Finite Group Databasebeta

← #13139607

Subgroups — #13139607

Order 1728.

650 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88534C211yesmaximal, characteristic, complemented
2C2×C3×(C2×D9)⋊C4C211yesmaximal, characteristic
3C2×(C2×D9)⋊Dic3C211yesmaximal, characteristic
4C2×(C2×D9)⋊Dic3C211yesmaximal, characteristic
5C2×#88442C211yesmaximal, characteristic, complemented
6C2×Dic9⋊Dic3C211yesmaximal, characteristic, complemented
7C9×C3⋊#95C211yesmaximal, characteristic, complemented
8#10591309—31nomaximal
9#10588356—31nomaximal
10C23×C3×D9C2211yescharacteristic
11C22×C3×C4×C9C2211yescharacteristic
12C22×C3⋊Dic9C2211yescharacteristic
13C22×C3⋊Dic9C2211yescharacteristic
14C22×C9×Dic3C2211yescharacteristic
15C22×C9×Dic3C2211yescharacteristic
16C22×C3×Dic9C2211yescharacteristic
17#88534—22no—
18C3×(C2×D9)⋊C4—22no—
19(C2×D9)⋊Dic3—22no—
20(C2×D9)⋊Dic3—22no—
21#88442—22no—
22Dic9⋊Dic3—22no—
23C2×#3750—31no—
24C2×C3×(C2×D3)⋊C4—31no—
25C2×#3724—31no—
26C2×Dic3⋊Dic3—31no—
27C2×(C2×D3)⋊Dic3—31no—
28C2×(C2×D3)⋊Dic3—31no—
29C3×C3⋊#95D311yescharacteristic
30C2×(C2×D9)⋊C4—31no—
31C2×(C2×D9)⋊C4—31no—
32C2×(C2×D9)⋊C4D311yescharacteristic
33C2×Dic9⋊C4—31no—
34C2×Dic9⋊C4—31no—
35C2×Dic9⋊C4—31no—
36C2×Dic9⋊C4—61no—
37C9×#95—31no—
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×D9Q812yes—
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⋊Dic9—21no—
52C2×C3⋊Dic9—21no—
53C2×C3⋊Dic9D412yes—
54C2×C3⋊Dic9D412yes—
55C2×C9×Dic3—21no—
56C2×C9×Dic3—21no—
57C2×C9×Dic3—21no—
58C2×C9×Dic3—21no—
59C2×C9×Dic3—21no—
60C2×C9×Dic3—21no—
61C2×C3×Dic9—21no—
62C2×C3×Dic9—21no—
63C2×C3×Dic9D412yes—
64#4948—91no—
65#4676—91no—
66C23×C3×D3—31no—
67C22×C32×C4C2×D311yescharacteristic
68C22×C3⋊Dic3—31no—
69C22×C3⋊Dic3—31no—
70C22×C3×Dic3—31no—
71C22×C3×Dic3C2×D311yescharacteristic
72C22×C3×Dic3C2×D311yescharacteristic
73C23×D9C2×D311yescharacteristic
74#3750—62no—
75C3×(C2×D3)⋊C4—62no—
76#3724—62no—
77Dic3⋊Dic3—62no—
78(C2×D3)⋊Dic3—62no—
79(C2×D3)⋊Dic3—62no—
80C22×C4×C9—21no—
81C22×C4×C9—31no—
82C22×C4×C9—31no—
83C22×C4×C9C2×D311yescharacteristic
84C22×Dic9—31no—
85C22×Dic9—31no—
86C22×Dic9—61no—
87C22×Dic9—61no—
88C22×Dic9C2×D311yescharacteristic
89(C2×D9)⋊C4—22no—
90(C2×D9)⋊C4—62no—
91(C2×D9)⋊C4—62no—
92Dic9⋊C4—62no—
93Dic9⋊C4—62no—
94Dic9⋊C4—64no—
95Dic9⋊C4—62no—
96C22×C3×C9C2×D411yescharacteristic
97C22×C3×C9C2×D411yescharacteristic
98C22×C3×C9C2×D411yescharacteristic
99C22×C3×C9C2×Q811yescharacteristic
100C22×C3×C9#4111yescharacteristic
101C22×C3×C9#4111yescharacteristic
102C22×C3×C9#4111yescharacteristic
103C2×C3×D9—22no—
104C2×C3×D9—22no—
105C2×C3×D9—22no—
106C2×C3×D9—41no—
107C2×C3×D9—41no—
108C2×C3×D9—41no—
109C2×C3×D9—41no—
110C3×C4×C9—41no—
111C3⋊Dic9—22no—
112C3⋊Dic9—22no—
113C9×Dic3—41no—
114C9×Dic3—41no—
115C3×Dic9—22no—
116C2×C3×C22⋊C4—91no—
117C2×#217—91no—
118C2×#217—91no—
119C2×(C2×D3)⋊C4—31no—
120C2×(C2×D3)⋊C4—91no—
121C2×(C2×D3)⋊C4—91no—
122C2×C4⋊Dic3—91no—
123C2×Dic3⋊C4—91no—
124C2×Dic3⋊C4—91no—
