Finite Group Databasebeta

← #13148426

Subgroups — #13148426

Order 1728.

1158 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88668C211yesmaximal, characteristic
2C22×(C3⋊D9)⋊C4C211yesmaximal, characteristic, complemented
3C22⋊C4×C3⋊D9C211yesmaximal, characteristic, complemented
4C9×#886C211yesmaximal, characteristic, complemented
5C3×#63448C211yesmaximal, characteristic, complemented
6#11721265C211yesmaximal, characteristic, complemented
7#11721259C211yesmaximal, characteristic, complemented
8#11721048C212yesmaximal, complemented
9#11721000C212yesmaximal, complemented
10#11720958C212yesmaximal, complemented
11#11720942C212yesmaximal, complemented
12#10594295—31nomaximal
13D9×#123—31nomaximal
14C23×C3⋊D9C2211yescharacteristic
15C22×D3×D9C2211yescharacteristic
16C2×C4×C3⋊D9C2212yes—
17C2×C9×#67C2211yescharacteristic
18C2×C3×#611C2211yescharacteristic
19C22×C9×Dic3C2211yescharacteristic, complemented
20C22×C3×Dic9C2211yescharacteristic, complemented
21#88668—24no—
22C2×#6195C2211yescharacteristic
23C2×#6192C2211yescharacteristic
24C2×(C3⋊D9)⋊C4—24no—
25C2×(C3⋊D9)⋊C4C2212yes—
26C2×(C3⋊D9)⋊C4C2211yescharacteristic
27C2×(C3⋊D9)⋊C4C2212yescomplemented
28C3×C9×C22⋊C4C2211yescharacteristic, complemented
29#88551C2211yescharacteristic, complemented
30#88536C2212yescomplemented
31C9×(C2×D3)⋊C4C2212yescomplemented
32C3×(C2×D9)⋊C4C2212yescomplemented
33C9×C4⋊Dic3C2212yescomplemented
34C3×C4⋊Dic9C2212yescomplemented
35(C2×D3)⋊Dic9C2212yescomplemented
36(C2×D9)⋊Dic3C2212yescomplemented
37#88442C2212yescomplemented
38#88442C2212yescomplemented
39C2×#3811—31no—
40C22×#624—31no—
41C22⋊C4×C3⋊D3—31no—
42C3×#886—31no—
43C3×#886D311yescharacteristic
44#63961—31no—
45#63961—31no—
46#63921—32no—
47#63900—32no—
48#63854—32no—
49#63854—32no—
50C2×D4×D9—31no—
51C22×C4×D9—31no—
52C9×#123—31no—
53#63497—31no—
54#63491—31no—
55#63456—32no—
56C4⋊C4×D9—32no—
57#63448D311yescharacteristic, complemented
58#63445—32no—
59C22⋊C4×D9—31no—
60C22⋊C4×D9—32no—
61C22⋊C4×D9—61no—
62C23×C3×C9C2311yescharacteristic
63C22×C3⋊D9—24no—
64C22×C3⋊D9D414yes—
65C22×C3⋊D9C2311yescharacteristic
66C22×C3⋊D9C2311yescharacteristic
67C22×D3×C9C2311yescharacteristic
68C22×C3×D9C2311yescharacteristic
69C2×D3×D9—22no—
70C2×D3×D9—24no—
71C2×C3×C4×C9C2312yes—
72C2×C3⋊Dic9C2312yes—
73C4×C3⋊D9—24no—
74C9×#67—22no—
75C3×#611—22no—
76C2×C9×Dic3—22no—
77C2×C9×Dic3C2312yes—
78C2×C9×Dic3C2311yescharacteristic
79C2×C3×Dic9—22no—
80C2×C3×Dic9C2312yes—
81C2×C3×Dic9C2311yescharacteristic
82#6195—22no—
83#6192—22no—
84(C3⋊D9)⋊C4—24no—
85(C3⋊D9)⋊C4—24no—
86(C3⋊D9)⋊C4—22no—
87D3×#123—91no—
88D3×#123—91no—
89C23×C3⋊D3—31no—
90C22×D32—31no—
91C2×C4×C3⋊D3—32no—
92C2×C3×#67—31no—
