Finite Group Databasebeta

← #13150545

Subgroups — #13150545

Order 1728.

1096 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C9×#235C211yesmaximal, characteristic, complemented
2C2×D9×D12C211yesmaximal, characteristic, complemented
3#11721819C211yesmaximal, characteristic, complemented
4C3×C4×#611C211yesmaximal, characteristic, complemented
5#11721292C212yesmaximal, complemented
6#11721282C212yesmaximal, complemented
7D3×(C2×D9)⋊C4C212yesmaximal, complemented
8D12×Dic9C211yesmaximal, characteristic, complemented
9#11720954C212yesmaximal, complemented
10#11720953C211yesmaximal, characteristic, complemented
11#11720944C211yesmaximal, characteristic, complemented
12#10595067—31nomaximal
13#10589931—31nomaximal
14C22×D3×D9C2212yes—
15C22×C3×C4×C9C2211yescharacteristic, complemented
16C2×#6233C2212yescomplemented
17C2×C4⋊(C3⋊D9)C2211yescharacteristic
18C2×C9×#67C2212yescomplemented
19C2×C3×#611C2211yescharacteristic
20C9×#235—24no—
21C2×C9×D12C2211yescharacteristic
22C2×C4×D3×C9C2212yescomplemented
23C2×C3×C4×D9C2211yescharacteristic
24C2×C9×Dic6C2211yescharacteristic, complemented
25C2×#6195C2212yes—
26C2×D3×Dic9C2212yescomplemented
27D9×D12—24no—
28#88536C2212yescomplemented
29C4⋊(C3⋊Dic9)C2211yescharacteristic, complemented
30C3×#3676C2211yescharacteristic, complemented
31C3×(C2×D9)⋊C4C2211yescharacteristic
32C3×Dic9⋊C4C2211yescharacteristic, complemented
33C3×C4×Dic9C2211yescharacteristic, complemented
34(C2×D3)⋊Dic9C2212yescomplemented
35#88445C2212yescomplemented
36#88445C2212yescomplemented
37(C2×D9)⋊Dic3C2212yes—
38#88442C2212yescomplemented
39C2×C3×#235D311yescharacteristic
40C2×D3×D12—31no—
41#64139—31no—
42C3×C4×#67—31no—
43#63983—32no—
44#63971—32no—
45D3×(C2×D3)⋊C4—32no—
46Dic3×D12—31no—
47#63864—31no—
48#63856—31no—
49#63850—32no—
50C2×C9×#41—31no—
51C2×D4×D9—31no—
52#63509—31no—
53#63508—31no—
54#63501—32no—
55#63499—32no—
56D4×Dic9—31no—
57#63492—31no—
58#63492—61no—
59#63491—32no—
60C4×#611D311yescharacteristic, complemented
61C2×(C2×D9)⋊C4—32no—
62C23×C3×C9C2311yescharacteristic
63C22×C3⋊D9C2312yes—
64C22×D3×C9C2312yes—
65C22×C3×D9C2311yescharacteristic
66C2×D3×D9—24no—
67C2×D3×D9—24no—
68C2×D3×D9—42no—
69C2×C3×C4×C9—22no—
70C2×C3×C4×C9C2311yescharacteristic
71C2×C3×C4×C9C2311yescharacteristic
72#6233—42no—
73C2×C3⋊Dic9C2312yes—
74C4⋊(C3⋊D9)—22no—
75C9×#67—24no—
76C3×#611—22no—
77C2×C9×Dic3C2312yes—
78C2×C3×Dic9C2311yescharacteristic
79C2×C3×Dic9C2311yescharacteristic
80C2×C3×Dic9C2311yescharacteristic
81C9×D12D414yescomplemented
82C4×D3×C9—24no—
83C3×C4×D9—22no—
84C9×Dic6—22no—
85#6195—24no—
86D3×Dic9—42no—
87#5829—91no—
88#5550—91no—
89C22×D32—32no—
90C22×C32×C4C2×D311yescharacteristic
91C2×#638—32no—
92C2×C4⋊(C3⋊D3)—31no—
93C2×C3×#67—31no—
94C2×C3×#67C2×D312yes—
95C3×#235—24no—
96C2×C3×D12C2×D311yescharacteristic
