Finite Group Databasebeta

← #13147601

Subgroups — #13147601

Order 1728.

964 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88639C211yesmaximal, characteristic, complemented
2C2×(C3⋊D9)⋊Q8C211yesmaximal, characteristic, complemented
3C4×C4⋊(C3⋊D9)C211yesmaximal, characteristic, complemented
4C9×#875C211yesmaximal, characteristic, complemented
5C3×C4×Dic18C211yesmaximal, characteristic, complemented
6#11721045C211yesmaximal, characteristic, complemented
7#11721039C212yesmaximal, complemented
8#11721023C212yesmaximal, complemented
9#11721013C212yesmaximal, complemented
10#11720956C212yesmaximal, complemented
11#11720929C211yesmaximal, characteristic, complemented
12#10593901—31nomaximal
13#10589779—31nomaximal
14C2×C4⋊(C3⋊D9)C2211yescharacteristic
15C2×C4×C3⋊D9C2211yescharacteristic
16C2×C4×C3⋊D9C2211yescharacteristic
17C2×C9×D12C2211yescharacteristic
18C2×C9×Dic6C2211yescharacteristic
19C2×C3×Dic18C2211yescharacteristic
20C2×#6195C2212yes—
21C2×#6195C2212yes—
22C2×D3×Dic9C2212yes—
23C2×(C3⋊D9)⋊C4C2212yes—
24C2×#6189C2212yes—
25#88639—24no—
26(C3⋊D9)⋊Q8—24no—
27C3×C42×C9C2211yescharacteristic
28#88536C2211yescharacteristic
29#88536C2211yescharacteristic
30C4⋊(C3⋊Dic9)C2211yescharacteristic
31C9×(C2×D3)⋊C4C2212yes—
32C9×(C2×D3)⋊C4C2212yes—
33C3×C4⋊Dic9C2211yescharacteristic
34C3×Dic9⋊C4C2212yes—
35C3×C4×Dic9C2212yes—
36(C2×D3)⋊Dic9C2212yes—
37#88445C2212yes—
38Dic9⋊Dic3C2212yes—
39C2×#3797—31no—
40C2×#3796—31no—
41C4×C4⋊(C3⋊D3)—31no—
42C3×#875D311yescharacteristic
43C3×C4×Dic6—31no—
44#63916—31no—
45#63910—32no—
46#63893—32no—
47#63880—32no—
48#63871—31no—
49#63852—32no—
50C2×Q8×D9—31no—
51C2×#3699—31no—
52C9×#124—31no—
53#63508—31no—
54#63500—31no—
55#63447—32no—
56#63446—32no—
57#611⋊C4—32no—
58#63440—32no—
59C4×D36—31no—
60C4×D36—61no—
61C4×Dic18D311yescharacteristic, complemented
62C22×C3⋊D9C2311yescharacteristic
63C22×C3⋊D9C2311yescharacteristic
64C22×D3×C9C2312yes—
65C2×C3×C4×C9C2311yescharacteristic
66C2×C3×C4×C9C2311yescharacteristic
67C2×C3×C4×C9C2311yescharacteristic
68C2×C3⋊Dic9C2311yescharacteristic
69C2×C3⋊Dic9C2311yescharacteristic
70C4⋊(C3⋊D9)—22no—
71C4×C3⋊D9—22no—
72C4×C3⋊D9—22no—
73C2×C9×Dic3C2312yes—
74C2×C3×Dic9C2312yes—
75C2×C3×Dic9C2312yes—
76C9×D12—22no—
77C9×Dic6—22no—
78C3×Dic18D414yescomplemented
79#6195—24no—
80#6195—24no—
81D3×Dic9—24no—
82(C3⋊D9)⋊C4—24no—
83#6189—24no—
84#5677—91no—
85#5576—91no—
86C2×C4⋊(C3⋊D3)—31no—
87C2×C4×C3⋊D3—31no—
88C2×C4×C3⋊D3—31no—
89C2×C3×D12C2×D311yescharacteristic
90C2×C3×Dic6—31no—
