Finite Group Databasebeta

← #29690271

Subgroups — #29690271

Order 15616.

1266 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C22⋊C8×D61C211yesmaximal, characteristic
2C2×#16839619C211yesmaximal, characteristic
3#380×D61C211yesmaximal, characteristic
4#21446014C211yesmaximal, characteristic, complemented
5#21446013C212yesmaximal, complemented
6#21446012C211yesmaximal, characteristic, complemented
7#21446011C211yesmaximal, characteristic, complemented
8#21446010C212yesmaximal, complemented
9#21445843C212yesmaximal, complemented
10#21445841C212yesmaximal, complemented
11C61×#1853C211yesmaximal, characteristic, complemented
12C23×C4×D61C2211yescharacteristic
13C22×C8×D61C2212yes—
14C22×C4⋊Dic61C2211yescharacteristic
15C2×C4⋊C4×D61C2211yescharacteristic
16C2×C4⋊C4×D61C2211yescharacteristic
17C22×#11758198—21no—
18C2×C22⋊C4×D61—21no—
19C2×#14519709C2212yescomplemented
20C2×#14519708C2212yescomplemented
21C2×#14519708C2212yescomplemented
22C2×C22⋊#92418C2211yescharacteristic
23C2×#14519689C2212yes—
24C22⋊C8×D61C418yescomplemented
25#16839622C2211yescharacteristic
26#16839621—21no—
27#16839620—21no—
28#16839619—21no—
29#16839619—21no—
30#16839619C2212yes—
31#16839619C2211yescharacteristic
32#16839619C2212yes—
33#16839619C2211yescharacteristic
34C2×#14519684—21no—
35#95×D61—21no—
36#16839525C2212yescomplemented
37#16839524C2212yescomplemented
38#16839524C2212yescomplemented
39C22×C4⋊C4×C61C2211yescharacteristic
40C2×#106×C61C2212yescomplemented
41C2×C22⋊C8×C61C2211yescharacteristic
42C61×#380C2211yescharacteristic
43C61×#340C2212yescomplemented
44C24×D61D411yescharacteristic
45C23×Dic61—21no—
46C23×Dic61D411yescharacteristic
47C22×C4×D61—21no—
48C22×C4×D61—21no—
49C22×C4×D61—42no—
50C22×C4×D61—42no—
51C22×C4×D61C2×C414yes—
52C22×C4×D61D411yescharacteristic
53C22×C4×D61D411yescharacteristic
54C22×C4×D61Q814yes—
55C22×C4×D61C2311yescharacteristic
56C22×C4×D61C2311yescharacteristic
57C22×#92418C2312yes—
58C2×C8×D61—42no—
59C2×C8×D61—42no—
60C2×C8×D61C2×C418yescomplemented
61C2×C4⋊Dic61—21no—
62C2×C4⋊Dic61—42no—
63C2×C4⋊Dic61C2311yescharacteristic
64C2×C4⋊Dic61C2311yescharacteristic
65C2×C4⋊Dic61C2312yes—
66C2×#11758196—21no—
67C2×#11758196—21no—
68C4⋊C4×D61—42no—
69C4⋊C4×D61—42no—
70C2×C22⋊Dic61—21no—
71C2×#11758198—21no—
72C2×#11758198—21no—
73C2×#11758198—22no—
74C2×#11758198—22no—
75C2×#11758198—44no—
76C22⋊C4×D61—44no—
77#14519709—42no—
78#14519708—42no—
79#14519708—42no—
80C22⋊#92418C2×C414yescomplemented
81#14519689C2×C418yescomplemented
82#14519684—21no—
83#14519684—21no—
84#14519684—21no—
85#14519684—22no—
86#14519684—22no—
87C23×C4×C61—21no—
88C23×C4×C61C2311yescharacteristic
89C22×C8×C61C2312yes—
90C2×C4⋊C4×C61—21no—
91C2×C4⋊C4×C61—42no—
92C2×C4⋊C4×C61C2311yescharacteristic
93C2×C4⋊C4×C61C2311yescharacteristic
94C2×C4⋊C4×C61C2312yes—
