Finite Group Databasebeta

← #29647756

Subgroups — #29647756

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#21459671C211yesmaximal, characteristic
2#21448792C211yesmaximal, characteristic, complemented
3#21444384C211yesmaximal, characteristic, complemented
4#21444380C211yesmaximal, characteristic, complemented
5#21444376C214yesmaximal, complemented
6#21444319C211yesmaximal, characteristic, complemented
7#21444315C211yesmaximal, characteristic, complemented
8#21444311C214yesmaximal, complemented
9C61×#2972C211yesmaximal, characteristic, complemented
10#16840495—22no—
11#16840482C412yescomplemented
12#16840477C412yescomplemented
13#16840462—22no—
14#16840433C412yescomplemented
15#16840417C412yescomplemented
16#16840350—22no—
17#16840329—21no—
18#16840319C2211yescharacteristic
19#16840312C2211yescharacteristic
20#16840273C2214yes—
21#16840243—21no—
22#16839665—21no—
23#16839655—22no—
24#16839628C2211yescharacteristic
25#16839625C2212yes—
26C4⋊C4×Dic61—21no—
27C4×C4⋊Dic61C2211yescharacteristic
28C4×C22⋊Dic61C2212yes—
29#16839500C2212yes—
30#16839500C2214yes—
31#16839500C2212yes—
32#16839497C2212yes—
33#16839497C2214yes—
34#16839497C2212yes—
35C2×C4×D4×C61C2211yescharacteristic
36C8×D4×C61—22no—
37C8×D4×C61—22no—
38C61×#431C2211yescharacteristic
39C61×#431C2211yescharacteristic
40C2×C22⋊C8×C61C2214yes—
41C2×#11758222—21no—
42C2×#11758222C2×C414yes—
43#14519777C2×C411yescharacteristic
44#14519777C2×C411yescharacteristic
45#14519777C2×C411yescharacteristic
46#14519774C2×C411yescharacteristic
47#14519773—22no—
48#14519770—24no—
49#14519769—21no—
50#14519765—21no—
51#14519764—22no—
52#14519763—24no—
53#14519760—24no—
54#14519759—24no—
55#14519756—21no—
56#14519756C2×C414yes—
57#14519756C2312yes—
58#14519751—21no—
59#14519751C2×C414yes—
60#14519751C2312yes—
61#14519750—24no—
62#14519748—22no—
63#14519748—22no—
64#14519746—24no—
65C4×Dic122—21no—
66C4×Dic122C2×C412yes—
67#14519742—22no—
68C4×D244—21no—
69C4×D244C2×C412yes—
70#14519739—22no—
71C4×C22⋊D61—24no—
72C4×C22⋊D61—24no—
73#14519736—21no—
74#14519736C2×C414yes—
75#14519735—22no—
76C2×C4⋊Dic61—21no—
77C2×C4⋊Dic61—21no—
78C4⋊C4×D61—22no—
79C2×C22⋊Dic61—22no—
80C2×C22⋊Dic61—22no—
81C2×C4×Dic61—21no—
82C2×C4×Dic61—22no—
83#14519684—22no—
84#14519684—21no—
85#14519684C2311yescharacteristic
86#14519684C2312yes—
87#14519684C2311yescharacteristic
88C22×D4×C61D411yescharacteristic
89C23×C4×C61C2312yes—
90C22×C8×C61—24no—
91C4×D4×C61C2×C412yes—
92C4×D4×C61C2×C412yes—
93C4×D4×C61D412yes—
94C4×D4×C61D412yes—
95C2×C4⋊C4×C61D411yescharacteristic
96C2×C22⋊C4×C61D412yes—
97C2×C42×C61C2311yescharacteristic
98C4⋊C8×C61—21no—
99C4⋊C8×C61—21no—
100C22⋊C8×C61—24no—
101C22⋊C8×C61C2312yes—
102C22⋊C8×C61C2312yes—
103C4×C8×C61—21no—
104C4×C8×C61—21no—
105C24×C61C2×D412yes—
106C2×D4×C61—22no—
107C2×D4×C61—22no—
108C2×D4×C61C22⋊C412yes—
109C2×D4×C61C22⋊C412yes—
