Finite Group Databasebeta

← #13147672

Subgroups — #13147672

Order 1728.

778 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11721712C211yesmaximal, characteristic, complemented
2C42×D3×C9C211yesmaximal, characteristic, complemented
3C3×#63432C211yesmaximal, characteristic, complemented
4#11721038C211yesmaximal, characteristic
5#11721030C212yesmaximal, complemented
6#11720993C211yesmaximal, characteristic, complemented
7#11720961C211yesmaximal, characteristic, complemented
8#11720954C212yesmaximal, complemented
9#11720926C212yesmaximal, complemented
10#11720912C211yesmaximal, characteristic, complemented
11#11720898C212yesmaximal, complemented
12#10593976—31nomaximal
13#10589581—31nomaximal
14C2×C4⋊(C3⋊D9)C2211yescharacteristic
15C2×#6226C2211yescharacteristic
16C2×C4×D3×C9C2211yescharacteristic
17C2×C4×D3×C9C2211yescharacteristic
18C2×C4×D3×C9C2211yescharacteristic
19C2×#6195C2212yes—
20C2×(C3⋊D9)⋊C4C2212yes—
21C3×C42×C9C2211yescharacteristic
22#88536C2212yes—
23#88536C2212yes—
24C4×C9×Dic3C2211yescharacteristic
25C4×C9×Dic3C2211yescharacteristic
26C4×C9×Dic3C2211yescharacteristic
27C3×C4⋊Dic9C2211yescharacteristic
28C3×C4⋊Dic9C2211yescharacteristic
29C3×Dic9⋊C4C2212yes—
30C3×Dic9⋊C4C2212yes—
31(C2×D3)⋊Dic9C2212yes—
32#88445C2212yes—
33#88445C2212yes—
34#88442C2212yes—
35Dic9⋊Dic3C2212yes—
36Dic3×Dic9C2212yes—
37#64084—31no—
38C3×C42×D3D311yescharacteristic
39C3×#870—31no—
40#63909—31no—
41#63904—32no—
42#63894—31no—
43#63868—32no—
44#63857—31no—
45#63850—32no—
46#63844—31no—
47#63836—32no—
48C2×C42×C9—31no—
49#63492—31no—
50#63491—32no—
51C4×#611—32no—
52#63486—31no—
53#63438—32no—
54#63437—31no—
55#63437—61no—
56C4×D36—31no—
57#63434—32no—
58#63432D311yescharacteristic, complemented
59C4×Dic18—31no—
60C22×C3⋊D9C2312yes—
61C22×D3×C9C2311yescharacteristic
62C2×C3×C4×C9C2311yescharacteristic
63C2×C3×C4×C9C2311yescharacteristic
64C2×C3×C4×C9C2311yescharacteristic
65C2×C3⋊Dic9C2312yes—
66C4⋊(C3⋊D9)—22no—
67#6226—22no—
68C2×C9×Dic3C2311yescharacteristic
69C2×C9×Dic3C2311yescharacteristic
70C2×C9×Dic3C2311yescharacteristic
71C2×C3×Dic9C2312yes—
72C2×C3×Dic9C2312yes—
73C4×D3×C9—22no—
74C4×D3×C9—22no—
75C4×D3×C9—22no—
76#6195—24no—
77(C3⋊D9)⋊C4—24no—
78#5691—91no—
79#5479—91no—
80C2×C4⋊(C3⋊D3)—31no—
81C2×#634—31no—
82C2×C3×C4×D3C2×D311yescharacteristic
83C2×C3×C4×D3C2×D311yescharacteristic
84C2×C3×C4×D3C2×D311yescharacteristic
85C2×#626—32no—
86C2×#624—32no—
87C32×C42C2×D311yescharacteristic
88#3752—32no—
89#3752—32no—
90C3×C4⋊Dic3—31no—
91C3×C4⋊Dic3—31no—
92C3×Dic3⋊C4—32no—
93C3×Dic3⋊C4—32no—
94C3×C4×Dic3C2×D311yescharacteristic
95C3×C4×Dic3C2×D311yescharacteristic
96C3×C4×Dic3C2×D311yescharacteristic
97#3724—32no—
98Dic3⋊Dic3—32no—
99#3722—32no—
100#3722—32no—
101(C2×D3)⋊Dic3—32no—
102Dic32—32no—
103C22×C4×C9—31no—
104C22×C4×C9—31no—
105C22×C4×C9—31no—
106C2×#611—32no—
