Finite Group Databasebeta

← #13177341

Subgroups — #13177341

Order 1728.

2604 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×#89039C212yesmaximal, complemented
2C2×#88859C211yesmaximal, characteristic, complemented
3C3×#64139C211yesmaximal, characteristic, complemented
4C32×#930C211yesmaximal, characteristic, complemented
5#11723637C212yesmaximal, complemented
6#11723541C212yesmaximal, complemented
7#11723540C212yesmaximal, complemented
8#11723504C212yesmaximal, complemented
9#11723422C212yesmaximal, complemented
10#10596076—31nomaximal
11#10595084—34nomaximal
12C23×C3⋊(C3⋊D3)C2211yescharacteristic
13C22×D3×C3⋊D3C2212yescomplemented
14C22×C33×C4C2211yescharacteristic, complemented
15C22×C3⋊(C3⋊Dic3)C2211yescharacteristic, complemented
16C2×C3×#638C2212yescomplemented
17C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic, complemented
18C2×C32×#67C2212yescomplemented
19C2×C32×D12C2211yescharacteristic, complemented
20#89039—28no—
21C2×#6292C2212yes—
22C2×#6291C2212yes—
23C2×#6290C2212yescomplemented
24C2×#6289C2212yescomplemented
25#88859C2212yescomplemented
26#88859C2212yescomplemented
27C3×#3752C2212yescomplemented
28C3×C4⋊(C3⋊Dic3)C2211yescharacteristic, complemented
29C32×(C2×D3)⋊C4C2212yescomplemented
30C32×C4⋊Dic3C2211yescharacteristic, complemented
31#88806C2212yescomplemented
32#88803C2212yescomplemented
33(C2×D3)⋊(C3⋊Dic3)C2212yescomplemented
34C2×D4×C3⋊D3—32no—
35C2×#3811—38no—
36C32×#121—31no—
37#64148—32no—
38#64146—32no—
39#64139D311yescharacteristic, complemented
40C2×#3752—31no—
41C2×#3752—34no—
42C2×#3752—64no—
43#64103—32no—
44#64092—32no—
45#64091—32no—
46C3×#930—34no—
47C3×#930D314yescomplemented
48#63980—38no—
49#63922—38no—
50#63922—38no—
51#63854—38no—
52#63854—38no—
53C23×C33C2311yescharacteristic
54C22×C3⋊(C3⋊D3)—41no—
55C22×C3⋊(C3⋊D3)—41no—
56C22×C3⋊(C3⋊D3)D414yescomplemented
57C22×C3⋊(C3⋊D3)C2312yes—
58C22×C3×C3⋊D3C2312yes—
59C22×C32×D3C2312yes—
60C2×D3×C3⋊D3—24no—
61C2×D3×C3⋊D3—42no—
62C2×D3×C3⋊D3—42no—
63C2×C33×C4—41no—
64C2×C33×C4C2311yescharacteristic
65C2×C33×C4C2311yescharacteristic
66C2×C3⋊(C3⋊Dic3)—41no—
67C2×C3⋊(C3⋊Dic3)C2312yes—
68C3×#638—24no—
69C2×C3×C3⋊Dic3C2312yes—
70C3×C4⋊(C3⋊D3)—41no—
71C32×#67—24no—
72C2×C32×Dic3C2312yes—
73C32×D12—41no—
74#6292—24no—
75#6291—24no—
76#6290—42no—
77#6289—24no—
78#5605—94no—
79#5605—91no—
80C23×C3⋊D3—31no—
81C23×C3⋊D3—34no—
82C23×C3⋊D3—32no—
83C23×C3⋊D3—64no—
84C22×D32—38no—
85C2×C32×D4—32no—
86C2×C32×D4—31no—
87C22×C32×C4—24no—
88C22×C32×C4C2×D314yescomplemented
89C22×C32×C4C2×D311yescharacteristic, complemented
90C2×#638—32no—
91C2×#638—32no—
92C2×#638C2×D312yescomplemented
