Finite Group Databasebeta

← #13177231

Subgroups — #13177231

Order 1728.

1796 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#11723965C211yesmaximal, characteristic, complemented
2C2×C3×#3752C211yesmaximal, characteristic, complemented
3C32×#930C211yesmaximal, characteristic, complemented
4#11723634C212yesmaximal, complemented
5#67×C3⋊Dic3C212yesmaximal, complemented
6#11723528C212yesmaximal, complemented
7#11723480C212yesmaximal, complemented
8#11723474C212yesmaximal, complemented
9#11723407C212yesmaximal, complemented
10#10596088—31nomaximal
11#10594977—34nomaximal
12C23×C3×C3⋊D3C2211yescharacteristic
13C22×C33×C4C2211yescharacteristic
14C2×#6308C2211yescharacteristic, complemented
15C22×C3×C3⋊Dic3C2211yescharacteristic
16C2×C32×#67C2212yes—
17C2×C32×D12C2211yescharacteristic, complemented
18C2×#6290C2212yes—
19C2×#6290C2212yes—
20C2×D3×C3⋊Dic3C2212yescomplemented
21#88864C2212yes—
22C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic, complemented
23#88857C2212yescomplemented
24C3×#3752C2212yes—
25C3×#3752C2212yes—
26C32×(C2×D3)⋊C4C2212yescomplemented
27C32×C4⋊Dic3C2211yescharacteristic, complemented
28Dic3⋊(C3⋊Dic3)C2212yescomplemented
29#88803C2212yes—
30#88803C2212yes—
31(C2×D3)⋊(C3⋊Dic3)C2212yescomplemented
32Dic3×C3⋊Dic3C2212yescomplemented
33C32×#121—31no—
34#64148—32no—
35#64146—32no—
36D4×C3⋊Dic3—32no—
37C2×#3752D311yescharacteristic, complemented
38#64133—31no—
39#64133—34no—
40#64133—64no—
41#64107—32no—
42#64095—32no—
43#64094—32no—
44C3×#930D314yescomplemented
45C2×C3×(C2×D3)⋊C4—34no—
46#63976—38no—
47Dic3×#67—38no—
48#63913—38no—
49#63891—38no—
50#63885—38no—
51#63858—38no—
52C23×C33C2311yescharacteristic
53C22×C3×C3⋊D3—22no—
54C22×C3×C3⋊D3—41no—
55C22×C3×C3⋊D3—41no—
56C22×C3×C3⋊D3C2312yes—
57C22×C32×D3C2312yes—
58C2×C33×C4—41no—
59C2×C33×C4C2311yescharacteristic
60C2×C33×C4C2311yescharacteristic
61C2×C3⋊(C3⋊Dic3)C2312yes—
62C2×C3⋊(C3⋊Dic3)C2312yes—
63#6308—41no—
64C2×C3×C3⋊Dic3—41no—
65C2×C3×C3⋊Dic3C2312yes—
66C32×#67—24no—
67C2×C32×Dic3C2312yes—
68C32×D12—41no—
69#6290—42no—
70#6290—42no—
71D3×C3⋊Dic3—42no—
72#5617—94no—
73#5494—91no—
74C23×C3⋊D3C2×D311yescharacteristic
75C23×C3×D3—34no—
76C2×C32×D4—32no—
77C2×C32×D4—31no—
78C22×C32×C4—24no—
79C22×C32×C4C2×D314yes—
80C22×C32×C4C2×D311yescharacteristic
81C2×#638—32no—
82C2×#638—32no—
83C22×C3⋊Dic3—32no—
84C22×C3⋊Dic3C2×D311yescharacteristic
85C2×#634—31no—
86C2×#634—34no—
87C2×#634—64no—
88C2×C3×#67C2×D318yes—
89C22×C3×Dic3—34no—
90C2×C3×D12C2×D314yescomplemented
91C2×#625—38no—
92C2×#625—38no—
93C2×D3×Dic3—38no—
94C32×C4⋊C4—31no—
95C32×C22⋊C4—32no—
96#3757—32no—
97#3757—38no—
98#3757—32no—
99#3757—68no—
100#3752—32no—
101#3752—32no—
102#3752C2×D312yes—
103#3752C2×D312yes—
104C4⋊(C3⋊Dic3)—31no—
105C4⋊(C3⋊Dic3)—34no—
106C4⋊(C3⋊Dic3)—32no—
107C4⋊(C3⋊Dic3)—64no—
108#3750—32no—
109#3750—38no—
