Finite Group Databasebeta

← #29843009

Subgroups — #29843009

Order 15616.

2134 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14519772C211yesmaximal, characteristic
2#21449009C214yesmaximal, complemented
3#21448938C214yesmaximal, complemented
4#21448694C212yesmaximal, complemented
5#21448683C212yesmaximal, complemented
6#21448671C212yesmaximal
7#21448661C212yesmaximal
8#21448599C214yesmaximal, complemented
9#21448597C214yesmaximal, complemented
10#21448437C211yesmaximal, characteristic, complemented
11#21448432C211yesmaximal, characteristic
12#21448416C211yesmaximal, characteristic
13#21448221C211yesmaximal, characteristic, complemented
14#21448214C211yesmaximal, characteristic, complemented
15C61×#2313C211yesmaximal, characteristic, complemented
16C2×#14519772C4116yescomplemented
17C2×#14519772C2212yes—
18C2×#14519772C2212yes—
19C2×#14519772C2212yes—
20C2×#14519772C2212yes—
21C2×#14519736C2212yes—
22C2×#14519733C2212yes—
23C22×C4⋊Dic61C2211yescharacteristic
24C22×#11758196C2211yescharacteristic
25C22×#11758198C2211yescharacteristic
26C22×#11758198C2211yescharacteristic
27C22×#11758198C2211yescharacteristic
28C2×C22⋊C4×D61C2212yes—
29C22×C4×Dic61C2211yescharacteristic
30#16839661C2214yescomplemented
31#16839661C2214yescomplemented
32#16839660C2214yescomplemented
33#16839659C2214yescomplemented
34#16839654C2214yes—
35#16839654C2214yes—
36#16839654C2214yes—
37#16839653C2214yes—
38#16839652C2214yes—
39#16839651C2214yes—
40#16839649C2214yescomplemented
41#16839645C2214yescomplemented
42#16839645C2214yescomplemented
43#16839644C2214yescomplemented
44#16839636C2214yes—
45#16839636C2214yes—
46#16839636C2214yes—
47#16839633C2214yes—
48#16839631C2212yescomplemented
49#16839629C2212yescomplemented
50#16839627C2212yescomplemented
51#16839626C2212yescomplemented
52#16839623C2212yes—
53#16839622C2211yescharacteristic
54#16839622C2211yescharacteristic
55#16839622C2212yes—
56#16839620C2212yes—
57#16839620C2212yes—
58#16839617C2212yes—
59C4⋊C4×Dic61C2212yescomplemented
60C4⋊C4×Dic61C2212yescomplemented
61C4×C4⋊Dic61C2214yescomplemented
62#16839613C2212yescomplemented
63#16839613C2214yescomplemented
64C22⋊C4×Dic61C2212yes—
65C22⋊C4×Dic61C2212yes—
66C4×#11758198C2214yes—
67#16839609C2214yes—
68#16839607C2212yescomplemented
69C22×C4⋊C4×C61C2211yescharacteristic
70C61×#387C2214yescomplemented
71C61×#387C2214yescomplemented
72C61×#381C2212yes—
73C61×#380C2211yescharacteristic
74C61×#380C2211yescharacteristic
75C4×C22⋊C4×C61C2212yes—
76C24×D61C2311yescharacteristic
77C23×Dic61C2311yescharacteristic
78C23×Dic61C2311yescharacteristic
