Finite Group Databasebeta

← #30616775

Subgroups — #30616775

Order 15616.

1138 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462501C211yesmaximal, characteristic, complemented
2#21462385C211yesmaximal, characteristic, complemented
3#21462013C212yesmaximal, complemented
4#21461961C212yesmaximal, complemented
5#21461845C212yesmaximal, complemented
6#21461845C212yesmaximal, complemented
7#21461569C212yesmaximal, complemented
8C4×#14519837C212yesmaximal, complemented
9C2×#16840364C211yesmaximal, characteristic
10C2×D61⋊M16C2212yes—
11#16840787—24no—
12#16840765C2212yes—
13C2×#14519837C2214yes—
14C2×#14519837C2214yes—
15#16840734C2212yes—
16#16840727—24no—
17C4×#11758217C2212yes—
18C4×#11758217C2212yes—
19C4×#11758217C2212yes—
20#16840642C2214yes—
21#16840642C2214yes—
22#16840364—24no—
23#16840364—24no—
24C2×#14519773C2211yescharacteristic
25C2×#14519767C2211yescharacteristic
26C2×#14519762C411yescharacteristic
27C2×#14519761C411yescharacteristic
28C2×#14519759C411yescharacteristic
29C2×#14519752C411yescharacteristic
30C2×#14519751C411yescharacteristic
31C2×#14519750C411yescharacteristic
32C2×#14519750C2211yescharacteristic
33C2×#14519750C2211yescharacteristic
34C2×#14519749C411yescharacteristic
35C2×C4×C22⋊D61C2211yescharacteristic
36C2×#14519737C2211yescharacteristic
37C2×#14519735C2211yescharacteristic
38C2×#123×C61C411yescharacteristic
39C2×#11758213—22no—
40D61⋊M16—28no—
41C2×#11758217C2314yes—
42C2×#11758217C2314yes—
43C2×D61⋊C8—22no—
44#14519844—22no—
45C4⋊#92420—22no—
46#14519837—28no—
47#14519837—28no—
48#14519834—24no—
49#14519832—22no—
50#14519832—22no—
51#14519832—22no—
52C4×#92420—22no—
53C4×#92420—22no—
54C4×#92420—22no—
55C22×D244C2×C411yescharacteristic
56C22×D244C2×C411yescharacteristic
57C22×D244C2×C411yescharacteristic
58C22×C22⋊D61C2×C411yescharacteristic
59C22×C22⋊D61C2×C411yescharacteristic
60C22×C22⋊D61C2×C411yescharacteristic
61C22×C22⋊D61C2×C411yescharacteristic
62C22×C22⋊D61C2311yescharacteristic
63C22×C22⋊D61C2311yescharacteristic
64C23×Dic61C2×C411yescharacteristic
65C22×C4×D61C2×C411yescharacteristic
66C22×C4×D61C2×C411yescharacteristic
67C22×C4×D61C2311yescharacteristic
68#14519773—22no—
69#14519773—22no—
70#14519767—22no—
71#14519767—22no—
72#14519762—24no—
73#14519761—24no—
74#14519759—24no—
75#14519752—24no—
76#14519751—24no—
77#14519750—22no—
78#14519750—22no—
79#14519750—22no—
80#14519750—22no—
81#14519750—24no—
82#14519749—24no—
83C4×C22⋊D61—22no—
84C4×C22⋊D61—22no—
85#14519737—22no—
86#14519737—22no—
87#14519735D414yes—
88#14519735D414yes—
89C2×C4⋊Dic61C2×C411yescharacteristic
90C2×C4⋊Dic61C2×C411yescharacteristic
91C2×#11758196C2×C411yescharacteristic
92C2×#11758196C2×C411yescharacteristic
