Finite Group Databasebeta

← #13150222

Subgroups — #13150222

Order 1728.

966 conjugacy classes of subgroups.

The subgroup lattice has 536 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6226C211yesmaximal, characteristic, complemented
2C2×#88663C211yesmaximal, characteristic
3C9×#929C211yesmaximal, characteristic, complemented
4C3×#63491C211yesmaximal, characteristic, complemented
5#11721266C211yesmaximal, characteristic, complemented
6#11721260C211yesmaximal, characteristic, complemented
7#11721248C211yesmaximal, characteristic, complemented
8#11721042C212yesmaximal, complemented
9#11721024C212yesmaximal, complemented
10#11720996C212yesmaximal, complemented
11#11720905C212yesmaximal, complemented
12#10594921—31nomaximal
13#10589750—31nomaximal
14C22×C3×C4×C9C2211yescharacteristic
15C22×C3⋊Dic9C2211yescharacteristic
16C22×C3⋊Dic9C2211yescharacteristic
17C2×#6226—24no—
18C2×#6226C2212yes—
19C2×#6226C2212yes—
20C2×C9×#67C2211yescharacteristic
21C2×C3×#611C2211yescharacteristic
22C2×#6194C2211yescharacteristic
23#88663—24no—
24C2×D3×Dic9C2211yescharacteristic
25C2×Dic3×D9C2211yescharacteristic
26C2×#6189C2211yescharacteristic
27C9×#217C2211yescharacteristic
28C3×#3676C2211yescharacteristic
29C9×(C2×D3)⋊C4C2212yes—
30C3×(C2×D9)⋊C4C2212yes—
31C9×Dic3⋊C4C2212yes—
32C3×Dic9⋊C4C2212yes—
33(C2×D3)⋊Dic9C2212yes—
34(C2×D9)⋊Dic3C2212yes—
35Dic3⋊Dic9C2212yes—
36#88442C2212yes—
37Dic9⋊Dic3C2212yes—
38Dic3×Dic9C2212yes—
39C22×#634—31no—
40C2×#3805—31no—
41C3×#929—31no—
42C3×#929D311yescharacteristic
43#63962—31no—
44#63962—31no—
45#63957—31no—
46#63913—32no—
47#63896—32no—
48#63896—32no—
49#63840—32no—
50C2×#3699—31no—
51C22×Dic18—31no—
52C22×Dic18—61no—
53C9×#123—31no—
54#63497—31no—
55#63491D311yescharacteristic, complemented
56#63458—32no—
57#63450—32no—
58#63448—31no—
59#63447—32no—
60#63440—32no—
61#63439—31no—
62C23×C3×C9C2311yescharacteristic
63C22×D3×C9C2311yescharacteristic
64C22×C3×D9C2311yescharacteristic
65C2×C3×C4×C9—22no—
66C2×C3×C4×C9C2312yes—
67C2×C3⋊Dic9—22no—
68C2×C3⋊Dic9D414yes—
69C2×C3⋊Dic9C2312yes—
70C2×C3⋊Dic9C2311yescharacteristic
71C2×C3⋊Dic9C2311yescharacteristic
72#6226—24no—
73#6226—24no—
74C9×#67—22no—
75C3×#611—22no—
76C2×C9×Dic3C2311yescharacteristic
77C2×C9×Dic3C2312yes—
78C2×C3×Dic9C2311yescharacteristic
79C2×C3×Dic9C2312yes—
80#6194—22no—
81D3×Dic9—22no—
82Dic3×D9—22no—
83#6189—22no—
84#5648—91no—
85#5648—91no—
86C22×C32×C4C2×D311yescharacteristic
87C22×C3⋊Dic3—31no—
88C22×C3⋊Dic3—31no—
89C2×#634—32no—
90C2×#634—32no—
91C2×#634—64no—
92C2×C3×#67—31no—
93C2×C3×#67C2×D311yescharacteristic
94C2×#627—31no—
95C2×#625—31no—
96#3805—64no—
97C2×D3×Dic3—31no—
98C2×D3×Dic3—31no—
99C3×#217—31no—
100C3×#217C2×D311yescharacteristic
101C3×(C2×D3)⋊C4—32no—
102C3×(C2×D3)⋊C4C2×D312yes—
103C3×Dic3⋊C4—32no—
104C3×Dic3⋊C4C2×D312yes—
105#3724—32no—
106Dic3⋊Dic3—32no—
107Dic3⋊Dic3—32no—
