Finite Group Databasebeta

← #29716401

Subgroups — #29716401

Order 15616.

1070 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
1C2×#16840489C211yesmaximal, characteristic, complemented
2C2×#16840041C211yesmaximal, characteristic
3C2×#16839849C211yesmaximal, characteristic
4#21450740C211yesmaximal, characteristic
5#21447159C211yesmaximal, characteristic, complemented
6#21446964C211yesmaximal, characteristic
7#21446962C211yesmaximal, characteristic
8#21446778C211yesmaximal, characteristic, complemented
9#21446753C211yesmaximal, characteristic, complemented
10#21446175C211yesmaximal, characteristic
11#21446173C211yesmaximal, characteristic
12#21446072C211yesmaximal, characteristic, complemented
13#21446070C211yesmaximal, characteristic, complemented
14#21445823C211yesmaximal, characteristic, complemented
15C61×#2059C211yesmaximal, characteristic, complemented
16C22×#11758190C2211yescharacteristic
17#16840489—24no—
18#16840489—24no—
19C2×#14519776—21no—
20C2×#14519776C2211yescharacteristic
21C2×#14519768C2211yescharacteristic
22C2×#14519767—21no—
23C2×#14519767—21no—
24C2×#14519755—21no—
25C2×C4×Dic122C2211yescharacteristic
26C2×C4×Dic122C2211yescharacteristic
27C2×#14519742—21no—
28C2×#14519742C2211yescharacteristic
29C2×#14519742C2211yescharacteristic
30C2×#14519739C2211yescharacteristic
31#16840041—24no—
32#16839849—24no—
33C2×#14519710C2211yescharacteristic
34C2×#14519706C2211yescharacteristic
35C2×#14519702C2211yescharacteristic
36C2×#14519700C2211yescharacteristic
37C2×#14519700C2211yescharacteristic
38C2×#14519699C2211yescharacteristic
39C2×#14519697C2211yescharacteristic
40C2×#14519697C2211yescharacteristic
41C2×C8×Dic61C2211yescharacteristic
42#16839686—21no—
43#16839684—21no—
44#16839664C2211yescharacteristic
45#16839663C2211yescharacteristic
46#16839658C2211yescharacteristic
47#16839631C2211yescharacteristic
48#16839621—21no—
49#16839525C2211yescharacteristic
50#16839524C2211yescharacteristic
51#16839524C2211yescharacteristic
52#16839524C2211yescharacteristic
53#16839517C2211yescharacteristic
54#16839516C2211yescharacteristic
55#16839516C2211yescharacteristic
56C2×#125×C61C2211yescharacteristic
57C2×#107×C61C2211yescharacteristic
58C2×#106×C61C2211yescharacteristic
59C61×#389C2211yescharacteristic
60C61×#340C2211yescharacteristic
61C61×#340C2211yescharacteristic
62C61×#336C2211yescharacteristic
63C22×Dic122—21no—
64C22×Dic122D411yescharacteristic
65C22×Dic122C2311yescharacteristic
66C22×D244C2311yescharacteristic
67C22×C4×D61—21no—
68C2×#11758190—22no—
69C2×#11758190—22no—
70C2×#11758190Q814yescomplemented
71C22×#92418C2311yescharacteristic
72C22×#92418C2311yescharacteristic
73#14519776—28no—
74#14519776—22no—
75#14519776D414yes—
76#14519768—22no—
77#14519768—22no—
78#14519767—28no—
79#14519767—28no—
80#14519755—44no—
81C4×Dic122—22no—
82C4×Dic122—22no—
83C4×Dic122—22no—
84C4×Dic122—22no—
