Finite Group Databasebeta

← #29648205

Subgroups — #29648205

Order 15616.

876 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459836C211yesmaximal, characteristic
2#21454833C211yesmaximal, characteristic, complemented
3#21444456C212yesmaximal, complemented
4#21444448C212yesmaximal, complemented
5#21444440C212yesmaximal, complemented
6#21444435C212yesmaximal, complemented
7#21444433C211yesmaximal, characteristic, complemented
8#21444424C211yesmaximal, characteristic, complemented
9#21444413C211yesmaximal, characteristic, complemented
10#21444405C211yesmaximal, characteristic, complemented
11C61×#2923C211yesmaximal, characteristic, complemented
12Q8×Dic122C411yescharacteristic
13#16840482C411yescharacteristic
14#16840482C412yes—
15#16840480—21no—
16#16840480—22no—
17#16840477C411yescharacteristic
18#16840477C412yes—
19Q8×D244C411yescharacteristic
20#16840419—21no—
21#16840377—21no—
22#16840377—22no—
23#16840319C2211yescharacteristic
24#16840319C2212yes—
25#16840296C2211yescharacteristic
26#16840296C2212yes—
27#16840238—21no—
28#16839893—21no—
29#16839834—22no—
30D4×#92418—21no—
31#16839819C2211yescharacteristic
32#16839819C2212yes—
33C4×#11758194C2211yescharacteristic
34C2×#14519688C2212yes—
35#16839503C2211yescharacteristic
36#16839503C2211yescharacteristic
37#16839503C2212yes—
38#16839503C2212yes—
39#16839503C2212yes—
40#16839503C2212yes—
41#16839503C2212yes—
42#16839501C2211yescharacteristic
43#16839501C2211yescharacteristic
44#16839501C2212yes—
45C4×#41×C61C2211yescharacteristic
46C61×#433—22no—
47C61×#432C2212yes—
48C61×#432C2212yes—
49C61×#424—21no—
50C61×#408—21no—
51C4×M16×C61C2211yescharacteristic
52C2×#97×C61C2211yescharacteristic
53C2×Q8×D61—21no—
54C2×#11758194—21no—
55C22×#92418—22no—
56#14519777C2×C411yescharacteristic
57#14519777C2×C412yes—
58#14519777C2×C411yescharacteristic
59#14519777C2×C412yes—
60#14519776—22no—
61#14519776—21no—
62#14519771—21no—
63#14519771—22no—
64#14519770—22no—
65#14519770—22no—
66#14519770—22no—
67#14519769C2311yescharacteristic
68#14519769C2312yes—
69#14519765C2311yescharacteristic
70#14519764—21no—
71#14519764—22no—
72#14519763—22no—
73#14519763—22no—
74#14519763—22no—
75#14519760—22no—
76#14519760—21no—
77#14519757—22no—
78#14519757—21no—
79#14519756—21no—
80#14519756—22no—
81#14519756—22no—
82#14519756—21no—
83#14519756C2×C411yescharacteristic
84#14519756C2×C412yes—
85#14519751C2×C411yescharacteristic
86#14519751C2×C412yes—
87#14519748—21no—
88#14519748—22no—
89Q8×Dic61—21no—
90C4×Dic122—21no—
91C4×Dic122—22no—
92C4×Dic122—21no—
93C4×Dic122—22no—
94C4×Dic122C2×C411yescharacteristic
95C4×Dic122C2×C412yes—
96#14519742—22no—
97#14519742—21no—
98C4×D244C2×C411yescharacteristic
