Finite Group Databasebeta

← #13156253

Subgroups — #13156253

Order 1728.

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

normal subgroups only

This group has 548 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#11723373C211yesmaximal, characteristic
2C2×C3×#3719C211yesmaximal, characteristic, complemented
3C3×#63827C211yesmaximal, characteristic
4C3×#63814C211yesmaximal, characteristic
5#11722074C211yesmaximal, characteristic, complemented
6#11722058C211yesmaximal, characteristic, complemented
7#11722028C211yesmaximal, characteristic, complemented
8#10591682—31nomaximal
9#10591641—31nomaximal
10#10590930—31nomaximal
11C2×C3×#634C2211yescharacteristic
12C32×#235C2211yescharacteristic
13C2×C32×Dic6C2211yescharacteristic
14#88855C2211yescharacteristic
15C3×#3748C2211yescharacteristic
16C3×#3748C2211yescharacteristic
17C2×C32×C3⋊C8C2211yescharacteristic
18#88798—21no—
19#88792—21no—
20C3×#3719—21no—
21C3×#3719—21no—
22C3×#3718—21no—
23C3×#3715—21no—
24C3×#3711—21no—
25C3×D3×C3⋊C8—21no—
26#64160—31no—
27C3×#951—31no—
28C3×#948—31no—
29C2×C3×#216—31no—
30C3×#942—31no—
31C2×C3×Dic12—31no—
32C3×#901—31no—
33C2×#3719D311yescharacteristic, complemented
34#63827—31no—
35#63827—31no—
36#63827—61no—
37#63827D311yescharacteristic
38#63814D311yescharacteristic
39#63663—31no—
40#63646—31no—
41#63577—31no—
42#63577—31no—
43#63577—61no—
44#63561—31no—
45#63560—31no—
46#63560—31no—
47#63560—31no—
48#63560—31no—
49#63560—61no—
50C2×C33×C4C2311yescharacteristic
51C2×C3×C3⋊Dic3—21no—
52C3×#634—21no—
53C3×#634—21no—
54C32×#67—21no—
55C2×C32×Dic3—21no—
56C32×D12D411yescharacteristic
57C32×C4×D3—21no—
58C32×Dic6—21no—
59C32×Dic6—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#5204—91no—
67#5192—91no—
68#5143—91no—
69C32×#41—31no—
70C2×C32×Q8—31no—
71C2×#634C2×D311yescharacteristic
72C3×#235—21no—
73C3×#235C2×D311yescharacteristic
74C3×#235C2×D311yescharacteristic
75C2×C3×Dic6—21no—
76C2×C3×Dic6—31no—
77C2×C3×Dic6—31no—
78C2×C3×Dic6—61no—
79C2×C3×Dic6C2×D311yescharacteristic
80C2×C3×Dic6C2×D311yescharacteristic
81C2×C32×C8—31no—
82#3756—61no—
83#3754—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×#216—61no—
94C3×#216—61no—
95C3×#216—61no—
96C3×#216—61no—
97C3×#215—61no—
98C3×#214—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×Dic12—61no—
111C3×Dic12—61no—
112C3×ΓL(1,𝔽25)—61no—
113C3×D3×C8—61no—
114#3719—21no—
115#3719—21no—
116#3718—21no—
117#3718—62no—
118#3718—61no—
119#3718—61no—
120#3715—21no—
121#3715—62no—
122#3715—61no—
123#3715—61no—
124#3711—21no—
125D3×C3⋊C8—21no—
126C22×C33#4111yescharacteristic
127C2×C32×D3—21no—
128C33×C4C2×D411yescharacteristic
129C33×C4C2×D411yescharacteristic
130C3×C3⋊Dic3—41no—
131C32×Dic3—21no—
132C32×Dic3—41no—
