Finite Group Databasebeta

← #13144268

Subgroups — #13144268

Order 1728.

1290 conjugacy classes of subgroups.

The subgroup lattice has 546 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6233C211yesmaximal, characteristic
2C2×C3×C9×C22⋊C4C211yesmaximal, characteristic, complemented
3C2×#88551C211yesmaximal, characteristic, complemented
4C2×#88536C211yesmaximal, characteristic, complemented
5C2×#88536C211yesmaximal, characteristic, complemented
6C2×#88534C211yesmaximal, characteristic, complemented
7C2×C4×C3⋊Dic9C211yesmaximal, characteristic, complemented
8#10592656—31nomaximal
9#10588637—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
20C3×C9×C22⋊C4—22no—
21#88551—22no—
22#88536—22no—
23#88536—22no—
24#88534—22no—
25C4×C3⋊Dic9—22no—
26C22×#638—31no—
27C2×C32×C22⋊C4D311yescharacteristic
28C2×#3757—31no—
29C2×#3752—31no—
30C2×#3752—31no—
31C2×#3750—31no—
32C2×C4×C3⋊Dic3—31no—
33C22×#611—33no—
34C2×C9×C22⋊C4D313yescomplemented
35C2×#3676—33no—
36C2×(C2×D9)⋊C4—33no—
37C2×(C2×D9)⋊C4—33no—
38C2×Dic9⋊C4—33no—
39C2×C4×Dic9—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×C412yes—
47C23×C3×C9C2311yescharacteristic
48C22×C3⋊D9—21no—
49C22×C3⋊D9—21no—
50C22×C3⋊D9—21no—
51C22×C3⋊D9—21no—
52C22×C3⋊D9—21no—
53C22×C3⋊D9—21no—
54C22×C3⋊D9C2×C412yes—
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⋊Dic9C2×C412yes—
68C2×C3⋊Dic9D412yes—
69C2×C3⋊Dic9D412yes—
70C2×C3⋊Dic9D412yescomplemented
71#4957—93no—
72#4957—91no—
73C24×C32C2×D311yescharacteristic
74C23×C3⋊D3—31no—
75C22×C32×C4C2×D311yescharacteristic
76C22×C32×C4C2×D311yescharacteristic
77C2×#638—34no—
78C2×#638—62no—
79C2×#638—62no—
80C22×C3⋊Dic3—31no—
81C22×C3⋊Dic3—31no—
82C22×C3⋊Dic3—31no—
83C24×C9C2×D313yes—
84C23×D9—33no—
85C32×C22⋊C4—22no—
86#3757—62no—
87#3752—62no—
88#3752—62no—
89#3750—62no—
90C4×C3⋊Dic3—62no—
91C22×C4×C9C2×D313yes—
92C22×C4×C9C2×D313yes—
93C2×#611—312no—
94C2×#611—66no—
95C2×#611—66no—
96C22×Dic9—33no—
97C22×Dic9—33no—
98C22×Dic9—33no—
99C9×C22⋊C4—26no—
100#3676—66no—
101(C2×D9)⋊C4—66no—
102(C2×D9)⋊C4—66no—
103Dic9⋊C4—66no—
104C4×Dic9—66no—
105C22×C3×C9—22no—
106C22×C3×C9—22no—
107C22×C3×C9—22no—
108C22×C3×C9—41no—
109C22×C3×C9—41no—
110C22×C3×C9—41no—
111C22×C3×C9—41no—
112C22×C3×C9C22×C411yescharacteristic
113C22×C3×C9C2×D411yescharacteristic
114C22×C3×C9C2×D411yescharacteristic
115C22×C3×C9C2×D411yescharacteristic
116C22×C3×C9C2×D411yescharacteristic
