Finite Group Databasebeta

← #29730849

Subgroups — #29730849

Order 15616.

1070 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×#16840007C212yesmaximal, complemented
2C2×#16839967C211yesmaximal, characteristic, complemented
3C2×#16839841C211yesmaximal, characteristic, complemented
4#21450768C211yesmaximal, characteristic, complemented
5#21450732C211yesmaximal, characteristic
6#21447025C212yesmaximal, complemented
7#21446823C212yesmaximal, complemented
8#21446621C211yesmaximal, characteristic, complemented
9#21446577C211yesmaximal, characteristic, complemented
10#21446251C212yesmaximal, complemented
11C4×#102×C61C211yesmaximal, characteristic, complemented
12C22×#11758200C2212yes—
13C22×#11758189C2212yes—
14C2×#14519777C2211yescharacteristic
15C2×#14519757C2212yes—
16C2×#14519757C2212yes—
17C2×#14519751—21no—
18C2×#14519748C2212yes—
19C2×C4×D244C2211yescharacteristic
20#16840007—28no—
21#16839967—24no—
22#16839841—24no—
23C2×#14519710C2212yes—
24C2×C4⋊#92418C2211yescharacteristic
25C2×#14519702C2212yes—
26C2×#14519702C2212yes—
27C2×#14519700C2212yes—
28C2×#14519689C2212yes—
29#16839699—22no—
30#16839678—22no—
31#16839664—21no—
32#16839632—21no—
33#16839632C2211yescharacteristic
34#16839622—21no—
35#16839619C2212yes—
36#16839524C2212yes—
37C2×C4×D4×C61C2211yescharacteristic
38C2×#102×C61C2211yescharacteristic
39C2×#102×C61C2211yescharacteristic
40C2×C4×C8×C61C2211yescharacteristic
41C4×C4⋊C4×C61C2211yescharacteristic
42C61×#340C2211yescharacteristic
43C61×#340C2211yescharacteristic
44C22×Dic122C2312yes—
45C22×D244D411yescharacteristic
46C22×C22⋊D61—22no—
47C22×C4×D61C2312yes—
48C2×#11758200—24no—
49C2×#11758200—24no—
50C2×#11758200—42no—
51C2×#11758189—24no—
52C2×#11758189—24no—
53C2×#11758189—42no—
54C22×#92418C2312yes—
55#14519777—22no—
56#14519777—22no—
57#14519757—24no—
58#14519757—24no—
59#14519757—42no—
60#14519757—42no—
61#14519751—44no—
62#14519748—24no—
63#14519748—42no—
64C4×D244—22no—
65C4×D244—22no—
66C2×C4⋊Dic61—21no—
67C2×C4⋊Dic61—21no—
68C2×C4⋊Dic61—21no—
69C2×C4⋊Dic61—22no—
70C2×C4⋊Dic61—22no—
71C2×C4⋊Dic61—21no—
72C2×C4⋊Dic61D411yescharacteristic
73C2×C4⋊Dic61C2312yes—
74C2×C22⋊Dic61—22no—
75C2×#11758198—22no—
76C2×#11758198—22no—
77C2×#11758198—22no—
78C2×#11758198D412yes—
79#14519710—24no—
80C4⋊#92418D414yescomplemented
81#14519702—24no—
82#14519702—42no—
83#14519700—24no—
84#14519689—24no—
85#14519684—22no—
86C22×D4×C61C2311yescharacteristic
87C23×C4×C61—21no—
88C22×C8×C61C2311yescharacteristic
89C22×C8×C61C2311yescharacteristic
90C4×D4×C61—22no—
91C4×D4×C61—22no—
92C2×C4⋊C4×C61C2311yescharacteristic
93C2×C4⋊C4×C61C2311yescharacteristic
94C2×C4⋊C4×C61C2311yescharacteristic
95C2×C22⋊C4×C61—21no—
96C2×C42×C61—21no—
97C2×C42×C61C2311yescharacteristic
98#102×C61—22no—
99#102×C61D414yescomplemented
100C4×C8×C61—22no—
101#95×C61—21no—
102C24×C61—21no—
103C23×D61C2×D412yes—
104C2×D4×C61—21no—
105C2×D4×C61—21no—
106C2×D4×C61—21no—
107C2×D4×C61—21no—
108C2×D4×C61—21no—
109C2×D4×C61C2×D412yes—
110C22×C4×C61—21no—
111C22×C4×C61—21no—
112C22×C4×C61—21no—
113C22×C4×C61—42no—
114C22×C4×C61—42no—
115C22×C4×C61—42no—
116C22×C4×C61C2×D411yescharacteristic
