Finite Group Databasebeta

← #29696433

Subgroups — #29696433

Order 15616.

1370 conjugacy classes of subgroups.

The subgroup lattice has 554 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840417C211yesmaximal, characteristic
2C2×#16839839C211yesmaximal, characteristic, complemented
3#21448770C211yesmaximal, characteristic, complemented
4#21446779C211yesmaximal, characteristic, complemented
5#21446770C211yesmaximal, characteristic, complemented
6#21446254C212yesmaximal, complemented
7#21446138C212yesmaximal, complemented
8#21445908C212yesmaximal, complemented
9C4×#14519698C211yesmaximal, characteristic, complemented
10#21445537C212yesmaximal, complemented
11C61×#1802C211yesmaximal, characteristic, complemented
12C22×#11758222—21no—
13C22×#11758189C2212yes—
14#16840417—24no—
15#16840417C418yescomplemented
16C2×#14519777C2211yescharacteristic
17C2×#14519759—22no—
18C2×#14519756—21no—
19C2×#14519748C2212yes—
20C2×C4×D244C2211yescharacteristic
21C2×#14519735C2212yes—
22C2×C4⋊C4×D61C2212yes—
23#16839839—24no—
24C2×#14519698C2211yescharacteristic
25C2×#14519698C2211yescharacteristic
26C2×#14519689C2212yes—
27C2×C4×#92418C2211yescharacteristic
28#16839663—21no—
29#16839652—22no—
30#16839619C2212yes—
31C4⋊C4×Dic61—21no—
32C4×C4⋊Dic61C2211yescharacteristic
33#16839525C2212yes—
34#16839525C2212yes—
35#16839524C2211yescharacteristic
36#16839524C2211yescharacteristic
37#16839524C2212yes—
38#16839524C2212yes—
39C2×C4×D4×C61C2211yescharacteristic
40C2×C4⋊C8×C61C2211yescharacteristic
41C2×#102×C61C2212yes—
42C61×#384C2211yescharacteristic
43C61×#340C2212yes—
44C2×#11758222—28no—
45C2×#11758222—44no—
46C2×#11758222—44no—
47C22×Dic122—21no—
48C22×D244D411yescharacteristic
49C22×C22⋊D61—22no—
50C22×C4×D61—22no—
51C22×C4×D61C2312yes—
52C2×#11758189—24no—
53C2×#11758189—24no—
54C2×#11758189—42no—
55C22×#92418C2311yescharacteristic
56C22×#92418C2311yescharacteristic
57#14519777—22no—
58#14519777C2×C414yes—
59#14519759—216no—
60#14519756—28no—
61#14519748—24no—
62#14519748C2×C418yes—
63C4×D244C2×C414yes—
64C4×D244D414yes—
65#14519735—24no—
66#14519735—42no—
67C2×C4⋊Dic61—21no—
68C2×C4⋊Dic61D411yescharacteristic
69C2×C4⋊Dic61C2311yescharacteristic
70C2×C4⋊Dic61C2312yes—
71C2×C4⋊Dic61C2311yescharacteristic
72C2×#11758196—22no—
73C4⋊C4×D61—24no—
74C4⋊C4×D61C2×C418yes—
75C2×C22⋊Dic61—22no—
76C2×#11758198—22no—
77C2×#11758198—22no—
78C2×#11758198D412yes—
79C2×C4×Dic61—21no—
80C2×C4×Dic61—22no—
81#14519698—22no—
82#14519698—22no—
83#14519689—24no—
84C4×#92418—22no—
85#14519684—21no—
86#14519684—22no—
87C22×D4×C61C2311yescharacteristic
88C23×C4×C61—21no—
