Finite Group Databasebeta

← #34258267

Subgroups — #34258267

Order 20224.

632 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23773467C211yesmaximal, characteristic
2C8×#11929623C211yesmaximal, characteristic
3#23772329C211yesmaximal, characteristic, complemented
4#23762274C211yesmaximal, characteristic, complemented
5#23762270C211yesmaximal, characteristic, complemented
6#23762258C211yesmaximal, characteristic, complemented
7#23762254C211yesmaximal, characteristic, complemented
8C8×Dic316C211yesmaximal, characteristic, complemented
9C8×#11929614C211yesmaximal, characteristic, complemented
10C8×#11929612C211yesmaximal, characteristic, complemented
11#23762187C211yesmaximal, characteristic, complemented
12#23761836C211yesmaximal, characteristic
13#23761819C211yesmaximal, characteristic, complemented
14C8×#11929607C211yesmaximal, characteristic, complemented
15C8×SD16×C79C211yesmaximal, characteristic, complemented
16#19099117C2211yescharacteristic
17C4×#11929623C2211yescharacteristic
18#19098907C411yescharacteristic
19#19098903C411yescharacteristic
20#19098847C411yescharacteristic
21#19098841C411yescharacteristic
22#19098772C411yescharacteristic
23#19098766C411yescharacteristic
24#19098726C411yescharacteristic
25#19098722C411yescharacteristic
26#19098694C2211yescharacteristic
27#19098692—21no—
28#19098690C2211yescharacteristic
29C8×Dic158—21no—
30C8×Dic158C2211yescharacteristic
31C8×Dic158C2211yescharacteristic
32#19098687C2211yescharacteristic
33C4×Dic316C2211yescharacteristic
34C4×#11929614C2211yescharacteristic
35C4×#11929612C2211yescharacteristic
36D4×C79⋊C8C2211yescharacteristic
37C8×C22⋊D79—21no—
38(C22⋊D79)⋊C8—21no—
39#19098615C2211yescharacteristic
40#19098614C2211yescharacteristic
41#19098613—21no—
42#19098588—21no—
43#19098585C2211yescharacteristic
44#19098517—21no—
45C4×#11929607C2211yescharacteristic
46C2×C8×Dic79—21no—
47C4×C8×D79C2211yescharacteristic
48#19098316C2211yescharacteristic
49#19098315C2211yescharacteristic
50#19098313C2211yescharacteristic
51#19098295C2211yescharacteristic
52#19098295C2211yescharacteristic
53Q8⋊(C79⋊C8)C2211yescharacteristic
54#19098293C2211yescharacteristic
55#19098293C2211yescharacteristic
56D4⋊(C79⋊C8)C2211yescharacteristic
57#19098285C2211yescharacteristic
58#19098284C2211yescharacteristic
59C8×C79⋊C8C2211yescharacteristic
60C8×Q8×C79C2211yescharacteristic
61C4×SD16×C79C2211yescharacteristic
62C8×D4×C79C2211yescharacteristic
63#334×C79C2211yescharacteristic
64Q8⋊C8×C79C2211yescharacteristic
65D4⋊C8×C79C2211yescharacteristic
66C82×C79C2211yescharacteristic
67C2×#11929623C2×C411yescharacteristic
68C2×Dic316C2×C411yescharacteristic
69C2×#11929614C2×C411yescharacteristic
70C2×#11929612C2×C411yescharacteristic
71C2×#11929607—21no—
72C2×#11929607—21no—
73C22×C79⋊C8—21no—
74C2×C8×D79D411yescharacteristic
75C2×C8×D79D411yescharacteristic