117C22×C4×C61C2×D411yescharacteristic
118C22×C4×C61C2411yescharacteristic
119C2×C22⋊D61—44no—
120C2×C22⋊D61—44no—
121C2×C22⋊D61—82no—
122C22×Dic61—22no—
123C22×Dic61—22no—
124C22×Dic61C2×D412yes—
125C2×D244—22no—
126C2×D244—22no—
127C2×D244—22no—
128C2×C4×D61—22no—
129C2×C4×D61—22no—
130C2×C4×D61—22no—
131C2×C4×D61—22no—
132C2×C4×D61—22no—
133C2×C4×D61—22no—
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—21no—
142C2×C8×C61—21no—
143C2×C8×C61—21no—
144C2×C8×C61—21no—
145C2×C8×C61C2×D412yes—
146C4⋊C4×C61—21no—
147C4⋊C4×C61—21no—
148C4⋊C4×C61—21no—
149C4⋊C4×C61—21no—
150C4⋊C4×C61C2×D412yes—
151C4⋊C4×C61#4112yes—
152C22⋊C4×C61—42no—
153C42×C61—42no—
154C42×C61C2×D412yes—
155C42×C61#4112yes—
156C22⋊Dic61—44no—
157#11758200—48no—
158#11758198—24no—
159#11758198—44no—
160#11758198—44no—
161#11758198—44no—
162C4⋊Dic61—22no—
163C4⋊Dic61—22no—
164C4⋊Dic61—22no—
165C4⋊Dic61—44no—
166C4⋊Dic61—44no—
167C4⋊Dic61—42no—
168C4⋊Dic61—42no—
169C4⋊Dic61—42no—
170C4⋊Dic61—81no—
171C2×#92418—22no—
172C2×#92418—22no—
173C2×#92418C2×D414yes—
174#11758189—48no—
175C23×C61—22no—
176C23×C61—41no—
177C23×C61—41no—
178C23×C61—41no—
179C23×C61—41no—
180C23×C61—41no—
181C23×C61—41no—
182C23×C61C22×D411yescharacteristic
183C22×D61—22no—
184C22×D61—22no—
185C22×D61—22no—
186C22×D61—22no—
187C22×D61—22no—
188C22×D61—22no—
189C22×D61—22no—
190D4×C61—22no—
191D4×C61—22no—
192D4×C61—41no—
193D4×C61—41no—
194C2×C4×C61—21no—
195C2×C4×C61—21no—
196C2×C4×C61—22no—
197C2×C4×C61—22no—
198C2×C4×C61—41no—
199C2×C4×C61—41no—
200C2×C4×C61—41no—
201C2×C4×C61—41no—
202C2×C4×C61—41no—
203C2×C4×C61—41no—
204C2×C4×C61—41no—
205C2×C4×C61—84no—
206C2×C4×C61#12112yes—
207C2×C4×C61#12112yes—
208C2×C4×C61AGL(1,ℤ/8)12yes—
209C2×C4×C61#13712yes—
210C2×C4×C61C22×D411yescharacteristic
211C2×C4×C61C22×D411yescharacteristic
212C2×C4×C61C2×#4111yescharacteristic
213C2×C4×C61C2×#4111yescharacteristic
214C2×C4×C61H(5,𝔽2)11yescharacteristic
215C2×C4×C61H(5,𝔽2)11yescharacteristic
216C22⋊D61—88no—
217C2×Dic61—22no—
218C2×Dic61—22no—
219C2×Dic61—22no—
220C2×Dic61—42no—
221C2×Dic61—42no—
222C2×Dic61—42no—
223C2×Dic61—42no—
224C2×Dic61—42no—
225C2×Dic61—42no—
226D244—44no—
227C4×D61—42no—
228C4×D61—42no—
229C4×D61—42no—
230C4×D61—42no—
231Dic122—42no—
232Dic122—42no—
233Dic122—42no—
234Dic122—42no—
235C8×C61—22no—
236C8×C61—41no—
237#92418—24no—
238#25393—611nomaximal, Sylow
239C22×C61—42no—
240C22×C61—42no—
241C22×C61—42no—
242C22×C61—81no—
243C22×C61—81no—
244C22×C61—81no—
245C22×C61—81no—
246C22×C61C2×#12111yescharacteristic
247C22×C61D4211yescharacteristic
248C22×C61#54611yescharacteristic
249C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
250C22×C61C2×#13711yescharacteristic
251C22×C61#57711yescharacteristic
252C22×C61#57811yescharacteristic
253C2×D61—42no—
254C2×D61—42no—
255C2×D61—42no—
256C2×D61—42no—
257C2×D61—42no—
258C2×D61—42no—
259C2×D61—42no—
260C4×C61—22no—
261C4×C61—42no—
262C4×C61—42no—
263C4×C61—81no—
264C4×C61#46814yescomplemented
265C4×C61#53211yescharacteristic
266C4×C61#54011yescharacteristic
267C4×C61D4211yescharacteristic
268C4×C61#55011yescharacteristic
269Dic61—42no—
