Finite Group Databasebeta

← #29647750

Subgroups — #29647750

Order 15616.

1018 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459668C211yesmaximal, characteristic
2#21454842C211yesmaximal, characteristic, complemented
3#21448790C211yesmaximal, characteristic, complemented
4#21444384C211yesmaximal, characteristic, complemented
5#21444379C211yesmaximal, characteristic, complemented
6#21444372C214yesmaximal, complemented
7#21444317C211yesmaximal, characteristic, complemented
8#21444307C214yesmaximal, complemented
9C61×#1502C211yesmaximal, characteristic, complemented
10#16840494—22no—
11#16840486C412yescomplemented
12#16840477C412yescomplemented
13#16840458—22no—
14#16840432C412yescomplemented
15#16840416C412yescomplemented
16#16840349—22no—
17#16840326—21no—
18#16840319C2211yescharacteristic
19#16840272C2214yes—
20#16840245—21no—
21C2×C4×D244C2211yescharacteristic
22C8×D244—22no—
23#16839827—22no—
24#16839815C2211yescharacteristic
25C22⋊C8×D61C2214yes—
26#16839664—21no—
27#16839652—22no—
28#16839628C2211yescharacteristic
29#16839624C2212yes—
30C4×C4⋊Dic61—21no—
31#16839609C2212yes—
32#16839500C2214yes—
33#16839498C2212yes—
34#16839498C2212yes—
35#16839496C2214yes—
36C61×#532C2211yescharacteristic
37C61×#431C2211yescharacteristic
38C4×C4⋊C4×C61C2211yescharacteristic
39C61×#327C2212yes—
40C61×#327C2212yes—
41C2×#11758224C2×C414yes—
42C22×D244D411yescharacteristic
43C22×C4×D61C2312yes—
44C2×C8×D61—24no—
45#14519777C2×C411yescharacteristic
46#14519777C2×C411yescharacteristic
47#14519773—22no—
48#14519772—24no—
49#14519769—21no—
50#14519766—22no—
51#14519763—24no—
52#14519761—24no—
53#14519759—24no—
54#14519757C2×C414yes—
55#14519756—21no—
56#14519756C2312yes—
57#14519751—22no—
58#14519751—22no—
59#14519749—24no—
60#14519748C2×C414yes—
61#14519745—24no—
62Q8×Dic61—22no—
63C4×Dic122—21no—
64C4×Dic122C2×C412yes—
65C4×D244C2×C412yes—
66C4×D244C2×C412yes—
67C4×D244D412yes—
68C4×D244D412yes—
69D4×Dic61—24no—
70D4×Dic61—22no—
71#14519737—24no—
72#14519736—21no—
73#14519736—22no—
74#14519735C2×C414yes—
75C2×C4⋊Dic61—22no—
76C2×C4⋊Dic61—21no—
77C2×C4⋊Dic61D411yescharacteristic
78C2×#11758198—22no—
79C2×#11758198—22no—
80C2×#11758198D412yes—
81C2×C4×Dic61—22no—
82C4⋊#92418—21no—
83C22⋊#92418C2312yes—
84#14519689—24no—
85C4×#92418—21no—
86#14519684—21no—
87#14519684—22no—
88#14519684C2312yes—
89C2×#41×C61—21no—
90C4⋊Q8×C61C2×C411yescharacteristic
91#127×C61C2×C411yescharacteristic
92#124×C61—21no—
93#121×C61—21no—
94#121×C61C2312yes—
95C4×D4×C61—21no—
96C4×D4×C61C2×C412yes—
97C2×C4⋊C4×C61—21no—
98C2×C42×C61—21no—
99C2×C42×C61C2311yescharacteristic
100C4⋊C8×C61—21no—
101C22⋊C8×C61C2312yes—
102C4×C8×C61—21no—
103#95×C61C2311yescharacteristic
104#95×C61C2311yescharacteristic
105C23×D61C2×D412yes—
106#41×C61—42no—
107#41×C61—42no—
108C2×Q8×C61—21no—
109C2×D4×C61—22no—
