Finite Group Databasebeta

← #29575576

Subgroups — #29575576

Order 15616.

1062 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C22⋊#92418C211yesmaximal, characteristic, complemented
2#21454101C211yesmaximal, characteristic
3C2×#16839495C212yesmaximal, complemented
4#21443698C212yesmaximal, complemented
5C61×C22⋊#95C211yesmaximal, characteristic, complemented
6C23×#92418—21no—
7C2×#14519751C411yescharacteristic, complemented
8C2×#14519751C411yescharacteristic, complemented
9C2×#14519751C2211yescharacteristic
10C2×#14519746—21no—
11C2×C22⋊#92418—24no—
12C2×C22⋊#92418—21no—
13C2×C22⋊#92418C2212yes—
14#16839641—21no—
15#16839641—21no—
16#16839641—21no—
17#16839622C411yescharacteristic, complemented
18#16839622C411yescharacteristic, complemented
19#16839622C2211yescharacteristic
20#16839495—24no—
21C24×C4×C61C2211yescharacteristic
22C22×C22⋊C4×C61—21no—
23C2×#95×C61—21no—
24C2×#95×C61C2212yes—
25C22×D244C2×C411yescharacteristic
26C22×C22⋊D61—21no—
27C22×C22⋊D61—21no—
28C22×C22⋊D61—21no—
29C22×#92418—24no—
30C22×#92418—22no—
31C22×#92418—42no—
32C22×#92418—42no—
33#14519751—22no—
34#14519751—22no—
35#14519751—42no—
36#14519751—42no—
37#14519746—42no—
38#14519746—42no—
39C2×C4⋊Dic61C2×C411yescharacteristic
40C2×C4⋊Dic61C2×C411yescharacteristic
41C2×C4⋊Dic61C2×C411yescharacteristic
42C2×C22⋊Dic61—21no—
43C2×C22⋊Dic61—21no—
44C2×C22⋊Dic61—21no—
45C2×#11758198—21no—
46C2×#11758198—21no—
47C2×#11758198—21no—
48C22⋊#92418—24no—
49C22⋊#92418—22no—
50C22⋊#92418D414yescomplemented
51#14519684—21no—
52#14519684—21no—
53#14519684—21no—
54C25×C61D411yescharacteristic
55C23×C4×C61—22no—
56C23×C4×C61—22no—
57C23×C4×C61—22no—
58C23×C4×C61—42no—
59C23×C4×C61—42no—
60C23×C4×C61C2×C411yescharacteristic
61C23×C4×C61C2×C411yescharacteristic
62C23×C4×C61D411yescharacteristic
63C23×C4×C61D411yescharacteristic
64C23×C4×C61D411yescharacteristic
65C23×C4×C61C2311yescharacteristic
66C2×C22⋊C4×C61—24no—
67C2×C22⋊C4×C61—42no—
68C2×C22⋊C4×C61—42no—
69#95×C61—22no—
70#95×C61—22no—
71#95×C61—42no—
72C24×C61—22no—
73C24×C61—22no—
74C24×C61—22no—
75C24×C61—22no—
76C24×C61—41no—
77C24×C61—41no—
78C24×C61C22⋊C411yescharacteristic
79C24×C61C22⋊C411yescharacteristic
80C24×C61D812yes—
81C24×C61SD1612yes—
82C24×C61C2×D411yescharacteristic
83C23×D61—21no—
84C22×C4×C61—22no—
85C22×C4×C61—22no—
86C22×C4×C61—21no—
87C22×C4×C61—21no—
88C22×C4×C61—22no—
89C22×C4×C61—22no—
90C22×C4×C61—22no—
91C22×C4×C61—22no—
92C22×C4×C61—22no—
93C22×C4×C61—22no—
94C22×C4×C61—21no—
95C22×C4×C61—21no—
96C22×C4×C61—24no—
97C22×C4×C61—41no—
98C22×C4×C61—41no—
99C22×C4×C61—41no—
100C22×C4×C61—41no—
101C22×C4×C61—41no—
102C22×C4×C61—41no—
103C22×C4×C61—41no—
104C22×C4×C61—48no—
105C22×C4×C61—48no—
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×C61C22⋊C411yescharacteristic
113C22×C4×C61C22⋊C411yescharacteristic
114C22×C4×C61C22×C411yescharacteristic
115C22×C4×C61C2×D411yescharacteristic
116C22×C4×C61C2×D412yes—
117C22×C4×C61#4112yes—
118C2×C22⋊D61—42no—
119C2×C22⋊D61—42no—
120C2×C22⋊D61—42no—
121C2×C22⋊D61—42no—
122C2×C22⋊D61—42no—
123C2×C22⋊D61—42no—
124C2×C22⋊D61—42no—
125C2×C22⋊D61—41no—
126C2×C22⋊D61—41no—
127C2×C22⋊D61—42no—
