Finite Group Databasebeta

← #13168715

Subgroups — #13168715

Order 1728.

1540 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D3×#634C212yesmaximal, complemented
2#11723678C211yesmaximal, characteristic, complemented
3#11723406C211yesmaximal, characteristic, complemented
4C3×C4×#625C211yesmaximal, characteristic
5C3×#63857C212yesmaximal, complemented
6#11723037C212yesmaximal, complemented
7#11722980C214yesmaximal, complemented
8#11722925C212yesmaximal
9#10594915—32nomaximal
10#10594526—31nomaximal
11C2×#6308C2211yescharacteristic
12C2×C3×#634C2212yes—
13C2×C32×C4×D3C2212yes—
14C2×#6293C2214yes—
15C2×D3×C3⋊Dic3C2214yes—
16D3×#634—28no—
17C2×C3×#625C2211yescharacteristic
18C3×C4×C3⋊Dic3C2211yescharacteristic
19C32×C4⋊Dic3C2211yescharacteristic
20#88815C2212yes—
21#88815C2212yes—
22#88806C2212yes—
23Dic3×C3⋊Dic3C2212yes—
24C3×#3724C2211yescharacteristic
25C3×Dic3⋊Dic3C2214yes—
26C3×(C2×D3)⋊Dic3C2214yes—
27C3×(C2×D3)⋊Dic3C2212yes—
28C22×#634—32no—
29C2×D3×Dic6—32no—
30C2×D3×Dic6—32no—
31C2×D3×Dic6—62no—
32#64096—31no—
33C3×C4×#67—32no—
34C3×#924—32no—
35C3×#897—32no—
36#63982—34no—
37#67⋊Dic3—34no—
38#63919—32no—
39#63902—32no—
40C4×#625D311yescharacteristic
41#63899—34no—
42#63896—34no—
43#63873—31no—
44#63866—32no—
45#63857D312yescomplemented
46#63844—32no—
47#63844—61no—
48#63840—34no—
49C22×C32×D3C2312yes—
50C2×C33×C4C2311yescharacteristic
51C2×C3⋊(C3⋊Dic3)C2312yes—
52#6308D414yescomplemented
53C2×C3×C3⋊Dic3C2311yescharacteristic
54C2×C3×C3⋊Dic3C2314yes—
55C2×C3×C3⋊Dic3C2311yescharacteristic
56C3×#634—24no—
57C2×C32×Dic3C2312yes—
58C2×C32×Dic3C2312yes—
59C32×C4×D3—24no—
60#6293—28no—
61D3×C3⋊Dic3—28no—
62C3×#625Q814yescomplemented
63#5632—92no—
64D4×Dic6—91no—
65C22×C32×C4—32no—
66C22×C3⋊Dic3—34no—
67C2×#634—31no—
68C2×#634—32no—
69C2×#634—34no—
70C2×#634—68no—
71C2×#634—62no—
72C2×#634—61no—
73C2×#634—121no—
74C2×#634C2×D312yes—
75C2×C3×#67—32no—
76C2×C3×C4×D3—22no—
77C2×C3×C4×D3C2×D312yes—
78C2×C3×C4×D3C2×D312yes—
79C2×C3×Dic6—32no—
80C2×C3×Dic6—32no—
81C2×C3×Dic6—62no—
82C2×#627—34no—
83C2×#627—34no—
84C2×#627—64no—
85C2×#625C2×D311yescharacteristic
86C2×D3×Dic3—34no—
87C2×D3×Dic3—34no—
88C2×D3×Dic3—64no—
89D3×Dic6—68no—
90D3×Dic6—68no—
91D3×Dic6—616no—
92C32×C4⋊C4—31no—
93#3750—32no—
94C4×C3⋊Dic3—32no—
95C4×C3⋊Dic3C2×D311yescharacteristic
96C3×#217—34no—
97C3×#217—32no—
98C3×(C2×D3)⋊C4—34no—
99C3×(C2×D3)⋊C4—32no—
100C3×C4⋊Dic3—21no—
101C3×C4⋊Dic3—34no—
102C3×C4⋊Dic3C2×D312yes—
103C3×Dic3⋊C4—34no—
104C3×Dic3⋊C4—32no—
105C3×C4×Dic3—32no—
106C3×C4×Dic3—61no—
107#3724—32no—
108#3724—32no—
109#3724—34no—
110#3724—62no—
111#3724C2×D311yescharacteristic
112Dic3⋊Dic3—34no—
113Dic3⋊Dic3—34no—
114Dic3⋊Dic3C2×D314yes—
115(C2×D3)⋊Dic3C2×D314yes—
116(C2×D3)⋊Dic3C2×D312yes—
117Dic32—34no—
