Finite Group Databasebeta

← #25104158

Subgroups — #25104158

Order 11008.

1000 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14583178C211yesmaximal, characteristic
2#19137166C211yesmaximal, characteristic, complemented
3C4×#13063287C211yesmaximal, characteristic
4#19135545C211yesmaximal, characteristic, complemented
5#19135352C211yesmaximal, characteristic, complemented
6#19135166C211yesmaximal, characteristic, complemented
7#19134845C211yesmaximal, characteristic
8#19134841C211yesmaximal, characteristic
9#19134594C211yesmaximal, characteristic
10#107×Dic43C211yesmaximal, characteristic, complemented
11#19134487C211yesmaximal, characteristic, complemented
12#19134443C211yesmaximal, characteristic, complemented
13#19134409C211yesmaximal, characteristic
14#19134409C211yesmaximal, characteristic
15C43×#1918C211yesmaximal, characteristic, complemented
16C2×#13063324C2211yescharacteristic
17C2×#13063290—21no—
18C2×C4×D172C2211yescharacteristic
19C2×#13063287C2211yescharacteristic
20C2×#13063287C2211yescharacteristic
21C2×C42×D43—21no—
22#14583178C418yescomplemented
23C2×#13063259C2211yescharacteristic
24C2×#13063258C2211yescharacteristic
25C2×#13063258C2211yescharacteristic
26C2×Dic43⋊C8C2211yescharacteristic
27C2×Dic43⋊C8C2211yescharacteristic
28C2×#13063247C2211yescharacteristic
29C2×#13063247C2211yescharacteristic
30C2×#13063245C2211yescharacteristic
31C2×#13063245C2211yescharacteristic
32C2×C8×Dic43C2211yescharacteristic
33C2×C8×Dic43C2211yescharacteristic
34#14582803—21no—
35#14582797C2211yescharacteristic
36#14582794—21no—
37#14582768C2211yescharacteristic
38C4⋊C4×Dic43C2211yescharacteristic
39C4⋊C4×Dic43C2211yescharacteristic
40C4⋊C4×Dic43C2211yescharacteristic
41C4⋊C4×Dic43C2211yescharacteristic
42#14582750—21no—
43C4×(C2×D43)⋊C4—21no—
44#14582746—21no—
45#14582746—21no—
46C42×Dic43C2211yescharacteristic
47#14582662C2211yescharacteristic
48#14582661C2211yescharacteristic
49#14582661C2211yescharacteristic
50#14582661C2211yescharacteristic
51#14582661C2211yescharacteristic
52#14582654C2211yescharacteristic
53C2×#107×C43C2211yescharacteristic
54C43×#389C2211yescharacteristic
55C4×C4⋊C4×C43C2211yescharacteristic
56C43×#340C2211yescharacteristic
57C43×#340C2211yescharacteristic
58C43×#340C2211yescharacteristic
59C43×#336C2211yescharacteristic
60C22×Dic86—21no—
61C22×D172C2311yescharacteristic
62C22×C4×D43—21no—
63C22×C4×D43—21no—
64C22×C4×D43—21no—
65C22×C43⋊C8C2311yescharacteristic
66C22×C43⋊C8C2311yescharacteristic
67#13063324—22no—
68#13063324C2×C414yescomplemented
69#13063290—44no—
70C4×D172—22no—
71C4×D172—22no—
72#13063287—22no—
73#13063287—22no—
74#13063287C2×C414yes—
75#13063287Q814yescomplemented
76C2×C4⋊Dic43C2311yescharacteristic
77C2×C4⋊Dic43C2311yescharacteristic
78C2×C4⋊Dic43C2311yescharacteristic
79C2×Dic43⋊C4—21no—
80C2×Dic43⋊C4—21no—
81C2×Dic43⋊C4—21no—
82C2×Dic43⋊C4—21no—
83C2×(C2×D43)⋊C4—21no—
84C2×(C2×D43)⋊C4—21no—
85C2×(C2×D43)⋊C4—21no—