93C22×C3⋊Dic3—31no—
94C22×C3⋊Dic3—34no—
95C22×C3⋊Dic3—64no—
96D4×C3⋊D3—68no—
97C2×C4⋊(C3⋊D3)—32no—
98C2×C4⋊(C3⋊D3)C2×D311yescharacteristic, complemented
99C2×C4×C3⋊D3—32no—
100C2×C3×#67—38no—
101C2×C3×#67C2×D318yescomplemented
102C2×C3×D12—34no—
103C2×C3×D12C2×D314yescomplemented
104#3811—632no—
105C2×#626—38no—
106C2×#626—38no—
107C2×#625—38no—
108C2×#624—38no—
109C32×C4⋊C4—31no—
110C32×C22⋊C4—32no—
111#3757—32no—
112#3752—32no—
113#3752—32no—
114#3752—32no—
115#3752—38no—
116#3752—38no—
117#3752—68no—
118#3752—68no—
119#3752C2×D312yescomplemented
120C4⋊(C3⋊Dic3)C2×D311yescharacteristic, complemented
121#3750—32no—
122C3×(C2×D3)⋊C4—38no—
123C3×(C2×D3)⋊C4C2×D318yescomplemented
124C3×C4⋊Dic3—34no—
125C3×C4⋊Dic3C2×D314yescomplemented
126#3724—38no—
127(C2×D3)⋊Dic3—38no—
128(C2×D3)⋊Dic3—38no—
129C22×C33—21no—
130C22×C33—21no—
131C22×C33C2×D412yes—
132C22×C33C2411yescharacteristic
133C2×C3⋊(C3⋊D3)—22no—
134C2×C3⋊(C3⋊D3)—22no—
135C2×C3⋊(C3⋊D3)—44no—
136C2×C3⋊(C3⋊D3)C2×D414yes—
137C2×C3×C3⋊D3—22no—
138C2×C3×C3⋊D3—22no—
139C2×C3×C3⋊D3—22no—
140C2×C32×D3—22no—
141C2×C32×D3—22no—
142C2×C32×D3—22no—
143D3×C3⋊D3—48no—
144C33×C4—21no—
145C33×C4—21no—
146C3⋊(C3⋊Dic3)—22no—
147C3×C3⋊Dic3—22no—
148C32×Dic3—22no—
149C2×D3×D4—98no—
150C2×D3×D4—92no—
151C3×#121—34no—
152C3×#121—91no—
153#939—98no—
154#939—92no—
155#937—98no—
156#937—92no—
157#930—34no—
158#930C3⋊D311yescharacteristic, complemented
159C2×(C2×D3)⋊C4—94no—
160C2×(C2×D3)⋊C4—91no—
161C2×(C2×D3)⋊C4—184no—
162#894—98no—
163#894—92no—
164#883—98no—
165#883—92no—
166#882—98no—
167#882—92no—
168C23×C32—24no—
169C23×C32—32no—
170C23×C32C22×D314yes—
171C23×C32C22×D311yescharacteristic
172C22×C3⋊D3—32no—
173C22×C3⋊D3—34no—
174C22×C3⋊D3—38no—
175C22×C3⋊D3—316no—
176C22×C3⋊D3—68no—
177C22×C3⋊D3—616no—
178C22×C3⋊D3—64no—
179C22×C3⋊D3—122no—
180C22×C3⋊D3—122no—
181C22×C3⋊D3—121no—
182C22×C3⋊D3—121no—
183C22×C3⋊D3—124no—
184C22×C3⋊D3—124no—
185C22×C3⋊D3—128no—
186C22×C3⋊D3—128no—
187C22×C3⋊D3C22×D312yes—
188C22×C3×D3—38no—
189C22×C3×D3C22×D318yes—
190C2×D32—616no—
191C2×D32—128no—
192C2×D32—128no—
193C32×D4—64no—
194C32×D4—121no—
195C2×C32×C4—24no—
196C2×C32×C4—24no—
197C2×C32×C4—32no—
198C2×C32×C4—41no—
199C2×C32×C4—44no—
200C2×C32×C4—48no—
201C2×C32×C4C22×D314yes—
202C2×C32×C4C22×D314yes—
203C2×C32×C4C22×D311yescharacteristic
204C2×C32×C4C22×D311yescharacteristic
205#638—24no—
206#638—64no—
207#638—122no—
208C2×C3⋊Dic3—32no—
209C2×C3⋊Dic3—38no—
210C2×C3⋊Dic3—68no—
211C2×C3⋊Dic3—121no—
212C2×C3⋊Dic3—124no—
213C2×C3⋊Dic3—128no—
214C2×C3⋊Dic3C22×D312yes—
215C4⋊(C3⋊D3)—41no—
216C4⋊(C3⋊D3)—64no—
217C4×C3⋊D3—64no—
218C3×#67—216no—
219C3×#67—616no—
