Finite Group Databasebeta

← #29845205

Subgroups — #29845205

Order 15616.

2132 conjugacy classes of subgroups.

The subgroup lattice has 676 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840469C211yesmaximal, characteristic
2#21448979C214yesmaximal, complemented
3#21448911C212yesmaximal, complemented
4#21448908C211yesmaximal, characteristic, complemented
5#21448846C212yesmaximal, complemented
6#21448767C212yesmaximal
7#21448731C211yesmaximal, characteristic, complemented
8#21448690C212yesmaximal, complemented
9#21448556C212yesmaximal, complemented
10#21448497C212yesmaximal, complemented
11#21448494C214yesmaximal, complemented
12#21448456C212yesmaximal
13#21448435C212yesmaximal, complemented
14#21448386C211yesmaximal, characteristic, complemented
15#21448293C211yesmaximal, characteristic, complemented
16#21448284C211yesmaximal, characteristic
17C61×#2354C211yesmaximal, characteristic
18#16840469C4116yescomplemented
19C2×#14519777C2211yescharacteristic
20C2×#14519768C2212yes—
21C2×#14519756C2212yes—
22C2×#14519754C2214yes—
23C2×C4×Dic122C2211yescharacteristic
24C2×#14519742C2212yes—
25C2×#14519742C2212yes—
26C2×D4×Dic61C2212yes—
27C2×#14519736C2212yes—
28C2×#14519733C2214yes—
29C22×C4⋊Dic61C2212yes—
30#16839667C2212yes—
31#16839667C2211yescharacteristic
32#16839667C2211yescharacteristic
33#16839666C2214yes—
34#16839665C2211yescharacteristic
35#16839665C2211yescharacteristic
36#16839661C2211yescharacteristic
37#16839661C2211yescharacteristic
38#16839660C2212yes—
39#16839660C2214yes—
40#16839660C2214yes—
41#16839659C2214yes—
42#16839658C2212yes—
43#16839658C2212yes—
44#16839658C2212yes—
45#16839658C2211yescharacteristic
46#16839658C2211yescharacteristic
47#16839658C2212yes—
48#16839657C2214yes—
49#16839656C2214yes—
50#16839656C2214yes—
51#16839655C2212yes—
52#16839655C2212yes—
53#16839650C2214yes—
54#16839650C2214yes—
55#16839649C2214yes—
56#16839646C2214yes—
57#16839645C2214yes—
58#16839645C2214yes—
59#16839644C2212yes—
60#16839644C2212yes—
61#16839631C2212yes—
62#16839623C2214yes—
63#16839620C2214yes—
64#16839620C2214yes—
65C4⋊C4×Dic61C2212yes—
66C4⋊C4×Dic61C2212yes—
67C4⋊C4×Dic61C2212yes—
68C4⋊C4×Dic61C2212yes—
69C4×#11758196C2214yes—
70C4×#11758196C2212yes—
71C4×C4⋊Dic61C2211yescharacteristic
72C4×C4⋊Dic61C2212yes—
73C4×C22⋊Dic61C2212yes—
74#16839608C2212yes—
75C42×Dic61C2211yescharacteristic
76C2×C4×D4×C61C2211yescharacteristic
77C2×#117×C61C2212yes—
78C61×#390C2212yes—
79C61×#388C2214yes—
80C61×#384C2211yescharacteristic
81C61×#380C2212yes—
82C4×C4⋊C4×C61C2211yescharacteristic
83C4×C4⋊C4×C61C2212yes—
84C22×Dic122C2311yescharacteristic
85C22×Dic122C2312yes—
