Finite Group Databasebeta

← #29651042

Subgroups — #29651042

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#21455864C211yesmaximal, characteristic
2#21454820C211yesmaximal, characteristic, complemented
3D4×#11758189C211yesmaximal, characteristic, complemented
4#21444730C211yesmaximal, characteristic, complemented
5#21444711C211yesmaximal, characteristic, complemented
6#21444696C211yesmaximal, characteristic, complemented
7#21444684C211yesmaximal, characteristic, complemented
8#21444655C211yesmaximal, characteristic, complemented
9#21444648C211yesmaximal, characteristic, complemented
10#21444628C211yesmaximal, characteristic, complemented
11#21444609C211yesmaximal, characteristic, complemented
12#21444281C211yesmaximal, characteristic, complemented
13#21444208C211yesmaximal, characteristic, complemented
14#21444113C211yesmaximal, characteristic, complemented
15C8×D8×C61C211yesmaximal, characteristic, complemented
16C22×#11758189—21no—
17#16840294C2211yescharacteristic
18C4×D4×D61C2211yescharacteristic
19#16840099C411yescharacteristic
20#16840079C411yescharacteristic
21#16840065C411yescharacteristic
22#16840064C411yescharacteristic
23#16839976C411yescharacteristic
24#16839975C411yescharacteristic
25#16839914C411yescharacteristic
26#16839894C411yescharacteristic
27#16839892C2211yescharacteristic, complemented
28#16839886C2211yescharacteristic, complemented
29C4×#11758190C2211yescharacteristic
30C4×#11758200C2211yescharacteristic
31C4×#11758199C2211yescharacteristic
32#16839841C2211yescharacteristic
33#16839840C2211yescharacteristic, complemented
34#16839839C2211yescharacteristic
35#16839837—21no—
36#16839833—21no—
37#16839833—21no—
38#16839830—21no—
39D4×#92418C2211yescharacteristic
40#16839820—21no—
41#16839814C2211yescharacteristic
42#16839813C2211yescharacteristic
43#16839813C2211yescharacteristic
44#16839784C2211yescharacteristic
45#16839724—21no—
46#16839717—21no—
47C4×#11758189C2211yescharacteristic
48#16839514C2211yescharacteristic, complemented
49#16839513C2211yescharacteristic, complemented
50#16839512C2211yescharacteristic, complemented
51#16839494C2211yescharacteristic, complemented
52#16839492C2211yescharacteristic, complemented
53#16839491C2211yescharacteristic
54D4⋊#92418C2211yescharacteristic
55D4⋊#92418C2211yescharacteristic, complemented
56#16839489C2211yescharacteristic
57#16839484C2211yescharacteristic, complemented
58#16839484C2211yescharacteristic, complemented
59#16839483C2211yescharacteristic, complemented
60C4×D8×C61C2211yescharacteristic
61C8×D4×C61C2211yescharacteristic
62C8×D4×C61C2211yescharacteristic, complemented
63C61×#335C2211yescharacteristic, complemented
64C61×D4⋊C8C2211yescharacteristic
65C61×D4⋊C8C2211yescharacteristic, complemented
66C82×C61C2211yescharacteristic, complemented
67C2×D4×D61C2×C411yescharacteristic
68C22×C4×D61—21no—
69C2×#11758190C2×C411yescharacteristic
70C2×#11758200C2×C411yescharacteristic
71C2×#11758199C2×C411yescharacteristic
72C2×#11758194—21no—
73C2×#11758189—21no—
74C2×#11758189—21no—
75C2×#11758189—21no—
76C2×#11758189—44no—
77C2×#11758189D411yescharacteristic
78C2×#11758189D411yescharacteristic
79C22×#92418—21no—
80#14519770—21no—
81#14519768C2×C411yescharacteristic
82#14519764C2×C411yescharacteristic
83#14519759—21no—
84#14519757C2×C411yescharacteristic
85#14519748C2×C411yescharacteristic
