Finite Group Databasebeta

← #29707516

Subgroups — #29707516

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×#14519757C211yesmaximal, characteristic
2C2×#16839772C211yesmaximal, characteristic, complemented
3C2×#16839770C211yesmaximal, characteristic, complemented
4C2×#16839725C211yesmaximal, characteristic
5#21448198C211yesmaximal, characteristic
6#21447024C214yesmaximal, complemented
7#21447017C214yesmaximal, complemented
8#21445538C211yesmaximal, characteristic, complemented
9C2×C61×#428C211yesmaximal, characteristic, complemented
10C23×Dic122C2211yescharacteristic
11C23×C4×D61C2211yescharacteristic
12C22×#11758194C2211yescharacteristic
13C2×#14519757—24no—
14C2×#14519757—24no—
15C2×#14519757—22no—
16C2×#14519757C2214yes—
17C2×#14519736C2211yescharacteristic
18C22×C4⋊Dic61—21no—
19C22×#11758198—21no—
20#16839772—24no—
21#16839770—24no—
22C2×#14519702C2214yescomplemented
23C2×#14519700C2212yescomplemented
24C2×#14519700C2212yescomplemented
25#16839725—24no—
26C2×#14519689C2214yes—
27#16839623—21no—
28#16839619C2212yes—
29#16839619C2212yes—
30#16839617—21no—
31C4×#11758198—22no—
32#16839524C2214yescomplemented
33C22×M16×C61C2211yescharacteristic
34C2×#117×C61C2211yescharacteristic
35C22×C4⋊C4×C61C2211yescharacteristic
36C61×#428—24no—
37C2×#107×C61C2212yescomplemented
38C2×#106×C61C2212yescomplemented
39C24×D61D411yescharacteristic
40C22×Dic122—22no—
41C22×Dic122—22no—
42C22×Dic122—22no—
43C22×Dic122—22no—
44C22×Dic122—21no—
45C22×Dic122—22no—
46C22×Dic122D414yes—
47C22×Dic122C2312yes—
48C23×Dic61—21no—
49C23×Dic61D411yescharacteristic
50C22×C4×D61—22no—
51C22×C4×D61—22no—
52C22×C4×D61—22no—
53C22×C4×D61—22no—
54C22×C4×D61—22no—
55C22×C4×D61—22no—
56C22×C4×D61D412yes—
57C22×C4×D61C2312yes—
58C2×#11758194—22no—
59C2×#11758194—22no—
60C2×#11758194—22no—
61C22×#92418C2312yes—
62#14519757—216no—
63#14519757—28no—
64#14519757—28no—
65#14519736—22no—
66#14519736—22no—
67C2×C4⋊Dic61—28no—
68C2×C4⋊Dic61—24no—
69C2×C4⋊Dic61C2312yes—
70C2×C22⋊Dic61—21no—
71C2×#11758198—28no—
72C2×#11758198—24no—
73C2×#11758198—22no—
74C2×#11758198—22no—
75C2×C4×Dic61—21no—
76#14519702—28no—
77#14519700—24no—
78#14519700—24no—
79#14519689—28no—
80#14519684—22no—
81#14519684—22no—
82C23×C4×C61—21no—
83C23×C4×C61C2311yescharacteristic
84C2×M16×C61—22no—
85C2×M16×C61—22no—
86C2×M16×C61D414yes—
87C22×C8×C61C2312yes—
88#117×C61—22no—
89#117×C61—22no—
90C2×C4⋊C4×C61—21no—
91C2×C4⋊C4×C61—22no—
92C2×C4⋊C4×C61D414yes—
93C2×C4⋊C4×C61C2311yescharacteristic
94C2×C4⋊C4×C61C2311yescharacteristic
95C2×C4⋊C4×C61C2312yes—
96C2×C22⋊C4×C61—21no—
97C2×C42×C61—21no—
98#107×C61—24no—
99#106×C61—24no—
100C24×C61C2×D411yescharacteristic
101C23×D61—22no—
102C23×D61—24no—
103C23×D61—22no—
104C23×D61—22no—
105C23×D61—42no—
106C23×D61C2×D412yes—
107C22×C4×C61—24no—
108C22×C4×C61—22no—
109C22×C4×C61—21no—
110C22×C4×C61—21no—
111C22×C4×C61—44no—
112C22×C4×C61C2×D411yescharacteristic
113C22×C4×C61C2×D412yes—
114C22×C4×C61C2×D412yes—
115C22×C4×C61C2×D412yes—
116C22×C4×C61#4112yes—
117C22×C4×C61#4112yes—
118C22×C4×C61#4112yes—
