Finite Group Databasebeta

← #29647751

Subgroups — #29647751

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#21454843C211yesmaximal, characteristic, complemented
3#21444382C211yesmaximal, characteristic, complemented
4#21444378C211yesmaximal, characteristic, complemented
5#21444373C214yesmaximal, complemented
6#21444317C211yesmaximal, characteristic, complemented
7#21444313C211yesmaximal, characteristic, complemented
8#21444308C214yesmaximal, complemented
9C61×#2357C211yesmaximal, 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—
22D4×#92418—22no—
23D4×#92418—22no—
24#16839815C2211yescharacteristic
25#16839815C2211yescharacteristic
26C2×C22⋊#92418C2214yes—
27#16839498C2212yes—
28#16839498C2214yes—
29#16839498C2212yes—
30#16839495C2212yes—
31#16839495C2214yes—
32#16839495C2212yes—
33C2×C4×D4×C61C2211yescharacteristic
34C61×#390—21no—
35C61×#388—22no—
36C61×#382C2211yescharacteristic
37C61×#381C2212yes—
38C4×C4⋊C4×C61—21no—
39C4×C4⋊C4×C61C2211yescharacteristic
40C4×C22⋊C4×C61C2212yes—
41C2×#11758222—21no—
42C2×#11758222C2×C414yes—
43C22×#92418—24no—
44#14519777C2×C411yescharacteristic
45#14519777C2×C411yescharacteristic
46#14519777C2×C411yescharacteristic
47#14519774C2×C411yescharacteristic
48#14519773—22no—
49#14519770—24no—
50#14519769—21no—
51#14519765—21no—
52#14519764—22no—
53#14519763—24no—
54#14519760—24no—
55#14519759—24no—
56#14519756—21no—
57#14519756C2×C414yes—
58#14519756C2312yes—
59#14519751—21no—
60#14519751C2×C414yes—
61#14519751C2312yes—
62#14519750—24no—
63#14519748—22no—
64#14519748—22no—
65#14519746—24no—
66C4×Dic122—21no—
67C4×Dic122C2×C412yes—
68#14519742—22no—
69C4×D244—21no—
70C4×D244C2×C412yes—
71#14519739—22no—
72C4×C22⋊D61—24no—
73C4×C22⋊D61—24no—
74#14519736—21no—
75#14519736C2×C414yes—
76#14519735—22no—
77C4⋊C4×D61—22no—
78C4⋊#92418—21no—
79C4⋊#92418—21no—
80C22⋊#92418—24no—
81C22⋊#92418C2312yes—
82C22⋊#92418C2312yes—
83C4×#92418—21no—
84C4×#92418—21no—
85C22×D4×C61D411yescharacteristic
86C23×C4×C61C2312yes—
87C4×D4×C61C2×C412yes—
88C4×D4×C61C2×C412yes—
89C4×D4×C61D412yes—
90C4×D4×C61D412yes—
91C2×C4⋊C4×C61—21no—
92C2×C4⋊C4×C61—21no—
93C2×C4⋊C4×C61D411yescharacteristic
94C2×C22⋊C4×C61—22no—
95C2×C22⋊C4×C61—22no—
96C2×C22⋊C4×C61D412yes—
97C2×C42×C61—21no—
98C2×C42×C61—22no—
99C2×C42×C61C2311yescharacteristic
100#95×C61—22no—
101#95×C61—21no—
102#95×C61C2311yescharacteristic
103#95×C61C2312yes—
104#95×C61C2311yescharacteristic
105C24×C61C2×D412yes—
106C2×D4×C61—22no—
107C2×D4×C61—22no—
108C2×D4×C61C22⋊C412yes—
109C2×D4×C61C22⋊C412yes—
110C22×C4×C61—21no—
111C22×C4×C61—22no—
112C22×C4×C61—21no—
113C22×C4×C61—82no—
114C22×C4×C61C22×C414yes—
115C22×C4×C61C2×D412yes—
116C22×C4×C61C2×D411yescharacteristic
117C22×C4×C61C2×D414yes—
118C22×C4×C61C2×D411yescharacteristic
119C22×C4×C61C2411yescharacteristic
120C2×C22⋊D61—22no—
121C2×C22⋊D61—24no—
122C2×C22⋊D61—24no—
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—
134C4⋊C4×C61—42no—
135C4⋊C4×C61—42no—
136C4⋊C4×C61C22⋊C412yes—
137C4⋊C4×C61C22⋊C412yes—
138C22⋊C4×C61—44no—
139C22⋊C4×C61—82no—
