Finite Group Databasebeta

← #29740166

Subgroups — #29740166

Order 15616.

868 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×SD16×Dic61C211yesmaximal, characteristic
2#21449591C211yesmaximal, characteristic, complemented
3#21448724C211yesmaximal, characteristic
4#21447369C211yesmaximal, characteristic, complemented
5#21447317C211yesmaximal, characteristic, complemented
6#21447240C211yesmaximal, characteristic, complemented
7#21447062C211yesmaximal, characteristic, complemented
8#21446492C211yesmaximal, characteristic, complemented
9#21446373C211yesmaximal, characteristic, complemented
10#21446155C211yesmaximal, characteristic
11#21446155C211yesmaximal, characteristic
12#21446155C211yesmaximal, characteristic
13#21446071C211yesmaximal, characteristic, complemented
14#21446066C211yesmaximal, characteristic
15C2×C61×#461C211yesmaximal, characteristic
16C2×Q8×Dic61C2211yescharacteristic
17C2×D4×Dic61C2211yescharacteristic
18C2×D4×Dic61C2211yescharacteristic
19SD16×Dic61—24no—
20C2×#14519710C2211yescharacteristic
21C2×#14519709C2211yescharacteristic
22C2×#14519701C2211yescharacteristic
23C2×#14519701C2211yescharacteristic
24C2×#14519701C2211yescharacteristic
25C2×#14519701C2211yescharacteristic
26C2×#14519698C2211yescharacteristic
27C2×#14519698C2211yescharacteristic
28C2×C8×Dic61C2211yescharacteristic
29#16839701—21no—
30#16839692—21no—
31#16839669C2211yescharacteristic
32#16839669C2211yescharacteristic
33#16839669C2211yescharacteristic
34#16839660C2211yescharacteristic
35#16839660C2211yescharacteristic
36#16839658C2211yescharacteristic
37#16839643—21no—
38#16839643—21no—
39#16839635—21no—
40#16839634—21no—
41#16839627C2211yescharacteristic
42#16839620—21no—
43C4×#11758196C2211yescharacteristic
44C2×#14519684—21no—
45#16839525C2211yescharacteristic
46#16839525C2211yescharacteristic
47#16839525C2211yescharacteristic
48#16839524C2211yescharacteristic
49#16839517C2211yescharacteristic
50#16839516C2211yescharacteristic
51#16839516C2211yescharacteristic
52#16839516C2211yescharacteristic
53C22×SD16×C61C2211yescharacteristic
54C22×SD16×C61C2211yescharacteristic
55C2×C4⋊Q8×C61C2211yescharacteristic
56C2×C4×D4×C61C2211yescharacteristic
57C61×#461—24no—
58C2×C4⋊C8×C61C2211yescharacteristic
59C2×#103×C61C2211yescharacteristic
60C2×#103×C61C2211yescharacteristic
61C23×Dic61—21no—
62C23×Dic61—21no—
63C22×#92418C2311yescharacteristic
64C22×#92418C2311yescharacteristic
65Q8×Dic61—22no—
66Q8×Dic61—41no—
67D4×Dic61—22no—
68D4×Dic61—22no—
69D4×Dic61—22no—
70D4×Dic61—41no—
71C2×C4⋊Dic61—21no—
72C2×C4⋊Dic61C2311yescharacteristic
73C2×C4⋊Dic61C2311yescharacteristic
74C2×C4⋊Dic61C2311yescharacteristic
75C2×C4⋊Dic61C2311yescharacteristic
76C2×#11758196—21no—
77C2×#11758196—21no—
78C2×#11758196—21no—
79C2×#11758196D411yescharacteristic
80C2×#11758196D411yescharacteristic
81C2×C22⋊Dic61—21no—
82C2×C22⋊Dic61—21no—
83C2×C4×Dic61—21no—
84C2×C4×Dic61C2311yescharacteristic
85C2×C4×Dic61C2311yescharacteristic
86#14519710—41no—
87#14519709—22no—
