Finite Group Databasebeta

← #29721315

Subgroups — #29721315

Order 15616.

912 conjugacy classes of subgroups.

The subgroup lattice has 566 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840043C211yesmaximal, characteristic, complemented
2C2×#16839818C211yesmaximal, characteristic, complemented
3#21454172C211yesmaximal, characteristic, complemented
4C4×#14519752C211yesmaximal, characteristic
5#21447249C212yesmaximal, complemented
6#21447010C212yesmaximal
7#21446771C211yesmaximal, characteristic, complemented
8#21446563C212yesmaximal, complemented
9#21446517C212yesmaximal, complemented
10#21445816C211yesmaximal, characteristic, complemented
11C61×#1948C211yesmaximal, characteristic, complemented
12C2×#14519777C2211yescharacteristic
13C2×#14519757—21no—
14C2×#14519752C2211yescharacteristic
15C2×C4×D244—21no—
16C2×C4×D244C2211yescharacteristic
17C2×C42×D61C2211yescharacteristic
18#16840043—24no—
19#16839818—24no—
20C2×#14519711C2212yes—
21C2×C4⋊#92418C2211yescharacteristic
22C2×#14519699C2212yes—
23C2×#14519697C2212yes—
24C2×#14519689C2212yes—
25C2×#14519687C2212yes—
26#16839667—21no—
27#16839663C2211yescharacteristic
28#16839661—21no—
29#16839661C2212yes—
30#16839661C2211yescharacteristic
31#16839654—21no—
32#16839640—21no—
33#16839619—21no—
34#16839619C2211yescharacteristic
35#16839619C2212yes—
36C4×#11758198—21no—
37#16839524C2212yes—
38#16839524C2212yes—
39#16839524C2212yes—
40C2×#106×C61C2212yes—
41C2×C4⋊C8×C61C2211yescharacteristic
42C61×#384C2211yescharacteristic
43C4×C4⋊C4×C61C2211yescharacteristic
44C61×#340C2212yes—
45C22×Dic122—21no—
46C22×D244—21no—
47C22×D244C2311yescharacteristic
48C22×C4×D61—21no—
49C22×C4×D61D411yescharacteristic
50C22×C4×D61D411yescharacteristic
51C22×C4×D61C2311yescharacteristic
52C22×#92418C2312yes—
53#14519777—22no—
54#14519777—22no—
55#14519757—44no—
56#14519752—22no—
57#14519752—22no—
58C4×D244—22no—
59C4×D244—22no—
60C4×D244—44no—
61C2×C4⋊Dic61—21no—
62C2×C4⋊Dic61—21no—
63C2×C4⋊Dic61—22no—
64C2×C4⋊Dic61C2311yescharacteristic
65C2×C4⋊Dic61C2312yes—
66C2×#11758198—21no—
67C2×#11758198—21no—
68C2×#11758198—22no—
69C2×#11758198—21no—
70C2×#11758198—21no—
71C2×#11758198—21no—
72C2×C4×Dic61D411yescharacteristic
73C2×C4×Dic61D411yescharacteristic
74C2×C4×Dic61C2311yescharacteristic
75C42×D61—22no—
76C42×D61—22no—
77#14519711—24no—
78C4⋊#92418—22no—
79#14519699—24no—
80#14519697—24no—
81#14519689—24no—
82#14519687—24no—
83#14519684—21no—
84#14519684—21no—
85#14519684—22no—
86#14519684—22no—
87C22×C8×C61C2312yes—
88C2×C4⋊C4×C61—21no—
89C2×C4⋊C4×C61—21no—
90C2×C4⋊C4×C61C2311yescharacteristic
