Finite Group Databasebeta

← #29682667

Subgroups — #29682667

Order 15616.

1074 conjugacy classes of subgroups.

The subgroup lattice has 676 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839769C211yesmaximal, characteristic
2C2×#103×D61C211yesmaximal, characteristic
3C2×#16839716C211yesmaximal, characteristic
4#21448434C211yesmaximal, characteristic
5#21448402C211yesmaximal, characteristic
6#21446512C211yesmaximal, characteristic, complemented
7#21446510C211yesmaximal, characteristic, complemented
8#21446508C211yesmaximal, characteristic, complemented
9#21446506C211yesmaximal, characteristic, complemented
10#21446497C211yesmaximal, characteristic, complemented
11#21446495C211yesmaximal, characteristic, complemented
12#21446493C211yesmaximal, characteristic, complemented
13#21445537C211yesmaximal, characteristic
14#21445534C211yesmaximal, characteristic
15C61×#1933C211yesmaximal, characteristic, complemented
16C22×Q8×D61C2211yescharacteristic
17C22×#11758189C2211yescharacteristic
18C22×C8×D61C2211yescharacteristic
19C2×#14519735C2211yescharacteristic
20C2×#14519735C2211yescharacteristic
21C2×#14519734C2211yescharacteristic
22C2×C4⋊C4×D61C2211yescharacteristic
23#16839769—24no—
24C2×#14519702C2211yescharacteristic
25C2×#14519702C2211yescharacteristic
26C2×#14519701C2211yescharacteristic
27C2×#14519701C2211yescharacteristic
28C2×#14519700C2211yescharacteristic
29C2×#14519700C2211yescharacteristic
30#103×D61C418yescomplemented
31#16839716—24no—
32C2×#14519688C2211yescharacteristic
33C2×#14519687C2211yescharacteristic
34C2×C8×Dic61C2211yescharacteristic
35#16839669C2211yescharacteristic
36#16839668C2211yescharacteristic
37#16839667C2211yescharacteristic
38#16839666—21no—
39#16839666—21no—
40#16839661C2211yescharacteristic
41#16839660C2211yescharacteristic
42#16839659—21no—
43#16839658C2211yescharacteristic
44#16839657—21no—
45#16839624—21no—
46#16839624—21no—
47#16839621—21no—
48#16839619—21no—
49#16839525C2211yescharacteristic
50#16839525C2211yescharacteristic
51#16839524C2211yescharacteristic
52#16839524C2211yescharacteristic
53#16839524C2211yescharacteristic
54#16839524C2211yescharacteristic
55C2×#103×C61C2211yescharacteristic
56C2×#103×C61C2211yescharacteristic
57C2×C4×C8×C61C2211yescharacteristic
58C61×#391C2211yescharacteristic
59C61×#389C2211yescharacteristic
60C61×#340C2211yescharacteristic
61C61×#340C2211yescharacteristic
62C22×Dic122—21no—
63C22×Dic122C2311yescharacteristic
64C2×Q8×D61—22no—
65C2×Q8×D61—22no—
66C2×Q8×D61—22no—
67C2×Q8×D61—22no—
68C2×Q8×D61—22no—
69C2×Q8×D61C2×C414yes—
70C22×C4×D61—21no—
71C22×C4×D61—21no—
72C22×C4×D61C2311yescharacteristic
73C2×#11758189—22no—
74C2×#11758189—22no—
75C2×#11758189D414yes—
76C22×#92418C2311yescharacteristic
77C22×#92418C2311yescharacteristic
78C2×C8×D61—22no—
79C2×C8×D61—22no—
80C2×C8×D61C2×C414yes—
81#14519735—22no—
82#14519735—22no—
83#14519735—22no—
84#14519735—22no—
85#14519734—22no—
86#14519734—22no—
87C2×C4⋊Dic61—21no—
88C2×C4⋊Dic61C2311yescharacteristic
