Finite Group Databasebeta

← #25102906

Subgroups — #25102906

Order 11008.

1004 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14583121C211yesmaximal, characteristic
2C2×#14583004C211yesmaximal, characteristic, complemented
3C2×#14582997C211yesmaximal, characteristic
4C4×#13063313C211yesmaximal, characteristic
5#19135527C211yesmaximal, characteristic
6#19135363C211yesmaximal, characteristic, complemented
7#19135236C211yesmaximal, characteristic, complemented
8#19135222C211yesmaximal, characteristic
9#19134950C211yesmaximal, characteristic
10#19134936C211yesmaximal, characteristic, complemented
11#19134610C211yesmaximal, characteristic, complemented
12#19134448C211yesmaximal, characteristic, complemented
13#19134428C211yesmaximal, characteristic, complemented
14#19134423C211yesmaximal, characteristic
15C42×Q8×C43C211yesmaximal, characteristic, complemented
16C22×#10621885C2211yescharacteristic
17C22×#10621884C2211yescharacteristic
18C2×#13063313C2211yescharacteristic
19C2×C4×Dic86C2211yescharacteristic
20C2×C4×D172C2211yescharacteristic
21#14583121C418yescomplemented
22#14583004—24no—
23#14582997—24no—
24C2×#13063258C2211yescharacteristic
25C2×#13063256C2211yescharacteristic
26C2×C4⋊(C43⋊C8)C2211yescharacteristic
27C2×C4⋊(C43⋊C8)C2211yescharacteristic
28C2×#13063249C2211yescharacteristic
29C2×#13063249C2211yescharacteristic
30C2×#13063248C2211yescharacteristic
31C2×#13063248C2211yescharacteristic
32C2×#13063248C2211yescharacteristic
33C2×#13063245C2211yescharacteristic
34C2×#13063245C2211yescharacteristic
35C2×#13063245C2211yescharacteristic
36C2×#13063236C2211yescharacteristic
37C2×#13063236C2211yescharacteristic
38#14582805C2211yescharacteristic
39#14582801C2211yescharacteristic
40#14582790—21no—
41#14582790—21no—
42#14582765C2211yescharacteristic
43C4×(C2×D43)⋊C4—21no—
44C4×(C2×D43)⋊C4—21no—
45#14582661C2211yescharacteristic
46#14582661C2211yescharacteristic
47#14582661C2211yescharacteristic
48#14582661C2211yescharacteristic
49#14582654C2211yescharacteristic
50#14582654C2211yescharacteristic
51C2×C4×Q8×C43C2211yescharacteristic
52C2×C4×Q8×C43C2211yescharacteristic
53C2×C4×Q8×C43C2211yescharacteristic
54C4×C4⋊C4×C43—21no—
55C4×C4⋊C4×C43—21no—
56C4×C4⋊C4×C43—21no—
57C4×C4⋊C4×C43C2211yescharacteristic
58C4×C4⋊C4×C43C2211yescharacteristic
59C4×C4⋊C4×C43C2211yescharacteristic
60C43×C43—21no—
61C43×C43C2211yescharacteristic
62C22×Dic86C2311yescharacteristic
63C22×D172C2311yescharacteristic
64C22×C4×D43—21no—
65C2×#10621885—22no—
66C2×#10621885—22no—
67C2×#10621885C2×C414yescomplemented
68C2×#10621884—22no—
69C2×#10621884—22no—
70C2×#10621884C2×C414yes—
71C22×C43⋊C8C2311yescharacteristic
