Finite Group Databasebeta

← #29840610

Subgroups — #29840610

Order 15616.

2194 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840316C211yesmaximal, characteristic
2#21448910C214yesmaximal, complemented
3#21448841C214yesmaximal
4#21448779C212yesmaximal, complemented
5#21448736C212yesmaximal, complemented
6#21448707C212yesmaximal
7#21448699C212yesmaximal
8#21448504C212yesmaximal
9#21448499C212yesmaximal
10#21448471C211yesmaximal, characteristic, complemented
11#21448390C212yesmaximal, complemented
12#21448388C212yesmaximal, complemented
13#21448223C211yesmaximal, characteristic
14#21448184C211yesmaximal, characteristic, complemented
15#21448183C211yesmaximal, characteristic, complemented
16#21448158C211yesmaximal, characteristic
17C61×#2320C211yesmaximal, characteristic, complemented
18#16840316C4116yescomplemented
19C2×#14519776C2211yescharacteristic
20C2×#14519775C2211yescharacteristic
21C2×#14519767C2212yes—
22C2×#14519757C2211yescharacteristic
23C2×#14519755C2212yes—
24C2×#14519753C2211yescharacteristic
25C2×Q8×Dic61C2212yes—
26C2×C4×Dic122C2211yescharacteristic
27C2×C4×Dic122C2212yes—
28C2×#14519742C2212yes—
29C2×#14519742C2214yes—
30C2×#14519734C2212yes—
31C2×#14519734C2212yes—
32C2×#14519734C2212yes—
33C2×C42×D61C2211yescharacteristic
34#16839668C2212yes—
35#16839666C2214yes—
36#16839660C2212yes—
37#16839659C2214yes—
38#16839659C2214yes—
39#16839659C2214yes—
40#16839658C2214yes—
41#16839657C2214yes—
42#16839654C2214yes—
43#16839652C2214yes—
44#16839651C2214yes—
45#16839651C2214yes—
46#16839645C2214yes—
47#16839644C2214yes—
48#16839644C2214yes—
49#16839631C2212yes—
50#16839629C2212yes—
51#16839626C2212yes—
52#16839626C2212yes—
53#16839621C2212yes—
54#16839619C2212yes—
55C4⋊C4×Dic61C2212yes—
56C4⋊C4×Dic61C2212yes—
57C4⋊C4×Dic61C2214yes—
58C4×#11758196C2212yes—
59C4×#11758196C2214yes—
60C4×C4⋊Dic61C2212yes—
61C4×C4⋊Dic61C2211yescharacteristic
62#16839613C2212yes—
63#16839613C2214yes—
64#16839613C2211yescharacteristic
65#16839613C2211yescharacteristic
66#16839613C2214yes—
67#16839613C2214yes—
68#16839613C2212yes—
69#16839609C2211yescharacteristic
70#16839609C2212yes—
71#16839609C2211yescharacteristic
72#16839608C2212yes—
73#16839608C2211yescharacteristic
74#16839607C2212yes—
75C42×Dic61C2212yes—
76C2×C4×Q8×C61C2211yescharacteristic
77C61×#391C2212yes—
78C61×#389C2212yes—
79C61×#387C2214yes—
80C61×#383C2212yes—
81C61×#382C2212yes—
82C4×C4⋊C4×C61C2211yescharacteristic
83C61×#376C2211yescharacteristic
84C22×Dic122C2312yes—
85C22×Dic122C2311yescharacteristic
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—
95Q8×Dic61—28no—
96C4×Dic122C2×C4116yes—
97C4×Dic122C2×C418yes—
98#14519742—28no—
99#14519742—48no—
100#14519734—28no—
101#14519734—44no—
102#14519734C2×C4116yes—
103C2×C4⋊Dic61C2312yes—
104C2×C4⋊Dic61C2314yes—
105C2×C4⋊Dic61C2311yescharacteristic
106C2×C4⋊Dic61C2312yes—
107C2×#11758196C2314yes—
108C2×#11758196C2312yes—
