Finite Group Databasebeta

← #29680082

Subgroups — #29680082

Order 15616.

1338 conjugacy classes of subgroups.

The subgroup lattice has 550 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840230C211yesmaximal, characteristic
2C2×#16839816C211yesmaximal, characteristic, complemented
3#21448214C211yesmaximal, characteristic, complemented
4#21445914C211yesmaximal, characteristic, complemented
5#21445833C212yesmaximal, complemented
6#21445828C212yesmaximal, complemented
7#21445760C211yesmaximal, characteristic
8#21445757C211yesmaximal, characteristic
9#21445648C212yesmaximal, complemented
10#21445643C212yesmaximal, complemented
11C61×#1825C211yesmaximal, characteristic, complemented
12C22×#11758222C2211yescharacteristic
13#16840230—24no—
14#16840230C418yescomplemented
15C2×#14519742C2211yescharacteristic
16C2×#14519742C2211yescharacteristic
17C2×#14519739C2211yescharacteristic
18C2×#14519739C2211yescharacteristic
19C2×#14519737—21no—
20C2×#14519737—21no—
21C2×#14519734—21no—
22C22×C4×Dic61C2211yescharacteristic
23C2×C42×D61—21no—
24#16839816—24no—
25C2×#14519706C2212yes—
26C2×#14519702C2212yes—
27C2×#14519700C2212yes—
28C2×#14519699C2212yes—
29C2×#14519697C2212yes—
30C2×C22⋊#92418C2211yescharacteristic
31C2×#14519687C2212yes—
32#16839629—21no—
33#16839627C2212yes—
34#16839623—21no—
35#16839622C2211yescharacteristic
36C4⋊C4×Dic61C2212yes—
37C22⋊C4×Dic61—21no—
38#16839607—21no—
39#16839524C2212yes—
40#16839524C2212yes—
41C2×#117×C61C2211yescharacteristic
42C2×C22⋊C8×C61C2211yescharacteristic
43C61×#380C2211yescharacteristic
44C61×#340C2212yes—
45C61×#340C2212yes—
46C2×#11758222—22no—
47C2×#11758222—24no—
48C2×#11758222—24no—
49C2×#11758222C2×C414yes—
50C22×Dic122C2311yescharacteristic
51C22×D244C2311yescharacteristic
52C22×C22⋊D61—21no—
53C23×Dic61D411yescharacteristic
54C23×Dic61D411yescharacteristic
55C22×C4×D61—21no—
56C22×C4×D61—21no—
57C22×C4×D61—21no—
58C22×#92418C2312yes—
59#14519742—22no—
60#14519742—22no—
61#14519742C2×C414yes—
62#14519742C2×C414yes—
63#14519739—22no—
64#14519739—22no—
65#14519739C2×C414yes—
66#14519739C2×C414yes—
67#14519737—28no—
68#14519737—28no—
69#14519734—28no—
70C2×C4⋊Dic61C2312yes—
71C2×#11758196—21no—
72C2×#11758196—21no—
73C2×C22⋊Dic61—21no—
74C2×#11758198—21no—
75C2×#11758198—21no—
76C2×C4×Dic61—21no—
77C2×C4×Dic61—21no—
78C2×C4×Dic61—22no—
79C2×C4×Dic61C2×C414yes—
80C2×C4×Dic61D411yescharacteristic
81C2×C4×Dic61D411yescharacteristic
82C2×C4×Dic61D414yes—
83C2×C4×Dic61C2311yescharacteristic
84C2×C4×Dic61C2311yescharacteristic
85C42×D61—28no—
86#14519706—24no—
87#14519702—24no—
88#14519700—24no—
89#14519699—24no—
90#14519697—24no—
91C22⋊#92418—22no—
92#14519687—24no—
93#14519684—21no—
94#14519684—22no—
95#14519684—22no—
96#14519684—22no—
97C23×C4×C61C2311yescharacteristic
98C22×C8×C61C2312yes—
99#117×C61—22no—
100#117×C61C2×C414yes—
101C2×C4⋊C4×C61C2312yes—
102C2×C4⋊C4×C61C2311yescharacteristic
103C2×C4⋊C4×C61C2311yescharacteristic
104C2×C22⋊C4×C61—21no—
105C2×C22⋊C4×C61—21no—
106C2×C42×C61—21no—
107C22⋊C8×C61—22no—
108#95×C61—21no—
109C24×C61C2×D411yescharacteristic
110C23×D61—21no—
