Finite Group Databasebeta

← #29842796

Subgroups — #29842796

Order 15616.

2458 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840274C211yesmaximal, characteristic
2#21448959C214yesmaximal, complemented
3#21448949C214yesmaximal, complemented
4#21448770C212yesmaximal, complemented
5#21448766C212yesmaximal, complemented
6#21448751C212yesmaximal, complemented
7#21448745C212yesmaximal, complemented
8#21448583C214yesmaximal, complemented
9#21448571C214yesmaximal, complemented
10#21448428C211yesmaximal, characteristic, complemented
11#21448408C212yesmaximal, complemented
12#21448201C212yesmaximal, complemented
13C61×#2314C211yesmaximal, characteristic, complemented
14C22×#11758222C2211yescharacteristic
15C23×C4×D61C2211yescharacteristic
16#16840274C4116yescomplemented
17C2×#14519761C2212yes—
18C2×#14519757C2212yes—
19C2×#14519752C2212yes—
20C2×#14519747C2212yes—
21C2×C4×D244C2212yes—
22C2×C4×D244C2212yes—
23C2×#14519735C2212yes—
24C2×#14519735C2212yes—
25C2×#14519735C2212yescomplemented
26C2×C4⋊C4×D61C2212yescomplemented
27#16839666C2214yescomplemented
28#16839664C2214yes—
29#16839663C2214yes—
30#16839662C2214yes—
31#16839655C2214yes—
32#16839654C2214yes—
33#16839654C2214yescomplemented
34#16839653C2214yes—
35#16839652C2214yes—
36#16839652C2214yescomplemented
37#16839651C2214yes—
38#16839648C2214yescomplemented
39#16839647C2214yescomplemented
40#16839645C2214yescomplemented
41#16839640C2214yes—
42#16839640C2214yes—
43#16839640C2214yescomplemented
44#16839638C2214yes—
45#16839632C2211yescharacteristic, complemented
46#16839628C2211yescharacteristic, complemented
47#16839624C2214yes—
48#16839622C2212yes—
49#16839619C2212yes—
50#16839619C2212yes—
51#16839619C2212yescomplemented
52#16839619C2212yescomplemented
53#16839617C2214yes—
54C4⋊C4×Dic61C2214yescomplemented
55C4×C4⋊Dic61C2212yescomplemented
56C4×C4⋊Dic61C2214yescomplemented
57C22⋊C4×Dic61C2214yes—
58C4×#11758198C2214yescomplemented
59#16839609C2214yes—
60#16839609C2214yescomplemented
61C2×#117×C61C2211yescharacteristic
62C61×#388C2214yes—
63C61×#387C2214yescomplemented
64C61×#384C2211yescharacteristic, complemented
65C61×#382C2211yescharacteristic, complemented
66C61×#380C2212yes—
67C4×C4⋊C4×C61C2212yescomplemented
68C24×D61C2311yescharacteristic
69C2×#11758222—42no—
70C2×#11758222—42no—
71C2×#11758222C2×C418yes—
72C22×Dic122C2311yescharacteristic
73C22×D244C2312yes—
74C22×D244C2312yes—
75C22×D244C2311yescharacteristic
76C22×C22⋊D61C2312yes—
77C23×Dic61C2311yescharacteristic
78C22×C4×D61—42no—
79C22×C4×D61—42no—
80C22×C4×D61C2×C418yescomplemented
81C22×C4×D61C2312yes—
82C22×C4×D61C2312yes—
83C22×C4×D61C2314yes—
