Finite Group Databasebeta

← #30622211

Subgroups — #30622211

Order 15616.

880 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×#16840809C212yesmaximal, complemented
2#21462562C211yesmaximal, characteristic, complemented
3#21462464C211yesmaximal, characteristic, complemented
4#21461953C211yesmaximal, characteristic, complemented
5#21461934C211yesmaximal, characteristic, complemented
6#21461915C212yesmaximal, complemented
7#21461810C212yesmaximal, complemented
8#21461788C212yesmaximal, complemented
9#21461606C211yesmaximal, characteristic, complemented
10C4×#14519841C211yesmaximal, characteristic, complemented
11#21455267C211yesmaximal, characteristic
12C2×#14519864C2212yes—
13C2×#14519864C2212yes—
14#16840809—28no—
15#16840797—21no—
16#16840793—22no—
17D4×#92420—21no—
18C2×#14519847C2212yes—
19#16840769C2212yes—
20#16840768—22no—
21#16840766C2211yescharacteristic
22#16840765—21no—
23D61⋊(C4⋊C8)C2211yescharacteristic
24C2×C4⋊#92420—21no—
25C2×#14519841C2211yescharacteristic
26C2×#14519841C2212yes—
27C2×#14519841C2212yes—
28C2×#14519841C2211yescharacteristic
29#16840736C2212yes—
30C4×#11758213C2211yescharacteristic
31C2×#14519831C2212yes—
32C42×#6923C2211yescharacteristic
33#16840638C2212yes—
34#16840632C2211yescharacteristic
35#16840632C2211yescharacteristic
36(C2×#6923)⋊C8C2212yes—
37C2×#14519796C2212yes—
38#16840310C2211yescharacteristic
39C4×#11758224C2211yescharacteristic
40#16839983C412yescomplemented
41#16839975C412yescomplemented
42#16839973C411yescharacteristic, complemented
43#16839967C411yescharacteristic, complemented
44#16839959C411yescharacteristic, complemented
45C4⋊C8×D61C2211yescharacteristic
46#16839755C2212yes—
47C61×#459C411yescharacteristic, complemented
48#14519864—24no—
49#14519864—24no—
50#14519864—42no—
51#14519864—42no—
52C2×#11758213C2311yescharacteristic
53C2×#11758213C2312yes—
54C2×#11758213C2311yescharacteristic
55C2×#11758217—22no—
56C2×#11758217—22no—
57C2×D61⋊C8C2312yes—
58C22×#92420—22no—
59C22×#92420—22no—
60C2×C4×#6923—21no—
61C2×C4×#6923—21no—
62C2×C4×#6923C2311yescharacteristic
63C2×C4×#6923C2311yescharacteristic
64#14519847—24no—
65#14519844—22no—
66#14519844D412yes—
67C4⋊#92420—22no—
68C4⋊#92420—21no—
69C4⋊#92420—21no—
70C4⋊#92420D411yescharacteristic
71C4⋊#92420D411yescharacteristic
72#14519841—22no—
73#14519841—24no—
74#14519841—42no—
75#14519841D414yescomplemented
76C22⋊#92420—22no—
77C22⋊#92420—22no—
78#14519832—21no—
79#14519831—24no—
80C4×#92420—21no—
81C2×#11758224C2312yes—
82C2×#11758224C2311yescharacteristic
83#14519796—24no—
84#14519796—24no—
85C2×Dic244C2×C412yes—
86C2×#11758200C2×C412yes—
87C2×#11758200C2×C412yes—
88C2×C8×D61C2312yes—
89#14519777C2×C411yescharacteristic
90#14519764C2×C412yes—
91#14519748—21no—
92#14519748—21no—
93C4×Dic122C2×C411yescharacteristic
94#14519742C2×C411yescharacteristic
95D4×Dic61—21no—
96D4×Dic61—21no—
97D4×Dic61C2×C411yescharacteristic
98C4×C22⋊D61—21no—
99#14519737—21no—
100#14519735C2311yescharacteristic
101#14519733—21no—
102C2×C4×Dic61—21no—
103C42×D61C2311yescharacteristic
104C4⋊#92418C2×C411yescharacteristic
105#14519706C2×C412yes—
106#14519702C2×C412yes—
107#14519700C2×C412yes—
108#14519698C2×C412yes—
