Finite Group Databasebeta

← #29667862

Subgroups — #29667862

Order 15616.

784 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456959C211yesmaximal, characteristic, complemented
2#21456740C211yesmaximal, characteristic
3#21456377C211yesmaximal, characteristic
4#21456362C211yesmaximal, characteristic, complemented
5#498×D61C211yesmaximal, characteristic, complemented
6#21456273C211yesmaximal, characteristic
7#21445452C211yesmaximal, characteristic, complemented
8#21445451C211yesmaximal, characteristic, complemented
9#21445440C211yesmaximal, characteristic, complemented
10#21445417C211yesmaximal, characteristic, complemented
11#21445415C211yesmaximal, characteristic, complemented
12#21445405C211yesmaximal, characteristic, complemented
13#21445405C211yesmaximal, characteristic, complemented
14#21444098C211yesmaximal, characteristic, complemented
15C61×#1714C211yesmaximal, characteristic, complemented
16C2×#14519806—22no—
17#16840317C2211yescharacteristic
18#16840310C2211yescharacteristic
19C4⋊Q8×D61C2211yescharacteristic
20C4⋊Q8×D61C2211yescharacteristic
21#16840170—22no—
22#16840165C2211yescharacteristic
23#16840160C2211yescharacteristic
24#16840160C2211yescharacteristic
25#16840160C2211yescharacteristic
26#16840159—22no—
27#16840158C2211yescharacteristic
28#16840158C2211yescharacteristic
29#16840158C2211yescharacteristic
30#16840158C2211yescharacteristic
31#16840158C2211yescharacteristic
32#16840153—22no—
33#16840142C2211yescharacteristic
34#16840141C2211yescharacteristic
35#16840134C2211yescharacteristic
36#16840134C2211yescharacteristic
37#16840134C2211yescharacteristic
38#16840113C2211yescharacteristic
39#16840113C2211yescharacteristic
40#16840112C2211yescharacteristic
41#16840112C2211yescharacteristic
42#16840091—22no—
43#16840045—22no—
44#16839990—22no—
45#16839965—22no—
46#16839899—22no—
47#16839853—22no—
48#16839813C2211yescharacteristic
49#16839803—22no—
50#106×D61—22no—
51C4×#11758189C2211yescharacteristic
52C4×C8×D61C2211yescharacteristic
53#16839484C2211yescharacteristic
54C8×#92418C2211yescharacteristic
55C8×#92418C2211yescharacteristic
56C61×#500C2211yescharacteristic
57C61×#498C2211yescharacteristic
58C61×#492C2211yescharacteristic
59C61×#492C2211yescharacteristic
60C61×#487C2211yescharacteristic
61C61×#486C2211yescharacteristic
62C61×#322C2211yescharacteristic
63C2×#11758224—21no—
64C2×Q8×D61—21no—
65C2×Q8×D61—21no—
66#14519806—48no—
67C2×Dic244—22no—
68C2×#11758202—22no—
69C2×#11758200—22no—
70C2×#11758200—22no—
71C2×#11758200—22no—
72C2×#11758189D412yes—
73C2×C8×D61D412yes—
74#14519777C2311yescharacteristic
75#14519777C2311yescharacteristic
76#14519777C2311yescharacteristic
77#14519777C2311yescharacteristic
78#14519776—21no—
79#14519776—21no—
80#14519776—21no—
81#14519776—21no—
82#14519775—21no—
83#14519775—21no—
84#14519768—21no—
85#14519764—21no—
86#14519757—21no—
87#14519753—21no—
88#14519748—21no—
89#14519748—21no—
90Q8×Dic61—21no—
91#14519742—21no—
92D4×Dic61—21no—
93D4×Dic61—21no—
94#14519735—21no—
95C4⋊C4×D61—21no—
96C4⋊C4×D61—21no—
97C4⋊C4×D61—21no—
98C4⋊C4×D61—21no—
