Finite Group Databasebeta

← #13148947

Subgroups — #13148947

Order 1728.

776 conjugacy classes of subgroups.

The subgroup lattice has 518 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11721744C211yesmaximal, characteristic, complemented
2C9×#892C211yesmaximal, characteristic, complemented
3C3×#63457C211yesmaximal, characteristic, complemented
4#11721042C212yesmaximal, complemented
5#11721037C211yesmaximal, characteristic, complemented
6C4×#6194C211yesmaximal, characteristic, complemented
7#11720998C211yesmaximal, characteristic, complemented
8#11720982C212yesmaximal, complemented
9#11720974C212yesmaximal, complemented
10#11720948C212yesmaximal, complemented
11#11720907C211yesmaximal, characteristic, complemented
12#10594486—31nomaximal
13#10589634—31nomaximal
14C2×#6226C2212yes—
15C2×C4×D3×C9C2211yescharacteristic
16C2×C3×D36C2211yescharacteristic
17C2×C3×D36C2212yes—
18C2×C3×C4×D9C2211yescharacteristic
19C2×#6194C2211yescharacteristic
20C2×#6194C2212yes—
21C2×Dic3×D9C2212yes—
22C3×C9×C4⋊C4C2211yescharacteristic
23C4⋊(C3⋊Dic9)C2211yescharacteristic
24#88534C2212yes—
25C4×C3⋊Dic9C2211yescharacteristic
26C9×(C2×D3)⋊C4C2212yes—
27C3×(C2×D9)⋊C4C2212yes—
28C9×C4⋊Dic3C2211yescharacteristic
29C4×C9×Dic3C2212yes—
30(C2×D3)⋊Dic9C2211yescharacteristic
31(C2×D9)⋊Dic3C2212yes—
32(C2×D9)⋊Dic3C2212yes—
33(C2×D9)⋊Dic3C2212yes—
34(C2×D9)⋊Dic3C2211yescharacteristic
35#88442C2211yescharacteristic
36Dic3×Dic9C2212yes—
37#64097—31no—
38C3×#895—31no—
39C3×#892D311yescharacteristic
40#63913—32no—
41#63908—31no—
42C4×#625—31no—
43#63898—31no—
44#63885—32no—
45#63877—32no—
46#63860—32no—
47#63841—31no—
48C9×#117—31no—
49#63492—31no—
50C4×#611—31no—
51#63457D311yescharacteristic, complemented
52#63455—31no—
53#63454—31no—
54#63450—31no—
55#63450—61no—
56#63447—32no—
57#63446—32no—
58#63437—32no—
59#63434—32no—
60C22×D3×C9C2311yescharacteristic
61C22×C3×D9C2312yes—
62C22×C3×D9C2311yescharacteristic
63C2×C3×C4×C9C2312yes—
64C2×C3×C4×C9C2311yescharacteristic
65C2×C3⋊Dic9C2312yes—
66C2×C3⋊Dic9C2311yescharacteristic
67C2×C3⋊Dic9C2311yescharacteristic
68#6226—42no—
69C2×C9×Dic3C2312yes—
70C2×C9×Dic3C2311yescharacteristic
71C2×C3×Dic9C2311yescharacteristic
72C4×D3×C9—22no—
73C3×D36—22no—
74C3×D36—42no—
75C3×C4×D9—22no—
76#6194—22no—
77#6194—42no—
78Dic3×D9—24no—
79#5616—91no—
80#5532—91no—
81C2×#634—32no—
82C2×C3×D12—31no—
83C2×C3×D12—32no—
84C2×C3×C4×D3—31no—
85C2×C3×C4×D3C2×D311yescharacteristic
86C2×#625—31no—
87C2×#625—32no—
88C2×D3×Dic3—32no—
89C32×C4⋊C4C2×D311yescharacteristic
90C4⋊(C3⋊Dic3)—31no—
91#3750—32no—
92C4×C3⋊Dic3—31no—
93C3×(C2×D3)⋊C4—32no—
94C3×(C2×D3)⋊C4C2×D312yes—
95C3×C4⋊Dic3C2×D311yescharacteristic
96C3×C4×Dic3C2×D312yes—
97#3724—31no—
98(C2×D3)⋊Dic3—31no—
99(C2×D3)⋊Dic3—32no—
100(C2×D3)⋊Dic3—31no—
101(C2×D3)⋊Dic3—32no—
102(C2×D3)⋊Dic3—32no—
103Dic32—32no—
104C22×C4×C9—31no—
