Finite Group Databasebeta

← #13156001

Subgroups — #13156001

Order 1728.

710 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11723499C211yesmaximal, characteristic
2#11723451C211yesmaximal, characteristic, complemented
3C2×C3×#3716C211yesmaximal, characteristic
4C2×C3×#3715C211yesmaximal, characteristic
5#11722076C211yesmaximal, characteristic, complemented
6#11722050C211yesmaximal, characteristic, complemented
7C3×#63566C211yesmaximal, characteristic, complemented
8#10591569—31nomaximal
9#10591497—31nomaximal
10#10590748—31nomaximal
11C2×C3×#634C2211yescharacteristic
12C2×C32×D12C2211yescharacteristic
13C2×C32×Dic6C2211yescharacteristic
14C2×D3×C3⋊Dic3—21no—
15C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
16C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
17C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
18C2×C32×C3⋊C8C2211yescharacteristic
19#88806—21no—
20#88806—21no—
21(C2×D3)⋊(C3⋊Dic3)—21no—
22C3×#3716—22no—
23C3×#3715—22no—
24#64153—31no—
25D4×C3⋊Dic3—31no—
26C2×C3×#215—31no—
27C2×C3×#214—31no—
28C2×C3×#214—31no—
29C2×C3×#204—31no—
30#63898—31no—
31#63898—31no—
32#63898—61no—
33#63873—31no—
34#63873—31no—
35#63873—61no—
36C2×#3716D311yescharacteristic
37C2×#3715D311yescharacteristic
38#63641—31no—
39C3×#835—31no—
40C3×#816—31no—
41#63572—31no—
42#63572—31no—
43#63572—61no—
44#63567—31no—
45#63566—31no—
46#63566D311yescharacteristic, complemented
47#63565—31no—
48C22×C32×D3—21no—
49C2×C33×C4C2311yescharacteristic
50C2×C3⋊(C3⋊Dic3)D411yescharacteristic
51C2×C3⋊(C3⋊Dic3)D411yescharacteristic
52C2×C3×C3⋊Dic3—21no—
53C2×C3×C3⋊Dic3—21no—
54C3×#634—21no—
55C3×#634—21no—
56C2×C32×Dic3—21no—
57C32×D12—21no—
58C32×D12D412yes—
59C32×Dic6—21no—
60C32×Dic6—21no—
61D3×C3⋊Dic3—42no—
62C3×C3⋊(C3⋊C8)—21no—
63C32×C3⋊C8—21no—
64#5162—91no—
65#5162—91no—
66#4818—91no—
67C2×C32×Q8—31no—
68C2×C32×D4—31no—
69C22×C3⋊Dic3—61no—
70C2×#634C2×D311yescharacteristic
71C2×C3×D12—21no—
72C2×C3×D12C2×D311yescharacteristic
73C2×C3×D12C2×D311yescharacteristic
74C2×C3×Dic6—21no—
75C2×C3×Dic6—31no—
76C2×C3×Dic6—31no—
77C2×C3×Dic6—61no—
78C2×C3×Dic6C2×D311yescharacteristic
79C2×C3×Dic6C2×D311yescharacteristic
80C2×D3×Dic3—61no—
81C2×D3×Dic3—61no—
82C2×D3×Dic3—62no—
83C2×C32×C8—31no—
84#3757—61no—
85C4⋊(C3⋊Dic3)C2×D311yescharacteristic
86#3750—61no—
87#3750—61no—
88C4×C3⋊Dic3—31no—
89C4×C3⋊Dic3—31no—
90C4×C3⋊Dic3—31no—
91C4×C3⋊Dic3—61no—
92C4×C3⋊Dic3—61no—
93C4×C3⋊Dic3—61no—
94C4×C3⋊Dic3—121no—
95C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
96C3×#215—62no—
97C3×#214—62no—
98C3×#214—62no—