72C22×C43⋊C8C2311yescharacteristic
73C22×C43⋊C8C2311yescharacteristic
74C22×C43⋊C8C2311yescharacteristic
75#13063313—22no—
76#13063313C2×C414yes—
77C4×Dic86—22no—
78C4×Dic86—22no—
79C4×D172—22no—
80C4×D172—22no—
81C2×C4⋊Dic43C2311yescharacteristic
82C2×C4⋊Dic43C2311yescharacteristic
83C2×Dic43⋊C4—21no—
84C2×(C2×D43)⋊C4—21no—
85C2×(C2×D43)⋊C4—21no—
86C2×(C2×D43)⋊C4—21no—
87C2×(C2×D43)⋊C4—21no—
88C2×(C2×D43)⋊C4—21no—
89C2×C4×Dic43—21no—
90#13063258—22no—
91#13063256—22no—
92C4⋊(C43⋊C8)—22no—
93C4⋊(C43⋊C8)C2×C414yescomplemented
94#13063249—22no—
95#13063249—22no—
96#13063248—22no—
97#13063248—22no—
98#13063248C2×C414yescomplemented
99#13063245—22no—
100#13063245—22no—
101#13063245C2×C414yes—
102#13063236—22no—
103#13063236—22no—
104C43⋊#95—21no—
105C43⋊#95—21no—
106C43⋊#95—21no—
107C43⋊#95—21no—
108C22×Q8×C43C2311yescharacteristic
109C4×Q8×C43—22no—
110C4×Q8×C43—22no—
111C4×Q8×C43—22no—
112C4×Q8×C43—22no—
113C4×Q8×C43C2×C414yescomplemented
114C4×Q8×C43D414yes—
115C2×C4⋊C4×C43—21no—
116C2×C4⋊C4×C43—21no—
117C2×C4⋊C4×C43—21no—
118C2×C4⋊C4×C43C2311yescharacteristic
119C2×C4⋊C4×C43C2311yescharacteristic
120C2×C4⋊C4×C43C2311yescharacteristic
121C2×C42×C43—21no—
122C2×C42×C43—21no—
123C2×C42×C43—21no—
124C2×C42×C43—21no—
125C2×C42×C43—21no—
126C2×C42×C43—21no—
127C2×C42×C43D411yescharacteristic
128C2×C42×C43D411yescharacteristic
129C2×C42×C43D411yescharacteristic
130C2×C42×C43D411yescharacteristic
131C2×C42×C43C2311yescharacteristic
132C2×C42×C43C2311yescharacteristic
133C2×C42×C43C2311yescharacteristic
134#95×C43—21no—
135#95×C43—21no—
136#95×C43—21no—
137C23×D43—21no—
138C2×Q8×C43—21no—
139C2×Q8×C43—21no—
140C2×Q8×C43—21no—
141C2×Q8×C43C22×C412yes—
142C2×Q8×C43C2×D412yes—
143C2×Q8×C43#4112yes—
144C22×C4×C43—21no—
145C22×C4×C43—21no—
146C22×C4×C43—21no—
147C22×C4×C43—21no—
148C22×C4×C43C2×D411yescharacteristic
149C22×C4×C43C2×D411yescharacteristic
150C22×C4×C43C2×D411yescharacteristic
151C22×C4×C43C2×D411yescharacteristic
152C22×C4×C43C2×D411yescharacteristic
153C22×C4×C43C2×D411yescharacteristic
154C22×C4×C43C2411yescharacteristic
155C22×Dic43—21no—
156C22×Dic43—21no—
157C22×Dic43—21no—
158C2×D172—21no—
159C2×D172—21no—
160C2×D172—21no—
161C2×D172—21no—
162C2×D172—21no—
163C2×D172C22×C412yes—
164C2×C4×D43—42no—
165C2×C4×D43—42no—
166C2×C4×D43—42no—
167C2×Dic86—21no—
168C2×Dic86—21no—
169C2×Dic86—21no—
170C2×Dic86—21no—
171C2×Dic86—21no—
172C2×Dic86C22×C412yes—
173C4⋊C4×C43—24no—
174C4⋊C4×C43—24no—
175C4⋊C4×C43—24no—
176C4⋊C4×C43—21no—
177C4⋊C4×C43—21no—
178C4⋊C4×C43—21no—
179C4⋊C4×C43C22×C412yes—
180C4⋊C4×C43C2×D412yes—
181C4⋊C4×C43#4112yes—
182C42×C43—24no—
183C42×C43—24no—
