Finite Group Databasebeta

← #29646294

Subgroups — #29646294

Order 15616.

922 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840453C211yesmaximal, characteristic
2#21454908C211yesmaximal, characteristic, complemented
3#21444189C212yesmaximal, complemented
4#21444186C212yesmaximal, complemented
5#21444176C211yesmaximal, characteristic, complemented
6#21444174C211yesmaximal, characteristic, complemented
7#21444159C211yesmaximal, characteristic
8#21444151C212yesmaximal, complemented
9#21444139C211yesmaximal, characteristic
10#21444131C212yesmaximal, complemented
11C61×#2984C211yesmaximal, characteristic, complemented
12#16840453—21no—
13#16840453—22no—
14#16840453—21no—
15#16840453—21no—
16#16840453—22no—
17#16840453—21no—
18C2×#14519771C2212yes—
19C2×#14519769C2211yescharacteristic
20C2×#14519767C412yes—
21C2×#14519762C412yes—
22C2×#14519753C411yescharacteristic
23C2×#14519753C411yescharacteristic
24C2×C4×D244C2211yescharacteristic
25C2×#14519735C411yescharacteristic
26C2×#14519734C2212yes—
27C2×C4⋊C4×D61C411yescharacteristic
28#16839893—22no—
29#16839893—21no—
30Q8×#92418—21no—
31#16839819C2212yes—
32#16839819C2211yescharacteristic
33#16839815C2212yes—
34#16839815C2211yescharacteristic
35#16839487C2212yes—
36#16839487C2212yes—
37#16839487C2212yes—
38#16839487C2212yes—
39#16839485C2211yescharacteristic
40#16839485C2212yes—
41#16839485C2211yescharacteristic
42#16839485C2211yescharacteristic
43#16839485C2211yescharacteristic
44#16839485C2212yes—
45C2×C4×Q8×C61C2211yescharacteristic
46C61×#446—22no—
47C61×#446—21no—
48C8×Q8×C61—21no—
49C61×#432C2212yes—
50C61×#432C2211yescharacteristic
51C61×#431C2212yes—
52C61×#431C2211yescharacteristic
53C22×D244C2×C411yescharacteristic
54C22×C4×D61C2×C412yes—
55C22×C4×D61C2×C411yescharacteristic
56C22×C4×D61C2×C411yescharacteristic
57#14519771—22no—
58#14519771—22no—
59#14519771—22no—
60#14519771—22no—
61#14519769—21no—
62#14519769—21no—
63#14519769—21no—
64#14519769—21no—
65#14519767—22no—
66#14519767—22no—
67#14519767—22no—
68#14519767—22no—
69#14519762—22no—
70#14519762—22no—
71#14519762—22no—
72#14519762—22no—
73#14519753—21no—
74#14519753—21no—
75#14519753—22no—
76#14519753—21no—
77#14519753—21no—
78#14519753—22no—
79C4×D244—21no—
80C4×D244—21no—
81C4×D244—21no—
82C4×D244—21no—
83#14519735—22no—
84#14519735—21no—
85#14519735—21no—
86#14519734—22no—
87#14519734—22no—
88#14519734—22no—
89#14519734—22no—
90C2×C4⋊Dic61C2×C411yescharacteristic
91C2×C4⋊Dic61C2×C411yescharacteristic
92C2×C4⋊Dic61C2×C411yescharacteristic
93C2×#11758196C2×C412yes—
94C2×#11758196C2×C412yes—
95C2×#11758196C2×C412yes—
96C2×#11758196C2×C412yes—
97C2×#11758196C2312yes—
98C4⋊C4×D61—22no—
99C4⋊C4×D61—21no—
100C4⋊C4×D61—21no—
101C2×#11758198C2×C412yes—
102C2×#11758198C2×C412yes—
103C2×#11758198C2×C412yes—
104C2×#11758198C2311yescharacteristic
105C2×#11758198C2311yescharacteristic
