Finite Group Databasebeta

← #29712810

Subgroups — #29712810

Order 15616.

1578 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455739C212yesmaximal, complemented
2C2×#16839665C211yesmaximal, characteristic, complemented
3#21447026C214yesmaximal, complemented
4#21446263C212yesmaximal, complemented
5#21446029C212yesmaximal, complemented
6#21445773C212yesmaximal, complemented
7#21445740C211yesmaximal, characteristic, complemented
8C61×#3484C211yesmaximal, characteristic
9C22×C4⋊Dic61C2211yescharacteristic, complemented
10C22×C22⋊Dic61—22no—
11C22×C4×Dic61C2211yescharacteristic, complemented
12#16840095—22no—
13#16840090—22no—
14#16840078—22no—
15#16840073—22no—
16#16839805C2214yescomplemented
17C2×#14519704C2214yescomplemented
18#16839763C2214yescomplemented
19C2×#14519694C2212yes—
20C2×#14519694C2212yescomplemented
21#16839726C2212yescomplemented
22#16839665—22no—
23#16839665C2212yes—
24#16839665C2212yescomplemented
25#16839521C2214yescomplemented
26C2×H(5,𝔽2)×C61C2211yescharacteristic
27C23×D4×C61C2211yescharacteristic
28C61×#546—22no—
29C61×#546—22no—
30D42×C61—22no—
31D42×C61—22no—
32C61×#535C2212yes—
33C61×#534—22no—
34C61×#525C2212yes—
35C2×#120×C61—22no—
36C61×#518C2211yescharacteristic
37C23×Dic61—21no—
38C23×Dic61—21no—
39C2×#11758194C2314yes—
40C2×C4⋊Dic61—22no—
41C2×C4⋊Dic61—22no—
42C2×C4⋊Dic61—21no—
43C2×C4⋊Dic61C2312yes—
44C2×C22⋊Dic61—24no—
45C2×C22⋊Dic61—24no—
46C2×C22⋊Dic61—48no—
47C2×C4×Dic61—21no—
48C2×C4×Dic61—42no—
49C2×C4×Dic61C2312yes—
50#14519710—24no—
51#14519708—24no—
52#14519704—44no—
53#14519701—24no—
54#14519698—24no—
55#14519694—24no—
56#14519694—42no—
57C22⋊#92418—22no—
58C22⋊#92418—22no—
59C25×C61D412yes—
60H(5,𝔽2)×C61—24no—
61H(5,𝔽2)×C61—42no—
62C2×#41×C61—22no—
63C2×#41×C61C2312yes—
64C22×D4×C61—22no—
65C22×D4×C61—22no—
66C22×D4×C61—22no—
67C22×D4×C61—22no—
68C22×D4×C61—24no—
69C22×D4×C61—22no—
70C22×D4×C61—22no—
71C22×D4×C61—42no—
72C22×D4×C61D412yes—
73C22×D4×C61C2311yescharacteristic
74C22×D4×C61C2311yescharacteristic
75C23×C4×C61C2311yescharacteristic
76#127×C61—22no—
77#124×C61—22no—
78#123×C61—24no—
79#123×C61—22no—
80C22⋊Q8×C61D412yes—
81C22⋊Q8×C61D412yes—
82#121×C61—24no—
83#121×C61—24no—
84#121×C61—22no—
85#121×C61—22no—
86#121×C61D412yes—
87#121×C61D412yes—
88#120×C61—24no—
89#120×C61—24no—
90#120×C61—24no—
91#120×C61—24no—
92#120×C61—22no—
93C4×D4×C61—22no—
94C4×D4×C61—22no—
95C4×D4×C61—22no—
96C4×D4×C61—22no—
97#117×C61C2312yes—
98C2×C22⋊C4×C61—22no—
99C2×C22⋊C4×C61—22no—
100C2×C22⋊C4×C61—22no—
101C24×C61—22no—
102C24×C61—24no—
103C24×C61—22no—
104C24×C61—22no—
105C24×C61—42no—
106C24×C61—44no—
107C24×C61—44no—
108C24×C61C2×D411yescharacteristic
109C24×C61C2×D412yes—
110C24×C61C2×D412yes—
111#41×C61—24no—
112#41×C61—24no—
113#41×C61—48no—
114C2×Q8×C61C2×D412yes—
115C2×D4×C61—24no—
116C2×D4×C61—22no—
117C2×D4×C61—22no—
118C2×D4×C61—24no—
119C2×D4×C61—22no—
120C2×D4×C61—24no—
121C2×D4×C61—24no—
122C2×D4×C61—24no—
123C2×D4×C61—22no—
124C2×D4×C61—22no—
125C2×D4×C61—22no—
126C2×D4×C61—22no—
