Finite Group Databasebeta

← #13153916

Subgroups — #13153916

Order 1728.

1690 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D4×C3⋊D9C211yesmaximal, characteristic
2C3×C4×D4×C9C211yesmaximal, characteristic, complemented
3D4×C3⋊Dic9C211yesmaximal, characteristic, complemented
4C4×#6233C212yesmaximal, complemented
5C2×#88536C212yesmaximal, complemented
6#11721749C211yesmaximal, characteristic, complemented
7#11721748C211yesmaximal, characteristic, complemented
8#11721727C212yesmaximal, complemented
9C22⋊C4×C3⋊D9C212yesmaximal, complemented
10C4×C4⋊(C3⋊D9)C211yesmaximal, characteristic, complemented
11#11721709C211yesmaximal, characteristic, complemented
12#10596016—31nomaximal
13#10589647—33nomaximal
14C23×C3⋊D9C2212yes—
15C2×C3×D4×C9C2211yescharacteristic
16C22×C3×C4×C9C2212yescomplemented
17C2×#6233C2212yes—
18C22×C3⋊Dic9C2212yescomplemented
19D4×C3⋊D9C418yescomplemented
20C2×C4⋊(C3⋊D9)C2211yescharacteristic
21C2×C4×C3⋊D9C2211yescharacteristic
22C2×C4×C3⋊D9C2212yescomplemented
23C2×C4×C3⋊D9C2212yescomplemented
24C3×C9×C4⋊C4C2211yescharacteristic, complemented
25C3×C9×C22⋊C4C2212yescomplemented
26C3×C42×C9C2211yescharacteristic, complemented
27#88551C2212yescomplemented
28#88536C2212yes—
29#88536C2211yescharacteristic
30#88536C2211yescharacteristic
31#88536C2212yescomplemented
32#88536C2212yescomplemented
33C4⋊(C3⋊Dic9)C2211yescharacteristic, complemented
34#88534C2211yescharacteristic, complemented
35#88534C2211yescharacteristic, complemented
36C4×C3⋊Dic9C2212yescomplemented
37C4×C3⋊Dic9C2211yescharacteristic, complemented
38C2×D4×C3⋊D3—31no—
39C32×C4×D4D311yescharacteristic
40D4×C3⋊Dic3—31no—
41C4×#638—32no—
42C2×#3752—32no—
43#64102—31no—
44#64101—31no—
45#64090—32no—
46C22⋊C4×C3⋊D3—32no—
47C4×C4⋊(C3⋊D3)—31no—
48#64081—31no—
49C2×D4×D9—33no—
50C4×D4×C9D313yescomplemented
51D4×Dic9—33no—
52C4×#611—36no—
53C2×(C2×D9)⋊C4—36no—
54#63455—33no—
55#63454—33no—
56#611⋊C4—36no—
57C22⋊C4×D9—36no—
58C4×D36—33no—
59#63434—33no—
60C23×C3×C9C2312yes—
61C22×C3⋊D9—42no—
62C22×C3⋊D9—42no—
63C22×C3⋊D9C2×C418yescomplemented
64C22×C3⋊D9C2311yescharacteristic
65C22×C3⋊D9C2312yes—
66C3×D4×C9C2×C414yescomplemented
67C2×C3×C4×C9—42no—
68C2×C3×C4×C9C2311yescharacteristic
69C2×C3×C4×C9C2312yes—
70C2×C3×C4×C9C2311yescharacteristic
71C2×C3×C4×C9C2311yescharacteristic
72#6233C2×C418yescomplemented
73C2×C3⋊Dic9—42no—
74C2×C3⋊Dic9C2311yescharacteristic
75C2×C3⋊Dic9C2312yes—
76C2×C3⋊Dic9C2311yescharacteristic
77C2×C3⋊Dic9C2311yescharacteristic
78C4⋊(C3⋊D9)C2×C414yescomplemented
79C4×C3⋊D9—42no—
80C4×C3⋊D9—42no—
81C4×C3⋊D9C2×C414yescomplemented
82#5545—93no—
83#5545—91no—
84C23×C3⋊D3—32no—
85C2×C32×D4C2×D311yescharacteristic
86C22×C32×C4C2×D312yes—
87C2×#638—32no—
88C22×C3⋊Dic3—32no—
89D4×C3⋊D3—38no—
90C2×C4⋊(C3⋊D3)—31no—
91C2×C4×C3⋊D3—31no—
92C2×C4×C3⋊D3—32no—
93C2×C4×C3⋊D3—32no—
