Finite Group Databasebeta

← #13150418

Subgroups — #13150418

Order 1728.

802 conjugacy classes of subgroups.

The subgroup lattice has 546 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#6233C211yesmaximal, characteristic, complemented
2C9×#930C211yesmaximal, characteristic, complemented
3C3×#63488C211yesmaximal, characteristic, complemented
4#11721282C212yesmaximal, complemented
5#11721263C212yesmaximal, complemented
6#11721045C211yesmaximal, characteristic, complemented
7#11721025C211yesmaximal, characteristic, complemented
8#11721010C212yesmaximal, complemented
9#11720971C211yesmaximal, characteristic, complemented
10#11720956C212yesmaximal, complemented
11#11720922C211yesmaximal, characteristic, complemented
12#10594961—31nomaximal
13#10589718—31nomaximal
14C22×C3×C4×C9C2211yescharacteristic
15C2×#6233C2211yescharacteristic
16C2×C4×C3⋊D9C2211yescharacteristic
17C2×C9×#67C2212yes—
18C2×C9×D12C2211yescharacteristic
19C2×#6195C2212yes—
20C2×D3×Dic9C2212yes—
21C2×#6189C2212yes—
22#88551C2211yescharacteristic
23#88536C2211yescharacteristic
24#88534C2211yescharacteristic
25C4×C3⋊Dic9C2211yescharacteristic
26C3×#3676C2212yes—
27C9×(C2×D3)⋊C4C2212yes—
28C9×C4⋊Dic3C2211yescharacteristic
29C3×C4⋊Dic9C2211yescharacteristic
30C3×C4⋊Dic9C2211yescharacteristic
31C3×C4×Dic9C2212yes—
32(C2×D3)⋊Dic9C2212yes—
33(C2×D3)⋊Dic9C2212yes—
34#88445C2212yes—
35#88442C2212yes—
36Dic3×Dic9C2212yes—
37C4×#638—31no—
38C3×#930D311yescharacteristic
39C3×#926—31no—
40#63971—32no—
41#63959—32no—
42#63919—31no—
43#63916—31no—
44#63898—31no—
45#63877—32no—
46#63852—32no—
47#63849—31no—
48C9×#121—31no—
49#63501—32no—
50#63500—31no—
51#63498—32no—
52D4×Dic9—31no—
53C4×#611—31no—
54C4×#611—61no—
55#63488D311yescharacteristic, complemented
56#63454—31no—
57#63450—31no—
58#63447—32no—
59#63439—32no—
60C23×C3×C9C2311yescharacteristic
61C22×C3⋊D9C2311yescharacteristic
62C22×D3×C9C2312yes—
63C2×C3×C4×C9—22no—
64C2×C3×C4×C9C2311yescharacteristic
65C2×C3×C4×C9C2311yescharacteristic
66#6233—22no—
67C2×C3⋊Dic9C2311yescharacteristic
68C2×C3⋊Dic9C2311yescharacteristic
69C2×C3⋊Dic9C2311yescharacteristic
70C4×C3⋊D9—22no—
71C9×#67—42no—
72C2×C9×Dic3C2312yes—
73C2×C3×Dic9C2312yes—
74C2×C3×Dic9C2312yes—
75C9×D12—22no—
76#6195—42no—
77D3×Dic9—24no—
78#6189—42no—
79#5616—91no—
80#5532—91no—
81C22×C32×C4C2×D311yescharacteristic
82C2×#638—31no—
83C2×C4×C3⋊D3—31no—
84C2×C3×#67C2×D312yes—
85C2×C3×D12C2×D311yescharacteristic
86C2×#627—32no—
87C2×#626—32no—
88C2×D3×Dic3—32no—
89#3757—31no—
90#3752—31no—
91#3750—31no—
92C4×C3⋊Dic3—31no—
93C3×#217—32no—
94C3×(C2×D3)⋊C4C2×D312yes—
95C3×C4⋊Dic3—31no—
96C3×C4⋊Dic3—31no—
97C3×C4⋊Dic3C2×D311yescharacteristic
98C3×C4×Dic3—32no—
99#3724—32no—
100#3722—32no—
101(C2×D3)⋊Dic3—32no—
102(C2×D3)⋊Dic3—32no—
103Dic32—32no—
104C2×D4×C9—32no—
105C2×D4×C9—31no—
106C22×C4×C9—21no—
107C22×C4×C9C2×D311yescharacteristic
108C2×#611—31no—
109C2×#611—32no—
110C2×#611—61no—
111C22×Dic9—32no—
112C2×C4×D9—31no—