76#14579160C2×C411yescharacteristic
77#14579159C2×C411yescharacteristic
78#14579151—21no—
79#14579141C2×C411yescharacteristic
80#14579139—21no—
81#14579137C2×C411yescharacteristic
82C4×Dic158—21no—
83C4×Dic158C2311yescharacteristic
84C4×Dic158C2311yescharacteristic
85#14579126C2×C411yescharacteristic
86D4×Dic79C2×C411yescharacteristic
87C4×C22⋊D79—21no—
88(C22⋊D79)⋊C4—21no—
89#14579119C2×C411yescharacteristic
90#14579118—21no—
91#14579117—21no—
92C2×C4×Dic79—21no—
93C42×D79C2311yescharacteristic
94#14579095C2×C411yescharacteristic
95#14579094C2×C411yescharacteristic
96#14579092C2×C411yescharacteristic
97C4⋊(C79⋊C8)—21no—
98C4⋊(C79⋊C8)C2311yescharacteristic
99C4⋊(C79⋊C8)C2311yescharacteristic
100Dic79⋊C8—21no—
101Dic79⋊C8—21no—
102Dic79⋊C8—21no—
103Dic79⋊C8—21no—
104Dic79⋊C8—21no—
105Dic79⋊C8C2×C411yescharacteristic
106Dic79⋊C8C2×C411yescharacteristic
107#14579086C2×C411yescharacteristic
108#14579086C2×C411yescharacteristic
109#14579085C2×C411yescharacteristic
110#14579084C2×C411yescharacteristic
111#14579084C2×C411yescharacteristic
112#14579082C2×C411yescharacteristic
113C22⋊(C79⋊C8)—21no—
114(C2×D79)⋊C8—21no—
115(C2×D79)⋊C8C2×C411yescharacteristic
116(C2×D79)⋊C8C2×C411yescharacteristic
117#14579072—21no—
118#14579071—21no—
119#14579071—21no—
120C4×C79⋊C8C2311yescharacteristic
121C4×C79⋊C8C2311yescharacteristic
122C4×C79⋊C8C2311yescharacteristic
123C8×Dic79—21no—
124C8×Dic79—21no—
125C8×Dic79—21no—
126C8×Dic79—21no—
127C8×Dic79D411yescharacteristic
128C8×Dic79D411yescharacteristic
129C2×SD16×C79C2×C411yescharacteristic
130C22×C8×C79—21no—
131C4×Q8×C79C2311yescharacteristic
132C4×D4×C79C2311yescharacteristic
133#106×C79C2×C411yescharacteristic
134C4⋊C8×C79—21no—
135C4⋊C8×C79C2311yescharacteristic
136C4⋊C8×C79C2311yescharacteristic
137#103×C79C2×C411yescharacteristic
138#102×C79C2×C411yescharacteristic
139C22⋊C8×C79—21no—
140C4×C8×C79—21no—
141C4×C8×C79C2311yescharacteristic
142C4×C8×C79C2311yescharacteristic
143C4×C8×C79C2311yescharacteristic
144#11929623—22no—
145#11929623—22no—
146C2×Dic158—21no—
147C2×Dic158C22×C411yescharacteristic
148C2×Dic158C22×C411yescharacteristic
149C2×C22⋊D79—21no—
150C22×Dic79—21no—
151C22×Dic79—21no—
152C2×C4×D79C22×C411yescharacteristic
153C2×C4×D79C2×D411yescharacteristic
154C2×C4×D79#4111yescharacteristic
155Dic316—22no—
156#11929614—22no—
157#11929612—22no—
158#11929607—24no—
159#11929607—42no—
160C2×C79⋊C8—21no—
161C2×C79⋊C8—21no—
162C2×C79⋊C8—21no—
163C2×C79⋊C8—42no—
164C2×C79⋊C8C22×C411yescharacteristic
165C2×C79⋊C8C22×C411yescharacteristic
166C2×C79⋊C8C2×D411yescharacteristic
167C2×C79⋊C8C2×D411yescharacteristic
168C8×D79—22no—
169C8×D79—41no—
170C4⋊Dic79—21no—
171C4⋊Dic79C22×C411yescharacteristic
172C4⋊Dic79C22×C411yescharacteristic
173Dic79⋊C4—21no—
174Dic79⋊C4—21no—
175Dic79⋊C4—21no—
176Dic79⋊C4—21no—
177Dic79⋊C4—21no—
178Dic79⋊C4—21no—
179Dic79⋊C4—21no—
180C22⋊Dic79—21no—
