Finite Group Databasebeta

← #33055993

Subgroups — #33055993

Order 18688.

712 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22619982C211yesmaximal, characteristic
2#22618908C211yesmaximal, characteristic, complemented
3C8×D4×D73C211yesmaximal, characteristic, complemented
4#22608770C211yesmaximal, characteristic, complemented
5#22608768C211yesmaximal, characteristic, complemented
6#22608759C211yesmaximal, characteristic, complemented
7#22608757C211yesmaximal, characteristic, complemented
8#22608756C211yesmaximal, characteristic, complemented
9#22608753C211yesmaximal, characteristic, complemented
10#22608741C211yesmaximal, characteristic, complemented
11#22608715C211yesmaximal, characteristic, complemented
12#22608372C211yesmaximal, characteristic, complemented
13#22608343C211yesmaximal, characteristic, complemented
14#22608219C211yesmaximal, characteristic, complemented
15C73×SD16⋊C8C211yesmaximal, characteristic, complemented
16C22×C8×D73—21no—
17#17989124C2211yescharacteristic
18C4×D4×D73C2211yescharacteristic
19#17988921C411yescharacteristic
20#17988907C411yescharacteristic
21#17988886C411yescharacteristic
22#17988885C411yescharacteristic
23#17988805C411yescharacteristic
24#17988804C411yescharacteristic
25#17988736C411yescharacteristic
26#17988722C411yescharacteristic
27#17988718C2211yescharacteristic, complemented
28C4×#11924015C2211yescharacteristic
29C4×D584C2211yescharacteristic
30C4×#11924013C2211yescharacteristic
31#17988664C2211yescharacteristic, complemented
32#17988657—21no—
33C8×D292—21no—
34C8×D292C2211yescharacteristic
35C8×D292C2211yescharacteristic, complemented
36D4×#10596831C2211yescharacteristic
37#17988652C2211yescharacteristic
38C8×C22⋊D73—21no—
39#17988650—21no—
40#17988647—21no—
41#17988643C2211yescharacteristic
42#17988638C2211yescharacteristic
43C4⋊C8×D73C2211yescharacteristic
44C22⋊C8×D73—21no—
45#17988541—21no—
46C4×#11924011C2211yescharacteristic
47C4×C8×D73C2211yescharacteristic
48#17988339C2211yescharacteristic, complemented
49#17988338C2211yescharacteristic, complemented
50#17988337C2211yescharacteristic, complemented
51Q8⋊#10596831C2211yescharacteristic, complemented
52#17988316C2211yescharacteristic
53#17988316C2211yescharacteristic, complemented
54D4⋊#10596831C2211yescharacteristic
55#17988314C2211yescharacteristic
56#17988314C2211yescharacteristic, complemented
57#17988309C2211yescharacteristic, complemented
58#17988309C2211yescharacteristic, complemented
59#17988309C2211yescharacteristic, complemented
60C8×Q8×C73C2211yescharacteristic, complemented
61C4×SD16×C73C2211yescharacteristic
62C8×D4×C73C2211yescharacteristic
63#335×C73C2211yescharacteristic, complemented
64Q8⋊C8×C73C2211yescharacteristic, complemented
65D4⋊C8×C73C2211yescharacteristic
66#322×C73C2211yescharacteristic, complemented
67C2×D4×D73C2×C411yescharacteristic
68C22×C4×D73—21no—
69C2×#11924015C2×C411yescharacteristic
70C2×D584C2×C411yescharacteristic
71C2×#11924013C2×C411yescharacteristic
72C2×#11924011—21no—
73C2×#11924011—21no—
74C22×#10596831—21no—
75C2×C8×D73—21no—
76C2×C8×D73—21no—
