Finite Group Databasebeta

← #29707513

Subgroups — #29707513

Order 15616.

1558 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14519753C211yesmaximal, characteristic
2C2×#16839804C211yesmaximal, characteristic, complemented
3C2×#16839771C211yesmaximal, characteristic, complemented
4C2×#16839725C211yesmaximal, characteristic
5#21448198C211yesmaximal, characteristic
6#21447021C214yesmaximal, complemented
7#21447014C214yesmaximal, complemented
8#21445540C211yesmaximal, characteristic, complemented
9C2×C61×#419C211yesmaximal, characteristic, complemented
10C23×C4×D61C2211yescharacteristic
11C22×#11758194C2211yescharacteristic
12C2×#14519753—24no—
13C2×#14519753—24no—
14C2×#14519753—22no—
15C2×#14519753C2214yes—
16C2×#14519736C2211yescharacteristic
17C22×C4⋊Dic61C2211yescharacteristic
18C22×#11758196—21no—
19C22×#11758198—21no—
20#16839804—24no—
21C2×#14519711C2212yescomplemented
22C2×#14519709C2212yescomplemented
23#16839771—24no—
24C2×#14519701C2212yescomplemented
25C2×#14519701C2212yescomplemented
26#16839725—24no—
27C2×#14519689C2214yes—
28#16839623—21no—
29#16839619C2212yes—
30#16839619C2212yes—
31#16839617—21no—
32C4×#11758198—22no—
33#16839524C2214yescomplemented
34C23×Q8×C61C2211yescharacteristic
35C22×M16×C61C2211yescharacteristic
36C2×#117×C61C2211yescharacteristic
37C61×#419—24no—
38C2×#103×C61C2214yescomplemented
39C24×D61D411yescharacteristic
40C23×Dic61—21no—
41C23×Dic61D411yescharacteristic
42C22×C4×D61—22no—
43C22×C4×D61—22no—
44C22×C4×D61—22no—
45C22×C4×D61—22no—
46C22×C4×D61—22no—
47C22×C4×D61—22no—
48C22×C4×D61D412yes—
49C22×C4×D61C2312yes—
50C2×#11758194—22no—
51C2×#11758194—22no—
52C2×#11758194—22no—
53C22×#92418C2312yes—
54#14519753—216no—
55#14519753—28no—
56#14519753—28no—
57#14519736—22no—
58#14519736—22no—
59C2×C4⋊Dic61—22no—
60C2×C4⋊Dic61—21no—
61C2×C4⋊Dic61D414yes—
62C2×C4⋊Dic61C2311yescharacteristic
63C2×C4⋊Dic61C2311yescharacteristic
64C2×C4⋊Dic61C2312yes—
65C2×#11758196—28no—
66C2×#11758196—24no—
67C2×C22⋊Dic61—21no—
68C2×#11758198—28no—
69C2×#11758198—24no—
70C2×#11758198—22no—
71C2×#11758198—22no—
72C2×C4×Dic61—21no—
73#14519711—24no—
74#14519709—24no—
75#14519701—24no—
76#14519701—24no—
77#14519689—28no—
78#14519684—22no—
79#14519684—22no—
80C22×Q8×C61—22no—
81C22×Q8×C61—21no—
82C22×Q8×C61—22no—
83C22×Q8×C61—22no—
84C22×Q8×C61—22no—
85C22×Q8×C61—22no—
86C22×Q8×C61D414yes—
87C22×Q8×C61C2312yes—
88C23×C4×C61—21no—
89C23×C4×C61C2311yescharacteristic
90C2×M16×C61—22no—
91C2×M16×C61—22no—
92C2×M16×C61D414yes—
93C22×C8×C61C2312yes—
