Finite Group Databasebeta

← #13141029

Subgroups — #13141029

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
1D3×C9×#41C211yesmaximal, characteristic
2#11720959C211yesmaximal, characteristic, complemented
3D3×#3667C211yesmaximal, characteristic, complemented
4#11720585C211yesmaximal, characteristic, complemented
5C3×#63396C211yesmaximal, characteristic, complemented
6#11720301C211yesmaximal, characteristic, complemented
7#11720300C211yesmaximal, characteristic, complemented
8#10591670—31nomaximal
9#10588914—31nomaximal
10C3×C9×#41C2211yescharacteristic
11C9×#239C411yescharacteristic, complemented
12D3×Q8×C9C411yescharacteristic, complemented
13C9×#237—21no—
14C9×#237C411yescharacteristic, complemented
15D3×D4×C9—21no—
16D3×D4×C9C411yescharacteristic, complemented
17C9×#235—21no—
18C9×#235C2211yescharacteristic
19C2×C4×D3×C9—21no—
20C2×C4×D3×C9C2211yescharacteristic
21C4×C3⋊Dic9C2211yescharacteristic, complemented
22#88532C2211yescharacteristic, complemented
23C3×C4×Dic9C2211yescharacteristic, complemented
24C3×#3667C2211yescharacteristic, complemented
25(C2×D3)⋊Dic9—21no—
26Dic3⋊Dic9—21no—
27#88420—21no—
28D3×C9⋊C8—21no—
29C3×D3×#41D311yescharacteristic
30#63855—31no—
31D3×#208—31no—
32C2×C9×#41—31no—
33#63664—31no—
34C3×#837—31no—
35#63569—31no—
36#63568—31no—
37#63488—31no—
38C2×#3667—31no—
39#63396—31no—
40#63396—31no—
41#63396—31no—
42#63396—61no—
43#63396D311yescharacteristic, complemented
44C22×D3×C9—21no—
45C22×D3×C9D411yescharacteristic
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—
52C9×#67—21no—
53C9×#67—21no—
54C9×#67—21no—
55C2×C9×Dic3—21no—
56C2×C9×Dic3D411yescharacteristic
57C2×C3×Dic9—21no—
58C9×D12—21no—
59C9×D12C2×C411yescharacteristic
60C4×D3×C9—21no—
61C4×D3×C9—21no—
62C4×D3×C9—21no—
63C4×D3×C9C2×C411yescharacteristic
64C4×D3×C9C2×C411yescharacteristic
65C4×D3×C9D411yescharacteristic
66C4×D3×C9D411yescharacteristic
67C9×Dic6—21no—
68C9×Dic6C2×C411yescharacteristic
69C3⋊(C9⋊C8)—21no—
70C3×C9⋊C8—21no—
71#5236—91no—
72#4821—91no—
73C32×#41C2×D311yescharacteristic
74C3×#239Dic311yescharacteristic
75C3×D3×Q8Dic311yescharacteristic
76C3×#237—21no—
77C3×#237Dic311yescharacteristic
78C3×D3×D4—21no—
79C3×D3×D4Dic311yescharacteristic
80C3×#235—21no—
81C3×#235C2×D311yescharacteristic
82C2×C3×C4×D3—21no—
83C2×C3×C4×D3C2×D311yescharacteristic
84C4×C3⋊Dic3—31no—
85#3748—31no—
86C3×C4×Dic3—31no—
87C3×#208—31no—
88Dic3⋊Dic3—61no—
89(C2×D3)⋊Dic3—61no—
90#3711—61no—
91D3×C3⋊C8—61no—
92C9×#41—21no—
93C9×#41—31no—
94C9×#41—31no—
95C9×#41—31no—
96C9×#41—61no—
97C9×#41—61no—
98C9×#41C2×D311yescharacteristic
99C2×Q8×C9—31no—
100C2×D4×C9—31no—
101C2×D4×C9—61no—
102C22×C4×C9—31no—
103C22×C4×C9—61no—
104#3676—61no—
105C4⋊Dic9—61no—
106C4×Dic9—31no—
107C4×Dic9—61no—
108C4×Dic9C2×D311yescharacteristic, complemented
109#3667—31no—
110#3667—61no—
111#3667—61no—
112#3667C2×D311yescharacteristic, complemented
113C2×C9⋊C8—61no—
