Finite Group Databasebeta

← #29657745

Subgroups — #29657745

Order 15616.

912 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21457131C211yesmaximal, characteristic
2#21456379C211yesmaximal, characteristic
3#21455291C211yesmaximal, characteristic
4#21445096C212yesmaximal, complemented
5#21445040C212yesmaximal, complemented
6#21444915C212yesmaximal, complemented
7#21444897C212yesmaximal, complemented
8#21444509C211yesmaximal, characteristic, complemented
9#21444274C211yesmaximal, characteristic, complemented
10C2×Q8⋊#92418C211yesmaximal, characteristic, complemented
11C61×#3502C211yesmaximal, characteristic, complemented
12C2×#14519808—21no—
13#16840312C2211yescharacteristic
14#16840158C2211yescharacteristic
15#16840136C2211yescharacteristic
16#16840112C2211yescharacteristic
17#16840110C2211yescharacteristic
18#16840049—21no—
19#16840047C2212yes—
20#16839993—22no—
21#16839993—22no—
22#16839984—22no—
23#16839972C2212yes—
24#16839964—21no—
25#16839962C2212yes—
26#16839962C2212yes—
27C4×#11758201—22no—
28#16839786C2211yescharacteristic
29C2×C4⋊#92418C2211yescharacteristic
30#16839761—21no—
31C2×C4×#92418C2211yescharacteristic
32#16839511C2212yes—
33#16839508C2212yes—
34Q8⋊#92418C2212yes—
35Q8⋊#92418C2212yes—
36Q8⋊#92418C2212yes—
37Q8⋊#92418C2212yes—
38Q82×C61—21no—
39Q82×C61C2212yes—
40C61×#557—21no—
41C61×#557—22no—
42C61×#557—22no—
43C61×#557C2212yes—
44C61×#544—22no—
45C61×#544—21no—
46C61×#533—22no—
47C61×#533C2211yescharacteristic
48C61×#533C2211yescharacteristic
49C61×#519—21no—
50C2×C4×Q8×C61C2211yescharacteristic
51C2×#11758222—21no—
52#14519808—44no—
53C2×#11758202—22no—
54C2×#11758201—22no—
55C2×#11758201—22no—
56C2×#11758201—22no—
57C2×#11758194—21no—
58C22×#92418—21no—
59C22×#92418—21no—
60#14519777C2311yescharacteristic
61#14519774C2311yescharacteristic
62#14519765—21no—
63#14519751—21no—
64#14519751—21no—
65C4×D244—21no—
66C4×D244—21no—
67#14519708—22no—
68#14519708—22no—
69C4⋊#92418—21no—
70C4⋊#92418—21no—
71C4⋊#92418C2312yes—
72C4⋊#92418C2312yes—
73#14519701—22no—
74#14519701—22no—
75#14519700—22no—
76#14519697—22no—
77#14519697—22no—
78#14519697—22no—
79C4×#92418—21no—
80C4×#92418C2311yescharacteristic
81C4×#92418C2311yescharacteristic
82C22×Q8×C61D411yescharacteristic
83C4⋊Q8×C61—24no—
84C4⋊Q8×C61—22no—
85C4⋊Q8×C61—22no—
86C4⋊Q8×C61—24no—
87C4⋊Q8×C61—22no—
88C4⋊Q8×C61—22no—
89C4⋊Q8×C61D411yescharacteristic
90C4⋊Q8×C61D411yescharacteristic
91C4⋊Q8×C61C2311yescharacteristic
92C4⋊Q8×C61C2311yescharacteristic
93#125×C61—24no—
94#125×C61—24no—
95#125×C61—22no—
96#125×C61—24no—
97#125×C61—22no—
98#125×C61—21no—
99#125×C61—21no—
100C22⋊Q8×C61—24no—
101C22⋊Q8×C61—21no—
102C22⋊Q8×C61—21no—
103C22⋊Q8×C61D412yes—
104C22⋊Q8×C61D412yes—
105C4×Q8×C61—24no—
106C4×Q8×C61—22no—
107C4×Q8×C61—22no—
108C4×Q8×C61—22no—
109C4×Q8×C61—22no—
110C4×Q8×C61—22no—
111C4×Q8×C61—22no—
112C4×Q8×C61—21no—
113C4×Q8×C61—21no—
114C4×Q8×C61—21no—
115C4×Q8×C61—21no—
116C4×Q8×C61D412yes—
117C4×Q8×C61D412yes—
118C4×Q8×C61C2312yes—
119C4×Q8×C61C2312yes—
