Finite Group Databasebeta

← #25105149

Subgroups — #25105149

Order 11008.

1094 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14583290C211yesmaximal, characteristic
2C2×#14582993C211yesmaximal, characteristic, complemented
3C2×#14582991C211yesmaximal, characteristic, complemented
4C2×#14582990C211yesmaximal, characteristic
5C2×SD16×Dic43C211yesmaximal, characteristic
6#19137164C211yesmaximal, characteristic, complemented
7#19137142C211yesmaximal, characteristic
8#19135475C211yesmaximal, characteristic
9#19135466C211yesmaximal, characteristic, complemented
10#19135465C211yesmaximal, characteristic, complemented
11#19135413C211yesmaximal, characteristic, complemented
12#19135410C211yesmaximal, characteristic
13#19134762C211yesmaximal, characteristic, complemented
14#97×Dic43C211yesmaximal, characteristic, complemented
15C43×#1973C211yesmaximal, characteristic
16C22×Dic172C2211yescharacteristic
17C22×#10621885C2211yescharacteristic
18C22×#10621883C2211yescharacteristic
19C2×#13063323C2211yescharacteristic
20C2×#13063312C2211yescharacteristic
21C2×Q8×Dic43C2211yescharacteristic
22C2×#13063290C2211yescharacteristic
23C2×#13063290C2211yescharacteristic
24C2×D4×Dic43C2211yescharacteristic
25#14583290C418yescomplemented
26#14582993—24no—
27#14582991—24no—
28#14582990—24no—
29SD16×Dic43—24no—
30C2×#13063258C2211yescharacteristic
31C2×#13063257C2211yescharacteristic
32C2×#13063256C2211yescharacteristic
33C2×#13063250C2211yescharacteristic
34C2×#13063250C2211yescharacteristic
35C2×#13063249C2211yescharacteristic
36C2×#13063248C2211yescharacteristic
37C2×#13063248C2211yescharacteristic
38C2×#13063246C2211yescharacteristic
39C2×#13063236C2211yescharacteristic
40C2×#13063235C2211yescharacteristic
41C2×#13063235C2211yescharacteristic
42C2×C8×Dic43C2211yescharacteristic
43C2×C8×Dic43C2211yescharacteristic
44#14582805C2211yescharacteristic
45#14582805C2211yescharacteristic
46#14582796—22no—
47#14582787—22no—
48#14582768C2211yescharacteristic
49#14582764C2211yescharacteristic
50#14582750—22no—
51C22⋊C4×Dic43—22no—
52C42×Dic43C2211yescharacteristic
53C22×SD16×C43C2211yescharacteristic
54C2×#107×C43C2211yescharacteristic
55C2×#103×C43C2211yescharacteristic
56C2×#102×C43C2211yescharacteristic
57C2×#97×C43C2211yescharacteristic
58C43×#391C2211yescharacteristic
59C43×#390C2211yescharacteristic
60C22×Dic86C2311yescharacteristic
61C22×Dic86C2311yescharacteristic
62C23×Dic43—21no—
63C2×Dic172—22no—
64C2×Dic172—22no—
65C2×Dic172C2×C414yescomplemented
66C2×#10621885—22no—
67C2×#10621885—22no—
68C2×#10621885C2×C414yescomplemented
69C2×#10621883—22no—
70C2×#10621883—22no—
71C2×#10621883C2×C414yes—
72C22×C43⋊C8C2311yescharacteristic
73C22×C43⋊C8C2311yescharacteristic
74#13063323—22no—
75#13063323C2×C414yescomplemented
76#13063312—22no—
77#13063312C2×C414yes—
78Q8×Dic43—22no—
79Q8×Dic43—22no—
80#13063290—22no—
81#13063290—22no—
