Finite Group Databasebeta

← #13161978

Subgroups — #13161978

Order 1728.

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

normal subgroups only

This group has 676 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×#64141C211yesmaximal, characteristic, complemented
2C2×C3×#3753C211yesmaximal, characteristic, complemented
3#11723661C211yesmaximal, characteristic, complemented
4#11723657C211yesmaximal, characteristic, complemented
5C3×#63660C211yesmaximal, characteristic, complemented
6#11722081C211yesmaximal, characteristic, complemented
7#11722073C211yesmaximal, characteristic, complemented
8#10591508—31nomaximal
9#10591505—31nomaximal
10#10591406—31nomaximal
11C2×C33×D4C2211yescharacteristic
12C3×#3828C2211yescharacteristic
13C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
14C3×#3754—21no—
15C3×#3753—21no—
16C3×#3753—21no—
17C3×#3753—21no—
18C3×#3748C2211yescharacteristic
19C3×#3748C2211yescharacteristic
20C3×#3748C2211yescharacteristic
21C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
22#88811—21no—
23#88810—21no—
24#88809—21no—
25#88808—21no—
26#64141D311yescharacteristic, complemented
27C2×#3753D311yescharacteristic, complemented
28C3×#932—31no—
29C3×#932—31no—
30C3×#932—61no—
31C2×C3×#213—31no—
32C2×C3×#213—31no—
33C2×C3×#213—61no—
34#63830—31no—
35#63825—31no—
36#63823—31no—
37#63817—31no—
38#63815—31no—
39#63814—31no—
40#63660D311yescharacteristic, complemented
41C3×#833—31no—
42C3×#833—31no—
43C3×#833—61no—
44#63578—31no—
45#63577—31no—
46#63577—31no—
47#63559—31no—
48#63558—31no—
49#63558—31no—
50C23×C33—21no—
51C22×C3×C3⋊D3—21no—
52C33×D4—21no—
53C33×D4—21no—
54C2×C33×C4C2311yescharacteristic
55C3×#638—21no—
56C3×C4⋊(C3⋊D3)—21no—
57C3×C4⋊(C3⋊D3)—21no—
58C3×C4⋊(C3⋊D3)D411yescharacteristic
59C3×C4×C3⋊D3—21no—
60C3×#634D411yescharacteristic
61C3×C3⋊(C3⋊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#5160—91no—
68#4749—91no—
69C2×C32×D4—21no—
70C2×C32×D4—21no—
71C2×C32×D4—21no—
72C2×C32×D4—41no—
73C2×C32×D4C2×D311yescharacteristic
74C2×C32×D4C2×D311yescharacteristic
75C2×C32×D4C2×D311yescharacteristic
76#3828C2×D311yescharacteristic
77C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
78C3×#235—31no—
79C3×#235—31no—
80C3×#235—61no—
81C2×C3×D12—31no—
82C2×C3×D12—31no—
83C2×C3×D12—61no—
84#3754—21no—
85#3753—21no—
86#3753—21no—
87#3753—21no—
88#3748C2×D311yescharacteristic
89#3748C2×D311yescharacteristic
90#3748C2×D311yescharacteristic
91C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
92C3×#214—61no—
93C3×#214—61no—
94C3×#214—62no—
95C3×#213—61no—
96C3×#213—61no—
97C3×#213—61no—
98C3×#213—61no—
99C3×#213—61no—
100C3×#213—61no—
101C3×#213—62no—
102C3×#213—62no—
103C3×#213—62no—
104C3×#208—31no—
105C3×#208—31no—
106C3×#208—31no—
107C3×#208—31no—
108C3×#208—31no—
109C3×#208—31no—
110C3×#208—61no—
111C3×#208—61no—
112C3×#208—61no—
113C2×C3×C3⋊C8—31no—
114C2×C3×C3⋊C8—31no—
115C2×C3×C3⋊C8—61no—
116#3717—61no—
117#3716—61no—
118#3715—61no—
119#3714—61no—
120#3714—61no—
121#3713—61no—
122#3712—61no—
123#3711—61no—
124#3711—61no—
