Finite Group Databasebeta

← #29585389

Subgroups — #29585389

Order 15616.

1222 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459020C211yesmaximal, characteristic
2C2×#16839735C211yesmaximal, characteristic
3C2×#16839730C211yesmaximal, characteristic
4#21445535C211yesmaximal, characteristic, complemented
5#21445535C211yesmaximal, characteristic, complemented
6#21445530C211yesmaximal, characteristic, complemented
7C61×#1733C211yesmaximal, characteristic, complemented
8C22×#11758222C2211yescharacteristic
9C23×#92418C2211yescharacteristic
10#16840243—22no—
11#16840242—22no—
12#16840239C412yescomplemented
13#16840236C412yescomplemented
14#16840235—22no—
15C4×#11758222—22no—
16#16840230—22no—
17C2×#14519736C2211yescharacteristic
18C2×#14519736C2211yescharacteristic
19#117×D61—22no—
20#16839735—22no—
21#16839735C414yescomplemented
22C2×#14519691—22no—
23#16839730—22no—
24#16839730C414yescomplemented
25C2×#14519689—22no—
26#16839524—22no—
27#16839524—22no—
28#16839523—22no—
29C22×M16×C61C2211yescharacteristic
30C2×#117×C61C2211yescharacteristic
31C2×#117×C61C2211yescharacteristic
32C61×#339—22no—
33C2×#11758222—21no—
34C2×#11758222—21no—
35C2×#11758222—21no—
36C2×#11758222—21no—
37C2×#11758222—21no—
38C2×#11758222—21no—
39C2×#11758222—21no—
40C2×#11758222—21no—
41C2×#11758222C2×C412yes—
42C2×#11758222C2×C412yes—
43C2×#11758222D412yes—
44C2×#11758222Q812yes—
45C22×Dic122C2×C411yescharacteristic
46C22×D244C2×C411yescharacteristic
47C22×C22⋊D61D412yes—
48C22×C4×D61D412yes—
49C22×#92418—21no—
50C22×#92418—21no—
51C22×#92418—21no—
52C22×#92418—21no—
53C22×#92418—21no—
54C22×#92418—21no—
55C22×#92418C2×C412yescomplemented
56C4×Dic122—24no—
57#14519742—24no—
58C4×D244—24no—
59#14519739—24no—
60C4×C22⋊D61—28no—
61#14519737—28no—
62#14519736—21no—
63#14519736—21no—
64#14519736—21no—
65#14519736—21no—
66#14519736—21no—
67#14519736C2×C412yes—
68#14519736C2×C412yescomplemented
69#14519736D412yes—
70#14519735—24no—
71#14519734—24no—
72#14519733—24no—
73C2×C4⋊Dic61—21no—
74C2×C4⋊Dic61D412yes—
75C4⋊C4×D61—24no—
76C2×C22⋊Dic61—21no—
77C2×C22⋊Dic61—21no—
78C22⋊C4×D61—24no—
79C2×C4×Dic61—21no—
80C2×C4×Dic61—21no—
81C42×D61—24no—
82#14519691—28no—
83#14519689—28no—
84C23×C4×C61C2311yescharacteristic
85C2×M16×C61—21no—
86C2×M16×C61—21no—
87C2×M16×C61—21no—
88C2×M16×C61—21no—
89C2×M16×C61—21no—
90C2×M16×C61C2×C412yescomplemented
91C22×C8×C61—21no—
92#117×C61—21no—
93#117×C61—21no—
94#117×C61—21no—
95#117×C61—21no—
96#117×C61—21no—
97#117×C61C2×C412yes—
98#117×C61C2×C412yescomplemented
99#117×C61D412yes—
100C2×C4⋊C4×C61—21no—
