Finite Group Databasebeta

← #29834376

Subgroups — #29834376

Order 15616.

694 conjugacy classes of subgroups.

The subgroup lattice has 554 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840201C211yesmaximal, characteristic, complemented
2C2×#16840191C211yesmaximal, characteristic
3#21456309C211yesmaximal, characteristic, complemented
4#21454689C211yesmaximal, characteristic, complemented
5#21447880C211yesmaximal, characteristic, complemented
6#21447878C211yesmaximal, characteristic, complemented
7#21447872C211yesmaximal, characteristic, complemented
8#21447870C211yesmaximal, characteristic, complemented
9#21447868C211yesmaximal, characteristic, complemented
10#21447864C211yesmaximal, characteristic, complemented
11#21447862C211yesmaximal, characteristic, complemented
12#21447576C211yesmaximal, characteristic, complemented
13#21447528C211yesmaximal, characteristic, complemented
14#358×D61C211yesmaximal, characteristic, complemented
15C61×#2210C211yesmaximal, characteristic, complemented
16C2×#14519806—21no—
17C2×#14519796C2211yescharacteristic
18C2×#14519796C2211yescharacteristic
19C2×Q16×D61C2211yescharacteristic
20#16840332—21no—
21#16840241—21no—
22#16840201—24no—
23#16840191—24no—
24C2×Dic488C2211yescharacteristic
25C2×Dic488C2211yescharacteristic
26C2×#14519723C2211yescharacteristic
27C2×#14519721C2211yescharacteristic
28C2×#14519721C2211yescharacteristic
29C2×#14519721C2211yescharacteristic
30C2×#14519716C2211yescharacteristic
31C2×C16×D61C2211yescharacteristic
32#16840157—21no—
33#16840153—21no—
34#16840145C2211yescharacteristic
35#16840142C2211yescharacteristic
36#16840140C2211yescharacteristic
37#16839862—21no—
38#16839859—21no—
39C4×Dic244C2211yescharacteristic
40#16839856C2211yescharacteristic
41D8×Dic61C2211yescharacteristic
42#107×D61C2211yescharacteristic
43#16839760—21no—
44#16839716C2211yescharacteristic
45C4⋊#11758185C2211yescharacteristic
46#16839580C2211yescharacteristic
47#16839580C2211yescharacteristic
48#16839567C2211yescharacteristic
49#16839567C2211yescharacteristic
50#16839566C2211yescharacteristic
51#16839565C2211yescharacteristic
52#16839565C2211yescharacteristic
53#16839563C2211yescharacteristic
54C2×SD32×C61C2211yescharacteristic
55C2×D16×C61C2211yescharacteristic
56C61×#495C2211yescharacteristic
57C4×D8×C61C2211yescharacteristic
58C61×C4⋊C16C2211yescharacteristic
59C61×#358C2211yescharacteristic
60C61×#357C2211yescharacteristic
61C2×#11758224—21no—
62C2×#11758224—21no—
63C2×Q8×D61—21no—
64#14519806—44no—
65#14519796—22no—
66#14519796—22no—
67#14519796—22no—
68#14519796D414yes—
69C2×Dic244—21no—
70C2×Dic244C2311yescharacteristic
71C2×Dic244C2311yescharacteristic
72C2×Dic244C2311yescharacteristic
73C2×#11758202—21no—
74C2×#11758202—21no—
75Q16×D61—22no—
76Q16×D61—22no—
77C2×#11758200—21no—
78C2×#11758200—21no—
79C2×#11758200—21no—
80C2×#11758189D411yescharacteristic
81C2×C8×D61C2311yescharacteristic
82#14519777—21no—
83#14519775—21no—
84#14519764—21no—
85#14519759—41no—
86#14519759—41no—
87#14519754—41no—
88#14519754—41no—
89C4×Dic122—21no—
90#14519742—21no—
91D4×Dic61—21no—
92C4×C22⋊D61—41no—
93#14519737—41no—
94#14519734D411yescharacteristic
95#14519733—41no—
96C2×#11758196—41no—
97C4⋊C4×D61—21no—
98Dic488—22no—
99Dic488—22no—
100#14519723—22no—
