Finite Group Databasebeta

← #13155681

Subgroups — #13155681

Order 1728.

646 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88793C211yesmaximal, characteristic
2C3×#63894C211yesmaximal, characteristic
3C3×Dic3×Dic6C211yesmaximal, characteristic, complemented
4#11722075C211yesmaximal, characteristic
5#11722056C211yesmaximal, characteristic, complemented
6#11722019C211yesmaximal, characteristic, complemented
7C3×#63565C211yesmaximal, characteristic, complemented
8#10591540—31nomaximal
9#10590478—31nomaximal
10#10590358—31nomaximal
11C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
12C2×C32×Dic6C2211yescharacteristic
13C2×C3×#624—21no—
14C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
15C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
16C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
17C32×C4⋊Dic3C2211yescharacteristic
18C32×C4×Dic3C2211yescharacteristic
19#88793C414yescomplemented
20C3×Dic3⋊Dic3—21no—
21C3×#3722—21no—
22C3×Dic32—21no—
23C2×#3755—31no—
24C3×Q8×Dic3—31no—
25C3×#893—31no—
26C3×C4×D12—31no—
27C3×C4×Dic6—31no—
28#63894D311yescharacteristic
29Dic3×Dic6D311yescharacteristic, complemented
30C2×#3715—31no—
31C2×#3715—31no—
32C2×#3715—61no—
33#63634—31no—
34#63630—31no—
35C3×#835—31no—
36C3×#810—31no—
37#63576—31no—
38#63576—31no—
39#63576—61no—
40#63565D311yescharacteristic, complemented
41#63564—31no—
42#63563—31no—
43#63562—31no—
44#63554—31no—
45#63554—31no—
46#63554—61no—
47C22×C3×C3⋊D3—21no—
48C2×C33×C4C2311yescharacteristic
49C2×C3×C3⋊Dic3—21no—
50C3×C4⋊(C3⋊D3)—21no—
51C3×C4⋊(C3⋊D3)C2×C412yes—
52C2×C32×Dic3—21no—
53C2×C32×Dic3—21no—
54C2×C32×Dic3D411yescharacteristic
55C2×C32×Dic3D411yescharacteristic
56C32×Dic6—21no—
57C32×Dic6C2×C412yescomplemented
58C3×#624—42no—
59C3⋊(C3⋊(C3⋊C8))C2×C412yescomplemented
60C3×C3⋊(C3⋊C8)—21no—
61#5026—91no—
62#5018—91no—
63#4817—91no—
64C2×C32×Q8—31no—
65C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
66C2×C3×D12—31no—
67C2×C3×D12—31no—
68C2×C3×D12—61no—
69C2×C3×C4×D3—61no—
70C2×C3×C4×D3—61no—
71C2×C3×Dic6—21no—
72C2×C3×Dic6C2×D311yescharacteristic
73C2×C3×Dic6C2×D311yescharacteristic
74C2×#624—21no—
75C32×C4⋊C4—31no—
76C32×C42—31no—
77#3755—34no—
78C4⋊(C3⋊Dic3)C2×D311yescharacteristic
79C2×C3⋊(C3⋊C8)—31no—
80C2×C3⋊(C3⋊C8)—31no—
81C2×C3⋊(C3⋊C8)—31no—
82C2×C3⋊(C3⋊C8)—61no—
83C2×C3⋊(C3⋊C8)—61no—
84C2×C3⋊(C3⋊C8)—61no—
85C2×C3⋊(C3⋊C8)—121no—
86C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
87C3×(C2×D3)⋊C4—61no—
88C3×(C2×D3)⋊C4—61no—