86C23×Dic61C2314yes—
87#14519777C2×C418yes—
88#14519768C2×C4116yes—
89#14519756C2×C4116yes—
90#14519754C2×C4132yes—
91C4×Dic122C2×C418yes—
92#14519742—44no—
93#14519742—44no—
94D4×Dic61C2×C4116yes—
95#14519736—44no—
96#14519733—48no—
97C2×C4⋊Dic61C2×C4116yes—
98C2×C4⋊Dic61C2311yescharacteristic
99C2×C4⋊Dic61C2311yescharacteristic
100C2×C4⋊Dic61C2312yes—
101C2×C4⋊Dic61C2312yes—
102C2×C4⋊Dic61C2311yescharacteristic
103C2×C4⋊Dic61C2314yes—
104C2×C4⋊Dic61C2312yes—
105C2×C4⋊Dic61C2312yes—
106C2×#11758196C2312yes—
107C2×#11758196C2314yes—
108C2×#11758196C2314yes—
109C2×#11758196C2314yes—
110C2×#11758196C2312yes—
111C2×#11758196C2314yes—
112C2×#11758196C2312yes—
113C2×#11758196C2314yes—
114C2×C22⋊Dic61C2312yes—
115C2×C22⋊Dic61C2312yes—
116C2×C22⋊Dic61C2312yes—
117C2×C22⋊Dic61C2312yes—
118C2×C22⋊Dic61C2314yes—
119C2×C4×Dic61C2314yes—
120C2×C4×Dic61C2311yescharacteristic
121C2×C4×Dic61C2312yes—
122C2×C4×Dic61C2311yescharacteristic
123C2×C4×Dic61C2311yescharacteristic
124C2×C4×Dic61C2311yescharacteristic
125C2×C4×Dic61C2311yescharacteristic
126C2×C4×Dic61C2314yes—
127C2×C4×Dic61C2314yes—
128C2×C4×Dic61C2311yescharacteristic
129C2×C4×Dic61C2312yes—
130#14519684C2312yes—
131#14519684C2312yes—
132#14519684C2312yes—
133#14519684C2312yes—
134#14519684C2312yes—
135#14519684C2312yes—
136#14519684C2314yes—
137#14519684C2312yes—
138#14519684C2312yes—
139#14519684C2312yes—
140#14519684C2314yes—
141#14519684C2314yes—
142#14519684C2314yes—
143#14519684C2314yes—
144C22×D4×C61C2311yescharacteristic
145C23×C4×C61C2312yes—
146C4×D4×C61C2×C418yes—
147#117×C61—44no—
148C2×C4⋊C4×C61C2312yes—
149C2×C4⋊C4×C61C2311yescharacteristic
150C2×C4⋊C4×C61C2312yes—
151C2×C4⋊C4×C61C2311yescharacteristic
152C2×C4⋊C4×C61C2312yes—
153C2×C4⋊C4×C61C2311yescharacteristic
154C2×C22⋊C4×C61C2314yes—
155C2×C22⋊C4×C61C2314yes—
156C2×C22⋊C4×C61C2312yes—
157C2×C42×C61C2312yes—
158C2×C42×C61C2314yes—
159C2×C42×C61C2311yescharacteristic
160#95×C61C2314yes—
161#95×C61C2312yes—
162C24×C61C2412yes—
163C2×D4×C61—22no—
164C2×D4×C61—22no—
165C2×D4×C61C22×C414yes—
166C22×C4×C61—24no—
167C22×C4×C61—24no—
168C22×C4×C61C22×C418yes—
169C22×C4×C61C2412yes—
170C22×C4×C61C2412yes—
171C22×C4×C61C2414yes—
172C22×C4×C61C2411yescharacteristic
173C22×C4×C61C2412yes—
174C22×C4×C61C2411yescharacteristic
175C22×C4×C61C2411yescharacteristic
176C22×Dic61—44no—
177C22×Dic61—44no—
178C22×Dic61C22×C4116yes—
179C22×Dic61C2411yescharacteristic
180C22×Dic61C2412yes—
181C22×Dic61C2411yescharacteristic
182C22×Dic61C2411yescharacteristic
183C22×Dic61C2412yes—
184C22×Dic61C2412yes—
185C22×Dic61C2414yes—
186C22×Dic61C2412yes—
