Finite Group Databasebeta

← #30617746

Subgroups — #30617746

Order 15616.

1318 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462418C212yesmaximal, complemented
2#21461868C214yesmaximal, complemented
3#21461829C214yesmaximal, complemented
4#21461730C212yesmaximal, complemented
5#21461649C212yesmaximal, complemented
6#21459460C211yesmaximal, characteristic
7#16840829C2214yes—
8#16840742—22no—
9#16840741—22no—
10D61⋊#99—22no—
11D61⋊#99C2214yes—
12C2×#14519835—22no—
13C2×#14519835C2214yes—
14C2×#14519835C2214yescomplemented
15#16840705C2214yescomplemented
16#16840639C2218yescomplemented
17C2×#14519822C2211yescharacteristic
18C22×#11758224C2211yescharacteristic
19#16840412—22no—
20#16840410—22no—
21#16840401—22no—
22#16840399—22no—
23#16840334C412yescomplemented
24#16840322C2212yes—
25#16840321C411yescharacteristic, complemented
26#16840268C412yescomplemented
27C2×#14519761C411yescharacteristic, complemented
28C2×#14519758C411yescharacteristic, complemented
29#123×D61C2212yes—
30#16840235C2211yescharacteristic
31C61×#534C411yescharacteristic, complemented
32D4×#6923—48no—
33#14519856—24no—
34C2×C22⋊#6923—24no—
35C2×C22⋊#6923C2318yes—
36#14519836—24no—
37#14519835—24no—
38#14519835—24no—
39#14519835—24no—
40#14519835—44no—
41(C2×#6923)⋊C4—44no—
42#14519822—44no—
43C2×#11758222D412yes—
44C2×#11758224—22no—
45C2×#11758224—22no—
46C2×#11758224—44no—
47C2×#11758224C2×C412yes—
48C2×#11758224D412yes—
49C22×Dic122C2×C411yescharacteristic, complemented
50C22×C22⋊D61C2×C411yescharacteristic
51C22×C22⋊D61D412yes—
52C22×C22⋊D61C2311yescharacteristic
53C22×C22⋊D61C2311yescharacteristic
54C2×D4×D61D412yes—
55C2×D4×D61C2312yes—
56C23×Dic61C2×C411yescharacteristic, complemented
57C23×Dic61C2×C411yescharacteristic, complemented
58C22×C4×D61C2311yescharacteristic
59#14519776—22no—
60#14519763—22no—
61#14519762—22no—
62#14519761—22no—
63#14519761C2×C412yescomplemented
64#14519761C2×C412yescomplemented
65#14519760C2×C412yescomplemented
66#14519759—22no—
67#14519759—22no—
68#14519758—22no—
69#14519758—21no—
70#14519758C2×C412yescomplemented
71#14519758C2×C412yescomplemented
72#14519755—22no—
73#14519754—22no—
74#14519754—22no—
75#14519750—22no—
76#14519749—22no—
77#14519749—22no—
78#14519748—22no—
79#14519746C2×C412yescomplemented
80#14519745—21no—
81#14519745—22no—
82D4×Dic61—22no—
83C4×C22⋊D61—22no—
84#14519737—22no—
85#14519737—22no—
86#14519733C2×C412yescomplemented
87C2×C4⋊Dic61—21no—
88C2×#11758196—21no—
89C4⋊C4×D61—22no—
90C2×C22⋊Dic61—21no—
91C2×C22⋊Dic61C2×C411yescharacteristic, complemented
92C2×#11758198—21no—
93C22⋊C4×D61—22no—
94C22⋊C4×D61C2312yes—
95C22×D4×C61C2×C411yescharacteristic
96C22×D4×C61C2×C411yescharacteristic, complemented
97#123×C61—21no—
98#123×C61C2×C412yescomplemented
99#121×C61—22no—
100#120×C61—21no—
101#120×C61C2×C412yescomplemented
102C2×C22⋊C4×C61C2×C411yescharacteristic, complemented
103C24×C61C22⋊C411yescharacteristic
104C24×C61C22×C411yescharacteristic
105C24×C61C22×C411yescharacteristic
