Finite Group Databasebeta

← #30617213

Subgroups — #30617213

Order 15616.

782 conjugacy classes of subgroups.

The subgroup lattice has 518 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462533C211yesmaximal, characteristic
2C2×#16840727C211yesmaximal, characteristic, complemented
3#21461901C212yesmaximal, complemented
4#21461887C212yesmaximal, complemented
5#21461802C211yesmaximal, characteristic
6#21461772C211yesmaximal, characteristic, complemented
7#21461725C211yesmaximal, characteristic, complemented
8#21461626C212yesmaximal, complemented
9#21461598C212yesmaximal, complemented
10#21461558C211yesmaximal, characteristic, complemented
11#21455727C211yesmaximal, characteristic
12C2×#14519864C2211yescharacteristic
13C22×#11758213C2211yescharacteristic
14C22×D61⋊C8C2211yescharacteristic
15#16840794—21no—
16#16840793—21no—
17#16840789—21no—
18D4×#92420—21no—
19C2×#14519847C2212yes—
20#16840769C2211yescharacteristic
21C2×#14519841C2212yes—
22#16840736C2211yescharacteristic
23#16840733C2211yescharacteristic
24#16840727—21no—
25#16840727—21no—
26#16840727C2211yescharacteristic
27#16840727C2212yes—
28#16840727C2212yes—
29C2×#14519832—21no—
30C2×#14519831C2212yes—
31C2×C4×#92420—21no—
32C4×C22⋊#6923C2211yescharacteristic
33#16840638C2212yes—
34#16840632C2212yes—
35#16840632C2212yes—
36#16840632C2212yes—
37(C2×#6923)⋊C8C2212yes—
38#16840272C2211yescharacteristic
39C2×#14519735C2211yescharacteristic
40#16840105C412yescomplemented
41#16840098C411yescharacteristic, complemented
42#16840092C411yescharacteristic, complemented
43#16840084C412yescomplemented
44#16840078C411yescharacteristic, complemented
45#16839799C2212yes—
46#16839755C2212yes—
47C22⋊C8×D61C2211yescharacteristic
48C61×#480C411yescharacteristic, complemented
49#14519864—24no—
50C2×#11758213—21no—
51C2×#11758213—24no—
52C2×#11758213C2311yescharacteristic
53C2×#11758213C2311yescharacteristic
54C2×#11758213C2312yes—
55C2×#11758217—21no—
56C2×D61⋊C8—24no—
57C2×D61⋊C8D411yescharacteristic
58C2×D61⋊C8D411yescharacteristic
59C2×D61⋊C8C2311yescharacteristic
60C2×D61⋊C8C2311yescharacteristic
61C22×#92420—21no—
62C22×#92420—21no—
63C22×#92420D411yescharacteristic
64C22×#92420D411yescharacteristic
65C2×C22⋊#6923—21no—
66C2×C4×#6923C2312yes—
67#14519847—24no—
68#14519844—21no—
69C4⋊#92420—21no—
70#14519841—24no—
71#14519834—21no—
72C22⋊#92420—21no—
73C22⋊#92420—21no—
74C22⋊#92420—21no—
75#14519832—22no—
76#14519832—22no—
77#14519832—21no—
78#14519831—24no—
79C4×#92420—22no—
80C4×#92420—22no—
81C4×#92420—21no—
82(C2×#6923)⋊C4—21no—
83C2×#11758224C2311yescharacteristic
84C22×C4×D61C2311yescharacteristic
85C2×C8×D61C2312yes—
86#14519759—21no—
87#14519757C2×C411yescharacteristic
88#14519756C2×C411yescharacteristic
89#14519748C2×C411yescharacteristic
90#14519745—21no—
91D4×Dic61—21no—
92#14519737—21no—
93#14519735—21no—
94#14519735—21no—
95#14519735C2×C412yes—
96#14519735C2312yes—
97#14519735C2311yescharacteristic
98C2×C4⋊Dic61C2×C411yescharacteristic
99C2×#11758198—21no—
100C2×C4×Dic61—21no—
