Finite Group Databasebeta

← #29679601

Subgroups — #29679601

Order 15616.

1438 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14519783C211yesmaximal, characteristic
2C2×#16839797C211yesmaximal, characteristic, complemented
3C2×#106×D61C211yesmaximal, characteristic, complemented
4#21448408C212yesmaximal
5#21445851C212yesmaximal, complemented
6#21445850C212yesmaximal, complemented
7#21445844C211yesmaximal, characteristic, complemented
8#21445841C211yesmaximal, characteristic, complemented
9#21445841C211yesmaximal, characteristic, complemented
10#21445537C212yesmaximal, complemented
11C61×#1807C211yesmaximal, characteristic, complemented
12C22×#11758222C2211yescharacteristic
13C2×#14519783—24no—
14C2×#14519783—24no—
15C2×#14519783C418yescomplemented
16C22×#11758194C2211yescharacteristic
17C22×#11758189C2211yescharacteristic
18C22×#11758189C2211yescharacteristic
19C22×C8×D61C2211yescharacteristic
20C22×C8×D61C2211yescharacteristic
21C2×#14519735C2212yes—
22C2×C4⋊C4×D61C2212yes—
23#16839797—24no—
24C2×#14519710C2212yes—
25C2×#14519709C2211yescharacteristic
26C2×#14519709C2211yescharacteristic
27C2×#14519708C2212yes—
28#106×D61—24no—
29#16839666—22no—
30#16839662—22no—
31#16839640—22no—
32#16839638—22no—
33#16839622C2212yes—
34#16839619C2212yes—
35#16839619C2212yes—
36#16839525C2212yes—
37#16839524C2212yes—
38#16839524C2212yes—
39C23×C8×C61C2211yescharacteristic
40C2×#106×C61C2211yescharacteristic
41C2×#106×C61C2211yescharacteristic
42C61×#380C2212yes—
43C61×#340C2212yes—
44C2×#11758222—22no—
45C2×#11758222—22no—
46C2×#11758222—22no—
47C2×#11758222—22no—
48C2×#11758222C2×C414yes—
49C2×#11758222Q814yes—
50C22×Dic122D411yescharacteristic
51C22×D244D411yescharacteristic
52C22×C22⋊D61D411yescharacteristic
53C22×C22⋊D61D411yescharacteristic
54C22×C4×D61—22no—
55C22×C4×D61C2311yescharacteristic
56C22×C4×D61C2311yescharacteristic
57#14519783—232no—
58C2×#11758194—22no—
59C2×#11758194—22no—
60C2×#11758194C2×C414yescomplemented
61C2×#11758189—22no—
62C2×#11758189—22no—
63C2×#11758189—22no—
64C2×#11758189—22no—
65C2×#11758189C2×C414yes—
66C2×#11758189C2×C414yescomplemented
67C22×#92418C2311yescharacteristic
68C22×#92418C2311yescharacteristic
69C2×C8×D61—22no—
70C2×C8×D61—22no—
71C2×C8×D61C2×C414yes—
72C2×C8×D61C2×C414yescomplemented
73C2×C8×D61D414yes—
74C2×C8×D61D414yes—
75#14519735—24no—
76#14519735—24no—
77C2×C4⋊Dic61C2312yes—
78C2×C4⋊Dic61C2312yes—
79C2×#11758196—22no—
80C4⋊C4×D61—24no—
81C4⋊C4×D61—24no—
82C2×C22⋊Dic61—22no—
83C2×#11758198—22no—
84C2×#11758198—22no—
85C2×#11758198—22no—
86C2×C4×Dic61—22no—
87#14519710—24no—
88#14519709—22no—
89#14519709—22no—
90#14519708—24no—
91#14519684—22no—
