Finite Group Databasebeta

← #29680030

Subgroups — #29680030

Order 15616.

1326 conjugacy classes of subgroups.

The subgroup lattice has 526 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×Q8×D61C211yesmaximal, characteristic
2C2×#16839813C211yesmaximal, characteristic
3#21448215C211yesmaximal, characteristic
4#21445904C212yesmaximal
5#21445833C212yesmaximal, complemented
6#21445832C212yesmaximal, complemented
7#21445752C212yesmaximal
8C4×#14519700C211yesmaximal, characteristic, complemented
9C4×#14519702C211yesmaximal, characteristic, complemented
10C4×#14519701C211yesmaximal, characteristic, complemented
11C4×#103×C61C211yesmaximal, characteristic, complemented
12C22×Q8×D61C2211yescharacteristic
13C2×Q8×Dic61C2211yescharacteristic
14C2×C4×Dic122—21no—
15C2×C4×Dic122C2211yescharacteristic
16C2×#14519742—21no—
17C2×#14519742C2211yescharacteristic
18C4×Q8×D61—24no—
19C4×Q8×D61C418yescomplemented
20C2×C4⋊C4×D61—21no—
21C2×C4⋊C4×D61C2211yescharacteristic
22C2×C42×D61—21no—
23C2×C42×D61C2211yescharacteristic
24#16839813—24no—
25C2×#14519706C2212yes—
26C2×#14519702C2212yes—
27C2×#14519701C2212yes—
28C2×#14519700C2212yes—
29C2×#14519689C2212yes—
30C2×C4×#92418C2211yescharacteristic
31#16839629—21no—
32#16839627C2212yes—
33#16839624—21no—
34#16839619C2212yes—
35C4×C4⋊Dic61C2211yescharacteristic
36C4×#11758198—21no—
37#16839608—21no—
38#16839525C2212yes—
39#16839524C2212yes—
40#16839524C2212yes—
41C2×C4×Q8×C61C2211yescharacteristic
42C2×#103×C61C2212yes—
43C2×C4×C8×C61C2211yescharacteristic
44C4×C4⋊C4×C61C2211yescharacteristic
45C61×#340C2212yes—
46C22×Dic122—21no—
47C22×Dic122C2311yescharacteristic
48C2×Q8×D61—24no—
49C2×Q8×D61—22no—
50C2×Q8×D61—24no—
51C2×Q8×D61C2×C414yes—
52C22×C4×D61—21no—
53C22×C4×D61—21no—
54C22×C4×D61D411yescharacteristic
55C22×C4×D61D411yescharacteristic
56C22×C4×D61C2311yescharacteristic
57C22×#92418C2312yes—
58Q8×Dic61—22no—
59Q8×Dic61C2×C414yes—
60C4×Dic122—22no—
61C4×Dic122—28no—
62C4×Dic122C2×C414yes—
63#14519742—28no—
64#14519742—22no—
65#14519742C2×C414yes—
66C2×C4⋊Dic61—21no—
67C2×C4⋊Dic61C2312yes—
68C2×C4⋊Dic61C2311yescharacteristic
69C2×#11758196—21no—
70C2×#11758196—21no—
71C2×#11758196—21no—
72C4⋊C4×D61—28no—
73C4⋊C4×D61—22no—
74C4⋊C4×D61C2×C414yes—
75C2×#11758198—22no—
76C2×C4×Dic61—21no—
77C2×C4×Dic61—21no—
78C2×C4×Dic61—21no—
79C2×C4×Dic61—21no—
80C2×C4×Dic61D411yescharacteristic
81C2×C4×Dic61D411yescharacteristic
82C2×C4×Dic61C2311yescharacteristic
83C42×D61—28no—
84C42×D61C2×C414yes—
85C42×D61D414yes—
86#14519706—24no—
87#14519702—24no—
88#14519701—24no—
89#14519700—24no—
90#14519689—24no—
91C4×#92418—22no—
92#14519684—21no—
93#14519684—22no—
94#14519684—22no—
95#14519684—22no—
96C22×Q8×C61C2311yescharacteristic
97C22×C8×C61C2312yes—
98C4×Q8×C61—22no—
99C4×Q8×C61C2×C414yes—
100C2×C4⋊C4×C61—21no—
101C2×C4⋊C4×C61C2312yes—
102C2×C4⋊C4×C61C2311yescharacteristic
103C2×C42×C61—21no—
104C2×C42×C61—21no—
105C2×C42×C61C2311yescharacteristic
106#103×C61—24no—
107C4×C8×C61—22no—
108#95×C61—21no—