105C2×#611—31no—
106C2×#611—32no—
107C2×D36C2×D311yescharacteristic
108C2×D36C2×D312yes—
109C2×C4×D9—32no—
110C2×C4×D9C2×D311yescharacteristic
111C2×Dic18—32no—
112C2×Dic18—62no—
113C9×C4⋊C4—21no—
114C9×C4⋊C4—31no—
115C9×C4⋊C4C2×D311yescharacteristic
116C9×C22⋊C4—32no—
117C42×C9—32no—
118#3676—31no—
119(C2×D9)⋊C4—31no—
120(C2×D9)⋊C4—32no—
121(C2×D9)⋊C4—32no—
122(C2×D9)⋊C4—32no—
123(C2×D9)⋊C4C2×D312yes—
124C4⋊Dic9—31no—
125C4⋊Dic9—61no—
126Dic9⋊C4—32no—
127Dic9⋊C4—31no—
128Dic9⋊C4—62no—
129C4×Dic9—31no—
130C4×Dic9—32no—
131C4×Dic9—61no—
132C22×C3×C9C2411yescharacteristic
133C2×D3×C9—21no—
134C2×D3×C9—21no—
135C2×D3×C9—21no—
136C2×C3×D9—22no—
137C2×C3×D9—21no—
138C2×C3×D9—21no—
139C2×C3×D9—22no—
140C2×C3×D9—22no—
141C2×C3×D9—21no—
142C3×C4×C9—22no—
143C3×C4×C9#4112yes—
144C3⋊Dic9—21no—
145C3⋊Dic9—22no—
146C3⋊Dic9#4112yes—
147C9×Dic3—22no—
148C9×Dic3—21no—
149C3×Dic9—21no—
150C3×#121—91no—
151C3×#117—31no—
152#939—92no—
153#938—91no—
154#936—92no—
155D4×Dic3—91no—
156#930—91no—
157C4×#67—91no—
158C4×#67—91no—
159#926—91no—
160#895—31no—
161#893—91no—
162#892—91no—
163#892D911yescharacteristic, complemented
164#888—91no—
165#888—91no—
166#888—181no—
167#885—92no—
168#885—92no—
169#884—92no—
170#877—92no—
171#875—92no—
172#872—92no—
173C22×C3×D3—31no—
174C22×C3×D3—32no—
175C22×C3×D3C22×D311yescharacteristic
176C2×C32×C4C22×D312yes—
177C2×C32×C4C22×D311yescharacteristic
178C2×C3⋊Dic3—31no—
179C2×C3⋊Dic3—32no—
180C2×C3⋊Dic3—31no—
181#634—122no—
182C2×C3×Dic3—31no—
183C2×C3×Dic3C22×D312yes—
184C2×C3×Dic3C22×D311yescharacteristic
185C3×D12—62no—
186C3×D12—122no—
187C3×C4×D3—22no—
188C3×C4×D3—62no—
189#625—62no—
190#625—122no—
191D3×Dic3—64no—
192C23×C9—31no—
193C22×D9C22×D311yescharacteristic
194C22×D9C22×D312yes—
195C2×C4×C9—21no—
196C2×C4×C9—22no—
197C2×C4×C9—32no—
198C2×C4×C9—31no—
199C2×C4×C9—62no—
200C2×C4×C9C22×D312yes—
201C2×C4×C9C22×D311yescharacteristic
202#611—62no—
203#611—122no—
204C2×Dic9—31no—
205C2×Dic9—32no—
206C2×Dic9—31no—
207C2×Dic9—61no—
208C2×Dic9—61no—
209C2×Dic9—62no—
210C2×Dic9C22×D311yescharacteristic
211D36—22no—
212D36—42no—
213C4×D9—22no—
214C4×D9—64no—
215Dic18—122no—
216Dic18—124no—
217#555—271noSylow
218C2×C3×C9C2×#4111yescharacteristic
219C2×C3×C9C2×#4111yescharacteristic
220C2×C3×C9H(5,𝔽2)11yescharacteristic
221D3×C9—41no—
222C3×D9—42no—
223C3×D9—41no—
224C2×C3×D4—91no—
225C2×C3×D4—92no—
226C22×C3×C4—31no—
227C22×C3×C4—91no—
228C2×#67—91no—
229C2×#67—91no—
230C2×#67—92no—
231C2×#67—92no—
232C22×Dic3—92no—
233C2×D12—31no—
234C2×D12—32no—
235C2×C4×D3—31no—
236C2×C4×D3—92no—
237C2×C4×D3C2×D911yescharacteristic
238C2×Dic6—92no—
239C2×Dic6—92no—
240C2×Dic6—182no—
241C3×C4⋊C4—21no—
242C3×C4⋊C4—31no—
243C3×C4⋊C4C2×D911yescharacteristic
244C3×C4⋊C4D3211yescharacteristic
245C3×C22⋊C4—32no—
246C3×C22⋊C4—92no—
247C3×C42—32no—
248#217—91no—
249#217—92no—
