Finite Group Databasebeta

← #33057528

Subgroups — #33057528

Order 18688.

642 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22618939C211yesmaximal, characteristic
2#22618883C211yesmaximal, characteristic, complemented
3C4×#14556620C211yesmaximal, characteristic
4#22608837C211yesmaximal, characteristic, complemented
5#22608826C211yesmaximal, characteristic, complemented
6#22608801C211yesmaximal, characteristic, complemented
7#22608789C211yesmaximal, characteristic, complemented
8#22608783C211yesmaximal, characteristic, complemented
9#22608740C211yesmaximal, characteristic, complemented
10C8×#11924016C211yesmaximal, characteristic, complemented
11#22608695C211yesmaximal, characteristic, complemented
12#22608376C211yesmaximal, characteristic, complemented
13#22608367C211yesmaximal, characteristic
14#22608235C211yesmaximal, characteristic, complemented
15C73×#1582C211yesmaximal, characteristic, complemented
16C2×#14556620C411yescharacteristic
17C4×#11924027C2211yescharacteristic
18C4×Q8×D73C2211yescharacteristic
19#17988718C2211yescharacteristic
20#17988717C2211yescharacteristic
21#17988716—21no—
22#17988715—21no—
23#17988714C2211yescharacteristic
24C8×Dic146C2211yescharacteristic
25#17988711C2211yescharacteristic
26#17988693C411yescharacteristic
27#17988691C411yescharacteristic
28#17988688C411yescharacteristic
29#17988684C411yescharacteristic
30C4×Dic292C2211yescharacteristic
31C4×#11924018C2211yescharacteristic
32C4×#11924016C2211yescharacteristic
33#17988658—21no—
34#17988655—21no—
35D4×#10596831C2211yescharacteristic
36#17988643C2211yescharacteristic
37#17988641—21no—
38#17988626C411yescharacteristic
39#17988609—21no—
40#17988609C2211yescharacteristic
41#17988594C411yescharacteristic
42#17988585C411yescharacteristic
43#17988539—21no—
44C4×#11924011C2211yescharacteristic
45C4×#11924011C2211yescharacteristic
46C4×#11924011C2211yescharacteristic
47C2×#14556501—21no—
48#17988340C2211yescharacteristic
49#17988339C2211yescharacteristic
50#17988337C2211yescharacteristic
51#17988319C2211yescharacteristic
52#17988319C2211yescharacteristic
53Q8⋊#10596831C2211yescharacteristic
54#17988317C2211yescharacteristic
55#17988317C2211yescharacteristic
56D4⋊#10596831C2211yescharacteristic
57#17988309C2211yescharacteristic
58#17988309C2211yescharacteristic
59C8×#10596831C2211yescharacteristic
60#446×C73C2211yescharacteristic
61C4×SD16×C73C2211yescharacteristic
62C8×D4×C73C2211yescharacteristic
63#334×C73C2211yescharacteristic
64Q8⋊C8×C73C2211yescharacteristic
65D4⋊C8×C73C2211yescharacteristic
66#322×C73C2211yescharacteristic
67C2×#11924027C2×C411yescharacteristic
68C2×Q8×D73C2×C411yescharacteristic
69#14556620—24no—
70C2×Dic292C2×C411yescharacteristic
71C2×#11924018C2×C411yescharacteristic
72C2×#11924016C2×C411yescharacteristic
73C2×#11924011—21no—
74C2×#11924011—21no—
