Finite Group Databasebeta

← #29575136

Subgroups — #29575136

Order 15616.

834 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839900C211yesmaximal, characteristic
2#21445753C211yesmaximal, characteristic, complemented
3C2×#16839496C211yesmaximal, characteristic
4#21444089C211yesmaximal, characteristic, complemented
5#21443697C211yesmaximal, characteristic, complemented
6#21443662C211yesmaximal, characteristic, complemented
7C61×#2325C211yesmaximal, characteristic
8C23×Dic122C2211yescharacteristic
9C22×#11758200—21no—
10#16839900—22no—
11#16839900C414yescomplemented
12C2×#14519700—21no—
13C2×#14519700—21no—
14C2×#14519700—21no—
15C2×C22⋊#92418C2211yescharacteristic
16C2×C22⋊#92418C2211yescharacteristic
17C2×#14519684—21no—
18C2×#14519684—21no—
19C2×#14519684C2211yescharacteristic
20#16839496—22no—
21C2×#121×C61C2211yescharacteristic
22C2×C4×D4×C61—21no—
23C61×#388—21no—
24C61×#387—21no—
25C61×#386—21no—
26C61×#386—21no—
27C61×#385—21no—
28C61×#380C2211yescharacteristic
29C2×#95×C61C2211yescharacteristic
30C22×Dic122—21no—
31C22×Dic122—21no—
32C22×Dic122—21no—
33C22×Dic122—21no—
34C22×Dic122—21no—
35C22×Dic122—42no—
36C22×Dic122—42no—
37C22×Dic122C2×C412yescomplemented
38C23×Dic61—21no—
39C23×Dic61—21no—
40C2×#11758200—24no—
41C2×#11758200—42no—
42C2×#11758200—42no—
43C22×#92418—21no—
44C22×#92418—21no—
45#14519700—24no—
46#14519700—42no—
47#14519700—42no—
48C22⋊#92418—21no—
49C22⋊#92418—21no—
50C22⋊#92418—21no—
51C22⋊#92418C2×C412yescomplemented
52#14519684—21no—
53#14519684—21no—
54#14519684—42no—
55#14519684—42no—
56C22×D4×C61—21no—
57C22×D4×C61D411yescharacteristic
58C23×C4×C61—21no—
59C23×C4×C61D411yescharacteristic
60C23×C4×C61D411yescharacteristic
61C23×C4×C61C2311yescharacteristic
62#121×C61—21no—
63#121×C61—21no—
64#121×C61C2×C412yes—
65#121×C61D412yes—
66C4×D4×C61—42no—
67C4×D4×C61—42no—
68C2×C4⋊C4×C61—21no—
69C2×C4⋊C4×C61D411yescharacteristic
70C2×C4⋊C4×C61D411yescharacteristic
71C2×C4⋊C4×C61D411yescharacteristic
72C2×C22⋊C4×C61—21no—
73C2×C22⋊C4×C61—21no—
74C2×C22⋊C4×C61—21no—
75C2×C22⋊C4×C61—21no—
76C2×C22⋊C4×C61—21no—
77C2×C42×C61—21no—
78#95×C61—21no—
79#95×C61—21no—
80#95×C61—21no—
81#95×C61—21no—
82#95×C61—21no—
83C24×C61—21no—
84C24×C61C2×D411yescharacteristic
85C2×D4×C61—22no—
86C2×D4×C61—22no—
87C2×D4×C61—21no—
88C2×D4×C61—41no—
89C2×D4×C61—41no—
90C2×D4×C61—41no—
91C2×D4×C61—41no—
92C2×D4×C61—41no—
93C2×D4×C61—41no—
94C2×D4×C61C22⋊C412yes—
95C22×C4×C61—21no—
96C22×C4×C61—21no—
97C22×C4×C61—21no—
98C22×C4×C61—21no—
99C22×C4×C61—21no—
100C22×C4×C61—21no—
101C22×C4×C61—21no—
102C22×C4×C61—21no—
103C22×C4×C61—41no—