184C42×C43—24no—
185C42×C43—22no—
186C42×C43—21no—
187C42×C43—21no—
188C42×C43—41no—
189C42×C43—41no—
190C42×C43—41no—
191C42×C43—42no—
192C42×C43—42no—
193C42×C43—42no—
194C42×C43C22×C412yes—
195C42×C43C2×D412yes—
196C42×C43#4112yes—
197C42×C43#4112yes—
198#10621885—28no—
199#10621884—28no—
200(C2×D43)⋊C4—24no—
201(C2×D43)⋊C4—24no—
202(C2×D43)⋊C4—42no—
203(C2×D43)⋊C4—42no—
204(C2×D43)⋊C4—42no—
205C4⋊Dic43—21no—
206C4⋊Dic43—21no—
207C4⋊Dic43—21no—
208C4⋊Dic43—21no—
209Dic43⋊C4—42no—
210C4×Dic43—42no—
211C2×C43⋊C8—21no—
212C2×C43⋊C8—21no—
213C2×C43⋊C8—21no—
214C2×C43⋊C8—21no—
215C2×C43⋊C8—21no—
216C2×C43⋊C8—21no—
217C2×C43⋊C8—21no—
218C2×C43⋊C8—21no—
219C2×C43⋊C8—21no—
220C2×C43⋊C8—21no—
221C2×C43⋊C8C22×C412yes—
222C2×C43⋊C8C22×C412yes—
223C23×C43C22×D411yescharacteristic
224C22×D43—22no—
225C22×D43—41no—
226C22×D43—41no—
227C22×D43—41no—
228C22×D43—41no—
229C22×D43—41no—
230C22×D43—41no—
231Q8×C43—22no—
232Q8×C43—22no—
233Q8×C43—22no—
234Q8×C43C4×D414yes—
235C2×C4×C43—22no—
236C2×C4×C43—22no—
237C2×C4×C43—21no—
238C2×C4×C43—22no—
239C2×C4×C43—21no—
240C2×C4×C43—21no—
241C2×C4×C43—21no—
242C2×C4×C43—21no—
243C2×C4×C43—21no—
244C2×C4×C43—21no—
245C2×C4×C43—21no—
246C2×C4×C43—21no—
247C2×C4×C43—21no—
248C2×C4×C43—22no—
249C2×C4×C43—22no—
250C2×C4×C43—21no—
251C2×C4×C43—22no—
252C2×C4×C43—21no—
253C2×C4×C43—21no—
254C2×C4×C43—41no—
255C2×C4×C43—41no—
256C2×C4×C43—41no—
257C2×C4×C43—41no—
258C2×C4×C43—41no—
259C2×C4×C43—41no—
260C2×C4×C43C4×D412yes—
261C2×C4×C43C4×D412yes—
262C2×C4×C43#12112yes—
263C2×C4×C43#12112yes—
264C2×C4×C43#13712yes—
265C2×C4×C43C23×C411yescharacteristic
266C2×C4×C43C22×D411yescharacteristic
267C2×C4×C43C2×#4111yescharacteristic
268C2×C4×C43C2×#4111yescharacteristic
269C2×C4×C43C2×#4111yescharacteristic
270C2×C4×C43C2×#4111yescharacteristic
271C2×Dic43—22no—
272C2×Dic43—41no—
273C2×Dic43—41no—
274C2×Dic43—41no—
275C2×Dic43—41no—
276C2×Dic43—41no—
277C2×Dic43—41no—
278C2×Dic43—41no—
279C2×Dic43—41no—
280D172—22no—
281D172—22no—
282D172—41no—
283D172—41no—
284C4×D43—84no—
285Dic86—22no—
286Dic86—22no—
287Dic86—41no—
288Dic86—41no—
289C43⋊C8—22no—
290C43⋊C8—22no—
291C43⋊C8—41no—
292C43⋊C8—41no—
293#21508—431nomaximal, Sylow
294C22×C43C2×C4×D411yescharacteristic
295C22×C43C2×#12111yescharacteristic
296C22×C43#52511yescharacteristic
297C22×C43C2×#13711yescharacteristic
298C22×C43#57511yescharacteristic
299C22×C43#57511yescharacteristic
300C22×C43#57511yescharacteristic
301C2×D43—42no—
302C2×D43—42no—
303C2×D43—42no—
304C2×D43—81no—
305C2×D43—81no—
306C2×D43—81no—
307C2×D43—81no—
308C4×C43—22no—
309C4×C43—22no—
310C4×C43—22no—
311C4×C43—22no—
312C4×C43—22no—