250#217—92no—
251#217—92no—
252#217—91no—
253(C2×D3)⋊C4—32no—
254(C2×D3)⋊C4—92no—
255(C2×D3)⋊C4—91no—
256(C2×D3)⋊C4—92no—
257(C2×D3)⋊C4—92no—
258(C2×D3)⋊C4—91no—
259(C2×D3)⋊C4C2×D912yes—
260C4⋊Dic3—91no—
261C4⋊Dic3—91no—
262C4⋊Dic3—181no—
263C4⋊Dic3C2×D911yescharacteristic
264Dic3⋊C4—91no—
265Dic3⋊C4—92no—
266Dic3⋊C4—92no—
267Dic3⋊C4—91no—
268Dic3⋊C4—182no—
269C4×Dic3—92no—
270C4×Dic3—91no—
271C4×Dic3—91no—
272C4×Dic3—92no—
273C4×Dic3—181no—
274C4×Dic3C2×D912yes—
275C22×C32C23×D311yescharacteristic
276C2×C3×D3—21no—
277C2×C3×D3—21no—
278C2×C3×D3—21no—
279C2×C3×D3—61no—
280C2×C3×D3—62no—
281C2×C3×D3—62no—
282C2×C3×D3—61no—
283C2×C3×D3—62no—
284C2×C3×D3—61no—
285C32×C4—22no—
286C32×C4#23512yes—
287C3⋊Dic3—32no—
288C3⋊Dic3—61no—
289C3⋊Dic3—62no—
290C3×Dic3—21no—
291C3×Dic3—22no—
292C3×Dic3—61no—
293C22×C9—21no—
294C22×C9—61no—
295C22×C9—61no—
296C22×C9—61no—
297C22×C9C23×D311yescharacteristic
298C2×D9—21no—
299C2×D9—22no—
300C2×D9—21no—
301C2×D9—22no—
302C2×D9—22no—
303C2×D9—21no—
304C4×C9—22no—
305C4×C9—22no—
306C4×C9—42no—
307C4×C9—61no—
308C4×C9—62no—
309C4×C9#23712yes—
310Dic9—21no—
311Dic9—32no—
312Dic9—62no—
313Dic9—62no—
314Dic9—61no—
315Dic9—62no—
316Dic9—122no—
317C4⋊Q8—271no—
318#124—272no—
319#124—272no—
320#121—91no—
321#121—272no—
322#121—271no—
323C4×D4—271no—
324C4×D4—271no—
325#117—31no—
326#117—272no—
327#117—271no—
328C3×C9#55511yescharacteristic, complemented, Sylow
329C23×C3—31no—
330C23×C3—91no—
331C23×C3—92no—
332C22×D3—32no—
333C22×D3—31no—
334C22×D3C22×D911yescharacteristic
335C3×D4—182no—
336C3×D4—362no—
337C2×C3×C4—21no—
338C2×C3×C4—22no—
339C2×C3×C4—32no—
340C2×C3×C4—31no—
341C2×C3×C4—62no—
342C2×C3×C4—91no—
343C2×C3×C4—182no—
344C2×C3×C4C22×D912yes—
345C2×C3×C4C22×D911yescharacteristic
346C2×C3×C4C2×D3212yes—
347C2×C3×C4C2×D3211yescharacteristic
348#67—182no—
349#67—182no—
350#67—362no—
351#67—362no—
352C2×Dic3—31no—
353C2×Dic3—91no—
354C2×Dic3—91no—
355C2×Dic3—92no—
356C2×Dic3—91no—
357C2×Dic3—92no—
358C2×Dic3—91no—
359C2×Dic3—181no—
360C2×Dic3—181no—
361C2×Dic3—182no—
362C2×Dic3—184no—
363C2×Dic3C22×D912yes—
364C2×Dic3C22×D911yescharacteristic
365D12—62no—
366D12—122no—
367C4×D3—22no—
368C4×D3—62no—
369C4×D3—184no—
370Dic6—362no—
371Dic6—362no—
372Dic6—364no—
373C2×C32C2×#23511yescharacteristic
374C2×C32D3×#4111yescharacteristic
375C2×C32#100911yescharacteristic
376C3×D3—41no—
377C3×D3—121no—
378C3×D3—122no—
379C2×C9—21no—
380C2×C9—21no—
381C2×C9—21no—
382C2×C9—121no—
383C2×C9C2×#23711yescharacteristic
384C2×C9#100411yescharacteristic
385C2×C9D3×#4111yescharacteristic
386D9—41no—
387D9—42no—
388C2×Q8—272no—
389C2×D4—91no—
390C2×D4—92no—
391C2×D4—271no—
392C2×D4—272no—
393C22×C4—31no—
394C22×C4—91no—
395C22×C4—272no—
396C4⋊C4—31no—
397C4⋊C4—271no—
398C4⋊C4—272no—
399C4⋊C4—271no—
400C4⋊C4D3×D911yescharacteristic
401C22⋊C4—32no—
402C22⋊C4—92no—
403C22⋊C4—272no—