84C22×C4×D61C2312yes—
85#14519761C2×C4116yescomplemented
86#14519757C2×C4116yescomplemented
87#14519752C2×C4116yes—
88#14519747C2×C4116yescomplemented
89C4×D244C2×C4116yescomplemented
90C4×D244C2×C4116yescomplemented
91#14519735—44no—
92#14519735—44no—
93#14519735—44no—
94C2×C4⋊Dic61C2312yes—
95C2×C4⋊Dic61C2312yes—
96C2×C4⋊Dic61C2314yes—
97C2×C4⋊Dic61C2312yes—
98C2×C4⋊Dic61C2312yes—
99C2×#11758196C2314yes—
100C4⋊C4×D61—44no—
101C2×C22⋊Dic61C2314yes—
102C2×#11758198C2314yes—
103C2×#11758198C2314yes—
104C2×#11758198C2314yes—
105C2×#11758198C2314yes—
106C2×#11758198C2314yes—
107C2×#11758198C2314yes—
108C2×#11758198C2314yes—
109C2×#11758198C2312yes—
110C2×#11758198C2312yes—
111C2×#11758198C2312yes—
112C2×#11758198C2312yes—
113C2×C4×Dic61C2314yes—
114C2×C4×Dic61C2314yes—
115C2×C4×Dic61C2314yes—
116C2×C4×Dic61C2314yes—
117#14519684C2314yes—
118#14519684C2314yes—
119#14519684C2314yes—
120#14519684C2314yes—
121#14519684C2314yes—
122#14519684C2314yes—
123#14519684C2314yes—
124C23×C4×C61C2311yescharacteristic
125#117×C61C2×C418yescomplemented
126C2×C4⋊C4×C61C2312yes—
127C2×C4⋊C4×C61C2312yes—
128C2×C4⋊C4×C61C2314yes—
129C2×C4⋊C4×C61C2312yes—
130C2×C22⋊C4×C61C2314yes—
131C2×C22⋊C4×C61C2312yes—
132C2×C42×C61C2314yes—
133C2×C42×C61C2311yescharacteristic
134C2×C42×C61C2311yescharacteristic
135#95×C61C2314yes—
136#95×C61C2314yes—
137#95×C61C2314yes—
138C24×C61C2411yescharacteristic
139C23×D61—41no—
140C23×D61—41no—
141C23×D61—42no—
142C23×D61—42no—
143C23×D61C22×C414yes—
144C23×D61C2412yes—
145C23×D61C2412yes—
146C22×C4×C61—22no—
147C22×C4×C61—22no—
148C22×C4×C61C22×C414yes—
149C22×C4×C61C2414yes—
150C22×C4×C61C2414yes—
151C22×C4×C61C2411yescharacteristic
152C22×C4×C61C2412yes—
153C22×C4×C61C2412yes—
154C22×C4×C61C2411yescharacteristic
155C2×C22⋊D61—42no—
156C2×C22⋊D61—42no—
157C2×C22⋊D61C22×C418yes—
158C22×Dic61—41no—
159C22×Dic61—41no—
160C22×Dic61C22×C414yes—
161C22×Dic61C2414yes—
162C22×Dic61C2414yes—
163C22×Dic61C2412yes—
164C22×Dic61C2412yes—
165#11758222—416no—
166C2×D244—42no—
167C2×D244—42no—
168C2×D244—42no—
169C2×D244—42no—
170C2×D244—41no—
171C2×D244—41no—
172C2×D244C22×C414yes—
173C2×D244C22×C418yes—
174C2×D244C22×C418yes—
175C2×C4×D61—24no—
176C2×C4×D61—24no—
177C2×C4×D61—24no—
178C2×C4×D61—24no—
179C2×C4×D61—416no—
180C2×C4×D61—42no—
181C2×C4×D61—42no—
182C2×C4×D61—44no—
183C2×C4×D61—44no—
184C2×C4×D61—44no—
185C2×C4×D61C22×C418yes—
186C2×C4×D61C22×C418yes—
187C2×C4×D61C22×C418yes—
188C2×Dic122—41no—
189C2×Dic122—41no—
190C2×Dic122C22×C414yes—
191C4⋊C4×C61—24no—
192C4⋊C4×C61—24no—
193C4⋊C4×C61—44no—
194C4⋊C4×C61C22×C418yes—
195C22⋊C4×C61—44no—
