Finite Group Databasebeta

← #29658695

Subgroups — #29658695

Order 15616.

1234 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456760C211yesmaximal, characteristic, complemented
2#21455827C211yesmaximal, characteristic, complemented
3#21455624C211yesmaximal, characteristic, complemented
4#21445019C214yesmaximal, complemented
5#21444946C214yesmaximal, complemented
6#21444487C211yesmaximal, characteristic, complemented
7#21444244C211yesmaximal, characteristic, complemented
8#21444214C211yesmaximal, characteristic, complemented
9C2×C61×#550C211yesmaximal, characteristic
10#16840319C2211yescharacteristic
11#16840134C2212yescomplemented
12#16840108C2212yescomplemented
13#16840092—21no—
14#16840069—22no—
15#16840057C2212yescomplemented
16#16840052C2212yescomplemented
17#16840047C2212yescomplemented
18#16839967C2212yescomplemented
19#16839959—22no—
20#16839913—22no—
21#16839913—21no—
22C4×#11758200—22no—
23#16839815C2211yescharacteristic
24#16839815C2211yescharacteristic
25#16839802—21no—
26#16839786C2211yescharacteristic
27#16839771—21no—
28#16839506C2214yescomplemented
29Q8⋊#92418C2214yescomplemented
30D4⋊#92418C2214yescomplemented
31C22×#41×C61—21no—
32C61×#550—22no—
33C61×#550—22no—
34C61×#550—22no—
35C61×#550C2214yes—
36C2×#127×C61—21no—
37C2×#127×C61C2211yescharacteristic
38C2×#121×C61—22no—
39C2×#121×C61—21no—
40C2×C4×Q8×C61C2211yescharacteristic
41C2×C4×D4×C61—21no—
42C2×C4×D4×C61C2211yescharacteristic
43C2×#11758200—22no—
44C2×#11758200—24no—
45C2×#11758194—21no—
46#14519777C2311yescharacteristic
47#14519777C2311yescharacteristic
48#14519769—21no—
49#14519756—21no—
50#14519756—21no—
51C4×Dic122—21no—
52C4×Dic122—21no—
53#14519736—21no—
54#14519710—22no—
55#14519708—22no—
56#14519708—22no—
57C4⋊#92418—21no—
58C4⋊#92418—21no—
59C4⋊#92418C2312yes—
60C4⋊#92418C2312yes—
61#14519701—22no—
62#14519701—22no—
63#14519700—22no—
64#14519698—24no—
65#14519698—22no—
66C22⋊#92418—21no—
67C22⋊#92418—21no—
68C4×#92418—21no—
69C4×#92418C2312yes—
70C2×#41×C61—24no—
71C2×#41×C61—24no—
72C2×#41×C61—44no—
73C2×#41×C61—44no—
74C22×Q8×C61D411yescharacteristic
75C22×D4×C61—22no—
76C22×D4×C61—22no—
77C22×D4×C61—21no—
78C22×D4×C61—21no—
79C22×D4×C61D411yescharacteristic
80C22×D4×C61D412yes—
81C23×C4×C61—22no—
82C23×C4×C61—21no—
83#127×C61—24no—
84#127×C61—24no—
85#127×C61—21no—
86#127×C61—21no—
87#127×C61D412yes—
88#127×C61C2312yes—
89#121×C61—24no—
90#121×C61—28no—
91#121×C61—28no—
92#121×C61—24no—
93C4×Q8×C61—21no—
94C4×Q8×C61—21no—
95C4×Q8×C61D412yes—
96C4×Q8×C61C2312yes—
97C4×D4×C61—24no—
98C4×D4×C61—24no—
99C4×D4×C61—21no—
100C4×D4×C61—21no—
101C4×D4×C61D412yes—
102C4×D4×C61C2312yes—
103C2×C4⋊C4×C61—21no—
104C2×C4⋊C4×C61D411yescharacteristic
105C2×C4⋊C4×C61D411yescharacteristic
106C2×C22⋊C4×C61—22no—
107C2×C22⋊C4×C61—21no—
108C2×C42×C61—21no—
109C2×C42×C61C2311yescharacteristic
110C24×C61—22no—
111C24×C61—21no—
112#41×C61—48no—
113#41×C61—416no—
114#41×C61—48no—
115C2×Q8×C61—21no—
