Finite Group Databasebeta

← #33239760

Subgroups — #33239760

Order 18688.

1150 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14556613C211yesmaximal, characteristic, complemented
2C2×#17989017C212yesmaximal, complemented
3#22611869C212yesmaximal, complemented
4#22611864C211yesmaximal, characteristic, complemented
5#22611859C211yesmaximal, characteristic, complemented
6#22611852C212yesmaximal, complemented
7#22611847C211yesmaximal, characteristic, complemented
8#22611842C211yesmaximal, characteristic, complemented
9#22611636C212yesmaximal, complemented
10C2×C22⋊#11924003C211yesmaximal, characteristic, complemented
11C73×#3050C211yesmaximal, characteristic, complemented
12C22×#11924028—21no—
13C2×#14556613—28no—
14C2×#14556613—21no—
15C2×#14556613—21no—
16C2×#14556613C2212yes—
17C2×#14556613C2212yes—
18C22×Dic292C2211yescharacteristic
19C22×#11924017—21no—
20C22×D584C2211yescharacteristic
21#17989017—28no—
22C2×#14556537C2212yescomplemented
23C2×#14556535C2212yescomplemented
24C2×#14556534C2212yescomplemented
25C2×#14556532C2212yescomplemented
26C22×#11924003C2212yescomplemented
27#17988390C2212yescomplemented
28#17988390C2212yescomplemented
29#17988387C2212yescomplemented
30#17988387C2212yescomplemented
31C22⋊#11924003C2211yescharacteristic
32C22⋊#11924003C2211yescharacteristic
33C22⋊#11924003C2212yescomplemented
34C2×#135×C73C2211yescharacteristic, complemented
35C23×C8×C73C2211yescharacteristic
36#525×C73—21no—
37#467×C73C2211yescharacteristic
38#467×C73C2211yescharacteristic, complemented
39#466×C73C2211yescharacteristic
40#466×C73C2211yescharacteristic, complemented
41#465×C73—21no—
42#465×C73—21no—
43#427×C73C2211yescharacteristic, complemented
44#416×C73—21no—
45C2×#11924028—28no—
46C2×#11924028—22no—
47C2×#11924028—22no—
48C2×#11924028—44no—
49C2×#11924028—41no—
50C2×#11924028—41no—
51C22×Dic146—21no—
52C22×D292—21no—
53C22×C22⋊D73—41no—
54C22×C4×D73—41no—
55#14556613—28no—
56#14556613—28no—
57#14556613—24no—
58#14556613—24no—
59#14556613—24no—
60#14556613D418yescomplemented
61C2×Dic292—24no—
62C2×Dic292—21no—
63C2×Dic292C2312yes—
64C2×#11924017—28no—
65C2×#11924017—22no—
66C2×#11924017—22no—
67C2×D584—24no—
68C2×D584—21no—
69C2×D584C2312yes—
70#14556537—24no—
71#14556535—24no—
72#14556534—24no—
73#14556532—24no—
74C2×#11924003—24no—
75C2×#11924003C2312yes—
76C2×#11924003C2312yes—
77C2×#41×C73—21no—
78C23×C4×C73D411yescharacteristic
79#135×C73—24no—
80C2×Q16×C73C2311yescharacteristic
81C2×SD16×C73—21no—
82C2×D8×C73C2311yescharacteristic
83C22×C8×C73—24no—
84C22×C8×C73D411yescharacteristic
85C22×C8×C73D411yescharacteristic
86C22×C8×C73D411yescharacteristic
87C22×C8×C73C2311yescharacteristic
88C22×C8×C73C2311yescharacteristic
89C22×C8×C73C2311yescharacteristic
90#123×C73—41no—
91C22⋊Q8×C73—21no—
92C22⋊Q8×C73—21no—
93#121×C73—21no—
94#121×C73—21no—
95#120×C73—41no—
96C4×D4×C73—41no—
97C4×D4×C73—41no—
98#117×C73—21no—
99#107×C73C2311yescharacteristic
100#107×C73C2311yescharacteristic
101#106×C73—21no—
102#103×C73—21no—
103#103×C73—21no—
104#102×C73—21no—
105#102×C73—21no—
106C23×D73—41no—
107#11924028—28no—
108#11924028—28no—
109#11924028—48no—
110#11924028—48no—
111#11924028—44no—
112#11924028—44no—
113C2×Dic146—24no—
114C2×Dic146—22no—
115C2×Dic146—42no—
116C2×Dic146—41no—
117C2×D292—24no—
118C2×D292—22no—
119C2×D292—42no—
120C2×D292—41no—
121C2×C22⋊D73—48no—
122C2×C22⋊D73—42no—
123C2×C22⋊D73—42no—
124C22×Dic73—41no—
125C2×C4×D73—48no—
126C2×C4×D73—42no—
127C2×C4×D73—42no—
128Dic292—24no—
129Dic292—22no—
130Dic292C2×D414yes—
131#11924017—28no—
132#11924017—28no—
133D584—24no—
134D584—22no—
135D584C2×D414yes—
136#11924003—22no—
137#11924003—22no—
138C24×C73D811yescharacteristic
139#41×C73—42no—
140#41×C73—42no—
141C2×Q8×C73—21no—
142C2×D4×C73—21no—
143C2×D4×C73—41no—
144C2×D4×C73—41no—
145C2×D4×C73—41no—
146C22×C4×C73—22no—
147C22×C4×C73—22no—
148C22×C4×C73—41no—
149C22×C4×C73D811yescharacteristic
