Finite Group Databasebeta

← #29707530

Subgroups — #29707530

Order 15616.

1610 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×#16839805C211yesmaximal, characteristic, complemented
2C2×#16839767C211yesmaximal, characteristic, complemented
3C22×#14519698C211yesmaximal, characteristic, complemented
4C2×#16839729C211yesmaximal, characteristic, complemented
5#21448411C211yesmaximal, characteristic, complemented
6#21447005C214yesmaximal, complemented
7#21446998C214yesmaximal, complemented
8#21445535C211yesmaximal, characteristic, complemented
9C2×C61×#535C211yesmaximal, characteristic
10C22×#11758194C2211yescharacteristic
11C23×#92418C2211yescharacteristic
12C2×#14519736C2211yescharacteristic
13C22×C4⋊Dic61C2211yescharacteristic
14#16839805—24no—
15C2×#14519710C2212yescomplemented
16C2×#14519708C2212yescomplemented
17#16839767—24no—
18C2×#14519698—24no—
19C2×#14519698C2214yes—
20C2×#14519698C2214yes—
21#16839729—24no—
22C2×C22⋊#92418C2214yescomplemented
23#16839669—21no—
24#16839665—21no—
25#16839642—22no—
26#16839622C2214yescomplemented
27#16839524C2212yescomplemented
28#16839524C2212yescomplemented
29C22×#41×C61C2211yescharacteristic
30C23×D4×C61C2211yescharacteristic
31C61×#535—24no—
32C61×#535—24no—
33C2×#127×C61—21no—
34C2×#124×C61—21no—
35C2×#121×C61C2212yes—
36C2×#121×C61C2212yes—
37C2×#120×C61—22no—
38C2×#117×C61C2211yescharacteristic
39C23×Dic61—21no—
40C2×#11758194—22no—
41C2×#11758194—22no—
42C2×#11758194D414yescomplemented
43C22×#92418—22no—
44C22×#92418—22no—
45C22×#92418—22no—
46C22×#92418C2312yes—
47C22×#92418C2312yes—
48#14519736—22no—
49#14519736—22no—
50C2×C4⋊Dic61—22no—
51C2×C4⋊Dic61—22no—
52C2×C4⋊Dic61—21no—
53C2×C4⋊Dic61C2312yes—
54C2×C4⋊Dic61C2312yes—
55C2×C22⋊Dic61—21no—
56C2×C22⋊Dic61—24no—
57C2×C4×Dic61—21no—
58#14519710—24no—
59#14519708—24no—
60#14519698—28no—
61#14519698—28no—
62C22⋊#92418—28no—
63#14519684—24no—
64C25×C61—21no—
65C2×#41×C61—22no—
66C2×#41×C61—22no—
67C2×#41×C61—22no—
68C2×#41×C61—22no—
69C2×#41×C61—22no—
70C2×#41×C61—22no—
71C22×Q8×C61D411yescharacteristic
72C22×D4×C61—22no—
73C22×D4×C61—22no—
74C22×D4×C61—22no—
75C22×D4×C61—21no—
76C22×D4×C61—22no—
77C22×D4×C61—22no—
78C22×D4×C61—22no—
79C22×D4×C61—22no—
80C22×D4×C61D411yescharacteristic
81C22×D4×C61D412yes—
82C22×D4×C61D414yes—
83C22×D4×C61C2312yes—
84C23×C4×C61C2312yes—
85C23×C4×C61C2311yescharacteristic
86#127×C61—28no—
87#124×C61—28no—
88#121×C61—24no—
89#121×C61—24no—
90#121×C61—24no—
91#121×C61—24no—
92#120×C61—216no—
93#117×C61—22no—
94#117×C61D414yes—
