Finite Group Databasebeta

← #13148905

Subgroups — #13148905

Order 1728.

796 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#11721749C211yesmaximal, characteristic, complemented
2C3×#63459C211yesmaximal, characteristic, complemented
3C9×#890C211yesmaximal, characteristic, complemented
4#11721004C211yesmaximal, characteristic
5#11720988C212yesmaximal, complemented
6#11720978C212yesmaximal, complemented
7C4×Dic3×D9C211yesmaximal, characteristic, complemented
8#11720952C211yesmaximal, characteristic, complemented
9#11720922C212yesmaximal, complemented
10#11720921C212yesmaximal, complemented
11#11720910C211yesmaximal, characteristic, complemented
12#10594453—31nomaximal
13#10589793—31nomaximal
14C2×C4⋊(C3⋊D9)C2211yescharacteristic
15C2×C4×C3⋊D9C2212yes—
16C2×C3×C4×D9C2211yescharacteristic
17C2×C3×Dic18C2211yescharacteristic
18C2×#6192C2212yes—
19C2×Dic3×D9C2211yescharacteristic
20C2×Dic3×D9C2211yescharacteristic
21C3×C9×C4⋊C4C2211yescharacteristic
22#88536C2212yes—
23C4×C3⋊Dic9C2211yescharacteristic
24C3×(C2×D9)⋊C4C2212yes—
25C9×C4⋊Dic3C2212yes—
26C9×C4⋊Dic3C2211yescharacteristic
27C9×Dic3⋊C4C2212yes—
28C4×C9×Dic3C2211yescharacteristic
29C3×C4⋊Dic9C2212yes—
30#88445C2212yes—
31#88445C2212yes—
32#88445C2212yes—
33Dic9⋊Dic3C2212yes—
34Dic3×Dic9C2211yescharacteristic
35Dic3×Dic9C2211yescharacteristic
36Dic3×Dic9C2212yes—
37#64102—31no—
38C3×#897—31no—
39C3×#890D311yescharacteristic
40#63909—31no—
41#63890—32no—
42#63881—32no—
43C4×D3×Dic3—31no—
44#63864—31no—
45#63849—32no—
46#63848—32no—
47#63843—31no—
48C9×#125—31no—
49#63460—32no—
50#63459D311yescharacteristic, complemented
51#63457—31no—
52#63456—32no—
53#63455—31no—
54#63455—32no—
55#63455—61no—
56#63454—32no—
57#63449—31no—
58#63437—31no—
59C42×D9—31no—
60C22×C3⋊D9C2312yes—
61C22×C3×D9C2311yescharacteristic
62C2×C3×C4×C9C2312yes—
63C2×C3×C4×C9C2311yescharacteristic
64C2×C3⋊Dic9C2311yescharacteristic
65C2×C3⋊Dic9C2311yescharacteristic
66C4⋊(C3⋊D9)—22no—
67C4×C3⋊D9—24no—
68C2×C9×Dic3C2311yescharacteristic
69C2×C9×Dic3C2311yescharacteristic
70C2×C9×Dic3C2312yes—
71C2×C3×Dic9C2312yes—
72C2×C3×Dic9C2311yescharacteristic
73C3×C4×D9—22no—
74C3×Dic18—22no—
75#6192—24no—
76Dic3×D9—22no—
77Dic3×D9—22no—
78#5691—91no—
79#5624—91no—
80C2×C4⋊(C3⋊D3)—31no—
81C2×C4×C3⋊D3—32no—
82C2×C3×C4×D3—31no—
83C2×C3×Dic6—31no—
84C2×#626—32no—
85C2×D3×Dic3—31no—
86C2×D3×Dic3—31no—
87C32×C4⋊C4C2×D311yescharacteristic
88#3752—32no—
89C4×C3⋊Dic3—31no—
90C3×(C2×D3)⋊C4—32no—
91C3×C4⋊Dic3—32no—
92C3×C4⋊Dic3C2×D311yescharacteristic
93C3×C4⋊Dic3C2×D312yes—
94C3×Dic3⋊C4C2×D312yes—
95C3×C4×Dic3C2×D311yescharacteristic
