Finite Group Databasebeta

← #13156232

Subgroups — #13156232

Order 1728.

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

normal subgroups only

This group has 572 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×#88811C211yesmaximal, characteristic, complemented
2#11723373C211yesmaximal, characteristic, complemented
3C3×#63825C211yesmaximal, characteristic
4C3×#63815C211yesmaximal, characteristic, complemented
5#11722060C211yesmaximal, characteristic, complemented
6#11722026C211yesmaximal, characteristic, complemented
7C3×#63561C211yesmaximal, characteristic, complemented
8#10591679—31nomaximal
9#10590998—31nomaximal
10#10590980—31nomaximal
11C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
12C2×C3×#634C2211yescharacteristic
13C32×#235C2211yescharacteristic
14#88855C2211yescharacteristic
15C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
16C32×#208C2211yescharacteristic
17C32×#208C2211yescharacteristic
18#88811—21no—
19#88811—21no—
20#88798—21no—
21#88792—21no—
22C3×#3717—21no—
23C3×#3714—21no—
24C3×#3711—21no—
25C3×D3×C3⋊C8—21no—
26#64160—31no—
27C3×#949—31no—
28C3×#948—31no—
29C3×#908—31no—
30C3×#907—31no—
31C2×#3717—31no—
32C2×#3716—31no—
33#63827—31no—
34#63827—31no—
35#63827—61no—
36C2×#3715—31no—
37#63825D311yescharacteristic
38#63815D311yescharacteristic, complemented
39#63650—31no—
40#63649—31no—
41C3×#824—31no—
42C3×#824—31no—
43#63577—31no—
44#63577—31no—
45#63577—61no—
46#63561D311yescharacteristic, complemented
47#63558—31no—
48#63558—31no—
49#63558—61no—
50C22×C3×C3⋊D3—21no—
51C2×C33×C4C2311yescharacteristic
52C2×C3×C3⋊Dic3—21no—
53C3×C4⋊(C3⋊D3)—21no—
54C3×C4⋊(C3⋊D3)—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#5200—91no—
67#5141—91no—
68#4754—91no—
69C32×#41—31no—
70C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
71C2×#634C2×D311yescharacteristic
72C3×#235—21no—
73C3×#235C2×D311yescharacteristic
74C3×#235C2×D311yescharacteristic
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—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×#215—61no—
94C3×#213—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×D24—61no—
107C3×#204—61no—
108C3×ΓL(1,𝔽25)—61no—
109C3×D3×C8—61no—
110#3718—62no—
111#3718—61no—
112#3718—61no—
113#3717—21no—
114#3717—61no—
115#3717—61no—
116#3716—61no—
117#3716—61no—
118#3715—62no—
119#3715—61no—
120#3715—61no—
121#3715—61no—
122#3715—61no—
123#3714—21no—
124#3711—21no—
125D3×C3⋊C8—21no—
126C22×C33#4111yescharacteristic
127C2×C3×C3⋊D3—41no—
128C2×C3×C3⋊D3—81no—
129C2×C32×D3—21no—
130C33×C4C2×D411yescharacteristic
131C33×C4C2×D411yescharacteristic
132C3×C3⋊Dic3—41no—
133C32×Dic3—21no—
134C3×AGL(1,ℤ/8)—91no—
