Finite Group Databasebeta

← #13141018

Subgroups — #13141018

Order 1728.

506 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#41×D9C211yesmaximal, characteristic
2#11720943C211yesmaximal, characteristic, complemented
3D9×#208C211yesmaximal, characteristic, complemented
4#11720585C211yesmaximal, characteristic, complemented
5C9×#837C211yesmaximal, characteristic, complemented
6#11720291C211yesmaximal, characteristic, complemented
7#11720260C211yesmaximal, characteristic, complemented
8#10591670—31nomaximal
9#10588501—31nomaximal
10C3×C9×#41C2211yescharacteristic
11C3×#3701C411yescharacteristic, complemented
12C3×Q8×D9C411yescharacteristic, complemented
13C3×#3699—21no—
14C3×#3699C411yescharacteristic, complemented
15C3×D4×D9—21no—
16C3×D4×D9C411yescharacteristic, complemented
17C3×#3697—21no—
18C3×#3697C2211yescharacteristic
19C2×C3×C4×D9—21no—
20C2×C3×C4×D9C2211yescharacteristic
21C4×C3⋊Dic9C2211yescharacteristic, complemented
22#88532C2211yescharacteristic, complemented
23C4×C9×Dic3C2211yescharacteristic, complemented
24C9×#208C2211yescharacteristic, complemented
25(C2×D9)⋊Dic3—21no—
26Dic9⋊Dic3—21no—
27#88413—21no—
28D9×C3⋊C8—21no—
29C3×D3×#41—31no—
30#63855—31no—
31D3×#208—31no—
32#41×D9D311yescharacteristic
33#63664—31no—
34C3×#837D311yescharacteristic
35#63569—31no—
36#63568—31no—
37M16×D9—31no—
38#63434—31no—
39C9×#104—31no—
40#63396—31no—
41#63396—61no—
42#63384—31no—
43#63364—31no—
44C22×C3×D9—21no—
45C22×C3×D9D411yescharacteristic
46C3×Q8×C9C2×C411yescharacteristic
47C3×D4×C9—21no—
48C3×D4×C9C2×C411yescharacteristic
49C2×C3×C4×C9—21no—
50C2×C3×C4×C9C2311yescharacteristic
51C2×C3⋊Dic9—21no—
52C3×#611—21no—
53C3×#611—21no—
54C3×#611—21no—
55C2×C9×Dic3—21no—
56C2×C3×Dic9—21no—
57C2×C3×Dic9D411yescharacteristic
58C3×D36—21no—
59C3×D36C2×C411yescharacteristic
60C3×C4×D9—21no—
61C3×C4×D9—21no—
62C3×C4×D9—21no—
63C3×C4×D9C2×C411yescharacteristic
64C3×C4×D9C2×C411yescharacteristic
65C3×C4×D9D411yescharacteristic
66C3×C4×D9D411yescharacteristic
67C3×Dic18—21no—
68C3×Dic18C2×C411yescharacteristic
69C3⋊(C9⋊C8)—21no—
70C9×C3⋊C8—21no—
71#5236—91no—
72#4821—91no—
73C32×#41C2×D311yescharacteristic
74C3×#239—31no—
75C3×D3×Q8—31no—
76C3×#237—31no—
77C3×#237—61no—
78C3×D3×D4—31no—
79C3×D3×D4—61no—
80C3×#235—31no—
81C3×#235—61no—
82C2×C3×C4×D3—31no—
83C2×C3×C4×D3—61no—
84C4×C3⋊Dic3—31no—
85#3748—31no—
86C3×C4×Dic3C2×D311yescharacteristic
87C3×#208C2×D311yescharacteristic
88Dic3⋊Dic3—61no—
89(C2×D3)⋊Dic3—61no—
90#3711—61no—
91D3×C3⋊C8—61no—
92C9×#41—21no—
93C9×#41C2×D311yescharacteristic
94#3701Dic311yescharacteristic, complemented
95Q8×D9Dic311yescharacteristic, complemented
96#3699—21no—
97#3699Dic311yescharacteristic, complemented
98D4×D9—21no—
99D4×D9Dic311yescharacteristic, complemented
100#3697—21no—
101#3697C2×D311yescharacteristic
102C2×C4×D9—21no—
103C2×C4×D9C2×D311yescharacteristic
104C9×M16—31no—
105C42×C9—31no—