110C2×D4×C61—21no—
111C2×D4×C61—21no—
112C2×D4×C61C22×C412yes—
113C22×C4×C61—21no—
114C22×C4×C61—22no—
115C22×C4×C61—21no—
116C22×C4×C61C2×D411yescharacteristic
117C22×C4×C61C2×D411yescharacteristic
118C22×C4×C61C2411yescharacteristic
119C2×C22⋊D61—24no—
120C2×C22⋊D61—24no—
121C22×Dic61—24no—
122C22×Dic61—22no—
123C22×Dic61C2×D412yes—
124#11758224—84no—
125C2×D244—22no—
126C2×D244—22no—
127C2×D244C22⋊C412yes—
128C2×D244C22⋊C412yes—
129C2×C4×D61—82no—
130C2×C4×D61C22×C414yes—
131C2×C4×D61C2×D414yes—
132C2×Dic122—21no—
133C2×Dic122C22×C412yes—
134C2×C8×C61—22no—
135C4⋊C4×C61—21no—
136C4⋊C4×C61—42no—
137C4⋊C4×C61C22×C412yes—
138C22⋊C4×C61—22no—
139C22⋊C4×C61—22no—
140C42×C61—42no—
141C42×C61C22×C411yescharacteristic
142C42×C61C22×C411yescharacteristic
143C42×C61C2×D411yescharacteristic
144C42×C61C2×D411yescharacteristic
145C22⋊Dic61—22no—
146C22⋊Dic61—24no—
147#11758198—24no—
148#11758198—24no—
149#11758198—44no—
150#11758198—82no—
151#11758198C22⋊C414yes—
152#11758198C22⋊C414yes—
153C4⋊Dic61—21no—
154C4⋊Dic61—21no—
155C4⋊Dic61—21no—
156C4⋊Dic61—24no—
157C4⋊Dic61—42no—
158C4⋊Dic61C22⋊C412yes—
159C4⋊Dic61C22⋊C412yes—
160C4⋊Dic61C22×C412yes—
161#11758196—22no—
162#11758196—22no—
163C4×Dic61—22no—
164C4×Dic61—24no—
165C4×Dic61—82no—
166C2×#92418—22no—
167C8×D61—84no—
168C23×C61—22no—
169C23×C61C22×D411yescharacteristic
170C22×D61—22no—
171C22×D61—22no—
172C22×D61—42no—
173C22×D61—42no—
174C22×D61C2×C22⋊C414yes—
175Q8×C61—42no—
176D4×C61—42no—
177D4×C61—42no—
178D4×C61—42no—
179D4×C61—42no—
180D4×C61—81no—
181C2×C4×C61—22no—
182C2×C4×C61—22no—
183C2×C4×C61—41no—
184C2×C4×C61—41no—
185C2×C4×C61—41no—
186C2×C4×C61—41no—
187C2×C4×C61—44no—
188C2×C4×C61—41no—
189C2×C4×C61—41no—
190C2×C4×C61—41no—
191C2×C4×C61C2×C22⋊C411yescharacteristic
192C2×C4×C61C2×C22⋊C411yescharacteristic
193C2×C4×C61C2×C22⋊C411yescharacteristic
194C2×C4×C61C2×C22⋊C411yescharacteristic
195C2×C4×C61#13512yes—
196C2×C4×C61#13512yes—
197C2×C4×C61C23×C411yescharacteristic
198C2×C4×C61C22×D411yescharacteristic
199C22⋊D61—84no—
200C22⋊D61—84no—
201C2×Dic61—22no—
202C2×Dic61—22no—
203C2×Dic61—42no—
204C2×Dic61—42no—
205C2×Dic61—42no—
206C2×Dic61—42no—
207C2×Dic61—84no—
208C2×Dic61C2×C22⋊C414yes—
209D244—24no—
210D244—24no—
211C4×D61—44no—
212C4×D61—44no—
213Dic122—42no—
214Dic122—81no—
215C8×C61—42no—
216#92418—42no—
217#17617—611nomaximal, Sylow
218C22×C61—21no—
219C22×C61—21no—
220C22×C61—42no—
221C22×C61—42no—
222C22×C61—42no—
223C22×C61C22×C22⋊C411yescharacteristic
224C22×C61C2×#13511yescharacteristic
225C22×C61C2×#13511yescharacteristic
226C2×D61—24no—
227C2×D61—24no—
228C2×D61—24no—
229C2×D61—82no—
230C2×D61—82no—
231C4×C61—21no—
232C4×C61—21no—
233C4×C61—21no—
234C4×C61—21no—
235C4×C61—42no—
236C4×C61—42no—
237C4×C61—81no—
238C4×C61#41612yes—
239C4×C61#41612yes—
240Dic61—24no—
241Dic61—42no—
