Finite Group Databasebeta

← #25012052

Subgroups — #25012052

Order 11008.

904 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14583664C211yesmaximal, characteristic
2#19136224C211yesmaximal, characteristic, complemented
3#19136223C211yesmaximal, characteristic, complemented
4#19136222C211yesmaximal, characteristic, complemented
5C2×#14582680C211yesmaximal, characteristic, complemented
6C2×#14582680C211yesmaximal, characteristic, complemented
7C43×#2117C211yesmaximal, characteristic, complemented
8C2×#41×D43C411yescharacteristic
9#14583664—22no—
10#14583664—22no—
11#14583664—22no—
12#14583664—22no—
13C2×#13063333—21no—
14C2×#13063333C411yescharacteristic
15C2×#13063332C411yescharacteristic
16C2×#13063331—21no—
17C2×#13063331C2211yescharacteristic
18C22×#10621872—21no—
19C22×#10621872C2211yescharacteristic
20C2×M16×D43—21no—
21C2×M16×D43C411yescharacteristic
22C2×#13063265C2211yescharacteristic, complemented
23C2×#13063265C2211yescharacteristic, complemented
24#14582684—22no—
25#14582683—22no—
26#14582682—22no—
27#14582680—21no—
28#14582680—21no—
29#14582680—21no—
30#14582680—21no—
31C2×#131×C43C2211yescharacteristic
32C2×M32×C43C2211yescharacteristic, complemented
33C2×M32×C43C2211yescharacteristic, complemented
34C43×#350—22no—
35C2×#10621902C811yescharacteristic
36C2×#10621898—21no—
37C2×#10621898C2×C411yescharacteristic
38C2×#10621900—21no—
39C2×#10621900C811yescharacteristic
40#41×D43—22no—
41#41×D43—22no—
42#41×D43—22no—
43#41×D43—22no—
44C2×Q8×D43C811yescharacteristic
45C2×D4×D43—21no—
46C2×D4×D43C811yescharacteristic
47C22×C4×D43—21no—
48C22×C4×D43C2×C411yescharacteristic
49#13063333—24no—
50#13063333—24no—
51#13063333—22no—
52#13063333—22no—
53#13063332—22no—
54#13063332—22no—
55#13063331—24no—
56#13063331—24no—
57#13063331—21no—
58#13063331—21no—
59#13063331D412yes—
60#13063331D412yes—
61C2×#10621877—21no—
62C2×#10621877C2×C411yescharacteristic
63C2×#10621872—24no—
64C2×#10621872—24no—
65C2×#10621872—21no—
66C2×#10621872—21no—
67C2×#10621872—21no—
68C2×#10621872—21no—
69C2×#10621872—21no—
70C2×#10621872—21no—
71C2×#10621872—21no—
72C2×#10621872D411yescharacteristic
73C2×#10621872D411yescharacteristic
74C2×#10621872D411yescharacteristic
75C2×#10621872D411yescharacteristic
76M16×D43—24no—
77M16×D43—24no—
78M16×D43—22no—
79M16×D43—22no—
80C22×C43⋊C8—21no—
81C22×C43⋊C8C2×C411yescharacteristic
82C2×C8×D43—21no—
83C2×C8×D43—21no—
84C2×C8×D43—21no—
85#13063265—21no—
86#13063265—21no—
87#13063265—21no—
88#13063265—21no—
89C2×C43⋊C16—21no—
90C2×C43⋊C16—21no—
91C2×#41×C43C2×C411yescharacteristic
92#131×C43—21no—
93#131×C43—21no—
94#131×C43D412yes—
95#131×C43D412yes—
