Finite Group Databasebeta

← #29706884

Subgroups — #29706884

Order 15616.

2014 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14519810C211yesmaximal, characteristic, complemented
2C2×#16840343C211yesmaximal, characteristic
3C2×#16840338C211yesmaximal, characteristic, complemented
4C2×#16839756C211yesmaximal, characteristic, complemented
5C22×#14519700C211yesmaximal, characteristic, complemented
6C2×#16839730C211yesmaximal, characteristic, complemented
7#21446830C214yesmaximal, complemented
8#21446825C214yesmaximal, complemented
9C61×#1736C211yesmaximal, characteristic, complemented
10C22×#11758222C2211yescharacteristic
11C22×#11758222C2211yescharacteristic
12C23×Dic122C2211yescharacteristic
13C2×#14519810—24no—
14C2×#14519810—24no—
15C2×#14519810—24no—
16C22×Dic244C2212yes—
17C22×#11758190C2212yes—
18C23×#92418C2211yescharacteristic
19#16840343—24no—
20#16840343—24no—
21#16840338—24no—
22#16840338—24no—
23C2×#14519777—21no—
24C2×#14519765—21no—
25C2×#14519761—22no—
26C2×#14519757—21no—
27C2×#14519757C2214yes—
28C2×#14519757C2212yes—
29C2×#14519752—21no—
30C2×#14519752C2212yes—
31C2×#14519751—22no—
32#16839756—24no—
33C2×#14519700—24no—
34C2×#14519700C2214yes—
35C2×#14519700C2212yes—
36C2×#14519697C2212yes—
37#16839730—24no—
38C2×#14519689C2214yes—
39C22×M16×C61C2211yescharacteristic
40C2×#117×C61C2211yescharacteristic
41C22×C4⋊C4×C61C2211yescharacteristic
42C61×#340C2214yes—
43C2×#11758222—22no—
44C2×#11758222—22no—
45C2×#11758222—22no—
46C2×#11758222—22no—
47C2×#11758222—22no—
48C2×#11758222—22no—
49C2×#11758222—22no—
50C2×#11758222—22no—
51C2×#11758222—22no—
52C2×#11758222—22no—
53C2×#11758222—22no—
54C2×#11758222D414yes—
55C22×Dic122—22no—
56C22×Dic122—22no—
57C22×Dic122—22no—
58C22×Dic122—22no—
59C22×Dic122—21no—
60C22×Dic122—22no—
61C22×Dic122D411yescharacteristic
62C22×Dic122D414yescomplemented
63C22×Dic122C2312yes—
64C22×Dic122C2311yescharacteristic
65C22×D244—22no—
66C22×D244D411yescharacteristic
67C22×D244C2311yescharacteristic
68C22×C22⋊D61—21no—
69C22×C22⋊D61D412yes—
70C23×Dic61—21no—
71C22×C4×D61—21no—
72C22×C4×D61C2312yes—
73#14519810—232no—
74C2×Dic244—24no—
75C2×Dic244—24no—
76C2×Dic244—24no—
77C2×#11758190—24no—
78C2×#11758190—24no—
79C2×#11758190—24no—
80C22×#92418—22no—
81C22×#92418—22no—
82C22×#92418—22no—
83C22×#92418C2312yes—
84#14519777—28no—
85#14519765—28no—
86#14519761—216no—
87#14519757—28no—
88#14519757—28no—
89#14519757—24no—
90#14519757—28no—
91#14519757—24no—
92#14519752—28no—
93#14519752—24no—
94#14519752—24no—
95#14519751—216no—
96C2×C4⋊Dic61—22no—
97C2×C4⋊Dic61—24no—
98C2×#11758198—22no—
99C2×#11758198—24no—
100C2×#11758198—22no—
101#14519700—28no—
102#14519700—24no—
103#14519697—24no—
104#14519689—28no—
105C23×C4×C61—21no—
106C23×C4×C61C2311yescharacteristic
107C2×M16×C61—22no—
108C2×M16×C61—22no—
109C2×M16×C61D414yescomplemented
110C22×C8×C61C2312yes—
111#117×C61—22no—
