Finite Group Databasebeta

← #29843193

Subgroups — #29843193

Order 15616.

1998 conjugacy classes of subgroups.

The subgroup lattice has 642 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840316C211yesmaximal, characteristic
2#21448981C214yesmaximal
3#21448905C212yesmaximal, complemented
4#21448873C212yesmaximal, complemented
5#21448784C211yesmaximal, characteristic, complemented
6#21448782C211yesmaximal, characteristic, complemented
7#21448736C211yesmaximal, characteristic, complemented
8#21448713C212yesmaximal
9#21448699C212yesmaximal
10#21448557C214yesmaximal, complemented
11#21448499C212yesmaximal
12#21448495C212yesmaximal
13#21448467C212yesmaximal, complemented
14#21448465C212yesmaximal, complemented
15#21448223C211yesmaximal, characteristic
16#21448212C211yesmaximal, characteristic
17C61×#2351C211yesmaximal, characteristic, complemented
18#16840316C4116yescomplemented
19C2×#14519776C2211yescharacteristic
20C2×#14519775C2211yescharacteristic
21C2×#14519767C2212yes—
22C2×#14519757C2211yescharacteristic
23C2×#14519755C2212yes—
24C2×#14519753C2211yescharacteristic
25C2×C4×Dic122C2212yes—
26C2×C4×Dic122C2211yescharacteristic
27C2×#14519742C2212yes—
28C2×#14519734C2212yes—
29C2×#14519734C2212yes—
30C2×C42×D61C2211yescharacteristic
31#16839669C2212yes—
32#16839668C2212yes—
33#16839667C2214yes—
34#16839666C2214yes—
35#16839659C2214yes—
36#16839658C2214yes—
37#16839658C2212yes—
38#16839657C2214yes—
39#16839657C2214yes—
40#16839657C2214yes—
41#16839653C2214yes—
42#16839652C2214yes—
43#16839652C2214yes—
44#16839651C2214yes—
45#16839648C2214yes—
46#16839646C2214yes—
47#16839646C2214yes—
48#16839645C2214yes—
49#16839643C2214yes—
50#16839643C2214yes—
51#16839631C2212yes—
52#16839630C2212yes—
53#16839629C2212yes—
54#16839629C2212yes—
55#16839628C2211yescharacteristic
56#16839626C2211yescharacteristic
57#16839626C2212yes—
58#16839626C2211yescharacteristic
59#16839624C2212yes—
60#16839621C2211yescharacteristic
61#16839621C2211yescharacteristic
62#16839619C2212yes—
63#16839619C2212yes—
64C4⋊C4×Dic61C2212yes—
65C4⋊C4×Dic61C2212yes—
66C4×#11758196C2214yes—
67C4×#11758196C2212yes—
68C4×C4⋊Dic61C2211yescharacteristic
69C4×C4⋊Dic61C2212yes—
70#16839613C2212yes—
71#16839613C2214yes—
72#16839609C2212yes—
73#16839608C2212yes—
74#16839607C2212yes—
75#16839607C2212yes—
76C2×C4×Q8×C61C2212yes—
77C2×C4×Q8×C61C2211yescharacteristic
78C61×#389C2214yes—
79C61×#389C2212yes—
80C61×#382C2212yes—
81C61×#382C2211yescharacteristic
82C4×C4⋊C4×C61C2212yes—
83C4×C4⋊C4×C61C2211yescharacteristic
84C22×Dic122C2311yescharacteristic
85C22×Dic122C2312yes—
86C22×C4×D61C2311yescharacteristic
87C22×C4×D61C2311yescharacteristic
88C22×C4×D61C2311yescharacteristic
89#14519776C2×C418yes—
90#14519775C2×C418yes—
91#14519767C2×C4116yes—
92#14519757C2×C418yes—
93#14519755C2×C4116yes—
94#14519753C2×C418yes—
95C4×Dic122C2×C4116yes—
96C4×Dic122C2×C418yes—
97#14519742—44no—
98#14519734—44no—
99#14519734C2×C4116yes—
