Finite Group Databasebeta

← #29679226

Subgroups — #29679226

Order 15616.

3966 conjugacy classes of subgroups.

The subgroup lattice has 630 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#11758226C211yesmaximal, characteristic
2C2×#16839771C211yesmaximal, characteristic, complemented
3C2×#16839768C211yesmaximal, characteristic, complemented
4C2×#16839766C211yesmaximal, characteristic, complemented
5C2×#16839725C211yesmaximal, characteristic, complemented
6#21448198C211yesmaximal, characteristic, complemented
7#21445758C218yesmaximal, complemented
8C2×C61×#419C211yesmaximal, characteristic, complemented
9C22×#11758226—24no—
10C22×#11758226—24no—
11C22×#11758226—24no—
12C22×#11758226—24no—
13C22×#11758226—24no—
14C22×#11758226—22no—
15C22×#11758226C418yescomplemented
16C22×#11758226C2214yes—
17C23×D244—21no—
18C23×D244C2211yescharacteristic
19C23×C4×D61—21no—
20C23×C4×D61C2211yescharacteristic
21C22×#11758194C2211yescharacteristic
22C2×#14519736C2211yescharacteristic
23#16839771—24no—
24#16839768—24no—
25#16839766—24no—
26C2×#14519701C2214yescomplemented
27C2×#14519699C2214yescomplemented
28C2×#14519697C2214yescomplemented
29#16839725—24no—
30C2×#14519689C2214yescomplemented
31#16839623—21no—
32#16839619C2214yescomplemented
33#16839617—21no—
34C4×#11758198—22no—
35C23×Q8×C61C2211yescharacteristic
36C22×M16×C61C2211yescharacteristic
37C2×#117×C61C2211yescharacteristic
38C61×#419—24no—
39C2×#103×C61C2214yescomplemented
40C24×D61—21no—
41C24×D61D411yescharacteristic
42C2×#11758226—216no—
43C2×#11758226—216no—
44C2×#11758226—232no—
45C2×#11758226—232no—
46C2×#11758226—232no—
47C2×#11758226—28no—
48C2×#11758226—28no—
49C2×#11758226—232no—
50C2×#11758226—28no—
51C2×#11758226—28no—
52C2×#11758226—216no—
53C2×#11758226—28no—
54C2×#11758226C2×C4116yes—
55C22×D244—28no—
56C22×D244—28no—
57C22×D244—28no—
58C22×D244—24no—
59C22×D244—22no—
60C22×D244—21no—
61C22×D244—22no—
62C22×D244—22no—
63C22×D244C2×C414yes—
64C22×D244D414yes—
65C22×D244D414yes—
66C22×D244C2312yes—
67C23×Dic61D411yescharacteristic
68C22×C4×D61—28no—
69C22×C4×D61—28no—
70C22×C4×D61—22no—
71C22×C4×D61—22no—
72C22×C4×D61—22no—
73C22×C4×D61—22no—
74C22×C4×D61—28no—
75C22×C4×D61—24no—
76C22×C4×D61C2×C414yes—
77C22×C4×D61D412yes—
78C22×C4×D61D414yes—
79C22×C4×D61C2312yes—
80C2×#11758194—22no—
81C2×#11758194—22no—
82C2×#11758194—22no—
83C22×#92418C2312yes—
84#14519736—22no—
85#14519736—22no—
86C2×C4⋊Dic61C2312yes—
87C2×C22⋊Dic61—21no—
88C2×#11758198—24no—
89C2×C4×Dic61—21no—
90#14519701—28no—
91#14519699—28no—
92#14519697—28no—
93#14519689—28no—
94#14519684—24no—
95C22×Q8×C61—22no—
96C22×Q8×C61—22no—
97C22×Q8×C61—22no—
98C22×Q8×C61—21no—
99C22×Q8×C61C2×C414yes—
100C22×Q8×C61D414yes—
101C22×Q8×C61D414yes—
102C22×Q8×C61C2312yes—
103C23×C4×C61—21no—
104C23×C4×C61C2311yescharacteristic
105C2×M16×C61—22no—
106C2×M16×C61—22no—
