Finite Group Databasebeta

← #13147592

Subgroups — #13147592

Order 1728.

958 conjugacy classes of subgroups.

The subgroup lattice has 524 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88639C211yesmaximal, characteristic
2#11721709C211yesmaximal, characteristic, complemented
3C4×C9×D12C211yesmaximal, characteristic, complemented
4C3×C4×Dic18C211yesmaximal, characteristic, complemented
5#11721030C212yesmaximal, complemented
6#11721025C211yesmaximal, characteristic, complemented
7#11721013C212yesmaximal, complemented
8#11721010C212yesmaximal, complemented
9D3×Dic9⋊C4C212yesmaximal, complemented
10#11721000C211yesmaximal, characteristic, complemented
11#11720910C211yesmaximal, characteristic, complemented
12#10593889—31nomaximal
13#10589648—31nomaximal
14C2×C4×C3⋊D9C2211yescharacteristic
15C2×C9×D12C2211yescharacteristic
16C2×C4×D3×C9C2212yes—
17C2×C3×Dic18C2211yescharacteristic
18C2×#6195C2212yes—
19C2×D3×Dic9C2212yes—
20C2×D3×Dic9C2212yes—
21C2×(C3⋊D9)⋊C4C2212yes—
22#88639C418yescomplemented
23C3×C42×C9C2211yescharacteristic
24#88536C2211yescharacteristic
25#88536C2211yescharacteristic
26#88534C2211yescharacteristic
27#88534C2211yescharacteristic
28C4×C3⋊Dic9C2211yescharacteristic
29C9×(C2×D3)⋊C4C2212yes—
30C9×C4⋊Dic3C2211yescharacteristic
31C3×C4⋊Dic9C2211yescharacteristic
32C3×Dic9⋊C4C2212yes—
33C3×C4×Dic9C2212yes—
34(C2×D3)⋊Dic9C2212yes—
35#88442C2212yes—
36Dic9⋊Dic3C2212yes—
37Dic3×Dic9C2212yes—
38C2×#3796—31no—
39#64081—31no—
40C3×C4×D12D311yescharacteristic
41C3×C4×Dic6—31no—
42#63904—32no—
43#63900—31no—
44#63898—31no—
45#63880—32no—
46#63877—32no—
47D3×Dic3⋊C4—32no—
48#63843—31no—
49C2×#3699—31no—
50C4×D4×C9—31no—
51D4×Dic9—31no—
52C4×#611—32no—
53C2×Dic9⋊C4—32no—
54C4⋊C4×D9—31no—
55#63449—31no—
56#611⋊C4—32no—
57#63439—32no—
58#63434—31no—
59#63434—61no—
60C4×Dic18D311yescharacteristic, complemented
61C22×C3⋊D9C2311yescharacteristic
62C22×D3×C9C2312yes—
63C2×C3×C4×C9C2311yescharacteristic
64C2×C3×C4×C9C2311yescharacteristic
65C2×C3×C4×C9C2311yescharacteristic
66C2×C3⋊Dic9C2311yescharacteristic
67C2×C3⋊Dic9C2311yescharacteristic
68C2×C3⋊Dic9C2311yescharacteristic
69C4×C3⋊D9C2×C414yescomplemented
70C2×C9×Dic3C2312yes—
71C2×C3×Dic9C2312yes—
72C2×C3×Dic9C2312yes—
73C9×D12C2×C414yescomplemented
74C4×D3×C9—42no—
75C3×Dic18C2×C414yescomplemented
76#6195C2×C418yescomplemented
77D3×Dic9—42no—
78D3×Dic9C2×C418yescomplemented
79(C3⋊D9)⋊C4—42no—
