Finite Group Databasebeta

← #29716096

Subgroups — #29716096

Order 15616.

1150 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840317C211yesmaximal, characteristic, complemented
2C2×#16840042C211yesmaximal, characteristic, complemented
3C4×#14519748C211yesmaximal, characteristic
4#21447159C212yesmaximal, complemented
5#21446963C212yesmaximal, complemented
6#21446770C211yesmaximal, characteristic, complemented
7#21446768C211yesmaximal, characteristic, complemented
8#21446138C212yesmaximal
9#21446017C212yesmaximal, complemented
10#21445796C211yesmaximal, characteristic, complemented
11C2×C61×#476C211yesmaximal, characteristic, complemented
12#16840317—24no—
13#16840317—24no—
14C2×#14519777C2211yescharacteristic
15C2×#14519768C2212yes—
16C2×#14519757—21no—
17C2×#14519753—21no—
18C2×#14519748C2211yescharacteristic
19C2×Q8×Dic61—21no—
20C2×#14519742—21no—
21C2×C4×C22⋊D61—21no—
22C2×#14519735C2212yes—
23C2×C42×D61C2211yescharacteristic
24#16840042—24no—
25C2×#14519710C2212yes—
26C2×C4⋊#92418C2211yescharacteristic
27C2×#14519698C2212yes—
28C2×#14519689C2212yes—
29#16839665C2211yescharacteristic
30#16839658C2211yescharacteristic
31#16839650—21no—
32#16839638—21no—
33#16839619C2211yescharacteristic
34C4×C4⋊Dic61C2211yescharacteristic
35C4×C22⋊Dic61—21no—
36#16839525C2212yes—
37#16839525C2212yes—
38#16839524C2212yes—
39#16839524C2212yes—
40#16839516C2212yes—
41C2×C4⋊Q8×C61C2211yescharacteristic
42C2×C4×D4×C61C2211yescharacteristic
43C61×#476—24no—
44C2×#106×C61C2212yes—
45C2×C4⋊C8×C61C2211yescharacteristic
46C2×#102×C61C2212yes—
47C22×Dic122—21no—
48C22×C22⋊D61—21no—
49C22×C4×D61D411yescharacteristic
50C22×C4×D61D411yescharacteristic
51C22×C4×D61C2311yescharacteristic
52C22×#92418C2312yes—
53#14519777—22no—
54#14519777D414yescomplemented
55#14519768—24no—
56#14519768—24no—
57#14519757—28no—
58#14519753—28no—
59#14519748—22no—
60#14519748Q814yescomplemented
61Q8×Dic61—28no—
62#14519742—28no—
63C4×C22⋊D61—44no—
64#14519735—24no—
65#14519735—24no—
66C2×C4⋊Dic61—21no—
67C2×C4⋊Dic61—22no—
68C2×C4⋊Dic61C2311yescharacteristic
69C2×C4⋊Dic61C2312yes—
70C2×C4⋊Dic61C2311yescharacteristic
71C2×#11758196—22no—
72C2×#11758196—21no—
73C2×C22⋊Dic61—21no—
74C2×C22⋊Dic61—21no—
75C2×C22⋊Dic61—21no—
76C2×#11758198—22no—
77C2×#11758198—21no—
78C2×C4×Dic61—22no—
79C2×C4×Dic61—21no—
80C2×C4×Dic61D411yescharacteristic
81C2×C4×Dic61D411yescharacteristic
82C2×C4×Dic61C2311yescharacteristic
83C42×D61—22no—
84C42×D61—22no—
85#14519710—24no—
86C4⋊#92418—22no—
87#14519698—24no—
88#14519689—24no—
89#14519684—21no—
90#14519684—21no—
91#14519684—21no—
92C22×Q8×C61—21no—
93C22×D4×C61C2311yescharacteristic
94C23×C4×C61—21no—
95C22×C8×C61C2312yes—
96C4⋊Q8×C61—22no—
97C4⋊Q8×C61D414yescomplemented
