Finite Group Databasebeta

← #13161764

Subgroups — #13161764

Order 1728.

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

normal subgroups only

This group has 596 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, complemented
2#11723661C211yesmaximal, characteristic, complemented
3C2×#88810C211yesmaximal, characteristic, complemented
4#11723657C211yesmaximal, characteristic, complemented
5C3×#63650C211yesmaximal, characteristic, complemented
6C3×#63649C211yesmaximal, characteristic, complemented
7#11722073C211yesmaximal, characteristic, complemented
8#10590987—31nomaximal
9#10590984—31nomaximal
10#10590982—31nomaximal
11C3×#3828C2211yescharacteristic
12C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
13C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
14C33×M16C2211yescharacteristic
15C3×#3748C2211yescharacteristic
16C3×#3748C2211yescharacteristic
17C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
18C3×#3745—21no—
19C3×#3744—21no—
20#88811—21no—
21#88810—21no—
22#88810—21no—
23#88810—21no—
24#88809—21no—
25#88808—21no—
26#64117D311yescharacteristic, complemented
27C3×#908—31no—
28C3×#908—31no—
29C3×#908—61no—
30#63830—31no—
31#63825—31no—
32C2×#3714—31no—
33C2×#3714—31no—
34#63823—31no—
35C2×#3713—31no—
36#63817—31no—
37#63815—31no—
38#63814—31no—
39#63650D311yescharacteristic, complemented
40#63649D311yescharacteristic, complemented
41C3×#823—31no—
42C3×#823—31no—
43C3×#823—61no—
44C3×#822—31no—
45C3×#822—31no—
46C3×#822—61no—
47#63559—31no—
48#63558—31no—
49#63558—31no—
50C22×C3×C3⋊D3—21no—
51C22×C3×C3⋊D3—21no—
52C2×C33×C4C2311yescharacteristic
53C3×#638—21no—
54C3×C4⋊(C3⋊D3)—21no—
55C3×C4⋊(C3⋊D3)—21no—
56C3×C4⋊(C3⋊D3)—21no—
57C3×C4⋊(C3⋊D3)—21no—
58C3×C4⋊(C3⋊D3)D411yescharacteristic
59C3×C4×C3⋊D3—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#5199—91no—
67#4897—91no—
68#4749—91no—
69#3828C2×D311yescharacteristic
70C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
71C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
72C3×#235—31no—
73C3×#235—31no—
74C3×#235—61no—
75C2×C3×D12—31no—
76C2×C3×D12—31no—
77C2×C3×D12—31no—
78C2×C3×D12—31no—
79C2×C3×D12—61no—
80C2×C3×D12—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#3717—61no—
109#3716—61no—
110#3715—61no—
111#3714—61no—
112#3714—61no—
113#3714—61no—
114#3714—61no—
115#3714—61no—
116#3714—61no—
117#3713—61no—
118#3713—61no—
119#3713—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—41no—
130C2×C3×C3⋊D3—81no—
131C2×C3×C3⋊D3—81no—
132C33×C4C2×D411yescharacteristic
133C33×C4C2×D411yescharacteristic
134C3×C3⋊Dic3—21no—
135C3×AGL(1,ℤ/8)—91no—
136#949—91no—
137#948—91no—
138#932—91no—
139C2×#213—91no—
140C2×#213—91no—
141#908—31no—
142#908—31no—
143#908—61no—
144#908—91no—
