Finite Group Databasebeta

← #13161995

Subgroups — #13161995

Order 1728.

650 conjugacy classes of subgroups.

The subgroup lattice has 524 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×Q8×C3⋊Dic3C211yesmaximal, characteristic
2C2×C3×#3756C211yesmaximal, characteristic
3#11723668C211yesmaximal, characteristic
4C3×#63662C211yesmaximal, characteristic
5#11722080C211yesmaximal, characteristic
6#11722076C211yesmaximal, characteristic
7#11722070C211yesmaximal, characteristic
8#10591581—31nomaximal
9Dic3×#216—31nomaximal
10#10591537—31nomaximal
11C2×C33×Q8C2211yescharacteristic
12C2×C3×#634C2211yescharacteristic
13C3×#3756C414yescomplemented
14C3×C4⋊(C3⋊Dic3)—21no—
15C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
16C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
17C3×C4×C3⋊Dic3—21no—
18C3×C4×C3⋊Dic3C2211yescharacteristic
19C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
20C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
21#88815—21no—
22#88813—21no—
23Q8×C3⋊Dic3D311yescharacteristic
24C2×#3756D311yescharacteristic
25C3×Q8×Dic3—31no—
26C3×Q8×Dic3—31no—
27C3×Q8×Dic3—61no—
28C2×C3×#216—31no—
29C2×C3×#216—31no—
30C2×C3×#216—61no—
31#63844—31no—
32#63843—31no—
33Dic3×Dic6—31no—
34#63662D311yescharacteristic
35C3×#835—31no—
36C3×#835—31no—
37C3×#835—61no—
38#63576—31no—
39#63575—31no—
40#63575—31no—
41#63567—31no—
42#63566—31no—
43#63565—31no—
44#63553—31no—
45Dic3×C3⋊C8—31no—
46Dic3×C3⋊C8—31no—
47C33×Q8—21no—
48C33×Q8C2×C412yes—
49C2×C33×C4—21no—
50C2×C33×C4C2311yescharacteristic
51C2×C3×C3⋊Dic3—21no—
52C2×C3×C3⋊Dic3—21no—
53C2×C3×C3⋊Dic3—21no—
54C2×C3×C3⋊Dic3D411yescharacteristic
55C2×C3×C3⋊Dic3D411yescharacteristic
56C3×#634—21no—
57C3×#634C2×C412yes—
58C3×C3⋊(C3⋊C8)—21no—
59C3×C3⋊(C3⋊C8)C2×C412yes—
60Dic3×Q16—91no—
61C4×#216—91no—
62#4790—91no—
63C2×C32×Q8—21no—
64C2×C32×Q8—21no—
65C2×C32×Q8—21no—
66C2×C32×Q8—41no—
67C2×C32×Q8C2×D311yescharacteristic
68C2×C32×Q8C2×D311yescharacteristic
69C2×C32×Q8C2×D311yescharacteristic
70C2×#634C2×D311yescharacteristic
71C2×C3×Dic6—31no—
72C2×C3×Dic6—31no—
73C2×C3×Dic6—61no—
74#3756Dic314yescomplemented
75C4⋊(C3⋊Dic3)—21no—
76C4⋊(C3⋊Dic3)C2×D311yescharacteristic
77C4⋊(C3⋊Dic3)C2×D311yescharacteristic
78C4×C3⋊Dic3—21no—
79C4×C3⋊Dic3C2×D311yescharacteristic
80C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
81C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
