Finite Group Databasebeta

← #29586524

Subgroups — #29586524

Order 15616.

1578 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
1C22×#14519756C211yesmaximal, characteristic
2C22×#14519700C212yesmaximal, complemented
3C22×C22⋊#92418C211yesmaximal, characteristic
4C2×#16839524C212yesmaximal, complemented
5C2×C61×#380C211yesmaximal, characteristic
6C23×Dic122C2211yescharacteristic
7C23×#92418C2212yes—
8C2×#14519756—24no—
9C2×#14519756—22no—
10C2×#14519756—24no—
11C2×#14519756C414yescomplemented
12C22×C4⋊Dic61C2211yescharacteristic
13C22×#11758196—21no—
14C22×C22⋊Dic61—21no—
15C2×#14519700—24no—
16C2×#14519700—24no—
17C2×#14519700—24no—
18C2×C22⋊#92418—24no—
19C2×C22⋊#92418—22no—
20#16839524—24no—
21C24×C4×C61C2211yescharacteristic
22C22×C4⋊C4×C61C2212yes—
23C22×C22⋊C4×C61—21no—
24C61×#380—24no—
25C2×#95×C61—21no—
26C22×Dic122—22no—
27C22×Dic122—22no—
28C22×Dic122—22no—
29C22×Dic122—22no—
30C22×Dic122—22no—
31C22×Dic122—21no—
32C22×Dic122C2×C412yes—
33C22×Dic122D414yes—
34C23×Dic61—21no—
35C23×Dic61—21no—
36C22×#92418—22no—
37C22×#92418—22no—
38C22×#92418—22no—
39C22×#92418—22no—
40C22×#92418—22no—
41C22×#92418—22no—
42C22×#92418D414yes—
43#14519756—216no—
44#14519756—28no—
45#14519756—28no—
46C2×C4⋊Dic61—21no—
47C2×C4⋊Dic61—22no—
48C2×C4⋊Dic61C2×C412yes—
49C2×C4⋊Dic61D414yes—
50C2×#11758196—28no—
51C2×#11758196—24no—
52C2×C22⋊Dic61—28no—
53C2×C22⋊Dic61—24no—
54#14519700—216no—
55C22⋊#92418—28no—
56C25×C61D411yescharacteristic
57C23×C4×C61—22no—
58C23×C4×C61—22no—
59C23×C4×C61—22no—
60C23×C4×C61—22no—
61C23×C4×C61—22no—
62C23×C4×C61—22no—
63C23×C4×C61—22no—
64C23×C4×C61C2×C412yes—
65C23×C4×C61D412yes—
66C23×C4×C61D411yescharacteristic
67C23×C4×C61C2311yescharacteristic
68C2×C4⋊C4×C61—22no—
69C2×C4⋊C4×C61—22no—
70C2×C4⋊C4×C61—22no—
71C2×C4⋊C4×C61—22no—
72C2×C4⋊C4×C61—22no—
73C2×C4⋊C4×C61D414yes—
74C2×C22⋊C4×C61—24no—
75C2×C22⋊C4×C61—42no—
76C2×C22⋊C4×C61—42no—
77#95×C61—24no—
78C24×C61—21no—
79C24×C61—22no—
80C24×C61—22no—
81C24×C61—21no—
82C24×C61—22no—
83C24×C61—21no—
84C24×C61—21no—
85C24×C61—22no—
86C24×C61C22⋊C412yes—
87C24×C61SD1612yes—
88C24×C61Q1612yes—
89C24×C61C2×D411yescharacteristic
90C22×C4×C61—21no—
91C22×C4×C61—22no—
92C22×C4×C61—22no—
93C22×C4×C61—21no—
94C22×C4×C61—22no—
95C22×C4×C61—22no—
96C22×C4×C61—24no—
97C22×C4×C61—24no—
98C22×C4×C61—22no—
99C22×C4×C61—24no—
100C22×C4×C61—24no—
101C22×C4×C61—22no—
102C22×C4×C61—24no—
103C22×C4×C61—42no—
104C22×C4×C61—42no—
105C22×C4×C61—42no—
106C22×C4×C61—42no—
107C22×C4×C61—42no—
108C22×C4×C61—42no—
109C22×C4×C61—42no—
110C22×C4×C61—42no—
111C22×C4×C61—42no—
112C22×C4×C61—42no—
113C22×C4×C61—42no—
114C22×C4×C61—42no—
115C22×C4×C61—42no—
