Finite Group Databasebeta

← #30620484

Subgroups — #30620484

Order 15616.

1172 conjugacy classes of subgroups.

The subgroup lattice has 558 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462328C212yesmaximal, complemented
2#21461959C212yesmaximal, complemented
3#21461959C214yesmaximal, complemented
4#21461682C212yesmaximal, complemented
5#21461680C211yesmaximal, characteristic, complemented
6C4×#14519836C211yesmaximal, characteristic, complemented
7#21461564C212yesmaximal, complemented
8C2×#16840465C211yesmaximal, characteristic
9C2×#14519836C2214yes—
10C2×#14519836C2212yes—
11C2×#14519836C2212yes—
12C2×#14519836C2214yes—
13#16840713—22no—
14#16840711—24no—
15#16840704C2212yes—
16#16840704C2214yes—
17C4⋊C4×#6923—22no—
18C4×C22⋊#6923C2214yes—
19C4×C22⋊#6923C2212yes—
20#16840640C2211yescharacteristic
21#16840640C2212yes—
22#16840640C2211yescharacteristic
23#16840465—24no—
24#16840465—22no—
25#16840465—22no—
26C2×#14519772C412yescomplemented
27C2×#14519762C412yescomplemented
28C2×#14519752C411yescharacteristic, complemented
29C2×#14519752C2211yescharacteristic
30C2×#14519752C2212yes—
31C2×C4×D244C411yescharacteristic, complemented
32C2×C4×D244C2211yescharacteristic
33C2×#14519739C2212yes—
34C2×#14519735C411yescharacteristic, complemented
35C2×C4⋊C4×D61C2211yescharacteristic
36C2×#125×C61C411yescharacteristic, complemented
37C2×C4⋊#6923—22no—
38C2×C4⋊#6923—24no—
39C2×C22⋊#6923C2314yes—
40C2×C22⋊#6923C2312yes—
41C2×C22⋊#6923C2312yes—
42C2×C4×#6923—22no—
43C2×C4×#6923—24no—
44#14519836—28no—
45#14519836—24no—
46#14519836—42no—
47#14519836—44no—
48(C2×#6923)⋊C4—22no—
49(C2×#6923)⋊C4—24no—
50(C2×#6923)⋊C4—22no—
51(C2×#6923)⋊C4—24no—
52C22×D244C2×C411yescharacteristic
53C22×D244C2×C412yes—
54C22×D244C2312yes—
55C22×D244C2311yescharacteristic
56C22×C4×D61C2×C411yescharacteristic
57C22×C4×D61C2×C412yes—
58C22×C4×D61C2311yescharacteristic
59C22×C4×D61C2312yes—
60#14519772—24no—
61#14519772—24no—
62#14519762—24no—
63#14519762—24no—
64#14519752—24no—
65#14519752—24no—
66#14519752—22no—
67#14519752—22no—
68#14519752—22no—
69#14519752—22no—
70C4×D244—22no—
71C4×D244—22no—
72C4×D244—22no—
73C4×D244—21no—
74C4×D244—21no—
75#14519739—24no—
76#14519739—24no—
77#14519735—22no—
78#14519735—21no—
79#14519735—21no—
80C2×C4⋊Dic61C2×C411yescharacteristic
81C2×C4⋊Dic61C2×C411yescharacteristic
82C2×#11758196C2×C412yes—
83C4⋊C4×D61D414yes—
84C4⋊C4×D61D412yes—
85C4⋊C4×D61D412yes—
86C2×#11758198C2×C412yes—
87C2×#11758198C2×C412yes—
88C2×#11758198C2×C412yes—
