Finite Group Databasebeta

← #30615490

Subgroups — #30615490

Order 15616.

1394 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462799C211yesmaximal, characteristic
2C2×#16840773C212yesmaximal, complemented
3#21462294C211yesmaximal, characteristic
4#21461775C214yesmaximal, complemented
5#21461774C214yesmaximal, complemented
6#21461501C212yesmaximal, complemented
7#413×D61C211yesmaximal, characteristic
8#16840831—22no—
9#16840831—22no—
10#16840830C2212yes—
11#16840829—22no—
12#16840829C2212yes—
13#41×#6923—24no—
14C2×#14519856—22no—
15C2×#14519856C2212yes—
16C2×#14519856C2211yescharacteristic
17C22×C4⋊#6923C2211yescharacteristic
18#16840773—24no—
19#16840773C2218yes—
20C2×#14519846—22no—
21C2×#14519845—22no—
22(C2×#6923)⋊Q8C2212yes—
23#16840714C2212yes—
24#16840710—22no—
25#16840704—22no—
26#16840634C2218yes—
27C2×#41×D61C2211yescharacteristic
28C2×M16×D61C2212yes—
29#16839746C411yescharacteristic, complemented
30#16839745C412yescomplemented
31#16839744C412yescomplemented
32#16839743C412yescomplemented
33#101×D61C2212yes—
34#100×D61C2212yes—
35C61×#413C411yescharacteristic, complemented
36C23×#6923—21no—
37Q8×#6923—24no—
38Q8×#6923—24no—
39D4×#6923—28no—
40D4×#6923—28no—
41D4×#6923—24no—
42D4×#6923—24no—
43#14519856—24no—
44#14519856—24no—
45#14519856—22no—
46#14519856—24no—
47#14519856—22no—
48#14519856—22no—
49#14519856—22no—
50#14519856D414yes—
51#14519856C2314yes—
52C2×C4⋊#6923—22no—
53C2×C4⋊#6923—22no—
54C2×C4⋊#6923—22no—
55C2×C4⋊#6923—22no—
56C2×C4⋊#6923—21no—
57C2×C4⋊#6923—21no—
58C2×C4⋊#6923D412yes—
59C2×C4⋊#6923D412yes—
60C2×C4⋊#6923C2312yes—
61C2×C22⋊#6923—24no—
62C2×C22⋊#6923—22no—
63C2×C22⋊#6923—21no—
64C2×C22⋊#6923—24no—
65C2×C4×#6923—24no—
66C2×C4×#6923—22no—
67C2×C4×#6923C2312yes—
68#14519846—28no—
69#14519845—28no—
70(C2×#6923)⋊C4—24no—
71C2×#11758226C2×C411yescharacteristic
72C2×#11758222C2×C411yescharacteristic
73C2×#11758222Q812yes—
74C2×#11758224C2×C411yescharacteristic
75C2×#11758224Q812yes—
76#41×D61—22no—
77#41×D61—22no—
78#41×D61—22no—
79#41×D61—22no—
80C2×Q8×D61C2311yescharacteristic
81C2×D4×D61D412yes—
82C2×D4×D61C2311yescharacteristic
83C22×C4×D61D412yes—
84C22×C4×D61C2311yescharacteristic
85C2×#11758194C2×C412yescomplemented
86C2×#11758189—22no—
87M16×D61—22no—
88M16×D61—22no—
89M16×D61C2×C414yescomplemented
90M16×D61C2314yes—
91C2×C8×D61—22no—
92#14519696C2×C412yescomplemented
93#14519695C2×C414yescomplemented
94#14519694C2×C412yescomplemented
95#14519693C2×C414yescomplemented
96C2×#41×C61C2×C411yescharacteristic
97C2×M16×C61C2×C412yescomplemented
98#101×C61C2×C412yescomplemented
99#100×C61C2×C412yescomplemented
100C23×D61C2×D412yes—