107C2×D36—31no—
108C2×D36—61no—
109C2×C4×D9—32no—
110C2×Dic18—31no—
111C2×Dic18—61no—
112C42×C9—21no—
113C42×C9—31no—
114C42×C9—31no—
115C42×C9—31no—
116C42×C9C2×D311yescharacteristic
117#3676—32no—
118(C2×D9)⋊C4—32no—
119(C2×D9)⋊C4—32no—
120(C2×D9)⋊C4—32no—
121(C2×D9)⋊C4—32no—
122(C2×D9)⋊C4—62no—
123(C2×D9)⋊C4—62no—
124C4⋊Dic9C2×D311yescharacteristic
125C4⋊Dic9C2×D311yescharacteristic
126Dic9⋊C4—32no—
127Dic9⋊C4—32no—
128Dic9⋊C4C2×D312yes—
129Dic9⋊C4C2×D312yes—
130C4×Dic9—32no—
131C22×C3×C9C2411yescharacteristic
132C2×C3⋊D9—22no—
133C2×C3⋊D9—22no—
134C2×C3⋊D9—22no—
135C2×D3×C9—21no—
136C2×D3×C9—21no—
137C2×D3×C9#4112yes—
138C3×C4×C9—21no—
139C3×C4×C9—21no—
140C3×C4×C9#4112yes—
141C3⋊Dic9—22no—
142C9×Dic3—21no—
143C9×Dic3—21no—
144C9×Dic3#4112yes—
145C3×Dic9—22no—
146C3×Dic9—22no—
147C3×#125—91no—
148C2×C3×C42—31no—
149#930—91no—
150#929—92no—
151C4×#67—92no—
152#924—91no—
153#898—92no—
154#897—91no—
155#895—91no—
156#894—92no—
157#893—92no—
158#893—91no—
159#892—92no—
160#887—91no—
161#876—92no—
162#875—91no—
163#875—91no—
164#875—181no—
165C4×D12—91no—
166#872—92no—
167C42×D3D911yescharacteristic, complemented
168#870—31no—
169C4×Dic6—91no—
170C22×C3⋊D3—32no—
171C22×C3×D3C22×D311yescharacteristic
172C2×C32×C4C22×D311yescharacteristic
173C2×C32×C4C22×D311yescharacteristic
174C2×C32×C4C22×D311yescharacteristic
175C2×C3⋊Dic3—32no—
176C4⋊(C3⋊D3)—62no—
177#634—62no—
178C2×C3×Dic3—32no—
179C2×C3×Dic3—32no—
180C2×C3×Dic3C22×D311yescharacteristic
181C2×C3×Dic3C22×D311yescharacteristic
182C2×C3×Dic3C22×D311yescharacteristic
183C3×C4×D3—22no—
184C3×C4×D3—22no—
185C3×C4×D3—22no—
186#626—64no—
187#624—64no—
188C23×C9—31no—
189C22×D9—32no—
190C22×D9—62no—
191C2×C4×C9—21no—
192C2×C4×C9—21no—
193C2×C4×C9—21no—
194C2×C4×C9—31no—
195C2×C4×C9—31no—
196C2×C4×C9—31no—
197C2×C4×C9—62no—
198C2×C4×C9—62no—
199C2×C4×C9—62no—
200C2×C4×C9C22×D311yescharacteristic
201C2×C4×C9C22×D311yescharacteristic
202C2×C4×C9C22×D311yescharacteristic
203#611—64no—
204C2×Dic9—32no—
205C2×Dic9—62no—
206C2×Dic9C22×D312yes—
207C2×Dic9C22×D312yes—
208D36—62no—
209D36—122no—
210C4×D9—64no—
211Dic18—62no—
212Dic18—122no—
213#529—271noSylow
214C2×C3×C9C2×#4111yescharacteristic
215C2×C3×C9C2×#4111yescharacteristic
216C2×C3×C9C2×#4111yescharacteristic
217C3⋊D9—42no—
218D3×C9—22no—
219C22×C3×C4—31no—
220C22×C3×C4—31no—
221C22×C3×C4—31no—
222C2×#67—92no—
223C2×D12—92no—
224C2×D12—91no—
225C2×D12—91no—
226C2×D12—181no—
227C2×C4×D3—92no—
228C2×C4×D3—92no—
229C2×C4×D3C2×D911yescharacteristic
230C2×C4×D3C2×D911yescharacteristic
231C2×C4×D3C2×D911yescharacteristic
232C2×Dic6—91no—
233C2×Dic6—91no—
234C2×Dic6—181no—
235C3×C4⋊C4—91no—
236C3×C4⋊C4—92no—
237C3×C4⋊C4—91no—
238C3×C4⋊C4—92no—
239C3×C42—21no—
240C3×C42—31no—
241C3×C42—31no—
242C3×C42—31no—
243C3×C42C2×D911yescharacteristic
244C3×C42D3211yescharacteristic
245#217—92no—
246(C2×D3)⋊C4—92no—
