Finite Group Databasebeta

← #29648244

Subgroups — #29648244

Order 15616.

896 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459741C211yesmaximal, characteristic
2#21454839C211yesmaximal, characteristic, complemented
3#21444453C211yesmaximal, characteristic, complemented
4#21444448C212yesmaximal, complemented
5#21444437C212yesmaximal, complemented
6#21444427C212yesmaximal, complemented
7#21444422C211yesmaximal, characteristic, complemented
8#21444420C211yesmaximal, characteristic, complemented
9#21444410C211yesmaximal, characteristic, complemented
10#21444405C212yesmaximal, complemented
11C61×#2970C211yesmaximal, characteristic, complemented
12#16840462—21no—
13#16840462—21no—
14#16840461—22no—
15#16840461—22no—
16#16840453—21no—
17#16840453—21no—
18#16840374C412yes—
19#16840367C412yes—
20#16840340C411yescharacteristic
21#16840340C411yescharacteristic
22#16840312C2211yescharacteristic
23#16840296C2212yes—
24#16840296C2212yes—
25C2×#14519769C2211yescharacteristic
26#16840242C411yescharacteristic
27#16840242C411yescharacteristic
28Q8×#92418—21no—
29#16839840—22no—
30#16839840—21no—
31#16839819C2211yescharacteristic
32#16839819C2212yes—
33C4×#11758194C2212yes—
34C4×#11758194C2211yescharacteristic
35#16839503C2212yes—
36#16839503C2212yes—
37#16839503C2212yes—
38#16839503C2212yes—
39#16839501C2211yescharacteristic
40#16839501C2212yes—
41#16839501C2211yescharacteristic
42#16839501C2211yescharacteristic
43#16839501C2211yescharacteristic
44#16839501C2212yes—
45C4×#41×C61C2211yescharacteristic
46C61×#446—21no—
47C61×#435—22no—
48C8×D4×C61—21no—
49C61×#432C2211yescharacteristic
50C61×#432C2212yes—
51C4×M16×C61C2211yescharacteristic
52C2×#97×C61C2212yes—
53C2×#11758222C2×C411yescharacteristic
54C2×#11758222C2×C411yescharacteristic
55C2×#11758194—21no—
56C2×#11758194—22no—
57#14519777C2×C411yescharacteristic
58#14519774C2×C411yescharacteristic
59#14519771—22no—
60#14519771—22no—
61#14519770—22no—
62#14519770—22no—
63#14519769—21no—
64#14519769—21no—
65#14519769C2×C412yes—
66#14519769C2312yes—
67#14519767—22no—
68#14519767—22no—
69#14519765C2311yescharacteristic
70#14519765C2311yescharacteristic
71#14519762—22no—
72#14519762—22no—
73#14519760—22no—
74#14519760—22no—
75#14519756C2×C412yes—
76#14519756C2×C412yes—
77#14519753—21no—
78#14519753—21no—
79#14519751—21no—
80#14519751—21no—
81#14519751C2×C412yes—
82#14519751C2×C412yes—
83#14519750—22no—
84#14519750—22no—
85#14519749—22no—
86#14519749—22no—
87#14519748—21no—
88#14519748—21no—
89C4×Dic122C2×C412yes—
90C4×Dic122C2×C412yes—
91C4×D244—21no—
92C4×D244—21no—
93C4×D244C2×C412yes—
94C4×D244C2×C412yes—
95C4×C22⋊D61—22no—
96C4×C22⋊D61—22no—
97#14519737—22no—
98#14519737—22no—
99#14519735—21no—
100#14519735—21no—
101#14519734—22no—
