Finite Group Databasebeta

← #29648285

Subgroups — #29648285

Order 15616.

900 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459713C211yesmaximal, characteristic
2#21454837C211yesmaximal, characteristic, complemented
3#21454831C211yesmaximal, characteristic, complemented
4#21444456C211yesmaximal, characteristic, complemented
5#21444447C212yesmaximal, complemented
6#21444436C212yesmaximal, complemented
7#21444423C212yesmaximal, complemented
8#21444417C211yesmaximal, characteristic, complemented
9#21444413C211yesmaximal, characteristic, complemented
10#21444404C212yesmaximal, complemented
11C61×#1527C211yesmaximal, characteristic, complemented
12#16840494—21no—
13#16840494—21no—
14#16840481—21no—
15#16840481—21no—
16#16840438—22no—
17#16840438—22no—
18#16840381C412yes—
19#16840381C412yes—
20#16840336C411yescharacteristic
21#16840329C411yescharacteristic
22#16840319C2211yescharacteristic
23#16840291C2212yes—
24#16840291C2212yes—
25#16840240C411yescharacteristic
26#16840237C411yescharacteristic
27C4×#11758222C2211yescharacteristic
28#16839889—21no—
29C8×Dic122—21no—
30#16839841—21no—
31#16839838—22no—
32#16839836—22no—
33#16839827—21no—
34#16839819C2211yescharacteristic
35#16839817C2212yescomplemented
36#16839817C2212yescomplemented
37C4×#11758194C2211yescharacteristic
38#16839714C2212yescomplemented
39#97×D61C2212yescomplemented
40#16839503C2211yescharacteristic, complemented
41#16839503C2212yescomplemented
42#16839503C2211yescharacteristic, complemented
43#16839503C2211yescharacteristic, complemented
44#16839503C2211yescharacteristic, complemented
45#16839503C2212yescomplemented
46#16839502C2212yes—
47#16839502C2212yes—
48C2×#124×C61C2211yescharacteristic
49C61×#432C2211yescharacteristic
50C4×M16×C61C2211yescharacteristic
51C61×#329C2212yes—
52C61×#329C2212yes—
53C2×#11758222C2×C411yescharacteristic
54C2×#11758194—21no—
55C2×#11758189—22no—
56C2×C8×D61—22no—
57#14519777C2×C411yescharacteristic
58#14519777C2×C411yescharacteristic
59#14519775—21no—
60#14519773—21no—
61#14519770—22no—
62#14519770—22no—
63#14519770—22no—
64#14519770—22no—
65#14519769C2311yescharacteristic
66#14519769C2311yescharacteristic
67#14519766—21no—
68#14519764—21no—
69#14519763—22no—
70#14519763—22no—
71#14519761—22no—
72#14519761—22no—
73#14519760—22no—
74#14519759—22no—
75#14519759—22no—
76#14519756—21no—
77#14519756—21no—
78#14519754—22no—
79#14519754—22no—
80#14519753C2×C412yes—
81#14519753C2×C412yes—
82#14519748C2×C412yes—
83#14519748C2×C412yes—
84#14519746—22no—
85Q8×Dic61—21no—
86Q8×Dic61—21no—
87Q8×Dic61C2×C412yes—
88Q8×Dic61C2×C412yes—
89C4×Dic122—21no—
90C4×Dic122—21no—
91C4×Dic122D411yescharacteristic
92C4×Dic122D411yescharacteristic
93C4×D244D411yescharacteristic
