Finite Group Databasebeta

← #13139855

Subgroups — #13139855

Order 1728.

738 conjugacy classes of subgroups.

The subgroup lattice has 550 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6189C211yesmaximal, characteristic, complemented
2C2×#88536C211yesmaximal, characteristic
3C2×#88445C211yesmaximal, characteristic
4C2×#88445C211yesmaximal, characteristic
5C9×C3⋊#95C211yesmaximal, characteristic, complemented
6C3×C9⋊#95C211yesmaximal, characteristic, complemented
7#11720331C211yesmaximal, characteristic, complemented
8#10591205—31nomaximal
9#10588353—31nomaximal
10C23×C3⋊D9C2211yescharacteristic
11C22×C3×C4×C9C2211yescharacteristic
12C22×C3⋊Dic9C2211yescharacteristic
13C22×C9×Dic3C2211yescharacteristic
14C22×C9×Dic3C2211yescharacteristic
15C22×C3×Dic9C2211yescharacteristic
16C22×C3×Dic9C2211yescharacteristic
17C2×#6189—22no—
18C2×#6189—22no—
19C2×#6189—22no—
20#88536—22no—
21#88445—22no—
22#88445—22no—
23C22×#627—31no—
24C2×#3752—31no—
25C2×#3722—31no—
26C2×#3722—31no—
27C22×Dic18—31no—
28C3×C3⋊#95—31no—
29C3×C3⋊#95D311yescharacteristic
30#63579—31no—
31C2×(C2×D9)⋊C4—31no—
32C2×(C2×D9)⋊C4—31no—
33C2×(C2×D9)⋊C4—31no—
34C2×(C2×D9)⋊C4—61no—
35C9×#95—31no—
36C9⋊#95—31no—
37C9⋊#95D311yescharacteristic, complemented
38C23×C3×C9C2311yescharacteristic
39C22×C3⋊D9—21no—
40C22×C3⋊D9—21no—
41C22×C3⋊D9—21no—
42C22×C3⋊D9—21no—
43C22×C3⋊D9—21no—
44C22×C3⋊D9—21no—
45C22×C3⋊D9Q812yescomplemented
46C2×C3×C4×C9—21no—
47C2×C3×C4×C9—21no—
48C2×C3×C4×C9—21no—
49C2×C3⋊Dic9—21no—
50C2×C3⋊Dic9—21no—
51C2×C3⋊Dic9D412yes—
52C2×C9×Dic3—21no—
53C2×C9×Dic3—21no—
54C2×C9×Dic3—21no—
55C2×C9×Dic3—21no—
56C2×C9×Dic3—21no—
57C2×C9×Dic3D412yes—
58C2×C3×Dic9—21no—
59C2×C3×Dic9—21no—
60C2×C3×Dic9—21no—
61C2×C3×Dic9—21no—
62C2×C3×Dic9—21no—
63C2×C3×Dic9D412yes—
64#6189—28no—
65#4673—91no—
66#4673—91no—
67C23×C3⋊D3—31no—
68C22×C32×C4C2×D311yescharacteristic
69C22×C3⋊Dic3—31no—
70C22×C3×Dic3—31no—
71C22×C3×Dic3—31no—
72C22×C3×Dic3C2×D311yescharacteristic
73C22×C3×Dic3C2×D311yescharacteristic
74C2×#627—62no—
75C2×#627—62no—
76C2×#627—62no—
77C23×D9—31no—
78C23×D9—61no—
79#3752—62no—
80#3722—62no—
81#3722—62no—
82C22×C4×C9—21no—
83C22×C4×C9—31no—
84C22×C4×C9—31no—
85C22×C4×C9C2×D311yescharacteristic
86C22×Dic9—31no—
87C22×Dic9—61no—
88C22×Dic9C2×D311yescharacteristic
89C22×Dic9C2×D311yescharacteristic
90C2×Dic18—62no—
91C2×Dic18—62no—
92C2×Dic18—62no—
93(C2×D9)⋊C4—62no—
94(C2×D9)⋊C4—62no—
95(C2×D9)⋊C4—64no—
96(C2×D9)⋊C4—62no—
97C22×C3×C9C2×D411yescharacteristic
98C22×C3×C9C2×D411yescharacteristic
