Finite Group Databasebeta

← #13150572

Subgroups — #13150572

Order 1728.

1170 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88668C212yesmaximal, complemented
2C2×#88536C211yesmaximal, characteristic, complemented
3C9×#930C211yesmaximal, characteristic, complemented
4C3×#63492C211yesmaximal, characteristic, complemented
5#11721290C212yesmaximal, complemented
6#11721050C212yesmaximal, complemented
7#11721049C212yesmaximal, complemented
8#11720958C212yesmaximal, complemented
9#11720942C212yesmaximal, complemented
10#10595084—31nomaximal
11#10589707—31nomaximal
12C23×C3⋊D9C2211yescharacteristic
13C22×D3×D9C2212yescomplemented
14C22×C3×C4×C9C2211yescharacteristic, complemented
15C22×C3⋊Dic9C2211yescharacteristic, complemented
16C2×C9×#67C2212yescomplemented
17C2×C3×#611C2212yescomplemented
18C2×C9×D12C2211yescharacteristic, complemented
19C2×C3×D36C2211yescharacteristic, complemented
20#88668—28no—
21C2×#6195C2212yes—
22C2×#6194C2212yescomplemented
23C2×#6192C2212yes—
24C2×(C3⋊D9)⋊C4C2212yescomplemented
25#88536C2212yescomplemented
26#88536C2212yescomplemented
27C9×(C2×D3)⋊C4C2212yescomplemented
28C3×(C2×D9)⋊C4C2212yescomplemented
29C9×C4⋊Dic3C2211yescharacteristic, complemented
30C3×C4⋊Dic9C2211yescharacteristic, complemented
31(C2×D3)⋊Dic9C2212yescomplemented
32(C2×D9)⋊Dic3C2212yescomplemented
33#88442C2212yescomplemented
34C2×#3811—32no—
35C2×#3752—31no—
36C3×#930—31no—
37C3×#930D311yescharacteristic
38#63980—32no—
39#63922—32no—
40#63922—32no—
41#63854—32no—
42#63854—32no—
43C2×D4×D9—32no—
44C9×#121—31no—
45#63501—32no—
46#63499—32no—
47#63492D311yescharacteristic, complemented
48C2×(C2×D9)⋊C4—31no—
49C2×(C2×D9)⋊C4—61no—
50#63456—32no—
51#63445—32no—
52#63444—32no—
53C23×C3×C9C2311yescharacteristic
54C22×C3⋊D9—41no—
55C22×C3⋊D9—41no—
56C22×C3⋊D9D414yescomplemented
57C22×C3⋊D9C2312yes—
58C22×D3×C9C2312yes—
59C22×C3×D9C2312yes—
60C2×D3×D9—24no—
61C2×D3×D9—42no—
62C2×D3×D9—42no—
63C2×C3×C4×C9—41no—
64C2×C3×C4×C9C2311yescharacteristic
65C2×C3×C4×C9C2311yescharacteristic
66C2×C3⋊Dic9—41no—
67C2×C3⋊Dic9C2312yes—
68C9×#67—24no—
69C3×#611—24no—
70C2×C9×Dic3C2312yes—
71C2×C3×Dic9C2312yes—
72C9×D12—41no—
73C3×D36—41no—
74#6195—24no—
75#6194—42no—
76#6192—24no—
77(C3⋊D9)⋊C4—24no—
78#5605—91no—
79#5605—91no—
80C23×C3⋊D3—31no—
81C22×D32—32no—
82C22×C32×C4C2×D311yescharacteristic
83C22×C3⋊Dic3—31no—
84C2×C3×#67—32no—
85C2×C3×#67C2×D312yes—
86C2×C3×D12—31no—
87C2×C3×D12C2×D311yescharacteristic
88#3811—68no—
89C2×#626—32no—
90C2×#626—32no—
91C2×#625—32no—
92C2×#624—32no—
93C23×D9—31no—
94C23×D9—32no—
95C23×D9—61no—
96#3752—32no—
97#3752—32no—
98C3×(C2×D3)⋊C4—32no—
99C3×(C2×D3)⋊C4C2×D312yes—