82#13063290—22no—
83#13063290—22no—
84D4×Dic43—22no—
85D4×Dic43—22no—
86C2×C4⋊Dic43—21no—
87C2×C4⋊Dic43C2311yescharacteristic
88C2×C4⋊Dic43C2311yescharacteristic
89C2×Dic43⋊C4—22no—
90C2×Dic43⋊C4—21no—
91C2×Dic43⋊C4—21no—
92C2×Dic43⋊C4—22no—
93C2×C22⋊Dic43—22no—
94C2×C22⋊Dic43—21no—
95C2×C4×Dic43—21no—
96C2×C4×Dic43—21no—
97C2×C4×Dic43—21no—
98C2×C4×Dic43D412yes—
99C2×C4×Dic43D412yes—
100C2×C4×Dic43C2311yescharacteristic
101C2×C4×Dic43C2311yescharacteristic
102#13063258—22no—
103#13063257—22no—
104#13063256—22no—
105#13063250—22no—
106#13063250—22no—
107#13063249—22no—
108#13063248—22no—
109#13063248—22no—
110#13063246—22no—
111#13063236—22no—
112#13063235—22no—
113#13063235C2×C414yescomplemented
114C8×Dic43D414yes—
115C8×Dic43D414yes—
116C43⋊#95—22no—
117C43⋊#95—22no—
118C43⋊#95—22no—
119C22×Q8×C43C2311yescharacteristic
120C22×D4×C43C2311yescharacteristic
121C2×SD16×C43—22no—
122C2×SD16×C43—22no—
123C2×SD16×C43C2×C414yes—
124C22×C8×C43C2311yescharacteristic
125C22×C8×C43C2311yescharacteristic
126C2×C4⋊C4×C43C2311yescharacteristic
127C2×C4⋊C4×C43C2311yescharacteristic
128C2×C22⋊C4×C43—22no—
129C2×C42×C43C2311yescharacteristic
130#107×C43—22no—
131#103×C43—22no—
132#102×C43—22no—
133#97×C43—22no—
134#95×C43—22no—
135C24×C43—21no—
136C2×Q8×C43—21no—
137C2×Q8×C43—21no—
138C2×Q8×C43—21no—
139C2×Q8×C43—21no—
140C2×Q8×C43—21no—
141C2×Q8×C43C22×C412yes—
142C2×D4×C43—21no—
143C2×D4×C43—21no—
144C2×D4×C43—21no—
145C2×D4×C43—21no—
146C2×D4×C43—21no—
147C2×D4×C43C22×C412yes—
148C22×C4×C43—21no—
149C22×C4×C43—21no—
150C22×C4×C43—21no—
151C22×C4×C43C2×D412yes—
152C22×C4×C43C2411yescharacteristic
153C22×Dic43—21no—
154C22×Dic43—21no—
155C22×Dic43—21no—
156C22×Dic43—21no—
157C22×Dic43—42no—
158C22×Dic43—44no—
159C22×Dic43C2×D412yes—
160C22×Dic43C2×D411yescharacteristic
161C22×Dic43C2×D411yescharacteristic
162C2×Dic86—21no—
163C2×Dic86—21no—
164C2×Dic86—21no—
165C2×Dic86—21no—
166C2×Dic86—21no—
167C2×Dic86—21no—
168C2×Dic86—21no—
169C2×Dic86—21no—
170C2×Dic86—21no—
171C2×Dic86—21no—
172C2×Dic86C22×C412yes—
173C2×Dic86C22×C412yes—
174SD16×C43—28no—
175C2×C8×C43—21no—
176C2×C8×C43—21no—
177C2×C8×C43C22×C412yes—
178C2×C8×C43C2×D412yes—
179C2×C8×C43C2×D412yes—
180C2×C8×C43#4112yes—
181C4⋊C4×C43—21no—
182C4⋊C4×C43—21no—
183C4⋊C4×C43—21no—
184C4⋊C4×C43—21no—
185C22⋊C4×C43—44no—
186C42×C43—21no—
187C42×C43—21no—
188C22⋊Dic43—28no—
189C22⋊Dic43—42no—
190#10621885—28no—
191#10621883—28no—
192C4⋊Dic43—21no—
193C4⋊Dic43—21no—
194C4⋊Dic43—21no—
195C4⋊Dic43—21no—
196C4⋊Dic43—42no—
197Dic43⋊C4—28no—
198Dic43⋊C4—44no—
199Dic43⋊C4—42no—
200Dic43⋊C4—42no—
201C4×Dic43—24no—
202C4×Dic43—42no—
203C4×Dic43—42no—
204C4×Dic43—42no—
205C4×Dic43—42no—
206C4×Dic43C22×C412yes—
207C4×Dic43C2×D412yes—
208C4×Dic43C2×D412yes—
209C4×Dic43#4112yes—
210C2×C43⋊C8—21no—
211C2×C43⋊C8—21no—
212C2×C43⋊C8C22×C412yes—
213C2×C43⋊C8C2×D412yes—
214C2×C43⋊C8C2×D412yes—
