Finite Group Databasebeta

← #25003116

Subgroups — #25003116

Order 11008.

1630 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
1C2×#14583410C211yesmaximal, characteristic
2C22×#13063253C212yesmaximal, complemented
3C2×#14582863C211yesmaximal, characteristic, complemented
4C2×#14582659C212yesmaximal, complemented
5C2×C4×C22⋊C4×C43C211yesmaximal, characteristic, complemented
6C22×#10621898C2211yescharacteristic
7C22×#10621877C2212yes—
8#14583410—24no—
9#14583410—24no—
10C2×#13063308—21no—
11C2×#13063304C412yescomplemented
12C2×#13063299C412yescomplemented
13C2×#13063294—21no—
14C2×C4×C22⋊D43—22no—
15C2×#13063284C2211yescharacteristic
16C2×#13063253—24no—
17C2×#13063253—24no—
18C2×#13063253—24no—
19#14582863—24no—
20C2×C22⋊(C43⋊C8)—22no—
21#14582659—24no—
22C24×C4×C43C2211yescharacteristic
23C22×C22⋊C4×C43—21no—
24C22×C42×C43C2212yes—
25C4×C22⋊C4×C43—24no—
26C2×#95×C43—21no—
27C2×#10621898—22no—
28C2×#10621898—22no—
29C2×#10621898—22no—
30C2×#10621898—22no—
31C2×#10621898—22no—
32C2×#10621898D414yes—
33C22×Dic86C2×C411yescharacteristic
34C22×D172C2×C411yescharacteristic
35C22×C22⋊D43—22no—
36C22×C22⋊D43—21no—
37C22×C4×D43—21no—
38C2×#10621877—22no—
39C2×#10621877—22no—
40C2×#10621877—22no—
41C2×#10621877—22no—
42C2×#10621877—22no—
43C2×#10621877D414yescomplemented
44C22×C43⋊C8—22no—
45#13063308—28no—
46#13063304—24no—
47#13063304—24no—
48#13063299—24no—
49#13063299—24no—
50#13063294—28no—
51C4×C22⋊D43—216no—
52#13063284—22no—
53#13063284D414yes—
54C2×C4⋊Dic43C2×C412yes—
55C2×Dic43⋊C4—22no—
56C2×C22⋊Dic43—22no—
57C2×C22⋊Dic43—21no—
58C2×(C2×D43)⋊C4—22no—
59C2×C4×Dic43—21no—
60#13063253—216no—
61C22⋊(C43⋊C8)—28no—
62C25×C43D411yescharacteristic
63C23×C4×C43—22no—
64C23×C4×C43—22no—
65C23×C4×C43—22no—
66C23×C4×C43—22no—
67C23×C4×C43—22no—
68C23×C4×C43—22no—
69C23×C4×C43—22no—
70C23×C4×C43C2×C412yes—
71C23×C4×C43D412yes—
72C23×C4×C43D411yescharacteristic
73C23×C4×C43C2311yescharacteristic
74C2×C22⋊C4×C43—28no—
75C2×C22⋊C4×C43—24no—
76C2×C42×C43—22no—
77C2×C42×C43—22no—
78C2×C42×C43—22no—
79C2×C42×C43—22no—
80C2×C42×C43—22no—
81C2×C42×C43D414yescomplemented
82#95×C43—24no—
83C24×C43—24no—
84C24×C43—21no—
85C24×C43—21no—
86C24×C43—22no—
87C24×C43—22no—
88C24×C43—42no—
89C24×C43C22⋊C412yes—
90C24×C43C2×D411yescharacteristic
91C23×D43—21no—
92C22×C4×C43—28no—
93C22×C4×C43—22no—
94C22×C4×C43—22no—
95C22×C4×C43—24no—
96C22×C4×C43—24no—
97C22×C4×C43—22no—
98C22×C4×C43—24no—
99C22×C4×C43—24no—
100C22×C4×C43—22no—
101C22×C4×C43—21no—
102C22×C4×C43—24no—
