Finite Group Databasebeta

← #13161991

Subgroups — #13161991

Order 1728.

706 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×Q8×C3⋊Dic3C211yesmaximal, characteristic, complemented
2C2×C3×#3755C211yesmaximal, characteristic
3#11723691C211yesmaximal, characteristic
4C3×#63662C211yesmaximal, characteristic, complemented
5#11722079C211yesmaximal, characteristic, complemented
6#11722075C211yesmaximal, characteristic
7#11722070C211yesmaximal, characteristic, complemented
8#10591571—31nomaximal
9Dic3×#215—31nomaximal
10#10591536—31nomaximal
11C2×C33×Q8C2211yescharacteristic
12C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
13C2×#6294—21no—
14C3×#3755C414yescomplemented
15C3×C4⋊(C3⋊Dic3)—21no—
16C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
17C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
18C3×C4×C3⋊Dic3—21no—
19C3×C4×C3⋊Dic3C2211yescharacteristic
20C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
21C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
22#88814—21no—
23Q8×C3⋊Dic3D311yescharacteristic, complemented
24C2×#3755D311yescharacteristic
25C3×Q8×Dic3—31no—
26C3×Q8×Dic3—31no—
27C3×Q8×Dic3—61no—
28C2×C3×#215—31no—
29C2×C3×#215—31no—
30C2×C3×#215—61no—
31#63898—31no—
32#63894—31no—
33Dic3×D12—31no—
34#63662D311yescharacteristic, complemented
35C3×#835—31no—
36C3×#835—31no—
37C3×#835—61no—
38#63574—31no—
39#63573—31no—
40#63573—31no—
41#63564—31no—
42#63563—31no—
43#63562—31no—
44#63553—31no—
45Dic3×C3⋊C8—31no—
46Dic3×C3⋊C8—31no—
47C22×C3×C3⋊D3—21no—
48C33×Q8—21no—
49C33×Q8C2×C412yescomplemented
50C2×C33×C4—21no—
51C2×C33×C4C2311yescharacteristic
52C2×C3×C3⋊Dic3—21no—
53C2×C3×C3⋊Dic3—21no—
54C2×C3×C3⋊Dic3D411yescharacteristic
55C2×C3×C3⋊Dic3D411yescharacteristic
56C3×C4⋊(C3⋊D3)—21no—
57C3×C4⋊(C3⋊D3)C2×C412yes—
58#6294—42no—
59C3×C3⋊(C3⋊C8)—21no—
60C3×C3⋊(C3⋊C8)C2×C412yescomplemented
61Dic3×SD16—91no—
62C4×#215—91no—
63#4788—91no—
64C2×C32×Q8—21no—
65C2×C32×Q8—21no—
66C2×C32×Q8—21no—
67C2×C32×Q8—41no—
68C2×C32×Q8C2×D311yescharacteristic
69C2×C32×Q8C2×D311yescharacteristic
70C2×C32×Q8C2×D311yescharacteristic
71C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
72C2×C3×D12—31no—
73C2×C3×D12—31no—
74C2×C3×D12—61no—
75C2×#624—61no—
76C2×D3×Dic3—61no—
77C2×D3×Dic3—61no—
78#3755Dic314yescomplemented
79C4⋊(C3⋊Dic3)—21no—
80C4⋊(C3⋊Dic3)C2×D311yescharacteristic
81C4⋊(C3⋊Dic3)C2×D311yescharacteristic
82C4×C3⋊Dic3—21no—
83C4×C3⋊Dic3C2×D311yescharacteristic
84C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
85C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
86C3×#215—34no—
87C3×#215—34no—
88C3×#215—64no—
89C3×C4⋊Dic3—31no—