135C3×#131—91no—
136#951—91no—
137#951—91no—
138#951—181no—
139#949—31no—
140#948—31no—
141#942—91no—
142C2×#215—91no—
143C2×#214—91no—
144#908—31no—
145#907—31no—
146C2×#204—91no—
147C3×#101—91no—
148#833—91no—
149#824—31no—
150#824—31no—
151#823—91no—
152#823—91no—
153#823—181no—
154#822—91no—
155#822—91no—
156#822—181no—
157#819—91no—
158C22×C3⋊D3—21no—
159C22×C3×D3—62no—
160C22×C3×D3—61no—
161C22×C3×D3—61no—
162C32×Q8—31no—
163C32×D4—31no—
164C32×D4—61no—
165C2×C32×C4—21no—
166C2×C32×C4—21no—
167C2×C32×C4—21no—
168C2×C32×C4—41no—
169C2×C32×C4—61no—
170C2×C32×C4C22×D311yescharacteristic
171C2×C32×C4C22×D311yescharacteristic
172C2×C32×C4C22×D311yescharacteristic
173C2×C3⋊Dic3—21no—
174C4⋊(C3⋊D3)—21no—
175C4⋊(C3⋊D3)—21no—
176#634—21no—
177#634—21no—
178C3×#67—21no—
179C3×#67—21no—
180C3×#67—41no—
181C2×C3×Dic3—61no—
182C2×C3×Dic3—61no—
183C2×C3×Dic3—121no—
184C3×D12—21no—
185C3×D12—61no—
186C3×D12—61no—
187C3×D12—61no—
188C3×D12—61no—
189C3×D12—121no—
190C3×D12—121no—
191C3×D12D1211yescharacteristic
192C3×D12#6711yescharacteristic
193C3×C4×D3—21no—
194C3×C4×D3—21no—
195C3×C4×D3—41no—
196C3×Dic6—21no—
197C3×Dic6—62no—
198C3×Dic6—61no—
199C3×Dic6—61no—
200C3×Dic6—61no—
201C3×Dic6—61no—
202C3×Dic6—121no—
203C3×Dic6D1211yescharacteristic
204C3×Dic6#6711yescharacteristic
205C32×C8—61no—
206C32×C8—61no—
207C3⋊(C3⋊C8)—61no—
208C3⋊(C3⋊C8)—62no—
209C3⋊(C3⋊C8)—61no—
210C3⋊(C3⋊C8)—61no—
211C3⋊(C3⋊C8)—121no—
212C3⋊(C3⋊C8)—121no—
213C3⋊(C3⋊C8)—122no—
214C3⋊(C3⋊C8)#6711yescharacteristic
215C3⋊(C3⋊C8)#6711yescharacteristic
216C3×C3⋊C8—21no—
217C3×C3⋊C8—21no—
218C3×C3⋊C8—21no—
219C3×C3⋊C8—21no—
220C3×C3⋊C8—31no—
221C3×C3⋊C8—31no—
222C3×C3⋊C8—31no—
223C3×C3⋊C8—31no—
224C3×C3⋊C8—41no—
225C3×C3⋊C8—41no—
226C3×C3⋊C8—61no—
227C3×C3⋊C8—61no—
228#471—271noSylow
229C2×C33—21no—
230C2×C33#12111yescharacteristic
231C3×C3⋊D3—81no—
232C32×D3—41no—
233C3×#41—31no—
234C3×#41—31no—
235C3×#41—61no—
236C2×C3×Q8—91no—
237C2×C3×D4—91no—
238#235D3211yescharacteristic
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—31no—
248C3×M16—31no—
249C3×M16—31no—
250C3×M16—31no—
251C3×M16—61no—
252C3×M16—61no—
253C3×M16—181no—
254C2×C3×C8—91no—
255C2×C3×C8—181no—
256#216—181no—
257#216—181no—
258#216—182no—
259#216—181no—
260#215—61no—
261#215—181no—
262#215—181no—
263#215—181no—
264#214—181no—
265#214—181no—
266#214—181no—
267#214—181no—
268#214—182no—
269#213—61no—
270#208—61no—
271#208—91no—
272#208—91no—
273#208—91no—
274#208—181no—
275#208—181no—
276#208—181no—
277#208—361no—
278#208D3211yescharacteristic
279#208D3211yescharacteristic
280C2×C3⋊C8—31no—
281C2×C3⋊C8—31no—
282C2×C3⋊C8—61no—
283C2×C3⋊C8—61no—
284D24—61no—