140C22⋊C4×C61C22⋊C414yes—
141C22⋊C4×C61C22⋊C414yes—
142C42×C61—42no—
143C42×C61—82no—
144C42×C61C22×C411yescharacteristic
145C42×C61C22×C411yescharacteristic
146C42×C61C2×D411yescharacteristic
147C42×C61C2×D411yescharacteristic
148C22⋊Dic61—24no—
149C22⋊Dic61—24no—
150C22⋊Dic61—22no—
151#11758198—24no—
152#11758198—22no—
153#11758198—22no—
154C4⋊Dic61—21no—
155C4⋊Dic61—21no—
156C4⋊Dic61—21no—
157C4⋊Dic61—21no—
158C4⋊Dic61C22×C412yes—
159C4⋊Dic61C22×C412yes—
160#11758196—24no—
161#11758196—22no—
162#11758196—22no—
163C4×Dic61—24no—
164C4×Dic61—22no—
165C2×#92418—22no—
166C2×#92418—22no—
167C2×#92418—84no—
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—22no—
178C2×C4×C61—44no—
179C2×C4×C61—42no—
180C2×C4×C61—41no—
181C2×C4×C61—44no—
182C2×C4×C61—41no—
183C2×C4×C61—41no—
184C2×C4×C61—41no—
185C2×C4×C61—42no—
186C2×C4×C61—42no—
187C2×C4×C61—41no—
188C2×C4×C61—41no—
189C2×C4×C61—41no—
190C2×C4×C61—42no—
191C2×C4×C61C2×C22⋊C414yes—
192C2×C4×C61C2×C22⋊C411yescharacteristic
193C2×C4×C61C2×C22⋊C411yescharacteristic
194C2×C4×C61C2×C22⋊C411yescharacteristic
195C2×C4×C61C2×C22⋊C411yescharacteristic
196C2×C4×C61#13512yes—
197C2×C4×C61#13512yes—
198C2×C4×C61C23×C411yescharacteristic
199C2×C4×C61C22×D411yescharacteristic
200C22⋊D61—42no—
201C22⋊D61—42no—
202C22⋊D61—84no—
203C22⋊D61—84no—
204C2×Dic61—22no—
205C2×Dic61—22no—
206C2×Dic61—22no—
207D244—42no—
208D244—42no—
209D244—81no—
210C4×D61—44no—
211C4×D61—84no—
212Dic122—42no—
213Dic122—42no—
214Dic122—81no—
215#92418—42no—
216#92418—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—42no—
237C4×C61—81no—
238C4×C61—82no—
239C4×C61#41612yes—
240C4×C61#41612yes—
241Dic61—42no—
242Dic61—42no—
243Dic61—42no—
244#3506—611no—
245#2972—611no—
246#2357D6111yescharacteristic, complemented
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—1222no—
274C8×D4—1222no—
275#431—611no—
276#431—611no—
277C2×C22⋊C8—614no—
278#390—21no—
279#388—22no—
280#382C2×D6111yescharacteristic
281#381C2×D6112yes—
282C4×C4⋊C4—21no—
283C4×C4⋊C4C2×D6111yescharacteristic
284C4×C22⋊C4C2×D6112yes—
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×D4C22⋊D6111yescharacteristic
295C23×C4C22×D6112yes—
296C22×C8—1224no—
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×D4D24412yes—
330C4×D4D24412yes—
331#117—614no—
332#117—1221no—
333#117—1222no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4—1222no—
337C2×C4⋊C4C22⋊D6111yescharacteristic
338C2×C22⋊C4—22no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4C22⋊D6112yes—
341C2×C42—21no—
342C2×C42—22no—
343C2×C42C22×D6111yescharacteristic
344C4⋊C8—1221no—
345C4⋊C8—1221no—
346C22⋊C8—612no—
347C22⋊C8—612no—
348C22⋊C8—1224no—
349C4×C8—1221no—
350C4×C8—1221no—
351#95—22no—
352#95—21no—
353#95C22×D6111yescharacteristic
354#95C22×D6112yes—
355#95C22×D6111yescharacteristic
356C24C2×C22⋊D6112yes—
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—21no—
374C22×C4—22no—
375C22×C4—21no—
376C22×C4—82no—
377C22×C4—1224no—
378C22×C4—1222no—
379C22×C4C2×C4×D6114yes—
380C22×C4C2×D24414yes—
381C22×C4C2×C22⋊D6112yes—
