Finite Group Databasebeta

← #29716674

Subgroups — #29716674

Order 15616.

1462 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21454103C212yesmaximal, complemented
2C4×#14519745C211yesmaximal, characteristic
3#21447171C212yesmaximal, complemented
4#21446982C212yesmaximal, complemented
5#21446805C212yesmaximal, complemented
6#21446227C212yesmaximal
7#21446100C212yesmaximal, complemented
8#21445807C211yesmaximal, characteristic, complemented
9C61×#3501C211yesmaximal, characteristic, complemented
10C23×C4×D61C2211yescharacteristic
11C2×#14519753—22no—
12C2×#14519745C2211yescharacteristic
13C2×C4×C22⋊D61—21no—
14C2×#14519692C2212yes—
15#16839642C2211yescharacteristic
16#16839638—21no—
17#16839634C2211yescharacteristic
18#16839634C2214yes—
19#16839633—21no—
20#16839633—24no—
21#16839622C2212yes—
22#16839619—22no—
23#16839617C2214yes—
24#16839617C2211yescharacteristic
25C4×#11758198—21no—
26C4×C22⋊Dic61C2211yescharacteristic
27#16839529C2212yes—
28#16839529C2214yes—
29#16839529C2214yes—
30C61×#554—22no—
31C61×#554—24no—
32D4×Q8×C61—22no—
33C61×#544—21no—
34C61×#543—21no—
35C61×#543C2212yes—
36C61×#533—21no—
37C2×C22⋊Q8×C61—21no—
38C2×C22⋊Q8×C61C2212yes—
39C61×#518C2212yes—
40C2×C4×D4×C61C2211yescharacteristic
41C24×D61C2311yescharacteristic
42C22×C22⋊D61—21no—
43C23×Dic61C2311yescharacteristic
44C22×C4×D61—21no—
45C22×C4×D61—22no—
46C22×C4×D61—22no—
47C22×C4×D61—21no—
48C22×C4×D61—44no—
49#14519753—48no—
50#14519745—22no—
51#14519745Q814yescomplemented
52C4×C22⋊D61—44no—
53C2×C4⋊Dic61—22no—
54C2×#11758196—21no—
55C2×#11758196—24no—
56C2×C22⋊Dic61—21no—
57C2×C22⋊Dic61C2314yes—
58C2×C22⋊Dic61C2311yescharacteristic
59C2×C22⋊Dic61C2311yescharacteristic
60C2×#11758198—21no—
61C2×#11758198—21no—
62C2×#11758198—24no—
63C2×#11758198—21no—
64C2×#11758198—24no—
65C2×C4×Dic61—21no—
66#14519692—24no—
67#14519684—21no—
68#14519684—21no—
69#14519684—21no—
70#14519684—24no—
71#14519684—22no—
72C22×Q8×C61—21no—
73C22×Q8×C61D412yescomplemented
74C22×D4×C61C2311yescharacteristic
75C23×C4×C61D411yescharacteristic
76C23×C4×C61C2311yescharacteristic
77C4⋊Q8×C61—21no—
78C4⋊Q8×C61—21no—
79C4⋊Q8×C61—24no—
80#125×C61—22no—
81#125×C61—24no—
82C22⋊Q8×C61—24no—
83C22⋊Q8×C61—24no—
84C22⋊Q8×C61—22no—
85C22⋊Q8×C61—22no—
86C22⋊Q8×C61—22no—
87C22⋊Q8×C61—22no—
88C22⋊Q8×C61—22no—
89C22⋊Q8×C61—22no—
90C22⋊Q8×C61—24no—
91C22⋊Q8×C61—24no—
92C22⋊Q8×C61—22no—
93C22⋊Q8×C61—22no—
94C4×Q8×C61—22no—
95C4×Q8×C61—22no—
96C4×D4×C61—22no—
97C4×D4×C61—24no—
98C4×D4×C61—22no—
99C4×D4×C61—24no—
100#117×C61—22no—
101C2×C4⋊C4×C61—21no—
102C2×C4⋊C4×C61—22no—
103C2×C4⋊C4×C61D411yescharacteristic
104C2×C4⋊C4×C61D412yescomplemented
105C2×C4⋊C4×C61D414yescomplemented
106C2×C22⋊C4×C61—22no—
107C2×C22⋊C4×C61D411yescharacteristic
108C2×C22⋊C4×C61C2311yescharacteristic
109C2×C22⋊C4×C61C2314yes—
110C2×C42×C61D411yescharacteristic