107C2×M16×C61—22no—
108C22×C8×C61C2312yes—
109#117×C61—22no—
110#117×C61—22no—
111C2×C4⋊C4×C61C2312yes—
112C2×C22⋊C4×C61—21no—
113C2×C42×C61—21no—
114#103×C61—28no—
115C24×C61C2×D411yescharacteristic
116C23×D61—24no—
117C23×D61—22no—
118C23×D61—22no—
119C23×D61—24no—
120C23×D61—24no—
121C23×D61—22no—
122C23×D61—44no—
123C23×D61—48no—
124C23×D61C22⋊C414yes—
125C23×D61C2×D412yes—
126C2×Q8×C61—24no—
127C2×Q8×C61—28no—
128C2×Q8×C61—24no—
129C2×Q8×C61—24no—
130C2×Q8×C61—28no—
131C2×Q8×C61—28no—
132C2×Q8×C61—22no—
133C2×Q8×C61—22no—
134C2×Q8×C61—22no—
135C2×Q8×C61C22⋊C4116yescomplemented
136C2×Q8×C61C22×C414yes—
137C2×Q8×C61C2×D414yes—
138C2×Q8×C61C2×D414yes—
139C22×C4×C61—24no—
140C22×C4×C61—24no—
141C22×C4×C61—24no—
142C22×C4×C61—22no—
143C22×C4×C61—22no—
144C22×C4×C61C22×C412yes—
145C22×C4×C61C2×D411yescharacteristic
146C22×C4×C61C2×D412yes—
147C22×C4×C61C2×D412yes—
148C22×C4×C61C2×D412yes—
149C22×C4×C61C2×D412yes—
150C22×C4×C61C2×D412yes—
151C22×C4×C61C2411yescharacteristic
152C22×Dic61—22no—
153C22×Dic61—22no—
154C22×Dic61—22no—
155C22×Dic61C22⋊C414yes—
156C22×Dic61C2×D412yes—
157#11758226—232no—
158#11758226—232no—
159#11758226—2128no—
160#11758226—232no—
161#11758226—232no—
162C2×D244—216no—
163C2×D244—216no—
164C2×D244—264no—
165C2×D244—216no—
166C2×D244—28no—
167C2×D244—24no—
168C2×D244—28no—
169C2×D244—28no—
170C2×D244—24no—
171C2×D244—24no—
172C2×D244—22no—
173C2×D244—22no—
174C2×D244—22no—
175C2×D244C22⋊C4116yescomplemented
176C2×D244C22×C414yes—
177C2×D244C2×D414yes—
178C2×D244C2×D414yes—
179C2×C4×D61—216no—
180C2×C4×D61—216no—
181C2×C4×D61—264no—
182C2×C4×D61—28no—
183C2×C4×D61—28no—
184C2×C4×D61—28no—
185C2×C4×D61—28no—
186C2×C4×D61—24no—
187C2×C4×D61—24no—
188C2×C4×D61—22no—
189C2×C4×D61—22no—
190C2×C4×D61—22no—
191C2×C4×D61—22no—
192C2×C4×D61—216no—
193C2×C4×D61C22⋊C418yes—
194C2×C4×D61C22×C414yes—
195C2×C4×D61C2×D414yes—
196M16×C61—44no—
197C2×C8×C61—22no—
198C2×C8×C61—22no—
199C2×C8×C61—22no—
200C4⋊C4×C61—22no—
201C4⋊C4×C61—22no—
202C22⋊C4×C61—42no—
203C42×C61—42no—
204C22⋊Dic61—42no—
205#11758198—48no—
206C4⋊Dic61—22no—
207C4⋊Dic61—22no—
208C4×Dic61—42no—
209#11758194—44no—
210C2×#92418—22no—
211C2×#92418—22no—
212C2×#92418—22no—
213C23×C61—24no—
214C23×C61C2×C22⋊C412yes—
215C23×C61#12012yes—
216C23×C61#12012yes—
217C23×C61C22×D411yescharacteristic
218C22×D61—24no—
219C22×D61—22no—
220C22×D61—22no—
221C22×D61—24no—
222C22×D61—24no—
223C22×D61—24no—
224C22×D61—24no—
225C22×D61—28no—
226C22×D61—216no—
227C22×D61—48no—
228C22×D61—416no—
229C22×D61—48no—
230C22×D61—48no—
231C22×D61—416no—
232C22×D61—416no—
233C22×D61—48no—
234C22×D61C2×C22⋊C414yes—
235Q8×C61—24no—
236Q8×C61—24no—
237Q8×C61—24no—
238Q8×C61—216no—
239Q8×C61C2×C22⋊C418yes—
240C2×C4×C61—24no—
241C2×C4×C61—24no—
242C2×C4×C61—24no—
243C2×C4×C61—216no—
244C2×C4×C61—42no—
245C2×C4×C61—44no—
246C2×C4×C61C2×C22⋊C414yes—