82C3×#216—34no—
83C3×#216—34no—
84C3×#216—64no—
85C3×C4⋊Dic3—31no—
86C3×C4⋊Dic3—31no—
87C3×C4⋊Dic3—31no—
88C3×C4⋊Dic3—31no—
89C3×C4⋊Dic3—61no—
90C3×C4⋊Dic3—61no—
91C3×C4⋊Dic3—61no—
92C3×C4⋊Dic3—62no—
93C3×C4⋊Dic3—61no—
94C3×C4×Dic3—31no—
95C3×C4×Dic3—31no—
96C3×C4×Dic3—61no—
97C3×C4×Dic3—61no—
98C3×C4×Dic3—61no—
99C3×C4×Dic3—62no—
100C2×C3×C3⋊C8—31no—
101C2×C3×C3⋊C8—31no—
102C2×C3×C3⋊C8—31no—
103C2×C3×C3⋊C8—31no—
104C2×C3×C3⋊C8—61no—
105C2×C3×C3⋊C8—61no—
106#3724—61no—
107Dic3⋊Dic3—61no—
108Dic3⋊Dic3—61no—
109Dic32—61no—
110Dic32—61no—
111Dic32—61no—
112C22×C33C2×D411yescharacteristic
113C33×C4—22no—
114C33×C4C22×C411yescharacteristic
115C33×C4#4111yescharacteristic
116C3×C3⋊Dic3—22no—
117C3×C3⋊Dic3—21no—
118C3×C3⋊Dic3—41no—
119C3×C3⋊Dic3—41no—
120C3×C3⋊Dic3Q1612yescomplemented
121C2×C3×Q16—91no—
122C3×C4×Q8—91no—
123Q8×Dic3—31no—
124Q8×Dic3—31no—
125Q8×Dic3—61no—
126Q8×Dic3—91no—
127C2×#216—31no—
128C2×#216—31no—
129C2×#216—61no—
130#887—91no—
131C4×Dic6—91no—
132C3×#103—91no—
133#835—31no—
134#835—31no—
135#835—61no—
136#835—91no—
137#818—91no—
138#816—91no—
139C8×Dic3—91no—
140C8×Dic3—91no—
141#810—91no—
142#807—91no—
143#807—91no—
144C4×C3⋊C8—91no—
145C32×Q8—21no—
146C32×Q8—22no—
147C32×Q8—21no—
148C32×Q8—21no—
149C32×Q8—22no—
150C32×Q8—22no—
151C32×Q8—22no—
152C32×Q8—42no—
153C32×Q8—41no—
154C32×Q8—41no—
155C32×Q8—42no—
156C32×Q8C4×D312yes—
157C32×Q8C4×D312yes—
158C32×Q8C2×Dic312yes—
159C2×C32×C4—21no—
160C2×C32×C4—22no—
161C2×C32×C4—21no—
162C2×C32×C4—21no—
163C2×C32×C4—21no—
164C2×C32×C4—21no—
165C2×C32×C4—21no—
166C2×C32×C4—41no—
167C2×C32×C4—41no—
168C2×C32×C4—41no—
169C2×C32×C4—42no—
170C2×C32×C4C22×D311yescharacteristic
171C2×C32×C4C22×D311yescharacteristic
172C2×C32×C4C22×D311yescharacteristic
173C2×C3⋊Dic3—21no—
174C2×C3⋊Dic3—21no—
175C2×C3⋊Dic3—21no—
176C2×C3⋊Dic3#6711yescharacteristic
177C2×C3⋊Dic3#6711yescharacteristic
178#634—21no—
179#634C2×Dic312yes—
180C2×C3×Dic3—31no—
181C2×C3×Dic3—31no—
182C2×C3×Dic3—31no—
183C2×C3×Dic3—31no—
184C2×C3×Dic3—61no—
185C2×C3×Dic3—61no—
186C2×C3×Dic3—61no—
187C2×C3×Dic3—61no—
188C2×C3×Dic3—62no—
189C2×C3×Dic3—61no—
190C2×C3×Dic3—61no—
191C2×C3×Dic3—61no—
192C2×C3×Dic3—61no—
193C2×C3×Dic3—121no—
194C2×C3×Dic3—121no—
195C3×Dic6—32no—
196C3×Dic6—32no—
197C3×Dic6—62no—
198C3×Dic6—61no—
199C3×Dic6—61no—
200C3×Dic6—121no—
201C3⋊(C3⋊C8)—21no—
202C3⋊(C3⋊C8)C2×Dic312yes—