120#117×C61—24no—
121#117×C61—21no—
122#117×C61—21no—
123C2×C4⋊C4×C61—21no—
124C2×C4⋊C4×C61D411yescharacteristic
125C2×C42×C61—21no—
126C2×C42×C61C2311yescharacteristic
127C2×Q8×C61—24no—
128C2×Q8×C61—21no—
129C2×Q8×C61—21no—
130C2×Q8×C61—22no—
131C2×Q8×C61—21no—
132C2×Q8×C61SD1614yescomplemented
133C2×Q8×C61C2×D412yes—
134C2×Q8×C61C2×D412yes—
135C2×Q8×C61C2×D412yes—
136C22×C4×C61—21no—
137C22×C4×C61—21no—
138C22×C4×C61C2×D411yescharacteristic
139C22×C4×C61C2×D411yescharacteristic
140C22×C4×C61C2×D411yescharacteristic
141C2×C22⋊D61—41no—
142#11758222—42no—
143#11758222—42no—
144C2×D244—21no—
145C2×D244—21no—
146C2×D244—21no—
147C2×C4×D61—41no—
148C2×Dic122—21no—
149C4⋊C4×C61—24no—
150C4⋊C4×C61—24no—
151C4⋊C4×C61—24no—
152C4⋊C4×C61—24no—
153C4⋊C4×C61—24no—
154C4⋊C4×C61—22no—
155C4⋊C4×C61—21no—
156C4⋊C4×C61—21no—
157C4⋊C4×C61—21no—
158C4⋊C4×C61—22no—
159C4⋊C4×C61—22no—
160C4⋊C4×C61—21no—
161C4⋊C4×C61—22no—
162C4⋊C4×C61—22no—
163C4⋊C4×C61—21no—
164C4⋊C4×C61—21no—
165C4⋊C4×C61—21no—
166C4⋊C4×C61—22no—
167C4⋊C4×C61C2×D412yes—
168C4⋊C4×C61C2×D412yes—
169C4⋊C4×C61C2×D412yes—
170C22⋊C4×C61—22no—
171C22⋊C4×C61—22no—
172C42×C61—24no—
173C42×C61—24no—
174C42×C61—22no—
175C42×C61—22no—
176C42×C61—22no—
177C42×C61—22no—
178C42×C61C2×D411yescharacteristic
179C42×C61C2×D411yescharacteristic
180C42×C61C2×D411yescharacteristic
181C42×C61C2411yescharacteristic
182#11758202—44no—
183#11758201—44no—
184#11758201—44no—
185#11758201—82no—
186#11758198—41no—
187#11758198—41no—
188C4⋊Dic61—21no—
189C4⋊Dic61—21no—
190#11758194—42no—
191C2×#92418—21no—
192C2×#92418—21no—
193C2×#92418—22no—
194C2×#92418—22no—
195C2×#92418—42no—
196C2×#92418—81no—
197C23×C61#12011yescharacteristic
198C22×D61—41no—
199Q8×C61—22no—
200Q8×C61—24no—
201Q8×C61—22no—
202Q8×C61—22no—
203Q8×C61—42no—
204Q8×C61—41no—
205Q8×C61—41no—
206Q8×C61—81no—
207Q8×C61—84no—
208Q8×C61C2×SD1614yes—
209C2×C4×C61—22no—
210C2×C4×C61—24no—
211C2×C4×C61—21no—
212C2×C4×C61—21no—
213C2×C4×C61—22no—
214C2×C4×C61—22no—
215C2×C4×C61—22no—
216C2×C4×C61—22no—
217C2×C4×C61—22no—
218C2×C4×C61—41no—
219C2×C4×C61—81no—
220C2×C4×C61#12011yescharacteristic
221C2×C4×C61#12011yescharacteristic
222C2×C4×C61#12011yescharacteristic
223C2×C4×C61C2×SD1612yes—
224C2×C4×C61#13712yes—
225C2×C4×C61C22×D411yescharacteristic
226C2×C4×C61C22×D411yescharacteristic
227C2×C4×C61C22×D411yescharacteristic
228C22⋊D61—82no—
229C2×Dic61—41no—
230D244—42no—
231D244—41no—
232D244—41no—
233D244—41no—
234D244—41no—
235C4×D61—82no—
236Dic122—41no—
237Dic122—41no—
238#92418—22no—
239#92418—41no—
240#92418—42no—
241#92418—42no—
242#18676—611nomaximal, Sylow
243C22×C61—21no—
244C22×C61—21no—
245C22×C61#44812yes—
246C22×C61C2×#12011yescharacteristic
247C2×D61—81no—
248C2×D61—81no—
249C2×D61—81no—
250C4×C61—24no—
251C4×C61—21no—
252C4×C61—21no—
253C4×C61—21no—
254C4×C61—42no—
255C4×C61—42no—
256C4×C61—42no—
257C4×C61—42no—
258C4×C61—42no—
259C4×C61#44812yes—
260C4×C61C22×SD1611yescharacteristic
261C4×C61C2×#13711yescharacteristic
