Finite Group Databasebeta

← #25132679

Subgroups — #25132679

Order 11008.

1150 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14583456C211yesmaximal, characteristic, complemented
2C2×#14583190C211yesmaximal, characteristic, complemented
3C2×#14583180C211yesmaximal, characteristic, complemented
4C4×#13063299C211yesmaximal, characteristic
5#19135934C212yesmaximal, complemented
6#19135742C212yesmaximal, complemented
7#19135546C211yesmaximal, characteristic, complemented
8#19135542C211yesmaximal, characteristic, complemented
9#19134915C212yesmaximal
10#19134796C212yesmaximal, complemented
11C43×#1797C211yesmaximal, characteristic, complemented
12#14583456—24no—
13#14583456—24no—
14C2×#13063325C2211yescharacteristic
15C2×#13063325C2211yescharacteristic
16C2×#13063317C2212yes—
17C2×#13063304—21no—
18C2×#13063304—21no—
19C2×#13063299C2211yescharacteristic
20C2×C4×Dic86—21no—
21C2×C4×Dic86—21no—
22C2×C4×D172C2211yescharacteristic
23C2×C4×C22⋊D43—21no—
24C2×#13063284C2212yes—
25#14583190—24no—
26#14583180—24no—
27C2×#13063259C2212yes—
28C2×#13063257C2212yes—
29C2×#13063247C2212yes—
30C2×#13063247C2212yes—
31#14582801C2211yescharacteristic
32#14582797C2211yescharacteristic
33#14582790—21no—
34#14582778—21no—
35#14582759C2211yescharacteristic
36C4×C4⋊Dic43C2211yescharacteristic
37C4×(C2×D43)⋊C4—21no—
38#14582662C2212yes—
39#14582662C2212yes—
40#14582662C2212yes—
41#14582662C2212yes—
42C22×C42×C43C2211yescharacteristic
43C2×C4⋊C8×C43C2211yescharacteristic
44C2×C4⋊C8×C43C2211yescharacteristic
45C2×C22⋊C8×C43C2212yes—
46C43×#336C2212yes—
47C22×Dic86—21no—
48C22×Dic86—21no—
49C22×D172C2311yescharacteristic
50C22×C22⋊D43—21no—
51C22×C4×D43—21no—
52#13063325—22no—
53#13063325—22no—
54#13063325D414yescomplemented
55#13063325D414yescomplemented
56#13063317—24no—
57#13063317—24no—
58#13063304—28no—
59#13063304—28no—
60#13063299—22no—
61#13063299Q814yescomplemented
62C4×Dic86—28no—
63C4×Dic86—28no—
64C4×D172—22no—
65C4×D172—22no—
66C4×C22⋊D43—44no—
67#13063284—24no—
68#13063284—24no—
69C2×C4⋊Dic43—21no—
70C2×C4⋊Dic43—21no—
71C2×C4⋊Dic43C2311yescharacteristic
72C2×C4⋊Dic43C2311yescharacteristic
73C2×C4⋊Dic43C2312yes—
74C2×C4⋊Dic43C2312yes—
75C2×C4⋊Dic43C2311yescharacteristic
76C2×Dic43⋊C4—22no—
77C2×Dic43⋊C4—22no—
78C2×Dic43⋊C4—21no—
79C2×C22⋊Dic43—22no—
80C2×C22⋊Dic43—21no—
81C2×(C2×D43)⋊C4—21no—
82C2×(C2×D43)⋊C4—21no—
83C2×(C2×D43)⋊C4—21no—
84C2×(C2×D43)⋊C4—21no—
85C2×C4×Dic43—22no—
86C2×C4×Dic43—21no—
87#13063259—24no—
88#13063257—24no—
89#13063247—24no—
90#13063247—24no—
91C43⋊#95—21no—
92C43⋊#95—21no—
93C43⋊#95—21no—
94C23×C4×C43D411yescharacteristic
95C23×C4×C43D411yescharacteristic
96C23×C4×C43C2311yescharacteristic
97C22×C8×C43C2312yes—
