Finite Group Databasebeta

← #13177489

Subgroups — #13177489

Order 1728.

1816 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D4×C3⋊(C3⋊Dic3)C211yesmaximal, characteristic, complemented
2C3×#64149C211yesmaximal, characteristic, complemented
3C32×#938C211yesmaximal, characteristic, complemented
4#11723637C212yesmaximal, complemented
5#11723631C212yesmaximal, complemented
6#11723621C212yesmaximal, complemented
7#11723609C212yesmaximal, complemented
8#11723515C211yesmaximal, characteristic, complemented
9#11723501C211yesmaximal, characteristic, complemented
10#11723447C211yesmaximal, characteristic, complemented
11#11723423C211yesmaximal, characteristic, complemented
12#10596090—31nomaximal
13#10595217—34nomaximal
14C2×C33×D4C2211yescharacteristic
15C22×C3⋊(C3⋊Dic3)C2212yes—
16C2×C3×#638C2212yes—
17C2×C3×#638C2212yes—
18C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
19C2×C32×#67C2212yes—
20C2×C32×C4×D3C2211yescharacteristic
21C2×#6290C2212yes—
22C2×Dic3×C3⋊D3C2212yes—
23#88864C2212yes—
24C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
25C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
26C3×C4×C3⋊Dic3C2211yescharacteristic
27C32×#217C2212yes—
28C32×C4⋊Dic3C2211yescharacteristic
29#88806C2212yes—
30Dic3⋊(C3⋊Dic3)C2211yescharacteristic
31Dic3⋊(C3⋊Dic3)C2212yes—
32Dic3⋊(C3⋊Dic3)C2211yescharacteristic
33#88803C2212yes—
34#88803C2212yes—
35(C2×D3)⋊(C3⋊Dic3)C2211yescharacteristic
36(C2×D3)⋊(C3⋊Dic3)C2211yescharacteristic
37C32×#121—31no—
38#64149D311yescharacteristic, complemented
39#64148—32no—
40#64147—32no—
41D4×C3⋊Dic3—31no—
42D4×C3⋊Dic3—34no—
43D4×C3⋊Dic3—64no—
44#64139—31no—
45#64135—31no—
46#64102—31no—
47#64099—31no—
48#64095—32no—
49#64092—32no—
50C3×#940—34no—
51C3×#938D314yescomplemented
52#63980—38no—
53#63970—38no—
54#63969—38no—
55#63962—38no—
56#63908—34no—
57#63898—34no—
58#63869—34no—
59#63855—34no—
60C23×C33C2312yes—
61C22×C3×C3⋊D3C2312yes—
62C22×C32×D3C2311yescharacteristic
63C33×D4—22no—
64C2×C33×C4C2311yescharacteristic
65C2×C3⋊(C3⋊Dic3)—42no—
66C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
67C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
68C2×C3⋊(C3⋊Dic3)C2312yes—
69C3×#638—42no—
70C3×#638—42no—
71C2×C3×C3⋊Dic3C2311yescharacteristic
72C2×C3×C3⋊Dic3C2311yescharacteristic
73C3×C4⋊(C3⋊D3)—22no—
74C32×#67—42no—
75C2×C32×Dic3C2311yescharacteristic
76C2×C32×Dic3C2312yes—
