Finite Group Databasebeta

← #13148749

Subgroups — #13148749

Order 1728.

632 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#11721746C211yesmaximal, characteristic, complemented
2C9×#897C211yesmaximal, characteristic
3C3×#63449C211yesmaximal, characteristic, complemented
4#11720965C212yesmaximal
5#11720963C212yesmaximal
6#11720960C211yesmaximal, characteristic
7#11720959C211yesmaximal, characteristic
8#11720935C212yesmaximal, complemented
9#11720930C211yesmaximal, characteristic, complemented
10#11720918C212yesmaximal, complemented
11Dic6×Dic9C211yesmaximal, characteristic, complemented
12#10594345—31nomaximal
13#10589727—31nomaximal
14C2×C4×D3×C9C2211yescharacteristic
15C2×C9×Dic6C2211yescharacteristic
16C2×C3×Dic18C2211yescharacteristic
17C3×C9×C4⋊C4C2211yescharacteristic
18C4⋊(C3⋊Dic9)C2211yescharacteristic
19C4⋊(C3⋊Dic9)C2212yes—
20#88534C2212yes—
21C4×C3⋊Dic9C2211yescharacteristic
22C9×(C2×D3)⋊C4C2212yes—
23C9×C4⋊Dic3C2212yes—
24C3×Dic9⋊C4C2212yes—
25C3×C4×Dic9C2212yes—
26C3×C4×Dic9C2211yescharacteristic
27(C2×D3)⋊Dic9C2212yes—
28(C2×D3)⋊Dic9C2211yescharacteristic
29(C2×D3)⋊Dic9C2211yescharacteristic
30Dic3⋊Dic9C2212yes—
31Dic3⋊Dic9C2211yescharacteristic
32Dic3⋊Dic9C2211yescharacteristic
33#88442C2212yes—
34#88442C2212yes—
35#88442C2212yes—
36Dic3×Dic9C2212yes—
37#64099—31no—
38C3×#897D311yescharacteristic
39C3×#887—31no—
40#63873—31no—
41#63869—32no—
42#63861—32no—
43#63859—32no—
44#63856—31no—
45#63855—31no—
46#63847—32no—
47Dic3×Dic6—31no—
48C9×C22⋊Q8—31no—
49Q8×Dic9—31no—
50#63506—31no—
51#63488—31no—
52#63486—31no—
53#63452—31no—
54#63452—32no—
55#63452—61no—
56#63451—32no—
57#63449D311yescharacteristic, complemented
58#63441—32no—
59#63441—32no—
60C22×D3×C9C2311yescharacteristic
61C2×C3×C4×C9C2312yes—
62C2×C3×C4×C9C2311yescharacteristic
63C2×C3⋊Dic9C2312yes—
64C2×C3⋊Dic9C2311yescharacteristic
65C2×C3⋊Dic9C2311yescharacteristic
66C2×C9×Dic3C2312yes—
67C2×C9×Dic3C2311yescharacteristic
68C2×C3×Dic9C2312yes—
69C2×C3×Dic9C2311yescharacteristic
70C2×C3×Dic9C2311yescharacteristic
71C4×D3×C9—22no—
72C9×Dic6—22no—
73C3×Dic18—22no—
74#5625—91no—
75#5583—91no—
76C2×C3×C4×D3C2×D311yescharacteristic
77C2×C3×Dic6—31no—
78C2×C3×Dic6C2×D311yescharacteristic
79C32×C4⋊C4C2×D311yescharacteristic
80C4⋊(C3⋊Dic3)—31no—
81C4⋊(C3⋊Dic3)—32no—
82#3750—32no—
83C4×C3⋊Dic3—31no—
84C3×(C2×D3)⋊C4C2×D312yes—
85C3×C4⋊Dic3C2×D312yes—
86C3×Dic3⋊C4—32no—
87C3×C4×Dic3—31no—
88C3×C4×Dic3—32no—
89#3724—32no—
90#3724—32no—
91#3724—32no—
92Dic3⋊Dic3—32no—
93Dic3⋊Dic3—31no—
