Finite Group Databasebeta

← #13150293

Subgroups — #13150293

Order 1728.

858 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88536C211yesmaximal, characteristic
2C9×#924C211yesmaximal, characteristic, complemented
3C3×#63486C211yesmaximal, characteristic, complemented
4#11721274C212yesmaximal
5#11721267C212yesmaximal, complemented
6#11720995C212yesmaximal, complemented
7#11720994C212yesmaximal, complemented
8#11720986C212yesmaximal, complemented
9#11720985C212yesmaximal, complemented
10#10594981—31nomaximal
11#10589736—31nomaximal
12C23×C3⋊D9C2211yescharacteristic
13C22×C3×C4×C9C2211yescharacteristic
14C22×C3⋊Dic9C2211yescharacteristic
15C2×C9×Dic6C2211yescharacteristic, complemented
16C2×C3×Dic18C2211yescharacteristic, complemented
17C2×(C3⋊D9)⋊C4C2212yes—
18C2×(C3⋊D9)⋊C4C2212yes—
19C2×#6189C2212yescomplemented
20#88536C2212yes—
21#88536C2212yes—
22C9×#217C2212yes—
23C3×#3676C2212yes—
24C9×C4⋊Dic3C2211yescharacteristic, complemented
25C9×Dic3⋊C4C2212yescomplemented
26C3×C4⋊Dic9C2211yescharacteristic, complemented
27C3×Dic9⋊C4C2212yescomplemented
28#88445C2212yes—
29#88445C2212yes—
30Dic3⋊Dic9C2212yescomplemented
31#88442C2212yescomplemented
32Dic9⋊Dic3C2212yescomplemented
33C2×#3752—31no—
34C3×#924—31no—
35C3×#924D311yescharacteristic
36#63982—32no—
37#63974—32no—
38#63895—32no—
39#63895—32no—
40#63888—32no—
41#63888—32no—
42C9×C22⋊Q8—31no—
43#63508—32no—
44C2×(C2×D9)⋊C4—31no—
45C2×(C2×D9)⋊C4—61no—
46#63486D311yescharacteristic, complemented
47#63459—32no—
48#63458—32no—
49#63458—32no—
50C22⋊C4×D9—32no—
51#63440—32no—
52C23×C3×C9C2311yescharacteristic
53C22×C3⋊D9—41no—
54C22×C3⋊D9—41no—
55C22×C3⋊D9Q814yescomplemented
56C22×C3⋊D9C2312yes—
57C2×C3×C4×C9—41no—
58C2×C3×C4×C9C2311yescharacteristic
59C2×C3×C4×C9C2311yescharacteristic
60C2×C3⋊Dic9—41no—
61C2×C3⋊Dic9C2312yes—
62C2×C9×Dic3C2312yes—
63C2×C9×Dic3C2312yes—
64C2×C3×Dic9C2312yes—
65C2×C3×Dic9C2312yes—
66C9×Dic6—41no—
67C3×Dic18—41no—
68(C3⋊D9)⋊C4—42no—
69(C3⋊D9)⋊C4—42no—
70#6189—42no—
71#5634—91no—
72#5634—91no—
73C23×C3⋊D3—31no—
74C22×C32×C4C2×D311yescharacteristic
75C22×C3⋊Dic3—31no—
76C2×C3×Dic6—31no—
77C2×C3×Dic6C2×D311yescharacteristic
78C2×#627—32no—
79C2×#624—32no—
80C2×#624—32no—
81C23×D9—31no—
82C23×D9—61no—
83#3752—32no—
84#3752—32no—
85C3×#217—32no—
86C3×#217C2×D312yes—
87C3×C4⋊Dic3—31no—
88C3×C4⋊Dic3C2×D311yescharacteristic
89C3×Dic3⋊C4—32no—
90C3×Dic3⋊C4C2×D312yes—
91#3724—32no—
92Dic3⋊Dic3—32no—
93Dic3⋊Dic3—32no—
94#3722—32no—
95#3722—32no—
96C2×Q8×C9—31no—
97C22×C4×C9—21no—
98C22×C4×C9C2×D311yescharacteristic
99C22×Dic9—31no—
100C22×Dic9—61no—
101C2×C4×D9—32no—
102C2×C4×D9—32no—
103C2×Dic18—32no—
104C2×Dic18C2×D311yescharacteristic, complemented
105C9×C4⋊C4—32no—
106C9×C4⋊C4—31no—
107C9×C22⋊C4—32no—
108#3676C2×D312yes—
109(C2×D9)⋊C4—32no—
110(C2×D9)⋊C4—32no—
111(C2×D9)⋊C4—32no—
112(C2×D9)⋊C4—32no—
113(C2×D9)⋊C4—62no—
