Finite Group Databasebeta

← D9×#890 #13148885

Subgroups — D9×#890 #13148885

Order 1728.

734 conjugacy classes of subgroups.

The subgroup lattice has 462 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11721746C211yesmaximal, characteristic, complemented
2C3×C4⋊C4×D9C211yesmaximal, characteristic
3C9×#890C211yesmaximal, characteristic, complemented
4D9×C4⋊Dic3C212yesmaximal
5D9×C4⋊Dic3C211yesmaximal, characteristic
6D9×Dic3⋊C4C212yesmaximal
7C4×Dic3×D9C211yesmaximal, characteristic
8#11720937C211yesmaximal, characteristic, complemented
9#11720927C211yesmaximal, characteristic, complemented
10#11720917C212yesmaximal, complemented
11#11720915C212yesmaximal, complemented
12D3×#890—31nomaximal
13D9×#125—31nomaximal
14C2×C3×C4×D9C2212yes—
15C2×C3×C4×D9C2211yescharacteristic
16C2×Dic3×D9C2211yescharacteristic
17C2×Dic3×D9C2211yescharacteristic
18C2×Dic3×D9C2212yes—
19C3×C9×C4⋊C4C2211yescharacteristic
20C4⋊(C3⋊Dic9)C2212yes—
21C4⋊(C3⋊Dic9)C2211yescharacteristic
22#88534C2212yes—
23C4×C3⋊Dic9C2211yescharacteristic
24C9×C4⋊Dic3C2212yes—
25C9×C4⋊Dic3C2211yescharacteristic
26C9×Dic3⋊C4C2212yes—
27C4×C9×Dic3C2211yescharacteristic
28C3×C4⋊Dic9C2211yescharacteristic
29C3×Dic9⋊C4C2212yes—
30Dic3⋊Dic9C2212yes—
31#88442C2212yes—
32Dic9⋊Dic3C2212yes—
33Dic9⋊Dic3C2212yes—
34Dic9⋊Dic3C2212yes—
35Dic3×Dic9C2211yescharacteristic
36Dic3×Dic9C2211yescharacteristic
37#64099—31no—
38C3×D3×C4⋊C4—31no—
39C3×#890D311yescharacteristic
40D3×C4⋊Dic3—31no—
41D3×C4⋊Dic3—32no—
42D3×Dic3⋊C4—32no—
43C4×D3×Dic3—31no—
44#63872—31no—
45#63869—31no—
46#63846—32no—
47#63845—32no—
48C9×#125—31no—
49C4⋊C4×D9—32no—
50C4⋊C4×D9—32no—
51C4⋊C4×D9—31no—
52C4⋊C4×D9D311yescharacteristic
53#63452—31no—
54#63452—31no—
55#63452—61no—
56#63451—32no—
57#63451—32no—
58C42×D9—31no—
59#63432—31no—
60C22×C3×D9C2311yescharacteristic
61C2×C3×C4×C9C2312yes—
62C2×C3×C4×C9C2311yescharacteristic
63C2×C3⋊Dic9C2311yescharacteristic
64C2×C3⋊Dic9C2311yescharacteristic
65C2×C3⋊Dic9C2312yes—
66C2×C9×Dic3C2311yescharacteristic
67C2×C9×Dic3C2311yescharacteristic
68C2×C9×Dic3C2312yes—
69C2×C3×Dic9C2312yes—
70C2×C3×Dic9C2311yescharacteristic
71C3×C4×D9—24no—
72C3×C4×D9Q814yes—
73Dic3×D9—22no—
74Dic3×D9—22no—
75Dic3×D9—24no—
76D3×#125—91no—
77C2×#890—91no—
78C2×C3×C4×D3—31no—
79C2×C3×C4×D3—32no—
80C2×D3×Dic3—31no—
81C2×D3×Dic3—31no—
82C2×D3×Dic3—32no—
83C32×C4⋊C4C2×D311yescharacteristic
84C4⋊(C3⋊Dic3)—32no—
85C4⋊(C3⋊Dic3)—31no—
86#3750—32no—
87C4×C3⋊Dic3—31no—
88C3×C4⋊Dic3—31no—
89C3×C4⋊Dic3C2×D312yes—
90C3×C4⋊Dic3C2×D311yescharacteristic
91C3×Dic3⋊C4—32no—
92C3×Dic3⋊C4C2×D312yes—
