Finite Group Databasebeta

← #30615151

Subgroups — #30615151

Order 15616.

1426 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840784C211yesmaximal, characteristic, complemented
2C2×#16840780C211yesmaximal, characteristic, complemented
3#21461711C212yesmaximal, complemented
4#21461678C212yesmaximal, complemented
5#21461677C212yesmaximal, complemented
6#21461642C212yesmaximal
7#21461549C212yesmaximal, complemented
8#21461548C212yesmaximal, complemented
9C2×#117×D61C211yesmaximal, characteristic
10#16840784—24no—
11#16840780—24no—
12C2×#14519844C2212yescomplemented
13C2×C4⋊#92420C2212yescomplemented
14C2×#14519834C2214yes—
15C2×#14519834C2212yes—
16C2×#14519834C2212yes—
17C2×C22⋊#92420C2214yes—
18C2×#14519832C2212yescomplemented
19C2×C4×#92420C2212yescomplemented
20#16840628C2214yescomplemented
21#16840628C2214yescomplemented
22C23×C4×D61C2211yescharacteristic
23C2×#14519736C411yescharacteristic
24C2×#14519735C2211yescharacteristic
25C2×#14519735C2211yescharacteristic
26C2×#14519734C411yescharacteristic
27C2×#14519734C411yescharacteristic
28C2×#14519733C2211yescharacteristic
29C2×#14519733C2211yescharacteristic
30#117×D61—24no—
31#117×D61—24no—
32C2×C4⋊C4×D61C411yescharacteristic
33C2×C4⋊C4×D61C411yescharacteristic
34C2×C22⋊C4×D61C411yescharacteristic
35C2×C22⋊C4×D61C411yescharacteristic
36C2×C42×D61C2211yescharacteristic
37C2×C42×D61C2211yescharacteristic
38C2×#117×C61C411yescharacteristic
39C22×#92420C2314yes—
40C22×#92420C2314yes—
41#14519844—24no—
42C4⋊#92420—24no—
43#14519834—24no—
44#14519834—24no—
45#14519834—28no—
46C22⋊#92420—28no—
47#14519832—24no—
48C4×#92420—24no—
49C24×D61C2×C411yescharacteristic
50C23×Dic61C2311yescharacteristic
51C22×C4×D61—24no—
52C22×C4×D61—22no—
53C22×C4×D61—22no—
54C22×C4×D61—24no—
55C22×C4×D61C2×C411yescharacteristic
56C22×C4×D61C2×C411yescharacteristic
57C22×C4×D61C2×C411yescharacteristic
58C22×C4×D61C2×C411yescharacteristic
59C22×C4×D61C2×C411yescharacteristic
60C22×C4×D61C2×C411yescharacteristic
61C22×C4×D61C2311yescharacteristic
62C22×C4×D61C2311yescharacteristic
63#14519736—24no—
64#14519735—22no—
65#14519735—22no—
66#14519735—22no—
67#14519735—22no—
68#14519734—24no—
69#14519734—24no—
70#14519733—22no—
71#14519733—22no—
72#14519733—22no—
73#14519733—22no—
74C2×C4⋊Dic61C2×C411yescharacteristic
75C2×C4⋊Dic61C2×C411yescharacteristic
76C2×#11758196C2×C411yescharacteristic
77C2×#11758196C2×C411yescharacteristic
78C2×#11758196C2×C411yescharacteristic
79C2×#11758196C2×C411yescharacteristic
80C4⋊C4×D61—24no—
81C4⋊C4×D61—24no—
82C2×C22⋊Dic61C2×C411yescharacteristic
83C2×C22⋊Dic61C2×C411yescharacteristic
84C2×#11758198C2×C411yescharacteristic
85C2×#11758198C2×C411yescharacteristic
86C2×#11758198C2×C411yescharacteristic
87C2×#11758198C2×C411yescharacteristic
