Finite Group Databasebeta

← #13147746

Subgroups — #13147746

Order 1728.

856 conjugacy classes of subgroups.

The subgroup lattice has 508 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11721712C211yesmaximal, characteristic, complemented
2C9×#875C211yesmaximal, characteristic, complemented
3C3×#63437C211yesmaximal, characteristic, complemented
4#11721051C212yesmaximal, complemented
5#11721050C212yesmaximal, complemented
6#11721049C212yesmaximal, complemented
7#11721042C212yesmaximal, complemented
8#11721035C212yesmaximal, complemented
9#11721003C212yesmaximal, complemented
10#10594021—31nomaximal
11#10589785—31nomaximal
12C22×D3×D9C2212yescomplemented
13C2×C4⋊(C3⋊D9)C2211yescharacteristic
14C2×#6226C2211yescharacteristic, complemented
15C2×C9×D12C2211yescharacteristic
16C2×C3×D36C2211yescharacteristic
17C2×C9×Dic6C2211yescharacteristic, complemented
18C2×C3×Dic18C2211yescharacteristic, complemented
19C2×#6195C2212yes—
20C2×#6194C2212yes—
21C2×#6192C2212yes—
22C3×C42×C9C2211yescharacteristic, complemented
23#88536C2212yescomplemented
24#88536C2212yescomplemented
25C9×(C2×D3)⋊C4C2212yescomplemented
26C9×(C2×D3)⋊C4C2212yescomplemented
27C3×(C2×D9)⋊C4C2212yescomplemented
28C3×(C2×D9)⋊C4C2212yescomplemented
29(C2×D3)⋊Dic9C2212yescomplemented
30#88445C2212yescomplemented
31(C2×D9)⋊Dic3C2212yescomplemented
32Dic3×Dic9C2212yescomplemented
33#64084—31no—
34C3×#875—31no—
35C3×#875D311yescharacteristic
36#63923—32no—
37#63922—32no—
38#63922—32no—
39#63913—32no—
40#63907—32no—
41#63907—32no—
42C9×#124—31no—
43#63509—32no—
44#63499—32no—
45#63447—32no—
46#63447—32no—
47#63444—32no—
48#63444—32no—
49#63437—31no—
50#63437—61no—
51#63437D311yescharacteristic, complemented
52C22×C3⋊D9C2312yes—
53C22×D3×C9C2312yes—
54C22×C3×D9C2312yes—
55C2×D3×D9—42no—
56C2×D3×D9—42no—
57C2×D3×D9—42no—
58C2×C3×C4×C9C2311yescharacteristic
59C2×C3×C4×C9C2311yescharacteristic
60C2×C3×C4×C9C2311yescharacteristic
61C2×C3⋊Dic9C2312yes—
62C4⋊(C3⋊D9)—41no—
63#6226—41no—
64C2×C9×Dic3C2312yes—
65C2×C3×Dic9C2312yes—
66C9×D12—41no—
67C3×D36—41no—
68C9×Dic6—41no—
69C3×Dic18—41no—
70#6195—42no—
71#6194—42no—
72#6192—42no—
73#5683—91no—
74#5683—91no—
75C22×D32—32no—
76C2×C4⋊(C3⋊D3)—31no—
77C2×#634—31no—
78C2×C3×D12—31no—
79C2×C3×D12C2×D311yescharacteristic
80C2×C3×Dic6—31no—
81C2×C3×Dic6C2×D311yescharacteristic
82C2×#626—32no—
83C2×#626—32no—
84C2×#625—32no—
85C23×D9—32no—
86C32×C42C2×D311yescharacteristic
87#3752—32no—
88#3752—32no—
89C3×(C2×D3)⋊C4—32no—
90C3×(C2×D3)⋊C4—32no—
91C3×(C2×D3)⋊C4C2×D312yes—
92C3×(C2×D3)⋊C4C2×D312yes—
93#3722—32no—
94(C2×D3)⋊Dic3—32no—
95(C2×D3)⋊Dic3—32no—
96Dic32—32no—
97C2×Q8×C9—31no—
98C2×D4×C9—31no—
99C2×#611—32no—
100C2×#611—32no—
101C2×D36—32no—
102C2×D36—31no—
103C2×D36—61no—
104C2×D36C2×D311yescharacteristic
105C2×Dic18—31no—
106C2×Dic18—61no—
107C2×Dic18C2×D311yescharacteristic, complemented
108C9×C22⋊C4—32no—
109C9×C22⋊C4—32no—
110C42×C9—21no—
111C42×C9C2×D311yescharacteristic, complemented
