Finite Group Databasebeta

← #13148908

Subgroups — #13148908

Order 1728.

698 conjugacy classes of subgroups.

The subgroup lattice has 480 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11721746C211yesmaximal, characteristic, complemented
2C9×#892C211yesmaximal, characteristic, complemented
3C3×#63459C211yesmaximal, characteristic, complemented
4C4×#6194C211yesmaximal, characteristic
5#11720989C212yesmaximal
6D9×C4⋊Dic3C211yesmaximal, characteristic, complemented
7#11720974C212yesmaximal, complemented
8#11720962C212yesmaximal, complemented
9#11720960C211yesmaximal, characteristic, complemented
10#11720927C212yesmaximal, complemented
11#11720910C211yesmaximal, characteristic, complemented
12#10594456—31nomaximal
13#10589631—31nomaximal
14C2×C4×D3×C9C2211yescharacteristic
15C2×C3×C4×D9C2211yescharacteristic
16C2×C3×Dic18C2211yescharacteristic
17C2×#6194C2211yescharacteristic
18C2×Dic3×D9C2212yes—
19C3×C9×C4⋊C4C2211yescharacteristic
20C4⋊(C3⋊Dic9)C2211yescharacteristic
21C4⋊(C3⋊Dic9)C2212yes—
22#88534C2212yes—
23C4×C3⋊Dic9C2211yescharacteristic
24C9×(C2×D3)⋊C4C2212yes—
25C3×(C2×D9)⋊C4C2212yes—
26C9×C4⋊Dic3C2211yescharacteristic
27C4×C9×Dic3C2212yes—
28C3×C4⋊Dic9C2212yes—
29(C2×D3)⋊Dic9C2212yes—
30(C2×D3)⋊Dic9C2211yescharacteristic
31(C2×D9)⋊Dic3C2211yescharacteristic
32#88442C2212yes—
33#88442C2211yescharacteristic
34Dic9⋊Dic3C2212yes—
35Dic9⋊Dic3C2212yes—
36Dic3×Dic9C2212yes—
37#64099—31no—
38C3×#897—31no—
39C3×#892D311yescharacteristic
40C4×#625—31no—
41#63891—32no—
42D3×C4⋊Dic3—31no—
43#63877—32no—
44#63869—32no—
45#63858—32no—
46#63856—31no—
47#63843—31no—
48C9×#117—31no—
49C4×#611—31no—
50#63486—31no—
51#63459D311yescharacteristic, complemented
52C4⋊C4×D9—31no—
53#63452—31no—
54#63452—61no—
55#63449—31no—
56#63445—32no—
57#63441—32no—
58#63434—32no—
59#63432—32no—
60C22×D3×C9C2311yescharacteristic
61C22×C3×D9C2311yescharacteristic
62C2×C3×C4×C9C2312yes—
63C2×C3×C4×C9C2311yescharacteristic
64C2×C3⋊Dic9C2312yes—
65C2×C3⋊Dic9C2311yescharacteristic
66C2×C3⋊Dic9C2311yescharacteristic
67C2×C9×Dic3C2312yes—
68C2×C9×Dic3C2311yescharacteristic
69C2×C3×Dic9C2312yes—
70C2×C3×Dic9C2311yescharacteristic
71C4×D3×C9—22no—
72C3×C4×D9—22no—
73C3×Dic18—22no—
74#6194—22no—
75Dic3×D9—24no—
76#5642—91no—
77#5529—91no—
78C2×C3×C4×D3—31no—
79C2×C3×C4×D3C2×D311yescharacteristic
80C2×C3×Dic6—31no—
81C2×#625—31no—
82C2×D3×Dic3—32no—
83C32×C4⋊C4C2×D311yescharacteristic
84C4⋊(C3⋊Dic3)—31no—
85C4⋊(C3⋊Dic3)—32no—
86#3750—32no—
87C4×C3⋊Dic3—31no—
88C3×(C2×D3)⋊C4—32no—
89C3×(C2×D3)⋊C4C2×D312yes—
90C3×C4⋊Dic3—32no—
91C3×C4⋊Dic3C2×D311yescharacteristic
92C3×C4×Dic3C2×D312yes—
93#3724—32no—
94#3724—31no—
95Dic3⋊Dic3—32no—
96Dic3⋊Dic3—32no—