110C22×C4×C61—82no—
111C22×C4×C61C22×C414yes—
112C22×C4×C61C2×D411yescharacteristic
113C22×C4×C61C2×D412yes—
114C22×C4×C61C2×D414yes—
115C22×C4×C61C2×D411yescharacteristic
116C22×C4×C61C2411yescharacteristic
117C2×C22⋊D61—22no—
118C2×C22⋊D61—24no—
119C2×C22⋊D61—24no—
120C22×Dic61—21no—
121C22×Dic61—22no—
122C22×Dic61—21no—
123#11758222—42no—
124#11758222—42no—
125#11758222—84no—
126C2×D244—21no—
127C2×D244—21no—
128C2×D244C22×C412yes—
129C2×C4×D61—22no—
130C2×C4×D61—24no—
131C2×Dic122—21no—
132C2×Dic122—21no—
133C2×Dic122C22×C412yes—
134C2×C8×C61—22no—
135C2×C8×C61—22no—
136C2×C8×C61—84no—
137C4⋊C4×C61C22⋊C412yes—
138C4⋊C4×C61C22⋊C412yes—
139C22⋊C4×C61C22⋊C414yes—
140C22⋊C4×C61C22⋊C414yes—
141C42×C61C22×C411yescharacteristic
142C42×C61C22×C411yescharacteristic
143C42×C61C2×D411yescharacteristic
144C42×C61C2×D411yescharacteristic
145C22⋊Dic61—24no—
146C22⋊Dic61—24no—
147C22⋊Dic61—22no—
148C22⋊Dic61—44no—
149C22⋊Dic61—82no—
150#11758198—24no—
151#11758198—22no—
152#11758198—22no—
153C4⋊Dic61—21no—
154C4⋊Dic61—21no—
155C4⋊Dic61—21no—
156C4⋊Dic61—21no—
157C4⋊Dic61—42no—
158C4⋊Dic61—42no—
159C4⋊Dic61C22×C412yes—
160C4⋊Dic61C22×C412yes—
161#11758196—24no—
162#11758196—22no—
163#11758196—22no—
164C4×Dic61—24no—
165C4×Dic61—22no—
166C4×Dic61—42no—
167C4×Dic61—82no—
168C23×C61—22no—
169C23×C61—22no—
170C23×C61—42no—
171C23×C61—42no—
172C23×C61C2×C22⋊C414yes—
173C23×C61C22×D411yescharacteristic
174C22×D61—22no—
175D4×C61—24no—
176D4×C61—24no—
177C2×C4×C61—42no—
178C2×C4×C61—41no—
179C2×C4×C61—44no—
180C2×C4×C61—44no—
181C2×C4×C61—41no—
182C2×C4×C61C2×C22⋊C414yes—
183C2×C4×C61C2×C22⋊C411yescharacteristic
184C2×C4×C61C2×C22⋊C411yescharacteristic
185C2×C4×C61C2×C22⋊C411yescharacteristic
186C2×C4×C61C2×C22⋊C411yescharacteristic
187C2×C4×C61#13512yes—
188C2×C4×C61#13512yes—
189C2×C4×C61C23×C411yescharacteristic
190C2×C4×C61C22×D411yescharacteristic
191C22⋊D61—42no—
192C22⋊D61—42no—
193C22⋊D61—84no—
194C22⋊D61—84no—
195C2×Dic61—22no—
196C2×Dic61—22no—
197C2×Dic61—22no—
198C2×Dic61—22no—
199C2×Dic61—41no—
200C2×Dic61—41no—
201C2×Dic61—42no—
202C2×Dic61—41no—
203C2×Dic61—42no—
204C2×Dic61—41no—
205C2×Dic61—41no—
206C2×Dic61—42no—
207D244—42no—
208D244—42no—
209D244—81no—
210C4×D61—44no—
211C4×D61—84no—
212Dic122—42no—
213Dic122—42no—
214Dic122—81no—
215C8×C61—42no—
216C8×C61—42no—
217#17617—611nomaximal, Sylow
218C22×C61—21no—
219C22×C61—21no—
220C22×C61—24no—
221C22×C61—24no—
222C22×C61—24no—
223C22×C61—82no—
224C22×C61—82no—
225C22×C61C22×C22⋊C411yescharacteristic
226C22×C61C2×#13511yescharacteristic
227C22×C61C2×#13511yescharacteristic
228C2×D61—42no—
229C2×D61—42no—
230C2×D61—42no—
231C4×C61—21no—
232C4×C61—21no—
233C4×C61—24no—
234C4×C61—21no—
235C4×C61—21no—
236C4×C61#41612yes—
237C4×C61#41612yes—
238Dic61—42no—
239Dic61—42no—
240Dic61—42no—
241Dic61—42no—
242Dic61—81no—
243Dic61—82no—
244#3506—611no—
