Finite Group Databasebeta

← #29707365

Subgroups — #29707365

Order 15616.

1250 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840390C211yesmaximal, characteristic, complemented
2C2×#16839764C211yesmaximal, characteristic
3C2×#16839759C211yesmaximal, characteristic
4#21448407C211yesmaximal, characteristic
5#21446971C212yesmaximal, complemented
6#21446966C212yesmaximal, complemented
7#21446961C212yesmaximal, complemented
8#21446956C212yesmaximal, complemented
9#21445539C211yesmaximal, characteristic, complemented
10C2×#16839525C211yesmaximal, characteristic, complemented
11C61×#1729C211yesmaximal, characteristic, complemented
12C22×#11758222C2211yescharacteristic
13#16840390—24no—
14#16840390—24no—
15C2×#14519777C2212yes—
16C2×#14519776—21no—
17C2×#14519769C2212yes—
18C2×#14519768—21no—
19C2×#14519754—22no—
20C2×#14519736C2211yescharacteristic
21C22×C4⋊Dic61C2211yescharacteristic
22C22×C4⋊Dic61C2211yescharacteristic
23#16839764—24no—
24#16839759—24no—
25C2×#14519702C2211yescharacteristic
26C2×#14519702C2212yes—
27C2×#14519702C2211yescharacteristic
28C2×#14519699C2211yescharacteristic
29C2×#14519699C2212yes—
30C2×#14519699C2211yescharacteristic
31#16839668C2212yes—
32#16839664C2212yes—
33#16839639—22no—
34#16839619—21no—
35#16839619—21no—
36#16839525C2212yes—
37#16839525C2212yes—
38#16839525C2212yes—
39#16839525C2212yes—
40C22×M16×C61C2211yescharacteristic
41C23×C8×C61C2211yescharacteristic
42C2×#117×C61C2211yescharacteristic
43C61×#336C2212yes—
44C61×#336C2212yes—
45C2×#11758222—22no—
46C2×#11758222—22no—
47C2×#11758222—22no—
48C2×#11758222—22no—
49C2×#11758222—22no—
50C2×#11758222Q814yescomplemented
51C22×Dic122—22no—
52C22×Dic122C2311yescharacteristic
53C22×D244C2311yescharacteristic
54C22×C22⋊D61—21no—
55C23×Dic61—21no—
56C23×Dic61—21no—
57C22×C4×D61—21no—
58#14519777—24no—
59#14519777—24no—
60#14519776—28no—
61#14519769—24no—
62#14519769—24no—
63#14519768—28no—
64#14519754—216no—
65#14519736—22no—
66#14519736D414yes—
67C2×C4⋊Dic61—21no—
68C2×C4⋊Dic61—22no—
69C2×C4⋊Dic61—22no—
70C2×C4⋊Dic61—22no—
71C2×C4⋊Dic61—22no—
72C2×C4⋊Dic61—21no—
73C2×C4⋊Dic61D414yes—
74C2×C4⋊Dic61C2312yes—
75C2×C4⋊Dic61C2312yes—
76C2×C4⋊Dic61C2311yescharacteristic
77C2×C4⋊Dic61C2311yescharacteristic
78C2×#11758196—22no—
79C2×#11758196—22no—
80C2×C22⋊Dic61—21no—
81C2×#11758198—22no—
82C2×#11758198—22no—
83C2×C4×Dic61—21no—
84#14519702—22no—
85#14519702—22no—
86#14519702—24no—
87#14519699—22no—
88#14519699—22no—
89#14519699—24no—
90#14519684—22no—
91#14519684—22no—
92C23×C4×C61C2311yescharacteristic
93C2×M16×C61—22no—
94C2×M16×C61—22no—
95C2×M16×C61—22no—
96C22×C8×C61—22no—
97C22×C8×C61—22no—
98C22×C8×C61—22no—
99C22×C8×C61C2311yescharacteristic
100C22×C8×C61C2311yescharacteristic
