Finite Group Databasebeta

← #29597507

Subgroups — #29597507

Order 15616.

818 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 818 conjugacy classes of subgroups — too many to draw. Only its 107 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
1#21449134C211yesmaximal, characteristic
2C2×#16839550C212yesmaximal, complemented
3#21445737C211yesmaximal, characteristic, complemented
4#21443938C212yesmaximal, complemented
5C2×C61×#413C211yesmaximal, characteristic, complemented
6C2×#14519745C411yescharacteristic, complemented
7C2×#14519745C2211yescharacteristic
8C22×#11758198C411yescharacteristic, complemented
9C2×#14519692—21no—
10C2×#14519692C2212yes—
11#16839673—21no—
12#16839670—21no—
13#16839642C411yescharacteristic, complemented
14#16839642C2211yescharacteristic
15#16839641—21no—
16#16839641—21no—
17#16839625—21no—
18#16839625C411yescharacteristic, complemented
19#16839550—24no—
20C22×#41×C61C2211yescharacteristic
21C22×M16×C61—21no—
22C61×#413—24no—
23C2×#101×C61—21no—
24C2×#100×C61C2212yes—
25C24×D61C2×C411yescharacteristic
26C22×C22⋊D61—21no—
27C22×C22⋊D61—21no—
28C23×Dic61C2×C411yescharacteristic
29#14519745—22no—
30#14519745—22no—
31#14519745—42no—
32C2×C22⋊Dic61—22no—
33C2×C22⋊Dic61—21no—
34C2×C22⋊Dic61—21no—
35C2×C22⋊Dic61C2×C411yescharacteristic
36C2×C22⋊Dic61C2×C411yescharacteristic
37C2×#11758198—21no—
38C2×#11758198—21no—
39C2×#11758198—21no—
40C2×#11758198—21no—
41C2×#11758198—41no—
42C2×#11758198—41no—
43#14519692—22no—
44#14519692—22no—
45#14519692—42no—
46#14519684—21no—
47#14519684—21no—
48#14519684—21no—
49#14519684—21no—
50#14519684—41no—
51#14519684—41no—
52C2×#41×C61—22no—
53C2×#41×C61—22no—
54C2×#41×C61—41no—
55C2×#41×C61—41no—
56C2×#41×C61—42no—
57C22×Q8×C61D411yescharacteristic
58C22×D4×C61C2×C411yescharacteristic
59C22×D4×C61D411yescharacteristic
60C22×D4×C61C2311yescharacteristic
61C23×C4×C61C2×C411yescharacteristic
62C23×C4×C61D411yescharacteristic
63C23×C4×C61D411yescharacteristic
64C2×M16×C61—24no—
65C2×M16×C61—22no—
66C2×M16×C61—42no—
67C2×M16×C61—41no—
68C22×C8×C61—41no—
69#101×C61—24no—
70#100×C61—22no—
71#100×C61D414yescomplemented
72C24×C61C22⋊C411yescharacteristic
73C24×C61C22×C411yescharacteristic
74C24×C61C2×D411yescharacteristic
75C23×D61—21no—
76C23×D61—22no—
77C23×D61—42no—
78C23×D61—41no—
79C23×D61—41no—
80C23×D61—81no—
81#41×C61—44no—
82#41×C61—44no—
83#41×C61—44no—
84#41×C61—44no—
85C2×Q8×C61—21no—
86C2×Q8×C61—21no—
87C2×Q8×C61—41no—
88C2×Q8×C61—41no—
89C2×D4×C61—22no—
90C2×D4×C61—22no—
91C2×D4×C61—22no—
92C2×D4×C61—22no—
93C2×D4×C61—41no—
94C2×D4×C61—41no—
95C2×D4×C61—41no—
96C2×D4×C61—41no—
97C2×D4×C61C2×D412yes—
98C2×D4×C61#4112yes—
99C22×C4×C61—22no—
100C22×C4×C61—21no—
101C22×C4×C61—21no—
102C22×C4×C61—21no—
103C22×C4×C61—21no—
104C22×C4×C61—22no—
105C22×C4×C61—22no—
106C22×C4×C61—21no—