112#3676—32no—
113(C2×D9)⋊C4—32no—
114(C2×D9)⋊C4—32no—
115(C2×D9)⋊C4—32no—
116(C2×D9)⋊C4—32no—
117(C2×D9)⋊C4—62no—
118(C2×D9)⋊C4—62no—
119(C2×D9)⋊C4C2×D312yescomplemented
120(C2×D9)⋊C4C2×D312yescomplemented
121C4×Dic9—32no—
122C22×C3×C9C2411yescharacteristic
123C2×C3⋊D9—22no—
124C2×C3⋊D9—22no—
125C2×C3⋊D9—22no—
126C2×D3×C9—22no—
127C2×D3×C9—22no—
128C2×D3×C9—22no—
129C2×C3×D9—22no—
130C2×C3×D9—22no—
131C2×C3×D9—22no—
132D3×D9—48no—
133C3×C4×C9—21no—
134C3×C4×C9—21no—
135C3×C4×C9—21no—
136C3⋊Dic9—22no—
137C9×Dic3—22no—
138C3×Dic9—22no—
139C3×#124—31no—
140C3×#124—91no—
141#947—92no—
142#947—92no—
143#937—92no—
144#937—92no—
145#885—92no—
146#885—92no—
147#885—92no—
148#885—92no—
149#882—92no—
150#882—92no—
151#882—92no—
152#882—92no—
153#875—31no—
154#875—91no—
155#875—91no—
156#875—181no—
157#875D911yescharacteristic, complemented
158C22×C3⋊D3—32no—
159C22×C3×D3—32no—
160C22×C3×D3C22×D312yes—
161C2×D32—122no—
162C2×D32—122no—
163C2×D32—122no—
164C2×C32×C4C22×D311yescharacteristic
165C2×C32×C4C22×D311yescharacteristic
166C2×C32×C4C22×D311yescharacteristic
167C2×C3⋊Dic3—32no—
168C4⋊(C3⋊D3)—121no—
169#634—121no—
170C2×C3×Dic3—32no—
171C2×C3×Dic3C22×D312yes—
172C3×D12—41no—
173C3×D12—121no—
174C3×Dic6—41no—
175C3×Dic6—121no—
176#626—122no—
177#626—122no—
178#625—122no—
179C23×C9—32no—
180C22×D9—32no—
181C22×D9—62no—
182C22×D9—122no—
183C22×D9—122no—
184C22×D9—122no—
185C22×D9C22×D312yes—
186Q8×C9—121no—
187D4×C9—121no—
188C2×C4×C9—21no—
189C2×C4×C9—21no—
190C2×C4×C9—21no—
191C2×C4×C9—32no—
192C2×C4×C9C22×D311yescharacteristic
193C2×C4×C9C22×D311yescharacteristic
194C2×C4×C9C22×D311yescharacteristic
195#611—122no—
196#611—122no—
197C2×Dic9—32no—
198C2×Dic9—62no—
199C2×Dic9C22×D312yes—
200D36—41no—
201D36—121no—
202D36—122no—
203D36—122no—
204Dic18—41no—
205Dic18—121no—
206Dic18—122no—
207H(3,𝔽4)—271noSylow
208C2×C3×C9H(5,𝔽2)11yescharacteristic
209C2×C3×C9H(5,𝔽2)11yescharacteristic
210C2×C3×C9H(5,𝔽2)11yescharacteristic
211C3⋊D9—42no—
212D3×C9—42no—
213C3×D9—42no—
214C23×D3—92no—
215C23×D3—92no—
216C2×C3×Q8—31no—
217C2×C3×Q8—91no—
218C2×C3×D4—31no—
219C2×C3×D4—91no—
220C2×#67—92no—
221C2×#67—92no—
222C2×#67—92no—
223C2×#67—92no—
224C2×D12—31no—
225C2×D12—92no—
226C2×D12—92no—
227C2×D12—91no—
228C2×D12—91no—
229C2×D12—181no—
230C2×D12C2×D911yescharacteristic
231C2×Dic6—31no—
232C2×Dic6—91no—
233C2×Dic6—91no—
234C2×Dic6—181no—
235C2×Dic6C2×D911yescharacteristic, complemented
236C3×C22⋊C4—32no—
237C3×C22⋊C4—32no—
238C3×C22⋊C4—92no—
239C3×C22⋊C4—92no—
240C3×C42—21no—
241C3×C42C2×D911yescharacteristic, complemented
242C3×C42D3211yescharacteristic
243#217—92no—
244#217—92no—
245(C2×D3)⋊C4—32no—
246(C2×D3)⋊C4—32no—
247(C2×D3)⋊C4—92no—
248(C2×D3)⋊C4—92no—
249(C2×D3)⋊C4—92no—
250(C2×D3)⋊C4—92no—
251(C2×D3)⋊C4—92no—
252(C2×D3)⋊C4—92no—
253(C2×D3)⋊C4—92no—
254(C2×D3)⋊C4—92no—
255(C2×D3)⋊C4—182no—
256(C2×D3)⋊C4—182no—
257(C2×D3)⋊C4C2×D912yescomplemented
258(C2×D3)⋊C4C2×D912yescomplemented
259C4×Dic3—92no—
260C4×Dic3—92no—