106C23×D61C22⋊C411yescharacteristic
107C23×D61C2×D412yes—
108C23×D61C2411yescharacteristic
109C22×#6923—28no—
110C22×#6923—44no—
111C2×D4×C61—21no—
112C2×D4×C61—21no—
113C2×D4×C61—22no—
114C2×D4×C61—42no—
115C2×D4×C61—42no—
116C2×D4×C61C22⋊C412yes—
117C2×D4×C61C22×C412yes—
118C22×C4×C61—21no—
119C22×C4×C61C22⋊C411yescharacteristic
120C22×C4×C61C22×C411yescharacteristic
121C2×C22⋊D61—24no—
122C2×C22⋊D61—22no—
123C2×C22⋊D61—24no—
124C2×C22⋊D61—22no—
125C2×C22⋊D61—44no—
126C2×C22⋊D61—42no—
127C2×C22⋊D61C22⋊C412yes—
128C2×C22⋊D61C22⋊C412yes—
129C2×C22⋊D61C22×C412yes—
130C2×C22⋊D61C2×D412yes—
131C2×C22⋊D61C2×D412yes—
132C2×C22⋊D61C2×D412yes—
133C22×Dic61—21no—
134C22×Dic61—21no—
135C22×Dic61—21no—
136C22×Dic61—22no—
137C22×Dic61—42no—
138C22×Dic61—42no—
139C22×Dic61C22⋊C411yescharacteristic
140C22×Dic61C22⋊C412yes—
141C22×Dic61C22×C411yescharacteristic
142C22×Dic61C22×C411yescharacteristic
143C22×Dic61C22×C411yescharacteristic
144#11758224—44no—
145#11758224—44no—
146#11758224—48no—
147#11758224—48no—
148D4×D61—28no—
149D4×D61—44no—
150#11758222—44no—
151C2×D244C2×D412yes—
152C2×C4×D61—22no—
153C2×C4×D61—24no—
154C2×C4×D61—22no—
155C2×C4×D61C22⋊C412yes—
156C2×C4×D61C2×D412yes—
157C2×Dic122—22no—
158C2×Dic122—42no—
159C2×Dic122C22⋊C412yes—
160C22⋊#6923—28no—
161C22⋊#6923—28no—
162C22⋊#6923—44no—
163C4⋊#6923—44no—
164C4×#6923—44no—
165C4⋊C4×C61—22no—
166C22⋊C4×C61—21no—
167C22⋊C4×C61—22no—
168C22⋊C4×C61—22no—
169C22⋊C4×C61C22×C412yes—
170C22⋊Dic61—22no—
171C22⋊Dic61—22no—
172C22⋊Dic61—21no—
173C22⋊Dic61—22no—
174C22⋊Dic61C22×C412yes—
175#11758198—22no—
176#11758198—22no—
177#11758198—22no—
178#11758198—22no—
179C4⋊Dic61—22no—
180C4⋊Dic61—22no—
181#11758196—22no—
182#11758196—22no—
183#11758196—22no—
184#11758196—22no—
185C4×Dic61—22no—
186C23×C61—22no—
187C23×C61—21no—
188C23×C61—22no—
189C23×C61—41no—
190C23×C61—81no—
191C23×C61—81no—
192C23×C61—81no—
193C23×C61C2×C22⋊C412yes—
194C23×C61C2×C22⋊C411yescharacteristic
195C23×C61C2×C22⋊C411yescharacteristic
196C23×C61C23×C411yescharacteristic
197C22×D61—24no—
198C22×D61—41no—
199C22×D61—81no—
200C22×D61—82no—
201C22×D61—81no—
202C22×D61C2×C22⋊C412yes—
203C22×D61C22×D412yes—
204C22×D61H(5,𝔽2)14yes—
205C2×#6923—48no—
206C2×#6923—88no—
207C2×#6923—164no—
208D4×C61—24no—
209D4×C61—42no—
210D4×C61—42no—
211D4×C61—42no—
212D4×C61—82no—
213C2×C4×C61—22no—
214C2×C4×C61—21no—
215C2×C4×C61—22no—
216C2×C4×C61—41no—
217C2×C4×C61—81no—
218C2×C4×C61C2×C22⋊C412yes—
219C22⋊D61—24no—
220C22⋊D61—24no—
221C22⋊D61—24no—
222C22⋊D61—24no—
223C22⋊D61—42no—
224C22⋊D61—42no—
225C22⋊D61—44no—
226C22⋊D61—82no—
227C2×Dic61—22no—
228C2×Dic61—22no—
229C2×Dic61—21no—
230C2×Dic61—21no—
231C2×Dic61—22no—
232C2×Dic61—22no—
233C2×Dic61—42no—
234C2×Dic61—41no—
235C2×Dic61—44no—
236C2×Dic61—42no—
237C2×Dic61—42no—
238C2×Dic61—81no—
239C2×Dic61C2×C22⋊C412yes—
240C2×Dic61C2×C22⋊C412yes—
241D244—24no—
242C4×D61—24no—
243C4×D61—42no—
244C4×D61—44no—
245C4×D61—82no—
246Dic122—42no—
247Dic122—44no—
248#21777—611nomaximal, Sylow
