Finite Group Databasebeta

← #29577999

Subgroups — #29577999

Order 15616.

716 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 716 conjugacy classes of subgroups — too many to draw. Only its 93 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#21454262C211yesmaximal, characteristic
2#21449154C211yesmaximal, characteristic
3#21445745C211yesmaximal, characteristic
4#21444149C211yesmaximal, characteristic, complemented
5#21443728C211yesmaximal, characteristic, complemented
6#21443728C211yesmaximal, characteristic, complemented
7C61×#1463C211yesmaximal, characteristic, complemented
8C22×#11758222C2211yescharacteristic
9C2×M16×D61—21no—
10C2×#14519760C411yescharacteristic
11C2×#14519760C411yescharacteristic
12C2×#14519695—21no—
13C2×#14519693—21no—
14#16839727—22no—
15#16839727—22no—
16#16839726C2211yescharacteristic
17#16839726C2211yescharacteristic
18#16839725—21no—
19#16839682—21no—
20#16839679—21no—
21#16839641C411yescharacteristic
22#16839641C411yescharacteristic
23#16839633—21no—
24#16839633—21no—
25#16839625C2211yescharacteristic
26#16839488—22no—
27C22×C4⋊C4×C61C2211yescharacteristic
28C61×#407C2211yescharacteristic
29C61×#407C2211yescharacteristic
30C61×#324—22no—
31C2×#11758222—21no—
32C2×#11758222—21no—
33C2×#11758222—21no—
34C2×#11758222—21no—
35C2×#11758222—42no—
36C2×#11758222—42no—
37C22×Dic122D411yescharacteristic
38C22×D244D411yescharacteristic
39C22×C22⋊D61C2×C411yescharacteristic
40C22×C22⋊D61C2×C411yescharacteristic
41C22×C4×D61D411yescharacteristic
42C22×C4×D61D411yescharacteristic
43C2×#11758194—21no—
44C2×#11758194—21no—
45C2×#11758189—22no—
46M16×D61—42no—
47M16×D61—42no—
48C2×C8×D61—22no—
49#14519760—42no—
50#14519760—42no—
51C2×#11758196—21no—
52C2×#11758196—21no—
53C2×C22⋊Dic61C2×C411yescharacteristic
54C2×C22⋊Dic61C2×C411yescharacteristic
55C2×#11758198—21no—
56C2×#11758198—21no—
57C2×#11758198—21no—
58C2×#11758198—21no—
59#14519695—42no—
60#14519693—42no—
61C22⋊#92418—21no—
62C22⋊#92418—21no—
63C22⋊#92418—21no—
64C22⋊#92418D412yes—
65#14519689—24no—
66#14519684—21no—
67#14519684—21no—
68#14519684—21no—
69#14519684—21no—
70C23×C4×C61C2×C411yescharacteristic
71C23×C4×C61C2×C411yescharacteristic
72C23×C4×C61C2311yescharacteristic
73C2×M16×C61—21no—
74C2×M16×C61—21no—
75C2×C4⋊C4×C61—21no—
76C2×C4⋊C4×C61—21no—
77C2×C4⋊C4×C61—21no—
78C2×C4⋊C4×C61—21no—
79C2×C4⋊C4×C61—21no—
80C2×C4⋊C4×C61—21no—
81C22⋊C8×C61—21no—
82C22⋊C8×C61—21no—
83C22⋊C8×C61—21no—
84C22⋊C8×C61D412yes—
85C24×C61C22×C411yescharacteristic
86C23×D61C22⋊C411yescharacteristic
87C23×D61C22⋊C411yescharacteristic
88C22×C4×C61—21no—
89C22×C4×C61—21no—
90C22×C4×C61—21no—
91C22×C4×C61—21no—
92C22×C4×C61—21no—
93C22×C4×C61—21no—
94C22×C4×C61—42no—
95C22×C4×C61—41no—
96C22×C4×C61—41no—
97C22×C4×C61—41no—
98C22×C4×C61—41no—
99C22×C4×C61C2×D411yescharacteristic
100C22×C4×C61C2×D411yescharacteristic
101C22×C4×C61C2×D411yescharacteristic
