Finite Group Databasebeta

← #29677426

Subgroups — #29677426

Order 15616.

2158 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14519785C211yesmaximal, characteristic
2C2×#16840243C211yesmaximal, characteristic
3C2×#16840242C211yesmaximal, characteristic
4#21445539C211yesmaximal, characteristic, complemented
5#21445537C212yesmaximal, complemented
6#21445537C212yesmaximal, complemented
7#21445535C211yesmaximal, characteristic, complemented
8#21445534C212yesmaximal, complemented
9C2×#16839524C211yesmaximal, characteristic, complemented
10#340×D61C212yesmaximal, complemented
11C61×#1736C211yesmaximal, characteristic, complemented
12C22×#11758222C2211yescharacteristic
13C2×#14519785—24no—
14C2×#14519785—24no—
15C2×#14519785C418yescomplemented
16C22×#11758189C2212yes—
17C23×#92418C2211yescharacteristic
18C22×C8×D61C2212yes—
19#16840243—24no—
20#16840243C418yescomplemented
21#16840242—24no—
22#16840242C418yescomplemented
23C2×C4×Dic122—21no—
24C2×#14519742—21no—
25C2×C4×D244—21no—
26C2×#14519739—21no—
27C2×C4×C22⋊D61—22no—
28C2×#14519737—22no—
29C2×#14519736C2211yescharacteristic
30C2×#14519735C2212yes—
31C2×#14519735C2212yes—
32C22×C4⋊Dic61C2211yescharacteristic
33C2×C4⋊C4×D61C2212yes—
34C2×C4⋊C4×D61C2212yes—
35#16839525C2212yes—
36#16839525C2212yes—
37#16839524C2211yescharacteristic
38#16839524C2212yes—
39#16839524C2212yes—
40#16839524C2211yescharacteristic
41#16839524C2211yescharacteristic
42#16839524C2211yescharacteristic
43C22×M16×C61C2211yescharacteristic
44C2×#117×C61C2211yescharacteristic
45C22×C4⋊C4×C61C2211yescharacteristic
46C61×#340C2212yes—
47C61×#340C2212yes—
48C2×#11758222C2×C414yes—
49C2×#11758222C2×C414yes—
50C2×#11758222C2×C414yes—
51C2×#11758222D414yes—
52C2×#11758222D414yes—
53C2×#11758222Q814yes—
54C22×Dic122D411yescharacteristic
55C22×D244D411yescharacteristic
56C22×C22⋊D61D412yes—
57C23×Dic61—21no—
58C22×C4×D61—22no—
59C22×C4×D61—22no—
60C22×C4×D61C2312yes—
61#14519785—232no—
62C2×#11758189—24no—
63C2×#11758189—24no—
64C2×#11758189C2×C418yescomplemented
65C22×#92418—22no—
66C22×#92418—22no—
67C22×#92418C2×C414yescomplemented
68C22×#92418C2311yescharacteristic
69C22×#92418C2311yescharacteristic
70C2×C8×D61—24no—
71C2×C8×D61—24no—
72C2×C8×D61C2×C418yescomplemented
73C4×Dic122—28no—
74#14519742—28no—
75C4×D244—28no—
76#14519739—28no—
77C4×C22⋊D61—216no—
78#14519737—216no—
79#14519736—22no—
80#14519736C2×C414yescomplemented
81#14519735—24no—
82#14519735—24no—
83#14519735C2×C418yescomplemented
84#14519735C2×C418yescomplemented
85C2×C4⋊Dic61—21no—
86C2×C4⋊Dic61—22no—