119C22×C4×C61C2411yescharacteristic
120C22×Dic61—22no—
121C22×Dic61—24no—
122C22×Dic61—22no—
123C22×Dic61—22no—
124C22×Dic61—22no—
125C22×Dic61—22no—
126C22×Dic61—42no—
127C22×Dic61—42no—
128C22×Dic61C2×D412yes—
129C2×C4×D61—24no—
130C2×C4×D61—24no—
131C2×C4×D61—24no—
132C2×C4×D61—22no—
133C2×C4×D61—22no—
134C2×C4×D61—22no—
135C2×C4×D61—22no—
136C2×C4×D61—22no—
137C2×C4×D61—22no—
138C2×C4×D61—44no—
139C2×C4×D61—44no—
140C2×C4×D61—44no—
141C2×C4×D61—44no—
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—
156C2×C8×C61—22no—
157C2×C8×C61—22no—
158C2×C8×C61C2×D414yes—
159C4⋊C4×C61—24no—
160C4⋊C4×C61—22no—
161C4⋊C4×C61—21no—
162C4⋊C4×C61—21no—
163C4⋊C4×C61—21no—
164C4⋊C4×C61—21no—
165C4⋊C4×C61C2×D414yes—
166C22⋊C4×C61—42no—
167C42×C61—42no—
168C22⋊Dic61—42no—
169#11758198—216no—
170#11758198—44no—
171#11758198—44no—
172C4⋊Dic61—216no—
173C4⋊Dic61—22no—
174C4⋊Dic61—22no—
175C4×Dic61—42no—
176#11758194—44no—
177C2×#92418—22no—
178C2×#92418—22no—
179C2×#92418—22no—
180C23×C61—22no—
181C23×C61—22no—
182C23×C61#12012yes—
183C23×C61#12112yes—
184C23×C61#12312yes—
185C23×C61C22×D411yescharacteristic
186C22×D61—28no—
187C22×D61—22no—
188C22×D61—22no—
189C22×D61—22no—
190C22×D61—22no—
191C22×D61—22no—
192C22×D61—22no—
193C22×D61—48no—
194C22×D61—48no—
195C22×D61—44no—
196C22×D61—44no—
197C22×D61#13714yes—
198C2×C4×C61—24no—
199C2×C4×C61—48no—
200C2×C4×C61—44no—
201C2×C4×C61—41no—
202C2×C4×C61—41no—
203C2×C4×C61—42no—
204C2×C4×C61—42no—
205C2×C4×C61#12012yes—
206C2×C4×C61#12114yes—
207C2×C4×C61#12114yes—
208C2×C4×C61#12112yes—
209C2×C4×C61#12314yes—
210C2×C4×C61#12312yes—
211C2×C4×C61#12414yes—
212C2×C4×C61C22×D411yescharacteristic
213C2×C4×C61C22×D411yescharacteristic
214C2×C4×C61C22×D411yescharacteristic
215C2×C4×C61C2×#4111yescharacteristic
216C2×C4×C61C2×#4111yescharacteristic
217C2×C4×C61C2×#4111yescharacteristic
218C2×Dic61—22no—
219C2×Dic61—24no—
220C2×Dic61—22no—
221C2×Dic61—22no—
222C2×Dic61—42no—
223C2×Dic61—42no—
224C2×Dic61—48no—
225C2×Dic61—44no—
226C2×Dic61—42no—
227C2×Dic61—42no—
228C2×Dic61—42no—
229C4×D61—42no—
230C4×D61—42no—
231C4×D61—42no—
232C4×D61—42no—
233C4×D61—48no—
234Dic122—24no—
235Dic122—24no—
236Dic122—42no—
237Dic122—42no—
238Dic122—48no—
239C8×C61—24no—
240#92418—42no—
241#23481—611nomaximal, Sylow
242C22×C61—24no—
243C22×C61—22no—
244C22×C61—22no—
245C22×C61—24no—
246C22×C61#39414yes—
247C22×C61C2×#12011yescharacteristic
248C22×C61C2×#12111yescharacteristic
249C22×C61C2×#12311yescharacteristic
250C22×C61C2×#13712yes—
251C22×C61#57512yes—
252C2×D61—24no—
253C2×D61—24no—
254C2×D61—24no—
255C2×D61—416no—
256C2×D61—42no—
257C2×D61—42no—
258C2×D61—42no—
259C2×D61—42no—
260C4×C61—44no—
261C4×C61—81no—
262C4×C61—81no—
263C4×C61#39414yes—
264C4×C61C2×#12111yescharacteristic
265C4×C61C2×#12111yescharacteristic
266C4×C61C2×#12311yescharacteristic
267C4×C61C2×#12411yescharacteristic
268Dic61—44no—
269Dic61—42no—
270Dic61—82no—
271C22×C22⋊Q8—611no—
272C2×#428D6111yescharacteristic, complemented
273C2×#419—611no—
274C2×#419—611no—
275C2×#407—611no—
276#2280—611no—
277#2036—614no—
