Finite Group Databasebeta

← #13161363

Subgroups — #13161363

Order 1728.

1100 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
1C32×D3×#41C211yesmaximal, characteristic
2#11723423C211yesmaximal, characteristic, complemented
3D3×#3748C211yesmaximal, characteristic, complemented
4#11722207C211yesmaximal, characteristic, complemented
5C3×#63664C211yesmaximal, characteristic, complemented
6#11722044C211yesmaximal, characteristic, complemented
7#11722043C211yesmaximal, characteristic, complemented
8#10592935—31nomaximal
9#10591670—34nomaximal
10C33×#41C2211yescharacteristic
11C32×#239C411yescharacteristic, complemented
12C32×D3×Q8C411yescharacteristic, complemented
13C32×#237—21no—
14C32×#237C411yescharacteristic, complemented
15C32×D3×D4—21no—
16C32×D3×D4C411yescharacteristic, complemented
17C32×#235—21no—
18C32×#235C2211yescharacteristic
19C2×C32×C4×D3—21no—
20C2×C32×C4×D3C2211yescharacteristic
21C4×C3⋊(C3⋊Dic3)C2211yescharacteristic, complemented
22#88855C2211yescharacteristic, complemented
23C3×C4×C3⋊Dic3C2211yescharacteristic, complemented
24C3×#3748C2211yescharacteristic, complemented
25Dic3⋊(C3⋊Dic3)—21no—
26(C2×D3)⋊(C3⋊Dic3)—21no—
27#88786—21no—
28D3×C3⋊(C3⋊C8)—21no—
29C2×C32×#41—31no—
30C3×D3×#41D314yes—
31#64135—31no—
32C2×#3748—31no—
33#63855—34no—
34D3×#208—34no—
35#63664—31no—
36#63664—31no—
37#63664—31no—
38#63664—34no—
39#63664—64no—
40#63664D311yescharacteristic, complemented
41C3×#837—34no—
42#63569—34no—
43#63568—34no—
44C22×C32×D3—21no—
45C22×C32×D3D411yescharacteristic
46C33×Q8C2×C411yescharacteristic
47C33×D4—21no—
48C33×D4C2×C411yescharacteristic
49C2×C33×C4—21no—
50C2×C33×C4C2311yescharacteristic
51C2×C3⋊(C3⋊Dic3)—21no—
52C2×C3×C3⋊Dic3—21no—
53C32×#67—21no—
54C32×#67—21no—
55C32×#67—21no—
56C2×C32×Dic3—21no—
57C2×C32×Dic3D411yescharacteristic
58C32×D12—21no—
59C32×D12C2×C411yescharacteristic
60C32×C4×D3—21no—
61C32×C4×D3—21no—
62C32×C4×D3—21no—
63C32×C4×D3C2×C411yescharacteristic
64C32×C4×D3C2×C411yescharacteristic
65C32×C4×D3D411yescharacteristic
66C32×C4×D3D411yescharacteristic
67C32×Dic6—21no—
68C32×Dic6C2×C411yescharacteristic
69C3⋊(C3⋊(C3⋊C8))—21no—
70C3×C3⋊(C3⋊C8)—21no—
71#5236—94no—
72#4821—91no—
73C32×#41—24no—
74C32×#41—31no—
75C32×#41—31no—
76C32×#41—31no—
77C32×#41—61no—
78C32×#41—61no—
79C32×#41C2×D311yescharacteristic
80C32×#41C2×D314yes—
81C2×C32×Q8—31no—
82C2×C32×D4—31no—
83C2×C32×D4—61no—
84C22×C32×C4—31no—
85C22×C32×C4—61no—
86C3×#239Dic314yescomplemented
87C3×D3×Q8Dic314yescomplemented
88C3×#237—24no—
89C3×#237Dic314yescomplemented
90C3×D3×D4—24no—
91C3×D3×D4Dic314yescomplemented
92C3×#235—24no—
93C3×#235C2×D314yes—
94C2×C3×C4×D3—24no—
95C2×C3×C4×D3C2×D314yes—
96#3757—61no—
97C4⋊(C3⋊Dic3)—61no—
98C4×C3⋊Dic3—31no—
99C4×C3⋊Dic3—34no—
100C4×C3⋊Dic3—64no—
101C4×C3⋊Dic3C2×D311yescharacteristic, complemented
102#3748—31no—
103#3748—34no—
104#3748—61no—
105#3748—64no—
106#3748C2×D311yescharacteristic, complemented
107C2×C3⋊(C3⋊C8)—61no—
108C3×C4×Dic3—34no—
109C3×#208—34no—
110Dic3⋊Dic3—64no—
111(C2×D3)⋊Dic3—64no—
112#3711—64no—
113D3×C3⋊C8—64no—
114C22×C33—21no—
115C22×C33C2×D411yescharacteristic
116C2×C32×D3—21no—
117C2×C32×D3—41no—
118C2×C32×D3—42no—
119C2×C32×D3C22⋊C411yescharacteristic
120C2×C32×D3C22⋊C411yescharacteristic
121C33×C4—21no—
122C33×C4C22×C411yescharacteristic
123C33×C4C2×D411yescharacteristic
124C3⋊(C3⋊Dic3)—41no—
125C3×C3⋊Dic3—41no—
126C32×Dic3—21no—
127C32×Dic3C22⋊C411yescharacteristic
128C32×Dic3C22⋊C411yescharacteristic
129C2×C3×#41—34no—
130D3×#41C3⋊D311yescharacteristic
