Finite Group Databasebeta

← #30602525

Subgroups — #30602525

Order 15616.

1254 conjugacy classes of subgroups.

The subgroup lattice has 566 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840831C211yesmaximal, characteristic
2D61⋊#410C211yesmaximal, characteristic
3#21461507C212yesmaximal, complemented
4#21461501C212yesmaximal, complemented
5#413×D61C211yesmaximal, characteristic
6#16840831—24no—
7#16840831—24no—
8C2×Q8×#6923—21no—
9C2×D4×#6923—22no—
10C2×D4×#6923—21no—
11C2×#14519856C412yescomplemented
12C2×#14519856C2212yes—
13C2×#14519856C2211yescharacteristic
14C22×C4⋊#6923C412yescomplemented
15C22×C4⋊#6923C2211yescharacteristic
16#16840743—21no—
17#16840742—21no—
18#16840741C414yescomplemented
19D61⋊#99—22no—
20#16840636—24no—
21#16840634—24no—
22C2×#41×D61C2211yescharacteristic
23C2×M16×D61C2212yes—
24#16839746C411yescharacteristic, complemented
25#16839745—21no—
26#16839744C412yescomplemented
27#16839743—21no—
28#101×D61—21no—
29#100×D61—21no—
30C61×#413C411yescharacteristic, complemented
31C23×#6923—22no—
32C23×#6923—21no—
33Q8×#6923—28no—
34D4×#6923—216no—
35D4×#6923—28no—
36#14519856—22no—
37#14519856—24no—
38#14519856—24no—
39#14519856—22no—
40#14519856—22no—
41#14519856—22no—
42#14519856C2×C414yescomplemented
43#14519856D414yes—
44C2×C4⋊#6923—22no—
45C2×C4⋊#6923—21no—
46C2×C4⋊#6923—22no—
47C2×C4⋊#6923—24no—
48C2×C4⋊#6923—24no—
49C2×C4⋊#6923—22no—
50C2×C4⋊#6923C2×C411yescharacteristic
51C2×C4⋊#6923C2×C412yes—
52C2×C4⋊#6923C2×C411yescharacteristic
53C2×C4⋊#6923D414yes—
54C2×C22⋊#6923—22no—
55C2×C22⋊#6923—21no—
56C2×C22⋊#6923—22no—
57C2×C4×#6923—22no—
58C2×C4×#6923—21no—
59C2×C4×#6923—22no—
60#14519836—24no—
61#14519835—24no—
62C2×#11758226D411yescharacteristic
63C2×#11758222C2×C411yescharacteristic
64C2×#11758222C2×C412yes—
65C2×#11758224D411yescharacteristic
66C2×#11758224Q812yes—
67#41×D61—22no—
68#41×D61—22no—
69#41×D61—22no—
70#41×D61—22no—
71C2×Q8×D61D411yescharacteristic
72C2×D4×D61D412yes—
73C2×D4×D61D411yescharacteristic
74C22×C4×D61C2×C412yes—
75C22×C4×D61C2311yescharacteristic
76C2×#11758194C2×C412yescomplemented
77C2×#11758189—22no—
78M16×D61—22no—
79M16×D61—22no—
80M16×D61—22no—
81M16×D61—22no—
82C2×C8×D61—22no—
83#14519696—21no—
84#14519695—22no—
85#14519694—21no—
86#14519693—22no—
87C2×#41×C61C2×C411yescharacteristic
88C2×M16×C61C2×C412yescomplemented
89#101×C61—21no—
90#100×C61—21no—
91C23×D61C22⋊C412yes—
92C23×D61C2×D411yescharacteristic
93C22×#6923—21no—
94C22×#6923—22no—
