Finite Group Databasebeta

← #29680048

Subgroups — #29680048

Order 15616.

1402 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#11758226C211yesmaximal, characteristic
2C2×#16839813C211yesmaximal, characteristic, complemented
3#21448223C211yesmaximal, characteristic, complemented
4#21445912C212yesmaximal, complemented
5#21445822C212yesmaximal, complemented
6#21445818C212yesmaximal, complemented
7#21445758C212yesmaximal
8C4×#14519701C211yesmaximal, characteristic, complemented
9C4×#14519697C211yesmaximal, characteristic, complemented
10C4×#14519699C211yesmaximal, characteristic, complemented
11C4×#103×C61C211yesmaximal, characteristic, complemented
12C22×#11758226C2211yescharacteristic
13C4×#11758226—24no—
14C4×#11758226C418yescomplemented
15C2×Q8×Dic61C2211yescharacteristic
16C2×C4×D244—21no—
17C2×C4×D244C2211yescharacteristic
18C2×#14519739—21no—
19C2×#14519739C2211yescharacteristic
20C2×#14519735—21no—
21C2×#14519735C2211yescharacteristic
22C2×#14519734—21no—
23C2×C42×D61—21no—
24C2×C42×D61C2211yescharacteristic
25#16839813—24no—
26C2×#14519706C2212yes—
27C2×#14519701C2212yes—
28C2×#14519699C2212yes—
29C2×#14519697C2212yes—
30C2×#14519689C2212yes—
31C2×C4×#92418C2211yescharacteristic
32#16839631C2212yes—
33#16839626—21no—
34#16839621—21no—
35#16839619C2212yes—
36C4×#11758196—21no—
37C4×C4⋊Dic61C2211yescharacteristic
38#16839525C2212yes—
39#16839524C2212yes—
40#16839524C2212yes—
41C2×C4×Q8×C61C2211yescharacteristic
42C2×#103×C61C2212yes—
43C2×C4×C8×C61C2211yescharacteristic
44C4×C4⋊C4×C61C2211yescharacteristic
45C61×#340C2212yes—
46C2×#11758226—24no—
47C2×#11758226—24no—
48C2×#11758226—22no—
49C2×#11758226C2×C414yes—
50C22×D244—21no—
51C22×D244C2311yescharacteristic
52C22×C4×D61—21no—
53C22×C4×D61—21no—
54C22×C4×D61—21no—
55C22×C4×D61D411yescharacteristic
56C22×C4×D61D411yescharacteristic
57C22×C4×D61C2311yescharacteristic
58C22×#92418C2312yes—
59Q8×Dic61—22no—
60Q8×Dic61C2×C414yes—
61C4×D244—28no—
62C4×D244—22no—
63C4×D244C2×C414yes—
64#14519739—28no—
65#14519739—22no—
66#14519739C2×C414yes—
67#14519735—22no—
68#14519735—28no—
69#14519735C2×C414yes—
70#14519734—44no—
71C2×C4⋊Dic61—21no—
72C2×C4⋊Dic61C2312yes—
73C2×C4⋊Dic61C2311yescharacteristic
74C2×#11758196—22no—
75C2×#11758196—22no—
76C2×#11758198—21no—
77C2×#11758198—21no—
78C2×#11758198—21no—
79C2×#11758198—22no—
80C2×C4×Dic61—21no—
81C2×C4×Dic61—21no—
82C2×C4×Dic61—21no—
83C2×C4×Dic61D411yescharacteristic
84C2×C4×Dic61D411yescharacteristic
85C2×C4×Dic61C2311yescharacteristic
86C42×D61—28no—
