Finite Group Databasebeta

← #29651044

Subgroups — #29651044

Order 15616.

670 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455870C211yesmaximal, characteristic
2#21454822C211yesmaximal, characteristic, complemented
3D4×#11758189C211yesmaximal, characteristic, complemented
4#21444734C211yesmaximal, characteristic, complemented
5#21444706C211yesmaximal, characteristic, complemented
6#21444697C211yesmaximal, characteristic, complemented
7#21444683C211yesmaximal, characteristic, complemented
8#21444656C211yesmaximal, characteristic, complemented
9#21444647C211yesmaximal, characteristic, complemented
10#21444611C211yesmaximal, characteristic, complemented
11#21444609C211yesmaximal, characteristic, complemented
12#21444283C211yesmaximal, characteristic, complemented
13#21444208C211yesmaximal, characteristic, complemented
14#21444115C211yesmaximal, characteristic, complemented
15C61×SD16⋊C8C211yesmaximal, characteristic, complemented
16C22×#11758189—21no—
17#16840299C2211yescharacteristic
18C4×D4×D61C2211yescharacteristic
19#16840096C411yescharacteristic
20#16840082C411yescharacteristic
21#16840061C411yescharacteristic
22#16840060C411yescharacteristic
23#16839980C411yescharacteristic
24#16839979C411yescharacteristic
25#16839911C411yescharacteristic
26#16839897C411yescharacteristic
27#16839893C2211yescharacteristic, complemented
28C4×#11758201C2211yescharacteristic
29C4×D488C2211yescharacteristic
30C4×#11758199C2211yescharacteristic
31#16839841—21no—
32#16839841C2211yescharacteristic
33#16839841C2211yescharacteristic, complemented
34#16839839C2211yescharacteristic
35#16839833—21no—
36#16839832—21no—
37#16839830—21no—
38D4×#92418C2211yescharacteristic
39#16839827C2211yescharacteristic, complemented
40#16839822—21no—
41#16839814C2211yescharacteristic
42#16839813C2211yescharacteristic
43#16839813C2211yescharacteristic
44#16839784C2211yescharacteristic
45#16839724—21no—
46#16839716—21no—
47C4×#11758189C2211yescharacteristic
48#16839514C2211yescharacteristic, complemented
49#16839513C2211yescharacteristic, complemented
50#16839512C2211yescharacteristic, complemented
51Q8⋊#92418C2211yescharacteristic, complemented
52#16839491C2211yescharacteristic
53#16839491C2211yescharacteristic, complemented
54D4⋊#92418C2211yescharacteristic
55#16839489C2211yescharacteristic
56#16839489C2211yescharacteristic, complemented
57#16839484C2211yescharacteristic, complemented
58#16839484C2211yescharacteristic, complemented
59C8×#92418C2211yescharacteristic, complemented
60C8×Q8×C61C2211yescharacteristic, complemented
61C4×SD16×C61C2211yescharacteristic
62C8×D4×C61C2211yescharacteristic
63C61×#335C2211yescharacteristic, complemented
64C61×Q8⋊C8C2211yescharacteristic, complemented
65C61×D4⋊C8C2211yescharacteristic
66C61×#322C2211yescharacteristic, complemented
67C2×D4×D61C2×C411yescharacteristic
68C22×C4×D61—21no—
69C2×#11758201C2×C411yescharacteristic
70C2×D488C2×C411yescharacteristic
71C2×#11758199C2×C411yescharacteristic
72C2×#11758189—21no—
73C2×#11758189—21no—
74C2×#11758189—21no—
75C2×#11758189—21no—
76C2×#11758189—44no—
77C2×#11758189D411yescharacteristic
78C2×#11758189D411yescharacteristic
79C22×#92418—21no—
80C2×C8×D61—21no—
81#14519772—21no—
82#14519768C2×C411yescharacteristic
83#14519766C2×C411yescharacteristic
84#14519762—21no—
85#14519753C2×C411yescharacteristic
86#14519752C2×C411yescharacteristic
87C4×D244—21no—
88C4×D244C2311yescharacteristic
89C4×D244C2311yescharacteristic
90D4×Dic61C2×C411yescharacteristic
91#14519739C2×C411yescharacteristic
92C4×C22⋊D61—21no—
93#14519737—21no—
94#14519734—21no—
95C4⋊C4×D61C2×C411yescharacteristic
96C22⋊C4×D61—21no—