93C2×#11758196C2311yescharacteristic
94C2×#11758196C2311yescharacteristic
95C2×C22⋊Dic61C2×C411yescharacteristic
96C2×#11758198C2×C411yescharacteristic
97C2×#11758198C2×C411yescharacteristic
98C2×#11758198C2×C411yescharacteristic
99C2×#11758198C2×C411yescharacteristic
100C2×#11758198C2×C411yescharacteristic
101C2×#11758198C2×C411yescharacteristic
102C2×C4×Dic61C2311yescharacteristic
103C2×C4×Dic61C2311yescharacteristic
104C22×D4×C61C2×C411yescharacteristic
105C23×C4×C61C2×C411yescharacteristic
106#123×C61—24no—
107C2×C4⋊C4×C61C2×C411yescharacteristic
108C2×C4⋊C4×C61C2×C411yescharacteristic
109C2×C22⋊C4×C61C2×C411yescharacteristic
110C2×C22⋊C4×C61C2×C411yescharacteristic
111C2×C22⋊C4×C61C2×C411yescharacteristic
112C24×C61C22×C411yescharacteristic
113C24×C61C22×C411yescharacteristic
114C23×D61C22×C411yescharacteristic
115C23×D61C22×C411yescharacteristic
116C23×D61C22×C411yescharacteristic
117C2×D4×C61—22no—
118C2×D4×C61—81no—
119C22×C4×C61—22no—
120C22×C4×C61—81no—
121C22×C4×C61C22×C411yescharacteristic
122C22×C4×C61C22×C411yescharacteristic
123C22×C4×C61C22×C411yescharacteristic
124C22×C4×C61C22×C411yescharacteristic
125C22×C4×C61C22×C411yescharacteristic
126C2×C22⋊D61—22no—
127C2×C22⋊D61—22no—
128C2×C22⋊D61—22no—
129C2×C22⋊D61—22no—
130C2×C22⋊D61—21no—
131C2×C22⋊D61—21no—
132C2×C22⋊D61—21no—
133C2×C22⋊D61—21no—
134C2×C22⋊D61—24no—
135C2×C22⋊D61—24no—
136C2×C22⋊D61—81no—
137C2×C22⋊D61—81no—
138C2×C22⋊D61—81no—
139C2×C22⋊D61—81no—
140C22×Dic61—22no—
141C22×Dic61—81no—
142C22×Dic61C22×C411yescharacteristic
143C22×Dic61C22×C411yescharacteristic
144C22×Dic61C22×C411yescharacteristic
145C22×Dic61C22×C411yescharacteristic
146C22×Dic61C2411yescharacteristic
147C2×D244—22no—
148C2×D244—22no—
149C2×D244—22no—
150C2×D244—81no—
151C2×D244—81no—
152C2×D244—81no—
153C2×C4×D61—24no—
154C2×C4×D61—22no—
155C2×C4×D61—22no—
156C2×C4×D61—81no—
157C2×C4×D61—81no—
158C2×C4×D61C2×D412yes—
159C2×C4×D61C2×D412yes—
160#11758217—24no—
161#11758217—24no—
162#11758217—24no—
163#11758217#4118yes—
164C2×#92420—24no—
165C2×#92420—24no—
166#11758213—28no—
167D61⋊C8—28no—
168C4⋊C4×C61—22no—
169C4⋊C4×C61C22⋊C414yes—
170C22⋊C4×C61—22no—
171C22⋊C4×C61—22no—
172C22⋊C4×C61—22no—
173C22⋊Dic61—22no—
174#11758198—22no—
175#11758198—22no—
176#11758198—22no—
177#11758198—22no—
178#11758198C22⋊C414yes—
179#11758198C22⋊C414yes—
180C4⋊Dic61—22no—
181C4⋊Dic61C22⋊C414yes—
182#11758196—22no—
183#11758196—22no—
184#11758196—21no—
185#11758196—21no—
186#11758196—21no—
187#11758196—21no—
188C4×Dic61C2×D412yes—
189C4×Dic61C2×D412yes—
190C4×Dic61C2×D412yes—
191C4×Dic61C2×D412yes—
192C23×C61—21no—
193C23×C61—21no—
194C23×C61—22no—
195C23×C61—22no—
