Finite Group Databasebeta

← #13147689

Subgroups — #13147689

Order 1728.

736 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11721712C211yesmaximal, characteristic, complemented
2C9×#875C211yesmaximal, characteristic, complemented
3C3×#63432C211yesmaximal, characteristic, complemented
4#11721039C212yesmaximal, complemented
5#11721024C212yesmaximal, complemented
6#11721020C212yesmaximal, complemented
7#11721017C212yesmaximal, complemented
8#11720986C212yesmaximal, complemented
9#11720958C212yesmaximal, complemented
10#10593953—31nomaximal
11#10589786—31nomaximal
12C2×C4⋊(C3⋊D9)C2211yescharacteristic
13C2×#6226C2211yescharacteristic
14C2×C9×D12C2211yescharacteristic
15C2×C9×Dic6C2211yescharacteristic
16C2×#6195C2212yes—
17C2×#6195C2212yes—
18C2×D3×Dic9C2212yes—
19C2×(C3⋊D9)⋊C4C2212yes—
20C3×C42×C9C2211yescharacteristic
21#88536C2212yes—
22#88536C2212yes—
23C9×(C2×D3)⋊C4C2212yes—
24C9×(C2×D3)⋊C4C2212yes—
25C3×C4⋊Dic9C2211yescharacteristic
26C3×C4⋊Dic9C2211yescharacteristic
27C3×Dic9⋊C4C2212yes—
28C3×Dic9⋊C4C2212yes—
29(C2×D3)⋊Dic9C2212yes—
30#88445C2212yes—
31Dic3⋊Dic9C2212yes—
32#88442C2212yes—
33#64084—31no—
34C3×#875D311yescharacteristic
35C3×#870—31no—
36#63910—32no—
37#63896—32no—
38#63890—32no—
39#63888—32no—
40#63886—32no—
41#63854—32no—
42C9×#124—31no—
43#63508—32no—
44#63501—32no—
45#63448—32no—
46#63446—32no—
47#63445—32no—
48#63440—32no—
49#63437—31no—
50#63437—61no—
51#63432D311yescharacteristic, complemented
52C22×C3⋊D9C2312yes—
53C22×D3×C9C2312yes—
54C2×C3×C4×C9C2311yescharacteristic
55C2×C3×C4×C9C2311yescharacteristic
56C2×C3×C4×C9C2311yescharacteristic
57C2×C3⋊Dic9C2312yes—
58C4⋊(C3⋊D9)—41no—
59#6226—41no—
60C2×C9×Dic3C2312yes—
61C2×C3×Dic9C2312yes—
62C2×C3×Dic9C2312yes—
63C9×D12—41no—
64C9×Dic6—41no—
65#6195—42no—
66#6195—42no—
67D3×Dic9—42no—
68(C3⋊D9)⋊C4—42no—
69#5700—91no—
70#5684—91no—
71C2×C4⋊(C3⋊D3)—31no—
72C2×#634—31no—
73C2×C3×D12C2×D311yescharacteristic
74C2×C3×Dic6C2×D311yescharacteristic
75C2×#626—32no—
76C2×#626—32no—
77C2×#624—32no—
78C2×D3×Dic3—32no—
79C32×C42C2×D311yescharacteristic
80#3752—32no—
81#3752—32no—
82C3×(C2×D3)⋊C4C2×D312yes—
83C3×(C2×D3)⋊C4C2×D312yes—
84C3×C4⋊Dic3—31no—
85C3×C4⋊Dic3—31no—
86C3×Dic3⋊C4—32no—
87C3×Dic3⋊C4—32no—
88#3724—32no—
89Dic3⋊Dic3—32no—
90#3722—32no—
91(C2×D3)⋊Dic3—32no—
92C2×Q8×C9—31no—
93C2×D4×C9—31no—
94C2×#611—32no—
95C2×#611—32no—
96C22×Dic9—32no—
97C2×D36—31no—
98C2×D36—61no—
99C2×C4×D9—32no—
100C2×Dic18—31no—
101C2×Dic18—61no—
102C9×C22⋊C4—32no—
103C9×C22⋊C4—32no—
104C42×C9—21no—
