Finite Group Databasebeta

← #33055608

Subgroups — #33055608

Order 18688.

700 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22619047C211yesmaximal, characteristic, complemented
2#22618952C211yesmaximal, characteristic, complemented
3C4×#14556618C211yesmaximal, characteristic
4#22608795C211yesmaximal, characteristic, complemented
5#22608794C211yesmaximal, characteristic, complemented
6#22608789C211yesmaximal, characteristic, complemented
7#22608786C211yesmaximal, characteristic, complemented
8#22608770C211yesmaximal, characteristic, complemented
9#22608762C211yesmaximal, characteristic, complemented
10#22608757C211yesmaximal, characteristic, complemented
11#22608680C211yesmaximal, characteristic, complemented
12#22608312C211yesmaximal, characteristic, complemented
13D4⋊C8×D73C211yesmaximal, characteristic, complemented
14#22608219C211yesmaximal, characteristic, complemented
15C73×#1582C211yesmaximal, characteristic, complemented
16C2×#14556618C411yescharacteristic
17C4×#11924029C2211yescharacteristic
18C4×D4×D73C2211yescharacteristic
19#17988718C2211yescharacteristic, complemented
20#17988698C411yescharacteristic
21#17988696C411yescharacteristic
22#17988690C411yescharacteristic
23#17988688C411yescharacteristic
24C4×#11924015C2211yescharacteristic
25C4×D584C2211yescharacteristic
26C4×#11924013C2211yescharacteristic
27#17988666C2211yescharacteristic, complemented
28#17988664C2211yescharacteristic, complemented
29#17988660—21no—
30#17988657—21no—
31C8×D292C2211yescharacteristic
32D4×#10596831C2211yescharacteristic
33#17988652C2211yescharacteristic
34C8×C22⋊D73—21no—
35#17988650—21no—
36#17988642—21no—
37#17988639C2211yescharacteristic
38#17988638C2211yescharacteristic
39#17988626C411yescharacteristic
40#17988609—21no—
41C4⋊C8×D73C2211yescharacteristic
42#17988590C411yescharacteristic
43#17988587C411yescharacteristic
44C2×#14556503—21no—
45C22⋊C8×D73—21no—
46C4×#11924011C2211yescharacteristic
47C4×C8×D73C2211yescharacteristic
48#17988340C2211yescharacteristic, complemented
49#17988339C2211yescharacteristic, complemented
50#17988337C2211yescharacteristic, complemented
51Q8⋊#10596831C2211yescharacteristic, complemented
52#17988316C2211yescharacteristic
53#17988316C2211yescharacteristic, complemented
54D4⋊#10596831C2211yescharacteristic
55#17988314C2211yescharacteristic
56#17988314C2211yescharacteristic, complemented
57#17988309C2211yescharacteristic, complemented
58#17988309C2211yescharacteristic, complemented
59#17988309C2211yescharacteristic, complemented
60#446×C73C2211yescharacteristic, complemented
61C4×SD16×C73C2211yescharacteristic
62C8×D4×C73C2211yescharacteristic
63#334×C73C2211yescharacteristic, complemented