101C23×D61C2×D411yescharacteristic
102C22×#6923—24no—
103C22×#6923—24no—
104C22×#6923—22no—
105C22×#6923—22no—
106C22×#6923—22no—
107C22×#6923—22no—
108C22×#6923—42no—
109C22×#6923—82no—
110#41×C61—22no—
111#41×C61—22no—
112C2×Q8×C61C22×C411yescharacteristic
113C2×D4×C61C4⋊C412yes—
114C2×D4×C61C22×C411yescharacteristic
115C22×C4×C61C4⋊C412yes—
116C22×C4×C61C22×C411yescharacteristic
117C2×C22⋊D61C4⋊C412yes—
118C2×C22⋊D61C4⋊C412yes—
119C2×C22⋊D61C2×Q812yes—
120C22×Dic61C4⋊C412yes—
121C22×Dic61C2×Q811yescharacteristic
122#11758226—22no—
123#11758226—22no—
124Q8×D61—22no—
125Q8×D61—22no—
126#11758224—24no—
127#11758224—24no—
128#11758224—22no—
129#11758224—22no—
130D4×D61—24no—
131D4×D61—24no—
132D4×D61—22no—
133D4×D61—22no—
134#11758222—22no—
135#11758222—22no—
136#11758222—24no—
137#11758222—24no—
138C2×D244C4⋊C412yes—
139C2×D244C22×C411yescharacteristic
140C2×C4×D61—22no—
141C2×C4×D61—22no—
142C2×C4×D61—22no—
143C2×C4×D61—22no—
144C2×C4×D61C4⋊C412yes—
145C2×C4×D61C4⋊C412yes—
146C2×C4×D61C22×C411yescharacteristic
147C2×C4×D61C2×D412yes—
148C2×C4×D61C2×D412yes—
149C2×C4×D61C2×D412yes—
150C2×C4×D61C2×D411yescharacteristic
151C2×C4×D61C2×Q811yescharacteristic
152C2×C4×D61#4112yes—
153C2×C4×D61#4112yes—
154C2×C4×D61C2411yescharacteristic
155C2×Dic122C22×C411yescharacteristic
156C2×Dic122C2×Q812yes—
157C22⋊#6923—28no—
158C22⋊#6923—24no—
159C22⋊#6923—24no—
160C22⋊#6923—42no—
161C22⋊#6923—84no—
162C4⋊#6923—24no—
163C4⋊#6923—24no—
164C4⋊#6923—22no—
165C4⋊#6923—22no—
166C4⋊#6923—24no—
167C4⋊#6923—41no—
168C4⋊#6923—41no—
169C4⋊#6923—41no—
170C4⋊#6923—41no—
171C4⋊#6923C2×D414yes—
172C4⋊#6923C2×D414yes—
173C4×#6923—28no—
174C4×#6923—24no—
175C4×#6923—24no—
176C4×#6923—24no—
177M16×C61—22no—
178M16×C61C22×C414yes—
179C2×C8×C61—22no—
180#11758194—22no—
181#11758194C22×C414yes—
182C2×#92418—22no—
183#11758189—24no—
184#11758189—24no—
185C8×D61—24no—
186C8×D61—24no—
187C23×C61C2×C4⋊C412yes—
188C23×C61C2×C4⋊C411yescharacteristic
189C22×D61—22no—
190C22×D61—22no—
191C22×D61—41no—
192C22×D61—41no—
193C22×D61—42no—
194C22×D61—42no—
195C22×D61C2×C4⋊C412yes—
196C22×D61C2×C4⋊C411yescharacteristic
197C22×D61#12012yes—
198C22×D61#12312yes—
199C22×D61C22×D411yescharacteristic
200C2×#6923—24no—
201C2×#6923—24no—
202C2×#6923—42no—
203C2×#6923—44no—
204C2×#6923—42no—
205C2×#6923—42no—
206C2×#6923—42no—
207C2×#6923—82no—
208C2×#6923—82no—
209C2×#6923—84no—
210C2×#6923—84no—
211C2×#6923—84no—
212Q8×C61—22no—
213D4×C61—24no—
214D4×C61—22no—
215C2×C4×C61—22no—
216C2×C4×C61—22no—
217C2×C4×C61C2×C4⋊C412yes—
218C2×C4×C61C2×C4⋊C411yescharacteristic
219C2×C4×C61C4×D412yes—
220C2×C4×C61C4×D412yes—
221C2×C4×C61C23×C411yescharacteristic
