Finite Group Databasebeta

← #13156237

Subgroups — #13156237

Order 1728.

596 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 596 conjugacy classes of subgroups — too many to draw. Only its 94 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11723365C211yesmaximal, characteristic, complemented
2C2×C3×#3714C211yesmaximal, characteristic, complemented
3C3×#63823C211yesmaximal, characteristic, complemented
4C3×#63814C211yesmaximal, characteristic, complemented
5#11722073C211yesmaximal, characteristic, complemented
6#11722058C211yesmaximal, characteristic, complemented
7#11722025C211yesmaximal, characteristic, complemented
8#10591677—31nomaximal
9#10591505—31nomaximal
10#10590926—31nomaximal
11C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
12C32×#235C2211yescharacteristic
13C2×C32×D12C2211yescharacteristic
14#88855C2211yescharacteristic
15C3×#3748C2211yescharacteristic
16C3×#3748C2211yescharacteristic
17C2×C32×C3⋊C8C2211yescharacteristic
18#88793—21no—
19#88789—21no—
20C3×#3715—21no—
21C3×#3714—21no—
22C3×#3714—21no—
23C3×#3713—21no—
24C3×#3711—21no—
25C3×D3×C3⋊C8—21no—
26#64158—31no—
27C3×#949—31no—
28C3×#948—31no—
29C3×#932—31no—
30C2×C3×#213—31no—
31C2×C3×D24—31no—
32C3×#901—31no—
33C2×#3714D311yescharacteristic, complemented
34#63823—31no—
35#63823—31no—
36#63823—61no—
37#63823D311yescharacteristic, complemented
38#63814D311yescharacteristic, complemented
39#63660—31no—
40#63646—31no—
41#63577—31no—
42#63577—31no—
43#63577—61no—
44#63559—31no—
45#63558—31no—
46#63558—31no—
47#63558—31no—
48#63558—31no—
49#63558—61no—
50C22×C3×C3⋊D3—21no—
51C22×C32×D3—21no—
52C2×C33×C4C2311yescharacteristic
53C3×C4⋊(C3⋊D3)—21no—
54C3×C4⋊(C3⋊D3)—21no—
55C32×#67—21no—
56C32×D12—21no—
57C32×D12—21no—
58C32×D12D411yescharacteristic
59C32×C4×D3—21no—
60C32×Dic6D411yescharacteristic
61C3⋊(C3⋊(C3⋊C8))—21no—
62C3×C3⋊(C3⋊C8)—21no—
63C3×C3⋊(C3⋊C8)—21no—
64C32×C3⋊C8D411yescharacteristic
65C32×C3⋊C8D411yescharacteristic
66#5199—91no—
67#5160—91no—
68#5142—91no—
69C32×#41—31no—
70C2×C32×D4—31no—
71C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
72C3×#235—21no—
73C3×#235C2×D311yescharacteristic
74C3×#235C2×D311yescharacteristic
75C2×C3×D12—21no—
76C2×C3×D12—31no—
77C2×C3×D12—31no—
78C2×C3×D12—61no—
79C2×C3×D12C2×D311yescharacteristic
80C2×C3×D12C2×D311yescharacteristic
81C2×C32×C8—31no—
82#3755—61no—
83#3753—61no—
84#3748—31no—
85#3748—31no—
86#3748—31no—
87#3748—61no—
88#3748—61no—
89#3748—61no—
90#3748—121no—
91#3748C2×D311yescharacteristic
92#3748C2×D311yescharacteristic
93C3×#215—61no—
94C3×#214—61no—
95C3×#213—61no—
96C3×#213—61no—
97C3×#213—61no—
98C3×#213—61no—
99C3×#208—31no—
100C3×#208—31no—
101C3×#208—31no—
102C3×#208—31no—
103C3×#208—61no—
104C3×#208—61no—
105C3×#208—61no—
106C2×C3×C3⋊C8—21no—
107C2×C3×C3⋊C8—61no—
108C2×C3×C3⋊C8C2×D311yescharacteristic
109C2×C3×C3⋊C8C2×D311yescharacteristic
110C3×D24—61no—
111C3×D24—61no—
112C3×ΓL(1,𝔽25)—61no—
113C3×D3×C8—61no—
114#3715—21no—
115#3715—62no—
116#3715—61no—
117#3715—61no—
118#3714—21no—
119#3714—21no—
120#3713—21no—
121#3713—62no—
