Finite Group Databasebeta

← #29648521

Subgroups — #29648521

Order 15616.

940 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#21459726C211yesmaximal, characteristic
2#21454928C211yesmaximal, characteristic, complemented
3#21454487C211yesmaximal, characteristic, complemented
4#21444521C212yesmaximal, complemented
5#21444516C212yesmaximal, complemented
6#21444505C211yesmaximal, characteristic, complemented
7#21444482C212yesmaximal, complemented
8#21444468C211yesmaximal, characteristic, complemented
9#21444465C212yesmaximal, complemented
10C2×#16839509C211yesmaximal, characteristic, complemented
11C61×#1547C211yesmaximal, characteristic, complemented
12#16840482—21no—
13#16840477C412yes—
14#16840469C412yes—
15#16840464C414yes—
16#16840418—22no—
17#16840416—21no—
18#16840381—22no—
19#16840340—21no—
20#16840317C2212yes—
21#16840310C2212yes—
22C2×#14519777C2211yescharacteristic
23#16840245—21no—
24C4×#11758222C2211yescharacteristic
25#16839886—21no—
26#16839827—21no—
27C8×C22⋊D61—22no—
28#16839822—22no—
29#16839818C2212yes—
30#16839815C2211yescharacteristic
31#16839814C2212yes—
32#16839814C2212yes—
33C2×C4⋊#92418C2211yescharacteristic
34C4⋊C8×D61C2212yes—
35#16839730—21no—
36#16839720—21no—
37#16839509C2212yes—
38#16839509C2212yes—
39#16839508C2211yescharacteristic
40#16839508C2211yescharacteristic
41#16839507C2212yes—
42#16839507C2212yes—
43#16839506C2211yescharacteristic
44#16839506C2211yescharacteristic
45C61×#532C2211yescharacteristic
46C61×#423C2211yescharacteristic
47C2×C4⋊C8×C61C2211yescharacteristic
48C61×#332C2212yes—
49C61×#331C2212yes—
50C2×#11758222D411yescharacteristic
51C2×#11758224—22no—
52C22×Dic122—21no—
53C22×#92418—21no—
54C2×C8×D61—22no—
55#14519777—21no—
56#14519777—21no—
57#14519777C2×C412yes—
58#14519777C2312yes—
59#14519770—24no—
60#14519768C2×C414yes—
61#14519764—22no—
62#14519763—24no—
63#14519761—24no—
64#14519757—22no—
65#14519756—22no—
66#14519754—24no—
67#14519753—22no—
68#14519751—22no—
69#14519749—24no—
70#14519748—22no—
71#14519748C2×C414yes—
72Q8×Dic61—22no—
73C4×Dic122C2×C411yescharacteristic
74C4×Dic122D411yescharacteristic
75#14519742—22no—
76C4×D244C2×C411yescharacteristic
77C4×D244D411yescharacteristic
78D4×Dic61—24no—
79D4×Dic61—22no—
80D4×Dic61C2×C414yes—
81C4×C22⋊D61C2×C412yes—
82C4×C22⋊D61D412yes—
83#14519736—22no—
84#14519736D411yescharacteristic
85#14519735C2×C414yes—
86#14519734D412yes—
87C2×C4⋊Dic61—21no—
88C2×C4⋊Dic61—22no—
89C2×C4⋊Dic61C2×C412yes—
90C42×D61C2312yes—
91C4⋊#92418—21no—
92C4⋊#92418C2311yescharacteristic
93C4⋊#92418C2311yescharacteristic
94C4⋊#92418C2311yescharacteristic
95C4⋊#92418C2311yescharacteristic
96#14519706—22no—
97C22⋊#92418—21no—
98#14519689—22no—
99#14519689—22no—
100#14519689—22no—
101#14519687—22no—
102C4×#92418—21no—
103C8×Dic61—22no—
104C8×Dic61—22no—
105C2×#41×C61—21no—
106C2×M16×C61—21no—
107C22×C8×C61—21no—
108C4⋊Q8×C61C2311yescharacteristic
109#127×C61C2311yescharacteristic
110#124×C61—21no—
111#121×C61—21no—
112#121×C61C2×C412yes—
113C4×D4×C61—21no—
114C4×D4×C61C2×C412yes—
