Finite Group Databasebeta

← #24991966

Subgroups — #24991966

Order 11008.

1136 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C22⋊(C43⋊C8)C211yesmaximal, characteristic, complemented
2C2×#14582778C211yesmaximal, characteristic
3C2×#14582622C212yesmaximal, complemented
4#19132437C212yesmaximal, complemented
5C43×#1779C211yesmaximal, characteristic, complemented
6C23×C22⋊D43C411yescharacteristic
7C23×C43⋊C8—21no—
8C22×C22⋊Dic43C411yescharacteristic
9C22×(C2×D43)⋊C4C411yescharacteristic
10C22×(C2×D43)⋊C4C2211yescharacteristic
11C2×C22⋊(C43⋊C8)—24no—
12C2×C22⋊(C43⋊C8)—21no—
13C2×C22⋊(C43⋊C8)C2212yes—
14#14582778—21no—
15#14582778—21no—
16#14582778—21no—
17#14582778—21no—
18C2×C43⋊#95C411yescharacteristic
19C2×C43⋊#95C2211yescharacteristic
20#14582622—24no—
21C24×C4×C43C2211yescharacteristic
22C2×C22⋊C8×C43—21no—
23C2×C22⋊C8×C43—21no—
24C2×C22⋊C8×C43C2212yes—
25C24×D43C2×C411yescharacteristic
26C22×C22⋊D43—21no—
27C22×C22⋊D43—21no—
28C22×C22⋊D43—22no—
29C22×C22⋊D43—42no—
30C22×C22⋊D43—42no—
31C22×C22⋊D43C814yescomplemented
32C23×Dic43C2×C411yescharacteristic
33C23×Dic43C2×C411yescharacteristic
34C23×Dic43C2×C411yescharacteristic
35C22×C43⋊C8—24no—
36C22×C43⋊C8—22no—
37C22×C43⋊C8—42no—
38C22×C43⋊C8—42no—
39C2×C22⋊Dic43—21no—
40C2×C22⋊Dic43—21no—
41C2×C22⋊Dic43—42no—
42C2×(C2×D43)⋊C4—21no—
43C2×(C2×D43)⋊C4—21no—
44C2×(C2×D43)⋊C4—22no—
45C2×(C2×D43)⋊C4—21no—
46C2×(C2×D43)⋊C4—21no—
47C2×(C2×D43)⋊C4—22no—
48C2×(C2×D43)⋊C4—42no—
49C22⋊(C43⋊C8)—24no—
50C22⋊(C43⋊C8)—22no—
51C22⋊(C43⋊C8)D414yescomplemented
52C43⋊#95—21no—
53C43⋊#95—21no—
54C43⋊#95—21no—
55C43⋊#95—21no—
56C25×C43C2×C411yescharacteristic
57C23×C4×C43—22no—
58C23×C4×C43—22no—
59C23×C4×C43—42no—
60C23×C4×C43—42no—
61C23×C4×C43C2×C411yescharacteristic
62C23×C4×C43D411yescharacteristic
63C23×C4×C43D411yescharacteristic
64C23×C4×C43D411yescharacteristic
65C23×C4×C43D411yescharacteristic
66C23×C4×C43C2311yescharacteristic
67C22×C8×C43—22no—
68C22⋊C8×C43—22no—
69C22⋊C8×C43—22no—
70C22⋊C8×C43—42no—
71C22⋊C8×C43—42no—
72C24×C43—22no—
73C24×C43—22no—
74C24×C43—22no—
75C24×C43—22no—
76C24×C43—41no—
77C24×C43—41no—
78C24×C43C22⋊C411yescharacteristic
79C24×C43C22⋊C411yescharacteristic
80C24×C43C2×C812yes—
81C24×C43M1612yes—
82C24×C43C22×C411yescharacteristic
83C23×D43—21no—
84C23×D43—21no—
85C23×D43—22no—
86C23×D43—22no—
87C23×D43—41no—
88C23×D43—41no—
89C23×D43—41no—
90C23×D43—41no—
91C23×D43C2×C812yescomplemented
92C22×C4×C43—22no—
93C22×C4×C43—21no—
94C22×C4×C43—21no—
95C22×C4×C43—22no—
96C22×C4×C43—22no—
97C22×C4×C43—21no—
98C22×C4×C43—22no—
99C22×C4×C43—22no—
100C22×C4×C43—21no—
101C22×C4×C43—22no—