196C22⋊C4×C61C22×C418yes—
197C42×C61—44no—
198C42×C61C22×C414yes—
199C42×C61C22×C414yes—
200C22⋊Dic61—44no—
201#11758198—44no—
202#11758198—44no—
203#11758198—44no—
204#11758198—44no—
205#11758198—44no—
206#11758198—44no—
207#11758198—44no—
208#11758198C22×C418yes—
209#11758198C22×C418yes—
210#11758198C22×C418yes—
211#11758198C22×C418yes—
212C4⋊Dic61—24no—
213C4⋊Dic61—24no—
214C4⋊Dic61—44no—
215C4⋊Dic61C22×C418yes—
216C4⋊Dic61C22×C418yes—
217#11758196—44no—
218C4×Dic61—44no—
219C4×Dic61—44no—
220C4×Dic61—44no—
221C4×Dic61—44no—
222C23×C61—21no—
223C23×C61—21no—
224C23×C61—22no—
225C23×C61—22no—
226C23×C61C23×C412yes—
227C23×C61C2511yescharacteristic
228C22×D61—22no—
229C22×D61—22no—
230C22×D61—22no—
231C22×D61—22no—
232C22×D61—22no—
233C22×D61—22no—
234C22×D61—22no—
235C22×D61—22no—
236C22×D61—22no—
237C22×D61—22no—
238C22×D61—22no—
239C22×D61—22no—
240C22×D61—44no—
241C22×D61—48no—
242C22×D61—44no—
243C22×D61—44no—
244C22×D61—48no—
245C22×D61C23×C414yes—
246C22×D61C23×C414yes—
247C2×C4×C61—21no—
248C2×C4×C61—21no—
249C2×C4×C61—22no—
250C2×C4×C61—22no—
251C2×C4×C61—22no—
252C2×C4×C61—22no—
253C2×C4×C61—28no—
254C2×C4×C61—24no—
255C2×C4×C61—24no—
256C2×C4×C61—24no—
257C2×C4×C61—24no—
258C2×C4×C61—24no—
259C2×C4×C61—24no—
260C2×C4×C61C23×C414yes—
261C2×C4×C61C23×C414yes—
262C2×C4×C61C23×C412yes—
263C2×C4×C61C23×C412yes—
264C2×C4×C61H(5,𝔽2)12yes—
265C2×C4×C61H(5,𝔽2)12yes—
266C22⋊D61—48no—
267C2×Dic61—22no—
268C2×Dic61—22no—
269C2×Dic61—22no—
270C2×Dic61—22no—
271C2×Dic61—24no—
272C2×Dic61—24no—
273C2×Dic61—24no—
274C2×Dic61—24no—
275C2×Dic61—24no—
276C2×Dic61—24no—
277C2×Dic61—44no—
278C2×Dic61C23×C414yes—
279C2×Dic61C23×C414yes—
280D244—48no—
281D244—48no—
282D244—44no—
283C4×D61—216no—
284C4×D61—216no—
285C4×D61—48no—
286C4×D61—88no—
287Dic122—44no—
288#34361—611nomaximal, Sylow
289C22×C61—22no—
290C22×C61—24no—
291C22×C61—41no—
292C22×C61—41no—
293C22×C61—42no—
294C22×C61C24×C411yescharacteristic
295C22×C61C2×H(5,𝔽2)11yescharacteristic
296C22×C61C2×H(5,𝔽2)11yescharacteristic
297C22×C61#58512yes—
298C22×C61#58512yes—
299C2×D61—24no—
300C2×D61—24no—
301C2×D61—24no—
302C2×D61—24no—
303C2×D61—24no—
304C2×D61—24no—
305C2×D61—416no—
306C2×D61—42no—
307C2×D61—42no—
308C2×D61—42no—
309C2×D61—42no—
310C2×D61—42no—
311C2×D61—42no—
312C2×D61—42no—
313C2×D61—42no—
314C4×C61—22no—
315C4×C61—24no—
316C4×C61—24no—
317C4×C61—44no—
318C4×C61—44no—
319C4×C61#51814yes—
320Dic61—24no—
321Dic61—24no—
322Dic61—44no—
323Dic61—44no—
324C2×#525—611no—
325#2375—614no—
326#2374—614no—
327#2354—612no—
328#2354—612no—
329#2352—612no—
330#2352—612no—
331#2333—614no—
332#2332—614no—
