Finite Group Databasebeta

← #29657153

Subgroups — #29657153

Order 15616.

1066 conjugacy classes of subgroups.

The subgroup lattice has 536 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×#16840134C211yesmaximal, characteristic, complemented
4#21444896C214yesmaximal, complemented
5#21444889C214yesmaximal, complemented
6#21444501C211yesmaximal, characteristic, complemented
7#21444273C211yesmaximal, characteristic, complemented
8#21444201C211yesmaximal, characteristic, complemented
9C61×#3038C211yesmaximal, characteristic
10C22×#11758200—22no—
11Q8×Dic122C2212yescomplemented
12#16840491—21no—
13#16840469—22no—
14#16840469C2212yescomplemented
15D4×Dic122—21no—
16D4×Dic122—22no—
17#16840390—22no—
18#16840390—21no—
19#16840319—21no—
20#16840319C2211yescharacteristic
21C2×#14519777—21no—
22C2×#14519777C2211yescharacteristic
23#16840245—21no—
24#16840134—22no—
25#16840134—21no—
26#16840134C2214yes—
27#16840108—21no—
28#16839959C2214yes—
29#16839913—22no—
30#16839815C2211yescharacteristic
31C2×C4×#92418C2211yescharacteristic
32#16839510C2214yescomplemented
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—22no—
46C22×Dic122D412yes—
47C2×#11758200—28no—
48C2×#11758200—24no—
49C2×#11758200—48no—
50C22×#92418—21no—
51#14519777—22no—
52#14519777—22no—
53#14519777—21no—
54#14519777—21no—
55#14519777—22no—
56#14519777D412yes—
57#14519777D411yescharacteristic
58#14519777D411yescharacteristic
59#14519777C2312yes—
60#14519776—24no—
61#14519776—22no—
62#14519769—22no—
63#14519769—21no—
64#14519768—24no—
65#14519768—22no—
66#14519756—22no—
67#14519756—21no—
68#14519756—22no—
69#14519756D412yes—
70#14519754—24no—
71#14519754—24no—
72#14519754—24no—
73Q8×Dic61—22no—
74C4×Dic122—22no—
75C4×Dic122—22no—
76C4×Dic122—21no—
77C4×Dic122C2312yes—
78D4×Dic61—24no—
79D4×Dic61—22no—
80#14519736—22no—
81#14519736—21no—
82C2×C4⋊Dic61—22no—
83C2×C4⋊Dic61—21no—
84C2×C4⋊Dic61—21no—
85C2×C4⋊Dic61—22no—
86C4⋊#92418C2312yes—
87#14519698—24no—
88C22⋊#92418—21no—
89C4×#92418—21no—
90C4×#92418—21no—
91C4×#92418C2312yes—
92C2×#41×C61D411yescharacteristic
93C22×D4×C61—21no—
94C22×D4×C61—21no—
95C22×D4×C61—21no—
96AGL(1,ℤ/8)×C61—44no—
97C2×SD16×C61—22no—
98C2×D8×C61—22no—
99C2×M16×C61—21no—
100#127×C61—21no—
101#127×C61—21no—
102#127×C61—21no—
103#127×C61C2312yes—
104C4×Q8×C61D411yescharacteristic
105C4×Q8×C61C2311yescharacteristic
106C4×D4×C61—21no—
107C4×D4×C61—21no—
108C4×D4×C61D411yescharacteristic
109C4×D4×C61C2311yescharacteristic
110#117×C61—21no—
111#117×C61D411yescharacteristic
112C2×C42×C61—21no—
113C2×C42×C61C2311yescharacteristic
114C4⋊C8×C61—21no—
115C4⋊C8×C61C2312yes—
116#102×C61—22no—
117#102×C61—22no—
118C24×C61—41no—
119#41×C61—22no—
120#41×C61SD1614yescomplemented
121C2×Q8×C61C2×D411yescharacteristic
122C2×D4×C61—21no—
123C2×D4×C61—22no—
124C2×D4×C61—22no—
125C2×D4×C61—22no—
126C2×D4×C61—44no—
127C2×D4×C61—41no—
128C2×D4×C61—41no—
129C2×D4×C61—41no—
130C2×D4×C61—42no—
131C2×D4×C61—42no—
132C2×D4×C61—82no—