100C3×C4⋊Dic3—31no—
101C3×C4⋊Dic3C2×D311yescharacteristic
102#3724—32no—
103(C2×D3)⋊Dic3—32no—
104(C2×D3)⋊Dic3—32no—
105C2×D4×C9—32no—
106C2×D4×C9—31no—
107C22×C4×C9—21no—
108C22×C4×C9C2×D311yescharacteristic, complemented
109C2×#611—32no—
110C2×#611—32no—
111C2×#611C2×D312yescomplemented
112C22×Dic9—31no—
113C22×Dic9—61no—
114D4×D9—68no—
115C2×D36—32no—
116C2×D36C2×D311yescharacteristic, complemented
117C2×C4×D9—32no—
118C9×C4⋊C4—31no—
119C9×C22⋊C4—32no—
120#3676—32no—
121(C2×D9)⋊C4—32no—
122(C2×D9)⋊C4—32no—
123(C2×D9)⋊C4—32no—
124(C2×D9)⋊C4—62no—
125(C2×D9)⋊C4—62no—
126(C2×D9)⋊C4C2×D312yescomplemented
127C4⋊Dic9C2×D311yescharacteristic, complemented
128Dic9⋊C4—32no—
129C22×C3×C9—21no—
130C22×C3×C9—21no—
131C22×C3×C9C2×D412yes—
132C22×C3×C9C2411yescharacteristic
133C2×C3⋊D9—22no—
134C2×C3⋊D9—22no—
135C2×C3⋊D9—44no—
136C2×C3⋊D9C2×D414yes—
137C2×D3×C9—22no—
138C2×D3×C9—22no—
139C2×D3×C9—22no—
140C2×C3×D9—22no—
141C2×C3×D9—22no—
142C2×C3×D9—22no—
143D3×D9—48no—
144C3×C4×C9—21no—
145C3×C4×C9—21no—
146C3⋊Dic9—22no—
147C9×Dic3—22no—
148C3×Dic9—22no—
149C2×D3×D4—92no—
150C2×D3×D4—92no—
151C3×#121—31no—
152C3×#121—91no—
153#939—92no—
154#939—92no—
155#937—92no—
156#937—92no—
157#930—31no—
158#930D911yescharacteristic, complemented
159C2×(C2×D3)⋊C4—91no—
160C2×(C2×D3)⋊C4—91no—
161C2×(C2×D3)⋊C4—181no—
162#894—92no—
163#894—92no—
164#883—92no—
165#883—92no—
166#882—92no—
167#882—92no—
168C23×C32C22×D311yescharacteristic
169C22×C3⋊D3—32no—
170C22×C3⋊D3—34no—
171C22×C3⋊D3—121no—
172C22×C3⋊D3—121no—
173C22×C3×D3—32no—
174C22×C3×D3C22×D312yes—
175C2×D32—64no—
176C2×D32—122no—
177C2×D32—122no—
178C2×C32×C4—41no—
179C2×C32×C4C22×D311yescharacteristic
180C2×C32×C4C22×D311yescharacteristic
181C2×C3⋊Dic3—32no—
182C2×C3⋊Dic3—121no—
183C3×#67—24no—
184C3×#67—64no—
185C2×C3×Dic3—32no—
186C2×C3×Dic3C22×D312yes—
187C3×D12—41no—
188C3×D12—121no—
189#626—64no—
190#626—64no—
191#625—122no—
192#624—64no—
193C23×C9—21no—
194C23×C9—32no—
195C23×C9C22×D311yescharacteristic
196C22×D9—32no—
197C22×D9—34no—
198C22×D9—62no—
199C22×D9—64no—
200C22×D9—64no—
201C22×D9—122no—
202C22×D9—122no—
203C22×D9—121no—
204C22×D9—121no—
205C22×D9—122no—
206C22×D9—122no—
207C22×D9C22×D312yes—
208D4×C9—64no—
209D4×C9—121no—
210C2×C4×C9—21no—
211C2×C4×C9—21no—
212C2×C4×C9—32no—
213C2×C4×C9—41no—
214C2×C4×C9—42no—
215C2×C4×C9C22×D311yescharacteristic
216C2×C4×C9C22×D311yescharacteristic
217#611—24no—
218#611—64no—
219#611—122no—
220C2×Dic9—32no—
221C2×Dic9—62no—
222C2×Dic9—121no—
223C2×Dic9—122no—
224C2×Dic9C22×D312yes—
225D36—41no—
226D36—64no—
227C4×D9—64no—
228#534—271noSylow
229C2×C3×C9—22no—
230C2×C3×C9C22×D411yescharacteristic