93C3×C4×Dic3C2×D311yescharacteristic
94#3724—32no—
95Dic3⋊Dic3—32no—
96Dic3⋊Dic3—32no—
97Dic3⋊Dic3—32no—
98Dic3⋊Dic3—32no—
99Dic32—31no—
100Dic32—31no—
101C2×C4×D9—31no—
102C2×C4×D9—31no—
103C2×C4×D9—32no—
104C2×C4×D9C2×D312yes—
105C2×C4×D9C2×D311yescharacteristic
106C9×C4⋊C4—21no—
107C9×C4⋊C4—32no—
108C9×C4⋊C4—32no—
109C9×C4⋊C4—31no—
110C9×C4⋊C4C2×D311yescharacteristic
111C42×C9—31no—
112C4⋊Dic9—31no—
113C4⋊Dic9—32no—
114C4⋊Dic9—32no—
115C4⋊Dic9—62no—
116C4⋊Dic9—61no—
117C4⋊Dic9C2×D311yescharacteristic
118Dic9⋊C4—32no—
119Dic9⋊C4—32no—
120Dic9⋊C4—32no—
121Dic9⋊C4—32no—
122Dic9⋊C4—32no—
123Dic9⋊C4—62no—
124Dic9⋊C4C2×D312yes—
125C4×Dic9—31no—
126C4×Dic9—31no—
127C4×Dic9—31no—
128C4×Dic9—61no—
129C22×C3×C9C2411yescharacteristic
130C2×C3×D9C2×Q812yes—
131C2×C3×D9#4112yes—
132C2×C3×D9#4112yes—
133C3×C4×C9—22no—
134C3×C4×C9C2×Q812yes—
135C3⋊Dic9—21no—
136C3⋊Dic9—21no—
137C3⋊Dic9—22no—
138C9×Dic3—21no—
139C9×Dic3—21no—
140C9×Dic3—22no—
141C3×Dic9—22no—
142C3×Dic9C2×Q812yes—
143C3×#125—31no—
144C2×C3×C4⋊C4—91no—
145C2×C4⋊Dic3—92no—
146C2×C4⋊Dic3—91no—
147C2×Dic3⋊C4—92no—
148C2×C4×Dic3—91no—
149D3×C4⋊C4—31no—
150D3×C4⋊C4—92no—
151D3×C4⋊C4—92no—
152D3×C4⋊C4—91no—
153#890—91no—
154#890—91no—
155#890—91no—
156#890—92no—
157#890—91no—
158#890—92no—
159#890—91no—
160#890—181no—
161#890D911yescharacteristic, direct product factor
162#889—92no—
163#889—92no—
164C42×D3—91no—
165#870—91no—
166C22×C3×D3—31no—
167C2×C32×C4C22×D312yes—
168C2×C32×C4C22×D311yescharacteristic
169C2×C3⋊Dic3—31no—
170C2×C3⋊Dic3—31no—
171C2×C3⋊Dic3—32no—
172C2×C3×Dic3—31no—
173C2×C3×Dic3—32no—
174C2×C3×Dic3C22×D311yescharacteristic
175C2×C3×Dic3C22×D311yescharacteristic
176C2×C3×Dic3C22×D312yes—
177C3×C4×D3—34no—
178C3×C4×D3—64no—
179D3×Dic3—62no—
180D3×Dic3—64no—
181D3×Dic3—62no—
182C22×D9C22×D311yescharacteristic
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—31no—
192C2×Dic9—32no—
193C2×Dic9—61no—
194C2×Dic9—61no—
195C2×Dic9—62no—
196C2×Dic9C22×D312yes—
197C2×Dic9C22×D311yescharacteristic
198C4×D9—24no—
199C4×D9—62no—
200C4×D9—62no—
201C4×D9—64no—
202C4×D9Dic614yes—
203C2×#125—271noSylow
204C2×C3×C9C22×Q811yescharacteristic
205C2×C3×C9C2×#4111yescharacteristic
206C2×C3×C9C2×#4111yescharacteristic
207C3×D9#12514yescomplemented
208C22×C3×C4—91no—
209C22×C3×C4—92no—
210C22×Dic3—91no—
211C22×Dic3—91no—
212C22×Dic3—92no—
213C2×C4×D3—31no—
214C2×C4×D3—32no—
215C2×C4×D3—91no—
216C2×C4×D3—91no—
217C2×C4×D3—92no—
218C3×C4⋊C4—21no—
219C3×C4⋊C4—32no—
220C3×C4⋊C4—32no—
221C3×C4⋊C4—31no—
222C3×C4⋊C4—92no—
223C3×C4⋊C4—91no—
224C3×C4⋊C4C2×D911yescharacteristic