92#14519684—22no—
93#14519684—22no—
94C23×C4×C61C2311yescharacteristic
95C22×C8×C61—22no—
96C22×C8×C61—22no—
97C22×C8×C61C2×C414yescomplemented
98C22×C8×C61C2311yescharacteristic
99C22×C8×C61C2311yescharacteristic
100C2×C4⋊C4×C61C2312yes—
101C2×C4⋊C4×C61C2312yes—
102C2×C22⋊C4×C61—22no—
103#106×C61—22no—
104#106×C61—22no—
105#95×C61—22no—
106C24×C61C2×D411yescharacteristic
107C23×D61C2×D411yescharacteristic
108C23×D61C2×D411yescharacteristic
109C22×C4×C61—21no—
110C22×C4×C61—21no—
111C22×C4×C61—21no—
112C22×C4×C61—21no—
113C22×C4×C61—22no—
114C22×C4×C61—22no—
115C22×C4×C61C22×C412yes—
116C22×C4×C61C2×D411yescharacteristic
117C22×C4×C61C2×Q812yes—
118C22×C4×C61C2411yescharacteristic
119C2×C22⋊D61—22no—
120C2×C22⋊D61—22no—
121C2×C22⋊D61—22no—
122C2×C22⋊D61—22no—
123C2×C22⋊D61C4⋊C414yes—
124C2×C22⋊D61C4⋊C414yes—
125C22×Dic61—22no—
126C22×Dic61—22no—
127C22×Dic61C2×D411yescharacteristic
128C22×Dic61C2×D411yescharacteristic
129#11758222—28no—
130#11758222—28no—
131#11758222—28no—
132#11758222—28no—
133C2×D244—22no—
134C2×D244—22no—
135C2×D244C4⋊C414yes—
136C2×C4×D61—21no—
137C2×C4×D61—21no—
138C2×C4×D61—21no—
139C2×C4×D61—21no—
140C2×C4×D61—44no—
141C2×C4×D61—82no—
142C2×C4×D61—82no—
143C2×C4×D61C22×C412yes—
144C2×C4×D61C22×C412yes—
145C2×C4×D61C2×D412yes—
146C2×C4×D61C2×D412yes—
147C2×C4×D61C2×Q812yes—
148C2×C4×D61C2×Q812yes—
149C2×C4×D61#4112yes—
150C2×C4×D61#4112yes—
151C2×Dic122—22no—
152C2×Dic122—22no—
153C2×Dic122C4⋊C414yes—
154C2×C8×C61—21no—
155C2×C8×C61—21no—
156C2×C8×C61—28no—
157C2×C8×C61C22×C412yes—
158C2×C8×C61C22×C412yes—
159C2×C8×C61C2×D412yes—
160C2×C8×C61C2×D412yes—
161C4⋊C4×C61—22no—
162C4⋊C4×C61—22no—
163C4⋊C4×C61—22no—
164C4⋊C4×C61—22no—
165C22⋊C4×C61—82no—
166C22⋊Dic61—82no—
167#11758198—44no—
168#11758198—82no—
169#11758198—82no—
170C4⋊Dic61—22no—
171C4⋊Dic61—22no—
172C4⋊Dic61—22no—
173C4⋊Dic61—22no—
174#11758196—44no—
175C4×Dic61—44no—
176#11758194—28no—
177C2×#92418—21no—
178C2×#92418—21no—
179C2×#92418C22×C412yes—
180C2×#92418C22×C412yes—
181C2×#92418C2×D412yes—
182C2×#92418C2×D412yes—
183#11758189—28no—
184#11758189—28no—
185C8×D61—28no—
186C8×D61C22⋊C4116yescomplemented
187C23×C61—21no—
188C23×C61—21no—
189C23×C61—22no—
190C23×C61—22no—
191C23×C61C2×C4⋊C412yes—
192C23×C61C22×D411yescharacteristic
193C22×D61—22no—
194C22×D61—22no—
195C22×D61—21no—
196C22×D61—21no—
197C22×D61—22no—
198C22×D61—22no—
199C22×D61C2×C4⋊C412yes—
200C22×D61C2×C4⋊C412yes—
201C22×D61#12112yes—
202C22×D61#12112yes—
203C2×C4×C61—21no—
204C2×C4×C61—21no—
205C2×C4×C61—22no—
206C2×C4×C61—22no—
207C2×C4×C61—22no—
208C2×C4×C61—22no—
209C2×C4×C61—42no—
210C2×C4×C61—42no—
211C2×C4×C61—42no—
212C2×C4×C61—42no—
