Finite Group Databasebeta

← C3×#10591416 #13158423

Subgroups — C3×#10591416 #13158423

Order 1728.

1198 conjugacy classes of subgroups.

The subgroup lattice has 526 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×D4×C3⋊D3C211yesmaximal, characteristic
2C3×#63900C211yesmaximal, characteristic, complemented
3C2×C3×#3710C211yesmaximal, characteristic, complemented
4C32×#832C212yesmaximal, complemented
5C3×#63564C212yesmaximal, complemented
6C3×D3×#102—32nomaximal
7#10591416C311yesmaximal, characteristic, direct product factor
8C23×C3×C3⋊D3—21no—
9C2×C33×D4C2211yescharacteristic
10C2×C3×#638—21no—
11C3×D4×C3⋊D3—22no—
12C3×D4×C3⋊D3C414yescomplemented
13C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
14C2×C3×C4×C3⋊D3C2211yescharacteristic
15C2×C3×#624—21no—
16C32×C4⋊Dic3C2212yescomplemented
17C2×C32×C3⋊C8C2212yescomplemented
18C3×#3724—21no—
19C3×#3710—22no—
20C2×D4×C3⋊D3C2×C311yescharacteristic
21C2×C3×D3×D4—32no—
22C2×C3×D3×D4—61no—
23C2×C3×D3×C8—32no—
24C3×D3×C4⋊C4—32no—
25#63900C2×C311yescharacteristic, complemented
26C2×#3710C2×C311yescharacteristic, complemented
27C32×#102—32no—
28C3×#832—22no—
29C3×#832D312yescomplemented
30C3×#832C2×C312yescomplemented
31C3×#821—32no—
32C3×#809—32no—
33#63564C2×C312yescomplemented
34C23×C33—21no—
35C22×C3×C3⋊D3—21no—
36C22×C3×C3⋊D3—24no—
37C22×C3×C3⋊D3—42no—
38C22×C3×C3⋊D3—42no—
39C22×C3×C3⋊D3D411yescharacteristic
40C33×D4—21no—
41C33×D4C2×C412yescomplemented
42C2×C33×C4C2311yescharacteristic
43C3×#638—24no—
44C2×C3×C3⋊Dic3D411yescharacteristic
45C3×C4⋊(C3⋊D3)—21no—
46C3×C4⋊(C3⋊D3)C2×C412yescomplemented
47C3×C4×C3⋊D3C2×C412yes—
48C3×C4×C3⋊D3D412yes—
49C2×C32×Dic3—22no—
50C3×#624—42no—
51C32×C3⋊C8—22no—
52C2×C3×#102—91no—
53D3×#102—32no—
54C23×C3⋊D3—21no—
55C23×C3×D3—62no—
56C23×C3×D3—62no—
57C2×C32×D4—21no—
58C2×C32×D4—22no—
59C2×C32×D4—41no—
60C2×C32×D4C2×D312yes—
61C2×C32×D4C22×C311yescharacteristic
62C2×#638—21no—
63D4×C3⋊D3—22no—
64D4×C3⋊D3C3×C414yescomplemented
65C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
66C2×C4×C3⋊D3C22×C311yescharacteristic
67C2×C3×#67—62no—
68C2×C3×#67—62no—
69C3×D3×D4—38no—
70C3×D3×D4—64no—
71C3×D3×D4—64no—
72C3×D3×D4—64no—
73C2×C3×D12—32no—
74C2×C3×D12—61no—
75C2×C3×C4×D3—32no—
76C2×C3×C4×D3—61no—
77C2×C3×C4×D3—62no—
78C2×#624—21no—
79C2×C32×C8—32no—
80C32×C4⋊C4—32no—
81C3×C4⋊Dic3—22no—
82C3×C4⋊Dic3C2×D312yescomplemented
83C3×C4⋊Dic3C22×C312yescomplemented
84C3×Dic3⋊C4—62no—
85C2×C3×C3⋊C8—22no—
86C2×C3×C3⋊C8C2×D312yescomplemented
87C2×C3×C3⋊C8C22×C312yescomplemented
88C3×D3×C8—64no—
89#3724—21no—
90#3710—22no—
91C22×C33—22no—
92C22×C33—41no—
93C22×C33—41no—
94C22×C33C2×D411yescharacteristic
95C2×C3×C3⋊D3—22no—
96C2×C3×C3⋊D3—41no—
97C2×C3×C3⋊D3—41no—
98C2×C3×C3⋊D3—48no—
99C2×C3×C3⋊D3C22⋊C412yes—
100C2×C3×C3⋊D3D812yes—
