Finite Group Databasebeta

← #13153906

Subgroups — #13153906

Order 1728.

1450 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88736C211yesmaximal, characteristic, complemented
2C22×#6226C211yesmaximal, characteristic, complemented
3C3×C4×D4×C9C211yesmaximal, characteristic, complemented
4#11721839C212yesmaximal, complemented
5#11721819C211yesmaximal, characteristic, complemented
6#11721813C211yesmaximal, characteristic, complemented
7#11721752C212yesmaximal, complemented
8#11721731C212yesmaximal, complemented
9#11721727C212yesmaximal, complemented
10#11721712C211yesmaximal, characteristic, complemented
11C4×#6226C211yesmaximal, characteristic, complemented
12#10596023—31nomaximal
13#10589654—33nomaximal
14C2×C3×D4×C9C2211yescharacteristic
15C22×C3×C4×C9C2211yescharacteristic
16C22×C3×C4×C9C2211yescharacteristic
17C2×#6233C2212yes—
18C2×#6233C2212yes—
19C22×C3⋊Dic9C2212yes—
20#88736—24no—
21C2×C4⋊(C3⋊D9)C2211yescharacteristic
22C2×C4×C3⋊D9C2212yes—
23C2×#6226—24no—
24C2×#6226C2212yes—
25C2×#6226C2211yescharacteristic
26C2×#6226C2211yescharacteristic
27C3×C9×C4⋊C4C2211yescharacteristic
28C3×C9×C22⋊C4C2211yescharacteristic
29C3×C9×C22⋊C4C2211yescharacteristic
30C3×C42×C9C2211yescharacteristic
31#88551C2212yes—
32#88536C2212yes—
33#88536C2212yes—
34#88536C2212yes—
35C4⋊(C3⋊Dic9)C2211yescharacteristic
36#88534C2212yes—
37#88534C2212yes—
38C4×C3⋊Dic9C2212yes—
39C2×#3828—31no—
40C22×#634—31no—
41C32×C4×D4D311yescharacteristic
42#64148—32no—
43#64139—31no—
44#64133—31no—
45#64105—32no—
46#64094—32no—
47#64090—32no—
48#64084—31no—
49C4×#634—31no—
50C2×#3697—33no—
51C22×Dic18—33no—
52C4×D4×C9D313yescomplemented
53#63501—36no—
54#63492—33no—
55#63486—33no—
56#63458—36no—
57#63447—36no—
58#611⋊C4—36no—
59#63437—33no—
60C4×Dic18—33no—
61C23×C3×C9C2311yescharacteristic
62C23×C3×C9C2311yescharacteristic
63C22×C3⋊D9C2312yes—
64C3×D4×C9—22no—
65C2×C3×C4×C9—22no—
66C2×C3×C4×C9—22no—
67C2×C3×C4×C9C2311yescharacteristic
68C2×C3×C4×C9C2311yescharacteristic
69C2×C3×C4×C9C2311yescharacteristic
70C2×C3×C4×C9C2311yescharacteristic
71C2×C3×C4×C9C2311yescharacteristic
72#6233—24no—
73#6233—24no—
74C2×C3⋊Dic9—24no—
75C2×C3⋊Dic9C2312yes—
76C2×C3⋊Dic9C2312yes—
77C2×C3⋊Dic9C2312yes—
78C4⋊(C3⋊D9)—22no—
79C4×C3⋊D9—24no—
80#6226—22no—
81#6226—24no—
82#6226D414yescomplemented
83#5552—93no—
84#5552—91no—
85C2×C32×D4C2×D311yescharacteristic
86C22×C32×C4C2×D311yescharacteristic