99C3×C4⋊Dic3—31no—
100C3×C4⋊Dic3—31no—
101C3×C4⋊Dic3—61no—
102C2×C3×C3⋊C8—21no—
103C2×C3×C3⋊C8—31no—
104C2×C3×C3⋊C8—31no—
105C2×C3×C3⋊C8—61no—
106C2×C3×C3⋊C8C2×D311yescharacteristic
107C2×C3×C3⋊C8C2×D311yescharacteristic
108C3×#204—62no—
109#3724—61no—
110#3724—61no—
111#3724—61no—
112#3724—61no—
113#3724—62no—
114#3724—62no—
115(C2×D3)⋊Dic3—61no—
116(C2×D3)⋊Dic3—61no—
117(C2×D3)⋊Dic3—62no—
118#3716—22no—
119#3715—22no—
120C22×C33C2×D411yescharacteristic
121C2×C32×D3—22no—
122C2×C32×D3—41no—
123C2×C32×D3—41no—
124C33×C4C2×D411yescharacteristic
125C33×C4#4111yescharacteristic
126C3⋊(C3⋊Dic3)—21no—
127C3⋊(C3⋊Dic3)SD1612yescomplemented
128C3×C3⋊Dic3—41no—
129C3×C3⋊Dic3—41no—
130C32×Dic3—41no—
131C2×C3×SD16—91no—
132C2×C3×SD16—91no—
133#944—91no—
134#944—91no—
135#944—91no—
136#944—181no—
137C2×#215—31no—
138D4×Dic3—91no—
139D4×Dic3—91no—
140D4×Dic3—181no—
141C2×#214—31no—
142C2×#214—31no—
143C2×#204—31no—
144#893—91no—
145C3×#103—91no—
146#835—31no—
147#835—91no—
148#816—31no—
149#816—91no—
150Dic3⋊C8—91no—
151Dic3⋊C8—91no—
152Dic3⋊C8—91no—
153Dic3⋊C8—181no—
154#810—91no—
155C23×C32—61no—
156C22×C3×D3—22no—
157C22×C3×D3—21no—
158C22×C3×D3—21no—
159C32×Q8—61no—
160C32×Q8—61no—
161C32×D4—32no—
162C32×D4—61no—
163C2×C32×C4—21no—
164C2×C32×C4—21no—
165C2×C32×C4—21no—
166C2×C32×C4—41no—
167C2×C32×C4—61no—
168C2×C32×C4C22×D311yescharacteristic
169C2×C32×C4C22×D311yescharacteristic
170C2×C32×C4C22×D311yescharacteristic
171C2×C3⋊Dic3—21no—
172C2×C3⋊Dic3—21no—
173C2×C3⋊Dic3—31no—
174C2×C3⋊Dic3—31no—
175C2×C3⋊Dic3—31no—
176C2×C3⋊Dic3—31no—
177C2×C3⋊Dic3—31no—
178C2×C3⋊Dic3—31no—
179C2×C3⋊Dic3—61no—
180C2×C3⋊Dic3—61no—
181C2×C3⋊Dic3—61no—
182C2×C3⋊Dic3—61no—
183C2×C3⋊Dic3—61no—
184C2×C3⋊Dic3—61no—
185C2×C3⋊Dic3—121no—
186C2×C3⋊Dic3—121no—
187C2×C3⋊Dic3—122no—
188#634—21no—
189#634—21no—
190C2×C3×Dic3—22no—
191C2×C3×Dic3—21no—
192C2×C3×Dic3—21no—
193C2×C3×Dic3—62no—
194C2×C3×Dic3—62no—
195C2×C3×Dic3—61no—
196C2×C3×Dic3—61no—
197C2×C3×Dic3—61no—
198C2×C3×Dic3—61no—
199C3×D12—22no—
200C3×D12—21no—
201C3×D12—21no—
202C3×D12—22no—
203C3×D12#6712yes—
204C3×D12#6712yes—
205C3×Dic6—21no—
206C3×Dic6—21no—
207C3×Dic6—21no—
208C3×Dic6—21no—
209C3×Dic6—41no—
210C3×Dic6—41no—
211C3×Dic6—61no—
212C3×Dic6—61no—
213C3×Dic6—61no—
214C3×Dic6—61no—
215C3×Dic6—121no—
216C3×Dic6—121no—
217D3×Dic3—122no—
218D3×Dic3—122no—
219D3×Dic3—124no—
220C32×C8—61no—
221C3⋊(C3⋊C8)—21no—
222C3×C3⋊C8—21no—
223C3×C3⋊C8—21no—
224C3×C3⋊C8—41no—
225C3×C3⋊C8—61no—
226C3×C3⋊C8—61no—
227C3×C3⋊C8—121no—
228#461—271noSylow
229C2×C33#12111yescharacteristic