109C2×#11758196C2312yes—
110C2×#11758196C2314yes—
111C2×#11758196C2314yes—
112C2×#11758196C2314yes—
113C2×#11758196C2312yes—
114C2×#11758196C2312yes—
115C2×#11758196C2312yes—
116C2×#11758196C2312yes—
117C2×#11758196C2312yes—
118C2×#11758198C2312yes—
119C2×#11758198C2312yes—
120C2×#11758198C2314yes—
121C2×#11758198C2312yes—
122C2×#11758198C2312yes—
123C2×C4×Dic61C2312yes—
124C2×C4×Dic61C2314yes—
125C2×C4×Dic61C2314yes—
126C2×C4×Dic61C2312yes—
127C2×C4×Dic61C2312yes—
128C2×C4×Dic61C2312yes—
129C2×C4×Dic61C2312yes—
130C2×C4×Dic61C2314yes—
131C2×C4×Dic61C2314yes—
132C2×C4×Dic61C2311yescharacteristic
133C2×C4×Dic61C2311yescharacteristic
134C2×C4×Dic61C2311yescharacteristic
135C2×C4×Dic61C2312yes—
136C2×C4×Dic61C2312yes—
137C42×D61C2×C418yes—
138#14519684C2311yescharacteristic
139#14519684C2314yes—
140#14519684C2311yescharacteristic
141#14519684C2312yes—
142#14519684C2312yes—
143#14519684C2314yes—
144#14519684C2314yes—
145#14519684C2314yes—
146#14519684C2314yes—
147#14519684C2314yes—
148C22×Q8×C61C2311yescharacteristic
149C4×Q8×C61C2×C418yes—
150C2×C4⋊C4×C61C2312yes—
151C2×C4⋊C4×C61C2314yes—
152C2×C4⋊C4×C61C2312yes—
153C2×C4⋊C4×C61C2311yescharacteristic
154C2×C42×C61C2312yes—
155C2×C42×C61C2312yes—
156C2×C42×C61C2311yescharacteristic
157C2×C42×C61C2312yes—
158#95×C61C2311yescharacteristic
159#95×C61C2311yescharacteristic
160#95×C61C2312yes—
161#95×C61C2312yes—
162#95×C61C2314yes—
163#95×C61C2314yes—
164#95×C61C2314yes—
165C23×D61C2411yescharacteristic
166C2×Q8×C61—24no—
167C2×Q8×C61C22×C414yes—
168C22×C4×C61C2412yes—
169C22×C4×C61C2414yes—
170C22×C4×C61C2412yes—
171C22×C4×C61C2411yescharacteristic
172C22×C4×C61C2411yescharacteristic
173C22×C4×C61C2412yes—
174C22×C4×C61C2412yes—
175C22×C4×C61C2411yescharacteristic
176C22×Dic61C2414yes—
177C22×Dic61C2411yescharacteristic
178C22×Dic61C2412yes—
179C22×Dic61C2412yes—
180C22×Dic61C2412yes—
181C22×Dic61C2412yes—
182C22×Dic61C2411yescharacteristic
183C22×Dic61C2411yescharacteristic
184C2×C4×D61—24no—
185C2×C4×D61—22no—
186C2×C4×D61—24no—
187C2×C4×D61—22no—
188C2×C4×D61C22×C414yes—
189C2×C4×D61C22×C414yes—
190C2×C4×D61C22×C414yes—
191C2×Dic122—24no—
192C2×Dic122—24no—
193C2×Dic122—24no—
194C2×Dic122C22×C418yes—
195C2×Dic122C22×C414yes—
196C4⋊C4×C61—24no—
197C4⋊C4×C61—44no—
198C4⋊C4×C61C22×C418yes—
199C4⋊C4×C61C22×C414yes—
200C42×C61—24no—
201C42×C61—42no—
202C42×C61C22×C414yes—
203C42×C61C22×C418yes—
204#11758198—24no—
205#11758198—24no—
206#11758198—44no—
207#11758198C22×C418yes—
208#11758198C22×C418yes—
209C4⋊Dic61—24no—
210C4⋊Dic61—28no—
211C4⋊Dic61C22×C414yes—
212C4⋊Dic61C22×C418yes—
213#11758196—24no—
214#11758196—24no—
215#11758196—28no—
216#11758196—28no—
217#11758196—28no—
218#11758196—44no—
219#11758196C22×C418yes—
220#11758196C22×C418yes—
221#11758196C22×C418yes—
222#11758196C22×C418yes—
223#11758196C22×C418yes—
224C4×Dic61—28no—
225C4×Dic61—28no—
226C4×Dic61—24no—
227C4×Dic61—24no—
228C4×Dic61—24no—