101#14519711C2×C412yes—
102#14519708C2×C412yes—
103#14519708C2×C412yes—
104#14519702C2×C412yes—
105#14519700C2×C412yes—
106#14519698C2×C412yes—
107C22⋊#92418C2×C411yescharacteristic
108#14519689C2×C412yes—
109#121×C61C2×C411yescharacteristic
110C2×C4⋊C4×C61C2×C411yescharacteristic
111#107×C61C2×C412yes—
112#102×C61C2×C412yes—
113C22⋊C8×C61C2×C411yescharacteristic
114C23×D61#4111yescharacteristic
115C22×#6923—22no—
116C2×D4×C61—21no—
117C2×D4×C61C22×C411yescharacteristic
118C22×C4×C61—21no—
119C22×C4×C61C22×C411yescharacteristic
120C2×C22⋊D61—21no—
121C2×C22⋊D61—21no—
122C22×Dic61—21no—
123C22×Dic61—21no—
124C22×Dic61C2×D411yescharacteristic
125#11758224—22no—
126#11758224—22no—
127C2×C4×D61—22no—
128C2×C4×D61—22no—
129C2×C4×D61C22×C411yescharacteristic
130C2×C4×D61C2×D411yescharacteristic
131C2×C4×D61#4111yescharacteristic
132C2×C4×D61C2411yescharacteristic
133C2×Dic122C22×C411yescharacteristic
134C22⋊#6923—42no—
135#11758217—42no—
136C2×#92420—24no—
137C2×#92420—22no—
138C2×#92420—22no—
139C2×#92420—22no—
140C2×#92420—21no—
141C2×#92420—21no—
142C2×#92420—42no—
143C2×#92420C2×D411yescharacteristic
144C2×#92420C2×D411yescharacteristic
145C2×#92420C2×D411yescharacteristic
146C2×#92420C2×D411yescharacteristic
147C4×#6923—22no—
148C4×#6923—22no—
149#11758213—24no—
150#11758213—22no—
151#11758213—22no—
152#11758213—22no—
153#11758213#4114yes—
154D61⋊C8—22no—
155D61⋊C8—22no—
156D61⋊C8—22no—
157D61⋊C8#4114yes—
158C2×C8×C61C22×C412yes—
159C4⋊C4×C61—21no—
160C4⋊C4×C61C22×C412yes—
161C4⋊C4×C61C22×C411yescharacteristic
162C22⋊C4×C61—21no—
163C22⋊Dic61—21no—
164C22⋊Dic61—21no—
165#11758198—22no—
166#11758198—21no—
167#11758198—22no—
168#11758198—21no—
169C4⋊Dic61—21no—
170C4⋊Dic61—21no—
171C4⋊Dic61C22×C412yes—
172C4⋊Dic61C22×C411yescharacteristic
173#11758196—21no—
174C4×Dic61—22no—
175C4×Dic61—22no—
176C4×Dic61—21no—
177C2×#92418C22×C412yes—
178C8×D61—22no—
179C8×D61—22no—
180C23×C61—21no—
181C23×C61C4×D411yescharacteristic
182C22×D61—22no—
183C22×D61—22no—
184C22×D61—41no—
185C22×D61C4×D411yescharacteristic
186C22×D61C2×#4111yescharacteristic
187C2×#6923—42no—
188C2×#6923—42no—
189C2×#6923—42no—
190D4×C61—22no—
191D4×C61—42no—
192C2×C4×C61—22no—
193C2×C4×C61—22no—
194C2×C4×C61—21no—
195C2×C4×C61—42no—
196C2×C4×C61C4×D411yescharacteristic
197C2×C4×C61C23×C411yescharacteristic
198C22⋊D61—42no—
199C22⋊D61—42no—
200C2×Dic61—21no—
201C2×Dic61—21no—
202C2×Dic61—22no—
203C2×Dic61—21no—
204C2×Dic61—21no—
205C2×Dic61—42no—
206C2×Dic61—41no—
207C2×Dic61—41no—
208C2×Dic61C4×D411yescharacteristic
209C2×Dic61C22×D411yescharacteristic
210C4×D61—22no—
211C4×D61—22no—
212C4×D61—21no—
213C4×D61—21no—
214C4×D61C2×#4112yes—
215C4×D61C2×#4112yes—
216Dic122—22no—
217#92420—21no—
218#92420—21no—
219#92420—24no—
220#92420—41no—
221#92420—41no—
222#92420#12312yes—
223#92420#12312yes—
224C8×C61—22no—
225#92418—22no—
226#21593—611nomaximal, Sylow
227C22×C61—21no—
228C22×C61—21no—
229C22×C61—41no—
230C22×C61—41no—
231C22×C61—41no—
232C22×C61#44412yes—