114(C2×D9)⋊C4—62no—
115C4⋊Dic9—32no—
116C4⋊Dic9C2×D311yescharacteristic, complemented
117Dic9⋊C4—32no—
118Dic9⋊C4—32no—
119Dic9⋊C4C2×D312yescomplemented
120C22×C3×C9—21no—
121C22×C3×C9—21no—
122C22×C3×C9C2×Q812yes—
123C22×C3×C9C2411yescharacteristic
124C2×C3⋊D9—22no—
125C2×C3⋊D9—22no—
126C2×C3⋊D9—44no—
127C2×C3⋊D9C2×Q814yes—
128C3×C4×C9—21no—
129C3×C4×C9—21no—
130C3⋊Dic9—22no—
131C9×Dic3—22no—
132C9×Dic3—22no—
133C3×Dic9—22no—
134C3×Dic9—22no—
135C3×C22⋊Q8—31no—
136C3×C22⋊Q8—91no—
137#946—92no—
138#946—92no—
139C2×(C2×D3)⋊C4—91no—
140C2×(C2×D3)⋊C4—91no—
141C2×(C2×D3)⋊C4—181no—
142#924—31no—
143#924D911yescharacteristic, complemented
144#897—92no—
145#897—92no—
146#896—92no—
147#896—92no—
148#896—92no—
149#896—92no—
150D3×C22⋊C4—92no—
151D3×C22⋊C4—92no—
152#878—92no—
153#878—92no—
154C23×C32C22×D311yescharacteristic
155C22×C3⋊D3—32no—
156C22×C3⋊D3—34no—
157C22×C3⋊D3—121no—
158C22×C3⋊D3—121no—
159C2×C32×C4—41no—
160C2×C32×C4C22×D311yescharacteristic
161C2×C32×C4C22×D311yescharacteristic
162C2×C3⋊Dic3—32no—
163C2×C3⋊Dic3—121no—
164C2×C3×Dic3—32no—
165C2×C3×Dic3—32no—
166C2×C3×Dic3C22×D312yes—
167C2×C3×Dic3C22×D312yes—
168C3×Dic6—41no—
169C3×Dic6—121no—
170#627—122no—
171#624—122no—
172#624—122no—
173C23×C9—21no—
174C23×C9C22×D311yescharacteristic
175C22×D9—32no—
176C22×D9—34no—
177C22×D9—62no—
178C22×D9—64no—
179C22×D9—121no—
180C22×D9—121no—
181C22×D9—122no—
182C22×D9—122no—
183Q8×C9—121no—
184C2×C4×C9—21no—
185C2×C4×C9—21no—
186C2×C4×C9—32no—
187C2×C4×C9—32no—
188C2×C4×C9—41no—
189C2×C4×C9—42no—
190C2×C4×C9C22×D311yescharacteristic
191C2×C4×C9C22×D311yescharacteristic
192C2×Dic9—32no—
193C2×Dic9—62no—
194C2×Dic9—121no—
195C2×Dic9—122no—
196C2×Dic9C22×D312yes—
197C2×Dic9C22×D312yes—
198C4×D9—122no—
199C4×D9—122no—
200Dic18—41no—
201Dic18—122no—
202#543—271noSylow
203C2×C3×C9—22no—
204C2×C3×C9C22×Q811yescharacteristic
205C2×C3×C9H(5,𝔽2)11yescharacteristic
206C2×C3×C9H(5,𝔽2)11yescharacteristic
207C3⋊D9—24no—
208C23×D3—91no—
209C23×D3—91no—
210C23×D3—181no—
211C2×C3×Q8—31no—
212C2×C3×Q8—91no—
213C22×C3×C4—21no—
214C22×C3×C4C2×D911yescharacteristic
215C22×C3×C4D3211yescharacteristic
216C22×Dic3—91no—
217C22×Dic3—91no—
218C22×Dic3—181no—
219C2×C4×D3—92no—
220C2×C4×D3—92no—
221C2×C4×D3—92no—
222C2×C4×D3—92no—
223C2×Dic6—31no—
224C2×Dic6—92no—
225C2×Dic6—92no—
226C2×Dic6C2×D911yescharacteristic, complemented
227C3×C4⋊C4—31no—
228C3×C4⋊C4—32no—
229C3×C4⋊C4—91no—
230C3×C4⋊C4—92no—
231C3×C22⋊C4—32no—
232C3×C22⋊C4—92no—
233#217—32no—
234#217C2×D912yes—
235(C2×D3)⋊C4—92no—
236(C2×D3)⋊C4—92no—
237(C2×D3)⋊C4—92no—
238(C2×D3)⋊C4—92no—
239(C2×D3)⋊C4—92no—
240(C2×D3)⋊C4—92no—
241(C2×D3)⋊C4—92no—
242(C2×D3)⋊C4—92no—
243(C2×D3)⋊C4—182no—
244(C2×D3)⋊C4—182no—
245C4⋊Dic3—31no—
246C4⋊Dic3—92no—
247C4⋊Dic3—92no—
248C4⋊Dic3C2×D911yescharacteristic, complemented
249Dic3⋊C4—32no—
250Dic3⋊C4—92no—
251Dic3⋊C4—92no—
