Finite Group Databasebeta

← #25086815

Subgroups — #25086815

Order 11008.

580 conjugacy classes of subgroups.

The subgroup lattice has 548 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#19147501C211yesmaximal, characteristic
2Q8×#10621884C211yesmaximal, characteristic
3#19146473C211yesmaximal, characteristic
4#19134271C211yesmaximal, characteristic, complemented
5#19134103C211yesmaximal, characteristic, complemented
6#19134053C211yesmaximal, characteristic, complemented
7#19133997C211yesmaximal, characteristic, complemented
8#19133991C211yesmaximal, characteristic
9#19133816C211yesmaximal, characteristic, complemented
10#19133758C211yesmaximal, characteristic
11#19133735C211yesmaximal, characteristic, complemented
12#19133728C211yesmaximal, characteristic
13#19133446C211yesmaximal, characteristic, complemented
14#19133405C211yesmaximal, characteristic
15Q8×Q16×C43C211yesmaximal, characteristic, complemented
16#14583609C2211yescharacteristic
17Q8×D172C2211yescharacteristic
18#14583305—21no—
19#14583302C2211yescharacteristic
20#14583295—21no—
21#14583295C2211yescharacteristic
22#14583278C2211yescharacteristic
23#14583250C2211yescharacteristic
24#14583249C2211yescharacteristic
25#14583236—21no—
26#14583206—21no—
27#14583194C2211yescharacteristic
28#14583194C2211yescharacteristic
29#14583186—21no—
30#14583186C2211yescharacteristic
31#14583186C2211yescharacteristic
32#14583184—21no—
33#14583184C2211yescharacteristic
34#14583176—21no—
35#14583139—21no—
36#14583109C2211yescharacteristic
37#14583053—21no—
38#14583030C2211yescharacteristic
39Q8×C43⋊C8C2211yescharacteristic
40#14582997—21no—
41C4×#10621884—21no—
42C4×#10621884C2211yescharacteristic
43C8×D172C2211yescharacteristic
44#103×D43—21no—
45#14582652C2211yescharacteristic
46#14582651C2211yescharacteristic
47#14582648C2211yescharacteristic
48#14582648C2211yescharacteristic
49#14582647C2211yescharacteristic
50#14582646C2211yescharacteristic
51Q8⋊(C43⋊C8)C2211yescharacteristic
52Q8⋊(C43⋊C8)C2211yescharacteristic
53Q8⋊(C43⋊C8)C2211yescharacteristic
54#14582626C2211yescharacteristic
55#14582621C2211yescharacteristic
56Q82×C43C2211yescharacteristic
57Q82×C43C2211yescharacteristic
58C43×#500—21no—
59C43×#500C2211yescharacteristic
60C43×#477—21no—
61C43×#477—21no—
62C43×#477C2211yescharacteristic
63C43×#477C2211yescharacteristic
64C8×Q8×C43C2211yescharacteristic
65C4×Q16×C43—21no—
66C4×Q16×C43C2211yescharacteristic
67C2×Q8×D43—21no—
68C2×#10621885—21no—
69C2×#10621885—41no—
70C2×#10621884D411yescharacteristic
71C2×C8×D43—21no—
72#13063325—21no—
