Finite Group Databasebeta

← #11870874

Subgroups — #11870874

Order 1152.

960 conjugacy classes of subgroups.

The subgroup lattice has 603 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#5624C211yesmaximal, characteristic, complemented
2C2×Dic3×D12C211yesmaximal, characteristic, complemented
3#10592743C211yesmaximal, characteristic, complemented
4C3×#5046C211yesmaximal, characteristic, complemented
5#10591567C212yesmaximal, complemented
6Dic3×#215C212yesmaximal, complemented
7#10591160C211yesmaximal, characteristic
8#10591092C211yesmaximal, characteristic, complemented
9#10591053C211yesmaximal, characteristic, complemented
10#10590467C212yesmaximal, complemented
11#10590452C212yesmaximal, complemented
12#83141—31nomaximal
13#80133—31nomaximal
14C22×C3×D12C2211yescharacteristic
15C22×D3×Dic3—21no—
16C32×C22⋊Q8C2211yescharacteristic
17C2×C4⋊(C3⋊Dic3)C2211yescharacteristic
18C3×#947C2211yescharacteristic, complemented
19C3×Q8×Dic3C2211yescharacteristic, complemented
20C2×C3×#215C2212yes—
21C3×#930C2211yescharacteristic
22C2×C3×C4×Dic3C2211yescharacteristic
23C3×#898—21no—
24C3×#893C2211yescharacteristic, complemented
25C3×#892—21no—
26C3×#890C2211yescharacteristic, complemented
27C3×#886—21no—
28C3×#67⋊C4—21no—
29C2×(C2×D3)⋊Dic3—21no—
30Dic3×D12—21no—
31Dic3×D12—21no—
32Dic3×D12C2212yes—
33Dic3×D12C2212yes—
34C2×#3713C2212yes—
35#63662C2212yescomplemented
36#63657C2211yescharacteristic
37#63634C2212yescomplemented
38C3×#830C2211yescharacteristic
39C3×#809C2212yes—
40#63573C2212yescomplemented
41Dic3⋊(C3⋊C8)C2212yescomplemented
42#63563C2212yescomplemented
43#63562C2212yes—
44Dic3×C3⋊C8C2212yescomplemented
45C3×#529—31no—
46C2×D4×Dic3—31no—
47#5624D311yescharacteristic, complemented
48C2×C4×D12—31no—
49C3×#450—31no—
50#5217—31no—
51#5165—32no—
52Dic3×SD16—32no—
53#5066—32no—
54#5057—32no—
55#5046D311yescharacteristic, complemented
56#5044—31no—
57#5044—31no—
58#5044—61no—
59#5036—31no—
60#5034—31no—
61C4×#215—32no—
62C4×#213—32no—
63#4925—31no—
64#4763—32no—
65#4756—32no—
66#4735—31no—
67#4718—31no—
68C23×C3×D3—21no—
69C2×C32×Q8C2311yescharacteristic
70C22×C32×C4C2311yescharacteristic
71C22×C3⋊Dic3—21no—
72C2×C3×#67—21no—
73C22×C3×Dic3D411yescharacteristic
74C22×C3×Dic3D411yescharacteristic
75C2×C3×D12—21no—
76C2×C3×D12—24no—
77C2×C3×D12C2311yescharacteristic
78C2×C3×D12C2312yes—
79C2×C3×C4×D3—21no—
80C2×D3×Dic3—22no—
81C2×D3×Dic3—22no—
82C2×D3×Dic3—44no—
83C32×C4⋊C4—21no—
84C32×C4⋊C4C2311yescharacteristic
85C32×C22⋊C4—21no—
86C4⋊(C3⋊Dic3)—21no—
87C4⋊(C3⋊Dic3)C2312yes—
88C2×C3⋊(C3⋊C8)C2312yes—
89C3×#215—24no—
90C3×(C2×D3)⋊C4—21no—
91C3×(C2×D3)⋊C4—21no—
92C3×(C2×D3)⋊C4—21no—
93C3×(C2×D3)⋊C4—21no—
94C3×C4⋊Dic3—21no—