247C2×C4×C61C2×C22⋊C414yes—
248C2×C4×C61C2×C22⋊C412yes—
249C2×C4×C61#12014yes—
250C2×C4×C61#12014yes—
251C2×C4×C61#12012yes—
252C2×C4×C61#12012yes—
253C2×C4×C61C23×C411yescharacteristic
254C2×C4×C61C22×D411yescharacteristic
255C2×C4×C61C22×D411yescharacteristic
256C2×C4×C61C22×D411yescharacteristic
257C2×C4×C61C22×D411yescharacteristic
258C2×C4×C61C22×D411yescharacteristic
259C2×Dic61—22no—
260C2×Dic61—22no—
261C2×Dic61—28no—
262C2×Dic61—42no—
263C2×Dic61—44no—
264C2×Dic61C2×C22⋊C414yes—
265D244—264no—
266D244—24no—
267D244—24no—
268D244—24no—
269D244—216no—
270D244C2×C22⋊C418yes—
271C4×D61—264no—
272C4×D61—24no—
273C4×D61—24no—
274C4×D61—24no—
275C4×D61—24no—
276C4×D61—216no—
277C8×C61—42no—
278#92418—42no—
279#20600—611nomaximal, Sylow
280C22×C61—24no—
281C22×C61—24no—
282C22×C61—24no—
283C22×C61#37914yes—
284C22×C61C22×C22⋊C411yescharacteristic
285C22×C61C2×#12011yescharacteristic
286C22×C61C2×#12011yescharacteristic
287C22×C61#57514yes—
288C2×D61—24no—
289C2×D61—24no—
290C2×D61—24no—
291C2×D61—28no—
292C2×D61—416no—
293C2×D61—432no—
294C2×D61—48no—
295C2×D61—44no—
296C2×D61—44no—
297C2×D61—48no—
298C2×D61—48no—
299C4×C61—28no—
300C4×C61—82no—
301C4×C61#37914yes—
302C4×C61C22×C22⋊C411yescharacteristic
303C4×C61C22×C22⋊C411yescharacteristic
304C4×C61C2×#12011yescharacteristic
305C4×C61C2×#12011yescharacteristic
306Dic61—24no—
307Dic61—82no—
308C23×#41—611no—
309C2×#419—611no—
310C2×#419D6111yescharacteristic, complemented
311C2×#418—611no—
312C2×#418—611no—
313C2×#407—611no—
314#2280—611no—
315#1788—618no—
316C2×C61—24no—
317C2×C61C2×#37911yescharacteristic
318C2×C61#289612yes—
319C2×C61#300512yes—
320C2×C61#300512yes—
321D61—44no—
322D61—48no—
323C22×#41—618no—
324C22×#41—614no—
325C22×#41—1222no—
326C22×#41—1224no—
327C22×#41—1224no—
328C22×#41—1224no—
329C22×#41—1224no—
330C22×#41—1224no—
331C23×Q8C2×D6111yescharacteristic
332C23×D4—611no—
333C23×D4—1221no—
334C24×C4—611no—
335C24×C4—1221no—
336C22×M16—611no—
337C22×M16C2×D6111yescharacteristic
338C2×#117—611no—
339C2×#117C2×D6111yescharacteristic
340#419—24no—
341#419—1224no—
342#418—1224no—
343#418—1224no—
344C2×#103—614no—
345C2×#103C2×D6114yescomplemented
346C2×#102—614no—
347C2×#102—614no—
348#407—1224no—
349C2×C22⋊C8—614no—
350#381—1221no—
351#380—614no—
352#379—1221no—
353C4×C22⋊C4—1222no—
354C61#2060011yescharacteristic, complemented, Sylow
355C25—611no—
356C25—1221no—
357C2×#41—6116no—
358C2×#41—1228no—
359C2×#41—12216no—
360C2×#41—1228no—
361C2×#41—1228no—
362C2×#41—12232no—
363C2×#41—12216no—
364C2×#41—12232no—
365C2×#41—12216no—
366C2×#41—12232no—
367C2×#41—12232no—
368C2×#41—1228no—
369C2×#41—1228no—
370C22×Q8—21no—
371C22×Q8—22no—
372C22×Q8—22no—
373C22×Q8—22no—
374C22×Q8C4×D6114yes—
375C22×Q8D24414yes—
376C22×Q8C22⋊D6114yes—
377C22×Q8C22×D6112yes—
378C22×D4—614no—
379C22×D4—614no—
380C22×D4—614no—
381C22×D4—612no—