101C2×C4⋊C4×C61D412yes—
102C2×C22⋊C4×C61—21no—
103C2×C22⋊C4×C61—21no—
104C2×C42×C61—21no—
105C2×C42×C61—21no—
106C24×C61C2×D411yescharacteristic
107C23×D61C22⋊C412yes—
108C22×C4×C61—22no—
109C22×C4×C61—41no—
110C22×C4×C61C22×C411yescharacteristic
111C22×C4×C61C22×C411yescharacteristic
112C22×C4×C61C22×C411yescharacteristic
113C22×C4×C61C2×D411yescharacteristic
114C22×C4×C61C2×D411yescharacteristic
115C22×C4×C61C2×D411yescharacteristic
116C22×C4×C61C2×D411yescharacteristic
117C22×C4×C61C2×D411yescharacteristic
118C22×C4×C61C2×D411yescharacteristic
119C22×C4×C61C2×Q811yescharacteristic
120C22×C4×C61#4111yescharacteristic
121C22×C4×C61#4111yescharacteristic
122C22×C4×C61#4111yescharacteristic
123C22×C4×C61#4111yescharacteristic
124C2×C22⋊D61—22no—
125C2×C22⋊D61—22no—
126C2×C22⋊D61—22no—
127C2×C22⋊D61—22no—
128C2×C22⋊D61C22⋊C414yes—
129C2×C22⋊D61C4⋊C414yes—
130C22×Dic61—22no—
131C22×Dic61—41no—
132C22×Dic61C22⋊C412yes—
133#11758222—22no—
134#11758222—22no—
135#11758222—22no—
136#11758222—22no—
137#11758222—22no—
138#11758222—22no—
139#11758222—22no—
140#11758222—22no—
141#11758222—22no—
142#11758222—22no—
143#11758222—22no—
144#11758222—22no—
145#11758222—22no—
146#11758222—22no—
147#11758222—22no—
148#11758222—22no—
149C2×D244—21no—
150C2×D244—21no—
151C2×D244—21no—
152C2×D244—21no—
153C2×D244C4212yes—
154C2×D244C4⋊C412yes—
155C2×C4×D61—22no—
156C2×C4×D61—22no—
157C2×C4×D61—22no—
158C2×C4×D61—22no—
159C2×C4×D61—28no—
160C2×C4×D61C22⋊C414yes—
161C2×C4×D61C4⋊C414yes—
162C2×Dic122—21no—
163C2×Dic122—21no—
164C2×Dic122—21no—
165C2×Dic122—21no—
166C2×Dic122C4212yes—
167C2×Dic122C4⋊C412yes—
168M16×C61—22no—
169M16×C61—22no—
170M16×C61—41no—
171M16×C61—41no—
172C2×C8×C61—22no—
173C2×C8×C61—41no—
174C2×C8×C61—41no—
175C4⋊C4×C61—22no—
176C4⋊C4×C61—22no—
177C4⋊C4×C61—41no—
178C4⋊C4×C61C22⋊C414yes—
179C22⋊C4×C61—22no—
180C22⋊C4×C61—22no—
181C22⋊C4×C61—22no—
182C22⋊C4×C61—41no—
183C42×C61—22no—
184C42×C61—22no—
185C42×C61—22no—
186C42×C61—41no—
187C22⋊Dic61—22no—
188C22⋊Dic61—22no—
189C22⋊Dic61—22no—
190C22⋊Dic61—41no—
191#11758198—28no—
192C4⋊Dic61—22no—
193C4⋊Dic61—22no—
194C4⋊Dic61—41no—
195C4⋊Dic61C22⋊C414yes—
196#11758196—28no—
197C4×Dic61—28no—
198C4×Dic61—22no—
199C4×Dic61—22no—
200C4×Dic61—22no—
201C4×Dic61—41no—
202C2×#92418—22no—
203C2×#92418—22no—
204C2×#92418—22no—
205C2×#92418—41no—
206C2×#92418—41no—
207C2×#92418—41no—
208C2×#92418—41no—
209C23×C61—42no—
210C23×C61C2×C22⋊C411yescharacteristic
211C23×C61C2×C22⋊C411yescharacteristic
212C23×C61C2×C4⋊C411yescharacteristic
213C23×C61#12011yescharacteristic
214C23×C61#12011yescharacteristic