99C4×D244C2×C412yes—
100C4×C22⋊D61—21no—
101C4×C22⋊D61—22no—
102#14519736C2×C412yes—
103#14519736C2×C412yes—
104#14519736C2×C412yes—
105#14519735—22no—
106#14519735—21no—
107#14519734—21no—
108#14519734—22no—
109C4⋊#92418—21no—
110C4⋊#92418—22no—
111C22⋊#92418—22no—
112C22⋊#92418—21no—
113#14519688—22no—
114#14519688C2311yescharacteristic
115#14519688C2312yes—
116#14519688C2311yescharacteristic
117C4×#92418—21no—
118C2×#41×C61D411yescharacteristic
119C2×M16×C61—21no—
120C22×C8×C61—21no—
121C4×Q8×C61C2×C411yescharacteristic
122C4×Q8×C61D411yescharacteristic
123C4×D4×C61C2×C411yescharacteristic
124C4×D4×C61C2×C412yes—
125C4×D4×C61D411yescharacteristic
126C4×D4×C61D412yes—
127#117×C61D411yescharacteristic
128#117×C61D412yes—
129C2×C42×C61C2312yes—
130C2×C42×C61C2311yescharacteristic
131C4⋊C8×C61—22no—
132C4⋊C8×C61—22no—
133C22⋊C8×C61—22no—
134C22⋊C8×C61—22no—
135#97×C61—21no—
136#97×C61C2311yescharacteristic
137#97×C61C2311yescharacteristic
138#97×C61C2312yes—
139C4×C8×C61—21no—
140#41×C61—42no—
141C2×Q8×C61C22⋊C411yescharacteristic
142C2×D4×C61C22⋊C411yescharacteristic
143C2×D4×C61C22⋊C412yes—
144C22×C4×C61C22×C412yes—
145C22×C4×C61C22×C411yescharacteristic
146C22×C4×C61C2×D411yescharacteristic
147C22×C4×C61C2×D412yes—
148C22×C4×C61C2×D411yescharacteristic
149C22×C4×C61C2×D412yes—
150C2×C22⋊D61—21no—
151C2×C22⋊D61—22no—
152Q8×D61—44no—
153C2×D244C22×C411yescharacteristic
154C2×C4×D61—22no—
155C2×C4×D61—21no—
156C2×Dic122—22no—
157C2×Dic122—21no—
158C2×Dic122—21no—
159C2×Dic122—22no—
160C2×Dic122C22×C411yescharacteristic
161M16×C61—42no—
162C2×C8×C61—22no—
163C2×C8×C61—21no—
164C2×C8×C61—21no—
165C2×C8×C61—42no—
166C4⋊C4×C61C22⋊C412yes—
167C4⋊C4×C61C22⋊C411yescharacteristic
168C4⋊C4×C61C22⋊C412yes—
169C4⋊C4×C61C22⋊C411yescharacteristic
170C22⋊C4×C61C22⋊C412yes—
171C22⋊C4×C61C22⋊C412yes—
172C22⋊C4×C61C22⋊C412yes—
173C42×C61C22×C412yes—
174C42×C61C22×C411yescharacteristic
175C42×C61C2×D411yescharacteristic
176C42×C61C2×D412yes—
177C42×C61C2×D411yescharacteristic
178C42×C61C2×D412yes—
179C42×C61C2411yescharacteristic
180C22⋊Dic61—22no—
181C22⋊Dic61—22no—
182C22⋊Dic61—22no—
183C22⋊Dic61—21no—
184C22⋊Dic61—22no—
185#11758198—21no—
186#11758198—22no—
187#11758198—21no—
188#11758198—22no—
189#11758198—21no—
190#11758198—21no—
191C4⋊Dic61—22no—
192C4⋊Dic61—21no—
193C4⋊Dic61—21no—
194C4⋊Dic61—22no—
195C4⋊Dic61C22×C412yes—
196C4⋊Dic61C22×C411yescharacteristic
197C4⋊Dic61C22×C412yes—
198C4⋊Dic61C22×C411yescharacteristic
199#11758196—21no—
200#11758196—22no—
201#11758196—21no—
202#11758196—22no—
203#11758196—21no—
204#11758196—22no—
205#11758196—21no—
206#11758196—22no—
207C4×Dic61—22no—
208C4×Dic61—21no—
