Finite Group Databasebeta

← #29657605

Subgroups — #29657605

Order 15616.

926 conjugacy classes of subgroups.

The subgroup lattice has 542 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459622C211yesmaximal, characteristic
2#21457161C211yesmaximal, characteristic, complemented
3#21456401C211yesmaximal, characteristic, complemented
4#21445067C212yesmaximal, complemented
5#21445032C212yesmaximal, complemented
6#21444983C212yesmaximal, complemented
7#21444951C212yesmaximal, complemented
8#21444510C211yesmaximal, characteristic, complemented
9#21444273C211yesmaximal, characteristic, complemented
10C2×D4⋊#92418C211yesmaximal, characteristic
11C61×#3077C211yesmaximal, characteristic
12#16840471—22no—
13#16840471—22no—
14#16840469—21no—
15#16840469C2212yes—
16D4×Dic122—21no—
17D4×Dic122C2212yes—
18#16840393—21no—
19#16840388—22no—
20#16840319C2211yescharacteristic
21#16840319C2211yescharacteristic
22C2×#14519756—22no—
23#16840237—21no—
24#16840165C2211yescharacteristic, complemented
25#16840141C2211yescharacteristic, complemented
26#16840114C2211yescharacteristic, complemented
27#16840112C2211yescharacteristic, complemented
28#16840057—21no—
29#16840054C2212yescomplemented
30#16840001—22no—
31#16840001—22no—
32#16839972—21no—
33#16839969C2212yescomplemented
34C4×#11758202—22no—
35C2×C4⋊#92418C2211yescharacteristic
36C2×C4×#92418C2211yescharacteristic
37#16839510C2212yescomplemented
38#16839507C2212yescomplemented
39#16839492C2212yescomplemented
40#16839492C2212yescomplemented
41D4⋊#92418C2212yes—
42D4⋊#92418C2212yes—
43C61×#533C2211yescharacteristic
44C2×C4×D4×C61C2211yescharacteristic
45C61×#478—22no—
46C61×#476C2212yes—
47C61×#474C2212yes—
48C61×#428—21no—
49C61×#423C2211yescharacteristic
50C61×#418—21no—
51C22×Dic122—22no—
52C2×#11758202—22no—
53C2×#11758202—22no—
54C22×#92418—21no—
55C22×#92418—21no—
56#14519777D411yescharacteristic
57#14519777D411yescharacteristic
58#14519777C2311yescharacteristic
59#14519777C2311yescharacteristic
60#14519776—22no—
61#14519776—22no—
62#14519769—21no—
63#14519769—21no—
64#14519768—22no—
65#14519768—22no—
66#14519756—24no—
67#14519756—24no—
68#14519756—21no—
69#14519756—21no—
70#14519756—22no—
71#14519756—22no—
72#14519756D412yes—
73#14519756D412yes—
74#14519754—24no—
75#14519754—24no—
76#14519754—24no—
77#14519754—24no—
78C4×Dic122—21no—
79C4×Dic122—21no—
80C4×Dic122D412yes—
81C4×Dic122C2312yes—
82D4×Dic61—22no—
83D4×Dic61—22no—
84D4×Dic61—22no—
85D4×Dic61—22no—
86#14519736—21no—
87#14519736—21no—
88C2×C4⋊Dic61—22no—
89C2×#11758196—24no—
90C2×C22⋊Dic61—24no—
91#14519710—22no—
92#14519710—22no—
93C4⋊#92418—21no—
94C4⋊#92418C2312yes—
95#14519701—22no—
96#14519701—22no—
97#14519700—22no—
98#14519700—22no—
99C4×#92418—21no—
100C4×#92418C2311yescharacteristic
101C4×#92418C2311yescharacteristic
102C22×D4×C61D411yescharacteristic
103C23×C4×C61—21no—
104C2×M16×C61—21no—
105C4⋊Q8×C61C2311yescharacteristic
106C4⋊Q8×C61C2311yescharacteristic
107#125×C61—21no—
108C22⋊Q8×C61—21no—
109C22⋊Q8×C61—21no—
110C4×Q8×C61—21no—
111C4×Q8×C61—21no—
112C4×D4×C61—21no—
113C4×D4×C61—21no—
114C4×D4×C61D412yes—
115C4×D4×C61C2312yes—
116#117×C61—21no—
