Finite Group Databasebeta

← #29576323

Subgroups — #29576323

Order 15616.

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

normal subgroups only

This group has 686 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×#16839922C211yesmaximal, characteristic
2#21454265C211yesmaximal, characteristic
3#21445762C211yesmaximal, characteristic
4#21444339C211yesmaximal, characteristic, complemented
5#21443695C211yesmaximal, characteristic, complemented
6#21443665C211yesmaximal, characteristic, complemented
7C61×#1472C211yesmaximal, characteristic, complemented
8C22×#11758222C2211yescharacteristic
9C22×#11758202C411yescharacteristic, complemented
10C22×#11758201C411yescharacteristic, complemented
11C2×#14519783—21no—
12#16839922—21no—
13#16839922—21no—
14#16839922—21no—
15#16839922—21no—
16C2×#14519703—21no—
17C2×#14519703—21no—
18C2×#14519700C411yescharacteristic, complemented
19C2×#14519697C411yescharacteristic, complemented
20#16839728—21no—
21#16839727—22no—
22#16839726C2211yescharacteristic
23#16839724—21no—
24C2×C22⋊#92418C2211yescharacteristic
25C22⋊C8×D61—21no—
26C4×C22⋊Dic61C2211yescharacteristic
27#16839499—22no—
28C2×C22⋊Q8×C61C2211yescharacteristic
29C61×#407C2211yescharacteristic
30C2×C22⋊C8×C61C2211yescharacteristic
31C61×#324—21no—
32C61×#324—21no—
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×#11758202—21no—
46C2×#11758202—21no—
47C2×#11758202—42no—
48C2×#11758201—21no—
49C2×#11758201—21no—
50C2×#11758201—42no—
51#14519783—42no—
52#14519783—42no—
53C2×#11758194—21no—
54C2×#11758189—22no—
55C22×#92418C2×C411yescharacteristic
56C22×#92418C2×C411yescharacteristic
57C2×C8×D61—22no—
58C2×C22⋊Dic61—21no—
59C2×C4×Dic61—21no—
60#14519703—42no—
61#14519703—42no—
62#14519700—21no—
63#14519700—21no—
64#14519697—21no—
65#14519697—21no—
66C22⋊#92418—21no—
67C22⋊#92418—21no—
68C22⋊#92418—21no—
69C22⋊#92418D412yes—
70#14519689—22no—
71#14519689—22no—
72#14519684—21no—
73C22×Q8×C61C2×C411yescharacteristic
74C23×C4×C61C2311yescharacteristic
75C2×M16×C61—21no—
76C22×C8×C61—21no—
77C22⋊Q8×C61—21no—
78C22⋊Q8×C61—21no—
79C22⋊Q8×C61—21no—
80C22⋊Q8×C61—21no—
81C2×C4⋊C4×C61—21no—
82C2×C4⋊C4×C61C2×C411yescharacteristic
83C2×C22⋊C4×C61—21no—
84C22⋊C8×C61—21no—
85C22⋊C8×C61—21no—
86C22⋊C8×C61—21no—
87C22⋊C8×C61D411yescharacteristic
88C22⋊C8×C61D411yescharacteristic
89C24×C61#4111yescharacteristic
90C23×D61—21no—
91C2×Q8×C61—21no—
92C2×Q8×C61—21no—
93C2×Q8×C61—41no—
94C2×Q8×C61—41no—
95C22×C4×C61—21no—
96C22×C4×C61—21no—
97C22×C4×C61—41no—
98C22×C4×C61—41no—
99C22×C4×C61C22×C411yescharacteristic
100C22×C4×C61C2×D411yescharacteristic
101C22×C4×C61C2×D411yescharacteristic
102C22×C4×C61C2×D411yescharacteristic
103C22×C4×C61C2×D411yescharacteristic
104C22×C4×C61#4111yescharacteristic
105C2×C22⋊D61—21no—
106C2×C22⋊D61—21no—
107C2×C22⋊D61—21no—
108C2×C22⋊D61—21no—
109C2×C22⋊D61—44no—
110C22×Dic61—21no—
111C22×Dic61—41no—
112#11758222—22no—
113#11758222—22no—
