Finite Group Databasebeta

← #25079377

Subgroups — #25079377

Order 11008.

760 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#19145054C211yesmaximal, characteristic, complemented
2#19144596C211yesmaximal, characteristic
3C2×#14583180C211yesmaximal, characteristic, complemented
4M16×D172C211yesmaximal, characteristic, complemented
5#19134085C212yesmaximal, complemented
6#19134061C212yesmaximal, complemented
7#19133961C212yesmaximal, complemented
8#19133937C212yesmaximal, complemented
9#19133270C211yesmaximal, characteristic, complemented
10#19133266C211yesmaximal, characteristic, complemented
11C43×#1565C211yesmaximal, characteristic, complemented
12C2×#13063358—21no—
13C2×#13063358—21no—
14C2×M16×D43—21no—
15#14583449C2211yescharacteristic
16C2×#13063325C2211yescharacteristic
17C2×C4×D172C2211yescharacteristic
18#14583293—22no—
19#14583279C2212yescomplemented
20#14583272C2212yescomplemented
21#14583213—21no—
22#14583180—21no—
23#14583180—21no—
24#14583180C2211yescharacteristic
25#14583180C2212yes—
26#14583180C2212yes—
27#14583180C2211yescharacteristic
28#14583095C2211yescharacteristic
29#14583095C2211yescharacteristic
30#14583035—21no—
31C4×D344—21no—
32C4×D344—21no—
33#14582978C2212yes—
34#14582972—22no—
35C8×D172C2212yes—
36C2×#13063259—21no—
37#14582923C2211yescharacteristic
38C2×#13063247—21no—
39#14582862—21no—
40#14582644C2212yescomplemented
41#14582644C2212yescomplemented
42#14582644C2212yescomplemented
43#14582642C2212yescomplemented
44C43×#431C2211yescharacteristic
45C2×C4⋊C8×C43C2211yescharacteristic
46C4×M16×C43C2211yescharacteristic
47C43×#335C2212yescomplemented
48C43×#334C2212yescomplemented
49C2×#10621898—21no—
50C22×Dic86—21no—
51C22×D172D411yescharacteristic
52C22×C4×D43—21no—
53#13063358—44no—
54#13063358—44no—
55C2×Dic172—22no—
56C2×Dic172—22no—
57C2×#10621873—22no—
58C2×#10621873—22no—
59C2×D344—21no—
60C2×D344—21no—
61C2×#10621877—21no—
62C2×#10621872—22no—
63M16×D43—44no—
64C2×C8×D43—22no—
65#13063325—21no—
66#13063325—21no—
67#13063325—21no—
68#13063325C2312yes—
69#13063325C2311yescharacteristic
70#13063322C2311yescharacteristic
71#13063313—21no—
72#13063299—21no—
73#13063299—21no—
74C4×D172—21no—
75C4×D172—21no—
76C4×D172—21no—
77C4×D172—21no—
78C4×D172D411yescharacteristic
79C4×D172D411yescharacteristic
80C4×D172C2311yescharacteristic
81C4×D172C2311yescharacteristic
82C2×C4⋊Dic43—21no—
83C2×C4⋊Dic43—21no—
84C2×C4⋊Dic43D411yescharacteristic
85C2×(C2×D43)⋊C4—21no—
86#13063259—22no—
87#13063259—22no—
88#13063259—21no—
89#13063259—21no—
90C4⋊(C43⋊C8)—21no—
91C4⋊(C43⋊C8)C2312yes—
92#13063247—22no—
93#13063247—22no—
94#13063247—21no—
95#13063247—21no—
96#13063247—21no—
97#13063247—21no—
98(C2×D43)⋊C8—22no—
99(C2×D43)⋊C8—22no—
100C2×M16×C43D411yescharacteristic
101C2×M16×C43D411yescharacteristic
102C22×C8×C43—21no—
103C2×C42×C43C2311yescharacteristic
104C4⋊C8×C43—21no—
105C4⋊C8×C43C2311yescharacteristic
106C4⋊C8×C43C2311yescharacteristic
107C4⋊C8×C43C2312yes—
108C22⋊C8×C43—21no—
109#97×C43D412yes—
110C4×C8×C43—21no—
111C4×C8×C43C2312yes—
112C23×D43—21no—
113C22×C4×C43C2×D411yescharacteristic
