Finite Group Databasebeta

← #29576331

Subgroups — #29576331

Order 15616.

712 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 712 conjugacy classes of subgroups — too many to draw. Only its 97 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839905C211yesmaximal, characteristic
2#21454269C211yesmaximal, characteristic, complemented
3#21445764C211yesmaximal, characteristic, complemented
4#21444179C211yesmaximal, characteristic, complemented
5#21443695C211yesmaximal, characteristic, complemented
6#21443667C211yesmaximal, characteristic, complemented
7C61×#1507C211yesmaximal, characteristic, complemented
8C22×#11758222C2211yescharacteristic
9C22×#11758200C411yescharacteristic, complemented
10C22×#11758199C411yescharacteristic, complemented
11C2×#14519785—21no—
12#16839905—21no—
13#16839905—21no—
14#16839905—21no—
15#16839905—21no—
16C2×#14519705—21no—
17C2×#14519705—21no—
18C2×#14519700C411yescharacteristic, complemented
19C2×#14519697C411yescharacteristic, complemented
20#16839730—21no—
21#16839727—22no—
22#16839726C2211yescharacteristic
23#16839724—21no—
24C2×C22⋊#92418C2211yescharacteristic
25C2×C22⋊#92418C2211yescharacteristic
26C22⋊C8×D61—21no—
27#16839486—21no—
28#16839486—21no—
29C2×#121×C61C2211yescharacteristic
30C61×#407C2211yescharacteristic
31C4×C22⋊C4×C61C2211yescharacteristic
32C61×#327—22no—
33C2×#11758222—21no—
34C2×#11758222—21no—
35C2×#11758222—42no—
36C2×#11758222—42no—
37C2×#11758222D411yescharacteristic
38C2×#11758222D411yescharacteristic
39C2×#11758222D411yescharacteristic
40C2×#11758222D411yescharacteristic
41C22×Dic122C2×C411yescharacteristic
42C22×D244C2×C411yescharacteristic
43C22×C22⋊D61—21no—
44C22×C4×D61—21no—
45C2×#11758200—21no—
46C2×#11758200—21no—
47C2×#11758200—42no—
48C2×#11758199—21no—
49C2×#11758199—21no—
50C2×#11758199—42no—
51#14519785—42no—
52#14519785—42no—
53C2×#11758194—21no—
54C2×#11758189—22no—
55C22×#92418—21no—
56C22×#92418C2×C411yescharacteristic
57C22×#92418C2×C411yescharacteristic
58C2×C8×D61—22no—
59#14519705—42no—
60#14519705—42no—
61#14519700—21no—
62#14519700—21no—
63#14519697—21no—
64#14519697—21no—
65C22⋊#92418—21no—
66C22⋊#92418—21no—
67C22⋊#92418—21no—
68C22⋊#92418—21no—
69C22⋊#92418—21no—
70C22⋊#92418D411yescharacteristic
71C22⋊#92418D411yescharacteristic
72#14519689—22no—
73#14519689—22no—
74C22×D4×C61—21no—
75C22×D4×C61C2×C411yescharacteristic
76C23×C4×C61C2311yescharacteristic
77C2×M16×C61—21no—
78#121×C61—21no—
79#121×C61—21no—
80#121×C61—21no—
81#121×C61—21no—
82C2×C4⋊C4×C61C2×C411yescharacteristic
83C2×C22⋊C4×C61—21no—
84C2×C22⋊C4×C61—21no—
85C2×C42×C61—21no—
86C22⋊C8×C61—21no—
87C22⋊C8×C61D412yes—
88#95×C61—21no—
89C24×C61—21no—
90C24×C61#4111yescharacteristic
91C23×D61—21no—
92C2×D4×C61—21no—
93C2×D4×C61—21no—
94C2×D4×C61—41no—
95C2×D4×C61—41no—
96C2×D4×C61—41no—
97C2×D4×C61—41no—
98C2×D4×C61—82no—
99C22×C4×C61—21no—
100C22×C4×C61—41no—
101C22×C4×C61—41no—
102C22×C4×C61—41no—
103C22×C4×C61C22×C411yescharacteristic
104C22×C4×C61C2×D411yescharacteristic
105C22×C4×C61C2×D411yescharacteristic
106C22×C4×C61C2×D411yescharacteristic