88#14519701—22no—
89#14519701—22no—
90#14519701—22no—
91#14519701—41no—
92#14519698—22no—
93#14519698—22no—
94C8×Dic61—22no—
95#14519684—21no—
96#14519684—21no—
97#14519684—21no—
98#14519684—21no—
99#14519684—21no—
100#14519684—21no—
101#14519684—21no—
102#14519684—21no—
103#14519684—21no—
104#14519684—21no—
105#14519684—21no—
106#14519684D411yescharacteristic
107#14519684D411yescharacteristic
108C22×Q8×C61C2311yescharacteristic
109C22×Q8×C61C2311yescharacteristic
110C22×D4×C61C2311yescharacteristic
111C23×C4×C61—21no—
112C2×SD16×C61—22no—
113C2×SD16×C61—22no—
114C2×SD16×C61—22no—
115C2×SD16×C61—22no—
116C2×SD16×C61—22no—
117C2×SD16×C61D414yes—
118C22×C8×C61C2311yescharacteristic
119C22×C8×C61C2311yescharacteristic
120C4⋊Q8×C61—22no—
121C4⋊Q8×C61—41no—
122C4×D4×C61—22no—
123C4×D4×C61—41no—
124C2×C4⋊C4×C61—21no—
125C2×C4⋊C4×C61—21no—
126C2×C4⋊C4×C61C2311yescharacteristic
127C2×C22⋊C4×C61—21no—
128C2×C42×C61C2311yescharacteristic
129C4⋊C8×C61—22no—
130#103×C61—22no—
131#103×C61—22no—
132C24×C61—21no—
133C2×Q8×C61—21no—
134C2×Q8×C61—21no—
135C2×Q8×C61—21no—
136C2×Q8×C61—21no—
137C2×Q8×C61—21no—
138C2×Q8×C61—21no—
139C2×Q8×C61—21no—
140C2×Q8×C61—21no—
141C2×Q8×C61—21no—
142C2×Q8×C61—21no—
143C2×Q8×C61—21no—
144C2×Q8×C61C2×D412yes—
145C2×D4×C61—21no—
146C2×D4×C61—21no—
147C2×D4×C61—21no—
148C2×D4×C61C2×D412yes—
149C2×D4×C61#4112yes—
150C2×D4×C61#4112yes—
151C22×C4×C61—21no—
152C22×C4×C61—21no—
153C22×C4×C61—21no—
154C22×C4×C61—42no—
155C22×C4×C61—42no—
156C22×C4×C61—81no—
157C22×C4×C61C2×D411yescharacteristic
158C22×C4×C61C2×D411yescharacteristic
159C22×C4×C61C2411yescharacteristic
160C22×Dic61—21no—
161C22×Dic61—24no—
162C22×Dic61—21no—
163C22×Dic61—21no—
164C22×Dic61—21no—
165C22×Dic61—42no—
166C22×Dic61—42no—
167C22×Dic61—81no—
168C22×Dic61—81no—
169C22×Dic61—81no—
170C22×Dic61C2×D411yescharacteristic
171C22×Dic61C2×D411yescharacteristic
172C22×Dic61C2×D411yescharacteristic
173C22×Dic61C2×D411yescharacteristic
174SD16×C61—28no—
175SD16×C61—44no—
176C2×C8×C61—21no—
177C2×C8×C61—21no—
178C2×C8×C61—21no—
179C2×C8×C61—21no—
180C2×C8×C61—21no—
181C2×C8×C61C2×D412yes—
182C4⋊C4×C61—21no—
183C4⋊C4×C61—21no—
184C4⋊C4×C61—42no—
185C4⋊C4×C61—42no—
186C22⋊C4×C61—42no—
187C42×C61—21no—
188C42×C61—21no—
189C22⋊Dic61—24no—
190C22⋊Dic61—42no—
191C4⋊Dic61—21no—
192C4⋊Dic61—21no—
193C4⋊Dic61—21no—
194C4⋊Dic61—21no—
195C4⋊Dic61—21no—
196C4⋊Dic61—21no—
197C4⋊Dic61—21no—
198C4⋊Dic61—42no—
199C4⋊Dic61#4112yes—
200#11758196—22no—
201#11758196—22no—
202#11758196—42no—
203#11758196—81no—
204#11758196—81no—
205C4×Dic61—21no—
206C4×Dic61—21no—
207C4×Dic61—21no—
208C4×Dic61—21no—
209C4×Dic61—42no—
210C2×#92418—21no—
211C2×#92418—21no—
212C2×#92418—21no—
213C2×#92418—21no—
214C2×#92418—21no—
215C2×#92418—21no—
216C23×C61—22no—
217C23×C61—22no—
218C23×C61—22no—
219C23×C61—41no—
220C23×C61—41no—
221C23×C61—41no—
222C23×C61—41no—
223C23×C61C22×D411yescharacteristic
224Q8×C61—22no—
225Q8×C61—22no—
226Q8×C61—41no—
227Q8×C61—41no—
228Q8×C61—41no—
229Q8×C61—41no—