233C22×C61C2×C4×D411yescharacteristic
234C2×D61—21no—
235C2×D61—21no—
236C2×D61—21no—
237C2×D61—21no—
238C2×D61—21no—
239C2×D61—44no—
240C2×D61#52912yes—
241#6923—82no—
242C4×C61—21no—
243C4×C61—42no—
244C4×C61—41no—
245C4×C61C4×#4112yes—
246Dic61—21no—
247Dic61—42no—
248Dic61—41no—
249Dic61—41no—
250Dic61C2×#12311yescharacteristic
251Dic61#52511yescharacteristic
252#3091—611no—
253#2929—611no—
254C2×#405—611no—
255#1929—612no—
256#1922—612no—
257#1871—611no—
258#1855—611no—
259#1826—611no—
260#1773—612no—
261#1765—612no—
262#1752—611no—
263C2×C61—22no—
264C2×C61—81no—
265C2×C61C4×#12311yescharacteristic
266C2×C61C2×#44411yescharacteristic
267C2×C61#295511yescharacteristic
268D61—22no—
269D61—41no—
270C2×#131—611no—
271C22×M16—611no—
272C23×C8—611no—
273#525—611no—
274C2×#117—611no—
275#485—612no—
276#484—611no—
277#483—611no—
278#482—612no—
279#481—611no—
280#480#692311yescharacteristic, complemented
281#435—1221no—
282#435—1221no—
283C8×D4—1221no—
284C8×D4—1221no—
285C2×#108—612no—
286#427—612no—
287#424—611no—
288C2×#104—612no—
289#416—612no—
290#408—611no—
291#407—611no—
292C2×C22⋊C8—611no—
293#405—612no—
294#405—612no—
295#405—611no—
296#405—1221no—
297#405—1221no—
298C2×#97—612no—
299C2×#97—1221no—
300C2×C4×C8—1221no—
301C4×C22⋊C4—611no—
302#341—612no—
303#339—612no—
304#339—612no—
305#339—612no—
306#336—612no—
307C61#2159311yescharacteristic, complemented, Sylow
308C2×#41—611no—
309C23×C4—611no—
310#131—1224no—
311C2×M16—612no—
312C2×M16—611no—
313C2×M16—611no—
314C2×M16—1221no—
315C2×M16—1224no—
316C2×M16—1221no—
317C22×C8—611no—
318C22×C8—611no—
319C22×C8—611no—
320C22×C8—611no—
321C22×C8—611no—
322C22×C8—611no—
323C22×C8—612no—
324C22×C8—1224no—
325C22×C8—1221no—
326C22×C8—1221no—
327#123—1221no—
328C22⋊Q8—611no—
329C22⋊Q8—611no—
330#121—611no—
331#121C2×#692311yescharacteristic
332#120—1221no—
333C4×D4—1221no—
334C4×D4—1221no—
335#117—612no—
336#117—612no—
337#117—611no—
338#117—1221no—
339#117—1221no—
340C2×C4⋊C4—611no—
341C2×C4⋊C4C2×#692311yescharacteristic
342C2×C22⋊C4—1221no—
343C2×C22⋊C4—1221no—
344C2×C42—612no—
345C2×C42—1221no—
346#108—1224no—
347#107—612no—
348#107C2×#692312yes—
349#106—612no—
350#106—612no—
351C4⋊C8—1221no—
352C4⋊C8—1221no—
353#104—1224no—
354#103—612no—
355#103—612no—
356#102—612no—
357#102C2×#692312yes—
358C22⋊C8—611no—
359C22⋊C8—612no—
360C22⋊C8—1221no—
361C22⋊C8—1221no—
362C22⋊C8—1221no—
363C22⋊C8—1221no—
364C22⋊C8C2×#692311yescharacteristic
365#97—1224no—
366#97—1222no—
367#97—1222no—
368#97—1221no—
369C4×C8—1222no—
370C4×C8—1222no—
371C4×C8—1221no—
372#95—1221no—
373C24—611no—
374#41—1222no—
375#41—1222no—
376C2×Q8—611no—
377C2×D4—21no—
378C2×D4—1221no—
379C2×D4—1221no—
380C2×D4C22×#692311yescharacteristic
381C22×C4—21no—
382C22×C4—611no—
383C22×C4—611no—
384C22×C4—611no—
385C22×C4—611no—
386C22×C4—611no—
387C22×C4—1222no—
388C22×C4—1222no—
389C22×C4—1221no—
390C22×C4—1221no—
391C22×C4—1222no—
392C22×C4C22×#692311yescharacteristic
393M16—614no—
394M16—1222no—
395M16—1224no—
396M16—1222no—
397M16—1222no—