245#2972D6111yescharacteristic, complemented
246#2357—611no—
247#1518—611no—
248#1517—611no—
249#1516—614no—
250#1503—611no—
251#1502—611no—
252#1501—614no—
253C2×C61—21no—
254C2×C61—21no—
255C2×C61—44no—
256C2×C61#288311yescharacteristic
257C2×C61C2×#41611yescharacteristic
258C2×C61#290111yescharacteristic
259D61—82no—
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#532—611no—
270#525—614no—
271#520—1221no—
272C2×C4×D4C2×D6111yescharacteristic
273C8×D4—22no—
274C8×D4—22no—
275#431C2×D6111yescharacteristic
276#431C2×D6111yescharacteristic
277C2×C22⋊C8C2×D6114yes—
278#390—1221no—
279#388—1222no—
280#382—611no—
281#381—612no—
282C4×C4⋊C4—611no—
283C4×C4⋊C4—1221no—
284C4×C22⋊C4—612no—
285#328—612no—
286#328—614no—
287#328—612no—
288#327—612no—
289#327—614no—
290#327—612no—
291C61#1761711yescharacteristic, complemented, Sylow
292C2×#41—614no—
293C2×#41—1221no—
294C22×D4D24411yescharacteristic
295C23×C4C22×D6112yes—
296C22×C8—24no—
297C4⋊Q8—611no—
298C4⋊Q8—611no—
299C4⋊Q8—611no—
300#127—611no—
301#127—1222no—
302#126—1224no—
303#125—1221no—
304#124—1222no—
305#124—1224no—
306#124—1221no—
307#123—1224no—
308#123—1224no—
309C22⋊Q8—614no—
310C22⋊Q8—612no—
311C22⋊Q8—1221no—
312#121—614no—
313#121—612no—
314#121—1222no—
315#121—1222no—
316#121—1221no—
317#121—1224no—
318#120—1224no—
319C4×Q8—612no—
320C4×Q8—1222no—
321C4×Q8—1221no—
322C4×D4—612no—
323C4×D4—1224no—
324C4×D4—1224no—
325C4×D4—1222no—
326C4×D4—1221no—
327C4×D4C4×D6112yes—
328C4×D4C4×D6112yes—
329C4×D4C22⋊D6112yes—
330C4×D4C22⋊D6112yes—
331#117—614no—
332#117—1221no—
333#117—1222no—
334C2×C4⋊C4—1222no—
335C2×C4⋊C4—1221no—
336C2×C4⋊C4—1221no—
337C2×C4⋊C4D24411yescharacteristic
338C2×C22⋊C4—1222no—
339C2×C22⋊C4—1222no—
340C2×C22⋊C4D24412yes—
341C2×C42—1222no—
342C2×C42—1221no—
343C2×C42C22×D6111yescharacteristic
344C4⋊C8—21no—
345C4⋊C8—21no—
346C22⋊C8—24no—
347C22⋊C8C22×D6112yes—
348C22⋊C8C22×D6112yes—
349C4×C8—21no—
350C4×C8—21no—
351#95—611no—
352#95—612no—
353#95—611no—
354#95—1222no—
355#95—1221no—
356C24C2×D24412yes—
357#41—4884no—
358#41—2442no—
359#41—2442no—
360C2×Q8—612no—
361C2×Q8—1221no—
362C2×Q8—1221no—
363C2×D4—22no—
364C2×D4—22no—
365C2×D4—612no—
366C2×D4—1224no—
367C2×D4—1224no—
368C2×D4—1222no—
369C2×D4—1221no—
370C2×D4—1221no—
371C2×D4#1175819812yes—
372C2×D4#1175819812yes—
373C22×C4—82no—
374C22×C4—1221no—
375C22×C4—1224no—
376C22×C4—1222no—
377C22×C4—1222no—
378C22×C4—1221no—
379C22×C4C2×C4×D6114yes—
380C22×C4C2×D24412yes—
381C22×C4C2×D24411yescharacteristic
382C22×C4C2×D24411yescharacteristic
383C22×C4C2×C22⋊D6114yes—
384C22×C4C23×D6111yescharacteristic
385C2×C8—22no—
386C2×C8—22no—
387C2×C8—84no—
388C4⋊C4—612no—
389C4⋊C4—612no—
390C4⋊C4—1222no—
391C4⋊C4—1221no—
392C4⋊C4—1224no—
393C4⋊C4—1222no—
394C4⋊C4—1221no—
395C4⋊C4—1221no—
396C4⋊C4—1221no—
397C4⋊C4—2442no—
398C4⋊C4—2442no—
399C4⋊C4#1175819812yes—
400C4⋊C4#1175819812yes—