96Dic3⋊Dic3—32no—
97#3722—32no—
98#3722—32no—
99#3722—32no—
100Dic32—32no—
101Dic32—31no—
102Dic32—31no—
103C2×D36—32no—
104C2×D36—31no—
105C2×D36—61no—
106C2×C4×D9—31no—
107C2×C4×D9—31no—
108C2×C4×D9—32no—
109C2×C4×D9—62no—
110C2×C4×D9C2×D311yescharacteristic
111C2×Dic18C2×D311yescharacteristic
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⋊Dic9C2×D312yes—
125Dic9⋊C4—32no—
126C4×Dic9—31no—
127C4×Dic9—31no—
128C4×Dic9—31no—
129C4×Dic9—32no—
130C4×Dic9—61no—
131C22×C3×C9C2411yescharacteristic
132C2×C3⋊D9—22no—
133C2×C3⋊D9—22no—
134C2×C3⋊D9—22no—
135C2×C3×D9—21no—
136C2×C3×D9—21no—
137C2×C3×D9#4112yes—
138C3×C4×C9—22no—
139C3×C4×C9#4112yes—
140C3⋊Dic9—21no—
141C3⋊Dic9#4112yes—
142C9×Dic3—21no—
143C9×Dic3—21no—
144C9×Dic3—22no—
145C3×Dic9—22no—
146C3×Dic9—21no—
147C3×#125—31no—
148C3×C22⋊Q8—91no—
149#947—91no—
150Q8×Dic3—91no—
151#930—91no—
152C2×C4×Dic3—91no—
153#898—92no—
154#898—92no—
155#897—31no—
156#895—91no—
157#894—92no—
158#893—92no—
159#893—91no—
160#893—91no—
161#893—181no—
162#892—92no—
163#892—92no—
164#890D911yescharacteristic, complemented
165#887—91no—
166#886—92no—
167#67⋊C4—92no—
168#875—91no—
169C42×D3—91no—
170C22×C3⋊D3—32no—
171C22×C3×D3—31no—
172C2×C32×C4C22×D312yes—
173C2×C32×C4C22×D311yescharacteristic
174C2×C3⋊Dic3—31no—
175C2×C3⋊Dic3—31no—
176C4⋊(C3⋊D3)—62no—
177C4×C3⋊D3—64no—
178C2×C3×Dic3—31no—
179C2×C3×Dic3—32no—
180C2×C3×Dic3C22×D311yescharacteristic
181C2×C3×Dic3C22×D311yescharacteristic
182C2×C3×Dic3C22×D312yes—
183C3×C4×D3—62no—
184C3×Dic6—62no—
185#626—64no—
186D3×Dic3—62no—
187D3×Dic3—62no—
188C22×D9—32no—
189C22×D9—62no—
190C22×D9C22×D311yescharacteristic
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—31no—
200C2×Dic9—61no—
201C2×Dic9—61no—
202C2×Dic9C22×D311yescharacteristic
203C2×Dic9C22×D312yes—
204D36—64no—
205D36—64no—
206D36—62no—
207C4×D9—22no—
208C4×D9—64no—
209C4×D9—62no—
210C4×D9—62no—
211C4×D9—124no—
212Dic18—22no—
213#529—271noSylow
214C2×C3×C9C2×#4111yescharacteristic
215C2×C3×C9C2×#4111yescharacteristic
216C2×C3×C9C2×#4111yescharacteristic
217C3⋊D9—42no—
218C3×D9—22no—
219C2×C3×Q8—91no—
220C22×C3×C4—91no—
221C2×#67—92no—
222C22×Dic3—91no—
223C22×Dic3—91no—
224C2×D12—92no—
225C2×D12—91no—
226C2×D12—91no—
227C2×D12—181no—
228C2×C4×D3—31no—
229C2×C4×D3—91no—
230C2×C4×D3—91no—
231C2×C4×D3—92no—
232C2×C4×D3—92no—
233C2×C4×D3—182no—
234C2×Dic6—31no—
235C3×C4⋊C4—21no—
236C3×C4⋊C4—31no—
237C3×C4⋊C4—32no—
238C3×C4⋊C4—32no—
239C3×C4⋊C4—92no—
240C3×C4⋊C4C2×D911yescharacteristic
241C3×C4⋊C4D3211yescharacteristic
242C3×C22⋊C4—92no—
243C3×C42—31no—
244(C2×D3)⋊C4—32no—
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—92no—
