Finite Group Databasebeta

← C2×D3×Dic9⋊C4 #13150353

Subgroups — C2×D3×Dic9⋊C4 #13150353

Order 1728.

1346 conjugacy classes of subgroups.

The subgroup lattice has 444 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×D3×C9C211yesmaximal, characteristic
2C22×D3×Dic9C211yesmaximal, characteristic
3C22×D3×Dic9C211yesmaximal, characteristic
4C2×#88534C211yesmaximal, characteristic, complemented
5C2×C3×Dic9⋊C4C211yesmaximal, characteristic, complemented
6C2×#88442C211yesmaximal, characteristic, complemented
7C2×Dic9⋊Dic3C211yesmaximal, characteristic, complemented
8D3×Dic9⋊C4C218yesmaximal, direct product factor
9C2×D3×Dic3⋊C4—31nomaximal
10C22×Dic9⋊C4—31nomaximal
11C23×D3×C9C2211yescharacteristic
12C22×C3×C4×C9C2211yescharacteristic
13C22×C3⋊Dic9C2211yescharacteristic
14C22×C3⋊Dic9C2211yescharacteristic
15C22×C9×Dic3C2211yescharacteristic
16C22×C3×Dic9C2211yescharacteristic
17C22×C3×Dic9C2211yescharacteristic
18C2×C4×D3×C9—24no—
19C2×C4×D3×C9C2214yes—
20C2×D3×Dic9—24no—
21C2×D3×Dic9C418yescomplemented
22C2×D3×Dic9C2214yes—
23C2×D3×Dic9C2214yes—
24#88534C2214yescomplemented
25C3×Dic9⋊C4C2214yescomplemented
26#88442C2214yescomplemented
27Dic9⋊Dic3C2214yescomplemented
28C22×C3×C4×D3D311yescharacteristic
29C22×D3×Dic3—31no—
30C22×D3×Dic3—31no—
31C2×#3750—31no—
32C2×C3×Dic3⋊C4—31no—
33C2×#3724—31no—
34C2×Dic3⋊Dic3—31no—
35D3×Dic3⋊C4—38no—
36C23×C4×C9—31no—
37C23×Dic9—31no—
38C23×Dic9—31no—
39C2×Dic9⋊C4—31no—
40C2×Dic9⋊C4—31no—
41C2×Dic9⋊C4—31no—
42C2×Dic9⋊C4—38no—
43C2×Dic9⋊C4—61no—
44C2×Dic9⋊C4D311yescharacteristic, direct product factor
45C23×C3×C9C2311yescharacteristic
46C22×D3×C9C2×C414yes—
47C22×D3×C9D414yes—
48C22×D3×C9Q814yes—
49C22×D3×C9C2312yes—
50C2×C3×C4×C9—22no—
51C2×C3×C4×C9C2312yes—
52C2×C3⋊Dic9—22no—
53C2×C3⋊Dic9C2×C414yescomplemented
54C2×C3⋊Dic9C2312yes—
55C2×C3⋊Dic9C2312yes—
56C2×C9×Dic3—22no—
57C2×C9×Dic3C2312yes—
58C2×C3×Dic9—22no—
59C2×C3×Dic9C2×C414yescomplemented
60C2×C3×Dic9C2312yes—
61C2×C3×Dic9C2312yes—
62C4×D3×C9—28no—
63D3×Dic9—28no—
64D3×Dic9C2×C4116yescomplemented
65C22×Dic3⋊C4—91no—
66C2×D3×C4⋊C4—91no—
67C23×C3×D3C2×D311yescharacteristic
68C22×C32×C4C2×D311yescharacteristic
69C22×C3⋊Dic3—31no—
70C22×C3⋊Dic3—31no—
71C22×C3×Dic3—31no—
72C22×C3×Dic3—31no—
73C22×C3×Dic3C2×D311yescharacteristic
