Finite Group Databasebeta

← #29679608

Subgroups — #29679608

Order 15616.

1438 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14519783C211yesmaximal, characteristic
2C2×#16839799C211yesmaximal, characteristic, complemented
3C2×#107×D61C211yesmaximal, characteristic, complemented
4#21448408C212yesmaximal
5#21445851C212yesmaximal, complemented
6#21445850C212yesmaximal, complemented
7#21445848C211yesmaximal, characteristic, complemented
8#21445845C211yesmaximal, characteristic, complemented
9#21445845C211yesmaximal, characteristic, complemented
10#21445537C212yesmaximal, complemented
11C61×#1808C211yesmaximal, 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#16839799—24no—
24C2×#14519711C2211yescharacteristic
25C2×#14519711C2211yescharacteristic
26C2×#14519710C2212yes—
27#107×D61—24no—
28C2×#14519708C2212yes—
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×#107×C61C2211yescharacteristic
41C2×#107×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#14519711—22no—
88#14519711—22no—
89#14519710—24no—
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#107×C61—22no—
104#107×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—21no—
190C23×C61—21no—
191C23×C61—21no—
192C23×C61—21no—
193C23×C61C2×C4⋊C412yes—
194C23×C61C22×D411yescharacteristic
195C22×D61—21no—
196C22×D61—21no—
197C22×D61—21no—
198C22×D61—21no—
199C22×D61—21no—
200C22×D61—21no—
201C22×D61—21no—
202C22×D61—21no—
203C22×D61—21no—
204C22×D61—21no—
205C22×D61C2×C4⋊C412yes—
206C22×D61C2×C4⋊C412yes—
207C22×D61#12112yes—
208C22×D61#12112yes—
209C2×C4×C61—21no—
210C2×C4×C61—21no—
211C2×C4×C61—22no—
212C2×C4×C61—22no—
213C2×C4×C61—22no—
214C2×C4×C61—22no—
215C2×C4×C61—42no—
216C2×C4×C61—42no—
217C2×C4×C61—42no—
218C2×C4×C61—42no—
219C2×C4×C61—42no—
220C2×C4×C61—42no—
221C2×C4×C61C2×C4⋊C412yes—
222C2×C4×C61C23×C411yescharacteristic
223C2×C4×C61C22×D411yescharacteristic
224C2×C4×C61C22×D411yescharacteristic
225C2×C4×C61C22×Q811yescharacteristic
226C2×C4×C61C2×#4111yescharacteristic
227C2×C4×C61C2×#4111yescharacteristic
228C22⋊D61—28no—
229C22⋊D61—28no—
230C2×Dic61—21no—
231C2×Dic61—21no—
232C2×Dic61—42no—
233C2×Dic61—42no—
234C2×Dic61—42no—
235C2×Dic61—42no—
236C2×Dic61—42no—
237C2×Dic61—42no—
238C2×Dic61C2×C4⋊C412yes—
239C2×Dic61C2×C4⋊C412yes—
240C2×Dic61#12112yes—
241C2×Dic61#12112yes—
242D244—28no—
243C4×D61—22no—
244C4×D61—22no—
245C4×D61—22no—
246C4×D61—22no—
247C4×D61—88no—
248C4×D61C2×C22⋊C414yes—
249C4×D61#11714yes—
250C4×D61C22⋊Q814yes—
251C4×D61C22⋊Q814yes—
252Dic122—28no—
253C8×C61—22no—
254C8×C61C2×C22⋊C414yes—
255#92418—22no—
256#92418C2×C22⋊C414yes—
257#20663—611nomaximal, Sylow
258C22×C61—22no—
259C22×C61—22no—
260C22×C61—22no—
261C22×C61—41no—
262C22×C61—41no—
263C22×C61—41no—
264C22×C61—41no—
265C22×C61C22×C4⋊C411yescharacteristic
266C22×C61C2×#12111yescharacteristic
267C22×C61C2×#12111yescharacteristic
268C22×C61#57611yescharacteristic
269C22×C61#57611yescharacteristic
270C22×C61#57811yescharacteristic
271C22×C61#57811yescharacteristic
272C2×D61—22no—
273C2×D61—22no—
274C2×D61—22no—
275C2×D61—22no—
276C2×D61—22no—
277C2×D61—22no—
278C2×D61—22no—
279C2×D61—22no—
280C2×D61—22no—
281C2×D61—41no—
282C2×D61—41no—
283C2×D61—41no—
284C2×D61—41no—
285C2×D61#38014yes—
286C4×C61—22no—
287C4×C61—82no—
288C4×C61—82no—
289C4×C61C22×C22⋊C411yescharacteristic
290C4×C61C2×#11711yescharacteristic
291C4×C61C2×C22⋊Q811yescharacteristic