230C2×C33C2×SD1611yescharacteristic
231C2×C33#13711yescharacteristic
232C32×D3—42no—
233C2×C3×Q8—31no—
234C2×C3×Q8—31no—
235C2×C3×Q8—61no—
236C2×C3×Q8—91no—
237C2×C3×D4—31no—
238C2×C3×D4—31no—
239C2×C3×D4—61no—
240C22×Dic3—181no—
241C22×Dic3—181no—
242C22×Dic3—182no—
243C2×D12D3211yescharacteristic
244C2×C4×D3—181no—
245C2×Dic6—31no—
246C2×Dic6—31no—
247C2×Dic6—61no—
248C2×Dic6D3211yescharacteristic
249C3×SD16—182no—
250C3×SD16—182no—
251C2×C3×C8—31no—
252C2×C3×C8—31no—
253C2×C3×C8—61no—
254C2×C3×C8—91no—
255C3×C4⋊C4—91no—
256#217—181no—
257#217—181no—
258#217—182no—
259#215—62no—
260#214—62no—
261#214—62no—
262(C2×D3)⋊C4—181no—
263C4⋊Dic3—31no—
264C4⋊Dic3—31no—
265C4⋊Dic3—61no—
266Dic3⋊C4—181no—
267Dic3⋊C4—181no—
268Dic3⋊C4—181no—
269Dic3⋊C4—181no—
270Dic3⋊C4—181no—
271Dic3⋊C4—181no—
272Dic3⋊C4—182no—
273Dic3⋊C4—182no—
274C4×Dic3—91no—
275C4×Dic3—91no—
276C4×Dic3—91no—
277C4×Dic3—181no—
278C4×Dic3—181no—
279C4×Dic3—181no—
280C4×Dic3—361no—
281C2×C3⋊C8—31no—
282C2×C3⋊C8—31no—
283C2×C3⋊C8—61no—
284C2×C3⋊C8D3211yescharacteristic
285#204—62no—
286C22×C32—21no—
287C22×C32—21no—
288C22×C32—21no—
289C22×C32—41no—
290C22×C32—62no—
291C22×C32—121no—
292C22×C32—121no—
293C22×C32D3×D411yescharacteristic
294C22×C32D3×D411yescharacteristic
295C22×C32D3×D411yescharacteristic
296C2×C3×D3—24no—
297C2×C3×D3—22no—
298C2×C3×D3—22no—
299C2×C3×D3—42no—
300C2×C3×D3—42no—
301C2×C3×D3—41no—
302C2×C3×D3—41no—
303C2×C3×D3—41no—
304C2×C3×D3—41no—
305C32×C4—21no—
306C32×C4—21no—
307C32×C4—21no—
308C32×C4—21no—
309C32×C4—21no—
310C32×C4—21no—
311C32×C4—41no—
312C32×C4—41no—
313C32×C4—121no—
314C32×C4#23511yescharacteristic
315C32×C4D3×D411yescharacteristic
316C32×C4#23711yescharacteristic
317C32×C4#23711yescharacteristic
318C32×C4C2×#6711yescharacteristic
319C32×C4C2×#6711yescharacteristic
320C3⋊Dic3—32no—
321C3⋊Dic3—32no—
322C3⋊Dic3—32no—
323C3⋊Dic3—41no—
324C3⋊Dic3—41no—
325C3⋊Dic3—62no—
326C3⋊Dic3—62no—
327C3⋊Dic3—62no—
328C3⋊Dic3—61no—
329C3⋊Dic3—62no—
330C3⋊Dic3—61no—
331C3⋊Dic3—61no—
332C3⋊Dic3—122no—
333C3⋊Dic3—121no—
334C3⋊Dic3—121no—
335C3⋊Dic3—122no—
336C3×Dic3—42no—
337C3×Dic3—41no—
338C3×Dic3—41no—
339C3×Dic3—122no—
340C3×Dic3—122no—
341C3×Dic3—121no—
342C3×Dic3—121no—
343C3×Dic3—121no—
344C3×Dic3—121no—
345C2×SD16—91no—
346C2×SD16—91no—
347C4⋊Q8—271no—
348C4×D4—271no—
349C4⋊C8—271no—
350#103—91no—
351#103—271no—
352C33#46111yescharacteristic, complemented, Sylow
353C23×C3—62no—
354C23×C3—61no—
355C23×C3—61no—
356C22×D3—21no—
357C3×Q8—61no—
358C3×Q8—61no—
359C3×Q8—61no—
360C3×Q8—61no—
361C3×Q8—121no—
