Finite Group Databasebeta

← #25086816

Subgroups — #25086816

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#19147503C211yesmaximal, characteristic
2Q8×Dic172C211yesmaximal, characteristic, complemented
3SD16×Dic86C211yesmaximal, characteristic
4#19146472C211yesmaximal, characteristic
5#19134272C211yesmaximal, characteristic, complemented
6#19134102C211yesmaximal, characteristic, complemented
7#19134053C211yesmaximal, characteristic, complemented
8#19134000C211yesmaximal, characteristic, complemented
9#19133994C211yesmaximal, characteristic
10#19133819C211yesmaximal, characteristic, complemented
11#19133763C211yesmaximal, characteristic
12#19133737C211yesmaximal, characteristic, complemented
13#19133731C211yesmaximal, characteristic
14#19133447C211yesmaximal, characteristic, complemented
15C43×SD16⋊C8C211yesmaximal, characteristic
16Q8×Dic86C2211yescharacteristic
17Q8×Dic86C2211yescharacteristic
18#14583606C2211yescharacteristic
19D4×Dic86C2211yescharacteristic
20#14583306—21no—
21#14583304—21no—
22#14583304C2211yescharacteristic
23#14583304C2211yescharacteristic
24#14583297C2211yescharacteristic
25#14583296—21no—
26#14583279C2211yescharacteristic
27#14583250C2211yescharacteristic
28#14583250C2211yescharacteristic
29#14583235—21no—
30#14583208—21no—
31#14583195—21no—
32#14583195—21no—
33#14583195C2211yescharacteristic
34#14583195C2211yescharacteristic
35#14583195C2211yescharacteristic
36#14583194C2211yescharacteristic
37#14583189C2211yescharacteristic
38#14583189C2211yescharacteristic
39#14583187—21no—
40#14583184C2211yescharacteristic
41#14583182—21no—
42#14583177—21no—
43#14583140—21no—
44#14583110C2211yescharacteristic
45#14583057—21no—
46#14583029C2211yescharacteristic
47C8×Dic86C2211yescharacteristic
48#14583000—21no—
49C4×Dic172—21no—
50C4×Dic172C2211yescharacteristic
51SD16×Dic43—21no—
52C2×#13063250—21no—
53#14582652C2211yescharacteristic
54#14582647C2211yescharacteristic
55#14582647C2211yescharacteristic
56#14582647C2211yescharacteristic
57#14582646C2211yescharacteristic
58#14582631C2211yescharacteristic
59#14582631C2211yescharacteristic
60C8×Q8×C43C2211yescharacteristic
61C4×SD16×C43C2211yescharacteristic
62C8×D4×C43C2211yescharacteristic
63C43×#335C2211yescharacteristic
64C43×Q8⋊C8C2211yescharacteristic
65C43×D4⋊C8C2211yescharacteristic
66C43×#322C2211yescharacteristic
67C22×Dic86—21no—
68C2×Dic172—21no—
69C2×Dic172—41no—
70C2×Dic172D411yescharacteristic
71#13063325—21no—
72#13063325—21no—
73#13063325C2311yescharacteristic
74#13063325C2311yescharacteristic
75#13063325C2311yescharacteristic
76#13063325C2311yescharacteristic
77#13063324—21no—
78#13063324—21no—
79#13063324—21no—
80#13063324—41no—
81#13063324—41no—
82#13063316—21no—
83#13063304—21no—
84#13063304—21no—
85#13063302—41no—
86#13063302—41no—
87Q8×Dic43—21no—
88Q8×Dic43—21no—
89C4×Dic86—21no—
90C4×Dic86—21no—
91C4×Dic86C2311yescharacteristic
92C4×Dic86C2311yescharacteristic
93D4×Dic43—21no—
94D4×Dic43—21no—
95C2×C4⋊Dic43—21no—
96#13063259—21no—
97#13063259—21no—
98#13063259—21no—
99#13063259—21no—
100#13063259—21no—
101#13063259—21no—
