Finite Group Databasebeta

← #30616492

Subgroups — #30616492

Order 15616.

1202 conjugacy classes of subgroups.

The subgroup lattice has 558 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462268C211yesmaximal, characteristic
2C4×#14519856C211yesmaximal, characteristic
3#21462009C214yesmaximal
4#21462008C214yesmaximal
5#21461519C214yesmaximal
6C2×#16840317C211yesmaximal, characteristic
7C2×#14519856C2212yes—
8C2×#14519856C2212yes—
9#16840708—21no—
10#16840706—21no—
11#16840704C2212yes—
12#16840704C2212yes—
13C4×C4⋊#6923—21no—
14C4×C4⋊#6923—21no—
15C4×C4⋊#6923—21no—
16C4×C4⋊#6923—21no—
17C4×C22⋊#6923C2212yes—
18C4×C22⋊#6923C2212yes—
19#16840698—21no—
20C42×#6923—21no—
21#16840640C22116yes—
22#16840317—21no—
23#16840317—21no—
24#16840317—21no—
25#16840317—21no—
26#16840317—21no—
27#16840317—21no—
28#16840317—21no—
29#16840317—21no—
30C2×#14519777C411yescharacteristic, complemented
31C2×#14519768C412yescomplemented
32C2×#14519757C2212yes—
33C2×#14519753C2212yes—
34C2×Q8×Dic61C412yescomplemented
35C2×#14519742C412yescomplemented
36C2×#14519735C2212yes—
37C2×C42×D61C2211yescharacteristic
38C2×C4⋊Q8×C61C411yescharacteristic, complemented
39#14519856—24no—
40#14519856—28no—
41#14519856Q818yescomplemented
42C2×C4⋊#6923—22no—
43C2×C4⋊#6923—22no—
44C2×C4⋊#6923—22no—
45C2×C4⋊#6923—22no—
46C2×C22⋊#6923C2314yes—
47C2×C22⋊#6923C2314yes—
48C2×C4×#6923—22no—
49C2×C4×#6923—22no—
50C2×C4×#6923—22no—
51C2×C4×#6923—22no—
52(C2×#6923)⋊C4—22no—
53(C2×#6923)⋊C4—22no—
54(C2×#6923)⋊C4—22no—
55(C2×#6923)⋊C4—22no—
56C22×Dic122C2×C412yes—
57C22×C4×D61C2311yescharacteristic
58C22×C4×D61C2311yescharacteristic
59C22×C4×D61C2311yescharacteristic
60#14519777—21no—
61#14519777—21no—
62#14519777—21no—
63#14519777—21no—
64#14519768—22no—
65#14519768—22no—
66#14519768—22no—
67#14519768—22no—
68#14519757—22no—
69#14519757—22no—
70#14519757—22no—
71#14519757—22no—
72#14519753—22no—
73#14519753—22no—
74#14519753—22no—
75#14519753—22no—
76Q8×Dic61—22no—
77Q8×Dic61—22no—
78Q8×Dic61—22no—
79Q8×Dic61—22no—
80#14519742—22no—
81#14519742—22no—
82#14519742—22no—
83#14519742—22no—
84#14519735—22no—
85#14519735—22no—
86#14519735—22no—
87#14519735—22no—
88C2×C4⋊Dic61C2×C412yes—
89C2×C4⋊Dic61C2×C412yes—
90C2×C4⋊Dic61C2×C414yes—
91C2×#11758196C2×C414yes—
92C2×#11758198C2314yes—
93C2×C4×Dic61C2×C414yes—
94C2×C4×Dic61C2×C411yescharacteristic
95C2×C4×Dic61C2×C411yescharacteristic
96C2×C4×Dic61C2×C411yescharacteristic
97C42×D61D411yescharacteristic
98C42×D61D411yescharacteristic
99C42×D61D411yescharacteristic
100C42×D61D411yescharacteristic
101C42×D61D411yescharacteristic