110#3750—68no—
111C4×C3⋊Dic3—32no—
112C3×(C2×D3)⋊C4—38no—
113C3×(C2×D3)⋊C4—38no—
114C3×(C2×D3)⋊C4C2×D318yescomplemented
115C3×C4⋊Dic3C2×D314yescomplemented
116Dic3⋊Dic3—38no—
117(C2×D3)⋊Dic3—38no—
118(C2×D3)⋊Dic3—38no—
119(C2×D3)⋊Dic3—38no—
120Dic32—38no—
121C22×C33—21no—
122C22×C33—21no—
123C22×C33#4112yes—
124C22×C33C2411yescharacteristic
125C2×C3×C3⋊D3—22no—
126C2×C3×C3⋊D3—22no—
127C2×C3×C3⋊D3—22no—
128C2×C3×C3⋊D3—44no—
129C2×C32×D3—22no—
130C2×C32×D3—22no—
131C2×C32×D3—22no—
132C33×C4—21no—
133C33×C4—21no—
134C3⋊(C3⋊Dic3)—22no—
135C3⋊(C3⋊Dic3)—22no—
136C3×C3⋊Dic3—22no—
137C32×Dic3—22no—
138C3×#121—34no—
139C2×C3×C22⋊C4—91no—
140#953—92no—
141#939—98no—
142#937—98no—
143D4×Dic3—98no—
144#930C3⋊D311yescharacteristic, complemented
145C4×#67—92no—
146C2×(C2×D3)⋊C4—34no—
147#924—94no—
148#924—91no—
149#924—184no—
150#898—98no—
151#886—98no—
152#885—98no—
153#885—92no—
154#884—92no—
155#883—92no—
156#879—92no—
157C23×C32—24no—
158C23×C32—32no—
159C23×C32C22×D314yes—
160C23×C32C22×D311yescharacteristic
161C22×C3⋊D3—22no—
162C22×C3⋊D3—41no—
163C22×C3⋊D3—41no—
164C22×C3⋊D3C22×D312yes—
165C22×C3×D3—38no—
166C22×C3×D3—68no—
167C22×C3×D3—124no—
168C22×C3×D3—124no—
169C22×C3×D3C22×D318yes—
170C32×D4—64no—
171C32×D4—121no—
172C2×C32×C4—24no—
173C2×C32×C4—24no—
174C2×C32×C4—32no—
175C2×C32×C4—41no—
176C2×C32×C4—44no—
177C2×C32×C4—48no—
178C2×C32×C4C22×D314yes—
179C2×C32×C4C22×D314yes—
180C2×C32×C4C22×D311yescharacteristic
181C2×C32×C4C22×D311yescharacteristic
182#638—122no—
183#638—122no—
184C2×C3⋊Dic3—32no—
185C2×C3⋊Dic3—32no—
186C2×C3⋊Dic3—38no—
187C2×C3⋊Dic3—38no—
188C2×C3⋊Dic3—41no—
189C2×C3⋊Dic3—68no—
190C2×C3⋊Dic3—68no—
191C2×C3⋊Dic3—122no—
192C2×C3⋊Dic3C22×D312yes—
193#634—121no—
194#634—124no—
195#634—128no—
196C3×#67—216no—
197C2×C3×Dic3—38no—
198C2×C3×Dic3—124no—
199C2×C3×Dic3C22×D318yes—
200C3×D12—44no—
201#625—128no—
202#625—128no—
203D3×Dic3—128no—
204#538—271noSylow
205C2×C33—22no—
206C2×C33C2×#4111yescharacteristic
207C2×C33H(5,𝔽2)11yescharacteristic
208C2×C33H(5,𝔽2)11yescharacteristic
209C3×C3⋊D3—42no—
210C32×D3—42no—
211C24×C3—91no—
212C23×D3—34no—
213C2×C3×D4—38no—
214C2×C3×D4—34no—
215C22×C3×C4—24no—
216C22×C3×C4—91no—
217C22×C3×C4D3214yes—
218C22×C3×C4C2×C3⋊D311yescharacteristic
219C2×#67—98no—
220C2×#67—98no—
221C2×#67—92no—
222C2×#67—92no—
223C2×#67C2×C3⋊D312yes—
224C22×Dic3—34no—
225C22×Dic3—98no—
226C2×D12C2×C3⋊D311yescharacteristic, complemented
227C2×C4×D3—92no—
228C2×Dic6—94no—
229C2×Dic6—91no—
230C2×Dic6—184no—
231C3×C4⋊C4—34no—
232C3×C22⋊C4—38no—
233C3×C22⋊C4—92no—
234C3×C22⋊C4—92no—
235#217—98no—
236#217—98no—
237#217—92no—
238#217—92no—
239#217—92no—
240#217—188no—
241(C2×D3)⋊C4—38no—
242(C2×D3)⋊C4—38no—
243(C2×D3)⋊C4—98no—
244(C2×D3)⋊C4—98no—
245(C2×D3)⋊C4—92no—
246(C2×D3)⋊C4C2×C3⋊D312yescomplemented
247C4⋊Dic3—98no—
248C4⋊Dic3—94no—
249C4⋊Dic3—91no—