249C22×C61—21no—
250C22×C61—21no—
251C22×C61—22no—
252C22×C61—41no—
253C22×C61—41no—
254C22×C61—41no—
255C22×C61—42no—
256C22×C61—42no—
257C22×C61—41no—
258C22×C61—82no—
259C22×C61—82no—
260C22×C61—161no—
261C22×C61C22×C22⋊C411yescharacteristic
262C22×C61#51812yes—
263C2×D61—22no—
264C2×D61—42no—
265C2×D61—44no—
266C2×D61—42no—
267C2×D61—84no—
268C2×D61—161no—
269C2×D61#53414yes—
270#6923—88no—
271C4×C61—22no—
272C4×C61—42no—
273C4×C61—41no—
274Dic61—22no—
275Dic61—22no—
276Dic61—42no—
277Dic61—41no—
278Dic61—41no—
279#3485—611no—
280#2883—612no—
281#1913—614no—
282#1890—614no—
283#1828—612no—
284#1782—612no—
285C2×C61—21no—
286C2×C61—22no—
287C2×C61—42no—
288C2×C61—41no—
289C2×C61—41no—
290C2×C61—81no—
291C2×C61#230811yescharacteristic
292D61—24no—
293D61—42no—
294C2×H(5,𝔽2)—611no—
295C22×#41—611no—
296#547—1222no—
297#547—1222no—
298#546—1222no—
299#546—1222no—
300#536—612no—
301#534—611no—
302#534—612no—
303#534#692311yescharacteristic, complemented
304#525—612no—
305C2×#123—611no—
306C2×#123—612no—
307C2×#123—611no—
308#518—611no—
309#518—614no—
310#410—1222no—
311#410—1222no—
312C2×#99—614no—
313C2×#99—614no—
314C2×#99—614no—
315C2×#99—1222no—
316C2×#99—1222no—
317#381—614no—
318#342—618no—
319C61#2177711yescharacteristic, complemented, Sylow
320H(5,𝔽2)—2444no—
321C2×#41—612no—
322C2×#41—612no—
323C2×#41—612no—
324C2×#41—1222no—
325C2×#41—1222no—
326C2×#41—2444no—
327C22×Q8—611no—
328C22×D4—611no—
329C22×D4—612no—
330C22×D4—611no—
331C22×D4—612no—
332C22×D4—612no—
333C22×D4—611no—
334C22×D4C2×#692311yescharacteristic
335C22×D4C2×#692311yescharacteristic, complemented
336C23×C4—611no—
337C23×C4—611no—
338C23×C4—611no—
339C4⋊Q8—1222no—
340#124—1222no—
341#123—21no—
342#123—612no—
343#123—612no—
344#123—612no—
345#123—612no—
346#123—612no—
347#123—1222no—
348#123—1222no—
349#123—1222no—
350#123—1222no—
351#123—1221no—
352#123—1222no—
353#123C2×#692312yescomplemented
354C22⋊Q8—1222no—
355C22⋊Q8—1222no—
356C22⋊Q8—1222no—
357#121—22no—
358#121—1222no—
359#121—1222no—
360#121—1222no—
361#121—1222no—
362#120—21no—
363#120—612no—
364#120—1222no—
365#120—1221no—
366#120C2×#692312yescomplemented
367C4×D4—1222no—
368C4×D4—1222no—
369C4×D4—1222no—
370C4×D4—1222no—
371C4×D4—2448no—
372#117—612no—
373#117—1224no—
374C2×C4⋊C4—1222no—
375C2×C4⋊C4—1221no—
376C2×C4⋊C4—1221no—
377C2×C22⋊C4—611no—
378C2×C22⋊C4—612no—
379C2×C22⋊C4—618no—
380C2×C22⋊C4—1222no—
381C2×C22⋊C4—1221no—
382C2×C22⋊C4—1221no—
383C2×C22⋊C4—1224no—
384C2×C22⋊C4C2×#692311yescharacteristic, complemented
385#99—1224no—
386#99—1224no—
387#99—1224no—
388#99—1224no—
389#99—2444no—
390#95—2444no—
391C24—611no—
392C24—612no—
393C24—611no—
394C24C22⋊#692311yescharacteristic
395C24C22×#692311yescharacteristic
396C24C22×#692311yescharacteristic
397#41—2444no—
398#41—2444no—
399#41—2444no—
400#41—2448no—
401#41—2448no—
402C2×Q8—612no—
403C2×Q8—1222no—
404C2×Q8—2442no—
405C2×D4—21no—
406C2×D4—21no—
407C2×D4—22no—
408C2×D4—42no—
409C2×D4—42no—
410C2×D4—612no—
411C2×D4—612no—