85#14519742—28no—
86#14519742—22no—
87#14519742—22no—
88#14519742—22no—
89#14519742D414yes—
90#14519739—22no—
91#14519739—22no—
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×#11758196—21no—
103C2×#11758196—21no—
104C2×#11758196D411yescharacteristic
105C2×#11758196D411yescharacteristic
106C2×#11758198—21no—
107C2×#11758198—21no—
108C2×#11758198—21no—
109C2×#11758198—21no—
110C2×#11758198—21no—
111C2×C4×Dic61—21no—
112C2×C4×Dic61—21no—
113C2×C4×Dic61—21no—
114C2×C4×Dic61C2311yescharacteristic
115C2×C4×Dic61C2311yescharacteristic
116#14519710—22no—
117#14519706—22no—
118#14519702—22no—
119#14519700—22no—
120#14519700—22no—
121#14519699—22no—
122#14519697—22no—
123#14519697—22no—
124C8×Dic61—22no—
125#14519684—21no—
126C22×C8×C61C2311yescharacteristic
127C22×C8×C61C2311yescharacteristic
128#125×C61—22no—
129#125×C61—22no—
130C2×C4⋊C4×C61—21no—
131C2×C4⋊C4×C61—21no—
132C2×C4⋊C4×C61—21no—
133C2×C4⋊C4×C61C2311yescharacteristic
134C2×C4⋊C4×C61C2311yescharacteristic
135C2×C4⋊C4×C61C2311yescharacteristic
136C2×C4⋊C4×C61C2311yescharacteristic
137C2×C42×C61C2311yescharacteristic
138#107×C61—22no—
139#106×C61—22no—
140#95×C61—21no—
141C23×D61—21no—
142C22×C4×C61—21no—
143C22×C4×C61—21no—
144C22×C4×C61—21no—
145C22×C4×C61—21no—
146C22×C4×C61C2×D411yescharacteristic
147C22×C4×C61C2×D411yescharacteristic
148C22×C4×C61C2411yescharacteristic
149C22×Dic61—21no—
150C22×Dic61—21no—
151C22×Dic61—21no—
152C22×Dic61C2×D411yescharacteristic
153C22×Dic61C2×D411yescharacteristic
154C22×Dic61C2×D411yescharacteristic
155C22×Dic61C2×D411yescharacteristic
156C2×D244—21no—
157C2×D244—21no—
158C2×D244—21no—
159C2×D244—21no—
160C2×D244—21no—
161C2×D244C2×Q812yes—
162C2×C4×D61—42no—
163C2×C4×D61—42no—
164C2×C4×D61—42no—
165C2×Dic122—21no—
166C2×Dic122—21no—
167C2×Dic122—21no—
168C2×Dic122—24no—
169C2×Dic122—22no—
170C2×Dic122—22no—
171C2×Dic122—22no—
172C2×Dic122—42no—
173C2×Dic122—42no—
174C2×Dic122C2×D412yes—
175C2×Dic122C2×Q812yes—
176C2×Dic122#4112yes—
177C2×C8×C61—21no—
178C2×C8×C61—21no—
179C2×C8×C61—21no—
180C2×C8×C61—21no—
181C2×C8×C61—21no—
182C2×C8×C61C2×Q812yes—
183C4⋊C4×C61—24no—
184C4⋊C4×C61—24no—
185C4⋊C4×C61—21no—
186C4⋊C4×C61—21no—
187C4⋊C4×C61—21no—
188C4⋊C4×C61—21no—
189C4⋊C4×C61—21no—
190C4⋊C4×C61—21no—
191C4⋊C4×C61—42no—
192C4⋊C4×C61C2×D412yes—
193C4⋊C4×C61C2×D412yes—
194C42×C61—21no—
195C42×C61—21no—
196#11758198—42no—
197#11758198—42no—
198#11758198—42no—
199#11758198—81no—
200#11758198—81no—
201C4⋊Dic61—24no—
202C4⋊Dic61—21no—
203C4⋊Dic61—22no—
204C4⋊Dic61—21no—
205C4⋊Dic61—21no—
206C4⋊Dic61C2×D412yes—
207#11758196—24no—
208#11758196—24no—
209#11758196—24no—
210#11758196—24no—
211#11758196—24no—
212#11758196—22no—
213#11758196—22no—
214#11758196—42no—
215#11758196—42no—
216#11758196—81no—
217C4×Dic61—24no—
218C4×Dic61—24no—
219C4×Dic61—21no—
220C4×Dic61—21no—
221C4×Dic61—24no—
222C4×Dic61C2×D412yes—
