Finite Group Databasebeta

← #13148964

Subgroups — #13148964

Order 1728.

674 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11721746C211yesmaximal, characteristic, complemented
2C3×#63454C211yesmaximal, characteristic
3C9×#890C211yesmaximal, characteristic, complemented
4#11720999C212yesmaximal
5#11720991C212yesmaximal
6D9×C4⋊Dic3C211yesmaximal, characteristic
7#11720971C211yesmaximal, characteristic, complemented
8#11720970C211yesmaximal, characteristic, complemented
9#11720969C212yesmaximal, complemented
10#11720943C211yesmaximal, characteristic
11#11720933C212yesmaximal, complemented
12#10594505—31nomaximal
13#10589791—31nomaximal
14C2×C3×C4×D9C2211yescharacteristic
15C2×Dic3×D9C2212yes—
16C2×#6189C2212yes—
17C2×#6189C2212yes—
18C3×C9×C4⋊C4C2211yescharacteristic
19C4⋊(C3⋊Dic9)C2212yes—
20C4⋊(C3⋊Dic9)C2211yescharacteristic
21#88534C2212yes—
22C4×C3⋊Dic9C2211yescharacteristic
23C3×(C2×D9)⋊C4C2212yes—
24C9×C4⋊Dic3C2212yes—
25C9×C4⋊Dic3C2211yescharacteristic
26C9×Dic3⋊C4C2212yes—
27C4×C9×Dic3C2211yescharacteristic
28C3×C4⋊Dic9C2211yescharacteristic
29C3×C4×Dic9C2212yes—
30(C2×D9)⋊Dic3C2211yescharacteristic
31(C2×D9)⋊Dic3C2211yescharacteristic
32Dic3⋊Dic9C2212yes—
33#88442C2212yes—
34Dic9⋊Dic3C2211yescharacteristic
35Dic9⋊Dic3C2212yes—
36Dic9⋊Dic3C2211yescharacteristic
37#64099—31no—
38C3×#892—31no—
39C3×#890D311yescharacteristic
40#63919—31no—
41#63918—32no—
42#63918—31no—
43#63899—32no—
44#63893—32no—
45D3×C4⋊Dic3—31no—
46#63867—32no—
47#63855—31no—
48C9×#125—31no—
49#63459—32no—
50#63458—32no—
51#63454D311yescharacteristic
52C4⋊C4×D9—31no—
53#63452—31no—
54#63452—61no—
55#63451—32no—
56#63450—31no—
57#63450—32no—
58#63434—31no—
59#63431—31no—
60C22×C3×D9C2311yescharacteristic
61C2×C3×C4×C9C2312yes—
62C2×C3×C4×C9C2311yescharacteristic
63C2×C3⋊Dic9C2311yescharacteristic
64C2×C3⋊Dic9C2312yes—
65C2×C3⋊Dic9C2311yescharacteristic
66C2×C9×Dic3C2311yescharacteristic
67C2×C9×Dic3C2312yes—
68C2×C9×Dic3C2311yescharacteristic
69C2×C3×Dic9C2312yes—
70C2×C3×Dic9C2311yescharacteristic
71C3×C4×D9Q814yescomplemented
72Dic3×D9—42no—
73#6189—42no—
74#6189—42no—
75#5689—91no—
76#5519—91no—
77C2×C3×C4×D3—31no—
78C2×#627—32no—
79C2×#627—32no—
80C2×D3×Dic3—32no—
81C32×C4⋊C4C2×D311yescharacteristic
82C4⋊(C3⋊Dic3)—31no—
83C4⋊(C3⋊Dic3)—32no—
84#3750—32no—
85C4×C3⋊Dic3—31no—
86C3×(C2×D3)⋊C4—32no—
87C3×C4⋊Dic3—31no—
88C3×C4⋊Dic3C2×D311yescharacteristic
89C3×C4⋊Dic3C2×D312yes—
90C3×Dic3⋊C4C2×D312yes—
91C3×C4×Dic3—32no—
