Finite Group Databasebeta

← #13148833

Subgroups — #13148833

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
1C2×(C3⋊D9)⋊Q8C211yesmaximal, characteristic, complemented
2C2×#88635C211yesmaximal, characteristic, complemented
3#11721751C211yesmaximal, characteristic, complemented
4C3×#63459C211yesmaximal, characteristic, complemented
5C9×#887C211yesmaximal, characteristic, complemented
6#11721005C212yesmaximal, complemented
7C4×#6192C211yesmaximal, characteristic, complemented
8#11720995C212yesmaximal, complemented
9#11720993C212yesmaximal, complemented
10#11720952C211yesmaximal, characteristic, complemented
11#11720940C212yesmaximal, complemented
12#10594329—31nomaximal
13#10589678—31nomaximal
14C2×C4⋊(C3⋊D9)C2212yes—
15C2×C4⋊(C3⋊D9)C2211yescharacteristic
16C2×C4×C3⋊D9C2211yescharacteristic
17C2×C3×C4×D9C2211yescharacteristic
18C2×C9×Dic6C2211yescharacteristic
19C2×C3×Dic18C2211yescharacteristic
20C2×#6192C2211yescharacteristic
21C2×#6192C2212yes—
22C2×(C3⋊D9)⋊C4C2212yes—
23C2×(C3⋊D9)⋊C4C2212yes—
24C2×#6189C2212yes—
25(C3⋊D9)⋊Q8—24no—
26#88635—24no—
27C3×C9×C4⋊C4C2211yescharacteristic
28#88536C2212yes—
29C3×(C2×D9)⋊C4C2212yes—
30C9×Dic3⋊C4C2212yes—
31C4×C9×Dic3C2212yes—
32C4×C9×Dic3C2211yescharacteristic
33C3×C4⋊Dic9C2212yes—
34#88445C2211yescharacteristic
35#88445C2212yes—
36#88445C2212yes—
37(C2×D9)⋊Dic3C2211yescharacteristic
38Dic3⋊Dic9C2211yescharacteristic
39C2×#3799—31no—
40C2×#3797—31no—
41#64104—31no—
42C3×#897—31no—
43C3×#887D311yescharacteristic
44#63910—32no—
45C4×#626—31no—
46#63895—32no—
47#63894—32no—
48#63864—31no—
49#63852—32no—
50C2×#3701—31no—
51C2×Q8×D9—31no—
52C4×Q8×C9—31no—
53#63459—32no—
54#63459D311yescharacteristic, complemented
55#63457—31no—
56#63457—32no—
57#63457—61no—
58#63437—31no—
59#63437—32no—
60C4×D36—31no—
61C4×D36—32no—
62C22×C3⋊D9C2312yes—
63C22×C3⋊D9C2311yescharacteristic
64C22×C3×D9C2311yescharacteristic
65C2×C3×C4×C9C2312yes—
66C2×C3×C4×C9C2311yescharacteristic
67C2×C3⋊Dic9C2311yescharacteristic
68C4⋊(C3⋊D9)—24no—
69C4⋊(C3⋊D9)—22no—
70C4×C3⋊D9—22no—
71C2×C9×Dic3C2311yescharacteristic
72C2×C9×Dic3C2311yescharacteristic
73C2×C9×Dic3C2312yes—
74C2×C3×Dic9C2312yes—
75C2×C3×Dic9C2311yescharacteristic
76C3×C4×D9—22no—
77C9×Dic6D414yescomplemented
78C3×Dic18—22no—
79#6192—22no—
80#6192—24no—
81(C3⋊D9)⋊C4—24no—
82(C3⋊D9)⋊C4—24no—
83#6189—24no—
84#5633—91no—
85#5576—91no—
86C2×C4⋊(C3⋊D3)—31no—
87C2×C4⋊(C3⋊D3)—32no—
88C2×C4×C3⋊D3—31no—