96C2×M16×C43—21no—
97C2×M16×C43C2×C411yescharacteristic
98C22×C8×C43—21no—
99C22×C8×C43C2311yescharacteristic
100M32×C43—21no—
101M32×C43—21no—
102M32×C43—21no—
103M32×C43—21no—
104C2×C16×C43—21no—
105C2×C16×C43—21no—
106C23×D43—21no—
107C23×D43M1611yescharacteristic
108#41×C43—21no—
109#41×C43—21no—
110#41×C43C22⋊C412yes—
111#41×C43M1612yes—
112C2×Q8×C43C2×C811yescharacteristic
113C2×D4×C43—21no—
114C2×D4×C43C2×C811yescharacteristic
115C22×C4×C43—21no—
116C22×C4×C43C22×C411yescharacteristic
117C2×C22⋊D43—21no—
118C2×C22⋊D43—21no—
119C2×C22⋊D43—21no—
120C22×Dic43—21no—
121C22×Dic43M1611yescharacteristic
122#10621902—22no—
123#10621902—22no—
124Q8×D43—22no—
125Q8×D43—22no—
126#10621900—24no—
127#10621900—24no—
128#10621900—22no—
129#10621900—22no—
130D4×D43—24no—
131D4×D43—24no—
132D4×D43—22no—
133D4×D43—22no—
134#10621898—24no—
135#10621898—24no—
136#10621898—21no—
137#10621898—21no—
138#10621898C22⋊C412yes—
139#10621898M1612yes—
140C2×D172—21no—
141C2×D172C2×C811yescharacteristic
142C2×C4×D43—24no—
143C2×C4×D43—24no—
144C2×C4×D43—21no—
145C2×C4×D43—21no—
146C2×C4×D43—21no—
147C2×C4×D43—21no—
148C2×C4×D43—21no—
149C2×C4×D43—21no—
150C2×C4×D43—21no—
151C2×C4×D43C22⋊C411yescharacteristic
152C2×C4×D43C22⋊C411yescharacteristic
153C2×C4×D43C2×C811yescharacteristic
154C2×C4×D43C2×C811yescharacteristic
155C2×Dic86—21no—
156C2×Dic86C2×C811yescharacteristic
157M16×C43—21no—
158M16×C43—22no—
159M16×C43—22no—
160M16×C43C22⋊C412yes—
161C2×C8×C43—21no—
162C2×C8×C43—22no—
163C2×C8×C43—22no—
164C2×C8×C43C2×D411yescharacteristic
165C2×C8×C43C2×D411yescharacteristic
166C2×C8×C43C2×D411yescharacteristic
167C2×C8×C43C2×D411yescharacteristic
168C2×C8×C43C2×D411yescharacteristic
169C2×C8×C43C2×D411yescharacteristic
170#10621877—24no—
171#10621877—21no—
172#10621877C22⋊C412yes—
173C2×C43⋊C8—24no—
174C2×C43⋊C8—21no—
175C2×C43⋊C8—21no—
176C2×C43⋊C8—21no—
177C2×C43⋊C8C22⋊C411yescharacteristic
178C2×C43⋊C8C22⋊C411yescharacteristic
179#10621872—24no—
180#10621872—24no—
181#10621872—21no—
182#10621872—21no—
183#10621872—21no—
184#10621872—21no—
185#10621872—22no—
186#10621872—22no—
187#10621872—21no—
188#10621872—21no—
189#10621872—21no—
190#10621872—21no—
191C8×D43—24no—
192C8×D43—24no—
193C8×D43—22no—
194C8×D43—22no—
195C16×C43—41no—
196C16×C43—41no—
197C43⋊C16—41no—
198C43⋊C16—41no—
199C23×C43—21no—
200C23×C43C2×M1611yescharacteristic
201C22×D43—21no—
202C22×D43—24no—
203C22×D43—21no—
204C22×D43—21no—
205C22×D43—82no—
206C22×D43—81no—
207C22×D43C22⋊C811yescharacteristic
208C22×D43C22⋊C811yescharacteristic
209Q8×C43—21no—
210Q8×C43C22⋊C812yes—
211D4×C43—22no—
212D4×C43—22no—
213D4×C43—21no—
214D4×C43C22⋊C812yes—