112#117×C61D414yescomplemented
113C2×C4⋊C4×C61—21no—
114C2×C4⋊C4×C61—22no—
115C2×C4⋊C4×C61D414yes—
116C2×C4⋊C4×C61C2312yes—
117C2×C4⋊C4×C61C2312yes—
118C2×C22⋊C4×C61—21no—
119C2×C42×C61—21no—
120C24×C61C2×D411yescharacteristic
121C23×D61—21no—
122C23×D61C2×D412yes—
123C22×C4×C61—22no—
124C22×C4×C61—22no—
125C22×C4×C61—24no—
126C22×C4×C61—82no—
127C22×C4×C61C2×D411yescharacteristic
128C22×C4×C61C2×D412yes—
129C22×C4×C61C2×D412yes—
130C22×C4×C61C2×D412yes—
131C22×C4×C61C2×D412yes—
132C22×C4×C61C2×D412yes—
133C22×C4×C61#4112yes—
134C22×C4×C61C2411yescharacteristic
135C2×C22⋊D61—24no—
136C2×C22⋊D61—24no—
137C2×C22⋊D61—24no—
138C2×C22⋊D61—24no—
139C2×C22⋊D61—42no—
140C2×C22⋊D61—42no—
141C22×Dic61—24no—
142C22×Dic61—21no—
143C22×Dic61—22no—
144C22×Dic61—42no—
145C22×Dic61—42no—
146C22×Dic61C2×D412yes—
147#11758222—28no—
148#11758222—28no—
149#11758222—44no—
150#11758222—44no—
151#11758222—44no—
152#11758222—44no—
153#11758222—44no—
154#11758222—44no—
155C2×D244—21no—
156C2×D244—21no—
157C2×D244—21no—
158C2×D244—21no—
159C2×D244—22no—
160C2×D244—22no—
161C2×D244—22no—
162C2×D244—21no—
163C2×D244—28no—
164C2×D244—84no—
165C2×D244C2×D412yes—
166C2×C4×D61—24no—
167C2×C4×D61—22no—
168C2×C4×D61—22no—
169C2×C4×D61—22no—
170C2×C4×D61—22no—
171C2×C4×D61—22no—
172C2×C4×D61—22no—
173C2×C4×D61—42no—
174C2×C4×D61—42no—
175C2×Dic122—28no—
176C2×Dic122—28no—
177C2×Dic122—22no—
178C2×Dic122—22no—
179C2×Dic122—22no—
180C2×Dic122—21no—
181C2×Dic122—21no—
182C2×Dic122—21no—
183C2×Dic122—22no—
184C2×Dic122—21no—
185C2×Dic122—24no—
186C2×Dic122—22no—
187C2×Dic122—22no—
188C2×Dic122—22no—
189C2×Dic122—22no—
190C2×Dic122—21no—
191C2×Dic122—42no—
192C2×Dic122—44no—
193C2×Dic122—42no—
194C2×Dic122—44no—
195C2×Dic122C2×D414yes—
196C2×Dic122C2×D412yes—
197M16×C61—28no—
198C2×C8×C61—22no—
199C2×C8×C61—22no—
200C2×C8×C61C2×D414yes—
201C4⋊C4×C61—24no—
202C4⋊C4×C61—22no—
203C4⋊C4×C61—22no—
204C4⋊C4×C61C2×D414yes—
205C4⋊C4×C61C2×D414yes—
206C22⋊C4×C61—24no—
207C42×C61—24no—
208#11758198—28no—
209#11758198—28no—
210#11758198—216no—
211C4⋊Dic61—28no—
212C4⋊Dic61—216no—
213C2×#92418—22no—
214C2×#92418—22no—
215C2×#92418—22no—
216C2×#92418—44no—
217Dic244—216no—
218#11758190—216no—
219C23×C61—22no—
220C23×C61—22no—
221C23×C61#12012yes—
222C23×C61#12012yes—
223C23×C61#12112yes—
224C23×C61C22×D411yescharacteristic
225C22×D61—22no—
226C22×D61—22no—
227C22×D61—22no—
228C22×D61—22no—
229C22×D61—24no—
230C22×D61—24no—
231C22×D61—41no—
232C22×D61—41no—
233C22×D61—42no—
234C22×D61—42no—
235C2×C4×C61—24no—
236C2×C4×C61—24no—
237C2×C4×C61—44no—
238C2×C4×C61—42no—
239C2×C4×C61—42no—
240C2×C4×C61—84no—
241C2×C4×C61#12014yes—
242C2×C4×C61#12012yes—
243C2×C4×C61#12012yes—
244C2×C4×C61#12114yes—
245C2×C4×C61#12114yes—
246C2×C4×C61#12112yes—
247C2×C4×C61#12714yes—
248C2×C4×C61C22×D411yescharacteristic
249C2×C4×C61C22×D411yescharacteristic
250C2×C4×C61C22×D411yescharacteristic