104C22×C4×C61—41no—
105C22×C4×C61—42no—
106C22×C4×C61—41no—
107C22×C4×C61—41no—
108C22×C4×C61—41no—
109C22×C4×C61—41no—
110C22×C4×C61—81no—
111C22×C4×C61—81no—
112C22×C4×C61SD1612yescomplemented
113C22×C4×C61C22×C411yescharacteristic
114C22×C4×C61C2×D411yescharacteristic
115C22×C4×C61C2×D411yescharacteristic
116C22×C4×C61C2×D411yescharacteristic
117C22×C4×C61#4111yescharacteristic
118C22×C4×C61#4111yescharacteristic
119C22×Dic61—22no—
120C22×Dic61—22no—
121C22×Dic61—22no—
122C22×Dic61—22no—
123C22×Dic61—41no—
124C22×Dic61—41no—
125C22×Dic61—41no—
126C22×Dic61—41no—
127C22×Dic61—41no—
128C22×Dic61—41no—
129C22×Dic61—41no—
130C22×Dic61—41no—
131C22×Dic61—41no—
132C22×Dic61—41no—
133C2×Dic122—22no—
134C2×Dic122—22no—
135C2×Dic122—22no—
136C2×Dic122—22no—
137C2×Dic122—22no—
138C2×Dic122—22no—
139C2×Dic122—44no—
140C2×Dic122—44no—
141C2×Dic122—44no—
142C2×Dic122—44no—
143C2×Dic122—41no—
144C2×Dic122—41no—
145C2×Dic122—41no—
146C2×Dic122—41no—
147C2×Dic122—41no—
148C2×Dic122—41no—
149C4⋊C4×C61—21no—
150C4⋊C4×C61—42no—
151C4⋊C4×C61—41no—
152C4⋊C4×C61—41no—
153C4⋊C4×C61C22⋊C412yes—
154C22⋊C4×C61—22no—
155C22⋊C4×C61—22no—
156C22⋊C4×C61—42no—
157C22⋊C4×C61—41no—
158C22⋊C4×C61—41no—
159C22⋊C4×C61—81no—
160C22⋊C4×C61—81no—
161C42×C61—42no—
162#11758200—48no—
163C2×#92418—22no—
164C2×#92418—41no—
165C2×#92418—41no—
166C2×#92418—41no—
167C2×#92418—41no—
168C2×#92418—41no—
169C23×C61—21no—
170C23×C61—21no—
171C23×C61—21no—
172C23×C61—21no—
173C23×C61—22no—
174C23×C61—41no—
175C23×C61—41no—
176C23×C61—41no—
177C23×C61—41no—
178C23×C61—41no—
179C23×C61—41no—
180C23×C61C4×D411yescharacteristic
181C23×C61#12011yescharacteristic
182C23×C61#12111yescharacteristic
183C23×C61C2×SD1611yescharacteristic
184C23×C61#13511yescharacteristic
185C23×C61AGL(1,ℤ/8)11yescharacteristic
186C23×C61#13711yescharacteristic
187D4×C61—42no—
188D4×C61—42no—
189D4×C61—42no—
190D4×C61—42no—
191D4×C61—42no—
192D4×C61—81no—
193C2×C4×C61—21no—
194C2×C4×C61—21no—
195C2×C4×C61—21no—
196C2×C4×C61—21no—
197C2×C4×C61—22no—
198C2×C4×C61—22no—
199C2×C4×C61—41no—
200C2×C4×C61—41no—
201C2×C4×C61—41no—
202C2×C4×C61—41no—
203C2×C4×C61—41no—
204C2×C4×C61—41no—
205C2×C4×C61—41no—
206C2×C4×C61—41no—
207C2×C4×C61—42no—
208C2×C4×C61—42no—
209C2×C4×C61—41no—
210C2×C4×C61—41no—
211C2×C4×C61—41no—
212C2×C4×C61—41no—
213C2×C4×C61—41no—
214C2×C4×C61—41no—
215C2×C4×C61—41no—
216C2×C4×C61—41no—
217C2×C4×C61—41no—
218C2×C4×C61—81no—
219C2×C4×C61—84no—
220C2×C4×C61—81no—
221C2×C4×C61C2×C22⋊C411yescharacteristic
222C2×C4×C61C4×D411yescharacteristic
223C2×C4×C61#12111yescharacteristic
224C2×C4×C61#12311yescharacteristic
