Finite Group Databasebeta

← #13178658

Subgroups — #13178658

Order 1728.

1504 conjugacy classes of subgroups.

The subgroup lattice has 544 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#89046C212yesmaximal, complemented
2C3×#64147C212yesmaximal, complemented
3C3×D4×C3⋊Dic3C211yesmaximal, characteristic, complemented
4#11723724C214yesmaximal, complemented
5#11723721C212yesmaximal, complemented
6C4⋊#6296C211yesmaximal, characteristic, complemented
7C4×#6295C211yesmaximal, characteristic, complemented
8#11723689C212yesmaximal, complemented
9#10595136—32nomaximal
10#10595134—31nomaximal
11C2×C33×D4C2211yescharacteristic
12C2×C3×#638C2214yes—
13C22×C3×C3⋊Dic3C2212yes—
14C2×C3×C4×C3⋊D3C2212yes—
15C2×#6296C2212yes—
16C2×#6295C2211yescharacteristic
17#89046—28no—
18C2×#6294C2214yes—
19C3×#3757C2212yes—
20C3×#3757C2214yes—
21C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
22C3×C4⋊(C3⋊Dic3)C2212yes—
23C3×C4×C3⋊Dic3C2211yescharacteristic
24#88815C2214yes—
25#88815C2211yescharacteristic
26#88815C2212yes—
27#88814C2212yes—
28C2×#3805—32no—
29C2×#3804—34no—
30#64147D312yescomplemented
31D4×C3⋊Dic3D311yescharacteristic, complemented
32C3×#938—32no—
33C3×#938—32no—
34C3×#938—62no—
35C3×D4×Dic3—32no—
36C3×D4×Dic3—61no—
37#63982—32no—
38#63975—34no—
39#63969—34no—
40#63969—34no—
41#63964—34no—
42#63919—31no—
43#63918—32no—
44C4×#626—32no—
45C4×#625—31no—
46#63900—32no—
47D3×C4⋊Dic3—32no—
48D3×C4⋊Dic3—32no—
49C23×C33C2312yes—
50C22×C3×C3⋊D3C2312yes—
51C33×D4—22no—
52C2×C33×C4C2311yescharacteristic
53C3×#638—28no—
54C2×C3×C3⋊Dic3—24no—
55C2×C3×C3⋊Dic3C2311yescharacteristic
56C2×C3×C3⋊Dic3C2312yes—
57C2×C3×C3⋊Dic3C2314yes—
58C2×C3×C3⋊Dic3C2312yes—
59C2×C3×C3⋊Dic3C2311yescharacteristic
60C3×C4×C3⋊D3—24no—
61#6296—24no—
62#6295D414yescomplemented
63#6294—28no—
64#5611—92no—
65#5551—91no—
66C2×C32×D4—21no—
67C2×C32×D4—22no—
68C2×C32×D4—41no—
69C2×C32×D4C2×D312yes—
70C2×C32×D4C2×D311yescharacteristic
71C2×#638C2×D314yes—
72C22×C3⋊Dic3C2×D312yes—
73C2×C4×C3⋊D3C2×D312yes—
74C2×C3×#67—34no—
75C2×C3×#67—34no—
76C2×C3×#67—64no—
77C22×C3×Dic3—34no—
78C22×C3×Dic3—62no—
79C2×C3×C4×D3—32no—
80C2×C3×C4×D3—32no—
81C2×C3×C4×D3—62no—
82C2×#627—34no—
83C2×#627—32no—
84C2×#626—32no—
85C2×#625—31no—
86C2×#624—34no—
87#3805—68no—
88#3804—616no—
89C2×D3×Dic3—34no—
90C2×D3×Dic3—34no—