382C22×C4C2×C22⋊D6111yescharacteristic
383C22×C4C2×C22⋊D6111yescharacteristic
384C22×C4C23×D6111yescharacteristic
385C2×C8—1222no—
386C2×C8—1222no—
387C2×C8—4884no—
388C4⋊C4—42no—
389C4⋊C4—42no—
390C4⋊C4—612no—
391C4⋊C4—612no—
392C4⋊C4—1222no—
393C4⋊C4—1221no—
394C4⋊C4—1224no—
395C4⋊C4—1222no—
396C4⋊C4—1221no—
397C4⋊C4—1221no—
398C4⋊C4—1221no—
399C4⋊C4#1175819812yes—
400C4⋊C4#1175819812yes—
401C22⋊C4—44no—
402C22⋊C4—82no—
403C22⋊C4—1222no—
404C22⋊C4—1224no—
405C22⋊C4—1224no—
406C22⋊C4—1222no—
407C22⋊C4—1224no—
408C22⋊C4—1222no—
409C22⋊C4#1175819814yes—
410C22⋊C4#1175819814yes—
411C42—42no—
412C42—82no—
413C42—1224no—
414C42—1222no—
415C42C2×C4×D6111yescharacteristic
416C42C2×C4×D6111yescharacteristic
417C42C2×D24411yescharacteristic
418C42C2×D24411yescharacteristic
419C23—22no—
420C23—22no—
421C23—42no—
422C23—42no—
423C23—1222no—
424C23C2×#1175819814yes—
425C23C22×C22⋊D6111yescharacteristic
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—22no—
439C2×C4—42no—
440C2×C4—41no—
441C2×C4—44no—
442C2×C4—41no—
443C2×C4—44no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—42no—
452C2×C4—1222no—
453C2×C4—1222no—
454C2×C4—1222no—
455C2×C4—4884no—
456C2×C4—2444no—
457C2×C4C2×#1175819814yes—
458C2×C4C2×#1175819811yescharacteristic
459C2×C4C2×#1175819811yescharacteristic
460C2×C4C2×#1175819811yescharacteristic
461C2×C4C2×#1175819811yescharacteristic
462C2×C4#1451979712yes—
463C2×C4#1451979712yes—
464C2×C4C22×C4×D6111yescharacteristic
465C2×C4C22×D24411yescharacteristic
466C8—2442no—
467C8—2442no—
468C22—21no—
469C22—21no—
470C22—24no—
471C22—24no—
472C22—24no—
473C22—82no—
474C22—82no—
475C22—2442no—
476C22—2442no—
477C22—2442no—
478C22C22×#1175819811yescharacteristic
479C22C2×#1451979711yescharacteristic
480C22C2×#1451979711yescharacteristic
481C4—21no—
482C4—21no—
483C4—21no—
484C4—21no—
485C4—24no—
486C4—42no—
487C4—81no—
488C4—82no—
489C4—2442no—
490C4—2442no—
491C4—2442no—
492C4#1683975612yes—
493C4#1683975612yes—
494C2—21no—
495C2—21no—
496C2—44no—
497C2—4882no—
498C2#2145427911yescharacteristic
499C2C2×#1683975611yescharacteristic
500C2#2145436711yescharacteristic
subgroupG/H
centerC22C22×#11758198
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#2357C2
socleC22×C61C22×C22⋊C4
prime radical#21459671C2
cosocleC22×C4×C61C24
hypercenter#2357D61
hypocenterC61#17617
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C4×D4 ◁ #2357

Complements

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

NH ≅ G/Ntype
2. #21454843C2semidirect
3. #21444382C2semidirect
4. #21444378C2semidirect
5. #21444373C2semidirect
6. #21444317C2semidirect
7. #21444313C2semidirect
8. #21444308C2semidirect
9. C61×#2357C2semidirect
11. #16840482C4semidirect
12. #16840477C4semidirect
14. #16840433C4semidirect
15. #16840417C4semidirect
246. #2357D61semidirect
291. C61#17617semidirect

Maximal subgroups

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

subgroupindex
#214596712
#214548432
#214443822
#214443782
#214443732
#214443172
#214443132
#214443082
C61×#23572
#1761761