Finite Group Databasebeta

← #13161663

Subgroups — #13161663

Order 1728.

658 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×C3×#3744C211yesmaximal, characteristic
2#11723691C211yesmaximal, characteristic
3#11723668C211yesmaximal, characteristic, complemented
4C3×#63644C211yesmaximal, characteristic, complemented
5C3×C8×C3⋊Dic3C211yesmaximal, characteristic, complemented
6#11722076C211yesmaximal, characteristic, complemented
7#11722075C211yesmaximal, characteristic
8#10590775—31nomaximal
9#10590758—31nomaximal
10Dic3×#204—31nomaximal
11C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
12C2×C3×#634C2211yescharacteristic
13C2×#6294—21no—
14C2×C33×C8C2211yescharacteristic
15C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
16C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
17C3×C4×C3⋊Dic3C2211yescharacteristic
18C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
19C3×#3744C414yescomplemented
20#88815—21no—
21#88814—21no—
22#88813—21no—
23C2×#3744D311yescharacteristic
24C2×C3×#204—31no—
25C2×C3×#204—31no—
26C2×C3×#204—61no—
27#63898—31no—
28#63894—31no—
29Dic3×D12—31no—
30#63844—31no—
31#63843—31no—
32Dic3×Dic6—31no—
33#63644D311yescharacteristic, complemented
34C8×C3⋊Dic3D311yescharacteristic, complemented
35C3×#817—31no—
36C3×#817—31no—
37C3×#817—61no—
38C3×C8×Dic3—31no—
39C3×C8×Dic3—31no—
40C3×C8×Dic3—61no—
41#63567—31no—
42#63566—31no—
43#63565—31no—
44#63564—31no—
45#63563—31no—
46#63562—31no—
47C22×C3×C3⋊D3—21no—
48C2×C33×C4C2311yescharacteristic
49C2×C3×C3⋊Dic3—21no—
50C2×C3×C3⋊Dic3—21no—
51C2×C3×C3⋊Dic3—21no—
52C2×C3×C3⋊Dic3D411yescharacteristic
53C2×C3×C3⋊Dic3D411yescharacteristic
54C3×C4⋊(C3⋊D3)—21no—
55C3×C4⋊(C3⋊D3)C2×C412yes—
56C3×#634—21no—
57C3×#634C2×C412yescomplemented
58#6294—42no—
59C33×C8C2×C412yescomplemented
60C3×C3⋊(C3⋊C8)—21no—
61Dic3×SD16—91no—
62#4852—91no—
63C4×#204—91no—
64C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
65C2×#634C2×D311yescharacteristic
66C2×C3×D12—31no—
67C2×C3×D12—31no—
68C2×C3×D12—61no—
69C2×C3×Dic6—31no—
70C2×C3×Dic6—31no—
71C2×C3×Dic6—61no—
72C2×#624—61no—
73C2×D3×Dic3—61no—
74C2×D3×Dic3—61no—
75C2×C32×C8—21no—
76C2×C32×C8—21no—
77C2×C32×C8—21no—
78C2×C32×C8—41no—
79C2×C32×C8C2×D311yescharacteristic
80C2×C32×C8C2×D311yescharacteristic
81C2×C32×C8C2×D311yescharacteristic
82C4⋊(C3⋊Dic3)C2×D311yescharacteristic
83C4⋊(C3⋊Dic3)C2×D311yescharacteristic
84C4×C3⋊Dic3C2×D311yescharacteristic
85C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
86#3744Dic314yescomplemented
87C3×C4⋊Dic3—31no—
88C3×C4⋊Dic3—31no—
89C3×C4⋊Dic3—31no—
90C3×C4⋊Dic3—31no—
91C3×C4⋊Dic3—61no—
92C3×C4⋊Dic3—61no—
93C3×C4×Dic3—31no—
94C3×C4×Dic3—31no—
95C3×C4×Dic3—61no—
96C2×C3×C3⋊C8—31no—
97C2×C3×C3⋊C8—31no—
98C2×C3×C3⋊C8—61no—
99C3×#204—34no—
100C3×#204—34no—
101C3×#204—64no—
102#3724—61no—
103Dic3⋊Dic3—61no—
104Dic3⋊Dic3—61no—
105#3722—61no—
106(C2×D3)⋊Dic3—61no—
107(C2×D3)⋊Dic3—61no—
108Dic32—61no—
109Dic32—61no—
