Finite Group Databasebeta

← #11869661

Subgroups — #11869661

Order 1152.

892 conjugacy classes of subgroups.

The subgroup lattice has 615 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#5627C211yesmaximal, characteristic
2C2×#63841C211yesmaximal, characteristic, complemented
3#10592742C211yesmaximal, characteristic, complemented
4C3×#5044C211yesmaximal, characteristic, complemented
5#10591558C212yesmaximal, complemented
6#10591556C212yesmaximal, complemented
7D3×#830C211yesmaximal, characteristic, complemented
8#10591094C211yesmaximal, characteristic, complemented
9#10591091C211yesmaximal, characteristic, complemented
10#10590571C212yesmaximal, complemented
11#10590446C212yesmaximal, complemented
12#82913—31nomaximal
13#81467—31nomaximal
14C2×C3×#239C2211yescharacteristic
15C22×C3×C4×D3C2211yescharacteristic
16C32×C22⋊Q8C2211yescharacteristic
17C2×C4⋊(C3⋊Dic3)C2211yescharacteristic
18C3×#946C2211yescharacteristic
19C3×#930C2211yescharacteristic
20C2×C3×C4⋊Dic3C2211yescharacteristic
21C3×#895C2211yescharacteristic
22C3×#894—21no—
23C3×#893—21no—
24C3×#892C2211yescharacteristic
25C3×#882—21no—
26C3×#67⋊C4—21no—
27C2×(C2×D3)⋊Dic3—21no—
28C2×Dic32—21no—
29#63841—21no—
30#63841—21no—
31#63841C2212yes—
32#63841C2212yes—
33C2×D3×C3⋊C8C2212yes—
34#63662C2212yescomplemented
35#63657C2211yescharacteristic
36#63635C2212yescomplemented
37C3×#835C2212yescomplemented
38C3×#830C2211yescharacteristic
39C3×#807C2212yescomplemented
40#63575C2212yescomplemented
41#63574C2212yescomplemented
42#63563C2212yescomplemented
43#63562C2212yescomplemented
44(C2×D3)⋊(C3⋊C8)C2212yes—
45C3×#525—31no—
46C2×#926—31no—
47#5627D311yescharacteristic
48C2×#892—31no—
49C3×#484—31no—
50#5233—32no—
51C2×#830—31no—
52#5045—32no—
53#5044D311yescharacteristic, complemented
54#5043—31no—
55#5043—31no—
56#5043—61no—
57#5035—32no—
58#5034—31no—
59#5033—31no—
60#4998—32no—
61#4968—31no—
62#4885—32no—
63#4880—32no—
64#4814—32no—
65#4804—32no—
66#4736—31no—
67D3×C22⋊C8—31no—
68C23×C3×D3D411yescharacteristic
69C2×C32×Q8C2311yescharacteristic
70C22×C32×C4C2311yescharacteristic
71C22×C3⋊Dic3—21no—
72C2×C3×#67—21no—
73C22×C3×Dic3—21no—
74C22×C3×Dic3D411yescharacteristic
75C3×#239—24no—
76C2×C3×D12—21no—
77C2×C3×D12—21no—
78C2×C3×D12C2311yescharacteristic
79C2×C3×C4×D3—24no—
80C2×C3×C4×D3—21no—
81C2×C3×C4×D3D411yescharacteristic
82C2×C3×C4×D3D411yescharacteristic
83C2×C3×C4×D3C2311yescharacteristic
84C2×C3×C4×D3C2311yescharacteristic
85C32×C4⋊C4—21no—
86C32×C4⋊C4C2311yescharacteristic
87C32×C22⋊C4—21no—
88C4⋊(C3⋊Dic3)—21no—
89C4⋊(C3⋊Dic3)C2312yes—
90C2×C3⋊(C3⋊C8)C2312yes—
91C3×(C2×D3)⋊C4—21no—
92C3×(C2×D3)⋊C4—21no—
93C3×(C2×D3)⋊C4—21no—
94C3×(C2×D3)⋊C4—21no—
95C3×C4⋊Dic3—21no—
96C3×C4⋊Dic3C2311yescharacteristic