118Dic32—62no—
119C22×C33C2411yescharacteristic
120C2×C32×D3—22no—
121C2×C32×D3—22no—
122C2×C32×D3C2×Q814yescomplemented
123C33×C4C2×D412yes—
124C3⋊(C3⋊Dic3)C2×D414yes—
125C3×C3⋊Dic3—24no—
126C3×C3⋊Dic3—21no—
127C3×C3⋊Dic3C2×Q812yescomplemented
128C32×Dic3—22no—
129C32×Dic3—22no—
130C2×D3×Q8—92no—
131C22×Dic6—92no—
132C22×Dic6—92no—
133C22×Dic6—182no—
134C3×C22⋊Q8—92no—
135C3×C4×D4—91no—
136#944—92no—
137D4×Dic3—92no—
138C4×#67—32no—
139#924—32no—
140#897—32no—
141#896—94no—
142#888—92no—
143#888—94no—
144#887—92no—
145#887—181no—
146#67⋊C4—94no—
147#878—94no—
148#878—94no—
149#869—91no—
150C4×Dic6—91no—
151C23×C32—32no—
152C22×C3×D3—22no—
153C22×C3×D3C22×D312yes—
154C22×C3×D3C22×D312yes—
155C2×C32×C4—21no—
156C2×C32×C4—22no—
157C2×C32×C4—32no—
158C2×C32×C4—32no—
159C2×C32×C4—41no—
160C2×C32×C4—64no—
161C2×C32×C4C22×D312yes—
162C2×C32×C4C22×D311yescharacteristic
163C2×C3⋊Dic3—34no—
164C2×C3⋊Dic3—32no—
165C2×C3⋊Dic3—62no—
166C2×C3⋊Dic3—64no—
167C2×C3⋊Dic3—68no—
168C2×C3⋊Dic3—122no—
169C2×C3⋊Dic3C22×D311yescharacteristic
170C2×C3⋊Dic3C22×D314yes—
171C2×C3⋊Dic3C22×D311yescharacteristic
172#634—24no—
173#634—34no—
174#634—38no—
175#634—68no—
176#634—64no—
177#634—68no—
178#634—124no—
179C3×#67—38no—
180C2×C3×Dic3—22no—
181C2×C3×Dic3—22no—
182C2×C3×Dic3—32no—
183C2×C3×Dic3—32no—
184C2×C3×Dic3—34no—
185C2×C3×Dic3—34no—
186C2×C3×Dic3—61no—
187C2×C3×Dic3—64no—
188C2×C3×Dic3—61no—
189C2×C3×Dic3C22×D314yes—
190C2×C3×Dic3C22×D312yes—
191C2×C3×Dic3C22×D312yes—
192C3×C4×D3—28no—
193C3×C4×D3—24no—
194C3×C4×D3—24no—
195C3×Dic6—68no—
196C3×Dic6—64no—
197C3×Dic6—64no—
198#627—68no—
199#627—68no—
200#627—616no—
201#625Dic614yescomplemented
202D3×Dic3—68no—
203D3×Dic3—68no—
204D3×Dic3—616no—
205D4×Q8—271noSylow
206C2×C33C22×D411yescharacteristic
207C2×C33C22×Q811yescharacteristic
208C2×C332−1+411yescharacteristic
209C32×D3—24no—
210C2×C3×Q8—92no—
211C2×C3×D4—91no—
212C22×C3×C4—32no—
213C22×C3×C4—32no—
214C22×C3×C4—62no—
215C2×#67—32no—
216C22×Dic3—94no—
217C22×Dic3—94no—
218C22×Dic3—184no—
219D3×Q8—188no—
220C2×C4×D3—94no—
221C2×C4×D3D3212yes—
222C2×Dic6—32no—
223C2×Dic6—32no—
224C2×Dic6—62no—
225C2×Dic6—91no—
226C2×Dic6—92no—
227C2×Dic6—94no—
228C2×Dic6—94no—
229C2×Dic6—94no—
230C2×Dic6—184no—
231C2×Dic6—181no—
232C2×Dic6—182no—
233C2×Dic6—188no—
234C2×Dic6—188no—
235C2×Dic6—1816no—
236C2×Dic6—361no—
237C3×C4⋊C4—32no—
238C3×C4⋊C4—61no—
239C3×C4⋊C4—91no—
240C3×C4⋊C4—94no—
241C3×C22⋊C4—92no—
242C3×C22⋊C4—94no—
243C3×C42—91no—
244#217—32no—
245#217—34no—
246(C2×D3)⋊C4—34no—
247(C2×D3)⋊C4—32no—
248C4⋊Dic3—34no—
249C4⋊Dic3—92no—
250C4⋊Dic3—92no—
251C4⋊Dic3D3211yescharacteristic
252Dic3⋊C4—34no—
253Dic3⋊C4—32no—
254Dic3⋊C4—94no—
255Dic3⋊C4—94no—
256Dic3⋊C4—94no—
257Dic3⋊C4—92no—
258Dic3⋊C4—182no—