122#3713—61no—
123#3713—61no—
124#3711—21no—
125D3×C3⋊C8—21no—
126C22×C33#4111yescharacteristic
127C2×C3×C3⋊D3—41no—
128C2×C3×C3⋊D3—81no—
129C2×C32×D3—21no—
130C2×C32×D3—41no—
131C2×C32×D3—81no—
132C33×C4C2×D411yescharacteristic
133C33×C4C2×D411yescharacteristic
134C32×Dic3—21no—
135C3×AGL(1,ℤ/8)—91no—
136C2×C3×D8—91no—
137C3×#131—91no—
138#949—31no—
139#949—91no—
140#949—91no—
141#949—181no—
142#948—31no—
143#932—31no—
144#932—91no—
145C2×#213—31no—
146C2×D24—31no—
147#901—31no—
148#833—91no—
149#833—91no—
150#833—91no—
151#833—181no—
152#823—91no—
153#823—91no—
154#823—91no—
155#822—91no—
156#819—91no—
157#819—91no—
158#819—181no—
159C23×C32—61no—
160C22×C3⋊D3—21no—
161C22×C3×D3—22no—
162C22×C3×D3—21no—
163C22×C3×D3—21no—
164C22×C3×D3—61no—
165C22×C3×D3—61no—
166C22×C3×D3—121no—
167C32×Q8—31no—
168C32×D4—31no—
169C32×D4—61no—
170C32×D4—61no—
171C32×D4—61no—
172C2×C32×C4—21no—
173C2×C32×C4—21no—
174C2×C32×C4—21no—
175C2×C32×C4—41no—
176C2×C32×C4—61no—
177C2×C32×C4C22×D311yescharacteristic
178C2×C32×C4C22×D311yescharacteristic
179C2×C32×C4C22×D311yescharacteristic
180C4⋊(C3⋊D3)—21no—
181C4⋊(C3⋊D3)—21no—
182C3×#67—21no—
183C3×#67—21no—
184C3×#67—41no—
185C3×D12—21no—
186C3×D12—21no—
187C3×D12—21no—
188C3×D12—21no—
189C3×D12—21no—
190C3×D12—41no—
191C3×D12—41no—
192C3×D12—62no—
193C3×D12—61no—
194C3×D12—61no—
195C3×D12—61no—
196C3×D12—61no—
197C3×D12—121no—
198C3×D12D1211yescharacteristic
199C3×D12#6711yescharacteristic
200C3×C4×D3—21no—
201C3×C4×D3—21no—
202C3×C4×D3—41no—
203C3×Dic6—21no—
204C3×Dic6D1211yescharacteristic
205C3×Dic6#6711yescharacteristic
206C32×C8—31no—
207C32×C8—31no—
208C3⋊(C3⋊C8)—21no—
209C3⋊(C3⋊C8)—21no—
210C3⋊(C3⋊C8)—61no—
211C3⋊(C3⋊C8)—62no—
212C3⋊(C3⋊C8)—61no—
213C3⋊(C3⋊C8)—61no—
214C3⋊(C3⋊C8)—121no—
215C3⋊(C3⋊C8)—121no—
216C3⋊(C3⋊C8)—122no—
217C3×C3⋊C8—21no—
218C3×C3⋊C8—21no—
219C3×C3⋊C8—61no—
220C3×C3⋊C8—61no—
221C3×C3⋊C8—61no—
222C3×C3⋊C8—61no—
223C3×C3⋊C8—121no—
224C3×C3⋊C8—121no—
225C3×C3⋊C8#6711yescharacteristic
226C3×C3⋊C8#6711yescharacteristic
227C3×C3⋊C8#6711yescharacteristic
228C3×C3⋊C8#6711yescharacteristic
229#472—271noSylow
230C2×C33—21no—
231C2×C33#12111yescharacteristic
232C3×C3⋊D3—81no—
233C32×D3—41no—
234C32×D3—81no—
235C3×#41—31no—
236C3×#41—31no—
237C3×#41—61no—
238C2×C3×D4—31no—
239C2×C3×D4—31no—
240C2×C3×D4—61no—
241C2×C3×D4—91no—
242#235D3211yescharacteristic
243C2×D12—31no—
244C2×D12—31no—
245C2×D12—61no—
246C2×D12D3211yescharacteristic
247C3×SD16—181no—
248C3×D8—181no—
249C3×D8—181no—
250C3×D8—181no—
251C3×M16—91no—
252C3×M16—91no—
253C3×M16—181no—
254C2×C3×C8—31no—
255C2×C3×C8—31no—
256C2×C3×C8—61no—
257C2×C3×C8—181no—
258#215—61no—
259#215—181no—
260#215—181no—
261#215—182no—
262#214—61no—
263#214—181no—
264#213—61no—
265#213—61no—
266#213—61no—
267#213—61no—
268#213—181no—
269#213—181no—
270#213—182no—
271#213—181no—
272#208—31no—
273#208—31no—
274#208—31no—
275#208—31no—
276#208—61no—
277#208—61no—
278#208—61no—
279#208—91no—
280#208—91no—
281#208—91no—
282#208—181no—
283#208—181no—
284#208—181no—
285#208—361no—
286C2×C3⋊C8—61no—
