Finite Group Databasebeta

← #13147660

Subgroups — #13147660

Order 1728.

772 conjugacy classes of subgroups.

The subgroup lattice has 494 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#6226C211yesmaximal, characteristic, complemented
2C42×D3×C9C211yesmaximal, characteristic, complemented
3C3×C4×D36C211yesmaximal, characteristic, complemented
4#11721037C211yesmaximal, characteristic
5#11721032C212yesmaximal, complemented
6C4×#6194C212yesmaximal, complemented
7#11720961C211yesmaximal, characteristic, complemented
8#11720948C211yesmaximal, characteristic, complemented
9#11720947C212yesmaximal, complemented
10#11720943C212yesmaximal, complemented
11#11720927C211yesmaximal, characteristic, complemented
12#10593974—31nomaximal
13#10589581—31nomaximal
14C2×#6226C2211yescharacteristic
15C2×C4×D3×C9C2211yescharacteristic
16C2×C4×D3×C9C2211yescharacteristic
17C2×C4×D3×C9C2211yescharacteristic
18C2×C3×D36C2211yescharacteristic
19C2×C3×C4×D9C2212yes—
20C2×#6194C2212yes—
21C3×C42×C9C2211yescharacteristic
22C4⋊(C3⋊Dic9)C2211yescharacteristic
23#88534C2212yes—
24C4×C3⋊Dic9C2212yes—
25C3×(C2×D9)⋊C4C2212yes—
26C4×C9×Dic3C2211yescharacteristic
27C4×C9×Dic3C2211yescharacteristic
28C4×C9×Dic3C2211yescharacteristic
29C3×C4⋊Dic9C2211yescharacteristic
30(C2×D3)⋊Dic9C2212yes—
31(C2×D9)⋊Dic3C2212yes—
32(C2×D9)⋊Dic3C2212yes—
33(C2×D9)⋊Dic3C2212yes—
34#88442C2212yes—
35Dic9⋊Dic3C2212yes—
36Dic9⋊Dic3C2212yes—
37C4×#634—31no—
38C3×C4×D12—31no—
39C3×C42×D3D311yescharacteristic
40#63908—31no—
41#63905—32no—
42C4×#625—32no—
43#63869—31no—
44#63860—31no—
45#63859—32no—
46#63857—31no—
47#63855—32no—
48C2×C42×C9—31no—
49#63492—31no—
50#63491—32no—
51C4×#611—32no—
52#63486—31no—
53#63438—32no—
54#63437—31no—
55C4×D36D311yescharacteristic, complemented
56#63434—32no—
57#63432—31no—
58C4×Dic18—31no—
59C4×Dic18—61no—
60C22×D3×C9C2311yescharacteristic
61C22×C3×D9C2312yes—
62C2×C3×C4×C9C2311yescharacteristic
63C2×C3×C4×C9C2311yescharacteristic
64C2×C3×C4×C9C2311yescharacteristic
65C2×C3⋊Dic9C2312yes—
66C2×C3⋊Dic9C2312yes—
67#6226—22no—
68C2×C9×Dic3C2311yescharacteristic
69C2×C9×Dic3C2311yescharacteristic
70C2×C9×Dic3C2311yescharacteristic
71C2×C3×Dic9C2312yes—
72C4×D3×C9—22no—
73C4×D3×C9—22no—
74C4×D3×C9—22no—
75C3×D36—22no—
76C3×C4×D9—24no—
77#6194—24no—
78#5557—91no—
79#5479—91no—
80C2×#634—31no—
81C2×C3×D12—31no—
82C2×C3×C4×D3—32no—
83C2×C3×C4×D3C2×D311yescharacteristic
84C2×C3×C4×D3C2×D311yescharacteristic
85C2×C3×C4×D3C2×D311yescharacteristic
86C2×#625—32no—
87C32×C42C2×D311yescharacteristic
88C4⋊(C3⋊Dic3)—31no—
89#3750—32no—
90C4×C3⋊Dic3—32no—
91C3×(C2×D3)⋊C4—32no—
92C3×C4⋊Dic3—31no—
