Finite Group Databasebeta

← #25106303

Subgroups — #25106303

Order 11008.

1386 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14583446C211yesmaximal, characteristic
2C2×#14582951C211yesmaximal, characteristic, complemented
3#19137001C211yesmaximal, characteristic, complemented
4#19135554C212yesmaximal, complemented
5#19135399C212yesmaximal, complemented
6#19134916C212yesmaximal, complemented
7#19134914C212yesmaximal, complemented
8#19134606C211yesmaximal, characteristic, complemented
9#19134605C211yesmaximal, characteristic, complemented
10#19134602C211yesmaximal, characteristic, complemented
11C43×#1804C211yesmaximal, characteristic, complemented
12C22×#10621898—21no—
13#14583446—24no—
14#14583446C418yescomplemented
15C2×#13063324C2211yescharacteristic
16C2×#13063321C2211yescharacteristic
17C2×#13063311—21no—
18C2×#13063300C2211yescharacteristic
19C2×#13063298—21no—
20C2×#13063296C2211yescharacteristic
21C2×C4×D172C2211yescharacteristic
22C2×C4×C22⋊D43—21no—
23C2×#13063283C2212yes—
24C2×C42×D43C2211yescharacteristic
25#14582951—24no—
26C2×C4⋊(C43⋊C8)C2211yescharacteristic
27C2×#13063247C2212yes—
28C2×#13063246C2212yes—
29C2×#13063245C2212yes—
30C2×(C2×D43)⋊C8C2212yes—
31C2×C8×Dic43C2212yes—
32#14582768C2212yes—
33#14582765C2211yescharacteristic
34#14582758—21no—
35#14582758—21no—
36#14582750—21no—
37#14582750—21no—
38#14582662C2212yes—
39#14582661C2212yes—
40#14582661C2212yes—
41C2×C4×D4×C43C2211yescharacteristic
42C2×C4⋊C8×C43C2211yescharacteristic
43C2×#102×C43C2212yes—
44C43×#382C2211yescharacteristic
45C43×#340C2212yes—
46C2×#10621898—28no—
47C2×#10621898—48no—
48C22×Dic86—21no—
49C22×D172—21no—
50C22×D172C2311yescharacteristic
51C22×C22⋊D43—21no—
52C22×C22⋊D43—21no—
53C22×C22⋊D43—21no—
54C22×C4×D43—21no—
55C22×C4×D43D411yescharacteristic
56C22×C4×D43D411yescharacteristic
57C22×C4×D43C2311yescharacteristic
58C22×C43⋊C8C2312yes—
59#13063324—22no—
60#13063324C2×C414yes—
61#13063321—22no—
62#13063321C2×C414yes—
63#13063311—28no—
64#13063300—22no—
65#13063300C2×C414yes—
66#13063298—28no—
67#13063296—22no—
68#13063296C2×C414yes—
69C4×D172—22no—
70C4×D172C2×C414yes—
71C4×C22⋊D43—28no—
72#13063283—42no—
73#13063283—42no—
74C2×C4⋊Dic43C2312yes—
75C2×C4⋊Dic43C2311yescharacteristic
76C2×Dic43⋊C4—21no—
77C2×Dic43⋊C4—22no—
78C2×Dic43⋊C4—22no—
79C2×C22⋊Dic43—21no—
80C2×(C2×D43)⋊C4—21no—
81C2×(C2×D43)⋊C4—21no—
82C2×(C2×D43)⋊C4—22no—
83C2×C4×Dic43—22no—
84C2×C4×Dic43D411yescharacteristic
85C2×C4×Dic43D411yescharacteristic
86C2×C4×Dic43C2311yescharacteristic
87C42×D43C2×C414yes—
88C42×D43D414yes—
89C4⋊(C43⋊C8)—22no—
90#13063247—42no—
91#13063246—42no—
92#13063245—42no—
93(C2×D43)⋊C8—24no—