247(C2×D3)⋊C4—92no—
248(C2×D3)⋊C4—92no—
249(C2×D3)⋊C4—92no—
250(C2×D3)⋊C4—92no—
251(C2×D3)⋊C4—92no—
252(C2×D3)⋊C4—92no—
253(C2×D3)⋊C4—92no—
254(C2×D3)⋊C4—92no—
255(C2×D3)⋊C4—182no—
256(C2×D3)⋊C4—182no—
257C4⋊Dic3—31no—
258C4⋊Dic3—31no—
259C4⋊Dic3—92no—
260Dic3⋊C4—32no—
261Dic3⋊C4—32no—
262Dic3⋊C4—92no—
263Dic3⋊C4—92no—
264Dic3⋊C4—92no—
265C4×Dic3—92no—
266C4×Dic3—92no—
267C4×Dic3C2×D911yescharacteristic
268C4×Dic3C2×D911yescharacteristic
269C4×Dic3C2×D911yescharacteristic
270C22×C32C23×D311yescharacteristic
271C2×C3⋊D3—62no—
272C2×C3⋊D3—62no—
273C2×C3⋊D3—62no—
274C2×C3×D3—21no—
275C2×C3×D3—21no—
276C2×C3×D3#23512yes—
277C32×C4—21no—
278C32×C4—21no—
279C32×C4#23512yes—
280C3⋊Dic3—62no—
281C3×Dic3—21no—
282C3×Dic3—21no—
283C3×Dic3—62no—
284C3×Dic3—62no—
285C3×Dic3#23512yes—
286C22×C9—21no—
287C22×C9—32no—
288C22×C9—61no—
289C22×C9—61no—
290C22×C9C23×D311yescharacteristic
291C2×D9—62no—
292C2×D9—62no—
293C2×D9—62no—
294C2×D9—122no—
295C2×D9—122no—
296C2×D9—122no—
297C4×C9—21no—
298C4×C9—21no—
299C4×C9—22no—
300C4×C9—32no—
301C4×C9—41no—
302C4×C9—41no—
303C4×C9—61no—
304C4×C9—61no—
305C4×C9#23912yes—
306Dic9—22no—
307Dic9—22no—
308Dic9—62no—
309Dic9—122no—
310#126—272no—
311#125—91no—
312#124—271no—
313#123—272no—
314C22⋊Q8—271no—
315#121—271no—
316C4×Q8—271no—
317C4×D4—271no—
318C4×D4—272no—
319#117—272no—
320C2×C42—31no—
321C3×C9#52911yescharacteristic, complemented, Sylow
322C23×C3—31no—
323C22×D3—92no—
324C22×D3—92no—
325C22×D3—182no—
326C22×D3C22×D911yescharacteristic
327C2×C3×C4—21no—
328C2×C3×C4—21no—
329C2×C3×C4—21no—
330C2×C3×C4—31no—
331C2×C3×C4—31no—
332C2×C3×C4—31no—
333C2×C3×C4—62no—
334C2×C3×C4—62no—
335C2×C3×C4—62no—
336C2×C3×C4—92no—
337C2×C3×C4—92no—
338C2×C3×C4C22×D911yescharacteristic
339C2×C3×C4C22×D911yescharacteristic
340C2×C3×C4C22×D911yescharacteristic
341C2×C3×C4C2×D3211yescharacteristic
342C2×C3×C4C2×D3211yescharacteristic
343C2×C3×C4C2×D3211yescharacteristic
344#67—184no—
345C2×Dic3—32no—
346C2×Dic3—32no—
347C2×Dic3—92no—
348C2×Dic3—92no—
349C2×Dic3—182no—
350C2×Dic3C22×D911yescharacteristic
351C2×Dic3C22×D911yescharacteristic
352C2×Dic3C22×D911yescharacteristic
353D12—184no—
354D12—182no—
355D12—182no—
356D12—362no—
357C4×D3—22no—
358C4×D3—22no—
359C4×D3—22no—
360C4×D3—184no—
361C4×D3—184no—
362Dic6—182no—
363Dic6—182no—
364Dic6—362no—
365C2×C32C2×#23511yescharacteristic
366C2×C32C2×#23511yescharacteristic
367C2×C32C2×#23511yescharacteristic
368C3⋊D3—122no—
369C3×D3—22no—
370C2×C9—21no—
371C2×C9—21no—
372C2×C9—21no—
373C2×C9—62no—
374C2×C9C2×#23911yescharacteristic
375C2×C9D3×#4111yescharacteristic
376C2×C9D3×#4111yescharacteristic
377D9—122no—
378D9—242no—
379C2×Q8—271no—
380C2×D4—272no—
381C2×D4—271no—
382C22×C4—31no—
383C22×C4—31no—
384C22×C4—31no—
385C22×C4—272no—
386C4⋊C4—92no—
387C4⋊C4—91no—
388C4⋊C4—92no—
389C4⋊C4—91no—
390C4⋊C4—272no—
391C4⋊C4—272no—