93C2×C3×#67C2×D311yescharacteristic
94C22×C3×Dic3—31no—
95C22×C3×Dic3C2×D311yescharacteristic
96#3811—64no—
97C2×#626—31no—
98C2×#626—31no—
99C2×#624—32no—
100C2×#624—32no—
101C2×#624—31no—
102C2×#624—64no—
103C23×D9—31no—
104C23×D9—31no—
105C23×D9—61no—
106C32×C22⋊C4C2×D311yescharacteristic
107#3757—31no—
108#3752—32no—
109C3×(C2×D3)⋊C4—32no—
110C3×(C2×D3)⋊C4C2×D312yes—
111C3×C4⋊Dic3—32no—
112C3×C4⋊Dic3C2×D312yes—
113#3724—32no—
114#3724—32no—
115(C2×D3)⋊Dic3—32no—
116(C2×D3)⋊Dic3—32no—
117C2×D4×C9—31no—
118C22×C4×C9—31no—
119C2×#611—31no—
120C2×#611C2×D311yescharacteristic
121C22×Dic9C2×D311yescharacteristic, complemented
122D4×D9—64no—
123C2×D36—31no—
124C2×C4×D9—32no—
125C2×C4×D9—32no—
126C2×C4×D9—31no—
127C2×C4×D9—32no—
128C2×C4×D9—64no—
129C2×C4×D9—62no—
130C9×C4⋊C4—32no—
131C9×C22⋊C4—21no—
132C9×C22⋊C4—32no—
133C9×C22⋊C4C2×D311yescharacteristic, complemented
134#3676—31no—
135#3676—32no—
136#3676—61no—
137(C2×D9)⋊C4—32no—
138(C2×D9)⋊C4—32no—
139(C2×D9)⋊C4—62no—
140(C2×D9)⋊C4C2×D312yescomplemented
141C4⋊Dic9C2×D312yescomplemented
142Dic9⋊C4—32no—
143Dic9⋊C4—32no—
144C22×C3×C9—22no—
145C22×C3×C9C2×D412yes—
146C22×C3×C9C2411yescharacteristic
147C2×C3⋊D9—22no—
148C2×C3⋊D9—28no—
149C2×C3⋊D9C2×D412yes—
150C2×C3⋊D9C2×D412yes—
151C2×C3⋊D9#4114yes—
152C2×D3×C9—21no—
153C2×D3×C9—22no—
154C2×C3×D9—21no—
155C2×C3×D9—22no—
156D3×D9—44no—
157C3×C4×C9—22no—
158C3⋊Dic9—22no—
159C9×Dic3—22no—
160C9×Dic3—21no—
161C3×Dic9—22no—
162C3×Dic9—21no—
163C2×D3×D4—91no—
164C2×D3×D4—91no—
165C22×C4×D3—91no—
166C22×C4×D3—91no—
167C3×#123—31no—
168C3×#123—91no—
169#935—91no—
170#935—91no—
171#929—91no—
172#929—91no—
173#894—92no—
174#894—92no—
175D3×C4⋊C4—92no—
176D3×C4⋊C4—92no—
177#886—31no—
178#886D911yescharacteristic, complemented
179#883—92no—
180#883—92no—
181D3×C22⋊C4—92no—
182D3×C22⋊C4—91no—
183D3×C22⋊C4—91no—
184D3×C22⋊C4—92no—
185D3×C22⋊C4—181no—
186C23×C32C22×D311yescharacteristic
187C22×C3⋊D3—31no—
188C22×C3⋊D3—34no—
189C22×C3⋊D3—31no—
190C22×C3⋊D3—64no—
191C22×C3×D3—31no—
192C22×C3×D3C22×D311yescharacteristic
193C2×D32—64no—
194C2×D32—62no—
195C2×C32×C4C22×D312yes—
196C2×C3⋊Dic3—32no—
197C4×C3⋊D3—64no—
198C3×#67—22no—
199C3×#67—62no—
200C2×C3×Dic3—22no—
201C2×C3×Dic3—32no—
202C2×C3×Dic3—31no—
203C2×C3×Dic3—62no—
204C2×C3×Dic3C22×D312yes—
205C2×C3×Dic3C22×D311yescharacteristic
206#626—62no—
207#626—62no—
208#624—64no—
209#624—64no—
210#624—62no—
211C23×C9—21no—
212C23×C9—31no—
213C23×C9C22×D311yescharacteristic
214C22×D9—31no—
215C22×D9—34no—
216C22×D9—31no—
217C22×D9—61no—
218C22×D9—61no—
219C22×D9—64no—
220C22×D9—62no—
221C22×D9—64no—
222C22×D9—64no—
223C22×D9—124no—
224C22×D9C22×D311yescharacteristic
225D4×C9—62no—