128C22×Dic61—21no—
129C22×Dic61—21no—
130C22×Dic61—21no—
131C2×D244—22no—
132C2×D244—41no—
133C2×D244—41no—
134C22⋊C4×C61—48no—
135C22⋊Dic61—42no—
136C22⋊Dic61—42no—
137C22⋊Dic61—41no—
138C22⋊Dic61—41no—
139#11758198—42no—
140#11758198—42no—
141#11758198—41no—
142#11758198—41no—
143C4⋊Dic61—22no—
144C4⋊Dic61—41no—
145C4⋊Dic61—41no—
146C2×#92418—24no—
147C2×#92418—48no—
148C2×#92418—42no—
149C2×#92418—42no—
150C23×C61—21no—
151C23×C61—21no—
152C23×C61—21no—
153C23×C61—21no—
154C23×C61—21no—
155C23×C61—21no—
156C23×C61—21no—
157C23×C61—21no—
158C23×C61—24no—
159C23×C61—24no—
160C23×C61—21no—
161C23×C61—21no—
162C23×C61—21no—
163C23×C61—21no—
164C23×C61—21no—
165C23×C61—21no—
166C23×C61—24no—
167C23×C61—24no—
168C23×C61—42no—
169C23×C61—42no—
170C23×C61—44no—
171C23×C61—44no—
172C23×C61—44no—
173C23×C61—44no—
174C23×C61#9912yes—
175C23×C61#10212yes—
176C23×C61#10212yes—
177C23×C61#10412yes—
178C23×C61C2×C22⋊C411yescharacteristic
179C23×C61#12112yes—
180C23×C61C2×D811yescharacteristic
181C23×C61C2×SD1611yescharacteristic
182C23×C61#13512yes—
183C22×D61—41no—
184C22×D61—41no—
185C22×D61—41no—
186C22×D61—41no—
187C22×D61—41no—
188C22×D61—41no—
189C22×D61—41no—
190C2×C4×C61—24no—
191C2×C4×C61—24no—
192C2×C4×C61—21no—
193C2×C4×C61—21no—
194C2×C4×C61—21no—
195C2×C4×C61—21no—
196C2×C4×C61—21no—
197C2×C4×C61—21no—
198C2×C4×C61—21no—
199C2×C4×C61—21no—
200C2×C4×C61—21no—
201C2×C4×C61—42no—
202C2×C4×C61—42no—
203C2×C4×C61—44no—
204C2×C4×C61—44no—
205C2×C4×C61—44no—
206C2×C4×C61—42no—
207C2×C4×C61—42no—
208C2×C4×C61—44no—
209C2×C4×C61—44no—
210C2×C4×C61—44no—
211C2×C4×C61—82no—
212C2×C4×C61—82no—
213C2×C4×C61—82no—
214C2×C4×C61—82no—
215C2×C4×C61#9912yes—
216C2×C4×C61C4×D412yes—
217C2×C4×C61#12011yescharacteristic
218C2×C4×C61#12311yescharacteristic
219C22⋊D61—84no—
220C22⋊D61—84no—
221C22⋊D61—82no—
222C22⋊D61—82no—
223C2×Dic61—41no—
224C2×Dic61—41no—
225C2×Dic61—41no—
226C2×Dic61—41no—
227C2×Dic61—41no—
228C2×Dic61—41no—
229C2×Dic61—41no—
230C2×Dic61—41no—
231C2×Dic61—41no—
232D244—82no—
233#92418—44no—
234#9113—611nomaximal, Sylow
235C22×C61—22no—
236C22×C61—22no—
237C22×C61—22no—
238C22×C61—22no—
239C22×C61—22no—
240C22×C61—21no—
241C22×C61—21no—
242C22×C61—22no—
243C22×C61—22no—
244C22×C61—22no—
245C22×C61—22no—
246C22×C61—21no—
247C22×C61—22no—
248C22×C61—21no—
249C22×C61—22no—
250C22×C61—22no—
251C22×C61—22no—
252C22×C61—48no—
253C22×C61—48no—
254C22×C61—41no—
255C22×C61—41no—
256C22×C61—41no—
257C22×C61—41no—
258C22×C61#32714yescomplemented
259C22×C61#38611yescharacteristic
260C22×C61C2×#9911yescharacteristic
261C22×C61C2×#9911yescharacteristic
262C22×C61C2×#10211yescharacteristic
263C22×C61#41611yescharacteristic
264C22×C61C2×#10411yescharacteristic
265C22×C61#42111yescharacteristic
266C22×C61#46512yes—
267C22×C61#46612yes—
268C2×D61—81no—
269C2×D61—81no—
270C2×D61—81no—
271C2×D61—81no—
272C2×D61—81no—
273C2×D61—81no—
274C2×D61—81no—
275C4×C61—21no—
276C4×C61—22no—
277C4×C61—22no—
278C4×C61—21no—
279C4×C61—41no—
280C4×C61—84no—
281Dic61—81no—
282Dic61—81no—
283Dic61—81no—
284C22×C22⋊C8—611no—
285#2847—611no—
286C2×#327—612no—
287C22⋊#95D6111yescharacteristic, complemented