87C42×D61C2×C414yes—
88C42×D61D414yes—
89#14519706—24no—
90#14519701—24no—
91#14519699—24no—
92#14519697—24no—
93#14519689—24no—
94C4×#92418—22no—
95#14519684—21no—
96#14519684—22no—
97C22×Q8×C61C2311yescharacteristic
98C22×C8×C61C2312yes—
99C4×Q8×C61—22no—
100C4×Q8×C61C2×C414yes—
101C2×C4⋊C4×C61—21no—
102C2×C4⋊C4×C61C2312yes—
103C2×C4⋊C4×C61C2311yescharacteristic
104C2×C42×C61—21no—
105C2×C42×C61—21no—
106C2×C42×C61C2311yescharacteristic
107#103×C61—24no—
108C4×C8×C61—22no—
109#95×C61—21no—
110C23×D61—21no—
111C23×D61C2×D411yescharacteristic
112C2×Q8×C61—21no—
113C2×Q8×C61—22no—
114C2×Q8×C61C22×C412yes—
115C2×Q8×C61#4114yes—
116C22×C4×C61—21no—
117C22×C4×C61—21no—
118C22×C4×C61—22no—
119C22×C4×C61C2×D411yescharacteristic
120C22×C4×C61C2×D411yescharacteristic
121C22×C4×C61C2411yescharacteristic
122C22×Dic61—21no—
123C22×Dic61—22no—
124C22×Dic61C2×D411yescharacteristic
125C22×Dic61C2×D411yescharacteristic
126C22×Dic61C2×D411yescharacteristic
127#11758226—216no—
128#11758226—28no—
129#11758226—28no—
130C2×D244—28no—
131C2×D244—24no—
132C2×D244—21no—
133C2×D244—22no—
134C2×D244C22×C412yes—
135C2×D244#4114yes—
136C2×C4×D61—28no—
137C2×C4×D61—22no—
138C2×C4×D61—22no—
139C2×C4×D61—22no—
140C2×C4×D61—22no—
141C2×C4×D61—24no—
142C2×C4×D61—24no—
143C2×C4×D61—24no—
144C2×C4×D61—24no—
145C2×C4×D61—42no—
146C2×C4×D61—42no—
147C2×C4×D61—44no—
148C2×C4×D61C22⋊C414yes—
149C2×C4×D61C22⋊C414yes—
150C2×C4×D61C22×C412yes—
151C2×C4×D61C2×D412yes—
152C2×C8×C61—22no—
153C2×C8×C61—22no—
154C2×C8×C61—22no—
155C4⋊C4×C61—24no—
156C4⋊C4×C61—21no—
157C4⋊C4×C61—22no—
158C4⋊C4×C61—22no—
159C4⋊C4×C61C22×C412yes—
160C42×C61—24no—
161C42×C61—42no—
162C42×C61C22×C412yes—
163C42×C61C2×D412yes—
164#11758198—24no—
165#11758198—24no—
166#11758198—24no—
167#11758198—44no—
168C4⋊Dic61—21no—
169C4⋊Dic61—24no—
170C4⋊Dic61—22no—
171C4⋊Dic61—22no—
172C4⋊Dic61C22×C412yes—
173#11758196—44no—
174#11758196—44no—
175C4×Dic61—24no—
176C4×Dic61—24no—
177C4×Dic61—42no—
178C4×Dic61C22⋊C414yes—
179C4×Dic61C22⋊C414yes—
180C4×Dic61C22×C412yes—
181C4×Dic61C2×D412yes—
182C2×#92418—22no—
183C2×#92418—22no—
184C2×#92418—22no—
185C23×C61C22×D411yescharacteristic
186C22×D61—22no—
187C22×D61—22no—
188C22×D61—24no—
189C22×D61—22no—
190C22×D61—21no—
191C22×D61—21no—
192C22×D61—42no—
193C22×D61—41no—
194C22×D61—41no—
195C22×D61C2×C22⋊C412yes—
196Q8×C61—22no—
197Q8×C61—24no—
198Q8×C61#11714yescomplemented
199C2×C4×C61—22no—
200C2×C4×C61—24no—
201C2×C4×C61—22no—
202C2×C4×C61—22no—
203C2×C4×C61—42no—
204C2×C4×C61—42no—
205C2×C4×C61—42no—