102C22×C4×C43—22no—
103C22×C4×C43—22no—
104C22×C4×C43—41no—
105C22×C4×C43—41no—
106C22×C4×C43—41no—
107C22×C4×C43—41no—
108C22×C4×C43—41no—
109C22×C4×C43—41no—
110C22×C4×C43—41no—
111C22×C4×C43—48no—
112C22×C4×C43—48no—
113C22×C4×C43C22⋊C411yescharacteristic
114C22×C4×C43C22⋊C411yescharacteristic
115C22×C4×C43C2×D411yescharacteristic
116C22×C4×C43C2×D411yescharacteristic
117C22×C4×C43C2×D412yes—
118C22×C4×C43#4112yes—
119C2×C22⋊D43—24no—
120C2×C22⋊D43—24no—
121C2×C22⋊D43—48no—
122C2×C22⋊D43—48no—
123C2×C22⋊D43—42no—
124C2×C22⋊D43—42no—
125C2×C22⋊D43—42no—
126C2×C22⋊D43—42no—
127C22×Dic43—21no—
128C22×Dic43—21no—
129C22×Dic43—21no—
130C22×Dic43—22no—
131C22×Dic43—21no—
132C22×Dic43—22no—
133C22×Dic43—41no—
134C22×Dic43—41no—
135C22×Dic43—41no—
136C22×Dic43—41no—
137C22×Dic43—41no—
138C22×Dic43—41no—
139C22×Dic43C2×C812yescomplemented
140C2×C8×C43—42no—
141C2×C8×C43—42no—
142C2×C8×C43—42no—
143C22⋊Dic43—42no—
144C22⋊Dic43—42no—
145(C2×D43)⋊C4—42no—
146(C2×D43)⋊C4—42no—
147(C2×D43)⋊C4—41no—
148(C2×D43)⋊C4—42no—
149(C2×D43)⋊C4—41no—
150C2×C43⋊C8—24no—
151C2×C43⋊C8—48no—
152C2×C43⋊C8—42no—
153C2×C43⋊C8—42no—
154C23×C43—21no—
155C23×C43—21no—
156C23×C43—21no—
157C23×C43—21no—
158C23×C43—21no—
159C23×C43—21no—
160C23×C43—21no—
161C23×C43—21no—
162C23×C43—24no—
163C23×C43—24no—
164C23×C43—21no—
165C23×C43—21no—
166C23×C43—21no—
167C23×C43—21no—
168C23×C43—21no—
169C23×C43—21no—
170C23×C43—24no—
171C23×C43—24no—
172C23×C43—42no—
173C23×C43—42no—
174C23×C43—44no—
175C23×C43—44no—
176C23×C43—44no—
177C23×C43—44no—
178C23×C43C22⋊C812yes—
179C23×C43C22⋊C812yes—
180C23×C43#9912yes—
181C23×C43#10012yes—
182C23×C43C2×C22⋊C411yescharacteristic
183C23×C43C4×D412yes—
184C23×C43C22×C811yescharacteristic
185C23×C43C2×M1611yescharacteristic
186C23×C43#13112yes—
187C22×D43—22no—
188C22×D43—24no—
189C22×D43—42no—
190C22×D43—42no—
191C22×D43—41no—
192C22×D43—41no—
193C22×D43—41no—
194C22×D43—41no—
195C22×D43—41no—
196C22×D43—41no—
197C22×D43—42no—
198C22×D43—42no—
199C22×D43—42no—
200C22×D43—82no—
201C22×D43—82no—
202C22×D43—82no—
203C22×D43—82no—
204C2×C4×C43—24no—
205C2×C4×C43—24no—
206C2×C4×C43—21no—
207C2×C4×C43—21no—
208C2×C4×C43—21no—
209C2×C4×C43—21no—
210C2×C4×C43—21no—
211C2×C4×C43—21no—
212C2×C4×C43—21no—
213C2×C4×C43—21no—
214C2×C4×C43—21no—
215C2×C4×C43—42no—
216C2×C4×C43—42no—
217C2×C4×C43—44no—
218C2×C4×C43—44no—
219C2×C4×C43—44no—
220C2×C4×C43—42no—
221C2×C4×C43—42no—
222C2×C4×C43#9912yes—
223C2×C4×C43#12011yescharacteristic
224C2×C4×C43#12112yes—
225C2×C4×C43#12311yescharacteristic
226C22⋊D43—48no—
227C22⋊D43—48no—
228C2×Dic43—22no—
229C2×Dic43—42no—
230C2×Dic43—41no—