101C2×C3×C3⋊D3SD1612yes—
102C33×C4C22×C411yescharacteristic
103C33×C4C2×D411yescharacteristic
104C3×C3⋊Dic3C22⋊C412yes—
105C32×Dic3—42no—
106C22×C3×D4—91no—
107C2×D3×D4—32no—
108C2×D3×D4—61no—
109C22×C3×C8—91no—
110C2×C3×C4⋊C4—91no—
111C2×D3×C8—32no—
112D3×C4⋊C4—32no—
113C3×#102—32no—
114C3×#102—32no—
115C3×#102—62no—
116C3×#102—92no—
117#832C3×D312yescomplemented
118#821—32no—
119#809—32no—
120C23×C32—21no—
121C23×C32—22no—
122C23×C32—22no—
123C23×C32—42no—
124C23×C32—42no—
125C22×C3⋊D3—21no—
126C22×C3⋊D3—24no—
127C22×C3⋊D3—42no—
128C22×C3⋊D3—42no—
129C22×C3⋊D3C3×D411yescharacteristic
130C22×C3×D3—32no—
131C22×C3×D3—61no—
132C22×C3×D3—68no—
133C22×C3×D3—62no—
134C22×C3×D3—68no—
135C22×C3×D3—62no—
136C22×C3×D3—124no—
137C22×C3×D3—124no—
138C22×C3×D3—124no—
139C22×C3×D3—124no—
140C32×D4—21no—
141C32×D4—22no—
142C32×D4—22no—
143C32×D4—22no—
144C32×D4—24no—
145C32×D4—42no—
146C32×D4—42no—
147C32×D4—42no—
148C32×D4C4×D314yescomplemented
149C32×D4C2×C3×C412yescomplemented
150C2×C32×C4—21no—
151C2×C32×C4—22no—
152C2×C32×C4—41no—
153C2×C32×C4—62no—
154C2×C32×C4C22×D312yes—
155C2×C32×C4C23×C311yescharacteristic
156#638—24no—
157C2×C3⋊Dic3C3×D411yescharacteristic
158C4⋊(C3⋊D3)—21no—
159C4⋊(C3⋊D3)C2×C3×C412yescomplemented
160C4×C3⋊D3C2×C3×C412yes—
161C4×C3⋊D3C3×D412yes—
162C3×#67—68no—
163C3×#67—68no—
164C2×C3×Dic3—22no—
165C2×C3×Dic3—22no—
166C2×C3×Dic3—32no—
167C2×C3×Dic3—42no—
168C2×C3×Dic3—61no—
169C3×D12—34no—
170C3×D12—62no—
171C3×D12—62no—
172C3×D12—62no—
173C3×C4×D3—34no—
174C3×C4×D3—34no—
175C3×C4×D3—62no—
176C3×C4×D3—62no—
177C3×C4×D3—124no—
178#624—42no—
179C32×C8—62no—
180C3×C3⋊C8—22no—
181C3×C3⋊C8—22no—
182C3×C3⋊C8—42no—
183C2×#102—91noSylow
184C2×C33—42no—
185C2×C33C2×C22⋊C411yescharacteristic
186C2×C33C2×D811yescharacteristic
187C2×C33C2×SD1611yescharacteristic
188C3×C3⋊D3—42no—
189C3×C3⋊D3#10214yescomplemented
190C24×C3—181no—
191C23×D3—62no—
192C23×D3—62no—
193C2×C3×D4—21no—
194C2×C3×D4—22no—
195C2×C3×D4—41no—
196C2×C3×D4—94no—
197C2×C3×D4—91no—
198C2×C3×D4—182no—
199C2×C3×D4—181no—
200C2×C3×D4D3211yescharacteristic
201C2×C3×D4C2×C3×D312yes—
202C22×C3×C4—91no—
203C22×C3×C4—181no—
204C2×#67—62no—
205C2×#67—62no—
206D3×D4—38no—
207D3×D4—64no—
208D3×D4—64no—
209D3×D4—64no—
210C2×D12—32no—
211C2×D12—61no—
212C2×C4×D3—32no—
213C2×C4×D3—61no—
214C2×C4×D3—62no—
215C2×C3×C8—32no—
216C2×C3×C8—32no—
217C2×C3×C8—62no—
218C2×C3×C8—182no—
219C3×C4⋊C4—32no—
220C3×C4⋊C4—32no—
221C3×C4⋊C4—62no—
222C3×C4⋊C4—181no—
223C4⋊Dic3C2×C3×D312yescomplemented
224Dic3⋊C4—62no—
225C2×C3⋊C8C2×C3×D312yescomplemented
226D3×C8—64no—
227C22×C32—22no—
228C22×C32—24no—
229C22×C32—24no—
230C22×C32—21no—
231C22×C32—22no—
232C22×C32—41no—
233C22×C32—44no—
234C22×C32—44no—
235C22×C32—41no—
236C22×C32—41no—
237C22×C32—42no—
238C22×C32—42no—
