Finite Group Databasebeta

← #30616667

Subgroups — #30616667

Order 15616.

1414 conjugacy classes of subgroups.

The subgroup lattice has 598 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840782C211yesmaximal, characteristic, complemented
2C2×C4×#11758217C211yesmaximal, characteristic, complemented
3#21461990C212yesmaximal, complemented
4#21461942C212yesmaximal, complemented
5#21461798C212yesmaximal, complemented
6#21461796C212yesmaximal, complemented
7#21461546C212yesmaximal, complemented
8#21461545C212yesmaximal, complemented
9C2×C4×#11758224C211yesmaximal, characteristic
10C22×#11758217C2212yes—
11#16840782—24no—
12C2×#14519844C2212yes—
13C2×#14519834C2214yes—
14C2×C22⋊#92420C2212yes—
15C2×C22⋊#92420C2214yes—
16C4×#11758217—24no—
17C2×#14519832C2212yes—
18C2×#14519832C2212yes—
19C2×C4×#92420C2212yes—
20#16840628C2214yes—
21#16840628C2214yes—
22C22×#11758224C411yescharacteristic
23C4×#11758224—24no—
24C4×#11758224—24no—
25C2×C4×Dic122C411yescharacteristic
26C2×#14519742C2211yescharacteristic
27C2×D4×Dic61C2211yescharacteristic
28C2×C4×C22⋊D61C411yescharacteristic
29C2×C4×C22⋊D61C411yescharacteristic
30C2×#14519737C2211yescharacteristic
31C2×#14519737C2211yescharacteristic
32C2×#14519735C411yescharacteristic
33C2×#14519733C411yescharacteristic
34C2×#14519733C411yescharacteristic
35C22×C4×Dic61C2211yescharacteristic
36C22×C4×Dic61C2211yescharacteristic
37C2×C42×D61C2211yescharacteristic
38C2×C4×D4×C61C411yescharacteristic
39C2×#11758217—24no—
40C2×#11758217—28no—
41C22×#92420C2314yes—
42C22×#92420C2314yes—
43#14519844—24no—
44#14519834—28no—
45C22⋊#92420—24no—
46C22⋊#92420—28no—
47#14519832—24no—
48#14519832—24no—
49C4×#92420—24no—
50C2×#11758224—24no—
51C2×#11758224—24no—
52C2×#11758224—24no—
53C22×Dic122C2×C411yescharacteristic
54C22×C22⋊D61C2×C411yescharacteristic
55C22×C22⋊D61C2×C411yescharacteristic
56C23×Dic61C2×C411yescharacteristic
57C23×Dic61C2×C411yescharacteristic
58C23×Dic61C2311yescharacteristic
59C23×Dic61C2311yescharacteristic
60C22×C4×D61C2×C411yescharacteristic
61C22×C4×D61C2×C411yescharacteristic
62C22×C4×D61C2311yescharacteristic
63C4×Dic122—24no—
64#14519742—22no—
65#14519742—22no—
66D4×Dic61—22no—
67D4×Dic61—22no—
68C4×C22⋊D61—24no—
69C4×C22⋊D61—24no—
70#14519737—22no—
71#14519737—22no—
72#14519737—22no—
73#14519737—22no—
74#14519735—24no—
75#14519733—24no—
76#14519733—24no—
77C2×C4⋊Dic61C2×C411yescharacteristic
78C2×#11758196C2×C411yescharacteristic
79C2×#11758196C2×C411yescharacteristic
80C2×#11758196C2×C411yescharacteristic
81C2×#11758196C2×C411yescharacteristic
82C2×C22⋊Dic61C2×C411yescharacteristic
83C2×C22⋊Dic61C2×C411yescharacteristic
84C2×#11758198C2×C411yescharacteristic
85C2×#11758198C2×C411yescharacteristic
86C2×C4×Dic61—22no—
87C2×C4×Dic61—22no—