231C2×Dic43—41no—
232C2×Dic43—42no—
233C2×Dic43—41no—
234C2×Dic43—41no—
235C2×Dic43—41no—
236C2×Dic43—41no—
237C2×Dic43—41no—
238C2×Dic43—41no—
239C2×Dic43—82no—
240C2×Dic43—82no—
241C8×C43—82no—
242C43⋊C8—44no—
243#9079—431nomaximal, Sylow
244C22×C43—22no—
245C22×C43—22no—
246C22×C43—22no—
247C22×C43—22no—
248C22×C43—22no—
249C22×C43—21no—
250C22×C43—21no—
251C22×C43—21no—
252C22×C43—21no—
253C22×C43—22no—
254C22×C43—22no—
255C22×C43—22no—
256C22×C43—22no—
257C22×C43—22no—
258C22×C43—22no—
259C22×C43—22no—
260C22×C43—22no—
261C22×C43—48no—
262C22×C43—48no—
263C22×C43—41no—
264C22×C43—41no—
265C22×C43—41no—
266C22×C43—41no—
267C22×C43#32314yescomplemented
268C22×C43#38611yescharacteristic
269C22×C43C2×C22⋊C811yescharacteristic
270C22×C43#40811yescharacteristic
271C22×C43C2×#9911yescharacteristic
272C22×C43C2×#9911yescharacteristic
273C22×C43C2×#10011yescharacteristic
274C22×C43#41311yescharacteristic
275C22×C43C8×D412yes—
276C22×C43#43512yes—
277C2×D43—42no—
278C2×D43—42no—
279C2×D43—42no—
280C2×D43—84no—
281C2×D43—81no—
282C2×D43—81no—
283C2×D43—81no—
284C2×D43—81no—
285C2×D43—84no—
286C4×C43—21no—
287C4×C43—22no—
288C4×C43—22no—
289C4×C43—21no—
290C4×C43—41no—
291Dic43—42no—
292Dic43—81no—
293Dic43—81no—
294C22×C22⋊C8—431no—
295C2×#386—431no—
296#1779D4311yescharacteristic, complemented
297C2×#323—432no—
298#1280—432no—
299C2×C43—22no—
300C2×C43—22no—
301C2×C43—24no—
302C2×C43—42no—
303C2×C43C2×#32311yescharacteristic
304C2×C43#145911yescharacteristic
305C2×C43#146811yescharacteristic
306C2×C43#147111yescharacteristic
307C2×C43#186911yescharacteristic
308C2×C43#189811yescharacteristic
309C2×C43#190011yescharacteristic
310D43—82no—
311C23×D4—431no—
312C24×C4C2×D4311yescharacteristic
313C23×C8—861no—
314C22×C22⋊C4—431no—
315C22×C22⋊C4—431no—
316C22×C22⋊C4—431no—
317C2×C22⋊C8—21no—
318C2×C22⋊C8—21no—
319C2×C22⋊C8—432no—
320C2×C22⋊C8—864no—
321C2×C22⋊C8—861no—
322C2×C22⋊C8C2×D4312yes—
323#386—861no—
324#386—861no—
325#386—861no—
326#386—861no—
327C2×#95—431no—
328C2×#95—431no—
329#323—864no—
330C43#907911yescharacteristic, complemented, Sylow
331C25—431no—
332C25C4×D4311yescharacteristic
333C22×D4—434no—
334C22×D4—862no—
335C22×D4—861no—
336C22×D4—861no—
337C22×D4—1722no—
338C22×D4—1722no—
339C23×C4—22no—
340C23×C4—22no—
341C23×C4—42no—
342C23×C4—42no—
343C23×C4—431no—
344C23×C4—431no—
345C23×C4—431no—
346C23×C4C4×D4311yescharacteristic
347C23×C4D17211yescharacteristic
348C23×C4D17211yescharacteristic
349C23×C4C22⋊D4311yescharacteristic
350C23×C4C22⋊D4311yescharacteristic
351C23×C4C22×D4311yescharacteristic
352C22×C8—22no—
353C22×C8—864no—
354C22×C8—862no—
355C22×C8—1722no—
