Finite Group Databasebeta

← #29740116

Subgroups — #29740116

Order 15616.

904 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839965C211yesmaximal, characteristic
2C2×#16839853C211yesmaximal, characteristic
3#21449598C211yesmaximal, characteristic, complemented
4D4×C4⋊Dic61C211yesmaximal, characteristic
5#21447362C211yesmaximal, characteristic, complemented
6#21447323C211yesmaximal, characteristic, complemented
7#21447229C211yesmaximal, characteristic, complemented
8#21447064C211yesmaximal, characteristic, complemented
9#21446497C211yesmaximal, characteristic, complemented
10#21446375C211yesmaximal, characteristic, complemented
11#21446155C211yesmaximal, characteristic
12#21446153C211yesmaximal, characteristic
13#21446088C211yesmaximal, characteristic, complemented
14#21446078C211yesmaximal, characteristic
15C61×#1877C211yesmaximal, characteristic
16C22×#11758200C2211yescharacteristic
17C2×#14519775C2211yescharacteristic
18C2×#14519742C2211yescharacteristic
19C2×D4×Dic61C2211yescharacteristic
20C2×D4×Dic61C2211yescharacteristic
21C22×C4⋊Dic61—21no—
22#16839965—24no—
23#16839853—24no—
24C2×#14519708C2211yescharacteristic
25C2×#14519708C2211yescharacteristic
26C2×#14519706C2211yescharacteristic
27C2×#14519702C2211yescharacteristic
28C2×#14519702C2211yescharacteristic
29C2×#14519701C2211yescharacteristic
30C2×#14519701C2211yescharacteristic
31C2×#14519700C2211yescharacteristic
32C2×#14519698C2211yescharacteristic
33C2×C8×Dic61C2211yescharacteristic
34#16839703—21no—
35#16839696—21no—
36#16839669C2211yescharacteristic
37#16839668C2211yescharacteristic
38#16839661C2211yescharacteristic
39#16839661C2211yescharacteristic
40#16839658C2211yescharacteristic
41#16839645—21no—
42#16839644—21no—
43#16839637—21no—
44#16839637—21no—
45#16839632C2211yescharacteristic
46#16839622—21no—
47C4×C4⋊Dic61C2211yescharacteristic
48#16839525C2211yescharacteristic
49#16839525C2211yescharacteristic
50#16839524C2211yescharacteristic
51#16839517C2211yescharacteristic
52#16839516C2211yescharacteristic
53#16839516C2211yescharacteristic
54C22×SD16×C61C2211yescharacteristic
55C2×C4×D4×C61C2211yescharacteristic
56C2×#103×C61C2211yescharacteristic
57C2×#102×C61C2211yescharacteristic
58C61×#391C2211yescharacteristic
59C61×#340C2211yescharacteristic
60C61×#336C2211yescharacteristic
61C22×Dic122C2311yescharacteristic
62C23×Dic61—21no—
63C23×Dic61—21no—
64C2×#11758200—22no—
65C2×#11758200—22no—
66C2×#11758200D414yes—
67C22×#92418C2311yescharacteristic
68C22×#92418C2311yescharacteristic
69#14519775—22no—
70#14519775—41no—
71#14519742—22no—
72#14519742—41no—
73D4×Dic61—22no—
74D4×Dic61—22no—
75D4×Dic61—41no—
76D4×Dic61—41no—
77C2×C4⋊Dic61—21no—
78C2×C4⋊Dic61—21no—
79C2×C4⋊Dic61—21no—
80C2×C4⋊Dic61—21no—
81C2×C4⋊Dic61—44no—
82C2×C4⋊Dic61D411yescharacteristic
83C2×C4⋊Dic61D411yescharacteristic
84C2×C4⋊Dic61C2311yescharacteristic
85C2×C4⋊Dic61C2311yescharacteristic
86C2×C4⋊Dic61C2311yescharacteristic