292C4×C61C2×C22⋊Q811yescharacteristic
293Dic61—22no—
294Dic61—82no—
295Dic61—82no—
296Dic61#38014yes—
297C22×#131—611no—
298C2×#427—611no—
299C22×#107—611no—
300#2311—612no—
301#1809—612no—
302#1809—612no—
303#1808—611no—
304#1808—611no—
305#1808—611no—
306#1808D6111yescharacteristic, complemented
307#1736—612no—
308C2×C61—42no—
309C2×C61C2×#38011yescharacteristic
310C2×C61#291311yescharacteristic
311C2×C61#291311yescharacteristic
312C2×C61#305811yescharacteristic
313C2×C61#305911yescharacteristic
314C2×C61#306111yescharacteristic
315C2×C61#306111yescharacteristic
316D61—24no—
317D61—42no—
318C22×#41—611no—
319C2×#131—618no—
320C2×#131—1224no—
321C2×#131—1224no—
322C22×M16—611no—
323C22×M16—611no—
324C22×M16—611no—
325C23×C8—611no—
326C23×C8—611no—
327C23×C8C2×D6111yescharacteristic
328C2×#117—612no—
329C22×C4⋊C4—612no—
330#427—1224no—
331C2×#107—611no—
332C2×#107—611no—
333C2×#107—612no—
334C2×#107—1224no—
335C2×#107C2×D6111yescharacteristic
336C2×#107C2×D6111yescharacteristic
337C2×#106—612no—
338#391—1222no—
339#390—1222no—
340#386—1222no—
341#386—1222no—
342#380—612no—
343#380—612no—
344#380—612no—
345#380C2×D6112yes—
346#340—612no—
347#340—612no—
348#340—612no—
349#340C2×D6112yes—
350C61#2066311yescharacteristic, complemented, Sylow
351C2×#41—614no—
352C2×#41—614no—
353C2×#41—1222no—
354C2×#41—1222no—
355C2×#41—1222no—
356C2×#41—1222no—
357C22×Q8—611no—
358C22×D4—611no—
359C22×D4—611no—
360C22×D4—611no—
361C23×C4—611no—
362C23×C4—611no—
363C23×C4—1222no—
364C23×C4C22×D6111yescharacteristic
365#131—12232no—
366C2×M16—614no—
367C2×M16—614no—
368C2×M16—614no—
369C2×M16—1222no—
370C2×M16—1222no—
371C2×M16—1222no—
372C2×M16—1222no—
373C2×M16—1222no—
374C2×M16—1222no—
375C22×C8—22no—
376C22×C8—22no—
377C22×C8—611no—
378C22×C8—614no—
379C22×C8—614no—
380C22×C8—614no—
381C22×C8—611no—
382C22×C8—614no—
383C22×C8—1222no—
384C22×C8—1222no—
385C22×C8C4×D6114yescomplemented
386C22×C8C22×D6111yescharacteristic
387C22×C8C22×D6111yescharacteristic
388#117—1224no—
389#117—1224no—
390C2×C4⋊C4—612no—
391C2×C4⋊C4—612no—
392C2×C4⋊C4—1224no—
393C2×C4⋊C4—1224no—
394C2×C4⋊C4—1222no—
395C2×C4⋊C4C22×D6112yes—
396C2×C4⋊C4C22×D6112yes—
397C2×C22⋊C4—22no—
398C2×C22⋊C4—1222no—
399C2×C22⋊C4—1222no—
400C2×C22⋊C4—1222no—
401C2×C22⋊C4—1222no—
402C2×C42—1222no—
403#107—22no—
404#107—22no—
405#107—1224no—
406#107—1222no—
407#107—1222no—
408#106—1224no—
409#95—22no—
410#95—1222no—
411#95—1222no—
412#95—1222no—
413C24—611no—
414C24—611no—
415C24D4×D6111yescharacteristic
416#41—1228no—
417#41—1228no—
418#41—1228no—
419#41—1228no—
420C2×Q8—614no—
421C2×Q8—1222no—
422C2×Q8—1222no—
423C2×D4—614no—
424C2×D4—614no—
425C2×D4—614no—
426C2×D4—1222no—
427C2×D4—1222no—
428C2×D4—1222no—
429C2×D4—1222no—
430C2×D4—1222no—
431C2×D4—1222no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—611no—
439C22×C4—612no—
440C22×C4—612no—
441C22×C4—612no—
442C22×C4—612no—
443C22×C4—612no—
444C22×C4—612no—
445C22×C4—611no—
446C22×C4—612no—
447C22×C4—612no—
448C22×C4—1221no—
449C22×C4—1221no—
450C22×C4—1221no—
451C22×C4—1221no—
452C22×C4—1222no—
453C22×C4—1222no—
454C22×C4—4882no—
455C22×C4—4882no—
456C22×C4—2444no—
457C22×C4C2×C4×D6112yes—
458C22×C4D4×D6111yescharacteristic
459C22×C4Q8×D6112yes—
460C22×C4C23×D6111yescharacteristic
461M16—1228no—
462M16—1228no—
463M16—1228no—
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—28no—
467C2×C8—612no—
468C2×C8—612no—
