Finite Group Databasebeta

← #29646954

Subgroups — #29646954

Order 15616.

994 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459414C211yesmaximal, characteristic
2#21454785C211yesmaximal, characteristic, complemented
3#21454384C211yesmaximal, characteristic, complemented
4#21444283C212yesmaximal, complemented
5#21444271C211yesmaximal, characteristic, complemented
6#21444257C212yesmaximal, complemented
7#21444248C211yesmaximal, characteristic, complemented
8#21444231C212yesmaximal, complemented
9#21444217C211yesmaximal, characteristic, complemented
10#21444205C212yesmaximal, complemented
11C2×C61×Q8⋊C8C211yesmaximal, characteristic, complemented
12C2×#14519783—21no—
13#16840497—21no—
14#16840497C411yescharacteristic
15#16840491C411yescharacteristic
16#16840485—21no—
17#16840485C411yescharacteristic
18#16840460—22no—
19#16840460C412yes—
20#16840434C411yescharacteristic
21#16840423C412yes—
22#16840312—21no—
23#16840312C2211yescharacteristic
24#16840299—22no—
25#16840299C2212yes—
26C4×#11758226C2212yes—
27C4×#11758222—21no—
28C4×#11758222C2211yescharacteristic
29#16839892—21no—
30#16839841—21no—
31#16839833—22no—
32#16839815C2211yescharacteristic
33#16839813C2212yes—
34#16839788—21no—
35#16839787—22no—
36#16839786C2211yescharacteristic
37#16839784C2212yes—
38C4⋊C8×D61C2212yes—
39C4×#11758189C2212yes—
40Q8⋊#92418C2212yes—
41Q8⋊#92418C2212yes—
42#16839491C2212yes—
43#16839491C2212yes—
44#16839489C2212yes—
45#16839489C2212yes—
46C2×C4×Q8×C61C2211yescharacteristic
47C2×C4⋊C8×C61C2211yescharacteristic
48C2×C4×C8×C61C2211yescharacteristic
49C61×Q8⋊C8C2211yescharacteristic
50C61×Q8⋊C8C2211yescharacteristic
51C61×Q8⋊C8C2211yescharacteristic
52C61×Q8⋊C8C2211yescharacteristic
53C2×#11758226C2×C412yes—
54C2×#11758222—21no—
55C2×#11758222D411yescharacteristic
56#14519783—44no—
57C2×#11758194—21no—
58C2×#11758189—22no—
59C2×#11758189—22no—
60C2×C8×D61—22no—
61#14519777—21no—
62#14519777C2×C411yescharacteristic
63#14519774—21no—
64#14519774C2×C411yescharacteristic
65#14519772—22no—
66#14519772—22no—
67#14519772—22no—
68#14519768—22no—
69#14519768C2×C412yes—
70#14519766—22no—
71#14519766C2×C412yes—
72#14519765—21no—
73#14519765—21no—
74#14519765—21no—
75#14519762—22no—
76#14519762—22no—
77#14519762—22no—
78#14519753—22no—
79#14519753C2×C412yes—
80#14519752—22no—
81#14519752C2×C412yes—
82#14519751—21no—
83#14519751—22no—
84#14519751—22no—
85#14519751C2×C412yes—
86Q8×Dic61C2×C412yes—
87C4×Dic122—21no—
88C4×Dic122—21no—
89C4×Dic122C2×C411yescharacteristic
90C4×Dic122D411yescharacteristic
91C4×D244—22no—
92C4×D244—22no—
93C4×D244—21no—
94C4×D244—21no—
95C4×D244—21no—
96C4×D244C2×C411yescharacteristic
97C4×D244D411yescharacteristic
98C4×D244C2312yes—
99#14519739—22no—
100#14519739C2×C412yes—
101C4×C22⋊D61—22no—
102C4×C22⋊D61—22no—
103C4×C22⋊D61C2×C412yes—
104C4×C22⋊D61D412yes—
105#14519736D411yescharacteristic