91C2×C3×Dic6C2×D311yescharacteristic
92C2×#627—32no—
93C2×#626—32no—
94C2×#626—32no—
95C2×#624—32no—
96C2×D3×Dic3—32no—
97#3797—64no—
98#3796—64no—
99C32×C42C2×D311yescharacteristic
100#3752—31no—
101#3752—31no—
102C4⋊(C3⋊Dic3)—31no—
103C3×(C2×D3)⋊C4C2×D312yes—
104C3×(C2×D3)⋊C4C2×D312yes—
105C3×C4⋊Dic3—31no—
106C3×Dic3⋊C4—32no—
107C3×C4×Dic3—32no—
108Dic3⋊Dic3—32no—
109#3722—32no—
110(C2×D3)⋊Dic3—32no—
111C2×Q8×C9—31no—
112C2×D4×C9—31no—
113C2×#611—32no—
114C2×#611—32no—
115C22×Dic9—32no—
116Q8×D9—64no—
117#3699—64no—
118C2×D36—31no—
119C2×D36—61no—
120C2×C4×D9—31no—
121C2×C4×D9—31no—
122C2×C4×D9—32no—
123C2×C4×D9—61no—
124C2×C4×D9—61no—
125C2×Dic18—32no—
126C2×Dic18C2×D311yescharacteristic
127C9×C22⋊C4—32no—
128C9×C22⋊C4—32no—
129C42×C9—21no—
130C42×C9C2×D311yescharacteristic
131#3676—32no—
132(C2×D9)⋊C4—31no—
133(C2×D9)⋊C4—31no—
134(C2×D9)⋊C4—32no—
135(C2×D9)⋊C4—61no—
136(C2×D9)⋊C4—61no—
137C4⋊Dic9—31no—
138C4⋊Dic9—61no—
139C4⋊Dic9C2×D311yescharacteristic
140Dic9⋊C4—32no—
141Dic9⋊C4C2×D312yes—
142C4×Dic9C2×D312yes—
143C22×C3×C9C2411yescharacteristic
144C2×C3⋊D9—21no—
145C2×C3⋊D9—21no—
146C2×C3⋊D9—21no—
147C2×C3⋊D9—21no—
148C2×C3⋊D9—21no—
149C2×C3⋊D9#4112yescomplemented
150C2×D3×C9—22no—
151C2×D3×C9—22no—
152C2×D3×C9—22no—
153C3×C4×C9—21no—
154C3×C4×C9—21no—
155C3×C4×C9C2×D412yes—
156C3⋊Dic9—21no—
157C3⋊Dic9—21no—
158C9×Dic3—22no—
159C3×Dic9—22no—
160C3×Dic9C2×D414yes—
161C2×#239—91no—
162C2×D3×Q8—91no—
163C2×D3×Q8—91no—
164C2×#237—91no—
165C3×#124—31no—
166C3×C4×Q8—91no—
167#946—91no—
168#938—91no—
169#897—91no—
170#897—92no—
171#895—91no—
172#895—92no—
173#885—92no—
174#884—92no—
175#67⋊C4—92no—
176#878—92no—
177#875—92no—
178#875D911yescharacteristic, complemented
179C4×D12—91no—
180C4×D12—91no—
181C4×D12—92no—
182C4×D12—181no—
183C4×Dic6—31no—
184C22×C3⋊D3—31no—
185C22×C3⋊D3—31no—
186C22×C3×D3C22×D312yes—
187C2×C32×C4C22×D311yescharacteristic
188C2×C32×C4C22×D311yescharacteristic
189C2×C32×C4C22×D311yescharacteristic
190C2×C3⋊Dic3—31no—
191C2×C3⋊Dic3—31no—
192C4⋊(C3⋊D3)—62no—
193C4×C3⋊D3—62no—
194C4×C3⋊D3—62no—
195C2×C3×Dic3—32no—
196C2×C3×Dic3—32no—
197C2×C3×Dic3C22×D312yes—
198C3×D12—22no—
199C3×Dic6—22no—
200C3×Dic6—34no—
201#627—64no—
202#626—64no—
203#626—64no—
204#624—64no—
205D3×Dic3—64no—
206C23×C9—32no—
207C22×D9—31no—
208C22×D9—31no—
209C22×D9—61no—
210C22×D9—61no—
211Q8×C9—62no—
212D4×C9—62no—
213C2×C4×C9—21no—
214C2×C4×C9—21no—
215C2×C4×C9—21no—
216C2×C4×C9—32no—
217C2×C4×C9C22×D311yescharacteristic
