Finite Group Databasebeta

← #13161766

Subgroups — #13161766

Order 1728.

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

normal subgroups only

This group has 572 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×#64117C211yesmaximal, characteristic
2#11723663C211yesmaximal, characteristic, complemented
3C2×#88811C211yesmaximal, characteristic, complemented
4#11723657C211yesmaximal, characteristic, complemented
5C3×#63651C211yesmaximal, characteristic, complemented
6C3×#63649C211yesmaximal, characteristic, complemented
7#11722073C211yesmaximal, characteristic, complemented
8#10590990—31nomaximal
9#10590986—31nomaximal
10#10590980—31nomaximal
11C3×#3828C2211yescharacteristic
12C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
13C2×C3×#634C2211yescharacteristic
14C33×M16C2211yescharacteristic
15C3×#3748C2211yescharacteristic
16C3×#3748C2211yescharacteristic
17C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
18C3×#3745—21no—
19C3×#3744—21no—
20#88812—21no—
21#88811—21no—
22#88811—21no—
23#88811—21no—
24#88809—21no—
25#88808—21no—
26#64117D311yescharacteristic
27C3×#908—31no—
28C3×#908—31no—
29C3×#908—61no—
30#63833—31no—
31C2×#3717—31no—
32#63831—31no—
33C2×#3716—31no—
34#63827—31no—
35C2×#3715—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#63559—31no—
48#63558—31no—
49#63558—31no—
50C22×C3×C3⋊D3—21no—
51C2×C33×C4C2311yescharacteristic
52C3×#638—21no—
53C2×C3×C3⋊Dic3—21no—
54C3×C4⋊(C3⋊D3)—21no—
55C3×C4⋊(C3⋊D3)—21no—
56C3×C4⋊(C3⋊D3)D411yescharacteristic
57C3×C4×C3⋊D3—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#5200—91no—
67#4898—91no—
68#4754—91no—
69#3828C2×D311yescharacteristic
70C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
71C2×#634C2×D311yescharacteristic
72C3×#235—31no—
73C3×#235—31no—
74C3×#235—61no—
75C2×C3×D12—31no—
76C2×C3×D12—31no—
77C2×C3×D12—61no—
78C2×C3×Dic6—31no—
79C2×C3×Dic6—31no—
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#3745—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×D24—61no—
103C3×D24—61no—
104C3×D24—62no—
105C3×#204—61no—
106C3×#204—61no—
107C3×#204—62no—
108#3719—61no—
109#3719—61no—
110#3718—61no—
111#3717—61no—
112#3717—61no—
113#3717—61no—
114#3716—61no—
115#3716—61no—
116#3716—61no—
117#3715—61no—
118#3715—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—
128C2×C3×C3⋊D3—41no—
129C2×C3×C3⋊D3—81no—
130C33×C4C2×D411yescharacteristic
131C33×C4C2×D411yescharacteristic
132C3×C3⋊Dic3—21no—
133C3×C3⋊Dic3—41no—
134C3×AGL(1,ℤ/8)—91no—
135#951—91no—
136#948—91no—
137#942—91no—
138C2×#215—91no—
139C2×#214—91no—
140#909—91no—
141#908—31no—
142#908—31no—
143#908—61no—
144#907—91no—
145C2×#204—91no—
146#901—91no—
147C3×#108—91no—
148C3×#101—91no—
149#833—91no—
150#824—31no—