94C4×D244D411yescharacteristic
95D4×Dic61—21no—
96D4×Dic61—21no—
97D4×Dic61C2×C412yes—
98D4×Dic61C2×C412yes—
99C4×C22⋊D61D412yes—
100C4×C22⋊D61D412yes—
101#14519737—22no—
102#14519737—22no—
103#14519736—22no—
104#14519736C2×C411yescharacteristic
105#14519736C2×C411yescharacteristic
106#14519736C2×C411yescharacteristic
107#14519734C2×C412yes—
108#14519733—22no—
109#14519733—22no—
110C2×C22⋊Dic61—22no—
111C42×D61C2312yes—
112C4⋊#92418—21no—
113#14519706—22no—
114#14519706—22no—
115C22⋊#92418—21no—
116#14519689—22no—
117#14519689—22no—
118#14519688C2311yescharacteristic
119#14519688C2312yes—
120#14519688C2311yescharacteristic
121#14519687—22no—
122C4×#92418—21no—
123C8×Dic61—22no—
124C22×Q8×C61C2×C411yescharacteristic
125C22×D4×C61C2×C411yescharacteristic
126C2×M16×C61—21no—
127#124×C61—21no—
128#124×C61—21no—
129#124×C61C2×C412yes—
130#124×C61C2312yes—
131C2×C22⋊C4×C61—21no—
132C2×C22⋊C4×C61—21no—
133C2×C42×C61C2311yescharacteristic
134C4⋊C8×C61—21no—
135C22⋊C8×C61—21no—
136#97×C61C2312yes—
137#97×C61C2311yescharacteristic
138#97×C61C2311yescharacteristic
139C4×C8×C61—21no—
140C24×C61—21no—
141C2×Q8×C61—21no—
142C2×Q8×C61—42no—
143C2×Q8×C61C22×C412yes—
144C2×D4×C61—21no—
145C2×D4×C61—42no—
146C2×D4×C61C22×C412yes—
147C22×C4×C61—21no—
148C22×C4×C61C22×C411yescharacteristic
149C22×C4×C61C2×D411yescharacteristic
150C22×C4×C61C2×D411yescharacteristic
151C2×C22⋊D61—22no—
152C2×C22⋊D61—22no—
153C2×C22⋊D61C22⋊C412yes—
154C22×Dic61—22no—
155C22×Dic61—22no—
156#11758222—42no—
157C2×D244C22⋊C411yescharacteristic
158C2×C4×D61C22×C412yes—
159C2×C4×D61C2×D412yes—
160C2×C4×D61C2×D412yes—
161C2×Dic122—21no—
162C2×Dic122—21no—
163C2×Dic122C22⋊C411yescharacteristic
164M16×C61—42no—
165C2×C8×C61—21no—
166C2×C8×C61—21no—
167C2×C8×C61—22no—
168C22⋊C4×C61—22no—
169C22⋊C4×C61—22no—
170C22⋊C4×C61—22no—
171C22⋊C4×C61—22no—
172C42×C61C22×C411yescharacteristic
173C42×C61C2×D411yescharacteristic
174C42×C61C2×D411yescharacteristic
175C42×C61C2411yescharacteristic
176C22⋊Dic61—21no—
177C22⋊Dic61—21no—
178C22⋊Dic61—22no—
179C22⋊Dic61—22no—
180C22⋊Dic61—22no—
181C22⋊Dic61—22no—
182C22⋊Dic61C22⋊C412yes—
183#11758198—22no—
184#11758198—22no—
185#11758198C22⋊C412yes—
186#11758198C22⋊C412yes—
187C4⋊Dic61—21no—
188C4⋊Dic61—21no—
189C4⋊Dic61—22no—
190C4⋊Dic61—22no—
191C4⋊Dic61C22⋊C411yescharacteristic
192C4⋊Dic61C22⋊C411yescharacteristic
193C4⋊Dic61C22×C411yescharacteristic
194C4⋊Dic61C22×C411yescharacteristic
195C4⋊Dic61C22×C411yescharacteristic
196C4⋊Dic61C22×C411yescharacteristic
197#11758196—21no—
198#11758196—21no—
199#11758196—21no—
200#11758196—21no—
201#11758196—22no—
202#11758196—22no—
203#11758196C22⋊C412yes—
204#11758196C22⋊C412yes—
205C4×Dic61—21no—
206C4×Dic61—21no—
207C4×Dic61—22no—
