Finite Group Databasebeta

← #13161767

Subgroups — #13161767

Order 1728.

548 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 548 conjugacy classes of subgroups — too many to draw. Only its 87 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
1C3×#64118C211yesmaximal, characteristic
2C2×#88812C211yesmaximal, characteristic, complemented
3#11723663C211yesmaximal, characteristic
4#11723657C211yesmaximal, characteristic
5C3×#63651C211yesmaximal, characteristic, complemented
6C3×#63649C211yesmaximal, characteristic, complemented
7#11722074C211yesmaximal, characteristic, complemented
8#10590992—31nomaximal
9#10590983—31nomaximal
10#10590981—31nomaximal
11C3×#3828C2211yescharacteristic
12C2×C3×#634C2211yescharacteristic
13C2×C3×#634C2211yescharacteristic
14C33×M16C2211yescharacteristic
15C3×#3748C2211yescharacteristic
16C3×#3748C2211yescharacteristic
17C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
18C3×#3746—21no—
19C3×#3744—21no—
20#88812—21no—
21#88812—21no—
22#88812—21no—
23#88811—21no—
24#88809—21no—
25#88808—21no—
26#64118D311yescharacteristic
27C3×#909—31no—
28C3×#909—31no—
29C3×#909—61no—
30C2×#3719—31no—
31C2×#3719—31no—
32C2×#3718—31no—
33#63833—31no—
34#63831—31no—
35#63827—31no—
36#63817—31no—
37#63815—31no—
38#63814—31no—
39#63651D311yescharacteristic, complemented
40#63649D311yescharacteristic, complemented
41C3×#824—31no—
42C3×#824—31no—
43C3×#824—61no—
44C3×#822—31no—
45C3×#822—31no—
46C3×#822—61no—
47#63561—31no—
48#63560—31no—
49#63560—31no—
50C2×C33×C4C2311yescharacteristic
51C3×#638—21no—
52C2×C3×C3⋊Dic3—21no—
53C2×C3×C3⋊Dic3—21no—
54C3×C4⋊(C3⋊D3)D411yescharacteristic
55C3×C4×C3⋊D3—21no—
56C3×#634—21no—
57C3×#634—21no—
58C3×#634—21no—
59C3×#634—21no—
60C3×#634D411yescharacteristic
61C33×C8—21no—
62C3×C3⋊(C3⋊C8)—21no—
63C3×C3⋊(C3⋊C8)—21no—
64C3×C3⋊(C3⋊C8)D411yescharacteristic
65C3×C3⋊(C3⋊C8)D411yescharacteristic
66#5204—91no—
67#4896—91no—
68#4755—91no—
69#3828C2×D311yescharacteristic
70C2×#634C2×D311yescharacteristic
71C2×#634C2×D311yescharacteristic
72C3×#235—31no—
73C3×#235—31no—
74C3×#235—61no—
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—21no—
82C32×M16—21no—
83C32×M16—21no—
84C32×M16—41no—
85C32×M16C2×D311yescharacteristic
86C32×M16C2×D311yescharacteristic
87C32×M16C2×D311yescharacteristic
88#3748C2×D311yescharacteristic
89#3748C2×D311yescharacteristic
90C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
91#3746—21no—
92#3744—21no—
93C3×#208—31no—
94C3×#208—31no—
95C3×#208—31no—
96C3×#208—31no—
97C3×#208—61no—
98C3×#208—61no—
99C2×C3×C3⋊C8—31no—
100C2×C3×C3⋊C8—31no—
101C2×C3×C3⋊C8—61no—
102C3×Dic12—61no—
103C3×Dic12—61no—
104C3×Dic12—62no—
105C3×#204—61no—
