Finite Group Databasebeta

← #29648249

Subgroups — #29648249

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#21459739C211yesmaximal, characteristic
2#21454836C211yesmaximal, characteristic, complemented
3#21454831C211yesmaximal, characteristic, complemented
4#21444447C212yesmaximal, complemented
5#21444436C212yesmaximal, complemented
6#21444423C212yesmaximal, complemented
7#21444422C211yesmaximal, characteristic, complemented
8#21444422C211yesmaximal, characteristic, complemented
9#21444411C211yesmaximal, characteristic, complemented
10#21444404C212yesmaximal, complemented
11C61×#1536C211yesmaximal, characteristic, complemented
12#16840458—21no—
13#16840458—21no—
14#16840455—22no—
15#16840455—22no—
16#16840452—21no—
17#16840452—21no—
18#16840373C412yes—
19#16840366C412yes—
20#16840340C411yescharacteristic
21#16840340C411yescharacteristic
22#16840291C2212yes—
23#16840291C2212yes—
24C2×#14519769C2211yescharacteristic
25#16840245C411yescharacteristic
26#16840245C411yescharacteristic
27C4×#11758222C2211yescharacteristic
28#16839889—21no—
29#16839886—21no—
30#16839839—21no—
31#16839837—22no—
32#16839836—22no—
33#16839827—21no—
34#16839817C2212yes—
35#16839817C2212yes—
36C4×#11758194C2211yescharacteristic
37C4×#11758194C2211yescharacteristic
38#16839714C2212yes—
39#97×D61C2212yes—
40#16839503C2212yes—
41#16839503C2212yes—
42#16839503C2212yes—
43#16839503C2212yes—
44#16839502C2211yescharacteristic
45#16839502C2211yescharacteristic
46#16839502C2211yescharacteristic
47#16839502C2211yescharacteristic
48C61×#532C2211yescharacteristic
49C61×#432C2211yescharacteristic
50C61×#432C2211yescharacteristic
51C61×#329C2212yes—
52C61×#329C2212yes—
53C2×#11758222C2×C411yescharacteristic
54C2×#11758194—21no—
55C2×#11758194—21no—
56C2×#11758189—22no—
57C2×C8×D61—22no—
58#14519772—22no—
59#14519772—22no—
60#14519770—22no—
61#14519770—22no—
62#14519769—21no—
63#14519769—21no—
64#14519769C2×C412yes—
65#14519769C2312yes—
66#14519767—22no—
67#14519767—22no—
68#14519759—22no—
69#14519759—22no—
70#14519758—22no—
71#14519758—22no—
72#14519756—21no—
73#14519756—21no—
74#14519753C2×C412yes—
75#14519753C2×C412yes—
76#14519751—21no—
77#14519751—21no—
78#14519749—22no—
79#14519749—22no—
80#14519749—22no—
81#14519749—22no—
82#14519748C2×C412yes—
83#14519748C2×C412yes—
84Q8×Dic61C2×C412yes—
85Q8×Dic61C2×C412yes—
86C4×Dic122D411yescharacteristic
87C4×Dic122D411yescharacteristic
88C4×D244D411yescharacteristic
89C4×D244D411yescharacteristic
90D4×Dic61—22no—
91D4×Dic61—22no—
92D4×Dic61C2×C412yes—
93D4×Dic61C2×C412yes—
94C4×C22⋊D61D412yes—
95C4×C22⋊D61D412yes—
96#14519737—22no—
97#14519737—22no—
98#14519736—21no—
99#14519736—21no—
100#14519736C2×C411yescharacteristic
101#14519734C2×C412yes—
102#14519733—22no—
103#14519733—22no—
104C2×C4⋊Dic61—21no—
105C2×C4⋊Dic61—21no—
106C2×C4⋊Dic61C2×C411yescharacteristic
107C2×C4⋊Dic61C2×C411yescharacteristic
108C2×#11758196—21no—
109C2×#11758196—21no—
110C42×D61C2312yes—
111#14519706—22no—
112#14519706—22no—
113#14519689—22no—
114#14519689—22no—
115#14519688C2311yescharacteristic
116#14519688C2311yescharacteristic
117#14519688C2311yescharacteristic
118#14519688C2311yescharacteristic
119#14519687—22no—
120C4×#92418—21no—
121C4×#92418—21no—
122C8×Dic61—22no—
123C2×#41×C61C2×C411yescharacteristic