117C2×C4⋊C4×C61D411yescharacteristic
118C2×C22⋊C4×C61—21no—
119C2×C42×C61C2311yescharacteristic
120#107×C61—22no—
121#106×C61—22no—
122C4⋊C8×C61—21no—
123C4⋊C8×C61C2312yes—
124#102×C61—22no—
125#102×C61—22no—
126C24×C61—21no—
127C2×Q8×C61—21no—
128C2×Q8×C61—21no—
129C2×D4×C61—21no—
130C2×D4×C61—22no—
131C2×D4×C61Q1614yescomplemented
132C2×D4×C61C2×D412yes—
133C22×C4×C61—21no—
134C22×C4×C61—22no—
135C22×C4×C61—22no—
136C22×C4×C61—81no—
137C22×C4×C61—81no—
138C22×C4×C61C2×D411yescharacteristic
139C22×C4×C61C2×D411yescharacteristic
140C22×C4×C61C2×D411yescharacteristic
141C22×Dic61—24no—
142C22×Dic61—24no—
143C2×Dic122—21no—
144C2×Dic122—24no—
145C2×Dic122—21no—
146C2×Dic122—82no—
147C2×Dic122—82no—
148C2×Dic122C2×D412yes—
149C2×Dic122C2×D412yes—
150M16×C61—42no—
151C2×C8×C61—22no—
152C4⋊C4×C61—21no—
153C4⋊C4×C61—21no—
154C4⋊C4×C61—21no—
155C4⋊C4×C61—21no—
156C4⋊C4×C61—21no—
157C4⋊C4×C61—41no—
158C4⋊C4×C61—41no—
159C4⋊C4×C61C2×D412yes—
160C22⋊C4×C61—22no—
161C22⋊C4×C61—22no—
162C22⋊C4×C61—41no—
163C42×C61—41no—
164C42×C61C2×D411yescharacteristic
165C42×C61C2×D411yescharacteristic
166C42×C61C2×D411yescharacteristic
167C42×C61C2411yescharacteristic
168C22⋊Dic61—24no—
169C22⋊Dic61—24no—
170C22⋊Dic61—22no—
171C22⋊Dic61—22no—
172#11758202—44no—
173#11758202—82no—
174C4⋊Dic61—24no—
175C4⋊Dic61—21no—
176C4⋊Dic61—21no—
177C4⋊Dic61—21no—
178C4⋊Dic61—21no—
179C4⋊Dic61—21no—
180C4⋊Dic61—21no—
181C4⋊Dic61C2×D412yes—
182C4⋊Dic61C2×D412yes—
183#11758196—24no—
184#11758196—24no—
185#11758196—22no—
186#11758196—22no—
187#11758196—22no—
188#11758196—22no—
189C4×Dic61—22no—
190C4×Dic61—22no—
191C2×#92418—21no—
192C2×#92418—21no—
193C2×#92418—22no—
194C2×#92418—42no—
195C2×#92418—81no—
196C23×C61—22no—
197C23×C61—24no—
198C23×C61—81no—
199C23×C61—81no—
200C23×C61#12011yescharacteristic
201Q8×C61—42no—
202Q8×C61—41no—
203Q8×C61—41no—
204D4×C61—22no—
205D4×C61—24no—
206D4×C61C2×Q1614yes—
207C2×C4×C61—21no—
208C2×C4×C61—21no—
209C2×C4×C61—22no—
210C2×C4×C61—41no—
211C2×C4×C61—42no—
212C2×C4×C61—42no—
213C2×C4×C61—41no—
214C2×C4×C61—41no—
215C2×C4×C61—82no—
216C2×C4×C61—81no—
217C2×C4×C61#12011yescharacteristic
218C2×C4×C61#12011yescharacteristic
219C2×C4×C61#12011yescharacteristic
220C2×C4×C61C2×Q1612yes—
221C2×C4×C61#13712yes—
222C2×C4×C61C22×D411yescharacteristic
223C2×C4×C61C22×D411yescharacteristic
224C2×C4×C61C22×D411yescharacteristic
225C2×Dic61—22no—
226C2×Dic61—22no—
227C2×Dic61—22no—
228C2×Dic61—22no—
229C2×Dic61—84no—
230C2×Dic61—84no—
231Dic122—22no—
232Dic122—22no—
233Dic122—41no—
234Dic122—41no—
235Dic122—42no—
236Dic122—81no—
237Dic122—84no—
238C8×C61—42no—
239#92418—22no—
240#92418—41no—
241#92418—42no—
242#18665—611nomaximal, Sylow
243C22×C61—21no—
244C22×C61—21no—
245C22×C61—24no—
246C22×C61—42no—
247C22×C61—42no—
248C22×C61—84no—
249C22×C61#45112yes—
250C22×C61C2×#12011yescharacteristic
251C4×C61—21no—
252C4×C61—21no—
253C4×C61—21no—
254C4×C61—42no—
255C4×C61—81no—
256C4×C61—81no—
257C4×C61#45112yes—
258C4×C61C22×Q1611yescharacteristic
259C4×C61C2×#13711yescharacteristic
260C4×C61#57611yescharacteristic
261C4×C61#57711yescharacteristic
262Dic61—42no—
263Dic61—42no—
264Dic61—42no—