145#907—91no—
146C2×D24—91no—
147#901—91no—
148C3×#108—91no—
149C3×#100—91no—
150#833—91no—
151#823—31no—
152#823—31no—
153#823—61no—
154#823—91no—
155#823—91no—
156#822—31no—
157#822—31no—
158#822—61no—
159C22×C3⋊D3—21no—
160C22×C3⋊D3—21no—
161C22×C3×D3—62no—
162C22×C3×D3—61no—
163C22×C3×D3—61no—
164C22×C3×D3—61no—
165C22×C3×D3—61no—
166C22×C3×D3—121no—
167C2×C32×C4—21no—
168C2×C32×C4—21no—
169C2×C32×C4—21no—
170C2×C32×C4—41no—
171C2×C32×C4C22×D311yescharacteristic
172C2×C32×C4C22×D311yescharacteristic
173C2×C32×C4C22×D311yescharacteristic
174#638—21no—
175C4⋊(C3⋊D3)—21no—
176C4⋊(C3⋊D3)—21no—
177C4⋊(C3⋊D3)—21no—
178C4⋊(C3⋊D3)—21no—
179C4⋊(C3⋊D3)#6711yescharacteristic
180C4×C3⋊D3—21no—
181#634#6711yescharacteristic
182C3×#67—61no—
183C3×#67—61no—
184C3×#67—121no—
185C3×D12—31no—
186C3×D12—31no—
187C3×D12—61no—
188C3×D12—62no—
189C3×D12—61no—
190C3×D12—61no—
191C3×D12—61no—
192C3×D12—61no—
193C3×D12—61no—
194C3×D12—61no—
195C3×D12—61no—
196C3×D12—61no—
197C3×D12—121no—
198C3×D12—121no—
199C3×D12—121no—
200C3×C4×D3—61no—
201C3×C4×D3—61no—
202C3×C4×D3—121no—
203C3×Dic6—31no—
204C3×Dic6—31no—
205C3×Dic6—61no—
206C32×C8—21no—
207C32×C8—22no—
208C32×C8—21no—
209C32×C8—21no—
210C32×C8—41no—
211C32×C8—41no—
212C32×C8—42no—
213C3⋊(C3⋊C8)—21no—
214C3⋊(C3⋊C8)—21no—
215C3⋊(C3⋊C8)D1211yescharacteristic
216C3⋊(C3⋊C8)D1211yescharacteristic
217C3×C3⋊C8—31no—
218C3×C3⋊C8—31no—
219C3×C3⋊C8—31no—
220C3×C3⋊C8—31no—
221C3×C3⋊C8—61no—
222C3×C3⋊C8—61no—
223C3×C3⋊C8—61no—
224C3×C3⋊C8—61no—
225C3×C3⋊C8—61no—
226C3×C3⋊C8—61no—
227C3×C3⋊C8—121no—
228C3×C3⋊C8—121no—
229#472—271noSylow
230C2×C33—21no—
231C2×C33#12111yescharacteristic
232C3×C3⋊D3—41no—
233C3×C3⋊D3—81no—
234C3×C3⋊D3—81no—
235C3×#41—91no—
236C2×C3×D4—91no—
237C2×C3×D4—91no—
238#235—31no—
239#235—31no—
240#235—61no—
241C2×D12—31no—
242C2×D12—31no—
243C2×D12—31no—
244C2×D12—31no—
245C2×D12—61no—
246C2×D12—61no—
247C3×SD16—181no—
248C3×D8—181no—
249C3×M16—21no—
250C3×M16—21no—
251C3×M16—21no—
252C3×M16—41no—
253C3×M16—91no—
254C3×M16—91no—
255C3×M16D3211yescharacteristic
256C3×M16D3211yescharacteristic
257C3×M16D3211yescharacteristic
258C2×C3×C8—91no—
259#215—181no—
260#214—181no—
261#213—181no—
262#213—181no—
263#213—181no—
264#213—181no—
265#213—181no—
266#213—181no—
267#208—31no—
268#208—31no—
269#208—31no—
270#208—31no—
271#208—61no—
272#208—61no—
273#208—181no—
274C2×C3⋊C8—31no—
275C2×C3⋊C8—31no—
276C2×C3⋊C8—61no—
277C2×C3⋊C8—181no—
278D24—61no—
279D24—61no—
280D24—62no—
281D24—181no—
282D24—181no—
283D24—181no—
284#204—61no—
285#204—61no—
286#204—62no—
287#204—181no—
288ΓL(1,𝔽25)—181no—
289ΓL(1,𝔽25)—181no—
290D3×C8—181no—
291D3×C8—181no—
292C22×C32—21no—
293C22×C32—21no—
294C22×C32—21no—
295C22×C32—41no—
296C22×C32#23511yescharacteristic
