Finite Group Databasebeta

← #29587158

Subgroups — #29587158

Order 15616.

938 conjugacy classes of subgroups.

The subgroup lattice has 634 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839778C211yesmaximal, characteristic
2C22×#14519705C211yesmaximal, characteristic
3C2×#16839735C211yesmaximal, characteristic
4#21448226C211yesmaximal, characteristic, complemented
5C2×#16839533C211yesmaximal, characteristic, complemented
6C2×#16839518C211yesmaximal, characteristic, complemented
7C2×C61×#339C211yesmaximal, characteristic, complemented
8C22×#11758222C2211yescharacteristic
9C22×#11758194C2211yescharacteristic
10C2×#14519736C2211yescharacteristic
11C2×#14519736C2211yescharacteristic
12#16839778—22no—
13#16839778—22no—
14C2×#14519705—21no—
15C2×#14519705—21no—
16C2×#14519705—22no—
17C2×#14519705C414yescomplemented
18C2×#14519703—22no—
19#16839735—22no—
20#16839735—22no—
21C2×#14519691—22no—
22#16839632—21no—
23#16839628—21no—
24#16839622—22no—
25C4×C4⋊Dic61—22no—
26#16839533—22no—
27#16839518—22no—
28C22×M16×C61C2211yescharacteristic
29C2×#117×C61C2211yescharacteristic
30C22×C42×C61C2211yescharacteristic
31C61×#339—22no—
32C2×#11758222—21no—
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×#11758222—21no—
42C2×#11758222C2×C412yes—
43C2×#11758222D412yes—
44C22×Dic122Q811yescharacteristic
45C22×D244Q811yescharacteristic
46C22×C22⋊D61—21no—
47C22×C4×D61—21no—
48C2×#11758194—21no—
49C2×#11758194—21no—
50C2×#11758194—21no—
51C2×#11758194—21no—
52C2×#11758194—21no—
53C2×#11758194C2×C412yes—
54C22×#92418—21no—
55#14519736—21no—
56#14519736—21no—
57#14519736—21no—
58#14519736—21no—
59#14519736—21no—
60#14519736—21no—
61#14519736—21no—
62#14519736—21no—
63C2×C4⋊Dic61—24no—
64C2×C4⋊Dic61—21no—
65C2×C4⋊Dic61—21no—
66C2×C22⋊Dic61—21no—
67C2×C22⋊Dic61D412yes—
68C2×C4×Dic61—21no—
69C2×C4×Dic61D412yes—
70#14519705—24no—
71#14519705—24no—
72#14519703—28no—
73#14519691—28no—
74#14519684—24no—
75C23×C4×C61—21no—
76C23×C4×C61C2311yescharacteristic
77C2×M16×C61—21no—
78C2×M16×C61—21no—
79C2×M16×C61—21no—
80C2×M16×C61—21no—
81C2×M16×C61—21no—
82C2×M16×C61D412yes—
83C22×C8×C61—21no—
84#117×C61—21no—
85#117×C61—21no—
86#117×C61—21no—
87#117×C61D412yes—
88C2×C4⋊C4×C61—21no—
89C2×C22⋊C4×C61—21no—
90C2×C42×C61—21no—
91C2×C42×C61—21no—
92C2×C42×C61—21no—
93C2×C42×C61—21no—
94C2×C42×C61C2×C412yes—
95C2×C42×C61D411yescharacteristic
96C2×C42×C61D411yescharacteristic
97C2×C42×C61D412yes—
98C24×C61C2×D411yescharacteristic
99C23×D61—21no—
100C22×C4×C61—21no—
101C22×C4×C61—22no—
102C22×C4×C61—21no—
103C22×C4×C61—22no—
104C22×C4×C61—44no—
105C22×C4×C61—41no—
106C22×C4×C61C22×C411yescharacteristic
107C22×C4×C61C2×D411yescharacteristic
108C22×C4×C61C2×D411yescharacteristic
109C22×C4×C61C2×D411yescharacteristic
110C22×C4×C61C2×D411yescharacteristic
111C22×C4×C61C2×D411yescharacteristic
112C22×C4×C61C2×D411yescharacteristic
113C22×C4×C61C2×Q811yescharacteristic
114C22×C4×C61#4111yescharacteristic
115C22×C4×C61#4111yescharacteristic
116C22×C4×C61#4111yescharacteristic