94#117×C61—22no—
95#117×C61—22no—
96C2×C4⋊C4×C61C2312yes—
97C2×C22⋊C4×C61—21no—
98C2×C42×C61—21no—
99#103×C61—28no—
100C24×C61C2×D411yescharacteristic
101C23×D61—22no—
102C23×D61—24no—
103C23×D61—22no—
104C23×D61—22no—
105C23×D61—42no—
106C23×D61C2×D412yes—
107C2×Q8×C61—24no—
108C2×Q8×C61—22no—
109C2×Q8×C61—28no—
110C2×Q8×C61—28no—
111C2×Q8×C61—22no—
112C2×Q8×C61—22no—
113C2×Q8×C61—22no—
114C2×Q8×C61—22no—
115C2×Q8×C61—42no—
116C2×Q8×C61—44no—
117C2×Q8×C61—42no—
118C2×Q8×C61—44no—
119C2×Q8×C61C2×D414yes—
120C22×C4×C61—24no—
121C22×C4×C61—22no—
122C22×C4×C61—22no—
123C22×C4×C61—42no—
124C22×C4×C61—42no—
125C22×C4×C61C2×D411yescharacteristic
126C22×C4×C61C2×D412yes—
127C22×C4×C61C2×D412yes—
128C22×C4×C61C2×D412yes—
129C22×C4×C61#4112yes—
130C22×C4×C61#4112yes—
131C22×C4×C61#4112yes—
132C22×C4×C61C2411yescharacteristic
133C22×Dic61—22no—
134C22×Dic61—24no—
135C22×Dic61—22no—
136C22×Dic61—22no—
137C22×Dic61—22no—
138C22×Dic61—21no—
139C22×Dic61—21no—
140C22×Dic61—44no—
141C22×Dic61C2×D412yes—
142C2×C4×D61—24no—
143C2×C4×D61—24no—
144C2×C4×D61—24no—
145C2×C4×D61—22no—
146C2×C4×D61—22no—
147C2×C4×D61—22no—
148C2×C4×D61—22no—
149C2×C4×D61—22no—
150C2×C4×D61—22no—
151C2×C4×D61—44no—
152C2×C4×D61—44no—
153C2×C4×D61—44no—
154C2×C4×D61—44no—
155M16×C61—28no—
156C2×C8×C61—22no—
157C2×C8×C61—22no—
158C2×C8×C61C2×D414yes—
159C4⋊C4×C61—22no—
160C4⋊C4×C61—22no—
161C22⋊C4×C61—42no—
162C42×C61—42no—
163C22⋊Dic61—42no—
164#11758198—216no—
165#11758198—44no—
166#11758198—44no—
167C4⋊Dic61—24no—
168C4⋊Dic61—22no—
169C4⋊Dic61—21no—
170C4⋊Dic61—21no—
171C4⋊Dic61—21no—
172C4⋊Dic61—21no—
173C4⋊Dic61C2×D414yes—
174#11758196—216no—
175C4×Dic61—42no—
176#11758194—44no—
177C2×#92418—22no—
178C2×#92418—22no—
179C2×#92418—22no—
180C23×C61—22no—
181C23×C61—22no—
182C23×C61#12012yes—
183C23×C61#12112yes—
184C23×C61#12312yes—
185C23×C61C22×D411yescharacteristic
186C22×D61—28no—
187C22×D61—22no—
188C22×D61—22no—
189C22×D61—22no—
190C22×D61—22no—
191C22×D61—22no—
192C22×D61—22no—
193C22×D61—48no—
194C22×D61—48no—
195C22×D61—44no—
196C22×D61—44no—
197C22×D61#13714yes—
198Q8×C61—24no—
199Q8×C61—24no—
200Q8×C61—42no—
201Q8×C61—42no—
202Q8×C61—48no—
203C2×C4×C61—24no—
204C2×C4×C61—42no—
205C2×C4×C61—42no—
206C2×C4×C61—48no—
207C2×C4×C61—42no—
208C2×C4×C61—42no—
209C2×C4×C61—42no—
210C2×C4×C61#12012yes—
211C2×C4×C61#12114yes—
212C2×C4×C61#12114yes—
213C2×C4×C61#12112yes—
214C2×C4×C61#12314yes—
215C2×C4×C61#12312yes—
216C2×C4×C61#12414yes—
217C2×C4×C61C22×D411yescharacteristic
218C2×C4×C61C22×D411yescharacteristic