87C22×C32×C4C2×D311yescharacteristic
88C2×#638—32no—
89C2×#638—32no—
90C22×C3⋊Dic3—32no—
91#3828—64no—
92C2×C4⋊(C3⋊D3)—31no—
93C2×C4×C3⋊D3—32no—
94C2×#634—31no—
95C2×#634—31no—
96C2×#634—32no—
97C2×#634—64no—
98C32×C4⋊C4C2×D311yescharacteristic
99C32×C22⋊C4C2×D311yescharacteristic
100C32×C22⋊C4C2×D311yescharacteristic
101C32×C42C2×D311yescharacteristic
102#3757—32no—
103#3752—32no—
104#3752—32no—
105#3752—32no—
106C4⋊(C3⋊Dic3)—31no—
107#3750—32no—
108#3750—32no—
109C4×C3⋊Dic3—32no—
110C2×D4×C9C2×D313yes—
111C22×C4×C9C2×D313yes—
112C22×C4×C9C2×D313yes—
113C2×#611—36no—
114C2×#611—36no—
115C22×Dic9—36no—
116#3697—612no—
117C2×D36—33no—
118C2×C4×D9—36no—
119C2×Dic18—33no—
120C2×Dic18—33no—
121C2×Dic18—36no—
122C2×Dic18—612no—
123C9×C4⋊C4C2×D313yes—
124C9×C22⋊C4C2×D313yes—
125C9×C22⋊C4C2×D313yes—
126C42×C9C2×D313yes—
127#3676—36no—
128(C2×D9)⋊C4—36no—
129(C2×D9)⋊C4—36no—
130(C2×D9)⋊C4—36no—
131C4⋊Dic9—33no—
132Dic9⋊C4—36no—
133Dic9⋊C4—36no—
134C4×Dic9—36no—
135C22×C3×C9—21no—
136C22×C3×C9—21no—
137C22×C3×C9—21no—
138C22×C3×C9—21no—
139C22×C3×C9—21no—
140C22×C3×C9#4112yescomplemented
141C22×C3×C9C2411yescharacteristic
142C2×C3⋊D9—22no—
143C2×C3⋊D9—22no—
144C2×C3⋊D9—22no—
145C3×C4×C9—21no—
146C3×C4×C9—21no—
147C3×C4×C9—21no—
148C3×C4×C9—21no—
149C3×C4×C9C2×D412yes—
150C3⋊Dic9—22no—
151C3⋊Dic9—22no—
152C3⋊Dic9C2×D414yes—
153C2×#235—93no—
154C2×#235—91no—
155C22×Dic6—93no—
156C22×Dic6—91no—
157C3×C4×D4D913yescomplemented
158C3×C4×D4C3⋊D311yescharacteristic
159#939—96no—
160#939—92no—
161#930—93no—
162#930—91no—
163#924—93no—
164#924—91no—
165#896—96no—
166#896—92no—
167#885—96no—
168#885—92no—
169#67⋊C4—96no—
170#67⋊C4—92no—
171#875—93no—
172#875—91no—
173C4×Dic6—93no—
174C4×Dic6—91no—
175C23×C32C22×D311yescharacteristic
176C23×C32C22×D311yescharacteristic
177C22×C3⋊D3—32no—
178C32×D4—22no—
179C2×C32×C4—22no—
180C2×C32×C4—22no—
181C2×C32×C4C22×D311yescharacteristic
182C2×C32×C4C22×D311yescharacteristic
183C2×C32×C4C22×D311yescharacteristic
184C2×C32×C4C22×D311yescharacteristic
185C2×C32×C4C22×D311yescharacteristic
186#638—64no—
187#638—64no—
188C2×C3⋊Dic3—32no—
189C2×C3⋊Dic3—32no—
190C2×C3⋊Dic3—32no—
191C2×C3⋊Dic3—64no—
192C4⋊(C3⋊D3)—62no—
193C4×C3⋊D3—64no—
194#634—34no—
195#634—64no—
196#634—62no—
197C23×C9C22×D313yes—
198C23×C9C22×D313yes—
199C22×D9—36no—
200D4×C9—26no—
201C2×C4×C9—26no—
202C2×C4×C9—26no—
203C2×C4×C9C22×D313yes—