105C42×C9C2×D311yescharacteristic
106#3676—32no—
107(C2×D9)⋊C4—32no—
108(C2×D9)⋊C4—32no—
109(C2×D9)⋊C4—32no—
110(C2×D9)⋊C4—62no—
111(C2×D9)⋊C4—62no—
112C4⋊Dic9—32no—
113C4⋊Dic9C2×D311yescharacteristic
114C4⋊Dic9C2×D311yescharacteristic
115Dic9⋊C4—32no—
116Dic9⋊C4C2×D312yes—
117Dic9⋊C4C2×D312yes—
118C22×C3×C9C2411yescharacteristic
119C2×C3⋊D9—22no—
120C2×C3⋊D9—22no—
121C2×C3⋊D9—22no—
122C2×D3×C9—22no—
123C2×D3×C9—22no—
124C2×D3×C9—22no—
125C3×C4×C9—21no—
126C3×C4×C9—21no—
127C3×C4×C9—21no—
128C3⋊Dic9—22no—
129C9×Dic3—22no—
130C3×Dic9—22no—
131C3×Dic9—22no—
132C3×#125—91no—
133C3×#124—31no—
134#946—92no—
135#939—92no—
136#896—92no—
137#896—92no—
138#895—92no—
139#895—92no—
140#894—92no—
141#894—92no—
142#886—92no—
143#884—92no—
144#883—92no—
145#878—92no—
146#875—91no—
147#875—91no—
148#875—181no—
149#875D911yescharacteristic, complemented
150#870—31no—
151C22×C3⋊D3—32no—
152C22×C3×D3C22×D312yes—
153C2×C32×C4C22×D311yescharacteristic
154C2×C32×C4C22×D311yescharacteristic
155C2×C32×C4C22×D311yescharacteristic
156C2×C3⋊Dic3—32no—
157C4⋊(C3⋊D3)—121no—
158#634—121no—
159C2×C3×Dic3—32no—
160C2×C3×Dic3—32no—
161C2×C3×Dic3C22×D312yes—
162C3×D12—41no—
163C3×Dic6—41no—
164#626—122no—
165#626—122no—
166#624—122no—
167D3×Dic3—122no—
168C23×C9—32no—
169C22×D9—32no—
170C22×D9—62no—
171Q8×C9—121no—
172D4×C9—121no—
173C2×C4×C9—21no—
174C2×C4×C9—21no—
175C2×C4×C9—21no—
176C2×C4×C9—32no—
177C2×C4×C9C22×D311yescharacteristic
178C2×C4×C9C22×D311yescharacteristic
179C2×C4×C9C22×D311yescharacteristic
180#611—122no—
181#611—122no—
182C2×Dic9—32no—
183C2×Dic9—62no—
184C2×Dic9—122no—
185C2×Dic9C22×D312yes—
186C2×Dic9C22×D312yes—
187D36—121no—
188D36—122no—
189C4×D9—122no—
190Dic18—121no—
191Dic18—122no—
192#562—271noSylow
193C2×C3×C9H(5,𝔽2)11yescharacteristic
194C2×C3×C9H(5,𝔽2)11yescharacteristic
195C2×C3×C92−1+411yescharacteristic
196C3⋊D9—42no—
197D3×C9—42no—
198C2×C3×Q8—31no—
199C2×C3×D4—31no—
200C2×#67—92no—
201C2×#67—92no—
202C22×Dic3—92no—
203C2×D12—92no—
204C2×D12—92no—
205C2×D12—91no—
206C2×D12—91no—
207C2×D12—181no—
208C2×D12C2×D911yescharacteristic
209C2×C4×D3—92no—
210C2×C4×D3—92no—
211C2×C4×D3—92no—
212C2×Dic6—91no—
213C2×Dic6—91no—
214C2×Dic6—181no—
215C2×Dic6C2×D911yescharacteristic
216C3×C4⋊C4—91no—
217C3×C4⋊C4—91no—
218C3×C4⋊C4—92no—
219C3×C4⋊C4—92no—
220C3×C22⋊C4—32no—
221C3×C22⋊C4—32no—
222C3×C42—21no—
223C3×C42C2×D911yescharacteristic
224C3×C42D3211yescharacteristic
225#217—92no—
226(C2×D3)⋊C4—92no—
227(C2×D3)⋊C4—92no—
228(C2×D3)⋊C4—92no—
229(C2×D3)⋊C4—92no—
230(C2×D3)⋊C4—92no—