106#14519735—22no—
107#14519735C2×C412yes—
108#14519734—22no—
109#14519734D412yes—
110C42×D61—22no—
111C42×D61C2312yes—
112C4⋊#92418—21no—
113C4⋊#92418—21no—
114C4⋊#92418C2312yes—
115#14519706—22no—
116#14519706—22no—
117#14519706—22no—
118C22⋊#92418—21no—
119#14519689—22no—
120#14519687—22no—
121C4×#92418C2312yes—
122C22×Q8×C61C2×C411yescharacteristic
123C22×C8×C61—21no—
124C22×C8×C61—21no—
125C4×Q8×C61—21no—
126C4×Q8×C61—21no—
127C4×Q8×C61C2×C411yescharacteristic
128C4×Q8×C61C2×C411yescharacteristic
129C4×Q8×C61C2311yescharacteristic
130C4×Q8×C61C2311yescharacteristic
131C2×C4⋊C4×C61—21no—
132C2×C4⋊C4×C61C2×C411yescharacteristic
133C2×C42×C61—21no—
134C2×C42×C61C2311yescharacteristic
135C4⋊C8×C61—21no—
136C4⋊C8×C61C2311yescharacteristic
137C4⋊C8×C61C2311yescharacteristic
138C4×C8×C61—21no—
139C4×C8×C61C2311yescharacteristic
140C4×C8×C61C2311yescharacteristic
141C2×Q8×C61—21no—
142C2×Q8×C61—22no—
143C2×Q8×C61—41no—
144C2×Q8×C61C22×C411yescharacteristic
145C2×Q8×C61C22×C411yescharacteristic
146C22×C4×C61—21no—
147C22×C4×C61—21no—
148C22×C4×C61C22×C411yescharacteristic
149C22×C4×C61C2×D411yescharacteristic
150C22×C4×C61C2×D411yescharacteristic
151C2×C22⋊D61—22no—
152C2×C22⋊D61C22⋊C412yes—
153#11758226—24no—
154#11758226—42no—
155#11758222—22no—
156#11758222—22no—
157#11758222—42no—
158#11758222—42no—
159C2×D244—22no—
160C2×D244—21no—
161C2×D244—22no—
162C2×D244—21no—
163C2×D244C22⋊C411yescharacteristic
164C2×D244C22×C412yes—
165C2×C4×D61—22no—
166C2×C4×D61—22no—
167C2×C4×D61—22no—
168C2×C4×D61C22×C412yes—
169C2×C4×D61C2×D412yes—
170C2×C4×D61C2×D412yes—
171C2×Dic122—21no—
172C2×Dic122C22⋊C411yescharacteristic
173C2×C8×C61—21no—
174C2×C8×C61—21no—
175C2×C8×C61—21no—
176C2×C8×C61—21no—
177C2×C8×C61—42no—
178C2×C8×C61—42no—
179C4⋊C4×C61—21no—
180C4⋊C4×C61—21no—
181C4⋊C4×C61—21no—
182C4⋊C4×C61—21no—
183C4⋊C4×C61—21no—
184C4⋊C4×C61C22×C411yescharacteristic
185C4⋊C4×C61C22×C411yescharacteristic
186C42×C61—21no—
187C42×C61—21no—
188C42×C61—21no—
189C42×C61—21no—
190C42×C61C22×C411yescharacteristic
191C42×C61C2×D411yescharacteristic
192C42×C61C2×D411yescharacteristic
193C42×C61C2411yescharacteristic
194C22⋊Dic61C22⋊C412yes—
195#11758198—22no—
196#11758198—22no—
197#11758198—22no—
198#11758198—22no—
199#11758198—22no—
200#11758198—22no—
201#11758198—22no—
202#11758198—22no—
203#11758198C22⋊C412yes—
204#11758198C22⋊C412yes—
205C4⋊Dic61—21no—
206C4⋊Dic61—22no—
207C4⋊Dic61—22no—
208C4⋊Dic61C22⋊C411yescharacteristic
209C4⋊Dic61C22⋊C411yescharacteristic
210C4⋊Dic61C22×C412yes—
211#11758196—22no—
212#11758196—22no—
213#11758196C22⋊C412yes—
214#11758196C22⋊C412yes—
215C4×Dic61—22no—
216C4×Dic61—22no—
217C4×Dic61C22×C412yes—
218C4×Dic61C2×D412yes—
219C4×Dic61C2×D412yes—
220#11758194—42no—
221C2×#92418—22no—
222C2×#92418—22no—
223#11758189—44no—
224#11758189—44no—
225C8×D61—44no—
226C23×C61C2×C22⋊C411yescharacteristic