114#11758222—22no—
115#11758222—22no—
116#11758222—41no—
117#11758222—41no—
118#11758222—41no—
119#11758222—41no—
120#11758222—44no—
121#11758222—44no—
122C2×D244—22no—
123C2×D244—21no—
124C2×D244—41no—
125C2×D244C22⋊C411yescharacteristic
126C2×D244C22⋊C411yescharacteristic
127C2×C4×D61—21no—
128C2×C4×D61—21no—
129C2×C4×D61—21no—
130C2×C4×D61—21no—
131C2×C4×D61—44no—
132C2×Dic122—22no—
133C2×Dic122—21no—
134C2×Dic122—41no—
135C2×Dic122C22⋊C411yescharacteristic
136C2×Dic122C22⋊C411yescharacteristic
137M16×C61—41no—
138M16×C61—41no—
139C2×C8×C61—21no—
140C2×C8×C61—21no—
141C2×C8×C61—41no—
142C2×C8×C61—41no—
143C2×C8×C61—41no—
144C4⋊C4×C61—21no—
145C4⋊C4×C61—21no—
146C4⋊C4×C61—41no—
147C4⋊C4×C61—41no—
148C22⋊C4×C61—41no—
149C22⋊C4×C61—41no—
150C22⋊Dic61—22no—
151C22⋊Dic61—41no—
152#11758202—42no—
153#11758202—42no—
154#11758201—42no—
155#11758201—42no—
156C4×Dic61—41no—
157C4×Dic61—41no—
158#11758194—41no—
159#11758194—41no—
160C2×#92418—21no—
161C2×#92418—21no—
162C2×#92418—22no—
163C2×#92418—41no—
164C2×#92418—41no—
165#11758189—44no—
166C8×D61—44no—
167C23×C61—21no—
168C23×C61—21no—
169C23×C61—41no—
170C23×C61—41no—
171C23×C61C4×D411yescharacteristic
172C23×C61#12111yescharacteristic
173C23×C61#12311yescharacteristic
174C22×D61—21no—
175C22×D61—21no—
176C22×D61—42no—
177C22×D61—41no—
178C22×D61—41no—
179C22×D61—81no—
180Q8×C61—41no—
181Q8×C61—41no—
182Q8×C61—81no—
183C2×C4×C61—21no—
184C2×C4×C61—21no—
185C2×C4×C61—21no—
186C2×C4×C61—21no—
187C2×C4×C61—21no—
188C2×C4×C61—21no—
189C2×C4×C61—41no—
190C2×C4×C61—41no—
191C2×C4×C61—41no—
192C2×C4×C61—41no—
193C2×C4×C61—41no—
194C2×C4×C61—41no—
195C2×C4×C61—41no—
196C2×C4×C61—81no—
197C2×C4×C61C2×C22⋊C411yescharacteristic
198C2×C4×C61C4×D411yescharacteristic
199C2×C4×C61#12011yescharacteristic
200C2×C4×C61#12111yescharacteristic
201C22⋊D61—24no—
202C22⋊D61—42no—
203C22⋊D61—42no—
204C22⋊D61—42no—
205C2×Dic61—21no—
206C2×Dic61—21no—
207C2×Dic61—42no—
208C2×Dic61—42no—
209C2×Dic61—81no—
210C2×Dic61—81no—
211D244—21no—
212D244—21no—
213D244—21no—
214D244—42no—
215D244#9912yes—
216C4×D61—24no—
217C4×D61—42no—
218C4×D61—42no—
219C4×D61—42no—
220Dic122—21no—
221Dic122—21no—
222Dic122—21no—
223Dic122—42no—
224Dic122#9912yes—
225C8×C61—41no—
226C8×C61—41no—
227C8×C61—81no—
228#92418—42no—
229#92418—41no—
230#92418—41no—
231#9239—611nomaximal, Sylow
232C22×C61—21no—
233C22×C61—21no—
234C22×C61—21no—
235C22×C61—21no—
236C22×C61—21no—
237C22×C61—21no—
238C22×C61—21no—
239C22×C61—21no—
240C22×C61—81no—
241C22×C61—81no—
242C22×C61#38611yescharacteristic
243C22×C61#44311yescharacteristic
244C22×C61#44411yescharacteristic
245C2×D61—22no—
246C2×D61—41no—
247C2×D61—41no—
248C2×D61—41no—
249C2×D61—41no—
250C2×D61—41no—
251C2×D61—82no—
252C2×D61—82no—
253C4×C61—21no—
254C4×C61—21no—
255C4×C61—21no—
256C4×C61—81no—
257C4×C61—81no—
258C4×C61C2×#9911yescharacteristic
259C4×C61C2×#9911yescharacteristic
260Dic61—22no—
