Finite Group Databasebeta

← #25159708

Subgroups — #25159708

Order 11008.

828 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14583209C211yesmaximal, characteristic, complemented
2C2×#14582986C211yesmaximal, characteristic
3#19138465C211yesmaximal, characteristic
4#19137507C211yesmaximal, characteristic, complemented
5#19136124C211yesmaximal, characteristic
6#19135984C211yesmaximal, characteristic
7#19135935C211yesmaximal, characteristic, complemented
8#19135327C211yesmaximal, characteristic, complemented
9#19135151C211yesmaximal, characteristic, complemented
10#19134951C211yesmaximal, characteristic
11#19134949C211yesmaximal, characteristic
12#19134853C211yesmaximal, characteristic
13#19134850C211yesmaximal, characteristic, complemented
14#19134848C211yesmaximal, characteristic, complemented
15C43×#2084C211yesmaximal, characteristic, complemented
16C22×#10621873C2211yescharacteristic
17C2×#13063316C2211yescharacteristic
18C2×C4×Dic86C2211yescharacteristic
19C2×#13063290C2211yescharacteristic
20C2×#13063290C2211yescharacteristic
21C2×#13063287C2211yescharacteristic
22#14583209—24no—
23#14582986—24no—
24C2×#13063258C2211yescharacteristic
25C2×#13063258C2211yescharacteristic
26C2×#13063258C2211yescharacteristic
27C2×Dic43⋊C8C2211yescharacteristic
28C2×#13063250C2211yescharacteristic
29C2×#13063248C2211yescharacteristic
30C2×#13063248C2211yescharacteristic
31C2×#13063247C2211yescharacteristic
32C2×#13063245C2211yescharacteristic
33C2×#13063245C2211yescharacteristic
34C2×C8×Dic43C2211yescharacteristic
35#14582836—21no—
36#14582821—21no—
37#14582801C2211yescharacteristic
38#14582800C2211yescharacteristic
39#14582798C2211yescharacteristic
40#14582795C2211yescharacteristic
41#14582795C2211yescharacteristic
42#14582794—21no—
43#14582794—21no—
44#14582773—21no—
45#14582768—21no—
46#14582756—21no—
47C4⋊C4×Dic43—21no—
48C4×C4⋊Dic43C2211yescharacteristic
49#14582662C2211yescharacteristic
50#14582661C2211yescharacteristic
51#14582654C2211yescharacteristic
52#14582653C2211yescharacteristic
53#14582653C2211yescharacteristic
54C2×#106×C43C2211yescharacteristic
55C2×#106×C43C2211yescharacteristic
56C43×#389C2211yescharacteristic
57C43×#384C2211yescharacteristic
58C43×#340C2211yescharacteristic
59C43×#340C2211yescharacteristic
60C43×#336C2211yescharacteristic
61C22×Dic86C2311yescharacteristic
62C22×D172C2311yescharacteristic
63C22×C4×D43—21no—
64C2×#10621873—22no—
65C2×#10621873—22no—
66C2×#10621873Q814yes—
67C22×C43⋊C8C2311yescharacteristic
68C22×C43⋊C8C2311yescharacteristic
69#13063316—22no—
70#13063316—41no—
71C4×Dic86—22no—
72C4×Dic86—22no—
73#13063290—22no—
74#13063290—22no—
75#13063290—41no—
76#13063290—41no—
77#13063287—22no—
78#13063287—41no—
79C2×C4⋊Dic43—21no—
80C2×C4⋊Dic43—21no—
81C2×C4⋊Dic43D411yescharacteristic
82C2×C4⋊Dic43D411yescharacteristic
83C2×C4⋊Dic43C2311yescharacteristic
84C2×C4⋊Dic43C2311yescharacteristic