313C4×C43—24no—
314C4×C43—41no—
315C4×C43—42no—
316C4×C43—42no—
317C4×C43—42no—
318C4×C43C2×C4×D411yescharacteristic
319C4×C43C4×#4111yescharacteristic
320C4×C43#52911yescharacteristic
321C4×C43#53211yescharacteristic
322Dic43—42no—
323Dic43—81no—
324Dic43—81no—
325C2×#464—431no—
326C2×#441—431no—
327C2×#440—431no—
328C4×#124—431no—
329C42×Q8D4311yescharacteristic, complemented
330#1983—431no—
331#1949—431no—
332#1928—431no—
333#1926—431no—
334#1880—431no—
335#1878—431no—
336#1805—431no—
337#1767—431no—
338#1763—431no—
339#1762—431no—
340C2×C43C4×#12111yescharacteristic
341C2×C43#294311yescharacteristic
342C2×C43C4×#13711yescharacteristic
343C2×C43#303711yescharacteristic
344C2×C43#304011yescharacteristic
345C2×C43#311111yescharacteristic
346C2×C43#311711yescharacteristic
347D43—82no—
348C22×Q16—431no—
349C22×SD16—431no—
350C2×#124—431no—
351C2×C4×Q8—431no—
352C2×C4×Q8C2×D4311yescharacteristic
353C2×C4×Q8C2×D4311yescharacteristic
354C2×C4×Q8C2×D4311yescharacteristic
355C2×C4×D4—431no—
356#464—438no—
357#441—864no—
358#440—864no—
359C2×#107—431no—
360C2×#106—431no—
361C2×C4⋊C8—431no—
362C2×C4⋊C8—431no—
363C2×#103—431no—
364C2×#103—431no—
365C2×#103—431no—
366C2×#103—431no—
367C2×#103—431no—
368C2×#102—431no—
369C2×#102—431no—
370C2×#102—431no—
371C2×#97—431no—
372C2×#97—431no—
373#391—431no—
374#390—431no—
375#388—861no—
376#388—861no—
377#382—431no—
378C4×C4⋊C4—21no—
379C4×C4⋊C4—21no—
380C4×C4⋊C4—21no—
381C4×C4⋊C4C2×D4311yescharacteristic
382C4×C4⋊C4C2×D4311yescharacteristic
383C4×C4⋊C4C2×D4311yescharacteristic
384C4×C22⋊C4—861no—
385C4×C22⋊C4—861no—
386C43—21no—
387C43C2×D4311yescharacteristic
388#340—431no—
389#340—431no—
390#340—431no—
391#340—431no—
392#336—431no—
393#336—431no—
394C43#2150811yescharacteristic, complemented, Sylow
395C22×Q8—431no—
396C22×Q8C22×D4311yescharacteristic
397C22×D4—431no—
398C23×C4—861no—
399C2×Q16—434no—
400C2×Q16—862no—
401C2×Q16—862no—
402C2×SD16—434no—
403C2×SD16—862no—
404C2×SD16—862no—
405C22×C8—431no—
406C22×C8—431no—
407C22×C8—431no—
408C22×C8—431no—
409#124—434no—
410#124—862no—
411C4×Q8—22no—
412C4×Q8—22no—
413C4×Q8—22no—
414C4×Q8—22no—
415C4×Q8—862no—
416C4×Q8—862no—
417C4×Q8C4×D4314yescomplemented
418C4×Q8D17214yes—
419C4×D4—862no—
420C4×D4—862no—
421C2×C4⋊C4—21no—
422C2×C4⋊C4—21no—
423C2×C4⋊C4—21no—
424C2×C4⋊C4—431no—
425C2×C4⋊C4—431no—
426C2×C4⋊C4—861no—
427C2×C4⋊C4C22×D4311yescharacteristic
428C2×C4⋊C4C22×D4311yescharacteristic
429C2×C4⋊C4C22×D4311yescharacteristic
430C2×C22⋊C4—861no—
431C2×C22⋊C4—861no—
432C2×C22⋊C4—861no—
433C2×C22⋊C4—861no—
434C2×C22⋊C4—861no—
435C2×C42—21no—
436C2×C42—21no—
437C2×C42—21no—
438C2×C42—21no—
