Finite Group Databasebeta

← #29585409

Subgroups — #29585409

Order 15616.

1192 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459020C211yesmaximal, characteristic
2C2×#16839744C211yesmaximal, characteristic
3#21454263C211yesmaximal, characteristic
4C2×#16839534C211yesmaximal, characteristic, complemented
5C2×#16839534C211yesmaximal, characteristic, complemented
6C2×#16839532C211yesmaximal, characteristic, complemented
7C2×C61×#344C211yesmaximal, characteristic, complemented
8C22×#11758222C2211yescharacteristic
9C22×#11758194C2211yescharacteristic
10C22×#11758194C2211yescharacteristic
11#16840243—22no—
12#16840242—22no—
13#16840239—21no—
14#16840236—21no—
15#16840235C414yes—
16C4×#11758222—22no—
17#16840230—22no—
18C2×#14519736C2211yescharacteristic
19#117×D61—22no—
20#16839744—22no—
21#16839744C414yescomplemented
22C2×#14519695—21no—
23C2×#14519693—21no—
24#16839730C414yescomplemented
25#16839728—22no—
26#16839725—22no—
27#16839534—22no—
28#16839534—22no—
29#16839532—22no—
30C22×M16×C61C2211yescharacteristic
31C22×M16×C61C2211yescharacteristic
32C2×#117×C61C2211yescharacteristic
33C61×#344—22no—
34C2×#11758222—21no—
35C2×#11758222—21no—
36C2×#11758222—21no—
37C2×#11758222—21no—
38C2×#11758222—21no—
39C2×#11758222—21no—
40C2×#11758222—21no—
41C2×#11758222—21no—
42C2×#11758222C2×C412yes—
43C2×#11758222C2×C412yes—
44C2×#11758222D412yes—
45C2×#11758222Q812yes—
46C22×Dic122D411yescharacteristic
47C22×D244D411yescharacteristic
48C22×C22⋊D61C2×C412yes—
49C22×C4×D61D412yes—
50C2×#11758194—21no—
51C2×#11758194—21no—
52C2×#11758194—21no—
53C2×#11758194—21no—
54C2×#11758194—21no—
55C2×#11758194—21no—
56C2×#11758194—21no—
57C2×#11758194—21no—
58C2×#11758194—21no—
59C2×#11758194—21no—
60C2×#11758194—21no—
61C2×#11758194C2×C412yescomplemented
62C22×#92418—21no—
63C22×#92418C2×C412yescomplemented
64C4×Dic122—24no—
65#14519742—24no—
66C4×D244—24no—
67#14519739—24no—
68C4×C22⋊D61—28no—
69#14519737—28no—
70#14519736—21no—
71#14519736—21no—
72#14519736D412yes—
73#14519736D412yes—
74#14519735—24no—
75#14519734—24no—
76#14519733—24no—
77C2×C4⋊Dic61—21no—
78C4⋊C4×D61—24no—
79C2×C22⋊Dic61C2×C412yes—
80C22⋊C4×D61—24no—
81C2×C4×Dic61—21no—
82C42×D61—24no—
83#14519695—24no—
84#14519693—24no—
85#14519689—28no—
86C23×C4×C61C2311yescharacteristic
87C2×M16×C61—21no—
88C2×M16×C61—21no—
89C2×M16×C61—21no—
90C2×M16×C61—21no—
91C2×M16×C61—21no—
92C2×M16×C61—21no—
93C2×M16×C61—21no—
94C2×M16×C61—21no—
95C2×M16×C61—21no—
96C2×M16×C61—21no—
97C2×M16×C61C2×C412yescomplemented
98C2×M16×C61C2×C412yescomplemented
99C22×C8×C61—21no—
100C22×C8×C61—21no—
101#117×C61—21no—
102#117×C61—21no—
103#117×C61D412yes—
104#117×C61D412yes—
105C2×C4⋊C4×C61—21no—
106C2×C22⋊C4×C61C2×C412yes—
107C2×C42×C61—21no—
