Finite Group Databasebeta

← #13148829

Subgroups — #13148829

Order 1728.

944 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88634C211yesmaximal, characteristic, complemented
2C2×D9×Dic6C211yesmaximal, characteristic, complemented
3#11721753C211yesmaximal, characteristic, complemented
4C3×#63457C211yesmaximal, characteristic, complemented
5C9×#887C211yesmaximal, characteristic, complemented
6C4×#6192C211yesmaximal, characteristic, complemented
7#11720991C212yesmaximal, complemented
8Dic3×D36C212yesmaximal, complemented
9#11720983C212yesmaximal, complemented
10#11720948C211yesmaximal, characteristic, complemented
11#11720940C212yesmaximal, complemented
12#10594382—31nomaximal
13#10589678—31nomaximal
14C2×C4×C3⋊D9C2211yescharacteristic
15C2×#6226C2211yescharacteristic
16C2×C3×D36C2212yes—
17C2×C3×D36C2211yescharacteristic
18C2×C3×C4×D9C2211yescharacteristic
19C2×C9×Dic6C2211yescharacteristic
20C2×#6192C2211yescharacteristic
21C2×#6192C2212yes—
22C2×Dic3×D9C2212yes—
23C2×Dic3×D9C2212yes—
24C2×#6189C2212yes—
25#88634—24no—
26D9×Dic6—24no—
27C3×C9×C4⋊C4C2211yescharacteristic
28#88536C2212yes—
29C4⋊(C3⋊Dic9)C2212yes—
30C3×(C2×D9)⋊C4C2212yes—
31C9×Dic3⋊C4C2212yes—
32C4×C9×Dic3C2212yes—
33C4×C9×Dic3C2211yescharacteristic
34#88445C2211yescharacteristic
35(C2×D9)⋊Dic3C2212yes—
36(C2×D9)⋊Dic3C2211yescharacteristic
37(C2×D9)⋊Dic3C2212yes—
38Dic3⋊Dic9C2211yescharacteristic
39C2×#3796—31no—
40C2×D3×Dic6—31no—
41#64106—31no—
42C3×#895—31no—
43C3×#887D311yescharacteristic
44C4×#626—31no—
45#63893—32no—
46Dic3×D12—32no—
47#63886—32no—
48#63860—31no—
49#63852—32no—
50C2×#3701—31no—
51C2×Q8×D9—31no—
52C4×Q8×C9—31no—
53#63459—31no—
54#63459—32no—
55#63459—61no—
56#63457—32no—
57#63457D311yescharacteristic, complemented
58#63437—31no—
59#63437—32no—
60C4×D36—32no—
61C4×D36—31no—
62C22×C3⋊D9C2311yescharacteristic
63C22×C3×D9C2312yes—
64C22×C3×D9C2311yescharacteristic
65C2×C3×C4×C9C2312yes—
66C2×C3×C4×C9C2311yescharacteristic
67C2×C3⋊Dic9C2312yes—
68C2×C3⋊Dic9C2311yescharacteristic
69C4×C3⋊D9—22no—
70#6226—22no—
71C2×C9×Dic3C2311yescharacteristic
72C2×C9×Dic3C2311yescharacteristic
73C2×C9×Dic3C2312yes—
74C2×C3×Dic9C2311yescharacteristic
75C3×D36—24no—
76C3×D36—22no—
77C3×C4×D9—22no—
78C9×Dic6D414yescomplemented
79#6192—22no—
80#6192—24no—
81Dic3×D9—24no—
82Dic3×D9—24no—
83#6189—24no—
84#5598—91no—
85#5576—91no—
86C2×C4×C3⋊D3—31no—
87C2×#634—31no—
88C2×C3×D12—31no—
89C2×C3×D12—32no—
