Finite Group Databasebeta

← #29647579

Subgroups — #29647579

Order 15616.

982 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459870C211yesmaximal, characteristic
2#21454856C211yesmaximal, characteristic, complemented
3#21448457C211yesmaximal, characteristic, complemented
4#21444389C212yesmaximal, complemented
5#21444372C212yesmaximal, complemented
6#21444359C212yesmaximal, complemented
7#21444350C212yesmaximal, complemented
8#21444332C211yesmaximal, characteristic, complemented
9#21444317C211yesmaximal, characteristic, complemented
10#21444305C211yesmaximal, characteristic, complemented
11C61×#1509C211yesmaximal, characteristic, complemented
12#16840485C411yescharacteristic, complemented
13#16840469—21no—
14#16840450C412yescomplemented
15#16840435—22no—
16Q8×D244C411yescharacteristic, complemented
17#16840420C412yescomplemented
18#16840417C411yescharacteristic, complemented
19#16840416C411yescharacteristic, complemented
20#16840408—21no—
21#16840383—21no—
22#16840353—22no—
23#16840319C2211yescharacteristic
24#16840272C2212yes—
25C22⋊Q8×D61C2212yes—
26#16840242—21no—
27C2×C4×D244C2211yescharacteristic
28#16839839—21no—
29#16839835—22no—
30#16839827—21no—
31#16839815C2211yescharacteristic
32#16839724C2212yes—
33C22⋊C8×D61C2212yes—
34#16839663—21no—
35#16839658—21no—
36#16839653—22no—
37#16839624C2212yes—
38#16839621C2212yes—
39#16839608C2211yescharacteristic
40#16839500C2212yes—
41#16839500C2212yes—
42#16839499C2211yescharacteristic
43#16839499C2211yescharacteristic
44#16839498C2212yes—
45#16839498C2212yes—
46#16839496C2211yescharacteristic
47#16839496C2211yescharacteristic
48C61×#529C2211yescharacteristic
49C61×#432C2211yescharacteristic
50C4×C4⋊C4×C61C2211yescharacteristic
51C61×#328C2212yes—
52C61×#327C2212yes—
53C2×#11758222—21no—
54C2×#11758224C2×C412yes—
55C2×Q8×D61C2×C412yes—
56C22×D244D411yescharacteristic
57C22×C4×D61C2312yes—
58C2×#11758189—22no—
59C2×C8×D61—22no—
60#14519777—21no—
61#14519777C2×C411yescharacteristic
62#14519777C2×C411yescharacteristic
63#14519772—22no—
64#14519772—22no—
65#14519769—21no—
66#14519768—22no—
67#14519765—21no—
68#14519763—22no—
69#14519762—22no—
70#14519761—22no—
71#14519759—22no—
72#14519759—22no—
73#14519757—22no—
74#14519757C2×C412yes—
75#14519757C2×C412yes—
76#14519756—21no—
77#14519756—21no—
78#14519756C2312yes—
79#14519755—22no—
80#14519754—22no—
81#14519754—22no—
82#14519753C2×C412yes—
83#14519751—21no—
84#14519750—22no—
85#14519748C2×C412yes—
86#14519745—22no—
87C4×Dic122—21no—
88C4×Dic122—21no—
89C4×Dic122C2×C412yes—
90C4×D244—21no—
91C4×D244C2×C412yes—
92C4×D244C2×C411yescharacteristic
93C4×D244C2×C411yescharacteristic
94C4×D244D411yescharacteristic
95C4×D244D411yescharacteristic
96C4×D244D412yes—
97D4×Dic61—22no—
98#14519737—22no—
99#14519737—22no—
100#14519736—21no—
101#14519735—22no—
102#14519735C2×C412yes—
103C2×C4⋊Dic61—21no—
104C2×C4⋊Dic61D411yescharacteristic
105C2×#11758196—22no—
