Finite Group Databasebeta

← #25148166

Subgroups — #25148166

Order 11008.

862 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14583303C211yesmaximal, characteristic, complemented
2C2×#14583177C211yesmaximal, characteristic
3#19139546C211yesmaximal, characteristic
4#19137648C211yesmaximal, characteristic, complemented
5#19136070C211yesmaximal, characteristic, complemented
6#19136066C211yesmaximal, characteristic, complemented
7#19135981C211yesmaximal, characteristic
8#19135545C211yesmaximal, characteristic, complemented
9#19135354C211yesmaximal, characteristic, complemented
10#19135350C211yesmaximal, characteristic, complemented
11#19135310C211yesmaximal, characteristic, complemented
12#19135218C211yesmaximal, characteristic
13#19135212C211yesmaximal, characteristic
14#102×Dic43C211yesmaximal, characteristic
15C43×#2067C211yesmaximal, characteristic
16C2×#13063324C2211yescharacteristic
17C2×#13063324C2211yescharacteristic
18C2×#13063302—21no—
19C2×#13063302—21no—
20C2×#13063290—21no—
21C2×D4×Dic43C2211yescharacteristic
22#14583303—24no—
23#14583177—24no—
24C2×#13063259C2211yescharacteristic
25C2×#13063259C2211yescharacteristic
26C2×#13063258C2211yescharacteristic
27C2×#13063258C2211yescharacteristic
28C2×#13063258C2211yescharacteristic
29C2×Dic43⋊C8C2211yescharacteristic
30C2×Dic43⋊C8C2211yescharacteristic
31C2×#13063246C2211yescharacteristic
32C2×#13063246C2211yescharacteristic
33C2×C8×Dic43C2211yescharacteristic
34#14582835—21no—
35#14582835—21no—
36#14582803—21no—
37#14582802C2211yescharacteristic
38#14582795C2211yescharacteristic
39#14582781—21no—
40#14582780—21no—
41#14582769C2211yescharacteristic
42#14582768C2211yescharacteristic
43C4⋊C4×Dic43C2211yescharacteristic
44C4⋊C4×Dic43C2211yescharacteristic
45C4×Dic43⋊C4C2211yescharacteristic
46C4×C4⋊Dic43C2211yescharacteristic
47#14582662C2211yescharacteristic
48#14582662C2211yescharacteristic
49#14582661C2211yescharacteristic
50#14582661C2211yescharacteristic
51#14582661C2211yescharacteristic
52#14582654C2211yescharacteristic
53#14582653C2211yescharacteristic
54C2×#107×C43C2211yescharacteristic
55C2×C4⋊C8×C43C2211yescharacteristic
56C2×#102×C43C2211yescharacteristic
57C2×#102×C43C2211yescharacteristic
58C43×#390C2211yescharacteristic
59C43×#389C2211yescharacteristic
60C43×#340C2211yescharacteristic
61C22×Dic86—21no—
62C22×Dic86—21no—
63C23×Dic43—21no—
64C22×C43⋊C8C2311yescharacteristic
65C22×C43⋊C8C2311yescharacteristic
66#13063324—22no—
67#13063324—22no—
68#13063324—22no—
69#13063324D414yes—
70#13063302—44no—
71#13063302—44no—
72#13063290—44no—
73D4×Dic43—22no—
74D4×Dic43—22no—
75C2×C4⋊Dic43—21no—
76C2×C4⋊Dic43—21no—
77C2×C4⋊Dic43C2311yescharacteristic
78C2×C4⋊Dic43C2311yescharacteristic
79C2×C4⋊Dic43C2311yescharacteristic
80C2×C4⋊Dic43C2311yescharacteristic
81C2×Dic43⋊C4—21no—
82C2×Dic43⋊C4—21no—
83C2×Dic43⋊C4—21no—
84C2×Dic43⋊C4—21no—
85C2×Dic43⋊C4—21no—
