Finite Group Databasebeta

← #29571608

Subgroups — #29571608

Order 15616.

852 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 852 conjugacy classes of subgroups — too many to draw. Only its 113 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×#16840459C211yesmaximal, characteristic
2#21445919C211yesmaximal, characteristic, complemented
3#21445916C211yesmaximal, characteristic, complemented
4C4×#14519691C211yesmaximal, characteristic, complemented
5#21444335C211yesmaximal, characteristic, complemented
6#21444155C211yesmaximal, characteristic, complemented
7C61×#1507C211yesmaximal, characteristic, complemented
8#16840459—22no—
9#16840459—22no—
10#16840459—22no—
11#16840459—22no—
12C2×#14519770—21no—
13C2×#14519768C411yescharacteristic, complemented
14C2×#14519758—21no—
15C2×#14519748—21no—
16C2×#14519748C411yescharacteristic, complemented
17C2×D4×Dic61C411yescharacteristic, complemented
18C2×C4×C22⋊D61C2211yescharacteristic
19C2×#14519737—21no—
20C2×#14519736C2211yescharacteristic
21C2×C4⋊C4×D61C411yescharacteristic, complemented
22C2×#14519691—21no—
23#16839726C2211yescharacteristic
24C4×C22⋊Dic61C2211yescharacteristic
25#16839528—21no—
26#16839523—22no—
27#16839522—22no—
28#16839496—22no—
29#16839487—21no—
30#16839485—21no—
31C2×#121×C61C2211yescharacteristic
32C61×#407C2211yescharacteristic
33C4×C22⋊C4×C61C2211yescharacteristic
34C61×#327—22no—
35C22×C22⋊D61—21no—
36C22×C22⋊D61D411yescharacteristic
37C23×Dic61—21no—
38C22×C4×D61—21no—
39C22×C4×D61C2×C411yescharacteristic
40C2×#11758194—21no—
41#14519770—24no—
42#14519770—24no—
43#14519768—22no—
44#14519768—22no—
45#14519758—24no—
46#14519758—24no—
47#14519748—24no—
48#14519748—24no—
49#14519748—22no—
50#14519748—22no—
51D4×Dic61—22no—
52D4×Dic61—22no—
53C4×C22⋊D61—21no—
54C4×C22⋊D61—21no—
55C4×C22⋊D61—21no—
56C4×C22⋊D61—21no—
57#14519737—24no—
58#14519737—24no—
59#14519736—21no—
60#14519736—21no—
61#14519736D412yes—
62#14519736D412yes—
63C2×C4⋊Dic61—21no—
64C2×C4⋊Dic61C2×C411yescharacteristic
65C2×C4⋊Dic61C2×C411yescharacteristic
66C2×#11758196—21no—
67C2×#11758196—21no—
68C2×#11758196D411yescharacteristic
69C4⋊C4×D61—22no—
70C4⋊C4×D61—22no—
71C2×C22⋊Dic61—21no—
72C2×C22⋊Dic61—21no—
73C2×C22⋊Dic61—21no—
74C2×C22⋊Dic61—21no—
75C2×C22⋊Dic61D411yescharacteristic
76C2×#11758198D411yescharacteristic
77C2×C4×Dic61—21no—
78C2×C4×Dic61—21no—
79C2×C4×Dic61C2×C411yescharacteristic
80#14519691—24no—
81C22⋊#92418—21no—
82C22⋊#92418—21no—
83#14519684—21no—
84C22×D4×C61—21no—
85C22×D4×C61C2×C411yescharacteristic
86C23×C4×C61C2311yescharacteristic
87C2×M16×C61—21no—
88#121×C61—21no—
89#121×C61—21no—
90#121×C61D412yes—
91#121×C61D412yes—
92C2×C4⋊C4×C61C2×C411yescharacteristic
93C2×C22⋊C4×C61—21no—
94C2×C22⋊C4×C61—21no—
95C2×C42×C61—21no—
96C22⋊C8×C61—21no—
97C22⋊C8×C61—21no—
98#95×C61—21no—
99C24×C61—21no—
100C24×C61C2×D411yescharacteristic
101C23×D61C22⋊C411yescharacteristic
102C2×D4×C61—22no—
103C2×D4×C61—22no—
104C2×D4×C61—21no—
105C2×D4×C61—42no—
106C2×D4×C61—82no—
107C2×D4×C61C22⋊C412yes—
108C22×C4×C61—21no—
109C22×C4×C61—42no—
