Finite Group Databasebeta

← #13156257

Subgroups — #13156257

Order 1728.

548 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 548 conjugacy classes of subgroups — too many to draw. Only its 90 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
1C2×#88812C211yesmaximal, characteristic, complemented
2#11723373C211yesmaximal, characteristic
3C3×#63831C211yesmaximal, characteristic
4C3×#63815C211yesmaximal, characteristic
5#11722060C211yesmaximal, characteristic, complemented
6#11722029C211yesmaximal, characteristic, complemented
7C3×#63561C211yesmaximal, characteristic, complemented
8#10591682—31nomaximal
9#10590999—31nomaximal
10#10590981—31nomaximal
11C2×C3×#634C2211yescharacteristic
12C2×C3×#634C2211yescharacteristic
13C32×#235C2211yescharacteristic
14#88855C2211yescharacteristic
15C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
16C32×#208C2211yescharacteristic
17C32×#208C2211yescharacteristic
18#88812—21no—
19#88812—21no—
20#88798—21no—
21#88792—21no—
22C3×#3719—21no—
23C3×#3716—21no—
24C3×#3711—21no—
25C3×D3×C3⋊C8—21no—
26#64160—31no—
27C3×#951—31no—
28C3×#948—31no—
29C3×#909—31no—
30C3×#907—31no—
31C2×#3719—31no—
32C2×#3719—31no—
33C2×#3718—31no—
34#63831D311yescharacteristic
35#63827—31no—
36#63827—31no—
37#63827—61no—
38#63815D311yescharacteristic
39#63651—31no—
40#63649—31no—
41C3×#824—31no—
42C3×#824—31no—
43#63577—31no—
44#63577—31no—
45#63577—61no—
46#63561D311yescharacteristic, complemented
47#63560—31no—
48#63560—31no—
49#63560—61no—
50C2×C33×C4C2311yescharacteristic
51C2×C3×C3⋊Dic3—21no—
52C2×C3×C3⋊Dic3—21no—
53C3×#634—21no—
54C3×#634—21no—
55C3×#634—21no—
56C3×#634—21no—
57C32×#67—21no—
58C32×D12D411yescharacteristic
59C32×C4×D3—21no—
60C32×Dic6D411yescharacteristic
61C3⋊(C3⋊(C3⋊C8))—21no—
62C3×C3⋊(C3⋊C8)D411yescharacteristic
63C3×C3⋊(C3⋊C8)D411yescharacteristic
64C32×C3⋊C8—21no—
65C32×C3⋊C8—21no—
66#5204—91no—
67#5143—91no—
68#4755—91no—
69C32×#41—31no—
70C2×#634C2×D311yescharacteristic
71C2×#634C2×D311yescharacteristic
72C3×#235—21no—
73C3×#235C2×D311yescharacteristic
74C3×#235C2×D311yescharacteristic
75C2×C3×Dic6—31no—
76C2×C3×Dic6—31no—
77C2×C3×Dic6—31no—
78C2×C3×Dic6—31no—
79C2×C3×Dic6—61no—
80C2×C3×Dic6—61no—
81C32×M16—31no—
82C32×M16—31no—
83#3756—61no—
84#3754—61no—
85#3748—31no—
86#3748—31no—
87#3748—31no—
88#3748—61no—
89#3748—61no—
90#3748—61no—
91#3748—121no—
92C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
93C3×#216—61no—
94C3×#214—61no—
95C3×#208—21no—
96C3×#208—21no—
97C3×#208—61no—
98C3×#208C2×D311yescharacteristic
99C3×#208C2×D311yescharacteristic
100C3×#208C2×D311yescharacteristic
101C3×#208C2×D311yescharacteristic
102C2×C3×C3⋊C8—31no—
103C2×C3×C3⋊C8—31no—
104C2×C3×C3⋊C8—61no—
105C2×C3×C3⋊C8—61no—
106C3×Dic12—61no—
107C3×#204—61no—
108C3×ΓL(1,𝔽25)—61no—
109C3×D3×C8—61no—
110#3719—21no—
111#3719—61no—
112#3719—61no—
113#3719—61no—
114#3719—61no—
115#3718—62no—
116#3718—61no—
117#3718—61no—
118#3718—61no—
119#3718—61no—
120#3716—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—
131C3×C3⋊Dic3—41no—
132C32×Dic3—21no—
133C3×#137—91no—
134C3×#131—91no—
135#951—31no—
136#951—91no—
137#951—91no—
138#951—181no—
139#948—31no—
140C2×#216—91no—
141C2×#216—91no—
142#942—91no—
143#909—31no—
144#907—31no—
145C2×Dic12—91no—
146C3×#101—91no—
147#836—91no—
148#824—31no—
149#824—31no—
150#824—91no—
