Finite Group Databasebeta

← #29707630

Subgroups — #29707630

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
1C2×#16840333C211yesmaximal, characteristic
2#21454409C211yesmaximal, characteristic, complemented
3#21454338C211yesmaximal, characteristic
4C22×#14519695C211yesmaximal, characteristic, complemented
5#21448403C211yesmaximal, characteristic, complemented
6#21447035C218yesmaximal, complemented
7#21445529C211yesmaximal, characteristic, complemented
8C2×C61×#421C211yesmaximal, characteristic
9C23×Dic122C2211yescharacteristic
10C22×#11758194C2211yescharacteristic
11#16840333—22no—
12#16840333—22no—
13#16840333—22no—
14#16840333C2214yes—
15C2×#14519775—21no—
16C2×#14519764—21no—
17C2×#14519758—22no—
18C2×#14519756—22no—
19C2×#14519736C2211yescharacteristic
20C2×#14519736C2211yescharacteristic
21#16839804—22no—
22#16839804C2214yescomplemented
23#16839800—22no—
24#16839771—21no—
25#16839767—21no—
26#16839763C2214yes—
27#16839758—22no—
28C2×#14519695—24no—
29C2×#14519695C2214yescomplemented
30#16839668C2214yescomplemented
31#16839667—22no—
32#16839623—22no—
33#16839522C2214yescomplemented
34C22×#41×C61C2211yescharacteristic
35C22×M16×C61C2211yescharacteristic
36C2×#117×C61C2211yescharacteristic
37C61×#421—24no—
38C2×#104×C61C2214yes—
39C22×Dic122—22no—
40C22×Dic122—21no—
41C22×Dic122—21no—
42C22×Dic122—21no—
43C22×Dic122—21no—
44C22×Dic122—22no—
45C22×Dic122—22no—
46C22×Dic122—22no—
47C22×Dic122D412yes—
48C22×Dic122C2312yes—
49C23×Dic61D412yes—
50C2×#11758194—21no—
51C2×#11758194—22no—
52C2×#11758194—22no—
53C2×#11758194C2312yes—
54C22×#92418—21no—
55#14519775—24no—
56#14519775—24no—
57#14519764—24no—
58#14519764—24no—
59#14519758—28no—
60#14519758—28no—
61#14519756—28no—
62#14519756—28no—
63#14519736—21no—
64#14519736—21no—
65#14519736—21no—
66#14519736—21no—
67#14519736—21no—
68#14519736—21no—
69#14519736D412yes—
70#14519736C2312yes—
71C2×C4⋊Dic61D412yes—
72C2×C4⋊Dic61C2312yes—
73C2×#11758196—24no—
74C2×C22⋊Dic61—24no—
75C2×C22⋊Dic61—21no—
76C2×C22⋊Dic61—21no—
77C2×C4×Dic61—21no—
78C2×C4×Dic61—21no—
79#14519711—24no—
80#14519709—24no—
81#14519701—24no—
82#14519698—24no—
83#14519695—28no—
84#14519684—28no—
85C2×#41×C61—21no—
86C2×#41×C61—21no—
87C2×#41×C61—22no—
88C2×#41×C61—22no—
89C2×#41×C61—22no—
90C2×#41×C61—22no—
91C2×#41×C61D412yes—
92C2×#41×C61C2312yes—
93C22×Q8×C61D411yescharacteristic
94C22×D4×C61—21no—
95C22×D4×C61D411yescharacteristic
96C23×C4×C61—21no—
97C23×C4×C61C2311yescharacteristic
98C2×M16×C61—21no—
99C2×M16×C61—22no—
100C2×M16×C61D414yes—
101C2×M16×C61C2312yes—
102C22×C8×C61—21no—
103#117×C61—22no—
104#117×C61—22no—
105C2×C4⋊C4×C61—21no—
106C2×C22⋊C4×C61—21no—
107C2×C42×C61C2312yes—
108#104×C61—28no—
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—21no—
122C2×Q8×C61—22no—
123C2×Q8×C61—22no—
124C2×Q8×C61C2×D412yes—
125C2×D4×C61—22no—
126C2×D4×C61—22no—
127C2×D4×C61—21no—
128C2×D4×C61—22no—
129C2×D4×C61—22no—
130C2×D4×C61—42no—
