Finite Group Databasebeta

← #13144282

Subgroups — #13144282

Order 1728.

1430 conjugacy classes of subgroups.

The subgroup lattice has 552 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6233C211yesmaximal, characteristic
2C22×C4×C3⋊D9C211yesmaximal, characteristic, complemented
3C2×C3×C9×C22⋊C4C211yesmaximal, characteristic, complemented
4C2×#88551C211yesmaximal, characteristic, complemented
5C2×#88536C211yesmaximal, characteristic, complemented
6#11720572C211yesmaximal, characteristic, complemented
7#11720572C211yesmaximal, characteristic, complemented
8#10592655—31nomaximal
9#10588636—33nomaximal
10C24×C3×C9C2211yescharacteristic
11C23×C3⋊D9C2211yescharacteristic
12C22×C3×C4×C9C2211yescharacteristic
13C22×C3×C4×C9C2211yescharacteristic
14C2×#6233—22no—
15C2×#6233—22no—
16C2×#6233C414yescomplemented
17C22×C3⋊Dic9C2211yescharacteristic
18C22×C3⋊Dic9C2211yescharacteristic
19C22×C3⋊Dic9C2211yescharacteristic
20C2×C4×C3⋊D9—22no—
21C2×C4×C3⋊D9—22no—
22C2×C4×C3⋊D9—22no—
23C3×C9×C22⋊C4—22no—
24#88551—22no—
25#88536—22no—
26C22×#638—31no—
27C22×C4×C3⋊D3—31no—
28C2×C32×C22⋊C4D311yescharacteristic
29C2×#3757—31no—
30C2×#3752—31no—
31C22×#611—33no—
32C22×C4×D9—33no—
33#63658—31no—
34#63658—31no—
35C2×C9×C22⋊C4D313yescomplemented
36C2×#3676—33no—
37C2×(C2×D9)⋊C4—33no—
38C9⋊#95—33no—
39C9⋊#95—33no—
40C23×C3×C9—21no—
41C23×C3×C9—21no—
42C23×C3×C9—21no—
43C23×C3×C9—21no—
44C23×C3×C9—21no—
45C23×C3×C9—21no—
46C23×C3×C9C2×C412yescomplemented
47C23×C3×C9C2311yescharacteristic
48C22×C3⋊D9—21no—
49C22×C3⋊D9—21no—
50C22×C3⋊D9—21no—
51C22×C3⋊D9—21no—
52C22×C3⋊D9C2×C412yes—
53C22×C3⋊D9D412yes—
54C22×C3⋊D9D412yes—
55C2×C3×C4×C9—21no—
56C2×C3×C4×C9—21no—
57C2×C3×C4×C9—21no—
58C2×C3×C4×C9—21no—
59C2×C3×C4×C9—21no—
60C2×C3×C4×C9D412yescomplemented
61#6233—28no—
62C2×C3⋊Dic9—21no—
63C2×C3⋊Dic9—21no—
64C2×C3⋊Dic9—21no—
65C2×C3⋊Dic9—21no—
66C2×C3⋊Dic9—21no—
67C2×C3⋊Dic9—21no—
68C2×C3⋊Dic9—21no—
69C2×C3⋊Dic9C2×C412yescomplemented
70C2×C3⋊Dic9D412yescomplemented
71C4×C3⋊D9—28no—
72#4956—93no—
73#4956—91no—
74C24×C32C2×D311yescharacteristic
75C23×C3⋊D3—31no—
76C22×C32×C4C2×D311yescharacteristic
77C22×C32×C4C2×D311yescharacteristic
78C2×#638—34no—
79C2×#638—62no—
80C2×#638—62no—
81C22×C3⋊Dic3—31no—
82C22×C3⋊Dic3—31no—
83C22×C3⋊Dic3—31no—
84C2×C4×C3⋊D3—62no—
85C2×C4×C3⋊D3—62no—
86C2×C4×C3⋊D3—62no—
87C24×C9C2×D313yes—
88C23×D9—33no—
89C32×C22⋊C4—22no—
90#3757—62no—
91#3752—62no—
92C22×C4×C9C2×D313yes—
93C22×C4×C9C2×D313yes—
94C2×#611—312no—
95C2×#611—66no—
96C2×#611—66no—
97C22×Dic9—33no—
98C22×Dic9—33no—
99C22×Dic9—33no—
100C2×C4×D9—66no—
