Finite Group Databasebeta

← #13147686

Subgroups — #13147686

Order 1728.

716 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11721707C211yesmaximal, characteristic, complemented
2C9×#875C211yesmaximal, characteristic, complemented
3C3×#63437C211yesmaximal, characteristic, complemented
4#11721021C212yesmaximal, complemented
5#11721018C212yesmaximal, complemented
6#11721016C212yesmaximal, complemented
7#11720989C212yesmaximal, complemented
8#11720984C212yesmaximal, complemented
9#11720982C212yesmaximal, complemented
10#10593985—31nomaximal
11#10589786—31nomaximal
12C2×C9×D12C2211yescharacteristic
13C2×C3×D36C2211yescharacteristic
14C2×C9×Dic6C2211yescharacteristic
15C2×C3×Dic18C2211yescharacteristic
16C2×#6194C2212yes—
17C2×#6194C2212yes—
18C2×D3×Dic9C2212yes—
19C2×Dic3×D9C2212yes—
20C3×C42×C9C2211yescharacteristic
21C4⋊(C3⋊Dic9)C2211yescharacteristic
22C4⋊(C3⋊Dic9)C2211yescharacteristic
23#88534C2212yes—
24#88534C2212yes—
25C9×(C2×D3)⋊C4C2212yes—
26C9×(C2×D3)⋊C4C2212yes—
27C3×(C2×D9)⋊C4C2212yes—
28C3×(C2×D9)⋊C4C2212yes—
29(C2×D3)⋊Dic9C2212yes—
30(C2×D9)⋊Dic3C2212yes—
31Dic3⋊Dic9C2212yes—
32Dic9⋊Dic3C2212yes—
33#64079—31no—
34C3×#875—31no—
35C3×#875D311yescharacteristic
36#63891—32no—
37#63891—32no—
38#63887—32no—
39#63887—32no—
40#63885—32no—
41#63885—32no—
42C9×#124—31no—
43#63508—32no—
44#63501—32no—
45#63448—32no—
46#63446—32no—
47#63445—32no—
48#63440—32no—
49#63437D311yescharacteristic, complemented
50#63432—31no—
51#63432—61no—
52C22×D3×C9C2312yes—
53C22×C3×D9C2312yes—
54C2×C3×C4×C9C2311yescharacteristic
55C2×C3×C4×C9C2311yescharacteristic
56C2×C3×C4×C9C2311yescharacteristic
57C2×C3⋊Dic9C2312yes—
58C2×C3⋊Dic9C2312yes—
59C2×C9×Dic3C2312yes—
60C2×C3×Dic9C2312yes—
61C9×D12—41no—
62C3×D36—41no—
63C9×Dic6—41no—
64C3×Dic18—41no—
65#6194—42no—
66#6194—42no—
67D3×Dic9—42no—
68Dic3×D9—42no—
69#5684—91no—
70#5684—91no—
71C2×C3×D12—31no—
72C2×C3×D12C2×D311yescharacteristic
73C2×C3×Dic6—31no—
74C2×C3×Dic6C2×D311yescharacteristic
75C2×#625—32no—
76C2×#625—32no—
77C2×D3×Dic3—32no—
78C2×D3×Dic3—32no—
79C32×C42C2×D311yescharacteristic
80C4⋊(C3⋊Dic3)—31no—
81C4⋊(C3⋊Dic3)—31no—
82#3750—32no—
83#3750—32no—
84C3×(C2×D3)⋊C4—32no—
85C3×(C2×D3)⋊C4—32no—
86C3×(C2×D3)⋊C4C2×D312yes—
87C3×(C2×D3)⋊C4C2×D312yes—
88Dic3⋊Dic3—32no—
89Dic3⋊Dic3—32no—
90(C2×D3)⋊Dic3—32no—
91(C2×D3)⋊Dic3—32no—
92C2×Q8×C9—31no—
93C2×D4×C9—31no—
94C2×#611—32no—
95C2×#611—32no—
96C22×Dic9—32no—
97C2×D36C2×D311yescharacteristic
98C2×C4×D9—32no—