218C2×C4×C9C22×D311yescharacteristic
219C2×C4×C9C22×D311yescharacteristic
220#611—64no—
221#611—64no—
222C2×Dic9—31no—
223C2×Dic9—31no—
224C2×Dic9—61no—
225C2×Dic9—61no—
226C2×Dic9—64no—
227C2×Dic9C22×D312yes—
228C2×Dic9C22×D312yes—
229D36—62no—
230D36—64no—
231C4×D9—62no—
232C4×D9—62no—
233C4×D9—64no—
234C4×D9—122no—
235C4×D9—122no—
236Dic18—64no—
237Dic18D1214yescomplemented
238#548—271noSylow
239C2×C3×C9C22×D411yescharacteristic
240C2×C3×C9C2×#4111yescharacteristic
241C2×C3×C92−1+411yescharacteristic
242C3⋊D9—22no—
243C3⋊D9—41no—
244D3×C9—42no—
245C2×C3×Q8—31no—
246C2×C3×Q8—91no—
247C2×C3×D4—31no—
248C2×#67—92no—
249C2×#67—92no—
250C22×Dic3—92no—
251#239—184no—
252D3×Q8—184no—
253D3×Q8—184no—
254#237—184no—
255C2×D12—92no—
256C2×D12—92no—
257C2×D12—91no—
258C2×D12—91no—
259C2×D12—181no—
260C2×D12C2×D911yescharacteristic
261C2×C4×D3—92no—
262C2×C4×D3—92no—
263C2×C4×D3—91no—
264C2×C4×D3—91no—
265C2×C4×D3—91no—
266C2×C4×D3—91no—
267C2×C4×D3—92no—
268C2×C4×D3—181no—
269C2×C4×D3—181no—
270C2×Dic6—31no—
271C2×Dic6—92no—
272C2×Dic6—92no—
273C2×Dic6C2×D911yescharacteristic
274C3×C4⋊C4—92no—
275C3×C4⋊C4—91no—
276C3×C22⋊C4—32no—
277C3×C22⋊C4—32no—
278C3×C42—21no—
279C3×C42—92no—
280C3×C42C2×D911yescharacteristic
281C3×C42D3211yescharacteristic
282#217—92no—
283(C2×D3)⋊C4—92no—
284(C2×D3)⋊C4—92no—
285(C2×D3)⋊C4—91no—
286(C2×D3)⋊C4—91no—
287(C2×D3)⋊C4—91no—
288(C2×D3)⋊C4—91no—
289(C2×D3)⋊C4—92no—
290(C2×D3)⋊C4—181no—
291(C2×D3)⋊C4—181no—
292(C2×D3)⋊C4C2×D912yes—
293(C2×D3)⋊C4C2×D912yes—
294C4⋊Dic3—31no—
295C4⋊Dic3—91no—
296C4⋊Dic3—91no—
297C4⋊Dic3—92no—
298C4⋊Dic3—181no—
299Dic3⋊C4—32no—
300Dic3⋊C4—92no—
301C4×Dic3—32no—
302C22×C32C23×D311yescharacteristic
303C2×C3⋊D3—32no—
304C2×C3⋊D3—61no—
305C2×C3⋊D3—61no—
306C2×C3⋊D3—61no—
307C2×C3⋊D3—61no—
308C2×C3⋊D3—61no—
309C2×C3×D3—22no—
310C2×C3×D3—22no—
311C2×C3×D3—22no—
312C32×C4—21no—
313C32×C4—21no—
314C32×C4D3×D412yes—
315C3⋊Dic3—61no—
316C3⋊Dic3—61no—
317C3×Dic3—22no—
318C3×Dic3—34no—
319C3×Dic3—62no—
320C22×C9—21no—
321C22×C9—62no—
322C22×C9—62no—
323C22×C9—62no—
324C22×C9C23×D311yescharacteristic
325C2×D9—32no—
326C2×D9—62no—
327C2×D9—61no—
328C2×D9—61no—
329C2×D9—62no—
330C2×D9—61no—
331C2×D9—61no—
332C2×D9—61no—
333C2×D9—121no—
334C2×D9—121no—
335C2×D9—121no—
336C2×D9—121no—
337C4×C9—22no—
338C4×C9—21no—
339C4×C9—21no—
340C4×C9—22no—
341C4×C9—41no—
342C4×C9—62no—
343C4×C9C2×D1212yes—
344Dic9—22no—
345Dic9—61no—
346Dic9—61no—
347Dic9—121no—
348Dic9—121no—
349Dic9C2×D1214yes—
350C2×#41—271no—
351C22×Q8—271no—