106C4⋊C4×D61—22no—
107C4⋊C4×D61C2×C412yes—
108C2×#11758198—22no—
109C2×#11758198—22no—
110C2×#11758198D412yes—
111C22⋊C4×D61—22no—
112C2×C4×Dic61—21no—
113C4⋊#92418—21no—
114C22⋊#92418C2311yescharacteristic
115C22⋊#92418C2311yescharacteristic
116#14519689—22no—
117#14519689—22no—
118C4×#92418—21no—
119#14519684—21no—
120#14519684—22no—
121#14519684C2311yescharacteristic
122#14519684C2311yescharacteristic
123#126×C61—21no—
124#125×C61C2×C411yescharacteristic
125#124×C61C2×C411yescharacteristic
126#123×C61—21no—
127C22⋊Q8×C61C2311yescharacteristic
128#121×C61C2311yescharacteristic
129C4×Q8×C61C2×C411yescharacteristic
130C4×D4×C61—21no—
131C4×D4×C61C2×C411yescharacteristic
132#117×C61—21no—
133C2×C4⋊C4×C61—21no—
134C2×C42×C61—21no—
135C2×C42×C61C2311yescharacteristic
136C4⋊C8×C61—21no—
137C22⋊C8×C61C2312yes—
138#97×C61—21no—
139#95×C61C2312yes—
140C23×D61C2×D412yes—
141C2×Q8×C61C22×C411yescharacteristic
142C2×D4×C61—21no—
143C2×D4×C61C22×C411yescharacteristic
144C22×C4×C61—21no—
145C22×C4×C61—22no—
146C22×C4×C61C2×D411yescharacteristic
147C22×C4×C61C2×D411yescharacteristic
148C22×C4×C61C2411yescharacteristic
149C2×C22⋊D61—22no—
150C2×C22⋊D61—22no—
151C22×Dic61—22no—
152C22×Dic61—21no—
153C22×Dic61—21no—
154C22×Dic61C2×D412yes—
155Q8×D61—82no—
156#11758224—82no—
157#11758222—44no—
158C2×D244—42no—
159C2×D244C22⋊C411yescharacteristic
160C2×D244C22⋊C411yescharacteristic
161C2×D244C22⋊C412yes—
162C2×C4×D61—22no—
163C2×C4×D61—22no—
164C2×C4×D61—82no—
165C2×C4×D61C22×C412yes—
166C2×C4×D61C22×C412yes—
167C2×C4×D61C2×D412yes—
168C2×C4×D61C2×D412yes—
169C2×Dic122—21no—
170C2×Dic122—22no—
171C2×Dic122C22×C412yes—
172C2×C8×C61—22no—
173C4⋊C4×C61—21no—
174C4⋊C4×C61—21no—
175C4⋊C4×C61—21no—
176C4⋊C4×C61—21no—
177C4⋊C4×C61—42no—
178C4⋊C4×C61C22×C411yescharacteristic
179C4⋊C4×C61C22×C411yescharacteristic
180C22⋊C4×C61—21no—
181C22⋊C4×C61—21no—
182C22⋊C4×C61—21no—
183C22⋊C4×C61—21no—
184C22⋊C4×C61—21no—
185C42×C61—21no—
186C42×C61—81no—
187C42×C61C22×C411yescharacteristic
188C42×C61C22×C411yescharacteristic
189C42×C61C2×D411yescharacteristic
190C42×C61C2×D411yescharacteristic
191C22⋊Dic61—22no—
192C22⋊Dic61—22no—
193#11758198—22no—
194#11758198—22no—
195#11758198—22no—
196#11758198—22no—
197#11758198—82no—
198#11758198—82no—
199#11758198C22⋊C412yes—
200#11758198C22⋊C412yes—
201#11758198C22⋊C412yes—
202#11758198C22⋊C412yes—
203C4⋊Dic61—21no—
204C4⋊Dic61—21no—
205C4⋊Dic61—21no—
206C4⋊Dic61—22no—
207C4⋊Dic61—22no—
208C4⋊Dic61C22⋊C411yescharacteristic
209C4⋊Dic61C22⋊C412yes—
210C4⋊Dic61C22⋊C411yescharacteristic
211C4⋊Dic61C22×C412yes—
212#11758196—22no—
213#11758196—22no—
214#11758196—22no—
215#11758196—22no—
216#11758196—82no—
217C4×Dic61—22no—
218C4×Dic61—22no—
219C4×Dic61—42no—
220C2×#92418—21no—
221C2×#92418—21no—
222#11758189—82no—
223C8×D61—82no—
224C23×C61—21no—
225C23×C61C22×D411yescharacteristic
226C22×D61—22no—
227C22×D61—22no—
228C22×D61—42no—