131C2×D4×C61—42no—
132C2×D4×C61C2×D412yes—
133C22×C4×C61—21no—
134C22×C4×C61—22no—
135C22×C4×C61—22no—
136C22×C4×C61—22no—
137C22×C4×C61—42no—
138C22×C4×C61—42no—
139C22×C4×C61C2×D411yescharacteristic
140C22×C4×C61C2×D411yescharacteristic
141C22×C4×C61C2×D411yescharacteristic
142C22×C4×C61C2×D411yescharacteristic
143C22×C4×C61C2×D411yescharacteristic
144C22×C4×C61C2×D412yes—
145C22×C4×C61#4112yes—
146C22×C4×C61#4112yes—
147C22×C4×C61#4112yes—
148C22×C4×C61C2411yescharacteristic
149C22×Dic61—22no—
150C22×Dic61—22no—
151C22×Dic61—22no—
152C22×Dic61—24no—
153C22×Dic61—24no—
154C22×Dic61—22no—
155C22×Dic61C2×D414yes—
156C2×Dic122—22no—
157C2×Dic122—22no—
158C2×Dic122—24no—
159C2×Dic122—24no—
160C2×Dic122—22no—
161C2×Dic122—22no—
162C2×Dic122—22no—
163C2×Dic122—22no—
164C2×Dic122—22no—
165C2×Dic122—22no—
166C2×Dic122—42no—
167C2×Dic122—42no—
168C2×Dic122—42no—
169C2×Dic122—42no—
170C2×Dic122—42no—
171C2×Dic122—42no—
172C2×Dic122—42no—
173C2×Dic122—42no—
174M16×C61—24no—
175M16×C61—22no—
176M16×C61C2×D414yes—
177C2×C8×C61—24no—
178C2×C8×C61—22no—
179C4⋊C4×C61—42no—
180C22⋊C4×C61—42no—
181C42×C61—22no—
182C42×C61—22no—
183C22⋊Dic61—216no—
184C22⋊Dic61—22no—
185C22⋊Dic61—22no—
186C22⋊Dic61—41no—
187C22⋊Dic61—41no—
188C4⋊Dic61—22no—
189C4⋊Dic61—22no—
190C4⋊Dic61—22no—
191C4⋊Dic61C2×D414yes—
192#11758196—216no—
193C4×Dic61—22no—
194C4×Dic61—22no—
195C4×Dic61—41no—
196C4×Dic61—41no—
197#11758194—22no—
198#11758194—22no—
199#11758194—42no—
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—
214D4×C61—22no—
215D4×C61—22no—
216D4×C61—42no—
217D4×C61—42no—
218D4×C61—44no—
219D4×C61—42no—
220C2×C4×C61—22no—
221C2×C4×C61—42no—
222C2×C4×C61—42no—
223C2×C4×C61—42no—
224C2×C4×C61—44no—
225C2×C4×C61—44no—
226C2×C4×C61—42no—
227C2×C4×C61#12011yescharacteristic
228C2×C4×C61#12011yescharacteristic
229C2×C4×C61#12112yes—
230C2×C4×C61#12112yes—
231C2×C4×C61#12112yes—
232C2×C4×C61#12112yes—
233C2×C4×C61#12112yes—
234C2×C4×C61#12312yes—
235C2×C4×C61#12312yes—
236C2×C4×C61#12312yes—
237C2×C4×C61#12412yes—
238C2×C4×C61#12412yes—
239C2×C4×C61C22×D411yescharacteristic
240C2×C4×C61C22×D411yescharacteristic
241C2×C4×C61C22×D411yescharacteristic
242C2×C4×C61C2×#4111yescharacteristic
243C2×C4×C61C2×#4111yescharacteristic
244C2×C4×C61C2×#4111yescharacteristic
245C2×Dic61—24no—
246C2×Dic61—24no—
247C2×Dic61—24no—
248C2×Dic61—24no—
249C2×Dic61—44no—
250C2×Dic61—44no—
251C2×Dic61—44no—
252C2×Dic61—44no—
253C2×Dic61—42no—
254Dic122—42no—
255Dic122—42no—
256Dic122—42no—
257Dic122—42no—
258Dic122—44no—
259Dic122—44no—
260C8×C61—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—42no—
277C4×C61—82no—
278C4×C61#39412yes—
279C4×C61#39412yes—
280C4×C61C2×#12111yescharacteristic
281C4×C61C2×#12111yescharacteristic
282C4×C61C2×#12311yescharacteristic
283C4×C61C2×#12411yescharacteristic
284Dic61—44no—
285Dic61—44no—
286Dic61—82no—
287C2×#536—611no—
288#2911—611no—
289C2×#421D6111yescharacteristic
