Finite Group Databasebeta

← #25106422

Subgroups — #25106422

Order 11008.

1098 conjugacy classes of subgroups.

The subgroup lattice has 546 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14583494C211yesmaximal, characteristic
2#19137212C211yesmaximal, characteristic, complemented
3#19137195C211yesmaximal, characteristic, complemented
4#19135580C211yesmaximal, characteristic, complemented
5#19135579C211yesmaximal, characteristic, complemented
6#19135432C211yesmaximal, characteristic
7#19135050C214yesmaximal, complemented
8#19134581C214yesmaximal, complemented
9C43×#1964C211yesmaximal, characteristic
10#14583494—22no—
11#14583494—22no—
12#14583494—22no—
13#14583494—22no—
14C2×#13063318C412yescomplemented
15C2×#13063315C412yescomplemented
16C2×#13063306C412yescomplemented
17C2×#13063306C2212yes—
18C2×#13063304C2211yescharacteristic
19C2×#13063302C412yescomplemented
20C2×D4×Dic43C2212yes—
21C2×#13063281C2212yes—
22C22×Dic43⋊C4C2211yescharacteristic
23C2×#13063240C2214yes—
24#14582796—22no—
25#14582794—22no—
26#14582783—22no—
27#14582781—22no—
28#14582760C2212yes—
29#14582759C2211yescharacteristic
30#14582757C2212yes—
31#14582757C2212yes—
32C4×C22⋊Dic43C2211yescharacteristic
33#14582665C2214yes—
34#14582665C2214yes—
35C2×#121×C43C2211yescharacteristic
36C2×#99×C43C2214yes—
37C43×#380C2211yescharacteristic
38C4×C22⋊C4×C43C2211yescharacteristic
39C22×Dic86C2×C411yescharacteristic
40C23×Dic43C2×C412yes—
41C23×Dic43C2311yescharacteristic
42C23×Dic43C2311yescharacteristic
43#13063318—24no—
44#13063318—24no—
45#13063315—24no—
46#13063315—24no—
47#13063306—24no—
48#13063306—24no—
49#13063306—24no—
50#13063306—24no—
51#13063304—21no—
52#13063304—21no—
53#13063304—21no—
54#13063304—21no—
55#13063302—24no—
56#13063302—24no—
57D4×Dic43—24no—
58D4×Dic43—24no—
59#13063281—44no—
60C2×C4⋊Dic43—21no—
61C2×C4⋊Dic43C2×C412yes—
62C2×C4⋊Dic43C2×C411yescharacteristic
63C2×Dic43⋊C4—22no—
64C2×Dic43⋊C4—22no—
65C2×Dic43⋊C4—22no—
66C2×Dic43⋊C4C2×C412yes—
67C2×Dic43⋊C4C2×C412yes—
68C2×Dic43⋊C4C2×C412yes—
69C2×Dic43⋊C4C2×C412yes—
70C2×Dic43⋊C4C2×C412yes—
71C2×Dic43⋊C4D412yes—
72C2×Dic43⋊C4D412yes—
73C2×Dic43⋊C4D412yes—
74C2×Dic43⋊C4D412yes—
75C2×C22⋊Dic43C2×C412yes—
76C2×C22⋊Dic43C2×C412yes—
77C2×C22⋊Dic43C2×C412yes—
78C2×C22⋊Dic43C2312yes—
79C2×C22⋊Dic43C2312yes—
80C2×C22⋊Dic43C2312yes—
81C2×C4×Dic43—22no—
82C2×C4×Dic43—21no—
83C2×C4×Dic43C2×C412yes—
84#13063240—44no—
85C43⋊#95—21no—
86C43⋊#95—21no—
87C43⋊#95—22no—
88C43⋊#95—22no—
89C43⋊#95—24no—
90C22×D4×C43C2×C411yescharacteristic
91C22×D4×C43C2312yes—
92C23×C4×C43C2311yescharacteristic
93#121×C43—21no—
94#121×C43—21no—
95#121×C43—21no—
96#121×C43—21no—
97C2×C4⋊C4×C43—21no—
98C2×C4⋊C4×C43C2×C411yescharacteristic
99C2×C22⋊C4×C43C2×C412yes—
100C2×C22⋊C4×C43C2312yes—
101C2×C22⋊C4×C43C2312yes—
102C2×C42×C43—21no—
103#99×C43—44no—
104#95×C43—21no—
105#95×C43—21no—
106C24×C43C22×C412yes—
107C24×C43C2411yescharacteristic
108C2×D4×C43—21no—
109C2×D4×C43—21no—
110C2×D4×C43—22no—
111C2×D4×C43—22no—
112C2×D4×C43—24no—
113C2×D4×C43—24no—