103C22×C4×C43—22no—
104C22×C4×C43—22no—
105C22×C4×C43—21no—
106C22×C4×C43—44no—
107C22×C4×C43—42no—
108C22×C4×C43—42no—
109C22×C4×C43—42no—
110C22×C4×C43—42no—
111C22×C4×C43—42no—
112C22×C4×C43—42no—
113C22×C4×C43—42no—
114C22×C4×C43—42no—
115C22×C4×C43—42no—
116C22×C4×C43—42no—
117C22×C4×C43C22⋊C412yes—
118C22×C4×C43C22×C411yescharacteristic
119C22×C4×C43C2×D411yescharacteristic
120C22×C4×C43C2×D412yes—
121C22×C4×C43C2×D412yes—
122C22×C4×C43C2×D412yes—
123C22×C4×C43C2×D412yes—
124C22×C4×C43#4112yes—
125C22×C4×C43#4112yes—
126C2×C22⋊D43—28no—
127C2×C22⋊D43—24no—
128C2×C22⋊D43—44no—
129C2×C22⋊D43—44no—
130C2×C22⋊D43—41no—
131C2×C22⋊D43—42no—
132C2×C22⋊D43—41no—
133C22×Dic43—22no—
134C22×Dic43—21no—
135#10621898—24no—
136#10621898—24no—
137#10621898—24no—
138#10621898—24no—
139#10621898—42no—
140#10621898—42no—
141#10621898—42no—
142#10621898—42no—
143C2×D172—21no—
144C2×D172—22no—
145C2×D172—22no—
146C2×D172C22⋊C412yescomplemented
147C2×C4×D43—24no—
148C2×C4×D43—41no—
149C2×C4×D43—42no—
150C2×C4×D43—41no—
151C2×Dic86—22no—
152C2×Dic86—22no—
153C2×Dic86—21no—
154C2×Dic86C22⋊C412yescomplemented
155C22⋊C4×C43—216no—
156C42×C43—24no—
157C42×C43—24no—
158C42×C43—42no—
159C42×C43—42no—
160C22⋊Dic43—28no—
161C22⋊Dic43—24no—
162(C2×D43)⋊C4—28no—
163C4⋊Dic43—22no—
164C4⋊Dic43C22⋊C414yescomplemented
165Dic43⋊C4—28no—
166C4×Dic43—24no—
167#10621877—24no—
168#10621877—24no—
169#10621877—42no—
170#10621877—42no—
171C2×C43⋊C8—24no—
172C2×C43⋊C8—42no—
173C2×C43⋊C8—42no—
174C23×C43—22no—
175C23×C43—22no—
176C23×C43—24no—
177C23×C43—24no—
178C23×C43—22no—
179C23×C43—22no—
180C23×C43—22no—
181C23×C43—22no—
182C23×C43—22no—
183C23×C43—22no—
184C23×C43—48no—
185C23×C43—48no—
186C23×C43—42no—
187C23×C43—42no—
188C23×C43—42no—
189C23×C43—42no—
190C23×C43#10414yescomplemented
191C23×C43C2×C22⋊C411yescharacteristic
192C23×C43#12012yes—
193C23×C43#12011yescharacteristic
194C23×C43#12112yes—
195C23×C43#12311yescharacteristic
196C22×D43—22no—
197C22×D43—41no—
198C22×D43—41no—
199C22×D43—42no—
200C22×D43—42no—
201C2×C4×C43—28no—
202C2×C4×C43—22no—
203C2×C4×C43—22no—
204C2×C4×C43—22no—
205C2×C4×C43—48no—
206C2×C4×C43—44no—
207C2×C4×C43—48no—
208C2×C4×C43—44no—
209C2×C4×C43—42no—
210C2×C4×C43—42no—
211C2×C4×C43—44no—
212C2×C4×C43—44no—
213C2×C4×C43—44no—
214C2×C4×C43—44no—
215C2×C4×C43C2×C22⋊C411yescharacteristic
216C2×C4×C43C2×C22⋊C411yescharacteristic
217C2×C4×C43C4×D412yes—
218C2×C4×C43C4×D412yes—
219C2×C4×C43#12012yes—
220C2×C4×C43#12011yescharacteristic
221C2×C4×C43#12012yes—
