Finite Group Databasebeta

← #33055856

Subgroups — #33055856

Order 18688.

686 conjugacy classes of subgroups.

The subgroup lattice has 598 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22619960C211yesmaximal, characteristic
2C8×Q8×D73C211yesmaximal, characteristic, complemented
3#22618638C211yesmaximal, characteristic
4#22608777C211yesmaximal, characteristic, complemented
5#22608775C211yesmaximal, characteristic, complemented
6#22608773C211yesmaximal, characteristic, complemented
7#22608771C211yesmaximal, characteristic, complemented
8C8×#11924017C211yesmaximal, characteristic, complemented
9C8×#11924015C211yesmaximal, characteristic, complemented
10C8×#11924016C211yesmaximal, characteristic, complemented
11#22608700C211yesmaximal, characteristic, complemented
12#22608358C211yesmaximal, characteristic, complemented
13#22608328C211yesmaximal, characteristic
14#22608213C211yesmaximal, characteristic, complemented
15C8×SD16×C73C211yesmaximal, characteristic, complemented
16C2×#14556597—21no—
17#17989134C2211yescharacteristic
18C4×Q8×D73C2211yescharacteristic
19#17988907C411yescharacteristic
20#17988904C411yescharacteristic
21#17988871C411yescharacteristic
22#17988870C411yescharacteristic
23#17988786C411yescharacteristic
24#17988785C411yescharacteristic
25#17988727C411yescharacteristic
26#17988724C411yescharacteristic
27Q8×#10596831C2211yescharacteristic
28C8×Dic146—21no—
29C8×Dic146C2211yescharacteristic
30#17988711—21no—
31#17988711C2211yescharacteristic
32C4×#11924017C2211yescharacteristic
33C4×#11924015C2211yescharacteristic
34C4×#11924016C2211yescharacteristic
35#17988665C2211yescharacteristic
36#17988664C2211yescharacteristic
37#17988662—21no—
38C8×D292C2211yescharacteristic
39C8×C22⋊D73—21no—
40#17988638C2211yescharacteristic
41#17988638C2211yescharacteristic
42#17988609C2211yescharacteristic
43C4⋊C8×D73—21no—
44C4⋊C8×D73C2211yescharacteristic
45#17988552—21no—
46C4×C8×D73—21no—
47C4×C8×D73C2211yescharacteristic
48#17988338C2211yescharacteristic
49#17988337C2211yescharacteristic
50#17988337C2211yescharacteristic
51#17988319C2211yescharacteristic
52Q8⋊#10596831C2211yescharacteristic
53#17988317C2211yescharacteristic
54#17988316C2211yescharacteristic
55D4⋊#10596831C2211yescharacteristic
56#17988314C2211yescharacteristic
57#17988309C2211yescharacteristic
58#17988309C2211yescharacteristic
59C8×#10596831C2211yescharacteristic
60C8×Q8×C73C2211yescharacteristic
61C4×SD16×C73C2211yescharacteristic
62C8×D4×C73C2211yescharacteristic
63#334×C73C2211yescharacteristic
64Q8⋊C8×C73C2211yescharacteristic
65D4⋊C8×C73C2211yescharacteristic
66C82×C73C2211yescharacteristic
67C2×#11924028—21no—
68C2×Q8×D73C2×C411yescharacteristic
69C2×#11924017C2×C411yescharacteristic
70C2×#11924016C2×C411yescharacteristic
71C2×#11924015C2×C411yescharacteristic