469C2×C8—612no—
470C2×C8—6116no—
471C2×C8—612no—
472C2×C8—1221no—
473C2×C8—1221no—
474C2×C8—1228no—
475C2×C8C2×C4×D6112yes—
476C2×C8C2×C4×D6112yes—
477C2×C8D4×D6112yes—
478C2×C8D4×D6112yes—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—1222no—
484C4⋊C4—1222no—
485C4⋊C4—1222no—
486C4⋊C4—1222no—
487C4⋊C4—2444no—
488C22⋊C4—82no—
489C22⋊C4—4882no—
490C22⋊C4—4882no—
491C22⋊C4—4882no—
492C22⋊C4—2444no—
493C42—2444no—
494C23—21no—
495C23—21no—
496C23—21no—
497C23—21no—
498C23—21no—
499C23—21no—
500C23—612no—
501C23—612no—
502C23—612no—
503C23—612no—
504C23—1221no—
505C23—1221no—
506C23—1221no—
507C23—1221no—
508C23—1221no—
509C23—1221no—
510C23—1221no—
511C23—1221no—
512C23—1221no—
513C23—1221no—
514C23C4⋊C4×D6112yes—
515C23C2×D4×D6111yescharacteristic
516Q8—1228no—
517D4—1228no—
518D4—1228no—
519D4—1228no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—612no—
533C2×C4—612no—
534C2×C4—612no—
535C2×C4—614no—
536C2×C4—614no—
537C2×C4—614no—
538C2×C4—614no—
539C2×C4—612no—
540C2×C4—1221no—
541C2×C4—1221no—
542C2×C4—1222no—
543C2×C4—1222no—
544C2×C4—1222no—
545C2×C4—1222no—
546C2×C4—4888no—
547C2×C4—2442no—
548C2×C4—2442no—
549C2×C4—2442no—
550C2×C4—2442no—
551C2×C4—2442no—
552C2×C4—2442no—
553C2×C4C4⋊C4×D6112yes—
554C2×C4C22×C4×D6111yescharacteristic
555C2×C4C2×D4×D6111yescharacteristic
556C2×C4C2×D4×D6111yescharacteristic
557C2×C4C2×Q8×D6111yescharacteristic
558C2×C4#41×D6111yescharacteristic
559C2×C4#41×D6111yescharacteristic
560C8—22no—
561C8—614no—
562C8—1222no—
563C8C22⋊C4×D6114yes—
564C22—22no—
565C22—22no—
566C22—22no—
567C22—41no—
568C22—41no—
569C22—41no—
570C22—41no—
571C22—614no—
572C22—1222no—
573C22—1222no—
574C22—1222no—
575C22—1222no—
576C22—1222no—
577C22—1222no—
578C22—1222no—
579C22—1222no—
580C22—1222no—
581C22—2441no—
582C22—2441no—
583C22—2441no—
584C22—2441no—
585C22C2×C4⋊C4×D6111yescharacteristic
586C22#121×D6111yescharacteristic
587C22#121×D6111yescharacteristic
588C22#1684056311yescharacteristic
589C22#1684056411yescharacteristic
590C22#1684057511yescharacteristic
591C22#1684057611yescharacteristic
592C4—22no—
593C4—82no—
594C4—82no—
595C4—614no—
596C4—1222no—
597C4—4882no—
598C4—4882no—
599C4C2×C22⋊C4×D6111yescharacteristic
600C4#117×D6111yescharacteristic
601C4C22⋊Q8×D6111yescharacteristic
602C4C22⋊Q8×D6111yescharacteristic
603C2—42no—
604C2—1224no—
605C2—2442no—
606C2#380×D6111yescharacteristic
607C2#2145441511yescharacteristic
608C2#2145441611yescharacteristic
609C2#2145546211yescharacteristic
610C2#2145546911yescharacteristic
611C2#2145548411yescharacteristic
612C2#2145549011yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1808C2
socleC23×C61C22×D4
prime radicalC22×#11758222C22
cosocleC22×C4×C61C24
hypercenter#1808D61
hypocenterC61#20663
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1808

Complements

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

NH ≅ G/Ntype
2. C2×#16839799C2semidirect
3. C2×#107×D61C2semidirect
5. #21445851C2semidirect
6. #21445850C2semidirect
7. #21445848C2semidirect
8. #21445845C2semidirect
9. #21445845C2semidirect
10. #21445537C2semidirect
11. C61×#1808C2semidirect
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
306. #1808D61semidirect
350. C61#20663semidirect
385. 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×#168397992
C2×#107×D612
#214484082
#214458512
#214458502
#214458482
#214458452
#214455372
C61×#18082
#2066361