213C2×C4×C61—42no—
214C2×C4×C61—42no—
215C2×C4×C61C2×C4⋊C412yes—
216C2×C4×C61C23×C411yescharacteristic
217C2×C4×C61C22×D411yescharacteristic
218C2×C4×C61C22×D411yescharacteristic
219C2×C4×C61C22×Q811yescharacteristic
220C2×C4×C61C2×#4111yescharacteristic
221C2×C4×C61C2×#4111yescharacteristic
222C22⋊D61—28no—
223C22⋊D61—28no—
224C2×Dic61—21no—
225C2×Dic61—21no—
226C2×Dic61—42no—
227C2×Dic61—42no—
228C2×Dic61—42no—
229C2×Dic61—42no—
230C2×Dic61—42no—
231C2×Dic61—42no—
232C2×Dic61C2×C4⋊C412yes—
233C2×Dic61C2×C4⋊C412yes—
234C2×Dic61#12112yes—
235C2×Dic61#12112yes—
236D244—28no—
237C4×D61—22no—
238C4×D61—22no—
239C4×D61—22no—
240C4×D61—22no—
241C4×D61—88no—
242C4×D61C2×C22⋊C414yes—
243C4×D61#11714yes—
244C4×D61C22⋊Q814yes—
245C4×D61C22⋊Q814yes—
246Dic122—28no—
247C8×C61—22no—
248C8×C61C2×C22⋊C414yes—
249#92418—22no—
250#92418C2×C22⋊C414yes—
251#20662—611nomaximal, Sylow
252C22×C61—22no—
253C22×C61—22no—
254C22×C61—22no—
255C22×C61—42no—
256C22×C61—42no—
257C22×C61C22×C4⋊C411yescharacteristic
258C22×C61C2×#12111yescharacteristic
259C22×C61C2×#12111yescharacteristic
260C22×C61#57712yes—
261C22×C61#57712yes—
262C2×D61—22no—
263C2×D61—22no—
264C2×D61—24no—
265C2×D61—24no—
266C2×D61—22no—
267C2×D61—22no—
268C2×D61—22no—
269C2×D61—42no—
270C2×D61—42no—
271C2×D61#38014yes—
272C4×C61—22no—
273C4×C61—82no—
274C4×C61—82no—
275C4×C61C22×C22⋊C411yescharacteristic
276C4×C61C2×#11711yescharacteristic
277C4×C61C2×C22⋊Q811yescharacteristic
278C4×C61C2×C22⋊Q811yescharacteristic
279Dic61—22no—
280Dic61—82no—
281Dic61—82no—
282Dic61#38014yes—
283C22×#131—611no—
284C2×#427—611no—
285C22×#106—611no—
286#2311—612no—
287#1809—612no—
288#1809—612no—
289#1807—611no—
290#1807—611no—
291#1807—611no—
292#1807D6111yescharacteristic, complemented
293#1736—612no—
294C2×C61—42no—
295C2×C61C2×#38011yescharacteristic
296C2×C61#291211yescharacteristic
297C2×C61#291211yescharacteristic
298C2×C61#305712yes—
299C2×C61#306012yes—
300D61—24no—
301D61—42no—
302C22×#41—611no—
303C2×#131—618no—
304C2×#131—1224no—
305C2×#131—1224no—
306C22×M16—611no—
307C22×M16—611no—
308C22×M16—611no—
309C23×C8—611no—
310C23×C8—611no—
311C23×C8C2×D6111yescharacteristic
312C2×#117—612no—
313C22×C4⋊C4—612no—
314#427—1224no—
315C2×#107—612no—
316C2×#106—611no—
317C2×#106—611no—
318C2×#106—612no—
319C2×#106—1224no—
320C2×#106C2×D6111yescharacteristic
321C2×#106C2×D6111yescharacteristic
322#391—1222no—
323#390—1222no—
324#386—1222no—
325#386—1222no—
326#380—612no—
327#380—612no—
328#380—612no—
329#380C2×D6112yes—
330#340—612no—
331#340—612no—
332#340—612no—
333#340C2×D6112yes—
334C61#2066211yescharacteristic, complemented, Sylow
335C2×#41—614no—
336C2×#41—614no—
