Finite Group Databasebeta

← #29716701

Subgroups — #29716701

Order 15616.

1078 conjugacy classes of subgroups.

The subgroup lattice has 554 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21451634C212yesmaximal, complemented
2#21448484C211yesmaximal, characteristic
3#21447171C212yesmaximal, complemented
4#21446985C212yesmaximal, complemented
5#21446805C212yesmaximal, complemented
6#21446232C212yesmaximal
7#21446100C212yesmaximal
8#21445807C211yesmaximal, characteristic
9C61×#3501C211yesmaximal, characteristic
10C2×#14519758C2211yescharacteristic
11C22×#11758196C2211yescharacteristic
12C2×#14519692C2212yes—
13#16839700—22no—
14#16839695—24no—
15#16839681—22no—
16#16839657—21no—
17#16839656—21no—
18#16839646—21no—
19#16839642C2211yescharacteristic
20#16839635—21no—
21#16839634C2211yescharacteristic
22#16839634C2214yes—
23#16839623C2211yescharacteristic
24#16839623C2214yes—
25#16839622C2212yes—
26C4×C22⋊Dic61C2211yescharacteristic
27#16839529C2212yes—
28#16839529C2214yes—
29#16839529C2214yes—
30C61×#554—24no—
31C61×#554—22no—
32D4×Q8×C61—22no—
33C61×#544—21no—
34C61×#543—21no—
35C61×#543C2212yes—
36C61×#533—21no—
37C2×C22⋊Q8×C61—21no—
38C2×C22⋊Q8×C61C2212yes—
39C61×#518C2212yes—
40C2×C4×D4×C61C2211yescharacteristic
41C23×Dic61C2311yescharacteristic
42C23×Dic61C2311yescharacteristic
43#14519758—22no—
44#14519758Q814yescomplemented
45C2×C4⋊Dic61—22no—
46C2×#11758196—21no—
47C2×#11758196—22no—
48C2×#11758196—22no—
49C2×#11758196—21no—
50C2×#11758196—21no—
51C2×#11758196—24no—
52C2×#11758196—21no—
53C2×C22⋊Dic61—21no—
54C2×C22⋊Dic61—21no—
55C2×C22⋊Dic61C2314yes—
56C2×C22⋊Dic61C2311yescharacteristic
57C2×C22⋊Dic61C2311yescharacteristic
58C2×C4×Dic61—21no—
59#14519692—24no—
60#14519684—24no—
61#14519684—22no—
62#14519684—24no—
63#14519684—24no—
64#14519684—21no—
65#14519684—21no—
66#14519684—21no—
67#14519684—21no—
68#14519684—21no—
69C22×Q8×C61—21no—
70C22×Q8×C61D412yes—
71C22×D4×C61C2311yescharacteristic
72C23×C4×C61D411yescharacteristic
73C23×C4×C61C2311yescharacteristic
74C4⋊Q8×C61—21no—
75C4⋊Q8×C61—21no—
76C4⋊Q8×C61—24no—
77#125×C61—22no—
78#125×C61—24no—
79C22⋊Q8×C61—24no—
80C22⋊Q8×C61—24no—
81C22⋊Q8×C61—22no—
82C22⋊Q8×C61—22no—
83C22⋊Q8×C61—22no—
84C22⋊Q8×C61—22no—
85C22⋊Q8×C61—22no—
86C22⋊Q8×C61—22no—
87C22⋊Q8×C61—24no—
88C22⋊Q8×C61—24no—
89C22⋊Q8×C61—22no—
90C22⋊Q8×C61—22no—
91C4×Q8×C61—22no—
92C4×Q8×C61—22no—
93C4×D4×C61—22no—
94C4×D4×C61—24no—
