Finite Group Databasebeta

← #30616409

Subgroups — #30616409

Order 15616.

746 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462504C211yesmaximal, characteristic, complemented
2#21462503C211yesmaximal, characteristic, complemented
3#21461975C212yesmaximal
4#21461929C212yesmaximal, complemented
5#21461915C212yesmaximal, complemented
6#21461914C212yesmaximal, complemented
7C4×#14519848C211yesmaximal, characteristic, complemented
8C4×#14519842C211yesmaximal, characteristic, complemented
9C4×#14519841C211yesmaximal, characteristic, complemented
10C4×#14519831C211yesmaximal, characteristic, complemented
11C4×#14519798C211yesmaximal, characteristic
12#16840787—22no—
13#16840786—22no—
14#16840781—21no—
15#16840780—21no—
16#16840779—21no—
17#16840778—21no—
18C2×#14519848C2212yes—
19#16840769C2212yes—
20#16840769C2212yes—
21C2×#14519842C2212yes—
22C2×#14519841C2212yes—
23#16840737C2212yes—
24#16840736C2212yes—
25C4×#11758213C2211yescharacteristic
26C4×#11758213C2211yescharacteristic
27C2×#14519831C2212yes—
28C2×#14519831C2212yes—
29C4×D61⋊C8C2211yescharacteristic
30C42×#6923C2211yescharacteristic
31#16840638C2212yes—
32#16840632C2212yes—
33#16840632C2212yes—
34C2×#14519798C2211yescharacteristic
35C4×#11758226C2211yescharacteristic
36C4×#11758224C2211yescharacteristic
37C4×#11758202C411yescharacteristic, complemented
38#16839854C411yescharacteristic, complemented
39C4×#11758190C411yescharacteristic, complemented
40SD16×Dic61C411yescharacteristic, complemented
41#16839849C411yescharacteristic, complemented
42C4×#11758199C411yescharacteristic, complemented
43#16839842C411yescharacteristic, complemented
44#16839797C2211yescharacteristic
45#16839757C2211yescharacteristic
46#16839755C2211yescharacteristic
47C4×C8×D61C2211yescharacteristic
48C4×SD16×C61C411yescharacteristic, complemented
49C2×#11758213C2312yes—
50C2×#11758213C2312yes—
51C2×D61⋊C8C2312yes—
52C2×C4×#6923—21no—
53C2×C4×#6923—21no—
54C2×C4×#6923C2312yes—
55#14519848—24no—
56#14519844—22no—
57#14519844—22no—
58C4⋊#92420—22no—
59C4⋊#92420—22no—
60#14519842—24no—
61#14519841—24no—
62#14519834—22no—
63#14519834—22no—
64C22⋊#92420—22no—
65C22⋊#92420—22no—
66#14519832—21no—
67#14519832—21no—
68#14519831—24no—
69#14519831—24no—
70C4×#92420—21no—
71C4×#92420—21no—
72C4×#92420D411yescharacteristic
73C4×#92420D411yescharacteristic
74C4×#92420D412yes—
75C2×#11758226C2311yescharacteristic
76C2×#11758224C2311yescharacteristic
77#14519798—24no—
78C2×#11758202C2×C411yescharacteristic
79C2×#11758190C2×C411yescharacteristic
80C2×#11758199C2×C411yescharacteristic
81C2×C8×D61C2311yescharacteristic
82C2×C8×D61C2311yescharacteristic
83Q8×Dic61C2×C411yescharacteristic
84C4×Dic122C2×C411yescharacteristic
85#14519742C2×C411yescharacteristic
86C4×D244—21no—
87C4×D244C2×C411yescharacteristic
88D4×Dic61C2×C411yescharacteristic
89#14519739—21no—
90#14519739C2×C411yescharacteristic
91C4×C22⋊D61—21no—
92#14519737—21no—
93#14519735—21no—
94#14519735C2311yescharacteristic
95#14519735C2311yescharacteristic
96#14519733—21no—
97C2×C4×Dic61—21no—
98C42×D61—21no—
99C42×D61C2311yescharacteristic
100#14519710C2×C411yescharacteristic
101#14519710C2×C411yescharacteristic
102#14519709C2×C411yescharacteristic