117C22×C4×C61#4111yescharacteristic
118C22×C4×C61#4111yescharacteristic
119C22×C4×C61#4111yescharacteristic
120C2×C22⋊D61—22no—
121C2×C22⋊D61—22no—
122C2×C22⋊D61—41no—
123C2×C22⋊D61—41no—
124C2×C22⋊D61—41no—
125C2×C22⋊D61—41no—
126C22×Dic61—21no—
127C22×Dic61—22no—
128C22×Dic61—22no—
129#11758222—22no—
130#11758222—22no—
131#11758222—22no—
132#11758222—22no—
133#11758222—22no—
134#11758222—22no—
135#11758222—22no—
136#11758222—22no—
137#11758222—41no—
138#11758222—41no—
139#11758222—41no—
140#11758222—41no—
141#11758222—41no—
142#11758222—41no—
143#11758222—41no—
144#11758222—41no—
145C2×D244—21no—
146C2×D244—21no—
147C2×D244—21no—
148C2×D244—21no—
149C2×D244—21no—
150C2×D244C4⋊C412yescomplemented
151C2×C4×D61—22no—
152C2×C4×D61—22no—
153C2×C4×D61—41no—
154C2×C4×D61—41no—
155C2×C4×D61—41no—
156C2×C4×D61—41no—
157C2×Dic122—21no—
158C2×Dic122—21no—
159C2×Dic122—21no—
160C2×Dic122—21no—
161C2×Dic122—21no—
162C2×Dic122C4⋊C412yescomplemented
163M16×C61—22no—
164M16×C61—22no—
165M16×C61—41no—
166M16×C61—41no—
167C2×C8×C61—22no—
168C2×C8×C61—41no—
169C2×C8×C61—41no—
170C4⋊C4×C61—22no—
171C4⋊C4×C61—41no—
172C22⋊C4×C61—22no—
173C22⋊C4×C61—41no—
174C42×C61—22no—
175C42×C61—22no—
176C42×C61—21no—
177C42×C61—21no—
178C42×C61—22no—
179C42×C61—41no—
180C42×C61C22⋊C412yes—
181C42×C61C22⋊C412yes—
182C22⋊Dic61—22no—
183C22⋊Dic61—22no—
184C22⋊Dic61—41no—
185C22⋊Dic61—41no—
186C4⋊Dic61—48no—
187C4⋊Dic61—41no—
188C4⋊Dic61—41no—
189C4⋊Dic61—41no—
190C4⋊Dic61—41no—
191C4×Dic61—22no—
192C4×Dic61—22no—
193C4×Dic61—41no—
194C4×Dic61—41no—
195#11758194—22no—
196#11758194—22no—
197#11758194—41no—
198#11758194—41no—
199C2×#92418—22no—
200C2×#92418—41no—
201C2×#92418—41no—
202C23×C61—42no—
203C23×C61C4×D411yescharacteristic
204C23×C61#12011yescharacteristic
205C23×C61#12011yescharacteristic
206C23×C61#12111yescharacteristic
207C23×C61#12111yescharacteristic
208C23×C61C22⋊Q811yescharacteristic
209C23×C61#12411yescharacteristic
210C22×D61—22no—
211C22×D61—41no—
212C22×D61—41no—
213C22×D61—82no—
214C2×C4×C61—24no—
215C2×C4×C61—22no—
216C2×C4×C61—22no—
217C2×C4×C61—44no—
218C2×C4×C61—44no—
219C2×C4×C61—42no—
220C2×C4×C61—42no—
221C2×C4×C61—82no—
222C2×C4×C61C2×C22⋊C411yescharacteristic
223C2×C4×C61C2×C4⋊C411yescharacteristic
224C2×C4×C61#11711yescharacteristic
225C2×C4×C61C4×D411yescharacteristic
226C2×C4×C61C4×D411yescharacteristic
227C2×C4×C61C4×D411yescharacteristic
228C2×C4×C61#12011yescharacteristic
229C2×C4×C61#12111yescharacteristic
230C2×C4×C61#12111yescharacteristic
231C2×C4×C61#12111yescharacteristic
232C2×C4×C61#12111yescharacteristic
233C2×C4×C61#12111yescharacteristic
234C2×C4×C61#12111yescharacteristic
235C2×C4×C61C22⋊Q811yescharacteristic
236C2×C4×C61C22⋊Q811yescharacteristic
237C2×C4×C61C22⋊Q811yescharacteristic
238C2×C4×C61C22⋊Q811yescharacteristic
239C2×C4×C61C22⋊Q811yescharacteristic
240C2×C4×C61#12311yescharacteristic
241C2×C4×C61#12311yescharacteristic
242C2×C4×C61#12311yescharacteristic
243C2×C4×C61#12311yescharacteristic