107C22×C4×C61C2×D411yescharacteristic
108C22×C4×C61#4111yescharacteristic
109C2×C22⋊D61—21no—
110C2×C22⋊D61—21no—
111C2×C22⋊D61—21no—
112C2×C22⋊D61—21no—
113C2×C22⋊D61—44no—
114C22×Dic61—21no—
115#11758222—22no—
116#11758222—22no—
117#11758222—22no—
118#11758222—22no—
119#11758222—41no—
120#11758222—41no—
121#11758222—41no—
122#11758222—41no—
123#11758222—44no—
124#11758222—44no—
125C2×D244—22no—
126C2×D244—21no—
127C2×D244—41no—
128C2×D244C22⋊C411yescharacteristic
129C2×D244C22⋊C411yescharacteristic
130C2×C4×D61—21no—
131C2×C4×D61—21no—
132C2×C4×D61—21no—
133C2×C4×D61—21no—
134C2×C4×D61—44no—
135C2×Dic122—22no—
136C2×Dic122—21no—
137C2×Dic122—41no—
138C2×Dic122C22⋊C411yescharacteristic
139C2×Dic122C22⋊C411yescharacteristic
140M16×C61—41no—
141M16×C61—41no—
142C2×C8×C61—22no—
143C4⋊C4×C61—21no—
144C4⋊C4×C61—21no—
145C22⋊C4×C61—22no—
146C22⋊C4×C61—41no—
147C22⋊C4×C61—41no—
148C22⋊C4×C61—41no—
149C42×C61—41no—
150C42×C61—41no—
151#11758200—42no—
152#11758200—42no—
153#11758199—42no—
154#11758199—42no—
155#11758194—41no—
156#11758194—41no—
157C2×#92418—21no—
158C2×#92418—21no—
159C2×#92418—21no—
160C2×#92418—21no—
161C2×#92418—41no—
162C2×#92418—41no—
163C2×#92418—41no—
164C2×#92418—41no—
165C2×#92418—41no—
166#11758189—44no—
167C8×D61—44no—
168C23×C61—21no—
169C23×C61—21no—
170C23×C61—41no—
171C23×C61—41no—
172C23×C61—41no—
173C23×C61—41no—
174C23×C61—41no—
175C23×C61—81no—
176C23×C61—81no—
177C23×C61C4×D411yescharacteristic
178C23×C61#12111yescharacteristic
179C23×C61#12311yescharacteristic
180C22×D61—21no—
181C22×D61—21no—
182C22×D61—42no—
183C22×D61—41no—
184C22×D61—41no—
185C22×D61—81no—
186D4×C61—41no—
187D4×C61—41no—
188D4×C61—81no—
189D4×C61—82no—
190D4×C61—82no—
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—41no—
198C2×C4×C61—41no—
199C2×C4×C61—41no—
200C2×C4×C61—42no—
201C2×C4×C61—41no—
202C2×C4×C61—81no—
203C2×C4×C61—81no—
204C2×C4×C61—81no—
205C2×C4×C61C2×C22⋊C411yescharacteristic
206C2×C4×C61C4×D411yescharacteristic
207C2×C4×C61#12011yescharacteristic
208C2×C4×C61#12111yescharacteristic
209C22⋊D61—24no—
210C22⋊D61—42no—
211C22⋊D61—42no—
212C22⋊D61—42no—
213C2×Dic61—21no—
214C2×Dic61—21no—
215C2×Dic61—42no—
216D244—21no—
217D244—21no—
218D244—21no—
219D244—42no—
220D244#9912yes—
221C4×D61—24no—
222C4×D61—42no—
223C4×D61—42no—
224C4×D61—42no—
225Dic122—21no—
226Dic122—21no—
227Dic122—21no—
228Dic122—42no—
229Dic122#9912yes—
230C8×C61—42no—
231#92418—41no—
232#92418—41no—
233#92418—41no—
234#92418—41no—
235#92418—81no—
236#9241—611nomaximal, Sylow
237C22×C61—21no—
238C22×C61—21no—
239C22×C61—21no—
240C22×C61—21no—
241C22×C61—21no—
242C22×C61—21no—
243C22×C61—21no—
244C22×C61—21no—
245C22×C61—81no—
246C22×C61—81no—
247C22×C61—81no—
248C22×C61—81no—
249C22×C61—81no—
250C22×C61—161no—
251C22×C61—161no—
252C22×C61#38611yescharacteristic
253C22×C61#44311yescharacteristic
254C22×C61#44411yescharacteristic
255C2×D61—22no—
256C2×D61—41no—
257C2×D61—41no—
258C2×D61—41no—
259C2×D61—41no—
260C2×D61—41no—
261C2×D61—82no—
262C2×D61—82no—
263C4×C61—21no—
264C4×C61—21no—
