Finite Group Databasebeta

← #25105705

Subgroups — #25105705

Order 11008.

1326 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14583470C211yesmaximal, characteristic
2#19137187C211yesmaximal, characteristic
3#19135028C212yesmaximal, complemented
4#19135018C212yesmaximal, complemented
5#19134905C212yesmaximal, complemented
6#19134900C212yesmaximal, complemented
7#19134691C212yesmaximal, complemented
8#19134531C212yesmaximal, complemented
9C2×C43×#408C211yesmaximal, characteristic
10C23×Dic86C2211yescharacteristic
11#14583470—24no—
12#14583470C418yescomplemented
13C2×#13063323—21no—
14C2×#13063312—21no—
15C2×#13063306—21no—
16C2×#13063306—21no—
17C2×#13063304C2212yes—
18C2×#13063304C2212yes—
19C2×#13063284C2211yescharacteristic
20C2×#13063284C2211yescharacteristic
21C22×C4⋊Dic43C2211yescharacteristic
22C2×#13063250C2212yes—
23C2×#13063250C2212yes—
24C2×#13063250C2212yes—
25C2×#13063250C2212yes—
26#14582806—21no—
27#14582802—21no—
28#14582779—21no—
29#14582779—21no—
30#14582759C2212yes—
31#14582759C2212yes—
32#14582662C2212yes—
33#14582662C2212yes—
34#14582662C2212yes—
35#14582662C2212yes—
36C22×#41×C43C2211yescharacteristic
37C22×M16×C43C2211yescharacteristic
38C23×C8×C43C2211yescharacteristic
39C43×#408—24no—
40C2×C22⋊C8×C43C2212yes—
41C2×C22⋊C8×C43C2212yes—
42C22×Dic86—21no—
43C22×Dic86—22no—
44C22×Dic86—41no—
45C22×Dic86—41no—
46C22×Dic86—41no—
47C22×Dic86—41no—
48C22×Dic86C2×C414yescomplemented
49C22×Dic86C2312yes—
50C23×Dic43—21no—
51C23×Dic43—21no—
52#13063323—28no—
53#13063312—28no—
54#13063306—28no—
55#13063306—28no—
56#13063304—24no—
57#13063304—24no—
58#13063304C2×C418yes—
59#13063304C2×C418yescomplemented
60#13063284—22no—
61#13063284—41no—
62#13063284—41no—
63#13063284C2×C414yescomplemented
64C2×C4⋊Dic43—21no—
65C2×C4⋊Dic43—41no—
66C2×C4⋊Dic43—41no—
67C2×C4⋊Dic43C2312yes—
68C2×C4⋊Dic43C2312yes—
69C2×C4⋊Dic43C2312yes—
70C2×Dic43⋊C4—22no—
71C2×Dic43⋊C4—22no—
72C2×C22⋊Dic43—21no—
73C2×C22⋊Dic43—22no—
74C2×C22⋊Dic43—22no—
75C2×C22⋊Dic43—21no—
76C2×C22⋊Dic43—22no—
77C2×C22⋊Dic43—22no—
78C2×C4×Dic43—21no—
79C2×C4×Dic43—21no—
80#13063250—42no—
81#13063250—42no—
82#13063250—42no—
83#13063250—42no—
84C43⋊#95—22no—
85C43⋊#95—22no—
86C2×#41×C43—41no—
87C2×#41×C43—41no—
88C2×#41×C43—41no—
89C2×#41×C43—41no—
90C2×#41×C43C2×C414yes—
91C2×#41×C43D414yes—
92C22×Q8×C43D411yescharacteristic
93C22×D4×C43D411yescharacteristic
94C22×D4×C43D411yescharacteristic
95C22×D4×C43D411yescharacteristic
96C23×C4×C43C2311yescharacteristic
97C23×C4×C43C2311yescharacteristic
98C23×C4×C43C2311yescharacteristic
99C2×M16×C43—22no—
100C2×M16×C43—41no—
101C2×M16×C43—41no—
102C22×C8×C43—22no—
