Finite Group Databasebeta

← #29681647

Subgroups — #29681647

Order 15616.

2126 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840237C211yesmaximal, characteristic, complemented
2C22×#14519692C211yesmaximal, characteristic, complemented
3#21448397C212yesmaximal, complemented
4#21446233C214yesmaximal, complemented
5#21446228C214yesmaximal, complemented
6C2×#16839528C212yesmaximal, complemented
7C2×C61×#534C211yesmaximal, characteristic
8#16840237—24no—
9#16840237—22no—
10#16840237C2214yes—
11C2×D4×Dic61—24no—
12C2×#14519736—21no—
13C22×C22⋊Dic61—21no—
14C22×C22⋊Dic61C2212yes—
15C22×C22⋊Dic61C2212yes—
16C2×#14519692—24no—
17C2×#14519692C2214yes—
18#16839642—24no—
19#16839642—24no—
20#16839625C2214yescomplemented
21#16839618C2214yescomplemented
22#16839528C2218yescomplemented
23C23×D4×C61C2211yescharacteristic
24C23×D4×C61C2211yescharacteristic
25C61×#534—24no—
26C61×#534C418yescomplemented
27C2×#123×C61—21no—
28C2×#123×C61C2212yes—
29C2×#121×C61—22no—
30C2×#120×C61—21no—
31C2×#120×C61C2212yes—
32C22×C22⋊C4×C61C2211yescharacteristic
33C23×Dic61—22no—
34C23×Dic61—22no—
35C23×Dic61D414yescomplemented
36D4×Dic61—216no—
37D4×Dic61—216no—
38#14519736—24no—
39#14519736—24no—
40C2×C4⋊Dic61—22no—
41C2×C22⋊Dic61—22no—
42C2×C22⋊Dic61—24no—
43C2×C22⋊Dic61—22no—
44C2×C22⋊Dic61—24no—
45C2×C22⋊Dic61—22no—
46C2×C22⋊Dic61—22no—
47C2×C22⋊Dic61—22no—
48C2×C22⋊Dic61—24no—
49C2×C22⋊Dic61—24no—
50C2×C22⋊Dic61—22no—
51C2×C22⋊Dic61—28no—
52C2×C22⋊Dic61D414yescomplemented
53C2×C22⋊Dic61C2314yes—
54C2×C22⋊Dic61C2314yes—
55C2×C4×Dic61—22no—
56#14519692—28no—
57#14519684—28no—
58C25×C61D411yescharacteristic
59C25×C61C2311yescharacteristic
60C25×C61C2311yescharacteristic
61C22×D4×C61—22no—
62C22×D4×C61—22no—
63C22×D4×C61—24no—
64C22×D4×C61—22no—
65C22×D4×C61—22no—
66C22×D4×C61—22no—
67C22×D4×C61—21no—
68C22×D4×C61—21no—
69C22×D4×C61—24no—
70C22×D4×C61—22no—
71C22×D4×C61C2×C414yes—
72C22×D4×C61C2×C414yescomplemented
73C22×D4×C61D414yes—
74C22×D4×C61D412yes—
75C22×D4×C61C2312yes—
76C23×C4×C61—21no—
77C23×C4×C61D411yescharacteristic
78C23×C4×C61C2311yescharacteristic
79#123×C61—24no—
80#123×C61—28no—
81#123×C61C2×C418yescomplemented
82#121×C61—216no—
83#120×C61—28no—
84#120×C61—24no—
85#120×C61C2×C418yescomplemented
86C2×C4⋊C4×C61—22no—
87C2×C22⋊C4×C61—22no—
88C2×C22⋊C4×C61—21no—
89C2×C22⋊C4×C61—22no—
90C2×C22⋊C4×C61—22no—
91C2×C22⋊C4×C61C2×C414yescomplemented
92C2×C22⋊C4×C61C2312yes—
93C24×C61—21no—
94C24×C61—21no—
95C24×C61—22no—
96C24×C61—21no—
97C24×C61—22no—
98C24×C61—21no—
99C24×C61—22no—
100C24×C61—82no—
101C24×C61—82no—
102C24×C61—82no—
103C24×C61C22⋊C414yescomplemented
104C24×C61C22×C412yes—
105C24×C61C22×C412yescomplemented
106C24×C61C2×D412yes—
107C24×C61C2×D411yescharacteristic
108C24×C61C2×D411yescharacteristic
109C24×C61C2×D412yes—
110C24×C61C2×D412yes—
111C24×C61C2×D412yes—
112C24×C61#4114yes—
113C24×C61C2411yescharacteristic
114C2×D4×C61—28no—
115C2×D4×C61—28no—
116C2×D4×C61—24no—
117C2×D4×C61—28no—
118C2×D4×C61—28no—
119C2×D4×C61—24no—
120C2×D4×C61—24no—
121C2×D4×C61—24no—
122C2×D4×C61—24no—
123C2×D4×C61—24no—
124C2×D4×C61—28no—