133C3×#137—91no—
134C2×C3×Q16—91no—
135C3×#131—91no—
136#951—31no—
137#951—91no—
138#951—91no—
139#951—181no—
140#948—31no—
141C2×#216—31no—
142#942—31no—
143#942—91no—
144C2×Dic12—31no—
145#901—31no—
146#836—91no—
147#836—91no—
148#836—91no—
149#836—181no—
150#824—91no—
151#824—91no—
152#824—91no—
153#822—91no—
154#819—91no—
155#819—91no—
156#819—181no—
157C32×Q8—31no—
158C32×Q8—61no—
159C32×Q8—61no—
160C32×D4—31no—
161C32×D4—61no—
162C2×C32×C4—21no—
163C2×C32×C4—21no—
164C2×C32×C4—21no—
165C2×C32×C4—41no—
166C2×C32×C4—61no—
167C2×C32×C4—61no—
168C2×C32×C4C22×D311yescharacteristic
169C2×C32×C4C22×D311yescharacteristic
170C2×C32×C4C22×D311yescharacteristic
171C2×C3⋊Dic3—21no—
172#634—21no—
173#634—21no—
174C3×#67—21no—
175C3×#67—21no—
176C3×#67—41no—
177C2×C3×Dic3—22no—
178C2×C3×Dic3—21no—
179C2×C3×Dic3—21no—
180C2×C3×Dic3—61no—
181C2×C3×Dic3—61no—
182C2×C3×Dic3—121no—
183C3×D12—21no—
184C3×D12D1211yescharacteristic
185C3×D12#6711yescharacteristic
186C3×C4×D3—21no—
187C3×C4×D3—21no—
188C3×C4×D3—41no—
189C3×Dic6—21no—
190C3×Dic6—21no—
191C3×Dic6—21no—
192C3×Dic6—21no—
193C3×Dic6—21no—
194C3×Dic6—41no—
195C3×Dic6—41no—
196C3×Dic6—62no—
197C3×Dic6—61no—
198C3×Dic6—61no—
199C3×Dic6—61no—
200C3×Dic6—61no—
201C3×Dic6—121no—
202C3×Dic6D1211yescharacteristic
203C3×Dic6#6711yescharacteristic
204C32×C8—31no—
205C32×C8—31no—
206C3⋊(C3⋊C8)—21no—
207C3⋊(C3⋊C8)—21no—
208C3⋊(C3⋊C8)—61no—
209C3⋊(C3⋊C8)—62no—
210C3⋊(C3⋊C8)—61no—
211C3⋊(C3⋊C8)—61no—
212C3⋊(C3⋊C8)—121no—
213C3⋊(C3⋊C8)—121no—
214C3⋊(C3⋊C8)—122no—
215C3×C3⋊C8—21no—
216C3×C3⋊C8—21no—
217C3×C3⋊C8—61no—
218C3×C3⋊C8—61no—
219C3×C3⋊C8—61no—
220C3×C3⋊C8—61no—
221C3×C3⋊C8—121no—
222C3×C3⋊C8—121no—
223C3×C3⋊C8#6711yescharacteristic
224C3×C3⋊C8#6711yescharacteristic
225C3×C3⋊C8#6711yescharacteristic
226C3×C3⋊C8#6711yescharacteristic
227#473—271noSylow
228C2×C33—21no—
229C2×C33#12111yescharacteristic
230C32×D3—41no—
231C3×#41—31no—
232C3×#41—31no—
233C3×#41—61no—
234C2×C3×Q8—31no—
235C2×C3×Q8—31no—
236C2×C3×Q8—61no—
237C2×C3×Q8—91no—
238#235D3211yescharacteristic
239C2×Dic6—31no—
240C2×Dic6—31no—
241C2×Dic6—61no—
242C2×Dic6D3211yescharacteristic
243C3×Q16—181no—
244C3×Q16—181no—
245C3×Q16—181no—
246C3×SD16—181no—
247C3×M16—91no—
248C3×M16—91no—
249C3×M16—181no—
250C2×C3×C8—31no—
251C2×C3×C8—31no—
252C2×C3×C8—61no—
253C2×C3×C8—181no—
254#216—61no—
255#216—61no—
256#216—61no—
257#216—61no—
258#216—181no—
259#216—181no—
260#216—182no—
261#216—181no—
262#215—61no—
263#215—181no—
264#214—61no—
265#214—181no—
266#214—181no—
267#214—182no—
268#208—31no—
269#208—31no—
270#208—31no—
271#208—31no—
272#208—61no—
273#208—61no—
274#208—61no—
275#208—91no—
276#208—91no—
277#208—91no—
278#208—181no—
279#208—181no—
280#208—181no—
281#208—361no—