398M16—2442no—
399C2×C8—611no—
400C2×C8—611no—
401C2×C8—614no—
402C2×C8—611no—
403C2×C8—611no—
404C2×C8—612no—
405C2×C8—1222no—
406C2×C8—1222no—
407C2×C8—1222no—
408C2×C8—1222no—
409C2×C8—1222no—
410C2×C8—1222no—
411C2×C8—1222no—
412C2×C8—1222no—
413C2×C8—1224no—
414C2×C8—1221no—
415C2×C8—1221no—
416C2×C8—2442no—
417C2×C8C22×#692312yes—
418C4⋊C4—21no—
419C4⋊C4—612no—
420C4⋊C4—611no—
421C4⋊C4—1221no—
422C4⋊C4—1221no—
423C4⋊C4—1221no—
424C4⋊C4C22×#692312yes—
425C4⋊C4C22×#692311yescharacteristic
426C22⋊C4—21no—
427C22⋊C4—1222no—
428C22⋊C4—1222no—
429C22⋊C4—1221no—
430C22⋊C4—1221no—
431C22⋊C4—1221no—
432C22⋊C4—1221no—
433C22⋊C4—2442no—
434C42—1222no—
435C42—1221no—
436C42—1222no—
437C42—1222no—
438C42—1222no—
439C23—21no—
440C23—611no—
441C23—611no—
442C23—1222no—
443C23—1222no—
444C23—2441no—
445C23D4×#692311yescharacteristic
446Q8—1222no—
447D4—22no—
448D4—42no—
449D4—2442no—
450D4—2442no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—21no—
454C2×C4—42no—
455C2×C4—611no—
456C2×C4—612no—
457C2×C4—612no—
458C2×C4—611no—
459C2×C4—1221no—
460C2×C4—1221no—
461C2×C4—1222no—
462C2×C4—1221no—
463C2×C4—1221no—
464C2×C4—1222no—
465C2×C4—1222no—
466C2×C4—1221no—
467C2×C4—1221no—
468C2×C4—2441no—
469C2×C4—2441no—
470C2×C4—2442no—
471C2×C4—2442no—
472C2×C4—2442no—
473C2×C4—2442no—
474C2×C4D4×#692311yescharacteristic
475C2×C4C23×#692311yescharacteristic
476C8—22no—
477C8—612no—
478C8—612no—
479C8—1221no—
480C8—1221no—
481C8—1224no—
482C8—1222no—
483C8—2441no—
484C8—2441no—
485C22—21no—
486C22—21no—
487C22—41no—
488C22—41no—
489C22—41no—
490C22—612no—
491C22—1221no—
492C22—1221no—
493C22—1221no—
494C22—1221no—
495C22—1221no—
496C22—2444no—
497C22#1684080912yes—
498C22C2×D4×#692311yescharacteristic
499C4—21no—
500C4—42no—
501C4—41no—
502C4—611no—
503C4—611no—
504C4—1221no—
505C4—4882no—
506C4—2441no—
507C4—2442no—
508C4—2441no—
509C4#41×#692312yes—
510C2—22no—
511C2—81no—
512C2—1222no—
513C2—2441no—
514C2#123×#692311yescharacteristic
515C2C2×#1684080911yescharacteristic
516C2#2146259211yescharacteristic
subgroupG/H
centerC22C2×D4×#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC61×#480C4
socleC22×C61C2×C4×D4
prime radical#16840272C22
cosocleC2×C4×D61C24
hypercenter#480#6923
hypocenterC61#21593
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #480

Complements

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

NH ≅ G/Ntype
2. C2×#16840727C2semidirect
3. #21461901C2semidirect
4. #21461887C2semidirect
6. #21461772C2semidirect
7. #21461725C2semidirect
8. #21461626C2semidirect
9. #21461598C2semidirect
10. #21461558C2semidirect
40. #16840105C4semidirect
41. #16840098C4semidirect
42. #16840092C4semidirect
43. #16840084C4semidirect
44. #16840078C4semidirect
48. C61×#480C4semidirect
280. #480#6923semidirect
307. C61#21593semidirect

Maximal subgroups

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

subgroupindex
#214625332
C2×#168407272
#214619012
#214618872
#214618022
#214617722
#214617252
#214616262
#214615982
#214615582
#214557272
#2159361