90C3×C4⋊Dic3—31no—
91C3×C4⋊Dic3—31no—
92C3×C4⋊Dic3—31no—
93C3×C4⋊Dic3—61no—
94C3×C4⋊Dic3—61no—
95C3×C4⋊Dic3—61no—
96C3×C4⋊Dic3—62no—
97C3×C4⋊Dic3—61no—
98C3×C4×Dic3—31no—
99C3×C4×Dic3—31no—
100C3×C4×Dic3—61no—
101C3×C4×Dic3—61no—
102C3×C4×Dic3—61no—
103C3×C4×Dic3—62no—
104C2×C3×C3⋊C8—31no—
105C2×C3×C3⋊C8—31no—
106C2×C3×C3⋊C8—31no—
107C2×C3×C3⋊C8—31no—
108C2×C3×C3⋊C8—61no—
109C2×C3×C3⋊C8—61no—
110#3722—61no—
111(C2×D3)⋊Dic3—61no—
112(C2×D3)⋊Dic3—61no—
113C22×C33C2×D411yescharacteristic
114C2×C3×C3⋊D3—22no—
115C2×C3×C3⋊D3—41no—
116C2×C3×C3⋊D3—41no—
117C33×C4—22no—
118C33×C4C22×C411yescharacteristic
119C33×C4#4111yescharacteristic
120C3×C3⋊Dic3—21no—
121C3×C3⋊Dic3—41no—
122C3×C3⋊Dic3—41no—
123C3×C3⋊Dic3SD1612yescomplemented
124C2×C3×SD16—91no—
125C3×C4×Q8—91no—
126Q8×Dic3—31no—
127Q8×Dic3—31no—
128Q8×Dic3—61no—
129C2×#215—31no—
130C2×#215—31no—
131C2×#215—61no—
132D4×Dic3—91no—
133#893—91no—
134C4×D12—91no—
135C3×#103—91no—
136#835—31no—
137#835—31no—
138#835—61no—
139#832—91no—
140#821—91no—
141#817—91no—
142C8×Dic3—91no—
143C8×Dic3—91no—
144#809—91no—
145#808—91no—
146#808—91no—
147C4×C3⋊C8—91no—
148C22×C3⋊D3—21no—
149C22×C3×D3—61no—
150C22×C3×D3—61no—
151C22×C3×D3—121no—
152C32×Q8—21no—
153C32×Q8—22no—
154C32×Q8—21no—
155C32×Q8—21no—
156C32×Q8—22no—
157C32×Q8—22no—
158C32×Q8—22no—
159C32×Q8—42no—
160C32×Q8—41no—
161C32×Q8—41no—
162C32×Q8—42no—
163C32×Q8C4×D312yescomplemented
164C32×Q8C4×D312yescomplemented
165C32×Q8C2×Dic312yescomplemented
166C2×C32×C4—21no—
167C2×C32×C4—22no—
168C2×C32×C4—21no—
169C2×C32×C4—21no—
170C2×C32×C4—21no—
171C2×C32×C4—21no—
172C2×C32×C4—21no—
173C2×C32×C4—41no—
174C2×C32×C4—41no—
175C2×C32×C4—41no—
176C2×C32×C4—42no—
177C2×C32×C4C22×D311yescharacteristic
178C2×C32×C4C22×D311yescharacteristic
179C2×C32×C4C22×D311yescharacteristic
180C2×C3⋊Dic3—21no—
181C2×C3⋊Dic3—21no—
182C2×C3⋊Dic3#6711yescharacteristic
183C2×C3⋊Dic3#6711yescharacteristic
184C4⋊(C3⋊D3)—21no—
185C4⋊(C3⋊D3)C2×Dic312yes—
186C2×C3×Dic3—31no—
187C2×C3×Dic3—31no—
188C2×C3×Dic3—31no—
189C2×C3×Dic3—31no—
190C2×C3×Dic3—61no—
191C2×C3×Dic3—61no—
192C2×C3×Dic3—62no—
193C2×C3×Dic3—61no—
194C2×C3×Dic3—61no—
195C2×C3×Dic3—61no—
196C2×C3×Dic3—61no—
197C2×C3×Dic3—121no—
198C3×D12—32no—
199C3×D12—32no—
200C3×D12—62no—
201C3×D12—61no—
202C3×D12—61no—
203C3×D12—121no—
204#624—122no—
205D3×Dic3—122no—
206D3×Dic3—122no—
207C3⋊(C3⋊C8)—21no—
208C3⋊(C3⋊C8)C2×Dic312yescomplemented
209C3×C3⋊C8—32no—
210C3×C3⋊C8—32no—
211C3×C3⋊C8—62no—
212C3×C3⋊C8—61no—
213C3×C3⋊C8—61no—
214C3×C3⋊C8—121no—
215C4×SD16—271noSylow
216C2×C33C4×D411yescharacteristic
217C2×C33C2×SD1611yescharacteristic
218C2×C33#13511yescharacteristic