229C4×Dic61—24no—
230C4×Dic61—42no—
231C4×Dic61—44no—
232C4×Dic61—44no—
233C4×Dic61C22×C414yes—
234C4×Dic61C22×C414yes—
235C4×Dic61C22×C414yes—
236C4×Dic61C22×C418yes—
237C4×Dic61#4118yes—
238C23×C61C2511yescharacteristic
239C22×D61—22no—
240C22×D61—21no—
241C22×D61—22no—
242C22×D61—21no—
243C22×D61C23×C412yes—
244Q8×C61—28no—
245C2×C4×C61—24no—
246C2×C4×C61—21no—
247C2×C4×C61—22no—
248C2×C4×C61—22no—
249C2×C4×C61—22no—
250C2×C4×C61—21no—
251C2×C4×C61—24no—
252C2×C4×C61—22no—
253C2×C4×C61—22no—
254C2×C4×C61—22no—
255C2×C4×C61—24no—
256C2×C4×C61—22no—
257C2×C4×C61—22no—
258C2×C4×C61—22no—
259C2×C4×C61—24no—
260C2×C4×C61C23×C412yes—
261C2×C4×C61C23×C412yes—
262C2×C4×C61C23×C414yes—
263C2×C4×C61C23×C414yes—
264C2×C4×C61C23×C412yes—
265C2×C4×C61C2×#4114yes—
266C2×Dic61—24no—
267C2×Dic61—22no—
268C2×Dic61—22no—
269C2×Dic61—22no—
270C2×Dic61—22no—
271C2×Dic61—22no—
272C2×Dic61—22no—
273C2×Dic61—22no—
274C2×Dic61—24no—
275C2×Dic61—24no—
276C2×Dic61—24no—
277C2×Dic61C23×C412yes—
278C2×Dic61C23×C414yes—
279C2×Dic61C23×C414yes—
280C2×Dic61C23×C412yes—
281C2×Dic61C23×C412yes—
282C2×Dic61C2×#4114yes—
283C2×Dic61C2×#4114yes—
284C2×Dic61H(5,𝔽2)12yes—
285C2×Dic612−1+414yes—
286C2×Dic612−1+412yes—
287C4×D61—28no—
288C4×D61—28no—
289C4×D61—28no—
290Dic122—28no—
291Dic122—216no—
292#34225—611nomaximal, Sylow
293C22×C61C24×C411yescharacteristic
294C22×C61C22×#4112yes—
295C22×C61C2×H(5,𝔽2)11yescharacteristic
296C22×C61C2×2−1+412yes—
297C22×C61C2×2−1+411yescharacteristic
298C2×D61—22no—
299C2×D61—22no—
300C2×D61—22no—
301C2×D61—42no—
302C2×D61—42no—
303C4×C61—24no—
304C4×C61—22no—
305C4×C61—24no—
306C4×C61—22no—
307C4×C61—42no—
308C4×C61—44no—
309C4×C61—42no—
310C4×C61C4×#4114yes—
311Dic61—24no—
312Dic61—22no—
313Dic61—24no—
314Dic61—24no—
315Dic61—24no—
316Dic61—44no—
317Dic61#51914yes—
318Dic61#52014yes—
319C2×#533—611no—
320#2370—614no—
321#2362—614no—
322#2356—612no—
323#2351—612no—
324#2347—612no—
325#2346—612no—
326#2323—612no—
327#2323—612no—
328#2320—611no—
329#2320D6111yescharacteristic, complemented
330C4×C4⋊Q8—612no—
331C4×C4⋊Q8—612no—
332#2285—611no—
333#2276—611no—
334#2276—611no—
335C4×#117—611no—
336C2×C61C2×C4×#4111yescharacteristic
337C2×C61C2×#51911yescharacteristic
338C2×C61C2×#52011yescharacteristic
339C2×C61#350912yes—
340C2×C61#351112yes—
341D61—42no—
342#533—6116no—
343C2×C4⋊Q8—611no—
344C2×C4⋊Q8—611no—
345C2×#125—612no—
346C2×C22⋊Q8—611no—
347C2×C22⋊Q8—611no—
348C2×C22⋊Q8—612no—
349C2×C4×Q8—612no—
350C2×C4×Q8—612no—
351C2×C4×Q8—614no—
352C2×C4×Q8—611no—
353C2×C4×Q8—612no—
354C2×C4×Q8C2×D6111yescharacteristic
355C2×#117—612no—
356C2×#117—612no—
357C2×#117—612no—
358C22×C42—611no—
359#391—612no—
360#391—614no—
361#391C2×D6112yes—
362#389—614no—
363#389—614no—
364#389—612no—
365#389—614no—
366#389—614no—
367#389—614no—
368#389C2×D6112yes—
369#388—614no—
370#388—614no—
371#388—614no—
372#388—614no—
373#387—614no—
374#387—614no—
375#387—614no—
