Finite Group Databasebeta

← #13156239

Subgroups — #13156239

Order 1728.

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

normal subgroups only

This group has 676 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
1#11723367C211yesmaximal, characteristic, complemented
2C2×#88790C211yesmaximal, characteristic, complemented
3C3×#63823C211yesmaximal, characteristic, complemented
4C3×#63815C211yesmaximal, characteristic, complemented
5#11722081C211yesmaximal, characteristic, complemented
6#11722027C211yesmaximal, characteristic, complemented
7C3×#63558C211yesmaximal, characteristic, complemented
8#10591677—31nomaximal
9#10591508—31nomaximal
10#10590997—31nomaximal
11C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
12C32×#235C2211yescharacteristic
13C2×C32×D12C2211yescharacteristic
14C3×#3748C2211yescharacteristic
15C3×#3748C2211yescharacteristic
16C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
17C32×#208C2211yescharacteristic
18#88795—21no—
19#88790—21no—
20#88790—21no—
21#88790—21no—
22C3×#3715—21no—
23C3×#3713—21no—
24C3×#3711—21no—
25C3×D3×C3⋊C8—21no—
26#64158—31no—
27C2×#3753—31no—
28C3×#949—31no—
29C3×#948—31no—
30C3×#932—31no—
31C3×#907—31no—
32#63825—31no—
33#63825—31no—
34#63825—61no—
35C2×#3714—31no—
36C2×#3714—31no—
37C2×#3714—61no—
38#63823D311yescharacteristic, complemented
39#63815D311yescharacteristic, complemented
40#63650—31no—
41C3×#833—31no—
42C3×#823—31no—
43#63578—31no—
44#63577—31no—
45#63577—31no—
46#63559—31no—
47#63559—31no—
48#63559—61no—
49#63558D311yescharacteristic, complemented
50C22×C3⋊(C3⋊D3)—21no—
51C22×C32×D3—21no—
52C2×C33×C4C2311yescharacteristic
53C4⋊(C3⋊(C3⋊D3))—21no—
54C4⋊(C3⋊(C3⋊D3))—21no—
55C32×#67—21no—
56C32×D12—21no—
57C32×D12—21no—
58C32×D12D411yescharacteristic
59C32×C4×D3—21no—
60C32×Dic6D411yescharacteristic
61C3×C3⋊(C3⋊C8)—21no—
62C3×C3⋊(C3⋊C8)—21no—
63C3×C3⋊(C3⋊C8)D411yescharacteristic
64C3×C3⋊(C3⋊C8)D411yescharacteristic
65C32×C3⋊C8—21no—
66#5199—91no—
67#5142—91no—
68#4749—91no—
69C32×#41—31no—
70C2×C32×D4—31no—
71C2×C4⋊(C3⋊D3)—31no—
72C2×C4⋊(C3⋊D3)—31no—
73C2×C4⋊(C3⋊D3)—31no—
74C2×C4⋊(C3⋊D3)—61no—
75C2×C4⋊(C3⋊D3)—61no—
76C2×C4⋊(C3⋊D3)—61no—
77C2×C4⋊(C3⋊D3)—121no—
78C3×#235—21no—
79C3×#235C2×D311yescharacteristic
80C3×#235C2×D311yescharacteristic
81C2×C3×D12—21no—
82C2×C3×D12C2×D311yescharacteristic
83C2×C3×D12C2×D311yescharacteristic
84C32×M16—31no—
85#3755—61no—
86#3753—61no—
87#3753—61no—
88#3753—61no—
89#3748C2×D311yescharacteristic
90#3748C2×D311yescharacteristic
91C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