79C23×Dic61C2311yescharacteristic
80C22×C4×D61C2314yes—
81#14519772C2×C4116yescomplemented
82#14519772C2×C4116yescomplemented
83#14519772C2×C4116yescomplemented
84#14519772C2×C4116yescomplemented
85#14519736—44no—
86#14519733—44no—
87C2×C4⋊Dic61C2×C418yescomplemented
88C2×C4⋊Dic61C2314yes—
89C2×C4⋊Dic61C2314yes—
90C2×C4⋊Dic61C2312yes—
91C2×C4⋊Dic61C2312yes—
92C2×C4⋊Dic61C2312yes—
93C2×C4⋊Dic61C2312yes—
94C2×#11758196C2×C418yescomplemented
95C2×#11758196C2314yes—
96C2×#11758196C2314yes—
97C2×#11758196C2314yes—
98C2×#11758196C2312yes—
99C2×#11758196C2312yes—
100C2×C22⋊Dic61C2312yes—
101C2×C22⋊Dic61C2312yes—
102C2×#11758198C2×C418yes—
103C2×#11758198C2×C418yes—
104C2×#11758198C2×C418yes—
105C2×#11758198C2314yes—
106C2×#11758198C2314yes—
107C2×#11758198C2314yes—
108C2×#11758198C2312yes—
109C2×#11758198C2312yes—
110C2×#11758198C2312yes—
111C2×#11758198C2312yes—
112C2×#11758198C2312yes—
113C2×#11758198C2312yes—
114C22⋊C4×D61—44no—
115C2×C4×Dic61C2×C418yescomplemented
116C2×C4×Dic61C2314yes—
117C2×C4×Dic61C2314yes—
118C2×C4×Dic61C2314yes—
119C2×C4×Dic61C2314yes—
120C2×C4×Dic61C2311yescharacteristic
121C2×C4×Dic61C2311yescharacteristic
122C2×C4×Dic61C2311yescharacteristic
123C2×C4×Dic61C2311yescharacteristic
124#14519684C2314yes—
125#14519684C2312yes—
126#14519684C2314yes—
127#14519684C2314yes—
128#14519684C2314yes—
129#14519684C2314yes—
130#14519684C2312yes—
131#14519684C2314yes—
132#14519684C2314yes—
133C23×C4×C61C2311yescharacteristic
134C23×C4×C61C2311yescharacteristic
135C23×C4×C61C2311yescharacteristic
136C2×C4⋊C4×C61C2×C418yescomplemented
137C2×C4⋊C4×C61C2312yes—
138C2×C4⋊C4×C61C2312yes—
139C2×C4⋊C4×C61C2312yes—
140C2×C4⋊C4×C61C2312yes—
141C2×C22⋊C4×C61C2312yes—
142C2×C22⋊C4×C61C2312yes—
143C2×C42×C61C2314yes—
144#95×C61C2314yes—
145#95×C61C2312yes—
146#95×C61C2312yes—
147#95×C61C2314yes—
148#95×C61C2314yes—
149C24×C61C2411yescharacteristic
150C23×D61—22no—
151C23×D61—22no—
152C23×D61—42no—
153C23×D61—42no—
154C23×D61C22×C414yes—
155C23×D61C2412yes—
156C22×C4×C61—22no—
157C22×C4×C61—22no—
158C22×C4×C61—42no—
159C22×C4×C61—42no—
160C22×C4×C61C22×C414yes—
161C22×C4×C61C22×C414yes—
162C22×C4×C61C22×C414yes—
163C22×C4×C61C2414yes—
164C22×C4×C61C2414yes—
165C22×C4×C61C2412yes—
166C22×C4×C61C2412yes—
167C22×C4×C61C2411yescharacteristic
168C22×C4×C61C2411yescharacteristic
169C22×Dic61—22no—
170C22×Dic61—22no—
171C22×Dic61—42no—
172C22×Dic61—42no—
173C22×Dic61C22×C414yes—
174C22×Dic61C22×C414yes—
175C22×Dic61C22×C414yes—
176C22×Dic61C2414yes—
177C22×Dic61C2414yes—
178C22×Dic61C2411yescharacteristic
179C22×Dic61C2411yescharacteristic