106C2×C4×Dic61C2×C412yes—
107C4⋊#92418—22no—
108C4⋊#92418—21no—
109C4⋊#92418—22no—
110C4⋊#92418—21no—
111C22⋊#92418C2311yescharacteristic
112C22⋊#92418C2312yes—
113C22⋊#92418C2311yescharacteristic
114#14519688—22no—
115#14519688—21no—
116C4×#92418—22no—
117C4×#92418—21no—
118C22×Q8×C61C2×C411yescharacteristic
119C4×Q8×C61D411yescharacteristic
120C4×Q8×C61D412yes—
121C4×Q8×C61D411yescharacteristic
122C4×Q8×C61D411yescharacteristic
123C4×Q8×C61D412yes—
124C4×Q8×C61D411yescharacteristic
125C2×C4⋊C4×C61C2×C411yescharacteristic
126C2×C4⋊C4×C61C2×C412yes—
127C2×C42×C61C2312yes—
128C2×C42×C61C2311yescharacteristic
129C4⋊C8×C61—22no—
130C4⋊C8×C61—21no—
131C4⋊C8×C61—22no—
132C4⋊C8×C61—21no—
133C22⋊C8×C61C2311yescharacteristic
134C22⋊C8×C61C2312yes—
135C22⋊C8×C61C2311yescharacteristic
136#97×C61—22no—
137#97×C61—21no—
138C4×C8×C61—22no—
139C4×C8×C61—21no—
140C23×D61C22×C411yescharacteristic
141C23×D61C22×C411yescharacteristic
142C2×Q8×C61—42no—
143C2×Q8×C61C22⋊C411yescharacteristic
144C2×Q8×C61C22⋊C411yescharacteristic
145C2×Q8×C61C22⋊C412yes—
146C22×C4×C61C22×C411yescharacteristic
147C22×C4×C61C22×C412yes—
148C22×C4×C61C22×C411yescharacteristic
149C22×C4×C61C22×C412yes—
150C22×C4×C61C2411yescharacteristic
151C22×Dic61C22×C412yes—
152C22×Dic61C22×C411yescharacteristic
153C22×Dic61C22×C411yescharacteristic
154C22×Dic61C22×C412yes—
155C2×D244—22no—
156C2×D244—21no—
157C2×D244—21no—
158C2×D244—41no—
159C2×C4×D61—24no—
160C2×C4×D61—22no—
161C2×C4×D61—22no—
162C2×C4×D61—21no—
163C2×C4×D61—21no—
164C2×C4×D61—21no—
165C2×C4×D61—21no—
166C2×C4×D61—42no—
167C2×C4×D61—42no—
168C2×C4×D61—42no—
169C2×C8×C61—21no—
170C2×C8×C61—22no—
171C2×C8×C61—21no—
172C4⋊C4×C61C22⋊C412yes—
173C4⋊C4×C61C22⋊C411yescharacteristic
174C4⋊C4×C61C22⋊C411yescharacteristic
175C4⋊C4×C61C22⋊C412yes—
176C4⋊C4×C61C22⋊C412yes—
177C4⋊C4×C61C22⋊C412yes—
178C4⋊C4×C61C22⋊C412yes—
179C42×C61C2×D412yes—
180C42×C61C2×D411yescharacteristic
181C42×C61C2×D412yes—
182C42×C61C2×D411yescharacteristic
183C42×C61C2×D412yes—
184C42×C61C2×D411yescharacteristic
185C42×C61C2×D412yes—
186C42×C61C2×D411yescharacteristic
187#11758198—22no—
188#11758198—22no—
189#11758198—22no—
190#11758198—22no—
191#11758198—22no—
192#11758198—22no—
193#11758198—21no—
194#11758198—21no—
195#11758198—21no—
196#11758198—21no—
197C4⋊Dic61—21no—
198C4⋊Dic61—21no—
199C4⋊Dic61—21no—
200C4⋊Dic61—21no—
201C4⋊Dic61—21no—
202C4⋊Dic61—21no—
203#11758196—22no—
204#11758196—22no—
205#11758196—22no—
206#11758196—22no—
207#11758196—22no—
208#11758196—22no—
209#11758196—22no—
210#11758196—22no—
211#11758196—22no—
212#11758196—22no—
213C4×Dic61—22no—
214C4×Dic61—22no—
215C2×#92418—21no—
216C2×#92418—22no—
217C2×#92418—21no—
218C23×C61C23×C411yescharacteristic
219C22×D61—21no—
220C22×D61—21no—
221C22×D61—21no—
222C22×D61—21no—
223C22×D61—21no—
224C22×D61—41no—
