Finite Group Databasebeta

← #24988719

Subgroups — #24988719

Order 11008.

868 conjugacy classes of subgroups.

The subgroup lattice has 536 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14583614C211yesmaximal, characteristic
2#19135397C211yesmaximal, characteristic, complemented
3#19134596C211yesmaximal, characteristic, complemented
4#19134595C211yesmaximal, characteristic, complemented
5C2×#14582647C211yesmaximal, characteristic, complemented
6C2×#14582628C211yesmaximal, characteristic, complemented
7C2×C43×D4⋊C8C211yesmaximal, characteristic, complemented
8#14583614—22no—
9#14583614—22no—
10#14583614—22no—
11#14583614C414yescomplemented
12C2×#13063325C2211yescharacteristic
13C2×#13063311—22no—
14C2×#13063299—21no—
15C2×#13063296—21no—
16C2×C4×D172C2211yescharacteristic
17C2×#13063284—21no—
18C2×#13063283—21no—
19C2×C4⋊(C43⋊C8)C2211yescharacteristic
20C2×#13063250—22no—
21C2×#13063247—21no—
22C2×#13063246—21no—
23#14582769C2211yescharacteristic
24#14582662—21no—
25#14582661—21no—
26#14582647—22no—
27#14582628—22no—
28C2×C4×D4×C43C2211yescharacteristic
29C2×C4⋊C8×C43C2211yescharacteristic
30C2×C4×C8×C43C2211yescharacteristic
31C43×D4⋊C8—22no—
32C22×Dic86D412yes—
33C22×D172D411yescharacteristic
34C22×C22⋊D43—22no—
35C22×C4×D43—21no—
36C22×C43⋊C8—21no—
37#13063325—21no—
38#13063325—21no—
39#13063325C2×C412yes—
40#13063325D412yes—
41#13063311—28no—
42#13063311—28no—
43#13063299—24no—
44#13063299—24no—
45#13063296—24no—
46#13063296—24no—
47C4×D172—21no—
48C4×D172—21no—
49C4×D172C2×C412yes—
50C4×D172D412yes—
51#13063284—24no—
52#13063284—24no—
53#13063283—24no—
54#13063283—24no—
55C2×C4⋊Dic43—21no—
56C2×C4⋊Dic43—21no—
57C2×C4⋊Dic43—21no—
58C2×C4⋊Dic43—21no—
59C2×C4⋊Dic43—21no—
60C2×C4⋊Dic43D411yescharacteristic
61C2×C22⋊Dic43—22no—
62C2×(C2×D43)⋊C4—22no—
63C2×(C2×D43)⋊C4—21no—
64C2×C4×Dic43—22no—
65C4⋊(C43⋊C8)—21no—
66C4⋊(C43⋊C8)—21no—
67#13063250—44no—
68#13063247—42no—
69#13063246—42no—
70C22×D4×C43D411yescharacteristic
71C23×C4×C43—21no—
72C22×C8×C43—21no—
73C22×C8×C43—21no—
74C4×D4×C43—21no—
75C4×D4×C43—21no—
76C4×D4×C43C2×C412yes—
77C4×D4×C43D412yes—
78C2×C4⋊C4×C43D411yescharacteristic
79C2×C22⋊C4×C43—21no—
80C2×C42×C43C2311yescharacteristic
81C4⋊C8×C43—21no—
82C4⋊C8×C43—21no—
83C4×C8×C43—21no—
84C4×C8×C43—21no—
85C24×C43—21no—
86C23×D43—21no—
87C2×D4×C43—21no—
88C2×D4×C43—21no—
89C2×D4×C43—21no—
90C2×D4×C43—21no—
91C2×D4×C43—21no—
92C2×D4×C43C22⋊C412yes—
93C22×C4×C43—22no—
94C22×C4×C43—22no—
95C22×C4×C43—21no—
96C22×C4×C43—41no—
97C22×C4×C43—41no—
98C22×C4×C43—41no—
99C22×C4×C43—41no—
100C22×C4×C43C2×D411yescharacteristic
101C22×C4×C43C2×D411yescharacteristic
102C22×C4×C43C2×D411yescharacteristic
103C2×C22⋊D43—28no—
104C2×C22⋊D43—82no—
105C2×C22⋊D43—82no—
106C22×Dic43—21no—
107C22×Dic43—22no—
108C22×Dic43—41no—
109C2×D172—21no—
110C2×D172—21no—
111C2×D172—21no—
112C2×D172—21no—
113C2×D172—21no—
114C2×D172C22⋊C412yes—
115C2×C4×D43—22no—
116C2×C4×D43—22no—
117C2×C4×D43—41no—
118C2×C4×D43—41no—
119C2×C4×D43—41no—
120C2×C4×D43—41no—
121C2×Dic86—22no—
122C2×Dic86—42no—
123C2×Dic86—42no—
124C2×Dic86C22⋊C414yes—
125C2×C8×C43—22no—
