Finite Group Databasebeta

← #13178133

Subgroups — #13178133

Order 1728.

1134 conjugacy classes of subgroups.

The subgroup lattice has 548 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#64081C211yesmaximal, characteristic
2C3×C4×#634C212yesmaximal, complemented
3C4×#6296C212yesmaximal, complemented
4#11723692C212yesmaximal
5#11723688C212yesmaximal
6C4×#6294C211yesmaximal, characteristic
7#11723673C211yesmaximal, characteristic
8#11723669C212yesmaximal, complemented
9#11723666C212yesmaximal, complemented
10#10593935—31nomaximal
11#10593926—32nomaximal
12C2×C3×C4×C3⋊D3C2211yescharacteristic
13C2×C3×#634C2212yes—
14C2×#6296C2212yes—
15C2×#6294C2212yes—
16C33×C42C2211yescharacteristic
17C3×#3752C2212yes—
18C3×C4⋊(C3⋊Dic3)C2212yes—
19C3×#3750C2212yes—
20C3×#3750C2212yes—
21C3×#3750C2212yes—
22C3×C4×C3⋊Dic3C2212yes—
23C3×C4×C3⋊Dic3C2212yes—
24C3×C4×C3⋊Dic3C2211yescharacteristic
25#88815C2212yes—
26#88815C2212yes—
27#88815C2212yes—
28#88815C2212yes—
29#88814C2211yescharacteristic
30#88814C2211yescharacteristic
31#88813C2211yescharacteristic
32#88813C2211yescharacteristic
33#64081D311yescharacteristic
34C4×#634D312yescomplemented
35C3×#872—32no—
36C3×#872—61no—
37C3×C4×Dic6—32no—
38C3×C4×Dic6—32no—
39C3×C4×Dic6—62no—
40C4×#627—32no—
41C4×#627—32no—
42C4×#627—32no—
43#63899—32no—
44#63896—32no—
45#63895—32no—
46#63893—32no—
47#63888—32no—
48#63887—32no—
49C4×#624—31no—
50C4×D3×Dic3—32no—
51#63874—31no—
52#63855—32no—
53#63847—32no—
54#63846—32no—
55#63845—32no—
56#63840—32no—
57#63839—32no—
58#63839—32no—
59C22×C3×C3⋊D3C2311yescharacteristic
60C2×C33×C4C2312yes—
61C2×C33×C4C2311yescharacteristic
62C2×C3×C3⋊Dic3C2312yes—
63C2×C3×C3⋊Dic3C2312yes—
64C2×C3×C3⋊Dic3C2312yes—
65C2×C3×C3⋊Dic3C2312yes—
66C2×C3×C3⋊Dic3C2311yescharacteristic
67C2×C3×C3⋊Dic3C2312yes—
68C3×C4×C3⋊D3Q814yescomplemented
69C3×#634—24no—
70#6296—24no—
71#6294—24no—
72#5578—92no—
73#5517—91no—
74C2×C4×C3⋊D3C2×D311yescharacteristic
75C2×#634C2×D312yes—
76C2×C3×C4×D3—32no—
77C2×C3×C4×D3—61no—
78C2×C3×Dic6—32no—
79C2×C3×Dic6—32no—
80C2×C3×Dic6—62no—
81C2×#627—32no—
82C2×#627—32no—
83C2×#627—32no—
84C2×#624—32no—
85C2×D3×Dic3—32no—
86C2×D3×Dic3—32no—
87C32×C42—22no—
88C32×C42—21no—
89C32×C42—41no—
90C32×C42C2×D312yes—
91C32×C42C2×D311yescharacteristic