227C22×D61—22no—
228C22×D61C2×C22⋊C412yes—
229Q8×C61—21no—
230Q8×C61—21no—
231Q8×C61—21no—
232Q8×C61—42no—
233Q8×C61#13112yescomplemented
234C2×C4×C61—21no—
235C2×C4×C61—21no—
236C2×C4×C61—21no—
237C2×C4×C61—21no—
238C2×C4×C61—21no—
239C2×C4×C61—21no—
240C2×C4×C61—41no—
241C2×C4×C61—42no—
242C2×C4×C61—81no—
243C2×C4×C61C2×C22⋊C411yescharacteristic
244C2×C4×C61C2×C22⋊C411yescharacteristic
245C2×C4×C61C2×C22⋊C411yescharacteristic
246C2×C4×C61AGL(1,ℤ/8)12yes—
247C2×C4×C61#13712yes—
248C2×C4×C61C23×C411yescharacteristic
249C2×C4×C61C22×D411yescharacteristic
250C2×C4×C61C22×D411yescharacteristic
251C22⋊D61—24no—
252C22⋊D61—44no—
253C2×Dic61—22no—
254C2×Dic61C2×C22⋊C412yes—
255C2×Dic61C2×C22⋊C412yes—
256C2×Dic61C2×C22⋊C412yes—
257D244—22no—
258D244—22no—
259D244—22no—
260D244—44no—
261D244—42no—
262D244—81no—
263D244—82no—
264C4×D61—22no—
265C4×D61—22no—
266C4×D61—22no—
267C4×D61—22no—
268C4×D61—42no—
269C4×D61—44no—
270C4×D61—42no—
271C4×D61—42no—
272C4×D61—82no—
273Dic122—22no—
274Dic122—42no—
275C8×C61—22no—
276C8×C61—41no—
277C8×C61—41no—
278C8×C61—41no—
279#92418—42no—
280#92418—42no—
281#17515—611nomaximal, Sylow
282C22×C61—21no—
283C22×C61—21no—
284C22×C61#41712yes—
285C22×C61C22×C22⋊C411yescharacteristic
286C2×D61—22no—
287C2×D61—22no—
288C2×D61—22no—
289C2×D61—42no—
290C2×D61—42no—
291C2×D61—42no—
292C4×C61—22no—
293C4×C61—21no—
294C4×C61—21no—
295C4×C61—21no—
296C4×C61—41no—
297C4×C61—41no—
298C4×C61—41no—
299C4×C61#41712yes—
300C4×C61C2×#13111yescharacteristic
301C4×C61#56811yescharacteristic
302C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
303C4×C61C2×#13711yescharacteristic
304Dic61—22no—
305Dic61—22no—
306Dic61—22no—
307Dic61—42no—
308#3480—611no—
309#2965—611no—
310#2906—611no—
311#1495—612no—
312#1492—611no—
313#1489—612no—
314#1487—611no—
315#1483—612no—
316#1480—611no—
317#1477—612no—
318C2×Q8⋊C8D6111yescharacteristic, complemented
319C2×C61—22no—
320C2×C61#288011yescharacteristic
321C2×C61C2×#41711yescharacteristic
322C2×C61#290111yescharacteristic
323D61—42no—
324D61—82no—
325C2×#131—1221no—
326#559—611no—
327#559—1221no—
328#557—611no—
329#556—611no—
330#556—1221no—
331#553—612no—
332#553—1222no—
333#550—611no—
334#548—612no—
335#532—611no—
336#532—1221no—
337#529—612no—
338#529—1222no—
339C4×#41—611no—
340C4×#41—612no—
341C4×#41—1221no—
342C2×C4×Q8C2×D6111yescharacteristic
343#446—1221no—
344#436—1221no—
345#435—1222no—
346#431—611no—
347#431—612no—
348#424—1222no—
349#424—1221no—
350#423—611no—
351#423—612no—
352C2×C4⋊C8—612no—
353C2×C4⋊C8C2×D6111yescharacteristic
354C4×M16—612no—
355C2×C4×C8C2×D6111yescharacteristic
356Q8⋊C8—612no—
357Q8⋊C8—612no—
358Q8⋊C8C2×D6111yescharacteristic
359Q8⋊C8C2×D6111yescharacteristic
360Q8⋊C8C2×D6111yescharacteristic
361Q8⋊C8C2×D6111yescharacteristic
362D4⋊C8—612no—
363D4⋊C8—612no—
364D4⋊C8—612no—
365D4⋊C8—612no—
366C61#1751511yescharacteristic, complemented, Sylow