101C2×C4×D9—66no—
102C2×C4×D9—66no—
103C9×C22⋊C4—26no—
104#3676—66no—
105(C2×D9)⋊C4—66no—
106C22×C3×C9—22no—
107C22×C3×C9—22no—
108C22×C3×C9—22no—
109C22×C3×C9—41no—
110C22×C3×C9—41no—
111C22×C3×C9—41no—
112C22×C3×C9—41no—
113C22×C3×C9C22×C411yescharacteristic
114C22×C3×C9C2×D411yescharacteristic
115C22×C3×C9C2×D411yescharacteristic
116C22×C3×C9C2×D411yescharacteristic
117C22×C3×C9C2×D411yescharacteristic
118C22×C3×C9#4111yescharacteristic
119C22×C3×C9#4111yescharacteristic
120C2×C3⋊D9—22no—
121C2×C3⋊D9—22no—
122C2×C3⋊D9—22no—
123C2×C3⋊D9—22no—
124C2×C3⋊D9—22no—
125C2×C3⋊D9—22no—
126C2×C3⋊D9C22⋊C414yescomplemented
127C3×C4×C9—22no—
128C3×C4×C9—41no—
129C3⋊Dic9—22no—
130C3⋊Dic9—22no—
131C3⋊Dic9—41no—
132C22×#67—93no—
133C22×#67—91no—
134C22×C4×D3—93no—
135C22×C4×D3—91no—
136C2×C3×C22⋊C4D913yescomplemented
137C2×C3×C22⋊C4C3⋊D311yescharacteristic
138C2×#217—93no—
139C2×#217—91no—
140C2×(C2×D3)⋊C4—93no—
141C2×(C2×D3)⋊C4—91no—
142C3⋊#95—93no—
143C3⋊#95—93no—
144C3⋊#95—91no—
145C3⋊#95—91no—
146C23×C32—21no—
147C23×C32—21no—
148C23×C32—21no—
149C23×C32—21no—
150C23×C32—21no—
151C23×C32—21no—
152C23×C32C4×D312yes—
153C23×C32C22×D311yescharacteristic
154C22×C3⋊D3—32no—
155C22×C3⋊D3—32no—
156C22×C3⋊D3—32no—
157C22×C3⋊D3—61no—
158C22×C3⋊D3—61no—
159C22×C3⋊D3—61no—
160C22×C3⋊D3—61no—
161C2×C32×C4—21no—
162C2×C32×C4—21no—
163C2×C32×C4—21no—
164C2×C32×C4—21no—
165C2×C32×C4—21no—
166C2×C32×C4#6712yes—
167#638—68no—
168C2×C3⋊Dic3—32no—
169C2×C3⋊Dic3—32no—
170C2×C3⋊Dic3—61no—
171C2×C3⋊Dic3—61no—
172C2×C3⋊Dic3—61no—
173C2×C3⋊Dic3—61no—
174C2×C3⋊Dic3—61no—
175C2×C3⋊Dic3—61no—
176C2×C3⋊Dic3—61no—
177C4×C3⋊D3—68no—
178C23×C9—23no—
179C23×C9—23no—
180C23×C9—23no—
181C23×C9—23no—
182C23×C9—23no—
183C23×C9—23no—
184C23×C9C4×D316yescomplemented
185C23×C9C22×D313yes—
186C22×D9—36no—
187C22×D9—36no—
188C22×D9—36no—
189C22×D9—63no—
190C22×D9—63no—
191C22×D9—63no—
192C22×D9—63no—
193C2×C4×C9—23no—
194C2×C4×C9—23no—
195C2×C4×C9—23no—
196C2×C4×C9—23no—
197C2×C4×C9—23no—
198C2×C4×C9#6716yescomplemented
199#611—624no—
200C2×Dic9—36no—
201C2×Dic9—36no—
202C2×Dic9—63no—
203C2×Dic9—63no—
204C2×Dic9—63no—
205C2×Dic9—63no—
206C2×Dic9—63no—
207C2×Dic9—63no—
208C2×Dic9—63no—
209C4×D9—624no—
210#386—271noSylow
211C2×C3×C9—42no—
212C2×C3×C9C2×C22⋊C411yescharacteristic
213C2×C3×C9C4×D411yescharacteristic
214C2×C3×C9C4×D411yescharacteristic
215C2×C3×C9#12011yescharacteristic
216C2×C3×C9#12111yescharacteristic
217C2×C3×C9#12111yescharacteristic
218C2×C3×C9#12311yescharacteristic
219C3⋊D9—24no—
220C24×C3C2×D913yes—
221C24×C3C2×C3⋊D311yescharacteristic
222C23×D3—93no—
223C23×D3—91no—