114C2×D4×C43—81no—
115C22×C4×C43—22no—
116C22×C4×C43—22no—
117C22×C4×C43—81no—
118C22×C4×C43C22×C412yes—
119C22×C4×C43C22×C411yescharacteristic
120C22×C4×C43C22×C411yescharacteristic
121C22×C4×C43C2×D411yescharacteristic
122C22×C4×C43C2×D411yescharacteristic
123C22×C4×C43C2×D411yescharacteristic
124C22×C4×C43C2×D411yescharacteristic
125C22×Dic43—24no—
126C22×Dic43—22no—
127C22×Dic43—22no—
128C22×Dic43—22no—
129C22×Dic43—22no—
130C22×Dic43—81no—
131C22×Dic43—82no—
132C22×Dic43C22×C412yes—
133C22×Dic43C22×C412yes—
134C22×Dic43C22×C412yes—
135C22×Dic43C22×C412yes—
136C22×Dic43C2×D412yes—
137C22×Dic43C2×D412yes—
138C22×Dic43C2×D412yes—
139C22×Dic43C2×D412yes—
140C2×Dic86—21no—
141C2×Dic86—21no—
142C2×Dic86—81no—
143C4⋊C4×C43—21no—
144C4⋊C4×C43—21no—
145C4⋊C4×C43—81no—
146C22⋊C4×C43—22no—
147C22⋊C4×C43—22no—
148C22⋊C4×C43—22no—
149C22⋊C4×C43—24no—
150C22⋊C4×C43—22no—
151C42×C43—81no—
152C22⋊Dic43—24no—
153C22⋊Dic43—24no—
154C22⋊Dic43—24no—
155C22⋊Dic43—22no—
156C22⋊Dic43—22no—
157C22⋊Dic43—22no—
158C22⋊Dic43—24no—
159C22⋊Dic43—22no—
160C4⋊Dic43—21no—
161C4⋊Dic43—21no—
162C4⋊Dic43—24no—
163C4⋊Dic43—81no—
164Dic43⋊C4—22no—
165Dic43⋊C4—22no—
166Dic43⋊C4—24no—
167Dic43⋊C4—24no—
168Dic43⋊C4—44no—
169Dic43⋊C4—82no—
170Dic43⋊C4—82no—
171Dic43⋊C4C22⋊C414yes—
172Dic43⋊C4C22⋊C414yes—
173Dic43⋊C4C22⋊C414yes—
174Dic43⋊C4C22⋊C414yes—
175C4×Dic43—24no—
176C4×Dic43—44no—
177C4×Dic43—81no—
178C23×C43—22no—
179C23×C43—24no—
180C23×C43—41no—
181C23×C43—41no—
182C23×C43—42no—
183C23×C43—42no—
184C23×C43C23×C411yescharacteristic
185C23×C43C22×D411yescharacteristic
186C23×C43C22×D411yescharacteristic
187C23×C43H(5,𝔽2)12yes—
188C23×C432−1+412yes—
189D4×C43—42no—
190D4×C43—42no—
191D4×C43—44no—
192D4×C43—81no—
193D4×C43—81no—
194C2×C4×C43—22no—
195C2×C4×C43—24no—
196C2×C4×C43—41no—
197C2×C4×C43—41no—
198C2×C4×C43—41no—
199C2×C4×C43—41no—
200C2×C4×C43—41no—
201C2×C4×C43—41no—
202C2×C4×C43—42no—
203C2×C4×C43—42no—
204C2×C4×C43—42no—
205C2×C4×C43—42no—
206C2×C4×C43—42no—
207C2×C4×C43—42no—
208C2×C4×C43C2×C22⋊C411yescharacteristic
209C2×C4×C43C2×C22⋊C411yescharacteristic
210C2×C4×C43C2×C22⋊C411yescharacteristic
211C2×C4×C43C2×C22⋊C411yescharacteristic
212C2×Dic43—24no—
213C2×Dic43—22no—
214C2×Dic43—22no—
215C2×Dic43—24no—
216C2×Dic43—22no—
217C2×Dic43—22no—
218C2×Dic43—42no—
219C2×Dic43—42no—
220C2×Dic43—42no—
221C2×Dic43—42no—
222C2×Dic43—42no—
223C2×Dic43—42no—
224C2×Dic43—42no—
225C2×Dic43—42no—
226C2×Dic43—42no—
227C2×Dic43—42no—
228C2×Dic43—42no—
229C2×Dic43—84no—
230C2×Dic43C2×C22⋊C414yes—
231C2×Dic43C2×C22⋊C414yes—
232Dic86—81no—
233Dic86—81no—
234#21877—431nomaximal, Sylow
235C22×C43—21no—
236C22×C43—21no—
237C22×C43—21no—
238C22×C43—21no—
239C22×C43—21no—
240C22×C43—21no—
241C22×C43—42no—
242C22×C43—42no—
243C22×C43—42no—
244C22×C43—42no—
245C22×C43—42no—
246C22×C43—84no—
247C22×C43C22×C22⋊C411yescharacteristic
248C22×C43#52012yes—
249C22×C43#53511yescharacteristic
250C22×C43#53611yescharacteristic
251C22×C43#53712yes—
252C4×C43—21no—
253C4×C43—21no—
254C4×C43—21no—