412C2×D4—612no—
413C2×D4—612no—
414C2×D4—612no—
415C2×D4—612no—
416C2×D4—612no—
417C2×D4—1228no—
418C2×D4—1224no—
419C2×D4—1222no—
420C2×D4—1224no—
421C2×D4—1222no—
422C2×D4—2444no—
423C2×D4—2442no—
424C2×D4—2444no—
425C2×D4C22⋊#692312yes—
426C2×D4C22×#692312yes—
427C22×C4—21no—
428C22×C4—611no—
429C22×C4—611no—
430C22×C4—612no—
431C22×C4—612no—
432C22×C4—612no—
433C22×C4—611no—
434C22×C4—611no—
435C22×C4—1222no—
436C22×C4—1224no—
437C22×C4—1222no—
438C22×C4—1222no—
439C22×C4—1221no—
440C22×C4—1221no—
441C22×C4—1221no—
442C22×C4—1228no—
443C22×C4—2442no—
444C22×C4—2442no—
445C22×C4—2444no—
446C22×C4C22⋊#692311yescharacteristic
447C22×C4C22×#692311yescharacteristic
448C4⋊C4—22no—
449C4⋊C4—1222no—
450C4⋊C4—1222no—
451C4⋊C4—1222no—
452C4⋊C4—1222no—
453C4⋊C4—1222no—
454C4⋊C4—1222no—
455C4⋊C4—2444no—
456C22⋊C4—22no—
457C22⋊C4—21no—
458C22⋊C4—22no—
459C22⋊C4—612no—
460C22⋊C4—1222no—
461C22⋊C4—1222no—
462C22⋊C4—1222no—
463C22⋊C4—1222no—
464C22⋊C4—1222no—
465C22⋊C4—1221no—
466C22⋊C4—1222no—
467C22⋊C4—1222no—
468C22⋊C4—1228no—
469C22⋊C4—1228no—
470C22⋊C4—2444no—
471C22⋊C4C22×#692312yes—
472C42—1222no—
473C42—2444no—
474C23—22no—
475C23—21no—
476C23—22no—
477C23—41no—
478C23—81no—
479C23—81no—
480C23—81no—
481C23—612no—
482C23—614no—
483C23—612no—
484C23—1224no—
485C23—4881no—
486C23—4882no—
487C23—4881no—
488C23—2441no—
489C23C2×C22⋊#692312yes—
490C23C2×C22⋊#692311yescharacteristic
491C23C2×C22⋊#692311yescharacteristic
492C23C23×#692311yescharacteristic
493Q8—2442no—
494Q8—2444no—
495D4—24no—
496D4—42no—
497D4—42no—
498D4—42no—
499D4—82no—
500D4—1224no—
501D4—1224no—
502D4—1224no—
503D4—1224no—
504D4—1224no—
505D4—4882no—
506D4—2442no—
507D4—2442no—
508D4—2444no—
509C2×C4—22no—
510C2×C4—21no—
511C2×C4—22no—
512C2×C4—41no—
513C2×C4—81no—
514C2×C4—612no—
515C2×C4—612no—
516C2×C4—1224no—
517C2×C4—1222no—
518C2×C4—1222no—
519C2×C4—1221no—
520C2×C4—1221no—
521C2×C4—1222no—
522C2×C4—1222no—
523C2×C4—9764no—
524C2×C4—4882no—
525C2×C4—4888no—
526C2×C4—4881no—
527C2×C4—2442no—
528C2×C4—2444no—
529C2×C4—2442no—
530C2×C4—2441no—
531C2×C4—2444no—
532C2×C4—2448no—
533C2×C4—2442no—
534C2×C4—2442no—
535C2×C4C2×C22⋊#692312yes—
536C22—21no—
537C22—21no—
538C22—22no—
539C22—41no—
540C22—41no—
541C22—41no—
542C22—42no—
543C22—42no—
544C22—41no—
545C22—82no—
546C22—82no—
547C22—161no—
548C22—614no—
549C22—1222no—
550C22—9761no—
551C22—4884no—
552C22—2444no—
553C22—2442no—
554C22—2442no—
555C22C22×C22⋊#692311yescharacteristic
556C22#1684082912yes—
557C4—22no—
558C4—42no—
559C4—41no—
560C4—1222no—
561C4—1222no—
562C4—4888no—
563C4—2442no—
564C4—2441no—
565C4—2441no—
566C2—21no—
567C2—22no—
568C2—41no—
569C2—41no—
570C2—42no—
571C2—81no—
572C2—1224no—
573C2—2442no—
574C2#2146229111yescharacteristic
subgroupG/H
centerC2#21462291
commutatorC23×C61C23×C4
FrattiniC23C23×#6923
FittingC61×#534C4
socleC2×C61#2308
prime radical#21459460C2
cosocleC23×D61C24
hypercenter#534#6923
hypocenterC61#21777
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C23 ◁ #534