95C2×C22⋊C4×C61—21no—
96#106×C61—42no—
97C22⋊C8×C61C2×C414yescomplemented
98#95×C61—21no—
99C24×C61C2×D411yescharacteristic
100C23×D61—41no—
101C23×D61—41no—
102C23×D61—41no—
103C23×D61—41no—
104C23×D61C4⋊C414yes—
105C23×D61SD1614yes—
106C23×D61SD1614yes—
107C23×D61C2×D411yescharacteristic
108C23×D61C2×D411yescharacteristic
109C22×C4×C61—22no—
110C22×C4×C61—22no—
111C22×C4×C61—21no—
112C22×C4×C61—21no—
113C22×C4×C61—22no—
114C22×C4×C61—42no—
115C22×C4×C61—42no—
116C22×C4×C61—42no—
117C22×C4×C61C22×C412yes—
118C22×C4×C61C2×D411yescharacteristic
119C22×C4×C61C2×Q812yes—
120C22×C4×C61C2411yescharacteristic
121C22×Dic61—21no—
122C22×Dic61—21no—
123C22×Dic61—22no—
124C22×Dic61—41no—
125C22×Dic61—41no—
126C22×Dic61—42no—
127C22×Dic61—42no—
128C22×Dic61—42no—
129C22×Dic61C4⋊C414yes—
130C22×Dic61C2×D411yescharacteristic
131C22×Dic61C2×D411yescharacteristic
132C2×C4×D61—22no—
133C2×C4×D61—22no—
134C2×C4×D61—22no—
135C2×C4×D61—22no—
136C2×C4×D61—48no—
137C2×C4×D61—48no—
138C2×C4×D61—41no—
139C2×C4×D61—41no—
140C2×C4×D61—41no—
141C2×C4×D61—41no—
142C2×C4×D61—42no—
143C2×C4×D61—42no—
144C2×C4×D61—42no—
145C2×C4×D61—42no—
146C2×C4×D61—42no—
147C2×C4×D61—42no—
148C2×C4×D61C4⋊C414yes—
149C2×C4×D61C4⋊C414yes—
150C2×C4×D61C22×C412yes—
151C2×C4×D61C22×C412yes—
152C2×C4×D61C2×Q812yes—
153C2×C4×D61C2×Q812yes—
154C2×C8×C61—22no—
155C2×C8×C61—22no—
156C2×C8×C61C22×C414yes—
157C4⋊C4×C61—22no—
158C4⋊C4×C61—22no—
159C4⋊C4×C61—22no—
160C4⋊C4×C61—22no—
161C4⋊C4×C61—42no—
162C22⋊C4×C61—42no—
163C22⋊Dic61—42no—
164#11758198—42no—
165#11758198—42no—
166#11758198—44no—
167#11758198—44no—
168C4⋊Dic61—22no—
169C4⋊Dic61—22no—
170C4⋊Dic61—22no—
171C4⋊Dic61—22no—
172C4⋊Dic61—42no—
173#11758196—42no—
174#11758196—42no—
175C2×#92418—22no—
176C2×#92418—22no—
177C2×#92418C22×C414yes—
178C8×D61—48no—
179C23×C61—21no—
180C23×C61—21no—
181C23×C61—21no—
182C23×C61—21no—
183C23×C61C2×C4⋊C412yes—
184C23×C61C2×SD1612yes—
185C23×C61C2×SD1612yes—
186C23×C61C22×D411yescharacteristic
187C22×D61—21no—
188C22×D61—21no—
189C22×D61—21no—
190C22×D61—21no—
191C22×D61—21no—
192C22×D61—21no—
193C22×D61—21no—
194C22×D61—21no—
195C22×D61—44no—
196C22×D61—44no—
197C22×D61—44no—
198C22×D61—44no—
199C22×D61—44no—
200C22×D61—44no—
201C22×D61#106116yescomplemented
202C22×D61C2×C4⋊C412yes—
203C22×D61C2×C4⋊C412yes—
204C22×D61C2×SD1612yes—
205C22×D61C2×SD1612yes—
206C22×D61C2×SD1612yes—
207C22×D61C2×SD1612yes—
208C2×C4×C61—21no—
209C2×C4×C61—21no—
210C2×C4×C61—24no—
211C2×C4×C61—24no—
212C2×C4×C61—41no—
213C2×C4×C61—41no—
214C2×C4×C61—42no—
215C2×C4×C61—42no—
216C2×C4×C61—41no—
217C2×C4×C61—41no—
218C2×C4×C61—41no—
219C2×C4×C61—41no—
220C2×C4×C61—42no—
221C2×C4×C61—84no—
222C2×C4×C61C2×C4⋊C412yes—
223C2×C4×C61C23×C411yescharacteristic
224C2×C4×C61C22×Q811yescharacteristic
225C2×C4×C61H(5,𝔽2)12yes—
