Finite Group Databasebeta

← #13150346

Subgroups — #13150346

Order 1728.

882 conjugacy classes of subgroups.

The subgroup lattice has 516 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88536C211yesmaximal, characteristic, complemented
2C3×#63492C211yesmaximal, characteristic, complemented
3C9×#924C211yesmaximal, characteristic, complemented
4#11721278C212yesmaximal, complemented
5#611⋊Dic3C212yesmaximal, complemented
6#11721003C212yesmaximal, complemented
7#11720988C212yesmaximal, complemented
8#11720983C212yesmaximal, complemented
9#11720921C212yesmaximal, complemented
10#10594979—31nomaximal
11#10589746—31nomaximal
12C23×C3⋊D9C2211yescharacteristic
13C22×C3×C4×C9C2211yescharacteristic
14C22×C3⋊Dic9C2211yescharacteristic
15C2×C3×#611C2212yes—
16C2×C3×D36C2211yescharacteristic, complemented
17C2×C9×Dic6C2211yescharacteristic, complemented
18C2×#6192C2212yes—
19C2×#6192C2212yes—
20C2×Dic3×D9C2212yescomplemented
21#88536C2212yes—
22#88536C2212yes—
23C9×#217C2212yes—
24C3×(C2×D9)⋊C4C2212yescomplemented
25C9×C4⋊Dic3C2211yescharacteristic, complemented
26C9×Dic3⋊C4C2212yescomplemented
27C3×C4⋊Dic9C2211yescharacteristic, complemented
28#88445C2212yes—
29#88445C2212yes—
30(C2×D9)⋊Dic3C2212yescomplemented
31Dic9⋊Dic3C2212yescomplemented
32Dic3×Dic9C2212yescomplemented
33C2×#3752—31no—
34C3×#930—31no—
35C3×#924D311yescharacteristic
36#63981—32no—
37#67⋊Dic3—32no—
38#63907—32no—
39#63890—32no—
40#63886—32no—
41#63848—32no—
42C9×C22⋊Q8—31no—
43#63509—32no—
44#63492D311yescharacteristic, complemented
45C2×(C2×D9)⋊C4—31no—
46C2×(C2×D9)⋊C4—61no—
47#63460—32no—
48#63457—32no—
49#63456—32no—
50#63444—32no—
51#611⋊C4—32no—
52C23×C3×C9C2311yescharacteristic
53C22×C3⋊D9—22no—
54C22×C3⋊D9—41no—
55C22×C3⋊D9—41no—
56C22×C3⋊D9C2312yes—
57C22×C3×D9C2312yes—
58C2×C3×C4×C9—41no—
59C2×C3×C4×C9C2311yescharacteristic
60C2×C3×C4×C9C2311yescharacteristic
61C2×C3⋊Dic9—41no—
62C2×C3⋊Dic9C2312yes—
63C3×#611—24no—
64C2×C9×Dic3C2312yes—
65C2×C9×Dic3C2312yes—
66C2×C3×Dic9C2312yes—
67C3×D36—41no—
68C9×Dic6—41no—
69#6192—42no—
70#6192—42no—
71Dic3×D9—42no—
72#5644—91no—
73#5617—91no—
74C23×C3⋊D3—31no—
75C22×C32×C4C2×D311yescharacteristic
76C22×C3⋊Dic3—31no—
77C2×C3×#67—32no—
78C2×C3×D12—31no—
79C2×C3×Dic6C2×D311yescharacteristic
80C2×#626—32no—
81C2×#626—32no—
82C2×D3×Dic3—32no—
83C23×D9—31no—
84C23×D9—61no—
85#3752—32no—
86#3752—32no—
87C3×#217C2×D312yes—
88C3×(C2×D3)⋊C4—32no—
89C3×C4⋊Dic3—31no—
90C3×C4⋊Dic3C2×D311yescharacteristic
91C3×Dic3⋊C4C2×D312yes—
92Dic3⋊Dic3—32no—
93#3722—32no—
94#3722—32no—
95(C2×D3)⋊Dic3—32no—
96Dic32—32no—
97C2×Q8×C9—31no—
98C22×C4×C9—21no—
99C22×C4×C9C2×D311yescharacteristic
100C2×#611C2×D312yes—
101C22×Dic9—31no—
102C22×Dic9—61no—
103C2×D36—32no—
104C2×D36—32no—
105C2×D36C2×D311yescharacteristic, complemented
106C2×C4×D9—32no—
107C9×C4⋊C4—32no—
108C9×C4⋊C4—31no—
109C9×C22⋊C4—32no—
110(C2×D9)⋊C4—32no—
111(C2×D9)⋊C4—32no—
112(C2×D9)⋊C4—32no—
113(C2×D9)⋊C4—32no—
