Finite Group Databasebeta

← #25102900

Subgroups — #25102900

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×#14583118C211yesmaximal, characteristic
2C2×#14583004C211yesmaximal, characteristic, complemented
3C2×#14582999C211yesmaximal, characteristic
4C42×Dic86C211yesmaximal, characteristic, complemented
5#19135528C211yesmaximal, characteristic
6#19135361C211yesmaximal, characteristic, complemented
7#19135233C211yesmaximal, characteristic, complemented
8#19135219C211yesmaximal, characteristic
9#19134953C211yesmaximal, characteristic
10#19134939C211yesmaximal, characteristic, complemented
11#19134607C211yesmaximal, characteristic, complemented
12#19134448C211yesmaximal, characteristic, complemented
13#19134425C211yesmaximal, characteristic, complemented
14#19134420C211yesmaximal, characteristic
15C4×#124×C43C211yesmaximal, characteristic
16C22×#10621885C2211yescharacteristic
17C22×#10621883C2211yescharacteristic
18C2×C4×Dic86C2211yescharacteristic
19C2×C4×Dic86C2211yescharacteristic
20C2×C4×Dic86C2211yescharacteristic
21#14583118C418yescomplemented
22#14583004—24no—
23#14582999—24no—
24C2×#13063258C2211yescharacteristic
25C2×#13063256C2211yescharacteristic
26C2×C4⋊(C43⋊C8)C2211yescharacteristic
27C2×C4⋊(C43⋊C8)C2211yescharacteristic
28C2×#13063249C2211yescharacteristic
29C2×#13063249C2211yescharacteristic
30C2×#13063249C2211yescharacteristic
31C2×#13063248C2211yescharacteristic
32C2×#13063248C2211yescharacteristic
33C2×#13063246C2211yescharacteristic
34C2×#13063246C2211yescharacteristic
35C2×#13063246C2211yescharacteristic
36C2×#13063236C2211yescharacteristic
37C2×#13063236C2211yescharacteristic
38C4×Dic43⋊C4—21no—
39C4×Dic43⋊C4—21no—
40C4×Dic43⋊C4—21no—
41C4×C4⋊Dic43C2211yescharacteristic
42C4×C4⋊Dic43C2211yescharacteristic
43C4×C4⋊Dic43C2211yescharacteristic
44C42×Dic43—21no—
45#14582661C2211yescharacteristic
46#14582661C2211yescharacteristic
47#14582661C2211yescharacteristic
48#14582661C2211yescharacteristic
49#14582654C2211yescharacteristic
50#14582654C2211yescharacteristic
51C2×#124×C43C2211yescharacteristic
52C2×C4×Q8×C43C2211yescharacteristic
53C2×C4×D4×C43C2211yescharacteristic
54C43×#391C2211yescharacteristic
55C43×#390C2211yescharacteristic
56C43×#388—21no—
57C43×#388—21no—
58C43×#382C2211yescharacteristic
59C4×C22⋊C4×C43—21no—
60C4×C22⋊C4×C43—21no—
61C43×C43C2211yescharacteristic
62C22×Dic86C2311yescharacteristic
63C2×#10621885—22no—
64C2×#10621885—22no—
65C2×#10621885C2×C414yescomplemented
66C2×#10621883—22no—
67C2×#10621883—22no—
68C2×#10621883C2×C414yes—
69C22×C43⋊C8C2311yescharacteristic
70C22×C43⋊C8C2311yescharacteristic
71C22×C43⋊C8C2311yescharacteristic
72C22×C43⋊C8C2311yescharacteristic
73C4×Dic86—22no—