251C2×C4×C61C22×D411yescharacteristic
252C2×C4×C61C22×D411yescharacteristic
253C2×C4×C61C2×#4111yescharacteristic
254C22⋊D61—48no—
255C22⋊D61—48no—
256C2×Dic61—22no—
257C2×Dic61—24no—
258C2×Dic61—22no—
259C2×Dic61—22no—
260C2×Dic61—22no—
261C2×Dic61—42no—
262C2×Dic61—41no—
263C2×Dic61—42no—
264C2×Dic61—41no—
265C2×Dic61—48no—
266D244—22no—
267D244—22no—
268D244—41no—
269D244—41no—
270D244—44no—
271D244—88no—
272C4×D61—42no—
273C4×D61—42no—
274C4×D61—42no—
275C4×D61—42no—
276C4×D61—48no—
277Dic122—24no—
278Dic122—22no—
279Dic122—24no—
280Dic122—22no—
281Dic122—42no—
282Dic122—41no—
283Dic122—42no—
284Dic122—41no—
285Dic122—48no—
286Dic122—44no—
287C8×C61—24no—
288#92418—42no—
289#23394—611nomaximal, Sylow
290C22×C61—22no—
291C22×C61—22no—
292C22×C61—24no—
293C22×C61—24no—
294C22×C61#39214yes—
295C22×C61C2×#12011yescharacteristic
296C22×C61C2×#12011yescharacteristic
297C22×C61C2×#12111yescharacteristic
298C22×C61#57712yes—
299C22×C61#57812yes—
300C2×D61—42no—
301C2×D61—42no—
302C2×D61—42no—
303C2×D61—42no—
304C2×D61—42no—
305C2×D61—42no—
306C2×D61—44no—
307C2×D61—44no—
308C2×D61—82no—
309C2×D61—82no—
310C4×C61—44no—
311C4×C61—44no—
312C4×C61#39214yes—
313C4×C61C2×#12011yescharacteristic
314C4×C61C2×#12111yescharacteristic
315C4×C61C2×#12111yescharacteristic
316C4×C61C2×#12711yescharacteristic
317Dic61—44no—
318Dic61—42no—
319Dic61—42no—
320C22×#137—611no—
321C2×#537—611no—
322C2×#536—611no—
323C2×#416—611no—
324C22×#103—611no—
325C2×#408—611no—
326#2001—614no—
327#2000—614no—
328#1736D6111yescharacteristic, complemented
329C2×C61—24no—
330C2×C61C2×#39211yescharacteristic
331C2×C61#300911yescharacteristic
332C2×C61#301111yescharacteristic
333C2×C61#301312yes—
334C2×C61#306312yes—
335D61—82no—
336D61—82no—
337C22×#41—611no—
338C22×#41—611no—
339C23×Q8—611no—
340C2×#137—1224no—
341C2×#137—1224no—
342C2×#137—1224no—
343C22×Q16—612no—
344C22×SD16—612no—
345C22×M16C2×D6111yescharacteristic
346C23×C8—611no—
347#537—1224no—
348#537—1224no—
349#536—1224no—
350#536—1224no—
351C2×C4⋊Q8—1221no—
352C2×#124—1221no—
353C2×#123—1222no—
354C2×C22⋊Q8—614no—
355C2×C22⋊Q8—612no—
356C2×C22⋊Q8—1221no—
357C2×#121—612no—
358C2×#121—1221no—
359C2×#121—1222no—
360C2×#117C2×D6111yescharacteristic
361C22×C4⋊C4C2×D6111yescharacteristic
362#416—1224no—
363C2×#103—614no—
364C2×#103—612no—
365C2×#103—1224no—
366C2×#102—612no—
367#408—1224no—
368C2×C22⋊C8—614no—
369#340C2×D6114yes—
370C61#2339411yescharacteristic, complemented, Sylow
371C2×#41—614no—
372C2×#41—1222no—
373C2×#41—1222no—
374C2×#41—1222no—
375C2×#41—1222no—
376C2×#41—1222no—
377C2×#41—1222no—
378C2×#41—1222no—
379C2×#41—1222no—
380C2×#41—1222no—
381C2×#41—1222no—
382C2×#41—1222no—
383C22×Q8—611no—
384C22×Q8—614no—
385C22×Q8—612no—
386C22×Q8—611no—
387C22×Q8—1222no—
388C22×Q8—1221no—
389C22×Q8—1222no—
390C22×Q8—1222no—
391C22×Q8—1222no—
392C22×Q8—1222no—
393C22×D4—612no—
394C22×D4—611no—
395C22×D4—611no—
