Finite Group Databasebeta

← #13139705

Subgroups — #13139705

Order 1728.

646 conjugacy classes of subgroups.

The subgroup lattice has 514 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×(C2×D9)⋊C4C211yesmaximal, characteristic
2C2×C9×Dic3⋊C4C211yesmaximal, characteristic, complemented
3C2×(C2×D9)⋊Dic3C211yesmaximal, characteristic
4C2×(C2×D9)⋊Dic3C211yesmaximal, characteristic
5C2×Dic3×Dic9C211yesmaximal, characteristic, complemented
6#11720572C211yesmaximal, characteristic, complemented
7#11720331C211yesmaximal, characteristic, complemented
8#10591285—31nomaximal
9#10588670—31nomaximal
10C23×C3×D9C2211yescharacteristic
11C22×C3×C4×C9C2211yescharacteristic
12C22×C3⋊Dic9C2211yescharacteristic
13C22×C3⋊Dic9C2211yescharacteristic
14C22×C9×Dic3C2211yescharacteristic
15C22×C9×Dic3C2211yescharacteristic
16C22×C3×Dic9C2211yescharacteristic
17C3×(C2×D9)⋊C4—22no—
18C9×Dic3⋊C4—22no—
19(C2×D9)⋊Dic3—22no—
20(C2×D9)⋊Dic3C414yescomplemented
21Dic3×Dic9—22no—
22C2×C3×(C2×D3)⋊C4—31no—
23C2×C3×Dic3⋊C4D311yescharacteristic
24C2×(C2×D3)⋊Dic3—31no—
25C2×(C2×D3)⋊Dic3—31no—
26C2×Dic32—31no—
27#63658—31no—
28#63579—31no—
29C2×C9×C4⋊C4—31no—
30C2×(C2×D9)⋊C4—31no—
31C2×(C2×D9)⋊C4—31no—
32C2×(C2×D9)⋊C4D311yescharacteristic
33C2×C4×Dic9—31no—
34C9⋊#95—31no—
35C9⋊#95—31no—
36C9⋊#95—61no—
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×C9—21no—
48C2×C3⋊Dic9—21no—
49C2×C3⋊Dic9—21no—
50C2×C3⋊Dic9—21no—
51C2×C3⋊Dic9—21no—
52C2×C3⋊Dic9—21no—
53C2×C3⋊Dic9C2×C412yes—
54C2×C9×Dic3—21no—
55C2×C9×Dic3—21no—
56C2×C9×Dic3—21no—
57C2×C9×Dic3—21no—
58C2×C9×Dic3C2×C412yes—
59C2×C9×Dic3D412yes—
60C2×C3×Dic9—21no—
61C2×C3×Dic9—21no—
62C2×C3×Dic9D412yes—
63#4990—91no—
64#4944—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×D9C2×D311yescharacteristic
73C3×(C2×D3)⋊C4—62no—
74C3×Dic3⋊C4—22no—
75(C2×D3)⋊Dic3—34no—
76(C2×D3)⋊Dic3—62no—
77Dic32—62no—
78C22×C4×C9—21no—
79C22×C4×C9—31no—
80C22×C4×C9—31no—
81C22×C4×C9C2×D311yescharacteristic
82C22×Dic9—31no—
83C22×Dic9—31no—
84C22×Dic9—61no—
85C22×Dic9—61no—
86C22×Dic9C2×D311yescharacteristic
87C9×C4⋊C4—62no—
88(C2×D9)⋊C4—22no—
89(C2×D9)⋊C4—34no—
90(C2×D9)⋊C4—62no—
91C4×Dic9—62no—
92C22×C3×C9C22×C411yescharacteristic
93C22×C3×C9C2×D411yescharacteristic
94C22×C3×C9C2×D411yescharacteristic
95C22×C3×C9#4111yescharacteristic
96C22×C3×C9#4111yescharacteristic
97C22×C3×C9#4111yescharacteristic
98C22×C3×C9#4111yescharacteristic
99C2×C3×D9—22no—
100C2×C3×D9—22no—
101C2×C3×D9—22no—
102C2×C3×D9—41no—
103C2×C3×D9—41no—
104C2×C3×D9—41no—
105C2×C3×D9—41no—
106C3×C4×C9—41no—
107C3⋊Dic9—22no—
108C3⋊Dic9—41no—
109C9×Dic3—22no—
110C9×Dic3—22no—
111C3×Dic9—22no—
112C2×C3×C4⋊C4—31no—
113C2×C3×C22⋊C4—91no—
114C2×#217—91no—
115C2×#217—91no—
116C2×(C2×D3)⋊C4—31no—
117C2×(C2×D3)⋊C4—91no—
118C2×(C2×D3)⋊C4—91no—
119C2×Dic3⋊C4D911yescharacteristic, complemented
120C2×C4×Dic3—91no—
121C2×C4×Dic3—91no—
122C3⋊#95—91no—
123C3⋊#95—91no—
