Finite Group Databasebeta

← #13161992

Subgroups — #13161992

Order 1728.

696 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#64153C211yesmaximal, characteristic, complemented
2C2×C3×#3755C211yesmaximal, characteristic
3#11723691C211yesmaximal, characteristic
4C2×#88811C211yesmaximal, characteristic
5C3×#63662C211yesmaximal, characteristic, complemented
6#11722078C211yesmaximal, characteristic, complemented
7#11722076C211yesmaximal, characteristic, complemented
8#10591569—31nomaximal
9#10591565—31nomaximal
10#10591564—31nomaximal
11C2×C33×Q8C2211yescharacteristic
12C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
13C2×C3×#634C2211yescharacteristic
14C2×#6294—21no—
15C3×#3755—22no—
16C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
17C3×#3750—21no—
18C3×#3750—21no—
19C3×C4×C3⋊Dic3C2211yescharacteristic
20C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
21C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
22#88814—21no—
23#88811—22no—
24#64153D311yescharacteristic, complemented
25C2×#3755D311yescharacteristic
26C3×#944—31no—
27C3×#944—31no—
28C3×#944—61no—
29C2×C3×#215—31no—
30C2×C3×#215—31no—
31C2×C3×#215—61no—
32#63898—31no—
33#63894—31no—
34Dic3×D12—31no—
35C2×#3717—31no—
36C2×#3716—31no—
37C2×#3715—31no—
38#63662D311yescharacteristic, complemented
39C3×#835—31no—
40C3×#835—31no—
41C3×#835—61no—
42#63572—31no—
43Dic3⋊(C3⋊C8)—31no—
44Dic3⋊(C3⋊C8)—31no—
45#63567—31no—
46#63566—31no—
47#63565—31no—
48C22×C3×C3⋊D3—21no—
49C33×Q8—21no—
50C33×Q8—21no—
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)D412yes—
59C3×#634—21no—
60C3×#634—21no—
61#6294—42no—
62C3×C3⋊(C3⋊C8)—21no—
63C3×C3⋊(C3⋊C8)—21no—
64#5162—91no—
65#5083—91no—
66#4818—91no—
67C2×C32×Q8—21no—
68C2×C32×Q8—21no—
69C2×C32×Q8—21no—
70C2×C32×Q8—41no—
71C2×C32×Q8C2×D311yescharacteristic
72C2×C32×Q8C2×D311yescharacteristic
73C2×C32×Q8C2×D311yescharacteristic
74C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
75C2×#634C2×D311yescharacteristic
76C2×C3×D12—31no—
77C2×C3×D12—31no—
78C2×C3×D12—61no—
79C2×C3×Dic6—31no—
80C2×C3×Dic6—31no—
81C2×C3×Dic6—61no—
82C2×#624—61no—
83C2×D3×Dic3—61no—
84C2×D3×Dic3—61no—
85#3755—22no—
86C4⋊(C3⋊Dic3)C2×D311yescharacteristic
87#3750—21no—
88#3750—21no—
89C4×C3⋊Dic3C2×D311yescharacteristic
90C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
91C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
92C3×#215—62no—
93C3×#215—62no—
94C3×#215—64no—
95C3×C4⋊Dic3—31no—
96C3×C4⋊Dic3—31no—
97C3×C4⋊Dic3—61no—
98C3×Dic3⋊C4—61no—
99C3×Dic3⋊C4—61no—
100C3×Dic3⋊C4—61no—
101C3×Dic3⋊C4—61no—
102C3×Dic3⋊C4—62no—
103C3×Dic3⋊C4—62no—
104C3×C4×Dic3—31no—
105C3×C4×Dic3—31no—
106C3×C4×Dic3—61no—
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(C2×D3)⋊Dic3—61no—
115(C2×D3)⋊Dic3—61no—
116#3717—62no—
117#3716—62no—
118#3715—62no—
119C22×C33C2×D411yescharacteristic
120C2×C3×C3⋊D3—22no—
121C2×C3×C3⋊D3—41no—
122C2×C3×C3⋊D3—41no—
123C33×C4—41no—
124C33×C4C2×D411yescharacteristic