88C2×C4×Dic61C2×C411yescharacteristic
89C2×C4×Dic61C2×C411yescharacteristic
90C2×C4×Dic61C2×C411yescharacteristic
91C2×C4×Dic61C2×C411yescharacteristic
92C2×C4×Dic61C2×C411yescharacteristic
93C2×C4×Dic61D414yes—
94C2×C4×Dic61D414yes—
95C2×C4×Dic61C2311yescharacteristic
96C2×C4×Dic61C2311yescharacteristic
97C2×C4×Dic61C2311yescharacteristic
98C2×C4×Dic61C2311yescharacteristic
99C42×D61—22no—
100C42×D61—22no—
101C22×D4×C61C2×C411yescharacteristic
102C23×C4×C61C2×C411yescharacteristic
103C23×C4×C61C2×C411yescharacteristic
104C4×D4×C61—24no—
105C2×C4⋊C4×C61C2×C411yescharacteristic
106C2×C22⋊C4×C61C2×C411yescharacteristic
107C2×C22⋊C4×C61C2×C411yescharacteristic
108C2×C42×C61C2×C411yescharacteristic
109C24×C61C22×C411yescharacteristic
110C24×C61C22×C411yescharacteristic
111C23×D61C22×C411yescharacteristic
112C2×D4×C61—22no—
113C2×D4×C61—22no—
114C2×D4×C61—22no—
115C22×C4×C61—22no—
116C22×C4×C61—42no—
117C22×C4×C61—42no—
118C22×C4×C61C22⋊C414yes—
119C22×C4×C61C22×C411yescharacteristic
120C22×C4×C61C22×C411yescharacteristic
121C22×C4×C61C22×C411yescharacteristic
122C22×C4×C61C22×C411yescharacteristic
123C22×C4×C61C22×C411yescharacteristic
124C2×C22⋊D61—22no—
125C2×C22⋊D61—22no—
126C2×C22⋊D61—22no—
127C2×C22⋊D61—22no—
128C2×C22⋊D61—22no—
129C2×C22⋊D61M1614yescomplemented
130C22×Dic61—22no—
131C22×Dic61—22no—
132C22×Dic61—22no—
133C22×Dic61—22no—
134C22×Dic61—22no—
135C22×Dic61—24no—
136C22×Dic61—21no—
137C22×Dic61—21no—
138C22×Dic61—24no—
139C22×Dic61C22⋊C414yes—
140C22×Dic61C22×C411yescharacteristic
141C22×Dic61C22×C411yescharacteristic
142C22×Dic61C22×C411yescharacteristic
143C22×Dic61C22×C411yescharacteristic
144C22×Dic61C22×C411yescharacteristic
145C22×Dic61C22×C411yescharacteristic
146C22×Dic61C2×D412yes—
147C22×Dic61C2×D412yes—
148C22×Dic61C2411yescharacteristic
149#11758224—232no—
150C2×C4×D61—22no—
151C2×C4×D61—22no—
152C2×C4×D61—22no—
153C2×C4×D61—22no—
154C2×C4×D61—21no—
155C2×C4×D61—21no—
156C2×C4×D61—24no—
157C2×C4×D61—42no—
158C2×Dic122—22no—
159C2×Dic122—22no—
160C2×Dic122M1614yescomplemented
161#11758217—216no—
162C2×#92420—24no—
163C2×#92420—24no—
164C2×#92420—24no—
165C2×#92420—24no—
166C2×#92420—24no—
167C2×#92420#4118yescomplemented
168C4⋊C4×C61—22no—
169C22⋊C4×C61—22no—
170C22⋊C4×C61—22no—
171C42×C61—22no—
172C22⋊Dic61—22no—
173C22⋊Dic61—22no—
174#11758198—22no—
175#11758198—22no—
176C4⋊Dic61—22no—
177#11758196—22no—
178#11758196—22no—
179#11758196—22no—
180#11758196—22no—
181C4×Dic61—22no—
182C4×Dic61—22no—
183C4×Dic61—22no—
184C4×Dic61—21no—
185C4×Dic61—21no—
186C4×Dic61—21no—
187C4×Dic61—21no—
188C4×Dic61C22⋊C414yes—
189C4×Dic61C22⋊C414yes—
190C4×Dic61C2×D412yes—
191C4×Dic61C2×D412yes—
192C4×Dic61C2×D412yes—
193C4×Dic61C2×D412yes—
194C23×C61—21no—
195C23×C61—22no—