89C3×C4⋊Dic3—21no—
90C3×C4⋊Dic3—31no—
91C3×C4⋊Dic3—31no—
92C3×C4⋊Dic3—61no—
93C3×C4⋊Dic3—61no—
94C3×C4⋊Dic3C2×D311yescharacteristic
95C3×C4⋊Dic3C2×D311yescharacteristic
96C3×Dic3⋊C4—61no—
97C3×C4×Dic3—21no—
98C3×C4×Dic3—61no—
99C3×C4×Dic3—61no—
100C3×C4×Dic3C2×D311yescharacteristic
101C3×C4×Dic3C2×D311yescharacteristic
102C2×C3×C3⋊C8—31no—
103C2×C3×C3⋊C8—31no—
104C2×C3×C3⋊C8—61no—
105Dic3⋊Dic3—21no—
106#3722—21no—
107Dic32—21no—
108#3715—34no—
109#3715—34no—
110#3715—64no—
111C22×C33C2×D411yescharacteristic
112C2×C3×C3⋊D3—22no—
113C2×C3×C3⋊D3—41no—
114C2×C3×C3⋊D3—41no—
115C33×C4C22×C411yescharacteristic
116C33×C4#4111yescharacteristic
117C3×C3⋊Dic3—41no—
118C32×Dic3—22no—
119C32×Dic3—21no—
120C32×Dic3—21no—
121C32×Dic3—41no—
122C3×C4×Q8—91no—
123C3×C4×D4—91no—
124Q8×Dic3—31no—
125C2×#215—91no—
126C2×#215—91no—
127C2×#215—181no—
128C2×#214—91no—
129#893—31no—
130C4×D12—31no—
131C4×Dic6—31no—
132C3×#103—91no—
133#835—31no—
134#832—91no—
135#821—91no—
136#815—91no—
137#810—31no—
138#809—91no—
139#807—91no—
140#807—91no—
141#807—91no—
142#807—181no—
143#803—91no—
144#803—91no—
145#803—181no—
146C22×C3⋊D3—21no—
147C22×C3×D3—61no—
148C22×C3×D3—61no—
149C22×C3×D3—121no—
150C32×Q8—32no—
151C32×Q8—61no—
152C2×C32×C4—21no—
153C2×C32×C4—21no—
154C2×C32×C4—21no—
155C2×C32×C4—31no—
156C2×C32×C4—31no—
157C2×C32×C4—41no—
158C2×C32×C4—61no—
159C2×C32×C4—61no—
160C2×C32×C4C22×D311yescharacteristic
161C2×C32×C4C22×D311yescharacteristic
162C2×C32×C4C22×D311yescharacteristic
163C2×C3⋊Dic3—21no—
164C4⋊(C3⋊D3)—21no—
165C4⋊(C3⋊D3)C4×D312yes—
166C2×C3×Dic3—21no—
167C2×C3×Dic3—21no—
168C2×C3×Dic3—21no—
169C2×C3×Dic3—21no—
170C2×C3×Dic3—21no—
171C2×C3×Dic3—21no—
172C2×C3×Dic3—41no—
173C2×C3×Dic3—41no—
174C2×C3×Dic3—61no—
175C2×C3×Dic3—61no—
176C2×C3×Dic3—121no—
177C2×C3×Dic3#6711yescharacteristic
178C2×C3×Dic3#6711yescharacteristic
179C2×C3×Dic3#6711yescharacteristic
180C2×C3×Dic3#6711yescharacteristic
181C3×D12—32no—
182C3×D12—32no—
183C3×D12—62no—
184C3×D12—61no—
185C3×D12—61no—
186C3×D12—121no—
187C3×C4×D3—122no—
188C3×C4×D3—122no—
189C3×Dic6—21no—
190C3×Dic6—21no—
191C3×Dic6—22no—
192C3×Dic6—41no—
193C3×Dic6C4×D312yescomplemented
194C3×Dic6C4×D312yescomplemented
195#624—42no—
196C3⋊(C3⋊C8)—21no—
197C3⋊(C3⋊C8)—32no—
198C3⋊(C3⋊C8)—32no—
199C3⋊(C3⋊C8)—32no—
200C3⋊(C3⋊C8)—62no—
201C3⋊(C3⋊C8)—62no—
202C3⋊(C3⋊C8)—62no—
203C3⋊(C3⋊C8)—122no—
204C3×C3⋊C8—62no—
205C3×C3⋊C8—61no—
206C3×C3⋊C8—61no—
207#440—271noSylow
208C2×C33C4×D411yescharacteristic
