Finite Group Databasebeta

← #13150192

Subgroups — #13150192

Order 1728.

964 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×Dic18C211yesmaximal, characteristic, complemented
2C2×#88642C211yesmaximal, characteristic, complemented
3#11721819C211yesmaximal, characteristic, complemented
4C4×C9×#67C211yesmaximal, characteristic, complemented
5#11721288C212yesmaximal, complemented
6#11721241C212yesmaximal, complemented
7#11721035C212yesmaximal, complemented
8#11720985C212yesmaximal, complemented
9#11720960C211yesmaximal, characteristic, complemented
10#11720952C211yesmaximal, characteristic, complemented
11Dic3×Dic18C211yesmaximal, characteristic, complemented
12#10594902—31nomaximal
13#10589654—31nomaximal
14C22×C3×C4×C9C2211yescharacteristic
15C2×#6233C2212yes—
16C2×C4⋊(C3⋊D9)C2211yescharacteristic
17C2×C9×#67C2211yescharacteristic
18C22×C3×Dic9C2212yes—
19C2×C4×D3×C9C2211yescharacteristic
20C2×C3×Dic18—24no—
21C2×C3×Dic18C2211yescharacteristic
22C2×C3×Dic18C2212yes—
23C2×C3×Dic18C2211yescharacteristic
24C2×#6195C2212yes—
25C2×(C3⋊D9)⋊C4C2212yes—
26#88642—24no—
27#88536C2212yes—
28C4⋊(C3⋊Dic9)C2211yescharacteristic
29C9×#217C2211yescharacteristic
30C9×(C2×D3)⋊C4C2211yescharacteristic
31C9×Dic3⋊C4C2211yescharacteristic
32C4×C9×Dic3C2211yescharacteristic
33(C2×D3)⋊Dic9C2212yes—
34#88445C2212yes—
35#88445C2212yes—
36#88442C2212yes—
37Dic9⋊Dic3C2212yes—
38Dic3×Dic9C2212yes—
39C22×C3×Dic6—31no—
40C2×#3799—31no—
41#64139—31no—
42C3×C4×#67D311yescharacteristic
43#63978—32no—
44#63952—32no—
45#63907—32no—
46#63888—32no—
47#63864—31no—
48#63856—31no—
49Dic3×Dic6—31no—
50C2×#3697—31no—
51C22×Dic18D311yescharacteristic, complemented
52C4×D4×C9—31no—
53#63501—32no—
54#63492—31no—
55#63492—61no—
56#63486—31no—
57#63458—32no—
58#63447—32no—
59#611⋊C4—32no—
60#63437—31no—
61C4×Dic18—31no—
62C23×C3×C9C2311yescharacteristic
63C22×C3⋊D9C2312yes—
64C22×D3×C9C2311yescharacteristic
65C2×C3×C4×C9—22no—
66C2×C3×C4×C9C2311yescharacteristic
67C2×C3×C4×C9C2311yescharacteristic
68#6233—24no—
69C2×C3⋊Dic9C2312yes—
70C4⋊(C3⋊D9)—22no—
71C9×#67—22no—
72C2×C9×Dic3C2311yescharacteristic
73C2×C9×Dic3C2311yescharacteristic
74C2×C9×Dic3C2311yescharacteristic
75C2×C3×Dic9—24no—
76C2×C3×Dic9C2312yes—
77C2×C3×Dic9C2312yes—
78C4×D3×C9—22no—
79C3×Dic18—22no—
80C3×Dic18—24no—
81C3×Dic18D414yescomplemented
82#6195—24no—
83(C3⋊D9)⋊C4—24no—
84#5817—91no—
85#5552—91no—
86C22×C32×C4C2×D311yescharacteristic
87C2×#638—32no—
88C2×C4⋊(C3⋊D3)—31no—
89C2×C3×#67C2×D311yescharacteristic
90C22×C3×Dic3—32no—