244C2×C4×C61#12311yescharacteristic
245C2×C4×C61#12411yescharacteristic
246C2×C4×C61#12411yescharacteristic
247C2×C4×C61#12611yescharacteristic
248C2×C4×C61#12611yescharacteristic
249C2×C4×C61#12711yescharacteristic
250C22⋊D61—42no—
251C22⋊D61—42no—
252C22⋊D61—42no—
253C22⋊D61—42no—
254C2×Dic61—22no—
255C2×Dic61—41no—
256C2×Dic61—41no—
257C2×Dic61—44no—
258C2×Dic61—44no—
259C2×Dic61—42no—
260C2×Dic61—42no—
261D244—22no—
262D244—22no—
263D244—41no—
264D244—41no—
265C4×D61—42no—
266C4×D61—42no—
267C4×D61—42no—
268C4×D61—42no—
269Dic122—22no—
270Dic122—22no—
271Dic122—41no—
272Dic122—41no—
273C8×C61—42no—
274#92418—42no—
275#11081—611nomaximal, Sylow
276C22×C61—22no—
277C22×C61—22no—
278C22×C61—22no—
279C22×C61—41no—
280C22×C61—41no—
281C22×C61—41no—
282C22×C61—41no—
283C22×C61#38511yescharacteristic
284C22×C61#38611yescharacteristic
285C22×C61#38811yescharacteristic
286C22×C61#39211yescharacteristic
287C22×C61#39311yescharacteristic
288C22×C61#39411yescharacteristic
289C22×C61#39611yescharacteristic
290C2×D61—42no—
291C2×D61—82no—
292C2×D61—82no—
293C2×D61—82no—
294C4×C61—24no—
295C4×C61—42no—
296C4×C61—82no—
297C4×C61#38011yescharacteristic
298C4×C61#38511yescharacteristic
299C4×C61#38811yescharacteristic
300C4×C61#39011yescharacteristic
301C4×C61#39411yescharacteristic
302C4×C61#39611yescharacteristic
303C4×C61#39711yescharacteristic
304C4×C61#39911yescharacteristic
305Dic61—42no—
306Dic61—82no—
307Dic61—82no—
308C2×#421—611no—
309C22×#104—611no—
310C2×#410—611no—
311#2285—611no—
312C2×#344—611no—
313C2×#339D6111yescharacteristic, complemented
314C2×H(3,ℤ/4)—611no—
315C2×C61—21no—
316C2×C61—21no—
317C2×C61—21no—
318C2×C61—21no—
319C2×C61—21no—
320C2×C61—21no—
321C2×C61#143911yescharacteristic
322C2×C61#203112yes—
323D61—82no—
324C22×#41—611no—
325C22×M16—611no—
326C22×M16C2×D6111yescharacteristic
327C2×#117—611no—
328C2×#117—611no—
329C2×#117C2×D6111yescharacteristic
330C22×C42C2×D6111yescharacteristic
331#421—1222no—
332#421—1222no—
333C2×#104—614no—
334C2×#104—1222no—
335C2×#104—1221no—
336C2×#104—1221no—
337C2×#104—1222no—
338#410—1222no—
339#410—1222no—
340C2×#99—1222no—
341#384—1221no—
342#382—1221no—
343#380—1222no—
344C4×C4⋊C4—1222no—
345#344—1222no—
346#339—22no—
347H(3,ℤ/4)—1222no—
348C61#1108111yescharacteristic, complemented, Sylow
349C2×#41—612no—
350C2×#41—612no—
351C2×#41—1221no—
352C2×#41—1221no—
353C2×#41—1221no—
354C2×#41—1221no—
355C2×#41—1221no—
356C2×#41—1221no—
357C2×#41—1221no—
358C2×#41—1221no—
359C2×#41—1221no—
360C2×#41—1221no—
361C22×Q8—611no—
362C22×D4—611no—
363C22×D4—1221no—
364C23×C4—21no—
365C23×C4—1221no—
366C23×C4C22×D6111yescharacteristic
367C2×M16—21no—
368C2×M16—21no—
369C2×M16—21no—
370C2×M16—21no—
371C2×M16—21no—
372C2×M16—612no—
373C2×M16—1221no—
374C2×M16—1221no—
375C2×M16—1221no—
376C2×M16—1221no—
377C2×M16—1221no—
378C2×M16D24412yes—
379C22×C8—21no—
380C22×C8—1221no—
381#117—21no—
382#117—21no—
383#117—21no—
384#117—1221no—
385#117—1221no—
386#117—1221no—
387#117—1221no—
388#117—1221no—