101#14519721—22no—
102#14519721—22no—
103#14519721—22no—
104#14519716—22no—
105C2×#11758185C2311yescharacteristic
106C2×#11758185C2311yescharacteristic
107C16×D61—22no—
108#14519711C2311yescharacteristic
109#14519710—21no—
110#14519702—21no—
111#14519700—21no—
112#14519698—21no—
113#14519688D411yescharacteristic
114#14519687D411yescharacteristic
115C8×Dic61C2311yescharacteristic
116C2×Q16×C61C2311yescharacteristic
117C2×SD16×C61—21no—
118C2×D8×C61C2311yescharacteristic
119C2×D8×C61C2311yescharacteristic
120#124×C61—21no—
121C4×D4×C61—21no—
122SD32×C61—22no—
123D16×C61—22no—
124C2×C16×C61C2311yescharacteristic
125C2×C16×C61C2311yescharacteristic
126#107×C61C2311yescharacteristic
127#102×C61—21no—
128C4×C8×C61C2311yescharacteristic
129C2×Q8×C61—21no—
130C2×D4×C61—21no—
131C2×D4×C61—21no—
132C22×C4×C61—41no—
133C2×C22⋊D61—41no—
134C2×C22⋊D61—41no—
135C22×Dic61—41no—
136C22×Dic61—41no—
137C22×Dic61—41no—
138Q8×D61—42no—
139Q8×D61—42no—
140#11758224—24no—
141#11758224—42no—
142#11758224—42no—
143#11758224—42no—
144C2×C4×D61—41no—
145C2×C4×D61—41no—
146C2×C4×D61C2×D411yescharacteristic
147C2×Dic122—21no—
148C2×Dic122—21no—
149C2×Dic122—21no—
150C2×Dic122—41no—
151Q16×C61—21no—
152Q16×C61—21no—
153SD16×C61—42no—
154D8×C61—21no—
155D8×C61—21no—
156D8×C61—21no—
157D8×C61C2×D412yes—
158C2×C8×C61C2×D411yescharacteristic
159C2×C8×C61C2411yescharacteristic
160C4⋊C4×C61—21no—
161C22⋊C4×C61—41no—
162C22⋊C4×C61—41no—
163C22⋊C4×C61—41no—
164C42×C61C2×D411yescharacteristic
165C22⋊Dic61—41no—
166C22⋊Dic61—41no—
167C22⋊Dic61—41no—
168#11758202—42no—
169#11758202—42no—
170#11758200—24no—
171#11758200—42no—
172#11758200—42no—
173#11758198D811yescharacteristic
174#11758198D811yescharacteristic
175C4⋊Dic61—21no—
176C4⋊Dic61—41no—
177C4⋊Dic61—41no—
178#11758196—41no—
179#11758196—41no—
180#11758196—41no—
181#11758196—41no—
182#11758196—41no—
183#11758196D811yescharacteristic
184#11758196D811yescharacteristic
185C4×Dic61—41no—
186C4×Dic61—41no—
187C4×Dic61C2×D411yescharacteristic
188C2×#92418C2×D411yescharacteristic
189C2×#92418C2×D411yescharacteristic
190Dic244—21no—
191Dic244—21no—
192Dic244—21no—
193Dic244—21no—
194Dic244—21no—
195Dic244—42no—
196Dic244C2×D412yes—
197#11758189—22no—
198C8×D61C2×D412yes—
199C8×D61#4112yes—
200C16×C61—21no—
201C16×C61—21no—
202#11758185—21no—
203#11758185—21no—
204C23×C61—41no—
205C23×C61—41no—
206C22×D61C2×D811yescharacteristic
207Q8×C61—41no—
208Q8×C61—41no—
209D4×C61—22no—
210D4×C61—41no—
211D4×C61—41no—
212D4×C61—41no—
213C2×C4×C61—41no—
214C2×C4×C61—41no—
215C2×C4×C61—161no—
216C2×C4×C61C2×D811yescharacteristic
217C2×C4×C61C2×D811yescharacteristic
218C2×C4×C61C22×D411yescharacteristic
219C22⋊D61—44no—
220C22⋊D61—82no—
221C2×Dic61—41no—
222C2×Dic61—44no—
223C2×Dic61—41no—
224C2×Dic61—41no—
225C2×Dic61—41no—
226C2×Dic61—82no—
227C2×Dic61—161no—
228C2×Dic61C2×D811yescharacteristic
229C2×Dic61C2×D811yescharacteristic
230C2×Dic61C2×D811yescharacteristic
231C4×D61—82no—
232C4×D61—161no—
233C4×D61#12112yes—
234C4×D61AGL(1,ℤ/8)12yes—
235Dic122—22no—
236Dic122—41no—
237Dic122—41no—
238Dic122—41no—
239Dic122—41no—
240Dic122—41no—
241Dic122—82no—
242C8×C61—21no—
243C8×C61C22×D411yescharacteristic
244C8×C61C2×#4111yescharacteristic
245#92418—21no—
246#92418#12112yes—
247#33590—611nomaximal, Sylow
