Finite Group Databasebeta

← #29716092

Subgroups — #29716092

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×#16840053C211yesmaximal, characteristic, complemented
3C2×#16840039C211yesmaximal, characteristic, complemented
4C4×#14519752C211yesmaximal, characteristic
5#21447156C212yesmaximal, complemented
6#21446964C212yesmaximal, complemented
7#21446767C211yesmaximal, characteristic, complemented
8#21446140C212yesmaximal
9#21446019C212yesmaximal, complemented
10#21445796C211yesmaximal, characteristic, complemented
11C61×#1988C211yesmaximal, characteristic, complemented
12#16840317—24no—
13#16840317—24no—
14C2×#14519777C2211yescharacteristic
15C2×#14519768C2212yes—
16C2×#14519757—21no—
17C2×#14519753—21no—
18C2×#14519752C2211yescharacteristic
19C2×Q8×Dic61—21no—
20C2×#14519742—21no—
21C2×C4×D244—21no—
22C2×C4×D244C2211yescharacteristic
23C2×#14519735C2212yes—
24C2×C42×D61C2211yescharacteristic
25#16840053—24no—
26#16840039—24no—
27C2×#14519710C2212yes—
28C2×#14519709C2212yes—
29C2×C4⋊#92418C2211yescharacteristic
30C2×#14519699C2212yes—
31C2×#14519697C2212yes—
32C2×#14519689C2212yes—
33#16839663C2211yescharacteristic
34#16839661C2211yescharacteristic
35#16839654—21no—
36#16839640—21no—
37#16839619C2211yescharacteristic
38C4×#11758198—21no—
39#16839524C2212yes—
40#16839524C2212yes—
41#16839516C2212yes—
42C2×C4⋊Q8×C61C2211yescharacteristic
43C2×C4⋊C8×C61C2211yescharacteristic
44C4×C4⋊C4×C61C2211yescharacteristic
45C61×#340C2212yes—
46C61×#340C2212yes—
47C22×Dic122—21no—
48C22×D244—21no—
49C22×D244C2311yescharacteristic
50C22×C4×D61—21no—
51C22×C4×D61D411yescharacteristic
52C22×C4×D61D411yescharacteristic
53C22×C4×D61C2311yescharacteristic
54C22×#92418C2312yes—
55#14519777—22no—
56#14519777D414yescomplemented
57#14519768—24no—
58#14519768—24no—
59#14519757—28no—
60#14519753—28no—
61#14519752—22no—
62#14519752Q814yescomplemented
63Q8×Dic61—28no—
64#14519742—28no—
65C4×D244—22no—
66C4×D244—22no—
67C4×D244—44no—
68#14519735—24no—
69#14519735—24no—
70C2×C4⋊Dic61—21no—
71C2×C4⋊Dic61—22no—
72C2×C4⋊Dic61—21no—
73C2×C4⋊Dic61C2311yescharacteristic
74C2×C4⋊Dic61C2312yes—
75C2×#11758196—22no—
76C2×#11758198—22no—
77C2×#11758198—21no—
78C2×#11758198—21no—
79C2×#11758198—21no—
80C2×#11758198—21no—
81C2×#11758198—21no—
82C2×C4×Dic61—22no—
83C2×C4×Dic61D411yescharacteristic
84C2×C4×Dic61D411yescharacteristic
85C2×C4×Dic61C2311yescharacteristic
86C42×D61—22no—
87C42×D61—22no—
88#14519710—24no—
89#14519709—24no—
90C4⋊#92418—22no—
91#14519699—24no—
92#14519697—24no—
93#14519689—24no—
94#14519684—21no—
95#14519684—21no—
96C22×Q8×C61—21no—