297C22×C32#23711yescharacteristic
298C22×C32#23911yescharacteristic
299C2×C3⋊D3—21no—
300C2×C3⋊D3—41no—
301C2×C3⋊D3—41no—
302C2×C3⋊D3—81no—
303C2×C3⋊D3—81no—
304C2×C3×D3—61no—
305C2×C3×D3—61no—
306C2×C3×D3—121no—
307C2×C3×D3—122no—
308C2×C3×D3—122no—
309C2×C3×D3—121no—
310C2×C3×D3—121no—
311C2×C3×D3—121no—
312C2×C3×D3—121no—
313C2×C3×D3—242no—
314C2×C3×D3—241no—
315C2×C3×D3—241no—
316C2×C3×D3—241no—
317C2×C3×D3—241no—
318C2×C3×D3—242no—
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×C4C2×D1211yescharacteristic
328C32×C4D3×D411yescharacteristic
329C32×C4D3×D411yescharacteristic
330C32×C4D3×D411yescharacteristic
331C32×C4D3×D411yescharacteristic
332C32×C4C2×#6711yescharacteristic
333C3⋊Dic3—21no—
334C3×Dic3—61no—
335C3×Dic3—61no—
336C3×Dic3—121no—
337AGL(1,ℤ/8)—91no—
338AGL(1,ℤ/8)—271no—
339C2×D8—271no—
340#131—271no—
341#108—91no—
342#100—91no—
343#100—271no—
344C33#47211yescharacteristic, complemented, Sylow
345C23×C3—181no—
346C23×C3—181no—
347C22×D3—62no—
348C22×D3—61no—
349C22×D3—61no—
350C22×D3—61no—
351C22×D3—61no—
352C22×D3—121no—
353C3×Q8—91no—
354C3×D4—91no—
355C3×D4—181no—
356C3×D4—181no—
357C3×D4—181no—
358C3×D4—181no—
359C3×D4—181no—
360C2×C3×C4—21no—
361C2×C3×C4—21no—
362C2×C3×C4—21no—
363C2×C3×C4—41no—
364C2×C3×C4—181no—
365C2×C3×C4C2×D3211yescharacteristic
366C2×C3×C4C2×D3211yescharacteristic
367C2×C3×C4C2×D3211yescharacteristic
368#67—61no—
369#67—61no—
370#67—121no—
371D12—31no—
372D12—31no—
373D12—61no—
374D12—62no—
375D12—61no—
376D12—61no—
377D12—61no—
378D12—61no—
379D12—61no—
380D12—61no—
381D12—61no—
382D12—61no—
383D12—121no—
384D12—121no—
385D12—121no—
386C4×D3—61no—
387C4×D3—61no—
388C4×D3—121no—
389Dic6—31no—
390Dic6—31no—
391Dic6—61no—
392C3×C8—22no—
393C3×C8—21no—
394C3×C8—21no—
395C3×C8—21no—
396C3×C8—41no—
397C3×C8—41no—
398C3×C8—42no—
399C3×C8—91no—
400C3×C8—91no—
401C3×C8—181no—
402C3×C8—181no—
403C3⋊C8—31no—
404C3⋊C8—31no—
405C3⋊C8—31no—
406C3⋊C8—31no—
407C3⋊C8—61no—
408C3⋊C8—61no—
409C3⋊C8—61no—
410C3⋊C8—61no—
411C3⋊C8—61no—
412C3⋊C8—61no—
413C3⋊C8—121no—
414C3⋊C8—121no—
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#88411yescharacteristic
427C2×C32#89511yescharacteristic
428C2×C32#93911yescharacteristic
429C3⋊D3—41no—
430C3⋊D3—81no—
431C3⋊D3—81no—
432C3×D3—121no—
433C3×D3—121no—
434C3×D3—241no—
435C3×D3—242no—
436C3×D3—242no—
437C3×D3—241no—
438C3×D3—241no—
439C3×D3—241no—
440C3×D3—241no—
441#41—91no—
442C2×D4—91no—
443C2×D4—91no—
444SD16—181no—
445SD16—541no—
446D8—181no—
447D8—541no—
448D8—541no—
449D8—541no—
450M16—91no—
451M16—91no—
452M16—541no—
453M16#111311yescharacteristic
454C2×C8—91no—
455C2×C8—541no—
456C22×C3—21no—
457C22×C3—21no—
458C22×C3—21no—
459C22×C3—41no—
460C22×C3—181no—
461C22×C3—361no—