72#14556597—44no—
73C2×#11924012—21no—
74C2×#11924011—21no—
75C2×#11924011—21no—
76C2×C8×D73—21no—
77C2×C8×D73—21no—
78C2×C8×D73D411yescharacteristic
79C2×C8×D73D411yescharacteristic
80#14556590C2×C411yescharacteristic
81#14556587C2×C411yescharacteristic
82#14556577—21no—
83#14556566C2×C411yescharacteristic
84#14556564—21no—
85#14556562C2×C411yescharacteristic
86Q8×Dic73C2×C411yescharacteristic
87C4×Dic146—21no—
88C4×Dic146C2311yescharacteristic
89#14556556—21no—
90#14556556C2×C411yescharacteristic
91C4×D292C2311yescharacteristic
92C4×C22⋊D73—21no—
93C4⋊C4×D73—21no—
94C4⋊C4×D73C2×C411yescharacteristic
95C42×D73—21no—
96C42×D73C2311yescharacteristic
97#14556523C2×C411yescharacteristic
98#14556522C2×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#14556520C2×C411yescharacteristic
109#14556520C2×C411yescharacteristic
110#14556516C2×C411yescharacteristic
111#14556515C2×C411yescharacteristic
112#14556514C2×C411yescharacteristic
113#14556513C2×C411yescharacteristic
114#14556512C2×C411yescharacteristic
115#14556511C2×C411yescharacteristic
116C22⋊#10596831—21no—
117#14556503—21no—
118#14556503—21no—
119#14556503—21no—
120#14556503C2×C411yescharacteristic
121#14556503C2×C411yescharacteristic
122C4×#10596831—21no—
123C4×#10596831C2311yescharacteristic
124C4×#10596831C2311yescharacteristic
125C4×#10596831C2311yescharacteristic
126C8×Dic73—21no—
127C8×Dic73—21no—
128C8×Dic73D411yescharacteristic
129C8×Dic73D411yescharacteristic
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—
141C4×C8×C73—21no—
142C4×C8×C73C2311yescharacteristic
143C4×C8×C73C2311yescharacteristic
144C4×C8×C73C2311yescharacteristic
145#11924028—42no—
146#11924028—42no—
147C2×Dic146—21no—
148C2×Dic146—21no—
149C2×Dic146C22×C411yescharacteristic
150Q8×D73—22no—
151Q8×D73—22no—
152C2×D292—21no—
153C2×D292C22×C411yescharacteristic
154C2×C22⋊D73—21no—
155C2×C22⋊D73—21no—
156C2×C22⋊D73—21no—
157C2×C4×D73—21no—
158C2×C4×D73—21no—
159C2×C4×D73—21no—
160C2×C4×D73C22×C411yescharacteristic
161C2×C4×D73C2×D411yescharacteristic
162C2×C4×D73#4111yescharacteristic
163#11924017—22no—
164#11924016—22no—
165#11924015—22no—
166#11924012—42no—
167#11924011—42no—
168#11924011—42no—
169C2×#10596831—21no—
170C2×#10596831—21no—
171C2×#10596831—21no—
172C2×#10596831C22×C411yescharacteristic
173C2×#10596831C22×C411yescharacteristic
174C2×#10596831C2×D411yescharacteristic
175C2×#10596831C2×D411yescharacteristic
176C8×D73—22no—
177C8×D73—42no—
178C8×D73—42no—
179C8×D73SD1614yescomplemented
180C4⋊Dic73—21no—
181C4⋊Dic73C22×C411yescharacteristic
182C4⋊Dic73C22×C411yescharacteristic
183#11924007—21no—
184#11924007—21no—
185#11924007—21no—
186#11924007—21no—
187C22⋊Dic73—21no—
188#11924005—21no—
189#11924005—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×D73—21no—
