Finite Group Databasebeta

← #33055783

Subgroups — #33055783

Order 18688.

648 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22618894C211yesmaximal, characteristic
2#446×D73C211yesmaximal, characteristic, complemented
3C4×#14556620C211yesmaximal, characteristic
4#22608841C211yesmaximal, characteristic, complemented
5#22608838C211yesmaximal, characteristic, complemented
6#22608833C211yesmaximal, characteristic, complemented
7#22608827C211yesmaximal, characteristic, complemented
8#22608825C211yesmaximal, characteristic, complemented
9#22608824C211yesmaximal, characteristic, complemented
10#22608822C211yesmaximal, characteristic, complemented
11#22608682C211yesmaximal, characteristic, complemented
12#22608324C211yesmaximal, characteristic, complemented
13#22608315C211yesmaximal, characteristic
14#22608210C211yesmaximal, characteristic, complemented
15C73×#1592C211yesmaximal, characteristic, complemented
16C2×#14556620C411yescharacteristic
17C4×#11924027C2211yescharacteristic
18C4×Q8×D73C2211yescharacteristic
19#17988718C2211yescharacteristic
20#17988717C2211yescharacteristic
21#17988717C2211yescharacteristic
22#17988716—21no—
23#17988715—21no—
24#17988714C2211yescharacteristic
25#17988714C2211yescharacteristic
26#17988693C411yescharacteristic
27#17988691C411yescharacteristic
28#17988688C411yescharacteristic
29#17988684C411yescharacteristic
30C4×Dic292C2211yescharacteristic
31C4×#11924018C2211yescharacteristic
32C4×#11924016C2211yescharacteristic
33#17988665C2211yescharacteristic
34#17988661—21no—
35#17988656—21no—
36#17988639C2211yescharacteristic
37#17988638C2211yescharacteristic
38#17988637—21no—
39#17988626C411yescharacteristic
40C4⋊C8×D73—21no—
41C4⋊C8×D73C2211yescharacteristic
42#17988594C411yescharacteristic
43#17988585C411yescharacteristic
44M16×Dic73—21no—
45C4×#11924011C2211yescharacteristic
46#97×D73—21no—
47C4×C8×D73C2211yescharacteristic
48#17988339C2211yescharacteristic
49#17988338C2211yescharacteristic
50#17988337C2211yescharacteristic
51#17988319C2211yescharacteristic
52#17988319C2211yescharacteristic
53Q8⋊#10596831C2211yescharacteristic
54#17988317C2211yescharacteristic
55#17988317C2211yescharacteristic
56D4⋊#10596831C2211yescharacteristic
57#17988309C2211yescharacteristic
58C8×#10596831C2211yescharacteristic
59C8×#10596831C2211yescharacteristic
60#446×C73C2211yescharacteristic
61C4×SD16×C73C2211yescharacteristic
62#436×C73C2211yescharacteristic
63#335×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×#11924012—21no—
74C2×#11924011C2×C411yescharacteristic
75C2×#11924011C2×C411yescharacteristic
76C2×C8×D73—21no—
77C2×C8×D73D411yescharacteristic
78C2×C8×D73D411yescharacteristic
79Q8×Dic73C2×C411yescharacteristic
80C4×Dic146—21no—
81C4×Dic146C2311yescharacteristic
82C4×Dic146C2311yescharacteristic
83#14556556—21no—
84#14556556C2×C411yescharacteristic
85#14556556C2×C411yescharacteristic
86D4×Dic73C2×C411yescharacteristic
87C4×C22⋊D73—21no—
88#14556551—21no—
89#14556549C2×C411yescharacteristic
90#14556547—21no—
91C4⋊C4×D73—21no—
92C4⋊C4×D73C2×C411yescharacteristic
93C2×C4×Dic73—21no—
94C42×D73—21no—
95C42×D73C2311yescharacteristic
96#14556525C2×C411yescharacteristic
97#14556524C2×C411yescharacteristic
98#14556522C2×C411yescharacteristic
99C4⋊#10596831—21no—
100C4⋊#10596831C2311yescharacteristic
101C4⋊#10596831C2311yescharacteristic
