Finite Group Databasebeta

← #29716224

Subgroups — #29716224

Order 15616.

1012 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840392C211yesmaximal, characteristic
2C2×#16840071C211yesmaximal, characteristic, complemented
3C2×#16840055C211yesmaximal, characteristic, complemented
4#21448504C211yesmaximal, characteristic
5#21447154C211yesmaximal, characteristic, complemented
6#21447148C211yesmaximal, characteristic, complemented
7#21446960C211yesmaximal, characteristic, complemented
8#21446775C211yesmaximal, characteristic, complemented
9#21446769C211yesmaximal, characteristic, complemented
10#21446128C211yesmaximal, characteristic
11#21446123C211yesmaximal, characteristic
12#21446021C211yesmaximal, characteristic
13#21446008C211yesmaximal, characteristic
14#21445802C211yesmaximal, characteristic
15C61×#2026C211yesmaximal, characteristic, complemented
16#16840392—24no—
17#16840392—24no—
18C2×#14519777C2211yescharacteristic
19C2×#14519776—21no—
20C2×#14519776C2211yescharacteristic
21C2×#14519768C2211yescharacteristic
22C2×#14519767—21no—
23C2×#14519757—21no—
24C2×#14519755—21no—
25C2×#14519753C2211yescharacteristic
26C2×Q8×Dic61C2211yescharacteristic
27C2×#14519735C2211yescharacteristic
28C2×#14519734C2211yescharacteristic
29C2×C4⋊C4×D61C2211yescharacteristic
30#16840071—24no—
31#16840055—24no—
32C2×#14519711C2211yescharacteristic
33C2×#14519711C2211yescharacteristic
34C2×#14519710C2211yescharacteristic
35C2×#14519708C2211yescharacteristic
36C2×#14519706C2211yescharacteristic
37C2×#14519706C2211yescharacteristic
38C2×#14519701C2211yescharacteristic
39C2×#14519701C2211yescharacteristic
40C2×#14519689C2211yescharacteristic
41C2×#14519689C2211yescharacteristic
42#16839660C2211yescharacteristic
43#16839659—21no—
44#16839654—21no—
45#16839651—21no—
46#16839644—21no—
47#16839619C2211yescharacteristic
48C4⋊C4×Dic61C2211yescharacteristic
49#16839525C2211yescharacteristic
50#16839525C2211yescharacteristic
51#16839524C2211yescharacteristic
52#16839524C2211yescharacteristic
53#16839524C2211yescharacteristic
54#16839524C2211yescharacteristic
55#16839517C2211yescharacteristic
56C2×#125×C61C2211yescharacteristic
57C2×#103×C61C2211yescharacteristic
58C2×#103×C61C2211yescharacteristic
59C61×#391C2211yescharacteristic
60C61×#340C2211yescharacteristic
61C61×#340C2211yescharacteristic
62C61×#336C2211yescharacteristic
63C22×Dic122—21no—
64C22×Dic122—21no—
65C22×C4×D61—21no—
66C22×C4×D61C2311yescharacteristic
67C22×#92418C2311yescharacteristic
68C22×#92418C2311yescharacteristic
69#14519777—22no—
70#14519777—22no—
71#14519776—28no—
72#14519776—22no—
73#14519776D414yes—
74#14519768—22no—
75#14519768D414yes—
76#14519767—28no—
77#14519757—28no—
78#14519755—28no—
79#14519753—22no—
80#14519753Q814yescomplemented
81Q8×Dic61—22no—
82Q8×Dic61—22no—
83#14519735—22no—
84#14519735—22no—
85#14519734—22no—
86#14519734—22no—
87C2×C4⋊Dic61—21no—
88C2×C4⋊Dic61—21no—
89C2×C4⋊Dic61—21no—
90C2×C4⋊Dic61C2311yescharacteristic