94C23×D9—36no—
95C32×C4⋊C4C2×D311yescharacteristic
96C32×C22⋊C4C2×D312yes—
97C32×C42C2×D311yescharacteristic
98#3757—32no—
99#3752—31no—
100#3752—31no—
101#3752—32no—
102#3752—32no—
103#3752—32no—
104C4⋊(C3⋊Dic3)—31no—
105#3750—31no—
106#3750—31no—
107C4×C3⋊Dic3—31no—
108C4×C3⋊Dic3—32no—
109C2×D4×C9C2×D313yes—
110C22×C4×C9C2×D316yescomplemented
111C2×#611—36no—
112C22×Dic9—36no—
113D4×D9—324no—
114C2×D36—33no—
115C2×C4×D9—33no—
116C2×C4×D9—36no—
117C2×C4×D9—36no—
118C9×C4⋊C4C2×D313yescomplemented
119C9×C22⋊C4C2×D316yescomplemented
120C42×C9C2×D313yescomplemented
121#3676—36no—
122(C2×D9)⋊C4—33no—
123(C2×D9)⋊C4—36no—
124(C2×D9)⋊C4—36no—
125(C2×D9)⋊C4—36no—
126(C2×D9)⋊C4—33no—
127C4⋊Dic9—33no—
128Dic9⋊C4—33no—
129Dic9⋊C4—33no—
130C4×Dic9—36no—
131C4×Dic9—33no—
132C22×C3×C9—22no—
133C22×C3×C9—22no—
134C22×C3×C9C22×C414yes—
135C22×C3×C9C2411yescharacteristic
136C2×C3⋊D9—22no—
137C2×C3⋊D9—22no—
138C2×C3⋊D9—21no—
139C2×C3⋊D9—21no—
140C2×C3⋊D9—48no—
141C2×C3⋊D9C22×C412yes—
142C2×C3⋊D9C22×C414yes—
143C3×C4×C9—21no—
144C3×C4×C9—22no—
145C3×C4×C9—21no—
146C3×C4×C9C22×C412yes—
147C3⋊Dic9—21no—
148C3⋊Dic9—22no—
149C3⋊Dic9—21no—
150C3⋊Dic9C22×C412yes—
151C2×D3×D4—93no—
152C2×D3×D4—91no—
153C3×C4×D4D913yescomplemented
154C3×C4×D4C3⋊D311yescharacteristic
155D4×Dic3—93no—
156D4×Dic3—91no—
157C4×#67—96no—
158C4×#67—92no—
159C2×(C2×D3)⋊C4—96no—
160C2×(C2×D3)⋊C4—92no—
161#893—93no—
162#893—91no—
163#892—93no—
164#892—91no—
165#67⋊C4—96no—
166#67⋊C4—92no—
167D3×C22⋊C4—96no—
168D3×C22⋊C4—92no—
169C4×D12—93no—
170C4×D12—91no—
171#872—93no—
172#872—91no—
173C23×C32C22×D312yes—
174C22×C3⋊D3—31no—
175C22×C3⋊D3—32no—
176C22×C3⋊D3—38no—
177C22×C3⋊D3—122no—
178C22×C3⋊D3—122no—
179C32×D4C4×D314yes—
180C2×C32×C4—42no—
181C2×C32×C4C22×D312yes—
182C2×C32×C4C22×D311yescharacteristic
183C2×C32×C4C22×D311yescharacteristic
184C2×C32×C4C22×D311yescharacteristic
185#638—38no—
186C2×C3⋊Dic3—31no—
187C2×C3⋊Dic3—31no—
188C2×C3⋊Dic3—31no—
189C2×C3⋊Dic3—32no—
190C2×C3⋊Dic3—122no—
191C4⋊(C3⋊D3)—34no—
192C4×C3⋊D3—34no—
193C4×C3⋊D3—122no—
194C4×C3⋊D3—122no—
195C23×C9C22×D316yes—
196C22×D9—33no—
197C22×D9—36no—
198C22×D9—324no—
199C22×D9—126no—
200C22×D9—126no—
201D4×C9C4×D3112yescomplemented
202C2×C4×C9—46no—
203C2×C4×C9C22×D316yes—
204C2×C4×C9C22×D313yes—
205C2×C4×C9C22×D313yes—
206C2×C4×C9C22×D313yes—
207#611—324no—
208C2×Dic9—33no—
209C2×Dic9—33no—
210C2×Dic9—33no—
211C2×Dic9—36no—
212C2×Dic9—126no—
213D36—312no—
214C4×D9—312no—
215C4×D9—126no—
216C4×D9—126no—
217#518—271noSylow
218C2×C3×C9—24no—
219C2×C3×C9C23×C411yescharacteristic
220C2×C3×C9H(5,𝔽2)11yescharacteristic
221C2×C3×C9H(5,𝔽2)11yescharacteristic
222C3⋊D9—22no—
223C3⋊D9—24no—
224C23×D3—96no—
225C23×D3—92no—
226C2×C3×D4C2×D913yes—
227C2×C3×D4C2×C3⋊D311yescharacteristic
228C22×C3×C4C2×D916yescomplemented
229C22×C3×C4C2×C3⋊D312yes—