203C3×C3⋊C8—32no—
204C3×C3⋊C8—32no—
205C3×C3⋊C8—62no—
206C3×C3⋊C8—61no—
207C3×C3⋊C8—61no—
208C3×C3⋊C8—121no—
209C4×Q16—271noSylow
210C2×C33C4×D411yescharacteristic
211C2×C33C2×Q1611yescharacteristic
212C2×C33#13511yescharacteristic
213C2×C3×Q8—21no—
214C2×C3×Q8—21no—
215C2×C3×Q8—21no—
216C2×C3×Q8—41no—
217C2×C3×Q8—91no—
218C2×C3×Q8D3211yescharacteristic
219C2×C3×Q8D3211yescharacteristic
220C2×C3×Q8D3211yescharacteristic
221C2×Dic6—31no—
222C2×Dic6—31no—
223C2×Dic6—61no—
224C3×Q16—94no—
225C2×C3×C8—91no—
226C2×C3×C8—91no—
227C3×C4⋊C4—91no—
228C3×C4⋊C4—91no—
229C3×C4⋊C4—181no—
230C3×C42—91no—
231C3×C42—181no—
232#216—34no—
233#216—34no—
234#216—64no—
235C4⋊Dic3—31no—
236C4⋊Dic3—31no—
237C4⋊Dic3—31no—
238C4⋊Dic3—31no—
239C4⋊Dic3—61no—
240C4⋊Dic3—61no—
241C4⋊Dic3—61no—
242C4⋊Dic3—61no—
243C4⋊Dic3—62no—
244C4⋊Dic3—181no—
245Dic3⋊C4—181no—
246Dic3⋊C4—181no—
247C4×Dic3—31no—
248C4×Dic3—31no—
249C4×Dic3—61no—
250C4×Dic3—61no—
251C4×Dic3—61no—
252C4×Dic3—62no—
253C4×Dic3—181no—
254C4×Dic3—181no—
255C4×Dic3—181no—
256C2×C3⋊C8—31no—
257C2×C3⋊C8—31no—
258C2×C3⋊C8—31no—
259C2×C3⋊C8—31no—
260C2×C3⋊C8—61no—
261C2×C3⋊C8—61no—
262C22×C32—21no—
263C22×C32—21no—
264C22×C32—21no—
265C22×C32—41no—
266C22×C32D3×D411yescharacteristic
267C22×C32D3×D411yescharacteristic
268C22×C32C2×#6711yescharacteristic
269C32×C4—22no—
270C32×C4—24no—
271C32×C4—22no—
272C32×C4—22no—
273C32×C4—21no—
274C32×C4—21no—
275C32×C4—21no—
276C32×C4—21no—
277C32×C4—21no—
278C32×C4—21no—
279C32×C4—41no—
280C32×C4—41no—
281C32×C4—42no—
282C32×C4—42no—
283C32×C4—44no—
284C32×C4C2×C4×D311yescharacteristic
285C32×C4C2×C4×D311yescharacteristic
286C32×C4#23511yescharacteristic
287C32×C4#23711yescharacteristic
288C32×C4#23711yescharacteristic
289C32×C4C22×Dic311yescharacteristic
290C3⋊Dic3—22no—
291C3⋊Dic3—21no—
292C3⋊Dic3—41no—
293C3⋊Dic3—41no—
294C3⋊Dic3#21612yescomplemented
295C3×Dic3—32no—
296C3×Dic3—32no—
297C3×Dic3—62no—
298C3×Dic3—62no—
299C3×Dic3—61no—
300C3×Dic3—61no—
301C3×Dic3—62no—
302C3×Dic3—122no—
303C3×Dic3—121no—
304C3×Dic3—122no—
305C3×Dic3—121no—
306C3×Dic3—121no—
307C3×Dic3—121no—
308C3×Dic3—121no—
309C3×Dic3—241no—
310C2×Q16—91no—
311C4×Q8—91no—
312C4×Q8—271no—
313#107—271no—
314#103—91no—
315#103—271no—
316C4×C8—271no—
317C33C4×Q1611yescharacteristic, complemented, Sylow
318C3×Q8—22no—
319C3×Q8—21no—
320C3×Q8—21no—
321C3×Q8—21no—
322C3×Q8—22no—
323C3×Q8—22no—