114C22×C3×C9—21no—
115C22×C3×C9C2×D411yescharacteristic
116C2×D3×C9—21no—
117C2×D3×C9—41no—
118C2×D3×C9—42no—
119C2×D3×C9C22⋊C411yescharacteristic
120C2×D3×C9C22⋊C411yescharacteristic
121C3×C4×C9—21no—
122C3×C4×C9C22×C411yescharacteristic
123C3×C4×C9C2×D411yescharacteristic
124C3⋊Dic9—41no—
125C9×Dic3—21no—
126C9×Dic3C22⋊C411yescharacteristic
127C9×Dic3C22⋊C411yescharacteristic
128C3×Dic9—41no—
129C2×C3×#41—31no—
130D3×#41D911yescharacteristic
131#926—91no—
132C2×#208—91no—
133D3×M16—91no—
134#872—91no—
135C3×#104—91no—
136#837—31no—
137#837—91no—
138#837—91no—
139#837—91no—
140#837—91no—
141#837—181no—
142#825—91no—
143#805—91no—
144C22×C3×D3—21no—
145C22×C3×D3#6711yescharacteristic
146C32×Q8C2×Dic311yescharacteristic
147C32×D4—21no—
148C32×D4C2×Dic311yescharacteristic
149C2×C32×C4—21no—
150C2×C32×C4C22×D311yescharacteristic
151C2×C3⋊Dic3—61no—
152C3×#67—21no—
153C3×#67—21no—
154C3×#67—21no—
155C2×C3×Dic3—21no—
156C2×C3×Dic3—61no—
157C2×C3×Dic3#6711yescharacteristic
158C3×D12—21no—
159C3×D12C2×Dic311yescharacteristic
160C3×C4×D3—21no—
161C3×C4×D3—21no—
162C3×C4×D3—21no—
163C3×C4×D3C2×Dic311yescharacteristic
164C3×C4×D3C2×Dic311yescharacteristic
165C3×C4×D3#6711yescharacteristic
166C3×C4×D3#6711yescharacteristic
167C3×Dic6—21no—
168C3×Dic6C2×Dic311yescharacteristic
169C23×C9—31no—
170C23×C9—61no—
171C3⋊(C3⋊C8)—61no—
172C3×C3⋊C8—61no—
173Q8×C9—21no—
174Q8×C9—31no—
175Q8×C9—61no—
176Q8×C9C4×D311yescharacteristic
177D4×C9—21no—
178D4×C9—21no—
179D4×C9—31no—
180D4×C9—41no—
181D4×C9—61no—
182D4×C9—61no—
183D4×C9—61no—
184D4×C9—61no—
185D4×C9C4×D311yescharacteristic
186C2×C4×C9—21no—
187C2×C4×C9—21no—
188C2×C4×C9—31no—
189C2×C4×C9—31no—
190C2×C4×C9—31no—
191C2×C4×C9—31no—
192C2×C4×C9—31no—
193C2×C4×C9—41no—
194C2×C4×C9—61no—
195C2×C4×C9—61no—
196C2×C4×C9—61no—
197C2×C4×C9—61no—
198C2×C4×C9C22×D311yescharacteristic
199C2×Dic9—21no—
200C2×Dic9—61no—
201C2×Dic9—121no—
202C9⋊C8—21no—
203C9⋊C8—61no—
204C9⋊C8—121no—
205#421—271noSylow
206C2×C3×C9—21no—
207C2×C3×C9—41no—
208C2×C3×C9C2×C22⋊C411yescharacteristic
209D3×C9—21no—
210D3×C9—21no—
211D3×C9—41no—
212C3×#41—21no—
213C3×#41—31no—
214C3×#41—31no—
215C3×#41—31no—
216C3×#41—61no—
217C3×#41—61no—
218C3×#41C2×D911yescharacteristic
219C3×#41D3211yescharacteristic
220C2×C3×Q8—31no—
221C2×C3×D4—31no—
222C2×C3×D4—61no—
223C22×C3×C4—31no—
224C22×C3×C4—61no—
225#239Dic911yescharacteristic, complemented
226D3×Q8Dic911yescharacteristic, complemented
227#237—21no—
228#237Dic911yescharacteristic, complemented
229D3×D4—21no—
230D3×D4Dic911yescharacteristic, complemented
231#235—21no—
232#235C2×D911yescharacteristic
233C2×C4×D3—21no—
234C2×C4×D3C2×D911yescharacteristic
235C3×M16—91no—
236C3×C42—91no—
237#217—181no—
238(C2×D3)⋊C4—181no—
239C4⋊Dic3—181no—
240Dic3⋊C4—181no—
241C4×Dic3—31no—
242C4×Dic3—91no—
243C4×Dic3—91no—
244C4×Dic3—181no—
245#208—31no—
246#208—91no—
247#208—91no—
248#208—181no—