91C2×C4⋊C4×C61C2312yes—
92C2×C4⋊C4×C61C2311yescharacteristic
93C2×C42×C61—21no—
94C2×C42×C61C2311yescharacteristic
95#106×C61—24no—
96C4⋊C8×C61D414yescomplemented
97#95×C61—21no—
98C23×D61—21no—
99C23×D61C2×D411yescharacteristic
100C22×C4×C61—21no—
101C22×C4×C61—21no—
102C22×C4×C61—22no—
103C22×C4×C61C2×D411yescharacteristic
104C22×C4×C61C2×D411yescharacteristic
105C22×C4×C61C2411yescharacteristic
106C22×Dic61—21no—
107C22×Dic61—22no—
108C22×Dic61C2×D411yescharacteristic
109C22×Dic61C2×D411yescharacteristic
110C22×Dic61C2×D411yescharacteristic
111C2×D244—21no—
112C2×D244—21no—
113C2×D244—21no—
114C2×D244—21no—
115C2×D244—24no—
116C2×D244—21no—
117C2×D244—42no—
118C2×D244—42no—
119C2×D244#4112yes—
120C2×C4×D61—22no—
121C2×C4×D61—22no—
122C2×C4×D61—22no—
123C2×C4×D61—22no—
124C2×C4×D61—22no—
125C2×C4×D61—21no—
126C2×C4×D61—21no—
127C2×C4×D61—21no—
128C2×C4×D61—21no—
129C2×C4×D61—21no—
130C2×C4×D61—21no—
131C2×C4×D61—41no—
132C2×C4×D61—42no—
133C2×C4×D61—42no—
134C2×C4×D61—42no—
135C2×Dic122—42no—
136C2×Dic122—42no—
137C2×Dic122—42no—
138C2×C8×C61—22no—
139C2×C8×C61—22no—
140C2×C8×C61C2×D414yes—
141C4⋊C4×C61—21no—
142C4⋊C4×C61—22no—
143C4⋊C4×C61—22no—
144C4⋊C4×C61—21no—
145C4⋊C4×C61—21no—
146C4⋊C4×C61—42no—
147C4⋊C4×C61—42no—
148C4⋊C4×C61#4112yes—
149C42×C61—42no—
150C42×C61C2×D412yes—
151C42×C61#4112yes—
152#11758198—24no—
153#11758198—42no—
154#11758198—42no—
155#11758198—42no—
156#11758198—44no—
157#11758198—42no—
158C4⋊Dic61—22no—
159C4⋊Dic61—21no—
160C4⋊Dic61—21no—
161C4⋊Dic61—22no—
162C4⋊Dic61—42no—
163C4⋊Dic61—44no—
164C4⋊Dic61—42no—
165C4×Dic61—22no—
166C4×Dic61—22no—
167C4×Dic61—21no—
168C4×Dic61—21no—
169C2×#92418—22no—
170C2×#92418—22no—
171C2×#92418—22no—
172C23×C61C22×D411yescharacteristic
173C22×D61—21no—
174C22×D61—21no—
175C22×D61—21no—
176C22×D61—21no—
177C22×D61—21no—
178C22×D61—21no—
179C22×D61—22no—
180C22×D61—41no—
181C22×D61—41no—
182C22×D61—41no—
183C22×D61—41no—
184C22×D61—41no—
185C22×D61—41no—
186C22×D61#13712yes—
187C2×C4×C61—21no—
188C2×C4×C61—21no—
189C2×C4×C61—22no—
190C2×C4×C61—41no—
191C2×C4×C61—41no—
192C2×C4×C61—42no—
193C2×C4×C61—41no—
194C2×C4×C61—42no—
195C2×C4×C61—41no—
196C2×C4×C61—41no—
197C2×C4×C61—42no—
198C2×C4×C61#12112yes—
199C2×C4×C61#12112yes—
200C2×C4×C61AGL(1,ℤ/8)12yes—
201C2×C4×C61AGL(1,ℤ/8)12yes—
202C2×C4×C61C22×D411yescharacteristic
203C2×C4×C61C2×#4111yescharacteristic
204C2×C4×C61C2×#4111yescharacteristic
205C2×C4×C61C2×#4111yescharacteristic
206C2×C4×C612−1+411yescharacteristic
207C2×C4×C612−1+411yescharacteristic
208C2×Dic61—21no—
209C2×Dic61—21no—
210C2×Dic61—21no—
211C2×Dic61—21no—
