Finite Group Databasebeta

← #13156526

Subgroups — #13156526

Order 1728.

542 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#63869C211yesmaximal, characteristic, complemented
2C3×#63860C211yesmaximal, characteristic
3#11722076C211yesmaximal, characteristic, complemented
4#11722039C211yesmaximal, characteristic, complemented
5#11722031C211yesmaximal, characteristic
6Dic3×C3⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
7C3×#63562C211yesmaximal, characteristic
8#10591478—31nomaximal
9#10590533—31nomaximal
10#10590278—31nomaximal
11C2×C3×#634C2211yescharacteristic
12C2×C32×D12C2211yescharacteristic
13C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
14C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
15C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
16C32×C4⋊Dic3C2211yescharacteristic
17C32×C4×Dic3C2211yescharacteristic
18C3×#3724—21no—
19C3×Dic3⋊Dic3—21no—
20C3×(C2×D3)⋊Dic3—21no—
21C3×(C2×D3)⋊Dic3—21no—
22C3×#936—31no—
23C3×#890—31no—
24C3×#875—31no—
25C3×#870—31no—
26#63869D311yescharacteristic, complemented
27#63860D311yescharacteristic
28#63659—31no—
29#63637—31no—
30C4×C3⋊(C3⋊C8)—31no—
31C3×#832—31no—
32C3×#809—31no—
33#63567—31no—
34#63566—31no—
35#63565—31no—
36#63565—31no—
37#63565—31no—
38#63565—61no—
39#63562—31no—
40#63562—31no—
41#63562—61no—
42#63562D311yescharacteristic
43Dic3×C3⋊C8—31no—
44Dic3×C3⋊C8—31no—
45Dic3×C3⋊C8—61no—
46C22×C32×D3—21no—
47C2×C33×C4C2311yescharacteristic
48C2×C3×C3⋊Dic3—21no—
49C2×C3×C3⋊Dic3—21no—
50C3×#634—21no—
51C3×#634—21no—
52C2×C32×Dic3—21no—
53C2×C32×Dic3D411yescharacteristic
54C2×C32×Dic3D411yescharacteristic
55C32×D12—21no—
56C32×D12—21no—
57C3⋊(C3⋊(C3⋊C8))—21no—
58C3×C3⋊(C3⋊C8)—21no—
59#5068—91no—
60#5056—91no—
61#4768—91no—
62C2×C32×D4—31no—
63C2×#634C2×D311yescharacteristic
64C2×C3×D12—21no—
65C2×C3×D12C2×D311yescharacteristic
66C2×C3×D12C2×D311yescharacteristic
67C2×C3×Dic6—31no—
68C2×C3×Dic6—31no—
69C2×C3×Dic6—61no—
70C32×C4⋊C4—31no—
71C32×C42—31no—
72C4⋊(C3⋊Dic3)C2×D311yescharacteristic
73C2×C3⋊(C3⋊C8)—31no—
74C2×C3⋊(C3⋊C8)—31no—
75C2×C3⋊(C3⋊C8)—31no—
76C2×C3⋊(C3⋊C8)—61no—
77C2×C3⋊(C3⋊C8)—61no—
78C2×C3⋊(C3⋊C8)—61no—
79C2×C3⋊(C3⋊C8)—121no—
80C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
81C3×#217—61no—
82C3×#217—61no—
83C3×(C2×D3)⋊C4—61no—
84C3×(C2×D3)⋊C4—61no—
85C3×C4⋊Dic3—21no—
86C3×C4⋊Dic3—31no—
87C3×C4⋊Dic3—31no—
88C3×C4⋊Dic3—61no—
89C3×C4⋊Dic3—61no—
90C3×C4⋊Dic3C2×D311yescharacteristic
91C3×C4⋊Dic3C2×D311yescharacteristic
92C3×Dic3⋊C4—61no—
93C3×Dic3⋊C4—61no—
94C3×Dic3⋊C4—61no—
95C3×C4×Dic3—21no—
96C3×C4×Dic3C2×D311yescharacteristic
97C3×C4×Dic3C2×D311yescharacteristic
98C2×C3×C3⋊C8—31no—
99C2×C3×C3⋊C8—31no—
100C2×C3×C3⋊C8—61no—
101#3724—21no—
102Dic3⋊Dic3—21no—
103(C2×D3)⋊Dic3—21no—
104(C2×D3)⋊Dic3—21no—
105C22×C33C2×D411yescharacteristic
106C2×C32×D3—41no—
107C2×C32×D3—41no—
108C2×C32×D3—41no—
109C33×C4#4111yescharacteristic
110C33×C4#4111yescharacteristic
