Finite Group Databasebeta

← #29680462

Subgroups — #29680462

Order 15616.

1070 conjugacy classes of subgroups.

The subgroup lattice has 554 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839803C211yesmaximal, characteristic
2C2×#16839801C211yesmaximal, characteristic
3C2×#16839797C211yesmaximal, characteristic
4C2×#107×D61C211yesmaximal, characteristic
5C2×#16839719C211yesmaximal, characteristic
6#21448444C211yesmaximal, characteristic
7#21448440C211yesmaximal, characteristic
8#21445987C211yesmaximal, characteristic, complemented
9#21445986C211yesmaximal, characteristic, complemented
10#21445984C211yesmaximal, characteristic, complemented
11#21445981C211yesmaximal, characteristic, complemented
12#21445980C211yesmaximal, characteristic, complemented
13#21445978C211yesmaximal, characteristic, complemented
14#21445977C211yesmaximal, characteristic, complemented
15C61×#1847C211yesmaximal, characteristic, complemented
16C22×#11758189C2211yescharacteristic
17C22×C8×D61C2211yescharacteristic
18C2×#14519735C2211yescharacteristic
19C2×#14519735C2211yescharacteristic
20C2×#14519734C2211yescharacteristic
21C2×C4⋊C4×D61C2211yescharacteristic
22C2×C4⋊C4×D61C2211yescharacteristic
23#16839803—24no—
24#16839801—24no—
25#16839797—24no—
26C2×#14519711C2211yescharacteristic
27C2×#14519711C2211yescharacteristic
28C2×#14519710C2211yescharacteristic
29C2×#14519710C2211yescharacteristic
30C2×#14519710C2211yescharacteristic
31C2×#14519710C2211yescharacteristic
32C2×#14519710C2211yescharacteristic
33C2×#14519710C2211yescharacteristic
34#107×D61—24no—
35C2×#14519709C2211yescharacteristic
36C2×#14519709C2211yescharacteristic
37C2×#14519708C2211yescharacteristic
38C2×#14519708C2211yescharacteristic
39#16839719C418yescomplemented
40C2×#14519687C2211yescharacteristic
41C2×C4×#92418C2211yescharacteristic
42C2×C8×Dic61C2211yescharacteristic
43#16839659—22no—
44#16839657—22no—
45#16839636—22no—
46#16839633—22no—
47#16839632C2211yescharacteristic
48#16839631C2211yescharacteristic
49#16839631C2211yescharacteristic
50#16839628C2211yescharacteristic
51#16839627C2211yescharacteristic
52#16839627C2211yescharacteristic
53C2×#107×C61C2211yescharacteristic
54C2×#107×C61C2211yescharacteristic
55C2×#106×C61C2211yescharacteristic
56C2×#106×C61C2211yescharacteristic
57C2×#97×C61C2211yescharacteristic
58C61×#384C2211yescharacteristic
59C61×#382C2211yescharacteristic
60C22×C4×D61—21no—
61C22×C4×D61—21no—
62C22×C4×D61C2311yescharacteristic
63C2×#11758189C2×C414yes—
64C2×#11758189D414yes—
65C2×#11758189Q814yes—
66C22×#92418C2311yescharacteristic
67C22×#92418C2311yescharacteristic
68C2×C8×D61C2×C414yes—
69C2×C8×D61D414yes—
70C2×C8×D61Q814yes—
71#14519735—22no—
72#14519735—22no—
73#14519735—22no—
74#14519735—22no—
