Finite Group Databasebeta

← #13150421

Subgroups — #13150421

Order 1728.

810 conjugacy classes of subgroups.

The subgroup lattice has 516 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×C3⋊Dic9C211yesmaximal, characteristic, complemented
2C9×#930C211yesmaximal, characteristic
3C3×#63486C211yesmaximal, characteristic, complemented
4#67⋊Dic9C212yesmaximal, complemented
5#11721266C212yesmaximal, complemented
6#11721025C211yesmaximal, characteristic, complemented
7#11721011C212yesmaximal, complemented
8#11720967C211yesmaximal, characteristic, complemented
9#11720966C212yesmaximal, complemented
10#11720937C211yesmaximal, characteristic, complemented
11#11720910C211yesmaximal, characteristic, complemented
12#10594965—31nomaximal
13#10589702—31nomaximal
14C22×C3×C4×C9C2211yescharacteristic
15C22×C3⋊Dic9C2211yescharacteristic
16C22×C3⋊Dic9C2211yescharacteristic
17C2×C9×#67C2212yes—
18C2×C9×D12C2211yescharacteristic
19C2×C3×Dic18C2211yescharacteristic
20C2×D3×Dic9C2212yes—
21C4×C3⋊Dic9C2211yescharacteristic
22C4×C3⋊Dic9C2211yescharacteristic
23C4×C3⋊Dic9C2211yescharacteristic
24C4×C3⋊Dic9C2211yescharacteristic
25C3×#3676C2212yes—
26C9×(C2×D3)⋊C4C2212yes—
27C9×C4⋊Dic3C2211yescharacteristic
28C3×C4⋊Dic9C2211yescharacteristic
29C3×Dic9⋊C4C2212yes—
30(C2×D3)⋊Dic9C2212yes—
31(C2×D3)⋊Dic9C2212yes—
32(C2×D3)⋊Dic9C2212yes—
33Dic3⋊Dic9C2212yes—
34Dic9⋊Dic3C2212yes—
35Dic9⋊Dic3C2212yes—
36Dic3×Dic9C2212yes—
37C2×C4×C3⋊Dic3—31no—
38C3×#930D311yescharacteristic
39C3×#924—31no—
40#67⋊Dic3—32no—
41#63962—32no—
42#63898—31no—
43#63878—32no—
44#63872—31no—
45#63863—31no—
46#63862—32no—
47#63843—31no—
48C9×#121—31no—
49#63498—31no—
50#63497—32no—
51D4×Dic9—32no—
52D4×Dic9—31no—
53#63486D311yescharacteristic, complemented
54C2×C4×Dic9—31no—
55C2×C4×Dic9—61no—
56#63452—31no—
57#63449—31no—
58#63441—32no—
59#63439—32no—
60C23×C3×C9C2311yescharacteristic
61C22×D3×C9C2312yes—
62C2×C3×C4×C9—22no—
63C2×C3×C4×C9C2311yescharacteristic
64C2×C3×C4×C9C2311yescharacteristic
65C2×C3⋊Dic9—22no—
66C2×C3⋊Dic9—22no—
67C2×C3⋊Dic9C2311yescharacteristic
68C2×C3⋊Dic9C2311yescharacteristic
69C2×C3⋊Dic9C2311yescharacteristic
70C2×C3⋊Dic9C2311yescharacteristic
71C9×#67—24no—
72C2×C9×Dic3C2312yes—
73C2×C3×Dic9C2312yes—
74C2×C3×Dic9C2312yes—
75C9×D12—22no—
76C3×Dic18—22no—
77D3×Dic9—24no—
78#5624—91no—
79#5600—91no—
80C22×C32×C4C2×D311yescharacteristic
81C22×C3⋊Dic3—31no—
82C22×C3⋊Dic3—31no—
83C2×C3×#67C2×D312yes—
84C2×C3×D12C2×D311yescharacteristic
85C2×C3×Dic6—31no—
86C2×D3×Dic3—32no—
87C4×C3⋊Dic3—31no—
88C4×C3⋊Dic3—31no—
89C4×C3⋊Dic3—31no—
90C4×C3⋊Dic3—31no—
91C3×#217—32no—
92C3×(C2×D3)⋊C4C2×D312yes—
93C3×C4⋊Dic3—31no—
94C3×C4⋊Dic3C2×D311yescharacteristic
95C3×Dic3⋊C4—32no—
96Dic3⋊Dic3—32no—
97Dic3⋊Dic3—32no—
98Dic3⋊Dic3—32no—
99(C2×D3)⋊Dic3—32no—
100(C2×D3)⋊Dic3—32no—
101(C2×D3)⋊Dic3—32no—
102Dic32—32no—
103C2×D4×C9—32no—
104C2×D4×C9—31no—
