Finite Group Databasebeta

← #25021900

Subgroups — #25021900

Order 11008.

2062 conjugacy classes of subgroups.

The subgroup lattice has 682 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×Dic86C211yesmaximal, characteristic, complemented
2C2×#14582812C211yesmaximal, characteristic
3C2×#14582773C212yesmaximal
4C2×C4×(C2×D43)⋊C4C211yesmaximal, characteristic
5#19132865C211yesmaximal, characteristic, complemented
6#19132860C211yesmaximal, characteristic, complemented
7#19132841C211yesmaximal, characteristic
8#19132831C212yesmaximal
9#19132830C212yesmaximal, complemented
10#19132829C212yesmaximal, complemented
11C43×#1434C211yesmaximal, characteristic, complemented
12C23×Dic86C2211yescharacteristic
13C23×C4×D43C2211yescharacteristic
14C2×C4×Dic86—28no—
15C2×C4×Dic86—24no—
16C22×C4⋊Dic43C2211yescharacteristic
17C22×C4⋊Dic43C2212yes—
18C22×Dic43⋊C4C2212yes—
19C22×(C2×D43)⋊C4C2212yes—
20C22×(C2×D43)⋊C4C2211yescharacteristic
21C22×(C2×D43)⋊C4C2212yes—
22C22×(C2×D43)⋊C4C2211yescharacteristic
23C22×C4×Dic43C2212yes—
24C22×C4×Dic43C2211yescharacteristic
25#14582812C418yescomplemented
26#14582773—28no—
27C4×(C2×D43)⋊C4—24no—
28C2×C43⋊#95C2211yescharacteristic
29C2×C43⋊#95C2212yes—
30C2×C43⋊#95C2211yescharacteristic
31C2×C43⋊#95C2212yes—
32C2×C43⋊#95C2212yes—
33C2×C43⋊#95C2212yes—
34C2×C43⋊#95C2212yes—
35C22×C42×C43C2211yescharacteristic
36C2×#95×C43C2212yes—
37C2×#95×C43C2212yes—
38C2×#95×C43C2211yescharacteristic
39C2×#95×C43C2211yescharacteristic
40C24×D43C2311yescharacteristic
41C22×Dic86—22no—
42C22×Dic86—22no—
43C22×Dic86—24no—
44C22×Dic86—24no—
45C22×Dic86C2×C414yescomplemented
46C23×Dic43C2311yescharacteristic
47C23×Dic43C2312yes—
48C23×Dic43C2311yescharacteristic
49C23×Dic43C2311yescharacteristic
50C23×Dic43C2312yes—
51C22×C4×D43—22no—
52C22×C4×D43—22no—
53C22×C4×D43—22no—
54C22×C4×D43—24no—
55C22×C4×D43—42no—
56C4×Dic86—232no—
57C2×C4⋊Dic43—22no—
58C2×C4⋊Dic43—24no—
59C2×C4⋊Dic43—24no—
60C2×C4⋊Dic43—24no—
61C2×C4⋊Dic43—42no—
62C2×Dic43⋊C4—24no—
63C2×Dic43⋊C4—24no—
64C2×Dic43⋊C4D418yes—
65C2×(C2×D43)⋊C4—22no—
66C2×(C2×D43)⋊C4—22no—
67C2×(C2×D43)⋊C4—22no—
68C2×(C2×D43)⋊C4—22no—
69C2×(C2×D43)⋊C4—24no—
70C2×(C2×D43)⋊C4—24no—
71C2×(C2×D43)⋊C4—24no—
72C2×(C2×D43)⋊C4—24no—
73C2×(C2×D43)⋊C4—42no—
74C2×(C2×D43)⋊C4C2×C414yes—
75C2×(C2×D43)⋊C4C2×C418yes—
76C2×(C2×D43)⋊C4Q814yescomplemented
77C2×C4×Dic43—24no—
78C2×C4×Dic43—24no—
79C2×C4×Dic43—24no—
80C2×C4×Dic43—22no—
81C2×C4×Dic43—42no—
82C43⋊#95—22no—