106(C2×D9)⋊C4—61no—
107Dic9⋊C4—61no—
108C4×Dic9—31no—
109C4×Dic9—61no—
110#3667—31no—
111#3667—61no—
112#3663—61no—
113C8×D9—61no—
114C22×C3×C9—21no—
115C22×C3×C9C2×D411yescharacteristic
116C2×C3×D9—21no—
117C2×C3×D9—41no—
118C2×C3×D9—42no—
119C2×C3×D9C22⋊C411yescharacteristic
120C2×C3×D9C22⋊C411yescharacteristic
121C3×C4×C9—21no—
122C3×C4×C9C22×C411yescharacteristic
123C3×C4×C9C2×D411yescharacteristic
124C3⋊Dic9—41no—
125C9×Dic3—41no—
126C3×Dic9—21no—
127C3×Dic9C22⋊C411yescharacteristic
128C3×Dic9C22⋊C411yescharacteristic
129C2×C3×#41—91no—
130D3×#41—31no—
131#926—91no—
132C2×#208—91no—
133D3×M16—91no—
134#872—91no—
135C3×#104—31no—
136#837—91no—
137#837—91no—
138#837—91no—
139#837—91no—
140#837—181no—
141#837D911yescharacteristic, complemented
142#825—91no—
143#805—91no—
144C22×C3×D3—31no—
145C22×C3×D3—61no—
146C32×Q8C4×D311yescharacteristic
147C32×D4—21no—
148C32×D4C4×D311yescharacteristic
149C2×C32×C4—21no—
150C2×C32×C4C22×D311yescharacteristic
151C2×C3⋊Dic3—61no—
152C3×#67—61no—
153C3×#67—61no—
154C3×#67—61no—
155C2×C3×Dic3—21no—
156C2×C3×Dic3—31no—
157C2×C3×Dic3—61no—
158C3×D12—31no—
159C3×D12—61no—
160C3×C4×D3—31no—
161C3×C4×D3—31no—
162C3×C4×D3—31no—
163C3×C4×D3—31no—
164C3×C4×D3—61no—
165C3×C4×D3—61no—
166C3×C4×D3—61no—
167C3×Dic6—31no—
168C3×Dic6—61no—
169C22×D9—21no—
170C22×D9#6711yescharacteristic
171C3⋊(C3⋊C8)—61no—
172C3×C3⋊C8—21no—
173Q8×C9—21no—
174Q8×C9C2×Dic311yescharacteristic
175D4×C9—21no—
176D4×C9—21no—
177D4×C9—41no—
178D4×C9C2×Dic311yescharacteristic
179C2×C4×C9—21no—
180C2×C4×C9—21no—
181C2×C4×C9—41no—
182C2×C4×C9—61no—
183C2×C4×C9C22×D311yescharacteristic
184#611—21no—
185#611—21no—
186#611—21no—
187C2×Dic9—21no—
188C2×Dic9—61no—
189C2×Dic9—121no—
190C2×Dic9#6711yescharacteristic
191D36—21no—
192D36C2×Dic311yescharacteristic
193C4×D9—21no—
194C4×D9—21no—
195C4×D9—21no—
196C4×D9C2×Dic311yescharacteristic
197C4×D9C2×Dic311yescharacteristic
198C4×D9#6711yescharacteristic
199C4×D9#6711yescharacteristic
200Dic18—21no—
201Dic18C2×Dic311yescharacteristic
202C8×C9—61no—
203C9⋊C8—61no—
204C9⋊C8—121no—
205#421—271noSylow
206C2×C3×C9—21no—
207C2×C3×C9—41no—
208C2×C3×C9C2×C22⋊C411yescharacteristic
209C3×D9—21no—
210C3×D9—21no—
211C3×D9—41no—
212C3×#41—21no—
213C3×#41—91no—
214C3×#41—91no—
215C3×#41—91no—
216C3×#41—181no—
217C3×#41—181no—
218C3×#41C2×D911yescharacteristic
219C3×#41D3211yescharacteristic
220C2×C3×Q8—91no—
221C2×C3×D4—91no—
222C2×C3×D4—181no—
223C22×C3×C4—91no—
224C22×C3×C4—181no—
225#239—31no—
226D3×Q8—31no—
227#237—31no—
228#237—61no—
229D3×D4—31no—
230D3×D4—61no—
231#235—31no—
232#235—61no—
233C2×C4×D3—31no—
234C2×C4×D3—61no—
235C3×M16—31no—
236C3×C42—31no—
237#217—181no—
238(C2×D3)⋊C4—181no—
239C4⋊Dic3—181no—
240Dic3⋊C4—181no—
241C4×Dic3—91no—
242C4×Dic3—91no—
243C4×Dic3—181no—
244C4×Dic3C2×D911yescharacteristic, complemented
245#208—91no—
246#208—91no—
247#208—181no—