98C4×D4×C61—22no—
99C4×D4×C61—22no—
100C2×C4⋊C4×C61—21no—
101C2×C4⋊C4×C61C2311yescharacteristic
102C2×C4⋊C4×C61C2312yes—
103C2×C22⋊C4×C61—21no—
104C2×C42×C61C2311yescharacteristic
105#106×C61—24no—
106C4⋊C8×C61—22no—
107#102×C61—24no—
108C24×C61—21no—
109C23×D61C2×D411yescharacteristic
110C2×Q8×C61—24no—
111C2×Q8×C61—42no—
112C2×Q8×C61—42no—
113C2×D4×C61—21no—
114C2×D4×C61—21no—
115C2×D4×C61—21no—
116C2×D4×C61—21no—
117C2×D4×C61—21no—
118C2×D4×C61C2×Q812yes—
119C22×C4×C61—22no—
120C22×C4×C61—21no—
121C22×C4×C61—42no—
122C22×C4×C61—42no—
123C22×C4×C61—42no—
124C22×C4×C61C2×D411yescharacteristic
125C22×C4×C61C2×D411yescharacteristic
126C22×C4×C61C2411yescharacteristic
127C2×C22⋊D61—24no—
128C2×C22⋊D61—42no—
129C2×C22⋊D61—42no—
130C22×Dic61—21no—
131C22×Dic61—22no—
132C22×Dic61—21no—
133C22×Dic61C2×D411yescharacteristic
134C22×Dic61C2×D411yescharacteristic
135C22×Dic61C2×D411yescharacteristic
136C2×C4×D61—21no—
137C2×C4×D61—21no—
138C2×C4×D61—21no—
139C2×C4×D61—21no—
140C2×C4×D61—22no—
141C2×C4×D61—22no—
142C2×C4×D61—22no—
143C2×C4×D61—22no—
144C2×C4×D61—41no—
145C2×C4×D61—41no—
146C2×C4×D61C2×Q812yes—
147C2×C4×D61#4112yes—
148C2×Dic122—24no—
149C2×Dic122—42no—
150C2×Dic122—42no—
151C2×C8×C61—22no—
152C2×C8×C61—22no—
153C2×C8×C61—22no—
154C4⋊C4×C61—24no—
155C4⋊C4×C61—22no—
156C4⋊C4×C61—21no—
157C4⋊C4×C61—21no—
158C4⋊C4×C61C2×D414yes—
159C22⋊C4×C61—42no—
160C42×C61C2×D412yes—
161C42×C61C2×D412yes—
162C22⋊Dic61—24no—
163C22⋊Dic61—42no—
164C22⋊Dic61—42no—
165#11758198—28no—
166#11758198—42no—
167C4⋊Dic61—28no—
168C4⋊Dic61—24no—
169C4⋊Dic61—22no—
170C4⋊Dic61—21no—
171C4⋊Dic61—21no—
172C4⋊Dic61—21no—
173C4⋊Dic61C2×D414yes—
174C4⋊Dic61C2×Q812yes—
175#11758196—28no—
176#11758196—42no—
177C4×Dic61—28no—
178C4×Dic61—22no—
179C4×Dic61—22no—
180C4×Dic61—21no—
181C4×Dic61—21no—
182C4×Dic61—42no—
183C2×#92418—22no—
184C2×#92418—22no—
185C2×#92418—22no—
186C23×C61—22no—
187C23×C61—41no—
188C23×C61—41no—
189C23×C61—41no—
190C23×C61—41no—
191C23×C61—41no—
192C23×C61—41no—
193C23×C61C22×D411yescharacteristic
194C22×D61—21no—
195C22×D61—21no—
196C22×D61—21no—
197C22×D61—21no—
198C22×D61—21no—
199C22×D61—21no—
200C22×D61C22⋊Q812yes—
201Q8×C61—48no—
202D4×C61—22no—
203D4×C61—22no—
204D4×C61—41no—
205D4×C61—41no—
206C2×C4×C61—21no—
207C2×C4×C61—21no—
208C2×C4×C61—24no—
209C2×C4×C61—42no—
210C2×C4×C61—42no—
211C2×C4×C61—41no—
212C2×C4×C61—41no—
213C2×C4×C61—41no—
214C2×C4×C61—84no—
215C2×C4×C61#12012yes—
216C2×C4×C61#12012yes—
217C2×C4×C61#13512yes—
218C2×C4×C61AGL(1,ℤ/8)12yes—
219C2×C4×C61C22×D411yescharacteristic
220C2×C4×C61C22×D411yescharacteristic
221C2×C4×C61C22×Q811yescharacteristic
222C2×C4×C61C2×#4111yescharacteristic
223C2×C4×C612−1+411yescharacteristic
