Finite Group Databasebeta

← #13148929

Subgroups — #13148929

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
2C9×#892C211yesmaximal, characteristic, complemented
3C3×#63450C211yesmaximal, characteristic, complemented
4#11721039C212yesmaximal, complemented
5#11721038C211yesmaximal, characteristic, complemented
6#11721035C212yesmaximal, complemented
7C4×#6195C211yesmaximal, characteristic, complemented
8#11720992C211yesmaximal, characteristic, complemented
9#11720952C212yesmaximal, complemented
10#11720922C211yesmaximal, characteristic, complemented
11#11720898C212yesmaximal, complemented
12#10594564—31nomaximal
13#10589634—31nomaximal
14C2×C4⋊(C3⋊D9)C2211yescharacteristic
15C2×C4⋊(C3⋊D9)C2212yes—
16C2×C4×C3⋊D9C2211yescharacteristic
17C2×C4×D3×C9C2211yescharacteristic
18C2×C3×Dic18C2212yes—
19C2×#6195C2211yescharacteristic
20C2×#6195C2212yes—
21C2×(C3⋊D9)⋊C4C2212yes—
22C3×C9×C4⋊C4C2211yescharacteristic
23#88536C2212yes—
24C9×(C2×D3)⋊C4C2212yes—
25C9×C4⋊Dic3C2211yescharacteristic
26C4×C9×Dic3C2212yes—
27C3×C4⋊Dic9C2211yescharacteristic
28C3×Dic9⋊C4C2212yes—
29C3×C4×Dic9C2211yescharacteristic
30(C2×D3)⋊Dic9C2211yescharacteristic
31#88445C2212yes—
32#88445C2211yescharacteristic
33#88445C2212yes—
34#88445C2212yes—
35Dic9⋊Dic3C2211yescharacteristic
36Dic3×Dic9C2212yes—
37#64104—31no—
38C3×#892D311yescharacteristic
39C3×#888—31no—
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×#117—31no—
49#63492—31no—
50C4×#611—31no—
51#63457—31no—
52#63457—61no—
53#63455—31no—
54#63454—31no—
55#63450D311yescharacteristic, complemented
56#63447—32no—
57#63446—32no—
58#63437—32no—
59#63434—32no—
60C22×C3⋊D9C2312yes—
61C22×C3⋊D9C2311yescharacteristic
62C22×D3×C9C2311yescharacteristic
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×C3×Dic9C2312yes—
72C2×C3×Dic9C2311yescharacteristic
73C2×C3×Dic9C2311yescharacteristic
74C4×D3×C9—22no—
75C3×Dic18—42no—
76#6195—22no—
77#6195—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×D3C2×D311yescharacteristic
85C2×C3×Dic6—32no—
86C2×#626—31no—
87C2×#626—32no—
88C2×#624—32no—
89C32×C4⋊C4C2×D311yescharacteristic
90#3752—32no—
91C3×(C2×D3)⋊C4C2×D312yes—
92C3×C4⋊Dic3—31no—
93C3×C4⋊Dic3C2×D311yescharacteristic
94C3×Dic3⋊C4—32no—
95C3×C4×Dic3—31no—
96C3×C4×Dic3C2×D312yes—
97Dic3⋊Dic3—31no—
98#3722—32no—
99#3722—32no—
100#3722—31no—
101#3722—32no—
102(C2×D3)⋊Dic3—31no—
103Dic32—32no—
104C22×C4×C9—31no—
105C2×#611—31no—
106C2×#611—32no—
107C2×D36—31no—
108C2×D36—32no—
109C2×D36—61no—
110C2×D36—62no—
111C2×C4×D9—31no—
112C2×C4×D9—32no—
113C2×C4×D9—61no—