204C2×C4×C9C22×D313yes—
205C2×C4×C9C22×D313yes—
206C2×C4×C9C22×D313yes—
207C2×C4×C9C22×D313yes—
208#611—612no—
209#611—612no—
210C2×Dic9—36no—
211C2×Dic9—36no—
212C2×Dic9—36no—
213C2×Dic9—612no—
214D36—66no—
215C4×D9—612no—
216Dic18—312no—
217Dic18—612no—
218Dic18—66no—
219#548—271noSylow
220C2×C3×C9—22no—
221C2×C3×C9—41no—
222C2×C3×C9C22×D411yescharacteristic
223C2×C3×C9C2×#4111yescharacteristic
224C2×C3×C92−1+411yescharacteristic
225C3⋊D9—42no—
226C2×C3×D4C2×D913yes—
227C2×C3×D4C2×C3⋊D311yescharacteristic
228C22×C3×C4C2×D913yes—
229C22×C3×C4C2×D913yes—
230C22×C3×C4C2×C3⋊D311yescharacteristic
231C22×C3×C4C2×C3⋊D311yescharacteristic
232C2×#67—96no—
233C2×#67—96no—
234C2×#67—92no—
235C2×#67—92no—
236C22×Dic3—96no—
237C22×Dic3—92no—
238#235—184no—
239#235—1812no—
240C2×D12—93no—
241C2×D12—91no—
242C2×C4×D3—96no—
243C2×C4×D3—92no—
244C2×Dic6—93no—
245C2×Dic6—91no—
246C2×Dic6—93no—
247C2×Dic6—96no—
248C2×Dic6—91no—
249C2×Dic6—92no—
250C2×Dic6—184no—
251C2×Dic6—1812no—
252C3×C4⋊C4C2×D913yes—
253C3×C4⋊C4C2×C3⋊D311yescharacteristic
254C3×C22⋊C4C2×D913yes—
255C3×C22⋊C4C2×D913yes—
256C3×C22⋊C4C2×C3⋊D311yescharacteristic
257C3×C22⋊C4C2×C3⋊D311yescharacteristic
258C3×C42C2×D913yes—
259C3×C42C2×C3⋊D311yescharacteristic
260#217—96no—
261#217—92no—
262(C2×D3)⋊C4—96no—
263(C2×D3)⋊C4—96no—
264(C2×D3)⋊C4—96no—
265(C2×D3)⋊C4—92no—
266(C2×D3)⋊C4—92no—
267(C2×D3)⋊C4—92no—
268C4⋊Dic3—93no—
269C4⋊Dic3—91no—
270Dic3⋊C4—96no—
271Dic3⋊C4—96no—
272Dic3⋊C4—92no—
273Dic3⋊C4—92no—
274C4×Dic3—96no—
275C4×Dic3—92no—
276C22×C32—21no—
277C22×C32—21no—
278C22×C32—21no—
279C22×C32—21no—
280C22×C32—21no—
281C22×C32#23512yes—
282C22×C32C23×D311yescharacteristic
283C2×C3⋊D3—62no—
284C2×C3⋊D3—62no—
285C2×C3⋊D3—62no—
286C32×C4—21no—
287C32×C4—21no—
288C32×C4—21no—
289C32×C4—21no—
290C32×C4D3×D412yes—
291C3⋊Dic3—34no—
292C3⋊Dic3—62no—
293C3⋊Dic3—62no—
294C22×C9—23no—
295C22×C9—23no—
296C22×C9—23no—
297C22×C9—23no—
298C22×C9—23no—
299C22×C9#23516yescomplemented
300C22×C9C23×D313yes—
301C2×D9—66no—
302C2×D9—66no—
303C2×D9—66no—
304C4×C9—23no—
305C4×C9—23no—
306C4×C9—23no—
307C4×C9—23no—
308C4×C9D3×D416yes—
309Dic9—312no—
310Dic9—66no—
311Dic9—66no—
312C2×#41—271no—
313C22×Q8—271no—
314#124—272no—
315#124—271no—
316C22⋊Q8—271no—
317C22⋊Q8—272no—
318#121—272no—
319#121—271no—
320C4×Q8—271no—
321C4×D4—272no—
322C4×D4C3⋊D911yescharacteristic, complemented
323C3×C9#54811yescharacteristic, complemented, Sylow
