Finite Group Databasebeta

← #29656904

Subgroups — #29656904

Order 15616.

910 conjugacy classes of subgroups.

The subgroup lattice has 516 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21457029C211yesmaximal, characteristic, complemented
2#21456560C211yesmaximal, characteristic, complemented
3#21455827C211yesmaximal, characteristic
4#21445082C212yesmaximal, complemented
5#21445026C212yesmaximal, complemented
6#21444984C212yesmaximal, complemented
7#21444953C212yesmaximal, complemented
8#21444529C211yesmaximal, characteristic, complemented
9#21444262C211yesmaximal, characteristic
10#21444240C211yesmaximal, characteristic, complemented
11C61×#3501C211yesmaximal, characteristic
12#16840319C2211yescharacteristic
13#16840168—21no—
14#16840158C2211yescharacteristic, complemented
15#16840141C2211yescharacteristic, complemented
16#16840130—21no—
17#16840114C2211yescharacteristic, complemented
18#16840112C2211yescharacteristic, complemented
19#16840100—22no—
20#16840092—21no—
21#16840057C2212yescomplemented
22#16840052C2212yes—
23#16840047C2212yescomplemented
24#16840018—22no—
25#16839967C2212yes—
26#16839913—21no—
27#16839867—22no—
28C4×#11758200—22no—
29#16839819C2211yescharacteristic
30#16839815C2211yescharacteristic
31C4×#11758194C2211yescharacteristic
32#16839511C2212yescomplemented
33#16839508C2212yescomplemented
34Q8⋊#92418C2212yescomplemented
35Q8⋊#92418C2212yescomplemented
36D4⋊#92418C2212yes—
37D4⋊#92418C2212yes—
38C61×#554—21no—
39C61×#554—22no—
40C61×#554—22no—
41C61×#554C2212yes—
42D4×Q8×C61—21no—
43D4×Q8×C61C2212yes—
44C61×#544—21no—
45C61×#543—22no—
46C61×#533C2211yescharacteristic
47C61×#533C2211yescharacteristic
48C2×C22⋊Q8×C61—22no—
49C61×#518—21no—
50C2×C4×D4×C61C2211yescharacteristic
51C2×#11758202—22no—
52C2×#11758200—22no—
53C2×#11758194—21no—
54#14519777C2311yescharacteristic
55#14519777C2311yescharacteristic
56#14519769—21no—
57#14519756—21no—
58#14519756—21no—
59C4×Dic122—21no—
60C4×Dic122—21no—
61#14519736—21no—
62#14519710—22no—
63#14519708—22no—
64#14519708—22no—
65#14519708—22no—
66C4⋊#92418C2312yes—
67C4⋊#92418C2312yes—
68#14519701—22no—
69#14519701—22no—
70#14519701—22no—
71#14519700—22no—
72#14519700—22no—
73#14519698—22no—
74C22⋊#92418—21no—
75C22⋊#92418—21no—
76#14519688—21no—
77#14519688—21no—
78C4×#92418—21no—
79C4×#92418C2311yescharacteristic
80C4×#92418C2311yescharacteristic
81C22×Q8×C61—22no—
82C22×D4×C61D411yescharacteristic
83C23×C4×C61—21no—
84C4⋊Q8×C61—22no—
85C4⋊Q8×C61—22no—
86C4⋊Q8×C61D411yescharacteristic
87C4⋊Q8×C61D411yescharacteristic
88C4⋊Q8×C61C2311yescharacteristic
89C4⋊Q8×C61C2311yescharacteristic
90#125×C61—22no—
91#125×C61—22no—
92#125×C61—21no—
93#125×C61—21no—
94C22⋊Q8×C61—24no—
95C22⋊Q8×C61—24no—
96C22⋊Q8×C61—24no—
97C22⋊Q8×C61—24no—
98C22⋊Q8×C61—24no—
99C22⋊Q8×C61—22no—
100C22⋊Q8×C61—21no—
101C22⋊Q8×C61—21no—
102C22⋊Q8×C61—24no—
103C22⋊Q8×C61—22no—
104C22⋊Q8×C61D412yes—
105C22⋊Q8×C61D412yes—
106C4×Q8×C61—21no—