216C22×D73C4×D411yescharacteristic
217Q8×C73—21no—
218Q8×C73#13112yescomplemented
219D4×C73—21no—
220D4×C73—21no—
221C2×C4×C73—21no—
222C2×C4×C73—21no—
223C2×C4×C73—21no—
224C2×C4×C73—41no—
225C2×C4×C73—41no—
226C2×C4×C73C23×C411yescharacteristic
227C2×C4×C73C22×D411yescharacteristic
228C2×C4×C73C2×#4111yescharacteristic
229C22⋊D73—42no—
230C22⋊D73—42no—
231C22⋊D73—42no—
232C2×Dic73—21no—
233C2×Dic73—21no—
234C2×Dic73—21no—
235C2×Dic73C4×D411yescharacteristic
236C2×Dic73C4×D411yescharacteristic
237C2×Dic73C4×D411yescharacteristic
238D292—21no—
239D292—21no—
240D292—42no—
241C4×D73—24no—
242C4×D73—21no—
243C4×D73—21no—
244C4×D73—41no—
245C4×D73—42no—
246C4×D73—41no—
247C4×D73—41no—
248C4×D73C2×SD1612yes—
249C4×D73#13512yes—
250Dic146—24no—
251Dic146—21no—
252Dic146—42no—
253Dic146#13112yescomplemented
254C8×C73—21no—
255C8×C73—21no—
256C8×C73—21no—
257C8×C73—41no—
258C8×C73—41no—
259C8×C73—41no—
260C8×C73C2×SD1612yes—
261#10596831—21no—
262#10596831—21no—
263#10596831—21no—
264#10596831—41no—
265#10596831—41no—
266#10596831—41no—
267#10596831C2×SD1612yes—
268C22×C73—41no—
269C22×C73—41no—
270C22×C73—41no—
271C22×C73C2×C4×D411yescharacteristic
272C2×D73—21no—
273C2×D73—21no—
274C2×D73—41no—
275C2×D73—41no—
276C2×D73—41no—
277C2×D73C4×SD1612yes—
278C4×C73—22no—
279C4×C73—21no—
280C4×C73—41no—
281C4×C73—41no—
282C4×C73C2×#13111yescharacteristic
283C4×C73C2×#13111yescharacteristic
284C4×C73C22×SD1611yescharacteristic
285C4×C73C2×#13511yescharacteristic
286Dic73—21no—
287Dic73—22no—
288Dic73—21no—
289Dic73—41no—
290Dic73—41no—
291Dic73C4×SD1612yes—
292#17901—731nomaximal, Sylow
293C2×C73—81no—
294C2×C73#292911yescharacteristic
295C2×C73C2×C4×SD1611yescharacteristic
296C2×C73#295511yescharacteristic
297D73—22no—
298D73—81no—
299#3103—731no—
300C2×C8×Q8—731no—
301#2931—731no—
302#1583—731no—
303#1583—731no—
304#1583—731no—
305#1583—731no—
306C8×SD16—731no—
307C8×SD16—731no—
308C8×SD16—731no—
309C8×SD16D7311yescharacteristic, complemented
310#1570—731no—
311#1487—731no—
312#1480—731no—
313#1450—731no—
314C73#1790111yescharacteristic, complemented, Sylow
315C2×#131—1461no—
316#532—731no—
317C2×C4×Q8—731no—
318#481—731no—
319#481—731no—
320#475—731no—
321#475—731no—
322#460—731no—
323#460—731no—
324#448—731no—
325#448—731no—
326C8×Q8—731no—
327C8×Q8—731no—
328C8×Q8—731no—
329C8×Q8—1461no—
330C8×Q8—1461no—
331C8×Q8C2×D7311yescharacteristic
332C4×SD16—731no—
333C4×SD16—731no—
334C4×SD16—731no—
335C4×SD16C2×D7311yescharacteristic
336#436—731no—
337#436—731no—
338#436—1461no—
339C8×D4—731no—
340C8×D4—1461no—
341C8×D4C2×D7311yescharacteristic
342#431—731no—
343#431—731no—
344#423—731no—
345C2×C4⋊C8—731no—
346C2×C4⋊C8—1461no—
347#408—1461no—
348C2×C4×C8—731no—
349C2×C4×C8—1461no—
350#334—731no—
351#334—731no—
352#334—731no—
353#334C2×D7311yescharacteristic