114C22×C4×C43C2×D411yescharacteristic
115C22×C4×C43#4111yescharacteristic
116C2×C22⋊D43—41no—
117C22×Dic43—21no—
118C22×Dic43—41no—
119#10621898—42no—
120#10621898—42no—
121C2×D172—21no—
122C2×D172—22no—
123C2×D172—21no—
124C2×D172—21no—
125C2×D172—21no—
126C2×D172—41no—
127C2×D172C2×D411yescharacteristic
128C2×D172C2×D411yescharacteristic
129C2×C4×D43—21no—
130C2×C4×D43—21no—
131C2×C4×D43—21no—
132C2×C4×D43—21no—
133C2×C4×D43—42no—
134C2×C4×D43—42no—
135C2×C4×D43—41no—
136C2×Dic86—21no—
137C2×Dic86—22no—
138C2×Dic86—42no—
139C2×Dic86—42no—
140C2×Dic86—41no—
141M16×C43—22no—
142M16×C43D814yescomplemented
143C2×C8×C43—21no—
144C2×C8×C43—21no—
145C2×C8×C43—22no—
146C2×C8×C43—42no—
147C2×C8×C43C2×D412yes—
148C2×C8×C43C2×D412yes—
149C42×C43C2×D411yescharacteristic
150C42×C43C2×D411yescharacteristic
151C42×C43#4111yescharacteristic
152C42×C43C2411yescharacteristic
153(C2×D43)⋊C4—21no—
154(C2×D43)⋊C4—21no—
155(C2×D43)⋊C4—21no—
156(C2×D43)⋊C4—21no—
157(C2×D43)⋊C4—41no—
158(C2×D43)⋊C4—41no—
159C4⋊Dic43—21no—
160C4⋊Dic43—22no—
161C4⋊Dic43—21no—
162C4⋊Dic43—22no—
163C4⋊Dic43—21no—
164C4⋊Dic43—41no—
165C4⋊Dic43—41no—
166C4⋊Dic43C2×D411yescharacteristic
167C4⋊Dic43C2×D411yescharacteristic
168#10621877—42no—
169C2×C43⋊C8—22no—
170Dic172—44no—
171Dic172—44no—
172D344—42no—
173D344—42no—
174#10621873—44no—
175#10621873—44no—
176#10621872—44no—
177C8×D43—44no—
178C23×C43#12111yescharacteristic
179C22×D43—21no—
180C22×D43—21no—
181C22×D43—42no—
182C22×D43—42no—
183C22×D43—41no—
184C22×D43—81no—
185C2×C4×C43—21no—
186C2×C4×C43—21no—
187C2×C4×C43—41no—
188C2×C4×C43#12111yescharacteristic
189C2×C4×C43#12111yescharacteristic
190C2×C4×C43#12111yescharacteristic
191C2×C4×C43C2×D812yes—
192C2×C4×C43AGL(1,ℤ/8)12yes—
193C2×C4×C43C22×D411yescharacteristic
194C2×C4×C43C22×D411yescharacteristic
195C2×C4×C43C2×#4111yescharacteristic
196C22⋊D43—82no—
197C2×Dic43—21no—
198C2×Dic43—21no—
199C2×Dic43—42no—
200C2×Dic43—42no—
201C2×Dic43—41no—
202C2×Dic43—82no—
203D172—21no—
204D172—21no—
205D172—21no—
206D172—42no—
207D172—41no—
208D172—41no—
209D172—41no—
210D172—41no—
211D172—81no—
212D172#13712yescomplemented
213C4×D43—41no—
214C4×D43—41no—
215C4×D43—41no—
216C4×D43—41no—
217C4×D43—42no—
218C4×D43—42no—
219C4×D43—82no—
220Dic86—42no—
221Dic86—41no—
222Dic86—42no—
223Dic86—41no—
224Dic86—82no—
225C8×C43—22no—
226C8×C43—41no—
227C8×C43—41no—
228C8×C43—42no—
229C8×C43C2×D814yes—
230C43⋊C8—42no—
231#19163—431nomaximal, Sylow
232C22×C43—21no—
233C22×C43—21no—
234C22×C43#45912yes—
235C22×C43C2×#12111yescharacteristic
236C2×D43—22no—
237C2×D43—41no—
238C2×D43—41no—
239C2×D43—41no—
240C2×D43—41no—
241C2×D43—41no—
242C2×D43—84no—
243C2×D43—81no—
244C2×D43—81no—
245C2×D43—81no—
246C4×C43—21no—
247C4×C43—21no—
248C4×C43—21no—
249C4×C43#45912yes—
250C4×C43C22×D811yescharacteristic
251C4×C43C2×AGL(1,ℤ/8)11yescharacteristic
252C4×C43C2×#13711yescharacteristic
253C4×C43#57511yescharacteristic
254Dic43—41no—
255Dic43—41no—
256Dic43—82no—
257Dic43—81no—
258#3152—431no—
259#3100—431no—
260C2×#474—431no—
261D4×M16—431no—
262#1689—432no—
263#1685—432no—
264#1669—432no—
