Finite Group Databasebeta

← #29707628

Subgroups — #29707628

Order 15616.

1362 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21454408C211yesmaximal, characteristic, complemented
2C2×#16839779C211yesmaximal, characteristic
3#21454338C211yesmaximal, characteristic
4C22×#14519696C211yesmaximal, characteristic, complemented
5#21448405C211yesmaximal, characteristic, complemented
6#21447033C218yesmaximal, complemented
7#21445528C211yesmaximal, characteristic, complemented
8C2×C61×#536C211yesmaximal, characteristic
9C22×#11758194C2211yescharacteristic
10C22×#11758194C2211yescharacteristic
11C2×#14519736C2211yescharacteristic
12C2×#14519736C2211yescharacteristic
13#16839805C2214yescomplemented
14#16839802—22no—
15#16839798—22no—
16#16839779—24no—
17C2×#14519704C2214yes—
18#16839771—21no—
19#16839767—21no—
20#16839763C2214yes—
21#16839758—22no—
22C2×#14519696—24no—
23C2×#14519696C2214yescomplemented
24#16839669—22no—
25#16839669C2214yescomplemented
26#16839625—22no—
27#16839521C2214yescomplemented
28C22×#41×C61C2211yescharacteristic
29C23×Q8×C61C2211yescharacteristic
30C61×#536—22no—
31C61×#536—22no—
32C61×#536—22no—
33C61×#536C2214yes—
34C2×C4⋊Q8×C61—21no—
35C2×#124×C61—21no—
36C2×#123×C61—22no—
37C2×C22⋊Q8×C61—22no—
38C2×#117×C61C2211yescharacteristic
39C2×#11758194—21no—
40C2×#11758194—22no—
41C2×#11758194—21no—
42C2×#11758194—22no—
43C2×#11758194—22no—
44C2×#11758194D414yes—
45C2×#11758194C2312yes—
46C2×#11758194C2312yes—
47C22×#92418—21no—
48C22×#92418—21no—
49#14519736—22no—
50#14519736—22no—
51#14519736—21no—
52#14519736—21no—
53#14519736—21no—
54#14519736—21no—
55C2×C4⋊Dic61—21no—
56C2×C4⋊Dic61C2312yes—
57C2×C22⋊Dic61—21no—
58C2×C22⋊Dic61—21no—
59C2×C4×Dic61—21no—
60C2×C4×Dic61C2312yes—
61#14519710—24no—
62#14519708—24no—
63#14519704—28no—
64#14519701—24no—
65#14519698—24no—
66#14519696—28no—
67#14519684—28no—
68C2×#41×C61—21no—
69C2×#41×C61—21no—
70C2×#41×C61—22no—
71C2×#41×C61—22no—
72C2×#41×C61—22no—
73C2×#41×C61—22no—
74C2×#41×C61D412yes—
75C2×#41×C61C2312yes—
76C22×Q8×C61—21no—
77C22×Q8×C61—21no—
78C22×Q8×C61—21no—
79C22×Q8×C61—22no—
80C22×Q8×C61—21no—
81C22×Q8×C61—22no—
82C22×Q8×C61—22no—
83C22×Q8×C61—22no—
84C22×Q8×C61D411yescharacteristic
85C22×Q8×C61D412yes—
86C22×Q8×C61C2312yes—
87C22×D4×C61—21no—
88C22×D4×C61D411yescharacteristic
89C23×C4×C61—21no—
90C23×C4×C61D412yes—
91C23×C4×C61C2311yescharacteristic
92C4⋊Q8×C61—24no—
93C4⋊Q8×C61—24no—
94#124×C61—24no—
95#124×C61—24no—
96#123×C61—28no—
97#123×C61—28no—
98C22⋊Q8×C61—28no—
99C22⋊Q8×C61—28no—
100#117×C61—21no—
101#117×C61—21no—
102#117×C61D412yes—
103#117×C61C2312yes—