110C22×C4×C61—41no—
111C22×C4×C61C22×C411yescharacteristic
112C22×C4×C61C2×D411yescharacteristic
113C22×C4×C61C2×D411yescharacteristic
114C22×C4×C61C2×D411yescharacteristic
115C22×C4×C61C2×D411yescharacteristic
116C22×C4×C61C2×D411yescharacteristic
117C2×C22⋊D61—24no—
118C2×C22⋊D61—21no—
119C2×C22⋊D61—21no—
120C2×C22⋊D61—22no—
121C2×C22⋊D61—41no—
122C2×C22⋊D61—42no—
123C2×C22⋊D61—42no—
124C22×Dic61—21no—
125C22×Dic61—21no—
126C22×Dic61—24no—
127C22×Dic61—41no—
128C22×Dic61—82no—
129C22×Dic61C22⋊C411yescharacteristic
130C22×Dic61C22⋊C411yescharacteristic
131C22×Dic61C22⋊C411yescharacteristic
132C2×C4×D61—21no—
133C2×C4×D61—21no—
134C2×C4×D61—24no—
135C2×C4×D61—42no—
136C2×C4×D61—82no—
137M16×C61—42no—
138C2×C8×C61—41no—
139C4⋊C4×C61—21no—
140C4⋊C4×C61C22⋊C412yes—
141C22⋊C4×C61—22no—
142C22⋊C4×C61—22no—
143C22⋊C4×C61—22no—
144C22⋊C4×C61—41no—
145C42×C61—42no—
146C22⋊Dic61—24no—
147C22⋊Dic61—24no—
148C22⋊Dic61—22no—
149C22⋊Dic61—22no—
150C22⋊Dic61—21no—
151C22⋊Dic61—21no—
152C22⋊Dic61—22no—
153C22⋊Dic61—41no—
154#11758198—21no—
155#11758198—21no—
156C4⋊Dic61—24no—
157C4⋊Dic61—21no—
158C4⋊Dic61—21no—
159C4⋊Dic61C22⋊C412yes—
160C4⋊Dic61C22⋊C412yes—
161#11758196—24no—
162#11758196—24no—
163#11758196—21no—
164#11758196—21no—
165C4×Dic61—21no—
166C4×Dic61—21no—
167C4×Dic61—22no—
168C4×Dic61—22no—
169C4×Dic61—42no—
170#11758194—42no—
171C2×#92418—41no—
172C23×C61—21no—
173C23×C61—21no—
174C23×C61—22no—
175C23×C61—41no—
176C23×C61—41no—
177C23×C61—42no—
178C23×C61—42no—
179C23×C61—41no—
180C23×C61—41no—
181C23×C61C2×C22⋊C411yescharacteristic
182C23×C61#12011yescharacteristic
183C23×C61#12011yescharacteristic
184C22×D61—21no—
185C22×D61—21no—
186C22×D61—41no—
187C22×D61—41no—
188C22×D61#10412yes—
189D4×C61—42no—
190D4×C61—42no—
191D4×C61—82no—
192D4×C61—82no—
193C2×C4×C61—22no—
194C2×C4×C61—22no—
195C2×C4×C61—22no—
196C2×C4×C61—22no—
197C2×C4×C61—41no—
198C2×C4×C61—41no—
199C2×C4×C61—41no—
200C2×C4×C61—42no—
201C2×C4×C61—42no—
202C2×C4×C61—81no—
203C2×C4×C61—81no—
204C2×C4×C61C2×C22⋊C411yescharacteristic
205C2×C4×C61C2×C22⋊C411yescharacteristic
206C2×C4×C61#12011yescharacteristic
207C2×C4×C61#12011yescharacteristic
208C22⋊D61—42no—
209C22⋊D61—42no—
210C22⋊D61—48no—
211C2×Dic61—21no—
212C2×Dic61—21no—
213C2×Dic61—22no—
214C2×Dic61—22no—
215C2×Dic61—21no—
216C2×Dic61—21no—
217C2×Dic61—21no—
218C2×Dic61—42no—
219C2×Dic61—41no—
220C2×Dic61—41no—
221C2×Dic61—41no—
222C2×Dic61—41no—
223C2×Dic61—42no—
224C2×Dic61—84no—
225C2×Dic61—81no—
226C2×Dic61—81no—
227C4×D61—42no—
228C4×D61—42no—
229C4×D61—84no—
230C8×C61—81no—
231#92418—81no—
232#8362—611nomaximal, Sylow
233C22×C61—21no—
234C22×C61—21no—
235C22×C61—21no—
236C22×C61—21no—
237C22×C61—21no—
238C22×C61—21no—
239C22×C61—21no—
240C22×C61—21no—
241C22×C61—42no—
242C22×C61—42no—
243C22×C61—42no—
244C22×C61—81no—
245C22×C61—81no—
246C22×C61—82no—
247C22×C61—82no—
248C22×C61#37911yescharacteristic
249C22×C61C2×#10411yescharacteristic
250C22×C61#42111yescharacteristic
251C2×D61—22no—
252C2×D61—42no—
253C2×D61—41no—
254C2×D61—41no—
255C2×D61—42no—