90C2×C3×C4×D3—31no—
91C2×C3×Dic6C2×D311yescharacteristic
92C2×#627—32no—
93C2×#626—32no—
94C2×#626—31no—
95C2×D3×Dic3—32no—
96C2×D3×Dic3—32no—
97#3796—64no—
98D3×Dic6—64no—
99C32×C4⋊C4C2×D311yescharacteristic
100#3752—32no—
101C4⋊(C3⋊Dic3)—32no—
102C3×(C2×D3)⋊C4—32no—
103C3×Dic3⋊C4C2×D312yes—
104C3×C4×Dic3C2×D312yes—
105C3×C4×Dic3C2×D311yescharacteristic
106Dic3⋊Dic3—31no—
107#3722—31no—
108(C2×D3)⋊Dic3—32no—
109(C2×D3)⋊Dic3—32no—
110(C2×D3)⋊Dic3—31no—
111C2×Q8×C9—31no—
112#3701—64no—
113Q8×D9—64no—
114C2×D36—31no—
115C2×D36—32no—
116C2×D36C2×D312yes—
117C2×D36C2×D311yescharacteristic
118C2×C4×D9—32no—
119C2×C4×D9—32no—
120C2×C4×D9—31no—
121C2×C4×D9—61no—
122C2×C4×D9C2×D311yescharacteristic
123C2×Dic18—31no—
124C2×Dic18—32no—
125C2×Dic18—61no—
126C9×C4⋊C4—21no—
127C9×C4⋊C4—32no—
128C9×C4⋊C4C2×D311yescharacteristic
129C42×C9—32no—
130C42×C9—31no—
131(C2×D9)⋊C4—31no—
132(C2×D9)⋊C4—32no—
133(C2×D9)⋊C4—32no—
134(C2×D9)⋊C4—32no—
135(C2×D9)⋊C4—31no—
136(C2×D9)⋊C4—62no—
137(C2×D9)⋊C4C2×D312yes—
138C4⋊Dic9—32no—
139C4⋊Dic9—31no—
140C4⋊Dic9—62no—
141C22×C3×C9C2411yescharacteristic
142C2×C3⋊D9—21no—
143C2×C3⋊D9—21no—
144C2×C3⋊D9—21no—
145C2×C3×D9—21no—
146C2×C3×D9—21no—
147C2×C3×D9—22no—
148C2×C3×D9—22no—
149C2×C3×D9—22no—
150C2×C3×D9#4112yescomplemented
151C3×C4×C9—22no—
152C3×C4×C9C2×D412yes—
153C3⋊Dic9—22no—
154C3⋊Dic9—21no—
155C9×Dic3—21no—
156C9×Dic3—21no—
157C9×Dic3C2×D414yes—
158C3×Dic9—21no—
159C2×#239—91no—
160C2×D3×Q8—91no—
161C2×#237—91no—
162C22×Dic6—91no—
163C3×#121—91no—
164C3×C4×Q8—31no—
165#939—92no—
166#936—91no—
167D4×Dic3—92no—
168C4×#67—91no—
169#897—92no—
170#897—91no—
171#897—91no—
172#897—181no—
173#895—31no—
174#895—92no—
175#887D911yescharacteristic, complemented
176#885—92no—
177#878—92no—
178#875—92no—
179#875—91no—
180C4×D12—91no—
181C4×D12—92no—
182C22×C3⋊D3—31no—
183C22×C3×D3—31no—
184C22×C3×D3—32no—
185C2×C32×C4C22×D312yes—
186C2×C32×C4C22×D311yescharacteristic
187C2×C3⋊Dic3—32no—
188C2×C3⋊Dic3—31no—
189C4×C3⋊D3—62no—
190#634—62no—
191C2×C3×Dic3—31no—
192C2×C3×Dic3C22×D311yescharacteristic
193C2×C3×Dic3C22×D311yescharacteristic
194C2×C3×Dic3C22×D312yes—
195C3×D12—64no—
196C3×D12—62no—
197C3×C4×D3—62no—
198C3×Dic6D1214yes—
199#627—64no—
200#626—62no—
201#626—64no—
202D3×Dic3—64no—
203D3×Dic3—64no—
204C22×D9—31no—
205C22×D9—61no—
206C22×D9C22×D312yes—
207C22×D9C22×D311yescharacteristic
208Q8×C9—34no—
209C2×C4×C9—21no—
210C2×C4×C9—22no—
