Finite Group Databasebeta

← #33056552

Subgroups — #33056552

Order 18688.

614 conjugacy classes of subgroups.

The subgroup lattice has 578 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22620497C211yesmaximal, characteristic
2#22618908C211yesmaximal, characteristic, complemented
3#22618903C211yesmaximal, characteristic, complemented
4#22608842C211yesmaximal, characteristic, complemented
5#22608837C211yesmaximal, characteristic, complemented
6#22608834C211yesmaximal, characteristic, complemented
7#22608832C211yesmaximal, characteristic, complemented
8#22608753C211yesmaximal, characteristic, complemented
9#22608748C211yesmaximal, characteristic, complemented
10#22608745C211yesmaximal, characteristic, complemented
11#22608693C211yesmaximal, characteristic, complemented
12#22608390C211yesmaximal, characteristic, complemented
13#22608388C211yesmaximal, characteristic, complemented
14#322×D73C211yesmaximal, characteristic, complemented
15C73×SD16⋊C8C211yesmaximal, characteristic, complemented
16#17989124C2211yescharacteristic
17#17989119C2211yescharacteristic
18#17988987C411yescharacteristic
19#17988968C411yescharacteristic
20#17988943C411yescharacteristic
21#17988940C411yescharacteristic
22#17988827C411yescharacteristic
23#17988822C411yescharacteristic
24#17988817C411yescharacteristic
25#17988813C411yescharacteristic
26#17988718C2211yescharacteristic
27#17988714C2211yescharacteristic
28C8×Dic146C2211yescharacteristic
29C4×#11924018C2211yescharacteristic
30C4×#11924017C2211yescharacteristic
31C4×#11924013C2211yescharacteristic
32#17988665C2211yescharacteristic
33#17988664C2211yescharacteristic
34#17988661—21no—
35#17988657—21no—
36C8×D292—21no—
37C8×D292C2211yescharacteristic
38C8×C22⋊D73—21no—
39#17988647—21no—
40#17988645—21no—
41#17988643C2211yescharacteristic
42#17988643C2211yescharacteristic
43#17988542—21no—
44#17988541—21no—
45C4×C8×D73C2211yescharacteristic
46C4×C8×D73C2211yescharacteristic
47C4×C8×D73C2211yescharacteristic
48#17988340C2211yescharacteristic
49#17988337C2211yescharacteristic
50#17988337C2211yescharacteristic
51#17988319C2211yescharacteristic
52Q8⋊#10596831C2211yescharacteristic
53#17988317C2211yescharacteristic
54#17988316C2211yescharacteristic
55D4⋊#10596831C2211yescharacteristic
56#17988314C2211yescharacteristic
57#17988309C2211yescharacteristic
58#17988309C2211yescharacteristic
59#17988308C2211yescharacteristic
60C8×Q8×C73C2211yescharacteristic
61C4×SD16×C73C2211yescharacteristic
62C8×D4×C73C2211yescharacteristic
63#335×C73C2211yescharacteristic
64Q8⋊C8×C73C2211yescharacteristic
65D4⋊C8×C73C2211yescharacteristic
66#322×C73C2211yescharacteristic
67C2×#11924018C2×C411yescharacteristic
68C2×#11924017C2×C411yescharacteristic
69C2×#11924013C2×C411yescharacteristic
70C2×#11924012—21no—
71C2×#11924011—21no—
72C2×C8×D73—21no—
73C2×C8×D73—21no—
74C2×C8×D73C2×C411yescharacteristic
75C2×C8×D73C2×C411yescharacteristic