107C22×C4×C61—21no—
108C22×C4×C61—21no—
109C22×C4×C61—21no—
110C22×C4×C61—41no—
111C22×C4×C61—41no—
112C22×C4×C61C22⋊C411yescharacteristic
113C22×C4×C61C22⋊C411yescharacteristic
114C22×C4×C61C22⋊C411yescharacteristic
115C22×C4×C61C2×D411yescharacteristic
116C2×C22⋊D61—42no—
117C2×C22⋊D61—42no—
118C2×C22⋊D61—42no—
119C2×C22⋊D61—42no—
120C2×C22⋊D61—81no—
121C2×C22⋊D61—81no—
122C22×Dic61—21no—
123C22×Dic61—21no—
124C22×Dic61—21no—
125C22×Dic61—41no—
126C22×Dic61—42no—
127C22×Dic61—81no—
128M16×C61—24no—
129M16×C61—44no—
130M16×C61—42no—
131C2×C8×C61—44no—
132C2×C8×C61—42no—
133C22⋊Dic61—22no—
134C22⋊Dic61—41no—
135C22⋊Dic61—42no—
136C22⋊Dic61—42no—
137C22⋊Dic61—81no—
138C22⋊Dic61—81no—
139#11758198—42no—
140#11758198—42no—
141#11758198—41no—
142#11758198—41no—
143#11758198—41no—
144#11758198—41no—
145C23×C61—21no—
146C23×C61—21no—
147C23×C61—21no—
148C23×C61—21no—
149C23×C61—21no—
150C23×C61—21no—
151C23×C61—41no—
152C23×C61—41no—
153C23×C61—41no—
154C23×C61—41no—
155C23×C61—41no—
156C23×C61—41no—
157C23×C61#9912yes—
158C23×C61C2×C22⋊C411yescharacteristic
159C23×C61C4×D412yes—
160C23×C61#12112yes—
161C22×D61—22no—
162C22×D61—41no—
163C22×D61—41no—
164C22×D61—41no—
165C22×D61—42no—
166C22×D61—41no—
167C22×D61—41no—
168C22×D61—41no—
169C22×D61—42no—
170C22×D61—84no—
171C22×D61—84no—
172C22×D61—82no—
173C22×D61—82no—
174Q8×C61—42no—
175Q8×C61—42no—
176D4×C61—42no—
177D4×C61—42no—
178D4×C61—44no—
179D4×C61—44no—
180C2×C4×C61—22no—
181C2×C4×C61—22no—
182C2×C4×C61—22no—
183C2×C4×C61—22no—
184C2×C4×C61—21no—
185C2×C4×C61—21no—
186C2×C4×C61—21no—
187C2×C4×C61—21no—
188C2×C4×C61—21no—
189C2×C4×C61—21no—
190C2×C4×C61—21no—
191C2×C4×C61—21no—
192C2×C4×C61—21no—
193C2×C4×C61—21no—
194C2×C4×C61—42no—
195C2×C4×C61—42no—
196C2×C4×C61—42no—
197C2×C4×C61—42no—
198C2×C4×C61#9911yescharacteristic
199C2×C4×C61#9911yescharacteristic
200C2×C4×C61#12011yescharacteristic
201C2×C4×C61#12311yescharacteristic
202C22⋊D61—84no—
203C22⋊D61—84no—
204C2×Dic61—41no—
205C2×Dic61—41no—
206C2×Dic61—41no—
207C2×Dic61—41no—
208C2×Dic61—41no—
209C2×Dic61—41no—
210C2×Dic61—41no—
211C2×Dic61—41no—
212C2×Dic61—41no—
213C2×Dic61—82no—
214C2×Dic61—82no—
215C2×Dic61—82no—
216C2×Dic61—82no—
217C8×C61—44no—
218#13083—611nomaximal, Sylow
219C22×C61—21no—
220C22×C61—21no—
221C22×C61—22no—
222C22×C61—22no—
223C22×C61—22no—
224C22×C61—42no—
225C22×C61—42no—
226C22×C61—42no—
227C22×C61—42no—
228C22×C61—41no—
229C22×C61—41no—
230C22×C61—41no—
231C22×C61—41no—
232C22×C61—41no—
233C22×C61—41no—
234C22×C61—81no—
235C22×C61—81no—
236C22×C61#38611yescharacteristic
237C22×C61C2×#9911yescharacteristic
238C22×C61C2×#9911yescharacteristic
239C2×D61—42no—
240C2×D61—42no—
241C2×D61—42no—
242C2×D61—81no—
243C2×D61—81no—
244C2×D61—84no—
245C2×D61—81no—
246C2×D61—81no—
247C2×D61—84no—
248C4×C61—21no—
249C4×C61—21no—