89C2×C4⋊Dic61C2311yescharacteristic
90C2×C4⋊Dic61C2311yescharacteristic
91C2×#11758196—21no—
92C2×#11758196—21no—
93C2×#11758196—21no—
94C2×#11758196D411yescharacteristic
95C2×#11758196D411yescharacteristic
96C4⋊C4×D61—22no—
97C4⋊C4×D61C2×C414yes—
98C2×#11758198—21no—
99C2×#11758198—21no—
100C2×#11758198D411yescharacteristic
101C2×#11758198D411yescharacteristic
102C2×C4×Dic61—21no—
103C2×C4×Dic61—21no—
104C2×C4×Dic61C2311yescharacteristic
105#14519702—22no—
106#14519702C2×C414yes—
107#14519701—22no—
108#14519701C2×C414yes—
109#14519700—22no—
110#14519700C2×C414yes—
111#14519688—22no—
112#14519687—22no—
113C8×Dic61—22no—
114#14519684—21no—
115#14519684—21no—
116#14519684—21no—
117#14519684—21no—
118#14519684—21no—
119#14519684—21no—
120#14519684—21no—
121#14519684—21no—
122#14519684—21no—
123C22×Q8×C61C2311yescharacteristic
124C22×C8×C61C2311yescharacteristic
125C22×C8×C61C2311yescharacteristic
126C2×C4⋊C4×C61—21no—
127C2×C4⋊C4×C61C2311yescharacteristic
128C2×C4⋊C4×C61C2311yescharacteristic
129C2×C4⋊C4×C61C2311yescharacteristic
130C2×C42×C61C2311yescharacteristic
131#103×C61—22no—
132#103×C61C2×C414yes—
133C4×C8×C61—22no—
134#95×C61—21no—
135#95×C61—21no—
136#95×C61—21no—
137C23×D61C2×D411yescharacteristic
138C2×Q8×C61—21no—
139C2×Q8×C61—21no—
140C2×Q8×C61—21no—
141C2×Q8×C61—21no—
142C2×Q8×C61—21no—
143C2×Q8×C61C22×C412yes—
144C22×C4×C61—21no—
145C22×C4×C61—21no—
146C22×C4×C61—21no—
147C22×C4×C61—21no—
148C22×C4×C61C2×D411yescharacteristic
149C22×C4×C61C2×D411yescharacteristic
150C22×C4×C61C2411yescharacteristic
151C22×Dic61—21no—
152C22×Dic61—21no—
153C22×Dic61—21no—
154C22×Dic61—21no—
155C22×Dic61C2×D411yescharacteristic
156C22×Dic61C2×D411yescharacteristic
157C22×Dic61C2×D411yescharacteristic
158Q8×D61—28no—
159Q8×D61—28no—
160Q8×D61—44no—
161Q8×D61—44no—
162C2×C4×D61—24no—
163C2×C4×D61—24no—
164C2×C4×D61—42no—
165C2×C4×D61—42no—
166C2×C4×D61—81no—
167C2×C4×D61—81no—
168C2×C4×D61C22×C412yes—
169C2×C4×D61C2×D412yes—
170C2×C4×D61#4112yes—
171C2×C4×D61#4112yes—
172C2×C4×D61#4112yes—
173C2×C4×D61#4112yes—
174C2×Dic122—21no—
175C2×Dic122—21no—
176C2×Dic122—21no—
177C2×Dic122—21no—
178C2×Dic122—21no—
179C2×Dic122—24no—
180C2×Dic122—42no—
181C2×Dic122—42no—
182C2×Dic122C22×C412yes—
183C2×C8×C61—21no—
184C2×C8×C61—21no—
185C2×C8×C61—21no—
186C2×C8×C61—21no—
187C2×C8×C61C22×C412yes—
188C2×C8×C61C2×D412yes—
189C4⋊C4×C61—21no—
190C4⋊C4×C61—21no—
191C4⋊C4×C61—21no—
192C4⋊C4×C61—21no—
193C4⋊C4×C61—21no—
194C4⋊C4×C61—81no—
195C4⋊C4×C61C22×C412yes—
196C42×C61—21no—
197C42×C61—21no—
198#11758198—22no—
199#11758198—22no—
200#11758198—42no—
201#11758198—42no—
202C4⋊Dic61—21no—
203C4⋊Dic61—21no—
204C4⋊Dic61—21no—
205C4⋊Dic61—21no—
206C4⋊Dic61—21no—
207C4⋊Dic61—81no—
208C4⋊Dic61C22×C412yes—
209#11758196—24no—
210#11758196—22no—
211#11758196—22no—
212#11758196—81no—
213#11758196—81no—
214C4×Dic61—21no—
215C4×Dic61—21no—
216C4×Dic61—42no—
217C4×Dic61—42no—
218C2×#92418—21no—
219C2×#92418—21no—
220C2×#92418—21no—
221C2×#92418—21no—
222C2×#92418C22×C412yes—
