Finite Group Databasebeta

← #13172189

Subgroups — #13172189

Order 1728.

1986 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6292C211yesmaximal, characteristic
2#11723723C211yesmaximal, characteristic, complemented
3#11723627C212yesmaximal, complemented
4C3×#63974C211yesmaximal, characteristic, complemented
5C3×#63955C212yesmaximal, complemented
6#11722937C214yesmaximal, complemented
7#11722933C214yesmaximal, complemented
8#10594996—31nomaximal
9#10594182—32nomaximal
10C23×C3⋊(C3⋊D3)C2211yescharacteristic
11C23×C3×C3⋊D3C2211yescharacteristic
12C22×C32×Dic3C2211yescharacteristic
13C2×#6294C2212yescomplemented
14C2×#6292C418yescomplemented
15C2×#6292C2212yes—
16C2×#6292C2212yes—
17C2×#6289C2214yescomplemented
18C2×C3×#624C2212yescomplemented
19C3×#3757C2212yes—
20C32×#217C2212yes—
21#88814C2212yescomplemented
22#88804C2214yescomplemented
23C3×Dic3⋊Dic3C2214yescomplemented
24C3×#3722C2212yescomplemented
25C3×Dic32C2214yescomplemented
26C22×C4⋊(C3⋊D3)—31no—
27C22×#626—32no—
28C22×#626—61no—
29C22⋊C4×C3⋊D3—32no—
30C3×#926—32no—
31C3×D3×C22⋊C4—32no—
32C3×#877—32no—
33#63974—32no—
34#63974—32no—
35#63974—31no—
36#63974—62no—
37#63974D311yescharacteristic, complemented
38D3×#217—32no—
39#63955D312yescomplemented
40#63904—34no—
41C4×#626—34no—
42#63894—34no—
43#63894—34no—
44#63881—34no—
45#63880—34no—
46C23×C33C2311yescharacteristic
47C22×C3⋊(C3⋊D3)—42no—
48C22×C3⋊(C3⋊D3)C2×C414yescomplemented
49C22×C3⋊(C3⋊D3)C2312yes—
50C22×C3×C3⋊D3—42no—
51C22×C3×C3⋊D3C2×C414yescomplemented
52C22×C3×C3⋊D3C2311yescharacteristic
53C22×C3×C3⋊D3C2311yescharacteristic
54C2×C3×C3⋊Dic3C2314yes—
55C2×C32×Dic3C2×C414yescomplemented
56C2×C32×Dic3C2314yes—
57C2×C32×Dic3C2312yes—
58#6294—42no—
59#6292C2×C418yescomplemented
60#6292C2×C418yescomplemented
61#6289—44no—
62C3×#624—42no—
63#5521—92no—
64#5486—91no—
65C23×C3⋊D3—32no—
66C23×C3⋊D3—31no—
67C23×C3⋊D3—61no—
68C23×C3⋊D3—62no—
69C23×C3⋊D3—121no—
70C23×C3⋊D3C2×D311yescharacteristic
71C23×C3×D3—32no—
72C23×C3×D3—61no—
73C22×C32×C4—31no—
74C2×C4⋊(C3⋊D3)—38no—
75C2×C4⋊(C3⋊D3)—32no—
76C2×C4⋊(C3⋊D3)—32no—
77C2×C4×C3⋊D3—34no—
78C22×C3×Dic3—21no—
79C22×C3×Dic3C2×D312yes—
80C2×C3×C4×D3—34no—
81C2×#626—316no—
82C2×#626—32no—
83C2×#626—32no—
84C2×#626—32no—
85C2×#626—32no—
86C2×#626—68no—
87C2×#626—62no—
88C2×#626—62no—
89C2×#624—32no—
90C2×#624—34no—
91C2×#624—34no—
92C2×#624—64no—
93C2×#624C2×D312yescomplemented
94C2×D3×Dic3—34no—
95C32×C22⋊C4—32no—
96#3757C2×D312yes—
97#3752—34no—
98C3×#217—22no—
99C3×#217—32no—
100C3×#217—32no—
101C3×#217—62no—
102C3×#217C2×D312yes—