97C22×C8×C61C2312yes—
98C4⋊Q8×C61—22no—
99C4⋊Q8×C61D414yescomplemented
100C2×C4⋊C4×C61—21no—
101C2×C4⋊C4×C61C2311yescharacteristic
102C2×C4⋊C4×C61C2312yes—
103C2×C4⋊C4×C61C2311yescharacteristic
104C2×C42×C61—21no—
105C2×C42×C61C2311yescharacteristic
106C4⋊C8×C61—22no—
107#95×C61—21no—
108C23×D61—21no—
109C23×D61C2×D411yescharacteristic
110C2×Q8×C61—24no—
111C2×Q8×C61—42no—
112C2×Q8×C61—42no—
113C22×C4×C61—21no—
114C22×C4×C61—22no—
115C22×C4×C61—21no—
116C22×C4×C61C2×D411yescharacteristic
117C22×C4×C61C2×D411yescharacteristic
118C22×C4×C61C2411yescharacteristic
119C22×Dic61—22no—
120C22×Dic61—21no—
121C22×Dic61C2×D411yescharacteristic
122C22×Dic61C2×D411yescharacteristic
123C22×Dic61C2×D411yescharacteristic
124C2×D244—21no—
125C2×D244—21no—
126C2×D244—21no—
127C2×D244—21no—
128C2×D244—24no—
129C2×D244—21no—
130C2×D244—42no—
131C2×D244—42no—
132C2×D244C2×Q812yes—
133C2×C4×D61—21no—
134C2×C4×D61—21no—
135C2×C4×D61—21no—
136C2×C4×D61—21no—
137C2×C4×D61—22no—
138C2×C4×D61—22no—
139C2×C4×D61—22no—
140C2×C4×D61—22no—
141C2×C4×D61—41no—
142C2×C4×D61—41no—
143C2×C4×D61—42no—
144C2×C4×D61—42no—
145C2×C4×D61—42no—
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×C61—21no—
159C4⋊C4×C61C2×D414yes—
160C4⋊C4×C61C2×Q812yes—
161C42×C61—42no—
162C42×C61C2×D412yes—
163C42×C61C2×D412yes—
164#11758198—28no—
165#11758198—24no—
166#11758198—42no—
167#11758198—42no—
168#11758198—42no—
169#11758198—42no—
170C4⋊Dic61—28no—
171C4⋊Dic61—24no—
172C4⋊Dic61—22no—
173C4⋊Dic61—21no—
174C4⋊Dic61—21no—
175C4⋊Dic61—42no—
176C4⋊Dic61C2×D414yes—
177#11758196—28no—
178C4×Dic61—28no—
179C4×Dic61—22no—
180C4×Dic61—22no—
181C4×Dic61—21no—
182C4×Dic61—21no—
183C2×#92418—22no—
184C2×#92418—22no—
185C2×#92418—22no—
186C23×C61C22×D411yescharacteristic
187C22×D61—21no—
188C22×D61—21no—
189C22×D61—21no—
190C22×D61—21no—
191C22×D61—21no—
192C22×D61—21no—
193C22×D61—22no—
194C22×D61—41no—
195C22×D61—41no—
196C22×D61—41no—
197C22×D61—41no—
198C22×D61—41no—
199C22×D61—41no—
200C22×D61C22⋊Q812yes—
201Q8×C61—48no—
202C2×C4×C61—21no—
203C2×C4×C61—21no—
204C2×C4×C61—24no—
205C2×C4×C61—22no—
206C2×C4×C61—42no—
207C2×C4×C61—41no—
208C2×C4×C61—42no—
209C2×C4×C61—41no—
210C2×C4×C61—41no—
211C2×C4×C61—41no—
212C2×C4×C61—41no—
213C2×C4×C61#12012yes—
214C2×C4×C61#12012yes—
215C2×C4×C61#13512yes—
216C2×C4×C61AGL(1,ℤ/8)12yes—
217C2×C4×C61C22×D411yescharacteristic
218C2×C4×C61C22×D411yescharacteristic
219C2×C4×C61C22×Q811yescharacteristic
220C2×C4×C61C2×#4111yescharacteristic
221C2×C4×C612−1+411yescharacteristic
222C2×C4×C612−1+411yescharacteristic
223C2×Dic61—21no—
224C2×Dic61—21no—
