Finite Group Databasebeta

← #29716586

Subgroups — #29716586

Order 15616.

1064 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×#16840471C211yesmaximal, characteristic
2C2×#16840046C211yesmaximal, characteristic, complemented
3C2×#16839859C211yesmaximal, characteristic
4#21451187C211yesmaximal, characteristic, complemented
5#21447159C211yesmaximal, characteristic, complemented
6#21447157C211yesmaximal, characteristic, complemented
7#21446971C211yesmaximal, characteristic, complemented
8#21446969C211yesmaximal, characteristic, complemented
9#21446790C211yesmaximal, characteristic, complemented
10#21446757C211yesmaximal, characteristic, complemented
11#21446153C211yesmaximal, characteristic
12#21446151C211yesmaximal, characteristic
13#21446078C211yesmaximal, characteristic
14#21445827C211yesmaximal, characteristic
15C61×#1864C211yesmaximal, characteristic
16C22×#11758200C2211yescharacteristic
17#16840471—24no—
18#16840471—24no—
19C2×#14519776C2211yescharacteristic
20C2×#14519769C2211yescharacteristic
21C2×#14519768C2211yescharacteristic
22C2×#14519756—21no—
23C2×#14519754—21no—
24C2×#14519754—21no—
25C2×C4×Dic122C2211yescharacteristic
26C2×#14519742C2211yescharacteristic
27C2×D4×Dic61C2211yescharacteristic
28C2×D4×Dic61C2211yescharacteristic
29C22×#11758196—21no—
30#16840046—24no—
31#16839859—24no—
32C2×#14519710C2211yescharacteristic
33C2×#14519708C2211yescharacteristic
34C2×#14519706C2211yescharacteristic
35C2×#14519702C2211yescharacteristic
36C2×#14519702C2211yescharacteristic
37C2×#14519700C2211yescharacteristic
38C2×#14519698C2211yescharacteristic
39C2×#14519687C2211yescharacteristic
40#16839703—21no—
41#16839682—21no—
42#16839668C2211yescharacteristic
43#16839667C2211yescharacteristic
44#16839658C2211yescharacteristic
45#16839648—21no—
46#16839644—21no—
47#16839627C2211yescharacteristic
48#16839525C2211yescharacteristic
49#16839525C2211yescharacteristic
50#16839524C2211yescharacteristic
51#16839524C2211yescharacteristic
52#16839524C2211yescharacteristic
53#16839517C2211yescharacteristic
54#16839516C2211yescharacteristic
55#16839516C2211yescharacteristic
56C2×C4×D4×C61C2211yescharacteristic
57C2×#106×C61C2211yescharacteristic
58C2×#102×C61C2211yescharacteristic
59C2×#102×C61C2211yescharacteristic
60C61×#389C2211yescharacteristic
61C61×#340C2211yescharacteristic
62C61×#336C2211yescharacteristic
63C22×Dic122—21no—
64C22×Dic122D411yescharacteristic
65C22×Dic122C2311yescharacteristic
66C23×Dic61—21no—
67C23×Dic61—21no—
68C2×#11758200—22no—
69C2×#11758200—22no—
70C2×#11758200Q814yescomplemented
71C22×#92418C2311yescharacteristic
