Finite Group Databasebeta

← #13139706

Subgroups — #13139706

Order 1728.

678 conjugacy classes of subgroups.

The subgroup lattice has 516 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×C3⋊Dic9C211yesmaximal, characteristic, complemented
2C2×C3×(C2×D9)⋊C4C211yesmaximal, characteristic
3C2×(C2×D9)⋊Dic3C211yesmaximal, characteristic
4C2×(C2×D9)⋊Dic3C211yesmaximal, characteristic
5C2×Dic9⋊Dic3C211yesmaximal, characteristic, complemented
6C9×C3⋊#95C211yesmaximal, characteristic, complemented
7#11720331C211yesmaximal, characteristic, complemented
8#10591286—31nomaximal
9#10588349—31nomaximal
10C23×C3×D9C2211yescharacteristic
11C22×C3×C4×C9C2211yescharacteristic
12C22×C3⋊Dic9C2211yescharacteristic
13C22×C3⋊Dic9C2211yescharacteristic
14C22×C9×Dic3C2211yescharacteristic
15C22×C9×Dic3C2211yescharacteristic
16C22×C3×Dic9C2211yescharacteristic
17C4×C3⋊Dic9—22no—
18C3×(C2×D9)⋊C4C414yescomplemented
19(C2×D9)⋊Dic3—22no—
20(C2×D9)⋊Dic3—22no—
21Dic9⋊Dic3—22no—
22C2×C4×C3⋊Dic3—31no—
23C2×C3×(C2×D3)⋊C4—31no—
24C2×Dic3⋊Dic3—31no—
25C2×(C2×D3)⋊Dic3—31no—
26C2×(C2×D3)⋊Dic3—31no—
27C3×C3⋊#95D311yescharacteristic
28#63579—31no—
29C2×(C2×D9)⋊C4—31no—
30C2×(C2×D9)⋊C4—31no—
31C2×(C2×D9)⋊C4D311yescharacteristic
32C2×Dic9⋊C4—31no—
33C2×C4×Dic9—31no—
34C2×C4×Dic9—61no—
35C9×#95—31no—
36C9⋊#95—31no—
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×C9C2×C412yes—
48C2×C3⋊Dic9—21no—
49C2×C3⋊Dic9—21no—
50C2×C3⋊Dic9—21no—
51C2×C3⋊Dic9—21no—
52C2×C3⋊Dic9D412yes—
53C2×C3⋊Dic9D412yes—
54C2×C9×Dic3—21no—
55C2×C9×Dic3—21no—
56C2×C9×Dic3—21no—
57C2×C9×Dic3—21no—
58C2×C9×Dic3—21no—
59C2×C9×Dic3—21no—
60C2×C3×Dic9—21no—
61C2×C3×Dic9—21no—
62C2×C3×Dic9C2×C412yes—
63#4951—91no—
64#4669—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
73C4×C3⋊Dic3—62no—
74C3×(C2×D3)⋊C4—34no—
75Dic3⋊Dic3—62no—
76(C2×D3)⋊Dic3—62no—
77(C2×D3)⋊Dic3—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
87(C2×D9)⋊C4—62no—
88(C2×D9)⋊C4—62no—
89(C2×D9)⋊C4Dic314yescomplemented
90Dic9⋊C4—62no—
91C4×Dic9—62no—
92C4×Dic9—64no—
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—22no—
109C3⋊Dic9—22no—
110C9×Dic3—41no—
111C9×Dic3—41no—
112C3×Dic9—22no—
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×C4⋊Dic3—91no—
120C2×Dic3⋊C4—91no—
121C2×C4×Dic3—91no—
122C2×C4×Dic3—91no—
123C2×C4×Dic3—181no—
124C3×#95—31no—
125C3⋊#95—91no—
126C3⋊#95—91no—
127C3⋊#95D911yescharacteristic, complemented
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×C4C4×D312yes—
139C2×C3⋊Dic3—32no—
140C2×C3⋊Dic3—32no—
141C2×C3⋊Dic3—61no—
142C2×C3⋊Dic3—61no—
143C2×C3⋊Dic3—61no—
144C2×C3⋊Dic3—61no—
145C2×C3×Dic3—21no—
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—
154C23×C9—21no—