95C4×D4×C61—22no—
96C4×D4×C61—24no—
97#117×C61—22no—
98C2×C4⋊C4×C61—21no—
99C2×C4⋊C4×C61—22no—
100C2×C4⋊C4×C61D411yescharacteristic
101C2×C4⋊C4×C61D414yes—
102C2×C4⋊C4×C61D412yes—
103C2×C22⋊C4×C61—22no—
104C2×C22⋊C4×C61D411yescharacteristic
105C2×C22⋊C4×C61C2311yescharacteristic
106C2×C22⋊C4×C61C2314yes—
107C2×C42×C61D411yescharacteristic
108C24×C61C2×D411yescharacteristic
109C24×C61C2411yescharacteristic
110C2×Q8×C61—22no—
111C2×Q8×C61—22no—
112C2×Q8×C61—22no—
113C2×Q8×C61—22no—
114C2×Q8×C61—82no—
115C2×Q8×C61—82no—
116C2×D4×C61—22no—
117C2×D4×C61—22no—
118C2×D4×C61C2×Q812yes—
119C2×D4×C61#4112yes—
120C22×C4×C61—22no—
121C22×C4×C61—22no—
122C22×C4×C61—21no—
123C22×C4×C61—21no—
124C22×C4×C61—81no—
125C22×C4×C61—81no—
126C22×C4×C61C2×D411yescharacteristic
127C22×C4×C61C2×D411yescharacteristic
128C22×C4×C61C2×D414yes—
129C22×C4×C61C2×D414yes—
130C22×C4×C61C2×D411yescharacteristic
131C22×C4×C61C2×D411yescharacteristic
132C22×C4×C61C2×D411yescharacteristic
133C22×C4×C61C2×D412yes—
134C22×C4×C61C2×D412yes—
135C22×Dic61—22no—
136C22×Dic61—21no—
137C22×Dic61—21no—
138C22×Dic61—21no—
139C22×Dic61—22no—
140C22×Dic61—21no—
141C22×Dic61—22no—
142C22×Dic61—21no—
143C22×Dic61—24no—
144C22×Dic61—81no—
145C22×Dic61C2×Q812yescomplemented
146C4⋊C4×C61—22no—
147C4⋊C4×C61—24no—
148C4⋊C4×C61—22no—
149C4⋊C4×C61—22no—
150C4⋊C4×C61—24no—
151C4⋊C4×C61—22no—
152C4⋊C4×C61—24no—
153C4⋊C4×C61—22no—
154C4⋊C4×C61—22no—
155C4⋊C4×C61—22no—
156C4⋊C4×C61—24no—
157C22⋊C4×C61—24no—
158C22⋊C4×C61—24no—
159C22⋊C4×C61—22no—
160C22⋊C4×C61—24no—
161C22⋊C4×C61—22no—
162C42×C61—24no—
163C42×C61—21no—
164C42×C61—21no—
165C22⋊Dic61—24no—
166C22⋊Dic61—21no—
167C22⋊Dic61—24no—
168C22⋊Dic61—22no—
169C22⋊Dic61—24no—
170C22⋊Dic61—81no—
171C22⋊Dic61C2×Q812yescomplemented
172C4⋊Dic61—82no—
173#11758196—24no—
174#11758196—41no—
175#11758196—41no—
176#11758196—42no—
177#11758196—81no—
178#11758196—84no—
179C4×Dic61—81no—
180C23×C61—21no—
181C23×C61—21no—
182C23×C61—41no—
183C23×C61—41no—
184C23×C61—41no—
185C23×C61—41no—
186C23×C61C22⋊Q812yes—
187C23×C61C22×D411yescharacteristic
188C23×C61C22×D412yes—
189C23×C61C22×Q811yescharacteristic
190C23×C61C2×#4111yescharacteristic
191C23×C612−1+412yes—
192Q8×C61—44no—
193Q8×C61—82no—
194Q8×C61—82no—
195Q8×C61—82no—
196D4×C61—24no—
197D4×C61—24no—
198C2×C4×C61—21no—
199C2×C4×C61—21no—
200C2×C4×C61—21no—
201C2×C4×C61—24no—
202C2×C4×C61—24no—
203C2×C4×C61—21no—
204C2×C4×C61—41no—
