Finite Group Databasebeta

← #25125688

Subgroups — #25125688

Order 11008.

950 conjugacy classes of subgroups.

The subgroup lattice has 634 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14582938C211yesmaximal, characteristic
2C2×#14582921C211yesmaximal, characteristic
3#19137076C211yesmaximal, characteristic
4#389×D43C211yesmaximal, characteristic
5#19136099C211yesmaximal, characteristic, complemented
6#19136096C211yesmaximal, characteristic, complemented
7#19136094C211yesmaximal, characteristic, complemented
8#19136091C211yesmaximal, characteristic, complemented
9#19136088C211yesmaximal, characteristic, complemented
10#19136087C211yesmaximal, characteristic, complemented
11#19136085C211yesmaximal, characteristic, complemented
12#19134313C211yesmaximal, characteristic
13#19134313C211yesmaximal, characteristic
14#19134313C211yesmaximal, characteristic
15C43×#2088C211yesmaximal, characteristic, complemented
16C22×#10621872C2211yescharacteristic
17C22×#10621872C2211yescharacteristic
18C2×#13063283C2211yescharacteristic
19C2×#13063283C2211yescharacteristic
20C2×C4⋊C4×D43—21no—
21C2×C4⋊C4×D43C2211yescharacteristic
22C2×C4⋊C4×D43C2211yescharacteristic
23C2×C42×D43C2211yescharacteristic
24#14582938—24no—
25C2×#13063259C2211yescharacteristic
26C2×#13063258C2211yescharacteristic
27C2×#13063256C2211yescharacteristic
28#14582921—24no—
29C2×C4⋊(C43⋊C8)C2211yescharacteristic
30C2×Dic43⋊C8C2211yescharacteristic
31C2×Dic43⋊C8C2211yescharacteristic
32#14582798C2211yescharacteristic
33#14582798C2211yescharacteristic
34#14582797C2211yescharacteristic
35#14582797C2211yescharacteristic
36#14582796—21no—
37#14582795C2211yescharacteristic
38#14582795C2211yescharacteristic
39#14582794—21no—
40#14582773—21no—
41#14582770—21no—
42C4⋊C4×Dic43—21no—
43#14582750—21no—
44#95×D43—21no—
45#14582662C2211yescharacteristic
46#14582662C2211yescharacteristic
47#14582662C2211yescharacteristic
48#14582661C2211yescharacteristic
49#14582661C2211yescharacteristic
50#14582661C2211yescharacteristic
51#14582661C2211yescharacteristic
52#14582661C2211yescharacteristic
53#14582661C2211yescharacteristic
54C2×#106×C43C2211yescharacteristic
55C2×C4⋊C8×C43C2211yescharacteristic
56C43×#389C2211yescharacteristic
57C43×#389C2211yescharacteristic
58C43×#340C2211yescharacteristic
59C43×#340C2211yescharacteristic
60C43×#340C2211yescharacteristic
61C22×C4×D43—21no—
62C22×C4×D43—21no—
63C22×C4×D43D411yescharacteristic
64C22×C4×D43D411yescharacteristic
65C22×C4×D43C2311yescharacteristic
66C2×#10621872—22no—
67C2×#10621872—22no—
68C2×#10621872—22no—
69C2×#10621872—22no—
70C2×#10621872—22no—
71C2×#10621872D414yes—
72C22×C43⋊C8C2311yescharacteristic
73C22×C43⋊C8C2311yescharacteristic
74#13063283—22no—
75#13063283—22no—
76#13063283—22no—
77#13063283Q814yes—
78C2×C4⋊Dic43—21no—
79C2×C4⋊Dic43—21no—
80C2×C4⋊Dic43C2311yescharacteristic
81C2×C4⋊Dic43C2311yescharacteristic
82C2×C4⋊Dic43C2311yescharacteristic