94C8×Dic43—24no—
95C43⋊#95—21no—
96C43⋊#95—21no—
97C22×D4×C43C2311yescharacteristic
98C23×C4×C43—21no—
99C22×C8×C43C2312yes—
100C4×D4×C43—22no—
101C4×D4×C43C2×C414yes—
102C2×C4⋊C4×C43C2312yes—
103C2×C4⋊C4×C43C2311yescharacteristic
104C2×C22⋊C4×C43—21no—
105C2×C42×C43C2311yescharacteristic
106C4⋊C8×C43—22no—
107#102×C43—42no—
108#95×C43—21no—
109#95×C43—21no—
110C24×C43—21no—
111C23×D43—21no—
112C23×D43C2×D411yescharacteristic
113C2×D4×C43—21no—
114C2×D4×C43—21no—
115C2×D4×C43—21no—
116C2×D4×C43—21no—
117C2×D4×C43—21no—
118C2×D4×C43C22×C412yes—
119C22×C4×C43—24no—
120C22×C4×C43—21no—
121C22×C4×C43—22no—
122C22×C4×C43—44no—
123C22×C4×C43C2×D411yescharacteristic
124C22×C4×C43C2×D411yescharacteristic
125C22×C4×C43C2411yescharacteristic
126C2×C22⋊D43—24no—
127C2×C22⋊D43—24no—
128C2×C22⋊D43—24no—
129C2×C22⋊D43—44no—
130C2×C22⋊D43—44no—
131C2×C22⋊D43—44no—
132C22×Dic43—21no—
133C22×Dic43—22no—
134C22×Dic43C2×D411yescharacteristic
135C22×Dic43C2×D411yescharacteristic
136C22×Dic43C2×D411yescharacteristic
137#10621898—432no—
138C2×D172—21no—
139C2×D172—24no—
140C2×D172—21no—
141C2×D172—21no—
142C2×D172—21no—
143C2×D172—21no—
144C2×D172—44no—
145C2×D172C22×C412yes—
146C2×C4×D43—21no—
147C2×C4×D43—21no—
148C2×C4×D43—21no—
149C2×C4×D43—21no—
150C2×C4×D43—24no—
151C2×C4×D43—24no—
152C2×C4×D43—24no—
153C2×C4×D43—44no—
154C2×C4×D43C22⋊C414yes—
155C2×C4×D43C22⋊C414yes—
156C2×C4×D43C22×C412yes—
157C2×C4×D43C2×D412yes—
158C2×Dic86—24no—
159C2×Dic86—44no—
160C2×C8×C43—22no—
161C2×C8×C43—22no—
162C2×C8×C43—22no—
163C4⋊C4×C43—21no—
164C4⋊C4×C43—22no—
165C4⋊C4×C43—22no—
166C4⋊C4×C43C22×C412yes—
167C22⋊C4×C43—24no—
168C42×C43C22×C412yes—
169C42×C43C2×D412yes—
170C22⋊Dic43—24no—
171(C2×D43)⋊C4—24no—
172(C2×D43)⋊C4—24no—
173(C2×D43)⋊C4—44no—
174C4⋊Dic43—21no—
175C4⋊Dic43—22no—
176C4⋊Dic43—22no—
177C4⋊Dic43C22×C412yes—
178Dic43⋊C4—24no—
179Dic43⋊C4—44no—
180Dic43⋊C4—44no—
181C4×Dic43—44no—
182C4×Dic43C22⋊C414yes—
183C4×Dic43C22⋊C414yes—
184C4×Dic43C22×C412yes—
185C4×Dic43C2×D412yes—
186C2×C43⋊C8—22no—
187C2×C43⋊C8—22no—
188C2×C43⋊C8—22no—
189C23×C43—22no—
190C23×C43—41no—
191C23×C43—41no—
192C23×C43—42no—
193C23×C43—42no—
194C23×C43C22×D411yescharacteristic
195C22×D43—21no—
196C22×D43—21no—
197C22×D43—22no—
198C22×D43—22no—
199C22×D43—22no—
200C22×D43—41no—
201C22×D43—41no—
202C22×D43—42no—
203C22×D43—42no—
204C22×D43C2×C22⋊C412yes—
205D4×C43—22no—
206D4×C43—22no—
207D4×C43—41no—
208D4×C43—41no—
209C2×C4×C43—22no—
210C2×C4×C43—22no—
211C2×C4×C43—22no—
212C2×C4×C43—48no—
213C2×C4×C43—42no—
214C2×C4×C43—42no—
215C2×C4×C43—42no—
216C2×C4×C43—42no—
217C2×C4×C43C2×C22⋊C412yes—
218C2×C4×C43C2×C22⋊C412yes—
