Finite Group Databasebeta

← #29679219

Subgroups — #29679219

Order 15616.

3838 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×Q8×D61C211yesmaximal, characteristic
2C2×#16839772C211yesmaximal, characteristic, complemented
3C2×#16839771C211yesmaximal, characteristic, complemented
4C2×#16839770C211yesmaximal, characteristic, complemented
5C2×#16839725C211yesmaximal, characteristic
6#21448198C211yesmaximal, characteristic
7#21445752C218yesmaximal, complemented
8C2×C61×#419C211yesmaximal, characteristic, complemented
9C23×Dic122—21no—
10C23×Dic122C2211yescharacteristic
11C22×Q8×D61—24no—
12C22×Q8×D61—24no—
13C22×Q8×D61—24no—
14C22×Q8×D61—24no—
15C22×Q8×D61—24no—
16C22×Q8×D61—22no—
17C22×Q8×D61C418yescomplemented
18C22×Q8×D61C2214yes—
19C23×C4×D61—21no—
20C23×C4×D61C2211yescharacteristic
21C22×#11758194C2211yescharacteristic
22C2×#14519736C2211yescharacteristic
23#16839772—24no—
24#16839771—24no—
25#16839770—24no—
26C2×#14519702C2214yescomplemented
27C2×#14519701C2214yescomplemented
28C2×#14519700C2214yescomplemented
29#16839725—24no—
30C2×#14519689C2214yes—
31#16839623—21no—
32#16839619C2214yes—
33#16839617—21no—
34C4×#11758198—22no—
35C23×Q8×C61C2211yescharacteristic
36C22×M16×C61C2211yescharacteristic
37C2×#117×C61C2211yescharacteristic
38C61×#419—24no—
39C2×#103×C61C2214yescomplemented
40C24×D61D411yescharacteristic
41C22×Dic122—28no—
42C22×Dic122—28no—
43C22×Dic122—21no—
44C22×Dic122—22no—
45C22×Dic122—22no—
46C22×Dic122—22no—
47C22×Dic122—28no—
48C22×Dic122—24no—
49C22×Dic122C2×C414yes—
50C22×Dic122D414yes—
51C22×Dic122D414yes—
52C22×Dic122C2312yes—
53C2×Q8×D61—216no—
54C2×Q8×D61—216no—
55C2×Q8×D61—232no—
56C2×Q8×D61—232no—
57C2×Q8×D61—232no—
58C2×Q8×D61—28no—
59C2×Q8×D61—28no—
60C2×Q8×D61—216no—
61C2×Q8×D61—28no—
62C2×Q8×D61—232no—
63C2×Q8×D61—28no—
64C2×Q8×D61—28no—
65C2×Q8×D61C2×C4116yes—
66C23×Dic61—21no—
67C23×Dic61D411yescharacteristic
68C22×C4×D61—28no—
69C22×C4×D61—28no—
70C22×C4×D61—28no—
71C22×C4×D61—24no—
72C22×C4×D61—22no—
73C22×C4×D61—22no—
74C22×C4×D61—22no—
75C22×C4×D61—22no—
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#14519702—28no—
91#14519701—28no—
92#14519700—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—22no—
117C23×D61—22no—
118C23×D61—28no—
119C23×D61C22⋊C414yes—
120C23×D61C2×D412yes—
121C2×Q8×C61—24no—
122C2×Q8×C61—28no—
123C2×Q8×C61—24no—
124C2×Q8×C61—24no—
125C2×Q8×C61—28no—
126C2×Q8×C61—28no—
127C2×Q8×C61—22no—
128C2×Q8×C61—22no—
129C2×Q8×C61—22no—
130C2×Q8×C61C22⋊C4116yescomplemented
131C2×Q8×C61C22×C414yes—
132C2×Q8×C61C2×D414yes—
133C2×Q8×C61C2×D414yes—
134C22×C4×C61—24no—
