Finite Group Databasebeta

← #29707515

Subgroups — #29707515

Order 15616.

1558 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14519756C211yesmaximal, characteristic
2C2×#16839804C211yesmaximal, characteristic, complemented
3C2×#16839772C211yesmaximal, characteristic, complemented
4C2×#16839772C211yesmaximal, characteristic, complemented
5#21448202C211yesmaximal, characteristic
6#21447023C214yesmaximal, complemented
7#21447016C214yesmaximal, complemented
8#21445539C211yesmaximal, characteristic, complemented
9C2×C61×#407C211yesmaximal, characteristic
10C23×Dic122C2211yescharacteristic
11C2×#14519756—24no—
12C2×#14519756—22no—
13C2×#14519756—24no—
14C2×#14519756C2214yes—
15C2×#14519736C2211yescharacteristic
16C2×#14519736C2211yescharacteristic
17C22×C4⋊Dic61C2211yescharacteristic
18C22×#11758196—21no—
19C22×C22⋊Dic61—21no—
20#16839804—24no—
21C2×#14519711C2212yescomplemented
22C2×#14519709C2212yescomplemented
23#16839772—24no—
24#16839772—24no—
25C2×#14519702C2212yescomplemented
26C2×#14519702C2214yescomplemented
27C2×#14519702C2212yescomplemented
28#16839625—21no—
29#16839622C2212yes—
30#16839622C2212yes—
31#16839618—21no—
32C4×C22⋊Dic61—22no—
33#16839525C2214yescomplemented
34C24×C4×C61C2211yescharacteristic
35C22×M16×C61C2211yescharacteristic
36C22×M16×C61C2211yescharacteristic
37C61×#407—24no—
38C2×C22⋊C8×C61C2214yes—
39C22×Dic122—22no—
40C22×Dic122—22no—
41C22×Dic122—22no—
42C22×Dic122—22no—
43C22×Dic122—21no—
44C22×Dic122—22no—
45C22×Dic122D414yes—
46C22×Dic122C2312yes—
47C23×Dic61—21no—
48C23×Dic61—21no—
49#14519756—216no—
50#14519756—28no—
51#14519756—28no—
52#14519736—22no—
53#14519736—22no—
54#14519736—22no—
55#14519736—22no—
56C2×C4⋊Dic61—21no—
57C2×C4⋊Dic61—22no—
58C2×C4⋊Dic61D414yes—
59C2×C4⋊Dic61C2312yes—
60C2×C4⋊Dic61C2311yescharacteristic
61C2×C4⋊Dic61C2311yescharacteristic
62C2×C4⋊Dic61C2312yes—
63C2×#11758196—28no—
64C2×#11758196—24no—
65C2×C22⋊Dic61—28no—
66C2×C22⋊Dic61—24no—
67C2×C22⋊Dic61—21no—
68C2×C22⋊Dic61—21no—
69C2×C22⋊Dic61—22no—
70C2×C22⋊Dic61—22no—
71C2×C4×Dic61—21no—
72C2×C4×Dic61—21no—
73#14519711—24no—
74#14519709—24no—
75#14519702—24no—
76#14519702—24no—
77#14519702—28no—
78#14519684—22no—
79#14519684—22no—
80C25×C61D411yescharacteristic
81C23×C4×C61—22no—
82C23×C4×C61—22no—
83C23×C4×C61—22no—
84C23×C4×C61—22no—
85C23×C4×C61—22no—
86C23×C4×C61—22no—
87C23×C4×C61D411yescharacteristic
88C23×C4×C61D412yes—
89C23×C4×C61C2312yes—
90C23×C4×C61C2311yescharacteristic
91C2×M16×C61—22no—
92C2×M16×C61—22no—
