Finite Group Databasebeta

← #29650093

Subgroups — #29650093

Order 15616.

596 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455903C211yesmaximal, characteristic
2#21454967C211yesmaximal, characteristic
3#21454898C211yesmaximal, characteristic, complemented
4#21444696C211yesmaximal, characteristic, complemented
5#21444691C211yesmaximal, characteristic, complemented
6#21444672C211yesmaximal, characteristic, complemented
7#21444659C211yesmaximal, characteristic, complemented
8#21444636C211yesmaximal, characteristic, complemented
9#21444629C211yesmaximal, characteristic, complemented
10C8×#11758190C211yesmaximal, characteristic, complemented
11#21444609C211yesmaximal, characteristic, complemented
12#21444246C211yesmaximal, characteristic, complemented
13#21444226C211yesmaximal, characteristic
14#21444118C211yesmaximal, characteristic, complemented
15C8×Q16×C61C211yesmaximal, characteristic, complemented
16#16840299C2211yescharacteristic
17C4×Q8×D61C2211yescharacteristic
18#16840091C411yescharacteristic
19#16840077C411yescharacteristic
20#16840068C411yescharacteristic
21#16840067C411yescharacteristic
22#16839987C411yescharacteristic
23#16839986C411yescharacteristic
24#16839916C411yescharacteristic
25#16839902C411yescharacteristic
26#16839893C2211yescharacteristic
27Q8×#92418C2211yescharacteristic
28C8×Dic122—21no—
29C8×Dic122C2211yescharacteristic
30#16839886—21no—
31#16839886C2211yescharacteristic
32C4×#11758202C2211yescharacteristic
33C4×#11758190C2211yescharacteristic
34C4×#11758201C2211yescharacteristic
35#16839841—21no—
36#16839839C2211yescharacteristic
37#16839832—21no—
38C8×D244C2211yescharacteristic
39#16839821—21no—
40#16839814—21no—
41#16839814C2211yescharacteristic
42#16839813—21no—
43#16839813C2211yescharacteristic
44#16839813C2211yescharacteristic
45#16839813C2211yescharacteristic
46#16839787—21no—
47#16839784C2211yescharacteristic
48#16839514C2211yescharacteristic
49#16839513C2211yescharacteristic
50#16839512C2211yescharacteristic
51#16839494C2211yescharacteristic
52Q8⋊#92418C2211yescharacteristic
53Q8⋊#92418C2211yescharacteristic
54#16839492C2211yescharacteristic
55#16839491C2211yescharacteristic
56#16839489C2211yescharacteristic
57#16839484C2211yescharacteristic
58#16839484C2211yescharacteristic
59#16839484C2211yescharacteristic
60C8×Q8×C61C2211yescharacteristic
61C8×Q8×C61C2211yescharacteristic
62C4×Q16×C61C2211yescharacteristic
63C61×#335C2211yescharacteristic
64C61×Q8⋊C8C2211yescharacteristic
65C61×Q8⋊C8C2211yescharacteristic
66C82×C61C2211yescharacteristic
67C2×Q8×D61C2×C411yescharacteristic
68C2×#11758202C2×C411yescharacteristic
69C2×#11758190C2×C411yescharacteristic
70C2×#11758201C2×C411yescharacteristic
71C2×#11758189—21no—
72C2×#11758189—21no—
73C2×C8×D61—21no—
74#14519772—21no—
75#14519768C2×C411yescharacteristic