219C2×C4×C43C23×C411yescharacteristic
220C2×C4×C43C22×D411yescharacteristic
221C2×C4×C43H(5,𝔽2)11yescharacteristic
222C2×C4×C43H(5,𝔽2)11yescharacteristic
223C2×C4×C432−1+411yescharacteristic
224C2×C4×C432−1+411yescharacteristic
225C22⋊D43—48no—
226C22⋊D43—48no—
227C22⋊D43—48no—
228C2×Dic43—21no—
229C2×Dic43—21no—
230C2×Dic43—22no—
231C2×Dic43—42no—
232C2×Dic43—42no—
233C2×Dic43—42no—
234C2×Dic43—42no—
235C2×Dic43C2×C22⋊C412yes—
236C2×Dic43C2×C22⋊C412yes—
237C2×Dic43C2×C22⋊C412yes—
238C2×Dic43#13514yes—
239C2×Dic43AGL(1,ℤ/8)14yes—
240D172—22no—
241D172—22no—
242D172—48no—
243D172—41no—
244D172—41no—
245C4×D43—22no—
246C4×D43—22no—
247C4×D43—22no—
248C4×D43—22no—
249C4×D43—28no—
250C4×D43—28no—
251C4×D43—48no—
252Dic86—48no—
253C8×C43—42no—
254C43⋊C8—42no—
255#21863—431nomaximal, Sylow
256C22×C43—42no—
257C22×C43—42no—
258C22×C43—42no—
259C22×C43—82no—
260C22×C43—82no—
261C22×C43C22×C22⋊C411yescharacteristic
262C22×C43#53511yescharacteristic
263C22×C43#53611yescharacteristic
264C22×C43C2×#13512yes—
265C22×C43C2×AGL(1,ℤ/8)12yes—
266C2×D43—22no—
267C2×D43—22no—
268C2×D43—22no—
269C2×D43—42no—
270C2×D43—42no—
271C2×D43—42no—
272C2×D43—42no—
273C2×D43—42no—
274C2×D43—82no—
275C2×D43—82no—
276C4×C43—22no—
277C4×C43—22no—
278C4×C43—42no—
279C4×C43—82no—
280C4×C43#52011yescharacteristic
281C4×C43#52011yescharacteristic
282C4×C43#53711yescharacteristic
283C4×C43#53711yescharacteristic
284Dic43—22no—
285Dic43—42no—
286Dic43—82no—
287Dic43#41614yes—
288Dic43#41814yes—
289C2×#532—431no—
290C2×#431—431no—
291#2285—431no—
292#1990—432no—
293#1957—432no—
294#1875—432no—
295#1875—432no—
296#1804—431no—
297#1804—431no—
298#1804—431no—
299#1804D4311yescharacteristic, complemented
300C2×C43—82no—
301C2×C43#231111yescharacteristic
302C2×C43C2×#41611yescharacteristic
303C2×C43C2×#41811yescharacteristic
304C2×C43#320712yes—
305C2×C43#322812yes—
306D43—42no—
307D43—82no—
308C22×#41—861no—
309#532—438no—
310#532—864no—
311C2×C4⋊Q8—431no—
312C2×#127—431no—
313C2×#124—861no—
314C2×#121—431no—
315C2×#121—431no—
316C2×#121—861no—
317C2×C4×D4—431no—
318C2×C4×D4—861no—
319C2×C4×D4C2×D4311yescharacteristic
320C2×#117—432no—
321C22×C42—431no—
322#431—864no—
323C2×C4⋊C8—431no—
324C2×C4⋊C8C2×D4311yescharacteristic
325C2×#102—432no—
326C2×#102—432no—
327C2×#102—432no—
328C2×#102C2×D4312yes—
329C2×C22⋊C8—432no—
330C2×C4×C8—432no—
331#384—432no—
332#382—431no—
333#382C2×D4311yescharacteristic
334#380—861no—
335#380—861no—
336C4×C4⋊C4—861no—
337C4×C4⋊C4—861no—
338#340—432no—
339#340—432no—
340#340—432no—
341#340C2×D4312yes—
342C43#2186311yescharacteristic, complemented, Sylow
343C2×#41—868no—
344C2×#41—1728no—
345C22×Q8—861no—
346C22×D4—431no—
