Finite Group Databasebeta

← #29708883

Subgroups — #29708883

Order 15616.

1166 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839725C211yesmaximal, characteristic
2#21448198C211yesmaximal, characteristic
3C2×#16839619C211yesmaximal, characteristic
4#21447253C212yesmaximal, complemented
5#21447253C212yesmaximal, complemented
6#21447248C212yesmaximal, complemented
7#21447243C212yesmaximal, complemented
8#21445540C211yesmaximal, characteristic, complemented
9#21445539C211yesmaximal, characteristic, complemented
10#21445538C211yesmaximal, characteristic, complemented
11C2×C61×#428C211yesmaximal, characteristic, complemented
12C23×C4×D61C2211yescharacteristic
13C22×#11758194C2211yescharacteristic
14C2×#14519736C2211yescharacteristic
15C22×C4⋊Dic61C2211yescharacteristic
16C22×#11758198—21no—
17#16839725—24no—
18C2×#14519689C2212yes—
19C2×#14519689C2212yes—
20#16839623—21no—
21#16839619—21no—
22#16839619—21no—
23#16839619C2211yescharacteristic
24#16839619C2211yescharacteristic
25#16839619C2211yescharacteristic
26#16839619C2211yescharacteristic
27#16839619C2212yes—
28#16839619C2212yes—
29#16839617—21no—
30C4×#11758198—21no—
31C4×#11758198—21no—
32C2×#14519684—21no—
33#16839525C2212yescomplemented
34#16839525C2212yescomplemented
35#16839524C2212yescomplemented
36#16839524C2212yescomplemented
37#16839524C2212yescomplemented
38#16839524C2212yescomplemented
39C22×M16×C61C2211yescharacteristic
40C2×#117×C61C2211yescharacteristic
41C22×C4⋊C4×C61C2211yescharacteristic
42C61×#428—24no—
43C2×#107×C61C2212yescomplemented
44C2×#106×C61C2212yescomplemented
45C24×D61D411yescharacteristic
46C23×Dic61—21no—
47C23×Dic61D411yescharacteristic
48C22×C4×D61—22no—
49C22×C4×D61—22no—
50C22×C4×D61—22no—
51C22×C4×D61—22no—
52C22×C4×D61—22no—
53C22×C4×D61—22no—
54C22×C4×D61D411yescharacteristic
55C22×C4×D61D411yescharacteristic
56C22×C4×D61C2311yescharacteristic
57C22×C4×D61C2311yescharacteristic
58C2×#11758194—22no—
59C2×#11758194—22no—
60C2×#11758194—22no—
61C22×#92418C2312yes—
62#14519736—22no—
63#14519736—22no—
64C2×C4⋊Dic61—22no—
65C2×C4⋊Dic61—22no—
66C2×C4⋊Dic61—21no—
67C2×C4⋊Dic61C2312yes—
68C2×C4⋊Dic61C2311yescharacteristic
69C2×C4⋊Dic61C2311yescharacteristic
70C2×C22⋊Dic61—21no—
71C2×#11758198—22no—
72C2×#11758198—22no—
73C2×#11758198—21no—
74C2×#11758198—21no—
75C2×#11758198—21no—
76C2×#11758198—21no—
77C2×#11758198—44no—
78C2×C4×Dic61—21no—
79#14519689—24no—
80#14519689—24no—
81#14519684—22no—
82#14519684—22no—
83#14519684—21no—
84#14519684—21no—
85#14519684—21no—
86#14519684—21no—
87C23×C4×C61—21no—
88C23×C4×C61C2311yescharacteristic
89C2×M16×C61—22no—
90C2×M16×C61—22no—
91C2×M16×C61D414yes—
92C22×C8×C61C2312yes—
93#117×C61—22no—
94#117×C61—22no—
95C2×C4⋊C4×C61—22no—
96C2×C4⋊C4×C61—22no—
97C2×C4⋊C4×C61—21no—
98C2×C4⋊C4×C61C2311yescharacteristic
99C2×C4⋊C4×C61C2312yes—
100C2×C4⋊C4×C61C2311yescharacteristic
101C2×C22⋊C4×C61—21no—
102C2×C42×C61—21no—
103#107×C61—24no—
104#106×C61—24no—
105C24×C61C2×D411yescharacteristic
106C23×D61—22no—