252Dic3⋊C4—92no—
253Dic3⋊C4—92no—
254Dic3⋊C4C2×D912yescomplemented
255C22×C32—21no—
256C22×C32—21no—
257C22×C32D3×Q812yes—
258C22×C32C23×D311yescharacteristic
259C2×C3⋊D3—34no—
260C2×C3⋊D3—62no—
261C2×C3⋊D3—62no—
262C2×C3⋊D3—124no—
263C32×C4—21no—
264C32×C4—21no—
265C3⋊Dic3—62no—
266C3×Dic3—22no—
267C3×Dic3—22no—
268C3×Dic3—62no—
269C3×Dic3—62no—
270C22×C9—22no—
271C22×C9—22no—
272C22×C9—21no—
273C22×C9—21no—
274C22×C9—21no—
275C22×C9—22no—
276C22×C9D3×Q812yes—
277C22×C9C23×D311yescharacteristic
278C2×D9—34no—
279C2×D9—64no—
280C2×D9—62no—
281C2×D9—62no—
282C2×D9—122no—
283C2×D9—122no—
284C2×D9—124no—
285C2×D9—128no—
286C4×C9—21no—
287C4×C9—21no—
288C4×C9—41no—
289C4×C9—41no—
290C4×C9—62no—
291C4×C9—62no—
292Dic9—22no—
293Dic9—22no—
294Dic9—62no—
295Dic9—122no—
296C22⋊Q8—31no—
297C22⋊Q8—91no—
298C22⋊Q8—272no—
299C22⋊Q8—272no—
300C22⋊Q8—272no—
301C22⋊Q8—272no—
302C22⋊Q8—272no—
303C2×C22⋊C4—272no—
304C2×C22⋊C4—271no—
305C3×C9#54311yescharacteristic, complemented, Sylow
306C23×C3—21no—
307C23×C3C22×D911yescharacteristic
308C23×C3C2×D3211yescharacteristic
309C22×D3—92no—
310C22×D3—92no—
311C22×D3—94no—
312C22×D3—94no—
313C22×D3—182no—
314C22×D3—184no—
315C22×D3—361no—
316C22×D3—361no—
317C22×D3—361no—
318C22×D3—361no—
319C22×D3—362no—
320C22×D3—362no—
321C3×Q8—121no—
322C3×Q8—361no—
323C2×C3×C4—21no—
324C2×C3×C4—21no—
325C2×C3×C4—32no—
326C2×C3×C4—32no—
327C2×C3×C4—41no—
328C2×C3×C4—41no—
329C2×C3×C4—42no—
330C2×C3×C4—92no—
331C2×C3×C4—92no—
332C2×C3×C4C22×D911yescharacteristic
333C2×C3×C4C22×D911yescharacteristic
334C2×C3×C4C2×D3211yescharacteristic
335C2×C3×C4C2×D3211yescharacteristic
336C2×Dic3—32no—
337C2×Dic3—32no—
338C2×Dic3—92no—
339C2×Dic3—92no—
340C2×Dic3—182no—
341C2×Dic3—361no—
342C2×Dic3—361no—
343C2×Dic3—362no—
344C2×Dic3C22×D912yes—
345C2×Dic3C22×D912yes—
346C4×D3—362no—
347C4×D3—362no—
348C4×D3—362no—
349C4×D3—362no—
350Dic6—41no—
351Dic6—121no—
352Dic6—362no—
353Dic6—362no—
354C2×C32—22no—
355C2×C32#100411yescharacteristic
356C2×C32C2×D3×Q811yescharacteristic
357C2×C32#100911yescharacteristic
358C3⋊D3—64no—
359C2×C9—24no—
360C2×C9—22no—
361C2×C9—21no—
362C2×C9—21no—
363C2×C9—21no—
364C2×C9#100411yescharacteristic
365C2×C9C2×D3×Q811yescharacteristic
366C2×C9#100911yescharacteristic
367D9—64no—
368D9—124no—
369C24—271no—
370C2×Q8—31no—
371C2×Q8—91no—
372C2×Q8—272no—
373C22×C4—272no—
374C22×C4—272no—
375C22×C4—271no—
376C22×C4D3×D911yescharacteristic
377C4⋊C4—32no—
378C4⋊C4—31no—
379C4⋊C4—92no—
380C4⋊C4—91no—
381C4⋊C4—272no—
382C4⋊C4—272no—
383C4⋊C4—272no—
384C22⋊C4—32no—
385C22⋊C4—92no—
386C22⋊C4—272no—
387C22⋊C4—272no—
388C22⋊C4—272no—
389C22⋊C4—272no—
390C22×C3—22no—
391C22×C3—22no—
392C22×C3—21no—
393C22×C3—21no—
394C22×C3—21no—
395C22×C3—21no—
396C22×C3—22no—
397C22×C3—21no—
398C22×C3Q8×D912yes—
399C22×C3C23×D911yescharacteristic
400C22×C3#379712yes—
401C22×C3C22×D3211yescharacteristic
