋊(C3⋊C8)—22no—
65C2×#837—91no—
66D3×#104—92no—
67C32×#41—21no—
68C32×#41—22no—
69C32×#41—41no—
70C32×#41C2×D312yes—
71C32×#41C2×D311yescharacteristic
72#3832Dic311yescharacteristic, complemented
73Q8×C3⋊D3Dic311yescharacteristic, complemented
74#3830—21no—
75#3830Dic311yescharacteristic, complemented
76D4×C3⋊D3—21no—
77D4×C3⋊D3Dic311yescharacteristic, complemented
78#3828—21no—
79#3828C2×D311yescharacteristic
80C2×C4×C3⋊D3—21no—
81C2×C4×C3⋊D3C2×D311yescharacteristic
82C3×#239—32no—
83C3×#239—61no—
84C3×D3×Q8—32no—
85C3×D3×Q8—61no—
86C3×#237—32no—
87C3×#237—62no—
88C3×#237—61no—
89C3×#237—62no—
90C3×D3×D4—32no—
91C3×D3×D4—62no—
92C3×D3×D4—61no—
93C3×D3×D4—62no—
94C3×#235—32no—
95C3×#235—62no—
96C3×#235—61no—
97C3×#235—62no—
98C2×C3×C4×D3—32no—
99C2×C3×C4×D3—62no—
100C2×C3×C4×D3—61no—
101C2×C3×C4×D3—62no—
102C2×#624—61no—
103C2×D3×Dic3—62no—
104C4×C3⋊Dic3C2×D312yescomplemented
105#3748C2×D312yescomplemented
106C3×C4×Dic3—32no—
107C3×C4×Dic3—32no—
108C3×C4×Dic3—62no—
109C3×#208—32no—
110C3×#208—32no—
111C3×#208—62no—
112Dic32—62no—
113Dic32—61no—
114#3712—61no—
115#3711—62no—
116#3710—61no—
117D3×C3⋊C8—62no—
118C22×C33—21no—
119C22×C33C2×D411yescharacteristic
120C2×C3×C3⋊D3—22no—
121C2×C3×C3⋊D3—21no—
122C2×C3×C3⋊D3—42no—
123C2×C3×C3⋊D3C22⋊C411yescharacteristic
124C2×C3×C3⋊D3C22⋊C411yescharacteristic
125C33×C4—21no—
126C33×C4C22×C411yescharacteristic
127C33×C4C2×D411yescharacteristic
128C3×C3⋊Dic3—24no—
129C3×C3⋊Dic3—21no—
130C3×C3⋊Dic3C22⋊C411yescharacteristic
131C3×C3⋊Dic3C22⋊C411yescharacteristic
132C2×C3×#41—91no—
133D3×#41—32no—
134D3×#41—61no—
135C2×C4×Dic3—91no—
136C2×#208—91no—
137D3×M16—92no—
138C42×D3—92no—
139C3×#104—92no—
140#837—32no—
141#837—32no—
142#837—62no—
143#837—92no—
144#825—92no—
145#805—92no—
146C22×C3⋊D3—21no—
147C22×C3⋊D3#6711yescharacteristic
148C22×C3×D3—32no—
149C22×C3×D3—61no—
150C22×C3×D3—62no—
151C22×C3×D3—62no—
152C32×Q8—22no—
153C32×Q8—21no—
154C32×Q8—41no—
155C32×Q8C4×D312yescomplemented
156C32×Q8C2×Dic311yescharacteristic, complemented
157C32×D4—21no—
158C32×D4—22no—
159C32×D4—22no—
160C32×D4—22no—
161C32×D4—21no—
162C32×D4—41no—
163C32×D4—42no—
164C32×D4—42no—
165C32×D4C4×D312yescomplemented
166C32×D4C2×Dic311yescharacteristic, complemented
167C2×C32×C4—21no—
168C2×C32×C4—22no—
169C2×C32×C4—22no—
170C2×C32×C4—22no—
171C2×C32×C4—21no—
172C2×C32×C4—41no—