108C24×C61C22×C411yescharacteristic
109C23×D61C22⋊C412yes—
110C22×C4×C61—22no—
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×D411yescharacteristic
120C22×C4×C61C2×Q811yescharacteristic
121C22×C4×C61#4111yescharacteristic
122C22×C4×C61#4111yescharacteristic
123C22×C4×C61#4111yescharacteristic
124C22×C4×C61#4111yescharacteristic
125C2×C22⋊D61—22no—
126C2×C22⋊D61—22no—
127C2×C22⋊D61—22no—
128C2×C22⋊D61—22no—
129C2×C22⋊D61C4214yes—
130C2×C22⋊D61C4⋊C414yes—
131C22×Dic61—22no—
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×D244C22⋊C412yes—
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×Dic122C22⋊C412yes—
167C2×Dic122C4⋊C412yes—
168M16×C61—22no—
169M16×C61—22no—
170M16×C61—22no—
171M16×C61—22no—
172M16×C61—41no—
173M16×C61—41no—
174M16×C61—41no—
175M16×C61—41no—
176C2×C8×C61—22no—
177C2×C8×C61—22no—
178C2×C8×C61—41no—
179C2×C8×C61—41no—
180C2×C8×C61—41no—
181C2×C8×C61—41no—
182C4⋊C4×C61—22no—
183C4⋊C4×C61—22no—
184C22⋊C4×C61—22no—
185C22⋊C4×C61C22⋊C414yes—
186C42×C61—22no—
187C42×C61—22no—
188C22⋊Dic61—22no—
189C22⋊Dic61C22⋊C414yes—
190#11758198—28no—
191C4⋊Dic61—22no—
192C4⋊Dic61—22no—
193#11758196—28no—
194C4×Dic61—28no—
195C4×Dic61—22no—
196C4×Dic61—22no—
197#11758194—22no—
198#11758194—22no—
199#11758194—41no—
200#11758194—41no—
201#11758194—41no—
202#11758194—41no—
203#11758194—41no—
204#11758194—41no—
205C2×#92418—22no—
206C2×#92418—22no—
207C2×#92418—22no—
208C2×#92418—22no—
209C2×#92418—41no—
210C2×#92418—41no—
211C23×C61—81no—
212C23×C61C2×C4211yescharacteristic
213C23×C61C2×C22⋊C411yescharacteristic
214C23×C61C2×C22⋊C411yescharacteristic
215C23×C61C2×C22⋊C411yescharacteristic
216C23×C61C2×C4⋊C411yescharacteristic
217C23×C61#11711yescharacteristic
218C23×C61#11711yescharacteristic
219C22×D61—22no—
220C22×D61—22no—
221C22×D61—82no—
222C22×D61#9514yes—
223C2×C4×C61—24no—
224C2×C4×C61—82no—
225C2×C4×C61C2×C22⋊C411yescharacteristic
226C2×C4×C61C2×C22⋊C411yescharacteristic
227C2×C4×C61C2×C4⋊C411yescharacteristic
228C2×C4×C61C2×C4⋊C411yescharacteristic
229C2×C4×C61C4×D411yescharacteristic
230C2×C4×C61C4×D411yescharacteristic
231C2×C4×C61C4×D411yescharacteristic
232C2×C4×C61C4×D411yescharacteristic
233C2×C4×C61C4×D411yescharacteristic
234C2×C4×C61C4×D411yescharacteristic
235C2×C4×C61C4×D411yescharacteristic
236C2×C4×C61C4×D411yescharacteristic
237C2×C4×C61#12011yescharacteristic
238C2×C4×C61#12011yescharacteristic
239C2×C4×C61#12011yescharacteristic
240C2×C4×C61#12011yescharacteristic
241C2×C4×C61#12111yescharacteristic
242C2×C4×C61#12111yescharacteristic
243C2×C4×C61#12111yescharacteristic
244C2×C4×C61#12111yescharacteristic
245C2×C4×C61C22⋊Q811yescharacteristic
246C2×C4×C61C22⋊Q811yescharacteristic
247C2×C4×C61C22⋊Q811yescharacteristic
248C2×C4×C61C22⋊Q811yescharacteristic