250C4⋊Dic3—184no—
251C4⋊Dic3C2×C3⋊D311yescharacteristic, complemented
252Dic3⋊C4—98no—
253Dic3⋊C4—92no—
254Dic3⋊C4—92no—
255Dic3⋊C4—188no—
256C4×Dic3—98no—
257C4×Dic3—92no—
258C22×C32—21no—
259C22×C32—21no—
260C22×C32—24no—
261C22×C32—28no—
262C22×C32—24no—
263C22×C32—24no—
264C22×C32—44no—
265C22×C32—44no—
266C22×C32—62no—
267C22×C32—62no—
268C22×C32—62no—
269C22×C32#23512yes—
270C22×C32#23718yes—
271C22×C32C23×D314yes—
272C22×C32C23×D311yescharacteristic
273C2×C3⋊D3—22no—
274C2×C3⋊D3—22no—
275C2×C3⋊D3—22no—
276C2×C3⋊D3—44no—
277C2×C3×D3—28no—
278C2×C3×D3—28no—
279C2×C3×D3—28no—
280C2×C3×D3—68no—
281C2×C3×D3—68no—
282C2×C3×D3—68no—
283C2×C3×D3—1216no—
284C32×C4—28no—
285C32×C4—24no—
286C32×C4—21no—
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#126—272no—
301#124—272no—
302#123—272no—
303C22⋊Q8—271no—
304#121—31no—
305#121—272no—
306#120—272no—
307C4×D4—272no—
308C2×C22⋊C4—91no—
309C33#53811yescharacteristic, complemented, Sylow
310C23×C3—24no—
311C23×C3—38no—
312C23×C3—92no—
313C23×C3—182no—
314C23×C3—361no—
315C23×C3—361no—
316C23×C3C2×D3214yes—
317C23×C3C22×C3⋊D311yescharacteristic
318C22×D3—38no—
319C22×D3—68no—
320C22×D3—124no—
321C22×D3—124no—
322C22×D3C22×C3⋊D312yes—
323C3×D4—616no—
324C3×D4—124no—
325C2×C3×C4—24no—
326C2×C3×C4—24no—
327C2×C3×C4—38no—
328C2×C3×C4—44no—
329C2×C3×C4—41no—
330C2×C3×C4—48no—
331C2×C3×C4—92no—
332C2×C3×C4—361no—
333C2×C3×C4C2×D3214yes—
334C2×C3×C4C2×D3214yes—
335C2×C3×C4C22×C3⋊D311yescharacteristic
336C2×C3×C4C22×C3⋊D311yescharacteristic
337#67—24no—
338#67—362no—
339#67—362no—
340#67—368no—
341#67—368no—
342C2×Dic3—38no—
343C2×Dic3—98no—
344C2×Dic3—98no—
345C2×Dic3—92no—
346C2×Dic3—92no—
347C2×Dic3—124no—
348C2×Dic3—188no—
349C2×Dic3—188no—
350C2×Dic3—368no—
351C2×Dic3C22×C3⋊D312yes—
352D12—41no—
353C4×D3—362no—
354Dic6—364no—
355Dic6—361no—
356Dic6—368no—
357C2×C32—22no—
358C2×C32—24no—
359C2×C32—24no—
360C2×C32—24no—
361C2×C32—28no—
362C2×C32—48no—
363C2×C32—122no—
364C2×C32C2×#23511yescharacteristic
365C2×C32C2×#23714yes—
366C2×C32#100414yes—
367C2×C32#100411yescharacteristic
368C2×C32#100411yescharacteristic
369C2×C32#100914yes—
370C3⋊D3—42no—
371C3×D3—48no—
372C3×D3—128no—
373C24—91no—
374C2×Q8—271no—
375C2×D4—32no—
376C2×D4—31no—
377C2×D4—272no—
378C2×D4—272no—
379C22×C4—91no—
380C22×C4—272no—
381C22×C4D3×C3⋊D311yescharacteristic
382C4⋊C4—31no—
383C4⋊C4—272no—
384C4⋊C4—272no—
385C4⋊C4—271no—
386C22⋊C4—32no—
387C22⋊C4—92no—
388C22⋊C4—92no—
389C22⋊C4—272no—
390C22⋊C4—272no—
391C22⋊C4—272no—
392C22⋊C4—272no—
393C42—272no—
394C22×C3—24no—
395C22×C3—24no—
396C22×C3—21no—
397C22×C3—21no—
398C22×C3—24no—
399C22×C3—28no—
400C22×C3—44no—
401C22×C3—44no—
402C22×C3—68no—
403C22×C3—68no—
404C22×C3—68no—
405C22×C3—182no—
406C22×C3—182no—
407C22×C3—182no—
408C22×C3—364no—
409C22×C3#379518yes—
410C22×C3#383012yes—