80#5574—91no—
81#5546—91no—
82C2×C4×C3⋊D3—31no—
83C2×C3×D12C2×D311yescharacteristic
84C2×C3×C4×D3C2×D312yes—
85C2×C3×Dic6—31no—
86C2×#626—32no—
87C2×#624—32no—
88C2×D3×Dic3—32no—
89C2×D3×Dic3—32no—
90#3796—38no—
91C32×C42C2×D311yescharacteristic
92#3752—31no—
93#3752—31no—
94#3750—31no—
95#3750—31no—
96C4×C3⋊Dic3—31no—
97C3×(C2×D3)⋊C4C2×D312yes—
98C3×C4⋊Dic3—31no—
99C3×C4⋊Dic3C2×D311yescharacteristic
100C3×Dic3⋊C4—32no—
101C3×C4×Dic3—32no—
102#3724—32no—
103Dic3⋊Dic3—32no—
104(C2×D3)⋊Dic3—32no—
105Dic32—32no—
106C2×D4×C9—31no—
107C22×C4×C9—32no—
108C2×#611—32no—
109C22×Dic9—32no—
110C22×Dic9—32no—
111#3699—38no—
112C2×C4×D9—31no—
113C2×C4×D9—32no—
114C2×C4×D9—61no—
115C2×Dic18C2×D311yescharacteristic
116C9×C4⋊C4—31no—
117C9×C22⋊C4—32no—
118C42×C9—21no—
119C42×C9C2×D311yescharacteristic
120#3676—32no—
121(C2×D9)⋊C4—31no—
122(C2×D9)⋊C4—31no—
123(C2×D9)⋊C4—61no—
124(C2×D9)⋊C4—61no—
125C4⋊Dic9C2×D311yescharacteristic
126Dic9⋊C4—31no—
127Dic9⋊C4—31no—
128Dic9⋊C4—32no—
129Dic9⋊C4—32no—
130Dic9⋊C4—61no—
131Dic9⋊C4—61no—
132Dic9⋊C4C2×D312yes—
133C4×Dic9—31no—
134C4×Dic9—32no—
135C4×Dic9—61no—
136C4×Dic9C2×D312yes—
137C22×C3×C9C2411yescharacteristic
138C2×C3⋊D9—21no—
139C2×C3⋊D9—21no—
140C2×C3⋊D9C22×C412yes—
141C2×D3×C9—22no—
142C2×D3×C9—22no—
143C2×D3×C9C22×C414yes—
144C3×C4×C9—21no—
145C3×C4×C9—21no—
146C3×C4×C9C22×C412yes—
147C3⋊Dic9—21no—
148C3⋊Dic9—21no—
149C3⋊Dic9C22×C412yes—
150C9×Dic3—22no—
151C3×Dic9—22no—
152C3×Dic9C22×C414yes—
153C2×#239—91no—
154C2×#237—91no—
155C3×C4×Q8—91no—
156C3×C4×D4—31no—
157Q8×Dic3—91no—
158D4×Dic3—91no—
159C4×#67—92no—
160C2×Dic3⋊C4—92no—
161#893—91no—
162#893—92no—
163D3×C4⋊C4—91no—
164D3×C4⋊C4—92no—
165D3×C4⋊C4—91no—
166#887—91no—
167#67⋊C4—92no—
168#877—92no—
169C4×D12—92no—
170C4×D12D911yescharacteristic, complemented
171#872—91no—
172#872—91no—
173#872—92no—
174#872—181no—
175C4×Dic6—31no—
176C22×C3⋊D3—31no—
177C22×C3×D3C22×D312yes—
178C2×C32×C4C22×D311yescharacteristic
179C2×C32×C4C22×D311yescharacteristic
180C2×C32×C4C22×D311yescharacteristic
181C2×C3⋊Dic3—31no—
182C2×C3⋊Dic3—31no—
183C2×C3⋊Dic3—31no—
184C4×C3⋊D3—34no—
185C2×C3×Dic3—32no—
186C2×C3×Dic3—32no—
187C2×C3×Dic3C22×D312yes—
188C3×D12C4×D314yes—
189C3×C4×D3—42no—
190C3×Dic6—34no—
191#626—38no—
192#624—122no—
193D3×Dic3—38no—
194D3×Dic3—122no—
195C23×C9—32no—
196C22×D9—31no—
197C22×D9—61no—
198D4×C9—34no—
199C2×C4×C9—21no—
200C2×C4×C9—21no—
201C2×C4×C9—21no—
202C2×C4×C9—32no—