86C2×C4×Dic43—21no—
87C2×C4×Dic43—21no—
88C2×C4×Dic43—21no—
89C2×C4×Dic43—21no—
90C2×C4×Dic43D411yescharacteristic
91C2×C4×Dic43D411yescharacteristic
92C2×C4×Dic43D411yescharacteristic
93C2×C4×Dic43D411yescharacteristic
94C2×C4×Dic43C2311yescharacteristic
95C2×C4×Dic43C2311yescharacteristic
96C42×D43—44no—
97#13063259—22no—
98#13063258—22no—
99#13063258C2×C414yescomplemented
100Dic43⋊C8—22no—
101Dic43⋊C8C2×C414yescomplemented
102#13063247—22no—
103#13063247C2×C414yes—
104#13063245—22no—
105#13063245C2×C414yes—
106C8×Dic43—22no—
107C8×Dic43—22no—
108C43⋊#95—21no—
109C43⋊#95—21no—
110C43⋊#95—21no—
111C43⋊#95—21no—
112C43⋊#95—21no—
113C22×C8×C43C2311yescharacteristic
114C22×C8×C43C2311yescharacteristic
115C2×C4⋊C4×C43—21no—
116C2×C4⋊C4×C43C2311yescharacteristic
117C2×C4⋊C4×C43C2311yescharacteristic
118C2×C4⋊C4×C43C2311yescharacteristic
119C2×C4⋊C4×C43C2311yescharacteristic
120C2×C42×C43—21no—
121C2×C42×C43C2311yescharacteristic
122#107×C43C2×C414yescomplemented
123#95×C43—21no—
124#95×C43—21no—
125C23×D43—21no—
126C22×C4×C43—21no—
127C22×C4×C43—21no—
128C22×C4×C43—21no—
129C22×C4×C43—21no—
130C22×C4×C43C2×D411yescharacteristic
131C22×C4×C43C2×D411yescharacteristic
132C22×C4×C43C2411yescharacteristic
133C22×Dic43—21no—
134C22×Dic43—21no—
135C22×Dic43—21no—
136C22×Dic43C2×D411yescharacteristic
137C22×Dic43C2×D411yescharacteristic
138C22×Dic43C2×D411yescharacteristic
139C22×Dic43C2×D411yescharacteristic
140C2×D172—21no—
141C2×D172—21no—
142C2×D172—21no—
143C2×D172C22×C412yes—
144C2×D172C2×Q812yes—
145C2×D172#4112yes—
146C2×C4×D43—24no—
147C2×C4×D43—24no—
148C2×C4×D43—24no—
149C2×C4×D43—42no—
150C2×C4×D43—42no—
151C2×C4×D43—42no—
152C2×C4×D43—42no—
153C2×C4×D43—42no—
154C2×C4×D43—42no—
155C2×Dic86—24no—
156C2×Dic86—42no—
157C2×Dic86—42no—
158C2×C8×C43—21no—
159C2×C8×C43—21no—
160C2×C8×C43—21no—
161C2×C8×C43—21no—
162C2×C8×C43—21no—
163C2×C8×C43C22×C412yes—
164C4⋊C4×C43—21no—
165C4⋊C4×C43—21no—
166C4⋊C4×C43—21no—
167C4⋊C4×C43—21no—
168C4⋊C4×C43—21no—
169C4⋊C4×C43—42no—
170C4⋊C4×C43C22×C412yes—
171C4⋊C4×C43C22×C412yes—
172C4⋊C4×C43C2×Q812yes—
173C42×C43—21no—
174C42×C43—21no—
175C42×C43—42no—
176(C2×D43)⋊C4—24no—
177(C2×D43)⋊C4—24no—
178(C2×D43)⋊C4—24no—
179C4⋊Dic43—21no—
180C4⋊Dic43—21no—
181C4⋊Dic43—21no—
182C4⋊Dic43—21no—
183C4⋊Dic43C22×C412yes—
184C4⋊Dic43#4112yes—
185Dic43⋊C4—24no—
186Dic43⋊C4—42no—
187Dic43⋊C4—42no—
188Dic43⋊C4—42no—
189C4×Dic43—22no—
190C4×Dic43—22no—
191C4×Dic43—41no—
192C4×Dic43—42no—
193C4×Dic43—42no—
194C4×Dic43—42no—
195C4×Dic43—42no—
196C4×Dic43D814yes—
197C4×Dic43C22×C412yes—
198C4×Dic43C2×Q812yes—
199C4×Dic43#4112yes—
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⋊C8C22×C412yes—
207C23×C43C22×D411yescharacteristic
208C22×D43—22no—
209C22×D43—22no—
210C22×D43—22no—
211C22×D43—41no—
212C22×D43—41no—
213C22×D43—41no—
214C22×D43—41no—
215C2×C4×C43—21no—
216C2×C4×C43—21no—
217C2×C4×C43—22no—
218C2×C4×C43—22no—
219C2×C4×C43—22no—