248#208—181no—
249#208C2×D911yescharacteristic, complemented
250C2×C3⋊C8—181no—
251ΓL(1,𝔽25)—181no—
252D3×C8—181no—
253C22×C32—21no—
254C22×C32D3×D411yescharacteristic
255C2×C3×D3—31no—
256C2×C3×D3—31no—
257C2×C3×D3—61no—
258C2×C3×D3—121no—
259C2×C3×D3—122no—
260C32×C4—21no—
261C32×C4C2×C4×D311yescharacteristic
262C32×C4D3×D411yescharacteristic
263C3⋊Dic3—121no—
264C3×Dic3—31no—
265C3×Dic3—31no—
266C3×Dic3—41no—
267C3×Dic3—61no—
268C22×C9—21no—
269C22×C9—21no—
270C22×C9—41no—
271C22×C9C2×#6711yescharacteristic
272C2×D9—21no—
273C2×D9—41no—
274C2×D9—42no—
275C2×D9#21711yescharacteristic
276C2×D9#21711yescharacteristic
277C4×C9—21no—
278C4×C9—21no—
279C4×C9—21no—
280C4×C9—41no—
281C4×C9—121no—
282C4×C9C22×Dic311yescharacteristic
283C4×C9C2×#6711yescharacteristic
284Dic9—21no—
285Dic9—121no—
286Dic9—122no—
287Dic9#21711yescharacteristic
288Dic9#21711yescharacteristic
289C2×#41—91no—
290C2×M16—271no—
291#117—271no—
292#104—31no—
293#104—271no—
294#104—271no—
295#104—271no—
296C3×C9#42111yescharacteristic, complemented, Sylow
297C23×C3—91no—
298C23×C3—181no—
299C22×D3—31no—
300C22×D3—61no—
301C3×Q8—21no—
302C3×Q8—91no—
303C3×Q8—181no—
304C3×Q8C4×D911yescharacteristic
305C3×Q8D3×Dic311yescharacteristic
306C3×D4—21no—
307C3×D4—21no—
308C3×D4—21no—
309C3×D4—41no—
310C3×D4—91no—
311C3×D4—181no—
312C3×D4—181no—
313C3×D4—181no—
314C3×D4—181no—
315C3×D4C4×D911yescharacteristic
316C3×D4D3×Dic311yescharacteristic
317C2×C3×C4—21no—
318C2×C3×C4—21no—
319C2×C3×C4—21no—
320C2×C3×C4—41no—
321C2×C3×C4—61no—
322C2×C3×C4—91no—
323C2×C3×C4—91no—
324C2×C3×C4—91no—
325C2×C3×C4—91no—
326C2×C3×C4—91no—
327C2×C3×C4—181no—
328C2×C3×C4—181no—
329C2×C3×C4—181no—
330C2×C3×C4—181no—
331C2×C3×C4C22×D911yescharacteristic
332C2×C3×C4C2×D3211yescharacteristic
333#67—61no—
334#67—61no—
335#67—61no—
336C2×Dic3—21no—
337C2×Dic3—31no—
338C2×Dic3—61no—
339C2×Dic3—181no—
340C2×Dic3—181no—
341C2×Dic3—361no—
342D12—31no—
343D12—61no—
344C4×D3—31no—
345C4×D3—31no—
346C4×D3—31no—
347C4×D3—31no—
348C4×D3—61no—
349C4×D3—61no—
350C4×D3—61no—
351Dic6—31no—
352Dic6—61no—
353C3×C8—61no—
354C3⋊C8—21no—
355C3⋊C8—181no—
356C3⋊C8—181no—
357C3⋊C8—361no—
358C2×C32—21no—
359C2×C32—41no—
360C2×C32D3×C22⋊C411yescharacteristic
361C3×D3—61no—
362C3×D3—61no—
363C3×D3—121no—
364C2×C9—21no—
365C2×C9—21no—
366C2×C9—41no—
367C2×C9—41no—
368C2×C9—81no—
369C2×C9C2×#21711yescharacteristic
370D9—21no—
371D9—21no—
372D9—41no—
373#41—91no—
374#41—91no—
375#41—91no—
376#41—181no—
377#41—181no—
378#41D3×D911yescharacteristic
379C2×Q8—91no—
380C2×D4—91no—
381C2×D4—181no—
382C22×C4—91no—
383C22×C4—181no—
384M16—31no—
385M16—271no—
386M16—541no—
387C2×C8—541no—
388C4⋊C4—541no—
389C22⋊C4—541no—
390C42—31no—
391C42—271no—
392C22×C3—21no—
393C22×C3—21no—
394C22×C3—21no—
395C22×C3—41no—
396C22×C3—91no—
397C22×C3—91no—
398C22×C3—181no—
399C22×C3—361no—
400C22×C3—362no—
401C22×C3D4×D911yescharacteristic
