Finite Group Databasebeta

← #29709167

Subgroups — #29709167

Order 15616.

924 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839853C211yesmaximal, characteristic
2C2×SD16×Dic61C211yesmaximal, characteristic
3#21448388C211yesmaximal, characteristic, complemented
4#21448281C211yesmaximal, characteristic
5#21447299C211yesmaximal, characteristic, complemented
6#21447225C211yesmaximal, characteristic
7#21447200C211yesmaximal, characteristic, complemented
8#21447199C211yesmaximal, characteristic, complemented
9#21447136C211yesmaximal, characteristic
10#21447063C211yesmaximal, characteristic, complemented
11#21446900C211yesmaximal, characteristic
12C4⋊C8×Dic61C211yesmaximal, characteristic, complemented
13#103×Dic61C211yesmaximal, characteristic, complemented
14#21445638C211yesmaximal, characteristic, complemented
15C61×#2055C211yesmaximal, characteristic
16C22×#11758200C2211yescharacteristic
17C2×#14519775C2211yescharacteristic
18C2×Q8×Dic61C2211yescharacteristic
19C2×#14519742C2211yescharacteristic
20C2×D4×Dic61C2211yescharacteristic
21C2×#14519733—21no—
22#16839853—24no—
23SD16×Dic61—24no—
24C2×#14519709C2211yescharacteristic
25C2×#14519708C2211yescharacteristic
26C2×C4⋊#92418C2211yescharacteristic
27C2×#14519702C2211yescharacteristic
28C2×#14519702C2211yescharacteristic
29C2×#14519701C2211yescharacteristic
30C2×#14519701C2211yescharacteristic
31C2×#14519700C2211yescharacteristic
32C2×#14519698C2211yescharacteristic
33C2×C8×Dic61C2211yescharacteristic
34C2×C8×Dic61C2211yescharacteristic
35#16839668C2211yescharacteristic
36#16839665C2211yescharacteristic
37#16839660C2211yescharacteristic
38#16839659—21no—
39#16839659—21no—
40#16839658C2211yescharacteristic
41#16839656—21no—
42#16839650—21no—
43#16839631C2211yescharacteristic
44C4⋊C4×Dic61C2211yescharacteristic
45#16839613—21no—
46#16839613—21no—
47C22⋊C4×Dic61—21no—
48C42×Dic61C2211yescharacteristic
49#16839525C2211yescharacteristic
50#16839524C2211yescharacteristic
51#16839524C2211yescharacteristic
52#16839516C2211yescharacteristic
53#16839516C2211yescharacteristic
54C22×SD16×C61C2211yescharacteristic
55C2×C4⋊C8×C61C2211yescharacteristic
56C2×#103×C61C2211yescharacteristic
57C2×#102×C61C2211yescharacteristic
58C61×#391C2211yescharacteristic
59C61×#390C2211yescharacteristic
60C61×#340C2211yescharacteristic
61C22×Dic122C2311yescharacteristic
62C23×Dic61—21no—
63C2×#11758200—22no—
64C2×#11758200—22no—
65C2×#11758200—22no—
66C22×#92418C2311yescharacteristic
67C22×#92418C2311yescharacteristic
68#14519775—22no—
69#14519775—22no—
70Q8×Dic61—22no—
71Q8×Dic61—22no—
72#14519742—22no—
73#14519742—22no—
74D4×Dic61—22no—
75D4×Dic61D414yes—
76#14519733—44no—
77C2×C4⋊Dic61—21no—
78C2×C4⋊Dic61C2311yescharacteristic
79C2×C4⋊Dic61C2311yescharacteristic
80C2×C4⋊Dic61C2311yescharacteristic
81C2×#11758196—21no—
82C2×#11758196—21no—
83C2×#11758196—21no—
84C2×#11758196—21no—
85C2×#11758196—21no—
86C2×#11758196—21no—
87C2×#11758196—21no—
88C2×C22⋊Dic61—21no—