396C22×D4—1222no—
397C22×D4—1221no—
398C23×C4—21no—
399C23×C4—612no—
400C23×C4—1221no—
401C23×C4—1221no—
402C23×C4C22×D6111yescharacteristic
403#137—12232no—
404C2×Q16—1224no—
405C2×Q16—1224no—
406C2×Q16—1224no—
407C2×SD16—1224no—
408C2×SD16—1224no—
409C2×SD16—1224no—
410C2×M16—22no—
411C2×M16—22no—
412C2×M16D24414yescomplemented
413C22×C8—612no—
414C22×C8—1222no—
415C22×C8—1222no—
416C22×C8—1222no—
417C22×C8C22×D6112yes—
418C4⋊Q8—1228no—
419#124—1228no—
420#123—12216no—
421C22⋊Q8—1228no—
422C22⋊Q8—1224no—
423C22⋊Q8—1228no—
424C22⋊Q8—1228no—
425C22⋊Q8—1224no—
426#121—1224no—
427#121—1228no—
428#121—12216no—
429#121—1224no—
430#117—22no—
431#117D24414yescomplemented
432C2×C4⋊C4—21no—
433C2×C4⋊C4—22no—
434C2×C4⋊C4—1222no—
435C2×C4⋊C4—1224no—
436C2×C4⋊C4D24414yes—
437C2×C4⋊C4C22×D6112yes—
438C2×C4⋊C4C22×D6112yes—
439C2×C22⋊C4—21no—
440C2×C22⋊C4—1222no—
441C2×C22⋊C4—1222no—
442C2×C22⋊C4—1224no—
443C2×C42—21no—
444#103—1228no—
445#103—1224no—
446#102—1224no—
447C22⋊C8—1228no—
448C24—612no—
449C24—1221no—
450C24D4×D6111yescharacteristic
451#41—1228no—
452#41—1228no—
453#41—2444no—
454#41—2444no—
455#41—2444no—
456#41—2444no—
457#41—2444no—
458#41—2444no—
459C2×Q8—614no—
460C2×Q8—612no—
461C2×Q8—1222no—
462C2×Q8—1221no—
463C2×Q8—1222no—
464C2×Q8—1224no—
465C2×Q8—1222no—
466C2×Q8—1222no—
467C2×Q8—1222no—
468C2×Q8—1221no—
469C2×Q8—1222no—
470C2×Q8—1221no—
471C2×Q8—1222no—
472C2×Q8—1221no—
473C2×Q8—1222no—
474C2×Q8—1221no—
475C2×Q8—1228no—
476C2×Q8—1228no—
477C2×Q8—2442no—
478C2×Q8—2442no—
479C2×Q8—2444no—
480C2×Q8—2444no—
481C2×D4—612no—
482C2×D4—1228no—
483C2×D4—1224no—
484C2×D4—1224no—
485C2×D4—1224no—
486C2×D4—1224no—
487C2×D4—1221no—
488C2×D4—1221no—
489C2×D4—1221no—
490C2×D4—1221no—
491C2×D4—1221no—
492C2×D4—1222no—
493C2×D4—1222no—
494C2×D4—1222no—
495C2×D4—4884no—
496C2×D4—2442no—
497C2×D4—2442no—
498C22×C4—22no—
499C22×C4—22no—
500C22×C4—24no—
501C22×C4—82no—
502C22×C4—612no—
503C22×C4—1221no—
504C22×C4—1222no—
505C22×C4—1224no—
506C22×C4—1222no—
507C22×C4—1222no—
508C22×C4—1222no—
509C22×C4—1222no—
510C22×C4—1222no—
511C22×C4—1222no—
512C22×C4—1224no—
513C22×C4—2442no—
514C22×C4—2442no—
515C22×C4—2442no—
516C22×C4—2442no—
517C22×C4C2×D24412yes—
518C22×C4C2×D24412yes—
519C22×C4C2×D24412yes—
520C22×C4D4×D6112yes—
521C22×C4D4×D6112yes—
522C22×C4D4×D6111yescharacteristic
523C22×C4#1175822612yes—
524C22×C4C23×D6111yescharacteristic
525Q16—12216no—
526SD16—12216no—
527M16—28no—
528C2×C8—22no—
529C2×C8—22no—
530C2×C8—1222no—
531C2×C8—1222no—
532C2×C8—1222no—
533C2×C8—2444no—
534C2×C8C2×D24414yes—
535C4⋊C4—22no—
536C4⋊C4—22no—
537C4⋊C4—24no—
538C4⋊C4—1228no—
539C4⋊C4—12216no—
540C4⋊C4C2×D24414yes—
541C4⋊C4C2×D24414yes—
542C22⋊C4—24no—
543C22⋊C4—1228no—
544C22⋊C4—1228no—
545C22⋊C4—12216no—
546C42—24no—
547C23—22no—
548C23—22no—
549C23—1222no—
550C23—1222no—
551C23—1222no—
552C23—1222no—
553C23—1224no—
554C23—1224no—
555C23—2441no—
