Finite Group Databasebeta

← #29646955

Subgroups — #29646955

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#21459412C211yesmaximal, characteristic
2#21454786C211yesmaximal, characteristic, complemented
3#21444282C212yesmaximal, complemented
4#21444269C211yesmaximal, characteristic, complemented
5#21444256C212yesmaximal, complemented
6#21444247C211yesmaximal, characteristic, complemented
7#21444230C212yesmaximal, complemented
8#21444217C211yesmaximal, characteristic, complemented
9#21444204C212yesmaximal, complemented
10C2×#16839492C211yesmaximal, characteristic, complemented
11C61×#2906C211yesmaximal, characteristic, complemented
12#16840497C411yescharacteristic
13#16840495—21no—
14#16840490C411yescharacteristic
15#16840484—21no—
16#16840482C411yescharacteristic
17#16840461—22no—
18#16840458C412yes—
19#16840433C411yescharacteristic
20#16840421C412yes—
21#16840312C2211yescharacteristic
22#16840308—21no—
23#16840296C2212yes—
24#16840290—22no—
25C4×#11758222C2212yes—
26#16840230—21no—
27C2×C4×Dic122C2211yescharacteristic
28#16839893—21no—
29#16839840—21no—
30#16839834—22no—
31#16839815C2212yes—
32#16839815C2211yescharacteristic
33C4×#11758194C2212yes—
34C2×C4×#92418C2211yescharacteristic
35#16839492C2211yescharacteristic
36#16839492C2211yescharacteristic
37#16839492C2211yescharacteristic
38#16839492C2211yescharacteristic
39#16839492C2212yes—
40#16839492C2212yes—
41#16839489C2212yes—
42#16839489C2212yes—
43#16839489C2212yes—
44#16839489C2212yes—
45C2×#131×C61—21no—
46C4×#41×C61C2211yescharacteristic
47C61×#424—21no—
48C61×#424—22no—
49C61×#423C2211yescharacteristic
50C61×#423C2212yes—
51C2×C4⋊C8×C61C2211yescharacteristic
52C2×C4⋊C8×C61C2212yes—
53C2×#11758222—21no—
54C2×#11758222C2×C412yes—
55C22×Dic122C2×C411yescharacteristic
56C2×#11758194—22no—
57C22×#92418—21no—
58#14519777C2×C411yescharacteristic
59#14519775—21no—
60#14519774C2×C411yescharacteristic
61#14519773—21no—
62#14519772—22no—
63#14519771—22no—
64#14519770—22no—
65#14519769C2×C412yes—
66#14519767—22no—
67#14519766—21no—
68#14519765—21no—
69#14519765C2×C412yes—
70#14519764—21no—
71#14519763—22no—
72#14519762—22no—
73#14519760—22no—
74#14519759—22no—
75#14519756C2×C412yes—
76#14519755—22no—
77#14519751—21no—
78#14519751C2×C412yes—
79#14519751C2×C412yes—
80#14519750—22no—
81#14519749—22no—
82#14519749—22no—
83C4×Dic122—21no—
84C4×Dic122—21no—
85C4×Dic122C2×C411yescharacteristic
86C4×Dic122C2×C411yescharacteristic
87C4×Dic122C2×C412yes—
88C4×Dic122C2311yescharacteristic
89C4×Dic122C2311yescharacteristic
90#14519742—21no—
91#14519742—21no—
92C4×D244—21no—
93C4×D244C2×C412yes—
94C4×D244C2312yes—
95#14519739—21no—
96#14519739—21no—
97C4×C22⋊D61—22no—
98C4×C22⋊D61—22no—
99C4×C22⋊D61—22no—
100#14519737—22no—
101#14519737—22no—
