Finite Group Databasebeta

← #29696230

Subgroups — #29696230

Order 15616.

1474 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×#16840415C211yesmaximal, characteristic
2C2×#16839827C211yesmaximal, characteristic, complemented
3C4⋊C4×D244C211yesmaximal, characteristic, complemented
4#21446781C211yesmaximal, characteristic, complemented
5#21446246C212yesmaximal, complemented
6#21446135C212yesmaximal, complemented
7#21445904C212yesmaximal, complemented
8#21445820C211yesmaximal, characteristic, complemented
9C4×#14519697C211yesmaximal, characteristic, complemented
10#340×D61C212yesmaximal, complemented
11C61×#1988C211yesmaximal, characteristic, complemented
12C22×#11758226—21no—
13C22×C8×D61C2212yes—
14#16840415—24no—
15#16840415C418yescomplemented
16C2×#14519762—22no—
17C2×#14519753—21no—
18C2×#14519752C2212yes—
19C2×C4×D244—21no—
20C2×C4×D244C2211yescharacteristic
21C2×C4×D244C2211yescharacteristic
22C2×C4⋊C4×D61C2212yes—
23C2×C4⋊C4×D61C2212yes—
24#16839827—24no—
25C2×#14519699C2212yes—
26C2×#14519697C2211yescharacteristic
27C2×#14519697C2211yescharacteristic
28C2×#14519689C2212yes—
29C2×C4×#92418C2211yescharacteristic
30#16839660—21no—
31#16839636—22no—
32#16839632C2211yescharacteristic
33#16839619C2212yes—
34#16839525C2212yes—
35#16839524C2211yescharacteristic
36#16839524C2211yescharacteristic
37#16839524C2212yes—
38#16839524C2212yes—
39C2×C4⋊Q8×C61C2211yescharacteristic
40C2×C4⋊C8×C61C2211yescharacteristic
41C4×C4⋊C4×C61C2211yescharacteristic
42C61×#340C2212yes—
43C61×#340C2212yes—
44C2×#11758226—28no—
45C2×#11758226—44no—
46C2×#11758226—44no—
47C22×D244—22no—
48C22×D244D411yescharacteristic
49C22×D244C2311yescharacteristic
50C22×C4×D61—22no—
51C22×C4×D61—21no—
52C22×C4×D61—22no—
53C22×C4×D61C2312yes—
54C22×#92418C2311yescharacteristic
55C22×#92418C2311yescharacteristic
56C2×C8×D61—24no—
57C2×C8×D61—24no—
58C2×C8×D61—42no—
59#14519762—216no—
60#14519753—28no—
61#14519752—24no—
62#14519752C2×C418yescomplemented
63C4×D244—22no—
64C4×D244—44no—
65C4×D244C2×C414yes—
66C4×D244C2×C414yescomplemented
67C4×D244D414yes—
68C2×C4⋊Dic61—21no—
69C2×C4⋊Dic61—21no—
70C2×C4⋊Dic61D411yescharacteristic
71C2×C4⋊Dic61C2312yes—
72C2×C4⋊Dic61C2311yescharacteristic
73C2×#11758196—22no—
74C2×#11758196—22no—
75C4⋊C4×D61—24no—
76C4⋊C4×D61—24no—
77C4⋊C4×D61—42no—
78C4⋊C4×D61C2×C418yescomplemented
79C2×#11758198—21no—
80C2×#11758198—22no—
81C2×#11758198—22no—
82C2×#11758198D412yes—
83#14519699—42no—
84#14519697—22no—
85#14519697—22no—
86#14519689—24no—
87C4×#92418—22no—
88#14519684—22no—
89C22×Q8×C61—21no—
90C22×C8×C61C2312yes—
91C4⋊Q8×C61—22no—
92C4⋊Q8×C61C2×C414yescomplemented
93C2×C4⋊C4×C61—21no—