222C22⋊D61—24no—
223C22⋊D61—24no—
224C22⋊D61—24no—
225C2×Dic61—22no—
226C2×Dic61—22no—
227C2×Dic61C2×C4⋊C412yes—
228C2×Dic61C2×C4⋊C411yescharacteristic
229C2×Dic61C22⋊Q812yes—
230C2×Dic61C22⋊Q812yes—
231C2×Dic61C22×Q811yescharacteristic
232D244—24no—
233D244—22no—
234C4×D61—22no—
235C4×D61—22no—
236C4×D61—22no—
237C4×D61—22no—
238C4×D61—22no—
239C4×D61—22no—
240C4×D61C4×D412yes—
241C4×D61C4×D412yes—
242C4×D61#12011yescharacteristic
243C4×D61#12011yescharacteristic
244C4×D61C22⋊Q812yes—
245C4×D61C22⋊Q812yes—
246C4×D61#12311yescharacteristic
247C4×D61#12311yescharacteristic
248C4×D61C22×D412yes—
249C4×D61C2×#4112yes—
250Dic122—22no—
251Dic122—24no—
252C8×C61—24no—
253#92418—24no—
254#20858—611nomaximal, Sylow
255C22×C61—22no—
256C22×C61—41no—
257C22×C61—42no—
258C22×C61#38512yes—
259C22×C61C22×C4⋊C411yescharacteristic
260C2×D61—21no—
261C2×D61—22no—
262C2×D61—21no—
263C2×D61—22no—
264C2×D61—22no—
265C2×D61—41no—
266C2×D61—42no—
267C2×D61—42no—
268C2×D61—42no—
269C2×D61—42no—
270C2×D61—82no—
271C2×D61#38512yes—
272C2×D61C2×#12011yescharacteristic
273C2×D61C2×#12311yescharacteristic
274#6923—44no—
275#6923—44no—
276#6923—82no—
277#6923—82no—
278C4×C61—22no—
279C4×C61#38511yescharacteristic
280C4×C61#38511yescharacteristic
281C4×C61C2×C4×D412yes—
282Dic61—22no—
283Dic61#38511yescharacteristic
284Dic61#38511yescharacteristic
285Dic61C2×C22⋊Q812yes—
286#3428—611no—
287C2×#428—612no—
288C2×#413—611no—
289#2311—611no—
290#1857—614no—
291#1856—614no—
292#1733—612no—
293C2×C61—21no—
294C2×C61—22no—
295C2×C61—42no—
296C2×C61C2×#38511yescharacteristic
297D61—21no—
298D61—22no—
299D61—42no—
300D61#302111yescharacteristic
301D61#302211yescharacteristic
302C22×#41—611no—
303C22×M16—612no—
304#520—1222no—
305#520—1222no—
306#519—612no—
307#518—612no—
308#518—1222no—
309C4×#41—1224no—
310C2×#117—611no—
311C2×#117—612no—
312C2×#117—1222no—
313C22×C4⋊C4—611no—
314#428—618no—
315#428—1224no—
316C2×#107—1222no—
317C2×#106—1222no—
318#413—611no—
319#413—612no—
320#413—612no—
321#413—612no—
322#413#692311yescharacteristic, complemented
323C2×#101—612no—
324C2×#100—612no—
325#391—612no—
326#390—612no—
327#386—1222no—
328#380—1222no—
329#339—618no—
330C61#2085811yescharacteristic, complemented, Sylow
331C2×#41—611no—
332C2×#41—612no—
333C2×#41—611no—
334C2×#41—612no—
335C2×#41—611no—
336C2×#41—1222no—
337C2×#41—1222no—
338C2×#41—1222no—
339C2×#41—1222no—
340C2×#41C2×#692311yescharacteristic
341C22×Q8—611no—
342C22×D4—612no—
343C22×D4—611no—
344C23×C4—611no—
345C23×C4—612no—
346C23×C4—1221no—
347C2×M16—612no—
348C2×M16—614no—
349C2×M16—614no—
350C2×M16—1222no—
351C2×M16—1222no—
352C2×M16—1222no—