347C22×D4—861no—
348C22×D4—861no—
349C22×D4—861no—
350C22×D4—861no—
351C22×D4C22×D4311yescharacteristic
352C23×C4—21no—
353C23×C4—431no—
354C23×C4—431no—
355C23×C4—431no—
356C23×C4—861no—
357C22×C8—432no—
358C22×C8C22×D4312yes—
359C4⋊Q8—434no—
360C4⋊Q8—862no—
361#127—434no—
362#127—862no—
363#124—868no—
364#121—434no—
365#121—434no—
366#121—868no—
367#121—862no—
368#121—862no—
369C4×D4—22no—
370C4×D4—434no—
371C4×D4—868no—
372C4×D4—862no—
373C4×D4C4×D4314yes—
374#117—1722no—
375#117—1722no—
376C2×C4⋊C4—431no—
377C2×C4⋊C4—432no—
378C2×C4⋊C4—861no—
379C2×C4⋊C4—862no—
380C2×C4⋊C4—862no—
381C2×C4⋊C4C22×D4312yes—
382C2×C4⋊C4C22×D4311yescharacteristic
383C2×C22⋊C4—21no—
384C2×C22⋊C4—861no—
385C2×C22⋊C4—861no—
386C2×C22⋊C4—861no—
387C2×C22⋊C4—862no—
388C2×C42—431no—
389C2×C42—431no—
390C2×C42—431no—
391C2×C42—434no—
392C2×C42—434no—
393C2×C42—862no—
394C2×C42C22×D4311yescharacteristic
395C4⋊C8—22no—
396C4⋊C8—862no—
397#102—42no—
398#102—1722no—
399#102—1722no—
400#102—1722no—
401C22⋊C8—864no—
402C4×C8—864no—
403#95—21no—
404#95—21no—
405#95—861no—
406#95—861no—
407C24—21no—
408C24—431no—
409C24—861no—
410#41—17232no—
411C2×Q8—864no—
412C2×Q8—1724no—
413C2×D4—21no—
414C2×D4—21no—
415C2×D4—21no—
416C2×D4—21no—
417C2×D4—21no—
418C2×D4—432no—
419C2×D4—864no—
420C2×D4—864no—
421C2×D4—864no—
422C2×D4—864no—
423C2×D4—861no—
424C2×D4—861no—
425C2×D4—861no—
426C2×D4—861no—
427C2×D4—861no—
428C2×D4—1724no—
429C2×D4—1724no—
430C2×D4—1724no—
431C2×D4—1724no—
432C2×D4C2×C4×D4312yes—
433C22×C4—21no—
434C22×C4—24no—
435C22×C4—22no—
436C22×C4—44no—
437C22×C4—431no—
438C22×C4—431no—
439C22×C4—431no—
440C22×C4—434no—
441C22×C4—434no—
442C22×C4—432no—
443C22×C4—432no—
444C22×C4—864no—
445C22×C4—864no—
446C22×C4—861no—
447C22×C4—861no—
448C22×C4—861no—
449C22×C4—861no—
450C22×C4—861no—
451C22×C4—864no—
452C22×C4—862no—
453C22×C4—1724no—
454C22×C4D4×D4311yescharacteristic
455C22×C4D4×D4311yescharacteristic
456C22×C4C23×D4311yescharacteristic
457C2×C8—22no—
458C2×C8—22no—
459C2×C8—22no—
460C2×C8—862no—
461C2×C8—862no—
462C2×C8—862no—
463C4⋊C4—21no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—432no—
467C4⋊C4—864no—
468C4⋊C4—861no—
469C4⋊C4—862no—
470C4⋊C4—862no—
471C4⋊C4—1724no—
472C4⋊C4—1724no—
473C4⋊C4C2×C4×D4312yes—
474C22⋊C4—24no—
475C22⋊C4—864no—
476C22⋊C4—864no—
477C22⋊C4—864no—
478C22⋊C4—1724no—
479C42—434no—
480C42—434no—
481C42—432no—
482C42—432no—
483C42—1724no—
484C42C2×C4×D4312yes—
485C42D4×D4312yes—
486C23—22no—
487C23—42no—
488C23—42no—
489C23—41no—
490C23—41no—
491C23—432no—
492C23—862no—
493C23—862no—
494C23—861no—
495C23—861no—
496C23—862no—
497C23—1722no—
498C23—1722no—
499C23—1721no—