101C22×C8×C61C2312yes—
102#117×C61—22no—
103#117×C61—22no—
104C2×C4⋊C4×C61D411yescharacteristic
105C2×C4⋊C4×C61D411yescharacteristic
106C2×C22⋊C4×C61D412yes—
107C2×C42×C61C2312yes—
108C24×C61C2×D411yescharacteristic
109C23×D61—21no—
110C22×C4×C61C2×D412yes—
111C22×C4×C61C2×D412yes—
112C22×C4×C61C2×D411yescharacteristic
113C22×C4×C61C2×D412yes—
114C22×C4×C61C2×D412yes—
115C22×C4×C61C2×Q812yes—
116C22×C4×C61#4112yes—
117C22×C4×C61#4112yes—
118C22×C4×C61#4112yes—
119C22×C4×C61C2411yescharacteristic
120C2×C22⋊D61—24no—
121C2×C22⋊D61—42no—
122C2×C22⋊D61—42no—
123C22×Dic61—21no—
124C22×Dic61—22no—
125C22×Dic61—21no—
126C22×Dic61—21no—
127C22×Dic61—24no—
128C22×Dic61—22no—
129C22×Dic61—42no—
130C22×Dic61—81no—
131C22×Dic61—81no—
132C22×Dic61—81no—
133C22×Dic61—81no—
134#11758222—28no—
135#11758222—28no—
136#11758222—44no—
137#11758222—44no—
138C2×D244—21no—
139C2×D244—21no—
140C2×D244—21no—
141C2×D244—21no—
142C2×D244—21no—
143C2×D244C2×Q812yes—
144C2×C4×D61—24no—
145C2×C4×D61—42no—
146C2×C4×D61—42no—
147C2×Dic122—21no—
148C2×Dic122—21no—
149C2×Dic122—21no—
150C2×Dic122—21no—
151C2×Dic122—21no—
152C2×Dic122—28no—
153C2×Dic122—82no—
154C2×Dic122—82no—
155C2×Dic122C2×Q812yes—
156M16×C61—44no—
157C2×C8×C61—21no—
158C2×C8×C61—21no—
159C2×C8×C61—22no—
160C2×C8×C61—21no—
161C2×C8×C61—21no—
162C2×C8×C61—21no—
163C2×C8×C61—21no—
164C2×C8×C61—22no—
165C2×C8×C61—22no—
166C2×C8×C61—44no—
167C4⋊C4×C61—22no—
168C4⋊C4×C61—22no—
169C22⋊C4×C61—24no—
170C42×C61—22no—
171C42×C61—22no—
172C22⋊Dic61—24no—
173#11758198—82no—
174#11758198—82no—
175C4⋊Dic61—28no—
176C4⋊Dic61—24no—
177C4⋊Dic61—22no—
178C4⋊Dic61—21no—
179C4⋊Dic61—21no—
180C4⋊Dic61—22no—
181C4⋊Dic61—22no—
182C4⋊Dic61—42no—
183C4⋊Dic61C2×D414yes—
184C4⋊Dic61C2×D412yes—
185C4⋊Dic61C2×D412yes—
186#11758196—28no—
187#11758196—28no—
188C4×Dic61—24no—
189C23×C61—21no—
190C23×C61—21no—
191C23×C61—21no—
192C23×C61—21no—
193C23×C61#12012yes—
194C23×C61C22⋊Q812yes—
195C23×C61#12412yes—
196C23×C61C22×D411yescharacteristic
197C22×D61—22no—
198C22×D61—41no—
199C22×D61—41no—
200C22×D61—41no—
201C22×D61—41no—
202C22×D61—41no—
203C22×D61—41no—
204C2×C4×C61—22no—
205C2×C4×C61—22no—
206C2×C4×C61—22no—
207C2×C4×C61—22no—
208C2×C4×C61—22no—
209C2×C4×C61—22no—
210C2×C4×C61#12012yes—
211C2×C4×C61C22⋊Q814yes—
212C2×C4×C61C22⋊Q814yes—
213C2×C4×C61C22⋊Q812yes—
214C2×C4×C61#12414yes—
215C2×C4×C61#12414yes—
216C2×C4×C61#12412yes—
217C2×C4×C61C22×D411yescharacteristic
218C2×C4×C61C22×D411yescharacteristic
219C2×C4×C61C22×Q811yescharacteristic
220C2×C4×C61C2×#4111yescharacteristic
221C2×C4×C61C2×#4111yescharacteristic
222C2×C4×C61C2×#4111yescharacteristic
223C22⋊D61—48no—
224C2×Dic61—22no—
225C2×Dic61—22no—
226C2×Dic61—24no—
