Finite Group Databasebeta

← #25102549

Subgroups — #25102549

Order 11008.

1322 conjugacy classes of subgroups.

The subgroup lattice has 554 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
12−1+4×Dic43C211yesmaximal, characteristic
2#137×Dic43C212yesmaximal, complemented
3#19137315C211yesmaximal, characteristic, complemented
4#19134989C211yesmaximal, characteristic, complemented
5#19134966C212yesmaximal, complemented
6#19134830C212yesmaximal, complemented
7#19134812C211yesmaximal, characteristic, complemented
8#19134554C211yesmaximal, characteristic
9#19134407C212yesmaximal, complemented
10#101×Dic43C211yesmaximal, characteristic, complemented
11C2×C43×#456C211yesmaximal, characteristic
12#14583382—22no—
13#14583382—22no—
14#14583382C2212yes—
15#14583377—22no—
16#14583377C2211yescharacteristic
17#41×Dic43—22no—
18#41×Dic43—22no—
19#41×Dic43C2212yes—
20C2×Q8×Dic43—22no—
21C2×Q8×Dic43C2211yescharacteristic
22C2×Q8×Dic43C2211yescharacteristic
23#14583006—22no—
24#14583000—22no—
25Q16×Dic43—22no—
26SD16×Dic43—22no—
27#14582941C2212yes—
28C2×#13063252C2212yes—
29#14582908C2212yes—
30#14582900C2212yes—
31C2×#13063244C2211yescharacteristic
32M16×Dic43C2212yes—
33#14582806C2211yescharacteristic
34#14582806C2211yescharacteristic
35#14582785—22no—
36#14582785—22no—
37#14582765C2211yescharacteristic
38#14582745C2211yescharacteristic
39#14582667C2212yes—
40#14582660C2212yes—
41#14582659C2212yes—
42C2×2−1+4×C43C2211yescharacteristic
43C2×#137×C43C2212yes—
44C2×C4⋊Q8×C43C2211yescharacteristic
45C43×#456C418yescomplemented
46C2×#104×C43C2212yes—
47C2×#101×C43C2211yescharacteristic
48C2×#10621877C2312yes—
49Q8×Dic43—22no—
50Q8×Dic43—24no—
51Q8×Dic43—22no—
52Q8×Dic43—22no—
53Q8×Dic43—22no—
54Q8×Dic43—22no—
55Q8×Dic43—22no—
56Q8×Dic43—22no—
57Q8×Dic43—22no—
58Q8×Dic43—22no—
59Q8×Dic43D412yes—
60Q8×Dic43D412yes—
61D4×Dic43—22no—
62D4×Dic43—24no—
63D4×Dic43—22no—
64D4×Dic43—22no—
65D4×Dic43—22no—
66D4×Dic43—24no—
67D4×Dic43—22no—
68D4×Dic43D412yes—
69D4×Dic43D412yes—
70#13063284—24no—
71#13063284—24no—
72#13063284—22no—
73#13063284—22no—
74#13063284—22no—
75#13063284C2312yes—
76C2×C4⋊Dic43—24no—
77C2×C4⋊Dic43—22no—
78C2×C4⋊Dic43—21no—
79C2×C4⋊Dic43D412yes—
80C2×C4⋊Dic43C2311yescharacteristic
81C2×C4⋊Dic43C2311yescharacteristic
82C2×C4×Dic43—24no—
83C2×C4×Dic43—22no—
84C2×C4×Dic43—21no—
85C2×C4×Dic43D412yes—
86C2×C4×Dic43C2311yescharacteristic
87C2×C4×Dic43C2311yescharacteristic
88#13063259—22no—