133C2×D4×C61C2×D411yescharacteristic
134C22×C4×C61—21no—
135C22×C4×C61—21no—
136C22×C4×C61—21no—
137C22×C4×C61C2×D411yescharacteristic
138C22×C4×C61C2×D411yescharacteristic
139C22×C4×C61C2×D411yescharacteristic
140C22×Dic61—24no—
141C22×Dic61—22no—
142C2×Dic122—21no—
143C2×Dic122—22no—
144C2×Dic122—24no—
145C2×Dic122—44no—
146C2×Dic122—84no—
147C2×Dic122C2×D412yes—
148C2×Dic122C2×D414yes—
149SD16×C61—44no—
150D8×C61—44no—
151M16×C61—42no—
152C2×C8×C61—22no—
153C4⋊C4×C61—21no—
154C4⋊C4×C61—21no—
155C4⋊C4×C61C2×D411yescharacteristic
156C4⋊C4×C61C2×D411yescharacteristic
157C22⋊C4×C61—21no—
158C22⋊C4×C61—21no—
159C22⋊C4×C61—21no—
160C42×C61—21no—
161C42×C61—21no—
162C42×C61—21no—
163C42×C61C2×D411yescharacteristic
164C42×C61C2×D411yescharacteristic
165C42×C61C2×D411yescharacteristic
166C42×C61C2411yescharacteristic
167C22⋊Dic61—24no—
168C22⋊Dic61—22no—
169#11758200—416no—
170#11758200—84no—
171C4⋊Dic61—24no—
172C4⋊Dic61—24no—
173C4⋊Dic61—22no—
174C4⋊Dic61—21no—
175C4⋊Dic61—22no—
176C4⋊Dic61—21no—
177C4⋊Dic61—21no—
178C4⋊Dic61—22no—
179C4⋊Dic61—22no—
180C4⋊Dic61C2×D412yes—
181#11758196—24no—
182#11758196—24no—
183#11758196—22no—
184#11758196—22no—
185C4×Dic61—24no—
186C4×Dic61—22no—
187C2×#92418—24no—
188C2×#92418—22no—
189C2×#92418—22no—
190C23×C61—21no—
191C23×C61—42no—
192C23×C61—82no—
193C23×C61—82no—
194C23×C61—82no—
195C23×C61#12011yescharacteristic
196Q8×C61—21no—
197Q8×C61C2×SD1612yescomplemented
198D4×C61—24no—
199D4×C61—21no—
200D4×C61—44no—
201D4×C61—42no—
202D4×C61—42no—
203D4×C61—44no—
204D4×C61—82no—
205D4×C61—82no—
206D4×C61—82no—
207D4×C61C2×SD1612yescomplemented
208C2×C4×C61—21no—
209C2×C4×C61—22no—
210C2×C4×C61—24no—
211C2×C4×C61—22no—
212C2×C4×C61—21no—
213C2×C4×C61—21no—
214C2×C4×C61—42no—
215C2×C4×C61—42no—
216C2×C4×C61#12011yescharacteristic
217C2×C4×C61#12011yescharacteristic
218C2×C4×C61#12011yescharacteristic
219C2×C4×C61C2×SD1612yes—
220C2×C4×C61#13712yes—
221C2×C4×C61C22×D411yescharacteristic
222C2×C4×C61C22×D411yescharacteristic
223C2×C4×C61C22×D411yescharacteristic
224C2×Dic61—22no—
225C2×Dic61—24no—
226C2×Dic61—22no—
227C2×Dic61—82no—
228C2×Dic61—84no—
229Dic122—24no—
230Dic122—44no—
231Dic122—42no—
232Dic122—44no—
233Dic122—84no—
234C8×C61—42no—
235#92418—24no—
236#92418—42no—
237#18622—611nomaximal, Sylow
238C22×C61—21no—
239C22×C61—21no—
240C22×C61—22no—
241C22×C61—41no—
242C22×C61—41no—
243C22×C61—82no—
244C22×C61—82no—
245C22×C61—82no—
246C22×C61—164no—
247C22×C61#44812yes—
248C22×C61C2×#12011yescharacteristic
249C4×C61—22no—
250C4×C61—21no—
251C4×C61—21no—
252C4×C61—41no—
253C4×C61—41no—
254C4×C61#44812yes—
255C4×C61#45312yes—
256C4×C61C22×SD1611yescharacteristic
257C4×C61C2×#13711yescharacteristic
258C4×C61#57712yes—
259Dic61—42no—
260Dic61—44no—
261Dic61—42no—
262#3503—611no—
263#3208—611no—
264C2×#492—611no—
265#3038D6111yescharacteristic
266#1623—614no—
267#1622—614no—
268#1548—611no—
269#1493—611no—
270#1476—611no—
271C2×C61—22no—
272C2×C61—42no—
273C2×C61—162no—