150C22×C4×C73D811yescharacteristic
151C22×C4×C73D811yescharacteristic
152C22×C4×C73C2×D411yescharacteristic
153C22×C4×C73C2×D411yescharacteristic
154C22×C4×C73C2×D411yescharacteristic
155Q16×C73—22no—
156SD16×C73—42no—
157D8×C73—22no—
158C2×C8×C73—22no—
159C2×C8×C73—22no—
160C2×C8×C73—22no—
161C2×C8×C73—22no—
162C2×C8×C73—22no—
163C2×C8×C73C2×D414yes—
164C2×C8×C73C2×D411yescharacteristic
165C2×C8×C73C2×D411yescharacteristic
166C2×C8×C73C2×D411yescharacteristic
167C2×C8×C73C2411yescharacteristic
168C4⋊C4×C73—21no—
169C4⋊C4×C73—21no—
170C4⋊C4×C73—41no—
171C4⋊C4×C73—41no—
172C22⋊C4×C73—41no—
173C22⋊C4×C73—41no—
174C22⋊C4×C73—41no—
175C22⋊C4×C73—41no—
176C22⋊C4×C73—41no—
177C42×C73—41no—
178C23×C73—22no—
179C23×C73—22no—
180C23×C73—41no—
181C23×C73—41no—
182C23×C73C2×D811yescharacteristic
183C23×C73C2×D811yescharacteristic
184C23×C73C2×D811yescharacteristic
185C22×D73—44no—
186C22×D73—42no—
187C22×D73—82no—
188C22×D73—82no—
189Q8×C73—41no—
190Q8×C73—41no—
191D4×C73—41no—
192D4×C73—41no—
193D4×C73—82no—
194D4×C73—82no—
195D4×C73—82no—
196C2×C4×C73—22no—
197C2×C4×C73—21no—
198C2×C4×C73—21no—
199C2×C4×C73—21no—
200C2×C4×C73—21no—
201C2×C4×C73—22no—
202C2×C4×C73—22no—
203C2×C4×C73—41no—
204C2×C4×C73—41no—
205C2×C4×C73—41no—
206C2×C4×C73—82no—
207C2×C4×C73#12012yes—
208C2×C4×C73C2×D811yescharacteristic
209C2×C4×C73C2×D811yescharacteristic
210C2×C4×C73C2×D811yescharacteristic
211C2×C4×C73AGL(1,ℤ/8)12yes—
212C2×C4×C73C22×D411yescharacteristic
213C22⋊D73—48no—
214C22⋊D73—48no—
215C2×Dic73—44no—
216C2×Dic73—42no—
217D292—24no—
218D292—44no—
219D292—42no—
220C4×D73—48no—
221C4×D73—48no—
222Dic146—24no—
223Dic146—44no—
224Dic146—42no—
225C8×C73—21no—
226C8×C73—21no—
227C8×C73#12012yes—
228C8×C73C22×D412yes—
229C22×C73—21no—
230C22×C73—21no—
231C22×C73—21no—
232C22×C73—21no—
233C22×C73—21no—
234C22×C73—21no—
235C22×C73—21no—
236C22×C73—44no—
237C22×C73—81no—
238C22×C73—81no—
239C22×C73—81no—
240C22×C73#44712yes—
241C22×C73#50812yescomplemented
242C22×C73C22×D811yescharacteristic
243C2×D73—44no—
244C2×D73—88no—
245C2×D73—82no—
246C2×D73—82no—
247C4×C73—21no—
248C4×C73—21no—
249C4×C73—81no—
250C4×C73—81no—
251C4×C73—81no—
252C4×C73#44712yes—
253C4×C73C2×#12011yescharacteristic
254C4×C73C2×AGL(1,ℤ/8)11yescharacteristic
255Dic73—44no—
256#33516—731nomaximal, Sylow
257C2×C73—22no—
258C2×C73—22no—
259C2×C73—41no—
260C2×C73—161no—
261C2×C73C2×#44711yescharacteristic
262C2×C73C2×#50811yescharacteristic
263C2×C73#341411yescharacteristic
264D73—84no—
265C22×#135—731no—
266C2×#508—732no—
267#3050D7311yescharacteristic, complemented
268#2189—732no—
269#2188—731no—
270#2187—731no—
271#2186—732no—
272#2185—731no—
273#2184—731no—
274#2138—732no—
275C2×C22⋊C16—731no—
276C73#3351611yescharacteristic, complemented, Sylow
277C22×#41—1461no—
278C2×#135—732no—
279C2×#135—732no—
280C2×#135—1468no—
281C2×#135—1461no—
282C2×#135—1461no—
283C2×#135C2×D7311yescharacteristic, complemented
284C22×Q16—731no—
285C22×SD16—1461no—
286C22×D8—731no—
287C23×C8C2×D7311yescharacteristic
288#525—21no—
289#508—1468no—
290C2×Q32—732no—
291C2×SD32—732no—
292C2×SD32—732no—
293C2×D16—732no—
294C22×C16—732no—
295#467C2×D7311yescharacteristic
296#467C2×D7311yescharacteristic, complemented
297#466C2×D7311yescharacteristic
298#466C2×D7311yescharacteristic, complemented
299#465—21no—
300#465—21no—
301#427C2×D7311yescharacteristic, complemented
302#416—21no—
303#358—732no—
304#358—732no—
305#357—732no—
306#357—732no—
307C22⋊C16—731no—
308C22⋊C16—731no—
309C22⋊C16—732no—
310C2×#41—21no—
311C2×#41—2924no—
312C2×#41—2921no—
313C2×#41—2921no—
314C2×#41—1468no—
315C2×#41—1462no—
316C2×#41—1462no—
317C22×Q8—1461no—
318C22×D4—2921no—
319C22×D4—1461no—
320C23×C4—2921no—
321C23×C4C22⋊D7311yescharacteristic
322#135—24no—
323#135—738no—
324#135—1464no—
325#135—1464no—
326#135—1468no—
327#135—1468no—
328#135—1464no—
329C2×Q16—732no—
330C2×Q16—1464no—
331C2×Q16—1461no—
