Finite Group Databasebeta

← #13139710

Subgroups — #13139710

Order 1728.

702 conjugacy classes of subgroups.

The subgroup lattice has 538 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88536C211yesmaximal, characteristic
2C2×C4×C9×Dic3C211yesmaximal, characteristic, complemented
3C2×#88445C211yesmaximal, characteristic
4C2×#88445C211yesmaximal, characteristic
5C2×Dic9⋊Dic3C211yesmaximal, characteristic, complemented
6C3×C9⋊#95C211yesmaximal, characteristic, complemented
7#11720331C211yesmaximal, characteristic, complemented
8#10591194—31nomaximal
9#10588620—31nomaximal
10C23×C3⋊D9C2211yescharacteristic
11C22×C3×C4×C9C2211yescharacteristic
12C22×C3⋊Dic9C2211yescharacteristic
13C22×C9×Dic3C2211yescharacteristic
14C22×C9×Dic3C2211yescharacteristic
15C22×C3×Dic9C2211yescharacteristic
16C22×C3×Dic9C2211yescharacteristic
17#88536—22no—
18C4×C9×Dic3—22no—
19#88445—22no—
20#88445C414yescomplemented
21Dic9⋊Dic3—22no—
22C2×#3752—31no—
23C2×C3×C4×Dic3D311yescharacteristic
24C2×Dic3⋊Dic3—31no—
25C2×#3722—31no—
26C2×#3722—31no—
27C3×C3⋊#95—31no—
28#63579—31no—
29C2×C42×C9—31no—
30C2×(C2×D9)⋊C4—31no—
31C2×(C2×D9)⋊C4—31no—
32C2×(C2×D9)⋊C4—31no—
33C2×(C2×D9)⋊C4—61no—
34C2×Dic9⋊C4—31no—
35C9⋊#95—31no—
36C9⋊#95D311yescharacteristic, complemented
37C23×C3×C9C2311yescharacteristic
38C22×C3⋊D9—21no—
39C22×C3⋊D9—21no—
40C22×C3⋊D9—21no—
41C22×C3⋊D9—21no—
42C22×C3⋊D9—21no—
43C22×C3⋊D9—21no—
44C22×C3⋊D9C2×C412yes—
45C2×C3×C4×C9—21no—
46C2×C3×C4×C9—21no—
47C2×C3×C4×C9D412yes—
48C2×C3⋊Dic9—21no—
49C2×C3⋊Dic9—21no—
50C2×C3⋊Dic9—21no—
51C2×C9×Dic3—21no—
52C2×C9×Dic3—21no—
53C2×C9×Dic3—21no—
54C2×C9×Dic3—21no—
55C2×C9×Dic3C2×C412yes—
56C2×C9×Dic3D412yes—
57C2×C3×Dic9—21no—
58C2×C3×Dic9—21no—
59C2×C3×Dic9—21no—
60C2×C3×Dic9—21no—
61C2×C3×Dic9—21no—
62C2×C3×Dic9C2×C412yes—
63#4940—91no—
64#4670—91no—
65C23×C3⋊D3—31no—
66C22×C32×C4C2×D311yescharacteristic
67C22×C3⋊Dic3—31no—
68C22×C3×Dic3—31no—
69C22×C3×Dic3—31no—
70C22×C3×Dic3C2×D311yescharacteristic
71C22×C3×Dic3C2×D311yescharacteristic
72C23×D9—31no—
73C23×D9—61no—
74#3752—62no—
75C3×C4×Dic3—22no—
76Dic3⋊Dic3—62no—
77#3722—34no—
78#3722—62no—
79C22×C4×C9—21no—
80C22×C4×C9—31no—
81C22×C4×C9—31no—
82C22×C4×C9C2×D311yescharacteristic
83C22×Dic9—31no—
84C22×Dic9—61no—
85C22×Dic9C2×D311yescharacteristic
86C22×Dic9C2×D311yescharacteristic
87C42×C9—62no—
88(C2×D9)⋊C4—34no—
89(C2×D9)⋊C4—62no—
90(C2×D9)⋊C4—64no—
91(C2×D9)⋊C4—62no—
92Dic9⋊C4—62no—
93C22×C3×C9C22×C411yescharacteristic
94C22×C3×C9C2×D411yescharacteristic