114(C2×D9)⋊C4—32no—
115(C2×D9)⋊C4—62no—
116(C2×D9)⋊C4—62no—
117(C2×D9)⋊C4C2×D312yescomplemented
118C4⋊Dic9C2×D311yescharacteristic, complemented
119Dic9⋊C4—32no—
120C4×Dic9—32no—
121C22×C3×C9—21no—
122C22×C3×C9—21no—
123C22×C3×C9#4112yes—
124C22×C3×C9C2411yescharacteristic
125C2×C3⋊D9—22no—
126C2×C3⋊D9—22no—
127C2×C3⋊D9—22no—
128C2×C3⋊D9—44no—
129C2×C3×D9—22no—
130C2×C3×D9—22no—
131C2×C3×D9—22no—
132C3×C4×C9—21no—
133C3×C4×C9—21no—
134C3⋊Dic9—22no—
135C9×Dic3—22no—
136C9×Dic3—22no—
137C3×Dic9—22no—
138C3×C22⋊Q8—31no—
139C3×#121—91no—
140#947—92no—
141#939—92no—
142#937—92no—
143D4×Dic3—92no—
144#930—31no—
145C2×(C2×D3)⋊C4—91no—
146C2×(C2×D3)⋊C4—91no—
147C2×(C2×D3)⋊C4—181no—
148#924D911yescharacteristic, complemented
149#898—92no—
150#898—92no—
151#895—92no—
152#894—92no—
153#886—92no—
154#885—92no—
155#882—92no—
156#67⋊C4—92no—
157C23×C32C22×D311yescharacteristic
158C22×C3⋊D3—32no—
159C22×C3⋊D3—62no—
160C22×C3⋊D3—121no—
161C22×C3⋊D3—121no—
162C22×C3×D3—32no—
163C2×C32×C4—41no—
164C2×C32×C4C22×D311yescharacteristic
165C2×C32×C4C22×D311yescharacteristic
166C2×C3⋊Dic3—32no—
167C2×C3⋊Dic3—121no—
168C3×#67—64no—
169C2×C3×Dic3—32no—
170C2×C3×Dic3C22×D312yes—
171C2×C3×Dic3C22×D312yes—
172C3×D12—121no—
173C3×Dic6—41no—
174#626—122no—
175#626—122no—
176D3×Dic3—122no—
177C23×C9—21no—
178C23×C9C22×D311yescharacteristic
179C22×D9—32no—
180C22×D9—62no—
181C22×D9—64no—
182C22×D9—62no—
183C22×D9—121no—
184C22×D9—121no—
185C22×D9—122no—
186C22×D9—122no—
187C22×D9C22×D312yes—
188Q8×C9—121no—
189C2×C4×C9—21no—
190C2×C4×C9—21no—
191C2×C4×C9—32no—
192C2×C4×C9—32no—
193C2×C4×C9—41no—
194C2×C4×C9—42no—
195C2×C4×C9C22×D311yescharacteristic
196C2×C4×C9C22×D311yescharacteristic
197#611—24no—
198C2×Dic9—32no—
199C2×Dic9—62no—
200C2×Dic9—121no—
201C2×Dic9—122no—
202C2×Dic9C22×D312yes—
203D36—41no—
204D36—122no—
205D36—122no—
206C4×D9—122no—
207#538—271noSylow
208C2×C3×C9—22no—
209C2×C3×C9C2×#4111yescharacteristic
210C2×C3×C9H(5,𝔽2)11yescharacteristic
211C2×C3×C9H(5,𝔽2)11yescharacteristic
212C3⋊D9—42no—
213C3×D9—42no—
214C23×D3—91no—
215C23×D3—91no—
216C23×D3—181no—
217C2×C3×Q8—31no—
218C2×C3×D4—92no—
219C2×C3×D4—91no—
220C22×C3×C4—21no—
221C22×C3×C4C2×D911yescharacteristic
222C22×C3×C4D3211yescharacteristic
223C2×#67—32no—
224C2×#67—92no—
225C2×#67—92no—
226C22×Dic3—91no—
227C22×Dic3—91no—
228C22×Dic3—92no—
229C22×Dic3—181no—
230C2×D12—31no—
231C2×D12—92no—
232C2×D12—92no—
233C2×C4×D3—92no—
234C2×Dic6C2×D911yescharacteristic, complemented
235C3×C4⋊C4—31no—
236C3×C4⋊C4—32no—
237C3×C4⋊C4—91no—
238C3×C22⋊C4—32no—
239C3×C22⋊C4—92no—
240#217—92no—
241#217C2×D912yes—
242(C2×D3)⋊C4—32no—
243(C2×D3)⋊C4—92no—
244(C2×D3)⋊C4—92no—
245(C2×D3)⋊C4—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—182no—
253(C2×D3)⋊C4—182no—
254C4⋊Dic3—31no—
255C4⋊Dic3—92no—
256C4⋊Dic3C2×D911yescharacteristic, complemented
257Dic3⋊C4—92no—
258Dic3⋊C4C2×D912yescomplemented
259C4×Dic3—92no—
260C4×Dic3—92no—
261C22×C32—21no—