102C22×C4×C61C2×D411yescharacteristic
103C22×C4×C61#4111yescharacteristic
104C22×C4×C61#4111yescharacteristic
105C2×C22⋊D61—21no—
106C2×C22⋊D61—21no—
107C2×C22⋊D61—21no—
108C2×C22⋊D61—21no—
109C2×C22⋊D61—22no—
110C2×C22⋊D61—22no—
111C2×C22⋊D61—41no—
112C2×C22⋊D61—41no—
113C22×Dic61—21no—
114C22×Dic61—21no—
115C22×Dic61C22⋊C411yescharacteristic
116C22×Dic61C22⋊C411yescharacteristic
117#11758222—42no—
118#11758222—42no—
119#11758222—42no—
120#11758222—42no—
121#11758222—44no—
122#11758222—44no—
123C2×D244—22no—
124C2×D244—21no—
125C2×D244—21no—
126C2×D244—41no—
127C2×C4×D61—22no—
128C2×C4×D61—21no—
129C2×C4×D61—21no—
130C2×C4×D61—22no—
131C2×C4×D61—21no—
132C2×C4×D61—21no—
133C2×C4×D61—41no—
134C2×C4×D61—41no—
135C2×Dic122—21no—
136C2×Dic122—21no—
137C2×Dic122—22no—
138C2×Dic122—41no—
139M16×C61—41no—
140M16×C61—41no—
141M16×C61—41no—
142M16×C61—41no—
143C2×C8×C61—22no—
144C2×C8×C61—41no—
145C4⋊C4×C61—41no—
146C4⋊C4×C61—41no—
147C4⋊C4×C61—41no—
148C4⋊C4×C61—41no—
149C22⋊Dic61—41no—
150C22⋊Dic61—41no—
151#11758198—42no—
152#11758198—42no—
153#11758198—42no—
154#11758198—42no—
155#11758196—42no—
156#11758196—42no—
157#11758194—41no—
158#11758194—41no—
159#11758194—41no—
160#11758194—41no—
161C2×#92418—22no—
162C2×#92418—41no—
163#11758189—44no—
164C8×D61—44no—
165C23×C61—21no—
166C23×C61—21no—
167C23×C61—41no—
168C23×C61—41no—
169C23×C61C2×C22⋊C411yescharacteristic
170C23×C61C4×D411yescharacteristic
171C23×C61C4×D411yescharacteristic
172C22×D61—21no—
173C22×D61—21no—
174C22×D61—21no—
175C22×D61—21no—
176C22×D61—21no—
177C22×D61—41no—
178C22×D61—41no—
179C22×D61—41no—
180C22×D61—41no—
181C22×D61#9912yes—
182C2×C4×C61—21no—
183C2×C4×C61—21no—
184C2×C4×C61—21no—
185C2×C4×C61—21no—
186C2×C4×C61—22no—
187C2×C4×C61—22no—
188C2×C4×C61—22no—
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—41no—
197C2×C4×C61—41no—
198C2×C4×C61—41no—
199C2×C4×C61—41no—
200C2×C4×C61—81no—
201C2×C4×C61—81no—
202C2×C4×C61#12011yescharacteristic
203C2×C4×C61#12111yescharacteristic
204C2×C4×C61#12111yescharacteristic
205C2×C4×C61#12311yescharacteristic
206C22⋊D61—42no—
207C22⋊D61—42no—
208C22⋊D61—42no—
209C22⋊D61—42no—
210C2×Dic61—21no—
211C2×Dic61—21no—
212C2×Dic61—21no—
213C2×Dic61—21no—
214C2×Dic61—21no—
215C2×Dic61—41no—
216C2×Dic61—41no—
217C2×Dic61—41no—
218C2×Dic61—41no—
219C2×Dic61—41no—
220C2×Dic61—41no—
221C2×Dic61#9912yes—
222D244—42no—
223D244—42no—
224C4×D61—22no—
225C4×D61—22no—
226C4×D61—24no—
227C4×D61—41no—
228C4×D61—41no—
229C4×D61—42no—
230Dic122—42no—
231Dic122—42no—
232C8×C61—42no—
233C8×C61—81no—
234#92418—42no—
235#92418—81no—
236#9513—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—42no—
246C22×C61—42no—
247C22×C61#38611yescharacteristic
248C22×C61C2×#9911yescharacteristic
249C22×C61C2×#9911yescharacteristic
250C2×D61—22no—
251C2×D61—41no—
252C2×D61—41no—
253C2×D61—42no—
254C2×D61—42no—
255C2×D61—41no—
256C2×D61—41no—
257C2×D61—41no—