77C32×C4×D3—22no—
78#6290—42no—
79Dic3×C3⋊D3—42no—
80#5719—91no—
81#5619—94no—
82C2×C32×D4—24no—
83C2×C32×D4—32no—
84C2×C32×D4C2×D314yes—
85C2×C32×D4C2×D311yescharacteristic
86C22×C32×C4—31no—
87C2×#638—32no—
88C2×#638C2×D312yes—
89C2×#638C2×D312yes—
90C22×C3⋊Dic3—32no—
91C22×C3⋊Dic3—38no—
92C22×C3⋊Dic3—68no—
93C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
94C2×C4×C3⋊D3—32no—
95C2×C3×#67—38no—
96C2×C3×#67—38no—
97C2×C3×#67C2×D318yes—
98C2×C3×D12—34no—
99C2×C3×C4×D3C2×D314yes—
100C2×#625—38no—
101C2×D3×Dic3—38no—
102C32×C4⋊C4—31no—
103C32×C22⋊C4—32no—
104#3757—32no—
105#3757—38no—
106#3757—31no—
107#3757—31no—
108#3757—68no—
109#3752—32no—
110#3752—32no—
111C4⋊(C3⋊Dic3)—31no—
112C4⋊(C3⋊Dic3)—34no—
113C4⋊(C3⋊Dic3)—31no—
114C4⋊(C3⋊Dic3)—31no—
115C4⋊(C3⋊Dic3)—32no—
116C4⋊(C3⋊Dic3)—64no—
117#3750—32no—
118C4×C3⋊Dic3—31no—
119C4×C3⋊Dic3—34no—
120C4×C3⋊Dic3—64no—
121C4×C3⋊Dic3C2×D311yescharacteristic
122C3×#217C2×D318yes—
123C3×C4⋊Dic3C2×D314yes—
124C3×C4×Dic3—34no—
125#3724—38no—
126Dic3⋊Dic3—34no—
127Dic3⋊Dic3—34no—
128Dic3⋊Dic3—38no—
129(C2×D3)⋊Dic3—34no—
130(C2×D3)⋊Dic3—34no—
131(C2×D3)⋊Dic3—38no—
132(C2×D3)⋊Dic3—38no—
133C22×C33—22no—
134C22×C33—22no—
135C22×C33—22no—
136C22×C33C2411yescharacteristic
137C2×C3×C3⋊D3—22no—
138C2×C3×C3⋊D3—22no—
139C2×C3×C3⋊D3—22no—
140C2×C32×D3—21no—
141C2×C32×D3—21no—
142C2×C32×D3—21no—
143C33×C4#4112yes—
144C3⋊(C3⋊Dic3)—21no—
145C3⋊(C3⋊Dic3)—21no—
146C3⋊(C3⋊Dic3)—22no—
147C3×C3⋊Dic3—21no—
148C3×C3⋊Dic3—21no—
149C32×Dic3—21no—
150C32×Dic3—22no—
151C3×#127—91no—
152C3×#121—34no—
153#940—34no—
154#939—98no—
155#939—92no—
156#939—92no—
157#938—98no—
158#938—91no—
159#938C3⋊D311yescharacteristic, complemented
160#935—92no—
161#935—92no—
162D4×Dic3—94no—
163D4×Dic3—91no—
164D4×Dic3—91no—
165D4×Dic3—184no—
166#930—94no—
167#926—94no—
168#893—94no—
169#890—94no—
170#886—98no—
171#883—98no—
172#872—91no—
173#870—91no—
174C23×C32—28no—
175C23×C32—31no—
176C23×C32C22×D318yes—
177C23×C32C22×D312yes—
178C22×C3⋊D3C22×D312yes—
179C22×C3×D3—38no—
180C22×C3×D3C22×D314yes—
181C32×D4—216no—
182C32×D4—28no—
183C32×D4—22no—
184C32×D4—122no—
185C2×C32×C4—24no—
186C2×C32×C4—32no—
187C2×C32×C4—31no—
188C2×C32×C4—62no—
189C2×C32×C4C22×D314yes—
190C2×C32×C4C22×D311yescharacteristic
191#638—42no—
192#638—42no—
193#638—122no—
194C2×C3⋊Dic3—31no—
195C2×C3⋊Dic3—31no—
196C2×C3⋊Dic3—32no—