231C2×C3×C9H(5,𝔽2)11yescharacteristic
232C2×C3×C9H(5,𝔽2)11yescharacteristic
233C3⋊D9—24no—
234D3×C9—42no—
235C3×D9—42no—
236C23×D3—92no—
237C23×D3—91no—
238C23×D3—91no—
239C23×D3—92no—
240C23×D3—181no—
241C2×C3×D4—32no—
242C2×C3×D4—31no—
243C2×C3×D4—92no—
244C2×C3×D4—91no—
245C22×C3×C4—21no—
246C22×C3×C4C2×D911yescharacteristic, complemented
247C22×C3×C4D3211yescharacteristic
248C2×#67—32no—
249C2×#67—92no—
250C2×#67—92no—
251C2×#67—92no—
252C2×#67—92no—
253C2×#67C2×D912yescomplemented
254C22×Dic3—91no—
255C22×Dic3—91no—
256C22×Dic3—181no—
257D3×D4—188no—
258D3×D4—188no—
259C2×D12—31no—
260C2×D12—92no—
261C2×D12—92no—
262C2×D12C2×D911yescharacteristic, complemented
263C2×C4×D3—92no—
264C2×C4×D3—92no—
265C3×C4⋊C4—31no—
266C3×C4⋊C4—91no—
267C3×C22⋊C4—32no—
268C3×C22⋊C4—92no—
269#217—92no—
270#217—92no—
271(C2×D3)⋊C4—32no—
272(C2×D3)⋊C4—92no—
273(C2×D3)⋊C4—92no—
274(C2×D3)⋊C4—92no—
275(C2×D3)⋊C4—92no—
276(C2×D3)⋊C4—92no—
277(C2×D3)⋊C4—92no—
278(C2×D3)⋊C4—182no—
279(C2×D3)⋊C4—182no—
280(C2×D3)⋊C4C2×D912yescomplemented
281C4⋊Dic3—31no—
282C4⋊Dic3C2×D911yescharacteristic, complemented
283Dic3⋊C4—92no—
284Dic3⋊C4—92no—
285C22×C32—21no—
286C22×C32—21no—
287C22×C32D3×D412yes—
288C22×C32C23×D311yescharacteristic
289C2×C3⋊D3—34no—
290C2×C3⋊D3—62no—
291C2×C3⋊D3—62no—
292C2×C3⋊D3—124no—
293C2×C3×D3—22no—
294C2×C3×D3—22no—
295C2×C3×D3—22no—
296C2×C3×D3—62no—
297C2×C3×D3—62no—
298C2×C3×D3—62no—
299D32—128no—
300C32×C4—21no—
301C32×C4—21no—
302C3⋊Dic3—62no—
303C3×Dic3—22no—
304C3×Dic3—62no—
305C22×C9—22no—
306C22×C9—22no—
307C22×C9—21no—
308C22×C9—21no—
309C22×C9—21no—
310C22×C9—22no—
311C22×C9—62no—
312C22×C9—62no—
313C22×C9—62no—
314C22×C9D3×D412yes—
315C22×C9C23×D311yescharacteristic
316C2×D9—22no—
317C2×D9—22no—
318C2×D9—22no—
319C2×D9—34no—
320C2×D9—64no—
321C2×D9—62no—
322C2×D9—62no—
323C2×D9—122no—
324C2×D9—122no—
325C2×D9—128no—
326C2×D9—124no—
327C2×D9—128no—
328C4×C9—21no—
329C4×C9—21no—
330C4×C9—41no—
331C4×C9—41no—
332C4×C9—62no—
333Dic9—22no—
334Dic9—62no—
335Dic9—122no—
336C22×D4—272no—
337#123—272no—
338#123—272no—
339#121—31no—
340#121—91no—
341#121—272no—
342#120—272no—
343#120—272no—
344C2×C22⋊C4—271no—
345C3×C9#53411yescharacteristic, complemented, Sylow
346C23×C3—21no—
347C23×C3—32no—
348C23×C3—92no—
349C23×C3C22×D911yescharacteristic
350C23×C3C2×D3211yescharacteristic
351C22×D3—32no—
352C22×D3—92no—
353C22×D3—92no—
354C22×D3—94no—
355C22×D3—94no—
356C22×D3—182no—
357C22×D3—184no—
358C22×D3—184no—
359C22×D3—184no—
360C22×D3—362no—
361C22×D3—362no—
362C22×D3—362no—
363C22×D3—362no—
364C22×D3—361no—
365C22×D3—361no—
366C22×D3—361no—
367C22×D3—361no—
368C22×D3—362no—
369C22×D3—362no—
370C22×D3C22×D912yes—
