Finite Group Databasebeta

← #29602242

Subgroups — #29602242

Order 15616.

2278 conjugacy classes of subgroups.

The subgroup lattice has 572 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#11758196C211yesmaximal, characteristic
2C2×#16839625C211yesmaximal, characteristic
3C2×#16839623C211yesmaximal, characteristic
4C2×#16839623C211yesmaximal, characteristic
5C2×#16839620C211yesmaximal, characteristic
6C2×C22⋊C4×Dic61C211yesmaximal, characteristic
7#21444085C214yesmaximal, complemented
8#21444047C214yesmaximal, complemented
9C2×C61×#381C211yesmaximal, characteristic
10C24×Dic61C2211yescharacteristic
11C24×Dic61C2211yescharacteristic
12C22×#11758196—24no—
13C22×#11758196—24no—
14C22×#11758196C418yescomplemented
15C22×#11758196C2212yes—
16C22×#11758196C2212yes—
17C22×#11758196C2212yes—
18C22×C22⋊Dic61C2211yescharacteristic
19C22×C22⋊Dic61C2211yescharacteristic
20C22×C4×Dic61C2212yes—
21#16839625—24no—
22#16839623—24no—
23#16839623—24no—
24#16839620—24no—
25C22⋊C4×Dic61—24no—
26C2×#14519684C2214yes—
27C2×#14519684C2214yes—
28C2×#14519684C2214yes—
29C2×#14519684C2212yes—
30C2×#14519684C2212yes—
31C24×C4×C61C2211yescharacteristic
32C22×C22⋊C4×C61C2211yescharacteristic
33C22×C22⋊C4×C61C2211yescharacteristic
34C61×#381—24no—
35C2×#95×C61C2214yes—
36C23×Dic61—28no—
37C23×Dic61—22no—
38C23×Dic61—22no—
39C23×Dic61—22no—
40C23×Dic61—22no—
41C23×Dic61—22no—
42C23×Dic61—22no—
43C23×Dic61C2×C414yes—
44C23×Dic61C2×C414yes—
45C23×Dic61D414yes—
46C23×Dic61D414yes—
47C23×Dic61C2312yes—
48C23×Dic61C2312yes—
49C23×Dic61C2312yes—
50C23×Dic61C2311yescharacteristic
51C23×Dic61C2311yescharacteristic
52C2×#11758196—232no—
53C2×#11758196—24no—
54C2×#11758196—24no—
55C2×#11758196—24no—
56C2×#11758196—24no—
57C2×#11758196—24no—
58C2×#11758196—28no—
59C2×#11758196C2×C418yes—
60C2×#11758196C2×C418yes—
61C2×C22⋊Dic61—22no—
62C2×C22⋊Dic61—22no—
63C2×C22⋊Dic61—22no—
64C2×C22⋊Dic61—22no—
65C2×C22⋊Dic61—22no—
66C2×C22⋊Dic61D414yes—
67C2×C4×Dic61—24no—
68C2×C4×Dic61—28no—
69#14519684—28no—
70#14519684—24no—
71#14519684—24no—
72#14519684—28no—
73#14519684—28no—
74C25×C61C2311yescharacteristic
75C23×C4×C61—22no—
76C23×C4×C61—22no—
77C23×C4×C61—22no—
78C23×C4×C61—22no—
79C23×C4×C61—22no—
80C23×C4×C61—22no—
81C23×C4×C61C2×C414yes—
82C23×C4×C61C2312yes—
83C23×C4×C61C2312yes—
84C23×C4×C61C2312yes—
85C2×C22⋊C4×C61—22no—
86C2×C22⋊C4×C61—22no—
87C2×C22⋊C4×C61—22no—
88C2×C22⋊C4×C61—22no—
89C2×C22⋊C4×C61—22no—
90C2×C22⋊C4×C61D414yes—
91#95×C61—28no—
92C24×C61—24no—
93C24×C61—24no—
94C24×C61C22×C412yes—