97(C2×D3)⋊Dic3—32no—
98(C2×D3)⋊Dic3—31no—
99(C2×D3)⋊Dic3—31no—
100Dic32—32no—
101C22×C4×C9—31no—
102C2×#611—31no—
103C2×C4×D9—32no—
104C2×C4×D9C2×D311yescharacteristic
105C2×Dic18C2×D311yescharacteristic
106C9×C4⋊C4—21no—
107C9×C4⋊C4—31no—
108C9×C4⋊C4C2×D311yescharacteristic
109C9×C22⋊C4—32no—
110C42×C9—32no—
111#3676—31no—
112#3676—32no—
113(C2×D9)⋊C4—31no—
114(C2×D9)⋊C4C2×D312yes—
115C4⋊Dic9—32no—
116C4⋊Dic9—31no—
117C4⋊Dic9—62no—
118C4⋊Dic9—61no—
119C4⋊Dic9C2×D312yes—
120Dic9⋊C4—32no—
121Dic9⋊C4—31no—
122Dic9⋊C4—32no—
123Dic9⋊C4—32no—
124Dic9⋊C4—32no—
125Dic9⋊C4—62no—
126C4×Dic9—31no—
127C4×Dic9—32no—
128C4×Dic9—61no—
129C22×C3×C9C2411yescharacteristic
130C2×D3×C9—21no—
131C2×D3×C9—21no—
132C2×D3×C9—21no—
133C2×C3×D9—21no—
134C2×C3×D9—21no—
135C2×C3×D9#4112yes—
136C3×C4×C9—22no—
137C3×C4×C9#4112yes—
138C3⋊Dic9—22no—
139C3⋊Dic9—21no—
140C3⋊Dic9—21no—
141C9×Dic3—22no—
142C9×Dic3—21no—
143C3×Dic9—22no—
144C3×Dic9—21no—
145C3×C22⋊Q8—91no—
146C3×#117—31no—
147#946—91no—
148Q8×Dic3—91no—
149C4×#67—91no—
150C4×#67—91no—
151C2×C4⋊Dic3—91no—
152#924—91no—
153#898—92no—
154#897—31no—
155#892D911yescharacteristic, complemented
156D3×C4⋊C4—91no—
157#890—91no—
158#890—91no—
159#890—92no—
160#890—181no—
161#887—91no—
162#886—92no—
163#883—92no—
164#879—92no—
165#877—92no—
166#872—92no—
167#870—92no—
168C22×C3×D3—31no—
169C22×C3×D3C22×D311yescharacteristic
170C2×C32×C4C22×D312yes—
171C2×C32×C4C22×D311yescharacteristic
172C2×C3⋊Dic3—31no—
173C2×C3⋊Dic3—32no—
174C2×C3⋊Dic3—31no—
175C2×C3×Dic3—32no—
176C2×C3×Dic3—31no—
177C2×C3×Dic3C22×D312yes—
178C2×C3×Dic3C22×D311yescharacteristic
179C3×C4×D3—22no—
180C3×C4×D3—62no—
181C3×Dic6—62no—
182#625—62no—
183D3×Dic3—64no—
184C23×C9—31no—
185C22×D9C22×D311yescharacteristic
186C2×C4×C9—21no—
187C2×C4×C9—22no—
188C2×C4×C9—32no—
189C2×C4×C9—31no—
190C2×C4×C9—62no—
191C2×C4×C9C22×D312yes—
192C2×C4×C9C22×D311yescharacteristic
193#611—62no—
194C2×Dic9—31no—
195C2×Dic9—31no—
196C2×Dic9—32no—
197C2×Dic9—61no—
198C2×Dic9—61no—
199C2×Dic9—62no—
200C2×Dic9C22×D312yes—
201C2×Dic9C22×D311yescharacteristic
202C4×D9—22no—
203C4×D9—64no—
204Dic18—22no—
205#552—271noSylow
206C2×C3×C9C2×#4111yescharacteristic
207C2×C3×C9C2×#4111yescharacteristic
208C2×C3×C92−1+411yescharacteristic
209D3×C9—41no—
210C3×D9—22no—
211C2×C3×Q8—91no—
212C22×C3×C4—31no—
213C22×C3×C4—91no—
214C2×#67—91no—
215C2×#67—91no—
216C22×Dic3—92no—
217C2×C4×D3—31no—
218C2×C4×D3—92no—
219C2×C4×D3C2×D911yescharacteristic
220C2×Dic6—31no—
221C3×C4⋊C4—21no—
222C3×C4⋊C4—31no—
223C3×C4⋊C4—92no—
224C3×C4⋊C4C2×D911yescharacteristic
225C3×C4⋊C4D3211yescharacteristic
226C3×C22⋊C4—32no—
227C3×C22⋊C4—92no—
228C3×C42—32no—
229#217—91no—
230#217—92no—
231#217—91no—
