Finite Group Databasebeta

← #13156544

Subgroups — #13156544

Order 1728.

630 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11723441C211yesmaximal, characteristic
2C3×#63918C211yesmaximal, characteristic, complemented
3#11722076C211yesmaximal, characteristic, complemented
4#11722036C211yesmaximal, characteristic
5#11722035C211yesmaximal, characteristic
6C3×#63567C211yesmaximal, characteristic, complemented
7C3×#63554C211yesmaximal, characteristic, complemented
8#10590893—31nomaximal
9#10590535—31nomaximal
10#10590294—31nomaximal
11C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
12C2×C3×#634C2211yescharacteristic
13C2×C3×#627—21no—
14C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
15C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
16C32×C4⋊Dic3C2211yescharacteristic
17C32×C4×Dic3C2211yescharacteristic
18C2×C32×C3⋊C8C2211yescharacteristic
19#88804—21no—
20#88804—21no—
21C3×Dic3⋊Dic3—21no—
22#64084—31no—
23C3×#888—31no—
24C3×#869—31no—
25#63918D311yescharacteristic, complemented
26#63864—31no—
27#63864—31no—
28#63864—61no—
29#63648—31no—
30#63636—31no—
31C3×#816—31no—
32C3×#815—31no—
33C3×#810—31no—
34C3×#803—31no—
35#63567—31no—
36#63567D311yescharacteristic, complemented
37#63566—31no—
38#63565—31no—
39#63564—31no—
40#63564—31no—
41#63564—31no—
42#63564—31no—
43#63564—61no—
44#63564—61no—
45#63554D311yescharacteristic, complemented
46C22×C3⋊(C3⋊D3)—21no—
47C2×C33×C4C2311yescharacteristic
48C4⋊(C3⋊(C3⋊D3))—21no—
49C4⋊(C3⋊(C3⋊D3))—21no—
50C2×C3×C3⋊Dic3—21no—
51C2×C3×C3⋊Dic3—21no—
52C3×#634—21no—
53C3×#634—21no—
54C2×C32×Dic3—21no—
55C2×C32×Dic3D411yescharacteristic
56C2×C32×Dic3D411yescharacteristic
57C3×#627—42no—
58C3×C3⋊(C3⋊C8)—21no—
59C32×C3⋊C8—21no—
60#5089—91no—
61#4794—91no—
62#4714—91no—
63C2×C4⋊(C3⋊D3)—31no—
64C2×C4⋊(C3⋊D3)—31no—
65C2×C4⋊(C3⋊D3)—31no—
66C2×C4⋊(C3⋊D3)—61no—
67C2×C4⋊(C3⋊D3)—61no—
68C2×C4⋊(C3⋊D3)—61no—
69C2×C4⋊(C3⋊D3)—121no—
70C2×#634C2×D311yescharacteristic
71C2×C3×Dic6—31no—
72C2×C3×Dic6—31no—
73C2×C3×Dic6—61no—
74C2×C3×Dic6—61no—
75C2×C3×Dic6—61no—
76C2×#627—21no—
77C2×C32×C8—31no—
78C32×C4⋊C4—31no—
79C32×C42—31no—
80#3752—61no—
81#3752—61no—
82C4⋊(C3⋊Dic3)C2×D311yescharacteristic
83C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
84C3×C4⋊Dic3—21no—
85C3×C4⋊Dic3—31no—
86C3×C4⋊Dic3—31no—
87C3×C4⋊Dic3—61no—
88C3×C4⋊Dic3—61no—
89C3×C4⋊Dic3C2×D311yescharacteristic
90C3×C4⋊Dic3C2×D311yescharacteristic
91C3×Dic3⋊C4—61no—
92C3×C4×Dic3—21no—
93C3×C4×Dic3C2×D311yescharacteristic
94C3×C4×Dic3C2×D311yescharacteristic