230C2×#67—96no—
231C2×#67—92no—
232C22×Dic3—96no—
233C22×Dic3—92no—
234D3×D4—924no—
235D3×D4—98no—
236C2×D12—93no—
237C2×D12—91no—
238C2×C4×D3—93no—
239C2×C4×D3—91no—
240C2×C4×D3—96no—
241C2×C4×D3—96no—
242C2×C4×D3—92no—
243C2×C4×D3—92no—
244C3×C4⋊C4C2×D913yescomplemented
245C3×C4⋊C4C2×C3⋊D311yescharacteristic
246C3×C22⋊C4C2×D916yescomplemented
247C3×C22⋊C4C2×C3⋊D312yes—
248C3×C42C2×D913yescomplemented
249C3×C42C2×C3⋊D311yescharacteristic
250#217—96no—
251#217—92no—
252(C2×D3)⋊C4—93no—
253(C2×D3)⋊C4—93no—
254(C2×D3)⋊C4—96no—
255(C2×D3)⋊C4—96no—
256(C2×D3)⋊C4—96no—
257(C2×D3)⋊C4—91no—
258(C2×D3)⋊C4—91no—
259(C2×D3)⋊C4—92no—
260(C2×D3)⋊C4—92no—
261(C2×D3)⋊C4—92no—
262C4⋊Dic3—93no—
263C4⋊Dic3—91no—
264Dic3⋊C4—93no—
265Dic3⋊C4—93no—
266Dic3⋊C4—91no—
267Dic3⋊C4—91no—
268C4×Dic3—93no—
269C4×Dic3—96no—
270C4×Dic3—91no—
271C4×Dic3—92no—
272C22×C32—22no—
273C22×C32—22no—
274C22×C32C2×C4×D314yes—
275C22×C32C23×D311yescharacteristic
276C2×C3⋊D3—32no—
277C2×C3⋊D3—34no—
278C2×C3⋊D3—62no—
279C2×C3⋊D3—61no—
280C2×C3⋊D3—62no—
281C2×C3⋊D3—61no—
282C2×C3⋊D3—128no—
283C32×C4—22no—
284C32×C4—21no—
285C32×C4—21no—
286C32×C4C2×C4×D312yes—
287C3⋊Dic3—32no—
288C3⋊Dic3—61no—
289C3⋊Dic3—62no—
290C3⋊Dic3—61no—
291C22×C9—26no—
292C22×C9—26no—
293C22×C9C2×C4×D3112yes—
294C22×C9C23×D313yes—
295C2×D9—36no—
296C2×D9—312no—
297C2×D9—66no—
298C2×D9—63no—
299C2×D9—66no—
300C2×D9—63no—
301C2×D9—1224no—
302C4×C9—26no—
303C4×C9—23no—
304C4×C9—23no—
305C4×C9C2×C4×D316yes—
306Dic9—36no—
307Dic9—63no—
308Dic9—66no—
309Dic9—63no—
310C22×D4—271no—
311C4×D4—271no—
312C4×D4—272no—
313C4×D4—271no—
314C4×D4—272no—
315C4×D4—271no—
316C4×D4C3⋊D911yescharacteristic, complemented
317#117—271no—
318#117—271no—
319C2×C22⋊C4—272no—
320C2×C22⋊C4—272no—
321C3×C9#51811yescharacteristic, complemented, Sylow
322C23×C3C22×D916yes—
323C23×C3C22×C3⋊D312yes—
324C22×D3—91no—
325C22×D3—92no—
326C22×D3—93no—
327C22×D3—96no—
328C22×D3—924no—
329C22×D3—98no—
330C22×D3—362no—
331C22×D3—362no—
332C22×D3—366no—
333C22×D3—366no—
334C3×D4C4×D9112yescomplemented
335C3×D4C4×C3⋊D314yes—
336C2×C3×C4—42no—
337C2×C3×C4—46no—
338C2×C3×C4C22×D916yes—
339C2×C3×C4C22×D913yes—
340C2×C3×C4C22×D913yes—
341C2×C3×C4C22×D913yes—
342C2×C3×C4C22×C3⋊D312yes—
343C2×C3×C4C22×C3⋊D311yescharacteristic
344C2×C3×C4C22×C3⋊D311yescharacteristic
345C2×C3×C4C22×C3⋊D311yescharacteristic
346#67—924no—
347#67—98no—
348C2×Dic3—91no—
349C2×Dic3—93no—
350C2×Dic3—91no—
351C2×Dic3—91no—
352C2×Dic3—92no—
353C2×Dic3—93no—
354C2×Dic3—93no—
355C2×Dic3—96no—
356C2×Dic3—362no—
357C2×Dic3—366no—
358D12—912no—
359D12—94no—
360C4×D3—912no—
361C4×D3—94no—
362C4×D3—362no—
363C4×D3—362no—
364C4×D3—366no—
365C4×D3—366no—
366C2×C32—24no—
367C2×C32C22×C4×D311yescharacteristic
368C2×C32#100411yescharacteristic
369C2×C32#100911yescharacteristic
370C3⋊D3—64no—
371C3⋊D3—62no—