89C22×C8×C61C2312yes—
90C4×D4×C61—22no—
91C4×D4×C61C2×C414yes—
92C2×C4⋊C4×C61—21no—
93C2×C4⋊C4×C61—21no—
94C2×C4⋊C4×C61C2312yes—
95C2×C4⋊C4×C61C2311yescharacteristic
96C2×C22⋊C4×C61—21no—
97C2×C42×C61C2311yescharacteristic
98C4⋊C8×C61—22no—
99#102×C61—42no—
100C24×C61—21no—
101C23×D61C2×D412yes—
102C2×D4×C61—21no—
103C2×D4×C61—21no—
104C2×D4×C61—21no—
105C2×D4×C61—21no—
106C2×D4×C61—21no—
107C2×D4×C61C22×C412yes—
108C22×C4×C61—21no—
109C22×C4×C61—24no—
110C22×C4×C61—22no—
111C22×C4×C61—42no—
112C22×C4×C61—42no—
113C22×C4×C61C2×D411yescharacteristic
114C22×C4×C61C2×D411yescharacteristic
115C22×C4×C61C2411yescharacteristic
116C2×C22⋊D61—28no—
117C2×C22⋊D61—44no—
118C2×C22⋊D61—44no—
119C22×Dic61—22no—
120C22×Dic61—21no—
121C22×Dic61—21no—
122C22×Dic61C2×D412yes—
123#11758222—432no—
124C2×D244—22no—
125C2×D244—22no—
126C2×D244C22⋊C414yes—
127C2×C4×D61—22no—
128C2×C4×D61—22no—
129C2×C4×D61—22no—
130C2×C4×D61—22no—
131C2×C4×D61—28no—
132C2×C4×D61—44no—
133C2×C4×D61—44no—
134C2×C4×D61C22×C414yes—
135C2×C4×D61C2×D414yes—
136C2×Dic122—24no—
137C2×Dic122—42no—
138C2×Dic122—42no—
139C2×C8×C61—22no—
140C2×C8×C61—22no—
141C2×C8×C61—22no—
142C4⋊C4×C61—21no—
143C4⋊C4×C61—22no—
144C4⋊C4×C61—22no—
145C4⋊C4×C61—42no—
146C4⋊C4×C61—81no—
147C4⋊C4×C61C22×C412yes—
148C22⋊C4×C61—24no—
149C42×C61C22×C412yes—
150C42×C61C2×D412yes—
151C22⋊Dic61—28no—
152#11758198—44no—
153#11758198—82no—
154#11758198C22⋊C418yes—
155C4⋊Dic61—24no—
156C4⋊Dic61—22no—
157C4⋊Dic61—21no—
158C4⋊Dic61—21no—
159C4⋊Dic61—21no—
160C4⋊Dic61C22⋊C414yes—
161C4⋊Dic61C22×C414yes—
162C4⋊Dic61#4112yescomplemented
163#11758196—28no—
164C4×Dic61—44no—
165C4×Dic61—42no—
166C2×#92418—21no—
167C2×#92418—21no—
168C2×#92418—21no—
169C2×#92418—21no—
170C2×#92418—21no—
171C2×#92418#4112yescomplemented
172#11758189—48no—
173C23×C61—22no—
174C23×C61—41no—
175C23×C61—41no—
176C23×C61—41no—
177C23×C61—41no—
178C23×C61—41no—
179C23×C61—41no—
180C23×C61C22×D411yescharacteristic
181C22×D61—22no—
182C22×D61—22no—
183C22×D61—22no—
184C22×D61—22no—
185C22×D61—22no—
186C22×D61—22no—
187C22×D61C2×C22⋊C414yes—
188D4×C61—22no—
189D4×C61—22no—
190D4×C61—41no—
191D4×C61—41no—
192C2×C4×C61—21no—
193C2×C4×C61—21no—
194C2×C4×C61—22no—
195C2×C4×C61—41no—
196C2×C4×C61—41no—
197C2×C4×C61—48no—
198C2×C4×C61—42no—
199C2×C4×C61—42no—
200C2×C4×C61—42no—
201C2×C4×C61C2×C22⋊C412yes—
202C2×C4×C61C2×C22⋊C412yes—
203C2×C4×C61AGL(1,ℤ/8)12yes—
204C2×C4×C61AGL(1,ℤ/8)12yes—
205C2×C4×C61C23×C411yescharacteristic
206C2×C4×C61C22×D411yescharacteristic
207C2×C4×C61C2×#4111yescharacteristic
208C2×C4×C61C2×#4111yescharacteristic
209C2×C4×C61H(5,𝔽2)11yescharacteristic