151#824—31no—
152#824—61no—
153#823—91no—
154#823—91no—
155#822—31no—
156#822—31no—
157#822—61no—
158C22×C3⋊D3—21no—
159C22×C3×D3—61no—
160C22×C3×D3—61no—
161C22×C3×D3—121no—
162C2×C32×C4—21no—
163C2×C32×C4—21no—
164C2×C32×C4—21no—
165C2×C32×C4—41no—
166C2×C32×C4C22×D311yescharacteristic
167C2×C32×C4C22×D311yescharacteristic
168C2×C32×C4C22×D311yescharacteristic
169#638—21no—
170C2×C3⋊Dic3—21no—
171C4⋊(C3⋊D3)—21no—
172C4⋊(C3⋊D3)—21no—
173C4⋊(C3⋊D3)#6711yescharacteristic
174C4×C3⋊D3—21no—
175#634—21no—
176#634—21no—
177#634#6711yescharacteristic
178C3×#67—61no—
179C3×#67—61no—
180C3×#67—121no—
181C2×C3×Dic3—62no—
182C2×C3×Dic3—61no—
183C2×C3×Dic3—61no—
184C3×D12—31no—
185C3×D12—31no—
186C3×D12—61no—
187C3×D12—62no—
188C3×D12—61no—
189C3×D12—61no—
190C3×D12—61no—
191C3×D12—61no—
192C3×D12—121no—
193C3×C4×D3—61no—
194C3×C4×D3—61no—
195C3×C4×D3—121no—
196C3×Dic6—31no—
197C3×Dic6—31no—
198C3×Dic6—61no—
199C3×Dic6—61no—
200C3×Dic6—61no—
201C3×Dic6—61no—
202C3×Dic6—61no—
203C3×Dic6—121no—
204C3×Dic6—121no—
205C32×C8—21no—
206C32×C8—22no—
207C32×C8—21no—
208C32×C8—21no—
209C32×C8—41no—
210C32×C8—41no—
211C32×C8—42no—
212C3⋊(C3⋊C8)—21no—
213C3⋊(C3⋊C8)—21no—
214C3⋊(C3⋊C8)D1211yescharacteristic
215C3⋊(C3⋊C8)D1211yescharacteristic
216C3×C3⋊C8—31no—
217C3×C3⋊C8—31no—
218C3×C3⋊C8—31no—
219C3×C3⋊C8—31no—
220C3×C3⋊C8—61no—
221C3×C3⋊C8—61no—
222C3×C3⋊C8—61no—
223C3×C3⋊C8—61no—
224C3×C3⋊C8—61no—
225C3×C3⋊C8—61no—
226C3×C3⋊C8—121no—
227C3×C3⋊C8—121no—
228#471—271noSylow
229C2×C33—21no—
230C2×C33#12111yescharacteristic
231C3×C3⋊D3—41no—
232C3×C3⋊D3—81no—
233C3×#41—91no—
234C2×C3×Q8—91no—
235C2×C3×D4—91no—
236#235—31no—
237#235—31no—
238#235—61no—
239C2×D12—31no—
240C2×D12—31no—
241C2×D12—61no—
242C2×Dic6—31no—
243C2×Dic6—31no—
244C2×Dic6—61no—
245C3×SD16—181no—
246C3×D8—181no—
247C3×M16—21no—
248C3×M16—21no—
249C3×M16—21no—
250C3×M16—41no—
251C3×M16—91no—
252C3×M16—91no—
253C3×M16D3211yescharacteristic
254C3×M16D3211yescharacteristic
255C3×M16D3211yescharacteristic
256C2×C3×C8—91no—
257#216—181no—
258#216—181no—
259#215—181no—
260#215—181no—
261#215—181no—
262#214—181no—
263#214—181no—
264#214—181no—
265#208—31no—
266#208—31no—
267#208—31no—
268#208—31no—
269#208—61no—
270#208—61no—
271#208—181no—
272C2×C3⋊C8—31no—
273C2×C3⋊C8—31no—
274C2×C3⋊C8—61no—
275C2×C3⋊C8—181no—
276Dic12—181no—
277D24—61no—
278D24—61no—
279D24—62no—
280#204—61no—
281#204—61no—
282#204—62no—
283#204—181no—
284#204—181no—
285#204—181no—
286ΓL(1,𝔽25)—181no—
287ΓL(1,𝔽25)—181no—
288D3×C8—181no—
289D3×C8—181no—
290C22×C32—21no—
291C22×C32—21no—
292C22×C32—21no—
293C22×C32—41no—
294C22×C32#23511yescharacteristic
295C22×C32#23711yescharacteristic
296C22×C32#23911yescharacteristic
297C2×C3⋊D3—21no—
298C2×C3⋊D3—41no—