181(C2×D79)⋊C4—21no—
182(C2×D79)⋊C4—21no—
183(C2×D79)⋊C4—21no—
184C4×Dic79—21no—
185C4×Dic79—21no—
186C4×Dic79—21no—
187C4×Dic79—21no—
188C4×Dic79C22×C411yescharacteristic
189C4×Dic79C2×D411yescharacteristic
190C4×Dic79#4111yescharacteristic
191C2×Q8×C79C22×C411yescharacteristic
192C2×D4×C79C22×C411yescharacteristic
193C22×C4×C79—21no—
194SD16×C79—22no—
195C2×C8×C79—21no—
196C2×C8×C79—21no—
197C2×C8×C79—21no—
198C2×C8×C79—42no—
199C2×C8×C79C22×C411yescharacteristic
200C2×C8×C79C22×C411yescharacteristic
201C2×C8×C79C2×D411yescharacteristic
202C2×C8×C79C2×D411yescharacteristic
203C4⋊C4×C79—21no—
204C4⋊C4×C79C22×C411yescharacteristic
205C4⋊C4×C79C22×C411yescharacteristic
206C22⋊C4×C79—21no—
207C42×C79—21no—
208C42×C79C2411yescharacteristic
209C23×C79—21no—
210C22×D79C4×D411yescharacteristic
211Q8×C79—21no—
212Q8×C79—21no—
213D4×C79—21no—
214D4×C79#13112yes—
215C2×C4×C79—21no—
216C2×C4×C79—21no—
217C2×C4×C79—21no—
218C2×C4×C79—41no—
219C2×C4×C79—41no—
220C2×C4×C79C23×C411yescharacteristic
221C2×C4×C79C22×D411yescharacteristic
222C2×C4×C79C2×#4111yescharacteristic
223C22⋊D79—24no—
224C2×Dic79—24no—
225C2×Dic79—21no—
226C2×Dic79—21no—
227C2×Dic79—21no—
228C2×Dic79—21no—
229C2×Dic79—42no—
230C2×Dic79C4×D411yescharacteristic
231C2×Dic79C4×D411yescharacteristic
232C2×Dic79C4×D411yescharacteristic
233C4×D79—21no—
234C4×D79—21no—
235C4×D79—21no—
236C4×D79—21no—
237C4×D79—22no—
238Dic158—21no—
239Dic158—21no—
240Dic158—21no—
241Dic158—42no—
242Dic158#13112yes—
243C8×C79—22no—
244C8×C79—21no—
245C8×C79—21no—
246C8×C79—21no—
247C8×C79—41no—
248C8×C79—41no—
249C8×C79#13712yes—
250C79⋊C8—22no—
251C79⋊C8—21no—
252C79⋊C8—21no—
253C79⋊C8—21no—
254C79⋊C8—21no—
255C79⋊C8—41no—
256C79⋊C8—41no—
257C22×C79—22no—
258C22×C79—41no—
259C22×C79—41no—
260C22×C79C2×C4×D411yescharacteristic
261C2×D79—21no—
262C2×D79—21no—
263C2×D79—21no—
264C4×C79—21no—
265C4×C79—41no—
266C4×C79—41no—
267C4×C79—41no—
268C4×C79C2×#13111yescharacteristic
269C4×C79C2×#13111yescharacteristic
270C4×C79C2×#13711yescharacteristic
271C4×C79#57511yescharacteristic
272Dic79—22no—
273Dic79—21no—
274Dic79—21no—
275Dic79—41no—
276Dic79—41no—
277Dic79—41no—
278Dic79#44312yes—
279#17913—791nomaximal, Sylow
280C2×C79—42no—
281C2×C79#292911yescharacteristic
282C2×C79C4×#13711yescharacteristic
283C2×C79C2×#44311yescharacteristic
284D79—41no—
285#3107—791no—
286C8×#41—791no—
287#2960—791no—
288#1587—791no—
289#1587—791no—
290#1585—791no—
291#1585—791no—
292C8×Q16—791no—
293C8×Q16—791no—
294C8×SD16—791no—
295C8×SD16D7911yescharacteristic, complemented
296#1574—791no—
297#1489—791no—
298#1485—791no—
299C8×M16—791no—
300C79#1791311yescharacteristic, complemented, Sylow
301#533—791no—
302C4×#41—791no—
303#485—791no—
304#485—791no—
305#475—791no—
306#474—791no—
307#462—791no—
308#461—791no—
309#452—791no—
310#452—791no—
311#446—791no—
312#446—791no—
313#446—1581no—
314C8×Q8—791no—