105C22×C4×C9—21no—
106C22×C4×C9C2×D311yescharacteristic
107C22×Dic9—31no—
108C22×Dic9—31no—
109C22×Dic9—32no—
110C22×Dic9—61no—
111C22×Dic9—61no—
112C2×Dic18C2×D311yescharacteristic
113C9×C4⋊C4—31no—
114C9×C22⋊C4—32no—
115#3676—32no—
116#3676—32no—
117#3676—32no—
118#3676C2×D312yes—
119C4⋊Dic9—32no—
120C4⋊Dic9C2×D311yescharacteristic
121Dic9⋊C4—32no—
122Dic9⋊C4—32no—
123Dic9⋊C4C2×D312yes—
124C4×Dic9—31no—
125C4×Dic9—31no—
126C4×Dic9—31no—
127C4×Dic9—31no—
128C4×Dic9—32no—
129C4×Dic9—61no—
130C4×Dic9—61no—
131C4×Dic9—61no—
132C4×Dic9—61no—
133C22×C3×C9—21no—
134C22×C3×C9—21no—
135C22×C3×C9#4112yes—
136C22×C3×C9C2411yescharacteristic
137C2×D3×C9—22no—
138C2×D3×C9—22no—
139C2×D3×C9—22no—
140C3×C4×C9—21no—
141C3×C4×C9#4112yes—
142C3⋊Dic9—21no—
143C3⋊Dic9—21no—
144C3⋊Dic9—21no—
145C3⋊Dic9#4112yes—
146C9×Dic3—22no—
147C3×Dic9—22no—
148C3×Dic9—22no—
149C3×C22⋊Q8—91no—
150C3×#121—31no—
151#947—91no—
152Q8×Dic3—91no—
153#936—91no—
154#935—92no—
155D4×Dic3—91no—
156D4×Dic3—92no—
157#930D911yescharacteristic
158#924—31no—
159C2×C4×Dic3—91no—
160C2×C4×Dic3—91no—
161C2×C4×Dic3—181no—
162#898—92no—
163#893—91no—
164#892—92no—
165#890—91no—
166#890—91no—
167#887—91no—
168#886—92no—
169#67⋊C4—92no—
170#879—92no—
171#877—92no—
172C23×C32C22×D311yescharacteristic
173C22×C3×D3C22×D312yes—
174C2×C32×C4—22no—
175C2×C32×C4C22×D311yescharacteristic
176C2×C32×C4C22×D311yescharacteristic
177C2×C3⋊Dic3—31no—
178C2×C3⋊Dic3—31no—
179C2×C3⋊Dic3—31no—
180C2×C3⋊Dic3—31no—
181C2×C3⋊Dic3—62no—
182C2×C3⋊Dic3—62no—
183C3×#67—24no—
184C2×C3×Dic3—32no—
185C2×C3×Dic3—32no—
186C2×C3×Dic3C22×D312yes—
187C3×D12—22no—
188C3×Dic6—62no—
189D3×Dic3—64no—
190C23×C9—21no—
191C23×C9—32no—
192C23×C9C22×D311yescharacteristic
193D4×C9—64no—
194D4×C9—62no—
195C2×C4×C9—24no—
196C2×C4×C9—22no—
197C2×C4×C9—21no—
198C2×C4×C9—21no—
199C2×C4×C9—32no—
200C2×C4×C9C22×D311yescharacteristic
201C2×C4×C9C22×D311yescharacteristic
202C2×Dic9—31no—
203C2×Dic9—31no—
204C2×Dic9—31no—
205C2×Dic9—31no—
206C2×Dic9—61no—
207C2×Dic9—61no—
208C2×Dic9—61no—
209C2×Dic9—61no—
210C2×Dic9—64no—
211C2×Dic9—62no—
212C2×Dic9—64no—
213C2×Dic9—64no—
214C2×Dic9—62no—
215C2×Dic9C22×D312yes—
216C2×Dic9C22×D312yes—
217Dic18—22no—
218#529—271noSylow
219C2×C3×C9—22no—
220C2×C3×C9C2×#4111yescharacteristic
221C2×C3×C9C2×#4111yescharacteristic
222C2×C3×C9C2×#4111yescharacteristic
223D3×C9—42no—
224C2×C3×Q8—91no—
225C2×C3×D4—32no—
226C2×C3×D4—31no—
227C22×C3×C4—21no—
228C22×C3×C4C2×D911yescharacteristic
229C22×C3×C4D3211yescharacteristic
230C2×#67C2×D912yes—
231C22×Dic3—92no—
232C22×Dic3—91no—
233C22×Dic3—91no—
234C22×Dic3—91no—
235C22×Dic3—91no—
236C22×Dic3—181no—
237C22×Dic3—181no—
238C2×D12C2×D911yescharacteristic
239C2×C4×D3—92no—
240C2×Dic6—31no—
241C3×C4⋊C4—31no—
242C3×C4⋊C4—91no—
243C3×C4⋊C4—92no—
244C3×C22⋊C4—32no—
245C3×C22⋊C4—92no—
246#217—32no—
247#217—92no—
248#217—92no—