220C2×C3×Dic3—38no—
221C2×C3×Dic3C22×D318yes—
222C3×D12—44no—
223C3×D12—124no—
224#626—616no—
225#626—616no—
226#625—128no—
227#624—616no—
228#534—271noSylow
229C2×C33—22no—
230C2×C33C22×D411yescharacteristic
231C2×C33H(5,𝔽2)11yescharacteristic
232C2×C33H(5,𝔽2)11yescharacteristic
233C3⋊(C3⋊D3)—24no—
234C3×C3⋊D3—42no—
235C32×D3—42no—
236C23×D3—98no—
237C23×D3—94no—
238C23×D3—91no—
239C23×D3—92no—
240C23×D3—184no—
241C2×C3×D4—38no—
242C2×C3×D4—34no—
243C2×C3×D4—92no—
244C2×C3×D4—91no—
245C22×C3×C4—24no—
246C22×C3×C4D3214yescomplemented
247C22×C3×C4C2×C3⋊D311yescharacteristic, complemented
248C2×#67—38no—
249C2×#67—98no—
250C2×#67—98no—
251C2×#67—92no—
252C2×#67—92no—
253C2×#67C2×C3⋊D312yescomplemented
254C22×Dic3—94no—
255C22×Dic3—91no—
256C22×Dic3—184no—
257D3×D4—188no—
258D3×D4—1832no—
259C2×D12—34no—
260C2×D12—98no—
261C2×D12—92no—
262C2×D12C2×C3⋊D311yescharacteristic, complemented
263C2×C4×D3—98no—
264C2×C4×D3—92no—
265C3×C4⋊C4—34no—
266C3×C4⋊C4—91no—
267C3×C22⋊C4—38no—
268C3×C22⋊C4—92no—
269#217—98no—
270#217—92no—
271(C2×D3)⋊C4—38no—
272(C2×D3)⋊C4—98no—
273(C2×D3)⋊C4—92no—
274(C2×D3)⋊C4—98no—
275(C2×D3)⋊C4—98no—
276(C2×D3)⋊C4—92no—
277(C2×D3)⋊C4—92no—
278(C2×D3)⋊C4—188no—
279(C2×D3)⋊C4—188no—
280(C2×D3)⋊C4C2×C3⋊D312yescomplemented
281C4⋊Dic3—34no—
282C4⋊Dic3C2×C3⋊D311yescharacteristic, complemented
283Dic3⋊C4—98no—
284Dic3⋊C4—92no—
285C22×C32—28no—
286C22×C32—28no—
287C22×C32—24no—
288C22×C32—21no—
289C22×C32—24no—
290C22×C32—21no—
291C22×C32—24no—
292C22×C32—28no—
293C22×C32—62no—
294C22×C32—62no—
295C22×C32—62no—
296C22×C32D3×D418yes—
297C22×C32D3×D412yes—
298C22×C32C23×D314yes—
299C22×C32C23×D311yescharacteristic
300C2×C3⋊D3—22no—
301C2×C3⋊D3—22no—
302C2×C3⋊D3—22no—
303C2×C3⋊D3—34no—
304C2×C3⋊D3—316no—
305C2×C3⋊D3—616no—
306C2×C3⋊D3—62no—
307C2×C3⋊D3—62no—
308C2×C3⋊D3—68no—
309C2×C3⋊D3—68no—
310C2×C3⋊D3—128no—
311C2×C3⋊D3—128no—
312C2×C3⋊D3—128no—
313C2×C3⋊D3—124no—
314C2×C3⋊D3—1216no—
315C2×C3⋊D3—1232no—
316C2×C3×D3—28no—
317C2×C3×D3—28no—
318C2×C3×D3—28no—
319C2×C3×D3—68no—
320C2×C3×D3—68no—
321C2×C3×D3—68no—
322D32—1232no—
323C32×C4—21no—
324C32×C4—21no—
325C32×C4—24no—
326C32×C4—24no—
327C32×C4—44no—
328C32×C4—44no—
329C32×C4—62no—
330C3⋊Dic3—22no—
331C3⋊Dic3—62no—
332C3⋊Dic3—68no—
333C3⋊Dic3—128no—
334C3×Dic3—28no—
335C3×Dic3—68no—
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—
345C33#53411yescharacteristic, complemented, Sylow
346C23×C3—24no—
347C23×C3—38no—
348C23×C3—92no—
349C23×C3C2×D3214yes—
350C23×C3C22×C3⋊D311yescharacteristic
351C22×D3—38no—
352C22×D3—98no—
353C22×D3—92no—
354C22×D3—916no—
355C22×D3—94no—
356C22×D3—188no—
357C22×D3—1816no—
