Finite Group Databasebeta

← #29646334

Subgroups — #29646334

Order 15616.

916 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840484C211yesmaximal, characteristic
2#21454908C211yesmaximal, characteristic, complemented
3#21444189C212yesmaximal, complemented
4#21444186C212yesmaximal, complemented
5#21444172C211yesmaximal, characteristic
6#21444170C211yesmaximal, characteristic
7#21444162C211yesmaximal, characteristic, complemented
8#21444151C212yesmaximal, complemented
9#21444142C211yesmaximal, characteristic, complemented
10#21444131C212yesmaximal, complemented
11C61×#2984C211yesmaximal, characteristic, complemented
12#16840484—21no—
13#16840484—22no—
14#16840484—21no—
15#16840484—21no—
16#16840484—22no—
17#16840484—21no—
18C2×#14519775C411yescharacteristic
19C2×#14519772C412yes—
20C2×#14519771C2212yes—
21C2×#14519766C411yescharacteristic
22C2×#14519765C2211yescharacteristic
23C2×#14519755C412yes—
24C2×C4×Dic122C2211yescharacteristic
25C2×#14519742C411yescharacteristic
26C2×#14519739C411yescharacteristic
27C2×#14519734C2212yes—
28#16839893—22no—
29#16839893—21no—
30Q8×#92418—21no—
31#16839819C2212yes—
32#16839819C2211yescharacteristic
33#16839815C2212yes—
34#16839815C2211yescharacteristic
35#16839487C2211yescharacteristic
36#16839487C2212yes—
37#16839487C2211yescharacteristic
38#16839487C2211yescharacteristic
39#16839487C2211yescharacteristic
40#16839487C2212yes—
41#16839485C2212yes—
42#16839485C2212yes—
43#16839485C2212yes—
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×Dic122C2×C411yescharacteristic
54C22×Dic122C2×C411yescharacteristic
55C22×D244C2×C411yescharacteristic
56C22×C4×D61C2×C412yes—
57#14519775—21no—
58#14519775—21no—
59#14519775—22no—
60#14519772—22no—
61#14519772—22no—
62#14519772—22no—
63#14519772—22no—
64#14519771—22no—
65#14519771—22no—
66#14519771—22no—
67#14519771—22no—
68#14519766—21no—
69#14519766—21no—
70#14519766—22no—
71#14519765—21no—
72#14519765—21no—
73#14519765—21no—
74#14519765—21no—
75#14519755—22no—
76#14519755—22no—
77#14519755—22no—
78#14519755—22no—
79C4×Dic122—21no—
80C4×Dic122—21no—
81C4×Dic122—21no—
82C4×Dic122—21no—
83#14519742—22no—
84#14519742—21no—
85#14519742—21no—
86#14519739—22no—
87#14519739—21no—
88#14519739—21no—
89#14519734—22no—
90#14519734—22no—
91#14519734—22no—
92#14519734—22no—
93C2×C4⋊Dic61C2×C411yescharacteristic
94C2×#11758196C2×C412yes—
95C2×#11758196C2×C412yes—
96C2×#11758196C2×C412yes—
97C2×#11758196C2311yescharacteristic
98C2×#11758196C2311yescharacteristic
99C2×#11758198C2×C412yes—
100C2×#11758198C2×C412yes—
101C2×#11758198C2×C412yes—
102C2×#11758198C2×C412yes—
103C2×#11758198C2312yes—
104C2×C4×Dic61C2×C412yes—
105C2×C4×Dic61C2×C411yescharacteristic
106C2×C4×Dic61C2×C411yescharacteristic
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×C412yes—
141C2×Q8×C61—42no—
142C2×Q8×C61C22⋊C411yescharacteristic
143C2×Q8×C61C22⋊C411yescharacteristic
144C2×Q8×C61C22⋊C412yes—
145C22×C4×C61C22×C412yes—
146C22×C4×C61C22×C411yescharacteristic
147C22×C4×C61C22×C411yescharacteristic