115C2×C42×C61C2311yescharacteristic
116C4⋊C8×C61—21no—
117C4⋊C8×C61—21no—
118C4⋊C8×C61C2312yes—
119C4⋊C8×C61C2312yes—
120#41×C61—42no—
121#41×C61—42no—
122C2×Q8×C61—21no—
123C2×D4×C61—22no—
124C2×D4×C61—21no—
125C2×D4×C61—21no—
126C2×D4×C61C22×C412yes—
127C22×C4×C61—22no—
128C22×C4×C61C22×C411yescharacteristic
129C22×C4×C61C2×D411yescharacteristic
130C22×C4×C61C2×D411yescharacteristic
131C2×C22⋊D61—24no—
132C2×C22⋊D61C22⋊C412yes—
133C22×Dic61—22no—
134C22×Dic61—24no—
135#11758224—82no—
136#11758224—82no—
137#11758222—22no—
138#11758222—22no—
139C2×D244C22⋊C411yescharacteristic
140C2×C4×D61C22×C412yes—
141C2×C4×D61C2×D412yes—
142C2×C4×D61C2×D412yes—
143C2×Dic122—22no—
144C2×Dic122—22no—
145C2×Dic122—81no—
146C2×Dic122—81no—
147C2×Dic122C22⋊C411yescharacteristic
148M16×C61—42no—
149C2×C8×C61—22no—
150C2×C8×C61—22no—
151C2×C8×C61—81no—
152C4⋊C4×C61—21no—
153C4⋊C4×C61C22×C412yes—
154C22⋊C4×C61—22no—
155C22⋊C4×C61—22no—
156C42×C61C22×C411yescharacteristic
157C42×C61C2×D411yescharacteristic
158C42×C61C2×D411yescharacteristic
159C42×C61C2411yescharacteristic
160C22⋊Dic61—24no—
161C22⋊Dic61—24no—
162C22⋊Dic61C22⋊C412yes—
163#11758198—24no—
164#11758198C22⋊C412yes—
165#11758198C22⋊C412yes—
166C4⋊Dic61—22no—
167C4⋊Dic61—22no—
168C4⋊Dic61—22no—
169C4⋊Dic61—24no—
170C4⋊Dic61C22⋊C411yescharacteristic
171C4⋊Dic61C22⋊C411yescharacteristic
172C4⋊Dic61C22×C414yes—
173#11758196—24no—
174#11758196C22⋊C412yes—
175#11758196C22⋊C412yes—
176C4×Dic61—24no—
177C4×Dic61C22×C412yes—
178C4×Dic61C2×D412yes—
179C4×Dic61C2×D412yes—
180C2×#92418—21no—
181C2×#92418—21no—
182C2×#92418—21no—
183C2×#92418—21no—
184C2×#92418—42no—
185C8×D61—82no—
186C23×C61—22no—
187C23×C61C2×C22⋊C411yescharacteristic
188C22×D61C2×C22⋊C412yes—
189Q8×C61—41no—
190Q8×C61—41no—
191D4×C61—42no—
192D4×C61—44no—
193D4×C61—41no—
194D4×C61—41no—
195D4×C61—41no—
196D4×C61—41no—
197C2×C4×C61—21no—
198C2×C4×C61—21no—
199C2×C4×C61—22no—
200C2×C4×C61—41no—
201C2×C4×C61—44no—
202C2×C4×C61C2×C22⋊C411yescharacteristic
203C2×C4×C61C2×C22⋊C411yescharacteristic
204C2×C4×C61C2×C22⋊C411yescharacteristic
205C2×C4×C61#13512yes—
206C2×C4×C61#13512yes—
207C2×C4×C61C23×C411yescharacteristic
208C2×C4×C61C22×D411yescharacteristic
209C2×C4×C61C22×D411yescharacteristic
210C22⋊D61—24no—
211C22⋊D61—84no—
212C2×Dic61—24no—
213C2×Dic61—84no—
214C2×Dic61—82no—
215C2×Dic61C2×C22⋊C412yes—
216C2×Dic61C2×C22⋊C412yes—
217C2×Dic61C2×C22⋊C412yes—
218D244—22no—
219C4×D61—22no—
220C4×D61—22no—
221C4×D61—22no—
222C4×D61—22no—
223C4×D61—42no—
224Dic122—22no—
225Dic122—42no—
226Dic122—42no—
227Dic122—82no—
228C8×C61—42no—
229C8×C61—42no—
230#92418—41no—
231#92418—41no—
232#92418—41no—
233#92418—41no—
234#17727—611nomaximal, Sylow
235C22×C61—21no—
236C22×C61—21no—
237C22×C61—42no—
238C22×C61—42no—
239C22×C61—42no—
240C22×C61#41612yes—
241C22×C61C22×C22⋊C411yescharacteristic
242C2×D61—22no—
243C2×D61—22no—
244C2×D61—22no—
245C4×C61—21no—
246C4×C61—21no—
247C4×C61—21no—
248C4×C61—42no—
249C4×C61#41612yes—
250C4×C61C2×#13511yescharacteristic
