Finite Group Databasebeta

← #13150408

Subgroups — #13150408

Order 1728.

776 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11721813C211yesmaximal, characteristic, complemented
2C9×#930C211yesmaximal, characteristic
3C2×C3×C4×Dic9C211yesmaximal, characteristic, complemented
4#67×Dic9C212yesmaximal, complemented
5#11721265C212yesmaximal, complemented
6D12×Dic9C211yesmaximal, characteristic, complemented
7#11721010C212yesmaximal, complemented
8#11720964C211yesmaximal, characteristic, complemented
9#11720963C212yesmaximal, complemented
10#11720935C211yesmaximal, characteristic, complemented
11#11720913C211yesmaximal, characteristic, complemented
12#10594949—31nomaximal
13#10589702—31nomaximal
14C22×C3×C4×C9C2211yescharacteristic
15C2×#6226C2211yescharacteristic
16C2×C9×#67C2212yes—
17C22×C3×Dic9C2211yescharacteristic
18C22×C3×Dic9C2211yescharacteristic
19C2×C9×D12C2211yescharacteristic
20C2×D3×Dic9C2212yes—
21#88551C2212yes—
22C4⋊(C3⋊Dic9)C2211yescharacteristic
23#88534C2212yes—
24C9×(C2×D3)⋊C4C2212yes—
25C9×C4⋊Dic3C2211yescharacteristic
26C3×C4×Dic9C2211yescharacteristic
27C3×C4×Dic9C2211yescharacteristic
28C3×C4×Dic9C2211yescharacteristic
29C3×C4×Dic9C2211yescharacteristic
30(C2×D3)⋊Dic9C2212yes—
31(C2×D3)⋊Dic9C2212yes—
32(C2×D3)⋊Dic9C2212yes—
33Dic3⋊Dic9C2212yes—
34#88442C2212yes—
35#88442C2212yes—
36Dic3×Dic9C2212yes—
37#64133—31no—
38C3×#930D311yescharacteristic
39C2×C3×C4×Dic3—31no—
40Dic3×#67—32no—
41#63961—32no—
42Dic3×D12—31no—
43#63877—32no—
44#63869—31no—
45#63860—31no—
46#63859—32no—
47#63844—31no—
48C9×#121—31no—
49#63498—31no—
50#63497—32no—
51D4×Dic9—32no—
52D4×Dic9—31no—
53#63486—31no—
54#63486—61no—
55C2×C4×Dic9D311yescharacteristic, complemented
56#63452—31no—
57#63449—31no—
58#63441—32no—
59#63439—32no—
60C23×C3×C9C2311yescharacteristic
61C22×D3×C9C2312yes—
62C2×C3×C4×C9—22no—
63C2×C3×C4×C9C2311yescharacteristic
64C2×C3×C4×C9C2311yescharacteristic
65C2×C3⋊Dic9C2312yes—
66C2×C3⋊Dic9C2312yes—
67#6226—22no—
68C9×#67—24no—
69C2×C9×Dic3C2312yes—
70C2×C3×Dic9—22no—
71C2×C3×Dic9—22no—
72C2×C3×Dic9C2311yescharacteristic
73C2×C3×Dic9C2311yescharacteristic
74C2×C3×Dic9C2311yescharacteristic
75C2×C3×Dic9C2311yescharacteristic
76C9×D12—22no—
77D3×Dic9—24no—
78#5600—91no—
79#5479—91no—
80C22×C32×C4C2×D311yescharacteristic
81C2×#634—31no—
82C2×C3×#67C2×D312yes—
83C22×C3×Dic3—31no—
84C22×C3×Dic3—31no—
85C2×C3×D12C2×D311yescharacteristic
86C2×D3×Dic3—32no—
87#3757—32no—
88C4⋊(C3⋊Dic3)—31no—
89#3750—32no—
90C3×(C2×D3)⋊C4C2×D312yes—
91C3×C4⋊Dic3C2×D311yescharacteristic
92C3×C4×Dic3—31no—
93C3×C4×Dic3—31no—
94C3×C4×Dic3—31no—
95C3×C4×Dic3—31no—
96#3724—32no—
97#3724—32no—
98Dic3⋊Dic3—32no—
99(C2×D3)⋊Dic3—32no—
100(C2×D3)⋊Dic3—32no—
101(C2×D3)⋊Dic3—32no—
102Dic32—32no—
103C2×D4×C9—32no—
104C2×D4×C9—31no—
105C22×C4×C9—21no—
106C22×C4×C9C2×D311yescharacteristic