91C2×C3×C4×D3C2×D311yescharacteristic
92C2×C3×Dic6—31no—
93C2×C3×Dic6—31no—
94C2×C3×Dic6—32no—
95C2×C3×Dic6—64no—
96C2×#626—32no—
97C2×#624—32no—
98#3799—64no—
99#3752—32no—
100C4⋊(C3⋊Dic3)—31no—
101C3×#217C2×D311yescharacteristic
102C3×(C2×D3)⋊C4C2×D311yescharacteristic
103C3×Dic3⋊C4C2×D311yescharacteristic
104C3×C4×Dic3C2×D311yescharacteristic
105#3724—32no—
106Dic3⋊Dic3—32no—
107#3722—32no—
108#3722—32no—
109(C2×D3)⋊Dic3—32no—
110Dic32—32no—
111C2×D4×C9—31no—
112C22×C4×C9—21no—
113C22×C4×C9—31no—
114C22×C4×C9C2×D311yescharacteristic
115C2×#611—32no—
116C2×#611—32no—
117C2×#611—62no—
118C22×Dic9C2×D312yes—
119#3697—64no—
120C2×D36—31no—
121C2×D36—61no—
122C2×C4×D9—32no—
123C2×Dic18—24no—
124C2×Dic18C2×D311yescharacteristic
125C2×Dic18C2×D312yes—
126C2×Dic18C2×D311yescharacteristic
127C9×C4⋊C4—31no—
128C9×C22⋊C4—31no—
129C9×C22⋊C4—31no—
130C42×C9—31no—
131#3676—32no—
132(C2×D9)⋊C4—32no—
133(C2×D9)⋊C4—32no—
134(C2×D9)⋊C4—32no—
135(C2×D9)⋊C4—62no—
136C4⋊Dic9—31no—
137C4⋊Dic9—61no—
138Dic9⋊C4—32no—
139Dic9⋊C4—32no—
140C4×Dic9—32no—
141C22×C3×C9—21no—
142C22×C3×C9—21no—
143C22×C3×C9#4112yescomplemented
144C22×C3×C9C2411yescharacteristic
145C2×C3⋊D9—22no—
146C2×C3⋊D9—22no—
147C2×C3⋊D9—22no—
148C2×D3×C9—21no—
149C2×D3×C9—21no—
150C2×D3×C9—21no—
151C3×C4×C9—21no—
152C3×C4×C9C2×D412yes—
153C3⋊Dic9—22no—
154C9×Dic3—21no—
155C9×Dic3—21no—
156C9×Dic3—21no—
157C3×Dic9—22no—
158C3×Dic9C2×D414yes—
159C22×C3×Q8—91no—
160C2×#239—91no—
161C2×#235—91no—
162C22×Dic6—31no—
163C3×C4×D4—31no—
164#947—92no—
165#947—91no—
166#946—91no—
167#946—92no—
168Q8×Dic3—91no—
169#939—92no—
170#930—91no—
171#930—91no—
172#930—92no—
173#930—181no—
174C4×#67—92no—
175C4×#67D911yescharacteristic, complemented
176#924—91no—
177#896—92no—
178#885—92no—
179#67⋊C4—92no—
180#875—91no—
181C4×Dic6—91no—
182C23×C32C22×D311yescharacteristic
183C22×C3⋊D3—32no—
184C22×C3×D3C22×D311yescharacteristic
185C2×C32×C4—22no—
186C2×C32×C4C22×D311yescharacteristic
187C2×C32×C4C22×D311yescharacteristic
188#638—64no—
189C2×C3⋊Dic3—32no—
190C4⋊(C3⋊D3)—62no—
191C3×#67—22no—
192C2×C3×Dic3—32no—
193C2×C3×Dic3—32no—
194C2×C3×Dic3—64no—
195C2×C3×Dic3C22×D311yescharacteristic
196C2×C3×Dic3C22×D311yescharacteristic
197C2×C3×Dic3C22×D311yescharacteristic
198C3×C4×D3—22no—
199C3×Dic6—34no—
200C3×Dic6—64no—
201C3×Dic6—62no—
202#626—64no—
203#624—64no—
204C23×C9—21no—
205C23×C9—31no—
206C23×C9C22×D311yescharacteristic
207C22×D9—32no—
208C22×D9—62no—
209D4×C9—62no—
210C2×C4×C9—22no—
211C2×C4×C9—21no—
212C2×C4×C9—21no—
213C2×C4×C9—31no—
214C2×C4×C9—31no—