102#14519734—22no—
103C2×C4⋊Dic61C2×C411yescharacteristic
104C2×C4⋊Dic61C2×C411yescharacteristic
105C2×#11758196—21no—
106C2×#11758196—21no—
107C4⋊C4×D61—21no—
108C4⋊C4×D61—21no—
109C4⋊#92418—22no—
110C4⋊#92418—21no—
111C22⋊#92418—21no—
112C22⋊#92418—22no—
113#14519688C2311yescharacteristic
114#14519688C2312yes—
115#14519688C2311yescharacteristic
116C4×#92418—22no—
117C4×#92418—21no—
118C2×#41×C61C2×C411yescharacteristic
119C2×M16×C61—21no—
120C22×C8×C61—22no—
121C4×Q8×C61D411yescharacteristic
122C4×Q8×C61D411yescharacteristic
123C4×D4×C61D412yes—
124C4×D4×C61D411yescharacteristic
125C4×D4×C61D412yes—
126C4×D4×C61D411yescharacteristic
127#117×C61C2×C411yescharacteristic
128#117×C61C2×C412yes—
129C2×C42×C61C2311yescharacteristic
130C2×C42×C61C2312yes—
131C4⋊C8×C61—21no—
132C4⋊C8×C61—22no—
133C22⋊C8×C61—22no—
134C22⋊C8×C61—21no—
135#97×C61—22no—
136#97×C61C2311yescharacteristic
137#97×C61C2312yes—
138#97×C61C2311yescharacteristic
139C4×C8×C61—21no—
140#41×C61—42no—
141C2×Q8×C61C22⋊C411yescharacteristic
142C2×D4×C61C22⋊C411yescharacteristic
143C2×D4×C61C22⋊C412yes—
144C22×C4×C61C22×C411yescharacteristic
145C22×C4×C61C22×C412yes—
146C22×C4×C61C2×D411yescharacteristic
147C22×C4×C61C2×D412yes—
148C22×C4×C61C2×D412yes—
149C22×C4×C61C2×D411yescharacteristic
150C2×C22⋊D61—21no—
151C2×C22⋊D61—21no—
152C2×C22⋊D61—22no—
153C2×C22⋊D61—22no—
154C22×Dic61—21no—
155C22×Dic61—21no—
156#11758222—42no—
157#11758222—42no—
158C2×D244—21no—
159C2×D244—21no—
160C2×D244C22×C411yescharacteristic
161C2×D244C22×C411yescharacteristic
162C2×C4×D61—21no—
163C2×C4×D61—21no—
164C2×C4×D61—22no—
165C2×C4×D61—22no—
166C2×Dic122C22×C411yescharacteristic
167C2×Dic122C22×C411yescharacteristic
168M16×C61—42no—
169C2×C8×C61—21no—
170C2×C8×C61—21no—
171C2×C8×C61—22no—
172C2×C8×C61—44no—
173C4⋊C4×C61C22⋊C411yescharacteristic
174C4⋊C4×C61C22⋊C411yescharacteristic
175C4⋊C4×C61C22⋊C412yes—
176C4⋊C4×C61C22⋊C412yes—
177C22⋊C4×C61C22⋊C412yes—
178C22⋊C4×C61C22⋊C412yes—
179C22⋊C4×C61C22⋊C412yes—
180C42×C61C22×C412yes—
181C42×C61C22×C411yescharacteristic
182C42×C61C2×D412yes—
183C42×C61C2×D411yescharacteristic
184C42×C61C2×D411yescharacteristic
185C42×C61C2×D412yes—
186C42×C61C2411yescharacteristic
187C22⋊Dic61—22no—
188C22⋊Dic61—22no—
189#11758198—22no—
190#11758198—22no—
191#11758198—22no—
192#11758198—22no—
193#11758198—21no—
194#11758198—21no—
195#11758198—21no—
196#11758198—21no—
197C4⋊Dic61—21no—
198C4⋊Dic61—21no—
199C4⋊Dic61—21no—
200C4⋊Dic61—21no—
201C4⋊Dic61C22×C412yes—
202C4⋊Dic61C22×C412yes—
203#11758196—22no—
204#11758196—22no—
205#11758196—22no—
206#11758196—22no—
207#11758196—22no—
208#11758196—22no—
209#11758196—22no—
210#11758196—22no—
211C4×Dic61—22no—
212C4×Dic61—22no—
213#11758194—44no—
214#11758194—42no—
215C2×#92418—21no—