227C2×Dic61—22no—
228C2×Dic61—41no—
229C2×Dic61—41no—
230C2×Dic61—41no—
231C2×Dic61—42no—
232C2×Dic61—41no—
233C2×Dic61—42no—
234C2×Dic61—41no—
235C2×Dic61—41no—
236C2×Dic61—42no—
237C2×Dic61—42no—
238C2×Dic61—42no—
239C2×Dic61—84no—
240C2×Dic61—84no—
241D244—22no—
242D244—22no—
243D244—41no—
244D244—41no—
245C4×D61—48no—
246Dic122—22no—
247Dic122—22no—
248Dic122—41no—
249Dic122—41no—
250Dic122—88no—
251C8×C61—41no—
252C8×C61—41no—
253C8×C61—42no—
254#23458—611nomaximal, Sylow
255C22×C61—22no—
256C22×C61—22no—
257C22×C61—22no—
258C22×C61—22no—
259C22×C61—22no—
260C22×C61—22no—
261C22×C61#39314yes—
262C22×C61C2×#12011yescharacteristic
263C22×C61C2×C22⋊Q811yescharacteristic
264C22×C61C2×#12411yescharacteristic
265C22×C61#57611yescharacteristic
266C22×C61#57711yescharacteristic
267C22×C61#57711yescharacteristic
268C22×C61#57811yescharacteristic
269C2×D61—42no—
270C2×D61—42no—
271C2×D61—42no—
272C2×D61—81no—
273C2×D61—81no—
274C2×D61—81no—
275C2×D61—81no—
276C4×C61—42no—
277C4×C61—42no—
278C4×C61#39314yes—
279C4×C61C2×C22⋊Q811yescharacteristic
280C4×C61C2×C22⋊Q811yescharacteristic
281C4×C61C2×#12411yescharacteristic
282C4×C61C2×#12411yescharacteristic
283Dic61—42no—
284Dic61—42no—
285Dic61—44no—
286Dic61—42no—
287Dic61—82no—
288C2×#544—611no—
289C2×#417—611no—
290C2×#416—611no—
291#2311—611no—
292#2026—612no—
293#2025—612no—
294#2024—612no—
295#2023—612no—
296#1736—611no—
297C2×#340—611no—
298#1729D6111yescharacteristic, complemented
299C2×C61—24no—
300C2×C61C2×#39311yescharacteristic
301C2×C61#301211yescharacteristic
302C2×C61#301311yescharacteristic
303C2×C61#308311yescharacteristic
304C2×C61#308411yescharacteristic
305C2×C61#313611yescharacteristic
306C2×C61#313711yescharacteristic
307D61—82no—
308C22×#41—611no—
309C22×M16C2×D6111yescharacteristic
310C23×C8C2×D6111yescharacteristic
311#544—1224no—
312#544—1224no—
313C2×C4⋊Q8—612no—
314C2×C4⋊Q8—1221no—
315C2×#125—612no—
316C2×#125—1221no—
317C2×C22⋊Q8—1222no—
318C2×#117—611no—
319C2×#117C2×D6111yescharacteristic
320C22×C4⋊C4—611no—
321C22×C4⋊C4—611no—
322#417—1224no—
323#416—1224no—
324C2×#103—611no—
325C2×#103—611no—
326C2×#103—612no—
327C2×#102—611no—
328C2×#102—611no—
329C2×#102—612no—
330#391—612no—
331#390—612no—
332#386—1222no—
333#380—1221no—
334#380—1221no—
335#340—612no—
336#340—612no—
337#340—612no—
338#340—612no—
339#336C2×D6112yes—
340#336C2×D6112yes—
341C61#2345811yescharacteristic, complemented, Sylow
342C2×#41—614no—
343C2×#41—1222no—
344C2×#41—1222no—
345C2×#41—1222no—
346C2×#41—1222no—
347C2×#41—1222no—
348C22×Q8—611no—
349C22×Q8—1222no—
350C22×D4—611no—
351C22×D4—1221no—
352C23×C4—1221no—
353C23×C4—1221no—
354C23×C4—1221no—
355C23×C4C22×D6111yescharacteristic
356C2×M16—22no—
357C2×M16—22no—
