Finite Group Databasebeta

← #29707777

Subgroups — #29707777

Order 15616.

1530 conjugacy classes of subgroups.

The subgroup lattice has 678 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459148C211yesmaximal, characteristic
2C2×#16839736C211yesmaximal, characteristic, complemented
3C2×#16839735C211yesmaximal, characteristic, complemented
4#21448309C211yesmaximal, characteristic, complemented
5#21448302C211yesmaximal, characteristic, complemented
6#21447068C218yesmaximal, complemented
7#21445551C211yesmaximal, characteristic, complemented
8C61×#2297C211yesmaximal, characteristic, complemented
9C22×#11758222C2211yescharacteristic
10#16840340—22no—
11#16840336—21no—
12#16840333—21no—
13#16840329—21no—
14#16840326—21no—
15#16840321C2214yes—
16#16840308—22no—
17#16840273—22no—
18#16840269—22no—
19C2×#14519736C2211yescharacteristic
20C2×#14519736C2211yescharacteristic
21C2×#14519736C2211yescharacteristic
22#16839736—24no—
23#16839735—24no—
24C2×#14519692C2214yes—
25C2×#14519691C2214yes—
26#16839668—21no—
27#16839664—21no—
28#16839660—22no—
29#16839641C2214yes—
30#16839635C2214yes—
31C4×#11758196—22no—
32C4×#11758198—22no—
33#16839528C2214yes—
34C22×#41×C61C2211yescharacteristic
35C2×#117×C61C2211yescharacteristic
36C22×C42×C61C2211yescharacteristic
37C61×#391—21no—
38C61×#390—21no—
39C61×#386C2214yes—
40C4×C22⋊C4×C61—22no—
41C2×#11758222—22no—
42C2×#11758222—21no—
43C2×#11758222—21no—
44C2×#11758222—21no—
45C2×#11758222—21no—
46C2×#11758222—22no—
47C2×#11758222—22no—
48C2×#11758222—22no—
49C22×Dic122D411yescharacteristic
50C22×D244D411yescharacteristic
51C22×C22⋊D61C2312yes—
52C22×C4×D61D412yes—
53#14519775—22no—
54#14519773—22no—
55#14519766—22no—
56#14519764—22no—
57#14519760—24no—
58#14519758—24no—
59#14519756—24no—
60#14519753—24no—
61#14519751—24no—
62#14519749—28no—
63#14519748—24no—
64#14519746—24no—
65#14519745—24no—
66C4×C22⋊D61—28no—
67#14519736—21no—
68#14519736—21no—
69#14519736—21no—
70#14519736—21no—
71#14519736—22no—
72#14519736—22no—
73#14519736—22no—
74#14519736D414yescomplemented
75C2×C4⋊Dic61—21no—
76C2×C4⋊Dic61—21no—
77C2×C4⋊Dic61D412yes—
78C2×#11758196—24no—
79C2×C22⋊Dic61C2312yes—
80C2×C22⋊Dic61C2312yes—
81C2×C22⋊Dic61C2312yes—
82C2×#11758198—24no—
83C2×C4×Dic61—21no—
84C2×C4×Dic61—21no—
85C2×C4×Dic61D412yes—
86#14519692—28no—
87#14519691—28no—
88#14519684—24no—
89#14519684—24no—
90C2×#41×C61—22no—
91C2×#41×C61—21no—
92C2×#41×C61—21no—
93C2×#41×C61—21no—
94C2×#41×C61—21no—
95C2×#41×C61—22no—
96C2×#41×C61—22no—
97C2×#41×C61—22no—
98C22×Q8×C61D411yescharacteristic
99C22×D4×C61D411yescharacteristic
100C22×D4×C61C2312yes—
101C23×C4×C61D412yes—
102C23×C4×C61C2312yes—
103C23×C4×C61C2311yescharacteristic
104#117×C61—22no—
105#117×C61—22no—
106C2×C4⋊C4×C61—21no—
107C2×C22⋊C4×C61—24no—
108C2×C22⋊C4×C61C2312yes—
109C2×C42×C61—21no—
110C2×C42×C61—21no—
111C2×C42×C61—21no—
112C2×C42×C61—21no—
113C2×C42×C61—21no—
114C2×C42×C61—21no—