180C22×Dic61C2411yescharacteristic
181C22×Dic61C2412yes—
182C22×Dic61C2411yescharacteristic
183C2×C4×D61—44no—
184C2×C4×D61—44no—
185C2×C4×D61—44no—
186C4⋊C4×C61—42no—
187C4⋊C4×C61—42no—
188C4⋊C4×C61C22×C418yes—
189C4⋊C4×C61C22×C418yes—
190C22⋊C4×C61—24no—
191C22⋊C4×C61—24no—
192C42×C61—44no—
193C22⋊Dic61—24no—
194C22⋊Dic61—24no—
195#11758198—44no—
196#11758198—44no—
197#11758198—44no—
198#11758198C22×C418yes—
199#11758198C22×C418yes—
200#11758198C22×C418yes—
201#11758198C22×C418yes—
202#11758198C22×C418yes—
203#11758198C22×C418yes—
204C4⋊Dic61—42no—
205C4⋊Dic61—44no—
206C4⋊Dic61—42no—
207C4⋊Dic61—44no—
208C4⋊Dic61C22×C418yes—
209C4⋊Dic61C22×C418yes—
210#11758196—44no—
211#11758196—44no—
212#11758196—44no—
213#11758196C22×C418yes—
214#11758196C22×C418yes—
215C4×Dic61—44no—
216C4×Dic61—44no—
217C4×Dic61—44no—
218C4×Dic61—44no—
219C4×Dic61C22×C414yes—
220C4×Dic61C22×C414yes—
221C4×Dic61C22×C414yes—
222C4×Dic61C22×C414yes—
223C23×C61—22no—
224C23×C61—22no—
225C23×C61C23×C412yes—
226C23×C61H(5,𝔽2)12yes—
227C23×C612−1+412yes—
228C23×C61C2511yescharacteristic
229C22×D61—22no—
230C22×D61—22no—
231C22×D61—24no—
232C22×D61—24no—
233C22×D61—28no—
234C22×D61—44no—
235C22×D61—44no—
236C22×D61—84no—
237C22×D61—84no—
238C22×D61C23×C414yes—
239C2×C4×C61—28no—
240C2×C4×C61—22no—
241C2×C4×C61—22no—
242C2×C4×C61—24no—
243C2×C4×C61—22no—
244C2×C4×C61—22no—
245C2×C4×C61—24no—
246C2×C4×C61—24no—
247C2×C4×C61—24no—
248C2×C4×C61—24no—
249C2×C4×C61—24no—
250C2×C4×C61—24no—
251C2×C4×C61—44no—
252C2×C4×C61—44no—
253C2×C4×C61C23×C412yes—
254C2×C4×C61C23×C412yes—
255C2×C4×C61C23×C414yes—
256C2×C4×C61C23×C414yes—
257C2×Dic61—28no—
258C2×Dic61—21no—
259C2×Dic61—21no—
260C2×Dic61—21no—
261C2×Dic61—21no—
262C2×Dic61—22no—
263C2×Dic61—24no—
264C2×Dic61—22no—
265C2×Dic61—24no—
266C2×Dic61—24no—
267C2×Dic61—24no—
268C2×Dic61—24no—
269C2×Dic61—24no—
270C2×Dic61—24no—
271C2×Dic61—44no—
272C2×Dic61—44no—
273C2×Dic61C23×C412yes—
274C2×Dic61C23×C412yes—
275C2×Dic61C23×C412yes—
276C2×Dic61C23×C412yes—
277C2×Dic61C23×C414yes—
278C4×D61—88no—
279#34381—611nomaximal, Sylow
280C22×C61—22no—
281C22×C61—22no—
282C22×C61—24no—
283C22×C61—24no—
284C22×C61#52014yes—
285C22×C61C24×C411yescharacteristic
286C22×C61C2×H(5,𝔽2)11yescharacteristic
287C22×C61C2×2−1+411yescharacteristic
288C22×C61#58512yes—
289C22×C61#58512yes—
290C2×D61—24no—
291C2×D61—24no—
292C2×D61—24no—
293C2×D61—44no—
294C2×D61—44no—
295C2×D61—88no—
296C4×C61—24no—
297C4×C61—24no—
298C4×C61—22no—
299C4×C61—22no—
300C4×C61—44no—
301C4×C61—44no—
302Dic61—22no—
303Dic61—22no—
304Dic61—22no—
305Dic61—24no—
306Dic61—22no—
