Finite Group Databasebeta

← #29707534

Subgroups — #29707534

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×#16839802C211yesmaximal, characteristic, complemented
3C2×#16839766C211yesmaximal, characteristic, complemented
4C22×#14519697C211yesmaximal, characteristic, complemented
5C2×#16839729C211yesmaximal, characteristic, complemented
6#21447008C214yesmaximal, complemented
7#21447001C214yesmaximal, complemented
8#21445535C211yesmaximal, characteristic, complemented
9C61×#2311C211yesmaximal, characteristic, complemented
10C23×D244C2211yescharacteristic
11C22×#11758194C2211yescharacteristic
12C23×#92418C2211yescharacteristic
13#16840326—24no—
14#16840326—24no—
15C2×#14519773—21no—
16C2×#14519766—21no—
17C2×#14519751C2212yes—
18C2×#14519751C2212yes—
19C2×#14519745—22no—
20C2×#14519736C2211yescharacteristic
21#16839802—24no—
22C2×#14519710C2212yescomplemented
23C2×#14519708C2212yescomplemented
24#16839766—24no—
25C2×#14519697—24no—
26C2×#14519697C2214yes—
27C2×#14519697C2214yes—
28#16839729—24no—
29C2×C22⋊#92418C2214yescomplemented
30#16839524C2212yescomplemented
31#16839524C2212yescomplemented
32C22×#41×C61C2211yescharacteristic
33C2×#117×C61C2211yescharacteristic
34C22×C4⋊C4×C61C2211yescharacteristic
35C61×#391—21no—
36C61×#390—21no—
37C61×#386—22no—
38C61×#380C2214yescomplemented
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—
50C2×#11758194—22no—
51C2×#11758194—22no—
52C2×#11758194D414yescomplemented
53C22×#92418—22no—
54C22×#92418—22no—
55C22×#92418—22no—
56C22×#92418C2312yes—
57C22×#92418C2312yes—
58#14519773—28no—
59#14519766—28no—
60#14519751—24no—
61#14519751—24no—
62#14519751—24no—
63#14519751—24no—
64#14519745—216no—
65#14519736—22no—
66#14519736D414yes—
67C2×C4⋊Dic61C2311yescharacteristic
68C2×C4⋊Dic61C2311yescharacteristic
69C2×C22⋊Dic61—21no—
70C2×#11758198—22no—
71C2×#11758198—22no—
72C2×C4×Dic61—21no—
73#14519710—24no—
74#14519708—24no—
75#14519697—28no—
76#14519697—28no—
77C22⋊#92418—28no—
78C2×#41×C61—22no—
79C2×#41×C61—22no—
80C2×#41×C61—22no—
81C2×#41×C61—22no—
82C2×#41×C61—22no—
83C2×#41×C61—22no—
84C22×Q8×C61D411yescharacteristic
85C22×D4×C61D412yes—
86C22×D4×C61D411yescharacteristic
87C23×C4×C61—21no—
88C23×C4×C61C2312yes—
89C23×C4×C61C2311yescharacteristic
90#117×C61—22no—
91#117×C61—22no—
92C2×C4⋊C4×C61—22no—
93C2×C4⋊C4×C61—22no—
94C2×C4⋊C4×C61—21no—
95C2×C4⋊C4×C61C2312yes—
96C2×C4⋊C4×C61C2312yes—
97C2×C22⋊C4×C61—21no—
98C2×C22⋊C4×C61—24no—
99C2×C42×C61—21no—
100#95×C61—24no—
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×C61—22no—
129C22×C4×C61—22no—
130C22×C4×C61—42no—
131C22×C4×C61—82no—
132C22×C4×C61C2×D412yes—
133C22×C4×C61C2×D411yescharacteristic
134C22×C4×C61C2×D412yes—
135C22×C4×C61C2×D412yes—
136C22×C4×C61C2×D412yes—
137C22×C4×C61#4112yes—
138C22×C4×C61#4112yes—
139C22×C4×C61#4112yes—
140C22×C4×C61C2411yescharacteristic
141C2×C22⋊D61—28no—
142C2×C22⋊D61—28no—
143C2×C22⋊D61—84no—
144C2×C22⋊D61—84no—
145C22×Dic61—21no—
146C22×Dic61—21no—
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—
160C4⋊C4×C61—22no—
161C4⋊C4×C61—22no—
162C4⋊C4×C61—22no—
163C4⋊C4×C61—22no—
164C4⋊C4×C61—42no—
165C22⋊C4×C61—42no—
166C22⋊C4×C61—84no—
167C42×C61—42no—
168C22⋊Dic61—24no—
169#11758198—28no—