107C22×Dic9—32no—
108C22×Dic9C2×D311yescharacteristic
109C22×Dic9C2×D311yescharacteristic
110C2×Dic18—31no—
111C2×Dic18—61no—
112C9×C4⋊C4—31no—
113C9×C22⋊C4—32no—
114#3676—32no—
115#3676—32no—
116#3676—32no—
117#3676—32no—
118#3676—62no—
119C4⋊Dic9—31no—
120C4⋊Dic9—32no—
121C4⋊Dic9—61no—
122Dic9⋊C4—32no—
123Dic9⋊C4—32no—
124Dic9⋊C4—32no—
125Dic9⋊C4—62no—
126C4×Dic9—32no—
127C4×Dic9C2×D311yescharacteristic
128C4×Dic9C2×D311yescharacteristic
129C4×Dic9C2×D311yescharacteristic
130C4×Dic9C2×D311yescharacteristic
131C22×C3×C9—21no—
132C22×C3×C9—21no—
133C22×C3×C9#4112yes—
134C22×C3×C9C2411yescharacteristic
135C2×D3×C9—22no—
136C2×D3×C9—22no—
137C2×D3×C9—22no—
138C3×C4×C9—21no—
139C3×C4×C9#4112yes—
140C3⋊Dic9—22no—
141C3⋊Dic9—22no—
142C9×Dic3—22no—
143C3×Dic9—21no—
144C3×Dic9—21no—
145C3×Dic9—21no—
146C3×Dic9#4112yes—
147C3×#121—31no—
148C2×C3×C42—91no—
149#936—91no—
150#935—92no—
151D4×Dic3—91no—
152D4×Dic3—92no—
153#930D911yescharacteristic
154#929—92no—
155C4×#67—92no—
156#924—91no—
157#924—91no—
158#924—181no—
159C2×C4×Dic3—31no—
160#890—91no—
161#887—91no—
162#879—92no—
163#877—92no—
164#876—92no—
165#875—91no—
166C4×D12—91no—
167#872—92no—
168#870—91no—
169C4×Dic6—91no—
170C23×C32C22×D311yescharacteristic
171C22×C3×D3C22×D312yes—
172C2×C32×C4—22no—
173C2×C32×C4C22×D311yescharacteristic
174C2×C32×C4C22×D311yescharacteristic
175C2×C3⋊Dic3—32no—
176C2×C3⋊Dic3—32no—
177#634—62no—
178C3×#67—24no—
179C2×C3×Dic3—31no—
180C2×C3×Dic3—31no—
181C2×C3×Dic3—31no—
182C2×C3×Dic3—31no—
183C2×C3×Dic3—62no—
184C2×C3×Dic3—62no—
185C2×C3×Dic3C22×D312yes—
186C3×D12—22no—
187D3×Dic3—64no—
188C23×C9—21no—
189C23×C9—32no—
190C23×C9C22×D311yescharacteristic
191D4×C9—64no—
192D4×C9—62no—
193C2×C4×C9—22no—
194C2×C4×C9—21no—
195C2×C4×C9—21no—
196C2×C4×C9—32no—
197C2×C4×C9—42no—
198C2×C4×C9C22×D311yescharacteristic
199C2×C4×C9C22×D311yescharacteristic
200C2×Dic9—22no—
201C2×Dic9—22no—
202C2×Dic9—32no—
203C2×Dic9—32no—
204C2×Dic9—62no—
205C2×Dic9—62no—
206C2×Dic9—64no—
207C2×Dic9C22×D311yescharacteristic
208C2×Dic9C22×D311yescharacteristic
209C2×Dic9C22×D311yescharacteristic
210C2×Dic9C22×D311yescharacteristic
211Dic18—62no—
212Dic18—122no—
213#529—271noSylow
214C2×C3×C9—22no—
215C2×C3×C9C2×#4111yescharacteristic
216C2×C3×C9C2×#4111yescharacteristic
217C2×C3×C9C2×#4111yescharacteristic
218D3×C9—42no—
219C2×C3×D4—32no—
220C2×C3×D4—31no—
221C22×C3×C4—21no—
222C22×C3×C4—91no—
223C22×C3×C4—91no—
224C22×C3×C4C2×D911yescharacteristic
225C22×C3×C4D3211yescharacteristic
226C2×#67C2×D912yes—
227C22×Dic3—31no—
228C22×Dic3—31no—
229C22×Dic3—92no—
230C2×D12C2×D911yescharacteristic
231C2×C4×D3—92no—
232C2×Dic6—91no—
233C2×Dic6—91no—
234C2×Dic6—181no—
235C3×C4⋊C4—31no—
236C3×C22⋊C4—32no—
237C3×C42—91no—
238C3×C42—91no—
239C3×C42—91no—
240C3×C42—91no—
241#217—92no—
242#217—92no—
243#217—92no—
