Finite Group Databasebeta

← #29679833

Subgroups — #29679833

Order 15616.

1374 conjugacy classes of subgroups.

The subgroup lattice has 574 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839729C211yesmaximal, characteristic
2#21448411C211yesmaximal, characteristic
3#21445914C212yesmaximal, complemented
4#21445914C212yesmaximal, complemented
5#21445909C212yesmaximal, complemented
6#21445905C212yesmaximal, complemented
7#21445540C211yesmaximal, characteristic, complemented
8#21445538C211yesmaximal, characteristic, complemented
9#21445535C211yesmaximal, characteristic, complemented
10C2×#16839524C211yesmaximal, characteristic, complemented
11C2×C61×#520C211yesmaximal, characteristic
12C22×#11758194C2211yescharacteristic
13C23×#92418C2211yescharacteristic
14C2×#14519736C2211yescharacteristic
15C22×C4⋊Dic61C2211yescharacteristic
16#16839729—24no—
17C2×C22⋊#92418C2212yes—
18C2×C22⋊#92418C2212yes—
19#16839669—21no—
20#16839665—21no—
21#16839642—22no—
22#16839622C2212yes—
23#16839622C2212yes—
24#16839524C2211yescharacteristic
25#16839524C2212yes—
26#16839524C2212yes—
27#16839524C2212yes—
28#16839524C2211yescharacteristic
29#16839524C2212yes—
30#16839524C2212yes—
31#16839524C2211yescharacteristic
32#16839524C2211yescharacteristic
33#16839524C2212yes—
34C22×#41×C61C2211yescharacteristic
35C61×#520—24no—
36C61×#520C418yescomplemented
37C2×C4×Q8×C61—21no—
38C2×C4×D4×C61—22no—
39C2×C4×D4×C61—21no—
40C2×#117×C61C2211yescharacteristic
41C2×#117×C61C2212yes—
42C22×C4⋊C4×C61C2211yescharacteristic
43C22×C4⋊C4×C61C2212yes—
44C23×Dic61—21no—
45C2×#11758194—22no—
46C2×#11758194—22no—
47C2×#11758194—22no—
48C22×#92418—22no—
49C22×#92418—22no—
50C22×#92418—22no—
51C22×#92418C2311yescharacteristic
52C22×#92418C2311yescharacteristic
53C22×#92418C2312yes—
54#14519736—22no—
55#14519736—22no—
56C2×C4⋊Dic61—22no—
57C2×C4⋊Dic61—22no—
58C2×C4⋊Dic61—21no—
59C2×C4⋊Dic61C2312yes—
60C2×C4⋊Dic61C2311yescharacteristic
61C2×C4⋊Dic61C2311yescharacteristic
62C2×C22⋊Dic61—21no—
63C2×C22⋊Dic61—22no—
64C2×C22⋊Dic61—22no—
65C2×C4×Dic61—21no—
66C22⋊#92418—24no—
67C22⋊#92418—24no—
68#14519684—22no—
69#14519684—22no—
70C2×#41×C61—22no—
71C2×#41×C61—22no—
72C2×#41×C61—22no—
73C2×#41×C61—22no—
74C2×#41×C61C2×C414yes—
75C2×#41×C61D414yes—
76C22×Q8×C61D411yescharacteristic
77C22×D4×C61D412yes—
78C22×D4×C61D411yescharacteristic
79C23×C4×C61—21no—
80C23×C4×C61—22no—
81C23×C4×C61C2312yes—
82C23×C4×C61C2311yescharacteristic
83C4×Q8×C61—28no—
84C4×D4×C61—28no—
85C4×D4×C61—216no—
86#117×C61—22no—
87#117×C61—24no—
88#117×C61C2×C414yes—
89#117×C61C2×C418yescomplemented
90C2×C4⋊C4×C61—21no—
91C2×C4⋊C4×C61—22no—
92C2×C4⋊C4×C61—22no—
93C2×C4⋊C4×C61—24no—
94C2×C4⋊C4×C61C2×C414yes—
95C2×C4⋊C4×C61C2×C418yescomplemented
