Finite Group Databasebeta

← #13162007

Subgroups — #13162007

Order 1728.

684 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×Q8×C3⋊Dic3C211yesmaximal, characteristic, complemented
2C2×C3×#3756C211yesmaximal, characteristic, complemented
3C2×C3×#3755C211yesmaximal, characteristic
4#11723677C211yesmaximal, characteristic
5#11722078C211yesmaximal, characteristic, complemented
6#11722076C211yesmaximal, characteristic, complemented
7#11722075C211yesmaximal, characteristic
8#10591612—31nomaximal
9#10591607—31nomaximal
10#10591604—31nomaximal
11C2×C33×Q8C2211yescharacteristic
12C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
13C2×C3×#634C2211yescharacteristic
14C3×#3756—22no—
15C3×#3755—22no—
16C3×C4⋊(C3⋊Dic3)—21no—
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—
23#88814—21no—
24Q8×C3⋊Dic3D311yescharacteristic, complemented
25C2×#3756D311yescharacteristic, complemented
26C2×#3755D311yescharacteristic
27C3×Q8×Dic3—31no—
28C3×Q8×Dic3—31no—
29C3×Q8×Dic3—61no—
30C2×C3×#216—31no—
31C2×C3×#216—31no—
32C2×C3×#216—61no—
33C2×C3×#215—31no—
34C2×C3×#215—31no—
35C2×C3×#215—61no—
36#63864—31no—
37#63863—31no—
38#63860—31no—
39#63572—31no—
40Dic3⋊(C3⋊C8)—31no—
41Dic3⋊(C3⋊C8)—31no—
42#63567—31no—
43#63566—31no—
44#63565—31no—
45#63564—31no—
46#63563—31no—
47#63562—31no—
48C22×C3×C3⋊D3—21no—
49C33×Q8—21no—
50C33×Q8D412yes—
51C2×C33×C4—21no—
52C2×C33×C4C2311yescharacteristic
53C2×C3×C3⋊Dic3—21no—
54C2×C3×C3⋊Dic3—21no—
55C2×C3×C3⋊Dic3D411yescharacteristic
56C2×C3×C3⋊Dic3D411yescharacteristic
57C3×C4⋊(C3⋊D3)—21no—
58C3×C4⋊(C3⋊D3)—21no—
59C3×#634—21no—
60C3×#634—21no—
61C3×C3⋊(C3⋊C8)—21no—
62C3×C3⋊(C3⋊C8)—21no—
63#5185—91no—
64#5166—91no—
65#5028—91no—
66C2×C32×Q8—21no—
67C2×C32×Q8—21no—
68C2×C32×Q8—21no—
69C2×C32×Q8—41no—
70C2×C32×Q8C2×D311yescharacteristic
71C2×C32×Q8C2×D311yescharacteristic
72C2×C32×Q8C2×D311yescharacteristic
73C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
74C2×#634C2×D311yescharacteristic
75C2×C3×D12—31no—
76C2×C3×D12—31no—
77C2×C3×D12—61no—
78C2×C3×Dic6—31no—
79C2×C3×Dic6—31no—
80C2×C3×Dic6—61no—
81#3756—22no—
82#3755—22no—
83C4⋊(C3⋊Dic3)—21no—
84C4⋊(C3⋊Dic3)C2×D311yescharacteristic
85C4×C3⋊Dic3—21no—
86C4×C3⋊Dic3C2×D311yescharacteristic
87C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
88C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
89C3×#216—64no—
90C3×#216—62no—
91C3×#216—62no—
92C3×#215—64no—
93C3×#215—62no—
94C3×#215—62no—
95C3×C4⋊Dic3—31no—
96C3×C4⋊Dic3—31no—
97C3×C4⋊Dic3—61no—
98C3×C4⋊Dic3—61no—
99C3×C4⋊Dic3—61no—