100C2×C4⋊Dic61C2312yes—
101C2×C4⋊Dic61C2312yes—
102C2×C4⋊Dic61C2311yescharacteristic
103C2×C4⋊Dic61C2312yes—
104C2×#11758196C2312yes—
105C2×#11758196C2312yes—
106C2×#11758196C2312yes—
107C2×#11758196C2312yes—
108C2×#11758196C2314yes—
109C2×#11758196C2314yes—
110C2×#11758196C2312yes—
111C2×#11758196C2312yes—
112C2×#11758196C2312yes—
113C2×#11758196C2312yes—
114C2×#11758196C2312yes—
115C2×#11758198C2312yes—
116C2×#11758198C2312yes—
117C2×#11758198C2314yes—
118C2×#11758198C2312yes—
119C2×#11758198C2312yes—
120C2×C4×Dic61C2312yes—
121C2×C4×Dic61C2314yes—
122C2×C4×Dic61C2314yes—
123C2×C4×Dic61C2314yes—
124C2×C4×Dic61C2314yes—
125C2×C4×Dic61C2311yescharacteristic
126C2×C4×Dic61C2311yescharacteristic
127C2×C4×Dic61C2311yescharacteristic
128C2×C4×Dic61C2312yes—
129C42×D61C2×C418yes—
130#14519684C2312yes—
131#14519684C2314yes—
132#14519684C2312yes—
133#14519684C2314yes—
134#14519684C2311yescharacteristic
135#14519684C2314yes—
136#14519684C2311yescharacteristic
137#14519684C2314yes—
138#14519684C2314yes—
139#14519684C2312yes—
140#14519684C2312yes—
141#14519684C2314yes—
142#14519684C2314yes—
143#14519684C2314yes—
144#14519684C2314yes—
145C22×Q8×C61C2311yescharacteristic
146C4×Q8×C61—44no—
147C4×Q8×C61C2×C418yes—
148C2×C4⋊C4×C61C2312yes—
149C2×C4⋊C4×C61C2312yes—
150C2×C4⋊C4×C61C2314yes—
151C2×C4⋊C4×C61C2314yes—
152C2×C4⋊C4×C61C2312yes—
153C2×C4⋊C4×C61C2311yescharacteristic
154C2×C42×C61C2312yes—
155C2×C42×C61C2314yes—
156C2×C42×C61C2311yescharacteristic
157C2×C42×C61C2312yes—
158#95×C61C2311yescharacteristic
159#95×C61C2311yescharacteristic
160#95×C61C2312yes—
161#95×C61C2312yes—
162#95×C61C2314yes—
163C23×D61C2411yescharacteristic
164C2×Q8×C61—22no—
165C2×Q8×C61—22no—
166C2×Q8×C61C22×C414yes—
167C22×C4×C61C2414yes—
168C22×C4×C61C2412yes—
169C22×C4×C61C2412yes—
170C22×C4×C61C2411yescharacteristic
171C22×C4×C61C2411yescharacteristic
172C22×C4×C61C2412yes—
173C22×C4×C61C2412yes—
174C22×C4×C61C2411yescharacteristic
175C22×Dic61C2414yes—
176C22×Dic61C2412yes—
177C22×Dic61C2412yes—
178C22×Dic61C2411yescharacteristic
179C22×Dic61C2411yescharacteristic
180C22×Dic61C2412yes—
181C22×Dic61C2412yes—
182C22×Dic61C2411yescharacteristic
183C2×C4×D61—22no—
184C2×C4×D61—22no—
185C2×C4×D61—24no—
186C2×C4×D61—42no—
187C2×C4×D61C22×C414yes—
188C2×C4×D61C22×C414yes—
189C2×C4×D61C22×C414yes—
190C2×Dic122—22no—
191C2×Dic122—22no—
192C2×Dic122—44no—
193C2×Dic122C22×C414yes—
194C2×Dic122C22×C418yes—
195C4⋊C4×C61—24no—
196C4⋊C4×C61—44no—
197C4⋊C4×C61—42no—
198C4⋊C4×C61—44no—
199C4⋊C4×C61C22×C418yes—
200C4⋊C4×C61C22×C414yes—
201C42×C61—24no—
202C42×C61—44no—
203C42×C61C22×C414yes—
204C42×C61C22×C418yes—
205#11758198—44no—
206#11758198—42no—
207#11758198—42no—
208#11758198C22×C418yes—
209#11758198C22×C418yes—
210C4⋊Dic61—24no—
211C4⋊Dic61—42no—
212C4⋊Dic61C22×C414yes—
213C4⋊Dic61C22×C418yes—
214#11758196—24no—
215#11758196—24no—
216#11758196—44no—
217#11758196—42no—
218#11758196—42no—
219#11758196—44no—