173C2×C32×C4—42no—
174C2×C32×C4—42no—
175C2×C32×C4C22×D312yes—
176C2×C32×C4C22×D311yescharacteristic
177#638—21no—
178#638—21no—
179#638—21no—
180C2×C3⋊Dic3—21no—
181C2×C3⋊Dic3—22no—
182C2×C3⋊Dic3#6711yescharacteristic
183C4⋊(C3⋊D3)—21no—
184C4⋊(C3⋊D3)C2×Dic311yescharacteristic, complemented
185C4×C3⋊D3—21no—
186C4×C3⋊D3—21no—
187C4×C3⋊D3—21no—
188C4×C3⋊D3C2×Dic311yescharacteristic
189C4×C3⋊D3C2×Dic311yescharacteristic
190C4×C3⋊D3#6711yescharacteristic
191C4×C3⋊D3#6711yescharacteristic
192#634—21no—
193#634C2×Dic311yescharacteristic, complemented
194C3×#67—62no—
195C3×#67—62no—
196C3×#67—62no—
197C3×#67—62no—
198C3×#67—62no—
199C3×#67—62no—
200C2×C3×Dic3—32no—
201C2×C3×Dic3—61no—
202C2×C3×Dic3—62no—
203C2×C3×Dic3—62no—
204C2×C3×Dic3—62no—
205C2×C3×Dic3—62no—
206C2×C3×Dic3—122no—
207C3×D12—32no—
208C3×D12—61no—
209C3×D12—62no—
210C3×D12—62no—
211C3×C4×D3—32no—
212C3×C4×D3—32no—
213C3×C4×D3—32no—
214C3×C4×D3—32no—
215C3×C4×D3—61no—
216C3×C4×D3—61no—
217C3×C4×D3—61no—
218C3×C4×D3—61no—
219C3×C4×D3—62no—
220C3×C4×D3—62no—
221C3×C4×D3—62no—
222C3×C4×D3—62no—
223C3×C4×D3—62no—
224C3×C4×D3—62no—
225C3×Dic6—32no—
226C3×Dic6—61no—
227C3×Dic6—62no—
228C3×Dic6—62no—
229#624—64no—
230D3×Dic3—68no—
231C3⋊(C3⋊C8)—22no—
232C3×C3⋊C8—62no—
233C3×C3⋊C8—62no—
234C3×C3⋊C8—122no—
235C2×#104—271noSylow
236C2×C33—21no—
237C2×C33—41no—
238C2×C33C2×C22⋊C411yescharacteristic
239C3×C3⋊D3—21no—
240C3×C3⋊D3—41no—
241C3×C3⋊D3#10412yescomplemented
242C3×#41—21no—
243C3×#41—22no—
244C3×#41—41no—
245C3×#41—91no—
246C3×#41—91no—
247C3×#41—91no—
248C3×#41—181no—
249C3×#41—181no—
250C3×#41D3212yes—
251C3×#41D3211yescharacteristic
252C2×C3×Q8—91no—
253C2×C3×D4—91no—
254C2×C3×D4—181no—
255C22×C3×C4—91no—
256C22×C3×C4—181no—
257C22×Dic3—181no—
258#239—32no—
259#239—61no—
260D3×Q8—32no—
261D3×Q8—61no—
262#237—32no—
263#237—61no—
264#237—62no—
265#237—62no—
266D3×D4—32no—
267D3×D4—61no—
268D3×D4—62no—
269D3×D4—62no—
270#235—32no—
271#235—61no—
272#235—62no—
273#235—62no—
274C2×C4×D3—32no—
275C2×C4×D3—61no—
276C2×C4×D3—62no—
277C2×C4×D3—62no—
278C2×C4×D3—182no—
279C3×M16—92no—
280C3×C42—92no—
281C4×Dic3—32no—
282C4×Dic3—32no—
283C4×Dic3—62no—
284C4×Dic3—182no—
285C4×Dic3—181no—
286#208—32no—
287#208—32no—
288#208—62no—
289#208—181no—
290C2×C3⋊C8—181no—
291ΓL(1,𝔽25)—182no—
292D3×C8—182no—
293C22×C32—21no—
294C22×C32—22no—
