Finite Group Databasebeta

← #29707535

Subgroups — #29707535

Order 15616.

1610 conjugacy classes of subgroups.

The subgroup lattice has 574 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840326C211yesmaximal, characteristic
2C2×#16839804C211yesmaximal, characteristic, complemented
3C2×#16839768C211yesmaximal, characteristic, complemented
4C22×#14519699C211yesmaximal, characteristic, complemented
5#21448411C211yesmaximal, characteristic, complemented
6#21447009C214yesmaximal, complemented
7#21447002C214yesmaximal, complemented
8#21445536C211yesmaximal, characteristic, complemented
9C2×C61×#408C211yesmaximal, characteristic, complemented
10C23×D244C2211yescharacteristic
11#16840326—24no—
12#16840326—24no—
13C2×#14519773—21no—
14C2×#14519766—21no—
15C2×#14519751C2212yes—
16C2×#14519751C2212yes—
17C2×#14519745—22no—
18C2×#14519736C2211yescharacteristic
19C2×#14519736C2211yescharacteristic
20C22×C4⋊Dic61C2211yescharacteristic
21#16839804—24no—
22C2×#14519711C2212yescomplemented
23C2×#14519709C2212yescomplemented
24#16839768—24no—
25C2×#14519699—24no—
26C2×#14519699C2214yes—
27C2×#14519699C2214yes—
28#16839669—21no—
29#16839665—21no—
30#16839642—22no—
31#16839622C2214yescomplemented
32#16839525C2212yescomplemented
33#16839525C2212yescomplemented
34C22×#41×C61C2211yescharacteristic
35C22×M16×C61C2211yescharacteristic
36C23×C8×C61C2211yescharacteristic
37C61×#408—24no—
38C2×C22⋊C8×C61C2214yescomplemented
39C24×D61—21no—
40C22×D244—22no—
41C22×D244—22no—
42C22×D244—22no—
43C22×D244—22no—
44C22×D244—21no—
45C22×D244—22no—
46C22×D244D414yes—
47C22×D244C2312yes—
48C22×C22⋊D61—22no—
49C22×C22⋊D61—22no—
50C23×Dic61—21no—
51#14519773—28no—
52#14519766—28no—
53#14519751—24no—
54#14519751—24no—
55#14519751—24no—
56#14519751—24no—
57#14519745—216no—
58#14519736—22no—
59#14519736—22no—
60#14519736—22no—
61#14519736D414yes—
62C2×C4⋊Dic61—22no—
63C2×C4⋊Dic61—22no—
64C2×C4⋊Dic61—21no—
65C2×C4⋊Dic61C2312yes—
66C2×C4⋊Dic61C2312yes—
67C2×C4⋊Dic61C2311yescharacteristic
68C2×C4⋊Dic61C2311yescharacteristic
69C2×C22⋊Dic61—21no—
70C2×C22⋊Dic61—21no—
71C2×C22⋊Dic61—24no—
72C2×#11758198—22no—
73C2×#11758198—22no—
74C2×C4×Dic61—21no—
75C2×C4×Dic61—21no—
76#14519711—24no—
77#14519709—24no—
78#14519699—28no—
79#14519699—28no—
80#14519684—24no—
81C2×#41×C61—22no—
82C2×#41×C61—22no—
83C2×#41×C61—22no—
84C2×#41×C61—22no—
85C2×#41×C61—22no—
86C2×#41×C61—22no—
87C22×Q8×C61D411yescharacteristic
88C22×D4×C61D412yes—
89C22×D4×C61D411yescharacteristic
90C23×C4×C61C2312yes—
91C23×C4×C61C2311yescharacteristic
92C2×M16×C61—22no—
93C2×M16×C61—22no—