411C22×C3C22×D3214yes—
412C22×C3C23×C3⋊D311yescharacteristic
413C2×D3—22no—
414C2×D3—22no—
415C2×D3—22no—
416C2×D3—68no—
417C2×D3—68no—
418C2×D3—68no—
419C2×D3—1216no—
420C3×C4—28no—
421C3×C4—24no—
422C3×C4—24no—
423C3×C4—21no—
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#549411yescharacteristic, complemented
438C32#561714yescomplemented
439C23—32no—
440C23—92no—
441C23—182no—
442C23—361no—
443C23—361no—
444C23C2×D3×C3⋊D311yescharacteristic
445Q8—1081no—
446D4—64no—
447D4—121no—
448D4—1082no—
449D4—1082no—
450C2×C4—32no—
451C2×C4—41no—
452C2×C4—92no—
453C2×C4—272no—
454C2×C4—272no—
455C2×C4—361no—
456C2×C4—1082no—
457C2×C4C2×D3×C3⋊D311yescharacteristic
458C2×C4C2×D3×C3⋊D311yescharacteristic
459C2×C3—28no—
460C2×C3—22no—
461C2×C3—24no—
462C2×C3—24no—
463C2×C3—24no—
464C2×C3—48no—
465C2×C3—128no—
466C2×C3—362no—
467C2×C3C2×#379514yes—
468C2×C3#6430714yes—
469C2×C3#6433014yes—
470C2×C3C2×#383011yescharacteristic
471C2×C3#6436111yescharacteristic
472C2×C3#6436611yescharacteristic
473D3—42no—
474D3—128no—
475C22—21no—
476C22—21no—
477C22—62no—
478C22—62no—
479C22—62no—
480C22—182no—
481C22—182no—
482C22—182no—
483C22—364no—
484C22#8901412yes—
485C22C22×D3×C3⋊D311yescharacteristic
486C4—21no—
487C4—21no—
488C4—62no—
489C4—182no—
490C4—542no—
491C4—542no—
492C3—24no—
493C3#1059497714yescomplemented
494C3#1059608811yescharacteristic, complemented
495C2—22no—
496C2—122no—
497C2—362no—
498C2C2×#8901411yescharacteristic
499C2#1172458011yescharacteristic
500C2#1172461111yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutator, socle, cosocleC22×C33C24
FittingC22×C33×C4C22
hypercenterC22×C4D3×C3⋊D3
hypocenterC33#538
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. #11723965C2semidirect
2. C2×C3×#3752C2semidirect
3. C32×#930C2semidirect
4. #11723634C2semidirect
5. #67×C3⋊Dic3C2semidirect
6. #11723528C2semidirect
7. #11723480C2semidirect
8. #11723474C2semidirect
9. #11723407C2semidirect
14. C2×#6308C22semidirect
17. C2×C32×D12C22semidirect
20. C2×D3×C3⋊Dic3C22semidirect
22. C4⋊(C3⋊(C3⋊Dic3))C22semidirect
23. #88857C22semidirect
26. C32×(C2×D3)⋊C4C22semidirect
27. C32×C4⋊Dic3C22semidirect
28. Dic3⋊(C3⋊Dic3)C22semidirect
31. (C2×D3)⋊(C3⋊Dic3)C22semidirect
32. Dic3×C3⋊Dic3C22semidirect
37. C2×#3752D3semidirect
44. C3×#930D3semidirect
90. C2×C3×D12C2×D3semidirect
114. C3×(C2×D3)⋊C4C2×D3semidirect
115. C3×C4⋊Dic3C2×D3semidirect
144. #930C3⋊D3semidirect
226. C2×D12C2×C3⋊D3semidirect
246. (C2×D3)⋊C4C2×C3⋊D3semidirect
251. C4⋊Dic3C2×C3⋊D3semidirect
309. C33#538semidirect
437. C32#5494semidirect
438. C32#5617semidirect
493. C3#10594977semidirect
494. C3#10596088semidirect

Maximal subgroups

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

subgroupindex
#117239652
C2×C3×#37522
C32×#9302
#117236342
#67×C3⋊Dic32
#117235282
#117234802
#117234742
#117234072
#105960883
#105949773