197C2×C3⋊Dic3—34no—
198C2×C3⋊Dic3—38no—
199C2×C3⋊Dic3—34no—
200C2×C3⋊Dic3—64no—
201C2×C3⋊Dic3—64no—
202C2×C3⋊Dic3—68no—
203C2×C3⋊Dic3—122no—
204C2×C3⋊Dic3—128no—
205C2×C3⋊Dic3—1216no—
206C2×C3⋊Dic3C22×D311yescharacteristic
207C2×C3⋊Dic3C22×D311yescharacteristic
208C4⋊(C3⋊D3)—22no—
209C4×C3⋊D3—122no—
210C3×#67—48no—
211C3×#67—128no—
212C3×#67—128no—
213C2×C3×Dic3—34no—
214C2×C3×Dic3—34no—
215C2×C3×Dic3C22×D314yes—
216C2×C3×Dic3C22×D318yes—
217C3×D12—68no—
218C3×C4×D3—28no—
219#625—128no—
220D3×Dic3—128no—
221#540—271noSylow
222C2×C33—42no—
223C2×C33C2×#4111yescharacteristic
224C2×C33H(5,𝔽2)11yescharacteristic
225C2×C33H(5,𝔽2)11yescharacteristic
226C3×C3⋊D3—42no—
227C32×D3—41no—
228C2×C3×D4—24no—
229C2×C3×D4—38no—
230C2×C3×D4—92no—
231C2×C3×D4—92no—
232C2×C3×D4—91no—
233C2×C3×D4D3214yes—
234C2×C3×D4C2×C3⋊D311yescharacteristic
235C22×C3×C4—34no—
236C2×#67—38no—
237C2×#67—38no—
238C2×#67—98no—
239C2×#67—92no—
240C2×#67C2×C3⋊D312yes—
241C22×Dic3—98no—
242C22×Dic3—92no—
243C22×Dic3—92no—
244C22×Dic3—188no—
245C2×D12—34no—
246C2×C4×D3—98no—
247C2×C4×D3C2×C3⋊D311yescharacteristic
248C3×C4⋊C4—34no—
249C3×C22⋊C4—38no—
250C3×C42—91no—
251#217—94no—
252#217—94no—
253#217—98no—
254#217—92no—
255#217—92no—
256#217—92no—
257#217—188no—
258#217C2×C3⋊D312yes—
259(C2×D3)⋊C4—98no—
260(C2×D3)⋊C4—98no—
261(C2×D3)⋊C4—91no—
262(C2×D3)⋊C4—91no—
263C4⋊Dic3—94no—
264C4⋊Dic3—94no—
265C4⋊Dic3—98no—
266C4⋊Dic3—94no—
267C4⋊Dic3—91no—
268C4⋊Dic3—184no—
269C4⋊Dic3C2×C3⋊D311yescharacteristic
270Dic3⋊C4—98no—
271Dic3⋊C4—92no—
272Dic3⋊C4—92no—
273Dic3⋊C4—91no—
274Dic3⋊C4—91no—
275C4×Dic3—34no—
276C4×Dic3—94no—
277C4×Dic3—91no—
278C4×Dic3—184no—
279C22×C32—216no—
280C22×C32—28no—
281C22×C32—22no—
282C22×C32—22no—
283C22×C32—22no—
284C22×C32—24no—
285C22×C32—28no—
286C22×C32—28no—
287C22×C32—48no—
288C22×C32—48no—
289C22×C32—61no—
290C22×C32—61no—
291C22×C32—61no—
292C22×C32C23×D314yes—
293C22×C32C23×D311yescharacteristic
294C2×C3⋊D3—22no—
295C2×C3⋊D3—22no—
296C2×C3⋊D3—22no—
297C2×C3×D3—24no—
298C2×C3×D3—24no—
299C2×C3×D3—24no—
300C2×C3×D3—68no—
301C2×C3×D3—68no—
302C2×C3×D3—68no—
303C32×C4—28no—
304C32×C4—61no—
305C32×C4—62no—
306C32×C4#23718yes—
307C32×C4#23712yes—
308C3⋊Dic3—21no—
309C3⋊Dic3—21no—
310C3⋊Dic3—68no—
311C3⋊Dic3—61no—
312C3⋊Dic3—61no—
313C3⋊Dic3—62no—
314C3⋊Dic3—64no—
315C3⋊Dic3—64no—
316C3⋊Dic3—68no—
317C3⋊Dic3—124no—
318C3⋊Dic3—128no—
