Finite Group Databasebeta

← #13148930

Subgroups — #13148930

Order 1728.

802 conjugacy classes of subgroups.

The subgroup lattice has 534 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11721751C211yesmaximal, characteristic, complemented
2C3×#63454C211yesmaximal, characteristic, complemented
3C9×#888C211yesmaximal, characteristic, complemented
4#11721005C212yesmaximal, complemented
5#11721004C211yesmaximal, characteristic, complemented
6#11721003C212yesmaximal, complemented
7C4×#6192C211yesmaximal, characteristic, complemented
8#11720993C211yesmaximal, characteristic, complemented
9#11720953C212yesmaximal, complemented
10#11720922C211yesmaximal, characteristic, complemented
11#11720899C212yesmaximal, complemented
12#10594564—31nomaximal
13#10589836—31nomaximal
14C2×C4⋊(C3⋊D9)C2211yescharacteristic
15C2×C4⋊(C3⋊D9)C2212yes—
16C2×C4×C3⋊D9C2211yescharacteristic
17C2×C3×C4×D9C2211yescharacteristic
18C2×C9×Dic6C2212yes—
19C2×#6192C2211yescharacteristic
20C2×#6192C2212yes—
21C2×(C3⋊D9)⋊C4C2212yes—
22C3×C9×C4⋊C4C2211yescharacteristic
23#88536C2212yes—
24C3×(C2×D9)⋊C4C2212yes—
25C9×C4⋊Dic3C2211yescharacteristic
26C9×Dic3⋊C4C2212yes—
27C4×C9×Dic3C2211yescharacteristic
28C3×C4⋊Dic9C2211yescharacteristic
29C3×C4×Dic9C2212yes—
30#88445C2212yes—
31#88445C2211yescharacteristic
32#88445C2212yes—
33#88445C2212yes—
34(C2×D9)⋊Dic3C2211yescharacteristic
35Dic3⋊Dic9C2211yescharacteristic
36Dic3×Dic9C2212yes—
37#64104—31no—
38C3×#892—31no—
39C3×#888D311yescharacteristic
40#63910—32no—
41#63909—31no—
42#63907—32no—
43C4×#626—31no—
44#63894—31no—
45#63864—32no—
46#63849—31no—
47#63836—32no—
48C9×C4⋊Q8—31no—
49#63509—32no—
50#63509—32no—
51#63457—31no—
52#63457—31no—
53#63457—32no—
54#63457—61no—
55#63454—31no—
56#63454—32no—
57#63454D311yescharacteristic, complemented
58C4×D36—31no—
59C4×D36—31no—
60C22×C3⋊D9C2312yes—
61C22×C3⋊D9C2311yescharacteristic
62C22×C3×D9C2311yescharacteristic
63C2×C3×C4×C9C2312yes—
64C2×C3×C4×C9C2311yescharacteristic
65C2×C3⋊Dic9C2311yescharacteristic
66C4⋊(C3⋊D9)—22no—
67C4⋊(C3⋊D9)—42no—
68C4×C3⋊D9—22no—
69C2×C9×Dic3C2312yes—
70C2×C9×Dic3C2311yescharacteristic
71C2×C9×Dic3C2311yescharacteristic
72C2×C3×Dic9C2312yes—
73C2×C3×Dic9C2311yescharacteristic
74C3×C4×D9—22no—
75C9×Dic6—42no—
76#6192—22no—
77#6192—42no—
78(C3⋊D9)⋊C4—24no—
79#5734—91no—
80#5532—91no—
81C2×C4⋊(C3⋊D3)—31no—
82C2×C4⋊(C3⋊D3)—32no—
83C2×C4×C3⋊D3—31no—
84C2×C3×C4×D3—31no—
85C2×C3×Dic6C2×D312yes—
86C2×#626—31no—
87C2×#626—32no—
88C2×#624—32no—
89C32×C4⋊C4C2×D311yescharacteristic
90#3752—32no—
91C3×(C2×D3)⋊C4—32no—
92C3×C4⋊Dic3—31no—
93C3×C4⋊Dic3C2×D311yescharacteristic
94C3×Dic3⋊C4C2×D312yes—
95C3×C4×Dic3—32no—
96C3×C4×Dic3C2×D311yescharacteristic
97Dic3⋊Dic3—31no—
98#3722—32no—
99#3722—32no—
100#3722—31no—
101#3722—32no—
102(C2×D3)⋊Dic3—31no—
103Dic32—32no—
104C2×Q8×C9—32no—
105C2×D36—31no—
106C2×D36—31no—
107C2×D36—32no—
108C2×D36—32no—
109C2×D36—61no—