376#387C2×D6114yes—
377#384—612no—
378#383—612no—
379#383C2×D6112yes—
380#382—612no—
381#382—612no—
382#382C2×D6112yes—
383#380—612no—
384#380—612no—
385C4×C4⋊C4—612no—
386C4×C4⋊C4—612no—
387C4×C4⋊C4—614no—
388C4×C4⋊C4—611no—
389C4×C4⋊C4—612no—
390C4×C4⋊C4—614no—
391C4×C4⋊C4—612no—
392C4×C4⋊C4—614no—
393C4×C4⋊C4—614no—
394C4×C4⋊C4—614no—
395C4×C4⋊C4—612no—
396C4×C4⋊C4—611no—
397C4×C4⋊C4—611no—
398C4×C4⋊C4—612no—
399C4×C4⋊C4C2×D6111yescharacteristic
400C4×C22⋊C4—611no—
401C4×C22⋊C4—611no—
402C4×C22⋊C4—612no—
403#376—612no—
404#376—611no—
405#376—612no—
406#376C2×D6111yescharacteristic
407C43—612no—
408C61#3422511yescharacteristic, complemented, Sylow
409C22×Q8—612no—
410C22×Q8—611no—
411C22×Q8C22×D6111yescharacteristic
412C23×C4—611no—
413C23×C4—611no—
414C23×C4—611no—
415C4⋊Q8—618no—
416C4⋊Q8—618no—
417#125—6116no—
418C22⋊Q8—618no—
419C22⋊Q8—6116no—
420C22⋊Q8—618no—
421C4×Q8—6116no—
422C4×Q8—618no—
423C4×Q8—1228no—
424C4×Q8—1228no—
425C4×Q8—2448no—
426C4×Q8C4×D6118yes—
427#117—6116no—
428#117—1228no—
429#117—2444no—
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—611no—
437C2×C4⋊C4—614no—
438C2×C4⋊C4—612no—
439C2×C4⋊C4—612no—
440C2×C4⋊C4—612no—
441C2×C4⋊C4—614no—
442C2×C4⋊C4—614no—
443C2×C4⋊C4—614no—
444C2×C4⋊C4—614no—
445C2×C4⋊C4C22×D6112yes—
446C2×C4⋊C4C22×D6114yes—
447C2×C4⋊C4C22×D6112yes—
448C2×C4⋊C4C22×D6111yescharacteristic
449C2×C22⋊C4—612no—
450C2×C22⋊C4—612no—
451C2×C22⋊C4—612no—
452C2×C22⋊C4—612no—
453C2×C22⋊C4—614no—
454C2×C42—612no—
455C2×C42—612no—
456C2×C42—611no—
457C2×C42—611no—
458C2×C42—611no—
459C2×C42—618no—
460C2×C42—612no—
461C2×C42—614no—
462C2×C42—614no—
463C2×C42—612no—
464C2×C42—612no—
465C2×C42—612no—
466C2×C42—612no—
467C2×C42—614no—
468C2×C42—614no—
469C2×C42C22×D6112yes—
470C2×C42C22×D6112yes—
471C2×C42C22×D6112yes—
472C2×C42C22×D6111yescharacteristic
473#95—611no—
474#95—612no—
475#95—611no—
476#95—612no—
477#95—614no—
478#95—614no—
479#95—614no—
480#95—614no—
481#95—614no—
482#95—614no—
483#95C22×D6112yes—
484#95C22×D6111yescharacteristic
485#95C22×D6111yescharacteristic
486#95C22×D6112yes—
487#95C22×D6114yes—
488#95C22×D6114yes—
489#95C22×D6114yes—
490C24—611no—
491C2×Q8—24no—
492C2×Q8—618no—
493C2×Q8—614no—
494C2×Q8—1224no—
495C2×Q8—1224no—
496C2×Q8—1224no—
497C2×Q8C2×C4×D6114yes—
498C22×C4—611no—
499C22×C4—611no—
500C22×C4—612no—
501C22×C4—612no—
502C22×C4—612no—
503C22×C4—611no—
504C22×C4—612no—
505C22×C4—614no—
506C22×C4—614no—
507C22×C4—614no—
508C22×C4—614no—
509C22×C4—1222no—
510C22×C4—1224no—
511C22×C4—1222no—
512C22×C4—1224no—
513C22×C4C23×D6112yes—
514C22×C4C23×D6112yes—
515C22×C4C23×D6114yes—
516C22×C4C23×D6111yescharacteristic
517C22×C4C23×D6111yescharacteristic
518C22×C4C23×D6112yes—
519C22×C4C23×D6112yes—
520C22×C4C23×D6111yescharacteristic
521C4⋊C4—24no—
522C4⋊C4—44no—
523C4⋊C4—618no—
524C4⋊C4—618no—
525C4⋊C4—618no—
526C4⋊C4—618no—
527C4⋊C4—618no—
528C4⋊C4—618no—
529C4⋊C4—614no—