261Dic61—41no—
262Dic61—82no—
263C2×#452—611no—
264#2880—611no—
265#1789—611no—
266#1508—611no—
267#1472D6111yescharacteristic, complemented
268#1293—611no—
269#1282—611no—
270C2×C61—21no—
271C2×C61—21no—
272C2×C61—41no—
273C2×C61—41no—
274C2×C61#187611yescharacteristic
275C2×C61#189811yescharacteristic
276C2×C61#190211yescharacteristic
277D61—42no—
278D61—81no—
279C22×#41—611no—
280C22×Q16—611no—
281C22×SD16—611no—
282C2×#131—1221no—
283C2×C22⋊Q8C2×D6111yescharacteristic
284#452—1221no—
285#452—1221no—
286#452—1221no—
287#452—1221no—
288C2×#104—1221no—
289C2×#104—1221no—
290C2×#103—611no—
291C2×#102—611no—
292#408—1222no—
293#408—1221no—
294#407—611no—
295#407—1221no—
296#407C2×D6111yescharacteristic
297C2×C22⋊C8—611no—
298C2×C22⋊C8—1221no—
299C2×C22⋊C8C2×D6111yescharacteristic
300C4×C22⋊C4—611no—
301#328—1222no—
302#324—21no—
303#324—21no—
304C61#923911yescharacteristic, complemented, Sylow
305C2×#41—611no—
306C2×#41—611no—
307C2×#41—611no—
308C2×#41—611no—
309C2×#41—1221no—
310C2×#41—1221no—
311C2×#41—2442no—
312C2×#41—2442no—
313C22×Q8—611no—
314C22×Q8C4×D6111yescharacteristic
315C22×D4—611no—
316C22×D4—1221no—
317C23×C4—1221no—
318C23×C4C22×D6111yescharacteristic
319C2×Q16—1221no—
320C2×Q16—1221no—
321C2×Q16—2442no—
322C2×SD16—1221no—
323C2×SD16—1221no—
324C2×SD16—2442no—
325#131—2442no—
326#131—2442no—
327C2×M16—21no—
328C2×M16—1222no—
329C2×M16—1221no—
330C22×C8—21no—
331C22×C8—611no—
332C22×C8—611no—
333C22×C8—1222no—
334C22⋊Q8—21no—
335C22⋊Q8—21no—
336C22⋊Q8—21no—
337C22⋊Q8—21no—
338C2×C4⋊C4—21no—
339C2×C4⋊C4C4×D6111yescharacteristic
340C2×C22⋊C4—21no—
341C2×C22⋊C4—1221no—
342C2×C42—1221no—
343#104—2442no—
344#104—2442no—
345#103—1221no—
346#103—1221no—
347#102—1221no—
348#102—1221no—
349C22⋊C8—21no—
350C22⋊C8—21no—
351C22⋊C8—21no—
352C22⋊C8—612no—
353C22⋊C8—1222no—
354C22⋊C8—1222no—
355C22⋊C8—1221no—
356C22⋊C8—1221no—
357C22⋊C8—1221no—
358C22⋊C8C22⋊D6111yescharacteristic
359C22⋊C8C22⋊D6111yescharacteristic
360#95—1221no—
361C24—1221no—
362C24#1175822211yescharacteristic
363#41—1222no—
364#41—1222no—
365#41—1222no—
366#41—1222no—
367#41—2441no—
368#41—2441no—
369#41—2441no—
370#41—2441no—
371#41—2444no—
372#41—2444no—
373C2×Q8—21no—
374C2×Q8—21no—
375C2×Q8—41no—
376C2×Q8—41no—
377C2×Q8—611no—
378C2×Q8—611no—
379C2×Q8—1222no—
380C2×Q8—1221no—
381C2×Q8—2441no—
382C2×D4—611no—
383C2×D4—611no—
384C2×D4—1222no—
385C2×D4—1221no—
386C2×D4—1221no—
387C2×D4—1221no—
388C2×D4—1221no—
389C2×D4—1221no—
390C2×D4—2441no—
391C2×D4—2444no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—41no—
395C22×C4—41no—
396C22×C4—1221no—
397C22×C4—1221no—
398C22×C4—1221no—
399C22×C4—1221no—
400C22×C4—1221no—
401C22×C4—2444no—
402C22×C4—2441no—
403C22×C4C2×C4×D6111yescharacteristic
404C22×C4D4×D6111yescharacteristic
405C22×C4D4×D6111yescharacteristic
406C22×C4D4×D6111yescharacteristic
407C22×C4#1175822411yescharacteristic
408C22×C4C2×C22⋊D6111yescharacteristic