210C2×C4×C612−1+411yescharacteristic
211C22⋊D61—416no—
212C2×Dic61—22no—
213C2×Dic61—22no—
214C2×Dic61—24no—
215C2×Dic61—22no—
216C2×Dic61—41no—
217C2×Dic61—42no—
218C2×Dic61—42no—
219C2×Dic61—41no—
220C2×Dic61—41no—
221C2×Dic61—41no—
222C2×Dic61—41no—
223C2×Dic61C2×C22⋊C414yes—
224D244—28no—
225C4×D61—24no—
226C4×D61—24no—
227C4×D61—24no—
228C4×D61—24no—
229C4×D61—416no—
230Dic122—48no—
231C8×C61—42no—
232#92418—22no—
233#92418—41no—
234#22500—611nomaximal, Sylow
235C22×C61—42no—
236C22×C61—42no—
237C22×C61—42no—
238C22×C61—81no—
239C22×C61—81no—
240C22×C61—81no—
241C22×C61—81no—
242C22×C61C22×C22⋊C411yescharacteristic
243C22×C61#54611yescharacteristic
244C22×C61#54711yescharacteristic
245C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
246C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
247C22×C61#57711yescharacteristic
248C22×C61#57811yescharacteristic
249C2×D61—24no—
250C2×D61—24no—
251C2×D61—24no—
252C2×D61—42no—
253C2×D61—42no—
254C2×D61—42no—
255C2×D61—42no—
256C4×C61—22no—
257C4×C61—42no—
258C4×C61—82no—
259C4×C61#41814yes—
260C4×C61C4×#4111yescharacteristic
261C4×C61#52011yescharacteristic
262C4×C61#54611yescharacteristic
263C4×C61#54711yescharacteristic
264Dic61—24no—
265Dic61—42no—
266Dic61—44no—
267Dic61—81no—
268C2×#547—611no—
269C2×#436—611no—
270#2354—611no—
271#1990—611no—
272#1988—611no—
273#1895—612no—
274#1875—612no—
275#1824—612no—
276#1802D6111yescharacteristic, complemented
277C4×#102—611no—
278#1736—612no—
279C2×C61—82no—
280C2×C61#233911yescharacteristic
281C2×C61C2×#41811yescharacteristic
282C2×C61#289811yescharacteristic
283C2×C61#333011yescharacteristic
284C2×C61#333211yescharacteristic
285C2×C61#335311yescharacteristic
286C2×C61#335411yescharacteristic
287D61—44no—
288C22×#41—1221no—
289C22×M16—612no—
290#547—618no—
291#547—1224no—
292C2×C4⋊Q8—611no—
293C2×#123—1222no—
294C2×C22⋊Q8—1221no—
295C2×#121—612no—
296C2×C4×D4—611no—
297C2×C4×D4C2×D6111yescharacteristic
298C2×#117—612no—
299C22×C4⋊C4—612no—
300#436—1224no—
301C2×C4⋊C8C2×D6111yescharacteristic
302C2×#102—611no—
303C2×#102—611no—
304C2×#102C2×D6112yes—
305C2×C22⋊C8—612no—
306C2×C4×C8—611no—
307#390—1221no—
308#388—1222no—
309#384C2×D6111yescharacteristic
310#380—612no—
311C4×C4⋊C4—611no—
312C4×C4⋊C4—1221no—
313#340—612no—
314#340—612no—
315#340—611no—
316#340—612no—
317#340—612no—
318#340—611no—
319#340C2×D6112yes—
320C61#2250011yescharacteristic, complemented, Sylow
321C2×#41—1228no—
322C2×#41—2444no—
323C2×#41—2444no—
324C22×Q8—1221no—
325C22×D4—611no—
326C22×D4—1222no—
327C22×D4C22×D6111yescharacteristic
328C23×C4—21no—
329C23×C4—612no—
