Finite Group Databasebeta

← #29689278

Subgroups — #29689278

Order 15616.

1334 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840339C211yesmaximal, characteristic
2#21448421C211yesmaximal, characteristic, complemented
3#21448419C211yesmaximal, characteristic, complemented
4#21446656C211yesmaximal, characteristic, complemented
5#21446655C211yesmaximal, characteristic, complemented
6#21446654C211yesmaximal, characteristic, complemented
7#21446653C214yesmaximal, complemented
8#21445735C214yesmaximal, complemented
9C61×#1964C211yesmaximal, characteristic, complemented
10C22×#11758224C2212yes—
11#16840339—22no—
12#16840339—22no—
13#16840339—22no—
14#16840339—22no—
15C2×#14519755C412yescomplemented
16C2×#14519754C412yescomplemented
17C2×#14519751C2211yescharacteristic
18C2×#14519750C412yescomplemented
19C2×#14519749C412yescomplemented
20C2×#14519749C2212yes—
21C22×#11758196C2211yescharacteristic
22C2×#14519692C2214yes—
23C2×#14519691C2214yes—
24C2×#14519691C2214yes—
25#16839649—22no—
26#16839646—22no—
27#16839644—22no—
28#16839643—22no—
29#16839623C2212yes—
30#16839623C2212yes—
31#16839622C2211yescharacteristic
32#16839620C2212yes—
33C22⋊C4×Dic61C2212yes—
34C4×C22⋊Dic61C2211yescharacteristic
35C2×#121×C61C2211yescharacteristic
36C2×#99×C61C2214yes—
37C61×#380C2211yescharacteristic
38C4×C22⋊C4×C61C2211yescharacteristic
39C2×#11758224—24no—
40C2×#11758224—24no—
41C2×#11758224—44no—
42C2×#11758224—44no—
43C22×Dic122C2×C412yes—
44C22×D244C2×C411yescharacteristic
45C22×C22⋊D61C2×C412yes—
46C22×C22⋊D61C2×C412yes—
47C22×C22⋊D61C2312yes—
48C23×Dic61C2×C412yes—
49C23×Dic61C2311yescharacteristic
50C23×Dic61C2311yescharacteristic
51C22×C4×D61C2×C412yes—
52#14519755—24no—
53#14519755—24no—
54#14519754—24no—
55#14519754—24no—
56#14519751—21no—
57#14519751—21no—
58#14519751—21no—
59#14519751—21no—
60#14519750—24no—
61#14519750—24no—
62#14519749—24no—
63#14519749—24no—
64#14519749—24no—
65#14519749—24no—
66C2×C4⋊Dic61—21no—
67C2×C4⋊Dic61C2×C411yescharacteristic
68C2×#11758196—22no—
69C2×#11758196C2×C412yes—
70C2×#11758196C2×C412yes—
71C2×#11758196C2×C412yes—
72C2×#11758196D412yes—
73C2×#11758196D412yes—
74C2×#11758196D412yes—
75C2×#11758196D412yes—
76C2×C22⋊Dic61C2×C412yes—
77C2×C22⋊Dic61C2312yes—
78C2×C22⋊Dic61C2312yes—
79C2×#11758198C2×C412yes—
80C2×#11758198C2×C412yes—
81C2×C4×Dic61—21no—
82C2×C4×Dic61—22no—
83#14519692—44no—
84#14519691—44no—
85#14519691—44no—
86#14519684—21no—
87#14519684—21no—
88#14519684—22no—
89#14519684—22no—
90#14519684—24no—
91#14519684—24no—
92C22×D4×C61C2×C411yescharacteristic
93C22×D4×C61C2312yes—
