Finite Group Databasebeta

← #13148895

Subgroups — #13148895

Order 1728.

776 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11721753C211yesmaximal, characteristic, complemented
2C3×#63455C211yesmaximal, characteristic, complemented
3C9×#890C211yesmaximal, characteristic, complemented
4#11721007C211yesmaximal, characteristic
5#11721002C212yesmaximal, complemented
6C4×(C3⋊D9)⋊C4C211yesmaximal, characteristic, complemented
7#11720949C212yesmaximal, complemented
8#11720948C211yesmaximal, characteristic, complemented
9#11720942C212yesmaximal, complemented
10#11720913C211yesmaximal, characteristic, complemented
11#11720907C212yesmaximal, complemented
12#10594447—31nomaximal
13#10589793—31nomaximal
14C2×C4×C3⋊D9C2211yescharacteristic
15C2×#6226C2211yescharacteristic
16C2×C3×D36C2211yescharacteristic
17C2×C3×C4×D9C2212yes—
18C2×#6192C2212yes—
19C2×(C3⋊D9)⋊C4C2211yescharacteristic
20C2×(C3⋊D9)⋊C4C2211yescharacteristic
21C3×C9×C4⋊C4C2211yescharacteristic
22#88536C2212yes—
23C4⋊(C3⋊Dic9)C2212yes—
24C3×(C2×D9)⋊C4C2212yes—
25C9×C4⋊Dic3C2211yescharacteristic
26C9×C4⋊Dic3C2212yes—
27C9×Dic3⋊C4C2212yes—
28C4×C9×Dic3C2211yescharacteristic
29C3×C4×Dic9C2211yescharacteristic
30(C2×D9)⋊Dic3C2212yes—
31(C2×D9)⋊Dic3C2212yes—
32(C2×D9)⋊Dic3C2212yes—
33#88442C2212yes—
34Dic3×Dic9C2211yescharacteristic
35Dic3×Dic9C2211yescharacteristic
36Dic3×Dic9C2212yes—
37#64106—31no—
38C3×#893—31no—
39C3×#890D311yescharacteristic
40#63916—31no—
41#63904—32no—
42C4×#624—31no—
43#63861—32no—
44#63860—31no—
45#63854—32no—
46#63844—31no—
47#63841—32no—
48C9×#125—31no—
49#63460—32no—
50#63459—31no—
51#63459—61no—
52#63457—31no—
53#63456—32no—
54#63455—32no—
55#63455D311yescharacteristic, complemented
56#63454—32no—
57#63449—31no—
58#63437—31no—
59C42×D9—31no—
60C22×C3⋊D9C2311yescharacteristic
61C22×C3×D9C2312yes—
62C2×C3×C4×C9C2312yes—
63C2×C3×C4×C9C2311yescharacteristic
64C2×C3⋊Dic9C2311yescharacteristic
65C2×C3⋊Dic9C2312yes—
66C4×C3⋊D9—22no—
67#6226—22no—
68C2×C9×Dic3C2311yescharacteristic
69C2×C9×Dic3C2311yescharacteristic
70C2×C9×Dic3C2312yes—
71C2×C3×Dic9C2311yescharacteristic
72C2×C3×Dic9C2311yescharacteristic
73C3×D36—22no—
74C3×C4×D9—24no—
75#6192—24no—
76(C3⋊D9)⋊C4—22no—
77(C3⋊D9)⋊C4—22no—
78#5691—91no—
79#5557—91no—
80C2×C4×C3⋊D3—31no—
81C2×#634—31no—
82C2×C3×D12—31no—
83C2×C3×C4×D3—32no—
84C2×#626—32no—
85C2×#624—31no—
86C2×#624—31no—
87C32×C4⋊C4C2×D311yescharacteristic
88#3752—32no—
89C4⋊(C3⋊Dic3)—32no—
90C3×(C2×D3)⋊C4—32no—
91C3×C4⋊Dic3C2×D312yes—
92C3×C4⋊Dic3C2×D311yescharacteristic
93C3×Dic3⋊C4C2×D312yes—
94C3×C4×Dic3—31no—
95C3×C4×Dic3C2×D311yescharacteristic
96#3724—32no—
97(C2×D3)⋊Dic3—32no—
98(C2×D3)⋊Dic3—32no—
99(C2×D3)⋊Dic3—32no—
100Dic32—31no—
101Dic32—32no—
102Dic32—31no—
103C2×D36—32no—
104C2×D36C2×D311yescharacteristic