89C2×C22⋊Dic61—21no—
90C2×C22⋊Dic61—21no—
91C2×C4×Dic61—21no—
92C2×C4×Dic61—21no—
93C2×C4×Dic61—21no—
94C2×C4×Dic61D411yescharacteristic
95C2×C4×Dic61D411yescharacteristic
96C2×C4×Dic61D411yescharacteristic
97C2×C4×Dic61D411yescharacteristic
98C2×C4×Dic61C2311yescharacteristic
99C2×C4×Dic61C2311yescharacteristic
100#14519709—22no—
101#14519708—22no—
102C4⋊#92418—22no—
103#14519702—22no—
104#14519702—22no—
105#14519701—22no—
106#14519701—22no—
107#14519700—22no—
108#14519698—22no—
109C8×Dic61—22no—
110C8×Dic61—22no—
111#14519684—21no—
112#14519684—21no—
113#14519684—21no—
114#14519684—21no—
115#14519684—21no—
116C22×Q8×C61C2311yescharacteristic
117C22×D4×C61C2311yescharacteristic
118C2×SD16×C61—22no—
119C2×SD16×C61—22no—
120C2×SD16×C61—22no—
121C22×C8×C61C2311yescharacteristic
122C22×C8×C61C2311yescharacteristic
123C2×C4⋊C4×C61C2311yescharacteristic
124C2×C4⋊C4×C61C2311yescharacteristic
125C2×C22⋊C4×C61—21no—
126C2×C22⋊C4×C61—21no—
127C2×C42×C61C2311yescharacteristic
128C4⋊C8×C61—22no—
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#4112yes—
140C2×D4×C61—21no—
141C2×D4×C61—21no—
142C2×D4×C61—21no—
143C2×D4×C61—21no—
144C2×D4×C61—21no—
145C2×D4×C61C2×D412yes—
146C22×C4×C61—21no—
147C22×C4×C61—21no—
148C22×C4×C61—21no—
149C22×C4×C61C2×D411yescharacteristic
150C22×C4×C61C2×D411yescharacteristic
151C22×C4×C61C2411yescharacteristic
152C22×Dic61—21no—
153C22×Dic61—21no—
154C22×Dic61—24no—
155C22×Dic61—21no—
156C22×Dic61—21no—
157C22×Dic61—42no—
158C22×Dic61—42no—
159C22×Dic61C2×D411yescharacteristic
160C22×Dic61C2×D411yescharacteristic
161C22×Dic61C2×D411yescharacteristic
162C22×Dic61C2×D411yescharacteristic
163C2×Dic122—21no—
164C2×Dic122—21no—
165C2×Dic122—21no—
166C2×Dic122—21no—
167C2×Dic122—21no—
168C2×Dic122#4112yes—
169SD16×C61—28no—
170C2×C8×C61—21no—
171C2×C8×C61—21no—
172C2×C8×C61—21no—
173C2×C8×C61—21no—
174C2×C8×C61—21no—
175C2×C8×C61#4112yes—
176C4⋊C4×C61—21no—
177C4⋊C4×C61—21no—
178C4⋊C4×C61—21no—
179C4⋊C4×C61—21no—
180C22⋊C4×C61—42no—
181C22⋊C4×C61—42no—
182C42×C61—21no—
183C42×C61—21no—
184C22⋊Dic61—24no—
185C22⋊Dic61—42no—
186C22⋊Dic61—42no—
187#11758200—28no—
188C4⋊Dic61—21no—
189C4⋊Dic61—21no—
190C4⋊Dic61—21no—
191C4⋊Dic61—21no—
192C4⋊Dic61—21no—
193C4⋊Dic61—42no—
194C4⋊Dic61C2×D412yes—
195#11758196—24no—
196#11758196—24no—
197#11758196—42no—
198#11758196—42no—
199#11758196—42no—
200#11758196—42no—
201#11758196—42no—
202C4×Dic61—21no—
203C4×Dic61—21no—
204C4×Dic61—22no—
205C4×Dic61—22no—
206C4×Dic61—41no—
207C4×Dic61—42no—
208C4×Dic61—42no—
209C4×Dic61—42no—
210C4×Dic61SD1614yescomplemented
211C4×Dic61C2×D412yes—
212C4×Dic61#4112yes—
213C2×#92418—21no—
214C2×#92418—21no—
215C2×#92418—21no—
216C2×#92418—21no—
217C2×#92418—21no—
218C2×#92418#4112yes—
219C23×C61—22no—
220C23×C61—41no—
221C23×C61—41no—
222C23×C61—41no—
223C23×C61—41no—
224C23×C61—41no—
225C23×C61—41no—
226C23×C61C22×D411yescharacteristic
227Q8×C61—22no—
228Q8×C61—22no—
229Q8×C61—41no—
230Q8×C61—41no—
231D4×C61—22no—