94Dic3⋊Dic3—31no—
95(C2×D3)⋊Dic3—31no—
96(C2×D3)⋊Dic3—31no—
97(C2×D3)⋊Dic3—32no—
98Dic32—32no—
99C2×Q8×C9—31no—
100C22×C4×C9—31no—
101C2×Dic18C2×D311yescharacteristic
102C9×C4⋊C4—21no—
103C9×C4⋊C4—32no—
104C9×C4⋊C4C2×D311yescharacteristic
105C9×C22⋊C4—32no—
106#3676—31no—
107#3676—31no—
108#3676—32no—
109C4⋊Dic9—31no—
110C4⋊Dic9—32no—
111C4⋊Dic9—31no—
112C4⋊Dic9—31no—
113C4⋊Dic9—32no—
114C4⋊Dic9—62no—
115C4⋊Dic9—61no—
116Dic9⋊C4—32no—
117Dic9⋊C4—32no—
118Dic9⋊C4—32no—
119Dic9⋊C4—32no—
120Dic9⋊C4—62no—
121Dic9⋊C4C2×D312yes—
122C4×Dic9—31no—
123C4×Dic9—32no—
124C4×Dic9—61no—
125C4×Dic9C2×D311yescharacteristic
126C4×Dic9C2×D312yes—
127C22×C3×C9C2411yescharacteristic
128C2×D3×C9—21no—
129C2×D3×C9—21no—
130C2×D3×C9—21no—
131C3×C4×C9—22no—
132C3×C4×C9#4112yes—
133C3⋊Dic9—22no—
134C3⋊Dic9—21no—
135C3⋊Dic9—21no—
136C9×Dic3—22no—
137C9×Dic3—21no—
138C3×Dic9—22no—
139C3×Dic9—21no—
140C3×Dic9—21no—
141C3×C22⋊Q8—31no—
142C3×C4×Q8—91no—
143#946—91no—
144Q8×Dic3—91no—
145#944—91no—
146#944—91no—
147#926—91no—
148#924—91no—
149#898—92no—
150#897D911yescharacteristic
151#890—92no—
152#890—91no—
153#890—91no—
154#890—181no—
155#889—92no—
156#889—92no—
157#887—31no—
158#879—92no—
159#879—92no—
160#876—92no—
161#872—91no—
162#870—92no—
163C4×Dic6—91no—
164C22×C3×D3C22×D311yescharacteristic
165C2×C32×C4C22×D312yes—
166C2×C32×C4C22×D311yescharacteristic
167C2×C3⋊Dic3—32no—
168C2×C3⋊Dic3—31no—
169C2×C3⋊Dic3—31no—
170C2×C3×Dic3—31no—
171C2×C3×Dic3—31no—
172C2×C3×Dic3—32no—
173C2×C3×Dic3C22×D312yes—
174C2×C3×Dic3C22×D311yescharacteristic
175C3×C4×D3—22no—
176C3×Dic6—22no—
177C3×Dic6—62no—
178C23×C9—31no—
179Q8×C9—62no—
180C2×C4×C9—21no—
181C2×C4×C9—22no—
182C2×C4×C9—32no—
183C2×C4×C9—31no—
184C2×C4×C9—62no—
185C2×C4×C9C22×D312yes—
186C2×C4×C9C22×D311yescharacteristic
187C2×Dic9—31no—
188C2×Dic9—31no—
189C2×Dic9—32no—
190C2×Dic9—61no—
191C2×Dic9—61no—
192C2×Dic9—62no—
193C2×Dic9C22×D311yescharacteristic
194C2×Dic9C22×D311yescharacteristic
195C2×Dic9C22×D312yes—
196Dic18—22no—
197#541—271noSylow
198C2×C3×C9C2×#4111yescharacteristic
199C2×C3×C92−1+411yescharacteristic
200C2×C3×C92−1+411yescharacteristic
201D3×C9—41no—
202C2×C3×Q8—31no—
203C2×C3×Q8—91no—
204C22×C3×C4—31no—
205C2×C4×D3C2×D911yescharacteristic
206C2×Dic6—31no—
207C2×Dic6C2×D911yescharacteristic
208C3×C4⋊C4—21no—
209C3×C4⋊C4—32no—
210C3×C4⋊C4—92no—
211C3×C4⋊C4C2×D911yescharacteristic
212C3×C4⋊C4D3211yescharacteristic
213C3×C22⋊C4—32no—
214C3×C42—91no—
215C3×C42—92no—
216#217—92no—
217#217—91no—
218#217—91no—