99C2×Dic18C2×D311yescharacteristic
100C9×C22⋊C4—32no—
101C9×C22⋊C4—32no—
102C42×C9—21no—
103C42×C9C2×D311yescharacteristic
104#3676—32no—
105(C2×D9)⋊C4—32no—
106(C2×D9)⋊C4C2×D312yes—
107(C2×D9)⋊C4C2×D312yes—
108C4⋊Dic9—31no—
109C4⋊Dic9—31no—
110C4⋊Dic9—32no—
111C4⋊Dic9—61no—
112C4⋊Dic9—61no—
113Dic9⋊C4—32no—
114Dic9⋊C4—32no—
115Dic9⋊C4—32no—
116Dic9⋊C4—62no—
117Dic9⋊C4—62no—
118C22×C3×C9C2411yescharacteristic
119C2×D3×C9—22no—
120C2×D3×C9—22no—
121C2×D3×C9—22no—
122C2×C3×D9—22no—
123C2×C3×D9—22no—
124C2×C3×D9—22no—
125C3×C4×C9—21no—
126C3×C4×C9—21no—
127C3×C4×C9—21no—
128C3⋊Dic9—22no—
129C3⋊Dic9—22no—
130C9×Dic3—22no—
131C3×Dic9—22no—
132C3×#124—31no—
133C3×#124—91no—
134#946—92no—
135#946—92no—
136#939—92no—
137#939—92no—
138#886—92no—
139#886—92no—
140#884—92no—
141#884—92no—
142#883—92no—
143#883—92no—
144#878—92no—
145#878—92no—
146#875—31no—
147#875D911yescharacteristic, complemented
148#870—91no—
149#870—91no—
150#870—181no—
151C22×C3×D3—32no—
152C22×C3×D3C22×D312yes—
153C2×C32×C4C22×D311yescharacteristic
154C2×C32×C4C22×D311yescharacteristic
155C2×C32×C4C22×D311yescharacteristic
156C2×C3⋊Dic3—32no—
157C2×C3⋊Dic3—32no—
158C2×C3×Dic3—32no—
159C2×C3×Dic3C22×D312yes—
160C3×D12—41no—
161C3×D12—121no—
162C3×Dic6—41no—
163C3×Dic6—121no—
164#625—122no—
165#625—122no—
166D3×Dic3—122no—
167D3×Dic3—122no—
168C23×C9—32no—
169C22×D9C22×D312yes—
170Q8×C9—121no—
171D4×C9—121no—
172C2×C4×C9—21no—
173C2×C4×C9—21no—
174C2×C4×C9—21no—
175C2×C4×C9—32no—
176C2×C4×C9C22×D311yescharacteristic
177C2×C4×C9C22×D311yescharacteristic
178C2×C4×C9C22×D311yescharacteristic
179#611—122no—
180#611—122no—
181C2×Dic9—32no—
182C2×Dic9—32no—
183C2×Dic9—62no—
184C2×Dic9—62no—
185C2×Dic9—122no—
186C2×Dic9C22×D312yes—
187D36—41no—
188C4×D9—122no—
189Dic18—41no—
190#562—271noSylow
191C2×C3×C9H(5,𝔽2)11yescharacteristic
192C2×C3×C9H(5,𝔽2)11yescharacteristic
193C2×C3×C92−1+411yescharacteristic
194D3×C9—42no—
195C3×D9—42no—
196C2×C3×Q8—31no—
197C2×C3×Q8—91no—
198C2×C3×D4—31no—
199C2×C3×D4—91no—
200C2×#67—92no—
201C2×#67—92no—
202C2×#67—92no—
203C2×#67—92no—
204C22×Dic3—92no—
205C22×Dic3—92no—
206C2×D12—31no—
207C2×D12C2×D911yescharacteristic
208C2×C4×D3—92no—
209C2×C4×D3—92no—
210C2×Dic6—31no—
211C2×Dic6C2×D911yescharacteristic
212C3×C22⋊C4—32no—
213C3×C22⋊C4—32no—
214C3×C22⋊C4—92no—
215C3×C22⋊C4—92no—
216C3×C42—21no—
217C3×C42C2×D911yescharacteristic
218C3×C42D3211yescharacteristic
219#217—92no—
220#217—92no—
221(C2×D3)⋊C4—32no—