220C2×C4×C43—21no—
221C2×C4×C43—21no—
222C2×C4×C43—21no—
223C2×C4×C43—41no—
224C2×C4×C43—41no—
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×C43C2×D812yes—
233C2×C4×C43C23×C411yescharacteristic
234C2×C4×C43C22×Q811yescharacteristic
235C2×C4×C43C2×#4111yescharacteristic
236C2×C4×C43C2×#4111yescharacteristic
237C2×C4×C43C2×#4111yescharacteristic
238C2×C4×C43C2×#4111yescharacteristic
239C2×Dic43—21no—
240C2×Dic43—21no—
241C2×Dic43—21no—
242C2×Dic43—21no—
243C2×Dic43—22no—
244C2×Dic43—22no—
245C2×Dic43—41no—
246C2×Dic43—41no—
247C2×Dic43—41no—
248C2×Dic43—41no—
249C2×Dic43—41no—
250C2×Dic43—41no—
251C2×Dic43—41no—
252C2×Dic43C4×D412yes—
253C2×Dic43C4×D412yes—
254C2×Dic43C22⋊Q812yes—
255C2×Dic43C22⋊Q812yes—
256C2×Dic43C2×D812yes—
257C2×Dic43C2×D812yes—
258C2×Dic43#13512yes—
259C2×Dic43#13712yes—
260D172—22no—
261D172—22no—
262D172—22no—
263D172C4×Q814yescomplemented
264C4×D43—48no—
265C4×D43—48no—
266C4×D43—48no—
267Dic86—48no—
268C8×C43—22no—
269C8×C43—41no—
270C43⋊C8—22no—
271C43⋊C8—41no—
272C4×#474—431nomaximal, Sylow
273C22×C43C2×C4×D411yescharacteristic
274C22×C43C2×C22⋊Q811yescharacteristic
275C22×C43#52511yescharacteristic
276C22×C43C22×D811yescharacteristic
277C22×C43C2×#13511yescharacteristic
278C22×C43C2×#13711yescharacteristic
279C22×C43#57511yescharacteristic
280C2×D43—24no—
281C2×D43—42no—
282C2×D43—42no—
283C2×D43—42no—
284C2×D43—42no—
285C2×D43—42no—
286C2×D43—42no—
287C4×C43—22no—
288C4×C43—41no—
289C4×C43—42no—
290C4×C43—42no—
291C4×C43—42no—
292C4×C43—81no—
293C4×C43C2×C4×Q811yescharacteristic
294C4×C43C4×#4111yescharacteristic
295C4×C43#52911yescharacteristic
296C4×C43#53311yescharacteristic
297Dic43—22no—
298Dic43—22no—
299Dic43—42no—
300Dic43—42no—
301Dic43—81no—
302Dic43C4×D814yes—
303Dic43#47414yescomplemented
304C2×#474—431no—
305C4×C4⋊Q8—431no—
306C42×D4—431no—
307#1988—431no—
308#1947—431no—
309#1918—431no—
310#1918D4311yescharacteristic, complemented
311#1862—431no—
312#1862—431no—
313#1802—431no—
314C4×#107—431no—
315C4×#107—431no—
316C4×C4⋊C8—431no—
317C4×#102—431no—
318C4×#102—431no—
319C2×C43C4×C22⋊Q811yescharacteristic
320C2×C43C2×C4×D811yescharacteristic
321C2×C43C4×#13711yescharacteristic
322C2×C43C2×#47411yescharacteristic
323C2×C43#307611yescharacteristic
324C2×C43#310011yescharacteristic
325C2×C43#310711yescharacteristic
326D43—44no—
327C2×C4⋊Q8—431no—
328C2×C4×Q8—861no—
329C2×C4×D4—431no—
330C2×C4×D4—431no—
331C2×C4×D4—431no—
332C22×C42—861no—
333#474—438no—
334C2×#107—431no—
335C2×#107—431no—
336C2×#107—431no—
337C2×#107C2×D4311yescharacteristic
338C2×C4⋊C8—431no—
339C2×C4⋊C8—431no—
340C2×#102—431no—
341C2×#102—431no—
342C2×#102—431no—
343C2×#102—431no—
344C2×C4×C8—431no—
345C2×C4×C8—431no—
346#391—861no—
347#389—431no—
348#389—861no—
349#389C2×D4311yescharacteristic
350#384—431no—
351C4×C4⋊C4—431no—
352C4×C4⋊C4—431no—
353C4×C4⋊C4—431no—
354C4×C4⋊C4—431no—
355C4×C4⋊C4—861no—
356C4×C4⋊C4C2×D4311yescharacteristic