87C2×C4⋊Dic61C2×C414yescomplemented
88C2×C4⋊Dic61C2311yescharacteristic
89C2×C4⋊Dic61C2312yes—
90C2×C4⋊Dic61C2311yescharacteristic
91C2×#11758196—22no—
92C2×#11758196—22no—
93C4⋊C4×D61—24no—
94C4⋊C4×D61—24no—
95C4⋊C4×D61C2×C418yescomplemented
96C4⋊C4×D61C2×C418yescomplemented
97C2×C22⋊Dic61—21no—
98C2×#11758198—22no—
99C2×#11758198—22no—
100C2×C4×Dic61—21no—
101C2×C4×Dic61—22no—
102C2×C4×Dic61—22no—
103C23×C4×C61—21no—
104C23×C4×C61C2311yescharacteristic
105C2×M16×C61—22no—
106C2×M16×C61—22no—
107C2×M16×C61C2×C414yescomplemented
108C22×C8×C61C2312yes—
109#117×C61—22no—
110#117×C61C2×C414yescomplemented
111C2×C4⋊C4×C61—21no—
112C2×C4⋊C4×C61—22no—
113C2×C4⋊C4×C61C2×C414yescomplemented
114C2×C4⋊C4×C61C2311yescharacteristic
115C2×C4⋊C4×C61C2312yes—
116C2×C4⋊C4×C61C2311yescharacteristic
117C2×C22⋊C4×C61—21no—
118C2×C42×C61—21no—
119C24×C61C2×D411yescharacteristic
120C23×D61C2×D412yes—
121C22×C4×C61—21no—
122C22×C4×C61—22no—
123C22×C4×C61—21no—
124C22×C4×C61—24no—
125C22×C4×C61—81no—
126C22×C4×C61—81no—
127C22×C4×C61C22×C412yes—
128C22×C4×C61C22×C412yes—
129C22×C4×C61C22×C412yes—
130C22×C4×C61C2×D411yescharacteristic
131C22×C4×C61C2×D412yes—
132C22×C4×C61C2×D412yes—
133C22×C4×C61C2×Q812yes—
134C22×C4×C61C2411yescharacteristic
135C2×C22⋊D61C22⋊C418yes—
136C2×C22⋊D61C22⋊C418yes—
137C2×C22⋊D61C4⋊C418yes—
138C22×Dic61—21no—
139C22×Dic61—22no—
140C22×Dic61—21no—
141C22×Dic61—24no—
142C22×Dic61—81no—
143C22×Dic61—81no—
144C22×Dic61C2×D412yes—
145#11758222C42116yes—
146#11758222C22⋊C4116yes—
147#11758222C4⋊C4116yes—
148#11758222C4⋊C4116yes—
149C2×D244C22⋊C414yes—
150C2×D244C22⋊C414yes—
151C2×D244C4⋊C414yes—
152C2×C4×D61—28no—
153C2×C4×D61—28no—
154C2×C4×D61—82no—
155C2×C4×D61—82no—
156C2×C4×D61—82no—
157C2×C4×D61—82no—
158C2×C4×D61C22×C414yes—
159C2×C4×D61C22×C414yes—
160C2×C4×D61C22×C414yes—
161C2×C4×D61C2×D414yes—
162C2×C4×D61C2×D414yes—
163C2×C4×D61C2×Q814yes—
164C2×Dic122C22⋊C414yes—
165C2×Dic122C22⋊C414yes—
166C2×Dic122C4⋊C414yes—
167M16×C61—28no—
168C2×C8×C61—22no—
169C2×C8×C61—22no—
170C2×C8×C61C22×C414yes—
171C4⋊C4×C61—24no—
172C4⋊C4×C61—22no—
173C4⋊C4×C61—21no—
174C4⋊C4×C61—21no—
175C4⋊C4×C61C22×C414yes—
176C4⋊C4×C61C22×C412yes—
177C4⋊C4×C61C22×C412yes—
178C22⋊C4×C61—24no—
179C42×C61—24no—
180C22⋊Dic61—24no—
181#11758198—28no—
182#11758198—28no—
183C4⋊Dic61—24no—
184C4⋊Dic61—21no—
185C4⋊Dic61—22no—
186C4⋊Dic61—21no—
187C4⋊Dic61C22×C412yes—
188C4⋊Dic61C22×C414yes—
189C4⋊Dic61C22×C412yes—
190#11758196—28no—
191#11758196—28no—
192C4×Dic61—24no—
193C4×Dic61—28no—
194C4×Dic61—28no—