91#3757C2×D312yes—
92#3757C2×D314yes—
93C4⋊(C3⋊Dic3)C2×D311yescharacteristic
94C4⋊(C3⋊Dic3)C2×D312yes—
95C4×C3⋊Dic3C2×D311yescharacteristic
96C3×#217—34no—
97C3×#217—34no—
98C3×#217—34no—
99C3×#217—62no—
100C3×#217—64no—
101C3×C4⋊Dic3—32no—
102C3×C4⋊Dic3—32no—
103C3×C4⋊Dic3—32no—
104C3×C4⋊Dic3—61no—
105C3×C4⋊Dic3—62no—
106C3×C4×Dic3—32no—
107C3×C4×Dic3—61no—
108#3724—34no—
109#3724—31no—
110#3724—32no—
111Dic3⋊Dic3—34no—
112Dic3⋊Dic3—34no—
113Dic3⋊Dic3—32no—
114Dic3⋊Dic3—34no—
115#3722—32no—
116(C2×D3)⋊Dic3—32no—
117(C2×D3)⋊Dic3—32no—
118C22×C33—22no—
119C22×C33—22no—
120C22×C33—22no—
121C22×C33C2411yescharacteristic
122C2×C3×C3⋊D3—22no—
123C2×C3×C3⋊D3—22no—
124C2×C3×C3⋊D3C2×D414yes—
125C33×C4#4112yescomplemented
126C3×C3⋊Dic3—22no—
127C3×C3⋊Dic3—24no—
128C3×C3⋊Dic3—22no—
129C3×C3⋊Dic3—21no—
130C3×C3⋊Dic3C2×D412yes—
131C2×#237—94no—
132C2×#235—92no—
133C3×#121—92no—
134C3×C4×D4—91no—
135#938—32no—
136#938—32no—
137#938—62no—
138#935—94no—
139D4×Dic3—32no—
140D4×Dic3—61no—
141C4×#67—92no—
142C2×C4⋊Dic3—92no—
143#924—92no—
144D3×C4⋊C4—92no—
145D3×C4⋊C4—92no—
146#888—92no—
147#883—94no—
148#883—94no—
149#878—94no—
150C4×D12—91no—
151#869—91no—
152C23×C32—22no—
153C23×C32—24no—
154C23×C32—42no—
155C23×C32C22×D314yes—
156C23×C32C22×D312yes—
157C22×C3⋊D3C22×D312yes—
158C22×C3×D3—32no—
159C22×C3×D3—32no—
160C22×C3×D3—62no—
161C32×D4—22no—
162C32×D4—24no—
163C32×D4—24no—
164C32×D4—44no—
165C32×D4—44no—
166C2×C32×C4—21no—
167C2×C32×C4—22no—
168C2×C32×C4—41no—
169C2×C32×C4C22×D311yescharacteristic
170C2×C32×C4C22×D312yes—
171#638—28no—
172C2×C3⋊Dic3—24no—
173C2×C3⋊Dic3C22×D311yescharacteristic
174C2×C3⋊Dic3C22×D312yes—
175C2×C3⋊Dic3C22×D312yes—
176C2×C3⋊Dic3C22×D314yes—
177C2×C3⋊Dic3C22×D311yescharacteristic
178C4×C3⋊D3—24no—
179C3×#67—68no—
180C3×#67—68no—
181C3×#67—128no—
182C2×C3×Dic3—32no—
183C2×C3×Dic3—32no—
184C2×C3×Dic3—32no—
185C2×C3×Dic3—32no—
186C2×C3×Dic3—34no—
187C2×C3×Dic3—34no—
188C2×C3×Dic3—34no—
189C2×C3×Dic3—61no—
190C2×C3×Dic3—61no—
191C2×C3×Dic3—62no—
192C2×C3×Dic3—64no—
193C2×C3×Dic3—62no—
194C2×C3×Dic3—68no—
195C2×C3×Dic3—124no—
196C3×C4×D3—64no—
197C3×C4×D3—64no—
198C3×C4×D3—124no—
199#627—68no—
200#627—64no—
201#626—38no—
202#625—34no—
203#624—68no—
204D3×Dic3—68no—
205D3×Dic3—68no—
206#547—271noSylow
207C2×C33—42no—
208C2×C33C22×D411yescharacteristic