108(C2×D3)⋊Dic3—32no—
109(C2×D3)⋊Dic3—32no—
110Dic32—32no—
111C2×D4×C9—31no—
112C22×C4×C9—21no—
113C22×C4×C9C2×D311yescharacteristic
114C2×#611—31no—
115C2×#611C2×D311yescharacteristic
116C22×Dic9—31no—
117C22×Dic9—31no—
118C22×Dic9—31no—
119C22×Dic9—61no—
120C22×Dic9—61no—
121#3699—64no—
122C2×C4×D9—31no—
123C2×Dic18—32no—
124C2×Dic18—32no—
125C2×Dic18—31no—
126C2×Dic18—64no—
127C2×Dic18—62no—
128C2×Dic18—62no—
129C2×Dic18—68no—
130C9×C4⋊C4—32no—
131C9×C22⋊C4—31no—
132C9×C22⋊C4—32no—
133#3676—32no—
134#3676C2×D311yescharacteristic
135(C2×D9)⋊C4—32no—
136(C2×D9)⋊C4C2×D312yes—
137C4⋊Dic9—32no—
138Dic9⋊C4—32no—
139Dic9⋊C4—32no—
140Dic9⋊C4C2×D312yes—
141C4×Dic9—32no—
142C22×C3×C9—22no—
143C22×C3×C9C2×D412yes—
144C22×C3×C9C2411yescharacteristic
145C2×D3×C9—21no—
146C2×D3×C9—22no—
147C2×C3×D9—21no—
148C2×C3×D9—22no—
149C3×C4×C9—22no—
150C3⋊Dic9—22no—
151C3⋊Dic9C2×D412yes—
152C3⋊Dic9C2×D412yes—
153C9×Dic3—21no—
154C9×Dic3—22no—
155C3×Dic9—21no—
156C3×Dic9—22no—
157C2×#237—91no—
158C2×#237—91no—
159C22×Dic6—91no—
160C22×Dic6—91no—
161C22×Dic6—181no—
162C3×#123—31no—
163C3×#123—91no—
164#935—91no—
165#935—91no—
166#929—31no—
167#929D911yescharacteristic, complemented
168#896—92no—
169#896—92no—
170#888—92no—
171#888—92no—
172#886—91no—
173#886—91no—
174#885—92no—
175#885—92no—
176#878—92no—
177#878—92no—
178#877—91no—
179#877—91no—
180C23×C32C22×D311yescharacteristic
181C22×C3×D3—31no—
182C22×C3×D3C22×D311yescharacteristic
183C2×C32×C4—22no—
184C2×C32×C4C22×D312yes—
185C2×C3⋊Dic3—34no—
186C2×C3⋊Dic3—31no—
187C2×C3⋊Dic3—31no—
188C2×C3⋊Dic3—32no—
189C2×C3⋊Dic3—62no—
190#634—64no—
191#634—64no—
192C3×#67—22no—
193C3×#67—62no—
194C2×C3×Dic3—32no—
195C2×C3×Dic3—31no—
196C2×C3×Dic3C22×D312yes—
197C2×C3×Dic3C22×D311yescharacteristic
198#627—62no—
199#625—62no—
200D3×Dic3—62no—
201D3×Dic3—62no—
202C23×C9—21no—
203C23×C9—31no—
204C23×C9C22×D311yescharacteristic
205C22×D9C22×D311yescharacteristic
206D4×C9—62no—
207C2×C4×C9—24no—
208C2×C4×C9—22no—
209C2×C4×C9—22no—
210C2×C4×C9—32no—
211C2×C4×C9—31no—
212C2×C4×C9C22×D312yes—
213#611—22no—
214#611—62no—
215C2×Dic9—34no—
216C2×Dic9—31no—
217C2×Dic9—31no—
218C2×Dic9—32no—
219C2×Dic9—64no—
220C2×Dic9—61no—
221C2×Dic9—61no—
222C2×Dic9—62no—
223C2×Dic9—62no—
224C2×Dic9—64no—
225C2×Dic9—62no—
226C2×Dic9C22×D312yes—
227C2×Dic9C22×D311yescharacteristic
228C4×D9—62no—
229Dic18—62no—
230Dic18—64no—
231Dic18—64no—
232Dic18—68no—
233Dic18—68no—
234#536—271noSylow
235C2×C3×C9—22no—
236C2×C3×C9C22×D411yescharacteristic
237C2×C3×C92−1+412yes—
238D3×C9—41no—
239C3×D9—41no—
240C2×C3×D4—31no—
241C2×C3×D4—91no—
242C22×C3×C4—21no—
243C22×C3×C4C2×D911yescharacteristic
244C22×C3×C4D3211yescharacteristic
245C2×#67—31no—
246C2×#67—91no—
247C2×#67—91no—
248C2×#67C2×D911yescharacteristic
249C22×Dic3—91no—
250C22×Dic3—91no—
251C22×Dic3—91no—
252C22×Dic3—91no—
253C22×Dic3—91no—
254C22×Dic3—91no—