232D4×C61—22no—
233D4×C61—41no—
234D4×C61—41no—
235C2×C4×C61—21no—
236C2×C4×C61—21no—
237C2×C4×C61—22no—
238C2×C4×C61—21no—
239C2×C4×C61—21no—
240C2×C4×C61—21no—
241C2×C4×C61—41no—
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×C61C2×SD1612yes—
250C2×C4×C61C22×D411yescharacteristic
251C2×C4×C61C2×#4111yescharacteristic
252C2×C4×C61C2×#4111yescharacteristic
253C2×C4×C61C2×#4111yescharacteristic
254C2×C4×C61C2×#4111yescharacteristic
255C2×C4×C61C2×#4111yescharacteristic
256C2×Dic61—21no—
257C2×Dic61—21no—
258C2×Dic61—21no—
259C2×Dic61—21no—
260C2×Dic61—21no—
261C2×Dic61—21no—
262C2×Dic61—22no—
263C2×Dic61—22no—
264C2×Dic61—41no—
265C2×Dic61—41no—
266C2×Dic61—41no—
267C2×Dic61—41no—
268C2×Dic61—48no—
269C2×Dic61—41no—
270C2×Dic61—41no—
271C2×Dic61—41no—
272C2×Dic61—41no—
273C2×Dic61—41no—
274C2×Dic61—41no—
275C2×Dic61#12112yes—
276C2×Dic61#12112yes—
277C2×Dic61C2×SD1612yes—
278C2×Dic61C2×SD1612yes—
279C2×Dic61#13512yes—
280C2×Dic61#13712yes—
281Dic122—22no—
282Dic122—22no—
283Dic122—41no—
284Dic122—41no—
285C8×C61—22no—
286C8×C61—41no—
287#92418—22no—
288#92418—41no—
289#23677—611nomaximal, Sylow
290C22×C61—42no—
291C22×C61—42no—
292C22×C61—42no—
293C22×C61—81no—
294C22×C61—81no—
295C22×C61—81no—
296C22×C61—81no—
297C22×C61C2×#12111yescharacteristic
298C22×C61#52511yescharacteristic
299C22×C61#53211yescharacteristic
300C22×C61C22×SD1611yescharacteristic
301C22×C61C2×#13511yescharacteristic
302C22×C61C2×#13711yescharacteristic
303C22×C61#57511yescharacteristic
304C4×C61—22no—
305C4×C61—42no—
306C4×C61—41no—
307C4×C61—81no—
308C4×C61—81no—
309C4×C61#52511yescharacteristic
310C4×C61#52511yescharacteristic
311C4×C61#52911yescharacteristic
312C4×C61#52911yescharacteristic
313Dic61—22no—
314Dic61—22no—
315Dic61—22no—
316Dic61—42no—
317Dic61—42no—
318Dic61—81no—
319Dic61—81no—
320Dic61#46114yes—
321C2×C4×SD16—611no—
322C2×C4×SD16—611no—
323C4×C4⋊Q8—611no—
324#2293—611no—
325#2084—611no—
326#2071—611no—
327#2066—611no—
328#2066—611no—
329#2055—611no—
330#2055D6111yescharacteristic
331#2043—611no—
332#2013—611no—
333C4×C4⋊C8—611no—
334C4×#103—611no—
335C4×#103—611no—
336C2×C61—82no—
337C2×C61#239211yescharacteristic
338C2×C61C2×#46111yescharacteristic
339C2×C61#303511yescharacteristic
340C2×C61#310311yescharacteristic
341C2×C61#310811yescharacteristic
342C2×C61#316711yescharacteristic
343C2×C61#317211yescharacteristic
344C22×SD16—611no—
345C22×SD16C2×D6111yescharacteristic
346C2×C4⋊Q8—611no—
347C2×C4×Q8—611no—
348C2×C4×Q8—611no—
349C2×C4×D4—611no—
350C2×#117—1221no—
351C4×SD16—1224no—
352C4×SD16—1224no—
353C2×#106—611no—
354C2×#106—611no—
355C2×C4⋊C8—611no—
356C2×C4⋊C8C2×D6111yescharacteristic
357C2×#103—611no—
358C2×#103—611no—
359C2×#103—611no—
360C2×#103—611no—
361C2×#103—611no—
362C2×#103C2×D6111yescharacteristic
363C2×#102—611no—
364C2×#102C2×D6111yescharacteristic
365C2×C4×C8—611no—
366C2×C4×C8—611no—
367#391—611no—
368#391C2×D6111yescharacteristic
369#390—611no—
370#390C2×D6111yescharacteristic
371#389—611no—