203C2×C4×C9—122no—
204C2×C4×C9C22×D311yescharacteristic
205C2×C4×C9C22×D311yescharacteristic
206C2×C4×C9C22×D311yescharacteristic
207#611—38no—
208C2×Dic9—31no—
209C2×Dic9—38no—
210C2×Dic9—31no—
211C2×Dic9—31no—
212C2×Dic9—61no—
213C2×Dic9—61no—
214C2×Dic9—61no—
215C2×Dic9—122no—
216C2×Dic9C22×D312yes—
217C2×Dic9C22×D312yes—
218C4×D9—34no—
219C4×D9—64no—
220C4×D9—122no—
221Dic18C4×D314yescomplemented
222#520—271noSylow
223C2×C3×C9C23×C411yescharacteristic
224C2×C3×C9H(5,𝔽2)11yescharacteristic
225C2×C3×C92−1+411yescharacteristic
226C3⋊D9—22no—
227D3×C9—24no—
228C2×C3×Q8—91no—
229C2×C3×D4—31no—
230C22×C3×C4—32no—
231C2×#67—92no—
232C22×Dic3—92no—
233C22×Dic3—92no—
234#239—98no—
235#237—98no—
236C2×D12—92no—
237C2×D12C2×D911yescharacteristic
238C2×C4×D3—92no—
239C2×C4×D3—91no—
240C2×C4×D3—91no—
241C2×C4×D3—92no—
242C2×C4×D3—92no—
243C2×C4×D3—92no—
244C2×C4×D3—181no—
245C2×C4×D3C2×D912yes—
246C2×Dic6—31no—
247C3×C4⋊C4—31no—
248C3×C4⋊C4—92no—
249C3×C4⋊C4—91no—
250C3×C22⋊C4—32no—
251C3×C42—21no—
252C3×C42—92no—
253C3×C42C2×D911yescharacteristic
254C3×C42D3211yescharacteristic
255#217—92no—
256(C2×D3)⋊C4—92no—
257(C2×D3)⋊C4—91no—
258(C2×D3)⋊C4—91no—
259(C2×D3)⋊C4—91no—
260(C2×D3)⋊C4—91no—
261(C2×D3)⋊C4—181no—
262(C2×D3)⋊C4—181no—
263(C2×D3)⋊C4C2×D912yes—
264C4⋊Dic3—31no—
265C4⋊Dic3—92no—
266C4⋊Dic3C2×D911yescharacteristic
267Dic3⋊C4—32no—
268Dic3⋊C4—92no—
269Dic3⋊C4—92no—
270Dic3⋊C4—91no—
271Dic3⋊C4—91no—
272Dic3⋊C4—91no—
273Dic3⋊C4—91no—
274Dic3⋊C4—92no—
275Dic3⋊C4—181no—
276Dic3⋊C4—181no—
277C4×Dic3—32no—
278C4×Dic3—92no—
279C4×Dic3—91no—
280C4×Dic3—91no—
281C4×Dic3—92no—
282C4×Dic3—181no—
283C22×C32C23×D311yescharacteristic
284C2×C3⋊D3—32no—
285C2×C3⋊D3—61no—
286C2×C3⋊D3—61no—
287C2×C3×D3—22no—
288C2×C3×D3—22no—
289C2×C3×D3C2×C4×D314yes—
290C32×C4—21no—
291C32×C4—21no—
292C32×C4C2×C4×D312yes—
293C3⋊Dic3—32no—
294C3⋊Dic3—61no—
295C3⋊Dic3—61no—
296C3×Dic3—22no—
297C3×Dic3—34no—
298C3×Dic3—62no—
299C22×C9—21no—
300C22×C9—34no—
301C22×C9—62no—
302C22×C9—62no—
303C22×C9C23×D311yescharacteristic
304C2×D9—32no—
305C2×D9—62no—
306C2×D9—61no—
307C2×D9—61no—
308C2×D9—121no—
309C2×D9—121no—
310C4×C9—21no—
311C4×C9—21no—
312C4×C9—22no—
313C4×C9—41no—
314C4×C9—41no—
315C4×C9—62no—
316C4×C9C2×C4×D312yes—
317Dic9—22no—
318Dic9—32no—
319Dic9—62no—
320Dic9—61no—
321Dic9—61no—
322Dic9—121no—
323Dic9—121no—
324Dic9C2×C4×D314yes—
325C2×#41—271no—
326C4×Q8—91no—
327C4×Q8—271no—