110Dic32—61no—
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—22no—
118C3×C3⋊Dic3—21no—
119C3×C3⋊Dic3—41no—
120C3×C3⋊Dic3—41no—
121C3×C3⋊Dic3SD1612yescomplemented
122C2×C3×SD16—91no—
123Q8×Dic3—91no—
124D4×Dic3—91no—
125C2×#204—31no—
126C2×#204—31no—
127C2×#204—61no—
128#893—91no—
129#887—91no—
130C4×D12—91no—
131C4×Dic6—91no—
132C3×#106—91no—
133C3×C4×C8—91no—
134#835—91no—
135#832—91no—
136#821—91no—
137#817—31no—
138#817—31no—
139#817—61no—
140#816—91no—
141C8×Dic3—31no—
142C8×Dic3—31no—
143C8×Dic3—61no—
144#810—91no—
145#809—91no—
146C22×C3⋊D3—21no—
147C22×C3×D3—61no—
148C22×C3×D3—61no—
149C22×C3×D3—121no—
150C2×C32×C4—21no—
151C2×C32×C4—21no—
152C2×C32×C4—21no—
153C2×C32×C4—41no—
154C2×C32×C4C22×D311yescharacteristic
155C2×C32×C4C22×D311yescharacteristic
156C2×C32×C4C22×D311yescharacteristic
157C2×C3⋊Dic3—21no—
158C2×C3⋊Dic3—21no—
159C2×C3⋊Dic3—21no—
160C2×C3⋊Dic3D1211yescharacteristic
161C2×C3⋊Dic3D1211yescharacteristic
162C4⋊(C3⋊D3)—21no—
163C4⋊(C3⋊D3)C2×Dic312yes—
164#634—21no—
165#634C2×Dic312yescomplemented
166C2×C3×Dic3—31no—
167C2×C3×Dic3—31no—
168C2×C3×Dic3—31no—
169C2×C3×Dic3—31no—
170C2×C3×Dic3—61no—
171C2×C3×Dic3—61no—
172C2×C3×Dic3—61no—
173C2×C3×Dic3—61no—
174C2×C3×Dic3—61no—
175C2×C3×Dic3—61no—
176C2×C3×Dic3—61no—
177C2×C3×Dic3—61no—
178C2×C3×Dic3—121no—
179C2×C3×Dic3—121no—
180C2×C3×Dic3—121no—
181C3×D12—32no—
182C3×D12—32no—
183C3×D12—62no—
184C3×D12—61no—
185C3×D12—61no—
186C3×D12—121no—
187C3×Dic6—32no—
188C3×Dic6—32no—
189C3×Dic6—62no—
190C3×Dic6—61no—
191C3×Dic6—61no—
192C3×Dic6—121no—
193#624—122no—
194D3×Dic3—122no—
195D3×Dic3—122no—
196C32×C8—22no—
197C32×C8—22no—
198C32×C8—22no—
199C32×C8—42no—
200C32×C8C4×D312yescomplemented
201C32×C8C4×D312yescomplemented
202C32×C8C2×Dic312yescomplemented
203C3⋊(C3⋊C8)—21no—
204C3×C3⋊C8—62no—
205C3×C3⋊C8—61no—
206C3×C3⋊C8—61no—
207C4×SD16—271noSylow
208C2×C33C4×D411yescharacteristic
209C2×C33C2×SD1611yescharacteristic
210C2×C33#13511yescharacteristic
211C3×C3⋊D3—42no—
212C2×C3×Q8—91no—
213C2×C3×D4—91no—
214C22×Dic3—181no—
215C2×D12—31no—
216C2×D12—31no—
217C2×D12—61no—
218C2×C4×D3—181no—
219C2×C4×D3—181no—
220C2×Dic6—31no—
221C2×Dic6—31no—
222C2×Dic6—61no—
223C3×SD16—94no—
224C2×C3×C8—21no—
225C2×C3×C8—21no—
226C2×C3×C8—21no—
227C2×C3×C8—41no—
228C2×C3×C8—91no—
229C2×C3×C8D3211yescharacteristic
230C2×C3×C8D3211yescharacteristic
231C2×C3×C8D3211yescharacteristic
232C3×C4⋊C4—91no—
233C3×C4⋊C4—91no—
234C3×C42—91no—
235#217—181no—
236(C2×D3)⋊C4—181no—
237(C2×D3)⋊C4—181no—
238C4⋊Dic3—31no—
239C4⋊Dic3—31no—
240C4⋊Dic3—31no—
241C4⋊Dic3—31no—
242C4⋊Dic3—61no—
243C4⋊Dic3—61no—
244C4⋊Dic3—181no—
245Dic3⋊C4—181no—
246Dic3⋊C4—181no—
247C4×Dic3—31no—
248C4×Dic3—31no—
249C4×Dic3—61no—
250C4×Dic3—181no—
251C4×Dic3—181no—
252C4×Dic3—181no—
253C2×C3⋊C8—31no—
254C2×C3⋊C8—31no—
255C2×C3⋊C8—61no—
256#204—34no—
257#204—34no—
258#204—64no—
259C22×C32—21no—
260C22×C32—21no—
261C22×C32—21no—
262C22×C32—41no—