372#389—611no—
373#389—1221no—
374#389—1221no—
375#388—1221no—
376#388—1221no—
377#384—611no—
378C4×C4⋊C4—611no—
379C4×C4⋊C4—1221no—
380C4×C4⋊C4—1221no—
381C4×C22⋊C4—1221no—
382C43—611no—
383#340—611no—
384#340—611no—
385#340—611no—
386#340C2×D6111yescharacteristic
387#336—611no—
388#336—611no—
389C61#2367711yescharacteristic, complemented, Sylow
390C22×Q8—611no—
391C22×Q8C22×D6111yescharacteristic
392C22×D4C22×D6111yescharacteristic
393C23×C4—1221no—
394C2×SD16—22no—
395C2×SD16—22no—
396C2×SD16—22no—
397C2×SD16—1222no—
398C2×SD16—1222no—
399C2×SD16—1222no—
400C22×C8—611no—
401C22×C8—611no—
402C22×C8C22×D6111yescharacteristic
403C22×C8C22×D6111yescharacteristic
404C4⋊Q8—1222no—
405C4⋊Q8—1222no—
406C4×Q8—1222no—
407C4×Q8—1222no—
408C4×Q8—1222no—
409C4×Q8—1222no—
410C4×D4—614no—
411C4×D4—1222no—
412#117—2444no—
413C2×C4⋊C4—611no—
414C2×C4⋊C4—611no—
415C2×C4⋊C4—611no—
416C2×C4⋊C4—1221no—
417C2×C4⋊C4—1221no—
418C2×C4⋊C4—1221no—
419C2×C4⋊C4—1221no—
420C2×C4⋊C4—1221no—
421C2×C4⋊C4—1221no—
422C2×C4⋊C4—1221no—
423C2×C4⋊C4—1221no—
424C2×C4⋊C4C22×D6111yescharacteristic
425C2×C4⋊C4C22×D6111yescharacteristic
426C2×C22⋊C4—21no—
427C2×C22⋊C4—21no—
428C2×C22⋊C4—1221no—
429C2×C22⋊C4—1221no—
430C2×C22⋊C4—1221no—
431C2×C42—611no—
432C2×C42—611no—
433C2×C42—611no—
434C2×C42—611no—
435C2×C42—611no—
436C2×C42—611no—
437C2×C42—1221no—
438C2×C42—1221no—
439C2×C42—1221no—
440C2×C42C22×D6111yescharacteristic
441#106—1222no—
442#106—1222no—
443C4⋊C8—22no—
444C4⋊C8—1222no—
445#103—22no—
446#103—1222no—
447#103—1222no—
448#103—1222no—
449#103—1222no—
450#103—1222no—
451#102—22no—
452#102—1222no—
453C4×C8—1222no—
454C4×C8—1222no—
455#95—21no—
456#95—21no—
457#95—1221no—
458#95—1221no—
459#95—1221no—
460#95—1221no—
461#95—1221no—
462C24—21no—
463C2×Q8—21no—
464C2×Q8—21no—
465C2×Q8—21no—
466C2×Q8—21no—
467C2×Q8—21no—
468C2×Q8—612no—
469C2×Q8—1221no—
470C2×Q8—1221no—
471C2×Q8—1221no—
472C2×Q8—1221no—
473C2×Q8—1221no—
474C2×Q8#1175822412yes—
475C2×D4—21no—
476C2×D4—21no—
477C2×D4—21no—
478C2×D4—21no—
479C2×D4—21no—
480C2×D4D4×D6112yes—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—611no—
485C22×C4—611no—
486C22×C4—611no—
487C22×C4—611no—
488C22×C4—1224no—
489C22×C4—1221no—
490C22×C4—1221no—
491C22×C4—1221no—
492C22×C4—1221no—
493C22×C4—2442no—
494C22×C4—2442no—
495C22×C4D4×D6111yescharacteristic
496C22×C4D4×D6111yescharacteristic
497C22×C4C23×D6111yescharacteristic
498SD16—28no—
499SD16—1228no—
500C2×C8—21no—
501C2×C8—21no—
502C2×C8—21no—
503C2×C8—21no—
504C2×C8—21no—
505C2×C8—612no—
506C2×C8—1221no—
507C2×C8—1221no—
508C2×C8—1221no—
509C2×C8—1221no—
510C2×C8—1221no—
511C2×C8#1175822412yes—
512C4⋊C4—21no—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—21no—
516C4⋊C4—612no—
517C4⋊C4—1224no—
518C4⋊C4—1224no—
519C4⋊C4—1221no—
520C4⋊C4—1221no—
521C4⋊C4—1221no—
522C4⋊C4—1221no—
523C4⋊C4—1221no—
524C4⋊C4—2442no—
525C4⋊C4—2442no—
526C4⋊C4—2442no—
527C4⋊C4—2442no—
528C4⋊C4—2442no—
529C4⋊C4—2442no—
530C22⋊C4—42no—