262C22×C32—21no—
263C22×C32#23912yes—
264C22×C32C23×D311yescharacteristic
265C2×C3⋊D3—62no—
266C2×C3⋊D3—62no—
267C2×C3⋊D3—62no—
268C2×C3⋊D3—124no—
269C2×C3×D3—62no—
270C2×C3×D3—62no—
271C2×C3×D3—62no—
272C32×C4—21no—
273C32×C4—21no—
274C3⋊Dic3—62no—
275C3×Dic3—22no—
276C3×Dic3—22no—
277C3×Dic3—62no—
278C22×C9—22no—
279C22×C9—22no—
280C22×C9—21no—
281C22×C9—21no—
282C22×C9—21no—
283C22×C9—22no—
284C22×C9#23712yes—
285C22×C9C23×D311yescharacteristic
286C2×D9—22no—
287C2×D9—22no—
288C2×D9—22no—
289C2×D9—64no—
290C2×D9—62no—
291C2×D9—62no—
292C2×D9—62no—
293C2×D9—122no—
294C2×D9—122no—
295C2×D9—128no—
296C2×D9—124no—
297C4×C9—21no—
298C4×C9—21no—
299C4×C9—41no—
300C4×C9—41no—
301C4×C9—62no—
302C4×C9—62no—
303Dic9—22no—
304Dic9—62no—
305Dic9—122no—
306#126—272no—
307#124—272no—
308#123—272no—
309C22⋊Q8—31no—
310#121—91no—
311#121—272no—
312#120—272no—
313C4×D4—272no—
314C2×C22⋊C4—271no—
315C3×C9#53811yescharacteristic, complemented, Sylow
316C23×C3—21no—
317C23×C3—92no—
318C23×C3C22×D911yescharacteristic
319C23×C3C2×D3211yescharacteristic
320C22×D3—32no—
321C22×D3—92no—
322C22×D3—92no—
323C22×D3—182no—
324C22×D3—184no—
325C22×D3—182no—
326C22×D3—182no—
327C22×D3—361no—
328C22×D3—361no—
329C22×D3—361no—
330C22×D3—361no—
331C22×D3—362no—
332C22×D3—362no—
333C3×Q8—121no—
334C3×D4—184no—
335C3×D4—361no—
336C2×C3×C4—21no—
337C2×C3×C4—21no—
338C2×C3×C4—32no—
339C2×C3×C4—32no—
340C2×C3×C4—41no—
341C2×C3×C4—41no—
342C2×C3×C4—42no—
343C2×C3×C4—92no—
344C2×C3×C4C22×D911yescharacteristic
345C2×C3×C4C22×D911yescharacteristic
346C2×C3×C4C2×D3211yescharacteristic
347C2×C3×C4C2×D3211yescharacteristic
348#67—64no—
349#67—362no—
350#67—362no—
351C2×Dic3—32no—
352C2×Dic3—92no—
353C2×Dic3—92no—
354C2×Dic3—182no—
355C2×Dic3—362no—
356C2×Dic3—361no—
357C2×Dic3—361no—
358C2×Dic3—362no—
359C2×Dic3C22×D912yes—
360C2×Dic3C22×D912yes—
361D12—121no—
362D12—362no—
363D12—362no—
364C4×D3—362no—
365Dic6—41no—
366C2×C32—22no—
367C2×C32#100411yescharacteristic
368C2×C32C2×#23911yescharacteristic
369C2×C32#100911yescharacteristic
370C3⋊D3—122no—
371C3×D3—122no—
372C2×C9—24no—
373C2×C9—22no—
374C2×C9—21no—
375C2×C9—21no—
376C2×C9—21no—
377C2×C9C2×#23711yescharacteristic
378C2×C9#100411yescharacteristic
379C2×C9#100911yescharacteristic
380D9—42no—
381D9—124no—
382D9—122no—
383C24—271no—
384C2×Q8—31no—
385C2×D4—92no—
386C2×D4—91no—
387C2×D4—272no—
388C2×D4—272no—
389C22×C4—272no—
390C22×C4—271no—
391C22×C4D3×D911yescharacteristic
392C4⋊C4—32no—
393C4⋊C4—31no—
394C4⋊C4—91no—
395C4⋊C4—272no—
396C22⋊C4—32no—
397C22⋊C4—92no—
398C22⋊C4—272no—
399C22⋊C4—272no—
400C22⋊C4—272no—
401C22⋊C4—272no—
402C22⋊C4—272no—
403C42—272no—
404C22×C3—22no—
405C22×C3—22no—
406C22×C3—21no—
407C22×C3—21no—
408C22×C3—21no—
409C22×C3—21no—
410C22×C3—22no—
411C22×C3—21no—
412C22×C3—182no—
413C22×C3—182no—
414C22×C3—182no—
415C22×C3#370112yes—
416C22×C3C23×D911yescharacteristic
417C22×C3#379612yes—
418C22×C3C22×D3211yescharacteristic