97C3×C4⋊Dic3C2312yes—
98C3×Dic3⋊C4—21no—
99C3×C4×Dic3—21no—
100C2×C3×C3⋊C8C2312yes—
101(C2×D3)⋊Dic3—22no—
102(C2×D3)⋊Dic3—22no—
103Dic32—22no—
104Dic32—22no—
105D3×C3⋊C8—24no—
106#3091—91noSylow
107C2×C3×#41—31no—
108C23×C3×C4—31no—
109C2×#239C2×D311yescharacteristic
110C22×C4×D3C2×D311yescharacteristic
111C3×#123—61no—
112C3×C22⋊Q8—21no—
113C3×C22⋊Q8—31no—
114C3×C22⋊Q8C2×D311yescharacteristic
115C3×C22⋊Q8C2×D311yescharacteristic
116C3×#121—31no—
117C3×#121—31no—
118C3×#120—61no—
119C3×C4×D4—61no—
120C3×C4×D4—61no—
121C3×#117—31no—
122C2×C3×C4⋊C4—31no—
123C2×#217—61no—
124#946C2×D311yescharacteristic
125#930C2×D311yescharacteristic
126C2×(C2×D3)⋊C4—61no—
127#926—32no—
128#926—32no—
129#926—61no—
130#926—61no—
131C2×C4⋊Dic3—31no—
132C2×C4⋊Dic3—31no—
133C2×C4⋊Dic3—61no—
134C2×C4⋊Dic3C2×D311yescharacteristic
135C2×C4×Dic3—61no—
136C2×C4×Dic3—61no—
137C22×C3⋊C8—32no—
138C2×D3×C8—32no—
139#895C2×D311yescharacteristic
140#894—21no—
141#893—21no—
142#892—32no—
143#892—32no—
144#892—61no—
145#892—61no—
146#892C2×D311yescharacteristic
147#882—21no—
148#67⋊C4—21no—
149C3×#107—32no—
150C3×#103—32no—
151C3×C22⋊C8—31no—
152#835—32no—
153#835—32no—
154#835—62no—
155#835C2×D312yescomplemented
156#832—32no—
157#832—32no—
158#830—31no—
159#830—31no—
160#830—32no—
161#830—61no—
162#830C2×D311yescharacteristic
163#821—32no—
164(C2×D3)⋊C8—32no—
165#818—32no—
166#809—32no—
167#808—32no—
168#808—32no—
169#808—32no—
170#808—32no—
171#808—62no—
172#807—32no—
173#807C2×D312yescomplemented
174C23×C32C2×D411yescharacteristic
175C22×C3×D3—22no—
176C22×C3×D3—22no—
177C22×C3×D3—21no—
178C22×C3×D3—41no—
179C22×C3×D3C2×D411yescharacteristic
180C22×C3×D3C2×D411yescharacteristic
181C32×Q8—22no—
182C2×C32×C4—22no—
183C2×C32×C4—21no—
184C2×C32×C4—21no—
185C2×C32×C4C2×D411yescharacteristic
186C2×C32×C4C2411yescharacteristic
187C2×C3⋊Dic3—22no—
188C2×C3⋊Dic3—42no—
189C3×#67—42no—
190C2×C3×Dic3—22no—
191C2×C3×Dic3—22no—
192C2×C3×Dic3—21no—
193C2×C3×Dic3—42no—
194C2×C3×Dic3C2×D411yescharacteristic
195C2×C3×Dic3C2×D411yescharacteristic
196C3×D12—22no—
197C3×D12—42no—
198C3×D12—42no—
199C3×C4×D3—22no—
200C3×C4×D3—22no—
201C3×C4×D3—22no—
202C3×C4×D3—42no—
203C3×C4×D3#4114yes—
204C3⋊(C3⋊C8)—22no—
205C3×C3⋊C8—22no—
206#525—31no—
207C2×#117—91no—
208#485—92no—
209#484—31no—
210#483—91no—
211#482—92no—
212#481—91no—
213#480—91no—
214#427—92no—
215#416—92no—
216C2×C22⋊C8—91no—
217C24×C3—31no—
218C23×D3#6711yescharacteristic
219C3×#41—64no—
220C2×C3×Q8—21no—
221C2×C3×Q8C22×D311yescharacteristic
222C2×C3×Q8C22×D311yescharacteristic
223C2×C3×D4—31no—
224C2×C3×D4—61no—
225C2×C3×D4—61no—
226C2×C3×D4—61no—
227C22×C3×C4—21no—
228C22×C3×C4—31no—
229C22×C3×C4—31no—
230C22×C3×C4—31no—
231C22×C3×C4—31no—
