Finite Group Databasebeta

← #13148021

Subgroups — #13148021

Order 1728.

888 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88658C211yesmaximal, characteristic
2C22×#6189C211yesmaximal, characteristic, complemented
3#11721732C211yesmaximal, characteristic, complemented
4C3×#63448C211yesmaximal, characteristic, complemented
5C9×#877C211yesmaximal, characteristic, complemented
6#11721260C211yesmaximal, characteristic, complemented
7#11721255C211yesmaximal, characteristic, complemented
8#11720995C212yesmaximal, complemented
9#11720991C212yesmaximal, complemented
10#11720970C212yesmaximal, complemented
11#11720940C212yesmaximal, complemented
12#10594081—31nomaximal
13#10589628—31nomaximal
14C2×#6233C2211yescharacteristic
15C22×C3⋊Dic9C2211yescharacteristic
16C2×C3×#611C2211yescharacteristic
17C22×C9×Dic3C2211yescharacteristic
18C22×C3×Dic9C2211yescharacteristic
19C2×#6192C2211yescharacteristic
20C2×(C3⋊D9)⋊C4C2211yescharacteristic
21#88658—24no—
22C2×Dic3×D9C2211yescharacteristic
23C2×#6189—24no—
24C2×#6189C2211yescharacteristic
25C2×#6189C2212yes—
26C2×#6189C2212yes—
27C3×C9×C22⋊C4C2211yescharacteristic
28#88536C2212yes—
29C4⋊(C3⋊Dic9)C2212yes—
30C9×#217C2211yescharacteristic
31C3×(C2×D9)⋊C4C2212yes—
32C9×Dic3⋊C4C2212yes—
33C4×C9×Dic3C2212yes—
34C3×C4⋊Dic9C2212yes—
35#88445C2212yes—
36(C2×D9)⋊Dic3C2212yes—
37Dic3⋊Dic9C2212yes—
38Dic3⋊Dic9C2212yes—
39C22×#627—31no—
40C2×#3804—31no—
41#64095—31no—
42C3×#886—31no—
43C3×#877D311yescharacteristic
44#63966—31no—
45#63962—31no—
46#63918—32no—
47#63895—32no—
48#63893—32no—
49#63852—32no—
50C2×#3697—31no—
51C22×Dic18—31no—
52C9×#117—31no—
53#63459—32no—
54#63459—32no—
55#63448—31no—
56#63448—31no—
57#63448—31no—
58#63448—61no—
59#63448D311yescharacteristic, complemented
60#63437—32no—
61#63431—32no—
62C23×C3×C9C2311yescharacteristic
63C22×C3⋊D9C2311yescharacteristic
64C22×C3×D9C2311yescharacteristic
65C2×C3×C4×C9C2312yes—
66#6233—22no—
67C2×C3⋊Dic9—22no—
68C2×C3⋊Dic9C2312yes—
69C2×C3⋊Dic9C2311yescharacteristic
70C3×#611—22no—
71C2×C9×Dic3D414yes—
72C2×C9×Dic3C2312yes—
73C2×C9×Dic3C2311yescharacteristic
74C2×C9×Dic3C2311yescharacteristic
75C2×C3×Dic9—22no—
76C2×C3×Dic9C2312yes—
77C2×C3×Dic9C2311yescharacteristic
78#6192—22no—
79(C3⋊D9)⋊C4—22no—
80Dic3×D9—22no—
81#6189—24no—
82#6189—24no—
83#6189—22no—
84#5648—91no—
85#5526—91no—
86C2×#638—31no—
87C22×C3⋊Dic3—31no—
88C2×C3×#67—31no—
89C22×C3×Dic3—31no—
90C22×C3×Dic3C2×D311yescharacteristic
91C2×#627—31no—
92C2×#627—32no—
93C2×#627—32no—
94C2×#627—64no—
95C2×#626—31no—
96C2×#624—31no—
97#3804—64no—
98C2×D3×Dic3—31no—
99C32×C22⋊C4C2×D311yescharacteristic
100#3752—32no—
101C4⋊(C3⋊Dic3)—32no—
102C3×#217C2×D311yescharacteristic
103C3×(C2×D3)⋊C4—32no—
104C3×C4⋊Dic3—32no—
105C3×Dic3⋊C4C2×D312yes—
106C3×C4×Dic3C2×D312yes—
107Dic3⋊Dic3—32no—
108Dic3⋊Dic3—32no—
109#3722—32no—
110(C2×D3)⋊Dic3—32no—