124C2×#41×C61C2×C411yescharacteristic
125C4⋊Q8×C61C2×C411yescharacteristic
126#127×C61C2×C411yescharacteristic
127#124×C61C2311yescharacteristic
128#124×C61C2311yescharacteristic
129#121×C61—21no—
130#121×C61—21no—
131C4×D4×C61—21no—
132C4×D4×C61—21no—
133C2×C42×C61C2311yescharacteristic
134C4⋊C8×C61—21no—
135C4⋊C8×C61—21no—
136C22⋊C8×C61—21no—
137C22⋊C8×C61—21no—
138#97×C61C2312yes—
139#97×C61C2312yes—
140#41×C61—42no—
141#41×C61—42no—
142C2×Q8×C61C22×C411yescharacteristic
143C2×Q8×C61C22×C411yescharacteristic
144C2×D4×C61—21no—
145C2×D4×C61—21no—
146C2×D4×C61—21no—
147C2×D4×C61—21no—
148C2×D4×C61C22×C411yescharacteristic
149C2×D4×C61C22×C411yescharacteristic
150C22×C4×C61—21no—
151C22×C4×C61—21no—
152C22×C4×C61C22×C411yescharacteristic
153C22×C4×C61C2×D411yescharacteristic
154C22×C4×C61C2×D411yescharacteristic
155C2×C22⋊D61—22no—
156C2×C22⋊D61—22no—
157C2×C22⋊D61C22⋊C412yes—
158C22×Dic61—21no—
159C22×Dic61—21no—
160C22×Dic61—22no—
161C22×Dic61—22no—
162#11758222—42no—
163C2×D244C22⋊C411yescharacteristic
164C2×C4×D61C22×C412yes—
165C2×C4×D61C2×D412yes—
166C2×C4×D61C2×D412yes—
167C2×Dic122C22⋊C411yescharacteristic
168C2×C8×C61—22no—
169C2×C8×C61—22no—
170C4⋊C4×C61—21no—
171C4⋊C4×C61—21no—
172C22⋊C4×C61—21no—
173C22⋊C4×C61—21no—
174C22⋊C4×C61—21no—
175C22⋊C4×C61—21no—
176C42×C61C22×C411yescharacteristic
177C42×C61C2×D411yescharacteristic
178C42×C61C2×D411yescharacteristic
179C42×C61C2411yescharacteristic
180C22⋊Dic61—22no—
181C22⋊Dic61—22no—
182C22⋊Dic61—22no—
183C22⋊Dic61—22no—
184C22⋊Dic61C22⋊C412yes—
185#11758198—22no—
186#11758198—22no—
187#11758198C22⋊C412yes—
188#11758198C22⋊C412yes—
189C4⋊Dic61—21no—
190C4⋊Dic61—21no—
191C4⋊Dic61—22no—
192C4⋊Dic61—22no—
193C4⋊Dic61C22⋊C411yescharacteristic
194C4⋊Dic61C22⋊C411yescharacteristic
195C4⋊Dic61C22×C412yes—
196C4⋊Dic61C22×C412yes—
197#11758196—22no—
198#11758196—22no—
199#11758196—22no—
200#11758196—22no—
201#11758196—22no—
202#11758196—22no—
203#11758196C22⋊C412yes—
204#11758196C22⋊C412yes—
205C4×Dic61—22no—
206C4×Dic61—22no—
207C4×Dic61C22×C412yes—
208C4×Dic61C2×D412yes—
209C4×Dic61C2×D412yes—
210#11758194—42no—
211#11758194—42no—
212C2×#92418—21no—
213C2×#92418—21no—
214C2×#92418—21no—
215C2×#92418—21no—
216#11758189—44no—
217C8×D61—44no—
218C23×C61—21no—
219C23×C61—21no—
220C23×C61C2×C22⋊C411yescharacteristic
221C22×D61C2×C22⋊C412yes—
222Q8×C61—41no—
223Q8×C61—41no—
224D4×C61—41no—
225D4×C61—41no—
226D4×C61—42no—
227D4×C61—42no—
228D4×C61—81no—
229D4×C61—81no—
230C2×C4×C61—22no—
231C2×C4×C61—21no—
232C2×C4×C61—21no—
233C2×C4×C61—41no—
234C2×C4×C61—41no—
235C2×C4×C61—42no—
236C2×C4×C61—42no—
237C2×C4×C61C2×C22⋊C411yescharacteristic
238C2×C4×C61C2×C22⋊C411yescharacteristic
239C2×C4×C61C2×C22⋊C411yescharacteristic
240C2×C4×C61C23×C411yescharacteristic
241C2×C4×C61C22×D411yescharacteristic
242C2×C4×C61C22×D411yescharacteristic
243C22⋊D61—42no—
244C22⋊D61—82no—
245C22⋊D61—82no—
246C2×Dic61—22no—
247C2×Dic61—22no—
248C2×Dic61—82no—
249C2×Dic61—82no—
250C2×Dic61—81no—
251C2×Dic61—81no—
252C2×Dic61C2×C22⋊C412yes—
253C2×Dic61C2×C22⋊C412yes—
254C2×Dic61C2×C22⋊C412yes—
255D244—22no—
256C4×D61—42no—
257C4×D61—42no—
258C4×D61—42no—
259Dic122—22no—
260C8×C61—42no—
261C8×C61—42no—
262#92418—22no—
263#92418—22no—
264#92418—41no—
265#92418—41no—
266#17687—611nomaximal, Sylow
