Finite Group Databasebeta

← #13150194

Subgroups — #13150194

Order 1728.

1030 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3697C211yesmaximal, characteristic, complemented
2C2×#88642C211yesmaximal, characteristic, complemented
3#11721819C211yesmaximal, characteristic, complemented
4C4×C9×#67C211yesmaximal, characteristic, complemented
5#11721289C212yesmaximal, complemented
6#11721256C212yesmaximal, complemented
7#11721037C211yesmaximal, characteristic, complemented
8#11721033C212yesmaximal, complemented
9#11721006C211yesmaximal, characteristic, complemented
10#11721002C212yesmaximal, complemented
11Dic3×Dic18C211yesmaximal, characteristic, complemented
12#10595007—31nomaximal
13#10589655—31nomaximal
14C22×C3×C4×C9C2211yescharacteristic
15C2×#6233C2212yes—
16C2×C4⋊(C3⋊D9)C2211yescharacteristic
17C2×C9×#67C2211yescharacteristic
18C2×C3×#611C2212yes—
19C2×C4×D3×C9C2211yescharacteristic
20C3×#3697—24no—
21C2×C3×D36C2211yescharacteristic
22C2×C3×C4×D9C2212yes—
23C2×C3×Dic18C2211yescharacteristic
24C2×#6195C2212yes—
25C2×#6194C2212yes—
26C2×#6192C2212yes—
27C2×#6192C2212yes—
28C2×(C3⋊D9)⋊C4C2212yes—
29#88642—24no—
30#88536C2212yes—
31C4⋊(C3⋊Dic9)C2211yescharacteristic
32C9×#217C2211yescharacteristic
33C9×(C2×D3)⋊C4C2211yescharacteristic
34C9×Dic3⋊C4C2211yescharacteristic
35C4×C9×Dic3C2211yescharacteristic
36(C2×D9)⋊Dic3C2212yes—
37Dic9⋊Dic3C2212yes—
38Dic3×Dic9C2212yes—
39C2×C3×#235—31no—
40C2×#3799—31no—
41#64139—31no—
42C3×C4×#67D311yescharacteristic
43#63979—32no—
44#63958—32no—
45#63911—31no—
46#63908—31no—
47#63906—32no—
48#63904—32no—
49Dic3×Dic6—31no—
50C2×#3697—31no—
51C2×#3697D311yescharacteristic, complemented
52C4×D4×C9—31no—
53#63502—32no—
54#63492—31no—
55#63492—31no—
56#63492—61no—
57#63455—32no—
58#63446—32no—
59#63446—32no—
60#63436—31no—
61C4×Dic18—31no—
62C23×C3×C9C2311yescharacteristic
63C22×C3⋊D9C2312yes—
64C22×D3×C9C2311yescharacteristic
65C22×C3×D9C2312yes—
66C2×C3×C4×C9—22no—
67C2×C3×C4×C9C2311yescharacteristic
68C2×C3×C4×C9C2311yescharacteristic
69#6233—42no—
70C2×C3⋊Dic9C2312yes—
71C4⋊(C3⋊D9)—22no—
72C9×#67—22no—
73C3×#611—24no—
74C2×C9×Dic3C2311yescharacteristic
75C2×C9×Dic3C2311yescharacteristic
76C2×C9×Dic3C2311yescharacteristic
77C2×C3×Dic9C2312yes—
78C4×D3×C9—22no—
79C3×D36—22no—
80C3×C4×D9—24no—
81C3×Dic18D414yescomplemented
82#6195—24no—
83#6194—42no—
84#6192—24no—
85#6192—42no—
86(C3⋊D9)⋊C4—24no—
87#5834—91no—
88#5553—91no—
89C22×C32×C4C2×D311yescharacteristic
90C2×#638—32no—
91C2×C4⋊(C3⋊D3)—31no—
92C2×C3×#67—32no—
93C2×C3×#67C2×D311yescharacteristic
94C3×#235—64no—
95C2×C3×D12—31no—