100C3×C4⋊Dic3—62no—
101C3×C4×Dic3—31no—
102C3×C4×Dic3—31no—
103C3×C4×Dic3—61no—
104C3×C4×Dic3—61no—
105C3×C4×Dic3—61no—
106C3×C4×Dic3—62no—
107C2×C3×C3⋊C8—31no—
108C2×C3×C3⋊C8—31no—
109C2×C3×C3⋊C8—31no—
110C2×C3×C3⋊C8—31no—
111C2×C3×C3⋊C8—61no—
112C2×C3×C3⋊C8—61no—
113#3722—61no—
114#3722—61no—
115(C2×D3)⋊Dic3—61no—
116(C2×D3)⋊Dic3—61no—
117(C2×D3)⋊Dic3—61no—
118(C2×D3)⋊Dic3—61no—
119C22×C33C2×D411yescharacteristic
120C2×C3×C3⋊D3—41no—
121C2×C3×C3⋊D3—41no—
122C2×C3×C3⋊D3—41no—
123C33×C4—22no—
124C33×C4C2×D411yescharacteristic
125C33×C4#4111yescharacteristic
126C3×C3⋊Dic3—21no—
127C3×C3⋊Dic3—21no—
128C3×C3⋊Dic3—41no—
129C3×C3⋊Dic3—41no—
130C2×C3×Q16—91no—
131C2×C3×SD16—91no—
132C3×C4×Q8—91no—
133#947—91no—
134Q8×Dic3—31no—
135Q8×Dic3—31no—
136Q8×Dic3—61no—
137C2×#216—31no—
138C2×#216—31no—
139C2×#216—61no—
140C2×#215—31no—
141C2×#215—31no—
142C2×#215—61no—
143#936—91no—
144#875—91no—
145#835—91no—
146#832—91no—
147#821—91no—
148#816—91no—
149Dic3⋊C8—91no—
150Dic3⋊C8—91no—
151#810—91no—
152#809—91no—
153C4⋊(C3⋊C8)—91no—
154C22×C3⋊D3—21no—
155C22×C3×D3—61no—
156C22×C3×D3—61no—
157C22×C3×D3—121no—
158C32×Q8—21no—
159C32×Q8—22no—
160C32×Q8—21no—
161C32×Q8—21no—
162C32×Q8—22no—
163C32×Q8—22no—
164C32×Q8—22no—
165C32×Q8—42no—
166C32×Q8—41no—
167C32×Q8—41no—
168C32×Q8—42no—
169C32×Q8D1212yes—
170C32×Q8#6712yes—
171C32×Q8#6712yes—
172C2×C32×C4—21no—
173C2×C32×C4—22no—
174C2×C32×C4—21no—
175C2×C32×C4—21no—
176C2×C32×C4—21no—
177C2×C32×C4—21no—
178C2×C32×C4—21no—
179C2×C32×C4—41no—
180C2×C32×C4—41no—
181C2×C32×C4—41no—
182C2×C32×C4—42no—
183C2×C32×C4C22×D311yescharacteristic
184C2×C32×C4C22×D311yescharacteristic
185C2×C32×C4C22×D311yescharacteristic
186C2×C3⋊Dic3—21no—
187C2×C3⋊Dic3—21no—
188C2×C3⋊Dic3#6711yescharacteristic
189C2×C3⋊Dic3#6711yescharacteristic
190C4⋊(C3⋊D3)—21no—
191C4⋊(C3⋊D3)—21no—
192#634—21no—
193#634—21no—
194C2×C3×Dic3—31no—
195C2×C3×Dic3—31no—
196C2×C3×Dic3—31no—
197C2×C3×Dic3—31no—
198C2×C3×Dic3—61no—
199C2×C3×Dic3—61no—
200C2×C3×Dic3—62no—
201C2×C3×Dic3—61no—
202C2×C3×Dic3—61no—
203C2×C3×Dic3—61no—
204C2×C3×Dic3—61no—
205C2×C3×Dic3—121no—
206C3×D12—62no—
207C3×D12—61no—
208C3×D12—61no—
209C3×D12—61no—
210C3×D12—61no—
211C3×D12—121no—
212C3×Dic6—62no—
213C3×Dic6—61no—
214C3×Dic6—61no—
215C3×Dic6—61no—
216C3×Dic6—61no—
217C3×Dic6—121no—
218C3⋊(C3⋊C8)—21no—
219C3⋊(C3⋊C8)—21no—
220C3×C3⋊C8—62no—
221C3×C3⋊C8—62no—
222C3×C3⋊C8—61no—
223C3×C3⋊C8—61no—
224C3×C3⋊C8—61no—
225C3×C3⋊C8—61no—
226#464—271noSylow
227C2×C33#12111yescharacteristic