148C22×C4×C61C22×C412yes—
149C22×C4×C61C2411yescharacteristic
150C22×Dic61C22×C411yescharacteristic
151C22×Dic61C22×C411yescharacteristic
152C22×Dic61C22×C412yes—
153C22×Dic61C22×C411yescharacteristic
154C22×Dic61C22×C411yescharacteristic
155C2×D244—21no—
156C2×D244—21no—
157C2×D244—42no—
158C2×C4×D61—22no—
159C2×C4×D61—22no—
160C2×C4×D61—24no—
161C2×C4×D61—42no—
162C2×Dic122—21no—
163C2×Dic122—21no—
164C2×Dic122—21no—
165C2×Dic122—21no—
166C2×Dic122—22no—
167C2×Dic122—41no—
168C2×Dic122—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—22no—
194#11758198—22no—
195#11758198—22no—
196#11758198—22no—
197C4⋊Dic61—21no—
198C4⋊Dic61—21no—
199#11758196—22no—
200#11758196—22no—
201#11758196—22no—
202#11758196—22no—
203#11758196—22no—
204#11758196—22no—
205#11758196—21no—
206#11758196—21no—
207#11758196—21no—
208#11758196—21no—
209C4×Dic61—22no—
210C4×Dic61—22no—
211C4×Dic61—21no—
212C4×Dic61—21no—
213C4×Dic61—21no—
214C4×Dic61—21no—
215C2×#92418—21no—
216C2×#92418—22no—
217C2×#92418—21no—
218C23×C61C23×C411yescharacteristic
219C22×D61—22no—
220C22×D61—22no—
221C22×D61—22no—
222C22×D61—42no—
223C22×D61—42no—
224Q8×C61—22no—
225Q8×C61—22no—
226Q8×C61—42no—
227C2×C4×C61—21no—
228C2×C4×C61—22no—
229C2×C4×C61—21no—
230C2×C4×C61—22no—
231C2×C4×C61—22no—
232C2×C4×C61—42no—
233C2×C4×C61—41no—
234C2×C4×C61C2×C22⋊C411yescharacteristic
235C2×C4×C61C2×C22⋊C412yes—
236C2×C4×C61C2×C22⋊C412yes—
237C2×C4×C61C2×C22⋊C412yes—
238C2×C4×C61C2×C22⋊C412yes—
239C2×C4×C61C2×C22⋊C411yescharacteristic
240C2×C4×C61C2×C22⋊C411yescharacteristic
241C2×C4×C61C2×C22⋊C411yescharacteristic
242C2×C4×C61C22×D411yescharacteristic
243C2×C4×C61C22×D411yescharacteristic
244C2×Dic61—21no—
245C2×Dic61—21no—
246C2×Dic61—21no—
247C2×Dic61—21no—
248C2×Dic61—21no—
249C2×Dic61—22no—
250C2×Dic61—21no—
251C2×Dic61—21no—
252C2×Dic61—42no—
253C2×Dic61—41no—
254C2×Dic61—41no—
255C2×Dic61#13112yes—
256D244—42no—
257D244—42no—
258C4×D61—44no—
259C4×D61—44no—
260Dic122—42no—
261Dic122—42no—
262Dic122—42no—
263Dic122—42no—
264C8×C61—41no—
265C8×C61—42no—
266C8×C61—41no—
267#92418—41no—
268#92418—42no—
269#92418—41no—
270#17428—611nomaximal, Sylow
271C22×C61—21no—
272C22×C61—21no—
273C22×C61C22×C22⋊C411yescharacteristic
274C22×C61C2×#13111yescharacteristic
275C22×C61#56811yescharacteristic
276C2×D61—42no—
277C2×D61—42no—
278C2×D61—42no—
279C2×D61—82no—
280C2×D61—82no—
281C4×C61—21no—
282C4×C61—21no—
283C4×C61—22no—
284C4×C61—22no—
285C4×C61—21no—
286C4×C61—21no—
287C4×C61—21no—
288C4×C61—22no—
289C4×C61—22no—
290C4×C61#41312yes—
291Dic61—22no—
292Dic61—41no—
293Dic61—41no—
294Dic61—41no—
295Dic61—42no—
296C2×#556—611no—
297#2984—611no—
298#2984D6111yescharacteristic, complemented
299#1473—612no—
300#1473—612no—
301#1469—611no—
302#1469—611no—
303#1467—611no—
304#1464—612no—
305#1461—611no—
306#1458—612no—
307C2×C61—21no—
308C2×C61—21no—
309C2×C61#288011yescharacteristic
310C2×C61#288411yescharacteristic
311C2×C61C2×#41311yescharacteristic
312D61—82no—
313#556—1221no—
314#556—1222no—
315#556—1221no—
316#556—1221no—