170#11758198—28no—
171C4⋊Dic61—21no—
172C4⋊Dic61—21no—
173C4⋊Dic61C2×D412yes—
174C4⋊Dic61C2×D412yes—
175C4×Dic61—24no—
176#11758194—28no—
177C2×#92418—22no—
178C2×#92418—22no—
179C2×#92418—22no—
180C2×#92418—22no—
181C2×#92418—22no—
182C2×#92418—44no—
183C2×#92418C2×D414yes—
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—42no—
216C2×C4×C61—42no—
217C2×C4×C61—42no—
218C2×C4×C61—44no—
219C2×C4×C61—42no—
220C2×C4×C61—84no—
221C2×C4×C61#12012yes—
222C2×C4×C61#12114yes—
223C2×C4×C61#12114yes—
224C2×C4×C61#12112yes—
225C2×C4×C61#12314yes—
226C2×C4×C61#12312yes—
227C2×C4×C61#12414yes—
228C2×C4×C61C22×D411yescharacteristic
229C2×C4×C61C22×D411yescharacteristic
230C2×C4×C61C22×D411yescharacteristic
231C2×C4×C61C2×#4111yescharacteristic
232C2×C4×C61C2×#4111yescharacteristic
233C2×C4×C61C2×#4111yescharacteristic
234C22⋊D61—88no—
235C22⋊D61—88no—
236C2×Dic61—22no—
237C2×Dic61—22no—
238C2×Dic61—42no—
239C2×Dic61—42no—
240D244—24no—
241D244—24no—
242D244—42no—
243D244—42no—
244D244—48no—
245#92418—24no—
246#92418—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—82no—
271C4×C61—82no—
272C4×C61#39414yes—
273C4×C61C2×#12111yescharacteristic
274C4×C61C2×#12111yescharacteristic
275C4×C61C2×#12311yescharacteristic
276C4×C61C2×#12411yescharacteristic
277Dic61—42no—
278Dic61—42no—
279C2×#535—611no—
280C2×#428—611no—
281C2×#418—611no—
282C22×#102—611no—
283C2×#408—611no—
284#2311D6111yescharacteristic, complemented
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×M16—611no—
299C23×C8—611no—
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×#117C2×D6111yescharacteristic
309C22×C4⋊C4C2×D6111yescharacteristic
310#428—1224no—
311C2×#107—612no—
312C2×#106—612no—
313#418—1224no—
314C2×#102—614no—
315C2×#102—614no—
316C2×#102—1224no—
317#408—1224no—
318C2×C22⋊C8—614no—
319#391—21no—
320#390—21no—
321#386—22no—
322#380C2×D6114yescomplemented
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×Q8C22⋊D6111yescharacteristic
334C22×D4—614no—
335C22×D4—612no—
336C22×D4—1222no—
337C22×D4—1222no—
338C22×D4—1222no—
339C22×D4—1222no—
340C22×D4—1222no—
341C22×D4—1222no—
342C22×D4—1222no—
343C22×D4—1221no—
344C22×D4C22⋊D6112yes—
345C22×D4C22⋊D6111yescharacteristic
346C23×C4—21no—
347C23×C4C22×D6112yes—
348C23×C4C22×D6111yescharacteristic
349C2×M16—614no—
350C2×M16—1222no—
351C2×M16—1222no—
352C22×C8—612no—
353C22×C8—612no—
354C22×C8—1222no—
355C22×C8—1222no—
356C22×C8—1222no—
357#127—1228no—
358#124—1228no—
359#121—1224no—
360#121—1224no—
361#121—1224no—
362#121—1224no—
363#120—12216no—
364#117—22no—
365#117—22no—
366#117—614no—
367#117—1222no—
368C2×C4⋊C4—22no—
369C2×C4⋊C4—22no—
370C2×C4⋊C4—21no—
371C2×C4⋊C4—611no—
372C2×C4⋊C4—611no—
373C2×C4⋊C4C22×D6112yes—
374C2×C4⋊C4C22×D6112yes—
375C2×C22⋊C4—21no—
376C2×C22⋊C4—24no—
377C2×C22⋊C4—1221no—
378C2×C22⋊C4—1222no—
379C2×C22⋊C4—1222no—
380C2×C42—21no—
381C2×C42—1221no—
382#107—1224no—
383#106—1224no—
384#102—1228no—
385#102—1228no—
386C22⋊C8—1228no—
387#95—24no—
388C24—1224no—
389C24—1222no—
390C24—4882no—
391C24—4882no—
392C24—2442no—
393C24—2442no—
394C24C2×C22⋊D6112yes—
395C24C2×C22⋊D6111yescharacteristic
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—2442no—
423C2×D4—2444no—