92C3×#215—61no—
93C3×#214—61no—
94C3×#213—61no—
95C3×#213—61no—
96C3×#208—21no—
97C3×#208—31no—
98C3×#208—31no—
99C3×#208—31no—
100C3×#208—31no—
101C3×#208—61no—
102C3×#208—61no—
103C3×#208—61no—
104C3×#208C2×D311yescharacteristic
105C3×#208C2×D311yescharacteristic
106C2×C3×C3⋊C8—31no—
107C2×C3×C3⋊C8—31no—
108C2×C3×C3⋊C8—61no—
109C2×C3×C3⋊C8—61no—
110C3×ΓL(1,𝔽25)—61no—
111C3×D3×C8—61no—
112#3717—62no—
113#3717—61no—
114#3717—61no—
115#3715—21no—
116#3714—61no—
117#3714—61no—
118#3714—61no—
119#3714—61no—
120#3714—62no—
121#3714—62no—
122#3714—62no—
123#3714—61no—
124#3714—61no—
125#3713—21no—
126#3711—21no—
127D3×C3⋊C8—21no—
128C22×C33#4111yescharacteristic
129C2×C3⋊(C3⋊D3)—41no—
130C2×C3⋊(C3⋊D3)—81no—
131C2×C32×D3—21no—
132C2×C32×D3—41no—
133C2×C32×D3—81no—
134C33×C4C2×D411yescharacteristic
135C33×C4C2×D411yescharacteristic
136C32×Dic3—21no—
137C3×AGL(1,ℤ/8)—91no—
138C3×#131—91no—
139#949—31no—
140#949—91no—
141#949—91no—
142#949—181no—
143#948—31no—
144#932—31no—
145C2×#213—91no—
146C2×#213—91no—
147C2×#213—181no—
148#908—91no—
149#907—31no—
150C2×D24—91no—
151C3×#100—91no—
152#833—31no—
153#823—31no—
154#823—91no—
155#823—91no—
156#823—91no—
157#823—181no—
158#822—91no—
159#822—91no—
160#819—91no—
161C23×C32—61no—
162C22×C3⋊D3—61no—
163C22×C3⋊D3—62no—
164C22×C3⋊D3—61no—
165C22×C3⋊D3—61no—
166C22×C3⋊D3—121no—
167C22×C3⋊D3—121no—
168C22×C3⋊D3—122no—
169C22×C3×D3—21no—
170C22×C3×D3—21no—
171C22×C3×D3—41no—
172C32×Q8—31no—
173C32×D4—31no—
174C32×D4—61no—
175C32×D4—61no—
176C32×D4—61no—
177C2×C32×C4—21no—
178C2×C32×C4—21no—
179C2×C32×C4—21no—
180C2×C32×C4—41no—
181C2×C32×C4—61no—
182C2×C32×C4C22×D311yescharacteristic
183C2×C32×C4C22×D311yescharacteristic
184C2×C32×C4C22×D311yescharacteristic
185C4⋊(C3⋊D3)—61no—
186C4⋊(C3⋊D3)—61no—
187C4⋊(C3⋊D3)—62no—
188C4⋊(C3⋊D3)—62no—
189C4⋊(C3⋊D3)—61no—
190C4⋊(C3⋊D3)—61no—
191C4⋊(C3⋊D3)—61no—
192C4⋊(C3⋊D3)—61no—
193C4⋊(C3⋊D3)—121no—
194C4⋊(C3⋊D3)—121no—
195C4⋊(C3⋊D3)—121no—
196C4⋊(C3⋊D3)—121no—
197C4⋊(C3⋊D3)—122no—
198C4⋊(C3⋊D3)—122no—
199C3×#67—21no—
200C3×#67—21no—
201C3×#67—41no—
202C3×D12—22no—
203C3×D12—21no—
204C3×D12—21no—
205C3×D12—21no—
206C3×D12—21no—
207C3×D12—21no—
208C3×D12—41no—
209C3×D12D1211yescharacteristic
210C3×D12#6711yescharacteristic
211C3×C4×D3—21no—
212C3×C4×D3—21no—
213C3×C4×D3—41no—
214C3×Dic6—21no—
215C3×Dic6D1211yescharacteristic
216C3×Dic6#6711yescharacteristic
217C32×C8—61no—
218C3⋊(C3⋊C8)—21no—
219C3⋊(C3⋊C8)—21no—
220C3⋊(C3⋊C8)#6711yescharacteristic
221C3⋊(C3⋊C8)#6711yescharacteristic
222C3×C3⋊C8—21no—
223C3×C3⋊C8—21no—
224C3×C3⋊C8—31no—