86C4×Dic122C2311yescharacteristic
87C4×D244C2311yescharacteristic
88D4×Dic61C2×C411yescharacteristic
89#14519739C2×C411yescharacteristic
90C4×C22⋊D61—21no—
91C4×C22⋊D61—21no—
92#14519737—21no—
93#14519736—21no—
94C4⋊C4×D61C2×C411yescharacteristic
95C22⋊C4×D61—21no—
96C42×D61C2311yescharacteristic
97#14519710C2×C411yescharacteristic
98#14519709C2×C411yescharacteristic
99#14519708C2×C411yescharacteristic
100C4⋊#92418C2311yescharacteristic
101C4⋊#92418C2311yescharacteristic
102#14519706—21no—
103#14519706—21no—
104#14519706—21no—
105#14519706—21no—
106#14519706C2×C411yescharacteristic
107#14519706C2×C411yescharacteristic
108#14519702C2×C411yescharacteristic
109#14519700C2×C411yescharacteristic
110#14519699C2×C411yescharacteristic
111#14519698C2×C411yescharacteristic
112#14519698C2×C411yescharacteristic
113#14519697C2×C411yescharacteristic
114C22⋊#92418—21no—
115C22⋊#92418—21no—
116#14519689—21no—
117#14519689—21no—
118#14519689—21no—
119#14519689—21no—
120#14519689C2×C411yescharacteristic
121#14519689C2×C411yescharacteristic
122#14519687—21no—
123#14519687D411yescharacteristic
124#14519687D411yescharacteristic
125C4×#92418C2311yescharacteristic
126C4×#92418C2311yescharacteristic
127C4×#92418C2311yescharacteristic
128C8×Dic61—21no—
129C8×Dic61—21no—
130C2×D8×C61C2×C411yescharacteristic
131C22×C8×C61—21no—
132C22×C8×C61—21no—
133C4×D4×C61C2311yescharacteristic
134C4×D4×C61C2311yescharacteristic
135#107×C61C2×C411yescharacteristic
136C4⋊C8×C61C2311yescharacteristic
137C4⋊C8×C61C2311yescharacteristic
138#102×C61C2×C411yescharacteristic
139#102×C61C2×C411yescharacteristic
140C22⋊C8×C61—21no—
141C22⋊C8×C61—21no—
142C4×C8×C61C2311yescharacteristic
143C4×C8×C61C2311yescharacteristic
144C4×C8×C61C2311yescharacteristic
145C23×D61—21no—
146C2×D4×C61C22×C411yescharacteristic
147C2×D4×C61C22×C411yescharacteristic
148C22×C4×C61—21no—
149C22×C4×C61—21no—
150C2×C22⋊D61—21no—
151C2×C22⋊D61—21no—
152C22×Dic61—21no—
153D4×D61—22no—
154D4×D61—41no—
155C2×D244C22×C411yescharacteristic
156C2×C4×D61—21no—
157C2×C4×D61—21no—
158C2×C4×D61—21no—
159C2×C4×D61—42no—
160C2×C4×D61—42no—
161C2×C4×D61C22×C411yescharacteristic
162C2×C4×D61C2×D411yescharacteristic
163C2×C4×D61#4111yescharacteristic
164C2×Dic122C22×C411yescharacteristic
165D8×C61—22no—
166C2×C8×C61—21no—
167C2×C8×C61—21no—
168C2×C8×C61—21no—
169C2×C8×C61—42no—
170C2×C8×C61—42no—
171C2×C8×C61C22×C411yescharacteristic
172C2×C8×C61C22×C411yescharacteristic
173C2×C8×C61C2×D411yescharacteristic
174C2×C8×C61C2×D411yescharacteristic
175C4⋊C4×C61C22×C411yescharacteristic
176C4⋊C4×C61C22×C411yescharacteristic
177C22⋊C4×C61—21no—
178C22⋊C4×C61—21no—
179C42×C61C2411yescharacteristic
180C22⋊Dic61—21no—
181C22⋊Dic61—21no—
182C22⋊Dic61—21no—
183C22⋊Dic61—21no—
184#11758200—22no—
185#11758199—41no—
186#11758198—21no—
187#11758198—21no—
188#11758198—21no—
189C4⋊Dic61—21no—
190C4⋊Dic61—21no—
191C4⋊Dic61C22×C411yescharacteristic
192C4⋊Dic61C22×C411yescharacteristic
193#11758196—21no—
194#11758196—21no—
195#11758196—21no—
196C4×Dic61—21no—
197C4×Dic61C22×C411yescharacteristic
198C4×Dic61C2×D411yescharacteristic
199C4×Dic61#4111yescharacteristic
200#11758194—42no—
201C2×#92418—21no—
202C2×#92418—21no—
203C2×#92418—21no—
204C2×#92418—42no—
205C2×#92418C22×C411yescharacteristic
206C2×#92418C22×C411yescharacteristic
207C2×#92418C2×D411yescharacteristic