224C2×C4×C612−1+411yescharacteristic
225C22⋊D61—48no—
226C2×Dic61—21no—
227C2×Dic61—21no—
228C2×Dic61—21no—
229C2×Dic61—21no—
230C2×Dic61—24no—
231C2×Dic61—22no—
232C2×Dic61—21no—
233C2×Dic61—21no—
234C2×Dic61—21no—
235C2×Dic61—21no—
236C2×Dic61—42no—
237C2×Dic61—41no—
238C2×Dic61—42no—
239C2×Dic61—41no—
240C2×Dic61—41no—
241C2×Dic61—41no—
242C2×Dic61—41no—
243C2×Dic61C22⋊Q812yes—
244C4×D61—22no—
245C4×D61—22no—
246C4×D61—22no—
247C4×D61—22no—
248C4×D61—44no—
249C4×D61—44no—
250Dic122—48no—
251C8×C61—42no—
252#92418—42no—
253#24300—611nomaximal, Sylow
254C22×C61—42no—
255C22×C61—42no—
256C22×C61—42no—
257C22×C61—81no—
258C22×C61—81no—
259C22×C61—81no—
260C22×C61—81no—
261C22×C61C2×#12011yescharacteristic
262C22×C61C2×C22⋊Q811yescharacteristic
263C22×C61#53611yescharacteristic
264C22×C61C2×#13511yescharacteristic
265C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
266C22×C61#57511yescharacteristic
267C22×C61#57511yescharacteristic
268C2×D61—22no—
269C2×D61—22no—
270C2×D61—22no—
271C2×D61—41no—
272C2×D61—41no—
273C2×D61—41no—
274C2×D61—41no—
275C4×C61—22no—
276C4×C61—44no—
277C4×C61—81no—
278C4×C61#44914yes—
279C4×C61#54811yescharacteristic
280C4×C61#54811yescharacteristic
281C4×C61D4×Q811yescharacteristic
282C4×C61D4×Q811yescharacteristic
283Dic61—22no—
284Dic61—42no—
285Dic61—44no—
286Dic61—41no—
287Dic61—41no—
288Dic61—81no—
289C2×#533—611no—
290C2×#476D6111yescharacteristic, complemented
291C2×#474—611no—
292C4×#121—611no—
293#2059—612no—
294#2025—612no—
295#1988—611no—
296#1987—611no—
297#1875—612no—
298#1854—612no—
299#1797—611no—
300C2×C61—82no—
301C2×C61#240911yescharacteristic
302C2×C61C2×#44911yescharacteristic
303C2×C61#300511yescharacteristic
304C2×C61#307611yescharacteristic
305C2×C61#307711yescharacteristic
306C2×C61#323311yescharacteristic
307C2×C61#323511yescharacteristic
308D61—42no—
309#533—1224no—
310#533—1224no—
311C2×C4⋊Q8—611no—
312C2×C4⋊Q8C2×D6111yescharacteristic
313C2×#125—612no—
314C2×C22⋊Q8—1221no—
315C2×C22⋊Q8—1221no—
316C2×#121—611no—
317C2×C4×Q8—1221no—
318C2×C4×Q8—1221no—
319C2×C4×D4—1221no—
320C2×C4×D4C2×D6111yescharacteristic
321C2×#117—612no—
322C22×C42—611no—
323#476—24no—
324#474—1224no—
325C2×#107—612no—
326C2×#106C2×D6112yes—
327C2×C4⋊C8—611no—
328C2×C4⋊C8C2×D6111yescharacteristic
329C2×#102—612no—
330C2×#102C2×D6112yes—
331C2×C22⋊C8—612no—
332#390—611no—
333#389—611no—
334#388—1221no—
335#386—1221no—
336#380—611no—
337C4×C4⋊C4—611no—
338C4×C22⋊C4—1221no—
339#340—612no—
340#340—612no—
341#340—612no—
342#340—612no—
343#336—612no—
344C61#2430011yescharacteristic, complemented, Sylow
345C22×Q8—21no—
346C22×Q8—1221no—
347C22×D4—1221no—
348C22×D4C22×D6111yescharacteristic
349C23×C4—21no—
350C23×C4—611no—
351C23×C4—611no—