324C3×Q8—22no—
325C3×Q8—41no—
326C3×Q8—41no—
327C3×Q8—42no—
328C3×Q8—42no—
329C3×Q8—92no—
330C3×Q8—181no—
331C3×Q8D3×Dic312yes—
332C3×Q8D3×Dic312yes—
333C3×Q8#62412yes—
334C2×C3×C4—22no—
335C2×C3×C4—21no—
336C2×C3×C4—21no—
337C2×C3×C4—21no—
338C2×C3×C4—21no—
339C2×C3×C4—21no—
340C2×C3×C4—21no—
341C2×C3×C4—41no—
342C2×C3×C4—41no—
343C2×C3×C4—42no—
344C2×C3×C4—41no—
345C2×C3×C4—91no—
346C2×C3×C4—91no—
347C2×C3×C4—181no—
348C2×C3×C4—181no—
349C2×C3×C4—181no—
350C2×C3×C4C2×D3211yescharacteristic
351C2×C3×C4C2×D3211yescharacteristic
352C2×C3×C4C2×D3211yescharacteristic
353C2×Dic3—31no—
354C2×Dic3—31no—
355C2×Dic3—31no—
356C2×Dic3—31no—
357C2×Dic3—61no—
358C2×Dic3—61no—
359C2×Dic3—61no—
360C2×Dic3—61no—
361C2×Dic3—62no—
362C2×Dic3—61no—
363C2×Dic3—61no—
364C2×Dic3—61no—
365C2×Dic3—61no—
366C2×Dic3—121no—
367C2×Dic3—121no—
368Dic6—32no—
369Dic6—32no—
370Dic6—62no—
371Dic6—61no—
372Dic6—61no—
373Dic6—121no—
374C3×C8—92no—
375C3×C8—181no—
376C3⋊C8—32no—
377C3⋊C8—32no—
378C3⋊C8—62no—
379C3⋊C8—61no—
380C3⋊C8—61no—
381C3⋊C8—121no—
382C2×C32—21no—
383C2×C32—21no—
384C2×C32—21no—
385C2×C32—21no—
386C2×C32—21no—
387C2×C32—21no—
388C2×C32—21no—
389C2×C32—21no—
390C2×C32—21no—
391C2×C32—41no—
392C2×C32—41no—
393C2×C32—41no—
394C2×C32#67⋊C411yescharacteristic
395C2×C32#91611yescharacteristic
396C2×C32D3×Q1611yescharacteristic
397C2×C32D3×Q1611yescharacteristic
398C2×C32#91911yescharacteristic
399C2×C32C4×#6711yescharacteristic
400C2×C32D4×Dic311yescharacteristic
401C2×C32C2×#21611yescharacteristic
402C2×C32#95011yescharacteristic
403C2×Q8—91no—
404C2×Q8#111311yescharacteristic
405Q16—94no—
406C2×C8—91no—
407C2×C8—91no—
408C4⋊C4—91no—
409C4⋊C4—91no—
410C4⋊C4—181no—
411C4⋊C4—541no—
412C42—91no—
413C42—181no—
414C42—541no—
415C22×C3—21no—
416C22×C3—21no—
417C22×C3—21no—
418C22×C3—41no—
419C22×C3D3×#6711yescharacteristic
420C22×C3D3×#6711yescharacteristic
421C22×C3#381111yescharacteristic
422C3×C4—24no—
423C3×C4—22no—
424C3×C4—22no—
425C3×C4—22no—
426C3×C4—21no—
427C3×C4—21no—
428C3×C4—21no—
429C3×C4—21no—
430C3×C4—21no—
431C3×C4—21no—
432C3×C4—42no—
433C3×C4—42no—
434C3×C4—44no—
435C3×C4—41no—
436C3×C4—41no—
437C3×C4—92no—
438C3×C4—181no—
439C3×C4—182no—
440C3×C4—361no—
441C3×C4—361no—
442C3×C4C2×D3×Dic311yescharacteristic
443C3×C4C2×D3×Dic311yescharacteristic
444C3×C4#380411yescharacteristic
445C3×C4#380411yescharacteristic