114C2×Dic18C2×D312yes—
115C9×C4⋊C4—21no—
116C9×C4⋊C4—31no—
117C9×C4⋊C4C2×D311yescharacteristic
118C9×C22⋊C4—32no—
119C42×C9—32no—
120#3676—31no—
121(C2×D9)⋊C4—32no—
122(C2×D9)⋊C4—31no—
123(C2×D9)⋊C4—32no—
124(C2×D9)⋊C4—32no—
125(C2×D9)⋊C4—32no—
126(C2×D9)⋊C4—62no—
127C4⋊Dic9C2×D311yescharacteristic
128Dic9⋊C4—31no—
129Dic9⋊C4C2×D312yes—
130C4×Dic9—32no—
131C4×Dic9C2×D311yescharacteristic
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×D3×C9—21no—
140C2×D3×C9—21no—
141C2×D3×C9—21no—
142C3×C4×C9—22no—
143C3×C4×C9#4112yes—
144C3⋊Dic9—21no—
145C9×Dic3—22no—
146C9×Dic3—21no—
147C3×Dic9—22no—
148C3×Dic9—21no—
149C3×Dic9#4112yes—
150C3×C4⋊Q8—91no—
151C3×#117—31no—
152#947—92no—
153#947—92no—
154#930—91no—
155C4×#67—91no—
156#895—91no—
157#895—91no—
158#895—91no—
159#895—92no—
160#895—181no—
161#893—91no—
162#892—91no—
163#892—91no—
164#892—92no—
165#892D911yescharacteristic, complemented
166#888—31no—
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×D3C22×D311yescharacteristic
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—31no—
183C2×C3×Dic3—32no—
184C2×C3×Dic3—31no—
185C2×C3×Dic3C22×D312yes—
186C2×C3×Dic3C22×D311yescharacteristic
187C3×C4×D3—22no—
188C3×Dic6—122no—
189#626—62no—
190#626—122no—
191#624—64no—
192C23×C9—31no—
193C22×D9—31no—
194C22×D9—32no—
195C22×D9—61no—
196C22×D9—62no—
197C2×C4×C9—21no—
198C2×C4×C9—22no—
199C2×C4×C9—32no—
200C2×C4×C9—31no—
201C2×C4×C9—62no—
202C2×C4×C9C22×D312yes—
203C2×C4×C9C22×D311yescharacteristic
204#611—62no—
205#611—122no—
206C2×Dic9—31no—
207C2×Dic9—61no—
208C2×Dic9C22×D312yes—
209C2×Dic9C22×D311yescharacteristic
210C2×Dic9C22×D311yescharacteristic
211D36—62no—
212D36—122no—
213D36—122no—
214D36—124no—
215C4×D9—62no—
216C4×D9—64no—
217C4×D9—122no—
218Dic18—42no—
219#555—271noSylow
220C2×C3×C9C2×#4111yescharacteristic
221C2×C3×C9C2×#4111yescharacteristic
222C2×C3×C9H(5,𝔽2)11yescharacteristic
223C3⋊D9—41no—
224C3⋊D9—42no—
225D3×C9—41no—
226C2×C3×Q8—92no—
227C22×C3×C4—31no—
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—92no—
239C2×C4×D3—91no—
240C2×C4×D3—91no—
241C2×C4×D3—92no—
242C2×C4×D3—181no—
243C2×C4×D3C2×D911yescharacteristic
244C2×Dic6—32no—
245C3×C4⋊C4—21no—
246C3×C4⋊C4—31no—
247C3×C4⋊C4—92no—
248C3×C4⋊C4—91no—
249C3×C4⋊C4C2×D911yescharacteristic
250C3×C4⋊C4D3211yescharacteristic
251C3×C22⋊C4—32no—
252C3×C42—32no—
253C3×C42—91no—
254#217—91no—
255(C2×D3)⋊C4—92no—
256(C2×D3)⋊C4—91no—
257(C2×D3)⋊C4—92no—
258(C2×D3)⋊C4—92no—
259(C2×D3)⋊C4—91no—
260(C2×D3)⋊C4—92no—
261(C2×D3)⋊C4—92no—
262(C2×D3)⋊C4—92no—
263(C2×D3)⋊C4—91no—
264(C2×D3)⋊C4—92no—