92C3×C4×Dic3C2×D311yescharacteristic
93#3724—32no—
94Dic3⋊Dic3—31no—
95Dic3⋊Dic3—31no—
96Dic3⋊Dic3—32no—
97Dic3⋊Dic3—32no—
98(C2×D3)⋊Dic3—31no—
99(C2×D3)⋊Dic3—31no—
100C2×C4×D9—32no—
101C2×C4×D9C2×D311yescharacteristic
102C2×Dic18—32no—
103C2×Dic18—32no—
104C9×C4⋊C4—21no—
105C9×C4⋊C4—32no—
106C9×C4⋊C4—31no—
107C9×C4⋊C4—32no—
108C9×C4⋊C4C2×D311yescharacteristic
109C42×C9—31no—
110(C2×D9)⋊C4—31no—
111(C2×D9)⋊C4—31no—
112(C2×D9)⋊C4C2×D312yes—
113C4⋊Dic9—31no—
114C4⋊Dic9—32no—
115C4⋊Dic9—32no—
116C4⋊Dic9—62no—
117C4⋊Dic9—61no—
118C4⋊Dic9C2×D311yescharacteristic
119Dic9⋊C4—32no—
120Dic9⋊C4—31no—
121Dic9⋊C4—31no—
122Dic9⋊C4—32no—
123Dic9⋊C4—32no—
124Dic9⋊C4—62no—
125C4×Dic9—31no—
126C4×Dic9—61no—
127C4×Dic9C2×D312yes—
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—21no—
135C3⋊Dic9—22no—
136C3⋊Dic9—21no—
137C9×Dic3—21no—
138C9×Dic3—22no—
139C9×Dic3—21no—
140C3×Dic9—22no—
141C3×Dic9C2×Q812yes—
142C3×#125—31no—
143C3×#117—91no—
144#926—91no—
145C2×C4⋊Dic3—91no—
146#897—92no—
147#896—92no—
148#892—31no—
149D3×C4⋊C4—91no—
150#890—91no—
151#890—91no—
152#890—181no—
153#890D911yescharacteristic, complemented
154#889—92no—
155#888—92no—
156#888—91no—
157#888—91no—
158#888—91no—
159#878—92no—
160#878—92no—
161#872—91no—
162#870—92no—
163#869—91no—
164#869—92no—
165C22×C3×D3—31no—
166C2×C32×C4C22×D312yes—
167C2×C32×C4C22×D311yescharacteristic
168C2×C3⋊Dic3—31no—
169C2×C3⋊Dic3—31no—
170C2×C3⋊Dic3—32no—
171C2×C3×Dic3—31no—
172C2×C3×Dic3—32no—
173C2×C3×Dic3C22×D311yescharacteristic
174C2×C3×Dic3C22×D311yescharacteristic
175C2×C3×Dic3C22×D312yes—
176C3×C4×D3—34no—
177#627—122no—
178#627—122no—
179D3×Dic3—122no—
180C22×D9C22×D311yescharacteristic
181C2×C4×C9—21no—
182C2×C4×C9—22no—
183C2×C4×C9—32no—
184C2×C4×C9—31no—
185C2×C4×C9—31no—
186C2×C4×C9C22×D312yes—
187C2×C4×C9C22×D311yescharacteristic
188C2×Dic9—31no—
189C2×Dic9—31no—
190C2×Dic9—32no—
191C2×Dic9—61no—
192C2×Dic9—61no—
193C2×Dic9—62no—
194C2×Dic9C22×D312yes—
195C2×Dic9C22×D311yescharacteristic
196C4×D9—122no—
197C4×D9Dic614yescomplemented
198Dic18—122no—
199Dic18—122no—
200#544—271noSylow
201C2×C3×C9C22×Q811yescharacteristic
202C2×C3×C9H(5,𝔽2)11yescharacteristic
203C2×C3×C92−1+411yescharacteristic
204C3×D9—22no—
205C22×C3×C4—91no—
206C22×Dic3—92no—
207C2×C4×D3—31no—
208C2×C4×D3—92no—
209C2×Dic6—92no—
210C2×Dic6—92no—
211C2×Dic6—92no—
212C2×Dic6—92no—
213C3×C4⋊C4—21no—
214C3×C4⋊C4—31no—
215C3×C4⋊C4—32no—
216C3×C4⋊C4—32no—
217C3×C4⋊C4—91no—