263C22×C32C2×D1211yescharacteristic
264C22×C32D3×D411yescharacteristic
265C22×C32D3×D411yescharacteristic
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×C4C2×C4×D311yescharacteristic
287C32×C4C2×C4×D311yescharacteristic
288C32×C4#23511yescharacteristic
289C32×C4#23711yescharacteristic
290C32×C4#23911yescharacteristic
291C32×C4C22×Dic311yescharacteristic
292C3⋊Dic3—21no—
293C3⋊Dic3—22no—
294C3⋊Dic3—41no—
295C3⋊Dic3—41no—
296C3⋊Dic3#20412yescomplemented
297C3×Dic3—32no—
298C3×Dic3—32no—
299C3×Dic3—62no—
300C3×Dic3—62no—
301C3×Dic3—62no—
302C3×Dic3—62no—
303C3×Dic3—61no—
304C3×Dic3—61no—
305C3×Dic3—122no—
306C3×Dic3—121no—
307C3×Dic3—121no—
308C3×Dic3—121no—
309C3×Dic3—121no—
310C3×Dic3—241no—
311C3×Dic3—241no—
312C2×SD16—91no—
313C4×Q8—271no—
314C4×D4—271no—
315#106—91no—
316#103—271no—
317#102—271no—
318C4×C8—91no—
319C33C4×SD1611yescharacteristic, complemented, Sylow
320C23×C3—181no—
321C22×D3—61no—
322C22×D3—61no—
323C22×D3—121no—
324C3×Q8—92no—
325C3×Q8—181no—
326C3×D4—92no—
327C3×D4—181no—
328C2×C3×C4—21no—
329C2×C3×C4—21no—
330C2×C3×C4—21no—
331C2×C3×C4—41no—
332C2×C3×C4—91no—
333C2×C3×C4—91no—
334C2×C3×C4—181no—
335C2×C3×C4—181no—
336C2×C3×C4—181no—
337C2×C3×C4C2×D3211yescharacteristic
338C2×C3×C4C2×D3211yescharacteristic
339C2×C3×C4C2×D3211yescharacteristic
340C2×Dic3—31no—
341C2×Dic3—31no—
342C2×Dic3—31no—
343C2×Dic3—31no—
344C2×Dic3—61no—
345C2×Dic3—61no—
346C2×Dic3—61no—
347C2×Dic3—61no—
348C2×Dic3—61no—
349C2×Dic3—61no—
350C2×Dic3—61no—
351C2×Dic3—61no—
352C2×Dic3—121no—
353C2×Dic3—121no—
354C2×Dic3—121no—
355C2×Dic3—362no—
356D12—32no—
357D12—32no—
358D12—62no—
359D12—61no—
360D12—61no—
361D12—121no—
362C4×D3—362no—
363C4×D3—362no—
364Dic6—32no—
365Dic6—32no—
366Dic6—62no—
367Dic6—61no—
368Dic6—61no—
369Dic6—121no—
370C3×C8—22no—
371C3×C8—22no—
372C3×C8—22no—
373C3×C8—42no—
374C3×C8—181no—
375C3×C8D3×Dic312yescomplemented
376C3×C8D3×Dic312yescomplemented
377C3×C8#62412yescomplemented
378C3⋊C8—62no—
379C3⋊C8—61no—
380C3⋊C8—61no—
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×C32C4×D1211yescharacteristic
394C2×C32#89311yescharacteristic
395C2×C32C2×#20411yescharacteristic
396C2×C32#90411yescharacteristic
397C2×C32D3×SD1611yescharacteristic
398C2×C32D3×SD1611yescharacteristic
399C2×C32#91611yescharacteristic
400C2×C32#91611yescharacteristic
401C2×C32D4×Dic311yescharacteristic
402C3⋊D3—42no—
403C3×D3—122no—
404C3×D3—122no—
405C3×D3—242no—
406C2×Q8—91no—
407C2×D4—91no—
408C22×C4—541no—
409SD16—94no—
410C2×C8—91no—
411C2×C8#111311yescharacteristic
412C4⋊C4—91no—
413C4⋊C4—91no—
414C4⋊C4—541no—
415C22⋊C4—541no—
416C42—91no—
417C42—541no—
418C22×C3—21no—
419C22×C3—21no—
420C22×C3—21no—
421C22×C3—41no—
422C22×C3—182no—
423C22×C3—361no—
424C22×C3—361no—
425C22×C3D3×D1211yescharacteristic
426C22×C3D3×D1211yescharacteristic
427C22×C3#380211yescharacteristic
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—92no—
446C3×C4—181no—
447C3×C4—182no—
448C3×C4—361no—