111C22×C4×C9—31no—
112C2×#611—31no—
113C2×#611—61no—
114C2×#611C2×D311yescharacteristic
115C22×Dic9—31no—
116C22×Dic9—61no—
117C22×Dic9C2×D311yescharacteristic
118#3697—64no—
119C2×D36—31no—
120C2×C4×D9—31no—
121C2×C4×D9—31no—
122C2×Dic18—31no—
123C2×Dic18—32no—
124C2×Dic18—32no—
125C2×Dic18—64no—
126C9×C4⋊C4—32no—
127C9×C22⋊C4—21no—
128C9×C22⋊C4—31no—
129C9×C22⋊C4C2×D311yescharacteristic
130C42×C9—32no—
131(C2×D9)⋊C4—32no—
132(C2×D9)⋊C4—32no—
133(C2×D9)⋊C4—32no—
134(C2×D9)⋊C4—62no—
135(C2×D9)⋊C4C2×D312yes—
136C4⋊Dic9—32no—
137C4⋊Dic9—32no—
138C4⋊Dic9—32no—
139C4⋊Dic9—62no—
140C4⋊Dic9C2×D312yes—
141C22×C3×C9—22no—
142C22×C3×C9C2×D412yes—
143C22×C3×C9C2411yescharacteristic
144C2×C3⋊D9—21no—
145C2×C3⋊D9—22no—
146C2×C3×D9—21no—
147C2×C3×D9—22no—
148C3×C4×C9—22no—
149C3⋊Dic9—22no—
150C3⋊Dic9—21no—
151C9×Dic3—22no—
152C9×Dic3C2×D412yes—
153C9×Dic3C2×D412yes—
154C3×Dic9—22no—
155C3×Dic9—21no—
156C2×#237—91no—
157C2×#235—91no—
158C22×Dic6—91no—
159C22×Dic6—91no—
160C3×#123—91no—
161C3×#117—31no—
162#935—91no—
163#929—91no—
164#897—92no—
165#897—92no—
166#896—92no—
167#888—92no—
168#886—31no—
169#886—91no—
170#886—91no—
171#886—91no—
172#886—91no—
173#886—181no—
174#885—92no—
175#878—92no—
176#877D911yescharacteristic, complemented
177#875—92no—
178#869—92no—
179C23×C32C22×D311yescharacteristic
180C22×C3⋊D3—31no—
181C22×C3×D3—31no—
182C2×C32×C4C22×D312yes—
183#638—62no—
184C2×C3⋊Dic3—32no—
185C2×C3⋊Dic3—31no—
186C2×C3⋊Dic3—62no—
187C3×#67—62no—
188C2×C3×Dic3—32no—
189C2×C3×Dic3—31no—
190C2×C3×Dic3—62no—
191C2×C3×Dic3D1214yes—
192C2×C3×Dic3C22×D312yes—
193C2×C3×Dic3C22×D311yescharacteristic
194C2×C3×Dic3C22×D311yescharacteristic
195#627—62no—
196#627—64no—
197#627—64no—
198#626—62no—
199#624—62no—
200D3×Dic3—62no—
201C23×C9—21no—
202C23×C9C22×D311yescharacteristic
203C22×D9—31no—
204C22×D9—61no—
205C22×D9C22×D311yescharacteristic
206C2×C4×C9—22no—
207C2×C4×C9—34no—
208C2×C4×C9—31no—
209C2×C4×C9—31no—
210C2×C4×C9—32no—
211C2×C4×C9C22×D312yes—
212#611—22no—
213#611—62no—
214#611—122no—
215C2×Dic9—22no—
216C2×Dic9—31no—
217C2×Dic9—32no—
218C2×Dic9—61no—
219C2×Dic9—62no—
220C2×Dic9—62no—
221C2×Dic9—122no—
222C2×Dic9C22×D311yescharacteristic
223C2×Dic9C22×D312yes—
224D36—62no—
225C4×D9—62no—
226C4×D9—62no—
227Dic18—64no—
228Dic18—64no—
229Dic18—62no—
230#536—271noSylow
231C2×C3×C9—22no—
232C2×C3×C9C22×D411yescharacteristic
233C2×C3×C92−1+412yes—
234C3⋊D9—41no—
235C3×D9—41no—
236C2×C3×D4—91no—
237C22×C3×C4—31no—
238C22×C3×C4—91no—
239C2×#67—31no—
240C2×#67—91no—
241C2×#67—91no—
242C2×#67—91no—
243C2×#67—181no—
244C22×Dic3—31no—
245C22×Dic3—91no—
246C22×Dic3—91no—
247C22×Dic3—91no—
248C22×Dic3—181no—
249C22×Dic3C2×D911yescharacteristic
250#237—184no—
251#235—184no—
252C2×D12—91no—
253C2×C4×D3—91no—
254C2×C4×D3—91no—
255C2×C4×D3—91no—
256C2×Dic6—91no—
257C2×Dic6—92no—
258C2×Dic6—92no—
259C2×Dic6—92no—