102#14556520—21no—
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#14556503—21no—
117#14556502—21no—
118#14556501—21no—
119#14556501—21no—
120#14556501—21no—
121#14556501—21no—
122#14556501C2×C411yescharacteristic
123#14556501C2×C411yescharacteristic
124C4×#10596831C2311yescharacteristic
125C4×#10596831C2311yescharacteristic
126C4×#10596831C2311yescharacteristic
127C8×Dic73—21no—
128C8×Dic73D411yescharacteristic
129C8×Dic73D411yescharacteristic
130C2×SD16×C73C2×C411yescharacteristic
131C2×M16×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—22no—
147C2×Dic146—21no—
148C2×Dic146C22×C411yescharacteristic
149C2×Dic146C22×C411yescharacteristic
150Q8×D73—22no—
151Q8×D73—22no—
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#11924016—22no—
163#11924012—42no—
164#11924011—22no—
165#11924011M1614yescomplemented
166C2×#10596831—21no—
167C2×#10596831—21no—
168C2×#10596831—21no—
169C2×#10596831C22×C411yescharacteristic
170C2×#10596831C22×C411yescharacteristic
171C2×#10596831C2×D411yescharacteristic
172C2×#10596831C2×D411yescharacteristic
173C8×D73—22no—
174C8×D73—41no—
175C8×D73—42no—
176C4⋊Dic73—21no—
177C4⋊Dic73C22×C411yescharacteristic
178C4⋊Dic73C22×C411yescharacteristic
179#11924007—21no—
180#11924007—21no—
181#11924007—21no—
182#11924007—21no—
183#11924007—21no—
184C22⋊Dic73—21no—
185#11924005—21no—
186C4×Dic73—21no—
187C4×Dic73—21no—
188C4×Dic73—21no—
189C4×Dic73—21no—
190C4×Dic73—21no—
191C4×Dic73—21no—
192C4×Dic73C22×C411yescharacteristic
193C4×Dic73C2×D411yescharacteristic
194C4×Dic73#4111yescharacteristic
195C2×Q8×C73C22×C411yescharacteristic
196C2×D4×C73C22×C411yescharacteristic
197C22×C4×C73—21no—
198SD16×C73—22no—
199M16×C73—42no—
200C2×C8×C73—21no—
201C2×C8×C73—21no—
202C2×C8×C73—21no—
203C2×C8×C73C22×C411yescharacteristic
204C2×C8×C73C22×C411yescharacteristic
205C2×C8×C73C2×D411yescharacteristic
206C2×C8×C73C2×D411yescharacteristic
207C4⋊C4×C73—21no—
208C4⋊C4×C73C22×C411yescharacteristic
209C4⋊C4×C73C22×C411yescharacteristic
210C22⋊C4×C73—21no—
211C42×C73—21no—
212C42×C73C2411yescharacteristic
213C23×C73—21no—
214C22×D73C4×D411yescharacteristic
215Q8×C73—21no—
216Q8×C73—21no—
217D4×C73—21no—
218D4×C73—21no—
219C2×C4×C73—21no—
220C2×C4×C73—21no—
221C2×C4×C73—21no—
222C2×C4×C73—41no—
223C2×C4×C73—41no—
224C2×C4×C73C23×C411yescharacteristic
225C2×C4×C73C22×D411yescharacteristic
226C2×C4×C73C2×#4111yescharacteristic
227C22⋊D73—42no—
228C2×Dic73—21no—
229C2×Dic73—21no—
230C2×Dic73—21no—
231C2×Dic73—21no—
232C2×Dic73—42no—
233C2×Dic73—42no—
234C2×Dic73C4×D411yescharacteristic
235C2×Dic73C4×D411yescharacteristic
236C2×Dic73C4×D411yescharacteristic
237C4×D73—21no—
238C4×D73—21no—
239C4×D73—41no—
240C4×D73—42no—
241C4×D73—42no—
242C4×D73C2×M1612yes—
243C4×D73#13112yes—
244Dic146—21no—
245Dic146—21no—
246Dic146—21no—
247Dic146—21no—
248Dic146—42no—
249C8×C73—21no—
250C8×C73—21no—
251C8×C73—21no—
252C8×C73—41no—
253C8×C73—41no—
254C8×C73—41no—
255C8×C73C2×M1612yes—
256#10596831—21no—
257#10596831—21no—
258#10596831—41no—
259#10596831—41no—
260#10596831—41no—
261#10596831C2×M1612yes—
262#10596831#13712yescomplemented
263C22×C73—41no—
264C22×C73—41no—
265C22×C73—41no—
266C22×C73C2×C4×D411yescharacteristic
267C2×D73—21no—
268C2×D73—21no—
269C2×D73#43612yes—
270C4×C73—21no—
271C4×C73—41no—
272C4×C73—41no—
273C4×C73—41no—
274C4×C73C22×M1611yescharacteristic
