Finite Group Databasebeta

← #29575596

Subgroups — #29575596

Order 15616.

1030 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839726C211yesmaximal, characteristic, complemented
2#21454101C211yesmaximal, characteristic
3C2×#16839495C212yesmaximal, complemented
4#21443698C212yesmaximal, complemented
5C61×C22⋊#95C211yesmaximal, characteristic, complemented
6C22×#11758194—21no—
7C2×#14519751C411yescharacteristic, complemented
8C2×#14519751C411yescharacteristic, complemented
9C2×#14519751C2211yescharacteristic
10C2×#14519746—21no—
11#16839726—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—
29C2×#11758194—24no—
30C2×#11758194—42no—
31C2×#11758194—42no—
32C22×#92418—22no—
33#14519751—22no—
34#14519751—22no—
35#14519751—42no—
36#14519751—42no—
37#14519746—44no—
38C2×C4⋊Dic61C2×C411yescharacteristic
39C2×C4⋊Dic61C2×C411yescharacteristic
40C2×C4⋊Dic61C2×C411yescharacteristic
41C2×C22⋊Dic61—21no—
42C2×C22⋊Dic61—21no—
43C2×C22⋊Dic61—21no—
44C2×#11758198—21no—
45C2×#11758198—21no—
46C2×#11758198—21no—
47C22⋊#92418—24no—
48C22⋊#92418—22no—
49C22⋊#92418D414yescomplemented
50#14519684—21no—
51#14519684—21no—
52#14519684—21no—
53C25×C61D411yescharacteristic
54C23×C4×C61—22no—
55C23×C4×C61—22no—
56C23×C4×C61—22no—
57C23×C4×C61—42no—
58C23×C4×C61—42no—
59C23×C4×C61C2×C411yescharacteristic
60C23×C4×C61C2×C411yescharacteristic
61C23×C4×C61D411yescharacteristic
62C23×C4×C61D411yescharacteristic
63C23×C4×C61D411yescharacteristic
64C23×C4×C61C2311yescharacteristic
65C2×C22⋊C4×C61—24no—
66C2×C22⋊C4×C61—42no—
67C2×C22⋊C4×C61—42no—
68#95×C61—22no—
69#95×C61—22no—
70#95×C61—42no—
71C24×C61—22no—
72C24×C61—22no—
73C24×C61—21no—
74C24×C61—21no—
75C24×C61—22no—
76C24×C61—22no—
77C24×C61—41no—
78C24×C61—41no—
79C24×C61C22⋊C411yescharacteristic
80C24×C61C22⋊C411yescharacteristic
81C24×C61C2×D411yescharacteristic
82C23×D61—21no—
83C22×C4×C61—22no—
84C22×C4×C61—22no—
85C22×C4×C61—21no—
86C22×C4×C61—21no—
87C22×C4×C61—22no—
88C22×C4×C61—22no—
89C22×C4×C61—22no—
90C22×C4×C61—22no—
91C22×C4×C61—22no—
92C22×C4×C61—22no—
93C22×C4×C61—21no—
94C22×C4×C61—21no—
95C22×C4×C61—24no—
96C22×C4×C61—41no—
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—48no—
104C22×C4×C61—48no—
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×C61C22⋊C411yescharacteristic
112C22×C4×C61C22⋊C411yescharacteristic
113C22×C4×C61C22×C411yescharacteristic
114C22×C4×C61C2×D411yescharacteristic
115C22×C4×C61C2×D412yes—
116C22×C4×C61#4112yes—
117C2×C22⋊D61—42no—
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—41no—
125C2×C22⋊D61—42no—
126C2×C22⋊D61—41no—
127C22×Dic61—21no—
128C22×Dic61—21no—
129C22×Dic61—21no—
130C2×D244—22no—
131C2×D244—41no—
132C2×D244—41no—
133C22⋊C4×C61—48no—
134C22⋊Dic61—42no—
135C22⋊Dic61—42no—
136C22⋊Dic61—42no—
137#11758198—42no—
138#11758198—42no—
139#11758198—41no—
140#11758198—41no—
141C4⋊Dic61—22no—