83C2×C4⋊Dic43C2311yescharacteristic
84C2×Dic43⋊C4—21no—
85C2×Dic43⋊C4—21no—
86C2×Dic43⋊C4—21no—
87C2×Dic43⋊C4—21no—
88C2×Dic43⋊C4—21no—
89C2×Dic43⋊C4—21no—
90C4⋊C4×D43—22no—
91C4⋊C4×D43—22no—
92C4⋊C4×D43—22no—
93C4⋊C4×D43—44no—
94C4⋊C4×D43Q814yes—
95C2×(C2×D43)⋊C4—21no—
96C2×(C2×D43)⋊C4—21no—
97C2×C4×Dic43—21no—
98C2×C4×Dic43—21no—
99C2×C4×Dic43D411yescharacteristic
100C2×C4×Dic43D411yescharacteristic
101C2×C4×Dic43C2311yescharacteristic
102C42×D43—22no—
103C42×D43—22no—
104#13063259—22no—
105#13063258—22no—
106#13063256—22no—
107C4⋊(C43⋊C8)—22no—
108Dic43⋊C8—22no—
109Dic43⋊C8—22no—
110C43⋊#95—21no—
111C43⋊#95—21no—
112C43⋊#95—21no—
113C43⋊#95—21no—
114C43⋊#95—21no—
115C43⋊#95—21no—
116C22×C8×C43C2311yescharacteristic
117C22×C8×C43C2311yescharacteristic
118C2×C4⋊C4×C43—21no—
119C2×C4⋊C4×C43—21no—
120C2×C4⋊C4×C43C2311yescharacteristic
121C2×C4⋊C4×C43C2311yescharacteristic
122C2×C4⋊C4×C43C2311yescharacteristic
123C2×C4⋊C4×C43C2311yescharacteristic
124C2×C42×C43C2311yescharacteristic
125#106×C43—22no—
126C4⋊C8×C43—22no—
127#95×C43—21no—
128#95×C43—21no—
129C23×D43C2×D411yescharacteristic
130C22×C4×C43—21no—
131C22×C4×C43—21no—
132C22×C4×C43—21no—
133C22×C4×C43—21no—
134C22×C4×C43C2×D411yescharacteristic
135C22×C4×C43C2×D411yescharacteristic
136C22×C4×C43C2411yescharacteristic
137C22×Dic43—21no—
138C22×Dic43—21no—
139C22×Dic43—21no—
140C22×Dic43—21no—
141C22×Dic43C2×D411yescharacteristic
142C22×Dic43C2×D411yescharacteristic
143C22×Dic43C2×D411yescharacteristic
144C2×C4×D43—22no—
145C2×C4×D43—22no—
146C2×C4×D43—22no—
147C2×C4×D43—22no—
148C2×C4×D43—24no—
149C2×C4×D43—41no—
150C2×C4×D43—41no—
151C2×C4×D43—42no—
152C2×C4×D43—42no—
153C2×C4×D43—42no—
154C2×C4×D43—42no—
155C2×C4×D43—42no—
156C2×C4×D43C2×D412yes—
157C2×C4×D43C2×Q812yes—
158C2×C4×D43C2×Q812yes—
159C2×C4×D43#4112yes—
160C2×C4×D43#4112yes—
161C2×C4×D43#4112yes—
162C2×C8×C43—21no—
163C2×C8×C43—21no—
164C2×C8×C43—21no—
165C2×C8×C43—21no—
166C2×C8×C43—21no—
167C2×C8×C43C2×D412yes—
168C4⋊C4×C43—21no—
169C4⋊C4×C43—21no—
170C4⋊C4×C43—21no—
171C4⋊C4×C43—21no—
172C4⋊C4×C43—21no—
173C4⋊C4×C43—21no—
174C4⋊C4×C43—42no—
175C4⋊C4×C43—42no—
176C4⋊C4×C43C2×Q812yes—
177C4⋊C4×C43C2×Q812yes—
178C42×C43—21no—
179C42×C43—21no—
180(C2×D43)⋊C4—24no—
181(C2×D43)⋊C4—42no—
182C4⋊Dic43—21no—
183C4⋊Dic43—21no—
184C4⋊Dic43—21no—
185C4⋊Dic43—21no—
186C4⋊Dic43—21no—
187C4⋊Dic43—21no—
188C4⋊Dic43—42no—
189C4⋊Dic43—42no—
190C4⋊Dic43C2×Q812yes—
191C4⋊Dic43C2×Q812yes—
192Dic43⋊C4—24no—
193Dic43⋊C4—42no—
194Dic43⋊C4—42no—
195Dic43⋊C4—42no—
196Dic43⋊C4—42no—
197Dic43⋊C4—42no—
198C4×Dic43—21no—
199C4×Dic43—21no—
200C4×Dic43—22no—
201C4×Dic43—22no—
202C4×Dic43—24no—
203C4×Dic43—42no—
204C2×C43⋊C8—21no—
205C2×C43⋊C8—21no—
206C2×C43⋊C8—21no—
207C2×C43⋊C8—21no—
208C2×C43⋊C8—21no—
209C2×C43⋊C8C2×D412yes—
210#10621872—28no—
211#10621872—44no—
212C23×C43C22×D411yescharacteristic
213C22×D43—21no—
214C22×D43—21no—