196C23×C61—41no—
197C23×C61—41no—
198C23×C61—41no—
199C23×C61—41no—
200C23×C61C23×C411yescharacteristic
201C22×D61—22no—
202C22×D61—21no—
203C22×D61—21no—
204C22×D61—22no—
205C22×D61—22no—
206C22×D61—41no—
207C22×D61—41no—
208C22×D61—41no—
209C22×D61—41no—
210C22×D61—41no—
211C22×D61—41no—
212C22×D61C2×C22⋊C412yes—
213D4×C61—82no—
214C2×C4×C61—22no—
215C2×C4×C61—21no—
216C2×C4×C61—21no—
217C2×C4×C61—41no—
218C2×C4×C61—41no—
219C2×C4×C61—41no—
220C2×C4×C61—41no—
221C2×C4×C61—82no—
222C2×C4×C61C2×C22⋊C412yes—
223C2×C4×C61C2×C22⋊C412yes—
224C2×C4×C61C2×C22⋊C412yes—
225C22⋊D61—42no—
226C22⋊D61—42no—
227C22⋊D61—42no—
228C22⋊D61—42no—
229C22⋊D61—82no—
230C22⋊D61—82no—
231C22⋊D61—82no—
232C22⋊D61—82no—
233C2×Dic61—21no—
234C2×Dic61—21no—
235C2×Dic61—41no—
236C2×Dic61—41no—
237C2×Dic61—41no—
238C2×Dic61—41no—
239C2×Dic61—82no—
240C2×Dic61C2×C22⋊C412yes—
241C2×Dic61C2×C22⋊C412yes—
242C2×Dic61C22×D411yescharacteristic
243C2×Dic61C22×D411yescharacteristic
244C2×Dic61C2×#4114yes—
245D244—82no—
246D244—82no—
247D244—82no—
248C4×D61—22no—
249C4×D61—22no—
250C4×D61—24no—
251C4×D61—82no—
252C4×D61—82no—
253#92420—28no—
254#92420—44no—
255#21428—611nomaximal, Sylow
256C22×C61—21no—
257C22×C61—21no—
258C22×C61—41no—
259C22×C61—41no—
260C22×C61—41no—
261C22×C61—41no—
262C22×C61—41no—
263C22×C61—41no—
264C22×C61—82no—
265C22×C61—82no—
266C22×C61C22×C22⋊C411yescharacteristic
267C22×C61C4×#4112yes—
268C2×D61—22no—
269C2×D61—22no—
270C2×D61—22no—
271C2×D61—41no—
272C2×D61—41no—
273C2×D61—41no—
274C2×D61—41no—
275C2×D61—41no—
276C2×D61—41no—
277C2×D61—82no—
278C2×D61—82no—
279C2×D61—82no—
280C4×C61—22no—
281C4×C61—22no—
282C4×C61—22no—
283C4×C61—41no—
284C4×C61—41no—
285Dic61—22no—
286Dic61—22no—
287Dic61—41no—
288Dic61—41no—
289Dic61#52512yes—
290Dic61#53212yes—
291C2×#540—611no—
292#2921—611no—
293#2873—611no—
294#1998—612no—
295#1965—612no—
296#1900—612no—
297#1900—612no—
298C4×#100—612no—
299C4×#100—612no—
300C2×C61—22no—
301C2×C61—81no—
302C2×C61—81no—
303C2×C61#229711yescharacteristic
304C2×C61#288811yescharacteristic
305C2×C61#288811yescharacteristic
306D61—42no—
307D61—81no—
308D61—81no—
309C22×M16—612no—
310#540—1224no—
311#540—1224no—
312C2×#127—611no—
313C2×#125—611no—
314C2×#123—611no—
315C2×#123—611no—
316C2×#123—611no—
317C2×#123#692311yescharacteristic
318C2×#121—611no—
319C2×#121—611no—
320C2×#121—611no—
321C2×#121—611no—
322C2×#121—611no—
323C2×#121—611no—
324C2×C4×D4—611no—
325C2×C4×D4—611no—
326C2×#117—611no—
327#433—1224no—
328#423—612no—
329C2×#100—614no—
330C2×#100—614no—
331#407—612no—
332#405—1224no—
333C4×M16—612no—
334C4×M16—612no—
335C4×M16—612no—
336#343—614no—
337#343—614no—
338C61#2142811yescharacteristic, complemented, Sylow