102#14519736C2×C412yes—
103#14519734—22no—
104#14519734—22no—
105C2×C4⋊Dic61C2×C411yescharacteristic
106C2×#11758196—21no—
107C2×C4×Dic61—21no—
108C42×D61—22no—
109C4⋊#92418—22no—
110C4⋊#92418—21no—
111C22⋊#92418—21no—
112C22⋊#92418—22no—
113#14519688—22no—
114C4×#92418—21no—
115C4×#92418C2311yescharacteristic
116C4×#92418C2311yescharacteristic
117C4×#92418C2312yes—
118C2×#41×C61D411yescharacteristic
119#131×C61—44no—
120C2×M16×C61—22no—
121C2×M16×C61—21no—
122C22×C8×C61—22no—
123C22×C8×C61—21no—
124C4×Q8×C61C2×C411yescharacteristic
125C4×Q8×C61D411yescharacteristic
126C4×D4×C61C2×C412yes—
127C4×D4×C61C2×C411yescharacteristic
128C4×D4×C61D412yes—
129C4×D4×C61D411yescharacteristic
130#117×C61D411yescharacteristic
131#117×C61D412yes—
132C2×C42×C61C2311yescharacteristic
133C2×C42×C61C2312yes—
134C4⋊C8×C61—22no—
135C4⋊C8×C61—21no—
136C4⋊C8×C61—22no—
137C4⋊C8×C61—21no—
138C4⋊C8×C61C2311yescharacteristic
139C4⋊C8×C61C2312yes—
140C4⋊C8×C61C2311yescharacteristic
141#41×C61—22no—
142#41×C61—22no—
143C2×Q8×C61C22⋊C411yescharacteristic
144C2×D4×C61C22⋊C412yes—
145C2×D4×C61C22⋊C411yescharacteristic
146C22×C4×C61C22×C411yescharacteristic
147C22×C4×C61C22×C412yes—
148C22×C4×C61C2×D412yes—
149C22×C4×C61C2×D411yescharacteristic
150C22×C4×C61C2×D412yes—
151C22×C4×C61C2×D411yescharacteristic
152C2×C22⋊D61—22no—
153C2×C22⋊D61—22no—
154C2×C22⋊D61—22no—
155C22×Dic61—21no—
156C22×Dic61—21no—
157#11758222—24no—
158#11758222—42no—
159#11758222—42no—
160#11758222—42no—
161C2×D244—21no—
162C2×D244—21no—
163C2×D244C22×C412yes—
164C2×C4×D61—22no—
165C2×C4×D61—22no—
166C2×C4×D61—22no—
167C2×Dic122—22no—
168C2×Dic122—21no—
169C2×Dic122—21no—
170C2×Dic122—41no—
171C2×Dic122C22×C411yescharacteristic
172C2×Dic122C22×C411yescharacteristic
173M16×C61—44no—
174M16×C61—42no—
175C2×C8×C61—21no—
176C2×C8×C61—22no—
177C2×C8×C61—21no—
178C2×C8×C61—44no—
179C2×C8×C61—42no—
180C4⋊C4×C61C22⋊C411yescharacteristic
181C4⋊C4×C61C22⋊C412yes—
182C4⋊C4×C61C22⋊C411yescharacteristic
183C4⋊C4×C61C22⋊C412yes—
184C22⋊C4×C61C22⋊C412yes—
185C22⋊C4×C61C22⋊C412yes—
186C22⋊C4×C61C22⋊C412yes—
187C42×C61C22×C412yes—
188C42×C61C22×C411yescharacteristic
189C42×C61C2×D411yescharacteristic
190C42×C61C2×D412yes—
191C42×C61C2×D412yes—
192C42×C61C2×D411yescharacteristic
193C42×C61C2411yescharacteristic
194C22⋊Dic61—22no—
195C22⋊Dic61—22no—
196#11758198—22no—
197#11758198—22no—
198#11758198—22no—
199#11758198—22no—
200#11758198—22no—
201#11758198—22no—
202C4⋊Dic61—21no—
203C4⋊Dic61—21no—
204C4⋊Dic61C22×C411yescharacteristic
205C4⋊Dic61C22×C411yescharacteristic
206C4⋊Dic61C22×C412yes—
207#11758196—21no—
208#11758196—21no—
209#11758196—22no—
210#11758196—22no—
211#11758196—22no—
212#11758196—22no—
213#11758196—21no—
214#11758196—21no—
215C4×Dic61—22no—
216C4×Dic61—22no—