94C2×C4⋊C4×C61C2311yescharacteristic
95C2×C4⋊C4×C61C2312yes—
96C2×C4⋊C4×C61C2311yescharacteristic
97C2×C42×C61—21no—
98C2×C42×C61C2311yescharacteristic
99C4⋊C8×C61—22no—
100#95×C61—21no—
101C23×D61—21no—
102C23×D61C2×D412yes—
103C2×Q8×C61—24no—
104C2×Q8×C61—42no—
105C2×Q8×C61—42no—
106C22×C4×C61—21no—
107C22×C4×C61—22no—
108C22×C4×C61—21no—
109C22×C4×C61C2×D411yescharacteristic
110C22×C4×C61C2×D411yescharacteristic
111C22×C4×C61C2411yescharacteristic
112C22×Dic61—21no—
113C22×Dic61—22no—
114C22×Dic61C2×D412yes—
115#11758226—432no—
116C2×D244—21no—
117C2×D244—21no—
118C2×D244—21no—
119C2×D244—21no—
120C2×D244—28no—
121C2×D244—21no—
122C2×D244—44no—
123C2×D244—44no—
124C2×D244C22⋊C414yes—
125C2×D244D814yes—
126C2×D244SD1614yes—
127C2×D244C22×C412yes—
128C2×C4×D61—22no—
129C2×C4×D61—22no—
130C2×C4×D61—22no—
131C2×C4×D61—22no—
132C2×C4×D61—28no—
133C2×C4×D61—24no—
134C2×C4×D61—44no—
135C2×C4×D61—44no—
136C2×C4×D61—42no—
137C2×C4×D61—42no—
138C2×C4×D61—44no—
139C2×C4×D61—44no—
140C2×C4×D61—82no—
141C2×C4×D61C22×C414yes—
142C2×C4×D61C2×D414yes—
143C2×C8×C61—22no—
144C2×C8×C61—22no—
145C2×C8×C61—22no—
146C4⋊C4×C61—24no—
147C4⋊C4×C61—22no—
148C4⋊C4×C61—21no—
149C4⋊C4×C61—21no—
150C4⋊C4×C61—21no—
151C4⋊C4×C61C22×C414yes—
152C4⋊C4×C61#4112yescomplemented
153C42×C61—42no—
154C42×C61C22×C412yes—
155C42×C61C2×D412yes—
156#11758198—24no—
157#11758198—28no—
158#11758198—44no—
159#11758198C22⋊C418yes—
160C4⋊Dic61—21no—
161C4⋊Dic61—22no—
162C4⋊Dic61—22no—
163C4⋊Dic61—42no—
164C4⋊Dic61—81no—
165C4⋊Dic61C22⋊C414yes—
166C4⋊Dic61C22×C412yes—
167#11758196—28no—
168#11758196—44no—
169C2×#92418—21no—
170C2×#92418—21no—
171C2×#92418—21no—
172C2×#92418—21no—
173C2×#92418—21no—
174C2×#92418#4112yescomplemented
175C8×D61—48no—
176C23×C61C22×D411yescharacteristic
177C22×D61—22no—
178C22×D61—22no—
179C22×D61—22no—
180C22×D61—22no—
181C22×D61—22no—
182C22×D61—41no—
183C22×D61—41no—
184C22×D61—41no—
185C22×D61—41no—
186C22×D61—41no—
187C22×D61—41no—
188C22×D61C2×C22⋊C414yes—
189C22×D61C2×D814yes—
190C22×D61C2×SD1614yes—
191Q8×C61—48no—
192C2×C4×C61—21no—
193C2×C4×C61—21no—
194C2×C4×C61—24no—
195C2×C4×C61—22no—
196C2×C4×C61—41no—
197C2×C4×C61—42no—
198C2×C4×C61—42no—
199C2×C4×C61—41no—
200C2×C4×C61—41no—
201C2×C4×C61—41no—
202C2×C4×C61—41no—
203C2×C4×C61C2×C22⋊C412yes—
204C2×C4×C61C2×C22⋊C412yes—
205C2×C4×C61C2×D812yes—
206C2×C4×C61C2×SD1612yes—
207C2×C4×C61C23×C411yescharacteristic
208C2×C4×C61C22×D411yescharacteristic
209C2×C4×C61C2×#4111yescharacteristic
210C2×C4×C61C2×#4111yescharacteristic
211C2×C4×C61H(5,𝔽2)11yescharacteristic
212C2×C4×C612−1+411yescharacteristic
213C2×Dic61—22no—
214C2×Dic61—22no—
215C2×Dic61—22no—
216C2×Dic61—41no—