91C2×C4⋊Dic61C2311yescharacteristic
92C2×C4⋊Dic61C2311yescharacteristic
93C2×C4⋊Dic61C2311yescharacteristic
94C2×#11758196—21no—
95C2×#11758196—21no—
96C2×#11758196—21no—
97C2×#11758196—21no—
98C2×#11758196—21no—
99C2×#11758196D411yescharacteristic
100C2×#11758196D411yescharacteristic
101C4⋊C4×D61—22no—
102C4⋊C4×D61—22no—
103C2×#11758198—21no—
104C2×#11758198—21no—
105C2×#11758198—21no—
106C2×#11758198D411yescharacteristic
107C2×#11758198D411yescharacteristic
108C2×C4×Dic61—21no—
109C2×C4×Dic61—21no—
110C2×C4×Dic61—21no—
111C2×C4×Dic61C2311yescharacteristic
112#14519711—22no—
113#14519711—22no—
114#14519710—22no—
115#14519708—22no—
116#14519706—22no—
117#14519706—22no—
118#14519701—22no—
119#14519701—22no—
120#14519689—22no—
121#14519689—22no—
122#14519684—21no—
123#14519684—21no—
124#14519684—21no—
125C22×Q8×C61C2311yescharacteristic
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×C61C2311yescharacteristic
133C2×C4⋊C4×C61C2311yescharacteristic
134C2×C4⋊C4×C61C2311yescharacteristic
135C2×C42×C61C2311yescharacteristic
136#103×C61—22no—
137#103×C61—22no—
138#95×C61—21no—
139#95×C61—21no—
140C23×D61C2×D411yescharacteristic
141C2×Q8×C61—21no—
142C2×Q8×C61—21no—
143C2×Q8×C61—21no—
144C2×Q8×C61—21no—
145C2×Q8×C61—21no—
146C2×Q8×C61C2×Q812yes—
147C22×C4×C61—21no—
148C22×C4×C61—21no—
149C22×C4×C61—21no—
150C22×C4×C61—21no—
151C22×C4×C61C2×D411yescharacteristic
152C22×C4×C61C2×D411yescharacteristic
153C22×C4×C61C2411yescharacteristic
154C22×Dic61—21no—
155C22×Dic61—21no—
156C22×Dic61—21no—
157C22×Dic61—21no—
158C22×Dic61C2×D411yescharacteristic
159C22×Dic61C2×D411yescharacteristic
160C22×Dic61C2×D411yescharacteristic
161C2×C4×D61—21no—
162C2×C4×D61—21no—
163C2×C4×D61—21no—
164C2×C4×D61—21no—
165C2×C4×D61—24no—
166C2×C4×D61—81no—
167C2×C4×D61—81no—
168C2×C4×D61C2×Q812yes—
169C2×C4×D61#4112yes—
170C2×Dic122—24no—
171C2×Dic122—24no—
172C2×Dic122—81no—
173C2×Dic122—81no—
174C2×Dic122—81no—
175C2×Dic122—81no—
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—24no—
183C4⋊C4×C61—24no—
184C4⋊C4×C61—21no—
185C4⋊C4×C61—21no—
186C4⋊C4×C61—21no—
187C4⋊C4×C61—21no—
188C4⋊C4×C61C2×D412yes—
189C4⋊C4×C61C2×D412yes—
190C42×C61—21no—
191C42×C61—21no—
192#11758198—24no—
193#11758198—24no—
194#11758198—22no—
195#11758198—22no—
196#11758198—81no—
197C4⋊Dic61—24no—
198C4⋊Dic61—21no—
199C4⋊Dic61—21no—
200C4⋊Dic61—21no—
201C4⋊Dic61—24no—
202C4⋊Dic61—21no—
203C4⋊Dic61—21no—
204C4⋊Dic61—42no—
205C4⋊Dic61C2×D412yes—
206C4⋊Dic61C2×D412yes—
207C4⋊Dic61C2×Q812yes—
208#11758196—24no—
209#11758196—24no—
210#11758196—24no—
211#11758196—22no—
212#11758196—22no—
213#11758196—24no—
214#11758196—81no—
215C4×Dic61—24no—
216C4×Dic61—42no—
217C4×Dic61—81no—
218C4×Dic61C2×D412yes—
219C4×Dic61C2×D412yes—
220C2×#92418—21no—
221C2×#92418—21no—
222C2×#92418—21no—
223C2×#92418—21no—
224C2×#92418—21no—
225C2×#92418—21no—
226C23×C61C22×D411yescharacteristic
227C22×D61—21no—
228C22×D61—21no—
229C22×D61—21no—
230C22×D61—21no—
231C22×D61—21no—