87C2×#11758196—21no—
88C2×#11758196—21no—
89C2×#11758196—21no—
90C2×#11758196—21no—
91C2×C22⋊Dic61—21no—
92C2×C22⋊Dic61—21no—
93C2×C4×Dic61—21no—
94C2×C4×Dic61C2311yescharacteristic
95C2×C4×Dic61C2311yescharacteristic
96#14519708—22no—
97#14519708—41no—
98#14519706—22no—
99#14519702—22no—
100#14519702—22no—
101#14519701—22no—
102#14519701—41no—
103#14519700—22no—
104#14519698—22no—
105C8×Dic61—22no—
106#14519684—21no—
107#14519684—21no—
108#14519684—21no—
109#14519684—21no—
110#14519684—21no—
111#14519684—21no—
112#14519684—21no—
113#14519684D411yescharacteristic
114#14519684D411yescharacteristic
115C22×Q8×C61C2311yescharacteristic
116C22×D4×C61C2311yescharacteristic
117C23×C4×C61—21no—
118C2×SD16×C61—22no—
119C2×SD16×C61—22no—
120C2×SD16×C61—22no—
121C22×C8×C61C2311yescharacteristic
122C22×C8×C61C2311yescharacteristic
123C4×D4×C61—22no—
124C4×D4×C61—22no—
125C2×C4⋊C4×C61C2311yescharacteristic
126C2×C4⋊C4×C61C2311yescharacteristic
127C2×C22⋊C4×C61—21no—
128C2×C42×C61C2311yescharacteristic
129#103×C61—22no—
130#102×C61—22no—
131#95×C61—21no—
132#95×C61—21no—
133C24×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×D4×C61—21no—
141C2×D4×C61—21no—
142C2×D4×C61—21no—
143C2×D4×C61C2×D412yes—
144C2×D4×C61#4112yes—
145C2×D4×C61#4112yes—
146C22×C4×C61—24no—
147C22×C4×C61—21no—
148C22×C4×C61—21no—
149C22×C4×C61—21no—
150C22×C4×C61—42no—
151C22×C4×C61—42no—
152C22×C4×C61C2×D411yescharacteristic
153C22×C4×C61C2×D411yescharacteristic
154C22×C4×C61C2411yescharacteristic
155C22×Dic61—21no—
156C22×Dic61—21no—
157C22×Dic61—21no—
158C22×Dic61—21no—
159C22×Dic61—42no—
160C22×Dic61—42no—
161C22×Dic61—42no—
162C22×Dic61—42no—
163C22×Dic61—81no—
164C22×Dic61—81no—
165C22×Dic61C2×D411yescharacteristic
166C22×Dic61C2×D411yescharacteristic
167C22×Dic61C2×D411yescharacteristic
168C22×Dic61C2×D411yescharacteristic
169C2×Dic122—21no—
170C2×Dic122—21no—
171C2×Dic122—21no—
172C2×Dic122—21no—
173C2×Dic122—21no—
174C2×Dic122C2×D412yes—
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×C61—21no—
182C4⋊C4×C61—21no—
183C4⋊C4×C61—21no—
184C4⋊C4×C61—21no—
185C4⋊C4×C61#4112yes—
186C22⋊C4×C61—24no—
187C42×C61—21no—
188C42×C61—21no—
189C22⋊Dic61—42no—
190C22⋊Dic61—42no—
191#11758200—28no—
192C4⋊Dic61—21no—
193C4⋊Dic61—21no—
194C4⋊Dic61—21no—
195C4⋊Dic61—21no—
196C4⋊Dic61—21no—
197C4⋊Dic61—21no—
198C4⋊Dic61—41no—
199C4⋊Dic61—42no—
200C4⋊Dic61—42no—
201C4⋊Dic61—42no—
202C4⋊Dic61—81no—
203C4⋊Dic61SD1614yescomplemented
204#11758196—42no—
205#11758196—42no—
206#11758196—42no—
207#11758196—42no—
208C4×Dic61—21no—
209C4×Dic61—21no—
210C4×Dic61—21no—
211C4×Dic61—21no—
212C4×Dic61—42no—
213C2×#92418—21no—
214C2×#92418—21no—
215C2×#92418—21no—
216C2×#92418—21no—
217C2×#92418—21no—
218C2×#92418C2×D412yes—
219C23×C61—22no—
220C23×C61—22no—
221C23×C61—22no—
222C23×C61—41no—
223C23×C61—41no—
224C23×C61—41no—
225C23×C61—41no—
226C23×C61C22×D411yescharacteristic
227Q8×C61—41no—
228Q8×C61—41no—
229Q8×C61—41no—
230Q8×C61—41no—