530C4⋊C4—1228no—
531C4⋊C4—1224no—
532C4⋊C4—1224no—
533C4⋊C4—1228no—
534C4⋊C4—1228no—
535C4⋊C4—1228no—
536C4⋊C4—1224no—
537C4⋊C4—2444no—
538C4⋊C4C2×C4×D6118yes—
539C4⋊C4C2×C4×D6114yes—
540C22⋊C4—618no—
541C22⋊C4—618no—
542C22⋊C4—1224no—
543C22⋊C4—1224no—
544C22⋊C4—2444no—
545C42—24no—
546C42—42no—
547C42—614no—
548C42—614no—
549C42—614no—
550C42—618no—
551C42—618no—
552C42—1224no—
553C42—1224no—
554C42—1224no—
555C42—1228no—
556C42—1228no—
557C42—1224no—
558C42—2442no—
559C42—2444no—
560C42—2444no—
561C42C2×C4×D6114yes—
562C42C2×C4×D6118yes—
563C23—612no—
564C23—1222no—
565C23—1221no—
566C23—1222no—
567C23—1221no—
568C23C24×D6111yescharacteristic
569Q8—28no—
570Q8—12216no—
571Q8—1228no—
572C2×C4—24no—
573C2×C4—21no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—21no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—24no—
587C2×C4—612no—
588C2×C4—612no—
589C2×C4—612no—
590C2×C4—614no—
591C2×C4—612no—
592C2×C4—612no—
593C2×C4—614no—
594C2×C4—614no—
595C2×C4—614no—
596C2×C4—614no—
597C2×C4—1228no—
598C2×C4—1228no—
599C2×C4—1224no—
600C2×C4—1222no—
601C2×C4—1222no—
602C2×C4—1222no—
603C2×C4—1222no—
604C2×C4—1222no—
605C2×C4—1222no—
606C2×C4—1222no—
607C2×C4—1228no—
608C2×C4—1224no—
609C2×C4—1224no—
610C2×C4—1224no—
611C2×C4C22×C4×D6112yes—
612C2×C4C22×C4×D6112yes—
613C2×C4C22×C4×D6114yes—
614C2×C4C22×C4×D6114yes—
615C2×C4C22×C4×D6112yes—
616C2×C4#41×D6114yes—
617C22—1222no—
618C22—1222no—
619C22—1222no—
620C22—2442no—
621C22—2442no—
622C22C23×C4×D6111yescharacteristic
623C22C2×#41×D6112yes—
624C22H(5,𝔽2)×D6111yescharacteristic
625C222−1+4×D6112yes—
626C222−1+4×D6111yescharacteristic
627C4—24no—
628C4—22no—
629C4—22no—
630C4—24no—
631C4—42no—
632C4—44no—
633C4—42no—
634C4—614no—
635C4—614no—
636C4—1224no—
637C4—1222no—
638C4—1224no—
639C4—1224no—
640C4—1224no—
641C4—2444no—
642C4#1684023014yes—
643C2—2442no—
644C2C2×#1684023011yescharacteristic
645C2#519×D6111yescharacteristic
646C2#520×D6111yescharacteristic
647C2#2145970112yes—
648C2#2145972412yes—
subgroupG/H
center, FrattiniC23C24×D61
commutatorC22×C61C24×C4
FittingC61×#2320C2
socle, cosocleC23×C61C25
prime radicalC23×D61C24
hypercenter#2320D61
hypocenterC61#34225
LCSG ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ #2320

Complements

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

NH ≅ G/Ntype
2. #21448910C2semidirect
4. #21448779C2semidirect
5. #21448736C2semidirect
10. #21448471C2semidirect
11. #21448390C2semidirect
12. #21448388C2semidirect
14. #21448184C2semidirect
15. #21448183C2semidirect
17. C61×#2320C2semidirect
18. #16840316C4semidirect
329. #2320D61semidirect
408. C61#34225semidirect

Maximal subgroups

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

subgroupindex
C2×#168403162
#214489102
#214488412
#214487792
#214487362
#214487072
#214486992
#214485042
#214484992
#214484712
#214483902
#214483882
#214482232
#214481842
#214481832
#214481582
C61×#23202
#3422561