219C2×C4×C61C22×D411yescharacteristic
220C2×C4×C61C2×#4111yescharacteristic
221C2×C4×C61C2×#4111yescharacteristic
222C2×C4×C61C2×#4111yescharacteristic
223C2×Dic61—22no—
224C2×Dic61—24no—
225C2×Dic61—22no—
226C2×Dic61—22no—
227C2×Dic61—48no—
228C2×Dic61—44no—
229C2×Dic61—44no—
230C2×Dic61—41no—
231C2×Dic61—41no—
232C2×Dic61—42no—
233C2×Dic61—42no—
234C4×D61—42no—
235C4×D61—42no—
236C4×D61—42no—
237C4×D61—42no—
238C4×D61—48no—
239C8×C61—24no—
240#92418—42no—
241#23481—611nomaximal, Sylow
242C22×C61—24no—
243C22×C61—22no—
244C22×C61—22no—
245C22×C61—24no—
246C22×C61#39414yes—
247C22×C61C2×#12011yescharacteristic
248C22×C61C2×#12111yescharacteristic
249C22×C61C2×#12311yescharacteristic
250C22×C61C2×#13712yes—
251C22×C61#57512yes—
252C2×D61—24no—
253C2×D61—24no—
254C2×D61—24no—
255C2×D61—416no—
256C2×D61—42no—
257C2×D61—42no—
258C2×D61—42no—
259C2×D61—42no—
260C4×C61—44no—
261C4×C61—82no—
262C4×C61#39414yes—
263C4×C61C2×#12111yescharacteristic
264C4×C61C2×#12111yescharacteristic
265C4×C61C2×#12311yescharacteristic
266C4×C61C2×#12411yescharacteristic
267Dic61—42no—
268Dic61—44no—
269Dic61—81no—
270Dic61—81no—
271C22×C22⋊Q8—611no—
272C2×#428—611no—
273C2×#419—611no—
274C2×#419D6111yescharacteristic, complemented
275C2×#407—611no—
276#2280—611no—
277#2036—614no—
278#2035—614no—
279#1736—611no—
280C2×C61—24no—
281C2×C61C2×#39411yescharacteristic
282C2×C61#300411yescharacteristic
283C2×C61#300511yescharacteristic
284C2×C61#305612yes—
285C2×C61#309512yes—
286D61—44no—
287C23×Q8C2×D6111yescharacteristic
288C24×C4—611no—
289C22×M16—611no—
290C22×M16C2×D6111yescharacteristic
291C2×C22⋊Q8—614no—
292C2×C22⋊Q8—1224no—
293C2×C22⋊Q8—1222no—
294C2×C22⋊Q8—1224no—
295C2×#117—611no—
296C2×#117C2×D6111yescharacteristic
297C22×C4⋊C4—611no—
298C22×C4⋊C4—1221no—
299C22×C22⋊C4—1221no—
300#428—1224no—
301C2×#107—612no—
302C2×#106—612no—
303#419—24no—
304#419—1224no—
305C2×#103—612no—
306C2×#103—612no—
307C2×#103C2×D6114yescomplemented
308#407—1224no—
309C2×C22⋊C8—614no—
310#381—1221no—
311#380—612no—
312#380—612no—
313#379—1221no—
314C4×C22⋊C4—1222no—
315#340—614no—
316C61#2348111yescharacteristic, complemented, Sylow
317C25—611no—
318C22×Q8—21no—
319C22×Q8—22no—
320C22×Q8—22no—
321C22×Q8—22no—
322C22×Q8—22no—
323C22×Q8—22no—
324C22×Q8D24414yes—
325C22×Q8C22×D6112yes—
326C23×C4—21no—
327C23×C4—611no—
328C23×C4—612no—
329C23×C4—612no—
330C23×C4—1221no—
331C23×C4—1222no—
332C23×C4—1222no—
333C23×C4—1222no—
334C23×C4—1222no—
335C23×C4—1222no—
336C23×C4—1222no—
337C23×C4C22×D6111yescharacteristic