317#556—1222no—
318#556—1221no—
319C2×C4⋊Q8—611no—
320C2×#126—612no—
321C2×#126—612no—
322C2×#124—611no—
323C2×#124—611no—
324C2×C22⋊Q8—612no—
325C2×C4×Q8—611no—
326C2×C4×Q8—611no—
327C2×C4×Q8C2×D6111yescharacteristic
328C2×C4×D4—611no—
329C2×#117—612no—
330#446—22no—
331#446—21no—
332#446—1222no—
333#446—1221no—
334C8×Q8—21no—
335C8×Q8—1221no—
336#432—612no—
337#432—611no—
338#432C2×D6112yes—
339#432C2×D6111yescharacteristic
340#431—612no—
341#431—611no—
342#431C2×D6112yes—
343#431C2×D6111yescharacteristic
344#324—611no—
345#324—612no—
346#324—611no—
347#324—611no—
348#324—611no—
349#324—612no—
350#323—612no—
351#323—612no—
352#323—612no—
353#323—612no—
354C61#1742811yescharacteristic, complemented, Sylow
355C22×Q8—611no—
356C22×Q8—611no—
357C22×Q8C4×D6111yescharacteristic
358C22×D4—611no—
359C23×C4—612no—
360C4⋊Q8—1221no—
361C4⋊Q8—1221no—
362C4⋊Q8—1222no—
363#126—1222no—
364#126—1222no—
365#126—1222no—
366#126—1222no—
367#126—1222no—
368#126—1222no—
369#126—1222no—
370#126—1222no—
371#124—1221no—
372#124—1221no—
373#124—1222no—
374#124—1221no—
375#124—1221no—
376#124—1221no—
377#124—1221no—
378C22⋊Q8—1222no—
379C22⋊Q8—1222no—
380C22⋊Q8—1222no—
381C22⋊Q8—1222no—
382C4×Q8—1222no—
383C4×Q8—1221no—
384C4×Q8—1221no—
385C4×Q8—1221no—
386C4×Q8—1221no—
387C4×Q8—1221no—
388C4×Q8—1221no—
389C4×Q8D24411yescharacteristic
390C4×Q8D24412yes—
391C4×Q8D24411yescharacteristic
392C4×Q8C22⋊D6111yescharacteristic
393C4×Q8C22⋊D6112yes—
394C4×Q8C22⋊D6111yescharacteristic
395C4×D4—1222no—
396C4×D4—1221no—
397C4×D4—1221no—
398#117—1222no—
399#117—1222no—
400#117—1222no—
401#117—1222no—
402C2×C4⋊C4—612no—
403C2×C4⋊C4—612no—
404C2×C4⋊C4—611no—
405C2×C4⋊C4—612no—
406C2×C4⋊C4—611no—
407C2×C4⋊C4—611no—
408C2×C4⋊C4C4×D6111yescharacteristic
409C2×C4⋊C4C4×D6112yes—
410C2×C22⋊C4—612no—
411C2×C22⋊C4—612no—
412C2×C22⋊C4—612no—
413C2×C22⋊C4—612no—
414C2×C22⋊C4—612no—
415C2×C42—612no—
416C2×C42—611no—
417C2×C42—611no—
418C2×C42C22×D6112yes—
419C2×C42C22×D6111yescharacteristic
420C4⋊C8—22no—
421C4⋊C8—21no—
422C4⋊C8—22no—
423C4⋊C8—21no—
424C4⋊C8—1222no—
425C4⋊C8—1221no—
426C4⋊C8—1222no—
427C4⋊C8—1221no—
428C22⋊C8—611no—
429C22⋊C8—612no—
430C22⋊C8—611no—
431C22⋊C8C22×D6111yescharacteristic
432C22⋊C8C22×D6112yes—
433C22⋊C8C22×D6111yescharacteristic
434#97—22no—
435#97—21no—
436#97—1222no—
437#97—1221no—
438C4×C8—22no—
439C4×C8—21no—
440C4×C8—1222no—
441C4×C8—1221no—
442C24—612no—
443C2×Q8—42no—
444C2×Q8—1222no—
445C2×Q8—1221no—
446C2×Q8—1221no—
447C2×Q8—1221no—
448C2×Q8—1221no—
449C2×Q8—2441no—
450C2×Q8—2442no—
451C2×Q8#1175819811yescharacteristic
452C2×Q8#1175819811yescharacteristic
453C2×Q8#1175819812yes—
454C2×D4—1221no—
455C2×D4—1221no—
456C2×D4—2442no—
457C22×C4—611no—
458C22×C4—611no—
459C22×C4—611no—
460C22×C4—611no—
461C22×C4—612no—
462C22×C4—1224no—
463C22×C4—1222no—
464C22×C4—1222no—
465C22×C4—2442no—
466C22×C4C2×C4×D6111yescharacteristic
467C22×C4C2×C4×D6112yes—
468C22×C4C2×C4×D6111yescharacteristic
469C22×C4C2×C4×D6112yes—
470C22×C4C23×D6111yescharacteristic
471C2×C8—21no—
