Finite Group Databasebeta

← #13168349

Subgroups — #13168349

Order 1728.

1036 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11723671C211yesmaximal, characteristic
2#11723435C211yesmaximal, characteristic
3#11723400C212yesmaximal, complemented
4C3×#63883C211yesmaximal, characteristic
5C3×#63839C212yesmaximal, complemented
6#11723035C212yesmaximal, complemented
7#11723011C212yesmaximal, complemented
8#11722970C212yesmaximal
9#11722915C212yesmaximal
10#10595375—31nomaximal
11#10594342—32nomaximal
12C2×C3×C4×C3⋊D3C2211yescharacteristic
13C2×C32×Dic6C2211yescharacteristic
14C2×#6296C2211yescharacteristic
15C2×C3×#627C2212yes—
16C2×C3×#624C2211yescharacteristic
17C2×C3×#624C2211yescharacteristic
18C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
19C3×#3750C2212yes—
20C32×Dic3⋊C4C2212yes—
21#88815C2211yescharacteristic
22#88814C2212yes—
23#88806C2212yes—
24#88806C2212yes—
25Dic3⋊(C3⋊Dic3)C2212yes—
26Dic3⋊(C3⋊Dic3)C2212yes—
27#88803C2212yes—
28Dic3×C3⋊Dic3C2212yes—
29C3×#3724C2212yes—
30C3×Dic3⋊Dic3C2212yes—
31C3×Dic3⋊Dic3C2212yes—
32C3×Dic32C2212yes—
33#64155—31no—
34#64099—32no—
35C3×#944—32no—
36C3×D3×C4⋊C4—32no—
37C3×#888—32no—
38#63919—32no—
39#63888—32no—
40#63883D311yescharacteristic
41#63869—32no—
42#63867—32no—
43#63865—31no—
44#63856—32no—
45#63856—32no—
46#63856—61no—
47#63855—32no—
48#63851—32no—
49#63851—32no—
50#63845—32no—
51#63845—32no—
52#63845—32no—
53#63845—62no—
54#63841—32no—
55#63839—32no—
56#63839—32no—
57#63839D312yescomplemented
58#63836—32no—
59C22×C3×C3⋊D3C2311yescharacteristic
60C2×C33×C4C2311yescharacteristic
61C2×C3⋊(C3⋊Dic3)C2312yes—
62C2×C3×C3⋊Dic3C2312yes—
63C2×C3×C3⋊Dic3C2312yes—
64C2×C3×C3⋊Dic3C2311yescharacteristic
65C3×C4×C3⋊D3—41no—
66C2×C32×Dic3C2312yes—
67C2×C32×Dic3C2312yes—
68C2×C32×Dic3C2312yes—
69C32×Dic6—41no—
70#6296—41no—
71C3×#627—42no—
72C3×#624—41no—
73C3×#624Q814yescomplemented
74#5729—92no—
75#5500—91no—
76C2×C32×Q8—31no—
77C2×C4×C3⋊D3C2×D311yescharacteristic
78C2×C3×C4×D3—32no—
79C2×C3×C4×D3—32no—
80C2×C3×C4×D3—32no—
81C2×C3×C4×D3—61no—
82C2×C3×Dic6—21no—
83C2×C3×Dic6—32no—
84C2×C3×Dic6—32no—
85C2×C3×Dic6C2×D312yes—
86C2×#627—31no—
87C2×#627—32no—
88C2×#627C2×D312yes—
89C2×#624C2×D311yescharacteristic
90C2×#624C2×D311yescharacteristic
91C32×C4⋊C4—32no—
92#3752—32no—
93C4⋊(C3⋊Dic3)—32no—
94C4⋊(C3⋊Dic3)—31no—
95C4⋊(C3⋊Dic3)—32no—
96C4⋊(C3⋊Dic3)—32no—