225C22×D61—41no—
226C22×D61—41no—
227C22×D61—41no—
228C22×D61#13112yes—
229Q8×C61—22no—
230Q8×C61—22no—
231Q8×C61—42no—
232C2×C4×C61—22no—
233C2×C4×C61—21no—
234C2×C4×C61—22no—
235C2×C4×C61—21no—
236C2×C4×C61—22no—
237C2×C4×C61—42no—
238C2×C4×C61—41no—
239C2×C4×C61C2×C22⋊C412yes—
240C2×C4×C61C2×C22⋊C412yes—
241C2×C4×C61C2×C22⋊C411yescharacteristic
242C2×C4×C61C2×C22⋊C411yescharacteristic
243C2×C4×C61C2×C22⋊C412yes—
244C2×C4×C61C2×C22⋊C412yes—
245C2×C4×C61C2×C22⋊C411yescharacteristic
246C2×C4×C61C2×C22⋊C411yescharacteristic
247C2×C4×C61C22×D411yescharacteristic
248C2×C4×C61C22×D411yescharacteristic
249C2×Dic61—21no—
250C2×Dic61—22no—
251C2×Dic61—22no—
252C2×Dic61—21no—
253C2×Dic61—22no—
254C2×Dic61—22no—
255C2×Dic61—41no—
256C2×Dic61—41no—
257C2×Dic61—42no—
258D244—42no—
259D244—42no—
260C4×D61—44no—
261C4×D61—44no—
262C4×D61—42no—
263C4×D61—42no—
264C4×D61—42no—
265C4×D61—42no—
266C8×C61—41no—
267C8×C61—42no—
268C8×C61—41no—
269#92418—41no—
270#92418—42no—
271#92418—41no—
272#17423—611nomaximal, Sylow
273C22×C61—21no—
274C22×C61—21no—
275C22×C61C22×C22⋊C411yescharacteristic
276C22×C61C2×#13111yescharacteristic
277C22×C61#56811yescharacteristic
278C2×D61—22no—
279C2×D61—42no—
280C2×D61—41no—
281C2×D61—41no—
282C2×D61—42no—
283C2×D61—41no—
284C2×D61—41no—
285C2×D61—41no—
286C2×D61—81no—
287C2×D61—81no—
288C4×C61—22no—
289C4×C61—22no—
290C4×C61—21no—
291C4×C61—21no—
292C4×C61—21no—
293C4×C61—21no—
294C4×C61—21no—
295C4×C61—22no—
296C4×C61—22no—
297C4×C61#41312yes—
298Dic61—42no—
299Dic61—42no—
300Dic61—41no—
301Dic61—41no—
302C2×#552—611no—
303#2984—611no—
304#2984D6111yescharacteristic, complemented
305#1473—612no—
306#1473—612no—
307#1470—611no—
308#1470—611no—
309#1466—611no—
310#1464—612no—
311#1460—611no—
312#1458—612no—
313C2×C61—21no—
314C2×C61—21no—
315C2×C61#288011yescharacteristic
316C2×C61#288411yescharacteristic
317C2×C61C2×#41311yescharacteristic
318D61—42no—
319D61—81no—
320#552—1221no—
321#552—1222no—
322#552—1221no—
323#552—1221no—
324#552—1222no—
325#552—1221no—
326C2×#126—612no—
327C2×#125—612no—
328C2×#125—611no—
329C2×#123—612no—
330C2×C22⋊Q8—611no—
331C2×C22⋊Q8—611no—
332C2×C4×Q8C2×D6111yescharacteristic
333C2×C4×D4—611no—
334C2×#117—611no—
335C2×#117—612no—
336C22×C4⋊C4—611no—
337#446—22no—
338#446—21no—
339#446—1222no—
340#446—1221no—
341C8×Q8—21no—
342C8×Q8—1221no—
343#432—612no—
344#432—611no—
345#432C2×D6112yes—
346#432C2×D6111yescharacteristic
347#431—612no—
348#431—611no—
349#431C2×D6112yes—
350#431C2×D6111yescharacteristic
351#324—612no—
352#324—612no—
353#324—612no—
354#324—612no—
355#323—611no—
356#323—612no—
357#323—611no—
358#323—611no—
359#323—611no—
360#323—612no—
361C61#1742311yescharacteristic, complemented, Sylow
362C22×Q8C4×D6111yescharacteristic
363C22×D4—611no—
364C23×C4—612no—
365C23×C4—611no—
366C23×C4—611no—
367#126—1222no—
368#126—1222no—