99C22×C3×C9C2×D411yescharacteristic
100C22×C3×C9C2×Q811yescharacteristic
101C22×C3×C9#4111yescharacteristic
102C22×C3×C9#4111yescharacteristic
103C22×C3×C9#4111yescharacteristic
104C2×C3⋊D9—22no—
105C2×C3⋊D9—22no—
106C2×C3⋊D9—22no—
107C2×C3⋊D9—41no—
108C2×C3⋊D9—41no—
109C2×C3⋊D9—41no—
110C2×C3⋊D9—41no—
111C3×C4×C9—41no—
112C3⋊Dic9—22no—
113C9×Dic3—22no—
114C9×Dic3—41no—
115C3×Dic9—22no—
116C3×Dic9—41no—
117C22×Dic6—91no—
118C22×Dic6—91no—
119C2×(C2×D3)⋊C4—91no—
120C2×(C2×D3)⋊C4—91no—
121C2×(C2×D3)⋊C4—91no—
122C2×(C2×D3)⋊C4—91no—
123C2×(C2×D3)⋊C4—91no—
124C2×(C2×D3)⋊C4—91no—
125C2×(C2×D3)⋊C4—181no—
126C3×#95—31no—
127C3×#95—91no—
128C3⋊#95—31no—
129C3⋊#95—91no—
130C3⋊#95—91no—
131C3⋊#95D911yescharacteristic, complemented
132C23×C32C22×D311yescharacteristic
133C22×C3⋊D3—32no—
134C22×C3⋊D3—61no—
135C22×C3⋊D3—61no—
136C22×C3⋊D3—61no—
137C22×C3⋊D3—61no—
138C22×C3⋊D3—61no—
139C22×C3⋊D3—61no—
140C2×C32×C4—21no—
141C2×C32×C4—21no—
142C2×C32×C4—21no—
143C2×C3⋊Dic3—32no—
144C2×C3⋊Dic3—61no—
145C2×C3⋊Dic3—61no—
146C2×C3×Dic3—21no—
147C2×C3×Dic3—21no—
148C2×C3×Dic3—21no—
149C2×C3×Dic3—21no—
150C2×C3×Dic3—21no—
151C2×C3×Dic3—32no—
152C2×C3×Dic3—61no—
153C2×C3×Dic3—61no—
154C2×C3×Dic3—61no—
155C2×C3×Dic3—61no—
156C2×C3×Dic3—61no—
157C2×C3×Dic3D1212yes—
158#627—68no—
159C23×C9—21no—
160C23×C9C22×D311yescharacteristic
161C22×D9—32no—
162C22×D9—62no—
163C22×D9—62no—
164C22×D9—62no—
165C22×D9—61no—
166C22×D9—61no—
167C22×D9—61no—
168C22×D9—61no—
169C22×D9—61no—
170C22×D9—61no—
171C22×D9—121no—
172C22×D9—121no—
173C22×D9—121no—
174C22×D9—121no—
175C2×C4×C9—22no—
176C2×C4×C9—21no—
177C2×C4×C9—21no—
178C2×C4×C9—21no—
179C2×C4×C9—32no—
180C2×C4×C9—41no—
181C2×C4×C9—41no—
182C2×C4×C9—61no—
183C2×C4×C9—61no—
184C2×C4×C9—61no—
185C2×C4×C9—61no—
186C2×C4×C9—61no—
187C2×Dic9—21no—
188C2×Dic9—21no—
189C2×Dic9—21no—
190C2×Dic9—21no—
191C2×Dic9—21no—
192C2×Dic9—32no—
193C2×Dic9—62no—
194C2×Dic9—61no—
195C2×Dic9—61no—
196C2×Dic9—121no—
197C2×Dic9—121no—
198C2×Dic9D1212yes—
199Dic18—68no—
200#393—271noSylow
201C2×C3×C9#12011yescharacteristic
202C2×C3×C9C22⋊Q811yescharacteristic
203C2×C3×C9C22⋊Q811yescharacteristic
204C2×C3×C9C22⋊Q811yescharacteristic
205C2×C3×C9#12411yescharacteristic
206C2×C3×C9#12411yescharacteristic
207C2×C3×C9#12411yescharacteristic
208C3⋊D9—42no—
209C23×D3—91no—
210C23×D3—91no—
211C23×D3—181no—
212C22×C3×C4—21no—
213C22×C3×C4—31no—
214C22×C3×C4—31no—
215C22×C3×C4—91no—
216C22×C3×C4—91no—
217C22×C3×C4C2×D911yescharacteristic
218C22×C3×C4D3211yescharacteristic