222C2×C4×C43#12112yes—
223C2×C4×C43#12112yes—
224C2×C4×C43#12112yes—
225C2×C4×C43#12112yes—
226C2×C4×C43#12112yes—
227C2×C4×C43#12312yes—
228C2×C4×C43#12311yescharacteristic
229C2×C4×C43#12412yes—
230C2×C4×C43#12712yes—
231C22⋊D43—416no—
232C22⋊D43—44no—
233C22⋊D43—44no—
234C2×Dic43—24no—
235C2×Dic43—22no—
236C2×Dic43—41no—
237C2×Dic43—41no—
238C2×Dic43—42no—
239C2×Dic43—42no—
240D172—24no—
241D172—42no—
242C4×D43—44no—
243C4×D43—44no—
244Dic86—24no—
245Dic86—42no—
246C43⋊C8—44no—
247#10971—431nomaximal, Sylow
248C22×C43—24no—
249C22×C43—21no—
250C22×C43—21no—
251C22×C43—21no—
252C22×C43—21no—
253C22×C43—24no—
254C22×C43—24no—
255C22×C43—24no—
256C22×C43—24no—
257C22×C43—48no—
258C22×C43—48no—
259C22×C43—42no—
260C22×C43—42no—
261C22×C43—42no—
262C22×C43—42no—
263C22×C43#37911yescharacteristic
264C22×C43#38611yescharacteristic
265C22×C43#38611yescharacteristic
266C22×C43#39212yes—
267C22×C43#39412yes—
268C22×C43C2×#10412yes—
269C22×C43#42112yes—
270C2×D43—42no—
271C2×D43—44no—
272C2×D43—82no—
273C2×D43—81no—
274C2×D43—81no—
275C4×C43—48no—
276C4×C43—42no—
277C4×C43#38612yes—
278C4×C43#39012yes—
279C4×C43#39212yes—
280C4×C43#39412yes—
281Dic43—42no—
282Dic43—44no—
283C2×#525—431no—
284C22×#104—432no—
285C2×#407—431no—
286C2×C4×C22⋊C4D4311yescharacteristic, complemented
287C2×#339—432no—
288C2×C43—22no—
289C2×C43—22no—
290C2×C43—44no—
291C2×C43#143811yescharacteristic
292C2×C43#178912yes—
293C2×C43#187112yes—
294C2×C43#189611yescharacteristic
295C2×C43#189711yescharacteristic
296D43—82no—
297C22×#41—431no—
298C24×C4C2×D4311yescharacteristic
299C22×M16—432no—
300#525—864no—
301#525—864no—
302C2×#123—861no—
303C2×C22⋊Q8—432no—
304C2×#121—432no—
305C2×#120—861no—
306C2×C4×D4—862no—
307C2×#117—431no—
308C22×C22⋊C4—21no—
309C22×C42C2×D4312yes—
310C2×#104—864no—
311C2×#104—864no—
312C2×#104—864no—
313#407—864no—
314C2×C22⋊C8—862no—
315C4×C22⋊C4—24no—
316C2×#95—21no—
317#339—864no—
318C43#1097111yescharacteristic, complemented, Sylow
319C25C22⋊D4311yescharacteristic
320C2×#41—434no—
321C2×#41—862no—
322C2×#41—862no—
323C2×#41—862no—
324C2×#41—862no—
325C2×#41—862no—
326C22×Q8—431no—
327C22×D4—431no—
328C22×D4—862no—
329C22×D4—861no—
330C23×C4—22no—
331C23×C4—22no—
332C23×C4—22no—
333C23×C4—22no—
334C23×C4—22no—
335C23×C4—22no—
336C23×C4—22no—
337C23×C4—861no—
338C23×C4C4×D4312yes—
339C23×C4D17212yes—
340C23×C4C22⋊D4311yescharacteristic
341C23×C4C22×D4311yescharacteristic
342C2×M16—434no—
343C2×M16—862no—
344C2×M16—862no—
345C2×M16—862no—
346C2×M16—862no—
347C2×M16—862no—