261C22×C32C23×D311yescharacteristic
262C2×C3⋊D3—62no—
263C2×C3⋊D3—62no—
264C2×C3⋊D3—62no—
265C2×C3×D3—22no—
266C2×C3×D3—22no—
267C2×C3×D3—22no—
268C2×C3×D3—62no—
269C2×C3×D3—62no—
270C2×C3×D3—62no—
271D32—128no—
272C32×C4—21no—
273C32×C4—21no—
274C32×C4—21no—
275C3⋊Dic3—62no—
276C3×Dic3—22no—
277C3×Dic3—62no—
278C22×C9—21no—
279C22×C9—62no—
280C22×C9—62no—
281C22×C9—62no—
282C22×C9C23×D311yescharacteristic
283C2×D9—22no—
284C2×D9—22no—
285C2×D9—22no—
286C2×D9—62no—
287C2×D9—62no—
288C2×D9—62no—
289C2×D9—122no—
290C2×D9—122no—
291C2×D9—122no—
292C2×D9—128no—
293C4×C9—22no—
294C4×C9—21no—
295C4×C9—21no—
296C4×C9—21no—
297C4×C9—41no—
298C4×C9—41no—
299C4×C9—62no—
300Dic9—22no—
301Dic9—62no—
302Dic9—122no—
303#124—31no—
304#124—91no—
305#124—272no—
306#124—272no—
307#124—272no—
308#124—271no—
309#120—272no—
310#120—272no—
311#120—272no—
312C3×C9H(3,𝔽4)11yescharacteristic, complemented, Sylow
313C23×C3—32no—
314C23×C3—92no—
315C22×D3—32no—
316C22×D3—92no—
317C22×D3—92no—
318C22×D3—182no—
319C22×D3—362no—
320C22×D3—362no—
321C22×D3—362no—
322C22×D3—362no—
323C22×D3—362no—
324C22×D3—362no—
325C22×D3C22×D912yes—
326C3×Q8—121no—
327C3×Q8—361no—
328C3×D4—121no—
329C3×D4—361no—
330C2×C3×C4—21no—
331C2×C3×C4—21no—
332C2×C3×C4—21no—
333C2×C3×C4—32no—
334C2×C3×C4—92no—
335C2×C3×C4C22×D911yescharacteristic
336C2×C3×C4C22×D911yescharacteristic
337C2×C3×C4C22×D911yescharacteristic
338C2×C3×C4C2×D3211yescharacteristic
339C2×C3×C4C2×D3211yescharacteristic
340C2×C3×C4C2×D3211yescharacteristic
341#67—362no—
342#67—362no—
343#67—362no—
344#67—362no—
345C2×Dic3—32no—
346C2×Dic3—92no—
347C2×Dic3—92no—
348C2×Dic3—182no—
349C2×Dic3C22×D912yes—
350D12—41no—
351D12—121no—
352D12—361no—
353D12—361no—
354D12—362no—
355D12—362no—
356D12—362no—
357Dic6—41no—
358Dic6—121no—
359Dic6—361no—
360Dic6—361no—
361Dic6—362no—
362C2×C32#100411yescharacteristic
363C2×C32#100911yescharacteristic
364C2×C32#100911yescharacteristic
365C3⋊D3—122no—
366C3×D3—42no—
367C3×D3—122no—
368C2×C9—21no—
369C2×C9—21no—
370C2×C9—21no—
371C2×C9—122no—
372C2×C9#100411yescharacteristic
373C2×C9#100911yescharacteristic
374C2×C9#100911yescharacteristic
375D9—42no—
376D9—122no—
377D9—242no—
378C24—272no—
379C2×Q8—31no—
380C2×Q8—91no—
381C2×Q8—271no—
382C2×D4—31no—
383C2×D4—91no—
384C2×D4—272no—
385C2×D4—272no—
386C2×D4—272no—
387C2×D4—271no—
388C22⋊C4—32no—
389C22⋊C4—32no—
390C22⋊C4—92no—
391C22⋊C4—92no—
392C22⋊C4—272no—
393C22⋊C4—272no—
394C22⋊C4—272no—
395C22⋊C4—272no—
396C22⋊C4—272no—
397C42—272no—
398C42D3×D911yescharacteristic, complemented
399C22×C3—21no—
400C22×C3—62no—
401C22×C3—62no—
402C22×C3—62no—
403C22×C3—182no—
404C22×C3—182no—
405C22×C3—182no—
406C22×C3C23×D911yescharacteristic
407C22×C3C22×D3211yescharacteristic
408C2×D3—22no—
409C2×D3—22no—
410C2×D3—22no—
411C2×D3—62no—
412C2×D3—62no—
413C2×D3—62no—
414C2×D3—182no—
415C2×D3—182no—
416C2×D3—182no—
417C2×D3—182no—
418C2×D3—182no—
419C2×D3—182no—
420C2×D3—362no—
421C2×D3—362no—
422C2×D3—362no—