104C2×C4⋊C4×C61—24no—
105C2×C4⋊C4×C61D412yes—
106C2×C22⋊C4×C61—24no—
107C2×C22⋊C4×C61—21no—
108C2×C42×C61—21no—
109C24×C61—21no—
110C24×C61C2×D411yescharacteristic
111#41×C61—24no—
112#41×C61—24no—
113#41×C61—22no—
114#41×C61—22no—
115#41×C61—22no—
116#41×C61—22no—
117#41×C61—42no—
118#41×C61—42no—
119#41×C61—42no—
120#41×C61—42no—
121C2×Q8×C61—22no—
122C2×Q8×C61—22no—
123C2×Q8×C61—21no—
124C2×Q8×C61—24no—
125C2×Q8×C61—24no—
126C2×Q8×C61—22no—
127C2×Q8×C61—22no—
128C2×Q8×C61—22no—
129C2×Q8×C61—22no—
130C2×Q8×C61—22no—
131C2×Q8×C61—22no—
132C2×Q8×C61—22no—
133C2×Q8×C61—22no—
134C2×Q8×C61—42no—
135C2×Q8×C61—42no—
136C2×Q8×C61—42no—
137C2×Q8×C61—42no—
138C2×Q8×C61—42no—
139C2×Q8×C61—42no—
140C2×Q8×C61—42no—
141C2×Q8×C61—42no—
142C2×Q8×C61C2×D412yes—
143C2×D4×C61—22no—
144C2×D4×C61—22no—
145C2×D4×C61—21no—
146C2×D4×C61—22no—
147C2×D4×C61—22no—
148C2×D4×C61—42no—
149C2×D4×C61—42no—
150C2×D4×C61C2×D412yes—
151C22×C4×C61—22no—
152C22×C4×C61—22no—
153C22×C4×C61—21no—
154C22×C4×C61—22no—
155C22×C4×C61—24no—
156C22×C4×C61—24no—
157C22×C4×C61—22no—
158C22×C4×C61—22no—
159C22×C4×C61—42no—
160C22×C4×C61—42no—
161C22×C4×C61C2×D414yes—
162C22×C4×C61C2×D411yescharacteristic
163C22×C4×C61C2×D411yescharacteristic
164C22×C4×C61C2×D411yescharacteristic
165C22×C4×C61C2×D411yescharacteristic
166C22×C4×C61C2×D411yescharacteristic
167C22×C4×C61C2×D412yes—
168C22×C4×C61#4112yes—
169C22×C4×C61#4112yes—
170C22×C4×C61#4112yes—
171C22×C4×C61C2411yescharacteristic
172C22×Dic61—22no—
173C22×Dic61—22no—
174C4⋊C4×C61—216no—
175C4⋊C4×C61—22no—
176C4⋊C4×C61C2×D414yes—
177C22⋊C4×C61—216no—
178C22⋊C4×C61—22no—
179C22⋊C4×C61—22no—
180C42×C61—22no—
181C42×C61—22no—
182C22⋊Dic61—41no—
183C22⋊Dic61—41no—
184C22⋊Dic61—42no—
185C4⋊Dic61—22no—
186C4⋊Dic61—22no—
187C4⋊Dic61—42no—
188C4×Dic61—22no—
189C4×Dic61—22no—
190C4×Dic61—41no—
191C4×Dic61—41no—
192#11758194—24no—
193#11758194—22no—
194#11758194—22no—
195#11758194—22no—
196#11758194—42no—
197#11758194C2×D414yes—
198C2×#92418—24no—
199C2×#92418—22no—
200C2×#92418—22no—
201C2×#92418—42no—
202C23×C61—22no—
203C23×C61—42no—
204C23×C61—44no—
205C23×C61—42no—
206C23×C61#12011yescharacteristic
207C23×C61#12011yescharacteristic
208C23×C61#12112yes—
209C23×C61#12312yes—
210C23×C61C22×D411yescharacteristic
211Q8×C61—22no—
212Q8×C61—22no—
213Q8×C61—42no—
214Q8×C61—42no—
215Q8×C61—42no—
216Q8×C61—42no—
217Q8×C61—44no—
218Q8×C61—44no—
219Q8×C61—42no—
220D4×C61—22no—
221D4×C61—22no—
222D4×C61—42no—
223D4×C61—42no—
224D4×C61—44no—
225D4×C61—42no—
226C2×C4×C61—24no—
227C2×C4×C61—24no—
228C2×C4×C61—22no—
229C2×C4×C61—24no—
230C2×C4×C61—24no—