253(C2×D3)⋊C4—182no—
254C4⋊Dic3—32no—
255C4⋊Dic3—92no—
256C4⋊Dic3C2×D912yes—
257C4⋊Dic3C2×D911yescharacteristic
258Dic3⋊C4—92no—
259Dic3⋊C4C2×D912yes—
260C4×Dic3—91no—
261C4×Dic3—91no—
262C4×Dic3—92no—
263C4×Dic3—91no—
264C4×Dic3—91no—
265C4×Dic3—92no—
266C4×Dic3—91no—
267C4×Dic3—91no—
268C4×Dic3—181no—
269C4×Dic3C2×D911yescharacteristic
270C22×C32C23×D311yescharacteristic
271C2×C3⋊D3—62no—
272C2×C3⋊D3—62no—
273C2×C3⋊D3—62no—
274C2×C3×D3—32no—
275C2×C3×D3—61no—
276C2×C3×D3—61no—
277C32×C4—22no—
278C32×C4#23912yes—
279C3⋊Dic3—32no—
280C3⋊Dic3—61no—
281C3×Dic3—21no—
282C3×Dic3—21no—
283C3×Dic3—22no—
284C3×Dic3—62no—
285C3×Dic3—61no—
286C22×C9—21no—
287C22×C9C23×D311yescharacteristic
288C2×D9—21no—
289C2×D9—21no—
290C2×D9—64no—
291C2×D9—62no—
292C2×D9—62no—
293C2×D9—62no—
294C2×D9—122no—
295C2×D9—122no—
296C2×D9#23912yes—
297C4×C9—22no—
298C4×C9—22no—
299C4×C9—42no—
300C4×C9—61no—
301C4×C9—62no—
302C4×C9—61no—
303C4×C9#23712yes—
304Dic9—22no—
305Dic9—21no—
306Dic9—32no—
307Dic9—62no—
308Dic9—61no—
309Dic9—121no—
310#126—272no—
311#125—31no—
312#124—271no—
313#123—272no—
314C22⋊Q8—91no—
315#121—271no—
316C4×Q8—271no—
317C4×D4—272no—
318C4×D4—271no—
319#117—272no—
320C2×C42—271no—
321C3×C9#52911yescharacteristic, complemented, Sylow
322C23×C3—91no—
323C22×D3—31no—
324C22×D3—92no—
325C22×D3—92no—
326C22×D3—182no—
327C3×Q8—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—92no—
335C2×C3×C4—182no—
336C2×C3×C4C22×D912yes—
337C2×C3×C4C22×D911yescharacteristic
338C2×C3×C4C2×D3212yes—
339C2×C3×C4C2×D3211yescharacteristic
340#67—184no—
341C2×Dic3—31no—
342C2×Dic3—32no—
343C2×Dic3—91no—
344C2×Dic3—91no—
345C2×Dic3—91no—
346C2×Dic3—91no—
347C2×Dic3—181no—
348C2×Dic3—181no—
349C2×Dic3—182no—
350C2×Dic3—182no—
351C2×Dic3C22×D911yescharacteristic
352C2×Dic3C22×D911yescharacteristic
353C2×Dic3C22×D912yes—
354D12—184no—
355D12—184no—
356D12—182no—
357D12—182no—
358C4×D3—62no—
359C4×D3—184no—
360C4×D3—184no—
361C4×D3—182no—
362C4×D3—182no—
363C4×D3—364no—
364Dic6—62no—
365C2×C32C2×#23911yescharacteristic
366C2×C32D3×#4111yescharacteristic
367C2×C32D3×#4111yescharacteristic
368C3⋊D3—122no—
369C3×D3—62no—
370C2×C9—21no—
371C2×C9—21no—
372C2×C9—21no—
373C2×C9C2×#23711yescharacteristic
374C2×C9C2×#23911yescharacteristic
375C2×C9D3×#4111yescharacteristic
376D9—22no—
377D9—124no—
378D9—122no—
379C2×Q8—91no—
380C2×D4—272no—
381C2×D4—271no—
382C22×C4—91no—
383C22×C4—271no—
384C22×C4—271no—
385C22×C4—272no—
386C4⋊C4—32no—
387C4⋊C4—32no—
388C4⋊C4—31no—
389C4⋊C4—92no—
390C4⋊C4—272no—
391C4⋊C4D3×D911yescharacteristic
392C22⋊C4—92no—
393C22⋊C4—272no—