332C2×Q16C22×D7311yescharacteristic
333C2×SD16—21no—
334C2×SD16—1468no—
335C2×SD16—1462no—
336C2×SD16—1462no—
337C2×D8—732no—
338C2×D8—1464no—
339C2×D8—1461no—
340C2×D8C22×D7311yescharacteristic
341C22×C8—24no—
342C22×C8C22⋊D7311yescharacteristic
343C22×C8C22⋊D7311yescharacteristic
344C22×C8C22⋊D7311yescharacteristic
345C22×C8C22×D7311yescharacteristic
346C22×C8C22×D7311yescharacteristic
347C22×C8C22×D7311yescharacteristic
348#123—41no—
349C22⋊Q8—21no—
350C22⋊Q8—21no—
351#121—21no—
352#121—21no—
353#120—41no—
354C4×D4—41no—
355C4×D4—41no—
356#117—21no—
357Q32—1464no—
358SD32—1464no—
359SD32—1464no—
360D16—1464no—
361C2×C16—732no—
362C2×C16—732no—
363C2×C16—1464no—
364#107C22×D7311yescharacteristic
365#107C22×D7311yescharacteristic
366#106—21no—
367#103—21no—
368#103—21no—
369#102—21no—
370#102—21no—
371C24—2921no—
372C24#1192401311yescharacteristic
373#41—42no—
374#41—42no—
375#41—2928no—
376#41—2928no—
377#41—2924no—
378#41—2924no—
379#41—1468no—
380#41—1468no—
381C2×Q8—21no—
382C2×Q8—2922no—
383C2×Q8—2921no—
384C2×Q8—1464no—
385C2×Q8—1462no—
386C2×D4—21no—
387C2×D4—41no—
388C2×D4—41no—
389C2×D4—41no—
390C2×D4—2922no—
391C2×D4—2928no—
392C2×D4—2922no—
393C2×D4—2922no—
394C2×D4—2921no—
395C2×D4—1464no—
396C2×D4—1462no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—41no—
400C22×C4—2928no—
401C22×C4—2921no—
402C22×C4—2922no—
403C22×C4—2922no—
404C22×C4#1192401311yescharacteristic
405C22×C4#1192401311yescharacteristic
406C22×C4#1192401311yescharacteristic
407C22×C4C2×C22⋊D7311yescharacteristic
408C22×C4C2×C22⋊D7311yescharacteristic
409C22×C4C2×C22⋊D7311yescharacteristic
410Q16—22no—
411Q16—734no—
412Q16—1464no—
413Q16—1462no—
414SD16—42no—
415SD16—1468no—
416SD16—1468no—
417D8—22no—
418D8—734no—
419D8—1464no—
420D8—1462no—
421C2×C8—22no—
422C2×C8—22no—
423C2×C8—22no—
424C2×C8—22no—
425C2×C8—22no—
426C2×C8D4×D7314yes—
427C2×C8C2×C22⋊D7311yescharacteristic
428C2×C8C2×C22⋊D7311yescharacteristic
429C2×C8C2×C22⋊D7311yescharacteristic
430C2×C8C23×D7311yescharacteristic
431C4⋊C4—21no—
432C4⋊C4—21no—
433C4⋊C4—41no—
434C4⋊C4—41no—
435C22⋊C4—41no—
436C22⋊C4—41no—
437C22⋊C4—41no—
438C22⋊C4—41no—
439C22⋊C4—41no—
440C42—41no—
441C16—1462no—
442C16—1462no—
443C23—22no—
444C23—22no—
445C23—41no—
446C23—41no—
447C23—2924no—
448C23—2922no—
449C23—5842no—
450C23—5842no—
451C23C2×#1192401311yescharacteristic
452C23C2×#1192401311yescharacteristic
453C23C2×#1192401311yescharacteristic
454Q8—41no—
455Q8—41no—
456Q8—2924no—
457Q8—2922no—
458Q8—1464no—
459D4—41no—
460D4—41no—
461D4—82no—
462D4—82no—
463D4—82no—
464D4—2924no—
465D4—2928no—
466D4—2928no—
467D4—2922no—
468D4—1464no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—82no—
480C2×C4—2928no—
481C2×C4—2924no—
482C2×C4—2922no—
483C2×C4—2928no—
484C2×C4#1455655912yes—
485C2×C4C2×#1192401311yescharacteristic
486C2×C4C2×#1192401311yescharacteristic
487C2×C4C2×#1192401311yescharacteristic
488C2×C4#1455661612yes—
489C2×C4C22×C22⋊D7311yescharacteristic
490C8—21no—
491C8—21no—
492C8#1455655912yes—
493C8C2×D4×D7312yes—
494C22—21no—
495C22—21no—
496C22—21no—
497C22—21no—
498C22—21no—
499C22—21no—
500C22—21no—
501C22—44no—
502C22—81no—
503C22—81no—
504C22—81no—
505C22—2924no—
506C22—5848no—
507C22—5842no—
508C22—5842no—
509C22#1798872012yes—
510C22#1798901712yescomplemented
511C22C22×#1192401311yescharacteristic
512C4—21no—
513C4—21no—
514C4—81no—
515C4—81no—
516C4—81no—
517C4—2924no—
518C4#1798872012yes—
519C4C2×#1455655911yescharacteristic
520C4C2×#1455661611yescharacteristic
521C2—22no—
522C2—22no—
523C2—41no—
524C2—161no—
525C2—5844no—
526C2C2×#1798872011yescharacteristic
527C2C2×#1798901711yescharacteristic
528C2#2262306511yescharacteristic
subgroupG/H
centerC2×C4C2×#11924013
commutator, cosocleC2×C8×C73C24
FrattiniC2×C8C23×D73
FittingC73×#3050C2
socleC22×C73C22×D8
hypercenter#3050D73
hypocenterC73#33516
LCSG ▷ C2×C8×C73 ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C23×C8 ◁ #3050