96C2×C4⋊C4×C61C2312yes—
97C2×C4⋊C4×C61C2311yescharacteristic
98C2×C4⋊C4×C61C2311yescharacteristic
99C2×C22⋊C4×C61—21no—
100C2×C22⋊C4×C61—22no—
101C2×C42×C61—21no—
102C2×C42×C61—22no—
103C24×C61C2×D412yes—
104C24×C61C2×D411yescharacteristic
105#41×C61—28no—
106#41×C61—28no—
107#41×C61—28no—
108#41×C61—28no—
109C2×Q8×C61—22no—
110C2×Q8×C61—22no—
111C2×Q8×C61C22⋊C414yes—
112C2×D4×C61—24no—
113C2×D4×C61—24no—
114C2×D4×C61—22no—
115C2×D4×C61—22no—
116C2×D4×C61C22⋊C418yes—
117C2×D4×C61C22⋊C414yes—
118C22×C4×C61—22no—
119C22×C4×C61—22no—
120C22×C4×C61—22no—
121C22×C4×C61—22no—
122C22×C4×C61—21no—
123C22×C4×C61—21no—
124C22×C4×C61—22no—
125C22×C4×C61—24no—
126C22×C4×C61—28no—
127C22×C4×C61—81no—
128C22×C4×C61—82no—
129C22×C4×C61—81no—
130C22×C4×C61—82no—
131C22×C4×C61C22×C414yes—
132C22×C4×C61C22×C412yes—
133C22×C4×C61C2×D412yes—
134C22×C4×C61C2×D414yes—
135C22×C4×C61C2×D411yescharacteristic
136C22×C4×C61C2×D412yes—
137C22×C4×C61#4112yes—
138C22×C4×C61#4112yes—
139C22×C4×C61#4112yes—
140C22×C4×C61#4112yes—
141C22×C4×C61C2411yescharacteristic
142C22×Dic61—22no—
143C22×Dic61—21no—
144C22×Dic61—21no—
145C22×Dic61—42no—
146C22×Dic61—81no—
147C22×Dic61—81no—
148C4⋊C4×C61—24no—
149C4⋊C4×C61—28no—
150C4⋊C4×C61—21no—
151C4⋊C4×C61—22no—
152C4⋊C4×C61—21no—
153C4⋊C4×C61C22×C414yes—
154C4⋊C4×C61C22×C412yes—
155C4⋊C4×C61C22×C412yes—
156C22⋊C4×C61—24no—
157C22⋊C4×C61—28no—
158C42×C61—24no—
159C42×C61—28no—
160C22⋊Dic61—42no—
161C22⋊Dic61—82no—
162C22⋊Dic61—82no—
163C4⋊Dic61—22no—
164C4⋊Dic61—22no—
165C4⋊Dic61—21no—
166C4⋊Dic61—21no—
167C4⋊Dic61—21no—
168C4⋊Dic61—21no—
169C4⋊Dic61—42no—
170C4×Dic61—42no—
171#11758194—44no—
172C2×#92418—21no—
173C2×#92418—21no—
174C2×#92418—22no—
175C2×#92418—21no—
176C2×#92418—21no—
177C2×#92418—21no—
178C2×#92418—21no—
179C2×#92418—22no—
180C2×#92418—22no—
181C2×#92418—44no—
182C23×C61—21no—
183C23×C61—21no—
184C23×C61—21no—
185C23×C61—21no—
186C23×C61—22no—
187C23×C61—22no—
188C23×C61—22no—
189C23×C61—22no—
190C23×C61—22no—
191C23×C61—22no—
192C23×C61C2×C22⋊C414yes—
193C23×C61C2×C22⋊C412yes—
194C23×C61#12312yes—
195C23×C61#12312yes—
196C23×C61C22×D411yescharacteristic
197Q8×C61—28no—
198D4×C61—216no—
199D4×C61—28no—
200C2×C4×C61—22no—
201C2×C4×C61—22no—
202C2×C4×C61—24no—
203C2×C4×C61—24no—
204C2×C4×C61—24no—
205C2×C4×C61—24no—
206C2×C4×C61—22no—
207C2×C4×C61—22no—
208C2×C4×C61—24no—
209C2×C4×C61—42no—
210C2×C4×C61—41no—
211C2×C4×C61—41no—
212C2×C4×C61—42no—
213C2×C4×C61—41no—
214C2×C4×C61—41no—
215C2×C4×C61—84no—
216C2×C4×C61—88no—
217C2×C4×C61C2×C22⋊C414yes—
218C2×C4×C61C2×C22⋊C412yes—
219C2×C4×C61#11714yes—
220C2×C4×C61#11714yes—
221C2×C4×C61#12314yes—
222C2×C4×C61#12314yes—
223C2×C4×C61#12312yes—
224C2×C4×C61#12312yes—