111C3×C3⋊Dic3—41no—
112C3×C3⋊Dic3—41no—
113C32×Dic3—21no—
114C32×Dic3—21no—
115C32×Dic3—41no—
116C3×#125—91no—
117C3×#124—91no—
118#936—31no—
119#890—31no—
120#875—31no—
121#870—31no—
122C3×#102—91no—
123#835—91no—
124#835—91no—
125#832—31no—
126#832—91no—
127#832—91no—
128#832—181no—
129#816—91no—
130C8×Dic3—91no—
131#810—91no—
132#810—91no—
133#810—91no—
134#810—181no—
135#809—31no—
136#809—91no—
137C4×C3⋊C8—91no—
138C4×C3⋊C8—91no—
139C4×C3⋊C8—181no—
140C23×C32—61no—
141C22×C3×D3—21no—
142C22×C3×D3—21no—
143C22×C3×D3—41no—
144C32×D4—61no—
145C32×D4—61no—
146C2×C32×C4—21no—
147C2×C32×C4—21no—
148C2×C32×C4—21no—
149C2×C32×C4—31no—
150C2×C32×C4—31no—
151C2×C32×C4—41no—
152C2×C32×C4—61no—
153C2×C32×C4C22×D311yescharacteristic
154C2×C32×C4C22×D311yescharacteristic
155C2×C32×C4C22×D311yescharacteristic
156C2×C3⋊Dic3—21no—
157C2×C3⋊Dic3—21no—
158#634—21no—
159#634—21no—
160C2×C3×Dic3—21no—
161C2×C3×Dic3—21no—
162C2×C3×Dic3—21no—
163C2×C3×Dic3—21no—
164C2×C3×Dic3—41no—
165C2×C3×Dic3—61no—
166C2×C3×Dic3—61no—
167C2×C3×Dic3—61no—
168C2×C3×Dic3—61no—
169C2×C3×Dic3—121no—
170C2×C3×Dic3—121no—
171C2×C3×Dic3#6711yescharacteristic
172C2×C3×Dic3#6711yescharacteristic
173C2×C3×Dic3#6711yescharacteristic
174C2×C3×Dic3#6711yescharacteristic
175C3×D12—22no—
176C3×D12—21no—
177C3×D12—21no—
178C3×D12—21no—
179C3×D12—21no—
180C3×D12—41no—
181C3×Dic6—62no—
182C3×Dic6—61no—
183C3×Dic6—61no—
184C3×Dic6—61no—
185C3×Dic6—61no—
186C3×Dic6—121no—
187C3⋊(C3⋊C8)—21no—
188C3⋊(C3⋊C8)—61no—
189C3⋊(C3⋊C8)—62no—
190C3⋊(C3⋊C8)—61no—
191C3⋊(C3⋊C8)—61no—
192C3⋊(C3⋊C8)—121no—
193C3⋊(C3⋊C8)—121no—
194C3⋊(C3⋊C8)—122no—
195C3×C3⋊C8—62no—
196C3×C3⋊C8—61no—
197C3×C3⋊C8—61no—
198#488—271noSylow
199C2×C33#12411yescharacteristic
200C2×C33#13511yescharacteristic
201C2×C33#13511yescharacteristic
202C32×D3—81no—
203C2×C3×Q8—91no—
204C2×C3×D4—31no—
205C2×C3×D4—31no—
206C2×C3×D4—61no—
207C2×D12D3211yescharacteristic
208C2×Dic6—31no—
209C2×Dic6—31no—
210C2×Dic6—61no—
211C2×C3×C8—91no—
212C3×C4⋊C4—31no—
213C3×C4⋊C4—31no—
214C3×C4⋊C4—61no—
215C3×C4⋊C4—91no—
216C3×C4⋊C4—181no—
217C3×C4⋊C4—181no—
218C3×C22⋊C4—181no—
219C3×C22⋊C4—181no—
220C3×C42—31no—
221C3×C42—31no—
222C3×C42—61no—
223#217—61no—
224#217—61no—
225(C2×D3)⋊C4—61no—
226(C2×D3)⋊C4—61no—
227C4⋊Dic3—31no—
228C4⋊Dic3—31no—
229C4⋊Dic3—61no—
230C4⋊Dic3—61no—
231C4⋊Dic3D3211yescharacteristic
232Dic3⋊C4—61no—
233Dic3⋊C4—61no—
234Dic3⋊C4—61no—
235C4×Dic3D3211yescharacteristic
236C2×C3⋊C8—31no—
237C2×C3⋊C8—31no—
238C2×C3⋊C8—61no—
239C2×C3⋊C8—91no—
240C2×C3⋊C8—91no—
241C2×C3⋊C8—91no—
242C2×C3⋊C8—181no—
243C2×C3⋊C8—181no—
244C2×C3⋊C8—181no—
245C2×C3⋊C8—361no—
246C22×C32—21no—
247C22×C32—21no—
248C22×C32—21no—
249C22×C32—41no—
250C22×C32—121no—
251C22×C32—121no—
252C22×C32—121no—
253C22×C32D3×D411yescharacteristic
254C22×C32C2×#6711yescharacteristic
255C22×C32C2×#6711yescharacteristic
256C2×C3×D3—42no—
257C2×C3×D3—41no—
258C2×C3×D3—41no—
259C2×C3×D3—41no—