330C23×C4—1222no—
331C2×M16—1224no—
332C2×M16—1224no—
333C2×M16—2442no—
334C22×C8—611no—
335C22×C8—611no—
336C22×C8C22×D6112yes—
337C4⋊Q8—614no—
338C4⋊Q8—1222no—
339#123—12216no—
340C22⋊Q8—1228no—
341#121—618no—
342#121—1224no—
343C4×D4—22no—
344C4×D4—614no—
345C4×D4—614no—
346C4×D4C4×D6114yes—
347#117—1224no—
348#117—2442no—
349C2×C4⋊C4—21no—
350C2×C4⋊C4—21no—
351C2×C4⋊C4—611no—
352C2×C4⋊C4—612no—
353C2×C4⋊C4—611no—
354C2×C4⋊C4—618no—
355C2×C4⋊C4—611no—
356C2×C4⋊C4—1224no—
357C2×C4⋊C4—1222no—
358C2×C4⋊C4—1221no—
359C2×C4⋊C4C22×D6112yes—
360C2×C4⋊C4C22×D6111yescharacteristic
361C2×C22⋊C4—21no—
362C2×C22⋊C4—612no—
363C2×C22⋊C4—1222no—
364C2×C22⋊C4—1222no—
365C2×C22⋊C4—1222no—
366C2×C42—1222no—
367C2×C42—1221no—
368C2×C42C22×D6111yescharacteristic
369C4⋊C8—22no—
370#102—42no—
371#102—1222no—
372#102—1222no—
373C22⋊C8—1224no—
374C4×C8—1222no—
375#95—1222no—
376#95—1221no—
377C24—21no—
378C24—612no—
379#41—24432no—
380C2×Q8—1224no—
381C2×Q8—2442no—
382C2×Q8—2442no—
383C2×D4—21no—
384C2×D4—21no—
385C2×D4—21no—
386C2×D4—21no—
387C2×D4—21no—
388C2×D4—614no—
389C2×D4—1228no—
390C2×D4—1222no—
391C2×D4—1222no—
392C2×D4—2444no—
393C2×D4—2444no—
394C2×D4C2×C4×D6112yes—
395C22×C4—21no—
396C22×C4—24no—
397C22×C4—22no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4—612no—
401C22×C4—614no—
402C22×C4—614no—
403C22×C4—1228no—
404C22×C4—1222no—
405C22×C4—1222no—
406C22×C4—1222no—
407C22×C4—1222no—
408C22×C4—1221no—
409C22×C4—1222no—
410C22×C4—1221no—
411C22×C4—2444no—
412C22×C4—2444no—
413C22×C4D4×D6111yescharacteristic
414C22×C4D4×D6111yescharacteristic
415C22×C4C23×D6111yescharacteristic
416M16—2448no—
417C2×C8—22no—
418C2×C8—22no—
419C2×C8—22no—
420C2×C8—612no—
421C2×C8—1221no—
422C2×C8—1221no—
423C2×C8—1221no—
424C2×C8—1221no—
425C2×C8—1221no—
426C4⋊C4—21no—
427C4⋊C4—22no—
428C4⋊C4—22no—
429C4⋊C4—42no—
430C4⋊C4—81no—
431C4⋊C4—614no—
432C4⋊C4—614no—
433C4⋊C4—612no—
434C4⋊C4—1228no—
435C4⋊C4—1222no—
436C4⋊C4—1221no—
437C4⋊C4—1224no—
438C4⋊C4—1221no—
439C4⋊C4—1221no—
440C4⋊C4C2×C4×D6112yes—
441C22⋊C4—24no—
442C22⋊C4—618no—
443C22⋊C4—1228no—
444C22⋊C4—4882no—
445C22⋊C4—2444no—
446C42—2444no—
447C42—2442no—
448C42C2×C4×D6112yes—
449C42D4×D6112yes—
450C23—22no—
451C23—41no—
452C23—41no—
453C23—41no—
454C23—41no—
455C23—41no—
456C23—41no—
457C23—614no—
458C23—1222no—
459C23—1222no—
460C23—1222no—
461C23—1222no—
462C23—1222no—
463C23—1222no—
464C23C2×D4×D6111yescharacteristic
465Q8—2448no—
466D4—22no—
467D4—22no—
468D4—41no—
469D4—41no—
470D4—1228no—
471D4—24416no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—22no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—48no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—614no—