288#1294—612no—
289C2×C61—22no—
290C2×C61—24no—
291C2×C61—22no—
292C2×C61—42no—
293C2×C61C2×#32711yescharacteristic
294C2×C61#150111yescharacteristic
295C2×C61#151611yescharacteristic
296C2×C61#151911yescharacteristic
297C2×C61#187111yescharacteristic
298C2×C61#189311yescharacteristic
299C2×C61#189811yescharacteristic
300D61—161no—
301C24×C4C2×D6111yescharacteristic
302C23×C8—1221no—
303C2×#121—611no—
304C2×#121—611no—
305C2×#121—611no—
306C2×#120—1221no—
307C22×C22⋊C4—21no—
308C2×C22⋊C8—612no—
309C2×C22⋊C8—1224no—
310C2×C22⋊C8—1221no—
311#386—1221no—
312#386—1221no—
313#386—1221no—
314#380—611no—
315#380—611no—
316#380—611no—
317C2×#95—21no—
318C2×#95C2×D6112yes—
319#327—1224no—
320C61#911311yescharacteristic, complemented, Sylow
321C25C22⋊D6111yescharacteristic
322C22×D4—611no—
323C22×D4—1221no—
324C22×D4—1221no—
325C22×D4—1221no—
326C23×C4—22no—
327C23×C4—22no—
328C23×C4—22no—
329C23×C4—42no—
330C23×C4—42no—
331C23×C4C4×D6111yescharacteristic
332C23×C4C4×D6111yescharacteristic
333C23×C4D24411yescharacteristic
334C23×C4D24411yescharacteristic
335C23×C4C22⋊D6111yescharacteristic
336C23×C4C22×D6111yescharacteristic
337C22×C8—1224no—
338C22×C8—1222no—
339C22×C8—2442no—
340C22×C8—2442no—
341#121—1222no—
342#121—1222no—
343#121—2442no—
344#121—2442no—
345#120—2442no—
346#120—2442no—
347C2×C4⋊C4—611no—
348C2×C4⋊C4—611no—
349C2×C4⋊C4—611no—
350C2×C22⋊C4—24no—
351C2×C22⋊C4—42no—
352C2×C22⋊C4—42no—
353C2×C22⋊C4—1221no—
354C2×C22⋊C4—1221no—
355C2×C22⋊C4—1221no—
356C2×C22⋊C4—1221no—
357C2×C22⋊C4—1221no—
358C2×C22⋊C4—1221no—
359C22⋊C8—614no—
360C22⋊C8—1224no—
361C22⋊C8—1222no—
362#95—22no—
363#95—22no—
364#95—42no—
365#95—1221no—
366#95—1221no—
367#95—1221no—
368C24—22no—
369C24—22no—
370C24—22no—
371C24—22no—
372C24—41no—
373C24—41no—
374C24—1221no—
375C24#1175819811yescharacteristic
376C24#1175819811yescharacteristic
377C24#1175819912yes—
378C24#1175820112yes—
379C24C2×C22⋊D6111yescharacteristic
380C2×D4—1222no—
381C2×D4—2441no—
382C2×D4—2441no—
383C2×D4—2442no—
384C2×D4—2442no—
385C2×D4—2442no—
386C2×D4—2442no—
387C2×D4—2442no—
388C2×D4—2442no—
389C2×D4—2442no—
390C2×D4—2441no—
391C2×D4—2441no—
392C2×D4—2442no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—24no—
406C22×C4—41no—
407C22×C4—41no—
408C22×C4—41no—
409C22×C4—41no—
410C22×C4—41no—
411C22×C4—41no—
412C22×C4—41no—
413C22×C4—48no—
414C22×C4—48no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—1221no—
422C22×C4—1221no—
423C22×C4—1221no—
424C22×C4#1175819811yescharacteristic
425C22×C4#1175819811yescharacteristic
426C22×C4C2×C4×D6111yescharacteristic
427C22×C4C2×D24411yescharacteristic
428C22×C4D4×D6112yes—
429C22×C4#1175822412yes—
430C2×C8—1224no—
431C2×C8—2448no—
432C2×C8—2442no—
433C2×C8—2442no—
434C4⋊C4—1222no—
435C4⋊C4—2441no—
436C4⋊C4—2441no—
437C22⋊C4—48no—
438C22⋊C4—2442no—
439C22⋊C4—2442no—
440C22⋊C4—2442no—
441C22⋊C4—2441no—
442C22⋊C4—2441no—
443C22⋊C4—2442no—
444C22⋊C4—2441no—
445C22⋊C4—2441no—
446C23—21no—
447C23—21no—
448C23—21no—
449C23—21no—
450C23—21no—
451C23—21no—
452C23—21no—
453C23—21no—
454C23—21no—
455C23—21no—
456C23—24no—
457C23—24no—
458C23—24no—
459C23—24no—
460C23—21no—