113C2×C4×D9—61no—
114C2×Dic18—32no—
115C9×C4⋊C4—31no—
116C9×C22⋊C4—32no—
117#3676—31no—
118#3676—32no—
119#3676—32no—
120#3676—61no—
121#3676C2×D312yes—
122(C2×D9)⋊C4—31no—
123(C2×D9)⋊C4—32no—
124(C2×D9)⋊C4—61no—
125C4⋊Dic9C2×D311yescharacteristic
126C4⋊Dic9C2×D311yescharacteristic
127Dic9⋊C4—31no—
128Dic9⋊C4—32no—
129Dic9⋊C4—61no—
130C4×Dic9—31no—
131C4×Dic9—32no—
132C4×Dic9—61no—
133C4×Dic9C2×D312yes—
134C22×C3×C9—21no—
135C22×C3×C9—21no—
136C22×C3×C9—21no—
137C22×C3×C9C2411yescharacteristic
138C2×C3⋊D9—21no—
139C2×C3⋊D9—21no—
140C2×C3⋊D9—21no—
141C2×D3×C9—22no—
142C2×D3×C9—22no—
143C2×D3×C9—22no—
144C3×C4×C9—21no—
145C3×C4×C9#4112yes—
146C3⋊Dic9—21no—
147C3⋊Dic9—21no—
148C3⋊Dic9#4112yes—
149C9×Dic3—22no—
150C3×Dic9—22no—
151C3×Dic9—22no—
152C3×#121—31no—
153C3×#117—91no—
154#939—92no—
155#938—91no—
156#936—92no—
157D4×Dic3—91no—
158#930D911yescharacteristic, complemented
159C4×#67—91no—
160C4×#67—91no—
161C4×#67—181no—
162#926—31no—
163#895—91no—
164#893—91no—
165#892—91no—
166#892—91no—
167#888—91no—
168#888—91no—
169#885—92no—
170#885—92no—
171#884—92no—
172#877—92no—
173#875—92no—
174#872—92no—
175C23×C32C22×D311yescharacteristic
176C22×C3⋊D3—31no—
177C22×C3×D3C22×D312yes—
178C2×C32×C4—22no—
179C2×C32×C4C22×D311yescharacteristic
180C2×C32×C4C22×D311yescharacteristic
181#638—62no—
182C2×C3⋊Dic3—31no—
183C2×C3⋊Dic3—31no—
184C2×C3⋊Dic3—31no—
185C4×C3⋊D3—62no—
186C3×#67—42no—
187C2×C3×Dic3—32no—
188C2×C3×Dic3—32no—
189C2×C3×Dic3C22×D312yes—
190C3×D12—22no—
191#627—122no—
192#626—122no—
193D3×Dic3—64no—
194C23×C9—21no—
195C23×C9—32no—
196C23×C9C22×D311yescharacteristic
197C22×D9—31no—
198C22×D9—61no—
199D4×C9—62no—
200D4×C9—122no—
201C2×C4×C9—22no—
202C2×C4×C9—21no—
203C2×C4×C9—21no—
204C2×C4×C9—32no—
205C2×C4×C9—42no—
206C2×C4×C9C22×D311yescharacteristic
207C2×C4×C9C22×D311yescharacteristic
208#611—62no—
209#611—64no—
210#611—122no—
211C2×Dic9—31no—
212C2×Dic9—31no—
213C2×Dic9—31no—
214C2×Dic9—61no—
215C2×Dic9—61no—
216C2×Dic9—61no—
217C2×Dic9—64no—
218C2×Dic9C22×D312yes—
219C2×Dic9C22×D312yes—
220C4×D9—62no—
221C4×D9—122no—
222Dic18—122no—
223#555—271noSylow
224C2×C3×C9—41no—
225C2×C3×C9C2×#4111yescharacteristic
226C2×C3×C9C2×#4111yescharacteristic
227C2×C3×C9H(5,𝔽2)11yescharacteristic
228C3⋊D9—41no—
229D3×C9—42no—
230C2×C3×D4—32no—
231C2×C3×D4—31no—
232C22×C3×C4—21no—
233C22×C3×C4C2×D911yescharacteristic
234C22×C3×C4D3211yescharacteristic
235C2×#67—92no—
236C2×#67—91no—
237C2×#67—91no—
238C2×#67—181no—
239C2×#67C2×D912yes—
240C22×Dic3—92no—
241C2×D12—92no—
242C2×D12C2×D911yescharacteristic
243C2×C4×D3—92no—
244C2×C4×D3—91no—
245C2×C4×D3—91no—
246C2×C4×D3—181no—
247C2×Dic6—92no—
248C2×Dic6—92no—
249C3×C4⋊C4—31no—
250C3×C4⋊C4—91no—
251C3×C4⋊C4—91no—
252C3×C22⋊C4—32no—
253C3×C22⋊C4—92no—
254C3×C42—92no—
255#217—32no—
256#217—92no—
257#217—92no—
258#217—91no—
259#217—91no—
260#217—181no—
261(C2×D3)⋊C4—92no—
262(C2×D3)⋊C4—92no—
263(C2×D3)⋊C4—92no—
264(C2×D3)⋊C4—91no—
265(C2×D3)⋊C4—91no—