217C4×Dic61—21no—
218C4×Dic61—21no—
219C4×Dic61—21no—
220C4×Dic61—21no—
221#11758194—44no—
222C2×#92418—21no—
223C2×#92418—21no—
224C2×#92418—22no—
225C2×#92418—42no—
226C23×C61C2×C22⋊C412yes—
227C23×C61C2×C22⋊C411yescharacteristic
228C22×D61—22no—
229Q8×C61—22no—
230D4×C61—24no—
231D4×C61—22no—
232C2×C4×C61—21no—
233C2×C4×C61—22no—
234C2×C4×C61—21no—
235C2×C4×C61—22no—
236C2×C4×C61—22no—
237C2×C4×C61—22no—
238C2×C4×C61—41no—
239C2×C4×C61—42no—
240C2×C4×C61C2×C22⋊C412yes—
241C2×C4×C61C2×C22⋊C412yes—
242C2×C4×C61C2×C22⋊C411yescharacteristic
243C2×C4×C61C2×C22⋊C412yes—
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—44no—
252C22⋊D61—44no—
253C22⋊D61—82no—
254C2×Dic61—21no—
255C2×Dic61—21no—
256C2×Dic61—21no—
257C2×Dic61—22no—
258C2×Dic61—21no—
259C2×Dic61—42no—
260C2×Dic61—81no—
261D244—22no—
262D244—22no—
263D244—42no—
264D244—81no—
265C4×D61—44no—
266C4×D61—42no—
267C4×D61—42no—
268C4×D61—82no—
269Dic122—21no—
270Dic122—21no—
271Dic122—21no—
272Dic122—42no—
273Dic122—42no—
274Dic122#13112yescomplemented
275C8×C61—41no—
276C8×C61—42no—
277C8×C61—41no—
278#92418—22no—
279#92418—41no—
280#92418—42no—
281#17515—611nomaximal, Sylow
282C22×C61—21no—
283C22×C61—21no—
284C22×C61—22no—
285C22×C61—22no—
286C22×C61—22no—
287C22×C61#41712yes—
288C22×C61C22×C22⋊C411yescharacteristic
289C2×D61—42no—
290C2×D61—42no—
291C2×D61—42no—
292C4×C61—21no—
293C4×C61—22no—
294C4×C61—22no—
295C4×C61—21no—
296C4×C61—22no—
297C4×C61—21no—
298C4×C61#41712yes—
299C4×C61C2×#13111yescharacteristic
300C4×C61#56811yescharacteristic
301C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
302C4×C61C2×#13711yescharacteristic
303Dic61—22no—
304Dic61—41no—
305Dic61—42no—
306Dic61—41no—
307Dic61—41no—
308#3480—611no—
309#2965—611no—
310#2906D6111yescharacteristic, complemented
311#1495—612no—
312#1492—611no—
313#1489—612no—
314#1487—611no—
315#1483—612no—
316#1480—611no—
317#1477—612no—
318C2×Q8⋊C8—611no—
319C2×C61—22no—
320C2×C61—42no—
321C2×C61#288011yescharacteristic
322C2×C61C2×#41711yescharacteristic
323C2×C61#290111yescharacteristic
324D61—82no—
325C2×#131—21no—
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—612no—
340C4×#41—1221no—
341C4×#41C2×D6111yescharacteristic
342C2×C4×Q8—611no—
343#446—1221no—
344#436—1221no—
345#435—1222no—
346#431—612no—
347#431—611no—
348#424—21no—
349#424—22no—
350#423C2×D6111yescharacteristic
351#423C2×D6112yes—
352C2×C4⋊C8C2×D6111yescharacteristic
353C2×C4⋊C8C2×D6112yes—
354C4×M16—612no—
355C2×C4×C8—611no—
356Q8⋊C8—611no—
357Q8⋊C8—611no—
358Q8⋊C8—611no—
359Q8⋊C8—611no—
360Q8⋊C8—612no—
361Q8⋊C8—612no—
362D4⋊C8—612no—
363D4⋊C8—612no—
364D4⋊C8—612no—
365D4⋊C8—612no—
366C61#1751511yescharacteristic, complemented, Sylow
367C2×#41—612no—
368C2×#41—1221no—
369C2×#41C22⋊D6111yescharacteristic