102C42×D61D411yescharacteristic
103C42×D61D411yescharacteristic
104C42×D61D411yescharacteristic
105C22×Q8×C61C2×C412yes—
106C4⋊Q8×C61—21no—
107C4⋊Q8×C61—21no—
108C4⋊Q8×C61—21no—
109C4⋊Q8×C61—21no—
110C2×C4⋊C4×C61C2×C412yes—
111C2×C4⋊C4×C61C2×C412yes—
112C2×C42×C61C2×C411yescharacteristic
113C23×D61C2411yescharacteristic
114C22×#6923—24no—
115C22×#6923—24no—
116C2×Q8×C61—22no—
117C2×Q8×C61—22no—
118C2×Q8×C61—44no—
119C22×C4×C61C22×C411yescharacteristic
120C22×C4×C61C22×C411yescharacteristic
121C22×C4×C61C22×C414yes—
122C22×C4×C61C22×C411yescharacteristic
123C22×Dic61C22×C414yes—
124C22×Dic61C22×C411yescharacteristic
125C22×Dic61C22×C411yescharacteristic
126C22×Dic61C22×C411yescharacteristic
127C2×C4×D61—42no—
128C2×C4×D61—42no—
129C2×C4×D61C2×D411yescharacteristic
130C2×C4×D61C2×D411yescharacteristic
131C2×C4×D61C2×D411yescharacteristic
132C2×C4×D61C2×D411yescharacteristic
133C2×C4×D61C2×D411yescharacteristic
134C2×C4×D61C2×D411yescharacteristic
135C2×C4×D61C2×D411yescharacteristic
136C2×C4×D61C2×D411yescharacteristic
137C2×C4×D61C2×D411yescharacteristic
138C2×C4×D61C2×D411yescharacteristic
139C2×C4×D61C2×D411yescharacteristic
140C2×C4×D61C2×D411yescharacteristic
141C2×C4×D61C2×Q814yes—
142C2×C4×D61#4114yes—
143C2×Dic122—22no—
144C2×Dic122—22no—
145C2×Dic122—44no—
146C22⋊#6923—24no—
147C22⋊#6923—28no—
148C22⋊#6923C2×Q818yes—
149C4⋊#6923—28no—
150C4⋊#6923—44no—
151C4⋊#6923—44no—
152C4⋊#6923—44no—
153C4×#6923—28no—
154C4×#6923—44no—
155C4×#6923—44no—
156C4×#6923—44no—
157C4⋊C4×C61—22no—
158C4⋊C4×C61—22no—
159C4⋊C4×C61—22no—
160C4⋊C4×C61—22no—
161C42×C61C22⋊C411yescharacteristic
162C42×C61C22⋊C411yescharacteristic
163C42×C61C22⋊C411yescharacteristic
164C42×C61C22⋊C411yescharacteristic
165#11758198—24no—
166#11758198—24no—
167C4⋊Dic61—22no—
168C4⋊Dic61—22no—
169C4⋊Dic61—22no—
170C4⋊Dic61—22no—
171C4⋊Dic61—24no—
172C4⋊Dic61—24no—
173#11758196—24no—
174#11758196—24no—
175C4×Dic61—24no—
176C4×Dic61—24no—
177C4×Dic61C22⋊C411yescharacteristic
178C4×Dic61C22⋊C411yescharacteristic
179C4×Dic61C22⋊C411yescharacteristic
180C4×Dic61C22⋊C411yescharacteristic
181C4×Dic61C22⋊C411yescharacteristic
182C4×Dic61C22⋊C411yescharacteristic
183C4×Dic61C22⋊C411yescharacteristic
184C4×Dic61C22⋊C411yescharacteristic
185C4×Dic61C22⋊C411yescharacteristic
186C4×Dic61C22⋊C411yescharacteristic
187C4×Dic61C22⋊C411yescharacteristic
188C4×Dic61C22⋊C411yescharacteristic
189C23×C61C23×C411yescharacteristic
190C22×D61—41no—
191C22×D61—41no—
192C22×D61C22×D411yescharacteristic
193C22×D61C22×D411yescharacteristic
194C22×D61C22×Q812yes—
195C22×D61C2×#4112yes—