89C2×C3×C4×D3—31no—
90C2×C3×Dic6—31no—
91C2×C3×Dic6C2×D311yescharacteristic
92C2×#627—32no—
93C2×#626—32no—
94C2×#626—31no—
95C2×#624—32no—
96C2×#624—32no—
97#3799—64no—
98#3797—64no—
99C32×C4⋊C4C2×D311yescharacteristic
100#3752—32no—
101C3×(C2×D3)⋊C4—32no—
102C3×C4⋊Dic3—32no—
103C3×Dic3⋊C4C2×D312yes—
104C3×C4×Dic3C2×D312yes—
105C3×C4×Dic3C2×D311yescharacteristic
106Dic3⋊Dic3—31no—
107#3722—32no—
108#3722—32no—
109#3722—31no—
110(C2×D3)⋊Dic3—31no—
111C2×Q8×C9—31no—
112#3701—64no—
113Q8×D9—64no—
114C2×D36—31no—
115C2×D36—32no—
116C2×D36—32no—
117C2×D36—31no—
118C2×D36—61no—
119C2×D36—62no—
120C2×C4×D9—31no—
121C2×C4×D9—32no—
122C2×C4×D9—32no—
123C2×C4×D9—61no—
124C2×C4×D9C2×D311yescharacteristic
125C2×Dic18—32no—
126C2×Dic18C2×D311yescharacteristic
127C9×C4⋊C4—21no—
128C9×C4⋊C4—32no—
129C9×C4⋊C4C2×D311yescharacteristic
130C42×C9—32no—
131C42×C9—31no—
132(C2×D9)⋊C4—31no—
133(C2×D9)⋊C4—32no—
134(C2×D9)⋊C4—31no—
135(C2×D9)⋊C4—32no—
136(C2×D9)⋊C4—32no—
137(C2×D9)⋊C4—62no—
138(C2×D9)⋊C4C2×D312yes—
139C4⋊Dic9—31no—
140C4⋊Dic9C2×D312yes—
141C22×C3×C9C2411yescharacteristic
142C2×C3⋊D9—21no—
143C2×C3⋊D9—21no—
144C2×C3⋊D9—22no—
145C2×C3⋊D9—22no—
146C2×C3⋊D9—22no—
147C2×C3⋊D9#4112yescomplemented
148C2×C3×D9—21no—
149C2×C3×D9—21no—
150C2×C3×D9—21no—
151C3×C4×C9—22no—
152C3×C4×C9C2×D412yes—
153C3⋊Dic9—21no—
154C9×Dic3—21no—
155C9×Dic3—21no—
156C9×Dic3C2×D414yes—
157C3×Dic9—22no—
158C3×Dic9—21no—
159C2×#239—91no—
160C2×D3×Q8—91no—
161C2×D3×Q8—91no—
162C2×#235—91no—
163C3×C22⋊Q8—91no—
164C3×C4×Q8—31no—
165#947—91no—
166C4×#67—91no—
167#897—31no—
168#897—92no—
169#896—92no—
170#895—92no—
171#895—91no—
172#895—91no—
173#895—181no—
174#893—92no—
175#887D911yescharacteristic, complemented
176#885—92no—
177#884—92no—
178#875—92no—
179#875—91no—
180C4×D12—92no—
181C4×D12—91no—
182C22×C3⋊D3—31no—
183C22×C3⋊D3—32no—
184C22×C3×D3—31no—
185C2×C32×C4C22×D312yes—
186C2×C32×C4C22×D311yescharacteristic
187C2×C3⋊Dic3—31no—
188C4⋊(C3⋊D3)—64no—
189C4⋊(C3⋊D3)—62no—
190C4×C3⋊D3—62no—
191C2×C3×Dic3—32no—
192C2×C3×Dic3—31no—
193C2×C3×Dic3C22×D311yescharacteristic
194C2×C3×Dic3C22×D311yescharacteristic
195C2×C3×Dic3C22×D312yes—
196C3×C4×D3—62no—
197C3×Dic6—62no—
198C3×Dic6D1214yes—
199#627—64no—
200#626—62no—
201#626—64no—
202#624—64no—
203#624—64no—
204C22×D9—31no—
205C22×D9—32no—
206C22×D9—61no—
207C22×D9—62no—
208C22×D9C22×D311yescharacteristic
209Q8×C9—34no—
210C2×C4×C9—21no—
211C2×C4×C9—22no—
212C2×C4×C9—32no—