219C22×Dic3—31no—
220C22×Dic3—31no—
221C22×Dic3—91no—
222C22×Dic3—91no—
223C22×Dic3—181no—
224C22×Dic3C2×D911yescharacteristic
225C22×Dic3C2×D911yescharacteristic
226C2×Dic6—182no—
227C2×Dic6—182no—
228C2×Dic6—182no—
229C2×Dic6—182no—
230C2×Dic6—182no—
231C2×Dic6—182no—
232(C2×D3)⋊C4—182no—
233(C2×D3)⋊C4—182no—
234(C2×D3)⋊C4—182no—
235(C2×D3)⋊C4—182no—
236(C2×D3)⋊C4—182no—
237(C2×D3)⋊C4—184no—
238(C2×D3)⋊C4—182no—
239C22×C32C2×D1211yescharacteristic
240C22×C32#23511yescharacteristic
241C22×C32#23511yescharacteristic
242C22×C32D3×D411yescharacteristic
243C22×C32D3×D411yescharacteristic
244C22×C32#23711yescharacteristic
245C22×C32D3×Q811yescharacteristic
246C2×C3⋊D3—62no—
247C2×C3⋊D3—62no—
248C2×C3⋊D3—62no—
249C2×C3⋊D3—121no—
250C2×C3⋊D3—121no—
251C2×C3⋊D3—121no—
252C2×C3⋊D3—121no—
253C32×C4—41no—
254C3⋊Dic3—62no—
255C3×Dic3—22no—
256C3×Dic3—41no—
257C3×Dic3—62no—
258C3×Dic3—121no—
259C22×C9—21no—
260C22×C9—21no—
261C22×C9—21no—
262C22×C9—21no—
263C22×C9—21no—
264C22×C9—21no—
265C22×C9—21no—
266C22×C9C2×D1211yescharacteristic
267C22×C9#23511yescharacteristic
268C22×C9#23511yescharacteristic
269C22×C9D3×D411yescharacteristic
270C22×C9D3×D411yescharacteristic
271C22×C9#23711yescharacteristic
272C22×C9D3×Q811yescharacteristic
273C2×D9—64no—
274C2×D9—62no—
275C2×D9—62no—
276C2×D9—62no—
277C2×D9—122no—
278C2×D9—122no—
279C2×D9—122no—
280C2×D9—122no—
281C2×D9—122no—
282C2×D9—122no—
283C2×D9—121no—
284C2×D9—121no—
285C2×D9—121no—
286C2×D9—121no—
287C4×C9—42no—
288C4×C9—41no—
289C4×C9—62no—
290C4×C9—121no—
291Dic9—22no—
292Dic9—41no—
293Dic9—62no—
294Dic9—122no—
295C22×Q8—271no—
296C2×C22⋊C4—271no—
297C2×C22⋊C4—271no—
298C2×C22⋊C4—271no—
299#95—31no—
300#95—91no—
301#95—271no—
302C3×C9#39311yescharacteristic, complemented, Sylow
303C23×C3—21no—
304C23×C3C22×D911yescharacteristic
305C23×C3C2×D3211yescharacteristic
306C22×D3—92no—
307C22×D3—92no—
308C22×D3—182no—
309C22×D3—182no—
310C22×D3—182no—
311C22×D3—181no—
312C22×D3—181no—
313C22×D3—181no—
314C22×D3—181no—
315C22×D3—181no—
316C22×D3—181no—
317C22×D3—181no—
318C22×D3—181no—
319C22×D3—181no—
320C22×D3—181no—
321C22×D3—181no—
322C22×D3—181no—
323C22×D3—361no—
324C22×D3—361no—
325C22×D3—361no—
326C22×D3—361no—
327C2×C3×C4—22no—
328C2×C3×C4—21no—
329C2×C3×C4—21no—
330C2×C3×C4—21no—
331C2×C3×C4—21no—
332C2×C3×C4—21no—
333C2×C3×C4—21no—
334C2×C3×C4—32no—
335C2×C3×C4—41no—
336C2×C3×C4—41no—
337C2×C3×C4—61no—
338C2×C3×C4—61no—
339C2×C3×C4—61no—
340C2×C3×C4—61no—
341C2×C3×C4—61no—
342C2×C3×C4—92no—
343C2×C3×C4—181no—
