Finite Group Databasebeta

← #13147590

Subgroups — #13147590

Order 1728.

952 conjugacy classes of subgroups.

The subgroup lattice has 506 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88638C211yesmaximal, characteristic
2C4×#6226C211yesmaximal, characteristic, complemented
3C4×C9×D12C211yesmaximal, characteristic, complemented
4C3×#63434C211yesmaximal, characteristic, complemented
5#11721028C212yesmaximal, complemented
6D12×Dic9C211yesmaximal, characteristic, complemented
7#11721011C212yesmaximal, complemented
8D3×Dic9⋊C4C212yesmaximal, complemented
9D9×C4⋊Dic3C211yesmaximal, characteristic, complemented
10#11720976C212yesmaximal, complemented
11#11720913C211yesmaximal, characteristic, complemented
12#10593887—31nomaximal
13#10589648—31nomaximal
14C2×#6226C2211yescharacteristic
15C2×C9×D12C2211yescharacteristic
16C2×C4×D3×C9C2212yes—
17C2×C3×C4×D9C2211yescharacteristic
18C2×#6194C2212yes—
19C2×D3×Dic9C2212yes—
20C2×D3×Dic9C2212yes—
21C2×Dic3×D9C2212yes—
22#88638C418yescomplemented
23C3×C42×C9C2211yescharacteristic
24C4⋊(C3⋊Dic9)C2211yescharacteristic
25#88534C2212yes—
26C4×C3⋊Dic9C2212yes—
27C9×(C2×D3)⋊C4C2212yes—
28C3×(C2×D9)⋊C4C2211yescharacteristic
29C3×(C2×D9)⋊C4C2211yescharacteristic
30C9×C4⋊Dic3C2211yescharacteristic
31C3×Dic9⋊C4C2211yescharacteristic
32C3×Dic9⋊C4C2211yescharacteristic
33C3×C4×Dic9C2211yescharacteristic
34(C2×D3)⋊Dic9C2212yes—
35#88442C2212yes—
36Dic9⋊Dic3C2212yes—
37Dic3×Dic9C2212yes—
38C2×#3795—31no—
39C4×#634—31no—
40C3×C4×D12D311yescharacteristic
41C3×#872—31no—
42#63902—32no—
43Dic3×D12—31no—
44D3×C4⋊Dic3—31no—
45#63879—32no—
46#63878—32no—
47D3×Dic3⋊C4—32no—
48#63844—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#63434D311yescharacteristic, complemented
59C4×Dic18—31no—
60C4×Dic18—61no—
61C22×D3×C9C2312yes—
62C22×C3×D9C2311yescharacteristic
63C2×C3×C4×C9C2311yescharacteristic
64C2×C3×C4×C9C2311yescharacteristic
65C2×C3×C4×C9C2311yescharacteristic
66C2×C3⋊Dic9C2312yes—
67C2×C3⋊Dic9C2312yes—
68#6226C2×C414yescomplemented
69C2×C9×Dic3C2312yes—
70C2×C3×Dic9C2311yescharacteristic
71C2×C3×Dic9C2311yescharacteristic
72C2×C3×Dic9C2311yescharacteristic
73C9×D12C2×C414yescomplemented
74C4×D3×C9—42no—
75C3×C4×D9C2×C414yescomplemented
76#6194C2×C418yescomplemented
77D3×Dic9—42no—
78D3×Dic9C2×C418yescomplemented
79Dic3×D9—42no—
80#5546—91no—
81#5523—91no—
82C2×#634—31no—
83C2×C3×D12C2×D311yescharacteristic
84C2×C3×C4×D3—31no—
85C2×C3×C4×D3C2×D312yes—
86C2×#625—32no—
87C2×D3×Dic3—32no—
88C2×D3×Dic3—32no—
89C2×D3×Dic3—32no—
90#3795—38no—
91C32×C42C2×D311yescharacteristic
92C4⋊(C3⋊Dic3)—31no—
93#3750—32no—
94C4×C3⋊Dic3—32no—
95C3×(C2×D3)⋊C4—31no—
96C3×(C2×D3)⋊C4—31no—
97C3×(C2×D3)⋊C4C2×D312yes—
98C3×C4⋊Dic3C2×D311yescharacteristic
99C3×Dic3⋊C4—31no—
100C3×Dic3⋊C4—31no—
101C3×C4×Dic3—31no—
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—32no—
113C2×C4×D9C2×D311yescharacteristic
114C2×Dic18—31no—