126C2×C8×C43—41no—
127C2×C8×C43—41no—
128C2×C8×C43—41no—
129C2×C8×C43—41no—
130C2×C8×C43—41no—
131C4⋊C4×C43—21no—
132C4⋊C4×C43C22⋊C412yes—
133C22⋊C4×C43—22no—
134C22⋊C4×C43—22no—
135C42×C43C22×C411yescharacteristic
136C42×C43C2×D411yescharacteristic
137C42×C43C2×D411yescharacteristic
138C42×C43C2×D411yescharacteristic
139C22⋊Dic43—28no—
140(C2×D43)⋊C4—28no—
141(C2×D43)⋊C4—22no—
142(C2×D43)⋊C4—22no—
143C4⋊Dic43—22no—
144C4⋊Dic43—22no—
145C4⋊Dic43—22no—
146C4⋊Dic43—22no—
147C4⋊Dic43—21no—
148C4⋊Dic43—42no—
149C4⋊Dic43—41no—
150C4⋊Dic43—41no—
151C4⋊Dic43—41no—
152C4⋊Dic43—41no—
153C4⋊Dic43C22⋊C412yes—
154C4×Dic43—28no—
155C2×C43⋊C8—41no—
156C2×C43⋊C8—41no—
157C2×C43⋊C8—41no—
158C23×C43—22no—
159C23×C43—41no—
160C23×C43—41no—
161C23×C43—41no—
162C23×C43—41no—
163C23×C43—41no—
164C23×C43—41no—
165C23×C43#12011yescharacteristic
166C22×D43—22no—
167C22×D43—41no—
168C22×D43—41no—
169C22×D43—41no—
170C22×D43—41no—
171C22×D43—41no—
172C22×D43—41no—
173D4×C43—22no—
174D4×C43—22no—
175D4×C43—41no—
176D4×C43—41no—
177C2×C4×C43—21no—
178C2×C4×C43—21no—
179C2×C4×C43—22no—
180C2×C4×C43—41no—
181C2×C4×C43—41no—
182C2×C4×C43—42no—
183C2×C4×C43—42no—
184C2×C4×C43—42no—
185C2×C4×C43—42no—
186C2×C4×C43C2×C22⋊C411yescharacteristic
187C2×C4×C43C2×C22⋊C411yescharacteristic
188C2×C4×C43C2×C22⋊C411yescharacteristic
189C2×C4×C43#12011yescharacteristic
190C2×C4×C43#12011yescharacteristic
191C2×C4×C43#12011yescharacteristic
192C2×C4×C43#13511yescharacteristic
193C2×C4×C43#13511yescharacteristic
194C2×C4×C43#13511yescharacteristic
195C2×C4×C43#13511yescharacteristic
196C2×C4×C43AGL(1,ℤ/8)11yescharacteristic
197C2×C4×C43AGL(1,ℤ/8)11yescharacteristic
198C2×C4×C43#13711yescharacteristic
199C2×C4×C43#13711yescharacteristic
200C22⋊D43—88no—
201C2×Dic43—22no—
202C2×Dic43—24no—
203C2×Dic43—41no—
204C2×Dic43—41no—
205C2×Dic43—42no—
206C2×Dic43—42no—
207C2×Dic43—81no—
208C2×Dic43—81no—
209C2×Dic43—81no—
210D172—22no—
211D172—22no—
212D172—41no—
213D172—41no—
214C4×D43—42no—
215C4×D43—42no—
216C4×D43—42no—
217C4×D43—42no—
218Dic86—44no—
219Dic86—82no—
220C8×C43—42no—
221C8×C43—81no—
222C43⋊C8—81no—
223#8452—431nomaximal, Sylow
224C22×C43—42no—
225C22×C43—42no—
226C22×C43—42no—
227C22×C43—81no—
228C22×C43—81no—
229C22×C43—81no—
230C22×C43—81no—
231C22×C43#37911yescharacteristic
232C22×C43#44911yescharacteristic
233C22×C43#44911yescharacteristic
234C22×C43#45211yescharacteristic
235C22×C43#45211yescharacteristic
236C22×C43#45311yescharacteristic
237C22×C43#45511yescharacteristic
238C2×D43—42no—
239C2×D43—42no—
240C2×D43—42no—
241C2×D43—81no—
242C2×D43—81no—
243C2×D43—81no—
244C2×D43—81no—
245C4×C43—21no—
246C4×C43—21no—
247C4×C43—42no—
248C4×C43#41611yescharacteristic
249C4×C43#41611yescharacteristic
250C4×C43#41711yescharacteristic
251C4×C43#41711yescharacteristic
252C4×C43#44911yescharacteristic
253C4×C43#44911yescharacteristic
254C4×C43#45211yescharacteristic
255C4×C43#45211yescharacteristic
256Dic43—44no—
257Dic43—42no—
258Dic43—161no—
259C2×#555—431no—
260#1956—431no—
261#1802—431no—
262#1802—431no—
263C2×#333—431no—
264C2×D4⋊C8—431no—
265C2×D4⋊C8D4311yescharacteristic, complemented