225C3×C3⋊C8—31no—
226C3×C3⋊C8—31no—
227C3×C3⋊C8—31no—
228C3×C3⋊C8—41no—
229C3×C3⋊C8—61no—
230C3×C3⋊C8—61no—
231C3×C3⋊C8—62no—
232C3×C3⋊C8—61no—
233C3×C3⋊C8—61no—
234C3×C3⋊C8—61no—
235C3×C3⋊C8—61no—
236C3×C3⋊C8—121no—
237#472—271noSylow
238C2×C33—21no—
239C2×C33#12111yescharacteristic
240C3⋊(C3⋊D3)—81no—
241C32×D3—41no—
242C32×D3—81no—
243C3×#41—31no—
244C3×#41—31no—
245C3×#41—61no—
246C2×C3×D4—31no—
247C2×C3×D4—31no—
248C2×C3×D4—61no—
249#235D3211yescharacteristic
250C2×D12—91no—
251C2×D12—91no—
252C2×D12—91no—
253C2×D12—181no—
254C2×D12—181no—
255C2×D12—181no—
256C2×D12—361no—
257C2×D12D3211yescharacteristic
258C3×SD16—181no—
259C3×D8—181no—
260C3×M16—31no—
261C3×M16—31no—
262C3×M16—61no—
263C3×M16—91no—
264C3×M16—91no—
265C3×M16—181no—
266C2×C3×C8—91no—
267C2×C3×C8—181no—
268#215—61no—
269#215—181no—
270#215—181no—
271#215—182no—
272#214—61no—
273#213—61no—
274#213—61no—
275#213—181no—
276#213—181no—
277#213—181no—
278#213—181no—
279#213—181no—
280#213—181no—
281#213—182no—
282#213—182no—
283#213—182no—
284#208—31no—
285#208—31no—
286#208—31no—
287#208—31no—
288#208—61no—
289#208—61no—
290#208—61no—
291#208D3211yescharacteristic
292C2×C3⋊C8—31no—
293C2×C3⋊C8—31no—
294C2×C3⋊C8—61no—
295C2×C3⋊C8—61no—
296D24—181no—
297D24—181no—
298D24—181no—
299#204—181no—
300ΓL(1,𝔽25)—61no—
301D3×C8—61no—
302C22×C32—21no—
303C22×C32—21no—
304C22×C32—21no—
305C22×C32—41no—
306C22×C32—61no—
307C22×C32—121no—
308C22×C32—241no—
309C22×C32#23511yescharacteristic
310C22×C32#23511yescharacteristic
311C22×C32#23711yescharacteristic
312C2×C3⋊D3—122no—
313C2×C3⋊D3—122no—
314C2×C3⋊D3—122no—
315C2×C3⋊D3—124no—
316C2×C3⋊D3—121no—
317C2×C3⋊D3—121no—
318C2×C3⋊D3—121no—
319C2×C3⋊D3—242no—
320C2×C3⋊D3—244no—
321C2×C3⋊D3—242no—
322C2×C3⋊D3—241no—
323C2×C3⋊D3—241no—
324C2×C3⋊D3—241no—
325C2×C3⋊D3—242no—
326C2×C3×D3—21no—
327C2×C3×D3—21no—
328C2×C3×D3—41no—
329C2×C3×D3—42no—
330C2×C3×D3—41no—
331C2×C3×D3—41no—
332C2×C3×D3—82no—
333C2×C3×D3—81no—
334C2×C3×D3—81no—
335C32×C4—21no—
336C32×C4—21no—
337C32×C4—21no—
338C32×C4—21no—
339C32×C4—21no—
340C32×C4—21no—
341C32×C4—41no—
342C32×C4—41no—
343C32×C4—61no—
344C32×C4C2×D1211yescharacteristic
345C32×C4D3×D411yescharacteristic
346C32×C4D3×D411yescharacteristic
347C32×C4D3×D411yescharacteristic
348C32×C4C2×#6711yescharacteristic
349C32×C4C2×#6711yescharacteristic
350C3×Dic3—21no—
351C3×Dic3—21no—
352C3×Dic3—41no—
353AGL(1,ℤ/8)—91no—
354AGL(1,ℤ/8)—271no—
355C2×D8—271no—
356#131—91no—
357#108—271no—
358#100—91no—
359#100—271no—
360C33#47211yescharacteristic, complemented, Sylow
