Finite Group Databasebeta

← #34258408

Subgroups — #34258408

Order 20224.

638 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23773406C211yesmaximal, characteristic
2#23772491C211yesmaximal, characteristic, complemented
3#23772421C211yesmaximal, characteristic
4#23762258C211yesmaximal, characteristic, complemented
5#23762256C211yesmaximal, characteristic, complemented
6#23762239C211yesmaximal, characteristic, complemented
7#23762233C211yesmaximal, characteristic, complemented
8#23762214C211yesmaximal, characteristic, complemented
9C8×#11929611C211yesmaximal, characteristic, complemented
10C8×#11929612C211yesmaximal, characteristic, complemented
11#23762179C211yesmaximal, characteristic, complemented
12#23761841C211yesmaximal, characteristic
13#23761819C211yesmaximal, characteristic, complemented
14#23761696C211yesmaximal, characteristic, complemented
15C79×SD16⋊C8C211yesmaximal, characteristic, complemented
16#19099117C2211yescharacteristic
17C4×D4×D79C2211yescharacteristic
18#19098895C411yescharacteristic
19#19098892C411yescharacteristic
20#19098852C411yescharacteristic
21#19098851C411yescharacteristic
22#19098767C411yescharacteristic
23#19098766C411yescharacteristic
24#19098715C411yescharacteristic
25#19098712C411yescharacteristic
26#19098694C2211yescharacteristic, complemented
27#19098693—21no—
28#19098692—21no—
29#19098690C2211yescharacteristic, complemented
30C8×Dic158C2211yescharacteristic, complemented
31C4×#11929613C2211yescharacteristic
32C4×#11929611C2211yescharacteristic
33C4×#11929612C2211yescharacteristic
34C8×D316C2211yescharacteristic
35D4×C79⋊C8C2211yescharacteristic
36#19098628C2211yescharacteristic
37C8×C22⋊D79—21no—
38(C22⋊D79)⋊C8—21no—
39#19098623—21no—
40#19098622—21no—
41#19098619C2211yescharacteristic
42#19098615C2211yescharacteristic
43#19098614C2211yescharacteristic
44#19098614C2211yescharacteristic
45C2×(C2×D79)⋊C8—21no—
46C22⋊C8×D79—21no—
47C4×#11929607C2211yescharacteristic
48#19098316C2211yescharacteristic, complemented
49#19098313C2211yescharacteristic, complemented
50#19098313C2211yescharacteristic, complemented
51#19098295C2211yescharacteristic, complemented
52Q8⋊(C79⋊C8)C2211yescharacteristic, complemented
53#19098293C2211yescharacteristic, complemented
54#19098292C2211yescharacteristic
55D4⋊(C79⋊C8)C2211yescharacteristic
56#19098290C2211yescharacteristic
57#19098285C2211yescharacteristic, complemented
58#19098285C2211yescharacteristic, complemented
59C8×C79⋊C8C2211yescharacteristic, complemented
60C8×Q8×C79C2211yescharacteristic, complemented
61C4×SD16×C79C2211yescharacteristic
62C8×D4×C79C2211yescharacteristic
63#335×C79C2211yescharacteristic, complemented
64Q8⋊C8×C79C2211yescharacteristic, complemented
65D4⋊C8×C79C2211yescharacteristic
66#322×C79C2211yescharacteristic, complemented
67C2×D4×D79C2×C411yescharacteristic
68C22×C4×D79—21no—
69C2×#11929613C2×C411yescharacteristic
70C2×#11929612C2×C411yescharacteristic
71C2×#11929611C2×C411yescharacteristic
72C2×#11929607—21no—
73C22×C79⋊C8—21no—
74C2×C8×D79—21no—
75C2×C8×D79—21no—
76#14579160C2×C411yescharacteristic
77#14579159C2×C411yescharacteristic
78#14579151—21no—
79#14579141C2×C411yescharacteristic
80#14579139—21no—
81#14579137C2×C411yescharacteristic
82C4×Dic158—21no—
83C4×Dic158C2311yescharacteristic
84C4×D316C2311yescharacteristic
85D4×Dic79C2×C411yescharacteristic
86#14579123C2×C411yescharacteristic
87C4×C22⋊D79—21no—
88(C22⋊D79)⋊C4—21no—
89#14579118—21no—
90C4⋊C4×D79C2×C411yescharacteristic
