Finite Group Databasebeta

← #13148750

Subgroups — #13148750

Order 1728.

646 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11721743C211yesmaximal, characteristic, complemented
2C9×#897C211yesmaximal, characteristic
3C3×#63452C211yesmaximal, characteristic, complemented
4#11720966C212yesmaximal
5#11720962C212yesmaximal
6#11720961C211yesmaximal, characteristic
7#11720959C211yesmaximal, characteristic
8#11720937C212yesmaximal, complemented
9#11720924C211yesmaximal, characteristic, complemented
10#11720917C212yesmaximal, complemented
11#11720911C211yesmaximal, characteristic, complemented
12#10594346—31nomaximal
13#10589727—31nomaximal
14C2×#6226C2211yescharacteristic
15C2×C4×D3×C9C2211yescharacteristic
16C2×C9×Dic6C2211yescharacteristic
17C3×C9×C4⋊C4C2211yescharacteristic
18#88534C2212yes—
19C4×C3⋊Dic9C2212yes—
20C4×C3⋊Dic9C2211yescharacteristic
21C9×(C2×D3)⋊C4C2212yes—
22C9×C4⋊Dic3C2212yes—
23C3×C4⋊Dic9C2211yescharacteristic
24C3×C4⋊Dic9C2212yes—
25C3×Dic9⋊C4C2212yes—
26C3×C4×Dic9C2211yescharacteristic
27(C2×D3)⋊Dic9C2212yes—
28(C2×D3)⋊Dic9C2211yescharacteristic
29(C2×D3)⋊Dic9C2211yescharacteristic
30Dic3⋊Dic9C2212yes—
31Dic3⋊Dic9C2211yescharacteristic
32Dic3⋊Dic9C2211yescharacteristic
33Dic9⋊Dic3C2212yes—
34Dic9⋊Dic3C2212yes—
35Dic9⋊Dic3C2212yes—
36Dic3×Dic9C2212yes—
37#64096—31no—
38C3×#897D311yescharacteristic
39C3×#890—31no—
40#63872—32no—
41#63866—31no—
42#63862—32no—
43#63858—32no—
44#63857—31no—
45#63855—31no—
46#63846—32no—
47#63843—31no—
48C9×C22⋊Q8—31no—
49Q8×Dic9—31no—
50#63506—31no—
51#63488—31no—
52#63486—31no—
53#63452—32no—
54#63452D311yescharacteristic, complemented
55#63451—32no—
56#63449—31no—
57#63449—61no—
58#63441—32no—
59#63441—32no—
60C22×D3×C9C2311yescharacteristic
61C2×C3×C4×C9C2312yes—
62C2×C3×C4×C9C2311yescharacteristic
63C2×C3⋊Dic9C2311yescharacteristic
64C2×C3⋊Dic9C2312yes—
65C2×C3⋊Dic9C2311yescharacteristic
66#6226—22no—
67C2×C9×Dic3C2312yes—
68C2×C9×Dic3C2311yescharacteristic
69C2×C3×Dic9C2312yes—
70C2×C3×Dic9C2311yescharacteristic
71C2×C3×Dic9C2311yescharacteristic
72C4×D3×C9—22no—
73C9×Dic6—22no—
74#5698—91no—
75#5625—91no—
76C2×#634—31no—
77C2×C3×C4×D3C2×D311yescharacteristic
78C2×C3×Dic6C2×D311yescharacteristic
79C32×C4⋊C4C2×D311yescharacteristic
80#3750—32no—
81C4×C3⋊Dic3—31no—
82C4×C3⋊Dic3—32no—
83C3×(C2×D3)⋊C4C2×D312yes—
84C3×C4⋊Dic3—31no—
85C3×C4⋊Dic3—32no—
86C3×C4⋊Dic3C2×D312yes—
87C3×Dic3⋊C4—32no—
88C3×C4×Dic3—31no—
89Dic3⋊Dic3—32no—
90Dic3⋊Dic3—31no—
91Dic3⋊Dic3—32no—
92Dic3⋊Dic3—32no—
93Dic3⋊Dic3—31no—
94Dic3⋊Dic3—32no—
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×Dic18—31no—
102C2×Dic18—61no—
103C9×C4⋊C4—21no—
104C9×C4⋊C4—32no—
105C9×C4⋊C4C2×D311yescharacteristic
106C9×C22⋊C4—32no—
107#3676—31no—
108#3676—31no—
109#3676—32no—
110C4⋊Dic9—31no—
111C4⋊Dic9—31no—
112C4⋊Dic9—32no—
113C4⋊Dic9C2×D311yescharacteristic
114C4⋊Dic9C2×D312yes—
115Dic9⋊C4—32no—
116Dic9⋊C4—32no—
117Dic9⋊C4—32no—
118Dic9⋊C4—32no—
119Dic9⋊C4—62no—
120Dic9⋊C4C2×D312yes—
121C4×Dic9—31no—
122C4×Dic9—32no—
123C4×Dic9—32no—
124C4×Dic9—62no—
125C4×Dic9—61no—
126C4×Dic9C2×D311yescharacteristic