117C22×C3×C9#4111yescharacteristic
118C22×C3×C9#4111yescharacteristic
119C2×C3⋊D9—22no—
120C2×C3⋊D9—22no—
121C2×C3⋊D9—22no—
122C2×C3⋊D9—41no—
123C2×C3⋊D9—41no—
124C2×C3⋊D9—41no—
125C2×C3⋊D9—41no—
126C3×C4×C9—22no—
127C3×C4×C9—41no—
128C3⋊Dic9—22no—
129C3⋊Dic9—41no—
130C3⋊Dic9C22⋊C414yescomplemented
131C22×#67—93no—
132C22×#67—91no—
133C2×C3×C22⋊C4D913yescomplemented
134C2×C3×C22⋊C4C3⋊D311yescharacteristic
135C2×#217—93no—
136C2×#217—91no—
137C2×(C2×D3)⋊C4—93no—
138C2×(C2×D3)⋊C4—93no—
139C2×(C2×D3)⋊C4—91no—
140C2×(C2×D3)⋊C4—91no—
141C2×Dic3⋊C4—93no—
142C2×Dic3⋊C4—91no—
143C2×C4×Dic3—93no—
144C2×C4×Dic3—91no—
145C23×C32—21no—
146C23×C32—21no—
147C23×C32—21no—
148C23×C32—21no—
149C23×C32—21no—
150C23×C32—21no—
151C23×C32C4×D312yes—
152C23×C32C22×D311yescharacteristic
153C22×C3⋊D3—32no—
154C22×C3⋊D3—61no—
155C22×C3⋊D3—61no—
156C22×C3⋊D3—61no—
157C22×C3⋊D3—61no—
158C22×C3⋊D3—61no—
159C22×C3⋊D3—61no—
160C2×C32×C4—21no—
161C2×C32×C4—21no—
162C2×C32×C4—21no—
163C2×C32×C4—21no—
164C2×C32×C4—21no—
165C2×C32×C4#6712yes—
166#638—68no—
167C2×C3⋊Dic3—32no—
168C2×C3⋊Dic3—32no—
169C2×C3⋊Dic3—32no—
170C2×C3⋊Dic3—32no—
171C2×C3⋊Dic3—61no—
172C2×C3⋊Dic3—61no—
173C2×C3⋊Dic3—61no—
174C2×C3⋊Dic3—61no—
175C2×C3⋊Dic3—61no—
176C23×C9—23no—
177C23×C9—23no—
178C23×C9—23no—
179C23×C9—23no—
180C23×C9—23no—
181C23×C9—23no—
182C23×C9C4×D316yes—
183C23×C9C22×D313yes—
184C22×D9—36no—
185C22×D9—63no—
186C22×D9—63no—
187C22×D9—63no—
188C22×D9—63no—
189C22×D9—63no—
190C22×D9—63no—
191C2×C4×C9—23no—
192C2×C4×C9—23no—
193C2×C4×C9—23no—
194C2×C4×C9—23no—
195C2×C4×C9—23no—
196C2×C4×C9#6716yescomplemented
197#611—624no—
198C2×Dic9—36no—
199C2×Dic9—36no—
200C2×Dic9—36no—
201C2×Dic9—36no—
202C2×Dic9—63no—
203C2×Dic9—63no—
204C2×Dic9—63no—
205C2×Dic9—63no—
206C2×Dic9—63no—
207#390—271noSylow
208C2×C3×C9—42no—
209C2×C3×C9C2×C22⋊C411yescharacteristic
210C2×C3×C9C4×D411yescharacteristic
211C2×C3×C9C4×D411yescharacteristic
212C2×C3×C9#12111yescharacteristic
213C2×C3×C9#12111yescharacteristic
214C2×C3×C9#12411yescharacteristic
215C2×C3×C9#12711yescharacteristic
216C3⋊D9—42no—
217C24×C3C2×D913yes—
218C24×C3C2×C3⋊D311yescharacteristic
219C23×D3—93no—
220C23×D3—91no—
221C22×C3×C4C2×D913yes—
222C22×C3×C4C2×D913yes—
223C22×C3×C4C2×C3⋊D311yescharacteristic
224C22×C3×C4C2×C3⋊D311yescharacteristic
225C2×#67—912no—
226C2×#67—94no—
227C2×#67—182no—
228C2×#67—182no—