209C2×C33AGL(1,ℤ/8)11yescharacteristic
210C2×C33#13711yescharacteristic
211C3×C3⋊D3—42no—
212C2×C3×Q8—31no—
213C2×C3×Q8—31no—
214C2×C3×Q8—61no—
215C2×C3×D4—91no—
216C22×C3×C4—181no—
217C2×D12—31no—
218C2×D12—31no—
219C2×D12—61no—
220C2×C4×D3—61no—
221C2×C4×D3—61no—
222C2×Dic6D3211yescharacteristic
223C2×C3×C8—91no—
224C3×C4⋊C4—31no—
225C3×C4⋊C4—31no—
226C3×C4⋊C4—61no—
227C3×C4⋊C4—91no—
228C3×C4⋊C4—181no—
229C3×C22⋊C4—181no—
230C3×C42—31no—
231C3×C42—31no—
232C3×C42—61no—
233C3×C42—181no—
234#215—94no—
235#215—94no—
236#215—184no—
237#214—94no—
238(C2×D3)⋊C4—61no—
239(C2×D3)⋊C4—61no—
240C4⋊Dic3—31no—
241C4⋊Dic3—31no—
242C4⋊Dic3—61no—
243C4⋊Dic3—61no—
244C4⋊Dic3D3211yescharacteristic
245Dic3⋊C4—61no—
246C4×Dic3—61no—
247C4×Dic3—61no—
248C4×Dic3D3211yescharacteristic
249C2×C3⋊C8—31no—
250C2×C3⋊C8—31no—
251C2×C3⋊C8—61no—
252C2×C3⋊C8—91no—
253C2×C3⋊C8—91no—
254C2×C3⋊C8—91no—
255C2×C3⋊C8—181no—
256C2×C3⋊C8—181no—
257C2×C3⋊C8—181no—
258C2×C3⋊C8—361no—
259C22×C32—21no—
260C22×C32—21no—
261C22×C32—21no—
262C22×C32—41no—
263C22×C32D3×D411yescharacteristic
264C22×C32C2×#6711yescharacteristic
265C22×C32C2×#6711yescharacteristic
266C2×C3⋊D3—22no—
267C2×C3⋊D3—41no—
268C2×C3⋊D3—41no—
269C2×C3×D3—62no—
270C2×C3×D3—62no—
271C2×C3×D3—122no—
272C2×C3×D3—121no—
273C2×C3×D3—121no—
274C2×C3×D3—121no—
275C2×C3×D3—121no—
276C2×C3×D3—241no—
277C2×C3×D3—241no—
278C32×C4—21no—
279C32×C4—21no—
280C32×C4—21no—
281C32×C4—21no—
282C32×C4—21no—
283C32×C4—21no—
284C32×C4—41no—
285C32×C4—41no—
286C32×C4—61no—
287C32×C4—61no—
288C32×C4—62no—
289C32×C4—121no—
290C32×C4C2×C4×D311yescharacteristic
291C32×C4C2×C4×D311yescharacteristic
292C32×C4C2×C4×D311yescharacteristic
293C32×C4#23511yescharacteristic
294C32×C4#23511yescharacteristic
295C32×C4#23711yescharacteristic
296C3⋊Dic3—41no—
297C3×Dic3—22no—
298C3×Dic3—22no—
299C3×Dic3—21no—
300C3×Dic3—21no—
301C3×Dic3—21no—
302C3×Dic3—21no—
303C3×Dic3—42no—
304C3×Dic3—41no—
305C3×Dic3—41no—
306C3×Dic3—41no—
307C3×Dic3—41no—
308C3×Dic3—81no—
309C3×Dic3—121no—
310C3×Dic3—121no—
311C3×Dic3—241no—
312C2×SD16—271no—
313C4×Q8—91no—
314C4×D4—91no—
315#107—271no—
316#103—91no—
317#102—271no—
318#97—271no—
319C33#44011yescharacteristic, complemented, Sylow
320C23×C3—181no—
321C22×D3—61no—
322C22×D3—61no—
323C22×D3—121no—
324C3×Q8—32no—
325C3×Q8—32no—
326C3×Q8—62no—
327C3×Q8—61no—
328C3×Q8—61no—
329C3×Q8—121no—
330C3×D4—92no—
331C3×D4—181no—
332C2×C3×C4—21no—
333C2×C3×C4—21no—
334C2×C3×C4—21no—
335C2×C3×C4—31no—
336C2×C3×C4—31no—