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×#125—91no—
142C3×C22⋊Q8—31no—
143Q8×Dic3—91no—
144#944—91no—
145#926—91no—
146#924—91no—
147#898—92no—
148#898—92no—
149#897—91no—
150#897D911yescharacteristic
151#890—31no—
152#890—92no—
153#890—92no—
154#890—92no—
155#889—92no—
156#887—91no—
157#887—91no—
158#887—91no—
159#887—181no—
160#879—92no—
161#879—92no—
162#872—91no—
163#869—91no—
164C22×C3×D3C22×D311yescharacteristic
165C2×C32×C4C22×D312yes—
166C2×C32×C4C22×D311yescharacteristic
167C2×C3⋊Dic3—31no—
168C2×C3⋊Dic3—31no—
169C2×C3⋊Dic3—32no—
170#634—62no—
171C2×C3×Dic3—32no—
172C2×C3×Dic3—31no—
173C2×C3×Dic3—31no—
174C2×C3×Dic3C22×D312yes—
175C2×C3×Dic3C22×D311yescharacteristic
176C3×C4×D3—22no—
177C3×Dic6—22no—
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—62no—
192C2×Dic9—61no—
193C2×Dic9C22×D312yes—
194C2×Dic9C22×D311yescharacteristic
195C2×Dic9C22×D311yescharacteristic
196Dic18—64no—
197Dic18—62no—
198#541—271noSylow
199C2×C3×C9C2×#4111yescharacteristic
200C2×C3×C92−1+411yescharacteristic
201C2×C3×C92−1+411yescharacteristic
202D3×C9—41no—
203C2×C3×Q8—31no—
204C22×C3×C4—31no—
205C2×C4×D3C2×D911yescharacteristic
206C2×Dic6—91no—
207C2×Dic6—91no—
208C2×Dic6—181no—
209C2×Dic6C2×D911yescharacteristic
210C3×C4⋊C4—21no—
211C3×C4⋊C4—32no—
212C3×C4⋊C4—91no—
213C3×C4⋊C4—92no—
214C3×C4⋊C4—92no—
215C3×C4⋊C4C2×D911yescharacteristic
216C3×C4⋊C4D3211yescharacteristic
217C3×C22⋊C4—32no—
218C3×C42—91no—
219#217—92no—
220#217—91no—
221#217—91no—
222(C2×D3)⋊C4—91no—
223(C2×D3)⋊C4—91no—
224(C2×D3)⋊C4—92no—
225(C2×D3)⋊C4C2×D912yes—
226C4⋊Dic3—32no—
227C4⋊Dic3—31no—
228C4⋊Dic3—92no—
229C4⋊Dic3—91no—
230C4⋊Dic3—91no—
231C4⋊Dic3—92no—
232C4⋊Dic3—92no—
233C4⋊Dic3—92no—
234C4⋊Dic3C2×D912yes—
235Dic3⋊C4—32no—
236Dic3⋊C4—92no—
237Dic3⋊C4—92no—
238Dic3⋊C4—92no—
239Dic3⋊C4—92no—
240Dic3⋊C4—92no—
241Dic3⋊C4—91no—
242Dic3⋊C4—91no—
243Dic3⋊C4—92no—
244Dic3⋊C4—182no—
245C4×Dic3—31no—
246C4×Dic3—92no—
247C4×Dic3—91no—
248C4×Dic3—92no—
249C4×Dic3—92no—
250C4×Dic3—91no—
251C4×Dic3—92no—
252C4×Dic3—181no—
253C4×Dic3—182no—
254C22×C32C23×D311yescharacteristic
255C2×C3×D3—21no—
256C2×C3×D3—21no—
257C2×C3×D3—21no—
258C32×C4—22no—
259C32×C4#23712yes—
260C3⋊Dic3—62no—
261C3⋊Dic3—61no—
262C3⋊Dic3—61no—
263C3×Dic3—22no—
264C3×Dic3—21no—
265C3×Dic3—62no—
266C3×Dic3—61no—
267C3×Dic3—61no—
268C22×C9—21no—
269C22×C9—61no—
270C22×C9—61no—
271C22×C9—61no—
272C22×C9C23×D311yescharacteristic
273C4×C9—22no—
274C4×C9—22no—
275C4×C9—42no—
276C4×C9—61no—
277C4×C9—62no—
278C4×C9#23912yes—
279Dic9—21no—
280Dic9—22no—
281Dic9—21no—
282Dic9—64no—
283Dic9—62no—
284Dic9—62no—
285Dic9—61no—
286Dic9—61no—
287Dic9—121no—
288C4⋊Q8—271no—
289#126—272no—
290#126—272no—
291#125—91no—
292#125—272no—
293#125—272no—
294C22⋊Q8—31no—
295C22⋊Q8—271no—
296C4×Q8—271no—
297C4×Q8—271no—
298#117—271no—
299C3×C9#54111yescharacteristic, complemented, Sylow
300C23×C3—31no—
301C22×D3C22×D911yescharacteristic
302C3×Q8—62no—
303C2×C3×C4—21no—
304C2×C3×C4—22no—
305C2×C3×C4—32no—
306C2×C3×C4—31no—
307C2×C3×C4—62no—
308C2×C3×C4—92no—
309C2×C3×C4—91no—