295C22×C32—22no—
296C22×C32—22no—
297C22×C32—21no—
298C22×C32—41no—
299C22×C32—42no—
300C22×C32—42no—
301C22×C32D3×D412yes—
302C22×C32C2×#6711yescharacteristic
303C2×C3⋊D3—21no—
304C2×C3⋊D3—22no—
305C2×C3⋊D3—42no—
306C2×C3⋊D3#21711yescharacteristic
307C2×C3⋊D3#21711yescharacteristic
308C2×C3×D3—32no—
309C2×C3×D3—32no—
310C2×C3×D3—61no—
311C2×C3×D3—61no—
312C2×C3×D3—64no—
313C2×C3×D3—62no—
314C2×C3×D3—62no—
315C2×C3×D3—122no—
316C2×C3×D3—124no—
317C2×C3×D3—124no—
318C32×C4—21no—
319C32×C4—22no—
320C32×C4—22no—
321C32×C4—22no—
322C32×C4—21no—
323C32×C4—22no—
324C32×C4—21no—
325C32×C4—41no—
326C32×C4—41no—
327C32×C4—42no—
328C32×C4—42no—
329C32×C4C2×C4×D312yes—
330C32×C4D3×D412yes—
331C32×C4C22×Dic311yescharacteristic
332C32×C4C2×#6711yescharacteristic
333C3⋊Dic3—21no—
334C3⋊Dic3—24no—
335C3⋊Dic3#21711yescharacteristic
336C3⋊Dic3#21711yescharacteristic
337C3×Dic3—32no—
338C3×Dic3—32no—
339C3×Dic3—61no—
340C3×Dic3—61no—
341C3×Dic3—64no—
342C3×Dic3—64no—
343C3×Dic3—62no—
344C3×Dic3—62no—
345C3×Dic3—124no—
346C2×#41—91no—
347C2×M16—271no—
348C2×C42—271no—
349#104—92no—
350#104—272no—
351C33C2×#10411yescharacteristic, complemented, Sylow
352C23×C3—91no—
353C23×C3—181no—
354C22×D3—32no—
355C22×D3—61no—
356C22×D3—62no—
357C22×D3—62no—
358C3×Q8—21no—
359C3×Q8—22no—
360C3×Q8—41no—
361C3×Q8—91no—
362C3×Q8—181no—
363C3×Q8D3×Dic312yescomplemented
364C3×Q8#62411yescharacteristic, complemented
365C3×D4—22no—
366C3×D4—22no—
367C3×D4—21no—
368C3×D4—21no—
369C3×D4—22no—
370C3×D4—42no—
371C3×D4—42no—
372C3×D4—41no—
373C3×D4—91no—
374C3×D4—181no—
375C3×D4—181no—
376C3×D4—181no—
377C3×D4—181no—
378C3×D4D3×Dic312yescomplemented
379C3×D4#62411yescharacteristic, complemented
380C2×C3×C4—22no—
381C2×C3×C4—22no—
382C2×C3×C4—21no—
383C2×C3×C4—22no—
384C2×C3×C4—21no—
385C2×C3×C4—42no—
386C2×C3×C4—42no—
387C2×C3×C4—41no—
388C2×C3×C4—91no—
389C2×C3×C4—91no—
390C2×C3×C4—91no—
391C2×C3×C4—91no—
392C2×C3×C4—91no—
393C2×C3×C4—181no—
394C2×C3×C4—181no—
395C2×C3×C4—181no—
396C2×C3×C4—182no—
397C2×C3×C4—181no—
398C2×C3×C4C2×D3212yes—
399C2×C3×C4C2×D3211yescharacteristic
400#67—62no—
401#67—62no—
402#67—62no—
403#67—62no—
404#67—62no—
405#67—62no—
406C2×Dic3—32no—
407C2×Dic3—61no—
408C2×Dic3—62no—
409C2×Dic3—62no—
410C2×Dic3—62no—
411C2×Dic3—62no—
412C2×Dic3—122no—
413C2×Dic3—184no—
414D12—32no—
415D12—61no—
416D12—62no—