76#14519766C2×C411yescharacteristic
77#14519762—21no—
78#14519753C2×C411yescharacteristic
79#14519752C2×C411yescharacteristic
80Q8×Dic61C2×C411yescharacteristic
81C4×Dic122—21no—
82C4×Dic122C2311yescharacteristic
83#14519742—21no—
84#14519742C2×C411yescharacteristic
85C4×D244—21no—
86C4×D244C2311yescharacteristic
87#14519734—21no—
88C4⋊C4×D61—21no—
89C4⋊C4×D61C2×C411yescharacteristic
90C42×D61—21no—
91C42×D61C2311yescharacteristic
92#14519710C2×C411yescharacteristic
93#14519709C2×C411yescharacteristic
94#14519708C2×C411yescharacteristic
95C4⋊#92418—21no—
96C4⋊#92418—21no—
97C4⋊#92418C2311yescharacteristic
98C4⋊#92418C2311yescharacteristic
99#14519706—21no—
100#14519706—21no—
101#14519706—21no—
102#14519706—21no—
103#14519706C2×C411yescharacteristic
104#14519706C2×C411yescharacteristic
105#14519706D411yescharacteristic
106#14519706D411yescharacteristic
107#14519702C2×C411yescharacteristic
108#14519701C2×C411yescharacteristic
109#14519701C2×C411yescharacteristic
110#14519700C2×C411yescharacteristic
111#14519699C2×C411yescharacteristic
112#14519697C2×C411yescharacteristic
113#14519689—21no—
114#14519689—21no—
115#14519689—21no—
116#14519689—21no—
117#14519689C2×C411yescharacteristic
118#14519689C2×C411yescharacteristic
119#14519689D411yescharacteristic
120#14519689D411yescharacteristic
121#14519688—21no—
122C4×#92418—21no—
123C4×#92418C2311yescharacteristic
124C4×#92418C2311yescharacteristic
125C4×#92418C2311yescharacteristic
126C8×Dic61—21no—
127C8×Dic61—21no—
128C8×Dic61—21no—
129C2×Q16×C61C2×C411yescharacteristic
130C4×Q8×C61C2311yescharacteristic
131C4×Q8×C61C2311yescharacteristic
132#107×C61C2×C411yescharacteristic
133C4⋊C8×C61—21no—
134C4⋊C8×C61—21no—
135C4⋊C8×C61C2311yescharacteristic
136C4⋊C8×C61C2311yescharacteristic
137#103×C61C2×C411yescharacteristic
138#103×C61C2×C411yescharacteristic
139C4×C8×C61—21no—
140C4×C8×C61—21no—
141C4×C8×C61C2311yescharacteristic
142C4×C8×C61C2311yescharacteristic
143C4×C8×C61C2311yescharacteristic
144C2×Q8×C61C22×C411yescharacteristic
145C2×Q8×C61C22×C411yescharacteristic
146Q8×D61—22no—
147Q8×D61—22no—
148C2×D244—21no—
149C2×D244C22×C411yescharacteristic
150C2×C4×D61—21no—
151C2×C4×D61—21no—
152C2×C4×D61—21no—
153C2×C4×D61C22×C411yescharacteristic
154C2×C4×D61C2×D411yescharacteristic
155C2×C4×D61#4111yescharacteristic
156C2×Dic122—21no—
157C2×Dic122C22×C411yescharacteristic
158Q16×C61—22no—
159C2×C8×C61—21no—
160C2×C8×C61—21no—
161C2×C8×C61—21no—
162C2×C8×C61C22×C411yescharacteristic
163C2×C8×C61C22×C411yescharacteristic
164C2×C8×C61C2×D411yescharacteristic
165C2×C8×C61C2×D411yescharacteristic
166C4⋊C4×C61—21no—
167C4⋊C4×C61—21no—
168C4⋊C4×C61C22×C411yescharacteristic
169C4⋊C4×C61C22×C411yescharacteristic
170C42×C61—21no—
171C42×C61—21no—
172C42×C61C2411yescharacteristic
173#11758202—22no—
174#11758201—22no—
175#11758198—21no—