83C43⋊#95—22no—
84C43⋊#95—24no—
85C43⋊#95—24no—
86C43⋊#95—42no—
87C43⋊#95—42no—
88C43⋊#95C2×C418yescomplemented
89C23×C4×C43C2312yes—
90C23×C4×C43C2312yes—
91C23×C4×C43C2311yescharacteristic
92C23×C4×C43C2311yescharacteristic
93C23×C4×C43C2311yescharacteristic
94C2×C42×C43—24no—
95C2×C42×C43D414yes—
96#95×C43—24no—
97#95×C43—22no—
98#95×C43—24no—
99#95×C43C2×C414yescomplemented
100C24×C43C2411yescharacteristic
101C23×D43—21no—
102C23×D43—22no—
103C23×D43—21no—
104C23×D43—22no—
105C23×D43—21no—
106C23×D43—21no—
107C23×D43—22no—
108C23×D43—22no—
109C23×D43C22×C412yes—
110C23×D43C2×Q812yes—
111C23×D43#4112yes—
112C22×C4×C43—22no—
113C22×C4×C43—22no—
114C22×C4×C43—22no—
115C22×C4×C43—22no—
116C22×C4×C43—21no—
117C22×C4×C43—21no—
118C22×C4×C43—22no—
119C22×C4×C43—22no—
120C22×C4×C43—22no—
121C22×C4×C43—22no—
122C22×C4×C43—22no—
123C22×C4×C43—22no—
124C22×C4×C43—22no—
125C22×C4×C43—22no—
126C22×C4×C43—22no—
127C22×C4×C43—22no—
128C22×C4×C43—22no—
129C22×C4×C43—22no—
130C22×C4×C43—22no—
131C22×C4×C43—22no—
132C22×C4×C43C22×C414yes—
133C22×C4×C43C22×C412yes—
134C22×C4×C43C2×D412yes—
135C22×C4×C43C2×D412yes—
136C22×C4×C43C2×D412yes—
137C22×C4×C43C2×D414yes—
138C22×C4×C43C2×Q812yes—
139C22×C4×C43#4112yes—
140C22×C4×C43#4112yes—
141C22×Dic43—22no—
142C22×Dic43—22no—
143C22×Dic43—21no—
144C22×Dic43—21no—
145C22×Dic43—21no—
146C22×Dic43—22no—
147C22×Dic43—22no—
148C22×Dic43—22no—
149C22×Dic43—22no—
150C22×Dic43—22no—
151C22×Dic43—22no—
152C22×Dic43—22no—
153C22×Dic43—21no—
154C22×Dic43—21no—
155C22×Dic43—22no—
156C22×Dic43—21no—
157C22×Dic43—22no—
158C22×Dic43—22no—
159C22×Dic43—22no—
160C22×Dic43—22no—
161C22×Dic43—22no—
162C22×Dic43—22no—
163C22×Dic43C22×C412yes—
164C22×Dic43C22×C414yes—
165C22×Dic43C2×D414yes—
166C22×Dic43C2×D414yes—
167C22×Dic43C2×Q812yes—
168C22×Dic43#4112yes—
169C22×Dic43#4114yes—
170C2×C4×D43—28no—
171C2×C4×D43—48no—
172C2×C4×D43—48no—
173C2×C4×D43—44no—
174C2×C4×D43—44no—
175C2×Dic86—28no—
176C2×Dic86—216no—
177C2×Dic86—28no—
178C2×Dic86—28no—
179C2×Dic86—216no—
180C42×C43—28no—
181(C2×D43)⋊C4—28no—
182(C2×D43)⋊C4—28no—
183(C2×D43)⋊C4—216no—
184(C2×D43)⋊C4—48no—
185C4⋊Dic43—28no—
186C4⋊Dic43—48no—
187Dic43⋊C4—216no—
188C4×Dic43—216no—
189C4×Dic43—44no—
190C23×C43C23×C411yescharacteristic
191C23×C43C22×D412yes—
192C23×C43C22×D411yescharacteristic
193C23×C43C22×Q811yescharacteristic
194C23×C43C2×#4112yes—
195C23×C43C2×#4111yescharacteristic
196C23×C43C2×#4111yescharacteristic
197C23×C43C2×#4111yescharacteristic
