Finite Group Databasebeta

← #29680045

Subgroups — #29680045

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×#16840230C211yesmaximal, characteristic
2C2×#16839812C211yesmaximal, characteristic, complemented
3#21448221C211yesmaximal, characteristic, complemented
4#21445914C211yesmaximal, characteristic, complemented
5#21445823C212yesmaximal, complemented
6#21445818C212yesmaximal, complemented
7#21445760C211yesmaximal, characteristic
8#21445757C211yesmaximal, characteristic
9#21445638C212yesmaximal, complemented
10#21445633C212yesmaximal, complemented
11C61×#1825C211yesmaximal, characteristic, complemented
12C22×#11758222C2211yescharacteristic
13#16840230—24no—
14#16840230C418yescomplemented
15C2×#14519742C2211yescharacteristic
16C2×#14519742C2211yescharacteristic
17C2×#14519739C2211yescharacteristic
18C2×#14519739C2211yescharacteristic
19C2×#14519737—21no—
20C2×#14519737—21no—
21C2×#14519734—21no—
22C2×#14519733—21no—
23C22×C4×Dic61C2211yescharacteristic
24C2×C42×D61—21no—
25#16839812—24no—
26C2×#14519706C2212yes—
27C2×#14519702C2212yes—
28C2×#14519700C2212yes—
29C2×#14519699C2212yes—
30C2×#14519697C2212yes—
31C2×C22⋊#92418C2211yescharacteristic
32C2×C8×Dic61C2212yes—
33#16839631C2212yes—
34#16839626—21no—
35#16839622C2211yescharacteristic
36#16839620—21no—
37C4⋊C4×Dic61C2212yes—
38#16839613—21no—
39#16839524C2212yes—
40#16839524C2212yes—
41C2×#117×C61C2211yescharacteristic
42C2×C22⋊C8×C61C2211yescharacteristic
43C61×#380C2211yescharacteristic
44C61×#340C2212yes—
45C61×#340C2212yes—
46C2×#11758222—22no—
47C2×#11758222—24no—
48C2×#11758222—24no—
49C2×#11758222C2×C414yes—
50C22×Dic122C2311yescharacteristic
51C22×D244C2311yescharacteristic
52C22×C22⋊D61—21no—
53C23×Dic61D411yescharacteristic
54C23×Dic61D411yescharacteristic
55C22×C4×D61—21no—
56C22×C4×D61—21no—
57C22×C4×D61—21no—
58C22×#92418C2312yes—
59#14519742—22no—
60#14519742—22no—
61#14519742C2×C414yes—
62#14519742C2×C414yes—
63#14519739—22no—
64#14519739—22no—
65#14519739C2×C414yes—
66#14519739C2×C414yes—
67#14519737—28no—
68#14519737—28no—
69#14519734—28no—
70#14519733—44no—
71C2×C4⋊Dic61C2312yes—
72C2×#11758196—21no—
73C2×#11758196—21no—
74C2×#11758196—22no—
75C2×#11758196—22no—
76C2×C22⋊Dic61—21no—
77C2×#11758198—21no—
78C2×#11758198—21no—
79C2×C4×Dic61—21no—
80C2×C4×Dic61—21no—
81C2×C4×Dic61—22no—
82C2×C4×Dic61C2×C414yes—
83C2×C4×Dic61D411yescharacteristic
84C2×C4×Dic61D411yescharacteristic
85C2×C4×Dic61D414yes—
86C2×C4×Dic61C2311yescharacteristic
87C2×C4×Dic61C2311yescharacteristic
88C42×D61—28no—
89#14519706—24no—
90#14519702—24no—
91#14519700—24no—
92#14519699—24no—
93#14519697—24no—
94C22⋊#92418—22no—
95C8×Dic61—24no—
96#14519684—21no—
97#14519684—22no—
98C23×C4×C61C2311yescharacteristic
99C22×C8×C61C2312yes—
100#117×C61—22no—
101#117×C61C2×C414yes—
102C2×C4⋊C4×C61C2312yes—
103C2×C4⋊C4×C61C2311yescharacteristic
104C2×C4⋊C4×C61C2311yescharacteristic
105C2×C22⋊C4×C61—21no—
106C2×C22⋊C4×C61—21no—
107C2×C42×C61—21no—
108C22⋊C8×C61—22no—
109#95×C61—21no—
110C24×C61C2×D411yescharacteristic
111C23×D61—21no—
112C22×C4×C61—22no—
113C22×C4×C61—22no—
114C22×C4×C61—21no—
