Finite Group Databasebeta

← #29730739

Subgroups — #29730739

Order 15616.

1102 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839991C212yesmaximal, complemented
2C2×#16839968C211yesmaximal, characteristic, complemented
3#21450772C211yesmaximal, characteristic, complemented
4#21450732C211yesmaximal, characteristic
5#21447016C212yesmaximal, complemented
6#21446824C212yesmaximal, complemented
7#21446621C211yesmaximal, characteristic, complemented
8#21446583C211yesmaximal, characteristic, complemented
9#21446242C212yesmaximal, complemented
10C4×#14519699C211yesmaximal, characteristic, complemented
11C2×C8×D4×C61C211yesmaximal, characteristic, complemented
12C22×#11758190C2212yes—
13C2×#14519777C2211yescharacteristic
14C2×#14519756C2212yes—
15C2×#14519756C2212yes—
16C2×#14519751C2212yes—
17C2×#14519748—21no—
18C2×C4×D244C2211yescharacteristic
19#16839991—28no—
20#16839968—24no—
21C2×#14519709C2212yes—
22C2×#14519702C2212yes—
23C2×#14519702C2212yes—
24C2×#14519702C2212yes—
25C2×#14519699C2211yescharacteristic
26C2×#14519699C2211yescharacteristic
27#16839698—22no—
28#16839678—22no—
29#16839665—21no—
30#16839632C2211yescharacteristic
31#16839631—21no—
32#16839622C2212yes—
33#16839619—21no—
34C4×C4⋊Dic61C2211yescharacteristic
35#16839525C2211yescharacteristic
36#16839525C2211yescharacteristic
37#16839525C2212yes—
38C23×C8×C61C2212yes—
39C2×C4×D4×C61C2211yescharacteristic
40C8×D4×C61—24no—
41C2×C4⋊C8×C61C2211yescharacteristic
42C2×C22⋊C8×C61C2212yes—
43C2×C4×C8×C61C2211yescharacteristic
44C22×Dic122C2312yes—
45C22×D244C2311yescharacteristic
46C22×C22⋊D61—22no—
47C22×C4×D61—21no—
48C2×#11758190—24no—
49C2×#11758190—24no—
50C2×#11758190—42no—
51#14519777—22no—
52#14519777—22no—
53#14519756—24no—
54#14519756—24no—
55#14519756—42no—
56#14519756—42no—
57#14519751—24no—
58#14519751—42no—
59#14519748—44no—
60C4×D244—22no—
61C4×D244—22no—
62C2×C4⋊Dic61—21no—
63C2×C4⋊Dic61—21no—
64C2×C4⋊Dic61—21no—
65C2×C4⋊Dic61—21no—
66C2×C4⋊Dic61C2311yescharacteristic
67C2×C4⋊Dic61C2312yes—
68C2×C4⋊Dic61C2311yescharacteristic
69C2×C4⋊Dic61C2311yescharacteristic
70C2×#11758196—22no—
71C2×#11758196—22no—
72C2×C22⋊Dic61—22no—
73C2×C22⋊Dic61—22no—
74C2×C22⋊Dic61—22no—
75C2×#11758198—21no—
76C2×#11758198—22no—
77C2×C4×Dic61—21no—
78#14519709—24no—
79#14519702—24no—
80#14519702—24no—
81#14519702—42no—
82#14519699—22no—
83#14519699D414yescomplemented
84#14519684—21no—
85#14519684—22no—
86C22×D4×C61D411yescharacteristic
87C23×C4×C61C2312yes—
88C22×C8×C61—24no—
89C22×C8×C61—24no—
90C22×C8×C61—42no—
91C22×C8×C61C2311yescharacteristic
92C22×C8×C61C2311yescharacteristic
93C22×C8×C61C2312yes—