230Q8×C61—41no—
231Q8×C61—41no—
232D4×C61—22no—
233D4×C61—22no—
234D4×C61—22no—
235D4×C61#12314yes—
236C2×C4×C61—21no—
237C2×C4×C61—21no—
238C2×C4×C61—22no—
239C2×C4×C61—21no—
240C2×C4×C61—21no—
241C2×C4×C61—21no—
242C2×C4×C61—41no—
243C2×C4×C61—41no—
244C2×C4×C61—41no—
245C2×C4×C61—41no—
246C2×C4×C61—41no—
247C2×C4×C61—41no—
248C2×C4×C61—41no—
249C2×C4×C61—41no—
250C2×C4×C61—84no—
251C2×C4×C61#13512yes—
252C2×C4×C61C22×D411yescharacteristic
253C2×C4×C61C2×#4111yescharacteristic
254C2×C4×C61C2×#4111yescharacteristic
255C2×C4×C61C2×#4111yescharacteristic
256C2×C4×C61H(5,𝔽2)11yescharacteristic
257C2×C4×C612−1+411yescharacteristic
258C2×Dic61—21no—
259C2×Dic61—21no—
260C2×Dic61—21no—
261C2×Dic61—21no—
262C2×Dic61—22no—
263C2×Dic61—21no—
264C2×Dic61—21no—
265C2×Dic61—21no—
266C2×Dic61—21no—
267C2×Dic61—21no—
268C2×Dic61—21no—
269C2×Dic61—21no—
270C2×Dic61—41no—
271C2×Dic61—41no—
272C2×Dic61—41no—
273C2×Dic61—41no—
274C2×Dic61—41no—
275C2×Dic61—41no—
276C2×Dic61—41no—
277C2×Dic61—41no—
278C2×Dic61—41no—
279C2×Dic61—41no—
280C2×Dic61—41no—
281C2×Dic61—84no—
282C2×Dic61—84no—
283C2×Dic61#13712yes—
284C8×C61—22no—
285C8×C61—41no—
286#92418—41no—
287#92418—41no—
288#26170—611nomaximal, Sylow
289C22×C61—24no—
290C22×C61—42no—
291C22×C61—42no—
292C22×C61—42no—
293C22×C61—42no—
294C22×C61—42no—
295C22×C61—42no—
296C22×C61#52511yescharacteristic
297C22×C61D4211yescharacteristic
298C22×C61#54811yescharacteristic
299C22×C61C2×#13511yescharacteristic
300C22×C61C2×#13711yescharacteristic
301C22×C61#57711yescharacteristic
302C22×C61#57711yescharacteristic
303C4×C61—22no—
304C4×C61—42no—
305C4×C61—41no—
306C4×C61—81no—
307C4×C61—81no—
308C4×C61C2×#12311yescharacteristic
309C4×C61#53911yescharacteristic
310C4×C61#54711yescharacteristic
311C4×C61#55311yescharacteristic
312Dic61—22no—
313Dic61—41no—
314Dic61—41no—
315Dic61—42no—
316Dic61—41no—
317Dic61—81no—
318Dic61—81no—
319Dic61—81no—
320C2×#461D6111yescharacteristic
321C2×C4×SD16—611no—
322#2446—611no—
323#2349—611no—
324#2097—611no—
325#2088—611no—
326#2074—611no—
327#2043—611no—
328#1933—611no—
329#1916—611no—
330#1877—611no—
331#1877—611no—
332#1877—611no—
333#1863—611no—
334#1862—611no—
335C2×C61—44no—
336C2×C61#244311yescharacteristic
337C2×C61#310711yescharacteristic
338C2×C61#312311yescharacteristic
339C2×C61#326611yescharacteristic
340C2×C61#328211yescharacteristic
341C2×C61#336011yescharacteristic
342C2×C61#337211yescharacteristic
343C22×SD16C2×D6111yescharacteristic
344C22×SD16C2×D6111yescharacteristic
345C2×C4⋊Q8C2×D6111yescharacteristic
346C2×C4×Q8—611no—
347C2×C4×D4—611no—
348C2×C4×D4—611no—
349C2×C4×D4C2×D6111yescharacteristic
350#461—24no—
351C4×SD16—1224no—
352C2×#107—611no—
353C2×#106—611no—
354C2×C4⋊C8C2×D6111yescharacteristic
355C2×#103—611no—
356C2×#103—611no—
357C2×#103—611no—
358C2×#103—611no—
359C2×#103C2×D6111yescharacteristic
360C2×#103C2×D6111yescharacteristic
361C2×#102—611no—
362C2×#102—611no—
363C2×C4×C8—611no—
364#400—1221no—
365#398—1221no—
366#391—611no—