86C2×Dic43⋊C4D411yescharacteristic
87C2×Dic43⋊C4D411yescharacteristic
88C2×C22⋊Dic43—21no—
89C2×C22⋊Dic43—21no—
90C2×C22⋊Dic43—21no—
91C2×C4×Dic43—21no—
92C2×C4×Dic43—21no—
93C2×C4×Dic43—21no—
94C2×C4×Dic43C2311yescharacteristic
95C2×C4×Dic43C2311yescharacteristic
96#13063259—22no—
97#13063259—22no—
98#13063258—22no—
99#13063258—22no—
100#13063258—22no—
101Dic43⋊C8—22no—
102Dic43⋊C8—22no—
103#13063246—22no—
104#13063246Q814yescomplemented
105C8×Dic43—22no—
106C43⋊#95—21no—
107C43⋊#95—21no—
108C43⋊#95—21no—
109C43⋊#95—21no—
110C43⋊#95D411yescharacteristic
111C43⋊#95D411yescharacteristic
112C22×D4×C43C2311yescharacteristic
113C22×C8×C43C2311yescharacteristic
114C22×C8×C43C2311yescharacteristic
115C2×C4⋊C4×C43—21no—
116C2×C4⋊C4×C43C2311yescharacteristic
117C2×C4⋊C4×C43C2311yescharacteristic
118C2×C4⋊C4×C43C2311yescharacteristic
119C2×C22⋊C4×C43—21no—
120C2×C22⋊C4×C43—21no—
121C2×C42×C43C2311yescharacteristic
122#107×C43—22no—
123C4⋊C8×C43—22no—
124#102×C43—22no—
125#102×C43—22no—
126#95×C43—21no—
127C24×C43—21no—
128C2×D4×C43—21no—
129C2×D4×C43—21no—
130C2×D4×C43—21no—
131C2×D4×C43—21no—
132C2×D4×C43—21no—
133C2×D4×C43C2×Q812yes—
134C22×C4×C43—21no—
135C22×C4×C43—21no—
136C22×C4×C43—21no—
137C22×C4×C43C2×D411yescharacteristic
138C22×C4×C43C2×D411yescharacteristic
139C22×C4×C43C2411yescharacteristic
140C22×Dic43—21no—
141C22×Dic43—21no—
142C22×Dic43—21no—
143C22×Dic43—21no—
144C22×Dic43—42no—
145C22×Dic43—42no—
146C22×Dic43—42no—
147C22×Dic43C2×D411yescharacteristic
148C22×Dic43C2×D411yescharacteristic
149C22×Dic43C2×D411yescharacteristic
150C22×Dic43C2×D411yescharacteristic
151C2×Dic86—24no—
152C2×Dic86—42no—
153C2×Dic86—42no—
154C2×Dic86—42no—
155C2×Dic86—42no—
156C2×Dic86—42no—
157C2×C8×C43—21no—
158C2×C8×C43—21no—
159C2×C8×C43—21no—
160C2×C8×C43—21no—
161C2×C8×C43—21no—
162C2×C8×C43#4112yes—
163C4⋊C4×C43—21no—
164C4⋊C4×C43—21no—
165C4⋊C4×C43—21no—
166C4⋊C4×C43—21no—
167C4⋊C4×C43—21no—
168C4⋊C4×C43—42no—
169C4⋊C4×C43C2×D412yes—
170C22⋊C4×C43—42no—
171C22⋊C4×C43—42no—
172C42×C43—21no—
173C42×C43—21no—
174C22⋊Dic43—42no—
175C22⋊Dic43—42no—
176C22⋊Dic43—42no—
177C4⋊Dic43—21no—
178C4⋊Dic43—21no—
179C4⋊Dic43—21no—
180C4⋊Dic43—21no—
181C4⋊Dic43—21no—
182C4⋊Dic43—21no—
183C4⋊Dic43—42no—
184C4⋊Dic43—42no—
185C4⋊Dic43C2×D412yes—
186C4⋊Dic43C2×Q812yes—
187Dic43⋊C4—24no—
188Dic43⋊C4—22no—
189Dic43⋊C4—42no—
190Dic43⋊C4—42no—
191Dic43⋊C4—42no—
192Dic43⋊C4—42no—
193Dic43⋊C4D814yes—
194C4×Dic43—21no—
195C4×Dic43—21no—
196C4×Dic43—42no—
197C4×Dic43—42no—
198C4×Dic43—42no—
199C4×Dic43C2×D412yes—
200C4×Dic43#4112yes—
201C2×C43⋊C8—21no—
202C2×C43⋊C8—21no—
203C2×C43⋊C8—21no—
204C2×C43⋊C8—21no—
205C2×C43⋊C8—21no—
206C2×C43⋊C8C2×Q812yes—
207C23×C43—22no—
208C23×C43—41no—
209C23×C43—41no—
210C23×C43—41no—
211C23×C43—41no—
212C23×C43—41no—
213C23×C43—41no—
214C23×C43C22×D411yescharacteristic
215D4×C43—22no—
216D4×C43—22no—
217D4×C43—41no—