77C2×C8×D73—21no—
78C2×C8×D73—21no—
79C2×C8×D73—44no—
80C2×C8×D73D411yescharacteristic
81C2×C8×D73D411yescharacteristic
82#14556586—21no—
83#14556582C2×C411yescharacteristic
84#14556580C2×C411yescharacteristic
85#14556576—21no—
86#14556567C2×C411yescharacteristic
87#14556566C2×C411yescharacteristic
88C4×D292—21no—
89C4×D292C2311yescharacteristic
90C4×D292C2311yescharacteristic
91D4×Dic73C2×C411yescharacteristic
92#14556553C2×C411yescharacteristic
93C4×C22⋊D73—21no—
94#14556551—21no—
95#14556548—21no—
96C4⋊C4×D73C2×C411yescharacteristic
97C22⋊C4×D73—21no—
98C42×D73C2311yescharacteristic
99#14556525C2×C411yescharacteristic
100#14556524C2×C411yescharacteristic
101#14556522C2×C411yescharacteristic
102C4⋊#10596831—21no—
103C4⋊#10596831C2311yescharacteristic
104C4⋊#10596831C2311yescharacteristic
105#14556520—21no—
106#14556520C2×C411yescharacteristic
107#14556520C2×C411yescharacteristic
108#14556515C2×C411yescharacteristic
109#14556513C2×C411yescharacteristic
110#14556513C2×C411yescharacteristic
111#14556512C2×C411yescharacteristic
112#14556511C2×C411yescharacteristic
113#14556511C2×C411yescharacteristic
114C22⋊#10596831—21no—
115#14556503—21no—
116#14556503—21no—
117#14556503—21no—
118#14556503—21no—
119#14556503—21no—
120#14556503C2×C411yescharacteristic
121#14556503C2×C411yescharacteristic
122#14556502—21no—
123#14556501—21no—
124#14556501—21no—
125C4×#10596831C2311yescharacteristic
126C4×#10596831C2311yescharacteristic
127C4×#10596831C2311yescharacteristic
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
145C23×D73—21no—
146C2×D292—21no—
147C2×D292C22×C411yescharacteristic
148C2×D292C22×C411yescharacteristic
149C2×C22⋊D73—21no—
150D4×D73—22no—
151D4×D73—22no—
152C22×Dic73—21no—
153C2×C4×D73—21no—
154C2×C4×D73—21no—
155C2×C4×D73—21no—
156C2×C4×D73—21no—
157C2×C4×D73—42no—
158C2×C4×D73—42no—
159C2×C4×D73C22×C411yescharacteristic
160C2×C4×D73C2×D411yescharacteristic
161C2×C4×D73#4111yescharacteristic
162#11924015—22no—
163D584—22no—
164#11924013—22no—
165#11924011—42no—
166#11924011—42no—
167C2×#10596831—21no—
168C2×#10596831—21no—
169C2×#10596831—21no—
170C2×#10596831—42no—
171C2×#10596831C22×C411yescharacteristic
172C2×#10596831C22×C411yescharacteristic
173C2×#10596831C2×D411yescharacteristic
174C2×#10596831C2×D411yescharacteristic
175C8×D73—22no—
176C8×D73—41no—
177C8×D73—42no—
178C8×D73—42no—
179C8×D73—42no—
180C8×D73—42no—
181C4⋊Dic73—21no—
182C4⋊Dic73C22×C411yescharacteristic
183C4⋊Dic73C22×C411yescharacteristic
184#11924007—21no—
185#11924007—21no—
186#11924007—21no—
187C22⋊Dic73—21no—
188#11924005—21no—
189#11924005—21no—
190#11924005—21no—
191#11924005—21no—
192#11924005—21no—
193#11924005—21no—
194#11924005—21no—
195C4×Dic73C22×C411yescharacteristic
196C4×Dic73C2×D411yescharacteristic
197C4×Dic73#4111yescharacteristic
198C2×Q8×C73C22×C411yescharacteristic
199C2×D4×C73C22×C411yescharacteristic
200C22×C4×C73—21no—
201SD16×C73—22no—
202C2×C8×C73—21no—
203C2×C8×C73—21no—
204C2×C8×C73—21no—
205C2×C8×C73—42no—
206C2×C8×C73C22×C411yescharacteristic
207C2×C8×C73C22×C411yescharacteristic