482C2×C4—1222no—
483C2×C4—1222no—
484C2×C4—1224no—
485C2×C4—1224no—
486C2×C4—1224no—
487C2×C4—1224no—
488C2×C4—1224no—
489C2×C4—1222no—
490C2×C4—24416no—
491C2×C4—2441no—
492C2×C4—2441no—
493C2×C4—2441no—
494C2×C4—2442no—
495C2×C4—2441no—
496C2×C4—2442no—
497C2×C4—2441no—
498C2×C4C22⋊C4×D6112yes—
499C2×C4C22⋊C4×D6112yes—
500C2×C4#1451980112yes—
501C2×C4#1451980412yes—
502C2×C4C22×C4×D6111yescharacteristic
503C2×C4C2×D4×D6111yescharacteristic
504C2×C4#41×D6111yescharacteristic
505C2×C4#41×D6111yescharacteristic
506C2×C4#1451982211yescharacteristic
507C2×C4#1451982511yescharacteristic
508C8—42no—
509C8—1222no—
510C8—2441no—
511C22—42no—
512C22—42no—
513C22—42no—
514C22—81no—
515C22—81no—
516C22—81no—
517C22—81no—
518C22—1224no—
519C22—1224no—
520C22—1224no—
521C22—2442no—
522C22—2442no—
523C22—2442no—
524C22—2442no—
525C22C2×C22⋊C4×D6111yescharacteristic
526C22#1684040911yescharacteristic
527C22#1684041411yescharacteristic
528C22C2×#1451980111yescharacteristic
529C22C2×#1451980411yescharacteristic
530C22#1684057211yescharacteristic
531C22#1684057811yescharacteristic
532C4—22no—
533C4—42no—
534C4—82no—
535C4—1224no—
536C4—4881no—
537C4—2442no—
538C4—2444no—
539C4#1683976514yes—
540C4#1684023011yescharacteristic
541C4#1684024311yescharacteristic
542C4#1684040911yescharacteristic
543C4#1684041411yescharacteristic
544C2—82no—
545C2—2444no—
546C2#2144864711yescharacteristic
547C2C2×#1683976511yescharacteristic
548C2#2145435011yescharacteristic
549C2#2145812711yescharacteristic
550C2#2145814911yescharacteristic
551C2#2145840011yescharacteristic
552C2#2145841111yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1802C2
socleC23×C61C22×D4
prime radicalC2×#16840417C2
cosocleC22×C4×C61C24
hypercenter#1802D61
hypocenterC61#22500
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1802

Complements

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

NH ≅ G/Ntype
2. C2×#16839839C2semidirect
3. #21448770C2semidirect
4. #21446779C2semidirect
5. #21446770C2semidirect
6. #21446254C2semidirect
7. #21446138C2semidirect
8. #21445908C2semidirect
9. C4×#14519698C2semidirect
10. #21445537C2semidirect
11. C61×#1802C2semidirect
15. #16840417C4semidirect
162. C4⋊Dic61#41semidirect
171. C2×#92418#41semidirect
276. #1802D61semidirect
320. C61#22500semidirect

Maximal subgroups

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

subgroupindex
C2×#168404172
C2×#168398392
#214487702
#214467792
#214467702
#214462542
#214461382
#214459082
C4×#145196982
#214455372
C61×#18022
#2250061