95C2×C3×C3⋊C8—21no—
96C2×C3×C3⋊C8—31no—
97C2×C3×C3⋊C8—31no—
98C2×C3×C3⋊C8—61no—
99C2×C3×C3⋊C8C2×D311yescharacteristic
100C2×C3×C3⋊C8C2×D311yescharacteristic
101Dic3⋊Dic3—21no—
102#3722—62no—
103#3722—62no—
104#3722—61no—
105#3722—61no—
106#3722—61no—
107#3722—61no—
108C22×C33C2×D411yescharacteristic
109C2×C3⋊(C3⋊D3)—41no—
110C2×C3⋊(C3⋊D3)—41no—
111C2×C3⋊(C3⋊D3)—41no—
112C33×C4#4111yescharacteristic
113C33×C4#4111yescharacteristic
114C3×C3⋊Dic3—41no—
115C3×C3⋊Dic3—41no—
116C32×Dic3—21no—
117C32×Dic3—21no—
118C32×Dic3—41no—
119C3×C4⋊Q8—91no—
120#947—91no—
121#888—31no—
122#875—91no—
123#875—91no—
124#875—181no—
125#869—31no—
126C3×#103—91no—
127C3×#97—91no—
128#835—91no—
129#821—91no—
130#821—91no—
131#821—91no—
132#821—181no—
133#816—31no—
134#816—91no—
135#815—31no—
136#810—31no—
137#810—91no—
138#809—91no—
139#809—91no—
140#809—91no—
141#809—181no—
142#803—31no—
143C22×C3⋊D3—61no—
144C22×C3⋊D3—62no—
145C22×C3⋊D3—61no—
146C22×C3⋊D3—61no—
147C22×C3⋊D3—121no—
148C22×C3⋊D3—121no—
149C22×C3⋊D3—122no—
150C2×C32×C4—21no—
151C2×C32×C4—21no—
152C2×C32×C4—21no—
153C2×C32×C4—31no—
154C2×C32×C4—31no—
155C2×C32×C4—41no—
156C2×C32×C4—61no—
157C2×C32×C4C22×D311yescharacteristic
158C2×C32×C4C22×D311yescharacteristic
159C2×C32×C4C22×D311yescharacteristic
160C2×C3⋊Dic3—21no—
161C2×C3⋊Dic3—21no—
162C4⋊(C3⋊D3)—61no—
163C4⋊(C3⋊D3)—61no—
164C4⋊(C3⋊D3)—62no—
165C4⋊(C3⋊D3)—62no—
166C4⋊(C3⋊D3)—61no—
167C4⋊(C3⋊D3)—61no—
168C4⋊(C3⋊D3)—61no—
169C4⋊(C3⋊D3)—61no—
170C4⋊(C3⋊D3)—121no—
171C4⋊(C3⋊D3)—121no—
172C4⋊(C3⋊D3)—121no—
173C4⋊(C3⋊D3)—121no—
174C4⋊(C3⋊D3)—122no—
175C4⋊(C3⋊D3)—122no—
176#634—21no—
177#634—21no—
178C2×C3×Dic3—21no—
179C2×C3×Dic3—21no—
180C2×C3×Dic3—21no—
181C2×C3×Dic3—21no—
182C2×C3×Dic3—41no—
183C2×C3×Dic3—62no—
184C2×C3×Dic3—61no—
185C2×C3×Dic3—61no—
186C2×C3×Dic3—61no—
187C2×C3×Dic3—61no—
188C2×C3×Dic3—121no—
189C2×C3×Dic3D1211yescharacteristic
190C2×C3×Dic3D1211yescharacteristic
191C2×C3×Dic3#6711yescharacteristic
192C2×C3×Dic3#6711yescharacteristic
193C3×Dic6—61no—
194C3×Dic6—61no—
195C3×Dic6—61no—
196C3×Dic6—61no—
197C3×Dic6—121no—
198C3×Dic6—121no—
199C3×Dic6—122no—
200C3×Dic6—122no—
201#627—42no—
202C32×C8—61no—
203C3⋊(C3⋊C8)—21no—
204C3×C3⋊C8—21no—
205C3×C3⋊C8—21no—
206C3×C3⋊C8—41no—
207C3×C3⋊C8—61no—
208C3×C3⋊C8—61no—
209C3×C3⋊C8—121no—
210#489—271noSylow
211C2×C33#12411yescharacteristic
212C2×C33AGL(1,ℤ/8)11yescharacteristic
213C2×C33#13711yescharacteristic
214C3⋊(C3⋊D3)—81no—
215C2×C3×Q8—91no—
216C2×C3×Q8—181no—
217C2×D12—91no—
218C2×D12—91no—