262C4×C61#57711yescharacteristic
263C4×C61#57811yescharacteristic
264Dic61—81no—
265#3502D6111yescharacteristic, complemented
266#3246—611no—
267#3167—611no—
268#3037—611no—
269#1653—612no—
270#1645—612no—
271#1626—612no—
272#1623—612no—
273#1550—611no—
274#1493—611no—
275C2×Q8⋊C8—611no—
276C2×C61—22no—
277C2×C61C2×#44811yescharacteristic
278C2×C61#301311yescharacteristic
279C2×C61#301811yescharacteristic
280D61—161no—
281C2×#137—1221no—
282Q82—21no—
283Q82C2×D6112yes—
284#557—21no—
285#557—22no—
286#557—22no—
287#557C2×D6112yes—
288#544—22no—
289#544—21no—
290#533—22no—
291#533C2×D6111yescharacteristic
292#533C2×D6111yescharacteristic
293#532—611no—
294#519—21no—
295C2×C4×Q8C2×D6111yescharacteristic
296#498—611no—
297#492—611no—
298#487—611no—
299#486—611no—
300#476—1221no—
301#475—612no—
302#465—1222no—
303#465—1222no—
304#464—1222no—
305#462—612no—
306#461—1221no—
307#460—612no—
308#460—612no—
309C4×SD16—1222no—
310#423—611no—
311C2×C4⋊C8—611no—
312#417—1221no—
313C2×C4×C8—611no—
314#333—612no—
315#332—612no—
316Q8⋊C8—612no—
317Q8⋊C8—612no—
318Q8⋊C8—612no—
319Q8⋊C8—612no—
320C61#1867611yescharacteristic, complemented, Sylow
321C2×#41—1221no—
322C22×Q8C22⋊D6111yescharacteristic
323#137—2444no—
324C2×Q16—1222no—
325C2×SD16—1222no—
326C2×SD16—1222no—
327C2×SD16—1222no—
328C2×M16—1221no—
329C22×C8—1221no—
330C22×C8—1221no—
331C4⋊Q8—24no—
332C4⋊Q8—22no—
333C4⋊Q8—22no—
334C4⋊Q8—24no—
335C4⋊Q8—22no—
336C4⋊Q8—22no—
337C4⋊Q8—611no—
338C4⋊Q8C22⋊D6111yescharacteristic
339C4⋊Q8C22⋊D6111yescharacteristic
340C4⋊Q8C22×D6111yescharacteristic
341C4⋊Q8C22×D6111yescharacteristic
342#127—611no—
343#125—24no—
344#125—24no—
345#125—22no—
346#125—24no—
347#125—22no—
348#125—21no—
349#125—21no—
350#124—1221no—
351C22⋊Q8—24no—
352C22⋊Q8—21no—
353C22⋊Q8—21no—
354C22⋊Q8C22⋊D6112yes—
355C22⋊Q8C22⋊D6112yes—
356#121—1221no—
357#121—1221no—
358C4×Q8—22no—
359C4×Q8—22no—
360C4×Q8—24no—
361C4×Q8—22no—
362C4×Q8—22no—
363C4×Q8—22no—
364C4×Q8—22no—
365C4×Q8—21no—
366C4×Q8—21no—
367C4×Q8—21no—
368C4×Q8—21no—
369C4×Q8C22⋊D6112yes—
370C4×Q8C22⋊D6112yes—
371C4×Q8C22×D6112yes—
372C4×Q8C22×D6112yes—
373C4×D4—1221no—
374C4×D4—1221no—
375#117—24no—
376#117—21no—
377#117—21no—
378C2×C4⋊C4—21no—
379C2×C4⋊C4C22⋊D6111yescharacteristic
380C2×C42—21no—
381C2×C42C22×D6111yescharacteristic
382#106—1222no—
383#106—1222no—
384C4⋊C8—612no—
385C4⋊C8—612no—
386C4⋊C8—1221no—
387C4⋊C8—1221no—
388#103—1222no—
389#103—1222no—
390#103—1222no—
391#102—1222no—
392#102—1222no—
393#102—1222no—
394C4×C8—611no—
395C4×C8—611no—
396C4×C8—1221no—
397#41—2442no—
398#41—2442no—
399C2×Q8—24no—
400C2×Q8—22no—
401C2×Q8—21no—
402C2×Q8—21no—
403C2×Q8—21no—
404C2×Q8—1221no—
405C2×Q8#1175820114yescomplemented
406C2×Q8C2×C22⋊D6112yes—
407C2×Q8C2×C22⋊D6112yes—
408C2×Q8C2×C22⋊D6112yes—
409C2×D4—1221no—
410C2×D4—1221no—
411C2×D4—1221no—
412C2×D4—2441no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—2441no—
416C22×C4C2×C22⋊D6111yescharacteristic
417C22×C4C2×C22⋊D6111yescharacteristic
418C22×C4C2×C22⋊D6111yescharacteristic