232C22×D61—21no—
233C22×D61C22⋊Q812yes—
234Q8×C61—22no—
235Q8×C61—22no—
236Q8×C61—41no—
237Q8×C61—41no—
238C2×C4×C61—22no—
239C2×C4×C61—22no—
240C2×C4×C61—21no—
241C2×C4×C61—21no—
242C2×C4×C61—22no—
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×C61C22×D411yescharacteristic
257C2×C4×C61C22×D411yescharacteristic
258C2×C4×C61C22×Q811yescharacteristic
259C2×C4×C61C2×#4111yescharacteristic
260C2×C4×C612−1+411yescharacteristic
261C2×C4×C612−1+411yescharacteristic
262C2×Dic61—21no—
263C2×Dic61—21no—
264C2×Dic61—21no—
265C2×Dic61—21no—
266C2×Dic61—21no—
267C2×Dic61—21no—
268C2×Dic61—22no—
269C2×Dic61—22no—
270C2×Dic61—22no—
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#12012yes—
281C2×Dic61#12012yes—
282C2×Dic61C22⋊Q812yes—
283C4×D61—22no—
284C4×D61—22no—
285C4×D61—22no—
286C4×D61—22no—
287C4×D61—84no—
288Dic122—84no—
289Dic122—84no—
290C8×C61—41no—
291C8×C61—41no—
292#92418—41no—
293#92418—41no—
294#24311—611nomaximal, Sylow
295C22×C61C2×#12011yescharacteristic
296C22×C61C2×C22⋊Q811yescharacteristic
297C22×C61#53611yescharacteristic
298C22×C61#57611yescharacteristic
299C22×C61#57711yescharacteristic
300C22×C61#57811yescharacteristic
301C22×C61#57811yescharacteristic
302C2×D61—22no—
303C2×D61—22no—
304C2×D61—22no—
305C2×D61—41no—
306C2×D61—41no—
307C2×D61—41no—
308C2×D61—41no—
309C4×C61—42no—
310C4×C61—41no—
311C4×C61—42no—
312C4×C61—41no—
313C4×C61—42no—
314C4×C61—81no—
315C4×C61#54811yescharacteristic
316C4×C61#54811yescharacteristic
317C4×C61D4×Q811yescharacteristic
318C4×C61D4×Q811yescharacteristic
319Dic61—22no—
320Dic61—22no—
321Dic61—22no—
322Dic61—42no—
323Dic61—42no—
324Dic61—42no—
325Dic61—81no—
326C2×#544—611no—
327C2×#479—611no—
328C2×#477—611no—
329#2323—611no—
330#2059—611no—
331#2057—611no—
332#2026D6111yescharacteristic, complemented
333#2024—611no—
334#1989—611no—
335#1988—611no—
336#1873—611no—
337#1872—611no—
338#1854—611no—
339#1852—611no—
340#1798—611no—
341C2×C61#240911yescharacteristic
342C2×C61#301111yescharacteristic
343C2×C61#301211yescharacteristic
344C2×C61#308211yescharacteristic
345C2×C61#308311yescharacteristic
346C2×C61#324211yescharacteristic
347C2×C61#324311yescharacteristic
348D61—42no—
349#544—1224no—
350#544—1224no—
351C2×C4⋊Q8—611no—
352C2×C4⋊Q8—611no—
353C2×C4⋊Q8—1221no—
354C2×#125—611no—
355C2×#125—1221no—
356C2×#125C2×D6111yescharacteristic
357C2×C22⋊Q8—611no—
358C2×C22⋊Q8—1221no—
359C2×C22⋊Q8—1221no—
360C2×C4×Q8—611no—
361C2×#117—611no—
362C2×#117—611no—
363C22×C4⋊C4—611no—
364#479—1224no—
365#477—1224no—
366C2×#107—611no—
367C2×#107—611no—
368C2×#107—611no—
369C2×#106—611no—
370C2×C4⋊C8—611no—
371C2×C4⋊C8—611no—
372C2×#103—611no—
373C2×#103—611no—
374C2×#103C2×D6111yescharacteristic
375C2×#103C2×D6111yescharacteristic
376C2×C22⋊C8—611no—
377C2×C22⋊C8—611no—
378#391C2×D6111yescharacteristic
379#389—611no—
380#389—1221no—
381#388—1221no—
382#388—1221no—
383#387—1221no—
384#380—611no—
385C4×C4⋊C4—611no—
386#340—611no—
387#340—611no—