220#11758196C22×C418yes—
221#11758196C22×C418yes—
222#11758196C22×C418yes—
223#11758196C22×C418yes—
224#11758196C22×C418yes—
225C4×Dic61—24no—
226C4×Dic61—28no—
227C4×Dic61—44no—
228C4×Dic61—44no—
229C4×Dic61—44no—
230C4×Dic61C22×C414yes—
231C4×Dic61C22×C414yes—
232C4×Dic61C22×C414yes—
233C4×Dic61C22×C418yes—
234C23×C61C2511yescharacteristic
235C22×D61—21no—
236C22×D61—22no—
237C22×D61—21no—
238C22×D61—22no—
239C22×D61C23×C412yes—
240Q8×C61—28no—
241C2×C4×C61—22no—
242C2×C4×C61—22no—
243C2×C4×C61—22no—
244C2×C4×C61—24no—
245C2×C4×C61—22no—
246C2×C4×C61—24no—
247C2×C4×C61—22no—
248C2×C4×C61—22no—
249C2×C4×C61—24no—
250C2×C4×C61—22no—
251C2×C4×C61—22no—
252C2×C4×C61—22no—
253C2×C4×C61—24no—
254C2×C4×C61—22no—
255C2×C4×C61C23×C412yes—
256C2×C4×C61C23×C412yes—
257C2×C4×C61C23×C414yes—
258C2×C4×C61C23×C414yes—
259C2×C4×C61C23×C412yes—
260C2×C4×C61H(5,𝔽2)12yes—
261C2×C4×C612−1+412yes—
262C2×Dic61—22no—
263C2×Dic61—21no—
264C2×Dic61—22no—
265C2×Dic61—21no—
266C2×Dic61—24no—
267C2×Dic61—22no—
268C2×Dic61—22no—
269C2×Dic61—22no—
270C2×Dic61—24no—
271C2×Dic61—22no—
272C2×Dic61—22no—
273C2×Dic61—24no—
274C2×Dic61—22no—
275C2×Dic61—24no—
276C2×Dic61—22no—
277C2×Dic61C23×C412yes—
278C2×Dic61C23×C412yes—
279C2×Dic61C23×C414yes—
280C2×Dic61C23×C414yes—
281C2×Dic61C23×C412yes—
282C2×Dic612−1+414yes—
283C4×D61—28no—
284C4×D61—28no—
285C4×D61—44no—
286Dic122—28no—
287Dic122—48no—
288#34394—611nomaximal, Sylow
289C22×C61C24×C411yescharacteristic
290C22×C61C2×H(5,𝔽2)11yescharacteristic
291C22×C61C2×2−1+412yes—
292C22×C61C2×2−1+411yescharacteristic
293C22×C61#58512yes—
294C2×D61—22no—
295C2×D61—22no—
296C2×D61—22no—
297C2×D61—42no—
298C2×D61—42no—
299C4×C61—22no—
300C4×C61—24no—
301C4×C61—24no—
302C4×C61—22no—
303C4×C61—44no—
304C4×C61—42no—
305C4×C61—42no—
306C4×C61#52014yes—
307Dic61—24no—
308Dic61—22no—
309Dic61—22no—
310Dic61—24no—
311Dic61—42no—
312Dic61—42no—
313Dic61—44no—
314Dic61#51914yes—
315C2×#533—611no—
316#2378—614no—
317#2369—612no—
318#2365—612no—
319#2356—611no—
320#2356—611no—
321#2351—611no—
322#2351D6111yescharacteristic, complemented
323#2347—612no—
324#2346—612no—
325#2330—614no—
326#2323—612no—
327#2323—612no—
328#2320—612no—
329#2320—612no—
330#2285—611no—
331#2283—611no—
332C2×C61C2×#51911yescharacteristic
333C2×C61C2×#52011yescharacteristic
334C2×C61#342811yescharacteristic
335C2×C61#354712yes—
336C2×C61#355512yes—
337D61—42no—
338#533—6116no—
339C2×C4⋊Q8—611no—
340C2×C4⋊Q8—611no—
341C2×#125—612no—
342C2×C22⋊Q8—611no—
343C2×C22⋊Q8—611no—
344C2×C22⋊Q8—612no—
345C2×C4×Q8—612no—
346C2×C4×Q8—612no—
347C2×C4×Q8—611no—
348C2×C4×Q8C2×D6112yes—
349C2×C4×Q8C2×D6111yescharacteristic
350C2×#117—612no—
351C2×#117—612no—
352C22×C42—611no—
353#391—612no—
354#391—614no—
355#391—612no—
356#391—614no—
357#389—614no—
358#389—614no—
359#389—612no—
360#389—614no—
361#389—614no—
362#389—614no—
363#389C2×D6112yes—
364#389C2×D6114yes—
365#388—614no—
366#388—614no—
367#388—614no—
