Finite Group Databasebeta

← #29845217

Subgroups — #29845217

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#21448986C214yesmaximal, complemented
3#21448920C212yesmaximal, complemented
4#21448908C211yesmaximal, characteristic, complemented
5#21448844C212yesmaximal, complemented
6#21448768C211yesmaximal, characteristic
7#21448767C212yesmaximal
8C4⋊C4×Dic122C211yesmaximal, characteristic, complemented
9#21448686C212yesmaximal, complemented
10#21448559C212yesmaximal, complemented
11#21448501C212yesmaximal, complemented
12#21448498C214yesmaximal, complemented
13#21448456C212yesmaximal
14#21448435C212yesmaximal, complemented
15C4×#14519777C211yesmaximal, characteristic, complemented
16#21448296C211yesmaximal, characteristic, complemented
17C61×#2293C211yesmaximal, characteristic
18#16840469C4116yescomplemented
19C2×#14519777C2211yescharacteristic
20C2×#14519768C2212yes—
21C2×#14519756C2212yes—
22C2×#14519754C2214yes—
23C2×C4×Dic122C2212yes—
24C2×C4×Dic122C2212yes—
25C2×C4×Dic122C2211yescharacteristic
26C2×D4×Dic61C2212yes—
27C2×#14519736C2212yes—
28C2×#14519733C2214yes—
29C22×C4⋊Dic61C2212yes—
30#16839668C2212yes—
31#16839668C2211yescharacteristic
32#16839668C2211yescharacteristic
33#16839666C2214yes—
34#16839665C2212yes—
35#16839660C2212yes—
36#16839660C2212yes—
37#16839660C2212yes—
38#16839660C2211yescharacteristic
39#16839660C2211yescharacteristic
40#16839660C2212yes—
41#16839659C2214yes—
42#16839658C2212yes—
43#16839658C2214yes—
44#16839658C2214yes—
45#16839657C2214yes—
46#16839656C2214yes—
47#16839656C2214yes—
48#16839655C2214yes—
49#16839650C2214yes—
50#16839650C2214yes—
51#16839647C2214yes—
52#16839647C2214yes—
53#16839646C2212yes—
54#16839646C2212yes—
55#16839644C2214yes—
56#16839643C2214yes—
57#16839632C2211yescharacteristic
58#16839631C2212yes—
59#16839623C2214yes—
60#16839622C2212yes—
61#16839620C2214yes—
62#16839620C2214yes—
63C4⋊C4×Dic61C2212yes—
64C4⋊C4×Dic61C2212yes—
65C4⋊C4×Dic61C2212yes—
66C4×#11758196C2212yes—
67C4×C4⋊Dic61C2212yes—
68C4×C4⋊Dic61C2212yes—
69C4×C4⋊Dic61C2211yescharacteristic
70C4×C4⋊Dic61C2212yes—
71#16839613C2214yes—
72#16839608C2212yes—
73C2×C4×D4×C61C2211yescharacteristic
74C2×#117×C61C2212yes—
75C61×#390C2211yescharacteristic
76C61×#390C2211yescharacteristic
77C61×#389C2211yescharacteristic
78C61×#389C2211yescharacteristic
79C61×#388C2212yes—
80C61×#388C2212yes—
81C4×C4⋊C4×C61C2211yescharacteristic
82C4×C22⋊C4×C61C2212yes—
83C43×C61C2211yescharacteristic
84C22×Dic122C2311yescharacteristic
85C22×Dic122C2312yes—
86C23×Dic61C2314yes—
87#14519777C2×C418yes—
88#14519768C2×C4116yes—
89#14519756C2×C4116yes—
90#14519754C2×C4132yes—
91C4×Dic122—44no—
92C4×Dic122—44no—