111C22×C4×C61—22no—
112C22×C4×C61—22no—
113C22×C4×C61—21no—
114C22×C4×C61—21no—
115C22×C4×C61—22no—
116C22×C4×C61C22×C412yes—
117C22×C4×C61C2×D411yescharacteristic
118C22×C4×C61C2×D412yes—
119C22×C4×C61C2411yescharacteristic
120C2×C22⋊D61—28no—
121C2×C22⋊D61—24no—
122C22×Dic61—21no—
123C22×Dic61—22no—
124C22×Dic61—42no—
125C22×Dic61—41no—
126C22×Dic61—41no—
127C22×Dic61C22⋊C414yes—
128C22×Dic61C22⋊C414yes—
129C22×Dic61C2×D411yescharacteristic
130C22×Dic61C2×D411yescharacteristic
131C22×Dic61C2×D411yescharacteristic
132C22×Dic61C2×D411yescharacteristic
133#11758222—216no—
134#11758222—28no—
135#11758222—28no—
136C2×D244—21no—
137C2×D244—22no—
138C2×D244C22×C412yes—
139C2×D244#4114yes—
140C2×C4×D61—28no—
141C2×C4×D61—24no—
142C2×C4×D61—24no—
143C2×C4×D61—24no—
144C2×C4×D61—42no—
145C2×C4×D61—44no—
146C2×C4×D61—42no—
147C2×Dic122—21no—
148C2×Dic122—22no—
149C2×Dic122C22×C412yes—
150C2×Dic122#4114yes—
151C2×C8×C61—22no—
152C2×C8×C61—22no—
153C2×C8×C61—22no—
154C4⋊C4×C61—21no—
155C4⋊C4×C61—21no—
156C4⋊C4×C61—22no—
157C4⋊C4×C61—22no—
158C4⋊C4×C61C22×C412yes—
159C4⋊C4×C61C22×C412yes—
160C22⋊C4×C61—24no—
161C22⋊C4×C61—42no—
162C42×C61—24no—
163C22⋊Dic61—42no—
164#11758198—24no—
165#11758198—24no—
166C4⋊Dic61—22no—
167C4⋊Dic61—22no—
168#11758196—24no—
169#11758196—24no—
170C4×Dic61—24no—
171C4×Dic61—24no—
172C4×Dic61—44no—
173C4×Dic61C22⋊C414yes—
174C4×Dic61C22⋊C414yes—
175C4×Dic61C22×C412yes—
176C4×Dic61C22×C412yes—
177C4×Dic61C2×D412yes—
178C4×Dic61C2×D412yes—
179C2×#92418—22no—
180C2×#92418—22no—
181C2×#92418—22no—
182C23×C61—22no—
183C23×C61—21no—
184C23×C61—21no—
185C23×C61—22no—
186C23×C61C2×C22⋊C412yes—
187C23×C61C22×D411yescharacteristic
188C22×D61—22no—
189C22×D61—24no—
190C22×D61—41no—
191C22×D61—42no—
192C22×D61—41no—
193C2×C4×C61—22no—
194C2×C4×C61—22no—
195C2×C4×C61—22no—
196C2×C4×C61—24no—
197C2×C4×C61—22no—
198C2×C4×C61—22no—
199C2×C4×C61—41no—
200C2×C4×C61—42no—
201C2×C4×C61—41no—
202C2×C4×C61—42no—
203C2×C4×C61—42no—
204C2×C4×C61—42no—
205C2×C4×C61C2×C22⋊C412yes—
206C2×C4×C61C23×C411yescharacteristic
207C2×C4×C61C22×D411yescharacteristic
208C2×C4×C61C2×#4112yes—
209C2×C4×C61C2×#4112yes—
210C22⋊D61—216no—
211C2×Dic61—22no—
212C2×Dic61—24no—
213C2×Dic61—22no—
214C2×Dic61—21no—
215C2×Dic61—22no—
216C2×Dic61—21no—
217C2×Dic61—21no—
218C2×Dic61—21no—
219C2×Dic61—44no—
220C2×Dic61—44no—
221C2×Dic61—42no—
222C2×Dic61—42no—
223C2×Dic61—42no—
224C2×Dic61C2×C22⋊C412yes—
225C2×Dic61C2×C22⋊C412yes—
226C2×Dic61C2×C22⋊C412yes—
227C2×Dic61C2×C22⋊C412yes—
228D244—22no—
229D244—24no—
230D244#11714yes—
231C4×D61—216no—
232C4×D61—48no—
233C4×D61—48no—
234Dic122—22no—
235Dic122—24no—
236Dic122#11714yes—
237C8×C61—42no—
238#92418—42no—
239#20742—611nomaximal, Sylow
240C22×C61—22no—
241C22×C61—22no—
242C22×C61—22no—
243C22×C61—42no—
244C22×C61—42no—
245C22×C61C22×C22⋊C411yescharacteristic
246C22×C61#52512yes—
247C22×C61#57511yescharacteristic
248C22×C61#57512yes—
249C22×C61#57511yescharacteristic
250C2×D61—24no—
251C2×D61—44no—
252C2×D61—44no—
253C2×D61—42no—
254C2×D61—42no—
255C4×C61—22no—
256C4×C61—42no—
257C4×C61—42no—
258C4×C61—82no—
