Finite Group Databasebeta

← #34258334

Subgroups — #34258334

Order 20224.

648 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23773451C211yesmaximal, characteristic
2#23772548C211yesmaximal, characteristic, complemented
3#23772421C211yesmaximal, characteristic, complemented
4#23762263C211yesmaximal, characteristic, complemented
5#23762258C211yesmaximal, characteristic, complemented
6#23762239C211yesmaximal, characteristic, complemented
7#23762226C211yesmaximal, characteristic, complemented
8#23762222C211yesmaximal, characteristic, complemented
9#23762210C211yesmaximal, characteristic, complemented
10C8×D632C211yesmaximal, characteristic, complemented
11#23762187C211yesmaximal, characteristic, complemented
12#23761823C211yesmaximal, characteristic, complemented
13#23761807C211yesmaximal, characteristic, complemented
14#23761699C211yesmaximal, characteristic, complemented
15C8×SD16×C79C211yesmaximal, characteristic, complemented
16#19099100C2211yescharacteristic
17C4×D4×D79C2211yescharacteristic
18#19098897C411yescharacteristic
19#19098883C411yescharacteristic
20#19098862C411yescharacteristic
21#19098861C411yescharacteristic
22#19098781C411yescharacteristic
23#19098780C411yescharacteristic
24#19098712C411yescharacteristic
25#19098698C411yescharacteristic
26#19098694C2211yescharacteristic, complemented
27C4×#11929611C2211yescharacteristic
28C4×D632C2211yescharacteristic
29C4×#11929609C2211yescharacteristic
30#19098642—21no—
31#19098640C2211yescharacteristic, complemented
32#19098633—21no—
33C8×D316C2211yescharacteristic
34C8×D316C2211yescharacteristic, complemented
35D4×C79⋊C8C2211yescharacteristic
36#19098628C2211yescharacteristic
37C8×C22⋊D79—21no—
38(C22⋊D79)⋊C8—21no—
39#19098622—21no—
40#19098615C2211yescharacteristic
41#19098614C2211yescharacteristic
42#19098614C2211yescharacteristic
43#19098614C2211yescharacteristic
44#19098588—21no—
45#19098585C2211yescharacteristic
46C2×(C2×D79)⋊C8—21no—
47C22⋊C8×D79—21no—
48#19098316C2211yescharacteristic, complemented
49#19098315C2211yescharacteristic, complemented
50#19098313C2211yescharacteristic, complemented
51Q8⋊(C79⋊C8)C2211yescharacteristic, complemented
52#19098292C2211yescharacteristic
53#19098292C2211yescharacteristic, complemented
54D4⋊(C79⋊C8)C2211yescharacteristic
55#19098290C2211yescharacteristic
56#19098290C2211yescharacteristic, complemented
57#19098285C2211yescharacteristic, complemented
58#19098285C2211yescharacteristic, complemented
59#19098285C2211yescharacteristic, complemented
60C8×Q8×C79C2211yescharacteristic, complemented
61C4×SD16×C79C2211yescharacteristic
62C8×D4×C79C2211yescharacteristic
63#334×C79C2211yescharacteristic, complemented
64Q8⋊C8×C79C2211yescharacteristic, complemented
65D4⋊C8×C79C2211yescharacteristic
66C82×C79C2211yescharacteristic, complemented
67C2×D4×D79C2×C411yescharacteristic
68C22×C4×D79—21no—
69C2×#11929611C2×C411yescharacteristic
70C2×D632C2×C411yescharacteristic
71C2×#11929609C2×C411yescharacteristic
72C2×#11929607—21no—
73C2×#11929607—21no—
74C22×C79⋊C8—21no—
75C2×C8×D79—21no—
76C2×C8×D79—21no—
77C2×C8×D79—21no—
78#14579156—21no—
79#14579152C2×C411yescharacteristic
80#14579150C2×C411yescharacteristic
81#14579146—21no—
82#14579137C2×C411yescharacteristic
83#14579136C2×C411yescharacteristic
84C4×D316—21no—
85C4×D316C2311yescharacteristic
86C4×D316C2311yescharacteristic
87D4×Dic79C2×C411yescharacteristic
88#14579123C2×C411yescharacteristic
89C4×C22⋊D79—21no—