216C2×#92418—22no—
217C2×#92418—21no—
218C23×C61C2×C22⋊C412yes—
219C23×C61C2×C22⋊C411yescharacteristic
220C22×D61—21no—
221C22×D61—21no—
222Q8×C61—22no—
223D4×C61—22no—
224D4×C61—42no—
225C2×C4×C61—22no—
226C2×C4×C61—42no—
227C2×C4×C61—41no—
228C2×C4×C61—42no—
229C2×C4×C61—41no—
230C2×C4×C61—42no—
231C2×C4×C61C2×C22⋊C412yes—
232C2×C4×C61C2×C22⋊C412yes—
233C2×C4×C61C2×C22⋊C411yescharacteristic
234C2×C4×C61C2×C22⋊C411yescharacteristic
235C2×C4×C61C2×C22⋊C412yes—
236C2×C4×C61C2×C22⋊C411yescharacteristic
237C2×C4×C61C23×C411yescharacteristic
238C2×C4×C61C22×D411yescharacteristic
239C2×C4×C61C22×D411yescharacteristic
240C22⋊D61—42no—
241C22⋊D61—42no—
242C22⋊D61—82no—
243C22⋊D61—82no—
244C2×Dic61—21no—
245C2×Dic61—21no—
246C2×Dic61—22no—
247C2×Dic61—22no—
248C2×Dic61—81no—
249C2×Dic61—81no—
250D244—41no—
251D244—41no—
252D244—81no—
253D244—81no—
254C4×D61—42no—
255C4×D61—42no—
256C4×D61—82no—
257C4×D61—82no—
258Dic122—41no—
259Dic122—41no—
260C8×C61—22no—
261C8×C61—42no—
262C8×C61—41no—
263#92418—22no—
264#92418—42no—
265#92418—41no—
266#17687—611nomaximal, Sylow
267C22×C61—21no—
268C22×C61—22no—
269C22×C61—21no—
270C22×C61—21no—
271C22×C61—22no—
272C22×C61—22no—
273C22×C61C22×C22⋊C411yescharacteristic
274C2×D61—41no—
275C2×D61—41no—
276C2×D61—41no—
277C2×D61—41no—
278C2×D61—41no—
279C2×D61—41no—
280C4×C61—21no—
281C4×C61—22no—
282C4×C61—21no—
283C4×C61—21no—
284C4×C61—22no—
285C4×C61—22no—
286C4×C61—21no—
287C4×C61—21no—
288C4×C61#41312yes—
289Dic61—41no—
290Dic61—41no—
291Dic61—42no—
292Dic61—42no—
293#3512—611no—
294#2971—611no—
295#2970D6111yescharacteristic, complemented
296#1536—611no—
297#1535—612no—
298#1532—612no—
299#1529—612no—
300#1528—611no—
301#1528—611no—
302#1525—611no—
303#1524—612no—
304C2×C61—41no—
305C2×C61—42no—
306C2×C61C2×#41311yescharacteristic
307C2×C61#290111yescharacteristic
308C2×C61#290111yescharacteristic
309D61—81no—
310D61—81no—
311#553—1222no—
312#553—1221no—
313#553—1222no—
314#553—1221no—
315#552—1221no—
316#552—1221no—
317#541—612no—
318#540—612no—
319#537—611no—
320#537—611no—
321#532—611no—
322#529—612no—
323#529—612no—
324C2×#125—611no—
325#520—611no—
326#520—611no—
327C4×#41C2×D6111yescharacteristic
328#446—21no—
329C8×Q8—1221no—
330#436—1222no—
331#436—1221no—
332#435—22no—
333C8×D4—21no—
334#432—611no—
335#432—612no—
336#432C2×D6111yescharacteristic
337#432C2×D6112yes—
338C4×M16—612no—
339C4×M16—611no—
340C4×M16C2×D6111yescharacteristic
341C2×#97C2×D6112yes—
342#330—612no—
343#330—612no—
344#330—612no—
345#330—612no—
346#329—612no—
347#329—612no—
348#329—611no—
349#329—611no—
350#329—611no—
351#329—611no—
352C61#1768711yescharacteristic, complemented, Sylow
353C2×#41—611no—
354C2×#41—611no—
355C2×#41C4×D6111yescharacteristic
356C2×M16—21no—