115C2×C42×C61—21no—
116#95×C61—24no—
117C24×C61C2×D412yes—
118C24×C61C2411yescharacteristic
119C23×D61C2×D412yes—
120#41×C61—42no—
121#41×C61—42no—
122#41×C61—42no—
123#41×C61—42no—
124#41×C61—42no—
125#41×C61—42no—
126#41×C61—42no—
127#41×C61—42no—
128C2×Q8×C61—21no—
129C2×Q8×C61—21no—
130C2×Q8×C61—22no—
131C2×Q8×C61—22no—
132C2×D4×C61—22no—
133C2×D4×C61—22no—
134C2×D4×C61—22no—
135C2×D4×C61—22no—
136C2×D4×C61—22no—
137C2×D4×C61—22no—
138C2×D4×C61—21no—
139C2×D4×C61—21no—
140C2×D4×C61—22no—
141C2×D4×C61—22no—
142C22×C4×C61—22no—
143C22×C4×C61—22no—
144C22×C4×C61—22no—
145C22×C4×C61—22no—
146C22×C4×C61—22no—
147C22×C4×C61—24no—
148C22×C4×C61—24no—
149C22×C4×C61—22no—
150C22×C4×C61—44no—
151C22×C4×C61C2×D412yes—
152C22×C4×C61C2×D411yescharacteristic
153C22×C4×C61C2×D411yescharacteristic
154C22×C4×C61C2×D411yescharacteristic
155C22×C4×C61C2×D411yescharacteristic
156C22×C4×C61C2×D411yescharacteristic
157C22×C4×C61C2×D411yescharacteristic
158C22×C4×C61C2×D412yes—
159C22×C4×C61#4112yes—
160C22×C4×C61#4112yes—
161C22×C4×C61#4112yes—
162C2×C22⋊D61—22no—
163C2×C22⋊D61—22no—
164C2×C22⋊D61—22no—
165C2×C22⋊D61—22no—
166C2×C22⋊D61—22no—
167C2×C22⋊D61—22no—
168C2×C22⋊D61—28no—
169C22×Dic61—22no—
170C22×Dic61—22no—
171C22×Dic61C2×D412yes—
172C22×Dic61C2×D414yes—
173#11758222—42no—
174#11758222—42no—
175#11758222—42no—
176#11758222—42no—
177#11758222—42no—
178#11758222—42no—
179#11758222—42no—
180#11758222—42no—
181C2×D244—21no—
182C2×D244—21no—
183C2×D244—22no—
184C2×D244—22no—
185C2×C4×D61—22no—
186C2×C4×D61—22no—
187C2×C4×D61—24no—
188C2×C4×D61—24no—
189C2×Dic122—21no—
190C2×Dic122—21no—
191C2×Dic122—22no—
192C2×Dic122—22no—
193C4⋊C4×C61—42no—
194C22⋊C4×C61—22no—
195C22⋊C4×C61—22no—
196C22⋊C4×C61—48no—
197C42×C61—41no—
198C42×C61—41no—
199C42×C61—41no—
200C42×C61—41no—
201C42×C61—42no—
202C22⋊Dic61—28no—
203C22⋊Dic61—22no—
204C22⋊Dic61—22no—
205C22⋊Dic61—22no—
206C22⋊Dic61—22no—
207C22⋊Dic61—22no—
208C22⋊Dic61C2×D414yes—
209#11758198—28no—
210#11758198—48no—
211C4⋊Dic61—22no—
212C4⋊Dic61—22no—
213C4⋊Dic61—24no—
214C4⋊Dic61—42no—
215#11758196—28no—
216#11758196—48no—
217C4×Dic61—22no—
218C4×Dic61—22no—
219C4×Dic61—24no—
220C4×Dic61—42no—
221C23×C61—22no—
222C23×C61—22no—
223C23×C61—22no—
224C23×C61—42no—
225C23×C61—44no—
226C23×C61C22×D411yescharacteristic
227C23×C61C22×D411yescharacteristic
228C23×C61C22×D411yescharacteristic
229C23×C61C22×D411yescharacteristic
230C23×C61C2×#4111yescharacteristic
231C23×C61C2×#4111yescharacteristic
232C23×C61C2×#4111yescharacteristic
233C22×D61—22no—
234C22×D61—22no—
235C22×D61—22no—
236C22×D61—44no—
237Q8×C61—42no—
238Q8×C61—42no—
239D4×C61—42no—
240D4×C61—42no—
241D4×C61—42no—
242D4×C61—42no—
243D4×C61—42no—
244D4×C61—42no—
245C2×C4×C61—24no—
246C2×C4×C61—24no—
247C2×C4×C61—24no—
248C2×C4×C61—22no—
249C2×C4×C61—22no—
250C2×C4×C61—22no—
251C2×C4×C61—42no—
252C2×C4×C61—42no—
253C2×C4×C61—42no—
254C2×C4×C61—42no—