93C3×C4×Dic3C2×D311yescharacteristic
94C3×C4×Dic3C2×D311yescharacteristic
95C3×C4×Dic3C2×D311yescharacteristic
96#3724—32no—
97Dic3⋊Dic3—32no—
98Dic3⋊Dic3—32no—
99(C2×D3)⋊Dic3—32no—
100(C2×D3)⋊Dic3—32no—
101(C2×D3)⋊Dic3—32no—
102(C2×D3)⋊Dic3—32no—
103C22×C4×C9—31no—
104C22×C4×C9—31no—
105C22×C4×C9—31no—
106C2×#611—32no—
107C2×D36C2×D311yescharacteristic
108C2×C4×D9C2×D312yes—
109C2×Dic18—31no—
110C2×Dic18—61no—
111C42×C9—21no—
112C42×C9—31no—
113C42×C9—31no—
114C42×C9—31no—
115C42×C9C2×D311yescharacteristic
116#3676—32no—
117(C2×D9)⋊C4—32no—
118(C2×D9)⋊C4—32no—
119(C2×D9)⋊C4—32no—
120(C2×D9)⋊C4C2×D312yes—
121C4⋊Dic9—31no—
122C4⋊Dic9—61no—
123C4⋊Dic9C2×D311yescharacteristic
124Dic9⋊C4—32no—
125Dic9⋊C4—32no—
126Dic9⋊C4—32no—
127Dic9⋊C4—32no—
128Dic9⋊C4—62no—
129C4×Dic9—32no—
130C4×Dic9—62no—
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#4112yes—
140C3×C4×C9#4112yes—
141C3⋊Dic9—22no—
142C3⋊Dic9—22no—
143C9×Dic3—21no—
144C9×Dic3—21no—
145C9×Dic3—21no—
146C3×Dic9—22no—
147C3×C4×D4—91no—
148C2×C3×C42—31no—
149#938—91no—
150#936—91no—
151#930—91no—
152#929—92no—
153C4×#67—92no—
154C4×#67—92no—
155#926—92no—
156#924—91no—
157#897—91no—
158#890—91no—
159#883—92no—
160#879—92no—
161#876—92no—
162#875—91no—
163C4×D12—31no—
164#872—92no—
165C42×D3D911yescharacteristic, complemented
166#870—91no—
167C4×Dic6—91no—
168C4×Dic6—91no—
169C4×Dic6—181no—
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—
177#634—62no—
178C2×C3×Dic3—32no—
179C2×C3×Dic3C22×D311yescharacteristic
180C2×C3×Dic3C22×D311yescharacteristic
181C2×C3×Dic3C22×D311yescharacteristic
182C3×D12—62no—
183C3×C4×D3—22no—
184C3×C4×D3—22no—
185C3×C4×D3—22no—
186C3×C4×D3—64no—
187#625—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—24no—
210Dic18—64no—
211Dic18—62no—
212#529—271noSylow
213C2×C3×C9C2×#4111yescharacteristic
214C2×C3×C9C2×#4111yescharacteristic
215C2×C3×C9C2×#4111yescharacteristic
216D3×C9—22no—
217C3×D9—42no—
218C2×C3×D4—91no—
219C22×C3×C4—31no—
220C22×C3×C4—31no—
221C22×C3×C4—31no—
222C22×C3×C4—92no—
223C2×#67—92no—
224C2×#67—92no—
225C2×D12—31no—
226C2×C4×D3—32no—
227C2×C4×D3C2×D911yescharacteristic
228C2×C4×D3C2×D911yescharacteristic
229C2×C4×D3C2×D911yescharacteristic
230C2×Dic6—91no—
231C2×Dic6—91no—
232C2×Dic6—181no—
233C3×C4⋊C4—91no—
234C3×C22⋊C4—92no—
235C3×C42—21no—
236C3×C42—31no—
237C3×C42—31no—
238C3×C42—31no—
239C3×C42C2×D911yescharacteristic
240C3×C42D3211yescharacteristic
241#217—92no—
242#217—92no—
243#217—92no—
244#217—92no—
245(C2×D3)⋊C4—32no—
246(C2×D3)⋊C4—92no—
247(C2×D3)⋊C4—92no—
248(C2×D3)⋊C4—92no—
249(C2×D3)⋊C4—92no—
250C4⋊Dic3—31no—
251C4⋊Dic3—91no—
252C4⋊Dic3—91no—
253C4⋊Dic3—92no—