198C23×C43C2×#4111yescharacteristic
199C23×C43H(5,𝔽2)12yes—
200C23×C432−1+412yes—
201C22×D43—24no—
202C22×D43—22no—
203C22×D43—22no—
204C22×D43—24no—
205C22×D43—22no—
206C22×D43—22no—
207C22×D43—22no—
208C22×D43—22no—
209C22×D43—24no—
210C22×D43—24no—
211C22×D43—24no—
212C22×D43—24no—
213C22×D43—41no—
214C22×D43—42no—
215C22×D43—41no—
216C22×D43—41no—
217C22×D43—41no—
218C22×D43—42no—
219C22×D43—42no—
220C22×D43—42no—
221C22×D43—42no—
222C22×D43—42no—
223C22×D43C4×Q814yescomplemented
224C2×C4×C43—24no—
225C2×C4×C43—22no—
226C2×C4×C43—22no—
227C2×C4×C43—22no—
228C2×C4×C43—24no—
229C2×C4×C43—22no—
230C2×C4×C43—22no—
231C2×C4×C43—22no—
232C2×C4×C43—24no—
233C2×C4×C43—24no—
234C2×C4×C43—24no—
235C2×C4×C43—24no—
236C2×C4×C43—24no—
237C2×C4×C43—24no—
238C2×C4×C43—24no—
239C2×C4×C43—42no—
240C2×C4×C43—42no—
241C2×C4×C43—42no—
242C2×C4×C43—42no—
243C2×C4×C43—42no—
244C2×C4×C43—42no—
245C2×C4×C43—42no—
246C2×C4×C43—42no—
247C2×C4×C43—42no—
248C2×C4×C43—42no—
249C2×C4×C43—42no—
250C2×C4×C43C4×D414yes—
251C2×C4×C43#12014yes—
252C2×C4×C43C22⋊Q814yes—
253C2×Dic43—24no—
254C2×Dic43—24no—
255C2×Dic43—24no—
256C2×Dic43—24no—
257C2×Dic43—24no—
258C2×Dic43—24no—
259C2×Dic43—22no—
260C2×Dic43—24no—
261C2×Dic43—24no—
262C2×Dic43—22no—
263C2×Dic43—22no—
264C2×Dic43—24no—
265C2×Dic43—24no—
266C2×Dic43—24no—
267C2×Dic43—24no—
268C2×Dic43—41no—
269C2×Dic43—41no—
270C2×Dic43—42no—
271C2×Dic43—41no—
272C2×Dic43—41no—
273C2×Dic43—41no—
274C2×Dic43—42no—
275C2×Dic43—41no—
276C2×Dic43—42no—
277C2×Dic43—42no—
278C2×Dic43—42no—
279C2×Dic43—42no—
280C2×Dic43—42no—
281C2×Dic43C4×D418yes—
282C4×D43—416no—
283Dic86—232no—
284#14172—431nomaximal, Sylow
285C22×C43C2×C4×D412yes—
286C22×C43C2×C4×D411yescharacteristic
287C22×C43C2×C4×Q811yescharacteristic
288C22×C43C4×#4111yescharacteristic
289C22×C43#52012yes—
290C22×C43C2×#12011yescharacteristic
291C22×C43C2×C22⋊Q811yescharacteristic
292C22×C43#52512yes—
293C22×C43#52511yescharacteristic
294C22×C43#52511yescharacteristic
295C22×C43#52911yescharacteristic
296C22×C43#53311yescharacteristic
297C22×C43#53511yescharacteristic
298C22×C43#53611yescharacteristic
299C22×C43#54212yes—
300C22×C43#54612yes—
301C22×C43#54612yes—
302C22×C43#54812yes—
303C22×C43D4×Q812yes—
304C22×C43#55212yes—
305C22×C43#55412yes—
306C22×C43#55612yes—
307C22×C43#55912yes—
308C2×D43—28no—
309C2×D43—24no—
310C2×D43—42no—
311C2×D43—42no—
312C2×D43—42no—
313C2×D43—44no—
314C2×D43—42no—
315C2×D43—42no—
316C2×D43—44no—