231D4×C61—22no—
232D4×C61—22no—
233D4×C61—22no—
234D4×C61#12314yes—
235C2×C4×C61—21no—
236C2×C4×C61—21no—
237C2×C4×C61—21no—
238C2×C4×C61—22no—
239C2×C4×C61—21no—
240C2×C4×C61—21no—
241C2×C4×C61—48no—
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×C61C2×SD1612yes—
251C2×C4×C61C22×D411yescharacteristic
252C2×C4×C61C2×#4111yescharacteristic
253C2×C4×C61C2×#4111yescharacteristic
254C2×C4×C61C2×#4111yescharacteristic
255C2×C4×C61H(5,𝔽2)11yescharacteristic
256C2×C4×C612−1+411yescharacteristic
257C2×Dic61—21no—
258C2×Dic61—21no—
259C2×Dic61—21no—
260C2×Dic61—21no—
261C2×Dic61—21no—
262C2×Dic61—21no—
263C2×Dic61—22no—
264C2×Dic61—21no—
265C2×Dic61—21no—
266C2×Dic61—21no—
267C2×Dic61—21no—
268C2×Dic61—41no—
269C2×Dic61—41no—
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—84no—
280C2×Dic61—84no—
281C2×Dic61C2×SD1612yes—
282C2×Dic61C2×SD1612yes—
283Dic122—22no—
284Dic122—22no—
285Dic122—41no—
286Dic122—41no—
287C8×C61—41no—
288C8×C61—41no—
289#92418—22no—
290#92418—41no—
291#26166—611nomaximal, Sylow
292C22×C61—24no—
293C22×C61—42no—
294C22×C61—42no—
295C22×C61—42no—
296C22×C61—42no—
297C22×C61—42no—
298C22×C61—42no—
299C22×C61#52511yescharacteristic
300C22×C61D4211yescharacteristic
301C22×C61#54811yescharacteristic
302C22×C61C22×SD1611yescharacteristic
303C22×C61#57511yescharacteristic
304C22×C61#57711yescharacteristic
305C22×C61#57811yescharacteristic
306C4×C61—22no—
307C4×C61—42no—
308C4×C61—41no—
309C4×C61—81no—
310C4×C61—81no—
311C4×C61C2×#12311yescharacteristic
312C4×C61#53911yescharacteristic
313C4×C61#54711yescharacteristic
314C4×C61#55311yescharacteristic
315Dic61—22no—
316Dic61—22no—
317Dic61—42no—
318Dic61—41no—
319Dic61—41no—
320Dic61—81no—
321Dic61—81no—
322Dic61—81no—
323C2×#461—611no—
324C2×C4×SD16—611no—
325#2446—611no—
326D4×C4⋊C4—611no—
327#2096—611no—
328#2088—611no—
329#2072—611no—
330#2043—611no—
331#1933—611no—
332#1916—611no—
333#1877—611no—
334#1877—611no—
335#1877D6111yescharacteristic
336#1865—611no—
337#1864—611no—
338C2×C61—44no—
339C2×C61#244311yescharacteristic
340C2×C61#310111yescharacteristic
341C2×C61#312011yescharacteristic
342C2×C61#326911yescharacteristic
343C2×C61D4×SD1611yescharacteristic
344C2×C61#335911yescharacteristic
345C2×C61#337611yescharacteristic
346C22×SD16—611no—
347C22×SD16C2×D6111yescharacteristic
348C2×C4⋊Q8—611no—
349C2×C4×Q8—611no—
350C2×C4×D4—611no—
351C2×C4×D4—611no—
352C2×C4×D4C2×D6111yescharacteristic
353C22×C4⋊C4—1221no—
354#461—1224no—
355C4×SD16—1224no—
356C2×#106—611no—
357C2×#106—611no—
358C2×C4⋊C8—611no—
359C2×#103—611no—
360C2×#103—611no—
361C2×#103—611no—
362C2×#103—611no—
363C2×#103—611no—
364C2×#103C2×D6111yescharacteristic
365C2×#102—611no—
366C2×#102C2×D6111yescharacteristic
367C2×C4×C8—611no—
368#400—1221no—
369#398—1221no—
370#391—611no—
371#391—611no—
372#391C2×D6111yescharacteristic
373#389—611no—
374#389—611no—
375#389—611no—
376#387—1221no—
377#387—1221no—
378#385—1221no—
379#385—1221no—
380#384—611no—