116C2×Q8×C61—22no—
117C2×Q8×C61SD1614yescomplemented
118C2×Q8×C61C2×D412yes—
119C2×D4×C61—28no—
120C2×D4×C61—28no—
121C2×D4×C61—21no—
122C2×D4×C61—22no—
123C2×D4×C61—22no—
124C2×D4×C61—22no—
125C2×D4×C61—22no—
126C2×D4×C61—22no—
127C2×D4×C61—42no—
128C2×D4×C61—44no—
129C2×D4×C61—48no—
130C2×D4×C61—42no—
131C2×D4×C61—42no—
132C2×D4×C61—82no—
133C2×D4×C61—84no—
134C2×D4×C61C2×D412yes—
135C2×D4×C61C2×D414yes—
136C22×C4×C61—21no—
137C22×C4×C61—28no—
138C22×C4×C61—21no—
139C22×C4×C61—21no—
140C22×C4×C61—22no—
141C22×C4×C61—22no—
142C22×C4×C61—48no—
143C22×C4×C61—42no—
144C22×C4×C61—42no—
145C22×C4×C61C2×D411yescharacteristic
146C22×C4×C61C2×D411yescharacteristic
147C22×C4×C61C2×D411yescharacteristic
148C2×Dic122—21no—
149C2×Dic122—21no—
150C4⋊C4×C61—22no—
151C4⋊C4×C61—21no—
152C4⋊C4×C61—21no—
153C4⋊C4×C61—22no—
154C4⋊C4×C61C2×D412yes—
155C4⋊C4×C61C2×D412yes—
156C22⋊C4×C61—28no—
157C22⋊C4×C61—22no—
158C22⋊C4×C61—22no—
159C42×C61—22no—
160C42×C61—22no—
161C42×C61C2×D411yescharacteristic
162C42×C61C2×D411yescharacteristic
163C42×C61C2×D411yescharacteristic
164C42×C61C2411yescharacteristic
165C22⋊Dic61—41no—
166#11758200—44no—
167#11758200—84no—
168C4⋊Dic61—21no—
169C4⋊Dic61—21no—
170C4⋊Dic61—21no—
171C4⋊Dic61—21no—
172#11758196—41no—
173#11758196—41no—
174C4×Dic61—41no—
175#11758194—42no—
176C2×#92418—22no—
177C2×#92418—22no—
178C2×#92418—22no—
179C23×C61—22no—
180C23×C61—24no—
181C23×C61—42no—
182C23×C61—44no—
183C23×C61—44no—
184C23×C61—42no—
185C23×C61—44no—
186C23×C61—42no—
187C23×C61#12011yescharacteristic
188Q8×C61—22no—
189Q8×C61—24no—
190Q8×C61C2×SD1614yes—
191D4×C61—22no—
192D4×C61—22no—
193D4×C61—44no—
194D4×C61—42no—
195D4×C61—44no—
196D4×C61—416no—
197D4×C61—42no—
198D4×C61—42no—
199D4×C61—44no—
200D4×C61—44no—
201D4×C61—82no—
202D4×C61—88no—
203C2×C4×C61—22no—
204C2×C4×C61—24no—
205C2×C4×C61—22no—
206C2×C4×C61—22no—
207C2×C4×C61—22no—
208C2×C4×C61—41no—
209C2×C4×C61—416no—
210C2×C4×C61—42no—
211C2×C4×C61—42no—
212C2×C4×C61—42no—
213C2×C4×C61—42no—
214C2×C4×C61—44no—
215C2×C4×C61#12011yescharacteristic
216C2×C4×C61#12011yescharacteristic
217C2×C4×C61#12011yescharacteristic
218C2×C4×C61C2×SD1612yes—
219C2×C4×C61AGL(1,ℤ/8)12yes—
220C2×C4×C61C22×D411yescharacteristic
221C2×C4×C61C22×D411yescharacteristic
222C2×C4×C61C22×D411yescharacteristic
223C2×Dic61—41no—
224C2×Dic61—41no—
225Dic122—42no—
226Dic122—42no—
227#92418—42no—
228#92418—42no—
229#92418—42no—
230#18767—611nomaximal, Sylow
231C22×C61—21no—
232C22×C61—21no—
233C22×C61—44no—
234C22×C61—42no—
235C22×C61—42no—
236C22×C61—44no—
237C22×C61—44no—
238C22×C61—42no—
239C22×C61—84no—
240C22×C61—88no—
241C22×C61#45012yes—
242C22×C61C2×#12011yescharacteristic
243C4×C61—24no—
244C4×C61—21no—
245C4×C61—21no—
246C4×C61—21no—
247C4×C61—42no—
248C4×C61—42no—
249C4×C61#45012yes—
250C4×C61C22×SD1611yescharacteristic
251C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
252C4×C61#57512yes—
253Dic61—81no—
254Dic61—81no—