115C22×C4×C61—21no—
116C22×C4×C61—22no—
117C22×C4×C61C22×C412yes—
118C22×C4×C61C2×D411yescharacteristic
119C22×C4×C61C2×D412yes—
120C22×C4×C61C2411yescharacteristic
121C2×C22⋊D61—28no—
122C2×C22⋊D61—24no—
123C22×Dic61—21no—
124C22×Dic61—24no—
125C22×Dic61—22no—
126C22×Dic61—22no—
127C22×Dic61—22no—
128C22×Dic61C22⋊C414yes—
129C22×Dic61C22⋊C414yes—
130C22×Dic61C2×D411yescharacteristic
131C22×Dic61C2×D411yescharacteristic
132C22×Dic61C2×D411yescharacteristic
133C22×Dic61C2×D411yescharacteristic
134#11758222—216no—
135#11758222—28no—
136#11758222—28no—
137C2×D244—21no—
138C2×D244—22no—
139C2×D244C22×C412yes—
140C2×D244#4114yes—
141C2×C4×D61—28no—
142C2×C4×D61—24no—
143C2×C4×D61—24no—
144C2×C4×D61—24no—
145C2×C4×D61—44no—
146C2×C4×D61—42no—
147C2×C4×D61—42no—
148C2×Dic122—21no—
149C2×Dic122—22no—
150C2×Dic122C22×C412yes—
151C2×Dic122#4114yes—
152C2×C8×C61—22no—
153C2×C8×C61—22no—
154C2×C8×C61—22no—
155C4⋊C4×C61—21no—
156C4⋊C4×C61—21no—
157C4⋊C4×C61—22no—
158C4⋊C4×C61—22no—
159C4⋊C4×C61C22×C412yes—
160C4⋊C4×C61C22×C412yes—
161C22⋊C4×C61—24no—
162C22⋊C4×C61—42no—
163C42×C61—24no—
164C22⋊Dic61—42no—
165#11758198—24no—
166#11758198—24no—
167C4⋊Dic61—22no—
168C4⋊Dic61—22no—
169#11758196—24no—
170#11758196—24no—
171#11758196—44no—
172#11758196—44no—
173C4×Dic61—24no—
174C4×Dic61—24no—
175C4×Dic61—44no—
176C4×Dic61C22⋊C414yes—
177C4×Dic61C22⋊C414yes—
178C4×Dic61C22×C412yes—
179C4×Dic61C22×C412yes—
180C4×Dic61C2×D412yes—
181C4×Dic61C2×D412yes—
182C2×#92418—22no—
183C2×#92418—22no—
184C2×#92418—22no—
185C23×C61—22no—
186C23×C61—22no—
187C23×C61—21no—
188C23×C61—21no—
189C23×C61C2×C22⋊C412yes—
190C23×C61C22×D411yescharacteristic
191C22×D61—22no—
192C22×D61—24no—
193C22×D61—42no—
194C22×D61—41no—
195C22×D61—41no—
196C2×C4×C61—22no—
197C2×C4×C61—22no—
198C2×C4×C61—22no—
199C2×C4×C61—24no—
200C2×C4×C61—22no—
201C2×C4×C61—22no—
202C2×C4×C61—42no—
203C2×C4×C61—41no—
204C2×C4×C61—41no—
205C2×C4×C61—42no—
206C2×C4×C61—42no—
207C2×C4×C61—42no—
208C2×C4×C61C2×C22⋊C412yes—
209C2×C4×C61C23×C411yescharacteristic
210C2×C4×C61C22×D411yescharacteristic
211C2×C4×C61C2×#4112yes—
212C2×C4×C61C2×#4112yes—
213C22⋊D61—216no—
214C2×Dic61—22no—
215C2×Dic61—24no—
216C2×Dic61—22no—
217C2×Dic61—21no—
218C2×Dic61—21no—
219C2×Dic61—28no—
220C2×Dic61—28no—
221C2×Dic61—42no—
222C2×Dic61—42no—
223C2×Dic61—42no—
224C2×Dic61C2×C22⋊C412yes—
225C2×Dic61C2×C22⋊C412yes—
226C2×Dic61C2×C22⋊C412yes—
227C2×Dic61C2×C22⋊C412yes—
228C2×Dic61#13514yes—
229C2×Dic61AGL(1,ℤ/8)12yes—
230C2×Dic61#13712yes—
231D244—22no—
232D244—24no—
233D244#11714yescomplemented
234C4×D61—216no—
235C4×D61—48no—
236C4×D61—48no—
237Dic122—22no—
238Dic122—24no—
239Dic122#11714yescomplemented
240C8×C61—42no—
241#92418—42no—
242#20737—611nomaximal, Sylow
243C22×C61—22no—
244C22×C61—22no—
245C22×C61—22no—
246C22×C61—42no—
247C22×C61—42no—
248C22×C61C22×C22⋊C411yescharacteristic
249C22×C61#52512yes—
250C22×C61C2×#13512yes—
251C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
252C22×C61C2×#13711yescharacteristic
253C2×D61—24no—
254C2×D61—44no—
255C2×D61—44no—
256C2×D61—42no—
257C2×D61—42no—
258C4×C61—22no—