229C2×#67—186no—
230C2×#67—186no—
231C22×Dic3—93no—
232C22×Dic3—91no—
233C22×Dic3—93no—
234C22×Dic3—91no—
235C22×Dic3—93no—
236C22×Dic3—91no—
237C3×C22⋊C4—26no—
238C3×C22⋊C4—22no—
239#217—182no—
240#217—186no—
241(C2×D3)⋊C4—182no—
242(C2×D3)⋊C4—186no—
243(C2×D3)⋊C4—186no—
244(C2×D3)⋊C4—182no—
245Dic3⋊C4—182no—
246Dic3⋊C4—186no—
247C4×Dic3—182no—
248C4×Dic3—186no—
249C22×C32—22no—
250C22×C32—22no—
251C22×C32—22no—
252C22×C32—41no—
253C22×C32—41no—
254C22×C32—41no—
255C22×C32—41no—
256C22×C32C2×C4×D311yescharacteristic
257C22×C32#23511yescharacteristic
258C22×C32D3×D411yescharacteristic
259C22×C32D3×D411yescharacteristic
260C22×C32D3×D411yescharacteristic
261C22×C32#23711yescharacteristic
262C22×C32C2×#6711yescharacteristic
263C2×C3⋊D3—62no—
264C2×C3⋊D3—62no—
265C2×C3⋊D3—62no—
266C2×C3⋊D3—121no—
267C2×C3⋊D3—121no—
268C2×C3⋊D3—121no—
269C2×C3⋊D3—121no—
270C32×C4—22no—
271C32×C4—41no—
272C3⋊Dic3—34no—
273C3⋊Dic3—62no—
274C3⋊Dic3—121no—
275C22×C9—26no—
276C22×C9—26no—
277C22×C9—26no—
278C22×C9—43no—
279C22×C9—43no—
280C22×C9—43no—
281C22×C9—43no—
282C22×C9C2×C4×D313yes—
283C22×C9#23513yes—
284C22×C9D3×D413yes—
285C22×C9D3×D413yes—
286C22×C9D3×D413yes—
287C22×C9#23713yes—
288C22×C9C2×#6713yes—
289C2×D9—66no—
290C2×D9—66no—
291C2×D9—66no—
292C2×D9—123no—
293C2×D9—123no—
294C2×D9—123no—
295C2×D9—123no—
296C4×C9—26no—
297C4×C9—43no—
298Dic9—312no—
299Dic9—66no—
300Dic9—123no—
301C22×D4—271no—
302C2×C4⋊C4—271no—
303C2×C22⋊C4—271no—
304C2×C22⋊C4—271no—
305C2×C22⋊C4—271no—
306C2×C22⋊C4C3⋊D911yescharacteristic, complemented
307C2×C42—271no—
308C3×C9#39011yescharacteristic, complemented, Sylow
309C23×C3—23no—
310C23×C3—23no—
311C23×C3—23no—
312C23×C3—23no—
313C23×C3—23no—
314C23×C3—23no—
315C23×C3—21no—
316C23×C3—21no—
317C23×C3—21no—
318C23×C3—21no—
319C23×C3—21no—
320C23×C3—21no—
321C23×C3C4×D916yes—
322C23×C3C22×D913yes—
323C23×C3C4×C3⋊D312yes—
324C23×C3C22×C3⋊D311yescharacteristic
325C22×D3—96no—
326C22×D3—92no—
327C22×D3—183no—
328C22×D3—183no—
329C22×D3—183no—
330C22×D3—183no—
331C22×D3—183no—
332C22×D3—183no—
333C22×D3—181no—
334C22×D3—181no—
335C22×D3—181no—
336C22×D3—181no—
337C22×D3—181no—
338C22×D3—181no—
339C2×C3×C4—23no—
340C2×C3×C4—23no—
341C2×C3×C4—21no—
342C2×C3×C4—21no—
343C2×C3×C4—23no—
344C2×C3×C4—23no—
345C2×C3×C4—23no—
346C2×C3×C4—21no—
347C2×C3×C4—21no—
348C2×C3×C4—21no—
349C2×C3×C4#61116yescomplemented
350C2×C3×C4#63812yes—