95C2×C4⋊C4×C61C2311yescharacteristic
96C2×C4⋊C4×C61C2311yescharacteristic
97C2×C22⋊C4×C61—21no—
98C2×C22⋊C4×C61—22no—
99C2×C22⋊C4×C61—22no—
100C2×C42×C61—21no—
101C24×C61—22no—
102C24×C61—24no—
103C24×C61—42no—
104C24×C61—42no—
105C24×C61—82no—
106C24×C61—82no—
107C24×C61C2×D412yes—
108C24×C61C2×D411yescharacteristic
109#41×C61—44no—
110#41×C61—44no—
111#41×C61—44no—
112#41×C61—44no—
113C2×Q8×C61—22no—
114C2×Q8×C61—22no—
115C2×Q8×C61—22no—
116C2×D4×C61—24no—
117C2×D4×C61—24no—
118C2×D4×C61—24no—
119C2×D4×C61—28no—
120C2×D4×C61—28no—
121C2×D4×C61—24no—
122C2×D4×C61—22no—
123C2×D4×C61—28no—
124C2×D4×C61—22no—
125C2×D4×C61—28no—
126C2×D4×C61—22no—
127C2×D4×C61—22no—
128C2×D4×C61—22no—
129C2×D4×C61—22no—
130C2×D4×C61—22no—
131C2×D4×C61—22no—
132C2×D4×C61—44no—
133C2×D4×C61—42no—
134C2×D4×C61—42no—
135C2×D4×C61—44no—
136C2×D4×C61—84no—
137C2×D4×C61—84no—
138C2×D4×C61C2×D414yes—
139C22×C4×C61—21no—
140C22×C4×C61—21no—
141C22×C4×C61—22no—
142C22×C4×C61—22no—
143C22×C4×C61—22no—
144C22×C4×C61—22no—
145C22×C4×C61—22no—
146C22×C4×C61—22no—
147C22×C4×C61C2×D412yes—
148C22×C4×C61C2×D411yescharacteristic
149C22×C4×C61C2×D412yes—
150C22×C4×C61C2×D412yes—
151C22×C4×C61C2×D412yes—
152C22×C4×C61#4112yes—
153C22×C4×C61#4112yes—
154C22×C4×C61#4112yes—
155C22×C4×C61C2411yescharacteristic
156C22×Dic61—22no—
157C22×Dic61—22no—
158C22×Dic61—42no—
159C22×Dic61—82no—
160C4⋊C4×C61—21no—
161C4⋊C4×C61—21no—
162C4⋊C4×C61C2×D412yes—
163C4⋊C4×C61C2×D412yes—
164C22⋊C4×C61—28no—
165C22⋊C4×C61—28no—
166C22⋊C4×C61—24no—
167C42×C61—24no—
168C22⋊Dic61—42no—
169C22⋊Dic61—84no—
170C4⋊Dic61—22no—
171C4⋊Dic61—22no—
172C4⋊Dic61—22no—
173C4⋊Dic61—22no—
174C4⋊Dic61—42no—
175C4×Dic61—42no—
176#11758194—28no—
177C2×#92418—22no—
178C2×#92418—22no—
179C2×#92418—22no—
180C2×#92418—22no—
181C2×#92418—22no—
182C2×#92418—44no—
183C2×#92418C2×D414yes—
184C23×C61—22no—
185C23×C61—22no—
186C23×C61—24no—
187C23×C61—22no—
188C23×C61—22no—
189C23×C61—22no—
190C23×C61—24no—
191C23×C61—24no—
192C23×C61—42no—
193C23×C61—42no—
194C23×C61—44no—
195C23×C61—44no—
196C23×C61—48no—
197C23×C61—84no—
198C23×C61—84no—
199C23×C61—88no—
200C23×C61—88no—
201C23×C61#12012yes—
202C23×C61#12112yes—
203C23×C61#12312yes—
204C23×C61C22×D411yescharacteristic
205Q8×C61—44no—
206D4×C61—24no—
207D4×C61—24no—
208D4×C61—48no—
209D4×C61—42no—
210D4×C61—42no—
211D4×C61—48no—
212D4×C61—44no—
213D4×C61—88no—
214D4×C61—88no—
215C2×C4×C61—22no—
216C2×C4×C61—22no—
217C2×C4×C61—22no—
218C2×C4×C61—22no—
219C2×C4×C61—22no—
220C2×C4×C61—42no—
221C2×C4×C61—42no—
222C2×C4×C61—42no—