224C22×C3×C4C2×D913yes—
225C22×C3×C4C2×D913yes—
226C22×C3×C4C2×C3⋊D311yescharacteristic
227C22×C3×C4C2×C3⋊D311yescharacteristic
228C2×#67—912no—
229C2×#67—94no—
230C2×#67—182no—
231C2×#67—182no—
232C2×#67—186no—
233C2×#67—186no—
234C22×Dic3—93no—
235C22×Dic3—93no—
236C22×Dic3—91no—
237C22×Dic3—91no—
238C22×Dic3—93no—
239C22×Dic3—91no—
240C2×C4×D3—182no—
241C2×C4×D3—186no—
242C2×C4×D3—182no—
243C2×C4×D3—186no—
244C2×C4×D3—186no—
245C2×C4×D3—182no—
246C3×C22⋊C4—26no—
247C3×C22⋊C4—22no—
248#217—182no—
249#217—186no—
250(C2×D3)⋊C4—182no—
251(C2×D3)⋊C4—186no—
252C22×C32—22no—
253C22×C32—22no—
254C22×C32—22no—
255C22×C32—41no—
256C22×C32—41no—
257C22×C32—41no—
258C22×C32—41no—
259C22×C32C2×C4×D311yescharacteristic
260C22×C32#23511yescharacteristic
261C22×C32D3×D411yescharacteristic
262C22×C32D3×D411yescharacteristic
263C22×C32D3×D411yescharacteristic
264C22×C32#23711yescharacteristic
265C22×C32C2×#6711yescharacteristic
266C2×C3⋊D3—34no—
267C2×C3⋊D3—62no—
268C2×C3⋊D3—62no—
269C2×C3⋊D3—62no—
270C2×C3⋊D3—62no—
271C2×C3⋊D3—62no—
272C2×C3⋊D3—62no—
273C32×C4—22no—
274C32×C4—41no—
275C3⋊Dic3—62no—
276C3⋊Dic3—62no—
277C3⋊Dic3—121no—
278C22×C9—26no—
279C22×C9—26no—
280C22×C9—26no—
281C22×C9—43no—
282C22×C9—43no—
283C22×C9—43no—
284C22×C9—43no—
285C22×C9C2×C4×D313yes—
286C22×C9#23513yes—
287C22×C9D3×D413yes—
288C22×C9D3×D413yes—
289C22×C9D3×D413yes—
290C22×C9#23713yes—
291C22×C9C2×#6713yes—
292C2×D9—312no—
293C2×D9—66no—
294C2×D9—66no—
295C2×D9—66no—
296C2×D9—66no—
297C2×D9—66no—
298C2×D9—66no—
299C4×C9—26no—
300C4×C9—43no—
301Dic9—66no—
302Dic9—66no—
303Dic9—123no—
304C22×D4—271no—
305C23×C4—271no—
306C2×C22⋊C4—271no—
307C2×C22⋊C4—271no—
308C2×C22⋊C4C3⋊D911yescharacteristic, complemented
309#95—271no—
310#95—271no—
311C3×C9#38611yescharacteristic, complemented, Sylow
312C23×C3—23no—
313C23×C3—23no—
314C23×C3—23no—
315C23×C3—23no—
316C23×C3—23no—
317C23×C3—23no—
318C23×C3—21no—
319C23×C3—21no—
320C23×C3—21no—
321C23×C3—21no—
322C23×C3—21no—
323C23×C3—21no—
324C23×C3C4×D916yescomplemented
325C23×C3C22×D913yes—
326C23×C3C4×C3⋊D312yes—
327C23×C3C22×C3⋊D311yescharacteristic
328C22×D3—96no—
329C22×D3—96no—
330C22×D3—96no—
331C22×D3—92no—
332C22×D3—92no—
333C22×D3—92no—
334C22×D3—183no—
335C22×D3—183no—
336C22×D3—183no—
337C22×D3—183no—
338C22×D3—181no—
339C22×D3—181no—
340C22×D3—181no—
341C22×D3—181no—
342C2×C3×C4—23no—
343C2×C3×C4—23no—
344C2×C3×C4—21no—
345C2×C3×C4—21no—
346C2×C3×C4—23no—
347C2×C3×C4—23no—
348C2×C3×C4—23no—
349C2×C3×C4—21no—
350C2×C3×C4—21no—
351C2×C3×C4—21no—
352C2×C3×C4#61116yescomplemented
353C2×C3×C4#63812yes—
