Finite Group Databasebeta

← #29657357

Subgroups — #29657357

Order 15616.

1002 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459638C211yesmaximal, characteristic, complemented
2#21456760C211yesmaximal, characteristic, complemented
3C2×#16840108C211yesmaximal, characteristic, complemented
4#21444921C214yesmaximal, complemented
5#21444914C214yesmaximal, complemented
6#21444505C211yesmaximal, characteristic, complemented
7#21444273C211yesmaximal, characteristic, complemented
8#21444201C211yesmaximal, characteristic, complemented
9C61×#3038C211yesmaximal, characteristic
10Q8×Dic122—21no—
11#16840491C2212yescomplemented
12#16840469—22no—
13#16840469—21no—
14D4×Dic122—22no—
15D4×Dic122C2212yescomplemented
16#16840390—22no—
17#16840390—21no—
18#16840319—21no—
19#16840319C2211yescharacteristic
20C2×#14519777—21no—
21C2×#14519777C2211yescharacteristic
22#16840245—21no—
23#16840134—21no—
24#16840108—21no—
25#16840108—22no—
26#16840108C2214yes—
27#16839959C2214yes—
28#16839913—22no—
29#16839815C2211yescharacteristic
30C2×#14519698—22no—
31C2×C4×#92418C2211yescharacteristic
32#16839507C2214yescomplemented
33Q8⋊#92418C2212yescomplemented
34#16839492C2214yescomplemented
35D4⋊#92418C2212yescomplemented
36C2×AGL(1,ℤ/8)×C61—21no—
37C2×#127×C61C2211yescharacteristic
38C4×#41×C61C2211yescharacteristic
39C61×#460—21no—
40C61×#460C2212yes—
41C61×#459—21no—
42C61×#459C2212yes—
43C61×#423C2211yescharacteristic
44C61×#418—21no—
45C22×Dic122—21no—
46C22×Dic122—22no—
47C2×#11758200—24no—
48C22×#92418—21no—
49#14519777—22no—
50#14519777—22no—
51#14519777—22no—
52#14519777—21no—
53#14519777—21no—
54#14519777D411yescharacteristic
55#14519777D412yes—
56#14519777D411yescharacteristic
57#14519777C2312yes—
58#14519776—24no—
59#14519776—22no—
60#14519769—22no—
61#14519769—21no—
62#14519768—22no—
63#14519768—24no—
64#14519756—22no—
65#14519756—21no—
66#14519756—22no—
67#14519756D412yes—
68#14519754—24no—
69#14519754—24no—
70#14519754—24no—
71Q8×Dic61—22no—
72C4×Dic122—22no—
73C4×Dic122—21no—
74C4×Dic122—22no—
75C4×Dic122C2312yes—
76D4×Dic61—24no—
77D4×Dic61—22no—
78#14519736—22no—
79#14519736—21no—
80C2×C4⋊Dic61—22no—
81C2×C4⋊Dic61—22no—
82C2×C4⋊Dic61—21no—
83C2×C4⋊Dic61D412yes—
84C4⋊#92418C2312yes—
85#14519698—24no—
86#14519698—28no—
87C22⋊#92418—21no—
88C4×#92418—21no—
89C4×#92418—21no—
90C4×#92418C2312yes—
91C2×#41×C61D411yescharacteristic
92C22×D4×C61—21no—
93C22×D4×C61—21no—
94C22×D4×C61—21no—
95AGL(1,ℤ/8)×C61—44no—
96C2×SD16×C61—22no—
97C2×D8×C61—22no—
98C2×M16×C61—21no—
99#127×C61—21no—
100#127×C61—21no—
101#127×C61—21no—
102#127×C61C2312yes—
103C4×Q8×C61D411yescharacteristic
104C4×Q8×C61C2311yescharacteristic
105C4×D4×C61—21no—
106C4×D4×C61—21no—
107C4×D4×C61D411yescharacteristic
108C4×D4×C61C2311yescharacteristic
109#117×C61—21no—
110#117×C61D411yescharacteristic
111C2×C42×C61—21no—
112C2×C42×C61C2311yescharacteristic
113C4⋊C8×C61—21no—
114C4⋊C8×C61C2312yes—
115#102×C61—22no—
116#102×C61—22no—
117C24×C61—41no—
118#41×C61—22no—
119#41×C61SD1614yescomplemented