348C22×C8—862no—
349#123—868no—
350C22⋊Q8—864no—
351C22⋊Q8—864no—
352#121—864no—
353#121—864no—
354#120—868no—
355C4×D4—8616no—
356#117—434no—
357#117—862no—
358C2×C4⋊C4—432no—
359C2×C4⋊C4—862no—
360C2×C22⋊C4—28no—
361C2×C22⋊C4—24no—
362C2×C22⋊C4—862no—
363C2×C22⋊C4—861no—
364C2×C22⋊C4—862no—
365C2×C42—22no—
366C2×C42—22no—
367C2×C42—22no—
368C2×C42—22no—
369C2×C42—22no—
370C2×C42—861no—
371C2×C42C22⋊D4314yescomplemented
372#104—8616no—
373C22⋊C8—868no—
374#95—24no—
375C24—24no—
376C24—21no—
377C24—21no—
378C24—22no—
379C24—22no—
380C24—42no—
381C24—861no—
382C24(C2×D43)⋊C412yes—
383C24C2×C22⋊D4311yescharacteristic
384#41—864no—
385#41—864no—
386#41—864no—
387#41—864no—
388#41—1722no—
389#41—1722no—
390#41—1722no—
391#41—1722no—
392C2×Q8—432no—
393C2×Q8—861no—
394C2×Q8—862no—
395C2×Q8—862no—
396C2×D4—432no—
397C2×D4—868no—
398C2×D4—864no—
399C2×D4—861no—
400C2×D4—862no—
401C2×D4—862no—
402C2×D4—1724no—
403C2×D4—1724no—
404C2×D4—1721no—
405C2×D4—1722no—
406C2×D4—1721no—
407C22×C4—28no—
408C22×C4—22no—
409C22×C4—21no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—21no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—44no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—862no—
433C22×C4—861no—
434C22×C4—864no—
435C22×C4—1721no—
436C22×C4—1722no—
437C22×C4—1721no—
438C22×C4(C2×D43)⋊C412yes—
439C22×C4C2×C4×D4311yescharacteristic
440C22×C4C2×D17211yescharacteristic
441C22×C4#1062189812yes—
442C22×C4D4×D4312yes—
443C22×C4D4×D4312yes—
444C22×C4D4×D4312yes—
445C22×C4#1062190012yes—
446C22×C4C2×C22⋊D4312yes—
447M16—864no—
448M16—864no—
449M16—1722no—
450M16—1722no—
451C2×C8—864no—
452C2×C8—1722no—
453C2×C8—1722no—
454C4⋊C4—434no—
455C4⋊C4—868no—
456C4⋊C4—862no—
457C22⋊C4—216no—
458C22⋊C4—868no—
459C22⋊C4—868no—
460C22⋊C4—864no—
461C42—24no—
462C42—24no—
463C42—42no—
464C42—42no—
465C42—864no—
466C23—22no—
467C23—22no—
468C23—24no—
469C23—24no—
470C23—22no—
471C23—22no—
472C23—22no—
473C23—22no—
474C23—22no—
475C23—22no—
476C23—48no—
477C23—48no—
478C23—42no—
479C23—42no—
480C23—42no—
481C23—42no—
482C23—862no—
483C23—1722no—
484C23—1722no—
485C23—1721no—
486C23—1721no—
487C23#1306325314yescomplemented
488C23C2×(C2×D43)⋊C411yescharacteristic
489C23#1306329312yes—
490C23#1306329411yescharacteristic
491C23#1306329712yes—
492C23#1306330811yescharacteristic
493Q8—864no—
494Q8—1722no—
495D4—864no—
496D4—17216no—
497D4—1724no—
498D4—1724no—
499D4—1722no—
500C2×C4—28no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—44no—
505C2×C4—48no—
506C2×C4—48no—