213C2×C4×C9—31no—
214C2×C4×C9—31no—
215C2×C4×C9C22×D312yes—
216C2×C4×C9C22×D311yescharacteristic
217C2×Dic9—31no—
218C2×Dic9—61no—
219C2×Dic9C22×D311yescharacteristic
220C2×Dic9C22×D312yes—
221D36—64no—
222D36—64no—
223D36—62no—
224D36—62no—
225D36—122no—
226D36—124no—
227C4×D9—22no—
228C4×D9—64no—
229C4×D9—64no—
230C4×D9—62no—
231C4×D9—122no—
232Dic18—22no—
233Dic18—64no—
234#548—271noSylow
235C2×C3×C9C22×D411yescharacteristic
236C2×C3×C9C2×#4111yescharacteristic
237C2×C3×C92−1+411yescharacteristic
238C3⋊D9—22no—
239C3⋊D9—42no—
240C3×D9—41no—
241C2×C3×Q8—31no—
242C2×C3×Q8—91no—
243C22×C3×C4—91no—
244C2×#67—91no—
245C2×#67—92no—
246#239—184no—
247D3×Q8—184no—
248D3×Q8—184no—
249#235—184no—
250C2×D12—91no—
251C2×D12—92no—
252C2×D12—92no—
253C2×D12—91no—
254C2×D12—92no—
255C2×D12—91no—
256C2×D12—182no—
257C2×D12—181no—
258C2×C4×D3—31no—
259C2×C4×D3—92no—
260C2×C4×D3—92no—
261C2×C4×D3—91no—
262C2×C4×D3—91no—
263C2×C4×D3—92no—
264C2×C4×D3—92no—
265C2×C4×D3—181no—
266C2×Dic6—31no—
267C2×Dic6—92no—
268C2×Dic6—92no—
269C2×Dic6C2×D911yescharacteristic
270C3×C4⋊C4—21no—
271C3×C4⋊C4—32no—
272C3×C4⋊C4—92no—
273C3×C4⋊C4C2×D911yescharacteristic
274C3×C4⋊C4D3211yescharacteristic
275C3×C22⋊C4—92no—
276C3×C42—32no—
277C3×C42—31no—
278#217—91no—
279(C2×D3)⋊C4—32no—
280(C2×D3)⋊C4—91no—
281(C2×D3)⋊C4—92no—
282(C2×D3)⋊C4—92no—
283(C2×D3)⋊C4—91no—
284(C2×D3)⋊C4—92no—
285(C2×D3)⋊C4—92no—
286(C2×D3)⋊C4—92no—
287(C2×D3)⋊C4—92no—
288(C2×D3)⋊C4—91no—
289(C2×D3)⋊C4—182no—
290C4⋊Dic3—32no—
291C4⋊Dic3—91no—
292Dic3⋊C4—91no—
293Dic3⋊C4C2×D912yes—
294C4×Dic3C2×D912yes—
295C4×Dic3C2×D911yescharacteristic
296C22×C32C23×D311yescharacteristic
297C2×C3⋊D3—32no—
298C2×C3⋊D3—61no—
299C2×C3⋊D3—61no—
300C2×C3⋊D3—62no—
301C2×C3⋊D3—62no—
302C2×C3⋊D3—62no—
303C2×C3×D3—61no—
304C2×C3×D3—61no—
305C2×C3×D3—61no—
306C32×C4—22no—
307C32×C4C2×D1212yes—
308C3⋊Dic3—61no—
309C3×Dic3—21no—
310C3×Dic3—21no—
311C3×Dic3—61no—
312C3×Dic3—62no—
313C3×Dic3C2×D1214yes—
314C22×C9—21no—
315C22×C9C23×D311yescharacteristic
316C2×D9—21no—
317C2×D9—21no—
318C2×D9—21no—
319C2×D9—32no—
320C2×D9—62no—
321C2×D9—61no—
322C2×D9—61no—
323C2×D9—62no—
324C2×D9—62no—
325C2×D9—62no—
326C2×D9—121no—
327C2×D9—121no—
328C2×D9—122no—
329C2×D9—122no—
330C2×D9—122no—
331C4×C9—22no—
332C4×C9—22no—
333C4×C9—34no—
334C4×C9—42no—
335C4×C9—61no—
336C4×C9—61no—
337C4×C9D3×D412yes—
338Dic9—22no—
339Dic9—21no—
340Dic9—61no—
341Dic9—121no—
342C2×#41—271no—
343C22×Q8—271no—