282C2×C3⋊C8—61no—
283C2×C3⋊C8D3211yescharacteristic
284Dic12—61no—
285Dic12—61no—
286ΓL(1,𝔽25)—61no—
287D3×C8—61no—
288C22×C32—21no—
289C22×C32—21no—
290C22×C32—21no—
291C22×C32—41no—
292C22×C32—61no—
293C22×C32#23511yescharacteristic
294C22×C32#23711yescharacteristic
295C22×C32#23711yescharacteristic
296C2×C3×D3—21no—
297C2×C3×D3—21no—
298C2×C3×D3—41no—
299C32×C4—21no—
300C32×C4—21no—
301C32×C4—21no—
302C32×C4—21no—
303C32×C4—21no—
304C32×C4—21no—
305C32×C4—41no—
306C32×C4—41no—
307C32×C4—61no—
308C32×C4—121no—
309C32×C4C2×D1211yescharacteristic
310C32×C4D3×D411yescharacteristic
311C32×C4D3×D411yescharacteristic
312C32×C4C2×#6711yescharacteristic
313C32×C4C2×#6711yescharacteristic
314C32×C4C2×#6711yescharacteristic
315C3⋊Dic3—41no—
316C3×Dic3—21no—
317C3×Dic3—21no—
318C3×Dic3—41no—
319C3×Dic3—42no—
320C3×Dic3—41no—
321C3×Dic3—41no—
322C3×Dic3—122no—
323C3×Dic3—121no—
324C3×Dic3—121no—
325#137—91no—
326#137—271no—
327C2×Q16—91no—
328#131—91no—
329#108—271no—
330#101—271no—
331#101—271no—
332C33#47311yescharacteristic, complemented, Sylow
333C3×Q8—31no—
334C3×Q8—31no—
335C3×Q8—61no—
336C3×Q8—61no—
337C3×Q8—61no—
338C3×Q8—61no—
339C3×Q8—61no—
340C3×Q8—121no—
341C3×Q8—121no—
342C3×Q8—181no—
343C3×Q8—181no—
344C3×D4—31no—
345C3×D4—31no—
346C3×D4—61no—
347C3×D4—61no—
348C3×D4—61no—
349C3×D4—121no—
350C2×C3×C4—21no—
351C2×C3×C4—21no—
352C2×C3×C4—21no—
353C2×C3×C4—41no—
354C2×C3×C4—61no—
355C2×C3×C4—61no—
356C2×C3×C4—62no—
357C2×C3×C4—61no—
358C2×C3×C4—61no—
359C2×C3×C4—121no—
360C2×C3×C4—181no—
361C2×C3×C4C2×D3211yescharacteristic
362C2×C3×C4C2×D3211yescharacteristic
363C2×C3×C4C2×D3211yescharacteristic
364#67—21no—
365C2×Dic3—21no—
366C2×Dic3—61no—
367C2×Dic3—61no—
368C2×Dic3—121no—
369D12#62611yescharacteristic
370C4×D3—21no—
371Dic6—21no—
372Dic6—21no—
373Dic6—62no—
374Dic6—61no—
375Dic6—61no—
376Dic6—61no—
377Dic6—61no—
378Dic6—121no—
379Dic6#62611yescharacteristic
380C3×C8—31no—
381C3×C8—31no—
382C3×C8—31no—
383C3×C8—31no—
384C3×C8—61no—
385C3×C8—61no—
386C3×C8—181no—
387C3×C8—181no—
388C3⋊C8—61no—
389C3⋊C8—61no—
390C3⋊C8—61no—
391C3⋊C8—61no—
392C3⋊C8—121no—
393C3⋊C8—121no—
394C3⋊C8—182no—
395C3⋊C8—181no—
396C3⋊C8—181no—
397C3⋊C8—181no—
398C3⋊C8—361no—
399C3⋊C8—361no—
400C3⋊C8—362no—
401C3⋊C8#62511yescharacteristic
402C3⋊C8#62511yescharacteristic
403C2×C32—21no—
404C2×C32—21no—
405C2×C32—21no—
406C2×C32—21no—
407C2×C32—21no—
408C2×C32—21no—
409C2×C32—41no—
410C2×C32—41no—
411C2×C32—41no—
412C2×C32—41no—
413C2×C32—81no—
414C2×C32—121no—
415C2×C32#93011yescharacteristic
416C2×C32#93811yescharacteristic
417C2×C32#93911yescharacteristic
418C3×D3—41no—
419C3×D3—41no—
420C3×D3—81no—
421#41—31no—
422C2×Q8—31no—
423C2×Q8—91no—
424Q16—181no—
425Q16—181no—
426Q16—181no—