255C2×#550D6111yescharacteristic
256#3208—611no—
257#3101—611no—
258#3078—611no—
259#1642—614no—
260#1631—614no—
261#1545—611no—
262#1486—611no—
263#1479—611no—
264C2×C61—22no—
265C2×C61—82no—
266C2×C61—84no—
267C2×C61C2×#45011yescharacteristic
268C2×C61#300511yescharacteristic
269C2×C61#301911yescharacteristic
270C22×#41—21no—
271#550—22no—
272#550—22no—
273#550—22no—
274#550C2×D6114yes—
275#533—611no—
276C2×#127—21no—
277C2×#127C2×D6111yescharacteristic
278C2×#121—22no—
279C2×#121—21no—
280C2×C4×Q8C2×D6111yescharacteristic
281C2×C4×D4—21no—
282C2×C4×D4C2×D6111yescharacteristic
283#492—612no—
284#486—612no—
285#483—1221no—
286#479—1222no—
287#477—612no—
288#476—612no—
289#475—612no—
290#461—612no—
291#460—1222no—
292#450—1221no—
293#450—1222no—
294C4×SD16—1222no—
295#431—611no—
296#431—611no—
297#428—1221no—
298#423—611no—
299#419—1221no—
300#331—614no—
301Q8⋊C8—614no—
302D4⋊C8—614no—
303C61#1876711yescharacteristic, complemented, Sylow
304C2×#41—24no—
305C2×#41—24no—
306C2×#41—44no—
307C2×#41—44no—
308C22×Q8C22⋊D6111yescharacteristic
309C22×D4—22no—
310C22×D4—22no—
311C22×D4—21no—
312C22×D4—21no—
313C22×D4C22⋊D6111yescharacteristic
314C22×D4C22⋊D6112yes—
315C23×C4—22no—
316C23×C4—21no—
317C2×SD16—1224no—
318C2×SD16—1222no—
319C2×M16—1221no—
320C4⋊Q8—611no—
321C4⋊Q8—611no—
322#127—24no—
323#127—24no—
324#127—21no—
325#127—21no—
326#127C22⋊D6112yes—
327#127C22×D6112yes—
328#125—1221no—
329C22⋊Q8—1221no—
330C22⋊Q8—1221no—
331#121—24no—
332#121—28no—
333#121—28no—
334#121—24no—
335C4×Q8—21no—
336C4×Q8—21no—
337C4×Q8—1221no—
338C4×Q8—1221no—
339C4×Q8C22⋊D6112yes—
340C4×Q8C22×D6112yes—
341C4×D4—24no—
342C4×D4—24no—
343C4×D4—21no—
344C4×D4—21no—
345C4×D4C22⋊D6112yes—
346C4×D4C22×D6112yes—
347#117—1221no—
348C2×C4⋊C4—21no—
349C2×C4⋊C4C22⋊D6111yescharacteristic
350C2×C4⋊C4C22⋊D6111yescharacteristic
351C2×C22⋊C4—22no—
352C2×C22⋊C4—21no—
353C2×C42—21no—
354C2×C42C22×D6111yescharacteristic
355#107—1222no—
356#106—1222no—
357#106—1222no—
358C4⋊C8—612no—
359C4⋊C8—612no—
360C4⋊C8—1221no—
361C4⋊C8—1221no—
362#103—1222no—
363#103—1222no—
364#103—1222no—
365#102—1222no—
366#102—1224no—
367C22⋊C8—1221no—
368C22⋊C8—1221no—
369C4×C8—612no—
370C4×C8—1221no—
371C24—22no—
372C24—21no—
373#41—48no—
374#41—416no—
375#41—48no—
376C2×Q8—21no—
377C2×Q8—22no—
378C2×Q8—1221no—
379C2×Q8—1221no—
380C2×Q8#1175820014yescomplemented
381C2×Q8C2×C22⋊D6112yes—
382C2×D4—28no—
383C2×D4—28no—
384C2×D4—22no—
385C2×D4—22no—
386C2×D4—21no—
387C2×D4—22no—
388C2×D4—22no—
389C2×D4—22no—
390C2×D4—42no—
391C2×D4—48no—
392C2×D4—44no—
393C2×D4—42no—
394C2×D4—42no—
395C2×D4—82no—
396C2×D4—84no—
397C2×D4C2×C22⋊D6112yes—
398C2×D4C2×C22⋊D6114yes—
399C22×C4—21no—
400C22×C4—28no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—48no—
406C22×C4—42no—
407C22×C4—42no—
408C22×C4C2×C22⋊D6111yescharacteristic
409C22×C4C2×C22⋊D6111yescharacteristic
410C22×C4C2×C22⋊D6111yescharacteristic
411SD16—4884no—
412SD16—2444no—
413M16—2442no—
414C2×C8—1222no—