225C2×C4×C61C23×C411yescharacteristic
226C2×C4×C61C22×D411yescharacteristic
227C2×C4×C61C2×#4111yescharacteristic
228C2×C4×C61C2×#4111yescharacteristic
229C2×C4×C61C2×#4111yescharacteristic
230C2×C4×C61C2×#4111yescharacteristic
231C2×Dic61—41no—
232C2×Dic61—41no—
233C2×Dic61—42no—
234C2×Dic61—41no—
235C2×Dic61—42no—
236C2×Dic61—41no—
237C2×Dic61—42no—
238C2×Dic61—41no—
239C2×Dic61—41no—
240C2×Dic61—84no—
241#92418—41no—
242#92418—41no—
243#92418—42no—
244#20703—611nomaximal, Sylow
245C22×C61—22no—
246C22×C61—22no—
247C22×C61—22no—
248C22×C61—22no—
249C22×C61—22no—
250C22×C61—22no—
251C22×C61—24no—
252C22×C61—24no—
253C22×C61—24no—
254C22×C61—42no—
255C22×C61—42no—
256C22×C61—42no—
257C22×C61—42no—
258C22×C61#38114yes—
259C22×C61C22×C22⋊C411yescharacteristic
260C22×C61C2×#12311yescharacteristic
261C22×C61C2×#12311yescharacteristic
262C22×C61#57611yescharacteristic
263C22×C61#57711yescharacteristic
264C22×C61#57711yescharacteristic
265C22×C61#57811yescharacteristic
266C4×C61—24no—
267C4×C61—42no—
268C4×C61—44no—
269C4×C61—42no—
270C4×C61#38114yes—
271C4×C61C2×#11711yescharacteristic
272C4×C61C2×#11711yescharacteristic
273C4×C61C2×#12311yescharacteristic
274C4×C61C2×#12311yescharacteristic
275Dic61—82no—
276Dic61—81no—
277Dic61—81no—
278C2×#520D6111yescharacteristic
279C2×#408—611no—
280#2311—611no—
281#1825—612no—
282#1825—612no—
283#1824—612no—
284#1823—612no—
285#1736—611no—
286#1736—611no—
287#1735—611no—
288C2×#340—611no—
289C2×C61—24no—
290C2×C61—44no—
291C2×C61C2×#38111yescharacteristic
292C2×C61#289711yescharacteristic
293C2×C61#289811yescharacteristic
294C2×C61#309611yescharacteristic
295C2×C61#309711yescharacteristic
296C2×C61#309811yescharacteristic
297C2×C61#309911yescharacteristic
298C22×#41C2×D6111yescharacteristic
299C22×M16—611no—
300C23×C8—611no—
301#520—24no—
302#520Dic6118yescomplemented
303C2×C4×Q8—21no—
304C2×C4×D4—22no—
305C2×C4×D4—21no—
306C2×#117—611no—
307C2×#117C2×D6111yescharacteristic
308C2×#117C2×D6112yes—
309C22×C4⋊C4—611no—
310C22×C4⋊C4C2×D6111yescharacteristic
311C22×C4⋊C4C2×D6112yes—
312#408—1224no—
313C2×C22⋊C8—612no—
314C2×C22⋊C8—612no—
315#391—1221no—
316#390—1221no—
317#386—1222no—
318#380—612no—
319#380—612no—
320#340—611no—
321#340—612no—
322#340—612no—
323#340—612no—
324#340—611no—
325#340—612no—
326#340—612no—
327#340—611no—
328#340—611no—
329#340—612no—
330C61#2070311yescharacteristic, complemented, Sylow
331C2×#41—22no—
332C2×#41—22no—
333C2×#41—22no—
334C2×#41—22no—
335C2×#41C2×Dic6114yes—
336C2×#41C22⋊D6114yes—
337C22×Q8C22⋊D6111yescharacteristic
338C22×D4C22⋊D6112yes—
339C22×D4C22⋊D6111yescharacteristic
340C23×C4—21no—
341C23×C4—22no—
342C23×C4—1221no—
343C23×C4C22×D6112yes—
344C23×C4C22×D6111yescharacteristic
345C2×M16—1222no—