225C2×Dic61—24no—
226C2×Dic61—42no—
227C2×Dic61—42no—
228C2×Dic61—42no—
229C2×Dic61—42no—
230C2×Dic61—42no—
231C2×Dic61—42no—
232C2×Dic61—42no—
233C2×Dic61—42no—
234C2×Dic61—42no—
235C2×Dic61—81no—
236C2×Dic61—81no—
237C2×Dic61—81no—
238C2×Dic61—81no—
239Dic122—24no—
240Dic122—24no—
241Dic122—42no—
242Dic122—42no—
243Dic122—42no—
244Dic122—42no—
245Dic122—42no—
246Dic122—42no—
247#92418—42no—
248#92418—81no—
249#9039—611nomaximal, Sylow
250C22×C61—21no—
251C22×C61—21no—
252C22×C61—21no—
253C22×C61—21no—
254C22×C61—21no—
255C22×C61—21no—
256C22×C61—21no—
257C22×C61—21no—
258C22×C61—21no—
259C22×C61—21no—
260C22×C61—21no—
261C22×C61—21no—
262C22×C61—42no—
263C22×C61—42no—
264C22×C61—42no—
265C22×C61—81no—
266C22×C61—81no—
267C22×C61—81no—
268C22×C61—81no—
269C22×C61#38611yescharacteristic
270C22×C61C4×SD1611yescharacteristic
271C22×C61#44011yescharacteristic
272C22×C61#45011yescharacteristic
273C22×C61#45211yescharacteristic
274C22×C61#45511yescharacteristic
275C22×C61#45611yescharacteristic
276C22×C61B(4,𝔽2)11yescharacteristic
277C22×C61B(4,𝔽2)11yescharacteristic
278C22×C61#46111yescharacteristic
279C22×C61#46311yescharacteristic
280C4×C61—21no—
281C4×C61—21no—
282C4×C61—21no—
283C4×C61—21no—
284C4×C61—42no—
285C4×C61—42no—
286C4×C61—81no—
287C4×C61—81no—
288C4×C61—81no—
289Dic61—44no—
290Dic61—82no—
291C2×#448—611no—
292#2325D6111yescharacteristic
293#1787—611no—
294C2×#327—611no—
295#1446—611no—
296#1294—611no—
297#1280—611no—
298C2×C61—21no—
299C2×C61—21no—
300C2×C61—21no—
301C2×C61—21no—
302C2×C61—82no—
303C2×C61#160111yescharacteristic
304C2×C61#160311yescharacteristic
305C2×C61#160411yescharacteristic
306C2×C61#160611yescharacteristic
307C2×C61#187711yescharacteristic
308C2×C61#188211yescharacteristic
309C2×C61#188911yescharacteristic
310C23×Q8—611no—
311C22×SD16—1221no—
312C2×#121C2×D6111yescharacteristic
313C2×C4×D4—21no—
314#448—614no—
315#448—1222no—
316C2×#103—1221no—
317C2×#103—1221no—
318C2×#103—1221no—
319C2×C22⋊C8—611no—
320C2×C22⋊C8—611no—
321#388—21no—
322#387—21no—
323#386—21no—
324#386—21no—
325#385—21no—
326#380C2×D6111yescharacteristic
327C2×#95—611no—
328C2×#95—1221no—
329C2×#95—1221no—
330C2×#95C2×D6111yescharacteristic
331#327—1222no—
332C61#903911yescharacteristic, complemented, Sylow
333C22×Q8—612no—
334C22×Q8—1221no—
335C22×Q8—1221no—
336C22×Q8—1221no—
337C22×Q8—1221no—
338C22×Q8—1221no—
339C22×Q8—2442no—
340C22×Q8—2442no—
341C22×D4—21no—
342C22×D4C22⋊D6111yescharacteristic
343C23×C4—21no—
344C23×C4—1221no—
345C23×C4—1221no—
346C23×C4C22⋊D6111yescharacteristic
347C23×C4C22⋊D6111yescharacteristic