223C2×#92418C2×D412yes—
224#11758189—28no—
225C8×D61—28no—
226C23×C61C22×D411yescharacteristic
227C22×D61—21no—
228C22×D61—21no—
229C22×D61—21no—
230C22×D61—21no—
231C22×D61C4×D412yes—
232C22×D61#12112yes—
233C22×D61#12412yes—
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—21no—
243C2×C4×C61—21no—
244C2×C4×C61—21no—
245C2×C4×C61—21no—
246C2×C4×C61—41no—
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×C61C23×C411yescharacteristic
257C2×C4×C61C22×D411yescharacteristic
258C2×C4×C61C2×#4111yescharacteristic
259C2×C4×C61C2×#4111yescharacteristic
260C2×C4×C61C2×#4111yescharacteristic
261C2×C4×C61C2×#4111yescharacteristic
262C2×Dic61—22no—
263C2×Dic61—22no—
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×Dic61C4×D412yes—
281C2×Dic61#12112yes—
282C2×Dic61#12412yes—
283C4×D61—48no—
284C4×D61—84no—
285C4×D61#11714yes—
286C4×D61C4×D414yes—
287C4×D61#12314yes—
288C4×D61#12614yes—
289Dic122—22no—
290Dic122—22no—
291Dic122—41no—
292Dic122—41no—
293Dic122—48no—
294C8×C61—22no—
295C8×C61—22no—
296#92418—22no—
297#92418—22no—
298#21088—611nomaximal, Sylow
299C22×C61C2×C4×D411yescharacteristic
300C22×C61C2×#12111yescharacteristic
301C22×C61C2×#12411yescharacteristic
302C22×C61#57711yescharacteristic
303C22×C61#57711yescharacteristic
304C22×C61#57811yescharacteristic
305C22×C61#57811yescharacteristic
306C2×D61—22no—
307C2×D61—22no—
308C2×D61—22no—
309C2×D61—22no—
310C2×D61—22no—
311C2×D61—22no—
312C2×D61#38814yes—
313C4×C61—42no—
314C4×C61—42no—
315C4×C61—41no—
316C4×C61—41no—
317C4×C61—81no—
318C4×C61—81no—
319C4×C61C2×#11711yescharacteristic
320C4×C61C2×C4×D411yescharacteristic
321C4×C61C2×#12311yescharacteristic
322C4×C61C2×#12611yescharacteristic
323Dic61—42no—
324Dic61—42no—
325Dic61—41no—
326Dic61—41no—
327Dic61—81no—
328Dic61—81no—
329Dic61#38814yes—
330C2×#419—611no—
331C22×#103—611no—
332C2×#405—611no—
333#2316—611no—
334#2310—611no—
335#1935—611no—
336#1935—611no—
337#1935—611no—
338#1935—611no—
339#1933—611no—
340#1933—611no—
341#1933—611no—
342#1933D6111yescharacteristic, complemented
343#1736—611no—
344#1735—611no—
345C2×C61C2×#38811yescharacteristic
346C2×C61#294911yescharacteristic
347C2×C61#295011yescharacteristic
348C2×C61#306311yescharacteristic
349C2×C61#306311yescharacteristic
350C2×C61#313711yescharacteristic
351C2×C61#313711yescharacteristic
352D61—24no—
353C23×Q8—611no—
354C22×M16—611no—
355C23×C8—611no—
356C2×#117—611no—
357C2×#117—611no—
358C2×#117—611no—
359C22×C4⋊C4—611no—
360#419—1224no—
361C2×#103—611no—
362C2×#103—611no—
363C2×#103—611no—
364C2×#103—618no—
365C2×#103—611no—
366C2×#103—611no—
367C2×#103—611no—
368C2×#103C2×D6111yescharacteristic
369C2×#103C2×D6111yescharacteristic
370#405—1224no—
371C2×#97—611no—
372C2×#97—611no—
373C2×C4×C8—611no—
374C2×C4×C8C2×D6111yescharacteristic
375#391—611no—
376#391—611no—
377#391—611no—
378#391—1221no—
379#391—1221no—
380#391C2×D6111yescharacteristic
381#389—611no—
382#389—611no—
383#389—611no—
384#389—1221no—
385#389—1221no—
386#389C2×D6111yescharacteristic
387#381—1221no—