423C2×D3—368no—
424C2×D3—368no—
425C3×C4—22no—
426C3×C4—21no—
427C3×C4—21no—
428C3×C4—21no—
429C3×C4—21no—
430C3×C4—21no—
431C3×C4—21no—
432C3×C4—41no—
433C3×C4—41no—
434C3×C4—62no—
435C3×C4—182no—
436Dic3—22no—
437Dic3—62no—
438Dic3—182no—
439Dic3—182no—
440Dic3—362no—
441C32#568311yescharacteristic
442C9—21no—
443C9#568311yescharacteristic, complemented
444C23—32no—
445C23—92no—
446C23—272no—
447C23—1082no—
448C23—1082no—
449C23—1082no—
450Q8—121no—
451Q8—361no—
452Q8—1081no—
453D4—121no—
454D4—361no—
455D4—1081no—
456D4—1082no—
457D4—1082no—
458D4—1082no—
459C2×C4—32no—
460C2×C4—92no—
461C2×C4—272no—
462C2×C4C2×D3×D911yescharacteristic
463C2×C4C2×D3×D911yescharacteristic
464C2×C4C2×D3×D911yescharacteristic
465C2×C3—21no—
466C2×C3—21no—
467C2×C3—21no—
468C2×C3—122no—
469C2×C3—362no—
470C2×C3#6371011yescharacteristic
471C2×C3#6371511yescharacteristic
472C2×C3#6371511yescharacteristic
473C2×C3#6430711yescharacteristic
474C2×C3#6430711yescharacteristic
475C2×C3#6430811yescharacteristic
476D3—42no—
477D3—122no—
478D3—362no—
479D3—362no—
480D3—722no—
481C22—62no—
482C22—62no—
483C22—62no—
484C22—182no—
485C22—182no—
486C22—182no—
487C22—542no—
488C22—542no—
489C22—542no—
490C22—1088no—
491C22C22×D3×D911yescharacteristic
492C4—21no—
493C4—21no—
494C4—21no—
495C4—62no—
496C4—182no—
497C4—542no—
498C3—21no—
499C3#1058978511yescharacteristic, complemented
500C3#1059402111yescharacteristic
501C2—122no—
502C2—362no—
503C2—1082no—
504C2#1172247611yescharacteristic
505C2#1172247711yescharacteristic
506C2#1172247811yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC3×C42×C9C22
socleC22×C32C23×D3
hypercenterC42D3×D9
hypocenterC3×C9H(3,𝔽4)
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
1. #11721712C2semidirect
2. C9×#875C2semidirect
3. C3×#63437C2semidirect
4. #11721051C2semidirect
5. #11721050C2semidirect
6. #11721049C2semidirect
7. #11721042C2semidirect
8. #11721035C2semidirect
9. #11721003C2semidirect
12. C22×D3×D9C22semidirect
14. C2×#6226C22semidirect
17. C2×C9×Dic6C22semidirect
18. C2×C3×Dic18C22semidirect
22. C3×C42×C9C22semidirect
23. #88536C22semidirect
24. #88536C22semidirect
25. C9×(C2×D3)⋊C4C22semidirect
26. C9×(C2×D3)⋊C4C22semidirect
27. C3×(C2×D9)⋊C4C22semidirect
28. C3×(C2×D9)⋊C4C22semidirect
29. (C2×D3)⋊Dic9C22semidirect
30. #88445C22semidirect
31. (C2×D9)⋊Dic3C22semidirect
32. Dic3×Dic9C22semidirect
51. #63437D3semidirect
107. C2×Dic18C2×D3semidirect
111. C42×C9C2×D3semidirect
119. (C2×D9)⋊C4C2×D3semidirect
120. (C2×D9)⋊C4C2×D3semidirect
157. #875D9semidirect
235. C2×Dic6C2×D9semidirect
241. C3×C42C2×D9semidirect
257. (C2×D3)⋊C4C2×D9semidirect
258. (C2×D3)⋊C4C2×D9semidirect
312. C3×C9H(3,𝔽4)semidirect
398. C42D3×D9semidirect
443. C9#5683semidirect
499. C3#10589785semidirect

Maximal subgroups

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

subgroupindex
#117217122
C9×#8752
C3×#634372
#117210512
#117210502
#117210492
#117210422
#117210352
#117210032
#105940213
#105897853