219(C2×D3)⋊C4—91no—
220(C2×D3)⋊C4—91no—
221(C2×D3)⋊C4—92no—
222(C2×D3)⋊C4C2×D912yes—
223C4⋊Dic3—92no—
224C4⋊Dic3—91no—
225C4⋊Dic3—91no—
226C4⋊Dic3—91no—
227C4⋊Dic3—92no—
228C4⋊Dic3—92no—
229C4⋊Dic3—91no—
230C4⋊Dic3—181no—
231C4⋊Dic3—182no—
232C4⋊Dic3C2×D912yes—
233Dic3⋊C4—32no—
234Dic3⋊C4—92no—
235Dic3⋊C4—92no—
236Dic3⋊C4—92no—
237Dic3⋊C4—92no—
238Dic3⋊C4—92no—
239Dic3⋊C4—92no—
240Dic3⋊C4—91no—
241Dic3⋊C4—92no—
242Dic3⋊C4—92no—
243Dic3⋊C4—91no—
244Dic3⋊C4—92no—
245Dic3⋊C4—182no—
246C4×Dic3—32no—
247C4×Dic3—31no—
248C4×Dic3—92no—
249C4×Dic3—91no—
250C4×Dic3—91no—
251C4×Dic3—92no—
252C4×Dic3—181no—
253C22×C32C23×D311yescharacteristic
254C2×C3×D3—21no—
255C2×C3×D3—21no—
256C2×C3×D3—21no—
257C32×C4—22no—
258C32×C4#23712yes—
259C3⋊Dic3—62no—
260C3⋊Dic3—61no—
261C3⋊Dic3—61no—
262C3×Dic3—22no—
263C3×Dic3—21no—
264C3×Dic3—62no—
265C3×Dic3—61no—
266C3×Dic3—61no—
267C22×C9—21no—
268C22×C9—61no—
269C22×C9—61no—
270C22×C9—61no—
271C22×C9C23×D311yescharacteristic
272C4×C9—22no—
273C4×C9—22no—
274C4×C9—42no—
275C4×C9—61no—
276C4×C9—62no—
277C4×C9#23512yes—
278Dic9—22no—
279Dic9—21no—
280Dic9—21no—
281Dic9—62no—
282Dic9—61no—
283Dic9—61no—
284Dic9—121no—
285Dic9—121no—
286Dic9—122no—
287C4⋊Q8—271no—
288#126—272no—
289#126—272no—
290#125—272no—
291#125—272no—
292#125—271no—
293C22⋊Q8—31no—
294C22⋊Q8—271no—
295C4×Q8—91no—
296C4×Q8—271no—
297#117—271no—
298C3×C9#54111yescharacteristic, complemented, Sylow
299C23×C3—31no—
300C22×D3C22×D911yescharacteristic
301C3×Q8—62no—
302C3×Q8—182no—
303C2×C3×C4—21no—
304C2×C3×C4—22no—
305C2×C3×C4—32no—
306C2×C3×C4—31no—
307C2×C3×C4—62no—
308C2×C3×C4—91no—
309C2×C3×C4—91no—
310C2×C3×C4—92no—
311C2×C3×C4C22×D912yes—
312C2×C3×C4C22×D911yescharacteristic
313C2×C3×C4C2×D3212yes—
314C2×C3×C4C2×D3211yescharacteristic
315C2×Dic3—31no—
316C2×Dic3—31no—
317C2×Dic3—32no—
318C2×Dic3—91no—
319C2×Dic3—91no—
320C2×Dic3—92no—
321C2×Dic3—91no—
322C2×Dic3—92no—
323C2×Dic3—91no—
324C2×Dic3—181no—
325C2×Dic3—181no—
326C2×Dic3—182no—
327C2×Dic3C22×D912yes—
328C2×Dic3C22×D911yescharacteristic
329C4×D3—22no—
330Dic6—22no—
331Dic6—62no—
332C2×C32C2×#23711yescharacteristic
333C2×C32#100711yescharacteristic
334C2×C32#101011yescharacteristic
335C3×D3—41no—
336C2×C9—21no—
337C2×C9—21no—
338C2×C9—21no—
339C2×C9—121no—
340C2×C9C2×#23511yescharacteristic
341C2×C9#100711yescharacteristic
342C2×C9#101011yescharacteristic
343C2×Q8—31no—
344C2×Q8—91no—
345C22×C4—31no—
346C4⋊C4—32no—
347C4⋊C4—92no—
348C4⋊C4—271no—
349C4⋊C4—271no—
350C4⋊C4—272no—
351C4⋊C4—272no—
