Finite Group Databasebeta

← #29713850

Subgroups — #29713850

Order 15616.

1088 conjugacy classes of subgroups.

The subgroup lattice has 546 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840451C211yesmaximal, characteristic
2C2×#16840097C211yesmaximal, characteristic, complemented
3#21450121C211yesmaximal, characteristic
4#21447159C212yesmaximal, complemented
5#21447017C212yesmaximal, complemented
6#21446252C211yesmaximal, characteristic, complemented
7#21446116C211yesmaximal, characteristic
8#21446070C212yesmaximal, complemented
9#21446005C212yesmaximal, complemented
10#21445914C211yesmaximal, characteristic, complemented
11C2×C61×#428C211yesmaximal, characteristic, complemented
12#16840451—24no—
13#16840451—24no—
14C2×#14519770—21no—
15C2×#14519769—21no—
16C2×#14519768C2211yescharacteristic
17C2×#14519760—21no—
18C2×#14519759—21no—
19C2×#14519757C2211yescharacteristic
20C2×#14519756C2211yescharacteristic
21C2×#14519742C2211yescharacteristic
22C2×C4×C22⋊D61C2211yescharacteristic
23C2×#14519734—21no—
24C2×C4⋊C4×D61C2211yescharacteristic
25#16840097—24no—
26C2×#14519710C2212yes—
27C2×#14519702C2212yes—
28C2×#14519700C2212yes—
29C2×C22⋊#92418C2211yescharacteristic
30C2×#14519689C2212yes—
31#16839693—21no—
32#16839688—21no—
33#16839658C2212yes—
34#16839625—21no—
35#16839622C2211yescharacteristic
36#16839619C2212yes—
37#16839525C2212yes—
38#16839524C2212yes—
39#16839524C2212yes—
40#16839516C2212yes—
41C22×M16×C61C2211yescharacteristic
42C2×#117×C61C2211yescharacteristic
43C22×C4⋊C4×C61C2211yescharacteristic
44C61×#428—24no—
45C2×#107×C61C2212yes—
46C2×#106×C61C2212yes—
47C22×Dic122C2311yescharacteristic
48C22×C22⋊D61D411yescharacteristic
49C22×C4×D61—21no—
50C22×C4×D61C2311yescharacteristic
51C22×#92418C2312yes—
52#14519770—28no—
53#14519769—28no—
54#14519768—22no—
55#14519768—22no—
56#14519760—44no—
57#14519759—28no—
58#14519757—22no—
59#14519757—22no—
60#14519756—22no—
61#14519756D414yes—
62#14519742—22no—
63#14519742—22no—
64C4×C22⋊D61—22no—
65C4×C22⋊D61—22no—
66#14519734—28no—
67C2×C4⋊Dic61—21no—
68C2×C4⋊Dic61—21no—
69C2×C4⋊Dic61C2312yes—
70C2×C4⋊Dic61C2311yescharacteristic
71C2×#11758196—21no—
72C2×#11758196—21no—
73C2×#11758196—21no—
74C2×#11758196—21no—
75C2×#11758196—22no—
76C2×#11758196D411yescharacteristic
77C4⋊C4×D61—22no—
78C4⋊C4×D61—22no—
79C2×C22⋊Dic61—21no—
80C2×C22⋊Dic61—21no—
81C2×C22⋊Dic61D411yescharacteristic
82C2×#11758198—21no—
83C2×#11758198—22no—
84C2×#11758198D411yescharacteristic
85C2×C4×Dic61—21no—
86C2×C4×Dic61C2311yescharacteristic
87#14519710—24no—
88#14519702—24no—
89#14519700—24no—
90C22⋊#92418—22no—
91#14519689—24no—
92#14519684—21no—
93#14519684—22no—
94#14519684—22no—
95C23×C4×C61—21no—
96C23×C4×C61C2311yescharacteristic
97C2×M16×C61—22no—
98C2×M16×C61—24no—
99C22×C8×C61C2312yes—
100#117×C61—22no—
101#117×C61D414yes—
102C2×C4⋊C4×C61—22no—
103C2×C4⋊C4×C61—22no—
104C2×C4⋊C4×C61—21no—
105C2×C4⋊C4×C61C2312yes—
106C2×C4⋊C4×C61C2311yescharacteristic
107C2×C4⋊C4×C61C2311yescharacteristic
108C2×C22⋊C4×C61—21no—
109C2×C42×C61—21no—