270Dic61—82no—
271Dic61—82no—
272C2×#468—612no—
273C2×#461—611no—
274C2×#436—611no—
275#2542—611no—
276#2537—611no—
277#2036—612no—
278#2000—612no—
279#1956—611no—
280#1947—611no—
281#1894—612no—
282C4×#102D6111yescharacteristic, complemented
283C2×C61—82no—
284C2×C61#242811yescharacteristic
285C2×C61C2×#46811yescharacteristic
286C2×C61#306311yescharacteristic
287C2×C61#327211yescharacteristic
288C2×C61#327611yescharacteristic
289C2×C61#332911yescharacteristic
290C2×C61#333011yescharacteristic
291D61—82no—
292C22×SD16—612no—
293C22×M16—612no—
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#468—1228no—
302#461—1224no—
303#436—1224no—
304C2×#107—612no—
305C2×C4⋊C8—611no—
306C2×#103—612no—
307C2×#103—612no—
308C2×#103—612no—
309C2×#102C2×D6111yescharacteristic
310C2×#102C2×D6111yescharacteristic
311C2×C22⋊C8—612no—
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⋊C4C2×D6111yescharacteristic
321#340—612no—
322#340C2×D6111yescharacteristic
323#340C2×D6111yescharacteristic
324C61#2539311yescharacteristic, complemented, Sylow
325C22×Q8—612no—
326C22×D4—611no—
327C22×D4—1222no—
328C22×D4C22×D6111yescharacteristic
329C23×C4—21no—
330C23×C4—612no—
331C2×SD16—1224no—
332C2×SD16—1224no—
333C2×SD16—2442no—
334C2×M16—1224no—
335C2×M16—1224no—
336C2×M16—2442no—
337C22×C8—612no—
338C22×C8C22×D6111yescharacteristic
339C22×C8C22×D6111yescharacteristic
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—612no—
355C2×C4⋊C4—1221no—
356C2×C4⋊C4—1222no—
357C2×C4⋊C4—1222no—
358C2×C4⋊C4—1221no—
359C2×C4⋊C4—1221no—
360C2×C4⋊C4—1221no—
361C2×C4⋊C4C22×D6111yescharacteristic
362C2×C4⋊C4C22×D6111yescharacteristic
363C2×C4⋊C4C22×D6111yescharacteristic
364C2×C22⋊C4—21no—
365C2×C22⋊C4—612no—
366C2×C22⋊C4—1222no—
367C2×C22⋊C4—1222no—
368C2×C22⋊C4—1222no—
369C2×C22⋊C4—1222no—
370C2×C42—21no—
371C2×C42C22×D6111yescharacteristic
372#107—1224no—
373C4⋊C8—614no—
374#103—1224no—
375#103—1224no—
376#103—2442no—
377#102—22no—
378#102D24414yescomplemented
379C22⋊C8—1224no—
380C4×C8—22no—
381#95—21no—
382#95—1222no—
383C24—21no—
384C24—612no—
385C2×Q8—1222no—
386C2×Q8—1222no—
387C2×Q8—1222no—
388C2×Q8—1222no—
389C2×Q8—1222no—
390C2×Q8—1222no—
391C2×D4—21no—
392C2×D4—21no—
393C2×D4—21no—
394C2×D4—21no—
395C2×D4—21no—
396C2×D4—1222no—
397C2×D4—1222no—
398C2×D4—1222no—
399C2×D4—4882no—
400C2×D4—2444no—
401C2×D4—2444no—
402C2×D4C2×D24412yes—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—42no—
407C22×C4—42no—
408C22×C4—42no—
409C22×C4—612no—
410C22×C4—1222no—
411C22×C4—1222no—
412C22×C4—1222no—
413C22×C4—1222no—
414C22×C4—1222no—
415C22×C4—1222no—
416C22×C4—1222no—
417C22×C4—1222no—
418C22×C4D4×D6111yescharacteristic
419C22×C4D4×D6111yescharacteristic
420C22×C4C23×D6111yescharacteristic
421SD16—2448no—
422M16—2448no—
423C2×C8—21no—
424C2×C8—21no—
425C2×C8—21no—
426C2×C8—21no—
427C2×C8—21no—
428C2×C8—614no—
429C2×C8—1222no—
430C2×C8—1222no—
431C2×C8C2×D24412yes—
432C4⋊C4—21no—
433C4⋊C4—21no—
434C4⋊C4—21no—
435C4⋊C4—21no—
436C4⋊C4—1222no—
437C4⋊C4—1222no—
438C4⋊C4—1222no—
439C4⋊C4—4881no—
440C4⋊C4—2444no—
441C4⋊C4—2444no—
442C4⋊C4—2442no—
443C4⋊C4—2442no—
444C4⋊C4—2442no—
445C4⋊C4C2×D24412yes—
