Finite Group Databasebeta

← #29695676

Subgroups — #29695676

Order 15616.

1582 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839840C211yesmaximal, characteristic, complemented
2C2×#16839763C212yesmaximal, complemented
3D4×C4⋊Dic61C211yesmaximal, characteristic, complemented
4#21446747C211yesmaximal, characteristic, complemented
5#21446626C211yesmaximal, characteristic, complemented
6#21446258C212yesmaximal, complemented
7#21446137C212yesmaximal, complemented
8#21445824C211yesmaximal, characteristic, complemented
9#21445759C212yesmaximal, complemented
10C4×#14519701C211yesmaximal, characteristic, complemented
11C2×C61×#550C211yesmaximal, characteristic
12C22×#11758194C2212yes—
13C2×D4×Dic61—21no—
14C22×C4⋊Dic61C2212yes—
15#16839840—24no—
16C2×C4⋊#92418C2211yescharacteristic
17#16839763—28no—
18C2×#14519701C2211yescharacteristic
19C2×#14519701C2212yes—
20C2×#14519701C2211yescharacteristic
21C2×#14519698C2212yes—
22C2×#14519698C2212yes—
23C2×#14519698C2212yes—
24C2×#14519698C2212yes—
25C2×C22⋊#92418C2212yes—
26C2×C4×#92418C2211yescharacteristic
27#16839661—21no—
28#16839637—22no—
29#16839632C2211yescharacteristic
30#16839622C2212yes—
31C4×C4⋊Dic61C2211yescharacteristic
32#16839524C2211yescharacteristic
33#16839524C2211yescharacteristic
34#16839524C2212yes—
35C22×#41×C61C2212yes—
36C61×#550—24no—
37C61×#550C418yescomplemented
38C2×#127×C61—21no—
39C2×#127×C61C2211yescharacteristic
40C2×#121×C61—22no—
41C2×#121×C61C2212yes—
42C2×C4×Q8×C61C2211yescharacteristic
43C2×C4×D4×C61—21no—
44C2×C4×D4×C61C2211yescharacteristic
45C23×Dic61—22no—
46C2×#11758194—24no—
47C2×#11758194—24no—
48C2×#11758194—42no—
49C22×#92418C2311yescharacteristic
50C22×#92418C2312yes—
51C22×#92418C2311yescharacteristic
52D4×Dic61—44no—
53C2×C4⋊Dic61—24no—
54C2×C4⋊Dic61—22no—
55C2×C4⋊Dic61—21no—
56C2×C4⋊Dic61—21no—
57C2×C4⋊Dic61—42no—
58C2×C4⋊Dic61C2312yes—
59C2×C4⋊Dic61C2311yescharacteristic
60C2×C4⋊Dic61C2311yescharacteristic
61C2×C22⋊Dic61—22no—
62C2×C4×Dic61—21no—
63C4⋊#92418—22no—
64#14519701—24no—
65#14519701—22no—
66#14519701D414yescomplemented
67#14519698—24no—
68#14519698—24no—
69#14519698—42no—
70#14519698—42no—
71C22⋊#92418—24no—
72C4×#92418—22no—
73#14519684—21no—
74#14519684—22no—
75C2×#41×C61—24no—
76C2×#41×C61—24no—
77C2×#41×C61—24no—
78C2×#41×C61—42no—
79C2×#41×C61—42no—
80C2×#41×C61C2×C418yescomplemented
81C22×Q8×C61C2311yescharacteristic
82C22×D4×C61—22no—
83C22×D4×C61—22no—
84C22×D4×C61—21no—
85C22×D4×C61—21no—
86C22×D4×C61D411yescharacteristic
87C22×D4×C61C2312yes—
88C23×C4×C61—22no—
89C23×C4×C61C2312yes—
90#127×C61—28no—
91#127×C61—22no—
92#127×C61C2×C414yescomplemented
93#121×C61—24no—
94#121×C61—216no—
95#121×C61C2×C418yescomplemented
96C4×Q8×C61—22no—
97C4×Q8×C61C2×C414yescomplemented
98C4×D4×C61—28no—
99C4×D4×C61C2×C414yes—
100C4×D4×C61D414yes—
101C2×C4⋊C4×C61—21no—
102C2×C4⋊C4×C61D411yescharacteristic
103C2×C4⋊C4×C61C2311yescharacteristic
104C2×C22⋊C4×C61—22no—
105C2×C22⋊C4×C61D412yes—
106C2×C42×C61—21no—
107C2×C42×C61C2311yescharacteristic
108C24×C61—22no—
109C24×C61C2×D412yes—
110#41×C61—216no—
111#41×C61—48no—
112#41×C61—48no—
113#41×C61—48no—
114C2×Q8×C61—21no—
115C2×Q8×C61—21no—
116C2×Q8×C61—21no—
117C2×Q8×C61C22×C412yes—
118C2×Q8×C61C2×D412yes—
119C2×Q8×C61#4112yes—
120C2×D4×C61—28no—
121C2×D4×C61—28no—
122C2×D4×C61—24no—