404C22⋊C4—272no—
405C22⋊C4—271no—
406C22⋊C4—272no—
407C22⋊C4—271no—
408C42—32no—
409C42—272no—
410C42—271no—
411C22×C3—21no—
412C22×C3—61no—
413C22×C3—61no—
414C22×C3—61no—
415C22×C3—181no—
416C22×C3—182no—
417C22×C3—182no—
418C22×C3—181no—
419C22×C3—182no—
420C22×C3—181no—
421C22×C3C23×D911yescharacteristic
422C22×C3C22×D3211yescharacteristic
423C2×D3—21no—
424C2×D3—21no—
425C2×D3—21no—
426C2×D3—61no—
427C2×D3—62no—
428C2×D3—61no—
429C2×D3—62no—
430C2×D3—62no—
431C2×D3—61no—
432C3×C4—22no—
433C3×C4—22no—
434C3×C4—22no—
435C3×C4—42no—
436C3×C4—61no—
437C3×C4—62no—
438C3×C4—181no—
439C3×C4#369712yes—
440C3×C4#379512yes—
441Dic3—21no—
442Dic3—22no—
443Dic3—61no—
444Dic3—92no—
445Dic3—92no—
446Dic3—182no—
447Dic3—182no—
448Dic3—181no—
449Dic3—182no—
450Dic3—181no—
451Dic3—182no—
452Dic3—362no—
453C32#553211yescharacteristic
454C9—21no—
455C9#561611yescharacteristic, complemented
456C23—31no—
457C23—92no—
458C23—91no—
459Q8—1082no—
460D4—182no—
461D4—362no—
462D4—542no—
463D4—1082no—
464C2×C4—31no—
465C2×C4—32no—
466C2×C4—62no—
467C2×C4—91no—
468C2×C4—182no—
469C2×C4—271no—
470C2×C4—271no—
471C2×C4—272no—
472C2×C4—544no—
473C2×C4C2×D3×D912yes—
474C2×C4C2×D3×D911yescharacteristic
475C2×C3—21no—
476C2×C3—21no—
477C2×C3—21no—
478C2×C3—121no—
479C2×C3—361no—
480C2×C3—362no—
481C2×C3C2×#369711yescharacteristic
482C2×C3#41×D911yescharacteristic
483C2×C3#6371511yescharacteristic
484C2×C3C2×#379511yescharacteristic
485C2×C3#6431411yescharacteristic
486C2×C3#6431911yescharacteristic
487D3—41no—
488D3—121no—
489D3—122no—
490C22—61no—
491C22—61no—
492C22—61no—
493C22—181no—
494C22—182no—
495C22—182no—
496C22—181no—
497C22—181no—
498C22—182no—
499C22C22×D3×D911yescharacteristic
500C4—22no—
501C4—62no—
502C4—61no—
503C4—181no—
504C4—272no—
505C4—542no—
506C4—541no—
507C4#8864112yes—
508C3—21no—
509C3#1058963411yescharacteristic, complemented
510C3#1059448611yescharacteristic
511C2—121no—
512C2—361no—
513C2—362no—
514C2C2×#8864111yescharacteristic
515C2#1172250111yescharacteristic
516C2#1172251411yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC3×C9×C4⋊C4C22
socleC22×C32C23×D3
hypercenterC4⋊C4D3×D9
hypocenterC3×C9#555
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. #11721744C2semidirect
2. C9×#892C2semidirect
3. C3×#63457C2semidirect
4. #11721042C2semidirect
5. #11721037C2semidirect
6. C4×#6194C2semidirect
7. #11720998C2semidirect
8. #11720982C2semidirect
9. #11720974C2semidirect
10. #11720948C2semidirect
11. #11720907C2semidirect
51. #63457D3semidirect
163. #892D9semidirect
328. C3×C9#555semidirect
455. C9#5616semidirect
509. C3#10589634semidirect

Maximal subgroups

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

subgroupindex
#117217442
C9×#8922
C3×#634572
#117210422
#117210372
C4×#61942
#117209982
#117209822
#117209742
#117209482
#117209072
#105944863
#105896343