125C33×C4#4111yescharacteristic
126C3×C3⋊Dic3—21no—
127C3×C3⋊Dic3—41no—
128C3×C3⋊Dic3—41no—
129C3×C3⋊Dic3SD1612yescomplemented
130C2×C3×SD16—91no—
131C3×C4⋊Q8—91no—
132#944—31no—
133#944—31no—
134#944—61no—
135C2×#215—31no—
136C2×#215—31no—
137C2×#215—61no—
138C2×#215—91no—
139D4×Dic3—91no—
140C2×#214—91no—
141C2×#204—91no—
142#893—91no—
143C4×D12—91no—
144C3×#103—91no—
145#835—31no—
146#835—31no—
147#835—61no—
148#835—91no—
149#816—91no—
150Dic3⋊C8—91no—
151Dic3⋊C8—91no—
152#810—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—21no—
160C32×Q8—22no—
161C32×Q8—22no—
162C32×Q8—21no—
163C32×Q8—21no—
164C32×Q8—21no—
165C32×Q8—21no—
166C32×Q8—41no—
167C32×Q8—41no—
168C32×Q8—41no—
169C32×Q8—41no—
170C32×Q8—42no—
171C32×Q8—42no—
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)#6712yes—
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—32no—
207C3×D12—32no—
208C3×D12—62no—
209C3×D12—61no—
210C3×D12—61no—
211C3×D12—121no—
212C3×Dic6—61no—
213C3×Dic6—61no—
214C3×Dic6—61no—
215C3×Dic6—61no—
216C3×Dic6—121no—
217C3×Dic6—121no—
218#624—122no—
219D3×Dic3—122no—
220D3×Dic3—122no—
221C3⋊(C3⋊C8)—21no—
222C3⋊(C3⋊C8)—21no—
223C3×C3⋊C8—62no—
224C3×C3⋊C8—61no—
225C3×C3⋊C8—61no—
226C3×C3⋊C8—61no—
227C3×C3⋊C8—61no—
228C3×C3⋊C8—121no—
229#461—271noSylow
230C2×C33#12111yescharacteristic
231C2×C33C2×SD1611yescharacteristic
232C2×C33#13711yescharacteristic
233C3×C3⋊D3—42no—
234C2×C3×Q8—21no—
235C2×C3×Q8—21no—
236C2×C3×Q8—21no—
237C2×C3×Q8—41no—
238C2×C3×Q8—91no—
239C2×C3×Q8D3211yescharacteristic
240C2×C3×Q8D3211yescharacteristic
241C2×C3×Q8D3211yescharacteristic
242C2×C3×D4—91no—
243C22×Dic3—181no—
244C2×D12—31no—
245C2×D12—31no—
246C2×D12—61no—
247C2×C4×D3—181no—
248C2×C4×D3—181no—
249C2×Dic6—31no—
250C2×Dic6—31no—
251C2×Dic6—61no—
252C3×SD16—182no—
253C2×C3×C8—91no—
254C2×C3×C8—91no—
255C3×C4⋊C4—91no—
256C3×C4⋊C4—181no—
257C3×C4⋊C4—181no—
258C3×C42—91no—
259#217—181no—
260#215—62no—
261#215—62no—
262#215—64no—
263#215—182no—
264#214—182no—
265(C2×D3)⋊C4—181no—
266(C2×D3)⋊C4—181no—
267C4⋊Dic3—31no—
268C4⋊Dic3—31no—
269C4⋊Dic3—61no—
270Dic3⋊C4—61no—
271Dic3⋊C4—61no—
272Dic3⋊C4—61no—
273Dic3⋊C4—61no—
274Dic3⋊C4—62no—
275Dic3⋊C4—62no—
276C4×Dic3—31no—
277C4×Dic3—31no—
278C4×Dic3—61no—
279C2×C3⋊C8—31no—
280C2×C3⋊C8—31no—
281C2×C3⋊C8—31no—
282C2×C3⋊C8—31no—
283C2×C3⋊C8—61no—
284C2×C3⋊C8—61no—
285#204—182no—
286C22×C32—21no—
287C22×C32—21no—
288C22×C32—21no—
289C22×C32—41no—
290C22×C32D3×D411yescharacteristic
291C22×C32D3×D411yescharacteristic
292C22×C32C2×#6711yescharacteristic
293C2×C3⋊D3—22no—
294C2×C3⋊D3—41no—
295C2×C3⋊D3—41no—
296C2×C3×D3—62no—
297C2×C3×D3—62no—
298C2×C3×D3—122no—
299C2×C3×D3—121no—
300C2×C3×D3—121no—
301C2×C3×D3—121no—
302C2×C3×D3—121no—
303C2×C3×D3—241no—
304C2×C3×D3—241no—
305C32×C4—21no—
306C32×C4—21no—
307C32×C4—21no—
308C32×C4—21no—
309C32×C4—21no—
310C32×C4—21no—
311C32×C4—41no—
312C32×C4—41no—
313C32×C4—42no—
314C32×C4—42no—
315C32×C4—42no—
316C32×C4—44no—
317C32×C4—41no—
318C32×C4—41no—
319C32×C4—41no—