109C23×D61C2×D411yescharacteristic
110C2×Q8×C61—21no—
111C2×Q8×C61—22no—
112C2×Q8×C61C22×C412yes—
113C2×Q8×C61#4114yes—
114C22×C4×C61—21no—
115C22×C4×C61—21no—
116C22×C4×C61—22no—
117C22×C4×C61C2×D411yescharacteristic
118C22×C4×C61C2×D411yescharacteristic
119C22×C4×C61C2411yescharacteristic
120C22×Dic61—21no—
121C22×Dic61—21no—
122C22×Dic61—22no—
123C22×Dic61C2×D411yescharacteristic
124C22×Dic61C2×D411yescharacteristic
125C22×Dic61C2×D411yescharacteristic
126Q8×D61—216no—
127Q8×D61—28no—
128Q8×D61—28no—
129C2×C4×D61—28no—
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—24no—
137C2×C4×D61—44no—
138C2×C4×D61C22⋊C414yes—
139C2×C4×D61C22⋊C414yes—
140C2×C4×D61C22×C412yes—
141C2×C4×D61C2×D412yes—
142C2×Dic122—28no—
143C2×Dic122—21no—
144C2×Dic122—22no—
145C2×Dic122—24no—
146C2×Dic122C22×C412yes—
147C2×Dic122#4114yes—
148C2×C8×C61—22no—
149C2×C8×C61—22no—
150C2×C8×C61—22no—
151C4⋊C4×C61—24no—
152C4⋊C4×C61—21no—
153C4⋊C4×C61—22no—
154C4⋊C4×C61—22no—
155C4⋊C4×C61C22×C412yes—
156C42×C61—24no—
157C42×C61—42no—
158C42×C61C22×C412yes—
159C42×C61C2×D412yes—
160#11758198—44no—
161C4⋊Dic61—21no—
162C4⋊Dic61—24no—
163C4⋊Dic61—22no—
164C4⋊Dic61—22no—
165C4⋊Dic61C22×C412yes—
166#11758196—24no—
167#11758196—24no—
168#11758196—24no—
169C4×Dic61—24no—
170C4×Dic61—24no—
171C4×Dic61—24no—
172C4×Dic61—42no—
173C4×Dic61C22⋊C414yes—
174C4×Dic61C22⋊C414yes—
175C4×Dic61C22×C412yes—
176C4×Dic61C2×D412yes—
177C2×#92418—22no—
178C2×#92418—22no—
179C2×#92418—22no—
180C23×C61C22×D411yescharacteristic
181C22×D61—22no—
182C22×D61—21no—
183C22×D61—21no—
184C22×D61C2×C22⋊C412yes—
185C22×D61#13514yes—
186Q8×C61—22no—
187Q8×C61—24no—
188Q8×C61#11714yes—
189C2×C4×C61—22no—
190C2×C4×C61—24no—
191C2×C4×C61—22no—
192C2×C4×C61—22no—
193C2×C4×C61—42no—
194C2×C4×C61—42no—
195C2×C4×C61—42no—
196C2×C4×C61—42no—
197C2×C4×C61C2×C22⋊C412yes—
198C2×C4×C61C2×C22⋊C412yes—
199C2×C4×C61#13712yes—
200C2×C4×C61#13712yes—
201C2×C4×C61C23×C411yescharacteristic
202C2×C4×C61C22×D411yescharacteristic
203C2×C4×C61C2×#4112yes—
204C2×C4×C61C2×#4112yes—
205C2×Dic61—22no—
206C2×Dic61—22no—
207C2×Dic61—24no—
208C2×Dic61—22no—
209C2×Dic61—21no—
210C2×Dic61—21no—
211C2×Dic61—22no—
212C2×Dic61—42no—
213C2×Dic61—42no—
214C2×Dic61—42no—
215C2×Dic61—42no—
216C2×Dic61C2×C22⋊C412yes—
217C2×Dic61C2×C22⋊C412yes—
218C2×Dic61C2×C22⋊C412yes—
219C4×D61—216no—
220C4×D61—28no—
221C4×D61—22no—
222C4×D61—22no—
223C4×D61—24no—
224C4×D61—28no—
225C4×D61—48no—
226Dic122—216no—
227Dic122—22no—
228Dic122—24no—
229Dic122#11714yes—
230C8×C61—42no—
231#92418—42no—
232#20735—611nomaximal, Sylow
233C22×C61C22×C22⋊C411yescharacteristic
234C22×C61#52512yes—
235C22×C61C2×#13512yes—
236C22×C61C2×#13711yescharacteristic
237C22×C61C2×#13711yescharacteristic
238C2×D61—24no—
239C2×D61—24no—
240C2×D61—22no—
241C2×D61—22no—
242C2×D61#41614yes—
243C4×C61—22no—
244C4×C61—24no—
245C4×C61—42no—
246C4×C61—82no—
247C4×C61#41914yes—
248C4×C61C2×#11711yescharacteristic
249C4×C61C2×#11711yescharacteristic
250C4×C61#52512yes—
251Dic61—22no—
252Dic61—22no—