232C22×C3×C4—31no—
233C22×C3×C4—64no—
234C22×C3×C4—61no—
235C22×C3×C4—61no—
236C22×C3×C4C22×D311yescharacteristic
237C22×C3×C4C22×D311yescharacteristic
238C2×#67—21no—
239C22×Dic3—21no—
240C22×Dic3—61no—
241C22×Dic3—61no—
242C22×Dic3—121no—
243C22×Dic3#6711yescharacteristic
244#239—24no—
245C2×D12—21no—
246C2×D12—21no—
247C2×D12C22×D311yescharacteristic
248C2×C4×D3—24no—
249C2×C4×D3—21no—
250C2×C4×D3#6711yescharacteristic
251C2×C4×D3#6711yescharacteristic
252C2×C4×D3C22×D311yescharacteristic
253C2×C4×D3C22×D311yescharacteristic
254C2×C3×C8—32no—
255C3×C4⋊C4—21no—
256C3×C4⋊C4—21no—
257C3×C4⋊C4—21no—
258C3×C4⋊C4—31no—
259C3×C4⋊C4—32no—
260C3×C4⋊C4—41no—
261C3×C4⋊C4—61no—
262C3×C4⋊C4—61no—
263C3×C4⋊C4C22×D311yescharacteristic
264C3×C4⋊C4C22×D311yescharacteristic
265C3×C22⋊C4—21no—
266C3×C22⋊C4—21no—
267C3×C22⋊C4—41no—
268C3×C22⋊C4—61no—
269C3×C22⋊C4—61no—
270C3×C22⋊C4—61no—
271C3×C22⋊C4—61no—
272C3×C42—61no—
273#217—62no—
274#217—62no—
275(C2×D3)⋊C4—21no—
276(C2×D3)⋊C4—21no—
277(C2×D3)⋊C4—21no—
278(C2×D3)⋊C4—21no—
279(C2×D3)⋊C4—62no—
280(C2×D3)⋊C4—62no—
281C4⋊Dic3—21no—
282C4⋊Dic3—32no—
283C4⋊Dic3—32no—
284C4⋊Dic3—62no—
285C4⋊Dic3—61no—
286C4⋊Dic3—61no—
287C4⋊Dic3—121no—
288C4⋊Dic3C22×D312yes—
289C4⋊Dic3C22×D311yescharacteristic
290Dic3⋊C4—21no—
291C4×Dic3—21no—
292C4×Dic3—62no—
293C4×Dic3—62no—
294C4×Dic3—62no—
295C4×Dic3—62no—
296C2×C3⋊C8—32no—
297C2×C3⋊C8—32no—
298C2×C3⋊C8—62no—
299C2×C3⋊C8—64no—
300C2×C3⋊C8C22×D312yes—
301D3×C8—64no—
302C22×C32—21no—
303C22×C32—21no—
304C22×C32#13512yes—
305C22×C32C22×D411yescharacteristic
306C2×C3×D3—21no—
307C2×C3×D3—21no—
308C2×C3×D3—21no—
309C2×C3×D3—21no—
310C2×C3×D3—21no—
311C2×C3×D3—44no—
312C2×C3×D3—41no—
313C2×C3×D3—41no—
314C2×C3×D3—41no—
315C2×C3×D3#12312yes—
316C32×C4—21no—
317C32×C4—41no—
318C32×C4—41no—
319C32×C4C2×#4112yes—
320C3⋊Dic3—42no—
321C3×Dic3—21no—
322C3×Dic3—41no—
323C3×Dic3—42no—
324C3×Dic3#12312yes—
325C2×#41—31no—
326C23×C4—31no—
327C22×C8—92no—
328#123—61no—
329C22⋊Q8—31no—
330C22⋊Q8D3211yescharacteristic
331#121—31no—
332#121—31no—
333#120—61no—
334C4×D4—61no—
335C4×D4—61no—
336#117—31no—
337#117—92no—
338#117—92no—
339#117—181no—
340#117—181no—
341C2×C4⋊C4—31no—
342C2×C4⋊C4—91no—
343C2×C22⋊C4—181no—
344C2×C42—181no—
345#107—32no—
346#107—92no—
347#106—92no—
348#106—92no—
349#103—32no—
350#103—92no—
351#102—92no—
352#102—92no—
353C22⋊C8—31no—
354C22⋊C8—92no—
355C22⋊C8—91no—
356C23×C3—21no—
357C23×C3—31no—
358C23×C3—31no—
359C23×C3—62no—
360C23×C3—62no—
361C23×C3—61no—
362C23×C3—121no—
363C23×C3D3×D411yescharacteristic
364C23×C3C2×#6711yescharacteristic
365C22×D3—22no—
366C22×D3—22no—
367C22×D3—21no—
368C22×D3—41no—
369C22×D3C2×#6711yescharacteristic
370C22×D3C2×#6711yescharacteristic
371C3×Q8—22no—