96C2×C3×C4×D3—32no—
97C2×C3×C4×D3C2×D311yescharacteristic
98C2×C3×Dic6—31no—
99C2×#626—32no—
100C2×#626—32no—
101C2×#626—32no—
102C2×#625—32no—
103C2×#624—32no—
104#3799—64no—
105#3752—32no—
106C4⋊(C3⋊Dic3)—31no—
107C3×#217C2×D311yescharacteristic
108C3×(C2×D3)⋊C4C2×D311yescharacteristic
109C3×Dic3⋊C4C2×D311yescharacteristic
110C3×C4×Dic3C2×D311yescharacteristic
111Dic3⋊Dic3—32no—
112(C2×D3)⋊Dic3—32no—
113Dic32—32no—
114C2×D4×C9—31no—
115C22×C4×C9—21no—
116C22×C4×C9—31no—
117C22×C4×C9C2×D311yescharacteristic
118C2×#611—32no—
119C2×#611—32no—
120C2×#611—32no—
121C2×#611—62no—
122C2×#611C2×D312yes—
123#3697—24no—
124#3697—64no—
125C2×D36—32no—
126C2×D36—31no—
127C2×D36—32no—
128C2×D36—61no—
129C2×D36C2×D311yescharacteristic
130C2×C4×D9—32no—
131C2×C4×D9C2×D312yes—
132C2×Dic18C2×D311yescharacteristic
133C9×C4⋊C4—31no—
134C9×C22⋊C4—31no—
135C9×C22⋊C4—31no—
136C42×C9—31no—
137(C2×D9)⋊C4—32no—
138(C2×D9)⋊C4—32no—
139(C2×D9)⋊C4—62no—
140C4⋊Dic9—31no—
141C4⋊Dic9—61no—
142Dic9⋊C4—32no—
143C4×Dic9—32no—
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—21no—
152C2×D3×C9—21no—
153C2×D3×C9—21no—
154C2×C3×D9—22no—
155C2×C3×D9—22no—
156C2×C3×D9—22no—
157C3×C4×C9—21no—
158C3×C4×C9C2×D412yes—
159C3⋊Dic9—22no—
160C9×Dic3—21no—
161C9×Dic3—21no—
162C9×Dic3#4112yescomplemented
163C3×Dic9C2×D414yes—
164C2×C3×#41—91no—
165C2×#239—91no—
166C2×#235—31no—
167C2×#235—91no—
168C3×C4×D4—31no—
169Q8×Dic3—91no—
170#940—92no—
171#940—92no—
172#940—91no—
173#938—91no—
174#938—92no—
175#930—91no—
176#930—91no—
177#930—91no—
178#930—92no—
179#930—181no—
180C4×#67—92no—
181C4×#67D911yescharacteristic, complemented
182#893—92no—
183#884—92no—
184#884—92no—
185#874—91no—
186C4×Dic6—91no—
187C23×C32C22×D311yescharacteristic
188C22×C3⋊D3—32no—
189C22×C3×D3—32no—
190C22×C3×D3C22×D311yescharacteristic
191C2×C32×C4—22no—
192C2×C32×C4C22×D311yescharacteristic
193C2×C32×C4C22×D311yescharacteristic
194#638—122no—
195C2×C3⋊Dic3—32no—
196C4⋊(C3⋊D3)—62no—
197C3×#67—22no—
198C3×#67—64no—
199C2×C3×Dic3—32no—
200C2×C3×Dic3C22×D311yescharacteristic
201C2×C3×Dic3C22×D311yescharacteristic
202C2×C3×Dic3C22×D311yescharacteristic
203C3×D12—62no—
204C3×C4×D3—22no—
205C3×C4×D3—64no—
206C3×Dic6—34no—
207#626—64no—
208#626—64no—
209#626—122no—
210#625—122no—
211#624—64no—
212C23×C9—21no—
213C23×C9—31no—
214C23×C9C22×D311yescharacteristic
215C22×D9—32no—
216C22×D9—62no—
217C22×D9C22×D312yes—
218D4×C9—62no—
219C2×C4×C9—22no—
220C2×C4×C9—21no—
221C2×C4×C9—21no—
222C2×C4×C9—31no—
223C2×C4×C9—31no—
224C2×C4×C9—31no—
225C2×C4×C9—42no—
226C2×C4×C9—62no—
227C2×C4×C9C22×D311yescharacteristic