255C2×C4×C61—44no—
256C2×C4×C61—44no—
257C2×C4×C61—44no—
258C2×C4×C61—42no—
259C2×C4×C61#12011yescharacteristic
260C2×C4×C61#12011yescharacteristic
261C2×C4×C61#12011yescharacteristic
262C2×C4×C61#12011yescharacteristic
263C2×C4×C61#12112yes—
264C2×C4×C61#12112yes—
265C2×C4×C61#12112yes—
266C2×C4×C61#12112yes—
267C2×C4×C61#12112yes—
268C2×C4×C61#12112yes—
269C2×C4×C61#12312yes—
270C2×C4×C61#12312yes—
271C2×C4×C61#12312yes—
272C2×C4×C61#12312yes—
273C2×C4×C61#12412yes—
274C2×C4×C61#12412yes—
275C22⋊D61—42no—
276C22⋊D61—42no—
277C22⋊D61—42no—
278C22⋊D61—42no—
279C22⋊D61—88no—
280C2×Dic61—24no—
281C2×Dic61—22no—
282C2×Dic61—24no—
283C2×Dic61—22no—
284C2×Dic61—22no—
285C2×Dic61—44no—
286C2×Dic61—44no—
287C2×Dic61—44no—
288C2×Dic61—42no—
289D244—42no—
290D244—42no—
291C4×D61—44no—
292C4×D61—44no—
293Dic122—42no—
294Dic122—42no—
295#23514—611nomaximal, Sylow
296C22×C61—22no—
297C22×C61—22no—
298C22×C61—22no—
299C22×C61—41no—
300C22×C61—41no—
301C22×C61—41no—
302C22×C61—41no—
303C22×C61—44no—
304C22×C61—42no—
305C22×C61—44no—
306C22×C61—82no—
307C22×C61C2×#12011yescharacteristic
308C22×C61C2×#12111yescharacteristic
309C22×C61C2×#12111yescharacteristic
310C22×C61C2×#12111yescharacteristic
311C22×C61C2×#12311yescharacteristic
312C22×C61C2×#12311yescharacteristic
313C22×C61C2×#12411yescharacteristic
314C2×D61—44no—
315C2×D61—42no—
316C2×D61—44no—
317C2×D61—82no—
318C4×C61—42no—
319C4×C61—44no—
320C4×C61—82no—
321C4×C61#39412yes—
322C4×C61#39412yes—
323C4×C61#39412yes—
324C4×C61#39412yes—
325Dic61—44no—
326Dic61—42no—
327Dic61—44no—
328Dic61—82no—
329#3449—611no—
330C2×#410—611no—
331C2×#410—611no—
332#2297—611no—
333#2297D6111yescharacteristic, complemented
334#2296—611no—
335#2044—618no—
336#1740—611no—
337C2×C61—21no—
338C2×C61—21no—
339C2×C61—21no—
340C2×C61—21no—
341C2×C61—21no—
342C2×C61—21no—
343C2×C61—82no—
344C2×C61C2×#39411yescharacteristic
345C2×C61#302512yes—
346D61—82no—
347C22×#41—611no—
348C22×#41C2×D6111yescharacteristic
349#537—1222no—
350#536—1221no—
351#536—1221no—
352#535—1221no—
353#535—1221no—
354#534—614no—
355#532—1222no—
356#525—1222no—
357#525—1222no—
358C2×#117—611no—
359C2×#117—611no—
360C2×#117—611no—
361C2×#117C2×D6111yescharacteristic
362C22×C42C2×D6111yescharacteristic
363#410—1224no—
364#410—1224no—
365C2×#99—614no—
366C2×#99—614no—
367#391—21no—
368#391—1221no—
369#390—21no—
370#390—1221no—
371#389—1222no—
372#386—614no—
373#386C2×D6114yes—
374#385—614no—
375C4×C4⋊C4—1222no—
376C4×C22⋊C4—22no—
377C4×C22⋊C4—1222no—
378#342—614no—
379C61#2351411yescharacteristic, complemented, Sylow
380C2×#41—22no—
381C2×#41—22no—
382C2×#41—21no—
383C2×#41—21no—
384C2×#41—21no—
385C2×#41—21no—
386C2×#41—22no—
387C2×#41—22no—
388C2×#41—1222no—
389C2×#41—1222no—
390C2×#41—1221no—
391C2×#41—1221no—
392C2×#41—1222no—
393C2×#41—1222no—
394C2×#41—1221no—
395C2×#41—1221no—
396C22×Q8—611no—
397C22×Q8C22⋊D6111yescharacteristic
398C22×D4—612no—