337C2×#41—1222no—
338C2×#41—1222no—
339C2×#41—1222no—
340C2×#41—1222no—
341C22×Q8—611no—
342C22×D4—611no—
343C22×D4—611no—
344C22×D4—611no—
345C23×C4—611no—
346C23×C4—611no—
347C23×C4—1222no—
348C23×C4C22×D6111yescharacteristic
349#131—12232no—
350C2×M16—614no—
351C2×M16—614no—
352C2×M16—614no—
353C2×M16—1222no—
354C2×M16—1222no—
355C2×M16—1222no—
356C2×M16—1222no—
357C2×M16—1222no—
358C2×M16—1222no—
359C22×C8—22no—
360C22×C8—22no—
361C22×C8—611no—
362C22×C8—614no—
363C22×C8—614no—
364C22×C8—614no—
365C22×C8—611no—
366C22×C8—614no—
367C22×C8—1222no—
368C22×C8—1222no—
369C22×C8C4×D6114yescomplemented
370C22×C8C22×D6111yescharacteristic
371C22×C8C22×D6111yescharacteristic
372#117—1224no—
373#117—1224no—
374C2×C4⋊C4—612no—
375C2×C4⋊C4—612no—
376C2×C4⋊C4—1224no—
377C2×C4⋊C4—1224no—
378C2×C4⋊C4—1222no—
379C2×C4⋊C4C22×D6112yes—
380C2×C4⋊C4C22×D6112yes—
381C2×C22⋊C4—22no—
382C2×C22⋊C4—1222no—
383C2×C22⋊C4—1222no—
384C2×C22⋊C4—1222no—
385C2×C22⋊C4—1222no—
386C2×C42—1222no—
387#107—1224no—
388#106—22no—
389#106—22no—
390#106—1224no—
391#106—1222no—
392#106—1222no—
393#95—22no—
394#95—1222no—
395#95—1222no—
396#95—1222no—
397C24—611no—
398C24—611no—
399C24D4×D6111yescharacteristic
400#41—1228no—
401#41—1228no—
402#41—1228no—
403#41—1228no—
404C2×Q8—614no—
405C2×Q8—1222no—
406C2×Q8—1222no—
407C2×D4—614no—
408C2×D4—614no—
409C2×D4—614no—
410C2×D4—1222no—
411C2×D4—1222no—
412C2×D4—1222no—
413C2×D4—1222no—
414C2×D4—1222no—
415C2×D4—1222no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—611no—
423C22×C4—612no—
424C22×C4—612no—
425C22×C4—612no—
426C22×C4—612no—
427C22×C4—612no—
428C22×C4—612no—
429C22×C4—611no—
430C22×C4—612no—
431C22×C4—612no—
432C22×C4—1221no—
433C22×C4—1221no—
434C22×C4—1221no—
435C22×C4—1221no—
436C22×C4—1222no—
437C22×C4—1222no—
438C22×C4—4882no—
439C22×C4—4882no—
440C22×C4—2444no—
441C22×C4C2×C4×D6112yes—
442C22×C4D4×D6111yescharacteristic
443C22×C4Q8×D6112yes—
444C22×C4C23×D6111yescharacteristic
445M16—1228no—
446M16—1228no—
447M16—1228no—
448C2×C8—21no—
449C2×C8—21no—
450C2×C8—28no—
451C2×C8—612no—
452C2×C8—612no—
453C2×C8—612no—
454C2×C8—6116no—
455C2×C8—612no—
456C2×C8—1221no—
457C2×C8—1221no—
458C2×C8—1228no—
459C2×C8C2×C4×D6112yes—
460C2×C8C2×C4×D6112yes—
461C2×C8D4×D6112yes—
462C2×C8D4×D6112yes—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—1222no—
468C4⋊C4—1222no—
469C4⋊C4—1222no—
470C4⋊C4—1222no—
471C4⋊C4—2444no—
472C22⋊C4—82no—
473C22⋊C4—4882no—
474C22⋊C4—4882no—
475C22⋊C4—4882no—
476C22⋊C4—2444no—
477C42—2444no—
478C23—21no—
479C23—21no—
480C23—22no—
481C23—22no—
482C23—612no—
483C23—612no—
484C23—612no—
485C23—612no—
486C23—1222no—
487C23—1222no—
488C23—1221no—