123C2×D4×C61—24no—
124C2×D4×C61—22no—
125C2×D4×C61—22no—
126C2×D4×C61—22no—
127C2×D4×C61—22no—
128C2×D4×C61—22no—
129C2×D4×C61—22no—
130C2×D4×C61—22no—
131C2×D4×C61—44no—
132C2×D4×C61—44no—
133C2×D4×C61—42no—
134C2×D4×C61—42no—
135C2×D4×C61—81no—
136C2×D4×C61—81no—
137C2×D4×C61—82no—
138C2×D4×C61—82no—
139C2×D4×C61C22⋊C414yes—
140C2×D4×C61C22×C414yes—
141C22×C4×C61—21no—
142C22×C4×C61—22no—
143C22×C4×C61—22no—
144C22×C4×C61—22no—
145C22×C4×C61—22no—
146C22×C4×C61—21no—
147C22×C4×C61—28no—
148C22×C4×C61—44no—
149C22×C4×C61—44no—
150C22×C4×C61C22×C414yes—
151C22×C4×C61C2×D411yescharacteristic
152C22×C4×C61C2×D412yes—
153C22×C4×C61C2×D414yes—
154C22×C4×C61C2×D411yescharacteristic
155C22×C4×C61C2411yescharacteristic
156C22×Dic61—21no—
157C22×Dic61—22no—
158C22×Dic61—21no—
159C22×Dic61—44no—
160C22×Dic61—44no—
161C22×Dic61—82no—
162C4⋊C4×C61—24no—
163C4⋊C4×C61—21no—
164C4⋊C4×C61C22⋊C414yes—
165C4⋊C4×C61C22×C412yes—
166C22⋊C4×C61—28no—
167C22⋊C4×C61C22⋊C418yes—
168C42×C61—24no—
169C42×C61C22×C412yes—
170C42×C61C2×D412yes—
171C22⋊Dic61—44no—
172C4⋊Dic61—21no—
173C4⋊Dic61—22no—
174C4⋊Dic61—21no—
175C4⋊Dic61—21no—
176C4⋊Dic61—22no—
177C4⋊Dic61—44no—
178C4⋊Dic61—42no—
179C4⋊Dic61—81no—
180C4⋊Dic61C2×D412yes—
181C4×Dic61—42no—
182#11758194—48no—
183C2×#92418—21no—
184C2×#92418—21no—
185C2×#92418—21no—
186C2×#92418—22no—
187C2×#92418—22no—
188C2×#92418—21no—
189C2×#92418—21no—
190C2×#92418—22no—
191C2×#92418C2×D412yes—
192C23×C61—22no—
193C23×C61—22no—
194C23×C61—22no—
195C23×C61—22no—
196C23×C61—22no—
197C23×C61—22no—
198C23×C61—24no—
199C23×C61—42no—
200C23×C61—42no—
201C23×C61—42no—
202C23×C61—42no—
203C23×C61—42no—
204C23×C61—42no—
205C23×C61C2×C22⋊C414yes—
206C23×C61C22×D411yescharacteristic
207Q8×C61—22no—
208Q8×C61—22no—
209Q8×C61—22no—
210Q8×C61C4×D414yescomplemented
211D4×C61—24no—
212D4×C61—24no—
213D4×C61—28no—
214D4×C61—416no—
215D4×C61—42no—
216D4×C61—42no—
217D4×C61—48no—
218D4×C61—84no—
219D4×C61—88no—
220C2×C4×C61—22no—
221C2×C4×C61—22no—
222C2×C4×C61—22no—
223C2×C4×C61—22no—
224C2×C4×C61—22no—
225C2×C4×C61—21no—
226C2×C4×C61—21no—
227C2×C4×C61—24no—
228C2×C4×C61—24no—
229C2×C4×C61—24no—
230C2×C4×C61—24no—
231C2×C4×C61—22no—
232C2×C4×C61—416no—
233C2×C4×C61—41no—
234C2×C4×C61—41no—
235C2×C4×C61C2×C22⋊C412yes—
236C2×C4×C61C2×C22⋊C414yes—
237C2×C4×C61C2×C22⋊C412yes—
238C2×C4×C61AGL(1,ℤ/8)12yes—
239C2×C4×C61#13712yes—
240C2×C4×C61C23×C411yescharacteristic
241C2×C4×C61C22×D411yescharacteristic
242C2×C4×C61C22×D411yescharacteristic
243C2×C4×C61C2×#4111yescharacteristic
244C2×C4×C61H(5,𝔽2)11yescharacteristic
245C2×C4×C61H(5,𝔽2)11yescharacteristic
246C2×Dic61—22no—
247C2×Dic61—41no—
248C2×Dic61—41no—
249C2×Dic61—42no—
250C2×Dic61—41no—
251C2×Dic61—41no—
252C2×Dic61—42no—
253C2×Dic61—42no—
254C2×Dic61—41no—
255C2×Dic61—88no—
256#92418—22no—
257#92418—42no—
258#92418—41no—
259#22432—611nomaximal, Sylow
260C22×C61—24no—
261C22×C61—24no—
262C22×C61—24no—
263C22×C61—42no—
264C22×C61—42no—
265C22×C61—42no—
266C22×C61—42no—
267C22×C61—44no—
268C22×C61—44no—
269C22×C61—44no—
270C22×C61—82no—
271C22×C61—82no—
272C22×C61—82no—
273C22×C61—82no—
274C22×C61C22×C22⋊C411yescharacteristic
275C22×C61D4211yescharacteristic
276C22×C61#54611yescharacteristic
277C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
278C22×C61C2×#13711yescharacteristic