388#381—1221no—
389#380—1221no—
390#380—1221no—
391#340—611no—
392#340—611no—
393#340—611no—
394#340—611no—
395#340—611no—
396#340—611no—
397#340C2×D6111yescharacteristic
398#340C2×D6111yescharacteristic
399C61#2108811yescharacteristic, complemented, Sylow
400C22×Q8—614no—
401C22×Q8—611no—
402C22×Q8—1221no—
403C22×Q8—1222no—
404C22×Q8—1222no—
405C22×Q8—1222no—
406C22×Q8—1222no—
407C22×Q8—1222no—
408C22×Q8C22×D6111yescharacteristic
409C23×C4—611no—
410C23×C4—1221no—
411C23×C4—1221no—
412C2×M16—614no—
413C2×M16—1222no—
414C2×M16—1222no—
415C22×C8—611no—
416C22×C8—611no—
417C22×C8—614no—
418C22×C8—1222no—
419C22×C8—1222no—
420C22×C8C22×D6111yescharacteristic
421C22×C8C22×D6111yescharacteristic
422#117—1222no—
423#117—1222no—
424#117—1222no—
425#117—1222no—
426#117—1222no—
427#117—1222no—
428C2×C4⋊C4—21no—
429C2×C4⋊C4—611no—
430C2×C4⋊C4—614no—
431C2×C4⋊C4—611no—
432C2×C4⋊C4—611no—
433C2×C4⋊C4—611no—
434C2×C4⋊C4—611no—
435C2×C4⋊C4—1222no—
436C2×C4⋊C4—1221no—
437C2×C4⋊C4—1221no—
438C2×C4⋊C4—1221no—
439C2×C4⋊C4—1221no—
440C2×C4⋊C4C22×D6111yescharacteristic
441C2×C4⋊C4C22×D6111yescharacteristic
442C2×C4⋊C4C22×D6111yescharacteristic
443C2×C22⋊C4—611no—
444C2×C22⋊C4—611no—
445C2×C22⋊C4—1221no—
446C2×C22⋊C4—1221no—
447C2×C42—611no—
448C2×C42—1221no—
449C2×C42—1221no—
450C2×C42C22×D6111yescharacteristic
451#103—22no—
452#103—614no—
453#103—614no—
454#103—614no—
455#103—1222no—
456#103—1222no—
457#103—1222no—
458#103C4×D6114yes—
459#97—1222no—
460#97—1222no—
461C4×C8—22no—
462C4×C8—1222no—
463#95—21no—
464#95—21no—
465#95—21no—
466#95—1221no—
467#95—1221no—
468#95—1221no—
469#95—1221no—
470#95—1221no—
471#95—1221no—
472#95—1221no—
473#95—1221no—
474#95—1221no—
475C24—611no—
476C2×Q8—21no—
477C2×Q8—21no—
478C2×Q8—21no—
479C2×Q8—21no—
480C2×Q8—21no—
481C2×Q8—612no—
482C2×Q8—1224no—
483C2×Q8—1228no—
484C2×Q8—1228no—
485C2×Q8—1221no—
486C2×Q8—1221no—
487C2×Q8—1221no—
488C2×Q8—1221no—
489C2×Q8—1221no—
490C2×Q8—2442no—
491C2×Q8—2442no—
492C2×Q8—2444no—
493C2×Q8—2444no—
494C2×Q8C2×C4×D6112yes—
495C22×C4—21no—
496C22×C4—21no—
497C22×C4—21no—
498C22×C4—21no—
499C22×C4—611no—
500C22×C4—612no—
501C22×C4—612no—
502C22×C4—612no—
503C22×C4—612no—
504C22×C4—612no—
505C22×C4—612no—
506C22×C4—611no—
507C22×C4—611no—
508C22×C4—1224no—
509C22×C4—1224no—
510C22×C4—1221no—
511C22×C4—1221no—
512C22×C4—1221no—
513C22×C4—1221no—
514C22×C4—4881no—
515C22×C4—4881no—
516C22×C4—2442no—
517C22×C4—2442no—
518C22×C4D4×D6111yescharacteristic
519C22×C4D4×D6111yescharacteristic
520C22×C4C23×D6111yescharacteristic
521M16—1228no—
522C2×C8—21no—
523C2×C8—21no—
524C2×C8—21no—
525C2×C8—21no—
526C2×C8—612no—
527C2×C8—612no—
528C2×C8—1228no—
529C2×C8—1221no—
530C2×C8—1221no—
531C2×C8—1221no—
532C2×C8—1221no—
533C2×C8C2×C4×D6112yes—
534C2×C8D4×D6112yes—
535C4⋊C4—21no—
536C4⋊C4—21no—
537C4⋊C4—21no—
538C4⋊C4—21no—
539C4⋊C4—21no—
540C4⋊C4—81no—
541C4⋊C4—612no—
542C4⋊C4—1224no—
543C4⋊C4—1222no—
544C4⋊C4—1222no—
545C4⋊C4—1221no—
546C4⋊C4—1221no—
547C4⋊C4—1221no—
548C4⋊C4—1221no—
549C4⋊C4—1221no—