357C4×C22⋊C4—861no—
358C4×C22⋊C4—861no—
359C4×C22⋊C4—861no—
360C43—431no—
361#340—431no—
362#340—431no—
363#340—431no—
364#340—431no—
365#340—431no—
366#340C2×D4311yescharacteristic
367#340C2×D4311yescharacteristic
368#340C2×D4311yescharacteristic
369#336—431no—
370#336C2×D4311yescharacteristic
371C43C4×#47411yescharacteristic, complemented, Sylow
372C22×Q8—861no—
373C22×D4—431no—
374C23×C4—861no—
375C23×C4—861no—
376C23×C4—861no—
377C22×C8—431no—
378C22×C8—431no—
379C22×C8C22×D4311yescharacteristic
380C22×C8C22×D4311yescharacteristic
381C4⋊Q8—434no—
382C4⋊Q8—862no—
383C4×Q8—1724no—
384C4×D4—434no—
385C4×D4—434no—
386C4×D4—862no—
387C4×D4—862no—
388C4×D4—862no—
389C4×D4—862no—
390C2×C4⋊C4—21no—
391C2×C4⋊C4—431no—
392C2×C4⋊C4—431no—
393C2×C4⋊C4—431no—
394C2×C4⋊C4—861no—
395C2×C4⋊C4—861no—
396C2×C4⋊C4—861no—
397C2×C4⋊C4—861no—
398C2×C4⋊C4C22×D4311yescharacteristic
399C2×C4⋊C4C22×D4311yescharacteristic
400C2×C4⋊C4C22×D4311yescharacteristic
401C2×C4⋊C4C22×D4311yescharacteristic
402C2×C22⋊C4—861no—
403C2×C22⋊C4—861no—
404C2×C22⋊C4—861no—
405C2×C42—21no—
406C2×C42—431no—
407C2×C42—431no—
408C2×C42—431no—
409C2×C42—431no—
410C2×C42—431no—
411C2×C42—431no—
412C2×C42—861no—
413C2×C42—861no—
414C2×C42—861no—
415C2×C42—861no—
416C2×C42—1724no—
417C2×C42C22×D4311yescharacteristic
418#107—434no—
419#107—862no—
420#107—862no—
421#107C4×D4314yescomplemented
422C4⋊C8—434no—
423C4⋊C8—862no—
424#102—434no—
425#102—434no—
426#102—862no—
427#102—862no—
428C4×C8—862no—
429C4×C8—862no—
430#95—21no—
431#95—21no—
432#95—861no—
433#95—861no—
434#95—861no—
435#95—861no—
436#95—861no—
437C24—861no—
438C2×Q8—864no—
439C2×Q8—1722no—
440C2×Q8—1722no—
441C2×D4—432no—
442C2×D4—432no—
443C2×D4—432no—
444C2×D4—861no—
445C2×D4—861no—
446C2×D4—861no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—431no—
452C22×C4—431no—
453C22×C4—431no—
454C22×C4—431no—
455C22×C4—864no—
456C22×C4—864no—
457C22×C4—864no—
458C22×C4—861no—
459C22×C4—861no—
460C22×C4—861no—
461C22×C4—1722no—
462C22×C4—1722no—
463C22×C4—1722no—
464C22×C4—1722no—
465C22×C4—1722no—
466C22×C4—1722no—
467C22×C4D4×D4311yescharacteristic
468C22×C4D4×D4311yescharacteristic
469C22×C4C23×D4311yescharacteristic
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—432no—
476C2×C8—861no—
477C2×C8—861no—
478C2×C8—861no—
479C2×C8—861no—
480C2×C8—861no—
481C2×C8C2×C4×D4312yes—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—42no—
488C4⋊C4—432no—
489C4⋊C4—432no—
490C4⋊C4—864no—
491C4⋊C4—861no—
492C4⋊C4—861no—
493C4⋊C4—861no—
494C4⋊C4—861no—
495C4⋊C4—1722no—
496C4⋊C4—1722no—
497C4⋊C4—1722no—
498C4⋊C4C2×C4×D4312yes—
499C4⋊C4C2×C4×D4312yes—
500C4⋊C4Q8×D4312yes—
501C22⋊C4—864no—
502C22⋊C4—864no—
503C22⋊C4—864no—
504C42—21no—
505C42—21no—
506C42—42no—
507C42—434no—
508C42—432no—
509C42—432no—
510C42—432no—
511C42—432no—
512C42—862no—
513C42—862no—
514C42—1721no—
515C42—1722no—
516C42—1722no—
517C42—1722no—
518C42—1722no—
519C23—862no—
520C23—862no—