299C2×C3⋊D3—81no—
300C2×C3×D3—61no—
301C2×C3×D3—61no—
302C2×C3×D3—121no—
303C2×C3×D3—122no—
304C2×C3×D3—121no—
305C2×C3×D3—121no—
306C2×C3×D3—242no—
307C2×C3×D3—241no—
308C2×C3×D3—241no—
309C32×C4—21no—
310C32×C4—21no—
311C32×C4—21no—
312C32×C4—21no—
313C32×C4—21no—
314C32×C4—21no—
315C32×C4—41no—
316C32×C4—41no—
317C32×C4C2×D1211yescharacteristic
318C32×C4D3×D411yescharacteristic
319C32×C4D3×D411yescharacteristic
320C32×C4D3×D411yescharacteristic
321C32×C4D3×D411yescharacteristic
322C32×C4C2×#6711yescharacteristic
323C3⋊Dic3—21no—
324C3⋊Dic3—41no—
325C3×Dic3—61no—
326C3×Dic3—61no—
327C3×Dic3—121no—
328C3×Dic3—122no—
329C3×Dic3—121no—
330C3×Dic3—121no—
331#137—271no—
332AGL(1,ℤ/8)—91no—
333C2×SD16—271no—
334#131—271no—
335#108—91no—
336#101—91no—
337#100—271no—
338C33#47111yescharacteristic, complemented, Sylow
339C23×C3—181no—
340C22×D3—61no—
341C22×D3—61no—
342C22×D3—121no—
343C3×Q8—91no—
344C3×Q8—181no—
345C3×Q8—181no—
346C3×D4—91no—
347C3×D4—181no—
348C3×D4—181no—
349C3×D4—181no—
350C2×C3×C4—21no—
351C2×C3×C4—21no—
352C2×C3×C4—21no—
353C2×C3×C4—41no—
354C2×C3×C4—181no—
355C2×C3×C4—181no—
356C2×C3×C4C2×D3211yescharacteristic
357C2×C3×C4C2×D3211yescharacteristic
358C2×C3×C4C2×D3211yescharacteristic
359#67—61no—
360#67—61no—
361#67—121no—
362C2×Dic3—62no—
363C2×Dic3—61no—
364C2×Dic3—61no—
365D12—31no—
366D12—31no—
367D12—61no—
368D12—62no—
369D12—61no—
370D12—61no—
371D12—61no—
372D12—61no—
373D12—121no—
374C4×D3—61no—
375C4×D3—61no—
376C4×D3—121no—
377Dic6—31no—
378Dic6—31no—
379Dic6—61no—
380Dic6—61no—
381Dic6—61no—
382Dic6—61no—
383Dic6—61no—
384Dic6—121no—
385Dic6—121no—
386C3×C8—22no—
387C3×C8—21no—
388C3×C8—21no—
389C3×C8—21no—
390C3×C8—41no—
391C3×C8—41no—
392C3×C8—42no—
393C3×C8—91no—
394C3×C8—91no—
395C3×C8—181no—
396C3×C8—181no—
397C3⋊C8—31no—
398C3⋊C8—31no—
399C3⋊C8—31no—
400C3⋊C8—31no—
401C3⋊C8—61no—
402C3⋊C8—61no—
403C3⋊C8—61no—
404C3⋊C8—61no—
405C3⋊C8—61no—
406C3⋊C8—61no—
407C3⋊C8—121no—
408C3⋊C8—121no—
409C2×C32—21no—
410C2×C32—21no—
411C2×C32—21no—
412C2×C32—21no—
413C2×C32—21no—
414C2×C32—21no—
415C2×C32—41no—
416C2×C32—41no—
417C2×C32—41no—
418C2×C32—41no—
419C2×C32—81no—
420C2×C32#88411yescharacteristic
421C2×C32#89511yescharacteristic
422C2×C32#93911yescharacteristic
423C3⋊D3—41no—
424C3⋊D3—81no—
425C3×D3—121no—
426C3×D3—121no—
427C3×D3—241no—
428C3×D3—242no—
429C3×D3—241no—
430C3×D3—241no—
431#41—91no—
432C2×Q8—91no—
433C2×D4—91no—
434Q16—541no—
435SD16—181no—
436SD16—541no—
437SD16—541no—
438SD16—541no—
439D8—181no—
440M16—91no—
441M16—91no—
442M16—541no—
443M16#111311yescharacteristic
444C2×C8—91no—
445C2×C8—541no—
446C22×C3—21no—
447C22×C3—21no—
448C22×C3—21no—
449C22×C3—41no—
450C22×C3—181no—
451C22×C3—361no—
452C22×C3—721no—