556C23—2441no—
557C23—2442no—
558C23—2442no—
559C23#1451974712yes—
560C23#1451974712yes—
561C23#1451975212yes—
562C23C2×D4×D6111yescharacteristic
563Q8—1224no—
564Q8—1222no—
565Q8—1224no—
566Q8—1222no—
567Q8—2442no—
568Q8—2441no—
569Q8—2442no—
570Q8—2441no—
571Q8—2448no—
572Q8—2444no—
573D4—1222no—
574D4—1222no—
575D4—4888no—
576D4—2448no—
577D4—2448no—
578D4—2441no—
579D4—2441no—
580D4—2444no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—42no—
584C2×C4—44no—
585C2×C4—42no—
586C2×C4—84no—
587C2×C4—1222no—
588C2×C4—1222no—
589C2×C4—1224no—
590C2×C4—1222no—
591C2×C4—1222no—
592C2×C4—2441no—
593C2×C4—2442no—
594C2×C4—2441no—
595C2×C4—2442no—
596C2×C4—2448no—
597C2×C4—2442no—
598C2×C4—2442no—
599C2×C4—2442no—
600C2×C4—2442no—
601C2×C4—2448no—
602C2×C4#1451974714yes—
603C2×C4#1451974712yes—
604C2×C4#1451974712yes—
605C2×C4#1451975214yes—
606C2×C4#1451975214yes—
607C2×C4#1451975212yes—
608C2×C4#1451977414yes—
609C2×C4C2×D4×D6111yescharacteristic
610C2×C4C2×D4×D6111yescharacteristic
611C2×C4C22×D24411yescharacteristic
612C2×C4C22×D24411yescharacteristic
613C2×C4C22×D24411yescharacteristic
614C2×C4C2×#1175822611yescharacteristic
615C8—24no—
616C8—2442no—
617C22—22no—
618C22—24no—
619C22—24no—
620C22—22no—
621C22—4882no—
622C22—4882no—
623C22—2442no—
624C22—2442no—
625C22—2444no—
626C22—2442no—
627C22—2442no—
628C22—2444no—
629C22—2442no—
630C22—2442no—
631C22#1683967214yes—
632C22C2×#1451974711yescharacteristic
633C22C2×#1451974711yescharacteristic
634C22C2×#1451975211yescharacteristic
635C22#1684057212yes—
636C22#1684057812yes—
637C4—44no—
638C4—44no—
639C4—2442no—
640C4—2442no—
641C4—2444no—
642C4#1683967214yes—
643C4C2×#1451974711yescharacteristic
644C4C2×#1451975211yescharacteristic
645C4C2×#1451975211yescharacteristic
646C4C2×#1451977411yescharacteristic
647C2—24no—
648C2—4882no—
649C2—4882no—
650C2C2×#1683967211yescharacteristic
651C2#2145509711yescharacteristic
652C2#2145511511yescharacteristic
653C2#2145513612yes—
654C2#2145551112yes—
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1736C2
socleC23×C61C22×D4
prime radicalC2×#16840343C2
hypercenter#1736D61
hypocenterC61#23394
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1736

Complements

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

NH ≅ G/Ntype
1. C22×#14519810C2semidirect
3. C2×#16840338C2semidirect
4. C2×#16839756C2semidirect
5. C22×#14519700C2semidirect
6. C2×#16839730C2semidirect
7. #21446830C2semidirect
8. #21446825C2semidirect
9. C61×#1736C2semidirect
62. C22×Dic122D4semidirect
109. C2×M16×C61D4semidirect
112. #117×C61D4semidirect
328. #1736D61semidirect
370. C61#23394semidirect
412. C2×M16D244semidirect
431. #117D244semidirect

Maximal subgroups

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

subgroupindex
C22×#145198102
C2×#168403432
C2×#168403382
C2×#168397562
C22×#145197002
C2×#168397302
#214468302
#214468252
C61×#17362
#2339461