339C22×D4—611no—
340C22×D4—611no—
341C22×D4—611no—
342C22×D4—611no—
343C22×D4—611no—
344C22×D4—611no—
345C22×D4—611no—
346C22×D4—611no—
347C22×D4—611no—
348C22×D4C2×#692311yescharacteristic
349C23×C4—611no—
350C23×C4—611no—
351C23×C4—611no—
352C23×C4—611no—
353C23×C4C2×#692311yescharacteristic
354C2×M16—614no—
355C2×M16—614no—
356C2×M16—1228no—
357C2×M16—1222no—
358C22×C8—1222no—
359#127—1222no—
360#127—1222no—
361#125—1222no—
362#125—1222no—
363#123—24no—
364#123—1224no—
365#123—1224no—
366#123—1224no—
367#121—1224no—
368#121—1222no—
369#121—1222no—
370#121—1222no—
371#121—1222no—
372#121—1224no—
373#121—1224no—
374#121—1224no—
375C4×D4—1222no—
376C4×D4—1222no—
377C4×D4—1222no—
378C4×D4—1222no—
379#117—614no—
380#117—614no—
381C2×C4⋊C4—611no—
382C2×C4⋊C4—611no—
383C2×C4⋊C4—611no—
384C2×C4⋊C4—611no—
385C2×C4⋊C4—611no—
386C2×C4⋊C4—611no—
387C2×C4⋊C4C2×#692311yescharacteristic
388C2×C4⋊C4C2×#692311yescharacteristic
389C2×C22⋊C4—611no—
390C2×C22⋊C4—611no—
391C2×C22⋊C4—611no—
392C2×C22⋊C4—611no—
393C2×C22⋊C4—611no—
394C2×C22⋊C4—611no—
395C2×C22⋊C4—611no—
396C2×C22⋊C4C2×#692311yescharacteristic
397C2×C22⋊C4C2×#692311yescharacteristic
398C2×C22⋊C4C2×#692311yescharacteristic
399C2×C42—611no—
400C2×C42—611no—
401C4⋊C8—1222no—
402C4⋊C8—1222no—
403#100—1228no—
404#100—1228no—
405C22⋊C8—1224no—
406#97—1222no—
407#97—1222no—
408#97—1222no—
409C4×C8—1222no—
410C4×C8—1222no—
411C4×C8—1222no—
412C24—611no—
413C24—611no—
414C24—611no—
415C24C22×#692311yescharacteristic
416C24C22×#692311yescharacteristic
417C2×D4—22no—
418C2×D4—81no—
419C2×D4—1222no—
420C2×D4—1222no—
421C2×D4—1222no—
422C2×D4—1222no—
423C2×D4—1224no—
424C2×D4—1224no—
425C2×D4—1221no—
426C2×D4—1221no—
427C2×D4—1221no—
428C2×D4—1221no—
429C2×D4—1222no—
430C2×D4—1222no—
431C2×D4—1222no—
432C2×D4—4881no—
433C2×D4—4881no—
434C2×D4—4881no—
435C2×D4—4881no—
436C2×D4—4881no—
437C2×D4—4881no—
438C2×D4—4881no—
439C22×C4—22no—
440C22×C4—81no—
441C22×C4—611no—
442C22×C4—611no—
443C22×C4—611no—
444C22×C4—612no—
445C22×C4—612no—
446C22×C4—611no—
447C22×C4—611no—
448C22×C4—1224no—
449C22×C4—1222no—
450C22×C4—1222no—
451C22×C4—1222no—
452C22×C4—4881no—
453C22×C4—4881no—
454C22×C4—4881no—
455C22×C4C22×#692311yescharacteristic
456C22×C4C22×#692311yescharacteristic
457C22×C4C22×#692311yescharacteristic
458C22×C4C22×#692311yescharacteristic
459C22×C4C22×#692311yescharacteristic
460M16—618no—
461M16—1224no—
462M16—1228no—
463M16—1224no—
464M16—1224no—
465C2×C8—1224no—
466C2×C8—1228no—
467C2×C8—1224no—
468C4⋊C4—22no—
469C4⋊C4—614no—
470C4⋊C4—1221no—
471C4⋊C4—1221no—
472C4⋊C4—1221no—
473C4⋊C4—1221no—
474C4⋊C4—1222no—
475C4⋊C4—1222no—
476C4⋊C4—1222no—
477C4⋊C4C22⋊#692314yes—