439C2×C42—21no—
440C2×C42—21no—
441C2×C42—861no—
442C2×C42C22⋊D4311yescharacteristic
443C2×C42C22⋊D4311yescharacteristic
444C2×C42C22⋊D4311yescharacteristic
445C2×C42C22⋊D4311yescharacteristic
446C2×C42C22×D4311yescharacteristic
447C2×C42C22×D4311yescharacteristic
448C2×C42C22×D4311yescharacteristic
449#107—862no—
450#106—862no—
451C4⋊C8—434no—
452C4⋊C8—862no—
453#103—434no—
454#103—862no—
455#103—862no—
456#103—862no—
457#103—862no—
458#102—434no—
459#102—862no—
460#102—862no—
461#97—862no—
462#97—862no—
463#95—21no—
464#95—21no—
465#95—21no—
466#95—861no—
467#95—861no—
468#95—861no—
469#95—861no—
470C24—861no—
471C2×Q8—21no—
472C2×Q8—21no—
473C2×Q8—21no—
474C2×Q8—432no—
475C2×Q8—861no—
476C2×Q8—861no—
477C2×Q8—861no—
478C2×Q8—861no—
479C2×Q8—861no—
480C2×Q8C2×C4×D4312yes—
481C2×Q8C2×D17212yes—
482C2×Q8#1062189812yes—
483C2×D4—432no—
484C2×D4—861no—
485C2×D4—861no—
486C2×D4—861no—
487C2×D4—861no—
488C2×D4—861no—
489C22×C4—21no—
490C22×C4—21no—
491C22×C4—21no—
492C22×C4—21no—
493C22×C4—861no—
494C22×C4—861no—
495C22×C4—861no—
496C22×C4—1722no—
497C22×C4—1722no—
498C22×C4—1722no—
499C22×C4C2×C22⋊D4311yescharacteristic
500C22×C4C2×C22⋊D4311yescharacteristic
501C22×C4C2×C22⋊D4311yescharacteristic
502C22×C4C2×C22⋊D4311yescharacteristic
503C22×C4C2×C22⋊D4311yescharacteristic
504C22×C4C2×C22⋊D4311yescharacteristic
505C22×C4C23×D4311yescharacteristic
506Q16—868no—
507SD16—868no—
508C2×C8—432no—
509C2×C8—432no—
510C2×C8—861no—
511C2×C8—861no—
512C2×C8—861no—
513C2×C8—861no—
514C2×C8—861no—
515C2×C8—861no—
516C2×C8—861no—
517C2×C8—861no—
518C2×C8—861no—
519C2×C8—861no—
520C4⋊C4—24no—
521C4⋊C4—24no—
522C4⋊C4—24no—
523C4⋊C4—21no—
524C4⋊C4—21no—
525C4⋊C4—21no—
526C4⋊C4—861no—
527C4⋊C4—861no—
528C4⋊C4—861no—
529C4⋊C4—861no—
530C4⋊C4—1722no—
531C4⋊C4C2×C4×D4312yes—
532C4⋊C4C2×D17212yes—
533C4⋊C4#1062189812yes—
534C22⋊C4—864no—
535C22⋊C4—864no—
536C22⋊C4—1722no—
537C22⋊C4—1722no—
538C22⋊C4—1722no—
539C42—24no—
540C42—24no—
541C42—24no—
542C42—22no—
543C42—21no—
544C42—21no—
545C42—41no—
546C42—41no—
547C42—41no—
548C42—42no—
549C42—42no—
550C42—42no—
551C42—1722no—
552C42C2×C4×D4312yes—
553C42C2×D17212yes—
554C42#1062189812yes—
555C42#1062189812yes—
556C23—862no—
557C23—1721no—
558C23—1721no—
559C23—1721no—
560C23—1721no—
561C23—1721no—
562C23—1721no—
563C23C22×C22⋊D4311yescharacteristic
564Q8—22no—
565Q8—22no—
566Q8—22no—
567Q8—862no—
568Q8—862no—
569Q8—1721no—
570Q8—1721no—
571Q8C4×D17214yes—
572D4—862no—
573D4—862no—
574D4—1721no—
575D4—1721no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—21no—
579C2×C4—22no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—21no—