98C22×C8×C43C2312yes—
99C2×C42×C43—22no—
100C2×C42×C43—22no—
101C2×C42×C43D411yescharacteristic
102C2×C42×C43D411yescharacteristic
103C2×C42×C43C2311yescharacteristic
104C2×C42×C43C2311yescharacteristic
105C4⋊C8×C43—22no—
106C4⋊C8×C43—22no—
107C22⋊C8×C43—24no—
108C24×C43C2×D411yescharacteristic
109C23×D43—21no—
110C22×C4×C43—21no—
111C22×C4×C43—21no—
112C22×C4×C43—21no—
113C22×C4×C43—21no—
114C22×C4×C43—22no—
115C22×C4×C43—22no—
116C22×C4×C43—22no—
117C22×C4×C43—22no—
118C22×C4×C43—41no—
119C22×C4×C43—41no—
120C22×C4×C43C2×D411yescharacteristic
121C22×C4×C43C2×D411yescharacteristic
122C22×C4×C43C2×D411yescharacteristic
123C22×C4×C43C2×D411yescharacteristic
124C22×C4×C43C2×D411yescharacteristic
125C22×C4×C43C2×Q812yes—
126C22×C4×C43#4112yes—
127C22×C4×C43C2411yescharacteristic
128C2×C22⋊D43—24no—
129C2×C22⋊D43—42no—
130C2×C22⋊D43—42no—
131C22×Dic43—21no—
132C22×Dic43—22no—
133C22×Dic43—22no—
134C22×Dic43—21no—
135C22×Dic43—21no—
136C2×D172—21no—
137C2×D172—21no—
138C2×D172—21no—
139C2×D172—21no—
140C2×D172—21no—
141C2×D172C2×Q812yes—
142C2×C4×D43—42no—
143C2×C4×D43—42no—
144C2×C4×D43—42no—
145C2×Dic86—24no—
146C2×Dic86—24no—
147C2×Dic86—42no—
148C2×Dic86—42no—
149C2×Dic86—42no—
150C2×Dic86—42no—
151C2×C8×C43—22no—
152C2×C8×C43—22no—
153C2×C8×C43—22no—
154C2×C8×C43—22no—
155C2×C8×C43—22no—
156C2×C8×C43—22no—
157C42×C43—22no—
158C42×C43—22no—
159C42×C43—21no—
160C42×C43—21no—
161C42×C43C2×D412yes—
162C42×C43C2×D412yes—
163C22⋊Dic43—28no—
164C22⋊Dic43—42no—
165(C2×D43)⋊C4—24no—
166(C2×D43)⋊C4—42no—
167(C2×D43)⋊C4—42no—
168(C2×D43)⋊C4—42no—
169C4⋊Dic43—24no—
170C4⋊Dic43—24no—
171C4⋊Dic43—22no—
172C4⋊Dic43—22no—
173C4⋊Dic43—21no—
174C4⋊Dic43—21no—
175C4⋊Dic43—21no—
176C4⋊Dic43—21no—
177C4⋊Dic43—21no—
178C4⋊Dic43C2×D414yes—
179C4⋊Dic43C2×D414yes—
180C4⋊Dic43C2×Q812yes—
181Dic43⋊C4—28no—
182Dic43⋊C4—28no—
183Dic43⋊C4—42no—
184C4×Dic43—28no—
185C4×Dic43—42no—
186C23×C43—21no—
187C23×C43—21no—
188C23×C43—21no—
189C23×C43—21no—
190C23×C43—21no—
191C23×C43—21no—
192C23×C43C22⋊Q812yes—
193C23×C43C22×D411yescharacteristic
194C22×D43—22no—
195C22×D43—41no—
196C22×D43—41no—
197C22×D43—41no—
198C22×D43—41no—
199C22×D43—41no—
200C22×D43—41no—
201C2×C4×C43—21no—
202C2×C4×C43—21no—
203C2×C4×C43—21no—
204C2×C4×C43—21no—
205C2×C4×C43—22no—
206C2×C4×C43—22no—
207C2×C4×C43—22no—
208C2×C4×C43—22no—
209C2×C4×C43—21no—
210C2×C4×C43—21no—
211C2×C4×C43—21no—
212C2×C4×C43—21no—
213C2×C4×C43—21no—
214C2×C4×C43—21no—
215C2×C4×C43—44no—
216C2×C4×C43—44no—
217C2×C4×C43#12012yes—
218C2×C4×C43#12012yes—
219C2×C4×C43C22⋊Q812yes—
220C2×C4×C43#13512yes—
221C2×C4×C43AGL(1,ℤ/8)12yes—
222C2×C4×C43C22×D411yescharacteristic
223C2×C4×C43C22×D411yescharacteristic