226C2×C4×C9—22no—
227C2×C4×C9—31no—
228C2×C4×C9—32no—
229C2×C4×C9—62no—
230C2×C4×C9C22×D312yes—
231#611—22no—
232#611—62no—
233C2×Dic9—22no—
234C2×Dic9—32no—
235C2×Dic9—62no—
236C2×Dic9C22×D311yescharacteristic
237C2×Dic9C22×D312yes—
238D36—62no—
239C4×D9—64no—
240C4×D9—64no—
241C4×D9—62no—
242C4×D9—64no—
243C4×D9—124no—
244C2×#123—271noSylow
245C2×C3×C9—22no—
246C2×C3×C9C22×D411yescharacteristic
247C2×C3×C9C2×#4112yes—
248C3⋊D9—22no—
249C3⋊D9#12314yescomplemented
250D3×C9—41no—
251C3×D9—41no—
252C23×D3—91no—
253C23×D3—91no—
254C23×D3—91no—
255C23×D3—91no—
256C23×D3—181no—
257C2×C3×D4—31no—
258C2×C3×D4—91no—
259C22×C3×C4—31no—
260C22×C3×C4—91no—
261C2×#67—31no—
262C2×#67—91no—
263C2×#67—91no—
264C2×#67C2×D911yescharacteristic
265C22×Dic3—31no—
266C22×Dic3C2×D911yescharacteristic, complemented
267D3×D4—184no—
268D3×D4—184no—
269C2×D12—91no—
270C2×D12—91no—
271C2×C4×D3—92no—
272C2×C4×D3—92no—
273C2×C4×D3—91no—
274C2×C4×D3—92no—
275C2×C4×D3—92no—
276C2×C4×D3—92no—
277C2×C4×D3—92no—
278C2×C4×D3—91no—
279C2×C4×D3—182no—
280C2×C4×D3—184no—
281C2×C4×D3—184no—
282C3×C4⋊C4—32no—
283C3×C4⋊C4—92no—
284C3×C22⋊C4—21no—
285C3×C22⋊C4—32no—
286C3×C22⋊C4—92no—
287C3×C22⋊C4C2×D911yescharacteristic, complemented
288C3×C22⋊C4D3211yescharacteristic
289#217—92no—
290#217—91no—
291#217—91no—
292#217—92no—
293#217—181no—
294(C2×D3)⋊C4—32no—
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—32no—
302C4⋊Dic3C2×D912yescomplemented
303Dic3⋊C4—92no—
304Dic3⋊C4—92no—
305Dic3⋊C4—92no—
306Dic3⋊C4—92no—
307C22×C32—22no—
308C22×C32D3×D412yes—
309C22×C32C23×D311yescharacteristic
310C2×C3⋊D3—32no—
311C2×C3⋊D3—34no—
312C2×C3⋊D3—32no—
313C2×C3⋊D3—68no—
314C2×C3⋊D3—62no—
315C2×C3×D3—21no—
316C2×C3×D3—22no—
317C2×C3×D3—61no—
318C2×C3×D3—62no—
319D32—124no—
320C32×C4—22no—
321C3⋊Dic3—62no—
322C3×Dic3—22no—
323C3×Dic3—21no—
324C3×Dic3—62no—
325C3×Dic3—61no—
326C22×C9—22no—
327C22×C9—21no—
328C22×C9—22no—
329C22×C9—42no—
330C22×C9—61no—
331C22×C9—62no—
332C22×C9D3×D412yes—
333C22×C9C23×D311yescharacteristic
334C2×D9—21no—
335C2×D9—22no—
336C2×D9—32no—
337C2×D9—34no—
338C2×D9—32no—
339C2×D9—62no—
340C2×D9—64no—
341C2×D9—62no—
342C2×D9—68no—
343C2×D9—62no—
344C2×D9—128no—
345C2×D9—122no—
346C2×D9—124no—
347C4×C9—22no—
348C4×C9—42no—
349C4×C9—62no—
350C4×C9—61no—
351Dic9—22no—
352Dic9—21no—
353Dic9—62no—
354Dic9—122no—
355C22×D4—271no—
356C23×C4—271no—
357#123—31no—
358#123—91no—
359#123—272no—
360#123—272no—
361#123—271no—
362#123—271no—
363C2×C4⋊C4—272no—
364C2×C22⋊C4—272no—
365C2×C22⋊C4—271no—
366C3×C9C2×#12311yescharacteristic, complemented, Sylow
367C23×C3—21no—
368C23×C3—31no—
369C23×C3—91no—
370C23×C3C22×D911yescharacteristic