74C4×Dic86—22no—
75C4×Dic86—22no—
76C4×Dic86—22no—
77C4×Dic86C2×C414yescomplemented
78C4×Dic86D414yes—
79C2×C4⋊Dic43C2311yescharacteristic
80C2×C4⋊Dic43C2311yescharacteristic
81C2×C4⋊Dic43C2311yescharacteristic
82C2×Dic43⋊C4—21no—
83C2×Dic43⋊C4—21no—
84C2×Dic43⋊C4—21no—
85C2×C4×Dic43—21no—
86C2×C4×Dic43—21no—
87C2×C4×Dic43—21no—
88C2×C4×Dic43—21no—
89C2×C4×Dic43—21no—
90C2×C4×Dic43—21no—
91#13063258—22no—
92#13063256—22no—
93C4⋊(C43⋊C8)—22no—
94C4⋊(C43⋊C8)C2×C414yescomplemented
95#13063249—22no—
96#13063249—22no—
97#13063249C2×C414yescomplemented
98#13063248—22no—
99#13063248—22no—
100#13063246—22no—
101#13063246—22no—
102#13063246C2×C414yes—
103#13063236—22no—
104#13063236—22no—
105C43⋊#95—21no—
106C43⋊#95—21no—
107C43⋊#95—21no—
108C22×Q8×C43C2311yescharacteristic
109C22×D4×C43C2311yescharacteristic
110C23×C4×C43—21no—
111#124×C43—22no—
112#124×C43C2×C414yes—
113C4×Q8×C43—22no—
114C4×Q8×C43—22no—
115C4×D4×C43—22no—
116C4×D4×C43—22no—
117C2×C4⋊C4×C43—21no—
118C2×C4⋊C4×C43C2311yescharacteristic
119C2×C4⋊C4×C43C2311yescharacteristic
120C2×C22⋊C4×C43—21no—
121C2×C22⋊C4×C43—21no—
122C2×C22⋊C4×C43—21no—
123C2×C22⋊C4×C43—21no—
124C2×C22⋊C4×C43—21no—
125C2×C42×C43—21no—
126C2×C42×C43D411yescharacteristic
127C2×C42×C43D411yescharacteristic
128C2×C42×C43D411yescharacteristic
129C2×C42×C43D411yescharacteristic
130C2×C42×C43C2311yescharacteristic
131C2×C42×C43C2311yescharacteristic
132C2×C42×C43C2311yescharacteristic
133#95×C43—21no—
134#95×C43—21no—
135#95×C43—21no—
136#95×C43—21no—
137C24×C43—21no—
138C2×Q8×C43—21no—
139C2×Q8×C43—21no—
140C2×Q8×C43—21no—
141C2×Q8×C43—21no—
142C2×Q8×C43—21no—
143C2×Q8×C43C22×C412yes—
144C2×D4×C43—21no—
145C2×D4×C43—21no—
146C2×D4×C43—21no—
147C2×D4×C43—21no—
148C2×D4×C43—21no—
149C2×D4×C43C22×C412yes—
150C22×C4×C43—21no—
151C22×C4×C43—21no—
152C22×C4×C43—21no—
153C22×C4×C43—42no—
154C22×C4×C43—42no—
155C22×C4×C43—42no—
156C22×C4×C43C2×D411yescharacteristic
157C22×C4×C43C2×D411yescharacteristic
158C22×C4×C43C2×D411yescharacteristic
159C22×C4×C43C2×D411yescharacteristic
160C22×C4×C43C2×D411yescharacteristic
161C22×C4×C43C2×D411yescharacteristic
162C22×C4×C43C2411yescharacteristic
163C22×Dic43—21no—
164C22×Dic43—21no—
165C22×Dic43—21no—
166C22×Dic43—21no—
167C2×Dic86—21no—
168C2×Dic86—21no—
169C2×Dic86—21no—
170C2×Dic86C22×C412yes—
171C2×Dic86C2×D412yes—
172C2×Dic86#4112yes—
173C4⋊C4×C43—21no—
174C4⋊C4×C43—21no—
175C4⋊C4×C43—21no—
176C4⋊C4×C43—21no—
177C4⋊C4×C43—42no—
178C22⋊C4×C43—24no—
179C22⋊C4×C43—24no—
180C22⋊C4×C43—42no—
181C22⋊C4×C43—42no—
182C22⋊C4×C43—42no—
183C42×C43—22no—
184C42×C43—21no—
185C42×C43—21no—
186C42×C43—41no—
187C42×C43—41no—
188C42×C43—41no—
189C42×C43—42no—
190C42×C43C22×C412yes—