95C22×#6923—21no—
96C22×#6923—28no—
97C22×#6923—24no—
98C22×#6923—42no—
99C22×#6923—82no—
100C22×#6923—84no—
101#41×C61—22no—
102#41×C61—22no—
103C2×Q8×C61C22⋊C411yescharacteristic
104C2×D4×C61C22⋊C411yescharacteristic
105C2×D4×C61C4⋊C412yes—
106C22×C4×C61C4212yes—
107C22×C4×C61C22×C411yescharacteristic
108C2×C22⋊D61C22⋊C412yes—
109C2×C22⋊D61C4⋊C412yes—
110C2×C22⋊D61C4⋊C412yes—
111C22×Dic61C4⋊C412yes—
112C22×Dic61C2×Q811yescharacteristic
113#11758226—22no—
114#11758226—22no—
115Q8×D61—22no—
116Q8×D61—22no—
117#11758224—24no—
118#11758224—24no—
119#11758224—22no—
120#11758224—22no—
121D4×D61—24no—
122D4×D61—24no—
123D4×D61—22no—
124D4×D61—22no—
125#11758222—22no—
126#11758222—22no—
127#11758222—24no—
128#11758222—24no—
129C2×D244C22⋊C412yes—
130C2×D244C22⋊C411yescharacteristic
131C2×C4×D61—22no—
132C2×C4×D61—22no—
133C2×C4×D61—22no—
134C2×C4×D61—22no—
135C2×C4×D61C4212yes—
136C2×C4×D61C22⋊C412yes—
137C2×C4×D61C4⋊C412yes—
138C2×C4×D61C22×C411yescharacteristic
139C2×C4×D61C22×C411yescharacteristic
140C2×C4×D61C2×D412yes—
141C2×C4×D61C2×D412yes—
142C2×C4×D61C2×D411yescharacteristic
143C2×C4×D61C2×D411yescharacteristic
144C2×C4×D61#4112yes—
145C2×C4×D61#4112yes—
146C2×Dic122C22⋊C411yescharacteristic
147C2×Dic122C4⋊C412yes—
148C22⋊#6923—28no—
149C22⋊#6923—24no—
150C22⋊#6923—24no—
151C22⋊#6923—42no—
152C4⋊#6923—28no—
153C4⋊#6923—22no—
154C4⋊#6923—21no—
155C4⋊#6923—21no—
156C4⋊#6923—24no—
157C4⋊#6923—24no—
158C4⋊#6923—42no—
159C4⋊#6923C22⋊C412yes—
160C4⋊#6923C22⋊C414yes—
161C4⋊#6923C22⋊C412yes—
162C4×#6923—28no—
163C4×#6923—24no—
164C4×#6923—24no—
165C4×#6923—42no—
166M16×C61—22no—
167M16×C61—22no—
168C2×C8×C61—22no—
169#11758194—22no—
170#11758194—22no—
171C2×#92418—22no—
172#11758189—42no—
173#11758189—42no—
174C8×D61—42no—
175C8×D61—42no—
176C23×C61#9512yes—
177C23×C61C2×C4⋊C411yescharacteristic
178C22×D61—22no—
179C22×D61—22no—
180C22×D61—41no—
181C22×D61—41no—
182C22×D61—42no—
183C22×D61—42no—
184C22×D61#9512yes—
185C22×D61C2×C22⋊C411yescharacteristic
186C22×D61C2×C22⋊C411yescharacteristic
187C22×D61#12012yes—
188C22×D61#12312yes—
189C2×#6923—22no—
190C2×#6923—24no—
191C2×#6923—22no—
192C2×#6923—42no—
193C2×#6923—44no—
194C2×#6923—42no—
195C2×#6923—44no—
196C2×#6923—84no—
197C2×#6923—88no—
198C2×#6923—82no—
199C2×#6923—82no—
200Q8×C61—22no—
201D4×C61—24no—
202D4×C61—22no—
203C2×C4×C61—22no—
204C2×C4×C61—22no—
205C2×C4×C61#9512yes—
206C2×C4×C61C2×C4211yescharacteristic
207C2×C4×C61C2×C22⋊C411yescharacteristic