239C22×C32—42no—
240C22×C32—42no—
241C22×C32—82no—
242C22×C32—82no—
243C22×C32—82no—
244C22×C32—82no—
245C22×C32D3×D412yes—
246C22×C32C2×C3×D411yescharacteristic
247C2×C3⋊D3—22no—
248C2×C3⋊D3—41no—
249C2×C3⋊D3—41no—
250C2×C3⋊D3—48no—
251C2×C3⋊D3C3×C22⋊C412yes—
252C2×C3⋊D3C3×D812yes—
253C2×C3⋊D3C3×SD1612yes—
254C2×C3×D3—34no—
255C2×C3×D3—34no—
256C2×C3×D3—34no—
257C2×C3×D3—62no—
258C2×C3×D3—62no—
259C2×C3×D3—62no—
260C2×C3×D3—64no—
261C2×C3×D3—64no—
262C2×C3×D3—1216no—
263C2×C3×D3—1216no—
264C2×C3×D3—122no—
265C2×C3×D3—122no—
266C2×C3×D3—122no—
267C2×C3×D3—122no—
268C32×C4—21no—
269C32×C4—21no—
270C32×C4—22no—
271C32×C4—22no—
272C32×C4—41no—
273C32×C4—41no—
274C32×C4—122no—
275C32×C4C2×C4×D312yes—
276C32×C4D3×D412yes—
277C32×C4C22×C3×C411yescharacteristic
278C32×C4C2×C3×D411yescharacteristic
279C3⋊Dic3C3×C22⋊C412yes—
280C3×Dic3—34no—
281C3×Dic3—42no—
282C3×Dic3—42no—
283C3×Dic3—62no—
284C3×Dic3—82no—
285C22×D4—91no—
286C22×C8—91no—
287C2×C4⋊C4—91no—
288#102—32no—
289#102—92no—
290C33C2×#10211yescharacteristic, complemented, Sylow
291C23×C3—22no—
292C23×C3—21no—
293C23×C3—22no—
294C23×C3—42no—
295C23×C3—42no—
296C23×C3—91no—
297C23×C3—184no—
298C23×C3—181no—
299C23×C3—362no—
300C23×C3—362no—
301C22×D3—32no—
302C22×D3—61no—
303C22×D3—68no—
304C22×D3—62no—
305C22×D3—68no—
306C22×D3—62no—
307C22×D3—124no—
308C22×D3—124no—
309C22×D3—124no—
310C22×D3—124no—
311C3×D4—22no—
312C3×D4—21no—
313C3×D4—22no—
314C3×D4—22no—
315C3×D4—24no—
316C3×D4—42no—
317C3×D4—42no—
318C3×D4—42no—
319C3×D4—92no—
320C3×D4—184no—
321C3×D4—181no—
322C3×D4#62412yescomplemented
323C3×D4C3×C4×D314yescomplemented
324C2×C3×C4—22no—
325C2×C3×C4—21no—
326C2×C3×C4—41no—
327C2×C3×C4—62no—
328C2×C3×C4—62no—
329C2×C3×C4—91no—
330C2×C3×C4—92no—
331C2×C3×C4—92no—
332C2×C3×C4—122no—
333C2×C3×C4—362no—
334C2×C3×C4C2×D3211yescharacteristic
335C2×C3×C4C22×C3×D312yes—
336#67—68no—
337#67—68no—
338C2×Dic3—22no—
339C2×Dic3—32no—
340C2×Dic3—61no—
341D12—34no—
342D12—62no—
343D12—62no—
344D12—62no—
345C4×D3—34no—
346C4×D3—34no—
347C4×D3—62no—
348C4×D3—62no—
349C4×D3—124no—
350C3×C8—62no—
351C3×C8—62no—
352C3×C8—122no—
353C3⋊C8—22no—
354C2×C32—22no—
355C2×C32—22no—
356C2×C32—22no—
357C2×C32—21no—
358C2×C32—21no—
359C2×C32—21no—
360C2×C32—41no—
361C2×C32—41no—
362C2×C32—41no—
363C2×C32—42no—
364C2×C32—44no—
365C2×C32—44no—
366C2×C32—84no—
367C2×C32—84no—
368C2×C32D3×C22⋊C412yes—
369C2×C32D3×D812yes—
370C2×C32D3×SD1612yes—
371C2×C32C2×C3×C22⋊C411yescharacteristic
372C2×C32C2×C3×D811yescharacteristic
373C2×C32C2×C3×SD1611yescharacteristic
374C3⋊D3—42no—
375C3⋊D3C3×#10214yescomplemented
376C3×D3—38no—
377C3×D3—64no—
378C3×D3—124no—
379C3×D3—124no—
380C24—181no—
381C2×D4—94no—
382C2×D4—91no—
383C2×D4—182no—
384C2×D4—181no—
385C2×D4C3×D3211yescharacteristic
386C22×C4—91no—