419Q16—2444no—
420SD16—4882no—
421SD16—2444no—
422SD16—2444no—
423M16—2442no—
424C2×C8—1221no—
425C2×C8—1221no—
426C2×C8—1222no—
427C2×C8—1222no—
428C2×C8—4881no—
429C2×C8—2442no—
430C4⋊C4—24no—
431C4⋊C4—24no—
432C4⋊C4—24no—
433C4⋊C4—24no—
434C4⋊C4—24no—
435C4⋊C4—22no—
436C4⋊C4—21no—
437C4⋊C4—21no—
438C4⋊C4—21no—
439C4⋊C4—22no—
440C4⋊C4—22no—
441C4⋊C4—21no—
442C4⋊C4—21no—
443C4⋊C4—21no—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—21no—
447C4⋊C4—22no—
448C4⋊C4—1221no—
449C4⋊C4—1221no—
450C4⋊C4C2×C22⋊D6112yes—
451C4⋊C4C2×C22⋊D6112yes—
452C4⋊C4C2×C22⋊D6112yes—
453C22⋊C4—22no—
454C22⋊C4—22no—
455C22⋊C4—2441no—
456C22⋊C4—2441no—
457C42—24no—
458C42—24no—
459C42—22no—
460C42—22no—
461C42—22no—
462C42—22no—
463C42C2×C22⋊D6111yescharacteristic
464C42C2×C22⋊D6111yescharacteristic
465C42C2×C22⋊D6111yescharacteristic
466C42C23×D6111yescharacteristic
467C23—2441no—
468C23#1451974611yescharacteristic
469Q8—22no—
470Q8—24no—
471Q8—22no—
472Q8—22no—
473Q8—42no—
474Q8—41no—
475Q8—41no—
476Q8—81no—
477Q8—84no—
478Q8—2441no—
479Q8—2441no—
480Q8C2×#1175820114yes—
481D4—4882no—
482D4—2442no—
483D4—2441no—
484D4—2441no—
485D4—2441no—
486D4—2441no—
487C2×C4—22no—
488C2×C4—24no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—41no—
497C2×C4—81no—
498C2×C4—4882no—
499C2×C4—2441no—
500C2×C4#1451974611yescharacteristic
501C2×C4#1451974611yescharacteristic
502C2×C4#1451974611yescharacteristic
503C2×C4C2×#1175820112yes—
504C2×C4#1451980812yes—
505C2×C4C22×C22⋊D6111yescharacteristic
506C2×C4C22×C22⋊D6111yescharacteristic
507C2×C4C22×C22⋊D6111yescharacteristic
508C8—1222no—
509C8—2441no—
510C8—2442no—
511C8—2442no—
512C22—21no—
513C22—21no—
514C22—4881no—
515C22—4881no—
516C22—4881no—
517C22#1683990112yes—
518C22C2×#1451974611yescharacteristic
519C4—21no—
520C4—21no—
521C4—21no—
522C4—24no—
523C4—42no—
524C4—42no—
525C4—42no—
526C4—42no—
527C4—42no—
528C4—4881no—
529C4#1683990112yes—
530C4C22×#1175820111yescharacteristic
531C4C2×#1451980811yescharacteristic
532C4#1684057011yescharacteristic
533C4#1684057711yescharacteristic
534C2—22no—
535C2—9761no—
536C2C2×#1683990111yescharacteristic
537C2#2145513411yescharacteristic
538C2#2145516811yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3502C2
socleC22×C61C2×#120
prime radical#16840312C22
hypercenter#3502D61
hypocenterC61#18676
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3502

Complements

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

NH ≅ G/Ntype
4. #21445096C2semidirect
5. #21445040C2semidirect
6. #21444915C2semidirect
7. #21444897C2semidirect
8. #21444509C2semidirect
9. #21444274C2semidirect
10. C2×Q8⋊#92418C2semidirect
11. C61×#3502C2semidirect
132. C2×Q8×C61SD16semidirect
265. #3502D61semidirect
320. C61#18676semidirect
405. C2×Q8#11758201semidirect

Maximal subgroups

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

subgroupindex
#214571312
#214563792
#214552912
#214450962
#214450402
#214449152
#214448972
#214445092
#214442742
C2×Q8⋊#924182
C61×#35022
#1867661