249#208—181no—
250C2×C3⋊C8—181no—
251ΓL(1,𝔽25)—181no—
252D3×C8—181no—
253C22×C32—21no—
254C22×C32C2×#6711yescharacteristic
255C2×C3×D3—21no—
256C2×C3×D3—41no—
257C2×C3×D3—42no—
258C2×C3×D3#21711yescharacteristic
259C2×C3×D3#21711yescharacteristic
260C32×C4—21no—
261C32×C4C22×Dic311yescharacteristic
262C32×C4C2×#6711yescharacteristic
263C3⋊Dic3—121no—
264C3×Dic3—21no—
265C3×Dic3—121no—
266C3×Dic3#21711yescharacteristic
267C3×Dic3#21711yescharacteristic
268C22×C9—21no—
269C22×C9—21no—
270C22×C9—31no—
271C22×C9—31no—
272C22×C9—41no—
273C22×C9—61no—
274C22×C9—121no—
275C22×C9—122no—
276C22×C9D3×D411yescharacteristic
277C4×C9—21no—
278C4×C9—21no—
279C4×C9—21no—
280C4×C9—31no—
281C4×C9—31no—
282C4×C9—41no—
283C4×C9—61no—
284C4×C9C2×C4×D311yescharacteristic
285C4×C9D3×D411yescharacteristic
286Dic9—41no—
287Dic9—121no—
288Dic9—122no—
289C2×#41—31no—
290C2×M16—271no—
291#117—271no—
292#104—91no—
293#104—271no—
294#104—271no—
295#104—271no—
296C3×C9#42111yescharacteristic, complemented, Sylow
297C23×C3—31no—
298C23×C3—61no—
299C22×D3—21no—
300C22×D3#61111yescharacteristic
301C3×Q8—21no—
302C3×Q8—31no—
303C3×Q8—61no—
304C3×Q8C2×Dic911yescharacteristic
305C3×Q8D3×Dic311yescharacteristic
306C3×D4—21no—
307C3×D4—21no—
308C3×D4—21no—
309C3×D4—31no—
310C3×D4—41no—
311C3×D4—61no—
312C3×D4—61no—
313C3×D4—61no—
314C3×D4—61no—
315C3×D4C2×Dic911yescharacteristic
316C3×D4D3×Dic311yescharacteristic
317C2×C3×C4—21no—
318C2×C3×C4—21no—
319C2×C3×C4—21no—
320C2×C3×C4—31no—
321C2×C3×C4—31no—
322C2×C3×C4—31no—
323C2×C3×C4—31no—
324C2×C3×C4—31no—
325C2×C3×C4—41no—
326C2×C3×C4—61no—
327C2×C3×C4—61no—
328C2×C3×C4—61no—
329C2×C3×C4—61no—
330C2×C3×C4—181no—
331C2×C3×C4C22×D911yescharacteristic
332C2×C3×C4C2×D3211yescharacteristic
333#67—21no—
334#67—21no—
335#67—21no—
336C2×Dic3—21no—
337C2×Dic3—61no—
338C2×Dic3—181no—
339C2×Dic3—181no—
340C2×Dic3—361no—
341C2×Dic3#61111yescharacteristic
342D12—21no—
343D12C2×Dic911yescharacteristic
344C4×D3—21no—
345C4×D3—21no—
346C4×D3—21no—
347C4×D3C2×Dic911yescharacteristic
348C4×D3C2×Dic911yescharacteristic
349C4×D3#61111yescharacteristic
350C4×D3#61111yescharacteristic
351Dic6—21no—
352Dic6C2×Dic911yescharacteristic
353C3×C8—181no—
354C3⋊C8—61no—
355C3⋊C8—181no—
356C3⋊C8—181no—
357C3⋊C8—361no—
358C2×C32—21no—
359C2×C32—41no—
360C2×C32C2×#21711yescharacteristic
361C3×D3—21no—
362C3×D3—21no—
363C3×D3—41no—
364C2×C9—21no—
365C2×C9—21no—
366C2×C9—41no—
367C2×C9—41no—
368C2×C9—61no—
369C2×C9—61no—
370C2×C9—81no—
371C2×C9—121no—
372C2×C9D3×C22⋊C411yescharacteristic
373#41—31no—
374#41—31no—
375#41—31no—
376#41—61no—
377#41—61no—
378#41D3×D911yescharacteristic
379C2×Q8—31no—
380C2×D4—31no—
381C2×D4—61no—
382C22×C4—31no—
383C22×C4—61no—
384M16—91no—
385M16—271no—
386M16—541no—
387C2×C8—541no—
388C4⋊C4—541no—
389C22⋊C4—541no—
390C42—91no—
391C42—271no—
392C22×C3—21no—
393C22×C3—21no—
394C22×C3—21no—
395C22×C3—31no—