387C22×C4—181no—
388C2×C8—32no—
389C2×C8—182no—
390C4⋊C4—32no—
391C4⋊C4—181no—
392C22×C3—24no—
393C22×C3—24no—
394C22×C3—22no—
395C22×C3—21no—
396C22×C3—22no—
397C22×C3—44no—
398C22×C3—44no—
399C22×C3—42no—
400C22×C3—42no—
401C22×C3—42no—
402C22×C3—42no—
403C22×C3—41no—
404C22×C3—41no—
405C22×C3—41no—
406C22×C3—82no—
407C22×C3—82no—
408C22×C3—82no—
409C22×C3—82no—
410C22×C3—92no—
411C22×C3—92no—
412C22×C3—92no—
413C22×C3—182no—
414C22×C3—368no—
415C22×C3—361no—
416C22×C3—361no—
417C22×C3#381111yescharacteristic
418C22×C3C3×D3×D412yes—
419C2×D3—34no—
420C2×D3—34no—
421C2×D3—34no—
422C2×D3—62no—
423C2×D3—62no—
424C2×D3—62no—
425C2×D3—64no—
426C2×D3—64no—
427C2×D3—1216no—
428C2×D3—1216no—
429C2×D3—122no—
430C2×D3—122no—
431C2×D3—122no—
432C2×D3—122no—
433C3×C4—21no—
434C3×C4—22no—
435C3×C4—22no—
436C3×C4—21no—
437C3×C4—41no—
438C3×C4—41no—
439C3×C4—92no—
440C3×C4—122no—
441C3×C4—122no—
442C3×C4—242no—
443C3×C4C2×#62411yescharacteristic
444C3×C4#381111yescharacteristic
445C3×C4C2×C3×C4×D312yes—
446C3×C4C3×D3×D412yes—
447Dic3—34no—
448Dic3—42no—
449Dic3—62no—
450C32—22no—
451C32—21no—
452C32—41no—
453C32D3×#10212yescomplemented
454C32C2×C3×#10211yescharacteristic, complemented
455C23—21no—
456C23—91no—
457C23—184no—
458C23—181no—
459C23—362no—
460C23—362no—
461D4—21no—
462D4—92no—
463D4—184no—
464D4—181no—
465D4C3×#62412yescomplemented
466C2×C4—62no—
467C2×C4—91no—
468C2×C4—92no—
469C2×C4—92no—
470C2×C4—362no—
471C2×C4C2×C3×D3211yescharacteristic
472C8—62no—
473C2×C3—21no—
474C2×C3—22no—
475C2×C3—22no—
476C2×C3—21no—
477C2×C3—21no—
478C2×C3—22no—
479C2×C3—44no—
480C2×C3—44no—
481C2×C3—42no—
482C2×C3—41no—
483C2×C3—41no—
484C2×C3—41no—
485C2×C3—84no—
486C2×C3—84no—
487C2×C3—94no—
488C2×C3—362no—
489C2×C3#6392611yescharacteristic
490C2×C3#6393011yescharacteristic
491C2×C3#6397411yescharacteristic
492C2×C3C3×D3×C22⋊C412yes—
493C2×C3C3×D3×D812yes—
494C2×C3C3×D3×SD1612yes—
495D3—38no—
496D3—64no—
497D3—124no—
498D3—124no—
499C22—22no—
500C22—41no—
501C22—41no—
502C22—92no—
503C22—92no—
504C22—92no—
505C22—182no—
506C22—368no—
507C22—361no—
508C22—361no—
509C22C3×#381111yescharacteristic
510C4—92no—
511C4—122no—
512C4C2×C3×#62411yescharacteristic
513C4C3×#381111yescharacteristic
514C3—21no—
515C3—22no—
516C3—41no—
517C3#1059141611yescharacteristic, direct product factor
518C3C3×D3×#10212yescomplemented
519C2—42no—
520C2—94no—
521C2—362no—
522C2C3×#6392611yescharacteristic
523C2C3×#6393011yescharacteristic
524C2C3×#6397411yescharacteristic
subgroupG/H
centerC22×C3#3811
commutatorC32×C4C22×C3×C4
FrattiniC2×C4C2×C3×D32
FittingC2×C33×D4C22
socleC22×C33C2×D4
prime radicalC2×C3×D4×C3⋊D3C2
cosocleC2×C32×C4C23×C3
hypercenterC2×C3×D4D32
hypocenterC32C2×C3×#102
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C2×C3×D4