196C23×C61—22no—
197C23×C61—21no—
198C23×C61—21no—
199C23×C61—21no—
200C23×C61—22no—
201C23×C61—22no—
202C23×C61C2×C22⋊C412yes—
203C23×C61C2×M1612yes—
204C23×C61C23×C411yescharacteristic
205C22×D61—21no—
206C22×D61—22no—
207C22×D61—21no—
208C22×D61—22no—
209C22×D61C2×M1612yes—
210D4×C61—28no—
211C2×C4×C61—21no—
212C2×C4×C61—21no—
213C2×C4×C61—22no—
214C2×C4×C61—22no—
215C2×C4×C61—21no—
216C2×C4×C61—21no—
217C2×C4×C61—22no—
218C2×C4×C61—22no—
219C2×C4×C61—44no—
220C2×C4×C61—44no—
221C2×C4×C61C2×C22⋊C412yes—
222C2×C4×C61C2×C22⋊C412yes—
223C2×C4×C61C2×M1612yes—
224C22⋊D61—28no—
225C22⋊D61—28no—
226C2×Dic61—28no—
227C2×Dic61—21no—
228C2×Dic61—22no—
229C2×Dic61—21no—
230C2×Dic61—21no—
231C2×Dic61—21no—
232C2×Dic61—24no—
233C2×Dic61—24no—
234C2×Dic61—28no—
235C2×Dic61—21no—
236C2×Dic61—21no—
237C2×Dic61—22no—
238C2×Dic61—22no—
239C2×Dic61—42no—
240C2×Dic61—42no—
241C2×Dic61C2×C22⋊C412yes—
242C2×Dic61C2×C22⋊C412yes—
243C2×Dic61C2×C22⋊C412yes—
244C2×Dic61C2×M1612yes—
245C2×Dic61C2×M1612yes—
246C2×Dic61C2×M1612yes—
247C2×Dic61C22×D411yescharacteristic
248C2×Dic61C22×D411yescharacteristic
249C2×Dic61C2×#4114yes—
250C4×D61—28no—
251C4×D61—44no—
252C4×D61—42no—
253C4×D61—42no—
254Dic122—28no—
255#92420—28no—
256#92420—44no—
257#21380—611nomaximal, Sylow
258C22×C61—22no—
259C22×C61—22no—
260C22×C61—22no—
261C22×C61—24no—
262C22×C61—22no—
263C22×C61—41no—
264C22×C61—42no—
265C22×C61—42no—
266C22×C61—42no—
267C22×C61—41no—
268C22×C61C22×C22⋊C411yescharacteristic
269C22×C61C4×#4112yes—
270C22×C61C22×M1611yescharacteristic
271C22×C61C22×M1611yescharacteristic
272C22×C61#56812yes—
273C2×D61—22no—
274C2×D61—24no—
275C2×D61—41no—
276C2×D61—42no—
277C2×D61—41no—
278C4×C61—22no—
279C4×C61—22no—
280C4×C61—22no—
281C4×C61—41no—
282C4×C61—41no—
283Dic61—22no—
284Dic61—22no—
285Dic61—22no—
286Dic61—41no—
287Dic61—41no—
288Dic61#40714yes—
289Dic61#52512yes—
290Dic61#53212yes—
291C2×C4×#41—611no—
292C2×#432—611no—
293C2×C4×M16—611no—
294#1984—612no—
295#1954—612no—
296#1870—612no—
297#1870—612no—
298#1749—612no—
299#1749—612no—
300C2×C61—42no—
301C2×C61—42no—
302C2×C61#229711yescharacteristic
303C2×C61C2×#40711yescharacteristic
304C2×C61#287911yescharacteristic
305C2×C61#297012yes—
306C2×C61#297112yes—
307D61—42no—
308C22×#41—611no—
309C22×M16—612no—
310C4×#41—1224no—
311C4×#41—1224no—
312C2×C4×Q8—611no—
313C2×C4×Q8—611no—
314C2×C4×D4—611no—
315C2×C4×D4—611no—
316C2×C4×D4—611no—
317C2×C4×D4—611no—
318C2×C4×D4—611no—
319C2×C4×D4#692311yescharacteristic
320C2×#117—611no—
321C2×#117—611no—
322C2×#117—611no—
323C22×C42—611no—
324C22×C42—611no—
325C22×C42—611no—
326#432—1224no—
327C2×C4⋊C8—612no—
328C2×C22⋊C8—612no—
329C2×C22⋊C8—614no—
330C2×C22⋊C8—614no—
331C4×M16—1224no—
332C2×#97—612no—
333C2×#97—612no—
334C2×C4×C8—612no—