208C2×C8×C73C2×D411yescharacteristic
209C2×C8×C73C2×D411yescharacteristic
210C4⋊C4×C73—21no—
211C4⋊C4×C73C22×C411yescharacteristic
212C4⋊C4×C73C22×C411yescharacteristic
213C22⋊C4×C73—21no—
214C42×C73—21no—
215C42×C73C2411yescharacteristic
216C23×C73—21no—
217C22×D73—24no—
218C22×D73—21no—
219C22×D73—21no—
220C22×D73—42no—
221C22×D73—42no—
222C22×D73C4×D411yescharacteristic
223Q8×C73—21no—
224Q8×C73—21no—
225D4×C73—21no—
226D4×C73#13112yes—
227C2×C4×C73—21no—
228C2×C4×C73—21no—
229C2×C4×C73—21no—
230C2×C4×C73—41no—
231C2×C4×C73—41no—
232C2×C4×C73C23×C411yescharacteristic
233C2×C4×C73C22×D411yescharacteristic
234C2×C4×C73C2×#4111yescharacteristic
235C22⋊D73—24no—
236C2×Dic73—21no—
237C2×Dic73—21no—
238C2×Dic73—42no—
239C2×Dic73C4×D411yescharacteristic
240C2×Dic73C4×D411yescharacteristic
241C2×Dic73C4×D411yescharacteristic
242D292—21no—
243D292—21no—
244D292—21no—
245D292—42no—
246D292#13112yes—
247C4×D73—22no—
248C4×D73—21no—
249C4×D73—21no—
250C4×D73—21no—
251C4×D73—21no—
252C4×D73—42no—
253C4×D73—42no—
254C4×D73—41no—
255C4×D73—41no—
256C4×D73—41no—
257C4×D73—41no—
258C8×C73—21no—
259C8×C73—21no—
260C8×C73—21no—
261C8×C73—21no—
262C8×C73—41no—
263C8×C73—41no—
264C8×C73—41no—
265#10596831—21no—
266#10596831—21no—
267#10596831—21no—
268#10596831—41no—
269#10596831—41no—
270#10596831—41no—
271#10596831AGL(1,ℤ/8)12yes—
272C22×C73—22no—
273C22×C73—41no—
274C22×C73—41no—
275C22×C73C2×C4×D411yescharacteristic
276C2×D73—21no—
277C2×D73—21no—
278C2×D73—22no—
279C2×D73—48no—
280C2×D73—41no—
281C2×D73—41no—
282C2×D73—41no—
283C2×D73—41no—
284C2×D73—41no—
285C2×D73#44412yes—
286C4×C73—21no—
287C4×C73—41no—
288C4×C73—41no—
289C4×C73—41no—
290C4×C73C2×#13111yescharacteristic
291C4×C73C2×#13111yescharacteristic
292C4×C73C2×AGL(1,ℤ/8)11yescharacteristic
293C4×C73#57511yescharacteristic
294Dic73—21no—
295Dic73—21no—
296Dic73—21no—
297Dic73—41no—
298Dic73—41no—
299#17914—731nomaximal, Sylow
300C2×C73—42no—
301C2×C73#292911yescharacteristic
302C2×C73C4×AGL(1,ℤ/8)11yescharacteristic
303C2×C73C2×#44411yescharacteristic
304D73—22no—
305D73—42no—
306D73—81no—
307#3105—731no—
308#2966—731no—
309C2×C8×D4—731no—
310#1582—731no—
311#1582—731no—
312#1581—731no—
313#1581—731no—
314#1580—731no—
315#1580—731no—
316SD16⋊C8—731no—
317SD16⋊C8D7311yescharacteristic, complemented
318#1573—731no—
319#1490—731no—
320#1483—731no—
321#1451—731no—
322C73#1791411yescharacteristic, complemented, Sylow
323C23×C8—1461no—
324#529—731no—
325C2×C4×D4—731no—
326#484—731no—
327#481—731no—
328#478—731no—
329#478—731no—
330#463—731no—
331#463—731no—
332#450—731no—
333#447—731no—
334#446—731no—
335C8×Q8C2×D7311yescharacteristic, complemented
336C4×SD16—731no—
337C4×SD16C2×D7311yescharacteristic
338C4×D8—731no—
339C4×D8—731no—
340#436—731no—
341#435—1461no—
342C8×D4—731no—
343C8×D4—731no—
344C8×D4—731no—
345C8×D4—731no—
346C8×D4—1461no—
347C8×D4—1461no—
348C8×D4—1461no—
349C8×D4C2×D7311yescharacteristic
350#433—1461no—
351#432—731no—
352#431—731no—
353C2×C4⋊C8—731no—
354C2×C22⋊C8—1461no—
355#405—1461no—
356C4×M16—731no—
357C2×C4×C8—731no—
358#335—731no—