229C22×D61—42no—
230C22×D61C2×C22⋊C412yes—
231C22×D61C2×C22⋊C412yes—
232Q8×C61—41no—
233D4×C61—41no—
234D4×C61—41no—
235D4×C61—41no—
236C2×C4×C61—21no—
237C2×C4×C61—21no—
238C2×C4×C61—22no—
239C2×C4×C61—21no—
240C2×C4×C61—21no—
241C2×C4×C61—21no—
242C2×C4×C61—41no—
243C2×C4×C61—42no—
244C2×C4×C61—42no—
245C2×C4×C61—42no—
246C2×C4×C61—42no—
247C2×C4×C61C2×C22⋊C411yescharacteristic
248C2×C4×C61C2×C22⋊C411yescharacteristic
249C2×C4×C61C2×C22⋊C411yescharacteristic
250C2×C4×C61C2×C22⋊C411yescharacteristic
251C2×C4×C61C23×C411yescharacteristic
252C2×C4×C61C22×D411yescharacteristic
253C22⋊D61—44no—
254C22⋊D61—82no—
255C2×Dic61—22no—
256C2×Dic61—22no—
257C2×Dic61—42no—
258C2×Dic61—41no—
259C2×Dic61—41no—
260C2×Dic61—41no—
261C2×Dic61—41no—
262C2×Dic61—41no—
263C2×Dic61—41no—
264C2×Dic61—82no—
265C2×Dic61C2×C22⋊C412yes—
266C2×Dic61C2×C22⋊C412yes—
267D244—22no—
268D244—22no—
269D244—42no—
270C4×D61—42no—
271C4×D61—42no—
272C4×D61—42no—
273C4×D61—42no—
274C4×D61—44no—
275C4×D61—82no—
276Dic122—42no—
277Dic122—42no—
278Dic122—82no—
279C8×C61—42no—
280#92418—41no—
281#92418—41no—
282#17600—611nomaximal, Sylow
283C22×C61—21no—
284C22×C61—21no—
285C22×C61—41no—
286C22×C61—41no—
287C22×C61—41no—
288C22×C61C22×C22⋊C411yescharacteristic
289C22×C61#57711yescharacteristic
290C22×C61#57811yescharacteristic
291C2×D61—22no—
292C2×D61—22no—
293C2×D61—22no—
294C2×D61—22no—
295C2×D61—22no—
296C2×D61—22no—
297C2×D61—82no—
298C2×D61—82no—
299C4×C61—21no—
300C4×C61—21no—
301C4×C61—21no—
302C4×C61—21no—
303C4×C61—21no—
304C4×C61—41no—
305C4×C61—41no—
306C4×C61—41no—
307C4×C61—82no—
308C4×C61#42112yes—
309Dic61—22no—
310Dic61—22no—
311Dic61—42no—
312Dic61—42no—
313Dic61—81no—
314Dic61—81no—
315#3520—611no—
316#2974—611no—
317#2319—611no—
318#1520—612no—
319#1516—612no—
320#1513—612no—
321#1511—612no—
322#1509D6111yescharacteristic, complemented
323#1506—611no—
324#1503—611no—
325#1500—611no—
326C2×C61—21no—
327C2×C61—21no—
328C2×C61—81no—
329C2×C61#288311yescharacteristic
330C2×C61#289811yescharacteristic
331C2×C61C2×#42111yescharacteristic
332D61—42no—
333D61—42no—
334#556—611no—
335#554—1221no—
336#552—612no—
337#551—1222no—
338D4×Q8—611no—
339#548—612no—
340#547—611no—
341#547—611no—
342#546—1221no—
343#542—1221no—
344#539—1222no—
345#533—611no—
346#529C2×D6111yescharacteristic
347#525—612no—
348C2×C22⋊Q8—612no—
349#520—1221no—
350C2×C4×D4—611no—
351#436—1221no—
352#435—1222no—
353C8×D4—1221no—
354#432C2×D6111yescharacteristic
355#431—611no—
356#407—612no—
357C2×C22⋊C8—612no—
358#390—1221no—
359#389—1221no—
360#388—1222no—
361#381—612no—
362#380—612no—
363C4×C4⋊C4C2×D6111yescharacteristic
364#376—611no—
365#328—611no—
366#328—611no—
367#328—612no—
368#328—612no—
369#328—612no—
370#328—612no—
371#328C2×D6112yes—
372#327—611no—
373#327—611no—
374#327C2×D6112yes—
375C61#1760011yescharacteristic, complemented, Sylow
376C2×#41—612no—
377C2×#41—1221no—
378C22×Q8—612no—
379C22×D4—611no—