99C42×D61C2311yescharacteristic
100#14519710—22no—
101#14519709—22no—
102#14519709—22no—
103#14519709—22no—
104#14519708—22no—
105#14519708—22no—
106#14519708—22no—
107#14519708—22no—
108#14519708—22no—
109#14519706D412yes—
110#14519702—22no—
111#14519702—22no—
112#14519701—22no—
113#14519701—22no—
114#14519689D412yes—
115#14519687D412yes—
116C4×#92418C2311yescharacteristic
117C4×#92418C2311yescharacteristic
118C4×#92418C2311yescharacteristic
119C8×Dic61D412yes—
120C2×SD16×C61—22no—
121C2×SD16×C61—22no—
122C4⋊Q8×C61C2311yescharacteristic
123C4⋊Q8×C61C2311yescharacteristic
124C4⋊Q8×C61C2311yescharacteristic
125#127×C61C2311yescharacteristic
126#107×C61—22no—
127#106×C61—22no—
128#103×C61—22no—
129#102×C61—22no—
130C4×C8×C61C2311yescharacteristic
131C4×C8×C61C2311yescharacteristic
132C4×C8×C61C2311yescharacteristic
133C2×Q8×C61—21no—
134C2×Q8×C61—21no—
135C2×Q8×C61—21no—
136C2×D4×C61—21no—
137C2×D4×C61—21no—
138C2×D4×C61—21no—
139C2×C22⋊D61—41no—
140C22×Dic61—41no—
141Q8×D61—44no—
142Q8×D61—82no—
143#11758224—44no—
144C2×C4×D61—41no—
145C2×C4×D61—41no—
146C2×C4×D61C2×D411yescharacteristic
147C2×C4×D61C2×D411yescharacteristic
148C2×C4×D61C2×D411yescharacteristic
149C2×Dic122—21no—
150C2×Dic122—21no—
151C2×Dic122—21no—
152C2×Dic122—21no—
153C2×Dic122—41no—
154C2×Dic122—41no—
155SD16×C61—44no—
156SD16×C61—82no—
157C2×C8×C61C2×D412yes—
158C2×C8×C61C2×D412yes—
159C2×C8×C61C2×D412yes—
160C4⋊C4×C61—21no—
161C4⋊C4×C61—21no—
162C4⋊C4×C61—21no—
163C4⋊C4×C61—21no—
164C4⋊C4×C61—21no—
165C4⋊C4×C61—21no—
166C42×C61C2411yescharacteristic
167C22⋊Dic61—41no—
168C22⋊Dic61—41no—
169#11758202—44no—
170#11758200—44no—
171#11758200—44no—
172#11758200—82no—
173#11758198—41no—
174C4⋊Dic61—21no—
175C4⋊Dic61—21no—
176C4⋊Dic61—21no—
177C4⋊Dic61—21no—
178C4⋊Dic61—21no—
179C4⋊Dic61—21no—
180C4⋊Dic61—21no—
181C4⋊Dic61—21no—
182C4⋊Dic61—41no—
183#11758196—41no—
184#11758196—41no—
185#11758196—41no—
186#11758196—41no—
187#11758196—41no—
188C4×Dic61—41no—
189C4×Dic61C2×D411yescharacteristic
190C4×Dic61C2×D411yescharacteristic
191C4×Dic61C2×D411yescharacteristic
192C2×#92418C2×D412yes—
193C2×#92418C2×D412yes—
194C2×#92418C2×D412yes—
195Dic244—44no—
196#11758189SD1618yescomplemented
197C8×D61—24no—
198C23×C61—41no—
199C22×D61#12711yescharacteristic
200Q8×C61—42no—
201Q8×C61—42no—
202Q8×C61—42no—
203D4×C61—42no—
204D4×C61—42no—
205D4×C61—42no—
206C2×C4×C61—41no—
207C2×C4×C61—41no—
208C2×C4×C61—41no—
209C2×C4×C61C22×D411yescharacteristic
210C2×C4×C61C22×D411yescharacteristic
211C2×C4×C61C22×D411yescharacteristic
212C22⋊D61—82no—
213C2×Dic61—41no—
214C2×Dic61—41no—
215C2×Dic61—41no—
216C2×Dic61—41no—
217C2×Dic61—82no—
218C2×Dic61#12711yescharacteristic
219C2×Dic61#12711yescharacteristic
220C2×Dic61#12711yescharacteristic
221C4×D61—22no—
222C4×D61—22no—
223C4×D61—82no—
224C4×D61—82no—
225C4×D61C2×SD1614yes—
226Dic122—42no—
227Dic122—42no—
228Dic122—42no—
229Dic122—42no—
230Dic122—82no—
231Dic122—82no—
232C8×C61—22no—
233C8×C61—22no—
234C8×C61C2×SD1614yes—
235#92418—22no—
236#92418C2×SD1614yes—
237#92418#13714yescomplemented
238#19622—611nomaximal, Sylow