116C22×C4×C61—42no—
117C22×C4×C61C22⋊C412yes—
118C22×C4×C61C22×C411yescharacteristic
119C22×C4×C61C2×D412yes—
120C22×C4×C61C2×D412yes—
121C22×C4×C61C2×D412yes—
122C22×C4×C61C2×D412yes—
123C22×C4×C61C2×D411yescharacteristic
124C22×C4×C61#4112yes—
125C22×C4×C61#4112yes—
126C22×Dic61—22no—
127C22×Dic61—22no—
128C22×Dic61—24no—
129C22×Dic61—24no—
130C22×Dic61—41no—
131C22×Dic61—42no—
132C22×Dic61—41no—
133C22×Dic61—42no—
134C22×Dic61—41no—
135C22×Dic61—41no—
136C2×Dic122—24no—
137C2×Dic122—24no—
138C2×Dic122—24no—
139C2×Dic122—24no—
140C2×Dic122—24no—
141C2×Dic122—24no—
142C2×Dic122—22no—
143C2×Dic122—24no—
144C2×Dic122—41no—
145C2×Dic122—42no—
146C2×Dic122—42no—
147C2×Dic122—42no—
148C2×Dic122—41no—
149C2×Dic122—42no—
150C2×Dic122—42no—
151C2×Dic122C22⋊C414yescomplemented
152C4⋊C4×C61—24no—
153C4⋊C4×C61—24no—
154C4⋊C4×C61—42no—
155C4⋊C4×C61—42no—
156C22⋊C4×C61—48no—
157C22⋊Dic61—216no—
158C4⋊Dic61—22no—
159C4⋊Dic61—24no—
160C4⋊Dic61C22⋊C414yescomplemented
161#11758196—216no—
162C2×#92418—24no—
163C2×#92418—24no—
164C2×#92418—24no—
165C2×#92418—42no—
166C2×#92418—42no—
167C2×#92418—42no—
168C2×#92418—42no—
169C23×C61—22no—
170C23×C61—22no—
171C23×C61—22no—
172C23×C61—22no—
173C23×C61—22no—
174C23×C61—22no—
175C23×C61—24no—
176C23×C61—22no—
177C23×C61—22no—
178C23×C61—24no—
179C23×C61—24no—
180C23×C61—24no—
181C23×C61—22no—
182C23×C61—22no—
183C23×C61—42no—
184C23×C61—42no—
185C23×C61—42no—
186C23×C61—42no—
187C23×C61—42no—
188C23×C61—42no—
189C23×C61—42no—
190C23×C61—42no—
191C23×C61#10314yescomplemented
192C23×C61C2×C22⋊C411yescharacteristic
193C23×C61#12012yes—
194C23×C61#12011yescharacteristic
195C23×C61#12112yes—
196C23×C61#12311yescharacteristic
197C23×C61C2×SD1611yescharacteristic
198C23×C61C2×Q1611yescharacteristic
199C23×C61#13512yes—
200C23×C61AGL(1,ℤ/8)11yescharacteristic
201C23×C61#13711yescharacteristic
202C23×C61#13712yes—
203C2×C4×C61—22no—
204C2×C4×C61—22no—
205C2×C4×C61—22no—
206C2×C4×C61—42no—
207C2×C4×C61—42no—
208C2×C4×C61—44no—
209C2×C4×C61—44no—
210C2×C4×C61—44no—
211C2×C4×C61—44no—
212C2×C4×C61—44no—
213C2×C4×C61—44no—
214C2×C4×C61—44no—
215C2×C4×C61—82no—
216C2×C4×C61—82no—
217C2×C4×C61—82no—
218C2×C4×C61—82no—
219C2×C4×C61C2×C22⋊C411yescharacteristic
220C2×C4×C61C2×C22⋊C411yescharacteristic
221C2×C4×C61C4×D412yes—
222C2×C4×C61C4×D412yes—
223C2×C4×C61#12012yes—
224C2×C4×C61#12011yescharacteristic
225C2×C4×C61#12012yes—
226C2×C4×C61#12112yes—
227C2×C4×C61#12112yes—
228C2×C4×C61#12112yes—
229C2×C4×C61#12112yes—
230C2×C4×C61#12112yes—
231C2×C4×C61#12312yes—
232C2×C4×C61#12311yescharacteristic
233C2×C4×C61#12412yes—
234C2×C4×C61#12712yes—
235C2×Dic61—24no—
236C2×Dic61—24no—
237C2×Dic61—42no—