135C22×C4×C61—24no—
136C22×C4×C61—24no—
137C22×C4×C61—22no—
138C22×C4×C61—22no—
139C22×C4×C61C22×C412yes—
140C22×C4×C61C2×D411yescharacteristic
141C22×C4×C61C2×D412yes—
142C22×C4×C61C2×D412yes—
143C22×C4×C61C2×D412yes—
144C22×C4×C61C2×D412yes—
145C22×C4×C61C2×D412yes—
146C22×C4×C61C2411yescharacteristic
147C22×Dic61—24no—
148C22×Dic61—24no—
149C22×Dic61—24no—
150C22×Dic61—22no—
151C22×Dic61—22no—
152C22×Dic61—22no—
153C22×Dic61—22no—
154C22×Dic61C22⋊C414yes—
155C22×Dic61C2×D412yes—
156Q8×D61—232no—
157Q8×D61—232no—
158Q8×D61—2128no—
159Q8×D61—232no—
160Q8×D61—232no—
161C2×C4×D61—216no—
162C2×C4×D61—216no—
163C2×C4×D61—264no—
164C2×C4×D61—216no—
165C2×C4×D61—28no—
166C2×C4×D61—28no—
167C2×C4×D61—28no—
168C2×C4×D61—28no—
169C2×C4×D61—24no—
170C2×C4×D61—24no—
171C2×C4×D61—22no—
172C2×C4×D61—22no—
173C2×C4×D61—22no—
174C2×C4×D61—22no—
175C2×C4×D61C22⋊C418yes—
176C2×C4×D61C22×C414yes—
177C2×C4×D61C2×D414yes—
178C2×Dic122—216no—
179C2×Dic122—216no—
180C2×Dic122—264no—
181C2×Dic122—28no—
182C2×Dic122—24no—
183C2×Dic122—28no—
184C2×Dic122—28no—
185C2×Dic122—24no—
186C2×Dic122—24no—
187C2×Dic122—22no—
188C2×Dic122—22no—
189C2×Dic122—22no—
190C2×Dic122—216no—
191C2×Dic122C22⋊C4116yescomplemented
192C2×Dic122C22×C414yes—
193C2×Dic122C2×D414yes—
194C2×Dic122C2×D414yes—
195M16×C61—44no—
196C2×C8×C61—22no—
197C2×C8×C61—22no—
198C2×C8×C61—22no—
199C4⋊C4×C61—22no—
200C4⋊C4×C61—22no—
201C22⋊C4×C61—42no—
202C42×C61—42no—
203C22⋊Dic61—42no—
204#11758198—48no—
205C4⋊Dic61—22no—
206C4⋊Dic61—22no—
207C4×Dic61—42no—
208#11758194—44no—
209C2×#92418—22no—
210C2×#92418—22no—
211C2×#92418—22no—
212C23×C61—24no—
213C23×C61C2×C22⋊C412yes—
214C23×C61#12012yes—
215C23×C61#12012yes—
216C23×C61C22×D411yescharacteristic
217C22×D61—216no—
218C22×D61—216no—
219C22×D61—22no—
220C22×D61—22no—
221C22×D61—24no—
222C22×D61—28no—
223C22×D61—216no—
224C22×D61C2×C22⋊C414yes—
225C22×D61#13718yes—
226Q8×C61—24no—
227Q8×C61—24no—
228Q8×C61—24no—
229Q8×C61—216no—
230Q8×C61C2×C22⋊C418yes—
231C2×C4×C61—24no—
232C2×C4×C61—24no—
233C2×C4×C61—24no—
234C2×C4×C61—216no—
235C2×C4×C61—42no—
236C2×C4×C61—44no—
237C2×C4×C61C2×C22⋊C414yes—
238C2×C4×C61C2×C22⋊C414yes—
239C2×C4×C61C2×C22⋊C412yes—
240C2×C4×C61#12014yes—
241C2×C4×C61#12014yes—
242C2×C4×C61#12012yes—
243C2×C4×C61#12012yes—
244C2×C4×C61C23×C411yescharacteristic
245C2×C4×C61C22×D411yescharacteristic
246C2×C4×C61C22×D411yescharacteristic
247C2×C4×C61C22×D411yescharacteristic
248C2×C4×C61C22×D411yescharacteristic
249C2×C4×C61C22×D411yescharacteristic
250C2×Dic61—24no—
251C2×Dic61—22no—
252C2×Dic61—22no—
253C2×Dic61—24no—
254C2×Dic61—24no—
255C2×Dic61—28no—