111C24×C61C2×D411yescharacteristic
112C24×C61C2411yescharacteristic
113C23×D61—21no—
114C23×D61—22no—
115C23×D61—21no—
116C23×D61—22no—
117C23×D61—42no—
118C23×D61—42no—
119C23×D61C2×Q812yescomplemented
120C2×Q8×C61—22no—
121C2×Q8×C61—22no—
122C2×Q8×C61—22no—
123C2×Q8×C61—22no—
124C2×Q8×C61—44no—
125C2×Q8×C61—82no—
126C2×D4×C61—22no—
127C2×D4×C61—22no—
128C2×D4×C61C2×Q812yes—
129C2×D4×C61#4112yes—
130C22×C4×C61—22no—
131C22×C4×C61—22no—
132C22×C4×C61—21no—
133C22×C4×C61—21no—
134C22×C4×C61—42no—
135C22×C4×C61—42no—
136C22×C4×C61C2×D411yescharacteristic
137C22×C4×C61C2×D412yes—
138C22×C4×C61C2×D411yescharacteristic
139C22×C4×C61C2×D411yescharacteristic
140C22×C4×C61C2×D411yescharacteristic
141C22×C4×C61C2×D414yes—
142C22×C4×C61C2×D414yes—
143C22×C4×C61C2×D411yescharacteristic
144C22×C4×C61C2×D412yes—
145C2×C22⋊D61—24no—
146C2×C22⋊D61—42no—
147C2×C22⋊D61—42no—
148C22×Dic61—21no—
149C22×Dic61—22no—
150C22×Dic61—21no—
151C22×Dic61—21no—
152C22×Dic61—24no—
153C22×Dic61—42no—
154C2×C4×D61—24no—
155C2×C4×D61—28no—
156C2×C4×D61—44no—
157C2×C4×D61—42no—
158C2×C4×D61—42no—
159C2×C4×D61—44no—
160C2×C4×D61—44no—
161C2×C4×D61—42no—
162C2×C4×D61—41no—
163C2×C4×D61—42no—
164C2×C4×D61—41no—
165C4⋊C4×C61—22no—
166C4⋊C4×C61—24no—
167C4⋊C4×C61—22no—
168C4⋊C4×C61—22no—
169C4⋊C4×C61—24no—
170C4⋊C4×C61—22no—
171C4⋊C4×C61—24no—
172C4⋊C4×C61—22no—
173C4⋊C4×C61—22no—
174C4⋊C4×C61—22no—
175C4⋊C4×C61—24no—
176C22⋊C4×C61—24no—
177C22⋊C4×C61—24no—
178C22⋊C4×C61—22no—
179C22⋊C4×C61—24no—
180C22⋊C4×C61—22no—
181C42×C61—24no—
182C42×C61—21no—
183C42×C61—21no—
184C22⋊Dic61—21no—
185C22⋊Dic61—24no—
186C22⋊Dic61—22no—
187C22⋊Dic61—24no—
188C22⋊Dic61—42no—
189C22⋊Dic61C2×Q812yescomplemented
190#11758198—24no—
191#11758198—48no—
192#11758198—42no—
193#11758198—42no—
194#11758198—48no—
195C4⋊Dic61—44no—
196#11758196—42no—
197#11758196—48no—
198C4×Dic61—42no—
199C23×C61—21no—
200C23×C61—21no—
201C23×C61—41no—
202C23×C61—41no—
203C23×C61—41no—
204C23×C61—41no—
205C23×C61C22⋊Q812yescomplemented
206C23×C61C22×D411yescharacteristic
207C23×C61C22×D412yes—
208C23×C61C22×Q811yescharacteristic
209C23×C61C2×#4111yescharacteristic
210C23×C612−1+412yes—
211C22×D61—24no—
212C22×D61—24no—
213C22×D61—22no—
214C22×D61—22no—
215C22×D61—22no—
216C22×D61—24no—
217C22×D61—44no—
218C22×D61—44no—
219C22×D61—48no—
220C22×D61—42no—
221C22×D61—41no—
222C22×D61—42no—
223C22×D61—41no—
224C22×D61—41no—
225C22×D61—41no—
226C22×D61—42no—
227Q8×C61—44no—
228Q8×C61—82no—
229Q8×C61—82no—
230Q8×C61—82no—
231D4×C61—24no—
232D4×C61—24no—
233C2×C4×C61—21no—
234C2×C4×C61—21no—
235C2×C4×C61—22no—
236C2×C4×C61—24no—
237C2×C4×C61—21no—
238C2×C4×C61—24no—
239C2×C4×C61—24no—
240C2×C4×C61—21no—
241C2×C4×C61—42no—
242C2×C4×C61—41no—
243C2×C4×C61—42no—
244C2×C4×C61—41no—
245C2×C4×C61—44no—
246C2×C4×C61—44no—
247C2×C4×C61—41no—
248C2×C4×C61—81no—
249C2×C4×C61—82no—