319C3×Dic3—24no—
320C3×Dic3—28no—
321C3×Dic3—64no—
322C3×Dic3—64no—
323#127—91no—
324#125—271no—
325#123—272no—
326#123—272no—
327#121—31no—
328#121—271no—
329#121—272no—
330#121—272no—
331C4×D4—271no—
332C4×D4—271no—
333#117—271no—
334C33#54011yescharacteristic, complemented, Sylow
335C23×C3—28no—
336C23×C3—34no—
337C23×C3—92no—
338C23×C3C2×D3218yes—
339C23×C3C22×C3⋊D312yes—
340C22×D3—38no—
341C22×D3C22×C3⋊D311yescharacteristic
342C3×D4—216no—
343C3×D4—28no—
344C3×D4—22no—
345C3×D4—128no—
346C3×D4—182no—
347C3×D4—362no—
348C3×D4—362no—
349C2×C3×C4—24no—
350C2×C3×C4—38no—
351C2×C3×C4—34no—
352C2×C3×C4—68no—
353C2×C3×C4—91no—
354C2×C3×C4—91no—
355C2×C3×C4C2×D3214yes—
356C2×C3×C4C22×C3⋊D311yescharacteristic
357#67—42no—
358#67—128no—
359#67—128no—
360#67—362no—
361#67—368no—
362C2×Dic3—34no—
363C2×Dic3—34no—
364C2×Dic3—94no—
365C2×Dic3—94no—
366C2×Dic3—98no—
367C2×Dic3—91no—
368C2×Dic3—91no—
369C2×Dic3—92no—
370C2×Dic3—184no—
371C2×Dic3—184no—
372C2×Dic3—188no—
373C2×Dic3—362no—
374C2×Dic3—368no—
375C2×Dic3—362no—
376C2×Dic3—3616no—
377C2×Dic3C22×C3⋊D312yes—
378C2×Dic3C22×C3⋊D311yescharacteristic
379D12—68no—
380C4×D3—22no—
381C4×D3—368no—
382C2×C32—24no—
383C2×C32—24no—
384C2×C32—24no—
385C2×C32—416no—
386C2×C32—42no—
387C2×C32—48no—
388C2×C32—121no—
389C2×C32C2×#23714yes—
390C2×C32C2×#23711yescharacteristic
391C2×C32#100414yes—
392C2×C32#100411yescharacteristic
393C2×C32#100411yescharacteristic
394C2×C32#100914yes—
395C3⋊D3—42no—
396C3×D3—44no—
397C3×D3—128no—
398C2×D4—32no—
399C2×D4—92no—
400C2×D4—92no—
401C2×D4—91no—
402C2×D4—272no—
403C2×D4D3×C3⋊D311yescharacteristic
404C22×C4—31no—
405C22×C4—272no—
406C22×C4—272no—
407C4⋊C4—31no—
408C4⋊C4—271no—
409C4⋊C4—271no—
410C4⋊C4—272no—
411C4⋊C4—272no—
412C4⋊C4—271no—
413C22⋊C4—32no—
414C22⋊C4—272no—
415C22⋊C4—272no—
416C22⋊C4—271no—
417C22⋊C4—271no—
418C22⋊C4—272no—
419C42—91no—
420C42—271no—
421C22×C3—216no—
422C22×C3—28no—
423C22×C3—28no—
424C22×C3—28no—
425C22×C3—22no—
426C22×C3—22no—
427C22×C3—22no—
428C22×C3—24no—
429C22×C3—48no—
430C22×C3—48no—
431C22×C3—64no—
432C22×C3—64no—
433C22×C3—64no—
434C22×C3—182no—
435C22×C3—182no—
436C22×C3—182no—
437C22×C3C22×D3214yes—
438C22×C3C23×C3⋊D311yescharacteristic
439C2×D3—21no—
440C2×D3—21no—
441C2×D3—21no—
442C2×D3—68no—
443C2×D3—68no—
444C2×D3—68no—
445C3×C4—28no—
446C3×C4—68no—
447C3×C4—64no—
448C3×C4—181no—
449C3×C4—181no—