224C2×C4×C43C22×Q811yescharacteristic
225C2×C4×C43C2×#4111yescharacteristic
226C2×C4×C432−1+411yescharacteristic
227C2×C4×C432−1+411yescharacteristic
228C22⋊D43—48no—
229C2×Dic43—24no—
230C2×Dic43—24no—
231C2×Dic43—22no—
232C2×Dic43—42no—
233C2×Dic43—41no—
234C2×Dic43—42no—
235C2×Dic43—42no—
236C2×Dic43—42no—
237C2×Dic43—41no—
238C2×Dic43—41no—
239C2×Dic43—41no—
240C2×Dic43—41no—
241C2×Dic43—41no—
242C2×Dic43—41no—
243C2×Dic43—41no—
244D172—22no—
245D172—22no—
246D172—41no—
247D172—41no—
248C4×D43—84no—
249Dic86—48no—
250Dic86—48no—
251C8×C43—42no—
252C8×C43—42no—
253#24300—431nomaximal, Sylow
254C22×C43—22no—
255C22×C43—22no—
256C22×C43—22no—
257C22×C43—41no—
258C22×C43—41no—
259C22×C43—41no—
260C22×C43—41no—
261C22×C43C2×#12011yescharacteristic
262C22×C43C2×C22⋊Q811yescharacteristic
263C22×C43#53611yescharacteristic
264C22×C43C2×#13511yescharacteristic
265C22×C43C2×AGL(1,ℤ/8)11yescharacteristic
266C22×C43#57511yescharacteristic
267C22×C43#57511yescharacteristic
268C2×D43—42no—
269C2×D43—42no—
270C2×D43—42no—
271C2×D43—81no—
272C2×D43—81no—
273C2×D43—81no—
274C2×D43—81no—
275C4×C43—22no—
276C4×C43—22no—
277C4×C43—41no—
278C4×C43—41no—
279C4×C43#44914yes—
280C4×C43#54811yescharacteristic
281C4×C43#54811yescharacteristic
282C4×C43D4×Q811yescharacteristic
283C4×C43D4×Q811yescharacteristic
284Dic43—42no—
285Dic43—44no—
286Dic43—44no—
287Dic43—81no—
288Dic43—81no—
289C2×#533—431no—
290C2×#476—431no—
291C2×#474—431no—
292C4×#121—431no—
293#2059—432no—
294#2025—432no—
295#1988—431no—
296#1987—431no—
297#1875—432no—
298#1854—432no—
299#1797D4311yescharacteristic, complemented
300C2×C43—42no—
301C2×C43#240911yescharacteristic
302C2×C43C2×#44911yescharacteristic
303C2×C43#300511yescharacteristic
304C2×C43#307611yescharacteristic
305C2×C43#307711yescharacteristic
306C2×C43#323311yescharacteristic
307C2×C43#323511yescharacteristic
308D43—82no—
309#533—864no—
310#533—864no—
311C2×C4⋊Q8—431no—
312C2×C4⋊Q8—431no—
313C2×#125—432no—
314C2×C22⋊Q8—861no—
315C2×C22⋊Q8—861no—
316C2×#121—431no—
317C2×C4×Q8—861no—
318C2×C4×Q8—861no—
319C2×C4×D4—431no—
320C2×C4×D4—861no—
321C2×#117—432no—
322C22×C42C2×D4311yescharacteristic
323#476—864no—
324#474—864no—
325C2×#107—432no—
326C2×#106—432no—
327C2×C4⋊C8C2×D4311yescharacteristic
328C2×C4⋊C8C2×D4311yescharacteristic
329C2×#102—432no—
330C2×#102—432no—
331C2×C22⋊C8C2×D4312yes—
332#390—431no—
333#389—431no—
334#388—861no—
335#386—861no—
336#380—431no—
337C4×C4⋊C4—431no—
338C4×C22⋊C4—861no—
339#340—432no—
340#340—432no—
341#340—432no—
342#340—432no—
343#336C2×D4312yes—
344C43#2430011yescharacteristic, complemented, Sylow
345C22×Q8—861no—
346C22×Q8—861no—
347C22×D4—431no—
348C22×D4—861no—
349C23×C4—861no—
350C23×C4D17211yescharacteristic