107C4×Q8×C61—21no—
108C4×Q8×C61D412yes—
109C4×Q8×C61C2312yes—
110C4×D4×C61—22no—
111C4×D4×C61—22no—
112C4×D4×C61—22no—
113C4×D4×C61—22no—
114C4×D4×C61—21no—
115C4×D4×C61—21no—
116C4×D4×C61D412yes—
117C4×D4×C61C2312yes—
118#117×C61—21no—
119#117×C61—21no—
120C2×C4⋊C4×C61—24no—
121C2×C4⋊C4×C61—22no—
122C2×C4⋊C4×C61D411yescharacteristic
123C2×C22⋊C4×C61—24no—
124C2×C22⋊C4×C61—21no—
125C2×C42×C61C2311yescharacteristic
126C24×C61—21no—
127C2×Q8×C61—24no—
128C2×Q8×C61—21no—
129C2×Q8×C61—21no—
130C2×Q8×C61—84no—
131C2×Q8×C61C2×D412yes—
132C2×Q8×C61C2×D412yes—
133C2×D4×C61—24no—
134C2×D4×C61—21no—
135C2×D4×C61C2×D412yes—
136C22×C4×C61—24no—
137C22×C4×C61—24no—
138C22×C4×C61—21no—
139C22×C4×C61—22no—
140C22×C4×C61—22no—
141C22×C4×C61—82no—
142C22×C4×C61C2×D411yescharacteristic
143C22×C4×C61C2×D411yescharacteristic
144C22×C4×C61C2×D411yescharacteristic
145C2×Dic122—21no—
146C2×Dic122—21no—
147C4⋊C4×C61—24no—
148C4⋊C4×C61—24no—
149C4⋊C4×C61—24no—
150C4⋊C4×C61—22no—
151C4⋊C4×C61—22no—
152C4⋊C4×C61—21no—
153C4⋊C4×C61—21no—
154C4⋊C4×C61—21no—
155C4⋊C4×C61—21no—
156C4⋊C4×C61—22no—
157C4⋊C4×C61—21no—
158C4⋊C4×C61—21no—
159C4⋊C4×C61—21no—
160C4⋊C4×C61—22no—
161C4⋊C4×C61C2×D412yes—
162C4⋊C4×C61C2×D412yes—
163C4⋊C4×C61C2×D412yes—
164C22⋊C4×C61—24no—
165C22⋊C4×C61—24no—
166C22⋊C4×C61—22no—
167C22⋊C4×C61—22no—
168C22⋊C4×C61—22no—
169C22⋊C4×C61—22no—
170C42×C61—22no—
171C42×C61—22no—
172C42×C61C2×D411yescharacteristic
173C42×C61C2×D411yescharacteristic
174C42×C61C2×D411yescharacteristic
175C42×C61C2411yescharacteristic
176C22⋊Dic61—41no—
177#11758202—44no—
178#11758200—44no—
179C4⋊Dic61—21no—
180C4⋊Dic61—21no—
181C4⋊Dic61—21no—
182C4⋊Dic61—21no—
183#11758196—41no—
184#11758196—41no—
185C4×Dic61—41no—
186#11758194—42no—
187C2×#92418—21no—
188C2×#92418—22no—
189C2×#92418—21no—
190C2×#92418—22no—
191C23×C61—24no—
192C23×C61—22no—
193C23×C61—81no—
194C23×C61—81no—
195C23×C61#12011yescharacteristic
196Q8×C61—24no—
197Q8×C61—42no—
198Q8×C61—42no—
199Q8×C61—81no—
200Q8×C61—84no—
201D4×C61—24no—
202D4×C61—22no—
203D4×C61#13714yes—
204C2×C4×C61—22no—
205C2×C4×C61—22no—
206C2×C4×C61—22no—
207C2×C4×C61—22no—
208C2×C4×C61—22no—
209C2×C4×C61—22no—
210C2×C4×C61—22no—
211C2×C4×C61—41no—
212C2×C4×C61—44no—
213C2×C4×C61—82no—
214C2×C4×C61—81no—
215C2×C4×C61—84no—
216C2×C4×C61—84no—
217C2×C4×C61#12011yescharacteristic
218C2×C4×C61#12011yescharacteristic
219C2×C4×C61#12011yescharacteristic
220C2×C4×C61C22×D411yescharacteristic
221C2×C4×C61C22×D411yescharacteristic
222C2×C4×C61C22×D411yescharacteristic
223C2×Dic61—41no—
224C2×Dic61—41no—
225Dic122—42no—
226Dic122—42no—
227#92418—22no—
228#92418—41no—
229#92418—42no—
230#92418—42no—
231#18601—611nomaximal, Sylow
232C22×C61—21no—
233C22×C61—24no—
234C22×C61—21no—
235C22×C61—21no—
236C22×C61—42no—
237C22×C61—42no—
238C22×C61—84no—
239C22×C61C2×#12011yescharacteristic
240C4×C61—21no—
241C4×C61—21no—
242C4×C61—21no—
243C4×C61—21no—
244C4×C61—42no—
245C4×C61—42no—
246C4×C61—42no—