208C2×C4×C61C4×D412yes—
209C2×C4×C61C4×D412yes—
210C22⋊D61—24no—
211C22⋊D61—24no—
212C22⋊D61—24no—
213C2×Dic61—22no—
214C2×Dic61—22no—
215C2×Dic61#9512yes—
216C2×Dic61C2×C4⋊C411yescharacteristic
217C2×Dic61C2×C4⋊C411yescharacteristic
218C2×Dic61C22⋊Q812yes—
219C2×Dic61C22⋊Q812yes—
220D244—24no—
221D244—22no—
222C4×D61—22no—
223C4×D61—22no—
224C4×D61—22no—
225C4×D61—22no—
226C4×D61—22no—
227C4×D61—22no—
228C4×D61C2×C22⋊C411yescharacteristic
229C4×D61C2×C22⋊C411yescharacteristic
230C4×D61#11711yescharacteristic
231C4×D61#11711yescharacteristic
232C4×D61C4×D412yes—
233C4×D61C4×D412yes—
234C4×D61#12012yes—
235C4×D61#12112yes—
236C4×D61#12112yes—
237C4×D61#12312yes—
238Dic122—22no—
239Dic122—24no—
240C8×C61—42no—
241#92418—42no—
242#10765—611nomaximal, Sylow
243C22×C61—22no—
244C22×C61—41no—
245C22×C61—42no—
246C22×C61C2×#9511yescharacteristic
247C22×C61#38512yes—
248C2×D61—21no—
249C2×D61—22no—
250C2×D61—21no—
251C2×D61—22no—
252C2×D61—22no—
253C2×D61—41no—
254C2×D61—42no—
255C2×D61—44no—
256C2×D61—42no—
257C2×D61—42no—
258C2×D61—42no—
259C2×D61#37911yescharacteristic
260C2×D61#38111yescharacteristic
261C2×D61#38612yes—
262#6923—44no—
263#6923—42no—
264#6923—42no—
265#6923—84no—
266C4×C61—22no—
267C4×C61C4×C22⋊C411yescharacteristic
268C4×C61C4×C22⋊C411yescharacteristic
269C4×C61#38612yes—
270Dic61—22no—
271Dic61#38011yescharacteristic
272Dic61#38011yescharacteristic
273Dic61#38512yes—
274C2×#520—611no—
275C2×#413—611no—
276C2×#410—611no—
277#1736—612no—
278#1733—612no—
279C2×C61—21no—
280C2×C61—22no—
281C2×C61—42no—
282C2×C61C22⋊#9511yescharacteristic
283D61—22no—
284D61—21no—
285D61—42no—
286D61#179511yescharacteristic
287D61#179611yescharacteristic
288C22×#41—611no—
289C22×M16—612no—
290#520—1224no—
291#520—1224no—
292C2×C4×Q8—1221no—
293C2×C4×D4—1222no—
294C2×C4×D4—1221no—
295C2×#117—611no—
296C2×#117—612no—
297C2×#117—612no—
298C22×C4⋊C4—611no—
299C22×C4⋊C4—612no—
300#413—611no—
301#413—612no—
302#413—1221no—
303#413—1221no—
304#413#692311yescharacteristic, complemented
305C2×#101—1221no—
306C2×#100—1221no—
307#410—614no—
308#410—1221no—
309#410—1221no—
310C2×#99—1222no—
311#340—1224no—
312#339—1224no—
313C61#1076511yescharacteristic, complemented, Sylow
314C2×#41—611no—
315C2×#41—612no—
316C2×#41—611no—
317C2×#41—612no—
318C2×#41—611no—
319C2×#41—1222no—
320C2×#41—1222no—
321C2×#41—1222no—
322C2×#41—1222no—
323C2×#41C2×#692311yescharacteristic
324C22×Q8—611no—
325C22×D4—612no—
326C22×D4—611no—
327C23×C4—611no—