228C2×C4×C9C22×D311yescharacteristic
229#611—24no—
230#611—64no—
231#611—122no—
232#611—122no—
233#611—124no—
234C2×Dic9—32no—
235C2×Dic9—62no—
236C2×Dic9C22×D312yes—
237D36—22no—
238D36—64no—
239D36—62no—
240D36—122no—
241D36—122no—
242C4×D9—24no—
243C4×D9—64no—
244Dic18#6714yescomplemented
245#550—271noSylow
246C2×C3×C9—41no—
247C2×C3×C9C22×D411yescharacteristic
248C2×C3×C9C2×#4111yescharacteristic
249C2×C3×C9H(5,𝔽2)11yescharacteristic
250C3⋊D9—42no—
251D3×C9—41no—
252C3×D9—42no—
253C3×#41—184no—
254C2×C3×Q8—91no—
255C2×C3×D4—31no—
256C2×C3×D4—92no—
257C2×C3×D4—91no—
258C22×C3×C4—21no—
259C22×C3×C4—31no—
260C22×C3×C4—92no—
261C22×C3×C4C2×D911yescharacteristic
262C22×C3×C4D3211yescharacteristic
263C2×#67—32no—
264C2×#67—92no—
265C2×#67—92no—
266C2×#67—92no—
267C2×#67—92no—
268C2×#67—92no—
269C2×#67—92no—
270C2×#67—92no—
271C2×#67—182no—
272C2×#67C2×D911yescharacteristic
273#239—184no—
274#235—64no—
275#235—184no—
276C2×D12—31no—
277C2×D12—92no—
278C2×D12—92no—
279C2×D12—91no—
280C2×D12—91no—
281C2×D12—92no—
282C2×D12—181no—
283C2×C4×D3—32no—
284C2×C4×D3—92no—
285C2×C4×D3—92no—
286C2×C4×D3C2×D911yescharacteristic
287C2×Dic6—31no—
288C3×C4⋊C4—31no—
289C3×C22⋊C4—31no—
290C3×C22⋊C4—31no—
291C3×C42—31no—
292#217—92no—
293#217C2×D911yescharacteristic
294(C2×D3)⋊C4—92no—
295(C2×D3)⋊C4—92no—
296(C2×D3)⋊C4—92no—
297(C2×D3)⋊C4—182no—
298(C2×D3)⋊C4C2×D911yescharacteristic
299C4⋊Dic3—91no—
300C4⋊Dic3—91no—
301C4⋊Dic3—92no—
302C4⋊Dic3—181no—
303Dic3⋊C4—92no—
304Dic3⋊C4C2×D911yescharacteristic
305C4×Dic3—92no—
306C4×Dic3—92no—
307C4×Dic3C2×D911yescharacteristic
308C22×C32—21no—
309C22×C32—21no—
310C22×C32—21no—
311C22×C32C23×D311yescharacteristic
312C2×C3⋊D3—62no—
313C2×C3⋊D3—62no—
314C2×C3⋊D3—62no—
315C2×C3×D3—21no—
316C2×C3×D3—21no—
317C2×C3×D3—21no—
318C2×C3×D3—62no—
319C2×C3×D3—62no—
320C2×C3×D3—62no—
321C32×C4—21no—
322C32×C4D3×D412yes—
323C3⋊Dic3—62no—
324C3×Dic3—21no—
325C3×Dic3—21no—
326C3×Dic3—34no—
327C3×Dic3#23512yes—
328C22×C9—22no—
329C22×C9—21no—
330C22×C9—21no—
331C22×C9—21no—
332C22×C9—21no—
333C22×C9—41no—
334C22×C9—41no—
335C22×C9—61no—
336C22×C9—61no—
337C22×C9—61no—
338C22×C9C23×D311yescharacteristic
339C2×D9—22no—
340C2×D9—22no—
341C2×D9—22no—
342C2×D9—62no—
343C2×D9—62no—
344C2×D9—62no—
345C2×D9—122no—
346C2×D9—122no—
347C2×D9—122no—
348C4×C9—21no—
349C4×C9—22no—
350C4×C9—32no—
351C4×C9—41no—
352C4×C9—61no—
353C4×C9—61no—
354C4×C9C2×#6712yes—
355Dic9—62no—
356Dic9—122no—
357Dic9C2×#6714yes—
358C2×#41—91no—
359C2×#41—271no—
360#127—271no—
361#127—272no—
362#121—272no—
363#121—272no—
364#121—271no—
365#121—271no—