353C2×M16C2×#692312yescomplemented
354C22×C8—1222no—
355C4×Q8—1224no—
356C4×Q8—1224no—
357C4×D4—1224no—
358C4×D4—1224no—
359C4×D4—1228no—
360C4×D4—1228no—
361#117—614no—
362#117—614no—
363#117—1222no—
364#117—1222no—
365#117—1224no—
366#117—1224no—
367#117—1224no—
368#117—1222no—
369#117—1222no—
370C2×C4⋊C4—612no—
371C2×C4⋊C4—612no—
372C2×C4⋊C4—612no—
373C2×C4⋊C4—1222no—
374C2×C4⋊C4—1222no—
375C2×C4⋊C4—1222no—
376C2×C4⋊C4—1222no—
377C2×C4⋊C4—1221no—
378C2×C4⋊C4—1221no—
379C2×C22⋊C4—1222no—
380C2×C22⋊C4—1224no—
381C2×C22⋊C4—1224no—
382C2×C22⋊C4—1221no—
383C2×C42—612no—
384C2×C42—1222no—
385C2×C42—1224no—
386#107—1228no—
387#106—1228no—
388#101—612no—
389#101—614no—
390#101C2×#692312yescomplemented
391#100—612no—
392#100—614no—
393#100C2×#692312yescomplemented
394#95—1224no—
395C24—611no—
396C24—612no—
397#41—22no—
398#41—22no—
399#41—1224no—
400#41—1224no—
401#41—1224no—
402#41—1224no—
403#41—1222no—
404#41—1222no—
405#41—1222no—
406#41—1222no—
407#41—1222no—
408#41—1222no—
409C2×Q8—612no—
410C2×Q8—611no—
411C2×Q8—1222no—
412C2×Q8—1222no—
413C2×Q8C22×#692311yescharacteristic
414C2×D4—612no—
415C2×D4—612no—
416C2×D4—612no—
417C2×D4—612no—
418C2×D4—611no—
419C2×D4—1222no—
420C2×D4—1222no—
421C2×D4—1224no—
422C2×D4—1224no—
423C2×D4C4⋊#692312yes—
424C2×D4C22×#692311yescharacteristic
425C22×C4—611no—
426C22×C4—611no—
427C22×C4—611no—
428C22×C4—612no—
429C22×C4—612no—
430C22×C4—612no—
431C22×C4—612no—
432C22×C4—611no—
433C22×C4—611no—
434C22×C4—612no—
435C22×C4—612no—
436C22×C4—612no—
437C22×C4—612no—
438C22×C4—1222no—
439C22×C4—1222no—
440C22×C4—1222no—
441C22×C4—1222no—
442C22×C4—1222no—
443C22×C4—1222no—
444C22×C4—1224no—
445C22×C4—1224no—
446C22×C4—1222no—
447C22×C4—1222no—
448C22×C4—4882no—
449C22×C4—2442no—
450C22×C4C4⋊#692312yes—
451C22×C4C22×#692311yescharacteristic
452M16—22no—
453M16—614no—
454M16—1224no—
455M16—1224no—
456M16—1222no—
457M16C22×#692314yes—
458C2×C8—22no—
459C2×C8—1224no—
460C2×C8—1224no—
461C2×C8—1222no—
462C4⋊C4—614no—
463C4⋊C4—614no—
464C4⋊C4—1222no—
465C4⋊C4—1222no—
466C4⋊C4—1224no—
467C4⋊C4—1224no—
468C4⋊C4—1224no—
469C4⋊C4—2441no—
470C4⋊C4—2441no—
471C4⋊C4—2441no—
472C4⋊C4—2441no—
473C22⋊C4—1224no—
474C22⋊C4—1224no—
475C22⋊C4—1228no—
476C22⋊C4—4884no—
477C22⋊C4—2442no—
478C42—1224no—
479C42—1224no—
480C42—1224no—
481C42—1228no—
482C23—612no—
483C23—611no—
484C23—612no—
485C23—611no—
486C23—612no—
487C23—1222no—
488C23—1222no—
489C23—2441no—
490C23—2441no—
491C23—2442no—
492C23—2442no—
493C23C2×C4⋊#692312yes—
494C23C2×C4⋊#692311yescharacteristic
495Q8—22no—
496Q8—1224no—
497Q8—1222no—
498D4—24no—
499D4—22no—
500D4—1224no—
501D4—1224no—
502D4—1224no—
503D4—1224no—