382C22×D4—1224no—
383C22×D4—1228no—
384C22×D4—1228no—
385C22×D4—1228no—
386C22×D4—1222no—
387C22×D4—1222no—
388C22×D4—1221no—
389C22×D4—1222no—
390C23×C4—21no—
391C23×C4—611no—
392C23×C4—614no—
393C23×C4—612no—
394C23×C4—614no—
395C23×C4—612no—
396C23×C4—1224no—
397C23×C4—1228no—
398C23×C4—1228no—
399C23×C4—1228no—
400C23×C4—1222no—
401C23×C4—1222no—
402C23×C4—1222no—
403C23×C4—1222no—
404C23×C4C22×D6111yescharacteristic
405C2×M16—22no—
406C2×M16—22no—
407C2×M16—22no—
408C2×M16—1222no—
409C2×M16—1222no—
410C2×M16—1222no—
411C22×C8—612no—
412C22×C8C22×D6112yes—
413#117—22no—
414#117—22no—
415#117—1222no—
416#117—1222no—
417C2×C4⋊C4—612no—
418C2×C4⋊C4C22×D6112yes—
419C2×C22⋊C4—21no—
420C2×C22⋊C4—1224no—
421C2×C22⋊C4—1221no—
422C2×C42—21no—
423C2×C42—1221no—
424#103—28no—
425#103—1228no—
426#102—1228no—
427#102—1228no—
428C22⋊C8—1228no—
429#95—1224no—
430C24—612no—
431C24—614no—
432C24—1222no—
433C24—1224no—
434C24—1224no—
435C24—1222no—
436C24—1224no—
437C24—1222no—
438C24—2444no—
439C24—2448no—
440C24D4×D6111yescharacteristic
441#41—12232no—
442#41—12232no—
443#41—12232no—
444#41—122128no—
445#41—12232no—
446C2×Q8—22no—
447C2×Q8—22no—
448C2×Q8—22no—
449C2×Q8—24no—
450C2×Q8—28no—
451C2×Q8—24no—
452C2×Q8—28no—
453C2×Q8—24no—
454C2×Q8—28no—
455C2×Q8#11758198116yescomplemented
456C2×Q8C2×C4×D6114yes—
457C2×Q8C2×D24414yes—
458C2×Q8C2×C22⋊D6114yes—
459C2×D4—614no—
460C2×D4—614no—
461C2×D4—614no—
462C2×D4—6116no—
463C2×D4—12216no—
464C2×D4—12216no—
465C2×D4—12216no—
466C2×D4—12264no—
467C2×D4—1222no—
468C2×D4—1222no—
469C2×D4—1222no—
470C2×D4—1224no—
471C2×D4—1228no—
472C2×D4—1228no—
473C2×D4—1224no—
474C2×D4—1228no—
475C2×D4—1224no—
476C22×C4—22no—
477C22×C4—24no—
478C22×C4—24no—
479C22×C4—24no—
480C22×C4—22no—
481C22×C4—612no—
482C22×C4—614no—
483C22×C4—614no—
484C22×C4—614no—
485C22×C4—618no—
486C22×C4—12216no—
487C22×C4—12216no—
488C22×C4—12216no—
489C22×C4—12264no—
490C22×C4—1222no—
491C22×C4—1222no—
492C22×C4—1222no—
493C22×C4—1222no—
494C22×C4—1222no—
495C22×C4—1222no—
496C22×C4—1228no—
497C22×C4—1228no—
498C22×C4—1224no—
499C22×C4—1228no—
500C22×C4—1224no—
501C22×C4—1228no—
502C22×C4—1222no—
503C22×C4C2×C4×D6112yes—
504C22×C4C2×D24412yes—
505C22×C4D4×D6112yes—
506C22×C4D4×D6112yes—
507C22×C4D4×D6112yes—
508C22×C4D4×D6111yescharacteristic
509C22×C4C2×C22⋊D6112yes—
510C22×C4C23×D6111yescharacteristic
511M16—44no—
512M16—2444no—
513C2×C8—22no—
514C2×C8—22no—
515C2×C8—22no—
516C2×C8—1222no—
517C2×C8—1222no—
518C2×C8—1222no—
519C4⋊C4—22no—
520C4⋊C4—22no—
521C4⋊C4—1222no—
522C4⋊C4—1222no—
523C22⋊C4—42no—
524C22⋊C4—2448no—
525C22⋊C4—2442no—
526C42—42no—
527C42—2442no—
528C23—24no—
529C23—614no—
530C23—1224no—
531C23—1224no—
532C23—1222no—
533C23—1224no—
534C23—1222no—
535C23—1224no—
536C23—1224no—
537C23—12216no—
538C23—1228no—
539C23—2448no—
540C23—24416no—
541C23—2448no—
542C23—24416no—
