Finite Group Databasebeta

← #29716229

Subgroups — #29716229

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×#16840066C211yesmaximal, characteristic, complemented
3C2×#16840058C211yesmaximal, characteristic, complemented
4#21448502C211yesmaximal, characteristic
5#21447160C211yesmaximal, characteristic, complemented
6#21447146C211yesmaximal, characteristic, complemented
7#21446969C211yesmaximal, characteristic, complemented
8#21446959C211yesmaximal, characteristic, complemented
9#21446771C211yesmaximal, characteristic, complemented
10#21446130C211yesmaximal, characteristic
11#21446125C211yesmaximal, characteristic
12#21446021C211yesmaximal, characteristic
13#21446008C211yesmaximal, characteristic
14#21445800C211yesmaximal, characteristic
15C61×#1989C211yesmaximal, characteristic, complemented
16#16840392—24no—
17#16840392—24no—
18C2×#14519777C2211yescharacteristic
19C2×#14519776—21no—
20C2×#14519776C2211yescharacteristic
21C2×#14519768C2211yescharacteristic
22C2×#14519767—21no—
23C2×#14519757—21no—
24C2×#14519757C2211yescharacteristic
25C2×#14519755—21no—
26C2×C4×Dic122C2211yescharacteristic
27C2×#14519735C2211yescharacteristic
28C2×#14519734C2211yescharacteristic
29C2×C4⋊C4×D61C2211yescharacteristic
30#16840066—24no—
31#16840058—24no—
32C2×#14519711C2211yescharacteristic
33C2×#14519711C2211yescharacteristic
34C2×#14519710C2211yescharacteristic
35C2×#14519708C2211yescharacteristic
36C2×C4⋊#92418C2211yescharacteristic
37C2×#14519702C2211yescharacteristic
38C2×#14519702C2211yescharacteristic
39C2×#14519700C2211yescharacteristic
40C2×#14519700C2211yescharacteristic
41C2×#14519689C2211yescharacteristic
42C2×#14519689C2211yescharacteristic
43#16839667C2211yescharacteristic
44#16839661C2211yescharacteristic
45#16839660—21no—
46#16839654—21no—
47#16839653—21no—
48#16839647—21no—
49#16839619C2211yescharacteristic
50#16839524C2211yescharacteristic
51#16839524C2211yescharacteristic
52#16839524C2211yescharacteristic
53#16839524C2211yescharacteristic
54#16839516C2211yescharacteristic
55#16839516C2211yescharacteristic
56C2×#125×C61C2211yescharacteristic
57C2×C4⋊C8×C61C2211yescharacteristic
58C4×C4⋊C4×C61C2211yescharacteristic
59C61×#340C2211yescharacteristic
60C61×#340C2211yescharacteristic
61C61×#340C2211yescharacteristic
62C61×#340C2211yescharacteristic
63C22×Dic122—21no—
64C22×Dic122—21no—
65C22×Dic122C2311yescharacteristic
66C22×C4×D61—21no—
67C22×C4×D61C2311yescharacteristic
68C22×#92418C2311yescharacteristic
69C22×#92418C2311yescharacteristic
70#14519777—22no—
71#14519777D414yes—
72#14519776—28no—
73#14519776—22no—
74#14519776—22no—
75#14519768—22no—
76#14519768—22no—
77#14519767—28no—
78#14519757—28no—
79#14519757—22no—
80#14519757Q814yescomplemented
81#14519755—28no—
82C4×Dic122—22no—
83C4×Dic122—22no—
84#14519735—22no—
85#14519735—22no—
86#14519734—22no—
87#14519734—22no—
88C2×C4⋊Dic61—21no—
89C2×C4⋊Dic61—21no—
90C2×C4⋊Dic61—21no—
91C2×C4⋊Dic61—21no—
92C2×C4⋊Dic61C2311yescharacteristic
93C2×C4⋊Dic61C2311yescharacteristic
94C2×C4⋊Dic61C2311yescharacteristic