265(C2×D3)⋊C4—92no—
266(C2×D3)⋊C4—182no—
267(C2×D3)⋊C4C2×D912yes—
268C4⋊Dic3—31no—
269C4⋊Dic3—91no—
270C4⋊Dic3C2×D911yescharacteristic
271Dic3⋊C4—32no—
272Dic3⋊C4—91no—
273C4×Dic3—31no—
274C4×Dic3—92no—
275C4×Dic3—92no—
276C4×Dic3C2×D912yes—
277C22×C32C23×D311yescharacteristic
278C2×C3⋊D3—61no—
279C2×C3⋊D3—61no—
280C2×C3⋊D3—62no—
281C2×C3⋊D3—62no—
282C2×C3⋊D3—61no—
283C2×C3⋊D3—62no—
284C2×C3×D3—21no—
285C2×C3×D3—21no—
286C2×C3×D3—21no—
287C32×C4—22no—
288C32×C4#23512yes—
289C3⋊Dic3—61no—
290C3×Dic3—21no—
291C3×Dic3—22no—
292C3×Dic3—32no—
293C3×Dic3—62no—
294C3×Dic3—61no—
295C22×C9—21no—
296C22×C9—61no—
297C22×C9—61no—
298C22×C9—61no—
299C22×C9C23×D311yescharacteristic
300C2×D9—62no—
301C2×D9—61no—
302C2×D9—62no—
303C2×D9—61no—
304C2×D9—62no—
305C2×D9—62no—
306C2×D9—61no—
307C2×D9—121no—
308C2×D9—121no—
309C2×D9—122no—
310C2×D9—122no—
311C2×D9—122no—
312C4×C9—22no—
313C4×C9—22no—
314C4×C9—42no—
315C4×C9—62no—
316C4×C9—61no—
317C4×C9#23912yes—
318Dic9—21no—
319Dic9—22no—
320Dic9—61no—
321Dic9—121no—
322Dic9#23912yes—
323C4⋊Q8—91no—
324#124—272no—
325#124—272no—
326#121—272no—
327#121—271no—
328#121—271no—
329C4×D4—271no—
330C4×D4—271no—
331#117—31no—
332#117—272no—
333#117—271no—
334C3×C9#55511yescharacteristic, complemented, Sylow
335C23×C3—31no—
336C22×D3—91no—
337C22×D3—92no—
338C22×D3—91no—
339C22×D3—92no—
340C22×D3—181no—
341C22×D3—182no—
342C22×D3C22×D911yescharacteristic
343C3×Q8—362no—
344C2×C3×C4—21no—
345C2×C3×C4—22no—
346C2×C3×C4—32no—
347C2×C3×C4—31no—
348C2×C3×C4—62no—
349C2×C3×C4—91no—
350C2×C3×C4—92no—
351C2×C3×C4—91no—
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—31no—
360C2×Dic3—32no—
361C2×Dic3—91no—
362C2×Dic3—91no—
363C2×Dic3—181no—
364C2×Dic3C22×D912yes—
365C2×Dic3C22×D911yescharacteristic
366D12—182no—
367D12—182no—
368D12—182no—
369D12—362no—
370D12—362no—
371D12—362no—
372D12—364no—
373D12—362no—
374C4×D3—22no—
375C4×D3—184no—
376C4×D3—184no—
377C4×D3—182no—
378C4×D3—182no—
379C4×D3—362no—
380Dic6—122no—
381C2×C32C2×#23511yescharacteristic
382C2×C32D3×#4111yescharacteristic
383C2×C32#100911yescharacteristic
384C3⋊D3—121no—
385C3⋊D3—122no—
386C3×D3—41no—
387C2×C9—21no—
388C2×C9—21no—
389C2×C9—21no—
390C2×C9—121no—
391C2×C9#100411yescharacteristic
392C2×C9C2×#23911yescharacteristic
393C2×C9C2×#23911yescharacteristic
394D9—122no—
395D9—121no—
396D9—122no—
397D9—242no—
398C2×Q8—92no—
399C2×D4—271no—
400C2×D4—272no—
401C2×D4—271no—
402C2×D4—272no—
403C22×C4—31no—
404C22×C4—272no—
405C22×C4—271no—
406C4⋊C4—31no—
407C4⋊C4—92no—
408C4⋊C4—91no—
409C4⋊C4—271no—