92#3752C2×D312yes—
93C4⋊(C3⋊Dic3)C2×D312yes—
94#3750C2×D312yes—
95#3750C2×D312yes—
96#3750C2×D312yes—
97C4×C3⋊Dic3C2×D312yes—
98C4×C3⋊Dic3C2×D312yes—
99C4×C3⋊Dic3C2×D311yescharacteristic
100C3×(C2×D3)⋊C4—32no—
101C3×(C2×D3)⋊C4—32no—
102C3×(C2×D3)⋊C4—62no—
103C3×C4⋊Dic3—32no—
104C3×C4⋊Dic3—32no—
105C3×C4⋊Dic3—62no—
106C3×Dic3⋊C4—32no—
107C3×Dic3⋊C4—32no—
108C3×Dic3⋊C4—32no—
109C3×Dic3⋊C4—32no—
110C3×Dic3⋊C4—32no—
111C3×Dic3⋊C4—32no—
112C3×Dic3⋊C4—62no—
113C3×Dic3⋊C4—62no—
114C3×Dic3⋊C4—62no—
115C3×C4×Dic3—32no—
116C3×C4×Dic3—32no—
117C3×C4×Dic3—32no—
118C3×C4×Dic3—32no—
119C3×C4×Dic3—32no—
120C3×C4×Dic3—62no—
121C3×C4×Dic3—62no—
122C3×C4×Dic3—61no—
123#3724—32no—
124#3724—32no—
125#3724—32no—
126#3724—32no—
127Dic3⋊Dic3—32no—
128Dic3⋊Dic3—32no—
129Dic3⋊Dic3—32no—
130Dic3⋊Dic3—32no—
131Dic3⋊Dic3—32no—
132Dic3⋊Dic3—32no—
133Dic3⋊Dic3—32no—
134Dic3⋊Dic3—32no—
135#3722—31no—
136#3722—31no—
137(C2×D3)⋊Dic3—32no—
138(C2×D3)⋊Dic3—32no—
139Dic32—31no—
140Dic32—32no—
141Dic32—32no—
142Dic32—31no—
143C22×C33C2411yescharacteristic
144C2×C3×C3⋊D3—22no—
145C2×C3×C3⋊D3C2×Q812yes—
146C33×C4—22no—
147C33×C4C2×Q812yes—
148C3×C3⋊Dic3—22no—
149C3×C3⋊Dic3—22no—
150C3×C3⋊Dic3—22no—
151C3×C3⋊Dic3—22no—
152C3×C3⋊Dic3C2×Q812yes—
153C3×C3⋊Dic3#4114yes—
154C3×C4×Q8—92no—
155C3×#117—91no—
156#946—92no—
157#944—92no—
158#926—91no—
159C2×C4×Dic3—91no—
160#897—92no—
161#896—92no—
162#896—92no—
163#890—92no—
164#889—92no—
165#889—92no—
166#888—92no—
167#888—92no—
168#878—92no—
169#878—92no—
170#872—32no—
171#872—61no—
172#872—92no—
173C42×D3—92no—
174C4×Dic6—32no—
175C4×Dic6—32no—
176C4×Dic6—62no—
177C4×Dic6—92no—
178C4×Dic6—92no—
179C4×Dic6—92no—
180C22×C3⋊D3C22×D311yescharacteristic
181C22×C3×D3—32no—
182C22×C3×D3—61no—
183C2×C32×C4—22no—
184C2×C32×C4—21no—
185C2×C32×C4—22no—
186C2×C32×C4—22no—
187C2×C32×C4—22no—
188C2×C32×C4—42no—
189C2×C32×C4—41no—
190C2×C32×C4C22×D312yes—
191C2×C32×C4C22×D312yes—
192C2×C32×C4C22×D312yes—
193C2×C32×C4C22×D311yescharacteristic
194C2×C32×C4C22×D312yes—
195C2×C3⋊Dic3C22×D312yes—
196C2×C3⋊Dic3C22×D312yes—
197C2×C3⋊Dic3C22×D312yes—
198C2×C3⋊Dic3C22×D312yes—
199C2×C3⋊Dic3C22×D311yescharacteristic
200C2×C3⋊Dic3C22×D312yes—
201C4×C3⋊D3Dic614yescomplemented
202#634—24no—
203C2×C3×Dic3—32no—
204C2×C3×Dic3—32no—