231C2×C4×C61—44no—
232C2×C4×C61—44no—
233C2×C4×C61—44no—
234C2×C4×C61—42no—
235C2×C4×C61—42no—
236C2×C4×C61—42no—
237C2×C4×C61—44no—
238C2×C4×C61#12011yescharacteristic
239C2×C4×C61#12011yescharacteristic
240C2×C4×C61#12112yes—
241C2×C4×C61#12112yes—
242C2×C4×C61#12112yes—
243C2×C4×C61#12112yes—
244C2×C4×C61#12112yes—
245C2×C4×C61#12312yes—
246C2×C4×C61#12312yes—
247C2×C4×C61#12312yes—
248C2×C4×C61#12412yes—
249C2×C4×C61#12412yes—
250C2×C4×C61C22×D411yescharacteristic
251C2×C4×C61C22×D411yescharacteristic
252C2×C4×C61C22×D411yescharacteristic
253C2×C4×C61C2×#4111yescharacteristic
254C2×C4×C61C2×#4111yescharacteristic
255C2×C4×C61C2×#4111yescharacteristic
256C2×Dic61—44no—
257C2×Dic61—44no—
258C2×Dic61—42no—
259C2×Dic61—42no—
260#92418—24no—
261#92418—42no—
262#23494—611nomaximal, Sylow
263C22×C61—22no—
264C22×C61—22no—
265C22×C61—22no—
266C22×C61—42no—
267C22×C61—42no—
268C22×C61—42no—
269C22×C61—44no—
270C22×C61—84no—
271C22×C61#39412yes—
272C22×C61#39412yes—
273C22×C61C2×#12011yescharacteristic
274C22×C61C2×#12111yescharacteristic
275C22×C61C2×#12311yescharacteristic
276C4×C61—44no—
277C4×C61—42no—
278C4×C61—44no—
279C4×C61#39412yes—
280C4×C61#39412yes—
281C4×C61C2×#12111yescharacteristic
282C4×C61C2×#12111yescharacteristic
283C4×C61C2×#12311yescharacteristic
284C4×C61C2×#12411yescharacteristic
285Dic61—82no—
286Dic61—82no—
287C2×#536D6111yescharacteristic
288#2911—611no—
289C2×#421—611no—
290#2896—611no—
291C22×#101—611no—
292#2310—611no—
293#2038—618no—
294#1733—611no—
295C2×C61—21no—
296C2×C61—22no—
297C2×C61—21no—
298C2×C61—22no—
299C2×C61—82no—
300C2×C61C2×#39411yescharacteristic
301C2×C61#302012yes—
302C22×#41C2×D6111yescharacteristic
303C23×Q8C2×D6111yescharacteristic
304C22×M16—611no—
305C22×M16—611no—
306#536—22no—
307#536—22no—
308#536—22no—
309#536C2×D6114yes—
310C2×C4⋊Q8—21no—
311C2×#124—21no—
312C2×#123—22no—
313C2×C22⋊Q8—22no—
314C2×#117—611no—
315C2×#117—611no—
316C2×#117C2×D6111yescharacteristic
317#428—614no—
318#428—1222no—
319#427—1222no—
320#421—1224no—
321C2×#104—614no—
322#419—1221no—
323#418—1221no—
324#417—614no—
325#416—1222no—
326C2×#101—614no—
327C2×#101—1224no—
328#391—614no—
329#391—1222no—
330#381—1222no—
331#339—614no—
332C61#2349411yescharacteristic, complemented, Sylow
333C2×#41—21no—
334C2×#41—21no—
335C2×#41—22no—
336C2×#41—22no—
337C2×#41—22no—
338C2×#41—22no—
339C2×#41C22⋊D6112yes—
340C2×#41C22×D6112yes—
341C22×Q8—22no—
342C22×Q8—22no—
343C22×Q8—22no—
344C22×Q8—22no—
345C22×Q8—21no—
346C22×Q8—21no—
347C22×Q8—21no—
348C22×Q8—21no—
349C22×Q8C22⋊D6111yescharacteristic
350C22×Q8C22⋊D6112yes—
