Finite Group Databasebeta

← #13150414

Subgroups — #13150414

Order 1728.

784 conjugacy classes of subgroups.

The subgroup lattice has 524 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11721815C211yesmaximal, characteristic, complemented
2C9×#930C211yesmaximal, characteristic, complemented
3C3×C4×#611C211yesmaximal, characteristic, complemented
4#11721281C212yesmaximal, complemented
5#11721263C212yesmaximal, complemented
6#11721044C212yesmaximal, complemented
7#11721036C211yesmaximal, characteristic, complemented
8D12×Dic9C211yesmaximal, characteristic, complemented
9#11721011C212yesmaximal, complemented
10#11720969C211yesmaximal, characteristic, complemented
11#11720907C211yesmaximal, characteristic, complemented
12#10595026—31nomaximal
13#10589718—31nomaximal
14C22×C3×C4×C9C2211yescharacteristic
15C2×C9×#67C2212yes—
16C2×C3×#611C2211yescharacteristic
17C2×C9×D12C2211yescharacteristic
18C2×C3×C4×D9C2211yescharacteristic
19C2×#6194C2212yes—
20C2×D3×Dic9C2212yes—
21C2×#6189C2212yes—
22#88551C2212yes—
23C4⋊(C3⋊Dic9)C2211yescharacteristic
24C4⋊(C3⋊Dic9)C2211yescharacteristic
25C4×C3⋊Dic9C2212yes—
26C3×#3676C2211yescharacteristic
27C9×(C2×D3)⋊C4C2212yes—
28C3×(C2×D9)⋊C4C2211yescharacteristic
29C9×C4⋊Dic3C2211yescharacteristic
30C3×Dic9⋊C4C2211yescharacteristic
31C3×C4×Dic9C2211yescharacteristic
32(C2×D3)⋊Dic9C2212yes—
33(C2×D3)⋊Dic9C2212yes—
34(C2×D9)⋊Dic3C2212yes—
35Dic9⋊Dic3C2212yes—
36Dic3×Dic9C2212yes—
37#64135—31no—
38C3×#930D311yescharacteristic
39C3×C4×#67—31no—
40#63970—32no—
41#63959—32no—
42#63918—31no—
43#63915—32no—
44#63908—31no—
45Dic3×D12—31no—
46#63878—32no—
47#63841—31no—
48C9×#121—31no—
49#63501—32no—
50#63500—31no—
51#63498—32no—
52D4×Dic9—31no—
53C4×#611D311yescharacteristic, complemented
54#63488—31no—
55#63488—61no—
56#63454—31no—
57#63450—31no—
58#63447—32no—
59#63439—32no—
60C23×C3×C9C2311yescharacteristic
61C22×D3×C9C2312yes—
62C22×C3×D9C2311yescharacteristic
63C2×C3×C4×C9—22no—
64C2×C3×C4×C9C2311yescharacteristic
65C2×C3×C4×C9C2311yescharacteristic
66C2×C3⋊Dic9C2312yes—
67C2×C3⋊Dic9C2312yes—
68C9×#67—42no—
69C3×#611—22no—
70C2×C9×Dic3C2312yes—
71C2×C3×Dic9C2311yescharacteristic
72C2×C3×Dic9C2311yescharacteristic
73C2×C3×Dic9C2311yescharacteristic
74C9×D12—22no—
75C3×C4×D9—22no—
76#6194—42no—
77D3×Dic9—24no—
78#6189—42no—
79#5616—91no—
80#5563—91no—
81C22×C32×C4C2×D311yescharacteristic
82C2×C3×#67—31no—
83C2×C3×#67C2×D312yes—
84C2×C3×D12C2×D311yescharacteristic
85C2×C3×C4×D3—31no—
86C2×#627—32no—
87C2×#625—32no—
88C2×D3×Dic3—32no—
89#3757—32no—
90C4⋊(C3⋊Dic3)—31no—
91C4⋊(C3⋊Dic3)—31no—
92C4×C3⋊Dic3—32no—
93C3×#217—31no—
94C3×(C2×D3)⋊C4—31no—
95C3×(C2×D3)⋊C4C2×D312yes—
96C3×C4⋊Dic3C2×D311yescharacteristic
97C3×Dic3⋊C4—31no—
98C3×C4×Dic3—31no—
99Dic3⋊Dic3—32no—
100(C2×D3)⋊Dic3—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—32no—
109C2×#611C2×D311yescharacteristic