351C23×C4D17211yescharacteristic
352C23×C4C22×D4311yescharacteristic
353C22×C8C22×D4312yes—
354C22×C8C22×D4312yes—
355C4⋊Q8—434no—
356C4⋊Q8—434no—
357C4⋊Q8—862no—
358C4⋊Q8—862no—
359#125—864no—
360#125—864no—
361C22⋊Q8—868no—
362C22⋊Q8—868no—
363#121—434no—
364#121—862no—
365C4×Q8—868no—
366C4×Q8—868no—
367C4×D4—862no—
368C4×D4—862no—
369C4×D4—1724no—
370#117—864no—
371#117—864no—
372C2×C4⋊C4—432no—
373C2×C4⋊C4—432no—
374C2×C4⋊C4—431no—
375C2×C4⋊C4—431no—
376C2×C4⋊C4—431no—
377C2×C4⋊C4—862no—
378C2×C4⋊C4—862no—
379C2×C4⋊C4—861no—
380C2×C4⋊C4—861no—
381C2×C4⋊C4—861no—
382C2×C22⋊C4—862no—
383C2×C22⋊C4—861no—
384C2×C22⋊C4—861no—
385C2×C22⋊C4—861no—
386C2×C22⋊C4—861no—
387C2×C22⋊C4—861no—
388C2×C42—22no—
389C2×C42—22no—
390C2×C42—862no—
391C2×C42—861no—
392C2×C42D17211yescharacteristic
393C2×C42D17211yescharacteristic
394C2×C42C22×D4311yescharacteristic
395C2×C42C22×D4311yescharacteristic
396#107—864no—
397#106—864no—
398C4⋊C8—22no—
399C4⋊C8—22no—
400#102—864no—
401#102—864no—
402C22⋊C8—24no—
403#95—861no—
404#95—861no—
405#95—861no—
406C24—861no—
407C24C2×D17211yescharacteristic
408C2×Q8—864no—
409C2×Q8—864no—
410C2×Q8—1722no—
411C2×Q8—1722no—
412C2×Q8—1722no—
413C2×Q8—1722no—
414C2×D4—432no—
415C2×D4—864no—
416C2×D4—861no—
417C2×D4—861no—
418C2×D4—861no—
419C2×D4—861no—
420C2×D4—861no—
421C2×D4—1722no—
422C2×D4—1722no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—41no—
432C22×C4—41no—
433C22×C4—861no—
434C22×C4—862no—
435C22×C4—862no—
436C22×C4—861no—
437C22×C4—861no—
438C22×C4—1722no—
439C22×C4—1722no—
440C22×C4—1722no—
441C22×C4C2×D17211yescharacteristic
442C22×C4C2×D17211yescharacteristic
443C22×C4C2×D17211yescharacteristic
444C22×C4C2×D17211yescharacteristic
445C22×C4C2×D17211yescharacteristic
446C22×C4#1062190012yes—
447C22×C4Q8×D4312yes—
448C22×C4C23×D4311yescharacteristic
449C2×C8—22no—
450C2×C8—22no—
451C2×C8—22no—
452C2×C8—22no—
453C2×C8—22no—
454C2×C8—22no—
455C4⋊C4—434no—
456C4⋊C4—434no—
457C4⋊C4—432no—
458C4⋊C4—868no—
459C4⋊C4—868no—
460C4⋊C4—864no—
461C4⋊C4—864no—
462C4⋊C4—862no—
463C4⋊C4—862no—
464C4⋊C4—861no—
465C4⋊C4—861no—
466C4⋊C4—861no—
467C4⋊C4—861no—
468C4⋊C4—861no—
469C4⋊C4—1722no—
470C22⋊C4—868no—
471C22⋊C4—864no—
472C22⋊C4—1722no—
473C22⋊C4—1722no—
474C22⋊C4—1722no—
475C22⋊C4—1722no—
476C42—22no—
477C42—22no—
478C42—21no—
479C42—21no—
480C42—868no—
481C42—1722no—
482C42D4×D4312yes—
483C42D4×D4312yes—
484C23—21no—
485C23—21no—
486C23—21no—
487C23—21no—
488C23—21no—
489C23—21no—
490C23—862no—
491C23—1721no—
492C23—1721no—
493C23—1721no—
494C23—1721no—
495C23—1721no—
496C23—1721no—
497C23#1306330512yes—
498C23C22×D17211yescharacteristic
499Q8—1728no—
500Q8—1728no—
501D4—862no—
502D4—862no—