75C2×#11924011C2×C411yescharacteristic
76C2×#11924011C2×C411yescharacteristic
77C2×#11924011D411yescharacteristic
78C2×#11924011D411yescharacteristic
79C22×#10596831—21no—
80Q8×Dic73C2×C411yescharacteristic
81C4×Dic146—21no—
82C4×Dic146C2311yescharacteristic
83C4×Dic146C2311yescharacteristic
84#14556556—21no—
85#14556556C2×C411yescharacteristic
86#14556556C2×C411yescharacteristic
87D4×Dic73C2×C411yescharacteristic
88C4×C22⋊D73—21no—
89#14556551—21no—
90#14556549C2×C411yescharacteristic
91#14556547—21no—
92C4⋊C4×D73—21no—
93C4⋊C4×D73C2×C411yescharacteristic
94C2×C4×Dic73—21no—
95C42×D73—21no—
96C42×D73C2311yescharacteristic
97#14556525C2×C411yescharacteristic
98#14556524C2×C411yescharacteristic
99#14556522C2×C411yescharacteristic
100C4⋊#10596831—21no—
101C4⋊#10596831C2311yescharacteristic
102C4⋊#10596831C2311yescharacteristic
103#14556520—21no—
104#14556520—21no—
105#14556520—21no—
106#14556520—21no—
107#14556520—21no—
108#14556516C2×C411yescharacteristic
109#14556516C2×C411yescharacteristic
110#14556515C2×C411yescharacteristic
111#14556514C2×C411yescharacteristic
112#14556514C2×C411yescharacteristic
113#14556512C2×C411yescharacteristic
114C22⋊#10596831—21no—
115#14556503—21no—
116#14556502—21no—
117#14556501—21no—
118#14556501—21no—
119#14556501—21no—
120#14556501—21no—
121#14556501C2×C411yescharacteristic
122#14556501C2×C411yescharacteristic
123#14556501D411yescharacteristic
124#14556501D411yescharacteristic
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
145#11924027—22no—
146#11924027—41no—
147C2×Dic146—21no—
148C2×Dic146C22×C411yescharacteristic
149C2×Dic146C22×C411yescharacteristic
150Q8×D73—22no—
151Q8×D73—41no—
152C2×C22⋊D73—21no—
153C22×Dic73—21no—
154C22×Dic73—21no—
155C2×C4×D73—21no—
156C2×C4×D73—21no—
157C2×C4×D73C22×C411yescharacteristic
158C2×C4×D73C2×D411yescharacteristic
159C2×C4×D73#4111yescharacteristic
160Dic292—22no—
161#11924018—22no—
162#11924016M1614yescomplemented
163#11924011—22no—
164#11924011—22no—
165#11924011—41no—
166#11924011—41no—
167#11924011—42no—
168#11924011—42no—
169C2×#10596831—21no—
170C2×#10596831—21no—
171C2×#10596831—21no—
172C2×#10596831—42no—
173C2×#10596831C22×C411yescharacteristic
174C2×#10596831C22×C411yescharacteristic
175C2×#10596831C2×D411yescharacteristic
176C2×#10596831C2×D411yescharacteristic
177C4⋊Dic73—21no—
178C4⋊Dic73C22×C411yescharacteristic
179C4⋊Dic73C22×C411yescharacteristic
180#11924007—21no—
181#11924007—21no—
182#11924007—21no—
183#11924007—21no—
184#11924007—21no—
185C22⋊Dic73—21no—
186#11924005—21no—
187C4×Dic73—21no—
188C4×Dic73—21no—
189C4×Dic73—21no—
190C4×Dic73—21no—
191C4×Dic73—21no—
192C4×Dic73—21no—
193C4×Dic73C22×C411yescharacteristic
194C4×Dic73C2×D411yescharacteristic
195C4×Dic73#4111yescharacteristic
196C2×Q8×C73C22×C411yescharacteristic
197C2×D4×C73C22×C411yescharacteristic