73#13063325C2311yescharacteristic
74#13063325C2311yescharacteristic
75#13063317—21no—
76#13063305—21no—
77#13063305—41no—
78#13063303—41no—
79#13063301—21no—
80C4×D172—21no—
81C4×D172—21no—
82C4×D172C2311yescharacteristic
83C4⋊C4×D43—21no—
84#13063258—21no—
85#13063258—21no—
86#13063258—21no—
87#13063258—21no—
88#13063256—21no—
89#13063256—21no—
90#13063256—21no—
91#13063256—21no—
92#13063256—21no—
93#13063256D411yescharacteristic
94C4⋊(C43⋊C8)—21no—
95C4⋊(C43⋊C8)—21no—
96C4⋊(C43⋊C8)C2311yescharacteristic
97C4⋊(C43⋊C8)C2311yescharacteristic
98C4⋊(C43⋊C8)C2311yescharacteristic
99#13063250—21no—
100#13063250—21no—
101#13063249—21no—
102#13063249—21no—
103#13063249—21no—
104#13063249—41no—
105#13063249D411yescharacteristic
106#13063248—21no—
107#13063245—21no—
108#13063245—21no—
109#13063245D411yescharacteristic
110(C2×D43)⋊C8—21no—
111#13063236—21no—
112C4×C43⋊C8—21no—
113C4×C43⋊C8C2311yescharacteristic
114C4×C43⋊C8C2311yescharacteristic
115C2×Q16×C43D411yescharacteristic
116C4⋊Q8×C43—21no—
117C4⋊Q8×C43—21no—
118C4⋊Q8×C43—21no—
119C4⋊Q8×C43—21no—
120C4⋊Q8×C43—41no—
121C4⋊Q8×C43—41no—
122C4⋊Q8×C43C2311yescharacteristic
123C4⋊Q8×C43C2311yescharacteristic
124C4×Q8×C43—21no—
125C4×Q8×C43—21no—
126C4×Q8×C43—21no—
127C4×Q8×C43—21no—
128C4×Q8×C43C2311yescharacteristic
129C4×Q8×C43C2311yescharacteristic
130C4×Q8×C43C2311yescharacteristic
131#107×C43—21no—
132#107×C43—21no—
133#107×C43—21no—
134#107×C43—21no—
135#107×C43D411yescharacteristic
136C4⋊C8×C43—21no—
137C4⋊C8×C43C2311yescharacteristic
138#103×C43—21no—
139#103×C43—21no—
140#103×C43D411yescharacteristic
141#103×C43D411yescharacteristic
142C4×C8×C43—21no—
143C4×C8×C43C2311yescharacteristic
144C2×Q8×C43—21no—
145C2×Q8×C43—21no—
146C2×Q8×C43—41no—
147C2×Q8×C43—41no—
148C2×Q8×C43C2×D411yescharacteristic
149C2×Q8×C43C2×D411yescharacteristic
150C2×Q8×C43C2×D411yescharacteristic
151Q8×D43—42no—
152Q8×D43—81no—
153C2×D172C2×D411yescharacteristic
154C2×C4×D43—21no—
155C2×C4×D43—41no—
156C2×C4×D43—41no—
157C2×Dic86—21no—
158C2×Dic86—21no—
159C2×Dic86—41no—
160Q16×C43—22no—
161C2×C8×C43—21no—
162C2×C8×C43C2×D411yescharacteristic
163C2×C8×C43C2×D411yescharacteristic
164C4⋊C4×C43—21no—
165C4⋊C4×C43—21no—
166C4⋊C4×C43—21no—
167C4⋊C4×C43—21no—
168C4⋊C4×C43—21no—
169C4⋊C4×C43—21no—
170C4⋊C4×C43—21no—
171C4⋊C4×C43—41no—
172C4⋊C4×C43—41no—
173C4⋊C4×C43—41no—
174C4⋊C4×C43—41no—
175C4⋊C4×C43C2×D411yescharacteristic
176C4⋊C4×C43C2×D411yescharacteristic
177C4⋊C4×C43C2×D411yescharacteristic
178C42×C43—21no—
179C42×C43—21no—
180C42×C43—21no—
181C42×C43—41no—