215C2×C43⋊C8#4112yes—
216Dic172—28no—
217C23×C43—22no—
218C23×C43—41no—
219C23×C43—41no—
220C23×C43—42no—
221C23×C43—42no—
222C23×C43C22×D411yescharacteristic
223Q8×C43—22no—
224Q8×C43—22no—
225Q8×C43—41no—
226Q8×C43—41no—
227D4×C43—22no—
228D4×C43—22no—
229D4×C43—41no—
230D4×C43—41no—
231C2×C4×C43—22no—
232C2×C4×C43—22no—
233C2×C4×C43—22no—
234C2×C4×C43—22no—
235C2×C4×C43—41no—
236C2×C4×C43—41no—
237C2×C4×C43—41no—
238C2×C4×C43—41no—
239C2×C4×C43—42no—
240C2×C4×C43—42no—
241C2×C4×C43C23×C411yescharacteristic
242C2×C4×C43C22×D411yescharacteristic
243C2×C4×C43C22×D411yescharacteristic
244C2×C4×C43C2×#4111yescharacteristic
245C2×C4×C43C2×#4111yescharacteristic
246C2×C4×C43C2×#4111yescharacteristic
247C2×Dic43—22no—
248C2×Dic43—22no—
249C2×Dic43—22no—
250C2×Dic43—22no—
251C2×Dic43—22no—
252C2×Dic43—41no—
253C2×Dic43—41no—
254C2×Dic43—41no—
255C2×Dic43—41no—
256C2×Dic43—41no—
257C2×Dic43—41no—
258C2×Dic43—42no—
259C2×Dic43—42no—
260C2×Dic43—84no—
261C2×Dic43C4×D414yes—
262C2×Dic43#12712yes—
263C2×Dic43#12712yes—
264C2×Dic43#13714yes—
265Dic86—22no—
266Dic86—22no—
267Dic86—22no—
268Dic86—22no—
269Dic86—41no—
270Dic86—41no—
271Dic86—41no—
272Dic86—41no—
273C8×C43—22no—
274C8×C43C4×D414yes—
275C43⋊C8—22no—
276C43⋊C8C4×D414yes—
277C4×#497—431nomaximal, Sylow
278C22×C43—42no—
279C22×C43—44no—
280C22×C43—81no—
281C22×C43—81no—
282C22×C43—82no—
283C22×C43C2×C4×D411yescharacteristic
284C22×C43C2×#12711yescharacteristic
285C22×C43#53211yescharacteristic
286C22×C43C2×#13712yes—
287C22×C43#57512yes—
288C4×C43—42no—
289C4×C43—42no—
290C4×C43—81no—
291C4×C43—81no—
292C4×C43C2×C4×D411yescharacteristic
293C4×C43C2×C4×D411yescharacteristic
294C4×C43#53211yescharacteristic
295C4×C43#53211yescharacteristic
296Dic43—22no—
297Dic43—24no—
298Dic43—42no—
299Dic43—42no—
300Dic43—81no—
301Dic43—81no—
302Dic43#49714yescomplemented
303C2×#497—431no—
304C2×C4×Q16—431no—
305C2×C4×Q16—431no—
306C2×C4×SD16—431no—
307C2×C4×SD16—431no—
308C4×C4⋊Q8—431no—
309C4×#124—431no—
310#1973—431no—
311#1973D4311yescharacteristic
312#1971—431no—
313#1971—431no—
314#1960—431no—
315#1959—431no—
316#1847—431no—
317C4×#97—431no—
318C2×C43—82no—
319C2×C43C4×#12711yescharacteristic
320C2×C43C4×#13712yes—
321C2×C43C2×#49711yescharacteristic
322C2×C43#315011yescharacteristic
323C2×C43#317212yes—
324C22×Q16—431no—
325C22×Q16—431no—
326C22×SD16—431no—
327C22×SD16C2×D4311yescharacteristic
328C2×C4⋊Q8—431no—
329C2×#124—431no—
330C2×C4×Q8—431no—
331C2×C4×Q8—431no—
332C2×C4×Q8—431no—
333C2×C4×D4—431no—
334#497—438no—
335C4×Q16—864no—
336C4×Q16—864no—
337C4×SD16—864no—
338C4×SD16—864no—
339C2×#107—431no—
340C2×#107C2×D4311yescharacteristic
341C2×#106—431no—
342C2×#106—431no—
343C2×#103—431no—
344C2×#103—431no—
345C2×#103—431no—
346C2×#103—431no—
347C2×#103—431no—
348C2×#103C2×D4311yescharacteristic
349C2×#102—431no—
350C2×#102C2×D4311yescharacteristic