238C2×Dic61—42no—
239C2×Dic61—44no—
240C2×Dic61—44no—
241C2×Dic61—42no—
242C2×Dic61—42no—
243C2×Dic61—44no—
244C2×Dic61—44no—
245Dic122—28no—
246Dic122—44no—
247Dic122—44no—
248Dic122—44no—
249#92418—44no—
250#10970—611nomaximal, Sylow
251C22×C61—24no—
252C22×C61—24no—
253C22×C61—24no—
254C22×C61—42no—
255C22×C61—42no—
256C22×C61—42no—
257C22×C61—44no—
258C22×C61—44no—
259C22×C61—44no—
260C22×C61—44no—
261C22×C61—42no—
262C22×C61#37911yescharacteristic
263C22×C61#38611yescharacteristic
264C22×C61#38611yescharacteristic
265C22×C61#39212yes—
266C22×C61#39412yes—
267C22×C61C2×#10311yescharacteristic
268C22×C61#41611yescharacteristic
269C22×C61#41711yescharacteristic
270C22×C61#41911yescharacteristic
271C22×C61#44812yes—
272C22×C61#44912yes—
273C22×C61#45011yescharacteristic
274C22×C61#45112yes—
275C22×C61#45111yescharacteristic
276C22×C61#45212yes—
277C22×C61#45212yes—
278C22×C61#46512yes—
279C22×C61#46712yes—
280C22×C61#46812yes—
281C22×C61#47012yes—
282C22×C61#48311yescharacteristic
283C22×C61#48411yescharacteristic
284C22×C61#48512yes—
285C4×C61—42no—
286C4×C61—84no—
287C4×C61#38612yes—
288C4×C61#39012yes—
289C4×C61#39212yes—
290C4×C61#39412yes—
291Dic61—44no—
292Dic61—44no—
293C22×C22⋊Q8—611no—
294C22×#103—612no—
295C22×C22⋊C8—611no—
296C2×#380D6111yescharacteristic
297C2×#340—612no—
298C2×C61—44no—
299C2×C61#143811yescharacteristic
300C2×C61#178711yescharacteristic
301C2×C61#178811yescharacteristic
302C2×C61#187211yescharacteristic
303C2×C61#187311yescharacteristic
304C2×C61#189211yescharacteristic
305C2×C61#189411yescharacteristic
306C2×C61#200012yes—
307C2×C61#200112yes—
308C2×C61#203512yes—
309C2×C61#203612yes—
310C23×Q8—611no—
311C24×C4C2×D6111yescharacteristic
312C23×C8—612no—
313C2×C22⋊Q8—614no—
314C2×C22⋊Q8—1224no—
315C2×C22⋊Q8—1224no—
316C2×C22⋊Q8—1222no—
317C22×C4⋊C4—611no—
318C22×C4⋊C4—1221no—
319C22×C4⋊C4C2×D6112yes—
320C22×C22⋊C4—21no—
321C22×C22⋊C4—1221no—
322C2×#103—1224no—
323C2×#103—1224no—
324C2×#103—1224no—
325C2×C22⋊C8—1224no—
326C2×C22⋊C8—1222no—
327#380—24no—
328C2×#95—21no—
329#340—1224no—
330C61#1097011yescharacteristic, complemented, Sylow
331C25C22⋊D6111yescharacteristic
332C22×Q8—612no—
333C22×Q8—614no—
334C22×Q8—1221no—
335C22×Q8—1222no—
336C22×Q8—1222no—
337C22×Q8—1222no—
338C22×Q8—1222no—
339C22×Q8—1222no—
340C23×C4—22no—
341C23×C4—22no—
342C23×C4—22no—
343C23×C4—22no—
344C23×C4—22no—
345C23×C4—22no—
346C23×C4—22no—
347C23×C4—1221no—
348C23×C4—1221no—
349C23×C4C4×D6112yes—
350C23×C4D24412yes—
351C23×C4C22⋊D6111yescharacteristic
352C23×C4C22×D6111yescharacteristic
353C22×C8—614no—
354C22×C8—1222no—
355C22×C8—1222no—
356C22×C8—1222no—
357C22×C8—1222no—
358C22×C8—1222no—