125C2×D4×C61—28no—
126C2×D4×C61—28no—
127C2×D4×C61—41no—
128C2×D4×C61—42no—
129C2×D4×C61—41no—
130C2×D4×C61—44no—
131C2×D4×C61—44no—
132C2×D4×C61—42no—
133C2×D4×C61—42no—
134C2×D4×C61—84no—
135C2×D4×C61C22⋊C418yescomplemented
136C2×D4×C61C22×C414yes—
137C2×D4×C61C2×D414yes—
138C22×C4×C61—21no—
139C22×C4×C61—21no—
140C22×C4×C61—22no—
141C22×C4×C61—22no—
142C22×C4×C61—21no—
143C22×C4×C61—21no—
144C22×C4×C61—22no—
145C22×C4×C61—22no—
146C22×C4×C61—24no—
147C22×C4×C61—82no—
148C22×C4×C61C22⋊C414yescomplemented
149C22×C4×C61C22×C412yescomplemented
150C22×C4×C61C2×D412yes—
151C22×Dic61—24no—
152C22×Dic61—28no—
153C22×Dic61—24no—
154C22×Dic61—24no—
155C22×Dic61—44no—
156C22×Dic61—84no—
157C22×Dic61—84no—
158C22×Dic61—84no—
159C22×Dic61C2×D418yes—
160C4⋊C4×C61—28no—
161C22⋊C4×C61—24no—
162C22⋊C4×C61—28no—
163C22⋊C4×C61—22no—
164C22⋊C4×C61—28no—
165C22⋊C4×C61C22×C414yescomplemented
166C22⋊Dic61—28no—
167C22⋊Dic61—24no—
168C22⋊Dic61—24no—
169C22⋊Dic61—24no—
170C22⋊Dic61—24no—
171C22⋊Dic61—24no—
172C22⋊Dic61—24no—
173C22⋊Dic61—44no—
174C22⋊Dic61—88no—
175C4⋊Dic61—28no—
176C4×Dic61—28no—
177C23×C61—22no—
178C23×C61—24no—
179C23×C61—22no—
180C23×C61—24no—
181C23×C61—22no—
182C23×C61—24no—
183C23×C61—44no—
184C23×C61—44no—
185C23×C61—42no—
186C23×C61—42no—
187C23×C61—42no—
188C23×C61—44no—
189C23×C61—44no—
190C23×C61—48no—
191C23×C61—44no—
192C23×C61—41no—
193C23×C61—42no—
194C23×C61—41no—
195C23×C61—82no—
196C23×C61—82no—
197C23×C61—82no—
198C23×C61—82no—
199C23×C61—84no—
200C23×C61—84no—
201C23×C61—84no—
202C23×C61—82no—
203C23×C61—84no—
204C23×C61C2×C22⋊C414yes—
205C23×C61C2×C22⋊C412yes—
206C23×C61C2×C22⋊C412yes—
207C23×C61C4×D414yes—
208C23×C61C4×D414yes—
209C23×C61#12011yescharacteristic
210C23×C61#12012yes—
211C23×C61#12011yescharacteristic
212C23×C61#12114yes—
213C23×C61#12114yes—
214C23×C61#12312yes—
215C23×C61#12312yes—
216C23×C61C23×C411yescharacteristic
217C23×C61C22×D411yescharacteristic
218C23×C61C22×D412yes—
219C23×C61C22×D411yescharacteristic
220C23×C61C2×#4112yes—
221D4×C61—28no—
222D4×C61—28no—
223D4×C61—216no—
224D4×C61—44no—
225D4×C61—44no—
226D4×C61—48no—
227D4×C61—88no—
228C2×C4×C61—22no—
229C2×C4×C61—22no—
230C2×C4×C61—24no—
231C2×C4×C61—24no—
232C2×C4×C61—24no—
233C2×C4×C61—22no—
234C2×C4×C61—24no—
235C2×C4×C61—44no—
236C2×C4×C61—41no—
237C2×C4×C61—42no—
238C2×C4×C61—41no—
239C2×C4×C61—84no—
240C2×C4×C61C2×C22⋊C414yes—
241C2×Dic61—28no—
242C2×Dic61—48no—
243C2×Dic61—48no—
244C2×Dic61—48no—
245C2×Dic61—44no—
246C2×Dic61—84no—
247C2×Dic61—88no—
248C2×Dic61—88no—
249#20960—611nomaximal, Sylow
250C22×C61—22no—
251C22×C61—24no—
252C22×C61—24no—
253C22×C61—22no—
254C22×C61—22no—
255C22×C61—24no—
256C22×C61—44no—
257C22×C61—42no—
258C22×C61—42no—
259C22×C61—42no—
260C22×C61—41no—
261C22×C61—41no—
262C22×C61—42no—
263C22×C61—44no—
264C22×C61—44no—
265C22×C61—44no—
266C22×C61—42no—
267C22×C61—44no—
268C22×C61—48no—
269C22×C61—44no—
270C22×C61—48no—
271C22×C61—44no—
272C22×C61—42no—
273C22×C61—82no—
274C22×C61—84no—
275C22×C61—84no—
276C22×C61—82no—
277C22×C61—82no—
278C22×C61#38612yes—
279C22×C61#38612yes—
280C22×C61#38614yes—