592C2×C4—22no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—862no—
602C2×C4—3444no—
603C2×C4—1721no—
604C2×C4—1721no—
605C2×C4—1721no—
606C2×C4—1721no—
607C2×C4—1721no—
608C2×C4—1721no—
609C2×C4—1721no—
610C2×C4—1721no—
611C2×C4C4×C22⋊D4312yes—
612C2×C4C4×C22⋊D4312yes—
613C2×C4#1306329912yes—
614C2×C4#1306329912yes—
615C2×C4#1306335612yes—
616C2×C4C22×C4×D4311yescharacteristic
617C2×C4C22×D17211yescharacteristic
618C2×C4C2×#1062189811yescharacteristic
619C2×C4C2×#1062189811yescharacteristic
620C2×C4C2×#1062189811yescharacteristic
621C2×C4C2×#1062189811yescharacteristic
622C8—862no—
623C8—862no—
624C8—1721no—
625C8—1721no—
626C22—3441no—
627C22—3441no—
628C22—3441no—
629C22—3441no—
630C22—1722no—
631C22—1722no—
632C22—1722no—
633C22C2×C4×C22⋊D4311yescharacteristic
634C22C2×#1306329911yescharacteristic
635C22#1458341011yescharacteristic
636C22C2×#1306335611yescharacteristic
637C22#1458369711yescharacteristic
638C22#1458369711yescharacteristic
639C22#1458369711yescharacteristic
640C4—22no—
641C4—24no—
642C4—22no—
643C4—22no—
644C4—22no—
645C4—22no—
646C4—41no—
647C4—42no—
648C4—42no—
649C4—42no—
650C4—3441no—
651C4—3441no—
652C4—1722no—
653C4C2×C4×D17211yescharacteristic
654C4C4×#1062189811yescharacteristic
655C4#1458343311yescharacteristic
656C4#1458344911yescharacteristic
657C2—3442no—
658C2C4×#1306329911yescharacteristic
659C2#1914339311yescharacteristic
660C2C4×#1306335611yescharacteristic
661C2#1914406711yescharacteristic
662C2#1914408911yescharacteristic
663C2#1914470411yescharacteristic
664C2#1914476611yescharacteristic
subgroupG/H
centerC23C22×C22⋊D43
commutatorC2×C4×C43C23×C4
FrattiniC22×C4C23×D43
FittingC42×Q8×C43C2
socleC23×C43C22×D4
prime radicalC22×D172C23
cosocleC22×C4×C43C24
hypercenterC42×Q8D43
hypocenterC43#21508
LCSG ▷ C2×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C23 ◁ C43 ◁ C42×Q8

Complements

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

NH ≅ G/Ntype
2. C2×#14583004C2semidirect
6. #19135363C2semidirect
7. #19135236C2semidirect
10. #19134936C2semidirect
11. #19134610C2semidirect
12. #19134448C2semidirect
13. #19134428C2semidirect
15. C42×Q8×C43C2semidirect
21. #14583121C4semidirect
67. C2×#10621885C2×C4semidirect
93. C4⋊(C43⋊C8)C2×C4semidirect
98. #13063248C2×C4semidirect
113. C4×Q8×C43C2×C4semidirect
329. C42×Q8D43semidirect
394. C43#21508semidirect
417. C4×Q8C4×D43semidirect

Maximal subgroups

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

subgroupindex
C2×#145831212
C2×#145830042
C2×#145829972
C4×#130633132
#191355272
#191353632
#191352362
#191352222
#191349502
#191349362
#191346102
#191344482
#191344282
#191344232
C42×Q8×C432
#2150843