97C2×C3×C4×D3—31no—
98C2×C3×C4×D3C2×D312yes—
99C2×C3×Dic6C2×D311yescharacteristic
100C2×#626—32no—
101C2×D3×Dic3—32no—
102D3×D12—64no—
103C23×D9—32no—
104#3752—32no—
105C4⋊(C3⋊Dic3)—31no—
106C3×#217—31no—
107C3×(C2×D3)⋊C4—31no—
108C3×Dic3⋊C4—31no—
109C3×C4×Dic3—31no—
110#3724—32no—
111#3722—32no—
112#3722—32no—
113(C2×D3)⋊Dic3—32no—
114(C2×D3)⋊Dic3—32no—
115C9×#41—64no—
116C2×Q8×C9—31no—
117C2×D4×C9—32no—
118C2×D4×C9—31no—
119C22×C4×C9—21no—
120C22×C4×C9—32no—
121C22×C4×C9C2×D311yescharacteristic, complemented
122C2×#611—32no—
123C2×#611—32no—
124C2×#611—62no—
125C2×#611C2×D311yescharacteristic
126C22×Dic9—32no—
127D4×D9—64no—
128C2×D36—31no—
129C2×D36—61no—
130C2×C4×D9C2×D311yescharacteristic
131#3676—32no—
132#3676C2×D311yescharacteristic, complemented
133(C2×D9)⋊C4—32no—
134(C2×D9)⋊C4—32no—
135(C2×D9)⋊C4—32no—
136(C2×D9)⋊C4—32no—
137(C2×D9)⋊C4—62no—
138(C2×D9)⋊C4C2×D311yescharacteristic
139C4⋊Dic9—31no—
140C4⋊Dic9—61no—
141Dic9⋊C4—32no—
142Dic9⋊C4C2×D311yescharacteristic, complemented
143C4×Dic9C2×D311yescharacteristic, complemented
144C22×C3×C9—21no—
145C22×C3×C9—21no—
146C22×C3×C9—21no—
147C22×C3×C9C2411yescharacteristic
148C2×C3⋊D9—22no—
149C2×C3⋊D9—22no—
150C2×C3⋊D9—22no—
151C2×D3×C9—22no—
152C2×D3×C9—22no—
153C2×D3×C9C2×D414yes—
154C2×C3×D9—21no—
155C2×C3×D9—21no—
156C2×C3×D9#4112yescomplemented
157D3×D9—48no—
158C3×C4×C9—21no—
159C3×C4×C9C2×D412yes—
160C3⋊Dic9—22no—
161C9×Dic3—22no—
162C3×Dic9—21no—
163C3×Dic9—21no—
164C3×Dic9—21no—
165C2×C3×#41—31no—
166C2×D3×D4—91no—
167C2×#235D911yescharacteristic, complemented
168C22×D12—91no—
169C3×C4×D4—91no—
170#947—91no—
171#946—91no—
172#939—92no—
173#937—92no—
174#937—92no—
175D4×Dic3—91no—
176#930—91no—
177#930—91no—
178#930—181no—
179#929—92no—
180C4×#67—31no—
181C2×(C2×D3)⋊C4—92no—
182#924—91no—
183#894—92no—
184#884—92no—
185D3×C22⋊C4—92no—
186#875—91no—
187C4×D12—91no—
188C23×C32C22×D311yescharacteristic
189C22×C3⋊D3—32no—
190C22×C3×D3—31no—
191C22×C3×D3C22×D312yes—
192C2×D32—64no—
193C2×D32—64no—
194C2×D32—122no—
195C2×C32×C4—22no—
196C2×C32×C4C22×D311yescharacteristic
197C2×C32×C4C22×D311yescharacteristic
198#638—122no—
199C2×C3⋊Dic3—32no—
200C4⋊(C3⋊D3)—62no—
201C3×#67—24no—
202C3×#67—62no—
203C2×C3×Dic3—31no—
204C2×C3×Dic3—31no—
205C2×C3×Dic3—31no—
206C2×C3×Dic3C22×D312yes—
207C3×D12#6714yes—
208C3×C4×D3—24no—
209C3×C4×D3—62no—
210C3×Dic6—22no—
211#626—64no—
212D3×Dic3—122no—
213C23×C9—21no—
214C23×C9—32no—
215C23×C9C22×D311yescharacteristic
216C22×D9—32no—
217C22×D9—62no—
218C22×D9—64no—
219C22×D9—64no—
220C22×D9—122no—
221C22×D9C22×D311yescharacteristic
222Q8×C9—62no—
223D4×C9—34no—
224D4×C9—64no—
225C2×C4×C9—22no—
226C2×C4×C9—21no—
227C2×C4×C9—21no—
228C2×C4×C9—32no—
229C2×C4×C9—42no—
230C2×C4×C9—64no—
231C2×C4×C9C22×D311yescharacteristic
232C2×C4×C9C22×D311yescharacteristic
233#611—22no—