219C2×D12—91no—
220C2×D12—181no—
221C2×D12—181no—
222C2×D12—181no—
223C2×D12—361no—
224C2×Dic6—31no—
225C2×Dic6—31no—
226C2×Dic6—61no—
227C2×Dic6—61no—
228C2×Dic6—61no—
229C2×C3×C8—31no—
230C2×C3×C8—31no—
231C2×C3×C8—61no—
232C2×C3×C8—91no—
233C3×C4⋊C4—31no—
234C3×C4⋊C4—31no—
235C3×C4⋊C4—61no—
236C3×C4⋊C4—91no—
237C3×C4⋊C4—181no—
238C3×C42—31no—
239C3×C42—31no—
240C3×C42—61no—
241(C2×D3)⋊C4—181no—
242(C2×D3)⋊C4—181no—
243(C2×D3)⋊C4—181no—
244(C2×D3)⋊C4—181no—
245(C2×D3)⋊C4—182no—
246(C2×D3)⋊C4—182no—
247(C2×D3)⋊C4—181no—
248(C2×D3)⋊C4—181no—
249C4⋊Dic3—31no—
250C4⋊Dic3—31no—
251C4⋊Dic3—61no—
252C4⋊Dic3—61no—
253C4⋊Dic3D3211yescharacteristic
254Dic3⋊C4—61no—
255C4×Dic3D3211yescharacteristic
256C2×C3⋊C8—31no—
257C2×C3⋊C8—31no—
258C2×C3⋊C8—61no—
259C2×C3⋊C8D3211yescharacteristic
260C22×C32—21no—
261C22×C32—21no—
262C22×C32—21no—
263C22×C32—41no—
264C22×C32C2×D1211yescharacteristic
265C22×C32D3×D411yescharacteristic
266C22×C32C2×#6711yescharacteristic
267C2×C3⋊D3—122no—
268C2×C3⋊D3—122no—
269C2×C3⋊D3—122no—
270C2×C3⋊D3—124no—
271C2×C3⋊D3—121no—
272C2×C3⋊D3—121no—
273C2×C3⋊D3—121no—
274C2×C3⋊D3—121no—
275C2×C3⋊D3—121no—
276C2×C3⋊D3—122no—
277C2×C3⋊D3—122no—
278C2×C3⋊D3—121no—
279C2×C3⋊D3—121no—
280C2×C3⋊D3—121no—
281C2×C3⋊D3—121no—
282C2×C3⋊D3—241no—
283C2×C3⋊D3—241no—
284C2×C3⋊D3—241no—
285C2×C3⋊D3—241no—
286C2×C3⋊D3—242no—
287C2×C3⋊D3—242no—
288C32×C4—21no—
289C32×C4—21no—
290C32×C4—21no—
291C32×C4—21no—
292C32×C4—21no—
293C32×C4—21no—
294C32×C4—41no—
295C32×C4—41no—
296C32×C4—61no—
297C32×C4—61no—
298C32×C4—121no—
299C32×C4#23511yescharacteristic
300C32×C4#23511yescharacteristic
301C32×C4#23511yescharacteristic
302C32×C4#23711yescharacteristic
303C32×C4#23911yescharacteristic
304C32×C4#23911yescharacteristic
305C3⋊Dic3—41no—
306C3⋊Dic3—41no—
307C3×Dic3—21no—
308C3×Dic3—21no—
309C3×Dic3—21no—
310C3×Dic3—21no—
311C3×Dic3—41no—
312C3×Dic3—41no—
313C3×Dic3—41no—
314C3×Dic3—41no—
315C3×Dic3—81no—
316C3×Dic3—122no—
317C3×Dic3—121no—
318C3×Dic3—121no—
319C3×Dic3—121no—
320C3×Dic3—121no—
321C3×Dic3—241no—
322C4⋊Q8—91no—
323#124—271no—
324#103—91no—
325#103—271no—
326#102—271no—
327#102—271no—
328#97—91no—
329C33#48911yescharacteristic, complemented, Sylow
330C22×D3—182no—
331C22×D3—181no—
332C22×D3—181no—
333C22×D3—181no—
334C22×D3—361no—
335C22×D3—361no—
336C22×D3—362no—
337C3×Q8—181no—
338C3×Q8—181no—
339C3×Q8—362no—
340C2×C3×C4—21no—
341C2×C3×C4—21no—
342C2×C3×C4—21no—
343C2×C3×C4—31no—
344C2×C3×C4—31no—
345C2×C3×C4—31no—
346C2×C3×C4—31no—
347C2×C3×C4—41no—