110C2×D36—62no—
111C2×C4×D9—31no—
112C2×C4×D9—32no—
113C2×C4×D9—61no—
114C2×C4×D9C2×D311yescharacteristic
115C9×C4⋊C4—21no—
116C9×C4⋊C4—32no—
117C9×C4⋊C4—31no—
118C9×C4⋊C4C2×D311yescharacteristic
119C42×C9—31no—
120(C2×D9)⋊C4—31no—
121(C2×D9)⋊C4—32no—
122(C2×D9)⋊C4—32no—
123(C2×D9)⋊C4—31no—
124(C2×D9)⋊C4—32no—
125(C2×D9)⋊C4—32no—
126(C2×D9)⋊C4—62no—
127(C2×D9)⋊C4C2×D312yes—
128C4⋊Dic9—31no—
129C4⋊Dic9C2×D311yescharacteristic
130C4×Dic9—32no—
131C4×Dic9C2×D312yes—
132C22×C3×C9C2411yescharacteristic
133C2×C3⋊D9—22no—
134C2×C3⋊D9—21no—
135C2×C3⋊D9—21no—
136C2×C3⋊D9—22no—
137C2×C3⋊D9—21no—
138C2×C3⋊D9—22no—
139C2×C3×D9—21no—
140C2×C3×D9—21no—
141C2×C3×D9—21no—
142C3×C4×C9—22no—
143C3×C4×C9#4112yes—
144C3⋊Dic9—21no—
145C9×Dic3—22no—
146C9×Dic3—21no—
147C9×Dic3#4112yes—
148C3×Dic9—22no—
149C3×Dic9—21no—
150C3×C4⋊Q8—31no—
151C3×#117—91no—
152#947—92no—
153#947—92no—
154#930—91no—
155C4×#67—91no—
156#895—92no—
157#895—91no—
158#895—91no—
159#895—91no—
160#895—181no—
161#893—91no—
162#892—31no—
163#892—92no—
164#892—91no—
165#892—91no—
166#888D911yescharacteristic, complemented
167#885—92no—
168#884—92no—
169#875—92no—
170C4×D12—91no—
171C4×D12—91no—
172#872—92no—
173C22×C3⋊D3—31no—
174C22×C3⋊D3—32no—
175C22×C3×D3—31no—
176C2×C32×C4C22×D312yes—
177C2×C32×C4C22×D311yescharacteristic
178C2×C3⋊Dic3—31no—
179C4⋊(C3⋊D3)—62no—
180C4⋊(C3⋊D3)—122no—
181C4×C3⋊D3—62no—
182C2×C3×Dic3—32no—
183C2×C3×Dic3—31no—
184C2×C3×Dic3C22×D312yes—
185C2×C3×Dic3C22×D311yescharacteristic
186C2×C3×Dic3C22×D311yescharacteristic
187C3×C4×D3—62no—
188C3×Dic6—42no—
189#626—62no—
190#626—122no—
191#624—64no—
192C22×D9—31no—
193C22×D9—32no—
194C22×D9—61no—
195C22×D9—62no—
196C22×D9C22×D311yescharacteristic
197Q8×C9—122no—
198C2×C4×C9—21no—
199C2×C4×C9—22no—
200C2×C4×C9—31no—
201C2×C4×C9—32no—
202C2×C4×C9—31no—
203C2×C4×C9C22×D312yes—
204C2×C4×C9C22×D311yescharacteristic
205C2×Dic9—31no—
206C2×Dic9—61no—
207C2×Dic9C22×D312yes—
208C2×Dic9C22×D311yescharacteristic
209D36—62no—
210D36—62no—
211D36—122no—
212D36—122no—
213D36—124no—
214D36—122no—
215C4×D9—22no—
216C4×D9—64no—
217C4×D9—62no—
218C4×D9—122no—
219#555—271noSylow
220C2×C3×C9C2×#4111yescharacteristic
221C2×C3×C9C2×#4111yescharacteristic
222C2×C3×C9H(5,𝔽2)11yescharacteristic
223C3⋊D9—41no—
224C3⋊D9—42no—
225C3×D9—41no—
226C2×C3×Q8—32no—
227C22×C3×C4—91no—
228C2×#67—91no—
229C2×#67—92no—
230C2×D12—91no—
231C2×D12—91no—
232C2×D12—91no—
233C2×D12—92no—
234C2×D12—92no—
235C2×D12—92no—
236C2×D12—181no—
237C2×D12—182no—
238C2×C4×D3—31no—
239C2×C4×D3—92no—
240C2×C4×D3—91no—
241C2×C4×D3—91no—
242C2×C4×D3—92no—
243C2×C4×D3—181no—
244C2×Dic6C2×D912yes—
245C3×C4⋊C4—21no—
246C3×C4⋊C4—32no—
247C3×C4⋊C4—31no—
248C3×C4⋊C4—91no—
249C3×C4⋊C4C2×D911yescharacteristic
250C3×C4⋊C4D3211yescharacteristic
251C3×C22⋊C4—92no—
252C3×C42—31no—
253C3×C42—92no—
254#217—91no—
255(C2×D3)⋊C4—32no—
256(C2×D3)⋊C4—91no—
257(C2×D3)⋊C4—92no—
258(C2×D3)⋊C4—92no—