249C2×C4×C61#12311yescharacteristic
250C2×C4×C61#12311yescharacteristic
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—82no—
261C2×Dic61#9514yes—
262D244—22no—
263D244—22no—
264D244—22no—
265D244—22no—
266C4×D61—24no—
267C4×D61—24no—
268C4×D61—24no—
269C4×D61—24no—
270C4×D61—88no—
271Dic122—22no—
272Dic122—22no—
273Dic122—22no—
274Dic122—22no—
275C8×C61—42no—
276C8×C61—42no—
277#92418—42no—
278#92418—42no—
279#10756—611nomaximal, Sylow
280C22×C61—41no—
281C22×C61—41no—
282C22×C61—41no—
283C22×C61—41no—
284C22×C61—41no—
285C22×C61—41no—
286C22×C61—41no—
287C22×C61C2×#9511yescharacteristic
288C22×C61C4×C22⋊C411yescharacteristic
289C22×C61C4×C22⋊C411yescharacteristic
290C22×C61#37911yescharacteristic
291C22×C61#38011yescharacteristic
292C22×C61#38011yescharacteristic
293C22×C61#38111yescharacteristic
294C2×D61—24no—
295C2×D61—44no—
296C2×D61—44no—
297C2×D61—82no—
298C4×C61—44no—
299C4×C61#38511yescharacteristic
300C4×C61#38511yescharacteristic
301C4×C61#38511yescharacteristic
302C4×C61#38511yescharacteristic
303C4×C61#38611yescharacteristic
304C4×C61#38611yescharacteristic
305C4×C61#38611yescharacteristic
306C4×C61#38611yescharacteristic
307Dic61—24no—
308Dic61—44no—
309#3428—611no—
310C2×#413—611no—
311#2879—611no—
312C2×#344—611no—
313C2×#344—611no—
314C2×#344—611no—
315C2×#344D6111yescharacteristic, complemented
316C2×C61—21no—
317C2×C61—21no—
318C2×C61—21no—
319C2×C61—21no—
320C2×C61—21no—
321C2×C61—21no—
322C2×C61—21no—
323C2×C61C22⋊#9511yescharacteristic
324D61—44no—
325C22×#41—611no—
326C22×M16—611no—
327C22×M16—611no—
328C22×M16C2×D6111yescharacteristic
329C22×M16C2×D6111yescharacteristic
330#520—1222no—
331#520—1222no—
332#519—1221no—
333#518—614no—
334#518—1221no—
335C4×#41—1222no—
336C4×#41—1222no—
337C2×#117—611no—
338C2×#117—1222no—
339C2×#117C2×D6111yescharacteristic
340#413—614no—
341#413—1222no—
342C2×#101—1221no—
343C2×#100—1221no—
344#408—614no—
345#408—1222no—
346#407—1222no—
347#344—22no—
348#344—1222no—
349#344—1222no—
350#344—1222no—
351C61#1075611yescharacteristic, complemented, Sylow
352C2×#41—612no—
353C2×#41—612no—
354C2×#41—612no—
355C2×#41—612no—
356C2×#41—1221no—
357C2×#41—1221no—
358C2×#41—1221no—
359C2×#41—1221no—
360C2×#41—1221no—
361C2×#41—1221no—
362C2×#41—1221no—
363C2×#41—1221no—
364C22×Q8—611no—
365C22×D4—612no—
366C22×D4—611no—
367C23×C4—612no—
368C23×C4C22×D6111yescharacteristic
369C2×M16—21no—
370C2×M16—21no—
371C2×M16—21no—
372C2×M16—21no—
373C2×M16—21no—
374C2×M16—21no—
375C2×M16—21no—
376C2×M16—21no—
377C2×M16—21no—
378C2×M16—21no—
379C2×M16—612no—
380C2×M16—1221no—
381C2×M16—1221no—
382C2×M16—1221no—
383C2×M16—1221no—
384C2×M16—1221no—
385C2×M16—1221no—
386C2×M16—1221no—
387C2×M16—1221no—
388C2×M16—1221no—
389C2×M16—1221no—
390C2×M16—1221no—