85C2×Dic43⋊C4—21no—
86C2×Dic43⋊C4—21no—
87C2×Dic43⋊C4—21no—
88C2×Dic43⋊C4—21no—
89C2×Dic43⋊C4—21no—
90C2×Dic43⋊C4—21no—
91C2×(C2×D43)⋊C4—21no—
92C2×(C2×D43)⋊C4—21no—
93C2×(C2×D43)⋊C4—21no—
94C2×(C2×D43)⋊C4—21no—
95C2×(C2×D43)⋊C4—21no—
96C2×C4×Dic43—21no—
97C2×C4×Dic43—21no—
98C2×C4×Dic43—21no—
99C2×C4×Dic43C2311yescharacteristic
100C2×C4×Dic43C2311yescharacteristic
101#13063258—22no—
102#13063258—22no—
103#13063258—41no—
104Dic43⋊C8—22no—
105#13063250—22no—
106#13063248—22no—
107#13063248D414yes—
108#13063247—22no—
109#13063245—22no—
110#13063245—41no—
111C8×Dic43—22no—
112C43⋊#95—21no—
113C43⋊#95—21no—
114C43⋊#95—21no—
115C43⋊#95D411yescharacteristic
116C43⋊#95D411yescharacteristic
117C22×C8×C43C2311yescharacteristic
118C22×C8×C43C2311yescharacteristic
119C2×C4⋊C4×C43—21no—
120C2×C4⋊C4×C43—21no—
121C2×C4⋊C4×C43—21no—
122C2×C4⋊C4×C43C2311yescharacteristic
123C2×C4⋊C4×C43C2311yescharacteristic
124C2×C4⋊C4×C43C2311yescharacteristic
125C2×C4⋊C4×C43C2311yescharacteristic
126C2×C42×C43C2311yescharacteristic
127#106×C43—22no—
128#106×C43—22no—
129#95×C43—21no—
130C23×D43—21no—
131C22×C4×C43—21no—
132C22×C4×C43—21no—
133C22×C4×C43—21no—
134C22×C4×C43—21no—
135C22×C4×C43C2×D411yescharacteristic
136C22×C4×C43C2×D411yescharacteristic
137C22×C4×C43C2411yescharacteristic
138C22×Dic43—21no—
139C22×Dic43—21no—
140C22×Dic43—21no—
141C22×Dic43C2×D411yescharacteristic
142C22×Dic43C2×D411yescharacteristic
143C22×Dic43C2×D411yescharacteristic
144C22×Dic43C2×D411yescharacteristic
145C2×D172—21no—
146C2×D172—21no—
147C2×D172—21no—
148C2×D172—21no—
149C2×D172—21no—
150C2×D172C2×Q812yes—
151C2×C4×D43—42no—
152C2×C4×D43—42no—
153C2×C4×D43—81no—
154C2×Dic86—21no—
155C2×Dic86—21no—
156C2×Dic86—21no—
157C2×Dic86C2×D412yes—
158C2×Dic86C2×Q812yes—
159C2×Dic86#4112yes—
160C2×C8×C43—21no—
161C2×C8×C43—21no—
162C2×C8×C43—21no—
163C2×C8×C43—21no—
164C2×C8×C43—21no—
165C2×C8×C43C2×Q812yes—
166C4⋊C4×C43—21no—
167C4⋊C4×C43—21no—
168C4⋊C4×C43—21no—
169C4⋊C4×C43—21no—
170C4⋊C4×C43—21no—
171C4⋊C4×C43—21no—
172C4⋊C4×C43—21no—
173C4⋊C4×C43—42no—
174C4⋊C4×C43—81no—
175C4⋊C4×C43—81no—
176C4⋊C4×C43C2×D412yes—
177C42×C43—21no—
178C42×C43—21no—
179(C2×D43)⋊C4—42no—
180(C2×D43)⋊C4—42no—
181(C2×D43)⋊C4—81no—
182(C2×D43)⋊C4—81no—
183(C2×D43)⋊C4—81no—
184C4⋊Dic43—22no—
185C4⋊Dic43—21no—
186C4⋊Dic43—22no—
187C4⋊Dic43—21no—
188C4⋊Dic43—21no—
189C4⋊Dic43—42no—
190C4⋊Dic43—81no—
191C4⋊Dic43#4112yes—
192Dic43⋊C4—24no—
193Dic43⋊C4—42no—
194Dic43⋊C4—42no—
195Dic43⋊C4—42no—
196Dic43⋊C4—81no—
197Dic43⋊C4—81no—
198C4×Dic43—24no—
199C4×Dic43—21no—
200C4×Dic43—21no—
201C4×Dic43—21no—
202C4×Dic43—21no—
203C4×Dic43—42no—
204C4×Dic43—42no—
205C2×C43⋊C8—21no—
206C2×C43⋊C8—21no—
207C2×C43⋊C8—21no—
208C2×C43⋊C8—21no—
209C2×C43⋊C8—21no—
210C2×C43⋊C8C2×D412yes—
211#10621873—28no—
212C23×C43C22×D411yescharacteristic
213C22×D43—22no—
214C22×D43—41no—
215C22×D43—41no—
216C22×D43—41no—