254C4⋊Dic3—92no—
255C4⋊Dic3—181no—
256Dic3⋊C4—92no—
257Dic3⋊C4—92no—
258Dic3⋊C4—92no—
259Dic3⋊C4—92no—
260Dic3⋊C4—92no—
261Dic3⋊C4—92no—
262Dic3⋊C4—182no—
263C4×Dic3—92no—
264C4×Dic3—92no—
265C4×Dic3—182no—
266C4×Dic3C2×D911yescharacteristic
267C4×Dic3C2×D911yescharacteristic
268C4×Dic3C2×D911yescharacteristic
269C22×C32C23×D311yescharacteristic
270C2×C3×D3—21no—
271C2×C3×D3—21no—
272C2×C3×D3—62no—
273C2×C3×D3—62no—
274C2×C3×D3—62no—
275C2×C3×D3#23512yes—
276C32×C4—21no—
277C32×C4#23512yes—
278C32×C4#23512yes—
279C3⋊Dic3—62no—
280C3⋊Dic3—62no—
281C3×Dic3—21no—
282C3×Dic3—21no—
283C3×Dic3—21no—
284C3×Dic3—62no—
285C22×C9—21no—
286C22×C9—32no—
287C22×C9—61no—
288C22×C9—61no—
289C22×C9C23×D311yescharacteristic
290C2×D9—22no—
291C2×D9—22no—
292C2×D9—22no—
293C4×C9—21no—
294C4×C9—22no—
295C4×C9—22no—
296C4×C9—41no—
297C4×C9—61no—
298C4×C9—61no—
299C4×C9—61no—
300C4×C9#23512yes—
301C4×C9#23712yes—
302Dic9—22no—
303Dic9—64no—
304Dic9—62no—
305Dic9—62no—
306Dic9—122no—
307#126—272no—
308#125—271no—
309#124—271no—
310#123—272no—
311C22⋊Q8—271no—
312#121—271no—
313C4×Q8—271no—
314C4×D4—91no—
315C4×D4—272no—
316#117—272no—
317C2×C42—31no—
318C3×C9#52911yescharacteristic, complemented, Sylow
319C23×C3—31no—
320C23×C3—92no—
321C22×D3—32no—
322C22×D3C22×D911yescharacteristic
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×C4—184no—
335C2×C3×C4C22×D911yescharacteristic
336C2×C3×C4C22×D911yescharacteristic
337C2×C3×C4C22×D911yescharacteristic
338C2×C3×C4C2×D3211yescharacteristic
339C2×C3×C4C2×D3211yescharacteristic
340C2×C3×C4C2×D3211yescharacteristic
341#67—184no—
342#67—184no—
343C2×Dic3—32no—
344C2×Dic3—92no—
345C2×Dic3—92no—
346C2×Dic3—92no—
347C2×Dic3—92no—
348C2×Dic3—182no—
349C2×Dic3—182no—
350C2×Dic3C22×D911yescharacteristic
351C2×Dic3C22×D911yescharacteristic
352C2×Dic3C22×D911yescharacteristic
353D12—62no—
354C4×D3—22no—
355C4×D3—22no—
356C4×D3—22no—
357C4×D3—64no—
358Dic6—184no—
359Dic6—182no—
360Dic6—182no—
361C2×C32C2×#23511yescharacteristic
362C2×C32C2×#23511yescharacteristic
363C2×C32C2×#23511yescharacteristic
364C3×D3—22no—
365C3×D3—122no—
366C2×C9—21no—
367C2×C9—21no—
368C2×C9—21no—
369C2×C9—62no—
370C2×C9C2×#23511yescharacteristic
371C2×C9C2×#23711yescharacteristic
372C2×C9D3×#4111yescharacteristic
373D9—42no—
374C2×Q8—271no—
375C2×D4—91no—
376C2×D4—272no—
377C22×C4—31no—
378C22×C4—31no—
379C22×C4—31no—
380C22×C4—92no—
381C4⋊C4—91no—
382C4⋊C4—272no—
383C4⋊C4—272no—
384C4⋊C4—272no—
385C4⋊C4—271no—
386C4⋊C4—272no—
387C22⋊C4—92no—
388C22⋊C4—272no—
389C22⋊C4—272no—
390C22⋊C4—272no—
391C22⋊C4—272no—
392C42—31no—
393C42—31no—
394C42—31no—
395C42—272no—
396C42D3×D911yescharacteristic