366C4×Q8—271no—
367C4×D4—31no—
368C4×D4—272no—
369C3×C9#55011yescharacteristic, complemented, Sylow
370C23×C3—21no—
371C23×C3—31no—
372C23×C3—92no—
373C23×C3C22×D911yescharacteristic
374C23×C3C2×D3211yescharacteristic
375C22×D3—32no—
376C22×D3—92no—
377C22×D3—92no—
378C22×D3—182no—
379C22×D3C22×D911yescharacteristic
380C3×Q8—94no—
381C3×D4—62no—
382C3×D4—184no—
383C3×D4—182no—
384C2×C3×C4—22no—
385C2×C3×C4—22no—
386C2×C3×C4—21no—
387C2×C3×C4—21no—
388C2×C3×C4—31no—
389C2×C3×C4—31no—
390C2×C3×C4—31no—
391C2×C3×C4—42no—
392C2×C3×C4—62no—
393C2×C3×C4—92no—
394C2×C3×C4—184no—
395C2×C3×C4C22×D911yescharacteristic
396C2×C3×C4C22×D911yescharacteristic
397C2×C3×C4C2×D3211yescharacteristic
398C2×C3×C4C2×D3211yescharacteristic
399#67—22no—
400#67—64no—
401#67—184no—
402#67—184no—
403#67—362no—
404#67—362no—
405#67—362no—
406#67—362no—
407#67—362no—
408#67—364no—
409C2×Dic3—32no—
410C2×Dic3—92no—
411C2×Dic3—92no—
412C2×Dic3—182no—
413C2×Dic3C22×D911yescharacteristic
414C2×Dic3C22×D911yescharacteristic
415C2×Dic3C22×D911yescharacteristic
416D12—62no—
417D12—184no—
418D12—184no—
419D12—182no—
420D12—182no—
421D12—362no—
422D12—362no—
423C4×D3—22no—
424C4×D3—64no—
425C4×D3—184no—
426C4×D3—184no—
427Dic6—34no—
428C2×C32—41no—
429C2×C32C2×#23511yescharacteristic
430C2×C32C2×D3×D411yescharacteristic
431C2×C32#100911yescharacteristic
432C3⋊D3—122no—
433C3×D3—41no—
434C3×D3—122no—
435C2×C9—21no—
436C2×C9—21no—
437C2×C9—21no—
438C2×C9—42no—
439C2×C9—41no—
440C2×C9—121no—
441C2×C9D3×#4111yescharacteristic
442C2×C9#100911yescharacteristic
443C2×C9C22×#6711yescharacteristic
444D9—42no—
445D9—122no—
446D9—242no—
447#41—184no—
448#41—544no—
449C2×Q8—91no—
450C2×D4—31no—
451C2×D4—92no—
452C2×D4—91no—
453C2×D4—272no—
454C2×D4—272no—
455C2×D4—272no—
456C2×D4—272no—
457C2×D4—272no—
458C2×D4—271no—
459C22×C4—31no—
460C22×C4—92no—
461C22×C4—272no—
462C22×C4D3×D911yescharacteristic
463C4⋊C4—31no—
464C4⋊C4—272no—
465C4⋊C4—271no—
466C22⋊C4—31no—
467C22⋊C4—31no—
468C22⋊C4—272no—
469C22⋊C4—272no—
470C42—31no—
471C42—272no—
472C22×C3—22no—
473C22×C3—21no—
474C22×C3—21no—
475C22×C3—21no—
476C22×C3—21no—
477C22×C3—21no—
478C22×C3—21no—
479C22×C3—21no—
480C22×C3—41no—
481C22×C3—41no—
482C22×C3—61no—
483C22×C3—61no—
484C22×C3—61no—
485C22×C3—182no—
486C22×C3—182no—
487C22×C3—182no—
488C22×C3C23×D911yescharacteristic
489C22×C3C22×D3211yescharacteristic
490C2×D3—21no—
491C2×D3—21no—
492C2×D3—21no—
493C2×D3—62no—
494C2×D3—62no—
495C2×D3—62no—
496C2×D3—182no—
497C2×D3—182no—
498C2×D3—182no—
499C2×D3—182no—
500C2×D3—182no—
501C2×D3—182no—
502C2×D3—362no—
503C2×D3—362no—