76C2×C8×D73D411yescharacteristic
77C2×C8×D73D411yescharacteristic
78#14556586—21no—
79#14556584—21no—
80#14556582C2×C411yescharacteristic
81#14556582C2×C411yescharacteristic
82#14556580C2×C411yescharacteristic
83#14556578C2×C411yescharacteristic
84#14556576—21no—
85#14556573—21no—
86#14556571C2×C411yescharacteristic
87#14556567C2×C411yescharacteristic
88#14556566C2×C411yescharacteristic
89#14556562C2×C411yescharacteristic
90C4×Dic146C2311yescharacteristic
91C4×D292—21no—
92C4×D292C2311yescharacteristic
93C4×C22⋊D73—21no—
94#14556550—21no—
95#14556548—21no—
96C42×D73C2311yescharacteristic
97#14556524C2×C411yescharacteristic
98#14556524C2×C411yescharacteristic
99#14556523C2×C411yescharacteristic
100C4⋊#10596831—21no—
101C4⋊#10596831C2311yescharacteristic
102C4⋊#10596831C2311yescharacteristic
103#14556520—21no—
104#14556520—21no—
105#14556516C2×C411yescharacteristic
106#14556515C2×C411yescharacteristic
107#14556514C2×C411yescharacteristic
108#14556513C2×C411yescharacteristic
109#14556512C2×C411yescharacteristic
110#14556511C2×C411yescharacteristic
111C22⋊#10596831—21no—
112#14556503—21no—
113#14556503—21no—
114#14556503—21no—
115#14556503—21no—
116#14556502—21no—
117#14556501—21no—
118#14556501—21no—
119#14556501—21no—
120C4×#10596831C2311yescharacteristic
121C4×#10596831C2311yescharacteristic
122C4×#10596831C2311yescharacteristic
123C8×Dic73—21no—
124C8×Dic73—21no—
125C8×Dic73C2×C411yescharacteristic
126C8×Dic73C2×C411yescharacteristic
127C8×Dic73D411yescharacteristic
128C8×Dic73D411yescharacteristic
129C2×SD16×C73C2×C411yescharacteristic
130C22×C8×C73—21no—
131C4×Q8×C73C2311yescharacteristic
132C4×D4×C73C2311yescharacteristic
133#106×C73C2×C411yescharacteristic
134C4⋊C8×C73—21no—
135C4⋊C8×C73C2311yescharacteristic
136C4⋊C8×C73C2311yescharacteristic
137#103×C73C2×C411yescharacteristic
138#102×C73C2×C411yescharacteristic
139C22⋊C8×C73—21no—
140C4×C8×C73—21no—
141C4×C8×C73C2311yescharacteristic
142C4×C8×C73C2311yescharacteristic
143C4×C8×C73C2311yescharacteristic
144C2×Dic146C22×C411yescharacteristic
145C2×D292—21no—
146C2×D292C22×C411yescharacteristic
147C2×C22⋊D73—21no—
148C2×C4×D73—21no—
149C2×C4×D73C22×C411yescharacteristic
150C2×C4×D73C2×D411yescharacteristic
151C2×C4×D73#4111yescharacteristic
152#11924018—22no—
153#11924017—22no—
154#11924013—22no—
155#11924012—42no—
156#11924011—42no—
157C2×#10596831—21no—
158C2×#10596831—21no—
159C2×#10596831—21no—
160C2×#10596831C22×C411yescharacteristic
161C2×#10596831C22×C411yescharacteristic
162C2×#10596831C2×D411yescharacteristic
163C2×#10596831C2×D411yescharacteristic
164C8×D73—22no—
165C8×D73—22no—
166C8×D73—41no—
167C8×D73—41no—
168C8×D73—42no—
169C8×D73—42no—
170C4⋊Dic73—21no—
171C4⋊Dic73—21no—
172C4⋊Dic73—21no—
173C4⋊Dic73C22×C411yescharacteristic
174C4⋊Dic73C22×C411yescharacteristic
175#11924007—21no—
176#11924007—21no—
177#11924007—21no—
178#11924007—21no—
179C22⋊Dic73—21no—
180C22⋊Dic73—21no—