95C3×C4⋊Dic3—21no—
96C3×C4⋊Dic3C2311yescharacteristic
97C3×Dic3⋊C4—21no—
98C3×C4×Dic3—21no—
99C3×C4×Dic3D411yescharacteristic
100C3×C4×Dic3D411yescharacteristic
101C3×C4×Dic3C2311yescharacteristic
102C3×C4×Dic3C2311yescharacteristic
103C2×C3×C3⋊C8C2312yes—
104(C2×D3)⋊Dic3—22no—
105(C2×D3)⋊Dic3—22no—
106#3713—24no—
107#3105—91noSylow
108C22×C3×D4—31no—
109C23×Dic3—61no—
110C22×D12C2×D311yescharacteristic
111C22×C4×D3—61no—
112C2×C3×SD16—32no—
113C3×#126—61no—
114C3×#125—31no—
115C3×#124—31no—
116C3×#123—61no—
117C3×C22⋊Q8—21no—
118C3×C22⋊Q8C2×D311yescharacteristic
119C3×C22⋊Q8C2×D311yescharacteristic
120C3×#121—31no—
121C3×C4×Q8—31no—
122C3×C4×D4—31no—
123C3×C4×D4—61no—
124C3×#117—61no—
125C2×C3×C42—31no—
126C2×#217—61no—
127#947C2×D311yescharacteristic, complemented
128Q8×Dic3C2×D311yescharacteristic, complemented
129C2×#215C2×D312yes—
130D4×Dic3—32no—
131D4×Dic3—32no—
132D4×Dic3—61no—
133D4×Dic3—61no—
134C2×#213—32no—
135C2×#213—32no—
136#930C2×D311yescharacteristic
137C2×(C2×D3)⋊C4—61no—
138C2×C4⋊Dic3—31no—
139C2×C4⋊Dic3—31no—
140C2×C4⋊Dic3—61no—
141C2×C4×Dic3C2×D311yescharacteristic
142#898—21no—
143#893C2×D311yescharacteristic, complemented
144#892—21no—
145#890C2×D311yescharacteristic, complemented
146#886—21no—
147#67⋊C4—21no—
148C4×D12—32no—
149C4×D12—32no—
150C4×D12—61no—
151C4×D12—61no—
152C3×#102—32no—
153C3×C22⋊C8—31no—
154#835—32no—
155#835—32no—
156#835—62no—
157#832—32no—
158#832—32no—
159#830—31no—
160#830—31no—
161#830—61no—
162#830C2×D311yescharacteristic
163#821—32no—
164#817—32no—
165Dic3⋊C8—32no—
166C8×Dic3—32no—
167#809—32no—
168#809C2×D312yes—
169#808—32no—
170#807—32no—
171#807—32no—
172#807—62no—
173C4⋊(C3⋊C8)—32no—
174C4×C3⋊C8—32no—
175C23×C32C2×D411yescharacteristic
176C22×C3×D3—24no—
177C22×C3×D3—22no—
178C22×C3×D3—21no—
179C22×C3×D3—42no—
180C22×C3×D3—42no—
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×Dic3C2×D411yescharacteristic
194C2×C3×Dic3C2×D411yescharacteristic
195C2×C3×Dic3C2×D411yescharacteristic
196C2×C3×Dic3C2×D411yescharacteristic
197C3×D12—24no—
198C3×D12—22no—
199C3×D12—22no—
200C3×D12#4114yes—
201C3×C4×D3—42no—
202D3×Dic3—44no—
203D3×Dic3—44no—
204C3⋊(C3⋊C8)—22no—
205C3×C3⋊C8—22no—
206#529—31no—
207C2×C4×D4—91no—
208#484—91no—
209#481—91no—
210#478—92no—
211#463—92no—
212#450—31no—
213#447—91no—
214C4×SD16—92no—
215C4×D8—92no—
216#431—91no—
217C24×C3—61no—
218C23×D3—21no—
219C2×C3×Q8—21no—
220C2×C3×Q8C22×D311yescharacteristic
221C2×C3×Q8C22×D311yescharacteristic
222C2×C3×D4—32no—
223C2×C3×D4—31no—
224C2×C3×D4—64no—
225C2×C3×D4—61no—
226C2×C3×D4—61no—
227C22×C3×C4—21no—
228C22×C3×C4—31no—
229C22×C3×C4—31no—
230C22×C3×C4—61no—
231C22×C3×C4C22×D311yescharacteristic