88C22⋊C4×D61—24no—
89C22⋊C4×D61—24no—
90C2×C4×Dic61C2×C411yescharacteristic
91C2×C4×Dic61C2×C411yescharacteristic
92C2×C4×Dic61C2311yescharacteristic
93C2×C4×Dic61C2311yescharacteristic
94C2×C4×Dic61C2311yescharacteristic
95C2×C4×Dic61C2311yescharacteristic
96C42×D61—22no—
97C42×D61—22no—
98C42×D61D414yes—
99C42×D61D414yes—
100C23×C4×C61C2×C411yescharacteristic
101#117×C61—24no—
102C2×C4⋊C4×C61C2×C411yescharacteristic
103C2×C4⋊C4×C61C2×C411yescharacteristic
104C2×C22⋊C4×C61C2×C411yescharacteristic
105C2×C22⋊C4×C61C2×C411yescharacteristic
106C2×C42×C61C2×C411yescharacteristic
107C2×C42×C61C2×C411yescharacteristic
108C24×C61C22×C411yescharacteristic
109C23×D61—22no—
110C23×D61—22no—
111C23×D61—22no—
112C23×D61—24no—
113C23×D61—42no—
114C23×D61C22×C411yescharacteristic
115C23×D61C22×C411yescharacteristic
116C22×C4×C61—22no—
117C22×C4×C61—22no—
118C22×C4×C61—22no—
119C22×C4×C61C22×C411yescharacteristic
120C22×C4×C61C22×C411yescharacteristic
121C22×C4×C61C22×C411yescharacteristic
122C22×C4×C61C22×C411yescharacteristic
123C22×C4×C61C22×C411yescharacteristic
124C22×C4×C61C22×C411yescharacteristic
125C22×Dic61—21no—
126C22×Dic61—21no—
127C22×Dic61—22no—
128C22×Dic61C22×C411yescharacteristic
129C22×Dic61C22×C411yescharacteristic
130C22×Dic61C22×C411yescharacteristic
131C22×Dic61C22×C411yescharacteristic
132C22×Dic61C22×C411yescharacteristic
133C22×Dic61#4114yes—
134C22×Dic61C2411yescharacteristic
135C2×C4×D61—216no—
136C2×C4×D61—216no—
137C2×C4×D61—22no—
138C2×C4×D61—22no—
139C2×C4×D61—24no—
140C2×C4×D61—22no—
141C2×C4×D61—22no—
142C2×C4×D61—22no—
143C2×C4×D61—22no—
144C2×C4×D61—21no—
145C2×C4×D61—21no—
146C2×C4×D61—22no—
147C2×C4×D61—22no—
148C2×C4×D61—22no—
149C2×C4×D61—22no—
150C2×C4×D61—22no—
151C2×C4×D61—24no—
152C2×C4×D61—42no—
153C2×C4×D61—42no—
154C2×C4×D61C22⋊C414yes—
155C2×C4×D61C22⋊C414yes—
156C2×C4×D61C2×D412yes—
157C2×C4×D61C2×D412yes—
158C2×C4×D61#4114yes—
159C2×#92420—24no—
160C2×#92420—24no—
161C2×#92420—24no—
162C2×#92420—24no—
163C2×#92420—24no—
164C2×#92420—24no—
165C4⋊C4×C61—22no—
166C4⋊C4×C61—22no—
167C22⋊C4×C61—22no—
168C22⋊C4×C61—22no—
169C42×C61—22no—
170C42×C61C22⋊C414yes—
171C22⋊Dic61—22no—
172C22⋊Dic61—22no—
173#11758198—22no—
174#11758198—22no—
175#11758198—22no—
176#11758198—22no—
177C4⋊Dic61—22no—
178C4⋊Dic61—22no—
179#11758196—22no—
180#11758196—22no—
181#11758196—22no—
182#11758196—22no—
183C4×Dic61—22no—
184C4×Dic61—21no—
185C4×Dic61—21no—
186C4×Dic61—21no—
187C4×Dic61—21no—
188C4×Dic61C22⋊C414yes—
189C4×Dic61C2×D412yes—
190C4×Dic61C2×D412yes—
191C4×Dic61C2×D412yes—
192C4×Dic61C2×D412yes—
193C23×C61—21no—
194C23×C61—21no—
195C23×C61—22no—
196C23×C61—22no—
197C23×C61#11712yes—
198C23×C61C23×C411yescharacteristic
199C22×D61—21no—
200C22×D61—21no—