354#67—188no—
355#67—1824no—
356C2×Dic3—96no—
357C2×Dic3—96no—
358C2×Dic3—92no—
359C2×Dic3—92no—
360C2×Dic3—183no—
361C2×Dic3—183no—
362C2×Dic3—183no—
363C2×Dic3—183no—
364C2×Dic3—181no—
365C2×Dic3—181no—
366C2×Dic3—181no—
367C2×Dic3—181no—
368C2×Dic3—183no—
369C2×Dic3—183no—
370C2×Dic3—183no—
371C2×Dic3—181no—
372C2×Dic3—181no—
373C2×Dic3—181no—
374C4×D3—188no—
375C4×D3—1824no—
376C2×C32—42no—
377C2×C32D3×C22⋊C411yescharacteristic
378C2×C32#67⋊C411yescharacteristic
379C2×C32#88311yescharacteristic
380C2×C32#88411yescharacteristic
381C2×C32C4×#6711yescharacteristic
382C2×C32#93711yescharacteristic
383C2×C32#93811yescharacteristic
384C3⋊D3—64no—
385C2×C9—46no—
386C2×C9D3×C22⋊C413yes—
387C2×C9#67⋊C413yes—
388C2×C9#88313yes—
389C2×C9#88413yes—
390C2×C9C4×#6713yes—
391C2×C9#93713yes—
392C2×C9#93813yes—
393D9—612no—
394C24—271no—
395C24C2×C3⋊D911yescharacteristic
396C2×D4—274no—
397C2×D4—542no—
398C2×D4—542no—
399C22×C4—271no—
400C22×C4—271no—
401C22×C4—271no—
402C22×C4—542no—
403C22×C4—542no—
404C22×C4—542no—
405C22×C4C2×C3⋊D911yescharacteristic
406C22×C4C2×C3⋊D911yescharacteristic
407C22⋊C4—22no—
408C22⋊C4—542no—
409C22⋊C4—542no—
410C22×C3—26no—
411C22×C3—22no—
412C22×C3—22no—
413C22×C3—22no—
414C22×C3—26no—
415C22×C3—26no—
416C22×C3—43no—
417C22×C3—43no—
418C22×C3—41no—
419C22×C3—41no—
420C22×C3—41no—
421C22×C3—41no—
422C22×C3—43no—
423C22×C3—43no—
424C22×C3C2×C4×D913yes—
425C22×C3#369713yes—
426C22×C3D4×D913yes—
427C22×C3D4×D913yes—
428C22×C3D4×D913yes—
429C22×C3#369913yes—
430C22×C3C2×#61113yes—
431C22×C3C2×C4×C3⋊D311yescharacteristic
432C22×C3#382811yescharacteristic
433C22×C3D4×C3⋊D311yescharacteristic
434C22×C3D4×C3⋊D311yescharacteristic
435C22×C3D4×C3⋊D311yescharacteristic
436C22×C3#383011yescharacteristic
437C22×C3C2×#63811yescharacteristic
438C2×D3—94no—
439C2×D3—912no—
440C2×D3—186no—
441C2×D3—186no—
442C2×D3—186no—
443C2×D3—182no—
444C2×D3—182no—
445C2×D3—182no—
446C2×D3—182no—
447C2×D3—182no—
448C2×D3—182no—
449C2×D3—186no—
450C2×D3—186no—
451C2×D3—186no—
452C3×C4—26no—
453C3×C4—22no—
454C3×C4—43no—
455C3×C4—41no—
456Dic3—186no—
457Dic3—182no—
458Dic3—182no—
459Dic3—186no—
460Dic3—361no—
461Dic3—363no—
462C32#495611yescharacteristic
463C9#495613yescomplemented
464C23—21no—
465C23—21no—
466C23—21no—
467C23—21no—
468C23—21no—
469C23—21no—
470C23—272no—
471C23—272no—
472C23—272no—
473C23—541no—
474C23—541no—
475C23—541no—
476C23—541no—
477C23C4×C3⋊D912yescomplemented
478C23C22×C3⋊D911yescharacteristic
479D4—548no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—272no—
486C2×C4—272no—
487C2×C4—548no—
488C2×C4—541no—