232C22×C3×C4C22×D311yescharacteristic
233C2×#67—21no—
234C22×Dic3—61no—
235C22×Dic3—61no—
236C22×Dic3—62no—
237C22×Dic3—62no—
238C22×Dic3—121no—
239C22×Dic3—124no—
240C22×Dic3#6711yescharacteristic
241C22×Dic3#6711yescharacteristic
242C2×D12—21no—
243C2×D12—24no—
244C2×D12C22×D311yescharacteristic
245C2×D12C22×D312yes—
246C2×C4×D3—21no—
247C2×C4×D3—62no—
248C2×C4×D3—62no—
249C2×C4×D3—124no—
250C3×SD16—64no—
251C2×C3×C8—32no—
252C3×C4⋊C4—21no—
253C3×C4⋊C4—21no—
254C3×C4⋊C4—21no—
255C3×C4⋊C4—31no—
256C3×C4⋊C4—41no—
257C3×C4⋊C4—61no—
258C3×C4⋊C4—61no—
259C3×C4⋊C4—61no—
260C3×C4⋊C4C22×D311yescharacteristic
261C3×C4⋊C4C22×D311yescharacteristic
262C3×C22⋊C4—21no—
263C3×C22⋊C4—21no—
264C3×C22⋊C4—41no—
265C3×C22⋊C4—61no—
266C3×C22⋊C4—61no—
267C3×C22⋊C4—61no—
268C3×C22⋊C4—61no—
269C3×C42—31no—
270C3×C42—31no—
271C3×C42—31no—
272C3×C42—31no—
273C3×C42—61no—
274#217—62no—
275#217—62no—
276#215—24no—
277#213—64no—
278#213—64no—
279(C2×D3)⋊C4—21no—
280(C2×D3)⋊C4—21no—
281(C2×D3)⋊C4—21no—
282(C2×D3)⋊C4—21no—
283(C2×D3)⋊C4—62no—
284(C2×D3)⋊C4—62no—
285C4⋊Dic3—21no—
286C4⋊Dic3—21no—
287C4⋊Dic3—32no—
288C4⋊Dic3—32no—
289C4⋊Dic3—62no—
290C4⋊Dic3—61no—
291C4⋊Dic3—61no—
292C4⋊Dic3—121no—
293C4⋊Dic3C22×D311yescharacteristic
294Dic3⋊C4—21no—
295C4×Dic3—21no—
296C4×Dic3#6711yescharacteristic
297C4×Dic3#6711yescharacteristic
298C4×Dic3C22×D311yescharacteristic
299C4×Dic3C22×D311yescharacteristic
300C2×C3⋊C8—32no—
301C2×C3⋊C8—32no—
302C2×C3⋊C8—62no—
303C2×C3⋊C8C22×D312yes—
304C22×C32—21no—
305C22×C32—21no—
306C22×C32#13512yes—
307C22×C32C22×D411yescharacteristic
308C2×C3×D3—24no—
309C2×C3×D3—48no—
310C2×C3×D3—42no—
311C2×C3×D3—42no—
312C2×C3×D3—41no—
313C2×C3×D3—41no—
314C2×C3×D3—41no—
315C32×C4—21no—
316C32×C4—41no—
317C32×C4—41no—
318C32×C4C2×#4112yes—
319C3⋊Dic3—42no—
320C3×Dic3—21no—
321C3×Dic3—21no—
322C3×Dic3—21no—
323C3×Dic3—41no—
324C3×Dic3AGL(1,ℤ/8)12yes—
325C22×D4—31no—
326C23×C4—181no—
327C2×SD16—32no—
328C2×D8—92no—
329#126—61no—
330#125—31no—
331#124—31no—
332#123—61no—
333C22⋊Q8D3211yescharacteristic
334#121—31no—
335C4×Q8—31no—
336C4×D4—31no—
337C4×D4—61no—
338C4×D4—92no—
339C4×D4—92no—
340C4×D4—181no—
341C4×D4—181no—
342#117—61no—
343C2×C4⋊C4—91no—
344C2×C22⋊C4—181no—
345C2×C42—31no—
346#107—92no—
347#106—92no—
348C4⋊C8—92no—
349#103—92no—
350#102—32no—
351#102—92no—
352#102—92no—
353C22⋊C8—31no—
354C22⋊C8—91no—
355C4×C8—92no—
356C23×C3—21no—
357C23×C3—64no—
358C23×C3—62no—
359C23×C3—61no—
360C23×C3—122no—
361C23×C3—122no—
362C23×C3D3×D411yescharacteristic
363C23×C3C2×#6711yescharacteristic
364C22×D3—24no—
365C22×D3—22no—
366C22×D3—21no—
367C22×D3—42no—
368C22×D3—42no—
369C3×Q8—22no—
370C3×Q8—22no—
371C3×Q8—42no—
372C3×D4—34no—