250C2×C4×C61—82no—
251C2×C4×C61#12011yescharacteristic
252C2×C4×C61#12012yes—
253C2×C4×C61#12011yescharacteristic
254C2×C4×C61C22⋊Q812yescomplemented
255C22⋊D61—48no—
256C2×Dic61—24no—
257C2×Dic61—22no—
258C2×Dic61—22no—
259C2×Dic61—41no—
260C2×Dic61—42no—
261C2×Dic61—41no—
262C2×Dic61—41no—
263C2×Dic61—41no—
264C2×Dic61—41no—
265C2×Dic61—44no—
266C2×Dic61—44no—
267C2×Dic61—41no—
268C2×Dic61—41no—
269C2×Dic61—44no—
270C4×D61—44no—
271C4×D61—48no—
272C4×D61—44no—
273#24350—611nomaximal, Sylow
274C22×C61—22no—
275C22×C61—21no—
276C22×C61—22no—
277C22×C61—21no—
278C22×C61—22no—
279C22×C61—41no—
280C22×C61—41no—
281C22×C61—42no—
282C22×C61—42no—
283C22×C61—41no—
284C22×C61—41no—
285C22×C61—81no—
286C22×C61C2×#12011yescharacteristic
287C22×C61C2×C22⋊Q811yescharacteristic
288C22×C61#53611yescharacteristic
289C22×C61#54812yes—
290C22×C61D4×Q812yes—
291C2×D61—24no—
292C2×D61—28no—
293C2×D61—42no—
294C2×D61—44no—
295C2×D61—44no—
296C2×D61—42no—
297C2×D61—42no—
298C2×D61—44no—
299C2×D61—42no—
300C2×D61—42no—
301C2×D61—42no—
302C4×C61—22no—
303C4×C61—21no—
304C4×C61—22no—
305C4×C61—41no—
306C4×C61—44no—
307C4×C61—44no—
308C4×C61—41no—
309C4×C61B(4,𝔽2)12yes—
310Dic61—42no—
311Dic61—42no—
312Dic61—84no—
313Dic61—81no—
314#3501D6111yescharacteristic, complemented
315#2848—612no—
316C4×#120—611no—
317#2062—612no—
318#2029—612no—
319#1996—612no—
320#1889—612no—
321#1867—612no—
322#1799—611no—
323C2×C61—21no—
324C2×C61—22no—
325C2×C61—21no—
326C2×C61—22no—
327C2×C61—42no—
328C2×C61#240911yescharacteristic
329C2×C61C2×B(4,𝔽2)11yescharacteristic
330C2×C61#302511yescharacteristic
331D61—44no—
332C24×C4—611no—
333#554—22no—
334#554—24no—
335D4×Q8—22no—
336#544—21no—
337#543—21no—
338#543C2×D6112yes—
339#533—21no—
340C2×C22⋊Q8—21no—
341C2×C22⋊Q8—1222no—
342C2×C22⋊Q8C2×D6112yes—
343C2×#120—611no—
344#518C2×D6112yes—
345C2×C4×D4—1221no—
346C2×C4×D4C2×D6111yescharacteristic
347C2×#99—612no—
348#386—611no—
349#386—1221no—
350#385—611no—
351#385—614no—
352#385—1221no—
353#385—1224no—
354#380—612no—
355#380—1222no—
356#379—614no—
357#379—611no—
358C4×C22⋊C4—611no—
359C4×C22⋊C4—1221no—
360#342—612no—
361#342—614no—
362#342—614no—
363C61#2435011yescharacteristic, complemented, Sylow
364C25—611no—
365C22×Q8—21no—
366C22×Q8C22⋊D6112yescomplemented
367C22×D4—1221no—
368C22×D4C22×D6111yescharacteristic
369C23×C4—611no—
370C23×C4—1221no—
371C23×C4—1222no—
372C23×C4—1222no—
373C23×C4—1221no—
374C23×C4—2444no—
375C23×C4C22⋊D6111yescharacteristic
376C23×C4C22×D6111yescharacteristic
377C4⋊Q8—21no—
378C4⋊Q8—21no—
379C4⋊Q8—24no—
380#125—22no—
381#125—24no—
382C22⋊Q8—22no—
383C22⋊Q8—22no—
384C22⋊Q8—24no—
385C22⋊Q8—24no—
386C22⋊Q8—22no—
387C22⋊Q8—22no—
388C22⋊Q8—22no—
389C22⋊Q8—22no—
390C22⋊Q8—22no—
391C22⋊Q8—22no—
392C22⋊Q8—24no—
393C22⋊Q8—24no—
394C22⋊Q8—2448no—
395#120—614no—
396#120—1222no—
397C4×Q8—22no—
398C4×Q8—22no—