187C22×Dic61C2411yescharacteristic
188C2×Dic122—22no—
189C2×Dic122—22no—
190C2×Dic122—42no—
191C2×Dic122—42no—
192C2×Dic122C22×C414yes—
193C2×Dic122C22×C418yes—
194C4⋊C4×C61—22no—
195C4⋊C4×C61—24no—
196C4⋊C4×C61—24no—
197C4⋊C4×C61—22no—
198C4⋊C4×C61—42no—
199C4⋊C4×C61C22×C414yes—
200C22⋊C4×C61—44no—
201C22⋊C4×C61—44no—
202C22⋊C4×C61C22×C418yes—
203C42×C61—24no—
204C42×C61—44no—
205C42×C61C22×C414yes—
206C22⋊Dic61—24no—
207C22⋊Dic61—42no—
208C22⋊Dic61—42no—
209C22⋊Dic61—42no—
210C22⋊Dic61C22×C4116yes—
211C4⋊Dic61—22no—
212C4⋊Dic61—22no—
213C4⋊Dic61—42no—
214C4⋊Dic61—42no—
215C4⋊Dic61C22×C414yes—
216C4⋊Dic61C22×C4116yes—
217C4⋊Dic61C22×C418yes—
218C4⋊Dic61C22×C418yes—
219#11758196—24no—
220#11758196—44no—
221#11758196—42no—
222#11758196—44no—
223#11758196—44no—
224#11758196—44no—
225#11758196C22×C418yes—
226#11758196C22×C4116yes—
227C4×Dic61—22no—
228C4×Dic61—24no—
229C4×Dic61—22no—
230C4×Dic61—22no—
231C4×Dic61—22no—
232C4×Dic61—28no—
233C4×Dic61—22no—
234C4×Dic61—41no—
235C4×Dic61—44no—
236C4×Dic61—44no—
237C4×Dic61C22×C418yes—
238C23×C61—22no—
239C23×C61—22no—
240C23×C61—22no—
241C23×C61—22no—
242C23×C61—22no—
243C23×C61—22no—
244C23×C61C23×C414yes—
245C23×C61C2511yescharacteristic
246D4×C61—28no—
247C2×C4×C61—21no—
248C2×C4×C61—22no—
249C2×C4×C61—22no—
250C2×C4×C61—21no—
251C2×C4×C61—216no—
252C2×C4×C61—24no—
253C2×C4×C61—22no—
254C2×C4×C61—24no—
255C2×C4×C61—22no—
256C2×C4×C61—22no—
257C2×C4×C61—22no—
258C2×C4×C61—22no—
259C2×C4×C61—24no—
260C2×C4×C61—22no—
261C2×C4×C61C23×C414yes—
262C2×C4×C61C23×C412yes—
263C2×C4×C61C23×C412yes—
264C2×C4×C61C23×C412yes—
265C2×C4×C61H(5,𝔽2)12yes—
266C2×C4×C612−1+412yes—
267C2×C4×C612−1+412yes—
268C2×C4×C612−1+412yes—
269C2×Dic61—24no—
270C2×Dic61—22no—
271C2×Dic61—22no—
272C2×Dic61—21no—
273C2×Dic61—21no—
274C2×Dic61—22no—
275C2×Dic61—24no—
276C2×Dic61—22no—
277C2×Dic61—22no—
278C2×Dic61—21no—
279C2×Dic61—22no—
280C2×Dic61—21no—
281C2×Dic61—21no—
282C2×Dic61—22no—
283C2×Dic61—21no—
284C2×Dic61—21no—
285C2×Dic61—21no—
286C2×Dic61—21no—
287C2×Dic61—22no—
288C2×Dic61—22no—
289C2×Dic61—21no—
290C2×Dic61—21no—
291C2×Dic61—22no—
292C2×Dic61—416no—
293C2×Dic61C23×C414yes—
294C2×Dic61C23×C414yes—
295C2×Dic61C23×C418yes—
296C2×Dic61H(5,𝔽2)12yes—
297Dic122—28no—
298Dic122—48no—
299#34484—611nomaximal, Sylow
300C22×C61—24no—
301C22×C61—24no—
302C22×C61—24no—
303C22×C61—42no—
304C22×C61—42no—
305C22×C61—42no—
306C22×C61—42no—
307C22×C61C24×C411yescharacteristic
308C22×C61C2×H(5,𝔽2)11yescharacteristic
309C22×C61C2×H(5,𝔽2)11yescharacteristic
310C22×C61C2×2−1+411yescharacteristic
311C22×C61C2×2−1+411yescharacteristic
312C22×C61C2×2−1+411yescharacteristic