335#336—614no—
336#336—614no—
337C61#2138011yescharacteristic, complemented, Sylow
338C2×#41—1224no—
339C2×#41—1224no—
340C2×#41—1224no—
341C22×Q8—611no—
342C22×D4—611no—
343C22×D4—611no—
344C22×D4C2×#692311yescharacteristic
345C23×C4—611no—
346C23×C4—611no—
347C23×C4—611no—
348C23×C4—611no—
349C23×C4—611no—
350C23×C4—611no—
351C23×C4—611no—
352C23×C4C2×#692311yescharacteristic
353C23×C4C2×#692311yescharacteristic
354C2×M16—1228no—
355C2×M16—1224no—
356C22×C8—614no—
357C22×C8—614no—
358C4×Q8—1224no—
359C4×Q8—1222no—
360C4×Q8—1222no—
361C4×D4—24no—
362C4×D4—1222no—
363C4×D4—1222no—
364C4×D4—1224no—
365C4×D4—1222no—
366C4×D4—1222no—
367C4×D4—1222no—
368C4×D4—1222no—
369C4×D4—1224no—
370#117—1224no—
371#117—1224no—
372#117—1224no—
373C2×C4⋊C4—611no—
374C2×C4⋊C4—611no—
375C2×C4⋊C4—611no—
376C2×C4⋊C4—611no—
377C2×C4⋊C4—611no—
378C2×C4⋊C4C2×#692311yescharacteristic
379C2×C22⋊C4—611no—
380C2×C22⋊C4—611no—
381C2×C22⋊C4—611no—
382C2×C22⋊C4—611no—
383C2×C22⋊C4C2×#692311yescharacteristic
384C2×C22⋊C4C2×#692311yescharacteristic
385C2×C42—614no—
386C2×C42—614no—
387C2×C42—611no—
388C2×C42—611no—
389C2×C42—611no—
390C2×C42—611no—
391C2×C42—611no—
392C2×C42—611no—
393C2×C42—611no—
394C2×C42—611no—
395C2×C42—611no—
396C2×C42—1222no—
397C2×C42—1222no—
398C2×C42—1222no—
399C2×C42—1222no—
400C2×C42C2×#692311yescharacteristic
401C4⋊C8—1224no—
402C22⋊C8—1228no—
403C22⋊C8—1228no—
404C22⋊C8—1224no—
405#97—1224no—
406#97—1224no—
407C4×C8—1224no—
408C24—611no—
409C24C22×#692311yescharacteristic
410C24C22×#692311yescharacteristic
411#41—12232no—
412C2×Q8—614no—
413C2×Q8—1222no—
414C2×Q8—1222no—
415C2×D4—22no—
416C2×D4—22no—
417C2×D4—22no—
418C2×D4—614no—
419C2×D4—1222no—
420C2×D4—1222no—
421C2×D4—1222no—
422C2×D4—1222no—
423C2×D4—1222no—
424C22×C4—22no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—611no—
428C22×C4—611no—
429C22×C4—612no—
430C22×C4—614no—
431C22×C4—612no—
432C22×C4—611no—
433C22×C4—611no—
434C22×C4—611no—
435C22×C4—611no—
436C22×C4—611no—
437C22×C4—1222no—
438C22×C4—1224no—
439C22×C4—1221no—
440C22×C4—1221no—
441C22×C4—1222no—
442C22×C4—1222no—
443C22×C4—1224no—
444C22×C4—1222no—
445C22×C4—1222no—
446C22×C4—1222no—
447C22×C4—1224no—
448C22×C4—1221no—
449C22×C4—1221no—
450C22×C4—1222no—
451C22×C4—1222no—
452C22×C4—1222no—
453C22×C4—2442no—
454C22×C4C22⋊#692314yes—
455C22×C4C22×#692311yescharacteristic
456C22×C4C22×#692311yescharacteristic
457C22×C4C22×#692311yescharacteristic
458C22×C4C22×#692311yescharacteristic
459C22×C4C22×#692311yescharacteristic
460M16—12216no—
461C2×C8—618no—
462C2×C8—1224no—
463C2×C8—1224no—
464C2×C8—1224no—
465C2×C8—1224no—
466C2×C8—1224no—
467C4⋊C4—22no—
468C4⋊C4—1222no—
469C4⋊C4—1222no—
470C4⋊C4—1222no—
471C4⋊C4—1222no—
472C4⋊C4—1222no—
473C22⋊C4—22no—
