Finite Group Databasebeta

← #29716668

Subgroups — #29716668

Order 15616.

1072 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840473C211yesmaximal, characteristic
2C2×#16840068C211yesmaximal, characteristic, complemented
3C2×#16839861C211yesmaximal, characteristic
4#21451191C211yesmaximal, characteristic, complemented
5#21447155C211yesmaximal, characteristic, complemented
6#21446969C211yesmaximal, characteristic, complemented
7#21446957C211yesmaximal, characteristic, complemented
8#21446786C211yesmaximal, characteristic, complemented
9#21446757C211yesmaximal, characteristic, complemented
10#21446154C211yesmaximal, characteristic
11#21446152C211yesmaximal, characteristic
12#21446079C211yesmaximal, characteristic
13#21446063C211yesmaximal, characteristic
14#21445818C211yesmaximal, characteristic
15C61×#2057C211yesmaximal, characteristic, complemented
16C22×#11758190C2211yescharacteristic
17#16840473—24no—
18#16840473—24no—
19C2×#14519768C2211yescharacteristic
20C2×#14519768C2211yescharacteristic
21C2×#14519757—21no—
22C2×#14519753—21no—
23C2×#14519753—21no—
24C2×C4×Dic122C2211yescharacteristic
25C2×#14519742C2211yescharacteristic
26C2×C4×D244C2211yescharacteristic
27C2×#14519739C2211yescharacteristic
28C2×#14519739C2211yescharacteristic
29C2×C4⋊C4×D61—21no—
30#16840068—24no—
31#16839861—24no—
32C2×#14519708C2211yescharacteristic
33C2×#14519706C2211yescharacteristic
34C2×#14519702C2211yescharacteristic
35C2×#14519700C2211yescharacteristic
36C2×#14519700C2211yescharacteristic
37C2×#14519699C2211yescharacteristic
38C2×#14519697C2211yescharacteristic
39C2×#14519697C2211yescharacteristic
40C2×#14519687C2211yescharacteristic
41#16839696—21no—
42#16839693—21no—
43#16839668C2211yescharacteristic
44#16839667C2211yescharacteristic
45#16839666—21no—
46#16839658C2211yescharacteristic
47#16839657—21no—
48#16839631C2211yescharacteristic
49#16839525C2211yescharacteristic
50#16839524C2211yescharacteristic
51#16839524C2211yescharacteristic
52#16839524C2211yescharacteristic
53#16839517C2211yescharacteristic
54#16839516C2211yescharacteristic
55#16839516C2211yescharacteristic
56C2×C4⋊Q8×C61C2211yescharacteristic
57C2×#107×C61C2211yescharacteristic
58C2×#107×C61C2211yescharacteristic
59C61×#389C2211yescharacteristic
60C61×#340C2211yescharacteristic
61C61×#340C2211yescharacteristic
62C61×#336C2211yescharacteristic
63C22×Dic122D411yescharacteristic
64C22×Dic122C2311yescharacteristic
65C22×D244C2311yescharacteristic
66C22×C4×D61—21no—
67C22×C4×D61—21no—
68C22×C4×D61—21no—
69C2×#11758190—22no—
70C2×#11758190—22no—
71C2×#11758190Q814yescomplemented
72C22×#92418C2311yescharacteristic
73C22×#92418C2311yescharacteristic
74#14519768—22no—
75#14519768—22no—
76#14519768—22no—
77#14519768D414yes—
78#14519757—28no—
79#14519753—28no—
80#14519753—28no—
81C4×Dic122—22no—
82C4×Dic122—22no—
83#14519742—22no—
84#14519742—22no—
85C4×D244—22no—
86C4×D244—22no—
87#14519739—22no—
88#14519739—22no—
89#14519739—22no—
90#14519739D414yes—
91C2×C4⋊Dic61—21no—
92C2×C4⋊Dic61—21no—
93C2×C4⋊Dic61D411yescharacteristic
94C2×C4⋊Dic61C2311yescharacteristic