232(C2×D3)⋊C4—32no—
233(C2×D3)⋊C4—91no—
234(C2×D3)⋊C4—92no—
235(C2×D3)⋊C4—91no—
236(C2×D3)⋊C4C2×D912yes—
237C4⋊Dic3—32no—
238C4⋊Dic3—92no—
239C4⋊Dic3—91no—
240C4⋊Dic3—92no—
241C4⋊Dic3—91no—
242C4⋊Dic3—92no—
243C4⋊Dic3—92no—
244C4⋊Dic3—181no—
245C4⋊Dic3—182no—
246C4⋊Dic3C2×D911yescharacteristic
247Dic3⋊C4—91no—
248Dic3⋊C4—92no—
249Dic3⋊C4—92no—
250Dic3⋊C4—92no—
251Dic3⋊C4—92no—
252Dic3⋊C4—92no—
253Dic3⋊C4—92no—
254Dic3⋊C4—91no—
255Dic3⋊C4—182no—
256C4×Dic3—92no—
257C4×Dic3—91no—
258C4×Dic3—91no—
259C4×Dic3—92no—
260C4×Dic3—181no—
261C4×Dic3C2×D912yes—
262C22×C32C23×D311yescharacteristic
263C2×C3×D3—21no—
264C2×C3×D3—21no—
265C2×C3×D3—21no—
266C2×C3×D3—32no—
267C2×C3×D3—61no—
268C2×C3×D3—61no—
269C32×C4—22no—
270C32×C4#23512yes—
271C3⋊Dic3—62no—
272C3⋊Dic3—61no—
273C3⋊Dic3—61no—
274C3×Dic3—21no—
275C3×Dic3—22no—
276C3×Dic3—62no—
277C3×Dic3—61no—
278C22×C9—21no—
279C22×C9—61no—
280C22×C9—61no—
281C22×C9—61no—
282C22×C9C23×D311yescharacteristic
283C2×D9—21no—
284C2×D9—21no—
285C2×D9#23912yes—
286C4×C9—22no—
287C4×C9—22no—
288C4×C9—42no—
289C4×C9—61no—
290C4×C9—62no—
291C4×C9#23712yes—
292Dic9—22no—
293Dic9—21no—
294Dic9—62no—
295Dic9—61no—
296Dic9—61no—
297Dic9—121no—
298Dic9—122no—
299Dic9—121no—
300#126—272no—
301#125—272no—
302#125—271no—
303#123—272no—
304C22⋊Q8—91no—
305C22⋊Q8—271no—
306C4×Q8—271no—
307C4×D4—271no—
308#117—31no—
309#117—272no—
310C2×C4⋊C4—271no—
311C3×C9#55211yescharacteristic, complemented, Sylow
312C23×C3—31no—
313C23×C3—91no—
314C22×D3—31no—
315C22×D3C22×D911yescharacteristic
316C3×Q8—182no—
317C2×C3×C4—21no—
318C2×C3×C4—22no—
319C2×C3×C4—32no—
320C2×C3×C4—31no—
321C2×C3×C4—62no—
322C2×C3×C4—92no—
323C2×C3×C4—91no—
324C2×C3×C4—182no—
325C2×C3×C4C22×D912yes—
326C2×C3×C4C22×D911yescharacteristic
327C2×C3×C4C2×D3212yes—
328C2×C3×C4C2×D3211yescharacteristic
329#67—182no—
330#67—182no—
331C2×Dic3—32no—
332C2×Dic3—31no—
333C2×Dic3—91no—
334C2×Dic3—91no—
335C2×Dic3—92no—
336C2×Dic3—91no—
337C2×Dic3—91no—
338C2×Dic3—92no—
339C2×Dic3—181no—
340C2×Dic3—181no—
341C2×Dic3—182no—
342C2×Dic3—184no—
343C2×Dic3C22×D912yes—
344C2×Dic3C22×D911yescharacteristic
345C4×D3—22no—
346C4×D3—62no—
347C4×D3—184no—
348Dic6—62no—
349C2×C32C2×#23511yescharacteristic
350C2×C32D3×#4111yescharacteristic
351C2×C32#101011yescharacteristic
352C3×D3—41no—
353C3×D3—62no—
354C2×C9—21no—
355C2×C9—21no—
356C2×C9—21no—
357C2×C9—121no—
358C2×C9C2×#23711yescharacteristic
359C2×C9C2×#23911yescharacteristic
360C2×C9#101011yescharacteristic
361D9—22no—
362C2×Q8—91no—
363C2×D4—271no—
364C22×C4—31no—
365C22×C4—91no—
366C22×C4—272no—
367C4⋊C4—31no—
368C4⋊C4—92no—
369C4⋊C4—272no—
370C4⋊C4—272no—
371C4⋊C4—271no—
372C4⋊C4—272no—
373C4⋊C4—271no—
374C4⋊C4—272no—