93C2×M16×C61—22no—
94C2×M16×C61—22no—
95C2×M16×C61—22no—
96C2×M16×C61D414yes—
97C22×C8×C61C2312yes—
98C22×C8×C61C2312yes—
99C22⋊C8×C61—28no—
100C24×C61—22no—
101C24×C61—24no—
102C24×C61—22no—
103C24×C61—22no—
104C24×C61—42no—
105C24×C61C2×D412yes—
106C24×C61C2×D411yescharacteristic
107C22×C4×C61—22no—
108C22×C4×C61—22no—
109C22×C4×C61—22no—
110C22×C4×C61—24no—
111C22×C4×C61—24no—
112C22×C4×C61—24no—
113C22×C4×C61—22no—
114C22×C4×C61—22no—
115C22×C4×C61—22no—
116C22×C4×C61—22no—
117C22×C4×C61—22no—
118C22×C4×C61—22no—
119C22×C4×C61—44no—
120C22×C4×C61—44no—
121C22×C4×C61—44no—
122C22×C4×C61—44no—
123C22×C4×C61C2×D412yes—
124C22×C4×C61C2×D411yescharacteristic
125C22×C4×C61C2×D412yes—
126C22×C4×C61C2×D412yes—
127C22×C4×C61C2×D412yes—
128C22×C4×C61#4112yes—
129C22×C4×C61#4112yes—
130C22×C4×C61#4112yes—
131C22×C4×C61C2411yescharacteristic
132C22×Dic61—24no—
133C22×Dic61—24no—
134C22×Dic61—22no—
135C22×Dic61—22no—
136C22×Dic61—21no—
137C22×Dic61—21no—
138C22×Dic61—22no—
139C22×Dic61—44no—
140C22×Dic61—42no—
141C22×Dic61—42no—
142C2×Dic122—28no—
143C2×Dic122—28no—
144C2×Dic122—22no—
145C2×Dic122—22no—
146C2×Dic122—22no—
147C2×Dic122—22no—
148C2×Dic122—24no—
149C2×Dic122—22no—
150C2×Dic122—44no—
151C2×Dic122—42no—
152C2×Dic122—42no—
153C2×Dic122—44no—
154C2×Dic122C2×D414yes—
155M16×C61—28no—
156M16×C61—44no—
157C2×C8×C61—22no—
158C2×C8×C61—22no—
159C2×C8×C61—22no—
160C2×C8×C61—22no—
161C2×C8×C61—22no—
162C2×C8×C61C2×D414yes—
163C22⋊Dic61—216no—
164C22⋊Dic61—44no—
165C22⋊Dic61—42no—
166C22⋊Dic61—42no—
167C22⋊Dic61—44no—
168C4⋊Dic61—24no—
169C4⋊Dic61—22no—
170C4⋊Dic61—21no—
171C4⋊Dic61—21no—
172C4⋊Dic61—22no—
173C4⋊Dic61—21no—
174C4⋊Dic61—21no—
175C4⋊Dic61—22no—
176C4⋊Dic61C2×D414yes—
177#11758196—216no—
178C4×Dic61—42no—
179C4×Dic61—42no—
180C23×C61—22no—
181C23×C61—22no—
182C23×C61—28no—
183C23×C61—22no—
184C23×C61—22no—
185C23×C61—22no—
186C23×C61—22no—
187C23×C61—22no—
188C23×C61—22no—
189C23×C61—48no—
190C23×C61—48no—
191C23×C61—44no—
192C23×C61—44no—
193C23×C61#12012yes—
194C23×C61#12112yes—
195C23×C61#12312yes—
196C23×C61#13714yes—
197C23×C61C22×D411yescharacteristic
198C2×C4×C61—22no—
199C2×C4×C61—22no—
200C2×C4×C61—22no—
201C2×C4×C61—44no—
202C2×C4×C61—42no—
203C2×C4×C61—42no—
204C2×C4×C61—42no—
205C2×C4×C61—42no—
206C2×C4×C61—48no—
207C2×C4×C61#12012yes—
208C2×C4×C61#12114yes—
209C2×C4×C61#12114yes—
210C2×C4×C61#12112yes—
211C2×C4×C61#12314yes—
212C2×C4×C61#12312yes—
213C2×C4×C61#12414yes—
214C2×C4×C61C22×D411yescharacteristic