198C22×C4×C73—21no—
199SD16×C73—22no—
200C2×C8×C73—21no—
201C2×C8×C73—21no—
202C2×C8×C73—21no—
203C2×C8×C73—42no—
204C2×C8×C73C22×C411yescharacteristic
205C2×C8×C73C22×C411yescharacteristic
206C2×C8×C73C2×D411yescharacteristic
207C2×C8×C73C2×D411yescharacteristic
208C4⋊C4×C73—21no—
209C4⋊C4×C73C22×C411yescharacteristic
210C4⋊C4×C73C22×C411yescharacteristic
211C22⋊C4×C73—21no—
212C42×C73—21no—
213C42×C73C2411yescharacteristic
214C23×C73—21no—
215C22×D73C4×D411yescharacteristic
216Q8×C73—21no—
217Q8×C73—21no—
218D4×C73—21no—
219D4×C73C2×M1612yes—
220C2×C4×C73—21no—
221C2×C4×C73—21no—
222C2×C4×C73—21no—
223C2×C4×C73—41no—
224C2×C4×C73—41no—
225C2×C4×C73C23×C411yescharacteristic
226C2×C4×C73C22×D411yescharacteristic
227C2×C4×C73C2×#4111yescharacteristic
228C22⋊D73—42no—
229C2×Dic73—21no—
230C2×Dic73—21no—
231C2×Dic73—21no—
232C2×Dic73—21no—
233C2×Dic73—42no—
234C2×Dic73—81no—
235C2×Dic73C4×D411yescharacteristic
236C2×Dic73C4×D411yescharacteristic
237C2×Dic73C4×D411yescharacteristic
238C4×D73—21no—
239C4×D73—21no—
240C4×D73—21no—
241C4×D73—21no—
242C4×D73—41no—
243C4×D73—42no—
244C4×D73—81no—
245Dic146—21no—
246Dic146—21no—
247Dic146—21no—
248Dic146—42no—
249Dic146C2×M1612yes—
250C8×C73—21no—
251C8×C73—21no—
252C8×C73—21no—
253C8×C73—41no—
254C8×C73—41no—
255C8×C73—41no—
256C8×C73#13712yescomplemented
257#10596831—22no—
258#10596831—21no—
259#10596831—21no—
260#10596831—21no—
261#10596831—41no—
262#10596831—41no—
263#10596831C2×M1612yes—
264C22×C73—22no—
265C22×C73—41no—
266C22×C73—41no—
267C22×C73C2×C4×D411yescharacteristic
268C2×D73—21no—
269C2×D73—21no—
270C2×D73—21no—
271C4×C73—21no—
272C4×C73—41no—
273C4×C73—41no—
274C4×C73—41no—
275C4×C73C22×M1611yescharacteristic
276C4×C73#56811yescharacteristic
277C4×C73C2×#13711yescharacteristic
278C4×C73#57511yescharacteristic
279Dic73—22no—
280Dic73—21no—
281Dic73—21no—
282Dic73—21no—
283Dic73—41no—
284Dic73—41no—
285Dic73—41no—
286#18047—731nomaximal, Sylow
287C2×C73—42no—
288C2×C73D4×M1611yescharacteristic
289C2×C73C4×#13711yescharacteristic
290C2×C73#295611yescharacteristic
291D73—41no—
292#2970—731no—
293Q8×M16—731no—
294C4×#137—731no—
295#1593—731no—
296#1591—731no—
297#1587—731no—
298#1585—731no—
299#1584—731no—
300#1582D7311yescharacteristic, complemented
301SD16⋊C8—731no—
302C8×SD16—731no—
303#1569—731no—
304#1491—731no—
305#1489—731no—
306#1455—731no—
307C73#1804711yescharacteristic, complemented, Sylow
308C2×#137—731no—
309C4×#41—731no—
310C2×C4×Q8—731no—
311#446—731no—
312#446—731no—
313#446—731no—
314#446—1461no—
315#446—1461no—
316#446C2×D7311yescharacteristic
317C8×Q8—731no—
318C8×Q8—731no—
319#441—731no—
320#441—731no—
321#440—731no—
322#440—731no—
323C4×Q16—731no—
324C4×Q16—731no—
325C4×SD16—731no—
326C4×SD16C2×D7311yescharacteristic
327#435—1461no—
328#435—1461no—
329C8×D4—731no—