504C2×D3—362no—
505C3×C4—21no—
506C3×C4—21no—
507C3×C4—22no—
508C3×C4—32no—
509C3×C4—41no—
510C3×C4—61no—
511C3×C4—61no—
512C3×C4—94no—
513C3×C4D4×D912yes—
514C3×C4D3×#6712yes—
515Dic3—21no—
516Dic3—21no—
517Dic3—34no—
518Dic3—182no—
519Dic3—182no—
520Dic3—362no—
521Dic3#369712yescomplemented
522C32#555311yescharacteristic
523C9—21no—
524C9#583411yescharacteristic, complemented
525C23—31no—
526C23—92no—
527C23—272no—
528C23C2×D3×D911yescharacteristic
529Q8—94no—
530D4—62no—
531D4—184no—
532D4—182no—
533D4—544no—
534D4—544no—
535D4—542no—
536D4—1082no—
537D4—1082no—
538D4—1082no—
539C2×C4—22no—
540C2×C4—31no—
541C2×C4—31no—
542C2×C4—31no—
543C2×C4—62no—
544C2×C4—92no—
545C2×C4—184no—
546C2×C4—272no—
547C2×C4—544no—
548C2×C4C2×D3×D911yescharacteristic
549C2×C4C2×D3×D911yescharacteristic
550C2×C3—21no—
551C2×C3—21no—
552C2×C3—21no—
553C2×C3—42no—
554C2×C3—41no—
555C2×C3—41no—
556C2×C3—121no—
557C2×C3—362no—
558C2×C3C2×#369711yescharacteristic
559C2×C3C2×D4×D911yescharacteristic
560C2×C3#6371511yescharacteristic
561C2×C3D3×#23511yescharacteristic
562C2×C3#6430811yescharacteristic
563C2×C3C2×D3×#6711yescharacteristic
564D3—41no—
565D3—122no—
566D3—362no—
567D3—362no—
568D3—722no—
569C22—21no—
570C22—21no—
571C22—21no—
572C22—61no—
573C22—61no—
574C22—61no—
575C22—182no—
576C22—182no—
577C22—182no—
578C22—542no—
579C22—542no—
580C22—542no—
581C22C22×D3×D911yescharacteristic
582C4—21no—
583C4—32no—
584C4—61no—
585C4—61no—
586C4—94no—
587C4—542no—
588C4D9×#6712yes—
589C3—21no—
590C3#1058965511yescharacteristic, complemented
591C3#1059500711yescharacteristic
592C2—41no—
593C2—121no—
594C2—362no—
595C2—1082no—
596C2D3×#369711yescharacteristic
597C2#1172247811yescharacteristic
598C2C2×D9×#6711yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC22×C3×C4×C9C22
socleC22×C32C23×D3
hypercenterC22×C4D3×D9
hypocenterC3×C9#550
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×C3×#3697C2semidirect
2. C2×#88642C2semidirect
3. #11721819C2semidirect
4. C4×C9×#67C2semidirect
5. #11721289C2semidirect
6. #11721256C2semidirect
7. #11721037C2semidirect
8. #11721033C2semidirect
9. #11721006C2semidirect
10. #11721002C2semidirect
11. Dic3×Dic18C2semidirect
51. C2×#3697D3semidirect
81. C3×Dic18D4semidirect
162. C9×Dic3#41semidirect
181. C4×#67D9semidirect
244. Dic18#67semidirect
369. C3×C9#550semidirect
521. Dic3#3697semidirect
524. C9#5834semidirect
590. C3#10589655semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#36972
C2×#886422
#117218192
C4×C9×#672
#117212892
#117212562
#117210372
#117210332
#117210062
#117210022
Dic3×Dic182
#105950073
#105896553