211C2×C4×C9—32no—
212C2×C4×C9—31no—
213C2×C4×C9—31no—
214C2×C4×C9C22×D312yes—
215C2×C4×C9C22×D311yescharacteristic
216C2×Dic9—31no—
217C2×Dic9—32no—
218C2×Dic9—61no—
219C2×Dic9—62no—
220C2×Dic9C22×D311yescharacteristic
221D36—24no—
222D36—22no—
223D36—64no—
224D36—62no—
225C4×D9—22no—
226C4×D9—64no—
227C4×D9—64no—
228C4×D9—62no—
229C4×D9—122no—
230Dic18—64no—
231Dic18—62no—
232Dic18—122no—
233#548—271noSylow
234C2×C3×C9C22×D411yescharacteristic
235C2×C3×C9C2×#4111yescharacteristic
236C2×C3×C92−1+411yescharacteristic
237C3⋊D9—41no—
238C3×D9—22no—
239C3×D9—42no—
240C2×C3×Q8—31no—
241C2×C3×D4—92no—
242C2×C3×D4—91no—
243C22×C3×C4—91no—
244C2×#67—92no—
245C2×#67—91no—
246C22×Dic3—92no—
247C22×Dic3—92no—
248#239—184no—
249D3×Q8—184no—
250#237—184no—
251C2×D12—32no—
252C2×D12—31no—
253C2×D12—91no—
254C2×D12—92no—
255C2×C4×D3—31no—
256C2×C4×D3—92no—
257C2×C4×D3—92no—
258C2×C4×D3—91no—
259C2×C4×D3—91no—
260C2×C4×D3—181no—
261C2×Dic6—92no—
262C2×Dic6—91no—
263C2×Dic6—91no—
264C2×Dic6—92no—
265C2×Dic6—181no—
266C2×Dic6—184no—
267C2×Dic6C2×D911yescharacteristic
268C3×C4⋊C4—21no—
269C3×C4⋊C4—32no—
270C3×C4⋊C4C2×D911yescharacteristic
271C3×C4⋊C4D3211yescharacteristic
272C3×C22⋊C4—92no—
273C3×C42—32no—
274C3×C42—31no—
275#217—92no—
276#217—92no—
277#217—91no—
278(C2×D3)⋊C4—32no—
279(C2×D3)⋊C4—91no—
280(C2×D3)⋊C4—91no—
281(C2×D3)⋊C4—92no—
282(C2×D3)⋊C4—92no—
283(C2×D3)⋊C4—92no—
284(C2×D3)⋊C4—92no—
285(C2×D3)⋊C4—91no—
286(C2×D3)⋊C4—182no—
287C4⋊Dic3—91no—
288C4⋊Dic3—92no—
289C4⋊Dic3—92no—
290C4⋊Dic3—182no—
291Dic3⋊C4—91no—
292Dic3⋊C4C2×D912yes—
293C4×Dic3C2×D912yes—
294C4×Dic3C2×D911yescharacteristic
295C22×C32C23×D311yescharacteristic
296C2×C3⋊D3—61no—
297C2×C3⋊D3—61no—
298C2×C3⋊D3—61no—
299C2×C3×D3—32no—
300C2×C3×D3—61no—
301C2×C3×D3—61no—
302C2×C3×D3—62no—
303C2×C3×D3—62no—
304C2×C3×D3—62no—
305C32×C4—22no—
306C32×C4C2×D1212yes—
307C3⋊Dic3—61no—
308C3⋊Dic3—62no—
309C3×Dic3—21no—
310C3×Dic3—21no—
311C3×Dic3—61no—
312C3×Dic3C2×D1214yes—
313C22×C9—21no—
314C22×C9C23×D311yescharacteristic
315C2×D9—21no—
316C2×D9—21no—
317C2×D9—22no—
318C2×D9—22no—
319C2×D9—22no—
320C2×D9—62no—
321C2×D9—61no—
322C2×D9—61no—
323C2×D9—61no—
324C2×D9—121no—
325C2×D9—121no—
326C2×D9#23712yescomplemented
327C4×C9—22no—
328C4×C9—22no—
329C4×C9—34no—
330C4×C9—42no—
331C4×C9—61no—
332C4×C9—61no—
333C4×C9D3×D412yes—
334Dic9—21no—
335Dic9—62no—
336Dic9—61no—
337Dic9—122no—
338Dic9—121no—
339C2×#41—271no—