372C3×Q8—22no—
373C3×Q8—42no—
374C3×D4—62no—
375C3×D4—122no—
376C3×D4—122no—
377C3×D4—122no—
378C2×C3×C4—22no—
379C2×C3×C4—22no—
380C2×C3×C4—21no—
381C2×C3×C4—21no—
382C2×C3×C4—21no—
383C2×C3×C4—21no—
384C2×C3×C4—21no—
385C2×C3×C4—21no—
386C2×C3×C4—31no—
387C2×C3×C4—34no—
388C2×C3×C4—31no—
389C2×C3×C4—42no—
390C2×C3×C4—41no—
391C2×C3×C4—41no—
392C2×C3×C4—62no—
393C2×C3×C4—62no—
394C2×C3×C4—62no—
395C2×C3×C4—62no—
396C2×C3×C4—61no—
397C2×C3×C4—62no—
398C2×C3×C4—122no—
399C2×C3×C4—122no—
400C2×C3×C4D3×D411yescharacteristic
401C2×C3×C4C2×#6711yescharacteristic
402C2×C3×C4C23×D311yescharacteristic
403C2×C3×C4C23×D311yescharacteristic
404#67—42no—
405C2×Dic3—22no—
406C2×Dic3—22no—
407C2×Dic3—21no—
408C2×Dic3—42no—
409C2×Dic3—62no—
410C2×Dic3—62no—
411C2×Dic3—122no—
412C2×Dic3—122no—
413C2×Dic3—122no—
414C2×Dic3—242no—
415C2×Dic3C2×#6711yescharacteristic
416C2×Dic3C2×#6711yescharacteristic
417D12—22no—
418D12—42no—
419D12—42no—
420C4×D3—22no—
421C4×D3—22no—
422C4×D3—22no—
423C4×D3—42no—
424C4×D3#23714yes—
425C3×C8—62no—
426C3⋊C8—22no—
427C3⋊C8—62no—
428C3⋊C8—62no—
429C3⋊C8—122no—
430C2×C32—22no—
431C2×C32C2×#12311yescharacteristic
432C2×C32C2×#13511yescharacteristic
433C2×C32#57511yescharacteristic
434C3×D3—22no—
435C3×D3—41no—
436C3×D3—81no—
437C24—31no—
438#41—64no—
439C2×Q8C2×D3211yescharacteristic
440C2×D4—31no—
441C2×D4—61no—
442C2×D4—61no—
443C2×D4—61no—
444C22×C4—31no—
445C22×C4—31no—
446C22×C4—31no—
447C22×C4—31no—
448C22×C4—31no—
449C22×C4—64no—
450C22×C4—61no—
451C22×C4—61no—
452C22×C4—181no—
453C22×C4C2×D3211yescharacteristic
454C2×C8—32no—
455C2×C8—92no—
456C2×C8—184no—
457C4⋊C4—21no—
458C4⋊C4—32no—
459C4⋊C4—31no—
460C4⋊C4—61no—
461C4⋊C4—61no—
462C4⋊C4—92no—
463C4⋊C4—181no—
464C4⋊C4C2×D3211yescharacteristic
465C22⋊C4—21no—
466C22⋊C4—61no—
467C22⋊C4—61no—
468C22⋊C4—61no—
469C22⋊C4—61no—
470C22⋊C4—182no—
471C22⋊C4—182no—
472C42—61no—
473C42—182no—
474C42—182no—
475C22×C3—21no—
476C22×C3—21no—
477C22×C3—21no—
478C22×C3—21no—
479C22×C3—21no—
480C22×C3—22no—
481C22×C3—32no—
482C22×C3—41no—
483C22×C3—41no—
484C22×C3—61no—
485C22×C3—61no—
486C22×C3—61no—
487C22×C3—61no—
488C22×C3—61no—
489C22×C3—124no—
490C22×C3—121no—
491C22×C3—121no—
492C22×C3—121no—
493C22×C3#91912yes—
494C22×C3#95012yes—
495C22×C3C2×D3×D411yescharacteristic
496C22×C3C22×#6711yescharacteristic
497C2×D3—21no—
498C2×D3—21no—
499C2×D3—21no—
500C2×D3—21no—
501C2×D3—21no—
502C2×D3—44no—
503C2×D3—41no—
504C2×D3—41no—
505C2×D3—41no—
506C2×D3#93512yes—
507C3×C4—21no—
508C3×C4—21no—
509C3×C4—22no—
510C3×C4—32no—
511C3×C4—41no—
512C3×C4—41no—
513C3×C4—41no—
514C3×C4—41no—
515C3×C4—41no—
516C3×C4—61no—
517C3×C4—81no—
518C3×C4—81no—
519C3×C4—121no—
520C3×C4—122no—
521C3×C4C2×#23712yes—