95C2×C4⋊Dic61C2311yescharacteristic
96C2×#11758196—21no—
97C2×#11758196—21no—
98C2×#11758196—21no—
99C2×#11758196—21no—
100C2×#11758196—21no—
101C2×#11758196—21no—
102C2×#11758196D411yescharacteristic
103C2×#11758196D411yescharacteristic
104C4⋊C4×D61—28no—
105C2×#11758198—21no—
106C2×#11758198—21no—
107C2×#11758198—21no—
108C2×C4×Dic61—21no—
109C2×C4×Dic61C2311yescharacteristic
110C2×C4×Dic61C2311yescharacteristic
111#14519708—22no—
112#14519706—22no—
113#14519702—22no—
114#14519700—22no—
115#14519700—22no—
116#14519699—22no—
117#14519697—22no—
118#14519697—22no—
119#14519687—22no—
120#14519684—21no—
121#14519684—21no—
122#14519684—21no—
123#14519684—21no—
124#14519684—21no—
125C22×Q8×C61—21no—
126C22×C8×C61C2311yescharacteristic
127C22×C8×C61C2311yescharacteristic
128C4⋊Q8×C61—22no—
129C4⋊Q8×C61—22no—
130C2×C4⋊C4×C61—21no—
131C2×C4⋊C4×C61—21no—
132C2×C4⋊C4×C61C2311yescharacteristic
133C2×C4⋊C4×C61C2311yescharacteristic
134C2×C4⋊C4×C61C2311yescharacteristic
135C2×C4⋊C4×C61C2311yescharacteristic
136C2×C42×C61C2311yescharacteristic
137#107×C61—22no—
138#107×C61—22no—
139#95×C61—21no—
140C23×D61—21no—
141C2×Q8×C61—24no—
142C2×Q8×C61—81no—
143C2×Q8×C61—81no—
144C22×C4×C61—21no—
145C22×C4×C61—21no—
146C22×C4×C61—21no—
147C22×C4×C61—21no—
148C22×C4×C61C2×D411yescharacteristic
149C22×C4×C61C2×D411yescharacteristic
150C22×C4×C61C2411yescharacteristic
151C22×Dic61—21no—
152C22×Dic61—21no—
153C22×Dic61—21no—
154C22×Dic61C2×D411yescharacteristic
155C22×Dic61C2×D411yescharacteristic
156C22×Dic61C2×D411yescharacteristic
157C22×Dic61C2×D411yescharacteristic
158C2×D244—21no—
159C2×D244—21no—
160C2×D244—21no—
161C2×D244C2×D412yes—
162C2×D244C2×Q812yes—
163C2×D244#4112yes—
164C2×C4×D61—24no—
165C2×C4×D61—24no—
166C2×C4×D61—24no—
167C2×C4×D61—42no—
168C2×C4×D61—42no—
169C2×C4×D61—81no—
170C2×C4×D61—81no—
171C2×C4×D61—81no—
172C2×C4×D61—81no—
173C2×Dic122—21no—
174C2×Dic122—21no—
175C2×Dic122—21no—
176C2×Dic122—21no—
177C2×Dic122—22no—
178C2×Dic122—21no—
179C2×Dic122—22no—
180C2×Dic122—41no—
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×C61C2×Q812yes—
188C4⋊C4×C61—24no—
189C4⋊C4×C61—21no—
190C4⋊C4×C61—21no—
191C4⋊C4×C61—21no—
192C4⋊C4×C61—21no—
193C4⋊C4×C61—21no—
194C4⋊C4×C61—21no—
195C4⋊C4×C61—81no—
196C4⋊C4×C61C2×D412yes—
197C4⋊C4×C61C2×D412yes—
198C42×C61—21no—
199C42×C61—21no—
200#11758198—24no—
201#11758198—24no—
202#11758198—24no—
203C4⋊Dic61—24no—
204C4⋊Dic61—21no—
205C4⋊Dic61—22no—
206C4⋊Dic61—24no—
207C4⋊Dic61—21no—
208C4⋊Dic61—21no—
209C4⋊Dic61C2×D412yes—
210#11758196—24no—
211#11758196—24no—
212#11758196—22no—
213#11758196—22no—
214#11758196—42no—
215#11758196—42no—
216#11758196—81no—
217#11758196—81no—
218C4×Dic61—21no—
219C4×Dic61—21no—
220C4×Dic61—42no—
221C4×Dic61C2×D412yes—
222C4×Dic61C2×D412yes—
223C2×#92418—21no—
224C2×#92418—21no—
225C2×#92418—21no—
226C2×#92418—21no—
227C2×#92418—21no—
228C2×#92418—21no—
229#11758190—28no—
230C23×C61C22×D411yescharacteristic
231C22×D61—22no—
232C22×D61—22no—
233C22×D61—22no—
234C22×D61—41no—
235C22×D61—41no—
236C22×D61—41no—
