Finite Group Databasebeta

← #29649697

Subgroups — #29649697

Order 15616.

678 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21454901C211yesmaximal, characteristic, complemented
2#21454840C211yesmaximal, characteristic
3C4×SD16×D61C211yesmaximal, characteristic
4#21444679C211yesmaximal, characteristic, complemented
5#21444676C211yesmaximal, characteristic, complemented
6#21444674C211yesmaximal, characteristic, complemented
7#21444673C211yesmaximal, characteristic, complemented
8#21444657C211yesmaximal, characteristic, complemented
9#21444656C211yesmaximal, characteristic, complemented
10#21444654C211yesmaximal, characteristic, complemented
11#21444564C211yesmaximal, characteristic, complemented
12#21444212C211yesmaximal, characteristic, complemented
13#21444208C211yesmaximal, characteristic
14#21444110C211yesmaximal, characteristic, complemented
15C61×#1582C211yesmaximal, characteristic, complemented
16C2×SD16×D61C411yescharacteristic
17C4×Q8×D61C2211yescharacteristic
18C4×D4×D61C2211yescharacteristic
19#16839893C2211yescharacteristic, complemented
20#16839892C2211yescharacteristic, complemented
21#16839891—21no—
22#16839890—21no—
23#16839889C2211yescharacteristic, complemented
24#16839853C411yescharacteristic
25C4×#11758190C2211yescharacteristic
26SD16×Dic61C411yescharacteristic
27C4×#11758201C2211yescharacteristic
28#16839849C411yescharacteristic
29C4×#11758200C2211yescharacteristic
30#16839847C411yescharacteristic
31C8×D244C2211yescharacteristic
32D4×#92418C2211yescharacteristic
33#16839827C2211yescharacteristic
34C8×C22⋊D61—21no—
35#16839825—21no—
36#16839818—21no—
37#16839817—21no—
38#16839814C2211yescharacteristic
39#16839814C2211yescharacteristic
40#16839813C2211yescharacteristic
41#16839813C2211yescharacteristic
42#106×D61C411yescharacteristic
43#103×D61C411yescharacteristic
44#102×D61C411yescharacteristic
45C2×#14519689—21no—
46C22⋊C8×D61—21no—
47C4×C8×D61C2211yescharacteristic
48#16839514C2211yescharacteristic, complemented
49#16839514C2211yescharacteristic, complemented
50#16839513C2211yescharacteristic, complemented
51#16839494C2211yescharacteristic, complemented
52Q8⋊#92418C2211yescharacteristic, complemented
53#16839492C2211yescharacteristic, complemented
54#16839491C2211yescharacteristic
55D4⋊#92418C2211yescharacteristic
56#16839489C2211yescharacteristic
57#16839484C2211yescharacteristic, complemented
58#16839484C2211yescharacteristic, complemented
59#16839483C2211yescharacteristic, complemented
60C61×#446C2211yescharacteristic, complemented
61C4×SD16×C61C2211yescharacteristic
62C8×D4×C61C2211yescharacteristic
63C61×#334C2211yescharacteristic, complemented
64C61×Q8⋊C8C2211yescharacteristic, complemented
65C61×D4⋊C8C2211yescharacteristic
66C61×#322C2211yescharacteristic, complemented
67C2×Q8×D61C2×C411yescharacteristic
68C2×D4×D61C2×C411yescharacteristic
69C22×C4×D61—21no—
70C2×#11758190C2×C411yescharacteristic
71C2×#11758200C2×C411yescharacteristic
72C2×#11758201C2×C411yescharacteristic
73SD16×D61—24no—
74C22×#92418—21no—
75C2×C8×D61—21no—
76C2×C8×D61—21no—
77C2×C8×D61C2×C411yescharacteristic
78C2×C8×D61C2×C411yescharacteristic
79Q8×Dic61C2×C411yescharacteristic
80C4×Dic122—21no—
81C4×Dic122C2311yescharacteristic
82#14519742—21no—
83#14519742C2×C411yescharacteristic
84C4×D244C2311yescharacteristic
85D4×Dic61C2×C411yescharacteristic
86#14519739C2×C411yescharacteristic
87C4×C22⋊D61—21no—
88#14519737—21no—
89C4⋊C4×D61—21no—
90C4⋊C4×D61C2×C411yescharacteristic