103C3×#217C2×D312yes—
104C3×(C2×D3)⋊C4—34no—
105C3×C4⋊Dic3—34no—
106C3×Dic3⋊C4—34no—
107C3×C4×Dic3—34no—
108C3×C4×Dic3—34no—
109Dic3⋊Dic3C2×D314yescomplemented
110#3722—32no—
111#3722—34no—
112#3722—34no—
113#3722—64no—
114#3722C2×D312yescomplemented
115(C2×D3)⋊Dic3—34no—
116Dic32C2×D314yescomplemented
117C22×C33—22no—
118C22×C33C22×C412yes—
119C22×C33C2411yescharacteristic
120C2×C3⋊(C3⋊D3)—24no—
121C2×C3⋊(C3⋊D3)—44no—
122C2×C3⋊(C3⋊D3)C22×C414yes—
123C2×C3×C3⋊D3—22no—
124C2×C3×C3⋊D3—22no—
125C2×C3×C3⋊D3—44no—
126C2×C3×C3⋊D3C22×C412yes—
127C2×C3×C3⋊D3C22×C412yes—
128C3×C3⋊Dic3—24no—
129C32×Dic3—24no—
130C32×Dic3C22×C414yes—
131C22×#67—91no—
132C22×D12—92no—
133C22×D12—181no—
134C3×#117—92no—
135C2×C3×C22⋊C4—91no—
136C2×#217—91no—
137C4×#67—94no—
138#926—32no—
139#893—94no—
140#893—94no—
141#67⋊C4—94no—
142D3×C22⋊C4—32no—
143D3×C22⋊C4—92no—
144D3×C22⋊C4—92no—
145D3×C22⋊C4—92no—
146D3×C22⋊C4—92no—
147D3×C22⋊C4—182no—
148#877—32no—
149C4×D12—94no—
150C4×D12—94no—
151C23×C32—21no—
152C23×C32—22no—
153C23×C32—41no—
154C23×C32C22×D312yes—
155C23×C32C22×D311yescharacteristic
156C22×C3⋊D3—32no—
157C22×C3⋊D3—32no—
158C22×C3⋊D3—32no—
159C22×C3⋊D3—38no—
160C22×C3⋊D3—34no—
161C22×C3⋊D3—42no—
162C22×C3⋊D3—62no—
163C22×C3⋊D3—62no—
164C22×C3⋊D3—68no—
165C22×C3⋊D3—64no—
166C22×C3⋊D3—62no—
167C22×C3⋊D3—124no—
168C22×C3⋊D3—122no—
169C22×C3⋊D3—124no—
170C22×C3⋊D3—122no—
171C22×C3⋊D3—128no—
172C22×C3⋊D3—124no—
173C22×C3⋊D3—128no—
174C22×C3⋊D3C4×D314yescomplemented
175C22×C3⋊D3C22×D311yescharacteristic
176C22×C3⋊D3C22×D311yescharacteristic
177C22×C3×D3—32no—
178C22×C3×D3—32no—
179C22×C3×D3—38no—
180C22×C3×D3—64no—
181C22×C3×D3—61no—
182C22×C3×D3—61no—
183C22×C3×D3—124no—
184C22×C3×D3—124no—
185C2×C32×C4—34no—
186C2×C32×C4—32no—
187C2×C32×C4—34no—
188C2×C3⋊Dic3C22×D314yes—
189C4⋊(C3⋊D3)—38no—
190C4⋊(C3⋊D3)—38no—
191C4×C3⋊D3—124no—
192C2×C3×Dic3—24no—
193C2×C3×Dic3—22no—
194C2×C3×Dic3—24no—
195C2×C3×Dic3—34no—
196C2×C3×Dic3—34no—
197C2×C3×Dic3—64no—
198C2×C3×Dic3C4×D318yescomplemented
199C2×C3×Dic3C22×D314yes—
200C2×C3×Dic3C22×D314yes—
201C2×C3×Dic3C22×D312yes—
202C2×C3×Dic3C22×D312yes—
203C3×C4×D3—124no—
204#626—38no—
205#626—38no—
206#626—38no—
207#626—38no—
208#626—68no—
209#626—68no—
210#624—42no—
211#624—124no—
212#624—124no—
213#624—128no—
214#624—122no—
215D3×Dic3—124no—
216#518—271noSylow
217C2×C33—22no—
218C2×C33C23×C411yescharacteristic
219C2×C33H(5,𝔽2)12yes—
220C3⋊(C3⋊D3)—24no—
221C3×C3⋊D3—22no—
222C3×C3⋊D3—22no—
223C24×C3—91no—
224C23×D3—32no—
225C23×D3—61no—
226C23×D3—92no—
227C23×D3—91no—
