Finite Group Databasebeta

← #29585513

Subgroups — #29585513

Order 15616.

1218 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459020C211yesmaximal, characteristic
2C2×#16839780C211yesmaximal, characteristic
3C2×#16839759C211yesmaximal, characteristic
4#21445551C211yesmaximal, characteristic, complemented
5#21445528C211yesmaximal, characteristic, complemented
6#21445515C211yesmaximal, characteristic, complemented
7C61×#1735C211yesmaximal, characteristic, complemented
8C22×#11758222C2211yescharacteristic
9C22×#11758194C2211yescharacteristic
10#16840243—22no—
11#16840242C414yescomplemented
12#16840239—21no—
13#16840236—21no—
14#16840235—22no—
15C4×#11758222—22no—
16#16840230—22no—
17C2×#14519736C2211yescharacteristic
18C2×#14519736C2211yescharacteristic
19#117×D61—22no—
20#16839780—22no—
21#16839780C414yescomplemented
22C2×#14519705—22no—
23#16839759—22no—
24#16839759C414yescomplemented
25C2×#14519702—21no—
26C2×#14519699—21no—
27#16839528—22no—
28#16839521—22no—
29#16839516—22no—
30C23×C8×C61C2211yescharacteristic
31C2×#117×C61C2211yescharacteristic
32C2×#117×C61C2211yescharacteristic
33C61×#340—22no—
34C2×#11758222—21no—
35C2×#11758222—21no—
36C2×#11758222—21no—
37C2×#11758222—21no—
38C2×#11758222—21no—
39C2×#11758222—21no—
40C2×#11758222C2×C412yes—
41C2×#11758222C2×C412yes—
42C2×#11758222C2×C412yes—
43C2×#11758222D412yes—
44C2×#11758222D412yes—
45C2×#11758222Q812yes—
46C22×Dic122D411yescharacteristic
47C22×D244D411yescharacteristic
48C22×C22⋊D61—21no—
49C22×C4×D61—21no—
50C2×#11758194—21no—
51C2×#11758194—21no—
52C2×#11758194—21no—
53C2×#11758194—21no—
54C2×#11758194—21no—
55C2×#11758194C2×C412yescomplemented
56C22×#92418—21no—
57C4×Dic122—24no—
58#14519742—24no—
59C4×D244—24no—
60#14519739—24no—
61C4×C22⋊D61—28no—
62#14519737—28no—
63#14519736—21no—
64#14519736—21no—
65#14519736—21no—
66#14519736—21no—
67#14519736—21no—
68#14519736—21no—
69#14519736—21no—
70#14519736C2×C412yescomplemented
71#14519735—24no—
72#14519734—24no—
73#14519733—24no—
74C2×C4⋊Dic61—21no—
75C2×C4⋊Dic61C2×C412yes—
76C4⋊C4×D61—24no—
77C2×C22⋊Dic61—21no—
78C2×C22⋊Dic61D412yes—
79C22⋊C4×D61—24no—
80C2×C4×Dic61—21no—
81C2×C4×Dic61D412yes—
82C42×D61—24no—
83#14519705—28no—
84#14519702—24no—
85#14519699—24no—
86C23×C4×C61C2311yescharacteristic
87C22×C8×C61—21no—
88C22×C8×C61—21no—
89C22×C8×C61—21no—
90C22×C8×C61—21no—
91C22×C8×C61—21no—
92C22×C8×C61—21no—
93C22×C8×C61C2×C412yescomplemented
94#117×C61—21no—
95#117×C61—21no—
96#117×C61—21no—
97#117×C61—21no—
98#117×C61—21no—
99#117×C61C2×C412yes—
100#117×C61C2×C412yescomplemented