217C22×D43—41no—
218C22×D43—41no—
219C22×D43—41no—
220C2×C4×C43—21no—
221C2×C4×C43—21no—
222C2×C4×C43—21no—
223C2×C4×C43—22no—
224C2×C4×C43—21no—
225C2×C4×C43—21no—
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×C43—41no—
234C2×C4×C43—41no—
235C2×C4×C43—41no—
236C2×C4×C43—41no—
237C2×C4×C43#13512yes—
238C2×C4×C43C22×D411yescharacteristic
239C2×C4×C43C22×Q811yescharacteristic
240C2×C4×C43C2×#4111yescharacteristic
241C2×C4×C43C2×#4111yescharacteristic
242C2×C4×C43H(5,𝔽2)11yescharacteristic
243C2×C4×C432−1+411yescharacteristic
244C2×Dic43—22no—
245C2×Dic43—21no—
246C2×Dic43—22no—
247C2×Dic43—22no—
248C2×Dic43—21no—
249C2×Dic43—22no—
250C2×Dic43—21no—
251C2×Dic43—21no—
252C2×Dic43—21no—
253C2×Dic43—21no—
254C2×Dic43—21no—
255C2×Dic43—21no—
256C2×Dic43—21no—
257C2×Dic43—21no—
258C2×Dic43—21no—
259C2×Dic43—41no—
260C2×Dic43—41no—
261C2×Dic43—41no—
262C2×Dic43—41no—
263C2×Dic43—41no—
264C2×Dic43#13712yes—
265D172—22no—
266D172—22no—
267D172—41no—
268D172—41no—
269C4×D43—84no—
270Dic86—22no—
271Dic86—22no—
272Dic86—22no—
273Dic86C22⋊Q814yes—
274C8×C43—22no—
275C8×C43—41no—
276C43⋊C8—22no—
277C43⋊C8—41no—
278#26379—431nomaximal, Sylow
279C22×C43#52511yescharacteristic
280C22×C43D4211yescharacteristic
281C22×C43D4×Q811yescharacteristic
282C22×C43C2×#13511yescharacteristic
283C22×C43C2×#13711yescharacteristic
284C22×C43#57611yescharacteristic
285C22×C43#57711yescharacteristic
286C2×D43—42no—
287C2×D43—42no—
288C2×D43—42no—
289C2×D43—81no—
290C2×D43—81no—
291C2×D43—81no—
292C2×D43—81no—
293C4×C43—22no—
294C4×C43—41no—
295C4×C43—42no—
296C4×C43—81no—
297C4×C43—81no—
298C4×C43—81no—
299C4×C43C2×C22⋊Q811yescharacteristic
300C4×C43#53911yescharacteristic
301C4×C43#54711yescharacteristic
302C4×C43#55411yescharacteristic
303Dic43—24no—
304Dic43—22no—
305Dic43—42no—
306Dic43—41no—
307Dic43—41no—
308Dic43—41no—
309Dic43—81no—
310C2×#479—431no—
311C2×C4×SD16—431no—
312#2454—431no—
313Q8×C4⋊C4—431no—
314#2093—431no—
315#2084D4311yescharacteristic, complemented
316#2067—431no—
317#2059—431no—
318#1942—431no—
319#1916—431no—
320#1880—431no—
321#1880—431no—
322#1864—431no—
323#1863—431no—
324#1863—431no—
325C2×C43#244911yescharacteristic
326C2×C43#310811yescharacteristic
327C2×C43#311911yescharacteristic
328C2×C43#326611yescharacteristic
329C2×C43#328711yescharacteristic
330C2×C43#338111yescharacteristic
331C2×C43#338511yescharacteristic
332D43—82no—
333C22×SD16—431no—
334C2×#125—431no—
335C2×C4×Q8—431no—
336C2×C4×Q8—431no—
337C2×C4×Q8—431no—
338C2×C4×D4—431no—
339#479—864no—
340C4×SD16—864no—
341C2×#107—431no—
342C2×#107—431no—
343C2×#107—431no—
344C2×#106C2×D4311yescharacteristic
345C2×#106C2×D4311yescharacteristic
346C2×C4⋊C8—431no—
347C2×#103—431no—
348C2×#103—431no—
349C2×#103—431no—
350C2×#102—431no—
351C2×#102—431no—
352C2×#102—431no—
353C2×C4×C8—431no—
354#399—861no—
355#396—861no—
356#390—431no—
357#390—431no—
358#389—431no—