522C3×C4D3×#4112yes—
523Dic3—21no—
524Dic3—42no—
525Dic3—41no—
526Dic3—122no—
527Dic3—122no—
528Dic3—242no—
529Dic3#93512yes—
530C32#309111yescharacteristic, complemented, Sylow
531C23—31no—
532C23—31no—
533C23—62no—
534C23—62no—
535C23—61no—
536C23—121no—
537C23D3×#6711yescharacteristic
538Q8—22no—
539D4—62no—
540D4—122no—
541D4—122no—
542D4—122no—
543C2×C4—22no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—31no—
547C2×C4—34no—
548C2×C4—31no—
549C2×C4—62no—
550C2×C4—62no—
551C2×C4—62no—
552C2×C4—62no—
553C2×C4—61no—
554C2×C4—62no—
555C2×C4—122no—
556C2×C4—122no—
557C2×C4—182no—
558C2×C4—362no—
559C2×C4D3×#6711yescharacteristic
560C2×C4C22×D3211yescharacteristic
561C8—62no—
562C8—182no—
563C2×C3—22no—
564C2×C3—22no—
565C2×C3—21no—
566C2×C3—21no—
567C2×C3—21no—
568C2×C3—42no—
569C2×C3—62no—
570C2×C3—121no—
571C2×C3—241no—
572C2×C3D3×#12311yescharacteristic
573C2×C3C2×#91911yescharacteristic
574C2×C3#577311yescharacteristic
575C2×C3C2×#93511yescharacteristic
576C2×C3C2×#95011yescharacteristic
577C2×C3#582211yescharacteristic
578D3—22no—
579D3—41no—
580D3—81no—
581C22—21no—
582C22—21no—
583C22—32no—
584C22—61no—
585C22—61no—
586C22—61no—
587C22—61no—
588C22—61no—
589C22—124no—
590C22—121no—
591C22—121no—
592C22—121no—
593C22#6394912yes—
594C22C2×D3×#6711yescharacteristic
595C4—21no—
596C4—32no—
597C4—41no—
598C4—41no—
599C4—61no—
600C4—121no—
601C4—122no—
602C4—362no—
603C4D3×#23712yes—
604C3—21no—
605C3#8146711yescharacteristic, complemented
606C3#8291311yescharacteristic, complemented
607C2—22no—
608C2—62no—
609C2—121no—
610C2—241no—
611C2D3×#93511yescharacteristic
612C2C2×#6394911yescharacteristic
613C2#1059546911yescharacteristic
subgroupG/H
centerC22C2×D3×#67
commutator, cosocleC2×C32×C4C24
FrattiniC2×C4C22×D32
FittingC32×C22⋊Q8C22
socleC22×C32C22×D4
prime radicalC3×#5627C2
hypercenterC22⋊Q8D32
hypocenterC32#3091
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. C2×#63841C2semidirect
3. #10592742C2semidirect
4. C3×#5044C2semidirect
5. #10591558C2semidirect
6. #10591556C2semidirect
7. D3×#830C2semidirect
8. #10591094C2semidirect
9. #10591091C2semidirect
10. #10590571C2semidirect
11. #10590446C2semidirect
34. #63662C22semidirect
36. #63635C22semidirect
37. C3×#835C22semidirect
39. C3×#807C22semidirect
40. #63575C22semidirect
41. #63574C22semidirect
42. #63563C22semidirect
43. #63562C22semidirect
53. #5044D3semidirect
155. #835C2×D3semidirect
173. #807C2×D3semidirect
530. C32#3091semidirect
605. C3#81467semidirect
606. C3#82913semidirect

Maximal subgroups

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

subgroupindex
C3×#56272
C2×#638412
#105927422
C3×#50442
#105915582
#105915562
D3×#8302
#105910942
#105910912
#105905712
#105904462
#829133
#814673