391C2×M16C4×D6112yescomplemented
392C2×M16C4×D6112yescomplemented
393C22×C8—21no—
394C22×C8—21no—
395C22×C8—612no—
396C22×C8—1221no—
397C4×Q8—1224no—
398C4×Q8—1224no—
399C4×D4—1224no—
400C4×D4—1228no—
401C4×D4—1224no—
402C4×D4—1228no—
403#117—21no—
404#117—21no—
405#117—612no—
406#117—612no—
407#117—1224no—
408#117—1224no—
409#117—1224no—
410#117—1221no—
411#117—1221no—
412#117D24412yes—
413#117C22⋊D6112yes—
414C2×C4⋊C4—21no—
415C2×C4⋊C4—1224no—
416C2×C4⋊C4—1221no—
417C2×C22⋊C4—612no—
418C2×C22⋊C4—1224no—
419C2×C22⋊C4C4×D6112yes—
420C2×C42—21no—
421C2×C42—1224no—
422C2×C42—1221no—
423#101—1224no—
424#100—1224no—
425C22⋊C8—1228no—
426C24—612no—
427C24C2×C4×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—612no—
464C22×C4—614no—
465C22×C4—614no—
466C22×C4—1228no—
467C22×C4—1222no—
468C22×C4—1222no—
469C22×C4—1222no—
470C22×C4—1222no—
471C22×C4—1222no—
472C22×C4C2×C4×D6111yescharacteristic
473C22×C4C2×C4×D6111yescharacteristic
474C22×C4C2×D24411yescharacteristic
475C22×C4#1175822211yescharacteristic
476C22×C4#1175822211yescharacteristic
477C22×C4D4×D6111yescharacteristic
478C22×C4D4×D6111yescharacteristic
479C22×C4D4×D6111yescharacteristic
480C22×C4D4×D6111yescharacteristic
481C22×C4D4×D6111yescharacteristic
482C22×C4#1175822411yescharacteristic
483C22×C4Q8×D6111yescharacteristic
484C22×C4#1175822611yescharacteristic
485C22×C4C2×C22⋊D6111yescharacteristic
486M16—22no—
487M16—22no—
488M16—22no—
489M16—22no—
490M16—41no—
491M16—41no—
492M16—41no—
493M16—41no—
494M16—1222no—
495M16—1222no—
496M16—2441no—
497M16—2441no—
498M16—2441no—
499M16—2441no—
500M16—2441no—
501M16—2441no—
502C2×C8—22no—
503C2×C8—22no—
504C2×C8—41no—
505C2×C8—41no—
506C2×C8—41no—
507C2×C8—41no—
508C2×C8—1222no—
509C2×C8—1222no—
510C2×C8—1222no—
511C2×C8—1222no—
512C2×C8—2441no—
513C2×C8—2441no—
514C4⋊C4—22no—
515C4⋊C4—22no—
516C4⋊C4—1228no—
517C4⋊C4—1222no—
518C4⋊C4—1222no—
519C22⋊C4—22no—
520C22⋊C4—614no—
521C22⋊C4—1228no—
522C22⋊C4—1222no—
523C22⋊C4#1175819814yes—
524C42—22no—
525C42—22no—
526C42—1228no—
527C42—1222no—
528C42—1222no—
529C23—81no—
530C23—614no—
531C23—1222no—
532C23—1222no—
533C23—4882no—
534C23C42×D6111yescharacteristic
535C23C22⋊C4×D6111yescharacteristic
536C23C22⋊C4×D6111yescharacteristic
537C23C2×#1175819811yescharacteristic
538C23C4⋊C4×D6111yescharacteristic
539C23#1451973411yescharacteristic
540C23#1451973511yescharacteristic
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—82no—
555C2×C4—614no—
556C2×C4—1222no—
557C2×C4—1222no—
558C2×C4—1224no—
559C2×C4—1224no—
560C2×C4—1224no—
561C2×C4—1224no—
562C2×C4—1224no—
563C2×C4—4888no—
564C2×C4—4882no—
565C2×C4C22⋊C4×D6111yescharacteristic
566C2×C4C22⋊C4×D6111yescharacteristic
567C2×C4C4⋊C4×D6111yescharacteristic
568C2×C4C4⋊C4×D6111yescharacteristic