89C2×#11758198C2×C411yescharacteristic
90C2×#11758198C2×C412yes—
91C2×#11758198C2311yescharacteristic
92C2×C4×Dic61C2×C412yes—
93#125×C61—22no—
94#125×C61—22no—
95C2×C4⋊C4×C61C2×C411yescharacteristic
96C2×C4⋊C4×C61C2×C412yes—
97C2×C4⋊C4×C61C2×C412yes—
98C2×C4⋊C4×C61C2×C411yescharacteristic
99C2×C42×C61C2×C411yescharacteristic
100C23×D61C22×C411yescharacteristic
101C23×D61C22×C412yes—
102C23×D61C2411yescharacteristic
103C22×#6923—22no—
104C22×#6923—22no—
105C22×#6923—24no—
106C22×C4×C61C22×C411yescharacteristic
107C22×C4×C61C22×C412yes—
108C22×C4×C61C22×C411yescharacteristic
109C22×C4×C61C22×C411yescharacteristic
110C22×C4×C61C22×C412yes—
111C22×Dic61C22×C411yescharacteristic
112C22×Dic61C22×C412yes—
113C22×Dic61C22×C411yescharacteristic
114C2×D244—24no—
115C2×D244—22no—
116C2×D244—24no—
117C2×D244—22no—
118C2×D244—22no—
119C2×D244—21no—
120C2×D244—22no—
121C2×D244—21no—
122C2×D244—21no—
123C2×D244—21no—
124C2×D244—24no—
125C2×D244—42no—
126C2×D244—82no—
127C2×C4×D61—21no—
128C2×C4×D61—21no—
129C2×C4×D61—22no—
130C2×C4×D61—22no—
131C2×C4×D61—42no—
132C2×C4×D61—42no—
133C2×C4×D61—82no—
134C2×C4×D61—82no—
135C2×C4×D61C2×D412yes—
136C2×C4×D61C2×D412yes—
137C2×C4×D61C2×D414yes—
138C2×C4×D61C2×D414yes—
139C22⋊#6923—22no—
140C22⋊#6923—24no—
141C22⋊#6923—22no—
142C22⋊#6923—24no—
143C22⋊#6923—22no—
144C22⋊#6923#4114yes—
145C4⋊#6923—44no—
146C4⋊#6923—84no—
147C4×#6923—44no—
148C4×#6923—84no—
149C4⋊C4×C61—21no—
150C4⋊C4×C61—21no—
151C4⋊C4×C61—24no—
152C4⋊C4×C61—24no—
153C4⋊C4×C61C22⋊C412yes—
154C4⋊C4×C61C22⋊C412yes—
155C42×C61—21no—
156C42×C61—21no—
157#11758198—22no—
158#11758198—24no—
159#11758198—22no—
160#11758198—22no—
161#11758198—22no—
162#11758198—22no—
163#11758198—22no—
164#11758198—24no—
165C4⋊Dic61—22no—
166C4⋊Dic61C22⋊C412yes—
167C4⋊Dic61C22⋊C411yescharacteristic
168C4⋊Dic61C22⋊C411yescharacteristic
169#11758196C22⋊C414yes—
170#11758196C22⋊C414yes—
171C4×Dic61—22no—
172C4×Dic61—22no—
173C23×C61C23×C411yescharacteristic
174C22×D61—21no—
175C22×D61—22no—
176C22×D61—21no—
177C22×D61—22no—
178C22×D61—21no—
179C22×D61—22no—
180C22×D61—41no—
181C22×D61—41no—
182C22×D61—41no—
183C22×D61—42no—
184C22×D61—42no—
185C22×D61—41no—
186C22×D61C22×D412yes—
187C22×D61C2×#4112yes—
188C22×D61H(5,𝔽2)11yescharacteristic
189C22×D612−1+411yescharacteristic
190C2×#6923—24no—
191C2×#6923—42no—
192C2×#6923—42no—
193C2×#6923—42no—
194C2×#6923—44no—
195C2×#6923—44no—
196C2×#6923—42no—
197C2×#6923—42no—
198C2×#6923—44no—
199C2×C4×C61—22no—