389#117—1221no—
390#117—1221no—
391#117—1221no—
392#117D24412yes—
393C2×C4⋊C4—21no—
394C2×C4⋊C4—1224no—
395C2×C4⋊C4—1221no—
396C2×C4⋊C4—1221no—
397C2×C22⋊C4—21no—
398C2×C22⋊C4—612no—
399C2×C22⋊C4—1221no—
400C2×C42—21no—
401C2×C42—21no—
402C2×C42—21no—
403C2×C42—21no—
404C2×C42—612no—
405C2×C42—1221no—
406C2×C42C4×D6112yes—
407C2×C42D24411yescharacteristic
408C2×C42D24411yescharacteristic
409C2×C42C22⋊D6112yes—
410#104—1228no—
411#104—1224no—
412#104—1224no—
413#99—1228no—
414#95—1224no—
415C24—1221no—
416C24D4×D6111yescharacteristic
417#41—1222no—
418#41—1222no—
419#41—1222no—
420#41—1222no—
421#41—1222no—
422#41—1222no—
423#41—1222no—
424#41—1222no—
425#41—2441no—
426#41—2441no—
427#41—2441no—
428#41—2441no—
429#41—2441no—
430#41—2441no—
431#41—2441no—
432#41—2441no—
433C2×Q8—612no—
434C2×Q8—1221no—
435C2×Q8—1221no—
436C2×Q8—1221no—
437C2×Q8—1221no—
438C2×Q8—1221no—
439C2×D4—612no—
440C2×D4—1222no—
441C2×D4—1222no—
442C2×D4—1221no—
443C2×D4—1221no—
444C2×D4—1221no—
445C2×D4—1221no—
446C2×D4—1221no—
447C2×D4—2441no—
448C2×D4—2441no—
449C2×D4—2441no—
450C2×D4—2441no—
451C22×C4—21no—
452C22×C4—22no—
453C22×C4—21no—
454C22×C4—22no—
455C22×C4—44no—
456C22×C4—41no—
457C22×C4—1221no—
458C22×C4—1222no—
459C22×C4—1222no—
460C22×C4—1222no—
461C22×C4—1222no—
462C22×C4—2441no—
463C22×C4—2441no—
464C22×C4—2441no—
465C22×C4—2441no—
466C22×C4C2×C4×D6111yescharacteristic
467C22×C4C2×D24411yescharacteristic
468C22×C4C2×D24411yescharacteristic
469C22×C4C2×D24411yescharacteristic
470C22×C4#1175822211yescharacteristic
471C22×C4#1175822211yescharacteristic
472C22×C4D4×D6111yescharacteristic
473C22×C4D4×D6111yescharacteristic
474C22×C4#1175822411yescharacteristic
475C22×C4#1175822411yescharacteristic
476C22×C4#1175822411yescharacteristic
477C22×C4Q8×D6111yescharacteristic
478C22×C4#1175822611yescharacteristic
479C22×C4C2×C22⋊D6111yescharacteristic
480M16—22no—
481M16—22no—
482M16—41no—
483M16—41no—
484M16—1222no—
485M16—1222no—
486M16—2441no—
487M16—2441no—
488C2×C8—22no—
489C2×C8—41no—
490C2×C8—41no—
491C2×C8—1222no—
492C2×C8—2441no—
493C2×C8—2441no—
494C4⋊C4—22no—
495C4⋊C4—41no—
496C4⋊C4—2448no—
497C4⋊C4—2441no—
498C4⋊C4—2441no—
499C4⋊C4—2441no—
500C4⋊C4—2441no—
501C22⋊C4—22no—
502C22⋊C4—41no—
503C22⋊C4—1222no—
504C22⋊C4—1222no—
505C22⋊C4—2441no—
506C22⋊C4—2441no—
507C42—22no—
508C42—22no—
509C42—21no—
510C42—21no—
511C42—22no—
512C42—41no—
513C42—1222no—
514C42—1222no—
515C42—2441no—
516C42—2441no—
517C42#1175819812yes—
518C42#1175819812yes—
519C23—42no—
520C23—1222no—
521C23—4882no—
522C23—2441no—
523C23—2441no—
524C23#1451973711yescharacteristic
525C23#1451974711yescharacteristic
526C23#1451974711yescharacteristic
527C23#1451974911yescharacteristic
528C23#1451975211yescharacteristic
529C23#1451975511yescharacteristic
530C23#1451976311yescharacteristic
531Q8—1222no—
532Q8—1222no—
533Q8—2441no—
534Q8—2441no—
535D4—1222no—
536D4—1222no—
537D4—2442no—
538D4—2442no—
539D4—2442no—
540D4—2442no—
541D4—2441no—
542D4—2441no—