250C4×C61—21no—
251C4×C61—21no—
252C4×C61—41no—
253C4×C61—41no—
254Dic61—81no—
255Dic61—81no—
256Dic61—81no—
257Dic61—162no—
258C2×#413D6111yescharacteristic, complemented
259#2404—611no—
260C2×AΣL(1,𝔽16)—612no—
261#1782—611no—
262#1390—612no—
263C2×C61—21no—
264C2×C61—21no—
265C2×C61—42no—
266C2×C61—42no—
267C2×C61—42no—
268C2×C61#189811yescharacteristic
269C2×C61#212011yescharacteristic
270C2×C61#212211yescharacteristic
271D61—82no—
272C22×#41C2×D6111yescharacteristic
273C22×M16—21no—
274C2×#120—611no—
275C2×#120—611no—
276C22×C22⋊C4—611no—
277#413—24no—
278C2×#101—21no—
279C2×#100C2×D6112yes—
280C2×#99—612no—
281C2×#99—1221no—
282#393—1221no—
283#392—1221no—
284#386—611no—
285#386—611no—
286#386—1221no—
287#386—1221no—
288#381—611no—
289#381—1221no—
290AΣL(1,𝔽16)—1224no—
291C61#1308311yescharacteristic, complemented, Sylow
292C25—611no—
293C2×#41—22no—
294C2×#41—22no—
295C2×#41—41no—
296C2×#41—41no—
297C2×#41—42no—
298C22×Q8C22⋊D6111yescharacteristic
299C22×D4—1221no—
300C22×D4—1221no—
301C22×D4C4×D6111yescharacteristic
302C22×D4D24411yescharacteristic
303C22×D4C22×D6111yescharacteristic
304C23×C4—611no—
305C23×C4C4×D6111yescharacteristic
306C23×C4D24411yescharacteristic
307C23×C4C22⋊D6111yescharacteristic
308C2×M16—24no—
309C2×M16—22no—
310C2×M16—42no—
311C2×M16—41no—
312C22×C8—41no—
313#120—1222no—
314#120—1222no—
315#120—2442no—
316C2×C22⋊C4—611no—
317C2×C22⋊C4—611no—
318C2×C22⋊C4—1221no—
319C2×C22⋊C4—1221no—
320C2×C22⋊C4—1221no—
321C2×C22⋊C4—1221no—
322C2×C22⋊C4—1221no—
323C2×C22⋊C4—1221no—
324C2×C22⋊C4—1222no—
325C2×C22⋊C4—2441no—
326C2×C22⋊C4—2441no—
327#101—24no—
328#100—22no—
329#100C22⋊D6114yescomplemented
330#99—1222no—
331#99—1222no—
332#99—2442no—
333#95—1221no—
334#95—1221no—
335#95—1221no—
336#95—1221no—
337#95—2441no—
338#95—2441no—
339C24—1221no—
340C24—1222no—
341C24—4881no—
342C24—2441no—
343C24—2441no—
344C24—2442no—
345C24#1175819811yescharacteristic
346C24C2×C4×D6111yescharacteristic
347C24C2×D24411yescharacteristic
348#41—44no—
349#41—44no—
350#41—44no—
351#41—44no—
352C2×Q8—21no—
353C2×Q8—21no—
354C2×Q8—41no—
355C2×Q8—41no—
356C2×D4—22no—
357C2×D4—22no—
358C2×D4—22no—
359C2×D4—22no—
360C2×D4—41no—
361C2×D4—41no—
362C2×D4—41no—
363C2×D4—41no—
364C2×D4—4881no—
365C2×D4—4881no—
366C2×D4—2442no—
367C2×D4—2442no—
368C2×D4—2442no—
369C2×D4—2442no—
370C2×D4#1175822212yes—
371C2×D4C2×C22⋊D6112yes—
372C22×C4—22no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—41no—
384C22×C4—41no—
385C22×C4—1221no—
386C22×C4—1221no—
387C22×C4—1221no—
388C22×C4—4881no—
389C22×C4—2442no—
390C22×C4—2441no—
391C22×C4#1175819811yescharacteristic
392C22×C4#1175819811yescharacteristic
393C22×C4#1175819811yescharacteristic
394C22×C4C2×C22⋊D6111yescharacteristic
395M16—24no—
396M16—44no—
397M16—42no—
398C2×C8—44no—
399C2×C8—42no—
400C22⋊C4—1222no—