317C2×D43—44no—
318C2×D43—42no—
319C4×C43—24no—
320C4×C43—24no—
321C4×C43—24no—
322C4×C43—44no—
323C4×C43—82no—
324Dic43—28no—
325Dic43—42no—
326Dic43—42no—
327Dic43—42no—
328Dic43—44no—
329C22×C4×Q8—431no—
330C2×#393—431no—
331C2×#385—432no—
332C2×C4×C22⋊C4—431no—
333#1446—431no—
334#1444—431no—
335#1439—431no—
336#1437—432no—
337#1436—432no—
338#1436—432no—
339#1434D4311yescharacteristic, complemented
340C2×C43C4×#12011yescharacteristic
341C2×C43C4×C22⋊Q811yescharacteristic
342C2×C43#232611yescharacteristic
343C2×C43#236012yes—
344C2×C43#236712yes—
345C2×C43#240411yescharacteristic
346C2×C43#240911yescharacteristic
347C2×C43#245812yes—
348C2×C43#246512yes—
349C2×C43#258011yescharacteristic
350C2×C43#259911yescharacteristic
351D43—44no—
352C23×Q8—431no—
353C24×C4—431no—
354C2×C4×Q8—868no—
355C2×C4×Q8—864no—
356C22×C4⋊C4—432no—
357C22×C4⋊C4—431no—
358C22×C4⋊C4—432no—
359C22×C22⋊C4—432no—
360C22×C22⋊C4—431no—
361C22×C22⋊C4—431no—
362C22×C22⋊C4—432no—
363C22×C42—432no—
364C22×C42—431no—
365C22×C42C2×D4311yescharacteristic
366#393—438no—
367#385—868no—
368C4×C22⋊C4—864no—
369C2×#95—432no—
370C2×#95—431no—
371C2×#95—432no—
372C2×#95—432no—
373C2×#95—431no—
374C2×#95—432no—
375C2×#95—432no—
376C2×#95C2×D4312yes—
377C2×#95C2×D4312yes—
378C2×#95C2×D4311yescharacteristic
379C2×#95C2×D4311yescharacteristic
380C43#1417211yescharacteristic, complemented, Sylow
381C25—431no—
382C22×Q8—434no—
383C22×Q8—864no—
384C22×Q8—864no—
385C22×Q8—862no—
386C22×Q8—862no—
387C23×C4—432no—
388C23×C4—432no—
389C23×C4—431no—
390C23×C4—431no—
391C23×C4—431no—
392C23×C4—862no—
393C23×C4—862no—
394C23×C4—862no—
395C23×C4—864no—
396C23×C4—1722no—
397C23×C4C22×D4312yes—
398C23×C4C22×D4312yes—
399C23×C4C22×D4311yescharacteristic
400C23×C4C22×D4311yescharacteristic
401C23×C4C22×D4311yescharacteristic
402C4×Q8—8632no—
403C2×C4⋊C4—438no—
404C2×C4⋊C4—864no—
405C2×C4⋊C4—864no—
406C2×C4⋊C4—862no—
407C2×C4⋊C4—864no—
408C2×C4⋊C4—864no—
409C2×C4⋊C4—864no—
410C2×C4⋊C4—1722no—
411C2×C22⋊C4—438no—
412C2×C22⋊C4—434no—
413C2×C22⋊C4—434no—
414C2×C22⋊C4—864no—
415C2×C22⋊C4—864no—
416C2×C22⋊C4—862no—
417C2×C22⋊C4—862no—
418C2×C22⋊C4—862no—
419C2×C22⋊C4—862no—
420C2×C22⋊C4—864no—
421C2×C22⋊C4—864no—
422C2×C22⋊C4—1722no—
423C2×C42—24no—
424C2×C42—864no—
425C2×C42—864no—
426C2×C42—864no—
427C2×C42—862no—
428C2×C42—1722no—
429C2×C42D17214yes—
430#95—22no—
431#95—24no—
432#95—24no—
433#95—438no—
434#95—864no—
435#95—864no—
436#95—862no—
437#95—862no—