209C4×Dic61—22no—
210C4×Dic61—22no—
211C4×Dic61—22no—
212#11758194—42no—
213C2×#92418—21no—
214C2×#92418—21no—
215C2×#92418—22no—
216C2×#92418—44no—
217C23×C61C2×C22⋊C411yescharacteristic
218C23×C61C2×C22⋊C412yes—
219C22×D61—21no—
220Q8×C61—22no—
221D4×C61—22no—
222D4×C61—42no—
223C2×C4×C61—22no—
224C2×C4×C61—42no—
225C2×C4×C61—41no—
226C2×C4×C61—42no—
227C2×C4×C61—41no—
228C2×C4×C61—42no—
229C2×C4×C61C2×C22⋊C411yescharacteristic
230C2×C4×C61C2×C22⋊C412yes—
231C2×C4×C61C2×C22⋊C412yes—
232C2×C4×C61C2×C22⋊C411yescharacteristic
233C2×C4×C61C2×C22⋊C412yes—
234C2×C4×C61C2×C22⋊C411yescharacteristic
235C2×C4×C61C23×C411yescharacteristic
236C2×C4×C61C22×D411yescharacteristic
237C2×C4×C61C22×D411yescharacteristic
238C22⋊D61—82no—
239C22⋊D61—81no—
240C2×Dic61—21no—
241C2×Dic61—22no—
242C2×Dic61—21no—
243C2×Dic61—21no—
244C2×Dic61—22no—
245D244—41no—
246C4×D61—44no—
247C4×D61—42no—
248Dic122—44no—
249Dic122—42no—
250Dic122—41no—
251Dic122—82no—
252Dic122—81no—
253C8×C61—41no—
254C8×C61—41no—
255C8×C61—42no—
256#92418—22no—
257#92418—42no—
258#92418—41no—
259#17683—611nomaximal, Sylow
260C22×C61—21no—
261C22×C61—22no—
262C22×C61—21no—
263C22×C61—22no—
264C22×C61—22no—
265C22×C61—21no—
266C22×C61C22×C22⋊C411yescharacteristic
267C2×D61—41no—
268C2×D61—41no—
269C2×D61—41no—
270C4×C61—21no—
271C4×C61—21no—
272C4×C61—22no—
273C4×C61—21no—
274C4×C61—22no—
275C4×C61—21no—
276C4×C61—22no—
277C4×C61—21no—
278C4×C61#42112yes—
279Dic61—41no—
280Dic61—41no—
281Dic61—42no—
282Dic61—41no—
283Dic61—42no—
284#3518—611no—
285#2970—611no—
286#2923D6111yescharacteristic, complemented
287#1537—612no—
288#1535—612no—
289#1532—612no—
290#1531—612no—
291#1531—611no—
292#1529—611no—
293#1526—611no—
294#1524—611no—
295C2×C61—41no—
296C2×C61—42no—
297C2×C61#288911yescharacteristic
298C2×C61C2×#42111yescharacteristic
299C2×C61#290111yescharacteristic
300D61—81no—
301Q82—611no—
302#556—611no—
303#556—612no—
304#556—1221no—
305#556—1222no—
306#555—611no—
307#555—612no—
308D4×Q8—611no—
309#548—1221no—
310#542—1221no—
311#542—1222no—
312#533—612no—
313#533—611no—
314#529—611no—
315#529—612no—
316#519—1221no—
317C4×#41C2×D6111yescharacteristic
318#446—1221no—
319#435—1222no—
320C8×D4—1221no—
321#433—22no—
322#432—612no—
323#432—611no—
324#432C2×D6112yes—
325#432C2×D6112yes—
326#424—21no—
327#408—21no—
328C4×M16—611no—
329C4×M16C2×D6111yescharacteristic
330C2×#97—612no—
331C2×#97C2×D6111yescharacteristic
332#330—611no—
333#330—612no—
334#330—612no—
335#330—612no—
336#330—611no—
337#330—612no—
338#330—612no—
339#329—611no—
340#329—611no—
341#329—612no—
342C61#1768311yescharacteristic, complemented, Sylow
343C2×#41C22⋊D6111yescharacteristic
344C22×Q8—1221no—