217C2×Dic61—41no—
218C2×Dic61—42no—
219C2×Dic61—42no—
220C2×Dic61—42no—
221C2×Dic61C2×C22⋊C414yes—
222D244—22no—
223D244—22no—
224D244—416no—
225D244—41no—
226D244—41no—
227D244#102116yescomplemented
228C4×D61—24no—
229C4×D61—24no—
230C4×D61—24no—
231C4×D61—24no—
232C4×D61—416no—
233C4×D61—48no—
234C4×D61—88no—
235C8×C61—42no—
236#92418—22no—
237#92418—41no—
238#22485—611nomaximal, Sylow
239C22×C61C22×C22⋊C411yescharacteristic
240C22×C61#54611yescharacteristic
241C22×C61#54711yescharacteristic
242C22×C61C22×D811yescharacteristic
243C22×C61C22×SD1611yescharacteristic
244C22×C61#57711yescharacteristic
245C22×C61#57811yescharacteristic
246C2×D61—24no—
247C2×D61—24no—
248C2×D61—24no—
249C2×D61—24no—
250C2×D61—24no—
251C2×D61—24no—
252C2×D61—42no—
253C2×D61—42no—
254C2×D61—42no—
255C2×D61—81no—
256C2×D61—81no—
257C2×D61—81no—
258C2×D61—81no—
259C2×D61C2×#10218yes—
260C4×C61—22no—
261C4×C61—44no—
262C4×C61—42no—
263C4×C61—81no—
264C4×C61C2×#10214yes—
265C4×C61C4×#4111yescharacteristic
266C4×C61#52011yescharacteristic
267C4×C61#54611yescharacteristic
268C4×C61#54711yescharacteristic
269Dic61—24no—
270Dic61—42no—
271Dic61—82no—
272C2×#547—611no—
273C2×C8×D4—611no—
274D4×C4⋊C4—611no—
275#1990—611no—
276#1988D6111yescharacteristic, complemented
277#1893—612no—
278#1874—612no—
279#1823—612no—
280#1802—611no—
281C4×#102—611no—
282C2×#340—612no—
283C2×C61#233911yescharacteristic
284C2×C61C22×#10211yescharacteristic
285C2×C61#289811yescharacteristic
286C2×C61#329411yescharacteristic
287C2×C61#329811yescharacteristic
288C2×C61#333411yescharacteristic
289C2×C61#333511yescharacteristic
290D61—28no—
291D61—82no—
292C22×#41—1221no—
293C23×C8—612no—
294#547—618no—
295#547—1224no—
296C2×C4⋊Q8C2×D6111yescharacteristic
297C2×#123—1222no—
298C2×C22⋊Q8—1221no—
299C2×#121—612no—
300C2×C4×D4—611no—
301C2×C4×D4—611no—
302C2×C4×D4—1221no—
303C22×C4⋊C4—612no—
304C22×C4⋊C4—612no—
305C8×D4—1224no—
306C2×C4⋊C8C2×D6111yescharacteristic
307C2×#102—611no—
308C2×#102—611no—
309C2×#102—612no—
310C2×C22⋊C8—612no—
311C2×C4×C8—611no—
312#389—1221no—
313#385—1222no—
314#384—611no—
315#380—612no—
316C4×C4⋊C4C2×D6111yescharacteristic
317#340—612no—
318#340—611no—
319#340—612no—
320#340—612no—
321#340—611no—
322#340C2×D6112yes—
323#340C2×D6112yes—
324C61#2248511yescharacteristic, complemented, Sylow
325C2×#41—1228no—
326C2×#41—2444no—
327C2×#41—2444no—
328C22×Q8—21no—
329C22×D4—611no—
330C22×D4—611no—
331C22×D4—1222no—
332C23×C4—612no—
333C23×C4—1222no—
334C23×C4—1222no—
335C23×C4—1221no—
336C22×C8—611no—
337C22×C8—611no—
338C22×C8—1224no—
339C22×C8—1224no—
340C22×C8—2442no—
341C22×C8C22×D6112yes—
342C4⋊Q8—22no—
343C4⋊Q8C4×D6114yescomplemented