489C23—1221no—
490C23—1222no—
491C23—1222no—
492C23C4⋊C4×D6112yes—
493C23C2×D4×D6111yescharacteristic
494Q8—1228no—
495D4—1228no—
496D4—1228no—
497D4—1228no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—612no—
511C2×C4—612no—
512C2×C4—612no—
513C2×C4—614no—
514C2×C4—614no—
515C2×C4—614no—
516C2×C4—614no—
517C2×C4—612no—
518C2×C4—1221no—
519C2×C4—1221no—
520C2×C4—1222no—
521C2×C4—1222no—
522C2×C4—1222no—
523C2×C4—1222no—
524C2×C4—4888no—
525C2×C4—2442no—
526C2×C4—2442no—
527C2×C4—2442no—
528C2×C4—2442no—
529C2×C4—2442no—
530C2×C4—2442no—
531C2×C4C4⋊C4×D6112yes—
532C2×C4C22×C4×D6111yescharacteristic
533C2×C4C2×D4×D6111yescharacteristic
534C2×C4C2×D4×D6111yescharacteristic
535C2×C4C2×Q8×D6111yescharacteristic
536C2×C4#41×D6111yescharacteristic
537C2×C4#41×D6111yescharacteristic
538C8—22no—
539C8—614no—
540C8—1222no—
541C8C22⋊C4×D6114yes—
542C22—22no—
543C22—22no—
544C22—22no—
545C22—42no—
546C22—42no—
547C22—614no—
548C22—1222no—
549C22—1222no—
550C22—1224no—
551C22—1224no—
552C22—1222no—
553C22—1222no—
554C22—1222no—
555C22—2442no—
556C22—2442no—
557C22C2×C4⋊C4×D6111yescharacteristic
558C22#121×D6111yescharacteristic
559C22#121×D6111yescharacteristic
560C22#1684056812yes—
561C22#1684056912yes—
562C4—22no—
563C4—82no—
564C4—82no—
565C4—614no—
566C4—1222no—
567C4—4882no—
568C4—4882no—
569C4C2×C22⋊C4×D6111yescharacteristic
570C4#117×D6111yescharacteristic
571C4C22⋊Q8×D6111yescharacteristic
572C4C22⋊Q8×D6111yescharacteristic
573C2—42no—
574C2—1224no—
575C2—2442no—
576C2#380×D6111yescharacteristic
577C2#2145441011yescharacteristic
578C2#2145441111yescharacteristic
579C2#2145545312yes—
580C2#2145547612yes—
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1807C2
socleC23×C61C22×D4
prime radicalC22×#11758222C22
cosocleC22×C4×C61C24
hypercenter#1807D61
hypocenterC61#20662
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1807

Complements

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

NH ≅ G/Ntype
2. C2×#16839797C2semidirect
3. C2×#106×D61C2semidirect
5. #21445851C2semidirect
6. #21445850C2semidirect
7. #21445844C2semidirect
8. #21445841C2semidirect
9. #21445841C2semidirect
10. #21445537C2semidirect
11. C61×#1807C2semidirect
15. C2×#14519783C4semidirect
60. C2×#11758194C2×C4semidirect
66. C2×#11758189C2×C4semidirect
72. C2×C8×D61C2×C4semidirect
97. C22×C8×C61C2×C4semidirect
186. C8×D61C22⋊C4semidirect
292. #1807D61semidirect
334. C61#20662semidirect
369. C22×C8C4×D61semidirect

Maximal subgroups

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

subgroupindex
C22×#145197832
C2×#168397972
C2×#106×D612
#214484082
#214458512
#214458502
#214458442
#214458412
#214455372
C61×#18072
#2066261