500C23—1721no—
501C23C2×D4×D4311yescharacteristic
502Q8—1728no—
503D4—22no—
504D4—22no—
505D4—41no—
506D4—41no—
507D4—862no—
508D4—862no—
509D4—1728no—
510D4—1728no—
511D4—1728no—
512D4—1728no—
513D4—1721no—
514D4—1721no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—48no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—432no—
524C2×C4—432no—
525C2×C4—434no—
526C2×C4—434no—
527C2×C4—432no—
528C2×C4—868no—
529C2×C4—868no—
530C2×C4—861no—
531C2×C4—861no—
532C2×C4—862no—
533C2×C4—862no—
534C2×C4—862no—
535C2×C4—862no—
536C2×C4—862no—
537C2×C4—1728no—
538C2×C4—1722no—
539C2×C4—1722no—
540C2×C4—1722no—
541C2×C4—1722no—
542C2×C4C22⋊C4×D4312yes—
543C2×C4C22⋊C4×D4312yes—
544C2×C4C22×C4×D4311yescharacteristic
545C2×C4C2×D4×D4311yescharacteristic
546C2×C4#1306337011yescharacteristic
547C2×C4#1306337111yescharacteristic
548C2×C4#1306337211yescharacteristic
549C2×C4#1306337311yescharacteristic
550C8—42no—
551C8—1722no—
552C22—42no—
553C22—42no—
554C22—42no—
555C22—82no—
556C22—82no—
557C22—862no—
558C22—862no—
559C22—862no—
560C22—3442no—
561C22—3442no—
562C22—1722no—
563C22—1722no—
564C22—1722no—
565C22—1722no—
566C22—1722no—
567C22C2×C22⋊C4×D4311yescharacteristic
568C22#1458346511yescharacteristic
569C22#1458347211yescharacteristic
570C22#135×D4312yes—
571C22AGL(1,ℤ/8)×D4312yes—
572C4—22no—
573C4—22no—
574C4—42no—
575C4—82no—
576C4—434no—
577C4—434no—
578C4—862no—
579C4—3442no—
580C4—1722no—
581C4#1458337911yescharacteristic
582C4#1458338011yescharacteristic
583C4#1458347811yescharacteristic
584C4#1458347911yescharacteristic
585C2—82no—
586C2—3442no—
587C2—1722no—
588C2#1913718411yescharacteristic
589C2#416×D4311yescharacteristic
590C2#418×D4311yescharacteristic
591C2#1914552712yes—
592C2#1914573012yes—
subgroupG/H
centerC23C2×D4×D43
commutatorC2×C4×C43C23×C4
FrattiniC22×C4C23×D43
FittingC43×#1804C2
socleC23×C43C22×D4
prime radicalC2×#14583446C2
cosocleC22×C4×C43C24
hypercenter#1804D43
hypocenterC43#21863
LCSG ▷ C2×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1804

Complements

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

NH ≅ G/Ntype
2. C2×#14582951C2semidirect
3. #19137001C2semidirect
4. #19135554C2semidirect
5. #19135399C2semidirect
6. #19134916C2semidirect
7. #19134914C2semidirect
8. #19134606C2semidirect
9. #19134605C2semidirect
10. #19134602C2semidirect
11. C43×#1804C2semidirect
14. #14583446C4semidirect
299. #1804D43semidirect
342. C43#21863semidirect

Maximal subgroups

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

subgroupindex
C2×#145834462
C2×#145829512
#191370012
#191355542
#191353992
#191349162
#191349142
#191346062
#191346052
#191346022
C43×#18042
#2186343