75#14519734C2×C414yes—
76#14519734Q814yes—
77C2×C4⋊Dic61—22no—
78C2×C4⋊Dic61C2311yescharacteristic
79C2×C4⋊Dic61C2311yescharacteristic
80C2×C4⋊Dic61C2311yescharacteristic
81C2×C4⋊Dic61C2311yescharacteristic
82C2×#11758196—21no—
83C2×#11758196—21no—
84C2×#11758196—22no—
85C2×#11758196—22no—
86C2×#11758196D412yes—
87C4⋊C4×D61—22no—
88C4⋊C4×D61—22no—
89C4⋊C4×D61—22no—
90C4⋊C4×D61—22no—
91C2×#11758198—21no—
92C2×#11758198—21no—
93C2×#11758198D412yes—
94C2×C4×Dic61—21no—
95C2×C4×Dic61—21no—
96C2×C4×Dic61C2311yescharacteristic
97#14519711—22no—
98#14519711—22no—
99#14519710—22no—
100#14519710—22no—
101#14519710—22no—
102#14519710—22no—
103#14519710—22no—
104#14519710—22no—
105#14519709—22no—
106#14519709—22no—
107#14519708—22no—
108#14519708—22no—
109#14519687C2×C414yescomplemented
110C4×#92418C2×C414yescomplemented
111C8×Dic61C2×C414yescomplemented
112#14519684—22no—
113#14519684—22no—
114#14519684—22no—
115C22×C8×C61C2311yescharacteristic
116C22×C8×C61C2311yescharacteristic
117C2×C4⋊C4×C61—22no—
118C2×C4⋊C4×C61C2311yescharacteristic
119C2×C4⋊C4×C61C2311yescharacteristic
120C2×C4⋊C4×C61C2311yescharacteristic
121C2×C4⋊C4×C61C2311yescharacteristic
122C2×C42×C61C2311yescharacteristic
123#107×C61—22no—
124#107×C61—22no—
125#106×C61—22no—
126#106×C61—22no—
127#97×C61C2×C414yescomplemented
128#95×C61—22no—
129C23×D61C2×D411yescharacteristic
130C22×C4×C61—21no—
131C22×C4×C61—21no—
132C22×C4×C61—21no—
133C22×C4×C61—21no—
134C22×C4×C61C2×D412yes—
135C22×C4×C61C2411yescharacteristic
136C22×Dic61—21no—
137C22×Dic61—21no—
138C22×Dic61—21no—
139C22×Dic61—21no—
140C22×Dic61C2×D412yes—
141C22×Dic61C2×D411yescharacteristic
142C2×C4×D61—42no—
143C2×C4×D61—42no—
144C2×C4×D61—82no—
145C2×C4×D61—82no—
146C2×C4×D61C22×C412yes—
147C2×C4×D61C2×D412yes—
148C2×C4×D61C2×D412yes—
149C2×C4×D61C2×Q812yes—
150C2×C4×D61C2×Q812yes—
151C2×C4×D61C2×Q812yes—
152C2×C8×C61C22×C412yes—
153C2×C8×C61C22×C412yes—
154C2×C8×C61C2×D412yes—
155C2×C8×C61C2×D412yes—
156C2×C8×C61C2×Q812yes—
157C2×C8×C61C2×Q812yes—
158C4⋊C4×C61—21no—
159C4⋊C4×C61—21no—
160C4⋊C4×C61—21no—
161C4⋊C4×C61—21no—
162C4⋊C4×C61—21no—
163C4⋊C4×C61—21no—
164C4⋊C4×C61—21no—
165C4⋊C4×C61—21no—
166C4⋊C4×C61—82no—
167C42×C61C22×C412yes—
168C42×C61C2×Q812yes—
169#11758198—42no—
170#11758198—42no—
171#11758198C4⋊C418yes—
172C4⋊Dic61—21no—
173C4⋊Dic61—21no—
174C4⋊Dic61—21no—
175C4⋊Dic61—21no—
176C4⋊Dic61—21no—
177C4⋊Dic61—21no—
178C4⋊Dic61—21no—
179C4⋊Dic61—21no—
180C4⋊Dic61—82no—
181#11758196—42no—
182#11758196—42no—
183#11758196—82no—
184#11758196—82no—
185#11758196C4⋊C418yes—
186C4×Dic61—42no—
187C4×Dic61—42no—
188C4×Dic61C22×C412yes—
189C4×Dic61C2×Q812yes—
190C2×#92418C22×C412yes—
191C2×#92418C22×C412yes—
192C2×#92418C2×D412yes—
193C2×#92418C2×D412yes—