215C2×C4×C43—21no—
216C2×C4×C43—22no—
217C2×C4×C43—22no—
218C2×C4×C43C2×C22⋊C411yescharacteristic
219C2×C4×C43C2×C22⋊C411yescharacteristic
220C2×C4×C43C2×C22⋊C411yescharacteristic
221C2×C4×C43C22×C811yescharacteristic
222C2×C4×C43C2×M1611yescharacteristic
223C2×C4×C43C2×M1611yescharacteristic
224C22⋊D43—24no—
225C22⋊D43—24no—
226C22⋊D43—22no—
227C22⋊D43—22no—
228C2×Dic43—24no—
229C2×Dic43—21no—
230C2×Dic43—21no—
231C2×Dic43—21no—
232C2×Dic43C22⋊C811yescharacteristic
233C2×Dic43C22⋊C811yescharacteristic
234D172—24no—
235D172—21no—
236D172C22⋊C812yes—
237C4×D43—24no—
238C4×D43—24no—
239C4×D43—22no—
240C4×D43—21no—
241C4×D43—21no—
242C4×D43—21no—
243C4×D43—21no—
244C4×D43—21no—
245C4×D43—21no—
246C4×D43—21no—
247C4×D43—22no—
248C4×D43—21no—
249Dic86—24no—
250Dic86—21no—
251Dic86C22⋊C812yes—
252C8×C43—22no—
253C8×C43#12011yescharacteristic
254C8×C43#12011yescharacteristic
255C8×C43#12011yescharacteristic
256C8×C43#12011yescharacteristic
257C43⋊C8—22no—
258C43⋊C8—21no—
259C43⋊C8—21no—
260#12686—431nomaximal, Sylow
261C22×C43—22no—
262C22×C43—41no—
263C22×C43—42no—
264C22×C43C2×C22⋊C811yescharacteristic
265C22×C43#40711yescharacteristic
266C22×C43#40711yescharacteristic
267C2×D43—21no—
268C2×D43—22no—
269C2×D43—21no—
270C2×D43—42no—
271C2×D43—41no—
272C2×D43—41no—
273C2×D43—81no—
274C2×D43—81no—
275C2×D43—84no—
276C4×C43—22no—
277C4×C43#37911yescharacteristic
278C4×C43C2×C22⋊C811yescharacteristic
279C4×C43C2×C22⋊C811yescharacteristic
280C4×C43#40711yescharacteristic
281Dic43—21no—
282Dic43—22no—
283Dic43—21no—
284C2×#568—431no—
285#2117—431no—
286#2117—431no—
287#2117—431no—
288#2117D4311yescharacteristic, complemented
289C2×#349—431no—
290C2×#349—431no—
291C2×C43—21no—
292C2×C43—21no—
293C2×C43—21no—
294C2×C43—42no—
295C2×C43#177911yescharacteristic
296D43—42no—
297D43—41no—
298D43—41no—
299C22×#41—431no—
300#568—862no—
301#568—862no—
302#568—862no—
303#568—862no—
304C2×#131—431no—
305C2×#131—431no—
306C2×#131—431no—
307C2×#131—861no—
308C2×#131—861no—
309C2×#131C2×D4311yescharacteristic
310C22×M16—431no—
311C22×M16—431no—
312C22×M16—861no—
313C22×M16—861no—
314C2×M32—431no—
315C2×M32—431no—
316C2×M32C2×D4311yescharacteristic, complemented
317C2×M32C2×D4311yescharacteristic, complemented
318#350—22no—
319#350—862no—
320#350—862no—
321#350—862no—
322#349—861no—
323#349—861no—
324#349—861no—
325#349—861no—
326C43#1268611yescharacteristic, complemented, Sylow
327C2×#41—431no—
328C2×#41—431no—
329C2×#41—431no—
330C2×#41—861no—
331C2×#41—861no—
332C2×#41—862no—
333C2×#41—862no—
334C2×#41—862no—
335C2×#41—862no—
336C2×#41C4×D4311yescharacteristic
337C22×Q8—431no—
338C22×D4—431no—
