Finite Group Databasebeta

← #13148955

Subgroups — #13148955

Order 1728.

694 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11721748C211yesmaximal, characteristic
2C9×#888C211yesmaximal, characteristic, complemented
3C3×#63450C211yesmaximal, characteristic, complemented
4#11721000C211yesmaximal, characteristic
5#11720986C212yesmaximal
6#11720985C212yesmaximal
7#11720935C211yesmaximal, characteristic, complemented
8#11720931C211yesmaximal, characteristic
9#11720930C212yesmaximal, complemented
10#11720928C212yesmaximal, complemented
11#11720927C211yesmaximal, characteristic, complemented
12#10594351—31nomaximal
13#10589831—31nomaximal
14C2×C4×C3⋊D9C2211yescharacteristic
15C2×C9×Dic6C2212yes—
16C2×C3×Dic18C2212yes—
17C2×(C3⋊D9)⋊C4C2212yes—
18C3×C9×C4⋊C4C2211yescharacteristic
19#88536C2212yes—
20C4⋊(C3⋊Dic9)C2211yescharacteristic
21C4×C3⋊Dic9C2212yes—
22C9×C4⋊Dic3C2211yescharacteristic
23C9×Dic3⋊C4C2212yes—
24C4×C9×Dic3C2211yescharacteristic
25C3×C4⋊Dic9C2211yescharacteristic
26C3×Dic9⋊C4C2212yes—
27C3×C4×Dic9C2211yescharacteristic
28#88445C2211yescharacteristic
29#88445C2211yescharacteristic
30Dic3⋊Dic9C2212yes—
31#88442C2212yes—
32#88442C2212yes—
33#88442C2212yes—
34#88442C2211yescharacteristic
35#88442C2211yescharacteristic
36Dic9⋊Dic3C2212yes—
37#64101—31no—
38C3×#888—31no—
39C3×#888D311yescharacteristic
40#63900—31no—
41#63888—32no—
42#63888—32no—
43#63874—31no—
44#63873—32no—
45#63870—32no—
46#63869—31no—
47#63869—31no—
48C9×C4⋊Q8—31no—
49#63508—32no—
50#63506—32no—
51#63458—32no—
52#63454—31no—
53#63454—61no—
54C4⋊C4×D9—31no—
55#63452—31no—
56#63451—32no—
57#63450D311yescharacteristic, complemented
58#63434—31no—
59#63432—31no—
60C22×C3⋊D9C2311yescharacteristic
61C2×C3×C4×C9C2312yes—
62C2×C3×C4×C9C2311yescharacteristic
63C2×C3⋊Dic9C2312yes—
64C2×C3⋊Dic9C2311yescharacteristic
65C4×C3⋊D9Q814yescomplemented
66C2×C9×Dic3C2312yes—
67C2×C9×Dic3C2311yescharacteristic
68C2×C9×Dic3C2311yescharacteristic
69C2×C3×Dic9C2312yes—
70C2×C3×Dic9C2311yescharacteristic
71C2×C3×Dic9C2311yescharacteristic
72C9×Dic6—42no—
73C3×Dic18—42no—
74(C3⋊D9)⋊C4—42no—
75#5729—91no—
76#5729—91no—
77C2×C4×C3⋊D3—31no—
78C2×C3×Dic6—32no—
79C2×C3×Dic6C2×D312yes—
80C2×#624—32no—
81C32×C4⋊C4C2×D311yescharacteristic
82#3752—32no—
83C4⋊(C3⋊Dic3)—31no—
84C4×C3⋊Dic3—32no—
85C3×C4⋊Dic3—31no—
86C3×C4⋊Dic3C2×D311yescharacteristic
87C3×Dic3⋊C4—32no—
88C3×Dic3⋊C4C2×D312yes—
89C3×C4×Dic3—31no—
90C3×C4×Dic3C2×D311yescharacteristic
91#3724—32no—
92#3724—32no—
93#3724—31no—
94#3724—32no—
95#3724—31no—
96Dic3⋊Dic3—32no—
97Dic3⋊Dic3—32no—
98#3722—31no—
99#3722—31no—