367C2×#41—611no—
368C2×#41—612no—
369C2×#41—1221no—
370C22×Q8C4×D6111yescharacteristic
371#131—2444no—
372C2×M16—1222no—
373C2×M16—1222no—
374C2×M16—1221no—
375C22×C8—21no—
376C22×C8—21no—
377C22×C8—1222no—
378C4⋊Q8—611no—
379C4⋊Q8—1221no—
380#127—611no—
381#127—1221no—
382#126—1222no—
383#126—1222no—
384#126—1222no—
385#125—612no—
386#125—1222no—
387#124—612no—
388#124—1221no—
389#124—1221no—
390#124—1222no—
391#124—1221no—
392#123—1222no—
393#123—1222no—
394#123—1222no—
395C22⋊Q8—612no—
396C22⋊Q8—1222no—
397#121—612no—
398#121—612no—
399#121—1222no—
400#121—1222no—
401#121—1222no—
402#121—1221no—
403C4×Q8—21no—
404C4×Q8—21no—
405C4×Q8—611no—
406C4×Q8—611no—
407C4×Q8—612no—
408C4×Q8—1221no—
409C4×Q8—1221no—
410C4×Q8C4×D6111yescharacteristic
411C4×Q8C4×D6111yescharacteristic
412C4×Q8C22×D6111yescharacteristic
413C4×Q8C22×D6111yescharacteristic
414C4×D4—612no—
415C4×D4—612no—
416C4×D4—612no—
417C4×D4—611no—
418C4×D4—611no—
419C4×D4—612no—
420C4×D4—1221no—
421C4×D4—1221no—
422C4×D4—1222no—
423C4×D4—1222no—
424C4×D4—1222no—
425C4×D4—1222no—
426C4×D4—1222no—
427C4×D4—1221no—
428#117—611no—
429#117—612no—
430#117—612no—
431#117—1222no—
432#117—1222no—
433C2×C4⋊C4—21no—
434C2×C4⋊C4C4×D6111yescharacteristic
435C2×C42—21no—
436C2×C42—612no—
437C2×C42—1222no—
438C2×C42C22×D6111yescharacteristic
439C4⋊C8—21no—
440C4⋊C8—612no—
441C4⋊C8—1222no—
442C4⋊C8—1222no—
443C4⋊C8—1222no—
444C4⋊C8—1221no—
445C4⋊C8—1221no—
446C4⋊C8C22×D6111yescharacteristic
447C4⋊C8C22×D6111yescharacteristic
448C22⋊C8—1222no—
449C22⋊C8—1221no—
450#97—1222no—
451C4×C8—21no—
452C4×C8—612no—
453C4×C8C22×D6111yescharacteristic
454C4×C8C22×D6111yescharacteristic
455#41—1222no—
456#41—1222no—
457#41—1224no—
458#41—2442no—
459#41—2442no—
460#41—2442no—
461C2×Q8—21no—
462C2×Q8—22no—
463C2×Q8—41no—
464C2×Q8—611no—
465C2×Q8—1221no—
466C2×Q8C2×C4×D6111yescharacteristic
467C2×Q8C2×C4×D6111yescharacteristic
468C2×D4—612no—
469C2×D4—612no—
470C2×D4—611no—
471C2×D4—1222no—
472C2×D4—1222no—
473C2×D4—1222no—
474C2×D4—1221no—
475C2×D4—1221no—
476C22×C4—21no—
477C22×C4—21no—
478C22×C4—612no—
479C22×C4—612no—
480C22×C4—612no—
481C22×C4—1222no—
482C22×C4—1222no—
483C22×C4—1222no—
484C22×C4C2×C4×D6111yescharacteristic
485C22×C4D4×D6111yescharacteristic
486C22×C4D4×D6111yescharacteristic
487M16—2444no—
488M16—2444no—
489M16—2442no—
490C2×C8—21no—
491C2×C8—21no—
492C2×C8—21no—
493C2×C8—21no—
494C2×C8—42no—
495C2×C8—42no—
496C2×C8—1222no—
497C2×C8—1222no—
498C2×C8—2444no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—21no—
504C4⋊C4—612no—
505C4⋊C4—612no—
506C4⋊C4—611no—
507C4⋊C4—611no—
508C4⋊C4—612no—
509C4⋊C4—1222no—
510C4⋊C4—1222no—
511C4⋊C4—1222no—
512C4⋊C4—1222no—
513C4⋊C4—1221no—
514C4⋊C4C2×C4×D6111yescharacteristic
515C4⋊C4C2×C4×D6111yescharacteristic
516C22⋊C4—612no—
517C22⋊C4—612no—
518C22⋊C4—612no—
519C22⋊C4—1222no—
520C22⋊C4—1222no—