372C2×C9—212no—
373C2×C9C22×C4×D313yes—
374C2×C9#100413yes—
375C2×C9#100913yes—
376D9—612no—
377D9—66no—
378C24—272no—
379C2×D4—272no—
380C2×D4—278no—
381C2×D4—271no—
382C2×D4C2×C3⋊D911yescharacteristic
383C22×C4—271no—
384C22×C4—272no—
385C22×C4—272no—
386C22×C4—272no—
387C22×C4C2×C3⋊D912yescomplemented
388C4⋊C4—271no—
389C4⋊C4—271no—
390C4⋊C4—271no—
391C4⋊C4C2×C3⋊D911yescharacteristic, complemented
392C22⋊C4—272no—
393C22⋊C4—272no—
394C22⋊C4—271no—
395C22⋊C4—271no—
396C22⋊C4—272no—
397C22⋊C4—272no—
398C22⋊C4C2×C3⋊D912yescomplemented
399C42—272no—
400C42—271no—
401C42C2×C3⋊D911yescharacteristic, complemented
402C22×C3—22no—
403C22×C3—22no—
404C22×C3—26no—
405C22×C3—26no—
406C22×C3C2×C4×D9112yes—
407C22×C3C23×D913yes—
408C22×C3C2×C4×C3⋊D314yes—
409C22×C3C23×C3⋊D311yescharacteristic
410C2×D3—92no—
411C2×D3—94no—
412C2×D3—96no—
413C2×D3—912no—
414C2×D3—182no—
415C2×D3—181no—
416C2×D3—182no—
417C2×D3—181no—
418C2×D3—186no—
419C2×D3—183no—
420C2×D3—186no—
421C2×D3—183no—
422C2×D3—368no—
423C2×D3—3624no—
424C3×C4—22no—
425C3×C4—21no—
426C3×C4—21no—
427C3×C4—26no—
428C3×C4—23no—
429C3×C4—23no—
430C3×C4C2×C4×D916yes—
431C3×C4C2×C4×C3⋊D312yes—
432Dic3—92no—
433Dic3—96no—
434Dic3—181no—
435Dic3—182no—
436Dic3—181no—
437Dic3—183no—
438Dic3—186no—
439Dic3—183no—
440C32#554511yescharacteristic
441C9#554513yescomplemented
442C23—271no—
443C23—272no—
444C23—278no—
445C23—1082no—
446C23—1082no—
447C23C22×C3⋊D912yes—
448D4—278no—
449D4—274no—
450D4C4×C3⋊D914yescomplemented
451C2×C4—42no—
452C2×C4—271no—
453C2×C4—274no—
454C2×C4—271no—
455C2×C4—272no—
456C2×C4—271no—
457C2×C4—1082no—
458C2×C4—1082no—
459C2×C4—1082no—
460C2×C4C22×C3⋊D912yes—
461C2×C4C22×C3⋊D911yescharacteristic
462C2×C4C22×C3⋊D911yescharacteristic
463C2×C4C22×C3⋊D911yescharacteristic
464C2×C3—24no—
465C2×C3—212no—
466C2×C3C22×C4×D913yes—
467C2×C3#6371013yes—
468C2×C3#6371513yes—
469C2×C3C22×C4×C3⋊D311yescharacteristic
470C2×C3#6436111yescharacteristic
471C2×C3#6436611yescharacteristic
472D3—182no—
473D3—184no—
474D3—186no—
475D3—1812no—
476C22—22no—
477C22—22no—
478C22—272no—
479C22—274no—
480C22—542no—
481C22—541no—
482C22—542no—
483C22—541no—
484C22—1088no—
485C22C2×C4×C3⋊D914yes—
486C22C23×C3⋊D911yescharacteristic
487C4—21no—
488C4—22no—
489C4—21no—
490C4—272no—
491C4—541no—
492C4—542no—
493C4—541no—
494C4C2×C4×C3⋊D912yes—
495C3#1058964713yescomplemented
496C3#1059601611yescharacteristic
497C2—24no—
498C2—542no—
499C2—544no—
500C2C22×C4×C3⋊D911yescharacteristic
501C2#1172266111yescharacteristic
502C2#1172267211yescharacteristic
subgroupG/H
centerC22C23×C3⋊D9
commutatorC2×C3×C9C23×C4
FrattiniC22×C3C23×C3⋊D3
FittingC3×C4×D4×C9C2
socleC22×C32C23×D3
prime radicalC2×D4×C3⋊D9C2
cosocleC22×C3×C9C24
hypercenterC4×D4C3⋊D9
hypocenterC3×C9#518
LCSG ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C4×D4

