Finite Group Databasebeta

← #13150994

Subgroups — #13150994

Order 1728.

1044 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C9×#237C211yesmaximal, characteristic, complemented
2C2×#88635C211yesmaximal, characteristic, complemented
3#11721840C211yesmaximal, characteristic, complemented
4C3×#63500C211yesmaximal, characteristic, complemented
5#11721289C212yesmaximal, complemented
6#11721277C212yesmaximal, complemented
7#11721262C212yesmaximal, complemented
8#11721240C212yesmaximal, complemented
9#11721038C211yesmaximal, characteristic, complemented
10C4×#6194C211yesmaximal, characteristic, complemented
11#11720911C211yesmaximal, characteristic, complemented
12#10595236—31nomaximal
13#10589936—31nomaximal
14C2×C3×D4×C9C2211yescharacteristic
15C2×#6233C2212yes—
16C2×#6233C2212yes—
17C2×C4⋊(C3⋊D9)C2211yescharacteristic
18C2×C9×#67C2212yes—
19C2×C3×#611C2212yes—
20C22×C9×Dic3C2212yes—
21C9×#237—24no—
22C2×C4×D3×C9C2211yescharacteristic
23C2×C3×C4×D9C2211yescharacteristic
24C2×C9×Dic6C2211yescharacteristic
25C2×#6195C2212yes—
26C2×#6194C2211yescharacteristic
27C2×#6192C2212yes—
28C2×(C3⋊D9)⋊C4C2212yes—
29#88635—24no—
30C4×C3⋊Dic9C2211yescharacteristic
31C3×#3676C2212yes—
32C3×C4⋊Dic9C2211yescharacteristic
33(C2×D3)⋊Dic9C2211yescharacteristic
34#88445C2212yes—
35(C2×D9)⋊Dic3C2211yescharacteristic
36Dic3⋊Dic9C2212yes—
37#88442C2211yescharacteristic
38Dic3×Dic9C2212yes—
39C2×C3×#237D311yescharacteristic
40C2×#3799—31no—
41#64149—31no—
42C3×#938—31no—
43#63979—32no—
44#63978—32no—
45#63958—32no—
46#63952—32no—
47#63909—31no—
48C4×#625—31no—
49#63843—31no—
50C2×C9×#41—31no—
51C2×#3701—31no—
52Q8×Dic9—31no—
53#63502—31no—
54#63502—32no—
55#63502—61no—
56#63500—32no—
57#63500D311yescharacteristic, complemented
58#63492—32no—
59#63492—31no—
60C4×#611—31no—
61C4×#611—32no—
62C23×C3×C9C2312yes—
63C22×C3⋊D9C2312yes—
64C22×D3×C9C2311yescharacteristic
65C22×C3×D9C2311yescharacteristic
66C3×D4×C9—22no—
67C2×C3×C4×C9C2311yescharacteristic
68#6233—24no—
69#6233—42no—
70C2×C3⋊Dic9C2311yescharacteristic
71C2×C3⋊Dic9C2311yescharacteristic
72C4⋊(C3⋊D9)—22no—
73C9×#67—24no—
74C3×#611—42no—
75C2×C9×Dic3—24no—
76C2×C9×Dic3C2311yescharacteristic
77C2×C9×Dic3C2312yes—
78C2×C3×Dic9C2311yescharacteristic
79C2×C3×Dic9C2312yes—
80C4×D3×C9—22no—
81C3×C4×D9—22no—
82C9×Dic6D414yescomplemented
83#6195—42no—
84#6194—22no—
85#6192—24no—
86(C3⋊D9)⋊C4—24no—
87#5834—91no—
88#5599—91no—
89C2×C32×D4C2×D311yescharacteristic
90C2×#638—32no—
91C2×#638—32no—
92C2×C4⋊(C3⋊D3)—31no—
93C2×C3×#67—32no—