410C4⋊C4D3×D911yescharacteristic
411C22⋊C4—32no—
412C22⋊C4—272no—
413C22⋊C4—272no—
414C22⋊C4—271no—
415C22⋊C4—272no—
416C22⋊C4—271no—
417C22⋊C4—272no—
418C42—32no—
419C42—91no—
420C42—272no—
421C22×C3—21no—
422C22×C3—61no—
423C22×C3—61no—
424C22×C3—61no—
425C22×C3C23×D911yescharacteristic
426C22×C3C22×D3211yescharacteristic
427C2×D3—21no—
428C2×D3—21no—
429C2×D3—21no—
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—181no—
442C2×D3—182no—
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—42no—
452C3×C4—61no—
453C3×C4—62no—
454C3×C4—92no—
455C3×C4—182no—
456C3×C4—181no—
457C3×C4#369712yes—
458C3×C4#379912yes—
459Dic3—21no—
460Dic3—22no—
461Dic3—32no—
462Dic3—61no—
463Dic3—62no—
464Dic3—181no—
465Dic3—181no—
466Dic3—361no—
467C32#553211yescharacteristic
468C9—21no—
469C9#573411yescharacteristic, complemented
470C23—31no—
471C23—271no—
472C23—272no—
473Q8—362no—
474D4—542no—
475D4—542no—
476D4—1082no—
477D4—1082no—
478C2×C4—31no—
479C2×C4—32no—
480C2×C4—62no—
481C2×C4—91no—
482C2×C4—91no—
483C2×C4—92no—
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—121no—
493C2×C3C2×#369711yescharacteristic
494C2×C3#41×D911yescharacteristic
495C2×C3#6371511yescharacteristic
496C2×C3C2×#379911yescharacteristic
497C2×C3#6431411yescharacteristic
498C2×C3D3×#23911yescharacteristic
499D3—41no—
500D3—362no—
501D3—361no—
502D3—362no—
503D3—361no—
504D3—362no—
505D3—722no—
506C22—61no—
507C22—61no—
508C22—61no—
509C22—541no—
510C22—542no—
511C22—542no—
512C22—541no—
513C22—541no—
514C22—542no—
515C22C22×D3×D911yescharacteristic
516C4—22no—
517C4—62no—
518C4—61no—
519C4—92no—
520C4—182no—
521C4—181no—
522C4—541no—
523C4#8864212yes—
524C3—21no—
525C3#1058963411yescharacteristic, complemented
526C3#1059456411yescharacteristic
527C2—121no—
528C2—1081no—
529C2—1082no—
530C2C2×#8864211yescharacteristic
531C2#1172250111yescharacteristic
532C2D9×#23911yescharacteristic
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. C9×#892C2semidirect
3. C3×#63450C2semidirect
4. #11721039C2semidirect
5. #11721038C2semidirect
6. #11721035C2semidirect
7. C4×#6195C2semidirect
8. #11720992C2semidirect
9. #11720952C2semidirect
10. #11720922C2semidirect
11. #11720898C2semidirect
55. #63450D3semidirect
165. #892D9semidirect
334. C3×C9#555semidirect
469. C9#5734semidirect
525. C3#10589634semidirect

Maximal subgroups

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

subgroupindex
#117217512
C9×#8922
C3×#634502
#117210392
#117210382
#117210352
C4×#61952
#117209922
#117209522
#117209222
#117208982
#105945643
#105896343