266(C2×D3)⋊C4—92no—
267(C2×D3)⋊C4—181no—
268(C2×D3)⋊C4C2×D912yes—
269C4⋊Dic3—31no—
270C4⋊Dic3—31no—
271C4⋊Dic3C2×D911yescharacteristic
272Dic3⋊C4—92no—
273Dic3⋊C4—91no—
274Dic3⋊C4—91no—
275Dic3⋊C4—92no—
276Dic3⋊C4—181no—
277C4×Dic3—32no—
278C4×Dic3—92no—
279C4×Dic3—91no—
280C4×Dic3—91no—
281C4×Dic3—92no—
282C4×Dic3—181no—
283C22×C32—21no—
284C22×C32—21no—
285C22×C32—21no—
286C22×C32C23×D311yescharacteristic
287C2×C3⋊D3—61no—
288C2×C3⋊D3—61no—
289C2×C3⋊D3—61no—
290C2×C3×D3—22no—
291C2×C3×D3—22no—
292C2×C3×D3—22no—
293C32×C4—21no—
294C32×C4#23712yes—
295C3⋊Dic3—32no—
296C3⋊Dic3—61no—
297C3⋊Dic3—61no—
298C3×Dic3—22no—
299C3×Dic3—62no—
300C3×Dic3—62no—
301C22×C9—22no—
302C22×C9—21no—
303C22×C9—21no—
304C22×C9—21no—
305C22×C9—21no—
306C22×C9—41no—
307C22×C9—41no—
308C22×C9—62no—
309C22×C9—62no—
310C22×C9—62no—
311C22×C9C23×D311yescharacteristic
312C2×D9—62no—
313C2×D9—61no—
314C2×D9—61no—
315C2×D9—61no—
316C2×D9—121no—
317C2×D9—121no—
318C4×C9—21no—
319C4×C9—22no—
320C4×C9—41no—
321C4×C9—62no—
322C4×C9#23512yes—
323Dic9—22no—
324Dic9—22no—
325Dic9—32no—
326Dic9—62no—
327Dic9—61no—
328Dic9—61no—
329Dic9—121no—
330Dic9—121no—
331C4⋊Q8—271no—
332#124—272no—
333#124—272no—
334#121—31no—
335#121—272no—
336#121—271no—
337C4×D4—271no—
338C4×D4—271no—
339#117—91no—
340#117—272no—
341#117—271no—
342C3×C9#55511yescharacteristic, complemented, Sylow
343C23×C3—21no—
344C23×C3—32no—
345C23×C3C22×D911yescharacteristic
346C23×C3C2×D3211yescharacteristic
347C22×D3—91no—
348C22×D3—91no—
349C22×D3—181no—
350C22×D3C22×D912yes—
351C3×D4—62no—
352C3×D4—122no—
353C2×C3×C4—22no—
354C2×C3×C4—22no—
355C2×C3×C4—21no—
356C2×C3×C4—21no—
357C2×C3×C4—32no—
358C2×C3×C4—42no—
359C2×C3×C4—92no—
360C2×C3×C4—92no—
361C2×C3×C4C22×D911yescharacteristic
362C2×C3×C4C22×D911yescharacteristic
363C2×C3×C4C2×D3211yescharacteristic
364C2×C3×C4C2×D3211yescharacteristic
365#67—42no—
366#67—182no—
367#67—182no—
368#67—184no—
369#67—362no—
370C2×Dic3—32no—
371C2×Dic3—32no—
372C2×Dic3—91no—
373C2×Dic3—91no—
374C2×Dic3—91no—
375C2×Dic3—91no—
376C2×Dic3—91no—
377C2×Dic3—91no—
378C2×Dic3—181no—
379C2×Dic3—181no—
380C2×Dic3—181no—
381C2×Dic3—184no—
382C2×Dic3C22×D912yes—
383D12—22no—
384D12—362no—
385C4×D3—184no—
386C4×D3—182no—
387C4×D3—182no—
388C4×D3—362no—
389Dic6—362no—
390Dic6—362no—
391C2×C32—41no—
392C2×C32C2×#23711yescharacteristic
393C2×C32#100411yescharacteristic
394C2×C32D3×#4111yescharacteristic
395C3⋊D3—121no—
396C3×D3—42no—
397C2×C9—21no—
398C2×C9—21no—
399C2×C9—21no—
400C2×C9—42no—
401C2×C9—41no—
402C2×C9—122no—
403C2×C9C2×#23511yescharacteristic
404C2×C9D3×#4111yescharacteristic
405C2×C9#100911yescharacteristic
406D9—122no—
407D9—121no—
408C2×Q8—272no—
409C2×D4—32no—
410C2×D4—31no—
411C2×D4—272no—
412C2×D4—271no—
413C22×C4—272no—
414C22×C4—271no—
415C22×C4D3×D911yescharacteristic
416C4⋊C4—31no—
417C4⋊C4—91no—
418C4⋊C4—91no—
419C4⋊C4—272no—
420C4⋊C4—271no—
421C22⋊C4—32no—
422C22⋊C4—92no—
423C22⋊C4—272no—
424C22⋊C4—272no—
425C22⋊C4—272no—