351#67—188no—
352#67—1824no—
353C2×Dic3—96no—
354C2×Dic3—96no—
355C2×Dic3—96no—
356C2×Dic3—92no—
357C2×Dic3—92no—
358C2×Dic3—92no—
359C2×Dic3—96no—
360C2×Dic3—92no—
361C2×Dic3—183no—
362C2×Dic3—183no—
363C2×Dic3—181no—
364C2×Dic3—181no—
365C2×Dic3—183no—
366C2×Dic3—183no—
367C2×Dic3—183no—
368C2×Dic3—181no—
369C2×Dic3—181no—
370C2×Dic3—181no—
371C2×C32—42no—
372C2×C32D3×C22⋊C411yescharacteristic
373C2×C32#67⋊C411yescharacteristic
374C2×C32#88411yescharacteristic
375C2×C32#88511yescharacteristic
376C2×C32C4×#6711yescharacteristic
377C2×C32#93911yescharacteristic
378C2×C32#94011yescharacteristic
379C3⋊D3—122no—
380C2×C9—46no—
381C2×C9D3×C22⋊C413yes—
382C2×C9#67⋊C413yes—
383C2×C9#88413yes—
384C2×C9#88513yes—
385C2×C9C4×#6713yes—
386C2×C9#93913yes—
387C2×C9#94013yes—
388D9—126no—
389C24—271no—
390C24C2×C3⋊D911yescharacteristic
391C2×D4—274no—
392C2×D4—542no—
393C2×D4—542no—
394C22×C4—271no—
395C22×C4—271no—
396C22×C4—271no—
397C22×C4C2×C3⋊D911yescharacteristic
398C22×C4C2×C3⋊D911yescharacteristic
399C4⋊C4—542no—
400C22⋊C4—22no—
401C22⋊C4—542no—
402C22⋊C4—542no—
403C22⋊C4—542no—
404C42—542no—
405C22×C3—26no—
406C22×C3—22no—
407C22×C3—22no—
408C22×C3—22no—
409C22×C3—26no—
410C22×C3—26no—
411C22×C3—43no—
412C22×C3—43no—
413C22×C3—41no—
414C22×C3—41no—
415C22×C3—41no—
416C22×C3—41no—
417C22×C3—43no—
418C22×C3—43no—
419C22×C3C2×C4×D913yes—
420C22×C3#369713yes—
421C22×C3D4×D913yes—
422C22×C3D4×D913yes—
423C22×C3D4×D913yes—
424C22×C3#369913yes—
425C22×C3C2×#61113yes—
426C22×C3C2×C4×C3⋊D311yescharacteristic
427C22×C3#382811yescharacteristic
428C22×C3D4×C3⋊D311yescharacteristic
429C22×C3D4×C3⋊D311yescharacteristic
430C22×C3D4×C3⋊D311yescharacteristic
431C22×C3#383011yescharacteristic
432C22×C3C2×#63811yescharacteristic
433C2×D3—186no—
434C2×D3—182no—
435C2×D3—182no—
436C2×D3—182no—
437C2×D3—186no—
438C2×D3—186no—
439C2×D3—363no—
440C2×D3—363no—
441C2×D3—361no—
442C2×D3—361no—
443C2×D3—361no—
444C2×D3—361no—
445C2×D3—363no—
446C2×D3—363no—
447C3×C4—26no—
448C3×C4—22no—
449C3×C4—43no—
450C3×C4—41no—
451Dic3—94no—
452Dic3—912no—
453Dic3—182no—
454Dic3—186no—
455Dic3—363no—
456Dic3—361no—
457C32#495711yescharacteristic
458C9#495713yescomplemented
459C23—21no—
460C23—21no—
461C23—21no—
462C23—21no—
463C23—21no—
464C23—21no—
465C23—272no—
466C23—541no—
467C23—541no—
468C23—541no—
469C23—541no—
470C23—541no—
471C23—541no—
472C23C4×C3⋊D912yes—