228C2×C33#13511yescharacteristic
229C2×C33#13711yescharacteristic
230C3×C3⋊D3—81no—
231C2×C3×Q8—21no—
232C2×C3×Q8—21no—
233C2×C3×Q8—21no—
234C2×C3×Q8—41no—
235C2×C3×Q8—91no—
236C2×C3×Q8D3211yescharacteristic
237C2×C3×Q8D3211yescharacteristic
238C2×C3×Q8D3211yescharacteristic
239C2×C3×D4—91no—
240C2×D12—31no—
241C2×D12—31no—
242C2×D12—61no—
243C2×Dic6—31no—
244C2×Dic6—31no—
245C2×Dic6—61no—
246C3×Q16—182no—
247C3×SD16—182no—
248C2×C3×C8—91no—
249C2×C3×C8—91no—
250C3×C4⋊C4—91no—
251C3×C4⋊C4—181no—
252C3×C42—91no—
253C3×C42—181no—
254#217—181no—
255#217—181no—
256#216—64no—
257#216—62no—
258#216—62no—
259#215—64no—
260#215—62no—
261#215—62no—
262(C2×D3)⋊C4—181no—
263(C2×D3)⋊C4—181no—
264(C2×D3)⋊C4—181no—
265(C2×D3)⋊C4—181no—
266C4⋊Dic3—31no—
267C4⋊Dic3—31no—
268C4⋊Dic3—61no—
269C4⋊Dic3—61no—
270C4⋊Dic3—61no—
271C4⋊Dic3—62no—
272C4×Dic3—31no—
273C4×Dic3—31no—
274C4×Dic3—61no—
275C4×Dic3—61no—
276C4×Dic3—61no—
277C4×Dic3—62no—
278C2×C3⋊C8—31no—
279C2×C3⋊C8—31no—
280C2×C3⋊C8—31no—
281C2×C3⋊C8—31no—
282C2×C3⋊C8—61no—
283C2×C3⋊C8—61no—
284C22×C32—21no—
285C22×C32—21no—
286C22×C32—21no—
287C22×C32—41no—
288C22×C32D3×D411yescharacteristic
289C22×C32D3×D411yescharacteristic
290C22×C32C2×#6711yescharacteristic
291C2×C3⋊D3—41no—
292C2×C3⋊D3—41no—
293C2×C3⋊D3—41no—
294C2×C3×D3—122no—
295C2×C3×D3—121no—
296C2×C3×D3—121no—
297C2×C3×D3—121no—
298C2×C3×D3—121no—
299C2×C3×D3—121no—
300C2×C3×D3—121no—
301C2×C3×D3—241no—
302C2×C3×D3—241no—
303C32×C4—22no—
304C32×C4—24no—
305C32×C4—22no—
306C32×C4—22no—
307C32×C4—21no—
308C32×C4—21no—
309C32×C4—21no—
310C32×C4—21no—
311C32×C4—21no—
312C32×C4—21no—
313C32×C4—41no—
314C32×C4—41no—
315C32×C4—42no—
316C32×C4—42no—
317C32×C4—44no—
318C32×C4C2×D1211yescharacteristic
319C32×C4#23511yescharacteristic
320C32×C4#23711yescharacteristic
321C32×C4#23711yescharacteristic
322C32×C4C2×#6711yescharacteristic
323C32×C4C2×#6711yescharacteristic
324C3⋊Dic3—21no—
325C3⋊Dic3—21no—
326C3⋊Dic3—41no—
327C3⋊Dic3—41no—
328C3×Dic3—62no—
329C3×Dic3—61no—
330C3×Dic3—61no—
331C3×Dic3—61no—
332C3×Dic3—61no—
333C3×Dic3—121no—
334C3×Dic3—122no—
335C3×Dic3—122no—
336C3×Dic3—121no—
337C3×Dic3—121no—
338C3×Dic3—121no—
339C3×Dic3—121no—
340C2×Q16—91no—
341C2×SD16—91no—
342#124—271no—
343C4×Q8—91no—
344C4⋊C8—271no—
345#103—271no—
346#102—271no—
347C33#46411yescharacteristic, complemented, Sylow
348C23×C3—181no—
349C22×D3—61no—
350C22×D3—61no—
351C22×D3—121no—
352C3×Q8—22no—
353C3×Q8—21no—
354C3×Q8—21no—
355C3×Q8—21no—
356C3×Q8—22no—
357C3×Q8—22no—
358C3×Q8—22no—
359C3×Q8—41no—
360C3×Q8—41no—
