Finite Group Databasebeta

← #13148211

Subgroups — #13148211

Order 1728.

1120 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88662C211yesmaximal, characteristic, complemented
2C22×(C3⋊D9)⋊C4C211yesmaximal, characteristic, complemented
3#11721728C211yesmaximal, characteristic, complemented
4C9×#886C211yesmaximal, characteristic, complemented
5C3×#63439C211yesmaximal, characteristic, complemented
6#11721291C211yesmaximal, characteristic, complemented
7#11721251C211yesmaximal, characteristic, complemented
8#11721039C212yesmaximal, complemented
9#11721013C212yesmaximal, complemented
10#11720994C212yesmaximal, complemented
11#11720992C212yesmaximal, complemented
12#10594079—31nomaximal
13#10589758—31nomaximal
14C23×C3⋊D9C2211yescharacteristic
15C2×#6233C2211yescharacteristic
16C2×C4⋊(C3⋊D9)C2212yes—
17C2×C9×#67C2211yescharacteristic
18C22×C9×Dic3C2211yescharacteristic
19C22×C3×Dic9C2211yescharacteristic
20C2×#6195C2212yes—
21C2×#6195C2211yescharacteristic, complemented
22#88662—24no—
23C2×D3×Dic9C2211yescharacteristic, complemented
24C2×(C3⋊D9)⋊C4—24no—
25C2×(C3⋊D9)⋊C4C2212yes—
26C2×(C3⋊D9)⋊C4C2212yes—
27C2×(C3⋊D9)⋊C4C2211yescharacteristic, complemented
28C2×#6189C2211yescharacteristic, complemented
29C3×C9×C22⋊C4C2211yescharacteristic
30#88536C2212yes—
31C3×#3676C2211yescharacteristic
32C9×(C2×D3)⋊C4C2212yescomplemented
33C9×C4⋊Dic3C2212yescomplemented
34C3×Dic9⋊C4C2212yescomplemented
35C3×C4×Dic9C2212yescomplemented
36#88445C2212yes—
37#88445C2212yescomplemented
38Dic9⋊Dic3C2212yescomplemented
39C22×#624—31no—
40C2×#3804—31no—
41#64091—31no—
42C3×#886D311yescharacteristic
43C3×#877—31no—
44#63981—31no—
45#63966—31no—
46#63910—32no—
47#63895—32no—
48#63894—32no—
49#63880—32no—
50C2×#3699—31no—
51C22×C4×D9—31no—
52C9×#123—31no—
53#63499—31no—
54#63491—31no—
55#63458—32no—
56#63455—32no—
57#63446—32no—
58#63444—31no—
59#63444—61no—
60#611⋊C4—32no—
61#63439D311yescharacteristic, complemented
62C23×C3×C9C2311yescharacteristic
63C22×C3⋊D9—22no—
64C22×C3⋊D9—24no—
65C22×C3⋊D9C2311yescharacteristic
66C22×C3⋊D9C2312yes—
67C22×D3×C9C2311yescharacteristic
68C2×C3×C4×C9C2312yes—
69#6233—22no—
70C2×C3⋊Dic9C2311yescharacteristic
71C4⋊(C3⋊D9)—24no—
72C9×#67—22no—
73C2×C9×Dic3—22no—
74C2×C9×Dic3C2311yescharacteristic
75C2×C9×Dic3C2312yes—
76C2×C3×Dic9D414yescomplemented
77C2×C3×Dic9C2312yes—
78C2×C3×Dic9C2311yescharacteristic
79C2×C3×Dic9C2311yescharacteristic
80#6195—24no—
81#6195—22no—
82D3×Dic9—22no—
83(C3⋊D9)⋊C4—24no—
84(C3⋊D9)⋊C4—24no—
85(C3⋊D9)⋊C4—22no—
86#6189—22no—
87#5656—91no—
88#5524—91no—
89C23×C3⋊D3—31no—
90C2×#638—31no—