74C2×C3×C4×D3—24no—
75C2×C3×C4×D3C2×D314yes—
76C2×D3×Dic3—38no—
77C2×D3×Dic3—34no—
78C2×D3×Dic3—34no—
79C2×D3×Dic3—64no—
80C24×C9—31no—
81#3750—34no—
82C3×Dic3⋊C4—34no—
83#3724—34no—
84Dic3⋊Dic3—34no—
85C22×C4×C9—21no—
86C22×C4×C9—31no—
87C22×C4×C9—34no—
88C22×C4×C9—64no—
89C22×C4×C9C2×D311yescharacteristic
90C22×Dic9—31no—
91C22×Dic9—38no—
92C22×Dic9—34no—
93C22×Dic9—31no—
94C22×Dic9—34no—
95C22×Dic9—64no—
96C22×Dic9—61no—
97C22×Dic9—61no—
98C22×Dic9C2×D311yescharacteristic
99C22×Dic9C2×D311yescharacteristic
100Dic9⋊C4—34no—
101Dic9⋊C4—34no—
102Dic9⋊C4—34no—
103Dic9⋊C4—64no—
104Dic9⋊C4C2×D314yesdirect product factor
105C22×C3×C9C22×C412yes—
106C22×C3×C9C2×D412yes—
107C22×C3×C9C2×Q812yes—
108C22×C3×C9C2411yescharacteristic
109C2×D3×C9C4⋊C4116yescomplemented
110C2×D3×C9C22×C414yes—
111C2×D3×C9C2×D414yes—
112C2×D3×C9C2×Q814yes—
113C3×C4×C9—22no—
114C3⋊Dic9—22no—
115C3⋊Dic9C22×C414yescomplemented
116C9×Dic3—22no—
117C3×Dic9—22no—
118C3×Dic9C22×C414yescomplemented
119C23×C3×C4—31no—
120C23×Dic3—91no—
121C23×Dic3—91no—
122C22×C4×D3—91no—
123C22×C4×D3—91no—
124C22×C4×D3D911yescharacteristic
125C2×C3×C4⋊C4—91no—
126C2×C4⋊Dic3—91no—
127C2×Dic3⋊C4—31no—
128C2×Dic3⋊C4—98no—
129C2×Dic3⋊C4—91no—
130C2×Dic3⋊C4—91no—
131C2×Dic3⋊C4—91no—
132C2×Dic3⋊C4—91no—
133C2×Dic3⋊C4—91no—
134C2×Dic3⋊C4—181no—
135D3×C4⋊C4—98no—
136C23×C32C22×D311yescharacteristic
137C22×C3×D3Dic614yes—
138C22×C3×D3C4×D314yes—
139C22×C3×D3#6714yes—
140C22×C3×D3C22×D312yes—
141C2×C32×C4—22no—
142C2×C32×C4C22×D312yes—
143C2×C3⋊Dic3—34no—
144C2×C3⋊Dic3—32no—
145C2×C3⋊Dic3—32no—
146C2×C3⋊Dic3—62no—
147C2×C3×Dic3—22no—
148C2×C3×Dic3—34no—
149C2×C3×Dic3—32no—
150C2×C3×Dic3—32no—
151C2×C3×Dic3—62no—
152C2×C3×Dic3C22×D312yes—
153C3×C4×D3—28no—
154D3×Dic3—316no—
155D3×Dic3—68no—
156C23×C9—21no—
157C23×C9—34no—
158C23×C9—34no—
159C23×C9—34no—
160C23×C9—32no—
161C23×C9C22×D311yescharacteristic
162C2×C4×C9—22no—
163C2×C4×C9—22no—
164C2×C4×C9—32no—
165C2×C4×C9—42no—
166C2×C4×C9—68no—
167C2×C4×C9—62no—
168C2×C4×C9C22×D312yes—
169C2×Dic9—22no—
170C2×Dic9—34no—
171C2×Dic9—32no—
172C2×Dic9—316no—
173C2×Dic9—32no—
174C2×Dic9—62no—
175C2×Dic9—64no—
176C2×Dic9—62no—
177C2×Dic9—68no—
178C2×Dic9—62no—
179C2×Dic9—122no—
180C2×Dic9C4×D314yescomplemented