324C23×C3C22×D913yes—
325C23×C3C22×D913yes—
326C23×C3C22×C3⋊D311yescharacteristic
327C23×C3C22×C3⋊D311yescharacteristic
328C22×D3—92no—
329C22×D3—96no—
330C3×D4—26no—
331C3×D4—22no—
332C2×C3×C4—26no—
333C2×C3×C4—26no—
334C2×C3×C4—22no—
335C2×C3×C4—22no—
336C2×C3×C4C22×D913yes—
337C2×C3×C4C22×D913yes—
338C2×C3×C4C22×D913yes—
339C2×C3×C4C22×D913yes—
340C2×C3×C4C22×D913yes—
341C2×C3×C4C22×C3⋊D311yescharacteristic
342C2×C3×C4C22×C3⋊D311yescharacteristic
343C2×C3×C4C22×C3⋊D311yescharacteristic
344C2×C3×C4C22×C3⋊D311yescharacteristic
345C2×C3×C4C22×C3⋊D311yescharacteristic
346#67—184no—
347#67—1812no—
348#67—1812no—
349#67—184no—
350C2×Dic3—92no—
351C2×Dic3—96no—
352C2×Dic3—92no—
353C2×Dic3—92no—
354C2×Dic3—96no—
355C2×Dic3—96no—
356C2×Dic3—184no—
357C2×Dic3—1812no—
358D12—186no—
359D12—182no—
360C4×D3—184no—
361C4×D3—1812no—
362Dic6—912no—
363Dic6—94no—
364Dic6—184no—
365Dic6—1812no—
366Dic6—186no—
367Dic6—182no—
368C2×C32—22no—
369C2×C32—41no—
370C2×C32C2×#23511yescharacteristic
371C2×C32C2×D3×D411yescharacteristic
372C2×C32#101011yescharacteristic
373C3⋊D3—122no—
374C2×C9—26no—
375C2×C9—43no—
376C2×C9C2×#23513yes—
377C2×C9C2×D3×D413yes—
378C2×C9#101013yes—
379D9—126no—
380#41—544no—
381C2×Q8—271no—
382C2×Q8—272no—
383C2×Q8—271no—
384C2×Q8—544no—
385C2×D4—272no—
386C2×D4—272no—
387C2×D4—271no—
388C2×D4C2×C3⋊D911yescharacteristic
389C22×C4—272no—
390C22×C4—272no—
391C22×C4C2×C3⋊D911yescharacteristic
392C22×C4C2×C3⋊D911yescharacteristic
393C4⋊C4—272no—
394C4⋊C4—272no—
395C4⋊C4—271no—
396C4⋊C4C2×C3⋊D911yescharacteristic
397C22⋊C4—272no—
398C22⋊C4—272no—
399C22⋊C4—272no—
400C22⋊C4—272no—
401C22⋊C4C2×C3⋊D911yescharacteristic
402C22⋊C4C2×C3⋊D911yescharacteristic
403C42—272no—
404C42C2×C3⋊D911yescharacteristic
405C22×C3—21no—
406C22×C3—21no—
407C22×C3—23no—
408C22×C3—23no—
409C22×C3—21no—
410C22×C3—21no—
411C22×C3—21no—
412C22×C3—23no—
413C22×C3—23no—
414C22×C3—23no—
415C22×C3#369716yescomplemented
416C22×C3C23×D913yes—
417C22×C3#382812yes—
418C22×C3C23×C3⋊D311yescharacteristic
419C2×D3—182no—
420C2×D3—182no—
421C2×D3—182no—
422C2×D3—186no—
423C2×D3—186no—
424C2×D3—186no—
425C3×C4—21no—
426C3×C4—21no—
427C3×C4—21no—
428C3×C4—21no—
429C3×C4—23no—
430C3×C4—23no—
431C3×C4—23no—
432C3×C4—23no—
433C3×C4D4×D916yes—
434C3×C4D4×C3⋊D312yes—
435Dic3—94no—
436Dic3—912no—
437Dic3—182no—
438Dic3—182no—
439Dic3—186no—
440Dic3—186no—