200C2×C4×C61—22no—
201C2×C4×C61—21no—
202C2×C4×C61—21no—
203C2×C4×C61—21no—
204C2×C4×C61—22no—
205C2×C4×C61—21no—
206C2×C4×C61—22no—
207C2×C4×C61—42no—
208C2×C4×C61—41no—
209C2×C4×C61C2×C22⋊C414yes—
210C2×C4×C61C2×C22⋊C411yescharacteristic
211C2×C4×C61C2×C22⋊C411yescharacteristic
212C2×Dic61—21no—
213C2×Dic61—41no—
214C2×Dic61—42no—
215C2×Dic61—41no—
216C2×Dic61C2×C22⋊C412yes—
217C2×Dic61C2×C22⋊C412yes—
218C2×Dic61C2×C22⋊C412yes—
219D244—42no—
220D244—44no—
221D244—42no—
222D244—44no—
223D244—42no—
224D244—42no—
225D244—84no—
226C4×D61—24no—
227C4×D61—44no—
228C4×D61—44no—
229C4×D61—42no—
230C4×D61—42no—
231C4×D61—42no—
232C4×D61—82no—
233C4×D61—82no—
234#22535—611nomaximal, Sylow
235C22×C61—21no—
236C22×C61—21no—
237C22×C61C22×C22⋊C411yescharacteristic
238C22×C61C4×#4111yescharacteristic
239C22×C61#52011yescharacteristic
240C2×D61—22no—
241C2×D61—22no—
242C2×D61—42no—
243C2×D61—41no—
244C2×D61—41no—
245C2×D61—41no—
246C2×D61—42no—
247C2×D61—41no—
248C2×D61—42no—
249C2×D61—42no—
250C2×D61—41no—
251C2×D61—81no—
252C2×D61—82no—
253C2×D61—81no—
254C2×D61—82no—
255C2×D61#54812yes—
256C2×D61#55012yes—
257#6923—44no—
258#6923—82no—
259#6923—84no—
260C4×C61—21no—
261C4×C61—24no—
262C4×C61—42no—
263C4×C61—41no—
264C4×C61—41no—
265C4×C61#41012yes—
266Dic61—22no—
267Dic61—22no—
268Dic61—22no—
269Dic61—41no—
270C2×#553—611no—
271#2345—612no—
272#1964—614no—
273#1964—612no—
274#1799—611no—
275#1799—612no—
276C4×#99—612no—
277C4×#99—611no—
278C2×C61—21no—
279C2×C61—21no—
280C2×C61#234111yescharacteristic
281C2×C61C2×#41011yescharacteristic
282C2×C61#288311yescharacteristic
283D61—22no—
284D61—22no—
285D61—81no—
286D61—82no—
287#553—1222no—
288#553—1224no—
289#553—1222no—
290C2×#126—612no—
291C2×#125#692311yescharacteristic, complemented
292C2×#123—612no—
293C2×#121—612no—
294C2×#121—611no—
295C2×#121—611no—
296C2×C4×D4—612no—
297C2×C4×D4—611no—
298C2×C4×D4—611no—
299C2×#117—611no—
300C22×C4⋊C4—611no—
301C2×#99—614no—
302C2×#99—612no—
303C2×#99—612no—
304C2×#99—614no—
305#389—1222no—
306#387—1224no—
307#380—614no—
308#380—612no—
309C4×C4⋊C4—1222no—
310C4×C22⋊C4—612no—
311C4×C22⋊C4—614no—
312#342—612no—
313#342—611no—
314#342—611no—
315C61#2253511yescharacteristic, complemented, Sylow
316C22×D4—612no—
317C22×D4—611no—
318C22×D4—612no—
319C22×D4—611no—
320C23×C4—612no—
321C23×C4—611no—
322C23×C4—612no—
323C23×C4—611no—
324#126—1224no—
325#126—1224no—
326#125—22no—
327#125—22no—
328#123—1224no—
329#123—1224no—
330#121—1224no—
331#121—1224no—