352C23×C4—611no—
353C22×C8—612no—
354C22×C8C22×D6112yes—
355C4⋊Q8—22no—
356C4⋊Q8—614no—
357C4⋊Q8—1222no—
358C4⋊Q8C22⋊D6114yescomplemented
359#125—1224no—
360#125—1224no—
361C22⋊Q8—1228no—
362C22⋊Q8—1228no—
363#121—614no—
364#121—1222no—
365C4×Q8—1228no—
366C4×Q8—1228no—
367C4×D4—22no—
368C4×D4—22no—
369C4×D4—2444no—
370#117—1224no—
371#117—1224no—
372C2×C4⋊C4—21no—
373C2×C4⋊C4—612no—
374C2×C4⋊C4—611no—
375C2×C4⋊C4—611no—
376C2×C4⋊C4—1222no—
377C2×C4⋊C4—1222no—
378C2×C4⋊C4—1221no—
379C2×C4⋊C4—1221no—
380C2×C4⋊C4C22×D6111yescharacteristic
381C2×C4⋊C4C22×D6112yes—
382C2×C22⋊C4—21no—
383C2×C22⋊C4—1222no—
384C2×C22⋊C4—1221no—
385C2×C22⋊C4—1221no—
386C2×C22⋊C4—1221no—
387C2×C22⋊C4—1221no—
388C2×C42—611no—
389C2×C42—611no—
390C2×C42—611no—
391C2×C42—1222no—
392C2×C42—1222no—
393C2×C42—1222no—
394C2×C42—1221no—
395C2×C42C22×D6111yescharacteristic
396#107—1224no—
397#106—24no—
398C4⋊C8—22no—
399C4⋊C8—1222no—
400#102—24no—
401#102—1224no—
402C22⋊C8—1224no—
403#95—1221no—
404#95—1221no—
405#95—1221no—
406C24—21no—
407C24—611no—
408C2×Q8—24no—
409C2×Q8—42no—
410C2×Q8—42no—
411C2×Q8—1224no—
412C2×Q8—2442no—
413C2×Q8—2442no—
414C2×D4—21no—
415C2×D4—21no—
416C2×D4—21no—
417C2×D4—21no—
418C2×D4—21no—
419C2×D4—1224no—
420C2×D4—2442no—
421C2×D4—2442no—
422C2×D4Q8×D6112yes—
423C22×C4—22no—
424C22×C4—21no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—611no—
429C22×C4—612no—
430C22×C4—612no—
431C22×C4—611no—
432C22×C4—611no—
433C22×C4—1221no—
434C22×C4—1221no—
435C22×C4—1221no—
436C22×C4—1221no—
437C22×C4—1221no—
438C22×C4—1222no—
439C22×C4—1222no—
440C22×C4—1222no—
441C22×C4—1222no—
442C22×C4—1222no—
443C22×C4—1221no—
444C22×C4—2441no—
445C22×C4—2441no—
446C22×C4D4×D6111yescharacteristic
447C22×C4D4×D6111yescharacteristic
448C22×C4C23×D6111yescharacteristic
449C2×C8—22no—
450C2×C8—22no—
451C2×C8—22no—
452C2×C8—1222no—
453C2×C8—1222no—
454C2×C8—1222no—
455C4⋊C4—22no—
456C4⋊C4—24no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—614no—
460C4⋊C4—612no—
461C4⋊C4—1228no—
462C4⋊C4—1228no—
463C4⋊C4—1222no—
464C4⋊C4—1221no—
465C4⋊C4—1224no—
466C4⋊C4—1221no—
467C4⋊C4—1221no—
468C4⋊C4—2442no—
469C4⋊C4C2×C22⋊D6114yes—
470C22⋊C4—42no—
471C22⋊C4—1228no—
472C22⋊C4—1224no—
473C22⋊C4—2442no—
474C22⋊C4—2442no—
475C22⋊C4—2442no—
476C42—1228no—
477C42—1221no—
478C42—1221no—
479C42—1222no—
480C42—1222no—
481C42—2442no—
482C42D4×D6112yes—
483C42C2×C22⋊D6112yes—
484C23—22no—
485C23—41no—
486C23—41no—
487C23—41no—
488C23—41no—
489C23—41no—
490C23—41no—
491C23—612no—
492C23—1221no—
493C23—1221no—
494C23—1221no—
495C23—1221no—
496C23—1221no—
497C23—1221no—
498C23C2×D4×D6111yescharacteristic
499Q8—48no—
500Q8—2448no—
501D4—22no—
502D4—22no—
503D4—41no—
504D4—41no—