259C4×C61C2×#11711yescharacteristic
260C4×C61C2×#11711yescharacteristic
261C4×C61#52512yes—
262Dic61—22no—
263Dic61—22no—
264Dic61—22no—
265Dic61—22no—
266Dic61—24no—
267Dic61—82no—
268C2×C4×#41—611no—
269C2×#432—611no—
270#2283—611no—
271#1825—611no—
272#1825D6111yescharacteristic, complemented
273#1805—612no—
274#1804—612no—
275#1788—611no—
276#1788—611no—
277#1763—612no—
278#1762—612no—
279C2×C61—42no—
280C2×C61#228011yescharacteristic
281C2×C61#289611yescharacteristic
282C2×C61#289611yescharacteristic
283C2×C61#311112yes—
284C2×C61#311112yes—
285D61—44no—
286C22×#41—611no—
287C4×#41—618no—
288C4×#41—1224no—
289C2×C4×Q8—611no—
290C2×C4×Q8—611no—
291C2×C4×D4—611no—
292C2×C4×D4—611no—
293C2×C4×D4—1221no—
294C2×C4×D4—1221no—
295C2×#117—1221no—
296C2×#117C2×D6111yescharacteristic
297C22×C42—611no—
298C22×C42—1221no—
299#432—1224no—
300C2×C4⋊C8—612no—
301C2×#103—612no—
302C2×#103—612no—
303C2×#102—612no—
304C2×#102—612no—
305C2×C22⋊C8—611no—
306C2×C22⋊C8C2×D6111yescharacteristic
307C2×#97—612no—
308#383—1221no—
309#382—612no—
310#381—1221no—
311#380—611no—
312#380C2×D6111yescharacteristic
313C4×C4⋊C4—612no—
314C4×C22⋊C4—1221no—
315#376—1221no—
316#340—612no—
317#340—612no—
318#340C2×D6112yes—
319#340C2×D6112yes—
320C61#2074211yescharacteristic, complemented, Sylow
321C2×#41—614no—
322C2×#41—1224no—
323C2×#41—1224no—
324C2×#41—1222no—
325C22×Q8—611no—
326C22×D4—611no—
327C22×D4—1221no—
328C23×C4—611no—
329C23×C4—611no—
330C23×C4—1221no—
331C23×C4—1221no—
332C23×C4—1221no—
333C23×C4C22×D6111yescharacteristic
334C22×C8—612no—
335C22×C8C22×D6112yes—
336C4×Q8—614no—
337C4×Q8—614no—
338C4×Q8—1222no—
339C4×Q8—1222no—
340C4×D4—614no—
341C4×D4—614no—
342C4×D4—1222no—
343C4×D4—1222no—
344C4×D4—1228no—
345C4×D4—1228no—
346#117—22no—
347#117—1228no—
348#117C4×D6114yes—
349C2×C4⋊C4—612no—
350C2×C4⋊C4—1221no—
351C2×C4⋊C4—1221no—
352C2×C4⋊C4C22×D6112yes—
353C2×C4⋊C4C22×D6111yescharacteristic
354C2×C4⋊C4C22×D6111yescharacteristic
355C2×C22⋊C4—21no—
356C2×C22⋊C4—21no—
357C2×C22⋊C4—1221no—
358C2×C22⋊C4—1221no—
359C2×C22⋊C4—1221no—
360C2×C42—21no—
361C2×C42—614no—
362C2×C42—611no—
363C2×C42—614no—
364C2×C42—611no—
365C2×C42—611no—
366C2×C42—611no—
367C2×C42—1228no—
368C2×C42—1221no—
369C2×C42—1221no—
370C2×C42—1222no—
371C4⋊C8—1224no—
372#103—1224no—
373#103—1224no—
374#102—1224no—
375#102—1224no—
376C22⋊C8—22no—
377C22⋊C8—1222no—
378#97—1224no—
379#95—21no—
380#95—1222no—
381#95—1222no—
382#95—1222no—
383#95—1221no—
384C24—1221no—
385C24D4×D6111yescharacteristic
386#41—1228no—
387#41—1228no—
388#41—12216no—
389C2×Q8—612no—
390C2×Q8—614no—
391C2×Q8—1221no—
392C2×Q8—1222no—
393C2×D4—612no—
394C2×D4—614no—
395C2×D4—1228no—
396C2×D4—1224no—
397C2×D4—1221no—
398C2×D4—1222no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—22no—
404C22×C4—611no—
405C22×C4—611no—
406C22×C4—611no—
407C22×C4—611no—
408C22×C4—614no—
409C22×C4—614no—
410C22×C4—1224no—
411C22×C4—1224no—
412C22×C4—1228no—
413C22×C4—1221no—
414C22×C4—1224no—
415C22×C4—1222no—
416C22×C4—2442no—
417C22×C4—2444no—
418C22×C4—2442no—
419C22×C4—2441no—
420C22×C4—2442no—
421C22×C4—2441no—