176#11758198—21no—
177#11758198—21no—
178#11758198—21no—
179#11758198—21no—
180C4⋊Dic61—21no—
181C4⋊Dic61—21no—
182C4⋊Dic61C22×C411yescharacteristic
183C4⋊Dic61C22×C411yescharacteristic
184#11758196—21no—
185#11758196—21no—
186#11758196—21no—
187#11758196—21no—
188#11758196—21no—
189C4×Dic61—21no—
190C4×Dic61—21no—
191C4×Dic61—21no—
192C4×Dic61C22×C411yescharacteristic
193C4×Dic61C2×D411yescharacteristic
194C4×Dic61#4111yescharacteristic
195C2×#92418—21no—
196C2×#92418—21no—
197C2×#92418—21no—
198C2×#92418C22×C411yescharacteristic
199C2×#92418C22×C411yescharacteristic
200C2×#92418C2×D411yescharacteristic
201C2×#92418C2×D411yescharacteristic
202#11758190—22no—
203#11758189—81no—
204#11758189—81no—
205C8×D61—81no—
206C22×D61—21no—
207C22×D61C4×D411yescharacteristic
208Q8×C61—21no—
209Q8×C61—21no—
210Q8×C61—21no—
211Q8×C61#13112yes—
212C2×C4×C61—21no—
213C2×C4×C61—21no—
214C2×C4×C61—21no—
215C2×C4×C61—21no—
216C2×C4×C61C23×C411yescharacteristic
217C2×C4×C61C22×D411yescharacteristic
218C2×C4×C61C2×#4111yescharacteristic
219C2×Dic61—21no—
220C2×Dic61—21no—
221C2×Dic61—21no—
222C2×Dic61C4×D411yescharacteristic
223C2×Dic61C4×D411yescharacteristic
224C2×Dic61C4×D411yescharacteristic
225D244—21no—
226D244—21no—
227D244—81no—
228C4×D61—24no—
229C4×D61—21no—
230C4×D61—21no—
231C4×D61—21no—
232C4×D61—21no—
233C4×D61—41no—
234C4×D61—41no—
235C4×D61—41no—
236C4×D61—81no—
237Dic122—24no—
238Dic122—21no—
239Dic122#13112yes—
240C8×C61—21no—
241C8×C61—21no—
242C8×C61—21no—
243C8×C61—21no—
244C8×C61—41no—
245C8×C61—41no—
246C8×C61—41no—
247#92418—21no—
248#92418—21no—
249#92418—21no—
250#92418—21no—
251#92418—41no—
252#92418—41no—
253#92418—41no—
254#17919—611nomaximal, Sylow
255C22×C61C2×C4×D411yescharacteristic
256C2×D61—21no—
257C2×D61—21no—
258C2×D61—21no—
259C2×D61—41no—
260C2×D61—41no—
261C2×D61—41no—
262C4×C61—22no—
263C4×C61—21no—
264C4×C61—41no—
265C4×C61—41no—
266C4×C61—41no—
267C4×C61C2×#13111yescharacteristic
268C4×C61C2×#13111yescharacteristic
269C4×C61#57711yescharacteristic
270C4×C61#57811yescharacteristic
271Dic61—21no—
272Dic61—22no—
273Dic61—21no—
274Dic61—21no—
275Dic61—41no—
276Dic61—41no—
277#3108—611no—
278#2993—611no—
279#2982—611no—
280#1588—611no—
281#1587—611no—
282#1584—611no—
283#1583—611no—
284#1579—611no—
285SD16⋊C8—611no—
286C8×Q16D6111yescharacteristic, complemented
287C8×SD16—611no—
288#1574—611no—
289#1487—611no—
290#1482—611no—
291#1454—611no—
292C2×C61#292911yescharacteristic
293C2×C61#294711yescharacteristic
294C2×C61#295711yescharacteristic
295D61—41no—
296D61—81no—
297#529—611no—
298C2×C4×Q8—611no—
299#483—611no—
300#480—611no—
301#479—611no—
302#479—611no—
303#464—611no—
304#464—611no—
305#451—611no—
306#448—611no—
307#446—611no—
308C8×Q8—611no—
309C8×Q8—611no—
310C8×Q8—611no—
311C8×Q8—1221no—
312C8×Q8—1221no—
313C8×Q8C2×D6111yescharacteristic