344C2×C3×C4—181no—
345C2×C3×C4—181no—
346C2×C3×C4—181no—
347C2×C3×C4—181no—
348C2×Dic3—21no—
349C2×Dic3—21no—
350C2×Dic3—21no—
351C2×Dic3—21no—
352C2×Dic3—21no—
353C2×Dic3—32no—
354C2×Dic3—61no—
355C2×Dic3—61no—
356C2×Dic3—61no—
357C2×Dic3—61no—
358C2×Dic3—61no—
359C2×Dic3—92no—
360C2×Dic3—92no—
361C2×Dic3—182no—
362C2×Dic3—181no—
363C2×Dic3—181no—
364C2×Dic3—181no—
365C2×Dic3—181no—
366C2×Dic3—361no—
367C2×Dic3—361no—
368C2×Dic3D3612yes—
369Dic6—188no—
370Dic6—188no—
371C2×C32#87511yescharacteristic
372C2×C32#88211yescharacteristic
373C2×C32#88511yescharacteristic
374C2×C32#88511yescharacteristic
375C2×C32#89611yescharacteristic
376C2×C32#89611yescharacteristic
377C2×C32#89711yescharacteristic
378C3⋊D3—122no—
379C2×C9—21no—
380C2×C9—21no—
381C2×C9—21no—
382C2×C9—21no—
383C2×C9—21no—
384C2×C9—21no—
385C2×C9—21no—
386C2×C9#87511yescharacteristic
387C2×C9#88211yescharacteristic
388C2×C9#88511yescharacteristic
389C2×C9#88511yescharacteristic
390C2×C9#89611yescharacteristic
391C2×C9#89611yescharacteristic
392C2×C9#89711yescharacteristic
393D9—124no—
394D9—122no—
395C24—271no—
396C2×Q8—542no—
397C2×Q8—542no—
398C2×Q8—542no—
399C22×C4—31no—
400C22×C4—31no—
401C22×C4—91no—
402C22×C4—91no—
403C22×C4—271no—
404C22×C4D3×D911yescharacteristic
405C22⋊C4—542no—
406C22⋊C4—542no—
407C22⋊C4—542no—
408C22×C3—21no—
409C22×C3—21no—
410C22×C3—21no—
411C22×C3—21no—
412C22×C3—21no—
413C22×C3—21no—
414C22×C3—21no—
415C22×C3C2×D3611yescharacteristic
416C22×C3#369711yescharacteristic
417C22×C3#369711yescharacteristic
418C22×C3D4×D911yescharacteristic
419C22×C3D4×D911yescharacteristic
420C22×C3#369911yescharacteristic
421C22×C3Q8×D911yescharacteristic
422C22×C3#379711yescharacteristic
423C22×C3#379811yescharacteristic
424C22×C3D3×D1211yescharacteristic
425C22×C3D3×D1211yescharacteristic
426C22×C3#380411yescharacteristic
427C22×C3#380411yescharacteristic
428C22×C3#381111yescharacteristic
429C2×D3—184no—
430C2×D3—182no—
431C2×D3—182no—
432C2×D3—182no—
433C2×D3—182no—
434C2×D3—182no—
435C2×D3—182no—
436C2×D3—362no—
437C2×D3—362no—
438C2×D3—362no—
439C2×D3—362no—
440C2×D3—362no—
441C2×D3—362no—
442C2×D3—361no—
443C2×D3—361no—
444C2×D3—361no—
445C2×D3—361no—
446C2×D3—361no—
447C2×D3—361no—
448C2×D3—361no—
449C2×D3—361no—
450C3×C4—42no—
451C3×C4—41no—
452C3×C4—41no—
453C3×C4—62no—
454C3×C4—121no—
455C3×C4—182no—
456C3×C4—361no—
457Dic3—22no—
458Dic3—41no—
459Dic3—62no—
460Dic3—121no—
461Dic3—182no—
462Dic3—182no—
463Dic3—362no—
464C32#467311yescharacteristic
465C9—21no—
466C9#467311yescharacteristic, complemented