332#121—1222no—
333#121—1222no—
334#121—1222no—
335#121—1222no—
336C4×D4—1222no—
337C4×D4—1221no—
338C4×D4—1221no—
339C4×D4—1222no—
340C4×D4—1222no—
341C4×D4—1224no—
342C4×D4—1224no—
343#117—1222no—
344#117—1221no—
345#117—1221no—
346C2×C4⋊C4—611no—
347C2×C4⋊C4—612no—
348C2×C4⋊C4—612no—
349C2×C4⋊C4—614no—
350C2×C4⋊C4—612no—
351C2×C4⋊C4—611no—
352C2×C4⋊C4—1224no—
353C2×C4⋊C4—1222no—
354C2×C4⋊C4C2×#692311yescharacteristic
355C2×C4⋊C4C2×#692312yes—
356C2×C4⋊C4C2×#692312yes—
357C2×C4⋊C4C2×#692311yescharacteristic
358C2×C22⋊C4—612no—
359C2×C22⋊C4—612no—
360C2×C22⋊C4—611no—
361C2×C22⋊C4—612no—
362C2×C22⋊C4—612no—
363C2×C22⋊C4—611no—
364C2×C22⋊C4—612no—
365C2×C22⋊C4—614no—
366C2×C22⋊C4—612no—
367C2×C42—612no—
368C2×C42—1224no—
369C2×C42—1222no—
370C2×C42C2×#692311yescharacteristic
371#99—1228no—
372#99—1224no—
373#99—2442no—
374#99—2444no—
375#95—1222no—
376#95—1224no—
377#95—1224no—
378#95—1222no—
379C24—611no—
380C24—612no—
381C24—611no—
382C2×D4—1224no—
383C2×D4—1222no—
384C2×D4—1222no—
385C2×D4—1221no—
386C2×D4—1222no—
387C2×D4—1224no—
388C2×D4—1222no—
389C2×D4—1221no—
390C2×D4—1224no—
391C2×D4—1221no—
392C2×D4—1221no—
393C2×D4—4882no—
394C2×D4—2442no—
395C22×C4—612no—
396C22×C4—611no—
397C22×C4—612no—
398C22×C4—612no—
399C22×C4—614no—
400C22×C4—614no—
401C22×C4—611no—
402C22×C4—1222no—
403C22×C4—1222no—
404C22×C4—1222no—
405C22×C4—1221no—
406C22×C4—1221no—
407C22×C4—1224no—
408C22×C4—1222no—
409C22×C4—4882no—
410C22×C4—4882no—
411C22×C4—2442no—
412C22×C4—2442no—
413C22×C4C22×#692312yes—
414C22×C4C22×#692311yescharacteristic
415C22×C4C22×#692311yescharacteristic
416C22×C4C22×#692311yescharacteristic
417C22×C4C22×#692312yes—
418C4⋊C4—21no—
419C4⋊C4—21no—
420C4⋊C4—24no—
421C4⋊C4—24no—
422C4⋊C4—614no—
423C4⋊C4—614no—
424C4⋊C4—612no—
425C4⋊C4—611no—
426C4⋊C4—611no—
427C4⋊C4—1222no—
428C4⋊C4—4884no—
429C4⋊C4—2444no—
430C4⋊C4C22⋊#692312yes—
431C4⋊C4C22⋊#692312yes—
432C22⋊C4—614no—
433C22⋊C4—1222no—
434C22⋊C4—1222no—
435C22⋊C4—1222no—
436C22⋊C4—1222no—
437C22⋊C4—1222no—
438C22⋊C4—1224no—
439C22⋊C4—1222no—
440C22⋊C4—1222no—
441C22⋊C4—1224no—
442C22⋊C4—1224no—
443C22⋊C4—1222no—
444C22⋊C4—1224no—
445C22⋊C4—1222no—
446C42—21no—
447C42—21no—
448C42—1222no—
449C42—1222no—
450C42—4884no—
451C42—2444no—
452C23—612no—
453C23—612no—
454C23—611no—
455C23—611no—
456C23—1222no—
457C23—1221no—
458C23—1222no—
459C23—1221no—
460C23—1221no—
461C23—1222no—
462C23—2441no—
463C23—2441no—
464C23—2442no—
465C23—2441no—
466C23—2441no—
467C23—2442no—