94C23×C4×C61C2311yescharacteristic
95#121×C61—21no—
96#121×C61—21no—
97#121×C61—21no—
98#121×C61—21no—
99C2×C4⋊C4×C61—21no—
100C2×C4⋊C4×C61C2×C411yescharacteristic
101C2×C22⋊C4×C61C2×C412yes—
102C2×C22⋊C4×C61C2312yes—
103C2×C22⋊C4×C61C2312yes—
104C2×C42×C61—21no—
105#99×C61—44no—
106#95×C61—21no—
107#95×C61—21no—
108C24×C61C22×C412yes—
109C24×C61C2411yescharacteristic
110C23×D61C22×C412yes—
111C2×D4×C61—22no—
112C2×D4×C61—22no—
113C2×D4×C61—24no—
114C2×D4×C61—24no—
115C2×D4×C61—21no—
116C2×D4×C61—21no—
117C2×D4×C61—81no—
118C22×C4×C61—22no—
119C22×C4×C61—22no—
120C22×C4×C61—81no—
121C22×C4×C61C22×C412yes—
122C22×C4×C61C22×C411yescharacteristic
123C22×C4×C61C22×C411yescharacteristic
124C22×C4×C61C2×D411yescharacteristic
125C22×C4×C61C2×D411yescharacteristic
126C22×C4×C61C2×D411yescharacteristic
127C22×C4×C61C2×D411yescharacteristic
128C2×C22⋊D61—22no—
129C2×C22⋊D61—22no—
130C2×C22⋊D61—24no—
131C2×C22⋊D61—24no—
132C2×C22⋊D61—24no—
133C2×C22⋊D61—24no—
134C2×C22⋊D61—44no—
135C2×C22⋊D61—82no—
136C22×Dic61—22no—
137C22×Dic61—22no—
138C22×Dic61—24no—
139C22×Dic61—22no—
140C22×Dic61—22no—
141C22×Dic61—44no—
142C22×Dic61—81no—
143C22×Dic61C22×C412yes—
144C22×Dic61C22×C412yes—
145C22×Dic61C22×C412yes—
146C22×Dic61C2×D412yes—
147C22×Dic61C2×D412yes—
148C22×Dic61C2×D412yes—
149C22×Dic61C2×D412yes—
150#11758224—48no—
151#11758224—416no—
152#11758224—48no—
153C2×D244—21no—
154C2×D244—21no—
155C2×D244—81no—
156C2×C4×D61—24no—
157C2×C4×D61—44no—
158C2×Dic122—24no—
159C2×Dic122—44no—
160C4⋊C4×C61—21no—
161C4⋊C4×C61—21no—
162C4⋊C4×C61—81no—
163C22⋊C4×C61—22no—
164C22⋊C4×C61—22no—
165C22⋊C4×C61—22no—
166C22⋊C4×C61—24no—
167C22⋊C4×C61—22no—
168C42×C61—81no—
169C22⋊Dic61—24no—
170C22⋊Dic61—22no—
171C22⋊Dic61—24no—
172C22⋊Dic61—22no—
173#11758198—22no—
174#11758198—22no—
175#11758198—24no—
176C4⋊Dic61—21no—
177C4⋊Dic61—21no—
178C4⋊Dic61—81no—
179#11758196—24no—
180#11758196—82no—
181#11758196C22⋊C414yes—
182#11758196C22⋊C414yes—
183#11758196C22⋊C414yes—
184#11758196C22⋊C414yes—
185C4×Dic61—82no—
186C4×Dic61—81no—
187C23×C61—22no—
188C23×C61—24no—
189C23×C61—41no—
190C23×C61—41no—
191C23×C61—42no—
192C23×C61—42no—
193C23×C61C23×C411yescharacteristic
194C23×C61C22×D411yescharacteristic
195C23×C61C22×D411yescharacteristic
196C23×C61H(5,𝔽2)12yes—
197C23×C61H(5,𝔽2)12yes—
198C22×D61—22no—
199C22×D61—24no—
200C22×D61—42no—
201C22×D61—42no—
202D4×C61—42no—
203D4×C61—44no—
204D4×C61—42no—
205D4×C61—81no—
206D4×C61—81no—
207C2×C4×C61—22no—
208C2×C4×C61—24no—
209C2×C4×C61—41no—
210C2×C4×C61—41no—
211C2×C4×C61—41no—
212C2×C4×C61—41no—
213C2×C4×C61—41no—
214C2×C4×C61—41no—