151#824—91no—
152#824—181no—
153#822—91no—
154#822—91no—
155#822—181no—
156#819—91no—
157C32×Q8—31no—
158C32×D4—31no—
159C32×D4—61no—
160C2×C32×C4—21no—
161C2×C32×C4—21no—
162C2×C32×C4—21no—
163C2×C32×C4—41no—
164C2×C32×C4—61no—
165C2×C32×C4C22×D311yescharacteristic
166C2×C32×C4C22×D311yescharacteristic
167C2×C32×C4C22×D311yescharacteristic
168C2×C3⋊Dic3—21no—
169C2×C3⋊Dic3—21no—
170#634—21no—
171#634—21no—
172#634—21no—
173#634—21no—
174C3×#67—21no—
175C3×#67—21no—
176C3×#67—41no—
177C2×C3×Dic3—62no—
178C2×C3×Dic3—61no—
179C2×C3×Dic3—61no—
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—62no—
191C3×Dic6—61no—
192C3×Dic6—61no—
193C3×Dic6—61no—
194C3×Dic6—61no—
195C3×Dic6—61no—
196C3×Dic6—61no—
197C3×Dic6—61no—
198C3×Dic6—61no—
199C3×Dic6—121no—
200C3×Dic6—121no—
201C3×Dic6—121no—
202C3×Dic6D1211yescharacteristic
203C3×Dic6#6711yescharacteristic
204C32×C8—61no—
205C32×C8—61no—
206C3⋊(C3⋊C8)—61no—
207C3⋊(C3⋊C8)—62no—
208C3⋊(C3⋊C8)—61no—
209C3⋊(C3⋊C8)—61no—
210C3⋊(C3⋊C8)—121no—
211C3⋊(C3⋊C8)—121no—
212C3⋊(C3⋊C8)—122no—
213C3⋊(C3⋊C8)#6711yescharacteristic
214C3⋊(C3⋊C8)#6711yescharacteristic
215C3×C3⋊C8—21no—
216C3×C3⋊C8—21no—
217C3×C3⋊C8—21no—
218C3×C3⋊C8—21no—
219C3×C3⋊C8—31no—
220C3×C3⋊C8—31no—
221C3×C3⋊C8—31no—
222C3×C3⋊C8—31no—
223C3×C3⋊C8—41no—
224C3×C3⋊C8—41no—
225C3×C3⋊C8—61no—
226C3×C3⋊C8—61no—
227#473—271noSylow
228C2×C33—21no—
229C2×C33#12111yescharacteristic
230C32×D3—41no—
231C3×#41—31no—
232C3×#41—31no—
233C3×#41—61no—
234C2×C3×Q8—91no—
235C2×C3×Q8—91no—
236#235D3211yescharacteristic
237C2×Dic6—31no—
238C2×Dic6—31no—
239C2×Dic6—31no—
240C2×Dic6—31no—
241C2×Dic6—61no—
242C2×Dic6—61no—
243C3×Q16—181no—
244C3×SD16—181no—
245C3×M16—31no—
246C3×M16—31no—
247C3×M16—31no—
248C3×M16—31no—
249C3×M16—61no—
250C3×M16—61no—
251C3×M16—181no—
252C2×C3×C8—91no—
253C2×C3×C8—181no—
254#216—61no—
255#216—181no—
256#216—181no—
257#216—181no—
258#216—181no—
259#216—181no—
260#216—181no—
261#216—182no—
262#216—181no—
263#215—181no—
264#214—61no—
265#214—181no—
266#214—181no—
267#214—182no—
268#208—61no—
269#208—91no—
270#208—91no—
271#208—91no—
272#208—181no—
273#208—181no—
274#208—181no—
275#208—361no—
276#208D3211yescharacteristic
277#208D3211yescharacteristic
278C2×C3⋊C8—31no—
279C2×C3⋊C8—31no—
280C2×C3⋊C8—61no—
281C2×C3⋊C8—61no—
282Dic12—61no—
283Dic12—181no—
284Dic12—181no—
285#204—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#23511yescharacteristic
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×C4C2×D1211yescharacteristic
309C32×C4D3×D411yescharacteristic
310C32×C4D3×D411yescharacteristic
311C32×C4D3×D411yescharacteristic
312C32×C4C2×#6711yescharacteristic
313C32×C4C2×#6711yescharacteristic
314C3⋊Dic3—41no—
315C3⋊Dic3—41no—
316C3×Dic3—21no—
317C3×Dic3—21no—
318C3×Dic3—41no—
319C3×Dic3—122no—
320C3×Dic3—122no—
321C3×Dic3—121no—
322C3×Dic3—121no—
323C3×Dic3—121no—
324C3×Dic3—121no—
325#137—91no—
326#137—271no—
327C2×Q16—271no—
328#131—91no—
329#108—271no—
330#101—91no—
331#101—271no—
332C33#47311yescharacteristic, complemented, Sylow
333C3×Q8—31no—
334C3×Q8—31no—
335C3×Q8—61no—
336C3×Q8—181no—
337C3×Q8—181no—
338C3×Q8—181no—
339C3×Q8—181no—
340C3×D4—31no—
341C3×D4—31no—
342C3×D4—61no—
343C3×D4—61no—
344C3×D4—61no—
345C3×D4—121no—
346C2×C3×C4—21no—
347C2×C3×C4—21no—
348C2×C3×C4—21no—
349C2×C3×C4—41no—
350C2×C3×C4—61no—
351C2×C3×C4—61no—