234#611—64no—
235#611—122no—
236#611—124no—
237C2×Dic9—32no—
238C2×Dic9—62no—
239C2×Dic9—122no—
240C2×Dic9C22×D311yescharacteristic
241C2×Dic9C22×D311yescharacteristic
242C2×Dic9C22×D311yescharacteristic
243D36—62no—
244D36—122no—
245C4×D9—22no—
246#546—271noSylow
247C2×C3×C9—41no—
248C2×C3×C9C22×D411yescharacteristic
249C2×C3×C9C2×#4111yescharacteristic
250C2×C3×C9H(5,𝔽2)11yescharacteristic
251C3⋊D9—42no—
252D3×C9—24no—
253C3×D9—22no—
254C23×D3—92no—
255C23×D3—92no—
256C3×#41—64no—
257C2×C3×Q8—31no—
258C2×C3×D4—32no—
259C2×C3×D4—31no—
260C2×C3×D4—91no—
261C22×C3×C4—21no—
262C22×C3×C4—32no—
263C22×C3×C4—91no—
264C22×C3×C4C2×D911yescharacteristic, complemented
265C22×C3×C4D3211yescharacteristic
266C2×#67—31no—
267C2×#67—92no—
268C2×#67—92no—
269C2×#67—92no—
270C2×#67—182no—
271C2×#67C2×D912yescomplemented
272C22×Dic3—92no—
273D3×D4—184no—
274#235—24no—
275C2×D12—92no—
276C2×D12—91no—
277C2×D12—91no—
278C2×D12—181no—
279C2×D12—184no—
280C2×D12C2×D911yescharacteristic
281C2×C4×D3—31no—
282C2×C4×D3—92no—
283C2×C4×D3C2×D912yescomplemented
284C2×Dic6C2×D911yescharacteristic, complemented
285C3×C4⋊C4—91no—
286C3×C22⋊C4—91no—
287C3×C22⋊C4—91no—
288C3×C42—91no—
289#217—31no—
290#217—92no—
291#217—92no—
292(C2×D3)⋊C4—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—92no—
300(C2×D3)⋊C4—92no—
301(C2×D3)⋊C4—182no—
302C4⋊Dic3—91no—
303C4⋊Dic3—91no—
304C4⋊Dic3—181no—
305Dic3⋊C4—31no—
306Dic3⋊C4—92no—
307Dic3⋊C4—92no—
308C4×Dic3—31no—
309C22×C32—21no—
310C22×C32—21no—
311C22×C32—21no—
312C22×C32C23×D311yescharacteristic
313C2×C3⋊D3—62no—
314C2×C3⋊D3—62no—
315C2×C3⋊D3—62no—
316C2×C3×D3—22no—
317C2×C3×D3—22no—
318C2×C3×D3—32no—
319C2×C3×D3—61no—
320C2×C3×D3—61no—
321C2×C3×D3C2×#6714yes—
322D32—128no—
323C32×C4—21no—
324C32×C4C2×#6712yes—
325C3⋊Dic3—62no—
326C3×Dic3—22no—
327C3×Dic3—61no—
328C3×Dic3—61no—
329C3×Dic3—61no—
330C22×C9—22no—
331C22×C9—21no—
332C22×C9—21no—
333C22×C9—21no—
334C22×C9—21no—
335C22×C9—34no—
336C22×C9—41no—
337C22×C9—41no—
338C22×C9—62no—
339C22×C9—62no—
340C22×C9C23×D311yescharacteristic
341C2×D9—21no—
342C2×D9—21no—
343C2×D9—62no—
344C2×D9—62no—
345C2×D9—62no—
346C2×D9—122no—
347C2×D9—122no—
348C2×D9—122no—
349C2×D9—128no—
350C2×D9#23512yescomplemented
351C4×C9—21no—
352C4×C9—22no—
353C4×C9—41no—
354C4×C9—62no—
355C4×C9D3×D412yes—
356Dic9—21no—
357Dic9—21no—
358Dic9—21no—
359Dic9—62no—
360Dic9—122no—
361C2×#41—31no—
362C22×D4—271no—
363#124—271no—
364#123—272no—
365C22⋊Q8—271no—
366#121—272no—
367#121—271no—
368#120—272no—
369C4×D4—91no—
370C4×D4—271no—
371C2×C22⋊C4—272no—
372C3×C9#54611yescharacteristic, complemented, Sylow
373C23×C3—21no—
374C23×C3—32no—
375C23×C3—91no—
376C23×C3C22×D911yescharacteristic
377C23×C3C2×D3211yescharacteristic
378C22×D3—31no—
379C22×D3—92no—
380C22×D3—92no—
381C22×D3—182no—
382C22×D3—184no—