90(C22⋊D79)⋊C4—21no—
91#14579118—21no—
92C4⋊C4×D79C2×C411yescharacteristic
93C22⋊C4×D79—21no—
94C42×D79C2311yescharacteristic
95#14579095C2×C411yescharacteristic
96#14579094C2×C411yescharacteristic
97#14579092C2×C411yescharacteristic
98C4⋊(C79⋊C8)—21no—
99C4⋊(C79⋊C8)C2311yescharacteristic
100C4⋊(C79⋊C8)C2311yescharacteristic
101Dic79⋊C8—21no—
102Dic79⋊C8—21no—
103Dic79⋊C8C2×C411yescharacteristic
104Dic79⋊C8C2×C411yescharacteristic
105Dic79⋊C8D411yescharacteristic
106Dic79⋊C8D411yescharacteristic
107#14579085C2×C411yescharacteristic
108#14579083C2×C411yescharacteristic
109#14579083C2×C411yescharacteristic
110#14579082C2×C411yescharacteristic
111#14579081C2×C411yescharacteristic
112#14579081C2×C411yescharacteristic
113C22⋊(C79⋊C8)—21no—
114(C2×D79)⋊C8—21no—
115(C2×D79)⋊C8—21no—
116(C2×D79)⋊C8—21no—
117(C2×D79)⋊C8—21no—
118(C2×D79)⋊C8—21no—
119(C2×D79)⋊C8—21no—
120(C2×D79)⋊C8C2×C411yescharacteristic
121(C2×D79)⋊C8C2×C411yescharacteristic
122(C2×D79)⋊C8D411yescharacteristic
123(C2×D79)⋊C8D411yescharacteristic
124#14579072—21no—
125C4×C79⋊C8C2311yescharacteristic
126C4×C79⋊C8C2311yescharacteristic
127C4×C79⋊C8C2311yescharacteristic
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×D316—21no—
146C2×D316C22×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—21no—
156C2×C4×D79—81no—
157C2×C4×D79—81no—
158C2×C4×D79C22×C411yescharacteristic
159C2×C4×D79C2×D411yescharacteristic
160C2×C4×D79#4111yescharacteristic
161#11929611—22no—
162D632—22no—
163#11929609—22no—
164#11929607—81no—
165#11929607—81no—
166C2×C79⋊C8—21no—
167C2×C79⋊C8—21no—
168C2×C79⋊C8—21no—
169C2×C79⋊C8—81no—
170C2×C79⋊C8C22×C411yescharacteristic
171C2×C79⋊C8C22×C411yescharacteristic
172C2×C79⋊C8C2×D411yescharacteristic
173C2×C79⋊C8C2×D411yescharacteristic
174C8×D79—81no—
175C8×D79—81no—
176C8×D79—81no—
177C4⋊Dic79—21no—
178C4⋊Dic79C22×C411yescharacteristic
179C4⋊Dic79C22×C411yescharacteristic
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—
188(C2×D79)⋊C4—21no—
189(C2×D79)⋊C4—21no—
190(C2×D79)⋊C4—21no—
191C4×Dic79C22×C411yescharacteristic
192C4×Dic79C2×D411yescharacteristic
193C4×Dic79#4111yescharacteristic
194C2×Q8×C79C22×C411yescharacteristic
195C2×D4×C79C22×C411yescharacteristic
196C22×C4×C79—21no—
197SD16×C79—22no—
198C2×C8×C79—21no—
199C2×C8×C79—21no—
200C2×C8×C79—21no—
201C2×C8×C79—81no—
202C2×C8×C79C22×C411yescharacteristic
203C2×C8×C79C22×C411yescharacteristic
204C2×C8×C79C2×D411yescharacteristic
205C2×C8×C79C2×D411yescharacteristic
206C4⋊C4×C79—21no—
207C4⋊C4×C79C22×C411yescharacteristic
208C4⋊C4×C79C22×C411yescharacteristic
209C22⋊C4×C79—21no—
210C42×C79—21no—
211C42×C79C2411yescharacteristic
212C23×C79—21no—
213C22×D79—21no—
214C22×D79—24no—
215C22×D79—21no—
216C22×D79—81no—
217C22×D79—81no—
218C22×D79C4×D411yescharacteristic
219Q8×C79—21no—
220Q8×C79—21no—
221D4×C79—21no—
222D4×C79#13112yes—
223C2×C4×C79—21no—
224C2×C4×C79—21no—
225C2×C4×C79—21no—
226C2×C4×C79—41no—
227C2×C4×C79—41no—
228C2×C4×C79C23×C411yescharacteristic
229C2×C4×C79C22×D411yescharacteristic
230C2×C4×C79C2×#4111yescharacteristic
231C22⋊D79—24no—
232C2×Dic79—21no—
233C2×Dic79—21no—
234C2×Dic79—81no—
235C2×Dic79C4×D411yescharacteristic
236C2×Dic79C4×D411yescharacteristic
237C2×Dic79C4×D411yescharacteristic
238D316—21no—
239D316—21no—
240D316—21no—
241D316—81no—
242D316#13112yes—
243C4×D79—21no—
244C4×D79—21no—
245C4×D79—21no—
246C4×D79—21no—
247C4×D79—41no—
248C4×D79—41no—
249C4×D79—41no—
250C4×D79—41no—