361C23×C3—61no—
362C23×C3—61no—
363C23×C3—121no—
364C22×D3—21no—
365C22×D3—182no—
366C22×D3—181no—
367C22×D3—181no—
368C22×D3—181no—
369C22×D3—361no—
370C22×D3—361no—
371C22×D3—362no—
372C3×Q8—31no—
373C3×Q8—31no—
374C3×Q8—61no—
375C3×D4—31no—
376C3×D4—31no—
377C3×D4—61no—
378C3×D4—62no—
379C3×D4—61no—
380C3×D4—61no—
381C3×D4—61no—
382C3×D4—61no—
383C3×D4—61no—
384C3×D4—61no—
385C3×D4—121no—
386C3×D4—121no—
387C2×C3×C4—21no—
388C2×C3×C4—21no—
389C2×C3×C4—21no—
390C2×C3×C4—41no—
391C2×C3×C4—61no—
392C2×C3×C4—61no—
393C2×C3×C4—121no—
394C2×C3×C4C2×D3211yescharacteristic
395C2×C3×C4C2×D3211yescharacteristic
396C2×C3×C4C2×D3211yescharacteristic
397#67—21no—
398D12—21no—
399D12—21no—
400D12—182no—
401D12—182no—
402D12—181no—
403D12—181no—
404D12—181no—
405D12—181no—
406D12—181no—
407D12—181no—
408D12—361no—
409D12—361no—
410D12—361no—
411D12—361no—
412D12—362no—
413D12—362no—
414D12#62611yescharacteristic
415C4×D3—21no—
416Dic6#62611yescharacteristic
417C3×C8—61no—
418C3×C8—61no—
419C3×C8—91no—
420C3×C8—91no—
421C3×C8—121no—
422C3×C8—181no—
423C3×C8—181no—
424C3⋊C8—21no—
425C3⋊C8—31no—
426C3⋊C8—31no—
427C3⋊C8—31no—
428C3⋊C8—31no—
429C3⋊C8—61no—
430C3⋊C8—61no—
431C3⋊C8—62no—
432C3⋊C8—61no—
433C3⋊C8—61no—
434C3⋊C8—61no—
435C3⋊C8—61no—
436C3⋊C8—121no—
437C2×C32—21no—
438C2×C32—22no—
439C2×C32—21no—
440C2×C32—21no—
441C2×C32—21no—
442C2×C32—21no—
443C2×C32—21no—
444C2×C32—41no—
445C2×C32—41no—
446C2×C32—41no—
447C2×C32—42no—
448C2×C32—121no—
449C2×C32—241no—
450C2×C32#88411yescharacteristic
451C2×C32#93011yescharacteristic
452C2×C32#93911yescharacteristic
453C3⋊D3—242no—
454C3⋊D3—242no—
455C3⋊D3—242no—
456C3⋊D3—244no—
457C3⋊D3—241no—
458C3⋊D3—241no—
459C3⋊D3—241no—
460C3×D3—41no—
461C3×D3—41no—
462C3×D3—81no—
463C3×D3—82no—
464C3×D3—81no—
465C3×D3—81no—
466#41—31no—
467C2×D4—31no—
468C2×D4—271no—
469SD16—181no—
470SD16—541no—
471D8—181no—
472D8—541no—
473D8—541no—
474D8—541no—
475M16—31no—
476M16—91no—
477M16—91no—
478M16—181no—
479C2×C8—91no—
480C2×C8—181no—
481C22×C3—21no—
482C22×C3—21no—
483C22×C3—21no—
484C22×C3—41no—
485C22×C3—61no—
486C22×C3—61no—
487C22×C3—121no—
488C22×C3—122no—
489C22×C3—121no—
490C22×C3—121no—
491C22×C3—241no—
492C22×C3—241no—
493C22×C3—242no—
494C22×C3#379811yescharacteristic
495C22×C3#380411yescharacteristic
496C22×C3#380411yescharacteristic
497C2×D3—21no—
498C2×D3—41no—
499C2×D3—81no—
500C2×D3—362no—
501C2×D3—362no—
502C2×D3—362no—
503C2×D3—364no—
504C2×D3—361no—
505C2×D3—361no—
506C2×D3—361no—
507C2×D3—722no—
508C2×D3—722no—
509C2×D3—724no—
510C2×D3—721no—
511C2×D3—721no—