124C3⋊#95—91no—
125C3⋊#95—91no—
126C3⋊#95—181no—
127C23×C32C22×D311yescharacteristic
128C22×C3×D3—32no—
129C22×C3×D3—61no—
130C22×C3×D3—61no—
131C22×C3×D3—61no—
132C22×C3×D3—61no—
133C22×C3×D3—61no—
134C22×C3×D3—61no—
135C2×C32×C4—21no—
136C2×C32×C4—21no—
137C2×C32×C4—21no—
138C2×C3⋊Dic3—32no—
139C2×C3⋊Dic3—61no—
140C2×C3⋊Dic3—61no—
141C2×C3⋊Dic3—61no—
142C2×C3⋊Dic3—61no—
143C2×C3⋊Dic3—61no—
144C2×C3×Dic3—21no—
145C2×C3×Dic3—21no—
146C2×C3×Dic3—21no—
147C2×C3×Dic3—21no—
148C2×C3×Dic3—32no—
149C2×C3×Dic3—61no—
150C2×C3×Dic3—61no—
151C2×C3×Dic3C4×D312yes—
152C2×C3×Dic3#6712yes—
153C23×C9—21no—
154C23×C9C22×D311yescharacteristic
155C22×D9—21no—
156C22×D9—21no—
157C22×D9—21no—
158C22×D9—21no—
159C22×D9—21no—
160C22×D9—21no—
161C22×D9C4×D312yes—
162C2×C4×C9—22no—
163C2×C4×C9—21no—
164C2×C4×C9—21no—
165C2×C4×C9—21no—
166C2×C4×C9—32no—
167C2×C4×C9—32no—
168C2×C4×C9—41no—
169C2×C4×C9—41no—
170C2×C4×C9—61no—
171C2×C4×C9—61no—
172C2×C4×C9—61no—
173C2×C4×C9—61no—
174C2×Dic9—21no—
175C2×Dic9—21no—
176C2×Dic9—32no—
177C2×Dic9—62no—
178C2×Dic9—61no—
179C2×Dic9—61no—
180C2×Dic9—62no—
181C2×Dic9—61no—
182C2×Dic9—61no—
183C2×Dic9—61no—
184C2×Dic9—121no—
185C2×Dic9—121no—
186C2×Dic9—121no—
187C2×Dic9—121no—
188C2×Dic9#6712yes—
189#388—271noSylow
190C2×C3×C9#11711yescharacteristic
191C2×C3×C9C4×D411yescharacteristic
192C2×C3×C9C4×D411yescharacteristic
193C2×C3×C9#12111yescharacteristic
194C2×C3×C9#12311yescharacteristic
195C2×C3×C9#12411yescharacteristic
196C2×C3×C9#12611yescharacteristic
197C3×D9—42no—
198C24×C3—91no—
199C23×D3—31no—
200C22×C3×C4—21no—
201C22×C3×C4—31no—
202C22×C3×C4—31no—
203C22×C3×C4—91no—
204C22×C3×C4C2×D911yescharacteristic
205C22×C3×C4D3211yescharacteristic
206C22×Dic3—31no—
207C22×Dic3—91no—
208C22×Dic3—91no—
209C22×Dic3—91no—
210C22×Dic3—91no—
211C22×Dic3—181no—
212C22×Dic3—181no—
213C22×Dic3C2×D911yescharacteristic
214C22×Dic3C2×D911yescharacteristic
215C3×C4⋊C4—62no—
216C3×C22⋊C4—182no—
217#217—94no—
218#217—182no—
219(C2×D3)⋊C4—62no—
220(C2×D3)⋊C4—94no—
221(C2×D3)⋊C4—182no—
222Dic3⋊C4—22no—
223C4×Dic3—182no—
224C4×Dic3—182no—
225C22×C32C2×C4×D311yescharacteristic
226C22×C32#23511yescharacteristic
227C22×C32D3×D411yescharacteristic
228C22×C32#23711yescharacteristic
229C22×C32#23911yescharacteristic
230C22×C32#23911yescharacteristic
231C22×C32C2×#6711yescharacteristic
232C2×C3×D3—62no—
233C2×C3×D3—62no—
234C2×C3×D3—62no—
235C2×C3×D3—121no—
236C2×C3×D3—121no—
237C2×C3×D3—121no—
238C2×C3×D3—121no—
239C32×C4—41no—
240C3⋊Dic3—62no—
241C3⋊Dic3—121no—
242C3×Dic3—22no—
243C3×Dic3—22no—
244C3×Dic3—62no—
245C22×C9—21no—
246C22×C9—21no—
247C22×C9—21no—
248C22×C9—21no—
249C22×C9—21no—
250C22×C9—21no—
251C22×C9—21no—
252C22×C9C2×C4×D311yescharacteristic
253C22×C9#23511yescharacteristic
254C22×C9D3×D411yescharacteristic
255C22×C9#23711yescharacteristic
256C22×C9#23711yescharacteristic
257C22×C9#23711yescharacteristic