251C4×D79—41no—
252C4×D79—81no—
253C4×D79—81no—
254C8×C79—21no—
255C8×C79—21no—
256C8×C79—21no—
257C8×C79—21no—
258C8×C79—41no—
259C8×C79—41no—
260C8×C79—41no—
261C79⋊C8—21no—
262C79⋊C8—21no—
263C79⋊C8—21no—
264C79⋊C8—21no—
265C79⋊C8—41no—
266C79⋊C8—41no—
267C79⋊C8—41no—
268C22×C79—22no—
269C22×C79—41no—
270C22×C79—41no—
271C22×C79C2×C4×D411yescharacteristic
272C2×D79—22no—
273C2×D79—21no—
274C2×D79—21no—
275C2×D79—21no—
276C2×D79—41no—
277C2×D79—41no—
278C2×D79—41no—
279C2×D79—41no—
280C2×D79—41no—
281C2×D79—84no—
282C4×C79—21no—
283C4×C79—41no—
284C4×C79—41no—
285C4×C79—41no—
286C4×C79C2×#13111yescharacteristic
287C4×C79C2×#13111yescharacteristic
288C4×C79#57611yescharacteristic
289C4×C79#57711yescharacteristic
290Dic79—21no—
291Dic79—21no—
292Dic79—21no—
293Dic79—41no—
294Dic79—41no—
295#17918—791nomaximal, Sylow
296C2×C79—42no—
297C2×C79#292911yescharacteristic
298C2×C79#294611yescharacteristic
299C2×C79#295711yescharacteristic
300D79—41no—
301D79—42no—
302D79—81no—
303#3105—791no—
304#2993—791no—
305#2972—791no—
306#1586—791no—
307#1585—791no—
308#1582—791no—
309#1581—791no—
310#1580—791no—
311SD16⋊C8—791no—
312C8×SD16D7911yescharacteristic, complemented
313C8×D8—791no—
314#1574—791no—
315#1486—791no—
316#1482—791no—
317#1454—791no—
318C79#1791811yescharacteristic, complemented, Sylow
319#529—791no—
320C2×C4×D4—791no—
321#484—791no—
322#481—791no—
323#478—791no—
324#478—791no—
325#463—791no—
326#463—791no—
327#450—791no—
328#447—791no—
329#446—791no—
330C8×Q8C2×D7911yescharacteristic, complemented
331C4×SD16—791no—
332C4×SD16C2×D7911yescharacteristic
333C4×D8—791no—
334C4×D8—791no—
335#436—791no—
336#436—1581no—
337#435—1581no—
338C8×D4—791no—
339C8×D4—791no—
340C8×D4—791no—
341C8×D4—791no—
342C8×D4—1581no—
343C8×D4—1581no—
344C8×D4C2×D7911yescharacteristic
345#433—1581no—
346#431—791no—
347#431—791no—
348#431—791no—
349#431—791no—
350#424—1581no—
351#423—791no—
352C2×C22⋊C8—1581no—
353C2×C22⋊C8—1581no—
354#335—791no—
355#335—791no—
356#334—791no—
357#334C2×D7911yescharacteristic, complemented
358Q8⋊C8—791no—
359Q8⋊C8C2×D7911yescharacteristic, complemented
360D4⋊C8—791no—
361D4⋊C8—791no—
362D4⋊C8—791no—
363D4⋊C8—791no—
364D4⋊C8—791no—
365D4⋊C8C2×D7911yescharacteristic
366#322—791no—
367#322—791no—
368#322—791no—
369C82C2×D7911yescharacteristic, complemented
370C22×D4—791no—
371C23×C4—1581no—
372C2×SD16—791no—
373C2×SD16C4×D7911yescharacteristic
374C2×D8—791no—
375C2×D8—791no—
376C2×M16—1581no—
377C2×M16—1581no—
378C22×C8—21no—
379C22×C8—1581no—
380C22×C8—1581no—
381C22×C8—1581no—
382C22×C8—1581no—
383#126—1581no—
384#125—791no—
385#124—791no—
386#123—1581no—
387C22⋊Q8—791no—
388#121—791no—
389C4×Q8C22×D7911yescharacteristic
390C4×D4—791no—
391C4×D4—791no—
392C4×D4—791no—
393C4×D4—791no—
394C4×D4—1581no—
395C4×D4—1581no—
396C4×D4—1581no—
397C4×D4C22×D7911yescharacteristic
398#117—1581no—
399C2×C4⋊C4—791no—
400C2×C22⋊C4—1581no—
401C2×C42—791no—
402#107—791no—
403#107—791no—
404#106—791no—
405#106C4×D7911yescharacteristic
406C4⋊C8—21no—
407C4⋊C8—791no—
408C4⋊C8—791no—
409C4⋊C8—791no—
410C4⋊C8—791no—
411C4⋊C8—791no—
412C4⋊C8—791no—
413C4⋊C8—1581no—
414C4⋊C8—1581no—
415C4⋊C8—1581no—
416C4⋊C8C22×D7911yescharacteristic
417C4⋊C8C22×D7911yescharacteristic
418#103—791no—
419#103C4×D7911yescharacteristic
420#102—791no—
421#102—791no—
422#102—791no—
423#102—791no—
424#102—791no—
425#102C4×D7911yescharacteristic
426C22⋊C8—21no—
427C22⋊C8—791no—
428C22⋊C8—791no—