215C22×D43—21no—
216C22×D43—21no—
217C22×D43#12112yes—
218C22×D43C22⋊Q812yes—
219C22×D43C22⋊Q812yes—
220C2×C4×C43—21no—
221C2×C4×C43—21no—
222C2×C4×C43—21no—
223C2×C4×C43—22no—
224C2×C4×C43—22no—
225C2×C4×C43—21no—
226C2×C4×C43—21no—
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×C43—41no—
234C2×C4×C43—41no—
235C2×C4×C43—41no—
236C2×C4×C43—41no—
237C2×C4×C43#13712yes—
238C2×C4×C43C22×D411yescharacteristic
239C2×C4×C43C22×Q811yescharacteristic
240C2×C4×C43C22×Q811yescharacteristic
241C2×C4×C43C2×#4111yescharacteristic
242C2×C4×C43C2×#4111yescharacteristic
243C2×C4×C43C2×#4111yescharacteristic
244C2×Dic43—21no—
245C2×Dic43—21no—
246C2×Dic43—21no—
247C2×Dic43—22no—
248C2×Dic43—22no—
249C2×Dic43—21no—
250C2×Dic43—21no—
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—41no—
260C2×Dic43—41no—
261C2×Dic43#12112yes—
262C2×Dic43C22⋊Q812yes—
263C2×Dic43C22⋊Q812yes—
264C2×Dic43AGL(1,ℤ/8)12yes—
265C4×D43—44no—
266C4×D43—44no—
267C4×D43—48no—
268C4×D43—84no—
269C4×D43#12314yes—
270C4×D43#12514yes—
271C4×D43#12514yes—
272C4×D43C4⋊Q814yes—
273C8×C43—22no—
274C8×C43—41no—
275C43⋊C8—22no—
276C43⋊C8—41no—
277#23670—431nomaximal, Sylow
278C22×C43C2×#12111yescharacteristic
279C22×C43C2×C22⋊Q811yescharacteristic
280C22×C43C2×C22⋊Q811yescharacteristic
281C22×C43C2×AGL(1,ℤ/8)11yescharacteristic
282C22×C43C2×#13711yescharacteristic
283C22×C43#57511yescharacteristic
284C22×C43#57511yescharacteristic
285C2×D43—22no—
286C2×D43—22no—
287C2×D43—22no—
288C2×D43—22no—
289C2×D43—22no—
290C2×D43—22no—
291C2×D43#39814yes—
292C4×C43—22no—
293C4×C43—42no—
294C4×C43—42no—
295C4×C43—41no—
296C4×C43—81no—
297C4×C43—81no—
298C4×C43C2×#12311yescharacteristic
299C4×C43C2×#12511yescharacteristic
300C4×C43C2×#12511yescharacteristic
301C4×C43C2×C4⋊Q811yescharacteristic
302Dic43—22no—
303Dic43—42no—
304Dic43—42no—
305Dic43—41no—
306Dic43—81no—
307Dic43—81no—
308Dic43#39814yes—
309C2×#428—431no—
310C2×#423—431no—
311#2296—431no—
312C2×#389—431no—
313#2088—431no—
314#2088—431no—
315#2088—431no—
316#2088D4311yescharacteristic, complemented
317#2087—431no—
318#2087—431no—
319#2086—431no—
320#2086—431no—
321#1736—431no—
322#1736—431no—
323#1736—431no—
324C2×C43C2×#39811yescharacteristic
325C2×C43#305511yescharacteristic
326C2×C43#305611yescharacteristic
327C2×C43#307711yescharacteristic
328C2×C43#307811yescharacteristic
329C2×C43#307911yescharacteristic
330C2×C43#308011yescharacteristic
331D43—24no—
332C22×M16—431no—
333C22×M16—431no—
334C2×#117—431no—
335C2×#117—431no—
336C22×C4⋊C4—431no—
337C22×C4⋊C4—431no—
338C22×C4⋊C4—861no—
339C22×C42—431no—
340#428—864no—
341C2×#107—431no—
342C2×#107—431no—
343C2×#106—431no—
344C2×#106C2×D4311yescharacteristic
345#423—864no—
346C2×C4⋊C8—431no—
347C2×C4⋊C8—431no—
348C2×C4⋊C8—431no—
349C2×C4⋊C8C2×D4311yescharacteristic
350#389—431no—
351#389—431no—
352#389—431no—
353#389—431no—
354#389—431no—
355#389—431no—
356#389—861no—
