Finite Group Databasebeta

← #29716095

Subgroups — #29716095

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×#16840319C211yesmaximal, characteristic, complemented
2C2×#16840053C211yesmaximal, characteristic, complemented
3C2×#16840043C211yesmaximal, characteristic, complemented
4C4×#14519751C211yesmaximal, characteristic
5#21447158C212yesmaximal, complemented
6#21446966C212yesmaximal, complemented
7#21446770C211yesmaximal, characteristic, complemented
8#21446766C211yesmaximal, characteristic, complemented
9#21446139C212yesmaximal
10#21446020C212yesmaximal, complemented
11C61×#1797C211yesmaximal, characteristic, complemented
12#16840319—24no—
13#16840319—24no—
14C2×#14519777C2211yescharacteristic
15C2×#14519777C2211yescharacteristic
16C2×#14519769C2212yes—
17C2×#14519756—21no—
18C2×#14519756—21no—
19C2×#14519751C2211yescharacteristic
20C2×C4×Dic122—21no—
21C2×C4×Dic122—21no—
22C2×C4×D244C2211yescharacteristic
23C2×C4×C22⋊D61—21no—
24C2×#14519736C2212yes—
25#16840053—24no—
26#16840043—24no—
27C2×#14519711C2212yes—
28C2×#14519709C2212yes—
29C2×#14519699C2212yes—
30C2×#14519699C2212yes—
31#16839664C2211yescharacteristic
32#16839660C2211yescharacteristic
33#16839653—21no—
34#16839641—21no—
35#16839622C2211yescharacteristic
36C4×C4⋊Dic61C2211yescharacteristic
37C4×#11758198—21no—
38#16839525C2212yes—
39#16839525C2212yes—
40#16839525C2212yes—
41#16839525C2212yes—
42C22×C42×C61C2211yescharacteristic
43C2×C4⋊C8×C61C2211yescharacteristic
44C2×C4⋊C8×C61C2211yescharacteristic
45C2×C22⋊C8×C61C2212yes—
46C61×#336C2212yes—
47C22×Dic122—21no—
48C22×Dic122—21no—
49C22×D244C2311yescharacteristic
50C22×C22⋊D61—21no—
51C22×C4×D61—21no—
52#14519777—22no—
53#14519777—22no—
54#14519777D414yescomplemented
55#14519777D414yescomplemented
56#14519769—24no—
57#14519769—24no—
58#14519756—28no—
59#14519756—28no—
60#14519751—22no—
61#14519751Q814yescomplemented
62C4×Dic122—28no—
63C4×Dic122—28no—
64C4×D244—22no—
65C4×D244—22no—
66C4×C22⋊D61—44no—
67#14519736—24no—
68#14519736—24no—
69C2×C4⋊Dic61—21no—
70C2×C4⋊Dic61—21no—
71C2×C4⋊Dic61C2311yescharacteristic
72C2×C4⋊Dic61C2311yescharacteristic
73C2×C4⋊Dic61C2312yes—
74C2×C4⋊Dic61C2312yes—
75C2×C4⋊Dic61C2311yescharacteristic
76C2×#11758196—22no—
77C2×#11758196—22no—
78C2×#11758196—21no—
79C2×C22⋊Dic61—22no—
80C2×C22⋊Dic61—21no—
81C2×#11758198—21no—
82C2×#11758198—21no—
83C2×#11758198—21no—
84C2×#11758198—21no—
85C2×C4×Dic61—22no—
86C2×C4×Dic61—21no—
87#14519711—24no—
88#14519709—24no—
89#14519699—24no—
90#14519699—24no—
91#14519684—21no—
92#14519684—21no—
93#14519684—21no—
94C23×C4×C61D411yescharacteristic
95C23×C4×C61D411yescharacteristic
96C23×C4×C61C2311yescharacteristic
97C22×C8×C61C2312yes—
98C22×C8×C61C2312yes—
99C2×C42×C61—22no—
100C2×C42×C61—22no—
101C2×C42×C61D411yescharacteristic
102C2×C42×C61D411yescharacteristic
103C2×C42×C61C2311yescharacteristic
104C2×C42×C61C2311yescharacteristic
105C4⋊C8×C61—22no—
106C4⋊C8×C61—22no—
107C22⋊C8×C61—24no—
108C24×C61C2×D411yescharacteristic
109C23×D61—21no—
110C22×C4×C61—21no—
111C22×C4×C61—21no—
112C22×C4×C61—21no—
113C22×C4×C61—21no—
114C22×C4×C61—22no—
115C22×C4×C61—22no—
116C22×C4×C61—22no—
117C22×C4×C61—22no—