125#3710—61no—
126D3×C3⋊C8—61no—
127D3×C3⋊C8—61no—
128C22×C33—41no—
129C22×C33—81no—
130C22×C33#4111yescharacteristic
131C2×C3×C3⋊D3—21no—
132C2×C3×C3⋊D3—41no—
133C2×C3×C3⋊D3—81no—
134C33×C4C2×D411yescharacteristic
135C33×C4C2×D411yescharacteristic
136C3×C3⋊Dic3—21no—
137C3×AGL(1,ℤ/8)—91no—
138C2×C3×D8—91no—
139#949—91no—
140#948—91no—
141#932—31no—
142#932—31no—
143#932—61no—
144#932—91no—
145C2×#213—31no—
146C2×#213—31no—
147C2×#213—61no—
148#908—91no—
149#907—91no—
150#901—91no—
151C3×#100—91no—
152#833—31no—
153#833—31no—
154#833—61no—
155#833—91no—
156#823—91no—
157#823—91no—
158#822—91no—
159#822—91no—
160#819—91no—
161C23×C32—21no—
162C23×C32—22no—
163C23×C32—21no—
164C23×C32—21no—
165C23×C32—41no—
166C23×C32—41no—
167C23×C32—42no—
168C22×C3⋊D3—21no—
169C22×C3×D3—61no—
170C22×C3×D3—61no—
171C22×C3×D3—121no—
172C32×D4—21no—
173C32×D4—21no—
174C32×D4—22no—
175C32×D4—22no—
176C32×D4—21no—
177C32×D4—21no—
178C32×D4—21no—
179C32×D4—21no—
180C32×D4—41no—
181C32×D4—41no—
182C32×D4—41no—
183C32×D4—41no—
184C32×D4—42no—
185C32×D4—42no—
186C2×C32×C4—21no—
187C2×C32×C4—21no—
188C2×C32×C4—21no—
189C2×C32×C4—41no—
190C2×C32×C4C22×D311yescharacteristic
191C2×C32×C4C22×D311yescharacteristic
192C2×C32×C4C22×D311yescharacteristic
193#638—21no—
194C4⋊(C3⋊D3)—21no—
195C4⋊(C3⋊D3)—21no—
196C4⋊(C3⋊D3)#6711yescharacteristic
197C4×C3⋊D3—21no—
198#634#6711yescharacteristic
199C3×#67—61no—
200C3×#67—61no—
201C3×#67—121no—
202C3×D12—31no—
203C3×D12—31no—
204C3×D12—61no—
205C3×D12—62no—
206C3×D12—61no—
207C3×D12—61no—
208C3×D12—61no—
209C3×D12—61no—
210C3×D12—121no—
211C3×C4×D3—61no—
212C3×C4×D3—61no—
213C3×C4×D3—121no—
214C3×Dic6—31no—
215C3×Dic6—31no—
216C3×Dic6—61no—
217C3⋊(C3⋊C8)—21no—
218C3⋊(C3⋊C8)—21no—
219C3⋊(C3⋊C8)—21no—
220C3⋊(C3⋊C8)#6711yescharacteristic
221C3⋊(C3⋊C8)#6711yescharacteristic
222C3×C3⋊C8—31no—
223C3×C3⋊C8—31no—
224C3×C3⋊C8—31no—
225C3×C3⋊C8—31no—
226C3×C3⋊C8—61no—
227C3×C3⋊C8—61no—
228C3×C3⋊C8—62no—
229C3×C3⋊C8—61no—
230C3×C3⋊C8—61no—
231C3×C3⋊C8—61no—
232C3×C3⋊C8—61no—
233C3×C3⋊C8—61no—
234C3×C3⋊C8—61no—
235C3×C3⋊C8—121no—
236C3×C3⋊C8—121no—
237#472—271noSylow
238C2×C33—21no—
239C2×C33—81no—
240C2×C33#12111yescharacteristic
241C3×C3⋊D3—41no—
242C3×C3⋊D3—81no—
243C3×#41—91no—
244C2×C3×D4—21no—
245C2×C3×D4—21no—
246C2×C3×D4—21no—
247C2×C3×D4—41no—
248C2×C3×D4—91no—
249C2×C3×D4D3211yescharacteristic
250C2×C3×D4D3211yescharacteristic
251C2×C3×D4D3211yescharacteristic
252#235—31no—
253#235—31no—
254#235—61no—
255C2×D12—31no—
256C2×D12—31no—
257C2×D12—61no—
258C3×SD16—181no—
259C3×D8—181no—
260C3×D8—181no—
261C3×D8—181no—
262C3×M16—91no—
263C3×M16—91no—
264C3×M16—91no—
265C2×C3×C8—91no—
266#215—181no—
267#214—61no—
268#214—61no—
269#214—62no—
270#214—181no—
271#213—61no—
272#213—61no—
273#213—61no—
274#213—61no—
275#213—61no—
276#213—61no—
277#213—62no—
278#213—62no—
279#213—62no—
280#213—181no—
281#213—181no—
282#208—31no—
283#208—31no—
284#208—31no—
285#208—31no—
286#208—31no—
287#208—31no—
288#208—61no—
289#208—61no—
290#208—61no—
291#208—181no—