91C4⋊C4×D61C2×C411yescharacteristic
92C22⋊C4×D61—21no—
93C42×D61—21no—
94C42×D61C2311yescharacteristic
95#14519709C2×C411yescharacteristic
96#14519708C2×C411yescharacteristic
97#14519708C2×C411yescharacteristic
98C4⋊#92418—21no—
99C4⋊#92418C2311yescharacteristic
100C4⋊#92418C2311yescharacteristic
101#14519706—21no—
102#14519706—21no—
103#14519706—21no—
104#14519706D411yescharacteristic
105#14519706D411yescharacteristic
106#14519702C2×C411yescharacteristic
107#14519701C2×C411yescharacteristic
108#14519700C2×C411yescharacteristic
109#14519699C2×C411yescharacteristic
110#14519698C2×C411yescharacteristic
111#14519697C2×C411yescharacteristic
112C22⋊#92418—21no—
113#14519689—21no—
114#14519689—21no—
115#14519689—21no—
116#14519689—21no—
117#14519689—21no—
118#14519689D411yescharacteristic
119#14519689D411yescharacteristic
120#14519688—21no—
121#14519687—21no—
122#14519687—21no—
123C4×#92418C2311yescharacteristic
124C4×#92418C2311yescharacteristic
125C4×#92418C2311yescharacteristic
126C8×Dic61—21no—
127C8×Dic61—21no—
128C8×Dic61C2×C411yescharacteristic
129C8×Dic61C2×C411yescharacteristic
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—
141#97×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—
151Q8×D61—22no—
152Q8×D61—22no—
153D4×D61—22no—
154D4×D61M1614yes—
155C2×D244C22×C411yescharacteristic
156C2×C4×D61—21no—
157C2×C4×D61—21no—
158C2×C4×D61—21no—
159C2×C4×D61—21no—
160C2×C4×D61—21no—
161C2×C4×D61—81no—
162C2×C4×D61—81no—
163C2×C4×D61C22×C411yescharacteristic
164C2×C4×D61C2×D411yescharacteristic
165C2×C4×D61#4111yescharacteristic
166C2×Dic122—21no—
167C2×Dic122C22×C411yescharacteristic
168SD16×C61—22no—
169C2×C8×C61—21no—
170C2×C8×C61—21no—
171C2×C8×C61—21no—
172C2×C8×C61—81no—
173C2×C8×C61C22×C411yescharacteristic
174C2×C8×C61C22×C411yescharacteristic
175C2×C8×C61C2×D411yescharacteristic
176C2×C8×C61C2×D411yescharacteristic
177C4⋊C4×C61—21no—
178C4⋊C4×C61C22×C411yescharacteristic
179C4⋊C4×C61C22×C411yescharacteristic
180C22⋊C4×C61—21no—
181C42×C61—21no—
182C42×C61C2411yescharacteristic
183C22⋊Dic61—21no—
184#11758201—22no—
185#11758200—22no—
186#11758198—21no—
187#11758198—21no—
188C4⋊Dic61—21no—
189C4⋊Dic61C22×C411yescharacteristic
190C4⋊Dic61C22×C411yescharacteristic
191#11758196—21no—
192#11758196—21no—
193#11758196—21no—
194#11758196—21no—
195C4×Dic61—21no—
196C4×Dic61—21no—
197C4×Dic61—21no—
198C4×Dic61C22×C411yescharacteristic
199C4×Dic61C2×D411yescharacteristic
200C4×Dic61#4111yescharacteristic
201C2×#92418—21no—
202C2×#92418—21no—
203C2×#92418—21no—
204C2×#92418—81no—
205C2×#92418C22×C411yescharacteristic
206C2×#92418C22×C411yescharacteristic
207C2×#92418C2×D411yescharacteristic
208C2×#92418C2×D411yescharacteristic
209#11758190—22no—
210C8×D61—22no—
211C8×D61—22no—
212C8×D61—81no—
213C8×D61—81no—
214C23×C61—21no—
215C22×D61—21no—
216C22×D61—24no—
217C22×D61—81no—
218C22×D61—81no—
219C22×D61C4×D411yescharacteristic
220Q8×C61—21no—
221Q8×C61—21no—
222D4×C61—21no—
223D4×C61C2×M1612yes—
224C2×C4×C61—21no—
225C2×C4×C61—21no—
226C2×C4×C61—21no—
227C2×C4×C61—41no—
228C2×C4×C61—41no—
229C2×C4×C61C23×C411yescharacteristic
230C2×C4×C61C22×D411yescharacteristic
231C2×C4×C61C2×#4111yescharacteristic
232C22⋊D61—24no—
233C2×Dic61—21no—
234C2×Dic61—21no—
235C2×Dic61—21no—
236C2×Dic61—81no—
237C2×Dic61C4×D411yescharacteristic
238C2×Dic61C4×D411yescharacteristic
239C2×Dic61C4×D411yescharacteristic
240D244—21no—
241D244C2×M1612yes—
242C4×D61—21no—
243C4×D61—21no—
244C4×D61—41no—
245C4×D61—42no—