359#389—431no—
360#389—431no—
361#389—861no—
362#389—861no—
363#389C2×D4311yescharacteristic
364#385—861no—
365#384—861no—
366#384C2×D4311yescharacteristic
367#380—861no—
368C4×C4⋊C4—431no—
369C4×C4⋊C4—861no—
370#340—431no—
371#340—431no—
372#340C2×D4311yescharacteristic
373#340C2×D4311yescharacteristic
374#336—431no—
375#336—431no—
376#336—431no—
377#336C2×D4311yescharacteristic
378C43#2637911yescharacteristic, complemented, Sylow
379C22×Q8—431no—
380C22×D4—431no—
381C23×C4—861no—
382C2×SD16—434no—
383C2×SD16—862no—
384C2×SD16—862no—
385C22×C8—431no—
386C22×C8—431no—
387C22×C8C22×D4311yescharacteristic
388C22×C8C22×D4311yescharacteristic
389#125—862no—
390#125—1721no—
391C4×Q8—862no—
392C4×Q8—862no—
393C4×Q8—862no—
394C4×Q8—862no—
395C4×Q8—1721no—
396C4×Q8—1721no—
397C4×D4—862no—
398C4×D4—1721no—
399C2×C4⋊C4—21no—
400C2×C4⋊C4—21no—
401C2×C4⋊C4—21no—
402C2×C4⋊C4—431no—
403C2×C4⋊C4—431no—
404C2×C4⋊C4—431no—
405C2×C4⋊C4—431no—
406C2×C4⋊C4—861no—
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⋊C4C22×D4311yescharacteristic
415C2×C4⋊C4C22×D4311yescharacteristic
416C2×C4⋊C4C22×D4311yescharacteristic
417C2×C4⋊C4C22×D4311yescharacteristic
418C2×C22⋊C4—861no—
419C2×C22⋊C4—861no—
420C2×C22⋊C4—861no—
421C2×C22⋊C4—861no—
422C2×C22⋊C4—861no—
423C2×C42—431no—
424C2×C42—431no—
425C2×C42—861no—
426C2×C42—861no—
427C2×C42—861no—
428C2×C42C22×D4311yescharacteristic
429#107—862no—
430#107—862no—
431#107—1721no—
432#106—22no—
433#106—22no—
434C4⋊C8—862no—
435#103—434no—
436#103—862no—
437#103—862no—
438#102—862no—
439#102—862no—
440#102—1721no—
441C4×C8—862no—
442#95—21no—
443#95—431no—
444#95—431no—
445#95—861no—
446#95—861no—
447#95—861no—
448C24—861no—
449C2×Q8—432no—
450C2×Q8—432no—
451C2×Q8—432no—
452C2×Q8—861no—
453C2×Q8—861no—
454C2×Q8—861no—
455C2×D4—432no—
456C2×D4—861no—
457C2×D4—861no—
458C2×D4—861no—
459C2×D4—861no—
460C2×D4—861no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—431no—
466C22×C4—431no—
467C22×C4—431no—
468C22×C4—431no—
469C22×C4—861no—
470C22×C4—861no—
471C22×C4—861no—
472C22×C4—3441no—
473C22×C4—1722no—
474C22×C4—1722no—
475C22×C4D4×D4311yescharacteristic
476C22×C4D4×D4311yescharacteristic
477C22×C4C23×D4311yescharacteristic
478SD16—868no—
479C2×C8—21no—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—21no—
483C2×C8—21no—
484C2×C8—432no—
485C2×C8—861no—
486C2×C8—861no—
487C2×C8—861no—
488C2×C8—861no—
489C2×C8—861no—
490C2×C8Q8×D4312yes—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—42no—
499C4⋊C4—81no—
500C4⋊C4—81no—
501C4⋊C4—432no—
502C4⋊C4—862no—
503C4⋊C4—864no—
504C4⋊C4—862no—
505C4⋊C4—861no—
506C4⋊C4—861no—
507C4⋊C4—861no—
508C4⋊C4—3441no—
509C4⋊C4—3441no—
510C4⋊C4—3441no—
511C4⋊C4—1722no—
512C4⋊C4—1722no—
513C4⋊C4—1722no—
514C4⋊C4—1722no—
515C4⋊C4D4×D4312yes—
516C22⋊C4—3441no—
517C22⋊C4—3441no—
518C22⋊C4—3441no—
519C22⋊C4—1722no—
520C22⋊C4—1722no—
521C42—21no—
522C42—21no—