512C2×D3—721no—
513C2×D3—722no—
514C3×C4—21no—
515C3×C4—21no—
516C3×C4—21no—
517C3×C4—21no—
518C3×C4—21no—
519C3×C4—21no—
520C3×C4—41no—
521C3×C4—41no—
522C3×C4—61no—
523C3×C4—61no—
524C3×C4—121no—
525C3×C4D3×D1211yescharacteristic
526C3×C4C2×#62611yescharacteristic
527C3×C4D3×#6711yescharacteristic
528C3×C4D3×#6711yescharacteristic
529C3×C4D3×#6711yescharacteristic
530C3×C4#381111yescharacteristic
531Dic3—21no—
532C32—21no—
533C32—21no—
534C32—21no—
535C32—41no—
536C32#474911yescharacteristic, complemented
537C32#514211yescharacteristic, complemented
538C32#519911yescharacteristic, complemented
539C23—61no—
540C23—541no—
541Q8—31no—
542D4—31no—
543D4—61no—
544D4—61no—
545D4—61no—
546D4—541no—
547D4—541no—
548C2×C4—61no—
549C2×C4D3311yescharacteristic
550C8—61no—
551C8—91no—
552C8—91no—
553C8—181no—
554C8—181no—
555C2×C3—22no—
556C2×C3—21no—
557C2×C3—21no—
558C2×C3—21no—
559C2×C3—21no—
560C2×C3—21no—
561C2×C3—21no—
562C2×C3—41no—
563C2×C3—41no—
564C2×C3—42no—
565C2×C3—41no—
566C2×C3—121no—
567C2×C3—121no—
568C2×C3—241no—
569C2×C3—242no—
570C2×C3—241no—
571C2×C3—241no—
572C2×C3#6390611yescharacteristic
573C2×C3#6397111yescharacteristic
574C2×C3#6397811yescharacteristic
575D3—41no—
576D3—81no—
577D3—722no—
578D3—722no—
579D3—722no—
580D3—724no—
581D3—721no—
582D3—721no—
583D3—721no—
584C22—61no—
585C22—121no—
586C22—241no—
587C22—1081no—
588C22—2161no—
589C22#8895911yescharacteristic
590C4—61no—
591C4D3×#62611yescharacteristic
592C4#8895211yescharacteristic
593C3—21no—
594C3—21no—
595C3—21no—
596C3—41no—
597C3#1059099711yescharacteristic, complemented
598C3#1059150811yescharacteristic, complemented
599C3#1059167711yescharacteristic, complemented
600C2—21no—
601C2—121no—
602C2—241no—
603C2—2161no—
604C2#1172295811yescharacteristic
subgroupG/H
centerC2#11722958
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socleC2×C33#121
hypocenterC33#472
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. #11723367C2semidirect
2. C2×#88790C2semidirect
3. C3×#63823C2semidirect
4. C3×#63815C2semidirect
5. #11722081C2semidirect
6. #11722027C2semidirect
7. C3×#63558C2semidirect
38. #63823D3semidirect
39. #63815D3semidirect
49. #63558D3semidirect
360. C33#472semidirect
536. C32#4749semidirect
537. C32#5142semidirect
538. C32#5199semidirect
597. C3#10590997semidirect
598. C3#10591508semidirect
599. C3#10591677semidirect

Maximal subgroups

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

subgroupindex
#117233672
C2×#887902
C3×#638232
C3×#638152
#117220812
#117220272
C3×#635582
#105916773
#105915083
#105909973