503D4—1728no—
504D4—1721no—
505D4—1721no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—44no—
521C2×C4—44no—
522C2×C4—864no—
523C2×C4—864no—
524C2×C4—862no—
525C2×C4—3444no—
526C2×C4—1721no—
527C2×C4—1721no—
528C2×C4—1721no—
529C2×C4—1721no—
530C2×C4—1721no—
531C2×C4—1721no—
532C2×C4—1722no—
533C2×C4—1721no—
534C2×C4—1722no—
535C2×C4—1721no—
536C2×C4—1722no—
537C2×C4—1722no—
538C2×C4#1306329512yes—
539C2×C4#1306329512yes—
540C2×C4#1306330512yes—
541C2×C4#1306334712yes—
542C2×C4#1306335312yes—
543C2×C4C2×D4×D4311yescharacteristic
544C2×C4C2×D4×D4311yescharacteristic
545C2×C4C2×Q8×D4311yescharacteristic
546C2×C4C2×#1062190011yescharacteristic
547C2×C4#1306337211yescharacteristic
548C2×C4#1306337211yescharacteristic
549C8—42no—
550C8—42no—
551C22—22no—
552C22—22no—
553C22—22no—
554C22—41no—
555C22—41no—
556C22—41no—
557C22—41no—
558C22—3441no—
559C22—3441no—
560C22—3441no—
561C22—3441no—
562C22—1722no—
563C22—1722no—
564C22—1722no—
565C22C2×#1306329511yescharacteristic
566C22C2×#1306330511yescharacteristic
567C22#1458347511yescharacteristic
568C22C2×#1306334711yescharacteristic
569C22C2×#1306335311yescharacteristic
570C22#1458369911yescharacteristic
571C22#1458369911yescharacteristic
572C4—22no—
573C4—22no—
574C4—41no—
575C4—41no—
576C4—3441no—
577C4—3441no—
578C4—1724no—
579C4—1724no—
580C4—1722no—
581C4#1458304614yes—
582C4#1458355511yescharacteristic
583C4#1458355511yescharacteristic
584C4#1458356211yescharacteristic
585C4#1458356211yescharacteristic
586C2—42no—
587C2—3442no—
588C2#1913796111yescharacteristic
589C2C2×#1458304611yescharacteristic
590C2#1914384011yescharacteristic
591C2#1914438511yescharacteristic
592C2#1914439411yescharacteristic
593C2#1914577611yescharacteristic
594C2#1914579511yescharacteristic
subgroupG/H
centerC23C22×D172
commutator, cosocleC22×C4×C43C24
FrattiniC22×C4C23×D43
FittingC43×#1797C2
socleC23×C43C22×D4
prime radicalC2×#13063299C22
hypercenter#1797D43
hypocenterC43#24300
LCSG ▷ C22×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1797

Complements

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

NH ≅ G/Ntype
1. C2×#14583456C2semidirect
2. C2×#14583190C2semidirect
3. C2×#14583180C2semidirect
5. #19135934C2semidirect
6. #19135742C2semidirect
7. #19135546C2semidirect
8. #19135542C2semidirect
10. #19134796C2semidirect
11. C43×#1797C2semidirect
54. #13063325D4semidirect
55. #13063325D4semidirect
61. #13063299Q8semidirect
299. #1797D43semidirect
344. C43#24300semidirect

Maximal subgroups

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

subgroupindex
C2×#145834562
C2×#145831902
C2×#145831802
C4×#130632992
#191359342
#191357422
#191355462
#191355422
#191349152
#191347962
C43×#17972
#2430043