265Dic61—42no—
266#3501—611no—
267#3249—611no—
268#3170—611no—
269#3077D6111yescharacteristic
270#1649—612no—
271#1644—612no—
272#1637—612no—
273#1632—612no—
274#1550—611no—
275#1493—611no—
276C2×D4⋊C8—611no—
277C2×C61—22no—
278C2×C61—44no—
279C2×C61C2×#45111yescharacteristic
280C2×C61#301211yescharacteristic
281C2×C61#301811yescharacteristic
282#554—612no—
283#554—1222no—
284#554—1221no—
285#554—1222no—
286D4×Q8—612no—
287D4×Q8—1221no—
288#544—1221no—
289#543—1222no—
290#533—611no—
291#533—611no—
292#533C2×D6111yescharacteristic
293C2×C22⋊Q8—1222no—
294#518—1221no—
295C2×C4×D4C2×D6111yescharacteristic
296#500—611no—
297#494—611no—
298#487—611no—
299#487—611no—
300#478—22no—
301#477—612no—
302#477—1221no—
303#476C2×D6112yes—
304#474C2×D6112yes—
305#467—1222no—
306#467—1222no—
307#462—612no—
308#462—1221no—
309C4×Q16—1222no—
310#428—21no—
311#423C2×D6111yescharacteristic
312C2×C4⋊C8—611no—
313#418—21no—
314C2×C4×C8—611no—
315#333—612no—
316#332—612no—
317Q8⋊C8—612no—
318Q8⋊C8—612no—
319D4⋊C8—612no—
320D4⋊C8—612no—
321C61#1866511yescharacteristic, complemented, Sylow
322C22×Q8—1222no—
323C22×D4C22⋊D6111yescharacteristic
324C23×C4—21no—
325C2×Q16—1222no—
326C2×Q16—1222no—
327C2×M16—21no—
328C22×C8—1221no—
329C22×C8—1221no—
330C4⋊Q8—611no—
331C4⋊Q8—611no—
332C4⋊Q8—611no—
333C4⋊Q8—611no—
334C4⋊Q8—1222no—
335C4⋊Q8—1222no—
336C4⋊Q8C22×D6111yescharacteristic
337C4⋊Q8C22×D6111yescharacteristic
338#125—21no—
339#125—1222no—
340#125—1222no—
341#125—1221no—
342#125—1221no—
343C22⋊Q8—21no—
344C22⋊Q8—21no—
345C22⋊Q8—612no—
346C22⋊Q8—612no—
347C22⋊Q8—1224no—
348C22⋊Q8—1224no—
349C22⋊Q8—1224no—
350C22⋊Q8—1224no—
351C22⋊Q8—1222no—
352C22⋊Q8—1222no—
353C22⋊Q8—1224no—
354C22⋊Q8—1224no—
355C22⋊Q8—1221no—
356C22⋊Q8—1221no—
357C4×Q8—21no—
358C4×Q8—21no—
359C4×Q8—612no—
360C4×Q8—612no—
361C4×Q8—1221no—
362C4×Q8—1221no—
363C4×D4—21no—
364C4×D4—21no—
365C4×D4—1222no—
366C4×D4—1222no—
367C4×D4—1222no—
368C4×D4—1222no—
369C4×D4C22⋊D6112yes—
370C4×D4C22×D6112yes—
371#117—21no—
372#117—1221no—
373#117—1221no—
374C2×C4⋊C4—1222no—
375C2×C4⋊C4—1224no—
376C2×C4⋊C4C22⋊D6111yescharacteristic
377C2×C22⋊C4—21no—
378C2×C22⋊C4—1224no—
379C2×C42C22×D6111yescharacteristic
380#107—22no—
381#107—1222no—
382#107—1222no—
383#106—22no—
384C4⋊C8—21no—
385C4⋊C8—612no—
386C4⋊C8—1221no—
387C4⋊C8C22×D6112yes—
388#103—1222no—
389#103—1222no—
390#103—1222no—
391#103—1222no—
392#102—22no—
393#102—22no—
394C4×C8—611no—
395C4×C8—611no—
396C4×C8—1221no—
397C24—21no—
398C2×Q8—21no—
399C2×Q8—21no—
400C2×Q8—612no—
401C2×Q8—612no—
402C2×Q8—1224no—
403C2×Q8—1221no—
404C2×Q8—1221no—
405C2×Q8—4882no—
406C2×Q8—4882no—
407C2×D4—21no—
408C2×D4—22no—
409C2×D4#1175820214yescomplemented
410C2×D4C2×C22⋊D6112yes—
411C22×C4—21no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—81no—
415C22×C4—81no—
416C22×C4—1224no—
417C22×C4—1224no—
418C22×C4D4×D6111yescharacteristic
419C22×C4D4×D6111yescharacteristic
420C22×C4C2×C22⋊D6111yescharacteristic
421Q16—4882no—
422Q16—2444no—
423M16—42no—
424C2×C8—22no—
425C2×C8—1221no—
426C2×C8—1221no—
427C2×C8—1222no—
428C2×C8—4881no—
429C2×C8—2442no—
430C4⋊C4—21no—
431C4⋊C4—21no—
432C4⋊C4—21no—
433C4⋊C4—21no—