361C3×Q8—42no—
362C3×Q8—42no—
363C3×Q8—181no—
364C3×Q8—181no—
365C3×Q8#62512yes—
366C3×Q8#62612yes—
367C3×Q8#62612yes—
368C3×D4—181no—
369C3×D4—181no—
370C2×C3×C4—22no—
371C2×C3×C4—21no—
372C2×C3×C4—21no—
373C2×C3×C4—21no—
374C2×C3×C4—21no—
375C2×C3×C4—21no—
376C2×C3×C4—21no—
377C2×C3×C4—41no—
378C2×C3×C4—41no—
379C2×C3×C4—42no—
380C2×C3×C4—41no—
381C2×C3×C4—91no—
382C2×C3×C4—91no—
383C2×C3×C4—181no—
384C2×C3×C4—181no—
385C2×C3×C4C2×D3211yescharacteristic
386C2×C3×C4C2×D3211yescharacteristic
387C2×C3×C4C2×D3211yescharacteristic
388C2×Dic3—31no—
389C2×Dic3—31no—
390C2×Dic3—31no—
391C2×Dic3—31no—
392C2×Dic3—61no—
393C2×Dic3—61no—
394C2×Dic3—62no—
395C2×Dic3—61no—
396C2×Dic3—61no—
397C2×Dic3—61no—
398C2×Dic3—61no—
399C2×Dic3—121no—
400D12—62no—
401D12—61no—
402D12—61no—
403D12—61no—
404D12—61no—
405D12—121no—
406Dic6—62no—
407Dic6—61no—
408Dic6—61no—
409Dic6—61no—
410Dic6—61no—
411Dic6—121no—
412C3×C8—181no—
413C3×C8—181no—
414C3⋊C8—62no—
415C3⋊C8—62no—
416C3⋊C8—61no—
417C3⋊C8—61no—
418C3⋊C8—61no—
419C3⋊C8—61no—
420C2×C32—21no—
421C2×C32—21no—
422C2×C32—21no—
423C2×C32—21no—
424C2×C32—21no—
425C2×C32—21no—
426C2×C32—21no—
427C2×C32—21no—
428C2×C32—21no—
429C2×C32—41no—
430C2×C32—41no—
431C2×C32—41no—
432C2×C32#91511yescharacteristic
433C2×C32#91611yescharacteristic
434C2×C32#91811yescharacteristic
435C2×C32#91911yescharacteristic
436C2×C32#93011yescharacteristic
437C2×C32#93911yescharacteristic
438C2×C32#93911yescharacteristic
439C2×C32#94211yescharacteristic
440C2×C32#95011yescharacteristic
441C3⋊D3—81no—
442C3×D3—242no—
443C3×D3—241no—
444C3×D3—241no—
445C2×Q8—91no—
446C2×Q8#111311yescharacteristic
447C2×D4—91no—
448Q16—182no—
449SD16—182no—
450C2×C8—91no—
451C2×C8—91no—
452C4⋊C4—91no—
453C4⋊C4—181no—
454C22⋊C4—541no—
455C22⋊C4—541no—
456C42—91no—
457C42—181no—
458C22×C3—21no—
459C22×C3—21no—
460C22×C3—21no—
461C22×C3—41no—
462C22×C3—361no—
463C22×C3—361no—
464C22×C3—361no—
465C22×C3D3×#6711yescharacteristic
466C22×C3D3×#6711yescharacteristic
467C22×C3#381111yescharacteristic
468C2×D3—122no—
469C2×D3—121no—
470C2×D3—121no—
471C2×D3—121no—
472C2×D3—121no—
473C2×D3—121no—
474C2×D3—121no—
475C2×D3—241no—
476C2×D3—241no—
477C3×C4—24no—
478C3×C4—22no—
479C3×C4—22no—
480C3×C4—22no—
481C3×C4—21no—
482C3×C4—21no—
483C3×C4—21no—
484C3×C4—21no—
485C3×C4—21no—
486C3×C4—21no—
487C3×C4—42no—
488C3×C4—42no—
489C3×C4—44no—
490C3×C4—41no—
491C3×C4—41no—
492C3×C4—181no—
493C3×C4—181no—
494C3×C4—361no—
495C3×C4—361no—
496C3×C4#380411yescharacteristic
497C3×C4#380411yescharacteristic