223C2×C4×C61—42no—
224C2×C4×C61—42no—
225C2×C4×C61—42no—
226C2×C4×C61#12012yes—
227C2×C4×C61#12114yes—
228C2×C4×C61#12114yes—
229C2×C4×C61#12112yes—
230C2×C4×C61#12314yes—
231C2×C4×C61#12312yes—
232C2×C4×C61#12414yes—
233C2×C4×C61C22×D411yescharacteristic
234C2×C4×C61C22×D411yescharacteristic
235C2×C4×C61C22×D411yescharacteristic
236C2×C4×C61C2×#4111yescharacteristic
237C2×C4×C61C2×#4111yescharacteristic
238C2×C4×C61C2×#4111yescharacteristic
239C2×Dic61—42no—
240C2×Dic61—42no—
241C2×Dic61—42no—
242C2×Dic61—44no—
243C2×Dic61—42no—
244C2×Dic61—84no—
245#92418—24no—
246#92418—42no—
247#23483—611nomaximal, Sylow
248C22×C61—24no—
249C22×C61—22no—
250C22×C61—22no—
251C22×C61—24no—
252C22×C61—44no—
253C22×C61—42no—
254C22×C61—44no—
255C22×C61—42no—
256C22×C61—44no—
257C22×C61—42no—
258C22×C61—44no—
259C22×C61—44no—
260C22×C61—816no—
261C22×C61—84no—
262C22×C61—84no—
263C22×C61#39414yes—
264C22×C61C2×#12011yescharacteristic
265C22×C61C2×#12111yescharacteristic
266C22×C61C2×#12311yescharacteristic
267C22×C61#57612yes—
268C22×C61#57712yes—
269C4×C61—42no—
270C4×C61—42no—
271C4×C61—42no—
272C4×C61#39414yes—
273C4×C61C2×#12111yescharacteristic
274C4×C61C2×#12111yescharacteristic
275C4×C61C2×#12311yescharacteristic
276C4×C61C2×#12411yescharacteristic
277Dic61—82no—
278Dic61—82no—
279C2×#535D6111yescharacteristic
280C2×#428—611no—
281C2×#418—611no—
282C22×#102—611no—
283C2×#408—611no—
284#2311—611no—
285#2034—614no—
286#2033—614no—
287#1735—611no—
288C2×C61—24no—
289C2×C61—82no—
290C2×C61—84no—
291C2×C61C2×#39411yescharacteristic
292C2×C61#300811yescharacteristic
293C2×C61#300911yescharacteristic
294C2×C61#306212yes—
295C2×C61#309612yes—
296C22×#41C2×D6111yescharacteristic
297C23×D4C2×D6111yescharacteristic
298C22×M16—611no—
299C23×C8—611no—
300#535—24no—
301#535—24no—
302C2×#127—21no—
303C2×#124—21no—
304C2×#121C2×D6112yes—
305C2×#121C2×D6112yes—
306C2×#120—22no—
307C2×#117—611no—
308C2×#117C2×D6111yescharacteristic
309C22×C4⋊C4—611no—
310#428—1224no—
311C2×#107—612no—
312C2×#106—612no—
313#418—1224no—
314C2×#102—614no—
315C2×#102—614no—
316C2×#102—1224no—
317#408—1224no—
318C2×C22⋊C8—614no—
319#391—1221no—
320#390—1221no—
321#386—1222no—
322#380—614no—
323#340—612no—
324#340—612no—
325C61#2348311yescharacteristic, complemented, Sylow
326C25—21no—
327C2×#41—22no—
328C2×#41—22no—
329C2×#41—22no—
330C2×#41—22no—
331C2×#41—22no—
332C2×#41—22no—
333C22×Q8C22⋊D6111yescharacteristic
334C22×D4—22no—
335C22×D4—22no—
336C22×D4—22no—
337C22×D4—22no—
338C22×D4—22no—
339C22×D4—22no—
340C22×D4—21no—
341C22×D4—22no—