333#2314—611no—
334#2314D6111yescharacteristic, complemented
335#2311—612no—
336#2280—612no—
337C2×C61—42no—
338C2×C61C2×#51811yescharacteristic
339C2×C61#342812yes—
340C2×C61#353911yescharacteristic
341C2×C61#353911yescharacteristic
342C2×C61#354012yes—
343D61—44no—
344D61—44no—
345C22×#41—611no—
346C24×C4—611no—
347#525—6116no—
348C2×#123—612no—
349C2×C22⋊Q8—612no—
350C2×#121—612no—
351C2×#120—612no—
352C2×C4×D4—612no—
353C2×C4×D4—612no—
354C2×#117—612no—
355C2×#117—612no—
356C2×#117—612no—
357C2×#117C2×D6111yescharacteristic
358C22×C4⋊C4—612no—
359#391—614no—
360#390—614no—
361#390—614no—
362#390—614no—
363#388—614no—
364#388—614no—
365#388—614no—
366#388—614no—
367#388—614no—
368#388—614no—
369#388—614no—
370#388C2×D6114yes—
371#387—614no—
372#387—614no—
373#387—614no—
374#387C2×D6114yescomplemented
375#386—614no—
376#386—614no—
377#386—614no—
378#386—614no—
379#384—611no—
380#384C2×D6111yescharacteristic, complemented
381#382—611no—
382#382C2×D6111yescharacteristic, complemented
383#381—614no—
384#380—612no—
385#380—612no—
386#380—612no—
387#380—612no—
388#380—612no—
389#380C2×D6112yes—
390#379—614no—
391C4×C4⋊C4—614no—
392C4×C4⋊C4—612no—
393C4×C4⋊C4—614no—
394C4×C4⋊C4C2×D6112yescomplemented
395C4×C22⋊C4—614no—
396C4×C22⋊C4—614no—
397C4×C22⋊C4—614no—
398C4×C22⋊C4—614no—
399C61#3436111yescharacteristic, complemented, Sylow
400C25—611no—
401C2×#41—618no—
402C2×#41—2442no—
403C2×#41—2442no—
404C22×Q8—611no—
405C22×D4—612no—
406C22×D4—612no—
407C22×D4—612no—
408C22×D4—611no—
409C23×C4—611no—
410C23×C4—612no—
411C23×C4—612no—
412C23×C4—618no—
413C23×C4—612no—
414C23×C4—614no—
415C23×C4—2442no—
416C23×C4—2442no—
417C23×C4C22×D6111yescharacteristic
418#123—6116no—
419C22⋊Q8—6116no—
420#121—6116no—
421#120—6116no—
422C4×D4—6116no—
423C4×D4—6116no—
424#117—2444no—
425#117—2444no—
426#117—2444no—
427#117C4×D6118yescomplemented
428C2×C4⋊C4—612no—
429C2×C4⋊C4—612no—
430C2×C4⋊C4—612no—
431C2×C4⋊C4—612no—
432C2×C4⋊C4—614no—
433C2×C4⋊C4—614no—
434C2×C4⋊C4—2444no—
435C2×C4⋊C4C22×D6112yes—
436C2×C4⋊C4C22×D6112yes—
437C2×C4⋊C4C22×D6114yes—
438C2×C4⋊C4C22×D6112yes—
439C2×C22⋊C4—612no—
440C2×C22⋊C4—612no—
441C2×C22⋊C4—612no—
442C2×C22⋊C4—612no—
443C2×C22⋊C4—614no—
444C2×C22⋊C4—614no—
445C2×C22⋊C4—614no—
446C2×C22⋊C4—614no—
447C2×C22⋊C4—614no—
448C2×C22⋊C4—614no—
449C2×C22⋊C4—614no—
450C2×C22⋊C4—614no—
451C2×C22⋊C4C22×D6114yes—
452C2×C22⋊C4C22×D6112yes—
453C2×C42—614no—
454C2×C42—614no—
455C2×C42—614no—
456C2×C42—614no—
457C2×C42C22×D6114yes—
458C2×C42C22×D6111yescharacteristic
459C2×C42C22×D6111yescharacteristic
460#95—614no—
461#95—614no—
462#95—614no—
463#95—614no—
464#95—614no—