358C2×M16—22no—
359C22×C8—22no—
360C22×C8—22no—
361C22×C8—22no—
362C22×C8C22×D6111yescharacteristic
363C22×C8C22×D6111yescharacteristic
364C22×C8C22×D6112yes—
365C4⋊Q8—1224no—
366C4⋊Q8—1224no—
367C4⋊Q8—1228no—
368#125—1224no—
369#125—1224no—
370#125—1228no—
371C22⋊Q8—12216no—
372#117—22no—
373#117—22no—
374#117—614no—
375#117—1222no—
376C2×C4⋊C4—614no—
377C2×C4⋊C4—612no—
378C2×C4⋊C4—611no—
379C2×C4⋊C4—611no—
380C2×C4⋊C4—612no—
381C2×C4⋊C4—1221no—
382C2×C4⋊C4—1222no—
383C2×C4⋊C4—1222no—
384C2×C4⋊C4—1222no—
385C2×C4⋊C4—1222no—
386C2×C4⋊C4—1222no—
387C2×C4⋊C4—1222no—
388C2×C4⋊C4—1221no—
389C2×C4⋊C4D24411yescharacteristic
390C2×C4⋊C4D24411yescharacteristic
391C2×C22⋊C4—1221no—
392C2×C22⋊C4—1222no—
393C2×C22⋊C4—1222no—
394C2×C22⋊C4D24412yes—
395C2×C42—1221no—
396C2×C42C22×D6112yes—
397#103—1222no—
398#103—1222no—
399#103—1224no—
400#102—1222no—
401#102—1222no—
402#102—1224no—
403#95—1222no—
404#95—1222no—
405C24—1221no—
406C24C2×D24411yescharacteristic
407#41—1228no—
408#41—1228no—
409#41—2444no—
410#41—2444no—
411C2×Q8—612no—
412C2×Q8—1228no—
413C2×Q8—1221no—
414C2×Q8—1221no—
415C2×Q8—1221no—
416C2×Q8—1221no—
417C2×Q8—1221no—
418C2×Q8—4882no—
419C2×Q8—4882no—
420C2×D4—612no—
421C2×D4—1224no—
422C2×D4—1221no—
423C2×D4—1221no—
424C2×D4—1221no—
425C2×D4—1221no—
426C2×D4—1221no—
427C2×D4—2442no—
428C2×D4—2442no—
429C22×C4—1224no—
430C22×C4—1221no—
431C22×C4—1221no—
432C22×C4—1221no—
433C22×C4—1222no—
434C22×C4—1224no—
435C22×C4—1222no—
436C22×C4—4881no—
437C22×C4—4881no—
438C22×C4—4881no—
439C22×C4—4881no—
440C22×C4—2442no—
441C22×C4—2442no—
442C22×C4—2442no—
443C22×C4C2×D24412yes—
444C22×C4C2×D24412yes—
445C22×C4C2×D24411yescharacteristic
446C22×C4#1175822212yes—
447C22×C4#1175822212yes—
448C22×C4D4×D6112yes—
449C22×C4D4×D6112yes—
450C22×C4#1175822412yes—
451C22×C4Q8×D6112yes—
452C22×C4C23×D6111yescharacteristic
453M16—44no—
454C2×C8—21no—
455C2×C8—21no—
456C2×C8—22no—
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—22no—
460C2×C8—21no—
461C2×C8—21no—
462C2×C8—22no—
463C2×C8—44no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—614no—
467C4⋊C4—612no—
468C4⋊C4—612no—
469C4⋊C4—1228no—
470C4⋊C4—1228no—
471C4⋊C4—1228no—
472C4⋊C4—1222no—
473C4⋊C4—1221no—
474C4⋊C4—1221no—
475C4⋊C4—1224no—
476C4⋊C4—1222no—
477C4⋊C4—1222no—
478C4⋊C4—2442no—
479C22⋊C4—24no—
480C22⋊C4—1224no—
481C22⋊C4—4882no—
482C22⋊C4—4882no—
483C42—22no—
484C42—22no—
485C42—1224no—
486C23—21no—
487C23—21no—
488C23—21no—
489C23—21no—
490C23—1222no—
491C23—2441no—
492C23—2441no—
493C23—2441no—
494C23—2441no—
495C23—2441no—
496C23—2441no—
497C23#1451974712yes—
498C23#1451975712yes—
499C23#1451976512yes—
500C23C22×D24411yescharacteristic
501Q8—1222no—
502Q8—1222no—
503Q8—4888no—
504Q8—2441no—