110C22×Dic9—32no—
111C2×C4×D9C2×D311yescharacteristic
112C2×Dic18—32no—
113C9×C4⋊C4—31no—
114C9×C22⋊C4—32no—
115#3676—32no—
116#3676—32no—
117#3676—32no—
118#3676—62no—
119#3676C2×D311yescharacteristic
120(C2×D9)⋊C4—32no—
121(C2×D9)⋊C4C2×D311yescharacteristic
122C4⋊Dic9—31no—
123C4⋊Dic9—31no—
124C4⋊Dic9—61no—
125C4⋊Dic9—61no—
126Dic9⋊C4—32no—
127Dic9⋊C4C2×D311yescharacteristic
128C4×Dic9—32no—
129C4×Dic9—32no—
130C4×Dic9—62no—
131C4×Dic9C2×D311yescharacteristic
132C22×C3×C9—21no—
133C22×C3×C9—21no—
134C22×C3×C9—21no—
135C22×C3×C9C2411yescharacteristic
136C2×D3×C9—22no—
137C2×D3×C9—22no—
138C2×D3×C9—22no—
139C2×C3×D9—21no—
140C2×C3×D9—21no—
141C2×C3×D9—21no—
142C3×C4×C9—21no—
143C3×C4×C9#4112yes—
144C3⋊Dic9—22no—
145C3⋊Dic9—22no—
146C9×Dic3—22no—
147C3×Dic9—21no—
148C3×Dic9—21no—
149C3×Dic9#4112yes—
150C3×#121—31no—
151C3×C4×D4—91no—
152#939—92no—
153#938—91no—
154#938—92no—
155#936—92no—
156D4×Dic3—91no—
157#930—91no—
158#930D911yescharacteristic, complemented
159C4×#67—31no—
160#926—91no—
161#926—91no—
162#926—91no—
163#926—181no—
164#892—91no—
165#892—92no—
166#888—91no—
167#885—92no—
168#885—92no—
169#885—92no—
170#877—92no—
171C4×D12—91no—
172#869—91no—
173C23×C32C22×D311yescharacteristic
174C22×C3×D3—31no—
175C22×C3×D3C22×D312yes—
176C2×C32×C4—22no—
177C2×C32×C4C22×D311yescharacteristic
178C2×C32×C4C22×D311yescharacteristic
179C2×C3⋊Dic3—32no—
180C2×C3⋊Dic3—32no—
181C3×#67—42no—
182C3×#67—62no—
183C2×C3×Dic3—31no—
184C2×C3×Dic3—31no—
185C2×C3×Dic3—31no—
186C2×C3×Dic3C22×D312yes—
187C3×D12—22no—
188C3×C4×D3—62no—
189#627—122no—
190#625—122no—
191D3×Dic3—64no—
192C23×C9—21no—
193C23×C9—32no—
194C23×C9C22×D311yescharacteristic
195C22×D9C22×D311yescharacteristic
196D4×C9—62no—
197D4×C9—122no—
198C2×C4×C9—24no—
199C2×C4×C9—22no—
200C2×C4×C9—21no—
201C2×C4×C9—21no—
202C2×C4×C9—32no—
203C2×C4×C9C22×D311yescharacteristic
204C2×C4×C9C22×D311yescharacteristic
205#611—22no—
206#611—122no—
207C2×Dic9—32no—
208C2×Dic9—32no—
209C2×Dic9—62no—
210C2×Dic9—62no—
211C2×Dic9—64no—
212C2×Dic9C22×D311yescharacteristic
213C2×Dic9C22×D311yescharacteristic
214C2×Dic9C22×D311yescharacteristic
215C4×D9—22no—
216Dic18—122no—
217#555—271noSylow
218C2×C3×C9—41no—
219C2×C3×C9C2×#4111yescharacteristic
220C2×C3×C9C2×#4111yescharacteristic
221C2×C3×C9H(5,𝔽2)11yescharacteristic
222D3×C9—42no—
223C3×D9—41no—
224C2×C3×D4—32no—
225C2×C3×D4—31no—
226C2×C3×D4—91no—
227C22×C3×C4—21no—
228C22×C3×C4—91no—
229C22×C3×C4C2×D911yescharacteristic
230C22×C3×C4D3211yescharacteristic
231C2×#67—31no—
232C2×#67—92no—
233C2×#67—92no—
234C2×#67C2×D912yes—
235C22×Dic3—92no—
236C2×D12C2×D911yescharacteristic
237C2×C4×D3—31no—
238C2×C4×D3—92no—
239C2×Dic6—92no—
240C2×Dic6—92no—
241C3×C4⋊C4—31no—
242C3×C4⋊C4—91no—
243C3×C22⋊C4—32no—
244C3×C22⋊C4—91no—
245C3×C22⋊C4—91no—
246C3×C42—91no—
247#217—31no—
248#217—92no—
249#217—92no—
250#217—92no—