155C23×C9C22×D311yescharacteristic
156C22×D9—21no—
157C22×D9—21no—
158C22×D9—21no—
159C22×D9—21no—
160C22×D9—21no—
161C22×D9—21no—
162C22×D9C2×Dic312yes—
163C2×C4×C9—22no—
164C2×C4×C9—22no—
165C2×C4×C9—21no—
166C2×C4×C9—21no—
167C2×C4×C9—22no—
168C2×C4×C9—61no—
169C2×C4×C9—61no—
170C2×C4×C9—61no—
171C2×C4×C9—61no—
172C2×C4×C9—61no—
173C2×C4×C9—61no—
174C2×C4×C9C2×Dic312yes—
175C2×Dic9—21no—
176C2×Dic9—21no—
177C2×Dic9—32no—
178C2×Dic9—32no—
179C2×Dic9—62no—
180C2×Dic9—62no—
181C2×Dic9—62no—
182C2×Dic9—62no—
183C2×Dic9—62no—
184C2×Dic9—62no—
185C2×Dic9—61no—
186C2×Dic9—61no—
187C2×Dic9—61no—
188C2×Dic9—61no—
189C2×Dic9C2×Dic312yes—
190#388—271noSylow
191C2×C3×C9#11711yescharacteristic
192C2×C3×C9C4×D411yescharacteristic
193C2×C3×C9C4×D411yescharacteristic
194C2×C3×C9#12111yescharacteristic
195C2×C3×C9#12311yescharacteristic
196C2×C3×C9#12411yescharacteristic
197C2×C3×C9#12611yescharacteristic
198C3×D9—42no—
199C24×C3—91no—
200C23×D3—31no—
201C22×C3×C4—21no—
202C22×C3×C4—31no—
203C22×C3×C4—31no—
204C22×C3×C4—91no—
205C22×C3×C4C2×D911yescharacteristic
206C22×C3×C4D3211yescharacteristic
207C22×Dic3—31no—
208C22×Dic3—91no—
209C22×Dic3—91no—
210C22×Dic3—91no—
211C22×Dic3—91no—
212C22×Dic3—181no—
213C22×Dic3—181no—
214C22×Dic3C2×D911yescharacteristic
215C22×Dic3C2×D911yescharacteristic
216C3×C22⋊C4—94no—
217#217—182no—
218#217—182no—
219(C2×D3)⋊C4—34no—
220(C2×D3)⋊C4—182no—
221(C2×D3)⋊C4—182no—
222C4⋊Dic3—182no—
223Dic3⋊C4—182no—
224C4×Dic3—182no—
225C4×Dic3—182no—
226C4×Dic3—184no—
227C22×C32C2×C4×D311yescharacteristic
228C22×C32#23511yescharacteristic
229C22×C32#23511yescharacteristic
230C22×C32D3×D411yescharacteristic
231C22×C32D3×D411yescharacteristic
232C22×C32#23711yescharacteristic
233C22×C32#23911yescharacteristic
234C2×C3×D3—62no—
235C2×C3×D3—62no—
236C2×C3×D3—62no—
237C2×C3×D3—121no—
238C2×C3×D3—121no—
239C2×C3×D3—121no—
240C2×C3×D3—121no—
241C32×C4—22no—
242C3⋊Dic3—62no—
243C3⋊Dic3—62no—
244C3×Dic3—41no—
245C3×Dic3—41no—
246C3×Dic3—62no—
247C22×C9—21no—
248C22×C9—21no—
249C22×C9—21no—
250C22×C9—21no—
251C22×C9—21no—
252C22×C9—21no—
253C22×C9—21no—
254C22×C9#23511yescharacteristic
255C22×C9#23511yescharacteristic
256C22×C9D3×D411yescharacteristic
257C22×C9D3×D411yescharacteristic
258C22×C9#23711yescharacteristic
259C22×C9#23711yescharacteristic
260C22×C9C22×Dic311yescharacteristic
261C2×D9—22no—
262C2×D9—22no—
263C2×D9—22no—
264C2×D9—41no—
265C2×D9—41no—
266C2×D9—41no—
267C2×D9—41no—
268C4×C9—24no—
269C4×C9—22no—
270C4×C9—121no—
271C4×C9—121no—
272Dic9—22no—
273Dic9—64no—
274Dic9—64no—
275Dic9—62no—
276Dic9—62no—
277C2×C4⋊C4—271no—
278C2×C22⋊C4—91no—
279C2×C22⋊C4—271no—
280C2×C22⋊C4—271no—
281C2×C42—271no—