97C4⋊(C3⋊Dic3)—62no—
98C4⋊(C3⋊Dic3)—61no—
99C4⋊(C3⋊Dic3)—121no—
100#3750—32no—
101#3750—32no—
102#3750C2×D312yes—
103C4×C3⋊Dic3—32no—
104C3×C4⋊Dic3—32no—
105C3×C4⋊Dic3—32no—
106C3×Dic3⋊C4—22no—
107C3×Dic3⋊C4—32no—
108C3×Dic3⋊C4—32no—
109C3×Dic3⋊C4—32no—
110C3×Dic3⋊C4—32no—
111C3×Dic3⋊C4—32no—
112C3×Dic3⋊C4—32no—
113C3×Dic3⋊C4—62no—
114C3×Dic3⋊C4C2×D312yes—
115C3×Dic3⋊C4C2×D312yes—
116C3×C4×Dic3—32no—
117C3×C4×Dic3—32no—
118#3724—31no—
119#3724—32no—
120#3724—32no—
121#3724—32no—
122#3724—32no—
123#3724—62no—
124#3724—62no—
125#3724C2×D312yes—
126Dic3⋊Dic3—32no—
127Dic3⋊Dic3—32no—
128Dic3⋊Dic3—32no—
129Dic3⋊Dic3—32no—
130Dic3⋊Dic3—32no—
131Dic3⋊Dic3—62no—
132Dic3⋊Dic3—62no—
133Dic3⋊Dic3C2×D312yes—
134Dic3⋊Dic3C2×D312yes—
135#3722—32no—
136(C2×D3)⋊Dic3—32no—
137(C2×D3)⋊Dic3—32no—
138(C2×D3)⋊Dic3—32no—
139(C2×D3)⋊Dic3—32no—
140(C2×D3)⋊Dic3—62no—
141Dic32—32no—
142Dic32—32no—
143Dic32—62no—
144Dic32C2×D312yes—
145C22×C33C2411yescharacteristic
146C2×C3×C3⋊D3—21no—
147C2×C3×C3⋊D3—21no—
148C2×C3×C3⋊D3C2×Q812yes—
149C33×C4—21no—
150C3⋊(C3⋊Dic3)—22no—
151C3×C3⋊Dic3—22no—
152C3×C3⋊Dic3—21no—
153C3×C3⋊Dic3—22no—
154C32×Dic3—22no—
155C32×Dic3—22no—
156C32×Dic3C2×Q814yes—
157C3×C4⋊Q8—92no—
158C2×C3×C4⋊C4—91no—
159#946—92no—
160#946—92no—
161#946—181no—
162#944—32no—
163#944—92no—
164#926—92no—
165#924—91no—
166#924—92no—
167#924—91no—
168#896—92no—
169#896—92no—
170#892—92no—
171D3×C4⋊C4—32no—
172#890—92no—
173#890—92no—
174#890—92no—
175#890—182no—
176#889—92no—
177#889—92no—
178#888—32no—
179#888—92no—
180#888—92no—
181#872—92no—
182#870—92no—
183C22×C3⋊D3C22×D311yescharacteristic
184C22×C3×D3—32no—
185C22×C3×D3—61no—
186C32×Q8—121no—
187C2×C32×C4—21no—
188C2×C32×C4—22no—
189C2×C32×C4—32no—
190C2×C32×C4—32no—
191C2×C32×C4—32no—
192C2×C32×C4—41no—
193C2×C32×C4C22×D311yescharacteristic
194C2×C32×C4C22×D312yes—
195C2×C3⋊Dic3—32no—
196C2×C3⋊Dic3—32no—
197C2×C3⋊Dic3—32no—
198C2×C3⋊Dic3—62no—
199C2×C3⋊Dic3—62no—
200C2×C3⋊Dic3—62no—
201C2×C3⋊Dic3—122no—
202C2×C3⋊Dic3C22×D312yes—
203C2×C3⋊Dic3C22×D312yes—
204C2×C3⋊Dic3C22×D311yescharacteristic
205C4×C3⋊D3—41no—
206C2×C3×Dic3—22no—
207C2×C3×Dic3—22no—
208C2×C3×Dic3—22no—
209C2×C3×Dic3—32no—
210C2×C3×Dic3—32no—
211C2×C3×Dic3—32no—
212C2×C3×Dic3—32no—
213C2×C3×Dic3—32no—
214C2×C3×Dic3—62no—
215C2×C3×Dic3—62no—
216C2×C3×Dic3—61no—
217C2×C3×Dic3C22×D312yes—
218C2×C3×Dic3C22×D312yes—