357C2×M16—1221no—
358C2×M16—1222no—
359C22×C8—22no—
360C4⋊Q8—611no—
361#127—611no—
362#126—1222no—
363#126—1222no—
364#126—1222no—
365#126—1222no—
366#125—612no—
367#125—612no—
368#125—1222no—
369#125—1222no—
370#125—1221no—
371#125—1221no—
372#124—611no—
373#124—611no—
374#123—1222no—
375#123—1222no—
376#123—1222no—
377#123—1222no—
378C22⋊Q8—612no—
379C22⋊Q8—612no—
380C22⋊Q8—1221no—
381C22⋊Q8—1221no—
382#121—612no—
383#121—612no—
384#121—1221no—
385#121—1221no—
386#121—1222no—
387#121—1222no—
388#121—1221no—
389#121—1221no—
390#121—1222no—
391#121—1222no—
392C4×Q8—612no—
393C4×Q8—612no—
394C4×Q8D24411yescharacteristic
395C4×Q8C22⋊D6111yescharacteristic
396C4×D4—612no—
397C4×D4—612no—
398C4×D4—1222no—
399C4×D4—1222no—
400C4×D4—1222no—
401C4×D4—1222no—
402C4×D4—1221no—
403C4×D4—1221no—
404C4×D4D24412yes—
405C4×D4D24411yescharacteristic
406C4×D4C22⋊D6111yescharacteristic
407C4×D4C22⋊D6112yes—
408#117—1221no—
409#117—1221no—
410#117—1222no—
411#117—1222no—
412#117C4×D6111yescharacteristic
413#117C4×D6112yes—
414C2×C4⋊C4—611no—
415C2×C4⋊C4—611no—
416C2×C4⋊C4—1221no—
417C2×C4⋊C4—1221no—
418C2×C4⋊C4—1221no—
419C2×C4⋊C4—1221no—
420C2×C42C22×D6111yescharacteristic
421C2×C42C22×D6112yes—
422C4⋊C8—21no—
423C4⋊C8—22no—
424C4⋊C8—1222no—
425C4⋊C8—1221no—
426C22⋊C8—22no—
427C22⋊C8—21no—
428C22⋊C8—1221no—
429C22⋊C8—1222no—
430#97—22no—
431#97—611no—
432#97—612no—
433#97—611no—
434#97C22×D6111yescharacteristic
435#97C22×D6112yes—
436#97C22×D6111yescharacteristic
437C4×C8—21no—
438C4×C8—1222no—
439C4×C8—1221no—
440#41—42no—
441#41—2442no—
442#41—2442no—
443C2×Q8—611no—
444C2×Q8—611no—
445C2×Q8#1175819811yescharacteristic
446C2×D4—611no—
447C2×D4—611no—
448C2×D4—1222no—
449C2×D4—1222no—
450C2×D4—1221no—
451C2×D4—1221no—
452C2×D4—1221no—
453C2×D4—1221no—
454C2×D4#1175819811yescharacteristic
455C2×D4#1175819812yes—
456C22×C4—1221no—
457C22×C4—1221no—
458C22×C4—1222no—
459C22×C4—1222no—
460C22×C4—1221no—
461C22×C4—1221no—
462C22×C4C2×C4×D6111yescharacteristic
463C22×C4C2×C4×D6112yes—
464C22×C4C2×D24411yescharacteristic
465C22×C4C2×D24412yes—
466C22×C4C2×C22⋊D6111yescharacteristic
467C22×C4C2×C22⋊D6112yes—
468M16—42no—
469M16—2444no—
470M16—2442no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—22no—
474C2×C8—44no—
475C2×C8—1221no—
476C2×C8—1222no—
477C2×C8—1221no—
478C4⋊C4—612no—
479C4⋊C4—612no—
480C4⋊C4—1222no—
481C4⋊C4—1222no—
482C4⋊C4—1221no—
483C4⋊C4—1221no—
484C4⋊C4—1222no—
485C4⋊C4—1222no—
486C4⋊C4—1222no—
487C4⋊C4—1222no—
488C4⋊C4—1222no—
489C4⋊C4—1222no—
490C4⋊C4—1221no—
491C4⋊C4—1221no—
492C4⋊C4#1175819811yescharacteristic
493C4⋊C4#1175819811yescharacteristic
494C4⋊C4#1175819812yes—
495C4⋊C4#1175819812yes—
496C22⋊C4—1222no—
497C22⋊C4—1222no—
498C22⋊C4—1221no—
499C22⋊C4—1221no—
500C22⋊C4—1221no—