259(C2×D3)⋊C4—92no—
260(C2×D3)⋊C4—91no—
261(C2×D3)⋊C4—92no—
262(C2×D3)⋊C4—92no—
263(C2×D3)⋊C4—92no—
264(C2×D3)⋊C4—91no—
265(C2×D3)⋊C4—92no—
266(C2×D3)⋊C4—92no—
267(C2×D3)⋊C4—182no—
268C4⋊Dic3—31no—
269C4⋊Dic3—91no—
270C4⋊Dic3C2×D911yescharacteristic
271Dic3⋊C4—91no—
272Dic3⋊C4C2×D912yes—
273C4×Dic3—32no—
274C4×Dic3—92no—
275C4×Dic3—92no—
276C4×Dic3C2×D911yescharacteristic
277C22×C32C23×D311yescharacteristic
278C2×C3⋊D3—61no—
279C2×C3⋊D3—62no—
280C2×C3⋊D3—62no—
281C2×C3⋊D3—61no—
282C2×C3⋊D3—61no—
283C2×C3⋊D3—62no—
284C2×C3×D3—61no—
285C2×C3×D3—61no—
286C2×C3×D3—61no—
287C32×C4—22no—
288C32×C4#23912yes—
289C3⋊Dic3—61no—
290C3×Dic3—22no—
291C3×Dic3—21no—
292C3×Dic3—62no—
293C3×Dic3—61no—
294C3×Dic3#23912yes—
295C22×C9—21no—
296C22×C9C23×D311yescharacteristic
297C2×D9—21no—
298C2×D9—21no—
299C2×D9—21no—
300C2×D9—62no—
301C2×D9—61no—
302C2×D9—62no—
303C2×D9—62no—
304C2×D9—61no—
305C2×D9—61no—
306C2×D9—62no—
307C2×D9—121no—
308C2×D9—121no—
309C2×D9—122no—
310C2×D9—122no—
311C2×D9—122no—
312C4×C9—22no—
313C4×C9—22no—
314C4×C9—32no—
315C4×C9—42no—
316C4×C9—62no—
317C4×C9—61no—
318C4×C9#23512yes—
319Dic9—21no—
320Dic9—22no—
321Dic9—61no—
322Dic9—121no—
323C4⋊Q8—31no—
324#124—272no—
325#124—272no—
326#121—272no—
327#121—271no—
328#121—271no—
329C4×D4—271no—
330C4×D4—271no—
331#117—91no—
332#117—272no—
333#117—271no—
334C3×C9#55511yescharacteristic, complemented, Sylow
335C23×C3—91no—
336C22×D3—31no—
337C22×D3—91no—
338C22×D3—92no—
339C22×D3—91no—
340C22×D3—92no—
341C22×D3—181no—
342C22×D3—182no—
343C3×Q8—122no—
344C2×C3×C4—21no—
345C2×C3×C4—22no—
346C2×C3×C4—31no—
347C2×C3×C4—32no—
348C2×C3×C4—31no—
349C2×C3×C4—92no—
350C2×C3×C4—91no—
351C2×C3×C4—182no—
352C2×C3×C4C22×D912yes—
353C2×C3×C4C22×D911yescharacteristic
354C2×C3×C4C2×D3212yes—
355C2×C3×C4C2×D3211yescharacteristic
356#67—182no—
357#67—362no—
358C2×Dic3—31no—
359C2×Dic3—32no—
360C2×Dic3—91no—
361C2×Dic3—91no—
362C2×Dic3—181no—
363C2×Dic3C22×D912yes—
364C2×Dic3C22×D911yescharacteristic
365C2×Dic3C22×D911yescharacteristic
366D12—182no—
367D12—182no—
368D12—182no—
369D12—362no—
370D12—362no—
371D12—362no—
372D12—364no—
373D12—362no—
374C4×D3—62no—
375C4×D3—184no—
376C4×D3—182no—
377C4×D3—182no—
378C4×D3—184no—
379C4×D3—362no—
380Dic6—42no—
381C2×C32#100411yescharacteristic
382C2×C32C2×#23911yescharacteristic
383C2×C32C2×#23911yescharacteristic
384C3⋊D3—121no—
385C3⋊D3—122no—
386C3×D3—121no—
387C2×C9—21no—
388C2×C9—21no—
389C2×C9—21no—
390C2×C9C2×#23511yescharacteristic
391C2×C9D3×#4111yescharacteristic
392C2×C9#100911yescharacteristic
393D9—41no—
394D9—122no—
395D9—121no—
396D9—122no—
397D9—242no—
398C2×Q8—32no—
399C2×D4—271no—
400C2×D4—272no—
401C2×D4—271no—
402C2×D4—272no—
403C22×C4—91no—
404C22×C4—272no—
405C22×C4—271no—
406C4⋊C4—32no—
407C4⋊C4—31no—
408C4⋊C4—91no—
409C4⋊C4—271no—
410C4⋊C4D3×D911yescharacteristic
411C22⋊C4—92no—
412C22⋊C4—272no—
413C22⋊C4—272no—
414C22⋊C4—271no—
415C22⋊C4—272no—