196C2×#6923—28no—
197C2×#6923—44no—
198C2×#6923—44no—
199C2×#6923—44no—
200C2×#6923—44no—
201C2×#6923—44no—
202Q8×C61—44no—
203Q8×C61—44no—
204C2×C4×C61—22no—
205C2×C4×C61—22no—
206C2×C4×C61—24no—
207C2×C4×C61—44no—
208C2×C4×C61C2×C22⋊C411yescharacteristic
209C2×C4×C61C2×C22⋊C411yescharacteristic
210C2×C4×C61C2×C22⋊C411yescharacteristic
211C2×C4×C61C2×C22⋊C411yescharacteristic
212C2×C4×C61C2×C22⋊C411yescharacteristic
213C2×C4×C61C2×C22⋊C411yescharacteristic
214C2×C4×C61C4×Q814yes—
215C2×Dic61—24no—
216C2×Dic61—41no—
217C2×Dic61—41no—
218C2×Dic61—44no—
219C2×Dic61C2×C22⋊C411yescharacteristic
220C2×Dic61C2×C22⋊C411yescharacteristic
221C2×Dic61C2×C22⋊C411yescharacteristic
222C2×Dic61C2×C22⋊C411yescharacteristic
223C2×Dic61C2×C22⋊C411yescharacteristic
224C2×Dic61C2×C22⋊C411yescharacteristic
225C2×Dic61C4×Q814yes—
226C4×D61—22no—
227C4×D61—22no—
228C4×D61—22no—
229C4×D61—22no—
230C4×D61—41no—
231C4×D61—41no—
232C4×D61—41no—
233C4×D61—41no—
234C4×D61C22⋊Q814yes—
235C4×D61C22⋊Q814yes—
236C4×D61#12412yes—
237C4×D61#12412yes—
238C4×D61C4⋊Q812yes—
239C4×D61C4⋊Q812yes—
240Dic122—44no—
241Dic122—44no—
242#21332—611nomaximal, Sylow
243C22×C61—21no—
244C22×C61—21no—
245C22×C61C22×C22⋊C411yescharacteristic
246C22×C61C2×C4×Q812yes—
247C2×D61—21no—
248C2×D61—21no—
249C2×D61—21no—
250C2×D61—21no—
251C2×D61—42no—
252C2×D61—42no—
253C2×D61C2×C22⋊Q812yes—
254C2×D61C2×#12411yescharacteristic
255C2×D61C2×C4⋊Q811yescharacteristic
256#6923—48no—
257#6923—84no—
258C4×C61—21no—
259C4×C61—21no—
260C4×C61—44no—
261C4×C61#39112yes—
262C4×C61#39112yes—
263C4×C61#41012yes—
264C4×C61#41012yes—
265Dic61—21no—
266Dic61—21no—
267Dic61—21no—
268Dic61—21no—
269Dic61—44no—
270Dic61#39112yes—
271Dic61#39112yes—
272C2×#533—611no—
273#2285—611no—
274C4×#117—611no—
275#1996—614no—
276#1996—614no—
277#1740—614no—
278C2×C61—22no—
279C2×C61C2×#39111yescharacteristic
280C2×C61C2×#41011yescharacteristic
281C2×C61C2×#41011yescharacteristic
282D61—22no—
283D61—21no—
284D61—21no—
285#533—1221no—
286#533—1221no—
287#533—1221no—
288#533—1221no—
289#533—1221no—
290#533—1221no—
291#533—1221no—
292#533—1221no—
293C2×C4⋊Q8—611no—
294C2×C4⋊Q8#692311yescharacteristic, complemented
295C2×#125—612no—
296C2×C22⋊Q8—612no—
297C2×C22⋊Q8—612no—
298C2×C4×Q8—612no—
299C2×C4×Q8—612no—
300C2×#117—612no—
301C2×#117—612no—
302C2×#117—612no—
303C22×C42—611no—
304#384—1221no—
305#382—1221no—
306#380—612no—
307#380—612no—
308C4×C4⋊C4—1221no—
309C4×C4⋊C4—1221no—
310C4×C4⋊C4—1221no—
311C4×C4⋊C4—1221no—
312C4×C22⋊C4—612no—
313C4×C22⋊C4—612no—
314#376—1221no—
315C43—1221no—
316#342—6116no—