369#126—1222no—
370#126—1222no—
371#125—1222no—
372#125—1222no—
373#125—1222no—
374#125—1222no—
375#125—1221no—
376#125—1221no—
377#125—1221no—
378#125—1221no—
379#123—1222no—
380#123—1222no—
381#123—1222no—
382#123—1222no—
383C22⋊Q8—1221no—
384C22⋊Q8—1221no—
385C22⋊Q8—1222no—
386C22⋊Q8—1221no—
387C22⋊Q8—1221no—
388C22⋊Q8—1222no—
389C4×Q8D24411yescharacteristic
390C4×Q8D24412yes—
391C4×Q8D24411yescharacteristic
392C4×Q8C22⋊D6111yescharacteristic
393C4×Q8C22⋊D6112yes—
394C4×Q8C22⋊D6111yescharacteristic
395C4×D4—1221no—
396C4×D4—1221no—
397C4×D4—1221no—
398C4×D4—1221no—
399#117—1222no—
400#117—1221no—
401#117—1221no—
402#117—1222no—
403#117—1222no—
404#117—1222no—
405#117—1222no—
406C2×C4⋊C4—611no—
407C2×C4⋊C4—612no—
408C2×C4⋊C4—612no—
409C2×C4⋊C4—611no—
410C2×C4⋊C4—611no—
411C2×C4⋊C4—612no—
412C2×C4⋊C4—612no—
413C2×C4⋊C4—612no—
414C2×C4⋊C4—1222no—
415C2×C4⋊C4—1221no—
416C2×C4⋊C4—1221no—
417C2×C4⋊C4C4×D6111yescharacteristic
418C2×C4⋊C4C4×D6112yes—
419C2×C22⋊C4—612no—
420C2×C22⋊C4—612no—
421C2×C22⋊C4—612no—
422C2×C22⋊C4—611no—
423C2×C22⋊C4—611no—
424C2×C42—612no—
425C2×C42C22×D6112yes—
426C2×C42C22×D6111yescharacteristic
427C4⋊C8—22no—
428C4⋊C8—21no—
429C4⋊C8—22no—
430C4⋊C8—21no—
431C4⋊C8—1222no—
432C4⋊C8—1221no—
433C4⋊C8—1222no—
434C4⋊C8—1221no—
435C22⋊C8—611no—
436C22⋊C8—612no—
437C22⋊C8—611no—
438C22⋊C8C22×D6111yescharacteristic
439C22⋊C8C22×D6112yes—
440C22⋊C8C22×D6111yescharacteristic
441#97—22no—
442#97—21no—
443#97—1222no—
444#97—1221no—
445C4×C8—22no—
446C4×C8—21no—
447C4×C8—1222no—
448C4×C8—1221no—
449C24—611no—
450C24—611no—
451C2×Q8—42no—
452C2×Q8#1175819811yescharacteristic
453C2×Q8#1175819811yescharacteristic
454C2×Q8#1175819812yes—
455C2×D4—1222no—
456C2×D4—1221no—
457C2×D4—1221no—
458C2×D4—2441no—
459C22×C4—612no—
460C22×C4—611no—
461C22×C4—612no—
462C22×C4—611no—
463C22×C4—1224no—
464C22×C4—1222no—
465C22×C4—1222no—
466C22×C4—1221no—
467C22×C4—1221no—
468C22×C4—1221no—
469C22×C4—1221no—
470C22×C4—2442no—
471C22×C4—2442no—
472C22×C4—2442no—
473C22×C4C2×C4×D6112yes—
474C22×C4C2×C4×D6111yescharacteristic
475C22×C4C2×C4×D6111yescharacteristic
476C22×C4C2×C4×D6112yes—
477C22×C4C23×D6111yescharacteristic
478C2×C8—21no—
479C2×C8—22no—
480C2×C8—21no—
481C2×C8—1221no—
482C2×C8—1222no—
483C2×C8—1221no—
484C4⋊C4—1222no—
485C4⋊C4—1222no—
486C4⋊C4—1222no—
487C4⋊C4—1222no—
488C4⋊C4—1221no—
489C4⋊C4—1221no—
490C4⋊C4—1222no—
491C4⋊C4—1222no—
492C4⋊C4—1222no—
493C4⋊C4—1222no—
494C4⋊C4—1222no—
495C4⋊C4—1222no—
496C4⋊C4—1221no—
497C4⋊C4—1221no—
498C4⋊C4—1221no—
499C4⋊C4—1221no—
500C4⋊C4#1175819812yes—
501C4⋊C4#1175819811yescharacteristic
502C4⋊C4#1175819811yescharacteristic
503C4⋊C4#1175819812yes—
504C4⋊C4#1175819812yes—
505C4⋊C4#1175819812yes—
506C4⋊C4#1175819812yes—
507C22⋊C4—1221no—
508C22⋊C4—1221no—
509C22⋊C4—1221no—
510C22⋊C4—1221no—
511C22⋊C4—1222no—
512C22⋊C4—1222no—