351C2×#97—431no—
352C2×#97—431no—
353C2×#97—431no—
354C2×#97C2×D4311yescharacteristic
355C2×C4×C8—431no—
356C2×C4×C8—431no—
357#391—431no—
358#391—431no—
359#391C2×D4311yescharacteristic
360#390C2×D4311yescharacteristic
361#389—862no—
362#388—862no—
363#384—431no—
364#382—431no—
365C4×C4⋊C4—862no—
366C4×C22⋊C4—862no—
367C43—431no—
368C43C4×#49711yescharacteristic, complemented, Sylow
369C22×Q8—431no—
370C22×Q8—431no—
371C22×Q8C22×D4311yescharacteristic
372C22×D4C22×D4311yescharacteristic
373C23×C4—861no—
374C2×Q16—434no—
375C2×Q16—434no—
376C2×Q16—862no—
377C2×Q16—862no—
378C2×Q16—862no—
379C2×Q16—862no—
380C2×SD16—22no—
381C2×SD16—22no—
382C2×SD16—434no—
383C2×SD16—862no—
384C2×SD16—862no—
385C2×SD16C4×D4314yes—
386C22×C8—431no—
387C22×C8—431no—
388C22×C8C22×D4311yescharacteristic
389C22×C8C22×D4311yescharacteristic
390C4⋊Q8—434no—
391C4⋊Q8—862no—
392#124—434no—
393#124—862no—
394C4×Q8—862no—
395C4×Q8—862no—
396C4×Q8—862no—
397C4×Q8—862no—
398C4×Q8—862no—
399C4×Q8—862no—
400C4×D4—862no—
401C4×D4—862no—
402C2×C4⋊C4—431no—
403C2×C4⋊C4—431no—
404C2×C4⋊C4—862no—
405C2×C4⋊C4—861no—
406C2×C4⋊C4—861no—
407C2×C4⋊C4—861no—
408C2×C4⋊C4—862no—
409C2×C4⋊C4C22×D4311yescharacteristic
410C2×C4⋊C4C22×D4311yescharacteristic
411C2×C22⋊C4—22no—
412C2×C22⋊C4—862no—
413C2×C22⋊C4—861no—
414C2×C42—431no—
415C2×C42—431no—
416C2×C42—432no—
417C2×C42—432no—
418C2×C42—861no—
419C2×C42—861no—
420C2×C42—861no—
421C2×C42C22×D4311yescharacteristic
422#107—22no—
423#107—862no—
424#106—862no—
425#106—862no—
426#103—22no—
427#103—862no—
428#103—862no—
429#103—862no—
430#103—862no—
431#103—862no—
432#102—22no—
433#102—862no—
434#97—22no—
435#97—434no—
436#97—862no—
437#97—862no—
438C4×C8—434no—
439C4×C8—434no—
440#95—22no—
441#95—862no—
442#95—862no—
443#95—862no—
444C24—21no—
445C2×Q8—21no—
446C2×Q8—21no—
447C2×Q8—21no—
448C2×Q8—21no—
449C2×Q8—21no—
450C2×Q8—432no—
451C2×Q8—432no—
452C2×Q8—861no—
453C2×Q8—861no—
454C2×Q8—861no—
455C2×Q8—861no—
456C2×Q8—861no—
457C2×Q8—861no—
458C2×Q8—861no—
459C2×Q8—861no—
460C2×Q8—861no—
461C2×Q8—861no—
462C2×Q8C2×C4×D4312yes—
463C2×D4—21no—
464C2×D4—21no—
465C2×D4—21no—
466C2×D4—21no—
467C2×D4—21no—
468C2×D4C2×C4×D4312yes—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—431no—
473C22×C4—432no—
474C22×C4—431no—
475C22×C4—861no—
476C22×C4—861no—
477C22×C4—861no—
478C22×C4—861no—
479C22×C4—1722no—
480C22×C4—1724no—
481C22×C4D4×D4312yes—
482C22×C4C23×D4311yescharacteristic
483Q16—868no—
484Q16—868no—
485SD16—28no—
486SD16—868no—
487C2×C8—21no—
488C2×C8—21no—
489C2×C8—432no—
490C2×C8—432no—
491C2×C8—432no—
492C2×C8—432no—
493C2×C8—861no—
494C2×C8—861no—
495C2×C8C2×C4×D4312yes—
496C2×C8D4×D4312yes—
497C2×C8D4×D4312yes—
498C2×C8#1062190012yes—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—868no—
504C4⋊C4—861no—
505C4⋊C4—861no—
506C4⋊C4—861no—
507C4⋊C4—861no—
508C4⋊C4—1722no—
509C4⋊C4—1722no—