89#13063257—22no—
90#13063252—24no—
91#13063249—22no—
92#13063249—22no—
93#13063249—22no—
94#13063246—22no—
95#13063244—22no—
96#13063235—22no—
97C8×Dic43—22no—
98C43⋊#95—22no—
99C43⋊#95—22no—
100C43⋊#95—21no—
101C43⋊#95—21no—
102C43⋊#95—22no—
1032−1+4×C43—24no—
1042−1+4×C43—22no—
1052−1+4×C43C2×C414yes—
106C2×#41×C43—22no—
107C2×#41×C43—22no—
108C2×#41×C43C2312yes—
109C22×Q8×C43—22no—
110C22×Q8×C43C2311yescharacteristic
111C22×Q8×C43C2311yescharacteristic
112#137×C43—24no—
113#137×C43C2×C418yescomplemented
114C2×Q16×C43—22no—
115C2×SD16×C43—22no—
116C2×M16×C43C2312yes—
117C4⋊Q8×C43—22no—
118C4⋊Q8×C43C2×C414yescomplemented
119C2×C4⋊C4×C43—22no—
120C2×C42×C43C2311yescharacteristic
121#104×C43C2×C418yescomplemented
122#101×C43C2×C414yescomplemented
123#41×C43—28no—
124#41×C43—28no—
125#41×C43—28no—
126#41×C43—28no—
127#41×C43—22no—
128#41×C43—22no—
129#41×C43C22×C414yes—
130#41×C43#4114yes—
131C2×Q8×C43—22no—
132C2×Q8×C43—28no—
133C2×Q8×C43—28no—
134C2×Q8×C43—22no—
135C2×Q8×C43—22no—
136C2×Q8×C43—21no—
137C2×Q8×C43—22no—
138C2×Q8×C43—22no—
139C2×Q8×C43C22×C412yes—
140C2×Q8×C43C22×C412yes—
141C2×Q8×C43C2×D412yes—
142C2×Q8×C43#4112yes—
143C2×D4×C43—22no—
144C2×D4×C43—22no—
145C2×D4×C43C2×D412yes—
146C22×C4×C43—24no—
147C22×C4×C43—22no—
148C22×C4×C43—21no—
149C22×C4×C43—21no—
150C22×C4×C43C2×D412yes—
151C22×C4×C43C2411yescharacteristic
152C22×Dic43—24no—
153C22×Dic43—22no—
154C22×Dic43—22no—
155C22×Dic43—21no—
156C22×Dic43—21no—
157C22×Dic43C2×D412yes—
158C22×Dic43C2×D411yescharacteristic
159C22×Dic43C2×D411yescharacteristic
160Q16×C43—28no—
161SD16×C43—28no—
162M16×C43—22no—
163M16×C43C22×C414yes—
164C2×C8×C43—22no—
165C4⋊C4×C43—28no—
166C42×C43—21no—
167C42×C43C22×C412yes—
168C22⋊Dic43—24no—
169C22⋊Dic43—22no—
170C22⋊Dic43—22no—
171C22⋊Dic43—22no—
172C4⋊Dic43—24no—
173C4⋊Dic43—22no—
174C4⋊Dic43—22no—
175C4⋊Dic43—22no—
176C4⋊Dic43—22no—
177C4⋊Dic43—24no—
178C4⋊Dic43—22no—
179C4⋊Dic43—22no—
180C4⋊Dic43—41no—
181C4⋊Dic43—41no—
182C4⋊Dic43C2×D412yes—
183C4⋊Dic43C2×D412yes—
184C4×Dic43—24no—
185C4×Dic43—24no—
186C4×Dic43—22no—
187C4×Dic43—22no—
188C4×Dic43—22no—
189C4×Dic43—22no—
190C4×Dic43—22no—
191C4×Dic43—22no—
192C4×Dic43—42no—
193C4×Dic43C2×D412yes—
194C4×Dic43C2×D412yes—
195#10621877—22no—
196#10621877—22no—
197C2×C43⋊C8—22no—
198C23×C43—22no—
199C23×C43C22×D411yescharacteristic
200Q8×C43—28no—
201Q8×C43—24no—
202Q8×C43—22no—
203Q8×C43—24no—
204Q8×C43—24no—
205Q8×C43—42no—
206Q8×C43C4×D414yes—