346C2×M16—1222no—
347C2×M16—1222no—
348C22×C8—611no—
349C22×C8—611no—
350C22×C8—612no—
351C22×C8—1222no—
352C22×C8—1222no—
353C22×C8—1222no—
354C4×Q8—28no—
355C4×D4—28no—
356C4×D4—216no—
357#117—22no—
358#117—24no—
359#117—1222no—
360#117—1222no—
361#117C2×Dic6114yes—
362#117C2×Dic6118yescomplemented
363C2×C4⋊C4—21no—
364C2×C4⋊C4—22no—
365C2×C4⋊C4—22no—
366C2×C4⋊C4—24no—
367C2×C4⋊C4—612no—
368C2×C4⋊C4—611no—
369C2×C4⋊C4—611no—
370C2×C4⋊C4—1222no—
371C2×C4⋊C4—1222no—
372C2×C4⋊C4—1221no—
373C2×C4⋊C4C2×Dic6114yes—
374C2×C4⋊C4C2×Dic6118yescomplemented
375C2×C4⋊C4C22×D6112yes—
376C2×C4⋊C4C22×D6111yescharacteristic
377C2×C4⋊C4C22×D6111yescharacteristic
378C2×C22⋊C4—21no—
379C2×C22⋊C4—22no—
380C2×C22⋊C4—1222no—
381C2×C22⋊C4—1222no—
382C2×C22⋊C4—1221no—
383C2×C42—21no—
384C2×C42—22no—
385C2×C42—1221no—
386C22⋊C8—1224no—
387C22⋊C8—1224no—
388#95—1222no—
389#95—1222no—
390C24C2×C22⋊D6112yes—
391C24C2×C22⋊D6111yescharacteristic
392#41—28no—
393#41—28no—
394#41—28no—
395#41—28no—
396C2×Q8—22no—
397C2×Q8—22no—
398C2×Q8C22⋊Dic6114yes—
399C2×D4—24no—
400C2×D4—24no—
401C2×D4—22no—
402C2×D4—22no—
403C2×D4C22⋊Dic6118yes—
404C2×D4C22⋊Dic6114yes—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—28no—
414C22×C4—81no—
415C22×C4—82no—
416C22×C4—81no—
417C22×C4—82no—
418C22×C4—1221no—
419C22×C4—1221no—
420C22×C4—1222no—
421C22×C4—4881no—
422C22×C4—4881no—
423C22×C4—2442no—
424C22×C4#1175822212yes—
425C22×C4#1175822212yes—
426C22×C4#1175822212yes—
427C22×C4#1175822212yes—
428C22×C4C22×Dic6114yes—
429C22×C4C22×Dic6112yes—
430C22×C4C2×C22⋊D6114yes—
431C22×C4C2×C22⋊D6112yes—
432C22×C4C2×C22⋊D6112yes—
433C22×C4C2×C22⋊D6111yescharacteristic
434C22×C4C23×D6111yescharacteristic
435M16—2444no—
436C2×C8—1221no—
437C2×C8—1221no—
438C2×C8—1222no—
439C2×C8—1221no—
440C2×C8—1221no—
441C2×C8—1221no—
442C2×C8—1221no—
443C2×C8—1222no—
444C2×C8—1222no—
445C2×C8—2444no—
446C4⋊C4—21no—
447C4⋊C4—22no—
448C4⋊C4—21no—
449C4⋊C4—24no—
450C4⋊C4—28no—
451C4⋊C4—1222no—
452C4⋊C4—1222no—
453C4⋊C4—1221no—
454C4⋊C4—1221no—
455C4⋊C4—1221no—
456C4⋊C4—1221no—
457C4⋊C4—2442no—
458C4⋊C4C22×Dic6114yes—
459C4⋊C4C22×Dic6112yes—
460C4⋊C4C22×Dic6112yes—
461C22⋊C4—24no—
462C22⋊C4—28no—
463C22⋊C4—4882no—
464C22⋊C4—4882no—
465C22⋊C4—2442no—
466C42—24no—
467C42—28no—
468C42—2442no—
469C23—21no—
470C23—21no—
471C23—21no—
472C23—21no—
473C23—22no—
474C23—22no—
475C23—22no—
476C23—22no—
477C23—22no—
478C23—22no—
479C23C2×C22⋊Dic6114yes—
480C23C2×C22⋊Dic6112yes—
481C23#1451976012yes—
482C23#1451976012yes—
483C23C22×C22⋊D6111yescharacteristic
484Q8—28no—
485D4—216no—
486D4—28no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—24no—
490C2×C4—24no—
491C2×C4—24no—
492C2×C4—24no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—24no—