91C2×C4⋊(C3⋊D3)—32no—
92C2×C3×#67C2×D311yescharacteristic
93C22×C3×Dic3—31no—
94C22×C3×Dic3C2×D311yescharacteristic
95C2×#627—31no—
96C2×#626—31no—
97C2×#626—32no—
98C2×#624—31no—
99C2×#624—32no—
100C2×#624—32no—
101C2×#624—64no—
102#3804—64no—
103C2×D3×Dic3—31no—
104C23×D9—31no—
105C23×D9—61no—
106C32×C22⋊C4C2×D311yescharacteristic
107#3752—32no—
108C3×#217—31no—
109C3×(C2×D3)⋊C4C2×D312yes—
110C3×C4⋊Dic3C2×D312yes—
111C3×Dic3⋊C4—32no—
112C3×C4×Dic3—32no—
113Dic3⋊Dic3—32no—
114#3722—32no—
115#3722—32no—
116C2×D4×C9—31no—
117C22×C4×C9—31no—
118C2×#611—31no—
119C2×#611—32no—
120C2×#611—31no—
121C2×#611—61no—
122C22×Dic9—31no—
123C22×Dic9C2×D311yescharacteristic
124#3699—64no—
125C2×D36—32no—
126C2×D36—62no—
127C2×C4×D9—32no—
128C2×C4×D9—32no—
129C2×C4×D9—31no—
130C2×C4×D9—64no—
131C2×Dic18—31no—
132C9×C4⋊C4—32no—
133C9×C22⋊C4—21no—
134C9×C22⋊C4—32no—
135C9×C22⋊C4C2×D311yescharacteristic
136#3676C2×D311yescharacteristic
137(C2×D9)⋊C4—32no—
138(C2×D9)⋊C4—32no—
139(C2×D9)⋊C4—32no—
140(C2×D9)⋊C4—62no—
141Dic9⋊C4—32no—
142Dic9⋊C4C2×D312yescomplemented
143C4×Dic9C2×D312yescomplemented
144C22×C3×C9—22no—
145C22×C3×C9C2×D412yes—
146C22×C3×C9C2411yescharacteristic
147C2×C3⋊D9—28no—
148C2×C3⋊D9—22no—
149C2×C3⋊D9—22no—
150C2×C3⋊D9—21no—
151C2×C3⋊D9—22no—
152C2×C3⋊D9#4114yescomplemented
153C2×D3×C9—21no—
154C2×D3×C9—22no—
155C3×C4×C9—22no—
156C3⋊Dic9—21no—
157C9×Dic3—21no—
158C9×Dic3—22no—
159C3×Dic9—22no—
160C3×Dic9C2×D412yes—
161C3×Dic9C2×D412yes—
162C2×#237—91no—
163C2×#235—91no—
164C22×C4×D3—91no—
165C22×C4×D3—91no—
166C3×#123—31no—
167C3×#117—91no—
168#937—91no—
169#929—91no—
170#897—92no—
171#896—92no—
172#895—92no—
173#893—92no—
174#886—91no—
175#886D911yescharacteristic, complemented
176#884—92no—
177#882—91no—
178#882—91no—
179#882—91no—
180#882—181no—
181#67⋊C4—92no—
182#877—31no—
183C4×D12—92no—
184C4×D12—92no—
185C23×C32C22×D311yescharacteristic
186C22×C3⋊D3—32no—
187C22×C3⋊D3—31no—
188C22×C3⋊D3—64no—
189C22×C3⋊D3—62no—
190C22×C3×D3C22×D311yescharacteristic
191C2×C32×C4C22×D312yes—
192#638—62no—
193C2×C3⋊Dic3—31no—
194C4⋊(C3⋊D3)—64no—
195C3×#67—22no—
196C2×C3×Dic3—22no—
197C2×C3×Dic3—34no—
198C2×C3×Dic3—31no—
199C2×C3×Dic3—31no—
200C2×C3×Dic3—32no—
201C2×C3×Dic3C22×D311yescharacteristic
202C2×C3×Dic3C22×D312yes—
203#627—62no—
204#626—62no—
205#626—64no—
206#624—62no—
207#624—64no—
208#624—64no—
209D3×Dic3—62no—
210C23×C9—21no—
211C23×C9—31no—
212C23×C9C22×D311yescharacteristic
213C22×D9—32no—
214C22×D9—31no—
215C22×D9—62no—
216C22×D9—61no—
217C22×D9—64no—
218C22×D9—62no—
219C22×D9—124no—
220C22×D9—122no—
221D4×C9—62no—
222C2×C4×C9—22no—