212C2×Dic61—21no—
213C2×Dic61—21no—
214C2×Dic61—21no—
215C2×Dic61—21no—
216C2×Dic61—21no—
217C2×Dic61—42no—
218C2×Dic61—41no—
219C2×Dic61—42no—
220C2×Dic61—41no—
221C2×Dic61—42no—
222C2×Dic61—41no—
223D244—22no—
224D244—22no—
225D244—48no—
226D244—41no—
227D244—41no—
228C4×D61—44no—
229C4×D61—41no—
230C4×D61—41no—
231C4×D61—41no—
232C4×D61—41no—
233C4×D61—44no—
234C4×D61—84no—
235Dic122—84no—
236C8×C61—24no—
237#92418—42no—
238#24709—611nomaximal, Sylow
239C22×C61C2×#12111yescharacteristic
240C22×C61#52511yescharacteristic
241C22×C61#53611yescharacteristic
242C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
243C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
244C22×C61C2×#13711yescharacteristic
245C22×C61#57511yescharacteristic
246C2×D61—22no—
247C2×D61—22no—
248C2×D61—22no—
249C2×D61—41no—
250C2×D61—41no—
251C2×D61—41no—
252C2×D61—41no—
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—82no—
263C4×C61—81no—
264C4×C61#46914yes—
265C4×C61#54711yescharacteristic
266C4×C61#54811yescharacteristic
267C4×C61#55211yescharacteristic
268C4×C61#55611yescharacteristic
269Dic61—41no—
270Dic61—41no—
271Dic61—41no—
272Dic61—82no—
273Dic61—81no—
274C2×#474—611no—
275C2×#432—611no—
276#2857—611no—
277C4×#121—611no—
278#2076—612no—
279#2034—612no—
280#1988—611no—
281#1948D6111yescharacteristic, complemented
282#1944—612no—
283#1936—612no—
284#1802—611no—
285C2×C61#241511yescharacteristic
286C2×C61C2×#46911yescharacteristic
287C2×C61#305611yescharacteristic
288C2×C61#311011yescharacteristic
289C2×C61#311511yescharacteristic
290C2×C61#323511yescharacteristic
291C2×C61#323911yescharacteristic
292D61—42no—
293D61—82no—
294C2×C4⋊Q8—611no—
295C2×C22⋊Q8—1221no—
296C2×#121—611no—
297C2×C4×D4—611no—
298C2×C4×D4—1221no—
299C22×C42—611no—
300#474—1224no—
301#432—1224no—
302C2×#107—612no—
303C2×#106C2×D6112yes—
304C2×C4⋊C8—611no—
305C2×C4⋊C8C2×D6111yescharacteristic
306C2×#102—612no—
307C2×#102—612no—
308C2×C22⋊C8—612no—
309C2×#97—612no—
310#391—1221no—
311#390—611no—
312#389—612no—
313#389—611no—
314#389—1221no—
315#388—1221no—
316#386—1221no—
317#384C2×D6111yescharacteristic
318#380—611no—
319#380—612no—
320#380—1221no—
321C4×C4⋊C4C2×D6111yescharacteristic
322C4×C22⋊C4—1221no—
323#340—612no—
324#340—612no—
325#340—612no—
326#340C2×D6112yes—
327C61#2470911yescharacteristic, complemented, Sylow
328C22×Q8—1221no—
329C22×D4—611no—
330C22×D4—1221no—
331C23×C4—611no—
332C23×C4—611no—
333C23×C4—611no—
334C23×C4—1221no—
335C22×C8—612no—
336C22×C8C22×D6112yes—
337C4⋊Q8—1222no—
338C4⋊Q8—1222no—
339C22⋊Q8—2444no—
340#121—1222no—
341#121—1222no—
342C4×D4—1222no—