218C3×C4⋊C4C2×D911yescharacteristic
219C3×C4⋊C4D3211yescharacteristic
220C3×C22⋊C4—92no—
221C3×C42—31no—
222C3×C42—92no—
223#217—91no—
224#217—91no—
225(C2×D3)⋊C4—32no—
226(C2×D3)⋊C4—91no—
227(C2×D3)⋊C4—91no—
228C4⋊Dic3—31no—
229C4⋊Dic3—92no—
230C4⋊Dic3—92no—
231C4⋊Dic3—91no—
232C4⋊Dic3—91no—
233C4⋊Dic3—92no—
234C4⋊Dic3—91no—
235C4⋊Dic3—92no—
236C4⋊Dic3—91no—
237C4⋊Dic3—181no—
238C4⋊Dic3—182no—
239C4⋊Dic3C2×D912yes—
240C4⋊Dic3C2×D911yescharacteristic
241Dic3⋊C4—91no—
242Dic3⋊C4—91no—
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—32no—
252C4×Dic3—91no—
253C4×Dic3—91no—
254C4×Dic3—181no—
255C4×Dic3C2×D911yescharacteristic
256C22×C32C23×D311yescharacteristic
257C2×C3×D3—32no—
258C2×C3×D3—61no—
259C2×C3×D3—61no—
260C32×C4—22no—
261C32×C4D3×Q812yes—
262C3⋊Dic3—62no—
263C3⋊Dic3—61no—
264C3⋊Dic3—61no—
265C3×Dic3—22no—
266C3×Dic3—21no—
267C3×Dic3—21no—
268C3×Dic3—32no—
269C3×Dic3—62no—
270C22×C9—21no—
271C22×C9C23×D311yescharacteristic
272C2×D9—21no—
273C2×D9—21no—
274C2×D9C2×Dic612yes—
275C4×C9—22no—
276C4×C9—22no—
277C4×C9—42no—
278C4×C9—61no—
279C4×C9—62no—
280C4×C9—61no—
281C4×C9C2×Dic612yes—
282Dic9—22no—
283Dic9—61no—
284Dic9—62no—
285Dic9—61no—
286Dic9—121no—
287Dic9—121no—
288Dic9—122no—
289Dic9C2×Dic612yes—
290C4⋊Q8—271no—
291C4⋊Q8—272no—
292C4⋊Q8—271no—
293#125—31no—
294#125—272no—
295#125—271no—
296C22⋊Q8—272no—
297C22⋊Q8—272no—
298#117—91no—
299#117—271no—
300C2×C4⋊C4—271no—
301C3×C9#54411yescharacteristic, complemented, Sylow
302C23×C3—91no—
303C22×D3—31no—
304C2×C3×C4—21no—
305C2×C3×C4—22no—
306C2×C3×C4—32no—
307C2×C3×C4—31no—
308C2×C3×C4—31no—
309C2×C3×C4—91no—
310C2×C3×C4—94no—
311C2×C3×C4—92no—
312C2×C3×C4C22×D912yes—
313C2×C3×C4C22×D911yescharacteristic
314C2×C3×C4C2×D3212yes—
315C2×C3×C4C2×D3211yescharacteristic
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—181no—
326C2×Dic3—182no—
327C2×Dic3—362no—
328C2×Dic3C22×D911yescharacteristic
329C2×Dic3C22×D911yescharacteristic
330C2×Dic3C22×D912yes—
331C4×D3—34no—
332C4×D3—362no—
333Dic6—362no—
334Dic6—362no—
335Dic6—362no—
336Dic6—362no—
337C2×C32C2×D3×Q811yescharacteristic
338C2×C32#100911yescharacteristic
339C2×C32#101011yescharacteristic
340C3×D3—62no—
341C2×C9—21no—
342C2×C9—21no—
343C2×C9—21no—
344C2×C9C22×Dic611yescharacteristic
345C2×C9#100911yescharacteristic
346C2×C9#101011yescharacteristic
347D9—22no—
348C2×Q8—272no—
349C2×Q8—272no—
350C22×C4—91no—
351C22×C4—272no—
352C4⋊C4—31no—
353C4⋊C4—32no—
354C4⋊C4—32no—
355C4⋊C4—91no—