265C4×C61—21no—
266C4×C61—82no—
267C4×C61—81no—
268C4×C61C2×#9911yescharacteristic
269C4×C61C2×#9911yescharacteristic
270Dic61—22no—
271Dic61—41no—
272C2×#449—611no—
273#2880—611no—
274#1789—611no—
275#1507D6111yescharacteristic, complemented
276#1471—611no—
277#1293—611no—
278#1282—611no—
279C2×C61—21no—
280C2×C61—21no—
281C2×C61—41no—
282C2×C61—41no—
283C2×C61—161no—
284C2×C61#187611yescharacteristic
285C2×C61#189811yescharacteristic
286C2×C61#190311yescharacteristic
287D61—42no—
288D61—81no—
289C22×#41—611no—
290C22×SD16—611no—
291C22×D8—611no—
292C2×#131—1221no—
293C2×#121C2×D6111yescharacteristic
294#449—1221no—
295#449—1221no—
296#449—1221no—
297#449—1221no—
298C2×#104—1221no—
299C2×#104—1221no—
300C2×#103—611no—
301C2×#102—611no—
302#408—1221no—
303#408—1222no—
304#407—611no—
305#407—1221no—
306#407C2×D6111yescharacteristic
307C2×C22⋊C8—611no—
308C2×C22⋊C8—611no—
309C2×C22⋊C8—1221no—
310C4×C22⋊C4C2×D6111yescharacteristic
311#327—22no—
312#323—1221no—
313#323—1221no—
314C61#924111yescharacteristic, complemented, Sylow
315C2×#41—611no—
316C2×#41—611no—
317C2×#41—611no—
318C2×#41—611no—
319C2×#41—1221no—
320C2×#41—1221no—
321C2×#41—2442no—
322C2×#41—2442no—
323C22×Q8—611no—
324C22×D4—21no—
325C22×D4—611no—
326C22×D4—1221no—
327C22×D4C4×D6111yescharacteristic
328C23×C4—1221no—
329C23×C4C22×D6111yescharacteristic
330C2×SD16—1221no—
331C2×SD16—1221no—
332C2×SD16—2442no—
333C2×D8—1221no—
334C2×D8—1221no—
335C2×D8—2442no—
336#131—2442no—
337#131—2442no—
338C2×M16—21no—
339C2×M16—1222no—
340C2×M16—1221no—
341C22×C8—611no—
342C22×C8—611no—
343C22×C8—1222no—
344C22×C8—1221no—
345#121—21no—
346#121—21no—
347#121—21no—
348#121—21no—
349C2×C4⋊C4C4×D6111yescharacteristic
350C2×C22⋊C4—21no—
351C2×C22⋊C4—21no—
352C2×C42—21no—
353#104—2442no—
354#104—2442no—
355#103—1221no—
356#103—1221no—
357#102—1221no—
358#102—1221no—
359C22⋊C8—21no—
360C22⋊C8—611no—
361C22⋊C8—611no—
362C22⋊C8—1222no—
363C22⋊C8—1222no—
364C22⋊C8—1221no—
365C22⋊C8—1221no—
366C22⋊C8—1221no—
367C22⋊C8—1221no—
368C22⋊C8—1221no—
369C22⋊C8C22⋊D6112yes—
370#95—21no—
371C24—21no—
372C24—1221no—
373C24#1175822411yescharacteristic
374#41—1222no—
375#41—1222no—
376#41—1222no—
377#41—1222no—
378#41—2441no—
379#41—2441no—
380#41—2441no—
381#41—2441no—
382#41—2444no—
383#41—2444no—
384C2×Q8—611no—
385C2×Q8—611no—
386C2×Q8—1222no—
387C2×Q8—1221no—
388C2×Q8—2441no—
389C2×D4—21no—
390C2×D4—21no—
391C2×D4—41no—
392C2×D4—41no—
393C2×D4—41no—
394C2×D4—41no—
395C2×D4—82no—
396C2×D4—611no—
397C2×D4—611no—
398C2×D4—1222no—
399C2×D4—1221no—
400C2×D4—1221no—
401C2×D4—1221no—
402C2×D4—1221no—
403C2×D4—1221no—
404C2×D4—2441no—
405C2×D4—2444no—
406C22×C4—21no—
407C22×C4—41no—
408C22×C4—41no—
409C22×C4—41no—
410C22×C4—1221no—
411C22×C4—1221no—
412C22×C4—1221no—
413C22×C4—1221no—
414C22×C4—1221no—
415C22×C4—2444no—
416C22×C4C2×C4×D6111yescharacteristic
417C22×C4#1175822211yescharacteristic
418C22×C4D4×D6111yescharacteristic
419C22×C4D4×D6111yescharacteristic
420C22×C4D4×D6111yescharacteristic
421C22×C4C2×C22⋊D6111yescharacteristic
422SD16—2442no—
423SD16—2442no—