343C4×D4—1222no—
344C4×D4—2444no—
345C2×C4⋊C4—21no—
346C2×C4⋊C4—21no—
347C2×C4⋊C4—611no—
348C2×C4⋊C4—612no—
349C2×C4⋊C4—1221no—
350C2×C4⋊C4—1222no—
351C2×C4⋊C4—1221no—
352C2×C4⋊C4C22×D6111yescharacteristic
353C2×C4⋊C4C22×D6112yes—
354C2×C4⋊C4C22×D6111yescharacteristic
355C2×C22⋊C4—1221no—
356C2×C22⋊C4—1221no—
357C2×C22⋊C4—1221no—
358C2×C22⋊C4—1221no—
359C2×C22⋊C4—1222no—
360C2×C22⋊C4—1221no—
361C2×C42—21no—
362C2×C42—611no—
363C2×C42—611no—
364C2×C42—611no—
365C2×C42—1222no—
366C2×C42—1222no—
367C2×C42C22×D6111yescharacteristic
368#107—1224no—
369#106—24no—
370C4⋊C8—1222no—
371C4⋊C8D24414yescomplemented
372#102—1224no—
373#102—1224no—
374C22⋊C8—1224no—
375#97—1224no—
376#95—21no—
377#95—1221no—
378#95—1221no—
379#95—1222no—
380#95—1222no—
381C24—611no—
382C24—1221no—
383C2×Q8—2442no—
384C2×Q8—2442no—
385C2×Q8—2442no—
386C2×D4—612no—
387C2×D4—1224no—
388C2×D4—1221no—
389C2×D4—1221no—
390C2×D4—1221no—
391C2×D4—1221no—
392C2×D4—1221no—
393C2×D4—2442no—
394C2×D4—2442no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—22no—
398C22×C4—611no—
399C22×C4—611no—
400C22×C4—611no—
401C22×C4—1222no—
402C22×C4—1221no—
403C22×C4—1221no—
404C22×C4—1221no—
405C22×C4—1221no—
406C22×C4—1221no—
407C22×C4—1221no—
408C22×C4—1222no—
409C22×C4—1222no—
410C22×C4—1222no—
411C22×C4—1222no—
412C22×C4—1221no—
413C22×C4—1222no—
414C22×C4—2441no—
415C22×C4—2442no—
416C22×C4—2442no—
417C22×C4—2442no—
418C22×C4D4×D6111yescharacteristic
419C22×C4D4×D6111yescharacteristic
420C22×C4C23×D6111yescharacteristic
421C2×C8—22no—
422C2×C8—22no—
423C2×C8—1222no—
424C2×C8—1222no—
425C2×C8—1222no—
426C2×C8C2×D24414yes—
427C4⋊C4—21no—
428C4⋊C4—22no—
429C4⋊C4—22no—
430C4⋊C4—21no—
431C4⋊C4—21no—
432C4⋊C4—42no—
433C4⋊C4—42no—
434C4⋊C4—1222no—
435C4⋊C4—1221no—
436C4⋊C4—1221no—
437C4⋊C4—1222no—
438C4⋊C4—2442no—
439C4⋊C4—2444no—
440C4⋊C4—2442no—
441C4⋊C4#1175822412yes—
442C22⋊C4—1224no—
443C22⋊C4—2442no—
444C22⋊C4—2442no—
445C22⋊C4—2442no—
446C22⋊C4—2444no—
447C22⋊C4—2442no—
448C42—42no—
449C42—1221no—
450C42—1221no—
451C42—1222no—
452C42—1222no—
453C42C2×D24412yes—
454C42#1175822612yes—
455C23—612no—
456C23—1221no—
457C23—1221no—
458C23—1221no—
459C23—1221no—
460C23—1221no—
461C23—1221no—
462C23—1222no—
463C23—2441no—
464C23—2441no—
465C23—2441no—
466C23—2441no—
467C23—2441no—
468C23—2441no—
469C23C2×D4×D6111yescharacteristic
470Q8—4884no—
471D4—1222no—
472D4—1222no—
473D4—2448no—
474D4—2441no—
475D4—2441no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—22no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—42no—
482C2×C4—41no—
483C2×C4—42no—
484C2×C4—41no—
485C2×C4—42no—