244#217—92no—
245#217—92no—
246#217—182no—
247(C2×D3)⋊C4—92no—
248(C2×D3)⋊C4—92no—
249(C2×D3)⋊C4—92no—
250(C2×D3)⋊C4C2×D912yes—
251C4⋊Dic3—92no—
252C4⋊Dic3—91no—
253C4⋊Dic3—91no—
254C4⋊Dic3—181no—
255C4⋊Dic3C2×D911yescharacteristic
256Dic3⋊C4—92no—
257Dic3⋊C4—92no—
258Dic3⋊C4—92no—
259Dic3⋊C4—92no—
260Dic3⋊C4—92no—
261Dic3⋊C4—92no—
262Dic3⋊C4—92no—
263Dic3⋊C4—182no—
264C4×Dic3—31no—
265C4×Dic3—31no—
266C4×Dic3—31no—
267C4×Dic3—31no—
268C4×Dic3—92no—
269C4×Dic3—92no—
270C22×C32—21no—
271C22×C32—21no—
272C22×C32#23712yes—
273C22×C32C23×D311yescharacteristic
274C2×C3×D3—22no—
275C2×C3×D3—22no—
276C2×C3×D3—22no—
277C32×C4—21no—
278C32×C4#23712yes—
279C3⋊Dic3—62no—
280C3⋊Dic3—62no—
281C3×Dic3—22no—
282C3×Dic3—32no—
283C3×Dic3—61no—
284C3×Dic3—61no—
285C3×Dic3—61no—
286C22×C9—21no—
287C22×C9—21no—
288C22×C9—21no—
289C22×C9—22no—
290C22×C9—41no—
291C22×C9—41no—
292C22×C9—62no—
293C22×C9—62no—
294C22×C9—62no—
295C22×C9#23512yes—
296C22×C9C23×D311yescharacteristic
297C4×C9—21no—
298C4×C9—22no—
299C4×C9—41no—
300C4×C9—62no—
301C4×C9#23512yes—
302Dic9—21no—
303Dic9—21no—
304Dic9—21no—
305Dic9—62no—
306Dic9—62no—
307Dic9—122no—
308Dic9—122no—
309Dic9#23512yes—
310#126—272no—
311#125—271no—
312#124—271no—
313#123—272no—
314C22⋊Q8—271no—
315#121—31no—
316C4×Q8—271no—
317C4×D4—271no—
318C4×D4—272no—
319#117—272no—
320C2×C42—91no—
321C3×C9#52911yescharacteristic, complemented, Sylow
322C23×C3—21no—
323C23×C3—32no—
324C23×C3C22×D911yescharacteristic
325C23×C3C2×D3211yescharacteristic
326C22×D3C22×D912yes—
327C3×D4—64no—
328C3×D4—62no—
329C2×C3×C4—22no—
330C2×C3×C4—22no—
331C2×C3×C4—21no—
332C2×C3×C4—21no—
333C2×C3×C4—32no—
334C2×C3×C4—42no—
335C2×C3×C4—91no—
336C2×C3×C4—91no—
337C2×C3×C4—91no—
338C2×C3×C4—91no—
339C2×C3×C4—182no—
340C2×C3×C4—182no—
341C2×C3×C4C22×D911yescharacteristic
342C2×C3×C4C22×D911yescharacteristic
343C2×C3×C4C2×D3211yescharacteristic
344C2×C3×C4C2×D3211yescharacteristic
345#67—24no—
346C2×Dic3—31no—
347C2×Dic3—31no—
348C2×Dic3—31no—
349C2×Dic3—31no—
350C2×Dic3—62no—
351C2×Dic3—62no—
352C2×Dic3—92no—
353C2×Dic3—92no—
354C2×Dic3—92no—
355C2×Dic3—92no—
356C2×Dic3—182no—
357C2×Dic3—182no—
358C2×Dic3—184no—
359C2×Dic3C22×D912yes—
360D12—22no—
361C4×D3—184no—
362Dic6—182no—
363Dic6—182no—
364Dic6—362no—
365C2×C32—22no—
366C2×C32C2×#23711yescharacteristic
367C2×C32C2×#23711yescharacteristic
368C2×C32D3×#4111yescharacteristic
369C3×D3—42no—
370C2×C9—22no—
371C2×C9—21no—
372C2×C9—21no—
373C2×C9—21no—
374C2×C9—42no—
375C2×C9—122no—
376C2×C9C2×#23511yescharacteristic
377C2×C9C2×#23511yescharacteristic
378C2×C9C2×#23511yescharacteristic
379C2×Q8—271no—
380C2×D4—32no—
381C2×D4—31no—
382C22×C4—91no—
383C22×C4—91no—
384C22×C4—272no—
385C22×C4D3×D911yescharacteristic
386C4⋊C4—31no—
387C4⋊C4—272no—
388C4⋊C4—272no—
389C4⋊C4—272no—
390C4⋊C4—272no—
391C4⋊C4—271no—