399C22×D4—611no—
400C22×D4C22⋊D6111yescharacteristic
401C22×D4C22×D6112yes—
402C23×C4—612no—
403C23×C4C22⋊D6112yes—
404C23×C4C22×D6112yes—
405C23×C4C22×D6111yescharacteristic
406C4⋊Q8—1222no—
407#127—1222no—
408#124—1222no—
409#124—1222no—
410#123—1224no—
411#123—1224no—
412C22⋊Q8—1224no—
413C22⋊Q8—1224no—
414#121—1224no—
415#121—1228no—
416#121—1224no—
417#120—1224no—
418#120—1224no—
419C4×D4—1228no—
420#117—22no—
421#117—22no—
422#117—614no—
423#117—1222no—
424#117—1221no—
425#117—1221no—
426#117—1221no—
427#117—1221no—
428#117—1222no—
429#117—1222no—
430C2×C4⋊C4—21no—
431C2×C4⋊C4—612no—
432C2×C4⋊C4—1221no—
433C2×C4⋊C4—1224no—
434C2×C4⋊C4—1221no—
435C2×C22⋊C4—24no—
436C2×C22⋊C4—612no—
437C2×C22⋊C4—612no—
438C2×C22⋊C4—612no—
439C2×C22⋊C4—1224no—
440C2×C22⋊C4C22×D6112yes—
441C2×C42—21no—
442C2×C42—21no—
443C2×C42—21no—
444C2×C42—21no—
445C2×C42—21no—
446C2×C42—21no—
447C2×C42—21no—
448C2×C42—612no—
449C2×C42—1221no—
450C2×C42—1221no—
451#99—1228no—
452#99—1228no—
453#95—24no—
454#95—1224no—
455#95—1224no—
456C24—612no—
457C24C2×C22⋊D6112yes—
458C24C23×D6111yescharacteristic
459#41—42no—
460#41—42no—
461#41—42no—
462#41—42no—
463#41—42no—
464#41—42no—
465#41—42no—
466#41—42no—
467#41—2442no—
468#41—2442no—
469#41—2442no—
470#41—2442no—
471#41—2442no—
472#41—2442no—
473#41—2442no—
474#41—2442no—
475C2×Q8—21no—
476C2×Q8—21no—
477C2×Q8—22no—
478C2×Q8—22no—
479C2×Q8—1221no—
480C2×Q8—1221no—
481C2×Q8—1222no—
482C2×Q8—1222no—
483C2×D4—22no—
484C2×D4—22no—
485C2×D4—22no—
486C2×D4—22no—
487C2×D4—22no—
488C2×D4—22no—
489C2×D4—21no—
490C2×D4—21no—
491C2×D4—22no—
492C2×D4—22no—
493C2×D4—1228no—
494C2×D4—1222no—
495C2×D4—1222no—
496C2×D4—1222no—
497C2×D4—1222no—
498C2×D4—1222no—
499C2×D4—1222no—
500C2×D4—1221no—
501C2×D4—1221no—
502C2×D4—1222no—
503C2×D4—1222no—
504C22×C4—22no—
505C22×C4—22no—
506C22×C4—22no—
507C22×C4—22no—
508C22×C4—22no—
509C22×C4—24no—
510C22×C4—24no—
511C22×C4—22no—
512C22×C4—44no—
513C22×C4—612no—
514C22×C4—614no—
515C22×C4—1222no—
516C22×C4—1222no—
517C22×C4—1222no—
518C22×C4—1224no—
519C22×C4—1224no—
520C22×C4—1222no—
521C22×C4#1175822212yes—
522C22×C4#1175822212yes—
523C22×C4D4×D6111yescharacteristic
524C22×C4D4×D6111yescharacteristic
525C22×C4D4×D6111yescharacteristic
526C22×C4D4×D6111yescharacteristic
527C22×C4D4×D6112yes—
528C22×C4#1175822412yes—
529C22×C4C2×C22⋊D6112yes—
530C22×C4C2×C22⋊D6111yescharacteristic
531C22×C4C2×C22⋊D6111yescharacteristic
532C4⋊C4—42no—
533C4⋊C4—1228no—
534C4⋊C4—1222no—
535C4⋊C4—1222no—
536C4⋊C4—1224no—
537C4⋊C4—2448no—
538C4⋊C4—2442no—
539C22⋊C4—22no—
540C22⋊C4—22no—
541C22⋊C4—48no—
542C22⋊C4—614no—
543C22⋊C4—1228no—
544C22⋊C4—1228no—
545C22⋊C4—1222no—
546C22⋊C4—1222no—
547C22⋊C4—1222no—
548C22⋊C4—1222no—
549C22⋊C4—1222no—
550C22⋊C4—2448no—
551C42—41no—
552C42—41no—
553C42—41no—
554C42—41no—
555C42—42no—
556C42—1222no—
557C42—1222no—
558C42—1224no—
559C42—2442no—
560C23—22no—
561C23—22no—
562C23—22no—