259C4×C61—42no—
260C4×C61—42no—
261C4×C61—82no—
262C4×C61C2×#11711yescharacteristic
263C4×C61C2×#11711yescharacteristic
264C4×C61#52512yes—
265Dic61—22no—
266Dic61—22no—
267Dic61—24no—
268Dic61—82no—
269Dic61#41614yes—
270Dic61#41714yes—
271C2×C4×#41—611no—
272C2×#431—611no—
273#2285—611no—
274#1825—611no—
275#1825D6111yescharacteristic, complemented
276#1803—612no—
277#1802—612no—
278#1788—611no—
279#1788—611no—
280C4×#103—612no—
281C4×#102—612no—
282C2×C61—42no—
283C2×C61#228011yescharacteristic
284C2×C61C2×#41611yescharacteristic
285C2×C61C2×#41711yescharacteristic
286C2×C61#310612yes—
287C2×C61#310712yes—
288D61—44no—
289C22×#41—611no—
290C4×#41—618no—
291C4×#41—1224no—
292C2×C4×Q8—611no—
293C2×C4×Q8—611no—
294C2×C4×D4—611no—
295C2×C4×D4—611no—
296C2×C4×D4—1221no—
297C2×C4×D4—1221no—
298C2×#117—1221no—
299C2×#117—1221no—
300C2×#117C2×D6111yescharacteristic
301C22×C42—611no—
302C22×C42—1221no—
303#431—1224no—
304C2×C4⋊C8—612no—
305C2×#103—612no—
306C2×#103—612no—
307C2×#102—612no—
308C2×#102—612no—
309C2×C22⋊C8—611no—
310C2×C22⋊C8C2×D6111yescharacteristic
311C2×C4×C8—612no—
312#384—612no—
313#382—1221no—
314#380—611no—
315#380—1221no—
316#380C2×D6111yescharacteristic
317C4×C4⋊C4—612no—
318C4×C4⋊C4—1221no—
319#340—612no—
320#340—612no—
321#340C2×D6112yes—
322#340C2×D6112yes—
323C61#2073711yescharacteristic, complemented, Sylow
324C2×#41—614no—
325C2×#41—1224no—
326C2×#41—1224no—
327C2×#41—1222no—
328C22×Q8—611no—
329C22×D4—611no—
330C22×D4—1221no—
331C23×C4—611no—
332C23×C4—611no—
333C23×C4—1221no—
334C23×C4—1221no—
335C23×C4—1221no—
336C23×C4C22×D6111yescharacteristic
337C22×C8—612no—
338C22×C8C22×D6112yes—
339C4×Q8—614no—
340C4×Q8—614no—
341C4×Q8—1222no—
342C4×Q8—1222no—
343C4×D4—614no—
344C4×D4—614no—
345C4×D4—1222no—
346C4×D4—1222no—
347C4×D4—1228no—
348C4×D4—1228no—
349#117—22no—
350#117—1228no—
351#117—2444no—
352#117C4×D6114yes—
353C2×C4⋊C4—612no—
354C2×C4⋊C4—1221no—
355C2×C4⋊C4—1221no—
356C2×C4⋊C4—1222no—
357C2×C4⋊C4—1222no—
358C2×C4⋊C4C22×D6112yes—
359C2×C4⋊C4C22×D6111yescharacteristic
360C2×C4⋊C4C22×D6111yescharacteristic
361C2×C22⋊C4—21no—
362C2×C22⋊C4—21no—
363C2×C22⋊C4—1221no—
364C2×C22⋊C4—1221no—
365C2×C22⋊C4—1221no—
366C2×C42—21no—
367C2×C42—614no—
368C2×C42—611no—
369C2×C42—614no—
370C2×C42—611no—
371C2×C42—611no—
372C2×C42—611no—
373C2×C42—1228no—
374C2×C42—1221no—
375C2×C42—1221no—
376C2×C42—1222no—
377C4⋊C8—1224no—
378#103—1224no—
379#103—1224no—
380#102—1224no—
381#102—1224no—
382C22⋊C8—22no—
383C22⋊C8—1222no—
384C4×C8—1224no—
385#95—21no—
386#95—1222no—
387#95—1221no—
388C24—1221no—
389C24D4×D6111yescharacteristic
390#41—1228no—
391#41—1228no—
392#41—12216no—
393C2×Q8—612no—
394C2×Q8—614no—
395C2×Q8—1221no—
396C2×Q8—1222no—
397C2×D4—612no—
398C2×D4—614no—
399C2×D4—1228no—
400C2×D4—1224no—
401C2×D4—1221no—
402C2×D4—1222no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—22no—
408C22×C4—611no—
409C22×C4—611no—
410C22×C4—611no—
411C22×C4—611no—
412C22×C4—614no—
413C22×C4—614no—
414C22×C4—1224no—
415C22×C4—1224no—
416C22×C4—1228no—
417C22×C4—1222no—
418C22×C4—1224no—
419C22×C4—1222no—
420C22×C4—1221no—
421C22×C4—1224no—