191C42×C43C2×D412yes—
192C42×C43#4112yes—
193C42×C43#4112yes—
194#10621885—28no—
195#10621883—28no—
196C4⋊Dic43—21no—
197C4⋊Dic43—21no—
198C4⋊Dic43—21no—
199C4⋊Dic43C22×C412yes—
200C4⋊Dic43C2×D412yes—
201C4⋊Dic43#4112yes—
202Dic43⋊C4—24no—
203Dic43⋊C4—24no—
204Dic43⋊C4—24no—
205C4×Dic43—24no—
206C4×Dic43—24no—
207C4×Dic43—24no—
208C4×Dic43—42no—
209C4×Dic43—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×C43—22no—
224C23×C43—41no—
225C23×C43—41no—
226C23×C43—41no—
227C23×C43—41no—
228C23×C43—41no—
229C23×C43—41no—
230C23×C43C22×D411yescharacteristic
231Q8×C43—22no—
232Q8×C43—22no—
233Q8×C43—41no—
234Q8×C43—41no—
235D4×C43—22no—
236D4×C43—22no—
237D4×C43—41no—
238D4×C43—41no—
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—21no—
249C2×C4×C43—22no—
250C2×C4×C43—21no—
251C2×C4×C43—21no—
252C2×C4×C43—21no—
253C2×C4×C43—41no—
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×C43—41no—
261C2×C4×C43—84no—
262C2×C4×C43C4×D412yes—
263C2×C4×C43C4×D412yes—
264C2×C4×C43#12112yes—
265C2×C4×C43#12112yes—
266C2×C4×C43#13712yes—
267C2×C4×C43C23×C411yescharacteristic
268C2×C4×C43C22×D411yescharacteristic
269C2×C4×C43C2×#4111yescharacteristic
270C2×C4×C43C2×#4111yescharacteristic
271C2×C4×C43C2×#4111yescharacteristic
272C2×C4×C43C2×#4111yescharacteristic
273C2×Dic43—22no—
274C2×Dic43—22no—
275C2×Dic43—22no—
276C2×Dic43—22no—
277C2×Dic43—22no—
278C2×Dic43—22no—
279C2×Dic43—41no—
280C2×Dic43—41no—
281C2×Dic43—41no—
282C2×Dic43—41no—
283C2×Dic43—41no—
284C2×Dic43—41no—
285Dic86—22no—
286Dic86—22no—
287Dic86—22no—
288Dic86C4×D414yes—
289C43⋊C8—22no—
290C43⋊C8—22no—
291C43⋊C8—41no—
292C43⋊C8—41no—
293#21508—431nomaximal, Sylow
294C22×C43—42no—
295C22×C43—42no—
296C22×C43—42no—
297C22×C43—81no—
298C22×C43—81no—
299C22×C43—81no—
300C22×C43—81no—
301C22×C43C2×C4×D411yescharacteristic
302C22×C43C2×#12111yescharacteristic
303C22×C43#52511yescharacteristic
304C22×C43C2×#13711yescharacteristic
305C22×C43#57511yescharacteristic
306C22×C43#57511yescharacteristic
307C22×C43#57511yescharacteristic
308C4×C43—22no—
309C4×C43—22no—
310C4×C43—22no—
311C4×C43—22no—
312C4×C43—22no—
313C4×C43—41no—
314C4×C43—42no—
315C4×C43—81no—
316C4×C43—81no—
317C4×C43C2×C4×D411yescharacteristic
318C4×C43C4×#4111yescharacteristic
319C4×C43#52911yescharacteristic
320C4×C43#53211yescharacteristic
321Dic43—24no—
322Dic43—42no—
323Dic43—42no—
324Dic43—42no—
325C2×#464—431no—
326C2×#441—431no—
327C2×#440—431no—
328C4×#124D4311yescharacteristic
329C42×Q8—431no—
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×C43—82no—
341C2×C43C4×#12111yescharacteristic
342C2×C43#294311yescharacteristic
343C2×C43C4×#13711yescharacteristic
344C2×C43#303711yescharacteristic
345C2×C43#304011yescharacteristic