344#124—272no—
345#124—271no—
346C22⋊Q8—91no—
347C22⋊Q8—272no—
348#121—272no—
349#121—271no—
350C4×Q8—31no—
351C4×D4—272no—
352C4×D4—271no—
353C3×C9#54811yescharacteristic, complemented, Sylow
354C23×C3—91no—
355C22×D3—31no—
356C22×D3—91no—
357C22×D3—91no—
358C22×D3—92no—
359C22×D3—92no—
360C22×D3—181no—
361C22×D3—182no—
362C3×Q8—34no—
363C3×Q8—182no—
364C2×C3×C4—21no—
365C2×C3×C4—22no—
366C2×C3×C4—32no—
367C2×C3×C4—31no—
368C2×C3×C4—31no—
369C2×C3×C4—91no—
370C2×C3×C4—92no—
371C2×C3×C4—182no—
372C2×C3×C4C22×D912yes—
373C2×C3×C4C22×D911yescharacteristic
374C2×C3×C4C2×D3212yes—
375C2×C3×C4C2×D3211yescharacteristic
376#67—182no—
377#67—184no—
378C2×Dic3—31no—
379C2×Dic3—32no—
380C2×Dic3—91no—
381C2×Dic3—91no—
382C2×Dic3—181no—
383C2×Dic3C22×D911yescharacteristic
384C2×Dic3C22×D911yescharacteristic
385C2×Dic3C22×D912yes—
386D12—184no—
387D12—184no—
388D12—184no—
389D12—182no—
390D12—182no—
391D12—182no—
392D12—362no—
393D12—364no—
394C4×D3—62no—
395C4×D3—184no—
396C4×D3—184no—
397C4×D3—182no—
398C4×D3—182no—
399C4×D3—184no—
400C4×D3—184no—
401C4×D3—362no—
402Dic6—62no—
403Dic6—184no—
404Dic6—184no—
405Dic6D3614yescomplemented
406C2×C32C22×D1211yescharacteristic
407C2×C32#100711yescharacteristic
408C2×C32D3×#4111yescharacteristic
409C3⋊D3—62no—
410C3⋊D3—122no—
411C3×D3—121no—
412C2×C9—21no—
413C2×C9—21no—
414C2×C9—21no—
415C2×C9C2×D3×D411yescharacteristic
416C2×C9#100711yescharacteristic
417C2×C9D3×#4111yescharacteristic
418D9—41no—
419D9—62no—
420D9—122no—
421D9—122no—
422D9—242no—
423#41—544no—
424C2×Q8—31no—
425C2×Q8—91no—
426C2×Q8—272no—
427C2×Q8—544no—
428C2×D4—271no—
429C2×D4—272no—
430C2×D4—272no—
431C2×D4—271no—
432C22×C4—91no—
433C22×C4—272no—
434C22×C4—272no—
435C22×C4—271no—
436C4⋊C4—32no—
437C4⋊C4—92no—
438C4⋊C4—271no—
439C4⋊C4D3×D911yescharacteristic
440C22⋊C4—92no—
441C22⋊C4—271no—
442C22⋊C4—272no—
443C22⋊C4—272no—
444C22⋊C4—271no—
445C22⋊C4—272no—
446C42—32no—
447C42—31no—
448C22×C3—21no—
449C22×C3—181no—
450C22×C3—181no—
451C22×C3—181no—
452C22×C3C23×D911yescharacteristic
453C22×C3C22×D3211yescharacteristic
454C2×D3—61no—
455C2×D3—61no—
456C2×D3—61no—
457C2×D3—92no—
458C2×D3—92no—
459C2×D3—182no—
460C2×D3—181no—
461C2×D3—181no—
462C2×D3—181no—
463C2×D3—181no—
464C2×D3—182no—
465C2×D3—182no—
466C2×D3—182no—
467C2×D3—182no—
468C2×D3—182no—
469C2×D3—182no—
470C2×D3—361no—
471C2×D3—361no—
472C2×D3—362no—
473C2×D3—362no—
474C2×D3—362no—
475C3×C4—22no—
476C3×C4—22no—