287C2×C3⋊C8D3211yescharacteristic
288D24—61no—
289D24—61no—
290ΓL(1,𝔽25)—61no—
291D3×C8—61no—
292C22×C32—21no—
293C22×C32—21no—
294C22×C32—21no—
295C22×C32—41no—
296C22×C32—61no—
297C22×C32—121no—
298C22×C32—241no—
299C22×C32#23511yescharacteristic
300C22×C32#23711yescharacteristic
301C22×C32#23711yescharacteristic
302C2×C3⋊D3—41no—
303C2×C3⋊D3—81no—
304C2×C3×D3—21no—
305C2×C3×D3—21no—
306C2×C3×D3—41no—
307C2×C3×D3—42no—
308C2×C3×D3—41no—
309C2×C3×D3—41no—
310C2×C3×D3—81no—
311C2×C3×D3—81no—
312C2×C3×D3—82no—
313C2×C3×D3—122no—
314C2×C3×D3—121no—
315C2×C3×D3—121no—
316C2×C3×D3—242no—
317C2×C3×D3—241no—
318C2×C3×D3—241no—
319C32×C4—21no—
320C32×C4—21no—
321C32×C4—21no—
322C32×C4—21no—
323C32×C4—21no—
324C32×C4—21no—
325C32×C4—41no—
326C32×C4—41no—
327C32×C4—61no—
328C32×C4C2×D1211yescharacteristic
329C32×C4D3×D411yescharacteristic
330C32×C4D3×D411yescharacteristic
331C32×C4C2×#6711yescharacteristic
332C32×C4C2×#6711yescharacteristic
333C32×C4C2×#6711yescharacteristic
334C3×Dic3—21no—
335C3×Dic3—21no—
336C3×Dic3—41no—
337AGL(1,ℤ/8)—91no—
338AGL(1,ℤ/8)—271no—
339C2×D8—91no—
340#131—91no—
341#108—271no—
342#100—271no—
343#100—271no—
344C33#47211yescharacteristic, complemented, Sylow
345C23×C3—62no—
346C23×C3—61no—
347C23×C3—61no—
348C23×C3—181no—
349C22×D3—21no—
350C22×D3—61no—
351C22×D3—61no—
352C22×D3—121no—
353C3×Q8—31no—
354C3×Q8—31no—
355C3×Q8—61no—
356C3×D4—31no—
357C3×D4—31no—
358C3×D4—61no—
359C3×D4—61no—
360C3×D4—61no—
361C3×D4—61no—
362C3×D4—61no—
363C3×D4—61no—
364C3×D4—61no—
365C3×D4—121no—
366C3×D4—121no—
367C3×D4—121no—
368C3×D4—181no—
369C3×D4—181no—
370C2×C3×C4—21no—
371C2×C3×C4—21no—
372C2×C3×C4—21no—
373C2×C3×C4—41no—
374C2×C3×C4—61no—
375C2×C3×C4—61no—
376C2×C3×C4—121no—
377C2×C3×C4C2×D3211yescharacteristic
378C2×C3×C4C2×D3211yescharacteristic
379C2×C3×C4C2×D3211yescharacteristic
380#67—21no—
381D12—21no—
382D12—21no—
383D12—62no—
384D12—61no—
385D12—61no—
386D12—61no—
387D12—61no—
388D12—121no—
389D12#62611yescharacteristic
390C4×D3—21no—
391Dic6#62611yescharacteristic
392C3×C8—31no—
393C3×C8—31no—
394C3×C8—31no—
395C3×C8—31no—
396C3×C8—61no—
397C3×C8—61no—
398C3×C8—181no—
399C3×C8—181no—
400C3⋊C8—61no—
401C3⋊C8—61no—
402C3⋊C8—61no—
403C3⋊C8—61no—
404C3⋊C8—121no—
405C3⋊C8—121no—
406C3⋊C8—182no—
407C3⋊C8—181no—
408C3⋊C8—181no—
409C3⋊C8—181no—
410C3⋊C8—361no—
411C3⋊C8—361no—
412C3⋊C8—362no—
413C3⋊C8#62511yescharacteristic
414C3⋊C8#62511yescharacteristic
415C2×C32—21no—
416C2×C32—21no—
417C2×C32—21no—
418C2×C32—21no—
419C2×C32—21no—
420C2×C32—21no—
421C2×C32—41no—
422C2×C32—41no—
423C2×C32—41no—
424C2×C32—41no—
425C2×C32—81no—
426C2×C32—121no—
427C2×C32—241no—
428C2×C32#93011yescharacteristic
429C2×C32#93811yescharacteristic
430C2×C32#93911yescharacteristic
431C3⋊D3—81no—
432C3×D3—41no—
433C3×D3—41no—
434C3×D3—81no—
435C3×D3—82no—
436C3×D3—81no—
437C3×D3—81no—
438C3×D3—242no—
439C3×D3—241no—
440C3×D3—241no—
441#41—31no—
442C2×D4—31no—
443C2×D4—91no—
444SD16—181no—
445SD16—541no—
446D8—181no—
447D8—181no—
448D8—181no—
449D8—541no—
450M16—91no—
451M16—91no—