543C2×C4—24no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—82no—
551C2×C4—1222no—
552C2×C4—2444no—
553C2×C4—2444no—
554C2×C4—2442no—
555C2×C4—2442no—
556C2×C4—2441no—
557C2×C4—2441no—
558C2×C4—2442no—
559C2×C4—2442no—
560C2×C4—2442no—
561C2×C4—2442no—
562C2×C4C2×#1175819811yescharacteristic
563C2×C4C4⋊C4×D6111yescharacteristic
564C2×C4#1451973511yescharacteristic
565C2×C4#1451973711yescharacteristic
566C2×C4C4×D24411yescharacteristic
567C2×C4C4×D24411yescharacteristic
568C2×C4#1451974711yescharacteristic
569C2×C4#1451974811yescharacteristic
570C2×C4#1451974911yescharacteristic
571C2×C4#1451975111yescharacteristic
572C2×C4#1451975111yescharacteristic
573C2×C4#1451975211yescharacteristic
574C2×C4#1451975211yescharacteristic
575C2×C4#1451975311yescharacteristic
576C2×C4#1451975511yescharacteristic
577C2×C4#1451975711yescharacteristic
578C2×C4#1451975711yescharacteristic
579C2×C4#1451975711yescharacteristic
580C2×C4#1451975911yescharacteristic
581C2×C4#1451975911yescharacteristic
582C2×C4#1451976111yescharacteristic
583C2×C4#1451976111yescharacteristic
584C2×C4#1451976111yescharacteristic
585C2×C4#1451976311yescharacteristic
586C2×C4#1451976511yescharacteristic
587C2×C4#1451977211yescharacteristic
588C2×C4#1451977211yescharacteristic
589C2×C4#1451977411yescharacteristic
590C8—42no—
591C8—2442no—
592C22—22no—
593C22—22no—
594C22—22no—
595C22—41no—
596C22—41no—
597C22—41no—
598C22—41no—
599C22—4882no—
600C22—4882no—
601C22—4882no—
602C22—2442no—
603C22#1683963611yescharacteristic
604C22#1683963911yescharacteristic
605C22#1683965411yescharacteristic
606C22#1683967211yescharacteristic
607C22#1683967511yescharacteristic
608C22#1683967811yescharacteristic
609C22#1683968411yescharacteristic
610C4—24no—
611C4—42no—
612C4—82no—
613C4—4882no—
614C4—4882no—
615C4—2442no—
616C4#1683961911yescharacteristic
617C4#1683963611yescharacteristic
618C4#1683965411yescharacteristic
619C4#1683966411yescharacteristic
620C4#1683967811yescharacteristic
621C4#1683968411yescharacteristic
622C4#1683968911yescharacteristic
623C4#1683969911yescharacteristic
624C2—21no—
625C2—21no—
626C2—21no—
627C2—21no—
628C2—21no—
629C2—21no—
630C2—4882no—
631C2#2144406511yescharacteristic
632C2#2144699212yes—
subgroupG/H
centerC2×C4#14519763
commutatorC22×C4×C61C22×C4
FrattiniC23×C4C22×D61
FittingC2×C61×#339C2
socleC22×C61#393
prime radicalC22×#11758222C22
cosocleC23×C4×C61C23
hypercenterC2×#339D61
hypocenterC61#11081
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C2×#339

Complements

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

NH ≅ G/Ntype
4. #21448226C2semidirect
5. C2×#16839533C2semidirect
6. C2×#16839518C2semidirect
7. C2×C61×#339C2semidirect
17. C2×#14519705C4semidirect
150. C2×D244C4⋊C4semidirect
162. C2×Dic122C4⋊C4semidirect
313. C2×#339D61semidirect
348. C61#11081semidirect

Maximal subgroups

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

subgroupindex
C2×#168397782
C22×#145197052
C2×#168397352
#214482262
C2×#168395332
C2×#168395182
C2×C61×#3392
#1108161