274C2×C61C2×#44811yescharacteristic
275C2×C61#301011yescharacteristic
276C2×C61C2×#45311yescharacteristic
277C2×AGL(1,ℤ/8)—21no—
278C22×SD16—1222no—
279Q82—612no—
280#557—1221no—
281#554—612no—
282#554—1222no—
283D4×Q8—1222no—
284D4×Q8—1221no—
285#544—1222no—
286#544—1221no—
287#533—611no—
288#533—1221no—
289C2×C4⋊Q8—611no—
290C2×C4⋊Q8—1221no—
291C2×#127C2×D6111yescharacteristic
292#520—1221no—
293C4×#41C2×D6111yescharacteristic
294#492—614no—
295#492—1222no—
296#492—1221no—
297#486—1221no—
298#460—21no—
299#460—614no—
300#460C2×D6112yes—
301#459—21no—
302#459C2×D6112yes—
303#450—1222no—
304#431—611no—
305#423C2×D6111yescharacteristic
306#418—21no—
307C2×C4×C8—611no—
308#333—614no—
309Q8⋊C8—612no—
310Q8⋊C8—614no—
311D4⋊C8—612no—
312C61#1862211yescharacteristic, complemented, Sylow
313C2×#41C22⋊D6111yescharacteristic
314C22×Q8—612no—
315C22×Q8—1222no—
316C22×D4—21no—
317C22×D4—21no—
318C22×D4—21no—
319AGL(1,ℤ/8)—44no—
320C2×SD16—22no—
321C2×SD16—1224no—
322C2×SD16—1228no—
323C2×SD16—2448no—
324C2×D8—22no—
325C2×M16—21no—
326C22×C8—1221no—
327C4⋊Q8—611no—
328C4⋊Q8—612no—
329C4⋊Q8—611no—
330C4⋊Q8—612no—
331C4⋊Q8—1222no—
332C4⋊Q8—1222no—
333C4⋊Q8—1222no—
334C4⋊Q8—1224no—
335C4⋊Q8—1222no—
336C4⋊Q8—1221no—
337C4⋊Q8—1221no—
338#127—21no—
339#127—21no—
340#127—21no—
341#127C22×D6112yes—
342#125—1222no—
343#125—1224no—
344#125—1222no—
345#125—1221no—
346C22⋊Q8—612no—
347C22⋊Q8—1224no—
348C22⋊Q8—1224no—
349C22⋊Q8—1224no—
350C22⋊Q8—1222no—
351C22⋊Q8—1222no—
352C22⋊Q8—1221no—
353C4×Q8—612no—
354C4×Q8—1222no—
355C4×Q8—1222no—
356C4×Q8—1222no—
357C4×Q8—1221no—
358C4×Q8C22⋊D6111yescharacteristic
359C4×Q8C22×D6111yescharacteristic
360C4×D4—21no—
361C4×D4—21no—
362C4×D4—1224no—
363C4×D4—1222no—
364C4×D4C22⋊D6111yescharacteristic
365C4×D4C22×D6111yescharacteristic
366#117—21no—
367#117—1222no—
368#117—1221no—
369#117C22⋊D6111yescharacteristic
370C2×C4⋊C4—1222no—
371C2×C4⋊C4—1222no—
372C2×C4⋊C4—1221no—
373C2×C4⋊C4—1221no—
374C2×C42—21no—
375C2×C42C22×D6111yescharacteristic
376C4⋊C8—21no—
377C4⋊C8—612no—
378C4⋊C8C22×D6112yes—
379#102—22no—
380#102—22no—
381#102—1224no—
382C22⋊C8—1221no—
383C4×C8—612no—
384C4×C8—1221no—
385C4×C8—1221no—
386C24—41no—
387#41—22no—
388#41#1175820014yescomplemented
389C2×Q8—612no—
390C2×Q8—614no—
391C2×Q8—1224no—
392C2×Q8—1221no—
393C2×Q8—1222no—
394C2×Q8—4884no—
395C2×Q8—2444no—
396C2×Q8C2×C22⋊D6111yescharacteristic
397C2×D4—21no—
398C2×D4—22no—
399C2×D4—22no—
400C2×D4—22no—
401C2×D4—44no—
402C2×D4—41no—
403C2×D4—41no—
404C2×D4—41no—
405C2×D4—42no—
406C2×D4—42no—
407C2×D4—82no—
408C2×D4C2×C22⋊D6111yescharacteristic
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—1224no—
413C22×C4—1222no—
414C22×C4D4×D6111yescharacteristic
415C22×C4D4×D6111yescharacteristic
416C22×C4C2×C22⋊D6111yescharacteristic
417SD16—44no—
418SD16—4884no—
419SD16—24416no—
420D8—44no—
421M16—42no—
422C2×C8—22no—
423C2×C8—1224no—
424C2×C8—1222no—
425C2×C8—1222no—
426C4⋊C4—21no—
427C4⋊C4—21no—
428C4⋊C4—612no—