258C2×D61—81no—
259C2×D61—81no—
260C4×C61—21no—
261C4×C61—21no—
262C4×C61—21no—
263C4×C61—21no—
264C4×C61—42no—
265C4×C61—81no—
266Dic61—22no—
267Dic61—41no—
268Dic61—81no—
269Dic61—81no—
270#2879—611no—
271#2406—611no—
272#1785—611no—
273#1463—611no—
274#1463D6111yescharacteristic, complemented
275#1306—611no—
276#1306—611no—
277C2×C61—21no—
278C2×C61—21no—
279C2×C61—41no—
280C2×C61—41no—
281C2×C61#189111yescharacteristic
282C2×C61#189811yescharacteristic
283C2×C61#191511yescharacteristic
284D61—42no—
285D61—81no—
286C22×#41—611no—
287C22×M16—1221no—
288C2×#123—611no—
289C2×#123—611no—
290C22×C4⋊C4C2×D6111yescharacteristic
291C2×#101—1221no—
292C2×#100—1221no—
293#408—1222no—
294#408—1222no—
295#407—611no—
296#407—611no—
297#407—1221no—
298#407C2×D6111yescharacteristic
299#407C2×D6111yescharacteristic
300#395—1221no—
301#394—1221no—
302#386—611no—
303#386—611no—
304#385—1221no—
305#385—1221no—
306#381—611no—
307#324—22no—
308#324—1222no—
309C61#951311yescharacteristic, complemented, Sylow
310C2×#41—1221no—
311C2×#41—1221no—
312C2×#41—1221no—
313C2×#41—1221no—
314C2×#41—2442no—
315C2×#41—2442no—
316C22×Q8—611no—
317C22×D4—611no—
318C22×D4—611no—
319C22×D4—611no—
320C23×C4—611no—
321C23×C4—611no—
322C23×C4C4×D6111yescharacteristic
323C23×C4C4×D6111yescharacteristic
324C23×C4C22×D6111yescharacteristic
325C2×M16—21no—
326C2×M16—21no—
327C2×M16—1222no—
328C2×M16—1221no—
329C2×M16—1221no—
330C2×M16—2442no—
331C2×M16—2442no—
332C22×C8—1222no—
333#123—2442no—
334#123—2442no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4—21no—
339C2×C4⋊C4—21no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—1221no—
342C2×C4⋊C4—1221no—
343C2×C22⋊C4—611no—
344C2×C22⋊C4—611no—
345C2×C22⋊C4—1221no—
346C2×C22⋊C4—1221no—
347C2×C22⋊C4—1221no—
348C2×C22⋊C4—1221no—
349#101—2442no—
350#100—2442no—
351C22⋊C8—21no—
352C22⋊C8—21no—
353C22⋊C8—21no—
354C22⋊C8—612no—
355C22⋊C8—1224no—
356C22⋊C8—1221no—
357C22⋊C8—1221no—
358C22⋊C8—1221no—
359C22⋊C8C22⋊D6112yes—
360#95—1221no—
361#95—1221no—
362#95—1221no—
363#95—1221no—
364C24—611no—
365C24—611no—
366C24C2×C4×D6111yescharacteristic
367#41—2444no—
368#41—2442no—
369#41—2442no—
370#41—2444no—
371#41—2442no—
372#41—2442no—
373C2×Q8—1222no—
374C2×Q8—1221no—
375C2×Q8—1221no—
376C2×Q8—2441no—
377C2×D4—1222no—
378C2×D4—1222no—
379C2×D4—1222no—
380C2×D4—1221no—
381C2×D4—1221no—
382C2×D4—1221no—
383C2×D4—1221no—
384C2×D4—1221no—
385C2×D4—1221no—
386C2×D4—2441no—
387C2×D4—2441no—
388C2×D4—2441no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—42no—
396C22×C4—41no—
397C22×C4—41no—
398C22×C4—41no—
399C22×C4—41no—
400C22×C4—611no—
401C22×C4—611no—
402C22×C4—1222no—
403C22×C4—1222no—
404C22×C4—1221no—
405C22×C4—1221no—
406C22×C4—1221no—
407C22×C4—1221no—
408C22×C4—1221no—
409C22×C4—1221no—
410C22×C4—2441no—
411C22×C4—2441no—
412C22×C4#1175822211yescharacteristic
413C22×C4D4×D6111yescharacteristic
414C22×C4D4×D6111yescharacteristic