72C22×#92418C2311yescharacteristic
73#14519776—22no—
74#14519776D414yes—
75#14519769—22no—
76#14519769—22no—
77#14519768—22no—
78#14519768—22no—
79#14519756—28no—
80#14519754—28no—
81#14519754—28no—
82C4×Dic122—22no—
83C4×Dic122—22no—
84#14519742—22no—
85#14519742—22no—
86D4×Dic61—22no—
87D4×Dic61—22no—
88D4×Dic61—22no—
89D4×Dic61D414yes—
90C2×C4⋊Dic61—21no—
91C2×C4⋊Dic61D411yescharacteristic
92C2×C4⋊Dic61C2311yescharacteristic
93C2×C4⋊Dic61C2311yescharacteristic
94C2×C4⋊Dic61C2311yescharacteristic
95C2×#11758196—28no—
96C2×#11758196—21no—
97C2×#11758196—21no—
98C2×#11758196—21no—
99C2×#11758196—21no—
100C2×#11758196—21no—
101C2×#11758196—21no—
102C2×#11758196D411yescharacteristic
103C2×#11758196D411yescharacteristic
104C2×C22⋊Dic61—21no—
105C2×C22⋊Dic61—21no—
106C2×C4×Dic61—21no—
107C2×C4×Dic61C2311yescharacteristic
108C2×C4×Dic61C2311yescharacteristic
109#14519710—22no—
110#14519708—22no—
111#14519706—22no—
112#14519702—22no—
113#14519702—22no—
114#14519700—22no—
115#14519698—22no—
116#14519687—22no—
117#14519684—21no—
118#14519684—21no—
119#14519684—21no—
120#14519684—21no—
121#14519684—21no—
122#14519684—21no—
123#14519684—21no—
124C22×D4×C61C2311yescharacteristic
125C23×C4×C61—21no—
126C22×C8×C61C2311yescharacteristic
127C22×C8×C61C2311yescharacteristic
128C4×D4×C61—22no—
129C4×D4×C61—22no—
130C2×C4⋊C4×C61—21no—
131C2×C4⋊C4×C61C2311yescharacteristic
132C2×C4⋊C4×C61C2311yescharacteristic
133C2×C4⋊C4×C61C2311yescharacteristic
134C2×C22⋊C4×C61—21no—
135C2×C42×C61C2311yescharacteristic
136#106×C61—22no—
137#102×C61—22no—
138#102×C61—22no—
139#95×C61—21no—
140C24×C61—21no—
141C2×D4×C61—21no—
142C2×D4×C61—21no—
143C2×D4×C61—21no—
144C2×D4×C61C2×D412yes—
145C2×D4×C61C2×Q812yes—
146C2×D4×C61#4112yes—
147C22×C4×C61—21no—
148C22×C4×C61—24no—
149C22×C4×C61—21no—
150C22×C4×C61—21no—
151C22×C4×C61—81no—
152C22×C4×C61—81no—
153C22×C4×C61C2×D411yescharacteristic
154C22×C4×C61C2×D411yescharacteristic
155C22×C4×C61C2411yescharacteristic
156C22×Dic61—21no—
157C22×Dic61—21no—
158C22×Dic61—21no—
159C22×Dic61—24no—
160C22×Dic61—24no—
161C22×Dic61—21no—
162C22×Dic61—42no—
163C22×Dic61—42no—
164C22×Dic61—81no—
165C22×Dic61—81no—
166C22×Dic61C2×D411yescharacteristic
167C22×Dic61C2×D411yescharacteristic
168C22×Dic61C2×D411yescharacteristic
169C22×Dic61C2×D411yescharacteristic
170C2×Dic122—21no—
171C2×Dic122—21no—
172C2×Dic122—21no—
173C2×Dic122—21no—
174C2×Dic122—21no—
175C2×Dic122—22no—
176C2×Dic122—24no—
177C2×Dic122—41no—
178C2×Dic122—41no—
179C2×Dic122—81no—
180C2×Dic122—81no—
181C2×Dic122C2×Q812yes—
182C2×C8×C61—21no—