285#204—61no—
286#204—181no—
287#204—181no—
288ΓL(1,𝔽25)—61no—
289D3×C8—61no—
290C22×C32—21no—
291C22×C32—21no—
292C22×C32—21no—
293C22×C32—41no—
294C22×C32—61no—
295C22×C32#23511yescharacteristic
296C22×C32#23511yescharacteristic
297C22×C32#23711yescharacteristic
298C2×C3⋊D3—41no—
299C2×C3⋊D3—81no—
300C2×C3×D3—21no—
301C2×C3×D3—21no—
302C2×C3×D3—41no—
303C2×C3×D3—122no—
304C2×C3×D3—121no—
305C2×C3×D3—121no—
306C2×C3×D3—241no—
307C2×C3×D3—241no—
308C2×C3×D3—242no—
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×C4—61no—
318C32×C4C2×D1211yescharacteristic
319C32×C4D3×D411yescharacteristic
320C32×C4D3×D411yescharacteristic
321C32×C4D3×D411yescharacteristic
322C32×C4C2×#6711yescharacteristic
323C32×C4C2×#6711yescharacteristic
324C3⋊Dic3—41no—
325C3×Dic3—21no—
326C3×Dic3—21no—
327C3×Dic3—41no—
328C3×Dic3—122no—
329C3×Dic3—121no—
330C3×Dic3—121no—
331#137—271no—
332AGL(1,ℤ/8)—91no—
333C2×SD16—271no—
334#131—91no—
335#108—271no—
336#101—91no—
337#100—271no—
338C33#47111yescharacteristic, complemented, Sylow
339C23×C3—181no—
340C22×D3—62no—
341C22×D3—61no—
342C22×D3—61no—
343C3×Q8—31no—
344C3×Q8—31no—
345C3×Q8—61no—
346C3×Q8—181no—
347C3×Q8—181no—
348C3×D4—31no—
349C3×D4—31no—
350C3×D4—61no—
351C3×D4—61no—
352C3×D4—61no—
353C3×D4—121no—
354C3×D4—181no—
355C3×D4—181no—
356C2×C3×C4—21no—
357C2×C3×C4—21no—
358C2×C3×C4—21no—
359C2×C3×C4—41no—
360C2×C3×C4—61no—
361C2×C3×C4—61no—
362C2×C3×C4—121no—
363C2×C3×C4—181no—
364C2×C3×C4C2×D3211yescharacteristic
365C2×C3×C4C2×D3211yescharacteristic
366C2×C3×C4C2×D3211yescharacteristic
367#67—21no—
368C2×Dic3—61no—
369C2×Dic3—61no—
370C2×Dic3—121no—
371D12—61no—
372D12—61no—
373D12—61no—
374D12—61no—
375D12—121no—
376D12—121no—
377D12#62611yescharacteristic
378C4×D3—21no—
379Dic6—62no—
380Dic6—61no—
381Dic6—61no—
382Dic6—61no—
383Dic6—61no—
384Dic6—121no—
385Dic6#62611yescharacteristic
386C3×C8—61no—
387C3×C8—61no—
388C3×C8—61no—
389C3×C8—61no—
390C3×C8—91no—
391C3×C8—91no—
392C3×C8—121no—
393C3×C8—121no—
394C3⋊C8—21no—
395C3⋊C8—21no—
396C3⋊C8—31no—
397C3⋊C8—31no—
398C3⋊C8—31no—
399C3⋊C8—31no—
400C3⋊C8—61no—
401C3⋊C8—61no—
402C3⋊C8—182no—
403C3⋊C8—181no—
404C3⋊C8—181no—
405C3⋊C8—181no—
406C3⋊C8—361no—
407C3⋊C8—361no—
408C3⋊C8—362no—
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—121no—
421C2×C32#88411yescharacteristic
422C2×C32#93011yescharacteristic
423C2×C32#93911yescharacteristic
424C3⋊D3—81no—
425C3×D3—41no—
426C3×D3—41no—
427C3×D3—81no—
428C3×D3—242no—
429C3×D3—241no—
430C3×D3—241no—
431#41—31no—
432C2×Q8—91no—
433C2×D4—91no—
434Q16—541no—
435SD16—181no—
436SD16—541no—
437SD16—541no—
438SD16—541no—
439D8—181no—
440M16—31no—
441M16—31no—
442M16—181no—