338C2×M16—22no—
339C2×M16—22no—
340C2×M16—1222no—
341C2×M16—1222no—
342C2×M16—1222no—
343C2×M16C22⋊D6114yes—
344C22×C8—612no—
345C22×C8C22×D6112yes—
346C22⋊Q8—1228no—
347C22⋊Q8—12216no—
348C22⋊Q8—1228no—
349#117—22no—
350#117—22no—
351#117—1222no—
352#117—1222no—
353C2×C4⋊C4—614no—
354C2×C4⋊C4—612no—
355C2×C4⋊C4—611no—
356C2×C4⋊C4—611no—
357C2×C4⋊C4—1224no—
358C2×C4⋊C4—1228no—
359C2×C4⋊C4—1222no—
360C2×C4⋊C4—1221no—
361C2×C4⋊C4C22×D6112yes—
362C2×C22⋊C4—21no—
363C2×C22⋊C4—1224no—
364C2×C22⋊C4—1228no—
365C2×C22⋊C4—1222no—
366C2×C22⋊C4—1222no—
367C2×C22⋊C4—1221no—
368C2×C42—21no—
369C2×C42—1221no—
370#107—1224no—
371#106—1224no—
372#103—28no—
373#103—1224no—
374#103—1224no—
375C22⋊C8—1228no—
376#95—1222no—
377#95—1222no—
378C24—612no—
379C24—1224no—
380C24—1222no—
381C24—1222no—
382C24—1222no—
383C24—2442no—
384C24D4×D6111yescharacteristic
385C2×Q8—22no—
386C2×Q8—24no—
387C2×Q8—22no—
388C2×Q8—22no—
389C2×Q8—22no—
390C2×Q8—22no—
391C2×Q8—28no—
392C2×Q8—28no—
393C2×Q8—42no—
394C2×Q8—44no—
395C2×Q8—42no—
396C2×Q8—44no—
397C2×Q8C2×D24414yes—
398C22×C4—22no—
399C22×C4—24no—
400C22×C4—22no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—612no—
404C22×C4—1222no—
405C22×C4—1224no—
406C22×C4—1222no—
407C22×C4—1222no—
408C22×C4—1222no—
409C22×C4—1222no—
410C22×C4—1222no—
411C22×C4—1222no—
412C22×C4—1222no—
413C22×C4—1222no—
414C22×C4—1222no—
415C22×C4—1224no—
416C22×C4—1224no—
417C22×C4—1224no—
418C22×C4—1221no—
419C22×C4—1221no—
420C22×C4—2444no—
421C22×C4—2444no—
422C22×C4—2444no—
423C22×C4—2444no—
424C22×C4—2444no—
425C22×C4C2×D24412yes—
426C22×C4#1175822212yes—
427C22×C4D4×D6112yes—
428C22×C4D4×D6111yescharacteristic
429C22×C4#1175822412yes—
430C22×C4#1175822412yes—
431C22×C4C2×C22⋊D6112yes—
432C22×C4C23×D6111yescharacteristic
433M16—28no—
434M16—2444no—
435C2×C8—22no—
436C2×C8—22no—
437C2×C8—1222no—
438C2×C8—1222no—
439C2×C8—1222no—
440C2×C8C2×C22⋊D6114yes—
441C4⋊C4—22no—
442C4⋊C4—22no—
443C4⋊C4—614no—
444C4⋊C4—12216no—
445C4⋊C4—1222no—
446C4⋊C4—1224no—
447C4⋊C4—1221no—
448C4⋊C4—1221no—
449C4⋊C4—1221no—
450C4⋊C4—1221no—
451C22⋊C4—42no—
452C22⋊C4—12216no—
453C22⋊C4—2444no—
454C22⋊C4—2444no—
455C22⋊C4—2442no—
456C42—42no—
457C42—2442no—
458C23—22no—
459C23—22no—
460C23—614no—
461C23—1228no—
462C23—1222no—
463C23—1222no—
464C23—1222no—
465C23—1222no—
466C23—1222no—
467C23—1222no—
468C23—2448no—
469C23—2448no—
470C23—2444no—
471C23—2444no—
472C23#1451974712yes—
473C23#1451974812yes—
474C23#1451975912yes—
475C23C2×D4×D6111yescharacteristic
476Q8—24no—
477Q8—24no—
478Q8—42no—
479Q8—42no—