95C24×C61C2×D412yes—
96C24×C61C2×D412yes—
97C24×C61C2×D412yes—
98C24×C61C2×D412yes—
99C24×C61#4112yes—
100C24×C61#4112yes—
101C24×C61C2411yescharacteristic
102C22×C4×C61—28no—
103C22×C4×C61—28no—
104C22×C4×C61—28no—
105C22×C4×C61—22no—
106C22×C4×C61—22no—
107C22×C4×C61—22no—
108C22×C4×C61—22no—
109C22×C4×C61—22no—
110C22×C4×C61—22no—
111C22×C4×C61—22no—
112C22×C4×C61—22no—
113C22×C4×C61—24no—
114C22×C4×C61—24no—
115C22×C4×C61—22no—
116C22×C4×C61—22no—
117C22×C4×C61—22no—
118C22×C4×C61—24no—
119C22×C4×C61—24no—
120C22×C4×C61—44no—
121C22×C4×C61—44no—
122C22×C4×C61—44no—
123C22×C4×C61—44no—
124C22×C4×C61C22×C414yes—
125C22×C4×C61C2×D414yes—
126C22×Dic61—24no—
127C22×Dic61—216no—
128C22×Dic61—216no—
129C22×Dic61—216no—
130C22×Dic61—28no—
131C22×Dic61—28no—
132C22×Dic61—28no—
133C22×Dic61—22no—
134C22×Dic61—22no—
135C22×Dic61—22no—
136C22×Dic61—22no—
137C22×Dic61—22no—
138C22×Dic61—22no—
139C22×Dic61—22no—
140C22×Dic61—24no—
141C22×Dic61—22no—
142C22×Dic61—24no—
143C22×Dic61—22no—
144C22×Dic61—22no—
145C22×Dic61—22no—
146C22×Dic61—24no—
147C22×Dic61—24no—
148C22×Dic61—44no—
149C22×Dic61—44no—
150C22×Dic61—44no—
151C22×Dic61—44no—
152C22×Dic61C22⋊C4116yes—
153C22×Dic61C22×C414yes—
154C22×Dic61C22×C412yes—
155C22×Dic61C22×C412yes—
156C22×Dic61C2×D414yes—
157C22×Dic61C2×D412yes—
158C22×Dic61C2×D412yes—
159C22×Dic61C2×D412yes—
160C22×Dic61C2×D412yes—
161C22×Dic61#4118yes—
162C22⋊C4×C61—28no—
163C22⋊C4×C61—44no—
164C22⋊Dic61—28no—
165C22⋊Dic61—44no—
166#11758196—216no—
167#11758196—216no—
168#11758196—48no—
169C4×Dic61—48no—
170C23×C61—24no—
171C23×C61—24no—
172C23×C61—24no—
173C23×C61—24no—
174C23×C61—24no—
175C23×C61—24no—
176C23×C61—28no—
177C23×C61—28no—
178C23×C61—28no—
179C23×C61—28no—
180C23×C61C2×C22⋊C414yes—
181C23×C61C4×D414yes—
182C23×C61C4×D414yes—
183C23×C61#12114yes—
184C23×C61#12114yes—
185C23×C61#12414yes—
186C23×C61#12714yes—
187C23×C61C23×C411yescharacteristic
188C23×C61C22×D411yescharacteristic
189C23×C61C22×D411yescharacteristic
190C23×C61C22×D411yescharacteristic
191C23×C61C22×D411yescharacteristic
192C23×C61C2×#4114yes—
193C23×C61C2×#4111yescharacteristic
194C23×C61C2×#4111yescharacteristic
195C23×C612−1+414yes—
196C2×C4×C61—24no—
197C2×C4×C61—24no—
198C2×C4×C61—24no—
199C2×C4×C61—24no—
200C2×C4×C61—28no—
201C2×C4×C61—42no—
202C2×C4×C61—42no—
203C2×C4×C61—42no—
204C2×C4×C61—42no—
205C2×C4×C61—42no—
206C2×C4×C61—42no—
207C2×C4×C61—44no—
208C2×C4×C61—416no—
209C2×C4×C61—42no—
210C2×C4×C61—42no—
211C2×C4×C61—42no—
212C2×C4×C61—44no—
213C2×C4×C61—44no—
214C2×Dic61—24no—