237C22×D61—41no—
238Q8×C61—84no—
239C2×C4×C61—21no—
240C2×C4×C61—21no—
241C2×C4×C61—21no—
242C2×C4×C61—22no—
243C2×C4×C61—22no—
244C2×C4×C61—21no—
245C2×C4×C61—21no—
246C2×C4×C61—41no—
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—41no—
256C2×C4×C61#13712yes—
257C2×C4×C61C22×D411yescharacteristic
258C2×C4×C61C22×D411yescharacteristic
259C2×C4×C61C22×Q811yescharacteristic
260C2×C4×C61C2×#4111yescharacteristic
261C2×C4×C612−1+411yescharacteristic
262C2×C4×C612−1+411yescharacteristic
263C2×Dic61—21no—
264C2×Dic61—21no—
265C2×Dic61—21no—
266C2×Dic61—21no—
267C2×Dic61—21no—
268C2×Dic61—21no—
269C2×Dic61—22no—
270C2×Dic61—22no—
271C2×Dic61—21no—
272C2×Dic61—21no—
273C2×Dic61—21no—
274C2×Dic61—21no—
275C2×Dic61—41no—
276C2×Dic61—41no—
277C2×Dic61—41no—
278C2×Dic61—41no—
279C2×Dic61—41no—
280C2×Dic61—41no—
281C2×Dic61—41no—
282C2×Dic61#12012yes—
283C2×Dic61#12012yes—
284D244—22no—
285D244—22no—
286D244—22no—
287D244C22⋊Q814yes—
288C4×D61—48no—
289C4×D61—84no—
290C4×D61—84no—
291Dic122—22no—
292Dic122—22no—
293Dic122—44no—
294Dic122—41no—
295Dic122—41no—
296C8×C61—22no—
297C8×C61—41no—
298#92418—41no—
299#92418—41no—
300#24349—611nomaximal, Sylow
301C22×C61C2×#12011yescharacteristic
302C22×C61#54811yescharacteristic
303C22×C61D4×Q811yescharacteristic
304C22×C61C2×#13711yescharacteristic
305C22×C61#57511yescharacteristic
306C22×C61#57611yescharacteristic
307C22×C61#57711yescharacteristic
308C2×D61—24no—
309C2×D61—42no—
310C2×D61—42no—
311C2×D61—42no—
312C2×D61—42no—
313C2×D61—42no—
314C2×D61—42no—
315C4×C61—22no—
316C4×C61—42no—
317C4×C61—42no—
318C4×C61—41no—
319C4×C61—81no—
320C4×C61—81no—
321C4×C61C2×C22⋊Q811yescharacteristic
322C4×C61#53611yescharacteristic
323C4×C61#54811yescharacteristic
324C4×C61D4×Q811yescharacteristic
325Dic61—22no—
326Dic61—22no—
327Dic61—42no—
328Dic61—41no—
329Dic61—42no—
330Dic61—41no—
331Dic61—81no—
332C2×#554—611no—
333C2×#479—611no—
334C2×#440—611no—
335#2582—611no—
336#2059—611no—
337#2057D6111yescharacteristic, complemented
338#2026—611no—
339#2024—611no—
340#1992—611no—
341#1985—611no—
342#1877—611no—
343#1877—611no—
344#1864—611no—
345#1862—611no—
346#1802—611no—
347C2×C61#240911yescharacteristic
348C2×C61#300711yescharacteristic
349C2×C61#301211yescharacteristic
350C2×C61#337011yescharacteristic
351C2×C61#337211yescharacteristic
352C2×C61#338511yescharacteristic
353C2×C61#338811yescharacteristic
354D61—44no—
355C22×SD16—611no—
356#554—1224no—
357#554—1224no—
358C2×C4⋊Q8C2×D6111yescharacteristic
359C2×#125—611no—
360C2×#125—611no—
361C2×C22⋊Q8—1221no—
362C2×C22⋊Q8—1221no—
363C2×C22⋊Q8—1221no—
364C2×C4×Q8—611no—
365C2×C4×Q8—611no—
366C2×C4×D4—611no—
367C2×C4×D4—611no—
368C2×C4×D4—611no—
369C22×C4⋊C4—1221no—
370#479—1224no—
371#440—1224no—
372C2×#107C2×D6111yescharacteristic
373C2×#107C2×D6111yescharacteristic
374C2×#106—611no—
375C2×C4⋊C8—611no—
376C2×#103—611no—
377C2×#103—611no—
378C2×#103—611no—
379C2×#102—611no—
380C2×#102—611no—
381C2×#102—611no—
382C2×#97—611no—
383#398—1221no—
384#398—1221no—
385#391—611no—
386#391—611no—
387#391—1221no—
388#389—611no—