330C8×D4C2×D7311yescharacteristic
331#432—731no—
332#432—1461no—
333#428—731no—
334#423—731no—
335#423—1461no—
336#419—731no—
337#417—731no—
338C4×M16—731no—
339C4×M16—731no—
340C4×M16—731no—
341C4×M16—1461no—
342C2×#97—1461no—
343#335—731no—
344#335—731no—
345#334—731no—
346#334C2×D7311yescharacteristic
347Q8⋊C8—731no—
348Q8⋊C8—731no—
349Q8⋊C8—731no—
350Q8⋊C8—731no—
351Q8⋊C8—731no—
352Q8⋊C8C2×D7311yescharacteristic
353D4⋊C8—731no—
354D4⋊C8C2×D7311yescharacteristic
355#322—731no—
356#322—731no—
357#322C2×D7311yescharacteristic
358C82—731no—
359C2×#41—731no—
360C22×Q8—731no—
361#137—1464no—
362C2×Q16—731no—
363C2×Q16—731no—
364C2×SD16—731no—
365C2×SD16C4×D7311yescharacteristic
366C2×M16—731no—
367C2×M16—731no—
368C2×M16—731no—
369C2×M16—731no—
370C2×M16—1461no—
371C2×M16—1461no—
372C22×C8—21no—
373C22×C8—1461no—
374C4×Q8—731no—
375C4×Q8—731no—
376C4×Q8—731no—
377C4×Q8—731no—
378C4×Q8—731no—
379C4×Q8—1461no—
380C4×Q8—1461no—
381C4×Q8C22×D7311yescharacteristic
382C4×D4—731no—
383C4×D4—1461no—
384C4×D4—1461no—
385C4×D4C22×D7311yescharacteristic
386#117—731no—
387#117—1461no—
388C2×C4⋊C4—731no—
389C2×C4⋊C4—1461no—
390C2×C42—731no—
391C2×C42—1461no—
392C2×C42—1461no—
393#107—731no—
394#107—731no—
395#106—731no—
396#106C4×D7311yescharacteristic
397C4⋊C8—21no—
398C4⋊C8—731no—
399C4⋊C8—731no—
400C4⋊C8—1461no—
401C4⋊C8—1461no—
402C4⋊C8—1461no—
403C4⋊C8—1461no—
404C4⋊C8—1461no—
405C4⋊C8—1461no—
406C4⋊C8C22×D7311yescharacteristic
407C4⋊C8C22×D7311yescharacteristic
408#103—731no—
409#103—731no—
410#103—731no—
411#103—731no—
412#103—731no—
413#103C4×D7311yescharacteristic
414#102—731no—
415#102C4×D7311yescharacteristic
416C22⋊C8—21no—
417C22⋊C8—1461no—
418C22⋊C8—1461no—
419#97—21no—
420#97—731no—
421#97—731no—
422#97—731no—
423#97—731no—
424#97—1461no—
425#97—1461no—
426#97—1461no—
427#97—1461no—
428#97—1461no—
429C4×C8—731no—
430C4×C8—731no—
431C4×C8—731no—
432C4×C8—1461no—
433C4×C8—1461no—
434C4×C8C22×D7311yescharacteristic
435C4×C8C22×D7311yescharacteristic
436C4×C8C22×D7311yescharacteristic
437#41—2921no—
438#41—1462no—
439C2×Q8—2921no—
440C2×Q8—731no—
441C2×Q8—731no—
442C2×Q8—1461no—
443C2×Q8—1462no—
444C2×Q8C2×C4×D7311yescharacteristic
445C2×D4—1461no—
446C2×D4C2×C4×D7311yescharacteristic
447C22×C4—21no—
448C22×C4—731no—
449C22×C4—731no—
450C22×C4—731no—
451C22×C4—1461no—
452C22×C4—1461no—
453C22×C4—1461no—
454C22×C4—1461no—
455Q16—1462no—
456Q16—1462no—
457SD16—22no—
458SD16—734no—
459M16—2921no—
460M16—2921no—
461M16—2922no—
462M16—2922no—
463M16—1462no—
464M16—1462no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—21no—
468C2×C8—42no—
469C2×C8—2922no—
470C2×C8—731no—
471C2×C8—731no—
472C2×C8—731no—
473C2×C8—731no—
474C2×C8—1461no—
475C2×C8—1461no—
476C2×C8—1461no—
477C2×C8C2×C4×D7311yescharacteristic
478C2×C8C2×C4×D7311yescharacteristic
479C2×C8D4×D7311yescharacteristic