247C4×C61—42no—
248C4×C61—42no—
249C4×C61C2×#13712yes—
250C4×C61#57712yes—
251Dic61—81no—
252Dic61—81no—
253#3501D6111yescharacteristic
254#3237—611no—
255#3185—611no—
256#3101—611no—
257#1651—612no—
258#1643—612no—
259#1637—612no—
260#1632—612no—
261#1554—611no—
262#1490—611no—
263#1485—611no—
264C2×C61—41no—
265C2×C61—44no—
266C2×C61#300411yescharacteristic
267C2×C61#300911yescharacteristic
268C2×C61#301811yescharacteristic
269#554—21no—
270#554—22no—
271#554—22no—
272#554C2×D6112yes—
273D4×Q8—21no—
274D4×Q8C2×D6112yes—
275#544—21no—
276#543—22no—
277#533—611no—
278#533C2×D6111yescharacteristic
279#533C2×D6111yescharacteristic
280C2×C22⋊Q8—22no—
281#518—21no—
282C2×C4×D4C2×D6111yescharacteristic
283#501—1221no—
284#498—611no—
285#494—611no—
286#491—1221no—
287#487—611no—
288#487—611no—
289#485—1222no—
290#483—1221no—
291#477—612no—
292#476—612no—
293#475—612no—
294#470—1222no—
295#461—612no—
296#450—1221no—
297#441—1222no—
298C4×SD16—1222no—
299#432—611no—
300#431—611no—
301C4×M16—611no—
302#333—612no—
303#332—612no—
304Q8⋊C8—612no—
305Q8⋊C8—612no—
306D4⋊C8—612no—
307D4⋊C8—612no—
308C61#1860111yescharacteristic, complemented, Sylow
309C22×Q8—22no—
310C22×D4C22⋊D6111yescharacteristic
311C23×C4—21no—
312C2×Q16—1222no—
313C2×SD16—1222no—
314C2×M16—1221no—
315C4⋊Q8—22no—
316C4⋊Q8—22no—
317C4⋊Q8—611no—
318C4⋊Q8—611no—
319C4⋊Q8C22⋊D6111yescharacteristic
320C4⋊Q8C22⋊D6111yescharacteristic
321C4⋊Q8C22×D6111yescharacteristic
322C4⋊Q8C22×D6111yescharacteristic
323#125—22no—
324#125—22no—
325#125—21no—
326#125—21no—
327#125—1221no—
328C22⋊Q8—24no—
329C22⋊Q8—24no—
330C22⋊Q8—24no—
331C22⋊Q8—24no—
332C22⋊Q8—22no—
333C22⋊Q8—24no—
334C22⋊Q8—21no—
335C22⋊Q8—21no—
336C22⋊Q8—22no—
337C22⋊Q8—24no—
338C22⋊Q8—1221no—
339C22⋊Q8—1221no—
340C22⋊Q8C22⋊D6112yes—
341C22⋊Q8C22⋊D6112yes—
342C4×Q8—21no—
343C4×Q8—21no—
344C4×Q8—1221no—
345C4×Q8—1221no—
346C4×Q8C22⋊D6112yes—
347C4×Q8C22×D6112yes—
348C4×D4—22no—
349C4×D4—22no—
350C4×D4—22no—
351C4×D4—22no—
352C4×D4—21no—
353C4×D4—21no—
354C4×D4C22⋊D6112yes—
355C4×D4C22×D6112yes—
356#117—21no—
357#117—21no—
358#117—1221no—
359C2×C4⋊C4—24no—
360C2×C4⋊C4—22no—
361C2×C4⋊C4C22⋊D6111yescharacteristic
362C2×C22⋊C4—24no—
363C2×C22⋊C4—21no—
364C2×C42C22×D6111yescharacteristic
365#107—1222no—
366#106—1222no—
367#106—1222no—
368#106—1222no—
369C4⋊C8—612no—
370C4⋊C8—612no—
371#103—1222no—
372#103—1222no—
373#103—1222no—
374#103—1222no—
375#103—1222no—
376#102—1222no—
377C22⋊C8—1221no—
378C22⋊C8—1221no—
379#97—1221no—
380#97—1221no—
381C4×C8—611no—
382C4×C8—611no—
383C4×C8—1221no—
384C24—21no—
385C2×Q8—24no—
386C2×Q8—21no—
387C2×Q8—21no—
388C2×Q8—84no—
389C2×Q8—1221no—
390C2×Q8—1221no—
391C2×Q8C2×C22⋊D6112yes—
392C2×Q8C2×C22⋊D6112yes—
393C2×D4—24no—
394C2×D4—21no—
395C2×D4C2×C22⋊D6112yes—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—21no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—82no—
402C22×C4C2×C22⋊D6111yescharacteristic
403C22×C4C2×C22⋊D6111yescharacteristic