473C23C22×C3⋊D911yescharacteristic
474D4—548no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—272no—
481C2×C4—272no—
482C2×C4—272no—
483C2×C4—272no—
484C2×C4—541no—
485C2×C4—541no—
486C2×C4—541no—
487C2×C4—541no—
488C2×C4—541no—
489C2×C4#623312yescomplemented
490C2×C3—42no—
491C2×C3—46no—
492C2×C3C22⋊C4×D913yes—
493C2×C3#611⋊C413yes—
494C2×C3#6344613yes—
495C2×C3#6344713yes—
496C2×C3C4×#61113yes—
497C2×C3#6350113yes—
498C2×C3#6350213yes—
499C2×C3C22⋊C4×C3⋊D311yescharacteristic
500C2×C3#6409011yescharacteristic
501C2×C3#6409311yescharacteristic
502C2×C3#6409411yescharacteristic
503C2×C3C4×#63811yescharacteristic
504C2×C3#6414811yescharacteristic
505C2×C3#6414911yescharacteristic
506D3—362no—
507D3—366no—
508C22—22no—
509C22—22no—
510C22—22no—
511C22—41no—
512C22—41no—
513C22—41no—
514C22—41no—
515C22—542no—
516C22—542no—
517C22—542no—
518C22—1081no—
519C22—1081no—
520C22—1081no—
521C22—1081no—
522C22C2×C4×C3⋊D911yescharacteristic
523C22#8873611yescharacteristic
524C22D4×C3⋊D911yescharacteristic
525C22D4×C3⋊D911yescharacteristic
526C22D4×C3⋊D911yescharacteristic
527C22#8874211yescharacteristic
528C22C2×#623311yescharacteristic
529C4—22no—
530C4—41no—
531C4—274no—
532C4—542no—
533C4—1081no—
534C3#1058863713yescomplemented
535C3#1059265611yescharacteristic
536C2—42no—
537C2—1082no—
538C2C22⋊C4×C3⋊D911yescharacteristic
539C2#1172172711yescharacteristic
540C2#1172173011yescharacteristic
541C2#1172173111yescharacteristic
542C2C4×#623311yescharacteristic
543C2#1172183911yescharacteristic
544C2#1172184011yescharacteristic
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#390
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. C2×C3×C9×C22⋊C4C2semidirect
3. C2×#88551C2semidirect
4. C2×#88536C2semidirect
5. C2×#88536C2semidirect
6. C2×#88534C2semidirect
7. C2×C4×C3⋊Dic9C2semidirect
16. C2×#6233C4semidirect
34. C2×C9×C22⋊C4D3semidirect
60. C2×C3×C4×C9D4semidirect
70. C2×C3⋊Dic9D4semidirect
130. C3⋊Dic9C22⋊C4semidirect
133. C2×C3×C22⋊C4D9semidirect
196. C2×C4×C9#67semidirect
306. C2×C22⋊C4C3⋊D9semidirect
308. C3×C9#390semidirect
349. C2×C3×C4#611semidirect
458. C9#4957semidirect
489. C2×C4#6233semidirect
534. C3#10588637semidirect

Maximal subgroups

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

subgroupindex
C22×#62332
C2×C3×C9×C22⋊C42
C2×#885512
C2×#885362
C2×#885342
C2×C4×C3⋊Dic92
#105926563
#105886373