182C42×C43—41no—
183C42×C43C2411yescharacteristic
184#10621885—81no—
185#10621885—82no—
186#10621884Q1614yescomplemented
187(C2×D43)⋊C4—21no—
188(C2×D43)⋊C4—41no—
189(C2×D43)⋊C4—41no—
190C4⋊Dic43—21no—
191C4⋊Dic43—21no—
192C4⋊Dic43—21no—
193C4⋊Dic43—21no—
194C4⋊Dic43—41no—
195C4⋊Dic43C2×D411yescharacteristic
196Dic43⋊C4—41no—
197Dic43⋊C4—41no—
198C2×C43⋊C8—21no—
199C2×C43⋊C8—21no—
200C2×C43⋊C8—21no—
201C2×C43⋊C8—21no—
202C2×C43⋊C8C2×D411yescharacteristic
203C2×C43⋊C8C2×D411yescharacteristic
204C8×D43—81no—
205C22×D43—21no—
206Q8×C43—21no—
207Q8×C43—21no—
208Q8×C43—21no—
209Q8×C43—21no—
210Q8×C43—81no—
211Q8×C43—81no—
212Q8×C43—161no—
213Q8×C43—161no—
214Q8×C43C2×Q1612yes—
215Q8×C43#13712yes—
216C2×C4×C43—21no—
217C2×C4×C43—21no—
218C2×C4×C43—21no—
219C2×C4×C43—21no—
220C2×C4×C43—21no—
221C2×C4×C43—21no—
222C2×C4×C43—41no—
223C2×C4×C43—41no—
224C2×C4×C43C22×D411yescharacteristic
225C2×C4×C43C22×D411yescharacteristic
226C2×C4×C43H(5,𝔽2)11yescharacteristic
227C2×Dic43—21no—
228C2×Dic43—41no—
229C2×Dic43—41no—
230D172—21no—
231D172C2×Q1612yes—
232C4×D43—41no—
233C4×D43—41no—
234C4×D43—82no—
235C4×D43—161no—
236Dic86—41no—
237Dic86—41no—
238Dic86—82no—
239Dic86—81no—
240C8×C43—21no—
241C8×C43—21no—
242C8×C43—41no—
243C43⋊C8—21no—
244C43⋊C8—41no—
245C43⋊C8—41no—
246C43⋊C8—41no—
247C43⋊C8—41no—
248C43⋊C8C2×Q1612yes—
249#19813—431nomaximal, Sylow
250C22×C43D4211yescharacteristic
251C2×D43—22no—
252C2×D43—41no—
253C2×D43—41no—
254C4×C43—22no—
255C4×C43—22no—
256C4×C43—21no—
257C4×C43—41no—
258C4×C43—41no—
259C4×C43—41no—
260C4×C43—41no—
261C4×C43—81no—
262C4×C43—81no—
263C4×C43C22×Q1611yescharacteristic
264C4×C43C2×#13711yescharacteristic
265C4×C43#57711yescharacteristic
266C4×C43#57811yescharacteristic
267Dic43—41no—
268Dic43—81no—
269Dic43—81no—
270#3385—431no—
271Q8×Q16D4311yescharacteristic, complemented
272Q8×SD16—431no—
273#3299—431no—
274#1723—431no—
275#1692—431no—
276#1684—431no—
277#1676—431no—
278#1675—431no—
279#1645—431no—
280#1637—431no—
281#1633—431no—
282#1632—431no—
283#1584—431no—
284SD16⋊C8—431no—
285C2×C43#326611yescharacteristic
286C2×C43D4×Q1611yescharacteristic
287C2×C43#328811yescharacteristic
288D43—42no—
289Q82C2×D4311yescharacteristic
290Q82C2×D4311yescharacteristic
291#554—431no—
292D4×Q8—431no—
293#501—861no—
294#500—21no—
295#500—431no—
296#500C2×D4311yescharacteristic
297#498—431no—
298#498—861no—
299#494—431no—
300#487—431no—
301#487—431no—
302#484—861no—
303#479—861no—
304#477—21no—
305#477—21no—
306#477—431no—
307#477—431no—
308#477C2×D4311yescharacteristic