219C3×C3⋊D3—42no—
220C2×C3×Q8—21no—
221C2×C3×Q8—21no—
222C2×C3×Q8—21no—
223C2×C3×Q8—41no—
224C2×C3×Q8D3211yescharacteristic
225C2×C3×Q8D3211yescharacteristic
226C2×C3×Q8D3211yescharacteristic
227C2×C3×D4—91no—
228C22×Dic3—181no—
229C2×D12—31no—
230C2×D12—31no—
231C2×D12—61no—
232C2×C4×D3—181no—
233C2×C4×D3—181no—
234C3×SD16—94no—
235C2×C3×C8—91no—
236C2×C3×C8—91no—
237C3×C4⋊C4—91no—
238C3×C4⋊C4—91no—
239C3×C4⋊C4—181no—
240C3×C42—91no—
241C3×C42—181no—
242#217—181no—
243#215—34no—
244#215—34no—
245#215—64no—
246(C2×D3)⋊C4—181no—
247(C2×D3)⋊C4—181no—
248C4⋊Dic3—31no—
249C4⋊Dic3—31no—
250C4⋊Dic3—31no—
251C4⋊Dic3—31no—
252C4⋊Dic3—61no—
253C4⋊Dic3—61no—
254C4⋊Dic3—61no—
255C4⋊Dic3—61no—
256C4⋊Dic3—62no—
257C4×Dic3—31no—
258C4×Dic3—31no—
259C4×Dic3—61no—
260C4×Dic3—61no—
261C4×Dic3—61no—
262C4×Dic3—62no—
263C2×C3⋊C8—31no—
264C2×C3⋊C8—31no—
265C2×C3⋊C8—31no—
266C2×C3⋊C8—31no—
267C2×C3⋊C8—61no—
268C2×C3⋊C8—61no—
269C22×C32—21no—
270C22×C32—21no—
271C22×C32—21no—
272C22×C32—41no—
273C22×C32D3×D411yescharacteristic
274C22×C32D3×D411yescharacteristic
275C22×C32C2×#6711yescharacteristic
276C2×C3⋊D3—22no—
277C2×C3⋊D3—41no—
278C2×C3⋊D3—41no—
279C2×C3×D3—62no—
280C2×C3×D3—62no—
281C2×C3×D3—122no—
282C2×C3×D3—121no—
283C2×C3×D3—121no—
284C2×C3×D3—121no—
285C2×C3×D3—121no—
286C2×C3×D3—241no—
287C2×C3×D3—241no—
288C32×C4—22no—
289C32×C4—24no—
290C32×C4—22no—
291C32×C4—22no—
292C32×C4—21no—
293C32×C4—21no—
294C32×C4—21no—
295C32×C4—21no—
296C32×C4—21no—
297C32×C4—21no—
298C32×C4—41no—
299C32×C4—41no—
300C32×C4—42no—
301C32×C4—42no—
302C32×C4—44no—
303C32×C4C2×C4×D311yescharacteristic
304C32×C4C2×C4×D311yescharacteristic
305C32×C4#23511yescharacteristic
306C32×C4#23711yescharacteristic
307C32×C4#23711yescharacteristic
308C32×C4C22×Dic311yescharacteristic
309C3⋊Dic3—21no—
310C3⋊Dic3—41no—
311C3⋊Dic3—41no—
312C3⋊Dic3#21512yescomplemented
313C3×Dic3—32no—
314C3×Dic3—32no—
315C3×Dic3—62no—
316C3×Dic3—61no—
317C3×Dic3—61no—
318C3×Dic3—121no—
319C3×Dic3—122no—
320C3×Dic3—121no—
321C3×Dic3—121no—
322C3×Dic3—121no—
323C3×Dic3—121no—
324C3×Dic3—241no—
325C2×SD16—91no—
326C4×Q8—91no—
327C4×D4—271no—
328#106—271no—
329#103—91no—
330#102—271no—
331C4×C8—271no—
332C33C4×SD1611yescharacteristic, complemented, Sylow
333C23×C3—181no—
334C22×D3—61no—
335C22×D3—61no—
336C22×D3—121no—
337C3×Q8—22no—
338C3×Q8—21no—
339C3×Q8—21no—
340C3×Q8—21no—
341C3×Q8—22no—
342C3×Q8—22no—
343C3×Q8—22no—
344C3×Q8—41no—
345C3×Q8—41no—
346C3×Q8—42no—
347C3×Q8—42no—
348C3×Q8D3×Dic312yescomplemented
349C3×Q8D3×Dic312yescomplemented
350C3×Q8#62412yescomplemented
351C3×D4—92no—
352C3×D4—181no—
353C2×C3×C4—22no—
354C2×C3×C4—21no—