97C42×D61C2311yescharacteristic
98#14519711C2×C411yescharacteristic
99#14519710C2×C411yescharacteristic
100#14519708C2×C411yescharacteristic
101C4⋊#92418—21no—
102C4⋊#92418C2311yescharacteristic
103C4⋊#92418C2311yescharacteristic
104#14519706—21no—
105#14519706—21no—
106#14519706C2×C411yescharacteristic
107#14519706C2×C411yescharacteristic
108#14519701C2×C411yescharacteristic
109#14519699C2×C411yescharacteristic
110#14519699C2×C411yescharacteristic
111#14519698C2×C411yescharacteristic
112#14519697C2×C411yescharacteristic
113#14519697C2×C411yescharacteristic
114C22⋊#92418—21no—
115#14519689—21no—
116#14519689—21no—
117#14519689—21no—
118#14519689—21no—
119#14519689—21no—
120#14519689—21no—
121#14519689C2×C411yescharacteristic
122#14519689C2×C411yescharacteristic
123#14519688—21no—
124#14519687D411yescharacteristic
125#14519687D411yescharacteristic
126C4×#92418C2311yescharacteristic
127C4×#92418C2311yescharacteristic
128C4×#92418C2311yescharacteristic
129C8×Dic61—21no—
130C2×SD16×C61C2×C411yescharacteristic
131C22×C8×C61—21no—
132C4×Q8×C61C2311yescharacteristic
133C4×D4×C61C2311yescharacteristic
134#106×C61C2×C411yescharacteristic
135C4⋊C8×C61—21no—
136C4⋊C8×C61C2311yescharacteristic
137C4⋊C8×C61C2311yescharacteristic
138#103×C61C2×C411yescharacteristic
139#102×C61C2×C411yescharacteristic
140C22⋊C8×C61—21no—
141C4×C8×C61—21no—
142C4×C8×C61C2311yescharacteristic
143C4×C8×C61C2311yescharacteristic
144C4×C8×C61C2311yescharacteristic
145C23×D61—21no—
146C2×Q8×C61C22×C411yescharacteristic
147C2×D4×C61C22×C411yescharacteristic
148C22×C4×C61—21no—
149C2×C22⋊D61—21no—
150C22×Dic61—21no—
151D4×D61—22no—
152D4×D61—41no—
153C2×D244—21no—
154C2×D244C22×C411yescharacteristic
155C2×D244C22×C411yescharacteristic
156C2×C4×D61—21no—
157C2×C4×D61—21no—
158C2×C4×D61—21no—
159C2×C4×D61—21no—
160C2×C4×D61—42no—
161C2×C4×D61—42no—
162C2×C4×D61C22×C411yescharacteristic
163C2×C4×D61C2×D411yescharacteristic
164C2×C4×D61#4111yescharacteristic
165SD16×C61—22no—
166C2×C8×C61—21no—
167C2×C8×C61—21no—
168C2×C8×C61—21no—
169C2×C8×C61—42no—
170C2×C8×C61C22×C411yescharacteristic
171C2×C8×C61C22×C411yescharacteristic
172C2×C8×C61C2×D411yescharacteristic
173C2×C8×C61C2×D411yescharacteristic
174C4⋊C4×C61—21no—
175C4⋊C4×C61C22×C411yescharacteristic
176C4⋊C4×C61C22×C411yescharacteristic
177C22⋊C4×C61—21no—
178C42×C61—21no—
179C42×C61C2411yescharacteristic
180C22⋊Dic61—21no—
181#11758201—41no—
182#11758199—22no—
183#11758198—21no—
184#11758198—21no—
185#11758198—21no—
186#11758198—21no—
187#11758198—21no—
188#11758198—21no—
189#11758198—21no—
190C4⋊Dic61—21no—
191C4⋊Dic61C22×C411yescharacteristic
192C4⋊Dic61C22×C411yescharacteristic
193#11758196—21no—
194#11758196—21no—
195#11758196—21no—
196C4×Dic61C22×C411yescharacteristic
197C4×Dic61C2×D411yescharacteristic
198C4×Dic61#4111yescharacteristic
199C2×#92418—21no—
200C2×#92418—21no—
201C2×#92418—21no—
202C2×#92418—42no—
203C2×#92418C22×C411yescharacteristic
204C2×#92418C22×C411yescharacteristic
205C2×#92418C2×D411yescharacteristic
206C2×#92418C2×D411yescharacteristic
207D488—41no—
208#11758189—22no—
209#11758189—41no—
210#11758189—42no—
211#11758189—42no—
212#11758189—42no—
213#11758189—42no—
214C8×D61—42no—
215C23×C61—21no—
216C22×D61—21no—
217C22×D61—21no—
218C22×D61—42no—
219C22×D61—42no—
220C22×D61—81no—
221C22×D61C4×D411yescharacteristic
222Q8×C61—21no—
223Q8×C61—21no—
224D4×C61—21no—
225D4×C61#13112yes—
226C2×C4×C61—21no—
227C2×C4×C61—21no—
228C2×C4×C61—21no—
229C2×C4×C61—41no—
230C2×C4×C61—41no—
231C2×C4×C61C23×C411yescharacteristic
232C2×C4×C61C22×D411yescharacteristic
233C2×C4×C61C2×#4111yescharacteristic
234C22⋊D61—42no—
235C2×Dic61—21no—