371C3×D4—64no—
372C3×D4—121no—
373C3×D4—184no—
374C3×D4—361no—
375C2×C3×C4—21no—
376C2×C3×C4—21no—
377C2×C3×C4—32no—
378C2×C3×C4—41no—
379C2×C3×C4—41no—
380C2×C3×C4—42no—
381C2×C3×C4—92no—
382C2×C3×C4C22×D911yescharacteristic
383C2×C3×C4C22×D911yescharacteristic
384C2×C3×C4C2×D3211yescharacteristic
385C2×C3×C4C2×D3211yescharacteristic
386#67—24no—
387#67—64no—
388#67—184no—
389#67—184no—
390#67—362no—
391#67—362no—
392C2×Dic3—32no—
393C2×Dic3—92no—
394C2×Dic3—92no—
395C2×Dic3—182no—
396C2×Dic3—361no—
397C2×Dic3—361no—
398C2×Dic3—362no—
399C2×Dic3C22×D912yes—
400D12—41no—
401D12—121no—
402D12—184no—
403D12—184no—
404C4×D3—184no—
405C4×D3—184no—
406C2×C32—22no—
407C2×C32C2×D3×D411yescharacteristic
408C2×C32#100411yescharacteristic
409C2×C32#100911yescharacteristic
410C3⋊D3—64no—
411C3×D3—42no—
412C3×D3—122no—
413C2×C9—24no—
414C2×C9—22no—
415C2×C9—21no—
416C2×C9—21no—
417C2×C9—21no—
418C2×C9—122no—
419C2×C9C2×D3×D411yescharacteristic
420C2×C9#100411yescharacteristic
421C2×C9#100911yescharacteristic
422D9—42no—
423D9—64no—
424D9—124no—
425C24—272no—
426C24—271no—
427C2×D4—32no—
428C2×D4—31no—
429C2×D4—92no—
430C2×D4—91no—
431C2×D4—272no—
432C2×D4—272no—
433C2×D4—272no—
434C2×D4—548no—
435C22×C4—272no—
436C22×C4—271no—
437C22×C4D3×D911yescharacteristic, complemented
438C4⋊C4—31no—
439C4⋊C4—91no—
440C4⋊C4—272no—
441C22⋊C4—32no—
442C22⋊C4—92no—
443C22⋊C4—272no—
444C22⋊C4—272no—
445C22⋊C4—272no—
446C22⋊C4—272no—
447C22×C3—22no—
448C22×C3—22no—
449C22×C3—21no—
450C22×C3—21no—
451C22×C3—21no—
452C22×C3—21no—
453C22×C3—21no—
454C22×C3—22no—
455C22×C3—62no—
456C22×C3—62no—
457C22×C3—62no—
458C22×C3—182no—
459C22×C3—182no—
460C22×C3—182no—
461C22×C3D4×D912yes—
462C22×C3C23×D911yescharacteristic
463C22×C3#380212yes—
464C22×C3C22×D3211yescharacteristic
465C2×D3—22no—
466C2×D3—22no—
467C2×D3—22no—
468C2×D3—62no—
469C2×D3—62no—
470C2×D3—62no—
471C2×D3—94no—
472C2×D3—94no—
473C2×D3—184no—
474C2×D3—182no—
475C2×D3—182no—
476C2×D3—182no—
477C2×D3—182no—
478C2×D3—362no—
479C2×D3—362no—
480C2×D3—368no—
481C2×D3—368no—
482C2×D3—364no—
483C2×D3—364no—
484C2×D3—368no—
485C3×C4—21no—
486C3×C4—21no—
487C3×C4—21no—
488C3×C4—21no—
489C3×C4—41no—
490C3×C4—41no—
491C3×C4—62no—
492C3×C4—182no—
493Dic3—22no—
494Dic3—62no—
495Dic3—182no—
496Dic3—182no—
497Dic3—362no—
498C32#560511yescharacteristic
499C9—21no—
500C9#560511yescharacteristic, complemented
501C23—32no—
502C23—92no—
503C23—272no—
504C23—274no—
505C23—544no—
506C23—1081no—
507C23—1081no—
508C23—1082no—
509C23—1082no—
510C23C2×D3×D911yescharacteristic
511D4—64no—
512D4—121no—
513D4—184no—
514D4—361no—
515D4—544no—
516D4—544no—
517D4—1082no—
518C2×C4—32no—