429C22⋊C8—791no—
430C22⋊C8—791no—
431C22⋊C8—1581no—
432C22⋊C8—1581no—
433C22⋊C8—1581no—
434C22⋊C8—1581no—
435C22⋊C8—1581no—
436C22⋊C8—1581no—
437C22⋊C8—1581no—
438#97—1581no—
439C4×C8—21no—
440C4×C8—791no—
441C4×C8—791no—
442C4×C8—791no—
443C4×C8—1581no—
444C4×C8C22×D7911yescharacteristic
445C4×C8C22×D7911yescharacteristic
446C4×C8C22×D7911yescharacteristic
447C24—1581no—
448C2×Q8C2×C4×D7911yescharacteristic
449C2×D4—791no—
450C2×D4—791no—
451C2×D4—1581no—
452C2×D4—1581no—
453C2×D4—1582no—
454C2×D4—1582no—
455C2×D4C2×C4×D7911yescharacteristic
456C22×C4—21no—
457C22×C4—791no—
458C22×C4—791no—
459C22×C4—791no—
460C22×C4—6321no—
461C22×C4—6321no—
462C22×C4—1581no—
463C22×C4—1581no—
464C22×C4—1581no—
465C22×C4—1581no—
466C22×C4—1581no—
467SD16—22no—
468SD16—1582no—
469D8—1582no—
470D8—1582no—
471M16—6321no—
472M16—6321no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—81no—
477C2×C8—791no—
478C2×C8—791no—
479C2×C8—791no—
480C2×C8—791no—
481C2×C8—6321no—
482C2×C8—6321no—
483C2×C8—6321no—
484C2×C8—6321no—
485C2×C8—1581no—
486C2×C8—1581no—
487C2×C8—1581no—
488C2×C8C2×C4×D7911yescharacteristic
489C2×C8C2×C4×D7911yescharacteristic
490C2×C8D4×D7911yescharacteristic
491C2×C8D4×D7911yescharacteristic
492C4⋊C4—21no—
493C4⋊C4—791no—
494C4⋊C4—791no—
495C4⋊C4—1581no—
496C4⋊C4—1581no—
497C4⋊C4—1581no—
498C4⋊C4—1581no—
499C4⋊C4C2×C4×D7911yescharacteristic
500C4⋊C4C2×C4×D7911yescharacteristic
501C22⋊C4—21no—
502C22⋊C4—1581no—
503C22⋊C4—1581no—
504C22⋊C4—1581no—
505C22⋊C4—1581no—
506C22⋊C4—1581no—
507C22⋊C4—1581no—
508C22⋊C4—1581no—
509C22⋊C4—1581no—
510C42—21no—
511C42—791no—
512C42—791no—
513C42—791no—
514C42C23×D7911yescharacteristic
515C23—21no—
516C23—791no—
517C23—6321no—
518C23—6321no—
519C23—1584no—
520C23—1581no—
521C23—1581no—
522Q8—21no—
523Q8—21no—
524D4—21no—
525D4—792no—
526D4—6321no—
527D4—1584no—
528D4—1581no—
529D4—1581no—
530D4—1581no—
531D4#1457916712yes—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—3161no—
538C2×C4—3161no—
539C2×C4—3161no—
540C2×C4—3161no—
541C2×C4—3161no—
542C2×C4—791no—
543C2×C4—791no—
544C2×C4—791no—
545C2×C4—6321no—
546C2×C4—6321no—
547C2×C4—6321no—
548C2×C4—1581no—
549C2×C4—1581no—
550C2×C4—1581no—
551C2×C4—1581no—
552C2×C4—1581no—
553C2×C4—1581no—
554C2×C4C22×C4×D7911yescharacteristic
555C2×C4C2×D4×D7911yescharacteristic
556C2×C4#41×D7911yescharacteristic
557C8—21no—
558C8—21no—
559C8—21no—
560C8—21no—
561C8—41no—
562C8—41no—
563C8—41no—
564C8—3161no—
565C8—3161no—
566C8—3161no—
567C8—1581no—
568C8—1581no—
569C8—1581no—
570C8—1581no—
571C22—22no—
572C22—41no—
573C22—41no—
574C22—3161no—
575C22—3161no—
576C22—3161no—
577C22—3161no—
578C22—3161no—
579C22—6324no—
580C22—1581no—
581C22—1581no—
582C22—1581no—
583C22—1582no—
584C22C4×D4×D7911yescharacteristic
585C4—21no—
586C4—41no—
587C4—41no—
588C4—41no—
589C4—3161no—
590C4—3161no—
591C4—1581no—
592C4—1581no—
593C4—1581no—
594C4#131×D7911yescharacteristic
595C4C2×#1457916711yescharacteristic
596C4#1909936511yescharacteristic
597C4#1909936911yescharacteristic
598C2—42no—
599C2—3162no—
600C2—3161no—
601C2—6321no—
602C2#2377208911yescharacteristic
603C2#2377222711yescharacteristic
604C2#2377231211yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D79
commutatorC2×C4×C79C23×C4
FrattiniC42C23×D79
FittingC8×SD16×C79C2
socleC22×C79C2×C4×D4
prime radical#19098698C4
cosocleC42×C79C24
hypercenterC8×SD16D79
hypocenterC79#17918
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ C8×SD16