267C22×C61—21no—
268C22×C61—21no—
269C22×C61—21no—
270C22×C61—41no—
271C22×C61—41no—
272C22×C61—41no—
273C22×C61—41no—
274C22×C61—41no—
275C22×C61—41no—
276C22×C61C22×C22⋊C411yescharacteristic
277C2×D61—22no—
278C2×D61—22no—
279C2×D61—22no—
280C4×C61—21no—
281C4×C61—21no—
282C4×C61—21no—
283C4×C61—21no—
284C4×C61—21no—
285C4×C61—41no—
286C4×C61—41no—
287C4×C61#41312yes—
288Dic61—22no—
289Dic61—22no—
290Dic61—22no—
291Dic61—42no—
292Dic61—42no—
293#3512—611no—
294#2971—611no—
295#2970—611no—
296#1536D6111yescharacteristic, complemented
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—81no—
306C2×C61—81no—
307C2×C61C2×#41311yescharacteristic
308C2×C61#290111yescharacteristic
309C2×C61#290111yescharacteristic
310D61—42no—
311#553—1221no—
312#553—1221no—
313#553—1222no—
314#553—1222no—
315#552—1221no—
316#552—1221no—
317#541—612no—
318#540—612no—
319#537—611no—
320#537—611no—
321#532C2×D6111yescharacteristic
322#529—612no—
323#529—612no—
324C2×#125—611no—
325#520—611no—
326#520—611no—
327C4×#41—611no—
328#446—1221no—
329C8×Q8—1221no—
330#436—1222no—
331#436—1221no—
332#435—1222no—
333C8×D4—1221no—
334#432—612no—
335#432—612no—
336#432C2×D6111yescharacteristic
337#432C2×D6111yescharacteristic
338C4×M16—611no—
339C4×M16—611no—
340C4×M16—612no—
341C2×#97—612no—
342#330—612no—
343#330—612no—
344#330—612no—
345#330—612no—
346#329—611no—
347#329—611no—
348#329—611no—
349#329—611no—
350#329C2×D6112yes—
351#329C2×D6112yes—
352C61#1768711yescharacteristic, complemented, Sylow
353C2×#41—611no—
354C2×#41C4×D6111yescharacteristic
355C2×#41C4×D6111yescharacteristic
356C2×M16—1222no—
357C2×M16—1221no—
358C2×M16—1221no—
359C22×C8—1222no—
360C4⋊Q8C4×D6111yescharacteristic
361#127C4×D6111yescharacteristic
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#124C22×D6111yescharacteristic
373#124C22×D6111yescharacteristic
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—21no—
383#121—21no—
384#121—612no—
385#121—612no—
386#121—1222no—
387#121—1222no—
388#121—1221no—
389#121—1221no—
390#121—1222no—
391#121—1222no—
392C4×Q8—611no—
393C4×Q8—611no—
394C4×Q8—612no—
395C4×Q8—612no—
396C4×D4—21no—
397C4×D4—21no—
398C4×D4—612no—
399C4×D4—612no—
400C4×D4—612no—
401C4×D4—612no—
402C4×D4—611no—
403C4×D4—611no—
404C4×D4—1222no—
405C4×D4—1222no—
406C4×D4—1222no—
407C4×D4—1222no—
408#117—611no—
409#117—612no—
410#117—1222no—
411#117—1222no—
412#117—1221no—
413#117—1221no—
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×C42—612no—
421C2×C42C22×D6111yescharacteristic
422C4⋊C8—21no—
423C4⋊C8—21no—
424C4⋊C8—1222no—
425C4⋊C8—1222no—
426C22⋊C8—21no—
427C22⋊C8—21no—
428C22⋊C8—1222no—
429C22⋊C8—1222no—
430#97—611no—
431#97—611no—
432#97—611no—
433#97—611no—
434#97—1222no—
435#97C22×D6112yes—
436#97C22×D6112yes—
437C4×C8—1222no—
438C4×C8—1221no—
439C4×C8—1221no—
440#41—42no—
441#41—42no—
442#41—2442no—
443C2×Q8—611no—
444C2×Q8C2×C4×D6111yescharacteristic
445C2×Q8C2×C4×D6111yescharacteristic
446C2×D4—21no—
447C2×D4—21no—
448C2×D4—21no—
449C2×D4—21no—
450C2×D4—612no—
451C2×D4—611no—
452C2×D4—1222no—
453C2×D4—1222no—
454C2×D4C2×C4×D6111yescharacteristic
455C2×D4C2×C4×D6111yescharacteristic
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—612no—
459C22×C4—612no—
460C22×C4—612no—
461C22×C4—1222no—
462C22×C4—1222no—
463C22×C4—1221no—