249#217—92no—
250(C2×D3)⋊C4—92no—
251(C2×D3)⋊C4—92no—
252(C2×D3)⋊C4—92no—
253(C2×D3)⋊C4C2×D912yes—
254C4⋊Dic3—31no—
255C4⋊Dic3—92no—
256C4⋊Dic3—92no—
257C4⋊Dic3—92no—
258C4⋊Dic3C2×D911yescharacteristic
259Dic3⋊C4—32no—
260Dic3⋊C4—92no—
261Dic3⋊C4—92no—
262Dic3⋊C4—92no—
263C4×Dic3—92no—
264C4×Dic3—91no—
265C4×Dic3—91no—
266C4×Dic3—91no—
267C4×Dic3—91no—
268C4×Dic3—91no—
269C4×Dic3—91no—
270C4×Dic3—91no—
271C4×Dic3—91no—
272C4×Dic3—92no—
273C4×Dic3—181no—
274C4×Dic3—181no—
275C4×Dic3—181no—
276C4×Dic3—181no—
277C22×C32—21no—
278C22×C32—21no—
279C22×C32#23712yes—
280C22×C32C23×D311yescharacteristic
281C2×C3×D3—22no—
282C2×C3×D3—22no—
283C2×C3×D3—22no—
284C32×C4—21no—
285C32×C4#23712yes—
286C3⋊Dic3—32no—
287C3⋊Dic3—61no—
288C3⋊Dic3—61no—
289C3⋊Dic3—61no—
290C3×Dic3—22no—
291C3×Dic3—62no—
292C3×Dic3—62no—
293C22×C9—22no—
294C22×C9—22no—
295C22×C9—21no—
296C22×C9—21no—
297C22×C9—21no—
298C22×C9—22no—
299C22×C9—62no—
300C22×C9—62no—
301C22×C9—62no—
302C22×C9#23912yes—
303C22×C9C23×D311yescharacteristic
304C4×C9—22no—
305C4×C9—21no—
306C4×C9—22no—
307C4×C9—62no—
308C4×C9#23712yes—
309Dic9—22no—
310Dic9—22no—
311Dic9—32no—
312Dic9—62no—
313Dic9—62no—
314Dic9—62no—
315Dic9—62no—
316Dic9—61no—
317Dic9—61no—
318Dic9—61no—
319#126—272no—
320#125—271no—
321#124—271no—
322#123—272no—
323C22⋊Q8—91no—
324#121—31no—
325C4×Q8—271no—
326C4×D4—272no—
327C4×D4—271no—
328#117—272no—
329C2×C42—271no—
330C3×C9#52911yescharacteristic, complemented, Sylow
331C23×C3—21no—
332C23×C3—32no—
333C23×C3C22×D911yescharacteristic
334C23×C3C2×D3211yescharacteristic
335C22×D3C22×D912yes—
336C3×Q8—182no—
337C3×D4—64no—
338C3×D4—62no—
339C2×C3×C4—24no—
340C2×C3×C4—22no—
341C2×C3×C4—22no—
342C2×C3×C4—21no—
343C2×C3×C4—21no—
344C2×C3×C4—32no—
345C2×C3×C4—92no—
346C2×C3×C4—92no—
347C2×C3×C4C22×D911yescharacteristic
348C2×C3×C4C22×D911yescharacteristic
349C2×C3×C4C2×D3211yescharacteristic
350C2×C3×C4C2×D3211yescharacteristic
351#67—24no—
352C2×Dic3—32no—
353C2×Dic3—32no—
354C2×Dic3—91no—
355C2×Dic3—91no—
356C2×Dic3—91no—
357C2×Dic3—91no—
358C2×Dic3—91no—
359C2×Dic3—91no—
360C2×Dic3—91no—
361C2×Dic3—91no—
362C2×Dic3—181no—
363C2×Dic3—181no—
364C2×Dic3—181no—
365C2×Dic3—181no—
366C2×Dic3—184no—
367C2×Dic3—182no—
368C2×Dic3—182no—
369C2×Dic3—184no—
370C2×Dic3—184no—
371C2×Dic3—182no—
372C2×Dic3—182no—
373C2×Dic3C22×D912yes—
374D12—22no—
375C4×D3—184no—
376Dic6—62no—
377C2×C32—22no—
378C2×C32C2×#23711yescharacteristic
379C2×C32C2×#23711yescharacteristic
380C2×C32D3×#4111yescharacteristic
381C3×D3—42no—
382C2×C9—24no—
383C2×C9—22no—
384C2×C9—21no—
385C2×C9—21no—
386C2×C9—21no—
387C2×C9—122no—
388C2×C9C2×#23711yescharacteristic
389C2×C9C2×#23911yescharacteristic
390C2×C9D3×#4111yescharacteristic
391C2×Q8—91no—
392C2×D4—32no—
393C2×D4—31no—
394C22×C4—272no—
395C22×C4—271no—
396C22×C4—271no—
397C22×C4D3×D911yescharacteristic
398C4⋊C4—31no—
399C4⋊C4—92no—