100C2×Q8×C9—32no—
101C2×C4×D9—31no—
102C2×C4×D9—32no—
103C2×C4×D9—61no—
104C2×Dic18C2×D312yes—
105C9×C4⋊C4—21no—
106C9×C4⋊C4—32no—
107C9×C4⋊C4—31no—
108C9×C4⋊C4C2×D311yescharacteristic
109C42×C9—31no—
110(C2×D9)⋊C4—32no—
111(C2×D9)⋊C4—31no—
112(C2×D9)⋊C4—31no—
113(C2×D9)⋊C4—62no—
114C4⋊Dic9—31no—
115C4⋊Dic9—32no—
116C4⋊Dic9—61no—
117C4⋊Dic9C2×D311yescharacteristic
118Dic9⋊C4—31no—
119Dic9⋊C4—32no—
120Dic9⋊C4—32no—
121Dic9⋊C4—32no—
122Dic9⋊C4—32no—
123Dic9⋊C4—31no—
124Dic9⋊C4C2×D312yes—
125C4×Dic9—32no—
126C4×Dic9—62no—
127C4×Dic9C2×D311yescharacteristic
128C22×C3×C9C2411yescharacteristic
129C2×C3⋊D9—21no—
130C2×C3⋊D9—21no—
131C2×C3⋊D9C2×Q812yes—
132C3×C4×C9—22no—
133C3×C4×C9C2×Q812yes—
134C3⋊Dic9—22no—
135C3⋊Dic9C2×Q812yes—
136C9×Dic3—22no—
137C9×Dic3—21no—
138C9×Dic3—21no—
139C3×Dic9—22no—
140C3×Dic9—21no—
141C3×Dic9—21no—
142C3×C4⋊Q8—31no—
143C3×C4⋊Q8—91no—
144#946—92no—
145#946—92no—
146#944—92no—
147#944—92no—
148#896—92no—
149#896—92no—
150#892—91no—
151#892—91no—
152#892—181no—
153D3×C4⋊C4—91no—
154D3×C4⋊C4—91no—
155#890—91no—
156#890—91no—
157#889—92no—
158#889—92no—
159#888—31no—
160#888D911yescharacteristic, complemented
161#872—91no—
162#872—91no—
163#870—91no—
164#870—91no—
165C22×C3⋊D3—31no—
166C2×C32×C4C22×D312yes—
167C2×C32×C4C22×D311yescharacteristic
168C2×C3⋊Dic3—31no—
169C2×C3⋊Dic3—32no—
170C4×C3⋊D3—34no—
171C2×C3×Dic3—31no—
172C2×C3×Dic3—31no—
173C2×C3×Dic3—32no—
174C2×C3×Dic3C22×D311yescharacteristic
175C2×C3×Dic3C22×D311yescharacteristic
176C2×C3×Dic3C22×D312yes—
177C3×Dic6—42no—
178C3×Dic6—122no—
179#624—122no—
180C22×D9—31no—
181C22×D9—61no—
182Q8×C9—122no—
183C2×C4×C9—21no—
184C2×C4×C9—22no—
185C2×C4×C9—31no—
186C2×C4×C9—31no—
187C2×C4×C9—32no—
188C2×C4×C9C22×D312yes—
189C2×C4×C9C22×D311yescharacteristic
190C2×Dic9—31no—
191C2×Dic9—32no—
192C2×Dic9—61no—
193C2×Dic9—62no—
194C2×Dic9C22×D311yescharacteristic
195C2×Dic9C22×D311yescharacteristic
196C2×Dic9C22×D312yes—
197C4×D9—34no—
198C4×D9—64no—
199C4×D9—122no—
200Dic18—42no—
201#544—271noSylow
202C2×C3×C9C22×Q811yescharacteristic
203C2×C3×C9H(5,𝔽2)11yescharacteristic
204C2×C3×C92−1+411yescharacteristic
205C3⋊D9—22no—
206C2×C3×Q8—32no—
207C2×C3×Q8—92no—
208C2×C4×D3—91no—
209C2×C4×D3—91no—
210C2×C4×D3—92no—
211C2×C4×D3—92no—
212C2×C4×D3—181no—
213C2×Dic6—32no—
214C2×Dic6C2×D912yes—
215C3×C4⋊C4—21no—
216C3×C4⋊C4—32no—
217C3×C4⋊C4—31no—
218C3×C4⋊C4—92no—
219C3×C4⋊C4—91no—
220C3×C4⋊C4C2×D911yescharacteristic
221C3×C4⋊C4D3211yescharacteristic
222C3×C42—31no—
223C3×C42—91no—
224(C2×D3)⋊C4—91no—
225(C2×D3)⋊C4—91no—
226(C2×D3)⋊C4—92no—
227(C2×D3)⋊C4—92no—