417D12—62no—
418C4×D3—32no—
419C4×D3—32no—
420C4×D3—32no—
421C4×D3—32no—
422C4×D3—61no—
423C4×D3—61no—
424C4×D3—61no—
425C4×D3—61no—
426C4×D3—62no—
427C4×D3—62no—
428C4×D3—62no—
429C4×D3—62no—
430C4×D3—62no—
431C4×D3—62no—
432C4×D3—188no—
433Dic6—32no—
434Dic6—61no—
435Dic6—62no—
436Dic6—62no—
437C3×C8—182no—
438C3⋊C8—62no—
439C3⋊C8—62no—
440C3⋊C8—122no—
441C2×C32—21no—
442C2×C32—22no—
443C2×C32—22no—
444C2×C32—21no—
445C2×C32—41no—
446C2×C32—42no—
447C2×C32—41no—
448C2×C32—41no—
449C2×C32—42no—
450C2×C32—42no—
451C2×C32—81no—
452C2×C32—82no—
453C2×C32—82no—
454C2×C32D3×C22⋊C412yes—
455C2×C32C2×#21711yescharacteristic
456C3⋊D3—21no—
457C3⋊D3—41no—
458C3⋊D3#83712yescomplemented
459C3×D3—34no—
460C3×D3—62no—
461C3×D3—62no—
462C3×D3—121no—
463C3×D3—122no—
464C3×D3—122no—
465#41—91no—
466#41—91no—
467#41—91no—
468#41—181no—
469#41—181no—
470#41#111311yescharacteristic
471C2×Q8—91no—
472C2×D4—91no—
473C2×D4—181no—
474C22×C4—91no—
475C22×C4—181no—
476C22×C4—541no—
477M16—92no—
478M16—541no—
479C2×C8—541no—
480C42—92no—
481C42—541no—
482C22×C3—22no—
483C22×C3—22no—
484C22×C3—21no—
485C22×C3—21no—
486C22×C3—22no—
487C22×C3—42no—
488C22×C3—42no—
489C22×C3—41no—
490C22×C3—91no—
491C22×C3—91no—
492C22×C3—182no—
493C22×C3—181no—
494C22×C3—362no—
495C22×C3D3×#6712yes—
496C22×C3#381111yescharacteristic
497C2×D3—32no—
498C2×D3—32no—
499C2×D3—61no—
500C2×D3—61no—
501C2×D3—64no—
502C2×D3—62no—
503C2×D3—62no—
504C2×D3—122no—
505C2×D3—124no—
506C2×D3—124no—
507C3×C4—22no—
508C3×C4—22no—
509C3×C4—21no—
510C3×C4—21no—
511C3×C4—22no—
512C3×C4—22no—
513C3×C4—21no—
514C3×C4—42no—
515C3×C4—42no—
516C3×C4—41no—
517C3×C4—41no—
518C3×C4—91no—
519C3×C4—91no—
520C3×C4—184no—
521C3×C4—181no—
522C3×C4C2×D3×Dic312yes—
523C3×C4C2×#62411yescharacteristic
524C3×C4D3×#6712yes—
525C3×C4#381111yescharacteristic
526Dic3—32no—
527Dic3—32no—
528Dic3—61no—
529Dic3—61no—
530Dic3—62no—
531Dic3—64no—
532Dic3—64no—
533Dic3—62no—
534Dic3—124no—
535C32—22no—
536C32—21no—
537C32—41no—
538C32D3×#10412yescomplemented
539C32C2×#83711yescharacteristic, complemented
540C23—91no—
541C23—181no—
542Q8—91no—
543Q8—181no—
544Q8#629411yescharacteristic, complemented
545D4—21no—
546D4—91no—
547D4—181no—
548D4—181no—
549D4—181no—
550D4—181no—
551D4#629411yescharacteristic, complemented
552C2×C4—21no—
553C2×C4—91no—
554C2×C4—91no—
555C2×C4—91no—