337C2×C3×C4—31no—
338C2×C3×C4—31no—
339C2×C3×C4—41no—
340C2×C3×C4—61no—
341C2×C3×C4—61no—
342C2×C3×C4—61no—
343C2×C3×C4—61no—
344C2×C3×C4—61no—
345C2×C3×C4—61no—
346C2×C3×C4—121no—
347C2×C3×C4—121no—
348C2×C3×C4—181no—
349C2×C3×C4—362no—
350C2×C3×C4C2×D3211yescharacteristic
351C2×C3×C4C2×D3211yescharacteristic
352C2×C3×C4C2×D3211yescharacteristic
353C2×Dic3—21no—
354C2×Dic3—21no—
355C2×Dic3—61no—
356C2×Dic3—61no—
357C2×Dic3—121no—
358C2×Dic3#62511yescharacteristic
359C2×Dic3#62511yescharacteristic
360D12—32no—
361D12—32no—
362D12—62no—
363D12—61no—
364D12—61no—
365D12—121no—
366C4×D3—122no—
367C4×D3—122no—
368Dic6—21no—
369Dic6#62412yescomplemented
370C3×C8—181no—
371C3⋊C8—62no—
372C3⋊C8—61no—
373C3⋊C8—61no—
374C3⋊C8—92no—
375C3⋊C8—92no—
376C3⋊C8—92no—
377C3⋊C8—182no—
378C3⋊C8—182no—
379C3⋊C8—182no—
380C3⋊C8—362no—
381C2×C32—21no—
382C2×C32—21no—
383C2×C32—21no—
384C2×C32—21no—
385C2×C32—21no—
386C2×C32—21no—
387C2×C32—21no—
388C2×C32—21no—
389C2×C32—21no—
390C2×C32—41no—
391C2×C32—41no—
392C2×C32—41no—
393C2×C32#67⋊C411yescharacteristic
394C2×C32#91411yescharacteristic
395C2×C32#91811yescharacteristic
396C2×C32C4×#6711yescharacteristic
397C2×C32C4×#6711yescharacteristic
398C2×C32#93211yescharacteristic
399C2×C32#94211yescharacteristic
400C2×C32#94911yescharacteristic
401C2×C32#95111yescharacteristic
402C3⋊D3—42no—
403C3×D3—122no—
404C3×D3—122no—
405C3×D3—242no—
406C2×Q8—31no—
407C2×D4—91no—
408C22×C4—181no—
409SD16—274no—
410C2×C8—91no—
411C2×C8—271no—
412C4⋊C4—31no—
413C4⋊C4—91no—
414C4⋊C4—181no—
415C22⋊C4—181no—
416C42—31no—
417C42—181no—
418C22×C3—21no—
419C22×C3—21no—
420C22×C3—21no—
421C22×C3—41no—
422C22×C3—182no—
423C22×C3—361no—
424C22×C3—361no—
425C22×C3C2×#62511yescharacteristic
426C22×C3D3×#6711yescharacteristic
427C22×C3D3×#6711yescharacteristic
428C2×D3—62no—
429C2×D3—62no—
430C2×D3—122no—
431C2×D3—121no—
432C2×D3—121no—
433C2×D3—121no—
434C2×D3—121no—
435C2×D3—241no—
436C2×D3—241no—
437C3×C4—21no—
438C3×C4—21no—
439C3×C4—21no—
440C3×C4—21no—
441C3×C4—21no—
442C3×C4—21no—
443C3×C4—41no—
444C3×C4—41no—
445C3×C4—62no—
446C3×C4—62no—
447C3×C4—61no—
448C3×C4—61no—
449C3×C4—61no—
450C3×C4—61no—
451C3×C4—121no—
452C3×C4—121no—
453C3×C4—122no—
454C3×C4—121no—
455C3×C4—121no—
456C3×C4—241no—
457C3×C4—361no—
458C3×C4#379511yescharacteristic
459C3×C4#379811yescharacteristic
460C3×C4C4×D3211yescharacteristic
461C3×C4C4×D3211yescharacteristic
462C3×C4#380411yescharacteristic
463C3×C4C2×#62411yescharacteristic
464Dic3—21no—