275C4×C73C2×#13111yescharacteristic
276C4×C73C2×#13711yescharacteristic
277C4×C73#57511yescharacteristic
278Dic73—21no—
279Dic73—21no—
280Dic73—41no—
281Dic73—41no—
282Dic73—41no—
283Dic73—41no—
284Dic73#43612yes—
285#17892—731nomaximal, Sylow
286C2×C73—81no—
287C2×C73C2×#43611yescharacteristic
288C2×C73C4×#13711yescharacteristic
289C2×C73#295511yescharacteristic
290D73—22no—
291#2965—731no—
292C2×#446—731no—
293C4×#137—731no—
294#1593—731no—
295#1593—731no—
296#1592—731no—
297#1592D7311yescharacteristic, complemented
298#1591—731no—
299#1591—731no—
300#1590—731no—
301#1590—731no—
302#1565—731no—
303#1479—731no—
304#1477—731no—
305C8×M16—731no—
306C73#1789211yescharacteristic, complemented, Sylow
307C2×#137—731no—
308C4×#41—731no—
309C2×C4×Q8—731no—
310#446—731no—
311#446—731no—
312#446—731no—
313#446—731no—
314#446—731no—
315#446—1461no—
316#446—1461no—
317#446C2×D7311yescharacteristic
318#441—731no—
319#441—731no—
320#440—731no—
321#440—731no—
322C4×Q16—731no—
323C4×Q16—731no—
324C4×SD16—731no—
325C4×SD16C2×D7311yescharacteristic
326#436—731no—
327#436C2×D7311yescharacteristic
328#435—1461no—
329#435—1461no—
330#431—731no—
331#431—731no—
332#431—1461no—
333#428—731no—
334C2×C4⋊C8—731no—
335C2×C4⋊C8—1461no—
336#419—731no—
337#417—731no—
338C4×M16—731no—
339C4×M16—1461no—
340C2×#97—1461no—
341C2×C4×C8—731no—
342#335—731no—
343#335C2×D7311yescharacteristic
344#334—731no—
345#334—731no—
346Q8⋊C8—731no—
347Q8⋊C8—731no—
348Q8⋊C8—731no—
349Q8⋊C8—731no—
350Q8⋊C8—731no—
351Q8⋊C8C2×D7311yescharacteristic
352D4⋊C8—731no—
353D4⋊C8C2×D7311yescharacteristic
354#322—731no—
355#322C2×D7311yescharacteristic
356C82—731no—
357C82—731no—
358C2×#41—731no—
359C22×Q8—731no—
360#137—1464no—
361C2×Q16—731no—
362C2×Q16—731no—
363C2×SD16—731no—
364C2×SD16C4×D7311yescharacteristic
365C2×M16—21no—
366C2×M16—731no—
367C2×M16—731no—
368C2×M16—1461no—
369C22×C8—731no—
370C22×C8—731no—
371C22×C8—1461no—
372C4×Q8—731no—
373C4×Q8—731no—
374C4×Q8—731no—
375C4×Q8—731no—
376C4×Q8—731no—
377C4×Q8—1461no—
378C4×Q8—1461no—
379C4×Q8C22×D7311yescharacteristic
380C4×D4—731no—
381C4×D4—1461no—
382C4×D4—1461no—
383C4×D4C22×D7311yescharacteristic
384#117—731no—
385#117—1461no—
386C2×C4⋊C4—731no—
387C2×C4⋊C4—1461no—
388C2×C42—731no—
389C2×C42—1461no—
390C2×C42—1461no—
391#107—731no—
392#107—731no—
393#106—731no—
394#106C4×D7311yescharacteristic
395C4⋊C8—21no—
396C4⋊C8—731no—
397C4⋊C8—731no—
398C4⋊C8—1461no—
399C4⋊C8—1461no—
400C4⋊C8—1461no—
401C4⋊C8—1461no—
402C4⋊C8—1461no—
403C4⋊C8—1461no—
404C4⋊C8—1461no—
405C4⋊C8C22×D7311yescharacteristic
406C4⋊C8C22×D7311yescharacteristic
407#103—731no—
408#103—731no—
409#103—731no—
410#103—731no—
411#103—731no—
412#103C4×D7311yescharacteristic
413#102—731no—
414#102C4×D7311yescharacteristic
415C22⋊C8—21no—
416C22⋊C8—1461no—
417C22⋊C8—1461no—
418C22⋊C8—1461no—
419#97—21no—
420#97—731no—
421#97—731no—
422#97—1461no—
423#97—1461no—
424#97—1461no—
425#97—1461no—
426#97—1461no—
427C4×C8—731no—
428C4×C8—731no—
429C4×C8—731no—
430C4×C8—731no—
431C4×C8—731no—
432C4×C8—1461no—
433C4×C8C22×D7311yescharacteristic
434C4×C8C22×D7311yescharacteristic
435C4×C8C22×D7311yescharacteristic
436#41—1462no—
437#41—1462no—
438C2×Q8—731no—
439C2×Q8—731no—
440C2×Q8—1461no—
441C2×Q8—1462no—
442C2×Q8—1462no—
443C2×Q8C2×C4×D7311yescharacteristic
444C2×D4—1461no—
445C2×D4C2×C4×D7311yescharacteristic