215C2×C4×C61C22×D411yescharacteristic
216C2×C4×C61C22×D411yescharacteristic
217C2×C4×C61C2×#4111yescharacteristic
218C2×C4×C61C2×#4111yescharacteristic
219C2×C4×C61C2×#4111yescharacteristic
220C2×Dic61—24no—
221C2×Dic61—24no—
222C2×Dic61—48no—
223C2×Dic61—42no—
224C2×Dic61—42no—
225C2×Dic61—44no—
226C2×Dic61—48no—
227C2×Dic61—41no—
228C2×Dic61—41no—
229C2×Dic61—42no—
230C2×Dic61—42no—
231C2×Dic61—42no—
232C2×Dic61—42no—
233C2×Dic61—42no—
234Dic122—24no—
235Dic122—24no—
236Dic122—42no—
237Dic122—42no—
238Dic122—48no—
239C8×C61—24no—
240C8×C61—42no—
241#23481—611nomaximal, Sylow
242C22×C61—24no—
243C22×C61—22no—
244C22×C61—22no—
245C22×C61—24no—
246C22×C61—24no—
247C22×C61—24no—
248C22×C61—24no—
249C22×C61—416no—
250C22×C61—42no—
251C22×C61—42no—
252C22×C61—42no—
253C22×C61—42no—
254C22×C61#39414yes—
255C22×C61C2×#12011yescharacteristic
256C22×C61C2×#12111yescharacteristic
257C22×C61C2×#12311yescharacteristic
258C22×C61C2×#13712yes—
259C22×C61#57512yes—
260C4×C61—42no—
261C4×C61#39414yes—
262C4×C61C2×#12111yescharacteristic
263C4×C61C2×#12111yescharacteristic
264C4×C61C2×#12311yescharacteristic
265C4×C61C2×#12411yescharacteristic
266Dic61—44no—
267Dic61—44no—
268Dic61—81no—
269Dic61—81no—
270Dic61—82no—
271C22×C22⋊Q8—611no—
272C2×#428—611no—
273C2×#419—611no—
274C2×#419—611no—
275C2×#407D6111yescharacteristic
276#2280—611no—
277#2036—614no—
278#2035—614no—
279#1736—611no—
280C2×C61—24no—
281C2×C61—44no—
282C2×C61C2×#39411yescharacteristic
283C2×C61#300411yescharacteristic
284C2×C61#300511yescharacteristic
285C2×C61#305612yes—
286C2×C61#309512yes—
287C23×Q8—611no—
288C24×C4C2×D6111yescharacteristic
289C22×M16C2×D6111yescharacteristic
290C22×M16C2×D6111yescharacteristic
291C2×C22⋊Q8—614no—
292C2×C22⋊Q8—1224no—
293C2×C22⋊Q8—1224no—
294C2×C22⋊Q8—1222no—
295C2×#117—611no—
296C2×#117—611no—
297C22×C4⋊C4—611no—
298C22×C4⋊C4—1221no—
299C22×C22⋊C4—1221no—
300#428—1224no—
301C2×#107—612no—
302C2×#106—612no—
303#419—1224no—
304#419—1224no—
305C2×#103—612no—
306C2×#103—614no—
307C2×#103—612no—
308#407—24no—
309C2×C22⋊C8C2×D6114yes—
310#381—1221no—
311#380—612no—
312#380—612no—
313#379—1221no—
314C4×C22⋊C4—1222no—
315#340—614no—
316C61#2348111yescharacteristic, complemented, Sylow
317C25D24411yescharacteristic
318C22×Q8—614no—
319C22×Q8—612no—
320C22×Q8—1221no—
321C22×Q8—1222no—
322C22×Q8—1222no—
323C22×Q8—1222no—
324C22×Q8—1222no—
325C22×Q8—1222no—
326C23×C4—22no—
327C23×C4—22no—
328C23×C4—22no—
329C23×C4—22no—
330C23×C4—22no—
331C23×C4—22no—