346C2×C43#311111yescharacteristic
347C2×C43#311711yescharacteristic
348C22×Q16—431no—
349C22×SD16—431no—
350C2×#124C2×D4311yescharacteristic
351C2×C4×Q8—431no—
352C2×C4×Q8—431no—
353C2×C4×Q8—431no—
354C2×C4×Q8C2×D4311yescharacteristic
355C2×C4×D4C2×D4311yescharacteristic
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#391C2×D4311yescharacteristic
374#390C2×D4311yescharacteristic
375#388—21no—
376#388—21no—
377#382C2×D4311yescharacteristic
378C4×C4⋊C4—431no—
379C4×C4⋊C4—431no—
380C4×C4⋊C4—431no—
381C4×C4⋊C4—861no—
382C4×C4⋊C4—861no—
383C4×C4⋊C4—861no—
384C4×C22⋊C4—21no—
385C4×C22⋊C4—21no—
386C43—861no—
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×D4C22×D4311yescharacteristic
398C23×C4—21no—
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—22no—
410#124C4×D4314yes—
411C4×Q8—22no—
412C4×Q8—22no—
413C4×Q8—434no—
414C4×Q8—434no—
415C4×Q8—862no—
416C4×Q8—862no—
417C4×Q8—862no—
418C4×Q8—862no—
419C4×D4—22no—
420C4×D4—22no—
421C2×C4⋊C4—21no—
422C2×C4⋊C4—431no—
423C2×C4⋊C4—431no—
424C2×C4⋊C4—431no—
425C2×C4⋊C4—861no—
426C2×C4⋊C4—861no—
427C2×C4⋊C4—861no—
428C2×C4⋊C4C22×D4311yescharacteristic
429C2×C4⋊C4C22×D4311yescharacteristic
430C2×C22⋊C4—21no—
431C2×C22⋊C4—21no—
432C2×C22⋊C4—21no—
433C2×C22⋊C4—21no—
434C2×C22⋊C4—21no—
435C2×C42—21no—
436C2×C42—861no—
437C2×C42—861no—
438C2×C42—861no—
439C2×C42—861no—
440C2×C42—861no—
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—21no—
467#95—861no—
468#95—861no—
469#95—861no—
470C24—21no—
471C2×Q8—21no—
472C2×Q8—21no—
473C2×Q8—21no—
474C2×Q8—21no—
475C2×Q8—21no—
476C2×Q8—432no—
477C2×Q8—432no—
478C2×Q8—432no—
479C2×Q8—861no—
480C2×Q8—861no—
481C2×Q8—861no—
482C2×Q8C2×C4×D4312yes—
483C2×D4—21no—
484C2×D4—21no—
485C2×D4—21no—
486C2×D4—21no—
487C2×D4—21no—
488C2×D4C2×C4×D4312yes—
489C22×C4—21no—
490C22×C4—21no—
491C22×C4—21no—
492C22×C4—42no—
493C22×C4—42no—
494C22×C4—42no—
495C22×C4—861no—
496C22×C4—861no—
497C22×C4—861no—
498C22×C4—861no—
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—21no—
521C4⋊C4—21no—
522C4⋊C4—21no—
523C4⋊C4—21no—
524C4⋊C4—42no—
525C4⋊C4—432no—
526C4⋊C4—432no—
527C4⋊C4—432no—
528C4⋊C4—864no—
529C4⋊C4—864no—
530C4⋊C4—864no—
531C4⋊C4—861no—
532C4⋊C4—861no—
533C4⋊C4—861no—
534C22⋊C4—24no—
535C22⋊C4—24no—
536C22⋊C4—42no—
537C22⋊C4—42no—
538C22⋊C4—42no—
539C42—22no—
540C42—21no—
541C42—21no—
542C42—41no—
543C42—41no—
544C42—41no—
545C42—42no—
546C42—864no—
547C42—864no—
548C42—864no—
549C42—1722no—
550C42—1722no—
551C42—1722no—
552C42C2×C4×D4312yes—
553C42#1062189812yes—
554C42D4×D4312yes—
555C42#1062190012yes—
556C23—22no—
557C23—41no—
558C23—41no—
559C23—41no—
560C23—41no—
561C23—41no—
562C23—41no—
563C23C22×C22⋊D4311yescharacteristic
564Q8—22no—