215C2×C4×C61—42no—
216C2×C4×C61—42no—
217C2×C4×C61—42no—
218C2×C4×C61—42no—
219C2×C4×C61—42no—
220C2×C4×C61—42no—
221C2×C4×C61C2×C22⋊C411yescharacteristic
222C2×C4×C61C2×C22⋊C411yescharacteristic
223C2×C4×C61C2×C22⋊C411yescharacteristic
224C2×C4×C61C2×C22⋊C411yescharacteristic
225C22⋊D61—48no—
226C22⋊D61—42no—
227C22⋊D61—44no—
228C22⋊D61—42no—
229C22⋊D61—84no—
230C2×Dic61—24no—
231C2×Dic61—22no—
232C2×Dic61—24no—
233C2×Dic61—22no—
234C2×Dic61—22no—
235C2×Dic61—24no—
236C2×Dic61—42no—
237C2×Dic61—42no—
238C2×Dic61—42no—
239C2×Dic61—48no—
240C2×Dic61—42no—
241C2×Dic61—42no—
242C2×Dic61—42no—
243C2×Dic61—42no—
244C2×Dic61—42no—
245C2×Dic61—42no—
246C2×Dic61C2×C22⋊C414yes—
247C2×Dic61C2×C22⋊C414yes—
248D244—81no—
249D244—81no—
250C4×D61—48no—
251Dic122—48no—
252#21796—611nomaximal, Sylow
253C22×C61—21no—
254C22×C61—21no—
255C22×C61—21no—
256C22×C61—21no—
257C22×C61—21no—
258C22×C61—21no—
259C22×C61—42no—
260C22×C61—42no—
261C22×C61—42no—
262C22×C61—42no—
263C22×C61—42no—
264C22×C61—84no—
265C22×C61C22×C22⋊C411yescharacteristic
266C22×C61#51812yes—
267C22×C61#53511yescharacteristic
268C22×C61#53512yes—
269C22×C61#53511yescharacteristic
270C2×D61—42no—
271C2×D61—42no—
272C2×D61—42no—
273C2×D61—84no—
274C4×C61—21no—
275C4×C61—21no—
276C4×C61—21no—
277C4×C61—21no—
278C4×C61—42no—
279C4×C61—82no—
280C4×C61—82no—
281Dic61—24no—
282Dic61—24no—
283Dic61—42no—
284Dic61—82no—
285Dic61—82no—
286C2×#537—611no—
287#2313—611no—
288#2313—611no—
289#1964—614no—
290#1964—611no—
291#1964—611no—
292#1964—611no—
293#1964D6111yescharacteristic, complemented
294#1782—614no—
295C2×C61—21no—
296C2×C61—22no—
297C2×C61—21no—
298C2×C61—22no—
299C2×C61—82no—
300C2×C61#230911yescharacteristic
301C2×C61#288311yescharacteristic
302C2×C61#288411yescharacteristic
303D61—82no—
304C22×#41—612no—
305#537—1222no—
306#537—1222no—
307#537—1222no—
308#537—1222no—
309C2×C22⋊Q8—612no—
310C2×C22⋊Q8—612no—
311C2×#121—612no—
312C2×#121—612no—
313C2×#121—611no—
314C2×#121—612no—
315C2×#121C2×D6111yescharacteristic
316C22×C4⋊C4—611no—
317C2×#99—614no—
318C2×#99—614no—
319C2×#99—614no—
320C2×#99C2×D6114yes—
321#387—1222no—
322#387—1222no—
323#387—1222no—
324#387—1222no—
325#381—612no—
326#381—612no—
327#380—611no—
328#380—612no—
329#380C2×D6111yescharacteristic
330C4×C22⋊C4—611no—
331C4×C22⋊C4—612no—
332C4×C22⋊C4C2×D6111yescharacteristic
333C61#2179611yescharacteristic, complemented, Sylow
334C2×#41—1224no—
335C2×#41—1224no—
336C2×#41—2444no—
337C2×#41—2444no—
338C22×Q8—612no—
339C22×D4—612no—
340C22×D4—612no—
341C22×D4—612no—
342C22×D4—611no—
343C22×D4C4×D6111yescharacteristic