205C2×C3×Dic3—32no—
206C2×C3×Dic3—32no—
207C2×C3×Dic3—32no—
208C2×C3×Dic3—32no—
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×Dic3—62no—
218C2×C3×Dic3—62no—
219C2×C3×Dic3—62no—
220C3×C4×D3—38no—
221C3×C4×D3—64no—
222C3×Dic6—64no—
223C3×Dic6—64no—
224C3×Dic6—68no—
225#627—64no—
226#627—64no—
227#627—64no—
228#624—64no—
229D3×Dic3—64no—
230D3×Dic3—64no—
231#533—271noSylow
232C2×C33C22×Q811yescharacteristic
233C2×C33C2×#4112yes—
234C3×C3⋊D3—22no—
235C2×C3×Q8—92no—
236C22×C3×C4—91no—
237C22×Dic3—92no—
238C2×C4×D3—32no—
239C2×C4×D3—61no—
240C2×C4×D3—92no—
241C2×C4×D3—92no—
242C2×Dic6—32no—
243C2×Dic6—32no—
244C2×Dic6—62no—
245C2×Dic6—92no—
246C2×Dic6—92no—
247C2×Dic6—92no—
248C3×C4⋊C4—92no—
249C3×C4⋊C4—92no—
250C3×C4⋊C4—92no—
251C3×C4⋊C4—92no—
252C3×C22⋊C4—92no—
253C3×C42—22no—
254C3×C42—21no—
255C3×C42—41no—
256C3×C42—91no—
257C3×C42—92no—
258C3×C42—92no—
259C3×C42D3211yescharacteristic
260C3×C42D3212yes—
261#217—91no—
262#217—91no—
263(C2×D3)⋊C4—32no—
264(C2×D3)⋊C4—32no—
265(C2×D3)⋊C4—62no—
266(C2×D3)⋊C4—92no—
267(C2×D3)⋊C4—92no—
268C4⋊Dic3—32no—
269C4⋊Dic3—32no—
270C4⋊Dic3—62no—
271C4⋊Dic3—92no—
272C4⋊Dic3—92no—
273C4⋊Dic3—92no—
274C4⋊Dic3—92no—
275Dic3⋊C4—32no—
276Dic3⋊C4—32no—
277Dic3⋊C4—32no—
278Dic3⋊C4—32no—
279Dic3⋊C4—32no—
280Dic3⋊C4—32no—
281Dic3⋊C4—62no—
282Dic3⋊C4—62no—
283Dic3⋊C4—62no—
284Dic3⋊C4—92no—
285Dic3⋊C4—92no—
286Dic3⋊C4—92no—
287Dic3⋊C4—92no—
288Dic3⋊C4—92no—
289Dic3⋊C4—92no—
290Dic3⋊C4—92no—
291Dic3⋊C4—92no—
292C4×Dic3—32no—
293C4×Dic3—32no—
294C4×Dic3—32no—
295C4×Dic3—32no—
296C4×Dic3—32no—
297C4×Dic3—62no—
298C4×Dic3—62no—
299C4×Dic3—61no—
300C4×Dic3—92no—
301C4×Dic3—92no—
302C4×Dic3—91no—
303C4×Dic3—91no—
304C22×C32—22no—
305C22×C32—21no—
306C22×C32—41no—
307C22×C32C23×D311yescharacteristic
308C22×C32C23×D312yes—
309C2×C3⋊D3—22no—
310C2×C3⋊D3C2×Dic612yes—
311C2×C3×D3—34no—
312C2×C3×D3—62no—
313C2×C3×D3—62no—
314C2×C3×D3—62no—
315C2×C3×D3—122no—
316C32×C4—22no—
317C32×C4—24no—
318C32×C4—22no—
319C32×C4—22no—
320C32×C4—24no—
321C32×C4—22no—
322C32×C4—42no—
323C32×C4—42no—
324C32×C4—42no—
325C32×C4—44no—
326C32×C4C2×Dic612yes—
327C32×C4D3×Q814yes—
328C3⋊Dic3—22no—
329C3⋊Dic3—22no—
330C3⋊Dic3—22no—
331C3⋊Dic3—22no—
332C3⋊Dic3C2×Dic612yes—
333C3⋊Dic3#23514yes—
334C3×Dic3—34no—
335C3×Dic3—34no—