251#217—92no—
252#217—92no—
253#217—182no—
254(C2×D3)⋊C4—31no—
255(C2×D3)⋊C4—92no—
256(C2×D3)⋊C4—92no—
257(C2×D3)⋊C4—92no—
258(C2×D3)⋊C4C2×D912yes—
259C4⋊Dic3—91no—
260C4⋊Dic3—91no—
261C4⋊Dic3—91no—
262C4⋊Dic3—91no—
263C4⋊Dic3—92no—
264C4⋊Dic3—181no—
265C4⋊Dic3—181no—
266C4⋊Dic3C2×D911yescharacteristic
267Dic3⋊C4—31no—
268Dic3⋊C4—92no—
269C4×Dic3—31no—
270C4×Dic3—92no—
271C4×Dic3—92no—
272C4×Dic3—92no—
273C4×Dic3—92no—
274C4×Dic3—182no—
275C22×C32—21no—
276C22×C32—21no—
277C22×C32—21no—
278C22×C32C23×D311yescharacteristic
279C2×C3×D3—22no—
280C2×C3×D3—22no—
281C2×C3×D3—22no—
282C2×C3×D3—61no—
283C2×C3×D3—61no—
284C2×C3×D3—61no—
285C32×C4—21no—
286C32×C4#23712yes—
287C3⋊Dic3—62no—
288C3⋊Dic3—62no—
289C3×Dic3—22no—
290C3×Dic3—32no—
291C3×Dic3—61no—
292C3×Dic3—61no—
293C22×C9—22no—
294C22×C9—21no—
295C22×C9—21no—
296C22×C9—21no—
297C22×C9—21no—
298C22×C9—41no—
299C22×C9—41no—
300C22×C9—62no—
301C22×C9—62no—
302C22×C9—62no—
303C22×C9C23×D311yescharacteristic
304C2×D9—21no—
305C2×D9—21no—
306C2×D9—21no—
307C4×C9—22no—
308C4×C9—21no—
309C4×C9—22no—
310C4×C9—62no—
311C4×C9#23712yes—
312Dic9—21no—
313Dic9—21no—
314Dic9—62no—
315Dic9—62no—
316Dic9—122no—
317Dic9—122no—
318Dic9#23512yes—
319C4⋊Q8—271no—
320#124—272no—
321#124—272no—
322#121—31no—
323#121—271no—
324#121—272no—
325C4×D4—91no—
326C4×D4—271no—
327#117—272no—
328#117—271no—
329#117—271no—
330C3×C9#55511yescharacteristic, complemented, Sylow
331C23×C3—21no—
332C23×C3—32no—
333C23×C3—91no—
334C23×C3C22×D911yescharacteristic
335C23×C3C2×D3211yescharacteristic
336C22×D3—31no—
337C22×D3C22×D912yes—
338C3×D4—62no—
339C3×D4—122no—
340C3×D4—182no—
341C2×C3×C4—24no—
342C2×C3×C4—22no—
343C2×C3×C4—22no—
344C2×C3×C4—21no—
345C2×C3×C4—21no—
346C2×C3×C4—32no—
347C2×C3×C4—91no—
348C2×C3×C4—91no—
349C2×C3×C4—91no—
350C2×C3×C4—182no—
351C2×C3×C4C22×D911yescharacteristic
352C2×C3×C4C22×D911yescharacteristic
353C2×C3×C4C2×D3211yescharacteristic
354C2×C3×C4C2×D3211yescharacteristic
355#67—42no—
356#67—62no—
357#67—362no—
358#67—362no—
359C2×Dic3—31no—
360C2×Dic3—31no—
361C2×Dic3—31no—
362C2×Dic3—92no—
363C2×Dic3—92no—
364C2×Dic3—92no—
365C2×Dic3—92no—
366C2×Dic3—182no—
367C2×Dic3—182no—
368C2×Dic3—184no—
369C2×Dic3C22×D912yes—
370D12—22no—
371C4×D3—62no—
372C4×D3—184no—
373Dic6—362no—
374Dic6—362no—
375C2×C32—41no—
376C2×C32C2×#23711yescharacteristic
377C2×C32#100411yescharacteristic
378C2×C32D3×#4111yescharacteristic
379C3×D3—42no—
380C3×D3—121no—
381C2×C9—21no—
382C2×C9—21no—
383C2×C9—21no—
384C2×C9—42no—
385C2×C9—41no—
386C2×C9—122no—
387C2×C9C2×#23511yescharacteristic
388C2×C9C2×#23711yescharacteristic
389C2×C9#100911yescharacteristic
390D9—41no—
391C2×Q8—272no—
392C2×D4—32no—
393C2×D4—31no—
394C2×D4—91no—
395C2×D4—272no—
396C22×C4—91no—
397C22×C4—272no—
398C22×C4D3×D911yescharacteristic
399C4⋊C4—31no—
400C4⋊C4—91no—
401C4⋊C4—272no—
402C4⋊C4—271no—