394C22⋊C4—272no—
395C22⋊C4—272no—
396C22⋊C4—272no—
397C42—31no—
398C42—271no—
399C42—271no—
400C42—272no—
401C42—271no—
402C22×C3—21no—
403C22×C3—92no—
404C22×C3—181no—
405C22×C3—181no—
406C22×C3C23×D911yescharacteristic
407C22×C3C22×D3211yescharacteristic
408C2×D3—32no—
409C2×D3—61no—
410C2×D3—61no—
411C2×D3—184no—
412C2×D3—182no—
413C2×D3—182no—
414C2×D3—182no—
415C2×D3—182no—
416C2×D3—182no—
417C2×D3—182no—
418C2×D3—362no—
419C2×D3—362no—
420C3×C4—22no—
421C3×C4—22no—
422C3×C4—22no—
423C3×C4—42no—
424C3×C4—61no—
425C3×C4—61no—
426C3×C4—62no—
427C3×C4—181no—
428C3×C4—182no—
429C3×C4#370112yes—
430C3×C4#379612yes—
431Dic3—21no—
432Dic3—21no—
433Dic3—22no—
434Dic3—62no—
435Dic3—61no—
436Dic3—92no—
437Dic3—92no—
438Dic3—182no—
439Dic3—181no—
440Dic3—181no—
441Dic3—361no—
442C32#569111yescharacteristic
443C9—21no—
444C9#562411yescharacteristic, complemented
445C23—91no—
446C23—272no—
447Q8—182no—
448D4—544no—
449D4—542no—
450C2×C4—31no—
451C2×C4—31no—
452C2×C4—32no—
453C2×C4—91no—
454C2×C4—92no—
455C2×C4—182no—
456C2×C4—271no—
457C2×C4—271no—
458C2×C4—544no—
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—182no—
467C2×C3C2×#370111yescharacteristic
468C2×C3#41×D911yescharacteristic
469C2×C3#41×D911yescharacteristic
470C2×C3C2×#379611yescharacteristic
471C2×C3#6431211yescharacteristic
472C2×C3D3×#23911yescharacteristic
473D3—62no—
474D3—364no—
475D3—362no—
476D3—362no—
477C22—92no—
478C22—181no—
479C22—181no—
480C22—542no—
481C22—542no—
482C22—542no—
483C22C22×D3×D911yescharacteristic
484C4—22no—
485C4—61no—
486C4—61no—
487C4—62no—
488C4—181no—
489C4—182no—
490C4—272no—
491C4—541no—
492C4#8863412yes—
493C3—21no—
494C3#1058979311yescharacteristic, complemented
495C3#1059445311yescharacteristic
496C2—182no—
497C2—1082no—
498C2C2×#8863411yescharacteristic
499C2#1172249811yescharacteristic
500C2D9×#23911yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC3×C9×C4⋊C4C22
socleC22×C32C23×D3
prime radical#11721004C2
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. #11721749C2semidirect
2. C3×#63459C2semidirect
3. C9×#890C2semidirect
5. #11720988C2semidirect
6. #11720978C2semidirect
7. C4×Dic3×D9C2semidirect
8. #11720952C2semidirect
9. #11720922C2semidirect
10. #11720921C2semidirect
11. #11720910C2semidirect
50. #63459D3semidirect
164. #890D9semidirect
321. C3×C9#529semidirect
444. C9#5624semidirect
494. C3#10589793semidirect

Maximal subgroups

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

subgroupindex
#117217492
C3×#634592
C9×#8902
#117210042
#117209882
#117209782
C4×Dic3×D92
#117209522
#117209222
#117209212
#117209102
#105944533
#105897933