550C4⋊C4—4881no—
551C4⋊C4—4881no—
552C4⋊C4—4881no—
553C4⋊C4C2×C4×D6112yes—
554C22⋊C4—1222no—
555C22⋊C4—1222no—
556C22⋊C4—2442no—
557C22⋊C4—2442no—
558C42—21no—
559C42—21no—
560C42—1221no—
561C42—1221no—
562C42—2442no—
563C42—2442no—
564C23—612no—
565C23—612no—
566C23—612no—
567C23—1221no—
568C23—1221no—
569C23—1221no—
570C23—1221no—
571C23C2×D4×D6111yescharacteristic
572Q8—22no—
573Q8—22no—
574Q8—41no—
575Q8—41no—
576Q8—1222no—
577Q8—1222no—
578Q8—2448no—
579Q8—2441no—
580Q8—2441no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—612no—
600C2×C4—612no—
601C2×C4—612no—
602C2×C4—614no—
603C2×C4—614no—
604C2×C4—614no—
605C2×C4—614no—
606C2×C4—1222no—
607C2×C4—1222no—
608C2×C4—1221no—
609C2×C4—1221no—
610C2×C4—1221no—
611C2×C4—1221no—
612C2×C4—1221no—
613C2×C4—1221no—
614C2×C4—4884no—
615C2×C4—2448no—
616C2×C4—2441no—
617C2×C4—2441no—
618C2×C4—2441no—
619C2×C4—2441no—
620C2×C4—2441no—
621C2×C4—2441no—
622C2×C4—2441no—
623C2×C4—2441no—
624C2×C4—2441no—
625C2×C4—2441no—
626C2×C4C22×C4×D6111yescharacteristic
627C2×C4C2×D4×D6111yescharacteristic
628C2×C4#41×D6111yescharacteristic
629C2×C4#41×D6111yescharacteristic
630C2×C4#41×D6111yescharacteristic
631C2×C4#41×D6111yescharacteristic
632C8—22no—
633C8—22no—
634C8—1222no—
635C8—1222no—
636C22—614no—
637C22—1222no—
638C22—1222no—
639C22—1222no—
640C22—1222no—
641C22—1222no—
642C22—1222no—
643C22C4×D4×D6111yescharacteristic
644C22#121×D6111yescharacteristic
645C22#124×D6111yescharacteristic
646C22#1684056811yescharacteristic
647C22#1684056911yescharacteristic
648C22#1684057511yescharacteristic
649C22#1684057611yescharacteristic
650C4—42no—
651C4—42no—
652C4—41no—
653C4—41no—
654C4—81no—
655C4—81no—
656C4—614no—
657C4—4881no—
658C4—4881no—
659C4—2442no—
660C4—2442no—
661C4—2441no—
662C4—2441no—
663C4#117×D6111yescharacteristic
664C4C4×D4×D6111yescharacteristic
665C4#123×D6111yescharacteristic
666C4#126×D6111yescharacteristic
667C2—1224no—
668C2#388×D6111yescharacteristic
669C2#2145467311yescharacteristic
670C2#2145468211yescharacteristic
671C2#2145550311yescharacteristic
672C2#2145550911yescharacteristic
673C2#2145617911yescharacteristic
674C2#2145618311yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1933C2
socleC23×C61C22×D4
prime radicalC23×D61C2×D4
cosocleC22×C4×C61C24
hypercenter#1933D61
hypocenterC61#21088
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1933

Complements

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

NH ≅ G/Ntype
6. #21446512C2semidirect
7. #21446510C2semidirect
8. #21446508C2semidirect
9. #21446506C2semidirect
10. #21446497C2semidirect
11. #21446495C2semidirect
12. #21446493C2semidirect
15. C61×#1933C2semidirect
30. #103×D61C4semidirect
342. #1933D61semidirect
399. C61#21088semidirect

Maximal subgroups

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

subgroupindex
C2×#168397692
C2×#103×D612
C2×#168397162
#214484342
#214484022
#214465122
#214465102
#214465082
#214465062
#214464972
#214464952
#214464932
#214455372
#214455342
C61×#19332
#2108861