438#95—1722no—
439#95—1722no—
440#95C4×D4314yescomplemented
441C24—432no—
442C24—432no—
443C24—432no—
444C24—862no—
445C24—861no—
446C24—862no—
447C24—861no—
448C24—861no—
449C24—861no—
450C24—862no—
451C24—862no—
452C24C23×D4311yescharacteristic
453C2×Q8—8616no—
454C2×Q8—868no—
455C2×Q8—8616no—
456C2×Q8—868no—
457C2×Q8—868no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—434no—
479C22×C4—434no—
480C22×C4—434no—
481C22×C4—432no—
482C22×C4—434no—
483C22×C4—432no—
484C22×C4—432no—
485C22×C4—868no—
486C22×C4—862no—
487C22×C4—862no—
488C22×C4—862no—
489C22×C4—862no—
490C22×C4—862no—
491C22×C4—862no—
492C22×C4—861no—
493C22×C4—861no—
494C22×C4—861no—
495C22×C4—862no—
496C22×C4—862no—
497C22×C4—862no—
498C22×C4—862no—
499C22×C4—861no—
500C22×C4—861no—
501C22×C4—861no—
502C22×C4—862no—
503C22×C4—862no—
504C22×C4—862no—
505C22×C4—862no—
506C22×C4—862no—
507C22×C4—862no—
508C22×C4—1728no—
509C22×C4—1728no—
510C22×C4—1724no—
511C22×C4—1724no—
512C22×C4C2×C4×D4314yes—
513C22×C4C2×C4×D4312yes—
514C22×C4C2×D17212yes—
515C22×C4C2×D17212yes—
516C22×C4C2×D17212yes—
517C22×C4#1062189812yes—
518C22×C4D4×D4314yes—
519C22×C4#1062190012yes—
520C22×C4Q8×D4312yes—
521C4⋊C4—868no—
522C4⋊C4—8616no—
523C4⋊C4—1728no—
524C22⋊C4—8616no—
525C22⋊C4—868no—
526C22⋊C4—868no—
527C22⋊C4—1728no—
528C42—28no—
529C42—8616no—
530C42—1724no—
531C23—434no—
532C23—864no—
533C23—864no—
534C23—864no—
535C23—864no—
536C23—862no—
537C23—862no—
538C23—864no—
539C23—862no—
540C23—862no—
541C23—862no—
542C23—862no—
543C23—864no—
544C23—1722no—
545C23—1722no—
546C23—1722no—
547C23—1721no—
548C23—1722no—
549C23—1721no—
550C23—1721no—
551C23—1721no—
552C23—1722no—
553C23—1722no—
554C23C22×C4×D4311yescharacteristic
555C23C2×D4×D4312yes—
556C23C22×D17211yescharacteristic
557C23C2×Q8×D4311yescharacteristic
558C23#41×D4311yescharacteristic
559C23#41×D4312yes—
560C23#41×D4311yescharacteristic
561C23C2×#1062190011yescharacteristic
562C23C2×#1062189811yescharacteristic
563C23#1306337112yes—
564C23#1306337212yes—
565Q8—8632no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—24no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—438no—
593C2×C4—864no—
594C2×C4—864no—
595C2×C4—864no—
596C2×C4—864no—
597C2×C4—864no—
598C2×C4—864no—
599C2×C4—864no—
600C2×C4—862no—
601C2×C4—864no—
602C2×C4—864no—
603C2×C4—862no—
604C2×C4—862no—
605C2×C4—864no—
606C2×C4—864no—
607C2×C4—864no—
608C2×C4—17216no—
609C2×C4—1722no—
610C2×C4—1721no—
611C2×C4—1721no—
612C2×C4—1722no—