131#926—94no—
132C2×#208—94no—
133D3×M16—91no—
134#872—91no—
135C3×#104—91no—
136#837—34no—
137#837—94no—
138#837—94no—
139#837—94no—
140#837—91no—
141#837—184no—
142#825—91no—
143#805—91no—
144C23×C32—31no—
145C23×C32—61no—
146C22×C3×D3—24no—
147C22×C3×D3#6714yes—
148C32×Q8—24no—
149C32×Q8—31no—
150C32×Q8—61no—
151C32×Q8C4×D311yescharacteristic
152C32×Q8C2×Dic314yes—
153C32×D4—21no—
154C32×D4—24no—
155C32×D4—24no—
156C32×D4—31no—
157C32×D4—44no—
158C32×D4—61no—
159C32×D4—61no—
160C32×D4—61no—
161C32×D4—61no—
162C32×D4C4×D311yescharacteristic
163C32×D4C2×Dic314yes—
164C2×C32×C4—21no—
165C2×C32×C4—24no—
166C2×C32×C4—24no—
167C2×C32×C4—31no—
168C2×C32×C4—31no—
169C2×C32×C4—31no—
170C2×C32×C4—31no—
171C2×C32×C4—31no—
172C2×C32×C4—44no—
173C2×C32×C4—61no—
174C2×C32×C4—61no—
175C2×C32×C4—61no—
176C2×C32×C4—61no—
177C2×C32×C4C22×D314yes—
178C2×C32×C4C22×D311yescharacteristic
179C2×C3⋊Dic3—21no—
180C2×C3⋊Dic3—61no—
181C2×C3⋊Dic3—64no—
182C2×C3⋊Dic3—124no—
183C3×#67—24no—
184C3×#67—24no—
185C3×#67—24no—
186C2×C3×Dic3—24no—
187C2×C3×Dic3—64no—
188C2×C3×Dic3#6714yes—
189C3×D12—24no—
190C3×D12C2×Dic314yes—
191C3×C4×D3—24no—
192C3×C4×D3—24no—
193C3×C4×D3—24no—
194C3×C4×D3C2×Dic314yes—
195C3×C4×D3C2×Dic314yes—
196C3×C4×D3#6714yes—
197C3×C4×D3#6714yes—
198C3×Dic6—24no—
199C3×Dic6C2×Dic314yes—
200C3⋊(C3⋊C8)—21no—
201C3⋊(C3⋊C8)—61no—
202C3⋊(C3⋊C8)—64no—
203C3⋊(C3⋊C8)—124no—
204C3×C3⋊C8—64no—
205#421—271noSylow
206C2×C33—21no—
207C2×C33—41no—
208C2×C33C2×C22⋊C411yescharacteristic
209C32×D3—21no—
210C32×D3—21no—
211C32×D3—41no—
212C3×#41—24no—
213C3×#41—34no—
214C3×#41—34no—
215C3×#41—34no—
216C3×#41—64no—
217C3×#41—64no—
218C3×#41D3214yes—
219C3×#41C2×C3⋊D311yescharacteristic
220C2×C3×Q8—34no—
221C2×C3×D4—34no—
222C2×C3×D4—64no—
223C22×C3×C4—34no—
224C22×C3×C4—64no—
225#239C3⋊Dic311yescharacteristic, complemented
226D3×Q8C3⋊Dic311yescharacteristic, complemented
227#237—21no—
228#237C3⋊Dic311yescharacteristic, complemented
229D3×D4—21no—
230D3×D4C3⋊Dic311yescharacteristic, complemented
231#235—21no—
232#235C2×C3⋊D311yescharacteristic
233C2×C4×D3—21no—
234C2×C4×D3C2×C3⋊D311yescharacteristic
235C3×M16—91no—
236C3×C42—91no—
237#217—184no—
238(C2×D3)⋊C4—181no—
239C4⋊Dic3—184no—
240Dic3⋊C4—181no—
241C4×Dic3—34no—
242C4×Dic3—94no—
243C4×Dic3—91no—
244C4×Dic3—184no—
245#208—34no—
246#208—94no—
247#208—91no—
248#208—184no—
249#208—184no—
250C2×C3⋊C8—184no—
251ΓL(1,𝔽25)—181no—
252D3×C8—181no—
253C22×C32—21no—
254C22×C32—24no—
255C22×C32—24no—
256C22×C32—31no—
257C22×C32—31no—
258C22×C32—44no—
259C22×C32—61no—
260C22×C32—121no—
261C22×C32—122no—
262C22×C32D3×D411yescharacteristic
263C22×C32C2×#6714yes—
264C2×C3×D3—24no—
265C2×C3×D3—44no—
266C2×C3×D3—48no—
267C2×C3×D3#21714yes—
268C2×C3×D3#21714yes—
269C32×C4—21no—
270C32×C4—24no—
271C32×C4—24no—
272C32×C4—24no—
273C32×C4—31no—
274C32×C4—31no—
275C32×C4—44no—
276C32×C4—61no—
277C32×C4C2×C4×D311yescharacteristic
278C32×C4D3×D411yescharacteristic
279C32×C4C22×Dic314yes—
280C32×C4C2×#6714yes—
281C3⋊Dic3—41no—
282C3⋊Dic3—121no—
283C3⋊Dic3—124no—
284C3⋊Dic3—128no—
285C3×Dic3—24no—
286C3×Dic3—124no—
287C3×Dic3#21714yes—
288C3×Dic3#21714yes—
289C2×#41—31no—
290C2×M16—271no—
291#117—271no—
292#104—91no—
293#104—271no—
294#104—271no—
295#104—271no—
296C33#42111yescharacteristic, complemented, Sylow
297C23×C3—34no—
298C23×C3—64no—
299C22×D3—21no—
300C22×D3#63811yescharacteristic
301C3×Q8—24no—
302C3×Q8—34no—
303C3×Q8—64no—
304C3×Q8D3×Dic314yes—
305C3×Q8C2×C3⋊Dic311yescharacteristic