480C2×C8D4×D7311yescharacteristic
481C4⋊C4—21no—
482C4⋊C4—731no—
483C4⋊C4—731no—
484C4⋊C4—1461no—
485C4⋊C4—1461no—
486C4⋊C4—1461no—
487C4⋊C4—1461no—
488C4⋊C4—1461no—
489C4⋊C4—1461no—
490C4⋊C4C2×C4×D7311yescharacteristic
491C4⋊C4C2×C4×D7311yescharacteristic
492C22⋊C4—21no—
493C22⋊C4—1461no—
494C22⋊C4—1461no—
495C42—21no—
496C42—731no—
497C42—731no—
498C42—731no—
499C42—1461no—
500C42—1461no—
501C42—1461no—
502C42—1461no—
503C42—1461no—
504C42—1461no—
505C42C23×D7311yescharacteristic
506C23—21no—
507C23—731no—
508Q8—21no—
509Q8—21no—
510Q8—2922no—
511Q8—732no—
512Q8—1461no—
513Q8—1461no—
514Q8—1461no—
515D4—21no—
516D4—2922no—
517D4M16×D7312yes—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—2921no—
524C2×C4—2922no—
525C2×C4—2922no—
526C2×C4—5841no—
527C2×C4—5841no—
528C2×C4—731no—
529C2×C4—731no—
530C2×C4—731no—
531C2×C4—1461no—
532C2×C4—1461no—
533C2×C4—1461no—
534C2×C4—1461no—
535C2×C4—1461no—
536C2×C4—1461no—
537C2×C4—1461no—
538C2×C4—1461no—
539C2×C4C22×C4×D7311yescharacteristic
540C2×C4C2×D4×D7311yescharacteristic
541C2×C4#41×D7311yescharacteristic
542C8—21no—
543C8—21no—
544C8—21no—
545C8—41no—
546C8—41no—
547C8—41no—
548C8—2921no—
549C8—2921no—
550C8—732no—
551C8—1462no—
552C8—1461no—
553C8—1461no—
554C8—1461no—
555C8#1455662012yescomplemented
556C22—22no—
557C22—41no—
558C22—41no—
559C22—1461no—
560C22—1461no—
561C22—1461no—
562C22C4×D4×D7311yescharacteristic
563C4—21no—
564C4—41no—
565C4—41no—
566C4—41no—
567C4—2921no—
568C4—2921no—
569C4—2921no—
570C4—1461no—
571C4—1462no—
572C4—1461no—
573C4—1461no—
574C4C2×M16×D7311yescharacteristic
575C4#1798935211yescharacteristic
576C4C2×#1455662011yescharacteristic
577C4#1798938411yescharacteristic
578C2—42no—
579C2—2921no—
580C2#2261866411yescharacteristic
581C2C4×#1455662011yescharacteristic
582C2#2261883911yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D73
commutatorC2×C4×C73C23×C4
FrattiniC42C23×D73
FittingC73×#1582C2
socleC22×C73C2×C4×D4
prime radicalC2×#11924027C2×C4
cosocleC42×C73C24
hypercenter#1582D73
hypocenterC73#18047
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
2. #22618883C2semidirect
4. #22608837C2semidirect
5. #22608826C2semidirect
6. #22608801C2semidirect
7. #22608789C2semidirect
8. #22608783C2semidirect
9. #22608740C2semidirect
10. C8×#11924016C2semidirect
11. #22608695C2semidirect
12. #22608376C2semidirect
14. #22608235C2semidirect
15. C73×#1582C2semidirect
162. #11924016M16semidirect
256. C8×C73#137semidirect
300. #1582D73semidirect
307. C73#18047semidirect
555. C8#14556620semidirect

Maximal subgroups

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

subgroupindex
#226189392
#226188832
C4×#145566202
#226088372
#226088262
#226088012
#226087892
#226087832
#226087402
C8×#119240162
#226086952
#226083762
#226083672
#226082352
C73×#15822
#1804773