101#117×C61D412yes—
102C2×C4⋊C4×C61—21no—
103C2×C4⋊C4×C61D412yes—
104C2×C22⋊C4×C61—21no—
105C2×C22⋊C4×C61—21no—
106C2×C42×C61—21no—
107C2×C42×C61—21no—
108C24×C61C2×D411yescharacteristic
109C23×D61—21no—
110C22×C4×C61—22no—
111C22×C4×C61—41no—
112C22×C4×C61C22×C411yescharacteristic
113C22×C4×C61C22×C411yescharacteristic
114C22×C4×C61C22×C411yescharacteristic
115C22×C4×C61C2×D411yescharacteristic
116C22×C4×C61C2×D411yescharacteristic
117C22×C4×C61C2×D411yescharacteristic
118C22×C4×C61C2×D411yescharacteristic
119C22×C4×C61C2×D411yescharacteristic
120C22×C4×C61C2×D411yescharacteristic
121C22×C4×C61C2×Q811yescharacteristic
122C22×C4×C61#4111yescharacteristic
123C22×C4×C61#4111yescharacteristic
124C22×C4×C61#4111yescharacteristic
125C22×C4×C61#4111yescharacteristic
126C2×C22⋊D61—22no—
127C2×C22⋊D61—22no—
128C2×C22⋊D61—22no—
129C2×C22⋊D61—22no—
130C2×C22⋊D61—22no—
131C2×C22⋊D61—22no—
132C22×Dic61—21no—
133C22×Dic61—41no—
134C22×Dic61C22⋊C414yes—
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#11758222C4214yes—
148#11758222C22⋊C414yes—
149#11758222C4⋊C414yes—
150#11758222C4⋊C414yes—
151C2×D244—21no—
152C2×D244—21no—
153C2×D244—21no—
154C2×D244C22⋊C412yes—
155C2×D244C22⋊C412yes—
156C2×D244C4⋊C412yes—
157C2×C4×D61—22no—
158C2×C4×D61—22no—
159C2×C4×D61—22no—
160C2×C4×D61—22no—
161C2×C4×D61—22no—
162C2×C4×D61—22no—
163C2×C4×D61—28no—
164C2×Dic122—21no—
165C2×Dic122—21no—
166C2×Dic122—21no—
167C2×Dic122C22⋊C412yes—
168C2×Dic122C22⋊C412yes—
169C2×Dic122C4⋊C412yes—
170C2×C8×C61—22no—
171C2×C8×C61—22no—
172C2×C8×C61—22no—
173C2×C8×C61—41no—
174C2×C8×C61—41no—
175C2×C8×C61—41no—
176C2×C8×C61—41no—
177C4⋊C4×C61—22no—
178C4⋊C4×C61—22no—
179C4⋊C4×C61—41no—
180C4⋊C4×C61C22⋊C414yes—
181C22⋊C4×C61—22no—
182C22⋊C4×C61—22no—
183C22⋊C4×C61—22no—
184C22⋊C4×C61—41no—
185C42×C61—22no—
186C42×C61—22no—
187C42×C61—22no—
188C42×C61—41no—
189C22⋊Dic61—22no—
190C22⋊Dic61—22no—
191C22⋊Dic61—22no—
192C22⋊Dic61—41no—
193#11758198—28no—
194C4⋊Dic61—22no—
195C4⋊Dic61—22no—
196C4⋊Dic61—22no—
197C4⋊Dic61—41no—
198#11758196—28no—
199C4×Dic61—28no—
200C4×Dic61—22no—
201C4×Dic61—22no—
202C4×Dic61—22no—
203C4×Dic61—41no—
204#11758194—22no—
205#11758194—22no—
206#11758194—41no—
207#11758194—41no—
208C2×#92418—22no—
209C2×#92418—41no—
210C2×#92418—41no—
211C23×C61—42no—
212C23×C61C2×C22⋊C411yescharacteristic
213C23×C61C4×D411yescharacteristic
214C23×C61C4×D411yescharacteristic
215C23×C61#12011yescharacteristic
216C23×C61#12011yescharacteristic
217C23×C61#12111yescharacteristic
218C23×C61C22⋊Q811yescharacteristic
219C22×D61—22no—