392C22⋊C4—272no—
393C22⋊C4—272no—
394C22⋊C4—272no—
395C22⋊C4—272no—
396C22⋊C4—272no—
397C42—31no—
398C42—31no—
399C42—31no—
400C42—272no—
401C42D3×D911yescharacteristic
402C22×C3—21no—
403C22×C3—32no—
404C22×C3—61no—
405C22×C3—61no—
406C22×C3C23×D911yescharacteristic
407C22×C3C22×D3211yescharacteristic
408C2×D3—21no—
409C2×D3—21no—
410C2×D3—182no—
411C2×D3—182no—
412C2×D3—182no—
413C2×D3—182no—
414C2×D3—182no—
415C2×D3—182no—
416C2×D3—362no—
417C2×D3—362no—
418C2×D3—362no—
419C2×D3#369712yes—
420C3×C4—21no—
421C3×C4—21no—
422C3×C4—21no—
423C3×C4—21no—
424C3×C4—22no—
425C3×C4—32no—
426C3×C4—41no—
427C3×C4—41no—
428C3×C4—61no—
429C3×C4—61no—
430C3×C4—182no—
431C3×C4—182no—
432C3×C4#369712yes—
433C3×C4#379912yes—
434Dic3—21no—
435Dic3—21no—
436Dic3—62no—
437Dic3—62no—
438Dic3—182no—
439Dic3—182no—
440Dic3—362no—
441Dic3#369712yes—
442C32#547911yescharacteristic
443C9—21no—
444C9#569111yescharacteristic, complemented
445C23—31no—
446C23—272no—
447Q8—542no—
448D4—544no—
449D4—542no—
450C2×C4—31no—
451C2×C4—31no—
452C2×C4—31no—
453C2×C4—62no—
454C2×C4—62no—
455C2×C4—62no—
456C2×C4—92no—
457C2×C4—92no—
458C2×C4—272no—
459C2×C4—544no—
460C2×C4C2×D3×D911yescharacteristic
461C2×C4C2×D3×D911yescharacteristic
462C2×C4C2×D3×D911yescharacteristic
463C2×C3—21no—
464C2×C3—21no—
465C2×C3—21no—
466C2×C3—62no—
467C2×C3C2×#369711yescharacteristic
468C2×C3C2×#369711yescharacteristic
469C2×C3C2×#369711yescharacteristic
470C2×C3C2×#379911yescharacteristic
471C2×C3D3×#23511yescharacteristic
472C2×C3D3×#23511yescharacteristic
473D3—22no—
474D3—362no—
475D3—362no—
476D3—722no—
477C22—32no—
478C22—61no—
479C22—61no—
480C22—542no—
481C22—542no—
482C22—542no—
483C22C22×D3×D911yescharacteristic
484C4—21no—
485C4—21no—
486C4—32no—
487C4—61no—
488C4—61no—
489C4—182no—
490C4—182no—
491C4—542no—
492C4#8864212yes—
493C3—21no—
494C3#1058958111yescharacteristic, complemented
495C3#1059397611yescharacteristic
496C2—62no—
497C2—1082no—
498C2C2×#8864211yescharacteristic
499C2D3×#369711yescharacteristic
500C2D3×#369711yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC3×C42×C9C22
socleC22×C32C23×D3
prime radical#11721038C2
hypercenterC42D3×D9
hypocenterC3×C9#529
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
1. #11721712C2semidirect
2. C42×D3×C9C2semidirect
3. C3×#63432C2semidirect
5. #11721030C2semidirect
6. #11720993C2semidirect
7. #11720961C2semidirect
8. #11720954C2semidirect
9. #11720926C2semidirect
10. #11720912C2semidirect
11. #11720898C2semidirect
58. #63432D3semidirect
167. C42×D3D9semidirect
321. C3×C9#529semidirect
444. C9#5691semidirect
494. C3#10589581semidirect

Maximal subgroups

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

subgroupindex
#117217122
C42×D3×C92
C3×#634322
#117210382
#117210302
#117209932
#117209612
#117209542
#117209262
#117209122
#117208982
#105939763
#105895813