362C3×Q8—121no—
363C3×Q8—181no—
364C3×Q8—181no—
365C3×D4—32no—
366C3×D4—32no—
367C3×D4—62no—
368C3×D4—62no—
369C3×D4—61no—
370C3×D4—61no—
371C2×C3×C4—21no—
372C2×C3×C4—21no—
373C2×C3×C4—21no—
374C2×C3×C4—41no—
375C2×C3×C4—62no—
376C2×C3×C4—61no—
377C2×C3×C4—61no—
378C2×C3×C4—181no—
379C2×C3×C4—181no—
380C2×C3×C4C2×D3211yescharacteristic
381C2×C3×C4C2×D3211yescharacteristic
382C2×C3×C4C2×D3211yescharacteristic
383C2×Dic3—21no—
384C2×Dic3—62no—
385C2×Dic3—62no—
386C2×Dic3—61no—
387C2×Dic3—61no—
388C2×Dic3—61no—
389C2×Dic3—61no—
390C2×Dic3—91no—
391C2×Dic3—91no—
392C2×Dic3—91no—
393C2×Dic3—91no—
394C2×Dic3—91no—
395C2×Dic3—91no—
396C2×Dic3—181no—
397C2×Dic3—181no—
398C2×Dic3—181no—
399C2×Dic3—181no—
400C2×Dic3—181no—
401C2×Dic3—181no—
402C2×Dic3—361no—
403C2×Dic3—361no—
404C2×Dic3—362no—
405C2×Dic3—362no—
406C2×Dic3—364no—
407D12—21no—
408D12#62512yes—
409C4×D3—362no—
410Dic6—21no—
411Dic6—21no—
412Dic6—61no—
413Dic6—61no—
414Dic6—61no—
415Dic6—61no—
416Dic6—121no—
417Dic6—121no—
418C3×C8—61no—
419C3×C8—61no—
420C3×C8—121no—
421C3×C8—181no—
422C3⋊C8—21no—
423C3⋊C8—61no—
424C3⋊C8—61no—
425C3⋊C8—121no—
426C2×C32—21no—
427C2×C32—21no—
428C2×C32—21no—
429C2×C32—21no—
430C2×C32—21no—
431C2×C32—21no—
432C2×C32—21no—
433C2×C32—21no—
434C2×C32—21no—
435C2×C32—41no—
436C2×C32—41no—
437C2×C32—41no—
438C2×C32—122no—
439C2×C32#88411yescharacteristic
440C2×C32D3×SD1611yescharacteristic
441C2×C32D3×SD1611yescharacteristic
442C2×C32D3×SD1611yescharacteristic
443C2×C32#91511yescharacteristic
444C2×C32#91511yescharacteristic
445C2×C32#91811yescharacteristic
446C2×C32#93911yescharacteristic
447C2×C32#93911yescharacteristic
448C3×D3—44no—
449C3×D3—42no—
450C3×D3—42no—
451C2×Q8—31no—
452C2×Q8—91no—
453C2×D4—31no—
454C22×C4—541no—
455SD16—182no—
456SD16—182no—
457C2×C8—31no—
458C2×C8—91no—
459C4⋊C4—91no—
460C4⋊C4—541no—
461C4⋊C4—541no—
462C22⋊C4—541no—
463C42—271no—
464C22×C3—21no—
465C22×C3—21no—
466C22×C3—21no—
467C22×C3—41no—
468C22×C3—64no—
469C22×C3—62no—
470C22×C3—62no—
471C22×C3—122no—
472C22×C3—122no—
473C22×C3—121no—
474C22×C3—121no—
475C22×C3—121no—
476C22×C3—121no—
477C22×C3#380211yescharacteristic
478C22×C3#381111yescharacteristic
479C22×C3#381111yescharacteristic
480C2×D3—22no—
481C2×D3—41no—
482C2×D3—41no—
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—122no—
492C3×C4—121no—
493C3×C4—121no—
494C3×C4—361no—
495C3×C4—361no—
496C3×C4#379511yescharacteristic
497C3×C4#380411yescharacteristic
498C3×C4#380511yescharacteristic
499C3×C4C2×#62511yescharacteristic
500C3×C4D3×#6711yescharacteristic