348C2×C3×C4—61no—
349C2×C3×C4—61no—
350C2×C3×C4—61no—
351C2×C3×C4—61no—
352C2×C3×C4—121no—
353C2×C3×C4—181no—
354C2×C3×C4—181no—
355C2×C3×C4C2×D3211yescharacteristic
356C2×C3×C4C2×D3211yescharacteristic
357C2×C3×C4C2×D3211yescharacteristic
358C2×Dic3—21no—
359C2×Dic3—62no—
360C2×Dic3—61no—
361C2×Dic3—61no—
362C2×Dic3—61no—
363C2×Dic3—61no—
364C2×Dic3—121no—
365C2×Dic3#62611yescharacteristic
366C2×Dic3#62611yescharacteristic
367D12—182no—
368D12—182no—
369D12—181no—
370D12—181no—
371D12—181no—
372D12—181no—
373D12—181no—
374D12—181no—
375D12—361no—
376D12—361no—
377D12—361no—
378D12—361no—
379D12—362no—
380D12—362no—
381Dic6—61no—
382Dic6—61no—
383Dic6—61no—
384Dic6—61no—
385Dic6—121no—
386Dic6—121no—
387Dic6—122no—
388Dic6—122no—
389C3×C8—61no—
390C3×C8—61no—
391C3×C8—121no—
392C3×C8—181no—
393C3⋊C8—21no—
394C3⋊C8—61no—
395C3⋊C8—61no—
396C3⋊C8—121no—
397C2×C32—21no—
398C2×C32—21no—
399C2×C32—21no—
400C2×C32—21no—
401C2×C32—21no—
402C2×C32—21no—
403C2×C32—21no—
404C2×C32—21no—
405C2×C32—21no—
406C2×C32—41no—
407C2×C32—41no—
408C2×C32—41no—
409C2×C32#87511yescharacteristic
410C2×C32#88511yescharacteristic
411C2×C32#90811yescharacteristic
412C2×C32#90911yescharacteristic
413C2×C32#91411yescharacteristic
414C2×C32#91811yescharacteristic
415C2×C32#93211yescharacteristic
416C2×C32#94211yescharacteristic
417C2×C32#94711yescharacteristic
418C3⋊D3—242no—
419C3⋊D3—242no—
420C3⋊D3—242no—
421C3⋊D3—244no—
422C3⋊D3—241no—
423C3⋊D3—241no—
424C3⋊D3—241no—
425C2×Q8—91no—
426C2×Q8—181no—
427C2×D4—271no—
428C2×C8—31no—
429C2×C8—91no—
430C4⋊C4—31no—
431C4⋊C4—91no—
432C4⋊C4—181no—
433C22⋊C4—541no—
434C22⋊C4—541no—
435C42—31no—
436C22×C3—21no—
437C22×C3—21no—
438C22×C3—21no—
439C22×C3—41no—
440C22×C3D3×D1211yescharacteristic
441C22×C3C2×#62611yescharacteristic
442C22×C3D3×#6711yescharacteristic
443C2×D3—362no—
444C2×D3—362no—
445C2×D3—362no—
446C2×D3—362no—
447C2×D3—362no—
448C2×D3—364no—
449C2×D3—361no—
450C2×D3—361no—
451C2×D3—361no—
452C2×D3—361no—
453C2×D3—361no—
454C2×D3—361no—
455C2×D3—361no—
456C2×D3—361no—
457C2×D3—361no—
458C2×D3—721no—
459C2×D3—721no—
460C2×D3—721no—
461C2×D3—721no—
462C2×D3—722no—
463C2×D3—722no—
464C3×C4—21no—
465C3×C4—21no—
466C3×C4—21no—
467C3×C4—21no—
468C3×C4—21no—
469C3×C4—21no—
470C3×C4—41no—
471C3×C4—41no—
472C3×C4—61no—
473C3×C4—61no—
474C3×C4—61no—
475C3×C4—61no—
476C3×C4—121no—
477C3×C4—121no—
478C3×C4—121no—
479C3×C4—121no—
480C3×C4—241no—
481C3×C4—361no—
482C3×C4—361no—
483C3×C4#379611yescharacteristic
484C3×C4#379811yescharacteristic