94C4×D4×C61—22no—
95C4×D4×C61—22no—
96C2×C4⋊C4×C61D411yescharacteristic
97C2×C22⋊C4×C61D412yes—
98C2×C42×C61C2311yescharacteristic
99C4⋊C8×C61D414yescomplemented
100C22⋊C8×C61—24no—
101C4×C8×C61—22no—
102C24×C61C2×D412yes—
103C23×D61—21no—
104C2×D4×C61—22no—
105C2×D4×C61—22no—
106C2×D4×C61SD1614yescomplemented
107C22×C4×C61—22no—
108C22×C4×C61—22no—
109C22×C4×C61—22no—
110C22×C4×C61—22no—
111C22×C4×C61—22no—
112C22×C4×C61—22no—
113C22×C4×C61C2×D412yes—
114C22×C4×C61C2×D411yescharacteristic
115C22×C4×C61C2×D411yescharacteristic
116C22×C4×C61C2411yescharacteristic
117C2×C22⋊D61—44no—
118C2×C22⋊D61—44no—
119C2×C22⋊D61—82no—
120C22×Dic61—21no—
121C22×Dic61—21no—
122C22×Dic61—21no—
123C22×Dic61—22no—
124C22×Dic61—22no—
125C2×D244—21no—
126C2×D244—21no—
127C2×D244—21no—
128C2×D244—21no—
129C2×D244—21no—
130C2×D244C2×D412yes—
131C2×C4×D61—42no—
132C2×C4×D61—42no—
133C2×C4×D61—42no—
134C2×Dic122—22no—
135C2×Dic122—22no—
136C2×Dic122—22no—
137C2×Dic122—22no—
138C2×Dic122—22no—
139C2×Dic122—22no—
140C2×C8×C61—21no—
141C2×C8×C61—22no—
142C2×C8×C61—21no—
143C2×C8×C61—22no—
144C2×C8×C61—21no—
145C2×C8×C61—21no—
146C2×C8×C61—21no—
147C2×C8×C61—48no—
148C2×C8×C61C2×D412yes—
149C2×C8×C61C2×D414yes—
150C4⋊C4×C61—22no—
151C22⋊C4×C61—24no—
152C42×C61C2×D412yes—
153C42×C61#4112yes—
154C22⋊Dic61—44no—
155C22⋊Dic61—44no—
156C22⋊Dic61—44no—
157#11758198—44no—
158#11758198—42no—
159C4⋊Dic61—21no—
160C4⋊Dic61—21no—
161C4⋊Dic61—21no—
162C4⋊Dic61—22no—
163C4⋊Dic61—21no—
164C4⋊Dic61—22no—
165C4⋊Dic61—42no—
166C4⋊Dic61—42no—
167C4⋊Dic61—42no—
168C4⋊Dic61—81no—
169C4⋊Dic61C2×D412yes—
170C4⋊Dic61#4112yes—
171#11758196—44no—
172#11758196—44no—
173C4×Dic61—42no—
174#11758190—48no—
175C23×C61—22no—
176C23×C61—22no—
177C23×C61—22no—
178C23×C61—22no—
179C23×C61—22no—
180C23×C61—22no—
181C23×C61C2×SD1614yes—
182C23×C61C22×D411yescharacteristic
183C22×D61—22no—
184C22×D61—41no—
185C22×D61—41no—
186C22×D61—41no—
187C22×D61—41no—
188C22×D61—41no—
189C22×D61—41no—
190D4×C61—28no—
191C2×C4×C61—21no—
192C2×C4×C61—21no—
193C2×C4×C61—22no—
194C2×C4×C61—22no—
195C2×C4×C61—22no—
196C2×C4×C61—42no—
197C2×C4×C61—42no—
198C2×C4×C61—42no—
199C2×C4×C61—42no—
200C2×C4×C61#12112yes—
201C2×C4×C61#12112yes—
202C2×C4×C61C2×SD1612yes—
203C2×C4×C61#13512yes—
204C2×C4×C61C22×D411yescharacteristic
205C2×C4×C61C22×D411yescharacteristic
206C2×C4×C61C2×#4111yescharacteristic
207C2×C4×C61C2×#4111yescharacteristic
208C2×C4×C61H(5,𝔽2)11yescharacteristic
209C2×C4×C61H(5,𝔽2)11yescharacteristic
210C22⋊D61—88no—
211C2×Dic61—22no—
212C2×Dic61—22no—
213C2×Dic61—41no—
214C2×Dic61—41no—
215C2×Dic61—41no—
216C2×Dic61—41no—