367#391—611no—
368#391—611no—
369#389—611no—
370#389—611no—
371#389—611no—
372#387—1221no—
373#387—1221no—
374#385—1221no—
375#385—1221no—
376#382—611no—
377#380—1221no—
378C4×C4⋊C4—611no—
379C2×#95—1221no—
380#340—611no—
381#340—611no—
382#340—611no—
383#340—611no—
384#336—611no—
385#336—611no—
386#336—611no—
387#336—611no—
388C61#2617011yescharacteristic, complemented, Sylow
389C22×Q8C22×D6111yescharacteristic
390C22×Q8C22×D6111yescharacteristic
391C22×D4C22×D6111yescharacteristic
392C23×C4—21no—
393C23×C4—1221no—
394C23×C4—1221no—
395C2×SD16—22no—
396C2×SD16—22no—
397C2×SD16—22no—
398C2×SD16—22no—
399C2×SD16—22no—
400C2×SD16C22⋊D6114yes—
401C22×C8—611no—
402C22×C8—611no—
403C22×C8C22×D6111yescharacteristic
404C22×C8C22×D6111yescharacteristic
405C4⋊Q8—22no—
406C4⋊Q8—41no—
407C4×Q8—1222no—
408C4×Q8—2441no—
409C4×D4—22no—
410C4×D4—41no—
411C4×D4—1222no—
412C4×D4—1222no—
413C4×D4—1222no—
414C4×D4—2441no—
415C2×C4⋊C4—21no—
416C2×C4⋊C4—21no—
417C2×C4⋊C4—611no—
418C2×C4⋊C4—611no—
419C2×C4⋊C4—611no—
420C2×C4⋊C4—611no—
421C2×C4⋊C4—611no—
422C2×C4⋊C4—611no—
423C2×C4⋊C4—1221no—
424C2×C4⋊C4—1221no—
425C2×C4⋊C4—1221no—
426C2×C4⋊C4—1221no—
427C2×C4⋊C4C22×D6111yescharacteristic
428C2×C22⋊C4—21no—
429C2×C22⋊C4—1221no—
430C2×C22⋊C4—1221no—
431C2×C42—611no—
432C2×C42—611no—
433C2×C42—1221no—
434C2×C42C22×D6111yescharacteristic
435#107—2441no—
436#106—1222no—
437C4⋊C8—22no—
438#103—22no—
439#103—22no—
440#103—1222no—
441#103—1222no—
442#103—1222no—
443#103—2441no—
444#102—1222no—
445#102—1222no—
446C4×C8—1222no—
447#95—611no—
448#95—611no—
449#95—1221no—
450#95—1221no—
451#95—1221no—
452#95—1221no—
453#95—1221no—
454#95—1221no—
455#95—1221no—
456#95—1221no—
457#95—1221no—
458#95—1221no—
459#95—1221no—
460C24—21no—
461C2×Q8—21no—
462C2×Q8—21no—
463C2×Q8—21no—
464C2×Q8—21no—
465C2×Q8—21no—
466C2×Q8—21no—
467C2×Q8—21no—
468C2×Q8—21no—
469C2×Q8—21no—
470C2×Q8—21no—
471C2×Q8—21no—
472C2×Q8C2×C22⋊D6112yes—
473C2×D4—21no—
474C2×D4—21no—
475C2×D4—21no—
476C2×D4#1175822412yes—
477C2×D4#1175822412yes—
478C2×D4C2×C22⋊D6112yes—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—42no—
483C22×C4—42no—
484C22×C4—81no—
485C22×C4—611no—
486C22×C4—611no—
487C22×C4—611no—
488C22×C4—611no—
489C22×C4—1224no—
490C22×C4—1221no—
491C22×C4—1221no—
492C22×C4—1221no—
493C22×C4—1221no—
494C22×C4—4881no—
495C22×C4—4881no—
496C22×C4—4881no—
497C22×C4—2442no—
498C22×C4—2442no—
499C22×C4D4×D6111yescharacteristic
500C22×C4D4×D6111yescharacteristic
501C22×C4C23×D6111yescharacteristic
502SD16—28no—
503SD16—44no—
504C2×C8—21no—
505C2×C8—21no—
506C2×C8—21no—
507C2×C8—21no—
508C2×C8—21no—
509C2×C8—1221no—
510C2×C8—1221no—
511C2×C8—1221no—
512C2×C8—1221no—
513C2×C8—1221no—
514C2×C8—1221no—
515C2×C8C2×C22⋊D6112yes—
516C4⋊C4—21no—
517C4⋊C4—21no—
518C4⋊C4—42no—
519C4⋊C4—42no—
520C4⋊C4—612no—
521C4⋊C4—1222no—
522C4⋊C4—1221no—
523C4⋊C4—1222no—
524C4⋊C4—1221no—
525C4⋊C4—1221no—
526C4⋊C4—1221no—
527C4⋊C4—1221no—
528C4⋊C4—1221no—
529C4⋊C4—1221no—
530C4⋊C4—4881no—
531C4⋊C4—4881no—