380C23×C4—612no—
381C2×M16—1222no—
382C22×C8—1222no—
383C4⋊Q8—611no—
384C4⋊Q8—611no—
385C4⋊Q8—1221no—
386#126—21no—
387#126—1222no—
388#126—1222no—
389#125—1222no—
390#125—1221no—
391#125C4×D6111yescharacteristic
392#124—1221no—
393#124—1222no—
394#124C4×D6111yescharacteristic
395#123—21no—
396#123—1222no—
397#123—1222no—
398#123—1222no—
399#123—1222no—
400C22⋊Q8—612no—
401C22⋊Q8—612no—
402C22⋊Q8—612no—
403C22⋊Q8—612no—
404C22⋊Q8—1222no—
405C22⋊Q8—1222no—
406C22⋊Q8—1222no—
407C22⋊Q8—1222no—
408C22⋊Q8—1221no—
409C22⋊Q8—1221no—
410C22⋊Q8C22×D6111yescharacteristic
411#121—612no—
412#121—1222no—
413#121—1221no—
414#121C22×D6111yescharacteristic
415#120—1222no—
416C4×Q8—612no—
417C4×Q8—1221no—
418C4×Q8—1221no—
419C4×Q8C4×D6111yescharacteristic
420C4×D4—21no—
421C4×D4—611no—
422C4×D4—611no—
423C4×D4—612no—
424C4×D4—611no—
425C4×D4—612no—
426C4×D4—611no—
427C4×D4—1221no—
428C4×D4—1222no—
429C4×D4—1222no—
430C4×D4—1222no—
431C4×D4C4×D6111yescharacteristic
432#117—21no—
433#117—612no—
434#117—1222no—
435#117—1221no—
436C2×C4⋊C4—21no—
437C2×C4⋊C4—611no—
438C2×C4⋊C4—612no—
439C2×C4⋊C4—1222no—
440C2×C4⋊C4—1221no—
441C2×C4⋊C4—1222no—
442C2×C22⋊C4—612no—
443C2×C22⋊C4—1222no—
444C2×C22⋊C4—1222no—
445C2×C22⋊C4—1222no—
446C2×C42—21no—
447C2×C42—1221no—
448C2×C42C22×D6111yescharacteristic
449C4⋊C8—21no—
450C4⋊C8—1221no—
451C22⋊C8—611no—
452C22⋊C8—611no—
453C22⋊C8—1222no—
454C22⋊C8—1222no—
455C22⋊C8C22×D6112yes—
456#97—21no—
457C4×C8—1221no—
458#95—611no—
459#95—611no—
460#95—1222no—
461#95—1221no—
462#95C22×D6112yes—
463C24—612no—
464#41—4882no—
465#41—2444no—
466C2×Q8—612no—
467C2×Q8—1222no—
468C2×Q8—1221no—
469C2×Q8—4882no—
470C2×Q8C2×C4×D6111yescharacteristic
471C2×D4—21no—
472C2×D4—611no—
473C2×D4—611no—
474C2×D4—612no—
475C2×D4—1222no—
476C2×D4—1222no—
477C2×D4—2442no—
478C2×D4C2×C4×D6111yescharacteristic
479C22×C4—21no—
480C22×C4—22no—
481C22×C4—612no—
482C22×C4—612no—
483C22×C4—612no—
484C22×C4—612no—
485C22×C4—612no—
486C22×C4—1222no—
487C22×C4—1222no—
488C22×C4—1222no—
489C22×C4—1221no—
490C22×C4—1221no—
491C22×C4—4882no—
492C22×C4D4×D6111yescharacteristic
493C22×C4D4×D6111yescharacteristic
494C22×C4C23×D6111yescharacteristic
495M16—4882no—
496C2×C8—22no—
497C2×C8—1221no—
498C2×C8—1221no—
499C2×C8—4882no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—21no—
504C4⋊C4—42no—
505C4⋊C4—611no—
506C4⋊C4—612no—
507C4⋊C4—611no—
508C4⋊C4—612no—
509C4⋊C4—1222no—
510C4⋊C4—1222no—
511C4⋊C4—1222no—
512C4⋊C4—1222no—
513C4⋊C4—1222no—
514C4⋊C4—1221no—
515C4⋊C4—1222no—
516C4⋊C4—1221no—
517C4⋊C4—1221no—
518C4⋊C4—4882no—
519C4⋊C4C2×C4×D6111yescharacteristic
520C4⋊C4C2×C4×D6111yescharacteristic
521C22⋊C4—21no—
522C22⋊C4—21no—
523C22⋊C4—21no—
524C22⋊C4—21no—
525C22⋊C4—21no—
526C22⋊C4—612no—
527C22⋊C4—612no—
528C22⋊C4—612no—
529C22⋊C4—612no—
530C22⋊C4—1222no—
531C22⋊C4—1222no—
532C22⋊C4—1222no—
533C22⋊C4—1222no—
534C22⋊C4—1222no—
535C22⋊C4—1222no—
536C22⋊C4—4882no—
