Finite Group Databasebeta

← #13152002

Subgroups — #13152002

Order 1728.

1250 conjugacy classes of subgroups.

The subgroup lattice has 448 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6233C211yesmaximal, characteristic
2C2×C9×#217C211yesmaximal, characteristic, complemented
3C2×C3×#3676C211yesmaximal, characteristic, complemented
4#11721274C212yesmaximal, complemented
5#11721248C212yesmaximal, complemented
6#11721241C214yesmaximal, complemented
7#11721240C214yesmaximal, complemented
8#10595510—31nomaximal
9#10589588—31nomaximal
10C24×C3×C9C2211yescharacteristic
11C23×C3⋊D9C2211yescharacteristic
12C2×#6233C418yescomplemented
13C2×#6233C2212yes—
14C2×#6233C2212yes—
15C22×C3⋊Dic9C2211yescharacteristic
16C22×C9×Dic3C2212yescomplemented
17C22×C3×Dic9C2212yescomplemented
18C2×(C3⋊D9)⋊C4C2214yescomplemented
19C9×#217C2212yes—
20C9×#217C2212yescomplemented
21C3×#3676C2212yes—
22C3×#3676C2212yescomplemented
23#88445C2214yescomplemented
24#88442C2214yescomplemented
25Dic3×Dic9C2214yescomplemented
26C22×#638—31no—
27C2×C3×#217—31no—
28C2×C3×#217D311yescharacteristic
29#63974—32no—
30#63957—32no—
31#63952—34no—
32#63952—34no—
33C22×#611—31no—
34C22×#611—61no—
35C2×C9×C22⋊C4—31no—
36C2×#3676D311yescharacteristic, complemented
37C4×#611—34no—
38#611⋊C4—34no—
39C22⋊C4×D9—32no—
40#63439—32no—
41C23×C3×C9—42no—
42C23×C3×C9C2×C414yescomplemented
43C23×C3×C9C2311yescharacteristic
44C23×C3×C9C2311yescharacteristic
45C23×C3×C9C2311yescharacteristic
46C22×C3⋊D9—42no—
47C22×C3⋊D9C2×C414yescomplemented
48C22×C3⋊D9C2312yes—
49#6233C2×C418yescomplemented
50#6233C2×C418yescomplemented
51C2×C3⋊Dic9C2×C414yescomplemented
52C2×C3⋊Dic9C2312yes—
53C2×C9×Dic3—42no—
54C2×C9×Dic3C2314yes—
55C2×C3×Dic9—42no—
56C2×C3×Dic9C2314yes—
57(C3⋊D9)⋊C4—44no—
58#5486—91no—
59#5486—91no—
60C24×C32C2×D311yescharacteristic
61C23×C3⋊D3—31no—
62C2×#638—38no—
63C2×#638—32no—
64C2×#638—32no—
65C22×C3⋊Dic3—31no—
66C22×C3×Dic3—32no—
67C22×C3×Dic3C2×D312yes—
68C2×#624—34no—
69C24×C9—21no—
70C24×C9C2×D311yescharacteristic
71C23×D9—31no—
72C23×D9—61no—
73C3×#217—32no—
74C3×#217—32no—
75C3×#217C2×D312yes—
76C3×#217C2×D312yes—
77#3724—34no—
78#3722—34no—
79Dic32—34no—
80C22×C4×C9—32no—
81C2×#611—38no—
82C2×#611—32no—
83C2×#611—32no—
84C2×#611—68no—
85C2×#611—62no—
86C2×#611—62no—
87C22×Dic9—31no—
88C22×Dic9—61no—
89C22×Dic9C2×D312yescomplemented
90C2×C4×D9—34no—
91C9×C22⋊C4—32no—
92C9×C22⋊C4—32no—
93#3676C2×D312yes—
94#3676C2×D312yescomplemented
95(C2×D9)⋊C4—34no—
96Dic9⋊C4—34no—
97C4×Dic9—34no—
98C22×C3×C9—22no—
99C22×C3×C9—22no—
100C22×C3×C9—22no—
101C22×C3×C9—44no—
102C22×C3×C9C22×C412yes—
103C22×C3×C9C22×C412yes—
104C22×C3×C9C22×C412yes—
105C22×C3×C9C2411yescharacteristic
106C2×C3⋊D9—24no—
107C2×C3⋊D9—44no—
108C2×C3⋊D9C22×C414yes—
109C3⋊Dic9C22×C414yes—
110C9×Dic3—24no—
111C3×Dic9—24no—
112C22×#67—91no—
113C22×#67—91no—
114C22×#67—181no—
115C2×C3×C22⋊C4—31no—
116C2×C3×C22⋊C4—91no—
117C2×#217—31no—
118C2×#217D911yescharacteristic, complemented
119C4×#67—94no—
120C4×#67—94no—
121#67⋊C4—94no—
122#67⋊C4—94no—
123D3×C22⋊C4—92no—
124D3×C22⋊C4—92no—
125#877—92no—
126#877—92no—
127C23×C32—42no—
128C23×C32C4×D314yes—
129C23×C32C22×D311yescharacteristic