389#389—1221no—
390#389C2×D6111yescharacteristic
391#384—611no—
392#340—611no—
393#340—611no—
394#340—611no—
395#340—611no—
396#340C2×D6111yescharacteristic
397#340C2×D6111yescharacteristic
398#336—611no—
399#336—611no—
400#336—611no—
401#336C2×D6111yescharacteristic
402C61#2434911yescharacteristic, complemented, Sylow
403C22×Q8—21no—
404C22×Q8—611no—
405C22×Q8—611no—
406C22×D4—611no—
407C23×C4—1221no—
408C23×C4—1221no—
409C23×C4—1221no—
410C2×SD16—614no—
411C2×SD16—1222no—
412C2×SD16—1222no—
413C22×C8—611no—
414C22×C8—611no—
415C22×C8C22×D6111yescharacteristic
416C22×C8C22×D6111yescharacteristic
417C4⋊Q8—22no—
418C4⋊Q8—22no—
419#125—614no—
420#125—1222no—
421#125—1222no—
422#125—1222no—
423C22⋊Q8—1228no—
424C22⋊Q8—1228no—
425C22⋊Q8—1228no—
426C4×Q8—1222no—
427C4×Q8—1222no—
428C4×Q8—1222no—
429C4×Q8—1222no—
430C4×D4—614no—
431C4×D4—1222no—
432C4×D4—1222no—
433C4×D4—1222no—
434C4×D4—1222no—
435C4×D4—1222no—
436C2×C4⋊C4—21no—
437C2×C4⋊C4—21no—
438C2×C4⋊C4—611no—
439C2×C4⋊C4—611no—
440C2×C4⋊C4—611no—
441C2×C4⋊C4—611no—
442C2×C4⋊C4—611no—
443C2×C4⋊C4—1228no—
444C2×C4⋊C4—1221no—
445C2×C4⋊C4—1221no—
446C2×C4⋊C4—1221no—
447C2×C4⋊C4—1221no—
448C2×C4⋊C4—1221no—
449C2×C4⋊C4—1221no—
450C2×C4⋊C4—1221no—
451C2×C4⋊C4—1221no—
452C2×C4⋊C4C22×D6111yescharacteristic
453C2×C4⋊C4C22×D6111yescharacteristic
454C2×C4⋊C4C22×D6111yescharacteristic
455C2×C4⋊C4C22×D6111yescharacteristic
456C2×C22⋊C4—1221no—
457C2×C22⋊C4—1221no—
458C2×C22⋊C4—1221no—
459C2×C42—611no—
460C2×C42—611no—
461C2×C42—1221no—
462C2×C42C22×D6111yescharacteristic
463#107—22no—
464#107—22no—
465#106—1222no—
466C4⋊C8—1222no—
467#103—1222no—
468#103—1222no—
469#103—1222no—
470#102—1222no—
471#102—1222no—
472#102—1222no—
473#97—1222no—
474#95—21no—
475#95—1221no—
476#95—1221no—
477#95—1221no—
478#95—1221no—
479#95—1221no—
480C24—1221no—
481C2×Q8—24no—
482C2×Q8—81no—
483C2×Q8—81no—
484C2×Q8—612no—
485C2×Q8—1222no—
486C2×Q8—1222no—
487C2×Q8—1221no—
488C2×Q8—1221no—
489C2×Q8—1221no—
490C2×Q8—1221no—
491C2×Q8—1221no—
492C2×Q8—2441no—
493C2×D4—612no—
494C2×D4—612no—
495C2×D4—612no—
496C2×D4—1221no—
497C2×D4—1221no—
498C2×D4—1221no—
499C22×C4—21no—
500C22×C4—21no—
501C22×C4—21no—
502C22×C4—21no—
503C22×C4—611no—
504C22×C4—611no—
505C22×C4—611no—
506C22×C4—611no—
507C22×C4—1224no—
508C22×C4—1224no—
509C22×C4—1224no—
510C22×C4—1221no—
511C22×C4—1221no—
512C22×C4—1221no—
513C22×C4—4881no—
514C22×C4—4881no—
515C22×C4—4881no—
516C22×C4—4881no—
517C22×C4—2442no—
518C22×C4—2442no—
519C22×C4D4×D6111yescharacteristic
520C22×C4D4×D6111yescharacteristic
521C22×C4C23×D6111yescharacteristic
522SD16—1228no—
523C2×C8—21no—
524C2×C8—21no—
525C2×C8—21no—
526C2×C8—21no—
527C2×C8—21no—
528C2×C8—1221no—
529C2×C8—1221no—
530C2×C8—1221no—
531C2×C8—1221no—
532C2×C8—1221no—
533C2×C8—1221no—
534C2×C8Q8×D6112yes—
535C4⋊C4—24no—
536C4⋊C4—21no—
537C4⋊C4—21no—
538C4⋊C4—21no—
539C4⋊C4—21no—
540C4⋊C4—21no—
541C4⋊C4—21no—
542C4⋊C4—81no—
543C4⋊C4—612no—
544C4⋊C4—1222no—
545C4⋊C4—1222no—
546C4⋊C4—1222no—
547C4⋊C4—1224no—
548C4⋊C4—1224no—
549C4⋊C4—1224no—
550C4⋊C4—1224no—
551C4⋊C4—1221no—