480Q8—48no—
481C2×C4—24no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—48no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—1224no—
489C2×C4—1222no—
490C2×C4—1222no—
491C2×C4—1222no—
492C2×C4—2441no—
493C2×C4—2441no—
494C2×C4—2442no—
495C2×C4—2442no—
496C2×C4—2448no—
497C2×C4—2444no—
498C2×C4—2444no—
499C2×C4—2442no—
500C2×C4—2442no—
501C2×C4—2442no—
502C2×C4—2442no—
503C2×C4—2448no—
504C2×C4#1451974712yes—
505C2×C4#1451974812yes—
506C2×C4#1451974914yes—
507C2×C4#1451975114yes—
508C2×C4#1451975912yes—
509C2×C4#1451976114yes—
510C2×C4#1451976314yes—
511C2×C4C2×D4×D6111yescharacteristic
512C2×C4C22×C22⋊D6111yescharacteristic
513C2×C4C22×D24411yescharacteristic
514C2×C4C2×#1175822411yescharacteristic
515C2×C4C2×#1175822411yescharacteristic
516C2×C4C2×#1175822211yescharacteristic
517C8—24no—
518C8—2442no—
519C22—24no—
520C22—24no—
521C22—22no—
522C22—22no—
523C22—1224no—
524C22—1224no—
525C22—1224no—
526C22—24416no—
527C22—2442no—
528C22—2442no—
529C22—2442no—
530C22—2442no—
531C22#1683967814yes—
532C22C2×#1451974711yescharacteristic
533C22C2×#1451974811yescharacteristic
534C22C2×#1451975911yescharacteristic
535C22#137×D6112yes—
536C22#1684056112yes—
537C4—44no—
538C4—82no—
539C4—4881no—
540C4—4881no—
541C4—2442no—
542C4—2444no—
543C4#1683967814yes—
544C4C2×#1451974911yescharacteristic
545C4C2×#1451975111yescharacteristic
546C4C2×#1451976111yescharacteristic
547C4C2×#1451976311yescharacteristic
548C2—24no—
549C2—2444no—
550C2C2×#1683967811yescharacteristic
551C2#2145505511yescharacteristic
552C2#2145506511yescharacteristic
553C2#2145544312yes—
554C2#2145576512yes—
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#419C2
socleC23×C61C22×D4
prime radicalC24×D61D4
hypercenterC2×#419D61
hypocenterC61#23481
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#419

Complements

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

NH ≅ G/Ntype
2. C2×#16839804C2semidirect
3. C2×#16839771C2semidirect
6. #21447021C2semidirect
7. #21447014C2semidirect
8. #21445540C2semidirect
9. C2×C61×#419C2semidirect
21. C2×#14519711C22semidirect
22. C2×#14519709C22semidirect
24. C2×#14519701C22semidirect
25. C2×#14519701C22semidirect
33. #16839524C22semidirect
38. C2×#103×C61C22semidirect
274. C2×#419D61semidirect
307. C2×#103C2×D61semidirect
316. C61#23481semidirect

Maximal subgroups

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

subgroupindex
C22×#145197532
C2×#168398042
C2×#168397712
C2×#168397252
#214481982
#214470212
#214470142
#214455402
C2×C61×#4192
#2348161