207D4×C43—28no—
208D4×C43—24no—
209D4×C43—24no—
210D4×C43—22no—
211D4×C43C4×D414yes—
212C2×C4×C43—216no—
213C2×C4×C43—22no—
214C2×C4×C43—22no—
215C2×C4×C43—22no—
216C2×C4×C43—24no—
217C2×C4×C43—24no—
218C2×C4×C43—22no—
219C2×C4×C43—22no—
220C2×C4×C43—41no—
221C2×C4×C43—41no—
222C2×C4×C43—41no—
223C2×C4×C43—41no—
224C2×C4×C43C4×D414yes—
225C2×C4×C43C23×C411yescharacteristic
226C2×C4×C43C22×D412yes—
227C2×C4×C43C2×#4112yes—
228C2×C4×C43C2×#4111yescharacteristic
229C2×Dic43—22no—
230C2×Dic43—22no—
231C2×Dic43—22no—
232C2×Dic43—41no—
233C2×Dic43—41no—
234C2×Dic43—44no—
235C2×Dic43—42no—
236C2×Dic43—41no—
237C2×Dic43—41no—
238C2×Dic43—42no—
239C2×Dic43—41no—
240C2×Dic43—41no—
241C2×Dic43—82no—
242C2×Dic43—84no—
243C2×Dic43#12011yescharacteristic
244C2×Dic43#12011yescharacteristic
245C2×Dic43#12012yes—
246C8×C43—24no—
247C43⋊C8—42no—
248#21476—431nomaximal, Sylow
249C22×C43—21no—
250C22×C43—21no—
251C22×C43—24no—
252C22×C43—42no—
253C22×C43—42no—
254C22×C43C2×C4×D411yescharacteristic
255C22×C43C2×#12011yescharacteristic
256C22×C43#52511yescharacteristic
257C4×C43—24no—
258C4×C43—24no—
259C4×C43—42no—
260C4×C43—42no—
261C4×C43C2×C4×D412yes—
262C4×C43#52512yes—
263Dic43—21no—
264Dic43—21no—
265Dic43—22no—
266Dic43—42no—
267Dic43—42no—
268Dic43—42no—
269Dic43—81no—
270Dic43—81no—
271C4×2−1+4—431no—
272C2×#456D4311yescharacteristic
273C4×#137—432no—
274#2327—431no—
275#1886—431no—
276#1882—432no—
277#1860—432no—
278#1857—431no—
279#1793—431no—
280#1759—432no—
281C4×#101—431no—
282C2×C43—21no—
283C2×C43—21no—
284C2×C43—44no—
285C2×C43C4×#12011yescharacteristic
286C2×C43#302011yescharacteristic
287C2×C43#302211yescharacteristic
288C2×2−1+4C2×D4311yescharacteristic
289C2×#137C2×D4312yes—
290C2×C4⋊Q8C2×D4311yescharacteristic
291#520—432no—
292#520—862no—
293#520—862no—
294#519—431no—
295#519—862no—
296C4×#41—432no—
297C4×#41—862no—
298C4×#41—862no—
299C2×C4×Q8—431no—
300C2×C4×Q8—431no—
301C2×C4×Q8—862no—
302#456Dic4318yescomplemented
303#441—862no—
304#440—862no—
305C4×Q16—862no—
306C4×SD16—862no—
307#428—432no—
308C2×#104—432no—
309C2×#104C2×D4312yes—
310#419—432no—
311#417—432no—
312C2×#101—431no—
313C2×#101C2×D4311yescharacteristic
314C4×M16—432no—
315#391—431no—
316#391—431no—
317#387—862no—
318#387—862no—
319#382—431no—
320#376—431no—
321#343—432no—
322#339—432no—
323#339—432no—
324C43#2147611yescharacteristic, complemented, Sylow
3252−1+4—24no—
3262−1+4—22no—
3272−1+4C2×Dic4314yes—