103C22×C8×C43—41no—
104C22×C8×C43—41no—
105C22×C8×C43C2312yes—
106C22×C8×C43C2312yes—
107C22⋊C8×C43—24no—
108C22⋊C8×C43—24no—
109C24×C43C2×D411yescharacteristic
110C24×C43C2×D411yescharacteristic
111C24×C43C2×D411yescharacteristic
112#41×C43—44no—
113#41×C43—44no—
114#41×C43—44no—
115#41×C43—44no—
116C2×Q8×C43—41no—
117C2×Q8×C43—41no—
118C2×Q8×C43C22⋊C414yes—
119C2×D4×C43—41no—
120C2×D4×C43—41no—
121C2×D4×C43—41no—
122C2×D4×C43—41no—
123C2×D4×C43—41no—
124C2×D4×C43—41no—
125C2×D4×C43C22⋊C414yes—
126C2×D4×C43C22⋊C414yes—
127C2×D4×C43C22⋊C414yes—
128C22×C4×C43—21no—
129C22×C4×C43—21no—
130C22×C4×C43—21no—
131C22×C4×C43—21no—
132C22×C4×C43—21no—
133C22×C4×C43—21no—
134C22×C4×C43—21no—
135C22×C4×C43—21no—
136C22×C4×C43—21no—
137C22×C4×C43—21no—
138C22×C4×C43—21no—
139C22×C4×C43—21no—
140C22×C4×C43C22×C412yes—
141C22×C4×C43C22×C412yes—
142C22×C4×C43C22×C412yes—
143C22×C4×C43C2×D412yes—
144C22×C4×C43C2×D412yes—
145C22×C4×C43C2×D412yes—
146C22×C4×C43C2×D411yescharacteristic
147C22×C4×C43C2×D411yescharacteristic
148C22×C4×C43C2×D411yescharacteristic
149C22×C4×C43C2411yescharacteristic
150C22×Dic43—24no—
151C22×Dic43—22no—
152C22×Dic43—22no—
153C22×Dic43—22no—
154C22×Dic43—22no—
155C22×Dic43—42no—
156C22×Dic43—42no—
157C22×Dic43—42no—
158C22×Dic43—82no—
159C2×Dic86—22no—
160C2×Dic86—22no—
161C2×Dic86—22no—
162C2×Dic86—22no—
163C2×Dic86—24no—
164C2×Dic86—22no—
165C2×Dic86—44no—
166C2×Dic86—44no—
167C2×Dic86—42no—
168C2×Dic86—42no—
169C2×Dic86—44no—
170C2×Dic86—44no—
171C2×Dic86C22×C414yes—
172M16×C43—44no—
173C2×C8×C43—22no—
174C2×C8×C43—22no—
175C2×C8×C43—22no—
176C2×C8×C43—22no—
177C2×C8×C43—22no—
178C2×C8×C43—22no—
179C2×C8×C43—44no—
180C22⋊Dic43—24no—
181C22⋊Dic43—28no—
182C22⋊Dic43—28no—
183C22⋊Dic43—42no—
184C22⋊Dic43—82no—
185C22⋊Dic43—82no—
186C4⋊Dic43—22no—
187C4⋊Dic43—22no—
188C4⋊Dic43—22no—
189C4⋊Dic43—22no—
190C4⋊Dic43—22no—
191C4⋊Dic43—42no—
192C4⋊Dic43C22×C414yes—
193Dic43⋊C4—28no—
194Dic43⋊C4—28no—
195C4×Dic43—24no—
196C4×Dic43—42no—
197C23×C43—21no—
198C23×C43—21no—
199C23×C43—21no—
200C23×C43—21no—
201C23×C43—21no—
202C23×C43—21no—
203C23×C43—22no—
204C23×C43—22no—
205C23×C43—22no—
206C23×C43—22no—
207C23×C43—22no—
208C23×C43—22no—
209C23×C43C2×C22⋊C412yes—
210C23×C43C2×C22⋊C412yes—
211C23×C43C2×C22⋊C412yes—
212C23×C43C22×D411yescharacteristic
213Q8×C43—44no—
214D4×C43—44no—
215D4×C43—44no—
216D4×C43—44no—
217C2×C4×C43—21no—
218C2×C4×C43—21no—
219C2×C4×C43—21no—
220C2×C4×C43—21no—
221C2×C4×C43—21no—
222C2×C4×C43—21no—
223C2×C4×C43—22no—
224C2×C4×C43—22no—
225C2×C4×C43—22no—
226C2×C4×C43—22no—
227C2×C4×C43—22no—
228C2×C4×C43—22no—
229C2×C4×C43—22no—
230C2×C4×C43—22no—