260C2×C3×D3—41no—
261C2×C3×D3—41no—
262C2×C3×D3—41no—
263C2×C3×D3—81no—
264C2×C3×D3—81no—
265C32×C4—21no—
266C32×C4—21no—
267C32×C4—21no—
268C32×C4—21no—
269C32×C4—21no—
270C32×C4—21no—
271C32×C4—41no—
272C32×C4—41no—
273C32×C4—61no—
274C32×C4—61no—
275C32×C4—121no—
276C32×C4#23511yescharacteristic
277C32×C4#23711yescharacteristic
278C32×C4#23711yescharacteristic
279C32×C4#23711yescharacteristic
280C32×C4#23911yescharacteristic
281C32×C4#23911yescharacteristic
282C3⋊Dic3—41no—
283C3⋊Dic3—41no—
284C3×Dic3—22no—
285C3×Dic3—22no—
286C3×Dic3—21no—
287C3×Dic3—21no—
288C3×Dic3—21no—
289C3×Dic3—21no—
290C3×Dic3—41no—
291C3×Dic3—41no—
292C3×Dic3—81no—
293C3×Dic3—122no—
294C3×Dic3—121no—
295C3×Dic3—121no—
296C3×Dic3—121no—
297C3×Dic3—121no—
298C3×Dic3—241no—
299#125—91no—
300#124—91no—
301#103—271no—
302#103—271no—
303#102—91no—
304#102—271no—
305C4×C8—271no—
306C33#48811yescharacteristic, complemented, Sylow
307C23×C3—61no—
308C23×C3—61no—
309C23×C3—121no—
310C22×D3—21no—
311C3×Q8—181no—
312C3×Q8—181no—
313C3×D4—62no—
314C3×D4—61no—
315C3×D4—61no—
316C3×D4—61no—
317C3×D4—61no—
318C3×D4—121no—
319C2×C3×C4—21no—
320C2×C3×C4—21no—
321C2×C3×C4—21no—
322C2×C3×C4—31no—
323C2×C3×C4—31no—
324C2×C3×C4—31no—
325C2×C3×C4—31no—
326C2×C3×C4—41no—
327C2×C3×C4—61no—
328C2×C3×C4—61no—
329C2×C3×C4—61no—
330C2×C3×C4—61no—
331C2×C3×C4—121no—
332C2×C3×C4—181no—
333C2×C3×C4—181no—
334C2×C3×C4C2×D3211yescharacteristic
335C2×C3×C4C2×D3211yescharacteristic
336C2×C3×C4C2×D3211yescharacteristic
337C2×Dic3—21no—
338C2×Dic3—61no—
339C2×Dic3—61no—
340C2×Dic3—61no—
341C2×Dic3—61no—
342C2×Dic3—121no—
343C2×Dic3—121no—
344C2×Dic3#62511yescharacteristic
345C2×Dic3#62511yescharacteristic
346D12—21no—
347D12—21no—
348Dic6—62no—
349Dic6—61no—
350Dic6—61no—
351Dic6—61no—
352Dic6—61no—
353Dic6—121no—
354C3×C8—181no—
355C3⋊C8—62no—
356C3⋊C8—61no—
357C3⋊C8—61no—
358C3⋊C8—182no—
359C3⋊C8—181no—
360C3⋊C8—181no—
361C3⋊C8—181no—
362C3⋊C8—361no—
363C3⋊C8—361no—
364C3⋊C8—362no—
365C2×C32—21no—
366C2×C32—21no—
367C2×C32—21no—
368C2×C32—21no—
369C2×C32—21no—
370C2×C32—21no—
371C2×C32—21no—
372C2×C32—21no—
373C2×C32—21no—
374C2×C32—41no—
375C2×C32—41no—
376C2×C32—41no—
377C2×C32—241no—
378C2×C32#88511yescharacteristic
379C2×C32#91211yescharacteristic
380C2×C32#91611yescharacteristic
381C2×C32#93611yescharacteristic
382C2×C32#94711yescharacteristic
383C2×C32#95011yescharacteristic
384C2×C32#95011yescharacteristic
385C2×C32#95011yescharacteristic
386C2×C32#95011yescharacteristic
387C3×D3—82no—
388C3×D3—81no—
389C3×D3—81no—
390C2×Q8—91no—
391C2×D4—31no—
392C2×C8—91no—
393C2×C8—271no—
394C4⋊C4—31no—
395C4⋊C4—91no—
396C4⋊C4—181no—
397C4⋊C4—181no—
398C22⋊C4—181no—
399C22⋊C4—181no—
400C42—31no—
401C22×C3—21no—
402C22×C3—21no—
403C22×C3—21no—
404C22×C3—41no—
405C22×C3—122no—
406C22×C3—121no—
407C22×C3—121no—
408C22×C3—121no—
409C22×C3—121no—
410C22×C3—121no—
411C22×C3—121no—
412C22×C3—241no—
413C22×C3—241no—