94C2×C3×#67C2×D312yes—
95C22×C3×Dic3C2×D312yes—
96C3×#237—24no—
97C2×C3×C4×D3—31no—
98C2×C3×C4×D3C2×D311yescharacteristic
99C2×C3×Dic6C2×D311yescharacteristic
100C2×#626—32no—
101C2×#626—32no—
102C2×#625—31no—
103C2×#624—32no—
104#3799—64no—
105C4×C3⋊Dic3—31no—
106C3×#217—32no—
107C3×C4⋊Dic3—31no—
108#3724—31no—
109Dic3⋊Dic3—32no—
110#3722—32no—
111(C2×D3)⋊Dic3—31no—
112(C2×D3)⋊Dic3—31no—
113Dic32—32no—
114C9×#41—64no—
115C2×Q8×C9—31no—
116C2×D4×C9—21no—
117C2×D4×C9—32no—
118C2×D4×C9C2×D311yescharacteristic
119C22×C4×C9—32no—
120C22×C4×C9—31no—
121C2×#611—32no—
122C2×#611—31no—
123C2×#611—32no—
124C2×#611—32no—
125C2×#611—62no—
126C2×#611—62no—
127C2×#611C2×D312yes—
128#3701—64no—
129C2×D36—32no—
130C2×D36—31no—
131C2×D36—61no—
132C2×C4×D9—32no—
133C2×C4×D9C2×D311yescharacteristic
134#3676—31no—
135#3676C2×D312yes—
136(C2×D9)⋊C4—31no—
137(C2×D9)⋊C4—32no—
138C4⋊Dic9—32no—
139C4⋊Dic9C2×D311yescharacteristic
140Dic9⋊C4—31no—
141C4×Dic9—31no—
142C4×Dic9—32no—
143C4×Dic9—61no—
144C22×C3×C9—22no—
145C22×C3×C9—22no—
146C22×C3×C9—22no—
147C22×C3×C9C2411yescharacteristic
148C2×C3⋊D9—22no—
149C2×C3⋊D9—22no—
150C2×C3⋊D9—22no—
151C2×D3×C9—21no—
152C2×D3×C9—21no—
153C2×D3×C9—21no—
154C2×C3×D9—21no—
155C2×C3×D9—21no—
156C2×C3×D9—21no—
157C3×C4×C9C2×D412yes—
158C3⋊Dic9—21no—
159C3⋊Dic9#4112yescomplemented
160C9×Dic3—21no—
161C9×Dic3C2×D414yes—
162C3×Dic9—21no—
163C3×Dic9—22no—
164C2×C3×#41—31no—
165C2×#239—91no—
166C2×#237D911yescharacteristic, complemented
167C2×#235—91no—
168C3×#121—91no—
169Q8×Dic3—91no—
170#940—92no—
171#940—91no—
172#940—91no—
173#940—92no—
174#940—181no—
175#939—92no—
176#938—31no—
177#938—92no—
178#930—91no—
179#930—92no—
180C4×#67—92no—
181C4×#67—91no—
182C4×#67—91no—
183#895—91no—
184#887—91no—
185#884—92no—
186#67⋊C4—92no—
187C23×C32C22×D312yes—
188C22×C3⋊D3—32no—
189C22×C3×D3—31no—
190C22×C3×D3C22×D311yescharacteristic
191C32×D4—22no—
192C2×C32×C4C22×D311yescharacteristic
193#638—64no—
194#638—122no—
195C2×C3⋊Dic3—31no—
196C2×C3⋊Dic3—31no—
197C4⋊(C3⋊D3)—62no—
198C3×#67—24no—
199C3×#67—122no—
200C2×C3×Dic3—24no—
201C2×C3×Dic3—32no—
202C2×C3×Dic3—31no—
203C2×C3×Dic3C22×D311yescharacteristic
204C2×C3×Dic3C22×D312yes—
205C3×C4×D3—22no—
206C3×C4×D3—62no—
207C3×Dic6#6714yes—
208#626—64no—
209#626—122no—
210#625—62no—
211#624—64no—
212C23×C9—22no—
213C23×C9—31no—
214C23×C9C22×D312yes—
215C22×D9—32no—
216C22×D9—62no—
217C22×D9C22×D311yescharacteristic
218Q8×C9—34no—
219D4×C9—22no—
220D4×C9—42no—
221D4×C9—64no—
222C2×C4×C9—21no—
223C2×C4×C9—32no—
224C2×C4×C9—31no—