256C4×C61—21no—
257C4×C61—21no—
258C4×C61—21no—
259C4×C61—42no—
260C4×C61—82no—
261C4×C61#41012yes—
262Dic61—42no—
263Dic61—41no—
264Dic61—41no—
265Dic61—41no—
266Dic61—42no—
267Dic61—82no—
268C2×#553—611no—
269#1826—611no—
270#1826—611no—
271C4×#99—611no—
272#1507—611no—
273#1507D6111yescharacteristic, complemented
274#1465—611no—
275C2×C61—21no—
276C2×C61—21no—
277C2×C61—41no—
278C2×C61—41no—
279C2×C61—82no—
280C2×C61#178211yescharacteristic
281C2×C61#178911yescharacteristic
282C2×C61#179511yescharacteristic
283D61—42no—
284#553—1222no—
285#553—1222no—
286#553—1222no—
287#553—1222no—
288C2×#126—1221no—
289C2×#125—611no—
290C2×#123—1221no—
291C2×#121—611no—
292C2×#121—1221no—
293C2×#121C2×D6111yescharacteristic
294C2×C4×D4—611no—
295C2×C4×D4—611no—
296C2×C4×D4—1221no—
297C2×#117—611no—
298C22×C4⋊C4—611no—
299C2×#99—1221no—
300#407—611no—
301#407C2×D6111yescharacteristic
302C4×C22⋊C4—611no—
303C4×C22⋊C4C2×D6111yescharacteristic
304#342—1221no—
305#339—1222no—
306#339—1222no—
307#327—22no—
308#327—1222no—
309#324—1221no—
310#323—1221no—
311C61#836211yescharacteristic, complemented, Sylow
312C22×D4—21no—
313C22×D4—611no—
314C22×D4—1221no—
315C22×D4C4×D6111yescharacteristic
316C23×C4—611no—
317C23×C4—1221no—
318C23×C4—1221no—
319C23×C4C22×D6111yescharacteristic
320C2×M16—21no—
321C2×M16—1221no—
322#126—1224no—
323#126—1224no—
324#125—1222no—
325#125—1222no—
326#123—1224no—
327#123—1224no—
328#121—21no—
329#121—21no—
330#121—1222no—
331#121—1224no—
332#121—1224no—
333#121—1222no—
334#121D24412yes—
335#121C22⋊D6112yes—
336C4×D4—1224no—
337C4×D4—1224no—
338C4×D4—1222no—
339C4×D4—1222no—
340C4×D4—1221no—
341C4×D4—1221no—
342C4×D4—1221no—
343C4×D4—1221no—
344#117—612no—
345#117—612no—
346#117—1221no—
347#117—1221no—
348C2×C4⋊C4—611no—
349C2×C4⋊C4—611no—
350C2×C4⋊C4—611no—
351C2×C4⋊C4—1222no—
352C2×C4⋊C4—1221no—
353C2×C4⋊C4—1221no—
354C2×C4⋊C4—1221no—
355C2×C4⋊C4—1222no—
356C2×C4⋊C4C4×D6111yescharacteristic
357C2×C22⋊C4—21no—
358C2×C22⋊C4—21no—
359C2×C22⋊C4—611no—
360C2×C22⋊C4—611no—
361C2×C22⋊C4—1221no—
362C2×C22⋊C4—1221no—
363C2×C22⋊C4—1221no—
364C2×C22⋊C4—1221no—
365C2×C42—21no—
366C2×C42—611no—
367C2×C42—1221no—
368C2×C42—1221no—
369#99—1224no—
370C22⋊C8—21no—
371C22⋊C8—21no—
372C22⋊C8—1221no—
373C22⋊C8—1221no—
374#95—21no—
375#95—1221no—
376C24—21no—
377C24—611no—
378C24D4×D6111yescharacteristic
379C2×D4—22no—
380C2×D4—22no—
381C2×D4—21no—
382C2×D4—42no—
383C2×D4—82no—
384C2×D4—1222no—
385C2×D4—1224no—
386C2×D4—1221no—
387C2×D4—1221no—
388C2×D4—2441no—
389C2×D4—2442no—
390C2×D4—2442no—
391C2×D4#1175819812yes—
392C22×C4—21no—
393C22×C4—42no—
394C22×C4—41no—
395C22×C4—611no—
396C22×C4—611no—
397C22×C4—611no—
398C22×C4—1224no—
399C22×C4—1224no—
400C22×C4—1221no—
401C22×C4—1221no—
402C22×C4—1221no—
403C22×C4—1221no—
404C22×C4—4882no—
405C22×C4—4882no—
406C22×C4—2442no—
407C22×C4—2441no—
408C22×C4C2×C4×D6111yescharacteristic
409C22×C4C2×D24411yescharacteristic
410C22×C4D4×D6111yescharacteristic
411C22×C4D4×D6111yescharacteristic
412C22×C4D4×D6111yescharacteristic