521C23—862no—
522C23—1721no—
523C23—1721no—
524C23—1721no—
525C23—1721no—
526C23C2×D4×D4311yescharacteristic
527Q8—1728no—
528D4—434no—
529D4—862no—
530D4—862no—
531D4—862no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—432no—
550C2×C4—432no—
551C2×C4—432no—
552C2×C4—432no—
553C2×C4—432no—
554C2×C4—432no—
555C2×C4—432no—
556C2×C4—432no—
557C2×C4—861no—
558C2×C4—861no—
559C2×C4—861no—
560C2×C4—861no—
561C2×C4—862no—
562C2×C4—862no—
563C2×C4—1728no—
564C2×C4—1728no—
565C2×C4—1728no—
566C2×C4—1721no—
567C2×C4—1721no—
568C2×C4—1721no—
569C2×C4—1721no—
570C2×C4—1721no—
571C2×C4—1721no—
572C2×C4—1721no—
573C2×C4D8×D4312yes—
574C2×C4C22×C4×D4311yescharacteristic
575C2×C4C2×Q8×D4311yescharacteristic
576C2×C4#41×D4311yescharacteristic
577C2×C4#41×D4311yescharacteristic
578C2×C4#41×D4311yescharacteristic
579C2×C4#41×D4311yescharacteristic
580C8—22no—
581C8—41no—
582C8—862no—
583C8—1721no—
584C22—864no—
585C22—1722no—
586C22—1722no—
587C22—1722no—
588C22—1722no—
589C22—1722no—
590C22—1722no—
591C22C4×D4×D4311yescharacteristic
592C22C22⋊Q8×D4311yescharacteristic
593C22#1458340711yescharacteristic
594C22C2×D8×D4311yescharacteristic
595C22#135×D4311yescharacteristic
596C22#137×D4311yescharacteristic
597C22#1458369811yescharacteristic
598C4—22no—
599C4—42no—
600C4—42no—
601C4—42no—
602C4—41no—
603C4—81no—
604C4—434no—
605C4—434no—
606C4—862no—
607C4—862no—
608C4—3441no—
609C4—1722no—
610C4—1722no—
611C4C4×Q8×D4311yescharacteristic
612C4#1458336711yescharacteristic
613C4#1458343211yescharacteristic
614C4#1458345511yescharacteristic
615C2—1724no—
616C2#1913712911yescharacteristic
617C2C4×D8×D4311yescharacteristic
618C2#1914341111yescharacteristic
619C2#474×D4311yescharacteristic
620C2#1914438211yescharacteristic
621C2#1914459211yescharacteristic
622C2#1914466111yescharacteristic
subgroupG/H
centerC23C2×D4×D43
commutatorC2×C4×C43C23×C4
FrattiniC22×C4C23×D43
FittingC43×#1918C2
socleC23×C43C22×D4
prime radicalC22×D172C23
cosocleC22×C4×C43C24
hypercenter#1918D43
hypocenterC43C4×#474
LCSG ▷ C2×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1918

Complements

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

NH ≅ G/Ntype
2. #19137166C2semidirect
4. #19135545C2semidirect
5. #19135352C2semidirect
6. #19135166C2semidirect
10. #107×Dic43C2semidirect
11. #19134487C2semidirect
12. #19134443C2semidirect
15. C43×#1918C2semidirect
22. #14583178C4semidirect
68. #13063324C2×C4semidirect
75. #13063287Q8semidirect
99. #13063258C2×C4semidirect
101. Dic43⋊C8C2×C4semidirect
122. #107×C43C2×C4semidirect
263. D172C4×Q8semidirect
303. Dic43#474semidirect
310. #1918D43semidirect
371. C43C4×#474semidirect
421. #107C4×D43semidirect

Maximal subgroups

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

subgroupindex
C2×#145831782
#191371662
C4×#130632872
#191355452
#191353522
#191351662
#191348452
#191348412
#191345942
#107×Dic432
#191344872
#191344432
#191344092
C43×#19182
C4×#47443