223C2×C4×C9—32no—
224C2×C4×C9—31no—
225C2×C4×C9—62no—
226C2×C4×C9C22×D312yes—
227#611—64no—
228#611—62no—
229#611—62no—
230#611—122no—
231C2×Dic9—31no—
232C2×Dic9—61no—
233C2×Dic9—62no—
234C2×Dic9D1214yescomplemented
235C2×Dic9C22×D312yes—
236C2×Dic9C22×D311yescharacteristic
237C2×Dic9C22×D311yescharacteristic
238D36—64no—
239D36—124no—
240C4×D9—64no—
241C4×D9—64no—
242C4×D9—62no—
243Dic18—62no—
244#525—271noSylow
245C2×C3×C9—22no—
246C2×C3×C9C22×D411yescharacteristic
247C2×C3×C9C2×#4112yes—
248C3⋊D9—24no—
249C3⋊D9—41no—
250D3×C9—41no—
251C23×D3—91no—
252C23×D3—91no—
253C23×D3—181no—
254C2×C3×D4—31no—
255C22×C3×C4—31no—
256C22×C3×C4—91no—
257C2×#67—92no—
258C2×#67—91no—
259C2×#67—91no—
260C2×#67—91no—
261C2×#67—181no—
262C2×#67C2×D911yescharacteristic
263C22×Dic3—31no—
264C22×Dic3—91no—
265C22×Dic3C2×D911yescharacteristic
266#237—184no—
267#235—184no—
268C2×D12—92no—
269C2×D12—91no—
270C2×D12—92no—
271C2×D12—92no—
272C2×D12—182no—
273C2×C4×D3—92no—
274C2×C4×D3—92no—
275C2×C4×D3—91no—
276C2×C4×D3—91no—
277C2×C4×D3—92no—
278C2×C4×D3—91no—
279C2×C4×D3—92no—
280C2×C4×D3—184no—
281C2×C4×D3—184no—
282C2×Dic6—91no—
283C2×Dic6—91no—
284C3×C4⋊C4—32no—
285C3×C4⋊C4—92no—
286C3×C22⋊C4—21no—
287C3×C22⋊C4—32no—
288C3×C22⋊C4—91no—
289C3×C22⋊C4C2×D911yescharacteristic
290C3×C22⋊C4D3211yescharacteristic
291C3×C42—92no—
292#217—31no—
293(C2×D3)⋊C4—92no—
294(C2×D3)⋊C4—92no—
295(C2×D3)⋊C4—92no—
296(C2×D3)⋊C4—92no—
297(C2×D3)⋊C4—92no—
298(C2×D3)⋊C4—92no—
299(C2×D3)⋊C4—182no—
300(C2×D3)⋊C4C2×D912yescomplemented
301C4⋊Dic3—92no—
302C4⋊Dic3C2×D912yescomplemented
303Dic3⋊C4—32no—
304Dic3⋊C4—92no—
305C4×Dic3—32no—
306C22×C32—22no—
307C22×C32D3×D412yes—
308C22×C32C23×D311yescharacteristic
309C2×C3⋊D3—34no—
310C2×C3⋊D3—68no—
311C2×C3⋊D3—62no—
312C2×C3⋊D3—62no—
313C2×C3⋊D3—61no—
314C2×C3⋊D3—62no—
315C2×C3×D3—21no—
316C2×C3×D3—22no—
317C32×C4—22no—
318C3⋊Dic3—61no—
319C3×Dic3—22no—
320C3×Dic3—21no—
321C3×Dic3—32no—
322C3×Dic3—32no—
323C3×Dic3—62no—
324C22×C9—22no—
325C22×C9—21no—
326C22×C9—22no—
327C22×C9—42no—
328C22×C9—61no—
329C22×C9—62no—
330C22×C9C2×D1212yes—
331C22×C9C23×D311yescharacteristic
332C2×D9—34no—
333C2×D9—64no—
334C2×D9—68no—
335C2×D9—62no—
336C2×D9—62no—
337C2×D9—61no—
338C2×D9—62no—
339C2×D9—122no—
340C2×D9—122no—
341C2×D9—128no—
342C2×D9—122no—
343C2×D9—121no—
344C4×C9—22no—
345C4×C9—42no—
346C4×C9—61no—
347C4×C9—62no—
348Dic9—22no—
349Dic9—61no—
350Dic9—121no—
351Dic9C2×D1212yes—
352Dic9C2×D1212yes—
353C2×#41—271no—
354C23×C4—271no—
355#123—31no—
356#123—271no—
357C22⋊Q8—272no—
358#121—272no—
359#120—271no—
360#120—271no—
361C4×D4—272no—