64Q8⋊C8×C73C2211yescharacteristic, complemented
65D4⋊C8×C73C2211yescharacteristic
66#322×C73C2211yescharacteristic, complemented
67C2×#11924029C2×C411yescharacteristic
68C2×D4×D73C2×C411yescharacteristic
69C22×C4×D73—21no—
70#14556618—24no—
71C2×#11924015C2×C411yescharacteristic
72C2×D584C2×C411yescharacteristic
73C2×#11924013C2×C411yescharacteristic
74C2×#11924011—21no—
75C2×#11924011—21no—
76C2×#11924011C2×C411yescharacteristic
77C2×#11924011C2×C411yescharacteristic
78C22×#10596831—21no—
79C2×C8×D73—21no—
80C2×C8×D73—21no—
81C2×C8×D73—21no—
82C2×C8×D73—21no—
83Q8×Dic73C2×C411yescharacteristic
84C4×D292—21no—
85C4×D292C2311yescharacteristic
86C4×D292C2311yescharacteristic
87D4×Dic73C2×C411yescharacteristic
88#14556553—21no—
89#14556553C2×C411yescharacteristic
90#14556553C2×C411yescharacteristic
91C4×C22⋊D73—21no—
92#14556551—21no—
93#14556549—21no—
94#14556549C2×C411yescharacteristic
95C4⋊C4×D73C2×C411yescharacteristic
96C22⋊C4×D73—21no—
97C42×D73—21no—
98C42×D73C2311yescharacteristic
99#14556525C2×C411yescharacteristic
100#14556524C2×C411yescharacteristic
101#14556522C2×C411yescharacteristic
102C4⋊#10596831—21no—
103C4⋊#10596831C2311yescharacteristic
104C4⋊#10596831C2311yescharacteristic
105#14556520—21no—
106#14556520D411yescharacteristic
107#14556520D411yescharacteristic
108#14556515C2×C411yescharacteristic
109#14556513C2×C411yescharacteristic
110#14556513C2×C411yescharacteristic
111#14556512C2×C411yescharacteristic
112#14556511C2×C411yescharacteristic
113#14556511C2×C411yescharacteristic
114C22⋊#10596831—21no—
115#14556503—21no—
116#14556503—21no—
117#14556503—21no—
118#14556503—21no—
119#14556503—21no—
120#14556503D411yescharacteristic
121#14556503D411yescharacteristic
122#14556502—21no—
123#14556501C2×C411yescharacteristic
124#14556501C2×C411yescharacteristic
125C4×#10596831C2311yescharacteristic
126C4×#10596831C2311yescharacteristic
127C4×#10596831C2311yescharacteristic
128C8×Dic73—21no—
129C8×Dic73—21no—
130C2×SD16×C73C2×C411yescharacteristic
131C22×C8×C73—21no—
132C4×Q8×C73C2311yescharacteristic
133C4×D4×C73C2311yescharacteristic
134#106×C73C2×C411yescharacteristic
135C4⋊C8×C73—21no—
136C4⋊C8×C73C2311yescharacteristic
137C4⋊C8×C73C2311yescharacteristic
138#103×C73C2×C411yescharacteristic
139#102×C73C2×C411yescharacteristic
140C22⋊C8×C73—21no—
141#97×C73—21no—
142C4×C8×C73C2311yescharacteristic
143C4×C8×C73C2311yescharacteristic
144C4×C8×C73C2311yescharacteristic
145C23×D73—21no—
146#11924029—22no—
147#11924029—22no—
148C2×D292—21no—
149C2×D292C22×C411yescharacteristic
150C2×D292C22×C411yescharacteristic
151C2×C22⋊D73—21no—
152D4×D73—22no—
153D4×D73M1614yescomplemented
154C22×Dic73—21no—
155C2×C4×D73—21no—
156C2×C4×D73—21no—