95C22×C3×C9C2×D411yescharacteristic
96C22×C3×C9#4111yescharacteristic
97C22×C3×C9#4111yescharacteristic
98C22×C3×C9#4111yescharacteristic
99C22×C3×C9#4111yescharacteristic
100C2×C3⋊D9—22no—
101C2×C3⋊D9—22no—
102C2×C3⋊D9—22no—
103C2×C3⋊D9—41no—
104C2×C3⋊D9—41no—
105C2×C3⋊D9—41no—
106C2×C3⋊D9—41no—
107C3×C4×C9—22no—
108C3⋊Dic9—41no—
109C9×Dic3—22no—
110C9×Dic3—22no—
111C3×Dic9—22no—
112C3×Dic9—41no—
113C2×C3×C42—31no—
114C2×(C2×D3)⋊C4—91no—
115C2×(C2×D3)⋊C4—91no—
116C2×(C2×D3)⋊C4—91no—
117C2×(C2×D3)⋊C4—91no—
118C2×(C2×D3)⋊C4—91no—
119C2×(C2×D3)⋊C4—91no—
120C2×(C2×D3)⋊C4—181no—
121C2×C4⋊Dic3—91no—
122C2×Dic3⋊C4—91no—
123C2×C4×Dic3D911yescharacteristic, complemented
124C3×#95—91no—
125C3⋊#95—31no—
126C3⋊#95—91no—
127C3⋊#95—91no—
128C23×C32C22×D311yescharacteristic
129C22×C3⋊D3—32no—
130C22×C3⋊D3—61no—
131C22×C3⋊D3—61no—
132C22×C3⋊D3—61no—
133C22×C3⋊D3—61no—
134C22×C3⋊D3—61no—
135C22×C3⋊D3—61no—
136C2×C32×C4—21no—
137C2×C32×C4—21no—
138C2×C32×C4#6712yes—
139C2×C3⋊Dic3—61no—
140C2×C3⋊Dic3—61no—
141C2×C3⋊Dic3—61no—
142C2×C3×Dic3—21no—
143C2×C3×Dic3—21no—
144C2×C3×Dic3—21no—
145C2×C3×Dic3—21no—
146C2×C3×Dic3—32no—
147C2×C3×Dic3—61no—
148C2×C3×Dic3—61no—
149C2×C3×Dic3—61no—
150C2×C3×Dic3—61no—
151C2×C3×Dic3—61no—
152C2×C3×Dic3C4×D312yes—
153C2×C3×Dic3D1212yes—
154C23×C9—21no—
155C23×C9C22×D311yescharacteristic
156C22×D9—32no—
157C22×D9—62no—
158C22×D9—62no—
159C22×D9—62no—
160C22×D9—61no—
161C22×D9—61no—
162C22×D9—61no—
163C22×D9—61no—
164C22×D9—61no—
165C22×D9—61no—
166C22×D9—121no—
167C22×D9—121no—
168C22×D9—121no—
169C22×D9—121no—
170C2×C4×C9—21no—
171C2×C4×C9—21no—
172C2×C4×C9—22no—
173C2×C4×C9—32no—
174C2×C4×C9—32no—
175C2×C4×C9—41no—
176C2×C4×C9—41no—
177C2×C4×C9—61no—
178C2×C4×C9—61no—
179C2×C4×C9—61no—
180C2×C4×C9—61no—
181C2×C4×C9D1212yes—
182C2×Dic9—21no—
183C2×Dic9—21no—
184C2×Dic9—21no—
185C2×Dic9—21no—
186C2×Dic9—21no—
187C2×Dic9—62no—
188C2×Dic9—61no—
189C2×Dic9—61no—
190C2×Dic9—61no—
191C2×Dic9—121no—
192C2×Dic9—121no—
193C2×Dic9C4×D312yes—
194#388—271noSylow
195C2×C3×C9#11711yescharacteristic
196C2×C3×C9C4×D411yescharacteristic
197C2×C3×C9C4×D411yescharacteristic
198C2×C3×C9#12111yescharacteristic
199C2×C3×C9#12311yescharacteristic
200C2×C3×C9#12411yescharacteristic
201C2×C3×C9#12611yescharacteristic
202C3⋊D9—42no—
203C23×D3—91no—
204C23×D3—91no—
205C23×D3—181no—
206C22×C3×C4—21no—
207C22×C3×C4—31no—
208C22×C3×C4—31no—
209C22×C3×C4—91no—
210C22×C3×C4—91no—
211C22×C3×C4C2×D911yescharacteristic
212C22×C3×C4D3211yescharacteristic
213C22×Dic3—31no—