Complements

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

NH ≅ G/Ntype
1. C22×#14556613C2semidirect
2. C2×#17989017C2semidirect
3. #22611869C2semidirect
4. #22611864C2semidirect
5. #22611859C2semidirect
6. #22611852C2semidirect
7. #22611847C2semidirect
8. #22611842C2semidirect
9. #22611636C2semidirect
10. C2×C22⋊#11924003C2semidirect
11. C73×#3050C2semidirect
22. C2×#14556537C22semidirect
23. C2×#14556535C22semidirect
24. C2×#14556534C22semidirect
25. C2×#14556532C22semidirect
26. C22×#11924003C22semidirect
27. #17988390C22semidirect
28. #17988390C22semidirect
29. #17988387C22semidirect
30. #17988387C22semidirect
33. C22⋊#11924003C22semidirect
34. C2×#135×C73C22semidirect
38. #467×C73C22semidirect
40. #466×C73C22semidirect
43. #427×C73C22semidirect
60. #14556613D4semidirect
241. C22×C73#508semidirect
267. #3050D73semidirect
276. C73#33516semidirect
283. C2×#135C2×D73semidirect
296. #467C2×D73semidirect
298. #466C2×D73semidirect
301. #427C2×D73semidirect
510. C22#17989017semidirect

Maximal subgroups

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

subgroupindex
C22×#145566132
C2×#179890172
#226118692
#226118642
#226118592
#226118522
#226118472
#226118422
#226116362
C2×C22⋊#119240032
C73×#30502
#3351673