183C2×C8×C61—21no—
184C2×C8×C61—21no—
185C2×C8×C61—21no—
186C2×C8×C61—21no—
187C2×C8×C61—21no—
188C4⋊C4×C61—21no—
189C4⋊C4×C61—21no—
190C4⋊C4×C61—21no—
191C4⋊C4×C61—21no—
192C4⋊C4×C61—21no—
193C4⋊C4×C61—42no—
194C4⋊C4×C61C2×D412yes—
195C22⋊C4×C61—24no—
196C42×C61—21no—
197C42×C61—21no—
198C22⋊Dic61—24no—
199C22⋊Dic61—24no—
200#11758200—28no—
201C4⋊Dic61—24no—
202C4⋊Dic61—21no—
203C4⋊Dic61—21no—
204C4⋊Dic61—22no—
205C4⋊Dic61—21no—
206C4⋊Dic61—21no—
207C4⋊Dic61C2×D412yes—
208C4⋊Dic61C2×D412yes—
209#11758196—24no—
210#11758196—24no—
211#11758196—24no—
212#11758196—22no—
213#11758196—24no—
214#11758196—22no—
215#11758196—42no—
216#11758196—42no—
217C4×Dic61—21no—
218C4×Dic61—21no—
219C4×Dic61—42no—
220C4×Dic61C2×D412yes—
221C4×Dic61C2×D412yes—
222C2×#92418—21no—
223C2×#92418—21no—
224C2×#92418—21no—
225C2×#92418—21no—
226C2×#92418—21no—
227C2×#92418C2×Q812yes—
228C23×C61—22no—
229C23×C61—22no—
230C23×C61—22no—
231C23×C61—41no—
232C23×C61—41no—
233C23×C61—41no—
234C23×C61—41no—
235C23×C61C22×D411yescharacteristic
236D4×C61—22no—
237D4×C61—22no—
238D4×C61—22no—
239D4×C61C22⋊Q814yes—
240C2×C4×C61—21no—
241C2×C4×C61—21no—
242C2×C4×C61—22no—
243C2×C4×C61—21no—
244C2×C4×C61—21no—
245C2×C4×C61—21no—
246C2×C4×C61—21no—
247C2×C4×C61—41no—
248C2×C4×C61—41no—
249C2×C4×C61—41no—
250C2×C4×C61—41no—
251C2×C4×C61—41no—
252C2×C4×C61—41no—
253C2×C4×C61—41no—
254C2×C4×C61—41no—
255C2×C4×C61—84no—
256C2×C4×C61C22×D411yescharacteristic
257C2×C4×C61C22×D411yescharacteristic
258C2×C4×C61C22×Q811yescharacteristic
259C2×C4×C61C2×#4111yescharacteristic
260C2×C4×C612−1+411yescharacteristic
261C2×C4×C612−1+411yescharacteristic
262C2×Dic61—21no—
263C2×Dic61—21no—
264C2×Dic61—21no—
265C2×Dic61—21no—
266C2×Dic61—21no—
267C2×Dic61—21no—
268C2×Dic61—22no—
269C2×Dic61—22no—
270C2×Dic61—22no—
271C2×Dic61—21no—
272C2×Dic61—21no—
273C2×Dic61—21no—
274C2×Dic61—21no—
275C2×Dic61—48no—
276C2×Dic61—41no—
277C2×Dic61—41no—
278C2×Dic61—41no—
279C2×Dic61—41no—
280C2×Dic61—41no—
281C2×Dic61—41no—
282C2×Dic61—41no—
283C2×Dic61—41no—
284C2×Dic61—41no—
285C2×Dic61—84no—
286C2×Dic61#12012yes—
287C2×Dic61#12012yes—
288Dic122—22no—
289Dic122—22no—
290Dic122—44no—
291Dic122—41no—
292Dic122—41no—
293Dic122—84no—
294C8×C61—41no—
295C8×C61—41no—
296#92418—22no—
297#92418—41no—
298#24343—611nomaximal, Sylow
299C22×C61—24no—
300C22×C61—42no—
301C22×C61—42no—
302C22×C61—42no—
303C22×C61—42no—
304C22×C61—42no—
305C22×C61—42no—
306C22×C61C2×#12011yescharacteristic
307C22×C61#54811yescharacteristic
308C22×C61D4×Q811yescharacteristic
309C22×C61#57511yescharacteristic