532C4⋊C4—2442no—
533C4⋊C4—2442no—
534C22⋊C4—42no—
535C22⋊C4—1224no—
536C22⋊C4—2442no—
537C42—21no—
538C42—21no—
539C42—1221no—
540C42—1221no—
541C42—1221no—
542C42—1221no—
543C42—2442no—
544C23—22no—
545C23—22no—
546C23—22no—
547C23—41no—
548C23—41no—
549C23—41no—
550C23—41no—
551C23C2×D4×D6111yescharacteristic
552Q8—22no—
553Q8—22no—
554Q8—41no—
555Q8—41no—
556Q8—41no—
557Q8—41no—
558Q8—41no—
559Q8—41no—
560D4—22no—
561D4—22no—
562D4—22no—
563D4#1451975814yes—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—22no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—84no—
579C2×C4—612no—
580C2×C4—1221no—
581C2×C4—1221no—
582C2×C4—1221no—
583C2×C4—1221no—
584C2×C4—1222no—
585C2×C4—1221no—
586C2×C4—1221no—
587C2×C4—1221no—
588C2×C4—1221no—
589C2×C4—1221no—
590C2×C4—1221no—
591C2×C4—1221no—
592C2×C4—4884no—
593C2×C4—4884no—
594C2×C4—2441no—
595C2×C4—2441no—
596C2×C4—2441no—
597C2×C4—2441no—
598C2×C4—2441no—
599C2×C4—2441no—
600C2×C4—2441no—
601C2×C4—2441no—
602C2×C4—2441no—
603C2×C4—2441no—
604C2×C4—2441no—
605C2×C4#1451979812yes—
606C2×C4C22×C22⋊D6111yescharacteristic
607C2×C4#41×D6111yescharacteristic
608C2×C4C2×#1175822411yescharacteristic
609C2×C4C2×#1175822411yescharacteristic
610C2×C4#1451982211yescharacteristic
611C2×C4#1451982411yescharacteristic
612C8—22no—
613C8—41no—
614C8—2441no—
615C8—2441no—
616C22—24no—
617C22—42no—
618C22—42no—
619C22—42no—
620C22—42no—
621C22—42no—
622C22—42no—
623C22#1684027211yescharacteristic
624C22D4×C22⋊D6111yescharacteristic
625C22#1684041811yescharacteristic
626C22C2×#1451979811yescharacteristic
627C22#137×D6111yescharacteristic
628C22#1684056811yescharacteristic
629C22#1684057111yescharacteristic
630C4—22no—
631C4—42no—
632C4—41no—
633C4—81no—
634C4—81no—
635C4—1222no—
636C4—4881no—
637C4—4881no—
638C4—4881no—
639C4—2441no—
640C4—2441no—
641C4—2441no—
642C4—2442no—
643C4C2×#1451975811yescharacteristic
644C4#1684035711yescharacteristic
645C4#1684041111yescharacteristic
646C4#1684045511yescharacteristic
647C2—44no—
648C2#2144955011yescharacteristic
649C2#2145588711yescharacteristic
650C2#2145606911yescharacteristic
651C2#2145732311yescharacteristic
652C2#2145752611yescharacteristic
653C2#2145847211yescharacteristic
654C2#2145859511yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#461C2
socleC23×C61C22×D4
prime radicalC22×D4×C61C23
hypercenterC2×#461D61
hypocenterC61#26170
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#461

Complements

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

NH ≅ G/Ntype
2. #21449591C2semidirect
4. #21447369C2semidirect
5. #21447317C2semidirect
6. #21447240C2semidirect
7. #21447062C2semidirect
8. #21446492C2semidirect
9. #21446373C2semidirect
13. #21446071C2semidirect
388. C61#26170semidirect

Maximal subgroups

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

subgroupindex
C2×SD16×Dic612
#214495912
#214487242
#214473692
#214473172
#214472402
#214470622
#214464922
#214463732
#214461552
#214460712
#214460662
C2×C61×#4612
#2617061