115C2×Dic18—61no—
116C9×C4⋊C4—31no—
117C9×C22⋊C4—32no—
118C42×C9—21no—
119C42×C9C2×D311yescharacteristic
120#3676—32no—
121(C2×D9)⋊C4C2×D311yescharacteristic
122(C2×D9)⋊C4C2×D311yescharacteristic
123C4⋊Dic9—31no—
124C4⋊Dic9—61no—
125Dic9⋊C4—32no—
126Dic9⋊C4—32no—
127Dic9⋊C4—32no—
128Dic9⋊C4—62no—
129Dic9⋊C4C2×D311yescharacteristic
130Dic9⋊C4C2×D311yescharacteristic
131C4×Dic9—32no—
132C4×Dic9—32no—
133C4×Dic9—62no—
134C4×Dic9C2×D311yescharacteristic
135C22×C3×C9C2411yescharacteristic
136C2×D3×C9—22no—
137C2×D3×C9—22no—
138C2×D3×C9C22×C414yes—
139C2×C3×D9—21no—
140C2×C3×D9—21no—
141C2×C3×D9C22×C412yes—
142C3×C4×C9—21no—
143C3×C4×C9—21no—
144C3×C4×C9C22×C412yes—
145C3⋊Dic9—22no—
146C3⋊Dic9C22×C414yes—
147C9×Dic3—22no—
148C3×Dic9—21no—
149C3×Dic9—21no—
150C3×Dic9C22×C412yes—
151C2×#237—91no—
152C2×#235—91no—
153C3×C4×D4—31no—
154C3×#117—91no—
155D4×Dic3—91no—
156C4×#67—92no—
157C2×C4⋊Dic3—91no—
158C2×Dic3⋊C4—92no—
159#892—92no—
160D3×C4⋊C4—91no—
161D3×C4⋊C4—92no—
162#887—91no—
163#67⋊C4—92no—
164#67⋊C4—92no—
165#67⋊C4—92no—
166#877—92no—
167C4×D12—91no—
168C4×D12D911yescharacteristic, complemented
169#872—31no—
170C4×Dic6—91no—
171C4×Dic6—91no—
172C4×Dic6—91no—
173C4×Dic6—181no—
174C22×C3×D3—31no—
175C22×C3×D3C22×D312yes—
176C2×C32×C4C22×D311yescharacteristic
177C2×C32×C4C22×D311yescharacteristic
178C2×C32×C4C22×D311yescharacteristic
179C2×C3⋊Dic3—32no—
180C2×C3⋊Dic3—32no—
181#634—34no—
182C2×C3×Dic3—31no—
183C2×C3×Dic3—31no—
184C2×C3×Dic3—31no—
185C2×C3×Dic3C22×D312yes—
186C3×D12C4×D314yes—
187C3×C4×D3—34no—
188C3×C4×D3—42no—
189#625—38no—
190D3×Dic3—38no—
191D3×Dic3—122no—
192D3×Dic3—122no—
193C23×C9—32no—
194C22×D9C22×D311yescharacteristic
195D4×C9—34no—
196C2×C4×C9—21no—
197C2×C4×C9—21no—
198C2×C4×C9—21no—
199C2×C4×C9—32no—
200C2×C4×C9—122no—
201C2×C4×C9C22×D311yescharacteristic
202C2×C4×C9C22×D311yescharacteristic
203C2×C4×C9C22×D311yescharacteristic
204#611—38no—
205C2×Dic9—32no—
206C2×Dic9—38no—
207C2×Dic9—32no—
208C2×Dic9—62no—
209C2×Dic9—62no—
210C2×Dic9—122no—
211C2×Dic9C22×D311yescharacteristic
212C2×Dic9C22×D311yescharacteristic
213C2×Dic9C22×D311yescharacteristic
214C4×D9—122no—
215C4×D9C4×D314yescomplemented
216Dic18—34no—
217Dic18—64no—
218#520—271noSylow
219C2×C3×C9C23×C411yescharacteristic
220C2×C3×C9H(5,𝔽2)11yescharacteristic
221C2×C3×C92−1+411yescharacteristic
222D3×C9—24no—
223C3×D9—22no—
224C2×C3×D4—31no—
225C22×C3×C4—32no—
226C22×C3×C4—91no—
227C2×#67—92no—
228C2×#67—92no—
229C22×Dic3—92no—
230C22×Dic3—92no—
231C22×Dic3—92no—
232#237—98no—
233#235—98no—
234C2×D12C2×D911yescharacteristic
235C2×C4×D3—31no—
236C2×C4×D3—92no—
237C2×C4×D3—92no—
238C2×C4×D3—92no—
239C2×C4×D3C2×D912yes—
240C2×Dic6—91no—
241C2×Dic6—91no—
242C2×Dic6—181no—
243C3×C4⋊C4—31no—
244C3×C4⋊C4—91no—
245C3×C4⋊C4—91no—
246C3×C22⋊C4—32no—
247C3×C22⋊C4—91no—
248C3×C22⋊C4—91no—
249C3×C42—21no—
250C3×C42—91no—
251C3×C42C2×D911yescharacteristic
252C3×C42D3211yescharacteristic