223C4×Dic61C2×D412yes—
224C2×#92418—21no—
225C2×#92418—21no—
226C2×#92418—21no—
227C2×#92418—21no—
228C2×#92418—21no—
229C2×#92418—21no—
230#11758190—28no—
231C23×C61C22×D411yescharacteristic
232C22×D61—22no—
233C22×D61—41no—
234C22×D61—41no—
235C22×D61—41no—
236C22×D61—41no—
237C22×D61—41no—
238C22×D61—41no—
239C2×C4×C61—21no—
240C2×C4×C61—22no—
241C2×C4×C61—22no—
242C2×C4×C61—21no—
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—41no—
256C2×C4×C61—41no—
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—22no—
264C2×Dic61—22no—
265C2×Dic61—22no—
266C2×Dic61—21no—
267C2×Dic61—21no—
268C2×Dic61—21no—
269C2×Dic61—21no—
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#12012yes—
281C2×Dic61#12012yes—
282C2×Dic61#13512yes—
283C2×Dic61AGL(1,ℤ/8)12yes—
284D244—22no—
285D244—22no—
286D244—41no—
287D244—41no—
288C4×D61—84no—
289Dic122—22no—
290Dic122—22no—
291Dic122—22no—
292Dic122—44no—
293Dic122—48no—
294Dic122C22⋊Q814yescomplemented
295C8×C61—22no—
296C8×C61—41no—
297#92418—41no—
298#92418—41no—
299#24328—611nomaximal, Sylow
300C22×C61C2×#12011yescharacteristic
301C22×C61#54811yescharacteristic
302C22×C61D4×Q811yescharacteristic
303C22×C61C2×#13511yescharacteristic
304C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
305C22×C61#57711yescharacteristic
306C22×C61#57811yescharacteristic
307C2×D61—42no—
308C2×D61—42no—
309C2×D61—42no—
310C2×D61—81no—
311C2×D61—81no—
312C2×D61—81no—
313C2×D61—81no—
314C4×C61—41no—
315C4×C61—42no—
316C4×C61—41no—
317C4×C61—42no—
318C4×C61—81no—
319C4×C61—81no—
320C4×C61C2×C22⋊Q811yescharacteristic
321C4×C61#53611yescharacteristic
322C4×C61#54811yescharacteristic
323C4×C61D4×Q811yescharacteristic
324Dic61—24no—
325Dic61—22no—
326Dic61—41no—
327Dic61—42no—
328Dic61—41no—
329Dic61—81no—
330Dic61#44914yes—
331C2×#557—611no—
332C2×#474—611no—
333C2×C4×SD16—611no—
334#2538—611no—
335#2059—611no—
336#2059D6111yescharacteristic, complemented
337#2025—611no—
338#2025—611no—
339#1990—611no—
340#1984—611no—
341#1880—611no—
342#1880—611no—
343#1863—611no—
344#1863—611no—
345#1803—611no—
346C2×C61#240911yescharacteristic
347C2×C61C2×#44911yescharacteristic
348C2×C61#301311yescharacteristic
349C2×C61#336411yescharacteristic
350C2×C61#336811yescharacteristic
351C2×C61#338111yescharacteristic
352C2×C61#338611yescharacteristic
353D61—82no—
354C22×SD16—611no—
355#557—1224no—
356#557—1224no—
357C2×C4⋊Q8—611no—
358C2×C4⋊Q8—1221no—
359C2×#125—611no—
360C2×#125—1221no—
361C2×#125—1221no—
362C2×#125C2×D6111yescharacteristic
363C2×C22⋊Q8—1221no—
364C2×C4×Q8—611no—
365C2×C4×Q8—611no—
366C2×C4×Q8—611no—
367C2×C4×Q8—611no—
368C2×C4×Q8—1221no—
369C2×C4×D4—611no—
370#474—1224no—
371C4×SD16—1224no—
372C2×#107—611no—
373C2×#107C2×D6111yescharacteristic
374C2×#106C2×D6111yescharacteristic
375C2×C4⋊C8—611no—
376C2×#103—611no—
377C2×#103—611no—
378C2×#103—611no—
379C2×#102—611no—
380C2×#102—611no—
381C2×#102—611no—