130C23×C32C22×D311yescharacteristic
131C23×C32C22×D311yescharacteristic
132C22×C3⋊D3—32no—
133C22×C3⋊D3—34no—
134C22×C3⋊D3—122no—
135#638—38no—
136#638—38no—
137C2×C3⋊Dic3—34no—
138C2×C3⋊Dic3—32no—
139C2×C3×Dic3—34no—
140C2×C3×Dic3—42no—
141C2×C3×Dic3—122no—
142C2×C3×Dic3C22×D314yes—
143#624—124no—
144C23×C9—24no—
145C23×C9—21no—
146C23×C9—21no—
147C23×C9—21no—
148C23×C9—42no—
149C23×C9—44no—
150C23×C9C4×D314yescomplemented
151C23×C9C22×D311yescharacteristic
152C23×C9C22×D311yescharacteristic
153C23×C9C22×D311yescharacteristic
154C22×D9—32no—
155C22×D9—34no—
156C22×D9—64no—
157C22×D9—62no—
158C22×D9—122no—
159C22×D9—124no—
160C2×C4×C9—34no—
161C2×C4×C9—122no—
162#611—38no—
163#611—38no—
164#611—68no—
165#611—68no—
166C2×Dic9—34no—
167C2×Dic9—32no—
168C2×Dic9—42no—
169C2×Dic9—64no—
170C2×Dic9—62no—
171C2×Dic9C22×D314yes—
172C4×D9—124no—
173#518—271noSylow
174C2×C3×C9—22no—
175C2×C3×C9—22no—
176C2×C3×C9—22no—
177C2×C3×C9C23×C411yescharacteristic
178C2×C3×C9H(5,𝔽2)12yes—
179C3⋊D9—24no—
180C24×C3—21no—
181C24×C3C2×D911yescharacteristic
182C24×C3D3211yescharacteristic
183C23×D3—91no—
184C23×D3—91no—
185C23×D3—181no—
186C22×C3×C4—32no—
187C22×C3×C4—92no—
188C2×#67—98no—
189C2×#67—92no—
190C2×#67—92no—
191C2×#67—98no—
192C2×#67—92no—
193C2×#67—92no—
194C2×#67—188no—
195C2×#67—182no—
196C2×#67—182no—
197C22×Dic3—32no—
198C22×Dic3—91no—
199C22×Dic3—91no—
200C22×Dic3—181no—
201C22×Dic3C2×D912yescomplemented
202C2×C4×D3—94no—
203C2×C4×D3—94no—
204C3×C22⋊C4—32no—
205C3×C22⋊C4—32no—
206C3×C22⋊C4—92no—
207C3×C22⋊C4—92no—
208#217—32no—
209#217—32no—
210#217C2×D912yes—
211#217C2×D912yescomplemented
212(C2×D3)⋊C4—94no—
213(C2×D3)⋊C4—94no—
214Dic3⋊C4—94no—
215Dic3⋊C4—94no—
216C4×Dic3—94no—
217C4×Dic3—94no—
218C22×C32—22no—
219C22×C32—22no—
220C22×C32—22no—
221C22×C32—44no—
222C22×C32C2×C4×D312yes—
223C22×C32C2×C4×D312yes—
224C22×C32C2×C4×D312yes—
225C22×C32C23×D311yescharacteristic
226C2×C3⋊D3—34no—
227C2×C3⋊D3—64no—
228C2×C3⋊D3—124no—
229C3⋊Dic3—34no—
230C3×Dic3—24no—
231C3×Dic3—64no—
232C22×C9—24no—
233C22×C9—22no—
234C22×C9—22no—
235C22×C9—22no—
236C22×C9—21no—
237C22×C9—22no—
238C22×C9—22no—
239C22×C9—22no—
240C22×C9—42no—
241C22×C9—42no—
242C22×C9—44no—
243C22×C9—48no—
244C22×C9C2×C4×D312yes—
245C22×C9C2×C4×D312yes—
246C22×C9C2×C4×D312yes—
247C22×C9C23×D311yescharacteristic
248C2×D9—34no—
249C2×D9—64no—
250C2×D9—64no—
251C2×D9—124no—
252C2×D9—124no—
253C2×D9—128no—
254C4×C9—64no—
255Dic9—24no—
256Dic9—34no—
257Dic9—64no—
258C22×D4—271no—
259C4×D4—274no—
260C4×D4—274no—
261#117—272no—
262C2×C22⋊C4—31no—
263C2×C22⋊C4—91no—
264C2×C22⋊C4—272no—
265C3×C9#51811yescharacteristic, complemented, Sylow
266C23×C3—24no—
267C23×C3—21no—
268C23×C3—21no—
269C23×C3—21no—
270C23×C3—42no—
271C23×C3—42no—
272C23×C3—44no—
273C23×C3C4×D914yescomplemented
274C23×C3C22×D911yescharacteristic
275C23×C3C22×D911yescharacteristic
276C23×C3C22×D911yescharacteristic
277C23×C3#62414yes—
278C23×C3C2×D3211yescharacteristic
279C23×C3C2×D3211yescharacteristic
280C23×C3C2×D3211yescharacteristic
281C22×D3—92no—
282C22×D3—92no—
283C22×D3—94no—
284C22×D3—94no—
285C22×D3—182no—
286C22×D3—184no—
287C22×D3—362no—
288C22×D3—362no—