Complements

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

NH ≅ G/Ntype
2. C3×#63900C2semidirect
3. C2×C3×#3710C2semidirect
4. C32×#832C2semidirect
5. C3×#63564C2semidirect
7. #10591416C3direct (N × H)
12. C3×D4×C3⋊D3C4semidirect
16. C32×C4⋊Dic3C22semidirect
17. C2×C32×C3⋊C8C22semidirect
25. #63900C2×C3semidirect
26. C2×#3710C2×C3semidirect
29. C3×#832D3semidirect
30. C3×#832C2×C3semidirect
33. #63564C2×C3semidirect
41. C33×D4C2×C4semidirect
46. C3×C4⋊(C3⋊D3)C2×C4semidirect
64. D4×C3⋊D3C3×C4semidirect
82. C3×C4⋊Dic3C2×D3semidirect
83. C3×C4⋊Dic3C22×C3semidirect
86. C2×C3×C3⋊C8C2×D3semidirect
87. C2×C3×C3⋊C8C22×C3semidirect
117. #832C3×D3semidirect
148. C32×D4C4×D3semidirect
149. C32×D4C2×C3×C4semidirect
159. C4⋊(C3⋊D3)C2×C3×C4semidirect
189. C3×C3⋊D3#102semidirect
223. C4⋊Dic3C2×C3×D3semidirect
225. C2×C3⋊C8C2×C3×D3semidirect
290. C33C2×#102semidirect
322. C3×D4#624semidirect
323. C3×D4C3×C4×D3semidirect
375. C3⋊D3C3×#102semidirect
453. C32D3×#102semidirect
454. C32C2×C3×#102semidirect
465. D4C3×#624semidirect
517. C3#10591416direct (N × H)
518. C3C3×D3×#102semidirect

Maximal subgroups

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

subgroupindex
C2×C3×D4×C3⋊D32
C3×#639002
C2×C3×#37102
C32×#8322
C3×#635642
C3×D3×#1023
#105914163