225C3×C4⋊C4D3211yescharacteristic
226C3×C42—31no—
227C4⋊Dic3—31no—
228C4⋊Dic3—92no—
229C4⋊Dic3—92no—
230C4⋊Dic3—91no—
231C4⋊Dic3—91no—
232C4⋊Dic3—92no—
233C4⋊Dic3—92no—
234C4⋊Dic3—92no—
235C4⋊Dic3—92no—
236C4⋊Dic3—182no—
237C4⋊Dic3—181no—
238C4⋊Dic3C2×D912yes—
239C4⋊Dic3C2×D911yescharacteristic
240Dic3⋊C4—32no—
241Dic3⋊C4—92no—
242Dic3⋊C4—92no—
243Dic3⋊C4—92no—
244Dic3⋊C4—92no—
245Dic3⋊C4—92no—
246Dic3⋊C4—92no—
247Dic3⋊C4—92no—
248Dic3⋊C4—92no—
249Dic3⋊C4—182no—
250Dic3⋊C4C2×D912yes—
251C4×Dic3—91no—
252C4×Dic3—91no—
253C4×Dic3—91no—
254C4×Dic3—91no—
255C4×Dic3—91no—
256C4×Dic3—91no—
257C4×Dic3—181no—
258C4×Dic3C2×D911yescharacteristic
259C22×C32C23×D311yescharacteristic
260C2×C3×D3—32no—
261C2×C3×D3—32no—
262C2×C3×D3—32no—
263C32×C4—22no—
264C32×C4D3×Q812yes—
265C3⋊Dic3—61no—
266C3⋊Dic3—61no—
267C3⋊Dic3—62no—
268C3×Dic3—21no—
269C3×Dic3—21no—
270C3×Dic3—22no—
271C3×Dic3—32no—
272C3×Dic3—62no—
273C22×C9—21no—
274C22×C9C23×D311yescharacteristic
275C2×D9C2×Dic612yes—
276C2×D9#23712yes—
277C2×D9#23912yes—
278C4×C9—22no—
279C4×C9—22no—
280C4×C9—42no—
281C4×C9—61no—
282C4×C9—61no—
283C4×C9—62no—
284C4×C9C2×Dic612yes—
285Dic9—22no—
286Dic9—61no—
287Dic9—61no—
288Dic9—62no—
289Dic9—121no—
290Dic9—121no—
291Dic9—122no—
292Dic9C2×Dic612yes—
293#125—31no—
294#125—271no—
295#125—272no—
296#125—272no—
297#125—271no—
298#125—271no—
299C2×C4⋊C4—91no—
300C2×C4⋊C4—272no—
301C2×C4⋊C4—271no—
302C2×C4⋊C4—272no—
303C2×C42—271no—
304C3×C9C2×#12511yescharacteristic, complemented, Sylow
305C23×C3—91no—
306C22×D3—31no—
307C2×C3×C4—21no—
308C2×C3×C4—22no—
309C2×C3×C4—31no—
310C2×C3×C4—31no—
311C2×C3×C4—32no—
312C2×C3×C4—91no—
313C2×C3×C4—94no—
314C2×C3×C4—92no—
315C2×C3×C4—184no—
316C2×C3×C4C22×D912yes—
317C2×C3×C4C22×D911yescharacteristic
318C2×C3×C4C2×D3212yes—
319C2×C3×C4C2×D3211yescharacteristic
320C2×Dic3—31no—
321C2×Dic3—32no—
322C2×Dic3—91no—
323C2×Dic3—91no—
324C2×Dic3—92no—
325C2×Dic3—91no—
326C2×Dic3—91no—
327C2×Dic3—92no—
328C2×Dic3—181no—
329C2×Dic3—181no—
330C2×Dic3—182no—
331C2×Dic3—182no—
332C2×Dic3—184no—
333C2×Dic3—182no—
334C2×Dic3C22×D911yescharacteristic
335C2×Dic3C22×D911yescharacteristic
336C2×Dic3C22×D912yes—
337C4×D3—34no—
338C4×D3—64no—
339C4×D3—182no—
340C4×D3—182no—
341C4×D3—184no—
342C2×C32C2×D3×Q811yescharacteristic
343C2×C32D3×#4111yescharacteristic
344C2×C32D3×#4111yescharacteristic
345C3×D3—34no—
346C2×C9—21no—
347C2×C9—21no—
348C2×C9—21no—
349C2×C9C22×Dic611yescharacteristic
350C2×C9C2×#23711yescharacteristic
351C2×C9C2×#23911yescharacteristic
352D9#89014yesdirect product factor
353C22×C4—91no—
354C22×C4—92no—
355C22×C4—271no—