381#380—1221no—
382C4×C4⋊C4—611no—
383#340—611no—
384#340—611no—
385#340—611no—
386#340C2×D6111yescharacteristic
387#336—611no—
388#336—611no—
389#336—611no—
390#336C2×D6111yescharacteristic
391C61#2616611yescharacteristic, complemented, Sylow
392C22×Q8—611no—
393C22×Q8C22×D6111yescharacteristic
394C22×D4C22×D6111yescharacteristic
395C23×C4—21no—
396C23×C4—1221no—
397C23×C4—1221no—
398C2×SD16—22no—
399C2×SD16—22no—
400C2×SD16—22no—
401C2×SD16—614no—
402C2×SD16—1222no—
403C2×SD16—1222no—
404C22×C8—611no—
405C22×C8—611no—
406C22×C8C22×D6111yescharacteristic
407C22×C8C22×D6111yescharacteristic
408C4⋊Q8—1222no—
409C4⋊Q8—2441no—
410C4×Q8—1222no—
411C4×Q8—2441no—
412C4×D4—22no—
413C4×D4—22no—
414C4×D4—1222no—
415C4×D4—1222no—
416C4×D4—2441no—
417C4×D4—2441no—
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⋊C4—1221no—
428C2×C4⋊C4—1221no—
429C2×C4⋊C4—1221no—
430C2×C4⋊C4—1221no—
431C2×C4⋊C4—2444no—
432C2×C4⋊C4C22×D6111yescharacteristic
433C2×C4⋊C4C22×D6111yescharacteristic
434C2×C22⋊C4—21no—
435C2×C22⋊C4—1221no—
436C2×C22⋊C4—1221no—
437C2×C42—611no—
438C2×C42—611no—
439C2×C42—1221no—
440C2×C42C22×D6111yescharacteristic
441#106—1222no—
442#106—2441no—
443C4⋊C8—1222no—
444#103—22no—
445#103—1222no—
446#103—1222no—
447#103—1222no—
448#103—1222no—
449#103—2441no—
450#102—22no—
451#102—1222no—
452C4×C8—1222no—
453#95—21no—
454#95—21no—
455#95—611no—
456#95—611no—
457#95—1221no—
458#95—1221no—
459#95—1221no—
460#95—1221no—
461#95—1221no—
462#95—1221no—
463#95—1221no—
464C24—21no—
465C2×Q8—21no—
466C2×Q8—21no—
467C2×Q8—21no—
468C2×Q8—21no—
469C2×Q8—21no—
470C2×Q8—21no—
471C2×Q8—612no—
472C2×Q8—1221no—
473C2×Q8—1221no—
474C2×Q8—1221no—
475C2×Q8—1221no—
476C2×Q8—1221no—
477C2×D4—21no—
478C2×D4—21no—
479C2×D4—21no—
480C2×D4#1175822212yes—
481C2×D4D4×D6112yes—
482C2×D4#1175822412yes—
483C22×C4—24no—
484C22×C4—21no—
485C22×C4—21no—
486C22×C4—21no—
487C22×C4—42no—
488C22×C4—42no—
489C22×C4—611no—
490C22×C4—611no—
491C22×C4—611no—
492C22×C4—611no—
493C22×C4—1221no—
494C22×C4—1221no—
495C22×C4—1221no—
496C22×C4—1221no—
497C22×C4—4881no—
498C22×C4—4881no—
499C22×C4—2442no—
500C22×C4—2442no—
501C22×C4—2442no—
502C22×C4—2442no—
503C22×C4D4×D6111yescharacteristic
504C22×C4D4×D6111yescharacteristic
505C22×C4C23×D6111yescharacteristic
506SD16—44no—
507SD16—1228no—
508C2×C8—21no—
509C2×C8—21no—
510C2×C8—21no—
511C2×C8—21no—
512C2×C8—21no—
513C2×C8—21no—
514C2×C8—612no—
515C2×C8—1221no—
516C2×C8—1221no—
517C2×C8—1221no—
518C2×C8—1221no—
519C2×C8—1221no—
520C4⋊C4—21no—
521C4⋊C4—21no—
522C4⋊C4—21no—
523C4⋊C4—614no—
524C4⋊C4—1221no—
525C4⋊C4—1221no—
526C4⋊C4—1221no—
527C4⋊C4—1221no—
528C4⋊C4—1221no—
529C4⋊C4—1221no—
530C4⋊C4—4881no—
531C4⋊C4—2441no—
532C4⋊C4—2442no—
533C4⋊C4—2442no—
534C4⋊C4—2442no—
535C4⋊C4—2442no—
536C4⋊C4—2442no—
537C4⋊C4—2442no—
538C4⋊C4—2442no—
539C4⋊C4#1175822412yes—
540C22⋊C4—24no—
541C22⋊C4—2442no—
542C22⋊C4—2442no—
543C42—21no—
544C42—21no—
545C42—1221no—
546C42—1221no—