206C2×C4×C61—42no—
207C2×C4×C61C2×C22⋊C412yes—
208C2×C4×C61C2×C22⋊C412yes—
209C2×C4×C61AGL(1,ℤ/8)12yes—
210C2×C4×C61#13712yes—
211C2×C4×C61C23×C411yescharacteristic
212C2×C4×C61C22×D411yescharacteristic
213C2×C4×C61C2×#4112yes—
214C2×C4×C61C2×#4112yes—
215C2×Dic61—22no—
216C2×Dic61—21no—
217C2×Dic61—21no—
218C2×Dic61—22no—
219C2×Dic61—41no—
220C2×Dic61—41no—
221C2×Dic61—42no—
222C2×Dic61—42no—
223C2×Dic61—42no—
224C2×Dic61C2×C22⋊C412yes—
225C2×Dic61C2×C22⋊C412yes—
226C2×Dic61C2×C22⋊C412yes—
227C2×Dic61#13514yes—
228D244—216no—
229D244—22no—
230D244—24no—
231D244#11714yescomplemented
232C4×D61—216no—
233C4×D61—22no—
234C4×D61—22no—
235C4×D61—24no—
236C4×D61—28no—
237C4×D61—28no—
238C4×D61—48no—
239C4×D61—48no—
240C8×C61—42no—
241#92418—42no—
242#20737—611nomaximal, Sylow
243C22×C61C22×C22⋊C411yescharacteristic
244C22×C61#52512yes—
245C22×C61C2×#13512yes—
246C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
247C22×C61C2×#13711yescharacteristic
248C2×D61—22no—
249C2×D61—22no—
250C2×D61—22no—
251C2×D61—24no—
252C2×D61—42no—
253C2×D61—44no—
254C2×D61—42no—
255C2×D61—44no—
256C2×D61—42no—
257C2×D61—42no—
258C4×C61—22no—
259C4×C61—24no—
260C4×C61—42no—
261C4×C61—82no—
262C4×C61#41714yes—
263C4×C61C2×#11711yescharacteristic
264C4×C61C2×#11711yescharacteristic
265C4×C61#52512yes—
266Dic61—22no—
267Dic61—22no—
268Dic61—42no—
269Dic61—82no—
270Dic61#41614yes—
271C2×C4×#41—611no—
272C2×#431—611no—
273#2285—611no—
274#1825—612no—
275#1803—612no—
276#1802—612no—
277#1788—612no—
278C4×#103—611no—
279C4×#103D6111yescharacteristic, complemented
280C4×#102—611no—
281C4×#102—611no—
282C2×C61#228011yescharacteristic
283C2×C61C2×#41611yescharacteristic
284C2×C61C2×#41711yescharacteristic
285C2×C61#310612yes—
286C2×C61#310712yes—
287D61—42no—
288D61—44no—
289C22×#41—611no—
290C4×#41—618no—
291C4×#41—1224no—
292C2×C4×Q8—611no—
293C2×C4×Q8C2×D6111yescharacteristic
294C2×C4×D4—611no—
295C2×C4×D4—611no—
296C2×C4×D4—1221no—
297C2×C4×D4—1221no—
298C2×#117—611no—
299C2×#117—1221no—
300C2×#117—1221no—
301C22×C42—611no—
302C22×C42—1221no—
303#431—1224no—
304C2×C4⋊C8—612no—
305C2×#103—612no—
306C2×#103C2×D6112yes—
307C2×#102—612no—
308C2×#102—612no—
309C2×C22⋊C8—612no—
310C2×C4×C8—611no—
311C2×C4×C8C2×D6111yescharacteristic
312#384—612no—
313#382—1221no—
314#380—612no—
315#380—1221no—
316C4×C4⋊C4—611no—
317C4×C4⋊C4—1221no—
318C4×C4⋊C4C2×D6111yescharacteristic
319#340—612no—
320#340—612no—
321#340—612no—
322#340C2×D6112yes—
323C61#2073711yescharacteristic, complemented, Sylow
324C2×#41—614no—