409Q16—2442no—
410Q16—2442no—
411SD16—2442no—
412SD16—2442no—
413M16—41no—
414M16—41no—
415M16—2444no—
416M16—2441no—
417M16—2441no—
418C2×C8—21no—
419C2×C8—21no—
420C2×C8—41no—
421C2×C8—41no—
422C2×C8—41no—
423C2×C8—1221no—
424C2×C8—1221no—
425C2×C8—1222no—
426C2×C8—2444no—
427C2×C8—2441no—
428C2×C8—2441no—
429C4⋊C4—21no—
430C4⋊C4—21no—
431C4⋊C4—41no—
432C4⋊C4—41no—
433C22⋊C4—41no—
434C22⋊C4—41no—
435C22⋊C4—1222no—
436C22⋊C4—2441no—
437C42—2441no—
438C42—2441no—
439C23—21no—
440C23—21no—
441C23—41no—
442C23—41no—
443C23—1221no—
444C23—1221no—
445C23—4881no—
446C23—2442no—
447C23—2441no—
448C23—2441no—
449C23C4×C22⋊D6111yescharacteristic
450C23#1451975011yescharacteristic
451C23#1451975911yescharacteristic
452Q8—41no—
453Q8—41no—
454Q8—81no—
455Q8—612no—
456Q8—1221no—
457Q8—1221no—
458Q8—1221no—
459Q8—2442no—
460D4—612no—
461D4—1224no—
462D4—1221no—
463D4—1221no—
464D4—1221no—
465D4—2442no—
466D4—2442no—
467D4—2442no—
468D4—2442no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—81no—
483C2×C4—1224no—
484C2×C4—1221no—
485C2×C4—1221no—
486C2×C4—4881no—
487C2×C4—4881no—
488C2×C4—2442no—
489C2×C4—2442no—
490C2×C4—2442no—
491C2×C4—2442no—
492C2×C4—2442no—
493C2×C4C22⋊C4×D6111yescharacteristic
494C2×C4#1451973711yescharacteristic
495C2×C4#1451974511yescharacteristic
496C2×C4#1451974811yescharacteristic
497C8—41no—
498C8—41no—
499C8—81no—
500C8—2442no—
501C8—2441no—
502C8—2441no—
503C22—21no—
504C22—21no—
505C22—21no—
506C22—21no—
507C22—21no—
508C22—21no—
509C22—21no—
510C22—21no—
511C22—81no—
512C22—81no—
513C22—1222no—
514C22—4882no—
515C22—4882no—
516C22—2441no—
517C22—2441no—
518C22—2441no—
519C22—2441no—
520C22—2441no—
521C22#1683963811yescharacteristic
522C22#1683987711yescharacteristic
523C22#1683988211yescharacteristic
524C4—21no—
525C4—21no—
526C4—21no—
527C4—81no—
528C4—81no—
529C4—1222no—
530C4—4882no—
531C4—2441no—
532C4#99×D6111yescharacteristic
533C4#99×D6111yescharacteristic
534C2—21no—
535C2—21no—
536C2—41no—
537C2—41no—
538C2—4881no—
539C2—2442no—
540C2#2144614911yescharacteristic
541C2#2144626911yescharacteristic
542C2#2144629511yescharacteristic
subgroupG/H
centerC22#16839638
commutatorC22×C4×C61C22×C4
FrattiniC23×C4C22×D61
FittingC61×#1472C2
socleC22×C61#386
prime radicalC22×#11758222C22
cosocleC23×C4×C61C23
hypercenter#1472D61
hypocenterC61#9239
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1472

Complements

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

NH ≅ G/Ntype
4. #21444339C2semidirect
5. #21443695C2semidirect
6. #21443665C2semidirect
7. C61×#1472C2semidirect
9. C22×#11758202C4semidirect
10. C22×#11758201C4semidirect
18. C2×#14519700C4semidirect
19. C2×#14519697C4semidirect
267. #1472D61semidirect
304. C61#9239semidirect

Maximal subgroups

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

subgroupindex
C2×#168399222
#214542652
#214457622
#214443392
#214436952
#214436652
C61×#14722
#923961