157C2×C4×D73—21no—
158C2×C4×D73—21no—
159C2×C4×D73—21no—
160C2×C4×D73—21no—
161C2×C4×D73—81no—
162C2×C4×D73—81no—
163C2×C4×D73C22×C411yescharacteristic
164C2×C4×D73C2×D411yescharacteristic
165C2×C4×D73#4111yescharacteristic
166#11924015—22no—
167D584—22no—
168#11924013—22no—
169#11924011—22no—
170#11924011—22no—
171#11924011—81no—
172#11924011—81no—
173C2×#10596831—21no—
174C2×#10596831—21no—
175C2×#10596831—21no—
176C2×#10596831—81no—
177C2×#10596831C22×C411yescharacteristic
178C2×#10596831C22×C411yescharacteristic
179C2×#10596831C2×D411yescharacteristic
180C2×#10596831C2×D411yescharacteristic
181C8×D73—42no—
182C8×D73—81no—
183C8×D73—81no—
184C8×D73—81no—
185C4⋊Dic73—21no—
186C4⋊Dic73C22×C411yescharacteristic
187C4⋊Dic73C22×C411yescharacteristic
188#11924007—21no—
189C22⋊Dic73—21no—
190#11924005—21no—
191#11924005—21no—
192#11924005—21no—
193#11924005—21no—
194#11924005—21no—
195C4×Dic73—21no—
196C4×Dic73—21no—
197C4×Dic73C22×C411yescharacteristic
198C4×Dic73C2×D411yescharacteristic
199C4×Dic73#4111yescharacteristic
200C2×Q8×C73C22×C411yescharacteristic
201C2×D4×C73C22×C411yescharacteristic
202C22×C4×C73—21no—
203SD16×C73—22no—
204C2×C8×C73—21no—
205C2×C8×C73—21no—
206C2×C8×C73—21no—
207C2×C8×C73—81no—
208C2×C8×C73C22×C411yescharacteristic
209C2×C8×C73C22×C411yescharacteristic
210C2×C8×C73C2×D411yescharacteristic
211C2×C8×C73C2×D411yescharacteristic
212C4⋊C4×C73—21no—
213C4⋊C4×C73C22×C411yescharacteristic
214C4⋊C4×C73C22×C411yescharacteristic
215C22⋊C4×C73—21no—
216C42×C73—21no—
217C42×C73C2411yescharacteristic
218C23×C73—21no—
219C22×D73—21no—
220C22×D73—24no—
221C22×D73—21no—
222C22×D73—81no—
223C22×D73—81no—
224C22×D73C4×D411yescharacteristic
225Q8×C73—21no—
226Q8×C73—21no—
227D4×C73—21no—
228D4×C73C2×M1612yes—
229C2×C4×C73—21no—
230C2×C4×C73—21no—
231C2×C4×C73—21no—
232C2×C4×C73—41no—
233C2×C4×C73—41no—
234C2×C4×C73C23×C411yescharacteristic
235C2×C4×C73C22×D411yescharacteristic
236C2×C4×C73C2×#4111yescharacteristic
237C22⋊D73—24no—
238C2×Dic73—21no—
239C2×Dic73—21no—
240C2×Dic73—81no—
241C2×Dic73C4×D411yescharacteristic
242C2×Dic73C4×D411yescharacteristic
243C2×Dic73C4×D411yescharacteristic
244D292—21no—
245D292—21no—
246D292—21no—
247D292—42no—
248D292C2×M1612yes—
249C4×D73—21no—
250C4×D73—21no—
251C4×D73—41no—
252C4×D73—41no—
253C4×D73—41no—
254C4×D73—41no—
255C4×D73—41no—
256C4×D73—42no—
257C4×D73—81no—
258C4×D73—81no—
259C4×D73—81no—
260C4×D73C2×M1612yes—
261C4×D73#13112yes—
262C8×C73—21no—
263C8×C73—21no—
264C8×C73—21no—
265C8×C73—21no—
266C8×C73—41no—
267C8×C73—41no—
268C8×C73—41no—
269#10596831—22no—
270#10596831—21no—
271#10596831—21no—
272#10596831—21no—
273#10596831—21no—
274#10596831—41no—
275#10596831—41no—