351C22×Q8C22×D6112yes—
352C22×D4—21no—
353C22×D4C22⋊D6111yescharacteristic
354C23×C4—21no—
355C23×C4C22⋊D6112yes—
356C23×C4C22×D6111yescharacteristic
357C2×M16—614no—
358C2×M16—612no—
359C2×M16—612no—
360C2×M16—1222no—
361C2×M16—1221no—
362C2×M16—1221no—
363C2×M16—1222no—
364C2×M16—1222no—
365C22×C8—1221no—
366C22×C8—1221no—
367C4⋊Q8—24no—
368C4⋊Q8—24no—
369#124—24no—
370#124—24no—
371#123—28no—
372#123—28no—
373C22⋊Q8—28no—
374C22⋊Q8—28no—
375#117—21no—
376#117—21no—
377#117—1222no—
378#117—1222no—
379#117—1221no—
380#117—1221no—
381#117—1221no—
382#117—1221no—
383#117C22⋊D6112yes—
384#117C22×D6112yes—
385C2×C4⋊C4—24no—
386C2×C4⋊C4—612no—
387C2×C4⋊C4—1221no—
388C2×C4⋊C4C22⋊D6112yes—
389C2×C22⋊C4—24no—
390C2×C22⋊C4—21no—
391C2×C22⋊C4—1221no—
392C2×C22⋊C4—1221no—
393C2×C42—21no—
394C2×C42—612no—
395C2×C42—1221no—
396#107—1224no—
397#106—1224no—
398#104—1228no—
399#103—1224no—
400#102—1224no—
401#101—1228no—
402#95—1228no—
403C24—21no—
404C24C2×C22⋊D6111yescharacteristic
405#41—22no—
406#41—22no—
407#41—22no—
408#41—22no—
409#41—24no—
410#41—24no—
411#41—42no—
412#41—42no—
413#41—42no—
414#41—42no—
415C2×Q8—22no—
416C2×Q8—22no—
417C2×Q8—22no—
418C2×Q8—22no—
419C2×Q8—22no—
420C2×Q8—22no—
421C2×Q8—22no—
422C2×Q8—22no—
423C2×Q8—21no—
424C2×Q8—24no—
425C2×Q8—24no—
426C2×Q8—22no—
427C2×Q8—22no—
428C2×Q8—42no—
429C2×Q8—42no—
430C2×Q8—42no—
431C2×Q8—42no—
432C2×Q8—42no—
433C2×Q8—42no—
434C2×Q8—42no—
435C2×Q8—42no—
436C2×Q8C2×C22⋊D6112yes—
437C2×D4—22no—
438C2×D4—22no—
439C2×D4—21no—
440C2×D4—22no—
441C2×D4—22no—
442C2×D4—42no—
443C2×D4—42no—
444C2×D4C2×C22⋊D6112yes—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—21no—
448C22×C4—22no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4—1222no—
456C22×C4—1222no—
457C22×C4D4×D6112yes—
458C22×C4#1175822412yes—
459C22×C4#1175822412yes—
460C22×C4#1175822412yes—
461C22×C4C2×C22⋊D6114yes—
462C22×C4C2×C22⋊D6111yescharacteristic
463C22×C4C2×C22⋊D6111yescharacteristic
464C22×C4C2×C22⋊D6111yescharacteristic
465C22×C4C2×C22⋊D6111yescharacteristic
466C22×C4C2×C22⋊D6111yescharacteristic
467C22×C4C23×D6111yescharacteristic
468M16—614no—
469M16—1222no—
470M16—1224no—
471M16—1222no—
472M16—1222no—
473M16—2442no—
474C2×C8—1222no—
475C2×C8—1224no—
476C2×C8—1222no—
477C2×C8—2442no—
478C4⋊C4—216no—
479C4⋊C4—22no—
480C4⋊C4—1222no—
481C4⋊C4—1222no—
482C4⋊C4—2442no—
483C4⋊C4C2×C22⋊D6114yes—
484C22⋊C4—216no—
485C22⋊C4—22no—
486C22⋊C4—22no—
487C22⋊C4—2441no—
488C22⋊C4—2441no—
489C22⋊C4—2442no—
490C42—22no—
491C42—22no—
492C42—1222no—
493C42—1222no—
494C42—2441no—
495C42—2441no—
496C23—22no—
497C23—42no—
498C23—44no—