344#123—12216no—
345C22⋊Q8—1228no—
346#121—618no—
347#121—1224no—
348C4×D4—614no—
349C4×D4—614no—
350C4×D4—614no—
351C4×D4—1222no—
352C4×D4—2444no—
353C2×C4⋊C4—21no—
354C2×C4⋊C4—611no—
355C2×C4⋊C4—611no—
356C2×C4⋊C4—618no—
357C2×C4⋊C4—612no—
358C2×C4⋊C4—1224no—
359C2×C4⋊C4—1222no—
360C2×C4⋊C4—1224no—
361C2×C4⋊C4—1221no—
362C2×C4⋊C4—1222no—
363C2×C4⋊C4—1221no—
364C2×C4⋊C4—2442no—
365C2×C4⋊C4C22×D6111yescharacteristic
366C2×C4⋊C4C22×D6112yes—
367C2×C4⋊C4C22×D6111yescharacteristic
368C2×C22⋊C4—612no—
369C2×C22⋊C4—1222no—
370C2×C22⋊C4—1221no—
371C2×C22⋊C4—1222no—
372C2×C42—21no—
373C2×C42C22×D6111yescharacteristic
374C4⋊C8—22no—
375#102—1222no—
376#102—1222no—
377#102—2442no—
378C22⋊C8—1224no—
379C4×C8—1222no—
380#95—21no—
381#95—1222no—
382C24—612no—
383C24—1221no—
384#41—24432no—
385C2×Q8—24no—
386C2×Q8—42no—
387C2×Q8—42no—
388C2×D4—612no—
389C2×D4—614no—
390C2×D4—614no—
391C2×D4—614no—
392C2×D4—1228no—
393C2×D4—1221no—
394C2×D4—1221no—
395C2×D4—1221no—
396C2×D4—1221no—
397C2×D4—1221no—
398C2×D4—2444no—
399C2×D4—2444no—
400C22×C4—21no—
401C22×C4—22no—
402C22×C4—21no—
403C22×C4—612no—
404C22×C4—614no—
405C22×C4—614no—
406C22×C4—1228no—
407C22×C4—1222no—
408C22×C4—1222no—
409C22×C4—1222no—
410C22×C4—1222no—
411C22×C4—1221no—
412C22×C4—1224no—
413C22×C4—1222no—
414C22×C4—4882no—
415C22×C4—2444no—
416C22×C4—2444no—
417C22×C4—2444no—
418C22×C4—2444no—
419C22×C4—2442no—
420C22×C4—2442no—
421C22×C4D4×D6111yescharacteristic
422C22×C4D4×D6111yescharacteristic
423C22×C4C23×D6111yescharacteristic
424C2×C8—22no—
425C2×C8—22no—
426C2×C8—22no—
427C2×C8—612no—
428C2×C8—1221no—
429C2×C8—1221no—
430C2×C8—1221no—
431C2×C8—1221no—
432C2×C8—1221no—
433C2×C8—2448no—
434C4⋊C4—22no—
435C4⋊C4—21no—
436C4⋊C4—24no—
437C4⋊C4—21no—
438C4⋊C4—21no—
439C4⋊C4—614no—
440C4⋊C4—612no—
441C4⋊C4—1228no—
442C4⋊C4—1221no—
443C4⋊C4—1222no—
444C4⋊C4—1222no—
445C4⋊C4—4881no—
446C4⋊C4—2444no—
447C4⋊C4—2442no—
448C4⋊C4C2×C4×D6114yes—
449C4⋊C4#1175822612yescomplemented
450C22⋊C4—618no—
451C22⋊C4—1228no—
452C22⋊C4—1224no—
453C22⋊C4—2444no—
454C42—42no—
455C42C2×C4×D6112yes—
456C42D4×D6112yes—
457C23—614no—
458C23—614no—
459C23—614no—
460C23—1222no—
461C23—1222no—
462C23—1222no—
463C23—1222no—
464C23—1222no—
465C23—2441no—
466C23—2441no—
467C23—2441no—
468C23—2441no—
469C23—2441no—
470C23—2441no—
471C23C2×D4×D6111yescharacteristic
472Q8—48no—
473D4—6116no—
474D4—1222no—
475D4—1222no—
476D4—24416no—
477D4—2441no—
478D4—2441no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—24no—
482C2×C4—22no—
483C2×C4—41no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—614no—
491C2×C4—1222no—
492C2×C4—1222no—
493C2×C4—1224no—