194C2×#92418C2×Q812yes—
195C2×#92418C2×Q812yes—
196#11758189C4⋊C4116yescomplemented
197C8×D61C4⋊C4116yescomplemented
198C23×C61C22×D411yescharacteristic
199C22×D61—22no—
200C22×D61—22no—
201C22×D61C2×C4⋊C412yes—
202C22×D61#12712yes—
203C22×D61C4⋊Q812yes—
204C2×C4×C61—22no—
205C2×C4×C61—22no—
206C2×C4×C61—41no—
207C2×C4×C61—41no—
208C2×C4×C61—41no—
209C2×C4×C61—41no—
210C2×C4×C61—42no—
211C2×C4×C61—42no—
212C2×C4×C61—42no—
213C2×C4×C61—42no—
214C2×C4×C61C2×C4⋊C414yes—
215C2×C4×C61C23×C411yescharacteristic
216C2×C4×C61C22×D411yescharacteristic
217C2×C4×C61C22×D411yescharacteristic
218C2×C4×C61C22×Q811yescharacteristic
219C2×C4×C61C22×Q811yescharacteristic
220C2×C4×C61C22×Q811yescharacteristic
221C2×Dic61—22no—
222C2×Dic61—22no—
223C2×Dic61—41no—
224C2×Dic61—41no—
225C2×Dic61—41no—
226C2×Dic61—41no—
227C2×Dic61—42no—
228C2×Dic61—42no—
229C2×Dic61—42no—
230C2×Dic61—42no—
231C2×Dic61C2×C4⋊C414yes—
232C2×Dic61C2×C4⋊C412yes—
233C2×Dic61#12712yes—
234C2×Dic61C4⋊Q812yes—
235C4×D61—84no—
236C4×D61—84no—
237C4×D61C2×C4⋊C414yes—
238C4×D61C2×C4⋊C414yes—
239C4×D61C4⋊Q814yes—
240C4×D61C4⋊Q814yes—
241C8×C61C2×C4⋊C414yes—
242C8×C61C2×C4⋊C414yes—
243#92418C2×C4⋊C414yes—
244#92418C2×C4⋊C414yes—
245#20803—611nomaximal, Sylow
246C22×C61C22×C4⋊C411yescharacteristic
247C22×C61C2×#12711yescharacteristic
248C22×C61C2×C4⋊Q811yescharacteristic
249C22×C61#57612yes—
250C22×C61#57812yes—
251C2×D61—22no—
252C2×D61—22no—
253C2×D61—24no—
254C2×D61—24no—
255C2×D61#38414yes—
256C4×C61—24no—
257C4×C61—81no—
258C4×C61—81no—
259C4×C61—81no—
260C4×C61—81no—
261C4×C61C22×C4⋊C411yescharacteristic
262C4×C61C22×C4⋊C411yescharacteristic
263C4×C61C2×C4⋊Q811yescharacteristic
264C4×C61C2×C4⋊Q811yescharacteristic
265Dic61—24no—
266Dic61—81no—
267Dic61—81no—
268Dic61—81no—
269Dic61—81no—
270Dic61#38414yes—
271C2×#428—611no—
272C2×#428—611no—
273C2×#427—611no—
274C22×#107—611no—
275C2×#405—611no—
276#2317—611no—
277#2317—611no—
278#1847—611no—
279#1847—611no—
280#1847—611no—
281#1847D6111yescharacteristic, complemented
282#1846—611no—
283#1846—611no—
284#1845—611no—
285#1845—611no—
286C2×C61C2×#38411yescharacteristic
287C2×C61#291312yes—
288C2×C61#315511yescharacteristic
289C2×C61#315611yescharacteristic
290C2×C61#316412yes—
291D61—24no—
292C22×M16—611no—
293C23×C8—611no—
294C2×#117—611no—
295C2×#117—611no—
296C2×#117—611no—
297C22×C4⋊C4—611no—
298C22×C4⋊C4—611no—
299#428—1224no—
300#428—1224no—
301#427—1224no—
302C2×#107—611no—
303C2×#107—611no—
304C2×#107—611no—
305C2×#107—611no—
306C2×#107—611no—
307C2×#107—611no—
308C2×#107—611no—
309C2×#107—611no—
310C2×#107—1224no—
311C2×#107C2×D6111yescharacteristic
312C2×#107C2×D6111yescharacteristic
313C2×#106—611no—
314C2×#106—611no—
315C2×#106—611no—
316C2×#106—611no—