356C22×C8—1722no—
357C2×C22⋊C4—861no—
358C2×C22⋊C4—861no—
359C2×C22⋊C4—861no—
360C2×C22⋊C4—861no—
361C2×C22⋊C4—861no—
362C2×C22⋊C4—862no—
363C2×C22⋊C4—862no—
364C2×C22⋊C4—861no—
365C2×C22⋊C4—1722no—
366C2×C22⋊C4—1722no—
367C22⋊C8—22no—
368C22⋊C8—22no—
369C22⋊C8—42no—
370C22⋊C8—42no—
371C22⋊C8—434no—
372C22⋊C8—864no—
373C22⋊C8—862no—
374#95—861no—
375#95—861no—
376#95—861no—
377#95—861no—
378C24—22no—
379C24—22no—
380C24—22no—
381C24—22no—
382C24—41no—
383C24—41no—
384C24—432no—
385C24—862no—
386C24—862no—
387C24—861no—
388C24—861no—
389C24—1721no—
390C24—1721no—
391C24—1721no—
392C24—1721no—
393C24C8×D4312yes—
394C24#1062187212yes—
395C24(C2×D43)⋊C411yescharacteristic
396C24(C2×D43)⋊C411yescharacteristic
397C24C2×C4×D4311yescharacteristic
398C2×D4—864no—
399C2×D4—864no—
400C2×D4—1728no—
401C2×D4—1722no—
402C2×D4—1722no—
403C2×D4—1722no—
404C2×D4—1722no—
405C2×D4—1728no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—21no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—21no—
418C22×C4—41no—
419C22×C4—41no—
420C22×C4—41no—
421C22×C4—41no—
422C22×C4—41no—
423C22×C4—41no—
424C22×C4—41no—
425C22×C4—48no—
426C22×C4—48no—
427C22×C4—432no—
428C22×C4—861no—
429C22×C4—861no—
430C22×C4—862no—
431C22×C4—862no—
432C22×C4—861no—
433C22×C4—861no—
434C22×C4—1721no—
435C22×C4—1721no—
436C22×C4—1721no—
437C22×C4—1721no—
438C22×C4—1721no—
439C22×C4—1721no—
440C22×C4(C2×D43)⋊C411yescharacteristic
441C22×C4(C2×D43)⋊C411yescharacteristic
442C22×C4C2×D17211yescharacteristic
443C22×C4D4×D4312yes—
444C22×C4#1062190012yes—
445C22×C4C2×C22⋊D4311yescharacteristic
446C2×C8—42no—
447C2×C8—42no—
448C2×C8—42no—
449C2×C8—864no—
450C2×C8—1728no—
451C2×C8—1722no—
452C2×C8—1722no—
453C22⋊C4—1722no—
454C22⋊C4—1722no—
455C22⋊C4—1722no—
456C22⋊C4—1722no—
457C22⋊C4—1721no—
458C22⋊C4—1722no—
459C22⋊C4—1721no—
460C23—21no—
461C23—21no—
462C23—21no—
463C23—21no—
464C23—21no—
465C23—21no—
466C23—21no—
467C23—21no—
468C23—21no—
469C23—21no—
470C23—21no—
471C23—21no—
472C23—24no—
473C23—24no—
474C23—21no—
475C23—24no—
476C23—24no—
477C23—21no—
478C23—42no—
479C23—42no—
480C23—44no—
481C23—44no—
482C23—44no—
483C23—44no—
484C23—864no—
485C23—862no—
486C23—3442no—
487C23—3442no—
488C23—3442no—
489C23—3442no—
490C23—1722no—
491C23—1722no—
492C23—1722no—
493C23—1722no—
494C23—1721no—
495C23—1721no—
496C23—1721no—
497C23—1721no—
498C23—1721no—
499C23—1722no—
500C23—1721no—
501C23(C2×D43)⋊C812yes—
502C23(C2×D43)⋊C812yes—
503C23#1306323912yes—
504C23#1306324112yes—
505C23C2×(C2×D43)⋊C411yescharacteristic
506C23(C22⋊D43)⋊C412yes—
507C23C2×C8×D4311yescharacteristic
508C23C2×#1062187211yescharacteristic