239C22×C61—81no—
240C22×C61—81no—
241C22×C61—81no—
242C22×C61C2×#12711yescharacteristic
243C2×D61—21no—
244C2×D61—21no—
245C2×D61#49212yes—
246C4×C61—81no—
247C4×C61—81no—
248C4×C61—81no—
249C4×C61C22×SD1612yes—
250C4×C61C2×#13712yes—
251C4×C61#57712yes—
252Dic61—21no—
253Dic61—21no—
254Dic61—81no—
255Dic61—81no—
256Dic61—81no—
257Dic61—81no—
258Dic61#49212yes—
259#3229—611no—
260#3206—611no—
261#3167—611no—
262#3165—611no—
263C2×#498—611no—
264#3152—611no—
265#1716—611no—
266#1716—611no—
267#1714—611no—
268#1714D6111yescharacteristic, complemented
269#1707—611no—
270#1707—611no—
271#1705—611no—
272#1705—611no—
273C8×M16—611no—
274C2×C61—161no—
275C2×C61C2×#49211yescharacteristic
276C2×C61#315211yescharacteristic
277C2×C61#315411yescharacteristic
278D61—22no—
279C2×#137—1222no—
280#533—611no—
281#532—611no—
282C2×C4⋊Q8—611no—
283C2×C4⋊Q8—611no—
284#501—1222no—
285#500—611no—
286#500C2×D6111yescharacteristic
287#498—611no—
288#498—611no—
289#498—611no—
290#498—611no—
291#498—611no—
292#498—611no—
293#498—611no—
294#498—611no—
295#498—1222no—
296#498C2×D6111yescharacteristic
297#497—1222no—
298#494—611no—
299#494—611no—
300#492—611no—
301#492—611no—
302#492—611no—
303#492C2×D6111yescharacteristic
304#492C2×D6111yescharacteristic
305#487—611no—
306#487—611no—
307#487—611no—
308#487—611no—
309#487C2×D6111yescharacteristic
310#486C2×D6111yescharacteristic
311#483—1222no—
312#475—1222no—
313#465—1222no—
314#461—1222no—
315#448—1222no—
316C4×SD16—1222no—
317#431—611no—
318#428—1222no—
319C2×#106—1222no—
320C4×M16—611no—
321C2×C4×C8—611no—
322#322—611no—
323#322C2×D6111yescharacteristic
324C82—611no—
325C82—611no—
326C61#1962211yescharacteristic, complemented, Sylow
327C2×#41—1221no—
328C22×Q8—1221no—
329C22×Q8—1221no—
330#137—2448no—
331C2×Q16—1222no—
332C2×Q16—1222no—
333C2×SD16—22no—
334C2×SD16—22no—
335C2×SD16—1222no—
336C2×SD16—1222no—
337C2×SD16—1222no—
338C2×M16—612no—
339C22×C8—612no—
340C4⋊Q8—611no—
341C4⋊Q8—611no—
342C4⋊Q8—611no—
343C4⋊Q8—611no—
344C4⋊Q8—1221no—
345C4⋊Q8—1221no—
346C4⋊Q8—1221no—
347C4⋊Q8—1221no—
348C4⋊Q8—1221no—
349C4⋊Q8—1221no—
350C4⋊Q8C22×D6111yescharacteristic
351C4⋊Q8C22×D6111yescharacteristic
352C4⋊Q8C22×D6111yescharacteristic
353#127C22×D6111yescharacteristic
354#125—1221no—
355#124—1221no—
356C22⋊Q8—1221no—
357C22⋊Q8—1221no—
358#121—1221no—
359#121—1221no—
360C4×Q8—1221no—
361C4×Q8—1221no—
362C4×D4—1221no—
363C4×D4—1221no—
364#117—1221no—
365C2×C4⋊C4—1221no—
366C2×C4⋊C4—1221no—
367C2×C4⋊C4—1221no—
368C2×C4⋊C4—1221no—
369C2×C42—611no—
370#107—22no—
371#107—1222no—
372#106—22no—
373#106—1222no—
374#106—1222no—
375#106—1222no—
376#106—1222no—
377#106—1222no—
378#106—1222no—
379#106—1222no—
380#106—1222no—
381C4⋊C8—612no—
382#103—22no—
383#103—1222no—
384#103—1222no—
385#103—1222no—
386#103—1222no—
387#102—22no—
388C22⋊C8—612no—
389#97—612no—
390C4×C8—612no—
391C4×C8—611no—
392C4×C8—611no—
393C4×C8—611no—
394C4×C8C22×D6111yescharacteristic
395C4×C8C22×D6111yescharacteristic
396C4×C8C22×D6111yescharacteristic
397#41—2444no—
398C2×Q8—21no—
399C2×Q8—21no—
400C2×Q8—21no—
401C2×Q8—1221no—
402C2×Q8—1221no—