278#2035—614no—
279#1736—611no—
280C2×C61—24no—
281C2×C61C2×#39411yescharacteristic
282C2×C61#300411yescharacteristic
283C2×C61#300511yescharacteristic
284C2×C61#305612yes—
285C2×C61#309512yes—
286D61—44no—
287C23×Q8—611no—
288C24×C4—611no—
289C22×M16—611no—
290C22×M16C2×D6111yescharacteristic
291C2×C22⋊Q8—614no—
292C2×C22⋊Q8—1222no—
293C2×C22⋊Q8—1224no—
294C2×C22⋊Q8—1224no—
295C2×#117—611no—
296C2×#117C2×D6111yescharacteristic
297C22×C4⋊C4—1221no—
298C22×C4⋊C4C2×D6111yescharacteristic
299C22×C22⋊C4—1221no—
300#428—24no—
301C2×#107C2×D6112yescomplemented
302C2×#106C2×D6112yescomplemented
303#419—1224no—
304#419—1224no—
305C2×#103—614no—
306C2×#103—612no—
307C2×#103—612no—
308#407—1224no—
309C2×C22⋊C8—614no—
310#381—1221no—
311#380—612no—
312#380—612no—
313#379—1221no—
314C4×C22⋊C4—1222no—
315#340—614no—
316C61#2348111yescharacteristic, complemented, Sylow
317C25—611no—
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—21no—
327C23×C4—611no—
328C23×C4—612no—
329C23×C4—612no—
330C23×C4—1221no—
331C23×C4—1222no—
332C23×C4—1222no—
333C23×C4—1222no—
334C23×C4—1222no—
335C23×C4—1222no—
336C23×C4—1222no—
337C23×C4C22×D6111yescharacteristic
338C2×M16—22no—
339C2×M16—22no—
340C2×M16—1222no—
341C2×M16—1222no—
342C2×M16—1222no—
343C2×M16D24414yes—
344C22×C8—612no—
345C22×C8C22×D6112yes—
346C22⋊Q8—1228no—
347C22⋊Q8—12216no—
348C22⋊Q8—1228no—
349#117—22no—
350#117—22no—
351#117—1222no—
352#117—1222no—
353C2×C4⋊C4—22no—
354C2×C4⋊C4—21no—
355C2×C4⋊C4—612no—
356C2×C4⋊C4—1224no—
357C2×C4⋊C4—1228no—
358C2×C4⋊C4C22⋊D6114yes—
359C2×C4⋊C4C22×D6111yescharacteristic
360C2×C4⋊C4C22×D6111yescharacteristic
361C2×C4⋊C4C22×D6112yes—
362C2×C22⋊C4—21no—
363C2×C22⋊C4—1224no—
364C2×C22⋊C4—1228no—
365C2×C22⋊C4—1222no—
366C2×C22⋊C4—1222no—
367C2×C22⋊C4—1221no—
368C2×C42—21no—
369C2×C42—1221no—
370#107—24no—
371#106—24no—
372#103—1224no—
373#103—1224no—
374#103—1228no—
375C22⋊C8—1228no—
376#95—1222no—
377#95—1222no—
378C24—612no—
379C24—1224no—
380C24—1222no—
381C24—1222no—
382C24—1222no—
383C24—2442no—
384C24D4×D6111yescharacteristic
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—24no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—44no—
403C22×C4—612no—
404C22×C4—1222no—
405C22×C4—1224no—
406C22×C4—1222no—
407C22×C4—1222no—
408C22×C4—1222no—
409C22×C4—1222no—
410C22×C4—1222no—
411C22×C4—1222no—
412C22×C4—1222no—
413C22×C4—1222no—
414C22×C4—1222no—
415C22×C4—1224no—
416C22×C4—1224no—
417C22×C4—1224no—
418C22×C4—1222no—
419C22×C4—2442no—
420C22×C4—2442no—
421C22×C4—2444no—
422C22×C4—2444no—
423C22×C4—2444no—
424C22×C4—2444no—
425C22×C4C2×D24412yes—
426C22×C4#1175822212yes—
427C22×C4D4×D6112yes—
428C22×C4D4×D6111yescharacteristic
429C22×C4#1175822612yes—
430C22×C4#1175822612yes—
431C22×C4C2×C22⋊D6112yes—
432C22×C4C23×D6111yescharacteristic
433M16—28no—
434M16—2444no—
435C2×C8—22no—
436C2×C8—22no—
437C2×C8—1222no—
438C2×C8—1222no—
439C2×C8—1222no—
440C2×C8C2×D24414yes—
441C4⋊C4—22no—
442C4⋊C4—24no—
443C4⋊C4—21no—
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—21no—
447C4⋊C4—12216no—
448C4⋊C4—1222no—