208C2×#92418C2×D411yescharacteristic
209#11758190—41no—
210#11758189—22no—
211#11758189—41no—
212#11758189—42no—
213#11758189—42no—
214#11758189—42no—
215C23×C61—21no—
216C23×C61—21no—
217C22×D61—21no—
218C22×D61—42no—
219C22×D61—42no—
220C22×D61—81no—
221C22×D61C4×D411yescharacteristic
222D4×C61—21no—
223D4×C61—21no—
224D4×C61—21no—
225D4×C61#13112yes—
226C2×C4×C61—21no—
227C2×C4×C61—21no—
228C2×C4×C61—41no—
229C2×C4×C61—41no—
230C2×C4×C61—41no—
231C2×C4×C61—41no—
232C2×C4×C61C23×C411yescharacteristic
233C2×C4×C61C22×D411yescharacteristic
234C2×C4×C61C2×#4111yescharacteristic
235C22⋊D61—42no—
236C22⋊D61—81no—
237C2×Dic61—21no—
238C2×Dic61—21no—
239C2×Dic61—21no—
240C2×Dic61—42no—
241C2×Dic61C4×D411yescharacteristic
242C2×Dic61C4×D411yescharacteristic
243C2×Dic61C4×D411yescharacteristic
244D244—21no—
245D244—21no—
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—
255Dic122—21no—
256Dic122—21no—
257C8×C61—21no—
258C8×C61—21no—
259C8×C61—21no—
260C8×C61—41no—
261C8×C61—41no—
262C8×C61—41no—
263C8×C61AGL(1,ℤ/8)12yes—
264#92418—21no—
265#92418—21no—
266#92418—21no—
267#92418—21no—
268#92418—41no—
269#92418—41no—
270#92418—41no—
271#18006—611nomaximal, Sylow
272C22×C61—22no—
273C22×C61—41no—
274C22×C61—41no—
275C22×C61—41no—
276C22×C61—41no—
277C22×C61—41no—
278C22×C61C2×C4×D411yescharacteristic
279C2×D61—21no—
280C2×D61—21no—
281C2×D61—21no—
282C2×D61—41no—
283C2×D61—41no—
284C2×D61—41no—
285C2×D61—84no—
286C4×C61—21no—
287C4×C61—41no—
288C4×C61—41no—
289C4×C61C2×#13111yescharacteristic
290C4×C61#56811yescharacteristic
291C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
292C4×C61#57511yescharacteristic
293Dic61—21no—
294Dic61—21no—
295Dic61—21no—
296Dic61—41no—
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⋊C8—611no—
309C8×D8D6111yescharacteristic, complemented
310#1574—611no—
311#1495—611no—
312#1478—611no—
313#1453—611no—
314C2×C61—42no—
315C2×C61—81no—
316C2×C61#293311yescharacteristic
317C2×C61C4×AGL(1,ℤ/8)11yescharacteristic
318C2×C61#295711yescharacteristic
319D61—41no—
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×Q8—611no—
334C4×SD16—611no—
335C4×SD16—611no—
336C4×D8—611no—
337C4×D8C2×D6111yescharacteristic
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×D4C2×D6111yescharacteristic
347C8×D4C2×D6111yescharacteristic, complemented
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⋊C8—611no—
362D4⋊C8—611no—
363D4⋊C8—611no—
364D4⋊C8—611no—
365D4⋊C8—611no—
366D4⋊C8C2×D6111yescharacteristic
367D4⋊C8C2×D6111yescharacteristic, complemented
368#322—611no—
369#322—611no—
370#322—611no—
371C82C2×D6111yescharacteristic, complemented
372C61#1800611yescharacteristic, complemented, Sylow
373C22×D4—611no—
374C23×C4—1221no—
375C2×SD16—611no—
376C2×SD16—611no—
377C2×D8—611no—
378C2×D8C4×D6111yescharacteristic
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—21no—
388C22×C8—1221no—
389#126—1221no—
390#125—611no—
391#124—611no—
392#123—1221no—
393C22⋊Q8—611no—
394#121—611no—
395C4×Q8—611no—
396C4×D4—611no—
397C4×D4—611no—
398C4×D4—611no—
399C4×D4—1221no—
400C4×D4—1221no—
401C4×D4—1221no—
402C4×D4C22×D6111yescharacteristic
403C4×D4C22×D6111yescharacteristic
404#117—1221no—
405C2×C4⋊C4—611no—
406C2×C22⋊C4—1221no—
407C2×C42—611no—
408#107—611no—
409#107C4×D6111yescharacteristic
410#106—611no—
411#106—611no—
412C4⋊C8—611no—
413C4⋊C8—611no—