106C3×#204—61no—
107C3×#204—62no—
108#3719—61no—
109#3719—61no—
110#3719—61no—
111#3719—61no—
112#3719—61no—
113#3719—61no—
114#3718—61no—
115#3718—61no—
116#3718—61no—
117#3717—61no—
118#3716—61no—
119#3715—61no—
120#3712—61no—
121#3711—61no—
122#3711—61no—
123#3710—61no—
124D3×C3⋊C8—61no—
125D3×C3⋊C8—61no—
126C22×C33#4111yescharacteristic
127C2×C3×C3⋊D3—21no—
128C33×C4C2×D411yescharacteristic
129C33×C4C2×D411yescharacteristic
130C3×C3⋊Dic3—21no—
131C3×C3⋊Dic3—41no—
132C3×C3⋊Dic3—41no—
133C3×#137—91no—
134#951—91no—
135#948—91no—
136C2×#216—91no—
137C2×#216—91no—
138#942—91no—
139#909—31no—
140#909—31no—
141#909—61no—
142#909—91no—
143#907—91no—
144C2×Dic12—91no—
145#901—91no—
146C3×#108—91no—
147C3×#101—91no—
148#836—91no—
149#824—31no—
150#824—31no—
151#824—61no—
152#824—91no—
153#824—91no—
154#822—31no—
155#822—31no—
156#822—61no—
157C2×C32×C4—21no—
158C2×C32×C4—21no—
159C2×C32×C4—21no—
160C2×C32×C4—41no—
161C2×C32×C4C22×D311yescharacteristic
162C2×C32×C4C22×D311yescharacteristic
163C2×C32×C4C22×D311yescharacteristic
164#638—21no—
165C2×C3⋊Dic3—21no—
166C2×C3⋊Dic3—21no—
167C4⋊(C3⋊D3)#6711yescharacteristic
168C4×C3⋊D3—21no—
169#634—21no—
170#634—21no—
171#634—21no—
172#634—21no—
173#634#6711yescharacteristic
174C3×#67—61no—
175C3×#67—61no—
176C3×#67—121no—
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—31no—
184C3×D12—31no—
185C3×D12—61no—
186C3×C4×D3—61no—
187C3×C4×D3—61no—
188C3×C4×D3—121no—
189C3×Dic6—31no—
190C3×Dic6—31no—
191C3×Dic6—61no—
192C3×Dic6—62no—
193C3×Dic6—61no—
194C3×Dic6—61no—
195C3×Dic6—61no—
196C3×Dic6—61no—
197C3×Dic6—61no—
198C3×Dic6—61no—
199C3×Dic6—61no—
200C3×Dic6—61no—
201C3×Dic6—121no—
202C3×Dic6—121no—
203C3×Dic6—121no—
204C32×C8—21no—
205C32×C8—22no—
206C32×C8—21no—
207C32×C8—21no—
208C32×C8—41no—
209C32×C8—41no—
210C32×C8—42no—
211C3⋊(C3⋊C8)—21no—
212C3⋊(C3⋊C8)—21no—
213C3⋊(C3⋊C8)D1211yescharacteristic
214C3⋊(C3⋊C8)D1211yescharacteristic
215C3×C3⋊C8—31no—
216C3×C3⋊C8—31no—
217C3×C3⋊C8—31no—
218C3×C3⋊C8—31no—
219C3×C3⋊C8—61no—
220C3×C3⋊C8—61no—
221C3×C3⋊C8—61no—
222C3×C3⋊C8—61no—
223C3×C3⋊C8—61no—
224C3×C3⋊C8—61no—
225C3×C3⋊C8—121no—
226C3×C3⋊C8—121no—
227#473—271noSylow
228C2×C33—21no—
229C2×C33#12111yescharacteristic
230C3×C3⋊D3—41no—
231C3×#41—91no—
232C2×C3×Q8—91no—
233C2×C3×Q8—91no—
234#235—31no—
235#235—31no—
236#235—61no—
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—21no—
246C3×M16—21no—
247C3×M16—21no—
248C3×M16—41no—
249C3×M16—91no—
250C3×M16—91no—
251C3×M16D3211yescharacteristic
252C3×M16D3211yescharacteristic
253C3×M16D3211yescharacteristic
254C2×C3×C8—91no—
255#216—181no—
256#216—181no—
257#216—181no—
258#216—181no—
259#216—181no—