328C2×#41—22no—
329C2×#41—22no—
330C2×#41C22×D4312yes—
331C22×Q8—22no—
332C22×Q8C22×D4311yescharacteristic
333C22×Q8C22×D4311yescharacteristic
334#137—24no—
335#137C2×Dic4318yescomplemented
336C2×Q16—22no—
337C2×SD16—22no—
338C2×M16—432no—
339C2×M16C22×D4312yes—
340C4⋊Q8—22no—
341C4⋊Q8C2×Dic4314yescomplemented
342C4×Q8—432no—
343C4×Q8—432no—
344C4×Q8—862no—
345C4×Q8—862no—
346C4×Q8—862no—
347C4×Q8—862no—
348C4×Q8—862no—
349C4×Q8—862no—
350C4×Q8—862no—
351C4×Q8—864no—
352C4×Q8—862no—
353C4×Q8—862no—
354C4×D4—432no—
355C4×D4—432no—
356C4×D4—862no—
357C4×D4—862no—
358C4×D4—864no—
359C4×D4—862no—
360C4×D4—862no—
361C4×D4—864no—
362C4×D4—862no—
363#117—432no—
364#117—862no—
365#117—862no—
366#117—864no—
367#117—864no—
368#117—862no—
369C2×C4⋊C4—22no—
370C2×C4⋊C4—432no—
371C2×C4⋊C4—431no—
372C2×C4⋊C4—431no—
373C2×C4⋊C4—862no—
374C2×C4⋊C4—864no—
375C2×C4⋊C4—861no—
376C2×C42—431no—
377C2×C42—432no—
378C2×C42—431no—
379C2×C42—862no—
380C2×C42—864no—
381C2×C42—861no—
382C2×C42C22×D4311yescharacteristic
383#107—862no—
384#106—862no—
385#104—864no—
386#104C2×Dic4318yescomplemented
387#103—862no—
388#103—862no—
389#103—862no—
390#102—862no—
391#101—862no—
392#101C2×Dic4314yescomplemented
393#97—862no—
394C4×C8—862no—
395#95—861no—
396#95—861no—
397#95—862no—
398#95—862no—
399#95—862no—
400#41—28no—
401#41—28no—
402#41—28no—
403#41—28no—
404#41—22no—
405#41—22no—
406#41#1062190014yes—
407#41C22×Dic4314yes—
408C2×Q8—22no—
409C2×Q8—28no—
410C2×Q8—28no—
411C2×Q8—22no—
412C2×Q8—22no—
413C2×Q8—21no—
414C2×Q8—22no—
415C2×Q8—22no—
416C2×Q8D4×D4312yes—
417C2×Q8#1062190012yes—
418C2×Q8C22×Dic4312yes—
419C2×Q8C22×Dic4312yes—
420C2×D4—22no—
421C2×D4—22no—
422C2×D4D4×D4312yes—
423C22×C4—24no—
424C22×C4—22no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—431no—
428C22×C4—431no—
429C22×C4—432no—
430C22×C4—862no—
431C22×C4—862no—
432C22×C4—864no—
433C22×C4—861no—
434C22×C4—861no—
435C22×C4D4×D4312yes—
436C22×C4C23×D4311yescharacteristic
437Q16—28no—
438SD16—28no—
439M16—22no—
440M16—862no—
441M16—862no—
442M16C22×Dic4314yes—
443C2×C8—22no—
444C2×C8—862no—
445C4⋊C4—28no—
446C4⋊C4—432no—
447C4⋊C4—432no—
448C4⋊C4—864no—
449C4⋊C4—862no—
450C4⋊C4—862no—
451C4⋊C4—864no—
452C4⋊C4—862no—
453C4⋊C4—862no—
454C4⋊C4—862no—
455C4⋊C4—862no—
456C4⋊C4—1721no—
457C4⋊C4—1721no—
458C22⋊C4—862no—
459C22⋊C4—862no—
460C22⋊C4—864no—
461C22⋊C4—862no—
462C42—21no—
463C42—432no—
464C42—432no—
465C42—862no—
466C42—862no—
467C42—862no—
468C42—862no—