388#340—611no—
389#340—611no—
390#340—611no—
391#340—611no—
392#340C2×D6111yescharacteristic
393#340C2×D6111yescharacteristic
394#336—611no—
395#336C2×D6111yescharacteristic
396C61#2431111yescharacteristic, complemented, Sylow
397C22×Q8—1221no—
398C22×Q8—1221no—
399C22×Q8C22×D6111yescharacteristic
400C23×C4—611no—
401C23×C4—1221no—
402C22×C8—611no—
403C22×C8—611no—
404C22×C8C22×D6111yescharacteristic
405C22×C8C22×D6111yescharacteristic
406C4⋊Q8—614no—
407C4⋊Q8—1222no—
408C4⋊Q8—1222no—
409C4⋊Q8—1222no—
410C4⋊Q8—1228no—
411#125—22no—
412#125—22no—
413#125—614no—
414#125—1222no—
415#125—1228no—
416C22⋊Q8—614no—
417C22⋊Q8—1228no—
418C22⋊Q8—1228no—
419C22⋊Q8—1222no—
420C4×Q8—1222no—
421C4×Q8—1222no—
422#117—1222no—
423#117—1222no—
424#117—1222no—
425#117—1222no—
426C2×C4⋊C4—21no—
427C2×C4⋊C4—21no—
428C2×C4⋊C4—611no—
429C2×C4⋊C4—611no—
430C2×C4⋊C4—611no—
431C2×C4⋊C4—611no—
432C2×C4⋊C4—611no—
433C2×C4⋊C4—611no—
434C2×C4⋊C4—1222no—
435C2×C4⋊C4—1221no—
436C2×C4⋊C4—1221no—
437C2×C4⋊C4—1221no—
438C2×C4⋊C4—1221no—
439C2×C4⋊C4—1221no—
440C2×C4⋊C4—1221no—
441C2×C4⋊C4—1222no—
442C2×C4⋊C4—1221no—
443C2×C4⋊C4—1221no—
444C2×C4⋊C4C22×D6111yescharacteristic
445C2×C4⋊C4C22×D6111yescharacteristic
446C2×C4⋊C4C22×D6111yescharacteristic
447C2×C22⋊C4—611no—
448C2×C22⋊C4—611no—
449C2×C22⋊C4—1221no—
450C2×C22⋊C4—1221no—
451C2×C22⋊C4—1221no—
452C2×C42—611no—
453C2×C42—1221no—
454C2×C42—1221no—
455C2×C42—1221no—
456C2×C42C22×D6111yescharacteristic
457#107—1222no—
458#107—1222no—
459#107—1222no—
460#106—1222no—
461C4⋊C8—1222no—
462C4⋊C8—1222no—
463#103—22no—
464#103—22no—
465#103—1222no—
466#103—1222no—
467C22⋊C8—1222no—
468C22⋊C8—1222no—
469#95—21no—
470#95—21no—
471#95—1221no—
472#95—1221no—
473#95—1221no—
474C24—611no—
475C2×Q8—21no—
476C2×Q8—21no—
477C2×Q8—21no—
478C2×Q8—21no—
479C2×Q8—21no—
480C2×Q8—1224no—
481C2×Q8—1224no—
482C2×Q8—4881no—
483C2×Q8—4881no—
484C2×Q8—4881no—
485C2×Q8—4881no—
486C2×Q8Q8×D6112yes—
487C22×C4—21no—
488C22×C4—21no—
489C22×C4—21no—
490C22×C4—21no—
491C22×C4—611no—
492C22×C4—611no—
493C22×C4—611no—
494C22×C4—612no—
495C22×C4—612no—
496C22×C4—1224no—
497C22×C4—1221no—
498C22×C4—1221no—
499C22×C4—1221no—
500C22×C4—1221no—
501C22×C4—1221no—
502C22×C4—1221no—
503C22×C4—1221no—
504C22×C4—1221no—
505C22×C4—4881no—
506C22×C4—4881no—
507C22×C4D4×D6111yescharacteristic
508C22×C4D4×D6111yescharacteristic
509C22×C4C23×D6111yescharacteristic
510C2×C8—21no—
511C2×C8—21no—
512C2×C8—21no—
513C2×C8—21no—
514C2×C8—21no—
515C2×C8—21no—
516C2×C8—1221no—
517C2×C8—1221no—
518C2×C8—1221no—
519C2×C8—1221no—
520C2×C8—1221no—
521C2×C8—1221no—
522C4⋊C4—24no—
523C4⋊C4—24no—
524C4⋊C4—21no—
525C4⋊C4—21no—
526C4⋊C4—21no—
527C4⋊C4—21no—
528C4⋊C4—612no—
529C4⋊C4—612no—
530C4⋊C4—612no—
531C4⋊C4—1222no—
532C4⋊C4—1222no—
533C4⋊C4—1224no—
534C4⋊C4—1224no—
535C4⋊C4—1224no—
536C4⋊C4—1224no—
537C4⋊C4—1224no—
538C4⋊C4—1224no—
539C4⋊C4—1221no—
540C4⋊C4—1221no—
541C4⋊C4—1221no—
542C4⋊C4—1221no—
543C4⋊C4—1221no—
544C4⋊C4—4881no—