255C4×C43—21no—
256C4×C43—42no—
257C4×C43—82no—
258C4×C43—82no—
259Dic43—24no—
260Dic43—24no—
261Dic43—42no—
262Dic43—42no—
263Dic43—82no—
264Dic43—82no—
265C2×#539—431no—
266#2316—431no—
267#2313—431no—
268#1996—431no—
269#1996—431no—
270#1964—431no—
271#1964D4311yescharacteristic
272#1899—434no—
273#1799—434no—
274C2×C43—21no—
275C2×C43—22no—
276C2×C43—22no—
277C2×C43—21no—
278C2×C43—82no—
279C2×C43#231111yescharacteristic
280C2×C43#288311yescharacteristic
281C2×C43#288411yescharacteristic
282#539—862no—
283#539—862no—
284#539—862no—
285#539—862no—
286C2×#126—432no—
287C2×#125—432no—
288C2×#123—432no—
289C2×#123—432no—
290C2×C22⋊Q8—431no—
291C2×C22⋊Q8—432no—
292C2×#121C2×D4311yescharacteristic
293C2×C4×D4—432no—
294C2×#117—432no—
295C22×C4⋊C4—431no—
296C2×#99—434no—
297C2×#99C2×D4314yes—
298#389—862no—
299#389—862no—
300#387—862no—
301#387—862no—
302#381—432no—
303#380—431no—
304#380—432no—
305#380—432no—
306#380C2×D4311yescharacteristic
307C4×C22⋊C4—431no—
308C4×C22⋊C4C2×D4311yescharacteristic
309#342—434no—
310#342—434no—
311C43#2187711yescharacteristic, complemented, Sylow
312C22×Q8—431no—
313C22×D4C4×D4311yescharacteristic
314C22×D4C22×D4312yes—
315C23×C4—431no—
316C23×C4—431no—
317C23×C4—432no—
318C23×C4C22×D4311yescharacteristic
319#126—864no—
320#126—864no—
321#125—864no—
322#125—864no—
323#123—864no—
324#123—864no—
325#123—864no—
326#123—864no—
327C22⋊Q8—864no—
328C22⋊Q8—864no—
329C22⋊Q8—861no—
330C22⋊Q8—861no—
331C22⋊Q8—861no—
332C22⋊Q8—861no—
333#121—21no—
334#121—21no—
335#121—21no—
336#121—21no—
337C4×D4—864no—
338C4×D4—864no—
339#117—1724no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—432no—
342C2×C4⋊C4—432no—
343C2×C4⋊C4—432no—
344C2×C4⋊C4—432no—
345C2×C4⋊C4—432no—
346C2×C4⋊C4—432no—
347C2×C4⋊C4—432no—
348C2×C4⋊C4—432no—
349C2×C4⋊C4—432no—
350C2×C4⋊C4—431no—
351C2×C4⋊C4—432no—
352C2×C4⋊C4—862no—
353C2×C4⋊C4—862no—
354C2×C4⋊C4—861no—
355C2×C4⋊C4—862no—
356C2×C4⋊C4C4×D4311yescharacteristic
357C2×C22⋊C4—432no—
358C2×C22⋊C4—432no—
359C2×C22⋊C4—432no—
360C2×C22⋊C4—432no—
361C2×C22⋊C4—432no—
362C2×C22⋊C4—432no—
363C2×C22⋊C4C4×D4312yes—
364C2×C22⋊C4C22×D4312yes—
365C2×C22⋊C4C22×D4312yes—
366C2×C42—21no—
367C2×C42—432no—
368C2×C42—861no—
369C2×C42—862no—
370#99—44no—
371#99—1724no—
372#95—21no—
373#95—21no—
374#95—862no—
375#95—862no—
376#95—864no—
377#95—861no—
378#95—861no—
379C24C2×C4×D4312yes—
380C24C23×D4311yescharacteristic
381C2×Q8—861no—
382C2×Q8—861no—
383C2×Q8—3441no—
384C2×D4—22no—
385C2×D4—22no—
386C2×D4—24no—
387C2×D4—24no—
388C2×D4—21no—
389C2×D4—21no—
390C2×D4—81no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—81no—
394C22×C4—432no—
395C22×C4—432no—
396C22×C4—432no—
397C22×C4—432no—
398C22×C4—432no—
399C22×C4—432no—
400C22×C4—432no—
401C22×C4—432no—
402C22×C4—864no—
403C22×C4—862no—
404C22×C4—862no—
405C22×C4—862no—
406C22×C4—862no—
407C22×C4—3442no—
408C22×C4—3441no—
409C22×C4C2×C4×D4312yes—
410C22×C4C2×C4×D4311yescharacteristic
411C22×C4C2×C4×D4311yescharacteristic
412C22×C4D4×D4311yescharacteristic
413C22×C4D4×D4311yescharacteristic