208C4×Dic61—22no—
209C4×Dic61C22×C412yes—
210C4×Dic61C2×D412yes—
211C4×Dic61C2×D412yes—
212#11758194—42no—
213C2×#92418—21no—
214C2×#92418—22no—
215C2×#92418—21no—
216#11758189—44no—
217C8×D61—44no—
218C23×C61—22no—
219C23×C61—42no—
220C23×C61—81no—
221C23×C61—81no—
222C23×C61C2×C22⋊C411yescharacteristic
223C22×D61C2×C22⋊C412yes—
224Q8×C61—42no—
225Q8×C61—42no—
226D4×C61—42no—
227D4×C61—42no—
228C2×C4×C61—22no—
229C2×C4×C61—22no—
230C2×C4×C61—41no—
231C2×C4×C61—41no—
232C2×C4×C61—42no—
233C2×C4×C61C2×C22⋊C411yescharacteristic
234C2×C4×C61C2×C22⋊C411yescharacteristic
235C2×C4×C61C2×C22⋊C411yescharacteristic
236C2×C4×C61C23×C411yescharacteristic
237C2×C4×C61C22×D411yescharacteristic
238C2×C4×C61C22×D411yescharacteristic
239C22⋊D61—42no—
240C22⋊D61—82no—
241C22⋊D61—82no—
242C2×Dic61—21no—
243C2×Dic61—21no—
244C2×Dic61—21no—
245C2×Dic61—21no—
246C2×Dic61—82no—
247C2×Dic61—82no—
248C2×Dic61C2×C22⋊C412yes—
249C2×Dic61C2×C22⋊C412yes—
250C2×Dic61C2×C22⋊C412yes—
251D244—22no—
252C4×D61—42no—
253C4×D61—42no—
254C4×D61—42no—
255Dic122—22no—
256Dic122—81no—
257Dic122—81no—
258C8×C61—22no—
259C8×C61—42no—
260C8×C61—41no—
261#92418—22no—
262#92418—42no—
263#92418—41no—
264#17691—611nomaximal, Sylow
265C22×C61—21no—
266C22×C61—21no—
267C22×C61—21no—
268C22×C61—42no—
269C22×C61—42no—
270C22×C61—42no—
271C22×C61—162no—
272C22×C61C22×C22⋊C411yescharacteristic
273C2×D61—22no—
274C2×D61—22no—
275C2×D61—22no—
276C4×C61—21no—
277C4×C61—21no—
278C4×C61—21no—
279C4×C61—21no—
280C4×C61—21no—
281C4×C61—42no—
282C4×C61#41312yes—
283Dic61—22no—
284Dic61—22no—
285Dic61—22no—
286Dic61—41no—
287Dic61—41no—
288Dic61—41no—
289Dic61—41no—
290#3510—611no—
291#2971—611no—
292#2970—611no—
293#1537—611no—
294#1535—612no—
295#1532—612no—
296#1529—612no—
297#1527—611no—
298#1527D6111yescharacteristic, complemented
299#1526—611no—
300#1524—612no—
301C2×C61—41no—
302C2×C61—82no—
303C2×C61C2×#41311yescharacteristic
304C2×C61#290111yescharacteristic
305C2×C61#290111yescharacteristic
306D61—42no—
307#559—1221no—
308#559—1221no—
309#556—1221no—
310#556—1221no—
311#551—1222no—
312#551—1222no—
313#542—612no—
314#542—612no—
315#536—611no—
316#535—611no—
317#533—611no—
318#529—612no—
319#529—612no—
320C2×#124C2×D6111yescharacteristic
321#519—611no—
322#518—611no—
323C4×#41—611no—
324#446—1221no—
325C8×Q8—1221no—
326#436—1221no—
327#436—1222no—
328#435—1222no—
329C8×D4—1221no—
330#432—611no—
331#432—612no—
332#432—612no—
333#432C2×D6111yescharacteristic
334C4×M16—611no—
335C4×M16—612no—
336C4×M16C2×D6111yescharacteristic
337C2×#97—612no—
338#330—612no—
339#330—612no—
340#330—611no—
341#330—611no—
342#330—611no—
343#330—611no—
344#329—612no—
345#329—612no—
346#329C2×D6112yes—
347#329C2×D6112yes—