443M16—271no—
444C2×C8—91no—
445C2×C8—181no—
446C22×C3—21no—
447C22×C3—21no—
448C22×C3—21no—
449C22×C3—41no—
450C22×C3—61no—
451C22×C3—61no—
452C22×C3—121no—
453C22×C3—361no—
454C22×C3—721no—
455C22×C3#379511yescharacteristic
456C22×C3#379811yescharacteristic
457C22×C3#380411yescharacteristic
458C2×D3—21no—
459C2×D3—122no—
460C2×D3—121no—
461C2×D3—121no—
462C2×D3—241no—
463C2×D3—241no—
464C2×D3—242no—
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—61no—
474C3×C4—61no—
475C3×C4—121no—
476C3×C4—361no—
477C3×C4D3×D1211yescharacteristic
478C3×C4#380211yescharacteristic
479C3×C4C2×#62611yescharacteristic
480C3×C4D3×#6711yescharacteristic
481C3×C4D3×#6711yescharacteristic
482C3×C4D3×#6711yescharacteristic
483Dic3—21no—
484Dic3—122no—
485Dic3—121no—
486Dic3—121no—
487C32—21no—
488C32—21no—
489C32—21no—
490C32—41no—
491C32#475411yescharacteristic, complemented
492C32#514111yescharacteristic, complemented
493C32#520011yescharacteristic, complemented
494C23—181no—
495Q8—31no—
496Q8—181no—
497Q8—181no—
498D4—31no—
499D4—61no—
500D4—181no—
501D4—181no—
502C2×C4—61no—
503C2×C4—181no—
504C2×C4D3311yescharacteristic
505C8—61no—
506C8—61no—
507C8—91no—
508C8—91no—
509C8—541no—
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—121no—
522C2×C3—121no—
523C2×C3—241no—
524C2×C3—721no—
525C2×C3#6388511yescharacteristic
526C2×C3#6390611yescharacteristic
527C2×C3#6397811yescharacteristic
528D3—41no—
529D3—242no—
530D3—241no—
531D3—241no—
532C22—61no—
533C22—361no—
534C22—721no—
535C22#8896011yescharacteristic
536C4—61no—
537C4—361no—
538C4D3×#62611yescharacteristic
539C4#8895311yescharacteristic
540C3—21no—
541C3—21no—
542C3—21no—
543C3—41no—
544C3#1059098011yescharacteristic, complemented
545C3#1059099811yescharacteristic, complemented
546C3#1059167911yescharacteristic, complemented
547C2—21no—
548C2—121no—
549C2—721no—
550C2#1172296011yescharacteristic
subgroupG/H
centerC2#11722960
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socleC2×C33#121
prime radicalC3×#63825C2
hypocenterC33#471
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×#88811C2semidirect
2. #11723373C2semidirect
4. C3×#63815C2semidirect
5. #11722060C2semidirect
6. #11722026C2semidirect
7. C3×#63561C2semidirect
38. #63815D3semidirect
46. #63561D3semidirect
338. C33#471semidirect
491. C32#4754semidirect
492. C32#5141semidirect
493. C32#5200semidirect
544. C3#10590980semidirect
545. C3#10590998semidirect
546. C3#10591679semidirect

Maximal subgroups

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

subgroupindex
C2×#888112
#117233732
C3×#638252
C3×#638152
#117220602
#117220262
C3×#635612
#105916793
#105909983
#105909803