107C23×D61—22no—
108C23×D61—22no—
109C23×D61—22no—
110C23×D61—22no—
111C23×D61—41no—
112C23×D61—41no—
113C23×D61C2×D411yescharacteristic
114C23×D61C2×D411yescharacteristic
115C22×C4×C61—21no—
116C22×C4×C61—21no—
117C22×C4×C61—22no—
118C22×C4×C61—42no—
119C22×C4×C61—42no—
120C22×C4×C61—42no—
121C22×C4×C61C2×D411yescharacteristic
122C22×C4×C61C2×D412yes—
123C22×C4×C61#4112yes—
124C22×C4×C61#4112yes—
125C22×C4×C61#4112yes—
126C22×C4×C61#4112yes—
127C22×C4×C61#4112yes—
128C22×C4×C61C2411yescharacteristic
129C22×Dic61—22no—
130C22×Dic61—22no—
131C22×Dic61—22no—
132C22×Dic61—22no—
133C22×Dic61—21no—
134C22×Dic61—21no—
135C22×Dic61—42no—
136C22×Dic61—42no—
137C22×Dic61—42no—
138C22×Dic61C2×D411yescharacteristic
139C22×Dic61C2×D411yescharacteristic
140C2×C4×D61—22no—
141C2×C4×D61—22no—
142C2×C4×D61—22no—
143C2×C4×D61—22no—
144C2×C4×D61—22no—
145C2×C4×D61—22no—
146C2×C4×D61—21no—
147C2×C4×D61—21no—
148C2×C4×D61—21no—
149C2×C4×D61—21no—
150C2×C4×D61—21no—
151C2×C4×D61—21no—
152C2×C4×D61—21no—
153C2×C4×D61—21no—
154C2×C4×D61—21no—
155C2×C4×D61—21no—
156C2×C4×D61—21no—
157C2×C4×D61—21no—
158C2×C4×D61—44no—
159C2×C4×D61—44no—
160C2×C4×D61—44no—
161C2×C4×D61—44no—
162M16×C61—28no—
163C2×C8×C61—22no—
164C2×C8×C61—22no—
165C2×C8×C61C2×D414yes—
166C4⋊C4×C61—21no—
167C4⋊C4×C61—22no—
168C4⋊C4×C61—21no—
169C4⋊C4×C61—22no—
170C4⋊C4×C61—21no—
171C4⋊C4×C61—21no—
172C4⋊C4×C61—42no—
173C22⋊C4×C61—42no—
174C42×C61—42no—
175C22⋊Dic61—42no—
176#11758198—42no—
177#11758198—42no—
178#11758198—44no—
179#11758198—42no—
180#11758198—44no—
181#11758198—42no—
182C4⋊Dic61—22no—
183C4⋊Dic61—22no—
184C4⋊Dic61—21no—
185C4⋊Dic61—21no—
186C4⋊Dic61—21no—
187C4⋊Dic61—21no—
188C4⋊Dic61—42no—
189C4×Dic61—42no—
190#11758194—44no—
191C2×#92418—22no—
192C2×#92418—22no—
193C2×#92418—22no—
194C23×C61—21no—
195C23×C61—21no—
196C23×C61—21no—
197C23×C61—21no—
198C23×C61#12112yes—
199C23×C61#12312yes—
200C23×C61#12312yes—
201C23×C61C22×D411yescharacteristic
202C22×D61—28no—
203C22×D61—21no—
204C22×D61—21no—
205C22×D61—21no—
206C22×D61—21no—
207C22×D61—21no—
208C22×D61—21no—
209C22×D61—21no—
210C22×D61—21no—
211C22×D61—21no—
212C22×D61—21no—
213C22×D61—21no—
214C22×D61—21no—
215C22×D61—44no—
216C22×D61—44no—
217C22×D61—44no—
218C22×D61—44no—
219C22×D61—44no—
220C22×D61—44no—
221C22×D61AGL(1,ℤ/8)12yes—
222C22×D61#13712yes—
223C2×C4×C61—41no—
224C2×C4×C61—41no—
225C2×C4×C61—42no—
226C2×C4×C61—42no—
227C2×C4×C61—41no—
228C2×C4×C61—41no—
229C2×C4×C61—41no—
230C2×C4×C61—41no—
231C2×C4×C61—42no—
232C2×C4×C61—84no—
233C2×C4×C61#12112yes—
234C2×C4×C61#12314yes—
235C2×C4×C61#12312yes—
236C2×C4×C61#12312yes—
237C2×C4×C61#12414yes—
238C2×C4×C61#12614yes—
239C2×C4×C61#12614yes—
240C2×C4×C61C22×D411yescharacteristic
241C2×C4×C61C2×#4111yescharacteristic
242C2×C4×C61C2×#4111yescharacteristic
243C2×C4×C61C2×#4111yescharacteristic
244C2×C4×C61C2×#4111yescharacteristic
245C2×C4×C61C2×#4111yescharacteristic
246C2×Dic61—21no—
247C2×Dic61—21no—
248C2×Dic61—21no—
249C2×Dic61—21no—