236C2×Dic61—21no—
237C2×Dic61—42no—
238C2×Dic61C4×D411yescharacteristic
239C2×Dic61C4×D411yescharacteristic
240C2×Dic61C4×D411yescharacteristic
241D244—21no—
242D244—21no—
243D244—21no—
244D244—21no—
245D244—81no—
246C4×D61—21no—
247C4×D61—21no—
248C4×D61—21no—
249C4×D61—21no—
250C4×D61—41no—
251C4×D61—42no—
252C4×D61—42no—
253C4×D61—41no—
254C4×D61—41no—
255C4×D61—41no—
256C4×D61—41no—
257C8×C61—21no—
258C8×C61—21no—
259C8×C61—21no—
260C8×C61—21no—
261C8×C61—41no—
262C8×C61—41no—
263C8×C61—41no—
264#92418—21no—
265#92418—21no—
266#92418—21no—
267#92418—41no—
268#92418—41no—
269#92418—41no—
270#92418AGL(1,ℤ/8)12yes—
271#18006—611nomaximal, Sylow
272C22×C61—22no—
273C22×C61—41no—
274C22×C61—41no—
275C22×C61C2×C4×D411yescharacteristic
276C2×D61—21no—
277C2×D61—21no—
278C2×D61—21no—
279C2×D61—41no—
280C2×D61—41no—
281C2×D61—41no—
282C2×D61—41no—
283C2×D61—41no—
284C2×D61—41no—
285C2×D61—84no—
286C4×C61—21no—
287C4×C61—41no—
288C4×C61—41no—
289C4×C61—41no—
290C4×C61C2×#13111yescharacteristic
291C4×C61#56811yescharacteristic
292C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
293C4×C61#57511yescharacteristic
294Dic61—21no—
295Dic61—21no—
296Dic61—21no—
297Dic61—41no—
298Dic61—41no—
299#3105—611no—
300#2969—611no—
301D4×M16—611no—
302#1594—611no—
303#1590—611no—
304#1588—611no—
305#1586—611no—
306#1582—611no—
307#1581—611no—
308SD16⋊C8D6111yescharacteristic, complemented
309C8×D8—611no—
310#1574—611no—
311#1495—611no—
312#1478—611no—
313#1453—611no—
314C2×C61—42no—
315C2×C61#293311yescharacteristic
316C2×C61C4×AGL(1,ℤ/8)11yescharacteristic
317C2×C61#295711yescharacteristic
318D61—41no—
319D61—81no—
320D61—81no—
321C22×M16—1221no—
322#529—611no—
323C2×C4×D4—611no—
324#484—611no—
325#481—611no—
326#478—611no—
327#478—611no—
328#463—611no—
329#463—611no—
330#450—611no—
331#447—611no—
332#446—611no—
333C8×Q8C2×D6111yescharacteristic, complemented
334C4×SD16—611no—
335C4×SD16C2×D6111yescharacteristic
336C4×D8—611no—
337C4×D8—611no—
338#436—611no—
339#436—611no—
340#436—611no—
341#436—1221no—
342#435—1221no—
343#435—1221no—
344#435—1221no—
345C8×D4—611no—
346C8×D4—611no—
347C8×D4C2×D6111yescharacteristic
348#433—1221no—
349#431—611no—
350#431—611no—
351#431—611no—
352#423—611no—
353#407—1221no—
354#405—1221no—
355C4×M16—611no—
356#335—611no—
357#335C2×D6111yescharacteristic, complemented
358#334—611no—
359#334—611no—
360Q8⋊C8—611no—
361Q8⋊C8C2×D6111yescharacteristic, complemented
362D4⋊C8—611no—
363D4⋊C8—611no—
364D4⋊C8—611no—
365D4⋊C8—611no—
366D4⋊C8—611no—
367D4⋊C8C2×D6111yescharacteristic
368#322—611no—
369#322—611no—
370#322C2×D6111yescharacteristic, complemented
371C82—611no—
372C61#1800611yescharacteristic, complemented, Sylow
373C22×D4—611no—
374C23×C4—1221no—
375C2×SD16—611no—
376C2×SD16C4×D6111yescharacteristic
377C2×D8—611no—
378C2×D8—611no—
379C2×M16—611no—
380C2×M16—611no—
381C2×M16—1221no—
382C2×M16—1221no—
383C2×M16—1221no—
384C2×M16—1221no—
385C2×M16—2444no—
386C22×C8—21no—
387C22×C8—1221no—
388C22×C8—1221no—
389#126—1221no—
390#125—611no—
391#124—611no—
392#123—1221no—
393C22⋊Q8—611no—
394#121—611no—
395C4×Q8C22×D6111yescharacteristic
396C4×D4—611no—
397C4×D4—611no—
398C4×D4—611no—
399C4×D4—611no—
400C4×D4—1221no—
401C4×D4—1221no—
402C4×D4—1221no—
403C4×D4C22×D6111yescharacteristic
404#117—1221no—
405C2×C4⋊C4—611no—
406C2×C22⋊C4—1221no—
407C2×C42—611no—
408#107—611no—
409#107—611no—
410#106—611no—
411#106C4×D6111yescharacteristic
412C4⋊C8—21no—
413C4⋊C8—611no—
414C4⋊C8—611no—
415C4⋊C8—611no—
416C4⋊C8—611no—
417C4⋊C8—1221no—
418C4⋊C8—1221no—
419C4⋊C8—1221no—
420C4⋊C8C22×D6111yescharacteristic
421C4⋊C8C22×D6111yescharacteristic
422#103—611no—