309#477C2×D4311yescharacteristic
310#476—431no—
311#476—431no—
312#476—861no—
313#475—431no—
314#475—861no—
315#474—861no—
316#467—861no—
317#462—431no—
318#451—861no—
319#446—431no—
320C8×Q8—431no—
321C8×Q8C2×D4311yescharacteristic
322#440—861no—
323C4×Q16—21no—
324C4×Q16C2×D4311yescharacteristic
325C4×SD16—431no—
326C4×SD16—861no—
327C8×D4—431no—
328C2×#103—861no—
329#335—431no—
330#335—431no—
331#333—431no—
332#333—431no—
333#333—431no—
334#332—431no—
335Q8⋊C8—431no—
336Q8⋊C8—431no—
337Q8⋊C8—431no—
338D4⋊C8—431no—
339#322—431no—
340C43#1981311yescharacteristic, complemented, Sylow
341C22×Q8—861no—
342C2×Q16—861no—
343C2×Q16—1721no—
344C2×Q16C22⋊D4311yescharacteristic
345C2×SD16—431no—
346C22×C8—861no—
347C4⋊Q8—21no—
348C4⋊Q8—21no—
349C4⋊Q8—21no—
350C4⋊Q8—21no—
351C4⋊Q8—41no—
352C4⋊Q8—41no—
353C4⋊Q8—431no—
354C4⋊Q8—431no—
355C4⋊Q8—861no—
356C4⋊Q8C22×D4311yescharacteristic
357C4⋊Q8C22×D4311yescharacteristic
358#125—861no—
359C22⋊Q8—861no—
360C22⋊Q8—861no—
361C22⋊Q8—1721no—
362C22⋊Q8—1721no—
363C4×Q8—21no—
364C4×Q8—21no—
365C4×Q8—21no—
366C4×Q8—21no—
367C4×Q8C22×D4311yescharacteristic
368C4×Q8C22×D4311yescharacteristic
369C4×Q8C22×D4311yescharacteristic
370C4×D4—431no—
371C4×D4—861no—
372C4×D4—861no—
373C2×C4⋊C4—861no—
374#107—21no—
375#107—21no—
376#107—21no—
377#107—21no—
378#107—861no—
379#107—861no—
380#107—861no—
381#107—861no—
382#107C22⋊D4311yescharacteristic
383#106—431no—
384#106—861no—
385#106—861no—
386#106—861no—
387#106—861no—
388#106—861no—
389C4⋊C8—21no—
390C4⋊C8—431no—
391C4⋊C8—431no—
392C4⋊C8—431no—
393C4⋊C8—861no—
394C4⋊C8—861no—
395C4⋊C8C22×D4311yescharacteristic
396#103—21no—
397#103—21no—
398#103—431no—
399#103—861no—
400#103—861no—
401#103—861no—
402#103—861no—
403#103—861no—
404#103—861no—
405#103—1721no—
406#103C22⋊D4311yescharacteristic
407#103C22⋊D4311yescharacteristic
408#102—431no—
409#102—861no—
410#102—861no—
411C22⋊C8—861no—
412#97—861no—
413C4×C8—21no—
414C4×C8—431no—
415C4×C8—431no—
416C4×C8—861no—
417C4×C8C22×D4311yescharacteristic
418C2×Q8—21no—
419C2×Q8—21no—
420C2×Q8—41no—
421C2×Q8—41no—
422C2×Q8—861no—
423C2×Q8—861no—
424C2×Q8—3441no—
425C2×Q8—1721no—
426C2×Q8—1722no—
427C2×Q8D4×D4311yescharacteristic
428C2×Q8C2×C22⋊D4311yescharacteristic
429C2×Q8C2×C22⋊D4311yescharacteristic
430C2×D4—431no—
431C22×C4—861no—
432C22×C4—1721no—
433C22×C4—1721no—
434Q16—22no—
435Q16—3441no—
436Q16—3442no—
437SD16—434no—
438C2×C8—21no—
439C2×C8—431no—
440C2×C8—431no—
441C2×C8—861no—
442C2×C8—861no—
443C2×C8—861no—
444C2×C8—861no—
445C2×C8—3441no—
446C2×C8C2×C22⋊D4311yescharacteristic