Complements

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

NH ≅ G/Ntype
1. #21462418C2semidirect
2. #21461868C2semidirect
3. #21461829C2semidirect
4. #21461730C2semidirect
5. #21461649C2semidirect
14. C2×#14519835C22semidirect
15. #16840705C22semidirect
16. #16840639C22semidirect
23. #16840334C4semidirect
25. #16840321C4semidirect
26. #16840268C4semidirect
27. C2×#14519761C4semidirect
28. C2×#14519758C4semidirect
31. C61×#534C4semidirect
49. C22×Dic122C2×C4semidirect
56. C23×Dic61C2×C4semidirect
57. C23×Dic61C2×C4semidirect
63. #14519761C2×C4semidirect
64. #14519761C2×C4semidirect
65. #14519760C2×C4semidirect
70. #14519758C2×C4semidirect
71. #14519758C2×C4semidirect
79. #14519746C2×C4semidirect
86. #14519733C2×C4semidirect
91. C2×C22⋊Dic61C2×C4semidirect
96. C22×D4×C61C2×C4semidirect
98. #123×C61C2×C4semidirect
101. #120×C61C2×C4semidirect
102. C2×C22⋊C4×C61C2×C4semidirect
303. #534#6923semidirect
319. C61#21777semidirect
335. C22×D4C2×#6923semidirect
353. #123C2×#6923semidirect
366. #120C2×#6923semidirect
384. C2×C22⋊C4C2×#6923semidirect

Maximal subgroups

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

subgroupindex
#214624182
#214618682
#214618292
#214617302
#214616492
#214594602
#2177761