523C42—864no—
524C42—861no—
525C42—861no—
526C42—861no—
527C42—861no—
528C42—1722no—
529C42—1722no—
530C23—862no—
531C23—1721no—
532C23—1721no—
533C23—1721no—
534C23—1721no—
535C23—1721no—
536C23—1721no—
537C23C2×D4×D4311yescharacteristic
538Q8—434no—
539Q8—862no—
540Q8—862no—
541Q8—862no—
542D4—862no—
543D4—862no—
544D4—1721no—
545D4—1721no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—22no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—432no—
564C2×C4—862no—
565C2×C4—862no—
566C2×C4—861no—
567C2×C4—861no—
568C2×C4—862no—
569C2×C4—862no—
570C2×C4—861no—
571C2×C4—861no—
572C2×C4—861no—
573C2×C4—861no—
574C2×C4—861no—
575C2×C4—861no—
576C2×C4—861no—
577C2×C4—861no—
578C2×C4—861no—
579C2×C4—3444no—
580C2×C4—1721no—
581C2×C4—1721no—
582C2×C4—1721no—
583C2×C4—1721no—
584C2×C4—1721no—
585C2×C4#1306334612yes—
586C2×C4C2×D4×D4311yescharacteristic
587C2×C4C2×Q8×D4311yescharacteristic
588C2×C4#41×D4311yescharacteristic
589C2×C4#41×D4311yescharacteristic
590C2×C4#1306337111yescharacteristic
591C2×C4#1306337311yescharacteristic
592C8—22no—
593C8—41no—
594C8—862no—
595C8—1721no—
596C22—3441no—
597C22—3441no—
598C22—3441no—
599C22—3441no—
600C22—1722no—
601C22—1722no—
602C22—1722no—
603C22#1458340711yescharacteristic
604C22#1458353311yescharacteristic
605C22#1458356411yescharacteristic
606C22C2×#1306334611yescharacteristic
607C22#137×D4311yescharacteristic
608C22#1458370311yescharacteristic
609C22#1458370511yescharacteristic
610C4—22no—
611C4—41no—
612C4—42no—
613C4—81no—
614C4—81no—
615C4—81no—
616C4—864no—
617C4—862no—
618C4—3441no—
619C4—1722no—
620C4—1721no—
621C4—1721no—
622C4—1721no—
623C4C22⋊Q8×D4311yescharacteristic
624C4#1458349711yescharacteristic
625C4#1458355111yescharacteristic
626C4#1458360511yescharacteristic
627C2—3442no—
628C2#1913839911yescharacteristic
629C2#1914467811yescharacteristic
630C2#1914478911yescharacteristic
631C2#1914611111yescharacteristic
632C2#1914635011yescharacteristic
633C2#1914747511yescharacteristic
634C2#1914750211yescharacteristic
subgroupG/H
centerC23C2×D4×D43
commutator, cosocleC22×C4×C43C24
FrattiniC22×C4C23×D43
FittingC43×#2084C2
socleC23×C43C22×D4
prime radicalC22×D172C23
hypercenter#2084D43
hypocenterC43#26379
LCSG ▷ C22×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2084

Complements

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

NH ≅ G/Ntype
1. C2×#14583209C2semidirect
4. #19137507C2semidirect
7. #19135935C2semidirect
8. #19135327C2semidirect
9. #19135151C2semidirect
13. #19134850C2semidirect
14. #19134848C2semidirect
15. C43×#2084C2semidirect
315. #2084D43semidirect
378. C43#26379semidirect

Maximal subgroups

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

subgroupindex
C2×#145832092
C2×#145829862
#191384652
#191375072
#191361242
#191359842
#191359352
#191353272
#191351512
#191349512
#191349492
#191348532
#191348502
#191348482
C43×#20842
#2637943