505D4—2448no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—24no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—84no—
515C2×C4—612no—
516C2×C4—1221no—
517C2×C4—1221no—
518C2×C4—1222no—
519C2×C4—1222no—
520C2×C4—1222no—
521C2×C4—1222no—
522C2×C4—1221no—
523C2×C4—1221no—
524C2×C4—1224no—
525C2×C4—1222no—
526C2×C4—1221no—
527C2×C4—1221no—
528C2×C4—1221no—
529C2×C4—1221no—
530C2×C4—2444no—
531C2×C4—2441no—
532C2×C4—2441no—
533C2×C4—2441no—
534C2×C4—2442no—
535C2×C4—2441no—
536C2×C4—2442no—
537C2×C4—2441no—
538C2×C4—2444no—
539C2×C4#1451974512yes—
540C2×C4#1451974512yes—
541C2×C4#1451979812yes—
542C2×C4#1451980412yes—
543C2×C4C2×D4×D6111yescharacteristic
544C2×C4C22×C22⋊D6111yescharacteristic
545C2×C4C2×Q8×D6111yescharacteristic
546C2×C4#41×D6111yescharacteristic
547C2×C4#1451982411yescharacteristic
548C2×C4#1451982511yescharacteristic
549C8—42no—
550C8—2442no—
551C22—42no—
552C22—42no—
553C22—42no—
554C22—81no—
555C22—81no—
556C22—81no—
557C22—81no—
558C22—1222no—
559C22—1222no—
560C22—1222no—
561C22—2441no—
562C22—2441no—
563C22—2441no—
564C22—2441no—
565C22C2×#1451974511yescharacteristic
566C22C22⋊Q8×D6111yescharacteristic
567C22#1684033511yescharacteristic
568C22C2×#1451979811yescharacteristic
569C22C2×#1451980411yescharacteristic
570C22#1684055811yescharacteristic
571C22#1684055811yescharacteristic
572C4—22no—
573C4—44no—
574C4—81no—
575C4—1222no—
576C4—4881no—
577C4—2442no—
578C4—2441no—
579C4—2444no—
580C4—2441no—
581C4#1683990614yes—
582C4#1684041911yescharacteristic
583C4#1684042211yescharacteristic
584C4#1684042511yescharacteristic
585C4Q8×C22⋊D6111yescharacteristic
586C2—82no—
587C2—2442no—
588C2#2144918011yescharacteristic
589C2C2×#1683990611yescharacteristic
590C2#2145505811yescharacteristic
591C2#2145560811yescharacteristic
592C2#2145561711yescharacteristic
593C2#2145699711yescharacteristic
594C2#2145701511yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#476C2
socleC23×C61C22×D4
prime radicalC2×#14519748C22
hypercenterC2×#476D61
hypocenterC61#24300
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#476

Complements

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

NH ≅ G/Ntype
1. C2×#16840317C2semidirect
2. C2×#16840042C2semidirect
4. #21447159C2semidirect
5. #21446963C2semidirect
6. #21446770C2semidirect
7. #21446768C2semidirect
9. #21446017C2semidirect
10. #21445796C2semidirect
11. C2×C61×#476C2semidirect
54. #14519777D4semidirect
60. #14519748Q8semidirect
97. C4⋊Q8×C61D4semidirect
290. C2×#476D61semidirect
344. C61#24300semidirect
358. C4⋊Q8C22⋊D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168403172
C2×#168400422
C4×#145197482
#214471592
#214469632
#214467702
#214467682
#214461382
#214460172
#214457962
C2×C61×#4762
#2430061