306C3×D4—24no—
307C3×D4—21no—
308C3×D4—24no—
309C3×D4—34no—
310C3×D4—44no—
311C3×D4—64no—
312C3×D4—64no—
313C3×D4—64no—
314C3×D4—64no—
315C3×D4D3×Dic314yes—
316C3×D4C2×C3⋊Dic311yescharacteristic
317C2×C3×C4—24no—
318C2×C3×C4—21no—
319C2×C3×C4—24no—
320C2×C3×C4—34no—
321C2×C3×C4—34no—
322C2×C3×C4—34no—
323C2×C3×C4—34no—
324C2×C3×C4—34no—
325C2×C3×C4—44no—
326C2×C3×C4—64no—
327C2×C3×C4—64no—
328C2×C3×C4—64no—
329C2×C3×C4—64no—
330C2×C3×C4—181no—
331C2×C3×C4C2×D3214yes—
332C2×C3×C4C22×C3⋊D311yescharacteristic
333#67—21no—
334#67—21no—
335#67—21no—
336C2×Dic3—21no—
337C2×Dic3—64no—
338C2×Dic3—184no—
339C2×Dic3—181no—
340C2×Dic3—364no—
341C2×Dic3#63811yescharacteristic
342D12—21no—
343D12C2×C3⋊Dic311yescharacteristic
344C4×D3—21no—
345C4×D3—21no—
346C4×D3—21no—
347C4×D3C2×C3⋊Dic311yescharacteristic
348C4×D3C2×C3⋊Dic311yescharacteristic
349C4×D3#63811yescharacteristic
350C4×D3#63811yescharacteristic
351Dic6—21no—
352Dic6C2×C3⋊Dic311yescharacteristic
353C3×C8—181no—
354C3⋊C8—64no—
355C3⋊C8—184no—
356C3⋊C8—181no—
357C3⋊C8—364no—
358C2×C32—21no—
359C2×C32—24no—
360C2×C32—24no—
361C2×C32—44no—
362C2×C32—41no—
363C2×C32—44no—
364C2×C32—61no—
365C2×C32—61no—
366C2×C32—84no—
367C2×C32—121no—
368C2×C32D3×C22⋊C411yescharacteristic
369C2×C32C2×#21714yes—
370C3×D3—24no—
371C3×D3—24no—
372C3×D3—44no—
373#41—31no—
374#41—31no—
375#41—31no—
376#41—61no—
377#41—61no—
378#41D3×C3⋊D311yescharacteristic
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—24no—
394C22×C3—24no—
395C22×C3—34no—
396C22×C3—34no—
397C22×C3—44no—
398C22×C3—64no—
399C22×C3—124no—
400C22×C3—128no—
401C22×C3D3×#6714yes—
402C22×C3C2×#63811yescharacteristic
403C2×D3—21no—
404C2×D3—41no—
405C2×D3—42no—
406C2×D3#375711yescharacteristic
407C2×D3#375711yescharacteristic
408C3×C4—21no—
409C3×C4—24no—
410C3×C4—24no—
411C3×C4—24no—
412C3×C4—34no—
413C3×C4—34no—
414C3×C4—44no—
415C3×C4—64no—
416C3×C4—361no—
417C3×C4C2×D3×Dic314yes—
418C3×C4D3×#6714yes—
419C3×C4C22×C3⋊Dic311yescharacteristic
420C3×C4C2×#63811yescharacteristic
421Dic3—21no—
422Dic3—124no—
423Dic3—361no—
424Dic3—364no—
425Dic3—368no—
426Dic3#375711yescharacteristic
427Dic3#375711yescharacteristic
428C32—24no—
429C32#482111yescharacteristic, complemented
430C32#523614yescomplemented
431C23—31no—
432C23—61no—
433Q8—31no—
434Q8—61no—
435Q8D3×C3⋊Dic311yescharacteristic
436D4—21no—
437D4—31no—
438D4—61no—
439D4—61no—
440D4—61no—
441D4—61no—
442D4D3×C3⋊Dic311yescharacteristic
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×C3⋊D311yescharacteristic
456C8—181no—
457C8—541no—
458C2×C3—21no—
459C2×C3—24no—
460C2×C3—24no—
461C2×C3—44no—
462C2×C3—41no—
463C2×C3—44no—
464C2×C3—64no—
465C2×C3—64no—
466C2×C3—84no—
467C2×C3—124no—
468C2×C3D3×#21714yes—
469C2×C3C2×#375711yescharacteristic
470D3—21no—
471D3—21no—
472D3—41no—
473C22—21no—
474C22—31no—
475C22—31no—
476C22—61no—
477C22—121no—
478C22—122no—
479C22D3×#63811yescharacteristic
480C4—21no—
481C4—31no—
482C4—31no—
483C4—61no—
484C4—361no—
485C4—1081no—
486C4C2×D3×C3⋊Dic311yescharacteristic
487C4D3×#63811yescharacteristic
488C3—24no—
489C3#1059167014yescomplemented
490C3#1059293511yescharacteristic, complemented
491C2—21no—
492C2—41no—
493C2—61no—
494C2—61no—
495C2—121no—
496C2D3×#375711yescharacteristic
subgroupG/H
centerC4D3×#638
commutatorC33×C4C22×C4
FrattiniC2×C4C2×D3×C3⋊D3
FittingC33×#41C22
socleC2×C33C2×C22⋊C4
prime radicalC32×D3×#41C2
cosocleC2×C33×C4C23
hypercenter#41D3×C3⋊D3
hypocenterC33#421
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C4 ◁ C2×C4 ◁ #41