225C2×Dic61—21no—
226C2×Dic61—21no—
227C2×Dic61—24no—
228C2×Dic61—21no—
229C2×Dic61—21no—
230C2×Dic61—21no—
231C2×Dic61—21no—
232C2×Dic61—42no—
233C2×Dic61—42no—
234C2×Dic61—41no—
235C2×Dic61—41no—
236C2×Dic61—41no—
237C2×Dic61C22⋊Q812yes—
238D244—22no—
239D244—22no—
240D244—48no—
241D244—41no—
242D244—41no—
243C4×D61—22no—
244C4×D61—22no—
245C4×D61—22no—
246C4×D61—22no—
247C4×D61—44no—
248C4×D61—44no—
249C4×D61—84no—
250Dic122—48no—
251C8×C61—42no—
252#92418—42no—
253#24300—611nomaximal, Sylow
254C22×C61C2×#12011yescharacteristic
255C22×C61C2×C22⋊Q811yescharacteristic
256C22×C61#53611yescharacteristic
257C22×C61C2×#13511yescharacteristic
258C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
259C22×C61#57511yescharacteristic
260C22×C61#57511yescharacteristic
261C2×D61—22no—
262C2×D61—22no—
263C2×D61—22no—
264C2×D61—41no—
265C2×D61—41no—
266C2×D61—41no—
267C2×D61—41no—
268C2×D61—42no—
269C2×D61—42no—
270C2×D61—42no—
271C2×D61—81no—
272C2×D61—81no—
273C2×D61—81no—
274C2×D61—81no—
275C4×C61—22no—
276C4×C61—42no—
277C4×C61—44no—
278C4×C61—81no—
279C4×C61#44914yes—
280C4×C61#54811yescharacteristic
281C4×C61#54811yescharacteristic
282C4×C61D4×Q811yescharacteristic
283C4×C61D4×Q811yescharacteristic
284Dic61—22no—
285Dic61—41no—
286Dic61—44no—
287Dic61—41no—
288Dic61—81no—
289C2×#533—611no—
290C2×#476—611no—
291C2×#474—611no—
292C4×#121—611no—
293#2059—612no—
294#2025—612no—
295#1988D6111yescharacteristic, complemented
296#1987—611no—
297#1875—612no—
298#1854—612no—
299#1797—611no—
300C2×C61#240911yescharacteristic
301C2×C61C2×#44911yescharacteristic
302C2×C61#300511yescharacteristic
303C2×C61#307611yescharacteristic
304C2×C61#307711yescharacteristic
305C2×C61#323311yescharacteristic
306C2×C61#323511yescharacteristic
307D61—42no—
308D61—82no—
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—611no—
320C2×C4×D4—1221no—
321C2×#117—612no—
322C22×C42—611no—
323#476—1224no—
324#474—1224no—
325C2×#107—612no—
326C2×#106—612no—
327C2×C4⋊C8—611no—
328C2×C4⋊C8C2×D6111yescharacteristic
329C2×#102—612no—
330C2×#102—612no—
331C2×C22⋊C8—612no—
332#390—611no—
333#389—611no—
334#388—1221no—
335#386—1221no—
336#380—611no—
337C4×C4⋊C4C2×D6111yescharacteristic
338C4×C22⋊C4—1221no—
339#340—612no—
340#340—612no—
341#340C2×D6112yes—
342#340C2×D6112yes—
343#336—612no—
344C61#2430011yescharacteristic, complemented, Sylow
345C22×Q8—21no—
346C22×Q8—1221no—
347C22×D4—611no—
348C22×D4—1221no—
349C23×C4—611no—
350C23×C4—611no—
351C23×C4—611no—
352C23×C4—1221no—
353C22×C8—612no—
354C22×C8C22×D6112yes—
355C4⋊Q8—22no—
356C4⋊Q8—614no—
357C4⋊Q8—1222no—