465Dic3—21no—
466Dic3—22no—
467Dic3—41no—
468Dic3—121no—
469Dic3—121no—
470Dic3—241no—
471C32—21no—
472C32—21no—
473C32—21no—
474C32—41no—
475C32#481711yescharacteristic, complemented
476C32#501811yescharacteristic, complemented
477C32#502611yescharacteristic, complemented
478C23—181no—
479Q8—32no—
480Q8—61no—
481D4—92no—
482D4—181no—
483C2×C4—31no—
484C2×C4—31no—
485C2×C4—61no—
486C2×C4—61no—
487C2×C4—181no—
488C2×C4—362no—
489C2×C4D3311yescharacteristic
490C8—181no—
491C8—272no—
492C2×C3—21no—
493C2×C3—21no—
494C2×C3—21no—
495C2×C3—21no—
496C2×C3—21no—
497C2×C3—21no—
498C2×C3—21no—
499C2×C3—21no—
500C2×C3—21no—
501C2×C3—41no—
502C2×C3—41no—
503C2×C3—41no—
504C2×C3—362no—
505C2×C3#6382311yescharacteristic
506C2×C3#6382711yescharacteristic
507C2×C3#6387911yescharacteristic
508C2×C3C4×#62511yescharacteristic
509C2×C3#6393411yescharacteristic
510C2×C3#6393911yescharacteristic
511C2×C3#6394311yescharacteristic
512C2×C3#6394811yescharacteristic
513C2×C3#6395211yescharacteristic
514D3—122no—
515D3—122no—
516D3—242no—
517C22—182no—
518C22—361no—
519C22—361no—
520C22D3×#62511yescharacteristic
521C4—61no—
522C4—61no—
523C4—62no—
524C4—121no—
525C4—361no—
526C4D3×#62411yescharacteristic
527C4#8896011yescharacteristic
528C3—21no—
529C3—21no—
530C3—21no—
531C3—41no—
532C3#1059035811yescharacteristic, complemented
533C3#1059047811yescharacteristic, complemented
534C3#1059154011yescharacteristic, complemented
535C2—362no—
536C2#1172286711yescharacteristic
537C2#1172288311yescharacteristic
538C2#1172293011yescharacteristic
subgroupG/H
centerC22D3×#625
commutatorC33×C4C22×C4
Frattini, hypercenterC2×C4D33
Fitting, cosocleC2×C33×C4C23
socleC22×C33C2×D4
prime radicalC2×C3×C4⋊(C3⋊D3)C22
hypocenterC33#440
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
3. C3×Dic3×Dic6C2semidirect
5. #11722056C2semidirect
6. #11722019C2semidirect
7. C3×#63565C2semidirect
19. #88793C4semidirect
29. Dic3×Dic6D3semidirect
40. #63565D3semidirect
57. C32×Dic6C2×C4semidirect
59. C3⋊(C3⋊(C3⋊C8))C2×C4semidirect
193. C3×Dic6C4×D3semidirect
194. C3×Dic6C4×D3semidirect
319. C33#440semidirect
369. Dic6#624semidirect
475. C32#4817semidirect
476. C32#5018semidirect
477. C32#5026semidirect
532. C3#10590358semidirect
533. C3#10590478semidirect
534. C3#10591540semidirect

Maximal subgroups

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

subgroupindex
C2×#887932
C3×#638942
C3×Dic3×Dic62
#117220752
#117220562
#117220192
C3×#635652
#105915403
#105904783
#105903583