118C22×C4×C61—41no—
119C22×C4×C61—41no—
120C22×C4×C61C2×D411yescharacteristic
121C22×C4×C61C2×D411yescharacteristic
122C22×C4×C61C2×D411yescharacteristic
123C22×C4×C61C2×D411yescharacteristic
124C22×C4×C61C2×D411yescharacteristic
125C22×C4×C61C2×Q812yes—
126C22×C4×C61#4112yes—
127C22×C4×C61C2411yescharacteristic
128C2×C22⋊D61—24no—
129C2×C22⋊D61—42no—
130C2×C22⋊D61—42no—
131C22×Dic61—21no—
132C22×Dic61—22no—
133C22×Dic61—22no—
134C22×Dic61—21no—
135C22×Dic61—21no—
136C2×D244—21no—
137C2×D244—21no—
138C2×D244—21no—
139C2×D244—21no—
140C2×D244—21no—
141C2×D244C2×Q812yes—
142C2×C4×D61—42no—
143C2×C4×D61—42no—
144C2×C4×D61—42no—
145C2×Dic122—24no—
146C2×Dic122—24no—
147C2×Dic122—42no—
148C2×Dic122—42no—
149C2×Dic122—42no—
150C2×Dic122—42no—
151C2×C8×C61—22no—
152C2×C8×C61—22no—
153C2×C8×C61—22no—
154C2×C8×C61—22no—
155C2×C8×C61—22no—
156C2×C8×C61—22no—
157C42×C61—22no—
158C42×C61—22no—
159C42×C61—21no—
160C42×C61—21no—
161C42×C61C2×D412yes—
162C42×C61C2×D412yes—
163C22⋊Dic61—28no—
164C22⋊Dic61—42no—
165#11758198—24no—
166#11758198—42no—
167#11758198—42no—
168#11758198—42no—
169C4⋊Dic61—24no—
170C4⋊Dic61—24no—
171C4⋊Dic61—22no—
172C4⋊Dic61—22no—
173C4⋊Dic61—21no—
174C4⋊Dic61—21no—
175C4⋊Dic61—21no—
176C4⋊Dic61—21no—
177C4⋊Dic61—21no—
178C4⋊Dic61C2×D414yes—
179C4⋊Dic61C2×D414yes—
180C4⋊Dic61C2×Q812yes—
181#11758196—28no—
182#11758196—28no—
183#11758196—42no—
184C4×Dic61—28no—
185C4×Dic61—42no—
186C23×C61—21no—
187C23×C61—21no—
188C23×C61—21no—
189C23×C61—21no—
190C23×C61—21no—
191C23×C61—21no—
192C23×C61C22⋊Q812yes—
193C23×C61C22×D411yescharacteristic
194C22×D61—22no—
195C22×D61—41no—
196C22×D61—41no—
197C22×D61—41no—
198C22×D61—41no—
199C22×D61—41no—
200C22×D61—41no—
201C2×C4×C61—21no—
202C2×C4×C61—21no—
203C2×C4×C61—21no—
204C2×C4×C61—21no—
205C2×C4×C61—22no—
206C2×C4×C61—22no—
207C2×C4×C61—22no—
208C2×C4×C61—22no—
209C2×C4×C61—21no—
210C2×C4×C61—21no—
211C2×C4×C61—21no—
212C2×C4×C61—21no—
213C2×C4×C61—21no—
214C2×C4×C61—21no—
215C2×C4×C61—44no—
216C2×C4×C61—44no—
217C2×C4×C61#12012yes—
218C2×C4×C61#12012yes—
219C2×C4×C61C22⋊Q812yes—
220C2×C4×C61#13512yes—
221C2×C4×C61AGL(1,ℤ/8)12yes—
222C2×C4×C61C22×D411yescharacteristic
223C2×C4×C61C22×D411yescharacteristic
224C2×C4×C61C22×Q811yescharacteristic
225C2×C4×C61C2×#4111yescharacteristic
226C2×C4×C612−1+411yescharacteristic
227C2×C4×C612−1+411yescharacteristic
228C22⋊D61—48no—
229C2×Dic61—24no—
230C2×Dic61—24no—
231C2×Dic61—22no—
232C2×Dic61—42no—
233C2×Dic61—41no—
234C2×Dic61—42no—
235C2×Dic61—42no—
236C2×Dic61—42no—
237C2×Dic61—41no—
238C2×Dic61—41no—
239C2×Dic61—41no—
240C2×Dic61—41no—
241C2×Dic61—41no—
242C2×Dic61—41no—
243C2×Dic61—41no—
244D244—22no—
245D244—22no—
246D244—41no—
247D244—41no—
248C4×D61—84no—
249Dic122—48no—
250Dic122—48no—
251C8×C61—42no—
252C8×C61—42no—
253#24300—611nomaximal, Sylow
254C22×C61—22no—
255C22×C61—22no—
256C22×C61—22no—
257C22×C61—41no—
258C22×C61—41no—
259C22×C61—41no—
260C22×C61—41no—
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—42no—
269C2×D61—42no—
270C2×D61—42no—
271C2×D61—81no—
272C2×D61—81no—
273C2×D61—81no—
274C2×D61—81no—
275C4×C61—22no—
276C4×C61—22no—
277C4×C61—41no—
278C4×C61—41no—
279C4×C61#44914yes—