250C2×Dic61—21no—
251C2×Dic61—21no—
252C2×Dic61—44no—
253C2×Dic61—41no—
254C2×Dic61—41no—
255C2×Dic61—42no—
256C2×Dic61—42no—
257C2×Dic61—41no—
258C2×Dic61—41no—
259C2×Dic61—41no—
260C2×Dic61—41no—
261C2×Dic61—42no—
262C2×Dic61—84no—
263C4×D61—41no—
264C4×D61—41no—
265C4×D61—41no—
266C4×D61—41no—
267C4×D61—41no—
268C4×D61—41no—
269C4×D61—41no—
270C4×D61—41no—
271C4×D61—44no—
272C4×D61—44no—
273C8×C61—24no—
274#92418—42no—
275#23642—611nomaximal, Sylow
276C22×C61—22no—
277C22×C61—22no—
278C22×C61—22no—
279C22×C61—22no—
280C22×C61—22no—
281C22×C61—22no—
282C22×C61#39714yes—
283C22×C61C2×#12111yescharacteristic
284C22×C61C2×#12311yescharacteristic
285C22×C61C2×#12311yescharacteristic
286C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
287C22×C61C2×#13711yescharacteristic
288C22×C61#57511yescharacteristic
289C22×C61#57511yescharacteristic
290C2×D61—22no—
291C2×D61—22no—
292C2×D61—22no—
293C2×D61—22no—
294C2×D61—22no—
295C2×D61—22no—
296C2×D61—416no—
297C2×D61—41no—
298C2×D61—41no—
299C2×D61—41no—
300C2×D61—41no—
301C2×D61—41no—
302C2×D61—41no—
303C2×D61—41no—
304C2×D61—41no—
305C4×C61—81no—
306C4×C61—82no—
307C4×C61—81no—
308C4×C61#39714yes—
309C4×C61C2×#12311yescharacteristic
310C4×C61C2×#12411yescharacteristic
311C4×C61C2×#12611yescharacteristic
312C4×C61C2×#12611yescharacteristic
313Dic61—41no—
314Dic61—41no—
315Dic61—81no—
316Dic61—81no—
317Dic61—82no—
318C2×#428D6111yescharacteristic, complemented
319C2×#407—611no—
320#2280—611no—
321C2×#380—611no—
322#2077—612no—
323#2077—612no—
324#2076—612no—
325#2075—612no—
326#1736—611no—
327#1736—611no—
328#1736—611no—
329C2×C61—24no—
330C2×C61C2×#39711yescharacteristic
331C2×C61#305511yescharacteristic
332C2×C61#305611yescharacteristic
333C2×C61#309211yescharacteristic
334C2×C61#309311yescharacteristic
335C2×C61#309411yescharacteristic
336C2×C61#309511yescharacteristic
337D61—42no—
338D61—42no—
339C24×C4—611no—
340C22×M16—611no—
341C22×M16C2×D6111yescharacteristic
342C2×#117—611no—
343C2×#117C2×D6111yescharacteristic
344C22×C4⋊C4—611no—
345C22×C4⋊C4C2×D6111yescharacteristic
346C22×C22⋊C4—1221no—
347#428—24no—
348C2×#107C2×D6112yescomplemented
349C2×#106C2×D6112yescomplemented
350#407—1224no—
351C2×C22⋊C8—612no—
352C2×C22⋊C8—612no—
353#381—1221no—
354#380—611no—
355#380—612no—
356#380—611no—
357#380—612no—
358#380—611no—
359#380—611no—
360#380—1221no—
361#380—1221no—
362#379—1221no—
363C4×C22⋊C4—1221no—
364C4×C22⋊C4—1221no—
365C2×#95—1221no—
366#340—612no—
367#340—612no—
368#340—612no—
369#340—612no—
370#340—612no—
371#340—612no—
372C61#2364211yescharacteristic, complemented, Sylow
373C25—611no—
374C23×C4—21no—
375C23×C4—611no—
376C23×C4—611no—
377C23×C4—611no—
378C23×C4—611no—
379C23×C4—611no—
380C23×C4—1222no—
381C23×C4—1222no—
382C23×C4—1222no—
383C23×C4—1222no—
384C23×C4—1222no—
385C23×C4—1222no—
386C23×C4—1221no—
387C23×C4C22×D6111yescharacteristic
388C2×M16—22no—
389C2×M16—22no—
390C2×M16—1222no—
391C2×M16—1222no—
392C2×M16—1222no—
393C2×M16C22⋊D6114yes—
394C22×C8—612no—