413C22×C4C2×C22⋊D6111yescharacteristic
414M16—42no—
415M16—2442no—
416C2×C8—41no—
417C2×C8—2441no—
418C4⋊C4—21no—
419C4⋊C4—612no—
420C4⋊C4—612no—
421C4⋊C4—1224no—
422C4⋊C4—1224no—
423C4⋊C4—1224no—
424C4⋊C4—1221no—
425C4⋊C4—1221no—
426C4⋊C4—1221no—
427C4⋊C4—1221no—
428C4⋊C4#1175819812yes—
429C22⋊C4—22no—
430C22⋊C4—22no—
431C22⋊C4—22no—
432C22⋊C4—41no—
433C22⋊C4—1224no—
434C22⋊C4—1224no—
435C22⋊C4—1221no—
436C22⋊C4—1222no—
437C22⋊C4—1221no—
438C22⋊C4—1222no—
439C22⋊C4—1221no—
440C22⋊C4—1221no—
441C22⋊C4—1222no—
442C22⋊C4—2441no—
443C42—42no—
444C42—1221no—
445C42—1221no—
446C42—1222no—
447C42—1222no—
448C42—2442no—
449C23—21no—
450C23—21no—
451C23—22no—
452C23—41no—
453C23—41no—
454C23—42no—
455C23—42no—
456C23—41no—
457C23—41no—
458C23—612no—
459C23—1221no—
460C23—1221no—
461C23—2441no—
462C23—2441no—
463C23C22⋊C4×D6111yescharacteristic
464C23#1451974511yescharacteristic
465C23#1451974711yescharacteristic
466D4—42no—
467D4—42no—
468D4—82no—
469D4—82no—
470D4—2442no—
471D4—2442no—
472D4—2448no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—81no—
483C2×C4—81no—
484C2×C4—1221no—
485C2×C4—1221no—
486C2×C4—1222no—
487C2×C4—1222no—
488C2×C4—1221no—
489C2×C4—1221no—
490C2×C4—1221no—
491C2×C4—4884no—
492C2×C4—4881no—
493C2×C4—4881no—
494C2×C4—4884no—
495C2×C4—2442no—
496C2×C4—2442no—
497C2×C4—2442no—
498C2×C4—2441no—
499C2×C4—2441no—
500C2×C4—2442no—
501C2×C4—2441no—
502C2×C4—2441no—
503C2×C4C22⋊C4×D6111yescharacteristic
504C2×C4C2×#1175819811yescharacteristic
505C2×C4#1451974511yescharacteristic
506C2×C4#1451974711yescharacteristic
507C8—81no—
508C8—4881no—
509C22—21no—
510C22—21no—
511C22—21no—
512C22—21no—
513C22—21no—
514C22—21no—
515C22—21no—
516C22—21no—
517C22—42no—
518C22—42no—
519C22—42no—
520C22—81no—
521C22—81no—
522C22—82no—
523C22—82no—
524C22—1222no—
525C22—2442no—
526C22—2442no—
527C22—2441no—
528C22—2441no—
529C22#1683961711yescharacteristic
530C22#104×D6111yescharacteristic
531C22#1683977711yescharacteristic
532C4—21no—
533C4—21no—
534C4—21no—
535C4—42no—
536C4—82no—
537C4—4882no—
538C4—2441no—
539C4—2441no—
540C4—2442no—
541C4—2441no—
542C4—2442no—
543C4#1683973412yes—
544C2—21no—
545C2—21no—
546C2—41no—
547C2—41no—
548C2—82no—
549C2—2442no—
550C2#2144573511yescharacteristic
551C2#2144576111yescharacteristic
552C2#2144578611yescharacteristic
subgroupG/H
centerC22#16839617
commutatorC22×C4×C61C22×C4
FrattiniC23×C4C22×D61
FittingC61×#1507C2
socleC22×C61#379
prime radicalC2×#16840459C2
cosocleC23×C4×C61C23
hypercenter#1507D61
hypocenterC61#8362
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. #21445919C2semidirect
3. #21445916C2semidirect
4. C4×#14519691C2semidirect
5. #21444335C2semidirect
6. #21444155C2semidirect
7. C61×#1507C2semidirect
13. C2×#14519768C4semidirect
16. C2×#14519748C4semidirect
17. C2×D4×Dic61C4semidirect
21. C2×C4⋊C4×D61C4semidirect
273. #1507D61semidirect
311. C61#8362semidirect

Maximal subgroups

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

subgroupindex
C2×#168404592
#214459192
#214459162
C4×#145196912
#214443352
#214441552
C61×#15072
#836261