358C4⋊Q8D24414yescomplemented
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—1222no—
368C4×D4—1222no—
369C4×D4—2444no—
370#117—1224no—
371#117—1224no—
372C2×C4⋊C4—21no—
373C2×C4⋊C4—612no—
374C2×C4⋊C4—611no—
375C2×C4⋊C4—1222no—
376C2×C4⋊C4—1222no—
377C2×C4⋊C4—1221no—
378C2×C4⋊C4—1221no—
379C2×C4⋊C4C22×D6111yescharacteristic
380C2×C4⋊C4C22×D6112yes—
381C2×C4⋊C4C22×D6111yescharacteristic
382C2×C22⋊C4—1221no—
383C2×C22⋊C4—1222no—
384C2×C22⋊C4—1221no—
385C2×C22⋊C4—1221no—
386C2×C22⋊C4—1221no—
387C2×C22⋊C4—1221no—
388C2×C42—21no—
389C2×C42—611no—
390C2×C42—611no—
391C2×C42—611no—
392C2×C42—1222no—
393C2×C42—1222no—
394C2×C42—1222no—
395C2×C42C22×D6111yescharacteristic
396#107—1224no—
397#106—1224no—
398C4⋊C8—22no—
399C4⋊C8—1222no—
400#102—1224no—
401#102—1224no—
402C22⋊C8—1224no—
403#95—21no—
404#95—1221no—
405#95—1221no—
406C24—611no—
407C24—1221no—
408C2×Q8—24no—
409C2×Q8—42no—
410C2×Q8—42no—
411C2×Q8—1224no—
412C2×Q8—2442no—
413C2×Q8—2442no—
414C2×D4—612no—
415C2×D4—1224no—
416C2×D4—1221no—
417C2×D4—1221no—
418C2×D4—1221no—
419C2×D4—1221no—
420C2×D4—1221no—
421C2×D4—2442no—
422C2×D4—2442no—
423C22×C4—21no—
424C22×C4—22no—
425C22×C4—21no—
426C22×C4—611no—
427C22×C4—612no—
428C22×C4—612no—
429C22×C4—611no—
430C22×C4—611no—
431C22×C4—1221no—
432C22×C4—1221no—
433C22×C4—1221no—
434C22×C4—1221no—
435C22×C4—1222no—
436C22×C4—1222no—
437C22×C4—1222no—
438C22×C4—1222no—
439C22×C4—1222no—
440C22×C4—1221no—
441C22×C4—2441no—
442C22×C4—2441no—
443C22×C4—2442no—
444C22×C4—2442no—
445C22×C4—2442no—
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—21no—
457C4⋊C4—24no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—614no—
461C4⋊C4—1228no—
462C4⋊C4—1228no—
463C4⋊C4—1222no—
464C4⋊C4—1224no—
465C4⋊C4—1221no—
466C4⋊C4—1221no—
467C4⋊C4—2442no—
468C4⋊C4C2×D24414yes—
469C4⋊C4Q8×D6112yes—
470C22⋊C4—1224no—
471C22⋊C4—1228no—
472C22⋊C4—2442no—
473C22⋊C4—2442no—
474C22⋊C4—2442no—
475C22⋊C4—2442no—
476C42—42no—
477C42—1228no—
478C42—1221no—
479C42—1221no—
480C42—1222no—
481C42—1222no—
482C42C2×D24412yes—
483C42D4×D6112yes—
484C23—612no—
485C23—1221no—
486C23—1221no—
487C23—1221no—
488C23—1221no—
489C23—1221no—
490C23—1221no—
491C23—1222no—
492C23—2441no—
493C23—2441no—
494C23—2441no—
495C23—2441no—
496C23—2441no—
497C23—2441no—
498C23C2×D4×D6111yescharacteristic
499Q8—48no—
500Q8—2448no—
501D4—1222no—
502D4—1222no—
503D4—2448no—
504D4—2441no—
505D4—2441no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—24no—
509C2×C4—22no—
510C2×C4—42no—
511C2×C4—41no—
512C2×C4—42no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—612no—
518C2×C4—1221no—