424D8—2442no—
425D8—2442no—
426M16—41no—
427M16—41no—
428M16—2444no—
429M16—2441no—
430M16—2441no—
431C2×C8—22no—
432C2×C8—1221no—
433C2×C8—1221no—
434C2×C8—1221no—
435C2×C8—1221no—
436C2×C8—2444no—
437C2×C8—2441no—
438C2×C8—2441no—
439C2×C8—2441no—
440C2×C8—2441no—
441C2×C8—2441no—
442C4⋊C4—21no—
443C4⋊C4—21no—
444C22⋊C4—22no—
445C22⋊C4—41no—
446C22⋊C4—41no—
447C22⋊C4—41no—
448C42—41no—
449C42—41no—
450C23—21no—
451C23—21no—
452C23—41no—
453C23—41no—
454C23—41no—
455C23—41no—
456C23—41no—
457C23—81no—
458C23—81no—
459C23—1221no—
460C23—1221no—
461C23—4881no—
462C23—2442no—
463C23—2441no—
464C23—2441no—
465C23#1451973711yescharacteristic
466C23#1451974811yescharacteristic
467C23#1451975911yescharacteristic
468Q8—612no—
469Q8—1221no—
470Q8—1221no—
471Q8—1221no—
472Q8—2442no—
473D4—41no—
474D4—41no—
475D4—81no—
476D4—82no—
477D4—82no—
478D4—612no—
479D4—1224no—
480D4—1221no—
481D4—1221no—
482D4—1221no—
483D4—2442no—
484D4—2442no—
485D4—2442no—
486D4—2442no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—42no—
497C2×C4—41no—
498C2×C4—81no—
499C2×C4—81no—
500C2×C4—81no—
501C2×C4—1224no—
502C2×C4—1221no—
503C2×C4—1221no—
504C2×C4—2442no—
505C2×C4—2442no—
506C2×C4—2442no—
507C2×C4—2442no—
508C2×C4C22⋊C4×D6111yescharacteristic
509C2×C4C4×C22⋊D6111yescharacteristic
510C2×C4#1451974511yescharacteristic
511C2×C4#1451975011yescharacteristic
512C8—42no—
513C8—4881no—
514C8—2441no—
515C8—2441no—
516C8—2441no—
517C8—2441no—
518C22—21no—
519C22—21no—
520C22—21no—
521C22—21no—
522C22—21no—
523C22—21no—
524C22—21no—
525C22—21no—
526C22—81no—
527C22—81no—
528C22—81no—
529C22—81no—
530C22—81no—
531C22—161no—
532C22—161no—
533C22—1222no—
534C22—4882no—
535C22—4882no—
536C22—2441no—
537C22—2441no—
538C22—2441no—
539C22—2441no—
540C22—2441no—
541C22#1683963811yescharacteristic
542C22#1683987611yescharacteristic
543C22#1683988111yescharacteristic
544C4—21no—
545C4—21no—
546C4—21no—
547C4—82no—
548C4—81no—
549C4—1222no—
550C4—2441no—
551C4#99×D6111yescharacteristic
552C4#99×D6111yescharacteristic
553C2—21no—
554C2—21no—
555C2—41no—
556C2—41no—
557C2—161no—
558C2—4881no—
559C2—2442no—
560C2#2144614311yescharacteristic
561C2#2144626911yescharacteristic
562C2#2144629611yescharacteristic
subgroupG/H
centerC22#16839638
commutatorC22×C4×C61C22×C4
FrattiniC23×C4C22×D61
FittingC61×#1507C2
socleC22×C61#386
prime radicalC2×#16839905C2
cosocleC23×C4×C61C23
hypercenter#1507D61
hypocenterC61#9241
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1507

Complements

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

NH ≅ G/Ntype
2. #21454269C2semidirect
3. #21445764C2semidirect
4. #21444179C2semidirect
5. #21443695C2semidirect
6. #21443667C2semidirect
7. C61×#1507C2semidirect
9. C22×#11758200C4semidirect
10. C22×#11758199C4semidirect
18. C2×#14519700C4semidirect
19. C2×#14519697C4semidirect
275. #1507D61semidirect
314. C61#9241semidirect

Maximal subgroups

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

subgroupindex
C2×#168399052
#214542692
#214457642
#214441792
#214436952
#214436672
C61×#15072
#924161