127C2×D4×C61—22no—
128C2×D4×C61—48no—
129C2×D4×C61—48no—
130C2×D4×C61—42no—
131C2×D4×C61—48no—
132C2×D4×C61—44no—
133C2×D4×C61—44no—
134C2×D4×C61—84no—
135C2×D4×C61—84no—
136C2×D4×C61C2×D412yes—
137C2×D4×C61#4114yes—
138C22×C4×C61—22no—
139C22×C4×C61—22no—
140C22×C4×C61—22no—
141C22×C4×C61—22no—
142C22×C4×C61—22no—
143C22×C4×C61—22no—
144C22×C4×C61C2×D411yescharacteristic
145C22×C4×C61C2×D412yes—
146C22×C4×C61C2×D412yes—
147C22×C4×C61C2411yescharacteristic
148C22×Dic61—24no—
149C22×Dic61—22no—
150C22×Dic61—22no—
151C22×Dic61—42no—
152C22×Dic61—42no—
153C22×Dic61—44no—
154C22×Dic61—41no—
155C22×Dic61—41no—
156C4⋊C4×C61—22no—
157C4⋊C4×C61—22no—
158C4⋊C4×C61C2×D412yes—
159C4⋊C4×C61C2×D412yes—
160C22⋊C4×C61—24no—
161C22⋊C4×C61—24no—
162C22⋊C4×C61—24no—
163C22⋊C4×C61—22no—
164C22⋊C4×C61—22no—
165C22⋊C4×C61—22no—
166C22⋊C4×C61—22no—
167C22⋊C4×C61—22no—
168C42×C61—22no—
169C22⋊Dic61—48no—
170C22⋊Dic61—48no—
171C4⋊Dic61—24no—
172C4⋊Dic61—42no—
173C4×Dic61—24no—
174C4×Dic61—42no—
175#11758194—24no—
176#11758194—24no—
177C2×#92418—24no—
178C23×C61—21no—
179C23×C61—21no—
180C23×C61—22no—
181C23×C61—22no—
182C23×C61—22no—
183C23×C61—22no—
184C23×C61—22no—
185C23×C61—22no—
186C23×C61—22no—
187C23×C61—22no—
188C23×C61—44no—
189C23×C61—44no—
190C23×C61—42no—
191C23×C61—42no—
192C23×C61—42no—
193C23×C61—42no—
194C23×C61—416no—
195C23×C61—44no—
196C23×C61—44no—
197C23×C61—44no—
198C23×C61—82no—
199C23×C61—82no—
200C23×C61—88no—
201C23×C61—82no—
202C23×C61—82no—
203C23×C61#12011yescharacteristic
204C23×C61#12011yescharacteristic
205C23×C61#12314yes—
206C23×C61C22×D411yescharacteristic
207Q8×C61—24no—
208D4×C61—28no—
209D4×C61—24no—
210D4×C61—48no—
211D4×C61—48no—
212D4×C61—42no—
213D4×C61—42no—
214D4×C61—44no—
215D4×C61—44no—
216D4×C61—44no—
217D4×C61—84no—
218D4×C61—84no—
219C2×C4×C61—21no—
220C2×C4×C61—21no—
221C2×C4×C61—24no—
222C2×C4×C61—24no—
223C2×C4×C61—22no—
224C2×C4×C61—22no—
225C2×C4×C61—22no—
226C2×C4×C61—44no—
227C2×C4×C61—44no—
228C2×C4×C61—82no—
229C2×C4×C61—82no—
230C2×C4×C61#12012yes—
231C2×C4×C61C22×D412yes—
232C2×C4×C61C2×#4112yes—
233C2×C4×C61H(5,𝔽2)12yes—
234C2×Dic61—24no—
235C2×Dic61—48no—
236C2×Dic61—42no—
237C2×Dic61—42no—
238C2×Dic61—44no—
239C2×Dic61—42no—
240C2×Dic61—82no—
241C2×Dic61—82no—
242#92418—44no—
243#24018—611nomaximal, Sylow
244C22×C61—24no—
245C22×C61—24no—
246C22×C61—21no—
247C22×C61—21no—
248C22×C61—21no—
249C22×C61—21no—
250C22×C61—21no—
251C22×C61—21no—
252C22×C61—24no—
253C22×C61—44no—
254C22×C61—44no—
255C22×C61—42no—
256C22×C61—42no—
257C22×C61—42no—
258C22×C61—42no—
259C22×C61—42no—
260C22×C61—42no—
261C22×C61—42no—
262C22×C61—42no—
263C22×C61—82no—
264C22×C61—88no—
265C22×C61—88no—
266C22×C61—82no—
267C22×C61—164no—
268C22×C61#45312yes—
269C22×C61#45512yes—
270C22×C61C2×#12011yescharacteristic
271C22×C61C2×#12311yescharacteristic
272C22×C61#53411yescharacteristic
273C4×C61—22no—
274C4×C61—42no—
275C4×C61—42no—
276C4×C61—42no—
277C4×C61#54614yes—
278Dic61—44no—
279Dic61—82no—
280#3484D6111yescharacteristic