452M16—181no—
453M16—271no—
454C2×C8—31no—
455C2×C8—181no—
456C22×C3—21no—
457C22×C3—21no—
458C22×C3—21no—
459C22×C3—41no—
460C22×C3—61no—
461C22×C3—61no—
462C22×C3—121no—
463C22×C3—122no—
464C22×C3—121no—
465C22×C3—121no—
466C22×C3—241no—
467C22×C3—241no—
468C22×C3—242no—
469C22×C3—361no—
470C22×C3—721no—
471C22×C3#379511yescharacteristic
472C22×C3#380411yescharacteristic
473C22×C3#380511yescharacteristic
474C2×D3—21no—
475C2×D3—41no—
476C2×D3—81no—
477C2×D3—122no—
478C2×D3—121no—
479C2×D3—121no—
480C2×D3—242no—
481C2×D3—241no—
482C2×D3—241no—
483C3×C4—21no—
484C3×C4—21no—
485C3×C4—21no—
486C3×C4—21no—
487C3×C4—21no—
488C3×C4—21no—
489C3×C4—41no—
490C3×C4—41no—
491C3×C4—61no—
492C3×C4—61no—
493C3×C4—121no—
494C3×C4D3×D1211yescharacteristic
495C3×C4C2×#62511yescharacteristic
496C3×C4C2×#62611yescharacteristic
497C3×C4D3×#6711yescharacteristic
498C3×C4D3×#6711yescharacteristic
499C3×C4D3×#6711yescharacteristic
500Dic3—21no—
501C32—21no—
502C32—21no—
503C32—21no—
504C32—41no—
505C32#514211yescharacteristic, complemented
506C32#516011yescharacteristic, complemented
507C32#519911yescharacteristic, complemented
508C23—61no—
509C23—181no—
510Q8—31no—
511D4—31no—
512D4—61no—
513D4—61no—
514D4—61no—
515D4—181no—
516D4—181no—
517C2×C4—61no—
518C2×C4D3311yescharacteristic
519C8—31no—
520C8—31no—
521C8—181no—
522C8—181no—
523C8—541no—
524C2×C3—21no—
525C2×C3—21no—
526C2×C3—21no—
527C2×C3—21no—
528C2×C3—21no—
529C2×C3—21no—
530C2×C3—41no—
531C2×C3—41no—
532C2×C3—41no—
533C2×C3—41no—
534C2×C3—81no—
535C2×C3—121no—
536C2×C3—121no—
537C2×C3—241no—
538C2×C3—242no—
539C2×C3—241no—
540C2×C3—241no—
541C2×C3—721no—
542C2×C3#6390811yescharacteristic
543C2×C3#6397011yescharacteristic
544C2×C3#6397811yescharacteristic
545D3—41no—
546D3—81no—
547D3—242no—
548D3—241no—
549D3—241no—
550C22—61no—
551C22—121no—
552C22—241no—
553C22—361no—
554C22—721no—
555C22#8896111yescharacteristic
556C4—61no—
557C4D3×#62511yescharacteristic
558C4D3×#62611yescharacteristic
559C3—21no—
560C3—21no—
561C3—21no—
562C3—41no—
563C3#1059092611yescharacteristic, complemented
564C3#1059150511yescharacteristic, complemented
565C3#1059167711yescharacteristic, complemented
566C2—21no—
567C2—121no—
568C2—241no—
569C2—721no—
570C2#1172295611yescharacteristic
subgroupG/H
centerC2#11722956
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socleC2×C33#121
hypocenterC33#472
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. #11723365C2semidirect
2. C2×C3×#3714C2semidirect
3. C3×#63823C2semidirect
4. C3×#63814C2semidirect
5. #11722073C2semidirect
6. #11722058C2semidirect
7. #11722025C2semidirect
33. C2×#3714D3semidirect
37. #63823D3semidirect
38. #63814D3semidirect
344. C33#472semidirect
505. C32#5142semidirect
506. C32#5160semidirect
507. C32#5199semidirect
563. C3#10590926semidirect
564. C3#10591505semidirect
565. C3#10591677semidirect

Maximal subgroups

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

subgroupindex
#117233652
C2×C3×#37142
C3×#638232
C3×#638142
#117220732
#117220582
#117220252
#105916773
#105915053
#105909263