447C2×C8C2×C22⋊D4311yescharacteristic
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—41no—
456C4⋊C4—41no—
457C4⋊C4—41no—
458C4⋊C4—41no—
459C4⋊C4—431no—
460C4⋊C4—861no—
461C4⋊C4—861no—
462C4⋊C4—861no—
463C4⋊C4—861no—
464C4⋊C4—1721no—
465C4⋊C4—1721no—
466C4⋊C4—1721no—
467C4⋊C4D4×D4311yescharacteristic
468C4⋊C4C2×C22⋊D4311yescharacteristic
469C4⋊C4C2×C22⋊D4311yescharacteristic
470C22⋊C4—861no—
471C22⋊C4—1721no—
472C22⋊C4—1721no—
473C42—21no—
474C42—21no—
475C42—21no—
476C42—41no—
477C42—41no—
478C42C23×D4311yescharacteristic
479C23—861no—
480Q8—21no—
481Q8—21no—
482Q8—21no—
483Q8—21no—
484Q8—81no—
485Q8—81no—
486Q8—161no—
487Q8—161no—
488Q8—3441no—
489Q8—3442no—
490Q8—1721no—
491Q8—1721no—
492Q8Q16×D4312yes—
493Q8#1306335612yes—
494D4—432no—
495D4—861no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—861no—
505C2×C4—3442no—
506C2×C4—1721no—
507C2×C4—1721no—
508C2×C4—1721no—
509C2×C4—1721no—
510C2×C4—6881no—
511C2×C4C2×D4×D4311yescharacteristic
512C2×C4C22×C22⋊D4311yescharacteristic
513C2×C4#1306337011yescharacteristic
514C8—21no—
515C8—21no—
516C8—41no—
517C8—432no—
518C8—861no—
519C8—1721no—
520C8—1721no—
521C8—1721no—
522C8—1721no—
523C22—862no—
524C22—1721no—
525C22—1721no—
526C22D4×C22⋊D4311yescharacteristic
527C4—22no—
528C4—21no—
529C4—22no—
530C4—41no—
531C4—41no—
532C4—41no—
533C4—41no—
534C4—81no—
535C4—81no—
536C4—3441no—
537C4—3441no—
538C4—1721no—
539C4C2×Q16×D4311yescharacteristic
540C4C2×#1306335611yescharacteristic
541C4#1458370711yescharacteristic
542C4#1458371211yescharacteristic
543C2—1722no—
544C2#1914610611yescharacteristic
545C2Q16×C22⋊D4311yescharacteristic
546C2#1914635911yescharacteristic
subgroupG/H
centerC22D4×C22⋊D43
commutator, cosocleC42×C43C24
FrattiniC42C23×D43
FittingQ8×Q16×C43C2
socleC22×C43D42
prime radicalC2×D172C2×D4
hypercenterQ8×Q16D43
hypocenterC43#19813
LCSG ▷ C42×C43 ▷ C22×C43 ▷ C43
UCSC1 ◁ C22 ◁ C42 ◁ Q8×Q16

Complements

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

NH ≅ G/Ntype
4. #19134271C2semidirect
5. #19134103C2semidirect
6. #19134053C2semidirect
7. #19133997C2semidirect
9. #19133816C2semidirect
11. #19133735C2semidirect
13. #19133446C2semidirect
15. Q8×Q16×C43C2semidirect
186. #10621884Q16semidirect
271. Q8×Q16D43semidirect
340. C43#19813semidirect

Maximal subgroups

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

subgroupindex
#191475012
Q8×#106218842
#191464732
#191342712
#191341032
#191340532
#191339972
#191339912
#191338162
#191337582
#191337352
#191337282
#191334462
#191334052
Q8×Q16×C432
#1981343