401C22⋊C4—1222no—
402C22⋊C4—1224no—
403C22⋊C4—1224no—
404C22⋊C4—1222no—
405C22⋊C4—1224no—
406C22⋊C4—1222no—
407C22⋊C4—4882no—
408C22⋊C4—2444no—
409C22⋊C4#1175819814yes—
410C22⋊C4#1175819814yes—
411C42—1224no—
412C42—1222no—
413C42—4882no—
414C42—2442no—
415C42C2×C4×D6111yescharacteristic
416C42C2×C4×D6111yescharacteristic
417C42C2×C22⋊D6111yescharacteristic
418C42C2×C22⋊D6111yescharacteristic
419C23—22no—
420C23—22no—
421C23—42no—
422C23—42no—
423C23—1222no—
424C23C2×#1175819814yes—
425C23C22×D24411yescharacteristic
426Q8—4881no—
427Q8—2442no—
428Q8—2442no—
429D4—24no—
430D4—24no—
431D4—4881no—
432D4—4884no—
433D4—4884no—
434D4—2442no—
435D4—2442no—
436D4—2442no—
437D4—2442no—
438C2×C4—42no—
439C2×C4—41no—
440C2×C4—44no—
441C2×C4—41no—
442C2×C4—44no—
443C2×C4—1222no—
444C2×C4—1222no—
445C2×C4—1222no—
446C2×C4—1222no—
447C2×C4—4884no—
448C2×C4—2442no—
449C2×C4—2441no—
450C2×C4—2442no—
451C2×C4—2441no—
452C2×C4—2441no—
453C2×C4—2442no—
454C2×C4—2441no—
455C2×C4—2441no—
456C2×C4—2444no—
457C2×C4C2×#1175819814yes—
458C2×C4C2×#1175819811yescharacteristic
459C2×C4C2×#1175819811yescharacteristic
460C2×C4C2×#1175819811yescharacteristic
461C2×C4C2×#1175819811yescharacteristic
462C2×C4#1451979912yes—
463C2×C4#1451979912yes—
464C2×C4C22×C4×D6111yescharacteristic
465C2×C4C22×C22⋊D6111yescharacteristic
466C8—42no—
467C8—42no—
468C22—21no—
469C22—21no—
470C22—24no—
471C22—24no—
472C22—24no—
473C22—82no—
474C22—82no—
475C22—2442no—
476C22—2442no—
477C22—2442no—
478C22C22×#1175819811yescharacteristic
479C22C2×#1451979911yescharacteristic
480C22C2×#1451979911yescharacteristic
481C4—21no—
482C4—21no—
483C4—21no—
484C4—21no—
485C4—24no—
486C4—4882no—
487C4—4881no—
488C4—2442no—
489C4—2442no—
490C4—2442no—
491C4—2442no—
492C4#1683975912yes—
493C4#1683975912yes—
494C2—21no—
495C2—21no—
496C2—44no—
497C2—4882no—
498C2#2145427911yescharacteristic
499C2C2×#1683975911yescharacteristic
500C2#2145436411yescharacteristic
subgroupG/H
centerC22C22×#11758198
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#2972C2
socleC22×C61C22×C22⋊C4
prime radical#21459671C2
cosocleC22×C4×C61C24
hypercenter#2972D61
hypocenterC61#17617
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C4×D4 ◁ #2972

Complements

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

NH ≅ G/Ntype
2. #21448792C2semidirect
3. #21444384C2semidirect
4. #21444380C2semidirect
5. #21444376C2semidirect
6. #21444319C2semidirect
7. #21444315C2semidirect
8. #21444311C2semidirect
9. C61×#2972C2semidirect
11. #16840482C4semidirect
12. #16840477C4semidirect
14. #16840433C4semidirect
15. #16840417C4semidirect
245. #2972D61semidirect
291. C61#17617semidirect

Maximal subgroups

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

subgroupindex
#214596712
#214487922
#214443842
#214443802
#214443762
#214443192
#214443152
#214443112
C61×#29722
#1761761