565Q8—22no—
566Q8—41no—
567Q8—41no—
568Q8—434no—
569Q8—862no—
570Q8—862no—
571Q8—862no—
572D4—22no—
573D4—22no—
574D4—41no—
575D4—41no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—22no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—84no—
599C2×C4—862no—
600C2×C4—862no—
601C2×C4—862no—
602C2×C4—862no—
603C2×C4—862no—
604C2×C4—862no—
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#1306329612yes—
614C2×C4#1306329612yes—
615C2×C4#1306335512yes—
616C2×C4C22×C4×D4311yescharacteristic
617C2×C4C2×D4×D4311yescharacteristic
618C2×C4#41×D4311yescharacteristic
619C2×C4#41×D4311yescharacteristic
620C2×C4C2×#1062190011yescharacteristic
621C2×C4C2×#1062189811yescharacteristic
622C8—862no—
623C8—862no—
624C8—1721no—
625C8—1721no—
626C22—42no—
627C22—42no—
628C22—42no—
629C22—81no—
630C22—81no—
631C22—81no—
632C22—81no—
633C22C2×C4×C22⋊D4311yescharacteristic
634C22C2×#1306329611yescharacteristic
635C22#1458340611yescharacteristic
636C22C2×#1306335511yescharacteristic
637C22#1458369711yescharacteristic
638C22#1458369711yescharacteristic
639C22#1458369711yescharacteristic
640C4—22no—
641C4—22no—
642C4—22no—
643C4—22no—
644C4—22no—
645C4—42no—
646C4—41no—
647C4—81no—
648C4—81no—
649C4—864no—
650C4—1722no—
651C4—1722no—
652C4—1722no—
653C4C4×D4×D4311yescharacteristic
654C4C4×#1062190011yescharacteristic
655C4#1458343111yescharacteristic
656C4#1458344611yescharacteristic
657C2—82no—
658C2C4×#1306329611yescharacteristic
659C2#1914339311yescharacteristic
660C2C4×#1306335511yescharacteristic
661C2#1914406311yescharacteristic
662C2#1914408511yescharacteristic
663C2#1914469811yescharacteristic
664C2#1914476211yescharacteristic
subgroupG/H
centerC23C22×C22⋊D43
commutatorC2×C4×C43C23×C4
FrattiniC22×C4C23×D43
FittingC4×#124×C43C2
socleC23×C43C22×D4
prime radicalC22×D4×C43C23
cosocleC22×C4×C43C24
hypercenterC4×#124D43
hypocenterC43#21508
LCSG ▷ C2×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C23 ◁ C43 ◁ C4×#124

Complements

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

NH ≅ G/Ntype
2. C2×#14583004C2semidirect
4. C42×Dic86C2semidirect
6. #19135361C2semidirect
7. #19135233C2semidirect
10. #19134939C2semidirect
11. #19134607C2semidirect
12. #19134448C2semidirect
13. #19134425C2semidirect
21. #14583118C4semidirect
65. C2×#10621885C2×C4semidirect
77. C4×Dic86C2×C4semidirect
94. C4⋊(C43⋊C8)C2×C4semidirect
97. #13063249C2×C4semidirect
394. C43#21508semidirect

Maximal subgroups

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

subgroupindex
C2×#145831182
C2×#145830042
C2×#145829992
C42×Dic862
#191355282
#191353612
#191352332
#191352192
#191349532
#191349392
#191346072
#191344482
#191344252
#191344202
C4×#124×C432
#2150843