352C4⋊C4—272no—
353C4⋊C4—272no—
354C4⋊C4—271no—
355C4⋊C4—272no—
356C4⋊C4—272no—
357C4⋊C4D3×D911yescharacteristic
358C22⋊C4—32no—
359C22⋊C4—271no—
360C22⋊C4—271no—
361C22⋊C4—272no—
362C42—91no—
363C42—92no—
364C42—272no—
365C42—271no—
366C22×C3—21no—
367C22×C3—61no—
368C22×C3—61no—
369C22×C3—61no—
370C22×C3C23×D911yescharacteristic
371C22×C3C22×D3211yescharacteristic
372C2×D3—21no—
373C2×D3—21no—
374C2×D3—21no—
375C3×C4—22no—
376C3×C4—22no—
377C3×C4—22no—
378C3×C4—42no—
379C3×C4—61no—
380C3×C4—62no—
381C3×C4—182no—
382C3×C4—181no—
383C3×C4—181no—
384C3×C4#369912yes—
385C3×C4#379512yes—
386Dic3—22no—
387Dic3—21no—
388Dic3—62no—
389Dic3—61no—
390Dic3—61no—
391Dic3—181no—
392Dic3—181no—
393Dic3—182no—
394Dic3—182no—
395Dic3—181no—
396Dic3—181no—
397Dic3—361no—
398Dic3—362no—
399Dic3—361no—
400C32#562511yescharacteristic
401C9—21no—
402C9#558311yescharacteristic, complemented
403C23—31no—
404Q8—62no—
405Q8—182no—
406C2×C4—32no—
407C2×C4—31no—
408C2×C4—62no—
409C2×C4—91no—
410C2×C4—91no—
411C2×C4—92no—
412C2×C4—272no—
413C2×C4—271no—
414C2×C4—271no—
415C2×C4C2×D3×D912yes—
416C2×C4C2×D3×D911yescharacteristic
417C2×C3—21no—
418C2×C3—21no—
419C2×C3—21no—
420C2×C3—121no—
421C2×C3C2×#369911yescharacteristic
422C2×C3#6371311yescharacteristic
423C2×C3#6371611yescharacteristic
424C2×C3C2×#379511yescharacteristic
425C2×C3#6430911yescharacteristic
426C2×C3#6431611yescharacteristic
427D3—41no—
428C22—61no—
429C22—61no—
430C22—61no—
431C22C22×D3×D911yescharacteristic
432C4—22no—
433C4—61no—
434C4—62no—
435C4—181no—
436C4—182no—
437C4—181no—
438C4—542no—
439C4—541no—
440C4—541no—
441C4#8863812yes—
442C3—21no—
443C3#1058972711yescharacteristic, complemented
444C3#1059434511yescharacteristic
445C2—121no—
446C2C2×#8863811yescharacteristic
447C2#1172249411yescharacteristic
448C2#1172250911yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC3×C9×C4⋊C4C22
socleC22×C32C23×D3
prime radicalC22×C3×D3C22×D3
hypercenterC4⋊C4D3×D9
hypocenterC3×C9#541
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. #11721746C2semidirect
3. C3×#63449C2semidirect
8. #11720935C2semidirect
9. #11720930C2semidirect
10. #11720918C2semidirect
11. Dic6×Dic9C2semidirect
57. #63449D3semidirect
298. C3×C9#541semidirect
402. C9#5583semidirect
443. C3#10589727semidirect

Maximal subgroups

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

subgroupindex
#117217462
C9×#8972
C3×#634492
#117209652
#117209632
#117209602
#117209592
#117209352
#117209302
#117209182
Dic6×Dic92
#105943453
#105897273