Finite Group Databasebeta

← #13147662

Subgroups — #13147662

Order 1728.

758 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11721707C211yesmaximal, characteristic, complemented
2C42×D3×C9C211yesmaximal, characteristic, complemented
3C3×#63437C211yesmaximal, characteristic, complemented
4#11721037C211yesmaximal, characteristic
5#11721032C212yesmaximal, complemented
6Dic3×D36C211yesmaximal, characteristic, complemented
7#11720976C212yesmaximal, complemented
8#11720974C212yesmaximal, complemented
9#11720960C211yesmaximal, characteristic, complemented
10#11720947C212yesmaximal, complemented
11Dic3×Dic18C211yesmaximal, characteristic, complemented
12#10593921—31nomaximal
13#10589581—31nomaximal
14C2×C4×D3×C9C2211yescharacteristic
15C2×C4×D3×C9C2211yescharacteristic
16C2×C4×D3×C9C2211yescharacteristic
17C2×C3×D36C2211yescharacteristic
18C2×C3×Dic18C2211yescharacteristic
19C2×#6194C2212yes—
20C2×Dic3×D9C2212yes—
21C3×C42×C9C2211yescharacteristic
22C4⋊(C3⋊Dic9)C2211yescharacteristic
23C4⋊(C3⋊Dic9)C2211yescharacteristic
24#88534C2212yes—
25#88534C2212yes—
26C3×(C2×D9)⋊C4C2212yes—
27C3×(C2×D9)⋊C4C2212yes—
28C4×C9×Dic3C2211yescharacteristic
29C4×C9×Dic3C2211yescharacteristic
30C4×C9×Dic3C2211yescharacteristic
31(C2×D3)⋊Dic9C2212yes—
32(C2×D9)⋊Dic3C2212yes—
33(C2×D9)⋊Dic3C2212yes—
34#88442C2212yes—
35Dic9⋊Dic3C2212yes—
36Dic3×Dic9C2212yes—
37#64079—31no—
38C3×#875—31no—
39C3×C42×D3D311yescharacteristic
40#63908—31no—
41#63905—32no—
42Dic3×D12—31no—
43#63879—32no—
44#63877—32no—
45#63859—32no—
46#63856—31no—
47Dic3×Dic6—31no—
48C2×C42×C9—31no—
49#63492—31no—
50#63491—32no—
51C4×#611—32no—
52#63486—31no—
53#63438—32no—
54#63437D311yescharacteristic, complemented
55C4×D36—31no—
56#63434—32no—
57#63432—31no—
58#63432—61no—
59C4×Dic18—31no—
60C22×D3×C9C2311yescharacteristic
61C22×C3×D9C2312yes—
62C2×C3×C4×C9C2311yescharacteristic
63C2×C3×C4×C9C2311yescharacteristic
64C2×C3×C4×C9C2311yescharacteristic
65C2×C3⋊Dic9C2312yes—
66C2×C3⋊Dic9C2312yes—
67C2×C9×Dic3C2311yescharacteristic
68C2×C9×Dic3C2311yescharacteristic
69C2×C9×Dic3C2311yescharacteristic
70C2×C3×Dic9C2312yes—
71C4×D3×C9—22no—
72C4×D3×C9—22no—
73C4×D3×C9—22no—
74C3×D36—22no—
75C3×Dic18—22no—
76#6194—24no—
77Dic3×D9—24no—
78#5680—91no—
79#5479—91no—
80C2×C3×D12—31no—
81C2×C3×C4×D3C2×D311yescharacteristic
82C2×C3×C4×D3C2×D311yescharacteristic
83C2×C3×C4×D3C2×D311yescharacteristic
84C2×C3×Dic6—31no—
85C2×#625—32no—
86C2×D3×Dic3—32no—
87C32×C42C2×D311yescharacteristic
88C4⋊(C3⋊Dic3)—31no—
89C4⋊(C3⋊Dic3)—31no—
90#3750—32no—
91#3750—32no—
92C3×(C2×D3)⋊C4—32no—
93C3×(C2×D3)⋊C4—32no—
94C3×C4×Dic3C2×D311yescharacteristic
95C3×C4×Dic3C2×D311yescharacteristic
96C3×C4×Dic3C2×D311yescharacteristic
97#3724—32no—
98Dic3⋊Dic3—32no—
99(C2×D3)⋊Dic3—32no—
100(C2×D3)⋊Dic3—32no—
101(C2×D3)⋊Dic3—32no—
102Dic32—32no—