215C2×Dic61—24no—
216C2×Dic61—24no—
217C2×Dic61—232no—
218C2×Dic61—24no—
219C2×Dic61—24no—
220C2×Dic61—24no—
221C2×Dic61—24no—
222C2×Dic61—28no—
223C2×Dic61—42no—
224C2×Dic61—42no—
225C2×Dic61—42no—
226C2×Dic61—42no—
227C2×Dic61—42no—
228C2×Dic61—42no—
229C2×Dic61—44no—
230C2×Dic61—416no—
231C2×Dic61—42no—
232C2×Dic61—42no—
233C2×Dic61—42no—
234C2×Dic61—44no—
235C2×Dic61—44no—
236C2×Dic61C2×C22⋊C414yes—
237C2×Dic61C2×C22⋊C414yes—
238C2×Dic61#11718yes—
239C2×Dic61#12318yes—
240C2×Dic61#12318yes—
241#13904—611nomaximal, Sylow
242C22×C61—24no—
243C22×C61—28no—
244C22×C61—28no—
245C22×C61—28no—
246C22×C61—28no—
247C22×C61—28no—
248C22×C61—28no—
249C22×C61—24no—
250C22×C61#39018yes—
251C22×C61C22×C22⋊C411yescharacteristic
252C22×C61C2×#11712yes—
253C22×C61C2×C4×D411yescharacteristic
254C22×C61C2×C4×D411yescharacteristic
255C22×C61#51912yes—
256C22×C61C2×#12111yescharacteristic
257C22×C61C2×#12111yescharacteristic
258C22×C61C2×#12312yes—
259C22×C61C2×#12312yes—
260C22×C61C2×#12411yescharacteristic
261C22×C61C2×#12711yescharacteristic
262C22×C61#53612yes—
263C22×C61#53612yes—
264C22×C61#54714yes—
265C22×C61#54714yes—
266C22×C61#55214yes—
267C22×C61#55214yes—
268C4×C61—44no—
269C4×C61—44no—
270C4×C61—82no—
271Dic61—24no—
272Dic61—44no—
273Dic61—44no—
274Dic61—82no—
275Dic61#38118yescomplemented
276C23×C4⋊C4—611no—
277C2×#381—611no—
278C2×#381—611no—
279C2×#381—611no—
280C2×#381D6111yescharacteristic
281C2×#380—611no—
282C2×C4×C22⋊C4—611no—
283#1445—614no—
284C4⋊#95—614no—
285C2×C61—28no—
286C2×C61C2×#38111yescharacteristic
287C2×C61C2×#39011yescharacteristic
288C2×C61#231011yescharacteristic
289C2×C61#234412yes—
290C2×C61#234412yes—
291C2×C61#248612yes—
292C2×C61#248612yes—
293C2×C61#250112yes—
294C2×C61#250312yes—
295C24×C4—611no—
296C24×C4—611no—
297C24×C4C2×D6111yescharacteristic
298C22×C4⋊C4—618no—
299C22×C4⋊C4—612no—
300C22×C4⋊C4—612no—
301C22×C4⋊C4—612no—
302C22×C4⋊C4—1224no—
303C22×C4⋊C4—1224no—
304C22×C22⋊C4—611no—
305C22×C22⋊C4—611no—
306C22×C22⋊C4C2×D6111yescharacteristic
307C22×C22⋊C4C2×D6111yescharacteristic
308C22×C42—612no—
309#381—24no—
310#381—1224no—
311#381—1224no—
312#381—1224no—
313#380—1224no—
314C4×C22⋊C4—1224no—
315C2×#95—614no—
316C2×#95—614no—
317C2×#95—614no—
318C2×#95—612no—
319C2×#95—612no—
320C2×#95C2×D6114yes—
321C61#1390411yescharacteristic, complemented, Sylow
322C25C22×D6111yescharacteristic
323C23×C4—22no—
324C23×C4—22no—
325C23×C4—22no—
326C23×C4—22no—
327C23×C4—22no—
328C23×C4—22no—
329C23×C4—611no—
330C23×C4—614no—
331C23×C4—614no—