251C4×C61C2×#13511yescharacteristic
252C4×C61#57611yescharacteristic
253C4×C61#57711yescharacteristic
254Dic61—22no—
255Dic61—22no—
256Dic61—22no—
257Dic61—44no—
258#3511—611no—
259#2988—611no—
260#2925—611no—
261#1553—612no—
262#1552—612no—
263#1549—611no—
264#1547D6111yescharacteristic, complemented
265#1544—612no—
266#1540—612no—
267#1540—611no—
268C2×#332—611no—
269C2×C61—22no—
270C2×C61—82no—
271C2×C61#288911yescharacteristic
272C2×C61C2×#41611yescharacteristic
273C2×C61#289711yescharacteristic
274D61—42no—
275#556—1221no—
276#555—612no—
277#554—612no—
278#553—614no—
279#548—1222no—
280#547—1221no—
281#542—1222no—
282#537—1221no—
283#533—612no—
284#532—612no—
285#532C2×D6111yescharacteristic
286C2×C4⋊Q8—611no—
287#520—1221no—
288C4×#41—611no—
289C8×Q8—1221no—
290C8×D4—1221no—
291C8×D4—1222no—
292#433—1222no—
293#432—612no—
294#431—611no—
295#431—612no—
296#431—612no—
297#423C2×D6111yescharacteristic
298C2×C4⋊C8—611no—
299C2×C4⋊C8—612no—
300C2×C4⋊C8C2×D6111yescharacteristic
301#408—1221no—
302#405—1221no—
303#332—611no—
304#332—611no—
305#332—612no—
306#332—612no—
307#332—612no—
308#332—612no—
309#332C2×D6112yes—
310#331—611no—
311#331—611no—
312#331C2×D6112yes—
313C61#1772711yescharacteristic, complemented, Sylow
314C2×#41—21no—
315C2×#41—611no—
316C2×#41—1222no—
317C22×Q8—1221no—
318C2×M16—21no—
319C22×C8—21no—
320C22×C8—1222no—
321C22×C8—1221no—
322C4⋊Q8—612no—
323C4⋊Q8—612no—
324C4⋊Q8—1221no—
325C4⋊Q8—1221no—
326C4⋊Q8C22×D6111yescharacteristic
327#127C22×D6111yescharacteristic
328#126—1224no—
329#125—614no—
330#124—21no—
331#124—1222no—
332#124—1224no—
333#123—1224no—
334C22⋊Q8—1224no—
335C22⋊Q8—1222no—
336C22⋊Q8—1222no—
337C22⋊Q8—1222no—
338#121—21no—
339#121—614no—
340#121—1222no—
341#121—1222no—
342#121—1224no—
343#121C4×D6112yes—
344C4×Q8—611no—
345C4×Q8—611no—
346C4×Q8—1222no—
347C4×Q8—1222no—
348C4×D4—21no—
349C4×D4—612no—
350C4×D4—612no—
351C4×D4—614no—
352C4×D4—611no—
353C4×D4—611no—
354C4×D4—1222no—
355C4×D4—1224no—
356C4×D4C4×D6112yes—
357#117—611no—
358#117—612no—
359#117—614no—
360#117—1222no—
361C2×C4⋊C4—612no—
362C2×C4⋊C4—1222no—
363C2×C4⋊C4—1221no—
364C2×C42—612no—
365C2×C42C22×D6111yescharacteristic
366C4⋊C8—21no—
367C4⋊C8—21no—
368C4⋊C8—611no—
369C4⋊C8—611no—
370C4⋊C8—611no—
371C4⋊C8—611no—
372C4⋊C8—1222no—
373C4⋊C8—1221no—
374C4⋊C8C22×D6112yes—
375C4⋊C8C22×D6112yes—
376C22⋊C8—1222no—
377C22⋊C8—1222no—
378C22⋊C8—1222no—
379C22⋊C8—1221no—
380#97—1222no—
381C4×C8—1222no—
382C4×C8—1222no—
383C4×C8—1221no—
384#41—42no—
385#41—42no—
386#41—1222no—
387#41—1222no—
388#41—4882no—
389#41—4882no—
390C2×Q8—21no—
391C2×Q8—611no—
392C2×Q8—1222no—
393C2×Q8—1222no—
394C2×Q8—4881no—
395C2×Q8—4881no—
396C2×D4—22no—
397C2×D4—21no—
398C2×D4—21no—
399C2×D4—612no—
400C2×D4—611no—
401C2×D4—1224no—
402C2×D4C2×C4×D6112yes—
403C22×C4—22no—
404C22×C4—612no—
405C22×C4—612no—
406C22×C4—612no—
407C22×C4—1224no—
408C22×C4—1222no—
409C22×C4C2×C4×D6111yescharacteristic
410C22×C4D4×D6111yescharacteristic
411C22×C4D4×D6111yescharacteristic
412M16—42no—
413C2×C8—22no—