368#388—614no—
369#387—614no—
370#387—614no—
371#387—614no—
372#387—614no—
373#387—614no—
374#387—614no—
375#384—612no—
376#383—612no—
377#383—612no—
378#383—612no—
379#382—611no—
380#382—611no—
381#382—611no—
382#382—612no—
383#382C2×D6112yes—
384#382C2×D6111yescharacteristic
385#381—612no—
386#380—612no—
387#380—612no—
388#380—611no—
389#380—611no—
390C4×C4⋊C4—612no—
391C4×C4⋊C4—612no—
392C4×C4⋊C4—612no—
393C4×C4⋊C4—614no—
394C4×C4⋊C4—611no—
395C4×C4⋊C4—612no—
396C4×C4⋊C4—612no—
397C4×C4⋊C4—614no—
398C4×C4⋊C4C2×D6112yes—
399C4×C4⋊C4C2×D6111yescharacteristic
400C4×C22⋊C4—612no—
401#376—612no—
402#376—612no—
403#376—612no—
404C61#3439411yescharacteristic, complemented, Sylow
405C22×Q8—611no—
406C22×Q8—612no—
407C22×Q8C22×D6111yescharacteristic
408C23×C4—611no—
409C23×C4—611no—
410C23×C4—611no—
411C4⋊Q8—618no—
412C4⋊Q8—618no—
413#125—6116no—
414C22⋊Q8—618no—
415C22⋊Q8—6116no—
416C22⋊Q8—618no—
417C4×Q8—44no—
418C4×Q8—6116no—
419C4×Q8—618no—
420C4×Q8—2444no—
421C4×Q8C4×D6118yes—
422#117—6116no—
423#117—2444no—
424C2×C4⋊C4—612no—
425C2×C4⋊C4—612no—
426C2×C4⋊C4—612no—
427C2×C4⋊C4—612no—
428C2×C4⋊C4—612no—
429C2×C4⋊C4—611no—
430C2×C4⋊C4—612no—
431C2×C4⋊C4—612no—
432C2×C4⋊C4—612no—
433C2×C4⋊C4—612no—
434C2×C4⋊C4—612no—
435C2×C4⋊C4—612no—
436C2×C4⋊C4—614no—
437C2×C4⋊C4—612no—
438C2×C4⋊C4—614no—
439C2×C4⋊C4C22×D6112yes—
440C2×C4⋊C4C22×D6114yes—
441C2×C4⋊C4C22×D6112yes—
442C2×C4⋊C4C22×D6114yes—
443C2×C4⋊C4C22×D6112yes—
444C2×C4⋊C4C22×D6111yescharacteristic
445C2×C22⋊C4—612no—
446C2×C22⋊C4—612no—
447C2×C22⋊C4—612no—
448C2×C22⋊C4—612no—
449C2×C22⋊C4—614no—
450C2×C42—612no—
451C2×C42—611no—
452C2×C42—611no—
453C2×C42—611no—
454C2×C42—618no—
455C2×C42—612no—
456C2×C42—614no—
457C2×C42—614no—
458C2×C42—614no—
459C2×C42—614no—
460C2×C42C22×D6112yes—
461C2×C42C22×D6114yes—
462C2×C42C22×D6112yes—
463C2×C42C22×D6111yescharacteristic
464#95—612no—
465#95—611no—
466#95—614no—
467#95—612no—
468#95—614no—
469#95—614no—
470#95—611no—
471#95—612no—
472#95—612no—
473#95—614no—
474#95—614no—
475#95—614no—
476#95—614no—
477#95—614no—
478#95—614no—
479#95C22×D6111yescharacteristic
480#95C22×D6112yes—
481#95C22×D6112yes—
482#95C22×D6114yes—
483#95C22×D6111yescharacteristic
484C24—611no—
485C2×Q8—22no—
486C2×Q8—22no—
487C2×Q8—614no—
488C2×Q8—618no—
489C2×Q8—1222no—
490C2×Q8—1222no—
491C2×Q8—2444no—
492C2×Q8C2×C4×D6114yes—
493C22×C4—611no—
494C22×C4—611no—
495C22×C4—612no—
496C22×C4—612no—
497C22×C4—611no—
498C22×C4—614no—
499C22×C4—614no—
500C22×C4—614no—
501C22×C4—614no—
502C22×C4—612no—
503C22×C4—612no—
504C22×C4—1224no—
505C22×C4—1222no—
506C22×C4—1222no—
507C22×C4—2442no—
508C22×C4C23×D6112yes—
509C22×C4C23×D6112yes—
510C22×C4C23×D6114yes—
511C22×C4C23×D6111yescharacteristic
512C22×C4C23×D6112yes—
513C22×C4C23×D6112yes—
514C22×C4C23×D6111yescharacteristic
515C22×C4C23×D6111yescharacteristic
516C4⋊C4—24no—
517C4⋊C4—44no—
518C4⋊C4—44no—
519C4⋊C4—42no—
520C4⋊C4—618no—
521C4⋊C4—618no—