403C4⋊C4—271no—
404C22⋊C4—32no—
405C22⋊C4—91no—
406C22⋊C4—91no—
407C22⋊C4—272no—
408C22⋊C4—272no—
409C22⋊C4—272no—
410C22⋊C4—272no—
411C42—91no—
412C42—272no—
413C42—272no—
414C22×C3—22no—
415C22×C3—21no—
416C22×C3—21no—
417C22×C3—21no—
418C22×C3—21no—
419C22×C3—21no—
420C22×C3—21no—
421C22×C3—21no—
422C22×C3—41no—
423C22×C3—41no—
424C22×C3—62no—
425C22×C3—62no—
426C22×C3—62no—
427C22×C3—181no—
428C22×C3—181no—
429C22×C3—181no—
430C22×C3C23×D911yescharacteristic
431C22×C3C22×D3211yescharacteristic
432C2×D3—22no—
433C2×D3—22no—
434C2×D3—22no—
435C2×D3—61no—
436C2×D3—61no—
437C2×D3—61no—
438C3×C4—22no—
439C3×C4—21no—
440C3×C4—21no—
441C3×C4—22no—
442C3×C4—62no—
443C3×C4—92no—
444C3×C4—181no—
445C3×C4—181no—
446C3×C4#369912yes—
447C3×C4#380512yes—
448Dic3—22no—
449Dic3—32no—
450Dic3—61no—
451Dic3—61no—
452Dic3—182no—
453Dic3—182no—
454Dic3—182no—
455Dic3—182no—
456Dic3—362no—
457Dic3—362no—
458C32#561611yescharacteristic
459C9—21no—
460C9#556311yescharacteristic, complemented
461C23—32no—
462C23—91no—
463C23C2×D3×D911yescharacteristic
464Q8—1082no—
465D4—62no—
466D4—122no—
467D4—182no—
468D4—1082no—
469C2×C4—22no—
470C2×C4—32no—
471C2×C4—91no—
472C2×C4—91no—
473C2×C4—91no—
474C2×C4—182no—
475C2×C4—272no—
476C2×C4—272no—
477C2×C4—544no—
478C2×C4C2×D3×D911yescharacteristic
479C2×C4C2×D3×D911yescharacteristic
480C2×C3—21no—
481C2×C3—21no—
482C2×C3—21no—
483C2×C3—42no—
484C2×C3—41no—
485C2×C3—41no—
486C2×C3—122no—
487C2×C3—361no—
488C2×C3C2×#369911yescharacteristic
489C2×C3#6371011yescharacteristic
490C2×C3#41×D911yescharacteristic
491C2×C3D3×#23511yescharacteristic
492C2×C3#6430711yescharacteristic
493C2×C3C2×#380511yescharacteristic
494D3—42no—
495D3—121no—
496C22—21no—
497C22—21no—
498C22—21no—
499C22—62no—
500C22—62no—
501C22—62no—
502C22—181no—
503C22—181no—
504C22—181no—
505C22C22×D3×D911yescharacteristic
506C4—21no—
507C4—62no—
508C4—92no—
509C4—181no—
510C4—181no—
511C4—542no—
512C4—542no—
513C4#8866312yes—
514C3—21no—
515C3#1058971811yescharacteristic, complemented
516C3#1059502611yescharacteristic
517C2—41no—
518C2—122no—
519C2—361no—
520C2D9×#23511yescharacteristic
521C2#1172247611yescharacteristic
522C2C2×#8866311yescharacteristic
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. #11721815C2semidirect
2. C9×#930C2semidirect
3. C3×C4×#611C2semidirect
4. #11721281C2semidirect
5. #11721263C2semidirect
6. #11721044C2semidirect
7. #11721036C2semidirect
8. D12×Dic9C2semidirect
9. #11721011C2semidirect
10. #11720969C2semidirect
11. #11720907C2semidirect
53. C4×#611D3semidirect
158. #930D9semidirect
330. C3×C9#555semidirect
460. C9#5563semidirect
515. C3#10589718semidirect

Maximal subgroups

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

subgroupindex
#117218152
C9×#9302
C3×C4×#6112
#117212812
#117212632
#117210442
#117210362
D12×Dic92
#117210112
#117209692
#117209072
#105950263
#105897183