260#216—181no—
261#215—181no—
262#214—181no—
263#208—31no—
264#208—31no—
265#208—31no—
266#208—31no—
267#208—61no—
268#208—61no—
269#208—181no—
270C2×C3⋊C8—31no—
271C2×C3⋊C8—31no—
272C2×C3⋊C8—61no—
273C2×C3⋊C8—181no—
274Dic12—61no—
275Dic12—61no—
276Dic12—62no—
277Dic12—181no—
278Dic12—181no—
279Dic12—181no—
280#204—61no—
281#204—61no—
282#204—62no—
283#204—181no—
284ΓL(1,𝔽25)—181no—
285ΓL(1,𝔽25)—181no—
286D3×C8—181no—
287D3×C8—181no—
288C22×C32—21no—
289C22×C32—21no—
290C22×C32—21no—
291C22×C32—41no—
292C22×C32#23511yescharacteristic
293C22×C32#23711yescharacteristic
294C22×C32#23911yescharacteristic
295C2×C3⋊D3—21no—
296C2×C3×D3—61no—
297C2×C3×D3—61no—
298C2×C3×D3—121no—
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×C4C2×D1211yescharacteristic
308C32×C4D3×D411yescharacteristic
309C32×C4D3×D411yescharacteristic
310C32×C4D3×D411yescharacteristic
311C32×C4D3×D411yescharacteristic
312C32×C4C2×#6711yescharacteristic
313C3⋊Dic3—21no—
314C3⋊Dic3—41no—
315C3⋊Dic3—41no—
316C3×Dic3—61no—
317C3×Dic3—61no—
318C3×Dic3—121no—
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—271no—
329#108—91no—
330#101—91no—
331#101—271no—
332C33#47311yescharacteristic, complemented, Sylow
333C3×Q8—91no—
334C3×Q8—181no—
335C3×Q8—181no—
336C3×Q8—181no—
337C3×Q8—181no—
338C3×D4—91no—
339C3×D4—181no—
340C2×C3×C4—21no—
341C2×C3×C4—21no—
342C2×C3×C4—21no—
343C2×C3×C4—41no—
344C2×C3×C4—181no—
345C2×C3×C4—181no—
346C2×C3×C4—181no—
347C2×C3×C4C2×D3211yescharacteristic
348C2×C3×C4C2×D3211yescharacteristic
349C2×C3×C4C2×D3211yescharacteristic
350#67—61no—
351#67—61no—
352#67—121no—
353C2×Dic3—62no—
354C2×Dic3—61no—
355C2×Dic3—61no—
356C2×Dic3—61no—
357C2×Dic3—61no—
358C2×Dic3—121no—
359D12—31no—
360D12—31no—
361D12—61no—
362C4×D3—61no—
363C4×D3—61no—
364C4×D3—121no—
365Dic6—31no—
366Dic6—31no—
367Dic6—61no—
368Dic6—62no—
369Dic6—61no—
370Dic6—61no—
371Dic6—61no—
372Dic6—61no—
373Dic6—61no—
374Dic6—61no—
375Dic6—61no—
376Dic6—61no—
377Dic6—121no—
378Dic6—121no—
379Dic6—121no—
380C3×C8—22no—
381C3×C8—21no—
382C3×C8—21no—
383C3×C8—21no—
384C3×C8—41no—
385C3×C8—41no—
386C3×C8—42no—
387C3×C8—91no—
388C3×C8—91no—
389C3×C8—181no—
390C3×C8—181no—
391C3⋊C8—31no—
392C3⋊C8—31no—
393C3⋊C8—31no—
394C3⋊C8—31no—
395C3⋊C8—61no—
396C3⋊C8—61no—
397C3⋊C8—61no—
398C3⋊C8—61no—
399C3⋊C8—61no—
400C3⋊C8—61no—
401C3⋊C8—121no—
402C3⋊C8—121no—
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#88411yescharacteristic
415C2×C32#89511yescharacteristic
416C2×C32#93911yescharacteristic
417C3⋊D3—41no—
418C3×D3—121no—
419C3×D3—121no—
420C3×D3—241no—
421#41—91no—
422C2×Q8—91no—
423C2×Q8—91no—
424Q16—181no—
425Q16—541no—
426Q16—541no—
427Q16—541no—
428SD16—181no—