357#389—861no—
358#389C2×D4311yescharacteristic
359#389C2×D4311yescharacteristic
360#385—861no—
361#385—861no—
362C4×C4⋊C4—861no—
363C4×C4⋊C4—861no—
364C2×#95—861no—
365#340—431no—
366#340—431no—
367#340—431no—
368#340—431no—
369#340—431no—
370#340—431no—
371#340—431no—
372#340—431no—
373#340—431no—
374#340C2×D4311yescharacteristic
375#340C2×D4311yescharacteristic
376#340C2×D4311yescharacteristic
377C43#2367011yescharacteristic, complemented, Sylow
378C23×C4—431no—
379C23×C4—431no—
380C23×C4—431no—
381C23×C4—861no—
382C23×C4—861no—
383C2×M16—434no—
384C2×M16—862no—
385C2×M16—862no—
386C2×M16—862no—
387C2×M16—862no—
388C2×M16—862no—
389C22×C8—431no—
390C22×C8—431no—
391C22×C8C22×D4311yescharacteristic
392C22×C8C22×D4311yescharacteristic
393#117—434no—
394#117—862no—
395#117—862no—
396#117—862no—
397C2×C4⋊C4—21no—
398C2×C4⋊C4—21no—
399C2×C4⋊C4—431no—
400C2×C4⋊C4—431no—
401C2×C4⋊C4—434no—
402C2×C4⋊C4—431no—
403C2×C4⋊C4—431no—
404C2×C4⋊C4—862no—
405C2×C4⋊C4—862no—
406C2×C4⋊C4—862no—
407C2×C4⋊C4—861no—
408C2×C4⋊C4—861no—
409C2×C4⋊C4—861no—
410C2×C4⋊C4—861no—
411C2×C4⋊C4—861no—
412C2×C4⋊C4—861no—
413C2×C4⋊C4—861no—
414C2×C4⋊C4—861no—
415C2×C4⋊C4—1724no—
416C2×C4⋊C4C22×D4311yescharacteristic
417C2×C4⋊C4C22×D4311yescharacteristic
418C2×C4⋊C4C22×D4311yescharacteristic
419C2×C4⋊C4C22×D4311yescharacteristic
420C2×C22⋊C4—861no—
421C2×C22⋊C4—861no—
422C2×C42—431no—
423C2×C42—431no—
424C2×C42—431no—
425C2×C42—862no—
426C2×C42—862no—
427C2×C42—861no—
428C2×C42—861no—
429C2×C42C22×D4311yescharacteristic
430#107—862no—
431#107—862no—
432#106—22no—
433#106—862no—
434C4⋊C8—22no—
435C4⋊C8—862no—
436C4⋊C8—862no—
437C4⋊C8—862no—
438#95—21no—
439#95—21no—
440#95—861no—
441#95—861no—
442#95—861no—
443#95—861no—
444#95—861no—
445#95—861no—
446C24—431no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—431no—
452C22×C4—432no—
453C22×C4—432no—
454C22×C4—432no—
455C22×C4—432no—
456C22×C4—432no—
457C22×C4—432no—
458C22×C4—431no—
459C22×C4—431no—
460C22×C4—862no—
461C22×C4—862no—
462C22×C4—862no—
463C22×C4—864no—
464C22×C4—861no—
465C22×C4—861no—
466C22×C4—862no—
467C22×C4—861no—
468C22×C4—861no—
469C22×C4—1721no—
470C22×C4—1721no—
471C22×C4—1722no—
472C22×C4—1722no—
473C22×C4—1722no—
474C22×C4—1722no—
475C22×C4—1722no—
476C22×C4D4×D4311yescharacteristic
477C22×C4D4×D4311yescharacteristic
478C22×C4C23×D4311yescharacteristic
479M16—868no—
480M16—1724no—
481C2×C8—21no—
482C2×C8—21no—
483C2×C8—21no—
484C2×C8—21no—
485C2×C8—21no—
486C2×C8—432no—
487C2×C8—861no—
488C2×C8—861no—
489C2×C8—861no—
490C2×C8—861no—
491C2×C8—861no—
492C2×C8D4×D4312yes—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—42no—
500C4⋊C4—42no—
501C4⋊C4—432no—
502C4⋊C4—432no—
503C4⋊C4—864no—
504C4⋊C4—861no—
505C4⋊C4—861no—
506C4⋊C4—861no—
507C4⋊C4—861no—
508C4⋊C4—861no—
509C4⋊C4—861no—
510C4⋊C4—1722no—
511C4⋊C4—1722no—
512C4⋊C4—1722no—
513C4⋊C4—1722no—
514C4⋊C4—1722no—
515C4⋊C4—1722no—