332C23×C4—1221no—
333C23×C4—1221no—
334C23×C4D24411yescharacteristic
335C23×C4D24412yes—
336C23×C4C22×D6112yes—
337C23×C4C22×D6111yescharacteristic
338C2×M16—22no—
339C2×M16—22no—
340C2×M16—22no—
341C2×M16—22no—
342C2×M16—22no—
343C2×M16C22⋊D6114yes—
344C22×C8C22×D6112yes—
345C22×C8C22×D6112yes—
346C22⋊Q8—1228no—
347C22⋊Q8—1228no—
348C22⋊Q8—12216no—
349#117—1222no—
350#117—1222no—
351#117—1222no—
352#117—1222no—
353C2×C4⋊C4—614no—
354C2×C4⋊C4—612no—
355C2×C4⋊C4—612no—
356C2×C4⋊C4—611no—
357C2×C4⋊C4—611no—
358C2×C4⋊C4—1228no—
359C2×C4⋊C4—1221no—
360C2×C4⋊C4—1224no—
361C2×C4⋊C4—1222no—
362C2×C22⋊C4—1228no—
363C2×C22⋊C4—1224no—
364C2×C22⋊C4—1221no—
365C2×C22⋊C4—1221no—
366C2×C22⋊C4—1222no—
367C2×C22⋊C4—1222no—
368C2×C42—1221no—
369C2×C42—1221no—
370#107—1224no—
371#106—1224no—
372#103—1224no—
373#103—1224no—
374#103—1228no—
375C22⋊C8—28no—
376#95—1222no—
377#95—1222no—
378C24—24no—
379C24—22no—
380C24—22no—
381C24—22no—
382C24—42no—
383C24C2×D24412yes—
384C24C2×D24411yescharacteristic
385C2×Q8—614no—
386C2×Q8—1222no—
387C2×Q8—1224no—
388C2×Q8—1222no—
389C2×Q8—1222no—
390C2×Q8—1222no—
391C2×Q8—1222no—
392C2×Q8—1228no—
393C2×Q8—1228no—
394C2×Q8—2444no—
395C2×Q8—2442no—
396C2×Q8—2442no—
397C2×Q8—2444no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—44no—
411C22×C4—44no—
412C22×C4—44no—
413C22×C4—44no—
414C22×C4—1222no—
415C22×C4—1222no—
416C22×C4—1224no—
417C22×C4—1224no—
418C22×C4—1221no—
419C22×C4—1221no—
420C22×C4—1222no—
421C22×C4—2444no—
422C22×C4—2442no—
423C22×C4—2442no—
424C22×C4C2×D24412yes—
425C22×C4C2×D24411yescharacteristic
426C22×C4#1175822212yes—
427C22×C4D4×D6112yes—
428C22×C4D4×D6112yes—
429C22×C4#1175822412yes—
430C22×C4#1175822412yes—
431C22×C4C2×C22⋊D6112yes—
432C22×C4C23×D6111yescharacteristic
433M16—28no—
434M16—44no—
435C2×C8—22no—
436C2×C8—22no—
437C2×C8—22no—
438C2×C8—22no—
439C2×C8—22no—
440C2×C8C2×C22⋊D6114yes—
441C4⋊C4—614no—
442C4⋊C4—12216no—
443C4⋊C4—1222no—
444C4⋊C4—1224no—
445C4⋊C4—1221no—
446C4⋊C4—1221no—
447C4⋊C4—1221no—
448C4⋊C4—1221no—
449C4⋊C4—1222no—
450C4⋊C4—1222no—
451C22⋊C4—12216no—
452C22⋊C4—2444no—
453C22⋊C4—2444no—
454C22⋊C4—2442no—
455C22⋊C4—2442no—
456C42—2442no—
457C42—2442no—
458C23—22no—
459C23—22no—
460C23—28no—
461C23—22no—
462C23—22no—
463C23—22no—
464C23—22no—
465C23—22no—
466C23—22no—
467C23—48no—
468C23—48no—
469C23—44no—
470C23—44no—
471C23#1451974712yes—
472C23#1451975112yes—
473C23#1451976112yes—
474C23#1451981014yes—
475C23C22×D24411yescharacteristic
476Q8—1224no—