314C8×Q8C2×D6111yescharacteristic
315C4×Q16—611no—
316C4×Q16C2×D6111yescharacteristic
317C4×SD16—611no—
318C4×SD16—611no—
319#436—611no—
320#436—1221no—
321#435—1221no—
322C8×D4—611no—
323#433—1221no—
324#431—611no—
325#431—611no—
326#431—611no—
327#431—611no—
328#431—1221no—
329#431—1221no—
330#424—1221no—
331#423—611no—
332#335—611no—
333#335C2×D6111yescharacteristic
334#334—611no—
335#334—611no—
336Q8⋊C8—611no—
337Q8⋊C8—611no—
338Q8⋊C8—611no—
339Q8⋊C8—611no—
340Q8⋊C8C2×D6111yescharacteristic
341Q8⋊C8C2×D6111yescharacteristic
342D4⋊C8—611no—
343D4⋊C8—611no—
344#322—611no—
345#322—611no—
346#322—611no—
347C82C2×D6111yescharacteristic
348C61#1791911yescharacteristic, complemented, Sylow
349C22×Q8—611no—
350C2×Q16—611no—
351C2×Q16C4×D6111yescharacteristic
352C2×SD16—611no—
353C2×SD16—611no—
354C2×M16—1221no—
355C2×M16—1221no—
356C22×C8—1221no—
357#126—1221no—
358#125—611no—
359#124—611no—
360#123—1221no—
361C22⋊Q8—611no—
362#121—611no—
363C4×Q8—611no—
364C4×Q8—611no—
365C4×Q8—611no—
366C4×Q8—1221no—
367C4×Q8—1221no—
368C4×Q8C22×D6111yescharacteristic
369C4×Q8C22×D6111yescharacteristic
370C4×D4—611no—
371C4×D4—1221no—
372#117—1221no—
373C2×C4⋊C4—611no—
374C2×C4⋊C4—1221no—
375C2×C42—611no—
376C2×C42—1221no—
377#107—611no—
378#107C4×D6111yescharacteristic
379#106—611no—
380#106—611no—
381C4⋊C8—21no—
382C4⋊C8—21no—
383C4⋊C8—611no—
384C4⋊C8—611no—
385C4⋊C8—611no—
386C4⋊C8—611no—
387C4⋊C8—611no—
388C4⋊C8—611no—
389C4⋊C8—1221no—
390C4⋊C8—1221no—
391C4⋊C8—1221no—
392C4⋊C8—1221no—
393C4⋊C8—1221no—
394C4⋊C8—1221no—
395C4⋊C8C22×D6111yescharacteristic
396C4⋊C8C22×D6111yescharacteristic
397#103—611no—
398#103—611no—
399#103—611no—
400#103—611no—
401#103C4×D6111yescharacteristic
402#103C4×D6111yescharacteristic
403#102—611no—
404#102—611no—
405C22⋊C8—611no—
406C22⋊C8—611no—
407C22⋊C8—611no—
408C22⋊C8—611no—
409C22⋊C8—1221no—
410C22⋊C8—1221no—
411C22⋊C8—1221no—
412C22⋊C8—1221no—
413#97—1221no—
414C4×C8—21no—
415C4×C8—21no—
416C4×C8—611no—
417C4×C8—611no—
418C4×C8—611no—
419C4×C8—1221no—
420C4×C8—1221no—
421C4×C8—1221no—
422C4×C8—1221no—
423C4×C8C22×D6111yescharacteristic
424C4×C8C22×D6111yescharacteristic
425C4×C8C22×D6111yescharacteristic
426C2×Q8—611no—
427C2×Q8—1221no—
428C2×Q8—1222no—
429C2×Q8—1222no—
430C2×Q8C2×C4×D6111yescharacteristic
431C2×Q8C2×C4×D6111yescharacteristic
432C2×D4—611no—
433C2×D4—1221no—
434C22×C4—611no—
435C22×C4—611no—
436C22×C4—611no—
437C22×C4—1221no—
438C22×C4—1221no—
439C22×C4—1221no—
440Q16—22no—
441Q16—1222no—
442SD16—1222no—
443SD16—1222no—
444M16—4881no—
445M16—4881no—
446C2×C8—21no—
447C2×C8—21no—
448C2×C8—21no—
449C2×C8—611no—
450C2×C8—611no—
451C2×C8—611no—
452C2×C8—611no—
453C2×C8—1221no—
454C2×C8—1221no—
455C2×C8—1221no—
456C2×C8—4881no—
457C2×C8C2×C4×D6111yescharacteristic
458C2×C8C2×C4×D6111yescharacteristic