373C3×D4—64no—
374C3×D4—62no—
375C3×D4—62no—
376C3×D4—122no—
377C2×C3×C4—22no—
378C2×C3×C4—22no—
379C2×C3×C4—21no—
380C2×C3×C4—21no—
381C2×C3×C4—21no—
382C2×C3×C4—21no—
383C2×C3×C4—21no—
384C2×C3×C4—21no—
385C2×C3×C4—31no—
386C2×C3×C4—31no—
387C2×C3×C4—31no—
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—61no—
395C2×C3×C4—122no—
396C2×C3×C4D3×D411yescharacteristic
397C2×C3×C4C2×#6711yescharacteristic
398C2×C3×C4C23×D311yescharacteristic
399C2×C3×C4C23×D311yescharacteristic
400#67—42no—
401C2×Dic3—22no—
402C2×Dic3—22no—
403C2×Dic3—21no—
404C2×Dic3—62no—
405C2×Dic3—62no—
406C2×Dic3—122no—
407C2×Dic3—124no—
408C2×Dic3—122no—
409C2×Dic3—122no—
410C2×Dic3—124no—
411C2×Dic3—242no—
412C2×Dic3C2×#6711yescharacteristic
413C2×Dic3C2×#6711yescharacteristic
414C2×Dic3C2×#6711yescharacteristic
415C2×Dic3C2×#6711yescharacteristic
416D12—24no—
417D12—22no—
418D12—22no—
419D12#23714yes—
420C4×D3—42no—
421C4×D3—124no—
422C4×D3—124no—
423C3×C8—62no—
424C3⋊C8—22no—
425C3⋊C8—62no—
426C3⋊C8—62no—
427C3⋊C8—122no—
428C2×C32—22no—
429C2×C32#52511yescharacteristic
430C2×C32C2×#13511yescharacteristic
431C2×C32C2×AGL(1,ℤ/8)11yescharacteristic
432C3×D3—44no—
433C3×D3—81no—
434C24—61no—
435C2×Q8C2×D3211yescharacteristic
436C2×D4—32no—
437C2×D4—31no—
438C2×D4—64no—
439C2×D4—61no—
440C2×D4—61no—
441C22×C4—31no—
442C22×C4—31no—
443C22×C4—61no—
444C22×C4—182no—
445C22×C4—182no—
446C22×C4—181no—
447C22×C4—364no—
448C22×C4C2×D3211yescharacteristic
449SD16—64no—
450D8—184no—
451C2×C8—32no—
452C2×C8—92no—
453C4⋊C4—21no—
454C4⋊C4—31no—
455C4⋊C4—61no—
456C4⋊C4—61no—
457C4⋊C4—61no—
458C4⋊C4—92no—
459C4⋊C4—181no—
460C4⋊C4C2×D3211yescharacteristic
461C22⋊C4—21no—
462C22⋊C4—61no—
463C22⋊C4—61no—
464C22⋊C4—61no—
465C22⋊C4—61no—
466C22⋊C4—182no—
467C22⋊C4—182no—
468C42—31no—
469C42—31no—
470C42—31no—
471C42—31no—
472C42—61no—
473C22×C3—21no—
474C22×C3—21no—
475C22×C3—21no—
476C22×C3—21no—
477C22×C3—21no—
478C22×C3—22no—
479C22×C3—41no—
480C22×C3—41no—
481C22×C3—64no—
482C22×C3—128no—
483C22×C3—122no—
484C22×C3—122no—
485C22×C3—121no—
486C22×C3—121no—
487C22×C3—121no—
488C22×C3#91612yes—
489C22×C3#95012yes—
490C22×C3C2×D3×D411yescharacteristic
491C22×C3C22×#6711yescharacteristic
492C2×D3—24no—
493C2×D3—48no—
494C2×D3—42no—
495C2×D3—42no—
496C2×D3—41no—
497C2×D3—41no—
498C2×D3—41no—
499C3×C4—21no—
500C3×C4—21no—
501C3×C4—22no—
502C3×C4—32no—
503C3×C4—41no—
504C3×C4—41no—
505C3×C4—41no—
506C3×C4—41no—
507C3×C4—41no—
508C3×C4—61no—
509C3×C4—61no—
510C3×C4—61no—
511C3×C4—81no—
512C3×C4—81no—
513C3×C4—121no—
514C3×C4C2×#23712yes—
515C3×C4D3×#4112yes—
516Dic3—21no—
517Dic3—21no—
518Dic3—21no—
519Dic3—41no—
520Dic3—122no—
521Dic3—122no—
522Dic3—242no—
523Dic3#94912yes—