248C22×C61—42no—
249C22×C61—81no—
250C22×C61—81no—
251C22×C61—81no—
252C22×C61—81no—
253C22×C61—81no—
254C22×C61C22×D811yescharacteristic
255C2×D61—21no—
256C2×D61—21no—
257C2×D61#45912yes—
258C4×C61—21no—
259C4×C61—21no—
260C4×C61—81no—
261C4×C61—81no—
262C4×C61C2×#12111yescharacteristic
263C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
264Dic61—21no—
265Dic61—21no—
266Dic61—42no—
267Dic61—81no—
268Dic61—81no—
269Dic61—81no—
270Dic61#45912yes—
271C2×#510—611no—
272C2×#508—611no—
273#3156—611no—
274#2951—611no—
275#2211—611no—
276#2211—611no—
277#2210—611no—
278#2210D6111yescharacteristic, complemented
279#2209—611no—
280#2209—611no—
281#2208—611no—
282#2208—611no—
283#2151—611no—
284#2139—611no—
285C2×#358—611no—
286C2×C61—82no—
287C2×C61—161no—
288C2×C61C2×#45911yescharacteristic
289C2×C61#341611yescharacteristic
290C2×C61#341711yescharacteristic
291D61—22no—
292C2×#137—1221no—
293C2×#135—611no—
294C2×#135—611no—
295C22×Q16—611no—
296#536—1221no—
297#520—1221no—
298#510—1224no—
299#508—1224no—
300C2×Q32—611no—
301C2×Q32—611no—
302C2×Q32—611no—
303C2×SD32—611no—
304C2×SD32—611no—
305C2×SD32—611no—
306C2×SD32C2×D6111yescharacteristic
307C2×D16C2×D6111yescharacteristic
308C2×M32—611no—
309C22×C16—611no—
310#497—1221no—
311#497—1221no—
312#495—611no—
313#495C2×D6111yescharacteristic
314#494—611no—
315#494—611no—
316#440—1221no—
317#440—1221no—
318C4×Q16—611no—
319C4×Q16—611no—
320C4×D8—611no—
321C4×D8C2×D6111yescharacteristic
322C2×#107—611no—
323#417—1221no—
324#405—611no—
325C4⋊C16—611no—
326C4⋊C16—611no—
327C4⋊C16—611no—
328C4⋊C16C2×D6111yescharacteristic
329#358—611no—
330#358—611no—
331#358—611no—
332#358—611no—
333#358—611no—
334#358C2×D6111yescharacteristic
335#357—611no—
336#357C2×D6111yescharacteristic
337C61#3359011yescharacteristic, complemented, Sylow
338C2×#41—1221no—
339C2×#41—1221no—
340C22×Q8—1221no—
341#137—2444no—
342#135—614no—
343#135—1222no—
344#135—1222no—
345#135—1222no—
346C2×Q16—611no—
347C2×Q16—611no—
348C2×Q16—611no—
349C2×Q16—1222no—
350C2×Q16—1222no—
351C2×Q16—1221no—
352C2×Q16—1221no—
353C2×Q16—1221no—
354C2×Q16C22×D6111yescharacteristic
355C2×SD16—21no—
356C2×SD16—1221no—
357C2×SD16—1221no—
358C2×SD16—1221no—
359C2×D8C22×D6111yescharacteristic
360C2×D8C22×D6111yescharacteristic
361C2×M16—611no—
362C22×C8—611no—
363C4⋊Q8—1221no—
364C4⋊Q8—1221no—
365#124—21no—
366#124—1221no—
367#123—2441no—
368#123—2441no—
369C22⋊Q8—2441no—
370C22⋊Q8—2441no—
371C4×Q8—1221no—
372C4×Q8—1221no—
373C4×D4—21no—
374C4×D4—1221no—
375C4×D4—2441no—
376C4×D4—2441no—
377#117—611no—
378#117—2441no—
379C2×C4⋊C4—1221no—
380C2×C4⋊C4—2441no—
381Q32—1222no—
382Q32—1222no—
383Q32—1222no—
384SD32—22no—
385SD32—1222no—
386SD32—1222no—
387SD32—1222no—
388D16—22no—
389M32—1222no—
390C2×C16—611no—
391C2×C16—611no—
392C2×C16—1222no—
393C2×C16C22×D6111yescharacteristic
394C2×C16C22×D6111yescharacteristic
395#107—611no—
396#107—1221no—
397#107C22×D6111yescharacteristic
398#103—1221no—
399#103—1221no—
400#102—21no—
401#102—1221no—
402#97—611no—
403#97—611no—
404C4×C8—611no—
405C4×C8C22×D6111yescharacteristic
406#41—1224no—
407#41—2442no—
408#41—2442no—
409#41—2442no—
410C2×Q8—21no—
411C2×Q8—1221no—
412C2×Q8—1221no—
413C2×Q8—1221no—
414C2×Q8—2441no—
415C2×Q8—2442no—
416C2×Q8—2442no—
417C2×D4—21no—