225C2×C4×C9—64no—
226C2×C4×C9—62no—
227C2×C4×C9C22×D311yescharacteristic
228#611—42no—
229#611—62no—
230#611—64no—
231#611—124no—
232#611—122no—
233#611—122no—
234#611—124no—
235C2×Dic9—31no—
236C2×Dic9—31no—
237C2×Dic9—61no—
238C2×Dic9—61no—
239C2×Dic9C22×D312yes—
240C2×Dic9C22×D311yescharacteristic
241D36—64no—
242D36—62no—
243D36—122no—
244C4×D9—22no—
245C4×D9—64no—
246#550—271noSylow
247C2×C3×C9—42no—
248C2×C3×C9C22×D411yescharacteristic
249C2×C3×C9C2×#4111yescharacteristic
250C2×C3×C9H(5,𝔽2)11yescharacteristic
251C3⋊D9—42no—
252D3×C9—41no—
253C3×D9—41no—
254C3×#41—64no—
255C2×C3×Q8—31no—
256C2×C3×D4—21no—
257C2×C3×D4—32no—
258C2×C3×D4—92no—
259C2×C3×D4C2×D911yescharacteristic
260C2×C3×D4D3211yescharacteristic
261C22×C3×C4—32no—
262C22×C3×C4—31no—
263C22×C3×C4—91no—
264C2×#67—32no—
265C2×#67—92no—
266C2×#67—91no—
267C2×#67—92no—
268C2×#67—92no—
269C2×#67—91no—
270C2×#67—92no—
271C2×#67—92no—
272C2×#67—92no—
273C2×#67—182no—
274C2×#67—182no—
275C2×#67C2×D912yes—
276C22×Dic3C2×D912yes—
277#239—184no—
278#237—24no—
279#235—184no—
280C2×D12—92no—
281C2×D12—91no—
282C2×D12—91no—
283C2×D12—92no—
284C2×D12—181no—
285C2×C4×D3—31no—
286C2×C4×D3—92no—
287C2×C4×D3—92no—
288C2×C4×D3C2×D911yescharacteristic
289C2×Dic6C2×D911yescharacteristic
290C3×C4⋊C4—91no—
291C3×C22⋊C4—92no—
292#217—32no—
293#217—91no—
294#217—91no—
295(C2×D3)⋊C4—92no—
296(C2×D3)⋊C4—91no—
297(C2×D3)⋊C4—91no—
298(C2×D3)⋊C4—92no—
299C4⋊Dic3—31no—
300C4⋊Dic3—92no—
301Dic3⋊C4—91no—
302Dic3⋊C4—91no—
303Dic3⋊C4—92no—
304C4×Dic3—92no—
305C4×Dic3—91no—
306C4×Dic3—91no—
307C4×Dic3—92no—
308C4×Dic3—181no—
309C22×C32—22no—
310C22×C32—22no—
311C22×C32—22no—
312C22×C32C23×D311yescharacteristic
313C2×C3⋊D3—62no—
314C2×C3⋊D3—62no—
315C2×C3⋊D3—62no—
316C2×C3×D3—21no—
317C2×C3×D3—21no—
318C2×C3×D3—21no—
319C2×C3×D3—61no—
320C2×C3×D3—61no—
321C2×C3×D3—61no—
322C32×C4C2×#6712yes—
323C3⋊Dic3—32no—
324C3⋊Dic3—61no—
325C3×Dic3—21no—
326C3×Dic3—62no—
327C3×Dic3—61no—
328C3×Dic3C2×#6714yes—
329C22×C9—22no—
330C22×C9—22no—
331C22×C9—22no—
332C22×C9—21no—
333C22×C9—42no—
334C22×C9—42no—
335C22×C9—42no—
336C22×C9—61no—
337C22×C9—61no—
338C22×C9—61no—
339C22×C9C23×D311yescharacteristic
340C2×D9—21no—
341C2×D9—21no—
342C2×D9—21no—
343C2×D9—62no—
344C2×D9—62no—
345C2×D9—62no—
346C2×D9—122no—
347C2×D9—122no—
348C2×D9—122no—
349C4×C9—22no—
350C4×C9—34no—
351C4×C9—61no—
352C4×C9D3×D412yes—
353Dic9—21no—
354Dic9—22no—
355Dic9—32no—
356Dic9—62no—
357Dic9—62no—
358Dic9—61no—
359C2×#41—31no—
360C2×#41—271no—
361#127—272no—