242Dic61—42no—
243Dic61—82no—
244#3506—611no—
245#2972—611no—
246#2357—611no—
247#1518—611no—
248#1517—611no—
249#1516—614no—
250#1503—611no—
251#1502D6111yescharacteristic, complemented
252#1501—614no—
253C2×C61—21no—
254C2×C61—21no—
255C2×C61—82no—
256C2×C61#288311yescharacteristic
257C2×C61C2×#41611yescharacteristic
258C2×C61#290111yescharacteristic
259D61—44no—
260#559—1222no—
261#556—612no—
262#555—612no—
263#553—1222no—
264#550—612no—
265#547—612no—
266#538—1222no—
267#535—1221no—
268#533—611no—
269#532C2×D6111yescharacteristic
270#525—614no—
271#520—1221no—
272C2×C4×D4—611no—
273C8×D4—1222no—
274C8×D4—1222no—
275#431—611no—
276#431C2×D6111yescharacteristic
277C2×C22⋊C8—614no—
278#390—1221no—
279#388—1222no—
280#382—611no—
281#381—612no—
282C4×C4⋊C4—1221no—
283C4×C4⋊C4C2×D6111yescharacteristic
284C4×C22⋊C4—612no—
285#328—614no—
286#328—612no—
287#328—612no—
288#327—614no—
289#327C2×D6112yes—
290#327C2×D6112yes—
291C61#1761711yescharacteristic, complemented, Sylow
292C2×#41—21no—
293C2×#41—614no—
294C22×D4—611no—
295C23×C4—612no—
296C22×C8—1224no—
297C4⋊Q8—611no—
298C4⋊Q8—611no—
299C4⋊Q8C4×D6111yescharacteristic
300#127—1222no—
301#127C4×D6111yescharacteristic
302#126—1224no—
303#125—1221no—
304#124—21no—
305#124—1224no—
306#124—1222no—
307#123—1224no—
308#123—1224no—
309C22⋊Q8—612no—
310C22⋊Q8—614no—
311C22⋊Q8—1221no—
312#121—21no—
313#121—614no—
314#121—1224no—
315#121—1222no—
316#121—1222no—
317#121C22×D6112yes—
318#120—1224no—
319C4×Q8—612no—
320C4×Q8—1222no—
321C4×Q8—1221no—
322C4×D4—21no—
323C4×D4—612no—
324C4×D4—612no—
325C4×D4—612no—
326C4×D4—612no—
327C4×D4—1224no—
328C4×D4—1222no—
329C4×D4—1224no—
330C4×D4C4×D6112yes—
331#117—614no—
332#117—1222no—
333#117—1221no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—611no—
336C2×C4⋊C4—1222no—
337C2×C4⋊C4—1221no—
338C2×C22⋊C4—612no—
339C2×C22⋊C4—1222no—
340C2×C22⋊C4—1222no—
341C2×C42—21no—
342C2×C42—1222no—
343C2×C42C22×D6111yescharacteristic
344C4⋊C8—21no—
345C4⋊C8—1221no—
346C22⋊C8—612no—
347C22⋊C8—1224no—
348C22⋊C8C22×D6112yes—
349C4×C8—21no—
350C4×C8—1221no—
351#95—612no—
352#95—1222no—
353#95—1221no—
354#95C22×D6111yescharacteristic
355#95C22×D6111yescharacteristic
356C24—612no—
357#41—42no—
358#41—42no—
359#41—4884no—
360C2×Q8—21no—
361C2×Q8—612no—
362C2×Q8—1221no—
363C2×D4—22no—
364C2×D4—21no—
365C2×D4—21no—
366C2×D4—612no—
367C2×D4—612no—
368C2×D4—1224no—
369C2×D4—1224no—
370C2×D4—1222no—
371C2×D4—1222no—
372C2×D4C2×C4×D6112yes—
373C22×C4—21no—
374C22×C4—22no—
375C22×C4—21no—
376C22×C4—612no—
377C22×C4—614no—
378C22×C4—614no—
379C22×C4—1224no—
380C22×C4—1222no—
381C22×C4—4882no—
382C22×C4D4×D6111yescharacteristic
383C22×C4D4×D6111yescharacteristic
384C22×C4C23×D6111yescharacteristic
385C2×C8—22no—
386C2×C8—1222no—
387C2×C8—4884no—
388C4⋊C4—21no—
389C4⋊C4—42no—
390C4⋊C4—612no—
391C4⋊C4—612no—