94C2×M16×C61D414yescomplemented
95C22×C8×C61—22no—
96C22×C8×C61—22no—
97C22×C8×C61—22no—
98C22×C8×C61C2312yes—
99C22×C8×C61C2312yes—
100C22⋊C8×C61—28no—
101C24×C61C2×D412yes—
102C24×C61C2×D411yescharacteristic
103C23×D61—22no—
104C23×D61—24no—
105C23×D61—42no—
106C23×D61—42no—
107C23×D61—82no—
108C23×D61—82no—
109#41×C61—44no—
110#41×C61—44no—
111#41×C61—44no—
112#41×C61—44no—
113C2×Q8×C61—22no—
114C2×Q8×C61—22no—
115C2×Q8×C61—22no—
116C2×D4×C61—24no—
117C2×D4×C61—24no—
118C2×D4×C61—24no—
119C2×D4×C61—22no—
120C2×D4×C61—22no—
121C2×D4×C61—22no—
122C22×C4×C61—22no—
123C22×C4×C61—22no—
124C22×C4×C61—22no—
125C22×C4×C61—22no—
126C22×C4×C61—22no—
127C22×C4×C61—22no—
128C22×C4×C61C2×D412yes—
129C22×C4×C61C2×D411yescharacteristic
130C22×C4×C61C2×D412yes—
131C22×C4×C61C2×D412yes—
132C22×C4×C61C2×D412yes—
133C22×C4×C61#4112yes—
134C22×C4×C61#4112yes—
135C22×C4×C61#4112yes—
136C22×C4×C61C2411yescharacteristic
137C2×C22⋊D61—28no—
138C2×C22⋊D61—28no—
139C2×C22⋊D61—84no—
140C2×C22⋊D61—84no—
141C22×Dic61—21no—
142C22×Dic61—21no—
143C22×Dic61—22no—
144C22×Dic61—22no—
145C22×Dic61—42no—
146C22×Dic61—82no—
147C2×D244—28no—
148C2×D244—28no—
149C2×D244—22no—
150C2×D244—22no—
151C2×D244—22no—
152C2×D244—22no—
153C2×D244—24no—
154C2×D244—22no—
155C2×D244—44no—
156C2×D244—42no—
157C2×D244—42no—
158C2×D244—44no—
159C2×D244C2×D414yes—
160M16×C61—28no—
161C2×C8×C61—22no—
162C2×C8×C61—22no—
163C2×C8×C61—22no—
164C2×C8×C61—22no—
165C2×C8×C61—22no—
166C2×C8×C61—44no—
167C2×C8×C61C2×D414yes—
168C22⋊Dic61—24no—
169C22⋊Dic61—42no—
170C22⋊Dic61—84no—
171#11758198—28no—
172#11758198—28no—
173C4⋊Dic61—21no—
174C4⋊Dic61—21no—
175C4⋊Dic61—22no—
176C4⋊Dic61—22no—
177C4⋊Dic61—22no—
178C4⋊Dic61—22no—
179C4⋊Dic61—42no—
180C4⋊Dic61C2×D412yes—
181C4⋊Dic61C2×D412yes—
182C4×Dic61—24no—
183C4×Dic61—42no—
184C23×C61—22no—
185C23×C61—22no—
186C23×C61—22no—
187C23×C61—22no—
188C23×C61—22no—
189C23×C61—24no—
190C23×C61—24no—
191C23×C61#12012yes—
192C23×C61#12112yes—
193C23×C61#12312yes—
194C23×C61C22×D411yescharacteristic
195C22×D61—24no—
196C22×D61—42no—
197C22×D61—42no—
198C22×D61—44no—
199C22×D61—44no—
200C22×D61—48no—
201C22×D61—84no—
202C22×D61—84no—
203C22×D61—88no—
204C22×D61—88no—
205Q8×C61—44no—
206D4×C61—48no—
207D4×C61—44no—
208C2×C4×C61—22no—
209C2×C4×C61—22no—
210C2×C4×C61—22no—
211C2×C4×C61—42no—
212C2×C4×C61—42no—
213C2×C4×C61—42no—
214C2×C4×C61—42no—
215C2×C4×C61#12012yes—
216C2×C4×C61#12114yes—
217C2×C4×C61#12114yes—
218C2×C4×C61#12112yes—
219C2×C4×C61#12314yes—
220C2×C4×C61#12312yes—
221C2×C4×C61#12414yes—
222C2×C4×C61C22×D411yescharacteristic