120C2×Q8×C61C2×D411yescharacteristic
121C2×D4×C61—21no—
122C2×D4×C61—22no—
123C2×D4×C61—22no—
124C2×D4×C61—22no—
125C2×D4×C61—44no—
126C2×D4×C61—41no—
127C2×D4×C61—41no—
128C2×D4×C61—41no—
129C2×D4×C61—42no—
130C2×D4×C61—42no—
131C2×D4×C61—82no—
132C2×D4×C61C2×D411yescharacteristic
133C22×C4×C61—21no—
134C22×C4×C61—21no—
135C22×C4×C61—21no—
136C22×C4×C61C2×D411yescharacteristic
137C22×C4×C61C2×D411yescharacteristic
138C22×C4×C61C2×D411yescharacteristic
139C22×Dic61—24no—
140C22×Dic61—22no—
141C2×Dic122—21no—
142C2×Dic122—22no—
143C2×Dic122—24no—
144C2×Dic122—22no—
145C2×Dic122—84no—
146C2×Dic122—82no—
147C2×Dic122C2×D412yes—
148SD16×C61—44no—
149D8×C61—44no—
150M16×C61—42no—
151C2×C8×C61—22no—
152C4⋊C4×C61—21no—
153C4⋊C4×C61—21no—
154C4⋊C4×C61C2×D411yescharacteristic
155C4⋊C4×C61C2×D411yescharacteristic
156C22⋊C4×C61—21no—
157C22⋊C4×C61—21no—
158C22⋊C4×C61—21no—
159C42×C61—21no—
160C42×C61—21no—
161C42×C61—21no—
162C42×C61C2×D411yescharacteristic
163C42×C61C2×D411yescharacteristic
164C42×C61C2×D411yescharacteristic
165C42×C61C2411yescharacteristic
166C22⋊Dic61—24no—
167C22⋊Dic61—22no—
168#11758200—84no—
169C4⋊Dic61—24no—
170C4⋊Dic61—24no—
171C4⋊Dic61—21no—
172C4⋊Dic61—22no—
173C4⋊Dic61—21no—
174C4⋊Dic61—22no—
175C4⋊Dic61—21no—
176C4⋊Dic61—22no—
177C4⋊Dic61C2×D414yes—
178C4⋊Dic61C2×D412yes—
179#11758196—24no—
180#11758196—24no—
181#11758196—22no—
182#11758196—22no—
183C4×Dic61—24no—
184C4×Dic61—22no—
185C2×#92418—22no—
186C2×#92418—22no—
187C2×#92418—42no—
188C23×C61—21no—
189C23×C61—42no—
190C23×C61—82no—
191C23×C61—82no—
192C23×C61—82no—
193C23×C61#12011yescharacteristic
194Q8×C61—21no—
195Q8×C61C2×SD1612yes—
196D4×C61—24no—
197D4×C61—21no—
198D4×C61—44no—
199D4×C61—42no—
200D4×C61—42no—
201D4×C61—44no—
202D4×C61—82no—
203D4×C61—82no—
204D4×C61—82no—
205D4×C61C2×SD1612yes—
206C2×C4×C61—21no—
207C2×C4×C61—22no—
208C2×C4×C61—24no—
209C2×C4×C61—22no—
210C2×C4×C61—21no—
211C2×C4×C61—21no—
212C2×C4×C61—42no—
213C2×C4×C61—42no—
214C2×C4×C61#12011yescharacteristic
215C2×C4×C61#12011yescharacteristic
216C2×C4×C61#12011yescharacteristic
217C2×C4×C61C2×SD1612yes—
218C2×C4×C61#13712yes—
219C2×C4×C61C22×D411yescharacteristic
220C2×C4×C61C22×D411yescharacteristic
221C2×C4×C61C22×D411yescharacteristic
222C2×Dic61—22no—
223C2×Dic61—22no—
224C2×Dic61—24no—
225C2×Dic61—82no—
226C2×Dic61—84no—
227Dic122—24no—
228Dic122—44no—
229Dic122—42no—
230Dic122—82no—
231Dic122—84no—
232C8×C61—42no—
233#92418—42no—
234#92418—42no—
235#18642—611nomaximal, Sylow
236C22×C61—21no—
237C22×C61—21no—
238C22×C61—22no—
239C22×C61—41no—
240C22×C61—41no—
241C22×C61—82no—
242C22×C61—82no—
243C22×C61—82no—
244C22×C61—164no—
245C22×C61#44812yes—
246C22×C61C2×#12011yescharacteristic
247C4×C61—22no—
248C4×C61—21no—
249C4×C61—21no—
250C4×C61—41no—
251C4×C61—41no—
252C4×C61#44812yes—
253C4×C61#45412yes—
254C4×C61C22×SD1611yescharacteristic
255C4×C61C2×#13711yescharacteristic
256C4×C61#57612yes—
257Dic61—42no—
258Dic61—44no—
259Dic61—42no—
260#3503—611no—
261#3208—611no—
262C2×#486—611no—