215C23×C61#12311yescharacteristic
216C23×C61#12311yescharacteristic
217C22×D61—22no—
218C22×D61—22no—
219C22×D61—44no—
220C22×D61#9514yes—
221C2×C4×C61—24no—
222C2×C4×C61—42no—
223C2×C4×C61—82no—
224C2×C4×C61—82no—
225C2×C4×C61C2×C4211yescharacteristic
226C2×C4×C61C2×C22⋊C411yescharacteristic
227C2×C4×C61C2×C22⋊C411yescharacteristic
228C2×C4×C61C2×C22⋊C411yescharacteristic
229C2×C4×C61C2×C4⋊C411yescharacteristic
230C2×C4×C61C2×C4⋊C411yescharacteristic
231C2×C4×C61#11711yescharacteristic
232C2×C4×C61#11711yescharacteristic
233C2×C4×C61C4×D411yescharacteristic
234C2×C4×C61C4×D411yescharacteristic
235C2×C4×C61C4×D411yescharacteristic
236C2×C4×C61C4×D411yescharacteristic
237C2×C4×C61C4×D411yescharacteristic
238C2×C4×C61C4×D411yescharacteristic
239C2×C4×C61C4×D411yescharacteristic
240C2×C4×C61C4×D411yescharacteristic
241C2×C4×C61#12011yescharacteristic
242C2×C4×C61#12011yescharacteristic
243C2×C4×C61#12111yescharacteristic
244C2×C4×C61#12111yescharacteristic
245C2×C4×C61#12111yescharacteristic
246C2×C4×C61#12111yescharacteristic
247C2×C4×C61C22⋊Q811yescharacteristic
248C2×C4×C61C22⋊Q811yescharacteristic
249C2×C4×C61C22⋊Q811yescharacteristic
250C2×C4×C61C22⋊Q811yescharacteristic
251C2×C4×C61#12311yescharacteristic
252C2×C4×C61#12311yescharacteristic
253C22⋊D61—24no—
254C22⋊D61—24no—
255C22⋊D61—24no—
256C22⋊D61—24no—
257C2×Dic61—22no—
258C2×Dic61—22no—
259C2×Dic61—24no—
260C2×Dic61—42no—
261C2×Dic61—82no—
262C2×Dic61—82no—
263C2×Dic61#9514yes—
264D244—22no—
265D244—22no—
266D244—22no—
267D244—22no—
268C4×D61—24no—
269C4×D61—24no—
270C4×D61—24no—
271C4×D61—24no—
272C4×D61—88no—
273Dic122—22no—
274Dic122—22no—
275Dic122—22no—
276Dic122—22no—
277C8×C61—42no—
278#92418—42no—
279#10752—611nomaximal, Sylow
280C22×C61—22no—
281C22×C61—22no—
282C22×C61—22no—
283C22×C61—41no—
284C22×C61—41no—
285C22×C61—41no—
286C22×C61—41no—
287C22×C61C2×#9511yescharacteristic
288C22×C61#37911yescharacteristic
289C22×C61#38111yescharacteristic
290C22×C61#38511yescharacteristic
291C22×C61#38511yescharacteristic
292C22×C61#38611yescharacteristic
293C22×C61#38611yescharacteristic
294C2×D61—24no—
295C2×D61—44no—
296C2×D61—44no—
297C2×D61—82no—
298C4×C61—44no—
299C4×C61—82no—
300C4×C61C4×C22⋊C411yescharacteristic
301C4×C61C4×C22⋊C411yescharacteristic
302C4×C61#38011yescharacteristic
303C4×C61#38011yescharacteristic
304C4×C61#38511yescharacteristic
305C4×C61#38511yescharacteristic
306C4×C61#38611yescharacteristic
307C4×C61#38611yescharacteristic
308Dic61—24no—
309Dic61—44no—
310Dic61—82no—
311#3428—611no—
312C2×#410—611no—
313C2×#408—611no—
314#1735—611no—