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×Dic61C2311yescharacteristic
111#14519711—22no—
112#14519711—22no—
113#14519710—22no—
114#14519708—22no—
115C4⋊#92418—22no—
116#14519702—22no—
117#14519702—22no—
118#14519700—22no—
119#14519700—22no—
120#14519689—22no—
121#14519689—22no—
122#14519684—21no—
123#14519684—21no—
124#14519684—21no—
125#14519684—21no—
126C22×C8×C61C2311yescharacteristic
127C22×C8×C61C2311yescharacteristic
128#125×C61—22no—
129#125×C61D414yes—
130C2×C4⋊C4×C61—21no—
131C2×C4⋊C4×C61—21no—
132C2×C4⋊C4×C61C2311yescharacteristic
133C2×C4⋊C4×C61C2311yescharacteristic
134C2×C4⋊C4×C61C2311yescharacteristic
135C2×C4⋊C4×C61C2311yescharacteristic
136C2×C42×C61—21no—
137C2×C42×C61C2311yescharacteristic
138C4⋊C8×C61—22no—
139#95×C61—21no—
140C23×D61C2×D411yescharacteristic
141C22×C4×C61—21no—
142C22×C4×C61—21no—
143C22×C4×C61—21no—
144C22×C4×C61—21no—
145C22×C4×C61C2×D411yescharacteristic
146C22×C4×C61C2×D411yescharacteristic
147C22×C4×C61C2411yescharacteristic
148C22×Dic61—21no—
149C22×Dic61—21no—
150C22×Dic61—21no—
151C22×Dic61—21no—
152C22×Dic61C2×D411yescharacteristic
153C22×Dic61C2×D411yescharacteristic
154C22×Dic61C2×D411yescharacteristic
155C2×C4×D61—21no—
156C2×C4×D61—21no—
157C2×C4×D61—21no—
158C2×C4×D61—21no—
159C2×C4×D61—24no—
160C2×C4×D61—81no—
161C2×C4×D61—81no—
162C2×C4×D61C2×Q812yes—
163C2×C4×D61#4112yes—
164C2×Dic122—21no—
165C2×Dic122—21no—
166C2×Dic122—21no—
167C2×Dic122—21no—
168C2×Dic122—21no—
169C2×Dic122—24no—
170C2×Dic122—24no—
171C2×Dic122—81no—
172C2×Dic122—81no—
173C2×Dic122—81no—
174C2×Dic122—81no—
175C2×Dic122C2×Q812yes—
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×C61—21no—
189C4⋊C4×C61C2×D412yes—
190C4⋊C4×C61C2×D412yes—
191C4⋊C4×C61C2×Q812yes—
192C42×C61—81no—
193C42×C61C2×D412yes—
194C42×C61C2×D412yes—
195#11758198—24no—
196#11758198—24no—
197#11758198—22no—
198#11758198—22no—
199#11758198—81no—
200C4⋊Dic61—24no—
201C4⋊Dic61—24no—
202C4⋊Dic61—21no—
203C4⋊Dic61—21no—
204C4⋊Dic61—24no—
205C4⋊Dic61—21no—
206C4⋊Dic61—21no—
207C4⋊Dic61—81no—
208C4⋊Dic61C2×D412yes—
209C4⋊Dic61C2×D412yes—
210#11758196—24no—
211#11758196—24no—
212#11758196—24no—
213#11758196—22no—
214#11758196—22no—
215#11758196—42no—
216C4×Dic61—24no—
217C4×Dic61—21no—
218C4×Dic61—21no—
219C4×Dic61—42no—
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—
234C2×C4×C61—21no—
235C2×C4×C61—21no—
236C2×C4×C61—21no—
237C2×C4×C61—21no—
238C2×C4×C61—22no—
239C2×C4×C61—22no—
240C2×C4×C61—22no—
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×C61—41no—
250C2×C4×C61#12012yes—
251C2×C4×C61#12012yes—
252C2×C4×C61C22×D411yescharacteristic
253C2×C4×C61C22×D411yescharacteristic
254C2×C4×C61C22×Q811yescharacteristic
255C2×C4×C61C2×#4111yescharacteristic
256C2×C4×C612−1+411yescharacteristic
257C2×C4×C612−1+411yescharacteristic
258C2×Dic61—21no—
259C2×Dic61—21no—
260C2×Dic61—21no—
261C2×Dic61—21no—
262C2×Dic61—22no—
263C2×Dic61—22no—
264C2×Dic61—22no—
265C2×Dic61—21no—
266C2×Dic61—21no—
267C2×Dic61—21no—
268C2×Dic61—21no—