255C22×Dic3—181no—
256C22×Dic3—181no—
257#237—184no—
258#237—184no—
259C2×C4×D3—91no—
260C2×C4×D3—91no—
261C2×Dic6—91no—
262C2×Dic6—92no—
263C2×Dic6—92no—
264C2×Dic6—92no—
265C2×Dic6—92no—
266C2×Dic6—91no—
267C2×Dic6—182no—
268C2×Dic6—182no—
269C2×Dic6—184no—
270C2×Dic6—184no—
271C2×Dic6—188no—
272C3×C4⋊C4—32no—
273C3×C4⋊C4—92no—
274C3×C22⋊C4—31no—
275C3×C22⋊C4—32no—
276C3×C22⋊C4—91no—
277C3×C22⋊C4—92no—
278#217—31no—
279#217—92no—
280#217—92no—
281#217C2×D911yescharacteristic
282(C2×D3)⋊C4—32no—
283(C2×D3)⋊C4—92no—
284(C2×D3)⋊C4—92no—
285(C2×D3)⋊C4C2×D912yes—
286C4⋊Dic3—92no—
287C4⋊Dic3—92no—
288Dic3⋊C4—32no—
289Dic3⋊C4—92no—
290Dic3⋊C4—92no—
291Dic3⋊C4—92no—
292Dic3⋊C4—92no—
293Dic3⋊C4C2×D912yes—
294C4×Dic3—92no—
295C4×Dic3—92no—
296C22×C32—22no—
297C22×C32D3×D412yes—
298C22×C32C23×D311yescharacteristic
299C2×C3×D3—21no—
300C2×C3×D3—22no—
301C2×C3×D3—61no—
302C2×C3×D3—62no—
303C32×C4—22no—
304C3⋊Dic3—32no—
305C3⋊Dic3—32no—
306C3⋊Dic3—62no—
307C3×Dic3—21no—
308C3×Dic3—22no—
309C3×Dic3—61no—
310C3×Dic3—62no—
311C22×C9—24no—
312C22×C9—22no—
313C22×C9—21no—
314C22×C9—22no—
315C22×C9—61no—
316C22×C9—62no—
317C22×C9D3×D412yes—
318C22×C9C23×D311yescharacteristic
319C2×D9—21no—
320C2×D9—22no—
321C4×C9—24no—
322C4×C9—22no—
323C4×C9—61no—
324C4×C9—62no—
325Dic9—21no—
326Dic9—22no—
327Dic9—32no—
328Dic9—32no—
329Dic9—62no—
330Dic9—62no—
331Dic9—64no—
332Dic9—62no—
333C2×#41—271no—
334C22×Q8—271no—
335C4⋊Q8—272no—
336#124—272no—
337#123—31no—
338#123—91no—
339#123—271no—
340#123—271no—
341C22⋊Q8—272no—
342C22⋊Q8—272no—
343#117—271no—
344C3×C9#53611yescharacteristic, complemented, Sylow
345C23×C3—21no—
346C23×C3—31no—
347C23×C3—91no—
348C23×C3C22×D911yescharacteristic
349C23×C3C2×D3211yescharacteristic
350C22×D3—31no—
351C22×D3C22×D911yescharacteristic
352C3×D4—62no—
353C3×D4—182no—
354C2×C3×C4—24no—
355C2×C3×C4—22no—
356C2×C3×C4—22no—
357C2×C3×C4—22no—
358C2×C3×C4—32no—
359C2×C3×C4—31no—
360C2×C3×C4—92no—
361C2×C3×C4—91no—
362C2×C3×C4C22×D912yes—
363C2×C3×C4C2×D3212yes—
364#67—22no—
365#67—62no—
366#67—182no—
367#67—182no—
368C2×Dic3—32no—
369C2×Dic3—31no—
370C2×Dic3—91no—
371C2×Dic3—91no—
372C2×Dic3—94no—
373C2×Dic3—91no—
374C2×Dic3—91no—
375C2×Dic3—94no—
376C2×Dic3—92no—
377C2×Dic3—92no—
378C2×Dic3—181no—
379C2×Dic3—181no—
380C2×Dic3—184no—
381C2×Dic3—182no—
382C2×Dic3—182no—
383C2×Dic3—182no—
384C2×Dic3—184no—
385C2×Dic3—182no—
386C2×Dic3—182no—
387C2×Dic3C22×D912yes—
388C2×Dic3C22×D911yescharacteristic
389C4×D3—182no—
390C4×D3—182no—
391Dic6—182no—
392Dic6—182no—
393Dic6—184no—
394Dic6—184no—
395Dic6—184no—
396Dic6—184no—
397Dic6—188no—
398Dic6—188no—
399C2×C32—22no—
400C2×C32C2×D3×D411yescharacteristic
401C2×C32#101012yes—
402C3×D3—41no—
403C3×D3—121no—
404C2×C9—24no—
405C2×C9—22no—
406C2×C9—22no—
407C2×C9—21no—
408C2×C9—121no—
409C2×C9C2×D3×D411yescharacteristic
410C2×C9#101012yes—
411D9—41no—
412#41—544no—