510C4⋊C4—1722no—
511C4⋊C4—1724no—
512C22⋊C4—44no—
513C22⋊C4—868no—
514C22⋊C4—1722no—
515C42—21no—
516C42—21no—
517C42—432no—
518C42—432no—
519C42—432no—
520C42—432no—
521C42—864no—
522C42—1722no—
523C42—1722no—
524C42—1722no—
525C42—1722no—
526C23—22no—
527C23—42no—
528C23—41no—
529C23—42no—
530C23—41no—
531C23C2×D4×D4311yescharacteristic
532Q8—22no—
533Q8—22no—
534Q8—41no—
535Q8—41no—
536Q8—862no—
537Q8—862no—
538Q8—862no—
539Q8—862no—
540Q8—1721no—
541Q8—1721no—
542Q8—1721no—
543Q8—1721no—
544D4—22no—
545D4—22no—
546D4—41no—
547D4—41no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—434no—
559C2×C4—432no—
560C2×C4—434no—
561C2×C4—432no—
562C2×C4—862no—
563C2×C4—862no—
564C2×C4—862no—
565C2×C4—862no—
566C2×C4—862no—
567C2×C4—3444no—
568C2×C4—1722no—
569C2×C4—1721no—
570C2×C4—1721no—
571C2×C4—1722no—
572C2×C4—1721no—
573C2×C4—1721no—
574C2×C4—1721no—
575C2×C4—1721no—
576C2×C4C22×C4×D4311yescharacteristic
577C2×C4C2×D4×D4311yescharacteristic
578C2×C4C2×D4×D4311yescharacteristic
579C2×C4#41×D4311yescharacteristic
580C2×C4#41×D4311yescharacteristic
581C2×C4C2×#1062190011yescharacteristic
582C8—22no—
583C8—434no—
584C8—862no—
585C8(C22⋊D43)⋊C414yes—
586C22—42no—
587C22—44no—
588C22—81no—
589C22—81no—
590C22—82no—
591C22C4×D4×D4311yescharacteristic
592C22#127×D4311yescharacteristic
593C22#1458344411yescharacteristic
594C22#137×D4312yes—
595C22#1458369512yes—
596C4—42no—
597C4—42no—
598C4—81no—
599C4—81no—
600C4—434no—
601C4—862no—
602C4—864no—
603C4—3441no—
604C4—3441no—
605C4—1722no—
606C4—1722no—
607C4C4×D4×D4311yescharacteristic
608C4C2×(C22⋊D43)⋊C411yescharacteristic
609C4#1458344411yescharacteristic
610C4#1458344811yescharacteristic
611C2—82no—
612C2#1913715911yescharacteristic
613C2#1914341712yes—
614C2#497×D4311yescharacteristic
615C2#1914503711yescharacteristic
616C2#1914519412yes—
subgroupG/H
centerC23C2×D4×D43
commutatorC2×C4×C43C23×C4
FrattiniC22×C4C23×D43
FittingC43×#1973C2
socleC23×C43C22×D4
prime radicalC22×D4×C43C23
cosocleC22×C4×C43C24
hypercenter#1973D43
hypocenterC43C4×#497
LCSG ▷ C2×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1973

Complements

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

NH ≅ G/Ntype
2. C2×#14582993C2semidirect
3. C2×#14582991C2semidirect
6. #19137164C2semidirect
9. #19135466C2semidirect
10. #19135465C2semidirect
11. #19135413C2semidirect
13. #19134762C2semidirect
14. #97×Dic43C2semidirect
25. #14583290C4semidirect
65. C2×Dic172C2×C4semidirect
68. C2×#10621885C2×C4semidirect
75. #13063323C2×C4semidirect
113. #13063235C2×C4semidirect
302. Dic43#497semidirect
368. C43C4×#497semidirect

Maximal subgroups

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

subgroupindex
C2×#145832902
C2×#145829932
C2×#145829912
C2×#145829902
C2×SD16×Dic432
#191371642
#191371422
#191354752
#191354662
#191354652
#191354132
#191354102
#191347622
#97×Dic432
C43×#19732
C4×#49743