Complements

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

NH ≅ G/Ntype
2. #23772548C2semidirect
3. #23772421C2semidirect
4. #23762263C2semidirect
5. #23762258C2semidirect
6. #23762239C2semidirect
7. #23762226C2semidirect
8. #23762222C2semidirect
9. #23762210C2semidirect
10. C8×D632C2semidirect
11. #23762187C2semidirect
12. #23761823C2semidirect
13. #23761807C2semidirect
14. #23761699C2semidirect
15. C8×SD16×C79C2semidirect
26. #19098694C22semidirect
31. #19098640C22semidirect
34. C8×D316C22semidirect
48. #19098316C22semidirect
49. #19098315C22semidirect
50. #19098313C22semidirect
51. Q8⋊(C79⋊C8)C22semidirect
53. #19098292C22semidirect
56. #19098290C22semidirect
57. #19098285C22semidirect
58. #19098285C22semidirect
59. #19098285C22semidirect
60. C8×Q8×C79C22semidirect
63. #334×C79C22semidirect
64. Q8⋊C8×C79C22semidirect
66. C82×C79C22semidirect
312. C8×SD16D79semidirect
318. C79#17918semidirect
330. C8×Q8C2×D79semidirect
357. #334C2×D79semidirect
359. Q8⋊C8C2×D79semidirect
369. C82C2×D79semidirect

Maximal subgroups

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

subgroupindex
#237734512
#237725482
#237724212
#237622632
#237622582
#237622392
#237622262
#237622222
#237622102
C8×D6322
#237621872
#237618232
#237618072
#237616992
C8×SD16×C792
#1791879