265#1665—432no—
266#1565D4311yescharacteristic, complemented
267#1547—431no—
268#1546—431no—
269C2×C43—22no—
270C2×C43C2×#45911yescharacteristic
271C2×C43#305611yescharacteristic
272C2×C43#306911yescharacteristic
273D43—42no—
274D43—81no—
275D43—161no—
276C2×#137—861no—
277C2×#137—861no—
278C22×M16—861no—
279#532—431no—
280C2×C4⋊Q8—431no—
281C2×C4×D4—431no—
282#497—862no—
283#494—432no—
284#492—432no—
285#480—861no—
286#474—431no—
287#474—431no—
288#474—432no—
289#474—432no—
290#474—861no—
291#474—861no—
292#459—431no—
293#459—431no—
294#447—861no—
295C4×D8—861no—
296C4×D8—861no—
297#436—432no—
298#435—862no—
299C8×D4—432no—
300#431C2×D4311yescharacteristic
301C2×#107—861no—
302#423—431no—
303C2×C4⋊C8C2×D4311yescharacteristic
304C2×#102—861no—
305#407—861no—
306C4×M16C2×D4311yescharacteristic
307#335C2×D4312yescomplemented
308#334C2×D4312yescomplemented
309#332—432no—
310#332—432no—
311#332—432no—
312#331—432no—
313C43#1916311yescharacteristic, complemented, Sylow
314C2×#41—861no—
315C22×Q8—861no—
316C22×D4—431no—
317C23×C4—861no—
318#137—1724no—
319#137—1724no—
320C2×Q16—862no—
321C2×Q16—862no—
322C2×SD16—862no—
323C2×SD16—862no—
324C2×D8—861no—
325C2×D8—861no—
326C2×M16—861no—
327C2×M16—862no—
328C2×M16—1724no—
329C2×M16D17211yescharacteristic
330C2×M16D17211yescharacteristic
331C22×C8—21no—
332C22×C8—862no—
333C4⋊Q8—432no—
334C4⋊Q8—431no—
335C4⋊Q8—861no—
336C4⋊Q8—861no—
337C4⋊Q8—861no—
338#127—431no—
339#124—861no—
340#121—861no—
341#121—861no—
342C4×D4—431no—
343C4×D4—431no—
344C4×D4—431no—
345C4×D4—431no—
346C4×D4—861no—
347C4×D4—861no—
348C4×D4—861no—
349C4×D4—861no—
350C2×C4⋊C4—431no—
351C2×C4⋊C4—861no—
352C2×C4⋊C4—861no—
353C2×C22⋊C4—861no—
354C2×C42C22×D4311yescharacteristic
355#107—862no—
356#107—862no—
357#107—861no—
358#107—861no—
359C4⋊C8—21no—
360C4⋊C8—432no—
361C4⋊C8—861no—
362C4⋊C8C22×D4312yes—
363C4⋊C8C22×D4311yescharacteristic
364C4⋊C8C22×D4311yescharacteristic
365#102—861no—
366#102—861no—
367#102—862no—
368#102—862no—
369#102—861no—
370#102—861no—
371C22⋊C8—21no—
372C22⋊C8—862no—
373C22⋊C8—862no—
374#97D17212yes—
375C4×C8—21no—
376C4×C8C22×D4312yes—
377C24—861no—
378#41—1722no—
379#41—1722no—
380C2×Q8—861no—
381C2×Q8—862no—
382C2×Q8—1722no—
383C2×Q8—1722no—
384C2×Q8—1721no—
385C2×D4—431no—
386C2×D4—431no—
387C2×D4—862no—
388C2×D4—861no—
389C2×D4—861no—
390C2×D4—861no—
391C2×D4—861no—
392C2×D4—1721no—
393C2×D4—1721no—
394C22×C4—861no—
395C22×C4—861no—
396C22×C4—861no—
397C22×C4—861no—
398C22×C4—861no—
399C22×C4—1721no—
400C22×C4—1721no—
401C22×C4—1722no—
402C22×C4—1722no—
403C22×C4C2×D17211yescharacteristic
404C22×C4D4×D4311yescharacteristic
405C22×C4#1062190211yescharacteristic
406Q16—1724no—
407Q16—1724no—
408SD16—1724no—
409SD16—1724no—
410D8—1722no—
411D8—1722no—
412M16—22no—
413M16—1722no—
414M16—1724no—
415M16D34414yescomplemented
416C2×C8—21no—
417C2×C8—21no—
418C2×C8—22no—
419C2×C8—42no—
420C2×C8—862no—
421C2×C8—1724no—
422C2×C8C2×D17212yes—
423C2×C8C2×D17212yes—
424C4⋊C4—431no—
425C4⋊C4—431no—
426C4⋊C4—861no—
427C4⋊C4—862no—
428C4⋊C4—861no—
429C4⋊C4—862no—
430C4⋊C4—861no—
431C4⋊C4—1721no—
432C4⋊C4—1721no—