315#1735—611no—
316#1733—611no—
317#1733D6111yescharacteristic, complemented
318C2×C61—21no—
319C2×C61—21no—
320C2×C61—21no—
321C2×C61—21no—
322C2×C61—21no—
323C2×C61—21no—
324C2×C61C22⋊#9511yescharacteristic
325C2×C61#179412yes—
326D61—44no—
327C22×#41—611no—
328C22×M16C2×D6111yescharacteristic
329C23×C8—611no—
330#520—1222no—
331#520—1222no—
332#519—612no—
333#518—612no—
334#518—1222no—
335C4×#41—1222no—
336C4×#41—1222no—
337C2×#117—611no—
338C2×#117—611no—
339C2×#117—1222no—
340C2×#117C2×D6111yescharacteristic
341C2×#117C2×D6111yescharacteristic
342#410—614no—
343#410—1222no—
344C2×#99—1222no—
345#408—614no—
346#408—1222no—
347C2×C22⋊C8—1222no—
348#340—1222no—
349#340—1222no—
350#339—22no—
351#339—1222no—
352C61#1075211yescharacteristic, complemented, Sylow
353C2×#41—612no—
354C2×#41—612no—
355C2×#41—612no—
356C2×#41—612no—
357C2×#41—1221no—
358C2×#41—1221no—
359C2×#41—1221no—
360C2×#41—1221no—
361C2×#41—1221no—
362C2×#41—1221no—
363C2×#41—1221no—
364C2×#41—1221no—
365C22×Q8—611no—
366C22×D4—612no—
367C22×D4—611no—
368C23×C4—612no—
369C23×C4C22×D6111yescharacteristic
370C2×M16—21no—
371C2×M16—21no—
372C2×M16—21no—
373C2×M16—21no—
374C2×M16—21no—
375C2×M16C4×D6112yescomplemented
376C22×C8—21no—
377C22×C8—612no—
378C22×C8—1221no—
379C22×C8—1221no—
380C22×C8—1221no—
381C22×C8—1221no—
382C22×C8—1221no—
383C22×C8—1221no—
384C4×Q8—1224no—
385C4×Q8—1224no—
386C4×D4—1224no—
387C4×D4—1228no—
388C4×D4—1228no—
389C4×D4—1224no—
390#117—21no—
391#117—21no—
392#117—21no—
393#117—21no—
394#117—21no—
395#117—612no—
396#117—612no—
397#117—612no—
398#117—1224no—
399#117—1224no—
400#117—1224no—
401#117—1221no—
402#117—1221no—
403#117—1221no—
404#117—1221no—
405#117—1221no—
406#117C4×D6112yes—
407#117C4×D6112yescomplemented
408#117C22⋊D6112yes—
409C2×C4⋊C4—21no—
410C2×C4⋊C4—612no—
411C2×C4⋊C4—1224no—
412C2×C4⋊C4—1221no—
413C2×C4⋊C4D24412yes—
414C2×C22⋊C4—21no—
415C2×C22⋊C4—21no—
416C2×C22⋊C4—1224no—
417C2×C22⋊C4—1221no—
418C2×C22⋊C4—1221no—
419C2×C42—21no—
420C2×C42—21no—
421C2×C42—1224no—
422C2×C42—1221no—
423C2×C42—1221no—
424#99—1228no—
425C22⋊C8—1228no—
426C24—612no—
427C24D4×D6111yescharacteristic
428#41—1222no—
429#41—1222no—
430#41—1222no—
431#41—1222no—
432#41—1222no—
433#41—1222no—
434#41—1222no—
435#41—1222no—
436#41—1222no—
437#41—1222no—
438#41—1222no—
439#41—1222no—
440#41—1222no—
441#41—1222no—
442#41—1222no—
443#41—1222no—
444C2×Q8—612no—
445C2×Q8—612no—
446C2×Q8—1221no—
447C2×Q8—1221no—
448C2×Q8—1221no—
449C2×Q8—1221no—
450C2×D4—614no—
451C2×D4—614no—
452C2×D4—612no—
453C2×D4—612no—
454C2×D4—1222no—
455C2×D4—1222no—