403C2×Q8—1221no—
404C2×Q8—1221no—
405C2×Q8—4882no—
406C2×Q8—2441no—
407C2×Q8—2441no—
408C2×Q8—2444no—
409C2×D4—21no—
410C2×D4—21no—
411C2×D4—21no—
412C2×D4—2441no—
413C22×C4—611no—
414C22×C4—611no—
415C22×C4—611no—
416C22×C4—2441no—
417C22×C4—2441no—
418C22×C4—2441no—
419Q16—2444no—
420Q16—2444no—
421SD16—44no—
422SD16—82no—
423SD16—4882no—
424SD16—2444no—
425SD16—2444no—
426M16—618no—
427C2×C8—612no—
428C2×C8—612no—
429C2×C8—612no—
430C2×C8—1224no—
431C2×C8D4×D6112yes—
432C2×C8D4×D6112yes—
433C2×C8D4×D6112yes—
434C4⋊C4—21no—
435C4⋊C4—21no—
436C4⋊C4—21no—
437C4⋊C4—21no—
438C4⋊C4—21no—
439C4⋊C4—21no—
440C4⋊C4—1221no—
441C4⋊C4—1221no—
442C4⋊C4—1221no—
443C4⋊C4—1221no—
444C4⋊C4—1221no—
445C4⋊C4—1221no—
446C4⋊C4—1221no—
447C4⋊C4—1221no—
448C4⋊C4—2441no—
449C4⋊C4—2441no—
450C4⋊C4—2441no—
451C4⋊C4—2441no—
452C4⋊C4—2441no—
453C4⋊C4—2441no—
454C22⋊C4—2441no—
455C22⋊C4—2441no—
456C22⋊C4—2441no—
457C42—611no—
458C42—611no—
459C42—611no—
460C42—2441no—
461C42C23×D6111yescharacteristic
462C23—41no—
463C23—611no—
464Q8—42no—
465Q8—42no—
466Q8—42no—
467Q8—4882no—
468Q8—4882no—
469Q8—2442no—
470Q8—2442no—
471Q8—2442no—
472Q8—2442no—
473D4—42no—
474D4—42no—
475D4—42no—
476D4—4882no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—611no—
481C2×C4—614no—
482C2×C4—611no—
483C2×C4—611no—
484C2×C4—1222no—
485C2×C4—1222no—
486C2×C4—4882no—
487C2×C4—4882no—
488C2×C4—4882no—
489C2×C4—2441no—
490C2×C4—2441no—
491C2×C4—2441no—
492C2×C4—2441no—
493C2×C4C2×D4×D6111yescharacteristic
494C2×C4C2×D4×D6111yescharacteristic
495C2×C4C2×D4×D6111yescharacteristic
496C8—22no—
497C8—22no—
498C8—614no—
499C8—614no—
500C8—1222no—
501C8SD16×D6114yes—
502C22—81no—
503C22—81no—
504C22—81no—
505C22—612no—
506C22—1221no—
507C22—1221no—
508C22#127×D6111yescharacteristic
509C4—81no—
510C4—81no—
511C4—81no—
512C4—612no—
513C4—1221no—
514C4—1221no—
515C4—4881no—
516C4—4881no—
517C4—4881no—
518C4—4881no—
519C4C2×SD16×D6112yes—
520C4#137×D6112yes—
521C4#1684057212yes—
522C2—161no—
523C2—1222no—
524C2#492×D6111yescharacteristic
525C2#2145627511yescharacteristic
526C2#2145629211yescharacteristic
subgroupG/H
centerC22#127×D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1714C2
socleC22×C61C2×#127
prime radical#16840310C22
hypercenter#1714D61
hypocenterC61#19622
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #1714

Complements

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

NH ≅ G/Ntype
1. #21456959C2semidirect
4. #21456362C2semidirect
5. #498×D61C2semidirect
7. #21445452C2semidirect
8. #21445451C2semidirect
9. #21445440C2semidirect
10. #21445417C2semidirect
11. #21445415C2semidirect
12. #21445405C2semidirect
13. #21445405C2semidirect
14. #21444098C2semidirect
15. C61×#1714C2semidirect
196. #11758189SD16semidirect
237. #92418#137semidirect
268. #1714D61semidirect
326. C61#19622semidirect

Maximal subgroups

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

subgroupindex
#214569592
#214567402
#214563772
#214563622
#498×D612
#214562732
#214454522
#214454512
#214454402
#214454172
#214454152
#214454052
#214440982
C61×#17142
#1962261