433C22⋊C4—861no—
434C22⋊C4—861no—
435C22⋊C4—861no—
436C22⋊C4—861no—
437C22⋊C4—1721no—
438C22⋊C4—1721no—
439C42C2×D17211yescharacteristic
440C42D4×D4311yescharacteristic
441C42#1062190211yescharacteristic
442C42C23×D4311yescharacteristic
443C23—861no—
444C23—861no—
445C23—3441no—
446C23—1722no—
447C23—1721no—
448C23—1722no—
449C23#1306330011yescharacteristic
450Q8—3442no—
451Q8—1722no—
452Q8—1721no—
453Q8—1722no—
454Q8—1721no—
455D4—432no—
456D4—861no—
457D4—861no—
458D4—861no—
459D4—3441no—
460D4—3442no—
461D4—1722no—
462D4—1721no—
463D4—1721no—
464D4—1721no—
465D4—1721no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—41no—
469C2×C4—861no—
470C2×C4—861no—
471C2×C4—3442no—
472C2×C4—3442no—
473C2×C4—1721no—
474C2×C4—1722no—
475C2×C4—1722no—
476C2×C4—1722no—
477C2×C4—1722no—
478C2×C4—1721no—
479C2×C4—1721no—
480C2×C4—1721no—
481C2×C4—1721no—
482C2×C4#1306330011yescharacteristic
483C2×C4#1306330011yescharacteristic
484C2×C4#1306330011yescharacteristic
485C2×C4C2×D34412yes—
486C2×C4#1306335312yes—
487C2×C4C2×D4×D4311yescharacteristic
488C2×C4C22×D17211yescharacteristic
489C2×C4C2×#1062190211yescharacteristic
490C8—22no—
491C8—41no—
492C8—42no—
493C8—41no—
494C8—1722no—
495C8C2×D34414yes—
496C22—21no—
497C22—21no—
498C22—862no—
499C22—3444no—
500C22—3441no—
501C22—3441no—
502C22—3441no—
503C22—1721no—
504C22—1721no—
505C22—1721no—
506C22—1721no—
507C22—1721no—
508C22#1458309512yes—
509C22C2×#1306330011yescharacteristic
510C4—21no—
511C4—21no—
512C4—21no—
513C4—3441no—
514C4—3442no—
515C4—1721no—
516C4—1721no—
517C4#1458309512yes—
518C4C22×D34411yescharacteristic
519C4C2×#1306335311yescharacteristic
520C4#137×D4311yescharacteristic
521C4#1458369511yescharacteristic
522C2—22no—
523C2—3441no—
524C2—1722no—
525C2—6881no—
526C2C2×#1458309511yescharacteristic
527C2#1914422611yescharacteristic
528C2#1914433311yescharacteristic
subgroupG/H
centerC22C2×#13063300
commutator, cosocleC42×C43C24
FrattiniC42C23×D43
FittingC43×#1565C2
socleC22×C43C2×#121
prime radical#19144596C2
hypercenter#1565D43
hypocenterC43#19163
LCSG ▷ C42×C43 ▷ C22×C43 ▷ C43
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1565

Complements

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

NH ≅ G/Ntype
1. #19145054C2semidirect
3. C2×#14583180C2semidirect
4. M16×D172C2semidirect
5. #19134085C2semidirect
6. #19134061C2semidirect
7. #19133961C2semidirect
8. #19133937C2semidirect
9. #19133270C2semidirect
10. #19133266C2semidirect
11. C43×#1565C2semidirect
19. #14583279C22semidirect
20. #14583272C22semidirect
40. #14582644C22semidirect
41. #14582644C22semidirect
42. #14582644C22semidirect
43. #14582642C22semidirect
47. C43×#335C22semidirect
48. C43×#334C22semidirect
142. M16×C43D8semidirect
212. D172#137semidirect
266. #1565D43semidirect
307. #335C2×D43semidirect
308. #334C2×D43semidirect
313. C43#19163semidirect
415. M16D344semidirect

Maximal subgroups

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

subgroupindex
#191450542
#191445962
C2×#145831802
M16×D1722
#191340852
#191340612
#191339612
#191339372
#191332702
#191332662
C43×#15652
#1916343