414C4⋊C8—611no—
415C4⋊C8—611no—
416C4⋊C8—1221no—
417C4⋊C8—1221no—
418C4⋊C8—1221no—
419C4⋊C8—1221no—
420C4⋊C8C22×D6111yescharacteristic
421C4⋊C8C22×D6111yescharacteristic
422#103—611no—
423#103—611no—
424#102—611no—
425#102—611no—
426#102—611no—
427#102—611no—
428#102C4×D6111yescharacteristic
429#102C4×D6111yescharacteristic
430C22⋊C8—21no—
431C22⋊C8—21no—
432C22⋊C8—611no—
433C22⋊C8—611no—
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—611no—
444C4×C8—611no—
445C4×C8—611no—
446C4×C8—1221no—
447C4×C8—1221no—
448C4×C8C22×D6111yescharacteristic
449C4×C8C22×D6111yescharacteristic
450C4×C8C22×D6111yescharacteristic
451C24—1221no—
452C2×Q8—611no—
453C2×D4—611no—
454C2×D4—1221no—
455C2×D4—1222no—
456C2×D4—1221no—
457C2×D4—2441no—
458C2×D4C2×C4×D6111yescharacteristic
459C2×D4C2×C4×D6111yescharacteristic
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—611no—
463C22×C4—611no—
464C22×C4—611no—
465C22×C4—1221no—
466C22×C4—1221no—
467C22×C4—1221no—
468C22×C4—1221no—
469C22×C4—2442no—
470C22×C4—2442no—
471SD16—1222no—
472SD16—2441no—
473D8—22no—
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—42no—
486C2×C8—611no—
487C2×C8—611no—
488C2×C8—611no—
489C2×C8—611no—
490C2×C8—1221no—
491C2×C8—1221no—
492C2×C8—1221no—
493C2×C8—2442no—
494C2×C8C2×C4×D6111yescharacteristic
495C2×C8C2×C4×D6111yescharacteristic
496C2×C8D4×D6111yescharacteristic
497C2×C8D4×D6111yescharacteristic
498C4⋊C4—611no—
499C4⋊C4—611no—
500C4⋊C4—1221no—
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—21no—
509C22⋊C4—1221no—
510C22⋊C4—1221no—
511C22⋊C4—1221no—
512C22⋊C4—1221no—
513C22⋊C4—1221no—
514C22⋊C4—1221no—
515C22⋊C4—1221no—
516C42—611no—
517C42—611no—
518C42—611no—
519C42—1221no—
520C42C23×D6111yescharacteristic
521C23—21no—
522C23—21no—
523C23—611no—
524C23—1221no—
525C23—4881no—
526C23—2442no—
527C23—2442no—
528Q8—1221no—
529Q8—1221no—
530D4—21no—
531D4—21no—
532D4—21no—
533D4—1221no—
534D4—1221no—
535D4—4881no—
536D4—2442no—
537D4#1451978312yes—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—611no—
545C2×C4—611no—
546C2×C4—611no—
547C2×C4—1221no—
548C2×C4—1221no—
549C2×C4—1221no—
550C2×C4—1221no—
551C2×C4—1221no—
552C2×C4—1221no—
553C2×C4—1221no—
554C2×C4—2441no—
555C2×C4—2442no—
556C2×C4—2442no—
557C2×C4—2442no—
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—41no—
567C8—41no—
568C8—41no—
569C8—1221no—
570C8—1221no—
571C8—1221no—
572C8—1221no—
573C8—2441no—
574C8—2441no—
575C8—2441no—
576C8#1451980112yes—
577C22—22no—
578C22—41no—
579C22—41no—
580C22—41no—
581C22—41no—
582C22—41no—
583C22—1221no—
584C22—1221no—
585C22—1221no—
586C22—4884no—
587C22—2441no—
588C22—2441no—
589C22—2441no—
590C22C4×D4×D6111yescharacteristic
591C4—21no—
592C4—41no—
593C4—41no—
594C4—1221no—
595C4—1221no—
596C4—1221no—
597C4—2441no—
598C4—2441no—
599C4—2441no—
600C4C2×#1451978311yescharacteristic
601C4#1684052711yescharacteristic
602C4C2×#1451980111yescharacteristic
603C4#1684055911yescharacteristic
604C2—42no—
605C2—81no—
606C2—4881no—
607C2—2441no—
608C2#2145453911yescharacteristic
609C2C4×#1451980111yescharacteristic
610C2#2145473111yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC42C23×D61
FittingC8×D8×C61C2
socleC22×C61C2×C4×D4
prime radical#16839894C4
cosocleC42×C61C24
hypercenterC8×D8D61
hypocenterC61#18006
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ C8×D8