279C22×C61#57611yescharacteristic
280C22×C61#57711yescharacteristic
281C4×C61—24no—
282C4×C61—24no—
283C4×C61—22no—
284C4×C61—42no—
285C4×C61#41714yescomplemented
286C4×C61C2×C4×D411yescharacteristic
287C4×C61#51811yescharacteristic
288C4×C61D4211yescharacteristic
289C4×C61#54611yescharacteristic
290Dic61—42no—
291Dic61—81no—
292Dic61—82no—
293C2×#550D6111yescharacteristic
294C2×#436—611no—
295C2×#417—612no—
296D4×C4⋊C4—611no—
297#1982—611no—
298#1957—611no—
299#1895—612no—
300#1874—612no—
301#1803—611no—
302#1788—612no—
303C4×#103—611no—
304C2×C61—44no—
305C2×C61—84no—
306C2×C61D4×C22⋊C411yescharacteristic
307C2×C61C2×#41711yescharacteristic
308C2×C61#289711yescharacteristic
309C2×C61#327411yescharacteristic
310C2×C61#327811yescharacteristic
311C2×C61#333211yescharacteristic
312C2×C61#333311yescharacteristic
313C22×#41C2×D6112yes—
314C22×M16—612no—
315#550—24no—
316#550Dic6118yescomplemented
317C2×#127—21no—
318C2×#127C2×D6111yescharacteristic
319C2×#121—22no—
320C2×#121C2×D6112yes—
321C2×C4×Q8C2×D6111yescharacteristic
322C2×C4×D4—21no—
323C2×C4×D4—1221no—
324C2×C4×D4C2×D6111yescharacteristic
325C22×C4⋊C4—612no—
326#436—1224no—
327C2×C4⋊C8—611no—
328#417—1228no—
329C2×#103—611no—
330C2×#103—612no—
331C2×#103—611no—
332C2×#102—612no—
333C2×#102—612no—
334C2×#102—612no—
335C2×#102—612no—
336C2×C22⋊C8—612no—
337C2×C4×C8—611no—
338#389—1221no—
339#385—1222no—
340#384—611no—
341#380—612no—
342C4×C4⋊C4—611no—
343#340—612no—
344#340—611no—
345#340—611no—
346C61#2243211yescharacteristic, complemented, Sylow
347C2×#41—24no—
348C2×#41—24no—
349C2×#41—24no—
350C2×#41—42no—
351C2×#41—42no—
352C2×#41C2×Dic6118yescomplemented
353C22×Q8C22×D6111yescharacteristic
354C22×D4—22no—
355C22×D4—22no—
356C22×D4—21no—
357C22×D4—21no—
358C22×D4C22⋊D6111yescharacteristic
359C22×D4C22×D6112yes—
360C23×C4—22no—
361C23×C4—1222no—
362C23×C4C22×D6112yes—
363C2×M16—1224no—
364C2×M16—1224no—
365C2×M16—2442no—
366C22×C8—611no—
367C22×C8—612no—
368C22×C8—611no—
369#127—28no—
370#127—22no—
371#127C2×Dic6114yescomplemented
372#121—24no—
373#121—216no—
374#121C2×Dic6118yescomplemented
375C4×Q8—22no—
376C4×Q8C2×Dic6114yescomplemented
377C4×D4—28no—
378C4×D4—2444no—
379C4×D4C2×Dic6114yes—
380C4×D4C22⋊D6114yes—
381C2×C4⋊C4—21no—
382C2×C4⋊C4—611no—
383C2×C4⋊C4—612no—
384C2×C4⋊C4—611no—
385C2×C4⋊C4—1224no—
386C2×C4⋊C4—1221no—
387C2×C4⋊C4—1222no—
388C2×C4⋊C4—1221no—
389C2×C4⋊C4—2442no—
390C2×C4⋊C4C22⋊D6111yescharacteristic
391C2×C4⋊C4C22×D6111yescharacteristic
392C2×C22⋊C4—22no—
393C2×C22⋊C4—1222no—
394C2×C22⋊C4C22⋊D6112yes—
395C2×C42—21no—
396C2×C42—1221no—
397C2×C42C22×D6111yescharacteristic
398C4⋊C8—1222no—
399#103—614no—
400#103—1224no—
401#103—1222no—
402#102—1224no—
403#102—1224no—
404#102—2442no—
405#102—2442no—
406C22⋊C8—1224no—
407C4×C8—1222no—
408#95—1221no—
409#95—1222no—
410C24—22no—
411C24C2×C22⋊D6112yes—
412#41—216no—
413#41—48no—
414#41—48no—
415#41—48no—
416C2×Q8—21no—
417C2×Q8—21no—
418C2×Q8—21no—
419C2×Q8D4×D6112yes—
420C2×Q8#1175822412yes—
421C2×Q8C22×Dic6112yes—
422C2×D4—28no—
423C2×D4—28no—
424C2×D4—22no—
425C2×D4—22no—
426C2×D4—22no—
427C2×D4—22no—
428C2×D4—22no—
429C2×D4—24no—
430C2×D4—24no—
431C2×D4—22no—
432C2×D4—22no—
433C2×D4—44no—
434C2×D4—44no—
435C2×D4—42no—
436C2×D4—42no—
437C2×D4—81no—
438C2×D4—81no—
439C2×D4—82no—
440C2×D4—82no—