354Q8⋊C8—731no—
355Q8⋊C8—731no—
356Q8⋊C8—731no—
357Q8⋊C8C2×D7311yescharacteristic
358D4⋊C8—731no—
359D4⋊C8—731no—
360D4⋊C8—731no—
361D4⋊C8C2×D7311yescharacteristic
362#322—731no—
363#322—731no—
364C82—731no—
365C82C2×D7311yescharacteristic
366C2×#41—1461no—
367C22×Q8—731no—
368C2×SD16—731no—
369C2×SD16—731no—
370C2×SD16—731no—
371C2×SD16C4×D7311yescharacteristic
372#131—2924no—
373C2×M16—1461no—
374C2×M16—1461no—
375C2×M16—1461no—
376C22×C8—21no—
377C22×C8—731no—
378C22×C8—731no—
379C22×C8—1461no—
380C22×C8—1461no—
381C4⋊Q8—731no—
382#127—731no—
383#124—1461no—
384#121—731no—
385#121—731no—
386#121—1461no—
387C4×Q8—731no—
388C4×Q8—731no—
389C4×Q8—731no—
390C4×Q8—1461no—
391C4×Q8—1461no—
392C4×Q8C22×D7311yescharacteristic
393C4×D4—731no—
394C4×D4—1461no—
395C4×D4C22×D7311yescharacteristic
396C2×C4⋊C4—731no—
397C2×C4⋊C4—1461no—
398C2×C42—731no—
399C2×C42—1461no—
400#106—731no—
401#106—731no—
402#106—731no—
403#106C4×D7311yescharacteristic
404C4⋊C8—21no—
405C4⋊C8—731no—
406C4⋊C8—731no—
407C4⋊C8—731no—
408C4⋊C8—731no—
409C4⋊C8—1461no—
410C4⋊C8—1461no—
411C4⋊C8—1461no—
412C4⋊C8—1461no—
413C4⋊C8—1461no—
414C4⋊C8—1461no—
415C4⋊C8C22×D7311yescharacteristic
416C4⋊C8C22×D7311yescharacteristic
417#103—731no—
418#103—731no—
419#103—731no—
420#103C4×D7311yescharacteristic
421#102—731no—
422#102—731no—
423#102—731no—
424#102C4×D7311yescharacteristic
425C22⋊C8—21no—
426C22⋊C8—731no—
427C22⋊C8—731no—
428C22⋊C8—1461no—
429C22⋊C8—1461no—
430C22⋊C8—1461no—
431C22⋊C8—1461no—
432C4×C8—21no—
433C4×C8—731no—
434C4×C8—731no—
435C4×C8—731no—
436C4×C8—731no—
437C4×C8—731no—
438C4×C8—1461no—
439C4×C8—1461no—
440C4×C8—1461no—
441C4×C8C22×D7311yescharacteristic
442C4×C8C22×D7311yescharacteristic
443C4×C8C22×D7311yescharacteristic
444#41—2922no—
445#41—2922no—
446C2×Q8—731no—
447C2×Q8—1461no—
448C2×Q8—1461no—
449C2×Q8—1462no—
450C2×Q8—1462no—
451C2×Q8C2×C4×D7311yescharacteristic
452C2×D4—731no—
453C2×D4—1461no—
454C2×D4—1461no—
455C2×D4—1461no—
456C2×D4—1461no—
457C2×D4C2×C4×D7311yescharacteristic
458C22×C4—21no—
459C22×C4—731no—
460C22×C4—731no—
461C22×C4—731no—
462C22×C4—1461no—
463C22×C4—1461no—
464C22×C4—1461no—
465SD16—22no—
466SD16—1462no—
467SD16—1462no—
468SD16—1462no—
469M16—2922no—
470M16—2922no—
471M16—2922no—
472C2×C8—21no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—42no—
476C2×C8—2922no—
477C2×C8—2922no—
478C2×C8—731no—
479C2×C8—734no—
480C2×C8—731no—
481C2×C8—731no—
482C2×C8—731no—
483C2×C8—1462no—
484C2×C8—1461no—
485C2×C8—1461no—
486C2×C8—1461no—
487C2×C8C2×C4×D7311yescharacteristic
488C2×C8C2×C4×D7311yescharacteristic
489C2×C8D4×D7311yescharacteristic
490C2×C8D4×D7311yescharacteristic
491C4⋊C4—21no—
492C4⋊C4—731no—
493C4⋊C4—731no—
494C4⋊C4—1461no—
495C4⋊C4—1461no—
496C4⋊C4—1461no—
497C4⋊C4—1461no—
498C4⋊C4—1461no—
499C4⋊C4C2×C4×D7311yescharacteristic