91C22⋊C4×D79—21no—
92C42×D79C2311yescharacteristic
93#14579093C2×C411yescharacteristic
94#14579092C2×C411yescharacteristic
95#14579092C2×C411yescharacteristic
96C4⋊(C79⋊C8)—21no—
97C4⋊(C79⋊C8)C2311yescharacteristic
98C4⋊(C79⋊C8)C2311yescharacteristic
99Dic79⋊C8—21no—
100Dic79⋊C8—21no—
101Dic79⋊C8C2×C411yescharacteristic
102Dic79⋊C8C2×C411yescharacteristic
103Dic79⋊C8D411yescharacteristic
104Dic79⋊C8D411yescharacteristic
105#14579086C2×C411yescharacteristic
106#14579085C2×C411yescharacteristic
107#14579084C2×C411yescharacteristic
108#14579083C2×C411yescharacteristic
109#14579082C2×C411yescharacteristic
110#14579081C2×C411yescharacteristic
111C22⋊(C79⋊C8)—21no—
112(C2×D79)⋊C8—21no—
113(C2×D79)⋊C8—21no—
114(C2×D79)⋊C8—21no—
115(C2×D79)⋊C8—21no—
116(C2×D79)⋊C8C2×C411yescharacteristic
117(C2×D79)⋊C8C2×C411yescharacteristic
118(C2×D79)⋊C8D411yescharacteristic
119(C2×D79)⋊C8D411yescharacteristic
120#14579072—21no—
121#14579071—21no—
122#14579071—21no—
123#14579071—21no—
124C4×C79⋊C8C2311yescharacteristic
125C4×C79⋊C8C2311yescharacteristic
126C4×C79⋊C8C2311yescharacteristic
127C8×Dic79—21no—
128C8×Dic79—21no—
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
144C23×D79—21no—
145C2×Dic158—21no—
146C2×Dic158C22×C411yescharacteristic
147C2×D316C22×C411yescharacteristic
148C2×C22⋊D79—21no—
149D4×D79—22no—
150D4×D79—22no—
151C22×Dic79—21no—
152C2×C4×D79—21no—
153C2×C4×D79—21no—
154C2×C4×D79—21no—
155C2×C4×D79—81no—
156C2×C4×D79—81no—
157C2×C4×D79C22×C411yescharacteristic
158C2×C4×D79C2×D411yescharacteristic
159C2×C4×D79#4111yescharacteristic
160#11929613—22no—
161#11929612—22no—
162#11929611—22no—
163#11929607—42no—
164C2×C79⋊C8—21no—
165C2×C79⋊C8—21no—
166C2×C79⋊C8—21no—
167C2×C79⋊C8—81no—
168C2×C79⋊C8C22×C411yescharacteristic
169C2×C79⋊C8C22×C411yescharacteristic
170C2×C79⋊C8C2×D411yescharacteristic
171C2×C79⋊C8C2×D411yescharacteristic
172C8×D79—81no—
173C8×D79—81no—
174C4⋊Dic79—21no—
175C4⋊Dic79C22×C411yescharacteristic
176C4⋊Dic79C22×C411yescharacteristic
177Dic79⋊C4—21no—
178Dic79⋊C4—21no—
179Dic79⋊C4—21no—
180Dic79⋊C4—21no—
181Dic79⋊C4—21no—
182Dic79⋊C4—21no—
183C22⋊Dic79—21no—
184(C2×D79)⋊C4—21no—
185(C2×D79)⋊C4—21no—
186(C2×D79)⋊C4—21no—
187(C2×D79)⋊C4—21no—
188C4×Dic79—21no—
189C4×Dic79C22×C411yescharacteristic
190C4×Dic79C2×D411yescharacteristic
191C4×Dic79#4111yescharacteristic
192C2×Q8×C79C22×C411yescharacteristic
193C2×D4×C79C22×C411yescharacteristic
194C22×C4×C79—21no—
195SD16×C79—22no—
196C2×C8×C79—21no—
197C2×C8×C79—21no—
198C2×C8×C79—21no—
199C2×C8×C79—81no—
200C2×C8×C79C22×C411yescharacteristic
201C2×C8×C79C22×C411yescharacteristic
202C2×C8×C79C2×D411yescharacteristic
203C2×C8×C79C2×D411yescharacteristic
204C4⋊C4×C79—21no—
205C4⋊C4×C79C22×C411yescharacteristic
206C4⋊C4×C79C22×C411yescharacteristic
207C22⋊C4×C79—21no—
208C42×C79—21no—
209C42×C79C2411yescharacteristic
210C23×C79—21no—
211C22×D79—21no—
212C22×D79—24no—
213C22×D79—81no—
214C22×D79—81no—
215C22×D79C4×D411yescharacteristic
216Q8×C79—21no—
217Q8×C79—21no—
218D4×C79—21no—
219D4×C79#13112yes—
220C2×C4×C79—21no—
221C2×C4×C79—21no—
222C2×C4×C79—21no—
223C2×C4×C79—41no—
224C2×C4×C79—41no—
225C2×C4×C79C23×C411yescharacteristic
226C2×C4×C79C22×D411yescharacteristic
227C2×C4×C79C2×#4111yescharacteristic
228C22⋊D79—24no—
229C2×Dic79—21no—
230C2×Dic79—21no—
231C2×Dic79—21no—
232C2×Dic79—81no—
233C2×Dic79C4×D411yescharacteristic
234C2×Dic79C4×D411yescharacteristic
235C2×Dic79C4×D411yescharacteristic