256C2×Dic61—216no—
257C2×Dic61—42no—
258C2×Dic61—44no—
259C2×Dic61C2×C22⋊C414yes—
260C4×D61—264no—
261C4×D61—24no—
262C4×D61—24no—
263C4×D61—24no—
264C4×D61—24no—
265C4×D61—216no—
266Dic122—264no—
267Dic122—24no—
268Dic122—24no—
269Dic122—24no—
270Dic122—216no—
271Dic122C2×C22⋊C418yes—
272C8×C61—42no—
273#92418—42no—
274#20599—611nomaximal, Sylow
275C22×C61—24no—
276C22×C61—24no—
277C22×C61—24no—
278C22×C61#37914yes—
279C22×C61C22×C22⋊C411yescharacteristic
280C22×C61C2×#12011yescharacteristic
281C22×C61C2×#12011yescharacteristic
282C22×C61C2×#13714yes—
283C2×D61—232no—
284C2×D61—24no—
285C2×D61—28no—
286C2×D61—28no—
287C2×D61—24no—
288C2×D61#41918yescomplemented
289C4×C61—28no—
290C4×C61—82no—
291C4×C61#37914yes—
292C4×C61C22×C22⋊C411yescharacteristic
293C4×C61C22×C22⋊C411yescharacteristic
294C4×C61C2×#12011yescharacteristic
295C4×C61C2×#12011yescharacteristic
296Dic61—24no—
297Dic61—28no—
298Dic61—82no—
299C24×Q8—611no—
300C2×#419—611no—
301C2×#419—611no—
302C2×#419—611no—
303C2×#419D6111yescharacteristic, complemented
304C2×#407—611no—
305#2280—611no—
306#1787—618no—
307C2×C61—24no—
308C2×C61C2×#37911yescharacteristic
309C2×C61C2×#41912yes—
310C2×C61#300412yes—
311C2×C61#300412yes—
312D61—28no—
313C23×Q8—618no—
314C23×Q8—614no—
315C23×Q8—611no—
316C23×Q8—1221no—
317C23×Q8—1222no—
318C23×Q8—1224no—
319C23×Q8—1224no—
320C23×Q8—1224no—
321C23×Q8—1224no—
322C23×Q8—1224no—
323C23×Q8C2×D6111yescharacteristic
324C24×C4—611no—
325C24×C4—1221no—
326C22×M16—611no—
327C22×M16C2×D6111yescharacteristic
328C2×#117—611no—
329C2×#117C2×D6111yescharacteristic
330#419—24no—
331#419—1224no—
332#419—1224no—
333#419—1224no—
334C2×#103—614no—
335C2×#103—614no—
336C2×#103—614no—
337C2×#103C2×D6114yescomplemented
338#407—1224no—
339C2×C22⋊C8—614no—
340#381—1221no—
341#380—614no—
342#379—1221no—
343C4×C22⋊C4—1222no—
344C61#2059911yescharacteristic, complemented, Sylow
345C25—611no—
346C22×Q8—21no—
347C22×Q8—22no—
348C22×Q8—22no—
349C22×Q8—22no—
350C22×Q8—6116no—
351C22×Q8—614no—
352C22×Q8—614no—
353C22×Q8—614no—
354C22×Q8—612no—
355C22×Q8—1224no—
356C22×Q8—1228no—
357C22×Q8—1228no—
358C22×Q8—1228no—
359C22×Q8—1228no—
360C22×Q8—12216no—
361C22×Q8—1228no—
362C22×Q8—1228no—
363C22×Q8—12232no—
364C22×Q8—12216no—
365C22×Q8—12232no—
366C22×Q8—12216no—
367C22×Q8—12232no—
368C22×Q8—12232no—
369C22×Q8—1228no—
370C22×Q8—1228no—
371C22×Q8—1222no—
372C22×Q8—1221no—
373C22×Q8—1222no—
374C22×Q8—1222no—
375C22×Q8C4×D6114yes—
376C22×Q8D24414yes—
377C22×Q8C22⋊D6114yes—
378C22×Q8C22×D6112yes—
379C23×C4—21no—
380C23×C4—611no—
381C23×C4—614no—
382C23×C4—612no—
383C23×C4—614no—