328C23×C4—612no—
329C23×C4—1221no—
330C23×C4—1222no—
331C2×M16—612no—
332C2×M16—1222no—
333C2×M16—1222no—
334C2×M16—1222no—
335C2×M16—1222no—
336C2×M16—1222no—
337C2×M16C2×#692312yescomplemented
338C22×C8—1222no—
339C4×Q8—1228no—
340C4×D4—1228no—
341C4×D4—12216no—
342#117—614no—
343#117—614no—
344#117—1224no—
345#117—1224no—
346#117—1222no—
347#117—1222no—
348#117—1222no—
349#117—1222no—
350C2×C4⋊C4—612no—
351C2×C4⋊C4—611no—
352C2×C4⋊C4—614no—
353C2×C4⋊C4—611no—
354C2×C4⋊C4—1224no—
355C2×C4⋊C4—1224no—
356C2×C4⋊C4—1221no—
357C2×C4⋊C4—1222no—
358C2×C4⋊C4—1222no—
359C2×C4⋊C4—1222no—
360C2×C22⋊C4—1221no—
361C2×C22⋊C4—1222no—
362C2×C22⋊C4—1222no—
363C2×C42—1221no—
364C2×C42—1222no—
365C2×C42—1222no—
366#101—21no—
367#101—1222no—
368#101—1221no—
369#100—21no—
370#100—1222no—
371#100—1221no—
372#99—1224no—
373#99—1224no—
374C24—611no—
375C24—612no—
376#41—22no—
377#41—22no—
378#41—1224no—
379#41—1224no—
380#41—1224no—
381#41—1224no—
382#41—1222no—
383#41—1222no—
384#41—1222no—
385#41—1222no—
386#41—1222no—
387#41—1222no—
388C2×Q8—612no—
389C2×Q8—611no—
390C2×Q8—1222no—
391C2×Q8—1222no—
392C2×Q8C22⋊#692311yescharacteristic
393C2×D4—612no—
394C2×D4—612no—
395C2×D4—612no—
396C2×D4—612no—
397C2×D4—611no—
398C2×D4—1222no—
399C2×D4—1222no—
400C2×D4—1224no—
401C2×D4—1224no—
402C2×D4C4⋊#692312yes—
403C2×D4C22⋊#692311yescharacteristic
404C22×C4—611no—
405C22×C4—611no—
406C22×C4—611no—
407C22×C4—611no—
408C22×C4—611no—
409C22×C4—612no—
410C22×C4—612no—
411C22×C4—612no—
412C22×C4—612no—
413C22×C4—612no—
414C22×C4—612no—
415C22×C4—612no—
416C22×C4—612no—
417C22×C4—1222no—
418C22×C4—1222no—
419C22×C4—1222no—
420C22×C4—1222no—
421C22×C4—1224no—
422C22×C4—1228no—
423C22×C4—1221no—
424C22×C4—1221no—
425C22×C4—1222no—
426C22×C4—4882no—
427C22×C4—4884no—
428C22×C4—2442no—
429C22×C4C4×#692312yes—
430C22×C4C22×#692311yescharacteristic
431M16—22no—
432M16—22no—
433M16—1222no—
434M16—1222no—
435M16—2442no—
436M16—2442no—
437C2×C8—22no—
438C2×C8—1222no—
439C2×C8—2442no—
440C2×C8—2442no—
441C4⋊C4—614no—
442C4⋊C4—612no—
443C4⋊C4—612no—
444C4⋊C4—1221no—
445C4⋊C4—1222no—
446C4⋊C4—1224no—
447C4⋊C4—1221no—
448C4⋊C4—1224no—
449C4⋊C4—1228no—
450C4⋊C4—2442no—
451C22⋊C4—1224no—
452C22⋊C4—1228no—
453C22⋊C4—1224no—
454C22⋊C4—2442no—
455C42—1224no—
456C42—1224no—
457C42—1228no—
458C42—2442no—
459C23—612no—
460C23—611no—
461C23—612no—
462C23—611no—
463C23—612no—
464C23—1222no—
465C23—1222no—
466C23—2441no—
467C23—2441no—
468C23—2442no—