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—22no—
433C22×C4—22no—
434C22×C4—42no—
435C22×C4—82no—
436C22×C4—1221no—
437C22×C4—1221no—
438C22×C4#1175822212yes—
439C22×C4#1175822212yes—
440C22×C4D4×D6112yes—
441C22×C4D4×D6112yes—
442C22×C4D4×D6112yes—
443C22×C4#1175822412yes—
444C22×C4C2×C22⋊D6112yes—
445C22×C4C2×C22⋊D6111yescharacteristic
446C22×C4C23×D6111yescharacteristic
447M16—1228no—
448C2×C8—614no—
449C2×C8—1222no—
450C2×C8—1222no—
451C2×C8—1222no—
452C2×C8—1222no—
453C2×C8—1222no—
454C2×C8—2444no—
455C4⋊C4—22no—
456C4⋊C4—22no—
457C4⋊C4—22no—
458C4⋊C4—22no—
459C4⋊C4—42no—
460C4⋊C4—612no—
461C4⋊C4—612no—
462C4⋊C4—1221no—
463C4⋊C4—1221no—
464C22⋊C4—42no—
465C22⋊C4—84no—
466C22⋊C4—1228no—
467C22⋊C4—1228no—
468C22⋊C4—1224no—
469C42—42no—
470C42—1224no—
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#1451974512yes—
489C23#1451974912yes—
490C23#1451976012yes—
491C23C22×C22⋊D6111yescharacteristic
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—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—44no—
513C2×C4—42no—
514C2×C4—84no—
515C2×C4—1222no—
516C2×C4—1222no—
517C2×C4—2442no—
518C2×C4—2442no—
519C2×C4#1451974512yes—
520C2×C4#1451974912yes—
521C2×C4#1451975014yes—
522C2×C4#1451975014yes—
523C2×C4#1451975914yes—
524C2×C4#1451976012yes—
525C2×C4#1451976314yes—
526C2×C4C2×D4×D6111yescharacteristic
527C2×C4C2×D4×D6111yescharacteristic
528C2×C4C2×D4×D6111yescharacteristic
529C2×C4C2×#1175822411yescharacteristic
530C2×C4C2×#1175822211yescharacteristic
531C2×C4C2×#1175822211yescharacteristic
532C8—1224no—
533C8—2442no—
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#1683967714yes—
550C22C2×#1451974511yescharacteristic
551C22C2×#1451974911yescharacteristic
552C22C2×#1451976011yescharacteristic
553C22#1684056512yes—
554C22#1684057012yes—
555C4—42no—
556C4—82no—
557C4—82no—
558C4—2442no—
559C4—2442no—
560C4#1683967714yes—
561C4C2×#1451975011yescharacteristic
562C4C2×#1451975011yescharacteristic
563C4C2×#1451975911yescharacteristic
564C4C2×#1451976311yescharacteristic
565C2—24no—
566C2—82no—
567C2—4884no—
568C2C2×#1683967711yescharacteristic
569C2#2145508511yescharacteristic
570C2#2145509411yescharacteristic
571C2#2145550112yes—
572C2#2145577712yes—
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2311C2
socleC23×C61C22×D4
prime radicalC2×#16840326C2
hypercenter#2311D61
hypocenterC61#23483
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C22×#41 ◁ #2311

Complements

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

NH ≅ G/Ntype
2. C2×#16839802C2semidirect
3. C2×#16839766C2semidirect
4. C22×#14519697C2semidirect
5. C2×#16839729C2semidirect
6. #21447008C2semidirect
7. #21447001C2semidirect
8. #21445535C2semidirect
9. C61×#2311C2semidirect
22. C2×#14519710C22semidirect
23. C2×#14519708C22semidirect
29. C2×C22⋊#92418C22semidirect
30. #16839524C22semidirect
31. #16839524C22semidirect
38. C61×#380C22semidirect
52. C2×#11758194D4semidirect
284. #2311D61semidirect
322. #380C2×D61semidirect
325. C61#23483semidirect

Maximal subgroups

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

subgroupindex
C2×#168403262
C2×#168398022
C2×#168397662
C22×#145196972
C2×#168397292
#214470082
#214470012
#214455352
C61×#23112
#2348361