429C4⋊C4—1224no—
430C4⋊C4—1224no—
431C4⋊C4—1224no—
432C4⋊C4—1224no—
433C4⋊C4—1222no—
434C4⋊C4—1222no—
435C4⋊C4—1222no—
436C4⋊C4—1221no—
437C4⋊C4—1222no—
438C4⋊C4—1222no—
439C4⋊C4—1222no—
440C4⋊C4—1221no—
441C4⋊C4—1221no—
442C4⋊C4C2×C22⋊D6111yescharacteristic
443C4⋊C4C2×C22⋊D6111yescharacteristic
444C22⋊C4—21no—
445C22⋊C4—21no—
446C22⋊C4—21no—
447C22⋊C4—1224no—
448C22⋊C4—1222no—
449C42—21no—
450C42—21no—
451C42—21no—
452C42—1224no—
453C42—1222no—
454C42D4×D6111yescharacteristic
455C42D4×D6111yescharacteristic
456C42C2×C22⋊D6111yescharacteristic
457C42C23×D6111yescharacteristic
458C23—21no—
459C23—42no—
460C23—82no—
461C23—82no—
462C23—82no—
463C23#1451974511yescharacteristic
464Q8—21no—
465Q8—1224no—
466Q8—4884no—
467Q8—2444no—
468Q8—2442no—
469Q8—2444no—
470Q8C2×#1175820012yescomplemented
471D4—24no—
472D4—21no—
473D4—44no—
474D4—42no—
475D4—42no—
476D4—44no—
477D4—82no—
478D4—82no—
479D4—82no—
480D4C2×#1175820012yescomplemented
481C2×C4—21no—
482C2×C4—22no—
483C2×C4—24no—
484C2×C4—22no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—1222no—
490C2×C4—1222no—
491C2×C4—1224no—
492C2×C4—4882no—
493C2×C4—4884no—
494C2×C4#1451974511yescharacteristic
495C2×C4#1451974511yescharacteristic
496C2×C4#1451974511yescharacteristic
497C2×C4C2×#1175820012yes—
498C2×C4#1451980712yes—
499C2×C4C2×D4×D6111yescharacteristic
500C2×C4C2×D4×D6111yescharacteristic
501C2×C4C22×C22⋊D6111yescharacteristic
502C8—42no—
503C8—1224no—
504C8—2442no—
505C22—21no—
506C22—21no—
507C22—22no—
508C22—41no—
509C22—41no—
510C22—82no—
511C22—82no—
512C22—82no—
513C22—164no—
514C22#1683990012yes—
515C22C2×#1451974511yescharacteristic
516C4—21no—
517C4—21no—
518C4—22no—
519C4—41no—
520C4—41no—
521C4—2442no—
522C4—2442no—
523C4—2444no—
524C4#1683990012yes—
525C4#1683992812yes—
526C4C22×#1175820011yescharacteristic
527C4C2×#1451980711yescharacteristic
528C4#1684057212yes—
529C2—22no—
530C2—42no—
531C2—162no—
532C2C2×#1683990011yescharacteristic
533C2#2145510411yescharacteristic
534C2C2×#1683992811yescharacteristic
subgroupG/H
centerC22C2×#14519745
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
Fitting, prime radicalC61×#3038C2
socleC22×C61C2×#120
hypercenter#3038D61
hypocenterC61#18622
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×#16840134C2semidirect
4. #21444896C2semidirect
5. #21444889C2semidirect
6. #21444501C2semidirect
7. #21444273C2semidirect
8. #21444201C2semidirect
11. Q8×Dic122C22semidirect
14. #16840469C22semidirect
32. #16839510C22semidirect
33. Q8⋊#92418C22semidirect
34. #16839492C22semidirect
35. D4⋊#92418C22semidirect
120. #41×C61SD16semidirect
197. Q8×C61C2×SD16semidirect
207. D4×C61C2×SD16semidirect
312. C61#18622semidirect
388. #41#11758200semidirect
470. Q8C2×#11758200semidirect
480. D4C2×#11758200semidirect

Maximal subgroups

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

subgroupindex
#214596382
#214567602
C2×#168401342
#214448962
#214448892
#214445012
#214442732
#214442012
C61×#30382
#1862261