263#3038D6111yescharacteristic
264#1627—614no—
265#1626—614no—
266#1549—611no—
267#1493—611no—
268#1476—611no—
269C2×C61—22no—
270C2×C61—42no—
271C2×C61—162no—
272C2×C61C2×#44811yescharacteristic
273C2×C61#300811yescharacteristic
274C2×C61C2×#45411yescharacteristic
275C2×AGL(1,ℤ/8)—21no—
276Q82—1221no—
277#557—612no—
278#554—1222no—
279#554—1221no—
280D4×Q8—612no—
281D4×Q8—1222no—
282#544—1222no—
283#544—1221no—
284#533—611no—
285#533—1221no—
286C2×C4⋊Q8—611no—
287C2×C4⋊Q8—1221no—
288C2×#127C2×D6111yescharacteristic
289#520—1221no—
290C4×#41C2×D6111yescharacteristic
291#492—1221no—
292#486—614no—
293#486—1222no—
294#486—1221no—
295#460—21no—
296#460—614no—
297#460C2×D6112yes—
298#459—21no—
299#459C2×D6112yes—
300#450—1222no—
301#431—611no—
302#423C2×D6111yescharacteristic
303#418—21no—
304C2×#102—1222no—
305C2×C4×C8—611no—
306#332—614no—
307Q8⋊C8—612no—
308Q8⋊C8—614no—
309D4⋊C8—612no—
310C61#1864211yescharacteristic, complemented, Sylow
311C2×#41C22⋊D6111yescharacteristic
312C22×Q8—1222no—
313C22×Q8—1221no—
314C22×D4—21no—
315C22×D4—21no—
316C22×D4—21no—
317AGL(1,ℤ/8)—44no—
318C2×SD16—22no—
319C2×SD16—1224no—
320C2×D8—22no—
321C2×M16—21no—
322C22×C8—1221no—
323C4⋊Q8—612no—
324C4⋊Q8—611no—
325C4⋊Q8—611no—
326C4⋊Q8—612no—
327C4⋊Q8—1222no—
328C4⋊Q8—1222no—
329C4⋊Q8—1222no—
330C4⋊Q8—1224no—
331C4⋊Q8—1222no—
332C4⋊Q8—1221no—
333C4⋊Q8—1221no—
334#127—21no—
335#127—21no—
336#127—21no—
337#127C22×D6112yes—
338#125—1222no—
339#125—1224no—
340#125—1222no—
341#125—1221no—
342C22⋊Q8—612no—
343C22⋊Q8—1224no—
344C22⋊Q8—1224no—
345C22⋊Q8—1224no—
346C22⋊Q8—1222no—
347C22⋊Q8—1222no—
348C22⋊Q8—1221no—
349C4×Q8—612no—
350C4×Q8—1222no—
351C4×Q8—1222no—
352C4×Q8—1222no—
353C4×Q8—1221no—
354C4×Q8C22⋊D6111yescharacteristic
355C4×Q8C22×D6111yescharacteristic
356C4×D4—21no—
357C4×D4—21no—
358C4×D4—1224no—
359C4×D4—1222no—
360C4×D4C22⋊D6111yescharacteristic
361C4×D4C22×D6111yescharacteristic
362#117—21no—
363#117—1222no—
364#117—1221no—
365#117C22⋊D6111yescharacteristic
366C2×C4⋊C4—612no—
367C2×C4⋊C4—1222no—
368C2×C4⋊C4—1222no—
369C2×C4⋊C4—1221no—
370C2×C42—21no—
371C2×C42C22×D6111yescharacteristic
372C4⋊C8—21no—
373C4⋊C8—612no—
374C4⋊C8C22×D6112yes—
375#102—22no—
376#102—22no—
377#102—1228no—
378#102—1224no—
379C22⋊C8—1221no—
380C4×C8—612no—
381C4×C8—1221no—
382C4×C8—1221no—
383C24—41no—
384#41—22no—
385#41#1175820014yescomplemented
386C2×Q8—612no—
387C2×Q8—1224no—
388C2×Q8—1222no—
389C2×Q8—1222no—
390C2×Q8—1221no—
391C2×Q8—4882no—
392C2×Q8—4884no—
393C2×Q8C2×C22⋊D6111yescharacteristic
394C2×D4—21no—
395C2×D4—22no—
396C2×D4—22no—
397C2×D4—22no—
398C2×D4—44no—
399C2×D4—41no—
400C2×D4—41no—
401C2×D4—41no—
402C2×D4—42no—
403C2×D4—42no—
404C2×D4—82no—
405C2×D4C2×C22⋊D6111yescharacteristic
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—1224no—
410C22×C4—1222no—
411C22×C4D4×D6111yescharacteristic
412C22×C4D4×D6111yescharacteristic
413C22×C4C2×C22⋊D6111yescharacteristic
414SD16—44no—
415SD16—4884no—
416D8—44no—
417M16—42no—
418C2×C8—22no—
419C2×C8—1222no—
420C2×C8—1222no—
421C2×C8—2442no—