105C2×C4×D9—31no—
106C2×C4×D9—31no—
107C2×C4×D9—31no—
108C2×C4×D9—61no—
109C2×C4×D9C2×D312yes—
110C2×Dic18—31no—
111C2×Dic18—61no—
112C9×C4⋊C4—21no—
113C9×C4⋊C4—32no—
114C9×C4⋊C4—31no—
115C9×C4⋊C4—32no—
116C9×C4⋊C4C2×D311yescharacteristic
117C42×C9—31no—
118(C2×D9)⋊C4—32no—
119(C2×D9)⋊C4—32no—
120(C2×D9)⋊C4—32no—
121(C2×D9)⋊C4—32no—
122(C2×D9)⋊C4—62no—
123(C2×D9)⋊C4C2×D312yes—
124C4⋊Dic9—32no—
125C4⋊Dic9—62no—
126Dic9⋊C4—32no—
127C4×Dic9—31no—
128C4×Dic9—31no—
129C4×Dic9—32no—
130C4×Dic9C2×D311yescharacteristic
131C22×C3×C9C2411yescharacteristic
132C2×C3⋊D9—21no—
133C2×C3⋊D9—21no—
134C2×C3⋊D9#4112yes—
135C2×C3×D9—22no—
136C2×C3×D9—22no—
137C2×C3×D9—22no—
138C3×C4×C9—22no—
139C3×C4×C9#4112yes—
140C3⋊Dic9—22no—
141C3⋊Dic9—21no—
142C9×Dic3—21no—
143C9×Dic3—21no—
144C9×Dic3—22no—
145C3×Dic9—21no—
146C3×Dic9#4112yes—
147C3×#125—31no—
148C3×C4×D4—91no—
149#938—91no—
150#936—91no—
151C4×#67—92no—
152#926—92no—
153#898—92no—
154#897—91no—
155#897—91no—
156#897—181no—
157#895—91no—
158#894—92no—
159#893—31no—
160#893—92no—
161#892—92no—
162#890D911yescharacteristic, complemented
163#887—91no—
164#883—92no—
165#879—92no—
166#875—91no—
167C42×D3—91no—
168C42×D3—91no—
169C4×Dic6—91no—
170C22×C3⋊D3—31no—
171C22×C3×D3—32no—
172C2×C32×C4C22×D312yes—
173C2×C32×C4C22×D311yescharacteristic
174C2×C3⋊Dic3—31no—
175C2×C3⋊Dic3—32no—
176C4×C3⋊D3—62no—
177#634—62no—
178C2×C3×Dic3—31no—
179C2×C3×Dic3—31no—
180C2×C3×Dic3C22×D311yescharacteristic
181C2×C3×Dic3C22×D311yescharacteristic
182C2×C3×Dic3C22×D312yes—
183C3×D12—62no—
184C3×C4×D3—64no—
185#626—64no—
186#624—62no—
187#624—62no—
188C22×D9—31no—
189C22×D9—61no—
190C22×D9C22×D312yes—
191C2×C4×C9—21no—
192C2×C4×C9—22no—
193C2×C4×C9—32no—
194C2×C4×C9—31no—
195C2×C4×C9—31no—
196C2×C4×C9C22×D312yes—
197C2×C4×C9C22×D311yescharacteristic
198C2×Dic9—31no—
199C2×Dic9—32no—
200C2×Dic9—61no—
201C2×Dic9—62no—
202C2×Dic9C22×D311yescharacteristic
203C2×Dic9C22×D311yescharacteristic
204D36—22no—
205D36—64no—
206C4×D9—24no—
207C4×D9—62no—
208C4×D9—62no—
209C4×D9—62no—
210C4×D9—122no—
211Dic18—62no—
212Dic18—122no—
213#529—271noSylow
214C2×C3×C9C2×#4111yescharacteristic
215C2×C3×C9C2×#4111yescharacteristic
216C2×C3×C9C2×#4111yescharacteristic
217C3⋊D9—22no—
218C3×D9—42no—
219C2×C3×D4—91no—
220C22×C3×C4—92no—
221C2×#67—92no—
222C2×D12—31no—
223C2×D12—92no—
224C2×C4×D3—32no—
225C2×C4×D3—91no—
226C2×C4×D3—91no—
227C2×C4×D3—91no—
228C2×C4×D3—91no—
229C2×C4×D3—91no—
230C2×C4×D3—91no—
231C2×C4×D3—181no—
232C2×Dic6—91no—
233C2×Dic6—91no—
234C2×Dic6—181no—
235C3×C4⋊C4—21no—
236C3×C4⋊C4—31no—
237C3×C4⋊C4—32no—
238C3×C4⋊C4—32no—
239C3×C4⋊C4C2×D911yescharacteristic
240C3×C4⋊C4D3211yescharacteristic
241C3×C22⋊C4—92no—
242C3×C42—31no—
243C3×C42—91no—
244#217—92no—
245#217—92no—
246#217—92no—