344C22×D4C22×D6112yes—
345C23×C4—611no—
346C23×C4—611no—
347C23×C4—612no—
348C23×C4—612no—
349C23×C4C22×D6111yescharacteristic
350C22⋊Q8—1224no—
351C22⋊Q8—1224no—
352C22⋊Q8—1224no—
353C22⋊Q8—1224no—
354#121—21no—
355#121—21no—
356#121—21no—
357#121—21no—
358#121—1224no—
359#121—1224no—
360#121—1224no—
361#121—1224no—
362#121—1224no—
363#121—1224no—
364#121—1221no—
365#121—1221no—
366#121—1221no—
367#121—1221no—
368C2×C4⋊C4—21no—
369C2×C4⋊C4—612no—
370C2×C4⋊C4—612no—
371C2×C4⋊C4—612no—
372C2×C4⋊C4—612no—
373C2×C4⋊C4—612no—
374C2×C4⋊C4—612no—
375C2×C4⋊C4—612no—
376C2×C4⋊C4—611no—
377C2×C4⋊C4—1222no—
378C2×C4⋊C4—1221no—
379C2×C4⋊C4C4×D6111yescharacteristic
380C2×C22⋊C4—612no—
381C2×C22⋊C4—612no—
382C2×C22⋊C4—612no—
383C2×C22⋊C4—612no—
384C2×C22⋊C4—612no—
385C2×C22⋊C4C4×D6112yes—
386C2×C22⋊C4C22×D6112yes—
387C2×C22⋊C4C22×D6112yes—
388C2×C42—21no—
389C2×C42—1222no—
390C2×C42—1221no—
391#99—44no—
392#99—2444no—
393#99—2444no—
394#99—2444no—
395#95—21no—
396#95—21no—
397#95—1222no—
398#95—1222no—
399#95—1224no—
400#95—1224no—
401#95—1221no—
402#95—1221no—
403C24—612no—
404C24C2×C4×D6112yes—
405C24C23×D6111yescharacteristic
406#41—2448no—
407#41—24416no—
408#41—2448no—
409C2×Q8—1224no—
410C2×Q8—2444no—
411C2×D4—22no—
412C2×D4—22no—
413C2×D4—24no—
414C2×D4—24no—
415C2×D4—21no—
416C2×D4—21no—
417C2×D4—81no—
418C2×D4—1224no—
419C2×D4—1224no—
420C2×D4—1222no—
421C2×D4—1222no—
422C2×D4—1224no—
423C2×D4—1224no—
424C2×D4—1221no—
425C2×D4—1221no—
426C2×D4—4881no—
427C2×D4—4882no—
428C2×D4—2444no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—81no—
432C22×C4—612no—
433C22×C4—612no—
434C22×C4—612no—
435C22×C4—612no—
436C22×C4—612no—
437C22×C4—612no—
438C22×C4—612no—
439C22×C4—1224no—
440C22×C4—1224no—
441C22×C4—1222no—
442C22×C4—1222no—
443C22×C4—1222no—
444C22×C4—1222no—
445C22×C4—4881no—
446C22×C4—2444no—
447C22×C4—2444no—
448C22×C4C2×C4×D6112yes—
449C22×C4C2×C4×D6111yescharacteristic
450C22×C4C2×C4×D6111yescharacteristic
451C22×C4D4×D6111yescharacteristic
452C22×C4D4×D6111yescharacteristic
453C22×C4D4×D6111yescharacteristic
454C22×C4D4×D6111yescharacteristic
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—81no—
458C4⋊C4—614no—
459C4⋊C4—614no—
460C4⋊C4—614no—
461C4⋊C4—614no—
462C4⋊C4—1224no—
463C4⋊C4—1221no—
464C4⋊C4—1221no—
465C4⋊C4—4882no—
466C4⋊C4—4881no—
467C22⋊C4—22no—
468C22⋊C4—22no—
469C22⋊C4—22no—
470C22⋊C4—24no—
471C22⋊C4—22no—
472C22⋊C4—1224no—
473C22⋊C4—1224no—
474C22⋊C4—1222no—
475C22⋊C4—1222no—
476C22⋊C4—1222no—
477C22⋊C4—1222no—
478C22⋊C4—1224no—
479C42—81no—
480C42—4881no—
481C42—4882no—
482C23—22no—
483C23—24no—
484C23—41no—