348C23×C4C22×D6111yescharacteristic
349C2×SD16—1224no—
350C2×SD16—2442no—
351C2×SD16—2442no—
352C22×C8—1221no—
353C22×C8—1221no—
354#121—21no—
355#121—21no—
356#121C4×D6112yes—
357#121D24412yes—
358C4×D4—42no—
359C4×D4—42no—
360C2×C4⋊C4—21no—
361C2×C4⋊C4C22⋊D6111yescharacteristic
362C2×C4⋊C4C22⋊D6111yescharacteristic
363C2×C4⋊C4C22⋊D6111yescharacteristic
364C2×C22⋊C4—21no—
365C2×C22⋊C4—21no—
366C2×C22⋊C4—21no—
367C2×C22⋊C4—21no—
368C2×C22⋊C4—21no—
369C2×C42—21no—
370#103—1224no—
371#103—2442no—
372#103—2442no—
373C22⋊C8—612no—
374C22⋊C8—1221no—
375C22⋊C8—1221no—
376C22⋊C8—1221no—
377#95—21no—
378#95—21no—
379#95—21no—
380#95—21no—
381#95—21no—
382#95—1221no—
383#95—1221no—
384#95—2442no—
385#95—2442no—
386C24—21no—
387C24C2×C22⋊D6111yescharacteristic
388C2×Q8—1222no—
389C2×Q8—1222no—
390C2×Q8—1222no—
391C2×Q8—1222no—
392C2×Q8—1222no—
393C2×Q8—1222no—
394C2×Q8—2441no—
395C2×Q8—2441no—
396C2×Q8—2441no—
397C2×Q8—2441no—
398C2×Q8—2444no—
399C2×Q8—2444no—
400C2×Q8—2444no—
401C2×Q8—2444no—
402C2×Q8—2441no—
403C2×Q8—2441no—
404C2×D4—22no—
405C2×D4—22no—
406C2×D4—21no—
407C2×D4—41no—
408C2×D4—41no—
409C2×D4—41no—
410C2×D4—41no—
411C2×D4—41no—
412C2×D4—41no—
413C2×D4#1175819812yes—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—41no—
423C22×C4—41no—
424C22×C4—42no—
425C22×C4—41no—
426C22×C4—41no—
427C22×C4—41no—
428C22×C4—41no—
429C22×C4—81no—
430C22×C4—81no—
431C22×C4—1222no—
432C22×C4—1222no—
433C22×C4—1222no—
434C22×C4—1222no—
435C22×C4—2441no—
436C22×C4—2441no—
437C22×C4—2441no—
438C22×C4—2441no—
439C22×C4—2441no—
440C22×C4—2441no—
441C22×C4—2441no—
442C22×C4—2441no—
443C22×C4—2441no—
444C22×C4—2441no—
445C22×C4#1175820012yescomplemented
446C22×C4C2×C4×D6111yescharacteristic
447C22×C4C2×D24411yescharacteristic
448C22×C4#1175822211yescharacteristic
449C22×C4#1175822211yescharacteristic
450C22×C4C2×C22⋊D6111yescharacteristic
451C22×C4C2×C22⋊D6111yescharacteristic
452SD16—2448no—
453C2×C8—1222no—
454C2×C8—2441no—
455C2×C8—2441no—
456C2×C8—2441no—
457C2×C8—2441no—
458C2×C8—2441no—
459C4⋊C4—21no—
460C4⋊C4—41no—
461C4⋊C4—41no—
462C4⋊C4—42no—
463C4⋊C4#1175819812yes—
464C22⋊C4—22no—
465C22⋊C4—22no—
466C22⋊C4—42no—
467C22⋊C4—41no—
468C22⋊C4—41no—
469C22⋊C4—81no—
470C22⋊C4—81no—
471C42—42no—
472C23—21no—
473C23—21no—
474C23—21no—
475C23—21no—
476C23—22no—
477C23—41no—
478C23—41no—
479C23—41no—
480C23—41no—
481C23—41no—
482C23—41no—
483C23C4×C22⋊D6111yescharacteristic
484C23#1451974611yescharacteristic
485C23#1451975111yescharacteristic
486C23C2×#1175820011yescharacteristic
487C23#1451979711yescharacteristic
488C23#1451980311yescharacteristic