222(C2×D3)⋊C4—32no—
223(C2×D3)⋊C4—92no—
224(C2×D3)⋊C4—92no—
225(C2×D3)⋊C4C2×D912yes—
226(C2×D3)⋊C4C2×D912yes—
227C4⋊Dic3—92no—
228C4⋊Dic3—91no—
229C4⋊Dic3—91no—
230C4⋊Dic3—91no—
231C4⋊Dic3—91no—
232C4⋊Dic3—92no—
233C4⋊Dic3—181no—
234C4⋊Dic3—181no—
235Dic3⋊C4—92no—
236Dic3⋊C4—92no—
237Dic3⋊C4—92no—
238Dic3⋊C4—92no—
239Dic3⋊C4—92no—
240Dic3⋊C4—92no—
241Dic3⋊C4—182no—
242Dic3⋊C4—182no—
243C22×C32C23×D311yescharacteristic
244C2×C3×D3—22no—
245C2×C3×D3—22no—
246C2×C3×D3—22no—
247C2×C3×D3—62no—
248C2×C3×D3—62no—
249C2×C3×D3—62no—
250C32×C4—21no—
251C32×C4—21no—
252C32×C4—21no—
253C3⋊Dic3—62no—
254C3⋊Dic3—62no—
255C3×Dic3—22no—
256C3×Dic3—62no—
257C22×C9—21no—
258C22×C9—62no—
259C22×C9—62no—
260C22×C9—62no—
261C22×C9C23×D311yescharacteristic
262C2×D9—22no—
263C2×D9—22no—
264C2×D9—22no—
265C4×C9—22no—
266C4×C9—21no—
267C4×C9—21no—
268C4×C9—21no—
269C4×C9—41no—
270C4×C9—41no—
271C4×C9—62no—
272Dic9—22no—
273Dic9—62no—
274Dic9—62no—
275Dic9—122no—
276Dic9—122no—
277#125—271no—
278#124—31no—
279#124—91no—
280#123—272no—
281#123—272no—
282C22⋊Q8—272no—
283C22⋊Q8—272no—
284#121—272no—
285#121—272no—
286C3×C9#56211yescharacteristic, complemented, Sylow
287C23×C3—32no—
288C23×C3—92no—
289C22×D3—32no—
290C22×D3C22×D912yes—
291C3×Q8—121no—
292C3×Q8—361no—
293C3×D4—121no—
294C3×D4—361no—
295C2×C3×C4—21no—
296C2×C3×C4—21no—
297C2×C3×C4—21no—
298C2×C3×C4—32no—
299C2×C3×C4—92no—
300C2×C3×C4C22×D911yescharacteristic
301C2×C3×C4C22×D911yescharacteristic
302C2×C3×C4C22×D911yescharacteristic
303C2×C3×C4C2×D3211yescharacteristic
304C2×C3×C4C2×D3211yescharacteristic
305C2×C3×C4C2×D3211yescharacteristic
306#67—362no—
307#67—362no—
308#67—362no—
309#67—362no—
310C2×Dic3—32no—
311C2×Dic3—92no—
312C2×Dic3—92no—
313C2×Dic3—92no—
314C2×Dic3—92no—
315C2×Dic3—182no—
316C2×Dic3—182no—
317C2×Dic3—362no—
318C2×Dic3—362no—
319C2×Dic3C22×D912yes—
320D12—41no—
321D12—121no—
322C4×D3—362no—
323C4×D3—362no—
324Dic6—41no—
325Dic6—121no—
326C2×C32#100411yescharacteristic
327C2×C32#100911yescharacteristic
328C2×C32#101011yescharacteristic
329C3×D3—42no—
330C3×D3—122no—
331C2×C9—21no—
332C2×C9—21no—
333C2×C9—21no—
334C2×C9—122no—
335C2×C9#100411yescharacteristic
336C2×C9#100911yescharacteristic
337C2×C9#101011yescharacteristic
338D9—42no—
339C2×Q8—31no—
340C2×Q8—91no—
341C2×D4—31no—
342C2×D4—91no—
343C2×D4—272no—
344C2×D4—272no—
345C22×C4—272no—
346C22×C4—272no—
347C4⋊C4—272no—
348C4⋊C4—272no—
349C4⋊C4—272no—
350C4⋊C4—272no—
351C4⋊C4—271no—