446C22×C4—21no—
447C22×C4—731no—
448C22×C4—731no—
449C22×C4—731no—
450C22×C4—1461no—
451C22×C4—1461no—
452C22×C4—1461no—
453C22×C4—1461no—
454Q16—1462no—
455Q16—1462no—
456SD16—22no—
457SD16—1462no—
458M16—42no—
459M16—2922no—
460M16—734no—
461M16—1462no—
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—2921no—
466C2×C8—2922no—
467C2×C8—731no—
468C2×C8—731no—
469C2×C8—731no—
470C2×C8—731no—
471C2×C8—1462no—
472C2×C8—1461no—
473C2×C8—1461no—
474C2×C8—1461no—
475C2×C8C2×C4×D7311yescharacteristic
476C2×C8C2×C4×D7311yescharacteristic
477C2×C8D4×D7311yescharacteristic
478C2×C8D4×D7311yescharacteristic
479C4⋊C4—21no—
480C4⋊C4—731no—
481C4⋊C4—731no—
482C4⋊C4—1461no—
483C4⋊C4—1461no—
484C4⋊C4—1461no—
485C4⋊C4—1461no—
486C4⋊C4—1461no—
487C4⋊C4—1461no—
488C4⋊C4C2×C4×D7311yescharacteristic
489C4⋊C4C2×C4×D7311yescharacteristic
490C22⋊C4—21no—
491C22⋊C4—1461no—
492C22⋊C4—1461no—
493C42—21no—
494C42—731no—
495C42—731no—
496C42—731no—
497C42—1461no—
498C42—1461no—
499C42—1461no—
500C42—1461no—
501C42—1461no—
502C42—1461no—
503C42C23×D7311yescharacteristic
504C23—21no—
505C23—731no—
506Q8—21no—
507Q8—21no—
508Q8—2922no—
509Q8—1461no—
510Q8—1461no—
511Q8—1461no—
512Q8—1461no—
513D4—21no—
514D4—21no—
515D4—2922no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—2921no—
522C2×C4—2922no—
523C2×C4—2922no—
524C2×C4—2922no—
525C2×C4—2922no—
526C2×C4—731no—
527C2×C4—732no—
528C2×C4—732no—
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×C4C22×C4×D7311yescharacteristic
538C2×C4C2×D4×D7311yescharacteristic
539C2×C4#41×D7311yescharacteristic
540C8—21no—
541C8—21no—
542C8—21no—
543C8—41no—
544C8—41no—
545C8—41no—
546C8—2921no—
547C8—2921no—
548C8—2921no—
549C8—732no—
550C8—732no—
551C8—1461no—
552C8—1461no—
553C8M16×D7312yes—
554C22—41no—
555C22—41no—
556C22—41no—
557C22—732no—
558C22—1461no—
559C22—1461no—
560C22C4×D4×D7311yescharacteristic
561C4—21no—
562C4—41no—
563C4—41no—
564C4—41no—
565C4—2921no—
566C4—2921no—
567C4—2921no—
568C4—2921no—
569C4—732no—
570C4—1461no—
571C4—1461no—
572C4C2×M16×D7311yescharacteristic
573C4#131×D7311yescharacteristic
574C4#137×D7311yescharacteristic
575C4#1798938311yescharacteristic
576C2—81no—
577C2—1462no—
578C2#436×D7311yescharacteristic
579C2#2261875311yescharacteristic
580C2#2261883111yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D73
commutatorC2×C4×C73C23×C4
FrattiniC42C23×D73
FittingC73×#1592C2
socleC22×C73C2×C4×D4
prime radicalC2×#11924027C2×C4
cosocleC42×C73C24
hypercenter#1592D73
hypocenterC73#17892
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ #1592

Complements

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

NH ≅ G/Ntype
2. #446×D73C2semidirect
4. #22608841C2semidirect
5. #22608838C2semidirect
6. #22608833C2semidirect
7. #22608827C2semidirect
8. #22608825C2semidirect
9. #22608824C2semidirect
10. #22608822C2semidirect
11. #22608682C2semidirect
12. #22608324C2semidirect
14. #22608210C2semidirect
15. C73×#1592C2semidirect
165. #11924011M16semidirect
262. #10596831#137semidirect
297. #1592D73semidirect
306. C73#17892semidirect

Maximal subgroups

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

subgroupindex
#226188942
#446×D732
C4×#145566202
#226088412
#226088382
#226088332
#226088272
#226088252
#226088242
#226088222
#226086822
#226083242
#226083152
#226082102
C73×#15922
#1789273