336C3×Dic3—34no—
337C3×Dic3—62no—
338C3×Dic3—64no—
339C3×Dic3—64no—
340C3×Dic3—62no—
341C3×Dic3—62no—
342C3×Dic3—62no—
343C3×Dic3—62no—
344C3×Dic3—62no—
345C3×Dic3—62no—
346C3×Dic3—62no—
347C3×Dic3—62no—
348C3×Dic3—122no—
349C3×Dic3—122no—
350C3×Dic3—122no—
351C4⋊Q8—272no—
352#125—272no—
353C22⋊Q8—272no—
354C22⋊Q8—272no—
355C4×Q8—92no—
356C4×Q8—272no—
357#117—91no—
358#117—271no—
359C2×C42—271no—
360C33#53311yescharacteristic, complemented, Sylow
361C23×C3—91no—
362C22×D3—32no—
363C22×D3—61no—
364C3×Q8—184no—
365C2×C3×C4—22no—
366C2×C3×C4—22no—
367C2×C3×C4—22no—
368C2×C3×C4—22no—
369C2×C3×C4—21no—
370C2×C3×C4—42no—
371C2×C3×C4—41no—
372C2×C3×C4—91no—
373C2×C3×C4—92no—
374C2×C3×C4—94no—
375C2×C3×C4—92no—
376C2×C3×C4—92no—
377C2×C3×C4—92no—
378C2×C3×C4—92no—
379C2×C3×C4C2×D3212yes—
380C2×C3×C4C2×D3212yes—
381C2×C3×C4C2×D3212yes—
382C2×C3×C4C2×D3212yes—
383C2×C3×C4C2×D3211yescharacteristic
384C2×Dic3—32no—
385C2×Dic3—32no—
386C2×Dic3—32no—
387C2×Dic3—32no—
388C2×Dic3—32no—
389C2×Dic3—32no—
390C2×Dic3—32no—
391C2×Dic3—32no—
392C2×Dic3—32no—
393C2×Dic3—32no—
394C2×Dic3—32no—
395C2×Dic3—62no—
396C2×Dic3—62no—
397C2×Dic3—61no—
398C2×Dic3—62no—
399C2×Dic3—62no—
400C2×Dic3—62no—
401C2×Dic3—184no—
402C4×D3—38no—
403C4×D3—64no—
404C4×D3—184no—
405C4×D3—184no—
406Dic6—64no—
407Dic6—64no—
408Dic6—68no—
409Dic6—184no—
410Dic6—184no—
411Dic6—184no—
412C2×C32—22no—
413C2×C32—21no—
414C2×C32—22no—
415C2×C32—22no—
416C2×C32—22no—
417C2×C32—41no—
418C2×C32—42no—
419C2×C32C22×Dic611yescharacteristic
420C2×C32C2×#23512yes—
421C2×C32C2×D3×Q812yes—
422C2×C32D3×#4112yes—
423C2×C32D3×#4112yes—
424C3⋊D3—22no—
425C3×D3—64no—
426C3×D3—122no—
427C2×Q8—92no—
428C2×Q8—272no—
429C22×C4—91no—
430C22×C4—272no—
431C4⋊C4—92no—
432C4⋊C4—92no—
433C4⋊C4—92no—
434C4⋊C4—92no—
435C4⋊C4—272no—
436C4⋊C4—272no—
437C4⋊C4—272no—
438C4⋊C4—272no—
439C22⋊C4—92no—
440C22⋊C4—271no—
441C22⋊C4—271no—
442C42—92no—
443C42—91no—
444C42—92no—
445C42—271no—
446C42—271no—
447C42#111311yescharacteristic
448C22×C3—22no—
449C22×C3—21no—
450C22×C3—41no—
451C22×C3—92no—
452C22×C3—182no—
453C22×C3C22×D3212yes—
454C22×C3C22×D3211yescharacteristic
455C2×D3—34no—
456C2×D3—62no—
457C2×D3—62no—
458C2×D3—62no—
459C2×D3—122no—
460C3×C4—24no—
461C3×C4—22no—
462C3×C4—22no—
463C3×C4—22no—