102#13063259—21no—
103#13063259D411yescharacteristic
104#13063258—21no—
105#13063257—21no—
106#13063257—21no—
107#13063256—21no—
108#13063256—21no—
109#13063256—21no—
110C4⋊(C43⋊C8)C2311yescharacteristic
111C4⋊(C43⋊C8)C2311yescharacteristic
112Dic43⋊C8—21no—
113Dic43⋊C8—21no—
114#13063250—21no—
115#13063250—21no—
116#13063250—21no—
117#13063250—21no—
118#13063250—21no—
119#13063250—21no—
120#13063250—41no—
121#13063250D411yescharacteristic
122#13063250D411yescharacteristic
123#13063249—21no—
124#13063249—21no—
125#13063246—21no—
126#13063246—21no—
127#13063235—21no—
128C8×Dic43—21no—
129C2×SD16×C43D411yescharacteristic
130C22×C8×C43—21no—
131C4×Q8×C43C2311yescharacteristic
132C4×D4×C43C2311yescharacteristic
133#106×C43D411yescharacteristic
134C4⋊C8×C43—21no—
135C4⋊C8×C43C2311yescharacteristic
136C4⋊C8×C43C2311yescharacteristic
137#103×C43D411yescharacteristic
138#102×C43D411yescharacteristic
139C22⋊C8×C43—21no—
140C4×C8×C43—21no—
141C4×C8×C43C2311yescharacteristic
142C4×C8×C43C2311yescharacteristic
143C4×C8×C43C2311yescharacteristic
144C2×Q8×C43C2×D411yescharacteristic
145C2×D4×C43C2×D411yescharacteristic
146C22×C4×C43—21no—
147C22×Dic43—41no—
148C22×Dic43—41no—
149C2×Dic86—21no—
150C2×Dic86—21no—
151C2×Dic86—21no—
152C2×Dic86—21no—
153C2×Dic86—42no—
154C2×Dic86—41no—
155C2×Dic86—41no—
156C2×Dic86—41no—
157C2×Dic86—81no—
158C2×Dic86C2×D411yescharacteristic
159C2×Dic86C2×D411yescharacteristic
160SD16×C43Q1614yescomplemented
161C2×C8×C43—21no—
162C2×C8×C43—21no—
163C2×C8×C43—21no—
164C2×C8×C43—81no—
165C2×C8×C43C2×D411yescharacteristic
166C2×C8×C43C2×D411yescharacteristic
167C2×C8×C43C2×D411yescharacteristic
168C2×C8×C43C2×D411yescharacteristic
169C4⋊C4×C43—21no—
170C4⋊C4×C43C2×D411yescharacteristic
171C4⋊C4×C43C2×D411yescharacteristic
172C22⋊C4×C43—21no—
173C42×C43—21no—
174C42×C43C2411yescharacteristic
175C22⋊Dic43—41no—
176C22⋊Dic43—41no—
177C4⋊Dic43—21no—
178C4⋊Dic43—21no—
179C4⋊Dic43—21no—
180C4⋊Dic43—21no—
181C4⋊Dic43—21no—
182C4⋊Dic43—21no—
183C4⋊Dic43—21no—
184C4⋊Dic43—21no—
185C4⋊Dic43—41no—
186C4⋊Dic43—41no—
187C4⋊Dic43—41no—
188C4⋊Dic43C2×D411yescharacteristic
189C4⋊Dic43C2×D411yescharacteristic
190Dic43⋊C4—21no—
191Dic43⋊C4—21no—
192Dic43⋊C4—41no—
193Dic43⋊C4—41no—
194Dic43⋊C4—41no—
195Dic43⋊C4—41no—
196C4×Dic43—21no—
197C4×Dic43—21no—
198C4×Dic43—41no—
199C4×Dic43—41no—
200C2×C43⋊C8—21no—
201C2×C43⋊C8—21no—
202Dic172—22no—
203Dic172—81no—
204Dic172—82no—
205C23×C43—21no—
206Q8×C43—21no—
207Q8×C43C2×Q1612yes—
208D4×C43—21no—
209D4×C43C2×Q1612yes—
210C2×C4×C43—21no—
211C2×C4×C43—21no—
212C2×C4×C43—21no—
213C2×C4×C43—41no—
214C2×C4×C43—41no—
215C2×C4×C43C22×D411yescharacteristic
216C2×C4×C43C22×D411yescharacteristic
217C2×C4×C43H(5,𝔽2)11yescharacteristic
218C2×Dic43—21no—
219C2×Dic43—21no—
220C2×Dic43—21no—
221C2×Dic43—21no—
222C2×Dic43—41no—
223C2×Dic43—41no—
224C2×Dic43—41no—
225C2×Dic43—41no—
226C2×Dic43—82no—
227C2×Dic43—161no—
228Dic86—21no—
229Dic86—21no—
230Dic86—21no—
231Dic86—41no—
232Dic86—41no—
233Dic86—82no—
234Dic86—81no—
235Dic86—81no—
236Dic86—81no—
237Dic86—161no—
238Dic86—161no—