181C2×Dic9C22×D312yes—
182C2×Dic9C22×D312yes—
183C22×C4⋊C4—271noSylow
184C2×C3×C9C2×C4⋊C414yescomplemented
185C2×C3×C9C23×C411yescharacteristic
186C2×C3×C9C22×D411yescharacteristic
187C2×C3×C9C22×Q811yescharacteristic
188D3×C9C2×C4⋊C418yescomplemented
189C24×C3—31no—
190C23×D3C2×D911yescharacteristic
191C22×C3×C4—21no—
192C22×C3×C4—31no—
193C22×C3×C4—34no—
194C22×C3×C4—64no—
195C22×C3×C4—91no—
196C22×C3×C4—91no—
197C22×C3×C4C2×D911yescharacteristic
198C22×C3×C4D3211yescharacteristic
199C22×Dic3—31no—
200C22×Dic3—31no—
201C22×Dic3—94no—
202C22×Dic3—98no—
203C22×Dic3—94no—
204C22×Dic3—91no—
205C22×Dic3—91no—
206C22×Dic3—91no—
207C22×Dic3—91no—
208C22×Dic3—181no—
209C22×Dic3—181no—
210C22×Dic3—184no—
211C22×Dic3C2×D911yescharacteristic
212C2×C4×D3—24no—
213C2×C4×D3—98no—
214C2×C4×D3—94no—
215C2×C4×D3—94no—
216C2×C4×D3—184no—
217C2×C4×D3C2×D914yes—
218C3×C4⋊C4—94no—
219C4⋊Dic3—94no—
220Dic3⋊C4—34no—
221Dic3⋊C4—94no—
222Dic3⋊C4—94no—
223Dic3⋊C4—94no—
224Dic3⋊C4—94no—
225Dic3⋊C4—94no—
226Dic3⋊C4—184no—
227C22×C32C2×Dic612yes—
228C22×C32C2×C4×D312yes—
229C22×C32C2×#6712yes—
230C22×C32C23×D311yescharacteristic
231C2×C3×D3Dic3⋊C4116yes—
232C2×C3×D3C2×Dic614yes—
233C2×C3×D3C2×C4×D314yes—
234C2×C3×D3C2×#6714yes—
235C32×C4—22no—
236C3⋊Dic3—34no—
237C3⋊Dic3—62no—
238C3×Dic3—22no—
239C3×Dic3—34no—
240C3×Dic3—62no—
241C22×C9—21no—
242C22×C9—22no—
243C22×C9—22no—
244C22×C9—22no—
245C22×C9—316no—
246C22×C9—34no—
247C22×C9—34no—
248C22×C9—34no—
249C22×C9C2×C4×D312yes—
250C22×C9D3×D412yes—
251C22×C9D3×Q812yes—
252C22×C9C23×D311yescharacteristic
253C4×C9—22no—
254C4×C9—42no—
255C4×C9—62no—
256Dic9—22no—
257Dic9—34no—
258Dic9—64no—
259Dic9—62no—
260Dic9—122no—
261Dic9C2×C4×D314yescomplemented
262C23×C4—31no—
263C23×C4—271no—
264C23×C4—271no—
265C2×C4⋊C4—91no—
266C2×C4⋊C4—278no—
267C2×C4⋊C4—271no—
268C2×C4⋊C4—271no—
269C2×C4⋊C4—271no—
270C3×C9C22×C4⋊C411yescharacteristic, complemented, Sylow
271C23×C3—21no—
272C23×C3—32no—
273C23×C3—34no—
274C23×C3—34no—
275C23×C3—34no—
276C23×C3C22×D911yescharacteristic
277C23×C3C2×D3211yescharacteristic
278C22×D3Dic1814yes—
279C22×D3C4×D914yes—
280C22×D3#61114yes—
281C22×D3C22×D912yes—
282C2×C3×C4—22no—
283C2×C3×C4—22no—
284C2×C3×C4—22no—
285C2×C3×C4—32no—
286C2×C3×C4—42no—
287C2×C3×C4—68no—
