Finite Group Databasebeta

← #13177625

Subgroups — #13177625

Order 1728.

1824 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11723976C211yesmaximal, characteristic, complemented
2C3×#64149C211yesmaximal, characteristic, complemented
3C32×#937C211yesmaximal, characteristic, complemented
4#11723637C212yesmaximal, complemented
5#11723631C212yesmaximal, complemented
6#11723629C212yesmaximal, complemented
7#11723621C212yesmaximal, complemented
8#11723519C212yesmaximal, complemented
9#11723428C212yesmaximal, complemented
10#10596066—31nomaximal
11#10595259—34nomaximal
12C23×C32×D3C2211yescharacteristic
13C2×C33×D4C2211yescharacteristic
14C22×C3⋊(C3⋊Dic3)C2211yescharacteristic, complemented
15C2×C3×#638C2212yes—
16C2×C3×#638C2212yescomplemented
17C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic, complemented
18C2×C32×#67C2212yes—
19C2×#6290C2212yes—
20C2×#6290C2212yes—
21C2×Dic3×C3⋊D3C2212yescomplemented
22#88864C2212yes—
23#88864C2211yescharacteristic, complemented
24#88857C2212yescomplemented
25C3×C4×C3⋊Dic3C2211yescharacteristic, complemented
26C32×#217C2211yescharacteristic
27C32×(C2×D3)⋊C4C2212yes—
28#88806C2212yescomplemented
29Dic3⋊(C3⋊Dic3)C2212yescomplemented
30#88803C2212yescomplemented
31(C2×D3)⋊(C3⋊Dic3)C2212yes—
32(C2×D3)⋊(C3⋊Dic3)C2212yes—
33C32×#120—31no—
34#64162—32no—
35#64149D311yescharacteristic, complemented
36#64148—32no—
37#64147—32no—
38#64144—31no—
39#64144—34no—
40#64144—64no—
41#64093—32no—
42#64092—32no—
43#64088—32no—
44C3×#940—34no—
45C3×#937D314yescomplemented
46#63980—38no—
47#63976—38no—
48#63970—38no—
49#63969—38no—
50#63885—38no—
51#63859—38no—
52C23×C33C2311yescharacteristic
53C23×C33C2311yescharacteristic
54C22×C3×C3⋊D3C2312yes—
55C22×C32×D3—41no—
56C22×C32×D3—42no—
57C22×C32×D3C2312yes—
58C33×D4—41no—
59C2×C33×C4C2311yescharacteristic
60C2×C3⋊(C3⋊Dic3)—41no—
61C2×C3⋊(C3⋊Dic3)C2312yes—
62C2×C3⋊(C3⋊Dic3)C2312yes—
63C3×#638—42no—
64C3×#638—42no—
65C2×C3×C3⋊Dic3C2312yes—
66C3×C4⋊(C3⋊D3)—41no—
67C32×#67—42no—
68C2×C32×Dic3C2312yes—
69#6290—42no—
70#6290—42no—
71Dic3×C3⋊D3—42no—
72#5720—91no—
73#5595—94no—
74C24×C32—31no—
75C23×C3×D3C2×D314yes—
76C2×C32×D4—24no—
77C2×C32×D4—32no—
78C2×C32×D4C2×D314yes—
79C2×C32×D4C2×D311yescharacteristic
80C2×#638—32no—
81C2×#638—32no—
82C2×#638C2×D312yes—
83C2×#638C2×D312yescomplemented
84C22×C3⋊Dic3—31no—
85C22×C3⋊Dic3—34no—
86C22×C3⋊Dic3—64no—
87C2×C4⋊(C3⋊D3)C2×D311yescharacteristic, complemented
88C2×C4×C3⋊D3—32no—
89C2×C3×#67—38no—
90C2×C3×#67—38no—
91C2×C3×#67C2×D318yes—
92C2×C3×D12—34no—
93C2×#625—38no—
94C2×#625—38no—
95C2×D3×Dic3—38no—
96C32×C22⋊C4—31no—
97C32×C22⋊C4—32no—
98#3757—32no—
99#3757—31no—
100#3757—38no—
101#3757—34no—
102#3757—32no—
103#3757—32no—
104#3757—68no—
105#3757—64no—
106#3752—32no—
107C4⋊(C3⋊Dic3)—32no—
108#3750—32no—