426C22⋊C4—271no—
427C22⋊C4—271no—
428C42—92no—
429C42—272no—
430C42—271no—
431C22×C3—22no—
432C22×C3—21no—
433C22×C3—21no—
434C22×C3—21no—
435C22×C3—21no—
436C22×C3—21no—
437C22×C3—21no—
438C22×C3—21no—
439C22×C3—41no—
440C22×C3—41no—
441C22×C3—62no—
442C22×C3—62no—
443C22×C3—62no—
444C22×C3C23×D911yescharacteristic
445C22×C3C22×D3211yescharacteristic
446C2×D3—22no—
447C2×D3—22no—
448C2×D3—22no—
449C2×D3—182no—
450C2×D3—181no—
451C2×D3—181no—
452C2×D3—181no—
453C2×D3—181no—
454C2×D3—181no—
455C2×D3—181no—
456C2×D3—361no—
457C2×D3—361no—
458C3×C4—21no—
459C3×C4—21no—
460C3×C4—22no—
461C3×C4—41no—
462C3×C4—62no—
463C3×C4—182no—
464C3×C4—182no—
465C3×C4#369912yes—
466C3×C4#380412yes—
467Dic3—22no—
468Dic3—62no—
469Dic3—62no—
470Dic3—92no—
471Dic3—92no—
472Dic3—182no—
473Dic3—181no—
474Dic3—181no—
475Dic3—181no—
476Dic3—181no—
477Dic3—361no—
478Dic3—361no—
479C32#561611yescharacteristic
480C9—21no—
481C9#553211yescharacteristic, complemented
482C23—32no—
483C23—271no—
484C23C2×D3×D911yescharacteristic
485Q8—1082no—
486D4—62no—
487D4—122no—
488D4—542no—
489D4—1082no—
490C2×C4—22no—
491C2×C4—32no—
492C2×C4—92no—
493C2×C4—92no—
494C2×C4—271no—
495C2×C4—271no—
496C2×C4—271no—
497C2×C4—544no—
498C2×C4—542no—
499C2×C4C2×D3×D911yescharacteristic
500C2×C4C2×D3×D911yescharacteristic
501C2×C3—21no—
502C2×C3—21no—
503C2×C3—21no—
504C2×C3—42no—
505C2×C3—41no—
506C2×C3—41no—
507C2×C3—122no—
508C2×C3C2×#369911yescharacteristic
509C2×C3#6371011yescharacteristic
510C2×C3#41×D911yescharacteristic
511C2×C3#6430611yescharacteristic
512C2×C3#6430711yescharacteristic
513C2×C3C2×#380411yescharacteristic
514D3—42no—
515D3—362no—
516D3—361no—
517D3—361no—
518C22—21no—
519C22—21no—
520C22—21no—
521C22—62no—
522C22—62no—
523C22—62no—
524C22—541no—
525C22—541no—
526C22—541no—
527C22C22×D3×D911yescharacteristic
528C4—21no—
529C4—62no—
530C4—182no—
531C4—182no—
532C4—272no—
533C4—541no—
534C4—541no—
535C4#8866212yes—
536C3—21no—
537C3#1058971811yescharacteristic, complemented
538C3#1059496111yescharacteristic
539C2—41no—
540C2—122no—
541C2—1081no—
542C2#1172247511yescharacteristic
543C2#1172247611yescharacteristic
544C2C2×#8866211yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC22×C3×C4×C9C22
socleC22×C32C23×D3
hypercenterC22×C4D3×D9
hypocenterC3×C9#555
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. C4×#6233C2semidirect
2. C9×#930C2semidirect
3. C3×#63488C2semidirect
4. #11721282C2semidirect
5. #11721263C2semidirect
6. #11721045C2semidirect
7. #11721025C2semidirect
8. #11721010C2semidirect
9. #11720971C2semidirect
10. #11720956C2semidirect
11. #11720922C2semidirect
55. #63488D3semidirect
158. #930D9semidirect
342. C3×C9#555semidirect
481. C9#5532semidirect
537. C3#10589718semidirect

Maximal subgroups

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

subgroupindex
C4×#62332
C9×#9302
C3×#634882
#117212822
#117212632
#117210452
#117210252
#117210102
#117209712
#117209562
#117209222
#105949613
#105897183