125C2×Dic3⋊C4—91no—
126C2×Dic3⋊C4—91no—
127C2×Dic3⋊C4—91no—
128C2×Dic3⋊C4—181no—
129C3×#95—31no—
130C3⋊#95D911yescharacteristic, complemented
131C23×C32C22×D311yescharacteristic
132C22×C3×D3—32no—
133C22×C3×D3—61no—
134C22×C3×D3—61no—
135C22×C3×D3—61no—
136C22×C3×D3—61no—
137C22×C3×D3—61no—
138C22×C3×D3—61no—
139C2×C32×C4—21no—
140C2×C32×C4—21no—
141C2×C32×C4—21no—
142C2×C3⋊Dic3—32no—
143C2×C3⋊Dic3—32no—
144C2×C3⋊Dic3—61no—
145C2×C3⋊Dic3—61no—
146C2×C3⋊Dic3—61no—
147C2×C3⋊Dic3—61no—
148C2×C3×Dic3—21no—
149C2×C3×Dic3—21no—
150C2×C3×Dic3—21no—
151C2×C3×Dic3—21no—
152C2×C3×Dic3—21no—
153C2×C3×Dic3—21no—
154C2×C3×Dic3—32no—
155C2×C3×Dic3—61no—
156C2×C3×Dic3—61no—
157C23×C9—21no—
158C23×C9C22×D311yescharacteristic
159C22×D9—21no—
160C22×D9—21no—
161C22×D9—21no—
162C22×D9—21no—
163C22×D9—21no—
164C22×D9—21no—
165C22×D9Dic612yes—
166C2×C4×C9—22no—
167C2×C4×C9—21no—
168C2×C4×C9—21no—
169C2×C4×C9—21no—
170C2×C4×C9—41no—
171C2×C4×C9—41no—
172C2×C4×C9—61no—
173C2×C4×C9—61no—
174C2×C4×C9—61no—
175C2×C4×C9—61no—
176C2×C4×C9—61no—
177C2×C4×C9—61no—
178C2×Dic9—21no—
179C2×Dic9—21no—
180C2×Dic9—32no—
181C2×Dic9—32no—
182C2×Dic9—62no—
183C2×Dic9—62no—
184C2×Dic9—62no—
185C2×Dic9—62no—
186C2×Dic9—61no—
187C2×Dic9—61no—
188C2×Dic9—61no—
189C2×Dic9—61no—
190C2×Dic9—121no—
191C2×Dic9—121no—
192C2×Dic9#6712yes—
193#396—271noSylow
194C2×C3×C9#12111yescharacteristic
195C2×C3×C9#12111yescharacteristic
196C2×C3×C9#12111yescharacteristic
197C2×C3×C9C22⋊Q811yescharacteristic
198C2×C3×C9C22⋊Q811yescharacteristic
199C2×C3×C9C22⋊Q811yescharacteristic
200C2×C3×C9#12611yescharacteristic
201C3×D9—42no—
202C24×C3—91no—
203C23×D3—31no—
204C22×C3×C4—21no—
205C22×C3×C4—31no—
206C22×C3×C4—31no—
207C22×C3×C4—91no—
208C22×C3×C4C2×D911yescharacteristic
209C22×C3×C4D3211yescharacteristic
210C22×Dic3—31no—
211C22×Dic3—91no—
212C22×Dic3—91no—
213C22×Dic3—91no—
214C22×Dic3—91no—
215C22×Dic3—181no—
216C22×Dic3—181no—
217C22×Dic3C2×D911yescharacteristic
218C22×Dic3C2×D911yescharacteristic
219C3×C22⋊C4—182no—
220#217—182no—
221#217—182no—
222(C2×D3)⋊C4—62no—
223(C2×D3)⋊C4—182no—
224(C2×D3)⋊C4—182no—
225C4⋊Dic3—182no—
226Dic3⋊C4—182no—
227Dic3⋊C4—182no—
228Dic3⋊C4—182no—
229Dic3⋊C4—182no—
230Dic3⋊C4—182no—
231Dic3⋊C4—184no—
232C22×C32#23511yescharacteristic
233C22×C32#23511yescharacteristic
234C22×C32#23511yescharacteristic
235C22×C32D3×D411yescharacteristic
236C22×C32D3×D411yescharacteristic
237C22×C32D3×D411yescharacteristic
238C22×C32D3×Q811yescharacteristic
239C2×C3×D3—62no—
240C2×C3×D3—62no—
241C2×C3×D3—62no—
242C2×C3×D3—121no—
243C2×C3×D3—121no—
244C2×C3×D3—121no—
245C2×C3×D3—121no—
246C32×C4—41no—
247C3⋊Dic3—62no—
248C3⋊Dic3—62no—
249C3×Dic3—41no—
250C3×Dic3—41no—
251C3×Dic3—62no—
252C22×C9—21no—
253C22×C9—21no—
254C22×C9—21no—
255C22×C9—21no—
256C22×C9—21no—
257C22×C9—21no—
258C22×C9—21no—
259C22×C9C2×Dic611yescharacteristic
260C22×C9#23511yescharacteristic
261C22×C9D3×D411yescharacteristic
262C22×C9D3×D411yescharacteristic