383C22×D3—184no—
384C22×D3—184no—
385C22×D3—184no—
386C22×D3—362no—
387C22×D3—362no—
388C22×D3C22×D912yes—
389C3×Q8—62no—
390C3×D4—34no—
391C3×D4—64no—
392C3×D4—182no—
393C2×C3×C4—22no—
394C2×C3×C4—22no—
395C2×C3×C4—21no—
396C2×C3×C4—21no—
397C2×C3×C4—32no—
398C2×C3×C4—42no—
399C2×C3×C4—64no—
400C2×C3×C4—91no—
401C2×C3×C4—91no—
402C2×C3×C4—91no—
403C2×C3×C4—182no—
404C2×C3×C4C22×D911yescharacteristic
405C2×C3×C4C22×D911yescharacteristic
406C2×C3×C4C2×D3211yescharacteristic
407C2×C3×C4C2×D3211yescharacteristic
408#67—24no—
409#67—62no—
410#67—184no—
411#67—362no—
412#67—362no—
413#67—364no—
414C2×Dic3—31no—
415C2×Dic3—31no—
416C2×Dic3—31no—
417C2×Dic3—92no—
418C2×Dic3—92no—
419C2×Dic3—182no—
420C2×Dic3—362no—
421C2×Dic3C22×D912yes—
422D12—184no—
423D12—182no—
424D12—182no—
425D12—362no—
426D12#61114yescomplemented
427C4×D3—24no—
428C4×D3—62no—
429C4×D3—362no—
430Dic6—22no—
431C2×C32—41no—
432C2×C32D3×#4111yescharacteristic
433C2×C32#100911yescharacteristic
434C2×C32C22×#6711yescharacteristic
435C3⋊D3—122no—
436C3×D3—24no—
437C3×D3—62no—
438C2×C9—21no—
439C2×C9—21no—
440C2×C9—21no—
441C2×C9—42no—
442C2×C9—41no—
443C2×C9—64no—
444C2×C9C2×#23511yescharacteristic
445C2×C9C2×D3×D411yescharacteristic
446C2×C9#100911yescharacteristic
447D9—22no—
448D9—122no—
449D9—242no—
450C24—272no—
451#41—64no—
452C2×Q8—31no—
453C2×D4—32no—
454C2×D4—31no—
455C2×D4—91no—
456C2×D4—272no—
457C2×D4—272no—
458C2×D4—271no—
459C2×D4—544no—
460C22×C4—32no—
461C22×C4—91no—
462C22×C4—272no—
463C22×C4D3×D911yescharacteristic, complemented
464C4⋊C4—91no—
465C4⋊C4—272no—
466C4⋊C4—271no—
467C22⋊C4—91no—
468C22⋊C4—91no—
469C22⋊C4—272no—
470C22⋊C4—272no—
471C22⋊C4—272no—
472C22⋊C4—272no—
473C22⋊C4—272no—
474C42—91no—
475C22×C3—22no—
476C22×C3—21no—
477C22×C3—21no—
478C22×C3—21no—
479C22×C3—21no—
480C22×C3—21no—
481C22×C3—21no—
482C22×C3—21no—
483C22×C3—34no—
484C22×C3—41no—
485C22×C3—41no—
486C22×C3—62no—
487C22×C3—62no—
488C22×C3—92no—
489C22×C3—181no—
490C22×C3—181no—
491C22×C3C23×D911yescharacteristic
492C22×C3C22×D3211yescharacteristic
493C2×D3—22no—
494C2×D3—22no—
495C2×D3—32no—
496C2×D3—61no—
497C2×D3—61no—
498C2×D3—182no—
499C2×D3—182no—
500C2×D3—182no—
501C2×D3—182no—
502C2×D3—182no—
503C2×D3—182no—
504C2×D3—362no—
505C2×D3—362no—
506C2×D3—362no—
507C2×D3—368no—
508C2×D3—368no—
509C2×D3C2×#61114yes—
510C3×C4—21no—
511C3×C4—21no—
512C3×C4—22no—
513C3×C4—41no—
514C3×C4—62no—
515C3×C4—181no—
516C3×C4—181no—
517C3×C4—181no—
518C3×C4C2×#61112yes—
519C3×C4D3×#6712yes—
520Dic3—22no—
521Dic3—61no—
522Dic3—61no—
523Dic3—61no—
524Dic3—182no—
525Dic3—182no—
526Dic3—362no—
527C32#582911yescharacteristic
528C9—21no—
529C9#555011yescharacteristic, complemented
530C23—32no—
531C23—91no—
532C23—272no—
533C23—544no—
534C23—544no—
535C23—1082no—
536C23C2×D3×D911yescharacteristic
537Q8—62no—