465#95—614no—
466#95—614no—
467#95C22×D6114yes—
468#95C22×D6114yes—
469#95C22×D6114yes—
470C24—612no—
471C24—612no—
472C24—614no—
473C24—2441no—
474C24—2441no—
475C24—2442no—
476C24—2442no—
477C24C23×D6111yescharacteristic
478#41—24416no—
479C2×Q8—614no—
480C2×Q8—2441no—
481C2×Q8—2441no—
482C2×D4—618no—
483C2×D4—618no—
484C2×D4—618no—
485C2×D4—614no—
486C2×D4—2441no—
487C2×D4—2441no—
488C2×D4—2442no—
489C2×D4—2442no—
490C2×D4—2442no—
491C2×D4—2442no—
492C2×D4—2442no—
493C2×D4—2442no—
494C22×C4—22no—
495C22×C4—22no—
496C22×C4—612no—
497C22×C4—612no—
498C22×C4—614no—
499C22×C4—618no—
500C22×C4—618no—
501C22×C4—618no—
502C22×C4—614no—
503C22×C4—614no—
504C22×C4—1224no—
505C22×C4—1224no—
506C22×C4—1224no—
507C22×C4—1224no—
508C22×C4—24416no—
509C22×C4—2444no—
510C22×C4—2444no—
511C22×C4—2444no—
512C22×C4—2442no—
513C22×C4—2442no—
514C22×C4—2441no—
515C22×C4—2441no—
516C22×C4C2×C4×D6114yes—
517C22×C4C23×D6114yes—
518C22×C4C23×D6114yes—
519C22×C4C23×D6111yescharacteristic
520C22×C4C23×D6112yes—
521C22×C4C23×D6112yes—
522C22×C4C23×D6111yescharacteristic
523C4⋊C4—24no—
524C4⋊C4—24no—
525C4⋊C4—44no—
526C4⋊C4—618no—
527C4⋊C4—618no—
528C4⋊C4—1224no—
529C4⋊C4—1224no—
530C4⋊C4—2444no—
531C4⋊C4—2444no—
532C4⋊C4C2×C4×D6118yes—
533C22⋊C4—44no—
534C22⋊C4—618no—
535C22⋊C4—618no—
536C22⋊C4—618no—
537C22⋊C4—618no—
538C22⋊C4—2444no—
539C22⋊C4—2444no—
540C22⋊C4—2444no—
541C22⋊C4—2444no—
542C22⋊C4—2444no—
543C22⋊C4—2444no—
544C22⋊C4—2444no—
545C22⋊C4—2444no—
546C22⋊C4C2×C4×D6118yes—
547C42—44no—
548C42—2444no—
549C42—2444no—
550C42—2444no—
551C42—2444no—
552C42C2×C4×D6114yes—
553C42C2×C4×D6114yes—
554C23—21no—
555C23—21no—
556C23—22no—
557C23—22no—
558C23—614no—
559C23—614no—
560C23—1222no—
561C23—1222no—
562C23—1222no—
563C23—1222no—
564C23—1222no—
565C23—1222no—
566C23—1222no—
567C23—1222no—
568C23—1222no—
569C23—1222no—
570C23—1222no—
571C23—1222no—
572C23—2444no—
573C23—2448no—
574C23—2444no—
575C23—2444no—
576C23—2448no—
577C23C22×C4×D6112yes—
578C23C24×D6111yescharacteristic
579Q8—2444no—
580D4—2448no—
581D4—2448no—
582D4—2444no—
583D4—2448no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—28no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—614no—
598C2×C4—614no—
599C2×C4—1222no—
600C2×C4—1222no—
601C2×C4—1222no—
602C2×C4—1222no—
603C2×C4—12216no—
604C2×C4—12216no—
605C2×C4—1224no—
606C2×C4—1224no—
607C2×C4—1224no—
608C2×C4—1224no—
609C2×C4—1224no—
610C2×C4—1224no—
611C2×C4—4888no—
612C2×C4—2448no—
613C2×C4—2444no—
614C2×C4C22×C4×D6114yes—
615C2×C4C22×C4×D6114yes—
616C2×C4C22×C4×D6112yes—
617C2×C4C22×C4×D6112yes—
618C2×C4#1451982312yes—
619C2×C4#1451982312yes—