362#127—271no—
363#121—91no—
364#121—272no—
365#121—271no—
366#121—272no—
367C4×Q8—271no—
368C4×D4—272no—
369C4×D4—271no—
370C3×C9#55011yescharacteristic, complemented, Sylow
371C23×C3—22no—
372C23×C3—31no—
373C23×C3—91no—
374C23×C3C22×D912yes—
375C23×C3C2×D3212yes—
376C22×D3—31no—
377C22×D3—92no—
378C22×D3—92no—
379C22×D3—182no—
380C22×D3C22×D911yescharacteristic
381C3×Q8—34no—
382C3×D4—22no—
383C3×D4—22no—
384C3×D4—42no—
385C3×D4—64no—
386C3×D4—362no—
387C2×C3×C4—21no—
388C2×C3×C4—32no—
389C2×C3×C4—31no—
390C2×C3×C4—64no—
391C2×C3×C4—62no—
392C2×C3×C4—92no—
393C2×C3×C4—91no—
394C2×C3×C4—182no—
395C2×C3×C4C22×D911yescharacteristic
396C2×C3×C4C2×D3211yescharacteristic
397#67—24no—
398#67—122no—
399#67—182no—
400#67—182no—
401#67—184no—
402#67—184no—
403#67—184no—
404#67—364no—
405#67—362no—
406#67—362no—
407#67—362no—
408#67—364no—
409C2×Dic3—24no—
410C2×Dic3—32no—
411C2×Dic3—31no—
412C2×Dic3—91no—
413C2×Dic3—91no—
414C2×Dic3—91no—
415C2×Dic3—91no—
416C2×Dic3—181no—
417C2×Dic3—181no—
418C2×Dic3C22×D911yescharacteristic
419C2×Dic3C22×D912yes—
420D12—184no—
421D12—182no—
422D12—182no—
423D12—362no—
424D12—362no—
425C4×D3—22no—
426C4×D3—62no—
427C4×D3—184no—
428C4×D3—184no—
429Dic6#61114yescomplemented
430C2×C32—42no—
431C2×C32D3×#4111yescharacteristic
432C2×C32#100911yescharacteristic
433C2×C32C22×#6711yescharacteristic
434C3⋊D3—122no—
435C3×D3—41no—
436C3×D3—121no—
437C2×C9—21no—
438C2×C9—21no—
439C2×C9—21no—
440C2×C9—42no—
441C2×C9—82no—
442C2×C9—121no—
443C2×C9C2×D3×D411yescharacteristic
444C2×C9#100411yescharacteristic
445C2×C9D3×#4111yescharacteristic
446D9—41no—
447D9—122no—
448D9—242no—
449#41—64no—
450#41—544no—
451C2×Q8—31no—
452C2×D4—32no—
453C2×D4—92no—
454C2×D4—272no—
455C2×D4—272no—
456C2×D4—271no—
457C2×D4—272no—
458C2×D4—272no—
459C2×D4—271no—
460C2×D4D3×D911yescharacteristic
461C22×C4—32no—
462C22×C4—31no—
463C22×C4—91no—
464C22×C4—272no—
465C4⋊C4—91no—
466C4⋊C4—271no—
467C4⋊C4—272no—
468C22⋊C4—92no—
469C22⋊C4—272no—
470C22⋊C4—271no—
471C22⋊C4—271no—
472C42—272no—
473C42—271no—
474C22×C3—22no—
475C22×C3—22no—
476C22×C3—22no—
477C22×C3—22no—
478C22×C3—22no—
479C22×C3—22no—
480C22×C3—21no—
481C22×C3—42no—
482C22×C3—42no—
483C22×C3—42no—
484C22×C3—61no—
485C22×C3—61no—
486C22×C3—61no—
487C22×C3—181no—
488C22×C3—181no—
489C22×C3—181no—
490C22×C3C23×D911yescharacteristic
491C22×C3C22×D3211yescharacteristic
492C2×D3—21no—
493C2×D3—21no—
494C2×D3—21no—
495C2×D3—61no—
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—
507C3×C4—22no—
508C3×C4—34no—