219C2×C3×Dic3C22×D312yes—
220C2×C3×Dic3C22×D312yes—
221C2×C3×Dic3C22×D312yes—
222C2×C3×Dic3C22×D312yes—
223C3×C4×D3—38no—
224C3×C4×D3—122no—
225C3×C4×D3—122no—
226C3×C4×D3—122no—
227C3×Dic6—42no—
228C3×Dic6—42no—
229C3×Dic6—122no—
230C3×Dic6—122no—
231#627—42no—
232#627—121no—
233#627—122no—
234#624—41no—
235#624Dic614yescomplemented
236#544—271noSylow
237C2×C33C22×Q811yescharacteristic
238C2×C33H(5,𝔽2)11yescharacteristic
239C2×C332−1+411yescharacteristic
240C3×C3⋊D3—22no—
241C2×C3×Q8—32no—
242C2×C3×Q8—61no—
243C2×C3×Q8—92no—
244C22×C3×C4—91no—
245C22×C3×C4—91no—
246C22×C3×C4—91no—
247C2×C4×D3—32no—
248C2×C4×D3—32no—
249C2×C4×D3—32no—
250C2×C4×D3—61no—
251C2×Dic6—32no—
252C2×Dic6—32no—
253C2×Dic6—91no—
254C2×Dic6—92no—
255C2×Dic6D3211yescharacteristic
256C3×C4⋊C4—32no—
257C3×C4⋊C4—32no—
258C3×C4⋊C4—62no—
259C3×C4⋊C4—92no—
260C3×C4⋊C4—92no—
261C3×C4⋊C4—92no—
262C3×C4⋊C4—92no—
263C3×C42—92no—
264#217—92no—
265#217—92no—
266(C2×D3)⋊C4—92no—
267(C2×D3)⋊C4—92no—
268(C2×D3)⋊C4—92no—
269(C2×D3)⋊C4—92no—
270(C2×D3)⋊C4—182no—
271C4⋊Dic3—32no—
272C4⋊Dic3—32no—
273C4⋊Dic3—91no—
274C4⋊Dic3—92no—
275C4⋊Dic3—92no—
276C4⋊Dic3—92no—
277C4⋊Dic3—92no—
278C4⋊Dic3—92no—
279C4⋊Dic3—91no—
280C4⋊Dic3—182no—
281C4⋊Dic3—182no—
282C4⋊Dic3—182no—
283C4⋊Dic3—181no—
284C4⋊Dic3—361no—
285Dic3⋊C4—32no—
286Dic3⋊C4—32no—
287Dic3⋊C4—32no—
288Dic3⋊C4—32no—
289Dic3⋊C4—32no—
290Dic3⋊C4—32no—
291Dic3⋊C4—62no—
292Dic3⋊C4—92no—
293Dic3⋊C4—92no—
294Dic3⋊C4—92no—
295Dic3⋊C4—92no—
296Dic3⋊C4—92no—
297Dic3⋊C4—92no—
298Dic3⋊C4—92no—
299Dic3⋊C4—92no—
300Dic3⋊C4—92no—
301Dic3⋊C4—182no—
302Dic3⋊C4—182no—
303Dic3⋊C4D3212yes—
304C4×Dic3—32no—
305C4×Dic3—32no—
306C4×Dic3—92no—
307C4×Dic3—92no—
308C4×Dic3—92no—
309C4×Dic3—182no—
310C22×C32—22no—
311C22×C32—21no—
312C22×C32—41no—
313C22×C32C23×D311yescharacteristic
314C22×C32C23×D312yes—
315C2×C3⋊D3—21no—
316C2×C3⋊D3—21no—
317C2×C3⋊D3C2×Dic612yes—
318C2×C3×D3—34no—
319C2×C3×D3—62no—
320C2×C3×D3—62no—
321C2×C3×D3—62no—
322C2×C3×D3—121no—
323C2×C3×D3—121no—
324C32×C4—21no—
325C32×C4—22no—
326C32×C4—22no—
327C32×C4—34no—
328C32×C4—42no—
329C32×C4—42no—
330C32×C4—62no—
331C32×C4—62no—
332C3⋊Dic3—22no—
333C3⋊Dic3—21no—
334C3⋊Dic3—22no—
335C3⋊Dic3—62no—
336C3⋊Dic3—62no—
337C3⋊Dic3—62no—
338C3⋊Dic3—122no—
339C3⋊Dic3—122no—
340C3⋊Dic3—122no—
341C3⋊Dic3—242no—
342C3×Dic3—22no—
343C3×Dic3—22no—
344C3×Dic3—22no—