402C22×C3D3×#6711yescharacteristic
403C2×D3—31no—
404C2×D3—31no—
405C2×D3—61no—
406C2×D3—121no—
407C2×D3—122no—
408C3×C4—21no—
409C3×C4—21no—
410C3×C4—21no—
411C3×C4—21no—
412C3×C4—41no—
413C3×C4—91no—
414C3×C4—91no—
415C3×C4—121no—
416C3×C4—181no—
417C3×C4C2×C4×D911yescharacteristic
418C3×C4D4×D911yescharacteristic
419C3×C4C2×D3×Dic311yescharacteristic
420C3×C4D3×#6711yescharacteristic
421Dic3—31no—
422Dic3—31no—
423Dic3—41no—
424Dic3—61no—
425Dic3—361no—
426Dic3—361no—
427Dic3—362no—
428C32#482111yescharacteristic
429C9—21no—
430C9#523611yescharacteristic, complemented
431C23—91no—
432C23—181no—
433Q8—91no—
434Q8—181no—
435Q8Dic3×D911yescharacteristic
436D4—21no—
437D4—91no—
438D4—181no—
439D4—181no—
440D4—181no—
441D4—181no—
442D4Dic3×D911yescharacteristic
443C2×C4—21no—
444C2×C4—61no—
445C2×C4—91no—
446C2×C4—91no—
447C2×C4—91no—
448C2×C4—91no—
449C2×C4—91no—
450C2×C4—181no—
451C2×C4—181no—
452C2×C4—181no—
453C2×C4—181no—
454C2×C4—541no—
455C2×C4C2×D3×D911yescharacteristic
456C8—61no—
457C8—541no—
458C2×C3—21no—
459C2×C3—21no—
460C2×C3—21no—
461C2×C3—41no—
462C2×C3—41no—
463C2×C3—41no—
464C2×C3—81no—
465C2×C3—181no—
466C2×C3—181no—
467C2×C3—361no—
468C2×C3C22⋊C4×D911yescharacteristic
469C2×C3D3×#21711yescharacteristic
470D3—61no—
471D3—61no—
472D3—121no—
473C22—21no—
474C22—91no—
475C22—91no—
476C22—181no—
477C22—361no—
478C22—362no—
479C22D9×#6711yescharacteristic
480C4—21no—
481C4—91no—
482C4—91no—
483C4—121no—
484C4—181no—
485C4—1081no—
486C4C2×Dic3×D911yescharacteristic
487C4D9×#6711yescharacteristic
488C3—21no—
489C3#1058850111yescharacteristic, complemented
490C3#1059167011yescharacteristic
491C2—21no—
492C2—41no—
493C2—181no—
494C2—181no—
495C2—361no—
496C2D9×#21711yescharacteristic
subgroupG/H
centerC4D9×#67
commutatorC3×C4×C9C22×C4
FrattiniC2×C3×C4C2×D32
FittingC3×C9×#41C22
socleC2×C32D3×C22⋊C4
prime radicalC3×#41×D9C2
cosocleC2×C3×C4×C9C23
hypercenter#41D3×D9
hypocenterC3×C9#421
LCSG ▷ C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C4 ◁ C2×C4 ◁ #41

Complements

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

NH ≅ G/Ntype
2. #11720943C2semidirect
3. D9×#208C2semidirect
4. #11720585C2semidirect
5. C9×#837C2semidirect
6. #11720291C2semidirect
7. #11720260C2semidirect
11. C3×#3701C4semidirect
12. C3×Q8×D9C4semidirect
14. C3×#3699C4semidirect
16. C3×D4×D9C4semidirect
21. C4×C3⋊Dic9C22semidirect
22. #88532C22semidirect
23. C4×C9×Dic3C22semidirect
24. C9×#208C22semidirect
94. #3701Dic3semidirect
95. Q8×D9Dic3semidirect
97. #3699Dic3semidirect
99. D4×D9Dic3semidirect
141. #837D9semidirect
244. C4×Dic3C2×D9semidirect
249. #208C2×D9semidirect
296. C3×C9#421semidirect
430. C9#5236semidirect
489. C3#10588501semidirect

Maximal subgroups

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

subgroupindex
C3×#41×D92
#117209432
D9×#2082
#117205852
C9×#8372
#117202912
#117202602
#105916703
#105885013