478C22⋊C4—22no—
479C22⋊C4—22no—
480C22⋊C4—22no—
481C22⋊C4—614no—
482C22⋊C4—614no—
483C22⋊C4—1222no—
484C22⋊C4—1222no—
485C22⋊C4—1222no—
486C22⋊C4—1222no—
487C22⋊C4—1222no—
488C42—612no—
489C42—612no—
490C42—612no—
491C42—612no—
492C23—21no—
493C23—21no—
494C23—22no—
495C23—22no—
496C23—41no—
497C23—41no—
498C23—41no—
499C23—41no—
500C23—612no—
501C23—1222no—
502C23—1222no—
503C23—1222no—
504C23—1221no—
505C23—1221no—
506C23—2441no—
507C23—2441no—
508C23—2441no—
509C23—2441no—
510C23—2441no—
511C23—2441no—
512C23C23×#692311yescharacteristic
513D4—82no—
514D4—4882no—
515D4—4882no—
516D4—4882no—
517D4—4882no—
518D4—4882no—
519D4—4882no—
520D4—4882no—
521D4—2442no—
522D4—2442no—
523D4—2442no—
524D4—2442no—
525C2×C4—22no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—82no—
533C2×C4—611no—
534C2×C4—611no—
535C2×C4—614no—
536C2×C4—612no—
537C2×C4—612no—
538C2×C4—1222no—
539C2×C4—1222no—
540C2×C4—1224no—
541C2×C4—1221no—
542C2×C4—1221no—
543C2×C4—4882no—
544C2×C4—4882no—
545C2×C4—4882no—
546C2×C4—2441no—
547C2×C4—2441no—
548C2×C4—2441no—
549C2×C4—2441no—
550C2×C4C2×C22⋊#692312yes—
551C2×C4C2×C22⋊#692312yes—
552C2×C4C2×C22⋊#692312yes—
553C8—1228no—
554C8—2444no—
555C22—21no—
556C22—21no—
557C22—41no—
558C22—41no—
559C22—41no—
560C22—41no—
561C22—41no—
562C22—41no—
563C22—82no—
564C22—82no—
565C22—1222no—
566C22—1222no—
567C22—1222no—
568C22—4882no—
569C22—4882no—
570C22—4882no—
571C22—2441no—
572C22—2441no—
573C22—2441no—
574C22—2441no—
575C22—2441no—
576C22—2441no—
577C22C22×C22⋊#692311yescharacteristic
578C22#41×#692312yes—
579C4—22no—
580C4—22no—
581C4—22no—
582C4—41no—
583C4—41no—
584C4—612no—
585C4—612no—
586C4—1222no—
587C4—1222no—
588C4—2441no—
589C4—2441no—
590C2—22no—
591C2—81no—
592C2—81no—
593C2—4881no—
594C2—4881no—
595C2—2442no—
596C2#2146228011yescharacteristic
597C2#2146242911yescharacteristic
598C2#2146243011yescharacteristic
subgroupG/H
centerC22C22×C22⋊#6923
commutatorC23×C61C23×C4
FrattiniC23C23×#6923
FittingC2×#123×C61C4
socleC22×C61C22×C22⋊C4
prime radicalC2×#16840364C2
cosocleC22×Dic61C24
hypercenterC2×#123#6923
hypocenterC61#21428
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C4⋊C4 ◁ C2×#123

Complements

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

NH ≅ G/Ntype
1. #21462501C2semidirect
2. #21462385C2semidirect
3. #21462013C2semidirect
4. #21461961C2semidirect
5. #21461845C2semidirect
6. #21461845C2semidirect
7. #21461569C2semidirect
8. C4×#14519837C2semidirect
338. C61#21428semidirect

Maximal subgroups

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

subgroupindex
#214625012
#214623852
#214620132
#214619612
#214618452
#214615692
C4×#145198372
C2×#168403642
#2142861