142C4⋊Dic61—41no—
143C4⋊Dic61—41no—
144#11758194—48no—
145C2×#92418—24no—
146C2×#92418—42no—
147C2×#92418—42no—
148C23×C61—21no—
149C23×C61—21no—
150C23×C61—21no—
151C23×C61—21no—
152C23×C61—24no—
153C23×C61—24no—
154C23×C61—21no—
155C23×C61—21no—
156C23×C61—21no—
157C23×C61—21no—
158C23×C61—21no—
159C23×C61—21no—
160C23×C61—21no—
161C23×C61—21no—
162C23×C61—21no—
163C23×C61—21no—
164C23×C61—21no—
165C23×C61—21no—
166C23×C61—44no—
167C23×C61—44no—
168C23×C61—44no—
169C23×C61—44no—
170C23×C61—42no—
171C23×C61—42no—
172C23×C61—42no—
173C23×C61—42no—
174C23×C61#9912yes—
175C23×C61#10412yes—
176C23×C61C2×C22⋊C411yescharacteristic
177C23×C61#12112yes—
178C23×C61AGL(1,ℤ/8)11yescharacteristic
179C23×C61AGL(1,ℤ/8)12yes—
180C23×C61#13711yescharacteristic
181C22×D61—41no—
182C22×D61—41no—
183C22×D61—41no—
184C22×D61—41no—
185C22×D61—41no—
186C22×D61—41no—
187C22×D61—41no—
188C2×C4×C61—24no—
189C2×C4×C61—24no—
190C2×C4×C61—21no—
191C2×C4×C61—21no—
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—42no—
200C2×C4×C61—42no—
201C2×C4×C61—44no—
202C2×C4×C61—44no—
203C2×C4×C61—44no—
204C2×C4×C61—42no—
205C2×C4×C61—42no—
206C2×C4×C61—44no—
207C2×C4×C61—44no—
208C2×C4×C61—44no—
209C2×C4×C61—82no—
210C2×C4×C61—82no—
211C2×C4×C61—82no—
212C2×C4×C61—82no—
213C2×C4×C61#9912yes—
214C2×C4×C61C4×D412yes—
215C2×C4×C61#12011yescharacteristic
216C2×C4×C61#12311yescharacteristic
217C22⋊D61—84no—
218C22⋊D61—84no—
219C22⋊D61—82no—
220C22⋊D61—82no—
221C2×Dic61—41no—
222C2×Dic61—41no—
223C2×Dic61—41no—
224C2×Dic61—41no—
225C2×Dic61—41no—
226C2×Dic61—41no—
227C2×Dic61—41no—
228C2×Dic61—41no—
229C2×Dic61—41no—
230D244—82no—
231#92418—44no—
232#9117—611nomaximal, Sylow
233C22×C61—22no—
234C22×C61—22no—
235C22×C61—21no—
236C22×C61—21no—
237C22×C61—22no—
238C22×C61—22no—
239C22×C61—21no—
240C22×C61—21no—
241C22×C61—22no—
242C22×C61—22no—
243C22×C61—22no—
244C22×C61—22no—
245C22×C61—22no—
246C22×C61—22no—
247C22×C61—48no—
248C22×C61—48no—
249C22×C61—41no—
250C22×C61—41no—
251C22×C61—41no—
252C22×C61—41no—
253C22×C61—41no—
254C22×C61—41no—
255C22×C61—41no—
256C22×C61—41no—
257C22×C61#38611yescharacteristic
258C22×C61C2×#9911yescharacteristic
259C22×C61C2×#9911yescharacteristic
260C22×C61#41711yescharacteristic
261C22×C61#41811yescharacteristic
262C22×C61C2×#10411yescharacteristic
263C22×C61#42111yescharacteristic
264C22×C61#46812yes—
265C22×C61#46912yes—
266C2×D61—81no—
267C2×D61—81no—
268C2×D61—81no—
269C2×D61—81no—
270C2×D61—81no—
271C2×D61—81no—
272C2×D61—81no—
273C4×C61—21no—
274C4×C61—22no—
275C4×C61—22no—
276C4×C61—21no—
277C4×C61—41no—
278C4×C61—84no—
279Dic61—81no—
280Dic61—81no—
281Dic61—81no—
282C2×#407—611no—
283#2847—611no—
284C2×#327—612no—
285C22⋊#95D6111yescharacteristic, complemented
286#1294—612no—
287C2×C61—22no—
288C2×C61—22no—
289C2×C61—42no—
290C2×C61—42no—
291C2×C61#150411yescharacteristic