231C2×C4×C43—22no—
232C2×C4×C43—22no—
233C2×C4×C43—22no—
234C2×C4×C43—22no—
235C2×C4×C43C2×C22⋊C412yes—
236C2×C4×C43C2×C22⋊C412yes—
237C2×C4×C43C2×C22⋊C412yes—
238C2×C4×C43C23×C411yescharacteristic
239C2×C4×C43C22×D411yescharacteristic
240C2×C4×C43H(5,𝔽2)11yescharacteristic
241C2×C4×C43H(5,𝔽2)11yescharacteristic
242C2×C4×C43H(5,𝔽2)11yescharacteristic
243C2×C4×C43H(5,𝔽2)11yescharacteristic
244C2×Dic43—24no—
245C2×Dic43—24no—
246C2×Dic43—48no—
247C2×Dic43—42no—
248C2×Dic43—42no—
249C2×Dic43—42no—
250C2×Dic43—42no—
251C2×Dic43—42no—
252C2×Dic43—42no—
253C2×Dic43—42no—
254C2×Dic43—42no—
255C2×Dic43—42no—
256C2×Dic43—42no—
257C2×Dic43—84no—
258Dic86—24no—
259Dic86—24no—
260Dic86—48no—
261Dic86—42no—
262Dic86—42no—
263C8×C43—42no—
264C8×C43—42no—
265#21771—431nomaximal, Sylow
266C22×C43—22no—
267C22×C43—22no—
268C22×C43—22no—
269C22×C43—24no—
270C22×C43—24no—
271C22×C43—24no—
272C22×C43—41no—
273C22×C43—41no—
274C22×C43—41no—
275C22×C43—42no—
276C22×C43—42no—
277C22×C43—42no—
278C22×C43—41no—
279C22×C43—41no—
280C22×C43—41no—
281C22×C43C22×C22⋊C411yescharacteristic
282C22×C43#53411yescharacteristic
283C22×C43#53411yescharacteristic
284C22×C43#57712yes—
285C22×C43#57812yes—
286C4×C43—22no—
287C4×C43—22no—
288C4×C43—22no—
289C4×C43#51811yescharacteristic
290C4×C43#51811yescharacteristic
291C4×C43#53411yescharacteristic
292C4×C43#53411yescharacteristic
293Dic43—44no—
294Dic43—44no—
295Dic43—82no—
296Dic43—82no—
297C2×#536—431no—
298C2×#408D4311yescharacteristic
299#2311—431no—
300#1894—432no—
301#1892—432no—
302#1873—432no—
303#1872—432no—
304#1825—432no—
305#1787—432no—
306C2×C43—42no—
307C2×C43—42no—
308C2×C43—42no—
309C2×C43#230811yescharacteristic
310C2×C43#289812yes—
311C2×C43#320011yescharacteristic
312C2×C43#320112yes—
313C2×C43#320211yescharacteristic
314C22×#41C2×D4311yescharacteristic
315C23×Q8—431no—
316C22×M16C2×D4311yescharacteristic
317C23×C8C2×D4311yescharacteristic
318#536—438no—
319#536—864no—
320C2×C4⋊Q8—861no—
321C2×#124—861no—
322C2×#123—861no—
323C2×#123—861no—
324C2×C22⋊Q8—432no—
325C2×C22⋊Q8—432no—
326C2×#117—431no—
327C2×#117—431no—
328C22×C4⋊C4—431no—
329C2×#103—432no—
330C2×#103—432no—
331C2×#103—432no—
332C2×#103—432no—
333#408—24no—
334C2×C22⋊C8C2×D4312yes—
335C2×C22⋊C8C2×D4312yes—
336#391—861no—
337#390—861no—
338#386—861no—
339#386—861no—
340#380—432no—
341#380—432no—
342#340—432no—
343#340—432no—
344#340—432no—
345#340—432no—
346C43#2177111yescharacteristic, complemented, Sylow
347C2×#41—41no—
348C2×#41—41no—
349C2×#41—41no—
350C2×#41—41no—
351C2×#41C4×D4314yes—
352C2×#41C22⋊D4314yes—
353C22×Q8—434no—
354C22×Q8—432no—
355C22×Q8—861no—