345C2×M16—21no—
346C2×M16—1221no—
347C22×C8—21no—
348C22×C8—1222no—
349C4⋊Q8—612no—
350C4⋊Q8—611no—
351C4⋊Q8—612no—
352C4⋊Q8—611no—
353C4⋊Q8—1222no—
354C4⋊Q8—1221no—
355#126—1222no—
356#126—1222no—
357#126—1222no—
358#126—1221no—
359#126—1222no—
360#125—611no—
361#125—612no—
362#124—611no—
363#124—1221no—
364#124—1222no—
365#124—1222no—
366#124—1222no—
367#124—1222no—
368#123—1222no—
369#123—1221no—
370C22⋊Q8—611no—
371C22⋊Q8—612no—
372C22⋊Q8—1222no—
373C22⋊Q8—1221no—
374C22⋊Q8—1221no—
375C22⋊Q8—1222no—
376C22⋊Q8—1222no—
377C22⋊Q8—1221no—
378#121—611no—
379#121—612no—
380#121—1221no—
381#121—1222no—
382C4×Q8—611no—
383C4×Q8—612no—
384C4×Q8—1221no—
385C4×Q8—1222no—
386C4×Q8—1221no—
387C4×Q8—1221no—
388C4×Q8—1222no—
389C4×Q8—1221no—
390C4×Q8—1222no—
391C4×Q8C4×D6111yescharacteristic
392C4×Q8D24411yescharacteristic
393C4×D4—611no—
394C4×D4—612no—
395C4×D4—1221no—
396C4×D4—1222no—
397C4×D4C4×D6111yescharacteristic
398C4×D4C4×D6112yes—
399C4×D4D24411yescharacteristic
400C4×D4D24412yes—
401#117—612no—
402#117—612no—
403#117—612no—
404#117—1222no—
405#117—1221no—
406#117—1221no—
407#117—1222no—
408#117C22⋊D6111yescharacteristic
409#117C22⋊D6112yes—
410C2×C42C22×D6112yes—
411C2×C42C22×D6111yescharacteristic
412C4⋊C8—22no—
413C4⋊C8—22no—
414C4⋊C8—1221no—
415C4⋊C8—1222no—
416C22⋊C8—22no—
417C22⋊C8—22no—
418C22⋊C8—1222no—
419C22⋊C8—1221no—
420#97—21no—
421#97—611no—
422#97—612no—
423#97—611no—
424#97—1222no—
425#97C22×D6111yescharacteristic
426#97C22×D6111yescharacteristic
427#97C22×D6112yes—
428C4×C8—21no—
429C4×C8—1221no—
430#41—42no—
431C2×Q8—611no—
432C2×Q8—1221no—
433C2×Q8—1222no—
434C2×Q8—1222no—
435C2×Q8—1221no—
436C2×Q8—2444no—
437C2×Q8#1175819811yescharacteristic
438C2×D4—611no—
439C2×D4—1221no—
440C2×D4—1222no—
441C2×D4#1175819811yescharacteristic
442C2×D4#1175819812yes—
443C22×C4—1222no—
444C22×C4—1221no—
445C22×C4C2×C4×D6112yes—
446C22×C4C2×C4×D6111yescharacteristic
447C22×C4C2×D24412yes—
448C22×C4C2×D24411yescharacteristic
449C22×C4C2×C22⋊D6111yescharacteristic
450C22×C4C2×C22⋊D6112yes—
451M16—42no—
452M16—2442no—
453C2×C8—22no—
454C2×C8—21no—
455C2×C8—21no—
456C2×C8—42no—
457C2×C8—1221no—
458C2×C8—1221no—
459C2×C8—1222no—
460C2×C8—2444no—
461C4⋊C4—611no—
462C4⋊C4—612no—
463C4⋊C4—611no—
464C4⋊C4—612no—
465C4⋊C4—1221no—
466C4⋊C4—1222no—
467C4⋊C4—1221no—
468C4⋊C4—1222no—
469C4⋊C4—1221no—
470C4⋊C4—1222no—
471C4⋊C4—1221no—
472C4⋊C4—1222no—
473C4⋊C4—1222no—
474C4⋊C4—1221no—
475C4⋊C4—1221no—
476C4⋊C4—1222no—
477C4⋊C4#1175819812yes—
478C4⋊C4#1175819811yescharacteristic
479C4⋊C4#1175819812yes—
480C4⋊C4#1175819811yescharacteristic
481C22⋊C4—1222no—
482C22⋊C4—1222no—
483C22⋊C4—1222no—
484C22⋊C4—1221no—
485C22⋊C4—1222no—