371C23×C3C2×D3211yescharacteristic
372C22×D3—31no—
373C22×D3—91no—
374C22×D3—91no—
375C22×D3—91no—
376C22×D3—94no—
377C22×D3—91no—
378C22×D3—94no—
379C22×D3—181no—
380C22×D3—181no—
381C22×D3—184no—
382C22×D3—182no—
383C22×D3—184no—
384C22×D3—182no—
385C22×D3—184no—
386C22×D3—184no—
387C22×D3—184no—
388C22×D3—364no—
389C22×D3C22×D911yescharacteristic
390C3×D4—62no—
391C3×D4—182no—
392C2×C3×C4—22no—
393C2×C3×C4—31no—
394C2×C3×C4—32no—
395C2×C3×C4—62no—
396C2×C3×C4—91no—
397C2×C3×C4—92no—
398C2×C3×C4—182no—
399C2×C3×C4C22×D912yes—
400C2×C3×C4C2×D3212yes—
401#67—22no—
402#67—62no—
403#67—182no—
404#67—182no—
405C2×Dic3—22no—
406C2×Dic3—31no—
407C2×Dic3—32no—
408C2×Dic3—62no—
409C2×Dic3—92no—
410C2×Dic3—92no—
411C2×Dic3—182no—
412C2×Dic3C22×D911yescharacteristic
413C2×Dic3C22×D912yes—
414D12—182no—
415D12—182no—
416C4×D3—184no—
417C4×D3—184no—
418C4×D3—182no—
419C4×D3—184no—
420C4×D3—184no—
421C4×D3—182no—
422C4×D3—184no—
423C4×D3—184no—
424C4×D3—364no—
425C2×C32—22no—
426C2×C32C2×D3×D411yescharacteristic
427C2×C32D3×#4112yes—
428C3⋊D3—34no—
429C3⋊D3—62no—
430C3×D3—41no—
431C3×D3—121no—
432C2×C9—22no—
433C2×C9—21no—
434C2×C9—22no—
435C2×C9—42no—
436C2×C9—121no—
437C2×C9C2×D3×D411yescharacteristic
438C2×C9D3×#4112yes—
439D9—34no—
440D9—41no—
441D9—64no—
442D9—62no—
443D9—122no—
444C24—271no—
445C24—271no—
446C2×D4—31no—
447C2×D4—91no—
448C2×D4—271no—
449C2×D4—271no—
450C2×D4—544no—
451C22×C4—31no—
452C22×C4—91no—
453C22×C4—272no—
454C22×C4—271no—
455C22×C4—272no—
456C22×C4—272no—
457C22×C4—544no—
458C4⋊C4—32no—
459C4⋊C4—92no—
460C4⋊C4—272no—
461C4⋊C4—272no—
462C22⋊C4—32no—
463C22⋊C4—92no—
464C22⋊C4—272no—
465C22⋊C4—272no—
466C22⋊C4—271no—
467C22⋊C4—272no—
468C22⋊C4D3×D911yescharacteristic, complemented
469C22×C3—22no—
470C22×C3—22no—
471C22×C3—21no—
472C22×C3—22no—
473C22×C3—42no—
474C22×C3—61no—
475C22×C3—62no—
476C22×C3—181no—
477C22×C3—182no—
478C22×C3D4×D912yes—
479C22×C3C23×D911yescharacteristic
480C22×C3#380212yes—
481C22×C3C22×D3211yescharacteristic
482C2×D3—21no—
483C2×D3—22no—
484C2×D3—61no—
485C2×D3—62no—
486C2×D3—92no—
487C2×D3—94no—
488C2×D3—92no—
489C2×D3—94no—
490C2×D3—92no—
491C2×D3—92no—
492C2×D3—184no—
493C2×D3—182no—
494C2×D3—182no—
495C2×D3—188no—
496C2×D3—188no—
497C2×D3—182no—
498C2×D3—182no—
499C2×D3—368no—
500C2×D3—362no—
501C2×D3—364no—
502C2×D3—364no—
503C3×C4—22no—
504C3×C4—22no—
505C3×C4—42no—
506C3×C4—62no—
507C3×C4—61no—
508C3×C4—182no—
509C3×C4—181no—
510Dic3—22no—
511Dic3—21no—
512Dic3—62no—
513Dic3—61no—
514Dic3—182no—
515Dic3—182no—
516Dic3—362no—
517C32D3×#12311yescharacteristic
518C9—21no—
519C9D3×#12311yescharacteristic, complemented
520C23—31no—
521C23—91no—
522C23—271no—
523C23—271no—