315C8×Q8—791no—
316C8×Q8—791no—
317C8×Q8—1581no—
318C8×Q8C2×D7911yescharacteristic
319C4×Q16—791no—
320C4×Q16—791no—
321C4×SD16—791no—
322C4×SD16C2×D7911yescharacteristic
323C8×D4—791no—
324C8×D4—1581no—
325C8×D4—1581no—
326C8×D4C2×D7911yescharacteristic
327#431—791no—
328#431—791no—
329#431—1581no—
330#424—1581no—
331#423—791no—
332#405—1581no—
333C4×M16—791no—
334C2×C4×C8—791no—
335C2×C4×C8—1581no—
336#335—791no—
337#335—791no—
338#334—791no—
339#334C2×D7911yescharacteristic
340Q8⋊C8—791no—
341Q8⋊C8—791no—
342Q8⋊C8—791no—
343Q8⋊C8—791no—
344Q8⋊C8—791no—
345Q8⋊C8C2×D7911yescharacteristic
346D4⋊C8—791no—
347D4⋊C8C2×D7911yescharacteristic
348#322—791no—
349#322—791no—
350C82—791no—
351C82C2×D7911yescharacteristic
352C2×#41—791no—
353C2×Q16—791no—
354C2×Q16—791no—
355C2×SD16—791no—
356C2×SD16C4×D7911yescharacteristic
357C2×M16—1581no—
358C2×M16—1581no—
359C22×C8—21no—
360C22×C8—791no—
361C22×C8—791no—
362C22×C8—1581no—
363C4⋊Q8—791no—
364C4⋊Q8—791no—
365#125—1581no—
366C22⋊Q8—791no—
367C22⋊Q8—791no—
368C22⋊Q8—1581no—
369C4×Q8—791no—
370C4×Q8—791no—
371C4×Q8—791no—
372C4×Q8—1581no—
373C4×Q8C22×D7911yescharacteristic
374C4×D4—791no—
375C4×D4—1581no—
376C4×D4—1581no—
377C4×D4C22×D7911yescharacteristic
378#117—791no—
379#117—1581no—
380#117—1581no—
381C2×C42—791no—
382C2×C42—1581no—
383#107—791no—
384#107—791no—
385#106—791no—
386#106C4×D7911yescharacteristic
387C4⋊C8—21no—
388C4⋊C8—791no—
389C4⋊C8—791no—
390C4⋊C8—791no—
391C4⋊C8—791no—
392C4⋊C8—1581no—
393C4⋊C8—1581no—
394C4⋊C8—1581no—
395C4⋊C8—1581no—
396C4⋊C8—1581no—
397C4⋊C8—1581no—
398C4⋊C8C22×D7911yescharacteristic
399C4⋊C8C22×D7911yescharacteristic
400#103—791no—
401#103—791no—
402#103—791no—
403#103—791no—
404#103—791no—
405#103C4×D7911yescharacteristic
406#102—791no—
407#102C4×D7911yescharacteristic
408C22⋊C8—21no—
409C22⋊C8—791no—
410C22⋊C8—791no—
411C22⋊C8—1581no—
412C22⋊C8—1581no—
413#97—1581no—
414#97—1581no—
415#97—1581no—
416C4×C8—21no—
417C4×C8—791no—
418C4×C8—791no—
419C4×C8—791no—
420C4×C8—791no—
421C4×C8—791no—
422C4×C8—1581no—
423C4×C8—1581no—
424C4×C8—1581no—
425C4×C8—1581no—
426C4×C8C22×D7911yescharacteristic
427C4×C8C22×D7911yescharacteristic
428C4×C8C22×D7911yescharacteristic
429#41—1582no—
430#41—1582no—
431C2×Q8—791no—
432C2×Q8—791no—
433C2×Q8—1581no—
434C2×Q8C2×C4×D7911yescharacteristic
435C2×D4—1581no—
436C2×D4C2×C4×D7911yescharacteristic
437C22×C4—21no—
438C22×C4—791no—
439C22×C4—791no—
440C22×C4—791no—
441C22×C4—1581no—
442C22×C4—1581no—
443Q16—1582no—
444Q16—1582no—
445SD16—22no—
446SD16—1582no—
447M16—3162no—
448M16—1584no—
449C2×C8—21no—
450C2×C8—21no—
451C2×C8—21no—
452C2×C8—42no—
453C2×C8—3161no—
454C2×C8—3162no—
455C2×C8—791no—
456C2×C8—791no—
457C2×C8—791no—
458C2×C8—791no—
459C2×C8—1581no—
460C2×C8—1581no—
461C2×C8—1581no—
462C2×C8—1582no—
463C2×C8C2×C4×D7911yescharacteristic
464C2×C8C2×C4×D7911yescharacteristic