513C22⋊C4—1222no—
514C22⋊C4—1222no—
515C22⋊C4—1222no—
516C22⋊C4—1222no—
517C42—1222no—
518C42—1222no—
519C42C2×D24412yes—
520C42C2×D24411yescharacteristic
521C42C2×D24412yes—
522C42C2×D24411yescharacteristic
523C42C2×C22⋊D6112yes—
524C42C2×C22⋊D6111yescharacteristic
525C42C2×C22⋊D6112yes—
526C42C2×C22⋊D6111yescharacteristic
527C23—612no—
528C23—1221no—
529C23—1221no—
530C23—1221no—
531C23—1221no—
532C23—1221no—
533C23—2441no—
534C23—2441no—
535C23—2441no—
536C23—2441no—
537C23C22×C4×D6111yescharacteristic
538Q8—22no—
539Q8—22no—
540Q8—42no—
541D4—2442no—
542D4—2442no—
543C2×C4—22no—
544C2×C4—21no—
545C2×C4—22no—
546C2×C4—21no—
547C2×C4—22no—
548C2×C4—42no—
549C2×C4—41no—
550C2×C4—1221no—
551C2×C4—1222no—
552C2×C4—1222no—
553C2×C4—1221no—
554C2×C4—1222no—
555C2×C4—1222no—
556C2×C4—2444no—
557C2×C4—2444no—
558C2×C4—2442no—
559C2×C4—2442no—
560C2×C4—2442no—
561C2×C4—2442no—
562C2×C4—2442no—
563C2×C4—2441no—
564C2×C4—2441no—
565C2×C4C2×#1175819812yes—
566C2×C4C2×#1175819812yes—
567C2×C4C2×#1175819812yes—
568C2×C4C2×#1175819812yes—
569C2×C4C2×#1175819811yescharacteristic
570C2×C4C2×#1175819811yescharacteristic
571C2×C4C2×#1175819811yescharacteristic
572C2×C4C2×#1175819811yescharacteristic
573C2×C4C22×C22⋊D6111yescharacteristic
574C2×C4C22×D24411yescharacteristic
575C8—41no—
576C8—42no—
577C8—41no—
578C8—2441no—
579C8—2442no—
580C8—2441no—
581C22—21no—
582C22—21no—
583C22—1222no—
584C22—4881no—
585C22—4881no—
586C22—2442no—
587C22—2442no—
588C22—2441no—
589C22—2441no—
590C22—2441no—
591C22—2441no—
592C22—2441no—
593C22C22×#1175819811yescharacteristic
594C22#131×D6111yescharacteristic
595C22#1684052711yescharacteristic
596C4—22no—
597C4—22no—
598C4—21no—
599C4—21no—
600C4—21no—
601C4—21no—
602C4—21no—
603C4—22no—
604C4—22no—
605C4—2442no—
606C4—2442no—
607C4—2441no—
608C4—2441no—
609C4#1683974412yes—
610C2—21no—
611C2—21no—
612C2—4881no—
613C2—2442no—
614C2#2145426611yescharacteristic
615C2#2145428211yescharacteristic
616C2C2×#1683974411yescharacteristic
subgroupG/H
centerC22C22×#11758198
commutatorC23×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#2984C2
socleC22×C61C22×C22⋊C4
prime radicalC22×D244C2×C4
cosocleC22×C4×C61C24
hypercenter#2984D61
hypocenterC61#17423
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C4×Q8 ◁ #2984

Complements

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

NH ≅ G/Ntype
2. #21454908C2semidirect
3. #21444189C2semidirect
4. #21444186C2semidirect
5. #21444176C2semidirect
6. #21444174C2semidirect
8. #21444151C2semidirect
10. #21444131C2semidirect
11. C61×#2984C2semidirect
304. #2984D61semidirect
361. C61#17423semidirect

Maximal subgroups

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

subgroupindex
C2×#168404532
#214549082
#214441892
#214441862
#214441762
#214441742
#214441592
#214441512
#214441392
#214441312
C61×#29842
#1742361