509C23#1306333312yes—
510D4—1728no—
511D4—1728no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—44no—
526C2×C4—44no—
527C2×C4—44no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—862no—
531C2×C4—3442no—
532C2×C4—3442no—
533C2×C4—1721no—
534C2×C4—1721no—
535C2×C4—1721no—
536C2×C4—1721no—
537C2×C4—1721no—
538C2×C4—1721no—
539C2×C4—1722no—
540C2×C4—1721no—
541C2×C4—1721no—
542C2×C4—1722no—
543C2×C4#1306323912yes—
544C2×C4#1306329511yescharacteristic
545C2×C4#1306329712yes—
546C2×C4#1306330911yescharacteristic
547C8—82no—
548C8—1724no—
549C22—22no—
550C22—22no—
551C22—22no—
552C22—22no—
553C22—22no—
554C22—22no—
555C22—22no—
556C22—22no—
557C22—22no—
558C22—21no—
559C22—21no—
560C22—21no—
561C22—21no—
562C22—22no—
563C22—22no—
564C22—22no—
565C22—22no—
566C22—48no—
567C22—48no—
568C22—41no—
569C22—41no—
570C22—41no—
571C22—41no—
572C22—3444no—
573C22—3444no—
574C22—3441no—
575C22—3441no—
576C22—3441no—
577C22—3441no—
578C22—1722no—
579C22—1722no—
580C22—1722no—
581C22#1458262214yescomplemented
582C22#1458277611yescharacteristic
583C22C2×(C2×D43)⋊C811yescharacteristic
584C22#1458286711yescharacteristic
585C22C2×#1306323911yescharacteristic
586C22C2×#1306323911yescharacteristic
587C22C2×#1306324111yescharacteristic
588C22#1458288111yescharacteristic
589C22(C22⋊D43)⋊C812yes—
590C22#1458296712yes—
591C4—22no—
592C4—21no—
593C4—22no—
594C4—21no—
595C4—41no—
596C4—3441no—
597C4—3441no—
598C4—1722no—
599C2—22no—
600C2—22no—
601C2—24no—
602C2—42no—
603C2—3442no—
604C2C2×#1458262211yescharacteristic
605C2#1913291111yescharacteristic
606C2#1913294411yescharacteristic
607C2#1913295611yescharacteristic
608C2#1913488511yescharacteristic
609C2#1913504611yescharacteristic
610C2#1913505611yescharacteristic
subgroupG/H
centerC23C2×(C2×D43)⋊C4
commutatorC23×C43C22×C8
FrattiniC23×C4C22×D43
FittingC43×#1779C2
socleC23×C43C2×C22⋊C4
prime radicalC23×C22⋊D43C4
cosocleC23×C4×C43C23
hypercenter#1779D43
hypocenterC43#9079
LCSG ▷ C23×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C23 ◁ C24×C4 ◁ #1779

Complements

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

NH ≅ G/Ntype
1. C22×C22⋊(C43⋊C8)C2semidirect
3. C2×#14582622C2semidirect
4. #19132437C2semidirect
5. C43×#1779C2semidirect
31. C22×C22⋊D43C8semidirect
51. C22⋊(C43⋊C8)D4semidirect
91. C23×D43C2×C8semidirect
139. C22×Dic43C2×C8semidirect
267. C22×C43#323semidirect
296. #1779D43semidirect
330. C43#9079semidirect
581. C22#14582622semidirect

Maximal subgroups

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

subgroupindex
C22×C22⋊(C43⋊C8)2
C2×#145827782
C2×#145826222
#191324372
C43×#17792
#907943