239Dic86#13712yes—
240C8×C43—21no—
241C8×C43—21no—
242C8×C43—21no—
243C8×C43—41no—
244C8×C43—41no—
245C8×C43—41no—
246C8×C43C2×Q1612yes—
247C43⋊C8—41no—
248C43⋊C8—41no—
249#19813—431nomaximal, Sylow
250C22×C43—22no—
251C22×C43—41no—
252C22×C43—41no—
253C22×C43D4211yescharacteristic
254C4×C43—22no—
255C4×C43—21no—
256C4×C43—41no—
257C4×C43—41no—
258C4×C43C22×Q1611yescharacteristic
259C4×C43C2×#13711yescharacteristic
260C4×C43#57711yescharacteristic
261C4×C43#57811yescharacteristic
262Dic43—22no—
263Dic43—41no—
264Dic43—41no—
265Dic43—41no—
266Dic43—81no—
267Dic43—81no—
268Dic43—81no—
269Dic43—81no—
270#3385—431no—
271Q8×Q16—431no—
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⋊C8D4311yescharacteristic
285C2×C43—42no—
286C2×C43#326611yescharacteristic
287C2×C43D4×Q1611yescharacteristic
288C2×C43#328811yescharacteristic
289Q82—431no—
290Q82—431no—
291#554—431no—
292D4×Q8—431no—
293#501—861no—
294#500—431no—
295#500—431no—
296#500—861no—
297#498—431no—
298#498—861no—
299#494—431no—
300#487—431no—
301#487—431no—
302#484—861no—
303#479—861no—
304#477—431no—
305#477—431no—
306#477—431no—
307#477—431no—
308#477—861no—
309#477—861no—
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—431no—
324C4×Q16—861no—
325C4×SD16—861no—
326C4×SD16C2×D4311yescharacteristic
327C8×D4C2×D4311yescharacteristic
328C2×#103—861no—
329#335—431no—
330#335C2×D4311yescharacteristic
331#333—431no—
332#333—431no—
333#333—431no—
334#332—431no—
335Q8⋊C8—431no—
336Q8⋊C8—431no—
337Q8⋊C8C2×D4311yescharacteristic
338D4⋊C8C2×D4311yescharacteristic
339#322C2×D4311yescharacteristic
340C43#1981311yescharacteristic, complemented, Sylow
341C22×Q8—861no—
342C2×Q16—431no—
343C2×Q16—861no—
344C2×Q16—1721no—
345C2×SD16D17211yescharacteristic
346C22×C8—21no—
347C4⋊Q8—431no—
348C4⋊Q8—431no—
349C4⋊Q8—431no—
350C4⋊Q8—431no—
351C4⋊Q8—861no—
352C4⋊Q8—861no—
353C4⋊Q8—861no—
354C4⋊Q8—861no—
355C4⋊Q8—861no—
356C4⋊Q8—1721no—
357C4⋊Q8—1721no—
358#125—861no—
359C22⋊Q8—861no—
360C22⋊Q8—861no—
361C22⋊Q8—1721no—
362C22⋊Q8—1721no—
363C4×Q8—431no—
364C4×Q8—431no—
365C4×Q8—861no—
366C4×Q8—861no—
367C4×Q8—861no—
368C4×Q8—861no—
369C4×Q8C22×D4311yescharacteristic
370C4×D4—861no—
371C4×D4—861no—
372C4×D4C22×D4311yescharacteristic
373C2×C4⋊C4—861no—
374#107—431no—
375#107—861no—
376#107—861no—
377#107—861no—
378#107—861no—
379#107—861no—
380#107—861no—
381#107—861no—
382#107—861no—
383#106—861no—
384#106—861no—
385#106—861no—
386#106—861no—
387#106—861no—
388#106D17211yescharacteristic
389C4⋊C8—21no—
390C4⋊C8—431no—
391C4⋊C8—431no—
392C4⋊C8—861no—
393C4⋊C8—861no—
394C4⋊C8C22×D4311yescharacteristic
395C4⋊C8C22×D4311yescharacteristic
396#103—431no—
397#103—431no—
398#103—861no—
399#103—861no—
400#103—861no—
401#103—861no—
402#103—861no—
403#103—861no—
404#103—861no—
405#103—861no—
406#103—1721no—
407#103D17211yescharacteristic
408#102—861no—
409#102—861no—
410#102D17211yescharacteristic
411C22⋊C8—21no—
412#97—861no—
413C4×C8—21no—
414C4×C8—861no—
415C4×C8C22×D4311yescharacteristic
416C4×C8C22×D4311yescharacteristic
417C4×C8C22×D4311yescharacteristic
418C2×Q8—431no—