498C3×C4#380511yescharacteristic
499C3×C4C2×#62511yescharacteristic
500C3×C4C2×#62611yescharacteristic
501C3×C4C2×#62611yescharacteristic
502Dic3—62no—
503Dic3—61no—
504Dic3—61no—
505Dic3—61no—
506Dic3—61no—
507Dic3—121no—
508Dic3—122no—
509Dic3—122no—
510Dic3—121no—
511Dic3—121no—
512Dic3—121no—
513Dic3—121no—
514C32—21no—
515C32—21no—
516C32—21no—
517C32—41no—
518C32#502811yescharacteristic, complemented
519C32#516611yescharacteristic, complemented
520C32#518511yescharacteristic, complemented
521C23—181no—
522Q8—21no—
523Q8—181no—
524Q8—181no—
525Q8#629512yes—
526D4—181no—
527D4—181no—
528C2×C4—21no—
529C2×C4—91no—
530C2×C4—91no—
531C2×C4—181no—
532C2×C4—181no—
533C2×C4C2×#111311yescharacteristic
534C8—181no—
535C8—181no—
536C2×C3—21no—
537C2×C3—21no—
538C2×C3—21no—
539C2×C3—21no—
540C2×C3—21no—
541C2×C3—21no—
542C2×C3—21no—
543C2×C3—21no—
544C2×C3—21no—
545C2×C3—41no—
546C2×C3—41no—
547C2×C3—41no—
548C2×C3—721no—
549C2×C3#6394611yescharacteristic
550C2×C3#6394711yescharacteristic
551C2×C3#6394811yescharacteristic
552C2×C3#6394911yescharacteristic
553C2×C3#6395011yescharacteristic
554C2×C3#6395111yescharacteristic
555C2×C3#6397811yescharacteristic
556C2×C3#6397811yescharacteristic
557C2×C3#6398011yescharacteristic
558D3—242no—
559D3—241no—
560D3—241no—
561C22—361no—
562C22—361no—
563C22—361no—
564C22#8904911yescharacteristic
565C4—22no—
566C4—181no—
567C4—181no—
568C4—361no—
569C4—361no—
570C4#8904611yescharacteristic
571C4C2×#629511yescharacteristic
572C3—21no—
573C3—21no—
574C3—21no—
575C3—41no—
576C3#1059160411yescharacteristic, complemented
577C3#1059160711yescharacteristic, complemented
578C3#1059161211yescharacteristic, complemented
579C2—721no—
580C2#1172371211yescharacteristic
581C2#1172371311yescharacteristic
582C2#1172371711yescharacteristic
subgroupG/H
centerC22#89049
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×#1113
FittingC2×C33×Q8C22
socleC22×C33C2×D4
prime radicalC2×C3×C4⋊(C3⋊D3)C22
hypercenterC2×Q8#1113
hypocenterC33#464
LCSG ▷ C2×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
2. C2×C3×#3756C2semidirect
5. #11722078C2semidirect
6. #11722076C2semidirect
24. Q8×C3⋊Dic3D3semidirect
25. C2×#3756D3semidirect
347. C33#464semidirect
518. C32#5028semidirect
519. C32#5166semidirect
520. C32#5185semidirect
576. C3#10591604semidirect
577. C3#10591607semidirect
578. C3#10591612semidirect

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
C2×C3×#37552
#117236772
#117220782
#117220762
#117220752
#105916123
#105916073
#105916043