110#107×C61—24no—
111#106×C61—24no—
112C24×C61C2×D411yescharacteristic
113C23×D61C2×D411yescharacteristic
114C22×C4×C61—22no—
115C22×C4×C61—22no—
116C22×C4×C61—21no—
117C22×C4×C61—21no—
118C22×C4×C61—22no—
119C22×C4×C61—42no—
120C22×C4×C61—42no—
121C22×C4×C61—42no—
122C22×C4×C61C2×D411yescharacteristic
123C22×C4×C61C2×D412yes—
124C22×C4×C61C2×D412yes—
125C22×C4×C61C2411yescharacteristic
126C2×C22⋊D61—22no—
127C2×C22⋊D61—22no—
128C2×C22⋊D61—41no—
129C22×Dic61—21no—
130C22×Dic61—21no—
131C22×Dic61—22no—
132C22×Dic61C2×D411yescharacteristic
133C22×Dic61C2×D411yescharacteristic
134C22×Dic61C2×D411yescharacteristic
135C2×C4×D61—24no—
136C2×C4×D61—21no—
137C2×C4×D61—21no—
138C2×C4×D61—22no—
139C2×C4×D61—22no—
140C2×C4×D61—42no—
141C2×C4×D61—42no—
142C2×Dic122—22no—
143C2×Dic122—22no—
144C2×Dic122—21no—
145C2×Dic122C2×D412yes—
146M16×C61—28no—
147C2×C8×C61—22no—
148C2×C8×C61—22no—
149C2×C8×C61#4114yes—
150C4⋊C4×C61—21no—
151C4⋊C4×C61—21no—
152C4⋊C4×C61—22no—
153C4⋊C4×C61—22no—
154C4⋊C4×C61—42no—
155C4⋊C4×C61C2×D412yes—
156C4⋊C4×C61C2×D412yes—
157C22⋊C4×C61—24no—
158C42×C61—24no—
159C22⋊Dic61—24no—
160C22⋊Dic61—22no—
161C22⋊Dic61—42no—
162#11758198—24no—
163#11758198—22no—
164#11758198—44no—
165C4⋊Dic61—24no—
166C4⋊Dic61—24no—
167C4⋊Dic61—21no—
168C4⋊Dic61—22no—
169C4⋊Dic61—22no—
170C4⋊Dic61C2×D412yes—
171#11758196—24no—
172#11758196—24no—
173#11758196—24no—
174#11758196—24no—
175#11758196—22no—
176#11758196—44no—
177C4×Dic61—24no—
178C4×Dic61—21no—
179C4×Dic61—21no—
180C2×#92418—22no—
181C2×#92418—22no—
182C2×#92418—22no—
183C23×C61—22no—
184C23×C61—21no—
185C23×C61—21no—
186C23×C61—22no—
187C23×C61#12012yes—
188C23×C61C22×D411yescharacteristic
189C22×D61—21no—
190C22×D61—22no—
191C22×D61—21no—
192C22×D61—22no—
193C22×D61AGL(1,ℤ/8)12yes—
194C2×C4×C61—22no—
195C2×C4×C61—22no—
196C2×C4×C61—22no—
197C2×C4×C61—24no—
198C2×C4×C61—22no—
199C2×C4×C61—22no—
200C2×C4×C61—41no—
201C2×C4×C61—41no—
202C2×C4×C61—42no—
203C2×C4×C61—42no—
204C2×C4×C61—42no—
205C2×C4×C61—42no—
206C2×C4×C61—84no—
207C2×C4×C61#12012yes—
208C2×C4×C61C22×D411yescharacteristic
209C2×C4×C61C22×D411yescharacteristic
210C2×C4×C61C2×#4112yes—
211C2×C4×C612−1+412yes—
212C22⋊D61—44no—
213C2×Dic61—21no—
214C2×Dic61—21no—
215C2×Dic61—22no—
216C2×Dic61—21no—
217C2×Dic61—22no—
218C2×Dic61—22no—
219C2×Dic61—21no—
220C2×Dic61—21no—
221C2×Dic61—22no—
222C2×Dic61—42no—
223C2×Dic61—41no—
224C2×Dic61—41no—
225C2×Dic61—42no—
226C2×Dic61—42no—
227C2×Dic61—42no—
228C4×D61—48no—
229C4×D61—41no—
230C4×D61—41no—
231C4×D61—42no—
232Dic122—24no—
233Dic122—41no—
234Dic122—41no—
235C8×C61—24no—
236#92418—42no—
237#24110—611nomaximal, Sylow
238C22×C61—22no—
239C22×C61—22no—
240C22×C61—22no—
241C22×C61—42no—
242C22×C61—42no—
243C22×C61C2×#12011yescharacteristic
244C22×C61#54712yes—
245C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
246C22×C61#57511yescharacteristic
247C22×C61#57812yes—
248C2×D61—24no—
249C2×D61—22no—
250C2×D61—42no—
251C2×D61—41no—
252C2×D61—41no—
253C4×C61—22no—
254C4×C61—42no—
255C4×C61—42no—
256C4×C61—82no—
257C4×C61#52511yescharacteristic