395C22×C8C22×D6112yes—
396#117—22no—
397#117—22no—
398#117—1222no—
399#117—1222no—
400C2×C4⋊C4—22no—
401C2×C4⋊C4—22no—
402C2×C4⋊C4—21no—
403C2×C4⋊C4—612no—
404C2×C4⋊C4—611no—
405C2×C4⋊C4—611no—
406C2×C4⋊C4—1222no—
407C2×C4⋊C4—1222no—
408C2×C4⋊C4—1221no—
409C2×C4⋊C4C22×D6112yes—
410C2×C4⋊C4C22×D6111yescharacteristic
411C2×C4⋊C4C22×D6111yescharacteristic
412C2×C22⋊C4—21no—
413C2×C22⋊C4—1222no—
414C2×C22⋊C4—1222no—
415C2×C22⋊C4—1221no—
416C2×C22⋊C4—1221no—
417C2×C22⋊C4—1221no—
418C2×C22⋊C4—1221no—
419C2×C22⋊C4—1221no—
420C2×C22⋊C4—2444no—
421C2×C42—21no—
422C2×C42—1221no—
423#107—24no—
424#106—24no—
425C22⋊C8—1224no—
426C22⋊C8—1224no—
427#95—1222no—
428#95—1222no—
429#95—1221no—
430#95—1221no—
431#95—1221no—
432#95—1221no—
433C24—611no—
434C24—611no—
435C24—1222no—
436C24—1222no—
437C24—1222no—
438C24—1222no—
439C24—1222no—
440C24—2441no—
441C24—2441no—
442C24D4×D6111yescharacteristic
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—22no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—611no—
450C22×C4—611no—
451C22×C4—1222no—
452C22×C4—1222no—
453C22×C4—1222no—
454C22×C4—1221no—
455C22×C4—1221no—
456C22×C4—1221no—
457C22×C4—1221no—
458C22×C4—1221no—
459C22×C4—1221no—
460C22×C4—1221no—
461C22×C4—1221no—
462C22×C4—1221no—
463C22×C4—1221no—
464C22×C4—1221no—
465C22×C4—1221no—
466C22×C4—1222no—
467C22×C4—1222no—
468C22×C4—1222no—
469C22×C4—1222no—
470C22×C4—1222no—
471C22×C4—1222no—
472C22×C4—1221no—
473C22×C4—1221no—
474C22×C4—1222no—
475C22×C4—2442no—
476C22×C4—2442no—
477C22×C4—2442no—
478C22×C4—2444no—
479C22×C4—2444no—
480C22×C4—2444no—
481C22×C4—2444no—
482C22×C4#1175822212yes—
483C22×C4#1175822212yes—
484C22×C4D4×D6111yescharacteristic
485C22×C4#1175822412yes—
486C22×C4#1175822612yes—
487C22×C4#1175822612yes—
488C22×C4C2×C22⋊D6112yes—
489C22×C4C23×D6111yescharacteristic
490M16—28no—
491M16—2444no—
492C2×C8—22no—
493C2×C8—22no—
494C2×C8—1222no—
495C2×C8—1222no—
496C2×C8—1222no—
497C2×C8C2×C22⋊D6114yes—
498C4⋊C4—21no—
499C4⋊C4—22no—
500C4⋊C4—21no—
501C4⋊C4—22no—
502C4⋊C4—21no—
503C4⋊C4—21no—
504C4⋊C4—42no—
505C4⋊C4—1222no—
506C4⋊C4—1221no—
507C4⋊C4—1221no—
508C4⋊C4—1222no—
509C4⋊C4—1221no—
510C4⋊C4—1221no—
511C4⋊C4—2442no—
512C22⋊C4—42no—
513C22⋊C4—2444no—
514C22⋊C4—2442no—
515C22⋊C4—2442no—
516C22⋊C4—2444no—
517C22⋊C4—2442no—
518C22⋊C4—2442no—
519C22⋊C4—2442no—
520C42—42no—
521C42—2442no—
522C23—21no—
523C23—21no—
524C23—21no—
525C23—21no—
526C23—612no—
527C23—612no—
528C23—1228no—
529C23—1221no—
530C23—1221no—
531C23—1221no—
532C23—1221no—
533C23—1221no—
534C23—1221no—
535C23—1221no—
536C23—1221no—
537C23—1221no—
538C23—1221no—
539C23—1221no—
540C23—1221no—
541C23—2444no—
542C23—2444no—
543C23—2444no—
544C23—2444no—
545C23—2444no—
546C23—2444no—
547C23#1451974812yes—
548C23#1451976212yes—
549C23#1451976212yes—
550C23C2×D4×D6111yescharacteristic
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—42no—
559C2×C4—41no—
560C2×C4—84no—
561C2×C4—1221no—
562C2×C4—1221no—
563C2×C4—1221no—
564C2×C4—1221no—