464C3×C4—22no—
465C3×C4—24no—
466C3×C4—42no—
467C3×C4—42no—
468C3×C4—44no—
469C3×C4—42no—
470C3×C4—92no—
471C3×C4—94no—
472C3×C4—182no—
473C3×C4—182no—
474C3×C4—182no—
475C3×C4—182no—
476C3×C4D3×Dic614yes—
477C3×C4#379712yes—
478Dic3—34no—
479Dic3—34no—
480Dic3—34no—
481Dic3—64no—
482Dic3—62no—
483Dic3—64no—
484Dic3—62no—
485Dic3—62no—
486Dic3—62no—
487Dic3—62no—
488Dic3—62no—
489Dic3—62no—
490Dic3—62no—
491Dic3—62no—
492Dic3—122no—
493Dic3—122no—
494Dic3—122no—
495C32—22no—
496C32—21no—
497C32—41no—
498C32#551711yescharacteristic, complemented
499C32#557812yescomplemented
500C23—91no—
501Q8—184no—
502Q8—544no—
503C2×C4—92no—
504C2×C4—91no—
505C2×C4—94no—
506C2×C4—92no—
507C2×C4—92no—
508C2×C4—92no—
509C2×C4—92no—
510C2×C4—544no—
511C2×C4C2×#111312yes—
512C2×C4C2×#111311yescharacteristic
513C2×C3—21no—
514C2×C3—22no—
515C2×C3—22no—
516C2×C3—22no—
517C2×C3—22no—
518C2×C3—41no—
519C2×C3—42no—
520C2×C3—182no—
521C2×C3C2×D3×Dic612yes—
522C2×C3C2×#379711yescharacteristic
523C2×C3D3×#23512yes—
524C2×C3D3×#23512yes—
525C2×C3#6430612yes—
526D3—64no—
527D3—122no—
528C22—92no—
529C22—182no—
530C22C22×#111311yescharacteristic
531C4—22no—
532C4—92no—
533C4—94no—
534C4—182no—
535C4—182no—
536C4—182no—
537C4—182no—
538C4#8904012yes—
539C3—21no—
540C3—22no—
541C3—41no—
542C3#1059392612yescomplemented
543C3#1059393511yescharacteristic, complemented
544C2—182no—
545C2C2×#8904011yescharacteristic
546C2#1172461812yes—
subgroupG/H
centerC2×C4C2×#1113
commutator, socle, cosocleC22×C33C24
FrattiniC22C22×#1113
FittingC33×C42C22
prime radicalC22×C3×C3⋊D3C23
hypercenterC42#1113
hypocenterC33#533
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C2×C4 ◁ C42

Complements

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

NH ≅ G/Ntype
2. C3×C4×#634C2semidirect
3. C4×#6296C2semidirect
8. #11723669C2semidirect
9. #11723666C2semidirect
34. C4×#634D3semidirect
68. C3×C4×C3⋊D3Q8semidirect
201. C4×C3⋊D3Dic6semidirect
360. C33#533semidirect
498. C32#5517semidirect
499. C32#5578semidirect
542. C3#10593926semidirect
543. C3#10593935semidirect

Maximal subgroups

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

subgroupindex
C3×#640812
C3×C4×#6342
C4×#62962
#117236922
#117236882
C4×#62942
#117236732
#117236692
#117236662
#105939353
#105939263