501C3×C4D3×#6711yescharacteristic
502Dic3—41no—
503Dic3—92no—
504Dic3—92no—
505Dic3—92no—
506Dic3—122no—
507Dic3—122no—
508Dic3—121no—
509Dic3—121no—
510Dic3—121no—
511Dic3—121no—
512Dic3—182no—
513Dic3—182no—
514Dic3—182no—
515Dic3—182no—
516Dic3—181no—
517Dic3—181no—
518Dic3—181no—
519Dic3—362no—
520Dic3—361no—
521Dic3—361no—
522Dic3—362no—
523C32—21no—
524C32—21no—
525C32—21no—
526C32—41no—
527C32#481811yescharacteristic, complemented
528C32#516211yescharacteristic, complemented
529C32#516211yescharacteristic, complemented
530C23—61no—
531Q8—61no—
532Q8—61no—
533Q8—181no—
534Q8—181no—
535D4—32no—
536D4—61no—
537C2×C4—61no—
538C2×C4—181no—
539C2×C4—181no—
540C2×C4—271no—
541C2×C4—271no—
542C2×C4—1082no—
543C2×C4D3311yescharacteristic
544C8—61no—
545C8—181no—
546C2×C3—21no—
547C2×C3—21no—
548C2×C3—21no—
549C2×C3—21no—
550C2×C3—21no—
551C2×C3—21no—
552C2×C3—21no—
553C2×C3—21no—
554C2×C3—21no—
555C2×C3—41no—
556C2×C3—41no—
557C2×C3—41no—
558C2×C3—124no—
559C2×C3—122no—
560C2×C3—122no—
561C2×C3#6379611yescharacteristic
562C2×C3#6381111yescharacteristic
563C2×C3#6388511yescharacteristic
564C2×C3#6393011yescharacteristic
565C2×C3#6393111yescharacteristic
566C2×C3#6394011yescharacteristic
567C2×C3#6395111yescharacteristic
568C2×C3#6397111yescharacteristic
569C2×C3#6398011yescharacteristic
570D3—42no—
571C22—62no—
572C22—121no—
573C22—121no—
574C22#8895411yescharacteristic
575C4—121no—
576C4—272no—
577C4—361no—
578C4—361no—
579C4—541no—
580C4D3×#62511yescharacteristic
581C4#8896111yescharacteristic
582C3—21no—
583C3—21no—
584C3—21no—
585C3—41no—
586C3#1059074811yescharacteristic, complemented
587C3#1059149711yescharacteristic, complemented
588C3#1059156911yescharacteristic, complemented
589C2—122no—
590C2#1172284111yescharacteristic
591C2#1172289311yescharacteristic
592C2#1172295411yescharacteristic
subgroupG/H
centerC22#88954
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socleC22×C33C2×D4
prime radicalC2×C32×D12C22
hypocenterC33#461
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. #11723451C2semidirect
5. #11722076C2semidirect
6. #11722050C2semidirect
7. C3×#63566C2semidirect
46. #63566D3semidirect
127. C3⋊(C3⋊Dic3)SD16semidirect
352. C33#461semidirect
527. C32#4818semidirect
528. C32#5162semidirect
529. C32#5162semidirect
586. C3#10590748semidirect
587. C3#10591497semidirect
588. C3#10591569semidirect

Maximal subgroups

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

subgroupindex
#117234992
#117234512
C2×C3×#37162
C2×C3×#37152
#117220762
#117220502
C3×#635662
#105915693
#105914973
#105907483