201C22×D61—22no—
202C22×D61—28no—
203C22×D61—21no—
204C22×D61—21no—
205C22×D61—22no—
206C22×D61—22no—
207C22×D61—48no—
208C22×D61—48no—
209C22×D61—44no—
210C22×D61—44no—
211C22×D61C2×C22⋊C412yes—
212C22×D61#11712yes—
213C22×D61#13114yes—
214C2×C4×C61—28no—
215C2×C4×C61—21no—
216C2×C4×C61—21no—
217C2×C4×C61—21no—
218C2×C4×C61—21no—
219C2×C4×C61—22no—
220C2×C4×C61—22no—
221C2×C4×C61—21no—
222C2×C4×C61—21no—
223C2×C4×C61—22no—
224C2×C4×C61—22no—
225C2×C4×C61C2×C22⋊C412yes—
226C2×C4×C61C2×C22⋊C412yes—
227C2×C4×C61C2×C22⋊C412yes—
228C2×C4×C61#11712yes—
229C2×Dic61—21no—
230C2×Dic61—21no—
231C2×Dic61—22no—
232C2×Dic61—22no—
233C2×Dic61—24no—
234C2×Dic61—24no—
235C2×Dic61—21no—
236C2×Dic61—21no—
237C2×Dic61—22no—
238C2×Dic61—22no—
239C2×Dic61C2×C22⋊C412yes—
240C2×Dic61C2×C22⋊C412yes—
241C2×Dic61#11712yes—
242C2×Dic61C22×D411yescharacteristic
243C2×Dic61C22×D411yescharacteristic
244C2×Dic61C2×#4112yes—
245C2×Dic61C2×#4112yes—
246C4×D61—28no—
247C4×D61—28no—
248C4×D61—28no—
249C4×D61—24no—
250C4×D61—24no—
251C4×D61—24no—
252C4×D61—24no—
253C4×D61—44no—
254C4×D61—44no—
255C4×D61—44no—
256#92420—44no—
257#92420—44no—
258#20717—611nomaximal, Sylow
259C22×C61—24no—
260C22×C61—22no—
261C22×C61—41no—
262C22×C61—41no—
263C22×C61—42no—
264C22×C61C22×C22⋊C411yescharacteristic
265C22×C61C2×#11711yescharacteristic
266C22×C61C2×#11711yescharacteristic
267C22×C61C2×#13112yes—
268C22×C61#56812yes—
269C2×D61—22no—
270C2×D61—24no—
271C2×D61—24no—
272C2×D61—22no—
273C2×D61—24no—
274C2×D61—22no—
275C2×D61—416no—
276C2×D61—41no—
277C2×D61—41no—
278C2×D61—42no—
279C2×D61#40814yes—
280C4×C61—22no—
281C4×C61—22no—
282C4×C61—22no—
283C4×C61—22no—
284C4×C61—41no—
285C4×C61—41no—
286Dic61—22no—
287Dic61—22no—
288Dic61—22no—
289Dic61—41no—
290Dic61—41no—
291Dic61#52512yes—
292Dic61#52512yes—
293C22×#117—611no—
294C2×#432—611no—
295C2×#431—611no—
296#1817—612no—
297#1798—612no—
298#1798—612no—
299#1779—612no—
300#1750—612no—
301#1750—612no—
302C2×C61—42no—
303C2×C61#228011yescharacteristic
304C2×C61C2×#40811yescharacteristic
305C2×C61#287911yescharacteristic
306C2×C61#292412yes—
307C2×C61#292412yes—
308D61—24no—
309D61—42no—
310C24×C4—611no—
311C2×#117—611no—
312C2×#117—611no—
313C2×#117—611no—
314C2×#117—611no—
315C2×#117—611no—
316C2×#117—611no—
317C2×#117—611no—
318C2×#117—1224no—
319C2×#117—1224no—
320C2×#117#692311yescharacteristic
321C22×C4⋊C4—611no—
322C22×C4⋊C4—611no—
323C22×C22⋊C4—611no—
324C22×C22⋊C4—611no—
325C22×C42—611no—
326C22×C42—611no—
327#432—1224no—
328#431—1224no—
329C2×C4⋊C8—612no—
330C2×C4⋊C8—612no—
331C2×C22⋊C8—614no—
332C2×C22⋊C8—614no—
333C2×C22⋊C8—612no—
334C2×C22⋊C8—612no—
335C2×#97—612no—
336C2×C4×C8—612no—
337#336—614no—
338#336—614no—
339C61#2071711yescharacteristic, complemented, Sylow
340C25—611no—