317C61#2133211yescharacteristic, complemented, Sylow
318C22×Q8—612no—
319C22×Q8C2×#692312yes—
320C23×C4—611no—
321C23×C4—611no—
322C23×C4—611no—
323C4⋊Q8—21no—
324C4⋊Q8—21no—
325C4⋊Q8—21no—
326C4⋊Q8—21no—
327C4⋊Q8—1221no—
328C4⋊Q8—1221no—
329C4⋊Q8—1221no—
330C4⋊Q8—1221no—
331#125—1222no—
332#125—1222no—
333#125—1222no—
334#125—1222no—
335C22⋊Q8—1222no—
336C22⋊Q8—1222no—
337C22⋊Q8—1222no—
338C22⋊Q8—1222no—
339C22⋊Q8—1222no—
340C22⋊Q8—1222no—
341C22⋊Q8—1222no—
342C22⋊Q8—1222no—
343C4×Q8—1222no—
344C4×Q8—1222no—
345C4×Q8—1222no—
346C4×Q8—1222no—
347C4×Q8—1222no—
348C4×Q8—1222no—
349C4×Q8—1222no—
350C4×Q8—1222no—
351#117—618no—
352#117—1222no—
353#117—1222no—
354#117—1222no—
355#117—1222no—
356#117—1224no—
357#117—1228no—
358C2×C4⋊C4—612no—
359C2×C4⋊C4—614no—
360C2×C4⋊C4—614no—
361C2×C4⋊C4—612no—
362C2×C4⋊C4—1222no—
363C2×C4⋊C4—1222no—
364C2×C4⋊C4—1222no—
365C2×C4⋊C4—1222no—
366C2×C4⋊C4C2×#692312yes—
367C2×C4⋊C4C2×#692312yes—
368C2×C22⋊C4—614no—
369C2×C22⋊C4—614no—
370C2×C22⋊C4—614no—
371C2×C42—611no—
372C2×C42—611no—
373C2×C42—611no—
374C2×C42—611no—
375C2×C42—611no—
376C2×C42—611no—
377C2×C42—611no—
378C2×C42—611no—
379C2×C42—611no—
380C2×C42—611no—
381C2×C42—611no—
382C2×C42—614no—
383C2×C42—1222no—
384C2×C42—1222no—
385C2×C42—1222no—
386C2×C42—1222no—
387C2×C42C2×#692311yescharacteristic
388#95—1222no—
389#95—1222no—
390#95—1222no—
391#95—1222no—
392C24—611no—
393C2×Q8—22no—
394C2×Q8—22no—
395C2×Q8—44no—
396C2×Q8—1222no—
397C2×Q8—1222no—
398C2×Q8—2444no—
399C22×C4—611no—
400C22×C4—611no—
401C22×C4—611no—
402C22×C4—611no—
403C22×C4—611no—
404C22×C4—614no—
405C22×C4—614no—
406C22×C4—611no—
407C22×C4—611no—
408C22×C4—611no—
409C22×C4—611no—
410C22×C4—611no—
411C22×C4—611no—
412C22×C4—611no—
413C22×C4—611no—
414C22×C4—611no—
415C22×C4—611no—
416C22×C4—614no—
417C22×C4—1224no—
418C22×C4—1224no—
419C22×C4—2442no—
420C22×C4—2442no—
421C22×C4C22×#692311yescharacteristic
422C22×C4C22×#692311yescharacteristic
423C22×C4C22×#692314yes—
424C22×C4C22×#692311yescharacteristic
425C4⋊C4—22no—
426C4⋊C4—22no—
427C4⋊C4—22no—
428C4⋊C4—22no—
429C4⋊C4—1224no—
430C4⋊C4—1224no—
431C4⋊C4—1224no—
432C4⋊C4—1224no—
433C4⋊C4—1228no—
434C4⋊C4—1222no—
435C4⋊C4—1222no—
436C4⋊C4—1222no—
437C4⋊C4—1222no—
438C4⋊C4—2444no—
439C4⋊C4—2444no—
440C4⋊C4—2444no—
441C22⋊C4—618no—
442C22⋊C4—1224no—
443C22⋊C4—1224no—
444C22⋊C4—1224no—
445C22⋊C4—1228no—
446C42—611no—
447C42—611no—
448C42—611no—
449C42—611no—
450C42—611no—
451C42—611no—
452C42—611no—
453C42—611no—
454C42—611no—
455C42—611no—
456C42—611no—
457C42—611no—
458C42—1224no—
459C42—1224no—
460C42—1228no—
461C42—2444no—