499C23—42no—
500C23#1451974611yescharacteristic
501C23#1451974611yescharacteristic
502C23#1451974912yes—
503C23#1451975812yes—
504C23C22×C22⋊D6111yescharacteristic
505Q8—22no—
506Q8—22no—
507Q8—42no—
508Q8—42no—
509Q8—42no—
510Q8—42no—
511Q8—44no—
512Q8—44no—
513Q8—42no—
514D4—22no—
515D4—22no—
516D4—42no—
517D4—42no—
518D4—44no—
519D4—42no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—22no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—44no—
526C2×C4—44no—
527C2×C4—44no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—44no—
532C2×C4—2444no—
533C2×C4—2444no—
534C2×C4—2442no—
535C2×C4—2442no—
536C2×C4#1451974611yescharacteristic
537C2×C4#1451974611yescharacteristic
538C2×C4#1451974812yes—
539C2×C4#1451974812yes—
540C2×C4#1451974912yes—
541C2×C4#1451974912yes—
542C2×C4#1451974912yes—
543C2×C4#1451975812yes—
544C2×C4#1451975812yes—
545C2×C4#1451975812yes—
546C2×C4#1451976412yes—
547C2×C4#1451976412yes—
548C2×C4C2×D4×D6111yescharacteristic
549C2×C4C22×C22⋊D6111yescharacteristic
550C2×C4C22×C22⋊D6111yescharacteristic
551C2×C4C2×#1175822411yescharacteristic
552C2×C4C2×#1175822411yescharacteristic
553C2×C4C2×#1175822411yescharacteristic
554C8—1224no—
555C8—2442no—
556C22—22no—
557C22—22no—
558C22—22no—
559C22—42no—
560C22—42no—
561C22—42no—
562C22—44no—
563C22—84no—
564C22#1683967612yes—
565C22#1683967612yes—
566C22C2×#1451974611yescharacteristic
567C22C2×#1451974911yescharacteristic
568C22C2×#1451975811yescharacteristic
569C4—44no—
570C4—42no—
571C4—44no—
572C4—4882no—
573C4—4882no—
574C4#1683967612yes—
575C4#1683967612yes—
576C4C2×#1451974811yescharacteristic
577C4C2×#1451974911yescharacteristic
578C4C2×#1451975811yescharacteristic
579C4C2×#1451976411yescharacteristic
580C2—21no—
581C2—22no—
582C2—21no—
583C2—22no—
584C2—82no—
585C2C2×#1683967611yescharacteristic
586C2#2145518312yes—
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#536C2
socleC22×C61C2×#120
prime radicalC22×#41×C61C22
hypercenterC2×#536D61
hypocenterC61#23494
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#536

Complements

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

NH ≅ G/Ntype
1. #21454408C2semidirect
4. C22×#14519696C2semidirect
5. #21448405C2semidirect
6. #21447033C2semidirect
7. #21445528C2semidirect
13. #16839805C22semidirect
23. C2×#14519696C22semidirect
25. #16839669C22semidirect
27. #16839521C22semidirect
332. C61#23494semidirect

Maximal subgroups

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

subgroupindex
#214544082
C2×#168397792
#214543382
C22×#145196962
#214484052
#214470332
#214455282
C2×C61×#5362
#2349461