507C2×C4—44no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—44no—
511C2×C4—44no—
512C2×C4—44no—
513C2×C4—44no—
514C2×C4—864no—
515C2×C4—862no—
516C2×C4—1722no—
517C2×C4—1722no—
518C2×C4—1721no—
519C2×C4—1721no—
520C2×C4—1724no—
521C2×C4—1724no—
522C2×C4C22⋊C4×D4311yescharacteristic
523C2×C4C22⋊C4×D4311yescharacteristic
524C2×C4(C22⋊D43)⋊C412yes—
525C2×C4C4×C22⋊D4312yes—
526C2×C4#1306329312yes—
527C2×C4#1306329511yescharacteristic
528C2×C4#1306329512yes—
529C2×C4#1306329612yes—
530C2×C4#1306329712yes—
531C2×C4#1306329812yes—
532C2×C4#1306329812yes—
533C2×C4#1306329912yes—
534C2×C4#1306330712yes—
535C2×C4#1306330911yescharacteristic
536C2×C4#1306331112yes—
537C2×C4#1306332112yes—
538C8—1724no—
539C22—21no—
540C22—21no—
541C22—21no—
542C22—21no—
543C22—24no—
544C22—24no—
545C22—24no—
546C22—24no—
547C22—24no—
548C22—48no—
549C22—48no—
550C22—42no—
551C22—42no—
552C22—42no—
553C22—42no—
554C22—3441no—
555C22—3441no—
556C22—3442no—
557C22—1724no—
558C22—1722no—
559C22#1458275411yescharacteristic
560C22#1458277611yescharacteristic
561C22#1458277811yescharacteristic
562C22#1458280712yes—
563C22#1458281412yes—
564C22C2×#1306325312yes—
565C22#1458291712yes—
566C4—48no—
567C4—42no—
568C4—1724no—
569C4—1722no—
570C4#1458277512yes—
571C4#1458279912yes—
572C4#1458280812yes—
573C4#1458281512yes—
574C2—22no—
575C2—22no—
576C2—44no—
577C2—3442no—
578C2#1913283411yescharacteristic
579C2#1913454012yes—
580C2#1913489512yes—
581C2#1913503811yescharacteristic
582C2#1913504311yescharacteristic
subgroupG/H
centerC22×C4C2×D172
commutatorC22×C4×C43C22×C4
FrattiniC23×C4C22×D43
FittingC2×C4×C22⋊C4×C43C2
socleC23×C43C2×C22⋊C4
prime radicalC2×#14583410C2
cosocleC23×C4×C43C23
hypercenterC2×C4×C22⋊C4D43
hypocenterC43#10971
LCSG ▷ C22×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C22×C4 ◁ C24×C4 ◁ C2×C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C22×#13063253C2semidirect
3. C2×#14582863C2semidirect
4. C2×#14582659C2semidirect
5. C2×C4×C22⋊C4×C43C2semidirect
11. C2×#13063304C4semidirect
12. C2×#13063299C4semidirect
43. C2×#10621877D4semidirect
81. C2×C42×C43D4semidirect
146. C2×D172C22⋊C4semidirect
154. C2×Dic86C22⋊C4semidirect
164. C4⋊Dic43C22⋊C4semidirect
190. C23×C43#104semidirect
286. C2×C4×C22⋊C4D43semidirect
318. C43#10971semidirect
371. C2×C42C22⋊D43semidirect
487. C23#13063253semidirect

Maximal subgroups

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

subgroupindex
C2×#145834102
C22×#130632532
C2×#145828632
C2×#145826592
C2×C4×C22⋊C4×C432
#1097143