620C22—22no—
621C22—24no—
622C22—41no—
623C22—41no—
624C22—42no—
625C22—1224no—
626C22—1224no—
627C22—1224no—
628C22—1224no—
629C22—1224no—
630C22—1224no—
631C22—24416no—
632C22—2442no—
633C22—2442no—
634C22—2442no—
635C22—2442no—
636C22—2442no—
637C22—2442no—
638C22—2442no—
639C22—2442no—
640C22C23×C4×D6111yescharacteristic
641C22C2×#1451982311yescharacteristic
642C22C2×#1451982311yescharacteristic
643C22#1684059712yes—
644C22#1684059912yes—
645C4—22no—
646C4—24no—
647C4—24no—
648C4—44no—
649C4—44no—
650C4—1224no—
651C4—1224no—
652C4—2444no—
653C4—2444no—
654C4#1684023614yes—
655C2—42no—
656C2—2444no—
657C2—2444no—
658C2C2×#1684023611yescharacteristic
659C2#2145901912yes—
660C2#2146007811yescharacteristic
661C2#2146008111yescharacteristic
662C2#2146009112yes—
subgroupG/H
center, FrattiniC23C24×D61
commutatorC22×C61C24×C4
FittingC61×#2314C2
socle, cosocleC23×C61C25
prime radicalC2×#16840274C2
hypercenter#2314D61
hypocenterC61#34361
LCSG ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ #2314

Complements

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

NH ≅ G/Ntype
2. #21448959C2semidirect
3. #21448949C2semidirect
4. #21448770C2semidirect
5. #21448766C2semidirect
6. #21448751C2semidirect
7. #21448745C2semidirect
8. #21448583C2semidirect
9. #21448571C2semidirect
10. #21448428C2semidirect
11. #21448408C2semidirect
12. #21448201C2semidirect
13. C61×#2314C2semidirect
16. #16840274C4semidirect
25. C2×#14519735C22semidirect
26. C2×C4⋊C4×D61C22semidirect
27. #16839666C22semidirect
33. #16839654C22semidirect
36. #16839652C22semidirect
38. #16839648C22semidirect
39. #16839647C22semidirect
40. #16839645C22semidirect
43. #16839640C22semidirect
45. #16839632C22semidirect
46. #16839628C22semidirect
51. #16839619C22semidirect
52. #16839619C22semidirect
54. C4⋊C4×Dic61C22semidirect
55. C4×C4⋊Dic61C22semidirect
56. C4×C4⋊Dic61C22semidirect
58. C4×#11758198C22semidirect
60. #16839609C22semidirect
63. C61×#387C22semidirect
64. C61×#384C22semidirect
65. C61×#382C22semidirect
67. C4×C4⋊C4×C61C22semidirect
80. C22×C4×D61C2×C4semidirect
85. #14519761C2×C4semidirect
86. #14519757C2×C4semidirect
88. #14519747C2×C4semidirect
89. C4×D244C2×C4semidirect
90. C4×D244C2×C4semidirect
125. #117×C61C2×C4semidirect
334. #2314D61semidirect
374. #387C2×D61semidirect
380. #384C2×D61semidirect
382. #382C2×D61semidirect
394. C4×C4⋊C4C2×D61semidirect
399. C61#34361semidirect
427. #117C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168402742
#214489592
#214489492
#214487702
#214487662
#214487512
#214487452
#214485832
#214485712
#214484282
#214484082
#214482012
C61×#23142
#3436161