209C2×C33C2×#4111yescharacteristic
210C2×C332−1+411yescharacteristic
211C3×C3⋊D3—24no—
212C2×C3×D4—21no—
213C2×C3×D4—22no—
214C2×C3×D4—41no—
215C2×C3×D4—94no—
216C2×C3×D4D3212yes—
217C2×C3×D4D3211yescharacteristic
218C22×C3×C4—92no—
219C22×C3×C4—92no—
220C2×#67—34no—
221C2×#67—34no—
222C2×#67—64no—
223C2×#67—92no—
224C22×Dic3—34no—
225C22×Dic3—62no—
226C22×Dic3—94no—
227#237—1816no—
228#235—188no—
229C2×D12—91no—
230C2×C4×D3—32no—
231C2×C4×D3—32no—
232C2×C4×D3—62no—
233C2×C4×D3—94no—
234C2×C4×D3—94no—
235C2×Dic6—92no—
236C2×Dic6—94no—
237C3×C4⋊C4—92no—
238C3×C4⋊C4—91no—
239C3×C22⋊C4—94no—
240C3×C22⋊C4—92no—
241C3×C42—91no—
242#217—34no—
243#217—34no—
244#217—34no—
245#217—62no—
246#217—64no—
247#217—92no—
248(C2×D3)⋊C4—92no—
249(C2×D3)⋊C4—92no—
250C4⋊Dic3—32no—
251C4⋊Dic3—32no—
252C4⋊Dic3—32no—
253C4⋊Dic3—61no—
254C4⋊Dic3—62no—
255C4⋊Dic3—92no—
256C4⋊Dic3—91no—
257C4⋊Dic3—94no—
258Dic3⋊C4—92no—
259Dic3⋊C4—94no—
260Dic3⋊C4—94no—
261Dic3⋊C4—94no—
262C4×Dic3—32no—
263C4×Dic3—61no—
264C22×C32—22no—
265C22×C32—24no—
266C22×C32—24no—
267C22×C32—22no—
268C22×C32—24no—
269C22×C32—22no—
270C22×C32—24no—
271C22×C32—21no—
272C22×C32—22no—
273C22×C32—41no—
274C22×C32—44no—
275C22×C32—44no—
276C22×C32—42no—
277C22×C32—42no—
278C22×C32—44no—
279C22×C32—44no—
280C22×C32—82no—
281C22×C32—82no—
282C22×C32C23×D311yescharacteristic
283C22×C32C23×D312yes—
284C2×C3⋊D3—22no—
285C2×C3⋊D3—22no—
286C2×C3⋊D3D3×D414yes—
287C2×C3×D3—34no—
288C2×C3×D3—34no—
289C2×C3×D3—64no—
290C2×C3×D3—62no—
291C2×C3×D3—62no—
292C2×C3×D3—62no—
293C2×C3×D3—62no—
294C2×C3×D3—122no—
295C2×C3×D3—122no—
296C32×C4—24no—
297C32×C4—22no—
298C32×C4—42no—
299C32×C4#23512yescomplemented
300C32×C4#23714yescomplemented
301C3⋊Dic3—22no—
302C3⋊Dic3—24no—
303C3⋊Dic3—22no—
304C3⋊Dic3—21no—
305C3⋊Dic3D3×D412yes—
306C3×Dic3—34no—
307C3×Dic3—62no—
308C3×Dic3—62no—
309C3×Dic3—64no—
310C3×Dic3—64no—
311C3×Dic3—64no—
312C3×Dic3—62no—
313C3×Dic3—62no—
314C3×Dic3—122no—
315C3×Dic3—121no—
316C3×Dic3—122no—
317C3×Dic3—124no—
318C2×#41—272no—
319C4⋊Q8—271no—
320#123—274no—
321C22⋊Q8—272no—
322#121—92no—
323C4×D4—91no—
324C4×D4—271no—
325C2×C4⋊C4—272no—
326C33#54711yescharacteristic, complemented, Sylow
327C23×C3—22no—
328C23×C3—24no—
329C23×C3—42no—
330C23×C3—92no—
331C23×C3C2×D3214yes—
332C23×C3C2×D3212yes—
333C22×D3—32no—