456C2×D4—1222no—
457C2×D4—1222no—
458C2×D4—1221no—
459C2×D4—1221no—
460C2×D4—1221no—
461C2×D4—1221no—
462C22×C4—22no—
463C22×C4—41no—
464C22×C4—612no—
465C22×C4—614no—
466C22×C4—614no—
467C22×C4—1228no—
468C22×C4—1222no—
469C22×C4—1222no—
470C22×C4—1222no—
471C22×C4—1222no—
472C22×C4—1222no—
473C22×C4—2441no—
474C22×C4C2×C4×D6111yescharacteristic
475C22×C4C2×C4×D6111yescharacteristic
476C22×C4C2×C4×D6111yescharacteristic
477C22×C4C2×D24411yescharacteristic
478C22×C4#1175822211yescharacteristic
479C22×C4#1175822211yescharacteristic
480C22×C4D4×D6111yescharacteristic
481C22×C4D4×D6111yescharacteristic
482C22×C4D4×D6111yescharacteristic
483C22×C4D4×D6111yescharacteristic
484C22×C4#1175822411yescharacteristic
485C22×C4Q8×D6111yescharacteristic
486C22×C4#1175822611yescharacteristic
487C22×C4C2×C22⋊D6111yescharacteristic
488M16—22no—
489M16—22no—
490M16—41no—
491M16—41no—
492C2×C8—22no—
493C2×C8—41no—
494C2×C8—41no—
495C2×C8—1222no—
496C2×C8—1222no—
497C2×C8—1222no—
498C2×C8—2441no—
499C2×C8—2441no—
500C2×C8—2441no—
501C2×C8—2441no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—41no—
505C4⋊C4—614no—
506C4⋊C4—1228no—
507C4⋊C4—1222no—
508C4⋊C4—1222no—
509C4⋊C4—2441no—
510C4⋊C4#1175819814yes—
511C22⋊C4—22no—
512C22⋊C4—22no—
513C22⋊C4—22no—
514C22⋊C4—41no—
515C22⋊C4—1228no—
516C22⋊C4—1222no—
517C22⋊C4—1222no—
518C22⋊C4—1222no—
519C22⋊C4—2441no—
520C42—22no—
521C42—22no—
522C42—22no—
523C42—41no—
524C42—1228no—
525C42—1222no—
526C42—1222no—
527C42—1222no—
528C42—2441no—
529C23—42no—
530C23—614no—
531C23—1222no—
532C23—1222no—
533C23—2444no—
534C23C22⋊C4×D6111yescharacteristic
535C23C22⋊C4×D6111yescharacteristic
536C23C4⋊C4×D6111yescharacteristic
537C23#1451974511yescharacteristic
538C23#1451974711yescharacteristic
539C23#1451975911yescharacteristic
540C23#1451976211yescharacteristic
541Q8—1222no—
542Q8—1222no—
543Q8—1222no—
544Q8—1222no—
545D4—1224no—
546D4—1224no—
547D4—1224no—
548D4—1224no—
549D4—1222no—
550D4—1222no—
551D4—1222no—
552D4—1222no—
553C2×C4—24no—
554C2×C4—42no—
555C2×C4—82no—
556C2×C4—82no—
557C2×C4—614no—
558C2×C4—1222no—
559C2×C4—1222no—
560C2×C4—1224no—
561C2×C4—1224no—
562C2×C4—1224no—
563C2×C4—1224no—
564C2×C4—1224no—
565C2×C4—4888no—
566C2×C4—4882no—
567C2×C4—4882no—
568C2×C4—2442no—
569C2×C4C42×D6111yescharacteristic
570C2×C4C22⋊C4×D6111yescharacteristic
571C2×C4C22⋊C4×D6111yescharacteristic
572C2×C4C2×#1175819811yescharacteristic
573C2×C4C4⋊C4×D6111yescharacteristic
574C2×C4C4⋊C4×D6111yescharacteristic
575C2×C4#1451973411yescharacteristic
576C2×C4#1451973511yescharacteristic
577C2×C4#1451973711yescharacteristic