352C2×C3×C4—121no—
353C2×C3×C4—181no—
354C2×C3×C4—181no—
355C2×C3×C4C2×D3211yescharacteristic
356C2×C3×C4C2×D3211yescharacteristic
357C2×C3×C4C2×D3211yescharacteristic
358#67—21no—
359C2×Dic3—62no—
360C2×Dic3—61no—
361C2×Dic3—61no—
362C2×Dic3—61no—
363C2×Dic3—61no—
364C2×Dic3—121no—
365D12#62611yescharacteristic
366C4×D3—21no—
367Dic6—62no—
368Dic6—61no—
369Dic6—61no—
370Dic6—61no—
371Dic6—61no—
372Dic6—61no—
373Dic6—61no—
374Dic6—61no—
375Dic6—61no—
376Dic6—121no—
377Dic6—121no—
378Dic6—121no—
379Dic6#62611yescharacteristic
380C3×C8—61no—
381C3×C8—61no—
382C3×C8—61no—
383C3×C8—61no—
384C3×C8—91no—
385C3×C8—91no—
386C3×C8—121no—
387C3×C8—121no—
388C3⋊C8—21no—
389C3⋊C8—21no—
390C3⋊C8—31no—
391C3⋊C8—31no—
392C3⋊C8—31no—
393C3⋊C8—31no—
394C3⋊C8—61no—
395C3⋊C8—61no—
396C3⋊C8—182no—
397C3⋊C8—181no—
398C3⋊C8—181no—
399C3⋊C8—181no—
400C3⋊C8—361no—
401C3⋊C8—361no—
402C3⋊C8—362no—
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#88411yescharacteristic
416C2×C32#93011yescharacteristic
417C2×C32#93911yescharacteristic
418C3×D3—41no—
419C3×D3—41no—
420C3×D3—81no—
421#41—31no—
422C2×Q8—91no—
423C2×Q8—91no—
424Q16—181no—
425Q16—541no—
426Q16—541no—
427Q16—541no—
428SD16—181no—
429SD16—541no—
430M16—31no—
431M16—31no—
432M16—181no—
433M16—271no—
434C2×C8—91no—
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#379811yescharacteristic
445C22×C3#380411yescharacteristic
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—361no—
459C3×C4—361no—
460C3×C4D3×D1211yescharacteristic
461C3×C4#380211yescharacteristic
462C3×C4C2×#62611yescharacteristic
463C3×C4D3×#6711yescharacteristic
464C3×C4D3×#6711yescharacteristic
465C3×C4D3×#6711yescharacteristic
466Dic3—21no—
467Dic3—122no—
468Dic3—122no—
469Dic3—121no—
470Dic3—121no—
471Dic3—121no—
472Dic3—121no—
473C32—21no—
474C32—21no—
475C32—21no—
476C32—41no—
477C32#475511yescharacteristic, complemented
478C32#514311yescharacteristic, complemented
479C32#520411yescharacteristic, complemented
480Q8—31no—
481Q8—181no—
482Q8—181no—
483Q8—181no—
484Q8—181no—
485D4—31no—
486D4—61no—
487C2×C4—61no—
488C2×C4—181no—
489C2×C4—181no—
490C2×C4D3311yescharacteristic
491C8—61no—
492C8—61no—
493C8—91no—
494C8—91no—
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#6388511yescharacteristic
511C2×C3#6390611yescharacteristic
512C2×C3#6397811yescharacteristic
513D3—41no—
514C22—61no—
515C22#8896011yescharacteristic
516C4—61no—
517C4—361no—
518C4—361no—
519C4D3×#62611yescharacteristic
520C4#8895311yescharacteristic
521C3—21no—
522C3—21no—
523C3—21no—
524C3—41no—
525C3#1059098111yescharacteristic, complemented
526C3#1059099911yescharacteristic, complemented
527C3#1059168211yescharacteristic, complemented
528C2—21no—
529C2—121no—
530C2#1172296011yescharacteristic
subgroupG/H
centerC2#11722960
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
1. C2×#88812C2semidirect
5. #11722060C2semidirect
6. #11722029C2semidirect
7. C3×#63561C2semidirect
46. #63561D3semidirect
332. C33#473semidirect
477. C32#4755semidirect
478. C32#5143semidirect
479. C32#5204semidirect
525. C3#10590981semidirect
526. C3#10590999semidirect
527. C3#10591682semidirect

Maximal subgroups

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

subgroupindex
C2×#888122
#117233732
C3×#638312
C3×#638152
#117220602
#117220292
C3×#635612
#105916823
#105909993
#105909813