307Dic61—44no—
308Dic61—44no—
309C22×#126—611no—
310#2383—614no—
311#2373—614no—
312#2345—612no—
313#2344—612no—
314#2342—612no—
315#2342—612no—
316#2334—614no—
317#2334—614no—
318#2316—611no—
319#2315—611no—
320#2313D6111yescharacteristic, complemented
321#2312—611no—
322#2285—611no—
323#2283—611no—
324C2×C61—24no—
325C2×C61C2×#52011yescharacteristic
326C2×C61#342811yescharacteristic
327C2×C61#342811yescharacteristic
328C2×C61#356012yes—
329C2×C61#356412yes—
330D61—44no—
331C2×#126—6116no—
332C2×#126—612no—
333C2×#126—612no—
334C2×#126—612no—
335C2×#126—612no—
336C2×#117—612no—
337C2×#117—612no—
338C22×C4⋊C4—611no—
339C22×C4⋊C4—611no—
340C22×C4⋊C4C2×D6111yescharacteristic
341C22×C22⋊C4—611no—
342C22×C22⋊C4—611no—
343C22×C22⋊C4—611no—
344C22×C22⋊C4—612no—
345C22×C42—611no—
346#389—614no—
347#389—614no—
348#389—614no—
349#389—614no—
350#388—614no—
351#388—614no—
352#388—614no—
353#388—614no—
354#388—614no—
355#388—614no—
356#387—614no—
357#387—614no—
358#387—614no—
359#387—614no—
360#387C2×D6114yescomplemented
361#387C2×D6114yescomplemented
362#385—614no—
363#385—614no—
364#385—614no—
365#385—614no—
366#384—612no—
367#383—612no—
368#382—612no—
369#382—612no—
370#381—612no—
371#381C2×D6112yes—
372#380—611no—
373#380—611no—
374#380—612no—
375#380—612no—
376#380—612no—
377#380C2×D6111yescharacteristic
378#380C2×D6111yescharacteristic
379#379—612no—
380C4×C4⋊C4—612no—
381C4×C4⋊C4—612no—
382C4×C4⋊C4—614no—
383C4×C4⋊C4—612no—
384C4×C4⋊C4—614no—
385C4×C22⋊C4—612no—
386C4×C22⋊C4—612no—
387C4×C22⋊C4—614no—
388C4×C22⋊C4—614no—
389C4×C22⋊C4C2×D6112yes—
390#376—612no—
391C61#3438111yescharacteristic, complemented, Sylow
392C25—611no—
393C23×C4—611no—
394C23×C4—611no—
395C23×C4—611no—
396C23×C4—614no—
397C23×C4C22×D6111yescharacteristic
398C23×C4C22×D6111yescharacteristic
399C23×C4C22×D6111yescharacteristic
400#126—6116no—
401#126—6116no—
402#126—6116no—
403#126—6116no—
404#117—2444no—
405#117—2444no—
406C2×C4⋊C4—618no—
407C2×C4⋊C4—618no—
408C2×C4⋊C4—612no—
409C2×C4⋊C4—612no—
410C2×C4⋊C4—612no—
411C2×C4⋊C4—612no—
412C2×C4⋊C4—614no—
413C2×C4⋊C4—614no—
414C2×C4⋊C4—612no—
415C2×C4⋊C4—614no—
416C2×C4⋊C4—612no—
417C2×C4⋊C4—614no—
418C2×C4⋊C4—614no—
419C2×C4⋊C4C4×D6118yescomplemented
420C2×C4⋊C4C22×D6112yes—
421C2×C4⋊C4C22×D6112yes—
422C2×C4⋊C4C22×D6112yes—
423C2×C4⋊C4C22×D6112yes—
424C2×C22⋊C4—618no—
425C2×C22⋊C4—618no—
426C2×C22⋊C4—618no—
427C2×C22⋊C4—612no—
428C2×C22⋊C4—612no—
429C2×C22⋊C4—612no—
430C2×C22⋊C4—612no—
431C2×C22⋊C4—612no—
432C2×C22⋊C4—612no—
433C2×C22⋊C4—612no—
434C2×C22⋊C4—612no—
435C2×C22⋊C4—614no—
436C2×C22⋊C4—614no—