320C32×C4#23511yescharacteristic
321C32×C4D3×D411yescharacteristic
322C32×C4D3×D411yescharacteristic
323C32×C4#23711yescharacteristic
324C32×C4#23711yescharacteristic
325C32×C4C2×#6711yescharacteristic
326C3⋊Dic3—21no—
327C3⋊Dic3—41no—
328C3⋊Dic3—41no—
329C3⋊Dic3#21512yescomplemented
330C3×Dic3—32no—
331C3×Dic3—32no—
332C3×Dic3—62no—
333C3×Dic3—62no—
334C3×Dic3—61no—
335C3×Dic3—61no—
336C3×Dic3—122no—
337C3×Dic3—121no—
338C3×Dic3—121no—
339C3×Dic3—121no—
340C3×Dic3—121no—
341C3×Dic3—241no—
342C2×SD16—91no—
343C2×SD16—271no—
344C4⋊Q8—91no—
345C4×D4—271no—
346C4⋊C8—271no—
347#103—91no—
348#103—271no—
349C33#46111yescharacteristic, complemented, Sylow
350C23×C3—181no—
351C22×D3—61no—
352C22×D3—61no—
353C22×D3—121no—
354C3×Q8—22no—
355C3×Q8—22no—
356C3×Q8—21no—
357C3×Q8—21no—
358C3×Q8—21no—
359C3×Q8—21no—
360C3×Q8—21no—
361C3×Q8—21no—
362C3×Q8—41no—
363C3×Q8—41no—
364C3×Q8—41no—
365C3×Q8—41no—
366C3×Q8—42no—
367C3×Q8—42no—
368C3×Q8—181no—
369C3×Q8—181no—
370C3×D4—92no—
371C3×D4—181no—
372C2×C3×C4—22no—
373C2×C3×C4—21no—
374C2×C3×C4—21no—
375C2×C3×C4—21no—
376C2×C3×C4—21no—
377C2×C3×C4—21no—
378C2×C3×C4—21no—
379C2×C3×C4—41no—
380C2×C3×C4—41no—
381C2×C3×C4—42no—
382C2×C3×C4—41no—
383C2×C3×C4—91no—
384C2×C3×C4—91no—
385C2×C3×C4—181no—
386C2×C3×C4—181no—
387C2×C3×C4C2×D3211yescharacteristic
388C2×C3×C4C2×D3211yescharacteristic
389C2×C3×C4C2×D3211yescharacteristic
390C2×Dic3—31no—
391C2×Dic3—31no—
392C2×Dic3—31no—
393C2×Dic3—31no—
394C2×Dic3—61no—
395C2×Dic3—61no—
396C2×Dic3—62no—
397C2×Dic3—61no—
398C2×Dic3—61no—
399C2×Dic3—61no—
400C2×Dic3—61no—
401C2×Dic3—121no—
402C2×Dic3—362no—
403D12—32no—
404D12—32no—
405D12—62no—
406D12—61no—
407D12—61no—
408D12—121no—
409C4×D3—362no—
410C4×D3—362no—
411Dic6—61no—
412Dic6—61no—
413Dic6—61no—
414Dic6—61no—
415Dic6—121no—
416Dic6—121no—
417C3×C8—181no—
418C3×C8—181no—
419C3⋊C8—62no—
420C3⋊C8—61no—
421C3⋊C8—61no—
422C3⋊C8—61no—
423C3⋊C8—61no—
424C3⋊C8—121no—
425C2×C32—21no—
426C2×C32—21no—
427C2×C32—21no—
428C2×C32—21no—
429C2×C32—21no—
430C2×C32—21no—
431C2×C32—21no—
432C2×C32—21no—
433C2×C32—21no—
434C2×C32—41no—
435C2×C32—41no—
436C2×C32—41no—
437C2×C32#88411yescharacteristic
438C2×C32D3×SD1611yescharacteristic
439C2×C32D3×SD1611yescharacteristic
440C2×C32#91511yescharacteristic
441C2×C32#91811yescharacteristic
442C2×C32#93811yescharacteristic
443C2×C32#93911yescharacteristic
444C2×C32C2×#21511yescharacteristic
445C2×C32#94211yescharacteristic
446C3⋊D3—42no—
447C3×D3—122no—
448C3×D3—122no—
449C3×D3—242no—
450C2×Q8—91no—
451C2×Q8#111311yescharacteristic
452C2×D4—91no—
453C22×C4—541no—
454SD16—182no—
455SD16—542no—
456C2×C8—91no—
457C2×C8—91no—
458C4⋊C4—91no—
459C4⋊C4—181no—
460C4⋊C4—181no—
461C22⋊C4—541no—
462C42—91no—
463C22×C3—21no—
464C22×C3—21no—
465C22×C3—21no—
466C22×C3—41no—
467C22×C3—182no—
468C22×C3—361no—
469C22×C3—361no—
470C22×C3D3×#6711yescharacteristic
471C22×C3D3×#6711yescharacteristic
472C22×C3#381111yescharacteristic
473C2×D3—62no—