569C2×C4#1451973711yescharacteristic
570C2×C4#1451973711yescharacteristic
571C2×C4C4×C22⋊D6111yescharacteristic
572C2×C4C4×C22⋊D6111yescharacteristic
573C2×C4#1451973911yescharacteristic
574C2×C4#1451973911yescharacteristic
575C2×C4C4×D24411yescharacteristic
576C2×C4C4×D24411yescharacteristic
577C2×C4#1451974511yescharacteristic
578C2×C4#1451974511yescharacteristic
579C2×C4#1451974711yescharacteristic
580C2×C4#1451974711yescharacteristic
581C2×C4#1451974811yescharacteristic
582C2×C4#1451975011yescharacteristic
583C2×C4#1451975011yescharacteristic
584C2×C4#1451975211yescharacteristic
585C2×C4#1451975311yescharacteristic
586C2×C4#1451975511yescharacteristic
587C2×C4#1451975511yescharacteristic
588C2×C4#1451975711yescharacteristic
589C2×C4#1451975911yescharacteristic
590C2×C4#1451975911yescharacteristic
591C2×C4#1451976211yescharacteristic
592C2×C4#1451976211yescharacteristic
593C8—42no—
594C8—42no—
595C8—2442no—
596C8—2442no—
597C22—41no—
598C22—41no—
599C22—41no—
600C22—41no—
601C22—41no—
602C22—41no—
603C22—41no—
604C22—1224no—
605C22—4882no—
606C22—2444no—
607C22—2444no—
608C22#95×D6111yescharacteristic
609C22#1683960911yescharacteristic
610C22C4×#1175819811yescharacteristic
611C22#1683961711yescharacteristic
612C22#1683961911yescharacteristic
613C22#1683962111yescharacteristic
614C22#1683962411yescharacteristic
615C4—44no—
616C4—1224no—
617C4—2444no—
618C4#1683963311yescharacteristic
619C4#1683963311yescharacteristic
620C4#1683963611yescharacteristic
621C4#1683963611yescharacteristic
622C4#1683963811yescharacteristic
623C4#1683963811yescharacteristic
624C4#1683964011yescharacteristic
625C4#1683964011yescharacteristic
626C2—21no—
627C2—21no—
628C2—21no—
629C2—21no—
630C2—21no—
631C2—21no—
632C2—21no—
633C2—2444no—
634C2#2144404011yescharacteristic
subgroupG/H
centerC4#16839640
commutatorC23×C61C2×C42
FrattiniC23×C4C22×D61
FittingC2×C61×#344C2
socleC2×C61C22⋊#95
prime radicalC22×#11758222C22
cosocleC23×C4×C61C23
hypercenterC2×#344D61
hypocenterC61#10756
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C4 ◁ C23×C4 ◁ C2×#344

Complements

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

NH ≅ G/Ntype
4. C2×#16839534C2semidirect
5. C2×#16839534C2semidirect
6. C2×#16839532C2semidirect
7. C2×C61×#344C2semidirect
21. #16839744C4semidirect
24. #16839730C4semidirect
61. C2×#11758194C2×C4semidirect
63. C22×#92418C2×C4semidirect
97. C2×M16×C61C2×C4semidirect
98. C2×M16×C61C2×C4semidirect
315. C2×#344D61semidirect
351. C61#10756semidirect
391. C2×M16C4×D61semidirect
392. C2×M16C4×D61semidirect

Maximal subgroups

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

subgroupindex
#214590202
C2×#168397442
#214542632
C2×#168395342
C2×#168395322
C2×C61×#3442
#1075661