359#335C2×D7311yescharacteristic, complemented
360#334—731no—
361#334—731no—
362Q8⋊C8—731no—
363Q8⋊C8C2×D7311yescharacteristic, complemented
364D4⋊C8—731no—
365D4⋊C8—731no—
366D4⋊C8—731no—
367D4⋊C8—731no—
368D4⋊C8—731no—
369D4⋊C8C2×D7311yescharacteristic
370#322—731no—
371#322—731no—
372#322—731no—
373#322C2×D7311yescharacteristic, complemented
374C22×D4—731no—
375C23×C4—1461no—
376C2×SD16—731no—
377C2×SD16C4×D7311yescharacteristic
378C2×D8—731no—
379C2×D8—731no—
380C2×M16—1461no—
381C2×M16—1461no—
382C22×C8—21no—
383C22×C8—2924no—
384C22×C8—731no—
385C22×C8—731no—
386C22×C8—1461no—
387C22×C8—1461no—
388C22×C8—1461no—
389C22×C8—1461no—
390C22×C8—1461no—
391#126—1461no—
392#125—731no—
393#124—731no—
394#123—1461no—
395C22⋊Q8—731no—
396#121—731no—
397C4×Q8C22×D7311yescharacteristic
398C4×D4—731no—
399C4×D4—731no—
400C4×D4—731no—
401C4×D4—731no—
402C4×D4—1461no—
403C4×D4—1461no—
404C4×D4—1461no—
405C4×D4C22×D7311yescharacteristic
406#117—1461no—
407C2×C4⋊C4—731no—
408C2×C22⋊C4—1461no—
409C2×C42—731no—
410#107—731no—
411#107—731no—
412#106—731no—
413#106C4×D7311yescharacteristic
414C4⋊C8—21no—
415C4⋊C8—731no—
416C4⋊C8—731no—
417C4⋊C8—731no—
418C4⋊C8—731no—
419C4⋊C8—1461no—
420C4⋊C8—1461no—
421C4⋊C8C22×D7311yescharacteristic
422C4⋊C8C22×D7311yescharacteristic
423#103—731no—
424#103C4×D7311yescharacteristic
425#102—731no—
426#102—731no—
427#102—731no—
428#102—731no—
429#102—731no—
430#102C4×D7311yescharacteristic
431C22⋊C8—21no—
432C22⋊C8—731no—
433C22⋊C8—731no—
434C22⋊C8—1461no—
435C22⋊C8—1461no—
436C22⋊C8—1461no—
437C22⋊C8—1461no—
438C22⋊C8—1461no—
439C22⋊C8—1461no—
440#97—1461no—
441#97—1461no—
442#97—1461no—
443C4×C8—21no—
444C4×C8—731no—
445C4×C8—731no—
446C4×C8—731no—
447C4×C8—731no—
448C4×C8—731no—
449C4×C8C22×D7311yescharacteristic
450C4×C8C22×D7311yescharacteristic
451C4×C8C22×D7311yescharacteristic
452C24—1461no—
453C2×Q8C2×C4×D7311yescharacteristic
454C2×D4—731no—
455C2×D4—731no—
456C2×D4—1461no—
457C2×D4—1461no—
458C2×D4—1462no—
459C2×D4—1462no—
460C2×D4C2×C4×D7311yescharacteristic
461C22×C4—21no—
462C22×C4—2922no—
463C22×C4—2922no—
464C22×C4—731no—
465C22×C4—731no—
466C22×C4—731no—
467C22×C4—1461no—
468C22×C4—1461no—
469C22×C4—1461no—
470C22×C4—1461no—
471C22×C4—1461no—
472SD16—22no—
473SD16—1462no—
474D8—1462no—
475D8—1462no—
476M16—2922no—
477M16—2922no—
478C2×C8—21no—
479C2×C8—21no—
480C2×C8—21no—
481C2×C8—42no—
482C2×C8—2922no—
483C2×C8—2922no—
484C2×C8—2921no—
485C2×C8—2922no—
486C2×C8—2922no—
487C2×C8—2922no—
488C2×C8—731no—
489C2×C8—731no—
490C2×C8—731no—
491C2×C8—731no—
492C2×C8—1462no—
493C2×C8—1461no—
494C2×C8—1461no—
495C2×C8—1461no—
496C2×C8C2×C4×D7311yescharacteristic
497C2×C8C2×C4×D7311yescharacteristic
498C2×C8D4×D7311yescharacteristic
499C2×C8D4×D7311yescharacteristic
500C4⋊C4—21no—
501C4⋊C4—731no—
502C4⋊C4—731no—
503C4⋊C4—1461no—
504C4⋊C4—1461no—
505C4⋊C4—1461no—
506C4⋊C4—1461no—
507C4⋊C4C2×C4×D7311yescharacteristic
508C4⋊C4C2×C4×D7311yescharacteristic
509C22⋊C4—21no—
510C22⋊C4—1461no—
511C22⋊C4—1461no—
512C22⋊C4—1461no—
513C22⋊C4—1461no—
514C22⋊C4—1461no—
515C22⋊C4—1461no—
516C22⋊C4—1461no—