215C2×C4×C9—31no—
216C2×C4×C9—42no—
217C2×C4×C9—62no—
218C2×C4×C9C22×D311yescharacteristic
219C2×C4×C9C22×D311yescharacteristic
220#611—64no—
221#611—64no—
222#611—124no—
223C2×Dic9—24no—
224C2×Dic9—32no—
225C2×Dic9—62no—
226C2×Dic9C22×D312yes—
227C2×Dic9C22×D312yes—
228D36—62no—
229D36—122no—
230C4×D9—64no—
231Dic18—22no—
232Dic18—24no—
233Dic18#6714yescomplemented
234#548—271noSylow
235C2×C3×C9—22no—
236C2×C3×C9C22×D411yescharacteristic
237C2×C3×C9C2×#4111yescharacteristic
238C2×C3×C92−1+411yescharacteristic
239C3⋊D9—42no—
240D3×C9—41no—
241C2×C3×Q8—91no—
242C2×C3×Q8—92no—
243C2×C3×Q8—91no—
244C2×C3×Q8—184no—
245C2×C3×D4—31no—
246C22×C3×C4—21no—
247C22×C3×C4—31no—
248C22×C3×C4—92no—
249C22×C3×C4C2×D911yescharacteristic
250C22×C3×C4D3211yescharacteristic
251C2×#67—92no—
252C2×#67—92no—
253C2×#67—92no—
254C2×#67—182no—
255C2×#67C2×D911yescharacteristic
256C22×Dic3—32no—
257#239—184no—
258#235—184no—
259C2×D12—92no—
260C2×D12—91no—
261C2×D12—91no—
262C2×D12—181no—
263C2×C4×D3—92no—
264C2×C4×D3—92no—
265C2×C4×D3C2×D911yescharacteristic
266C2×Dic6—31no—
267C2×Dic6—31no—
268C2×Dic6—32no—
269C2×Dic6—64no—
270C3×C4⋊C4—31no—
271C3×C22⋊C4—31no—
272C3×C22⋊C4—31no—
273C3×C42—31no—
274#217—92no—
275#217C2×D911yescharacteristic
276(C2×D3)⋊C4—92no—
277(C2×D3)⋊C4—92no—
278(C2×D3)⋊C4—92no—
279(C2×D3)⋊C4—92no—
280(C2×D3)⋊C4—92no—
281(C2×D3)⋊C4—92no—
282(C2×D3)⋊C4—92no—
283(C2×D3)⋊C4—182no—
284(C2×D3)⋊C4C2×D911yescharacteristic
285C4⋊Dic3—91no—
286C4⋊Dic3—91no—
287C4⋊Dic3—92no—
288C4⋊Dic3—181no—
289Dic3⋊C4—92no—
290Dic3⋊C4—92no—
291Dic3⋊C4—92no—
292Dic3⋊C4C2×D911yescharacteristic
293C4×Dic3—92no—
294C4×Dic3—92no—
295C4×Dic3C2×D911yescharacteristic
296C22×C32—21no—
297C22×C32—21no—
298C22×C32#23512yes—
299C22×C32C23×D311yescharacteristic
300C2×C3⋊D3—62no—
301C2×C3⋊D3—62no—
302C2×C3⋊D3—62no—
303C2×C3×D3—21no—
304C2×C3×D3—21no—
305C2×C3×D3—21no—
306C32×C4—21no—
307C32×C4D3×D412yes—
308C3⋊Dic3—62no—
309C3×Dic3—21no—
310C3×Dic3—21no—
311C3×Dic3—21no—
312C3×Dic3—34no—
313C3×Dic3—62no—
314C22×C9—21no—
315C22×C9—21no—
316C22×C9—21no—
317C22×C9—22no—
318C22×C9—41no—
319C22×C9—41no—
320C22×C9—61no—
321C22×C9—61no—
322C22×C9—61no—
323C22×C9#23912yescomplemented
324C22×C9C23×D311yescharacteristic
325C2×D9—62no—
326C2×D9—62no—
327C2×D9—62no—
328C2×D9—122no—
329C2×D9—122no—
330C2×D9—122no—
331C4×C9—21no—
332C4×C9—22no—
333C4×C9—41no—
334C4×C9—61no—
335C4×C9—61no—
336C4×C9—61no—
337C4×C9C2×#6712yes—
338Dic9—22no—
339Dic9—62no—
340Dic9—122no—
341Dic9C2×#6714yes—
342C2×#41—271no—
343C22×Q8—91no—