228C23×D3—182no—
229C23×D3—181no—
230C23×D3—361no—
231C22×C3×C4—32no—
232C22×C3×C4—61no—
233C22×C3×C4—92no—
234C2×#67—98no—
235C2×#67—92no—
236C2×#67—92no—
237C22×Dic3—92no—
238C22×Dic3D3211yescharacteristic
239C2×D12—92no—
240C2×D12—92no—
241C2×D12—92no—
242C2×D12—92no—
243C2×D12—916no—
244C2×D12—182no—
245C2×D12—182no—
246C2×D12—188no—
247C2×C4×D3—34no—
248C2×C4×D3—94no—
249C2×C4×D3—94no—
250C2×C4×D3—94no—
251C2×C4×D3—94no—
252C2×C4×D3—184no—
253C3×C4⋊C4—94no—
254C3×C22⋊C4—32no—
255C3×C22⋊C4—32no—
256C3×C22⋊C4—62no—
257C3×C22⋊C4—92no—
258C3×C22⋊C4—92no—
259C3×C42—94no—
260#217—32no—
261#217—32no—
262#217—62no—
263#217—92no—
264#217D3212yes—
265(C2×D3)⋊C4—34no—
266(C2×D3)⋊C4—94no—
267(C2×D3)⋊C4—94no—
268(C2×D3)⋊C4—94no—
269(C2×D3)⋊C4—94no—
270(C2×D3)⋊C4—184no—
271C4⋊Dic3—34no—
272Dic3⋊C4—34no—
273C4×Dic3—34no—
274C4×Dic3—34no—
275C22×C32—22no—
276C22×C32—24no—
277C22×C32—24no—
278C22×C32—22no—
279C22×C32—21no—
280C22×C32—24no—
281C22×C32—22no—
282C22×C32—42no—
283C22×C32—41no—
284C22×C32—44no—
285C22×C32—44no—
286C22×C32C2×C4×D312yes—
287C22×C32C2×C4×D314yes—
288C22×C32C23×D311yescharacteristic
289C22×C32C23×D312yes—
290C2×C3⋊D3—22no—
291C2×C3⋊D3—22no—
292C2×C3⋊D3—34no—
293C2×C3⋊D3—34no—
294C2×C3⋊D3—34no—
295C2×C3⋊D3—44no—
296C2×C3⋊D3—64no—
297C2×C3⋊D3—64no—
298C2×C3⋊D3—64no—
299C2×C3⋊D3—64no—
300C2×C3⋊D3—68no—
301C2×C3⋊D3—64no—
302C2×C3⋊D3—64no—
303C2×C3⋊D3—124no—
304C2×C3⋊D3—124no—
305C2×C3⋊D3—124no—
306C2×C3⋊D3—128no—
307C2×C3⋊D3—128no—
308C2×C3⋊D3—124no—
309C2×C3⋊D3—1216no—
310C2×C3⋊D3—128no—
311C2×C3⋊D3—1216no—
312C2×C3⋊D3C2×C4×D312yes—
313C2×C3⋊D3C2×C4×D312yes—
314C2×C3×D3—34no—
315C2×C3×D3—34no—
316C2×C3×D3—62no—
317C2×C3×D3—62no—
318C2×C3×D3—64no—
319C2×C3×D3—64no—
320C2×C3×D3—122no—
321C2×C3×D3—122no—
322C2×C3×D3—128no—
323C2×C3×D3—128no—
324C32×C4—34no—
325C32×C4—64no—
326C3⋊Dic3—24no—
327C3×Dic3—24no—
328C3×Dic3—24no—
329C3×Dic3—24no—
330C3×Dic3—44no—
331C3×Dic3—64no—
332C3×Dic3—64no—
333C3×Dic3—124no—
334C3×Dic3C2×C4×D314yes—
335C3×Dic3C2×C4×D314yes—
336C22×D4—271no—
337C4×D4—274no—
338C4×D4—274no—
339#117—92no—
340C2×C22⋊C4—91no—
341C2×C22⋊C4—272no—
342C2×C22⋊C4—271no—
343C33#51811yescharacteristic, complemented, Sylow
344C23×C3—21no—
345C23×C3—22no—
346C23×C3—41no—
347C23×C3—91no—
348C23×C3—91no—
349C23×C3—94no—
350C23×C3—362no—
351C23×C3C2×D3212yes—
352C23×C3C2×D3211yescharacteristic
353C22×D3—32no—
354C22×D3—32no—
355C22×D3—38no—
356C22×D3—64no—
357C22×D3—61no—
358C22×D3—61no—
359C22×D3—92no—
360C22×D3—92no—
361C22×D3—92no—
362C22×D3—98no—
363C22×D3—94no—