282#95—31no—
283#95—271no—
284C3×C9#38811yescharacteristic, complemented, Sylow
285C23×C3—21no—
286C23×C3—92no—
287C23×C3—181no—
288C23×C3—181no—
289C23×C3—181no—
290C23×C3—181no—
291C23×C3—181no—
292C23×C3—181no—
293C23×C3C22×D911yescharacteristic
294C23×C3C2×D3211yescharacteristic
295C22×D3—32no—
296C22×D3—61no—
297C22×D3—61no—
298C22×D3—61no—
299C22×D3—61no—
300C22×D3—61no—
301C22×D3—61no—
302C2×C3×C4—22no—
303C2×C3×C4—22no—
304C2×C3×C4—21no—
305C2×C3×C4—21no—
306C2×C3×C4—21no—
307C2×C3×C4—21no—
308C2×C3×C4—22no—
309C2×C3×C4—61no—
310C2×C3×C4—61no—
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×C3×C4C4×D912yes—
319C2×C3×C4D3×Dic312yes—
320C2×Dic3—21no—
321C2×Dic3—21no—
322C2×Dic3—21no—
323C2×Dic3—21no—
324C2×Dic3—21no—
325C2×Dic3—21no—
326C2×Dic3—32no—
327C2×Dic3—61no—
328C2×Dic3—61no—
329C2×Dic3—92no—
330C2×Dic3—92no—
331C2×Dic3—92no—
332C2×Dic3—92no—
333C2×Dic3—182no—
334C2×Dic3—182no—
335C2×Dic3—182no—
336C2×Dic3—182no—
337C2×Dic3—182no—
338C2×Dic3—182no—
339C2×Dic3—181no—
340C2×Dic3—181no—
341C2×Dic3—181no—
342C2×Dic3—181no—
343C2×Dic3—181no—
344C2×Dic3—181no—
345C2×Dic3—181no—
346C2×Dic3—181no—
347C2×C32#87211yescharacteristic
348C2×C32#67⋊C411yescharacteristic
349C2×C32#88411yescharacteristic
350C2×C32#88511yescharacteristic
351C2×C32#89311yescharacteristic
352C2×C32#89411yescharacteristic
353C2×C32#89811yescharacteristic
354C3×D3—122no—
355C2×C9—21no—
356C2×C9—21no—
357C2×C9—21no—
358C2×C9—21no—
359C2×C9—21no—
360C2×C9—21no—
361C2×C9—21no—
362C2×C9#87911yescharacteristic
363C2×C9#88311yescharacteristic
364C2×C9#88411yescharacteristic
365C2×C9#88511yescharacteristic
366C2×C9#92611yescharacteristic
367C2×C9D4×Dic311yescharacteristic
368C2×C9D4×Dic311yescharacteristic
369D9—42no—
370C24—91no—
371C22×C4—31no—
372C22×C4—31no—
373C22×C4—91no—
374C22×C4—271no—
375C22×C4—271no—
376C22×C4D3×D911yescharacteristic
377C4⋊C4—542no—
378C22⋊C4—94no—
379C22⋊C4—542no—
380C22⋊C4—542no—
381C42—542no—
382C22×C3—21no—
383C22×C3—21no—
384C22×C3—21no—
385C22×C3—21no—
386C22×C3—21no—
387C22×C3—21no—
388C22×C3—21no—
389C22×C3—182no—
390C22×C3—182no—
391C22×C3—182no—
392C22×C3—361no—
393C22×C3—361no—
394C22×C3—361no—
395C22×C3—361no—
396C22×C3C2×C4×D911yescharacteristic
397C22×C3#369711yescharacteristic
398C22×C3#369711yescharacteristic
399C22×C3D4×D911yescharacteristic
400C22×C3D4×D911yescharacteristic
401C22×C3#369911yescharacteristic
402C22×C3#370111yescharacteristic
403C22×C3#379611yescharacteristic
404C22×C3#379811yescharacteristic
405C22×C3#379811yescharacteristic
406C22×C3#380211yescharacteristic
407C22×C3C2×D3×Dic311yescharacteristic
408C22×C3#380511yescharacteristic
409C22×C3#381111yescharacteristic
410C2×D3—62no—