253Dic61—22no—
254Dic61—24no—
255Dic61—42no—
256Dic61—82no—
257C22×C4×Q8—611no—
258C2×#431—611no—
259#2283—611no—
260#1823—612no—
261#1805—612no—
262#1805—612no—
263#1787—612no—
264C4×#103—611no—
265C4×#103—611no—
266C4×#103—611no—
267C4×#103D6111yescharacteristic, complemented
268C2×C61#228011yescharacteristic
269C2×C61C2×#41611yescharacteristic
270C2×C61C2×#41911yescharacteristic
271C2×C61#310812yes—
272C2×C61#310812yes—
273D61—24no—
274C23×Q8—611no—
275C2×C4×Q8—611no—
276C2×C4×Q8—618no—
277C2×C4×Q8—611no—
278C2×C4×Q8—611no—
279C2×C4×Q8—1221no—
280C2×C4×Q8—1221no—
281C2×C4×Q8—1224no—
282C2×C4×Q8C2×D6111yescharacteristic
283C22×C4⋊C4—611no—
284C22×C4⋊C4—1221no—
285C22×C42—611no—
286C22×C42—1221no—
287#431—1224no—
288C2×C4⋊C8—612no—
289C2×#103—612no—
290C2×#103—612no—
291C2×#103—612no—
292C2×#103C2×D6112yes—
293C2×C22⋊C8—612no—
294C2×C4×C8—611no—
295C2×C4×C8C2×D6111yescharacteristic
296#383—1221no—
297#382—612no—
298#381—1221no—
299#380—612no—
300C4×C4⋊C4—611no—
301C4×C4⋊C4C2×D6111yescharacteristic
302C4×C22⋊C4—1221no—
303#376—1221no—
304#340—612no—
305#340—612no—
306#340—612no—
307#340C2×D6112yes—
308C61#2073511yescharacteristic, complemented, Sylow
309C22×Q8—614no—
310C22×Q8—611no—
311C22×Q8—1221no—
312C22×Q8—1222no—
313C22×Q8—1224no—
314C22×Q8—1224no—
315C22×Q8C22×D6111yescharacteristic
316C23×C4—611no—
317C23×C4—611no—
318C23×C4—611no—
319C23×C4—1221no—
320C23×C4—1221no—
321C22×C8—612no—
322C22×C8C22×D6112yes—
323C4×Q8—22no—
324C4×Q8—614no—
325C4×Q8—614no—
326C4×Q8—614no—
327C4×Q8—1228no—
328C4×Q8—1222no—
329C4×Q8—1228no—
330C4×Q8—1222no—
331C4×Q8—1222no—
332C4×Q8C4×D6114yes—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—611no—
335C2×C4⋊C4—614no—
336C2×C4⋊C4—612no—
337C2×C4⋊C4—1222no—
338C2×C4⋊C4—1228no—
339C2×C4⋊C4—1221no—
340C2×C4⋊C4—1221no—
341C2×C4⋊C4—1221no—
342C2×C4⋊C4—1221no—
343C2×C4⋊C4C22×D6112yes—
344C2×C4⋊C4C22×D6111yescharacteristic
345C2×C22⋊C4—1222no—
346C2×C42—21no—
347C2×C42—21no—
348C2×C42—611no—
349C2×C42—611no—
350C2×C42—611no—
351C2×C42—614no—
352C2×C42—614no—
353C2×C42—1228no—
354C2×C42—1221no—
355C2×C42—1221no—
356C2×C42—1221no—
357C2×C42—1221no—
358C2×C42C22×D6111yescharacteristic
359C4⋊C8—1224no—
360#103—24no—
361#103—1224no—
362#103—1224no—
363#103—1224no—
364C22⋊C8—1224no—
365C4×C8—22no—
366C4×C8—1222no—
367#95—21no—
368#95—1222no—
369#95—1222no—
370#95—1222no—
371#95—1221no—
372C24—611no—
373C2×Q8—21no—
374C2×Q8—22no—
375C2×Q8—612no—
376C2×Q8—614no—
377C2×Q8—1224no—
378C2×Q8—1228no—
379C2×Q8—1228no—
380C2×Q8—12216no—
381C2×Q8—1228no—
382C2×Q8—1221no—
383C2×Q8—1222no—
384C2×Q8C2×C4×D6112yes—
385C2×Q8#1175822214yes—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—22no—
389C22×C4—611no—
390C22×C4—611no—
391C22×C4—611no—
392C22×C4—612no—
393C22×C4—612no—
394C22×C4—614no—
395C22×C4—614no—
396C22×C4—1224no—
397C22×C4—1224no—
398C22×C4—1228no—
399C22×C4—1224no—
400C22×C4—1221no—
401C22×C4—1222no—
402C22×C4—1222no—
403C22×C4—1222no—
404C22×C4—1222no—
405C22×C4—1221no—
406C22×C4—1222no—