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×Q8—611no—
304C22×M16—611no—
305C22×M16C2×D6111yescharacteristic
306#536—614no—
307#536—1222no—
308#536—1222no—
309#536—1222no—
310C2×C4⋊Q8—1221no—
311C2×#124—1221no—
312C2×#123—1222no—
313C2×C22⋊Q8—1222no—
314C2×#117—611no—
315C2×#117—611no—
316C2×#117C2×D6111yescharacteristic
317#428—614no—
318#428—1222no—
319#427—1222no—
320#421—24no—
321C2×#104C2×D6114yes—
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×#41D24412yes—
340C2×#41C22×D6112yes—
341C22×Q8—612no—
342C22×Q8—612no—
343C22×Q8—1221no—
344C22×Q8—1221no—
345C22×Q8—1222no—
346C22×Q8—1221no—
347C22×Q8—1221no—
348C22×Q8—1222no—
349C22×Q8—1222no—
350C22×Q8—1222no—
351C22×Q8D24411yescharacteristic
352C22×D4—21no—
353C22×D4D24411yescharacteristic
354C23×C4—21no—
355C23×C4—612no—
356C23×C4C22×D6111yescharacteristic
357C2×M16—21no—
358C2×M16—22no—
359C2×M16—612no—
360C2×M16—1221no—
361C2×M16—1222no—
362C2×M16—1222no—
363C2×M16C22⋊D6114yes—
364C2×M16C22×D6112yes—
365C22×C8—21no—
366C22×C8—1221no—
367C4⋊Q8—1224no—
368C4⋊Q8—1224no—
369#124—1224no—
370#124—1224no—
371#123—1228no—
372#123—1228no—
373C22⋊Q8—1228no—
374C22⋊Q8—1228no—
375#117—22no—
376#117—22no—
377#117—612no—
378#117—612no—
379#117—1221no—
380#117—1221no—
381#117—1221no—
382#117—1221no—
383#117—1221no—
384#117—1221no—
385C2×C4⋊C4—21no—
386C2×C4⋊C4—612no—
387C2×C4⋊C4—612no—
388C2×C4⋊C4—1224no—
389C2×C22⋊C4—21no—
390C2×C22⋊C4—1224no—
391C2×C22⋊C4—1221no—
392C2×C22⋊C4—1221no—
393C2×C42—1221no—
394C2×C42—1221no—
395C2×C42C22×D6112yes—
396#107—1224no—
397#106—1224no—
398#104—28no—
399#103—1224no—
400#102—1224no—
401#101—1228no—
402#95—1228no—
403C24—21no—
404C24D4×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—21no—
416C2×Q8—22no—
417C2×Q8—22no—
418C2×Q8—1222no—
419C2×Q8—1222no—
420C2×Q8—1222no—
421C2×Q8—1222no—
422C2×Q8—1222no—
423C2×Q8—1222no—
424C2×Q8—1222no—
425C2×Q8—1222no—
426C2×Q8—1224no—
427C2×Q8—1224no—
428C2×Q8—2442no—
429C2×Q8—2442no—
430C2×Q8—2442no—
431C2×Q8—2442no—
432C2×Q8—2442no—
433C2×Q8—2442no—
434C2×Q8—2442no—
435C2×Q8—2442no—
436C2×Q8C2×D24412yes—
437C2×D4—22no—
438C2×D4—22no—
439C2×D4—21no—
440C2×D4—22no—
441C2×D4—22no—
442C2×D4—42no—
443C2×D4—42no—
444C2×D4C2×D24412yes—
445C22×C4—21no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—614no—
452C22×C4—1222no—
453C22×C4—1222no—
454C22×C4—1222no—
455C22×C4—1224no—
456C22×C4—1224no—
457C22×C4—1222no—
458C22×C4C2×D24411yescharacteristic
459C22×C4C2×D24411yescharacteristic
460C22×C4#1175822212yes—
461C22×C4D4×D6111yescharacteristic
462C22×C4D4×D6111yescharacteristic
463C22×C4D4×D6111yescharacteristic
464C22×C4#1175822412yes—
465C22×C4#1175822412yes—
466C22×C4C2×C22⋊D6112yes—
467C22×C4C23×D6111yescharacteristic
468M16—22no—
469M16—24no—
470M16—1222no—
471M16—1222no—
472M16—2442no—
473M16C2×C22⋊D6114yes—
474C2×C8—22no—
475C2×C8—24no—
476C2×C8—1222no—
477C2×C8—2442no—
478C4⋊C4—42no—
479C4⋊C4—614no—