413C2×Q8—271no—
414C2×Q8—272no—
415C2×Q8—272no—
416C2×Q8—544no—
417C2×D4—31no—
418C2×D4—91no—
419C2×D4—271no—
420C22×C4—271no—
421C22×C4—271no—
422C22×C4—271no—
423C22×C4—271no—
424C22×C4D3×D911yescharacteristic
425C4⋊C4—32no—
426C4⋊C4—92no—
427C4⋊C4—272no—
428C4⋊C4—272no—
429C4⋊C4—272no—
430C22⋊C4—31no—
431C22⋊C4—32no—
432C22⋊C4—91no—
433C22⋊C4—92no—
434C22⋊C4—272no—
435C22⋊C4—272no—
436C42—272no—
437C22×C3—24no—
438C22×C3—22no—
439C22×C3—22no—
440C22×C3—21no—
441C22×C3—22no—
442C22×C3—62no—
443C22×C3—61no—
444C22×C3—181no—
445C22×C3—182no—
446C22×C3D4×D912yes—
447C22×C3C23×D911yescharacteristic
448C22×C3#381112yes—
449C22×C3C22×D3211yescharacteristic
450C2×D3—21no—
451C2×D3—22no—
452C2×D3—62no—
453C2×D3—61no—
454C3×C4—24no—
455C3×C4—22no—
456C3×C4—22no—
457C3×C4—62no—
458C3×C4—61no—
459C3×C4—181no—
460C3×C4—182no—
461Dic3—21no—
462Dic3—22no—
463Dic3—62no—
464Dic3—61no—
465Dic3—92no—
466Dic3—92no—
467Dic3—92no—
468Dic3—92no—
469Dic3—182no—
470Dic3—182no—
471Dic3—184no—
472Dic3—182no—
473Dic3—182no—
474C32#564811yescharacteristic
475C9—21no—
476C9#564811yescharacteristic, complemented
477C23—31no—
478C23—91no—
479C23C2×D3×D911yescharacteristic
480Q8—544no—
481Q8—544no—
482Q8—542no—
483D4—62no—
484D4—182no—
485D4—542no—
486C2×C4—22no—
487C2×C4—32no—
488C2×C4—31no—
489C2×C4—92no—
490C2×C4—91no—
491C2×C4—271no—
492C2×C4—271no—
493C2×C4—274no—
494C2×C4—272no—
495C2×C4—542no—
496C2×C4—542no—
497C2×C4—542no—
498C2×C4C2×D3×D912yes—
499C2×C3—24no—
500C2×C3—22no—
501C2×C3—22no—
502C2×C3—21no—
503C2×C3—22no—
504C2×C3—121no—
505C2×C3—361no—
506C2×C3C2×D4×D911yescharacteristic
507C2×C3#6371612yes—
508C2×C3#6429812yes—
509C2×C3C2×#381111yescharacteristic
510D3—41no—
511D3—121no—
512C22—22no—
513C22—61no—
514C22—62no—
515C22—181no—
516C22—182no—
517C22#8866812yes—
518C22C22×D3×D911yescharacteristic
519C4—22no—
520C4—61no—
521C4—62no—
522C4—181no—
523C4—182no—
524C4—272no—
525C4—272no—
526C4—542no—
527C3—21no—
528C3#1058975011yescharacteristic, complemented
529C3#1059492111yescharacteristic
530C2—22no—
531C2—121no—
532C2—361no—
533C2#1172246512yes—
534C2C2×#8866811yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC22×C3×C4×C9C22
socleC22×C32C23×D3
prime radicalC2×#88663C2
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
1. C22×#6226C2semidirect
3. C9×#929C2semidirect
4. C3×#63491C2semidirect
5. #11721266C2semidirect
6. #11721260C2semidirect
7. #11721248C2semidirect
8. #11721042C2semidirect
9. #11721024C2semidirect
10. #11720996C2semidirect
11. #11720905C2semidirect
55. #63491D3semidirect
167. #929D9semidirect
344. C3×C9#536semidirect
476. C9#5648semidirect
528. C3#10589750semidirect

Maximal subgroups

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

subgroupindex
C22×#62262
C2×#886632
C9×#9292
C3×#634912
#117212662
#117212602
#117212482
#117210422
#117210242
#117209962
#117209052
#105949213
#105897503