362C4×D4—272no—
363#117—91no—
364C3×C9#52511yescharacteristic, complemented, Sylow
365C23×C3—21no—
366C23×C3—31no—
367C23×C3C22×D911yescharacteristic
368C23×C3C2×D3211yescharacteristic
369C22×D3—92no—
370C22×D3—92no—
371C22×D3—91no—
372C22×D3—91no—
373C22×D3—182no—
374C22×D3—181no—
375C22×D3—184no—
376C22×D3—184no—
377C22×D3—182no—
378C22×D3—182no—
379C22×D3—364no—
380C22×D3—362no—
381C22×D3C22×D911yescharacteristic
382C3×D4—62no—
383C2×C3×C4—22no—
384C2×C3×C4—32no—
385C2×C3×C4—31no—
386C2×C3×C4—62no—
387C2×C3×C4—94no—
388C2×C3×C4—91no—
389C2×C3×C4—91no—
390C2×C3×C4—92no—
391C2×C3×C4C22×D912yes—
392C2×C3×C4C2×D3212yes—
393#67—22no—
394#67—182no—
395#67—184no—
396#67—182no—
397#67—182no—
398#67—362no—
399C2×Dic3—22no—
400C2×Dic3—31no—
401C2×Dic3—31no—
402C2×Dic3—34no—
403C2×Dic3—32no—
404C2×Dic3—91no—
405C2×Dic3—91no—
406C2×Dic3—181no—
407C2×Dic3—182no—
408C2×Dic3C22×D912yes—
409C2×Dic3C22×D911yescharacteristic
410D12—184no—
411D12—184no—
412D12—182no—
413D12—184no—
414D12—364no—
415C4×D3—184no—
416C4×D3—184no—
417C4×D3—182no—
418C4×D3—184no—
419C4×D3—184no—
420C4×D3—182no—
421C4×D3—182no—
422Dic6—182no—
423Dic6—182no—
424C2×C32—22no—
425C2×C32C2×D3×D411yescharacteristic
426C2×C32D3×#4112yes—
427C3⋊D3—64no—
428C3⋊D3—121no—
429C3×D3—41no—
430C2×C9—22no—
431C2×C9—21no—
432C2×C9—22no—
433C2×C9—42no—
434C2×C9—121no—
435C2×C9C22×D1211yescharacteristic
436C2×C9D3×#4112yes—
437D9—64no—
438D9—124no—
439D9—121no—
440D9—241no—
441C24—271no—
442#41—544no—
443C2×Q8—271no—
444C2×D4—31no—
445C2×D4—272no—
446C2×D4—271no—
447C2×D4—271no—
448C2×D4—272no—
449C22×C4—31no—
450C22×C4—91no—
451C22×C4—272no—
452C22×C4—271no—
453C22×C4—272no—
454C22×C4—271no—
455C22×C4—544no—
456C4⋊C4—32no—
457C4⋊C4—92no—
458C4⋊C4—272no—
459C22⋊C4—32no—
460C22⋊C4—91no—
461C22⋊C4—272no—
462C22⋊C4—272no—
463C22⋊C4—272no—
464C22⋊C4D3×D911yescharacteristic
465C42—92no—
466C22×C3—22no—
467C22×C3—22no—
468C22×C3—21no—
469C22×C3—22no—
470C22×C3—42no—
471C22×C3—61no—
472C22×C3—62no—
473C22×C3D4×D912yes—
474C22×C3C23×D911yescharacteristic
475C22×C3D3×D1212yes—
476C22×C3C22×D3211yescharacteristic
477C2×D3—21no—
478C2×D3—22no—
479C2×D3—94no—
480C2×D3—94no—
481C2×D3—184no—
482C2×D3—188no—
483C2×D3—188no—
484C2×D3—182no—
485C2×D3—182no—
486C2×D3—182no—
487C2×D3—182no—
488C2×D3—181no—
489C2×D3—182no—
490C2×D3—181no—
491C2×D3—182no—
492C2×D3—362no—
493C2×D3—362no—
494C2×D3—368no—
495C2×D3—362no—
496C2×D3—361no—
497C3×C4—22no—
498C3×C4—22no—
499C3×C4—42no—
500C3×C4—61no—
501C3×C4—62no—
502C3×C4—92no—
503C3×C4—92no—
504C3×C4—182no—
505Dic3—21no—
506Dic3—22no—
507Dic3—32no—
508Dic3—32no—