205C2×C4×C61—42no—
206C2×C4×C61—41no—
207C2×C4×C61—42no—
208C2×C4×C61—42no—
209C2×C4×C61—41no—
210C2×C4×C61—44no—
211C2×C4×C61—44no—
212C2×C4×C61—41no—
213C2×C4×C61—81no—
214C2×C4×C61—82no—
215C2×C4×C61—82no—
216C2×C4×C61#12011yescharacteristic
217C2×C4×C61#12011yescharacteristic
218C2×C4×C61#12012yes—
219C2×C4×C61C22⋊Q812yes—
220C2×Dic61—22no—
221C2×Dic61—22no—
222C2×Dic61—24no—
223C2×Dic61—41no—
224C2×Dic61—41no—
225C2×Dic61—41no—
226C2×Dic61—41no—
227C2×Dic61—42no—
228C2×Dic61—41no—
229C2×Dic61—44no—
230C2×Dic61—44no—
231C2×Dic61—41no—
232C2×Dic61—41no—
233C2×Dic61—44no—
234C2×Dic61—41no—
235C2×Dic61—41no—
236C2×Dic61—41no—
237C2×Dic61—44no—
238#24353—611nomaximal, Sylow
239C22×C61—22no—
240C22×C61—21no—
241C22×C61—22no—
242C22×C61—21no—
243C22×C61—22no—
244C22×C61—41no—
245C22×C61—41no—
246C22×C61—42no—
247C22×C61—42no—
248C22×C61—41no—
249C22×C61—41no—
250C22×C61—81no—
251C22×C61C2×#12011yescharacteristic
252C22×C61C2×C22⋊Q811yescharacteristic
253C22×C61#53611yescharacteristic
254C22×C61#54812yes—
255C22×C61D4×Q812yes—
256C4×C61—22no—
257C4×C61—21no—
258C4×C61—21no—
259C4×C61—22no—
260C4×C61—41no—
261C4×C61—44no—
262C4×C61—44no—
263C4×C61—41no—
264Dic61—42no—
265Dic61—42no—
266Dic61—84no—
267Dic61—81no—
268Dic61—81no—
269#3501D6111yescharacteristic
270#2627—612no—
271#2322—611no—
272#2062—612no—
273#2030—612no—
274#1996—612no—
275#1890—612no—
276#1867—612no—
277#1799—611no—
278C2×C61—21no—
279C2×C61—22no—
280C2×C61—21no—
281C2×C61—22no—
282C2×C61—42no—
283C2×C61#240911yescharacteristic
284C2×C61#302611yescharacteristic
285C2×C61#302811yescharacteristic
286#554—24no—
287#554—22no—
288D4×Q8—22no—
289#544—21no—
290#543—21no—
291#543C2×D6112yes—
292#533—21no—
293C2×#123—611no—
294C2×C22⋊Q8—21no—
295C2×C22⋊Q8C2×D6112yes—
296#518C2×D6112yes—
297C2×C4×D4C2×D6111yescharacteristic
298C22×C4⋊C4—611no—
299C2×#99—612no—
300#400—1222no—
301#398—1224no—
302#395—1222no—
303#389—1221no—
304#388—1221no—
305#387—1221no—
306#386—611no—
307#385—611no—
308#385—614no—
309#385—1221no—
310#381—611no—
311#381—614no—
312#380—612no—
313C4×C22⋊C4—611no—
314#342—612no—
315#342—614no—
316#342—614no—
317C61#2435311yescharacteristic, complemented, Sylow
318C22×Q8—21no—
319C22×Q8C22⋊D6112yes—
320C22×D4C22×D6111yescharacteristic
321C23×C4—611no—
322C23×C4—611no—
323C23×C4C22⋊D6111yescharacteristic
324C23×C4C22×D6111yescharacteristic
325C4⋊Q8—21no—
326C4⋊Q8—21no—
327C4⋊Q8—24no—
328#125—22no—