109C2×D8×C61C2×C412yes—
110#127×C61C2×C411yescharacteristic
111C4×D4×C61C2×C411yescharacteristic
112C4⋊C8×C61C2×C411yescharacteristic
113#102×C61C2×C412yes—
114C22×#6923—21no—
115C22×#6923—21no—
116C2×D4×C61—21no—
117C2×D4×C61—21no—
118C2×D4×C61C22×C412yes—
119C2×D4×C61C22×C411yescharacteristic
120C22×C4×C61—21no—
121C2×C22⋊D61—21no—
122C2×C22⋊D61—22no—
123C22×Dic61—21no—
124C22×Dic61—22no—
125C22×Dic61—21no—
126#11758224—22no—
127#11758224—22no—
128#11758224—21no—
129#11758224—22no—
130#11758224—21no—
131#11758224—22no—
132#11758224C2×D412yes—
133#11758224#4112yes—
134C2×C4×D61#4111yescharacteristic
135C2×C4×D61#4111yescharacteristic
136C2×C4×D61C2411yescharacteristic
137C2×Dic122C22×C412yes—
138C2×Dic122C22×C411yescharacteristic
139#11758217—44no—
140#11758217—44no—
141C2×#92420—21no—
142C2×#92420—21no—
143C2×#92420—22no—
144C2×#92420—44no—
145C2×#92420—44no—
146C2×#92420C2×D412yes—
147C2×#92420C2×D412yes—
148C4×#6923—21no—
149C4×#6923—21no—
150C4×#6923—21no—
151C4×#6923—42no—
152C4×#6923—42no—
153C4×#6923C2×D412yes—
154#11758213—21no—
155#11758213—22no—
156#11758213—21no—
157#11758213—22no—
158#11758213—21no—
159#11758213C2×D412yes—
160D61⋊C8—22no—
161D61⋊C8—22no—
162D8×C61—24no—
163C2×C8×C61C22×C412yes—
164C4⋊C4×C61C22×C411yescharacteristic
165C22⋊C4×C61—21no—
166C42×C61C22×C411yescharacteristic
167C22⋊Dic61—21no—
168C22⋊Dic61—22no—
169C22⋊Dic61—22no—
170#11758200—24no—
171#11758200—24no—
172#11758198—21no—
173C4⋊Dic61—21no—
174C4⋊Dic61—21no—
175C4⋊Dic61C22×C411yescharacteristic
176#11758196—21no—
177#11758196—21no—
178C4×Dic61—21no—
179C4×Dic61—21no—
180C4×Dic61—21no—
181C4×Dic61C22×C411yescharacteristic
182C4×Dic61C2×D411yescharacteristic
183C4×Dic61C2×D411yescharacteristic
184C2×#92418C22×C412yes—
185Dic244—24no—
186C8×D61—22no—
187C8×D61—22no—
188C23×C61—21no—
189C23×C61—22no—
190C22×D61C2×#4111yescharacteristic
191C2×#6923—22no—
192C2×#6923—22no—
193C2×#6923—41no—
194C2×#6923—41no—
195C2×#6923—41no—
196C2×#6923—41no—
197D4×C61—22no—
198D4×C61—21no—
199D4×C61—22no—
200D4×C61—42no—
201D4×C61—81no—
202D4×C61C4×D412yes—
203C2×C4×C61—21no—
204C2×C4×C61—42no—
205C2×C4×C61C4×D411yescharacteristic
206C2×C4×C61C4×D411yescharacteristic
207C2×C4×C61C23×C411yescharacteristic
208C22⋊D61—22no—
209C22⋊D61—22no—
210C22⋊D61—42no—
211C22⋊D61—42no—
212C2×Dic61—21no—
213C2×Dic61—22no—
214C2×Dic61—22no—
215C2×Dic61—21no—
216C2×Dic61—22no—
217C2×Dic61—42no—
218C2×Dic61—42no—
219C2×Dic61—81no—
220C2×Dic61C4×D411yescharacteristic
221C2×Dic61C4×D411yescharacteristic
222C2×Dic61C22×D411yescharacteristic
223C4×D61—22no—
224C4×D61—41no—
225C4×D61C22×D411yescharacteristic
226C4×D61C2×#4111yescharacteristic
227C4×D61H(5,𝔽2)11yescharacteristic
228C4×D612−1+411yescharacteristic
229Dic122—21no—
230Dic122—22no—
231Dic122—22no—
232Dic122C4×D412yes—
233#92420—22no—
234#92420—22no—
235#92420—22no—
236#92420—42no—
237#92420—41no—
238C8×C61—22no—
239#92418—22no—
240#23016—611nomaximal, Sylow
241C22×C61—22no—
242C22×C61—41no—
243C22×C61—41no—
244C22×C61—42no—
245C22×C61—42no—
246C22×C61—42no—
247C22×C61C2×C4×D411yescharacteristic
248C2×D61—21no—
249C2×D61—21no—
250C2×D61#54711yescharacteristic