399C4×D4—22no—
400C4×D4—22no—
401C4×D4—24no—
402C4×D4—24no—
403C4×D4—2444no—
404#117—22no—
405C2×C4⋊C4—21no—
406C2×C4⋊C4—22no—
407C2×C4⋊C4—1221no—
408C2×C4⋊C4—1222no—
409C2×C4⋊C4—1224no—
410C2×C4⋊C4C22⋊D6111yescharacteristic
411C2×C4⋊C4C22⋊D6112yescomplemented
412C2×C4⋊C4C22⋊D6114yescomplemented
413C2×C22⋊C4—22no—
414C2×C22⋊C4—611no—
415C2×C22⋊C4—611no—
416C2×C22⋊C4—614no—
417C2×C22⋊C4—1221no—
418C2×C22⋊C4—1221no—
419C2×C22⋊C4—1221no—
420C2×C22⋊C4—1224no—
421C2×C22⋊C4—1221no—
422C2×C22⋊C4—1224no—
423C2×C22⋊C4C22⋊D6111yescharacteristic
424C2×C22⋊C4C22×D6111yescharacteristic
425C2×C22⋊C4C22×D6114yes—
426C2×C42—1221no—
427C2×C42C22⋊D6111yescharacteristic
428#99—1224no—
429#95—1221no—
430#95—1221no—
431#95—1221no—
432#95—1222no—
433#95—1224no—
434C24—612no—
435C24—1222no—
436C24—1221no—
437C24—1221no—
438C24—1222no—
439C24—2442no—
440C24—2442no—
441C24C2×C22⋊D6111yescharacteristic
442C24C23×D6111yescharacteristic
443C2×Q8—22no—
444C2×Q8—22no—
445C2×Q8—22no—
446C2×Q8—22no—
447C2×Q8—44no—
448C2×Q8—82no—
449C2×D4—22no—
450C2×D4—22no—
451C2×D4—1224no—
452C2×D4—2442no—
453C2×D4—2442no—
454C2×D4Q8×D6112yes—
455C2×D4#1175822612yes—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—42no—
461C22×C4—42no—
462C22×C4—1224no—
463C22×C4—1228no—
464C22×C4—1221no—
465C22×C4—1221no—
466C22×C4—1222no—
467C22×C4—1221no—
468C22×C4—1224no—
469C22×C4—2442no—
470C22×C4—2444no—
471C22×C4—2442no—
472C22×C4—2444no—
473C22×C4—2444no—
474C22×C4—2442no—
475C22×C4—2442no—
476C22×C4—2441no—
477C22×C4—2442no—
478C22×C4—2441no—
479C22×C4C2×C22⋊D6111yescharacteristic
480C22×C4C2×C22⋊D6112yes—
481C22×C4C2×C22⋊D6111yescharacteristic
482C22×C4C2×C22⋊D6111yescharacteristic
483C22×C4C2×C22⋊D6114yes—
484C22×C4C2×C22⋊D6111yescharacteristic
485C22×C4C2×C22⋊D6114yes—
486C22×C4C2×C22⋊D6111yescharacteristic
487C22×C4C2×C22⋊D6112yes—
488C4⋊C4—22no—
489C4⋊C4—24no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—24no—
493C4⋊C4—22no—
494C4⋊C4—24no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—24no—
499C4⋊C4—2444no—
500C4⋊C4—2442no—
501C4⋊C4—2448no—
502C22⋊C4—24no—
503C22⋊C4—24no—
504C22⋊C4—22no—
505C22⋊C4—24no—
506C22⋊C4—22no—
507C22⋊C4—612no—
508C22⋊C4—1221no—
509C22⋊C4—1224no—
510C22⋊C4—1224no—
511C22⋊C4—1222no—
512C22⋊C4—1224no—
513C22⋊C4—2442no—
514C22⋊C4—2448no—
515C22⋊C4—2442no—
516C22⋊C4—2448no—
517C22⋊C4—2442no—
518C42—24no—
519C42—21no—
520C42—21no—
521C42—2442no—
522C23—21no—
523C23—21no—
524C23—41no—
525C23—41no—
526C23—41no—
527C23—41no—
528C23—1224no—
529C23—1224no—
530C23—1222no—
531C23—1222no—
532C23—1222no—
533C23—1224no—
534C23—2442no—
535C23—2441no—
536C23—2442no—
537C23—2441no—
538C23—2444no—
539C23—2444no—
540C23—2448no—
541C23—2441no—
542C23—2441no—
543C23—2442no—
544C23#1451975312yescomplemented
545C23C22×C22⋊D6111yescharacteristic
546C23C22×C22⋊D6112yes—
547C23C2×Q8×D6111yescharacteristic