489C23#1451980711yescharacteristic
490Q8—1224no—
491Q8—1224no—
492Q8—2442no—
493Q8—2442no—
494Q8—2442no—
495Q8—2442no—
496Q8—2442no—
497Q8—2442no—
498D4—42no—
499D4—42no—
500D4—42no—
501D4—42no—
502D4—42no—
503D4—81no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—81no—
530C2×C4—84no—
531C2×C4—81no—
532C2×C4—1224no—
533C2×C4—4881no—
534C2×C4—4881no—
535C2×C4—4881no—
536C2×C4—4881no—
537C2×C4—2442no—
538C2×C4—2442no—
539C2×C4—2442no—
540C2×C4—2442no—
541C2×C4—2442no—
542C2×C4—2442no—
543C2×C4—2442no—
544C2×C4—2442no—
545C2×C4—2442no—
546C2×C4C2×#1175819811yescharacteristic
547C2×C4C4×C22⋊D6111yescharacteristic
548C2×C4#1451975111yescharacteristic
549C2×C4#1451976011yescharacteristic
550C8—4881no—
551C8—2442no—
552C22—21no—
553C22—21no—
554C22—21no—
555C22—21no—
556C22—21no—
557C22—21no—
558C22—21no—
559C22—21no—
560C22—21no—
561C22—21no—
562C22—21no—
563C22—21no—
564C22—42no—
565C22—42no—
566C22—42no—
567C22—81no—
568C22—81no—
569C22—81no—
570C22—81no—
571C22#1683964111yescharacteristic
572C22C4×#1175820011yescharacteristic
573C22#1683986211yescharacteristic
574C22#1683991311yescharacteristic
575C22#1683992411yescharacteristic
576C22#1683994011yescharacteristic
577C22#1683994511yescharacteristic
578C22#1683994911yescharacteristic
579C22#1683994911yescharacteristic
580C22#1683996711yescharacteristic
581C22#1683997711yescharacteristic
582C4—21no—
583C4—21no—
584C4—21no—
585C4—21no—
586C4—42no—
587C4—42no—
588C4—81no—
589C4—81no—
590C4—81no—
591C4—4882no—
592C4—2444no—
593C2—21no—
594C2—21no—
595C2—21no—
596C2—21no—
597C2—82no—
598C2#2144477111yescharacteristic
599C2#2144478211yescharacteristic
600C2#2144478811yescharacteristic
601C2#2144479911yescharacteristic
602C2#2144615311yescharacteristic
603C2#2144618811yescharacteristic
604C2#2144622711yescharacteristic
subgroupG/H
centerC23#14519746
commutatorC22×C4×C61C22×C4
FrattiniC23×C4C22×D61
FittingC61×#2325C2
socleC23×C61#120
prime radicalC2×#121×C61C22
cosocleC23×C4×C61C23
hypercenter#2325D61
hypocenterC61#9039
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2325

Complements

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

NH ≅ G/Ntype
2. #21445753C2semidirect
4. #21444089C2semidirect
5. #21443697C2semidirect
6. #21443662C2semidirect
11. #16839900C4semidirect
37. C22×Dic122C2×C4semidirect
51. C22⋊#92418C2×C4semidirect
112. C22×C4×C61SD16semidirect
332. C61#9039semidirect
445. C22×C4#11758200semidirect

Maximal subgroups

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

subgroupindex
C2×#168399002
#214457532
C2×#168394962
#214440892
#214436972
#214436622
C61×#23252
#903961