251C2×D61#55311yescharacteristic
252#6923—42no—
253#6923—42no—
254C4×C61—21no—
255C4×C61—41no—
256C4×C61#44412yescomplemented
257C4×C61C2×C4×D411yescharacteristic
258C4×C61#52011yescharacteristic
259Dic61—22no—
260Dic61—21no—
261Dic61—21no—
262Dic61—42no—
263Dic61—41no—
264Dic61D4211yescharacteristic
265Dic61#54811yescharacteristic
266#3035—611no—
267C2×#444—612no—
268#2935—611no—
269#2906—611no—
270#1962—611no—
271#1950—611no—
272#1938—612no—
273#1876—612no—
274#1866—612no—
275#1767—611no—
276C4×#104—611no—
277C2×C61—42no—
278C2×C61—82no—
279C2×C61#236111yescharacteristic
280C2×C61C2×#44411yescharacteristic
281C2×C61#295611yescharacteristic
282D61—21no—
283D61—21no—
284C2×#135—612no—
285C2×#131—612no—
286C2×#131—612no—
287#532—611no—
288C4×#41—611no—
289#464—612no—
290#463—612no—
291#462—611no—
292#461—611no—
293#460—611no—
294#459#692311yescharacteristic, complemented
295#444—1228no—
296#436—1221no—
297#435—1222no—
298C8×D4—1221no—
299C2×#108—612no—
300#424—612no—
301#424—1222no—
302#423—611no—
303#423—1221no—
304C2×C4⋊C8—611no—
305C2×C4⋊C8—611no—
306C2×C4⋊C8—1221no—
307C2×#104—611no—
308C2×#104—611no—
309C2×#104—612no—
310C2×#104—612no—
311#416—612no—
312#408—612no—
313C4×M16—611no—
314C2×#97—612no—
315C43—611no—
316#341—612no—
317#339—611no—
318#339—611no—
319#336—612no—
320C61#2301611yescharacteristic, complemented, Sylow
321C2×#41—611no—
322C2×#41—612no—
323#135—1224no—
324#135—1224no—
325C2×Q16—612no—
326C2×SD16—612no—
327C2×SD16—612no—
328C2×D8C2×#692312yes—
329#131—1224no—
330#131—1224no—
331#131—2442no—
332#131—2442no—
333C2×M16—611no—
334C2×M16—612no—
335C2×M16—611no—
336C2×M16—1222no—
337C2×M16—1222no—
338C22×C8—612no—
339C22×C8—612no—
340C22×C8—1222no—
341C22×C8—1222no—
342C4⋊Q8—611no—
343#127C2×#692311yescharacteristic
344#124—612no—
345#121—1221no—
346#121—1221no—
347C4×Q8—611no—
348C4×Q8—611no—
349C4×D4—611no—
350C4×D4—1221no—
351C4×D4—1221no—
352C4×D4—1221no—
353C4×D4—1221no—
354C4×D4C2×#692311yescharacteristic
355#117—611no—
356#117—1221no—
357C2×C42—611no—
358C2×C42—611no—
359C2×C42—611no—
360C2×C42—1221no—
361C2×C42—1221no—
362C2×C42—1221no—
363#108—1224no—
364C4⋊C8—611no—
365C4⋊C8—611no—
366C4⋊C8—612no—
367C4⋊C8—612no—
368C4⋊C8—611no—
369C4⋊C8—1221no—
370C4⋊C8—1221no—
371C4⋊C8—1222no—
372C4⋊C8—1222no—
373C4⋊C8C2×#692311yescharacteristic
374#104—614no—
375#104—1224no—
376#104—1222no—
377#104—2442no—
378#103—612no—
379#103—612no—
380#102—612no—
381#102C2×#692312yes—
382C22⋊C8—1222no—
383C22⋊C8—1222no—
384#97—1224no—
385#97—1221no—
386C4×C8—1221no—
387#41—612no—
388#41—612no—
389#41—1221no—
390#41—1221no—
391#41—1222no—
392#41—1222no—
393#41—1222no—
394#41—1222no—
395C2×Q8—611no—
396C2×Q8—612no—
397C2×D4—21no—
398C2×D4—21no—
399C2×D4—1221no—
400C2×D4—1222no—
401C2×D4C22×#692312yes—
402C2×D4C22×#692311yescharacteristic
403C22×C4—21no—
404C22×C4—611no—
405C22×C4—611no—
406C22×C4—611no—
407C22×C4—1221no—
408C22×C4—1221no—
409C22×C4—1222no—
410C22×C4—1221no—
411C22×C4—1221no—
412Q16—1224no—
413SD16—1224no—
414SD16—1224no—
415D8—24no—
416M16—612no—
417M16—1221no—
418M16—1222no—
419M16—1221no—
420M16—1222no—
421M16—1221no—
422M16—2444no—
423M16—2444no—
424C2×C8—612no—
425C2×C8—612no—