223C2×C4×C61C22×D411yescharacteristic
224C2×C4×C61C22×D411yescharacteristic
225C2×C4×C61C2×#4111yescharacteristic
226C2×C4×C61C2×#4111yescharacteristic
227C2×C4×C61C2×#4111yescharacteristic
228C22⋊D61—88no—
229C22⋊D61—88no—
230C2×Dic61—22no—
231C2×Dic61—22no—
232C2×Dic61—42no—
233C2×Dic61—42no—
234C2×Dic61—42no—
235C2×Dic61—42no—
236C2×Dic61—42no—
237C2×Dic61—44no—
238C2×Dic61—42no—
239C2×Dic61—84no—
240D244—24no—
241D244—24no—
242D244—42no—
243D244—42no—
244D244—48no—
245C8×C61—24no—
246C8×C61—42no—
247#23483—611nomaximal, Sylow
248C22×C61—24no—
249C22×C61—22no—
250C22×C61—22no—
251C22×C61—24no—
252C22×C61—44no—
253C22×C61—42no—
254C22×C61—42no—
255C22×C61—42no—
256C22×C61—44no—
257C22×C61#39414yes—
258C22×C61C2×#12011yescharacteristic
259C22×C61C2×#12111yescharacteristic
260C22×C61C2×#12311yescharacteristic
261C22×C61#57612yes—
262C22×C61#57712yes—
263C2×D61—44no—
264C2×D61—44no—
265C2×D61—44no—
266C2×D61—816no—
267C2×D61—84no—
268C2×D61—84no—
269C4×C61—42no—
270C4×C61#39414yes—
271C4×C61C2×#12111yescharacteristic
272C4×C61C2×#12111yescharacteristic
273C4×C61C2×#12311yescharacteristic
274C4×C61C2×#12411yescharacteristic
275Dic61—42no—
276Dic61—42no—
277Dic61—82no—
278Dic61—82no—
279C2×#535—611no—
280C2×#428—611no—
281C2×#418—611no—
282C22×#102—611no—
283C2×#408D6111yescharacteristic, complemented
284#2311—611no—
285#2034—614no—
286#2033—614no—
287#1735—611no—
288C2×C61—24no—
289C2×C61—82no—
290C2×C61C2×#39411yescharacteristic
291C2×C61#300811yescharacteristic
292C2×C61#300911yescharacteristic
293C2×C61#306212yes—
294C2×C61#309612yes—
295D61—84no—
296C22×#41C2×D6111yescharacteristic
297C23×D4—611no—
298C22×M16C2×D6111yescharacteristic
299C23×C8C2×D6111yescharacteristic
300#535—1224no—
301#535—1224no—
302C2×#127—1221no—
303C2×#124—1221no—
304C2×#121—612no—
305C2×#121—612no—
306C2×#120—1222no—
307C2×#117—611no—
308C2×#117—611no—
309C22×C4⋊C4—611no—
310#428—1224no—
311C2×#107—612no—
312C2×#106—612no—
313#418—1224no—
314C2×#102—614no—
315C2×#102—614no—
316C2×#102—1224no—
317#408—24no—
318C2×C22⋊C8C2×D6114yescomplemented
319#391—1221no—
320#390—1221no—
321#386—1222no—
322#380—614no—
323#340—612no—
324#340—612no—
325C61#2348311yescharacteristic, complemented, Sylow
326C25—1221no—
327C2×#41—22no—
328C2×#41—22no—
329C2×#41—22no—
330C2×#41—22no—
331C2×#41—22no—
332C2×#41—22no—
333C22×Q8D24411yescharacteristic
334C22×D4—614no—
335C22×D4—612no—
336C22×D4—1222no—
337C22×D4—1222no—
338C22×D4—1221no—
339C22×D4—1222no—
340C22×D4—1222no—
341C22×D4—1222no—
342C22×D4—1222no—
343C22×D4—1222no—
344C22×D4D24412yes—
345C22×D4D24411yescharacteristic