356C22×Q8—862no—
357C22×Q8—1721no—
358C22×Q8—1721no—
359C22×Q8—1721no—
360C22×Q8—1721no—
361C22×Q8D17211yescharacteristic
362C22×D4D17211yescharacteristic
363C22×D4D17211yescharacteristic
364C22×D4D17211yescharacteristic
365C23×C4—861no—
366C23×C4—861no—
367C23×C4C22×D4311yescharacteristic
368C23×C4C22×D4311yescharacteristic
369C23×C4C22×D4311yescharacteristic
370C2×M16—22no—
371C2×M16—41no—
372C2×M16—41no—
373C22×C8—22no—
374C22×C8—41no—
375C22×C8—41no—
376C22×C8C22×D4312yes—
377C22×C8C22×D4312yes—
378C4⋊Q8—868no—
379#124—868no—
380#123—868no—
381#123—868no—
382C22⋊Q8—438no—
383C22⋊Q8—438no—
384C22⋊Q8—864no—
385C22⋊Q8—864no—
386#117—434no—
387#117—862no—
388#117—1721no—
389#117—1721no—
390C2×C4⋊C4—432no—
391C2×C4⋊C4—432no—
392C2×C4⋊C4—432no—
393C2×C4⋊C4—862no—
394C2×C4⋊C4—862no—
395C2×C4⋊C4—861no—
396C2×C4⋊C4—1721no—
397C2×C4⋊C4—1721no—
398C2×C22⋊C4—861no—
399C2×C22⋊C4—862no—
400C2×C22⋊C4—862no—
401C2×C22⋊C4—862no—
402C2×C22⋊C4—862no—
403C2×C22⋊C4—861no—
404C2×C42—861no—
405C2×C42—861no—
406#103—1722no—
407#103—1722no—
408#103—1722no—
409#103—1722no—
410C22⋊C8—24no—
411C22⋊C8—24no—
412#95—862no—
413#95—862no—
414C24C2×D17211yescharacteristic
415C24C2×D17211yescharacteristic
416C24C2×D17211yescharacteristic
417#41—44no—
418#41—44no—
419#41—44no—
420#41—44no—
421C2×Q8—41no—
422C2×Q8—41no—
423C2×Q8—434no—
424C2×Q8—864no—
425C2×Q8—862no—
426C2×Q8—862no—
427C2×Q8—862no—
428C2×Q8—862no—
429C2×Q8—862no—
430C2×Q8—1724no—
431C2×Q8—1724no—
432C2×Q8—1722no—
433C2×Q8—1722no—
434C2×Q8—1724no—
435C2×Q8—1724no—
436C2×Q8(C2×D43)⋊C414yes—
437C2×D4—41no—
438C2×D4—41no—
439C2×D4—41no—
440C2×D4—41no—
441C2×D4—41no—
442C2×D4—41no—
443C2×D4(C2×D43)⋊C414yes—
444C2×D4(C2×D43)⋊C414yes—
445C2×D4(C2×D43)⋊C414yes—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—864no—
459C22×C4—862no—
460C22×C4—862no—
461C22×C4—862no—
462C22×C4—862no—
463C22×C4—3442no—
464C22×C4—1722no—
465C22×C4—1722no—
466C22×C4—1722no—
467C22×C4C2×C4×D4312yes—
468C22×C4C2×C4×D4312yes—
469C22×C4C2×C4×D4312yes—
470C22×C4C2×D17211yescharacteristic
471C22×C4C2×D17211yescharacteristic
472C22×C4C2×D17211yescharacteristic
473C22×C4C2×C22⋊D4312yes—
474C22×C4C2×C22⋊D4312yes—
475C22×C4C2×C22⋊D4312yes—
476C22×C4C23×D4311yescharacteristic
477M16—44no—
478C2×C8—22no—
479C2×C8—22no—
480C2×C8—22no—
481C2×C8—22no—
482C2×C8—22no—
483C2×C8—22no—
484C2×C8—44no—
485C4⋊C4—434no—
486C4⋊C4—868no—
487C4⋊C4—868no—
488C4⋊C4—862no—
489C4⋊C4—862no—
490C4⋊C4—862no—
491C4⋊C4—862no—
492C4⋊C4—862no—
493C4⋊C4—1722no—
494C22⋊C4—864no—
495C22⋊C4—868no—
496C22⋊C4—868no—
497C22⋊C4—3442no—
498C22⋊C4—3442no—
499C22⋊C4—1722no—
500C42—864no—
501C42—1722no—
502C23—22no—