329#125—24no—
330#123—614no—
331#123—1222no—
332C22⋊Q8—22no—
333C22⋊Q8—22no—
334C22⋊Q8—24no—
335C22⋊Q8—24no—
336C22⋊Q8—22no—
337C22⋊Q8—22no—
338C22⋊Q8—22no—
339C22⋊Q8—22no—
340C22⋊Q8—22no—
341C22⋊Q8—22no—
342C22⋊Q8—24no—
343C22⋊Q8—24no—
344C4×Q8—22no—
345C4×Q8—22no—
346C4×D4—22no—
347C4×D4—22no—
348C4×D4—24no—
349C4×D4—24no—
350#117—22no—
351C2×C4⋊C4—21no—
352C2×C4⋊C4—22no—
353C2×C4⋊C4—1221no—
354C2×C4⋊C4—1222no—
355C2×C4⋊C4—1221no—
356C2×C4⋊C4—1224no—
357C2×C4⋊C4—1221no—
358C2×C4⋊C4—1221no—
359C2×C4⋊C4—1222no—
360C2×C4⋊C4—1222no—
361C2×C4⋊C4C22⋊D6111yescharacteristic
362C2×C4⋊C4C22⋊D6114yes—
363C2×C4⋊C4C22⋊D6112yes—
364C2×C22⋊C4—22no—
365C2×C22⋊C4—611no—
366C2×C22⋊C4—611no—
367C2×C22⋊C4—614no—
368C2×C22⋊C4—1221no—
369C2×C22⋊C4—1221no—
370C2×C22⋊C4C22⋊D6111yescharacteristic
371C2×C22⋊C4C22×D6111yescharacteristic
372C2×C22⋊C4C22×D6114yes—
373C2×C42—1221no—
374C2×C42C22⋊D6111yescharacteristic
375#99—1224no—
376#95—1221no—
377#95—1222no—
378#95—1221no—
379#95—1221no—
380#95—1224no—
381#95—1224no—
382#95—1221no—
383#95—1224no—
384#95—1221no—
385C24C2×C22⋊D6111yescharacteristic
386C24C23×D6111yescharacteristic
387C2×Q8—22no—
388C2×Q8—22no—
389C2×Q8—22no—
390C2×Q8—22no—
391C2×Q8—82no—
392C2×Q8—82no—
393C2×D4—22no—
394C2×D4—22no—
395C2×D4Q8×D6112yes—
396C2×D4#1175822612yes—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—81no—
402C22×C4—81no—
403C22×C4—612no—
404C22×C4—1222no—
405C22×C4—1221no—
406C22×C4—1221no—
407C22×C4—1221no—
408C22×C4—1222no—
409C22×C4—1221no—
410C22×C4—1224no—
411C22×C4—1221no—
412C22×C4—1222no—
413C22×C4—4881no—
414C22×C4C2×C22⋊D6111yescharacteristic
415C22×C4C2×C22⋊D6111yescharacteristic
416C22×C4C2×C22⋊D6114yes—
417C22×C4C2×C22⋊D6114yes—
418C22×C4C2×C22⋊D6111yescharacteristic
419C22×C4C2×C22⋊D6111yescharacteristic
420C22×C4C2×C22⋊D6112yes—
421C22×C4C2×C22⋊D6111yescharacteristic
422C22×C4C2×C22⋊D6112yes—
423C4⋊C4—22no—
424C4⋊C4—24no—
425C4⋊C4—22no—
426C4⋊C4—22no—
427C4⋊C4—24no—
428C4⋊C4—22no—
429C4⋊C4—24no—
430C4⋊C4—22no—
431C4⋊C4—22no—
432C4⋊C4—22no—
433C4⋊C4—24no—
434C4⋊C4—1224no—
435C4⋊C4—4881no—
436C4⋊C4—4884no—
437C4⋊C4—4882no—
438C4⋊C4—2441no—
439C4⋊C4—2441no—
440C4⋊C4—2442no—
441C22⋊C4—24no—
442C22⋊C4—24no—
443C22⋊C4—22no—
444C22⋊C4—24no—
445C22⋊C4—22no—
446C22⋊C4—612no—
447C22⋊C4—1221no—
448C22⋊C4—1224no—
449C22⋊C4—1224no—
450C22⋊C4—1222no—
451C22⋊C4—1224no—
452C22⋊C4—4881no—
453C42—24no—
454C42—21no—
455C42—21no—
456C42—4881no—