352#124—31no—
353#124—272no—
354C22⋊Q8—272no—
355C22⋊Q8—271no—
356#121—272no—
357#121—271no—
358C4×Q8—91no—
359C4×D4—272no—
360C4×D4—271no—
361C3×C9#54811yescharacteristic, complemented, Sylow
362C23×C3—32no—
363C22×D3—91no—
364C22×D3—91no—
365C22×D3—91no—
366C22×D3—91no—
367C22×D3—181no—
368C22×D3—181no—
369C22×D3C22×D912yes—
370C3×Q8—62no—
371C3×Q8—94no—
372C3×D4—62no—
373C2×C3×C4—21no—
374C2×C3×C4—21no—
375C2×C3×C4—21no—
376C2×C3×C4—32no—
377C2×C3×C4—92no—
378C2×C3×C4—92no—
379C2×C3×C4C22×D911yescharacteristic
380C2×C3×C4C22×D911yescharacteristic
381C2×C3×C4C22×D911yescharacteristic
382C2×C3×C4C2×D3211yescharacteristic
383C2×C3×C4C2×D3211yescharacteristic
384C2×C3×C4C2×D3211yescharacteristic
385#67—184no—
386#67—184no—
387C2×Dic3—32no—
388C2×Dic3—32no—
389C2×Dic3—91no—
390C2×Dic3—91no—
391C2×Dic3—91no—
392C2×Dic3—91no—
393C2×Dic3—181no—
394C2×Dic3—181no—
395C2×Dic3—184no—
396C2×Dic3C22×D912yes—
397D12—22no—
398D12—182no—
399D12—182no—
400D12—184no—
401D12—184no—
402D12—184no—
403C4×D3—184no—
404C4×D3—184no—
405C4×D3—184no—
406C4×D3—182no—
407C4×D3—182no—
408C4×D3—182no—
409C4×D3—182no—
410C4×D3—362no—
411C4×D3—362no—
412Dic6—22no—
413Dic6—34no—
414Dic6—184no—
415Dic6—184no—
416C2×C32C2×D3×D411yescharacteristic
417C2×C32D3×#4111yescharacteristic
418C2×C32#101011yescharacteristic
419C3⋊D3—62no—
420C3⋊D3—121no—
421C3×D3—42no—
422C2×C9—21no—
423C2×C9—21no—
424C2×C9—21no—
425C2×C9—122no—
426C2×C9C22×D1211yescharacteristic
427C2×C9#100711yescharacteristic
428C2×C9D3×#4111yescharacteristic
429D9—62no—
430D9—122no—
431D9—122no—
432D9—121no—
433#41—544no—
434C2×Q8—31no—
435C2×Q8—91no—
436C2×Q8—272no—
437C2×Q8—544no—
438C2×D4—31no—
439C2×D4—272no—
440C2×D4—272no—
441C2×D4—271no—
442C22×C4—272no—
443C22×C4—272no—
444C22×C4—271no—
445C22×C4—271no—
446C4⋊C4—92no—
447C4⋊C4—91no—
448C4⋊C4—272no—
449C4⋊C4—271no—
450C22⋊C4—32no—
451C22⋊C4—32no—
452C22⋊C4—272no—
453C22⋊C4—272no—
454C22⋊C4—271no—
455C22⋊C4—271no—
456C42—92no—
457C42D3×D911yescharacteristic
458C22×C3—21no—
459C22×C3—62no—
460C22×C3—62no—
461C22×C3—62no—
462C22×C3C23×D911yescharacteristic
463C22×C3C22×D3211yescharacteristic
464C2×D3—22no—
465C2×D3—22no—
466C2×D3—22no—
467C2×D3—92no—
468C2×D3—92no—
469C2×D3—182no—
470C2×D3—181no—
471C2×D3—181no—
472C2×D3—181no—
473C2×D3—181no—
474C2×D3—182no—
475C2×D3—181no—
476C2×D3—181no—
477C2×D3—181no—
478C2×D3—181no—
479C2×D3—181no—
480C2×D3—181no—
481C2×D3—361no—
482C2×D3—361no—
483C2×D3—361no—
484C2×D3—361no—
485C3×C4—22no—
486C3×C4—21no—
487C3×C4—21no—