292C2×C61#150711yescharacteristic
293C2×C61#151011yescharacteristic
294C2×C61#151311yescharacteristic
295C2×C61#187111yescharacteristic
296C2×C61#189511yescharacteristic
297C2×C61#189811yescharacteristic
298D61—161no—
299C24×C4C2×D6111yescharacteristic
300C22×M16—1221no—
301C2×#121—611no—
302C2×#121—611no—
303C2×#121—611no—
304C2×#120—1221no—
305C22×C22⋊C4—21no—
306#407—1224no—
307C2×C22⋊C8—612no—
308C2×C22⋊C8—1221no—
309#386—1221no—
310#386—1221no—
311#386—1221no—
312#380—611no—
313#380—611no—
314#380—611no—
315C2×#95—21no—
316C2×#95C2×D6112yes—
317#327—1224no—
318C61#911711yescharacteristic, complemented, Sylow
319C25C22⋊D6111yescharacteristic
320C22×D4—611no—
321C22×D4—1221no—
322C22×D4—1221no—
323C22×D4—1221no—
324C23×C4—22no—
325C23×C4—22no—
326C23×C4—22no—
327C23×C4—42no—
328C23×C4—42no—
329C23×C4C4×D6111yescharacteristic
330C23×C4C4×D6111yescharacteristic
331C23×C4D24411yescharacteristic
332C23×C4D24411yescharacteristic
333C23×C4C22⋊D6111yescharacteristic
334C23×C4C22×D6111yescharacteristic
335C2×M16—1224no—
336C2×M16—2442no—
337C2×M16—2442no—
338C22×C8—1222no—
339#121—1222no—
340#121—1222no—
341#121—2442no—
342#121—2442no—
343#120—2444no—
344C2×C4⋊C4—611no—
345C2×C4⋊C4—611no—
346C2×C4⋊C4—611no—
347C2×C22⋊C4—24no—
348C2×C22⋊C4—42no—
349C2×C22⋊C4—42no—
350C2×C22⋊C4—1221no—
351C2×C22⋊C4—1221no—
352C2×C22⋊C4—1221no—
353C2×C22⋊C4—1221no—
354C2×C22⋊C4—1221no—
355C2×C22⋊C4—1221no—
356C22⋊C8—614no—
357C22⋊C8—1224no—
358C22⋊C8—1222no—
359#95—22no—
360#95—22no—
361#95—42no—
362#95—1221no—
363#95—1221no—
364#95—1221no—
365C24—21no—
366C24—22no—
367C24—22no—
368C24—21no—
369C24—22no—
370C24—22no—
371C24—41no—
372C24—41no—
373C24—1221no—
374C24#1175819811yescharacteristic
375C24#1175819811yescharacteristic
376C24C2×C22⋊D6111yescharacteristic
377C2×D4—1222no—
378C2×D4—2441no—
379C2×D4—2441no—
380C2×D4—2442no—
381C2×D4—2442no—
382C2×D4—2442no—
383C2×D4—2442no—
384C2×D4—2441no—
385C2×D4—2442no—
386C2×D4—2442no—
387C2×D4—2442no—
388C2×D4—2442no—
389C2×D4—2441no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—24no—
403C22×C4—41no—
404C22×C4—41no—
405C22×C4—41no—
406C22×C4—41no—
407C22×C4—41no—
408C22×C4—41no—
409C22×C4—41no—
410C22×C4—48no—
411C22×C4—48no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—1221no—
419C22×C4—1221no—
420C22×C4—1221no—
421C22×C4#1175819811yescharacteristic
422C22×C4#1175819811yescharacteristic
423C22×C4C2×C4×D6111yescharacteristic
424C22×C4C2×D24411yescharacteristic
425C22×C4D4×D6112yes—
426C22×C4#1175822412yes—
427M16—2448no—
428C2×C8—1224no—
429C2×C8—2442no—
430C2×C8—2442no—
431C4⋊C4—1222no—
432C4⋊C4—2441no—
433C4⋊C4—2441no—
434C22⋊C4—48no—
435C22⋊C4—2442no—
436C22⋊C4—2442no—
437C22⋊C4—2442no—
438C22⋊C4—2442no—
439C22⋊C4—2442no—
440C22⋊C4—2441no—
441C22⋊C4—2441no—
442C23—21no—
443C23—21no—
444C23—21no—
445C23—21no—
446C23—24no—
447C23—24no—
448C23—21no—