228(C2×D3)⋊C4—91no—
229(C2×D3)⋊C4—91no—
230(C2×D3)⋊C4—182no—
231C4⋊Dic3—31no—
232C4⋊Dic3—92no—
233C4⋊Dic3—91no—
234C4⋊Dic3—91no—
235C4⋊Dic3—92no—
236C4⋊Dic3—181no—
237C4⋊Dic3C2×D911yescharacteristic
238Dic3⋊C4—32no—
239Dic3⋊C4—92no—
240Dic3⋊C4—92no—
241Dic3⋊C4—91no—
242Dic3⋊C4—91no—
243Dic3⋊C4—92no—
244Dic3⋊C4—92no—
245Dic3⋊C4—92no—
246Dic3⋊C4—92no—
247Dic3⋊C4—91no—
248Dic3⋊C4—91no—
249Dic3⋊C4—92no—
250Dic3⋊C4—92no—
251Dic3⋊C4C2×D912yes—
252C4×Dic3—31no—
253C4×Dic3—92no—
254C4×Dic3—92no—
255C4×Dic3—182no—
256C4×Dic3C2×D911yescharacteristic
257C22×C32C23×D311yescharacteristic
258C2×C3⋊D3—32no—
259C2×C3⋊D3—61no—
260C2×C3⋊D3—61no—
261C32×C4—22no—
262C32×C4D3×Q812yes—
263C3⋊Dic3—32no—
264C3⋊Dic3—62no—
265C3×Dic3—22no—
266C3×Dic3—21no—
267C3×Dic3—21no—
268C3×Dic3—62no—
269C3×Dic3—61no—
270C3×Dic3—61no—
271C22×C9—21no—
272C22×C9C23×D311yescharacteristic
273C2×D9—32no—
274C2×D9—62no—
275C2×D9—61no—
276C2×D9—61no—
277C2×D9—121no—
278C2×D9—121no—
279C4×C9—22no—
280C4×C9—22no—
281C4×C9—42no—
282C4×C9—62no—
283C4×C9—61no—
284C4×C9—61no—
285C4×C9D3×Q812yes—
286Dic9—21no—
287Dic9—22no—
288Dic9—21no—
289Dic9—32no—
290Dic9—62no—
291Dic9—62no—
292Dic9—122no—
293C4⋊Q8—31no—
294C4⋊Q8—91no—
295C4⋊Q8—272no—
296#125—271no—
297#125—271no—
298#125—272no—
299C22⋊Q8—272no—
300C22⋊Q8—272no—
301#117—271no—
302#117—271no—
303C2×C4⋊C4—271no—
304C3×C9#54411yescharacteristic, complemented, Sylow
305C22×D3—91no—
306C22×D3—91no—
307C22×D3—181no—
308C3×Q8—122no—
309C3×Q8—362no—
310C2×C3×C4—21no—
311C2×C3×C4—22no—
312C2×C3×C4—31no—
313C2×C3×C4—31no—
314C2×C3×C4—32no—
315C2×C3×C4—91no—
316C2×C3×C4—91no—
317C2×C3×C4—92no—
318C2×C3×C4C22×D912yes—
319C2×C3×C4C22×D911yescharacteristic
320C2×C3×C4C2×D3212yes—
321C2×C3×C4C2×D3211yescharacteristic
322C2×Dic3—31no—
323C2×Dic3—31no—
324C2×Dic3—32no—
325C2×Dic3—91no—
326C2×Dic3—91no—
327C2×Dic3—92no—
328C2×Dic3—92no—
329C2×Dic3—181no—
330C2×Dic3—182no—
331C2×Dic3C22×D911yescharacteristic
332C2×Dic3C22×D911yescharacteristic
333C2×Dic3C22×D912yes—
334C4×D3—94no—
335C4×D3—94no—
336C4×D3—184no—
337C4×D3—362no—
338C4×D3—362no—
339Dic6—42no—
340Dic6—122no—
341C2×C32#100411yescharacteristic
342C2×C32C2×D3×Q811yescharacteristic
343C2×C32#100711yescharacteristic
344C3⋊D3—62no—
345C2×C9—21no—
346C2×C9—21no—
347C2×C9—21no—
348C2×C9#100411yescharacteristic
349C2×C9C2×D3×Q811yescharacteristic
350C2×C9#100711yescharacteristic
351D9—62no—
352D9—122no—
353C2×Q8—32no—
354C2×Q8—92no—
355C22×C4—272no—
356C22×C4—271no—
357C4⋊C4—31no—
358C4⋊C4—32no—
359C4⋊C4—91no—
360C4⋊C4—92no—
361C4⋊C4—272no—
362C4⋊C4—271no—
363C4⋊C4—271no—