500C4⋊C4C2×C4×D7311yescharacteristic
501C22⋊C4—21no—
502C22⋊C4—1461no—
503C22⋊C4—1461no—
504C22⋊C4—1461no—
505C42—21no—
506C42—731no—
507C42—731no—
508C42—731no—
509C42—1461no—
510C42—1461no—
511C42—1461no—
512C42C23×D7311yescharacteristic
513C23—21no—
514C23—731no—
515C23—1461no—
516Q8—21no—
517Q8—2922no—
518Q8—732no—
519Q8—1464no—
520Q8—1461no—
521Q8#1455659712yescomplemented
522D4—21no—
523D4—21no—
524D4—2922no—
525D4—2922no—
526D4—2922no—
527D4—2922no—
528D4—1461no—
529D4—1461no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—2921no—
536C2×C4—2922no—
537C2×C4—2921no—
538C2×C4—2921no—
539C2×C4—731no—
540C2×C4—732no—
541C2×C4—732no—
542C2×C4—731no—
543C2×C4—731no—
544C2×C4—1464no—
545C2×C4—1461no—
546C2×C4—1461no—
547C2×C4—1461no—
548C2×C4—1461no—
549C2×C4—1461no—
550C2×C4C22×C4×D7311yescharacteristic
551C2×C4C2×D4×D7311yescharacteristic
552C2×C4#41×D7311yescharacteristic
553C8—21no—
554C8—21no—
555C8—21no—
556C8—41no—
557C8—41no—
558C8—41no—
559C8—2921no—
560C8—2921no—
561C8—2921no—
562C8—732no—
563C8—1461no—
564C8—1461no—
565C8—1461no—
566C8SD16×D7312yes—
567C22—41no—
568C22—41no—
569C22—41no—
570C22—2921no—
571C22—2921no—
572C22—2921no—
573C22—732no—
574C22—1461no—
575C22—1461no—
576C22C4×D4×D7311yescharacteristic
577C4—21no—
578C4—22no—
579C4—41no—
580C4—41no—
581C4—2921no—
582C4—2921no—
583C4—732no—
584C4—1461no—
585C4—1461no—
586C4—1462no—
587C4#131×D7311yescharacteristic
588C4C2×#1455659711yescharacteristic
589C4C2×SD16×D7311yescharacteristic
590C4#135×D7311yescharacteristic
591C2—81no—
592C2—5841no—
593C2—1462no—
594C2#2261862011yescharacteristic
595C2C4×SD16×D7311yescharacteristic
596C2#2261882911yescharacteristic
subgroupG/H
centerC2×C8D4×D73
commutatorC2×C4×C73C23×C4
FrattiniC42C23×D73
FittingC8×SD16×C73C2
socleC22×C73C2×C4×D4
prime radical#17989134C22
cosocleC42×C73C24
hypercenterC8×SD16D73
hypocenterC73#17901
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C2×C8 ◁ C4×C8 ◁ C8×SD16

Complements

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

NH ≅ G/Ntype
2. C8×Q8×D73C2semidirect
4. #22608777C2semidirect
5. #22608775C2semidirect
6. #22608773C2semidirect
7. #22608771C2semidirect
8. C8×#11924017C2semidirect
9. C8×#11924015C2semidirect
10. C8×#11924016C2semidirect
11. #22608700C2semidirect
12. #22608358C2semidirect
14. #22608213C2semidirect
15. C8×SD16×C73C2semidirect
179. C8×D73SD16semidirect
218. Q8×C73#131semidirect
253. Dic146#131semidirect
309. C8×SD16D73semidirect
314. C73#17901semidirect
521. Q8#14556597semidirect

Maximal subgroups

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

subgroupindex
#226199602
C8×Q8×D732
#226186382
#226087772
#226087752
#226087732
#226087712
C8×#119240172
C8×#119240152
C8×#119240162
#226087002
#226083582
#226083282
#226082132
C8×SD16×C732
#1790173