402C2×D3—94no—
403C2×D3—94no—
404C2×D3—184no—
405C2×D3—182no—
406C2×D3—182no—
407C2×D3—182no—
408C2×D3—182no—
409C2×D3—362no—
410C2×D3—362no—
411C2×D3—364no—
412C2×D3—364no—
413C2×D3—368no—
414C3×C4—21no—
415C3×C4—21no—
416C3×C4—21no—
417C3×C4—21no—
418C3×C4—41no—
419C3×C4—41no—
420C3×C4—62no—
421C3×C4—62no—
422C3×C4—182no—
423C3×C4—182no—
424Dic3—22no—
425Dic3—22no—
426Dic3—62no—
427Dic3—62no—
428Dic3—182no—
429Dic3—182no—
430Dic3—362no—
431C32#563411yescharacteristic
432C9—21no—
433C9#563411yescharacteristic, complemented
434C23—272no—
435C23—274no—
436C23—1081no—
437C23—1081no—
438C23C2×D3×D911yescharacteristic
439Q8—121no—
440Q8—361no—
441Q8—1082no—
442C2×C4—32no—
443C2×C4—32no—
444C2×C4—41no—
445C2×C4—92no—
446C2×C4—92no—
447C2×C4—272no—
448C2×C4—1082no—
449C2×C4—1082no—
450C2×C4—1081no—
451C2×C4C2×D3×D911yescharacteristic
452C2×C4C2×D3×D911yescharacteristic
453C2×C3—24no—
454C2×C3—22no—
455C2×C3—22no—
456C2×C3—21no—
457C2×C3—21no—
458C2×C3—21no—
459C2×C3#6371011yescharacteristic
460C2×C3C2×Q8×D911yescharacteristic
461C2×C3#6371511yescharacteristic
462C2×C3C2×#379711yescharacteristic
463C2×C3#6430811yescharacteristic
464C2×C3#6433011yescharacteristic
465D3—184no—
466D3—184no—
467D3—364no—
468C22—21no—
469C22—21no—
470C22—274no—
471C22—542no—
472C22—542no—
473C22—1084no—
474C22(C3⋊D9)⋊Q812yes—
475C22C22×D3×D911yescharacteristic
476C4—21no—
477C4—21no—
478C4—62no—
479C4—62no—
480C4—182no—
481C4—182no—
482C4—542no—
483C3—21no—
484C3#1058973611yescharacteristic, complemented
485C3#1059498111yescharacteristic
486C2—22no—
487C2—544no—
488C2C2×(C3⋊D9)⋊Q811yescharacteristic
489C2#1172247811yescharacteristic
490C2#1172253511yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC22×C3×C4×C9C22
socleC22×C32C23×D3
prime radicalC23×C3⋊D9C22
hypercenterC22×C4D3×D9
hypocenterC3×C9#543
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. C9×#924C2semidirect
3. C3×#63486C2semidirect
5. #11721267C2semidirect
6. #11720995C2semidirect
7. #11720994C2semidirect
8. #11720986C2semidirect
9. #11720985C2semidirect
15. C2×C9×Dic6C22semidirect
16. C2×C3×Dic18C22semidirect
19. C2×#6189C22semidirect
24. C9×C4⋊Dic3C22semidirect
25. C9×Dic3⋊C4C22semidirect
26. C3×C4⋊Dic9C22semidirect
27. C3×Dic9⋊C4C22semidirect
30. Dic3⋊Dic9C22semidirect
31. #88442C22semidirect
32. Dic9⋊Dic3C22semidirect
46. #63486D3semidirect
55. C22×C3⋊D9Q8semidirect
104. C2×Dic18C2×D3semidirect
116. C4⋊Dic9C2×D3semidirect
119. Dic9⋊C4C2×D3semidirect
143. #924D9semidirect
226. C2×Dic6C2×D9semidirect
248. C4⋊Dic3C2×D9semidirect
254. Dic3⋊C4C2×D9semidirect
305. C3×C9#543semidirect
433. C9#5634semidirect
484. C3#10589736semidirect

Maximal subgroups

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

subgroupindex
C2×#885362
C9×#9242
C3×#634862
#117212742
#117212672
#117209952
#117209942
#117209862
#117209852
#105949813
#105897363