195C2×#92418—28no—
196C2×#92418—21no—
197C2×#92418—21no—
198C2×#92418—21no—
199C2×#92418—21no—
200C2×#92418C22×C412yes—
201C2×#92418C22×C412yes—
202#11758189—216no—
203C8×D61—216no—
204C23×C61—21no—
205C23×C61—21no—
206C23×C61—21no—
207C23×C61—21no—
208C23×C61C2×C22⋊C412yes—
209C23×C61C2×C22⋊C412yes—
210C23×C61C2×C4⋊C412yes—
211C23×C61C22×D411yescharacteristic
212C22×D61—22no—
213C22×D61—22no—
214C22×D61—22no—
215C22×D61—22no—
216C22×D61C2×C22⋊C414yes—
217C22×D61C2×C22⋊C414yes—
218C22×D61C2×C4⋊C414yes—
219C2×C4×C61—22no—
220C2×C4×C61—24no—
221C2×C4×C61—22no—
222C2×C4×C61—41no—
223C2×C4×C61—42no—
224C2×C4×C61—41no—
225C2×C4×C61—42no—
226C2×C4×C61—41no—
227C2×C4×C61—41no—
228C2×C4×C61—84no—
229C2×C4×C61C2×C4214yes—
230C2×C4×C61C2×C22⋊C414yes—
231C2×C4×C61C2×C22⋊C412yes—
232C2×C4×C61C2×C22⋊C412yes—
233C2×C4×C61C2×C4⋊C414yes—
234C2×C4×C61C2×C4⋊C414yes—
235C2×C4×C61C2×C4⋊C412yes—
236C2×C4×C61C23×C411yescharacteristic
237C2×C4×C61C23×C411yescharacteristic
238C2×C4×C61C23×C411yescharacteristic
239C2×C4×C61C22×D411yescharacteristic
240C2×C4×C61C22×D411yescharacteristic
241C2×C4×C61C22×Q811yescharacteristic
242C22⋊D61#95132yes—
243C2×Dic61—22no—
244C2×Dic61—22no—
245C2×Dic61—24no—
246C2×Dic61—41no—
247C2×Dic61—42no—
248C2×Dic61—41no—
249C2×Dic61—42no—
250C2×Dic61—41no—
251C2×Dic61—41no—
252C2×Dic61—84no—
253C2×Dic61C2×C22⋊C414yes—
254C2×Dic61C2×C22⋊C414yes—
255C2×Dic61C2×C4⋊C414yes—
256D244#95116yes—
257C4×D61—88no—
258C4×D61—88no—
259C4×D61C2×C4218yes—
260C4×D61C2×C22⋊C418yes—
261C4×D61C2×C4⋊C418yes—
262C4×D61C2×C4⋊C418yes—
263Dic122#95116yes—
264C8×C61—24no—
265#92418—22no—
266#92418—22no—
267#20348—611nomaximal, Sylow
268C22×C61—22no—
269C22×C61—22no—
270C22×C61—22no—
271C22×C61—22no—
272C22×C61—22no—
273C22×C61—22no—
274C22×C61C2×#9514yes—
275C22×C61C22×C22⋊C411yescharacteristic
276C22×C61C22×C22⋊C411yescharacteristic
277C22×C61C22×C4⋊C411yescharacteristic
278C22×C61#57611yescharacteristic
279C22×C61#57711yescharacteristic
280C22×C61#57711yescharacteristic
281C22×C61#57811yescharacteristic
282C2×D61—24no—
283C2×D61—24no—
284C2×D61—24no—
285C2×D61—24no—
286C2×D61—24no—
287C2×D61—24no—
288C2×D61C2×#9518yes—
289C4×C61—42no—
290C4×C61—44no—
291C4×C61—42no—
292C4×C61C2×#9514yes—
293C4×C61C22×C4211yescharacteristic
294C4×C61C22×C22⋊C411yescharacteristic
295C4×C61C22×C4⋊C411yescharacteristic
296C4×C61C22×C4⋊C411yescharacteristic
297Dic61—42no—
298Dic61—44no—
299Dic61—42no—
300Dic61C2×#9518yes—
301C22×#131—611no—
302C2×#520—611no—
303C2×#520—611no—
304#1736—612no—
305#1736—612no—
306#1736—611no—