340C22×Q8—271no—
341#124—272no—
342#124—271no—
343C22⋊Q8—272no—
344C22⋊Q8—271no—
345#121—91no—
346#121—272no—
347C4×Q8—31no—
348C4×D4—271no—
349C4×D4—272no—
350C3×C9#54811yescharacteristic, complemented, Sylow
351C23×C3—91no—
352C23×C3—92no—
353C22×D3—31no—
354C22×D3—32no—
355C22×D3—91no—
356C22×D3—91no—
357C22×D3—181no—
358C3×Q8—34no—
359C3×D4—184no—
360C3×D4—182no—
361C2×C3×C4—21no—
362C2×C3×C4—22no—
363C2×C3×C4—32no—
364C2×C3×C4—31no—
365C2×C3×C4—31no—
366C2×C3×C4—91no—
367C2×C3×C4—182no—
368C2×C3×C4C22×D912yes—
369C2×C3×C4C22×D911yescharacteristic
370C2×C3×C4C2×D3212yes—
371C2×C3×C4C2×D3211yescharacteristic
372#67—182no—
373#67—184no—
374C2×Dic3—31no—
375C2×Dic3—91no—
376C2×Dic3—92no—
377C2×Dic3—91no—
378C2×Dic3—92no—
379C2×Dic3—181no—
380C2×Dic3—182no—
381C2×Dic3—184no—
382C2×Dic3—184no—
383C2×Dic3C22×D911yescharacteristic
384C2×Dic3C22×D911yescharacteristic
385C2×Dic3C22×D912yes—
386D12—64no—
387D12—62no—
388D12—184no—
389D12—182no—
390C4×D3—62no—
391C4×D3—184no—
392C4×D3—184no—
393C4×D3—182no—
394C4×D3—182no—
395C4×D3—362no—
396Dic6—184no—
397Dic6—182no—
398Dic6—182no—
399Dic6—184no—
400Dic6—362no—
401Dic6D3614yescomplemented
402C2×C32C22×D1211yescharacteristic
403C2×C32#100711yescharacteristic
404C2×C32D3×#4111yescharacteristic
405C3⋊D3—121no—
406C3×D3—62no—
407C3×D3—122no—
408C2×C9—21no—
409C2×C9—21no—
410C2×C9—21no—
411C2×C9C2×D3×D411yescharacteristic
412C2×C9C2×#23711yescharacteristic
413C2×C9#101011yescharacteristic
414D9—22no—
415D9—42no—
416D9—122no—
417D9—121no—
418#41—544no—
419C2×Q8—31no—
420C2×Q8—272no—
421C2×Q8—271no—
422C2×Q8—544no—
423C2×D4—92no—
424C2×D4—91no—
425C2×D4—271no—
426C2×D4—272no—
427C22×C4—91no—
428C22×C4—272no—
429C22×C4—272no—
430C22×C4—271no—
431C4⋊C4—32no—
432C4⋊C4—271no—
433C4⋊C4—272no—
434C4⋊C4D3×D911yescharacteristic
435C22⋊C4—92no—
436C22⋊C4—271no—
437C22⋊C4—271no—
438C22⋊C4—272no—
439C22⋊C4—272no—
440C22⋊C4—272no—
441C42—32no—
442C42—31no—
443C22×C3—21no—
444C22×C3—92no—
445C22×C3—181no—
446C22×C3—181no—
447C22×C3—182no—
448C22×C3—182no—
449C22×C3—182no—
450C22×C3C23×D911yescharacteristic
451C22×C3C22×D3211yescharacteristic
452C2×D3—32no—
453C2×D3—61no—
454C2×D3—61no—
455C2×D3—62no—
456C2×D3—62no—
457C2×D3—62no—
458C2×D3—182no—
459C2×D3—181no—
460C2×D3—181no—
461C2×D3—181no—
462C2×D3—181no—
463C2×D3—181no—
464C2×D3—181no—
465C2×D3—361no—
466C2×D3—361no—
467C3×C4—22no—
468C3×C4—22no—
469C3×C4—22no—
470C3×C4—34no—
471C3×C4—42no—