276C22×C73—22no—
277C22×C73—41no—
278C22×C73—41no—
279C22×C73C2×C4×D411yescharacteristic
280C2×D73—21no—
281C2×D73—21no—
282C2×D73—22no—
283C2×D73—41no—
284C2×D73—41no—
285C2×D73—41no—
286C2×D73—41no—
287C2×D73—41no—
288C2×D73—84no—
289C2×D73#43512yes—
290C4×C73—21no—
291C4×C73—41no—
292C4×C73—41no—
293C4×C73—41no—
294C4×C73C22×M1611yescharacteristic
295C4×C73C2×#13111yescharacteristic
296C4×C73#57611yescharacteristic
297C4×C73#57711yescharacteristic
298Dic73—21no—
299Dic73—21no—
300Dic73—41no—
301Dic73—41no—
302Dic73#43512yes—
303#17873—731nomaximal, Sylow
304C2×C73—42no—
305C2×C73C2×#43511yescharacteristic
306C2×C73#294611yescharacteristic
307C2×C73#295611yescharacteristic
308D73—22no—
309D73—42no—
310D73—81no—
311#2989—731no—
312#2972—731no—
313C4×AGL(1,ℤ/8)—731no—
314#1586—731no—
315#1586—731no—
316#1585—731no—
317#1585—731no—
318#1582—731no—
319#1582D7311yescharacteristic, complemented
320#1581—731no—
321#1581—731no—
322#1565—731no—
323#1476—731no—
324C2×D4⋊C8—731no—
325#1451—731no—
326C73#1787311yescharacteristic, complemented, Sylow
327C2×AGL(1,ℤ/8)—731no—
328C4×#41—731no—
329C2×C4×D4—731no—
330#446—731no—
331#446C2×D7311yescharacteristic, complemented
332#442—731no—
333#442—731no—
334#440—731no—
335#440—731no—
336C4×SD16—731no—
337C4×SD16C2×D7311yescharacteristic
338C4×D8—731no—
339C4×D8—731no—
340#436—731no—
341#436—731no—
342#435—1461no—
343#435—1461no—
344C8×D4—731no—
345C8×D4—731no—
346C8×D4—731no—
347C8×D4—1461no—
348C8×D4—1461no—
349C8×D4C2×D7311yescharacteristic
350#432—1461no—
351#431—731no—
352#431—731no—
353#428—731no—
354#423—1461no—
355C2×C4⋊C8—731no—
356#418—731no—
357#417—731no—
358C2×C22⋊C8—1461no—
359C2×C22⋊C8—1461no—
360C4×M16—731no—
361C2×C4×C8—731no—
362#335—731no—
363#335—731no—
364#334—731no—
365#334C2×D7311yescharacteristic, complemented
366Q8⋊C8—731no—
367Q8⋊C8C2×D7311yescharacteristic, complemented
368D4⋊C8—731no—
369D4⋊C8—731no—
370D4⋊C8—731no—
371D4⋊C8—731no—
372D4⋊C8—731no—
373D4⋊C8C2×D7311yescharacteristic
374#322—731no—
375#322—731no—
376#322—731no—
377#322C2×D7311yescharacteristic, complemented
378C2×#41—731no—
379C22×D4—731no—
380C23×C4—1461no—
381AGL(1,ℤ/8)—1464no—
382C2×SD16—731no—
383C2×SD16C4×D7311yescharacteristic
384C2×D8—731no—
385C2×D8—731no—
386C2×M16—731no—
387C2×M16—731no—
388C2×M16—1461no—
389C2×M16—1461no—
390C22×C8—21no—
391C22×C8—1461no—
392C22×C8—1461no—
393C22×C8—1461no—
394C22×C8—1461no—
395C22×C8—1461no—
396C4×Q8—731no—
397C4×Q8C22×D7311yescharacteristic
398C4×D4—731no—
399C4×D4—731no—
400C4×D4—731no—
401C4×D4—731no—
402C4×D4—731no—
403C4×D4—1461no—
404C4×D4—1461no—
405C4×D4—1461no—
406C4×D4—1461no—
407C4×D4C22×D7311yescharacteristic
408#117—731no—
409#117—1461no—
410C2×C4⋊C4—731no—