281#3092—612no—
282C2×#390—611no—
283#2037—614no—
284#1896—612no—
285#1856—612no—
286#1792—612no—
287#1783—611no—
288C2×C61—21no—
289C2×C61—21no—
290C2×C61—22no—
291C2×C61—22no—
292C2×C61—44no—
293C2×C61—82no—
294C2×C61—82no—
295C2×C61#240511yescharacteristic
296C2×C61C2×#45311yescharacteristic
297C2×C61C2×#45511yescharacteristic
298C2×H(5,𝔽2)C2×D6111yescharacteristic
299C23×D4C2×D6111yescharacteristic
300#546—22no—
301#546—22no—
302D42—22no—
303D42—22no—
304#535C2×D6112yes—
305#534—22no—
306#525C2×D6112yes—
307C2×#120—22no—
308#518C2×D6111yescharacteristic
309C22×C4⋊C4—611no—
310C22×C22⋊C4—1222no—
311C22×C42—611no—
312#484—1222no—
313#483—1222no—
314#481—1222no—
315#480—1222no—
316#428—614no—
317C2×#104—614no—
318#417—614no—
319C2×#100—612no—
320C2×#100—612no—
321#407—612no—
322#390—612no—
323#390—612no—
324#390—1222no—
325#339—614no—
326C61#2401811yescharacteristic, complemented, Sylow
327C25C22⋊D6112yes—
328H(5,𝔽2)—24no—
329H(5,𝔽2)—42no—
330C2×#41—22no—
331C2×#41C22×D6112yes—
332C22×D4—22no—
333C22×D4—22no—
334C22×D4—22no—
335C22×D4—24no—
336C22×D4—22no—
337C22×D4—22no—
338C22×D4—22no—
339C22×D4—42no—
340C22×D4C22⋊D6112yes—
341C22×D4C22×D6111yescharacteristic
342C22×D4C22×D6111yescharacteristic
343C23×C4—1221no—
344C23×C4—1221no—
345C23×C4C22×D6111yescharacteristic
346C2×M16—614no—
347#127—22no—
348#124—22no—
349#123—24no—
350#123—22no—
351C22⋊Q8C22⋊D6112yes—
352C22⋊Q8C22⋊D6112yes—
353#121—24no—
354#121—24no—
355#121—22no—
356#121—22no—
357#121C22⋊D6112yes—
358#121C22⋊D6112yes—
359#120—24no—
360#120—24no—
361#120—24no—
362#120—22no—
363#120—24no—
364C4×D4—22no—
365C4×D4—22no—
366C4×D4—22no—
367C4×D4—22no—
368#117C22×D6112yes—
369C2×C4⋊C4—612no—
370C2×C4⋊C4—1222no—
371C2×C4⋊C4—1222no—
372C2×C4⋊C4—1221no—
373C2×C22⋊C4—22no—
374C2×C22⋊C4—22no—
375C2×C22⋊C4—22no—
376C2×C22⋊C4—1224no—
377C2×C22⋊C4—1224no—
378C2×C22⋊C4—2448no—
379C2×C42—612no—
380C2×C42—1221no—
381C2×C42—2442no—
382#107—1224no—
383#106—1224no—
384#104—2444no—
385#103—1224no—
386#102—1224no—
387#100—1224no—
388#100—2442no—
389C22⋊C8—1222no—
390C22⋊C8—1222no—
391C24—22no—
392C24—24no—
393C24—22no—
394C24—22no—
395C24—42no—
396C24—44no—
397C24—44no—
398C24C2×C22⋊D6111yescharacteristic
399C24C2×C22⋊D6112yes—
400C24C2×C22⋊D6112yes—
401#41—24no—
402#41—24no—
403#41—48no—
404C2×Q8C2×C22⋊D6112yes—
405C2×D4—24no—
406C2×D4—22no—
407C2×D4—22no—
408C2×D4—22no—
409C2×D4—24no—
410C2×D4—24no—
411C2×D4—24no—
412C2×D4—22no—
413C2×D4—22no—
414C2×D4—22no—
415C2×D4—22no—
416C2×D4—22no—
417C2×D4—24no—
418C2×D4—48no—
419C2×D4—48no—
420C2×D4—42no—
421C2×D4—48no—
422C2×D4—44no—
423C2×D4—44no—
424C2×D4—84no—
425C2×D4—84no—
426C2×D4#1175822414yes—
427C2×D4C2×C22⋊D6112yes—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—1224no—
435C22×C4—1222no—
436C22×C4—1222no—
437C22×C4—2444no—
438C22×C4—2441no—
439C22×C4—2441no—
440C22×C4—2442no—
441C22×C4—2442no—
442C22×C4C2×C22⋊D6112yes—
443C22×C4C2×C22⋊D6112yes—
444C22×C4C2×C22⋊D6111yescharacteristic
445C22×C4C23×D6111yescharacteristic
446M16—1224no—