103#14519702C2×C411yescharacteristic
104#14519701C2×C411yescharacteristic
105#14519700C2×C411yescharacteristic
106#14519699C2×C411yescharacteristic
107#14519698C2×C411yescharacteristic
108#14519697C2×C411yescharacteristic
109C4×#92418C2×C411yescharacteristic
110C8×Dic61C2×C411yescharacteristic
111C8×Dic61C2×C411yescharacteristic
112C2×SD16×C61C2×C411yescharacteristic
113C4×Q8×C61C2×C411yescharacteristic
114C4×D4×C61C2×C411yescharacteristic
115#106×C61C2×C411yescharacteristic
116#103×C61C2×C411yescharacteristic
117#102×C61C2×C411yescharacteristic
118C4×C8×C61C2×C411yescharacteristic
119C22×#6923—22no—
120C2×Q8×C61C22×C411yescharacteristic
121C2×D4×C61C22×C411yescharacteristic
122C22×C4×C61—21no—
123C2×C22⋊D61—21no—
124C22×Dic61—21no—
125C22×Dic61—21no—
126#11758226—22no—
127#11758226—22no—
128#11758224—22no—
129#11758224—22no—
130C2×D244—21no—
131C2×D244C22×C411yescharacteristic
132C2×C4×D61—21no—
133C2×C4×D61—21no—
134C2×C4×D61—21no—
135C2×C4×D61#4111yescharacteristic
136C2×C4×D61#4111yescharacteristic
137C2×C4×D61C2411yescharacteristic
138C2×Dic122C22×C411yescharacteristic
139C2×#92420—22no—
140C2×#92420—22no—
141C2×#92420C2×D412yes—
142C2×#92420C2×D412yes—
143C4×#6923—22no—
144C4×#6923—42no—
145C4×#6923—42no—
146C4×#6923#4114yes—
147#11758213—22no—
148#11758213—22no—
149#11758213—22no—
150#11758213—22no—
151D61⋊C8—22no—
152D61⋊C8—22no—
153SD16×C61—22no—
154C2×C8×C61C22×C411yescharacteristic
155C2×C8×C61C22×C411yescharacteristic
156C4⋊C4×C61—21no—
157C4⋊C4×C61C22×C411yescharacteristic
158C4⋊C4×C61C22×C411yescharacteristic
159C22⋊C4×C61—21no—
160C42×C61—21no—
161C42×C61C22×C411yescharacteristic
162C22⋊Dic61—21no—
163#11758202—22no—
164#11758199—22no—
165#11758198—21no—
166#11758198—21no—
167#11758198—21no—
168#11758198—21no—
169C4⋊Dic61—21no—
170C4⋊Dic61C22×C411yescharacteristic
171C4⋊Dic61C22×C411yescharacteristic
172#11758196—21no—
173#11758196—21no—
174C4×Dic61—21no—
175C4×Dic61—21no—
176C4×Dic61—21no—
177C4×Dic61—21no—
178C4×Dic61—21no—
179C4×Dic61C22×C411yescharacteristic
180C4×Dic61C2×D411yescharacteristic
181C4×Dic61C2×D411yescharacteristic
182C2×#92418C22×C411yescharacteristic
183C2×#92418C22×C411yescharacteristic
184#11758190—22no—
185C8×D61—22no—
186C8×D61#4114yes—
187C23×C61—21no—
188C22×D61—21no—
189C22×D61C2×#4111yescharacteristic
190C2×#6923—24no—
191C2×#6923—42no—
192C2×#6923—42no—
193Q8×C61—21no—
194Q8×C61—21no—
195D4×C61—21no—
196D4×C61—21no—
197C2×C4×C61—21no—
198C2×C4×C61—21no—
199C2×C4×C61—21no—
200C2×C4×C61—42no—
201C2×C4×C61C4×D411yescharacteristic
202C2×C4×C61C4×D411yescharacteristic
203C2×C4×C61C23×C411yescharacteristic
204C22⋊D61—42no—
205C2×Dic61—21no—
206C2×Dic61—21no—
207C2×Dic61—21no—
208C2×Dic61—42no—
209C2×Dic61—42no—
210C2×Dic61C4×D411yescharacteristic
211C2×Dic61C4×D411yescharacteristic
212C2×Dic61C22×D411yescharacteristic
213D244—21no—
214D244—21no—
215D244—42no—
216C4×D61—22no—
217C4×D61—22no—
218C4×D61—42no—
219C4×D61—42no—
220C4×D61—42no—
221C4×D61C2×#4112yes—
222C4×D61C2×#4112yes—
223Dic122—21no—
224Dic122—21no—
225#92420—22no—
226#92420—22no—
227#92420—42no—
228#92420—42no—
229C8×C61—21no—
230C8×C61#11712yes—
231#92418—21no—
232#92418#11712yes—
233#21293—611nomaximal, Sylow
234C22×C61—41no—
235C22×C61—41no—