468C23C23×#692311yescharacteristic
469D4—4884no—
470D4—2442no—
471D4—2442no—
472D4—2444no—
473D4—2442no—
474D4—2444no—
475D4—2442no—
476C2×C4—22no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—21no—
482C2×C4—22no—
483C2×C4—21no—
484C2×C4—42no—
485C2×C4—41no—
486C2×C4—612no—
487C2×C4—612no—
488C2×C4—612no—
489C2×C4—1224no—
490C2×C4—1224no—
491C2×C4—1221no—
492C2×C4—4882no—
493C2×C4—4882no—
494C2×C4—2444no—
495C2×C4—2444no—
496C2×C4—2442no—
497C2×C4—2442no—
498C2×C4—2442no—
499C2×C4—2441no—
500C2×C4—2444no—
501C2×C4—2442no—
502C2×C4—2444no—
503C2×C4—2442no—
504C2×C4—2442no—
505C2×C4—2444no—
506C2×C4—2442no—
507C2×C4—2442no—
508C2×C4—2442no—
509C2×C4—2441no—
510C2×C4C2×C22⋊#692311yescharacteristic
511C2×C4C2×C22⋊#692314yes—
512C2×C4C2×C22⋊#692311yescharacteristic
513C22—21no—
514C22—21no—
515C22—612no—
516C22—612no—
517C22—1222no—
518C22—1222no—
519C22—4882no—
520C22—4881no—
521C22—4882no—
522C22—4881no—
523C22—2442no—
524C22—2441no—
525C22—2441no—
526C22—2441no—
527C22—2442no—
528C22—2441no—
529C22—2442no—
530C22—2442no—
531C22—2441no—
532C22C22×C22⋊#692311yescharacteristic
533C22#41×#692311yescharacteristic
534C22#1684083111yescharacteristic
535C4—21no—
536C4—24no—
537C4—41no—
538C4—42no—
539C4—41no—
540C4—1222no—
541C4—1222no—
542C4—1222no—
543C4—4884no—
544C4—4882no—
545C4—2444no—
546C4—2441no—
547C4#1684074312yes—
548C2—21no—
549C2—21no—
550C2—1222no—
551C2—1222no—
552C2—4881no—
553C2—4882no—
554C2#2146232411yescharacteristic
555C2C2×#1684074311yescharacteristic
556C2#2146241911yescharacteristic
subgroupG/H
centerC22C22×C22⋊#6923
commutatorC23×C61C23×C4
FrattiniC23C23×#6923
FittingC2×#125×C61C4
socleC22×C61C22×C22⋊C4
prime radicalC2×#16840465C2
cosocleC23×D61C24
hypercenterC2×#125#6923
hypocenterC61#22535
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C4⋊C4 ◁ C2×#125

Complements

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

NH ≅ G/Ntype
1. #21462328C2semidirect
2. #21461959C2semidirect
3. #21461959C2semidirect
4. #21461682C2semidirect
5. #21461680C2semidirect
6. C4×#14519836C2semidirect
7. #21461564C2semidirect
26. C2×#14519772C4semidirect
27. C2×#14519762C4semidirect
28. C2×#14519752C4semidirect
31. C2×C4×D244C4semidirect
34. C2×#14519735C4semidirect
36. C2×#125×C61C4semidirect
291. C2×#125#6923semidirect
315. C61#22535semidirect

Maximal subgroups

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

subgroupindex
#214623282
#214619592
#214616822
#214616802
C4×#145198362
#214615642
C2×#168404652
#2253561