307#1736D6111yescharacteristic, complemented
308#1735—612no—
309#1735—611no—
310C2×#340—612no—
311C2×#340—611no—
312C2×C61—24no—
313C2×C61C22×#9511yescharacteristic
314C2×C61#289711yescharacteristic
315C2×C61#289711yescharacteristic
316C2×C61#289811yescharacteristic
317C2×C61#289811yescharacteristic
318C2×C61#291211yescharacteristic
319C2×C61#291311yescharacteristic
320D61—28no—
321C22×#41—611no—
322C2×#131—618no—
323C2×#131—1224no—
324C2×#131—1224no—
325C22×M16—612no—
326C22×M16C2×D6111yescharacteristic
327C23×C8—611no—
328C23×C8—612no—
329#520—618no—
330#520—618no—
331#520—1224no—
332#520—1224no—
333C2×C4×Q8—1221no—
334C2×C4×Q8—1221no—
335C2×C4×D4—1221no—
336C2×C4×D4—1222no—
337C2×C4×D4—1221no—
338C2×C4×D4—1222no—
339C2×#117—611no—
340C2×#117—612no—
341C2×#117—612no—
342C2×#117C2×D6111yescharacteristic
343C22×C4⋊C4—611no—
344C22×C4⋊C4—612no—
345C22×C4⋊C4—612no—
346C22×C4⋊C4C2×D6111yescharacteristic
347#340—612no—
348#340—612no—
349#340—611no—
350#340—612no—
351#340—612no—
352#340—611no—
353#340—611no—
354#340—611no—
355#340C2×D6112yes—
356#340C2×D6112yes—
357C61#2034811yescharacteristic, complemented, Sylow
358C2×#41—614no—
359C2×#41—614no—
360C2×#41—614no—
361C2×#41—614no—
362C2×#41—614no—
363C2×#41—614no—
364C22×Q8—611no—
365C22×D4—612no—
366C22×D4—611no—
367C23×C4—21no—
368C23×C4—612no—
369C23×C4—1222no—
370C23×C4—1222no—
371C23×C4—1221no—
372C23×C4C22×D6111yescharacteristic
373#131—12232no—
374C2×M16—22no—
375C2×M16—22no—
376C2×M16—618no—
377C2×M16—1224no—
378C2×M16—1224no—
379C2×M16C4×D6114yescomplemented
380C22×C8—614no—
381C22×C8—611no—
382C22×C8—611no—
383C22×C8—618no—
384C22×C8—1224no—
385C22×C8—1224no—
386C22×C8—1222no—
387C22×C8—1222no—
388C22×C8C22×D6112yes—
389C4×Q8—1228no—
390C4×Q8—1228no—
391C4×D4—1228no—
392C4×D4—12216no—
393C4×D4—1228no—
394C4×D4—12216no—
395#117—22no—
396#117—614no—
397#117—618no—
398#117—618no—
399#117—1224no—
400#117—1224no—
401#117—1222no—
402#117C4×D6114yescomplemented
403C2×C4⋊C4—21no—
404C2×C4⋊C4—22no—
405C2×C4⋊C4—614no—
406C2×C4⋊C4—612no—
407C2×C4⋊C4—611no—
408C2×C4⋊C4—611no—
409C2×C4⋊C4—618no—
410C2×C4⋊C4—618no—
411C2×C4⋊C4—1224no—
412C2×C4⋊C4—1224no—
413C2×C4⋊C4—1222no—
414C2×C4⋊C4—1222no—
415C2×C4⋊C4—1221no—
416C2×C4⋊C4—1222no—
417C2×C4⋊C4C4×D6114yescomplemented
418C2×C4⋊C4C22×D6111yescharacteristic
419C2×C4⋊C4C22×D6112yes—
420C2×C4⋊C4C22×D6111yescharacteristic
421C2×C22⋊C4—21no—
422C2×C22⋊C4—1222no—
423C2×C22⋊C4—1222no—
424C2×C22⋊C4—1221no—
425C2×C42—21no—
426C2×C42—1222no—
427C2×C42—1222no—
428C2×C42—1221no—
429C24—612no—