342C22×D4C22⋊D6112yes—
343C22×D4C22⋊D6111yescharacteristic
344C22×D4C22⋊D6114yes—
345C22×D4C22×D6112yes—
346C23×C4—1221no—
347C23×C4C22×D6112yes—
348C23×C4C22×D6111yescharacteristic
349C2×M16—614no—
350C2×M16—1222no—
351C2×M16—1222no—
352C22×C8—612no—
353C22×C8—612no—
354C22×C8—1222no—
355C22×C8—1222no—
356C22×C8—1222no—
357#127—28no—
358#124—28no—
359#121—24no—
360#121—24no—
361#121—24no—
362#121—24no—
363#120—216no—
364#117—22no—
365#117—1222no—
366#117—1222no—
367#117C22⋊D6114yes—
368C2×C4⋊C4—612no—
369C2×C4⋊C4—612no—
370C2×C4⋊C4—1222no—
371C2×C4⋊C4—1222no—
372C2×C4⋊C4—1221no—
373C2×C4⋊C4C22×D6111yescharacteristic
374C2×C4⋊C4C22×D6111yescharacteristic
375C2×C22⋊C4—21no—
376C2×C22⋊C4—22no—
377C2×C22⋊C4—22no—
378C2×C22⋊C4—1224no—
379C2×C22⋊C4—1221no—
380C2×C42—21no—
381C2×C42—1221no—
382#107—1224no—
383#106—1224no—
384#102—1228no—
385#102—1228no—
386C22⋊C8—1228no—
387#95—1224no—
388C24—22no—
389C24—24no—
390C24—42no—
391C24—42no—
392C24—82no—
393C24—82no—
394C24C2×C22⋊D6112yes—
395C24C2×C22⋊D6111yescharacteristic
396#41—44no—
397#41—44no—
398#41—44no—
399#41—44no—
400C2×Q8—22no—
401C2×Q8—22no—
402C2×Q8—22no—
403C2×D4—24no—
404C2×D4—24no—
405C2×D4—24no—
406C2×D4—28no—
407C2×D4—28no—
408C2×D4—22no—
409C2×D4—22no—
410C2×D4—22no—
411C2×D4—22no—
412C2×D4—22no—
413C2×D4—24no—
414C2×D4—22no—
415C2×D4—28no—
416C2×D4—22no—
417C2×D4—28no—
418C2×D4—22no—
419C2×D4—44no—
420C2×D4—42no—
421C2×D4—42no—
422C2×D4—44no—
423C2×D4—84no—
424C2×D4—84no—
425C2×D4C2×C22⋊D6114yes—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—1222no—
435C22×C4—1222no—
436C22×C4—4882no—
437C22×C4—2442no—
438C22×C4D4×D6112yes—
439C22×C4#1175822412yes—
440C22×C4#1175822412yes—
441C22×C4#1175822412yes—
442C22×C4C2×C22⋊D6112yes—
443C22×C4C2×C22⋊D6112yes—
444C22×C4C2×C22⋊D6112yes—
445C22×C4C2×C22⋊D6111yescharacteristic
446C22×C4C23×D6111yescharacteristic
447M16—1228no—
448C2×C8—614no—
449C2×C8—1222no—
450C2×C8—1222no—
451C2×C8—1222no—
452C2×C8—1222no—
453C2×C8—1222no—
454C2×C8—2444no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—1222no—
458C4⋊C4—1222no—
459C4⋊C4—1222no—
460C4⋊C4—1222no—
461C4⋊C4—2442no—
462C4⋊C4C2×C22⋊D6112yes—
463C4⋊C4C2×C22⋊D6112yes—
464C22⋊C4—28no—
465C22⋊C4—28no—
466C22⋊C4—24no—
467C22⋊C4—4884no—
468C22⋊C4—2442no—
469C42—24no—
470C42—2442no—
471C23—22no—
472C23—22no—
473C23—24no—
474C23—22no—
475C23—22no—
476C23—22no—
477C23—24no—
478C23—24no—
479C23—42no—
480C23—42no—
481C23—44no—
482C23—44no—
483C23—48no—
484C23—84no—
485C23—84no—
486C23—88no—
487C23—88no—
488C23#1451974612yes—