545C4⋊C4—2442no—
546C4⋊C4D4×D6112yes—
547C4⋊C4D4×D6112yes—
548C22⋊C4—1222no—
549C22⋊C4—1222no—
550C22⋊C4—1224no—
551C22⋊C4—1224no—
552C22⋊C4—4881no—
553C42—21no—
554C42—21no—
555C42—612no—
556C42—612no—
557C42—1224no—
558C42—4881no—
559C42—2442no—
560C23—612no—
561C23—1221no—
562C23—1221no—
563C23—1221no—
564C23—1221no—
565C23—1221no—
566C23—1221no—
567C23C2×D4×D6111yescharacteristic
568Q8—22no—
569Q8—22no—
570Q8—41no—
571Q8—41no—
572Q8—4884no—
573Q8—4884no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—612no—
593C2×C4—612no—
594C2×C4—612no—
595C2×C4—1221no—
596C2×C4—1221no—
597C2×C4—1221no—
598C2×C4—1221no—
599C2×C4—1221no—
600C2×C4—1221no—
601C2×C4—1222no—
602C2×C4—1222no—
603C2×C4—1222no—
604C2×C4—1222no—
605C2×C4—1222no—
606C2×C4—1222no—
607C2×C4—1222no—
608C2×C4—4884no—
609C2×C4—2441no—
610C2×C4—2441no—
611C2×C4—2441no—
612C2×C4—2441no—
613C2×C4—2441no—
614C2×C4—2441no—
615C2×C4—2441no—
616C2×C4—2441no—
617C2×C4—2441no—
618C2×C4C2×D4×D6111yescharacteristic
619C2×C4C2×D4×D6111yescharacteristic
620C2×C4C2×Q8×D6111yescharacteristic
621C2×C4#41×D6111yescharacteristic
622C2×C4#1451982411yescharacteristic
623C2×C4#1451982411yescharacteristic
624C8—41no—
625C8—41no—
626C8—2441no—
627C8—2441no—
628C22—1222no—
629C22—1222no—
630C22—1222no—
631C22—2441no—
632C22—2441no—
633C22—2441no—
634C22—2441no—
635C22#120×D6111yescharacteristic
636C22C22⋊Q8×D6111yescharacteristic
637C22#1684033211yescharacteristic
638C22#1684056411yescharacteristic
639C22#1684057111yescharacteristic
640C22#1684057611yescharacteristic
641C22#1684057811yescharacteristic
642C4—41no—
643C4—42no—
644C4—41no—
645C4—42no—
646C4—42no—
647C4—81no—
648C4—1222no—
649C4—1222no—
650C4—1222no—
651C4—4881no—
652C4—2442no—
653C4—2442no—
654C4—2442no—
655C4#1684042011yescharacteristic
656C4#1684042011yescharacteristic
657C4#1684042511yescharacteristic
658C4#1684042511yescharacteristic
659C2—2442no—
660C2#2144917711yescharacteristic
661C2#2145511111yescharacteristic
662C2#2145512711yescharacteristic
663C2#2145566011yescharacteristic
664C2#2145567311yescharacteristic
665C2#2145708211yescharacteristic
666C2#2145709511yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2026C2
socleC23×C61C22×D4
prime radicalC23×D61C2×D4
hypercenter#2026D61
hypocenterC61#24311
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2026

Complements

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

NH ≅ G/Ntype
2. C2×#16840071C2semidirect
3. C2×#16840055C2semidirect
5. #21447154C2semidirect
6. #21447148C2semidirect
7. #21446960C2semidirect
8. #21446775C2semidirect
9. #21446769C2semidirect
15. C61×#2026C2semidirect
80. #14519753Q8semidirect
332. #2026D61semidirect
396. C61#24311semidirect

Maximal subgroups

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

subgroupindex
C2×#168403922
C2×#168400712
C2×#168400552
#214485042
#214471542
#214471482
#214469602
#214467752
#214467692
#214461282
#214461232
#214460212
#214460082
#214458022
C61×#20262
#2431161