Complements

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

NH ≅ G/Ntype
2. #21454820C2semidirect
3. D4×#11758189C2semidirect
4. #21444730C2semidirect
5. #21444711C2semidirect
6. #21444696C2semidirect
7. #21444684C2semidirect
8. #21444655C2semidirect
9. #21444648C2semidirect
10. #21444628C2semidirect
11. #21444609C2semidirect
12. #21444281C2semidirect
13. #21444208C2semidirect
14. #21444113C2semidirect
15. C8×D8×C61C2semidirect
27. #16839892C22semidirect
28. #16839886C22semidirect
33. #16839840C22semidirect
48. #16839514C22semidirect
49. #16839513C22semidirect
50. #16839512C22semidirect
51. #16839494C22semidirect
52. #16839492C22semidirect
55. D4⋊#92418C22semidirect
57. #16839484C22semidirect
58. #16839484C22semidirect
59. #16839483C22semidirect
62. C8×D4×C61C22semidirect
63. C61×#335C22semidirect
65. C61×D4⋊C8C22semidirect
66. C82×C61C22semidirect
309. C8×D8D61semidirect
347. C8×D4C2×D61semidirect
357. #335C2×D61semidirect
367. D4⋊C8C2×D61semidirect
371. C82C2×D61semidirect
372. C61#18006semidirect

Maximal subgroups

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

subgroupindex
#214558642
#214548202
D4×#117581892
#214447302
#214447112
#214446962
#214446842
#214446552
#214446482
#214446282
#214446092
#214442812
#214442082
#214441132
C8×D8×C612
#1800661