422C4⋊C4—21no—
423C4⋊C4—21no—
424C4⋊C4—614no—
425C4⋊C4—612no—
426C4⋊C4—1224no—
427C4⋊C4—1224no—
428C4⋊C4—1224no—
429C4⋊C4—1224no—
430C4⋊C4—1222no—
431C4⋊C4—1222no—
432C4⋊C4—1222no—
433C4⋊C4—1222no—
434C4⋊C4—1221no—
435C4⋊C4—1221no—
436C4⋊C4—1222no—
437C4⋊C4—1221no—
438C4⋊C4C2×C22⋊D6111yescharacteristic
439C4⋊C4C2×C22⋊D6111yescharacteristic
440C22⋊C4—21no—
441C22⋊C4—21no—
442C22⋊C4—21no—
443C22⋊C4—1224no—
444C22⋊C4—1222no—
445C42—21no—
446C42—21no—
447C42—21no—
448C42—1224no—
449C42—1222no—
450C42D4×D6111yescharacteristic
451C42D4×D6111yescharacteristic
452C42C2×C22⋊D6111yescharacteristic
453C42C23×D6111yescharacteristic
454C23—21no—
455C23—42no—
456C23—82no—
457C23—82no—
458C23—82no—
459C23#1451974511yescharacteristic
460Q8—21no—
461Q8—1224no—
462Q8—4882no—
463Q8—4884no—
464Q8—2444no—
465Q8—2442no—
466Q8C2×#1175820012yes—
467D4—24no—
468D4—21no—
469D4—44no—
470D4—42no—
471D4—42no—
472D4—44no—
473D4—82no—
474D4—82no—
475D4—82no—
476D4C2×#1175820012yes—
477C2×C4—21no—
478C2×C4—22no—
479C2×C4—24no—
480C2×C4—22no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—1222no—
486C2×C4—1224no—
487C2×C4—1222no—
488C2×C4—4882no—
489C2×C4—4884no—
490C2×C4#1451974511yescharacteristic
491C2×C4#1451974511yescharacteristic
492C2×C4#1451974511yescharacteristic
493C2×C4C2×#1175820012yes—
494C2×C4#1451980712yes—
495C2×C4C2×D4×D6111yescharacteristic
496C2×C4C2×D4×D6111yescharacteristic
497C2×C4C22×C22⋊D6111yescharacteristic
498C8—42no—
499C8—2442no—
500C8—2442no—
501C22—21no—
502C22—21no—
503C22—22no—
504C22—41no—
505C22—41no—
506C22—82no—
507C22—82no—
508C22—82no—
509C22—164no—
510C22#1683990012yes—
511C22C2×#1451974511yescharacteristic
512C4—21no—
513C4—22no—
514C4—21no—
515C4—41no—
516C4—41no—
517C4—2442no—
518C4—2442no—
519C4—2444no—
520C4#1683990012yes—
521C4#1683993312yes—
522C4C22×#1175820011yescharacteristic
523C4C2×#1451980711yescharacteristic
524C4#1684056612yes—
525C2—22no—
526C2—42no—
527C2—162no—
528C2C2×#1683990011yescharacteristic
529C2#2145508611yescharacteristic
530C2C2×#1683993311yescharacteristic
subgroupG/H
centerC22C2×#14519745
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
Fitting, prime radicalC61×#3038C2
socleC22×C61C2×#120
hypercenter#3038D61
hypocenterC61#18642
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3038

Complements

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

NH ≅ G/Ntype
1. #21459638C2semidirect
2. #21456760C2semidirect
3. C2×#16840108C2semidirect
4. #21444921C2semidirect
5. #21444914C2semidirect
6. #21444505C2semidirect
7. #21444273C2semidirect
8. #21444201C2semidirect
11. #16840491C22semidirect
15. D4×Dic122C22semidirect
32. #16839507C22semidirect
33. Q8⋊#92418C22semidirect
34. #16839492C22semidirect
35. D4⋊#92418C22semidirect
119. #41×C61SD16semidirect
310. C61#18642semidirect
385. #41#11758200semidirect

Maximal subgroups

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

subgroupindex
#214596382
#214567602
C2×#168401082
#214449212
#214449142
#214445052
#214442732
#214442012
C61×#30382
#1864261