259C4×Dic3—32no—
260C4×Dic3—61no—
261C4×Dic3—94no—
262C4×Dic3—92no—
263C4×Dic3—182no—
264C22×C32—21no—
265C22×C32—22no—
266C22×C32—34no—
267C22×C32—41no—
268C22×C32—62no—
269C22×C32—62no—
270C22×C32C23×D311yescharacteristic
271C22×C32C23×D312yes—
272C2×C3×D3—24no—
273C2×C3×D3—24no—
274C2×C3×D3—22no—
275C2×C3×D3—22no—
276C2×C3×D3—22no—
277C2×C3×D3—22no—
278C2×C3×D3—24no—
279C2×C3×D3C2×Dic614yescomplemented
280C2×C3×D3D3×Q814yescomplemented
281C32×C4—22no—
282C32×C4—24no—
283C32×C4—42no—
284C32×C4—62no—
285C32×C4—62no—
286C32×C4D3×D414yes—
287C32×C4D3×D412yes—
288C3⋊Dic3—24no—
289C3⋊Dic3—21no—
290C3⋊Dic3—34no—
291C3⋊Dic3—38no—
292C3⋊Dic3—68no—
293C3⋊Dic3—64no—
294C3⋊Dic3—124no—
295C3⋊Dic3C2×Dic612yescomplemented
296C3×Dic3—24no—
297C3×Dic3—22no—
298C3×Dic3—22no—
299C3×Dic3—24no—
300C3×Dic3—34no—
301C3×Dic3—42no—
302C3×Dic3—62no—
303C3×Dic3—68no—
304C3×Dic3—64no—
305C3×Dic3—64no—
306C3×Dic3—62no—
307C3×Dic3—121no—
308C22×Q8—272no—
309C4⋊Q8—271no—
310C4⋊Q8—272no—
311C22⋊Q8—92no—
312C22⋊Q8—274no—
313C4×Q8—271no—
314C4×D4—91no—
315C4×D4—272no—
316C33D4×Q811yescharacteristic, complemented, Sylow
317C23×C3—32no—
318C23×C3—32no—
319C23×C3—62no—
320C22×D3C2×D3212yes—
321C3×Q8—184no—
322C3×D4—94no—
323C2×C3×C4—21no—
324C2×C3×C4—22no—
325C2×C3×C4—34no—
326C2×C3×C4—32no—
327C2×C3×C4—32no—
328C2×C3×C4—41no—
329C2×C3×C4—62no—
330C2×C3×C4—62no—
331C2×C3×C4—68no—
332C2×C3×C4—64no—
333C2×C3×C4—64no—
334C2×C3×C4—91no—
335C2×C3×C4—91no—
336C2×C3×C4—94no—
337C2×C3×C4C2×D3211yescharacteristic
338C2×C3×C4C2×D3212yes—
339#67—38no—
340C2×Dic3—32no—
341C2×Dic3—32no—
342C2×Dic3—34no—
343C2×Dic3—34no—
344C2×Dic3—61no—
345C2×Dic3—64no—
346C2×Dic3—61no—
347C2×Dic3—94no—
348C2×Dic3—92no—
349C2×Dic3—182no—
350C2×Dic3—184no—
351C2×Dic3—188no—
352C2×Dic3—188no—
353C2×Dic3—1816no—
354C2×Dic3—362no—
355C2×Dic3C2×D3212yes—
356C2×Dic3C2×D3212yes—
357C4×D3—24no—
358C4×D3—188no—
359Dic6—68no—
360Dic6—64no—
361Dic6—64no—
362Dic6—94no—
363Dic6—98no—
364Dic6—184no—
365Dic6—188no—
366Dic6—188no—
367Dic6—188no—
368Dic6—188no—
369Dic6—1816no—
370Dic6—364no—
371C2×C32—22no—
372C2×C32—22no—
373C2×C32—21no—
374C2×C32—21no—
375C2×C32—21no—
376C2×C32—22no—
377C2×C32—41no—
378C2×C32—41no—
379C2×C32—41no—
380C2×C32—64no—
381C2×C32C22×Dic611yescharacteristic
382C2×C32C2×D3×D412yes—
383C2×C32C2×D3×D411yescharacteristic
384C2×C32C2×D3×Q812yes—
385C2×C32#101011yescharacteristic
386C2×C32#101012yes—
387C3×D3—28no—
388C3×D3—24no—
389C3×D3—24no—
390C2×Q8—92no—
391C2×Q8—274no—
392C2×Q8—271no—
393C2×Q8—548no—
394C2×D4—91no—
395C22×C4—32no—
396C22×C4—274no—
397C4⋊C4—31no—
398C4⋊C4—91no—
399C4⋊C4—94no—
400C4⋊C4—272no—
401C4⋊C4—272no—
402C4⋊C4—272no—
403C22⋊C4—92no—
404C22⋊C4—94no—
405C42—91no—
406C42—272no—
407C22×C3—21no—
408C22×C3—22no—
409C22×C3—34no—