310C22×C61#57511yescharacteristic
311C22×C61#57711yescharacteristic
312C22×C61#57711yescharacteristic
313C4×C61—41no—
314C4×C61—42no—
315C4×C61—41no—
316C4×C61—81no—
317C4×C61—81no—
318C4×C61C2×C22⋊Q811yescharacteristic
319C4×C61#53611yescharacteristic
320C4×C61#54811yescharacteristic
321C4×C61D4×Q811yescharacteristic
322Dic61—22no—
323Dic61—22no—
324Dic61—42no—
325Dic61—41no—
326Dic61—42no—
327Dic61—42no—
328Dic61—41no—
329Dic61—81no—
330C2×#554—611no—
331C2×#475—611no—
332C2×#440—611no—
333#2581—611no—
334#2059—611no—
335#2059—611no—
336#2026—611no—
337#2026—611no—
338#1993—611no—
339#1985—611no—
340#1877—611no—
341#1877—611no—
342#1864—611no—
343#1864D6111yescharacteristic
344#1804—611no—
345C2×C61—44no—
346C2×C61#240911yescharacteristic
347C2×C61#300511yescharacteristic
348C2×C61#300911yescharacteristic
349C2×C61#337611yescharacteristic
350C2×C61#337611yescharacteristic
351C2×C61#338811yescharacteristic
352C2×C61#338811yescharacteristic
353C22×SD16—611no—
354#554—1224no—
355#554—1224no—
356C2×C4⋊Q8—611no—
357C2×#125—611no—
358C2×#125—611no—
359C2×C22⋊Q8—1221no—
360C2×C22⋊Q8—1221no—
361C2×C22⋊Q8—1221no—
362C2×C4×Q8—611no—
363C2×C4×Q8—611no—
364C2×C4×D4—611no—
365C2×C4×D4—611no—
366C2×C4×D4C2×D6111yescharacteristic
367C22×C4⋊C4—1221no—
368#475—1224no—
369#440—1224no—
370C2×#107—611no—
371C2×#106—611no—
372C2×#106C2×D6111yescharacteristic
373C2×C4⋊C8—611no—
374C2×#103—611no—
375C2×#103—611no—
376C2×#103—611no—
377C2×#102—611no—
378C2×#102C2×D6111yescharacteristic
379C2×#102C2×D6111yescharacteristic
380C2×#97—611no—
381#400—1221no—
382#395—1221no—
383#391—611no—
384#391—611no—
385#389—611no—
386#389C2×D6111yescharacteristic
387#387—1221no—
388#387—1221no—
389#382—611no—
390#340—611no—
391#340—611no—
392#340—611no—
393#340—611no—
394#340—611no—
395#340C2×D6111yescharacteristic
396#336—611no—
397#336—611no—
398#336—611no—
399#336C2×D6111yescharacteristic
400C61#2434311yescharacteristic, complemented, Sylow
401C22×Q8—611no—
402C22×Q8—611no—
403C22×Q8—1221no—
404C22×D4C22×D6111yescharacteristic
405C23×C4—21no—
406C23×C4—1221no—
407C23×C4—1221no—
408C2×SD16—614no—
409C2×SD16—1222no—
410C2×SD16—1222no—
411C22×C8—611no—
412C22×C8—611no—
413C22×C8C22×D6111yescharacteristic
414C22×C8C22×D6111yescharacteristic
415C4⋊Q8—614no—
416C4⋊Q8—1222no—
417#125—1222no—
418#125—1222no—
419#125—1222no—
420#125—1222no—
421C22⋊Q8—1228no—
422C22⋊Q8—1228no—
423C22⋊Q8—1228no—
424C4×Q8—1222no—
425C4×Q8—1222no—
426C4×Q8—1222no—
427C4×Q8—1222no—
428C4×D4—22no—
429C4×D4—22no—
430C4×D4—614no—
431C4×D4—1222no—
432C4×D4—1222no—
433C4×D4—1222no—
434C2×C4⋊C4—21no—