339C22×D4—861no—
340C23×C4—431no—
341C23×C4—861no—
342#131—21no—
343#131—21no—
344#131—432no—
345#131—432no—
346#131—864no—
347#131—864no—
348#131—862no—
349#131—864no—
350#131—864no—
351#131—862no—
352#131—862no—
353#131—862no—
354#131—861no—
355#131—861no—
356#131D17212yes—
357#131C22⋊D4312yes—
358C2×M16—21no—
359C2×M16—431no—
360C2×M16—431no—
361C2×M16—431no—
362C2×M16—431no—
363C2×M16—431no—
364C2×M16—861no—
365C2×M16—861no—
366C2×M16—864no—
367C2×M16—864no—
368C2×M16—861no—
369C2×M16—862no—
370C2×M16—864no—
371C2×M16—864no—
372C2×M16—862no—
373C2×M16—861no—
374C2×M16—861no—
375C2×M16—861no—
376C2×M16—861no—
377C2×M16—861no—
378C2×M16C4×D4311yescharacteristic
379C22×C8—21no—
380C22×C8—431no—
381C22×C8—861no—
382C22×C8—861no—
383C22×C8—861no—
384C22×C8—861no—
385C22×C8C22×D4311yescharacteristic
386M32—21no—
387M32—21no—
388M32—21no—
389M32—21no—
390M32—861no—
391M32—861no—
392M32—861no—
393M32—861no—
394C2×C16—21no—
395C2×C16—21no—
396C2×C16—861no—
397C2×C16—861no—
398C24—431no—
399C24—861no—
400#41—21no—
401#41—21no—
402#41—432no—
403#41—432no—
404#41—864no—
405#41—864no—
406#41—862no—
407#41—864no—
408#41—864no—
409#41—862no—
410#41—862no—
411#41—862no—
412#41—861no—
413#41—861no—
414#41#1062187212yes—
415#41(C2×D43)⋊C412yes—
416C2×Q8—431no—
417C2×Q8—861no—
418C2×Q8—862no—
419C2×Q8—862no—
420C2×Q8C8×D4311yescharacteristic
421C2×D4—21no—
422C2×D4—431no—
423C2×D4—861no—
424C2×D4—861no—
425C2×D4—862no—
426C2×D4—864no—
427C2×D4—864no—
428C2×D4—861no—
429C2×D4—861no—
430C2×D4—862no—
431C2×D4C8×D4311yescharacteristic
432C22×C4—21no—
433C22×C4—431no—
434C22×C4—431no—
435C22×C4—431no—
436C22×C4—431no—
437C22×C4—431no—
438C22×C4—861no—
439C22×C4—861no—
440C22×C4—864no—
441C22×C4—864no—
442C22×C4—861no—
443C22×C4—861no—
444C22×C4—861no—
445C22×C4—861no—
446C22×C4—861no—
447C22×C4—861no—
448C22×C4C2×C4×D4311yescharacteristic
449M16—21no—
450M16—22no—
451M16—22no—
452M16—432no—
453M16—864no—
454M16—864no—
455M16—864no—
456M16—861no—
457M16—861no—
458M16—861no—
459M16—861no—
460M16—861no—
461M16—862no—
462M16—861no—
463M16—861no—
464M16—861no—
465M16—861no—
466M16—862no—
467M16(C2×D43)⋊C412yes—
468C2×C8—21no—
469C2×C8—22no—
470C2×C8—22no—
471C2×C8—431no—
472C2×C8—431no—
473C2×C8—864no—
474C2×C8—864no—
475C2×C8—864no—
476C2×C8—861no—
477C2×C8—861no—
478C2×C8—861no—
479C2×C8—862no—
480C2×C8—862no—
481C2×C8C2×D17211yescharacteristic
482C2×C8D4×D4311yescharacteristic
483C2×C8D4×D4311yescharacteristic
484C2×C8D4×D4311yescharacteristic
485C2×C8D4×D4311yescharacteristic
486C2×C8C2×C22⋊D4311yescharacteristic
487C16—41no—
488C16—41no—
489C16—1721no—
490C16—1721no—
491C23—21no—
492C23—431no—