348C61#1769111yescharacteristic, complemented, Sylow
349C2×#41—611no—
350C22×Q8C4×D6111yescharacteristic
351C22×D4C4×D6111yescharacteristic
352C2×M16—21no—
353C2×M16—1222no—
354C2×M16—1221no—
355C22×C8—1222no—
356C4⋊Q8—611no—
357C4⋊Q8—611no—
358C4⋊Q8—1221no—
359#127—1221no—
360#126—1222no—
361#126—1222no—
362#126—1222no—
363#126—1222no—
364#125—611no—
365#125—611no—
366#124—21no—
367#124—21no—
368#124—1222no—
369#124—1222no—
370#124—1221no—
371#124—1221no—
372#124C4×D6112yes—
373#124C22×D6112yes—
374#123—1222no—
375#123—1222no—
376#123—1222no—
377#123—1222no—
378#123—1222no—
379C22⋊Q8—612no—
380C22⋊Q8—612no—
381C22⋊Q8—1222no—
382C22⋊Q8—1222no—
383C22⋊Q8—1221no—
384C22⋊Q8—1221no—
385#121—612no—
386#121—612no—
387#120—1222no—
388C4×Q8—611no—
389C4×Q8—611no—
390C4×Q8—612no—
391C4×Q8—612no—
392C4×Q8—1221no—
393C4×Q8—1221no—
394C4×Q8—1221no—
395C4×Q8—1221no—
396C4×D4—612no—
397C4×D4—612no—
398C4×D4—612no—
399C4×D4—612no—
400C4×D4—611no—
401C4×D4—611no—
402C4×D4—1222no—
403C4×D4—1222no—
404C4×D4—1221no—
405C4×D4—1221no—
406#117—611no—
407#117—612no—
408#117—611no—
409#117—611no—
410#117—1222no—
411#117—1222no—
412#117—1222no—
413C2×C22⋊C4—21no—
414C2×C22⋊C4—21no—
415C2×C22⋊C4—1222no—
416C2×C42—612no—
417C2×C42C22×D6111yescharacteristic
418C4⋊C8—21no—
419C4⋊C8—1222no—
420C4⋊C8—1222no—
421C4⋊C8—1221no—
422C22⋊C8—21no—
423C22⋊C8—1222no—
424C22⋊C8—1222no—
425C22⋊C8—1221no—
426#97—611no—
427#97—612no—
428#97—611no—
429#97—1222no—
430#97C22×D6112yes—
431#97C22×D6111yescharacteristic
432#97C22×D6111yescharacteristic
433C4×C8—21no—
434C4×C8—1222no—
435C4×C8—1221no—
436C24—21no—
437#41—2442no—
438C2×Q8—21no—
439C2×Q8—42no—
440C2×Q8—611no—
441C2×Q8—1221no—
442C2×Q8—1221no—
443C2×Q8C2×C4×D6112yes—
444C2×D4—21no—
445C2×D4—42no—
446C2×D4—612no—
447C2×D4—611no—
448C2×D4—1222no—
449C2×D4—1222no—
450C2×D4C2×C4×D6112yes—
451C22×C4—21no—
452C22×C4—612no—
453C22×C4—612no—
454C22×C4—612no—
455C22×C4—1222no—
456C22×C4—1222no—
457C22×C4C2×C4×D6111yescharacteristic
458C22×C4D4×D6111yescharacteristic
459C22×C4D4×D6111yescharacteristic
460M16—42no—
461M16—2444no—
462M16—2442no—
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—22no—
466C2×C8—1221no—
467C2×C8—1222no—
468C2×C8—1221no—
469C2×C8—2444no—
470C4⋊C4—612no—
471C4⋊C4—612no—
472C4⋊C4—611no—
473C4⋊C4—611no—
474C4⋊C4—611no—
475C4⋊C4—611no—
476C4⋊C4—611no—
477C4⋊C4—611no—
478C4⋊C4—1222no—
479C4⋊C4—1222no—
480C4⋊C4—1222no—
481C4⋊C4—1222no—
482C4⋊C4—1221no—
483C4⋊C4—1221no—
484C4⋊C4—1221no—
485C4⋊C4—1221no—
486C4⋊C4—1221no—
487C4⋊C4—1221no—
488C22⋊C4—22no—
489C22⋊C4—22no—
490C22⋊C4—22no—
491C22⋊C4—22no—
492C22⋊C4—612no—
493C22⋊C4—612no—
494C22⋊C4—612no—
495C22⋊C4—1222no—
496C22⋊C4—1222no—
497C22⋊C4—1222no—
498C22⋊C4—1222no—
499C22⋊C4—1222no—