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×Dic61C22⋊Q812yes—
279C4×D61—22no—
280C4×D61—22no—
281C4×D61—22no—
282C4×D61—22no—
283C4×D61—84no—
284Dic122—22no—
285Dic122—22no—
286Dic122—41no—
287Dic122—41no—
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—22no—
310C4×C61—22no—
311C4×C61—42no—
312C4×C61—42no—
313C4×C61—42no—
314C4×C61—81no—
315C4×C61#54811yescharacteristic
316C4×C61#54811yescharacteristic
317C4×C61D4×Q811yescharacteristic
318C4×C61D4×Q811yescharacteristic
319Dic61—22no—
320Dic61—42no—
321Dic61—41no—
322Dic61—41no—
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#2026—611no—
333#2024—611no—
334#1989D6111yescharacteristic, complemented
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⋊C8C2×D6111yescharacteristic
372C2×#103—611no—
373C2×#103—611no—
374C2×#103—611no—
375C2×#103—611no—
376C2×C22⋊C8—611no—
377C2×C22⋊C8—611no—
378#391—611no—
379#389—611no—
380#389—1221no—
381#388—1221no—
382#388—1221no—
383#387—1221no—
384#380—611no—
385C4×C4⋊C4C2×D6111yescharacteristic
386#340—611no—
387#340—611no—
388#340—611no—
389#340—611no—
390#340C2×D6111yescharacteristic
391#340C2×D6111yescharacteristic
392#340C2×D6111yescharacteristic
393#340C2×D6111yescharacteristic
394#336—611no—
395#336—611no—
396C61#2431111yescharacteristic, complemented, Sylow
397C22×Q8—611no—
398C22×Q8—1221no—
399C22×Q8—1221no—
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—1228no—
409C4⋊Q8—1222no—
410C4⋊Q8—1222no—
411#125—22no—
412#125—1222no—
413#125—1228no—
414#125—1222no—
415#125D24414yes—
416C22⋊Q8—614no—
417C22⋊Q8—1222no—
418C22⋊Q8—1228no—
419C22⋊Q8—1228no—
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—1222no—
434C2×C4⋊C4—1221no—
435C2×C4⋊C4—1221no—
436C2×C4⋊C4—1221no—
437C2×C4⋊C4—1221no—
438C2×C4⋊C4—1221no—
439C2×C4⋊C4—1221no—
440C2×C4⋊C4—1222no—
441C2×C4⋊C4—1221no—
442C2×C4⋊C4—1221no—
443C2×C4⋊C4C22×D6111yescharacteristic
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—21no—
453C2×C42—611no—
454C2×C42—1221no—
455C2×C42—1221no—
456C2×C42C22×D6111yescharacteristic
457#107—1222no—
458#107—1222no—
459#107—1222no—
460#106—1222no—
461C4⋊C8—22no—
462C4⋊C8—1222no—
463#103—1222no—
464#103—1222no—
465#103—1222no—
466#103—1222no—
467C22⋊C8—1222no—
468C22⋊C8—1222no—
469#95—21no—
470#95—1221no—
471#95—1221no—
472#95—1221no—
473#95—1221no—
474C24—611no—
475C2×Q8—612no—
476C2×Q8—1224no—
477C2×Q8—1221no—
478C2×Q8—1224no—
479C2×Q8—1221no—
480C2×Q8—1221no—
481C2×Q8—1221no—
482C2×Q8—1221no—
483C2×Q8—4881no—
484C2×Q8—4881no—
485C2×Q8—4881no—
486C2×Q8—4881no—
487C22×C4—21no—
488C22×C4—21no—
489C22×C4—21no—
490C22×C4—21no—
491C22×C4—611no—
492C22×C4—612no—
493C22×C4—612no—
494C22×C4—611no—
495C22×C4—611no—
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—21no—
523C4⋊C4—21no—
524C4⋊C4—21no—
525C4⋊C4—24no—
526C4⋊C4—24no—
527C4⋊C4—21no—
528C4⋊C4—21no—
529C4⋊C4—612no—
530C4⋊C4—612no—
531C4⋊C4—1224no—
532C4⋊C4—1224no—
533C4⋊C4—1224no—
534C4⋊C4—1224no—
535C4⋊C4—1224no—
536C4⋊C4—1221no—
537C4⋊C4—1221no—
538C4⋊C4—1222no—
539C4⋊C4—1222no—
540C4⋊C4—1224no—
541C4⋊C4—1221no—