503C23—22no—
504C23—22no—
505C23—22no—
506C23—22no—
507C23—22no—
508C23—21no—
509C23—21no—
510C23—21no—
511C23—21no—
512C23—21no—
513C23—21no—
514C23C2×(C2×D43)⋊C412yes—
515C23C2×(C2×D43)⋊C412yes—
516C23C2×(C2×D43)⋊C412yes—
517C23C22×D17211yescharacteristic
518Q8—44no—
519Q8—864no—
520Q8—864no—
521Q8—1728no—
522Q8—1722no—
523Q8—1722no—
524D4—44no—
525D4—44no—
526D4—44no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—864no—
546C2×C4—864no—
547C2×C4—3444no—
548C2×C4—1722no—
549C2×C4—1722no—
550C2×C4—1722no—
551C2×C4—1722no—
552C2×C4—1722no—
553C2×C4—1722no—
554C2×C4—1728no—
555C2×C4—1722no—
556C2×C4—1722no—
557C2×C4—1722no—
558C2×C4—1722no—
559C2×C4C2×(C2×D43)⋊C412yes—
560C2×C4C2×(C2×D43)⋊C412yes—
561C2×C4C2×(C2×D43)⋊C412yes—
562C2×C4C22×C4×D4311yescharacteristic
563C2×C4C22×C22⋊D4311yescharacteristic
564C2×C4#1306337011yescharacteristic
565C2×C4#1306337011yescharacteristic
566C2×C4#1306337011yescharacteristic
567C2×C4#1306337011yescharacteristic
568C8—42no—
569C8—42no—
570C22—22no—
571C22—22no—
572C22—22no—
573C22—24no—
574C22—24no—
575C22—24no—
576C22—42no—
577C22—42no—
578C22—42no—
579C22—41no—
580C22—41no—
581C22—41no—
582C22—41no—
583C22—41no—
584C22—41no—
585C22C22×(C2×D43)⋊C411yescharacteristic
586C22#1458346111yescharacteristic
587C22#1458346111yescharacteristic
588C22#1458371112yes—
589C22#1458371612yes—
590C4—22no—
591C4—22no—
592C4—22no—
593C4—3442no—
594C4—3442no—
595C4—1724no—
596C4—1724no—
597C4#1458337211yescharacteristic
598C4#1458337211yescharacteristic
599C4#1458345811yescharacteristic
600C4#1458345811yescharacteristic
601C2—42no—
602C2—42no—
603C2—42no—
604C2#1913717011yescharacteristic
605C2#1914312912yes—
606C2#1914546811yescharacteristic
607C2#1914547912yes—
608C2#1914549011yescharacteristic
subgroupG/H
centerC23C22×D172
commutatorC2×C4×C43C23×C4
FrattiniC22×C4C23×D43
FittingC2×C43×#408C2
socleC23×C43C22×D4
prime radicalC22×#41×C43C22
cosocleC22×C4×C43C24
hypercenterC2×#408D43
hypocenterC43#21771
LCSG ▷ C2×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C23 ◁ C22×#41 ◁ C2×#408

Complements

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

NH ≅ G/Ntype
3. #19135028C2semidirect
4. #19135018C2semidirect
5. #19134905C2semidirect
6. #19134900C2semidirect
7. #19134691C2semidirect
8. #19134531C2semidirect
12. #14583470C4semidirect
48. C22×Dic86C2×C4semidirect
59. #13063304C2×C4semidirect
63. #13063284C2×C4semidirect
346. C43#21771semidirect

Maximal subgroups

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

subgroupindex
C2×#145834702
#191371872
#191350282
#191350182
#191349052
#191349002
#191346912
#191345312
C2×C43×#4082
#2177143