485C23—41no—
486C23—42no—
487C23—42no—
488C23—1224no—
489C23—1222no—
490C23—2442no—
491C23—2442no—
492C23C22×C4×D6111yescharacteristic
493C23C2×D4×D6111yescharacteristic
494C23C2×D4×D6111yescharacteristic
495C23#1451982212yes—
496C23#1451982312yes—
497Q8—2448no—
498D4—42no—
499D4—44no—
500D4—42no—
501D4—81no—
502D4—81no—
503D4—4881no—
504D4—4881no—
505D4—4884no—
506D4—2448no—
507D4—2442no—
508D4—2444no—
509D4—2442no—
510C2×C4—22no—
511C2×C4—24no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—614no—
525C2×C4—614no—
526C2×C4—1224no—
527C2×C4—1222no—
528C2×C4—1224no—
529C2×C4—1222no—
530C2×C4—1222no—
531C2×C4—1224no—
532C2×C4—2448no—
533C2×C4—2442no—
534C2×C4—2442no—
535C2×C4—2442no—
536C2×C4—2448no—
537C2×C4—2442no—
538C2×C4—2442no—
539C2×C4—2442no—
540C2×C4—2442no—
541C2×C4—2442no—
542C2×C4—2442no—
543C2×C4C22⋊C4×D6111yescharacteristic
544C2×C4C22⋊C4×D6111yescharacteristic
545C2×C4C22⋊C4×D6111yescharacteristic
546C2×C4C22⋊C4×D6111yescharacteristic
547C22—21no—
548C22—21no—
549C22—21no—
550C22—21no—
551C22—21no—
552C22—21no—
553C22—42no—
554C22—42no—
555C22—42no—
556C22—42no—
557C22—42no—
558C22—84no—
559C22—4884no—
560C22—2442no—
561C22—2442no—
562C22—2442no—
563C22C2×C22⋊C4×D6111yescharacteristic
564C22#1684023412yes—
565C22#1684032511yescharacteristic
566C22#1684032812yes—
567C22#1684033011yescharacteristic
568C4—21no—
569C4—21no—
570C4—21no—
571C4—21no—
572C4—42no—
573C4—82no—
574C4—82no—
575C4—1224no—
576C4—1224no—
577C4—4882no—
578C4—4882no—
579C4—2442no—
580C2—21no—
581C2—22no—
582C2—21no—
583C2—22no—
584C2—82no—
585C2—4882no—
586C2#2144839811yescharacteristic
587C2#2145427711yescharacteristic
588C2#2145428111yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D61
commutatorC23×C61C23×C4
FrattiniC24C23×D61
FittingC61×#1964C2
socleC22×C61C22×C22⋊C4
prime radicalC2×#16840339C2
cosocleC24×C61C24
hypercenter#1964D61
hypocenterC61#21796
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1964

Complements

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

NH ≅ G/Ntype
2. #21448421C2semidirect
3. #21448419C2semidirect
4. #21446656C2semidirect
5. #21446655C2semidirect
6. #21446654C2semidirect
7. #21446653C2semidirect
8. #21445735C2semidirect
9. C61×#1964C2semidirect
15. C2×#14519755C4semidirect
16. C2×#14519754C4semidirect
18. C2×#14519750C4semidirect
19. C2×#14519749C4semidirect
293. #1964D61semidirect
333. C61#21796semidirect

Maximal subgroups

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

subgroupindex
C2×#168403392
#214484212
#214484192
#214466562
#214466552
#214466542
#214466532
#214457352
C61×#19642
#2179661