477C3×C4—22no—
478C3×C4—34no—
479C3×C4—42no—
480C3×C4—61no—
481C3×C4—61no—
482C3×C4—182no—
483C3×C4—181no—
484C3×C4C2×D3612yes—
485C3×C4D3×D1212yes—
486Dic3—21no—
487Dic3—21no—
488Dic3—62no—
489Dic3—61no—
490Dic3—181no—
491Dic3—181no—
492Dic3—361no—
493Dic3C2×D3614yes—
494C32#557611yescharacteristic
495C9—21no—
496C9#563311yescharacteristic, complemented
497C23—91no—
498C23—271no—
499C23—272no—
500Q8—34no—
501Q8—182no—
502Q8—544no—
503D4—544no—
504D4—544no—
505D4—542no—
506D4—542no—
507C2×C4—32no—
508C2×C4—31no—
509C2×C4—31no—
510C2×C4—92no—
511C2×C4—91no—
512C2×C4—182no—
513C2×C4—271no—
514C2×C4—544no—
515C2×C4—544no—
516C2×C4—542no—
517C2×C4C2×D3×D912yes—
518C2×C4C2×D3×D911yescharacteristic
519C2×C3—21no—
520C2×C3—21no—
521C2×C3—21no—
522C2×C3—361no—
523C2×C3C22×D3611yescharacteristic
524C2×C3#6371311yescharacteristic
525C2×C3#41×D911yescharacteristic
526C2×C3C2×D3×D1211yescharacteristic
527C2×C3#6431211yescharacteristic
528C2×C3#6431611yescharacteristic
529D3—121no—
530D3—182no—
531D3—182no—
532D3—362no—
533D3—362no—
534D3—362no—
535D3—722no—
536C22—181no—
537C22—181no—
538C22—181no—
539C22—272no—
540C22—541no—
541C22—541no—
542C22—542no—
543C22—542no—
544C22—542no—
545C22C22×D3×D911yescharacteristic
546C4—22no—
547C4—34no—
548C4—61no—
549C4—61no—
550C4—181no—
551C4—182no—
552C4—541no—
553C4D3×D3612yes—
554C3—21no—
555C3#1058967811yescharacteristic, complemented
556C3#1059432911yescharacteristic
557C2—361no—
558C2—542no—
559C2—1082no—
560C2C2×D3×D3611yescharacteristic
561C2#1172249811yescharacteristic
562C2#1172250911yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC3×C9×C4⋊C4C22
socleC22×C32C23×D3
hypercenterC4⋊C4D3×D9
hypocenterC3×C9#548
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×(C3⋊D9)⋊Q8C2semidirect
2. C2×#88635C2semidirect
3. #11721751C2semidirect
4. C3×#63459C2semidirect
5. C9×#887C2semidirect
6. #11721005C2semidirect
7. C4×#6192C2semidirect
8. #11720995C2semidirect
9. #11720993C2semidirect
10. #11720952C2semidirect
11. #11720940C2semidirect
54. #63459D3semidirect
77. C9×Dic6D4semidirect
147. C2×C3⋊D9#41semidirect
175. #887D9semidirect
353. C3×C9#548semidirect
405. Dic6D36semidirect
496. C9#5633semidirect
555. C3#10589678semidirect

Maximal subgroups

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

subgroupindex
C2×(C3⋊D9)⋊Q82
C2×#886352
#117217512
C3×#634592
C9×#8872
#117210052
C4×#61922
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