414C2×C8—22no—
415C2×C8—81no—
416C2×C8—1221no—
417C2×C8—1221no—
418C2×C8—1221no—
419C2×C8—1221no—
420C2×C8—4882no—
421C2×C8—2442no—
422C4⋊C4—21no—
423C4⋊C4—611no—
424C4⋊C4—612no—
425C4⋊C4—612no—
426C4⋊C4—611no—
427C4⋊C4—614no—
428C4⋊C4—1224no—
429C4⋊C4—1224no—
430C4⋊C4—1222no—
431C4⋊C4—1222no—
432C4⋊C4—1222no—
433C4⋊C4C2×C4×D6112yes—
434C22⋊C4—22no—
435C22⋊C4—22no—
436C22⋊C4—612no—
437C22⋊C4—612no—
438C22⋊C4—612no—
439C22⋊C4—1224no—
440C22⋊C4—1224no—
441C22⋊C4—1224no—
442C42—612no—
443C42—612no—
444C42—612no—
445C42—1224no—
446C42C2×C4×D6111yescharacteristic
447C42D4×D6111yescharacteristic
448C42D4×D6111yescharacteristic
449C42C23×D6111yescharacteristic
450C23—22no—
451C23—612no—
452C23C22⋊C4×D6111yescharacteristic
453Q8—41no—
454Q8—41no—
455Q8—1222no—
456Q8—4882no—
457Q8—2442no—
458Q8—2442no—
459D4—42no—
460D4—44no—
461D4—41no—
462D4—41no—
463D4—41no—
464D4—41no—
465D4—1224no—
466D4—1222no—
467D4—4884no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—22no—
471C2×C4—41no—
472C2×C4—44no—
473C2×C4—612no—
474C2×C4—612no—
475C2×C4—612no—
476C2×C4—1222no—
477C2×C4—1222no—
478C2×C4—1222no—
479C2×C4—1222no—
480C2×C4—1224no—
481C2×C4—4884no—
482C2×C4—4882no—
483C2×C4—2442no—
484C2×C4C22⋊C4×D6111yescharacteristic
485C2×C4C22⋊C4×D6111yescharacteristic
486C2×C4C22⋊C4×D6111yescharacteristic
487C2×C4#1451979612yes—
488C2×C4#1451979812yes—
489C2×C4C22×C4×D6111yescharacteristic
490C2×C4C2×D4×D6111yescharacteristic
491C2×C4C2×D4×D6111yescharacteristic
492C8—42no—
493C8—42no—
494C8—2441no—
495C8—2441no—
496C8—2441no—
497C8—2441no—
498C22—21no—
499C22—21no—
500C22—42no—
501C22—42no—
502C22—42no—
503C22—1222no—
504C22—1222no—
505C22—1222no—
506C22#1683975512yes—
507C22C2×C22⋊C4×D6111yescharacteristic
508C4—21no—
509C4—21no—
510C4—21no—
511C4—42no—
512C4—1222no—
513C4—1222no—
514C4—1222no—
515C4—2444no—
516C4#1683975512yes—
517C4C2×#1451979611yescharacteristic
518C4C2×#1451979811yescharacteristic
519C4#1684056611yescharacteristic
520C4#1684057111yescharacteristic
521C2—22no—
522C2—82no—
523C2—2442no—
524C2#2145430011yescharacteristic
525C2C2×#1683975511yescharacteristic
526C2#2145434311yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D61
commutatorC2×C4×C61C23×C4
FrattiniC42C23×D61
FittingC61×#1547C2
socleC22×C61C22×C22⋊C4
prime radical#21459726C2
cosocleC42×C61C24
hypercenter#1547D61
hypocenterC61#17727
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1547

Complements

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

NH ≅ G/Ntype
2. #21454928C2semidirect
3. #21454487C2semidirect
4. #21444521C2semidirect
5. #21444516C2semidirect
6. #21444505C2semidirect
7. #21444482C2semidirect
8. #21444468C2semidirect
9. #21444465C2semidirect
10. C2×#16839509C2semidirect
11. C61×#1547C2semidirect
264. #1547D61semidirect
313. C61#17727semidirect

Maximal subgroups

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

subgroupindex
#214597262
#214549282
#214544872
#214445212
#214445162
#214445052
#214444822
#214444682
#214444652
C2×#168395092
C61×#15472
#1772761