423#103C4×D6111yescharacteristic
424#102—611no—
425#102—611no—
426#102—611no—
427#102—611no—
428#102—611no—
429#102C4×D6111yescharacteristic
430C22⋊C8—21no—
431C22⋊C8—611no—
432C22⋊C8—611no—
433C22⋊C8—1221no—
434C22⋊C8—1221no—
435C22⋊C8—1221no—
436C22⋊C8—1221no—
437C22⋊C8—1221no—
438C22⋊C8—1221no—
439C22⋊C8—1221no—
440#97—611no—
441#97—611no—
442#97—1221no—
443C4×C8—21no—
444C4×C8—611no—
445C4×C8—611no—
446C4×C8—611no—
447C4×C8—1221no—
448C4×C8C22×D6111yescharacteristic
449C4×C8C22×D6111yescharacteristic
450C4×C8C22×D6111yescharacteristic
451C24—1221no—
452C2×Q8C2×C4×D6111yescharacteristic
453C2×D4—611no—
454C2×D4—611no—
455C2×D4—1221no—
456C2×D4—1221no—
457C2×D4—1222no—
458C2×D4—2441no—
459C2×D4C2×C4×D6111yescharacteristic
460C22×C4—21no—
461C22×C4—611no—
462C22×C4—611no—
463C22×C4—611no—
464C22×C4—1221no—
465C22×C4—1221no—
466C22×C4—1221no—
467C22×C4—1221no—
468C22×C4—1221no—
469C22×C4—2442no—
470C22×C4—2442no—
471SD16—22no—
472SD16—2441no—
473D8—1222no—
474D8—2441no—
475M16—1222no—
476M16—2441no—
477M16—2442no—
478M16—2442no—
479M16—2442no—
480M16—2442no—
481C2×C8—21no—
482C2×C8—21no—
483C2×C8—21no—
484C2×C8—42no—
485C2×C8—611no—
486C2×C8—611no—
487C2×C8—611no—
488C2×C8—611no—
489C2×C8—1221no—
490C2×C8—1221no—
491C2×C8—1221no—
492C2×C8—2442no—
493C2×C8—2442no—
494C2×C8C2×C4×D6111yescharacteristic
495C2×C8C2×C4×D6111yescharacteristic
496C2×C8D4×D6111yescharacteristic
497C2×C8D4×D6111yescharacteristic
498C4⋊C4—21no—
499C4⋊C4—611no—
500C4⋊C4—611no—
501C4⋊C4—1221no—
502C4⋊C4—1221no—
503C4⋊C4—1221no—
504C4⋊C4—1221no—
505C4⋊C4C2×C4×D6111yescharacteristic
506C4⋊C4C2×C4×D6111yescharacteristic
507C22⋊C4—21no—
508C22⋊C4—1221no—
509C22⋊C4—1221no—
510C22⋊C4—1221no—
511C22⋊C4—1221no—
512C22⋊C4—1221no—
513C22⋊C4—1221no—
514C22⋊C4—1221no—
515C22⋊C4—1221no—
516C42—21no—
517C42—611no—
518C42—611no—
519C42—611no—
520C42C23×D6111yescharacteristic
521C23—21no—
522C23—611no—
523C23—1221no—
524C23—1221no—
525C23—4881no—
526C23—2442no—
527C23—2442no—
528Q8—21no—
529Q8—21no—
530D4—21no—
531D4—1221no—
532D4—1221no—
533D4—1221no—
534D4—1221no—
535D4—4881no—
536D4—2442no—
537D4#1451978312yes—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—611no—
544C2×C4—611no—
545C2×C4—611no—
546C2×C4—1221no—
547C2×C4—1221no—
548C2×C4—1221no—
549C2×C4—1221no—
550C2×C4—1221no—
551C2×C4—1221no—
552C2×C4—2441no—
553C2×C4—2442no—
554C2×C4—2442no—
555C2×C4—2442no—
556C2×C4—2441no—
557C2×C4—2441no—
558C2×C4—2441no—
559C2×C4—2441no—
560C2×C4C22×C4×D6111yescharacteristic
561C2×C4C2×D4×D6111yescharacteristic
562C2×C4#41×D6111yescharacteristic
563C8—21no—
564C8—21no—
565C8—21no—
566C8—21no—
567C8—41no—
568C8—41no—
569C8—41no—
570C8—612no—
571C8—1221no—
572C8—1221no—
573C8—1221no—
574C8—2441no—
575C8—2441no—
576C8—2441no—
577C22—22no—
578C22—41no—
579C22—41no—
580C22—1221no—
581C22—1221no—
582C22—1221no—
583C22—4884no—
584C22—2441no—
585C22—2441no—
586C22—2441no—
587C22—2441no—
588C22—2441no—
589C22—2441no—
590C22C4×D4×D6111yescharacteristic
591C4—21no—
592C4—41no—
593C4—41no—
594C4—41no—
595C4—1221no—
596C4—1221no—
597C4—1221no—
598C4—2441no—
599C4—2441no—
600C4C2×#1451978311yescharacteristic
601C4#1684052711yescharacteristic
602C4AGL(1,ℤ/8)×D6111yescharacteristic
603C4#1684056111yescharacteristic
604C2—42no—
605C2—4881no—
606C2—4881no—
607C2—2441no—
608C2#2145453911yescharacteristic
609C2#2145462411yescharacteristic
610C2#2145473311yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC42C23×D61
FittingC61×SD16⋊C8C2
socleC22×C61C2×C4×D4
prime radical#16839897C4
cosocleC42×C61C24
hypercenterSD16⋊C8D61
hypocenterC61#18006
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ SD16⋊C8