281C22×C61C22×C22⋊C411yescharacteristic
282C22×C61C2×C4×D412yes—
283C22×C61C2×#12011yescharacteristic
284C22×C61C2×#12112yes—
285C22×C61C2×#12311yescharacteristic
286C4×C61—24no—
287C4×C61—44no—
288C4×C61—42no—
289Dic61—48no—
290Dic61—84no—
291C2×#534D6111yescharacteristic
292C2×#518—611no—
293C22×#99—611no—
294#2308—612no—
295#1890—614no—
296#1889—614no—
297C2×#342—612no—
298C2×C61—22no—
299C2×C61—21no—
300C2×C61—22no—
301C2×C61—22no—
302C2×C61—21no—
303C2×C61—22no—
304C2×C61—24no—
305C2×C61—44no—
306C2×C61—82no—
307C2×C61C2×#38611yescharacteristic
308C2×C61#302612yes—
309C23×D4C2×D6111yescharacteristic
310C23×D4C2×D6111yescharacteristic
311#534—24no—
312#534Dic6118yescomplemented
313C2×#123—21no—
314C2×#123C2×D6112yes—
315C2×#121—22no—
316C2×#120—21no—
317C2×#120C2×D6112yes—
318#518—614no—
319#518—1224no—
320#518—1222no—
321C2×C4×D4—1224no—
322C2×#117—1221no—
323C22×C22⋊C4—612no—
324C22×C22⋊C4—612no—
325C22×C22⋊C4—1221no—
326C22×C22⋊C4C2×D6111yescharacteristic
327C2×#99—614no—
328C2×#99—1224no—
329#386—1224no—
330#386—1224no—
331#381—614no—
332#379—614no—
333#342—618no—
334C61#2096011yescharacteristic, complemented, Sylow
335C25C22⋊D6111yescharacteristic
336C25C22×D6111yescharacteristic
337C25C22×D6111yescharacteristic
338C22×D4—22no—
339C22×D4—22no—
340C22×D4—24no—
341C22×D4—22no—
342C22×D4—22no—
343C22×D4—24no—
344C22×D4—21no—
345C22×D4—21no—
346C22×D4—22no—
347C22×D4—22no—
348C22×D4C2×Dic6114yes—
349C22×D4C2×Dic6114yescomplemented
350C22×D4C22⋊D6114yes—
351C22×D4C22⋊D6112yes—
352C22×D4C22×D6112yes—
353C23×C4—21no—
354C23×C4—614no—
355C23×C4—1222no—
356C23×C4—1222no—
357C23×C4C22⋊D6111yescharacteristic
358C23×C4C22×D6111yescharacteristic
359#123—24no—
360#123—28no—
361#123C2×Dic6118yescomplemented
362#121—216no—
363#120—28no—
364#120—24no—
365#120C2×Dic6118yescomplemented
366C4×D4—12216no—
367C4×D4—12216no—
368#117—1224no—
369#117—1224no—
370C2×C4⋊C4—22no—
371C2×C4⋊C4—1222no—
372C2×C22⋊C4—21no—
373C2×C22⋊C4—22no—
374C2×C22⋊C4—22no—
375C2×C22⋊C4—22no—
376C2×C22⋊C4—614no—
377C2×C22⋊C4—614no—
378C2×C22⋊C4—614no—
379C2×C22⋊C4—1222no—
380C2×C22⋊C4—1224no—
381C2×C22⋊C4—1222no—
382C2×C22⋊C4—1222no—
383C2×C22⋊C4—1222no—
384C2×C22⋊C4—1222no—
385C2×C22⋊C4—1224no—
386C2×C22⋊C4—1224no—
387C2×C22⋊C4—1224no—
388C2×C22⋊C4—1228no—
389C2×C22⋊C4—1222no—
390C2×C22⋊C4C2×Dic6114yescomplemented
391C2×C22⋊C4C22×D6112yes—
392C2×C42—1222no—
393#99—1228no—
394#95—1228no—
395C24—21no—
396C24—21no—
397C24—22no—
398C24—22no—
399C24—21no—
400C24—21no—
401C24—22no—
402C24—82no—
403C24—82no—
404C24—82no—
405C24C22⋊Dic6114yescomplemented
406C24D4×D6112yes—
407C24D4×D6112yes—
408C24#1175822414yes—
409C24C22×Dic6112yes—
410C24C22×Dic6112yescomplemented
411C24C2×C22⋊D6112yes—
412C24C2×C22⋊D6111yescharacteristic
413C24C2×C22⋊D6111yescharacteristic
414C24C2×C22⋊D6112yes—
415C24C23×D6111yescharacteristic
416C2×D4—24no—
417C2×D4—24no—
418C2×D4—28no—
419C2×D4—28no—
420C2×D4—24no—
421C2×D4—28no—
422C2×D4—28no—
423C2×D4—24no—
424C2×D4—24no—
425C2×D4—24no—
426C2×D4—28no—
427C2×D4—28no—
428C2×D4—28no—
429C2×D4—41no—
430C2×D4—42no—
431C2×D4—41no—
432C2×D4—44no—