288C2×C3×C4—62no—
289C2×C3×C4—94no—
290C2×C3×C4—92no—
291C2×C3×C4—92no—
292C2×C3×C4—182no—
293C2×C3×C4C22×D912yes—
294C2×C3×C4C2×D3212yes—
295C2×Dic3—22no—
296C2×Dic3—32no—
297C2×Dic3—34no—
298C2×Dic3—32no—
299C2×Dic3—62no—
300C2×Dic3—916no—
301C2×Dic3—92no—
302C2×Dic3—94no—
303C2×Dic3—92no—
304C2×Dic3—94no—
305C2×Dic3—92no—
306C2×Dic3—92no—
307C2×Dic3—182no—
308C2×Dic3—184no—
309C2×Dic3—182no—
310C2×Dic3—188no—
311C2×Dic3—182no—
312C2×Dic3—182no—
313C2×Dic3—362no—
314C2×Dic3C22×D912yes—
315C4×D3—28no—
316C4×D3—916no—
317C4×D3—188no—
318C2×C32C2×Dic3⋊C414yes—
319C2×C32C22×Dic611yescharacteristic
320C2×C32C22×C4×D311yescharacteristic
321C2×C32C22×#6711yescharacteristic
322C3×D3C2×Dic3⋊C418yes—
323C2×C9—21no—
324C2×C9—21no—
325C2×C9—21no—
326C2×C9—24no—
327C2×C9—38no—
328C2×C9D3×C4⋊C414yescomplemented
329C2×C9C22×C4×D311yescharacteristic
330C2×C9C2×D3×D411yescharacteristic
331C2×C9C2×D3×Q811yescharacteristic
332C24—31no—
333C22×C4—31no—
334C22×C4—34no—
335C22×C4—64no—
336C22×C4—91no—
337C22×C4—91no—
338C22×C4—274no—
339C22×C4—278no—
340C22×C4—274no—
341C22×C4—271no—
342C22×C4—271no—
343C22×C4—544no—
344C22×C4D3×D911yescharacteristic
345C4⋊C4—94no—
346C4⋊C4—274no—
347C4⋊C4—274no—
348C4⋊C4—274no—
349C22×C3—22no—
350C22×C3—21no—
351C22×C3—22no—
352C22×C3—22no—
353C22×C3—34no—
354C22×C3—34no—
355C22×C3—34no—
356C22×C3—316no—
357C22×C3C2×Dic1812yes—
358C22×C3C2×C4×D912yes—
359C22×C3C2×#61112yes—
360C22×C3C23×D911yescharacteristic
361C22×C3D3×Dic612yes—
362C22×C3C4×D3212yes—
363C22×C3D3×#6712yes—
364C22×C3C22×D3211yescharacteristic
365C2×D3Dic9⋊C4116yesdirect product factor
366C2×D3C2×Dic1814yes—
367C2×D3C2×C4×D914yes—
368C2×D3C2×#61114yes—
369C3×C4—22no—
370C3×C4—22no—
371C3×C4—42no—
372C3×C4—62no—
373C3×C4—94no—
374C3×C4—182no—
375Dic3—22no—
376Dic3—34no—
377Dic3—62no—
378Dic3—94no—
379Dic3—94no—
380Dic3—184no—
381Dic3—182no—
382Dic3—182no—
383Dic3—362no—
384C32C22×Dic3⋊C411yescharacteristic
385C9—21no—
386C9C2×D3×C4⋊C411yescharacteristic, complemented
387C23—34no—
388C23—32no—
389C23—34no—
390C23—34no—
391C23C2×D3×D911yescharacteristic
392C2×C4—22no—
393C2×C4—32no—
394C2×C4—68no—
395C2×C4—62no—
396C2×C4—92no—
397C2×C4—94no—
398C2×C4—92no—
399C2×C4—182no—
400C2×C4—2716no—
401C2×C4—272no—
402C2×C4—274no—