231(C2×D3)⋊C4—92no—
232(C2×D3)⋊C4—92no—
233(C2×D3)⋊C4—182no—
234(C2×D3)⋊C4—182no—
235(C2×D3)⋊C4C2×D912yes—
236(C2×D3)⋊C4C2×D912yes—
237C4⋊Dic3—31no—
238C4⋊Dic3—31no—
239C4⋊Dic3—92no—
240Dic3⋊C4—32no—
241Dic3⋊C4—32no—
242Dic3⋊C4—92no—
243Dic3⋊C4—92no—
244Dic3⋊C4—92no—
245C22×C32C23×D311yescharacteristic
246C2×C3⋊D3—62no—
247C2×C3⋊D3—62no—
248C2×C3⋊D3—62no—
249C2×C3×D3—22no—
250C2×C3×D3—22no—
251C2×C3×D3—22no—
252C32×C4—21no—
253C32×C4—21no—
254C32×C4—21no—
255C3⋊Dic3—62no—
256C3×Dic3—22no—
257C3×Dic3—62no—
258C3×Dic3—62no—
259C22×C9—21no—
260C22×C9—62no—
261C22×C9—62no—
262C22×C9—62no—
263C22×C9C23×D311yescharacteristic
264C2×D9—62no—
265C2×D9—62no—
266C2×D9—62no—
267C2×D9—122no—
268C2×D9—122no—
269C2×D9—122no—
270C4×C9—22no—
271C4×C9—21no—
272C4×C9—21no—
273C4×C9—21no—
274C4×C9—41no—
275C4×C9—41no—
276C4×C9—62no—
277Dic9—22no—
278Dic9—22no—
279Dic9—62no—
280Dic9—122no—
281#125—91no—
282#124—31no—
283#124—271no—
284#123—272no—
285#123—272no—
286C22⋊Q8—272no—
287C22⋊Q8—272no—
288#121—272no—
289#121—272no—
290C3×C9#56211yescharacteristic, complemented, Sylow
291C23×C3—32no—
292C22×D3—92no—
293C22×D3—92no—
294C22×D3—182no—
295C22×D3C22×D912yes—
296C3×Q8—121no—
297C3×D4—121no—
298C2×C3×C4—21no—
299C2×C3×C4—21no—
300C2×C3×C4—21no—
301C2×C3×C4—32no—
302C2×C3×C4—92no—
303C2×C3×C4—92no—
304C2×C3×C4C22×D911yescharacteristic
305C2×C3×C4C22×D911yescharacteristic
306C2×C3×C4C22×D911yescharacteristic
307C2×C3×C4C2×D3211yescharacteristic
308C2×C3×C4C2×D3211yescharacteristic
309C2×C3×C4C2×D3211yescharacteristic
310#67—362no—
311#67—362no—
312C2×Dic3—32no—
313C2×Dic3—32no—
314C2×Dic3—92no—
315C2×Dic3—92no—
316C2×Dic3—182no—
317C2×Dic3—362no—
318C2×Dic3C22×D912yes—
319D12—41no—
320D12—361no—
321D12—361no—
322D12—362no—
323D12—362no—
324D12—362no—
325C4×D3—362no—
326C4×D3—362no—
327C4×D3—362no—
328Dic6—41no—
329Dic6—361no—
330Dic6—361no—
331Dic6—362no—
332C2×C32#100411yescharacteristic
333C2×C32#100911yescharacteristic
334C2×C32#101011yescharacteristic
335C3⋊D3—122no—
336C3×D3—42no—
337C2×C9—21no—
338C2×C9—21no—
339C2×C9—21no—
340C2×C9—122no—
341C2×C9#100711yescharacteristic
342C2×C9#100911yescharacteristic
343C2×C9#100911yescharacteristic
344D9—122no—
345D9—242no—
346C2×Q8—31no—
347C2×Q8—271no—
348C2×D4—31no—
349C2×D4—272no—
350C2×D4—272no—
351C2×D4—271no—
352C22×C4—272no—
353C22×C4—272no—
354C4⋊C4—92no—
355C4⋊C4—92no—
356C4⋊C4—91no—
357C4⋊C4—91no—
358C4⋊C4—272no—
359C4⋊C4—272no—
360C22⋊C4—32no—
361C22⋊C4—32no—
362C22⋊C4—272no—
363C22⋊C4—272no—
364C22⋊C4—272no—
365C22⋊C4—272no—
366C42D3×D911yescharacteristic