462C42—2444no—
463C42—2444no—
464C42C22⋊#692311yescharacteristic
465C42C22⋊#692311yescharacteristic
466C42C22⋊#692311yescharacteristic
467C42C22⋊#692311yescharacteristic
468C23—611no—
469C23—611no—
470C23—612no—
471C23—612no—
472C23—2441no—
473C23—2441no—
474C23C23×#692311yescharacteristic
475Q8—44no—
476Q8—44no—
477Q8—2444no—
478Q8—2444no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—24no—
482C2×C4—44no—
483C2×C4—611no—
484C2×C4—611no—
485C2×C4—614no—
486C2×C4—614no—
487C2×C4—614no—
488C2×C4—612no—
489C2×C4—612no—
490C2×C4—612no—
491C2×C4—612no—
492C2×C4—611no—
493C2×C4—611no—
494C2×C4—611no—
495C2×C4—611no—
496C2×C4—1222no—
497C2×C4—1222no—
498C2×C4—1222no—
499C2×C4—1222no—
500C2×C4—1228no—
501C2×C4—1224no—
502C2×C4—2441no—
503C2×C4—2441no—
504C2×C4—2441no—
505C2×C4—2441no—
506C2×C4—2441no—
507C2×C4—2441no—
508C2×C4—2444no—
509C2×C4—2444no—
510C2×C4—2444no—
511C2×C4—2444no—
512C2×C4—2444no—
513C2×C4—2444no—
514C2×C4C2×C22⋊#692311yescharacteristic
515C2×C4C2×C22⋊#692311yescharacteristic
516C2×C4C2×C22⋊#692311yescharacteristic
517C2×C4C2×C22⋊#692311yescharacteristic
518C2×C4C2×C22⋊#692311yescharacteristic
519C2×C4C2×C22⋊#692311yescharacteristic
520C2×C4Q8×#692314yes—
521C22—21no—
522C22—21no—
523C22—612no—
524C22—611no—
525C22—611no—
526C22—1221no—
527C22—1221no—
528C22—1221no—
529C22—1221no—
530C22—2442no—
531C22—2442no—
532C22C22×C22⋊#692311yescharacteristic
533C22C2×Q8×#692312yes—
534C4—21no—
535C4—21no—
536C4—44no—
537C4—612no—
538C4—612no—
539C4—1221no—
540C4—1221no—
541C4—1221no—
542C4—1221no—
543C4—4884no—
544C4—2448no—
545C4—2444no—
546C4(C2×#6923)⋊Q812yes—
547C4(C2×#6923)⋊Q812yes—
548C4#1684074212yes—
549C4#1684074312yes—
550C2—22no—
551C2—1221no—
552C2—1222no—
553C2—1221no—
554C2C2×(C2×#6923)⋊Q811yescharacteristic
555C2C2×#1684074211yescharacteristic
556C2C2×#1684074311yescharacteristic
subgroupG/H
centerC22C22×C22⋊#6923
commutatorC23×C61C23×C4
FrattiniC23C23×#6923
FittingC2×C4⋊Q8×C61C4
socleC22×C61C22×C22⋊C4
prime radical, cosocleC23×D61C24
hypercenterC2×C4⋊Q8#6923
hypocenterC61#21332
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C4⋊Q8

Complements

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

NH ≅ G/Ntype
30. C2×#14519777C4semidirect
31. C2×#14519768C4semidirect
34. C2×Q8×Dic61C4semidirect
35. C2×#14519742C4semidirect
38. C2×C4⋊Q8×C61C4semidirect
41. #14519856Q8semidirect
294. C2×C4⋊Q8#6923semidirect
317. C61#21332semidirect

Maximal subgroups

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

subgroupindex
#214622682
C4×#145198562
#214620092
#214620082
#214615192
C2×#168403172
#2133261