489C2×C4—541no—
490C2×C4—541no—
491C2×C4—541no—
492C2×C4—541no—
493C2×C4—541no—
494C2×C4—541no—
495C2×C4#623312yescomplemented
496C2×C3—42no—
497C2×C3—46no—
498C2×C3C22⋊C4×D913yes—
499C2×C3#611⋊C413yes—
500C2×C3#6344513yes—
501C2×C3#6344613yes—
502C2×C3C4×#61113yes—
503C2×C3#6349913yes—
504C2×C3#6350013yes—
505C2×C3C22⋊C4×C3⋊D311yescharacteristic
506C2×C3#6409011yescharacteristic
507C2×C3#6409211yescharacteristic
508C2×C3#6409311yescharacteristic
509C2×C3C4×#63811yescharacteristic
510C2×C3#6414611yescharacteristic
511C2×C3#6414711yescharacteristic
512D3—184no—
513D3—1812no—
514C22—22no—
515C22—22no—
516C22—22no—
517C22—41no—
518C22—41no—
519C22—41no—
520C22—41no—
521C22—274no—
522C22—542no—
523C22—542no—
524C22—542no—
525C22—542no—
526C22—542no—
527C22—542no—
528C22C2×C4×C3⋊D911yescharacteristic
529C22#8873611yescharacteristic
530C22D4×C3⋊D911yescharacteristic
531C22D4×C3⋊D911yescharacteristic
532C22D4×C3⋊D911yescharacteristic
533C22#8874211yescharacteristic
534C22C2×#623311yescharacteristic
535C4—22no—
536C4—41no—
537C4—542no—
538C4—542no—
539C4—1081no—
540C3#1058863613yescomplemented
541C3#1059265511yescharacteristic
542C2—42no—
543C2—544no—
544C2C22⋊C4×C3⋊D911yescharacteristic
545C2#1172172711yescharacteristic
546C2#1172172911yescharacteristic
547C2#1172173011yescharacteristic
548C2C4×#623311yescharacteristic
549C2#1172183711yescharacteristic
550C2#1172183811yescharacteristic
subgroupG/H
centerC23C22×C3⋊D9
commutatorC22×C3×C9C22×C4
FrattiniC23×C3C22×C3⋊D3
FittingC2×C3×C9×C22⋊C4C2
socleC23×C32C22×D3
prime radicalC22×#6233C2
cosocleC23×C3×C9C23
hypercenterC2×C22⋊C4C3⋊D9
hypocenterC3×C9#386
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C23 ◁ C2×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C22×C4×C3⋊D9C2semidirect
3. C2×C3×C9×C22⋊C4C2semidirect
4. C2×#88551C2semidirect
5. C2×#88536C2semidirect
6. #11720572C2semidirect
7. #11720572C2semidirect
16. C2×#6233C4semidirect
35. C2×C9×C22⋊C4D3semidirect
46. C23×C3×C9C2×C4semidirect
60. C2×C3×C4×C9D4semidirect
69. C2×C3⋊Dic9C2×C4semidirect
70. C2×C3⋊Dic9D4semidirect
126. C2×C3⋊D9C22⋊C4semidirect
136. C2×C3×C22⋊C4D9semidirect
184. C23×C9C4×D3semidirect
198. C2×C4×C9#67semidirect
308. C2×C22⋊C4C3⋊D9semidirect
311. C3×C9#386semidirect
324. C23×C3C4×D9semidirect
352. C2×C3×C4#611semidirect
463. C9#4956semidirect
477. C23C4×C3⋊D9semidirect
495. C2×C4#6233semidirect
540. C3#10588636semidirect

Maximal subgroups

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

subgroupindex
C22×#62332
C22×C4×C3⋊D92
C2×C3×C9×C22⋊C42
C2×#885512
C2×#885362
#117205722
#105926553
#105886363