93C4×Dic122C2×C418yes—
94D4×Dic61C2×C4116yes—
95#14519736—44no—
96#14519733—48no—
97C2×C4⋊Dic61C2×C4116yes—
98C2×C4⋊Dic61C2311yescharacteristic
99C2×C4⋊Dic61C2312yes—
100C2×C4⋊Dic61C2311yescharacteristic
101C2×C4⋊Dic61C2312yes—
102C2×C4⋊Dic61C2312yes—
103C2×C4⋊Dic61C2311yescharacteristic
104C2×C4⋊Dic61C2314yes—
105C2×C4⋊Dic61C2312yes—
106C2×C4⋊Dic61C2312yes—
107C2×#11758196C2312yes—
108C2×#11758196C2314yes—
109C2×#11758196C2314yes—
110C2×#11758196C2314yes—
111C2×#11758196C2314yes—
112C2×#11758196C2312yes—
113C2×#11758196C2312yes—
114C2×#11758196C2314yes—
115C2×C22⋊Dic61C2314yes—
116C2×C22⋊Dic61C2314yes—
117C2×C22⋊Dic61C2314yes—
118C2×C4×Dic61C2314yes—
119C2×C4×Dic61C2312yes—
120C2×C4×Dic61C2314yes—
121C2×C4×Dic61C2314yes—
122C2×C4×Dic61C2314yes—
123C2×C4×Dic61C2312yes—
124#14519684C2312yes—
125#14519684C2312yes—
126#14519684C2312yes—
127#14519684C2312yes—
128#14519684C2312yes—
129#14519684C2314yes—
130#14519684C2314yes—
131#14519684C2314yes—
132#14519684C2312yes—
133#14519684C2312yes—
134#14519684C2314yes—
135#14519684C2314yes—
136#14519684C2314yes—
137C22×D4×C61C2311yescharacteristic
138C23×C4×C61C2312yes—
139C4×D4×C61C2×C418yes—
140#117×C61—44no—
141C2×C4⋊C4×C61C2311yescharacteristic
142C2×C4⋊C4×C61C2311yescharacteristic
143C2×C4⋊C4×C61C2312yes—
144C2×C4⋊C4×C61C2312yes—
145C2×C4⋊C4×C61C2311yescharacteristic
146C2×C22⋊C4×C61C2312yes—
147C2×C22⋊C4×C61C2312yes—
148C2×C22⋊C4×C61C2312yes—
149C2×C22⋊C4×C61C2312yes—
150C2×C22⋊C4×C61C2312yes—
151C2×C42×C61C2311yescharacteristic
152C2×C42×C61C2312yes—
153C2×C42×C61C2311yescharacteristic
154C2×C42×C61C2311yescharacteristic
155C2×C42×C61C2311yescharacteristic
156C2×C42×C61C2311yescharacteristic
157C2×C42×C61C2311yescharacteristic
158C2×C42×C61C2311yescharacteristic
159#95×C61C2312yes—
160#95×C61C2312yes—
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×C61C2411yescharacteristic
170C22×C4×C61C2412yes—
171C22×C4×C61C2412yes—
172C22×C4×C61C2411yescharacteristic
173C22×C4×C61C2411yescharacteristic
174C22×C4×C61C2411yescharacteristic
175C22×C4×C61C2412yes—
176C22×C4×C61C2411yescharacteristic
177C22×C4×C61C2411yescharacteristic
178C22×C4×C61C2411yescharacteristic
179C22×Dic61—44no—
180C22×Dic61—44no—
181C22×Dic61C22×C4116yes—
182C22×Dic61C2412yes—
183C22×Dic61C2412yes—
184C22×Dic61C2414yes—
185C22×Dic61C2412yes—
186C22×Dic61C2414yes—
187C22×Dic61C2412yes—
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—22no—
196C4⋊C4×C61—42no—
197C4⋊C4×C61—42no—
198C4⋊C4×C61C22×C414yes—
199C22⋊C4×C61—24no—
200C22⋊C4×C61—42no—
201C22⋊C4×C61—42no—
202C22⋊C4×C61—42no—
203C22⋊C4×C61C22×C418yes—
204C42×C61—22no—
205C42×C61—24no—
206C42×C61—22no—
207C42×C61—22no—
208C42×C61—22no—