435C2×C4⋊C4—611no—
436C2×C4⋊C4—611no—
437C2×C4⋊C4—611no—
438C2×C4⋊C4—611no—
439C2×C4⋊C4—611no—
440C2×C4⋊C4—611no—
441C2×C4⋊C4—1221no—
442C2×C4⋊C4—1221no—
443C2×C4⋊C4—1221no—
444C2×C4⋊C4—1228no—
445C2×C4⋊C4—1221no—
446C2×C4⋊C4—1221no—
447C2×C4⋊C4—1221no—
448C2×C4⋊C4—1221no—
449C2×C4⋊C4C22×D6111yescharacteristic
450C2×C4⋊C4C22×D6111yescharacteristic
451C2×C4⋊C4C22×D6111yescharacteristic
452C2×C22⋊C4—21no—
453C2×C22⋊C4—1221no—
454C2×C22⋊C4—1221no—
455C2×C42—611no—
456C2×C42—611no—
457C2×C42—1221no—
458C2×C42C22×D6111yescharacteristic
459#107—1222no—
460#106—22no—
461#106—1222no—
462C4⋊C8—1222no—
463#103—1222no—
464#103—1222no—
465#103—1222no—
466#102—22no—
467#102—22no—
468#102—1222no—
469#97—1222no—
470#95—21no—
471#95—1221no—
472#95—1221no—
473#95—1221no—
474#95—1221no—
475#95—1221no—
476#95—1221no—
477#95—1221no—
478C24—21no—
479C2×Q8—612no—
480C2×Q8—1224no—
481C2×Q8—1222no—
482C2×Q8—1221no—
483C2×Q8—1221no—
484C2×Q8—1221no—
485C2×Q8—1221no—
486C2×Q8—1221no—
487C2×Q8—4881no—
488C2×Q8—4881no—
489C2×Q8—2441no—
490C2×Q8—2441no—
491C2×D4—21no—
492C2×D4—21no—
493C2×D4—21no—
494C2×D4#1175822212yes—
495C2×D4D4×D6112yes—
496C2×D4Q8×D6112yes—
497C22×C4—21no—
498C22×C4—24no—
499C22×C4—21no—
500C22×C4—21no—
501C22×C4—81no—
502C22×C4—81no—
503C22×C4—611no—
504C22×C4—611no—
505C22×C4—611no—
506C22×C4—611no—
507C22×C4—1224no—
508C22×C4—1224no—
509C22×C4—1221no—
510C22×C4—1221no—
511C22×C4—1221no—
512C22×C4—1221no—
513C22×C4—4881no—
514C22×C4—4881no—
515C22×C4—2442no—
516C22×C4—2442no—
517C22×C4D4×D6111yescharacteristic
518C22×C4D4×D6111yescharacteristic
519C22×C4C23×D6111yescharacteristic
520SD16—1228no—
521C2×C8—21no—
522C2×C8—21no—
523C2×C8—21no—
524C2×C8—21no—
525C2×C8—21no—
526C2×C8—21no—
527C2×C8—612no—
528C2×C8—1221no—
529C2×C8—1221no—
530C2×C8—1221no—
531C2×C8—1221no—
532C2×C8—1221no—
533C4⋊C4—21no—
534C4⋊C4—21no—
535C4⋊C4—21no—
536C4⋊C4—21no—
537C4⋊C4—21no—
538C4⋊C4—42no—
539C4⋊C4—612no—
540C4⋊C4—612no—
541C4⋊C4—1222no—
542C4⋊C4—1222no—
543C4⋊C4—1222no—
544C4⋊C4—1224no—
545C4⋊C4—1224no—
546C4⋊C4—1224no—
547C4⋊C4—1224no—
548C4⋊C4—1224no—
549C4⋊C4—1221no—
550C4⋊C4—1221no—
551C4⋊C4—1221no—
552C4⋊C4—1221no—
553C4⋊C4—2442no—
554C4⋊C4—2442no—
555C4⋊C4D4×D6112yes—
556C22⋊C4—24no—
557C22⋊C4—1224no—
558C22⋊C4—1224no—
559C42—21no—
560C42—21no—
561C42—612no—
562C42—612no—
563C42—1221no—
564C42—1221no—
565C42—2442no—
566C23—22no—
567C23—22no—
568C23—22no—
569C23—41no—
570C23—41no—
571C23—41no—
572C23—41no—
573C23C2×D4×D6111yescharacteristic
574Q8—1222no—
575Q8—1222no—
576Q8—4884no—
577Q8—2444no—
578Q8—2441no—
579Q8—2441no—
580D4—22no—
581D4—22no—