519C2×C4—1221no—
520C2×C4—1222no—
521C2×C4—1222no—
522C2×C4—1222no—
523C2×C4—1222no—
524C2×C4—1221no—
525C2×C4—1221no—
526C2×C4—1224no—
527C2×C4—1221no—
528C2×C4—1221no—
529C2×C4—1221no—
530C2×C4—1221no—
531C2×C4—4884no—
532C2×C4—2444no—
533C2×C4—2441no—
534C2×C4—2441no—
535C2×C4—2441no—
536C2×C4—2442no—
537C2×C4—2442no—
538C2×C4—2444no—
539C2×C4#1451974712yes—
540C2×C4#1451974712yes—
541C2×C4#1451979812yes—
542C2×C4#1451980112yes—
543C2×C4C2×D4×D6111yescharacteristic
544C2×C4C22×D24411yescharacteristic
545C2×C4C2×Q8×D6111yescharacteristic
546C2×C4#41×D6111yescharacteristic
547C2×C4#1451982411yescharacteristic
548C2×C4#1451982511yescharacteristic
549C8—42no—
550C8—2442no—
551C22—1222no—
552C22—1222no—
553C22—1222no—
554C22—4881no—
555C22—4881no—
556C22—4881no—
557C22—4881no—
558C22—2442no—
559C22—2442no—
560C22—2442no—
561C22—2441no—
562C22—2441no—
563C22—2441no—
564C22—2441no—
565C22C2×#1451974711yescharacteristic
566C22C22⋊Q8×D6111yescharacteristic
567C22#1684033511yescharacteristic
568C22C2×#1451979811yescharacteristic
569C22C2×#1451980111yescharacteristic
570C22#1684056111yescharacteristic
571C22#1684056111yescharacteristic
572C4—22no—
573C4—44no—
574C4—42no—
575C4—81no—
576C4—1222no—
577C4—4881no—
578C4—2441no—
579C4—2441no—
580C4—2444no—
581C4#1683990814yes—
582C4#1684042011yescharacteristic
583C4#1684042411yescharacteristic
584C4#1684042711yescharacteristic
585C4Q8×D24411yescharacteristic
586C2—4882no—
587C2—2442no—
588C2#2144918211yescharacteristic
589C2C2×#1683990811yescharacteristic
590C2#2145506511yescharacteristic
591C2#2145560211yescharacteristic
592C2#2145561111yescharacteristic
593C2#2145700211yescharacteristic
594C2#2145701611yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1988C2
socleC23×C61C22×D4
prime radicalC2×#14519752C22
hypercenter#1988D61
hypocenterC61#24300
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1988

Complements

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

NH ≅ G/Ntype
1. C2×#16840317C2semidirect
2. C2×#16840053C2semidirect
3. C2×#16840039C2semidirect
5. #21447156C2semidirect
6. #21446964C2semidirect
7. #21446767C2semidirect
9. #21446019C2semidirect
10. #21445796C2semidirect
11. C61×#1988C2semidirect
56. #14519777D4semidirect
62. #14519752Q8semidirect
99. C4⋊Q8×C61D4semidirect
295. #1988D61semidirect
344. C61#24300semidirect
358. C4⋊Q8D244semidirect

Maximal subgroups

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

subgroupindex
C2×#168403172
C2×#168400532
C2×#168400392
C4×#145197522
#214471562
#214469642
#214467672
#214461402
#214460192
#214457962
C61×#19882
#2430061