446C4⋊C4#1175822212yes—
447C22⋊C4—42no—
448C22⋊C4—1224no—
449C22⋊C4—2444no—
450C22⋊C4—2444no—
451C22⋊C4—2444no—
452C22⋊C4—2444no—
453C42—42no—
454C42#1175822212yes—
455C42D4×D6112yes—
456C23—22no—
457C23—41no—
458C23—41no—
459C23—41no—
460C23—41no—
461C23—41no—
462C23—41no—
463C23—1222no—
464C23—1222no—
465C23—1222no—
466C23—1222no—
467C23—1222no—
468C23—1222no—
469C23—1222no—
470C23C2×D4×D6111yescharacteristic
471Q8—2442no—
472Q8—2442no—
473Q8—2442no—
474Q8—2442no—
475D4—22no—
476D4—22no—
477D4—41no—
478D4—41no—
479D4—4888no—
480D4—2444no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—84no—
493C2×C4—1222no—
494C2×C4—1222no—
495C2×C4—1222no—
496C2×C4—2442no—
497C2×C4—2442no—
498C2×C4—2442no—
499C2×C4—2442no—
500C2×C4—2442no—
501C2×C4—2442no—
502C2×C4—2442no—
503C2×C4—2442no—
504C2×C4—2442no—
505C2×C4—2442no—
506C2×C4#1451975012yes—
507C2×C4#1451975012yes—
508C2×C4#1451980112yes—
509C2×C4#1451980612yes—
510C2×C4C2×D4×D6111yescharacteristic
511C2×C4C22×D24411yescharacteristic
512C2×C4C2×#1175822211yescharacteristic
513C2×C4C2×#1175822211yescharacteristic
514C2×C4#1451982311yescharacteristic
515C2×C4#1451982311yescharacteristic
516C8—22no—
517C8—41no—
518C8—1224no—
519C22—42no—
520C22—42no—
521C22—42no—
522C22—81no—
523C22—81no—
524C22—81no—
525C22—81no—
526C22—2442no—
527C22—2442no—
528C22—2442no—
529C22—2442no—
530C22—2442no—
531C22—2442no—
532C22—2442no—
533C22C2×#1451975011yescharacteristic
534C22D4×D24411yescharacteristic
535C22#1684040811yescharacteristic
536C22C2×#1451980111yescharacteristic
537C22C2×#1451980611yescharacteristic
538C22#1684056911yescharacteristic
539C22#1684057611yescharacteristic
540C4—22no—
541C4—42no—
542C4—42no—
543C4—81no—
544C4—4882no—
545C4—4882no—
546C4—2442no—
547C4#1684000714yescomplemented
548C4#1684031211yescharacteristic
549C4#1684036711yescharacteristic
550C4D4×D24411yescharacteristic
551C4#1684043311yescharacteristic
552C2—82no—
553C2—4882no—
554C2#2144940711yescharacteristic
555C2C2×#1684000711yescharacteristic
556C2#2145551311yescharacteristic
557C2#2145742011yescharacteristic
558C2#2145746611yescharacteristic
559C2#2145811511yescharacteristic
560C2#2145812611yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC4×#102×C61C2
socleC23×C61C22×D4
prime radical#21450732C2
hypercenterC4×#102D61
hypocenterC61#25393
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C4×#102

Complements

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

NH ≅ G/Ntype
1. C2×#16840007C2semidirect
2. C2×#16839967C2semidirect
3. C2×#16839841C2semidirect
4. #21450768C2semidirect
6. #21447025C2semidirect
7. #21446823C2semidirect
8. #21446621C2semidirect
9. #21446577C2semidirect
10. #21446251C2semidirect
11. C4×#102×C61C2semidirect
80. C4⋊#92418D4semidirect
99. #102×C61D4semidirect
264. C4×C61#468semidirect
282. C4×#102D61semidirect
324. C61#25393semidirect
378. #102D244semidirect
547. C4#16840007semidirect

Maximal subgroups

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

subgroupindex
C2×#168400072
C2×#168399672
C2×#168398412
#214507682
#214507322
#214470252
#214468232
#214466212
#214465772
#214462512
C4×#102×C612
#2539361