489C23#1451974912yes—
490C23#1451975812yes—
491C23C22×C22⋊D6111yescharacteristic
492Q8—44no—
493D4—24no—
494D4—24no—
495D4—48no—
496D4—42no—
497D4—42no—
498D4—48no—
499D4—44no—
500D4—88no—
501D4—88no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—4884no—
514C2×C4—2442no—
515C2×C4—2442no—
516C2×C4—2442no—
517C2×C4—2444no—
518C2×C4—2442no—
519C2×C4#1451974612yes—
520C2×C4#1451974814yes—
521C2×C4#1451974914yes—
522C2×C4#1451974912yes—
523C2×C4#1451975814yes—
524C2×C4#1451975812yes—
525C2×C4#1451976414yes—
526C2×C4C2×D4×D6111yescharacteristic
527C2×C4C22×C22⋊D6111yescharacteristic
528C2×C4C22×C22⋊D6111yescharacteristic
529C2×C4C2×#1175822411yescharacteristic
530C2×C4C2×#1175822411yescharacteristic
531C2×C4C2×#1175822411yescharacteristic
532C8—1224no—
533C8—2442no—
534C22—24no—
535C22—24no—
536C22—22no—
537C22—22no—
538C22—42no—
539C22—44no—
540C22—44no—
541C22—44no—
542C22—42no—
543C22—44no—
544C22—42no—
545C22—44no—
546C22—816no—
547C22—84no—
548C22—84no—
549C22#1683967614yes—
550C22C2×#1451974611yescharacteristic
551C22C2×#1451974911yescharacteristic
552C22C2×#1451975811yescharacteristic
553C22#1684056512yes—
554C22#1684057312yes—
555C4—42no—
556C4—42no—
557C4—42no—
558C4—4882no—
559C4—4882no—
560C4#1683967614yes—
561C4C2×#1451974811yescharacteristic
562C4C2×#1451974911yescharacteristic
563C4C2×#1451975811yescharacteristic
564C4C2×#1451976411yescharacteristic
565C2—24no—
566C2—82no—
567C2—84no—
568C2C2×#1683967611yescharacteristic
569C2#2145508711yescharacteristic
570C2#2145509611yescharacteristic
571C2#2145549512yes—
572C2#2145577112yes—
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
Fitting, prime radicalC2×C61×#535C2
socleC23×C61C22×D4
hypercenterC2×#535D61
hypocenterC61#23483
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C22×#41 ◁ C2×#535

Complements

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

NH ≅ G/Ntype
1. C2×#16839805C2semidirect
2. C2×#16839767C2semidirect
3. C22×#14519698C2semidirect
4. C2×#16839729C2semidirect
5. #21448411C2semidirect
6. #21447005C2semidirect
7. #21446998C2semidirect
8. #21445535C2semidirect
15. C2×#14519710C22semidirect
16. C2×#14519708C22semidirect
22. C2×C22⋊#92418C22semidirect
26. #16839622C22semidirect
27. #16839524C22semidirect
28. #16839524C22semidirect
42. C2×#11758194D4semidirect
325. C61#23483semidirect

Maximal subgroups

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

subgroupindex
C2×#168398052
C2×#168397672
C22×#145196982
C2×#168397292
#214484112
#214470052
#214469982
#214455352
C2×C61×#5352
#2348361