341C23×C4—611no—
342C23×C4—611no—
343C23×C4—611no—
344C23×C4—611no—
345C23×C4—611no—
346C23×C4—611no—
347C23×C4—611no—
348C23×C4—611no—
349C23×C4—611no—
350C23×C4—1222no—
351C23×C4—1222no—
352C23×C4—1224no—
353C23×C4—1224no—
354C23×C4C2×#692311yescharacteristic
355C22×C8—614no—
356C22×C8—614no—
357#117—24no—
358#117—1224no—
359#117—1222no—
360#117—1222no—
361#117—1222no—
362#117—1222no—
363#117—1222no—
364#117—1222no—
365#117—1222no—
366#117—1222no—
367#117—1224no—
368#117—1224no—
369C2×C4⋊C4—611no—
370C2×C4⋊C4—611no—
371C2×C4⋊C4—611no—
372C2×C4⋊C4—611no—
373C2×C4⋊C4—611no—
374C2×C4⋊C4—611no—
375C2×C4⋊C4—1224no—
376C2×C4⋊C4—1224no—
377C2×C4⋊C4C2×#692311yescharacteristic
378C2×C4⋊C4C2×#692311yescharacteristic
379C2×C22⋊C4—611no—
380C2×C22⋊C4—611no—
381C2×C22⋊C4—611no—
382C2×C22⋊C4—611no—
383C2×C22⋊C4—611no—
384C2×C22⋊C4—611no—
385C2×C22⋊C4—1224no—
386C2×C22⋊C4—1224no—
387C2×C22⋊C4C2×#692311yescharacteristic
388C2×C22⋊C4C2×#692311yescharacteristic
389C2×C42—611no—
390C2×C42—611no—
391C2×C42—611no—
392C2×C42—614no—
393C2×C42—614no—
394C2×C42—611no—
395C2×C42—611no—
396C2×C42—611no—
397C2×C42—1222no—
398C2×C42—1222no—
399C2×C42C2×#692311yescharacteristic
400C2×C42C2×#692311yescharacteristic
401C4⋊C8—1224no—
402C4⋊C8—1224no—
403C22⋊C8—1228no—
404C22⋊C8—1228no—
405C22⋊C8—1224no—
406C22⋊C8—1224no—
407#97—1224no—
408C4×C8—1224no—
409C24—611no—
410C24—611no—
411C24—1222no—
412C24—1222no—
413C24—1224no—
414C24—1222no—
415C24—2442no—
416C24C22×#692311yescharacteristic
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—611no—
421C22×C4—614no—
422C22×C4—611no—
423C22×C4—611no—
424C22×C4—611no—
425C22×C4—612no—
426C22×C4—612no—
427C22×C4—614no—
428C22×C4—614no—
429C22×C4—614no—
430C22×C4—611no—
431C22×C4—611no—
432C22×C4—1221no—
433C22×C4—1221no—
434C22×C4—1222no—
435C22×C4—1222no—
436C22×C4—1222no—
437C22×C4—1222no—
438C22×C4—1222no—
439C22×C4—1222no—
440C22×C4—1222no—
441C22×C4—1224no—
442C22×C4—12216no—
443C22×C4—12216no—
444C22×C4—1222no—
445C22×C4—1222no—
446C22×C4—1224no—
447C22×C4—1221no—
448C22×C4—1221no—
449C22×C4—1222no—
450C22×C4—1222no—
451C22×C4—1222no—
452C22×C4—2442no—
453C22×C4—2442no—
454C22×C4C22×#692311yescharacteristic
455C22×C4C22×#692311yescharacteristic
456C22×C4C22×#692311yescharacteristic
457C22×C4C22×#692311yescharacteristic
458C22×C4C22×#692311yescharacteristic
459C22×C4C22×#692311yescharacteristic
460C2×C8—1224no—
461C2×C8—1224no—
462C2×C8—1224no—
463C2×C8—1224no—
464C2×C8—1224no—
465C2×C8—1224no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—1222no—
469C4⋊C4—1222no—
470C4⋊C4—1222no—
471C4⋊C4—1222no—
472C4⋊C4—1222no—
473C4⋊C4—1222no—
474C22⋊C4—22no—
475C22⋊C4—22no—
476C22⋊C4—1222no—
477C22⋊C4—1222no—
478C22⋊C4—1222no—
479C22⋊C4—1222no—
480C22⋊C4—1222no—