109#3750—38no—
110#3750—32no—
111#3750—68no—
112C4×C3⋊Dic3C2×D311yescharacteristic, complemented
113C3×#217C2×D314yes—
114C3×(C2×D3)⋊C4C2×D318yes—
115C3×C4×Dic3—34no—
116#3724—38no—
117Dic3⋊Dic3—38no—
118(C2×D3)⋊Dic3—38no—
119(C2×D3)⋊Dic3—38no—
120(C2×D3)⋊Dic3—38no—
121C22×C33—21no—
122C22×C33—22no—
123C22×C33—22no—
124C22×C33—21no—
125C22×C33C2411yescharacteristic
126C2×C3×C3⋊D3—22no—
127C2×C3×C3⋊D3—22no—
128C2×C3×C3⋊D3—22no—
129C2×C32×D3—22no—
130C2×C32×D3—22no—
131C2×C32×D3—22no—
132C2×C32×D3—44no—
133C33×C4—21no—
134C3⋊(C3⋊Dic3)—22no—
135C3⋊(C3⋊Dic3)—22no—
136C3×C3⋊Dic3—22no—
137C32×Dic3—22no—
138C3×#127—91no—
139C3×#120—34no—
140#953—98no—
141#940—34no—
142#939—98no—
143#939—92no—
144#939—92no—
145#939—92no—
146#938—98no—
147#937—92no—
148#937C3⋊D311yescharacteristic, complemented
149#935—94no—
150#935—91no—
151#935—92no—
152#935—184no—
153#884—98no—
154#883—98no—
155#879—98no—
156#876—92no—
157C23×C32—24no—
158C23×C32—24no—
159C23×C32—32no—
160C23×C32—122no—
161C23×C32—121no—
162C23×C32C22×D314yes—
163C23×C32C22×D314yes—
164C23×C32C22×D311yescharacteristic
165C23×C32C22×D311yescharacteristic
166C22×C3⋊D3C22×D312yes—
167C22×C3×D3—38no—
168C22×C3×D3—44no—
169C22×C3×D3—48no—
170C22×C3×D3C22×D318yes—
171C32×D4—41no—
172C32×D4—44no—
173C32×D4—48no—
174C32×D4—122no—
175C2×C32×C4—24no—
176C2×C32×C4—32no—
177C2×C32×C4C22×D314yes—
178C2×C32×C4C22×D311yescharacteristic
179#638—42no—
180#638—42no—
181#638—122no—
182#638—122no—
183C2×C3⋊Dic3—32no—
184C2×C3⋊Dic3—32no—
185C2×C3⋊Dic3—38no—
186C2×C3⋊Dic3—38no—
187C2×C3⋊Dic3—68no—
188C2×C3⋊Dic3—68no—
189C2×C3⋊Dic3—121no—
190C2×C3⋊Dic3—124no—
191C2×C3⋊Dic3—128no—
192C2×C3⋊Dic3C22×D312yes—
193C4⋊(C3⋊D3)—41no—
194C4×C3⋊D3—122no—
195C3×#67—48no—
196C3×#67—128no—
197C3×#67—128no—
198C2×C3×Dic3—38no—
199C2×C3×Dic3C22×D318yes—
200C3×D12—124no—
201#625—128no—
202#625—128no—
203D3×Dic3—128no—
204#560—271noSylow
205C2×C33—41no—
206C2×C33—41no—
207C2×C33H(5,𝔽2)12yes—
208C2×C33H(5,𝔽2)11yescharacteristic
209C3×C3⋊D3—42no—
210C32×D3—42no—
211C24×C3—34no—
212C23×D3C2×C3⋊D311yescharacteristic
213C2×C3×D4—24no—
214C2×C3×D4—38no—
215C2×C3×D4—92no—
216C2×C3×D4—92no—
217C2×C3×D4—91no—
218C2×C3×D4D3214yes—
219C2×C3×D4C2×C3⋊D311yescharacteristic
220C2×#67—38no—
221C2×#67—38no—
222C2×#67—98no—
223C2×#67—98no—
224C2×#67—92no—
225C2×#67—92no—
226C2×#67C2×C3⋊D312yes—
227C22×Dic3—94no—
228C22×Dic3—91no—
229C22×Dic3—92no—
230C22×Dic3—184no—
231C2×D12—34no—
232C2×C4×D3—98no—
233C3×C22⋊C4—34no—
234C3×C22⋊C4—38no—
235C3×C42—91no—
236#217—98no—
237#217—98no—
238#217—98no—
239#217—94no—
240#217—92no—
241#217—91no—
242#217—92no—
243#217—188no—
244#217—184no—
245#217C2×C3⋊D311yescharacteristic
246(C2×D3)⋊C4—98no—
247(C2×D3)⋊C4—92no—
248(C2×D3)⋊C4—92no—
249(C2×D3)⋊C4C2×C3⋊D312yes—
250C4⋊Dic3—98no—
251Dic3⋊C4—98no—
252Dic3⋊C4—98no—