509Dic3—62no—
510Dic3—181no—
511Dic3—181no—
512Dic3—361no—
513C32#565611yescharacteristic
514C9—21no—
515C9#552411yescharacteristic, complemented
516C23—31no—
517C23—272no—
518C23—271no—
519C23—544no—
520C23—542no—
521C23C2×D3×D911yescharacteristic
522Q8—542no—
523D4—62no—
524D4—544no—
525D4—542no—
526D4—544no—
527D4—542no—
528C2×C4—32no—
529C2×C4—31no—
530C2×C4—62no—
531C2×C4—91no—
532C2×C4—91no—
533C2×C4—94no—
534C2×C4—92no—
535C2×C4—271no—
536C2×C4—544no—
537C2×C4—544no—
538C2×C4—542no—
539C2×C4—542no—
540C2×C4C2×D3×D912yes—
541C2×C3—22no—
542C2×C3—22no—
543C2×C3—22no—
544C2×C3—21no—
545C2×C3—42no—
546C2×C3—121no—
547C2×C3C2×D4×D911yescharacteristic
548C2×C3#41×D912yes—
549C2×C3C2×D3×D1211yescharacteristic
550C2×C3#6431212yes—
551D3—41no—
552D3—184no—
553D3—184no—
554D3—364no—
555D3—361no—
556D3—361no—
557D3—721no—
558C22—22no—
559C22—62no—
560C22—61no—
561C22—274no—
562C22—548no—
563C22—542no—
564C22—542no—
565C22—542no—
566C22—541no—
567C22D9×D1212yes—
568C22C22×D3×D911yescharacteristic
569C4—22no—
570C4—62no—
571C4—61no—
572C4—92no—
573C4—92no—
574C4—182no—
575C4—541no—
576C3—21no—
577C3#1058975811yescharacteristic, complemented
578C3#1059407911yescharacteristic
579C2—22no—
580C2—121no—
581C2—544no—
582C2—1081no—
583C2C2×D9×D1211yescharacteristic
584C2#1172249812yes—
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC3×C9×C22⋊C4C22
socleC22×C32C23×D3
hypercenterC22⋊C4D3×D9
hypocenterC3×C9#525
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. C2×#88662C2semidirect
2. C22×(C3⋊D9)⋊C4C2semidirect
3. #11721728C2semidirect
4. C9×#886C2semidirect
5. C3×#63439C2semidirect
6. #11721291C2semidirect
7. #11721251C2semidirect
8. #11721039C2semidirect
9. #11721013C2semidirect
10. #11720994C2semidirect
11. #11720992C2semidirect
21. C2×#6195C22semidirect
23. C2×D3×Dic9C22semidirect
27. C2×(C3⋊D9)⋊C4C22semidirect
28. C2×#6189C22semidirect
32. C9×(C2×D3)⋊C4C22semidirect
33. C9×C4⋊Dic3C22semidirect
34. C3×Dic9⋊C4C22semidirect
35. C3×C4×Dic9C22semidirect
37. #88445C22semidirect
38. Dic9⋊Dic3C22semidirect
61. #63439D3semidirect
76. C2×C3×Dic9D4semidirect
142. Dic9⋊C4C2×D3semidirect
143. C4×Dic9C2×D3semidirect
152. C2×C3⋊D9#41semidirect
175. #886D9semidirect
234. C2×Dic9D12semidirect
300. (C2×D3)⋊C4C2×D9semidirect
302. C4⋊Dic3C2×D9semidirect
364. C3×C9#525semidirect
515. C9#5524semidirect
577. C3#10589758semidirect

Maximal subgroups

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

subgroupindex
C2×#886622
C22×(C3⋊D9)⋊C42
#117217282
C9×#8862
C3×#634392
#117212912
#117212512
#117210392
#117210132
#117209942
#117209922
#105940793
#105897583