517C22⋊C4—1461no—
518C42—21no—
519C42—731no—
520C42—731no—
521C42—731no—
522C42C23×D7311yescharacteristic
523C23—21no—
524C23—2922no—
525C23—2922no—
526C23—731no—
527C23—1464no—
528C23—1461no—
529C23—1461no—
530Q8—21no—
531Q8—21no—
532D4—21no—
533D4—2922no—
534D4—732no—
535D4—1464no—
536D4—1461no—
537D4—1461no—
538D4—1461no—
539D4#1455659712yes—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—2922no—
546C2×C4—2922no—
547C2×C4—2922no—
548C2×C4—2921no—
549C2×C4—2921no—
550C2×C4—2921no—
551C2×C4—2921no—
552C2×C4—731no—
553C2×C4—731no—
554C2×C4—731no—
555C2×C4—1462no—
556C2×C4—1461no—
557C2×C4—1461no—
558C2×C4—1461no—
559C2×C4—1461no—
560C2×C4—1461no—
561C2×C4—1461no—
562C2×C4C22×C4×D7311yescharacteristic
563C2×C4C2×D4×D7311yescharacteristic
564C2×C4#41×D7311yescharacteristic
565C8—21no—
566C8—21no—
567C8—21no—
568C8—21no—
569C8—41no—
570C8—41no—
571C8—41no—
572C8—2921no—
573C8—2921no—
574C8—2921no—
575C8—732no—
576C8—1461no—
577C8—1461no—
578C8—1461no—
579C22—22no—
580C22—41no—
581C22—41no—
582C22—2928no—
583C22—2921no—
584C22—2921no—
585C22—2921no—
586C22—2921no—
587C22—2921no—
588C22—732no—
589C22—1462no—
590C22—1461no—
591C22—1461no—
592C22C4×D4×D7311yescharacteristic
593C4—21no—
594C4—41no—
595C4—41no—
596C4—41no—
597C4—2921no—
598C4—2921no—
599C4—1461no—
600C4—1461no—
601C4—1461no—
602C4#131×D7311yescharacteristic
603C4C2×#1455659711yescharacteristic
604C4AGL(1,ℤ/8)×D7311yescharacteristic
605C4#1798938611yescharacteristic
606C2—42no—
607C2—2922no—
608C2—5841no—
609C2—1462no—
610C2#2261862011yescharacteristic
611C2#2261873611yescharacteristic
612C2#444×D7311yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D73
commutatorC2×C4×C73C23×C4
FrattiniC42C23×D73
FittingC73×SD16⋊C8C2
socleC22×C73C2×C4×D4
prime radical#17988722C4
cosocleC42×C73C24
hypercenterSD16⋊C8D73
hypocenterC73#17914
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. C8×D4×D73C2semidirect
4. #22608770C2semidirect
5. #22608768C2semidirect
6. #22608759C2semidirect
7. #22608757C2semidirect
8. #22608756C2semidirect
9. #22608753C2semidirect
10. #22608741C2semidirect
11. #22608715C2semidirect
12. #22608372C2semidirect
13. #22608343C2semidirect
14. #22608219C2semidirect
15. C73×SD16⋊C8C2semidirect
27. #17988718C22semidirect
31. #17988664C22semidirect
35. C8×D292C22semidirect
48. #17988339C22semidirect
49. #17988338C22semidirect
50. #17988337C22semidirect
51. Q8⋊#10596831C22semidirect
53. #17988316C22semidirect
56. #17988314C22semidirect
57. #17988309C22semidirect
58. #17988309C22semidirect
59. #17988309C22semidirect
60. C8×Q8×C73C22semidirect
63. #335×C73C22semidirect
64. Q8⋊C8×C73C22semidirect
66. #322×C73C22semidirect
317. SD16⋊C8D73semidirect
322. C73#17914semidirect
335. C8×Q8C2×D73semidirect
359. #335C2×D73semidirect
363. Q8⋊C8C2×D73semidirect
373. #322C2×D73semidirect

Maximal subgroups

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

subgroupindex
#226199822
#226189082
C8×D4×D732
#226087702
#226087682
#226087592
#226087572
#226087562
#226087532
#226087412
#226087152
#226083722
#226083432
#226082192
C73×SD16⋊C82
#1791473