Complements

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

NH ≅ G/Ntype
2. #21454822C2semidirect
3. D4×#11758189C2semidirect
4. #21444734C2semidirect
5. #21444706C2semidirect
6. #21444697C2semidirect
7. #21444683C2semidirect
8. #21444656C2semidirect
9. #21444647C2semidirect
10. #21444611C2semidirect
11. #21444609C2semidirect
12. #21444283C2semidirect
13. #21444208C2semidirect
14. #21444115C2semidirect
15. C61×SD16⋊C8C2semidirect
27. #16839893C22semidirect
33. #16839841C22semidirect
39. #16839827C22semidirect
48. #16839514C22semidirect
49. #16839513C22semidirect
50. #16839512C22semidirect
51. Q8⋊#92418C22semidirect
53. #16839491C22semidirect
56. #16839489C22semidirect
57. #16839484C22semidirect
58. #16839484C22semidirect
59. C8×#92418C22semidirect
60. C8×Q8×C61C22semidirect
63. C61×#335C22semidirect
64. C61×Q8⋊C8C22semidirect
66. C61×#322C22semidirect
308. SD16⋊C8D61semidirect
333. C8×Q8C2×D61semidirect
357. #335C2×D61semidirect
361. Q8⋊C8C2×D61semidirect
370. #322C2×D61semidirect
372. C61#18006semidirect

Maximal subgroups

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

subgroupindex
#214558702
#214548222
D4×#117581892
#214447342
#214447062
#214446972
#214446832
#214446562
#214446472
#214446112
#214446092
#214442832
#214442082
#214441152
C61×SD16⋊C82
#1800661