253#217—92no—
254(C2×D3)⋊C4—31no—
255(C2×D3)⋊C4—31no—
256(C2×D3)⋊C4—92no—
257(C2×D3)⋊C4C2×D912yes—
258C4⋊Dic3—91no—
259C4⋊Dic3—91no—
260C4⋊Dic3—92no—
261C4⋊Dic3—181no—
262C4⋊Dic3C2×D911yescharacteristic
263Dic3⋊C4—31no—
264Dic3⋊C4—31no—
265Dic3⋊C4—92no—
266Dic3⋊C4—92no—
267Dic3⋊C4—92no—
268Dic3⋊C4—92no—
269Dic3⋊C4—92no—
270Dic3⋊C4—182no—
271C4×Dic3—31no—
272C4×Dic3—92no—
273C4×Dic3—92no—
274C4×Dic3—92no—
275C4×Dic3—92no—
276C4×Dic3—182no—
277C22×C32C23×D311yescharacteristic
278C2×C3×D3—22no—
279C2×C3×D3—22no—
280C2×C3×D3—32no—
281C2×C3×D3—61no—
282C2×C3×D3—61no—
283C2×C3×D3C2×C4×D314yes—
284C32×C4—21no—
285C32×C4—21no—
286C32×C4C2×C4×D312yes—
287C3⋊Dic3—34no—
288C3⋊Dic3—62no—
289C3×Dic3—22no—
290C3×Dic3—32no—
291C3×Dic3—61no—
292C3×Dic3—61no—
293C22×C9—21no—
294C22×C9—34no—
295C22×C9—62no—
296C22×C9—62no—
297C22×C9C23×D311yescharacteristic
298C2×D9—21no—
299C2×D9—21no—
300C2×D9C2×C4×D312yes—
301C4×C9—22no—
302C4×C9—21no—
303C4×C9—21no—
304C4×C9—22no—
305C4×C9—41no—
306C4×C9—62no—
307C4×C9C2×C4×D312yes—
308Dic9—21no—
309Dic9—21no—
310Dic9—34no—
311Dic9—64no—
312Dic9—62no—
313Dic9—122no—
314Dic9C2×C4×D312yes—
315C2×#41—271no—
316C4×Q8—271no—
317C4×Q8—271no—
318C4×D4—31no—
319C4×D4—271no—
320C4×D4—272no—
321C4×D4—272no—
322#117—91no—
323#117—272no—
324C2×C4⋊C4—271no—
325C2×C4⋊C4—272no—
326C3×C9#52011yescharacteristic, complemented, Sylow
327C23×C3—32no—
328C23×C3—91no—
329C22×D3—31no—
330C22×D3C22×D912yes—
331C3×D4—34no—
332C2×C3×C4—21no—
333C2×C3×C4—21no—
334C2×C3×C4—21no—
335C2×C3×C4—32no—
336C2×C3×C4—91no—
337C2×C3×C4—94no—
338C2×C3×C4—91no—
339C2×C3×C4—91no—
340C2×C3×C4—122no—
341C2×C3×C4C22×D911yescharacteristic
342C2×C3×C4C22×D911yescharacteristic
343C2×C3×C4C22×D911yescharacteristic
344C2×C3×C4C2×D3211yescharacteristic
345C2×C3×C4C2×D3211yescharacteristic
346C2×C3×C4C2×D3211yescharacteristic
347#67—98no—
348#67—98no—
349C2×Dic3—31no—
350C2×Dic3—31no—
351C2×Dic3—31no—
352C2×Dic3—98no—
353C2×Dic3—92no—
354C2×Dic3—92no—
355C2×Dic3—92no—
356C2×Dic3—92no—
357C2×Dic3—182no—
358C2×Dic3—182no—
359C2×Dic3—362no—
360C2×Dic3—362no—
361C2×Dic3C22×D912yes—
362D12C4×D914yescomplemented
363C4×D3—34no—
364C4×D3—42no—
365C4×D3—98no—
366C4×D3—362no—
367C4×D3—362no—
368Dic6—94no—
369Dic6—94no—
370Dic6—184no—
371C2×C32C22×C4×D311yescharacteristic
372C2×C32#100411yescharacteristic
373C2×C32#101011yescharacteristic
374C3×D3—24no—
375C3×D3—62no—
376C2×C9—21no—
377C2×C9—21no—
378C2×C9—21no—
379C2×C9—64no—
380C2×C9C22×C4×D311yescharacteristic
381C2×C9#100911yescharacteristic
382C2×C9#101011yescharacteristic
383D9—22no—
384#41—278no—
385C2×Q8—271no—
386C2×D4—31no—
387C2×D4—272no—
388C22×C4—32no—
389C22×C4—91no—
390C22×C4—272no—
391C22×C4—272no—
392C22×C4—272no—
393C4⋊C4—31no—
394C4⋊C4—91no—
395C4⋊C4—91no—
396C4⋊C4—272no—
397C4⋊C4—272no—
398C4⋊C4—272no—
399C4⋊C4—271no—
400C22⋊C4—32no—
401C22⋊C4—91no—
402C22⋊C4—91no—
403C22⋊C4—272no—
404C42—91no—
405C42—272no—