522C4⋊C4—618no—
523C4⋊C4—618no—
524C4⋊C4—618no—
525C4⋊C4—618no—
526C4⋊C4—614no—
527C4⋊C4—1224no—
528C4⋊C4—1224no—
529C4⋊C4—1224no—
530C4⋊C4—2444no—
531C4⋊C4—2442no—
532C4⋊C4—2442no—
533C4⋊C4—2444no—
534C4⋊C4—2442no—
535C4⋊C4C2×C4×D6118yes—
536C4⋊C4C2×C4×D6114yes—
537C22⋊C4—618no—
538C22⋊C4—618no—
539C22⋊C4—2442no—
540C22⋊C4—2442no—
541C22⋊C4—2444no—
542C42—24no—
543C42—44no—
544C42—614no—
545C42—614no—
546C42—614no—
547C42—618no—
548C42—1228no—
549C42—1224no—
550C42—2444no—
551C42—2444no—
552C42—2444no—
553C42C2×C4×D6114yes—
554C42C2×C4×D6118yes—
555C23—612no—
556C23—1221no—
557C23—1222no—
558C23—1221no—
559C23—1222no—
560C23C24×D6111yescharacteristic
561Q8—28no—
562Q8—1228no—
563Q8—2448no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—24no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—24no—
577C2×C4—22no—
578C2×C4—612no—
579C2×C4—614no—
580C2×C4—612no—
581C2×C4—612no—
582C2×C4—614no—
583C2×C4—614no—
584C2×C4—1228no—
585C2×C4—1222no—
586C2×C4—1221no—
587C2×C4—1221no—
588C2×C4—1222no—
589C2×C4—1224no—
590C2×C4—1222no—
591C2×C4—1228no—
592C2×C4—1222no—
593C2×C4—1222no—
594C2×C4—1224no—
595C2×C4—1222no—
596C2×C4—1222no—
597C2×C4—1224no—
598C2×C4—1224no—
599C2×C4—1222no—
600C2×C4—1222no—
601C2×C4—2444no—
602C2×C4C22×C4×D6112yes—
603C2×C4C22×C4×D6112yes—
604C2×C4C22×C4×D6114yes—
605C2×C4C22×C4×D6114yes—
606C2×C4C22×C4×D6112yes—
607C2×C4#1451982212yes—
608C2×C4#1451982512yes—
609C22—1222no—
610C22—1222no—
611C22—1222no—
612C22—2442no—
613C22—2442no—
614C22C23×C4×D6111yescharacteristic
615C22C2×#1451982211yescharacteristic
616C222−1+4×D6112yes—
617C22C2×#1451982511yescharacteristic
618C22#1684059812yes—
619C4—22no—
620C4—24no—
621C4—22no—
622C4—24no—
623C4—42no—
624C4—42no—
625C4—44no—
626C4—614no—
627C4—1222no—
628C4—1224no—
629C4—1222no—
630C4—1224no—
631C4—2444no—
632C4—2442no—
633C4—2442no—
634C4#1684024314yes—
635C2—2442no—
636C2#519×D6111yescharacteristic
637C2C2×#1684024311yescharacteristic
638C2#2145901811yescharacteristic
639C2#2146020012yes—
640C2#2146029912yes—
subgroupG/H
center, FrattiniC23C24×D61
commutatorC22×C61C24×C4
FittingC61×#2351C2
socle, cosocleC23×C61C25
prime radicalC23×D61C24
hypercenter#2351D61
hypocenterC61#34394
LCSG ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ #2351

Complements

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

NH ≅ G/Ntype
3. #21448905C2semidirect
4. #21448873C2semidirect
5. #21448784C2semidirect
6. #21448782C2semidirect
7. #21448736C2semidirect
10. #21448557C2semidirect
13. #21448467C2semidirect
14. #21448465C2semidirect
17. C61×#2351C2semidirect
18. #16840316C4semidirect
322. #2351D61semidirect
404. C61#34394semidirect

Maximal subgroups

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

subgroupindex
C2×#168403162
#214489812
#214489052
#214488732
#214487842
#214487822
#214487362
#214487132
#214486992
#214485572
#214484992
#214484952
#214484672
#214484652
#214482232
#214482122
C61×#23512
#3439461