481C22⋊C4—1222no—
482C42—22no—
483C42—612no—
484C42—612no—
485C42—612no—
486C42—612no—
487C42—614no—
488C42—1221no—
489C42—1221no—
490C42—1221no—
491C42—1221no—
492C42—1222no—
493C42C22⋊#692314yes—
494C23—21no—
495C23—21no—
496C23—22no—
497C23—22no—
498C23—612no—
499C23—614no—
500C23—612no—
501C23—1228no—
502C23—1221no—
503C23—1221no—
504C23—1222no—
505C23—1221no—
506C23—1221no—
507C23—1222no—
508C23—1222no—
509C23—2448no—
510C23—2444no—
511C23—2448no—
512C23—2444no—
513C23#1451985612yes—
514C23C23×#692311yescharacteristic
515C2×C4—28no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—611no—
527C2×C4—611no—
528C2×C4—612no—
529C2×C4—612no—
530C2×C4—612no—
531C2×C4—612no—
532C2×C4—612no—
533C2×C4—1224no—
534C2×C4—1224no—
535C2×C4—1224no—
536C2×C4—1224no—
537C2×C4—1228no—
538C2×C4—1228no—
539C2×C4—1228no—
540C2×C4—1224no—
541C2×C4—1224no—
542C2×C4—1221no—
543C2×C4—1221no—
544C2×C4—1222no—
545C2×C4—1222no—
546C2×C4—1221no—
547C2×C4—1221no—
548C2×C4—1222no—
549C2×C4—1222no—
550C2×C4—2444no—
551C2×C4—2444no—
552C2×C4—2444no—
553C2×C4C2×C22⋊#692312yes—
554C2×C4C2×C22⋊#692312yes—
555C2×C4C2×C22⋊#692312yes—
556C2×C4#1451985612yes—
557C8—2444no—
558C8—2444no—
559C22—24no—
560C22—22no—
561C22—41no—
562C22—41no—
563C22—42no—
564C22—614no—
565C22—1222no—
566C22—1224no—
567C22—1224no—
568C22—1222no—
569C22—1224no—
570C22—1222no—
571C22—24416no—
572C22—2441no—
573C22—2441no—
574C22—2442no—
575C22C22×C22⋊#692311yescharacteristic
576C22C2×#1451985611yescharacteristic
577C22C2×#1451985611yescharacteristic
578C22D61⋊#13112yes—
579C22#1684084112yes—
580C4—22no—
581C4—22no—
582C4—22no—
583C4—22no—
584C4—41no—
585C4—41no—
586C4—612no—
587C4—612no—
588C4—1222no—
589C4—1222no—
590C4—1222no—
591C4—2441no—
592C4—2441no—
593C2—42no—
594C2—1224no—
595C2—2442no—
596C2#2146226311yescharacteristic
597C2D61⋊#40811yescharacteristic
598C2#2146240411yescharacteristic
599C2#2146251112yes—
600C2#2146251212yes—
subgroupG/H
center, FrattiniC23C23×#6923
commutator, socleC23×C61C23×C4
FittingC2×#117×C61C4
prime radicalC24×D61C2×C4
cosocleC22×Dic61C24
hypercenterC2×#117#6923
hypocenterC61#20717
LCSG ▷ C23×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
1. C2×#16840784C2semidirect
2. C2×#16840780C2semidirect
3. #21461711C2semidirect
4. #21461678C2semidirect
5. #21461677C2semidirect
7. #21461549C2semidirect
8. #21461548C2semidirect
12. C2×#14519844C22semidirect
13. C2×C4⋊#92420C22semidirect
18. C2×#14519832C22semidirect
19. C2×C4×#92420C22semidirect
20. #16840628C22semidirect
21. #16840628C22semidirect
339. C61#20717semidirect

Maximal subgroups

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

subgroupindex
C2×#168407842
C2×#168407802
#214617112
#214616782
#214616772
#214616422
#214615492
#214615482
C2×#117×D612
#2071761