236C22×C61—41no—
237C22×C61C2×C4×D411yescharacteristic
238C2×D61—21no—
239C2×D61—21no—
240C2×D61—41no—
241C2×D61—41no—
242C2×D61—41no—
243C2×D61#52912yes—
244#6923—44no—
245C4×C61—21no—
246C4×C61—41no—
247C4×C61—41no—
248C4×C61—41no—
249C4×C61#44412yes—
250C4×C61C2×#11711yescharacteristic
251C4×C61C4×#4111yescharacteristic
252Dic61—21no—
253Dic61—41no—
254Dic61—41no—
255Dic61—41no—
256Dic61#44412yes—
257Dic61#53212yes—
258C4×#135—611no—
259#2922—611no—
260#2922—611no—
261#1975—612no—
262#1946—612no—
263#1938—612no—
264#1938—612no—
265C4×#108—611no—
266C4×#104—611no—
267C4×#104—611no—
268C4×#97—611no—
269C2×C61—81no—
270C2×C61#229311yescharacteristic
271C2×C61C2×#44411yescharacteristic
272C2×C61C2×#44411yescharacteristic
273D61—22no—
274D61—81no—
275C2×#135—611no—
276C4×#41—611no—
277C4×#41—611no—
278C4×Q16—611no—
279C4×Q16—611no—
280C4×SD16—611no—
281C4×SD16—611no—
282C4×SD16—611no—
283C4×SD16#692311yescharacteristic, complemented
284C4×D8—611no—
285C4×D8—611no—
286#433—1222no—
287#433—1222no—
288#431—1221no—
289#431—1221no—
290#431—1221no—
291#431—1221no—
292C2×#108—612no—
293#427—611no—
294#424—612no—
295#424—612no—
296C2×#104—612no—
297C2×#104—612no—
298#416—611no—
299#416—611no—
300#408—612no—
301#408—612no—
302C4×M16—611no—
303C4×M16—611no—
304C2×#97—612no—
305C2×#97—612no—
306C2×C4×C8—611no—
307C2×C4×C8—611no—
308C43—611no—
309#341—612no—
310#339—612no—
311#339—612no—
312C61#2129311yescharacteristic, complemented, Sylow
313C2×#41—611no—
314C2×#41—611no—
315#135—1224no—
316C2×Q16—611no—
317C2×SD16—611no—
318C2×SD16C2×#692311yescharacteristic
319C2×D8—611no—
320C2×M16—612no—
321C2×M16—612no—
322C22×C8—611no—
323C22×C8—612no—
324C22×C8—611no—
325C4×Q8—611no—
326C4×Q8—611no—
327C4×Q8—611no—
328C4×Q8C2×#692311yescharacteristic
329C4×D4—611no—
330C4×D4—611no—
331C4×D4—611no—
332C4×D4—1221no—
333C4×D4—1221no—
334C4×D4—1221no—
335C4×D4—1221no—
336C4×D4C2×#692311yescharacteristic
337#117—611no—
338#117—611no—
339#117—1221no—
340#117—1221no—
341C2×C42—611no—
342C2×C42—612no—
343C2×C42—1221no—
344C2×C42—1221no—
345C2×C42—1221no—
346C2×C42—1221no—
347#108—1224no—
348#107—611no—
349#107—611no—
350#106—611no—
351#106C2×#692311yescharacteristic
352C4⋊C8—1222no—
353C4⋊C8—1222no—
354C4⋊C8—1222no—
355C4⋊C8—1222no—
356#104—1224no—
357#104—1224no—
358#103—611no—
359#103—611no—
360#103—611no—
361#103C2×#692311yescharacteristic
362#102—611no—
363#102—611no—
364#102—611no—
365#102C2×#692311yescharacteristic
366C22⋊C8—1222no—
367C22⋊C8—1222no—
368C22⋊C8—1222no—
369C22⋊C8—1222no—
370#97—1224no—
371#97—1221no—
372#97—1221no—
373#97—1224no—
374C4×C8—611no—
375C4×C8—611no—
376C4×C8—612no—
377C4×C8—611no—
378C4×C8—611no—
379C4×C8—611no—
380C4×C8—1221no—
381C4×C8—1221no—
382C4×C8C2×#692311yescharacteristic
383#41—1222no—
384#41—1222no—
385#41—1222no—
386#41—1222no—
387C2×Q8—611no—
388C2×Q8C22×#692311yescharacteristic
389C2×D4—611no—
390C2×D4—1221no—
391C2×D4—1221no—
392C2×D4C22×#692311yescharacteristic
393C22×C4—21no—
394C22×C4—611no—
395C22×C4—611no—
396C22×C4—611no—
397C22×C4—1221no—
398C22×C4—1221no—
399C22×C4—1221no—
400C22×C4—1221no—
401C22×C4—1222no—
402C22×C4—1221no—
403Q16—1222no—
404SD16—22no—
405SD16—1222no—
406D8—1222no—
407M16—1222no—