407C22×C4—2444no—
408C22×C4D4×D6111yescharacteristic
409C22×C4D4×D6111yescharacteristic
410C22×C4C23×D6111yescharacteristic
411C2×C8—22no—
412C2×C8—22no—
413C2×C8—22no—
414C2×C8—1222no—
415C2×C8—1222no—
416C2×C8—1222no—
417C4⋊C4—21no—
418C4⋊C4—24no—
419C4⋊C4—22no—
420C4⋊C4—22no—
421C4⋊C4—612no—
422C4⋊C4—1224no—
423C4⋊C4—1224no—
424C4⋊C4—1224no—
425C4⋊C4—1221no—
426C4⋊C4—1224no—
427C4⋊C4—1222no—
428C4⋊C4—1222no—
429C4⋊C4C2×C4×D6112yes—
430C22⋊C4—2444no—
431C42—24no—
432C42—42no—
433C42—612no—
434C42—612no—
435C42—614no—
436C42—614no—
437C42—1224no—
438C42—1224no—
439C42—1224no—
440C42—2442no—
441C42C2×C4×D6112yes—
442C42D4×D6112yes—
443C23—612no—
444C23—614no—
445C23—1222no—
446C23—1221no—
447C23—1221no—
448C23C2×D4×D6111yescharacteristic
449Q8—22no—
450Q8—24no—
451Q8—614no—
452Q8—12216no—
453Q8—1222no—
454Q8—1224no—
455Q8#1451973414yes—
456C2×C4—22no—
457C2×C4—24no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—612no—
465C2×C4—612no—
466C2×C4—612no—
467C2×C4—1228no—
468C2×C4—12216no—
469C2×C4—1222no—
470C2×C4—1222no—
471C2×C4—1224no—
472C2×C4—1222no—
473C2×C4—1221no—
474C2×C4—1221no—
475C2×C4—1222no—
476C2×C4—1222no—
477C2×C4—1224no—
478C2×C4—1228no—
479C2×C4—1222no—
480C2×C4—2448no—
481C2×C4—2442no—
482C2×C4—2442no—
483C2×C4—2442no—
484C2×C4—2442no—
485C2×C4C22⋊C4×D6112yes—
486C2×C4C22⋊C4×D6112yes—
487C2×C4#1451980612yes—
488C2×C4#1451980912yes—
489C2×C4C22×C4×D6111yescharacteristic
490C2×C4C2×D4×D6111yescharacteristic
491C2×C4#41×D6112yes—
492C2×C4C2×#1175822212yes—
493C8—42no—
494C8—2442no—
495C22—614no—
496C22—1222no—
497C22—1224no—
498C22—1224no—
499C22—1222no—
500C22C2×C22⋊C4×D6111yescharacteristic
501C22#1684027112yes—
502C22#135×D6112yes—
503C22C2×#1451980611yescharacteristic
504C22C2×#1451980911yescharacteristic
505C4—24no—
506C4—22no—
507C4—42no—
508C4—82no—
509C4—1222no—
510C4—1224no—
511C4—1222no—
512C4—1222no—
513C4—4882no—
514C4—2442no—
515C4#1683976914yes—
516C4#117×D6111yescharacteristic
517C4C2×#1451973411yescharacteristic
518C4#1684027112yes—
519C2—1224no—
520C2#2144820011yescharacteristic
521C2#416×D6111yescharacteristic
522C2C2×#1683976911yescharacteristic
523C2#2145589712yes—
524C2#2145590312yes—
subgroupG/H
centerC22×C4D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC4×#103×C61C2
socleC23×C61C22×D4
prime radicalC23×D61C2×D4
cosocleC22×C4×C61C24
hypercenterC4×#103D61
hypocenterC61#20735
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22×C4 ◁ C2×C42 ◁ C4×#103

Complements

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

NH ≅ G/Ntype
5. #21445833C2semidirect
6. #21445832C2semidirect
8. C4×#14519700C2semidirect
9. C4×#14519702C2semidirect
10. C4×#14519701C2semidirect
11. C4×#103×C61C2semidirect
19. C4×Q8×D61C4semidirect
267. C4×#103D61semidirect
308. C61#20735semidirect

Maximal subgroups

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

subgroupindex
C2×C4×Q8×D612
C2×#168398132
#214482152
#214459042
#214458332
#214458322
#214457522
C4×#145197002
C4×#145197022
C4×#145197012
C4×#103×C612
#2073561