263C22×C9#23711yescharacteristic
264C22×C9#23711yescharacteristic
265C22×C9C2×#6711yescharacteristic
266C2×D9—22no—
267C2×D9—22no—
268C2×D9—22no—
269C2×D9—41no—
270C2×D9—41no—
271C2×D9—41no—
272C2×D9—41no—
273C4×C9—42no—
274C4×C9—41no—
275C4×C9—121no—
276C4×C9—121no—
277Dic9—22no—
278Dic9—64no—
279Dic9—62no—
280Dic9—62no—
281Dic9—122no—
282C2×C4⋊C4—271no—
283C2×C4⋊C4—271no—
284C2×C4⋊C4—271no—
285C2×C22⋊C4—91no—
286C2×C22⋊C4—271no—
287C2×C22⋊C4—271no—
288#95—31no—
289C3×C9#39611yescharacteristic, complemented, Sylow
290C23×C3—21no—
291C23×C3—92no—
292C23×C3—181no—
293C23×C3—181no—
294C23×C3—181no—
295C23×C3—181no—
296C23×C3—181no—
297C23×C3—181no—
298C23×C3C22×D911yescharacteristic
299C23×C3C2×D3211yescharacteristic
300C22×D3—32no—
301C22×D3—61no—
302C22×D3—61no—
303C22×D3—61no—
304C22×D3—61no—
305C22×D3—61no—
306C22×D3—61no—
307C2×C3×C4—22no—
308C2×C3×C4—21no—
309C2×C3×C4—21no—
310C2×C3×C4—21no—
311C2×C3×C4—21no—
312C2×C3×C4—21no—
313C2×C3×C4—21no—
314C2×C3×C4—41no—
315C2×C3×C4—41no—
316C2×C3×C4—61no—
317C2×C3×C4—61no—
318C2×C3×C4—61no—
319C2×C3×C4—61no—
320C2×C3×C4—61no—
321C2×C3×C4—61no—
322C2×C3×C4—92no—
323C2×C3×C4—181no—
324C2×C3×C4—181no—
325C2×Dic3—21no—
326C2×Dic3—21no—
327C2×Dic3—21no—
328C2×Dic3—21no—
329C2×Dic3—21no—
330C2×Dic3—21no—
331C2×Dic3—32no—
332C2×Dic3—61no—
333C2×Dic3—61no—
334C2×Dic3—92no—
335C2×Dic3—92no—
336C2×Dic3—92no—
337C2×Dic3—92no—
338C2×Dic3—182no—
339C2×Dic3—182no—
340C2×Dic3—182no—
341C2×Dic3—182no—
342C2×Dic3—181no—
343C2×Dic3—181no—
344C2×Dic3—181no—
345C2×Dic3—181no—
346C2×Dic3—181no—
347C2×Dic3—181no—
348C2×Dic3—181no—
349C2×Dic3—181no—
350C2×Dic3—361no—
351C2×Dic3—361no—
352C2×C32#87611yescharacteristic
353C2×C32#88411yescharacteristic
354C2×C32#88411yescharacteristic
355C2×C32#88411yescharacteristic
356C2×C32#89611yescharacteristic
357C2×C32#89611yescharacteristic
358C2×C32#89611yescharacteristic
359C3×D3—122no—
360C2×C9—21no—
361C2×C9—21no—
362C2×C9—21no—
363C2×C9—21no—
364C2×C9—21no—
365C2×C9—21no—
366C2×C9—21no—
367C2×C9#87811yescharacteristic
368C2×C9#87811yescharacteristic
369C2×C9#87911yescharacteristic
370C2×C9#88411yescharacteristic
371C2×C9#92411yescharacteristic
372C2×C9#93811yescharacteristic
373C2×C9#93911yescharacteristic
374D9—42no—
375C24—91no—
376C22×C4—31no—
377C22×C4—31no—
378C22×C4—91no—
379C22×C4—271no—
380C22×C4—271no—
381C22×C4D3×D911yescharacteristic
382C4⋊C4—542no—
383C4⋊C4—542no—
384C4⋊C4—542no—
385C22⋊C4—182no—
386C22⋊C4—542no—
387C22⋊C4—542no—
388C22×C3—21no—
389C22×C3—21no—
390C22×C3—21no—
391C22×C3—21no—
392C22×C3—21no—
393C22×C3—21no—
394C22×C3—21no—
395C22×C3—182no—
396C22×C3—182no—
397C22×C3—182no—
398C22×C3—361no—
399C22×C3—361no—
400C22×C3—361no—
401C22×C3—361no—
402C22×C3#369711yescharacteristic
403C22×C3#369711yescharacteristic
404C22×C3#369711yescharacteristic
405C22×C3D4×D911yescharacteristic
406C22×C3D4×D911yescharacteristic
407C22×C3D4×D911yescharacteristic
408C22×C3Q8×D911yescharacteristic