317C2×#106C2×D6111yescharacteristic
318C2×#106C2×D6111yescharacteristic
319#405—618no—
320C2×#97—611no—
321C2×#97C2×D6111yescharacteristic
322C2×C4×C8—611no—
323C2×C4×C8—611no—
324#389—1222no—
325#389—1222no—
326#385—1222no—
327#385—1222no—
328#384—611no—
329#384—611no—
330#384—611no—
331#384C2×D6111yescharacteristic
332#382—611no—
333#382—611no—
334#382—611no—
335#382C2×D6111yescharacteristic
336C61#2080311yescharacteristic, complemented, Sylow
337C23×C4—611no—
338C23×C4—1221no—
339C23×C4—1221no—
340C2×M16—614no—
341C2×M16—614no—
342C2×M16—614no—
343C22×C8—611no—
344C22×C8—611no—
345C22×C8—614no—
346C22×C8—614no—
347C22×C8—614no—
348C22×C8C22×D6111yescharacteristic
349C22×C8C22×D6111yescharacteristic
350#117—614no—
351#117—614no—
352#117—1222no—
353#117—1222no—
354#117—1222no—
355#117—1222no—
356C2×C4⋊C4—22no—
357C2×C4⋊C4—612no—
358C2×C4⋊C4—611no—
359C2×C4⋊C4—611no—
360C2×C4⋊C4—611no—
361C2×C4⋊C4—611no—
362C2×C4⋊C4—1222no—
363C2×C4⋊C4—1222no—
364C2×C4⋊C4—1222no—
365C2×C4⋊C4—1222no—
366C2×C4⋊C4—1222no—
367C2×C4⋊C4—1222no—
368C2×C4⋊C4—1222no—
369C2×C4⋊C4—1221no—
370C2×C4⋊C4—1221no—
371C2×C4⋊C4C22×D6111yescharacteristic
372C2×C4⋊C4C22×D6111yescharacteristic
373C2×C4⋊C4C22×D6111yescharacteristic
374C2×C4⋊C4C22×D6111yescharacteristic
375C2×C22⋊C4—612no—
376C2×C22⋊C4—1221no—
377C2×C22⋊C4—1221no—
378C2×C42—611no—
379C2×C42—1221no—
380C2×C42—1221no—
381C2×C42C22×D6111yescharacteristic
382#107—22no—
383#107—22no—
384#107—1222no—
385#107—1222no—
386#107—1222no—
387#107—1222no—
388#107—1222no—
389#107—1222no—
390#107—1222no—
391#107—1222no—
392#106—22no—
393#106—22no—
394#106—1222no—
395#106—1222no—
396#106—1222no—
397#106—1222no—
398#97—614no—
399#97C4×D6114yescomplemented
400C4×C8—614no—
401C4×C8—614no—
402#95—22no—
403#95—1222no—
404#95—1222no—
405#95—1222no—
406C24—611no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—611no—
412C22×C4—612no—
413C22×C4—612no—
414C22×C4—612no—
415C22×C4—612no—
416C22×C4—612no—
417C22×C4—612no—
418C22×C4—612no—
419C22×C4—1221no—
420C22×C4—1221no—
421C22×C4—1221no—
422C22×C4—1221no—
423C22×C4—4882no—
424C22×C4—4882no—
425C22×C4—2442no—
426C22×C4—2442no—
427C22×C4D4×D6112yes—
428C22×C4C23×D6111yescharacteristic
429M16—6116no—
430C2×C8—612no—
431C2×C8—612no—
432C2×C8—612no—
433C2×C8—612no—
434C2×C8—612no—
435C2×C8—612no—
436C2×C8—6116no—
437C2×C8C2×C4×D6112yes—
438C2×C8C2×C4×D6112yes—
439C2×C8D4×D6112yes—
440C2×C8D4×D6112yes—
441C2×C8Q8×D6112yes—
442C2×C8Q8×D6112yes—
443C4⋊C4—21no—
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—21no—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—82no—
452C4⋊C4—618no—
453C4⋊C4—1221no—
454C4⋊C4—1221no—
455C4⋊C4—1221no—
456C4⋊C4—1221no—
457C4⋊C4—1221no—
458C4⋊C4—1221no—
459C4⋊C4—1221no—
460C4⋊C4—1221no—