382C2×C4×C8—611no—
383#396—1221no—
384#396—1221no—
385#390—611no—
386#390—611no—
387#389—611no—
388#389C2×D6111yescharacteristic
389#384—611no—
390#380—1221no—
391#340—611no—
392#340—611no—
393#340—611no—
394#340—611no—
395#340C2×D6111yescharacteristic
396#340C2×D6111yescharacteristic
397#336—611no—
398#336—611no—
399#336—611no—
400#336C2×D6111yescharacteristic
401C61#2432811yescharacteristic, complemented, Sylow
402C22×Q8—611no—
403C22×Q8—611no—
404C22×Q8—1221no—
405C22×D4—611no—
406C23×C4—1221no—
407C2×SD16—614no—
408C2×SD16—1222no—
409C2×SD16—1222no—
410C22×C8—611no—
411C22×C8—611no—
412C22×C8C22×D6111yescharacteristic
413C22×C8C22×D6111yescharacteristic
414C4⋊Q8—614no—
415C4⋊Q8—1222no—
416C4⋊Q8—1228no—
417#125—22no—
418#125—22no—
419#125—1222no—
420#125—1228no—
421#125—1228no—
422#125—1222no—
423C22⋊Q8—2444no—
424C4×Q8—614no—
425C4×Q8—1222no—
426C4×Q8—1222no—
427C4×Q8—1222no—
428C4×Q8—1222no—
429C4×Q8—1222no—
430C4×Q8—1222no—
431C4×Q8—1228no—
432C4×Q8—1222no—
433C4×D4—1222no—
434C4×D4—1222no—
435C2×C4⋊C4—21no—
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—1221no—
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×C22⋊C4—1221no—
460C2×C22⋊C4—1221no—
461C2×C42—611no—
462C2×C42—611no—
463C2×C42—1221no—
464C2×C42—1221no—
465C2×C42—1221no—
466C2×C42C22×D6111yescharacteristic
467#107—22no—
468#107—1222no—
469#106—22no—
470C4⋊C8—1222no—
471#103—1222no—
472#103—1222no—
473#103—1222no—
474#102—1222no—
475#102—1222no—
476#102—1222no—
477C4×C8—1222no—
478#95—21no—
479#95—1221no—
480C24—1221no—
481C2×Q8—612no—
482C2×Q8—612no—
483C2×Q8—612no—
484C2×Q8—1224no—
485C2×Q8—1222no—
486C2×Q8—1222no—
487C2×Q8—1222no—
488C2×Q8—1221no—
489C2×Q8—1221no—
490C2×Q8—1221no—
491C2×Q8—2442no—
492C2×Q8—2442no—
493C2×D4—612no—
494C2×D4—1221no—
495C2×D4—1221no—
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—1221no—
508C22×C4—1221no—
509C22×C4—1221no—
510C22×C4—2442no—
511C22×C4—2442no—
512C22×C4—2442no—
513C22×C4D4×D6111yescharacteristic
514C22×C4D4×D6111yescharacteristic
515C22×C4C23×D6111yescharacteristic
516SD16—1228no—
517C2×C8—21no—
518C2×C8—21no—
519C2×C8—21no—
520C2×C8—21no—
521C2×C8—21no—
522C2×C8—1221no—
523C2×C8—1221no—
524C2×C8—1221no—
525C2×C8—1221no—
526C2×C8—1221no—
527C2×C8—1221no—
528C2×C8Q8×D6112yes—
529C4⋊C4—24no—
530C4⋊C4—24no—
531C4⋊C4—21no—
532C4⋊C4—21no—
533C4⋊C4—21no—
534C4⋊C4—21no—
535C4⋊C4—21no—
536C4⋊C4—21no—
537C4⋊C4—42no—
538C4⋊C4—612no—
539C4⋊C4—1222no—
540C4⋊C4—1222no—
541C4⋊C4—1222no—
542C4⋊C4—1224no—
543C4⋊C4—1224no—
544C4⋊C4—1224no—
545C4⋊C4—1224no—
546C4⋊C4—1224no—
547C4⋊C4—1224no—
548C4⋊C4—1221no—
549C4⋊C4—1221no—
550C4⋊C4—1221no—
551C4⋊C4—4881no—
552C4⋊C4—2442no—
553C4⋊C4—2442no—
554C4⋊C4D4×D6112yes—
555C4⋊C4D4×D6112yes—
556C22⋊C4—4881no—
557C22⋊C4—4881no—
558C22⋊C4—2442no—
559C22⋊C4—2442no—
560C22⋊C4—2442no—
561C42—21no—
562C42—21no—
563C42—612no—
564C42—612no—
565C42—1224no—