332C23×C4—614no—
333C23×C4—611no—
334C23×C4—614no—
335C23×C4—612no—
336C23×C4—612no—
337C23×C4—612no—
338C23×C4—1228no—
339C23×C4—1222no—
340C23×C4—1222no—
341C23×C4—1222no—
342C23×C4—1222no—
343C23×C4—1222no—
344C23×C4—1222no—
345C23×C4C4×D6114yes—
346C23×C4C22×D6112yes—
347C23×C4C22×D6112yes—
348C23×C4C22×D6112yes—
349C2×C4⋊C4—618no—
350C2×C4⋊C4—618no—
351C2×C4⋊C4—1224no—
352C2×C4⋊C4—1224no—
353C2×C4⋊C4—1224no—
354C2×C4⋊C4—1224no—
355C2×C4⋊C4—12232no—
356C2×C4⋊C4—1224no—
357C2×C4⋊C4—1228no—
358C2×C22⋊C4—22no—
359C2×C22⋊C4—22no—
360C2×C22⋊C4—22no—
361C2×C22⋊C4—22no—
362C2×C22⋊C4—22no—
363C2×C22⋊C4—614no—
364C2×C22⋊C4—1222no—
365C2×C22⋊C4—1222no—
366C2×C22⋊C4—1222no—
367C2×C22⋊C4—1222no—
368C2×C22⋊C4—1222no—
369C2×C22⋊C4C22⋊D6114yes—
370C2×C42—1224no—
371C2×C42—1228no—
372#95—28no—
373#95—1224no—
374#95—1224no—
375#95—1228no—
376#95—1228no—
377#95—1228no—
378C24—24no—
379C24—24no—
380C24C2×C4×D6112yes—
381C24#1175822212yes—
382C24D4×D6112yes—
383C24D4×D6112yes—
384C24D4×D6112yes—
385C24#1175822412yes—
386C24C2×C22⋊D6112yes—
387C24C23×D6111yescharacteristic
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—24no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—24no—
398C22×C4—28no—
399C22×C4—28no—
400C22×C4—28no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—44no—
407C22×C4—44no—
408C22×C4—44no—
409C22×C4—44no—
410C22×C4—612no—
411C22×C4—612no—
412C22×C4—612no—
413C22×C4—618no—
414C22×C4—612no—
415C22×C4—612no—
416C22×C4—612no—
417C22×C4—6116no—
418C22×C4—614no—
419C22×C4—614no—
420C22×C4—12216no—
421C22×C4—12216no—
422C22×C4—12216no—
423C22×C4—1224no—
424C22×C4—1222no—
425C22×C4—1222no—
426C22×C4—1222no—
427C22×C4—1222no—
428C22×C4—1222no—
429C22×C4—1224no—
430C22×C4—1222no—
431C22×C4—1222no—
432C22×C4—1222no—
433C22×C4—1224no—
434C22×C4—1228no—
435C22×C4—1228no—
436C22×C4—1228no—
437C22×C4—1222no—
438C22×C4—1222no—
439C22×C4—1222no—
440C22×C4—1224no—
441C22×C4—1224no—
442C22×C4—2444no—
443C22×C4—2444no—
444C22×C4—2444no—
445C22×C4—2444no—
446C22×C4C2×C4×D6114yes—
447C22×C4C2×C22⋊D6114yes—
448C4⋊C4—12216no—
449C4⋊C4—12216no—
450C4⋊C4—2448no—
451C22⋊C4—28no—
452C22⋊C4—44no—
453C22⋊C4—1228no—
454C22⋊C4—2444no—
455C42—2448no—
456C23—28no—
457C23—24no—
458C23—24no—
459C23—24no—
460C23—24no—
461C23—24no—
462C23—24no—
463C23—28no—
464C23—28no—
465C23—28no—
466C23C22⋊C4×D6114yes—
467C23#1451973714yes—
468C23C4×C22⋊D6114yes—
469C23#1451974914yes—
470C23#1451975014yes—
471C23#1451976314yes—
472C23#1451977314yes—
473C23C22×C4×D6111yescharacteristic
474C23C2×D4×D6111yescharacteristic