258C4×C61#53611yescharacteristic
259C4×C61#54712yes—
260Dic61—42no—
261Dic61—41no—
262Dic61—42no—
263Dic61—41no—
264Dic61—41no—
265Dic61—82no—
266C2×#552—611no—
267C2×#484—611no—
268C2×#428D6111yescharacteristic, complemented
269#2486—611no—
270#2059—612no—
271#2035—612no—
272#1894—611no—
273#1870—611no—
274#1863—612no—
275#1852—612no—
276#1825—611no—
277C2×C61—42no—
278C2×C61#240611yescharacteristic
279C2×C61#300611yescharacteristic
280C2×C61#301111yescharacteristic
281C2×C61#334312yes—
282C2×C61#335212yes—
283D61—42no—
284C22×M16C2×D6111yescharacteristic
285#552—1224no—
286#552—1224no—
287C2×#126—1221no—
288C2×#125—611no—
289C2×#125—1221no—
290C2×#123—1221no—
291C2×#123—1221no—
292C2×C22⋊Q8—611no—
293C2×C22⋊Q8—611no—
294C2×C4×Q8—611no—
295C2×C4×D4—611no—
296C2×#117—1221no—
297C2×#117C2×D6111yescharacteristic
298C22×C4⋊C4—611no—
299C22×C4⋊C4C2×D6111yescharacteristic
300#484—1224no—
301#428—24no—
302C2×#107—612no—
303C2×#107C2×D6112yes—
304C2×#106C2×D6112yes—
305C2×#103—612no—
306C2×#103—612no—
307C2×C22⋊C8—611no—
308C2×C22⋊C8—612no—
309#398—1221no—
310#397—1221no—
311#389—612no—
312#381—1221no—
313#380—611no—
314#380—612no—
315#340—612no—
316#340—612no—
317#340—612no—
318#336—612no—
319C61#2411011yescharacteristic, complemented, Sylow
320C22×Q8—611no—
321C22×D4—611no—
322C23×C4—21no—
323C23×C4—611no—
324C23×C4—1221no—
325C23×C4C22×D6111yescharacteristic
326C2×M16—22no—
327C2×M16—24no—
328C22×C8—612no—
329C22×C8C22×D6112yes—
330#126—1228no—
331#125—1228no—
332#125—1222no—
333#125—1222no—
334#123—1228no—
335#123—2444no—
336C22⋊Q8—614no—
337C22⋊Q8—1222no—
338C22⋊Q8—1222no—
339C22⋊Q8—1222no—
340C4×Q8—1222no—
341C4×Q8—1222no—
342C4×D4—1222no—
343C4×D4—1222no—
344#117—22no—
345#117—1228no—
346#117C22⋊D6114yes—
347C2×C4⋊C4—22no—
348C2×C4⋊C4—22no—
349C2×C4⋊C4—21no—
350C2×C4⋊C4—611no—
351C2×C4⋊C4—611no—
352C2×C4⋊C4—612no—
353C2×C4⋊C4—1222no—
354C2×C4⋊C4—1221no—
355C2×C4⋊C4—1221no—
356C2×C4⋊C4—1221no—
357C2×C4⋊C4—1221no—
358C2×C4⋊C4—1221no—
359C2×C4⋊C4—1221no—
360C2×C4⋊C4—1222no—
361C2×C4⋊C4—1222no—
362C2×C4⋊C4C22×D6112yes—
363C2×C4⋊C4C22×D6111yescharacteristic
364C2×C4⋊C4C22×D6111yescharacteristic
365C2×C22⋊C4—21no—
366C2×C22⋊C4—611no—
367C2×C22⋊C4—611no—
368C2×C22⋊C4—1221no—
369C2×C22⋊C4—1221no—
370C2×C22⋊C4—1222no—
371C2×C22⋊C4—1221no—
372C2×C42—21no—
373C2×C42—611no—
374C2×C42—1221no—
375#107—24no—
376#107—1224no—
377#106—24no—
378#103—1224no—
379#103—1224no—
380C22⋊C8—1222no—
381C22⋊C8—1224no—
382#95—1222no—
383#95—1222no—
384#95—1221no—
385C24—611no—
386C24D4×D6111yescharacteristic
387C2×Q8—612no—
388C2×Q8—1221no—
389C2×Q8—1222no—
390C2×Q8—1222no—
391C2×D4—1222no—
392C2×D4—1222no—
393C2×D4—2441no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—22no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4—611no—
403C22×C4—611no—
404C22×C4—611no—
405C22×C4—1224no—
406C22×C4—1221no—
407C22×C4—1221no—
408C22×C4—1221no—
409C22×C4—1221no—
410C22×C4—1222no—
411C22×C4—1222no—
412C22×C4—1222no—
413C22×C4—2442no—
414C22×C4—2442no—
415C22×C4D4×D6112yes—