346C23×C4—1221no—
347C23×C4C22×D6112yes—
348C23×C4C22×D6111yescharacteristic
349C2×M16—22no—
350C2×M16—22no—
351C2×M16C22⋊D6114yescomplemented
352C22×C8—22no—
353C22×C8—22no—
354C22×C8—22no—
355C22×C8C22×D6112yes—
356C22×C8C22×D6112yes—
357#127—1228no—
358#124—1228no—
359#121—1224no—
360#121—1224no—
361#121—1224no—
362#121—1224no—
363#120—12216no—
364#117—614no—
365#117—1222no—
366#117—1222no—
367#117—1222no—
368C2×C4⋊C4—611no—
369C2×C4⋊C4—611no—
370C2×C4⋊C4—612no—
371C2×C4⋊C4—612no—
372C2×C4⋊C4—1222no—
373C2×C4⋊C4—1222no—
374C2×C4⋊C4—1221no—
375C2×C22⋊C4—1221no—
376C2×C22⋊C4—1222no—
377C2×C22⋊C4—1222no—
378C2×C22⋊C4—1224no—
379C2×C22⋊C4—1221no—
380C2×C42—1221no—
381C2×C42—1221no—
382#107—1224no—
383#106—1224no—
384#102—1228no—
385#102—1228no—
386C22⋊C8—28no—
387#95—1224no—
388C24—1224no—
389C24—1222no—
390C24—4882no—
391C24—4882no—
392C24—2442no—
393C24—2442no—
394C24C2×D24412yes—
395C24C2×D24411yescharacteristic
396#41—44no—
397#41—44no—
398#41—44no—
399#41—44no—
400C2×Q8—22no—
401C2×Q8—22no—
402C2×Q8—22no—
403C2×D4—24no—
404C2×D4—24no—
405C2×D4—24no—
406C2×D4—22no—
407C2×D4—22no—
408C2×D4—22no—
409C2×D4—614no—
410C2×D4—1228no—
411C2×D4—1228no—
412C2×D4—1222no—
413C2×D4—1222no—
414C2×D4—1222no—
415C2×D4—1222no—
416C2×D4—1222no—
417C2×D4—1224no—
418C2×D4—1228no—
419C2×D4—1228no—
420C2×D4—4884no—
421C2×D4—4884no—
422C2×D4—2444no—
423C2×D4—2442no—
424C2×D4—2442no—
425C2×D4—2444no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—1221no—
433C22×C4—1221no—
434C22×C4—1222no—
435C22×C4—1222no—
436C22×C4—4882no—
437C22×C4—2442no—
438C22×C4C2×D24412yes—
439C22×C4C2×D24411yescharacteristic
440C22×C4#1175822212yes—
441C22×C4D4×D6112yes—
442C22×C4D4×D6112yes—
443C22×C4#1175822412yes—
444C22×C4#1175822412yes—
445C22×C4C2×C22⋊D6112yes—
446C22×C4C23×D6111yescharacteristic
447M16—28no—
448C2×C8—22no—
449C2×C8—22no—
450C2×C8—22no—
451C2×C8—22no—
452C2×C8—22no—
453C2×C8—44no—
454C2×C8C2×C22⋊D6114yes—
455C4⋊C4—612no—
456C4⋊C4—612no—
457C4⋊C4—1221no—
458C4⋊C4—1221no—
459C4⋊C4—1222no—
460C4⋊C4—1222no—
461C4⋊C4—1222no—
462C4⋊C4—1222no—
463C4⋊C4—2442no—
464C22⋊C4—1228no—
465C22⋊C4—1228no—
466C22⋊C4—1224no—
467C22⋊C4—4884no—
468C22⋊C4—2442no—
469C42—1224no—
470C42—2442no—
471C23—22no—
472C23—22no—
473C23—22no—
474C23—22no—
475C23—22no—
476C23—24no—
477C23—24no—
478C23—1224no—
479C23—4888no—
480C23—4888no—
481C23—4884no—
482C23—4884no—
483C23—2442no—
484C23—2442no—
485C23—2444no—
486C23—2444no—
487C23—2448no—
488C23#1451974712yes—
489C23#1451975112yes—
490C23#1451976112yes—
491C23C22×D24411yescharacteristic
492Q8—44no—
493D4—48no—
494D4—44no—