419C2×Q8—431no—
420C2×Q8—861no—
421C2×Q8—861no—
422C2×Q8—861no—
423C2×Q8—861no—
424C2×Q8—3441no—
425C2×Q8—1721no—
426C2×Q8—1721no—
427C2×Q8—1721no—
428C2×Q8—1722no—
429C2×Q8C2×D17211yescharacteristic
430C2×D4C2×D17211yescharacteristic
431C22×C4—21no—
432C22×C4—1721no—
433C22×C4—1721no—
434Q16—862no—
435Q16—3441no—
436Q16—3442no—
437SD16Dic17214yescomplemented
438C2×C8—21no—
439C2×C8—21no—
440C2×C8—21no—
441C2×C8—81no—
442C2×C8—861no—
443C2×C8—861no—
444C2×C8C2×D17211yescharacteristic
445C2×C8C2×D17211yescharacteristic
446C2×C8D4×D4311yescharacteristic
447C2×C8D4×D4311yescharacteristic
448C4⋊C4—21no—
449C4⋊C4—431no—
450C4⋊C4—431no—
451C4⋊C4—861no—
452C4⋊C4—861no—
453C4⋊C4—861no—
454C4⋊C4—861no—
455C4⋊C4—861no—
456C4⋊C4—861no—
457C4⋊C4—861no—
458C4⋊C4—861no—
459C4⋊C4—861no—
460C4⋊C4—861no—
461C4⋊C4—1721no—
462C4⋊C4—1721no—
463C4⋊C4—1721no—
464C4⋊C4—1721no—
465C4⋊C4—1721no—
466C4⋊C4—1721no—
467C4⋊C4—1721no—
468C4⋊C4C2×D17211yescharacteristic
469C4⋊C4C2×D17211yescharacteristic
470C22⋊C4—21no—
471C22⋊C4—1721no—
472C22⋊C4—1721no—
473C42—21no—
474C42—861no—
475C42—861no—
476C42—1721no—
477C42—1721no—
478C42C23×D4311yescharacteristic
479C23—21no—
480Q8—21no—
481Q8—432no—
482Q8—861no—
483Q8—861no—
484Q8—861no—
485Q8—3441no—
486Q8—3441no—
487Q8—3441no—
488Q8—3442no—
489Q8—1721no—
490Q8—1721no—
491Q8—6881no—
492Q8—6881no—
493Q8C2×Dic17212yes—
494D4—21no—
495D4C2×Dic17212yes—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—861no—
502C2×C4—861no—
503C2×C4—861no—
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×D17211yescharacteristic
513C2×C4#1306337111yescharacteristic
514C8—21no—
515C8—21no—
516C8—21no—
517C8—41no—
518C8—41no—
519C8—41no—
520C8—1721no—
521C8—1721no—
522C8C2×Dic17212yes—
523C22—22no—
524C22—41no—
525C22—41no—
526C22D4×D17211yescharacteristic
527C4—22no—
528C4—21no—
529C4—41no—
530C4—41no—
531C4—862no—
532C4—3441no—
533C4—3441no—
534C4—3441no—
535C4—3441no—
536C4—1721no—
537C4—1721no—
538C4—1721no—
539C4C22×Dic17211yescharacteristic
540C4#137×D4311yescharacteristic
541C4#1458370911yescharacteristic
542C4#1458371611yescharacteristic
543C2—42no—
544C2#1914610711yescharacteristic
545C2D4×Dic17211yescharacteristic
546C2#1914636211yescharacteristic
subgroupG/H
centerC22D4×D172
commutator, cosocleC42×C43C24
FrattiniC42C23×D43
FittingC43×SD16⋊C8C2
socleC22×C43D42
prime radicalC2×D4×C43C2×D4
hypercenterSD16⋊C8D43
hypocenterC43#19813
LCSG ▷ C42×C43 ▷ C22×C43 ▷ C43
UCSC1 ◁ C22 ◁ C42 ◁ SD16⋊C8

Complements

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

NH ≅ G/Ntype
2. Q8×Dic172C2semidirect
5. #19134272C2semidirect
6. #19134102C2semidirect
7. #19134053C2semidirect
8. #19134000C2semidirect
10. #19133819C2semidirect
12. #19133737C2semidirect
14. #19133447C2semidirect
160. SD16×C43Q16semidirect
340. C43#19813semidirect
437. SD16Dic172semidirect

Maximal subgroups

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

subgroupindex
#191475032
Q8×Dic1722
SD16×Dic862
#191464722
#191342722
#191341022
#191340532
#191340002
#191339942
#191338192
#191337632
#191337372
#191337312
#191334472
C43×SD16⋊C82
#1981343