313C22×C61#58511yescharacteristic
314C4×C61—24no—
315C4×C61—22no—
316C4×C61—42no—
317C4×C61—42no—
318C4×C61—44no—
319C4×C61#51914yes—
320C4×C61#52014yes—
321Dic61—24no—
322Dic61—28no—
323Dic61—24no—
324Dic61—22no—
325Dic61—42no—
326Dic61—41no—
327Dic61—41no—
328Dic61—42no—
329Dic61—41no—
330C2×#554—611no—
331#2378—614no—
332#2370—612no—
333#2369—611no—
334#2362—612no—
335#2354—612no—
336#2354D6111yescharacteristic
337Q8×C4⋊C4—611no—
338#2344—612no—
339#2330—612no—
340#2323—614no—
341#2323—612no—
342#2319—612no—
343#2316—612no—
344C4×C4⋊Q8—611no—
345#2295—611no—
346#2293—611no—
347C2×C61—44no—
348C2×C61C2×#51911yescharacteristic
349C2×C61C2×#52011yescharacteristic
350C2×C61C4×H(5,𝔽2)11yescharacteristic
351C2×C61#354411yescharacteristic
352C2×C61#354711yescharacteristic
353C2×C61#354811yescharacteristic
354C2×C61#355511yescharacteristic
355#554—6116no—
356C2×C4⋊Q8—611no—
357C2×#125—612no—
358C2×C22⋊Q8—612no—
359C2×C22⋊Q8—614no—
360C2×C4×Q8—612no—
361C2×C4×Q8—612no—
362C2×C4×Q8—611no—
363C2×C4×D4—612no—
364C2×C4×D4C2×D6111yescharacteristic
365C2×#117—612no—
366C2×#117—614no—
367C2×#117C2×D6112yes—
368C22×C4⋊C4—612no—
369#391—612no—
370#391—611no—
371#391—611no—
372#391—614no—
373#390—611no—
374#390—611no—
375#390C2×D6112yes—
376#389—612no—
377#389—612no—
378#389—611no—
379#389—612no—
380#389—612no—
381#389—614no—
382#389—611no—
383#389—611no—
384#389—611no—
385#389—614no—
386#389—614no—
387#389—612no—
388#389—614no—
389#388—614no—
390#388—614no—
391#388—612no—
392#388—614no—
393#388—614no—
394#388—612no—
395#388C2×D6114yes—
396#387—614no—
397#387—612no—
398#387—614no—
399#387—614no—
400#387—614no—
401#387—612no—
402#384—612no—
403#384C2×D6111yescharacteristic
404#381—614no—
405#380—614no—
406#380—614no—
407#380C2×D6112yes—
408C4×C4⋊C4—612no—
409C4×C4⋊C4—612no—
410C4×C4⋊C4—612no—
411C4×C4⋊C4—612no—
412C4×C4⋊C4—614no—
413C4×C4⋊C4—612no—
414C4×C4⋊C4—611no—
415C4×C4⋊C4—612no—
416C4×C4⋊C4C2×D6112yes—
417C4×C4⋊C4C2×D6111yescharacteristic
418C4×C22⋊C4—612no—
419#376—612no—
420C43—611no—
421C61#3448411yescharacteristic, complemented, Sylow
422C22×Q8—611no—
423C22×Q8—612no—
424C22×D4C22×D6111yescharacteristic
425C23×C4—614no—
426C23×C4C22×D6112yes—
427C4⋊Q8—618no—
428#125—6116no—
429C22⋊Q8—6116no—
430C22⋊Q8—6132no—
431C4×Q8—618no—
432C4×Q8—2444no—
433C4×Q8—2444no—
434C4×D4—6116no—
435C4×D4C4×D6118yes—
436#117—44no—
437#117—2444no—
438#117—2448no—
439C2×C4⋊C4—6116no—
440C2×C4⋊C4—611no—
441C2×C4⋊C4—614no—
442C2×C4⋊C4—612no—
443C2×C4⋊C4—612no—
444C2×C4⋊C4—612no—
445C2×C4⋊C4—614no—
446C2×C4⋊C4—612no—
447C2×C4⋊C4—611no—
448C2×C4⋊C4—611no—
449C2×C4⋊C4—614no—
450C2×C4⋊C4—612no—