441C2×D4C22⋊Dic6114yes—
442C2×D4C22×Dic6114yes—
443C22×C4—21no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—21no—
449C22×C4—28no—
450C22×C4—44no—
451C22×C4—44no—
452C22×C4—1221no—
453C22×C4—1221no—
454C22×C4—1222no—
455C22×C4—4882no—
456C22×C4—2444no—
457C22×C4—2444no—
458C22×C4C22×Dic6114yes—
459C22×C4C2×C22⋊D6114yes—
460C22×C4C2×C22⋊D6111yescharacteristic
461C22×C4C2×C22⋊D6112yes—
462C22×C4C2×C22⋊D6111yescharacteristic
463C22×C4C23×D6111yescharacteristic
464M16—2448no—
465C2×C8—612no—
466C2×C8—1221no—
467C2×C8—1221no—
468C2×C8—1222no—
469C2×C8—1221no—
470C2×C8—1222no—
471C2×C8—1221no—
472C2×C8—1221no—
473C2×C8—1222no—
474C4⋊C4—21no—
475C4⋊C4—24no—
476C4⋊C4—612no—
477C4⋊C4—1221no—
478C4⋊C4—1222no—
479C4⋊C4—1221no—
480C4⋊C4—1221no—
481C4⋊C4—1222no—
482C4⋊C4—4881no—
483C4⋊C4—2444no—
484C4⋊C4—2442no—
485C4⋊C4C22⋊Dic6114yes—
486C4⋊C4C22×Dic6112yes—
487C22⋊C4—28no—
488C22⋊C4—2444no—
489C22⋊C4C22⋊Dic6118yes—
490C42—24no—
491C42—2442no—
492C42C22×Dic6112yes—
493C42C2×C22⋊D6112yes—
494C23—22no—
495C23—22no—
496C23—22no—
497C23—22no—
498C23—22no—
499C23—22no—
500C23—24no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23—42no—
506C23—42no—
507C23C2×C22⋊Dic6114yes—
508C23C22×C22⋊D6111yescharacteristic
509Q8—22no—
510Q8—22no—
511Q8—22no—
512Q8D4×Dic6114yescomplemented
513D4—24no—
514D4—24no—
515D4—28no—
516D4—416no—
517D4—48no—
518D4—42no—
519D4—42no—
520D4—84no—
521D4—88no—
522C2×C4—22no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—22no—
534C2×C4—416no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—1222no—
538C2×C4—4888no—
539C2×C4—2441no—
540C2×C4—2442no—
541C2×C4—2442no—
542C2×C4—2441no—
543C2×C4—2442no—
544C2×C4—2441no—
545C2×C4—2441no—
546C2×C4—2441no—
547C2×C4C2×C22⋊Dic6112yes—
548C2×C4C2×C22⋊Dic6114yes—
549C2×C4C2×C22⋊Dic6112yes—
550C2×C4#1451980212yes—
551C2×C4#1451980712yes—
552C2×C4C23×Dic6111yescharacteristic
553C2×C4C2×D4×D6111yescharacteristic
554C2×C4C22×C22⋊D6111yescharacteristic
555C2×C4C2×#1175822411yescharacteristic
556C2×C4#1451982211yescharacteristic
557C2×C4#1451982211yescharacteristic
558C8—1222no—
559C8—2442no—
560C8—2441no—
561C22—24no—
562C22—24no—
563C22—24no—
564C22—42no—
565C22—42no—
566C22—42no—
567C22—42no—
568C22—44no—
569C22—44no—
570C22—44no—
571C22—82no—
572C22—82no—
573C22—82no—
574C22—82no—
575C22C22×C22⋊Dic6111yescharacteristic
576C22D4×C22⋊D6111yescharacteristic
577C22#1684040611yescharacteristic
578C22C2×#1451980211yescharacteristic
579C22C2×#1451980711yescharacteristic
580C22#1684056511yescharacteristic
581C22#1684057311yescharacteristic
582C4—22no—
583C4—24no—
584C4—24no—
585C4—42no—
586C4—4882no—
587C4—4881no—
588C4—2442no—
589C4#1683976314yescomplemented
590C4C2×D4×Dic6111yescharacteristic
591C4#1684023711yescharacteristic
592C4D4×C22⋊D6111yescharacteristic
593C4#1684040611yescharacteristic
594C2—44no—
595C2—84no—
596C2D4×C22⋊Dic6111yescharacteristic
597C2C2×#1683976311yescharacteristic
598C2#2145434511yescharacteristic
599C2#2145743911yescharacteristic
600C2#2145748511yescharacteristic
601C2#2145814611yescharacteristic
602C2#2145815711yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
Fitting, prime radicalC2×C61×#550C2
socleC23×C61C22×D4
cosocleC22×C4×C61C24
hypercenterC2×#550D61
hypocenterC61#22432
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#550