356C22×C4—271no—
357C22×C4—272no—
358C4⋊C4—32no—
359C4⋊C4—32no—
360C4⋊C4—31no—
361C4⋊C4—92no—
362C4⋊C4—91no—
363C4⋊C4—272no—
364C4⋊C4—272no—
365C4⋊C4—272no—
366C4⋊C4—272no—
367C4⋊C4—272no—
368C4⋊C4—272no—
369C4⋊C4—272no—
370C4⋊C4—271no—
371C4⋊C4D3×D911yescharacteristic
372C42—31no—
373C42—271no—
374C42—271no—
375C42—271no—
376C22×C3—21no—
377C22×C3—92no—
378C22×C3—92no—
379C22×C3—92no—
380C22×C3C23×D911yescharacteristic
381C22×C3C22×D3211yescharacteristic
382C2×D3—32no—
383C2×D3—32no—
384C2×D3—32no—
385C3×C4—22no—
386C3×C4—22no—
387C3×C4—22no—
388C3×C4—42no—
389C3×C4—61no—
390C3×C4—61no—
391C3×C4—62no—
392C3×C4—92no—
393C3×C4—182no—
394C3×C4Q8×D912yes—
395C3×C4D3×Dic612yes—
396Dic3—21no—
397Dic3—21no—
398Dic3—22no—
399Dic3—32no—
400Dic3—62no—
401Dic3—181no—
402Dic3—181no—
403Dic3—182no—
404Dic3—181no—
405Dic3—181no—
406Dic3—182no—
407Dic3—361no—
408Dic3—361no—
409Dic3—362no—
410C32D3×#12511yescharacteristic
411C9—21no—
412C9C2×#89011yescharacteristic, complemented
413C23—91no—
414C2×C4—31no—
415C2×C4—31no—
416C2×C4—32no—
417C2×C4—91no—
418C2×C4—94no—
419C2×C4—92no—
420C2×C4—184no—
421C2×C4—271no—
422C2×C4—271no—
423C2×C4—272no—
424C2×C4—542no—
425C2×C4—542no—
426C2×C4—544no—
427C2×C4C2×D3×D912yes—
428C2×C4C2×D3×D911yescharacteristic
429C2×C3—21no—
430C2×C3—21no—
431C2×C3—21no—
432C2×C3—94no—
433C2×C3C2×Q8×D911yescharacteristic
434C2×C3#41×D911yescharacteristic
435C2×C3#41×D911yescharacteristic
436C2×C3C2×D3×Dic611yescharacteristic
437C2×C3D3×#23711yescharacteristic
438C2×C3D3×#23911yescharacteristic
439D3—34no—
440C22—92no—
441C22—92no—
442C22—92no—
443C22C22×D3×D911yescharacteristic
444C4—22no—
445C4—61no—
446C4—61no—
447C4—62no—
448C4—92no—
449C4—182no—
450C4—541no—
451C4—541no—
452C4—542no—
453C4D9×Dic612yes—
454C3—21no—
455C3D9×#12511yescharacteristic, complemented
456C3D3×#89011yescharacteristic
457C2—94no—
458C2C2×D9×Dic611yescharacteristic
459C2D9×#23711yescharacteristic
460C2D9×#23911yescharacteristic
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×C9C2×#125
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. C9×#890C2semidirect
8. #11720937C2semidirect
9. #11720927C2semidirect
10. #11720917C2semidirect
11. #11720915C2semidirect
161. #890D9direct (N × H)
207. C3×D9#125semidirect
304. C3×C9C2×#125semidirect
352. D9#890direct (N × H)
412. C9C2×#890semidirect
455. C3D9×#125semidirect

Maximal subgroups

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

subgroupindex
#117217462
C3×C4⋊C4×D92
C9×#8902
D9×C4⋊Dic32
D9×Dic3⋊C42
C4×Dic3×D92
#117209372
#117209272
#117209172
#117209152
D3×#8903
D9×#1253