358C22×D3—184no—
359C22×D3—1816no—
360C22×D3—362no—
361C22×D3—362no—
362C22×D3—368no—
363C22×D3—368no—
364C22×D3—364no—
365C22×D3—364no—
366C22×D3—361no—
367C22×D3—361no—
368C22×D3—368no—
369C22×D3—368no—
370C22×D3C22×C3⋊D312yes—
371C3×D4—616no—
372C3×D4—124no—
373C3×D4—184no—
374C3×D4—361no—
375C2×C3×C4—24no—
376C2×C3×C4—24no—
377C2×C3×C4—38no—
378C2×C3×C4—44no—
379C2×C3×C4—41no—
380C2×C3×C4—48no—
381C2×C3×C4—92no—
382C2×C3×C4C2×D3214yes—
383C2×C3×C4C2×D3214yes—
384C2×C3×C4C22×C3⋊D311yescharacteristic
385C2×C3×C4C22×C3⋊D311yescharacteristic
386#67—24no—
387#67—616no—
388#67—184no—
389#67—1816no—
390#67—362no—
391#67—368no—
392C2×Dic3—38no—
393C2×Dic3—98no—
394C2×Dic3—92no—
395C2×Dic3—188no—
396C2×Dic3—364no—
397C2×Dic3—361no—
398C2×Dic3—368no—
399C2×Dic3C22×C3⋊D312yes—
400D12—41no—
401D12—124no—
402D12—1816no—
403D12—184no—
404C4×D3—184no—
405C4×D3—1816no—
406C2×C32—216no—
407C2×C32—28no—
408C2×C32—22no—
409C2×C32—24no—
410C2×C32—24no—
411C2×C32—24no—
412C2×C32—122no—
413C2×C32C2×D3×D414yes—
414C2×C32C2×D3×D411yescharacteristic
415C2×C32#100414yes—
416C2×C32#100411yescharacteristic
417C2×C32#100914yes—
418C2×C32#100911yescharacteristic
419C3⋊D3—42no—
420C3⋊D3—64no—
421C3⋊D3—616no—
422C3⋊D3—1216no—
423C3×D3—48no—
424C3×D3—128no—
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×C3⋊D311yescharacteristic, 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—28no—
448C22×C3—28no—
449C22×C3—24no—
450C22×C3—24no—
451C22×C3—21no—
452C22×C3—21no—
453C22×C3—24no—
454C22×C3—28no—
455C22×C3—68no—
456C22×C3—68no—
457C22×C3—68no—
458C22×C3—182no—
459C22×C3—182no—
460C22×C3—182no—
461C22×C3#380218yes—
462C22×C3D4×C3⋊D312yes—
463C22×C3C22×D3214yes—
464C22×C3C23×C3⋊D311yescharacteristic
465C2×D3—22no—
466C2×D3—22no—
467C2×D3—22no—
468C2×D3—68no—
469C2×D3—68no—
470C2×D3—68no—
471C2×D3—916no—
472C2×D3—94no—
473C2×D3—1816no—
474C2×D3—188no—
475C2×D3—188no—
476C2×D3—182no—
477C2×D3—182no—
478C2×D3—368no—
479C2×D3—368no—
480C2×D3—368no—
481C2×D3—3632no—
482C2×D3—3616no—
483C2×D3—364no—
484C2×D3—3632no—
485C3×C4—24no—
486C3×C4—24no—
487C3×C4—21no—
488C3×C4—21no—
489C3×C4—44no—
490C3×C4—44no—
491C3×C4—68no—
492C3×C4—182no—
493Dic3—22no—
494Dic3—68no—
495Dic3—188no—
496Dic3—182no—
497Dic3—368no—
498C32—24no—
499C32#560514yescomplemented
500C32#560511yescharacteristic, complemented
501C23—32no—
502C23—92no—
503C23—272no—
504C23—274no—
505C23—544no—
506C23—1081no—
507C23—1081no—
508C23—1082no—
509C23—1082no—
510C23C2×D3×C3⋊D311yescharacteristic
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×C3⋊D311yescharacteristic
525C2×C4C2×D3×C3⋊D311yescharacteristic
526C2×C3—216no—
527C2×C3—28no—