563C23—42no—
564C23—44no—
565C23—1222no—
566C23—1222no—
567C23—1222no—
568C23—2444no—
569C23C2×D4×D6111yescharacteristic
570C23C2×D4×D6111yescharacteristic
571C23C2×D4×D6111yescharacteristic
572C23C22×C22⋊D6111yescharacteristic
573C23C2×#1175822411yescharacteristic
574C23C2×#1175822211yescharacteristic
575C23C2×#1175822211yescharacteristic
576Q8—42no—
577Q8—42no—
578Q8—2442no—
579Q8—2442no—
580D4—42no—
581D4—42no—
582D4—42no—
583D4—42no—
584D4—42no—
585D4—42no—
586D4—4888no—
587D4—2442no—
588D4—2442no—
589D4—2442no—
590D4—2442no—
591D4—2442no—
592D4—2442no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—24no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—42no—
607C2×C4—1224no—
608C2×C4—1222no—
609C2×C4—1224no—
610C2×C4—1222no—
611C2×C4—1222no—
612C2×C4—2444no—
613C2×C4—2442no—
614C2×C4—2444no—
615C2×C4—2444no—
616C2×C4—2444no—
617C2×C4—2444no—
618C2×C4#1451974511yescharacteristic
619C2×C4#1451974511yescharacteristic
620C2×C4#1451974511yescharacteristic
621C2×C4#1451974511yescharacteristic
622C2×C4#1451974912yes—
623C2×C4#1451974912yes—
624C2×C4#1451975012yes—
625C2×C4#1451975012yes—
626C2×C4#1451975012yes—
627C2×C4#1451975012yes—
628C2×C4#1451975912yes—
629C2×C4#1451975912yes—
630C2×C4#1451976012yes—
631C2×C4#1451976012yes—
632C2×C4#1451976312yes—
633C2×C4#1451976312yes—
634C22—22no—
635C22—22no—
636C22—22no—
637C22—41no—
638C22—41no—
639C22—41no—
640C22—41no—
641C22—44no—
642C22—42no—
643C22—44no—
644C22—82no—
645C22—4882no—
646C22—2444no—
647C22—2442no—
648C22—2444no—
649C22C2×#1451974511yescharacteristic
650C22C2×#1451974911yescharacteristic
651C22C2×#1451975011yescharacteristic
652C22C2×#1451975011yescharacteristic
653C22C2×#1451975911yescharacteristic
654C22C2×#1451976011yescharacteristic
655C22C2×#1451976311yescharacteristic
656C4—42no—
657C4—44no—
658C4—82no—
659C4—4882no—
660C4—2444no—
661C4—2442no—
662C4—2444no—
663C4#1683967712yes—
664C4#1683967712yes—
665C4#1683967712yes—
666C4#1683967712yes—
667C2—21no—
668C2—21no—
669C2—21no—
670C2—21no—
671C2—21no—
672C2—21no—
673C2—82no—
674C2—4882no—
675C2C2×#1683967711yescharacteristic
676C2#2145521012yes—
subgroupG/H
centerC2×C4#14519745
commutator, cosocleC24×C61C24
FrattiniC24C23×D61
FittingC61×#2297C2
socleC22×C61C2×#120
prime radical#21459148C2
hypercenter#2297D61
hypocenterC61#23514
LCSG ▷ C24×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #2297

Complements

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

NH ≅ G/Ntype
2. C2×#16839736C2semidirect
3. C2×#16839735C2semidirect
4. #21448309C2semidirect
5. #21448302C2semidirect
6. #21447068C2semidirect
7. #21445551C2semidirect
8. C61×#2297C2semidirect
74. #14519736D4semidirect
333. #2297D61semidirect
379. C61#23514semidirect

Maximal subgroups

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

subgroupindex
#214591482
C2×#168397362
C2×#168397352
#214483092
#214483022
#214470682
#214455512
C61×#22972
#2351461