355C2×C3×C4—21no—
356C2×C3×C4—21no—
357C2×C3×C4—21no—
358C2×C3×C4—21no—
359C2×C3×C4—21no—
360C2×C3×C4—41no—
361C2×C3×C4—41no—
362C2×C3×C4—42no—
363C2×C3×C4—41no—
364C2×C3×C4—91no—
365C2×C3×C4—91no—
366C2×C3×C4—181no—
367C2×C3×C4—181no—
368C2×C3×C4C2×D3211yescharacteristic
369C2×C3×C4C2×D3211yescharacteristic
370C2×C3×C4C2×D3211yescharacteristic
371C2×Dic3—31no—
372C2×Dic3—31no—
373C2×Dic3—31no—
374C2×Dic3—31no—
375C2×Dic3—61no—
376C2×Dic3—61no—
377C2×Dic3—62no—
378C2×Dic3—61no—
379C2×Dic3—61no—
380C2×Dic3—61no—
381C2×Dic3—61no—
382C2×Dic3—121no—
383C2×Dic3—362no—
384D12—32no—
385D12—32no—
386D12—62no—
387D12—61no—
388D12—61no—
389D12—121no—
390C4×D3—362no—
391C4×D3—362no—
392C3×C8—92no—
393C3×C8—181no—
394C3⋊C8—32no—
395C3⋊C8—32no—
396C3⋊C8—62no—
397C3⋊C8—61no—
398C3⋊C8—61no—
399C3⋊C8—121no—
400C2×C32—21no—
401C2×C32—21no—
402C2×C32—21no—
403C2×C32—21no—
404C2×C32—21no—
405C2×C32—21no—
406C2×C32—21no—
407C2×C32—21no—
408C2×C32—21no—
409C2×C32—41no—
410C2×C32—41no—
411C2×C32—41no—
412C2×C32#67⋊C411yescharacteristic
413C2×C32D3×SD1611yescharacteristic
414C2×C32D3×SD1611yescharacteristic
415C2×C32#91611yescharacteristic
416C2×C32#91911yescharacteristic
417C2×C32C4×#6711yescharacteristic
418C2×C32D4×Dic311yescharacteristic
419C2×C32C2×#21511yescharacteristic
420C2×C32#95011yescharacteristic
421C3⋊D3—42no—
422C3×D3—122no—
423C3×D3—122no—
424C3×D3—242no—
425C2×Q8#111311yescharacteristic
426C2×D4—91no—
427C22×C4—541no—
428SD16—94no—
429C2×C8—91no—
430C2×C8—91no—
431C4⋊C4—91no—
432C4⋊C4—91no—
433C4⋊C4—181no—
434C22⋊C4—541no—
435C42—91no—
436C42—181no—
437C22×C3—21no—
438C22×C3—21no—
439C22×C3—21no—
440C22×C3—41no—
441C22×C3—182no—
442C22×C3—361no—
443C22×C3—361no—
444C22×C3D3×#6711yescharacteristic
445C22×C3D3×#6711yescharacteristic
446C22×C3#381111yescharacteristic
447C2×D3—62no—
448C2×D3—62no—
449C2×D3—122no—
450C2×D3—121no—
451C2×D3—121no—
452C2×D3—121no—
453C2×D3—121no—
454C2×D3—241no—
455C2×D3—241no—
456C3×C4—24no—
457C3×C4—22no—
458C3×C4—22no—
459C3×C4—22no—
460C3×C4—21no—
461C3×C4—21no—
462C3×C4—21no—
463C3×C4—21no—
464C3×C4—21no—
465C3×C4—21no—
466C3×C4—42no—
467C3×C4—42no—
468C3×C4—44no—
469C3×C4—41no—
470C3×C4—41no—
471C3×C4—92no—
472C3×C4—181no—
473C3×C4—361no—
474C3×C4—361no—
475C3×C4C2×D3×Dic311yescharacteristic
476C3×C4C2×D3×Dic311yescharacteristic
477C3×C4#380411yescharacteristic
478C3×C4#380411yescharacteristic
479C3×C4#380511yescharacteristic
480C3×C4C2×#62411yescharacteristic
481Dic3—32no—
482Dic3—32no—
483Dic3—62no—
484Dic3—61no—
485Dic3—61no—
486Dic3—121no—
487Dic3—122no—
488Dic3—121no—
489Dic3—121no—
490Dic3—121no—
491Dic3—121no—
492Dic3—241no—