556C2×C4—91no—
557C2×C4—91no—
558C2×C4—181no—
559C2×C4—181no—
560C2×C4—181no—
561C2×C4—182no—
562C2×C4—181no—
563C2×C4—544no—
564C2×C4C2×#111311yescharacteristic
565C8—182no—
566C2×C3—22no—
567C2×C3—21no—
568C2×C3—21no—
569C2×C3—22no—
570C2×C3—42no—
571C2×C3—41no—
572C2×C3—42no—
573C2×C3—42no—
574C2×C3—41no—
575C2×C3—41no—
576C2×C3—81no—
577C2×C3—82no—
578C2×C3—82no—
579C2×C3—92no—
580C2×C3—181no—
581C2×C3—361no—
582C2×C3D3×#21712yes—
583C2×C3#6397411yescharacteristic
584D3—34no—
585D3—62no—
586D3—62no—
587D3—121no—
588D3—122no—
589D3—122no—
590C22—21no—
591C22—91no—
592C22—91no—
593C22—182no—
594C22—181no—
595C22—362no—
596C22#8904911yescharacteristic
597C4—21no—
598C4—91no—
599C4—91no—
600C4—184no—
601C4—181no—
602C4C2×#629411yescharacteristic
603C4#8904911yescharacteristic
604C3—21no—
605C3—22no—
606C3—41no—
607C3D3×#83712yescomplemented
608C3#1059167111yescharacteristic, complemented
609C2—21no—
610C2—41no—
611C2—92no—
612C2—181no—
613C2—361no—
614C2#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
hypocenterC33C2×#104
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. C4×#6294C2semidirect
3. #11723656C2semidirect
4. C3×#63664C2semidirect
5. #11722077C2semidirect
9. C3×#3832C4semidirect
10. C3×Q8×C3⋊D3C4semidirect
12. C3×#3830C4semidirect
14. C3×D4×C3⋊D3C4semidirect
20. C3×C4×C3⋊Dic3C22semidirect
21. C3×#3748C22semidirect
32. #63664D3semidirect
41. C33×Q8C2×C4semidirect
43. C33×D4C2×C4semidirect
53. C3×C4⋊(C3⋊D3)C2×C4semidirect
62. C3×#634C2×C4semidirect
72. #3832Dic3semidirect
73. Q8×C3⋊D3Dic3semidirect
75. #3830Dic3semidirect
77. D4×C3⋊D3Dic3semidirect
104. C4×C3⋊Dic3C2×D3semidirect
105. #3748C2×D3semidirect
155. C32×Q8C4×D3semidirect
156. C32×Q8C2×Dic3semidirect
165. C32×D4C4×D3semidirect
166. C32×D4C2×Dic3semidirect
184. C4⋊(C3⋊D3)C2×Dic3semidirect
193. #634C2×Dic3semidirect
241. C3×C3⋊D3#104semidirect
351. C33C2×#104semidirect
363. C3×Q8D3×Dic3semidirect
364. C3×Q8#624semidirect
378. C3×D4D3×Dic3semidirect
379. C3×D4#624semidirect
458. C3⋊D3#837semidirect
538. C32D3×#104semidirect
539. C32C2×#837semidirect
544. Q8#6294semidirect
551. D4#6294semidirect
607. C3D3×#837semidirect
608. C3#10591671semidirect

Maximal subgroups

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

subgroupindex
C3×#41×C3⋊D32
C4×#62942
#117236562
C3×#636642
#117220772
#105916713
D3×#8373