258C22×C9C2×#6711yescharacteristic
259C2×D9—22no—
260C2×D9—22no—
261C2×D9—22no—
262C2×D9—41no—
263C2×D9—41no—
264C2×D9—41no—
265C2×D9—41no—
266C4×C9—42no—
267C4×C9—41no—
268C4×C9—62no—
269C4×C9—62no—
270Dic9—22no—
271Dic9—62no—
272Dic9—122no—
273Dic9—122no—
274Dic9—121no—
275C2×C4⋊C4—31no—
276C2×C22⋊C4—91no—
277C2×C22⋊C4—271no—
278C2×C22⋊C4—271no—
279C2×C42—271no—
280#95—271no—
281#95—271no—
282C3×C9#38811yescharacteristic, complemented, Sylow
283C23×C3—21no—
284C23×C3—92no—
285C23×C3—181no—
286C23×C3—181no—
287C23×C3—181no—
288C23×C3—181no—
289C23×C3—181no—
290C23×C3—181no—
291C23×C3C22×D911yescharacteristic
292C23×C3C2×D3211yescharacteristic
293C22×D3—32no—
294C22×D3—61no—
295C22×D3—61no—
296C22×D3—61no—
297C22×D3—61no—
298C22×D3—61no—
299C22×D3—61no—
300C2×C3×C4—22no—
301C2×C3×C4—21no—
302C2×C3×C4—21no—
303C2×C3×C4—21no—
304C2×C3×C4—21no—
305C2×C3×C4—21no—
306C2×C3×C4—21no—
307C2×C3×C4—32no—
308C2×C3×C4—32no—
309C2×C3×C4—41no—
310C2×C3×C4—41no—
311C2×C3×C4—61no—
312C2×C3×C4—61no—
313C2×C3×C4—61no—
314C2×C3×C4—61no—
315C2×C3×C4—92no—
316C2×C3×C4—181no—
317C2×C3×C4—181no—
318C2×Dic3—21no—
319C2×Dic3—21no—
320C2×Dic3—21no—
321C2×Dic3—21no—
322C2×Dic3—32no—
323C2×Dic3—61no—
324C2×Dic3—61no—
325C2×Dic3—92no—
326C2×Dic3—92no—
327C2×Dic3—182no—
328C2×Dic3—181no—
329C2×Dic3—181no—
330C2×Dic3—181no—
331C2×Dic3—181no—
332C2×Dic3—182no—
333C2×Dic3—181no—
334C2×Dic3—181no—
335C2×Dic3—181no—
336C2×Dic3—181no—
337C2×Dic3—181no—
338C2×Dic3—181no—
339C2×Dic3—361no—
340C2×Dic3—361no—
341C2×Dic3—361no—
342C2×Dic3—361no—
343C2×Dic3C4×D912yes—
344C2×Dic3#61112yes—
345C2×C32#89211yescharacteristic
346C2×C32#89311yescharacteristic
347C2×C32#89411yescharacteristic
348C2×C32#89811yescharacteristic
349C2×C32C4×#6711yescharacteristic
350C2×C32#93911yescharacteristic
351C2×C32#94711yescharacteristic
352C3×D3—122no—
353C2×C9—21no—
354C2×C9—21no—
355C2×C9—21no—
356C2×C9—21no—
357C2×C9—21no—
358C2×C9—21no—
359C2×C9—21no—
360C2×C9#87711yescharacteristic
361C2×C9#87911yescharacteristic
362C2×C9#67⋊C411yescharacteristic
363C2×C9#88511yescharacteristic
364C2×C9C4×#6711yescharacteristic
365C2×C9#93511yescharacteristic
366C2×C9#93811yescharacteristic
367D9—42no—
368C24—91no—
369C22×C4—31no—
370C22×C4—31no—
371C22×C4—91no—
372C22×C4—271no—
373C22×C4—271no—
374C22×C4D3×D911yescharacteristic
375C4⋊C4—62no—
376C22⋊C4—182no—
377C22⋊C4—274no—
378C22⋊C4—542no—
379C42—542no—
380C22×C3—21no—
381C22×C3—21no—
382C22×C3—21no—
383C22×C3—21no—
384C22×C3—21no—
385C22×C3—21no—
386C22×C3—21no—
387C22×C3—182no—
388C22×C3—182no—
389C22×C3—182no—
390C22×C3—361no—
391C22×C3—361no—
392C22×C3—361no—
393C22×C3—361no—
394C22×C3C2×C4×D911yescharacteristic
395C22×C3#369711yescharacteristic
396C22×C3D4×D911yescharacteristic
397C22×C3#369911yescharacteristic
398C22×C3#370111yescharacteristic
399C22×C3#370111yescharacteristic
400C22×C3C2×#61111yescharacteristic
401C22×C3#379511yescharacteristic