462C22×C3—361no—
463C22×C3—721no—
464C22×C3—721no—
465C22×C3#379511yescharacteristic
466C22×C3#379611yescharacteristic
467C22×C3#379911yescharacteristic
468C2×D3—61no—
469C2×D3—61no—
470C2×D3—121no—
471C2×D3—122no—
472C2×D3—122no—
473C2×D3—121no—
474C2×D3—121no—
475C2×D3—121no—
476C2×D3—121no—
477C2×D3—242no—
478C2×D3—241no—
479C2×D3—241no—
480C2×D3—241no—
481C2×D3—241no—
482C2×D3—242no—
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—181no—
492C3×C4D3×D1211yescharacteristic
493C3×C4D3×D1211yescharacteristic
494C3×C4#380211yescharacteristic
495C3×C4D3×#6711yescharacteristic
496C3×C4D3×#6711yescharacteristic
497C3×C4#381111yescharacteristic
498Dic3—61no—
499Dic3—61no—
500Dic3—121no—
501C32—21no—
502C32—21no—
503C32—21no—
504C32—41no—
505C32#474911yescharacteristic, complemented
506C32#489711yescharacteristic, complemented
507C32#519911yescharacteristic, complemented
508C23—181no—
509C23—181no—
510Q8—91no—
511D4—91no—
512D4—181no—
513D4—181no—
514D4—181no—
515D4—181no—
516D4—181no—
517C2×C4—181no—
518C2×C4C2×#111311yescharacteristic
519C8—21no—
520C8—91no—
521C8—91no—
522C8—181no—
523C8—181no—
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—361no—
536C2×C3—721no—
537C2×C3—721no—
538C2×C3#6388511yescharacteristic
539C2×C3#6388611yescharacteristic
540C2×C3#6391011yescharacteristic
541D3—121no—
542D3—121no—
543D3—241no—
544D3—242no—
545D3—242no—
546D3—241no—
547D3—241no—
548D3—241no—
549D3—241no—
550C22—181no—
551C22—361no—
552C22—361no—
553C22—721no—
554C22—721no—
555C22#8904111yescharacteristic
556C4—181no—
557C4C4⋊#111311yescharacteristic
558C4#8904911yescharacteristic
559C3—21no—
560C3—21no—
561C3—21no—
562C3—41no—
563C3#1059098211yescharacteristic, complemented
564C3#1059098411yescharacteristic, complemented
565C3#1059098711yescharacteristic, complemented
566C2—21no—
567C2—361no—
568C2—721no—
569C2—721no—
570C2#1172368711yescharacteristic
subgroupG/H
centerC2#11723687
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×#1113
FittingC33×M16C22
socleC2×C33#121
hypercenterM16#1113
hypocenterC33#472
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
1. C3×#64117C2semidirect
2. #11723661C2semidirect
3. C2×#88810C2semidirect
4. #11723657C2semidirect
5. C3×#63650C2semidirect
6. C3×#63649C2semidirect
7. #11722073C2semidirect
26. #64117D3semidirect
39. #63650D3semidirect
40. #63649D3semidirect
344. C33#472semidirect
505. C32#4749semidirect
506. C32#4897semidirect
507. C32#5199semidirect
563. C3#10590982semidirect
564. C3#10590984semidirect
565. C3#10590987semidirect

Maximal subgroups

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

subgroupindex
C3×#641172
#117236612
C2×#888102
#117236572
C3×#636502
C3×#636492
#117220732
#105909873
#105909843
#105909823