253Dic3⋊C4—92no—
254Dic3⋊C4—92no—
255Dic3⋊C4—92no—
256Dic3⋊C4—188no—
257C4×Dic3—34no—
258C22×C32—28no—
259C22×C32—24no—
260C22×C32—21no—
261C22×C32—22no—
262C22×C32—22no—
263C22×C32—21no—
264C22×C32—24no—
265C22×C32—28no—
266C22×C32—28no—
267C22×C32—24no—
268C22×C32—48no—
269C22×C32—44no—
270C22×C32—48no—
271C22×C32—62no—
272C22×C32—62no—
273C22×C32—62no—
274C22×C32—124no—
275C22×C32C23×D314yes—
276C22×C32C23×D311yescharacteristic
277C2×C3⋊D3—22no—
278C2×C3⋊D3—22no—
279C2×C3⋊D3—22no—
280C2×C3×D3—28no—
281C2×C3×D3—28no—
282C2×C3×D3—28no—
283C2×C3×D3—416no—
284C2×C3×D3—68no—
285C2×C3×D3—68no—
286C2×C3×D3—68no—
287C32×C4—21no—
288C32×C4—24no—
289C32×C4—44no—
290C32×C4—62no—
291C3⋊Dic3—22no—
292C3⋊Dic3—62no—
293C3⋊Dic3—62no—
294C3⋊Dic3—68no—
295C3⋊Dic3—68no—
296C3⋊Dic3—128no—
297C3⋊Dic3—128no—
298C3×Dic3—28no—
299C3×Dic3—68no—
300#127—91no—
301#126—272no—
302#123—272no—
303#123—271no—
304#121—272no—
305#121—272no—
306#121—272no—
307#120—31no—
308#120—272no—
309C33#56011yescharacteristic, complemented, Sylow
310C23×C3—24no—
311C23×C3—24no—
312C23×C3—38no—
313C23×C3—92no—
314C23×C3—128no—
315C23×C3—124no—
316C23×C3C2×D3214yes—
317C23×C3C2×D3214yes—
318C23×C3C22×C3⋊D311yescharacteristic
319C23×C3C22×C3⋊D311yescharacteristic
320C22×D3—38no—
321C22×D3—42no—
322C22×D3—41no—
323C22×D3C22×C3⋊D312yes—
324C3×D4—44no—
325C3×D4—41no—
326C3×D4—48no—
327C3×D4—128no—
328C3×D4—361no—
329C3×D4—362no—
330C3×D4—362no—
331C2×C3×C4—24no—
332C2×C3×C4—38no—
333C2×C3×C4—92no—
334C2×C3×C4C2×D3214yes—
335C2×C3×C4C22×C3⋊D311yescharacteristic
336#67—42no—
337#67—128no—
338#67—128no—
339#67—362no—
340#67—362no—
341#67—368no—
342#67—368no—
343C2×Dic3—38no—
344C2×Dic3—98no—
345C2×Dic3—98no—
346C2×Dic3—92no—
347C2×Dic3—92no—
348C2×Dic3—188no—
349C2×Dic3—188no—
350C2×Dic3—362no—
351C2×Dic3—364no—
352C2×Dic3—361no—
353C2×Dic3—368no—
354C2×Dic3C22×C3⋊D312yes—
355D12—124no—
356C4×D3—368no—
357C2×C32—28no—
358C2×C32—24no—
359C2×C32—48no—
360C2×C32—41no—
361C2×C32—41no—
362C2×C32—44no—
363C2×C32—44no—
364C2×C32—84no—
365C2×C32—122no—
366C2×C32#100418yes—
367C2×C32#100414yes—
368C2×C32#100412yes—
369C2×C32#100411yescharacteristic
370C3⋊D3—42no—
371C3×D3—48no—
372C3×D3—128no—
373C24—31no—
374C2×D4—32no—
375C2×D4—92no—
376C2×D4—92no—
377C2×D4—91no—
378C2×D4—272no—
379C2×D4—272no—
380C2×D4D3×C3⋊D311yescharacteristic
381C22×C4—272no—
382C22×C4—271no—
383C4⋊C4—272no—
384C4⋊C4—272no—
385C4⋊C4—272no—
386C22⋊C4—31no—
387C22⋊C4—32no—
388C22⋊C4—272no—
389C22⋊C4—272no—
390C22⋊C4—272no—
391C22⋊C4—272no—
392C22⋊C4—271no—
393C42—91no—
394C22×C3—28no—
395C22×C3—28no—
396C22×C3—28no—
397C22×C3—24no—
398C22×C3—24no—
399C22×C3—21no—
400C22×C3—22no—
401C22×C3—21no—
402C22×C3—22no—
403C22×C3—24no—
404C22×C3—48no—
405C22×C3—44no—
406C22×C3—48no—
407C22×C3—68no—
408C22×C3—68no—
409C22×C3—68no—