328C4×D4—31no—
329C4×D4—272no—
330C4×D4—272no—
331C4×D4—271no—
332#117—272no—
333#117—271no—
334C2×C4⋊C4—271no—
335C2×C4⋊C4—272no—
336C3×C9#52011yescharacteristic, complemented, Sylow
337C23×C3—32no—
338C22×D3—91no—
339C22×D3—91no—
340C22×D3—181no—
341C22×D3C22×D912yes—
342C3×Q8—94no—
343C3×D4—34no—
344C2×C3×C4—21no—
345C2×C3×C4—21no—
346C2×C3×C4—21no—
347C2×C3×C4—32no—
348C2×C3×C4—92no—
349C2×C3×C4—92no—
350C2×C3×C4—122no—
351C2×C3×C4C22×D911yescharacteristic
352C2×C3×C4C22×D911yescharacteristic
353C2×C3×C4C22×D911yescharacteristic
354C2×C3×C4C2×D3211yescharacteristic
355C2×C3×C4C2×D3211yescharacteristic
356C2×C3×C4C2×D3211yescharacteristic
357#67—98no—
358C2×Dic3—32no—
359C2×Dic3—32no—
360C2×Dic3—98no—
361C2×Dic3—91no—
362C2×Dic3—91no—
363C2×Dic3—91no—
364C2×Dic3—91no—
365C2×Dic3—91no—
366C2×Dic3—91no—
367C2×Dic3—181no—
368C2×Dic3—181no—
369C2×Dic3—181no—
370C2×Dic3—362no—
371C2×Dic3C22×D912yes—
372D12—98no—
373D12C4×D914yescomplemented
374C4×D3—42no—
375C4×D3—98no—
376C4×D3—94no—
377C4×D3—94no—
378C4×D3—184no—
379C4×D3—362no—
380C4×D3—362no—
381C4×D3—362no—
382Dic6—34no—
383C2×C32C22×C4×D311yescharacteristic
384C2×C32#100411yescharacteristic
385C2×C32#101011yescharacteristic
386C3⋊D3—62no—
387C3×D3—24no—
388C2×C9—21no—
389C2×C9—21no—
390C2×C9—21no—
391C2×C9—64no—
392C2×C9C22×C4×D311yescharacteristic
393C2×C9#100711yescharacteristic
394C2×C9#100911yescharacteristic
395D9—62no—
396D9—122no—
397#41—278no—
398C2×Q8—91no—
399C2×D4—31no—
400C2×D4—272no—
401C22×C4—32no—
402C22×C4—272no—
403C22×C4—272no—
404C22×C4—272no—
405C22×C4—271no—
406C4⋊C4—31no—
407C4⋊C4—92no—
408C4⋊C4—91no—
409C4⋊C4—272no—
410C4⋊C4—272no—
411C4⋊C4—271no—
412C4⋊C4—271no—
413C22⋊C4—32no—
414C22⋊C4—272no—
415C22⋊C4—271no—
416C22⋊C4—271no—
417C42—92no—
418C42—272no—
419C42—271no—
420C42D3×D911yescharacteristic
421C22×C3—21no—
422C22×C3—34no—
423C22×C3—62no—
424C22×C3—62no—
425C22×C3C23×D911yescharacteristic
426C22×C3C22×D3211yescharacteristic
427C2×D3—22no—
428C2×D3—22no—
429C2×D3—92no—
430C2×D3—92no—
431C2×D3—182no—
432C2×D3—181no—
433C2×D3—181no—
434C2×D3—181no—
435C2×D3—181no—
436C2×D3—361no—
437C2×D3—361no—
438C2×D3C2×C4×D914yes—
439C3×C4—21no—
440C3×C4—21no—
441C3×C4—21no—
442C3×C4—21no—
443C3×C4—22no—
444C3×C4—41no—
445C3×C4—41no—
446C3×C4—62no—
447C3×C4—94no—
448C3×C4—182no—
449C3×C4C2×C4×D912yes—
450C3×C4C4×D3212yes—
451Dic3—22no—
452Dic3—34no—
453Dic3—62no—
454Dic3—92no—
455Dic3—92no—