565C2×C4—1221no—
566C2×C4—1221no—
567C2×C4—4884no—
568C2×C4—2441no—
569C2×C4—2441no—
570C2×C4—2442no—
571C2×C4—2442no—
572C2×C4—2441no—
573C2×C4—2441no—
574C2×C4—2441no—
575C2×C4—2441no—
576C2×C4—2442no—
577C2×C4—2444no—
578C2×C4—2444no—
579C2×C4—2444no—
580C2×C4—2441no—
581C2×C4—2441no—
582C2×C4—2441no—
583C2×C4—2441no—
584C2×C4—2441no—
585C2×C4—2441no—
586C2×C4—2441no—
587C2×C4—2441no—
588C2×C4#1451974812yes—
589C2×C4#1451976014yes—
590C2×C4#1451976212yes—
591C2×C4#1451976212yes—
592C2×C4#1451976614yes—
593C2×C4#1451977214yes—
594C2×C4#1451977214yes—
595C2×C4C22×C22⋊D6111yescharacteristic
596C2×C4C2×#1175822411yescharacteristic
597C2×C4C2×#1175822211yescharacteristic
598C2×C4C2×#1175822211yescharacteristic
599C2×C4C2×#1175822611yescharacteristic
600C2×C4C2×#1175822611yescharacteristic
601C8—24no—
602C8—2442no—
603C22—22no—
604C22—22no—
605C22—22no—
606C22—22no—
607C22—22no—
608C22—22no—
609C22—1222no—
610C22—1222no—
611C22—1222no—
612C22—1222no—
613C22—1222no—
614C22—1222no—
615C22—24416no—
616C22—2441no—
617C22—2441no—
618C22—2441no—
619C22—2441no—
620C22—2441no—
621C22—2441no—
622C22—2441no—
623C22—2441no—
624C22#1683968814yes—
625C22C2×#1451974811yescharacteristic
626C22C2×#1451976211yescharacteristic
627C22C2×#1451976211yescharacteristic
628C22AGL(1,ℤ/8)×D6111yescharacteristic
629C22#137×D6111yescharacteristic
630C22#1684055811yescharacteristic
631C22#1684056111yescharacteristic
632C4—81no—
633C4—82no—
634C4—81no—
635C4—4881no—
636C4—4881no—
637C4—4882no—
638C4—2441no—
639C4—2441no—
640C4#1683968814yes—
641C4C2×#1451976011yescharacteristic
642C4C2×#1451976611yescharacteristic
643C4C2×#1451977211yescharacteristic
644C4C2×#1451977211yescharacteristic
645C2—24no—
646C2—2442no—
647C2—2442no—
648C2C2×#1683968811yescharacteristic
649C2#2145543211yescharacteristic
650C2#2145544311yescharacteristic
651C2#2145574511yescharacteristic
652C2#2145575311yescharacteristic
653C2#2145576111yescharacteristic
654C2#2145576911yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#428C2
socleC23×C61C22×D4
prime radicalC24×D61D4
hypercenterC2×#428D61
hypocenterC61#23642
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#428

Complements

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

NH ≅ G/Ntype
4. #21447253C2semidirect
5. #21447253C2semidirect
6. #21447248C2semidirect
7. #21447243C2semidirect
8. #21445540C2semidirect
9. #21445539C2semidirect
10. #21445538C2semidirect
11. C2×C61×#428C2semidirect
33. #16839525C22semidirect
34. #16839525C22semidirect
35. #16839524C22semidirect
36. #16839524C22semidirect
37. #16839524C22semidirect
38. #16839524C22semidirect
43. C2×#107×C61C22semidirect
44. C2×#106×C61C22semidirect
318. C2×#428D61semidirect
348. C2×#107C2×D61semidirect
349. C2×#106C2×D61semidirect
372. C61#23642semidirect

Maximal subgroups

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

subgroupindex
C2×#168397252
#214481982
C2×#168396192
#214472532
#214472482
#214472432
#214455402
#214455392
#214455382
C2×C61×#4282
#2364261