422C22×C4C2×C4×D6112yes—
423C22×C4D4×D6112yes—
424C22×C4D4×D6111yescharacteristic
425C22×C4C23×D6111yescharacteristic
426C2×C8—22no—
427C2×C8—22no—
428C2×C8—22no—
429C2×C8—1222no—
430C2×C8—1222no—
431C2×C8—1222no—
432C4⋊C4—21no—
433C4⋊C4—21no—
434C4⋊C4—22no—
435C4⋊C4—22no—
436C4⋊C4—1224no—
437C4⋊C4—1224no—
438C4⋊C4—1222no—
439C4⋊C4—1222no—
440C4⋊C4C2×C4×D6112yes—
441C4⋊C4C2×C4×D6112yes—
442C22⋊C4—24no—
443C22⋊C4—42no—
444C22⋊C4—1224no—
445C22⋊C4—1224no—
446C22⋊C4—2442no—
447C42—24no—
448C42—612no—
449C42—612no—
450C42—612no—
451C42—612no—
452C42—614no—
453C42—614no—
454C42—1224no—
455C42—1224no—
456C42—2444no—
457C23—22no—
458C23—21no—
459C23—22no—
460C23—21no—
461C23—1222no—
462C23—1224no—
463C23—2441no—
464C23—2442no—
465C23—2441no—
466C23C22⋊C4×D6112yes—
467C23C2×D4×D6111yescharacteristic
468Q8—614no—
469Q8—1222no—
470Q8—1224no—
471D4—614no—
472D4—12216no—
473D4—1222no—
474D4—1224no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—24no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—41no—
482C2×C4—42no—
483C2×C4—41no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—612no—
488C2×C4—612no—
489C2×C4—612no—
490C2×C4—612no—
491C2×C4—12216no—
492C2×C4—1222no—
493C2×C4—1224no—
494C2×C4—1221no—
495C2×C4—1221no—
496C2×C4—1222no—
497C2×C4—1222no—
498C2×C4—1221no—
499C2×C4—1221no—
500C2×C4—2448no—
501C2×C4—2448no—
502C2×C4—2444no—
503C2×C4—2444no—
504C2×C4—2442no—
505C2×C4—2442no—
506C2×C4—2442no—
507C2×C4C22⋊C4×D6112yes—
508C2×C4C22×C4×D6111yescharacteristic
509C2×C4C2×D4×D6111yescharacteristic
510C2×C4#41×D6112yes—
511C2×C4#41×D6112yes—
512C8—42no—
513C8—2442no—
514C22—22no—
515C22—22no—
516C22—22no—
517C22—42no—
518C22—42no—
519C22—1224no—
520C22—2444no—
521C22—2444no—
522C22—2442no—
523C22—2442no—
524C22C2×C22⋊C4×D6111yescharacteristic
525C22#1684027012yes—
526C22#1684055811yescharacteristic
527C22#1684055912yes—
528C22#1684056111yescharacteristic
529C4—22no—
530C4—42no—
531C4—42no—
532C4—82no—
533C4—1224no—
534C4—1222no—
535C4—1222no—
536C4—1222no—
537C4—1222no—
538C4—4882no—
539C4#117×D6111yescharacteristic
540C4#117×D6111yescharacteristic
541C4#1684027012yes—
542C2—42no—
543C2—2444no—
544C2#2144819911yescharacteristic
545C2#2145433411yescharacteristic
546C2#2145433511yescharacteristic
547C2#2145592312yes—
548C2#2145593112yes—
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1825C2
socleC23×C61C22×D4
prime radicalC22×#11758222C22
cosocleC22×C4×C61C24
hypercenter#1825D61
hypocenterC61#20742
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1825

Complements

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

NH ≅ G/Ntype
2. C2×#16839816C2semidirect
3. #21448214C2semidirect
4. #21445914C2semidirect
5. #21445833C2semidirect
6. #21445828C2semidirect
9. #21445648C2semidirect
10. #21445643C2semidirect
11. C61×#1825C2semidirect
14. #16840230C4semidirect
272. #1825D61semidirect
320. C61#20742semidirect

Maximal subgroups

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

subgroupindex
C2×#168402302
C2×#168398162
#214482142
#214459142
#214458332
#214458282
#214457602
#214457572
#214456482
#214456432
C61×#18252
#2074261