325C2×#41—1222no—
326C2×#41—1224no—
327C2×#41—1224no—
328C22×Q8C22×D6111yescharacteristic
329C22×D4—611no—
330C22×D4—1221no—
331C23×C4—611no—
332C23×C4—611no—
333C23×C4—611no—
334C23×C4—1221no—
335C23×C4—1221no—
336C23×C4—1221no—
337C22×C8—612no—
338C22×C8C22×D6112yes—
339C4×Q8—22no—
340C4×Q8—614no—
341C4×Q8—1222no—
342C4×Q8C4×D6114yes—
343C4×D4—614no—
344C4×D4—614no—
345C4×D4—1228no—
346C4×D4—1222no—
347C4×D4—1228no—
348C4×D4—1222no—
349#117—614no—
350#117—1228no—
351#117—1222no—
352#117—2444no—
353C2×C4⋊C4—21no—
354C2×C4⋊C4—611no—
355C2×C4⋊C4—612no—
356C2×C4⋊C4—1221no—
357C2×C4⋊C4—1222no—
358C2×C4⋊C4—1222no—
359C2×C4⋊C4C22×D6112yes—
360C2×C4⋊C4C22×D6111yescharacteristic
361C2×C22⋊C4—1221no—
362C2×C22⋊C4—1221no—
363C2×C22⋊C4—1221no—
364C2×C22⋊C4—1222no—
365C2×C42—21no—
366C2×C42—21no—
367C2×C42—611no—
368C2×C42—611no—
369C2×C42—611no—
370C2×C42—614no—
371C2×C42—614no—
372C2×C42—1228no—
373C2×C42—1221no—
374C2×C42—1221no—
375C2×C42—1221no—
376C2×C42C22×D6111yescharacteristic
377C4⋊C8—1224no—
378#103—24no—
379#103—1224no—
380#102—1224no—
381#102—1224no—
382C22⋊C8—1224no—
383C4×C8—22no—
384C4×C8—1222no—
385#95—21no—
386#95—1222no—
387#95—1221no—
388C24—611no—
389C24—1221no—
390#41—1228no—
391#41—12216no—
392#41—1228no—
393C2×Q8—21no—
394C2×Q8—22no—
395C2×Q8C2×C4×D6112yes—
396C2×Q8#1175822214yes—
397C2×D4—612no—
398C2×D4—614no—
399C2×D4—1224no—
400C2×D4—1228no—
401C2×D4—1221no—
402C2×D4—1222no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—22no—
406C22×C4—611no—
407C22×C4—611no—
408C22×C4—611no—
409C22×C4—612no—
410C22×C4—612no—
411C22×C4—614no—
412C22×C4—614no—
413C22×C4—1224no—
414C22×C4—1224no—
415C22×C4—1224no—
416C22×C4—1228no—
417C22×C4—1222no—
418C22×C4—1222no—
419C22×C4—1222no—
420C22×C4—1222no—
421C22×C4—1221no—
422C22×C4—1224no—
423C22×C4—1222no—
424C22×C4—2444no—
425C22×C4—2442no—
426C22×C4—2442no—
427C22×C4D4×D6111yescharacteristic
428C22×C4D4×D6111yescharacteristic
429C22×C4C23×D6111yescharacteristic
430C2×C8—22no—
431C2×C8—22no—
432C2×C8—22no—
433C2×C8—1222no—
434C2×C8—1222no—
435C2×C8—1222no—
436C4⋊C4—21no—
437C4⋊C4—24no—
438C4⋊C4—22no—
439C4⋊C4—22no—
440C4⋊C4—612no—
441C4⋊C4—1224no—
442C4⋊C4—1221no—
443C4⋊C4—1222no—
444C4⋊C4—1222no—
445C4⋊C4—2444no—
446C4⋊C4—2444no—
447C4⋊C4C2×C4×D6112yes—
448C22⋊C4—1224no—
449C22⋊C4—1224no—
450C22⋊C4—1224no—
451C22⋊C4—2444no—
452C42—24no—
453C42—42no—
454C42—612no—
455C42—612no—
456C42—614no—
457C42—614no—
458C42—1224no—
459C42—1224no—
460C42—2442no—
461C42C2×C4×D6112yes—
462C42D4×D6112yes—
463C23—612no—