220C22×D61—22no—
221C22×D61—22no—
222C22×D61—82no—
223C2×C4×C61—24no—
224C2×C4×C61—42no—
225C2×C4×C61—82no—
226C2×C4×C61—82no—
227C2×C4×C61C2×C4211yescharacteristic
228C2×C4×C61C2×C22⋊C411yescharacteristic
229C2×C4×C61C2×C22⋊C411yescharacteristic
230C2×C4×C61C2×C22⋊C411yescharacteristic
231C2×C4×C61C2×C22⋊C411yescharacteristic
232C2×C4×C61C2×C4⋊C411yescharacteristic
233C2×C4×C61C2×C4⋊C411yescharacteristic
234C2×C4×C61C2×C4⋊C411yescharacteristic
235C2×C4×C61#11711yescharacteristic
236C2×C4×C61#11711yescharacteristic
237C2×C4×C61C4×D411yescharacteristic
238C2×C4×C61C4×D411yescharacteristic
239C2×C4×C61C4×D411yescharacteristic
240C2×C4×C61C4×D411yescharacteristic
241C2×C4×C61C4×D411yescharacteristic
242C2×C4×C61C4×D411yescharacteristic
243C2×C4×C61#12011yescharacteristic
244C2×C4×C61#12011yescharacteristic
245C2×C4×C61#12111yescharacteristic
246C2×C4×C61#12111yescharacteristic
247C2×C4×C61#12111yescharacteristic
248C2×C4×C61C22⋊Q811yescharacteristic
249C2×C4×C61C22⋊Q811yescharacteristic
250C2×C4×C61C22⋊Q811yescharacteristic
251C2×C4×C61#12311yescharacteristic
252C2×C4×C61#12311yescharacteristic
253C2×C4×C61#12311yescharacteristic
254C2×C4×C61#12311yescharacteristic
255C22⋊D61—24no—
256C22⋊D61—24no—
257C22⋊D61—24no—
258C22⋊D61—24no—
259C2×Dic61—22no—
260C2×Dic61—22no—
261C2×Dic61—22no—
262C2×Dic61—24no—
263C2×Dic61—44no—
264C2×Dic61—42no—
265C2×Dic61—82no—
266D244—22no—
267D244—22no—
268D244—22no—
269D244#9514yes—
270C4×D61—24no—
271C4×D61—24no—
272C4×D61—24no—
273C4×D61—24no—
274C4×D61—88no—
275Dic122—22no—
276Dic122—22no—
277Dic122—22no—
278Dic122#9514yes—
279C8×C61—42no—
280#92418—42no—
281#10779—611nomaximal, Sylow
282C22×C61—22no—
283C22×C61—22no—
284C22×C61—22no—
285C22×C61—41no—
286C22×C61—41no—
287C22×C61—41no—
288C22×C61—41no—
289C22×C61C4×C22⋊C411yescharacteristic
290C22×C61#37911yescharacteristic
291C22×C61#38011yescharacteristic
292C22×C61#38511yescharacteristic
293C22×C61#38511yescharacteristic
294C22×C61#38611yescharacteristic
295C22×C61#38611yescharacteristic
296C2×D61—24no—
297C2×D61—44no—
298C2×D61—82no—
299C2×D61—82no—
300C4×C61—44no—
301C4×C61—82no—
302C4×C61C2×#9511yescharacteristic
303C4×C61C4×C22⋊C411yescharacteristic
304C4×C61#38011yescharacteristic
305C4×C61#38111yescharacteristic
306C4×C61#38511yescharacteristic
307C4×C61#38511yescharacteristic
308C4×C61#38611yescharacteristic
309C4×C61#38611yescharacteristic
310Dic61—24no—
311Dic61—44no—
312Dic61—82no—
313#3428—611no—
314C2×#421—611no—
315C2×#416—611no—
316#1740—611no—
317#1735D6111yescharacteristic, complemented
318#1733—611no—