494C2×C4—1224no—
495C2×C4—1224no—
496C2×C4—1224no—
497C2×C4—1222no—
498C2×C4—4888no—
499C2×C4—24416no—
500C2×C4—2442no—
501C2×C4—2442no—
502C2×C4—2442no—
503C2×C4—2441no—
504C2×C4—2441no—
505C2×C4—2448no—
506C2×C4C22⋊C4×D6112yes—
507C2×C4C22⋊C4×D6112yes—
508C2×C4D8×D6112yes—
509C2×C4SD16×D6112yes—
510C2×C4C22×C4×D6111yescharacteristic
511C2×C4C2×D4×D6111yescharacteristic
512C2×C4#41×D6111yescharacteristic
513C2×C4C2×#1175822611yescharacteristic
514C2×C4#1451982311yescharacteristic
515C2×C4#1451982511yescharacteristic
516C8—42no—
517C8—1222no—
518C8—2441no—
519C22—618no—
520C22—1224no—
521C22—1224no—
522C22—1224no—
523C22—1224no—
524C22—1224no—
525C22—1224no—
526C22—4881no—
527C22—4881no—
528C22—4881no—
529C22—4881no—
530C22—2442no—
531C22—2442no—
532C22—2442no—
533C22C2×C22⋊C4×D6111yescharacteristic
534C22#1684040311yescharacteristic
535C22#1684041411yescharacteristic
536C22C2×D8×D6111yescharacteristic
537C22C2×SD16×D6111yescharacteristic
538C22#1684057111yescharacteristic
539C22#1684057811yescharacteristic
540C4—22no—
541C4—42no—
542C4—44no—
543C4—81no—
544C4—1224no—
545C4—4882no—
546C4—2442no—
547C4#102×D6114yes—
548C4C4×#1175822611yescharacteristic
549C4#1684024411yescharacteristic
550C4#1684040511yescharacteristic
551C4#1684041511yescharacteristic
552C2—1228no—
553C2—4882no—
554C2#2144864811yescharacteristic
555C2C2×#102×D6111yescharacteristic
556C2#2145435111yescharacteristic
557C2#2145762911yescharacteristic
558C2#2145768111yescharacteristic
559C2#2145816511yescharacteristic
560C2#2145817911yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1988C2
socleC23×C61C22×D4
prime radicalC2×#16840415C2
cosocleC22×C4×C61C24
hypercenter#1988D61
hypocenterC61#22485
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1988

Complements

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

NH ≅ G/Ntype
2. C2×#16839827C2semidirect
3. C4⋊C4×D244C2semidirect
4. #21446781C2semidirect
5. #21446246C2semidirect
6. #21446135C2semidirect
7. #21445904C2semidirect
8. #21445820C2semidirect
9. C4×#14519697C2semidirect
10. #340×D61C2semidirect
11. C61×#1988C2semidirect
15. #16840415C4semidirect
62. #14519752C2×C4semidirect
66. C4×D244C2×C4semidirect
78. C4⋊C4×D61C2×C4semidirect
92. C4⋊Q8×C61C2×C4semidirect
152. C4⋊C4×C61#41semidirect
174. C2×#92418#41semidirect
227. D244#102semidirect
276. #1988D61semidirect
324. C61#22485semidirect
343. C4⋊Q8C4×D61semidirect
449. C4⋊C4#11758226semidirect

Maximal subgroups

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

subgroupindex
C2×#168404152
C2×#168398272
C4⋊C4×D2442
#214467812
#214462462
#214461352
#214459042
#214458202
C4×#145196972
#340×D612
C61×#19882
#2248561