422C22×C4—1222no—
423C22×C4—2444no—
424C22×C4—2442no—
425C22×C4—2442no—
426C22×C4C2×C4×D6112yes—
427C22×C4D4×D6112yes—
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—21no—
438C4⋊C4—22no—
439C4⋊C4—22no—
440C4⋊C4—1224no—
441C4⋊C4—1224no—
442C4⋊C4—1222no—
443C4⋊C4—1222no—
444C4⋊C4—2444no—
445C4⋊C4—2444no—
446C4⋊C4C2×C4×D6112yes—
447C4⋊C4C2×C4×D6112yes—
448C22⋊C4—24no—
449C22⋊C4—42no—
450C22⋊C4—1224no—
451C22⋊C4—1224no—
452C22⋊C4—2442no—
453C42—24no—
454C42—612no—
455C42—612no—
456C42—612no—
457C42—612no—
458C42—614no—
459C42—614no—
460C42—1224no—
461C42—1224no—
462C42—2444no—
463C23—22no—
464C23—22no—
465C23—21no—
466C23—21no—
467C23—1222no—
468C23—1224no—
469C23—2442no—
470C23—2441no—
471C23—2441no—
472C23C22⋊C4×D6112yes—
473C23C2×D4×D6111yescharacteristic
474Q8—614no—
475Q8—1222no—
476Q8—1224no—
477D4—614no—
478D4—12216no—
479D4—1222no—
480D4—1224no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—24no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—42no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—612no—
494C2×C4—612no—
495C2×C4—614no—
496C2×C4—612no—
497C2×C4—612no—
498C2×C4—612no—
499C2×C4—612no—
500C2×C4—12216no—
501C2×C4—1228no—
502C2×C4—1228no—
503C2×C4—1222no—
504C2×C4—1224no—
505C2×C4—1222no—
506C2×C4—1221no—
507C2×C4—1221no—
508C2×C4—2448no—
509C2×C4—2448no—
510C2×C4—2442no—
511C2×C4—2442no—
512C2×C4—2442no—
513C2×C4C22⋊C4×D6112yes—
514C2×C4C22×C4×D6111yescharacteristic
515C2×C4C2×D4×D6111yescharacteristic
516C2×C4#41×D6112yes—
517C2×C4#41×D6112yes—
518C8—42no—
519C8—2442no—
520C22—22no—
521C22—22no—
522C22—22no—
523C22—42no—
524C22—42no—
525C22—1224no—
526C22—2444no—
527C22—2444no—
528C22—2442no—
529C22—2442no—
530C22C2×C22⋊C4×D6111yescharacteristic
531C22#1684027012yes—
532C22#135×D6112yes—
533C22AGL(1,ℤ/8)×D6111yescharacteristic
534C22#137×D6111yescharacteristic
535C4—22no—
536C4—42no—
537C4—42no—
538C4—82no—
539C4—614no—
540C4—614no—
541C4—1224no—
542C4—1222no—
543C4—1222no—
544C4—4882no—
545C4#117×D6111yescharacteristic
546C4#117×D6111yescharacteristic
547C4#1684027012yes—
548C2—42no—
549C2—2444no—
550C2#2144819911yescharacteristic
551C2#416×D6111yescharacteristic
552C2#417×D6111yescharacteristic
553C2#2145587412yes—
554C2#2145588512yes—
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1825C2
socleC23×C61C22×D4
prime radicalC22×#11758222C22
cosocleC22×C4×C61C24
hypercenter#1825D61
hypocenterC61#20737
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1825

Complements

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

NH ≅ G/Ntype
2. C2×#16839812C2semidirect
3. #21448221C2semidirect
4. #21445914C2semidirect
5. #21445823C2semidirect
6. #21445818C2semidirect
9. #21445638C2semidirect
10. #21445633C2semidirect
11. C61×#1825C2semidirect
14. #16840230C4semidirect
233. D244#117semidirect
239. Dic122#117semidirect
275. #1825D61semidirect
323. C61#20737semidirect

Maximal subgroups

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

subgroupindex
C2×#168402302
C2×#168398122
#214482212
#214459142
#214458232
#214458182
#214457602
#214457572
#214456382
#214456332
C61×#18252
#2073761