415C22×C4D4×D6111yescharacteristic
416C22×C4#1175822411yescharacteristic
417C22×C4C2×C22⋊D6111yescharacteristic
418M16—41no—
419M16—41no—
420M16—41no—
421M16—41no—
422M16—2444no—
423M16—2441no—
424M16—2441no—
425M16—2441no—
426M16—2441no—
427C2×C8—22no—
428C2×C8—41no—
429C2×C8—1222no—
430C2×C8—2444no—
431C2×C8—2441no—
432C4⋊C4—41no—
433C4⋊C4—41no—
434C4⋊C4—41no—
435C4⋊C4—41no—
436C4⋊C4—2442no—
437C4⋊C4—2442no—
438C22⋊C4—2442no—
439C22⋊C4—2442no—
440C22⋊C4—2442no—
441C22⋊C4—2442no—
442C22⋊C4—2441no—
443C22⋊C4—2441no—
444C23—21no—
445C23—21no—
446C23—41no—
447C23—41no—
448C23—612no—
449C23—1221no—
450C23—1221no—
451C23—1221no—
452C23—1221no—
453C23—1221no—
454C23—2441no—
455C23—2441no—
456C23—2441no—
457C23—2441no—
458C23C22⋊C4×D6111yescharacteristic
459C23#1451973711yescharacteristic
460C23C4×C22⋊D6111yescharacteristic
461Q8—2442no—
462Q8—2442no—
463D4—2442no—
464D4—2442no—
465D4—2442no—
466D4—2442no—
467D4—2442no—
468D4—2442no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—81no—
488C2×C4—81no—
489C2×C4—612no—
490C2×C4—1222no—
491C2×C4—1222no—
492C2×C4—1224no—
493C2×C4—1221no—
494C2×C4—1221no—
495C2×C4—1221no—
496C2×C4—1221no—
497C2×C4—1221no—
498C2×C4—2441no—
499C2×C4—2441no—
500C2×C4—2442no—
501C2×C4—2441no—
502C2×C4—2441no—
503C2×C4—2441no—
504C2×C4—2441no—
505C2×C4—2441no—
506C2×C4—2441no—
507C2×C4#1451974511yescharacteristic
508C2×C4#1451974811yescharacteristic
509C2×C4#1451975011yescharacteristic
510C2×C4#1451975911yescharacteristic
511C8—42no—
512C8—81no—
513C8—4881no—
514C8—2442no—
515C22—21no—
516C22—21no—
517C22—21no—
518C22—21no—
519C22—21no—
520C22—21no—
521C22—21no—
522C22—21no—
523C22—42no—
524C22—42no—
525C22—1222no—
526C22—4881no—
527C22—4881no—
528C22—2442no—
529C22—2442no—
530C22—2441no—
531C22—2441no—
532C22—2441no—
533C22—2441no—
534C22—2441no—
535C22#1683963811yescharacteristic
536C22#99×D6111yescharacteristic
537C22#99×D6111yescharacteristic
538C4—21no—
539C4—21no—
540C4—21no—
541C4—21no—
542C4—42no—
543C4—81no—
544C4—1222no—
545C4—4881no—
546C4—4881no—
547C4—2441no—
548C2—21no—
549C2—21no—
550C2—41no—
551C2—41no—
552C2—4881no—
553C2—2442no—
554C2#2144623411yescharacteristic
555C2#2144626911yescharacteristic
556C2#2144636811yescharacteristic
subgroupG/H
centerC22#16839638
commutatorC24×C61C22×C4
FrattiniC23×C4C22×D61
FittingC61×#1463C2
socleC22×C61#386
prime radicalC22×#11758222C22
cosocleC23×C4×C61C23
hypercenter#1463D61
hypocenterC61#9513
LCSG ▷ C24×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1463

Complements

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

NH ≅ G/Ntype
4. #21444149C2semidirect
5. #21443728C2semidirect
6. #21443728C2semidirect
7. C61×#1463C2semidirect
274. #1463D61semidirect
309. C61#9513semidirect

Maximal subgroups

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

subgroupindex
#214542622
#214491542
#214457452
#214441492
#214437282
C61×#14632
#951361