246C4×D61—41no—
247C4×D61—41no—
248C4×D61—81no—
249C4×D61—81no—
250C4×D61—81no—
251C4×D61C2×M1612yes—
252C4×D61#13112yes—
253Dic122—21no—
254Dic122—21no—
255Dic122—42no—
256C8×C61—21no—
257C8×C61—21no—
258C8×C61—21no—
259C8×C61—21no—
260C8×C61—41no—
261C8×C61—41no—
262C8×C61—41no—
263#92418—21no—
264#92418—21no—
265#92418—21no—
266#92418—21no—
267#92418—41no—
268#92418—41no—
269#92418—41no—
270#17879—611nomaximal, Sylow
271C22×C61—22no—
272C22×C61—41no—
273C22×C61—41no—
274C22×C61C2×C4×D411yescharacteristic
275C2×D61—21no—
276C2×D61—21no—
277C2×D61—22no—
278C2×D61—41no—
279C2×D61—41no—
280C2×D61—84no—
281C2×D61#43512yes—
282C4×C61—21no—
283C4×C61—41no—
284C4×C61—41no—
285C4×C61—41no—
286C4×C61C22×M1611yescharacteristic
287C4×C61C2×#13111yescharacteristic
288C4×C61#57711yescharacteristic
289C4×C61#57711yescharacteristic
290Dic61—21no—
291Dic61—21no—
292Dic61—41no—
293Dic61—41no—
294Dic61—41no—
295Dic61#43512yes—
296#2983—611no—
297#2972—611no—
298C2×C4×SD16—611no—
299#1585—611no—
300#1585—611no—
301#1585—611no—
302#1585—611no—
303#1582—611no—
304#1582—611no—
305#1582—611no—
306#1582D6111yescharacteristic, complemented
307#1564—611no—
308#1479—611no—
309#1478—611no—
310#1452—611no—
311C2×C61—42no—
312C2×C61C2×#43511yescharacteristic
313C2×C61#294911yescharacteristic
314C2×C61#295711yescharacteristic
315D61—22no—
316D61—42no—
317C22×SD16—611no—
318C2×C4×Q8—611no—
319C2×C4×D4—611no—
320#446—611no—
321#446—611no—
322#446—611no—
323#446—1221no—
324#446—1221no—
325#446C2×D6111yescharacteristic, complemented
326C4×SD16—611no—
327C4×SD16—611no—
328C4×SD16—611no—
329C4×SD16—611no—
330C4×SD16—611no—
331C4×SD16—611no—
332C4×SD16—611no—
333C4×SD16C2×D6111yescharacteristic
334C8×D4—611no—
335C8×D4—611no—
336C8×D4—611no—
337C8×D4—1221no—
338C8×D4—1221no—
339C8×D4C2×D6111yescharacteristic
340#432—1221no—
341#432—1221no—
342#431—611no—
343#431—611no—
344#431—611no—
345#431—611no—
346C2×#106—611no—
347C2×#103—611no—
348C2×#102—611no—
349C2×C22⋊C8—1221no—
350C2×C22⋊C8—1221no—
351C2×C4×C8—611no—
352#335—611no—
353#335—611no—
354#334—611no—
355#334C2×D6111yescharacteristic, complemented
356Q8⋊C8—611no—
357Q8⋊C8—611no—
358Q8⋊C8—611no—
359Q8⋊C8C2×D6111yescharacteristic, complemented
360D4⋊C8—611no—
361D4⋊C8—611no—
362D4⋊C8—611no—
363D4⋊C8C2×D6111yescharacteristic
364#322—611no—
365#322—611no—
366#322—611no—
367#322C2×D6111yescharacteristic, complemented
368C61#1787911yescharacteristic, complemented, Sylow
369C22×Q8—611no—
370C22×D4—611no—
371C23×C4—1221no—
372C2×SD16—611no—
373C2×SD16—611no—
374C2×SD16—611no—
375C2×SD16—1224no—
376C2×SD16C4×D6111yescharacteristic
377C22×C8—21no—
378C22×C8—611no—
379C22×C8—611no—
380C22×C8—1221no—
381C22×C8—1221no—
382C22×C8—1221no—
383C4×Q8—611no—
384C4×Q8—611no—
385C4×Q8—611no—
386C4×Q8—1221no—
387C4×Q8—1221no—
388C4×Q8C22×D6111yescharacteristic
389C4×D4—611no—
390C4×D4—611no—
391C4×D4—611no—
392C4×D4—1221no—
393C4×D4—1221no—
394C4×D4C22×D6111yescharacteristic
395C2×C4⋊C4—611no—
396C2×C4⋊C4—611no—
397C2×C4⋊C4—1221no—
398C2×C22⋊C4—1221no—
399C2×C42—611no—
400C2×C42—1221no—
401#106—611no—
402#106—611no—
403#106—611no—
404#106C4×D6111yescharacteristic
405C4⋊C8—21no—
406C4⋊C8—611no—
407C4⋊C8—611no—
408C4⋊C8—611no—
409C4⋊C8—611no—
410C4⋊C8—1221no—
411C4⋊C8—1221no—
412C4⋊C8—1221no—
413C4⋊C8—1221no—
414C4⋊C8C22×D6111yescharacteristic
415C4⋊C8C22×D6111yescharacteristic
416#103—611no—
417#103—611no—
418#103—611no—
419#103C4×D6111yescharacteristic
420#102—611no—
421#102—611no—
422#102—611no—