488C3×C4—21no—
489C3×C4—21no—
490C3×C4—22no—
491C3×C4—41no—
492C3×C4—62no—
493C3×C4—94no—
494C3×C4—182no—
495C3×C4D4×D912yes—
496C3×C4D3×D1212yes—
497Dic3—22no—
498Dic3—34no—
499Dic3—62no—
500Dic3—181no—
501Dic3—181no—
502Dic3—181no—
503Dic3—181no—
504Dic3—361no—
505Dic3—361no—
506C32#567711yescharacteristic
507C9—21no—
508C9#557611yescharacteristic, complemented
509C23—32no—
510C23—271no—
511C23—271no—
512Q8—62no—
513Q8—94no—
514Q8—544no—
515D4—62no—
516D4—542no—
517D4—544no—
518D4—544no—
519C2×C4—32no—
520C2×C4—92no—
521C2×C4—92no—
522C2×C4—271no—
523C2×C4—271no—
524C2×C4—544no—
525C2×C4—544no—
526C2×C4—542no—
527C2×C4—542no—
528C2×C4C2×D3×D911yescharacteristic
529C2×C4C2×D3×D911yescharacteristic
530C2×C4C2×D3×D911yescharacteristic
531C2×C3—21no—
532C2×C3—21no—
533C2×C3—21no—
534C2×C3—122no—
535C2×C3C2×D4×D911yescharacteristic
536C2×C3#41×D911yescharacteristic
537C2×C3#6371611yescharacteristic
538C2×C3#6429711yescharacteristic
539C2×C3C2×D3×D1211yescharacteristic
540C2×C3#6430611yescharacteristic
541D3—42no—
542D3—182no—
543D3—182no—
544D3—362no—
545D3—362no—
546D3—361no—
547D3—361no—
548C22—62no—
549C22—62no—
550C22—62no—
551C22—272no—
552C22—541no—
553C22—541no—
554C22—541no—
555C22—541no—
556C22—541no—
557C22C22×D3×D911yescharacteristic
558C4—21no—
559C4—21no—
560C4—62no—
561C4—94no—
562C4—182no—
563C4—541no—
564C4—541no—
565C4D9×D1212yes—
566C3—21no—
567C3#1058977911yescharacteristic, complemented
568C3#1059390111yescharacteristic
569C2—122no—
570C2—542no—
571C2—1081no—
572C2#1172246411yescharacteristic
573C2C2×D9×D1211yescharacteristic
574C2#1172247511yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC3×C42×C9C22
socleC22×C32C23×D3
hypercenterC42D3×D9
hypocenterC3×C9#548
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
1. C2×#88639C2semidirect
2. C2×(C3⋊D9)⋊Q8C2semidirect
3. C4×C4⋊(C3⋊D9)C2semidirect
4. C9×#875C2semidirect
5. C3×C4×Dic18C2semidirect
6. #11721045C2semidirect
7. #11721039C2semidirect
8. #11721023C2semidirect
9. #11721013C2semidirect
10. #11720956C2semidirect
11. #11720929C2semidirect
61. C4×Dic18D3semidirect
78. C3×Dic18D4semidirect
149. C2×C3⋊D9#41semidirect
178. #875D9semidirect
237. Dic18D12semidirect
361. C3×C9#548semidirect
508. C9#5576semidirect
567. C3#10589779semidirect

Maximal subgroups

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

subgroupindex
C2×#886392
C2×(C3⋊D9)⋊Q82
C4×C4⋊(C3⋊D9)2
C9×#8752
C3×C4×Dic182
#117210452
#117210392
#117210232
#117210132
#117209562
#117209292
#105939013
#105897793