411C2×C22⋊C4—1461no—
412C2×C42—731no—
413C2×C42—1461no—
414#107—731no—
415#107—731no—
416#106—731no—
417#106C4×D7311yescharacteristic
418C4⋊C8—21no—
419C4⋊C8—731no—
420C4⋊C8—731no—
421C4⋊C8—731no—
422C4⋊C8—731no—
423C4⋊C8—1461no—
424C4⋊C8—1461no—
425C4⋊C8C22×D7311yescharacteristic
426C4⋊C8C22×D7311yescharacteristic
427#103—731no—
428#103C4×D7311yescharacteristic
429#102—731no—
430#102—731no—
431#102—731no—
432#102—731no—
433#102—731no—
434#102C4×D7311yescharacteristic
435C22⋊C8—21no—
436C22⋊C8—731no—
437C22⋊C8—731no—
438C22⋊C8—1461no—
439C22⋊C8—1461no—
440C22⋊C8—1461no—
441C22⋊C8—1461no—
442C22⋊C8—1461no—
443C22⋊C8—1461no—
444#97—21no—
445#97—731no—
446#97—731no—
447#97—1461no—
448C4×C8—731no—
449C4×C8—731no—
450C4×C8—731no—
451C4×C8—1461no—
452C4×C8—1461no—
453C4×C8C22×D7311yescharacteristic
454C4×C8C22×D7311yescharacteristic
455C4×C8C22×D7311yescharacteristic
456C24—1461no—
457#41—1462no—
458#41—1462no—
459C2×Q8C2×C4×D7311yescharacteristic
460C2×D4—734no—
461C2×D4—731no—
462C2×D4—731no—
463C2×D4—1461no—
464C2×D4—1462no—
465C2×D4—1461no—
466C2×D4C2×C4×D7311yescharacteristic
467C22×C4—21no—
468C22×C4—5841no—
469C22×C4—5841no—
470C22×C4—731no—
471C22×C4—731no—
472C22×C4—731no—
473C22×C4—1461no—
474C22×C4—1461no—
475C22×C4—1461no—
476C22×C4—1461no—
477C22×C4—1461no—
478C22×C4—1461no—
479C22×C4—1461no—
480SD16—22no—
481SD16—1462no—
482D8—1462no—
483D8—1462no—
484M16—5841no—
485M16—5841no—
486M16—1462no—
487M16—1462no—
488C2×C8—21no—
489C2×C8—21no—
490C2×C8—21no—
491C2×C8—81no—
492C2×C8—2922no—
493C2×C8—5841no—
494C2×C8—5841no—
495C2×C8—5841no—
496C2×C8—5841no—
497C2×C8—731no—
498C2×C8—731no—
499C2×C8—731no—
500C2×C8—731no—
501C2×C8—1461no—
502C2×C8—1461no—
503C2×C8—1461no—
504C2×C8C2×C4×D7311yescharacteristic
505C2×C8C2×C4×D7311yescharacteristic
506C2×C8D4×D7311yescharacteristic
507C2×C8D4×D7311yescharacteristic
508C4⋊C4—21no—
509C4⋊C4—731no—
510C4⋊C4—731no—
511C4⋊C4—1461no—
512C4⋊C4—1461no—
513C4⋊C4C2×C4×D7311yescharacteristic
514C4⋊C4C2×C4×D7311yescharacteristic
515C22⋊C4—21no—
516C22⋊C4—1461no—
517C22⋊C4—1461no—
518C22⋊C4—1461no—
519C22⋊C4—1461no—
520C22⋊C4—1461no—
521C22⋊C4—1461no—
522C42—21no—
523C42—731no—
524C42—731no—
525C42—731no—
526C42—1461no—
527C42—1461no—
528C42C23×D7311yescharacteristic
529C23—21no—
530C23—5841no—
531C23—5841no—
532C23—731no—
533C23—1464no—
534C23—1461no—
535C23—1461no—
536Q8—21no—
537Q8—21no—
538D4—21no—
539D4—2922no—
540D4—732no—
541D4—1464no—
542D4—1461no—
543D4—1461no—
544D4—1461no—
545D4M16×D7312yes—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—2921no—
552C2×C4—2922no—
553C2×C4—2921no—
554C2×C4—2921no—
555C2×C4—2921no—
556C2×C4—2921no—
557C2×C4—5841no—
558C2×C4—5841no—
559C2×C4—5841no—