280C4×C61#54811yescharacteristic
281C4×C61#54811yescharacteristic
282C4×C61D4×Q811yescharacteristic
283C4×C61D4×Q811yescharacteristic
284Dic61—42no—
285Dic61—44no—
286Dic61—44no—
287Dic61—81no—
288Dic61—81no—
289C2×#533—611no—
290C2×#476—611no—
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#1797D6111yescharacteristic, complemented
300C2×C61—42no—
301C2×C61#240911yescharacteristic
302C2×C61C2×#44911yescharacteristic
303C2×C61#300511yescharacteristic
304C2×C61#307611yescharacteristic
305C2×C61#307711yescharacteristic
306C2×C61#323311yescharacteristic
307C2×C61#323511yescharacteristic
308D61—82no—
309#533—1224no—
310#533—1224no—
311C2×C4⋊Q8—611no—
312C2×C4⋊Q8—611no—
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×C42C2×D6111yescharacteristic
323#476—1224no—
324#474—1224no—
325C2×#107—612no—
326C2×#106—612no—
327C2×C4⋊C8C2×D6111yescharacteristic
328C2×C4⋊C8C2×D6111yescharacteristic
329C2×#102—612no—
330C2×#102—612no—
331C2×C22⋊C8C2×D6112yes—
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#336C2×D6112yes—
344C61#2430011yescharacteristic, complemented, Sylow
345C22×Q8—1221no—
346C22×Q8—1221no—
347C22×D4—611no—
348C22×D4—1221no—
349C23×C4—1221no—
350C23×C4D24411yescharacteristic
351C23×C4D24411yescharacteristic
352C23×C4C22×D6111yescharacteristic
353C22×C8C22×D6112yes—
354C22×C8C22×D6112yes—
355C4⋊Q8—614no—
356C4⋊Q8—614no—
357C4⋊Q8—1222no—
358C4⋊Q8—1222no—
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—612no—
373C2×C4⋊C4—612no—
374C2×C4⋊C4—611no—
375C2×C4⋊C4—611no—
376C2×C4⋊C4—611no—
377C2×C4⋊C4—1222no—
378C2×C4⋊C4—1222no—
379C2×C4⋊C4—1221no—
380C2×C4⋊C4—1221no—
381C2×C4⋊C4—1221no—
382C2×C22⋊C4—1222no—
383C2×C22⋊C4—1221no—
384C2×C22⋊C4—1221no—
385C2×C22⋊C4—1221no—
386C2×C22⋊C4—1221no—
387C2×C22⋊C4—1221no—
388C2×C42—22no—
389C2×C42—22no—
390C2×C42—1222no—
391C2×C42—1221no—
392C2×C42D24411yescharacteristic
393C2×C42D24411yescharacteristic
394C2×C42C22×D6111yescharacteristic
395C2×C42C22×D6111yescharacteristic
396#107—1224no—
397#106—1224no—
398C4⋊C8—22no—
399C4⋊C8—22no—
400#102—1224no—
401#102—1224no—
402C22⋊C8—24no—
403#95—1221no—
404#95—1221no—
405#95—1221no—
406C24—1221no—
407C24C2×D24411yescharacteristic
408C2×Q8—1224no—
409C2×Q8—1224no—
410C2×Q8—2442no—
411C2×Q8—2442no—
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—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—41no—
432C22×C4—41no—
433C22×C4—1221no—
434C22×C4—1222no—
435C22×C4—1222no—
436C22×C4—1221no—
437C22×C4—1221no—
438C22×C4—2442no—
439C22×C4—2442no—
440C22×C4—2442no—
441C22×C4C2×D24411yescharacteristic
442C22×C4C2×D24411yescharacteristic
443C22×C4C2×D24411yescharacteristic
444C22×C4C2×D24411yescharacteristic
445C22×C4C2×D24411yescharacteristic
446C22×C4#1175822412yes—
447C22×C4Q8×D6112yes—
448C22×C4C23×D6111yescharacteristic
449C2×C8—22no—
450C2×C8—22no—
451C2×C8—22no—
452C2×C8—22no—
453C2×C8—22no—
454C2×C8—22no—
455C4⋊C4—614no—
456C4⋊C4—614no—
457C4⋊C4—612no—
458C4⋊C4—1228no—
459C4⋊C4—1228no—
460C4⋊C4—1222no—
461C4⋊C4—1222no—
462C4⋊C4—1221no—
463C4⋊C4—1224no—
464C4⋊C4—1224no—
465C4⋊C4—1221no—
466C4⋊C4—1221no—
467C4⋊C4—1221no—
468C4⋊C4—1221no—
469C4⋊C4—2442no—
470C22⋊C4—1228no—
471C22⋊C4—1224no—