260C2×Dic6—92no—
261C2×Dic6—91no—
262C2×Dic6—184no—
263C2×Dic6—184no—
264C3×C4⋊C4—32no—
265C3×C4⋊C4—92no—
266C3×C22⋊C4—21no—
267C3×C22⋊C4—31no—
268C3×C22⋊C4—92no—
269C3×C22⋊C4C2×D911yescharacteristic
270C3×C22⋊C4D3211yescharacteristic
271C3×C42—32no—
272#217—92no—
273#217C2×D911yescharacteristic
274(C2×D3)⋊C4—32no—
275(C2×D3)⋊C4—92no—
276(C2×D3)⋊C4—92no—
277(C2×D3)⋊C4—92no—
278(C2×D3)⋊C4—92no—
279(C2×D3)⋊C4—92no—
280(C2×D3)⋊C4—182no—
281C4⋊Dic3—32no—
282C4⋊Dic3—92no—
283C4⋊Dic3—92no—
284C4⋊Dic3—92no—
285C4⋊Dic3—92no—
286C4⋊Dic3—182no—
287Dic3⋊C4—92no—
288Dic3⋊C4—92no—
289Dic3⋊C4C2×D912yes—
290C4×Dic3C2×D912yes—
291C22×C32—22no—
292C22×C32C2×D1212yes—
293C22×C32C23×D311yescharacteristic
294C2×C3⋊D3—61no—
295C2×C3⋊D3—62no—
296C2×C3×D3—61no—
297C2×C3×D3—62no—
298C32×C4—22no—
299C3⋊Dic3—61no—
300C3⋊Dic3—62no—
301C3×Dic3—22no—
302C3×Dic3—61no—
303C3×Dic3—62no—
304C3×Dic3C2×D1212yes—
305C3×Dic3C2×D1212yes—
306C22×C9—22no—
307C22×C9—21no—
308C22×C9—22no—
309C22×C9—42no—
310C22×C9D3×D412yes—
311C22×C9C23×D311yescharacteristic
312C2×D9—21no—
313C2×D9—22no—
314C2×D9—61no—
315C2×D9—62no—
316C2×D9—121no—
317C2×D9—122no—
318C4×C9—22no—
319C4×C9—32no—
320C4×C9—32no—
321C4×C9—42no—
322C4×C9—62no—
323Dic9—22no—
324Dic9—21no—
325Dic9—62no—
326Dic9—61no—
327Dic9—122no—
328Dic9—121no—
329C2×#41—271no—
330C22×Q8—271no—
331C4⋊Q8—272no—
332#124—272no—
333#123—91no—
334#123—271no—
335#123—271no—
336#123—271no—
337C22⋊Q8—272no—
338C22⋊Q8—272no—
339#117—31no—
340C3×C9#53611yescharacteristic, complemented, Sylow
341C23×C3—21no—
342C23×C3—91no—
343C23×C3C22×D911yescharacteristic
344C23×C3C2×D3211yescharacteristic
345C22×D3—31no—
346C22×D3—91no—
347C22×D3—91no—
348C22×D3—181no—
349C3×D4—182no—
350C2×C3×C4—22no—
351C2×C3×C4—31no—
352C2×C3×C4—31no—
353C2×C3×C4—34no—
354C2×C3×C4—32no—
355C2×C3×C4—91no—
356C2×C3×C4—92no—
357C2×C3×C4—182no—
358C2×C3×C4C22×D912yes—
359C2×C3×C4C2×D3212yes—
360#67—62no—
361#67—182no—
362#67—182no—
363#67—182no—
364#67—362no—
365C2×Dic3—31no—
366C2×Dic3—32no—
367C2×Dic3—62no—
368C2×Dic3—91no—
369C2×Dic3—92no—
370C2×Dic3—91no—
371C2×Dic3—92no—
372C2×Dic3—181no—
373C2×Dic3—182no—
374C2×Dic3—182no—
375C2×Dic3—182no—
376C2×Dic3—182no—
377C2×Dic3—362no—
378C2×Dic3D3614yes—
379C2×Dic3C22×D912yes—
380C2×Dic3C22×D911yescharacteristic
381C2×Dic3C22×D911yescharacteristic
382D12—182no—
383C4×D3—182no—
384C4×D3—182no—
385C4×D3—182no—
386Dic6—184no—
387Dic6—184no—
388Dic6—182no—
389Dic6—184no—
390Dic6—184no—
391Dic6—182no—
392C2×C32—22no—
393C2×C32C22×D1211yescharacteristic
394C2×C32#101012yes—
395C3⋊D3—121no—
396C3×D3—121no—
397C2×C9—22no—
398C2×C9—21no—
399C2×C9—22no—
400C2×C9—42no—
401C2×C9C2×D3×D411yescharacteristic
402C2×C9#101012yes—
403D9—41no—
404D9—121no—
405D9—241no—
406#41—544no—
407C2×Q8—272no—
408C2×Q8—272no—
409C2×Q8—271no—
410C2×Q8—544no—
411C2×D4—91no—
412C2×D4—271no—
413C2×D4—271no—
414C22×C4—31no—