486C2×C4—41no—
487C2×C4—1221no—
488C2×C4—1221no—
489C2×C4—1221no—
490C2×C4—1221no—
491C2×C4—1221no—
492C2×C4—1221no—
493C2×C4—1221no—
494C2×C4—1221no—
495C2×C4—1221no—
496C2×C4—4884no—
497C2×C4—2444no—
498C2×C4—2441no—
499C2×C4—2442no—
500C2×C4—2441no—
501C2×C4—2442no—
502C2×C4—2441no—
503C2×C4—2442no—
504C2×C4—2441no—
505C2×C4—2441no—
506C2×C4—2444no—
507C2×C4—2441no—
508C2×C4—2441no—
509C2×C4#1451975212yes—
510C2×C4#1451975212yes—
511C2×C4#1451980412yes—
512C2×C4#1451980412yes—
513C2×C4C22×D24411yescharacteristic
514C2×C4#41×D6111yescharacteristic
515C2×C4C2×#1175822411yescharacteristic
516C2×C4C2×#1175822611yescharacteristic
517C2×C4#1451982411yescharacteristic
518C2×C4#1451982511yescharacteristic
519C8—24no—
520C8—2442no—
521C22—1222no—
522C22—1222no—
523C22—1222no—
524C22—4881no—
525C22—4881no—
526C22—4881no—
527C22—4881no—
528C22—2442no—
529C22—2442no—
530C22—2442no—
531C22—2441no—
532C22—2441no—
533C22—2441no—
534C22—2441no—
535C22C2×#1451975211yescharacteristic
536C22#1684027211yescharacteristic
537C22#1684033511yescharacteristic
538C22C2×#1451980411yescharacteristic
539C22C2×#1451980411yescharacteristic
540C22#137×D6111yescharacteristic
541C22#1684055811yescharacteristic
542C4—22no—
543C4—42no—
544C4—82no—
545C4—81no—
546C4—4881no—
547C4—4882no—
548C4—2441no—
549C4—2441no—
550C4—2441no—
551C4#1684001614yes—
552C4#1684041611yescharacteristic
553C4#1684042411yescharacteristic
554C4#1684045011yescharacteristic
555C4#1684048611yescharacteristic
556C2—4882no—
557C2—2442no—
558C2#2144924611yescharacteristic
559C2C2×#1684001611yescharacteristic
560C2#2145545011yescharacteristic
561C2#2145591711yescharacteristic
562C2#2145596911yescharacteristic
563C2#2145701511yescharacteristic
564C2#2145705311yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1948C2
socleC23×C61C22×D4
prime radicalC2×#14519752C22
hypercenter#1948D61
hypocenterC61#24709
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1948

Complements

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

NH ≅ G/Ntype
1. C2×#16840043C2semidirect
2. C2×#16839818C2semidirect
3. #21454172C2semidirect
5. #21447249C2semidirect
7. #21446771C2semidirect
8. #21446563C2semidirect
9. #21446517C2semidirect
10. #21445816C2semidirect
11. C61×#1948C2semidirect
96. C4⋊C8×C61D4semidirect
281. #1948D61semidirect
327. C61#24709semidirect
371. C4⋊C8D244semidirect

Maximal subgroups

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

subgroupindex
C2×#168400432
C2×#168398182
#214541722
C4×#145197522
#214472492
#214470102
#214467712
#214465632
#214465172
#214458162
C61×#19482
#2470961