434C4⋊C4—21no—
435C4⋊C4—41no—
436C4⋊C4—41no—
437C4⋊C4—612no—
438C4⋊C4—612no—
439C4⋊C4—1224no—
440C4⋊C4—1224no—
441C4⋊C4—1224no—
442C4⋊C4—1221no—
443C4⋊C4—1222no—
444C4⋊C4—1221no—
445C4⋊C4—1221no—
446C4⋊C4—1222no—
447C4⋊C4—1222no—
448C4⋊C4—1222no—
449C4⋊C4—1221no—
450C4⋊C4—1221no—
451C4⋊C4—1221no—
452C4⋊C4C2×C22⋊D6112yes—
453C22⋊C4—22no—
454C22⋊C4—22no—
455C22⋊C4—41no—
456C22⋊C4—1224no—
457C22⋊C4—1224no—
458C22⋊C4—1222no—
459C22⋊C4—1222no—
460C42—41no—
461C42—1222no—
462C42—1222no—
463C42D4×D6111yescharacteristic
464C42D4×D6111yescharacteristic
465C42C2×C22⋊D6111yescharacteristic
466C42C23×D6111yescharacteristic
467C23—22no—
468C23—24no—
469C23—81no—
470C23—81no—
471C23#1451974511yescharacteristic
472Q8—42no—
473Q8—41no—
474Q8—41no—
475Q8—1222no—
476Q8—1222no—
477Q8—4881no—
478Q8—4884no—
479Q8—2442no—
480Q8—2441no—
481Q8—2441no—
482D4—22no—
483D4—24no—
484D4C2×#1175820214yes—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—22no—
488C2×C4—41no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—82no—
494C2×C4—81no—
495C2×C4—1222no—
496C2×C4—1222no—
497C2×C4—1222no—
498C2×C4—1222no—
499C2×C4—4884no—
500C2×C4—4884no—
501C2×C4#1451974511yescharacteristic
502C2×C4#1451974511yescharacteristic
503C2×C4#1451974511yescharacteristic
504C2×C4C2×#1175820212yes—
505C2×C4#1451980712yes—
506C2×C4C2×D4×D6111yescharacteristic
507C2×C4C2×D4×D6111yescharacteristic
508C2×C4C22×C22⋊D6111yescharacteristic
509C8—42no—
510C8—1222no—
511C8—2441no—
512C8—2442no—
513C22—21no—
514C22—21no—
515C22—24no—
516C22—42no—
517C22—42no—
518C22—84no—
519C22#1683991712yes—
520C22C2×#1451974511yescharacteristic
521C4—21no—
522C4—21no—
523C4—21no—
524C4—42no—
525C4—81no—
526C4—81no—
527C4—2442no—
528C4—2442no—
529C4—2442no—
530C4—2442no—
531C4#1683991712yes—
532C4C22×#1175820211yescharacteristic
533C4C2×#1451980711yescharacteristic
534C4#1684056611yescharacteristic
535C4#1684057211yescharacteristic
536C2—22no—
537C2—44no—
538C2C2×#1683991711yescharacteristic
539C2#2145512211yescharacteristic
540C2#2145516511yescharacteristic
subgroupG/H
centerC22C2×#14519745
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3077C2
socleC22×C61C2×#120
prime radicalC22×D4×C61D4
hypercenter#3077D61
hypocenterC61#18665
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3077

Complements

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

NH ≅ G/Ntype
2. #21457161C2semidirect
3. #21456401C2semidirect
4. #21445067C2semidirect
5. #21445032C2semidirect
6. #21444983C2semidirect
7. #21444951C2semidirect
8. #21444510C2semidirect
9. #21444273C2semidirect
24. #16840165C22semidirect
25. #16840141C22semidirect
26. #16840114C22semidirect
27. #16840112C22semidirect
29. #16840054C22semidirect
33. #16839969C22semidirect
37. #16839510C22semidirect
38. #16839507C22semidirect
39. #16839492C22semidirect
40. #16839492C22semidirect
131. C2×D4×C61Q16semidirect
321. C61#18665semidirect
409. C2×D4#11758202semidirect

Maximal subgroups

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

subgroupindex
#214596222
#214571612
#214564012
#214450672
#214450322
#214449832
#214449512
#214445102
#214442732
C2×D4⋊#924182
C61×#30772
#1866561