486C22⋊C4—1221no—
487C22⋊C4—1221no—
488C22⋊C4—1221no—
489C22⋊C4—1222no—
490C22⋊C4—1221no—
491C22⋊C4—1222no—
492C22⋊C4#1175819812yes—
493C22⋊C4#1175819812yes—
494C22⋊C4#1175819812yes—
495C42—1222no—
496C42—1222no—
497C42—1222no—
498C42—1222no—
499C42—1221no—
500C42C2×C4×D6112yes—
501C42C2×C4×D6111yescharacteristic
502C42C2×D24411yescharacteristic
503C42C2×D24412yes—
504C42C2×C22⋊D6111yescharacteristic
505C42C2×C22⋊D6112yes—
506C42C23×D6111yescharacteristic
507C23—1221no—
508C23C2×#1175819811yescharacteristic
509C23C2×#1175819812yes—
510Q8—22no—
511Q8—4882no—
512Q8—4881no—
513Q8—2441no—
514Q8—2444no—
515Q8—2442no—
516D4—22no—
517D4—42no—
518D4—4882no—
519D4—4881no—
520D4—2441no—
521C2×C4—22no—
522C2×C4—42no—
523C2×C4—41no—
524C2×C4—42no—
525C2×C4—41no—
526C2×C4—42no—
527C2×C4—1221no—
528C2×C4—1222no—
529C2×C4—1221no—
530C2×C4—1222no—
531C2×C4—1221no—
532C2×C4—2444no—
533C2×C4—2442no—
534C2×C4C2×#1175819811yescharacteristic
535C2×C4C2×#1175819812yes—
536C2×C4C2×#1175819812yes—
537C2×C4C2×#1175819811yescharacteristic
538C2×C4C2×#1175819812yes—
539C2×C4C2×#1175819811yescharacteristic
540C2×C4C22×C4×D6111yescharacteristic
541C2×C4C22×C22⋊D6111yescharacteristic
542C2×C4C22×D24411yescharacteristic
543C8—41no—
544C8—41no—
545C8—42no—
546C8—1222no—
547C8—2442no—
548C8—2441no—
549C22—21no—
550C22—22no—
551C22—22no—
552C22—21no—
553C22—21no—
554C22—22no—
555C22—2441no—
556C22—2441no—
557C22—2441no—
558C22C22×#1175819811yescharacteristic
559C4—21no—
560C4—22no—
561C4—21no—
562C4—22no—
563C4—21no—
564C4—21no—
565C4—21no—
566C4—22no—
567C4—2442no—
568C4—2441no—
569C4—2442no—
570C4—2441no—
571C4—2441no—
572C4#1683978012yes—
573C2—41no—
574C2—42no—
575C2—4881no—
576C2#2145430111yescharacteristic
577C2C2×#1683978011yescharacteristic
578C2#2145436411yescharacteristic
subgroupG/H
centerC22C22×#11758198
commutatorC2×C4×C61C23×C4
FrattiniC42C23×D61
FittingC61×#2923C2
socleC22×C61C22×C22⋊C4
prime radical#21459836C2
cosocleC42×C61C24
hypercenter#2923D61
hypocenterC61#17683
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C4×#41 ◁ #2923

Complements

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

NH ≅ G/Ntype
2. #21454833C2semidirect
3. #21444456C2semidirect
4. #21444448C2semidirect
5. #21444440C2semidirect
6. #21444435C2semidirect
7. #21444433C2semidirect
8. #21444424C2semidirect
9. #21444413C2semidirect
10. #21444405C2semidirect
11. C61×#2923C2semidirect
286. #2923D61semidirect
342. C61#17683semidirect

Maximal subgroups

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

subgroupindex
#214598362
#214548332
#214444562
#214444482
#214444402
#214444352
#214444332
#214444242
#214444132
#214444052
C61×#29232
#1768361