433C2×D4—44no—
434C2×D4—42no—
435C2×D4—42no—
436C2×D4—84no—
437C2×D4C22⋊Dic6118yescomplemented
438C2×D4C22×Dic6114yes—
439C2×D4C2×C22⋊D6114yes—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—22no—
448C22×C4—24no—
449C22×C4—82no—
450C22×C4—618no—
451C22×C4—1228no—
452C22×C4—1224no—
453C22×C4—1224no—
454C22×C4—1224no—
455C22×C4—4884no—
456C22×C4—4884no—
457C22×C4—4884no—
458C22×C4—2444no—
459C22×C4C22⋊Dic6114yescomplemented
460C22×C4C22×Dic6112yescomplemented
461C22×C4C2×C22⋊D6112yes—
462C4⋊C4—28no—
463C4⋊C4—1228no—
464C22⋊C4—22no—
465C22⋊C4—24no—
466C22⋊C4—28no—
467C22⋊C4—28no—
468C22⋊C4—1228no—
469C22⋊C4—1224no—
470C22⋊C4—1224no—
471C22⋊C4—1224no—
472C22⋊C4—1224no—
473C22⋊C4—1224no—
474C22⋊C4—1224no—
475C22⋊C4—4888no—
476C22⋊C4—2444no—
477C22⋊C4C22×Dic6114yescomplemented
478C42—1228no—
479C23—22no—
480C23—24no—
481C23—22no—
482C23—24no—
483C23—22no—
484C23—24no—
485C23—42no—
486C23—44no—
487C23—44no—
488C23—42no—
489C23—42no—
490C23—44no—
491C23—44no—
492C23—48no—
493C23—44no—
494C23—41no—
495C23—42no—
496C23—41no—
497C23—82no—
498C23—84no—
499C23—82no—
500C23—82no—
501C23—82no—
502C23—84no—
503C23—82no—
504C23—84no—
505C23—84no—
506C23C2×C22⋊Dic6114yes—
507C23C2×C22⋊Dic6112yes—
508C23C2×C22⋊Dic6112yes—
509C23D4×Dic6114yes—
510C23D4×Dic6114yes—
511C23#1451974511yescharacteristic
512C23#1451974512yes—
513C23#1451974511yescharacteristic
514C23#1451974914yes—
515C23#1451974914yes—
516C23#1451975812yes—
517C23#1451975812yes—
518C23C23×Dic6111yescharacteristic
519C23C2×D4×D6112yes—
520C23C22×C22⋊D6111yescharacteristic
521C23C22×C22⋊D6111yescharacteristic
522C23C2×#1175822412yes—
523D4—28no—
524D4—28no—
525D4—216no—
526D4—44no—
527D4—44no—
528D4—48no—
529D4—88no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—22no—
536C2×C4—24no—
537C2×C4—44no—
538C2×C4—41no—
539C2×C4—42no—
540C2×C4—41no—
541C2×C4—84no—
542C2×C4—1228no—
543C2×C4—4888no—
544C2×C4—4884no—
545C2×C4—4888no—
546C2×C4—2448no—
547C2×C4—2444no—
548C2×C4—2448no—
549C2×C4—2448no—
550C2×C4C2×C22⋊Dic6114yes—
551C22—22no—
552C22—24no—
553C22—24no—
554C22—22no—
555C22—22no—
556C22—24no—
557C22—42no—
558C22—44no—
559C22—41no—
560C22—41no—
561C22—42no—
562C22—42no—
563C22—42no—
564C22—44no—
565C22—44no—
566C22—44no—
567C22—42no—
568C22—44no—
569C22—48no—
570C22—44no—
571C22—48no—
572C22—44no—
573C22—42no—
574C22—82no—
575C22—84no—
576C22—84no—
577C22—82no—
578C22—82no—
579C22#1683964212yes—
580C22#1683964212yes—
581C22#1683964214yes—
582C22C22×C22⋊Dic6111yescharacteristic
583C22C2×D4×Dic6112yes—
584C22C2×#1451974511yescharacteristic
585C22C2×#1451974912yes—
586C22C2×#1451975811yescharacteristic
587C4—24no—
588C4—44no—
589C4—42no—
590C4—4884no—
591C4—2448no—
592C2—21no—
593C2—22no—
594C2—22no—
595C2—22no—
596C2—21no—
597C2—22no—
598C2—24no—
599C2—44no—
600C2—82no—
601C2C2×#1683964211yescharacteristic
602C2#2145521512yes—
subgroupG/H
centerC22C2×#14519745
commutatorC23×C61C23×C4
FrattiniC24C23×D61
Fitting, prime radicalC2×C61×#534C2
socleC22×C61C2×#120
cosocleC24×C61C24
hypercenterC2×#534D61
hypocenterC61#20960
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C25 ◁ C2×#534