423#102C4×D6111yescharacteristic
424C22⋊C8—21no—
425C22⋊C8—611no—
426C22⋊C8—611no—
427C22⋊C8—1221no—
428C22⋊C8—1221no—
429C22⋊C8—1221no—
430C22⋊C8—1221no—
431C22⋊C8—1221no—
432C22⋊C8—1221no—
433#97—21no—
434#97—1221no—
435#97—1221no—
436#97—1221no—
437C4×C8—611no—
438C4×C8—611no—
439C4×C8—611no—
440C4×C8—611no—
441C4×C8—611no—
442C4×C8—1221no—
443C4×C8—1221no—
444C4×C8C22×D6111yescharacteristic
445C4×C8C22×D6111yescharacteristic
446C4×C8C22×D6111yescharacteristic
447C24—1221no—
448C2×Q8—611no—
449C2×Q8—1221no—
450C2×Q8—1222no—
451C2×Q8—1222no—
452C2×Q8C2×C4×D6111yescharacteristic
453C2×D4—614no—
454C2×D4—611no—
455C2×D4—1222no—
456C2×D4—1221no—
457C2×D4C2×C4×D6111yescharacteristic
458C22×C4—21no—
459C22×C4—611no—
460C22×C4—611no—
461C22×C4—611no—
462C22×C4—1221no—
463C22×C4—1221no—
464C22×C4—1221no—
465C22×C4—1221no—
466C22×C4—1221no—
467C22×C4—1221no—
468C22×C4—4881no—
469C22×C4—4881no—
470SD16—22no—
471SD16—1222no—
472SD16—1222no—
473SD16—1222no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—81no—
478C2×C8—611no—
479C2×C8—611no—
480C2×C8—611no—
481C2×C8—611no—
482C2×C8—1222no—
483C2×C8—1222no—
484C2×C8—1221no—
485C2×C8—1221no—
486C2×C8—1221no—
487C2×C8—4881no—
488C2×C8—4881no—
489C2×C8—4881no—
490C2×C8C2×C4×D6111yescharacteristic
491C2×C8C2×C4×D6111yescharacteristic
492C2×C8D4×D6111yescharacteristic
493C2×C8D4×D6111yescharacteristic
494C4⋊C4—21no—
495C4⋊C4—611no—
496C4⋊C4—611no—
497C4⋊C4—1221no—
498C4⋊C4—1221no—
499C4⋊C4—1221no—
500C4⋊C4—1221no—
501C4⋊C4—1221no—
502C4⋊C4C2×C4×D6111yescharacteristic
503C4⋊C4C2×C4×D6111yescharacteristic
504C22⋊C4—21no—
505C22⋊C4—1221no—
506C22⋊C4—1221no—
507C22⋊C4—1221no—
508C42—21no—
509C42—611no—
510C42—611no—
511C42—611no—
512C42—1221no—
513C42—1221no—
514C42—1221no—
515C42C23×D6111yescharacteristic
516C23—21no—
517C23—611no—
518C23—1224no—
519C23—1221no—
520C23—4881no—
521C23—4881no—
522Q8—21no—
523Q8—21no—
524Q8—1221no—
525Q8—1221no—
526Q8—2442no—
527D4—21no—
528D4—612no—
529D4—1224no—
530D4—1221no—
531D4M16×D6112yes—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—611no—
538C2×C4—612no—
539C2×C4—612no—
540C2×C4—611no—
541C2×C4—611no—
542C2×C4—1221no—
543C2×C4—1221no—
544C2×C4—1221no—
545C2×C4—1221no—
546C2×C4—1221no—
547C2×C4—4881no—
548C2×C4—4881no—
549C2×C4—4881no—
550C2×C4—4881no—
551C2×C4—2441no—
552C2×C4—2442no—
553C2×C4—2441no—
554C2×C4—2441no—
555C2×C4C22×C4×D6111yescharacteristic
556C2×C4C2×D4×D6111yescharacteristic
557C2×C4#41×D6111yescharacteristic
558C8—21no—
559C8—21no—
560C8—21no—
561C8—21no—
562C8—41no—
563C8—41no—
564C8—41no—
565C8—1221no—
566C8—1221no—
567C8—1221no—
568C8—1221no—
569C8—2441no—
570C8—2441no—
571C8—2441no—
572C22—22no—
573C22—41no—
574C22—41no—
575C22—612no—
576C22—1221no—
577C22—1221no—
578C22—1222no—
579C22—4884no—
580C22—2441no—
581C22—2441no—
582C22C4×D4×D6111yescharacteristic
583C4—21no—
584C4—41no—
585C4—41no—
586C4—41no—
587C4—612no—
588C4—1221no—
589C4—1221no—
590C4—2441no—
591C4—2441no—
592C4—2441no—
593C4C2×M16×D6111yescharacteristic
594C4#131×D6111yescharacteristic
595C4#1684056811yescharacteristic
596C4#1684056911yescharacteristic
597C2—42no—
598C2—1222no—
599C2—2442no—
600C2#435×D6111yescharacteristic
601C2#2145467311yescharacteristic
602C2#2145473311yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC42C23×D61
FittingC61×#1582C2
socleC22×C61C2×C4×D4
prime radicalC2×D4×D61C2×C4
cosocleC42×C61C24
hypercenter#1582D61
hypocenterC61#17879
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ #1582