334C22×D3—32no—
335C22×D3—62no—
336C3×D4—24no—
337C3×D4—24no—
338C3×D4—22no—
339C3×D4—44no—
340C3×D4—44no—
341C3×D4—188no—
342C2×C3×C4—21no—
343C2×C3×C4—22no—
344C2×C3×C4—41no—
345C2×C3×C4—91no—
346C2×C3×C4—94no—
347C2×C3×C4—92no—
348C2×C3×C4—92no—
349C2×C3×C4—91no—
350C2×C3×C4—184no—
351C2×C3×C4—184no—
352C2×C3×C4C2×D3212yes—
353C2×C3×C4C2×D3211yescharacteristic
354#67—68no—
355#67—68no—
356#67—98no—
357#67—128no—
358C2×Dic3—32no—
359C2×Dic3—32no—
360C2×Dic3—32no—
361C2×Dic3—32no—
362C2×Dic3—34no—
363C2×Dic3—34no—
364C2×Dic3—34no—
365C2×Dic3—61no—
366C2×Dic3—61no—
367C2×Dic3—62no—
368C2×Dic3—64no—
369C2×Dic3—62no—
370C2×Dic3—68no—
371C2×Dic3—124no—
372C2×Dic3—188no—
373D12—94no—
374C4×D3—64no—
375C4×D3—64no—
376C4×D3—124no—
377C4×D3—188no—
378C4×D3—188no—
379Dic6—188no—
380Dic6—184no—
381C2×C32—21no—
382C2×C32—22no—
383C2×C32—21no—
384C2×C32—22no—
385C2×C32—22no—
386C2×C32—21no—
387C2×C32—41no—
388C2×C32—41no—
389C2×C32—41no—
390C2×C32—42no—
391C2×C32—44no—
392C2×C32—44no—
393C2×C32—84no—
394C2×C32—84no—
395C2×C32C2×#23511yescharacteristic
396C2×C32C2×D3×D411yescharacteristic
397C2×C32C2×D3×D412yes—
398C2×C32C2×#23712yes—
399C2×C32#101011yescharacteristic
400C2×C32#101012yes—
401C3⋊D3—24no—
402C3×D3—64no—
403C3×D3—64no—
404C3×D3—124no—
405#41—548no—
406C2×Q8—272no—
407C2×D4—94no—
408C2×D4—271no—
409C2×D4#111311yescharacteristic
410C22×C4—92no—
411C22×C4—92no—
412C22×C4—274no—
413C4⋊C4—91no—
414C4⋊C4—92no—
415C4⋊C4—274no—
416C4⋊C4—271no—
417C4⋊C4—272no—
418C22⋊C4—92no—
419C22⋊C4—94no—
420C22⋊C4—272no—
421C42—91no—
422C22×C3—24no—
423C22×C3—22no—
424C22×C3—24no—
425C22×C3—24no—
426C22×C3—22no—
427C22×C3—24no—
428C22×C3—22no—
429C22×C3—21no—
430C22×C3—22no—
431C22×C3—42no—
432C22×C3—42no—
433C22×C3—44no—
434C22×C3—44no—
435C22×C3—44no—
436C22×C3—44no—
437C22×C3—41no—
438C22×C3—82no—
439C22×C3—82no—
440C22×C3—94no—
441C22×C3—182no—
442C22×C3—182no—
443C22×C3C22×D3212yes—
444C22×C3C22×D3211yescharacteristic
445C2×D3—34no—
446C2×D3—34no—
447C2×D3—64no—
448C2×D3—62no—
449C2×D3—62no—
450C2×D3—62no—
451C2×D3—62no—
452C2×D3—122no—
453C2×D3—122no—
454C3×C4—24no—
455C3×C4—22no—
456C3×C4—42no—
457C3×C4—92no—
458C3×C4—181no—
459C3×C4—184no—
460C3×C4—182no—
461C3×C4—182no—
462C3×C4#380414yescomplemented
463C3×C4#380512yescomplemented
464Dic3—34no—
465Dic3—62no—
466Dic3—62no—
467Dic3—64no—