414C22×C4D4×D4311yescharacteristic
415C22×C4D4×D4311yescharacteristic
416C4⋊C4—21no—
417C4⋊C4—21no—
418C4⋊C4—81no—
419C4⋊C4—434no—
420C4⋊C4—434no—
421C4⋊C4—434no—
422C4⋊C4—434no—
423C4⋊C4—864no—
424C4⋊C4—864no—
425C4⋊C4—864no—
426C4⋊C4—862no—
427C4⋊C4—862no—
428C4⋊C4—861no—
429C4⋊C4—861no—
430C4⋊C4—3442no—
431C4⋊C4—3442no—
432C4⋊C4—3441no—
433C4⋊C4—1724no—
434C22⋊C4—22no—
435C22⋊C4—22no—
436C22⋊C4—22no—
437C22⋊C4—24no—
438C22⋊C4—22no—
439C22⋊C4—864no—
440C22⋊C4—864no—
441C22⋊C4—864no—
442C22⋊C4—862no—
443C22⋊C4—862no—
444C22⋊C4—862no—
445C22⋊C4—864no—
446C22⋊C4—862no—
447C42—81no—
448C42—864no—
449C42—3441no—
450C42—1724no—
451C23—22no—
452C23—24no—
453C23—41no—
454C23—41no—
455C23—42no—
456C23—42no—
457C23C22×C4×D4311yescharacteristic
458C23C2×D4×D4311yescharacteristic
459C23C2×D4×D4311yescharacteristic
460C23#1306337012yes—
461C23#1306337212yes—
462Q8—3441no—
463Q8—3441no—
464D4—42no—
465D4—42no—
466D4—44no—
467D4—81no—
468D4—81no—
469C2×C4—22no—
470C2×C4—24no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—434no—
484C2×C4—434no—
485C2×C4—864no—
486C2×C4—862no—
487C2×C4—862no—
488C2×C4—864no—
489C2×C4—862no—
490C2×C4—862no—
491C2×C4—3444no—
492C2×C4—1722no—
493C2×C4—1722no—
494C2×C4—1722no—
495C2×C4—1722no—
496C2×C4—1722no—
497C2×C4—1722no—
498C2×C4—1722no—
499C2×C4—1722no—
500C2×C4—1722no—
501C2×C4—1722no—
502C2×C4—1722no—
503C2×C4C22⋊C4×D4311yescharacteristic
504C2×C4C22⋊C4×D4311yescharacteristic
505C2×C4C22⋊C4×D4311yescharacteristic
506C2×C4C22⋊C4×D4311yescharacteristic
507C22—21no—
508C22—21no—
509C22—21no—
510C22—21no—
511C22—21no—
512C22—21no—
513C22—42no—
514C22—42no—
515C22—42no—
516C22—42no—
517C22—42no—
518C22—84no—
519C22C2×C22⋊C4×D4311yescharacteristic
520C22#1458337812yes—
521C22#1458346211yescharacteristic
522C22#1458346911yescharacteristic
523C22#1458347812yes—
524C4—21no—
525C4—21no—
526C4—21no—
527C4—21no—
528C4—42no—
529C4—82no—
530C4—82no—
531C4—864no—
532C4—864no—
533C4—3442no—
534C4—3442no—
535C4—1722no—
536C4—1722no—
537C2—21no—
538C2—22no—
539C2—22no—
540C2—21no—
541C2—82no—
542C2#1913718511yescharacteristic
543C2#1914305311yescharacteristic
544C2#1914305711yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D43
commutatorC23×C43C23×C4
FrattiniC24C23×D43
FittingC43×#1964C2
socleC22×C43C22×C22⋊C4
prime radicalC2×#121×C43C22
cosocleC24×C43C24
hypercenter#1964D43
hypocenterC43#21877
LCSG ▷ C23×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1964

Complements

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

NH ≅ G/Ntype
2. #19137212C2semidirect
3. #19137195C2semidirect
4. #19135580C2semidirect
5. #19135579C2semidirect
7. #19135050C2semidirect
8. #19134581C2semidirect
14. C2×#13063318C4semidirect
15. C2×#13063315C4semidirect
16. C2×#13063306C4semidirect
19. C2×#13063302C4semidirect
311. C43#21877semidirect

Maximal subgroups

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

subgroupindex
C2×#145834942
#191372122
#191371952
#191355802
#191355792
#191354322
#191350502
#191345812
C43×#19642
#2187743