218D4×C43—41no—
219C2×C4×C43—21no—
220C2×C4×C43—21no—
221C2×C4×C43—22no—
222C2×C4×C43—21no—
223C2×C4×C43—21no—
224C2×C4×C43—21no—
225C2×C4×C43—41no—
226C2×C4×C43—41no—
227C2×C4×C43—41no—
228C2×C4×C43—41no—
229C2×C4×C43—41no—
230C2×C4×C43—41no—
231C2×C4×C43—41no—
232C2×C4×C43—41no—
233C2×C4×C43C2×D812yes—
234C2×C4×C43C22×D411yescharacteristic
235C2×C4×C43C22×Q811yescharacteristic
236C2×C4×C43C2×#4111yescharacteristic
237C2×C4×C43C2×#4111yescharacteristic
238C2×C4×C432−1+411yescharacteristic
239C2×C4×C432−1+411yescharacteristic
240C2×Dic43—21no—
241C2×Dic43—21no—
242C2×Dic43—21no—
243C2×Dic43—21no—
244C2×Dic43—22no—
245C2×Dic43—21no—
246C2×Dic43—22no—
247C2×Dic43—21no—
248C2×Dic43—21no—
249C2×Dic43—41no—
250C2×Dic43—41no—
251C2×Dic43—41no—
252C2×Dic43—41no—
253C2×Dic43—41no—
254C2×Dic43—41no—
255C2×Dic43—41no—
256C2×Dic43—41no—
257C2×Dic43—41no—
258C2×Dic43—41no—
259C2×Dic43—84no—
260C2×Dic43#12112yes—
261C2×Dic43#12112yes—
262C2×Dic43C2×D812yes—
263C2×Dic43C2×D812yes—
264C2×Dic43AGL(1,ℤ/8)12yes—
265Dic86—48no—
266Dic86—84no—
267C8×C43—22no—
268C8×C43—41no—
269C43⋊C8—22no—
270C43⋊C8—41no—
271#25449—431nomaximal, Sylow
272C22×C43—42no—
273C22×C43—42no—
274C22×C43—42no—
275C22×C43—81no—
276C22×C43—81no—
277C22×C43—81no—
278C22×C43—81no—
279C22×C43C2×#12111yescharacteristic
280C22×C43#54711yescharacteristic
281C22×C43D4×Q811yescharacteristic
282C22×C43C22×D811yescharacteristic
283C22×C43C2×AGL(1,ℤ/8)11yescharacteristic
284C22×C43#57811yescharacteristic
285C22×C43#57811yescharacteristic
286C4×C43—22no—
287C4×C43—41no—
288C4×C43—42no—
289C4×C43—81no—
290C4×C43—81no—
291C4×C43#53311yescharacteristic
292C4×C43#54111yescharacteristic
293C4×C43#54811yescharacteristic
294C4×C43D4×Q811yescharacteristic
295Dic43—22no—
296Dic43—22no—
297Dic43—41no—
298Dic43—42no—
299Dic43—42no—
300Dic43—81no—
301Dic43—81no—
302Dic43#45914yes—
303C2×#500—431no—
304C2×#474—431no—
305#2542—431no—
306#2365—431no—
307#2083—431no—
308#2083—431no—
309#2067—431no—
310#2067D4311yescharacteristic
311#1988—431no—
312#1947—431no—
313#1947—431no—
314#1939—431no—
315#1925—431no—
316#1925—431no—
317C4×#102—431no—
318C2×C43—82no—
319C2×C43#242911yescharacteristic
320C2×C43C2×#45911yescharacteristic
321C2×C43#304411yescharacteristic
322C2×C43#333411yescharacteristic
323C2×C43#335311yescharacteristic
324C2×C43Q8×D811yescharacteristic
325C2×C43#338611yescharacteristic
326C2×C4⋊Q8—431no—
327C2×C4⋊Q8—431no—
328C2×C22⋊Q8—861no—
329C2×C22⋊Q8—861no—
330C2×C4×Q8—861no—
331C2×C4×D4—431no—
332#500—864no—
333#474—864no—
334C2×#107—431no—
335C2×#107—431no—
336C2×#107—431no—
337C2×#107—431no—
338C2×#107—431no—
339C2×#107C2×D4311yescharacteristic
340C2×C4⋊C8—431no—
341C2×C4⋊C8—431no—
342C2×C4⋊C8C2×D4311yescharacteristic
343C2×#102—431no—
344C2×#102—431no—
345C2×#102C2×D4311yescharacteristic
346C2×#102C2×D4311yescharacteristic
347C2×C4×C8—431no—
348#399—861no—
349#399—861no—