501C22⋊C4—1221no—
502C22⋊C4—1222no—
503C22⋊C4—1222no—
504C22⋊C4—1222no—
505C22⋊C4—1222no—
506C22⋊C4#1175819812yes—
507C22⋊C4#1175819812yes—
508C22⋊C4#1175819812yes—
509C42—1222no—
510C42—1222no—
511C42C2×C4×D6112yes—
512C42C2×C4×D6111yescharacteristic
513C42C2×D24412yes—
514C42C2×D24411yescharacteristic
515C42C2×C22⋊D6111yescharacteristic
516C42C2×C22⋊D6112yes—
517C42C23×D6111yescharacteristic
518C23—1221no—
519C23—1221no—
520C23C2×#1175819812yes—
521C23C2×#1175819811yescharacteristic
522Q8—22no—
523Q8—2441no—
524Q8—2441no—
525D4—22no—
526D4—42no—
527D4—4881no—
528D4—4881no—
529D4—4882no—
530D4—4882no—
531D4—2441no—
532D4—2441no—
533D4—2442no—
534D4—2442no—
535C2×C4—22no—
536C2×C4—42no—
537C2×C4—41no—
538C2×C4—42no—
539C2×C4—41no—
540C2×C4—42no—
541C2×C4—1221no—
542C2×C4—1221no—
543C2×C4—1222no—
544C2×C4—1222no—
545C2×C4—4882no—
546C2×C4—4882no—
547C2×C4—4881no—
548C2×C4—4881no—
549C2×C4—2442no—
550C2×C4—2442no—
551C2×C4C2×#1175819812yes—
552C2×C4C2×#1175819812yes—
553C2×C4C2×#1175819811yescharacteristic
554C2×C4C2×#1175819811yescharacteristic
555C2×C4C2×#1175819812yes—
556C2×C4C2×#1175819811yescharacteristic
557C2×C4C22×C4×D6111yescharacteristic
558C2×C4C22×C22⋊D6111yescharacteristic
559C2×C4C22×D24411yescharacteristic
560C8—22no—
561C8—42no—
562C8—41no—
563C8—1222no—
564C8—2442no—
565C8—2441no—
566C22—21no—
567C22—22no—
568C22—21no—
569C22—22no—
570C22—21no—
571C22—22no—
572C22—2441no—
573C22—2441no—
574C22—2441no—
575C22—2441no—
576C22—2441no—
577C22—2441no—
578C22C22×#1175819811yescharacteristic
579C4—21no—
580C4—22no—
581C4—21no—
582C4—21no—
583C4—22no—
584C4—21no—
585C4—21no—
586C4—22no—
587C4—2442no—
588C4—2442no—
589C4—2441no—
590C4—2441no—
591C4#1683974412yes—
592C2—41no—
593C2—42no—
594C2—4881no—
595C2—4881no—
596C2C2×#1683974411yescharacteristic
597C2#2145436411yescharacteristic
598C2#2145436711yescharacteristic
subgroupG/H
centerC22C22×#11758198
commutatorC2×C4×C61C23×C4
FrattiniC42C23×D61
FittingC61×#2970C2
socleC22×C61C22×C22⋊C4
prime radical#21459741C2
cosocleC42×C61C24
hypercenter#2970D61
hypocenterC61#17687
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C4×#41 ◁ #2970

Complements

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

NH ≅ G/Ntype
2. #21454839C2semidirect
3. #21444453C2semidirect
4. #21444448C2semidirect
5. #21444437C2semidirect
6. #21444427C2semidirect
7. #21444422C2semidirect
8. #21444420C2semidirect
9. #21444410C2semidirect
10. #21444405C2semidirect
11. C61×#2970C2semidirect
295. #2970D61semidirect
352. C61#17687semidirect

Maximal subgroups

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

subgroupindex
#214597412
#214548392
#214444532
#214444482
#214444372
#214444272
#214444222
#214444202
#214444102
#214444052
C61×#29702
#1768761