364C4⋊C4—272no—
365C4⋊C4—272no—
366C4⋊C4—272no—
367C4⋊C4—272no—
368C4⋊C4—271no—
369C4⋊C4D3×D911yescharacteristic
370C22⋊C4—271no—
371C22⋊C4—271no—
372C22⋊C4—272no—
373C42—31no—
374C42—91no—
375C42—272no—
376C22×C3—21no—
377C22×C3C23×D911yescharacteristic
378C22×C3C22×D3211yescharacteristic
379C2×D3—92no—
380C2×D3—92no—
381C2×D3—182no—
382C2×D3—181no—
383C2×D3—181no—
384C2×D3—181no—
385C2×D3—181no—
386C2×D3—361no—
387C2×D3—361no—
388C3×C4—22no—
389C3×C4—22no—
390C3×C4—22no—
391C3×C4—42no—
392C3×C4—62no—
393C3×C4—61no—
394C3×C4—61no—
395C3×C4—182no—
396C3×C4—181no—
397C3×C4—181no—
398C3×C4Q8×D912yes—
399C3×C4#379712yes—
400Dic3—21no—
401Dic3—22no—
402Dic3—21no—
403Dic3—61no—
404Dic3—62no—
405Dic3—61no—
406Dic3—92no—
407Dic3—92no—
408Dic3—182no—
409Dic3—182no—
410Dic3—182no—
411Dic3—362no—
412C32#572911yescharacteristic
413C9—21no—
414C9#572911yescharacteristic, complemented
415C23—271no—
416Q8—122no—
417Q8—362no—
418C2×C4—31no—
419C2×C4—31no—
420C2×C4—32no—
421C2×C4—91no—
422C2×C4—91no—
423C2×C4—92no—
424C2×C4—271no—
425C2×C4—274no—
426C2×C4—272no—
427C2×C4—1082no—
428C2×C4C2×D3×D912yes—
429C2×C4C2×D3×D911yescharacteristic
430C2×C3—21no—
431C2×C3—21no—
432C2×C3—21no—
433C2×C3#6371011yescharacteristic
434C2×C3C2×Q8×D911yescharacteristic
435C2×C3#6371311yescharacteristic
436C2×C3C2×#379711yescharacteristic
437C2×C3#6431311yescharacteristic
438C2×C3#6431611yescharacteristic
439D3—182no—
440D3—182no—
441D3—362no—
442C22—272no—
443C22—541no—
444C22—541no—
445C22C22×D3×D911yescharacteristic
446C4—22no—
447C4—62no—
448C4—61no—
449C4—61no—
450C4—182no—
451C4—181no—
452C4—181no—
453C4—272no—
454C4—542no—
455C4(C3⋊D9)⋊Q812yes—
456C3—21no—
457C3#1058983111yescharacteristic, complemented
458C3#1059435111yescharacteristic
459C2—542no—
460C2C2×(C3⋊D9)⋊Q811yescharacteristic
461C2#1172249911yescharacteristic
462C2#1172250911yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC3×C9×C4⋊C4C22
socleC22×C32C23×D3
prime radicalC22×C3⋊D9C23
hypercenterC4⋊C4D3×D9
hypocenterC3×C9#544
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C9×#888C2semidirect
3. C3×#63450C2semidirect
7. #11720935C2semidirect
9. #11720930C2semidirect
10. #11720928C2semidirect
11. #11720927C2semidirect
57. #63450D3semidirect
65. C4×C3⋊D9Q8semidirect
160. #888D9semidirect
304. C3×C9#544semidirect
414. C9#5729semidirect
457. C3#10589831semidirect

Maximal subgroups

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

subgroupindex
#117217482
C9×#8882
C3×#634502
#117210002
#117209862
#117209852
#117209352
#117209312
#117209302
#117209282
#117209272
#105943513
#105898313