437C2×C22⋊C4—614no—
438C2×C22⋊C4—2444no—
439C2×C22⋊C4C22×D6112yes—
440C2×C22⋊C4C22×D6112yes—
441C2×C42—618no—
442C2×C42—611no—
443C2×C42—611no—
444C2×C42—611no—
445C2×C42—611no—
446C2×C42—614no—
447C2×C42—614no—
448C2×C42—614no—
449C2×C42—614no—
450C2×C42C22×D6114yes—
451#95—614no—
452#95—612no—
453#95—614no—
454#95—614no—
455#95—614no—
456#95—614no—
457#95—612no—
458#95—614no—
459#95—614no—
460#95C22×D6112yes—
461#95C22×D6114yes—
462#95C22×D6114yes—
463#95C22×D6112yes—
464#95C22×D6114yes—
465C24—612no—
466C24—614no—
467C24—1222no—
468C24—1222no—
469C24—2442no—
470C24—2442no—
471C24C23×D6111yescharacteristic
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—42no—
475C22×C4—42no—
476C22×C4—614no—
477C22×C4—611no—
478C22×C4—611no—
479C22×C4—611no—
480C22×C4—611no—
481C22×C4—612no—
482C22×C4—614no—
483C22×C4—614no—
484C22×C4—614no—
485C22×C4—614no—
486C22×C4—1222no—
487C22×C4—1222no—
488C22×C4—2444no—
489C22×C4—2444no—
490C22×C4—2444no—
491C22×C4—2442no—
492C22×C4—2442no—
493C22×C4C2×C4×D6114yes—
494C22×C4C2×C4×D6114yes—
495C22×C4C2×C4×D6114yes—
496C22×C4C23×D6114yes—
497C22×C4C23×D6114yes—
498C22×C4C23×D6111yescharacteristic
499C22×C4C23×D6111yescharacteristic
500C22×C4C23×D6112yes—
501C22×C4C23×D6112yes—
502C4⋊C4—42no—
503C4⋊C4—42no—
504C4⋊C4—618no—
505C4⋊C4—618no—
506C4⋊C4—618no—
507C4⋊C4—618no—
508C4⋊C4—2444no—
509C4⋊C4—2444no—
510C4⋊C4—2444no—
511C4⋊C4—2444no—
512C4⋊C4—2444no—
513C4⋊C4—2442no—
514C4⋊C4—2442no—
515C4⋊C4C2×C4×D6118yes—
516C4⋊C4C2×C4×D6118yes—
517C22⋊C4—24no—
518C22⋊C4—24no—
519C22⋊C4—618no—
520C22⋊C4—618no—
521C22⋊C4—618no—
522C22⋊C4—618no—
523C22⋊C4—618no—
524C22⋊C4—618no—
525C22⋊C4—1224no—
526C22⋊C4—1224no—
527C22⋊C4—2444no—
528C22⋊C4—2444no—
529C22⋊C4—2444no—
530C42—44no—
531C42—614no—
532C42—614no—
533C42—614no—
534C42—614no—
535C42—2444no—
536C42—2444no—
537C42—2444no—
538C42—2444no—
539C23—22no—
540C23—22no—
541C23—614no—
542C23—1228no—
543C23—1222no—
544C23—1222no—
545C23—1224no—
546C23—1224no—
547C23—4884no—
548C23—4884no—
549C23—2444no—
550C23—2444no—
551C23C22×C4×D6112yes—
552C23#1451982312yes—
553C23#1451982412yes—
554C23C24×D6111yescharacteristic
555C2×C4—28no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—24no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—612no—
570C2×C4—612no—
571C2×C4—612no—
572C2×C4—612no—
573C2×C4—614no—
574C2×C4—1228no—
575C2×C4—1221no—
576C2×C4—1221no—
577C2×C4—1221no—
578C2×C4—1221no—
579C2×C4—1222no—
580C2×C4—1224no—
581C2×C4—1222no—
582C2×C4—1224no—
583C2×C4—1224no—
584C2×C4—1224no—
585C2×C4—1224no—
586C2×C4—1224no—
587C2×C4—1224no—
588C2×C4—4888no—
589C2×C4—2444no—