226C2×C4×C61H(5,𝔽2)12yes—
227C2×Dic61—21no—
228C2×Dic61—21no—
229C2×Dic61—21no—
230C2×Dic61—21no—
231C2×Dic61—44no—
232C2×Dic61—41no—
233C2×Dic61—41no—
234C2×Dic61—42no—
235C2×Dic61—42no—
236C2×Dic61—41no—
237C2×Dic61—41no—
238C2×Dic61—41no—
239C2×Dic61—41no—
240C2×Dic61—42no—
241C2×Dic61—84no—
242C2×Dic61C2×C4⋊C412yes—
243C2×Dic61C2×C4⋊C412yes—
244C4×D61—24no—
245C4×D61—24no—
246C4×D61—24no—
247C4×D61—24no—
248C4×D61—44no—
249C4×D61—44no—
250C4×D61—84no—
251C4×D61—84no—
252C8×C61—24no—
253#92418—24no—
254#21938—611nomaximal, Sylow
255C22×C61—22no—
256C22×C61—22no—
257C22×C61—22no—
258C22×C61—22no—
259C22×C61—22no—
260C22×C61—22no—
261C22×C61C2×#10614yes—
262C22×C61C22×C4⋊C411yescharacteristic
263C22×C61#53411yescharacteristic
264C22×C61#53411yescharacteristic
265C22×C61C22×SD1611yescharacteristic
266C22×C61C22×SD1611yescharacteristic
267C22×C61#57511yescharacteristic
268C22×C61#57511yescharacteristic
269C2×D61—22no—
270C2×D61—22no—
271C2×D61—22no—
272C2×D61—22no—
273C2×D61—22no—
274C2×D61—22no—
275C2×D61—22no—
276C2×D61—22no—
277C2×D61—22no—
278C2×D61—22no—
279C2×D61—22no—
280C2×D61—22no—
281C2×D61—416no—
282C2×D61C2×#10614yes—
283C2×D61C2×#10614yes—
284C4×C61—22no—
285C4×C61—81no—
286C4×C61—81no—
287C4×C61—82no—
288C4×C61#51812yes—
289C4×C61#54312yes—
290Dic61—22no—
291Dic61—22no—
292Dic61—81no—
293Dic61—81no—
294Dic61—82no—
295C22×C22⋊C8—611no—
296C2×#380—611no—
297C2×#380—611no—
298#1853—612no—
299#1853—611no—
300#1853—611no—
301#1853—612no—
302#1853—611no—
303#1853D6111yescharacteristic, complemented
304#1807—612no—
305#1807—612no—
306C2×C61—24no—
307C2×C61#231511yescharacteristic
308C2×C61C22×#10611yescharacteristic
309C2×C61#291111yescharacteristic
310C2×C61#318711yescharacteristic
311C2×C61#318711yescharacteristic
312C2×C61#318711yescharacteristic
313C2×C61#318711yescharacteristic
314D61—24no—
315D61—24no—
316C24×C4—611no—
317C23×C8—612no—
318C22×C4⋊C4—611no—
319C22×C4⋊C4—611no—
320C22×C4⋊C4—611no—
321C22×C4⋊C4C2×D6111yescharacteristic
322C22×C22⋊C4—1221no—
323C22×C22⋊C4—1221no—
324C2×#106—612no—
325C2×#106—612no—
326C2×#106—612no—
327C2×#106C2×D6112yescomplemented
328C2×C22⋊C8—611no—
329C2×C22⋊C8—612no—
330C2×C22⋊C8—618no—
331C2×C22⋊C8C2×D6111yescharacteristic
332#380—611no—
333#380—611no—
334#380—611no—
335#380—612no—
336#380—612no—
337#380—1221no—
338#380—1221no—
339#380—1221no—
340#380—1221no—
341#380C2×D6111yescharacteristic
342C2×#95—1221no—
343C2×#95—1221no—
344#340—612no—
345#340—612no—
346#340—612no—
347#340C2×D6112yescomplemented
348C61#2193811yescharacteristic, complemented, Sylow
349C25—611no—
350C23×C4—21no—
351C23×C4—611no—
352C23×C4—611no—
353C23×C4—611no—
354C23×C4—614no—
355C23×C4—611no—
356C23×C4—611no—
357C23×C4—614no—
358C23×C4—1221no—
359C23×C4—1221no—
360C23×C4—1221no—
361C23×C4—2442no—
362C23×C4—2442no—