410C22×C3—1216no—
411C22×C3—182no—
412C22×C3—182no—
413C22×C3—182no—
414C22×C3C22×D3214yes—
415C22×C3C23×C3⋊D311yescharacteristic
416C2×D3—22no—
417C2×D3—22no—
418C2×D3—22no—
419C2×D3—44no—
420C2×D3—68no—
421C2×D3—68no—
422C2×D3—68no—
423C3×C4—24no—
424C3×C4—21no—
425C3×C4—44no—
426C3×C4—68no—
427C3×C4—182no—
428Dic3—22no—
429Dic3—68no—
430Dic3—188no—
431Dic3—188no—
432Dic3—182no—
433Dic3—182no—
434Dic3—368no—
435Dic3—368no—
436C32—24no—
437C32#559514yescomplemented
438C32#572011yescharacteristic, complemented
439C23—32no—
440C23—92no—
441C23—122no—
442C23—121no—
443C23C2×D3×C3⋊D311yescharacteristic
444C23C2×D3×C3⋊D311yescharacteristic
445D4—41no—
446D4—122no—
447D4—361no—
448D4—362no—
449D4—362no—
450D4—1082no—
451D4—1082no—
452C2×C4—32no—
453C2×C4—92no—
454C2×C4—272no—
455C2×C4—272no—
456C2×C4—1082no—
457C2×C4—1081no—
458C2×C4C2×D3×C3⋊D311yescharacteristic
459C2×C3—28no—
460C2×C3—24no—
461C2×C3—48no—
462C2×C3—44no—
463C2×C3—44no—
464C2×C3—41no—
465C2×C3—41no—
466C2×C3—84no—
467C2×C3—128no—
468C2×C3—362no—
469C2×C3#6431318yes—
470C2×C3#6433014yes—
471C2×C3#6436112yes—
472C2×C3#6436111yescharacteristic
473D3—42no—
474D3—128no—
475C22—22no—
476C22—22no—
477C22—21no—
478C22—21no—
479C22—62no—
480C22—62no—
481C22—62no—
482C22—124no—
483C22—182no—
484C22—182no—
485C22—182no—
486C22C22×D3×C3⋊D311yescharacteristic
487C4—21no—
488C4—62no—
489C4—182no—
490C4—542no—
491C4—542no—
492C3—24no—
493C3#1059525914yescomplemented
494C3#1059606611yescharacteristic, complemented
495C2—41no—
496C2—41no—
497C2—122no—
498C2—362no—
499C2#1172458712yes—
500C2#1172461111yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutator, socle, cosocleC22×C33C24
FittingC2×C33×D4C22
hypercenterC2×D4D3×C3⋊D3
hypocenterC33#560
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. #11723976C2semidirect
2. C3×#64149C2semidirect
3. C32×#937C2semidirect
4. #11723637C2semidirect
5. #11723631C2semidirect
6. #11723629C2semidirect
7. #11723621C2semidirect
8. #11723519C2semidirect
9. #11723428C2semidirect
14. C22×C3⋊(C3⋊Dic3)C22semidirect
16. C2×C3×#638C22semidirect
17. C2×C3×C4⋊(C3⋊D3)C22semidirect
21. C2×Dic3×C3⋊D3C22semidirect
23. #88864C22semidirect
24. #88857C22semidirect
25. C3×C4×C3⋊Dic3C22semidirect
28. #88806C22semidirect
29. Dic3⋊(C3⋊Dic3)C22semidirect
30. #88803C22semidirect
35. #64149D3semidirect
45. C3×#937D3semidirect
83. C2×#638C2×D3semidirect
87. C2×C4⋊(C3⋊D3)C2×D3semidirect
112. C4×C3⋊Dic3C2×D3semidirect
148. #937C3⋊D3semidirect
309. C33#560semidirect
437. C32#5595semidirect
438. C32#5720semidirect
493. C3#10595259semidirect
494. C3#10596066semidirect

Maximal subgroups

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

subgroupindex
#117239762
C3×#641492
C32×#9372
#117236372
#117236312
#117236292
#117236212
#117235192
#117234282
#105960663
#105952593