430C24D4×D6111yescharacteristic
431#41—6116no—
432#41—6116no—
433#41—6116no—
434#41—6116no—
435C2×Q8—614no—
436C2×Q8—614no—
437C2×Q8—614no—
438C2×D4—618no—
439C2×D4—618no—
440C2×D4—618no—
441C2×D4—614no—
442C2×D4—614no—
443C2×D4—614no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—22no—
447C22×C4—24no—
448C22×C4—81no—
449C22×C4—81no—
450C22×C4—612no—
451C22×C4—614no—
452C22×C4—614no—
453C22×C4—614no—
454C22×C4—614no—
455C22×C4—614no—
456C22×C4—614no—
457C22×C4—1228no—
458C22×C4—1228no—
459C22×C4—1224no—
460C22×C4—1221no—
461C22×C4—1222no—
462C22×C4—1221no—
463C22×C4—4882no—
464C22×C4—4882no—
465C22×C4—4882no—
466C22×C4—4882no—
467C22×C4—4881no—
468C22×C4—4881no—
469C22×C4C2×C4×D6112yes—
470C22×C4C2×C4×D6112yes—
471C22×C4C2×C4×D6112yes—
472C22×C4D4×D6112yes—
473C22×C4D4×D6112yes—
474C22×C4D4×D6111yescharacteristic
475C22×C4Q8×D6112yes—
476C22×C4C23×D6111yescharacteristic
477M16—28no—
478M16—12216no—
479C2×C8—22no—
480C2×C8—22no—
481C2×C8—612no—
482C2×C8—612no—
483C2×C8—12216no—
484C2×C8—1221no—
485C2×C8—1221no—
486C2×C8—1221no—
487C2×C8—1221no—
488C2×C8—1228no—
489C2×C8C2×C4×D6114yes—
490C4⋊C4—22no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—24no—
494C4⋊C4—614no—
495C4⋊C4—612no—
496C4⋊C4—612no—
497C4⋊C4—1228no—
498C4⋊C4—1228no—
499C4⋊C4—1221no—
500C4⋊C4—1222no—
501C4⋊C4—1221no—
502C4⋊C4—1224no—
503C4⋊C4C2×C4×D6112yes—
504C4⋊C4C2×C4×D6114yes—
505C4⋊C4C2×C4×D6112yes—
506C22⋊C4—24no—
507C22⋊C4—1228no—
508C22⋊C4—1228no—
509C22⋊C4—1224no—
510C42—24no—
511C42—1228no—
512C42—1228no—
513C42—1224no—
514C23—21no—
515C23—21no—
516C23—21no—
517C23—21no—
518C23—614no—
519C23—614no—
520C23—614no—
521C23—1222no—
522C23—1222no—
523C23—1222no—
524C23—1222no—
525C23C22⋊C4×D6112yes—
526C23C22⋊C4×D6112yes—
527C23C4⋊C4×D6112yes—
528C23C2×D4×D6111yescharacteristic
529Q8—6116no—
530D4—6132no—
531D4—6116no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—24no—
535C2×C4—42no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—42no—
540C2×C4—41no—
541C2×C4—84no—
542C2×C4—614no—
543C2×C4—614no—
544C2×C4—614no—
545C2×C4—618no—
546C2×C4—618no—
547C2×C4—618no—
548C2×C4—618no—
549C2×C4—1222no—
550C2×C4—1224no—
551C2×C4—1222no—
552C2×C4—4888no—
553C2×C4—4888no—
554C2×C4—4884no—
555C2×C4—2442no—
556C2×C4—2441no—
557C2×C4—2441no—
558C2×C4—2442no—
559C2×C4—2441no—
560C2×C4—2441no—
561C2×C4C42×D6114yes—
562C2×C4C22⋊C4×D6114yes—
563C2×C4C22⋊C4×D6112yes—
564C2×C4C22⋊C4×D6112yes—
565C2×C4C4⋊C4×D6114yes—
566C2×C4C4⋊C4×D6114yes—
567C2×C4C4⋊C4×D6112yes—
568C2×C4C22×C4×D6111yescharacteristic
569C2×C4C22×C4×D6111yescharacteristic