524C32#310511yescharacteristic, complemented, Sylow
525C23—64no—
526C23—62no—
527C23—61no—
528C23—122no—
529C23—122no—
530C23D3×#6711yescharacteristic
531Q8—22no—
532D4—34no—
533D4—64no—
534D4—62no—
535D4—62no—
536D4—122no—
537C2×C4—22no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—31no—
541C2×C4—31no—
542C2×C4—31no—
543C2×C4—31no—
544C2×C4—62no—
545C2×C4—62no—
546C2×C4—61no—
547C2×C4—122no—
548C2×C4—182no—
549C2×C4—364no—
550C2×C4—364no—
551C2×C4—362no—
552C2×C4D3×#6711yescharacteristic
553C2×C4C22×D3211yescharacteristic
554C8—62no—
555C8—182no—
556C2×C3—22no—
557C2×C3—22no—
558C2×C3—21no—
559C2×C3—21no—
560C2×C3—21no—
561C2×C3—42no—
562C2×C3—124no—
563C2×C3—241no—
564C2×C3#558611yescharacteristic
565C2×C3C2×#91611yescharacteristic
566C2×C3D3×AGL(1,ℤ/8)11yescharacteristic
567C2×C3C2×#94911yescharacteristic
568C2×C3C2×#95011yescharacteristic
569C2×C3#582811yescharacteristic
570D3—44no—
571D3—81no—
572C22—21no—
573C22—21no—
574C22—64no—
575C22—128no—
576C22—122no—
577C22—122no—
578C22—121no—
579C22—121no—
580C22—121no—
581C22#6394712yes—
582C22C2×D3×#6711yescharacteristic
583C4—21no—
584C4—32no—
585C4—41no—
586C4—41no—
587C4—61no—
588C4—61no—
589C4—61no—
590C4—121no—
591C4—362no—
592C4D3×#23712yes—
593C3—21no—
594C3#8013311yescharacteristic, complemented
595C3#8314111yescharacteristic, complemented
596C2—22no—
597C2—124no—
598C2—241no—
599C2#1059515411yescharacteristic
600C2C2×#6394711yescharacteristic
601C2D3×#94911yescharacteristic
subgroupG/H
centerC22C2×D3×#67
commutator, cosocleC2×C32×C4C24
FrattiniC2×C4C22×D32
FittingC32×C22⋊Q8C22
socleC22×C32C22×D4
prime radical#10591160C2
hypercenterC22⋊Q8D32
hypocenterC32#3105
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
1. C3×#5624C2semidirect
2. C2×Dic3×D12C2semidirect
3. #10592743C2semidirect
4. C3×#5046C2semidirect
5. #10591567C2semidirect
6. Dic3×#215C2semidirect
8. #10591092C2semidirect
9. #10591053C2semidirect
10. #10590467C2semidirect
11. #10590452C2semidirect
18. C3×#947C22semidirect
19. C3×Q8×Dic3C22semidirect
24. C3×#893C22semidirect
26. C3×#890C22semidirect
35. #63662C22semidirect
37. #63634C22semidirect
40. #63573C22semidirect
41. Dic3⋊(C3⋊C8)C22semidirect
42. #63563C22semidirect
44. Dic3×C3⋊C8C22semidirect
47. #5624D3semidirect
55. #5046D3semidirect
127. #947C2×D3semidirect
128. Q8×Dic3C2×D3semidirect
143. #893C2×D3semidirect
145. #890C2×D3semidirect
524. C32#3105semidirect
594. C3#80133semidirect
595. C3#83141semidirect

Maximal subgroups

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

subgroupindex
C3×#56242
C2×Dic3×D122
#105927432
C3×#50462
#105915672
Dic3×#2152
#105911602
#105910922
#105910532
#105904672
#105904522
#831413
#801333