310C2×C3×C4—91no—
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—91no—
323C2×Dic3—92no—
324C2×Dic3—181no—
325C2×Dic3—182no—
326C2×Dic3—181no—
327C2×Dic3C22×D912yes—
328C2×Dic3C22×D911yescharacteristic
329C4×D3—22no—
330Dic6—22no—
331Dic6—184no—
332Dic6—182no—
333Dic6—182no—
334C2×C32C2×#23711yescharacteristic
335C2×C32#100711yescharacteristic
336C2×C32#101011yescharacteristic
337C3×D3—41no—
338C2×C9—21no—
339C2×C9—21no—
340C2×C9—21no—
341C2×C9—121no—
342C2×C9C2×#23911yescharacteristic
343C2×C9#101011yescharacteristic
344C2×C9#101011yescharacteristic
345C2×Q8—31no—
346C2×Q8—271no—
347C22×C4—31no—
348C4⋊C4—32no—
349C4⋊C4—91no—
350C4⋊C4—92no—
351C4⋊C4—92no—
352C4⋊C4—271no—
353C4⋊C4—271no—
354C4⋊C4—272no—
355C4⋊C4—272no—
356C4⋊C4—272no—
357C4⋊C4—272no—
358C4⋊C4—272no—
359C4⋊C4D3×D911yescharacteristic
360C22⋊C4—32no—
361C22⋊C4—271no—
362C22⋊C4—271no—
363C22⋊C4—272no—
364C42—91no—
365C42—272no—
366C42—271no—
367C42—272no—
368C22×C3—21no—
369C22×C3—61no—
370C22×C3—61no—
371C22×C3—61no—
372C22×C3C23×D911yescharacteristic
373C22×C3C22×D3211yescharacteristic
374C2×D3—21no—
375C2×D3—21no—
376C2×D3—21no—
377C3×C4—22no—
378C3×C4—22no—
379C3×C4—22no—
380C3×C4—42no—
381C3×C4—61no—
382C3×C4—62no—
383C3×C4—182no—
384C3×C4—181no—
385C3×C4—181no—
386C3×C4#369912yes—
387C3×C4#379612yes—
388Dic3—22no—
389Dic3—21no—
390Dic3—61no—
391Dic3—61no—
392Dic3—62no—
393Dic3—184no—
394Dic3—182no—
395Dic3—182no—
396Dic3—181no—
397Dic3—181no—
398Dic3—182no—
399Dic3—181no—
400Dic3—181no—
401Dic3—361no—
402C32#562511yescharacteristic
403C9—21no—
404C9#569811yescharacteristic, complemented
405C23—31no—
406Q8—62no—
407Q8—542no—
408C2×C4—32no—
409C2×C4—31no—
410C2×C4—62no—
411C2×C4—91no—
412C2×C4—91no—
413C2×C4—92no—
414C2×C4—271no—
415C2×C4—271no—
416C2×C4—272no—
417C2×C4C2×D3×D912yes—
418C2×C4C2×D3×D911yescharacteristic
419C2×C3—21no—
420C2×C3—21no—
421C2×C3—21no—
422C2×C3—121no—
423C2×C3C2×#369911yescharacteristic
424C2×C3#6371311yescharacteristic
425C2×C3#6371611yescharacteristic
426C2×C3C2×#379611yescharacteristic
427C2×C3#6430911yescharacteristic
428C2×C3#6431511yescharacteristic
429D3—41no—
430C22—61no—
431C22—61no—
432C22—61no—
433C22C22×D3×D911yescharacteristic
434C4—22no—
435C4—61no—
436C4—62no—
437C4—181no—
438C4—182no—
439C4—181no—
440C4—541no—
441C4—542no—
442C4—541no—
443C4#8863912yes—
444C3—21no—
445C3#1058972711yescharacteristic, complemented
446C3#1059434611yescharacteristic
447C2—121no—
448C2C2×#8863911yescharacteristic
449C2#1172249411yescharacteristic
450C2#1172251011yescharacteristic
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. #11721743C2semidirect
3. C3×#63452C2semidirect
8. #11720937C2semidirect
9. #11720924C2semidirect
10. #11720917C2semidirect
11. #11720911C2semidirect
54. #63452D3semidirect
299. C3×C9#541semidirect
404. C9#5698semidirect
445. C3#10589727semidirect

Maximal subgroups

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

subgroupindex
#117217432
C9×#8972
C3×#634522
#117209662
#117209622
#117209612
#117209592
#117209372
#117209242
#117209172
#117209112
#105943463
#105897273