504D4—1222no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—612no—
508C2×C4—612no—
509C2×C4—611no—
510C2×C4—611no—
511C2×C4—611no—
512C2×C4—611no—
513C2×C4—612no—
514C2×C4—612no—
515C2×C4—611no—
516C2×C4—611no—
517C2×C4—612no—
518C2×C4—612no—
519C2×C4—612no—
520C2×C4—612no—
521C2×C4—612no—
522C2×C4—1222no—
523C2×C4—1222no—
524C2×C4—1222no—
525C2×C4—1222no—
526C2×C4—1222no—
527C2×C4—1222no—
528C2×C4—1222no—
529C2×C4—1222no—
530C2×C4—1224no—
531C2×C4—1224no—
532C2×C4—4884no—
533C2×C4—4882no—
534C2×C4—4882no—
535C2×C4—4884no—
536C2×C4—4884no—
537C2×C4—2442no—
538C2×C4—2442no—
539C2×C4—2442no—
540C2×C4—2444no—
541C2×C4—2442no—
542C2×C4C2×C4⋊#692312yes—
543C2×C4C2×C4⋊#692311yescharacteristic
544C2×C4D4×#692312yes—
545C2×C4D4×#692312yes—
546C2×C4C23×#692311yescharacteristic
547C8—24no—
548C8—1224no—
549C22—22no—
550C22—41no—
551C22—42no—
552C22—611no—
553C22—611no—
554C22—612no—
555C22—1222no—
556C22—1222no—
557C22—1221no—
558C22—1221no—
559C22—1222no—
560C22—4882no—
561C22—2441no—
562C22—2442no—
563C22—2442no—
564C22—2442no—
565C22—2442no—
566C22#1684070912yes—
567C22C22×C4⋊#692311yescharacteristic
568C4—22no—
569C4—612no—
570C4—611no—
571C4—611no—
572C4—1222no—
573C4—4882no—
574C4—4882no—
575C4—2444no—
576C4—2444no—
577C4#1684070911yescharacteristic
578C4#1684070911yescharacteristic
579C4C2×D4×#692312yes—
580C2—21no—
581C2—22no—
582C2—42no—
583C2—611no—
584C2—611no—
585C2—1221no—
586C2—1222no—
587C2—2442no—
588C2C2×#1684070911yescharacteristic
subgroupG/H
centerC2C2×#16840709
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC61×#413C4
socleC2×C61C2×#385
prime radicalC2×#41×D61C22
cosocleC2×C4×D61C24
hypercenter#413#6923
hypocenterC61#20858
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C22×C4 ◁ #413

Complements

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

NH ≅ G/Ntype
2. C2×#16840773C2semidirect
4. #21461775C2semidirect
5. #21461774C2semidirect
6. #21461501C2semidirect
29. #16839746C4semidirect
30. #16839745C4semidirect
31. #16839744C4semidirect
32. #16839743C4semidirect
35. C61×#413C4semidirect
85. C2×#11758194C2×C4semidirect
89. M16×D61C2×C4semidirect
92. #14519696C2×C4semidirect
93. #14519695C2×C4semidirect
94. #14519694C2×C4semidirect
95. #14519693C2×C4semidirect
97. C2×M16×C61C2×C4semidirect
98. #101×C61C2×C4semidirect
99. #100×C61C2×C4semidirect
322. #413#6923semidirect
330. C61#20858semidirect
353. C2×M16C2×#6923semidirect
390. #101C2×#6923semidirect
393. #100C2×#6923semidirect

Maximal subgroups

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

subgroupindex
#214627992
C2×#168407732
#214622942
#214617752
#214617742
#214615012
#413×D612
#2085861