582D4—22no—
583D4#1451975514yes—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—22no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—84no—
600C2×C4—612no—
601C2×C4—612no—
602C2×C4—1221no—
603C2×C4—1221no—
604C2×C4—1221no—
605C2×C4—1221no—
606C2×C4—1221no—
607C2×C4—1221no—
608C2×C4—1222no—
609C2×C4—1222no—
610C2×C4—1221no—
611C2×C4—1221no—
612C2×C4—1221no—
613C2×C4—1221no—
614C2×C4—1222no—
615C2×C4—4884no—
616C2×C4—2448no—
617C2×C4—2441no—
618C2×C4—2441no—
619C2×C4—2441no—
620C2×C4—2441no—
621C2×C4—2441no—
622C2×C4—2441no—
623C2×C4—2441no—
624C2×C4—2441no—
625C2×C4—2441no—
626C2×C4C2×D4×D6111yescharacteristic
627C2×C4C2×D4×D6111yescharacteristic
628C2×C4C2×Q8×D6111yescharacteristic
629C2×C4C2×#1175822211yescharacteristic
630C2×C4#1451982411yescharacteristic
631C2×C4#1451982411yescharacteristic
632C8—41no—
633C8—41no—
634C8—1222no—
635C8—2441no—
636C22—24no—
637C22—42no—
638C22—42no—
639C22—42no—
640C22—42no—
641C22—42no—
642C22—42no—
643C22#120×D6111yescharacteristic
644C22#1684042111yescharacteristic
645C22#1684042511yescharacteristic
646C22#1684055811yescharacteristic
647C22#1684055911yescharacteristic
648C22#1684056811yescharacteristic
649C22#1684057211yescharacteristic
650C4—41no—
651C4—41no—
652C4—42no—
653C4—81no—
654C4—81no—
655C4—1222no—
656C4—1222no—
657C4—4881no—
658C4—2441no—
659C4—2442no—
660C4—2442no—
661C4—2441no—
662C4—2442no—
663C4C2×#1451975511yescharacteristic
664C4#1684033411yescharacteristic
665C4#1684042111yescharacteristic
666C4#1684042511yescharacteristic
667C2—44no—
668C2#2144918311yescharacteristic
669C2#2145505711yescharacteristic
670C2#2145509111yescharacteristic
671C2#2145864211yescharacteristic
672C2#2145864711yescharacteristic
673C2#2145875711yescharacteristic
674C2#2145876811yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1864C2
socleC23×C61C22×D4
prime radicalC22×D4×C61C23
hypercenter#1864D61
hypocenterC61#24343
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1864

Complements

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

NH ≅ G/Ntype
2. C2×#16840046C2semidirect
4. #21451187C2semidirect
5. #21447159C2semidirect
6. #21447157C2semidirect
7. #21446971C2semidirect
8. #21446969C2semidirect
9. #21446790C2semidirect
10. #21446757C2semidirect
70. C2×#11758200Q8semidirect
400. C61#24343semidirect

Maximal subgroups

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

subgroupindex
C2×#168404712
C2×#168400462
C2×#168398592
#214511872
#214471592
#214471572
#214469712
#214469692
#214467902
#214467572
#214461532
#214461512
#214460782
#214458272
C61×#18642
#2434361