370C22×Q8—611no—
371#131—44no—
372C2×M16—22no—
373C2×M16—21no—
374C2×M16—1222no—
375C22×C8—22no—
376C22×C8—21no—
377C22×C8—1221no—
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—1221no—
402#121—1222no—
403C4×Q8—611no—
404C4×Q8—611no—
405C4×Q8—612no—
406C4×Q8—611no—
407C4×Q8—611no—
408C4×Q8—1221no—
409C4×Q8—1221no—
410C4×Q8—1221no—
411C4×Q8—1221no—
412C4×Q8C4×D6111yescharacteristic
413C4×Q8D24411yescharacteristic
414C4×D4—612no—
415C4×D4—612no—
416C4×D4—1222no—
417C4×D4—1222no—
418C4×D4—1222no—
419C4×D4—1221no—
420C4×D4—1221no—
421C4×D4—1222no—
422C4×D4—1222no—
423C4×D4—1221no—
424C4×D4C4×D6112yes—
425C4×D4C4×D6111yescharacteristic
426C4×D4D24412yes—
427C4×D4D24411yescharacteristic
428#117—612no—
429#117—1222no—
430#117—1222no—
431#117C22⋊D6111yescharacteristic
432#117C22⋊D6112yes—
433C2×C4⋊C4—611no—
434C2×C4⋊C4—1221no—
435C2×C42—1221no—
436C2×C42—1222no—
437C2×C42C22×D6111yescharacteristic
438C2×C42C22×D6112yes—
439C4⋊C8—22no—
440C4⋊C8—21no—
441C4⋊C8—22no—
442C4⋊C8—21no—
443C4⋊C8—1222no—
444C4⋊C8—1221no—
445C4⋊C8C22×D6111yescharacteristic
446C4⋊C8C22×D6112yes—
447C4⋊C8C22×D6111yescharacteristic
448C22⋊C8—1221no—
449C22⋊C8—1222no—
450#97—1222no—
451C4×C8—611no—
452C4×C8—611no—
453C4×C8—612no—
454C4×C8—1221no—
455#41—22no—
456#41—22no—
457#41—1224no—
458#41—2442no—
459#41—2442no—
460#41—2442no—
461C2×Q8—611no—
462C2×Q8—611no—
463C2×Q8—1222no—
464C2×Q8—1221no—
465C2×Q8—1221no—
466C2×Q8—2441no—
467C2×Q8#1175819811yescharacteristic
468C2×D4—612no—
469C2×D4—1222no—
470C2×D4—1222no—
471C2×D4—1222no—
472C2×D4—1221no—
473C2×D4—1221no—
474C2×D4#1175819812yes—
475C2×D4#1175819811yescharacteristic
476C22×C4—1221no—
477C22×C4—1221no—
478C22×C4—1222no—
479C22×C4—1222no—
480C22×C4—1222no—
481C22×C4C2×C4×D6111yescharacteristic
482C22×C4C2×C4×D6112yes—
483C22×C4C2×D24411yescharacteristic
484C22×C4C2×D24412yes—
485C22×C4C2×C22⋊D6112yes—
486C22×C4C2×C22⋊D6111yescharacteristic
487M16—44no—
488M16—42no—
489M16—2444no—
490C2×C8—21no—
491C2×C8—22no—
492C2×C8—21no—
493C2×C8—44no—
494C2×C8—42no—
495C2×C8—1221no—
496C2×C8—1221no—
497C2×C8—1222no—
498C2×C8—2442no—
499C4⋊C4—611no—
500C4⋊C4—611no—
501C4⋊C4—612no—
502C4⋊C4—1221no—
503C4⋊C4—1221no—
504C4⋊C4—1221no—
505C4⋊C4—1221no—
506C4⋊C4—1221no—
507C4⋊C4—1222no—
508C4⋊C4—1222no—
509C4⋊C4—1222no—
510C4⋊C4—1222no—
511C4⋊C4—1221no—
512C4⋊C4#1175819811yescharacteristic
513C4⋊C4#1175819811yescharacteristic
514C4⋊C4#1175819812yes—
515C4⋊C4#1175819812yes—
516C22⋊C4—1222no—
517C22⋊C4—1222no—
518C22⋊C4—1222no—
519C22⋊C4—1222no—
520C22⋊C4—1222no—
521C22⋊C4—1222no—
522C22⋊C4—1222no—
523C22⋊C4—1222no—
524C22⋊C4#1175819812yes—
525C22⋊C4#1175819812yes—
526C22⋊C4#1175819812yes—
527C42—1221no—
528C42—1221no—
529C42—1222no—
530C42—1222no—
531C42—1221no—
532C42—1221no—
533C42C2×C4×D6112yes—