415C2×C8—1222no—
416C2×C8—1222no—
417C4⋊C4—22no—
418C4⋊C4—22no—
419C4⋊C4—21no—
420C4⋊C4—21no—
421C4⋊C4—1221no—
422C4⋊C4—1221no—
423C4⋊C4—1221no—
424C4⋊C4—1221no—
425C4⋊C4—2441no—
426C4⋊C4—2441no—
427C4⋊C4C2×C22⋊D6112yes—
428C4⋊C4C2×C22⋊D6112yes—
429C22⋊C4—28no—
430C22⋊C4—22no—
431C22⋊C4—22no—
432C22⋊C4—2441no—
433C42—22no—
434C42—22no—
435C42—2441no—
436C42C2×C22⋊D6111yescharacteristic
437C42C2×C22⋊D6111yescharacteristic
438C42C2×C22⋊D6111yescharacteristic
439C42C23×D6111yescharacteristic
440C23—24no—
441C23—22no—
442C23—42no—
443C23—44no—
444C23—44no—
445C23—42no—
446C23—44no—
447C23—42no—
448C23#1451974611yescharacteristic
449Q8—22no—
450Q8—24no—
451Q8—2442no—
452Q8—2442no—
453Q8C2×#1175820014yes—
454D4—22no—
455D4—22no—
456D4—44no—
457D4—42no—
458D4—44no—
459D4—416no—
460D4—42no—
461D4—42no—
462D4—44no—
463D4—44no—
464D4—82no—
465D4—88no—
466C2×C4—22no—
467C2×C4—24no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—41no—
472C2×C4—416no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—44no—
478C2×C4—2441no—
479C2×C4—2441no—
480C2×C4#1451974611yescharacteristic
481C2×C4#1451974611yescharacteristic
482C2×C4#1451974611yescharacteristic
483C2×C4C2×#1175820012yes—
484C2×C4#1451980312yes—
485C2×C4C22×C22⋊D6111yescharacteristic
486C2×C4C22×C22⋊D6111yescharacteristic
487C2×C4C22×C22⋊D6111yescharacteristic
488C8—2442no—
489C8—2442no—
490C8—2442no—
491C22—21no—
492C22—21no—
493C22—44no—
494C22—42no—
495C22—44no—
496C22—42no—
497C22—42no—
498C22—44no—
499C22—88no—
500C22—84no—
501C22#1683991312yes—
502C22C2×#1451974611yescharacteristic
503C4—21no—
504C4—21no—
505C4—24no—
506C4—21no—
507C4—42no—
508C4—42no—
509C4—4881no—
510C4—4881no—
511C4#1683991312yes—
512C4C22×#1175820011yescharacteristic
513C4C2×#1451980311yescharacteristic
514C4#1684056012yes—
515C2—22no—
516C2—84no—
517C2—82no—
518C2C2×#1683991311yescharacteristic
519C2#2145506111yescharacteristic
520C2#2145517611yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
Fitting, prime radicalC2×C61×#550C2
socleC22×C61C2×#120
hypercenterC2×#550D61
hypocenterC61#18767
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#550

Complements

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

NH ≅ G/Ntype
1. #21456760C2semidirect
2. #21455827C2semidirect
3. #21455624C2semidirect
4. #21445019C2semidirect
5. #21444946C2semidirect
6. #21444487C2semidirect
7. #21444244C2semidirect
8. #21444214C2semidirect
11. #16840134C22semidirect
12. #16840108C22semidirect
15. #16840057C22semidirect
16. #16840052C22semidirect
17. #16840047C22semidirect
18. #16839967C22semidirect
28. #16839506C22semidirect
29. Q8⋊#92418C22semidirect
30. D4⋊#92418C22semidirect
117. C2×Q8×C61SD16semidirect
303. C61#18767semidirect
380. C2×Q8#11758200semidirect

Maximal subgroups

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

subgroupindex
#214567602
#214558272
#214556242
#214450192
#214449462
#214444872
#214442442
#214442142
C2×C61×#5502
#1876761