493C32—21no—
494C32—21no—
495C32—21no—
496C32—41no—
497C32#478811yescharacteristic, complemented
498C32C4×#21511yescharacteristic, complemented
499C32Dic3×SD1611yescharacteristic, complemented
500C23—181no—
501Q8—21no—
502Q8#629412yescomplemented
503D4—92no—
504D4—181no—
505C2×C4—21no—
506C2×C4—91no—
507C2×C4—91no—
508C2×C4—181no—
509C2×C4—181no—
510C2×C4—1082no—
511C2×C4C2×#111311yescharacteristic
512C8—92no—
513C8—181no—
514C2×C3—21no—
515C2×C3—21no—
516C2×C3—21no—
517C2×C3—21no—
518C2×C3—21no—
519C2×C3—21no—
520C2×C3—21no—
521C2×C3—21no—
522C2×C3—21no—
523C2×C3—41no—
524C2×C3—41no—
525C2×C3—41no—
526C2×C3—362no—
527C2×C3D3×#21511yescharacteristic
528C2×C3D3×#21511yescharacteristic
529C2×C3#6394011yescharacteristic
530C2×C3#6394711yescharacteristic
531C2×C3#6394911yescharacteristic
532C2×C3#6395011yescharacteristic
533C2×C3#6395211yescharacteristic
534C2×C3Dic3×#6711yescharacteristic
535C2×C3#67⋊Dic311yescharacteristic
536D3—122no—
537D3—122no—
538D3—242no—
539C22—182no—
540C22—361no—
541C22—361no—
542C22#8904911yescharacteristic
543C4—22no—
544C4—92no—
545C4—181no—
546C4—361no—
547C4—361no—
548C4C2×#629411yescharacteristic
549C4#8904611yescharacteristic
550C3—21no—
551C3—21no—
552C3—21no—
553C3—41no—
554C3#1059153611yescharacteristic, complemented
555C3Dic3×#21511yescharacteristic, complemented
556C3#1059157111yescharacteristic, complemented
557C2—362no—
558C2#1172370811yescharacteristic
559C2#1172371311yescharacteristic
560C2#1172372611yescharacteristic
subgroupG/H
centerC22#89049
commutatorC33×C4C22×C4
FrattiniC2×C4C2×#1113
FittingC2×C33×Q8C22
socleC22×C33C2×D4
prime radicalC2×C3×C4⋊(C3⋊D3)C22
cosocleC2×C33×C4C23
hypercenterC2×Q8#1113
hypocenterC33C4×SD16
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
1. C3×Q8×C3⋊Dic3C2semidirect
4. C3×#63662C2semidirect
5. #11722079C2semidirect
7. #11722070C2semidirect
14. C3×#3755C4semidirect
23. Q8×C3⋊Dic3D3semidirect
34. #63662D3semidirect
49. C33×Q8C2×C4semidirect
60. C3×C3⋊(C3⋊C8)C2×C4semidirect
78. #3755Dic3semidirect
123. C3×C3⋊Dic3SD16semidirect
163. C32×Q8C4×D3semidirect
164. C32×Q8C4×D3semidirect
165. C32×Q8C2×Dic3semidirect
208. C3⋊(C3⋊C8)C2×Dic3semidirect
312. C3⋊Dic3#215semidirect
332. C33C4×SD16semidirect
348. C3×Q8D3×Dic3semidirect
349. C3×Q8D3×Dic3semidirect
350. C3×Q8#624semidirect
497. C32#4788semidirect
498. C32C4×#215semidirect
499. C32Dic3×SD16semidirect
502. Q8#6294semidirect
554. C3#10591536semidirect
555. C3Dic3×#215semidirect
556. C3#10591571semidirect

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×#37552
#117236912
C3×#636622
#117220792
#117220752
#117220702
#105915713
Dic3×#2153
#105915363