493C23—431no—
494C23—864no—
495C23—861no—
496C23—861no—
497C23—861no—
498C23—3442no—
499C23—3441no—
500C23M16×D4311yescharacteristic
501Q8—21no—
502Q8—432no—
503Q8—864no—
504Q8—861no—
505Q8(C2×D43)⋊C812yes—
506D4—22no—
507D4—22no—
508D4—21no—
509D4—432no—
510D4—864no—
511D4—864no—
512D4—864no—
513D4—862no—
514D4—862no—
515D4—861no—
516D4(C2×D43)⋊C812yes—
517C2×C4—21no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—431no—
521C2×C4—431no—
522C2×C4—864no—
523C2×C4—864no—
524C2×C4—864no—
525C2×C4—861no—
526C2×C4—861no—
527C2×C4—861no—
528C2×C4—861no—
529C2×C4—861no—
530C2×C4—861no—
531C2×C4—861no—
532C2×C4—861no—
533C2×C4—861no—
534C2×C4—862no—
535C2×C4—861no—
536C2×C4—862no—
537C2×C4—861no—
538C2×C4C22⋊C4×D4311yescharacteristic
539C2×C4C22⋊C4×D4311yescharacteristic
540C2×C4C2×(C2×D43)⋊C411yescharacteristic
541C2×C4C2×C8×D4311yescharacteristic
542C2×C4M16×D4311yescharacteristic
543C2×C4C2×#1062187211yescharacteristic
544C8—22no—
545C8—861no—
546C8—861no—
547C8—862no—
548C8#1306329311yescharacteristic
549C8#1306329311yescharacteristic
550C8#1306329511yescharacteristic
551C8#1306329511yescharacteristic
552C22—22no—
553C22—41no—
554C22—42no—
555C22—861no—
556C22—862no—
557C22—861no—
558C22—3441no—
559C22—3441no—
560C22—3444no—
561C22—1721no—
562C22—1721no—
563C22—1722no—
564C22C22⋊C8×D4311yescharacteristic
565C22#1458286111yescharacteristic
566C22#1458286211yescharacteristic
567C4—22no—
568C4—861no—
569C4—861no—
570C4—862no—
571C4#1458275411yescharacteristic
572C4C22⋊C8×D4311yescharacteristic
573C4C2×(C2×D43)⋊C811yescharacteristic
574C4#1458286111yescharacteristic
575C2—21no—
576C2—21no—
577C2—21no—
578C2—42no—
579C2—1721no—
580C2—1722no—
581C2—1721no—
582C2#1913450411yescharacteristic
subgroupG/H
centerC4#14582754
commutatorC2×C4×C43C22×C8
FrattiniC22×C8C22×D43
FittingC43×#2117C2
socleC2×C43#1779
prime radicalC2×#41×D43C4
cosocleC22×C8×C43C23
hypercenter#2117D43
hypocenterC43#12686
LCSG ▷ C2×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C4 ◁ C22×C8 ◁ #2117

Complements

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

NH ≅ G/Ntype
2. #19136224C2semidirect
3. #19136223C2semidirect
4. #19136222C2semidirect
5. C2×#14582680C2semidirect
6. C2×#14582680C2semidirect
7. C43×#2117C2semidirect
22. C2×#13063265C22semidirect
23. C2×#13063265C22semidirect
32. C2×M32×C43C22semidirect
33. C2×M32×C43C22semidirect
288. #2117D43semidirect
316. C2×M32C2×D43semidirect
317. C2×M32C2×D43semidirect
326. C43#12686semidirect

Maximal subgroups

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

subgroupindex
C2×#145836642
#191362242
#191362232
#191362222
C2×#145826802
C43×#21172
#1268643