359C22×C8—1222no—
360C22⋊Q8—1228no—
361C22⋊Q8—1228no—
362C22⋊Q8—12216no—
363C2×C4⋊C4—22no—
364C2×C4⋊C4—22no—
365C2×C4⋊C4—22no—
366C2×C4⋊C4—22no—
367C2×C4⋊C4—22no—
368C2×C4⋊C4—612no—
369C2×C4⋊C4—614no—
370C2×C4⋊C4—1228no—
371C2×C4⋊C4—1224no—
372C2×C4⋊C4—1221no—
373C2×C4⋊C4—1222no—
374C2×C4⋊C4C22⋊D6114yes—
375C2×C22⋊C4—24no—
376C2×C22⋊C4—42no—
377C2×C22⋊C4—42no—
378C2×C22⋊C4—1228no—
379C2×C22⋊C4—1224no—
380#103—12216no—
381C22⋊C8—1228no—
382#95—24no—
383C24—21no—
384C24—22no—
385C24—21no—
386C24—22no—
387C24—21no—
388C24—22no—
389C24—21no—
390C24—22no—
391C24#1175819812yes—
392C24#1175820012yes—
393C24#1175820212yes—
394C24C2×C22⋊D6111yescharacteristic
395C2×Q8—614no—
396C2×Q8—1222no—
397C2×Q8—1224no—
398C2×Q8—1224no—
399C2×Q8—1224no—
400C2×Q8—1224no—
401C2×Q8—1224no—
402C2×Q8—1224no—
403C2×Q8—1224no—
404C2×Q8—2441no—
405C2×Q8—2442no—
406C2×Q8—2442no—
407C2×Q8—2442no—
408C2×Q8—2441no—
409C2×Q8—2442no—
410C2×Q8—2442no—
411C22×C4—22no—
412C22×C4—21no—
413C22×C4—22no—
414C22×C4—21no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—1222no—
439C22×C4—1222no—
440C22×C4—1224no—
441C22×C4—1224no—
442C22×C4—2441no—
443C22×C4—2441no—
444C22×C4—2442no—
445C22×C4—2441no—
446C22×C4—2442no—
447C22×C4—2441no—
448C22×C4#1175819812yes—
449C22×C4C2×C4×D6111yescharacteristic
450C22×C4C2×D24411yescharacteristic
451C22×C4#1175822212yes—
452C22×C4D4×D6112yes—
453C22×C4D4×D6112yes—
454C22×C4D4×D6112yes—
455C22×C4#1175822412yes—
456C22×C4C2×C22⋊D6112yes—
457C2×C8—1224no—
458C2×C8—1224no—
459C2×C8—1224no—
460C2×C8—2442no—
461C2×C8—2442no—
462C2×C8—2442no—
463C2×C8—2442no—
464C4⋊C4—24no—
465C4⋊C4—24no—
466C4⋊C4—42no—
467C4⋊C4—42no—
468C4⋊C4—614no—
469C4⋊C4—12216no—
470C4⋊C4—1222no—
471C4⋊C4—1224no—
472C22⋊C4—48no—
473C22⋊C4—12216no—
474C23—24no—
475C23—24no—
476C23—22no—
477C23—22no—
478C23—22no—
479C23—22no—
480C23—22no—
481C23—22no—
482C23—22no—
483C23—22no—
484C23—24no—
485C23—22no—
486C23—24no—
487C23—22no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23—42no—
493C23—42no—
494C23—42no—
495C23—42no—
496C23#1451970014yescomplemented
497C23C2×#1175819811yescharacteristic
498C23#1451974512yes—
499C23#1451974611yescharacteristic
500C23#1451974912yes—
501C23#1451976011yescharacteristic
502C23C2×#1175820011yescharacteristic
503C23C2×#1175820211yescharacteristic
504C23#1451979712yes—
505C23#1451980311yescharacteristic
506C23#1451980712yes—
507C23#1451980811yescharacteristic
508Q8—1228no—
509Q8—2444no—
510Q8—2444no—
511Q8—2444no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—44no—
518C2×C4—44no—
519C2×C4—44no—
520C2×C4—44no—
521C2×C4—44no—
522C2×C4—44no—
523C2×C4—44no—
524C2×C4—82no—
525C2×C4—82no—
526C2×C4—82no—
527C2×C4—82no—