392C4⋊C4—612no—
393C4⋊C4—1224no—
394C4⋊C4—1222no—
395C4⋊C4—1221no—
396C4⋊C4—1222no—
397C4⋊C4—1221no—
398C4⋊C4—1221no—
399C4⋊C4—2442no—
400C4⋊C4C2×C4×D6112yes—
401C22⋊C4—22no—
402C22⋊C4—22no—
403C22⋊C4—614no—
404C22⋊C4—614no—
405C22⋊C4—1224no—
406C22⋊C4—1224no—
407C22⋊C4—1224no—
408C22⋊C4—1222no—
409C22⋊C4—4882no—
410C22⋊C4—2444no—
411C42—42no—
412C42—1224no—
413C42—1222no—
414C42—4882no—
415C42C2×C4×D6111yescharacteristic
416C42C2×C4×D6111yescharacteristic
417C42D4×D6111yescharacteristic
418C42D4×D6111yescharacteristic
419C23—22no—
420C23—614no—
421C23—1222no—
422C23—1222no—
423C23—2442no—
424C23—2442no—
425C23C2×D4×D6111yescharacteristic
426Q8—42no—
427Q8—4881no—
428Q8—2442no—
429D4—42no—
430D4—42no—
431D4—42no—
432D4—42no—
433D4—81no—
434D4—1224no—
435D4—1224no—
436D4—4884no—
437D4—4884no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—44no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—614no—
449C2×C4—1222no—
450C2×C4—1222no—
451C2×C4—4884no—
452C2×C4—2444no—
453C2×C4—2444no—
454C2×C4—2442no—
455C2×C4—2442no—
456C2×C4—2442no—
457C2×C4—2442no—
458C2×C4C22⋊C4×D6111yescharacteristic
459C2×C4C22⋊C4×D6111yescharacteristic
460C2×C4C22⋊C4×D6111yescharacteristic
461C2×C4C22⋊C4×D6111yescharacteristic
462C2×C4#1451979612yes—
463C2×C4#1451979812yes—
464C2×C4C22×C4×D6111yescharacteristic
465C2×C4C2×D4×D6111yescharacteristic
466C8—42no—
467C8—2442no—
468C22—21no—
469C22—21no—
470C22—42no—
471C22—42no—
472C22—42no—
473C22—1224no—
474C22—1224no—
475C22—1224no—
476C22—4882no—
477C22—4882no—
478C22C2×C22⋊C4×D6111yescharacteristic
479C22C2×#1451979611yescharacteristic
480C22C2×#1451979811yescharacteristic
481C4—21no—
482C4—21no—
483C4—21no—
484C4—21no—
485C4—42no—
486C4—42no—
487C4—81no—
488C4—1224no—
489C4—4882no—
490C4—2442no—
491C4—2442no—
492C4#1683975512yes—
493C4#1683975512yes—
494C2—21no—
495C2—21no—
496C2—82no—
497C2—2444no—
498C2#2145427811yescharacteristic
499C2C2×#1683975511yescharacteristic
500C2#2145436311yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1502C2
socleC22×C61C22×C22⋊C4
prime radical#21459668C2
cosocleC22×C4×C61C24
hypercenter#1502D61
hypocenterC61#17617
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1502

Complements

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

NH ≅ G/Ntype
2. #21454842C2semidirect
3. #21448790C2semidirect
4. #21444384C2semidirect
5. #21444379C2semidirect
6. #21444372C2semidirect
7. #21444317C2semidirect
8. #21444307C2semidirect
9. C61×#1502C2semidirect
11. #16840486C4semidirect
12. #16840477C4semidirect
14. #16840432C4semidirect
15. #16840416C4semidirect
251. #1502D61semidirect
291. C61#17617semidirect

Maximal subgroups

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

subgroupindex
#214596682
#214548422
#214487902
#214443842
#214443792
#214443722
#214443172
#214443072
C61×#15022
#1761761