364C22×D3—124no—
365C22×D3—124no—
366C22×D3—182no—
367C22×D3—182no—
368C22×D3—182no—
369C22×D3—184no—
370C22×D3—188no—
371C22×D3—362no—
372C22×D3—364no—
373C22×D3—362no—
374C22×D3—364no—
375C22×D3—368no—
376C22×D3—364no—
377C22×D3—368no—
378C2×C3×C4—32no—
379C2×C3×C4—32no—
380C2×C3×C4—38no—
381C2×C3×C4—34no—
382C2×C3×C4—34no—
383C2×C3×C4—62no—
384C2×C3×C4—64no—
385C2×C3×C4—64no—
386C2×C3×C4—94no—
387C2×C3×C4—362no—
388#67—98no—
389#67—98no—
390C2×Dic3—34no—
391C2×Dic3—34no—
392C2×Dic3—64no—
393C2×Dic3—362no—
394C2×Dic3#62414yescomplemented
395C2×Dic3C2×D3214yes—
396C2×Dic3C2×D3212yes—
397D12—98no—
398D12—98no—
399D12—98no—
400D12—98no—
401D12—188no—
402D12—188no—
403C4×D3—124no—
404C4×D3—364no—
405C4×D3—364no—
406C4×D3—364no—
407C4×D3—364no—
408C4×D3—368no—
409C2×C32—22no—
410C2×C32—24no—
411C2×C32—24no—
412C2×C32—22no—
413C2×C32—22no—
414C2×C32—24no—
415C2×C32—21no—
416C2×C32—41no—
417C2×C32—42no—
418C2×C32—44no—
419C2×C32—44no—
420C2×C32C22×C4×D311yescharacteristic
421C2×C32C22×C4×D312yes—
422C2×C32#100412yes—
423C2×C32#100914yes—
424C3⋊D3—22no—
425C3⋊D3—22no—
426C3⋊D3—64no—
427C3⋊D3—68no—
428C3⋊D3—64no—
429C3⋊D3—64no—
430C3⋊D3—124no—
431C3⋊D3—124no—
432C3⋊D3—128no—
433C3×D3—64no—
434C3×D3—64no—
435C3×D3—122no—
436C3×D3—122no—
437C24—91no—
438C24—271no—
439C2×D4—272no—
440C2×D4—272no—
441C2×D4—278no—
442C22×C4—31no—
443C22×C4—92no—
444C22×C4—274no—
445C22×C4—272no—
446C4⋊C4—94no—
447C22⋊C4—32no—
448C22⋊C4—92no—
449C22⋊C4—92no—
450C22⋊C4—274no—
451C22⋊C4—272no—
452C42—94no—
453C22×C3—24no—
454C22×C3—22no—
455C22×C3—24no—
456C22×C3—22no—
457C22×C3—21no—
458C22×C3—22no—
459C22×C3—24no—
460C22×C3—44no—
461C22×C3—44no—
462C22×C3—41no—
463C22×C3—42no—
464C22×C3—92no—
465C22×C3—92no—
466C22×C3—182no—
467C22×C3—182no—
468C22×C3—364no—
469C22×C3C4×D3214yes—
470C22×C3C2×#62412yes—
471C22×C3C22×D3212yes—
472C22×C3C22×D3211yescharacteristic
473C2×D3—34no—
474C2×D3—34no—
475C2×D3—62no—
476C2×D3—62no—
477C2×D3—64no—
478C2×D3—64no—
479C2×D3—94no—
480C2×D3—94no—
481C2×D3—94no—
482C2×D3—122no—
483C2×D3—122no—
484C2×D3—128no—
485C2×D3—128no—
486C2×D3—184no—
487C2×D3—184no—
488C2×D3—184no—
489C2×D3—188no—
490C2×D3—184no—
491C2×D3—184no—
492C2×D3—184no—
493C2×D3—364no—
494C2×D3—364no—
495C2×D3—364no—
496C2×D3—368no—
497C2×D3—368no—
498C2×D3—364no—
499C2×D3—3616no—
500C2×D3—368no—
501C2×D3—3616no—
502C3×C4—34no—
503C3×C4—34no—
504C3×C4—64no—
505C3×C4—64no—
506C3×C4—64no—
507C3×C4—124no—
508C3×C4—184no—
509Dic3—24no—
510Dic3—64no—
511Dic3—64no—
512Dic3—124no—
513Dic3C2×#62414yes—
514C32—22no—
515C32—21no—