505Q8—2441no—
506D4—1222no—
507D4—1222no—
508D4—2448no—
509D4—2441no—
510D4—2441no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—1222no—
518C2×C4—1222no—
519C2×C4—1224no—
520C2×C4—1222no—
521C2×C4—4884no—
522C2×C4—4884no—
523C2×C4—2442no—
524C2×C4—2442no—
525C2×C4—2442no—
526C2×C4—2441no—
527C2×C4—2441no—
528C2×C4—2442no—
529C2×C4—2441no—
530C2×C4—2441no—
531C2×C4—2442no—
532C2×C4—2441no—
533C2×C4—2441no—
534C2×C4—2448no—
535C2×C4#1451974712yes—
536C2×C4#1451975514yes—
537C2×C4#1451975514yes—
538C2×C4#1451975712yes—
539C2×C4#1451976314yes—
540C2×C4#1451976314yes—
541C2×C4#1451976512yes—
542C2×C4C2×D4×D6111yescharacteristic
543C2×C4C2×D4×D6111yescharacteristic
544C2×C4C2×Q8×D6111yescharacteristic
545C2×C4C2×#1175822411yescharacteristic
546C2×C4C2×#1175822211yescharacteristic
547C2×C4C2×#1175822211yescharacteristic
548C8—41no—
549C8—41no—
550C8—42no—
551C22—22no—
552C22—22no—
553C22—22no—
554C22—22no—
555C22—22no—
556C22—22no—
557C22—4881no—
558C22—4881no—
559C22—4881no—
560C22—4881no—
561C22—2442no—
562C22—2442no—
563C22—2442no—
564C22#1683967514yes—
565C22C2×#1451974711yescharacteristic
566C22C2×#1451975711yescharacteristic
567C22C2×#1451976511yescharacteristic
568C22#1684056711yescharacteristic
569C22#1684057411yescharacteristic
570C22#1684057411yescharacteristic
571C22#1684057911yescharacteristic
572C4—42no—
573C4—42no—
574C4—4882no—
575C4—2444no—
576C4—2442no—
577C4—2442no—
578C4—2442no—
579C4#1683967514yes—
580C4C2×#1451975511yescharacteristic
581C4C2×#1451975511yescharacteristic
582C4C2×#1451976311yescharacteristic
583C4C2×#1451976311yescharacteristic
584C2—24no—
585C2—4882no—
586C2C2×#1683967511yescharacteristic
587C2#2145512611yescharacteristic
588C2#2145513711yescharacteristic
589C2#2145567211yescharacteristic
590C2#2145568311yescharacteristic
591C2#2145617511yescharacteristic
592C2#2145618611yescharacteristic
subgroupG/H
centerC23C22×D244
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1729C2
socleC23×C61C22×D4
prime radicalC22×#11758222C22
hypercenter#1729D61
hypocenterC61#23458
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×#117 ◁ #1729

Complements

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

NH ≅ G/Ntype
1. C2×#16840390C2semidirect
5. #21446971C2semidirect
6. #21446966C2semidirect
7. #21446961C2semidirect
8. #21446956C2semidirect
9. #21445539C2semidirect
10. C2×#16839525C2semidirect
11. C61×#1729C2semidirect
50. C2×#11758222Q8semidirect
298. #1729D61semidirect
341. C61#23458semidirect

Maximal subgroups

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

subgroupindex
C2×#168403902
C2×#168397642
C2×#168397592
#214484072
#214469712
#214469662
#214469612
#214469562
#214455392
C2×#168395252
C61×#17292
#2345861