538D4—34no—
539D4—64no—
540D4—182no—
541D4—544no—
542D4—542no—
543D4—1082no—
544C2×C4—22no—
545C2×C4—32no—
546C2×C4—64no—
547C2×C4—91no—
548C2×C4—91no—
549C2×C4—91no—
550C2×C4—182no—
551C2×C4—272no—
552C2×C4—1082no—
553C2×C4C2×D3×D911yescharacteristic
554C2×C4C2×D3×D911yescharacteristic
555C2×C3—21no—
556C2×C3—21no—
557C2×C3—21no—
558C2×C3—42no—
559C2×C3—41no—
560C2×C3—41no—
561C2×C3—64no—
562C2×C3—182no—
563C2×C3#41×D911yescharacteristic
564C2×C3#6371511yescharacteristic
565C2×C3C22×#61111yescharacteristic
566C2×C3D3×#23511yescharacteristic
567C2×C3#6430811yescharacteristic
568C2×C3C2×D3×#6711yescharacteristic
569D3—24no—
570D3—62no—
571D3—362no—
572D3—362no—
573D3—722no—
574C22—21no—
575C22—21no—
576C22—21no—
577C22—34no—
578C22—62no—
579C22—62no—
580C22—92no—
581C22—181no—
582C22—181no—
583C22—542no—
584C22—542no—
585C22—542no—
586C22—1088no—
587C22C22×D3×D911yescharacteristic
588C4—21no—
589C4—62no—
590C4—181no—
591C4—181no—
592C4—181no—
593C4—542no—
594C4D3×#61112yes—
595C3—21no—
596C3#1058993111yescharacteristic, complemented
597C3#1059506711yescharacteristic
598C2—41no—
599C2—64no—
600C2—182no—
601C2—1082no—
602C2D9×#23511yescharacteristic
603C2#1172247811yescharacteristic
604C2C2×D3×#61111yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC22×C3×C4×C9C22
socleC22×C32C23×D3
hypercenterC22×C4D3×D9
hypocenterC3×C9#546
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×C9×#235C2semidirect
2. C2×D9×D12C2semidirect
3. #11721819C2semidirect
4. C3×C4×#611C2semidirect
5. #11721292C2semidirect
6. #11721282C2semidirect
7. D3×(C2×D9)⋊C4C2semidirect
8. D12×Dic9C2semidirect
9. #11720954C2semidirect
10. #11720953C2semidirect
11. #11720944C2semidirect
15. C22×C3×C4×C9C22semidirect
16. C2×#6233C22semidirect
18. C2×C9×#67C22semidirect
22. C2×C4×D3×C9C22semidirect
24. C2×C9×Dic6C22semidirect
26. C2×D3×Dic9C22semidirect
28. #88536C22semidirect
29. C4⋊(C3⋊Dic9)C22semidirect
30. C3×#3676C22semidirect
32. C3×Dic9⋊C4C22semidirect
33. C3×C4×Dic9C22semidirect
34. (C2×D3)⋊Dic9C22semidirect
35. #88445C22semidirect
36. #88445C22semidirect
38. #88442C22semidirect
60. C4×#611D3semidirect
81. C9×D12D4semidirect
121. C22×C4×C9C2×D3semidirect
132. #3676C2×D3semidirect
142. Dic9⋊C4C2×D3semidirect
143. C4×Dic9C2×D3semidirect
156. C2×C3×D9#41semidirect
167. C2×#235D9semidirect
264. C22×C3×C4C2×D9semidirect
271. C2×#67C2×D9semidirect
283. C2×C4×D3C2×D9semidirect
284. C2×Dic6C2×D9semidirect
350. C2×D9#235semidirect
372. C3×C9#546semidirect
426. D12#611semidirect
463. C22×C4D3×D9semidirect
529. C9#5550semidirect
596. C3#10589931semidirect

Maximal subgroups

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

subgroupindex
C2×C9×#2352
C2×D9×D122
#117218192
C3×C4×#6112
#117212922
#117212822
D3×(C2×D9)⋊C42
D12×Dic92
#117209542
#117209532
#117209442
#105950673
#105899313