414C22×C3C2×#62511yescharacteristic
415C22×C3D3×#6711yescharacteristic
416C22×C3D3×#6711yescharacteristic
417C2×D3—41no—
418C2×D3—41no—
419C2×D3—41no—
420C3×C4—21no—
421C3×C4—21no—
422C3×C4—21no—
423C3×C4—21no—
424C3×C4—21no—
425C3×C4—21no—
426C3×C4—41no—
427C3×C4—41no—
428C3×C4—62no—
429C3×C4—62no—
430C3×C4—61no—
431C3×C4—61no—
432C3×C4—61no—
433C3×C4—61no—
434C3×C4—121no—
435C3×C4—121no—
436C3×C4—241no—
437C3×C4—361no—
438C3×C4—361no—
439C3×C4#379611yescharacteristic
440C3×C4#379611yescharacteristic
441C3×C4#379611yescharacteristic
442C3×C4#379911yescharacteristic
443C3×C4#380411yescharacteristic
444C3×C4#380511yescharacteristic
445Dic3—21no—
446Dic3—21no—
447Dic3—41no—
448Dic3—122no—
449Dic3—121no—
450Dic3—121no—
451Dic3—121no—
452Dic3—121no—
453Dic3—241no—
454C32—21no—
455C32—21no—
456C32—21no—
457C32—41no—
458C32#476811yescharacteristic, complemented
459C32#505611yescharacteristic, complemented
460C32#506811yescharacteristic, complemented
461C23—61no—
462Q8—181no—
463Q8—181no—
464D4—61no—
465D4—61no—
466C2×C4—31no—
467C2×C4—31no—
468C2×C4—61no—
469C2×C4—181no—
470C2×C4—181no—
471C2×C4D3311yescharacteristic
472C8—181no—
473C8—541no—
474C2×C3—21no—
475C2×C3—21no—
476C2×C3—21no—
477C2×C3—21no—
478C2×C3—21no—
479C2×C3—21no—
480C2×C3—21no—
481C2×C3—21no—
482C2×C3—21no—
483C2×C3—41no—
484C2×C3—41no—
485C2×C3—41no—
486C2×C3—242no—
487C2×C3—241no—
488C2×C3—241no—
489C2×C3#6382211yescharacteristic
490C2×C3#6382211yescharacteristic
491C2×C3#6386311yescharacteristic
492C2×C3#6390711yescharacteristic
493C2×C3#6393311yescharacteristic
494C2×C3#6393311yescharacteristic
495C2×C3#6393511yescharacteristic
496C2×C3#6394711yescharacteristic
497C2×C3#6395911yescharacteristic
498D3—81no—
499C22—121no—
500C22—121no—
501C22—121no—
502C22D3×#62511yescharacteristic
503C4—61no—
504C4—61no—
505C4—121no—
506C4—361no—
507C4—361no—
508C4#8896211yescharacteristic
509C4#8896411yescharacteristic
510C3—21no—
511C3—21no—
512C3—21no—
513C3—41no—
514C3#1059027811yescharacteristic, complemented
515C3#1059053311yescharacteristic, complemented
516C3#1059147811yescharacteristic, complemented
517C2—241no—
518C2#1172285011yescharacteristic
519C2#1172285111yescharacteristic
520C2#1172300111yescharacteristic
subgroupG/H
centerC22D3×#625
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socleC22×C33C2×D4
prime radicalC2×C32×D12C22
hypocenterC33#488
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
1. C3×#63869C2semidirect
3. #11722076C2semidirect
4. #11722039C2semidirect
6. Dic3×C3⋊(C3⋊C8)C2semidirect
26. #63869D3semidirect
306. C33#488semidirect
458. C32#4768semidirect
459. C32#5056semidirect
460. C32#5068semidirect
514. C3#10590278semidirect
515. C3#10590533semidirect
516. C3#10591478semidirect

Maximal subgroups

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

subgroupindex
C3×#638692
C3×#638602
#117220762
#117220392
#117220312
Dic3×C3⋊(C3⋊C8)2
C3×#635622
#105914783
#105905333
#105902783