590C2×C4—2444no—
591C2×C4C22×C4×D6112yes—
592C2×C4C22×C4×D6112yes—
593C2×C4C22×C4×D6114yes—
594C2×C4C22×C4×D6114yes—
595C22—22no—
596C22—22no—
597C22—24no—
598C22—24no—
599C22—1224no—
600C22—1224no—
601C22—1224no—
602C22—4888no—
603C22—2444no—
604C22—2444no—
605C22#1684024214yes—
606C22C23×C4×D6111yescharacteristic
607C22C2×#1451982311yescharacteristic
608C22C2×#1451982411yescharacteristic
609C22#1684059712yes—
610C22#1684059912yes—
611C4—24no—
612C4—24no—
613C4—22no—
614C4—22no—
615C4—44no—
616C4—44no—
617C4—1222no—
618C4—1222no—
619C4—1222no—
620C4—1222no—
621C4—1224no—
622C4—2444no—
623C4—2444no—
624C2—24no—
625C2—2444no—
626C2C2×#1684024211yescharacteristic
627C2#2145901711yescharacteristic
628C2#2145902111yescharacteristic
629C2#2146037612yes—
630C2#2146044812yes—
subgroupG/H
center, FrattiniC23C24×D61
commutatorC22×C61C24×C4
FittingC61×#2313C2
socle, cosocleC23×C61C25
prime radicalC24×D61C23
hypercenter#2313D61
hypocenterC61#34381
LCSG ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ #2313

Complements

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

NH ≅ G/Ntype
2. #21449009C2semidirect
3. #21448938C2semidirect
4. #21448694C2semidirect
5. #21448683C2semidirect
8. #21448599C2semidirect
9. #21448597C2semidirect
10. #21448437C2semidirect
13. #21448221C2semidirect
14. #21448214C2semidirect
15. C61×#2313C2semidirect
16. C2×#14519772C4semidirect
30. #16839661C22semidirect
31. #16839661C22semidirect
32. #16839660C22semidirect
33. #16839659C22semidirect
40. #16839649C22semidirect
41. #16839645C22semidirect
42. #16839645C22semidirect
43. #16839644C22semidirect
48. #16839631C22semidirect
49. #16839629C22semidirect
50. #16839627C22semidirect
51. #16839626C22semidirect
59. C4⋊C4×Dic61C22semidirect
60. C4⋊C4×Dic61C22semidirect
61. C4×C4⋊Dic61C22semidirect
62. #16839613C22semidirect
63. #16839613C22semidirect
68. #16839607C22semidirect
70. C61×#387C22semidirect
71. C61×#387C22semidirect
81. #14519772C2×C4semidirect
82. #14519772C2×C4semidirect
83. #14519772C2×C4semidirect
84. #14519772C2×C4semidirect
87. C2×C4⋊Dic61C2×C4semidirect
94. C2×#11758196C2×C4semidirect
115. C2×C4×Dic61C2×C4semidirect
136. C2×C4⋊C4×C61C2×C4semidirect
320. #2313D61semidirect
360. #387C2×D61semidirect
361. #387C2×D61semidirect
391. C61#34381semidirect
419. C2×C4⋊C4C4×D61semidirect

Maximal subgroups

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

subgroupindex
C22×#145197722
#214490092
#214489382
#214486942
#214486832
#214486712
#214486612
#214485992
#214485972
#214484372
#214484322
#214484162
#214482212
#214482142
C61×#23132
#3438161