247(C2×D3)⋊C4—32no—
248(C2×D3)⋊C4—92no—
249(C2×D3)⋊C4—92no—
250(C2×D3)⋊C4—92no—
251(C2×D3)⋊C4—92no—
252(C2×D3)⋊C4—92no—
253(C2×D3)⋊C4—182no—
254C4⋊Dic3—92no—
255C4⋊Dic3—92no—
256C4⋊Dic3—182no—
257C4⋊Dic3C2×D911yescharacteristic
258C4⋊Dic3C2×D912yes—
259Dic3⋊C4—92no—
260Dic3⋊C4—92no—
261Dic3⋊C4C2×D912yes—
262C4×Dic3—31no—
263C4×Dic3—91no—
264C4×Dic3—91no—
265C4×Dic3—92no—
266C4×Dic3—92no—
267C4×Dic3—91no—
268C4×Dic3—91no—
269C4×Dic3C2×D911yescharacteristic
270C22×C32C23×D311yescharacteristic
271C2×C3⋊D3—32no—
272C2×C3⋊D3—61no—
273C2×C3⋊D3—61no—
274C2×C3×D3—62no—
275C2×C3×D3—62no—
276C2×C3×D3—62no—
277C32×C4—22no—
278C32×C4#23912yes—
279C3⋊Dic3—62no—
280C3⋊Dic3—61no—
281C3×Dic3—21no—
282C3×Dic3—21no—
283C3×Dic3—22no—
284C3×Dic3—32no—
285C3×Dic3—61no—
286C22×C9—21no—
287C22×C9C23×D311yescharacteristic
288C2×D9—22no—
289C2×D9—22no—
290C2×D9—22no—
291C2×D9—32no—
292C2×D9—62no—
293C2×D9—61no—
294C2×D9—61no—
295C2×D9—121no—
296C2×D9—121no—
297C4×C9—22no—
298C4×C9—22no—
299C4×C9—42no—
300C4×C9—61no—
301C4×C9—61no—
302C4×C9—62no—
303C4×C9#23512yes—
304Dic9—21no—
305Dic9—62no—
306Dic9—61no—
307Dic9—122no—
308Dic9—121no—
309Dic9#23712yes—
310#126—272no—
311#125—31no—
312#124—271no—
313#123—272no—
314C22⋊Q8—271no—
315#121—271no—
316C4×Q8—271no—
317C4×D4—91no—
318C4×D4—272no—
319#117—272no—
320C2×C42—271no—
321C3×C9#52911yescharacteristic, complemented, Sylow
322C23×C3—92no—
323C22×D3—32no—
324C22×D3—91no—
325C22×D3—91no—
326C22×D3—181no—
327C3×D4—182no—
328C2×C3×C4—21no—
329C2×C3×C4—22no—
330C2×C3×C4—32no—
331C2×C3×C4—31no—
332C2×C3×C4—31no—
333C2×C3×C4—91no—
334C2×C3×C4—91no—
335C2×C3×C4—184no—
336C2×C3×C4C22×D912yes—
337C2×C3×C4C22×D911yescharacteristic
338C2×C3×C4C2×D3212yes—
339C2×C3×C4C2×D3211yescharacteristic
340#67—184no—
341C2×Dic3—31no—
342C2×Dic3—31no—
343C2×Dic3—91no—
344C2×Dic3—92no—
345C2×Dic3—91no—
346C2×Dic3—92no—
347C2×Dic3—181no—
348C2×Dic3—182no—
349C2×Dic3C22×D911yescharacteristic
350C2×Dic3C22×D911yescharacteristic
351C2×Dic3C22×D912yes—
352D12—62no—
353D12—184no—
354C4×D3—64no—
355C4×D3—182no—
356C4×D3—182no—
357C4×D3—182no—
358C4×D3—182no—
359C4×D3—182no—
360C4×D3—182no—
361C4×D3—362no—
362Dic6—182no—
363Dic6—182no—
364Dic6—362no—
365C2×C32C2×#23911yescharacteristic
366C2×C32D3×#4111yescharacteristic
367C2×C32D3×#4111yescharacteristic
368C3⋊D3—62no—
369C3×D3—122no—
370C2×C9—21no—
371C2×C9—21no—
372C2×C9—21no—
373C2×C9C2×#23511yescharacteristic
374C2×C9C2×#23711yescharacteristic
375C2×C9D3×#4111yescharacteristic
376D9—42no—
377D9—62no—
378D9—122no—
379C2×Q8—271no—
380C2×D4—91no—
381C2×D4—272no—
382C22×C4—92no—
383C22×C4—271no—
384C22×C4—271no—
385C22×C4—271no—
386C4⋊C4—32no—
387C4⋊C4—32no—
388C4⋊C4—31no—
389C4⋊C4—272no—
390C4⋊C4—272no—
391C4⋊C4D3×D911yescharacteristic
392C22⋊C4—92no—