519C2×C4—41no—
520C2×C4—92no—
521C2×C4—272no—
522C2×C4—544no—
523C2×C4—1081no—
524C2×C4C2×D3×D911yescharacteristic
525C2×C4C2×D3×D911yescharacteristic
526C2×C3—24no—
527C2×C3—22no—
528C2×C3—22no—
529C2×C3—21no—
530C2×C3—21no—
531C2×C3—21no—
532C2×C3—122no—
533C2×C3—362no—
534C2×C3C2×D4×D911yescharacteristic
535C2×C3#6371011yescharacteristic
536C2×C3#6371511yescharacteristic
537C2×C3C2×#380211yescharacteristic
538C2×C3#6430811yescharacteristic
539C2×C3#6433011yescharacteristic
540D3—42no—
541D3—122no—
542D3—184no—
543D3—184no—
544D3—364no—
545C22—21no—
546C22—21no—
547C22—62no—
548C22—62no—
549C22—62no—
550C22—182no—
551C22—182no—
552C22—182no—
553C22—274no—
554C22—542no—
555C22—542no—
556C22—1084no—
557C22—1088no—
558C22#8864612yes—
559C22C22×D3×D911yescharacteristic
560C4—21no—
561C4—21no—
562C4—62no—
563C4—182no—
564C4—542no—
565C3—21no—
566C3#1058970711yescharacteristic, complemented
567C3#1059508411yescharacteristic
568C2—22no—
569C2—122no—
570C2—362no—
571C2—544no—
572C2C2×#8864611yescharacteristic
573C2#1172247811yescharacteristic
574C2#1172253511yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC22×C3×C4×C9C22
socleC22×C32C23×D3
hypercenterC22×C4D3×D9
hypocenterC3×C9#534
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C2×#88668C2semidirect
2. C2×#88536C2semidirect
3. C9×#930C2semidirect
4. C3×#63492C2semidirect
5. #11721290C2semidirect
6. #11721050C2semidirect
7. #11721049C2semidirect
8. #11720958C2semidirect
9. #11720942C2semidirect
13. C22×D3×D9C22semidirect
14. C22×C3×C4×C9C22semidirect
15. C22×C3⋊Dic9C22semidirect
16. C2×C9×#67C22semidirect
17. C2×C3×#611C22semidirect
18. C2×C9×D12C22semidirect
19. C2×C3×D36C22semidirect
22. C2×#6194C22semidirect
24. C2×(C3⋊D9)⋊C4C22semidirect
25. #88536C22semidirect
26. #88536C22semidirect
27. C9×(C2×D3)⋊C4C22semidirect
28. C3×(C2×D9)⋊C4C22semidirect
29. C9×C4⋊Dic3C22semidirect
30. C3×C4⋊Dic9C22semidirect
31. (C2×D3)⋊Dic9C22semidirect
32. (C2×D9)⋊Dic3C22semidirect
33. #88442C22semidirect
47. #63492D3semidirect
56. C22×C3⋊D9D4semidirect
108. C22×C4×C9C2×D3semidirect
111. C2×#611C2×D3semidirect
116. C2×D36C2×D3semidirect
126. (C2×D9)⋊C4C2×D3semidirect
127. C4⋊Dic9C2×D3semidirect
158. #930D9semidirect
246. C22×C3×C4C2×D9semidirect
253. C2×#67C2×D9semidirect
262. C2×D12C2×D9semidirect
280. (C2×D3)⋊C4C2×D9semidirect
282. C4⋊Dic3C2×D9semidirect
345. C3×C9#534semidirect
437. C22×C4D3×D9semidirect
500. C9#5605semidirect
566. C3#10589707semidirect

Maximal subgroups

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

subgroupindex
C2×#886682
C2×#885362
C9×#9302
C3×#634922
#117212902
#117210502
#117210492
#117209582
#117209422
#105950843
#105897073