217C2×Dic61—41no—
218C2×Dic61—42no—
219C2×Dic61—42no—
220C2×Dic61—42no—
221C2×Dic61—42no—
222C2×Dic61—42no—
223C2×Dic61—41no—
224C2×Dic61—41no—
225C2×Dic61—42no—
226D244—22no—
227D244—22no—
228D244—41no—
229D244—41no—
230C4×D61—84no—
231Dic122—42no—
232Dic122—42no—
233Dic122—42no—
234Dic122—42no—
235C8×C61—24no—
236C8×C61—22no—
237C8×C61—41no—
238#25385—611nomaximal, Sylow
239C22×C61—24no—
240C22×C61—24no—
241C22×C61—24no—
242C22×C61—42no—
243C22×C61—42no—
244C22×C61—42no—
245C22×C61—42no—
246C22×C61C2×#12111yescharacteristic
247C22×C61D4211yescharacteristic
248C22×C61#54611yescharacteristic
249C22×C61C22×SD1611yescharacteristic
250C22×C61C2×#13511yescharacteristic
251C22×C61#57711yescharacteristic
252C22×C61#57811yescharacteristic
253C2×D61—42no—
254C2×D61—42no—
255C2×D61—42no—
256C2×D61—81no—
257C2×D61—81no—
258C2×D61—81no—
259C2×D61—81no—
260C4×C61—22no—
261C4×C61—42no—
262C4×C61#46514yescomplemented
263C4×C61#53211yescharacteristic
264C4×C61#54011yescharacteristic
265C4×C61D4211yescharacteristic
266C4×C61#55011yescharacteristic
267Dic61—42no—
268Dic61—42no—
269Dic61—82no—
270Dic61—82no—
271Dic61—81no—
272C2×#465—612no—
273C2×#461—611no—
274C2×C8×D4D6111yescharacteristic, complemented
275#2542—611no—
276#2537—611no—
277#2035—612no—
278#2000—612no—
279#1956—611no—
280#1948—611no—
281#1892—612no—
282C4×#102—611no—
283C2×C61—44no—
284C2×C61#242811yescharacteristic
285C2×C61C2×#46511yescharacteristic
286C2×C61#306311yescharacteristic
287C2×C61D4×SD1611yescharacteristic
288C2×C61#328311yescharacteristic
289C2×C61#329811yescharacteristic
290C2×C61#330311yescharacteristic
291D61—82no—
292C22×SD16—612no—
293C23×C8C2×D6112yes—
294C2×C4⋊Q8—611no—
295C2×C22⋊Q8—612no—
296C2×C22⋊Q8—612no—
297C2×#121—612no—
298C2×#121—1221no—
299C2×C4×D4—611no—
300C2×C4×D4C2×D6111yescharacteristic
301#465—1228no—
302#461—1224no—
303C8×D4—24no—
304C2×#106—612no—
305C2×C4⋊C8C2×D6111yescharacteristic
306C2×#103—612no—
307C2×#103—612no—
308C2×#103—612no—
309C2×#102—611no—
310C2×#102—611no—
311C2×C22⋊C8C2×D6112yes—
312C2×C4×C8C2×D6111yescharacteristic
313#399—1222no—
314#394—1222no—
315#390—1221no—
316#384—611no—
317#384—1221no—
318#380—612no—
319#380—1221no—
320C4×C4⋊C4—611no—
321#340—611no—
322#340—612no—
323#340—611no—
324C61#2538511yescharacteristic, complemented, Sylow
325C22×Q8—612no—
326C22×D4—611no—
327C22×D4—1222no—
328C22×D4D24411yescharacteristic
329C23×C4—1221no—
330C23×C4C22×D6112yes—
331C2×SD16—1224no—
332C2×SD16—1224no—
333C2×SD16—2442no—
334C22×C8—24no—
335C22×C8—24no—
336C22×C8—42no—
337C22×C8C22×D6111yescharacteristic
338C22×C8C22×D6111yescharacteristic
339C22×C8C22×D6112yes—