451C2×C4⋊C4—612no—
452C2×C4⋊C4—614no—
453C2×C4⋊C4—614no—
454C2×C4⋊C4—612no—
455C2×C4⋊C4—614no—
456C2×C4⋊C4C22×D6111yescharacteristic
457C2×C4⋊C4C22×D6112yes—
458C2×C4⋊C4C22×D6111yescharacteristic
459C2×C4⋊C4C22×D6112yes—
460C2×C4⋊C4C22×D6112yes—
461C2×C4⋊C4C22×D6111yescharacteristic
462C2×C22⋊C4—614no—
463C2×C22⋊C4—612no—
464C2×C22⋊C4—612no—
465C2×C22⋊C4—612no—
466C2×C22⋊C4—612no—
467C2×C22⋊C4C22×D6114yes—
468C2×C22⋊C4C22×D6114yes—
469C2×C22⋊C4C22×D6112yes—
470C2×C42—612no—
471C2×C42—611no—
472C2×C42—612no—
473C2×C42—614no—
474C2×C42—611no—
475C2×C42—611no—
476C2×C42—611no—
477C2×C42—611no—
478C2×C42—614no—
479C2×C42—614no—
480C2×C42—611no—
481C2×C42C22×D6112yes—
482C2×C42C22×D6114yes—
483C2×C42C22×D6111yescharacteristic
484#95—612no—
485#95—612no—
486#95—612no—
487#95—612no—
488#95—612no—
489#95—612no—
490#95—612no—
491#95—614no—
492#95—614no—
493#95—614no—
494#95—614no—
495#95—614no—
496#95—612no—
497#95—612no—
498#95C22×D6114yes—
499#95C22×D6112yes—
500C24C23×D6112yes—
501C2×Q8—614no—
502C2×Q8—618no—
503C2×Q8—1222no—
504C2×Q8—1222no—
505C2×Q8—2442no—
506C2×Q8—2442no—
507C2×D4—22no—
508C2×D4—22no—
509C2×D4C2×C4×D6114yes—
510C22×C4—24no—
511C22×C4—24no—
512C22×C4—611no—
513C22×C4—612no—
514C22×C4—614no—
515C22×C4—612no—
516C22×C4—6116no—
517C22×C4—611no—
518C22×C4—612no—
519C22×C4—611no—
520C22×C4—611no—
521C22×C4—612no—
522C22×C4—2444no—
523C22×C4—2444no—
524C22×C4C2×C4×D6118yes—
525C22×C4C23×D6112yes—
526C22×C4C23×D6112yes—
527C22×C4C23×D6114yes—
528C22×C4C23×D6111yescharacteristic
529C22×C4C23×D6112yes—
530C22×C4C23×D6111yescharacteristic
531C22×C4C23×D6111yescharacteristic
532C4⋊C4—22no—
533C4⋊C4—24no—
534C4⋊C4—24no—
535C4⋊C4—22no—
536C4⋊C4—42no—
537C4⋊C4—614no—
538C4⋊C4—6116no—
539C4⋊C4—618no—
540C4⋊C4—618no—
541C4⋊C4—6116no—
542C4⋊C4—618no—
543C4⋊C4—1224no—
544C4⋊C4—1222no—
545C4⋊C4—1222no—
546C4⋊C4—2442no—
547C4⋊C4—2444no—
548C4⋊C4—2444no—
549C4⋊C4—2444no—
550C4⋊C4—2444no—
551C4⋊C4—2442no—
552C4⋊C4—2442no—
553C4⋊C4C2×C4×D6114yes—
554C22⋊C4—44no—
555C22⋊C4—44no—
556C22⋊C4—6116no—
557C22⋊C4—1224no—
558C22⋊C4—2442no—
559C22⋊C4—2442no—
560C22⋊C4—2442no—
561C22⋊C4C2×C4×D6118yes—
562C42—24no—
563C42—44no—
564C42—618no—
565C42—1228no—
566C42—1222no—
567C42—1222no—
568C42—1224no—
569C42—1222no—
570C42—1222no—
571C42—1222no—
572C42—2441no—
573C42—2444no—
574C42—2444no—
575C42C2×C4×D6114yes—
576C23—22no—
577C23—22no—
578C23—22no—
579C23—22no—
580C23—22no—
581C23—22no—
582C23C22×C4×D6114yes—
583C23C24×D6111yescharacteristic
584Q8—1228no—
585Q8—2448no—
586D4—28no—
587C2×C4—21no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—21no—
591C2×C4—216no—