480C4⋊C4—12216no—
481C4⋊C4—1222no—
482C4⋊C4—1222no—
483C4⋊C4—1222no—
484C22⋊C4—42no—
485C22⋊C4—12216no—
486C22⋊C4—1222no—
487C22⋊C4—1222no—
488C22⋊C4—2441no—
489C22⋊C4—2441no—
490C42—22no—
491C42—22no—
492C42—1222no—
493C42—1222no—
494C42—2441no—
495C42—2441no—
496C23—22no—
497C23—42no—
498C23—44no—
499C23—42no—
500C23#1451974711yescharacteristic
501C23#1451974711yescharacteristic
502C23#1451974812yes—
503C23#1451975912yes—
504C23C2×D4×D6111yescharacteristic
505Q8—22no—
506Q8—22no—
507Q8—42no—
508Q8—2442no—
509Q8—2442no—
510Q8—2442no—
511Q8—2442no—
512Q8—2444no—
513Q8—2444no—
514D4—22no—
515D4—22no—
516D4—42no—
517D4—42no—
518D4—44no—
519D4—42no—
520C2×C4—22no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—44no—
525C2×C4—44no—
526C2×C4—42no—
527C2×C4—1224no—
528C2×C4—1224no—
529C2×C4—1224no—
530C2×C4—1224no—
531C2×C4—2444no—
532C2×C4—2442no—
533C2×C4—2444no—
534C2×C4—2444no—
535C2×C4—2444no—
536C2×C4#1451974711yescharacteristic
537C2×C4#1451974711yescharacteristic
538C2×C4#1451974812yes—
539C2×C4#1451974912yes—
540C2×C4#1451974912yes—
541C2×C4#1451975112yes—
542C2×C4#1451975112yes—
543C2×C4#1451975912yes—
544C2×C4#1451976112yes—
545C2×C4#1451976112yes—
546C2×C4#1451976312yes—
547C2×C4#1451976312yes—
548C2×C4C2×D4×D6111yescharacteristic
549C2×C4C22×C22⋊D6111yescharacteristic
550C2×C4C22×D24411yescharacteristic
551C2×C4C2×#1175822411yescharacteristic
552C2×C4C2×#1175822411yescharacteristic
553C2×C4C2×#1175822211yescharacteristic
554C8—24no—
555C8—2442no—
556C22—22no—
557C22—22no—
558C22—22no—
559C22—42no—
560C22—42no—
561C22—42no—
562C22—44no—
563C22—84no—
564C22#1683967812yes—
565C22#1683967812yes—
566C22C2×#1451974711yescharacteristic
567C22C2×#1451974811yescharacteristic
568C22C2×#1451975911yescharacteristic
569C4—42no—
570C4—82no—
571C4—4882no—
572C4—2444no—
573C4—2444no—
574C4#1683967812yes—
575C4#1683967812yes—
576C4C2×#1451974911yescharacteristic
577C4C2×#1451975111yescharacteristic
578C4C2×#1451976111yescharacteristic
579C4C2×#1451976311yescharacteristic
580C2—21no—
581C2—22no—
582C2—21no—
583C2—22no—
584C2—82no—
585C2C2×#1683967811yescharacteristic
586C2#2145518512yes—
subgroupG/H
centerC22C2×#14519747
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#421C2
socleC22×C61C2×#120
prime radicalC22×#41×C61C22
hypercenterC2×#421D61
hypocenterC61#23494
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#421

Complements

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

NH ≅ G/Ntype
2. #21454409C2semidirect
4. C22×#14519695C2semidirect
5. #21448403C2semidirect
6. #21447035C2semidirect
7. #21445529C2semidirect
22. #16839804C22semidirect
29. C2×#14519695C22semidirect
30. #16839668C22semidirect
33. #16839522C22semidirect
332. C61#23494semidirect

Maximal subgroups

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

subgroupindex
C2×#168403332
#214544092
#214543382
C22×#145196952
#214484032
#214470352
#214455292
C2×C61×#4212
#2349461