340C4⋊Q8—1222no—
341C4⋊Q8—1222no—
342C22⋊Q8—1224no—
343C22⋊Q8—1224no—
344C22⋊Q8—2442no—
345C22⋊Q8—2442no—
346#121—1224no—
347#121—2442no—
348#121—2444no—
349C4×D4—22no—
350C4×D4—22no—
351C4×D4—1222no—
352C4×D4—1222no—
353C2×C4⋊C4—611no—
354C2×C4⋊C4—611no—
355C2×C4⋊C4—611no—
356C2×C4⋊C4—612no—
357C2×C4⋊C4—1221no—
358C2×C4⋊C4—1221no—
359C2×C4⋊C4—1222no—
360C2×C4⋊C4—1222no—
361C2×C4⋊C4—1221no—
362C2×C4⋊C4—1221no—
363C2×C4⋊C4D24411yescharacteristic
364C2×C22⋊C4—1222no—
365C2×C22⋊C4—1222no—
366C2×C22⋊C4—1222no—
367C2×C22⋊C4—1221no—
368C2×C22⋊C4—1222no—
369C2×C22⋊C4D24412yes—
370C2×C42—1221no—
371C2×C42C22×D6111yescharacteristic
372#106—1224no—
373C4⋊C8C22⋊D6114yescomplemented
374#103—1224no—
375#103—1224no—
376#103—2442no—
377#102—614no—
378#102—1222no—
379C22⋊C8—24no—
380C4×C8—22no—
381#95—1221no—
382#95—1222no—
383C24—1221no—
384C24C2×D24412yes—
385C2×Q8—1222no—
386C2×Q8—1222no—
387C2×Q8—1222no—
388C2×Q8—1222no—
389C2×Q8—1222no—
390C2×Q8—1222no—
391C2×D4—22no—
392C2×D4—22no—
393C2×D4—612no—
394C2×D4—1221no—
395C2×D4—1221no—
396C2×D4—1221no—
397C2×D4—1221no—
398C2×D4—1221no—
399C2×D4—4882no—
400C2×D4—2444no—
401C2×D4—2444no—
402C2×D4#1175819014yescomplemented
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—1221no—
410C22×C4—1221no—
411C22×C4—1221no—
412C22×C4—1222no—
413C22×C4—1222no—
414C22×C4—2442no—
415C22×C4—2442no—
416C22×C4—2442no—
417C22×C4C2×D24412yes—
418C22×C4C2×D24411yescharacteristic
419C22×C4C2×D24411yescharacteristic
420C22×C4C23×D6111yescharacteristic
421SD16—2448no—
422C2×C8—22no—
423C2×C8—21no—
424C2×C8—21no—
425C2×C8—22no—
426C2×C8—21no—
427C2×C8—21no—
428C2×C8—21no—
429C2×C8—48no—
430C2×C8D4×D6112yes—
431C2×C8C2×C22⋊D6114yes—
432C4⋊C4—22no—
433C4⋊C4—612no—
434C4⋊C4—612no—
435C4⋊C4—1221no—
436C4⋊C4—1221no—
437C4⋊C4—1222no—
438C4⋊C4—1221no—
439C4⋊C4—1221no—
440C4⋊C4—1222no—
441C4⋊C4—4881no—
442C4⋊C4—2444no—
443C4⋊C4—2444no—
444C4⋊C4—2442no—
445C4⋊C4—2442no—
446C4⋊C4—2442no—
447C22⋊C4—24no—
448C22⋊C4—2444no—
449C22⋊C4—2444no—
450C22⋊C4—2444no—
451C22⋊C4—2444no—
452C22⋊C4—2442no—
453C42—2442no—
454C42#1175822212yes—
455C42C2×C22⋊D6112yes—
456C23—22no—
457C23—22no—
458C23—22no—
459C23—22no—
460C23—22no—
461C23—22no—
462C23—1222no—
463C23—2441no—
464C23—2441no—
465C23—2441no—
466C23—2441no—
467C23—2441no—
468C23—2441no—
469C23C2×#1175819014yes—
470C23C22×D24411yescharacteristic
471Q8—2442no—
472Q8—2442no—
473Q8—2442no—
474Q8—2442no—
475D4—28no—
476D4—1222no—
477D4—1222no—
478D4—4888no—
479D4—2441no—
480D4—2441no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—1222no—
491C2×C4—1222no—
492C2×C4—4884no—
493C2×C4—2441no—
494C2×C4—2442no—