350#391—861no—
351#390—431no—
352#390C2×D4311yescharacteristic
353#389—431no—
354#389C2×D4311yescharacteristic
355#387—861no—
356#387—861no—
357#384—431no—
358#384—431no—
359C4×C4⋊C4—431no—
360C4×C4⋊C4—431no—
361C4×C4⋊C4—431no—
362C4×C4⋊C4—431no—
363#340—431no—
364#340—431no—
365#340—431no—
366#340—431no—
367#340—431no—
368#340C2×D4311yescharacteristic
369#336—431no—
370#336—431no—
371C43#2544911yescharacteristic, complemented, Sylow
372C22×Q8—861no—
373C22×Q8—861no—
374C22×D4C22×D4311yescharacteristic
375C23×C4—861no—
376C22×C8—431no—
377C22×C8—431no—
378C22×C8C22×D4311yescharacteristic
379C22×C8C22×D4311yescharacteristic
380C4⋊Q8—434no—
381C4⋊Q8—862no—
382C4⋊Q8—862no—
383C4⋊Q8—862no—
384C22⋊Q8—1724no—
385C22⋊Q8—1724no—
386C4×Q8—1724no—
387C4×D4—862no—
388C4×D4—862no—
389C2×C4⋊C4—21no—
390C2×C4⋊C4—431no—
391C2×C4⋊C4—431no—
392C2×C4⋊C4—431no—
393C2×C4⋊C4—431no—
394C2×C4⋊C4—431no—
395C2×C4⋊C4—431no—
396C2×C4⋊C4—861no—
397C2×C4⋊C4—861no—
398C2×C4⋊C4—861no—
399C2×C4⋊C4—861no—
400C2×C4⋊C4—861no—
401C2×C4⋊C4—861no—
402C2×C4⋊C4—861no—
403C2×C4⋊C4C22×D4311yescharacteristic
404C2×C4⋊C4C22×D4311yescharacteristic
405C2×C4⋊C4C22×D4311yescharacteristic
406C2×C22⋊C4—21no—
407C2×C22⋊C4—21no—
408C2×C22⋊C4—861no—
409C2×C22⋊C4—861no—
410C2×C22⋊C4—861no—
411C2×C42—431no—
412C2×C42—431no—
413C2×C42—861no—
414C2×C42—861no—
415C2×C42—861no—
416C2×C42C22×D4311yescharacteristic
417#107—22no—
418#107—862no—
419#107—862no—
420#107—862no—
421#107—862no—
422#107—862no—
423C4⋊C8—22no—
424C4⋊C8—862no—
425C4⋊C8—862no—
426#102—22no—
427#102—22no—
428#102—434no—
429#102—862no—
430C4×C8—862no—
431#95—21no—
432#95—431no—
433#95—431no—
434#95—861no—
435#95—861no—
436#95—861no—
437#95—861no—
438C24—21no—
439C2×Q8—864no—
440C2×Q8—1722no—
441C2×Q8—1722no—
442C2×Q8—1722no—
443C2×Q8—1722no—
444C2×Q8—1722no—
445C2×D4—21no—
446C2×D4—21no—
447C2×D4—21no—
448C2×D4—21no—
449C2×D4—21no—
450C2×D4Q8×D4312yes—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—431no—
455C22×C4—431no—
456C22×C4—431no—
457C22×C4—431no—
458C22×C4—861no—
459C22×C4—861no—
460C22×C4—861no—
461C22×C4—861no—
462C22×C4—1722no—
463C22×C4—1722no—
464C22×C4—1722no—
465C22×C4D4×D4311yescharacteristic
466C22×C4D4×D4311yescharacteristic
467C22×C4C23×D4311yescharacteristic
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—432no—
474C2×C8—861no—
475C2×C8—861no—
476C2×C8—861no—
477C2×C8—861no—
478C2×C8—861no—
479C2×C8#1062190012yes—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—42no—
486C4⋊C4—434no—
487C4⋊C4—432no—
488C4⋊C4—432no—
489C4⋊C4—862no—
490C4⋊C4—864no—
491C4⋊C4—861no—
492C4⋊C4—861no—
493C4⋊C4—861no—
494C4⋊C4—861no—
495C4⋊C4—861no—
496C4⋊C4—861no—
497C4⋊C4—1722no—
498C4⋊C4—1722no—
499C4⋊C4—1722no—
500C4⋊C4—1722no—
501C4⋊C4—1722no—
502C4⋊C4—1722no—
503C4⋊C4D4×D4312yes—
504C22⋊C4—42no—
505C22⋊C4—42no—
506C22⋊C4—1722no—
507C22⋊C4—1722no—
508C22⋊C4—1722no—
509C42—21no—
510C42—21no—