524C23—274no—
525C23—544no—
526C23—542no—
527C23—544no—
528C23C2×D3×D911yescharacteristic
529D4—62no—
530D4—182no—
531D4—542no—
532D4—542no—
533C2×C4—32no—
534C2×C4—31no—
535C2×C4—62no—
536C2×C4—92no—
537C2×C4—91no—
538C2×C4—182no—
539C2×C4—272no—
540C2×C4—544no—
541C2×C4—544no—
542C2×C4—544no—
543C2×C4—542no—
544C2×C4C2×D3×D912yes—
545C2×C3—22no—
546C2×C3—22no—
547C2×C3—22no—
548C2×C3—21no—
549C2×C3—42no—
550C2×C3—121no—
551C2×C3—361no—
552C2×C3C2×D4×D911yescharacteristic
553C2×C3#41×D912yes—
554C2×C3C2×#380211yescharacteristic
555C2×C3#6431212yes—
556D3—41no—
557D3—94no—
558D3—94no—
559D3—121no—
560D3—184no—
561D3—182no—
562D3—182no—
563D3—362no—
564C22—22no—
565C22—62no—
566C22—61no—
567C22—182no—
568C22—181no—
569C22—274no—
570C22—272no—
571C22—272no—
572C22—548no—
573C22—542no—
574C22—1084no—
575C22#8864612yes—
576C22C22×D3×D911yescharacteristic
577C4—22no—
578C4—62no—
579C4—61no—
580C4—182no—
581C4—181no—
582C4—542no—
583C3—21no—
584C3D9×#12311yescharacteristic, complemented
585C3#1059429511yescharacteristic
586C2—22no—
587C2—121no—
588C2—274no—
589C2—361no—
590C2—542no—
591C2C2×#8864611yescharacteristic
592C2#1172249812yes—
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC3×C9×C22⋊C4C22
socleC22×C32C23×D3
prime radicalC2×#88668C2
hypercenterC22⋊C4D3×D9
hypocenterC3×C9C2×#123
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C22×(C3⋊D9)⋊C4C2semidirect
3. C22⋊C4×C3⋊D9C2semidirect
4. C9×#886C2semidirect
5. C3×#63448C2semidirect
6. #11721265C2semidirect
7. #11721259C2semidirect
8. #11721048C2semidirect
9. #11721000C2semidirect
10. #11720958C2semidirect
11. #11720942C2semidirect
19. C22×C9×Dic3C22semidirect
20. C22×C3×Dic9C22semidirect
27. C2×(C3⋊D9)⋊C4C22semidirect
28. C3×C9×C22⋊C4C22semidirect
29. #88551C22semidirect
30. #88536C22semidirect
31. C9×(C2×D3)⋊C4C22semidirect
32. C3×(C2×D9)⋊C4C22semidirect
33. C9×C4⋊Dic3C22semidirect
34. C3×C4⋊Dic9C22semidirect
35. (C2×D3)⋊Dic9C22semidirect
36. (C2×D9)⋊Dic3C22semidirect
37. #88442C22semidirect
38. #88442C22semidirect
57. #63448D3semidirect
121. C22×Dic9C2×D3semidirect
133. C9×C22⋊C4C2×D3semidirect
140. (C2×D9)⋊C4C2×D3semidirect
141. C4⋊Dic9C2×D3semidirect
178. #886D9semidirect
249. C3⋊D9#123semidirect
266. C22×Dic3C2×D9semidirect
287. C3×C22⋊C4C2×D9semidirect
300. (C2×D3)⋊C4C2×D9semidirect
302. C4⋊Dic3C2×D9semidirect
366. C3×C9C2×#123semidirect
468. C22⋊C4D3×D9semidirect
519. C9D3×#123semidirect
584. C3D9×#123semidirect

Maximal subgroups

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

subgroupindex
C2×#886682
C22×(C3⋊D9)⋊C42
C22⋊C4×C3⋊D92
C9×#8862
C3×#634482
#117212652
#117212592
#117210482
#117210002
#117209582
#117209422
#105942953
D9×#1233