547C42—1221no—
548C42—1221no—
549C42—2442no—
550C23—22no—
551C23—22no—
552C23—22no—
553C23—41no—
554C23—41no—
555C23—41no—
556C23—41no—
557C23C2×D4×D6111yescharacteristic
558Q8—41no—
559Q8—41no—
560Q8—41no—
561Q8—41no—
562Q8—1222no—
563Q8—1222no—
564Q8—2441no—
565Q8—2441no—
566D4—22no—
567D4—22no—
568D4—22no—
569D4#1451975914yes—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—22no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—48no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—612no—
586C2×C4—612no—
587C2×C4—1221no—
588C2×C4—1221no—
589C2×C4—1221no—
590C2×C4—1221no—
591C2×C4—1221no—
592C2×C4—1221no—
593C2×C4—1221no—
594C2×C4—1221no—
595C2×C4—1221no—
596C2×C4—1221no—
597C2×C4—1222no—
598C2×C4—4884no—
599C2×C4—4884no—
600C2×C4—2441no—
601C2×C4—2441no—
602C2×C4—2441no—
603C2×C4—2441no—
604C2×C4—2441no—
605C2×C4—2441no—
606C2×C4—2441no—
607C2×C4—2441no—
608C2×C4—2441no—
609C2×C4—2441no—
610C2×C4—2441no—
611C2×C4SD16×D6112yes—
612C2×C4C2×D4×D6111yescharacteristic
613C2×C4C2×#1175822411yescharacteristic
614C2×C4C2×#1175822411yescharacteristic
615C2×C4C2×#1175822211yescharacteristic
616C2×C4#1451982311yescharacteristic
617C2×C4#1451982411yescharacteristic
618C8—41no—
619C8—41no—
620C8—1222no—
621C8—2441no—
622C22—24no—
623C22—42no—
624C22—42no—
625C22—42no—
626C22—42no—
627C22—42no—
628C22—42no—
629C22#1684026811yescharacteristic
630C22#1684039611yescharacteristic
631C22#1684042111yescharacteristic
632C22C2×SD16×D6111yescharacteristic
633C22#1684055811yescharacteristic
634C22#1684057111yescharacteristic
635C22#1684057611yescharacteristic
636C4—22no—
637C4—41no—
638C4—42no—
639C4—81no—
640C4—81no—
641C4—1222no—
642C4—1222no—
643C4—4881no—
644C4—4881no—
645C4—4881no—
646C4—2442no—
647C4—2441no—
648C4—2441no—
649C4C2×#1451975911yescharacteristic
650C4#1684035611yescharacteristic
651C4#1684041211yescharacteristic
652C4#1684045811yescharacteristic
653C2—44no—
654C2#2144956011yescharacteristic
655C2#2145582211yescharacteristic
656C2#2145603411yescharacteristic
657C2#2145738011yescharacteristic
658C2#2145751811yescharacteristic
659C2#2145845911yescharacteristic
660C2#2145865111yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1877C2
socleC23×C61C22×D4
prime radicalC22×D4×C61C23
hypercenter#1877D61
hypocenterC61#26166
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1877

Complements

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

NH ≅ G/Ntype
3. #21449598C2semidirect
5. #21447362C2semidirect
6. #21447323C2semidirect
7. #21447229C2semidirect
8. #21447064C2semidirect
9. #21446497C2semidirect
10. #21446375C2semidirect
13. #21446088C2semidirect
203. C4⋊Dic61SD16semidirect
391. C61#26166semidirect

Maximal subgroups

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

subgroupindex
C2×#168399652
C2×#168398532
#214495982
D4×C4⋊Dic612
#214473622
#214473232
#214472292
#214470642
#214464972
#214463752
#214461552
#214461532
#214460882
#214460782
C61×#18772
#2616661