266C2×C43—82no—
267C2×C43#162411yescharacteristic
268C2×C43#163411yescharacteristic
269C2×C43#164611yescharacteristic
270C2×C43#164611yescharacteristic
271C2×C43#178811yescharacteristic
272C2×C43#178811yescharacteristic
273C2×C43#179211yescharacteristic
274D43—82no—
275#555—434no—
276#555—862no—
277#555—862no—
278#555—862no—
279C2×C4⋊Q8—431no—
280C2×#124—862no—
281C2×#121—861no—
282C2×#121—861no—
283C2×C4×D4—431no—
284C2×C4×D4C2×D4311yescharacteristic
285C2×#117—861no—
286C2×#117—861no—
287C2×C4⋊C8—431no—
288C2×C4⋊C8C2×D4311yescharacteristic
289C2×#103—862no—
290C2×#102—861no—
291C2×#102—861no—
292C2×C4×C8C2×D4311yescharacteristic
293#384—431no—
294#340—861no—
295#340—861no—
296#333—862no—
297D4⋊C8—22no—
298D4⋊C8—862no—
299C43#845211yescharacteristic, complemented, Sylow
300C22×Q8—432no—
301C22×D4—431no—
302C22×D4—862no—
303C22×D4D17211yescharacteristic
304C23×C4—21no—
305C23×C4—861no—
306C22×C8—21no—
307C22×C8—21no—
308C22×C8—861no—
309C4⋊Q8—432no—
310C4⋊Q8—432no—
311C4⋊Q8—861no—
312C4⋊Q8—861no—
313#124—868no—
314#124—868no—
315#121—864no—
316#121—864no—
317#121—864no—
318#121—864no—
319C4×D4—21no—
320C4×D4—21no—
321C4×D4—432no—
322C4×D4—432no—
323C4×D4—861no—
324C4×D4—861no—
325C4×D4C4×D4312yes—
326C4×D4C22⋊D4312yes—
327#117—864no—
328#117—864no—
329#117—864no—
330#117—864no—
331C2×C4⋊C4—431no—
332C2×C4⋊C4—861no—
333C2×C4⋊C4—861no—
334C2×C4⋊C4—861no—
335C2×C4⋊C4—861no—
336C2×C4⋊C4—861no—
337C2×C4⋊C4D17211yescharacteristic
338C2×C22⋊C4—21no—
339C2×C22⋊C4—862no—
340C2×C22⋊C4—862no—
341C2×C22⋊C4—861no—
342C2×C42—862no—
343C2×C42C22×D4311yescharacteristic
344C4⋊C8—21no—
345C4⋊C8—21no—
346C4⋊C8—861no—
347C4⋊C8—861no—
348#103—1724no—
349#102—1722no—
350#102—1722no—
351C4×C8—21no—
352C4×C8—21no—
353C24—21no—
354C24—861no—
355C2×Q8—434no—
356C2×Q8—862no—
357C2×Q8—1722no—
358C2×Q8—1722no—
359C2×D4—21no—
360C2×D4—21no—
361C2×D4—21no—
362C2×D4—21no—
363C2×D4—21no—
364C2×D4—432no—
365C2×D4—868no—
366C2×D4—861no—
367C2×D4—861no—
368C2×D4—861no—
369C2×D4—861no—
370C2×D4—861no—
371C2×D4—3442no—
372C2×D4—3442no—
373C2×D4(C2×D43)⋊C412yes—
374C22×C4—21no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—41no—
378C22×C4—41no—
379C22×C4—41no—
380C22×C4—41no—
381C22×C4—861no—
382C22×C4—862no—
383C22×C4—862no—
384C22×C4—862no—
385C22×C4—1721no—
386C22×C4—1721no—
387C22×C4—1721no—
388C22×C4—1721no—
389C22×C4—1721no—
390C22×C4C2×D17211yescharacteristic
391C22×C4D4×D4311yescharacteristic
392C22×C4D4×D4311yescharacteristic
393C2×C8—22no—
394C2×C8—41no—
395C2×C8—41no—
396C2×C8—41no—
397C2×C8—41no—
398C2×C8—41no—
399C2×C8—1721no—
400C2×C8—1721no—
401C2×C8—1721no—
402C4⋊C4—21no—
403C4⋊C4—432no—
404C4⋊C4—862no—
405C4⋊C4—862no—
406C4⋊C4—861no—
407C4⋊C4—862no—
408C4⋊C4—862no—
409C4⋊C4—1722no—
410C4⋊C4—1721no—
411C4⋊C4—1721no—
412C4⋊C4—1721no—
413C4⋊C4—1721no—
414C4⋊C4(C2×D43)⋊C412yes—
415C22⋊C4—22no—
416C22⋊C4—22no—
417C22⋊C4—868no—
418C22⋊C4—868no—
419C22⋊C4—862no—
420C22⋊C4—862no—
421C42—868no—
422C42C2×C4×D4311yescharacteristic
423C42D4×D4311yescharacteristic
424C42D4×D4311yescharacteristic
425C42C2×C22⋊D4311yescharacteristic
426C23—22no—
427C23—41no—