409C22×C3D3×Dic611yescharacteristic
410C22×C3#379511yescharacteristic
411C22×C3#379811yescharacteristic
412C22×C3#380211yescharacteristic
413C22×C3#380411yescharacteristic
414C22×C3D3×#6711yescharacteristic
415C22×C3#381111yescharacteristic
416C2×D3—62no—
417C2×D3—62no—
418C2×D3—62no—
419C2×D3—121no—
420C2×D3—121no—
421C2×D3—121no—
422C2×D3—121no—
423C3×C4—42no—
424C3×C4—41no—
425C3×C4—41no—
426C3×C4—121no—
427C3×C4—121no—
428C3×C4—182no—
429Dic3—41no—
430Dic3—41no—
431Dic3—62no—
432Dic3—184no—
433Dic3—182no—
434Dic3—182no—
435Dic3—182no—
436Dic3—182no—
437Dic3—362no—
438C32#467611yescharacteristic
439C9—21no—
440C9#494811yescharacteristic, complemented
441C23—92no—
442C23—181no—
443C23—181no—
444C23—181no—
445C23—181no—
446C23—181no—
447C23—181no—
448C23C2×D3×D911yescharacteristic
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—61no—
453C2×C4—61no—
454C2×C4—61no—
455C2×C4—61no—
456C2×C4—61no—
457C2×C4—61no—
458C2×C4—92no—
459C2×C4—181no—
460C2×C4—181no—
461C2×C4—272no—
462C2×C4—272no—
463C2×C4—541no—
464C2×C4—541no—
465C2×C4—541no—
466C2×C4—541no—
467C2×C3—21no—
468C2×C3—21no—
469C2×C3—21no—
470C2×C3—21no—
471C2×C3—21no—
472C2×C3—21no—
473C2×C3—21no—
474C2×C3—362no—
475C2×C3#6343811yescharacteristic
476C2×C3#6344611yescharacteristic
477C2×C3#6344611yescharacteristic
478C2×C3#6344611yescharacteristic
479C2×C3#6345811yescharacteristic
480C2×C3#6345811yescharacteristic
481C2×C3#6345811yescharacteristic
482C2×C3#6385111yescharacteristic
483C2×C3#6385911yescharacteristic
484C2×C3#6388511yescharacteristic
485C2×C3#6389611yescharacteristic
486C2×C3#6389911yescharacteristic
487C2×C3#6391211yescharacteristic
488C2×C3#6395811yescharacteristic
489D3—122no—
490C22—182no—
491C22—182no—
492C22—182no—
493C22—361no—
494C22—361no—
495C22—361no—
496C22—361no—
497C22D9×Dic611yescharacteristic
498C22#8864011yescharacteristic
499C22#8864111yescharacteristic
500C22#8864611yescharacteristic
501C22#8865811yescharacteristic
502C22D9×#6711yescharacteristic
503C22#8866811yescharacteristic
504C4—41no—
505C4—121no—
506C4—121no—
507C4—182no—
508C4—542no—
509C4—542no—
510C3—21no—
511C3#1058835611yescharacteristic, complemented
512C3#1059130911yescharacteristic
513C2—362no—
514C2#1172093911yescharacteristic
515C2#1172094711yescharacteristic
516C2#1172098211yescharacteristic
517C2#1172099611yescharacteristic
518C2#1172099911yescharacteristic
519C2#1172104111yescharacteristic
520C2#1172125611yescharacteristic
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#396
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. C2×#88534C2semidirect
5. C2×#88442C2semidirect
6. C2×Dic9⋊Dic3C2semidirect
7. C9×C3⋊#95C2semidirect
130. C3⋊#95D9semidirect
289. C3×C9#396semidirect
440. C9#4948semidirect
511. C3#10588356semidirect

Maximal subgroups

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

subgroupindex
C2×#885342
C2×C3×(C2×D9)⋊C42
C2×(C2×D9)⋊Dic32
C2×#884422
C2×Dic9⋊Dic32
C9×C3⋊#952
#105913093
#105883563