592C2×C4—24no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—24no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—24no—
600C2×C4—22no—
601C2×C4—614no—
602C2×C4—614no—
603C2×C4—618no—
604C2×C4—612no—
605C2×C4—1224no—
606C2×C4—1222no—
607C2×C4—1222no—
608C2×C4—1221no—
609C2×C4—1221no—
610C2×C4—1222no—
611C2×C4—1222no—
612C2×C4—1224no—
613C2×C4—1221no—
614C2×C4—1222no—
615C2×C4—1222no—
616C2×C4—1221no—
617C2×C4—1221no—
618C2×C4—1222no—
619C2×C4—1221no—
620C2×C4—1221no—
621C2×C4—1221no—
622C2×C4—1221no—
623C2×C4—1222no—
624C2×C4—1222no—
625C2×C4—1221no—
626C2×C4—1221no—
627C2×C4—1222no—
628C2×C4—24416no—
629C2×C4C22×C4×D6112yes—
630C2×C4C22×C4×D6114yes—
631C2×C4C22×C4×D6112yes—
632C2×C4C22×C4×D6112yes—
633C2×C4#1451982212yes—
634C2×C4#1451982412yes—
635C2×C4#1451982412yes—
636C2×C4#1451982412yes—
637C22—24no—
638C22—24no—
639C22—24no—
640C22—42no—
641C22—42no—
642C22—42no—
643C22—42no—
644C22C23×C4×D6111yescharacteristic
645C22H(5,𝔽2)×D6111yescharacteristic
646C22C2×#1451982211yescharacteristic
647C22C2×#1451982411yescharacteristic
648C22C2×#1451982411yescharacteristic
649C22C2×#1451982411yescharacteristic
650C22#1684059911yescharacteristic
651C4—24no—
652C4—22no—
653C4—42no—
654C4—42no—
655C4—44no—
656C4—1224no—
657C4—1224no—
658C4—1228no—
659C4—1222no—
660C4—2441no—
661C4—2442no—
662C4—2441no—
663C4—2441no—
664C4—2442no—
665C4#1684023914yes—
666C4#1684024114yes—
667C2—44no—
668C2C2×#1684023911yescharacteristic
669C2C2×#1684024111yescharacteristic
670C2#2145902511yescharacteristic
671C2#2146015411yescharacteristic
672C2#2146019611yescharacteristic
673C2#2146021211yescharacteristic
674C2#2146029511yescharacteristic
subgroupG/H
center, FrattiniC23C24×D61
commutatorC22×C61C24×C4
FittingC61×#2354C2
socle, cosocleC23×C61C25
prime radicalC22×D4×C61C23
hypercenter#2354D61
hypocenterC61#34484
LCSG ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ #2354

Complements

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

NH ≅ G/Ntype
2. #21448979C2semidirect
3. #21448911C2semidirect
4. #21448908C2semidirect
5. #21448846C2semidirect
7. #21448731C2semidirect
8. #21448690C2semidirect
9. #21448556C2semidirect
10. #21448497C2semidirect
11. #21448494C2semidirect
13. #21448435C2semidirect
14. #21448386C2semidirect
15. #21448293C2semidirect
18. #16840469C4semidirect
421. C61#34484semidirect

Maximal subgroups

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

subgroupindex
C2×#168404692
#214489792
#214489112
#214489082
#214488462
#214487672
#214487312
#214486902
#214485562
#214484972
#214484942
#214484562
#214484352
#214483862
#214482932
#214482842
C61×#23542
#3448461