236D316—21no—
237D316#13112yes—
238C4×D79—21no—
239C4×D79—21no—
240C4×D79—21no—
241C4×D79—21no—
242C4×D79—41no—
243C4×D79—41no—
244C4×D79—41no—
245C4×D79—81no—
246C4×D79—81no—
247Dic158—21no—
248Dic158—21no—
249Dic158—81no—
250C8×C79—21no—
251C8×C79—21no—
252C8×C79—21no—
253C8×C79—21no—
254C8×C79—41no—
255C8×C79—41no—
256C8×C79—41no—
257C79⋊C8—22no—
258C79⋊C8—21no—
259C79⋊C8—21no—
260C79⋊C8—21no—
261C79⋊C8—21no—
262C79⋊C8—41no—
263C79⋊C8—41no—
264C22×C79—22no—
265C22×C79—41no—
266C22×C79—41no—
267C22×C79C2×C4×D411yescharacteristic
268C2×D79—22no—
269C2×D79—21no—
270C2×D79—21no—
271C2×D79—21no—
272C2×D79—41no—
273C2×D79—41no—
274C2×D79—84no—
275C4×C79—21no—
276C4×C79—41no—
277C4×C79—41no—
278C4×C79—41no—
279C4×C79C2×#13111yescharacteristic
280C4×C79C2×#13111yescharacteristic
281C4×C79#57711yescharacteristic
282C4×C79#57711yescharacteristic
283Dic79—21no—
284Dic79—21no—
285Dic79—21no—
286Dic79—41no—
287Dic79—41no—
288Dic79—41no—
289#17924—791nomaximal, Sylow
290C2×C79—42no—
291C2×C79#292911yescharacteristic
292C2×C79#294911yescharacteristic
293C2×C79#295611yescharacteristic
294D79—41no—
295D79—42no—
296#3101—791no—
297#2985—791no—
298#2972—791no—
299#1585—791no—
300#1585—791no—
301#1582—791no—
302#1582—791no—
303SD16⋊C8—791no—
304SD16⋊C8D7911yescharacteristic, complemented
305C8×SD16—791no—
306C8×SD16—791no—
307#1572—791no—
308#1490—791no—
309#1485—791no—
310#1453—791no—
311C79#1792411yescharacteristic, complemented, Sylow
312#533—791no—
313C2×C4×D4—791no—
314#483—791no—
315#483—791no—
316#476—791no—
317#476—791no—
318#461—791no—
319#461—791no—
320#450—791no—
321#450—791no—
322#446—791no—
323#446—791no—
324#446—1581no—
325#446—1581no—
326C8×Q8—791no—
327C8×Q8C2×D7911yescharacteristic, complemented
328C4×SD16—791no—
329C4×SD16—791no—
330C4×SD16—791no—
331C4×SD16C2×D7911yescharacteristic
332C8×D4—791no—
333C8×D4—791no—
334C8×D4—791no—
335C8×D4—1581no—
336C8×D4—1581no—
337C8×D4C2×D7911yescharacteristic
338#433—1581no—
339#433—1581no—
340#432—791no—
341#431—791no—
342#431—791no—
343#431—791no—
344C2×C22⋊C8—1581no—
345C2×C22⋊C8—1581no—
346C4×M16—791no—
347#335—791no—
348#335C2×D7911yescharacteristic, complemented
349#334—791no—
350#334—791no—
351Q8⋊C8—791no—
352Q8⋊C8—791no—
353Q8⋊C8—791no—
354Q8⋊C8C2×D7911yescharacteristic, complemented
355D4⋊C8—791no—
356D4⋊C8—791no—
357D4⋊C8—791no—
358D4⋊C8C2×D7911yescharacteristic
359#322—791no—
360#322—791no—
361#322C2×D7911yescharacteristic, complemented
362C82—791no—
363C22×D4—791no—
364C23×C4—1581no—
365C2×SD16—791no—
366C2×SD16—791no—
367C2×SD16—791no—
368C2×SD16C4×D7911yescharacteristic
369C2×M16—1581no—
370C22×C8—21no—
371C22×C8—1581no—
372C22×C8—1581no—
373C22×C8—1581no—
374C4⋊Q8—791no—
375C4⋊Q8—791no—
376#125—1581no—
377C22⋊Q8—791no—
378C22⋊Q8—791no—
379C22⋊Q8—1581no—
380C4×Q8—791no—
381C4×Q8—1581no—
382C4×Q8C22×D7911yescharacteristic
383C4×D4—791no—
384C4×D4—791no—
385C4×D4—791no—
386C4×D4—1581no—
387C4×D4—1581no—
388C4×D4C22×D7911yescharacteristic
389#117—1581no—
390C2×C4⋊C4—791no—
391C2×C22⋊C4—1581no—
392C2×C42—791no—
393#106—791no—
394#106—791no—
395#106—791no—
396#106C4×D7911yescharacteristic
397C4⋊C8—21no—
398C4⋊C8—791no—
399C4⋊C8—791no—
400C4⋊C8—791no—
401C4⋊C8—791no—
402C4⋊C8—791no—
403C4⋊C8—791no—
404C4⋊C8—1581no—
405C4⋊C8—1581no—
406C4⋊C8—1581no—
407C4⋊C8C22×D7911yescharacteristic
408C4⋊C8C22×D7911yescharacteristic
409#103—791no—
410#103—791no—
411#103—791no—
412#103C4×D7911yescharacteristic
413#102—791no—
414#102—791no—