472C2×C8—22no—
473C2×C8—21no—
474C2×C8—1221no—
475C2×C8—1222no—
476C2×C8—1221no—
477C4⋊C4—1221no—
478C4⋊C4—1221no—
479C4⋊C4—1221no—
480C4⋊C4—1221no—
481C4⋊C4—1222no—
482C4⋊C4—1222no—
483C4⋊C4—1222no—
484C4⋊C4—1222no—
485C4⋊C4—1222no—
486C4⋊C4—1222no—
487C4⋊C4—1221no—
488C4⋊C4—1221no—
489C4⋊C4#1175819812yes—
490C4⋊C4#1175819811yescharacteristic
491C4⋊C4#1175819811yescharacteristic
492C4⋊C4#1175819812yes—
493C4⋊C4#1175819812yes—
494C4⋊C4#1175819812yes—
495C4⋊C4#1175819812yes—
496C22⋊C4—1222no—
497C22⋊C4—1222no—
498C22⋊C4—1222no—
499C22⋊C4—1222no—
500C22⋊C4—1222no—
501C22⋊C4—1222no—
502C22⋊C4—1222no—
503C22⋊C4—1222no—
504C22⋊C4—1222no—
505C22⋊C4—1222no—
506C42—1222no—
507C42—1222no—
508C42—1221no—
509C42—1221no—
510C42—1221no—
511C42—1221no—
512C42C2×D24412yes—
513C42C2×D24411yescharacteristic
514C42C2×D24412yes—
515C42C2×D24411yescharacteristic
516C42C2×C22⋊D6112yes—
517C42C2×C22⋊D6111yescharacteristic
518C42C2×C22⋊D6112yes—
519C42C2×C22⋊D6111yescharacteristic
520C23—1222no—
521C23—1222no—
522C23—1222no—
523C23—2442no—
524C23—2442no—
525C23C22×C4×D6111yescharacteristic
526Q8—22no—
527Q8—22no—
528Q8—42no—
529Q8—2442no—
530Q8—2442no—
531Q8—2442no—
532Q8—2442no—
533D4—2442no—
534D4—2442no—
535C2×C4—22no—
536C2×C4—21no—
537C2×C4—22no—
538C2×C4—21no—
539C2×C4—22no—
540C2×C4—42no—
541C2×C4—41no—
542C2×C4—612no—
543C2×C4—1221no—
544C2×C4—1221no—
545C2×C4—1221no—
546C2×C4—1221no—
547C2×C4—1221no—
548C2×C4—1222no—
549C2×C4—1221no—
550C2×C4—1221no—
551C2×C4—2444no—
552C2×C4—2444no—
553C2×C4—2441no—
554C2×C4—2441no—
555C2×C4—2442no—
556C2×C4C2×#1175819811yescharacteristic
557C2×C4C2×#1175819812yes—
558C2×C4C2×#1175819812yes—
559C2×C4C2×#1175819812yes—
560C2×C4C2×#1175819812yes—
561C2×C4C2×#1175819811yescharacteristic
562C2×C4C2×#1175819811yescharacteristic
563C2×C4C2×#1175819811yescharacteristic
564C2×C4C22×C22⋊D6111yescharacteristic
565C2×C4C22×D24411yescharacteristic
566C8—41no—
567C8—42no—
568C8—41no—
569C8—2441no—
570C8—2442no—
571C8—2441no—
572C22—21no—
573C22—21no—
574C22—4882no—
575C22—4882no—
576C22—2442no—
577C22—2442no—
578C22—2442no—
579C22C22×#1175819811yescharacteristic
580C22#131×D6111yescharacteristic
581C22#1684052711yescharacteristic
582C4—22no—
583C4—22no—
584C4—21no—
585C4—21no—
586C4—21no—
587C4—21no—
588C4—21no—
589C4—22no—
590C4—22no—
591C4—1222no—
592C4—2441no—
593C4—2441no—
594C4—2441no—
595C4—2442no—
596C4#1683974412yes—
597C2—21no—
598C2—21no—
599C2—4882no—
600C2#2145426611yescharacteristic
601C2#2145428211yescharacteristic
602C2C2×#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#17428
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
7. #21444162C2semidirect
8. #21444151C2semidirect
9. #21444142C2semidirect
10. #21444131C2semidirect
11. C61×#2984C2semidirect
298. #2984D61semidirect
354. C61#17428semidirect

Maximal subgroups

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

subgroupindex
C2×#168404842
#214549082
#214441892
#214441862
#214441722
#214441702
#214441622
#214441512
#214441422
#214441312
C61×#29842
#1742861