403C2×C4—272no—
404C2×C4—548no—
405C2×C4—542no—
406C2×C4C2×D3×D912yes—
407C2×C3—21no—
408C2×C3—24no—
409C2×C3—21no—
410C2×C3—21no—
411C2×C3—38no—
412C2×C3C2×Dic9⋊C414yescomplemented
413C2×C3C22×Dic1811yescharacteristic
414C2×C3C22×C4×D911yescharacteristic
415C2×C3C22×#61111yescharacteristic
416C2×C3D3×Dic3⋊C414yes—
417C2×C3C2×D3×Dic611yescharacteristic
418C2×C3C2×C4×D3211yescharacteristic
419C2×C3C2×D3×#6711yescharacteristic
420D3C2×Dic9⋊C418yesdirect product factor
421C22—34no—
422C22—316no—
423C22—34no—
424C22—34no—
425C22D3×Dic1812yes—
426C22C4×D3×D912yes—
427C22D3×#61112yes—
428C22C22×D3×D911yescharacteristic
429C4—22no—
430C4—62no—
431C4—94no—
432C4—182no—
433C4—274no—
434C4—542no—
435C3—21no—
436C3C22×Dic9⋊C411yescharacteristic, complemented
437C3C2×D3×Dic3⋊C411yescharacteristic
438C2—38no—
439C2D3×Dic9⋊C414yesdirect product factor
440C2C2×D3×Dic1811yescharacteristic
441C2C2×C4×D3×D911yescharacteristic
442C2C2×D3×#61111yescharacteristic
subgroupG/H
centerC23C2×D3×D9
commutatorC2×C3×C9C23×C4
FrattiniC22×C3C22×D32
FittingC22×C3×C4×C9C22
socleC23×C32C22×D3
prime radicalC23×C3×D3C2×D3
cosocleC22×C3×C9C24
hypercenterC22×C4D3×D9
hypocenterC3×C9C22×C4⋊C4
LCSG ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
4. C2×#88534C2semidirect
5. C2×C3×Dic9⋊C4C2semidirect
6. C2×#88442C2semidirect
7. C2×Dic9⋊Dic3C2semidirect
8. D3×Dic9⋊C4C2direct (N × H)
21. C2×D3×Dic9C4semidirect
24. #88534C22semidirect
25. C3×Dic9⋊C4C22semidirect
26. #88442C22semidirect
27. Dic9⋊Dic3C22semidirect
44. C2×Dic9⋊C4D3direct (N × H)
53. C2×C3⋊Dic9C2×C4semidirect
59. C2×C3×Dic9C2×C4semidirect
64. D3×Dic9C2×C4semidirect
104. Dic9⋊C4C2×D3direct (N × H)
109. C2×D3×C9C4⋊C4semidirect
115. C3⋊Dic9C22×C4semidirect
118. C3×Dic9C22×C4semidirect
180. C2×Dic9C4×D3semidirect
184. C2×C3×C9C2×C4⋊C4semidirect
188. D3×C9C2×C4⋊C4semidirect
261. Dic9C2×C4×D3semidirect
270. C3×C9C22×C4⋊C4semidirect
328. C2×C9D3×C4⋊C4semidirect
365. C2×D3Dic9⋊C4direct (N × H)
386. C9C2×D3×C4⋊C4semidirect
412. C2×C3C2×Dic9⋊C4semidirect
420. D3C2×Dic9⋊C4direct (N × H)
436. C3C22×Dic9⋊C4semidirect
439. C2D3×Dic9⋊C4direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C4×D3×C92
C22×D3×Dic92
C2×#885342
C2×C3×Dic9⋊C42
C2×#884422
C2×Dic9⋊Dic32
D3×Dic9⋊C42
C2×D3×Dic3⋊C43
C22×Dic9⋊C43