447M16—1224no—
448C2×C8—1224no—
449C4⋊C4—22no—
450C4⋊C4—22no—
451C4⋊C4—1224no—
452C4⋊C4—2442no—
453C4⋊C4C2×C22⋊D6112yes—
454C4⋊C4C2×C22⋊D6112yes—
455C22⋊C4—24no—
456C22⋊C4—24no—
457C22⋊C4—24no—
458C22⋊C4—22no—
459C22⋊C4—22no—
460C22⋊C4—22no—
461C22⋊C4—22no—
462C22⋊C4—22no—
463C22⋊C4—2448no—
464C22⋊C4—2448no—
465C42—22no—
466C42—1224no—
467C42—2442no—
468C23—22no—
469C23—21no—
470C23—21no—
471C23—22no—
472C23—22no—
473C23—22no—
474C23—22no—
475C23—22no—
476C23—22no—
477C23—22no—
478C23—44no—
479C23—44no—
480C23—42no—
481C23—42no—
482C23—42no—
483C23—42no—
484C23—416no—
485C23—44no—
486C23—44no—
487C23—44no—
488C23—82no—
489C23—82no—
490C23—88no—
491C23—82no—
492C23—82no—
493C23#1451974611yescharacteristic
494C23#1451974611yescharacteristic
495C23#1451975814yes—
496C23C22×C22⋊D6111yescharacteristic
497Q8—24no—
498D4—28no—
499D4—24no—
500D4—48no—
501D4—48no—
502D4—44no—
503D4—42no—
504D4—42no—
505D4—44no—
506D4—44no—
507D4—84no—
508D4—84no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—44no—
517C2×C4—44no—
518C2×C4—82no—
519C2×C4—82no—
520C2×C4—1224no—
521C2×C4—4882no—
522C2×C4—4882no—
523C2×C4—2444no—
524C2×C4—2442no—
525C2×C4—2448no—
526C2×C4—2442no—
527C2×C4—2442no—
528C2×C4#1451974612yes—
529C2×C4C22×C22⋊D6112yes—
530C2×C4C2×#1175822412yes—
531C2×C4#1451982212yes—
532C8—2444no—
533C22—24no—
534C22—24no—
535C22—21no—
536C22—21no—
537C22—21no—
538C22—21no—
539C22—21no—
540C22—21no—
541C22—24no—
542C22—44no—
543C22—44no—
544C22—42no—
545C22—42no—
546C22—42no—
547C22—42no—
548C22—42no—
549C22—42no—
550C22—42no—
551C22—42no—
552C22—82no—
553C22—88no—
554C22—88no—
555C22—82no—
556C22—164no—
557C22#1683993012yes—
558C22#1683994012yes—
559C22C2×#1451974611yescharacteristic
560C22C2×#1451975811yescharacteristic
561C22#1684032111yescharacteristic
562C4—22no—
563C4—42no—
564C4—42no—
565C4—42no—
566C4—4882no—
567C4—2444no—
568C4#1684040614yes—
569C2—21no—
570C2—21no—
571C2—22no—
572C2—22no—
573C2—44no—
574C2—82no—
575C2—82no—
576C2#2144914211yescharacteristic
577C2C2×#1683993011yescharacteristic
578C2C2×#1683994011yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
Fitting, prime radicalC61×#3484C2
socleC22×C61C2×#120
hypercenter#3484D61
hypocenterC61#24018
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3484

Complements

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

NH ≅ G/Ntype
1. #21455739C2semidirect
2. C2×#16839665C2semidirect
3. #21447026C2semidirect
4. #21446263C2semidirect
5. #21446029C2semidirect
6. #21445773C2semidirect
7. #21445740C2semidirect
9. C22×C4⋊Dic61C22semidirect
11. C22×C4×Dic61C22semidirect
16. #16839805C22semidirect
17. C2×#14519704C22semidirect
18. #16839763C22semidirect
20. C2×#14519694C22semidirect
21. #16839726C22semidirect
24. #16839665C22semidirect
25. #16839521C22semidirect
326. C61#24018semidirect

Maximal subgroups

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

subgroupindex
#214557392
C2×#168396652
#214470262
#214462632
#214460292
#214457732
#214457402
C61×#34842
#2401861