214C22×Dic3—31no—
215C22×Dic3—91no—
216C22×Dic3—91no—
217C22×Dic3—181no—
218C22×Dic3C2×D911yescharacteristic
219C22×Dic3C2×D911yescharacteristic
220C3×C42—62no—
221(C2×D3)⋊C4—94no—
222(C2×D3)⋊C4—94no—
223(C2×D3)⋊C4—182no—
224(C2×D3)⋊C4—184no—
225(C2×D3)⋊C4—182no—
226(C2×D3)⋊C4—182no—
227(C2×D3)⋊C4—182no—
228C4⋊Dic3—182no—
229Dic3⋊C4—182no—
230C4×Dic3—22no—
231C22×C32C2×C4×D311yescharacteristic
232C22×C32C2×D1211yescharacteristic
233C22×C32#23511yescharacteristic
234C22×C32#23511yescharacteristic
235C22×C32#23511yescharacteristic
236C22×C32#23511yescharacteristic
237C22×C32C2×#6711yescharacteristic
238C2×C3⋊D3—62no—
239C2×C3⋊D3—62no—
240C2×C3⋊D3—62no—
241C2×C3⋊D3—121no—
242C2×C3⋊D3—121no—
243C2×C3⋊D3—121no—
244C2×C3⋊D3—121no—
245C32×C4—22no—
246C3⋊Dic3—121no—
247C3×Dic3—22no—
248C3×Dic3—22no—
249C3×Dic3—62no—
250C3×Dic3—121no—
251C22×C9—21no—
252C22×C9—21no—
253C22×C9—21no—
254C22×C9—21no—
255C22×C9—21no—
256C22×C9—21no—
257C22×C9—21no—
258C22×C9C2×C4×D311yescharacteristic
259C22×C9C2×D1211yescharacteristic
260C22×C9#23511yescharacteristic
261C22×C9D3×D411yescharacteristic
262C22×C9#23711yescharacteristic
263C22×C9#23711yescharacteristic
264C22×C9#23911yescharacteristic
265C2×D9—64no—
266C2×D9—62no—
267C2×D9—62no—
268C2×D9—62no—
269C2×D9—122no—
270C2×D9—122no—
271C2×D9—122no—
272C2×D9—122no—
273C2×D9—122no—
274C2×D9—122no—
275C2×D9—121no—
276C2×D9—121no—
277C2×D9—121no—
278C2×D9—121no—
279C4×C9—22no—
280C4×C9—42no—
281C4×C9—62no—
282C4×C9—62no—
283Dic9—22no—
284Dic9—41no—
285Dic9—122no—
286Dic9—121no—
287C2×C4⋊C4—271no—
288C2×C22⋊C4—271no—
289C2×C22⋊C4—271no—
290C2×C22⋊C4—271no—
291C2×C42—31no—
292#95—91no—
293#95—271no—
294C3×C9#38811yescharacteristic, complemented, Sylow
295C23×C3—21no—
296C23×C3C22×D911yescharacteristic
297C23×C3C2×D3211yescharacteristic
298C22×D3—92no—
299C22×D3—92no—
300C22×D3—182no—
301C22×D3—182no—
302C22×D3—182no—
303C22×D3—181no—
304C22×D3—181no—
305C22×D3—181no—
306C22×D3—181no—
307C22×D3—181no—
308C22×D3—181no—
309C22×D3—181no—
310C22×D3—181no—
311C22×D3—181no—
312C22×D3—181no—
313C22×D3—181no—
314C22×D3—181no—
315C22×D3—361no—
316C22×D3—361no—
317C22×D3—361no—
318C22×D3—361no—
319C2×C3×C4—21no—
320C2×C3×C4—21no—
321C2×C3×C4—21no—
322C2×C3×C4—21no—
323C2×C3×C4—22no—
324C2×C3×C4—32no—
325C2×C3×C4—32no—
326C2×C3×C4—41no—
327C2×C3×C4—41no—
328C2×C3×C4—61no—
329C2×C3×C4—61no—
330C2×C3×C4—61no—
331C2×C3×C4—61no—
332C2×C3×C4—92no—
333C2×C3×C4—181no—
334C2×C3×C4—181no—
335C2×C3×C4—181no—
336C2×C3×C4—181no—
337C2×C3×C4—181no—