528C2×C3—22no—
529C2×C3—24no—
530C2×C3—24no—
531C2×C3—24no—
532C2×C3—128no—
533C2×C3—362no—
534C2×C3C2×#380214yes—
535C2×C3#6430814yes—
536C2×C3#6433014yes—
537C2×C3C2×D4×C3⋊D311yescharacteristic
538C2×C3#6436111yescharacteristic
539C2×C3#6436611yescharacteristic
540D3—42no—
541D3—128no—
542D3—1816no—
543D3—184no—
544D3—3616no—
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#8902612yes—
559C22C22×D3×C3⋊D311yescharacteristic
560C4—21no—
561C4—21no—
562C4—62no—
563C4—182no—
564C4—542no—
565C3—24no—
566C3#1059508414yescomplemented
567C3#1059607611yescharacteristic, complemented
568C2—22no—
569C2—122no—
570C2—362no—
571C2—544no—
572C2C2×#8902611yescharacteristic
573C2#1172458111yescharacteristic
574C2#1172461111yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutator, socle, cosocleC22×C33C24
FittingC22×C33×C4C22
hypercenterC22×C4D3×C3⋊D3
hypocenterC33#534
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C2×#89039C2semidirect
2. C2×#88859C2semidirect
3. C3×#64139C2semidirect
4. C32×#930C2semidirect
5. #11723637C2semidirect
6. #11723541C2semidirect
7. #11723540C2semidirect
8. #11723504C2semidirect
9. #11723422C2semidirect
13. C22×D3×C3⋊D3C22semidirect
14. C22×C33×C4C22semidirect
15. C22×C3⋊(C3⋊Dic3)C22semidirect
16. C2×C3×#638C22semidirect
17. C2×C3×C4⋊(C3⋊D3)C22semidirect
18. C2×C32×#67C22semidirect
19. C2×C32×D12C22semidirect
23. C2×#6290C22semidirect
24. C2×#6289C22semidirect
25. #88859C22semidirect
26. #88859C22semidirect
27. C3×#3752C22semidirect
28. C3×C4⋊(C3⋊Dic3)C22semidirect
29. C32×(C2×D3)⋊C4C22semidirect
30. C32×C4⋊Dic3C22semidirect
31. #88806C22semidirect
32. #88803C22semidirect
33. (C2×D3)⋊(C3⋊Dic3)C22semidirect
39. #64139D3semidirect
47. C3×#930D3semidirect
56. C22×C3⋊(C3⋊D3)D4semidirect
88. C22×C32×C4C2×D3semidirect
89. C22×C32×C4C2×D3semidirect
92. C2×#638C2×D3semidirect
98. C2×C4⋊(C3⋊D3)C2×D3semidirect
101. C2×C3×#67C2×D3semidirect
103. C2×C3×D12C2×D3semidirect
119. #3752C2×D3semidirect
120. C4⋊(C3⋊Dic3)C2×D3semidirect
123. C3×(C2×D3)⋊C4C2×D3semidirect
125. C3×C4⋊Dic3C2×D3semidirect
158. #930C3⋊D3semidirect
246. C22×C3×C4D32semidirect
247. C22×C3×C4C2×C3⋊D3semidirect
253. C2×#67C2×C3⋊D3semidirect
262. C2×D12C2×C3⋊D3semidirect
280. (C2×D3)⋊C4C2×C3⋊D3semidirect
282. C4⋊Dic3C2×C3⋊D3semidirect
345. C33#534semidirect
437. C22×C4D3×C3⋊D3semidirect
499. C32#5605semidirect
500. C32#5605semidirect
566. C3#10595084semidirect
567. C3#10596076semidirect

Maximal subgroups

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

subgroupindex
C2×#890392
C2×#888592
C3×#641392
C32×#9302
#117236372
#117235412
#117235402
#117235042
#117234222
#105960763
#105950843