472C3×C4—61no—
473C3×C4—61no—
474C3×C4—181no—
475C3×C4C2×D3612yes—
476C3×C4D3×D1212yes—
477Dic3—21no—
478Dic3—21no—
479Dic3—61no—
480Dic3—182no—
481Dic3—181no—
482Dic3—182no—
483Dic3—181no—
484Dic3—362no—
485Dic3—361no—
486Dic3C2×D3614yes—
487C32#557611yescharacteristic
488C9—21no—
489C9#559811yescharacteristic, complemented
490C23—91no—
491C23—92no—
492C23—271no—
493Q8—34no—
494Q8—544no—
495Q8—542no—
496D4—184no—
497D4—182no—
498D4—544no—
499D4—542no—
500C2×C4—32no—
501C2×C4—31no—
502C2×C4—31no—
503C2×C4—91no—
504C2×C4—182no—
505C2×C4—272no—
506C2×C4—271no—
507C2×C4—544no—
508C2×C4—544no—
509C2×C4—542no—
510C2×C4C2×D3×D912yes—
511C2×C4C2×D3×D911yescharacteristic
512C2×C3—21no—
513C2×C3—21no—
514C2×C3—21no—
515C2×C3—182no—
516C2×C3—362no—
517C2×C3C22×D3611yescharacteristic
518C2×C3#6371311yescharacteristic
519C2×C3#41×D911yescharacteristic
520C2×C3C2×D3×D1211yescharacteristic
521C2×C3D3×#23711yescharacteristic
522C2×C3#6431511yescharacteristic
523D3—62no—
524D3—122no—
525D3—362no—
526D3—361no—
527D3—361no—
528C22—92no—
529C22—181no—
530C22—181no—
531C22—182no—
532C22—182no—
533C22—182no—
534C22—541no—
535C22—541no—
536C22—541no—
537C22C22×D3×D911yescharacteristic
538C4—22no—
539C4—34no—
540C4—61no—
541C4—61no—
542C4—181no—
543C4—541no—
544C4—542no—
545C4D3×D3612yes—
546C3—21no—
547C3#1058967811yescharacteristic, complemented
548C3#1059438211yescharacteristic
549C2—182no—
550C2—362no—
551C2—1081no—
552C2C2×D3×D3611yescharacteristic
553C2D9×#23711yescharacteristic
554C2#1172251011yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC3×C9×C4⋊C4C22
socleC22×C32C23×D3
hypercenterC4⋊C4D3×D9
hypocenterC3×C9#548
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. C2×#88634C2semidirect
2. C2×D9×Dic6C2semidirect
3. #11721753C2semidirect
4. C3×#63457C2semidirect
5. C9×#887C2semidirect
6. C4×#6192C2semidirect
7. #11720991C2semidirect
8. Dic3×D36C2semidirect
9. #11720983C2semidirect
10. #11720948C2semidirect
11. #11720940C2semidirect
57. #63457D3semidirect
78. C9×Dic6D4semidirect
150. C2×C3×D9#41semidirect
175. #887D9semidirect
326. C2×D9#237semidirect
350. C3×C9#548semidirect
401. Dic6D36semidirect
489. C9#5598semidirect
547. C3#10589678semidirect

Maximal subgroups

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

subgroupindex
C2×#886342
C2×D9×Dic62
#117217532
C3×#634572
C9×#8872
C4×#61922
#117209912
Dic3×D362
#117209832
#117209482
#117209402
#105943823
#105896783