537C22⋊C4—4882no—
538C42—21no—
539C42—81no—
540C42—1222no—
541C42—1222no—
542C42—2442no—
543C42C2×C4×D6111yescharacteristic
544C42C2×C4×D6111yescharacteristic
545C42D4×D6111yescharacteristic
546C42D4×D6111yescharacteristic
547C23—21no—
548C23—612no—
549C23—612no—
550C23—1222no—
551C23—1222no—
552C23—2442no—
553C23—2442no—
554C23C2×D4×D6111yescharacteristic
555Q8—41no—
556Q8—4882no—
557Q8—2442no—
558Q8—2442no—
559D4—41no—
560D4—41no—
561D4—41no—
562D4—1222no—
563D4—1222no—
564D4—4882no—
565D4—2442no—
566D4—2444no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—22no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—41no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—612no—
579C2×C4—612no—
580C2×C4—1222no—
581C2×C4—1222no—
582C2×C4—4882no—
583C2×C4—4882no—
584C2×C4—2442no—
585C2×C4—2442no—
586C2×C4—2442no—
587C2×C4—2442no—
588C2×C4—2444no—
589C2×C4—2442no—
590C2×C4—2441no—
591C2×C4—2441no—
592C2×C4—2441no—
593C2×C4—2441no—
594C2×C4—2441no—
595C2×C4—2441no—
596C2×C4C22⋊C4×D6111yescharacteristic
597C2×C4C22⋊C4×D6111yescharacteristic
598C2×C4C22⋊C4×D6111yescharacteristic
599C2×C4C22⋊C4×D6111yescharacteristic
600C2×C4C22×C4×D6111yescharacteristic
601C2×C4C2×D4×D6111yescharacteristic
602C8—42no—
603C8—2441no—
604C8—2441no—
605C22—21no—
606C22—21no—
607C22—41no—
608C22—41no—
609C22—41no—
610C22—1222no—
611C22—1222no—
612C22—1222no—
613C22—1222no—
614C22—1222no—
615C22—1222no—
616C22—4882no—
617C22—4882no—
618C22C2×C22⋊C4×D6111yescharacteristic
619C22#1684057211yescharacteristic
620C22#1684057811yescharacteristic
621C4—21no—
622C4—21no—
623C4—21no—
624C4—21no—
625C4—21no—
626C4—41no—
627C4—41no—
628C4—41no—
629C4—82no—
630C4—1222no—
631C4—1222no—
632C4—4881no—
633C4—4881no—
634C4—2442no—
635C4—2442no—
636C4#1683977712yes—
637C2—21no—
638C2—21no—
639C2—81no—
640C2—2442no—
641C2—2442no—
642C2#2145427811yescharacteristic
643C2#2145435011yescharacteristic
644C2C2×#1683977711yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1509C2
socleC22×C61C22×C22⋊C4
prime radical#21459870C2
cosocleC22×C4×C61C24
hypercenter#1509D61
hypocenterC61#17600
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1509

Complements

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

NH ≅ G/Ntype
2. #21454856C2semidirect
3. #21448457C2semidirect
4. #21444389C2semidirect
5. #21444372C2semidirect
6. #21444359C2semidirect
7. #21444350C2semidirect
8. #21444332C2semidirect
9. #21444317C2semidirect
10. #21444305C2semidirect
11. C61×#1509C2semidirect
12. #16840485C4semidirect
14. #16840450C4semidirect
16. Q8×D244C4semidirect
17. #16840420C4semidirect
18. #16840417C4semidirect
19. #16840416C4semidirect
322. #1509D61semidirect
375. C61#17600semidirect

Maximal subgroups

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

subgroupindex
#214598702
#214548562
#214484572
#214443892
#214443722
#214443592
#214443502
#214443322
#214443172
#214443052
C61×#15092
#1760061