363C23×C4C22×D6111yescharacteristic
364C22×C8—612no—
365C22×C8—618no—
366C22×C8—2442no—
367C22×C8—2442no—
368C22×C8C22×D6112yes—
369C2×C4⋊C4—21no—
370C2×C4⋊C4—42no—
371C2×C4⋊C4—611no—
372C2×C4⋊C4—612no—
373C2×C4⋊C4—611no—
374C2×C4⋊C4—1221no—
375C2×C4⋊C4—1221no—
376C2×C4⋊C4—1221no—
377C2×C4⋊C4—2442no—
378C2×C4⋊C4—2442no—
379C2×C4⋊C4—2442no—
380C2×C4⋊C4C22×D6111yescharacteristic
381C2×C4⋊C4C22×D6111yescharacteristic
382C2×C4⋊C4C22×D6112yes—
383C2×C22⋊C4—21no—
384C2×C22⋊C4—1221no—
385C2×C22⋊C4—1221no—
386C2×C22⋊C4—1222no—
387C2×C22⋊C4—1222no—
388C2×C22⋊C4—1221no—
389C2×C22⋊C4—2444no—
390C2×C22⋊C4—2444no—
391#106—42no—
392#106—2442no—
393#106—2442no—
394#106—2442no—
395C22⋊C8—614no—
396C22⋊C8—618no—
397C22⋊C8C4×D6114yescomplemented
398#95—21no—
399#95—1221no—
400#95—1221no—
401#95—1222no—
402#95—1222no—
403#95—1221no—
404C24—611no—
405C24—611no—
406C24—614no—
407C24—614no—
408C24—614no—
409C24—2441no—
410C24—2441no—
411C24—2441no—
412C24—2441no—
413C24D4×D6111yescharacteristic
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—22no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—611no—
423C22×C4—611no—
424C22×C4—614no—
425C22×C4—612no—
426C22×C4—612no—
427C22×C4—612no—
428C22×C4—612no—
429C22×C4—614no—
430C22×C4—614no—
431C22×C4—1222no—
432C22×C4—1222no—
433C22×C4—1222no—
434C22×C4—1222no—
435C22×C4—1221no—
436C22×C4—1221no—
437C22×C4—1222no—
438C22×C4—2448no—
439C22×C4—2448no—
440C22×C4—2441no—
441C22×C4—2441no—
442C22×C4—2441no—
443C22×C4—2441no—
444C22×C4—2442no—
445C22×C4—2442no—
446C22×C4—2442no—
447C22×C4—2442no—
448C22×C4—2442no—
449C22×C4—2442no—
450C22×C4—2441no—
451C22×C4—2441no—
452C22×C4—2442no—
453C22×C4—2442no—
454C22×C4—2442no—
455C22×C4C2×C4×D6112yes—
456C22×C4D4×D6111yescharacteristic
457C22×C4Q8×D6112yes—
458C22×C4C23×D6111yescharacteristic
459C2×C8—22no—
460C2×C8—22no—
461C2×C8—614no—
462C2×C8—1222no—
463C2×C8—1222no—
464C2×C8—2448no—
465C2×C8C2×C4×D6114yes—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—42no—
471C4⋊C4—1222no—
472C4⋊C4—1222no—
473C4⋊C4—1222no—
474C4⋊C4—1222no—
475C4⋊C4—2442no—
476C4⋊C4—2442no—
477C4⋊C4—2442no—
478C22⋊C4—42no—
479C22⋊C4—2442no—
480C22⋊C4—2442no—
481C22⋊C4—2444no—
482C22⋊C4—2444no—
483C22⋊C4—2442no—
484C23—21no—
485C23—21no—
486C23—21no—
487C23—21no—
488C23—612no—
489C23—612no—
490C23—612no—
491C23—612no—
492C23—612no—
493C23—612no—
494C23—6116no—
495C23—1221no—
496C23—1221no—
497C23—1221no—
498C23—1221no—
499C23—1221no—
500C23—1221no—
501C23—1221no—
502C23—1221no—
503C23—2444no—
504C23—2444no—
505C23—2444no—
506C23—2444no—
507C23—2444no—
508C23—2444no—
509C23C4⋊C4×D6112yes—
510C23SD16×D6112yes—
511C23SD16×D6112yes—
512C23C2×D4×D6111yescharacteristic
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—42no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—84no—
527C2×C4—612no—