472C22⋊C4—2442no—
473C22⋊C4—2442no—
474C22⋊C4—2442no—
475C22⋊C4—2442no—
476C42—21no—
477C42—21no—
478C42—22no—
479C42—22no—
480C42—1228no—
481C42—2442no—
482C42D4×D6112yes—
483C42D4×D6112yes—
484C23—21no—
485C23—21no—
486C23—21no—
487C23—21no—
488C23—21no—
489C23—21no—
490C23—1222no—
491C23—2441no—
492C23—2441no—
493C23—2441no—
494C23—2441no—
495C23—2441no—
496C23—2441no—
497C23#1451975712yes—
498C23C22×D24411yescharacteristic
499Q8—2448no—
500Q8—2448no—
501D4—1222no—
502D4—1222no—
503D4—2448no—
504D4—2441no—
505D4—2441no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—44no—
521C2×C4—44no—
522C2×C4—1224no—
523C2×C4—1224no—
524C2×C4—1222no—
525C2×C4—4884no—
526C2×C4—2441no—
527C2×C4—2441no—
528C2×C4—2441no—
529C2×C4—2441no—
530C2×C4—2441no—
531C2×C4—2441no—
532C2×C4—2442no—
533C2×C4—2441no—
534C2×C4—2442no—
535C2×C4—2441no—
536C2×C4—2442no—
537C2×C4—2442no—
538C2×C4#1451974712yes—
539C2×C4#1451974712yes—
540C2×C4#1451975712yes—
541C2×C4#1451979912yes—
542C2×C4#1451980512yes—
543C2×C4C2×D4×D6111yescharacteristic
544C2×C4C2×D4×D6111yescharacteristic
545C2×C4C2×Q8×D6111yescharacteristic
546C2×C4C2×#1175822411yescharacteristic
547C2×C4#1451982411yescharacteristic
548C2×C4#1451982411yescharacteristic
549C8—42no—
550C8—42no—
551C22—22no—
552C22—22no—
553C22—22no—
554C22—41no—
555C22—41no—
556C22—41no—
557C22—41no—
558C22—4881no—
559C22—4881no—
560C22—4881no—
561C22—4881no—
562C22—2442no—
563C22—2442no—
564C22—2442no—
565C22C2×#1451974711yescharacteristic
566C22C2×#1451975711yescharacteristic
567C22#1684033811yescharacteristic
568C22C2×#1451979911yescharacteristic
569C22C2×#1451980511yescharacteristic
570C22#1684056211yescharacteristic
571C22#1684056211yescharacteristic
572C4—22no—
573C4—22no—
574C4—41no—
575C4—41no—
576C4—4881no—
577C4—4881no—
578C4—2442no—
579C4—2444no—
580C4—2444no—
581C4#1683990914yes—
582C4#1684041811yescharacteristic
583C4#1684041811yescharacteristic
584C4#1684042511yescharacteristic
585C4#1684042511yescharacteristic
586C2—42no—
587C2—4882no—
588C2#2144918511yescharacteristic
589C2C2×#1683990911yescharacteristic
590C2#2145506411yescharacteristic
591C2#2145560911yescharacteristic
592C2#2145561811yescharacteristic
593C2#2145700011yescharacteristic
594C2#2145701911yescharacteristic
subgroupG/H
centerC23C22×D244
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1797C2
socleC23×C61C22×D4
prime radicalC2×#14519751C22
hypercenter#1797D61
hypocenterC61#24300
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1797

Complements

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

NH ≅ G/Ntype
1. C2×#16840319C2semidirect
2. C2×#16840053C2semidirect
3. C2×#16840043C2semidirect
5. #21447158C2semidirect
6. #21446966C2semidirect
7. #21446770C2semidirect
8. #21446766C2semidirect
10. #21446020C2semidirect
11. C61×#1797C2semidirect
54. #14519777D4semidirect
55. #14519777D4semidirect
61. #14519751Q8semidirect
299. #1797D61semidirect
344. C61#24300semidirect

Maximal subgroups

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

subgroupindex
C2×#168403192
C2×#168400532
C2×#168400432
C4×#145197512
#214471582
#214469662
#214467702
#214467662
#214461392
#214460202
C61×#17972
#2430061