103C22×C4×C9—31no—
104C22×C4×C9—31no—
105C22×C4×C9—31no—
106C2×#611—32no—
107C2×D36C2×D311yescharacteristic
108C2×C4×D9—32no—
109C2×Dic18C2×D311yescharacteristic
110C42×C9—21no—
111C42×C9—31no—
112C42×C9—31no—
113C42×C9—31no—
114C42×C9C2×D311yescharacteristic
115#3676—32no—
116(C2×D9)⋊C4—32no—
117(C2×D9)⋊C4—32no—
118(C2×D9)⋊C4C2×D312yes—
119(C2×D9)⋊C4C2×D312yes—
120C4⋊Dic9—31no—
121C4⋊Dic9—31no—
122C4⋊Dic9—61no—
123C4⋊Dic9—61no—
124Dic9⋊C4—32no—
125Dic9⋊C4—32no—
126Dic9⋊C4—32no—
127Dic9⋊C4—32no—
128Dic9⋊C4—62no—
129Dic9⋊C4—62no—
130C4×Dic9—32no—
131C22×C3×C9C2411yescharacteristic
132C2×D3×C9—21no—
133C2×D3×C9—21no—
134C2×D3×C9#4112yes—
135C2×C3×D9—22no—
136C2×C3×D9—22no—
137C2×C3×D9—22no—
138C3×C4×C9—21no—
139C3×C4×C9—21no—
140C3×C4×C9#4112yes—
141C3⋊Dic9—22no—
142C3⋊Dic9—22no—
143C9×Dic3—21no—
144C9×Dic3—21no—
145C9×Dic3#4112yes—
146C3×Dic9—22no—
147C3×#124—91no—
148C2×C3×C42—31no—
149#946—91no—
150Q8×Dic3—91no—
151#938—91no—
152D4×Dic3—91no—
153#930—91no—
154#929—92no—
155C4×#67—92no—
156#924—91no—
157#883—92no—
158#67⋊C4—92no—
159#879—92no—
160#877—92no—
161#876—92no—
162#875—31no—
163C4×D12—91no—
164#872—92no—
165C42×D3D911yescharacteristic, complemented
166#870—91no—
167#870—91no—
168#870—181no—
169C4×Dic6—91no—
170C22×C3×D3—32no—
171C22×C3×D3C22×D311yescharacteristic
172C2×C32×C4C22×D311yescharacteristic
173C2×C32×C4C22×D311yescharacteristic
174C2×C32×C4C22×D311yescharacteristic
175C2×C3⋊Dic3—32no—
176C2×C3⋊Dic3—32no—
177C2×C3×Dic3—32no—
178C2×C3×Dic3C22×D311yescharacteristic
179C2×C3×Dic3C22×D311yescharacteristic
180C2×C3×Dic3C22×D311yescharacteristic
181C3×D12—62no—
182C3×C4×D3—22no—
183C3×C4×D3—22no—
184C3×C4×D3—22no—
185C3×Dic6—62no—
186#625—64no—
187D3×Dic3—64no—
188C23×C9—31no—
189C22×D9C22×D312yes—
190C2×C4×C9—21no—
191C2×C4×C9—21no—
192C2×C4×C9—21no—
193C2×C4×C9—31no—
194C2×C4×C9—31no—
195C2×C4×C9—31no—
196C2×C4×C9—62no—
197C2×C4×C9—62no—
198C2×C4×C9—62no—
199C2×C4×C9C22×D311yescharacteristic
200C2×C4×C9C22×D311yescharacteristic
201C2×C4×C9C22×D311yescharacteristic
202#611—64no—
203C2×Dic9—32no—
204C2×Dic9—32no—
205C2×Dic9—62no—
206C2×Dic9—62no—
207C2×Dic9C22×D312yes—
208D36—22no—
209C4×D9—64no—
210Dic18—22no—
211#529—271noSylow
212C2×C3×C9C2×#4111yescharacteristic
213C2×C3×C9C2×#4111yescharacteristic
214C2×C3×C9C2×#4111yescharacteristic
215D3×C9—22no—
216C3×D9—42no—
217C2×C3×Q8—91no—
218C2×C3×D4—91no—
219C22×C3×C4—31no—
220C22×C3×C4—31no—
221C22×C3×C4—31no—
222C2×#67—92no—
223C2×#67—92no—
224C22×Dic3—92no—
225C2×D12—31no—
226C2×C4×D3—92no—
227C2×C4×D3C2×D911yescharacteristic
228C2×C4×D3C2×D911yescharacteristic
229C2×C4×D3C2×D911yescharacteristic
230C2×Dic6—31no—
231C3×C22⋊C4—92no—
232C3×C22⋊C4—92no—
233C3×C42—21no—
234C3×C42—31no—
235C3×C42—31no—
236C3×C42—31no—
237C3×C42C2×D911yescharacteristic
238C3×C42D3211yescharacteristic
239#217—92no—