542C4⋊C4—1221no—
543C4⋊C4—4881no—
544C4⋊C4—2442no—
545C4⋊C4C2×D24412yes—
546C4⋊C4C2×D24412yes—
547C4⋊C4Q8×D6112yes—
548C22⋊C4—1224no—
549C22⋊C4—1224no—
550C22⋊C4—1222no—
551C22⋊C4—1222no—
552C22⋊C4—4881no—
553C42—81no—
554C42—1224no—
555C42—1221no—
556C42—1221no—
557C42—2442no—
558C42C2×D24412yes—
559C42D4×D6112yes—
560C23—612no—
561C23—1221no—
562C23—1221no—
563C23—1221no—
564C23—1221no—
565C23—1221no—
566C23—1221no—
567C23C2×D4×D6111yescharacteristic
568Q8—1222no—
569Q8—1222no—
570Q8—4884no—
571Q8—4884no—
572Q8—2441no—
573Q8—2441no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—612no—
591C2×C4—1221no—
592C2×C4—1221no—
593C2×C4—1222no—
594C2×C4—1222no—
595C2×C4—1222no—
596C2×C4—1222no—
597C2×C4—1222no—
598C2×C4—1221no—
599C2×C4—1221no—
600C2×C4—1222no—
601C2×C4—1222no—
602C2×C4—1221no—
603C2×C4—1221no—
604C2×C4—1221no—
605C2×C4—1221no—
606C2×C4—4884no—
607C2×C4—2441no—
608C2×C4—2441no—
609C2×C4—2441no—
610C2×C4—2441no—
611C2×C4—2441no—
612C2×C4—2441no—
613C2×C4—2441no—
614C2×C4—2441no—
615C2×C4—2441no—
616C2×C4#1451974712yes—
617C2×C4#1451974712yes—
618C2×C4C2×D4×D6111yescharacteristic
619C2×C4C22×D24411yescharacteristic
620C2×C4C2×Q8×D6111yescharacteristic
621C2×C4#41×D6111yescharacteristic
622C2×C4#1451982411yescharacteristic
623C2×C4#1451982511yescharacteristic
624C8—41no—
625C8—41no—
626C8—2441no—
627C8—2441no—
628C22—1222no—
629C22—1222no—
630C22—1222no—
631C22—2441no—
632C22—2441no—
633C22—2441no—
634C22—2441no—
635C22C2×#1451974711yescharacteristic
636C22C22⋊Q8×D6111yescharacteristic
637C22#1684033511yescharacteristic
638C22#1684056611yescharacteristic
639C22#1684057111yescharacteristic
640C22#1684057611yescharacteristic
641C22#1684057611yescharacteristic
642C4—22no—
643C4—22no—
644C4—42no—
645C4—42no—
646C4—42no—
647C4—81no—
648C4—1222no—
649C4—4881no—
650C4—2441no—
651C4—2441no—
652C4—2442no—
653C4—2442no—
654C4—2442no—
655C4#1684042011yescharacteristic
656C4#1684042411yescharacteristic
657C4#1684042711yescharacteristic
658C4Q8×D24411yescharacteristic
659C2—2442no—
660C2#2144918211yescharacteristic
661C2#2145511711yescharacteristic
662C2#2145512511yescharacteristic
663C2#2145566011yescharacteristic
664C2#2145567111yescharacteristic
665C2#2145709211yescharacteristic
666C2#2145710111yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1989C2
socleC23×C61C22×D4
prime radicalC23×D61C2×D4
hypercenter#1989D61
hypocenterC61#24311
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1989

Complements

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

NH ≅ G/Ntype
2. C2×#16840066C2semidirect
3. C2×#16840058C2semidirect
5. #21447160C2semidirect
6. #21447146C2semidirect
7. #21446969C2semidirect
8. #21446959C2semidirect
9. #21446771C2semidirect
15. C61×#1989C2semidirect
80. #14519757Q8semidirect
334. #1989D61semidirect
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×#168400662
C2×#168400582
#214485022
#214471602
#214471462
#214469692
#214469592
#214467712
#214461302
#214461252
#214460212
#214460082
#214458002
C61×#19892
#2431161