456Dic3—182no—
457Dic3—181no—
458Dic3—181no—
459Dic3—181no—
460Dic3—181no—
461Dic3—361no—
462Dic3—361no—
463C32#554611yescharacteristic
464C9—21no—
465C9#557411yescharacteristic, complemented
466C23—32no—
467C23—271no—
468Q8—94no—
469D4—34no—
470D4—278no—
471C2×C4—32no—
472C2×C4—92no—
473C2×C4—92no—
474C2×C4—122no—
475C2×C4—278no—
476C2×C4—271no—
477C2×C4—274no—
478C2×C4—271no—
479C2×C4—271no—
480C2×C4—1082no—
481C2×C4—1082no—
482C2×C4C2×D3×D911yescharacteristic
483C2×C4C2×D3×D911yescharacteristic
484C2×C4C2×D3×D911yescharacteristic
485C2×C3—21no—
486C2×C3—21no—
487C2×C3—21no—
488C2×C3—64no—
489C2×C3C22×C4×D911yescharacteristic
490C2×C3#6371011yescharacteristic
491C2×C3#6371611yescharacteristic
492C2×C3#6429711yescharacteristic
493C2×C3C2×C4×D3211yescharacteristic
494C2×C3#6430711yescharacteristic
495D3—24no—
496D3—182no—
497D3—182no—
498D3—362no—
499C22—34no—
500C22—62no—
501C22—62no—
502C22—272no—
503C22—541no—
504C22—541no—
505C22C22×D3×D911yescharacteristic
506C4—21no—
507C4—21no—
508C4—62no—
509C4—94no—
510C4—182no—
511C4—272no—
512C4—541no—
513C4—541no—
514C4C4×D3×D912yes—
515C3—21no—
516C3#1058964811yescharacteristic, complemented
517C3#1059388911yescharacteristic
518C2—64no—
519C2—542no—
520C2#1172246411yescharacteristic
521C2C2×C4×D3×D911yescharacteristic
522C2#1172247611yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutatorC2×C3×C9C23×C4
FrattiniC22×C3C22×D32
FittingC3×C42×C9C22
socleC22×C32C23×D3
prime radicalC2×#88639C2
cosocleC22×C3×C9C24
hypercenterC42D3×D9
hypocenterC3×C9#520
LCSG ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
2. #11721709C2semidirect
3. C4×C9×D12C2semidirect
4. C3×C4×Dic18C2semidirect
5. #11721030C2semidirect
6. #11721025C2semidirect
7. #11721013C2semidirect
8. #11721010C2semidirect
9. D3×Dic9⋊C4C2semidirect
10. #11721000C2semidirect
11. #11720910C2semidirect
22. #88639C4semidirect
60. C4×Dic18D3semidirect
69. C4×C3⋊D9C2×C4semidirect
73. C9×D12C2×C4semidirect
75. C3×Dic18C2×C4semidirect
76. #6195C2×C4semidirect
78. D3×Dic9C2×C4semidirect
170. C4×D12D9semidirect
221. Dic18C4×D3semidirect
336. C3×C9#520semidirect
373. D12C4×D9semidirect
465. C9#5574semidirect
516. C3#10589648semidirect

Maximal subgroups

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

subgroupindex
C2×#886392
#117217092
C4×C9×D122
C3×C4×Dic182
#117210302
#117210252
#117210132
#117210102
D3×Dic9⋊C42
#117210002
#117209102
#105938893
#105896483