457C23—21no—
458C23—21no—
459C23—41no—
460C23—41no—
461C23—41no—
462C23—41no—
463C23#1451975312yes—
464C23C22×C22⋊D6111yescharacteristic
465C23C22×C22⋊D6112yes—
466C23C2×Q8×D6111yescharacteristic
467C23C2×#1175822611yescharacteristic
468C23#1451982512yes—
469Q8—44no—
470Q8—82no—
471Q8—82no—
472Q8—82no—
473D4—24no—
474D4—24no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—24no—
479C2×C4—24no—
480C2×C4—21no—
481C2×C4—41no—
482C2×C4—42no—
483C2×C4—41no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—41no—
487C2×C4—44no—
488C2×C4—44no—
489C2×C4—41no—
490C2×C4—81no—
491C2×C4—82no—
492C2×C4—82no—
493C2×C4—1222no—
494C2×C4—1222no—
495C2×C4—1224no—
496C2×C4—2441no—
497C2×C4—2441no—
498C2×C4—2444no—
499C2×C4—2444no—
500C2×C4—2441no—
501C2×C4—2444no—
502C2×C4—2441no—
503C2×C4—2441no—
504C2×C4—2441no—
505C2×C4—2444no—
506C2×C4—2441no—
507C2×C4—2441no—
508C2×C4—2441no—
509C2×C4—2441no—
510C2×C4—2442no—
511C2×C4#1451974611yescharacteristic
512C2×C4#1451974611yescharacteristic
513C2×C4#1451974612yes—
514C2×C4#1451975312yes—
515C22—22no—
516C22—21no—
517C22—22no—
518C22—21no—
519C22—22no—
520C22—41no—
521C22—41no—
522C22—42no—
523C22—42no—
524C22—41no—
525C22—41no—
526C22—81no—
527C22C2×#1451974611yescharacteristic
528C22C2×#1451975311yescharacteristic
529C22#1684033611yescharacteristic
530C22#1684042312yes—
531C22Q8×C22⋊D6112yes—
532C4—21no—
533C4—21no—
534C4—22no—
535C4—22no—
536C4—41no—
537C4—44no—
538C4—44no—
539C4—41no—
540C4—4884no—
541C4—4881no—
542C4—4881no—
543C4—2442no—
544C4—2442no—
545C2—22no—
546C2—21no—
547C2—21no—
548C2—22no—
549C2—42no—
550C2#2144917811yescharacteristic
551C2#2145521611yescharacteristic
552C2#2145522511yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC24×C61C24
FrattiniC24C23×D61
FittingC61×#3501C2
socleC22×C61C2×#120
prime radicalC22×D4×C61C23
hypercenter#3501D61
hypocenterC61#24353
LCSG ▷ C24×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3501

Complements

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

NH ≅ G/Ntype
1. #21451634C2semidirect
3. #21447171C2semidirect
4. #21446985C2semidirect
5. #21446805C2semidirect
44. #14519758Q8semidirect
145. C22×Dic61C2×Q8semidirect
171. C22⋊Dic61C2×Q8semidirect
317. C61#24353semidirect

Maximal subgroups

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

subgroupindex
#214516342
#214484842
#214471712
#214469852
#214468052
#214462322
#214461002
#214458072
C61×#35012
#2435361