408M16—1222no—
409M16—1222no—
410M16—1222no—
411C2×C8—611no—
412C2×C8—614no—
413C2×C8—612no—
414C2×C8—612no—
415C2×C8—611no—
416C2×C8—1222no—
417C2×C8—1222no—
418C2×C8—1222no—
419C2×C8—1222no—
420C2×C8—1222no—
421C2×C8C22×#692311yescharacteristic
422C2×C8C22×#692311yescharacteristic
423C4⋊C4—21no—
424C4⋊C4—611no—
425C4⋊C4—611no—
426C4⋊C4—1221no—
427C4⋊C4—1221no—
428C4⋊C4—1221no—
429C4⋊C4C22×#692311yescharacteristic
430C4⋊C4C22×#692311yescharacteristic
431C22⋊C4—21no—
432C22⋊C4—1221no—
433C22⋊C4—1221no—
434C22⋊C4—1221no—
435C22⋊C4—1221no—
436C22⋊C4—1221no—
437C42—21no—
438C42—611no—
439C42—611no—
440C42—611no—
441C42—614no—
442C42—1221no—
443C42—1221no—
444C42—1221no—
445C42—1221no—
446C42—1222no—
447C42—1221no—
448C42—2442no—
449C42—2442no—
450C42C22×#692311yescharacteristic
451C23—21no—
452C23—611no—
453C23—1221no—
454Q8—21no—
455Q8—21no—
456Q8—1221no—
457Q8—1221no—
458D4—21no—
459D4—21no—
460D4—1221no—
461D4—1221no—
462D4—2442no—
463D4—2442no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—42no—
468C2×C4—611no—
469C2×C4—611no—
470C2×C4—612no—
471C2×C4—612no—
472C2×C4—611no—
473C2×C4—1222no—
474C2×C4—1222no—
475C2×C4—1224no—
476C2×C4—1221no—
477C2×C4—1221no—
478C2×C4—1221no—
479C2×C4—2442no—
480C2×C4—2442no—
481C2×C4—2442no—
482C2×C4—2442no—
483C2×C4—2442no—
484C2×C4—2442no—
485C2×C4—2442no—
486C2×C4D4×#692311yescharacteristic
487C2×C4D4×#692311yescharacteristic
488C2×C4C23×#692311yescharacteristic
489C8—21no—
490C8—612no—
491C8—1222no—
492C8—1222no—
493C8—1221no—
494C8—2442no—
495C8—2442no—
496C8#1451985612yes—
497C22—41no—
498C22—41no—
499C22—41no—
500C22—612no—
501C22—1221no—
502C22—1221no—
503C22—2441no—
504C22—2441no—
505C22—2441no—
506C22C2×D4×#692311yescharacteristic
507C4—21no—
508C4—41no—
509C4—41no—
510C4—41no—
511C4—612no—
512C4—612no—
513C4—1221no—
514C4—2444no—
515C4—2441no—
516C4—2441no—
517C4—2441no—
518C4#1684080712yes—
519C4C2×#1451985611yescharacteristic
520C4#41×#692311yescharacteristic
521C2—81no—
522C2—1222no—
523C2—4881no—
524C2#2146227611yescharacteristic
525C2C2×#1684080711yescharacteristic
526C2D61⋊#44411yescharacteristic
subgroupG/H
centerC2×C4D4×#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC4×SD16×C61C4
socleC22×C61C2×C4×D4
prime radicalC2×#14519798C22
cosocleC2×C4×D61C24
hypercenterC4×SD16#6923
hypocenterC61#21293
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C42 ◁ C4×SD16

Complements

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

NH ≅ G/Ntype
1. #21462504C2semidirect
2. #21462503C2semidirect
4. #21461929C2semidirect
5. #21461915C2semidirect
6. #21461914C2semidirect
7. C4×#14519848C2semidirect
8. C4×#14519842C2semidirect
9. C4×#14519841C2semidirect
10. C4×#14519831C2semidirect
37. C4×#11758202C4semidirect
38. #16839854C4semidirect
39. C4×#11758190C4semidirect
40. SD16×Dic61C4semidirect
41. #16839849C4semidirect
42. C4×#11758199C4semidirect
43. #16839842C4semidirect
48. C4×SD16×C61C4semidirect
283. C4×SD16#6923semidirect
312. C61#21293semidirect

Maximal subgroups

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

subgroupindex
#214625042
#214625032
#214619752
#214619292
#214619152
#214619142
C4×#145198482
C4×#145198422
C4×#145198412
C4×#145198312
C4×#145197982
#2129361