461C23—21no—
462C23—21no—
463C23—21no—
464C23—42no—
465C23—42no—
466C23—44no—
467C23—44no—
468C23—44no—
469C23—44no—
470C23—2441no—
471C23—2441no—
472C23—2441no—
473C23—2441no—
474C23—2441no—
475C23—2441no—
476C23—2441no—
477C23#1451969112yes—
478C23#1451969712yes—
479C23#1451969712yes—
480C23#1451970512yes—
481C23C2×#1175819811yescharacteristic
482C23#1451974912yes—
483C23C2×#1175819911yescharacteristic
484C23C2×#1175820111yescharacteristic
485C23#1451979712yes—
486D4—4882no—
487D4—4884no—
488D4—4882no—
489D4—4882no—
490D4—4884no—
491C2×C4—24no—
492C2×C4—24no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—44no—
505C2×C4—44no—
506C2×C4—44no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—44no—
510C2×C4—44no—
511C2×C4—44no—
512C2×C4—82no—
513C2×C4—82no—
514C2×C4—82no—
515C2×C4—82no—
516C2×C4—2441no—
517C2×C4—2441no—
518C2×C4—2441no—
519C2×C4—2441no—
520C2×C4—2441no—
521C2×C4—2441no—
522C2×C4—2441no—
523C2×C4—2441no—
524C2×C4—2441no—
525C2×C4#1451969112yes—
526C2×C4#1451973712yes—
527C2×C4#1451974711yescharacteristic
528C2×C4#1451976111yescharacteristic
529C8—2444no—
530C22—22no—
531C22—22no—
532C22—21no—
533C22—22no—
534C22—21no—
535C22—22no—
536C22—22no—
537C22—22no—
538C22—21no—
539C22—21no—
540C22—22no—
541C22—22no—
542C22—22no—
543C22—22no—
544C22—22no—
545C22—22no—
546C22—22no—
547C22—48no—
548C22—48no—
549C22—41no—
550C22—41no—
551C22—41no—
552C22—41no—
553C22—4881no—
554C22—4881no—
555C22—4881no—
556C22—4881no—
557C22—4881no—
558C22—4881no—
559C22—4881no—
560C22#1683949514yescomplemented
561C22#1683963911yescharacteristic
562C22C2×#1451969111yescharacteristic
563C22C2×#1451969111yescharacteristic
564C22C2×#1451969711yescharacteristic
565C22#1683975611yescharacteristic
566C22C2×#1451970511yescharacteristic
567C22#1683978011yescharacteristic
568C22#1683999312yes—
569C22#1683999612yes—
570C4—21no—
571C4—22no—
572C4—21no—
573C4—22no—
574C4—41no—
575C4—84no—
576C4—4881no—
577C4—4881no—
578C4—4881no—
579C2—22no—
580C2—22no—
581C2—24no—
582C2—42no—
583C2—9761no—
584C2C2×#1683949511yescharacteristic
585C2#2144430811yescharacteristic
586C2#2144437311yescharacteristic
587C2#2144438611yescharacteristic
588C2#2144611911yescharacteristic
589C2#2144624511yescharacteristic
590C2#2144627011yescharacteristic
subgroupG/H
centerC23C2×#11758198
commutatorC22×C4×C61C22×C4
FrattiniC23×C4C22×D61
FittingC61×C22⋊#95C2
socleC23×C61C2×C22⋊C4
prime radical#21454101C2
cosocleC23×C4×C61C23
hypercenterC22⋊#95D61
hypocenterC61#9113
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C24×C4 ◁ C22⋊#95

Complements

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

NH ≅ G/Ntype
1. C22×C22⋊#92418C2semidirect
3. C2×#16839495C2semidirect
4. #21443698C2semidirect
5. C61×C22⋊#95C2semidirect
7. C2×#14519751C4semidirect
8. C2×#14519751C4semidirect
17. #16839622C4semidirect
18. #16839622C4semidirect
50. C22⋊#92418D4semidirect
258. C22×C61#327semidirect
287. C22⋊#95D61semidirect
320. C61#9113semidirect
560. C22#16839495semidirect

Maximal subgroups

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

subgroupindex
C22×C22⋊#924182
#214541012
C2×#168394952
#214436982
C61×C22⋊#952
#911361