338C2×C3×C4#61112yes—
339C2×C3×C4#62612yes—
340C2×Dic3—21no—
341C2×Dic3—21no—
342C2×Dic3—21no—
343C2×Dic3—21no—
344C2×Dic3—32no—
345C2×Dic3—61no—
346C2×Dic3—61no—
347C2×Dic3—61no—
348C2×Dic3—61no—
349C2×Dic3—61no—
350C2×Dic3—182no—
351C2×Dic3—181no—
352C2×Dic3—181no—
353C2×Dic3—181no—
354C2×Dic3—181no—
355C2×Dic3—181no—
356C2×Dic3—181no—
357C2×Dic3—361no—
358C2×Dic3—361no—
359C2×Dic3C4×D912yes—
360C2×Dic3D3612yes—
361C2×C32#87211yescharacteristic
362C2×C32C4×D1211yescharacteristic
363C2×C32#87511yescharacteristic
364C2×C32#87611yescharacteristic
365C2×C32C4×#6711yescharacteristic
366C2×C32#92911yescharacteristic
367C2×C32#93011yescharacteristic
368C3⋊D3—122no—
369C2×C9—21no—
370C2×C9—21no—
371C2×C9—21no—
372C2×C9—21no—
373C2×C9—21no—
374C2×C9—21no—
375C2×C9—21no—
376C2×C9C4×D1211yescharacteristic
377C2×C9#67⋊C411yescharacteristic
378C2×C9#88511yescharacteristic
379C2×C9#88611yescharacteristic
380C2×C9#89211yescharacteristic
381C2×C9#89511yescharacteristic
382C2×C9#89811yescharacteristic
383D9—124no—
384D9—122no—
385C24—271no—
386C22×C4—31no—
387C22×C4—31no—
388C22×C4—91no—
389C22×C4—91no—
390C22×C4—271no—
391C22×C4D3×D911yescharacteristic
392C4⋊C4—542no—
393C22⋊C4—274no—
394C22⋊C4—542no—
395C22⋊C4—542no—
396C42—62no—
397C22×C3—21no—
398C22×C3—21no—
399C22×C3—21no—
400C22×C3—21no—
401C22×C3—21no—
402C22×C3—21no—
403C22×C3—21no—
404C22×C3C2×C4×D911yescharacteristic
405C22×C3C2×D3611yescharacteristic
406C22×C3#369711yescharacteristic
407C22×C3#369711yescharacteristic
408C22×C3#369711yescharacteristic
409C22×C3#369711yescharacteristic
410C22×C3C2×#61111yescharacteristic
411C22×C3#379811yescharacteristic
412C22×C3#379911yescharacteristic
413C22×C3C4×D3211yescharacteristic
414C22×C3D3×D1211yescharacteristic
415C22×C3#380411yescharacteristic
416C22×C3#380411yescharacteristic
417C22×C3C2×#62611yescharacteristic
418C2×D3—184no—
419C2×D3—182no—
420C2×D3—182no—
421C2×D3—182no—
422C2×D3—182no—
423C2×D3—182no—
424C2×D3—182no—
425C2×D3—362no—
426C2×D3—362no—
427C2×D3—362no—
428C2×D3—362no—
429C2×D3—362no—
430C2×D3—362no—
431C2×D3—361no—
432C2×D3—361no—
433C2×D3—361no—
434C2×D3—361no—
435C2×D3—361no—
436C2×D3—361no—
437C2×D3—361no—
438C2×D3—361no—
439C3×C4—22no—
440C3×C4—22no—
441C3×C4—42no—
442C3×C4—62no—
443C3×C4—62no—
444C3×C4—182no—
445C3×C4—361no—
446Dic3—22no—
447Dic3—22no—
448Dic3—62no—
449Dic3—121no—
450Dic3—362no—
451Dic3—361no—
452Dic3—361no—
453C32#494011yescharacteristic
454C9—21no—
455C9#467011yescharacteristic, complemented
456C23—272no—