465C2×C8D4×D7911yescharacteristic
466C2×C8D4×D7911yescharacteristic
467C4⋊C4—21no—
468C4⋊C4—791no—
469C4⋊C4—791no—
470C4⋊C4—1581no—
471C4⋊C4—1581no—
472C4⋊C4—1581no—
473C4⋊C4—1581no—
474C4⋊C4—1581no—
475C4⋊C4—1581no—
476C4⋊C4—1581no—
477C4⋊C4—1581no—
478C4⋊C4C2×C4×D7911yescharacteristic
479C4⋊C4C2×C4×D7911yescharacteristic
480C22⋊C4—21no—
481C22⋊C4—1581no—
482C22⋊C4—1581no—
483C22⋊C4—1581no—
484C22⋊C4—1581no—
485C42—21no—
486C42—791no—
487C42—791no—
488C42—791no—
489C42—1581no—
490C42—1581no—
491C42—1581no—
492C42—1581no—
493C42C23×D7911yescharacteristic
494C23—21no—
495C23—791no—
496Q8—21no—
497Q8—21no—
498Q8—3162no—
499Q8—792no—
500Q8—1581no—
501Q8—1581no—
502Q8—1581no—
503D4—21no—
504D4—1584no—
505D4#1457916712yes—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—3162no—
512C2×C4—791no—
513C2×C4—791no—
514C2×C4—791no—
515C2×C4—1582no—
516C2×C4—1581no—
517C2×C4—1581no—
518C2×C4—1581no—
519C2×C4—1584no—
520C2×C4—1581no—
521C2×C4—1581no—
522C2×C4—1581no—
523C2×C4—1581no—
524C2×C4—1581no—
525C2×C4C22×C4×D7911yescharacteristic
526C2×C4C2×D4×D7911yescharacteristic
527C2×C4#41×D7911yescharacteristic
528C8—22no—
529C8—21no—
530C8—21no—
531C8—21no—
532C8—41no—
533C8—41no—
534C8—3161no—
535C8—3161no—
536C8—1582no—
537C8—1581no—
538C8—1581no—
539C8—1581no—
540C8—1581no—
541C8#1457919012yes—
542C22—22no—
543C22—41no—
544C22—41no—
545C22—1581no—
546C22—1581no—
547C22—1581no—
548C22C4×D4×D7911yescharacteristic
549C4—21no—
550C4—41no—
551C4—41no—
552C4—41no—
553C4—3161no—
554C4—3161no—
555C4—3161no—
556C4—792no—
557C4—1581no—
558C4—1581no—
559C4—1582no—
560C4#131×D7911yescharacteristic
561C4C2×#1457916711yescharacteristic
562C4C2×#1457919011yescharacteristic
563C4#1909936011yescharacteristic
564C2—42no—
565C2—3161no—
566C2#2377208911yescharacteristic
567C2C4×#1457919011yescharacteristic
568C2#443×D7911yescharacteristic
subgroupG/H
centerC2×C8D4×D79
commutatorC2×C4×C79C23×C4
FrattiniC42C23×D79
FittingC8×SD16×C79C2
socleC22×C79C2×C4×D4
prime radicalC2×#11929623C2×C4
cosocleC42×C79C24
hypercenterC8×SD16D79
hypocenterC79#17913
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C2×C8 ◁ C4×C8 ◁ C8×SD16

Complements

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

NH ≅ G/Ntype
3. #23772329C2semidirect
4. #23762274C2semidirect
5. #23762270C2semidirect
6. #23762258C2semidirect
7. #23762254C2semidirect
8. C8×Dic316C2semidirect
9. C8×#11929614C2semidirect
10. C8×#11929612C2semidirect
11. #23762187C2semidirect
13. #23761819C2semidirect
14. C8×#11929607C2semidirect
15. C8×SD16×C79C2semidirect
295. C8×SD16D79semidirect
300. C79#17913semidirect

Maximal subgroups

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

subgroupindex
#237734672
C8×#119296232
#237723292
#237622742
#237622702
#237622582
#237622542
C8×Dic3162
C8×#119296142
C8×#119296122
#237621872
#237618362
#237618192
C8×#119296072
C8×SD16×C792
#1791379