411C2×D3—62no—
412C2×D3—62no—
413C2×D3—121no—
414C2×D3—121no—
415C2×D3—121no—
416C2×D3—121no—
417C3×C4—24no—
418C3×C4—22no—
419C3×C4—22no—
420C3×C4—121no—
421C3×C4—121no—
422C3×C4—182no—
423Dic3—41no—
424Dic3—41no—
425Dic3—62no—
426Dic3—184no—
427Dic3—184no—
428Dic3—182no—
429Dic3—182no—
430Dic3—182no—
431Dic3—182no—
432C32#466911yescharacteristic
433C9—21no—
434C9#495111yescharacteristic, complemented
435C23—92no—
436C23—181no—
437C23—181no—
438C23—181no—
439C23—181no—
440C23—181no—
441C23—181no—
442C23C2×D3×D911yescharacteristic
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—61no—
446C2×C4—61no—
447C2×C4—61no—
448C2×C4—61no—
449C2×C4—61no—
450C2×C4—61no—
451C2×C4—92no—
452C2×C4—181no—
453C2×C4—181no—
454C2×C4—272no—
455C2×C4—272no—
456C2×C4—541no—
457C2×C4—541no—
458C2×C4—541no—
459C2×C4—541no—
460C2×C4Dic3×D912yes—
461C2×C3—21no—
462C2×C3—21no—
463C2×C3—21no—
464C2×C3—21no—
465C2×C3—21no—
466C2×C3—21no—
467C2×C3—21no—
468C2×C3—362no—
469C2×C3#6343411yescharacteristic
470C2×C3#611⋊C411yescharacteristic
471C2×C3#6344611yescharacteristic
472C2×C3#6344711yescharacteristic
473C2×C3#6345511yescharacteristic
474C2×C3#6345611yescharacteristic
475C2×C3#6346011yescharacteristic
476C2×C3#6385311yescharacteristic
477C2×C3#6385511yescharacteristic
478C2×C3#6386211yescharacteristic
479C2×C3#6389811yescharacteristic
480C2×C3#6391211yescharacteristic
481C2×C3#6391511yescharacteristic
482C2×C3#67⋊Dic311yescharacteristic
483D3—122no—
484C22—182no—
485C22—182no—
486C22—182no—
487C22—361no—
488C22—361no—
489C22—361no—
490C22—361no—
491C22#8863411yescharacteristic
492C22#8864011yescharacteristic
493C22#8864011yescharacteristic
494C22#8864611yescharacteristic
495C22C2×Dic3×D911yescharacteristic
496C22#8866311yescharacteristic
497C22#8866811yescharacteristic
498C4—22no—
499C4—121no—
500C4—121no—
501C4—182no—
502C4—542no—
503C4—542no—
504C3—21no—
505C3#1058834911yescharacteristic, complemented
506C3#1059128611yescharacteristic
507C2—362no—
508C2#1172094111yescharacteristic
509C2#1172094311yescharacteristic
510C2#1172095011yescharacteristic
511C2#1172099811yescharacteristic
512C2#1172104111yescharacteristic
513C2#1172104411yescharacteristic
514C2#611⋊Dic311yescharacteristic
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
1. C2×C4×C3⋊Dic9C2semidirect
5. C2×Dic9⋊Dic3C2semidirect
6. C9×C3⋊#95C2semidirect
7. #11720331C2semidirect
18. C3×(C2×D9)⋊C4C4semidirect
89. (C2×D9)⋊C4Dic3semidirect
127. C3⋊#95D9semidirect
284. C3×C9#388semidirect
434. C9#4951semidirect
505. C3#10588349semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C3⋊Dic92
C2×C3×(C2×D9)⋊C42
C2×(C2×D9)⋊Dic32
C2×Dic9⋊Dic32
C9×C3⋊#952
#117203312
#105912863
#105883493