402C22×C3#379611yescharacteristic
403C22×C3#379911yescharacteristic
404C22×C3C4×D3211yescharacteristic
405C22×C3#380511yescharacteristic
406C22×C3D3×#6711yescharacteristic
407C22×C3D3×#6711yescharacteristic
408C2×D3—62no—
409C2×D3—62no—
410C2×D3—62no—
411C2×D3—121no—
412C2×D3—121no—
413C2×D3—121no—
414C2×D3—121no—
415C3×C4—42no—
416C3×C4—41no—
417C3×C4—41no—
418C3×C4—62no—
419C3×C4—62no—
420C3×C4—182no—
421Dic3—22no—
422Dic3—22no—
423Dic3—62no—
424Dic3—182no—
425Dic3—182no—
426Dic3—362no—
427Dic3—362no—
428Dic3—361no—
429Dic3—361no—
430C32#499011yescharacteristic
431C9—21no—
432C9#494411yescharacteristic, complemented
433C23—92no—
434C23—181no—
435C23—181no—
436C23—181no—
437C23—181no—
438C23—181no—
439C23—181no—
440C23C2×D3×D911yescharacteristic
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—32no—
445C2×C4—32no—
446C2×C4—61no—
447C2×C4—61no—
448C2×C4—61no—
449C2×C4—61no—
450C2×C4—92no—
451C2×C4—181no—
452C2×C4—181no—
453C2×C4—272no—
454C2×C4—541no—
455C2×C4—541no—
456C2×C4—541no—
457C2×C4—541no—
458C2×C4—541no—
459C2×C3—21no—
460C2×C3—21no—
461C2×C3—21no—
462C2×C3—21no—
463C2×C3—21no—
464C2×C3—21no—
465C2×C3—21no—
466C2×C3—362no—
467C2×C3#6345411yescharacteristic
468C2×C3#6345511yescharacteristic
469C2×C3#6345611yescharacteristic
470C2×C3#6346011yescharacteristic
471C2×C3C4×#61111yescharacteristic
472C2×C3#6350111yescharacteristic
473C2×C3#6350911yescharacteristic
474C2×C3#6386111yescharacteristic
475C2×C3#6387811yescharacteristic
476C2×C3#6387911yescharacteristic
477C2×C3#6388411yescharacteristic
478C2×C3#6390411yescharacteristic
479C2×C3#6390711yescharacteristic
480C2×C3#6397011yescharacteristic
481D3—122no—
482C22—182no—
483C22—182no—
484C22—182no—
485C22—361no—
486C22—361no—
487C22—361no—
488C22—361no—
489C22#8863411yescharacteristic
490C22#8863511yescharacteristic
491C22#8864111yescharacteristic
492C22C4×D3×D911yescharacteristic
493C22#8866311yescharacteristic
494C22D3×#61111yescharacteristic
495C22D9×#6711yescharacteristic
496C4—41no—
497C4—62no—
498C4—62no—
499C4—182no—
500C4—542no—
501C4—1081no—
502C3—21no—
503C3#1058867011yescharacteristic, complemented
504C3#1059128511yescharacteristic
505C2—362no—
506C2#1172094911yescharacteristic
507C2#1172097511yescharacteristic
508C2#1172097611yescharacteristic
509C2#1172098111yescharacteristic
510C2#1172100211yescharacteristic
511C2#1172100311yescharacteristic
512C2#1172128111yescharacteristic
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×C9×Dic3⋊C4C2semidirect
5. C2×Dic3×Dic9C2semidirect
6. #11720572C2semidirect
7. #11720331C2semidirect
20. (C2×D9)⋊Dic3C4semidirect
119. C2×Dic3⋊C4D9semidirect
282. C3×C9#388semidirect
432. C9#4944semidirect
503. C3#10588670semidirect

Maximal subgroups

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

subgroupindex
C2×C3×(C2×D9)⋊C42
C2×C9×Dic3⋊C42
C2×(C2×D9)⋊Dic32
C2×Dic3×Dic92
#117205722
#117203312
#105912853
#105886703