467C23—272no—
468C23—541no—
469C23—541no—
470C23—541no—
471C23—541no—
472C23—541no—
473C23—541no—
474C23C2×D3×D911yescharacteristic
475Q8—548no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—32no—
480C2×C4—61no—
481C2×C4—61no—
482C2×C4—61no—
483C2×C4—61no—
484C2×C4—61no—
485C2×C4—92no—
486C2×C4—181no—
487C2×C4—181no—
488C2×C4—181no—
489C2×C4—181no—
490C2×C4—181no—
491C2×C4—272no—
492C2×C4—541no—
493C2×C4—541no—
494C2×C3—21no—
495C2×C3—21no—
496C2×C3—21no—
497C2×C3—21no—
498C2×C3—21no—
499C2×C3—21no—
500C2×C3—21no—
501C2×C3#6343711yescharacteristic
502C2×C3#6344411yescharacteristic
503C2×C3#6344711yescharacteristic
504C2×C3#6344711yescharacteristic
505C2×C3#6345811yescharacteristic
506C2×C3#6345811yescharacteristic
507C2×C3#6345911yescharacteristic
508C2×C3#6385211yescharacteristic
509C2×C3#6385211yescharacteristic
510C2×C3#6386511yescharacteristic
511C2×C3#6389511yescharacteristic
512C2×C3#6389511yescharacteristic
513C2×C3#6392311yescharacteristic
514C2×C3#6395311yescharacteristic
515D3—364no—
516D3—362no—
517D3—362no—
518C22—542no—
519C22—542no—
520C22—542no—
521C22—1081no—
522C22—1081no—
523C22—1081no—
524C22—1081no—
525C22(C3⋊D9)⋊Q811yescharacteristic
526C22#8864011yescharacteristic
527C22D3×D3611yescharacteristic
528C22D9×D1211yescharacteristic
529C22#8865811yescharacteristic
530C22#8866211yescharacteristic
531C22#8866811yescharacteristic
532C4—41no—
533C4—62no—
534C4—121no—
535C4—182no—
536C4—361no—
537C4—542no—
538C3—21no—
539C3#1058835311yescharacteristic, complemented
540C3#1059120511yescharacteristic
541C2—1082no—
542C2#1172092311yescharacteristic
543C2#1172094011yescharacteristic
544C2#1172095611yescharacteristic
545C2#1172099411yescharacteristic
546C2#1172099511yescharacteristic
547C2#1172105111yescharacteristic
548C2#1172124211yescharacteristic
subgroupG/H
centerC23C2×D3×D9
commutator, cosocleC23×C3×C9C23
FrattiniC23×C3C2×D32
FittingC22×C3×C4×C9C22
socleC23×C32C22×D3
prime radicalC23×C3⋊D9C22
hypercenterC22×C4D3×D9
hypocenterC3×C9#393
LCSG ▷ C23×C3×C9 ▷ C3×C9
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C22×#6189C2semidirect
5. C9×C3⋊#95C2semidirect
6. C3×C9⋊#95C2semidirect
7. #11720331C2semidirect
37. C9⋊#95D3semidirect
45. C22×C3⋊D9Q8semidirect
131. C3⋊#95D9semidirect
302. C3×C9#393semidirect
466. C9#4673semidirect
539. C3#10588353semidirect

Maximal subgroups

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

subgroupindex
C22×#61892
C2×#885362
C2×#884452
C9×C3⋊#952
C3×C9⋊#952
#117203312
#105912053
#105883533