384C23×C4—612no—
385C23×C4—1224no—
386C23×C4—1228no—
387C23×C4—1228no—
388C23×C4—1228no—
389C23×C4—1221no—
390C23×C4—1222no—
391C23×C4—1222no—
392C23×C4—1222no—
393C23×C4—1222no—
394C23×C4C22×D6111yescharacteristic
395C2×M16—22no—
396C2×M16—22no—
397C2×M16—22no—
398C2×M16—1222no—
399C2×M16—1222no—
400C2×M16—1222no—
401C22×C8—612no—
402C22×C8C22×D6112yes—
403#117—22no—
404#117—22no—
405#117—1222no—
406#117—1222no—
407C2×C4⋊C4—612no—
408C2×C4⋊C4C22×D6112yes—
409C2×C22⋊C4—21no—
410C2×C22⋊C4—1224no—
411C2×C22⋊C4—1221no—
412C2×C42—21no—
413C2×C42—1221no—
414#103—28no—
415#103—1228no—
416#103—1228no—
417#103—1228no—
418C22⋊C8—1228no—
419#95—1224no—
420C24—612no—
421C24—614no—
422C24—1228no—
423C24—1222no—
424C24—1222no—
425C24D4×D6111yescharacteristic
426C2×Q8—22no—
427C2×Q8—22no—
428C2×Q8—22no—
429C2×Q8—24no—
430C2×Q8—28no—
431C2×Q8—24no—
432C2×Q8—28no—
433C2×Q8—24no—
434C2×Q8—28no—
435C2×Q8—614no—
436C2×Q8—614no—
437C2×Q8—614no—
438C2×Q8—6116no—
439C2×Q8—12216no—
440C2×Q8—12216no—
441C2×Q8—12216no—
442C2×Q8—12264no—
443C2×Q8—12232no—
444C2×Q8—12232no—
445C2×Q8—12232no—
446C2×Q8—122128no—
447C2×Q8—12232no—
448C2×Q8—1222no—
449C2×Q8—1222no—
450C2×Q8—1222no—
451C2×Q8—1224no—
452C2×Q8—1228no—
453C2×Q8—1224no—
454C2×Q8—1228no—
455C2×Q8—1224no—
456C2×Q8—1228no—
457C2×Q8#11758198116yescomplemented
458C2×Q8C2×C4×D6114yes—
459C2×Q8C2×D24414yes—
460C2×Q8C2×C22⋊D6114yes—
461C22×C4—22no—
462C22×C4—24no—
463C22×C4—24no—
464C22×C4—24no—
465C22×C4—22no—
466C22×C4—612no—
467C22×C4—614no—
468C22×C4—614no—
469C22×C4—614no—
470C22×C4—618no—
471C22×C4—12216no—
472C22×C4—12216no—
473C22×C4—12216no—
474C22×C4—12264no—
475C22×C4—1222no—
476C22×C4—1224no—
477C22×C4—1222no—
478C22×C4—1224no—
479C22×C4—1222no—
480C22×C4—1224no—
481C22×C4—1222no—
482C22×C4—1222no—
483C22×C4—1222no—
484C22×C4—1222no—
485C22×C4—1228no—
486C22×C4—1228no—
487C22×C4—1224no—
488C22×C4—1228no—
489C22×C4—1224no—
490C22×C4—1228no—
491C22×C4—1222no—
492C22×C4C2×C4×D6112yes—
493C22×C4C2×D24412yes—
494C22×C4D4×D6112yes—
495C22×C4D4×D6112yes—
496C22×C4D4×D6112yes—
497C22×C4D4×D6111yescharacteristic
498C22×C4C2×C22⋊D6112yes—
499C22×C4C23×D6111yescharacteristic
500M16—44no—
501M16—2444no—
502C2×C8—22no—
503C2×C8—22no—
504C2×C8—22no—
505C2×C8—1222no—
506C2×C8—1222no—
507C2×C8—1222no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—1222no—
511C4⋊C4—1222no—
512C22⋊C4—42no—
513C22⋊C4—2448no—
514C22⋊C4—2442no—
515C42—42no—
516C42—2442no—
517C23—24no—
518C23—614no—
519C23—618no—
520C23—12216no—
521C23—12216no—
522C23—12216no—
523C23—1222no—
524C23—1222no—
525C23—1224no—
526C23—1228no—
527C23C22⋊C4×D6112yes—