495C2×C4—2441no—
496C2×C4—2442no—
497C2×C4—2441no—
498C2×C4—2442no—
499C2×C4—2442no—
500C2×C4—2442no—
501C2×C4—2442no—
502C2×C4—2441no—
503C2×C4—2441no—
504C2×C4—2441no—
505C2×C4—2441no—
506C2×C4#1451975112yes—
507C2×C4#1451975112yes—
508C2×C4C2×#1175819012yes—
509C2×C4#1451979912yes—
510C2×C4C2×D4×D6111yescharacteristic
511C2×C4C22×C22⋊D6111yescharacteristic
512C2×C4#41×D6111yescharacteristic
513C2×C4C2×#1175822211yescharacteristic
514C2×C4#1451982211yescharacteristic
515C2×C4#1451982311yescharacteristic
516C8—22no—
517C8—24no—
518C8—41no—
519C22—24no—
520C22—24no—
521C22—24no—
522C22—42no—
523C22—42no—
524C22—42no—
525C22—42no—
526C22—4881no—
527C22—4881no—
528C22—4881no—
529C22—4881no—
530C22—2442no—
531C22—2442no—
532C22—2442no—
533C22C2×#1451975111yescharacteristic
534C22D4×D24411yescharacteristic
535C22#1684040711yescharacteristic
536C22C22×#1175819011yescharacteristic
537C22C2×#1451979911yescharacteristic
538C22#1684057411yescharacteristic
539C22#1684057911yescharacteristic
540C4—22no—
541C4—42no—
542C4—4881no—
543C4—4882no—
544C4—4882no—
545C4—2442no—
546C4—2442no—
547C4#1683999114yescomplemented
548C4#1684030911yescharacteristic
549C4#1684036311yescharacteristic
550C4D4×C22⋊D6111yescharacteristic
551C4#1684043111yescharacteristic
552C2—44no—
553C2—4882no—
554C2#2144940111yescharacteristic
555C2C2×#1683999111yescharacteristic
556C2#2145551011yescharacteristic
557C2D4×#1175819011yescharacteristic
558C2#2145754111yescharacteristic
559C2#2145768511yescharacteristic
560C2#2145774811yescharacteristic
subgroupG/H
centerC23C22×D244
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C8×D4×C61C2
socleC23×C61C22×D4
prime radical#21450732C2
hypercenterC2×C8×D4D61
hypocenterC61#25385
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×C8×D4

Complements

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

NH ≅ G/Ntype
1. C2×#16839991C2semidirect
2. C2×#16839968C2semidirect
3. #21450772C2semidirect
5. #21447016C2semidirect
6. #21446824C2semidirect
7. #21446621C2semidirect
8. #21446583C2semidirect
9. #21446242C2semidirect
10. C4×#14519699C2semidirect
11. C2×C8×D4×C61C2semidirect
83. #14519699D4semidirect
99. C4⋊C8×C61D4semidirect
106. C2×D4×C61SD16semidirect
262. C4×C61#465semidirect
274. C2×C8×D4D61semidirect
324. C61#25385semidirect
373. C4⋊C8C22⋊D61semidirect
402. C2×D4#11758190semidirect
547. C4#16839991semidirect

Maximal subgroups

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

subgroupindex
C2×#168399912
C2×#168399682
#214507722
#214507322
#214470162
#214468242
#214466212
#214465832
#214462422
C4×#145196992
C2×C8×D4×C612
#2538561