475C23C2×D4×D6111yescharacteristic
476C23C2×D4×D6111yescharacteristic
477C23C22×C22⋊D6111yescharacteristic
478C23#41×D6114yes—
479C23C2×#1175822411yescharacteristic
480C23C2×#1175822211yescharacteristic
481C23#1451982414yes—
482C2×C4—24no—
483C2×C4—24no—
484C2×C4—24no—
485C2×C4—24no—
486C2×C4—28no—
487C2×C4—44no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—416no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—44no—
499C2×C4—44no—
500C2×C4—614no—
501C2×C4—618no—
502C2×C4—618no—
503C2×C4—618no—
504C2×C4—614no—
505C2×C4—12232no—
506C2×C4—1224no—
507C2×C4—1224no—
508C2×C4—1224no—
509C2×C4—1224no—
510C2×C4—1224no—
511C2×C4—1224no—
512C2×C4—1224no—
513C2×C4—1228no—
514C2×C4—2442no—
515C2×C4—2442no—
516C2×C4—2442no—
517C2×C4—2444no—
518C2×C4—2444no—
519C2×C4—2442no—
520C2×C4—2442no—
521C2×C4—2442no—
522C2×C4—2442no—
523C2×C4—2442no—
524C2×C4—2442no—
525C2×C4—24416no—
526C2×C4—2444no—
527C22—24no—
528C22—28no—
529C22—28no—
530C22—28no—
531C22—28no—
532C22—24no—
533C22—28no—
534C22—28no—
535C22#1683966218yes—
536C22C2×C22⋊C4×D6111yescharacteristic
537C22#117×D6112yes—
538C22C2×#1451973711yescharacteristic
539C22C2×C4×C22⋊D6111yescharacteristic
540C22#1684023912yes—
541C22C2×#1451974911yescharacteristic
542C22C2×#1451975011yescharacteristic
543C22#123×D6112yes—
544C22#123×D6112yes—
545C22C2×#1451976311yescharacteristic
546C22C2×#1451977311yescharacteristic
547C22#1684033412yes—
548C22#1684033412yes—
549C22#1684041014yes—
550C22#1684041114yes—
551C22#1684044514yes—
552C22#1684044914yes—
553C4—44no—
554C4—44no—
555C4—82no—
556C4—618no—
557C4—1224no—
558C4—4882no—
559C4—2444no—
560C4—2444no—
561C2—28no—
562C2#381×D6111yescharacteristic
563C2C2×#1683966211yescharacteristic
564C2#2144840411yescharacteristic
565C2#2144868312yes—
566C2#2144868912yes—
567C2#2145012412yes—
568C2#2145012612yes—
569C2#2145028812yes—
570C2#2145030212yes—
subgroupG/H
center, FrattiniC24C23×D61
commutatorC23×C61C23×C4
FittingC2×C61×#381C2
socle, cosocleC24×C61C24
prime radicalC25×C61C23
hypercenterC2×#381D61
hypocenterC61#13904
LCSG ▷ C23×C61 ▷ C61
UCSC1 ◁ C24 ◁ C2×#381

Complements

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

NH ≅ G/Ntype
7. #21444085C2semidirect
8. #21444047C2semidirect
14. C22×#11758196C4semidirect
275. Dic61#381semidirect
321. C61#13904semidirect

Maximal subgroups

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

subgroupindex
C23×#117581962
C2×#168396252
C2×#168396232
C2×#168396202
C2×C22⋊C4×Dic612
#214440852
#214440472
C2×C61×#3812
#1390461