Complements

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

NH ≅ G/Ntype
1. C2×#16840237C2semidirect
2. C22×#14519692C2semidirect
3. #21448397C2semidirect
4. #21446233C2semidirect
5. #21446228C2semidirect
6. C2×#16839528C2semidirect
20. #16839625C22semidirect
21. #16839618C22semidirect
22. #16839528C22semidirect
26. C61×#534C4semidirect
35. C23×Dic61D4semidirect
52. C2×C22⋊Dic61D4semidirect
72. C22×D4×C61C2×C4semidirect
81. #123×C61C2×C4semidirect
85. #120×C61C2×C4semidirect
91. C2×C22⋊C4×C61C2×C4semidirect
103. C24×C61C22⋊C4semidirect
105. C24×C61C22×C4semidirect
135. C2×D4×C61C22⋊C4semidirect
148. C22×C4×C61C22⋊C4semidirect
149. C22×C4×C61C22×C4semidirect
165. C22⋊C4×C61C22×C4semidirect
312. #534Dic61semidirect
334. C61#20960semidirect
349. C22×D4C2×Dic61semidirect
361. #123C2×Dic61semidirect
365. #120C2×Dic61semidirect
390. C2×C22⋊C4C2×Dic61semidirect
405. C24C22⋊Dic61semidirect
410. C24C22×Dic61semidirect
437. C2×D4C22⋊Dic61semidirect
459. C22×C4C22⋊Dic61semidirect
460. C22×C4C22×Dic61semidirect
477. C22⋊C4C22×Dic61semidirect

Maximal subgroups

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

subgroupindex
C2×#168402372
C22×#145196922
#214483972
#214462332
#214462282
C2×#168395282
C2×C61×#5342
#2096061