469C23—2442no—
470C23(C2×#6923)⋊C412yes—
471C23C2×C4⋊#692311yescharacteristic
472Q8—22no—
473Q8—1224no—
474Q8—1222no—
475D4—24no—
476D4—22no—
477D4—1224no—
478D4—1224no—
479D4—1224no—
480D4—1224no—
481D4—1222no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—612no—
485C2×C4—612no—
486C2×C4—611no—
487C2×C4—611no—
488C2×C4—612no—
489C2×C4—612no—
490C2×C4—612no—
491C2×C4—612no—
492C2×C4—611no—
493C2×C4—611no—
494C2×C4—611no—
495C2×C4—611no—
496C2×C4—612no—
497C2×C4—612no—
498C2×C4—612no—
499C2×C4—1222no—
500C2×C4—1222no—
501C2×C4—1222no—
502C2×C4—1222no—
503C2×C4—1222no—
504C2×C4—1222no—
505C2×C4—1222no—
506C2×C4—1222no—
507C2×C4—1222no—
508C2×C4—1222no—
509C2×C4—1224no—
510C2×C4—4882no—
511C2×C4—4882no—
512C2×C4—4884no—
513C2×C4—4888no—
514C2×C4—2444no—
515C2×C4—2442no—
516C2×C4—2444no—
517C2×C4—2442no—
518C2×C4(C2×#6923)⋊C412yes—
519C2×C4C2×C4×#692311yescharacteristic
520C2×C4C2×C22⋊#692311yescharacteristic
521C2×C4D4×#692312yes—
522C2×C4D4×#692312yes—
523C8—42no—
524C8—2442no—
525C22—22no—
526C22—41no—
527C22—42no—
528C22—611no—
529C22—611no—
530C22—612no—
531C22—1222no—
532C22—1222no—
533C22—1221no—
534C22—1221no—
535C22—1222no—
536C22—2441no—
537C22—2442no—
538C22—2442no—
539C22—2442no—
540C22—2442no—
541C22—2444no—
542C22C2×(C2×#6923)⋊C411yescharacteristic
543C22#1684070912yes—
544C4—22no—
545C4—611no—
546C4—611no—
547C4—612no—
548C4—1222no—
549C4—4884no—
550C4—2444no—
551C4—2442no—
552C4—2442no—
553C4C22⋊C4×#692311yescharacteristic
554C4C22⋊C4×#692311yescharacteristic
555C4#1684071012yes—
556C2—21no—
557C2—22no—
558C2—42no—
559C2—611no—
560C2—611no—
561C2—1222no—
562C2—1221no—
563C2—2442no—
564C2#2146099211yescharacteristic
subgroupG/H
centerC2#21460992
commutatorC2×C4×C61C2×C42
FrattiniC22×C4C22×#6923
FittingC61×#413C4
socleC2×C61C22⋊#95
prime radicalC2×#41×D61C22
cosocleC22×C4×D61C23
hypercenter#413#6923
hypocenterC61#10765
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
3. #21461507C2semidirect
4. #21461501C2semidirect
11. C2×#14519856C4semidirect
14. C22×C4⋊#6923C4semidirect
18. #16840741C4semidirect
24. #16839746C4semidirect
26. #16839744C4semidirect
30. C61×#413C4semidirect
42. #14519856C2×C4semidirect
76. C2×#11758194C2×C4semidirect
88. C2×M16×C61C2×C4semidirect
304. #413#6923semidirect
313. C61#10765semidirect
337. C2×M16C2×#6923semidirect

Maximal subgroups

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

subgroupindex
C2×#168408312
D61⋊#4102
#214615072
#214615012
#413×D612
#1076561