457C23—541no—
458C23—541no—
459C23—541no—
460C23—541no—
461C23—541no—
462C23—541no—
463C23C2×D3×D911yescharacteristic
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—32no—
467C2×C4—32no—
468C2×C4—61no—
469C2×C4—61no—
470C2×C4—61no—
471C2×C4—61no—
472C2×C4—92no—
473C2×C4—181no—
474C2×C4—181no—
475C2×C4—181no—
476C2×C4—181no—
477C2×C4—181no—
478C2×C4—541no—
479C2×C4—541no—
480C2×C4—541no—
481C2×C4#619512yes—
482C2×C3—21no—
483C2×C3—21no—
484C2×C3—21no—
485C2×C3—21no—
486C2×C3—21no—
487C2×C3—21no—
488C2×C3—21no—
489C2×C3#6343411yescharacteristic
490C2×C3C4×D3611yescharacteristic
491C2×C3#6343711yescharacteristic
492C2×C3#6343811yescharacteristic
493C2×C3C4×#61111yescharacteristic
494C2×C3#6349111yescharacteristic
495C2×C3#6349211yescharacteristic
496C2×C3#6383611yescharacteristic
497C2×C3#6385211yescharacteristic
498C2×C3#6386811yescharacteristic
499C2×C3#6388011yescharacteristic
500C2×C3C4×#62611yescharacteristic
501C2×C3#6390911yescharacteristic
502C2×C3#6396611yescharacteristic
503D3—364no—
504D3—362no—
505D3—362no—
506C22—542no—
507C22—542no—
508C22—542no—
509C22—1081no—
510C22—1081no—
511C22—1081no—
512C22—1081no—
513C22#8864011yescharacteristic
514C22#8864211yescharacteristic
515C22C4×D3×D911yescharacteristic
516C22D3×D3611yescharacteristic
517C22#8865811yescharacteristic
518C22#8865811yescharacteristic
519C22C2×#619511yescharacteristic
520C4—22no—
521C4—62no—
522C4—62no—
523C4—182no—
524C4—361no—
525C4—1081no—
526C3—21no—
527C3#1058862011yescharacteristic, complemented
528C3#1059119411yescharacteristic
529C2—1082no—
530C2#1172089811yescharacteristic
531C2#1172092611yescharacteristic
532C2#1172094011yescharacteristic
533C2#1172097711yescharacteristic
534C2C4×#619511yescharacteristic
535C2#1172103811yescharacteristic
536C2#1172125111yescharacteristic
subgroupG/H
centerC23C2×D3×D9
commutatorC22×C3×C9C22×C4
FrattiniC23×C3C2×D32
FittingC22×C3×C4×C9C22
socleC23×C32C22×D3
prime radicalC23×C3⋊D9C22
cosocleC23×C3×C9C23
hypercenterC22×C4D3×D9
hypocenterC3×C9#388
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. C2×C4×C9×Dic3C2semidirect
5. C2×Dic9⋊Dic3C2semidirect
6. C3×C9⋊#95C2semidirect
7. #11720331C2semidirect
20. #88445C4semidirect
36. C9⋊#95D3semidirect
123. C2×C4×Dic3D9semidirect
294. C3×C9#388semidirect
455. C9#4670semidirect
527. C3#10588620semidirect

Maximal subgroups

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

subgroupindex
C2×#885362
C2×C4×C9×Dic32
C2×#884452
C2×Dic9⋊Dic32
C3×C9⋊#952
#117203312
#105911943
#105886203