396C22×C3—31no—
397C22×C3—41no—
398C22×C3—61no—
399C22×C3—121no—
400C22×C3—122no—
401C22×C3C2×#61111yescharacteristic
402C22×C3D3×#6711yescharacteristic
403C2×D3—21no—
404C2×D3—41no—
405C2×D3—42no—
406C2×D3#367611yescharacteristic
407C2×D3#367611yescharacteristic
408C3×C4—21no—
409C3×C4—21no—
410C3×C4—21no—
411C3×C4—21no—
412C3×C4—31no—
413C3×C4—31no—
414C3×C4—41no—
415C3×C4—61no—
416C3×C4—361no—
417C3×C4C22×Dic911yescharacteristic
418C3×C4C2×#61111yescharacteristic
419C3×C4C2×D3×Dic311yescharacteristic
420C3×C4D3×#6711yescharacteristic
421Dic3—21no—
422Dic3—121no—
423Dic3—361no—
424Dic3—361no—
425Dic3—362no—
426Dic3#367611yescharacteristic
427Dic3#367611yescharacteristic
428C32#523611yescharacteristic
429C9—21no—
430C9#482111yescharacteristic, complemented
431C23—31no—
432C23—61no—
433Q8—31no—
434Q8—61no—
435Q8D3×Dic911yescharacteristic
436D4—21no—
437D4—31no—
438D4—61no—
439D4—61no—
440D4—61no—
441D4—61no—
442D4D3×Dic911yescharacteristic
443C2×C4—21no—
444C2×C4—31no—
445C2×C4—31no—
446C2×C4—31no—
447C2×C4—31no—
448C2×C4—31no—
449C2×C4—61no—
450C2×C4—61no—
451C2×C4—61no—
452C2×C4—61no—
453C2×C4—181no—
454C2×C4—541no—
455C2×C4C2×D3×D911yescharacteristic
456C8—181no—
457C8—541no—
458C2×C3—21no—
459C2×C3—21no—
460C2×C3—21no—
461C2×C3—41no—
462C2×C3—41no—
463C2×C3—41no—
464C2×C3—61no—
465C2×C3—61no—
466C2×C3—81no—
467C2×C3—121no—
468C2×C3C2×#367611yescharacteristic
469C2×C3D3×#21711yescharacteristic
470D3—21no—
471D3—21no—
472D3—41no—
473C22—21no—
474C22—31no—
475C22—31no—
476C22—61no—
477C22—121no—
478C22—122no—
479C22D3×#61111yescharacteristic
480C4—21no—
481C4—31no—
482C4—31no—
483C4—61no—
484C4—361no—
485C4—1081no—
486C4C2×D3×Dic911yescharacteristic
487C4D3×#61111yescharacteristic
488C3—21no—
489C3#1058891411yescharacteristic, complemented
490C3#1059167011yescharacteristic
491C2—21no—
492C2—41no—
493C2—61no—
494C2—61no—
495C2—121no—
496C2D3×#367611yescharacteristic
subgroupG/H
centerC4D3×#611
commutatorC3×C4×C9C22×C4
FrattiniC2×C3×C4C2×D32
FittingC3×C9×#41C22
socleC2×C32C2×#217
prime radicalC3×D3×#41D3
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. #11720959C2semidirect
3. D3×#3667C2semidirect
4. #11720585C2semidirect
5. C3×#63396C2semidirect
6. #11720301C2semidirect
7. #11720300C2semidirect
11. C9×#239C4semidirect
12. D3×Q8×C9C4semidirect
14. C9×#237C4semidirect
16. D3×D4×C9C4semidirect
21. C4×C3⋊Dic9C22semidirect
22. #88532C22semidirect
23. C3×C4×Dic9C22semidirect
24. C3×#3667C22semidirect
43. #63396D3semidirect
108. C4×Dic9C2×D3semidirect
112. #3667C2×D3semidirect
225. #239Dic9semidirect
226. D3×Q8Dic9semidirect
228. #237Dic9semidirect
230. D3×D4Dic9semidirect
296. C3×C9#421semidirect
430. C9#4821semidirect
489. C3#10588914semidirect

Maximal subgroups

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

subgroupindex
D3×C9×#412
#117209592
D3×#36672
#117205852
C3×#633962
#117203012
#117203002
#105916703
#105889143