367C22×C3—21no—
368C22×C3—62no—
369C22×C3—62no—
370C22×C3—62no—
371C22×C3C23×D911yescharacteristic
372C22×C3C22×D3211yescharacteristic
373C2×D3—22no—
374C2×D3—22no—
375C2×D3—22no—
376C2×D3—182no—
377C2×D3—182no—
378C2×D3—182no—
379C2×D3—182no—
380C2×D3—182no—
381C2×D3—182no—
382C2×D3—362no—
383C2×D3—362no—
384C2×D3—362no—
385C3×C4—22no—
386C3×C4—21no—
387C3×C4—21no—
388C3×C4—21no—
389C3×C4—21no—
390C3×C4—21no—
391C3×C4—21no—
392C3×C4—41no—
393C3×C4—41no—
394C3×C4—62no—
395C3×C4—182no—
396C3×C4—182no—
397Dic3—22no—
398Dic3—62no—
399Dic3—62no—
400Dic3—182no—
401Dic3—182no—
402Dic3—362no—
403C32#568411yescharacteristic
404C9—21no—
405C9#570011yescharacteristic, complemented
406C23—32no—
407C23—272no—
408Q8—121no—
409Q8—1081no—
410D4—121no—
411D4—1081no—
412D4—1082no—
413D4—1082no—
414C2×C4—32no—
415C2×C4—92no—
416C2×C4—92no—
417C2×C4—272no—
418C2×C4—1082no—
419C2×C4—1082no—
420C2×C4C2×D3×D911yescharacteristic
421C2×C4C2×D3×D911yescharacteristic
422C2×C4C2×D3×D911yescharacteristic
423C2×C3—21no—
424C2×C3—21no—
425C2×C3—21no—
426C2×C3—122no—
427C2×C3#6371011yescharacteristic
428C2×C3#6371511yescharacteristic
429C2×C3#6371611yescharacteristic
430C2×C3#6429711yescharacteristic
431C2×C3#6430711yescharacteristic
432C2×C3#6430811yescharacteristic
433D3—42no—
434D3—362no—
435D3—362no—
436D3—722no—
437C22—62no—
438C22—62no—
439C22—62no—
440C22—542no—
441C22—542no—
442C22—542no—
443C22C22×D3×D911yescharacteristic
444C4—21no—
445C4—21no—
446C4—21no—
447C4—62no—
448C4—182no—
449C4—182no—
450C4—542no—
451C3—21no—
452C3#1058978611yescharacteristic, complemented
453C3#1059395311yescharacteristic
454C2—122no—
455C2—1082no—
456C2#1172246411yescharacteristic
457C2#1172247611yescharacteristic
458C2#1172247811yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC3×C42×C9C22
socleC22×C32C23×D3
hypercenterC42D3×D9
hypocenterC3×C9#562
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C42

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #11721712C2semidirect
2. C9×#875C2semidirect
3. C3×#63432C2semidirect
4. #11721039C2semidirect
5. #11721024C2semidirect
6. #11721020C2semidirect
7. #11721017C2semidirect
8. #11720986C2semidirect
9. #11720958C2semidirect
51. #63432D3semidirect
149. #875D9semidirect
290. C3×C9#562semidirect
405. C9#5700semidirect
452. C3#10589786semidirect

Maximal subgroups

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

subgroupindex
#117217122
C9×#8752
C3×#634322
#117210392
#117210242
#117210202
#117210172
#117209862
#117209582
#105939533
#105897863