406C42—272no—
407C42D3×D911yescharacteristic
408C22×C3—21no—
409C22×C3—34no—
410C22×C3—62no—
411C22×C3—62no—
412C22×C3—92no—
413C22×C3—181no—
414C22×C3—181no—
415C22×C3C23×D911yescharacteristic
416C22×C3C22×D3211yescharacteristic
417C2×D3—22no—
418C2×D3—22no—
419C2×D3—32no—
420C2×D3—61no—
421C2×D3—61no—
422C2×D3C2×C4×D914yes—
423C3×C4—22no—
424C3×C4—21no—
425C3×C4—21no—
426C3×C4—21no—
427C3×C4—21no—
428C3×C4—22no—
429C3×C4—41no—
430C3×C4—62no—
431C3×C4—92no—
432C3×C4—181no—
433C3×C4—181no—
434C3×C4C2×C4×D912yes—
435C3×C4C4×D3212yes—
436Dic3—22no—
437Dic3—32no—
438Dic3—61no—
439Dic3—61no—
440Dic3—94no—
441Dic3—94no—
442Dic3—184no—
443Dic3—182no—
444Dic3—182no—
445Dic3—362no—
446C32#554611yescharacteristic
447C9—21no—
448C9#552311yescharacteristic, complemented
449C23—32no—
450C23—91no—
451Q8—274no—
452D4—34no—
453D4—278no—
454C2×C4—32no—
455C2×C4—91no—
456C2×C4—94no—
457C2×C4—91no—
458C2×C4—91no—
459C2×C4—122no—
460C2×C4—278no—
461C2×C4—272no—
462C2×C4—272no—
463C2×C4—1082no—
464C2×C4—1082no—
465C2×C4C2×D3×D911yescharacteristic
466C2×C4C2×D3×D911yescharacteristic
467C2×C4C2×D3×D911yescharacteristic
468C2×C3—21no—
469C2×C3—21no—
470C2×C3—21no—
471C2×C3—64no—
472C2×C3—182no—
473C2×C3C22×C4×D911yescharacteristic
474C2×C3#6371011yescharacteristic
475C2×C3#6371611yescharacteristic
476C2×C3#6429811yescharacteristic
477C2×C3C2×C4×D3211yescharacteristic
478C2×C3#6430711yescharacteristic
479D3—24no—
480D3—62no—
481C22—34no—
482C22—62no—
483C22—62no—
484C22—92no—
485C22—181no—
486C22—181no—
487C22C22×D3×D911yescharacteristic
488C4—21no—
489C4—21no—
490C4—62no—
491C4—92no—
492C4—181no—
493C4—181no—
494C4—274no—
495C4—542no—
496C4C4×D3×D912yes—
497C3—21no—
498C3#1058964811yescharacteristic, complemented
499C3#1059388711yescharacteristic
500C2—64no—
501C2—182no—
502C2#1172246511yescharacteristic
503C2C2×C4×D3×D911yescharacteristic
504C2#1172247611yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutatorC2×C3×C9C23×C4
FrattiniC22×C3C22×D32
FittingC3×C42×C9C22
socleC22×C32C23×D3
prime radicalC2×#88638C2
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. C4×#6226C2semidirect
3. C4×C9×D12C2semidirect
4. C3×#63434C2semidirect
5. #11721028C2semidirect
6. D12×Dic9C2semidirect
7. #11721011C2semidirect
8. D3×Dic9⋊C4C2semidirect
9. D9×C4⋊Dic3C2semidirect
10. #11720976C2semidirect
11. #11720913C2semidirect
22. #88638C4semidirect
58. #63434D3semidirect
68. #6226C2×C4semidirect
73. C9×D12C2×C4semidirect
75. C3×C4×D9C2×C4semidirect
76. #6194C2×C4semidirect
78. D3×Dic9C2×C4semidirect
168. C4×D12D9semidirect
215. C4×D9C4×D3semidirect
326. C3×C9#520semidirect
362. D12C4×D9semidirect
448. C9#5523semidirect
498. C3#10589648semidirect

Maximal subgroups

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

subgroupindex
C2×#886382
C4×#62262
C4×C9×D122
C3×#634342
#117210282
D12×Dic92
#117210112
D3×Dic9⋊C42
D9×C4⋊Dic32
#117209762
#117209132
#105938873
#105896483