319#1729—611no—
320C2×C61—21no—
321C2×C61—21no—
322C2×C61—21no—
323C2×C61—21no—
324C2×C61—21no—
325C2×C61—21no—
326C2×C61C22⋊#9511yescharacteristic
327C2×C61#179412yes—
328D61—44no—
329C22×#41—611no—
330C22×M16—611no—
331C23×C8C2×D6111yescharacteristic
332#520—614no—
333#520—1222no—
334#519—1221no—
335#518—1222no—
336#518—1221no—
337C4×#41—1222no—
338C4×#41—1222no—
339C2×#117—611no—
340C2×#117—611no—
341C2×#117—1222no—
342C2×#117C2×D6111yescharacteristic
343C2×#117C2×D6111yescharacteristic
344#421—614no—
345#421—1222no—
346C2×#104—1222no—
347#416—614no—
348#416—1222no—
349C2×#103—1221no—
350C2×#102—1221no—
351#342—1222no—
352#340—22no—
353#339—1222no—
354#336—1222no—
355C61#1077911yescharacteristic, complemented, Sylow
356C2×#41—612no—
357C2×#41—612no—
358C2×#41—612no—
359C2×#41—612no—
360C2×#41—612no—
361C2×#41—612no—
362C2×#41—1221no—
363C2×#41—1221no—
364C2×#41—1221no—
365C2×#41—1221no—
366C2×#41—1221no—
367C2×#41—1221no—
368C22×Q8—611no—
369C22×D4—611no—
370C22×D4—1221no—
371C23×C4—1221no—
372C23×C4C22×D6111yescharacteristic
373C2×M16—612no—
374C2×M16—1221no—
375C2×M16—1221no—
376C2×M16—1221no—
377C2×M16—1221no—
378C2×M16—1221no—
379C22×C8—21no—
380C22×C8—21no—
381C22×C8—21no—
382C22×C8—21no—
383C22×C8—21no—
384C22×C8—21no—
385C22×C8—1221no—
386C22×C8C4×D6112yescomplemented
387C4×Q8—1224no—
388C4×Q8—1224no—
389C4×D4—1224no—
390C4×D4—1228no—
391C4×D4—1224no—
392C4×D4—1228no—
393#117—21no—
394#117—21no—
395#117—21no—
396#117—21no—
397#117—21no—
398#117—612no—
399#117—1224no—
400#117—1224no—
401#117—1221no—
402#117—1221no—
403#117—1221no—
404#117—1221no—
405#117—1224no—
406#117—1221no—
407#117—1221no—
408#117—1221no—
409#117C4×D6112yes—
410#117C4×D6112yescomplemented
411#117D24412yes—
412C2×C4⋊C4—21no—
413C2×C4⋊C4—612no—
414C2×C4⋊C4—1224no—
415C2×C4⋊C4—1221no—
416C2×C4⋊C4C22⋊D6112yes—
417C2×C22⋊C4—21no—
418C2×C22⋊C4—21no—
419C2×C22⋊C4—612no—
420C2×C22⋊C4—1224no—
421C2×C22⋊C4—1221no—
422C2×C42—21no—
423C2×C42—21no—
424C2×C42—612no—
425C2×C42—1224no—
426C2×C42—1221no—
427#104—1228no—
428#103—1224no—
429#102—1224no—
430C24—1221no—
431C24D4×D6111yescharacteristic
432#41—614no—
433#41—614no—
434#41—614no—
435#41—614no—
436#41—1222no—
437#41—1222no—
438#41—1222no—
439#41—1222no—
440#41—1222no—
441#41—1222no—
442#41—1222no—
443#41—1222no—
444#41—1222no—
445#41—1222no—
446#41—1222no—
447#41—1222no—
448C2×Q8—612no—
449C2×Q8—612no—
450C2×Q8—612no—
451C2×Q8—1221no—
452C2×Q8—1221no—
453C2×Q8—1221no—
454C2×D4—612no—
455C2×D4—612no—
456C2×D4—612no—
457C2×D4—1222no—
458C2×D4—1222no—
459C2×D4—1222no—
460C2×D4—1222no—