427Q16—541no—
428SD16—181no—
429SD16—541no—
430M16—91no—
431M16—91no—
432M16—181no—
433M16—271no—
434C2×C8—31no—
435C2×C8—181no—
436C22×C3—21no—
437C22×C3—21no—
438C22×C3—21no—
439C22×C3—41no—
440C22×C3—61no—
441C22×C3—61no—
442C22×C3—121no—
443C22×C3#379511yescharacteristic
444C22×C3#380411yescharacteristic
445C22×C3#380511yescharacteristic
446C2×D3—21no—
447C3×C4—21no—
448C3×C4—21no—
449C3×C4—21no—
450C3×C4—21no—
451C3×C4—21no—
452C3×C4—21no—
453C3×C4—41no—
454C3×C4—41no—
455C3×C4—61no—
456C3×C4—61no—
457C3×C4—121no—
458C3×C4—122no—
459C3×C4—121no—
460C3×C4—121no—
461C3×C4—361no—
462C3×C4D3×D1211yescharacteristic
463C3×C4C2×#62511yescharacteristic
464C3×C4C2×#62611yescharacteristic
465C3×C4D3×#6711yescharacteristic
466C3×C4D3×#6711yescharacteristic
467C3×C4D3×#6711yescharacteristic
468Dic3—21no—
469Dic3—41no—
470Dic3—122no—
471Dic3—121no—
472Dic3—121no—
473C32—21no—
474C32—21no—
475C32—21no—
476C32—41no—
477C32#514311yescharacteristic, complemented
478C32#519211yescharacteristic, complemented
479C32#520411yescharacteristic, complemented
480Q8—31no—
481Q8—61no—
482Q8—61no—
483Q8—181no—
484Q8—181no—
485D4—31no—
486D4—61no—
487C2×C4—61no—
488C2×C4—61no—
489C2×C4—181no—
490C2×C4D3311yescharacteristic
491C8—31no—
492C8—31no—
493C8—181no—
494C8—181no—
495C8—541no—
496C2×C3—21no—
497C2×C3—21no—
498C2×C3—21no—
499C2×C3—21no—
500C2×C3—21no—
501C2×C3—21no—
502C2×C3—41no—
503C2×C3—41no—
504C2×C3—41no—
505C2×C3—41no—
506C2×C3—81no—
507C2×C3—121no—
508C2×C3—121no—
509C2×C3—241no—
510C2×C3#6390811yescharacteristic
511C2×C3#6397011yescharacteristic
512C2×C3#6397811yescharacteristic
513D3—41no—
514C22—61no—
515C22#8896111yescharacteristic
516C4—61no—
517C4—121no—
518C4—361no—
519C4D3×#62511yescharacteristic
520C4D3×#62611yescharacteristic
521C3—21no—
522C3—21no—
523C3—21no—
524C3—41no—
525C3#1059093011yescharacteristic, complemented
526C3#1059164111yescharacteristic, complemented
527C3#1059168211yescharacteristic, complemented
528C2—21no—
529C2—121no—
530C2#1172295611yescharacteristic
subgroupG/H
centerC2#11722956
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socleC2×C33#121
prime radicalC32×#235C22
hypocenterC33#473
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
2. C2×C3×#3719C2semidirect
5. #11722074C2semidirect
6. #11722058C2semidirect
7. #11722028C2semidirect
33. C2×#3719D3semidirect
332. C33#473semidirect
477. C32#5143semidirect
478. C32#5192semidirect
479. C32#5204semidirect
525. C3#10590930semidirect
526. C3#10591641semidirect
527. C3#10591682semidirect

Maximal subgroups

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

subgroupindex
#117233732
C2×C3×#37192
C3×#638272
C3×#638142
#117220742
#117220582
#117220282
#105916823
#105916413
#105909303