240#217—92no—
241#217—92no—
242(C2×D3)⋊C4—32no—
243(C2×D3)⋊C4—32no—
244(C2×D3)⋊C4—92no—
245(C2×D3)⋊C4—92no—
246(C2×D3)⋊C4—92no—
247C4⋊Dic3—91no—
248C4⋊Dic3—91no—
249C4⋊Dic3—91no—
250C4⋊Dic3—91no—
251C4⋊Dic3—92no—
252C4⋊Dic3—181no—
253C4⋊Dic3—181no—
254Dic3⋊C4—92no—
255Dic3⋊C4—92no—
256Dic3⋊C4—92no—
257Dic3⋊C4—92no—
258Dic3⋊C4—92no—
259Dic3⋊C4—92no—
260Dic3⋊C4—92no—
261Dic3⋊C4—182no—
262Dic3⋊C4—182no—
263C4×Dic3—92no—
264C4×Dic3—92no—
265C4×Dic3C2×D911yescharacteristic
266C4×Dic3C2×D911yescharacteristic
267C4×Dic3C2×D911yescharacteristic
268C22×C32C23×D311yescharacteristic
269C2×C3×D3—21no—
270C2×C3×D3—21no—
271C2×C3×D3—62no—
272C2×C3×D3—62no—
273C2×C3×D3—62no—
274C2×C3×D3#23512yes—
275C32×C4—21no—
276C32×C4—21no—
277C32×C4#23512yes—
278C3⋊Dic3—62no—
279C3⋊Dic3—62no—
280C3×Dic3—21no—
281C3×Dic3—21no—
282C3×Dic3—62no—
283C3×Dic3#23512yes—
284C22×C9—21no—
285C22×C9—32no—
286C22×C9—61no—
287C22×C9—61no—
288C22×C9C23×D311yescharacteristic
289C2×D9—22no—
290C2×D9—22no—
291C2×D9—22no—
292C4×C9—21no—
293C4×C9—21no—
294C4×C9—22no—
295C4×C9—32no—
296C4×C9—41no—
297C4×C9—41no—
298C4×C9—61no—
299C4×C9—61no—
300C4×C9#23712yes—
301Dic9—22no—
302Dic9—62no—
303Dic9—62no—
304Dic9—122no—
305Dic9—122no—
306#126—272no—
307#125—271no—
308#124—91no—
309#123—272no—
310C22⋊Q8—271no—
311#121—271no—
312C4×Q8—271no—
313C4×D4—271no—
314C4×D4—272no—
315#117—272no—
316C2×C42—31no—
317C3×C9#52911yescharacteristic, complemented, Sylow
318C23×C3—31no—
319C23×C3—92no—
320C22×D3—32no—
321C22×D3C22×D911yescharacteristic
322C3×Q8—182no—
323C3×D4—182no—
324C2×C3×C4—21no—
325C2×C3×C4—21no—
326C2×C3×C4—21no—
327C2×C3×C4—31no—
328C2×C3×C4—31no—
329C2×C3×C4—31no—
330C2×C3×C4—62no—
331C2×C3×C4—62no—
332C2×C3×C4—62no—
333C2×C3×C4—92no—
334C2×C3×C4C22×D911yescharacteristic
335C2×C3×C4C22×D911yescharacteristic
336C2×C3×C4C22×D911yescharacteristic
337C2×C3×C4C2×D3211yescharacteristic
338C2×C3×C4C2×D3211yescharacteristic
339C2×C3×C4C2×D3211yescharacteristic
340#67—184no—
341#67—184no—
342C2×Dic3—32no—
343C2×Dic3—92no—
344C2×Dic3—92no—
345C2×Dic3—92no—
346C2×Dic3—92no—
347C2×Dic3—182no—
348C2×Dic3—182no—
349C2×Dic3—184no—
350C2×Dic3C22×D911yescharacteristic
351C2×Dic3C22×D911yescharacteristic
352C2×Dic3C22×D911yescharacteristic
353D12—62no—
354C4×D3—22no—
355C4×D3—22no—
356C4×D3—22no—
357C4×D3—184no—
358Dic6—62no—
359C2×C32C2×#23511yescharacteristic
360C2×C32C2×#23511yescharacteristic
361C2×C32C2×#23511yescharacteristic
362C3×D3—22no—
363C3×D3—122no—
364C2×C9—21no—
365C2×C9—21no—
366C2×C9—21no—
367C2×C9—62no—
368C2×C9C2×#23711yescharacteristic
369C2×C9D3×#4111yescharacteristic
370C2×C9D3×#4111yescharacteristic
371D9—42no—
372C2×Q8—91no—
373C2×D4—91no—
374C2×D4—272no—
375C22×C4—31no—
376C22×C4—31no—
377C22×C4—31no—
378C22×C4—272no—
379C4⋊C4—272no—
380C4⋊C4—272no—
381C4⋊C4—272no—
382C4⋊C4—271no—