416C22×C4D4×D6111yescharacteristic
417C22×C4C2×C22⋊D6112yes—
418C22×C4C23×D6111yescharacteristic
419M16—28no—
420C2×C8—22no—
421C2×C8—22no—
422C2×C8—1222no—
423C2×C8—1222no—
424C2×C8—1222no—
425C2×C8#1175822214yes—
426C4⋊C4—21no—
427C4⋊C4—21no—
428C4⋊C4—22no—
429C4⋊C4—22no—
430C4⋊C4—42no—
431C4⋊C4—612no—
432C4⋊C4—1224no—
433C4⋊C4—1224no—
434C4⋊C4—1224no—
435C4⋊C4—1224no—
436C4⋊C4—1224no—
437C4⋊C4—1221no—
438C4⋊C4—1222no—
439C4⋊C4—1224no—
440C4⋊C4—1222no—
441C4⋊C4—1222no—
442C4⋊C4—2444no—
443C4⋊C4C2×C22⋊D6112yes—
444C4⋊C4C2×C22⋊D6112yes—
445C22⋊C4—24no—
446C22⋊C4—1224no—
447C22⋊C4—1222no—
448C22⋊C4—1224no—
449C22⋊C4—1222no—
450C22⋊C4—2444no—
451C22⋊C4—2442no—
452C42—24no—
453C42—1224no—
454C42—1221no—
455C42—1221no—
456C23—21no—
457C23—21no—
458C23—22no—
459C23—22no—
460C23—612no—
461C23—1221no—
462C23—1221no—
463C23—1222no—
464C23—1222no—
465C23#1451974512yes—
466C23C2×D4×D6111yescharacteristic
467Q8—1224no—
468Q8—2441no—
469Q8—2441no—
470D4—2444no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—24no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—84no—
484C2×C4—1221no—
485C2×C4—1222no—
486C2×C4—1221no—
487C2×C4—1222no—
488C2×C4—1221no—
489C2×C4—1221no—
490C2×C4—1221no—
491C2×C4—1222no—
492C2×C4—1222no—
493C2×C4—2448no—
494C2×C4—2442no—
495C2×C4—2442no—
496C2×C4—2442no—
497C2×C4—2441no—
498C2×C4—2441no—
499C2×C4—2442no—
500C2×C4—2441no—
501C2×C4—2441no—
502C2×C4—2442no—
503C2×C4#1451974512yes—
504C2×C4C2×D4×D6111yescharacteristic
505C2×C4C22×C22⋊D6111yescharacteristic
506C2×C4C2×#1175822212yes—
507C2×C4#1451982512yes—
508C8—24no—
509C8—2442no—
510C22—22no—
511C22—22no—
512C22—22no—
513C22—42no—
514C22—42no—
515C22—1224no—
516C22—1222no—
517C22—2441no—
518C22—2441no—
519C22—2442no—
520C22C2×#1451974511yescharacteristic
521C22#1684041312yes—
522C22AGL(1,ℤ/8)×D6111yescharacteristic
523C22#1684055811yescharacteristic
524C22#1684057812yes—
525C4—22no—
526C4—42no—
527C4—42no—
528C4—82no—
529C4—4882no—
530C4—2441no—
531C4—2441no—
532C4—2442no—
533C4—2441no—
534C4—2442no—
535C4#1684027311yescharacteristic
536C4#1684033611yescharacteristic
537C4#1684041312yes—
538C2—42no—
539C2—2442no—
540C2#2144915411yescharacteristic
541C2#2145506711yescharacteristic
542C2#2145511311yescharacteristic
543C2#2145826012yes—
544C2#2145839012yes—
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#428C2
socleC23×C61C22×D4
prime radicalC22×C22⋊D61D4
hypercenterC2×#428D61
hypocenterC61#24110
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#428

Complements

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

NH ≅ G/Ntype
2. C2×#16840097C2semidirect
4. #21447159C2semidirect
5. #21447017C2semidirect
6. #21446252C2semidirect
8. #21446070C2semidirect
9. #21446005C2semidirect
10. #21445914C2semidirect
11. C2×C61×#428C2semidirect
268. C2×#428D61semidirect
319. C61#24110semidirect

Maximal subgroups

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

subgroupindex
C2×#168404512
C2×#168400972
#214501212
#214471592
#214470172
#214462522
#214461162
#214460702
#214460052
#214459142
C2×C61×#4282
#2411061