528C23#1451974512yes—
529C23#1451974712yes—
530C23C2×D4×D6111yescharacteristic
531Q8—24no—
532Q8—24no—
533Q8—24no—
534Q8—216no—
535Q8—618no—
536Q8—12264no—
537Q8—1224no—
538Q8—1224no—
539Q8—1224no—
540Q8—12216no—
541Q8C2×#1175819818yes—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—216no—
546C2×C4—42no—
547C2×C4—44no—
548C2×C4—614no—
549C2×C4—12264no—
550C2×C4—1224no—
551C2×C4—1222no—
552C2×C4—1224no—
553C2×C4—1222no—
554C2×C4—1224no—
555C2×C4—1228no—
556C2×C4—12216no—
557C2×C4—1224no—
558C2×C4—1224no—
559C2×C4—1224no—
560C2×C4—1224no—
561C2×C4—12216no—
562C2×C4—2442no—
563C2×C4—2444no—
564C2×C4C22⋊C4×D6114yes—
565C2×C4C22⋊C4×D6112yes—
566C2×C4C2×#1175819814yes—
567C2×C4#1451974514yes—
568C2×C4#1451974512yes—
569C2×C4#1451974714yes—
570C2×C4#1451974712yes—
571C2×C4C22×C4×D6111yescharacteristic
572C2×C4C2×D4×D6111yescharacteristic
573C2×C4C2×D4×D6111yescharacteristic
574C2×C4C2×D4×D6111yescharacteristic
575C2×C4C22×C22⋊D6111yescharacteristic
576C2×C4C22×D24411yescharacteristic
577C8—42no—
578C8—2442no—
579C22—24no—
580C22—24no—
581C22—24no—
582C22—618no—
583C22—12232no—
584C22—1228no—
585C22—1228no—
586C22—1224no—
587C22—1224no—
588C22#1683961714yes—
589C22C2×C22⋊C4×D6111yescharacteristic
590C22C2×#1451974511yescharacteristic
591C22C2×#1451974711yescharacteristic
592C22#137×D6114yes—
593C4—28no—
594C4—82no—
595C4—1224no—
596C4—1228no—
597C4—4882no—
598C4#1683961714yes—
599C4C2×C22⋊C4×D6111yescharacteristic
600C4C22×#1175819811yescharacteristic
601C4C2×#1451974511yescharacteristic
602C4C2×#1451974711yescharacteristic
603C2—24no—
604C2—1228no—
605C2C2×#1683961711yescharacteristic
606C2#419×D6112yes—
607C2#2145505112yes—
608C2#2145505512yes—
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC2×C61×#419C2
socleC23×C61C22×D4
prime radicalC24×D61D4
cosocleC22×C4×C61C24
hypercenterC2×#419D61
hypocenterC61#20599
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×#16839772C2semidirect
3. C2×#16839771C2semidirect
4. C2×#16839770C2semidirect
7. #21445752C2semidirect
8. C2×C61×#419C2semidirect
17. C22×Q8×D61C4semidirect
26. C2×#14519702C22semidirect
27. C2×#14519701C22semidirect
28. C2×#14519700C22semidirect
39. C2×#103×C61C22semidirect
130. C2×Q8×C61C22⋊C4semidirect
191. C2×Dic122C22⋊C4semidirect
288. C2×D61#419semidirect
303. C2×#419D61semidirect
337. C2×#103C2×D61semidirect
344. C61#20599semidirect
457. C2×Q8#11758198semidirect

Maximal subgroups

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

subgroupindex
C23×Q8×D612
C2×#168397722
C2×#168397712
C2×#168397702
C2×#168397252
#214481982
#214457522
C2×C61×#4192
#2059961