446C3×C4#380511yescharacteristic
447C3×C4C2×#62411yescharacteristic
448Dic3—32no—
449Dic3—32no—
450Dic3—62no—
451Dic3—61no—
452Dic3—61no—
453Dic3—62no—
454Dic3—62no—
455Dic3—122no—
456Dic3—121no—
457Dic3—122no—
458Dic3—121no—
459Dic3—121no—
460Dic3—121no—
461Dic3—121no—
462Dic3—241no—
463C32—21no—
464C32—21no—
465C32—21no—
466C32—41no—
467C32#479011yescharacteristic, complemented
468C32C4×#21611yescharacteristic, complemented
469C32Dic3×Q1611yescharacteristic, complemented
470Q8—21no—
471Q8—92no—
472Q8—181no—
473Q8#629412yes—
474C2×C4—21no—
475C2×C4—91no—
476C2×C4—91no—
477C2×C4—181no—
478C2×C4—181no—
479C2×C4—181no—
480C2×C4C2×#111311yescharacteristic
481C8—92no—
482C8—181no—
483C2×C3—21no—
484C2×C3—21no—
485C2×C3—21no—
486C2×C3—21no—
487C2×C3—21no—
488C2×C3—21no—
489C2×C3—21no—
490C2×C3—21no—
491C2×C3—21no—
492C2×C3—41no—
493C2×C3—41no—
494C2×C3—41no—
495C2×C3D3×#21611yescharacteristic
496C2×C3D3×#21611yescharacteristic
497C2×C3#6394411yescharacteristic
498C2×C3#6394711yescharacteristic
499C2×C3#6394911yescharacteristic
500C2×C3#6395011yescharacteristic
501C2×C3#6395211yescharacteristic
502C2×C3Dic3×#6711yescharacteristic
503C2×C3#67⋊Dic311yescharacteristic
504C22#8904911yescharacteristic
505C4—22no—
506C4—92no—
507C4—181no—
508C4—182no—
509C4—361no—
510C4—361no—
511C4C2×#629411yescharacteristic
512C4#8904611yescharacteristic
513C3—21no—
514C3—21no—
515C3—21no—
516C3—41no—
517C3#1059153711yescharacteristic, complemented
518C3Dic3×#21611yescharacteristic, complemented
519C3#1059158111yescharacteristic, complemented
520C2#1172371011yescharacteristic
521C2#1172371311yescharacteristic
522C2#1172372611yescharacteristic
subgroupG/H
centerC22#89049
commutatorC33×C4C22×C4
FrattiniC2×C4C2×#1113
FittingC2×C33×Q8C22
socle, prime radicalC22×C33C2×D4
cosocleC2×C33×C4C23
hypercenterC2×Q8#1113
hypocenterC33C4×Q16
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
13. C3×#3756C4semidirect
74. #3756Dic3semidirect
120. C3×C3⋊Dic3Q16semidirect
294. C3⋊Dic3#216semidirect
317. C33C4×Q16semidirect
467. C32#4790semidirect
468. C32C4×#216semidirect
469. C32Dic3×Q16semidirect
517. C3#10591537semidirect
518. C3Dic3×#216semidirect
519. C3#10591581semidirect

Maximal subgroups

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

subgroupindex
C3×Q8×C3⋊Dic32
C2×C3×#37562
#117236682
C3×#636622
#117220802
#117220762
#117220702
#105915813
Dic3×#2163
#105915373