181C22⋊Dic73—21no—
182#11924005—21no—
183#11924005—21no—
184#11924005—21no—
185#11924005—21no—
186#11924005—21no—
187#11924005—21no—
188C4×Dic73—21no—
189C4×Dic73C22×C411yescharacteristic
190C4×Dic73C2×D411yescharacteristic
191C4×Dic73#4111yescharacteristic
192C2×Q8×C73C22×C411yescharacteristic
193C2×D4×C73C22×C411yescharacteristic
194C22×C4×C73—21no—
195SD16×C73—22no—
196C2×C8×C73—21no—
197C2×C8×C73—21no—
198C2×C8×C73—21no—
199C2×C8×C73—42no—
200C2×C8×C73C22×C411yescharacteristic
201C2×C8×C73C22×C411yescharacteristic
202C2×C8×C73C2×D411yescharacteristic
203C2×C8×C73C2×D411yescharacteristic
204C4⋊C4×C73—21no—
205C4⋊C4×C73C22×C411yescharacteristic
206C4⋊C4×C73C22×C411yescharacteristic
207C22⋊C4×C73—21no—
208C42×C73—21no—
209C42×C73C2411yescharacteristic
210C23×C73—21no—
211C22×D73—21no—
212C22×D73C4×D411yescharacteristic
213Q8×C73—21no—
214Q8×C73—21no—
215D4×C73—21no—
216D4×C73—21no—
217C2×C4×C73—21no—
218C2×C4×C73—21no—
219C2×C4×C73—21no—
220C2×C4×C73—41no—
221C2×C4×C73—41no—
222C2×C4×C73C23×C411yescharacteristic
223C2×C4×C73C22×D411yescharacteristic
224C2×C4×C73C2×#4111yescharacteristic
225C22⋊D73—42no—
226C2×Dic73—21no—
227C2×Dic73—21no—
228C2×Dic73—21no—
229C2×Dic73C4×D411yescharacteristic
230C2×Dic73C4×D411yescharacteristic
231C2×Dic73C4×D411yescharacteristic
232D292—21no—
233D292—21no—
234D292—42no—
235C4×D73—22no—
236C4×D73—21no—
237C4×D73—21no—
238C4×D73—21no—
239C4×D73—21no—
240C4×D73—41no—
241C4×D73—41no—
242Dic146—21no—
243Dic146—21no—
244C8×C73—21no—
245C8×C73—21no—
246C8×C73—21no—
247C8×C73—41no—
248C8×C73—41no—
249C8×C73—41no—
250C8×C73#13112yes—
251#10596831—21no—
252#10596831—21no—
253#10596831—21no—
254#10596831—41no—
255#10596831—41no—
256#10596831—41no—
257#10596831#13112yes—
258C22×C73—41no—
259C22×C73—41no—
260C22×C73—41no—
261C22×C73C2×C4×D411yescharacteristic
262C2×D73—21no—
263C2×D73—21no—
264C2×D73—41no—
265C2×D73—41no—
266C2×D73—41no—
267C2×D73#44412yes—
268C4×C73—21no—
269C4×C73—41no—
270C4×C73—41no—
271C4×C73—41no—
272C4×C73C2×#13111yescharacteristic
273C4×C73C2×#13111yescharacteristic
274C4×C73#57511yescharacteristic
275C4×C73#57511yescharacteristic
276Dic73—21no—
277Dic73—21no—
278Dic73—21no—
279Dic73—41no—
280Dic73—41no—
281Dic73—41no—
282#17963—731nomaximal, Sylow
283C2×C73—81no—
284C2×C73#293111yescharacteristic
285C2×C73#294311yescharacteristic
286C2×C73C2×#44411yescharacteristic
287D73—22no—
288D73—81no—
289#3168—731no—
290#2966—731no—
291#2966—731no—
292#1594—731no—
293#1593—731no—
294#1592—731no—
295#1592—731no—
296#1580—731no—
297#1579—731no—
298SD16⋊C8—731no—
299SD16⋊C8D7311yescharacteristic, complemented
300#1568—731no—
301#1494—731no—
302#1494—731no—
303C2×#322—731no—
304C73#1796311yescharacteristic, complemented, Sylow
305#529—731no—
306#529—731no—
307#499—731no—
308#495—731no—
309#488—731no—
310#488—731no—
311#467—731no—
312#466—731no—
313#465—731no—
314#465—731no—
315#446—731no—
316#446—731no—