352C4⋊C4—271no—
353C22⋊C4—32no—
354C22⋊C4—32no—
355C22⋊C4—92no—
356C22⋊C4—92no—
357C22⋊C4—272no—
358C22⋊C4—272no—
359C42D3×D911yescharacteristic
360C22×C3—21no—
361C22×C3—62no—
362C22×C3—62no—
363C22×C3—62no—
364C22×C3—182no—
365C22×C3—182no—
366C22×C3—182no—
367C22×C3C23×D911yescharacteristic
368C22×C3C22×D3211yescharacteristic
369C2×D3—22no—
370C2×D3—22no—
371C2×D3—22no—
372C2×D3—62no—
373C2×D3—62no—
374C2×D3—62no—
375C3×C4—22no—
376C3×C4—21no—
377C3×C4—21no—
378C3×C4—21no—
379C3×C4—21no—
380C3×C4—21no—
381C3×C4—21no—
382C3×C4—41no—
383C3×C4—41no—
384C3×C4—62no—
385C3×C4—182no—
386Dic3—22no—
387Dic3—62no—
388Dic3—182no—
389Dic3—182no—
390Dic3—182no—
391Dic3—182no—
392Dic3—362no—
393Dic3—362no—
394C32#568411yescharacteristic
395C9—21no—
396C9#568411yescharacteristic, complemented
397C23—32no—
398C23—92no—
399Q8—121no—
400Q8—361no—
401D4—121no—
402D4—361no—
403D4—1082no—
404D4—1082no—
405C2×C4—32no—
406C2×C4—92no—
407C2×C4—272no—
408C2×C4—272no—
409C2×C4—1082no—
410C2×C4—1082no—
411C2×C4C2×D3×D911yescharacteristic
412C2×C4C2×D3×D911yescharacteristic
413C2×C4C2×D3×D911yescharacteristic
414C2×C3—21no—
415C2×C3—21no—
416C2×C3—21no—
417C2×C3—122no—
418C2×C3—362no—
419C2×C3#6371011yescharacteristic
420C2×C3#6371511yescharacteristic
421C2×C3#6371611yescharacteristic
422C2×C3#6429811yescharacteristic
423C2×C3#6430711yescharacteristic
424C2×C3#6430711yescharacteristic
425D3—42no—
426D3—122no—
427C22—62no—
428C22—62no—
429C22—62no—
430C22—182no—
431C22—182no—
432C22—182no—
433C22C22×D3×D911yescharacteristic
434C4—21no—
435C4—21no—
436C4—21no—
437C4—62no—
438C4—182no—
439C4—542no—
440C4—542no—
441C3—21no—
442C3#1058978611yescharacteristic, complemented
443C3#1059398511yescharacteristic
444C2—122no—
445C2—362no—
446C2#1172246511yescharacteristic
447C2#1172247611yescharacteristic
448C2#1172247711yescharacteristic
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. #11721707C2semidirect
2. C9×#875C2semidirect
3. C3×#63437C2semidirect
4. #11721021C2semidirect
5. #11721018C2semidirect
6. #11721016C2semidirect
7. #11720989C2semidirect
8. #11720984C2semidirect
9. #11720982C2semidirect
49. #63437D3semidirect
147. #875D9semidirect
286. C3×C9#562semidirect
396. C9#5684semidirect
442. C3#10589786semidirect

Maximal subgroups

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

subgroupindex
#117217072
C9×#8752
C3×#634372
#117210212
#117210182
#117210162
#117209892
#117209842
#117209822
#105939853
#105897863