392C22⋊C4—32no—
393C22⋊C4—272no—
394C22⋊C4—272no—
395C22⋊C4—272no—
396C22⋊C4—272no—
397C42—91no—
398C42—91no—
399C42—91no—
400C42—91no—
401C42—272no—
402C22×C3—21no—
403C22×C3—21no—
404C22×C3—21no—
405C22×C3—21no—
406C22×C3—22no—
407C22×C3—21no—
408C22×C3—41no—
409C22×C3—41no—
410C22×C3—62no—
411C22×C3—62no—
412C22×C3—62no—
413C22×C3#369912yes—
414C22×C3C23×D911yescharacteristic
415C22×C3#379512yes—
416C22×C3C22×D3211yescharacteristic
417C2×D3—22no—
418C2×D3—22no—
419C2×D3—22no—
420C3×C4—21no—
421C3×C4—21no—
422C3×C4—22no—
423C3×C4—41no—
424C3×C4—62no—
425C3×C4—92no—
426C3×C4—181no—
427C3×C4—181no—
428C3×C4—181no—
429C3×C4#369912yes—
430C3×C4#380412yes—
431Dic3—22no—
432Dic3—32no—
433Dic3—61no—
434Dic3—61no—
435Dic3—61no—
436Dic3—182no—
437Dic3—182no—
438Dic3—182no—
439Dic3—182no—
440Dic3—362no—
441Dic3—362no—
442C32#560011yescharacteristic
443C9—21no—
444C9#547911yescharacteristic, complemented
445C23—32no—
446C23C2×D3×D911yescharacteristic
447Q8—542no—
448D4—64no—
449D4—62no—
450C2×C4—22no—
451C2×C4—32no—
452C2×C4—91no—
453C2×C4—91no—
454C2×C4—91no—
455C2×C4—91no—
456C2×C4—182no—
457C2×C4—182no—
458C2×C4—272no—
459C2×C4—272no—
460C2×C4—544no—
461C2×C4C2×D3×D911yescharacteristic
462C2×C4C2×D3×D911yescharacteristic
463C2×C3—22no—
464C2×C3—22no—
465C2×C3—21no—
466C2×C3—21no—
467C2×C3—21no—
468C2×C3—42no—
469C2×C3—122no—
470C2×C3C2×#369911yescharacteristic
471C2×C3C2×#369911yescharacteristic
472C2×C3#41×D911yescharacteristic
473C2×C3C2×#379511yescharacteristic
474C2×C3D3×#23511yescharacteristic
475C2×C3C2×#380411yescharacteristic
476D3—42no—
477C22—21no—
478C22—21no—
479C22—62no—
480C22—62no—
481C22—62no—
482C22#8863812yes—
483C22C22×D3×D911yescharacteristic
484C4—21no—
485C4—62no—
486C4—92no—
487C4—181no—
488C4—181no—
489C4—181no—
490C4—542no—
491C4—542no—
492C4#8866212yes—
493C3—21no—
494C3#1058970211yescharacteristic, complemented
495C3#1059494911yescharacteristic
496C2—22no—
497C2—122no—
498C2C2×#8863811yescharacteristic
499C2D9×#23511yescharacteristic
500C2C2×#8866211yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC22×C3×C4×C9C22
socleC22×C32C23×D3
prime radicalC3×#930D3
hypercenterC22×C4D3×D9
hypocenterC3×C9#529
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. #11721813C2semidirect
3. C2×C3×C4×Dic9C2semidirect
4. #67×Dic9C2semidirect
5. #11721265C2semidirect
6. D12×Dic9C2semidirect
7. #11721010C2semidirect
8. #11720964C2semidirect
9. #11720963C2semidirect
10. #11720935C2semidirect
11. #11720913C2semidirect
55. C2×C4×Dic9D3semidirect
321. C3×C9#529semidirect
444. C9#5479semidirect
494. C3#10589702semidirect

Maximal subgroups

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

subgroupindex
#117218132
C9×#9302
C2×C3×C4×Dic92
#67×Dic92
#117212652
D12×Dic92
#117210102
#117209642
#117209632
#117209352
#117209132
#105949493
#105897023