552C4⋊C4—1221no—
553C4⋊C4—1221no—
554C4⋊C4—4881no—
555C4⋊C4—4881no—
556C4⋊C4—2442no—
557C4⋊C4—2442no—
558C4⋊C4D4×D6112yes—
559C4⋊C4D4×D6112yes—
560C22⋊C4—1224no—
561C22⋊C4—1224no—
562C22⋊C4—1224no—
563C42—21no—
564C42—21no—
565C42—612no—
566C42—612no—
567C42—1221no—
568C42—1221no—
569C42—2442no—
570C23—1222no—
571C23—1222no—
572C23—1222no—
573C23—2441no—
574C23—2441no—
575C23—2441no—
576C23—2441no—
577C23C2×D4×D6111yescharacteristic
578Q8—84no—
579Q8—1222no—
580Q8—1222no—
581Q8—2444no—
582Q8—2441no—
583Q8—2441no—
584D4—614no—
585D4—1222no—
586D4—1222no—
587D4—1222no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—612no—
606C2×C4—612no—
607C2×C4—1221no—
608C2×C4—1221no—
609C2×C4—1221no—
610C2×C4—1221no—
611C2×C4—1221no—
612C2×C4—1221no—
613C2×C4—1222no—
614C2×C4—1221no—
615C2×C4—1221no—
616C2×C4—1221no—
617C2×C4—1221no—
618C2×C4—1222no—
619C2×C4—4884no—
620C2×C4—4884no—
621C2×C4—2448no—
622C2×C4—2441no—
623C2×C4—2441no—
624C2×C4—2441no—
625C2×C4—2441no—
626C2×C4—2441no—
627C2×C4—2441no—
628C2×C4—2441no—
629C2×C4#1451980912yes—
630C2×C4C2×D4×D6111yescharacteristic
631C2×C4C2×D4×D6111yescharacteristic
632C2×C4C2×Q8×D6111yescharacteristic
633C2×C4#41×D6111yescharacteristic
634C2×C4#1451982511yescharacteristic
635C2×C4#1451982511yescharacteristic
636C8—22no—
637C8—41no—
638C8—2441no—
639C8—2441no—
640C22—1224no—
641C22—2442no—
642C22—2442no—
643C22—2442no—
644C22—2442no—
645C22—2442no—
646C22—2442no—
647C22#120×D6111yescharacteristic
648C22#1684042211yescharacteristic
649C22#1684042711yescharacteristic
650C22C2×#1451980911yescharacteristic
651C22#1684056111yescharacteristic
652C22#1684056311yescharacteristic
653C22#1684057211yescharacteristic
654C4—22no—
655C4—41no—
656C4—42no—
657C4—42no—
658C4—81no—
659C4—81no—
660C4—1222no—
661C4—1222no—
662C4—4881no—
663C4—2441no—
664C4—2442no—
665C4—2441no—
666C4—2442no—
667C4C22⋊Q8×D6111yescharacteristic
668C4#1684033711yescharacteristic
669C4#1684042211yescharacteristic
670C4#1684042711yescharacteristic
671C2—2444no—
672C2#2144917911yescharacteristic
673C2#2145507811yescharacteristic
674C2#2145511811yescharacteristic
675C2#2145857911yescharacteristic
676C2#2145859911yescharacteristic
677C2#2145873411yescharacteristic
678C2#2145876311yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2057C2
socleC23×C61C22×D4
prime radicalC22×D244C23
hypercenter#2057D61
hypocenterC61#24349
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2057

Complements

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

NH ≅ G/Ntype
2. C2×#16840068C2semidirect
4. #21451191C2semidirect
5. #21447155C2semidirect
6. #21446969C2semidirect
7. #21446957C2semidirect
8. #21446786C2semidirect
9. #21446757C2semidirect
15. C61×#2057C2semidirect
71. C2×#11758190Q8semidirect
337. #2057D61semidirect
402. C61#24349semidirect

Maximal subgroups

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

subgroupindex
C2×#168404732
C2×#168400682
C2×#168398612
#214511912
#214471552
#214469692
#214469572
#214467862
#214467572
#214461542
#214461522
#214460792
#214460632
#214458182
C61×#20572
#2434961