534C42C2×C4×D6111yescharacteristic
535C42C2×D24411yescharacteristic
536C42C2×D24412yes—
537C42C2×C22⋊D6112yes—
538C42C2×C22⋊D6111yescharacteristic
539C42C23×D6111yescharacteristic
540C23—1222no—
541C23C2×#1175819812yes—
542C23C2×#1175819811yescharacteristic
543Q8—22no—
544Q8—612no—
545Q8—1221no—
546Q8—1221no—
547Q8—1221no—
548Q8—2442no—
549Q8—2442no—
550D4—24no—
551D4—22no—
552D4—1222no—
553D4—1222no—
554D4—4881no—
555D4—4882no—
556D4—2444no—
557D4—2444no—
558D4—2442no—
559C2×C4—21no—
560C2×C4—22no—
561C2×C4—21no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—41no—
566C2×C4—42no—
567C2×C4—1221no—
568C2×C4—1221no—
569C2×C4—1221no—
570C2×C4—1222no—
571C2×C4—1221no—
572C2×C4—4882no—
573C2×C4—4881no—
574C2×C4—2444no—
575C2×C4—2442no—
576C2×C4—2442no—
577C2×C4—2442no—
578C2×C4C2×#1175819812yes—
579C2×C4C2×#1175819812yes—
580C2×C4C2×#1175819812yes—
581C2×C4C2×#1175819811yescharacteristic
582C2×C4C2×#1175819811yescharacteristic
583C2×C4C2×#1175819811yescharacteristic
584C2×C4#1451980312yes—
585C2×C4#1451980812yes—
586C2×C4C22×C4×D6111yescharacteristic
587C2×C4C22×C22⋊D6111yescharacteristic
588C2×C4C22×D24411yescharacteristic
589C8—41no—
590C8—42no—
591C8—41no—
592C8—1222no—
593C8—2441no—
594C8—2442no—
595C22—21no—
596C22—21no—
597C22—22no—
598C22—22no—
599C22—22no—
600C22—2442no—
601C22—2442no—
602C22—2442no—
603C22#1683976112yes—
604C22C22×#1175819811yescharacteristic
605C4—21no—
606C4—22no—
607C4—21no—
608C4—22no—
609C4—21no—
610C4—22no—
611C4—1222no—
612C4—2441no—
613C4—2441no—
614C4—2441no—
615C4—2442no—
616C4#1683976112yes—
617C4#131×D6111yescharacteristic
618C4#1684052711yescharacteristic
619C4C2×#1451980311yescharacteristic
620C4C2×#1451980811yescharacteristic
621C2—22no—
622C2—42no—
623C2—4882no—
624C2#2145426611yescharacteristic
625C2C2×#1683976111yescharacteristic
626C2#2145436411yescharacteristic
subgroupG/H
centerC2×C4C22×C22⋊D61
commutatorC2×C4×C61C23×C4
FrattiniC42C23×D61
FittingC61×#2906C2
socleC22×C61C22×C22⋊C4
prime radical#21459412C2
cosocleC42×C61C24
hypercenter#2906D61
hypocenterC61#17515
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C4×#41 ◁ #2906

Complements

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

NH ≅ G/Ntype
2. #21454786C2semidirect
3. #21444282C2semidirect
4. #21444269C2semidirect
5. #21444256C2semidirect
6. #21444247C2semidirect
7. #21444230C2semidirect
8. #21444217C2semidirect
9. #21444204C2semidirect
10. C2×#16839492C2semidirect
11. C61×#2906C2semidirect
274. Dic122#131semidirect
310. #2906D61semidirect
366. C61#17515semidirect

Maximal subgroups

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

subgroupindex
#214594122
#214547862
#214442822
#214442692
#214442562
#214442472
#214442302
#214442172
#214442042
C2×#168394922
C61×#29062
#1751561