560C2×C4—5841no—
561C2×C4—731no—
562C2×C4—732no—
563C2×C4—732no—
564C2×C4—731no—
565C2×C4—731no—
566C2×C4—1461no—
567C2×C4—1461no—
568C2×C4—1461no—
569C2×C4—1461no—
570C2×C4C22×C4×D7311yescharacteristic
571C2×C4C2×D4×D7311yescharacteristic
572C2×C4#41×D7311yescharacteristic
573C8—21no—
574C8—21no—
575C8—21no—
576C8—21no—
577C8—41no—
578C8—41no—
579C8—41no—
580C8—2921no—
581C8—2921no—
582C8—1462no—
583C8—1461no—
584C8—1461no—
585C8—1461no—
586C8—1461no—
587C22—22no—
588C22—41no—
589C22—41no—
590C22—2921no—
591C22—2921no—
592C22—2921no—
593C22—2921no—
594C22—2921no—
595C22—5844no—
596C22—732no—
597C22—1462no—
598C22—1461no—
599C22—1461no—
600C22C4×D4×D7311yescharacteristic
601C4—21no—
602C4—41no—
603C4—41no—
604C4—41no—
605C4—2921no—
606C4—2921no—
607C4—732no—
608C4—1461no—
609C4—1461no—
610C4C2×M16×D7311yescharacteristic
611C4#131×D7311yescharacteristic
612C4#1798938911yescharacteristic
613C4#1798939311yescharacteristic
614C2—42no—
615C2—2922no—
616C2—5841no—
617C2—1462no—
618C2#435×D7311yescharacteristic
619C2#2261875811yescharacteristic
620C2#2261883911yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D73
commutatorC2×C4×C73C23×C4
FrattiniC42C23×D73
FittingC73×#1582C2
socleC22×C73C2×C4×D4
prime radicalC2×#14556618C4
cosocleC42×C73C24
hypercenter#1582D73
hypocenterC73#17873
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ #1582

Complements

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

NH ≅ G/Ntype
1. #22619047C2semidirect
2. #22618952C2semidirect
4. #22608795C2semidirect
5. #22608794C2semidirect
6. #22608789C2semidirect
7. #22608786C2semidirect
8. #22608770C2semidirect
9. #22608762C2semidirect
10. #22608757C2semidirect
11. #22608680C2semidirect
12. #22608312C2semidirect
13. D4⋊C8×D73C2semidirect
14. #22608219C2semidirect
15. C73×#1582C2semidirect
19. #17988718C22semidirect
27. #17988666C22semidirect
28. #17988664C22semidirect
48. #17988340C22semidirect
49. #17988339C22semidirect
50. #17988337C22semidirect
51. Q8⋊#10596831C22semidirect
53. #17988316C22semidirect
56. #17988314C22semidirect
57. #17988309C22semidirect
58. #17988309C22semidirect
59. #17988309C22semidirect
60. #446×C73C22semidirect
63. #334×C73C22semidirect
64. Q8⋊C8×C73C22semidirect
66. #322×C73C22semidirect
153. D4×D73M16semidirect
319. #1582D73semidirect
326. C73#17873semidirect
331. #446C2×D73semidirect
365. #334C2×D73semidirect
367. Q8⋊C8C2×D73semidirect
377. #322C2×D73semidirect

Maximal subgroups

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

subgroupindex
#226190472
#226189522
C4×#145566182
#226087952
#226087942
#226087892
#226087862
#226087702
#226087622
#226087572
#226086802
#226083122
D4⋊C8×D732
#226082192
C73×#15822
#1787373