415C22×C4—91no—
416C22×C4—271no—
417C22×C4—271no—
418C22×C4—271no—
419C4⋊C4—32no—
420C4⋊C4—92no—
421C4⋊C4—272no—
422C4⋊C4—272no—
423C4⋊C4—272no—
424C22⋊C4—31no—
425C22⋊C4—92no—
426C22⋊C4—272no—
427C22⋊C4—272no—
428C22⋊C4—272no—
429C22⋊C4D3×D911yescharacteristic
430C42—32no—
431C22×C3—22no—
432C22×C3—22no—
433C22×C3—21no—
434C22×C3—22no—
435C22×C3—42no—
436C22×C3—181no—
437C22×C3—182no—
438C22×C3C2×D3612yes—
439C22×C3C23×D911yescharacteristic
440C22×C3D3×D1212yes—
441C22×C3C22×D3211yescharacteristic
442C2×D3—61no—
443C2×D3—62no—
444C2×D3—181no—
445C2×D3—182no—
446C2×D3—181no—
447C2×D3—182no—
448C2×D3—361no—
449C2×D3—362no—
450C3×C4—22no—
451C3×C4—22no—
452C3×C4—32no—
453C3×C4—32no—
454C3×C4—42no—
455C3×C4—62no—
456C3×C4—182no—
457C3×C4—181no—
458Dic3—22no—
459Dic3—62no—
460Dic3—61no—
461Dic3—182no—
462Dic3—181no—
463Dic3—182no—
464Dic3—181no—
465Dic3—362no—
466Dic3—361no—
467Dic3C2×D3612yes—
468Dic3C2×D3612yes—
469C32#552611yescharacteristic
470C9—21no—
471C9#564811yescharacteristic, complemented
472C23—91no—
473C23—271no—
474C23C2×D3×D911yescharacteristic
475Q8—544no—
476Q8—544no—
477Q8—542no—
478D4—182no—
479D4—542no—
480D4—542no—
481C2×C4—31no—
482C2×C4—31no—
483C2×C4—34no—
484C2×C4—32no—
485C2×C4—92no—
486C2×C4—91no—
487C2×C4—182no—
488C2×C4—272no—
489C2×C4—271no—
490C2×C4—542no—
491C2×C4—542no—
492C2×C4—542no—
493C2×C4C2×D3×D912yes—
494C2×C3—22no—
495C2×C3—22no—
496C2×C3—21no—
497C2×C3—22no—
498C2×C3—42no—
499C2×C3—361no—
500C2×C3C22×D3611yescharacteristic
501C2×C3#6371612yes—
502C2×C3C2×D3×D1211yescharacteristic
503C2×C3#6430912yes—
504D3—121no—
505D3—361no—
506D3—361no—
507D3—721no—
508C22—22no—
509C22—181no—
510C22—182no—
511C22—541no—
512C22—542no—
513C22D3×D3612yes—
514C22C22×D3×D911yescharacteristic
515C4—22no—
516C4—32no—
517C4—32no—
518C4—62no—
519C4—182no—
520C4—181no—
521C4—541no—
522C4—542no—
523C3—21no—
524C3#1058962811yescharacteristic, complemented
525C3#1059408111yescharacteristic
526C2—22no—
527C2—361no—
528C2—1081no—
529C2C2×D3×D3611yescharacteristic
530C2#1172249412yes—
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC3×C9×C22⋊C4C22
socleC22×C32C23×D3
prime radicalC2×#88658C2
hypercenterC22⋊C4D3×D9
hypocenterC3×C9#536
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C22×#6189C2semidirect
3. #11721732C2semidirect
4. C3×#63448C2semidirect
5. C9×#877C2semidirect
6. #11721260C2semidirect
7. #11721255C2semidirect
8. #11720995C2semidirect
9. #11720991C2semidirect
10. #11720970C2semidirect
11. #11720940C2semidirect
59. #63448D3semidirect
176. #877D9semidirect
340. C3×C9#536semidirect
471. C9#5648semidirect
524. C3#10589628semidirect

Maximal subgroups

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

subgroupindex
C2×#886582
C22×#61892
#117217322
C3×#634482
C9×#8772
#117212602
#117212552
#117209952
#117209912
#117209702
#117209402
#105940813
#105896283