449C23—21no—
450C23—21no—
451C23—21no—
452C23—21no—
453C23—21no—
454C23—21no—
455C23—21no—
456C23—21no—
457C23—21no—
458C23—21no—
459C23—21no—
460C23—44no—
461C23—44no—
462C23—44no—
463C23—44no—
464C23—42no—
465C23—42no—
466C23—42no—
467C23—42no—
468C23—2441no—
469C23—2441no—
470C23—2441no—
471C23—2441no—
472C23—2441no—
473C23—2441no—
474C23—2441no—
475C23#1451969112yes—
476C23#1451970512yes—
477C23C2×#1175819811yescharacteristic
478C23#1451974912yes—
479C23#1451980212yes—
480C23#1451980311yescharacteristic
481C23#1451980811yescharacteristic
482D4—4882no—
483D4—4884no—
484D4—4884no—
485D4—4882no—
486D4—4882no—
487C2×C4—24no—
488C2×C4—24no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—44no—
501C2×C4—44no—
502C2×C4—44no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—44no—
506C2×C4—44no—
507C2×C4—44no—
508C2×C4—82no—
509C2×C4—82no—
510C2×C4—82no—
511C2×C4—82no—
512C2×C4—2441no—
513C2×C4—2441no—
514C2×C4—2441no—
515C2×C4—2441no—
516C2×C4—2441no—
517C2×C4—2441no—
518C2×C4—2441no—
519C2×C4—2441no—
520C2×C4—2441no—
521C2×C4#1451969112yes—
522C2×C4#1451973712yes—
523C2×C4#1451974711yescharacteristic
524C2×C4#1451976111yescharacteristic
525C8—2444no—
526C22—21no—
527C22—21no—
528C22—22no—
529C22—21no—
530C22—21no—
531C22—22no—
532C22—22no—
533C22—22no—
534C22—22no—
535C22—22no—
536C22—22no—
537C22—22no—
538C22—22no—
539C22—22no—
540C22—48no—
541C22—48no—
542C22—41no—
543C22—41no—
544C22—41no—
545C22—41no—
546C22—41no—
547C22—41no—
548C22—41no—
549C22—41no—
550C22—4881no—
551C22—4881no—
552C22—4881no—
553C22—4881no—
554C22—4881no—
555C22—4881no—
556C22—4881no—
557C22#1683963911yescharacteristic
558C22C2×#1451969111yescharacteristic
559C22C2×#1451969111yescharacteristic
560C22#1683976111yescharacteristic
561C22#1683976611yescharacteristic
562C22C2×#1451970511yescharacteristic
563C22#1683978011yescharacteristic
564C22#1684001012yes—
565C22#1684001312yes—
566C4—21no—
567C4—22no—
568C4—21no—
569C4—22no—
570C4—41no—
571C4—84no—
572C4—4881no—
573C4—4881no—
574C4—4881no—
575C2—22no—
576C2—22no—
577C2—42no—
578C2—42no—
579C2—9761no—
580C2#2144432111yescharacteristic
581C2#2144433411yescharacteristic
582C2#2144434711yescharacteristic
583C2#2144436011yescharacteristic
584C2#2144611911yescharacteristic
585C2#2144625511yescharacteristic
586C2#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#9117
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. C2×#16839726C2semidirect
3. C2×#16839495C2semidirect
4. #21443698C2semidirect
5. C61×C22⋊#95C2semidirect
7. C2×#14519751C4semidirect
8. C2×#14519751C4semidirect
17. #16839622C4semidirect
18. #16839622C4semidirect
49. C22⋊#92418D4semidirect
285. C22⋊#95D61semidirect
318. C61#9117semidirect

Maximal subgroups

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

subgroupindex
C2×#168397262
#214541012
C2×#168394952
#214436982
C61×C22⋊#952
#911761