613C2×C4—1721no—
614C2×C4—1721no—
615C2×C4—1721no—
616C2×C4—1722no—
617C2×C4—1721no—
618C2×C4—1722no—
619C2×C4—1722no—
620C2×C4—1722no—
621C2×C4—1722no—
622C2×C4C4×D17214yes—
623C2×C4#1306329514yes—
624C2×C4#1306330514yes—
625C22—868no—
626C22—864no—
627C22—1722no—
628C22—1724no—
629C22—1722no—
630C22—1722no—
631C22—1722no—
632C22—1722no—
633C22—1724no—
634C22—1722no—
635C22—1724no—
636C22C4×D4×D4312yes—
637C22C2×C4×D17211yescharacteristic
638C22C4×Q8×D4311yescharacteristic
639C22C4×#1062190011yescharacteristic
640C22#1458337912yes—
641C22C2×#1306329511yescharacteristic
642C22C2×#1306330511yescharacteristic
643C22#1458340812yes—
644C22#1458341111yescharacteristic
645C22#1458341111yescharacteristic
646C22#1458343111yescharacteristic
647C22#1458345311yescharacteristic
648C22#1458346811yescharacteristic
649C22#1458347511yescharacteristic
650C22#1458352012yes—
651C22#1458354112yes—
652C22#1458354112yes—
653C22#1458355512yes—
654C22#1458356212yes—
655C22#1458358512yes—
656C22#1458360512yes—
657C22#1458361612yes—
658C22#1458363112yes—
659C4—24no—
660C4—24no—
661C4—24no—
662C4—44no—
663C4—82no—
664C4—868no—
665C4—1722no—
666C4—1722no—
667C4—1722no—
668C4—1724no—
669C2—1724no—
670C2C4×#1306329511yescharacteristic
671C2C4×#1306330511yescharacteristic
672C2#1913730611yescharacteristic
673C2#1913760312yes—
674C2#1913766912yes—
675C2#1913791611yescharacteristic
676C2#1913796111yescharacteristic
677C2#1913851912yes—
678C2#1913861812yes—
679C2#1913995411yescharacteristic
680C2#1914017411yescharacteristic
subgroupG/H
center, FrattiniC24C23×D43
commutatorC23×C43C23×C4
FittingC43×#1434C2
socle, cosocleC24×C43C24
prime radicalC24×D43C23
hypercenter#1434D43
hypocenterC43#14172
LCSG ▷ C23×C43 ▷ C43
UCSC1 ◁ C24 ◁ #1434

Complements

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

NH ≅ G/Ntype
1. C22×C4×Dic86C2semidirect
5. #19132865C2semidirect
6. #19132860C2semidirect
9. #19132830C2semidirect
10. #19132829C2semidirect
11. C43×#1434C2semidirect
25. #14582812C4semidirect
45. C22×Dic86C2×C4semidirect
76. C2×(C2×D43)⋊C4Q8semidirect
88. C43⋊#95C2×C4semidirect
99. #95×C43C2×C4semidirect
223. C22×D43C4×Q8semidirect
339. #1434D43semidirect
380. C43#14172semidirect
440. #95C4×D43semidirect

Maximal subgroups

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

subgroupindex
C22×C4×Dic862
C2×#145828122
C2×#145827732
C2×C4×(C2×D43)⋊C42
#191328652
#191328602
#191328412
#191328312
#191328302
#191328292
C43×#14342
#1417243