426C2×C8—612no—
427C2×C8—1222no—
428C2×C8—1222no—
429C2×C8—1221no—
430C2×C8—1222no—
431C2×C8—1222no—
432C2×C8—1221no—
433C2×C8—1222no—
434C2×C8—2444no—
435C2×C8—2444no—
436C2×C8C22×#692312yes—
437C4⋊C4—611no—
438C4⋊C4—1221no—
439C4⋊C4—1221no—
440C4⋊C4—1221no—
441C4⋊C4—1221no—
442C4⋊C4C22×#692311yescharacteristic
443C22⋊C4—21no—
444C22⋊C4—1221no—
445C22⋊C4—1221no—
446C22⋊C4—1222no—
447C22⋊C4—1222no—
448C42—611no—
449C42—611no—
450C42—611no—
451C42—612no—
452C42—1221no—
453C42—1221no—
454C42—1221no—
455C42—1221no—
456C42—1221no—
457C42—1221no—
458C42—2442no—
459C42—2442no—
460C42C22×#692311yescharacteristic
461C23—21no—
462C23—22no—
463C23—611no—
464Q8—612no—
465Q8—1221no—
466Q8—1222no—
467Q8—1222no—
468D4—22no—
469D4—21no—
470D4—22no—
471D4—42no—
472D4—81no—
473D4—1222no—
474D4—1222no—
475D4—2442no—
476D4—2442no—
477D4D4×#692312yes—
478C2×C4—21no—
479C2×C4—42no—
480C2×C4—611no—
481C2×C4—611no—
482C2×C4—611no—
483C2×C4—611no—
484C2×C4—611no—
485C2×C4—611no—
486C2×C4—611no—
487C2×C4—1222no—
488C2×C4—1222no—
489C2×C4—1221no—
490C2×C4—1222no—
491C2×C4—1222no—
492C2×C4—1222no—
493C2×C4—1221no—
494C2×C4—1222no—
495C2×C4—4881no—
496C2×C4—2441no—
497C2×C4—2442no—
498C2×C4—2442no—
499C2×C4—2441no—
500C2×C4—2441no—
501C2×C4—2441no—
502C2×C4—2441no—
503C2×C4D4×#692311yescharacteristic
504C2×C4D4×#692311yescharacteristic
505C2×C4C23×#692311yescharacteristic
506C8—22no—
507C8—1222no—
508C8—1222no—
509C8—1222no—
510C8—1222no—
511C8—2442no—
512C8—2441no—
513C22—22no—
514C22—41no—
515C22—41no—
516C22—42no—
517C22—42no—
518C22—42no—
519C22—611no—
520C22—611no—
521C22—1221no—
522C22—1221no—
523C22C2×D4×#692311yescharacteristic
524C4—21no—
525C4—41no—
526C4—611no—
527C4—611no—
528C4—1221no—
529C4—1221no—
530C4—1222no—
531C4—2442no—
532C4—2442no—
533C4—2441no—
534C4—2442no—
535C4#1684080912yescomplemented
536C4C2×D4×#692311yescharacteristic
537C4#1684083111yescharacteristic
538C2—42no—
539C2—82no—
540C2—1221no—
541C2—1221no—
542C2#2146234411yescharacteristic
543C2C2×#1684080911yescharacteristic
544C2#2146259311yescharacteristic
subgroupG/H
centerC22C2×D4×#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC61×#459C4
socleC22×C61C2×C4×D4
prime radical#21455267C2
cosocleC2×C4×D61C24
hypercenter#459#6923
hypocenterC61#23016
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #459

Complements

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

NH ≅ G/Ntype
1. C2×#16840809C2semidirect
2. #21462562C2semidirect
3. #21462464C2semidirect
4. #21461953C2semidirect
5. #21461934C2semidirect
6. #21461915C2semidirect
7. #21461810C2semidirect
8. #21461788C2semidirect
9. #21461606C2semidirect
10. C4×#14519841C2semidirect
40. #16839983C4semidirect
41. #16839975C4semidirect
42. #16839973C4semidirect
43. #16839967C4semidirect
44. #16839959C4semidirect
47. C61×#459C4semidirect
75. #14519841D4semidirect
256. C4×C61#444semidirect
294. #459#6923semidirect
320. C61#23016semidirect
535. C4#16840809semidirect

Maximal subgroups

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

subgroupindex
C2×#168408092
#214625622
#214624642
#214619532
#214619342
#214619152
#214618102
#214617882
#214616062
C4×#145198412
#214552672
#2301661