428C23—41no—
429C23—41no—
430C23—41no—
431C23—41no—
432C23—41no—
433C23—862no—
434C23—1721no—
435C23—1721no—
436C23—1721no—
437C23—1721no—
438C23—1721no—
439C23—1721no—
440C23#1306329511yescharacteristic
441Q8—3442no—
442Q8—1724no—
443D4—22no—
444D4—22no—
445D4—41no—
446D4—41no—
447D4—862no—
448D4—862no—
449D4—3448no—
450D4—1721no—
451D4—1721no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—22no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—862no—
462C2×C4—864no—
463C2×C4—3441no—
464C2×C4—3441no—
465C2×C4—3441no—
466C2×C4—1721no—
467C2×C4—1721no—
468C2×C4—1722no—
469C2×C4—1722no—
470C2×C4—1722no—
471C2×C4—1722no—
472C2×C4—1722no—
473C2×C4—1722no—
474C2×C4C22⋊C4×D4311yescharacteristic
475C2×C4C22⋊C4×D4311yescharacteristic
476C2×C4C2×(C2×D43)⋊C411yescharacteristic
477C2×C4#1306329311yescharacteristic
478C2×C4#1306329311yescharacteristic
479C2×C4#1306329511yescharacteristic
480C2×C4#1306334411yescharacteristic
481C2×C4#1306334611yescharacteristic
482C2×C4#1306334711yescharacteristic
483C2×C4#1306334711yescharacteristic
484C2×C4#1306334911yescharacteristic
485C2×C4#1306335311yescharacteristic
486C2×C4#1306335411yescharacteristic
487C2×C4#1306335811yescharacteristic
488C8—42no—
489C8—81no—
490C8—3441no—
491C22—42no—
492C22—42no—
493C22—42no—
494C22—81no—
495C22—81no—
496C22—81no—
497C22—81no—
498C22—3441no—
499C22—3441no—
500C22—3441no—
501C22—3441no—
502C22—1722no—
503C22—1722no—
504C22—1722no—
505C22#1458275411yescharacteristic
506C22#1458304511yescharacteristic
507C22#1458304611yescharacteristic
508C22#1458306211yescharacteristic
509C22#1458306311yescharacteristic
510C22#1458306811yescharacteristic
511C22#1458307811yescharacteristic
512C4—21no—
513C4—21no—
514C4—42no—
515C4—1722no—
516C4—1724no—
517C4—6881no—
518C4#1458289211yescharacteristic
519C4#1458289611yescharacteristic
520C4#1458289711yescharacteristic
521C4#1458290111yescharacteristic
522C4#1458304111yescharacteristic
523C4#1458304611yescharacteristic
524C4#1458305811yescharacteristic
525C4#1458306311yescharacteristic
526C2—82no—
527C2—3442no—
528C2#1913368011yescharacteristic
529C2#1913374211yescharacteristic
530C2#1913382511yescharacteristic
531C2#1913382611yescharacteristic
532C2#1913453211yescharacteristic
533C2#1913453611yescharacteristic
534C2#1913455011yescharacteristic
subgroupG/H
centerC23#13063295
commutatorC42×C43C22×C4
FrattiniC2×C42C22×D43
FittingC2×C43×D4⋊C8C2
socleC23×C43#120
prime radicalC2×#14583614C2
cosocleC2×C42×C43C23
hypercenterC2×D4⋊C8D43
hypocenterC43#8452
LCSG ▷ C42×C43 ▷ C22×C43 ▷ C43
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×D4⋊C8

Complements

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

NH ≅ G/Ntype
2. #19135397C2semidirect
3. #19134596C2semidirect
4. #19134595C2semidirect
5. C2×#14582647C2semidirect
6. C2×#14582628C2semidirect
7. C2×C43×D4⋊C8C2semidirect
11. #14583614C4semidirect
265. C2×D4⋊C8D43semidirect
299. C43#8452semidirect

Maximal subgroups

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

subgroupindex
C2×#145836142
#191353972
#191345962
#191345952
C2×#145826472
C2×#145826282
C2×C43×D4⋊C82
#845243