500C22⋊C4—1222no—
501C22⋊C4—1221no—
502C22⋊C4—1221no—
503C42—612no—
504C42—612no—
505C42—612no—
506C42—1222no—
507C42—1222no—
508C42—1221no—
509C42—1221no—
510C42C2×C4×D6111yescharacteristic
511C42D4×D6111yescharacteristic
512C42D4×D6111yescharacteristic
513C42C23×D6111yescharacteristic
514C23—22no—
515C23—42no—
516C23—81no—
517C23—81no—
518C23—612no—
519C23C22⋊C4×D6111yescharacteristic
520Q8—42no—
521Q8—42no—
522Q8—1222no—
523Q8—4881no—
524Q8—4881no—
525D4—42no—
526D4—42no—
527D4—1222no—
528D4—4882no—
529D4—4882no—
530D4—2442no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—42no—
536C2×C4—612no—
537C2×C4—612no—
538C2×C4—612no—
539C2×C4—1221no—
540C2×C4—1221no—
541C2×C4—1221no—
542C2×C4—1221no—
543C2×C4—4882no—
544C2×C4—4882no—
545C2×C4—2442no—
546C2×C4—2442no—
547C2×C4—2442no—
548C2×C4C22⋊C4×D6111yescharacteristic
549C2×C4C22⋊C4×D6111yescharacteristic
550C2×C4C22⋊C4×D6111yescharacteristic
551C2×C4C22×C4×D6111yescharacteristic
552C2×C4C2×D4×D6111yescharacteristic
553C2×C4C2×D4×D6111yescharacteristic
554C8—22no—
555C8—42no—
556C8—41no—
557C8—1222no—
558C8—2442no—
559C8—2441no—
560C22—21no—
561C22—21no—
562C22—21no—
563C22—42no—
564C22—42no—
565C22—42no—
566C22—162no—
567C22—1222no—
568C22—1222no—
569C22—1222no—
570C22C2×C22⋊C4×D6111yescharacteristic
571C4—21no—
572C4—21no—
573C4—21no—
574C4—21no—
575C4—21no—
576C4—42no—
577C4—1222no—
578C4—1222no—
579C4—1222no—
580C4—2441no—
581C4—2441no—
582C4—2441no—
583C4—2441no—
584C4#1683974312yes—
585C2—41no—
586C2—82no—
587C2—2442no—
588C2C2×#1683974311yescharacteristic
589C2#2145436311yescharacteristic
590C2#2145436511yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D61
commutatorC2×C4×C61C23×C4
FrattiniC42C23×D61
FittingC61×#1527C2
socleC22×C61C22×C22⋊C4
prime radical#16840329C4
cosocleC42×C61C24
hypercenter#1527D61
hypocenterC61#17691
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1527

Complements

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

NH ≅ G/Ntype
2. #21454837C2semidirect
3. #21454831C2semidirect
4. #21444456C2semidirect
5. #21444447C2semidirect
6. #21444436C2semidirect
7. #21444423C2semidirect
8. #21444417C2semidirect
9. #21444413C2semidirect
10. #21444404C2semidirect
11. C61×#1527C2semidirect
35. #16839817C22semidirect
36. #16839817C22semidirect
38. #16839714C22semidirect
39. #97×D61C22semidirect
40. #16839503C22semidirect
41. #16839503C22semidirect
42. #16839503C22semidirect
43. #16839503C22semidirect
44. #16839503C22semidirect
45. #16839503C22semidirect
298. #1527D61semidirect
348. C61#17691semidirect

Maximal subgroups

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

subgroupindex
#214597132
#214548372
#214548312
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