397C22×C3—21no—
398C22×C3—32no—
399C22×C3—61no—
400C22×C3—61no—
401C22×C3—182no—
402C22×C3—182no—
403C22×C3—182no—
404C22×C3C23×D911yescharacteristic
405C22×C3C22×D3211yescharacteristic
406C2×D3—21no—
407C2×D3—21no—
408C2×D3—62no—
409C2×D3—62no—
410C2×D3—62no—
411C2×D3#369712yes—
412C3×C4—21no—
413C3×C4—21no—
414C3×C4—22no—
415C3×C4—22no—
416C3×C4—41no—
417C3×C4—61no—
418C3×C4—61no—
419C3×C4—61no—
420C3×C4—182no—
421C3×C4#369712yes—
422C3×C4#369712yes—
423C3×C4#379512yes—
424C3×C4#379812yes—
425Dic3—21no—
426Dic3—21no—
427Dic3—21no—
428Dic3—62no—
429Dic3—184no—
430Dic3—182no—
431Dic3—182no—
432Dic3—182no—
433Dic3—182no—
434Dic3—362no—
435C32#547911yescharacteristic
436C9—21no—
437C9#555711yescharacteristic, complemented
438C23—31no—
439C23—92no—
440Q8—542no—
441D4—182no—
442D4—544no—
443C2×C4—31no—
444C2×C4—31no—
445C2×C4—31no—
446C2×C4—62no—
447C2×C4—62no—
448C2×C4—62no—
449C2×C4—92no—
450C2×C4—184no—
451C2×C4—272no—
452C2×C4—272no—
453C2×C4C2×D3×D911yescharacteristic
454C2×C4C2×D3×D911yescharacteristic
455C2×C4C2×D3×D911yescharacteristic
456C2×C3—21no—
457C2×C3—21no—
458C2×C3—21no—
459C2×C3—62no—
460C2×C3—362no—
461C2×C3C2×#369711yescharacteristic
462C2×C3C2×#369711yescharacteristic
463C2×C3C2×#369711yescharacteristic
464C2×C3C2×#379511yescharacteristic
465C2×C3C2×#379811yescharacteristic
466C2×C3D3×#23511yescharacteristic
467D3—22no—
468D3—122no—
469C22—32no—
470C22—61no—
471C22—61no—
472C22—182no—
473C22—182no—
474C22—182no—
475C22C22×D3×D911yescharacteristic
476C4—21no—
477C4—61no—
478C4—61no—
479C4—61no—
480C4—182no—
481C4—542no—
482C4—542no—
483C4#8864012yes—
484C4#8864112yes—
485C3—21no—
486C3#1058958111yescharacteristic, complemented
487C3#1059397411yescharacteristic
488C2—62no—
489C2—362no—
490C2C2×#8864011yescharacteristic
491C2C2×#8864111yescharacteristic
492C2D3×#369711yescharacteristic
subgroupG/H
centerC2×C4C2×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 ◁ C2×C4 ◁ C42

Complements

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

NH ≅ G/Ntype
1. C4×#6226C2semidirect
2. C42×D3×C9C2semidirect
3. C3×C4×D36C2semidirect
5. #11721032C2semidirect
6. C4×#6194C2semidirect
7. #11720961C2semidirect
8. #11720948C2semidirect
9. #11720947C2semidirect
10. #11720943C2semidirect
11. #11720927C2semidirect
55. C4×D36D3semidirect
165. C42×D3D9semidirect
318. C3×C9#529semidirect
437. C9#5557semidirect
486. C3#10589581semidirect

Maximal subgroups

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

subgroupindex
C4×#62262
C42×D3×C92
C3×C4×D362
#117210372
#117210322
C4×#61942
#117209612
#117209482
#117209472
#117209432
#117209272
#105939743
#105895813