495D4—1224no—
496D4—1224no—
497D4—4888no—
498D4—4888no—
499D4—2442no—
500D4—2442no—
501D4—2448no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—1222no—
510C2×C4—1222no—
511C2×C4—4884no—
512C2×C4—2442no—
513C2×C4—2442no—
514C2×C4—2442no—
515C2×C4—2444no—
516C2×C4—2442no—
517C2×C4—2442no—
518C2×C4—2442no—
519C2×C4#1451974712yes—
520C2×C4#1451974814yes—
521C2×C4#1451974914yes—
522C2×C4#1451975112yes—
523C2×C4#1451975914yes—
524C2×C4#1451976112yes—
525C2×C4#1451976314yes—
526C2×C4C2×D4×D6111yescharacteristic
527C2×C4C2×D4×D6111yescharacteristic
528C2×C4C22×C22⋊D6111yescharacteristic
529C2×C4C2×#1175822411yescharacteristic
530C2×C4C2×#1175822411yescharacteristic
531C2×C4C2×#1175822211yescharacteristic
532C8—24no—
533C8—42no—
534C22—24no—
535C22—24no—
536C22—22no—
537C22—22no—
538C22—42no—
539C22—44no—
540C22—44no—
541C22—42no—
542C22—42no—
543C22—48816no—
544C22—4884no—
545C22—4884no—
546C22—2444no—
547C22—2444no—
548C22—2444no—
549C22#1683967814yes—
550C22C2×#1451974711yescharacteristic
551C22C2×#1451975111yescharacteristic
552C22C2×#1451976111yescharacteristic
553C22#1684056712yes—
554C22#1684057412yes—
555C4—42no—
556C4—4882no—
557C4—4882no—
558C4—2442no—
559C4—2442no—
560C4#1683967814yes—
561C4C2×#1451974811yescharacteristic
562C4C2×#1451974911yescharacteristic
563C4C2×#1451975911yescharacteristic
564C4C2×#1451976311yescharacteristic
565C2—24no—
566C2—82no—
567C2—4884no—
568C2C2×#1683967811yescharacteristic
569C2#2145508911yescharacteristic
570C2#2145509811yescharacteristic
571C2#2145549912yes—
572C2#2145577812yes—
subgroupG/H
centerC23C22×D244
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#408C2
socleC23×C61C22×D4
prime radicalC2×#16840326C2
hypercenterC2×#408D61
hypocenterC61#23483
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C22×#41 ◁ C2×#408

Complements

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

NH ≅ G/Ntype
2. C2×#16839804C2semidirect
3. C2×#16839768C2semidirect
4. C22×#14519699C2semidirect
5. #21448411C2semidirect
6. #21447009C2semidirect
7. #21447002C2semidirect
8. #21445536C2semidirect
9. C2×C61×#408C2semidirect
22. C2×#14519711C22semidirect
23. C2×#14519709C22semidirect
31. #16839622C22semidirect
32. #16839525C22semidirect
33. #16839525C22semidirect
38. C2×C22⋊C8×C61C22semidirect
94. C2×M16×C61D4semidirect
283. C2×#408D61semidirect
318. C2×C22⋊C8C2×D61semidirect
325. C61#23483semidirect
351. C2×M16C22⋊D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168403262
C2×#168398042
C2×#168397682
C22×#145196992
#214484112
#214470092
#214470022
#214455362
C2×C61×#4082
#2348361