477Q8—1224no—
478Q8—2442no—
479Q8—2442no—
480Q8—2448no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—44no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—48no—
490C2×C4—1224no—
491C2×C4—1224no—
492C2×C4—2441no—
493C2×C4—2441no—
494C2×C4—2442no—
495C2×C4—2442no—
496C2×C4—2442no—
497C2×C4—2442no—
498C2×C4—2442no—
499C2×C4—2448no—
500C2×C4—2442no—
501C2×C4—2442no—
502C2×C4—2444no—
503C2×C4—2448no—
504C2×C4#1451974712yes—
505C2×C4#1451974814yes—
506C2×C4#1451974914yes—
507C2×C4#1451975112yes—
508C2×C4#1451975914yes—
509C2×C4#1451976112yes—
510C2×C4#1451976314yes—
511C2×C4C2×D4×D6111yescharacteristic
512C2×C4C2×D4×D6111yescharacteristic
513C2×C4C22×C22⋊D6111yescharacteristic
514C2×C4C2×#1175822411yescharacteristic
515C2×C4C2×#1175822411yescharacteristic
516C2×C4C2×#1175822211yescharacteristic
517C8—24no—
518C8—42no—
519C22—24no—
520C22—24no—
521C22—22no—
522C22—22no—
523C22—24no—
524C22—24no—
525C22—24no—
526C22—416no—
527C22—42no—
528C22—42no—
529C22—42no—
530C22—42no—
531C22#1683967814yes—
532C22C2×#1451974711yescharacteristic
533C22C2×#1451975111yescharacteristic
534C22C2×#1451976111yescharacteristic
535C22C2×#1451981012yes—
536C22#1684056212yes—
537C4—42no—
538C4—4882no—
539C4—4881no—
540C4—4881no—
541C4—2444no—
542C4—2444no—
543C4#1683967814yes—
544C4C2×#1451974811yescharacteristic
545C4C2×#1451974911yescharacteristic
546C4C2×#1451975911yescharacteristic
547C4C2×#1451976311yescharacteristic
548C2—24no—
549C2—44no—
550C2C2×#1683967811yescharacteristic
551C2#2145505611yescharacteristic
552C2#2145506411yescharacteristic
553C2#2145544912yes—
554C2#2145576812yes—
subgroupG/H
centerC23C22×D244
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#407C2
socleC23×C61C22×D4
prime radicalC25×C61D4
hypercenterC2×#407D61
hypocenterC61#23481
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C24×C4 ◁ C2×#407

Complements

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

NH ≅ G/Ntype
2. C2×#16839804C2semidirect
3. C2×#16839772C2semidirect
4. C2×#16839772C2semidirect
6. #21447023C2semidirect
7. #21447016C2semidirect
8. #21445539C2semidirect
21. C2×#14519711C22semidirect
22. C2×#14519709C22semidirect
25. C2×#14519702C22semidirect
26. C2×#14519702C22semidirect
27. C2×#14519702C22semidirect
33. #16839525C22semidirect
316. C61#23481semidirect

Maximal subgroups

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

subgroupindex
C22×#145197562
C2×#168398042
C2×#168397722
#214482022
#214470232
#214470162
#214455392
C2×C61×#4072
#2348161