528C2×C4—1224no—
529C2×C4—1224no—
530C2×C4—2442no—
531C2×C4—2442no—
532C2×C4—2442no—
533C2×C4—2442no—
534C2×C4—2444no—
535C2×C4—2444no—
536C2×C4—2444no—
537C2×C4—2444no—
538C2×C4C22⋊C4×D6111yescharacteristic
539C2×C4C22⋊C4×D6111yescharacteristic
540C2×C4#1451973712yes—
541C2×C4C4×C22⋊D6112yes—
542C2×C4#1451974512yes—
543C2×C4#1451974711yescharacteristic
544C2×C4#1451974712yes—
545C2×C4#1451974812yes—
546C2×C4#1451974912yes—
547C2×C4#1451975012yes—
548C2×C4#1451975012yes—
549C2×C4#1451975112yes—
550C2×C4#1451975912yes—
551C2×C4#1451976111yescharacteristic
552C2×C4#1451976312yes—
553C2×C4#1451977312yes—
554C8—2444no—
555C22—24no—
556C22—24no—
557C22—24no—
558C22—44no—
559C22—42no—
560C22—44no—
561C22—42no—
562C22—44no—
563C22—44no—
564C22—42no—
565C22—42no—
566C22#1683961711yescharacteristic
567C22#1683963911yescharacteristic
568C22#1683964111yescharacteristic
569C22#1683967012yes—
570C22#1683967712yes—
571C22C2×#1451970011yescharacteristic
572C22#1683975611yescharacteristic
573C22#1683976111yescharacteristic
574C22#1683977011yescharacteristic
575C22#1683990012yes—
576C22#1683990512yes—
577C22#1683991311yescharacteristic
578C22#1683991712yes—
579C22#1683991811yescharacteristic
580C22#1683992212yes—
581C22#1683992412yes—
582C22#1683998912yes—
583C22#1684000112yes—
584C22#1684000612yes—
585C22#1684001812yes—
586C22#1684009211yescharacteristic
587C22#1684009711yescharacteristic
588C22#1684010312yes—
589C4—42no—
590C4—84no—
591C4—2444no—
592C4—2444no—
593C4#1683963812yes—
594C4#1683966212yes—
595C4#1683967112yes—
596C4#1683967812yes—
597C2—44no—
598C2#2144405811yescharacteristic
599C2#2144575311yescharacteristic
600C2#2144575711yescharacteristic
601C2#2144612611yescharacteristic
602C2#2144613111yescharacteristic
603C2#2144624011yescharacteristic
604C2#2144625011yescharacteristic
605C2#2144682212yes—
606C2#2144682712yes—
607C2#2144701512yes—
608C2#2144702212yes—
subgroupG/H
centerC24C2×C22⋊D61
commutatorC22×C4×C61C22×C4
FrattiniC23×C4C22×D61
FittingC2×C61×#380C2
socleC24×C61C2×D4
prime radicalC25×C61D4
cosocleC23×C4×C61C23
hypercenterC2×#380D61
hypocenterC61#10970
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C24 ◁ C24×C4 ◁ C2×#380

Complements

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

NH ≅ G/Ntype
2. C22×#14519700C2semidirect
4. C2×#16839524C2semidirect
11. C2×#14519756C4semidirect
151. C2×Dic122C22⋊C4semidirect
160. C4⋊Dic61C22⋊C4semidirect
191. C23×C61#103semidirect
330. C61#10970semidirect
496. C23#14519700semidirect

Maximal subgroups

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

subgroupindex
C22×#145197562
C22×#145197002
C22×C22⋊#924182
C2×#168395242
C2×C61×#3802
#1097061