Complements

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

NH ≅ G/Ntype
2. C3×C4×D4×C9C2semidirect
3. D4×C3⋊Dic9C2semidirect
4. C4×#6233C2semidirect
5. C2×#88536C2semidirect
6. #11721749C2semidirect
7. #11721748C2semidirect
8. #11721727C2semidirect
9. C22⋊C4×C3⋊D9C2semidirect
10. C4×C4⋊(C3⋊D9)C2semidirect
11. #11721709C2semidirect
16. C22×C3×C4×C9C22semidirect
18. C22×C3⋊Dic9C22semidirect
19. D4×C3⋊D9C4semidirect
22. C2×C4×C3⋊D9C22semidirect
23. C2×C4×C3⋊D9C22semidirect
24. C3×C9×C4⋊C4C22semidirect
25. C3×C9×C22⋊C4C22semidirect
26. C3×C42×C9C22semidirect
27. #88551C22semidirect
31. #88536C22semidirect
32. #88536C22semidirect
33. C4⋊(C3⋊Dic9)C22semidirect
34. #88534C22semidirect
35. #88534C22semidirect
36. C4×C3⋊Dic9C22semidirect
37. C4×C3⋊Dic9C22semidirect
50. C4×D4×C9D3semidirect
63. C22×C3⋊D9C2×C4semidirect
66. C3×D4×C9C2×C4semidirect
72. #6233C2×C4semidirect
78. C4⋊(C3⋊D9)C2×C4semidirect
81. C4×C3⋊D9C2×C4semidirect
110. C22×C4×C9C2×D3semidirect
118. C9×C4⋊C4C2×D3semidirect
119. C9×C22⋊C4C2×D3semidirect
120. C42×C9C2×D3semidirect
153. C3×C4×D4D9semidirect
201. D4×C9C4×D3semidirect
228. C22×C3×C4C2×D9semidirect
244. C3×C4⋊C4C2×D9semidirect
246. C3×C22⋊C4C2×D9semidirect
248. C3×C42C2×D9semidirect
316. C4×D4C3⋊D9semidirect
321. C3×C9#518semidirect
334. C3×D4C4×D9semidirect
387. C22×C4C2×C3⋊D9semidirect
391. C4⋊C4C2×C3⋊D9semidirect
398. C22⋊C4C2×C3⋊D9semidirect
401. C42C2×C3⋊D9semidirect
441. C9#5545semidirect
450. D4C4×C3⋊D9semidirect
495. C3#10589647semidirect

Maximal subgroups

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

subgroupindex
C2×D4×C3⋊D92
C3×C4×D4×C92
D4×C3⋊Dic92
C4×#62332
C2×#885362
#117217492
#117217482
#117217272
C22⋊C4×C3⋊D92
C4×C4⋊(C3⋊D9)2
#117217092
#105960163
#105896473