416C22⋊C4—271no—
417C22⋊C4—272no—
418C42—31no—
419C42—92no—
420C42—272no—
421C22×C3—21no—
422C22×C3—181no—
423C22×C3—181no—
424C22×C3—181no—
425C22×C3C23×D911yescharacteristic
426C22×C3C22×D3211yescharacteristic
427C2×D3—61no—
428C2×D3—61no—
429C2×D3—61no—
430C2×D3—182no—
431C2×D3—181no—
432C2×D3—182no—
433C2×D3—181no—
434C2×D3—182no—
435C2×D3—182no—
436C2×D3—181no—
437C2×D3—181no—
438C2×D3—182no—
439C2×D3—182no—
440C2×D3—181no—
441C2×D3—182no—
442C2×D3—181no—
443C2×D3—361no—
444C2×D3—361no—
445C2×D3—362no—
446C2×D3—362no—
447C2×D3—362no—
448C3×C4—22no—
449C3×C4—22no—
450C3×C4—22no—
451C3×C4—32no—
452C3×C4—42no—
453C3×C4—62no—
454C3×C4—61no—
455C3×C4—182no—
456C3×C4—181no—
457C3×C4#370112yes—
458C3×C4#379912yes—
459Dic3—21no—
460Dic3—22no—
461Dic3—61no—
462Dic3—62no—
463Dic3—181no—
464Dic3—181no—
465Dic3—361no—
466Dic3#370112yes—
467C32#573411yescharacteristic
468C9—21no—
469C9#553211yescharacteristic, complemented
470C23—91no—
471C23—271no—
472C23—272no—
473Q8—122no—
474D4—542no—
475D4—542no—
476D4—1082no—
477D4—1082no—
478C2×C4—31no—
479C2×C4—31no—
480C2×C4—32no—
481C2×C4—91no—
482C2×C4—92no—
483C2×C4—182no—
484C2×C4—271no—
485C2×C4—544no—
486C2×C4—542no—
487C2×C4C2×D3×D912yes—
488C2×C4C2×D3×D911yescharacteristic
489C2×C3—21no—
490C2×C3—21no—
491C2×C3—21no—
492C2×C3—361no—
493C2×C3#6371011yescharacteristic
494C2×C3C2×#370111yescharacteristic
495C2×C3C2×#370111yescharacteristic
496C2×C3C2×#379911yescharacteristic
497C2×C3#6431411yescharacteristic
498C2×C3D3×#23911yescharacteristic
499D3—121no—
500D3—362no—
501D3—361no—
502D3—362no—
503D3—361no—
504D3—362no—
505D3—722no—
506C22—181no—
507C22—181no—
508C22—181no—
509C22—541no—
510C22—542no—
511C22—542no—
512C22—541no—
513C22—541no—
514C22—542no—
515C22C22×D3×D911yescharacteristic
516C4—22no—
517C4—32no—
518C4—62no—
519C4—61no—
520C4—182no—
521C4—181no—
522C4—541no—
523C4#8863512yes—
524C3—21no—
525C3#1058983611yescharacteristic, complemented
526C3#1059456411yescharacteristic
527C2—361no—
528C2—1081no—
529C2—1082no—
530C2C2×#8863511yescharacteristic
531C2#1172250011yescharacteristic
532C2D3×#370111yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC3×C9×C4⋊C4C22
socleC22×C32C23×D3
hypercenterC4⋊C4D3×D9
hypocenterC3×C9#555
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. #11721751C2semidirect
2. C3×#63454C2semidirect
3. C9×#888C2semidirect
4. #11721005C2semidirect
5. #11721004C2semidirect
6. #11721003C2semidirect
7. C4×#6192C2semidirect
8. #11720993C2semidirect
9. #11720953C2semidirect
10. #11720922C2semidirect
11. #11720899C2semidirect
57. #63454D3semidirect
166. #888D9semidirect
334. C3×C9#555semidirect
469. C9#5532semidirect
525. C3#10589836semidirect

Maximal subgroups

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

subgroupindex
#117217512
C3×#634542
C9×#8882
#117210052
#117210042
#117210032
C4×#61922
#117209932
#117209532
#117209222
#117208992
#105945643
#105898363