566C42—1224no—
567C42—1224no—
568C42—1221no—
569C42—1221no—
570C23—1222no—
571C23—2441no—
572C23—2441no—
573C23—2441no—
574C23—2441no—
575C23—2441no—
576C23—2441no—
577C23C2×D4×D6111yescharacteristic
578Q8—614no—
579Q8—1222no—
580Q8—1222no—
581Q8—1222no—
582Q8—2444no—
583Q8—2448no—
584D4—1222no—
585D4—1222no—
586D4—2441no—
587D4—2441no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
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—41no—
606C2×C4—612no—
607C2×C4—612no—
608C2×C4—612no—
609C2×C4—612no—
610C2×C4—1222no—
611C2×C4—1221no—
612C2×C4—1221no—
613C2×C4—1222no—
614C2×C4—1222no—
615C2×C4—1221no—
616C2×C4—1222no—
617C2×C4—1221no—
618C2×C4—1221no—
619C2×C4—1221no—
620C2×C4—1221no—
621C2×C4—1221no—
622C2×C4—4884no—
623C2×C4—2441no—
624C2×C4—2441no—
625C2×C4—2441no—
626C2×C4—2441no—
627C2×C4—2441no—
628C2×C4C2×D4×D6111yescharacteristic
629C2×C4C2×D4×D6111yescharacteristic
630C2×C4C2×Q8×D6111yescharacteristic
631C2×C4#41×D6111yescharacteristic
632C2×C4#1451982511yescharacteristic
633C2×C4#1451982511yescharacteristic
634C8—22no—
635C8—41no—
636C8—2441no—
637C8—2441no—
638C22—4881no—
639C22—4881no—
640C22—4881no—
641C22—4881no—
642C22—2442no—
643C22—2442no—
644C22—2442no—
645C22#120×D6111yescharacteristic
646C22#1684042211yescharacteristic
647C22#1684042711yescharacteristic
648C22#135×D6111yescharacteristic
649C22AGL(1,ℤ/8)×D6111yescharacteristic
650C22#1684056911yescharacteristic
651C22#1684057811yescharacteristic
652C4—41no—
653C4—42no—
654C4—41no—
655C4—42no—
656C4—81no—
657C4—81no—
658C4—614no—
659C4—1222no—
660C4—1224no—
661C4—4881no—
662C4—2441no—
663C4—2441no—
664C4—2442no—
665C4C22⋊Q8×D6111yescharacteristic
666C4#1684033711yescharacteristic
667C4#1684042211yescharacteristic
668C4#1684042711yescharacteristic
669C2—4882no—
670C2#2144917911yescharacteristic
671C2#449×D6111yescharacteristic
672C2#2145513811yescharacteristic
673C2#2145850711yescharacteristic
674C2#2145855111yescharacteristic
675C2#2145869111yescharacteristic
676C2#2145873711yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2059C2
socleC23×C61C22×D4
prime radicalC22×D244C23
hypercenter#2059D61
hypocenterC61#24328
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2059

Complements

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

NH ≅ G/Ntype
1. C2×#16840489C2semidirect
5. #21447159C2semidirect
8. #21446778C2semidirect
9. #21446753C2semidirect
12. #21446072C2semidirect
13. #21446070C2semidirect
14. #21445823C2semidirect
15. C61×#2059C2semidirect
70. C2×#11758190Q8semidirect
294. Dic122C22⋊Q8semidirect
336. #2059D61semidirect
401. C61#24328semidirect

Maximal subgroups

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

subgroupindex
C2×#168404892
C2×#168400412
C2×#168398492
#214507402
#214471592
#214469642
#214469622
#214467782
#214467532
#214461752
#214461732
#214460722
#214460702
#214458232
C61×#20592
#2432861