Complements

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

NH ≅ G/Ntype
1. C2×#16839840C2semidirect
2. C2×#16839763C2semidirect
3. D4×C4⋊Dic61C2semidirect
4. #21446747C2semidirect
5. #21446626C2semidirect
6. #21446258C2semidirect
7. #21446137C2semidirect
8. #21445824C2semidirect
9. #21445759C2semidirect
10. C4×#14519701C2semidirect
37. C61×#550C4semidirect
66. #14519701D4semidirect
80. C2×#41×C61C2×C4semidirect
92. #127×C61C2×C4semidirect
95. #121×C61C2×C4semidirect
97. C4×Q8×C61C2×C4semidirect
210. Q8×C61C4×D4semidirect
285. C4×C61#417semidirect
316. #550Dic61semidirect
346. C61#22432semidirect
352. C2×#41C2×Dic61semidirect
371. #127C2×Dic61semidirect
374. #121C2×Dic61semidirect
376. C4×Q8C2×Dic61semidirect
512. Q8D4×Dic61semidirect
589. C4#16839763semidirect

Maximal subgroups

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

subgroupindex
C2×#168398402
C2×#168397632
D4×C4⋊Dic612
#214467472
#214466262
#214462582
#214461372
#214458242
#214457592
C4×#145197012
C2×C61×#5502
#2243261