496C2×C4—42no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—42no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—84no—
503C2×C4—88no—
504C2×C4—4884no—
505C2×C4—2441no—
506C2×C4—2441no—
507C2×C4—2442no—
508C2×C4—2441no—
509C2×C4—2442no—
510C2×C4—2441no—
511C2×C4—2442no—
512C2×C4—2441no—
513C2×C4—2441no—
514C2×C4C2×C22⋊Dic6114yes—
515C2×C4C2×C22⋊Dic6112yes—
516C2×C4#1451973614yes—
517C2×C4#1451973614yes—
518C2×C4#1451976014yes—
519C2×C4#1451976014yes—
520C2×C4#1451976012yes—
521C2×C4#1451976012yes—
522C2×C4C23×Dic6111yescharacteristic
523C2×C4C22×C22⋊D6111yescharacteristic
524C2×C4C2×#1175822211yescharacteristic
525C2×C4C2×#1175822211yescharacteristic
526C2×C4C2×#1175822211yescharacteristic
527C2×C4C2×#1175822211yescharacteristic
528C8—2441no—
529C8—2441no—
530C8—2442no—
531C22—22no—
532C22—22no—
533C22—22no—
534C22—22no—
535C22—22no—
536C22—22no—
537C22—24no—
538C22—24no—
539C22—24no—
540C22—42no—
541C22—42no—
542C22—42no—
543C22—42no—
544C22#1683962514yes—
545C22C22×C22⋊Dic6111yescharacteristic
546C22C2×#1451976011yescharacteristic
547C22C2×#1451976011yescharacteristic
548C22#1684056511yescharacteristic
549C22#1684057011yescharacteristic
550C22#1684057311yescharacteristic
551C22#1684057711yescharacteristic
552C4—24no—
553C4—42no—
554C4—44no—
555C4—42no—
556C4—4882no—
557C4—4881no—
558C4—4881no—
559C4#1683962514yes—
560C4C2×#1451973611yescharacteristic
561C4C2×#1451973611yescharacteristic
562C4C2×#1451976011yescharacteristic
563C4C2×#1451976011yescharacteristic
564C2—24no—
565C2—44no—
566C2C2×#1683962511yescharacteristic
567C2#2145434511yescharacteristic
568C2#2145435211yescharacteristic
569C2#2145577711yescharacteristic
570C2#2145578811yescharacteristic
571C2#2145579711yescharacteristic
572C2#2145580811yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC2×C61×#520C2
socleC23×C61C22×D4
prime radicalC22×#41×C61C22
cosocleC22×C4×C61C24
hypercenterC2×#520D61
hypocenterC61#20703
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C22×#41 ◁ C2×#520

Complements

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

NH ≅ G/Ntype
3. #21445914C2semidirect
4. #21445914C2semidirect
5. #21445909C2semidirect
6. #21445905C2semidirect
7. #21445540C2semidirect
8. #21445538C2semidirect
9. #21445535C2semidirect
10. C2×#16839524C2semidirect
36. C61×#520C4semidirect
89. #117×C61C2×C4semidirect
95. C2×C4⋊C4×C61C2×C4semidirect
302. #520Dic61semidirect
330. C61#20703semidirect
362. #117C2×Dic61semidirect
374. C2×C4⋊C4C2×Dic61semidirect

Maximal subgroups

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

subgroupindex
C2×#168397292
#214484112
#214459142
#214459092
#214459052
#214455402
#214455382
#214455352
C2×#168395242
C2×C61×#5202
#2070361