464C23—1222no—
465C23—1224no—
466C23—1222no—
467C23—1222no—
468C23—1221no—
469C23—1221no—
470C23—2442no—
471C23—2441no—
472C23—2441no—
473C23C2×D4×D6111yescharacteristic
474Q8—22no—
475Q8—24no—
476Q8#1451973414yescomplemented
477D4—614no—
478D4—12216no—
479D4—1222no—
480D4—1224no—
481C2×C4—22no—
482C2×C4—24no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—612no—
490C2×C4—614no—
491C2×C4—612no—
492C2×C4—612no—
493C2×C4—1228no—
494C2×C4—12216no—
495C2×C4—1222no—
496C2×C4—1221no—
497C2×C4—1221no—
498C2×C4—1222no—
499C2×C4—1222no—
500C2×C4—1224no—
501C2×C4—1228no—
502C2×C4—1222no—
503C2×C4—2448no—
504C2×C4—2442no—
505C2×C4—2442no—
506C2×C4—2442no—
507C2×C4—2441no—
508C2×C4—2441no—
509C2×C4—2448no—
510C2×C4C22⋊C4×D6112yes—
511C2×C4C22⋊C4×D6112yes—
512C2×C4#1451980412yes—
513C2×C4#1451980912yes—
514C2×C4C22×C4×D6111yescharacteristic
515C2×C4C2×D4×D6111yescharacteristic
516C2×C4#41×D6112yes—
517C2×C4C2×#1175822212yes—
518C8—42no—
519C8—2442no—
520C22—1224no—
521C22—1222no—
522C22—1222no—
523C22—1222no—
524C22—2442no—
525C22—2444no—
526C22—2442no—
527C22—2444no—
528C22—2442no—
529C22—2442no—
530C22C2×C22⋊C4×D6111yescharacteristic
531C22#1684027112yes—
532C22#135×D6112yes—
533C22C2×#1451980411yescharacteristic
534C22C2×#1451980911yescharacteristic
535C4—24no—
536C4—22no—
537C4—42no—
538C4—82no—
539C4—614no—
540C4—1222no—
541C4—1222no—
542C4—4882no—
543C4—2442no—
544C4#1683976214yes—
545C4#117×D6111yescharacteristic
546C4C2×#1451973411yescharacteristic
547C4#1684027112yes—
548C2—2442no—
549C2—2444no—
550C2#2144820011yescharacteristic
551C2#416×D6111yescharacteristic
552C2C2×#1683976211yescharacteristic
553C2#2145587712yes—
554C2#2145588812yes—
subgroupG/H
centerC22×C4D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC4×#103×C61C2
socleC23×C61C22×D4
prime radicalC22×#11758226C22
cosocleC22×C4×C61C24
hypercenterC4×#103D61
hypocenterC61#20737
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
2. C2×#16839813C2semidirect
3. #21448223C2semidirect
4. #21445912C2semidirect
5. #21445822C2semidirect
6. #21445818C2semidirect
8. C4×#14519701C2semidirect
9. C4×#14519697C2semidirect
10. C4×#14519699C2semidirect
11. C4×#103×C61C2semidirect
14. C4×#11758226C4semidirect
198. Q8×C61#117semidirect
231. D244#117semidirect
279. C4×#103D61semidirect
323. C61#20737semidirect
476. Q8#14519734semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#117582262
C2×#168398132
#214482232
#214459122
#214458222
#214458182
#214457582
C4×#145197012
C4×#145196972
C4×#145196992
C4×#103×C612
#2073761