464C22×C4—1221no—
465C22×C4C2×C4×D6111yescharacteristic
466C22×C4D4×D6111yescharacteristic
467C22×C4D4×D6111yescharacteristic
468M16—2444no—
469M16—2442no—
470M16—2442no—
471C2×C8—22no—
472C2×C8—22no—
473C2×C8—1221no—
474C2×C8—1221no—
475C2×C8—1221no—
476C2×C8—1221no—
477C2×C8—2444no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—611no—
481C4⋊C4—611no—
482C4⋊C4—612no—
483C4⋊C4—612no—
484C4⋊C4—612no—
485C4⋊C4—612no—
486C4⋊C4—1222no—
487C4⋊C4—1222no—
488C4⋊C4—1222no—
489C4⋊C4—1222no—
490C4⋊C4—1222no—
491C4⋊C4—1222no—
492C4⋊C4—1221no—
493C4⋊C4—1221no—
494C4⋊C4—1222no—
495C4⋊C4—1222no—
496C22⋊C4—21no—
497C22⋊C4—21no—
498C22⋊C4—21no—
499C22⋊C4—21no—
500C22⋊C4—612no—
501C22⋊C4—612no—
502C22⋊C4—612no—
503C22⋊C4—1222no—
504C22⋊C4—1222no—
505C22⋊C4—1222no—
506C22⋊C4—1222no—
507C22⋊C4—1222no—
508C22⋊C4—1222no—
509C42—612no—
510C42—612no—
511C42—612no—
512C42—1222no—
513C42—1222no—
514C42C2×C4×D6111yescharacteristic
515C42D4×D6111yescharacteristic
516C42D4×D6111yescharacteristic
517C42C23×D6111yescharacteristic
518C23—21no—
519C23—21no—
520C23—612no—
521C23C22⋊C4×D6111yescharacteristic
522Q8—41no—
523Q8—41no—
524Q8—1222no—
525D4—41no—
526D4—41no—
527D4—42no—
528D4—42no—
529D4—81no—
530D4—81no—
531D4—1222no—
532D4—4882no—
533D4—4882no—
534D4—2442no—
535C2×C4—22no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—612no—
543C2×C4—612no—
544C2×C4—612no—
545C2×C4—1222no—
546C2×C4—1222no—
547C2×C4—4882no—
548C2×C4—4882no—
549C2×C4—4881no—
550C2×C4—4881no—
551C2×C4—2442no—
552C2×C4—2442no—
553C2×C4—2442no—
554C2×C4C22⋊C4×D6111yescharacteristic
555C2×C4C22⋊C4×D6111yescharacteristic
556C2×C4C22⋊C4×D6111yescharacteristic
557C2×C4C22×C4×D6111yescharacteristic
558C2×C4C2×D4×D6111yescharacteristic
559C2×C4C2×D4×D6111yescharacteristic
560C8—42no—
561C8—42no—
562C8—1222no—
563C8—1222no—
564C8—2441no—
565C8—2441no—
566C22—21no—
567C22—21no—
568C22—21no—
569C22—41no—
570C22—41no—
571C22—41no—
572C22—41no—
573C22—41no—
574C22—41no—
575C22—1222no—
576C22—1222no—
577C22—1222no—
578C22C2×C22⋊C4×D6111yescharacteristic
579C4—21no—
580C4—21no—
581C4—21no—
582C4—21no—
583C4—21no—
584C4—41no—
585C4—41no—
586C4—1222no—
587C4—1222no—
588C4—1222no—
589C4—2442no—
590C4—2442no—
591C4#1683974312yes—
592C2—41no—
593C2—81no—
594C2—81no—
595C2—2442no—
596C2C2×#1683974311yescharacteristic
597C2#2145436511yescharacteristic
598C2#2145436511yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D61
commutatorC2×C4×C61C23×C4
FrattiniC42C23×D61
FittingC61×#1536C2
socleC22×C61C22×C22⋊C4
prime radical#21459739C2
cosocleC42×C61C24
hypercenter#1536D61
hypocenterC61#17687
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1536

Complements

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

NH ≅ G/Ntype
2. #21454836C2semidirect
3. #21454831C2semidirect
4. #21444447C2semidirect
5. #21444436C2semidirect
6. #21444423C2semidirect
7. #21444422C2semidirect
8. #21444422C2semidirect
9. #21444411C2semidirect
10. #21444404C2semidirect
11. C61×#1536C2semidirect
296. #1536D61semidirect
352. C61#17687semidirect

Maximal subgroups

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

subgroupindex
#214597392
#214548362
#214548312
#214444472
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C61×#15362
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