521C22⋊C4—1222no—
522C22⋊C4—1222no—
523C22⋊C4—1222no—
524C22⋊C4—1222no—
525C22⋊C4—1222no—
526C22⋊C4—1222no—
527C42—21no—
528C42—21no—
529C42—21no—
530C42—21no—
531C42—612no—
532C42—612no—
533C42—612no—
534C42—1222no—
535C42—1222no—
536C42C2×C4×D6111yescharacteristic
537C42D4×D6111yescharacteristic
538C42D4×D6111yescharacteristic
539C42C23×D6111yescharacteristic
540C23—612no—
541C23—1222no—
542C23C22⋊C4×D6111yescharacteristic
543Q8—21no—
544Q8—21no—
545Q8—21no—
546Q8—42no—
547Q8—1222no—
548Q8—2442no—
549Q8#1451978312yescomplemented
550D4—1224no—
551D4—1222no—
552D4—1222no—
553D4—1222no—
554D4—4881no—
555D4—4882no—
556D4—2444no—
557D4—2444no—
558D4—2442no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—41no—
566C2×C4—42no—
567C2×C4—81no—
568C2×C4—612no—
569C2×C4—612no—
570C2×C4—612no—
571C2×C4—1222no—
572C2×C4—1222no—
573C2×C4—1222no—
574C2×C4—1222no—
575C2×C4—1222no—
576C2×C4—4882no—
577C2×C4—2442no—
578C2×C4—2444no—
579C2×C4—2442no—
580C2×C4—2442no—
581C2×C4C22⋊C4×D6111yescharacteristic
582C2×C4C22⋊C4×D6111yescharacteristic
583C2×C4C22⋊C4×D6111yescharacteristic
584C2×C4#1451980412yes—
585C2×C4#1451980912yes—
586C2×C4C22×C4×D6111yescharacteristic
587C2×C4C2×D4×D6111yescharacteristic
588C2×C4C2×D4×D6111yescharacteristic
589C8—22no—
590C8—41no—
591C8—41no—
592C8—41no—
593C8—2442no—
594C8—2442no—
595C22—21no—
596C22—21no—
597C22—1222no—
598C22—1222no—
599C22—1222no—
600C22—2442no—
601C22—2442no—
602C22—2442no—
603C22#1683976212yes—
604C22C2×C22⋊C4×D6111yescharacteristic
605C4—21no—
606C4—21no—
607C4—21no—
608C4—22no—
609C4—41no—
610C4—41no—
611C4—41no—
612C4—1222no—
613C4—1222no—
614C4—1222no—
615C4—2442no—
616C4#1683976212yes—
617C4C2×#1451978311yescharacteristic
618C4#1684052811yescharacteristic
619C4C2×#1451980411yescharacteristic
620C4C2×#1451980911yescharacteristic
621C2—22no—
622C2—4882no—
623C2—2442no—
624C2#2145426511yescharacteristic
625C2C2×#1683976211yescharacteristic
626C2#2145436311yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC42C23×D61
FittingC2×C61×Q8⋊C8C2
socleC22×C61C22×C22⋊C4
prime radical#21459414C2
cosocleC42×C61C24
hypercenterC2×Q8⋊C8D61
hypocenterC61#17515
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C2×Q8⋊C8

Complements

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

NH ≅ G/Ntype
2. #21454785C2semidirect
3. #21454384C2semidirect
4. #21444283C2semidirect
5. #21444271C2semidirect
6. #21444257C2semidirect
7. #21444248C2semidirect
8. #21444231C2semidirect
9. #21444217C2semidirect
10. #21444205C2semidirect
11. C2×C61×Q8⋊C8C2semidirect
233. Q8×C61#131semidirect
318. C2×Q8⋊C8D61semidirect
366. C61#17515semidirect
549. Q8#14519783semidirect

Maximal subgroups

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

subgroupindex
#214594142
#214547852
#214543842
#214442832
#214442712
#214442572
#214442482
#214442312
#214442172
#214442052
C2×C61×Q8⋊C82
#1751561