415#102—791no—
416#102C4×D7911yescharacteristic
417C22⋊C8—21no—
418C22⋊C8—791no—
419C22⋊C8—791no—
420C22⋊C8—791no—
421C22⋊C8—791no—
422C22⋊C8—1581no—
423C22⋊C8—1581no—
424C22⋊C8—1581no—
425C22⋊C8—1581no—
426C22⋊C8—1581no—
427#97—1581no—
428#97—1581no—
429#97—1581no—
430#97—1581no—
431C4×C8—21no—
432C4×C8—791no—
433C4×C8—791no—
434C4×C8—791no—
435C4×C8—1581no—
436C4×C8—1581no—
437C4×C8C22×D7911yescharacteristic
438C4×C8C22×D7911yescharacteristic
439C4×C8C22×D7911yescharacteristic
440C24—1581no—
441C2×Q8—791no—
442C2×Q8—1581no—
443C2×Q8C2×C4×D7911yescharacteristic
444C2×D4—791no—
445C2×D4—1581no—
446C2×D4—1582no—
447C2×D4—1582no—
448C2×D4C2×C4×D7911yescharacteristic
449C22×C4—21no—
450C22×C4—791no—
451C22×C4—791no—
452C22×C4—791no—
453C22×C4—6321no—
454C22×C4—6321no—
455C22×C4—1581no—
456C22×C4—1581no—
457C22×C4—1581no—
458C22×C4—1581no—
459SD16—22no—
460SD16—1582no—
461SD16—1582no—
462SD16—1582no—
463M16—3162no—
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—81no—
468C2×C8—791no—
469C2×C8—791no—
470C2×C8—791no—
471C2×C8—791no—
472C2×C8—6321no—
473C2×C8—6321no—
474C2×C8—6321no—
475C2×C8—1581no—
476C2×C8—1581no—
477C2×C8—1581no—
478C2×C8C2×C4×D7911yescharacteristic
479C2×C8C2×C4×D7911yescharacteristic
480C2×C8D4×D7911yescharacteristic
481C2×C8D4×D7911yescharacteristic
482C4⋊C4—21no—
483C4⋊C4—791no—
484C4⋊C4—791no—
485C4⋊C4—1581no—
486C4⋊C4—1581no—
487C4⋊C4—1581no—
488C4⋊C4—1581no—
489C4⋊C4—1581no—
490C4⋊C4—1581no—
491C4⋊C4—1581no—
492C4⋊C4C2×C4×D7911yescharacteristic
493C4⋊C4C2×C4×D7911yescharacteristic
494C22⋊C4—21no—
495C22⋊C4—1581no—
496C22⋊C4—1581no—
497C22⋊C4—1581no—
498C22⋊C4—1581no—
499C22⋊C4—1581no—
500C42—21no—
501C42—791no—
502C42—791no—
503C42—791no—
504C42—1581no—
505C42C23×D7911yescharacteristic
506C23—21no—
507C23—791no—
508C23—6321no—
509C23—6321no—
510C23—1584no—
511C23—1581no—
512Q8—21no—
513Q8—21no—
514Q8—6321no—
515Q8—1581no—
516Q8—1581no—
517D4—21no—
518D4—792no—
519D4—1584no—
520D4—1581no—
521D4#1457916712yes—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—3161no—
528C2×C4—3161no—
529C2×C4—3161no—
530C2×C4—791no—
531C2×C4—791no—
532C2×C4—791no—
533C2×C4—6321no—
534C2×C4—6321no—
535C2×C4—6321no—
536C2×C4—1581no—
537C2×C4—1581no—
538C2×C4—1581no—
539C2×C4—1581no—
540C2×C4—1581no—
541C2×C4—1581no—
542C2×C4—1581no—
543C2×C4C22×C4×D7911yescharacteristic
544C2×C4C2×D4×D7911yescharacteristic
545C2×C4#41×D7911yescharacteristic
546C8—21no—
547C8—21no—
548C8—21no—
549C8—21no—
550C8—41no—
551C8—41no—
552C8—41no—
553C8—3161no—
554C8—3161no—
555C8—1582no—
556C8—1581no—
557C8—1581no—
558C8—1581no—
559C8—1581no—
560C22—22no—
561C22—41no—
562C22—41no—
563C22—3161no—
564C22—3161no—
565C22—6324no—
566C22—1581no—
567C22—1581no—
568C22—1581no—
569C22—1582no—
570C22C4×D4×D7911yescharacteristic
571C4—21no—
572C4—41no—
573C4—41no—
574C4—41no—
575C4—3161no—
576C4—3161no—
577C4—3161no—
578C4—1581no—
579C4—1581no—
580C4—1581no—
581C4#131×D7911yescharacteristic
582C4C2×#1457916711yescharacteristic
583C4#1909937211yescharacteristic
584C4#1909937311yescharacteristic
585C2—42no—
586C2—3162no—
587C2—3161no—
588C2#2377208911yescharacteristic
589C2#2377225611yescharacteristic
590C2#2377230811yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D79
commutatorC2×C4×C79C23×C4
FrattiniC42C23×D79
FittingC79×SD16⋊C8C2
socleC22×C79C2×C4×D4
prime radicalC2×D4×D79C2×C4
cosocleC42×C79C24
hypercenterSD16⋊C8D79
hypocenterC79#17924
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ SD16⋊C8