209C42×C61—22no—
210C42×C61—41no—
211C42×C61C22×C414yes—
212C22⋊Dic61—44no—
213C22⋊Dic61—44no—
214C22⋊Dic61C22×C4116yes—
215C4⋊Dic61—22no—
216C4⋊Dic61—24no—
217C4⋊Dic61—24no—
218C4⋊Dic61—22no—
219C4⋊Dic61—42no—
220C4⋊Dic61C22×C414yes—
221C4⋊Dic61C22×C4116yes—
222C4⋊Dic61C22×C418yes—
223C4⋊Dic61C22×C418yes—
224#11758196—24no—
225#11758196—44no—
226#11758196—42no—
227#11758196—44no—
228#11758196—44no—
229#11758196—44no—
230#11758196C22×C418yes—
231#11758196C22×C4116yes—
232C4×Dic61—28no—
233C4×Dic61—24no—
234C4×Dic61—44no—
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—22no—
248C2×C4×C61—21no—
249C2×C4×C61—21no—
250C2×C4×C61—21no—
251C2×C4×C61—22no—
252C2×C4×C61—21no—
253C2×C4×C61—216no—
254C2×C4×C61—22no—
255C2×C4×C61—22no—
256C2×C4×C61—21no—
257C2×C4×C61—22no—
258C2×C4×C61—21no—
259C2×C4×C61—21no—
260C2×C4×C61—21no—
261C2×C4×C61—21no—
262C2×C4×C61—21no—
263C2×C4×C61—21no—
264C2×C4×C61—22no—
265C2×C4×C61—22no—
266C2×C4×C61—21no—
267C2×C4×C61—22no—
268C2×C4×C61—21no—
269C2×C4×C61C23×C414yes—
270C2×C4×C61C23×C412yes—
271C2×C4×C61C23×C412yes—
272C2×C4×C61C23×C412yes—
273C2×C4×C61H(5,𝔽2)12yes—
274C2×C4×C61H(5,𝔽2)12yes—
275C2×C4×C612−1+412yes—
276C2×C4×C612−1+412yes—
277C2×C4×C612−1+412yes—
278C2×Dic61—24no—
279C2×Dic61—22no—
280C2×Dic61—22no—
281C2×Dic61—22no—
282C2×Dic61—24no—
283C2×Dic61—22no—
284C2×Dic61—24no—
285C2×Dic61—22no—
286C2×Dic61—24no—
287C2×Dic61—22no—
288C2×Dic61—22no—
289C2×Dic61—22no—
290C2×Dic61—22no—
291C2×Dic61—24no—
292C2×Dic61—22no—
293C2×Dic61—416no—
294C2×Dic61C23×C414yes—
295C2×Dic61C23×C414yes—
296C2×Dic61C23×C418yes—
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—22no—
317C4×C61—42no—
318C4×C61—41no—
319C4×C61—42no—
320C4×C61—41no—
321C4×C61—41no—
322C4×C61#51914yes—
323C4×C61#52014yes—
324Dic61—24no—
325Dic61—28no—
326Dic61—24no—
327Dic61—42no—
328Dic61—42no—
329Dic61—44no—
330C2×#554—611no—
331#2378—614no—
332#2370—612no—
333#2369—611no—
334#2362—612no—
335#2354—612no—
336#2354—611no—
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#2293D6111yescharacteristic
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—611no—
361C2×C4×Q8—612no—
362C2×C4×Q8—612no—
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—612no—
374#390C2×D6111yescharacteristic
375#390C2×D6111yescharacteristic
376#389—614no—
377#389—614no—
378#389—612no—
379#389—614no—
380#389—612no—
381#389—612no—
382#389—611no—
383#389—612no—
384#389—612no—
385#389—611no—
386#389—614no—
387#389C2×D6111yescharacteristic
388#389C2×D6111yescharacteristic
389#388—614no—
390#388—614no—
391#388—614no—
392#388—614no—
393#388—614no—
394#388C2×D6112yes—
395#388C2×D6112yes—
396#387—614no—
397#387—612no—