509C3×C4—61no—
510C3×C4—182no—
511C3×C4—181no—
512C3×C4C2×#61112yes—
513C3×C4D3×#6712yes—
514Dic3—21no—
515Dic3—62no—
516Dic3—61no—
517Dic3—92no—
518Dic3—92no—
519Dic3—182no—
520Dic3—182no—
521Dic3—181no—
522Dic3—181no—
523Dic3C2×#61114yes—
524C32#583411yescharacteristic
525C9—21no—
526C9#559911yescharacteristic, complemented
527C23—31no—
528C23—91no—
529C23—272no—
530C23C2×D3×D912yes—
531Q8—34no—
532D4—22no—
533D4—64no—
534D4—362no—
535D4—544no—
536D4—544no—
537D4—542no—
538D4—542no—
539D4—1082no—
540D4—1082no—
541C2×C4—32no—
542C2×C4—31no—
543C2×C4—64no—
544C2×C4—62no—
545C2×C4—92no—
546C2×C4—91no—
547C2×C4—182no—
548C2×C4—271no—
549C2×C4—271no—
550C2×C4—544no—
551C2×C4C2×D3×D911yescharacteristic
552C2×C3—21no—
553C2×C3—21no—
554C2×C3—21no—
555C2×C3—42no—
556C2×C3—42no—
557C2×C3—82no—
558C2×C3—121no—
559C2×C3—361no—
560C2×C3#41×D911yescharacteristic
561C2×C3#6371511yescharacteristic
562C2×C3C22×#61111yescharacteristic
563C2×C3#6431211yescharacteristic
564C2×C3#6431411yescharacteristic
565C2×C3C2×D3×#6711yescharacteristic
566D3—41no—
567D3—121no—
568D3—362no—
569D3—362no—
570D3—722no—
571C22—22no—
572C22—22no—
573C22—22no—
574C22—61no—
575C22—61no—
576C22—61no—
577C22—181no—
578C22—181no—
579C22—181no—
580C22—542no—
581C22—542no—
582C22—542no—
583C22C22×D3×D911yescharacteristic
584C4—34no—
585C4—61no—
586C4—182no—
587C4—181no—
588C4—272no—
589C4—541no—
590C4D3×#61112yes—
591C3—21no—
592C3#1058993611yescharacteristic, complemented
593C3#1059523611yescharacteristic
594C2—42no—
595C2—121no—
596C2—361no—
597C2—1082no—
598C2#1172249811yescharacteristic
599C2#1172250111yescharacteristic
600C2C2×D3×#61111yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC2×C3×D4×C9C22
socleC22×C32C23×D3
hypercenterC2×D4D3×D9
hypocenterC3×C9#550
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C2×C9×#237C2semidirect
2. C2×#88635C2semidirect
3. #11721840C2semidirect
4. C3×#63500C2semidirect
5. #11721289C2semidirect
6. #11721277C2semidirect
7. #11721262C2semidirect
8. #11721240C2semidirect
9. #11721038C2semidirect
10. C4×#6194C2semidirect
11. #11720911C2semidirect
57. #63500D3semidirect
82. C9×Dic6D4semidirect
159. C3⋊Dic9#41semidirect
166. C2×#237D9semidirect
370. C3×C9#550semidirect
429. Dic6#611semidirect
526. C9#5599semidirect
592. C3#10589936semidirect

Maximal subgroups

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

subgroupindex
C2×C9×#2372
C2×#886352
#117218402
C3×#635002
#117212892
#117212772
#117212622
#117212402
#117210382
C4×#61942
#117209112
#105952363
#105899363