345C3×Dic3—22no—
346C3×Dic3—24no—
347C3×Dic3—42no—
348C3×Dic3—42no—
349C3×Dic3—62no—
350C3×Dic3—62no—
351C3×Dic3—62no—
352C3×Dic3—62no—
353C3×Dic3—62no—
354C3×Dic3—121no—
355C3×Dic3—122no—
356C3×Dic3—122no—
357C3×Dic3C2×Dic614yes—
358C3×Dic3D3×Q814yes—
359C4⋊Q8—92no—
360C4⋊Q8—272no—
361#125—272no—
362#125—272no—
363C22⋊Q8—271no—
364C22⋊Q8—272no—
365C22⋊Q8—271no—
366#117—272no—
367C2×C4⋊C4—91no—
368C33#54411yescharacteristic, complemented, Sylow
369C23×C3—91no—
370C22×D3—32no—
371C22×D3—61no—
372C3×Q8—122no—
373C3×Q8—122no—
374C3×Q8—362no—
375C2×C3×C4—21no—
376C2×C3×C4—22no—
377C2×C3×C4—32no—
378C2×C3×C4—32no—
379C2×C3×C4—32no—
380C2×C3×C4—32no—
381C2×C3×C4—32no—
382C2×C3×C4—32no—
383C2×C3×C4—41no—
384C2×C3×C4—62no—
385C2×C3×C4—62no—
386C2×C3×C4—62no—
387C2×C3×C4—94no—
388C2×C3×C4—92no—
389C2×C3×C4—92no—
390C2×C3×C4—91no—
391C2×C3×C4—361no—
392C2×C3×C4—361no—
393C2×C3×C4C2×D3211yescharacteristic
394C2×C3×C4C2×D3212yes—
395C2×Dic3—32no—
396C2×Dic3—32no—
397C2×Dic3—32no—
398C2×Dic3—32no—
399C2×Dic3—32no—
400C2×Dic3—62no—
401C2×Dic3—62no—
402C2×Dic3—61no—
403C2×Dic3—92no—
404C2×Dic3—92no—
405C2×Dic3—92no—
406C2×Dic3—182no—
407C2×Dic3—182no—
408C2×Dic3—182no—
409C2×Dic3—362no—
410C2×Dic3C2×D3212yes—
411C2×Dic3C2×D3212yes—
412C2×Dic3C2×D3212yes—
413C4×D3—38no—
414C4×D3—122no—
415C4×D3—122no—
416C4×D3—122no—
417Dic6—41no—
418Dic6—122no—
419Dic6—122no—
420Dic6—362no—
421Dic6—361no—
422C2×C32—22no—
423C2×C32—22no—
424C2×C32—22no—
425C2×C32—21no—
426C2×C32—21no—
427C2×C32—21no—
428C2×C32—41no—
429C2×C32—41no—
430C2×C32—41no—
431C2×C32C22×Dic611yescharacteristic
432C2×C32#100411yescharacteristic
433C2×C32#100412yes—
434C2×C32C2×D3×Q812yes—
435C2×C32#100712yes—
436C2×C32#100711yescharacteristic
437C3⋊D3—22no—
438C3×D3—64no—
439C3×D3—122no—
440C2×Q8—31no—
441C2×Q8—92no—
442C2×Q8—271no—
443C22×C4—91no—
444C22×C4—91no—
445C22×C4—91no—
446C4⋊C4—32no—
447C4⋊C4—92no—
448C4⋊C4—92no—
449C4⋊C4—92no—
450C4⋊C4—92no—
451C4⋊C4—272no—
452C4⋊C4—272no—
453C4⋊C4—272no—
454C4⋊C4—272no—
455C4⋊C4—271no—
456C4⋊C4—271no—
457C22⋊C4—272no—
458C22⋊C4—272no—
459C42—92no—
460C42—272no—
461C22×C3—22no—
462C22×C3—21no—
463C22×C3—41no—
464C22×C3—92no—
465C22×C3—181no—
466C22×C3—181no—
467C22×C3C22×D3212yes—
468C22×C3C22×D3211yescharacteristic
469C2×D3—34no—
470C2×D3—62no—
471C2×D3—62no—
472C2×D3—62no—
473C2×D3—121no—
474C2×D3—121no—