400C4⋊C4—91no—
401C4⋊C4—272no—
402C4⋊C4—272no—
403C4⋊C4—272no—
404C22⋊C4—32no—
405C22⋊C4—92no—
406C22⋊C4—272no—
407C22⋊C4—272no—
408C22⋊C4—272no—
409C42—272no—
410C42—271no—
411C42—271no—
412C42—271no—
413C42—271no—
414C22×C3—22no—
415C22×C3—22no—
416C22×C3—21no—
417C22×C3—21no—
418C22×C3—21no—
419C22×C3—21no—
420C22×C3—22no—
421C22×C3—21no—
422C22×C3—62no—
423C22×C3—62no—
424C22×C3—62no—
425C22×C3#369912yes—
426C22×C3C23×D911yescharacteristic
427C22×C3#379612yes—
428C22×C3C22×D3211yescharacteristic
429C2×D3—22no—
430C2×D3—22no—
431C2×D3—22no—
432C3×C4—22no—
433C3×C4—21no—
434C3×C4—21no—
435C3×C4—22no—
436C3×C4—62no—
437C3×C4—182no—
438C3×C4—182no—
439C3×C4#369912yes—
440C3×C4#380512yes—
441Dic3—22no—
442Dic3—62no—
443Dic3—62no—
444Dic3—92no—
445Dic3—92no—
446Dic3—182no—
447Dic3—182no—
448Dic3—182no—
449Dic3—182no—
450Dic3—181no—
451Dic3—181no—
452Dic3—181no—
453Dic3—181no—
454Dic3—181no—
455Dic3—181no—
456C32#560011yescharacteristic
457C9—21no—
458C9#562411yescharacteristic, complemented
459C23—32no—
460C23C2×D3×D911yescharacteristic
461Q8—182no—
462D4—64no—
463D4—62no—
464C2×C4—22no—
465C2×C4—32no—
466C2×C4—92no—
467C2×C4—92no—
468C2×C4—271no—
469C2×C4—271no—
470C2×C4—271no—
471C2×C4—271no—
472C2×C4—544no—
473C2×C4—542no—
474C2×C4—542no—
475C2×C4C2×D3×D911yescharacteristic
476C2×C4C2×D3×D911yescharacteristic
477C2×C3—24no—
478C2×C3—22no—
479C2×C3—22no—
480C2×C3—21no—
481C2×C3—21no—
482C2×C3—21no—
483C2×C3—122no—
484C2×C3C2×#369911yescharacteristic
485C2×C3C2×#369911yescharacteristic
486C2×C3#41×D911yescharacteristic
487C2×C3C2×#379611yescharacteristic
488C2×C3#6430611yescharacteristic
489C2×C3C2×#380511yescharacteristic
490D3—42no—
491C22—21no—
492C22—21no—
493C22—62no—
494C22—62no—
495C22—62no—
496C22#8863912yes—
497C22C22×D3×D911yescharacteristic
498C4—21no—
499C4—62no—
500C4—182no—
501C4—182no—
502C4—272no—
503C4—541no—
504C4—541no—
505C4—541no—
506C4#8866312yes—
507C3—21no—
508C3#1058970211yescharacteristic, complemented
509C3#1059496511yescharacteristic
510C2—22no—
511C2—122no—
512C2C2×#8863911yescharacteristic
513C2#1172247511yescharacteristic
514C2C2×#8866311yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC22×C3×C4×C9C22
socleC22×C32C23×D3
prime radicalC3×#930D3
hypercenterC22×C4D3×D9
hypocenterC3×C9#529
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C2×C4×C3⋊Dic9C2semidirect
3. C3×#63486C2semidirect
4. #67⋊Dic9C2semidirect
5. #11721266C2semidirect
6. #11721025C2semidirect
7. #11721011C2semidirect
8. #11720967C2semidirect
9. #11720966C2semidirect
10. #11720937C2semidirect
11. #11720910C2semidirect
53. #63486D3semidirect
330. C3×C9#529semidirect
458. C9#5624semidirect
508. C3#10589702semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C3⋊Dic92
C9×#9302
C3×#634862
#67⋊Dic92
#117212662
#117210252
#117210112
#117209672
#117209662
#117209372
#117209102
#105949653
#105897023