344#124—271no—
345#124—272no—
346C22⋊Q8—272no—
347C22⋊Q8—271no—
348#121—272no—
349#121—271no—
350C4×Q8—271no—
351C4×D4—31no—
352C4×D4—272no—
353C3×C9#54811yescharacteristic, complemented, Sylow
354C23×C3—21no—
355C23×C3—31no—
356C23×C3C22×D911yescharacteristic
357C23×C3C2×D3211yescharacteristic
358C22×D3—92no—
359C22×D3—92no—
360C22×D3—182no—
361C22×D3C22×D911yescharacteristic
362C3×Q8—94no—
363C3×Q8—184no—
364C3×Q8—182no—
365C3×D4—62no—
366C2×C3×C4—22no—
367C2×C3×C4—22no—
368C2×C3×C4—21no—
369C2×C3×C4—21no—
370C2×C3×C4—31no—
371C2×C3×C4—31no—
372C2×C3×C4—31no—
373C2×C3×C4—42no—
374C2×C3×C4—62no—
375C2×C3×C4—92no—
376C2×C3×C4—92no—
377C2×C3×C4—184no—
378C2×C3×C4C22×D911yescharacteristic
379C2×C3×C4C22×D911yescharacteristic
380C2×C3×C4C2×D3211yescharacteristic
381C2×C3×C4C2×D3211yescharacteristic
382#67—22no—
383#67—184no—
384#67—184no—
385#67—184no—
386#67—364no—
387C2×Dic3—32no—
388C2×Dic3—32no—
389C2×Dic3—64no—
390C2×Dic3—92no—
391C2×Dic3—92no—
392C2×Dic3—182no—
393C2×Dic3C22×D911yescharacteristic
394C2×Dic3C22×D911yescharacteristic
395C2×Dic3C22×D911yescharacteristic
396D12—184no—
397D12—182no—
398D12—182no—
399D12—362no—
400C4×D3—22no—
401C4×D3—184no—
402C4×D3—184no—
403Dic6—34no—
404Dic6—64no—
405Dic6—62no—
406C2×C32—22no—
407C2×C32C2×#23511yescharacteristic
408C2×C32C2×D3×D411yescharacteristic
409C2×C32#101011yescharacteristic
410C3⋊D3—122no—
411C3×D3—41no—
412C2×C9—22no—
413C2×C9—21no—
414C2×C9—21no—
415C2×C9—21no—
416C2×C9—42no—
417C2×C9—121no—
418C2×C9C2×#23911yescharacteristic
419C2×C9#100711yescharacteristic
420C2×C9C22×#6711yescharacteristic
421D9—122no—
422D9—242no—
423#41—544no—
424C2×Q8—91no—
425C2×Q8—92no—
426C2×Q8—91no—
427C2×Q8—184no—
428C2×D4—31no—
429C2×D4—272no—
430C2×D4—272no—
431C2×D4—271no—
432C22×C4—31no—
433C22×C4—92no—
434C22×C4—272no—
435C22×C4D3×D911yescharacteristic
436C4⋊C4—31no—
437C4⋊C4—272no—
438C4⋊C4—272no—
439C4⋊C4—271no—
440C22⋊C4—31no—
441C22⋊C4—31no—
442C22⋊C4—272no—
443C22⋊C4—272no—
444C22⋊C4—272no—
445C22⋊C4—272no—
446C42—31no—
447C42—272no—
448C22×C3—21no—
449C22×C3—21no—
450C22×C3—21no—
451C22×C3—21no—
452C22×C3—21no—
453C22×C3—22no—
454C22×C3—41no—
455C22×C3—41no—
456C22×C3—61no—
457C22×C3—61no—
458C22×C3—61no—
459C22×C3#369712yescomplemented
460C22×C3C23×D911yescharacteristic
461C22×C3#379912yes—
462C22×C3C22×D3211yescharacteristic
463C2×D3—21no—
464C2×D3—21no—
465C2×D3—21no—
466C2×D3—182no—
467C2×D3—182no—
468C2×D3—182no—
469C2×D3—182no—
470C2×D3—182no—
471C2×D3—182no—
472C2×D3—362no—
473C2×D3—362no—
474C2×D3—362no—