453C22×C3#379511yescharacteristic
454C22×C3#379611yescharacteristic
455C22×C3#379911yescharacteristic
456C2×D3—61no—
457C2×D3—61no—
458C2×D3—121no—
459C2×D3—122no—
460C2×D3—121no—
461C2×D3—121no—
462C2×D3—242no—
463C2×D3—241no—
464C2×D3—241no—
465C3×C4—21no—
466C3×C4—21no—
467C3×C4—21no—
468C3×C4—21no—
469C3×C4—21no—
470C3×C4—21no—
471C3×C4—41no—
472C3×C4—41no—
473C3×C4—181no—
474C3×C4—361no—
475C3×C4D3×D1211yescharacteristic
476C3×C4D3×D1211yescharacteristic
477C3×C4#380211yescharacteristic
478C3×C4D3×#6711yescharacteristic
479C3×C4D3×#6711yescharacteristic
480C3×C4#381111yescharacteristic
481Dic3—61no—
482Dic3—61no—
483Dic3—121no—
484Dic3—122no—
485Dic3—121no—
486Dic3—121no—
487C32—21no—
488C32—21no—
489C32—21no—
490C32—41no—
491C32#475411yescharacteristic, complemented
492C32#489811yescharacteristic, complemented
493C32#520011yescharacteristic, complemented
494C23—181no—
495Q8—91no—
496Q8—181no—
497Q8—181no—
498D4—91no—
499D4—181no—
500D4—181no—
501D4—181no—
502C2×C4—181no—
503C2×C4—181no—
504C2×C4C2×#111311yescharacteristic
505C8—21no—
506C8—91no—
507C8—91no—
508C8—181no—
509C8—181no—
510C2×C3—21no—
511C2×C3—21no—
512C2×C3—21no—
513C2×C3—21no—
514C2×C3—21no—
515C2×C3—21no—
516C2×C3—41no—
517C2×C3—41no—
518C2×C3—41no—
519C2×C3—41no—
520C2×C3—81no—
521C2×C3—361no—
522C2×C3—721no—
523C2×C3#6388511yescharacteristic
524C2×C3#6388611yescharacteristic
525C2×C3#6391011yescharacteristic
526D3—121no—
527D3—121no—
528D3—241no—
529D3—242no—
530D3—241no—
531D3—241no—
532C22—181no—
533C22—361no—
534C22—721no—
535C22#8904111yescharacteristic
536C4—181no—
537C4—361no—
538C4C4⋊#111311yescharacteristic
539C4#8904911yescharacteristic
540C3—21no—
541C3—21no—
542C3—21no—
543C3—41no—
544C3#1059098011yescharacteristic, complemented
545C3#1059098611yescharacteristic, complemented
546C3#1059099011yescharacteristic, complemented
547C2—21no—
548C2—361no—
549C2—721no—
550C2#1172368711yescharacteristic
subgroupG/H
centerC2#11723687
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×#1113
FittingC33×M16C22
socleC2×C33#121
prime radicalC3×#64117C2
hypercenterM16#1113
hypocenterC33#471
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. #11723663C2semidirect
3. C2×#88811C2semidirect
4. #11723657C2semidirect
5. C3×#63651C2semidirect
6. C3×#63649C2semidirect
7. #11722073C2semidirect
39. #63651D3semidirect
40. #63649D3semidirect
338. C33#471semidirect
491. C32#4754semidirect
492. C32#4898semidirect
493. C32#5200semidirect
544. C3#10590980semidirect
545. C3#10590986semidirect
546. C3#10590990semidirect

Maximal subgroups

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

subgroupindex
C3×#641172
#117236632
C2×#888112
#117236572
C3×#636512
C3×#636492
#117220732
#105909903
#105909863
#105909803