289C22×D3—364no—
290C2×C3×C4—34no—
291C2×C3×C4—94no—
292C2×C3×C4—122no—
293C2×C3×C4—362no—
294#67—98no—
295#67—98no—
296#67—98no—
297#67—98no—
298#67—188no—
299#67—188no—
300C2×Dic3—34no—
301C2×Dic3—42no—
302C2×Dic3—92no—
303C2×Dic3—94no—
304C2×Dic3—92no—
305C2×Dic3—94no—
306C2×Dic3—122no—
307C2×Dic3—182no—
308C2×Dic3—184no—
309C2×Dic3C22×D914yes—
310C4×D3—364no—
311C4×D3—364no—
312C2×C32—22no—
313C2×C32—22no—
314C2×C32—22no—
315C2×C32C22×C4×D311yescharacteristic
316C2×C32#100412yes—
317C3⋊D3—64no—
318C2×C9—24no—
319C2×C9—22no—
320C2×C9—22no—
321C2×C9—22no—
322C2×C9—21no—
323C2×C9—22no—
324C2×C9—42no—
325C2×C9—42no—
326C2×C9C22×C4×D311yescharacteristic
327C2×C9#100412yes—
328D9—64no—
329D9—124no—
330C24—271no—
331C24D3×D911yescharacteristic
332C2×D4—278no—
333C2×D4—272no—
334C2×D4—272no—
335C22×C4—32no—
336C22×C4—92no—
337C22×C4—274no—
338C22×C4—271no—
339C4⋊C4—274no—
340C22⋊C4—32no—
341C22⋊C4—32no—
342C22⋊C4—92no—
343C22⋊C4—92no—
344C22⋊C4—274no—
345C42—274no—
346C22×C3—24no—
347C22×C3—22no—
348C22×C3—22no—
349C22×C3—22no—
350C22×C3—22no—
351C22×C3—22no—
352C22×C3—22no—
353C22×C3—21no—
354C22×C3—22no—
355C22×C3—22no—
356C22×C3—22no—
357C22×C3—42no—
358C22×C3—42no—
359C22×C3—44no—
360C22×C3—44no—
361C22×C3—48no—
362C22×C3C2×C4×D912yes—
363C22×C3C2×C4×D912yes—
364C22×C3C2×C4×D912yes—
365C22×C3C23×D911yescharacteristic
366C22×C3C2×#62412yes—
367C22×C3C2×#62412yes—
368C22×C3C2×#62412yes—
369C22×C3C22×D3211yescharacteristic
370C2×D3—94no—
371C2×D3—94no—
372C2×D3—184no—
373C2×D3—184no—
374C2×D3—184no—
375C2×D3—364no—
376C2×D3—364no—
377C2×D3—364no—
378C2×D3—368no—
379C3×C4—64no—
380C3×C4—184no—
381Dic3—24no—
382Dic3—64no—
383Dic3—94no—
384Dic3—94no—
385Dic3—184no—
386C32#548611yescharacteristic
387C9—21no—
388C9#548611yescharacteristic, complemented
389C23—42no—
390C23—272no—
391C23—274no—
392C23—1082no—
393C23(C3⋊D9)⋊C414yescomplemented
394C23C2×D3×D911yescharacteristic
395C23C2×D3×D911yescharacteristic
396C23C2×D3×D911yescharacteristic
397D4—278no—
398D4—278no—
399C2×C4—34no—
400C2×C4—94no—
401C2×C4—122no—
402C2×C4—272no—
403C2×C4—274no—
404C2×C4—362no—
405C2×C4—1084no—
406C2×C3—24no—
407C2×C3—22no—
408C2×C3—22no—
409C2×C3—22no—
410C2×C3—22no—
411C2×C3—22no—
412C2×C3—22no—
413C2×C3—21no—
414C2×C3—22no—
415C2×C3—42no—
416C2×C3—42no—
417C2×C3C22×C4×D911yescharacteristic
418C2×C3#6371012yes—
419C2×C3C22×#62411yescharacteristic
420C2×C3#6433012yes—
421D3—184no—
422D3—184no—
423D3—364no—
424C22—22no—
425C22—22no—
426C22—22no—
427C22—44no—
428C22—274no—
429C22—544no—
430C22—1084no—
431C22C2×(C3⋊D9)⋊C412yes—
432C22C2×(C3⋊D9)⋊C412yes—
433C22C2×(C3⋊D9)⋊C412yes—
434C22C22×D3×D911yescharacteristic
435C4—64no—
436C4—184no—
437C4—274no—
438C3—21no—
439C3#1058958811yescharacteristic, complemented
440C3#1059551011yescharacteristic
441C2—22no—
442C2—22no—
443C2—22no—
444C2—544no—
445C2C22×(C3⋊D9)⋊C411yescharacteristic
446C2#1172253512yes—
subgroupG/H
centerC22C22×D3×D9
commutatorC2×C3×C9C23×C4
FrattiniC22×C3C22×D32
FittingC24×C3×C9C22
socleC22×C32C23×D3
prime radicalC22×#6233C2
cosocleC22×C3×C9C24
hypercenterC24D3×D9
hypocenterC3×C9#518
LCSG ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C24