Complements

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

NH ≅ G/Ntype
2. #23772491C2semidirect
4. #23762258C2semidirect
5. #23762256C2semidirect
6. #23762239C2semidirect
7. #23762233C2semidirect
8. #23762214C2semidirect
9. C8×#11929611C2semidirect
10. C8×#11929612C2semidirect
11. #23762179C2semidirect
13. #23761819C2semidirect
14. #23761696C2semidirect
15. C79×SD16⋊C8C2semidirect
26. #19098694C22semidirect
29. #19098690C22semidirect
30. C8×Dic158C22semidirect
48. #19098316C22semidirect
49. #19098313C22semidirect
50. #19098313C22semidirect
51. #19098295C22semidirect
52. Q8⋊(C79⋊C8)C22semidirect
53. #19098293C22semidirect
57. #19098285C22semidirect
58. #19098285C22semidirect
59. C8×C79⋊C8C22semidirect
60. C8×Q8×C79C22semidirect
63. #335×C79C22semidirect
64. Q8⋊C8×C79C22semidirect
66. #322×C79C22semidirect
304. SD16⋊C8D79semidirect
311. C79#17924semidirect
327. C8×Q8C2×D79semidirect
348. #335C2×D79semidirect
354. Q8⋊C8C2×D79semidirect
361. #322C2×D79semidirect

Maximal subgroups

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

subgroupindex
#237734062
#237724912
#237724212
#237622582
#237622562
#237622392
#237622332
#237622142
C8×#119296112
C8×#119296122
#237621792
#237618412
#237618192
#237616962
C79×SD16⋊C82
#1792479