578C2×C4#1451973711yescharacteristic
579C2×C4C4×C22⋊D6111yescharacteristic
580C2×C4C4×C22⋊D6111yescharacteristic
581C2×C4#1451973911yescharacteristic
582C2×C4#1451973911yescharacteristic
583C2×C4C4×D24411yescharacteristic
584C2×C4C4×D24411yescharacteristic
585C2×C4#1451974511yescharacteristic
586C2×C4#1451974711yescharacteristic
587C2×C4#1451974811yescharacteristic
588C2×C4#1451975011yescharacteristic
589C2×C4#1451975011yescharacteristic
590C2×C4#1451975211yescharacteristic
591C2×C4#1451975311yescharacteristic
592C2×C4#1451975511yescharacteristic
593C2×C4#1451975511yescharacteristic
594C2×C4#1451975711yescharacteristic
595C2×C4#1451975911yescharacteristic
596C2×C4#1451976211yescharacteristic
597C8—42no—
598C8—2442no—
599C22—22no—
600C22—22no—
601C22—22no—
602C22—41no—
603C22—41no—
604C22—41no—
605C22—41no—
606C22—1224no—
607C22—4882no—
608C22—2444no—
609C22—2444no—
610C22#95×D6111yescharacteristic
611C22#1683961711yescharacteristic
612C22#1683962411yescharacteristic
613C22#1683963311yescharacteristic
614C22#1683963611yescharacteristic
615C22#1683963811yescharacteristic
616C22#1683964011yescharacteristic
617C4—44no—
618C4—82no—
619C4—1224no—
620C4—4882no—
621C4—2444no—
622C4#1683960911yescharacteristic
623C4C4×#1175819811yescharacteristic
624C4#1683961911yescharacteristic
625C4#1683962111yescharacteristic
626C4#1683963311yescharacteristic
627C4#1683963611yescharacteristic
628C4#1683963811yescharacteristic
629C4#1683964011yescharacteristic
630C2—21no—
631C2—21no—
632C2—21no—
633C2—21no—
634C2—21no—
635C2—21no—
636C2—2444no—
637C2#2144404011yescharacteristic
638C2#2144578112yes—
subgroupG/H
centerC2×C4#14519745
commutatorC2×C4×C61C2×C42
FrattiniC23×C4C22×D61
FittingC61×#1733C2
socleC22×C61#379
prime radicalC22×#11758222C22
cosocleC23×C4×C61C23
hypercenter#1733D61
hypocenterC61#10752
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #1733

Complements

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

NH ≅ G/Ntype
4. #21445535C2semidirect
5. #21445535C2semidirect
6. #21445530C2semidirect
7. C61×#1733C2semidirect
12. #16840239C4semidirect
13. #16840236C4semidirect
21. #16839735C4semidirect
24. #16839730C4semidirect
55. C22×#92418C2×C4semidirect
68. #14519736C2×C4semidirect
90. C2×M16×C61C2×C4semidirect
98. #117×C61C2×C4semidirect
317. #1733D61semidirect
352. C61#10752semidirect
375. C2×M16C4×D61semidirect
407. #117C4×D61semidirect

Maximal subgroups

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

subgroupindex
#214590202
C2×#168397352
C2×#168397302
#214455352
#214455302
C61×#17332
#1075261