570C2×C4C22×C4×D6111yescharacteristic
571C2×C4C2×D4×D6111yescharacteristic
572C2×C4C2×D4×D6111yescharacteristic
573C2×C4C2×Q8×D6111yescharacteristic
574C8—24no—
575C8—1222no—
576C8—1222no—
577C22—22no—
578C22—22no—
579C22—22no—
580C22—22no—
581C22—22no—
582C22—22no—
583C22—618no—
584C22—1224no—
585C22—1224no—
586C22—1224no—
587C22—1224no—
588C22—1224no—
589C22—1224no—
590C22#95×D6114yes—
591C22C2×C22⋊C4×D6111yescharacteristic
592C22C2×C22⋊C4×D6111yescharacteristic
593C22C2×C4⋊C4×D6111yescharacteristic
594C22#1684056611yescharacteristic
595C22#1684057111yescharacteristic
596C22#1684057211yescharacteristic
597C22#1684057811yescharacteristic
598C4—42no—
599C4—44no—
600C4—42no—
601C4—618no—
602C4—2444no—
603C4—2442no—
604C4—2442no—
605C4#95×D6114yes—
606C4C2×C42×D6111yescharacteristic
607C4C2×C22⋊C4×D6111yescharacteristic
608C4C2×C4⋊C4×D6111yescharacteristic
609C4C2×C4⋊C4×D6111yescharacteristic
610C2—24no—
611C2—1228no—
612C2C2×#95×D6111yescharacteristic
613C2#2145434311yescharacteristic
614C2#2145434411yescharacteristic
615C2#2145435011yescharacteristic
616C2#2145435111yescharacteristic
617C2#2145441311yescharacteristic
618C2#2145441811yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC4×C61C22×C42
FrattiniC22×C4C23×D61
FittingC61×#1736C2
socleC23×C61C22×D4
prime radicalC22×#11758222C22
cosocleC22×C4×C61C24
hypercenter#1736D61
hypocenterC61#20348
LCSG ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1736

Complements

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

NH ≅ G/Ntype
4. #21445539C2semidirect
5. #21445537C2semidirect
6. #21445537C2semidirect
7. #21445535C2semidirect
8. #21445534C2semidirect
9. C2×#16839524C2semidirect
10. #340×D61C2semidirect
11. C61×#1736C2semidirect
15. C2×#14519785C4semidirect
20. #16840243C4semidirect
22. #16840242C4semidirect
64. C2×#11758189C2×C4semidirect
67. C22×#92418C2×C4semidirect
72. C2×C8×D61C2×C4semidirect
80. #14519736C2×C4semidirect
83. #14519735C2×C4semidirect
84. #14519735C2×C4semidirect
87. C2×C4⋊Dic61C2×C4semidirect
95. C4⋊C4×D61C2×C4semidirect
96. C4⋊C4×D61C2×C4semidirect
107. C2×M16×C61C2×C4semidirect
110. #117×C61C2×C4semidirect
113. C2×C4⋊C4×C61C2×C4semidirect
307. #1736D61semidirect
357. C61#20348semidirect
379. C2×M16C4×D61semidirect
402. #117C4×D61semidirect
417. C2×C4⋊C4C4×D61semidirect

Maximal subgroups

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

subgroupindex
C22×#145197852
C2×#168402432
C2×#168402422
#214455392
#214455372
#214455352
#214455342
C2×#168395242
#340×D612
C61×#17362
#2034861