410C22×C3—34no—
411C22×C3—41no—
412C22×C3—64no—
413C22×C3—64no—
414C22×C3—64no—
415C22×C3—62no—
416C22×C3—62no—
417C22×C3—62no—
418C22×C3—62no—
419C22×C3C22×D3212yes—
420C22×C3C22×D3211yescharacteristic
421C2×D3—22no—
422C2×D3—22no—
423C2×D3D3×Dic614yescomplemented
424C3×C4—24no—
425C3×C4—22no—
426C3×C4—42no—
427C3×C4—64no—
428C3×C4—64no—
429C3×C4—62no—
430C3×C4—62no—
431C3×C4—92no—
432C3×C4—122no—
433C3×C4—181no—
434C3×C4—184no—
435C3×C4#380212yes—
436C3×C4#381114yes—
437Dic3—22no—
438Dic3—22no—
439Dic3—34no—
440Dic3—62no—
441Dic3—68no—
442Dic3—62no—
443Dic3—64no—
444Dic3—64no—
445Dic3—94no—
446Dic3—98no—
447Dic3—121no—
448Dic3—188no—
449Dic3—184no—
450Dic3—364no—
451C32—22no—
452C32—21no—
453C32—41no—
454C32D4×Dic611yescharacteristic, complemented
455C32#563212yescomplemented
456C23—32no—
457Q8—184no—
458Q8—274no—
459Q8—548no—
460D4—94no—
461C2×C4—32no—
462C2×C4—32no—
463C2×C4—64no—
464C2×C4—91no—
465C2×C4—91no—
466C2×C4—94no—
467C2×C4—272no—
468C2×C4—548no—
469C2×C4D3311yescharacteristic
470C2×C3—21no—
471C2×C3—22no—
472C2×C3—21no—
473C2×C3—22no—
474C2×C3—22no—
475C2×C3—21no—
476C2×C3—41no—
477C2×C3—41no—
478C2×C3—41no—
479C2×C3—68no—
480C2×C3—64no—
481C2×C3—64no—
482C2×C3C2×D3×Dic612yes—
483C2×C3#6429812yes—
484C2×C3C2×#380211yescharacteristic
485C2×C3#6430911yescharacteristic
486C2×C3D32×Q811yescharacteristic
487C2×C3C2×#381112yes—
488D3—24no—
489C22—34no—
490C22—62no—
491C22—62no—
492C22C2×D3311yescharacteristic
493C4—62no—
494C4—62no—
495C4—92no—
496C4—181no—
497C4—184no—
498C4—274no—
499C4#8895412yes—
500C3—22no—
501C3—21no—
502C3—41no—
503C3#1059452611yescharacteristic, complemented
504C3#1059491512yescomplemented
505C2—64no—
506C2D32×Dic611yescharacteristic
507C2#1172439211yescharacteristic
508C2C2×#8895411yescharacteristic
subgroupG/H
center, FrattiniC22C2×D33
commutator, socle, cosocleC22×C33C24
FittingC2×C33×C4C23
prime radicalC2×C3×#625C22
hypercenterC2×C4D33
hypocenterC33D4×Q8
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. C2×D3×#634C2semidirect
2. #11723678C2semidirect
3. #11723406C2semidirect
5. C3×#63857C2semidirect
6. #11723037C2semidirect
7. #11722980C2semidirect
45. #63857D3semidirect
52. #6308D4semidirect
62. C3×#625Q8semidirect
122. C2×C32×D3C2×Q8semidirect
127. C3×C3⋊Dic3C2×Q8semidirect
201. #625Dic6semidirect
279. C2×C3×D3C2×Dic6semidirect
280. C2×C3×D3D3×Q8semidirect
295. C3⋊Dic3C2×Dic6semidirect
316. C33D4×Q8semidirect
423. C2×D3D3×Dic6semidirect
454. C32D4×Dic6semidirect
455. C32#5632semidirect
503. C3#10594526semidirect
504. C3#10594915semidirect

Maximal subgroups

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

subgroupindex
C2×D3×#6342
#117236782
#117234062
C3×C4×#6252
C3×#638572
#117230372
#117229802
#117229252
#105949153
#105945263