Complements

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

NH ≅ G/Ntype
2. C2×C9×#217C2semidirect
3. C2×C3×#3676C2semidirect
4. #11721274C2semidirect
5. #11721248C2semidirect
6. #11721241C2semidirect
7. #11721240C2semidirect
12. C2×#6233C4semidirect
16. C22×C9×Dic3C22semidirect
17. C22×C3×Dic9C22semidirect
18. C2×(C3⋊D9)⋊C4C22semidirect
20. C9×#217C22semidirect
22. C3×#3676C22semidirect
23. #88445C22semidirect
24. #88442C22semidirect
25. Dic3×Dic9C22semidirect
36. C2×#3676D3semidirect
42. C23×C3×C9C2×C4semidirect
47. C22×C3⋊D9C2×C4semidirect
49. #6233C2×C4semidirect
50. #6233C2×C4semidirect
51. C2×C3⋊Dic9C2×C4semidirect
89. C22×Dic9C2×D3semidirect
94. #3676C2×D3semidirect
118. C2×#217D9semidirect
150. C23×C9C4×D3semidirect
201. C22×Dic3C2×D9semidirect
211. #217C2×D9semidirect
265. C3×C9#518semidirect
273. C23×C3C4×D9semidirect
388. C9#5486semidirect
393. C23(C3⋊D9)⋊C4semidirect
439. C3#10589588semidirect

Maximal subgroups

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

subgroupindex
C22×#62332
C2×C9×#2172
C2×C3×#36762
#117212742
#117212482
#117212412
#117212402
#105955103
#105895883