461C2×D4—1222no—
462C2×D4—1222no—
463C2×D4—1221no—
464C2×D4—1221no—
465C2×D4—1221no—
466C22×C4—22no—
467C22×C4—41no—
468C22×C4—614no—
469C22×C4—1228no—
470C22×C4—1221no—
471C22×C4—1222no—
472C22×C4—1222no—
473C22×C4—1222no—
474C22×C4—1222no—
475C22×C4—1222no—
476C22×C4—1222no—
477C22×C4—2441no—
478C22×C4C2×C4×D6111yescharacteristic
479C22×C4C2×C4×D6111yescharacteristic
480C22×C4C2×C4×D6111yescharacteristic
481C22×C4C2×D24411yescharacteristic
482C22×C4#1175822211yescharacteristic
483C22×C4#1175822211yescharacteristic
484C22×C4D4×D6111yescharacteristic
485C22×C4D4×D6111yescharacteristic
486C22×C4D4×D6111yescharacteristic
487C22×C4D4×D6111yescharacteristic
488C22×C4#1175822411yescharacteristic
489C22×C4Q8×D6111yescharacteristic
490C22×C4#1175822611yescharacteristic
491C22×C4C2×C22⋊D6111yescharacteristic
492M16—1222no—
493M16—1222no—
494M16—2441no—
495M16—2441no—
496C2×C8—22no—
497C2×C8—22no—
498C2×C8—22no—
499C2×C8—41no—
500C2×C8—41no—
501C2×C8—41no—
502C2×C8—41no—
503C2×C8—1222no—
504C2×C8—2441no—
505C2×C8—2441no—
506C4⋊C4—22no—
507C4⋊C4—22no—
508C4⋊C4—41no—
509C4⋊C4—1228no—
510C4⋊C4—1222no—
511C4⋊C4—1222no—
512C4⋊C4—1222no—
513C4⋊C4—2441no—
514C4⋊C4#1175819814yes—
515C22⋊C4—22no—
516C22⋊C4—22no—
517C22⋊C4—22no—
518C22⋊C4—41no—
519C22⋊C4—1228no—
520C22⋊C4—1222no—
521C22⋊C4—1222no—
522C22⋊C4—1222no—
523C22⋊C4—2441no—
524C42—22no—
525C42—22no—
526C42—22no—
527C42—41no—
528C42—1228no—
529C42—1222no—
530C42—1222no—
531C42—1222no—
532C42—2441no—
533C23—42no—
534C23—1222no—
535C23—1222no—
536C23—1222no—
537C23—4882no—
538C23C22⋊C4×D6111yescharacteristic
539C23#1451973711yescharacteristic
540C23#1451973911yescharacteristic
541C23#1451974511yescharacteristic
542C23#1451974711yescharacteristic
543C23#1451975011yescharacteristic
544C23#1451975511yescharacteristic
545Q8—614no—
546Q8—1222no—
547Q8—1222no—
548Q8—1222no—
549D4—614no—
550D4—1224no—
551D4—1224no—
552D4—1224no—
553D4—1224no—
554D4—1222no—
555D4—1222no—
556D4—1222no—
557C2×C4—24no—
558C2×C4—42no—
559C2×C4—82no—
560C2×C4—82no—
561C2×C4—1222no—
562C2×C4—1222no—
563C2×C4—1222no—
564C2×C4—1224no—
565C2×C4—1224no—
566C2×C4—1224no—
567C2×C4—1224no—
568C2×C4—1224no—
569C2×C4—4888no—
570C2×C4—4882no—
571C2×C4—2442no—
572C2×C4—2444no—
573C2×C4C42×D6111yescharacteristic
574C2×C4C22⋊C4×D6111yescharacteristic
575C2×C4C22⋊C4×D6111yescharacteristic
576C2×C4C22⋊C4×D6111yescharacteristic
577C2×C4C2×#1175819811yescharacteristic
578C2×C4C4⋊C4×D6111yescharacteristic
579C2×C4C4⋊C4×D6111yescharacteristic
580C2×C4C4⋊C4×D6111yescharacteristic
581C2×C4#1451973411yescharacteristic