511C42—432no—
512C42—432no—
513C42—861no—
514C42—861no—
515C42—1722no—
516C42—1722no—
517C42—1722no—
518C23—22no—
519C23—41no—
520C23—41no—
521C23—41no—
522C23—41no—
523C23—41no—
524C23—41no—
525C23C2×D4×D4311yescharacteristic
526Q8—3444no—
527Q8—1728no—
528D4—22no—
529D4—22no—
530D4—41no—
531D4—41no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—22no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—432no—
547C2×C4—432no—
548C2×C4—432no—
549C2×C4—432no—
550C2×C4—432no—
551C2×C4—861no—
552C2×C4—861no—
553C2×C4—861no—
554C2×C4—861no—
555C2×C4—862no—
556C2×C4—862no—
557C2×C4—861no—
558C2×C4—861no—
559C2×C4—861no—
560C2×C4—3444no—
561C2×C4—1721no—
562C2×C4—1721no—
563C2×C4—1721no—
564C2×C4—1721no—
565C2×C4—1721no—
566C2×C4—1721no—
567C2×C4—1721no—
568C2×C4—1721no—
569C2×C4—1721no—
570C2×C4—1721no—
571C2×C4D8×D4312yes—
572C2×C4C2×D4×D4311yescharacteristic
573C2×C4C2×Q8×D4311yescharacteristic
574C2×C4#41×D4311yescharacteristic
575C2×C4C2×#1062190011yescharacteristic
576C2×C4#1306337211yescharacteristic
577C2×C4#1306337211yescharacteristic
578C8—22no—
579C8—41no—
580C8—862no—
581C8—1721no—
582C22—42no—
583C22—42no—
584C22—42no—
585C22—81no—
586C22—81no—
587C22—81no—
588C22—81no—
589C22#121×D4311yescharacteristic
590C22#1458354911yescharacteristic
591C22#1458356211yescharacteristic
592C22C2×D8×D4311yescharacteristic
593C22AGL(1,ℤ/8)×D4311yescharacteristic
594C22#1458371311yescharacteristic
595C22#1458371511yescharacteristic
596C4—22no—
597C4—42no—
598C4—41no—
599C4—81no—
600C4—81no—
601C4—434no—
602C4—862no—
603C4—862no—
604C4—3441no—
605C4—3441no—
606C4—1722no—
607C4—1721no—
608C4—1722no—
609C4#1458345111yescharacteristic
610C4#1458350611yescharacteristic
611C4#1458355511yescharacteristic
612C4#1458356211yescharacteristic
613C2—82no—
614C2#1913818711yescharacteristic
615C2#459×D4311yescharacteristic
616C2#1914412311yescharacteristic
617C2#1914693911yescharacteristic
618C2#1914717411yescharacteristic
619C2#1914744511yescharacteristic
620C2#1914751111yescharacteristic
subgroupG/H
centerC23C2×D4×D43
commutator, cosocleC22×C4×C43C24
FrattiniC22×C4C23×D43
FittingC43×#2067C2
socleC23×C43C22×D4
prime radicalC22×D4×C43C23
hypercenter#2067D43
hypocenterC43#25449
LCSG ▷ C22×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2067

Complements

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

NH ≅ G/Ntype
1. C2×#14583303C2semidirect
4. #19137648C2semidirect
5. #19136070C2semidirect
6. #19136066C2semidirect
8. #19135545C2semidirect
9. #19135354C2semidirect
10. #19135350C2semidirect
11. #19135310C2semidirect
104. #13063246Q8semidirect
371. C43#25449semidirect

Maximal subgroups

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

subgroupindex
C2×#145833032
C2×#145831772
#191395462
#191376482
#191360702
#191360662
#191359812
#191355452
#191353542
#191353502
#191353102
#191352182
#191352122
#102×Dic432
C43×#20672
#2544943