516C4⋊C4—1722no—
517C4⋊C4Q8×D4312yes—
518C4⋊C4Q8×D4312yes—
519C22⋊C4—864no—
520C22⋊C4—1722no—
521C42—21no—
522C42—21no—
523C42—862no—
524C42—862no—
525C42—864no—
526C42—861no—
527C42—861no—
528C42—1722no—
529C23—432no—
530C23—432no—
531C23—432no—
532C23—861no—
533C23—861no—
534C23—861no—
535C23—861no—
536C23C2×D4×D4311yescharacteristic
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—432no—
555C2×C4—432no—
556C2×C4—432no—
557C2×C4—434no—
558C2×C4—434no—
559C2×C4—434no—
560C2×C4—434no—
561C2×C4—432no—
562C2×C4—861no—
563C2×C4—861no—
564C2×C4—861no—
565C2×C4—862no—
566C2×C4—862no—
567C2×C4—861no—
568C2×C4—861no—
569C2×C4—3444no—
570C2×C4—1724no—
571C2×C4—1724no—
572C2×C4—1728no—
573C2×C4—1721no—
574C2×C4—1721no—
575C2×C4—1721no—
576C2×C4—1721no—
577C2×C4—1721no—
578C2×C4—1721no—
579C2×C4—1721no—
580C2×C4—1721no—
581C2×C4—1721no—
582C2×C4—1721no—
583C2×C4#1306335412yes—
584C2×C4C2×D4×D4311yescharacteristic
585C2×C4C2×Q8×D4311yescharacteristic
586C2×C4C2×Q8×D4311yescharacteristic
587C2×C4#41×D4311yescharacteristic
588C2×C4#41×D4311yescharacteristic
589C2×C4#41×D4311yescharacteristic
590C8—22no—
591C8—41no—
592C8—862no—
593C8—1721no—
594C22—434no—
595C22—862no—
596C22—862no—
597C22—862no—
598C22—862no—
599C22—862no—
600C22—862no—
601C22#121×D4311yescharacteristic
602C22C22⋊Q8×D4311yescharacteristic
603C22C22⋊Q8×D4311yescharacteristic
604C22AGL(1,ℤ/8)×D4311yescharacteristic
605C22C2×#1306335411yescharacteristic
606C22#1458369611yescharacteristic
607C22#1458369811yescharacteristic
608C4—22no—
609C4—41no—
610C4—42no—
611C4—42no—
612C4—81no—
613C4—81no—
614C4—434no—
615C4—862no—
616C4—3441no—
617C4—3441no—
618C4—1721no—
619C4—1722no—
620C4—1722no—
621C4#123×D4311yescharacteristic
622C4#125×D4311yescharacteristic
623C4#125×D4311yescharacteristic
624C4C4⋊Q8×D4311yescharacteristic
625C2—864no—
626C2#398×D4311yescharacteristic
627C2#1914420911yescharacteristic
628C2#1914421811yescharacteristic
629C2#1914439011yescharacteristic
630C2#1914439511yescharacteristic
631C2#1914440711yescharacteristic
632C2#1914441411yescharacteristic
subgroupG/H
centerC23C2×D4×D43
commutator, cosocleC22×C4×C43C24
FrattiniC22×C4C23×D43
FittingC43×#2088C2
socleC23×C43C22×D4
prime radicalC23×D43C2×D4
hypercenter#2088D43
hypocenterC43#23670
LCSG ▷ C22×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2088

Complements

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

NH ≅ G/Ntype
5. #19136099C2semidirect
6. #19136096C2semidirect
7. #19136094C2semidirect
8. #19136091C2semidirect
9. #19136088C2semidirect
10. #19136087C2semidirect
11. #19136085C2semidirect
15. C43×#2088C2semidirect
316. #2088D43semidirect
377. C43#23670semidirect

Maximal subgroups

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

subgroupindex
C2×#145829382
C2×#145829212
#191370762
#389×D432
#191360992
#191360962
#191360942
#191360912
#191360882
#191360872
#191360852
#191343132
C43×#20882
#2367043