356C4⋊C4—272no—
357C4⋊C4—271no—
358C4⋊C4—271no—
359C4⋊C4—272no—
360C4⋊C4—272no—
361C4⋊C4—271no—
362C4⋊C4—272no—
363C4⋊C4—272no—
364C4⋊C4D3×D911yescharacteristic
365C22⋊C4—92no—
366C22⋊C4—271no—
367C22⋊C4—271no—
368C42—31no—
369C42—92no—
370C42—271no—
371C22×C3—21no—
372C22×C3—92no—
373C22×C3—181no—
374C22×C3—181no—
375C22×C3C23×D911yescharacteristic
376C22×C3C22×D3211yescharacteristic
377C2×D3—32no—
378C2×D3—61no—
379C2×D3—61no—
380C3×C4—22no—
381C3×C4—22no—
382C3×C4—22no—
383C3×C4—42no—
384C3×C4—61no—
385C3×C4—61no—
386C3×C4—62no—
387C3×C4—92no—
388C3×C4—182no—
389C3×C4Q8×D912yes—
390C3×C4D3×Dic612yes—
391Dic3—21no—
392Dic3—22no—
393Dic3—21no—
394Dic3—32no—
395Dic3—62no—
396Dic3—181no—
397Dic3—181no—
398Dic3—182no—
399Dic3—181no—
400Dic3—182no—
401Dic3—181no—
402Dic3—362no—
403Dic3—361no—
404Dic3—361no—
405C32#568911yescharacteristic
406C9—21no—
407C9#551911yescharacteristic, complemented
408C23—91no—
409Q8—1082no—
410Q8—1082no—
411C2×C4—31no—
412C2×C4—31no—
413C2×C4—32no—
414C2×C4—91no—
415C2×C4—94no—
416C2×C4—92no—
417C2×C4—271no—
418C2×C4—271no—
419C2×C4—272no—
420C2×C4—1082no—
421C2×C4C2×D3×D912yes—
422C2×C4C2×D3×D911yescharacteristic
423C2×C3—21no—
424C2×C3—21no—
425C2×C3—21no—
426C2×C3—182no—
427C2×C3C2×Q8×D911yescharacteristic
428C2×C3#6371511yescharacteristic
429C2×C3#6371611yescharacteristic
430C2×C3C2×D3×Dic611yescharacteristic
431C2×C3#6430911yescharacteristic
432C2×C3#6432011yescharacteristic
433D3—62no—
434C22—92no—
435C22—181no—
436C22—181no—
437C22C22×D3×D911yescharacteristic
438C4—22no—
439C4—62no—
440C4—61no—
441C4—61no—
442C4—92no—
443C4—182no—
444C4—542no—
445C4—541no—
446C4—541no—
447C4D9×Dic612yes—
448C3—21no—
449C3#1058979111yescharacteristic, complemented
450C3#1059450511yescharacteristic
451C2—182no—
452C2C2×D9×Dic611yescharacteristic
453C2#1172249411yescharacteristic
454C2#1172251511yescharacteristic
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
1. #11721746C2semidirect
3. C9×#890C2semidirect
7. #11720971C2semidirect
8. #11720970C2semidirect
9. #11720969C2semidirect
11. #11720933C2semidirect
71. C3×C4×D9Q8semidirect
153. #890D9semidirect
197. C4×D9Dic6semidirect
301. C3×C9#544semidirect
407. C9#5519semidirect
449. C3#10589791semidirect

Maximal subgroups

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

subgroupindex
#117217462
C3×#634542
C9×#8902
#117209992
#117209912
D9×C4⋊Dic32
#117209712
#117209702
#117209692
#117209432
#117209332
#105945053
#105897913