485C3×C4#379911yescharacteristic
486C3×C4#379911yescharacteristic
487C3×C4#379911yescharacteristic
488C3×C4#380411yescharacteristic
489Dic3—21no—
490Dic3—21no—
491Dic3—41no—
492Dic3—122no—
493Dic3—121no—
494Dic3—121no—
495Dic3—121no—
496Dic3—121no—
497Dic3—241no—
498C32—21no—
499C32—21no—
500C32—21no—
501C32—41no—
502C32#471411yescharacteristic, complemented
503C32#479411yescharacteristic, complemented
504C32#508911yescharacteristic, complemented
505C23—541no—
506Q8—181no—
507Q8—181no—
508Q8—362no—
509D4—541no—
510D4—541no—
511C2×C4—31no—
512C2×C4—31no—
513C2×C4—61no—
514C2×C4—181no—
515C2×C4—181no—
516C2×C4D3311yescharacteristic
517C8—61no—
518C8—181no—
519C2×C3—21no—
520C2×C3—21no—
521C2×C3—21no—
522C2×C3—21no—
523C2×C3—21no—
524C2×C3—21no—
525C2×C3—21no—
526C2×C3—21no—
527C2×C3—21no—
528C2×C3—41no—
529C2×C3—41no—
530C2×C3—41no—
531C2×C3#6379411yescharacteristic
532C2×C3#6380111yescharacteristic
533C2×C3#6383011yescharacteristic
534C2×C3#6383311yescharacteristic
535C2×C3#6385211yescharacteristic
536C2×C3#6386411yescharacteristic
537C2×C3#6390711yescharacteristic
538C2×C3#6393411yescharacteristic
539C2×C3#6394811yescharacteristic
540D3—722no—
541D3—722no—
542D3—722no—
543D3—724no—
544D3—721no—
545D3—721no—
546D3—721no—
547C22—1081no—
548C22—1081no—
549C22—1081no—
550C22D3×#62611yescharacteristic
551C4—61no—
552C4—61no—
553C4—121no—
554C4—361no—
555C4—361no—
556C4#8896211yescharacteristic
557C4#8896511yescharacteristic
558C3—21no—
559C3—21no—
560C3—21no—
561C3—41no—
562C3#1059029411yescharacteristic, complemented
563C3#1059053511yescharacteristic, complemented
564C3#1059089311yescharacteristic, complemented
565C2—2161no—
566C2#1172286811yescharacteristic
567C2#1172288511yescharacteristic
568C2#1172300311yescharacteristic
subgroupG/H
centerC22D3×#626
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socleC22×C33C2×D4
prime radicalC2×C4⋊(C3⋊(C3⋊D3))C22
hypocenterC33#489
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. C3×#63918C2semidirect
3. #11722076C2semidirect
6. C3×#63567C2semidirect
7. C3×#63554C2semidirect
25. #63918D3semidirect
36. #63567D3semidirect
45. #63554D3semidirect
329. C33#489semidirect
502. C32#4714semidirect
503. C32#4794semidirect
504. C32#5089semidirect
562. C3#10590294semidirect
563. C3#10590535semidirect
564. C3#10590893semidirect

Maximal subgroups

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

subgroupindex
#117234412
C3×#639182
#117220762
#117220362
#117220352
C3×#635672
C3×#635542
#105908933
#105905353
#105902943