419C2×D3—62no—
420C2×D3—62no—
421C2×D3—62no—
422C2×D3—184no—
423C2×D3—182no—
424C2×D3—182no—
425C2×D3—182no—
426C2×D3—182no—
427C2×D3—182no—
428C2×D3—182no—
429C2×D3—362no—
430C2×D3—362no—
431C2×D3—364no—
432C2×D3—368no—
433C2×D3—364no—
434C3×C4—21no—
435C3×C4—21no—
436C3×C4—21no—
437C3×C4—21no—
438C3×C4—41no—
439C3×C4—41no—
440C3×C4—62no—
441C3×C4—62no—
442C3×C4—182no—
443Dic3—22no—
444Dic3—22no—
445Dic3—62no—
446Dic3—182no—
447Dic3—182no—
448Dic3—362no—
449C32#564411yescharacteristic
450C9—21no—
451C9#561711yescharacteristic, complemented
452C23—92no—
453C23—272no—
454C23—542no—
455C23—1081no—
456C23—1081no—
457C23C2×D3×D911yescharacteristic
458Q8—121no—
459D4—184no—
460D4—361no—
461D4—1082no—
462D4—1082no—
463C2×C4—32no—
464C2×C4—32no—
465C2×C4—41no—
466C2×C4—92no—
467C2×C4—272no—
468C2×C4—1082no—
469C2×C4—1081no—
470C2×C4C2×D3×D911yescharacteristic
471C2×C4C2×D3×D911yescharacteristic
472C2×C3—24no—
473C2×C3—22no—
474C2×C3—22no—
475C2×C3—21no—
476C2×C3—21no—
477C2×C3—21no—
478C2×C3—362no—
479C2×C3#6371011yescharacteristic
480C2×C3C2×#370111yescharacteristic
481C2×C3#6371511yescharacteristic
482C2×C3C2×#379611yescharacteristic
483C2×C3#6430811yescharacteristic
484C2×C3#6433011yescharacteristic
485D3—122no—
486D3—364no—
487D3—362no—
488D3—362no—
489C22—21no—
490C22—21no—
491C22—182no—
492C22—182no—
493C22—182no—
494C22—542no—
495C22—542no—
496C22—542no—
497C22—1084no—
498C22#8863412yes—
499C22C22×D3×D911yescharacteristic
500C4—21no—
501C4—21no—
502C4—62no—
503C4—62no—
504C4—182no—
505C4—542no—
506C3—21no—
507C3#1058974611yescharacteristic, complemented
508C3#1059497911yescharacteristic
509C2—22no—
510C2—362no—
511C2—1082no—
512C2C2×#8863411yescharacteristic
513C2#1172247811yescharacteristic
514C2#1172253511yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC22×C3×C4×C9C22
socleC22×C32C23×D3
hypercenterC22×C4D3×D9
hypocenterC3×C9#538
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C2×#88536C2semidirect
2. C3×#63492C2semidirect
3. C9×#924C2semidirect
4. #11721278C2semidirect
5. #611⋊Dic3C2semidirect
6. #11721003C2semidirect
7. #11720988C2semidirect
8. #11720983C2semidirect
9. #11720921C2semidirect
16. C2×C3×D36C22semidirect
17. C2×C9×Dic6C22semidirect
20. C2×Dic3×D9C22semidirect
24. C3×(C2×D9)⋊C4C22semidirect
25. C9×C4⋊Dic3C22semidirect
26. C9×Dic3⋊C4C22semidirect
27. C3×C4⋊Dic9C22semidirect
30. (C2×D9)⋊Dic3C22semidirect
31. Dic9⋊Dic3C22semidirect
32. Dic3×Dic9C22semidirect
44. #63492D3semidirect
105. C2×D36C2×D3semidirect
117. (C2×D9)⋊C4C2×D3semidirect
118. C4⋊Dic9C2×D3semidirect
148. #924D9semidirect
234. C2×Dic6C2×D9semidirect
256. C4⋊Dic3C2×D9semidirect
258. Dic3⋊C4C2×D9semidirect
315. C3×C9#538semidirect
451. C9#5617semidirect
507. C3#10589746semidirect

Maximal subgroups

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

subgroupindex
C2×#885362
C3×#634922
C9×#9242
#117212782
#611⋊Dic32
#117210032
#117209882
#117209832
#117209212
#105949793
#105897463