398#387—612no—
399#387—614no—
400#387—614no—
401#387—614no—
402#384—612no—
403#384—611no—
404#381—614no—
405#380—612no—
406#380—614no—
407#380—614no—
408C4×C4⋊C4—612no—
409C4×C4⋊C4—612no—
410C4×C4⋊C4—612no—
411C4×C4⋊C4—612no—
412C4×C4⋊C4—612no—
413C4×C4⋊C4—612no—
414C4×C4⋊C4—611no—
415C4×C4⋊C4—612no—
416C4×C4⋊C4—614no—
417C4×C4⋊C4C2×D6111yescharacteristic
418C4×C22⋊C4C2×D6112yes—
419#376—612no—
420C43C2×D6111yescharacteristic
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—2448no—
438#117—2444no—
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—612no—
449C2×C4⋊C4—611no—
450C2×C4⋊C4—612no—
451C2×C4⋊C4—614no—
452C2×C4⋊C4—614no—
453C2×C4⋊C4—614no—
454C2×C4⋊C4—612no—
455C2×C4⋊C4—612no—
456C2×C4⋊C4—614no—
457C2×C4⋊C4C22×D6111yescharacteristic
458C2×C4⋊C4C22×D6111yescharacteristic
459C2×C4⋊C4C22×D6112yes—
460C2×C4⋊C4C22×D6112yes—
461C2×C4⋊C4C22×D6111yescharacteristic
462C2×C22⋊C4—614no—
463C2×C22⋊C4—614no—
464C2×C22⋊C4—614no—
465C2×C22⋊C4C22×D6112yes—
466C2×C22⋊C4C22×D6112yes—
467C2×C22⋊C4C22×D6112yes—
468C2×C22⋊C4C22×D6112yes—
469C2×C22⋊C4C22×D6112yes—
470C2×C42—612no—
471C2×C42—614no—
472C2×C42—612no—
473C2×C42—614no—
474C2×C42—614no—
475C2×C42—614no—
476C2×C42C22×D6111yescharacteristic
477C2×C42C22×D6112yes—
478C2×C42C22×D6111yescharacteristic
479C2×C42C22×D6111yescharacteristic
480C2×C42C22×D6111yescharacteristic
481C2×C42C22×D6111yescharacteristic
482C2×C42C22×D6111yescharacteristic
483C2×C42C22×D6111yescharacteristic
484#95—612no—
485#95—612no—
486#95—612no—
487#95—612no—
488#95—612no—
489#95—614no—
490#95—614no—
491#95—614no—
492#95—614no—
493#95—614no—
494#95—614no—
495#95—612no—
496#95—612no—
497#95C22×D6112yes—
498#95C22×D6112yes—
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—612no—
513C22×C4—614no—
514C22×C4—612no—
515C22×C4—6116no—
516C22×C4—612no—
517C22×C4—612no—
518C22×C4—614no—
519C22×C4—2444no—
520C22×C4—2444no—
521C22×C4C2×C4×D6118yes—
522C22×C4C23×D6111yescharacteristic
523C22×C4C23×D6111yescharacteristic
524C22×C4C23×D6112yes—
525C22×C4C23×D6111yescharacteristic
526C22×C4C23×D6112yes—
527C22×C4C23×D6112yes—
528C22×C4C23×D6111yescharacteristic
529C22×C4C23×D6111yescharacteristic
530C22×C4C23×D6111yescharacteristic
531C22×C4C23×D6111yescharacteristic
532C4⋊C4—22no—
533C4⋊C4—22no—
534C4⋊C4—42no—
535C4⋊C4—42no—
536C4⋊C4—614no—
537C4⋊C4—6116no—
538C4⋊C4—618no—
539C4⋊C4—618no—
540C4⋊C4—6116no—
541C4⋊C4—618no—
542C4⋊C4—1222no—
543C4⋊C4—1224no—
544C4⋊C4—1224no—
545C4⋊C4—1224no—
546C4⋊C4—1222no—
547C4⋊C4—2442no—
548C4⋊C4—2444no—
549C4⋊C4—2444no—
550C4⋊C4—2444no—
551C4⋊C4—2444no—
552C4⋊C4—2442no—
553C4⋊C4C2×C4×D6114yes—
554C22⋊C4—24no—
555C22⋊C4—42no—
556C22⋊C4—42no—
557C22⋊C4—42no—