Complements

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

NH ≅ G/Ntype
1. #21454901C2semidirect
4. #21444679C2semidirect
5. #21444676C2semidirect
6. #21444674C2semidirect
7. #21444673C2semidirect
8. #21444657C2semidirect
9. #21444656C2semidirect
10. #21444654C2semidirect
11. #21444564C2semidirect
12. #21444212C2semidirect
14. #21444110C2semidirect
15. C61×#1582C2semidirect
19. #16839893C22semidirect
20. #16839892C22semidirect
23. #16839889C22semidirect
48. #16839514C22semidirect
49. #16839514C22semidirect
50. #16839513C22semidirect
51. #16839494C22semidirect
52. Q8⋊#92418C22semidirect
53. #16839492C22semidirect
57. #16839484C22semidirect
58. #16839484C22semidirect
59. #16839483C22semidirect
60. C61×#446C22semidirect
63. C61×#334C22semidirect
64. C61×Q8⋊C8C22semidirect
66. C61×#322C22semidirect
306. #1582D61semidirect
325. #446C2×D61semidirect
355. #334C2×D61semidirect
359. Q8⋊C8C2×D61semidirect
367. #322C2×D61semidirect
368. C61#17879semidirect

Maximal subgroups

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

subgroupindex
#214549012
#214548402
C4×SD16×D612
#214446792
#214446762
#214446742
#214446732
#214446572
#214446562
#214446542
#214445642
#214442122
#214442082
#214441102
C61×#15822
#1787961