317C8×Q8—731no—
318C8×Q8C2×D7311yescharacteristic
319C4×Q16—731no—
320C4×SD16—731no—
321C4×SD16C2×D7311yescharacteristic
322C4×D8—731no—
323#436—731no—
324#436—731no—
325#435—1461no—
326#435—1461no—
327C8×D4—731no—
328C8×D4—1461no—
329C8×D4—1461no—
330C8×D4C2×D7311yescharacteristic
331#433—1461no—
332#433—1461no—
333#432—731no—
334#432—731no—
335#405—1461no—
336#405—1461no—
337C2×C4×C8—731no—
338C2×C4×C8—731no—
339C2×C4×C8—731no—
340#335—731no—
341#335C2×D7311yescharacteristic
342#334—731no—
343#334—731no—
344Q8⋊C8—731no—
345Q8⋊C8—731no—
346Q8⋊C8—731no—
347Q8⋊C8C2×D7311yescharacteristic
348D4⋊C8—731no—
349D4⋊C8—731no—
350D4⋊C8—731no—
351D4⋊C8C2×D7311yescharacteristic
352#322—731no—
353#322—731no—
354#322—731no—
355#322C2×D7311yescharacteristic
356C2×Q16—731no—
357C2×SD16—731no—
358C2×SD16C4×D7311yescharacteristic
359C2×D8—731no—
360C2×M16—1461no—
361C2×M16—1461no—
362C22×C8—21no—
363C22×C8—731no—
364C22×C8—731no—
365C22×C8—731no—
366C22×C8—731no—
367C22×C8—1461no—
368C22×C8—1461no—
369#126—1461no—
370#126—1461no—
371#125—731no—
372#125—731no—
373#124—731no—
374#124—731no—
375#123—1461no—
376#123—1461no—
377C22⋊Q8—731no—
378C22⋊Q8—731no—
379#121—731no—
380#121—731no—
381C4×Q8—731no—
382C4×Q8C22×D7311yescharacteristic
383C4×D4—731no—
384C4×D4—1461no—
385C4×D4—1461no—
386C4×D4C22×D7311yescharacteristic
387#117—1461no—
388#117—1461no—
389C2×C42—731no—
390#107—731no—
391#107—731no—
392#106—731no—
393#106C4×D7311yescharacteristic
394C4⋊C8—21no—
395C4⋊C8—731no—
396C4⋊C8—731no—
397C4⋊C8—1461no—
398C4⋊C8—1461no—
399C4⋊C8—1461no—
400C4⋊C8C22×D7311yescharacteristic
401C4⋊C8C22×D7311yescharacteristic
402#103—731no—
403#103—731no—
404#103—731no—
405#103C4×D7311yescharacteristic
406#102—731no—
407#102—731no—
408#102—731no—
409#102C4×D7311yescharacteristic
410C22⋊C8—21no—
411C22⋊C8—1461no—
412C22⋊C8—1461no—
413C22⋊C8—1461no—
414C22⋊C8—1461no—
415C22⋊C8—1461no—
416#97—1461no—
417#97—1461no—
418#97—1461no—
419#97—1461no—
420C4×C8—21no—
421C4×C8—731no—
422C4×C8—731no—
423C4×C8—731no—
424C4×C8—731no—
425C4×C8—731no—
426C4×C8—731no—
427C4×C8—731no—
428C4×C8—1461no—
429C4×C8—1461no—
430C4×C8C22×D7311yescharacteristic
431C4×C8C22×D7311yescharacteristic
432C4×C8C22×D7311yescharacteristic
433C2×Q8—731no—
434C2×Q8C2×C4×D7311yescharacteristic
435C2×D4—731no—
436C2×D4—1461no—
437C2×D4—1461no—
438C2×D4C2×C4×D7311yescharacteristic
439C22×C4—21no—
440C22×C4—731no—
441C22×C4—731no—
442C22×C4—731no—
443C22×C4—1461no—
444Q16—1462no—
445SD16—22no—
446SD16—1462no—
447D8—1462no—
448M16—2922no—
449M16—2922no—
450C2×C8—21no—
451C2×C8—21no—
452C2×C8—21no—
453C2×C8—42no—
454C2×C8—2922no—
455C2×C8—2921no—
456C2×C8—2921no—
457C2×C8—2922no—
458C2×C8—731no—
459C2×C8—731no—
460C2×C8—731no—
461C2×C8—731no—
462C2×C8—1462no—
463C2×C8—1461no—
464C2×C8—1461no—
465C2×C8—1461no—
466C2×C8—1462no—
467C2×C8C2×C4×D7311yescharacteristic
468C2×C8C2×C4×D7311yescharacteristic
469C2×C8D4×D7311yescharacteristic