393C22⋊C4—272no—
394C22⋊C4—272no—
395C22⋊C4—272no—
396C22⋊C4—272no—
397C42—31no—
398C42—91no—
399C42—271no—
400C42—271no—
401C42—272no—
402C22×C3—21no—
403C22×C3—182no—
404C22×C3—182no—
405C22×C3—182no—
406C22×C3C23×D911yescharacteristic
407C22×C3C22×D3211yescharacteristic
408C2×D3—62no—
409C2×D3—62no—
410C2×D3—62no—
411C2×D3—92no—
412C2×D3—92no—
413C2×D3—182no—
414C2×D3—181no—
415C2×D3—181no—
416C2×D3—181no—
417C2×D3—181no—
418C2×D3—361no—
419C2×D3—361no—
420C3×C4—22no—
421C3×C4—22no—
422C3×C4—22no—
423C3×C4—42no—
424C3×C4—61no—
425C3×C4—61no—
426C3×C4—62no—
427C3×C4—92no—
428C3×C4—181no—
429C3×C4#370112yes—
430C3×C4#379912yes—
431Dic3—21no—
432Dic3—21no—
433Dic3—22no—
434Dic3—32no—
435Dic3—61no—
436Dic3—182no—
437Dic3—181no—
438Dic3—182no—
439Dic3—181no—
440Dic3—362no—
441Dic3—361no—
442C32#569111yescharacteristic
443C9—21no—
444C9#555711yescharacteristic, complemented
445C23—92no—
446C23—271no—
447Q8—542no—
448D4—182no—
449D4—544no—
450C2×C4—31no—
451C2×C4—31no—
452C2×C4—32no—
453C2×C4—91no—
454C2×C4—91no—
455C2×C4—184no—
456C2×C4—271no—
457C2×C4—272no—
458C2×C4—542no—
459C2×C4—542no—
460C2×C4—542no—
461C2×C4C2×D3×D912yes—
462C2×C4C2×D3×D911yescharacteristic
463C2×C3—21no—
464C2×C3—21no—
465C2×C3—21no—
466C2×C3—362no—
467C2×C3C2×#370111yescharacteristic
468C2×C3#41×D911yescharacteristic
469C2×C3#41×D911yescharacteristic
470C2×C3C2×#379911yescharacteristic
471C2×C3D3×#23711yescharacteristic
472C2×C3#6431911yescharacteristic
473D3—122no—
474D3—182no—
475D3—182no—
476D3—362no—
477C22—182no—
478C22—182no—
479C22—182no—
480C22—272no—
481C22—541no—
482C22—541no—
483C22C22×D3×D911yescharacteristic
484C4—22no—
485C4—61no—
486C4—61no—
487C4—62no—
488C4—92no—
489C4—181no—
490C4—541no—
491C4—542no—
492C4#8863512yes—
493C3—21no—
494C3#1058979311yescharacteristic, complemented
495C3#1059444711yescharacteristic
496C2—362no—
497C2—542no—
498C2C2×#8863511yescharacteristic
499C2D9×#23711yescharacteristic
500C2#1172251411yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC3×C9×C4⋊C4C22
socleC22×C32C23×D3
prime radical#11721007C2
hypercenterC4⋊C4D3×D9
hypocenterC3×C9#529
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. #11721753C2semidirect
2. C3×#63455C2semidirect
3. C9×#890C2semidirect
5. #11721002C2semidirect
6. C4×(C3⋊D9)⋊C4C2semidirect
7. #11720949C2semidirect
8. #11720948C2semidirect
9. #11720942C2semidirect
10. #11720913C2semidirect
11. #11720907C2semidirect
55. #63455D3semidirect
162. #890D9semidirect
321. C3×C9#529semidirect
444. C9#5557semidirect
494. C3#10589793semidirect

Maximal subgroups

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

subgroupindex
#117217532
C3×#634552
C9×#8902
#117210072
#117210022
C4×(C3⋊D9)⋊C42
#117209492
#117209482
#117209422
#117209132
#117209072
#105944473
#105897933