474C2×D3—62no—
475C2×D3—122no—
476C2×D3—121no—
477C2×D3—121no—
478C2×D3—121no—
479C2×D3—121no—
480C2×D3—241no—
481C2×D3—241no—
482C3×C4—21no—
483C3×C4—21no—
484C3×C4—21no—
485C3×C4—21no—
486C3×C4—21no—
487C3×C4—21no—
488C3×C4—42no—
489C3×C4—42no—
490C3×C4—42no—
491C3×C4—44no—
492C3×C4—41no—
493C3×C4—41no—
494C3×C4—41no—
495C3×C4—41no—
496C3×C4—41no—
497C3×C4—92no—
498C3×C4—181no—
499C3×C4—361no—
500C3×C4—361no—
501C3×C4#380411yescharacteristic
502C3×C4#380411yescharacteristic
503C3×C4#380511yescharacteristic
504C3×C4D3×#6711yescharacteristic
505C3×C4D3×#6711yescharacteristic
506C3×C4#381111yescharacteristic
507Dic3—32no—
508Dic3—32no—
509Dic3—62no—
510Dic3—62no—
511Dic3—61no—
512Dic3—61no—
513Dic3—122no—
514Dic3—121no—
515Dic3—121no—
516Dic3—121no—
517Dic3—121no—
518Dic3—241no—
519C32—21no—
520C32—21no—
521C32—21no—
522C32—41no—
523C32#481811yescharacteristic, complemented
524C32#508311yescharacteristic, complemented
525C32#516211yescharacteristic, complemented
526C23—181no—
527Q8—21no—
528Q8—21no—
529Q8—181no—
530Q8—181no—
531D4—92no—
532D4—181no—
533C2×C4—21no—
534C2×C4—91no—
535C2×C4—91no—
536C2×C4—181no—
537C2×C4—181no—
538C2×C4—1082no—
539C2×C4C2×#111311yescharacteristic
540C8—181no—
541C8—181no—
542C2×C3—21no—
543C2×C3—21no—
544C2×C3—21no—
545C2×C3—21no—
546C2×C3—21no—
547C2×C3—21no—
548C2×C3—21no—
549C2×C3—21no—
550C2×C3—21no—
551C2×C3—41no—
552C2×C3—41no—
553C2×C3—41no—
554C2×C3—362no—
555C2×C3D3×#21511yescharacteristic
556C2×C3D3×#21511yescharacteristic
557C2×C3#6394011yescharacteristic
558C2×C3#6394611yescharacteristic
559C2×C3#6394811yescharacteristic
560C2×C3#6395111yescharacteristic
561C2×C3#6395811yescharacteristic
562C2×C3#6397011yescharacteristic
563C2×C3#6397111yescharacteristic
564D3—122no—
565D3—122no—
566D3—242no—
567C22—182no—
568C22—361no—
569C22—361no—
570C22#8904911yescharacteristic
571C4—41no—
572C4—92no—
573C4—181no—
574C4—361no—
575C4—361no—
576C4#8904611yescharacteristic
577C4#8904911yescharacteristic
578C3—21no—
579C3—21no—
580C3—21no—
581C3—41no—
582C3#1059156411yescharacteristic, complemented
583C3#1059156511yescharacteristic, complemented
584C3#1059156911yescharacteristic, complemented
585C2—362no—
586C2#1172370811yescharacteristic
587C2#1172371211yescharacteristic
588C2#1172372511yescharacteristic
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#461
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×#64153C2semidirect
5. C3×#63662C2semidirect
6. #11722078C2semidirect
7. #11722076C2semidirect
24. #64153D3semidirect
38. #63662D3semidirect
129. C3×C3⋊Dic3SD16semidirect
329. C3⋊Dic3#215semidirect
349. C33#461semidirect
523. C32#4818semidirect
524. C32#5083semidirect
525. C32#5162semidirect
582. C3#10591564semidirect
583. C3#10591565semidirect
584. C3#10591569semidirect

Maximal subgroups

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

subgroupindex
C3×#641532
C2×C3×#37552
#117236912
C2×#888112
C3×#636622
#117220782
#117220762
#105915693
#105915653
#105915643