469C42—864no—
470C42—864no—
471C42—862no—
472C42—862no—
473C42—1722no—
474C42C22×Dic4312yes—
475C23—22no—
476C23C2×D4×D4311yescharacteristic
477Q8—28no—
478Q8—24no—
479Q8—22no—
480Q8—24no—
481Q8—24no—
482Q8—42no—
483Q8D4×Dic4314yes—
484D4—28no—
485D4—24no—
486D4—24no—
487D4—22no—
488D4D4×Dic4314yes—
489C2×C4—216no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—431no—
502C2×C4—431no—
503C2×C4—432no—
504C2×C4—862no—
505C2×C4—862no—
506C2×C4—862no—
507C2×C4—3442no—
508C2×C4—3444no—
509C2×C4—1721no—
510C2×C4—1721no—
511C2×C4—1724no—
512C2×C4—1721no—
513C2×C4—1721no—
514C2×C4—1722no—
515C2×C4—1721no—
516C2×C4—1721no—
517C2×C4—1722no—
518C2×C4D4×Dic4314yes—
519C2×C4C23×Dic4311yescharacteristic
520C2×C4C2×D4×D4312yes—
521C2×C4C2×#1062190012yes—
522C2×C4C2×#1062190011yescharacteristic
523C8—24no—
524C8—1722no—
525C22—21no—
526C22—21no—
527C22—24no—
528C22—42no—
529C22—42no—
530C22C2×D4×Dic4311yescharacteristic
531C22#120×D4311yescharacteristic
532C22#1458340511yescharacteristic
533C4—24no—
534C4—24no—
535C4—42no—
536C4—42no—
537C4—861no—
538C4—861no—
539C4—862no—
540C4—3441no—
541C4—3441no—
542C4—1722no—
543C4—1722no—
544C4—1722no—
545C4C2×D4×Dic4312yes—
546C4#1458340512yes—
547C2—21no—
548C2—21no—
549C2—44no—
550C2#120×Dic4311yescharacteristic
551C2#1914396011yescharacteristic
552C2#1914397211yescharacteristic
subgroupG/H
centerC22#120×D43
commutatorC2×C4×C43C23×C4
FrattiniC22×C4C23×D43
FittingC2×C43×#456C2
socleC22×C43C2×#120
prime radicalC2×2−1+4×C43C22
cosocleC22×C4×C43C24
hypercenterC2×#456D43
hypocenterC43#21476
LCSG ▷ C2×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#456

Complements

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

NH ≅ G/Ntype
2. #137×Dic43C2semidirect
3. #19137315C2semidirect
4. #19134989C2semidirect
5. #19134966C2semidirect
6. #19134830C2semidirect
7. #19134812C2semidirect
9. #19134407C2semidirect
10. #101×Dic43C2semidirect
45. C43×#456C4semidirect
113. #137×C43C2×C4semidirect
118. C4⋊Q8×C43C2×C4semidirect
121. #104×C43C2×C4semidirect
122. #101×C43C2×C4semidirect
302. #456Dic43semidirect
324. C43#21476semidirect
335. #137C2×Dic43semidirect
341. C4⋊Q8C2×Dic43semidirect
386. #104C2×Dic43semidirect
392. #101C2×Dic43semidirect

Maximal subgroups

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

subgroupindex
2−1+4×Dic432
#137×Dic432
#191373152
#191349892
#191349662
#191348302
#191348122
#191345542
#191344072
#101×Dic432
C2×C43×#4562
#2147643