516C32—41no—
517C32#548611yescharacteristic, complemented
518C32#552112yescomplemented
519C23—91no—
520C23—91no—
521C23—94no—
522C23—272no—
523C23—274no—
524C23—362no—
525C23—1082no—
526C23D3311yescharacteristic
527D4—278no—
528D4—278no—
529C2×C4—32no—
530C2×C4—34no—
531C2×C4—34no—
532C2×C4—94no—
533C2×C4—362no—
534C2×C4—1084no—
535C2×C4—1082no—
536C2×C3—24no—
537C2×C3—24no—
538C2×C3—22no—
539C2×C3—21no—
540C2×C3—22no—
541C2×C3—24no—
542C2×C3—22no—
543C2×C3—44no—
544C2×C3—44no—
545C2×C3—41no—
546C2×C3—42no—
547C2×C3—182no—
548C2×C3—182no—
549C2×C3C2×C4×D3212yes—
550C2×C3#6430812yes—
551C2×C3#6431414yes—
552C2×C3C22×#62411yescharacteristic
553D3—64no—
554D3—64no—
555D3—122no—
556D3—122no—
557D3—188no—
558D3—184no—
559D3—184no—
560D3—184no—
561D3—364no—
562D3—364no—
563D3—368no—
564C22—22no—
565C22—92no—
566C22—92no—
567C22—182no—
568C22—182no—
569C22—274no—
570C22—364no—
571C22—544no—
572C22—1084no—
573C22D3×#62412yes—
574C22C2×D3311yescharacteristic
575C4—34no—
576C4—64no—
577C4—184no—
578C3—21no—
579C3—22no—
580C3—41no—
581C3#1059418212yescomplemented
582C3#1059499611yescharacteristic, complemented
583C2—22no—
584C2—182no—
585C2—182no—
586C2—544no—
587C2C2×D3×#62411yescharacteristic
588C2#1172442012yes—
subgroupG/H
center, FrattiniC22C2×D33
commutatorC2×C33C23×C4
FittingC23×C33C23
socle, cosocleC22×C33C24
prime radicalC22×#6292C2
hypercenterC23D33
hypocenterC33#518
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
2. #11723723C2semidirect
3. #11723627C2semidirect
4. C3×#63974C2semidirect
5. C3×#63955C2semidirect
6. #11722937C2semidirect
7. #11722933C2semidirect
13. C2×#6294C22semidirect
14. C2×#6292C4semidirect
17. C2×#6289C22semidirect
18. C2×C3×#624C22semidirect
21. #88814C22semidirect
22. #88804C22semidirect
23. C3×Dic3⋊Dic3C22semidirect
24. C3×#3722C22semidirect
25. C3×Dic32C22semidirect
37. #63974D3semidirect
39. #63955D3semidirect
48. C22×C3⋊(C3⋊D3)C2×C4semidirect
51. C22×C3×C3⋊D3C2×C4semidirect
55. C2×C32×Dic3C2×C4semidirect
59. #6292C2×C4semidirect
60. #6292C2×C4semidirect
93. C2×#624C2×D3semidirect
109. Dic3⋊Dic3C2×D3semidirect
114. #3722C2×D3semidirect
116. Dic32C2×D3semidirect
174. C22×C3⋊D3C4×D3semidirect
198. C2×C3×Dic3C4×D3semidirect
343. C33#518semidirect
394. C2×Dic3#624semidirect
517. C32#5486semidirect
518. C32#5521semidirect
581. C3#10594182semidirect
582. C3#10594996semidirect

Maximal subgroups

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

subgroupindex
C22×#62922
#117237232
#117236272
C3×#639742
C3×#639552
#117229372
#117229332
#105949963
#105941823