582C2×C4#1451973511yescharacteristic
583C2×C4#1451973711yescharacteristic
584C2×C4C4×C22⋊D6111yescharacteristic
585C2×C4C4×C22⋊D6111yescharacteristic
586C2×C4#1451973911yescharacteristic
587C2×C4C4×D24411yescharacteristic
588C2×C4C4×D24411yescharacteristic
589C2×C4#1451974511yescharacteristic
590C2×C4#1451974711yescharacteristic
591C2×C4#1451974811yescharacteristic
592C2×C4#1451975011yescharacteristic
593C2×C4#1451975211yescharacteristic
594C2×C4#1451975311yescharacteristic
595C2×C4#1451975511yescharacteristic
596C2×C4#1451975711yescharacteristic
597C2×C4#1451975911yescharacteristic
598C2×C4#1451975911yescharacteristic
599C2×C4#1451976211yescharacteristic
600C2×C4#1451976211yescharacteristic
601C8—42no—
602C8—2442no—
603C22—22no—
604C22—22no—
605C22—22no—
606C22—41no—
607C22—41no—
608C22—41no—
609C22—41no—
610C22—1224no—
611C22—4882no—
612C22—4882no—
613C22—2444no—
614C22#1683960911yescharacteristic
615C22#1683961711yescharacteristic
616C22#1683962111yescharacteristic
617C22#1683963311yescharacteristic
618C22#1683963611yescharacteristic
619C22#1683963811yescharacteristic
620C22#1683964011yescharacteristic
621C4—44no—
622C4—82no—
623C4—1224no—
624C4—4882no—
625C4—2444no—
626C4#95×D6111yescharacteristic
627C4C4×#1175819811yescharacteristic
628C4#1683961911yescharacteristic
629C4#1683962411yescharacteristic
630C4#1683963311yescharacteristic
631C4#1683963611yescharacteristic
632C4#1683963811yescharacteristic
633C4#1683964011yescharacteristic
634C2—21no—
635C2—21no—
636C2—21no—
637C2—21no—
638C2—21no—
639C2—21no—
640C2—2444no—
641C2#2144404011yescharacteristic
642C2#2144578412yes—
subgroupG/H
centerC2×C4#14519747
commutatorC2×C4×C61C2×C42
FrattiniC23×C4C22×D61
FittingC61×#1735C2
socleC22×C61#379
prime radicalC22×#11758222C22
cosocleC23×C4×C61C23
hypercenter#1735D61
hypocenterC61#10779
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #1735

Complements

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

NH ≅ G/Ntype
4. #21445551C2semidirect
5. #21445528C2semidirect
6. #21445515C2semidirect
7. C61×#1735C2semidirect
11. #16840242C4semidirect
21. #16839780C4semidirect
24. #16839759C4semidirect
55. C2×#11758194C2×C4semidirect
70. #14519736C2×C4semidirect
93. C22×C8×C61C2×C4semidirect
100. #117×C61C2×C4semidirect
317. #1735D61semidirect
355. C61#10779semidirect
386. C22×C8C4×D61semidirect
410. #117C4×D61semidirect

Maximal subgroups

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

subgroupindex
#214590202
C2×#168397802
C2×#168397592
#214455512
#214455282
#214455152
C61×#17352
#1077961