418C2×D4—21no—
419C2×D4—2441no—
420C2×D4—2441no—
421C22×C4—41no—
422C22×C4—611no—
423C22×C4—2441no—
424C22×C4—2441no—
425C22×C4—2441no—
426C22×C4—2441no—
427C22×C4—2441no—
428Q16—21no—
429Q16—21no—
430Q16—612no—
431Q16—1221no—
432Q16—1221no—
433Q16—1221no—
434Q16—1221no—
435Q16—1221no—
436Q16—2442no—
437Q16—2442no—
438Q16—2442no—
439SD16—42no—
440SD16—1224no—
441SD16—2442no—
442SD16—2442no—
443D8—21no—
444D8—21no—
445D8—21no—
446D8D4×D6112yes—
447M16—1222no—
448C2×C8—611no—
449C2×C8—612no—
450C2×C8—612no—
451C2×C8—611no—
452C2×C8D4×D6111yescharacteristic
453C2×C8C23×D6111yescharacteristic
454C4⋊C4—21no—
455C4⋊C4—611no—
456C4⋊C4—611no—
457C4⋊C4—1221no—
458C4⋊C4—2441no—
459C4⋊C4—2441no—
460C4⋊C4—2441no—
461C4⋊C4—2441no—
462C4⋊C4—2441no—
463C4⋊C4—2441no—
464C4⋊C4—2441no—
465C22⋊C4—41no—
466C22⋊C4—41no—
467C22⋊C4—41no—
468C22⋊C4—611no—
469C22⋊C4—611no—
470C22⋊C4—2441no—
471C22⋊C4—2441no—
472C22⋊C4—2441no—
473C42—611no—
474C42—2441no—
475C42—2441no—
476C42D4×D6111yescharacteristic
477C16—21no—
478C16—21no—
479C16—1221no—
480C16—1221no—
481C23—41no—
482C23—41no—
483C23—611no—
484Q8—41no—
485Q8—41no—
486Q8—1222no—
487Q8—4882no—
488Q8—2441no—
489Q8—2441no—
490Q8—2441no—
491Q8—2441no—
492Q8—2441no—
493D4—22no—
494D4—41no—
495D4—41no—
496D4—41no—
497D4—4882no—
498D4—2444no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—161no—
502C2×C4—611no—
503C2×C4—612no—
504C2×C4—612no—
505C2×C4—611no—
506C2×C4—611no—
507C2×C4—9761no—
508C2×C4—9761no—
509C2×C4—4882no—
510C2×C4—4882no—
511C2×C4—2444no—
512C2×C4—2441no—
513C2×C4—2441no—
514C2×C4—2441no—
515C2×C4—2441no—
516C2×C4D8×D6111yescharacteristic
517C2×C4D8×D6111yescharacteristic
518C2×C4C2×D4×D6111yescharacteristic
519C8—21no—
520C8—612no—
521C8—1221no—
522C8C2×D4×D6111yescharacteristic
523C8#41×D6111yescharacteristic
524C22—42no—
525C22—81no—
526C22—81no—
527C22—81no—
528C22—81no—
529C22—81no—
530C22—612no—
531C22—1221no—
532C22—1221no—
533C22C2×D8×D6111yescharacteristic
534C4—21no—
535C4—21no—
536C4—81no—
537C4—81no—
538C4—612no—
539C4—1221no—
540C4—1221no—
541C4—4881no—
542C4—4881no—
543C4—4881no—
544C4—2442no—
545C4#121×D6111yescharacteristic
546C4AGL(1,ℤ/8)×D6111yescharacteristic
547C2—82no—
548C2—161no—
549C2—1222no—
550C2#459×D6111yescharacteristic
551C2#2145896511yescharacteristic
552C2#2145896911yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2210C2
socleC22×C61C22×D8
prime radicalC2×#16840191C2
hypercenter#2210D61
hypocenterC61#33590
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2210

Complements

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

NH ≅ G/Ntype
1. C2×#16840201C2semidirect
3. #21456309C2semidirect
4. #21454689C2semidirect
5. #21447880C2semidirect
6. #21447878C2semidirect
7. #21447872C2semidirect
8. #21447870C2semidirect
9. #21447868C2semidirect
10. #21447864C2semidirect
11. #21447862C2semidirect
12. #21447576C2semidirect
13. #21447528C2semidirect
14. #358×D61C2semidirect
15. C61×#2210C2semidirect
278. #2210D61semidirect
337. C61#33590semidirect

Maximal subgroups

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

subgroupindex
C2×#168402012
C2×#168401912
#214563092
#214546892
#214478802
#214478782
#214478722
#214478702
#214478682
#214478642
#214478622
#214475762
#214475282
#358×D612
C61×#22102
#3359061