459C2×C8D4×D6111yescharacteristic
460C2×C8D4×D6111yescharacteristic
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—611no—
464C4⋊C4—611no—
465C4⋊C4—1221no—
466C4⋊C4—1221no—
467C4⋊C4—1221no—
468C4⋊C4—1221no—
469C4⋊C4—1221no—
470C4⋊C4—1221no—
471C4⋊C4—1221no—
472C4⋊C4C2×C4×D6111yescharacteristic
473C4⋊C4C2×C4×D6111yescharacteristic
474C22⋊C4—1221no—
475C22⋊C4—1221no—
476C22⋊C4—1221no—
477C22⋊C4—1221no—
478C22⋊C4—1221no—
479C42—21no—
480C42—21no—
481C42—611no—
482C42—611no—
483C42—611no—
484C42—1221no—
485C42—1221no—
486C42—1221no—
487C42C23×D6111yescharacteristic
488C23—611no—
489C23—1221no—
490Q8—21no—
491Q8—21no—
492Q8—21no—
493Q8—612no—
494Q8—1224no—
495Q8—1221no—
496Q8#1451978312yes—
497D4—1221no—
498D4—1221no—
499D4—4881no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—611no—
505C2×C4—611no—
506C2×C4—611no—
507C2×C4—1224no—
508C2×C4—1221no—
509C2×C4—1221no—
510C2×C4—1221no—
511C2×C4—1221no—
512C2×C4—1221no—
513C2×C4—1221no—
514C2×C4—1221no—
515C2×C4—4881no—
516C2×C4—2441no—
517C2×C4—2441no—
518C2×C4—2441no—
519C2×C4C22×C4×D6111yescharacteristic
520C2×C4C2×D4×D6111yescharacteristic
521C2×C4#41×D6111yescharacteristic
522C8—21no—
523C8—21no—
524C8—21no—
525C8—21no—
526C8—41no—
527C8—41no—
528C8—41no—
529C8—1221no—
530C8—1221no—
531C8—1221no—
532C8—1221no—
533C8—2441no—
534C8—2441no—
535C8—2441no—
536C22—1221no—
537C22—1221no—
538C22—1221no—
539C22—2441no—
540C22—2441no—
541C22—2441no—
542C22C4×D4×D6111yescharacteristic
543C4—21no—
544C4—22no—
545C4—41no—
546C4—41no—
547C4—41no—
548C4—1221no—
549C4—1221no—
550C4—1221no—
551C4—1222no—
552C4—2441no—
553C4—2441no—
554C4#131×D6111yescharacteristic
555C4C2×#1451978311yescharacteristic
556C4#1684056911yescharacteristic
557C4#1684057511yescharacteristic
558C2—4881no—
559C2—2441no—
560C2#2145450811yescharacteristic
561C2#2145465711yescharacteristic
562C2#2145473111yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC42C23×D61
FittingC8×Q16×C61C2
socleC22×C61C2×C4×D4
prime radical#14519752C2×C4
cosocleC42×C61C24
hypercenterC8×Q16D61
hypocenterC61#17919
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ C8×Q16

Complements

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

NH ≅ G/Ntype
3. #21454898C2semidirect
4. #21444696C2semidirect
5. #21444691C2semidirect
6. #21444672C2semidirect
7. #21444659C2semidirect
8. #21444636C2semidirect
9. #21444629C2semidirect
10. C8×#11758190C2semidirect
11. #21444609C2semidirect
12. #21444246C2semidirect
14. #21444118C2semidirect
15. C8×Q16×C61C2semidirect
286. C8×Q16D61semidirect
348. C61#17919semidirect

Maximal subgroups

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

subgroupindex
#214559032
#214549672
#214548982
#214446962
#214446912
#214446722
#214446592
#214446362
#214446292
C8×#117581902
#214446092
#214442462
#214442262
#214441182
C8×Q16×C612
#1791961