292C2×C3⋊C8—31no—
293C2×C3⋊C8—31no—
294C2×C3⋊C8—61no—
295C2×C3⋊C8—181no—
296D24—181no—
297#204—181no—
298ΓL(1,𝔽25)—181no—
299ΓL(1,𝔽25)—181no—
300D3×C8—181no—
301D3×C8—181no—
302C22×C32—21no—
303C22×C32—21no—
304C22×C32—21no—
305C22×C32—41no—
306C22×C32—42no—
307C22×C32—42no—
308C22×C32—42no—
309C22×C32—44no—
310C22×C32—41no—
311C22×C32—41no—
312C22×C32—41no—
313C22×C32—81no—
314C22×C32—81no—
315C22×C32—81no—
316C22×C32—82no—
317C22×C32—82no—
318C22×C32—82no—
319C22×C32—84no—
320C22×C32#23511yescharacteristic
321C22×C32#23711yescharacteristic
322C22×C32#23711yescharacteristic
323C2×C3⋊D3—21no—
324C2×C3⋊D3—41no—
325C2×C3⋊D3—81no—
326C2×C3×D3—61no—
327C2×C3×D3—61no—
328C2×C3×D3—121no—
329C2×C3×D3—122no—
330C2×C3×D3—121no—
331C2×C3×D3—121no—
332C2×C3×D3—242no—
333C2×C3×D3—241no—
334C2×C3×D3—241no—
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×C4D3×D411yescharacteristic
344C32×C4D3×D411yescharacteristic
345C32×C4D3×D411yescharacteristic
346C32×C4D3×D411yescharacteristic
347C32×C4C2×#6711yescharacteristic
348C32×C4C2×#6711yescharacteristic
349C3⋊Dic3—21no—
350C3×Dic3—61no—
351C3×Dic3—61no—
352C3×Dic3—121no—
353AGL(1,ℤ/8)—91no—
354AGL(1,ℤ/8)—271no—
355C2×D8—91no—
356#131—271no—
357#108—271no—
358#100—91no—
359#100—271no—
360C33#47211yescharacteristic, complemented, Sylow
361C23×C3—22no—
362C23×C3—21no—
363C23×C3—21no—
364C23×C3—21no—
365C23×C3—41no—
366C23×C3—41no—
367C23×C3—42no—
368C23×C3—181no—
369C22×D3—61no—
370C22×D3—61no—
371C22×D3—121no—
372C3×Q8—91no—
373C3×D4—22no—
374C3×D4—22no—
375C3×D4—21no—
376C3×D4—21no—
377C3×D4—21no—
378C3×D4—21no—
379C3×D4—21no—
380C3×D4—21no—
381C3×D4—41no—
382C3×D4—41no—
383C3×D4—41no—
384C3×D4—41no—
385C3×D4—42no—
386C3×D4—42no—
387C3×D4—91no—
388C3×D4—181no—
389C3×D4—181no—
390C3×D4—181no—
391C2×C3×C4—21no—
392C2×C3×C4—21no—
393C2×C3×C4—21no—
394C2×C3×C4—41no—
395C2×C3×C4—181no—
396C2×C3×C4C2×D3211yescharacteristic
397C2×C3×C4C2×D3211yescharacteristic
398C2×C3×C4C2×D3211yescharacteristic
399#67—61no—
400#67—61no—
401#67—121no—
402D12—31no—
403D12—31no—
404D12—61no—
405D12—62no—
406D12—61no—
407D12—61no—
408D12—61no—
409D12—61no—
410D12—121no—
411C4×D3—61no—
412C4×D3—61no—
413C4×D3—121no—
414Dic6—31no—
415Dic6—31no—
416Dic6—61no—
417C3×C8—91no—
418C3×C8—91no—
419C3×C8—181no—
420C3×C8—181no—
421C3×C8—181no—
422C3⋊C8—31no—
423C3⋊C8—31no—
424C3⋊C8—31no—
425C3⋊C8—31no—
426C3⋊C8—61no—
427C3⋊C8—61no—
428C3⋊C8—62no—
429C3⋊C8—61no—
430C3⋊C8—61no—
431C3⋊C8—61no—
432C3⋊C8—61no—
433C3⋊C8—61no—
434C3⋊C8—61no—
435C3⋊C8—121no—
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—82no—
449C2×C32—82no—
450C2×C32—82no—
451C2×C32—84no—
452C2×C32—81no—
453C2×C32—81no—
454C2×C32—81no—
455C2×C32#88411yescharacteristic
456C2×C32#93811yescharacteristic
457C2×C32#93911yescharacteristic
458C3⋊D3—41no—
459C3⋊D3—81no—
460C3×D3—121no—
461C3×D3—121no—
462C3×D3—241no—
463C3×D3—242no—
464C3×D3—241no—
465C3×D3—241no—
466#41—91no—
467C2×D4—91no—
468C2×D4#111311yescharacteristic
469SD16—181no—
470SD16—541no—
471D8—181no—
472D8—181no—
473D8—181no—
474D8—541no—