429SD16—541no—
430M16—91no—
431M16—91no—
432M16—541no—
433M16#111311yescharacteristic
434C2×C8—91no—
435C2×C8—541no—
436C22×C3—21no—
437C22×C3—21no—
438C22×C3—21no—
439C22×C3—41no—
440C22×C3—181no—
441C22×C3#379511yescharacteristic
442C22×C3#379611yescharacteristic
443C22×C3#379911yescharacteristic
444C2×D3—61no—
445C2×D3—61no—
446C2×D3—121no—
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—181no—
456C3×C4—361no—
457C3×C4—361no—
458C3×C4D3×D1211yescharacteristic
459C3×C4D3×D1211yescharacteristic
460C3×C4#380211yescharacteristic
461C3×C4D3×#6711yescharacteristic
462C3×C4D3×#6711yescharacteristic
463C3×C4#381111yescharacteristic
464Dic3—61no—
465Dic3—61no—
466Dic3—121no—
467Dic3—122no—
468Dic3—122no—
469Dic3—121no—
470Dic3—121no—
471Dic3—121no—
472Dic3—121no—
473C32—21no—
474C32—21no—
475C32—21no—
476C32—41no—
477C32#475511yescharacteristic, complemented
478C32#489611yescharacteristic, complemented
479C32#520411yescharacteristic, complemented
480Q8—91no—
481Q8—181no—
482Q8—181no—
483Q8—181no—
484Q8—181no—
485D4—91no—
486D4—181no—
487C2×C4—181no—
488C2×C4—181no—
489C2×C4—181no—
490C2×C4C2×#111311yescharacteristic
491C8—21no—
492C8—91no—
493C8—91no—
494C8—181no—
495C8—181no—
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—361no—
508C2×C3#6388511yescharacteristic
509C2×C3#6388611yescharacteristic
510C2×C3#6391011yescharacteristic
511D3—121no—
512D3—121no—
513D3—241no—
514C22—181no—
515C22#8904111yescharacteristic
516C4—181no—
517C4—361no—
518C4—361no—
519C4C4⋊#111311yescharacteristic
520C4#8904911yescharacteristic
521C3—21no—
522C3—21no—
523C3—21no—
524C3—41no—
525C3#1059098111yescharacteristic, complemented
526C3#1059098311yescharacteristic, complemented
527C3#1059099211yescharacteristic, complemented
528C2—21no—
529C2—361no—
530C2#1172368711yescharacteristic
subgroupG/H
centerC2#11723687
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×#1113
FittingC33×M16C22
socleC2×C33#121
prime radicalC3×#3828C22
hypercenterM16#1113
hypocenterC33#473
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C2×C4 ◁ M16

Complements

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

NH ≅ G/Ntype
2. C2×#88812C2semidirect
5. C3×#63651C2semidirect
6. C3×#63649C2semidirect
7. #11722074C2semidirect
39. #63651D3semidirect
40. #63649D3semidirect
332. C33#473semidirect
477. C32#4755semidirect
478. C32#4896semidirect
479. C32#5204semidirect
525. C3#10590981semidirect
526. C3#10590983semidirect
527. C3#10590992semidirect

Maximal subgroups

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

subgroupindex
C3×#641182
C2×#888122
#117236632
#117236572
C3×#636512
C3×#636492
#117220742
#105909923
#105909833
#105909813