441C32#555211yescharacteristic
442C9#555213yescomplemented
443C23—272no—
444C23C22×C3⋊D911yescharacteristic
445C23C22×C3⋊D911yescharacteristic
446Q8—274no—
447Q8—544no—
448Q8—542no—
449D4—22no—
450D4—544no—
451D4—544no—
452D4—542no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—272no—
456C2×C4—272no—
457C2×C4—272no—
458C2×C4—544no—
459C2×C4—544no—
460C2×C4C22×C3⋊D911yescharacteristic
461C2×C4C22×C3⋊D911yescharacteristic
462C2×C4C22×C3⋊D911yescharacteristic
463C2×C4C22×C3⋊D911yescharacteristic
464C2×C4C22×C3⋊D911yescharacteristic
465C2×C3—22no—
466C2×C3—26no—
467C2×C3—41no—
468C2×C3—43no—
469C2×C3C2×#369713yes—
470C2×C3C2×D4×D913yes—
471C2×C3#6371613yes—
472C2×C3C2×#382811yescharacteristic
473C2×C3C2×D4×C3⋊D311yescharacteristic
474C2×C3#6436711yescharacteristic
475D3—362no—
476D3—366no—
477C22—21no—
478C22—21no—
479C22—21no—
480C22—21no—
481C22—21no—
482C22—542no—
483C22—542no—
484C22—542no—
485C22#8873612yescomplemented
486C22C23×C3⋊D911yescharacteristic
487C4—21no—
488C4—21no—
489C4—21no—
490C4—21no—
491C4—274no—
492C4—542no—
493C4—542no—
494C4D4×C3⋊D912yes—
495C3#1058965413yescomplemented
496C3#1059602311yescharacteristic
497C2—22no—
498C2—41no—
499C2—1082no—
500C2C2×#8873611yescharacteristic
501C2C2×D4×C3⋊D911yescharacteristic
502C2#1172267311yescharacteristic
subgroupG/H
centerC22C23×C3⋊D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C23×C3⋊D3
FittingC3×C4×D4×C9C2
socleC22×C32C23×D3
hypercenterC4×D4C3⋊D9
hypocenterC3×C9#548
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C4×D4

Complements

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

NH ≅ G/Ntype
1. C2×#88736C2semidirect
2. C22×#6226C2semidirect
3. C3×C4×D4×C9C2semidirect
4. #11721839C2semidirect
5. #11721819C2semidirect
6. #11721813C2semidirect
7. #11721752C2semidirect
8. #11721731C2semidirect
9. #11721727C2semidirect
10. #11721712C2semidirect
11. C4×#6226C2semidirect
52. C4×D4×C9D3semidirect
82. #6226D4semidirect
140. C22×C3×C9#41semidirect
157. C3×C4×D4D9semidirect
299. C22×C9#235semidirect
322. C4×D4C3⋊D9semidirect
323. C3×C9#548semidirect
415. C22×C3#3697semidirect
442. C9#5552semidirect
485. C22#88736semidirect
495. C3#10589654semidirect

Maximal subgroups

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

subgroupindex
C2×#887362
C22×#62262
C3×C4×D4×C92
#117218392
#117218192
#117218132
#117217522
#117217312
#117217272
#117217122
C4×#62262
#105960233
#105896543