475M16—91no—
476M16—91no—
477M16—91no—
478M16—541no—
479C2×C8—91no—
480C2×C8—541no—
481C22×C3—21no—
482C22×C3—21no—
483C22×C3—21no—
484C22×C3—42no—
485C22×C3—42no—
486C22×C3—42no—
487C22×C3—44no—
488C22×C3—41no—
489C22×C3—41no—
490C22×C3—41no—
491C22×C3—41no—
492C22×C3—81no—
493C22×C3—81no—
494C22×C3—81no—
495C22×C3—82no—
496C22×C3—82no—
497C22×C3—82no—
498C22×C3—84no—
499C22×C3—181no—
500C22×C3—361no—
501C22×C3—721no—
502C22×C3#380411yescharacteristic
503C22×C3#380411yescharacteristic
504C22×C3#380511yescharacteristic
505C2×D3—61no—
506C2×D3—61no—
507C2×D3—121no—
508C2×D3—122no—
509C2×D3—121no—
510C2×D3—121no—
511C2×D3—242no—
512C2×D3—241no—
513C2×D3—241no—
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—181no—
523C3×C4D3×#6711yescharacteristic
524C3×C4D3×#6711yescharacteristic
525C3×C4D3×#6711yescharacteristic
526C3×C4D3×#6711yescharacteristic
527C3×C4#381111yescharacteristic
528C3×C4#381111yescharacteristic
529Dic3—61no—
530Dic3—61no—
531Dic3—121no—
532C32—21no—
533C32—21no—
534C32—21no—
535C32—41no—
536C32#474911yescharacteristic, complemented
537C32#516011yescharacteristic, complemented
538C32#519911yescharacteristic, complemented
539C23—21no—
540C23—181no—
541Q8—91no—
542D4—21no—
543D4—21no—
544D4—91no—
545D4—181no—
546D4—181no—
547D4—181no—
548C2×C4—181no—
549C2×C4C2×#111311yescharacteristic
550C8—91no—
551C8—91no—
552C8—181no—
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—82no—
567C2×C3—82no—
568C2×C3—82no—
569C2×C3—84no—
570C2×C3—81no—
571C2×C3—81no—
572C2×C3—81no—
573C2×C3—361no—
574C2×C3—721no—
575C2×C3#6395811yescharacteristic
576C2×C3#6397011yescharacteristic
577C2×C3#6397111yescharacteristic
578D3—121no—
579D3—121no—
580D3—241no—
581D3—242no—
582D3—241no—
583D3—241no—
584C22—41no—
585C22—81no—
586C22—181no—
587C22—361no—
588C22—721no—
589C22#8904611yescharacteristic
590C4—181no—
591C4#8904911yescharacteristic
592C4#8904911yescharacteristic
593C3—21no—
594C3—21no—
595C3—21no—
596C3—41no—
597C3#1059140611yescharacteristic, complemented
598C3#1059150511yescharacteristic, complemented
599C3#1059150811yescharacteristic, complemented
600C2—21no—
601C2—81no—
602C2—361no—
603C2—721no—
604C2#1172372511yescharacteristic
subgroupG/H
centerC2#11723725
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×#1113
FittingC2×C33×D4C22
socleC2×C33#121
hypercenterC2×D4#1113
hypocenterC33#472
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C3×#64141C2semidirect
2. C2×C3×#3753C2semidirect
3. #11723661C2semidirect
4. #11723657C2semidirect
5. C3×#63660C2semidirect
6. #11722081C2semidirect
7. #11722073C2semidirect
26. #64141D3semidirect
27. C2×#3753D3semidirect
40. #63660D3semidirect
360. C33#472semidirect
536. C32#4749semidirect
537. C32#5160semidirect
538. C32#5199semidirect
597. C3#10591406semidirect
598. C3#10591505semidirect
599. C3#10591508semidirect

Maximal subgroups

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

subgroupindex
C3×#641412
C2×C3×#37532
#117236612
#117236572
C3×#636602
#117220812
#117220732
#105915083
#105915053
#105914063