404C22×C4C2×C22⋊D6111yescharacteristic
405Q16—2444no—
406SD16—2444no—
407M16—2442no—
408C2×C8—1221no—
409C2×C8—1221no—
410C2×C8—1222no—
411C2×C8—1222no—
412C4⋊C4—24no—
413C4⋊C4—24no—
414C4⋊C4—24no—
415C4⋊C4—21no—
416C4⋊C4—21no—
417C4⋊C4—21no—
418C4⋊C4—22no—
419C4⋊C4—22no—
420C4⋊C4—22no—
421C4⋊C4—21no—
422C4⋊C4—21no—
423C4⋊C4—21no—
424C4⋊C4—21no—
425C4⋊C4—22no—
426C4⋊C4—1221no—
427C4⋊C4—1221no—
428C4⋊C4—1221no—
429C4⋊C4—1221no—
430C4⋊C4—2441no—
431C4⋊C4—2441no—
432C4⋊C4C2×C22⋊D6112yes—
433C4⋊C4C2×C22⋊D6112yes—
434C4⋊C4C2×C22⋊D6112yes—
435C22⋊C4—24no—
436C22⋊C4—24no—
437C22⋊C4—22no—
438C22⋊C4—22no—
439C22⋊C4—22no—
440C22⋊C4—22no—
441C22⋊C4—2441no—
442C42—22no—
443C42—22no—
444C42—2441no—
445C42C2×C22⋊D6111yescharacteristic
446C42C2×C22⋊D6111yescharacteristic
447C42C2×C22⋊D6111yescharacteristic
448C42C23×D6111yescharacteristic
449C23—24no—
450C23—22no—
451C23—81no—
452C23—81no—
453C23#1451974611yescharacteristic
454Q8—24no—
455Q8—42no—
456Q8—42no—
457Q8—81no—
458Q8—84no—
459Q8—2442no—
460Q8—2442no—
461D4—24no—
462D4—22no—
463D4#1451980814yes—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—41no—
472C2×C4—44no—
473C2×C4—82no—
474C2×C4—81no—
475C2×C4—84no—
476C2×C4—84no—
477C2×C4—2441no—
478C2×C4—2441no—
479C2×C4#1451974611yescharacteristic
480C2×C4#1451974611yescharacteristic
481C2×C4#1451974611yescharacteristic
482C2×C4C22×C22⋊D6111yescharacteristic
483C2×C4C22×C22⋊D6111yescharacteristic
484C2×C4C22×C22⋊D6111yescharacteristic
485C8—1222no—
486C8—2441no—
487C8—2442no—
488C8—2442no—
489C22—21no—
490C22—21no—
491C22—24no—
492C22—21no—
493C22—42no—
494C22—42no—
495C22—84no—
496C22C2×#1451974611yescharacteristic
497C4—21no—
498C4—21no—
499C4—21no—
500C4—21no—
501C4—42no—
502C4—42no—
503C4—42no—
504C4—42no—
505C4—42no—
506C4—4881no—
507C4—4881no—
508C4C2×#1451980812yes—
509C4#1684057312yes—
510C2—41no—
511C2—44no—
512C2#2145505311yescharacteristic
513C2#2145509611yescharacteristic
514C2#2145516811yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3501C2
socleC22×C61C2×#120
prime radicalC22×D4×C61D4
hypercenter#3501D61
hypocenterC61#18601
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3501

Complements

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

NH ≅ G/Ntype
1. #21457029C2semidirect
2. #21456560C2semidirect
4. #21445082C2semidirect
5. #21445026C2semidirect
6. #21444984C2semidirect
7. #21444953C2semidirect
8. #21444529C2semidirect
10. #21444240C2semidirect
14. #16840158C22semidirect
15. #16840141C22semidirect
17. #16840114C22semidirect
18. #16840112C22semidirect
21. #16840057C22semidirect
23. #16840047C22semidirect
32. #16839511C22semidirect
33. #16839508C22semidirect
34. Q8⋊#92418C22semidirect
35. Q8⋊#92418C22semidirect
308. C61#18601semidirect

Maximal subgroups

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

subgroupindex
#214570292
#214565602
#214558272
#214450822
#214450262
#214449842
#214449532
#214445292
#214442622
#214442402
C61×#35012
#1860161