475C3×C4—21no—
476C3×C4—21no—
477C3×C4—22no—
478C3×C4—41no—
479C3×C4—61no—
480C3×C4—61no—
481C3×C4—61no—
482C3×C4—94no—
483C3×C4—182no—
484C3×C4D4×D912yes—
485C3×C4D3×#6712yes—
486Dic3—21no—
487Dic3—21no—
488Dic3—21no—
489Dic3—34no—
490Dic3—62no—
491Dic3—182no—
492Dic3—182no—
493Dic3—362no—
494C32#555211yescharacteristic
495C9—21no—
496C9#581711yescharacteristic, complemented
497C23—31no—
498C23—272no—
499C23C2×D3×D911yescharacteristic
500Q8—94no—
501Q8—184no—
502Q8—182no—
503D4—62no—
504D4—544no—
505D4—544no—
506D4—542no—
507C2×C4—22no—
508C2×C4—31no—
509C2×C4—31no—
510C2×C4—31no—
511C2×C4—62no—
512C2×C4—92no—
513C2×C4—92no—
514C2×C4—184no—
515C2×C4—272no—
516C2×C4—544no—
517C2×C4C2×D3×D911yescharacteristic
518C2×C4C2×D3×D911yescharacteristic
519C2×C3—22no—
520C2×C3—22no—
521C2×C3—21no—
522C2×C3—21no—
523C2×C3—21no—
524C2×C3—42no—
525C2×C3—121no—
526C2×C3C2×#369711yescharacteristic
527C2×C3C2×D4×D911yescharacteristic
528C2×C3#6371611yescharacteristic
529C2×C3#6429711yescharacteristic
530C2×C3C2×#379911yescharacteristic
531C2×C3C2×D3×#6711yescharacteristic
532D3—41no—
533D3—362no—
534D3—362no—
535D3—722no—
536C22—21no—
537C22—21no—
538C22—61no—
539C22—61no—
540C22—61no—
541C22—542no—
542C22—542no—
543C22—542no—
544C22#8864212yescomplemented
545C22C22×D3×D911yescharacteristic
546C4—21no—
547C4—61no—
548C4—61no—
549C4—61no—
550C4—94no—
551C4—182no—
552C4—542no—
553C4D9×#6712yes—
554C3—21no—
555C3#1058965411yescharacteristic, complemented
556C3#1059490211yescharacteristic
557C2—22no—
558C2—121no—
559C2—1082no—
560C2#1172246411yescharacteristic
561C2C2×#8864211yescharacteristic
562C2C2×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#548
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. C22×C3×Dic18C2semidirect
2. C2×#88642C2semidirect
3. #11721819C2semidirect
4. C4×C9×#67C2semidirect
5. #11721288C2semidirect
6. #11721241C2semidirect
7. #11721035C2semidirect
8. #11720985C2semidirect
9. #11720960C2semidirect
10. #11720952C2semidirect
11. Dic3×Dic18C2semidirect
51. C22×Dic18D3semidirect
81. C3×Dic18D4semidirect
143. C22×C3×C9#41semidirect
175. C4×#67D9semidirect
233. Dic18#67semidirect
323. C22×C9#239semidirect
353. C3×C9#548semidirect
459. C22×C3#3697semidirect
496. C9#5817semidirect
544. C22#88642semidirect
555. C3#10589654semidirect

Maximal subgroups

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

subgroupindex
C22×C3×Dic182
C2×#886422
#117218192
C4×C9×#672
#117212882
#117212412
#117210352
#117209852
#117209602
#117209522
Dic3×Dic182
#105949023
#105896543