383C4⋊C4—272no—
384C4⋊C4—271no—
385C22⋊C4—92no—
386C22⋊C4—92no—
387C22⋊C4—272no—
388C22⋊C4—272no—
389C22⋊C4—272no—
390C42—31no—
391C42—31no—
392C42—31no—
393C42—272no—
394C42D3×D911yescharacteristic
395C22×C3—21no—
396C22×C3—32no—
397C22×C3—61no—
398C22×C3—61no—
399C22×C3—182no—
400C22×C3—182no—
401C22×C3—182no—
402C22×C3C23×D911yescharacteristic
403C22×C3C22×D3211yescharacteristic
404C2×D3—21no—
405C2×D3—21no—
406C2×D3—62no—
407C2×D3—62no—
408C2×D3—62no—
409C2×D3#369712yes—
410C3×C4—21no—
411C3×C4—21no—
412C3×C4—21no—
413C3×C4—21no—
414C3×C4—22no—
415C3×C4—32no—
416C3×C4—41no—
417C3×C4—41no—
418C3×C4—61no—
419C3×C4—61no—
420C3×C4—182no—
421C3×C4#369712yes—
422C3×C4#379512yes—
423Dic3—21no—
424Dic3—21no—
425Dic3—62no—
426Dic3—182no—
427Dic3—182no—
428Dic3—182no—
429Dic3—182no—
430Dic3—362no—
431Dic3—362no—
432Dic3#369712yes—
433C32#547911yescharacteristic
434C9—21no—
435C9#568011yescharacteristic, complemented
436C23—31no—
437C23—92no—
438Q8—182no—
439D4—182no—
440D4—544no—
441C2×C4—31no—
442C2×C4—31no—
443C2×C4—31no—
444C2×C4—62no—
445C2×C4—62no—
446C2×C4—62no—
447C2×C4—92no—
448C2×C4—272no—
449C2×C4—272no—
450C2×C4—544no—
451C2×C4C2×D3×D911yescharacteristic
452C2×C4C2×D3×D911yescharacteristic
453C2×C4C2×D3×D911yescharacteristic
454C2×C3—21no—
455C2×C3—21no—
456C2×C3—21no—
457C2×C3—62no—
458C2×C3—362no—
459C2×C3C2×#369711yescharacteristic
460C2×C3C2×#369711yescharacteristic
461C2×C3C2×#369711yescharacteristic
462C2×C3C2×#379511yescharacteristic
463C2×C3D3×#23511yescharacteristic
464C2×C3D3×#23511yescharacteristic
465D3—22no—
466D3—122no—
467C22—32no—
468C22—61no—
469C22—61no—
470C22—182no—
471C22—182no—
472C22—182no—
473C22C22×D3×D911yescharacteristic
474C4—21no—
475C4—21no—
476C4—32no—
477C4—61no—
478C4—61no—
479C4—182no—
480C4—542no—
481C4—542no—
482C4#8864112yes—
483C3—21no—
484C3#1058958111yescharacteristic, complemented
485C3#1059392111yescharacteristic
486C2—62no—
487C2—362no—
488C2C2×#8864111yescharacteristic
489C2D3×#369711yescharacteristic
490C2D3×#369711yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC3×C42×C9C22
socleC22×C32C23×D3
prime radical#11721037C2
hypercenterC42D3×D9
hypocenterC3×C9#529
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
1. #11721707C2semidirect
2. C42×D3×C9C2semidirect
3. C3×#63437C2semidirect
5. #11721032C2semidirect
6. Dic3×D36C2semidirect
7. #11720976C2semidirect
8. #11720974C2semidirect
9. #11720960C2semidirect
10. #11720947C2semidirect
11. Dic3×Dic18C2semidirect
54. #63437D3semidirect
165. C42×D3D9semidirect
317. C3×C9#529semidirect
435. C9#5680semidirect
484. C3#10589581semidirect

Maximal subgroups

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

subgroupindex
#117217072
C42×D3×C92
C3×#634372
#117210372
#117210322
Dic3×D362
#117209762
#117209742
#117209602
#117209472
Dic3×Dic182
#105939213
#105895813