528C2×C4—612no—
529C2×C4—1221no—
530C2×C4—1221no—
531C2×C4—1221no—
532C2×C4—1221no—
533C2×C4—1224no—
534C2×C4—1224no—
535C2×C4—1224no—
536C2×C4—1224no—
537C2×C4—4884no—
538C2×C4—4884no—
539C2×C4—4884no—
540C2×C4—2444no—
541C2×C4—2444no—
542C2×C4—2444no—
543C2×C4—2441no—
544C2×C4—2442no—
545C2×C4—2441no—
546C2×C4—2441no—
547C2×C4—2441no—
548C2×C4—2442no—
549C2×C4—2441no—
550C2×C4—2442no—
551C2×C4—2441no—
552C2×C4C4⋊C4×D6112yes—
553C2×C4C22×C4×D6111yescharacteristic
554C2×C4C2×Q8×D6111yescharacteristic
555C2×C4#1451982212yes—
556C2×C4#1451982312yes—
557C8—24no—
558C8—1224no—
559C22—22no—
560C22—22no—
561C22—22no—
562C22—22no—
563C22—22no—
564C22—22no—
565C22—614no—
566C22—614no—
567C22—1222no—
568C22—1222no—
569C22—1222no—
570C22—1222no—
571C22—1222no—
572C22—1222no—
573C22—1222no—
574C22—1222no—
575C22—1222no—
576C22—1222no—
577C22—1222no—
578C22—1222no—
579C22—24416no—
580C22#106×D6114yes—
581C22C2×C4⋊C4×D6111yescharacteristic
582C22#1684032211yescharacteristic
583C22#1684032311yescharacteristic
584C22C2×SD16×D6111yescharacteristic
585C22C2×SD16×D6111yescharacteristic
586C22#1684055811yescharacteristic
587C22#1684056111yescharacteristic
588C4—22no—
589C4—82no—
590C4—81no—
591C4—81no—
592C4—1222no—
593C4—1222no—
594C4—4881no—
595C4—4882no—
596C4—4881no—
597C4#1684023412yes—
598C4#1684038712yes—
599C2—24no—
600C2—1224no—
601C2—1224no—
602C2#2144843011yescharacteristic
603C2C2×#106×D6111yescharacteristic
604C2#2145440611yescharacteristic
605C2#2145657311yescharacteristic
606C2#2145657411yescharacteristic
607C2#2145657511yescharacteristic
608C2#2145657811yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1853C2
socleC23×C61C22×D4
prime radicalC24×D61D4
cosocleC22×C4×C61C24
hypercenter#1853D61
hypocenterC61#21938
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1853

Complements

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

NH ≅ G/Ntype
4. #21446014C2semidirect
5. #21446013C2semidirect
6. #21446012C2semidirect
7. #21446011C2semidirect
8. #21446010C2semidirect
9. #21445843C2semidirect
10. #21445841C2semidirect
11. C61×#1853C2semidirect
19. C2×#14519709C22semidirect
20. C2×#14519708C22semidirect
21. C2×#14519708C22semidirect
24. C22⋊C8×D61C4semidirect
36. #16839525C22semidirect
37. #16839524C22semidirect
38. #16839524C22semidirect
40. C2×#106×C61C22semidirect
43. C61×#340C22semidirect
60. C2×C8×D61C2×C4semidirect
80. C22⋊#92418C2×C4semidirect
81. #14519689C2×C4semidirect
97. C22⋊C8×C61C2×C4semidirect
201. C22×D61#106semidirect
303. #1853D61semidirect
327. C2×#106C2×D61semidirect
347. #340C2×D61semidirect
348. C61#21938semidirect
397. C22⋊C8C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×C22⋊C8×D612
C2×#168396192
#380×D612
#214460142
#214460132
#214460122
#214460112
#214460102
#214458432
#214458412
C61×#18532
#2193861