Complements

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

NH ≅ G/Ntype
2. #11723423C2semidirect
3. D3×#3748C2semidirect
4. #11722207C2semidirect
5. C3×#63664C2semidirect
6. #11722044C2semidirect
7. #11722043C2semidirect
11. C32×#239C4semidirect
12. C32×D3×Q8C4semidirect
14. C32×#237C4semidirect
16. C32×D3×D4C4semidirect
21. C4×C3⋊(C3⋊Dic3)C22semidirect
22. #88855C22semidirect
23. C3×C4×C3⋊Dic3C22semidirect
24. C3×#3748C22semidirect
40. #63664D3semidirect
86. C3×#239Dic3semidirect
87. C3×D3×Q8Dic3semidirect
89. C3×#237Dic3semidirect
91. C3×D3×D4Dic3semidirect
101. C4×C3⋊Dic3C2×D3semidirect
106. #3748C2×D3semidirect
225. #239C3⋊Dic3semidirect
226. D3×Q8C3⋊Dic3semidirect
228. #237C3⋊Dic3semidirect
230. D3×D4C3⋊Dic3semidirect
296. C33#421semidirect
429. C32#4821semidirect
430. C32#5236semidirect
489. C3#10591670semidirect
490. C3#10592935semidirect

Maximal subgroups

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

subgroupindex
C32×D3×#412
#117234232
D3×#37482
#117222072
C3×#636642
#117220442
#117220432
#105929353
#105916703