449C4⋊C4—1222no—
450C4⋊C4C2×C22⋊D6114yes—
451C22⋊C4—42no—
452C22⋊C4—12216no—
453C22⋊C4—2444no—
454C22⋊C4—2444no—
455C22⋊C4—2442no—
456C42—42no—
457C42—2442no—
458C23—22no—
459C23—22no—
460C23—614no—
461C23—1228no—
462C23—1222no—
463C23—1222no—
464C23—1222no—
465C23—1222no—
466C23—1222no—
467C23—1222no—
468C23—2448no—
469C23—2448no—
470C23—2444no—
471C23—2444no—
472C23#1451974512yes—
473C23#1451975212yes—
474C23#1451976212yes—
475C23C2×D4×D6111yescharacteristic
476Q8—1224no—
477Q8—1224no—
478Q8—2442no—
479Q8—2442no—
480Q8—2448no—
481C2×C4—24no—
482C2×C4—48no—
483C2×C4—44no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—1224no—
489C2×C4—1222no—
490C2×C4—1222no—
491C2×C4—1222no—
492C2×C4—2442no—
493C2×C4—2442no—
494C2×C4—2442no—
495C2×C4—2442no—
496C2×C4—2442no—
497C2×C4—2444no—
498C2×C4—2448no—
499C2×C4—2442no—
500C2×C4—2442no—
501C2×C4—2442no—
502C2×C4—2442no—
503C2×C4—2448no—
504C2×C4#1451974512yes—
505C2×C4#1451975114yes—
506C2×C4#1451975214yes—
507C2×C4#1451975212yes—
508C2×C4#1451976214yes—
509C2×C4#1451976212yes—
510C2×C4#1451976614yes—
511C2×C4C2×D4×D6111yescharacteristic
512C2×C4C22×C22⋊D6111yescharacteristic
513C2×C4C22×D24411yescharacteristic
514C2×C4C2×#1175822211yescharacteristic
515C2×C4C2×#1175822611yescharacteristic
516C2×C4C2×#1175822611yescharacteristic
517C8—24no—
518C8—2442no—
519C22—24no—
520C22—24no—
521C22—22no—
522C22—22no—
523C22—1224no—
524C22—1224no—
525C22—1224no—
526C22—24416no—
527C22—2442no—
528C22—2442no—
529C22—2442no—
530C22—2442no—
531C22#1683967914yes—
532C22C2×#1451974511yescharacteristic
533C22C2×#1451975211yescharacteristic
534C22C2×#1451976211yescharacteristic
535C22#137×D6112yes—
536C22#1684055812yes—
537C4—44no—
538C4—81no—
539C4—81no—
540C4—4882no—
541C4—2442no—
542C4—2444no—
543C4#1683967914yes—
544C4C2×#1451975111yescharacteristic
545C4C2×#1451975211yescharacteristic
546C4C2×#1451976211yescharacteristic
547C4C2×#1451976611yescharacteristic
548C2—24no—
549C2—2444no—
550C2C2×#1683967911yescharacteristic
551C2#2145505111yescharacteristic
552C2#2145505811yescharacteristic
553C2#2145545012yes—
554C2#2145576912yes—
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#428C2
socleC23×C61C22×D4
prime radicalC24×D61D4
hypercenterC2×#428D61
hypocenterC61#23481
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#428

Complements

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

NH ≅ G/Ntype
2. C2×#16839772C2semidirect
3. C2×#16839770C2semidirect
6. #21447024C2semidirect
7. #21447017C2semidirect
8. #21445538C2semidirect
9. C2×C61×#428C2semidirect
22. C2×#14519702C22semidirect
23. C2×#14519700C22semidirect
24. C2×#14519700C22semidirect
32. #16839524C22semidirect
37. C2×#107×C61C22semidirect
38. C2×#106×C61C22semidirect
272. C2×#428D61semidirect
301. C2×#107C2×D61semidirect
302. C2×#106C2×D61semidirect
316. C61#23481semidirect

Maximal subgroups

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

subgroupindex
C22×#145197572
C2×#168397722
C2×#168397702
C2×#168397252
#214481982
#214470242
#214470172
#214455382
C2×C61×#4282
#2348161