468Dic3—64no—
469Dic3—64no—
470Dic3—62no—
471Dic3—62no—
472Dic3—122no—
473Dic3—122no—
474Dic3—124no—
475Dic3—121no—
476C32—21no—
477C32—22no—
478C32—41no—
479C32#555111yescharacteristic, complemented
480C32#561112yescomplemented
481C23—92no—
482C23C2×#111312yes—
483Q8—544no—
484D4—22no—
485D4—188no—
486D4—274no—
487C2×C4—91no—
488C2×C4—91no—
489C2×C4—92no—
490C2×C4—92no—
491C2×C4—94no—
492C2×C4—184no—
493C2×C4—184no—
494C2×C4—548no—
495C2×C4C2×#111311yescharacteristic
496C2×C3—21no—
497C2×C3—22no—
498C2×C3—21no—
499C2×C3—22no—
500C2×C3—22no—
501C2×C3—21no—
502C2×C3—44no—
503C2×C3—42no—
504C2×C3—44no—
505C2×C3—41no—
506C2×C3—41no—
507C2×C3—41no—
508C2×C3—84no—
509C2×C3—84no—
510C2×C3—184no—
511C2×C3#6430911yescharacteristic
512C2×C3#6430912yes—
513C2×C3D32×D411yescharacteristic
514C2×C3D32×D412yes—
515C2×C3C2×#380412yes—
516C2×C3C2×#380511yescharacteristic
517D3—64no—
518D3—64no—
519D3—124no—
520C22—22no—
521C22—22no—
522C22—22no—
523C22—94no—
524C22—182no—
525C22—182no—
526C22C22×#111311yescharacteristic
527C4—92no—
528C4—181no—
529C4—182no—
530C4—184no—
531C4—182no—
532C4#8904612yescomplemented
533C3—22no—
534C3—21no—
535C3—41no—
536C3#1059513411yescharacteristic, complemented
537C3#1059513612yescomplemented
538C2—42no—
539C2—184no—
540C2#1172462011yescharacteristic
541C2D4×#111311yescharacteristic
542C2C2×#8904611yescharacteristic
subgroupG/H
center, FrattiniC22C22×#1113
commutator, socle, cosocleC22×C33C24
FittingC2×C33×D4C22
hypercenterC2×D4#1113
hypocenterC33#547
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C2×#89046C2semidirect
2. C3×#64147C2semidirect
3. C3×D4×C3⋊Dic3C2semidirect
4. #11723724C2semidirect
5. #11723721C2semidirect
6. C4⋊#6296C2semidirect
7. C4×#6295C2semidirect
8. #11723689C2semidirect
30. #64147D3semidirect
31. D4×C3⋊Dic3D3semidirect
62. #6295D4semidirect
125. C33×C4#41semidirect
299. C32×C4#235semidirect
300. C32×C4#237semidirect
326. C33#547semidirect
462. C3×C4#3804semidirect
463. C3×C4#3805semidirect
479. C32#5551semidirect
480. C32#5611semidirect
532. C4#89046semidirect
536. C3#10595134semidirect
537. C3#10595136semidirect

Maximal subgroups

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

subgroupindex
C2×#890462
C3×#641472
C3×D4×C3⋊Dic32
#117237242
#117237212
C4⋊#62962
C4×#62952
#117236892
#105951363
#105951343