474C22⋊C4—22no—
475C22⋊C4—1222no—
476C22⋊C4—1222no—
477C22⋊C4—1222no—
478C22⋊C4—1222no—
479C42—22no—
480C42—612no—
481C42—612no—
482C42—612no—
483C42—612no—
484C42—614no—
485C42—614no—
486C42—1222no—
487C42—1222no—
488C42—1221no—
489C42—1221no—
490C42—1221no—
491C42—1221no—
492C42—1222no—
493C23—21no—
494C23—22no—
495C23—22no—
496C23—21no—
497C23—21no—
498C23—21no—
499C23—22no—
500C23—22no—
501C23—612no—
502C23—1221no—
503C23—1222no—
504C23—1221no—
505C23—1222no—
506C23C2×C22⋊#692312yes—
507C23D61⋊M1612yes—
508C23C23×#692311yescharacteristic
509Q8—1228no—
510D4—28no—
511D4—1228no—
512D4—1228no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—44no—
522C2×C4—44no—
523C2×C4—612no—
524C2×C4—611no—
525C2×C4—611no—
526C2×C4—614no—
527C2×C4—612no—
528C2×C4—612no—
529C2×C4—612no—
530C2×C4—612no—
531C2×C4—612no—
532C2×C4—1228no—
533C2×C4—1224no—
534C2×C4—1224no—
535C2×C4—1228no—
536C2×C4—1228no—
537C2×C4—1221no—
538C2×C4—1222no—
539C2×C4—1221no—
540C2×C4—1221no—
541C2×C4—1221no—
542C2×C4—1221no—
543C2×C4—1221no—
544C2×C4—1222no—
545C2×C4—1222no—
546C2×C4—2444no—
547C2×C4—2442no—
548C2×C4—2442no—
549C2×C4—2442no—
550C2×C4—2442no—
551C2×C4C2×C22⋊#692312yes—
552C2×C4C2×C22⋊#692312yes—
553C2×C4D61⋊M1612yes—
554C8—1228no—
555C8—2444no—
556C22—22no—
557C22—22no—
558C22—22no—
559C22—22no—
560C22—24no—
561C22—41no—
562C22—41no—
563C22—42no—
564C22—42no—
565C22—42no—
566C22—1222no—
567C22—1224no—
568C22—2441no—
569C22—2441no—
570C22—2442no—
571C22C22×C22⋊#692311yescharacteristic
572C22#41×#692312yes—
573C22C2×D61⋊M1611yescharacteristic
574C22C2×D61⋊M1611yescharacteristic
575C22#1684084012yes—
576C4—22no—
577C4—22no—
578C4—22no—
579C4—41no—
580C4—41no—
581C4—612no—
582C4—612no—
583C4—614no—
584C4—1222no—
585C4—1222no—
586C4—1222no—
587C4—2441no—
588C4—2441no—
589C2—42no—
590C2—42no—
591C2—2442no—
592C2#2146228011yescharacteristic
593C2D61⋊#40711yescharacteristic
594C2#2146240311yescharacteristic
595C2#2146265912yes—
596C2#2146266412yes—
subgroupG/H
center, FrattiniC23C23×#6923
commutator, socleC23×C61C23×C4
FittingC2×C4×D4×C61C4
prime radicalC22×#11758224C4
cosocleC22×Dic61C24
hypercenterC2×C4×D4#6923
hypocenterC61#21380
LCSG ▷ C23×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
1. C2×#16840782C2semidirect
2. C2×C4×#11758217C2semidirect
3. #21461990C2semidirect
4. #21461942C2semidirect
5. #21461798C2semidirect
6. #21461796C2semidirect
7. #21461546C2semidirect
8. #21461545C2semidirect
129. C2×C22⋊D61M16semidirect
160. C2×Dic122M16semidirect
167. C2×#92420#41semidirect
337. C61#21380semidirect

Maximal subgroups

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

subgroupindex
C2×#168407822
C2×C4×#117582172
#214619902
#214619422
#214617982
#214617962
#214615462
#214615452
C2×C4×#117582242
#2138061