449C3×C4—361no—
450C3×C4#379511yescharacteristic
451C3×C4#379611yescharacteristic
452C3×C4#379911yescharacteristic
453C3×C4C2×D3×Dic311yescharacteristic
454C3×C4C2×D3×Dic311yescharacteristic
455C3×C4C2×#62411yescharacteristic
456Dic3—32no—
457Dic3—32no—
458Dic3—62no—
459Dic3—62no—
460Dic3—61no—
461Dic3—61no—
462Dic3—62no—
463Dic3—62no—
464Dic3—122no—
465Dic3—121no—
466Dic3—121no—
467Dic3—121no—
468Dic3—121no—
469Dic3—241no—
470Dic3—241no—
471C32—21no—
472C32—21no—
473C32—21no—
474C32—41no—
475C32C4×#20411yescharacteristic, complemented
476C32#485211yescharacteristic, complemented
477C32Dic3×SD1611yescharacteristic, complemented
478C23—181no—
479Q8—92no—
480Q8—181no—
481D4—92no—
482D4—181no—
483C2×C4—91no—
484C2×C4—91no—
485C2×C4—181no—
486C2×C4—181no—
487C2×C4—181no—
488C2×C4—1082no—
489C2×C4C2×#111311yescharacteristic
490C8—181no—
491C8#629412yescomplemented
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×C3D3×#20411yescharacteristic
506C2×C3D3×#20411yescharacteristic
507C2×C3#6379611yescharacteristic
508C2×C3#6380611yescharacteristic
509C2×C3#6380611yescharacteristic
510C2×C3#6380911yescharacteristic
511C2×C3Dic3×D1211yescharacteristic
512C2×C3#6389411yescharacteristic
513C2×C3#6389811yescharacteristic
514D3—122no—
515D3—122no—
516D3—242no—
517C22—182no—
518C22—361no—
519C22—361no—
520C22C4⋊#111311yescharacteristic
521C4—92no—
522C4—181no—
523C4—182no—
524C4—361no—
525C4—361no—
526C4#8904111yescharacteristic
527C4C2×#629411yescharacteristic
528C3—21no—
529C3—21no—
530C3—21no—
531C3—41no—
532C3Dic3×#20411yescharacteristic, complemented
533C3#1059075811yescharacteristic, complemented
534C3#1059077511yescharacteristic, complemented
535C2—362no—
536C2#1172364611yescharacteristic
537C2#1172365311yescharacteristic
538C2#1172369111yescharacteristic
subgroupG/H
centerC22C4⋊#1113
commutatorC33×C4C22×C4
FrattiniC2×C4C2×#1113
FittingC2×C33×C8C22
socleC22×C33C2×D4
prime radicalC2×C3×C4⋊(C3⋊D3)C22
cosocleC2×C33×C4C23
hypercenterC2×C8#1113
hypocenterC33C4×SD16
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
3. #11723668C2semidirect
4. C3×#63644C2semidirect
5. C3×C8×C3⋊Dic3C2semidirect
6. #11722076C2semidirect
19. C3×#3744C4semidirect
33. #63644D3semidirect
34. C8×C3⋊Dic3D3semidirect
57. C3×#634C2×C4semidirect
59. C33×C8C2×C4semidirect
86. #3744Dic3semidirect
121. C3×C3⋊Dic3SD16semidirect
165. #634C2×Dic3semidirect
200. C32×C8C4×D3semidirect
201. C32×C8C4×D3semidirect
202. C32×C8C2×Dic3semidirect
296. C3⋊Dic3#204semidirect
319. C33C4×SD16semidirect
375. C3×C8D3×Dic3semidirect
376. C3×C8D3×Dic3semidirect
377. C3×C8#624semidirect
475. C32C4×#204semidirect
476. C32#4852semidirect
477. C32Dic3×SD16semidirect
491. C8#6294semidirect
532. C3Dic3×#204semidirect
533. C3#10590758semidirect
534. C3#10590775semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#37442
#117236912
#117236682
C3×#636442
C3×C8×C3⋊Dic32
#117220762
#117220752
#105907753
#105907583
Dic3×#2043