543C23—2448no—
544C23—24416no—
545C23—2448no—
546C23C22⋊C4×D6112yes—
547C23#1451974512yes—
548C23#1451974712yes—
549C23C2×D4×D6111yescharacteristic
550Q8—24no—
551Q8—24no—
552Q8—24no—
553Q8—216no—
554Q8C2×#1175819818yes—
555D4—618no—
556D4—12264no—
557D4—1224no—
558D4—1224no—
559D4—1224no—
560D4—12216no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—216no—
565C2×C4—42no—
566C2×C4—44no—
567C2×C4—614no—
568C2×C4—12264no—
569C2×C4—1222no—
570C2×C4—1222no—
571C2×C4—1228no—
572C2×C4—1224no—
573C2×C4—1224no—
574C2×C4—1224no—
575C2×C4—1224no—
576C2×C4—12216no—
577C2×C4—2442no—
578C2×C4—2444no—
579C2×C4C22⋊C4×D6114yes—
580C2×C4C22⋊C4×D6112yes—
581C2×C4C2×#1175819814yes—
582C2×C4#1451974514yes—
583C2×C4#1451974512yes—
584C2×C4#1451974714yes—
585C2×C4#1451974712yes—
586C2×C4C22×C4×D6111yescharacteristic
587C2×C4C2×D4×D6111yescharacteristic
588C2×C4C2×D4×D6111yescharacteristic
589C2×C4C2×D4×D6111yescharacteristic
590C2×C4C22×C22⋊D6111yescharacteristic
591C2×C4C22×D24411yescharacteristic
592C8—42no—
593C8—2442no—
594C22—24no—
595C22—24no—
596C22—24no—
597C22—1224no—
598C22—1224no—
599C22—1228no—
600C22—1224no—
601C22—24416no—
602C22—24432no—
603C22—2448no—
604C22—2444no—
605C22—2448no—
606C22—2444no—
607C22—2448no—
608C22#1683961714yes—
609C22C2×C22⋊C4×D6111yescharacteristic
610C22C2×#1451974511yescharacteristic
611C22C2×#1451974711yescharacteristic
612C22#1684056114yes—
613C4—28no—
614C4—82no—
615C4—1224no—
616C4—4882no—
617C4#1683961714yes—
618C4C2×C22⋊C4×D6111yescharacteristic
619C4C22×#1175819811yescharacteristic
620C4C2×#1451974511yescharacteristic
621C4C2×#1451974711yescharacteristic
622C2—24no—
623C2—2444no—
624C2—2448no—
625C2C2×#1683961711yescharacteristic
626C2#2145433712yes—
627C2#2145506012yes—
628C2#2145506512yes—
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC2×C61×#419C2
socleC23×C61C22×D4
prime radicalC23×#11758226C2
cosocleC22×C4×C61C24
hypercenterC2×#419D61
hypocenterC61#20600
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#419

Complements

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

NH ≅ G/Ntype
2. C2×#16839771C2semidirect
3. C2×#16839768C2semidirect
4. C2×#16839766C2semidirect
5. C2×#16839725C2semidirect
6. #21448198C2semidirect
7. #21445758C2semidirect
8. C2×C61×#419C2semidirect
15. C22×#11758226C4semidirect
26. C2×#14519701C22semidirect
27. C2×#14519699C22semidirect
28. C2×#14519697C22semidirect
30. C2×#14519689C22semidirect
32. #16839619C22semidirect
39. C2×#103×C61C22semidirect
135. C2×Q8×C61C22⋊C4semidirect
175. C2×D244C22⋊C4semidirect
310. C2×#419D61semidirect
345. C2×#103C2×D61semidirect
354. C61#20600semidirect
455. C2×Q8#11758198semidirect

Maximal subgroups

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

subgroupindex
C23×#117582262
C2×#168397712
C2×#168397682
C2×#168397662
C2×#168397252
#214481982
#214457582
C2×C61×#4192
#2060061