401C22⋊C4—4881no—
402C22⋊C4—4881no—
403C22⋊C4—2442no—
404C22⋊C4—2442no—
405C22⋊C4—2442no—
406C22⋊C4—2442no—
407C22⋊C4—2441no—
408C22⋊C4—2441no—
409C22⋊C4—2441no—
410C22⋊C4—2441no—
411C22⋊C4—2441no—
412C23—21no—
413C23—21no—
414C23—21no—
415C23—21no—
416C23—21no—
417C23—21no—
418C23—41no—
419C23—41no—
420C23—41no—
421C23—41no—
422C23—41no—
423C23—41no—
424C23—1222no—
425C23—4882no—
426C23—4882no—
427C23—4884no—
428C23—4884no—
429C23—2441no—
430C23—2441no—
431C23—2441no—
432C23—2442no—
433C23—2442no—
434C23—2441no—
435C23—2441no—
436C23—2441no—
437C23#1451969112yes—
438C23C2×#1175819811yescharacteristic
439C23C4×C22⋊D6112yes—
440C23#1451975112yes—
441Q8—42no—
442Q8—42no—
443D4—42no—
444D4—42no—
445D4—44no—
446D4—44no—
447D4—4884no—
448D4—4884no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—4882no—
468C2×C4—4882no—
469C2×C4—4882no—
470C2×C4—4882no—
471C2×C4—2441no—
472C2×C4—2441no—
473C2×C4—2441no—
474C2×C4—2441no—
475C2×C4—2441no—
476C2×C4—2441no—
477C2×C4—2441no—
478C2×C4—2441no—
479C2×C4—2441no—
480C2×C4#1451969111yescharacteristic
481C2×C4#1451969111yescharacteristic
482C2×C4#1451974611yescharacteristic
483C2×C4#1451976011yescharacteristic
484C8—44no—
485C22—21no—
486C22—21no—
487C22—22no—
488C22—22no—
489C22—22no—
490C22—42no—
491C22—42no—
492C22—42no—
493C22—42no—
494C22—41no—
495C22—41no—
496C22—41no—
497C22—41no—
498C22—41no—
499C22—41no—
500C22—81no—
501C22—81no—
502C22—4881no—
503C22—4881no—
504C22—4884no—
505C22—4884no—
506C22—4881no—
507C22—4881no—
508C22—2442no—
509C22—2442no—
510C22—2442no—
511C22#1683964111yescharacteristic
512C22C2×#1451969111yescharacteristic
513C22C2×#1451969111yescharacteristic
514C4—21no—
515C4—21no—
516C4—21no—
517C4—21no—
518C4—41no—
519C4—41no—
520C4—9762no—
521C4—4881no—
522C4—4881no—
523C4—4881no—
524C2—21no—
525C2—21no—
526C2—42no—
527C2—42no—
528C2—42no—
529C2—4882no—
530C2#2144627211yescharacteristic
531C2#2144746111yescharacteristic
532C2#2144747011yescharacteristic
subgroupG/H
centerC22C2×#14519691
commutatorC24×C61C22×C4
FrattiniC22×D4C22×D61
FittingC2×C61×#413C2
socleC22×C61C2×#99
prime radical#21449134C2
cosocleC22×D4×C61C23
hypercenterC2×#413D61
hypocenterC61#13083
LCSG ▷ C24×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ C2×#413

Complements

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

NH ≅ G/Ntype
2. C2×#16839550C2semidirect
3. #21445737C2semidirect
4. #21443938C2semidirect
5. C2×C61×#413C2semidirect
6. C2×#14519745C4semidirect
8. C22×#11758198C4semidirect
13. #16839642C4semidirect
18. #16839625C4semidirect
71. #100×C61D4semidirect
258. C2×#413D61semidirect
291. C61#13083semidirect
329. #100C22⋊D61semidirect

Maximal subgroups

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

subgroupindex
#214491342
C2×#168395502
#214457372
#214439382
C2×C61×#4132
#1308361