461C4⋊C4—4882no—
462C4⋊C4—4882no—
463C4⋊C4—4882no—
464C4⋊C4—2442no—
465C4⋊C4—2442no—
466C22⋊C4—618no—
467C22⋊C4—2442no—
468C22⋊C4—2442no—
469C42—612no—
470C42—612no—
471C42—2442no—
472C42—2442no—
473C42C2×C4×D6112yes—
474C42Q8×D6112yes—
475C23—612no—
476C23—612no—
477C23—612no—
478C23—1222no—
479C23—1222no—
480C23C2×D4×D6111yescharacteristic
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—612no—
492C2×C4—612no—
493C2×C4—612no—
494C2×C4—614no—
495C2×C4—614no—
496C2×C4—614no—
497C2×C4—614no—
498C2×C4—614no—
499C2×C4—1222no—
500C2×C4—1222no—
501C2×C4—4884no—
502C2×C4—4884no—
503C2×C4—2441no—
504C2×C4—2441no—
505C2×C4—2441no—
506C2×C4—2441no—
507C2×C4—2442no—
508C2×C4—2442no—
509C2×C4—2442no—
510C2×C4—2442no—
511C2×C4C4⋊C4×D6114yes—
512C2×C4C22×C4×D6111yescharacteristic
513C2×C4C2×D4×D6111yescharacteristic
514C2×C4C2×D4×D6111yescharacteristic
515C2×C4C2×Q8×D6111yescharacteristic
516C2×C4C2×Q8×D6111yescharacteristic
517C2×C4C2×Q8×D6111yescharacteristic
518C8—614no—
519C8—614no—
520C8C4⋊C4×D6114yes—
521C8C4⋊C4×D6114yes—
522C22—614no—
523C22—1224no—
524C22—1222no—
525C22—1222no—
526C22—1224no—
527C22C2×C4⋊C4×D6111yescharacteristic
528C22#127×D6111yescharacteristic
529C22C4⋊Q8×D6111yescharacteristic
530C22#1684056612yes—
531C22#1684057812yes—
532C4—24no—
533C4—81no—
534C4—81no—
535C4—81no—
536C4—81no—
537C4—614no—
538C4—1224no—
539C4—4881no—
540C4—4881no—
541C4—4881no—
542C4—4881no—
543C4C2×C4⋊C4×D6111yescharacteristic
544C4C2×C4⋊C4×D6111yescharacteristic
545C4C4⋊Q8×D6111yescharacteristic
546C4C4⋊Q8×D6111yescharacteristic
547C2—1224no—
548C2#384×D6111yescharacteristic
549C2#2145441812yes—
550C2#2145630311yescharacteristic
551C2#2145631111yescharacteristic
552C2#2145635412yes—
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1847C2
socleC23×C61C22×D4
prime radicalC23×D61C2×D4
cosocleC22×C4×C61C24
hypercenter#1847D61
hypocenterC61#20803
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1847

Complements

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

NH ≅ G/Ntype
8. #21445987C2semidirect
9. #21445986C2semidirect
10. #21445984C2semidirect
11. #21445981C2semidirect
12. #21445980C2semidirect
13. #21445978C2semidirect
14. #21445977C2semidirect
15. C61×#1847C2semidirect
39. #16839719C4semidirect
109. #14519687C2×C4semidirect
110. C4×#92418C2×C4semidirect
111. C8×Dic61C2×C4semidirect
127. #97×C61C2×C4semidirect
196. #11758189C4⋊C4semidirect
197. C8×D61C4⋊C4semidirect
281. #1847D61semidirect
336. C61#20803semidirect
399. #97C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168398032
C2×#168398012
C2×#168397972
C2×#107×D612
C2×#168397192
#214484442
#214484402
#214459872
#214459862
#214459842
#214459812
#214459802
#214459782
#214459772
C61×#18472
#2080361