531C22⋊C4—42no—
532C22⋊C4—1224no—
533C22⋊C4—2442no—
534C22⋊C4—2442no—
535C42—21no—
536C42—21no—
537C42—612no—
538C42—612no—
539C42—614no—
540C42—1222no—
541C42—1222no—
542C42—1221no—
543C42—1221no—
544C42—2441no—
545C42—2442no—
546C42—2442no—
547C42—2442no—
548C23—22no—
549C23—41no—
550C23—41no—
551C23—41no—
552C23—41no—
553C23—41no—
554C23—41no—
555C23C2×D4×D6111yescharacteristic
556Q8—22no—
557Q8—22no—
558Q8—41no—
559Q8—41no—
560Q8—1222no—
561Q8—1222no—
562Q8—2441no—
563Q8—2441no—
564D4—22no—
565D4—22no—
566D4—41no—
567D4—41no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—22no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—612no—
583C2×C4—612no—
584C2×C4—612no—
585C2×C4—612no—
586C2×C4—612no—
587C2×C4—612no—
588C2×C4—1221no—
589C2×C4—1221no—
590C2×C4—1221no—
591C2×C4—1221no—
592C2×C4—1221no—
593C2×C4—1221no—
594C2×C4—1222no—
595C2×C4—1222no—
596C2×C4—2448no—
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—2441no—
606C2×C4—2441no—
607C2×C4SD16×D6112yes—
608C2×C4C2×D4×D6111yescharacteristic
609C2×C4#41×D6111yescharacteristic
610C2×C4#41×D6111yescharacteristic
611C2×C4#41×D6111yescharacteristic
612C2×C4C2×#1175822411yescharacteristic
613C2×C4C2×#1175822411yescharacteristic
614C8—22no—
615C8—41no—
616C8—1222no—
617C8—2441no—
618C22—42no—
619C22—42no—
620C22—42no—
621C22—81no—
622C22—81no—
623C22—81no—
624C22—81no—
625C22#121×D6111yescharacteristic
626C22#1684027211yescharacteristic
627C22#1684030711yescharacteristic
628C22C2×SD16×D6111yescharacteristic
629C22#135×D6111yescharacteristic
630C22#137×D6111yescharacteristic
631C22#1684055811yescharacteristic
632C4—22no—
633C4—41no—
634C4—42no—
635C4—81no—
636C4—81no—
637C4—614no—
638C4—1222no—
639C4—1222no—
640C4—1222no—
641C4—4881no—
642C4—4881no—
643C4—2442no—
644C4—2442no—
645C4#1684026811yescharacteristic
646C4#1684027011yescharacteristic
647C4#1684028911yescharacteristic
648C4#1684029111yescharacteristic
649C2—82no—
650C2#2144906211yescharacteristic
651C2#461×D6111yescharacteristic
652C2#2145527111yescharacteristic
653C2#2145584911yescharacteristic
654C2#2145589811yescharacteristic
655C2#2145637311yescharacteristic
656C2#2145641811yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2055C2
socleC23×C61C22×D4
prime radicalC22×D4×C61C23
hypercenter#2055D61
hypocenterC61#23677
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2055

Complements

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

NH ≅ G/Ntype
3. #21448388C2semidirect
5. #21447299C2semidirect
7. #21447200C2semidirect
8. #21447199C2semidirect
10. #21447063C2semidirect
12. C4⋊C8×Dic61C2semidirect
13. #103×Dic61C2semidirect
14. #21445638C2semidirect
210. C4×Dic61SD16semidirect
389. C61#23677semidirect

Maximal subgroups

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

subgroupindex
C2×#168398532
C2×SD16×Dic612
#214483882
#214482812
#214472992
#214472252
#214472002
#214471992
#214471362
#214470632
#214469002
C4⋊C8×Dic612
#103×Dic612
#214456382
C61×#20552
#2367761