548C23C2×#1175822611yescharacteristic
549C23#1451982512yes—
550Q8—44no—
551Q8—82no—
552Q8—82no—
553Q8—82no—
554D4—24no—
555D4—24no—
556D4—2448no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—22no—
560C2×C4—24no—
561C2×C4—21no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—21no—
565C2×C4—42no—
566C2×C4—41no—
567C2×C4—42no—
568C2×C4—41no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—41no—
572C2×C4—81no—
573C2×C4—82no—
574C2×C4—82no—
575C2×C4—1222no—
576C2×C4—1224no—
577C2×C4—1222no—
578C2×C4—2444no—
579C2×C4—2448no—
580C2×C4—2441no—
581C2×C4—2444no—
582C2×C4—2444no—
583C2×C4—2441no—
584C2×C4—2441no—
585C2×C4—2444no—
586C2×C4—2441no—
587C2×C4—2442no—
588C2×C4—2441no—
589C2×C4—2441no—
590C2×C4—2441no—
591C2×C4—2444no—
592C2×C4#1451974611yescharacteristic
593C2×C4#1451974612yes—
594C2×C4#1451974611yescharacteristic
595C2×C4#1451975312yescomplemented
596C22—22no—
597C22—21no—
598C22—22no—
599C22—21no—
600C22—22no—
601C22—41no—
602C22—41no—
603C22—42no—
604C22—42no—
605C22—41no—
606C22—41no—
607C22—81no—
608C22—1224no—
609C22—1228no—
610C22—2444no—
611C22—2444no—
612C22—2442no—
613C22—2442no—
614C22—2442no—
615C22—2444no—
616C22—2442no—
617C22—2442no—
618C22—2442no—
619C22C2×#1451974611yescharacteristic
620C22C2×#1451975311yescharacteristic
621C22#1684033611yescharacteristic
622C22#1684042312yes—
623C22Q8×C22⋊D6112yes—
624C4—21no—
625C4—22no—
626C4—22no—
627C4—41no—
628C4—44no—
629C4—44no—
630C4—41no—
631C4—4884no—
632C4—4881no—
633C4—2442no—
634C4—2442no—
635C4#1683994912yes—
636C2—22no—
637C2—21no—
638C2—21no—
639C2—22no—
640C2—42no—
641C2—2444no—
642C2#2144917811yescharacteristic
643C2C2×#1683994911yescharacteristic
644C2#2145521111yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC24×C61C24
FrattiniC24C23×D61
FittingC61×#3501C2
socleC22×C61C2×#120
prime radicalC2×#14519745C22
hypercenter#3501D61
hypocenterC61#24350
LCSG ▷ C24×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3501

Complements

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

NH ≅ G/Ntype
1. #21454103C2semidirect
3. #21447171C2semidirect
4. #21446982C2semidirect
5. #21446805C2semidirect
7. #21446100C2semidirect
8. #21445807C2semidirect
9. C61×#3501C2semidirect
51. #14519745Q8semidirect
73. C22×Q8×C61D4semidirect
104. C2×C4⋊C4×C61D4semidirect
105. C2×C4⋊C4×C61D4semidirect
119. C23×D61C2×Q8semidirect
189. C22⋊Dic61C2×Q8semidirect
205. C23×C61C22⋊Q8semidirect
254. C2×C4×C61C22⋊Q8semidirect
314. #3501D61semidirect
363. C61#24350semidirect
366. C22×Q8C22⋊D61semidirect
411. C2×C4⋊C4C22⋊D61semidirect
412. C2×C4⋊C4C22⋊D61semidirect
544. C23#14519753semidirect
595. C2×C4#14519753semidirect

Maximal subgroups

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

subgroupindex
#214541032
C4×#145197452
#214471712
#214469822
#214468052
#214462272
#214461002
#214458072
C61×#35012
#2435061