558C22⋊C4—6116no—
559C22⋊C4—2444no—
560C22⋊C4—2444no—
561C22⋊C4C2×C4×D6118yes—
562C42—22no—
563C42—24no—
564C42—22no—
565C42—22no—
566C42—22no—
567C42—22no—
568C42—41no—
569C42—618no—
570C42—1228no—
571C42—1224no—
572C42—2444no—
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—22no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—22no—
592C2×C4—21no—
593C2×C4—216no—
594C2×C4—22no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—22no—
598C2×C4—21no—
599C2×C4—22no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—22no—
609C2×C4—614no—
610C2×C4—614no—
611C2×C4—618no—
612C2×C4—1222no—
613C2×C4—1224no—
614C2×C4—1222no—
615C2×C4—1222no—
616C2×C4—1222no—
617C2×C4—1224no—
618C2×C4—1222no—
619C2×C4—1224no—
620C2×C4—1224no—
621C2×C4—1222no—
622C2×C4—1222no—
623C2×C4—1222no—
624C2×C4—1222no—
625C2×C4—1224no—
626C2×C4—1222no—
627C2×C4—24416no—
628C2×C4C22×C4×D6112yes—
629C2×C4C22×C4×D6114yes—
630C2×C4C22×C4×D6112yes—
631C2×C4C22×C4×D6112yes—
632C2×C4#1451982212yes—
633C2×C4#1451982312yes—
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
645C22C2×#1451982211yescharacteristic
646C22C2×#1451982311yescharacteristic
647C22C2×#1451982411yescharacteristic
648C22C2×#1451982411yescharacteristic
649C22C2×#1451982411yescharacteristic
650C22#1684059811yescharacteristic
651C4—24no—
652C4—22no—
653C4—22no—
654C4—41no—
655C4—42no—
656C4—41no—
657C4—41no—
658C4—42no—
659C4—1224no—
660C4—1224no—
661C4—1228no—
662C4—2442no—
663C4—2442no—
664C4—2444no—
665C4#1684023914yes—
666C4#1684024114yes—
667C2—44no—
668C2C2×#1684023911yescharacteristic
669C2C2×#1684024111yescharacteristic
670C2C4×#1451982311yescharacteristic
671C2#2146015911yescharacteristic
672C2#2146020811yescharacteristic
673C2#2146021711yescharacteristic
674C2#2146030911yescharacteristic
subgroupG/H
center, FrattiniC23C24×D61
commutatorC22×C61C24×C4
FittingC61×#2293C2
socle, cosocleC23×C61C25
prime radicalC22×D4×C61C23
hypercenter#2293D61
hypocenterC61#34484
LCSG ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ #2293

Complements

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

NH ≅ G/Ntype
2. #21448986C2semidirect
3. #21448920C2semidirect
4. #21448908C2semidirect
5. #21448844C2semidirect
8. C4⋊C4×Dic122C2semidirect
9. #21448686C2semidirect
10. #21448559C2semidirect
11. #21448501C2semidirect
12. #21448498C2semidirect
14. #21448435C2semidirect
15. C4×#14519777C2semidirect
16. #21448296C2semidirect
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
#214489862
#214489202
#214489082
#214488442
#214487682
#214487672
C4⋊C4×Dic1222
#214486862
#214485592
#214485012
#214484982
#214484562
#214484352
C4×#145197772
#214482962
C61×#22932
#3448461