450C3×C4#380518yes—
451C3×C4#383012yes—
452Dic3—21no—
453Dic3—22no—
454Dic3—64no—
455Dic3—64no—
456Dic3—188no—
457Dic3—184no—
458Dic3—184no—
459Dic3—188no—
460Dic3—181no—
461Dic3—181no—
462Dic3—182no—
463Dic3—368no—
464Dic3—364no—
465C32—24no—
466C32#561914yescomplemented
467C32#571911yescharacteristic, complemented
468C23—31no—
469C23—92no—
470C23C2×D3×C3⋊D312yes—
471D4—22no—
472D4—122no—
473D4—182no—
474D4—362no—
475D4—362no—
476D4—1082no—
477C2×C4—31no—
478C2×C4—32no—
479C2×C4—62no—
480C2×C4—91no—
481C2×C4—91no—
482C2×C4—271no—
483C2×C4—271no—
484C2×C4—272no—
485C2×C4—1082no—
486C2×C4—1082no—
487C2×C4C2×D3×C3⋊D311yescharacteristic
488C2×C3—24no—
489C2×C3—24no—
490C2×C3—24no—
491C2×C3—416no—
492C2×C3—48no—
493C2×C3—42no—
494C2×C3—124no—
495C2×C3—362no—
496C2×C3#6431314yes—
497C2×C3#6431414yes—
498C2×C3C2×#380514yes—
499C2×C3C2×#383011yescharacteristic
500C2×C3#6436111yescharacteristic
501C2×C3#6436611yescharacteristic
502D3—41no—
503D3—128no—
504C22—22no—
505C22—22no—
506C22—22no—
507C22—61no—
508C22—61no—
509C22—61no—
510C22—182no—
511C22—182no—
512C22—182no—
513C22C22×D3×C3⋊D311yescharacteristic
514C4—62no—
515C4—61no—
516C4—181no—
517C4—181no—
518C4—541no—
519C4—541no—
520C4—542no—
521C4#8902912yes—
522C3—24no—
523C3#1059521714yescomplemented
524C3#1059609011yescharacteristic, complemented
525C2—42no—
526C2—121no—
527C2—362no—
528C2#1172458711yescharacteristic
529C2#1172458911yescharacteristic
530C2C2×#8902911yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutator, socle, cosocleC22×C33C24
FittingC2×C33×D4C22
hypercenterC2×D4D3×C3⋊D3
hypocenterC33#540
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. D4×C3⋊(C3⋊Dic3)C2semidirect
2. C3×#64149C2semidirect
3. C32×#938C2semidirect
4. #11723637C2semidirect
5. #11723631C2semidirect
6. #11723621C2semidirect
7. #11723609C2semidirect
8. #11723515C2semidirect
9. #11723501C2semidirect
10. #11723447C2semidirect
11. #11723423C2semidirect
38. #64149D3semidirect
51. C3×#938D3semidirect
159. #938C3⋊D3semidirect
334. C33#540semidirect
466. C32#5619semidirect
467. C32#5719semidirect
523. C3#10595217semidirect
524. C3#10596090semidirect

Maximal subgroups

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

subgroupindex
D4×C3⋊(C3⋊Dic3)2
C3×#641492
C32×#9382
#117236372
#117236312
#117236212
#117236092
#117235152
#117235012
#117234472
#117234232
#105960903
#105952173