470C2×C8D4×D7311yescharacteristic
471C4⋊C4—21no—
472C4⋊C4—731no—
473C4⋊C4—731no—
474C4⋊C4—1461no—
475C4⋊C4—1461no—
476C4⋊C4—1461no—
477C4⋊C4—1461no—
478C4⋊C4—1461no—
479C4⋊C4—1461no—
480C4⋊C4—1461no—
481C4⋊C4C2×C4×D7311yescharacteristic
482C4⋊C4C2×C4×D7311yescharacteristic
483C22⋊C4—21no—
484C22⋊C4—1461no—
485C22⋊C4—1461no—
486C22⋊C4—1461no—
487C22⋊C4—1461no—
488C22⋊C4—1461no—
489C22⋊C4—1461no—
490C22⋊C4—1461no—
491C22⋊C4—1461no—
492C22⋊C4—1461no—
493C42—21no—
494C42—731no—
495C42—731no—
496C42—731no—
497C42—1461no—
498C42C23×D7311yescharacteristic
499C23—21no—
500C23—731no—
501C23—1461no—
502Q8—21no—
503Q8—21no—
504Q8—1461no—
505Q8—1461no—
506D4—21no—
507D4—21no—
508D4—2922no—
509D4—2922no—
510D4—1461no—
511D4—1461no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—2921no—
518C2×C4—2921no—
519C2×C4—731no—
520C2×C4—731no—
521C2×C4—731no—
522C2×C4—1462no—
523C2×C4—1461no—
524C2×C4—1461no—
525C2×C4—1461no—
526C2×C4—1461no—
527C2×C4—1461no—
528C2×C4—1461no—
529C2×C4—1461no—
530C2×C4C22×C4×D7311yescharacteristic
531C2×C4C2×D4×D7311yescharacteristic
532C2×C4#41×D7311yescharacteristic
533C8—21no—
534C8—21no—
535C8—21no—
536C8—41no—
537C8—41no—
538C8—41no—
539C8—2921no—
540C8—2921no—
541C8—2921no—
542C8—732no—
543C8—1461no—
544C8—1461no—
545C8—1461no—
546C8#1455659712yes—
547C22—41no—
548C22—41no—
549C22—41no—
550C22—2921no—
551C22—2921no—
552C22—2921no—
553C22—732no—
554C22—1461no—
555C22—1461no—
556C22C4×D4×D7311yescharacteristic
557C4—21no—
558C4—41no—
559C4—41no—
560C4—41no—
561C4—2921no—
562C4—2921no—
563C4—2921no—
564C4—1461no—
565C4—1461no—
566C4—1461no—
567C4#131×D7311yescharacteristic
568C4C2×#1455659711yescharacteristic
569C4#1798938311yescharacteristic
570C4#1798938611yescharacteristic
571C2—81no—
572C2—5841no—
573C2—1462no—
574C2#2261863811yescharacteristic
575C2#2261872611yescharacteristic
576C2#444×D7311yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D73
commutatorC2×C4×C73C23×C4
FrattiniC42C23×D73
FittingC73×SD16⋊C8C2
socleC22×C73C2×C4×D4
prime radical#17988822C4
cosocleC42×C73C24
hypercenterSD16⋊C8D73
hypocenterC73#17963
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ SD16⋊C8

Complements

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

NH ≅ G/Ntype
2. #22618908C2semidirect
3. #22618903C2semidirect
4. #22608842C2semidirect
5. #22608837C2semidirect
6. #22608834C2semidirect
7. #22608832C2semidirect
8. #22608753C2semidirect
9. #22608748C2semidirect
10. #22608745C2semidirect
11. #22608693C2semidirect
12. #22608390C2semidirect
13. #22608388C2semidirect
14. #322×D73C2semidirect
15. C73×SD16⋊C8C2semidirect
299. SD16⋊C8D73semidirect
304. C73#17963semidirect

Maximal subgroups

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

subgroupindex
#226204972
#226189082
#226189032
#226088422
#226088372
#226088342
#226088322
#226087532
#226087482
#226087452
#226086932
#226083902
#226083882
#322×D732
C73×SD16⋊C82
#1796373