475C3×C4—22no—
476C3×C4—22no—
477C3×C4—21no—
478C3×C4—34no—
479C3×C4—34no—
480C3×C4—42no—
481C3×C4—42no—
482C3×C4—64no—
483C3×C4—62no—
484C3×C4—62no—
485C3×C4—62no—
486C3×C4—62no—
487C3×C4—122no—
488C3×C4—122no—
489C3×C4—182no—
490C3×C4—182no—
491C3×C4—181no—
492Dic3—22no—
493Dic3—22no—
494Dic3—62no—
495Dic3—62no—
496Dic3—62no—
497Dic3—62no—
498Dic3—62no—
499Dic3—121no—
500Dic3—122no—
501Dic3—122no—
502Dic3—182no—
503Dic3—182no—
504Dic3—182no—
505Dic3—362no—
506Dic3—362no—
507Dic3—362no—
508Dic3—722no—
509Dic3D3×Dic614yes—
510C32—22no—
511C32—21no—
512C32—41no—
513C32#550011yescharacteristic, complemented
514C32#572912yescomplemented
515C23—91no—
516Q8—121no—
517Q8—362no—
518Q8—1081no—
519C2×C4—32no—
520C2×C4—32no—
521C2×C4—32no—
522C2×C4—94no—
523C2×C4—92no—
524C2×C4—92no—
525C2×C4—91no—
526C2×C4—272no—
527C2×C4—361no—
528C2×C4—361no—
529C2×C4D3311yescharacteristic
530C2×C3—22no—
531C2×C3—21no—
532C2×C3—22no—
533C2×C3—21no—
534C2×C3—21no—
535C2×C3—22no—
536C2×C3—41no—
537C2×C3—41no—
538C2×C3—41no—
539C2×C3—182no—
540C2×C3C2×D3×Dic612yes—
541C2×C3#6431312yes—
542C2×C3#6431612yes—
543C2×C3#6431611yescharacteristic
544C2×C3D32×Q811yescharacteristic
545C2×C3#6433011yescharacteristic
546D3—64no—
547D3—122no—
548C22—92no—
549C22—181no—
550C22—181no—
551C22C2×D3311yescharacteristic
552C4—21no—
553C4—34no—
554C4—62no—
555C4—62no—
556C4—182no—
557C4—181no—
558C4—182no—
559C4—542no—
560C3—21no—
561C3—22no—
562C3—41no—
563C3#1059434212yescomplemented
564C3#1059537511yescharacteristic, complemented
565C2—182no—
566C2D32×Dic611yescharacteristic
567C2#1172439911yescharacteristic
568C2#1172441511yescharacteristic
subgroupG/H
center, FrattiniC22C2×D33
commutator, socle, cosocleC22×C33C24
FittingC2×C33×C4C23
prime radicalC22×C3×C3⋊D3C23
hypercenterC2×C4D33
hypocenterC33#544
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
3. #11723400C2semidirect
5. C3×#63839C2semidirect
6. #11723035C2semidirect
7. #11723011C2semidirect
57. #63839D3semidirect
73. C3×#624Q8semidirect
235. #624Dic6semidirect
368. C33#544semidirect
513. C32#5500semidirect
514. C32#5729semidirect
563. C3#10594342semidirect
564. C3#10595375semidirect

Maximal subgroups

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

subgroupindex
#117236712
#117234352
#117234002
C3×#638832
C3×#638392
#117230352
#117230112
#117229702
#117229152
#105953753
#105943423