375C4⋊C4—272no—
376C4⋊C4D3×D911yescharacteristic
377C22⋊C4—32no—
378C22⋊C4—92no—
379C22⋊C4—271no—
380C22⋊C4—271no—
381C22⋊C4—272no—
382C42—32no—
383C42—272no—
384C42—271no—
385C22×C3—21no—
386C22×C3—61no—
387C22×C3—61no—
388C22×C3—61no—
389C22×C3—92no—
390C22×C3—181no—
391C22×C3—181no—
392C22×C3C23×D911yescharacteristic
393C22×C3C22×D3211yescharacteristic
394C2×D3—21no—
395C2×D3—21no—
396C2×D3—21no—
397C2×D3—32no—
398C2×D3—61no—
399C2×D3—61no—
400C3×C4—22no—
401C3×C4—22no—
402C3×C4—22no—
403C3×C4—42no—
404C3×C4—61no—
405C3×C4—62no—
406C3×C4—181no—
407C3×C4—182no—
408C3×C4#369712yes—
409C3×C4#379512yes—
410Dic3—21no—
411Dic3—22no—
412Dic3—62no—
413Dic3—61no—
414Dic3—181no—
415Dic3—182no—
416Dic3—181no—
417Dic3—182no—
418Dic3—181no—
419Dic3—181no—
420Dic3—362no—
421Dic3—361no—
422Dic3—361no—
423C32#552911yescharacteristic
424C9—21no—
425C9#564211yescharacteristic, complemented
426C23—31no—
427C23—91no—
428Q8—182no—
429D4—542no—
430C2×C4—31no—
431C2×C4—32no—
432C2×C4—62no—
433C2×C4—92no—
434C2×C4—91no—
435C2×C4—182no—
436C2×C4—271no—
437C2×C4—271no—
438C2×C4—272no—
439C2×C4—544no—
440C2×C4C2×D3×D912yes—
441C2×C4C2×D3×D911yescharacteristic
442C2×C3—21no—
443C2×C3—21no—
444C2×C3—21no—
445C2×C3—121no—
446C2×C3—182no—
447C2×C3C2×#369711yescharacteristic
448C2×C3#41×D911yescharacteristic
449C2×C3#6371611yescharacteristic
450C2×C3C2×#379511yescharacteristic
451C2×C3#6430911yescharacteristic
452C2×C3D3×#23911yescharacteristic
453D3—41no—
454D3—62no—
455C22—61no—
456C22—61no—
457C22—61no—
458C22—92no—
459C22—181no—
460C22—181no—
461C22C22×D3×D911yescharacteristic
462C4—22no—
463C4—62no—
464C4—61no—
465C4—181no—
466C4—182no—
467C4—542no—
468C4—541no—
469C4—541no—
470C4#8864112yes—
471C3—21no—
472C3#1058963111yescharacteristic, complemented
473C3#1059445611yescharacteristic
474C2—121no—
475C2—182no—
476C2C2×#8864111yescharacteristic
477C2#1172249411yescharacteristic
478C2D9×#23911yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC3×C9×C4⋊C4C22
socleC22×C32C23×D3
prime radicalC2×#6194C22
hypercenterC4⋊C4D3×D9
hypocenterC3×C9#552
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. #11721746C2semidirect
2. C9×#892C2semidirect
3. C3×#63459C2semidirect
6. D9×C4⋊Dic3C2semidirect
7. #11720974C2semidirect
8. #11720962C2semidirect
9. #11720960C2semidirect
10. #11720927C2semidirect
11. #11720910C2semidirect
51. #63459D3semidirect
155. #892D9semidirect
311. C3×C9#552semidirect
425. C9#5642semidirect
472. C3#10589631semidirect

Maximal subgroups

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

subgroupindex
#117217462
C9×#8922
C3×#634592
C4×#61942
#117209892
D9×C4⋊Dic32
#117209742
#117209622
#117209602
#117209272
#117209102
#105944563
#105896313