Finite Group Databasebeta

← #29572087

Subgroups — #29572087

Order 15616.

1116 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840434C211yesmaximal, characteristic
2#21446624C211yesmaximal, characteristic, complemented
3#21445816C211yesmaximal, characteristic, complemented
4C2×#16839505C211yesmaximal, characteristic, complemented
5C2×Q8⋊#92418C211yesmaximal, characteristic, complemented
6C2×#16839489C211yesmaximal, characteristic, complemented
7C61×#1803C211yesmaximal, characteristic, complemented
8C22×#11758226—21no—
9#16840434—22no—
10#16840434—22no—
11#16840434—22no—
12#16840434C414yescomplemented
13C2×#14519774—21no—
14C2×#14519774C2211yescharacteristic
15C2×#14519752—21no—
16C2×#14519752—22no—
17C2×C4×D244—21no—
18C2×C4×D244C2211yescharacteristic
19C2×C4⋊#92418C2211yescharacteristic
20C2×#14519697—21no—
21C2×#14519697—22no—
22C2×C4×#92418C2211yescharacteristic
23#16839524—21no—
24#16839505—22no—
25Q8⋊#92418—22no—
26#16839489—22no—
27C2×C4×Q8×C61C2211yescharacteristic
28C2×C4⋊C8×C61C2211yescharacteristic
29C2×#103×C61—21no—
30C61×#384C2211yescharacteristic
31C61×#340—21no—
32C2×#11758226—24no—
33C2×#11758226—24no—
34C2×#11758226—42no—
35C2×#11758226—42no—
36C2×#11758226—42no—
37C2×#11758226—42no—
38C22×D244—22no—
39C22×D244—21no—
40C22×D244—21no—
41C22×D244—22no—
42C22×D244D411yescharacteristic
43C22×D244D412yes—
44C22×C4×D61—22no—
45C22×C4×D61—21no—
46C22×#92418—21no—
47C22×#92418—21no—
48#14519774—24no—
49#14519774—24no—
50#14519774—21no—
51#14519774—21no—
52#14519774C2×C412yes—
53#14519774D412yes—
54#14519752—24no—
55#14519752—28no—
56#14519752—28no—
57#14519752—24no—
58C4×D244—24no—
59C4×D244—21no—
60C4×D244—21no—
61C4×D244—24no—
62C4×D244C2×C412yes—
63C4×D244D412yes—
64C2×C4⋊Dic61D411yescharacteristic
65C2×#11758198—22no—
66C2×#11758198—21no—
67C4⋊#92418—21no—
68C4⋊#92418—21no—
69#14519697—44no—
70#14519697—42no—
71C4×#92418—21no—
72C4×#92418—21no—
73C22×Q8×C61D411yescharacteristic
74C22×C8×C61—21no—
75C4×Q8×C61—21no—
76C4×Q8×C61—21no—
77C4×Q8×C61C2×C412yes—
78C4×Q8×C61D412yes—
79C2×C4⋊C4×C61—21no—
80C2×C4⋊C4×C61—21no—
81C2×C4⋊C4×C61—21no—
82C2×C4⋊C4×C61—21no—
83C2×C4⋊C4×C61D411yescharacteristic
84C2×C42×C61—21no—
85C2×C42×C61C2311yescharacteristic
86C4⋊C8×C61—21no—
87C4⋊C8×C61—21no—
88#103×C61—42no—
89C23×D61—21no—
90C23×D61—22no—
91C2×Q8×C61—21no—
92C2×Q8×C61—21no—
93C2×Q8×C61—21no—
94C2×Q8×C61C22⋊C412yes—
95C2×Q8×C61D812yes—
96C2×Q8×C61SD1612yes—
97C22×C4×C61—21no—
98C22×C4×C61—21no—
99C22×C4×C61—41no—
100C22×C4×C61C2×D411yescharacteristic
101C22×C4×C61C2×D411yescharacteristic
102C22×C4×C61C2×D411yescharacteristic
103C22×Dic61—21no—
104#11758226—48no—
105#11758226—48no—
106#11758226—48no—
107#11758226—48no—
108C2×D244—21no—
109C2×D244—21no—
110C2×D244—21no—
111C2×D244—21no—
112C2×D244—28no—
113C2×D244—22no—
114C2×D244—22no—
115C2×D244—22no—
116C2×D244—22no—
117C2×D244—21no—
118C2×D244—28no—
119C2×D244—22no—
120C2×D244—44no—
121C2×D244—44no—
122C2×D244—41no—
123C2×D244—41no—
124C2×D244—41no—
125C2×D244—41no—
126C2×D244—41no—
127C2×D244—41no—
128C2×D244—41no—
129C2×D244—41no—
130C2×D244—42no—
131C2×D244—42no—
132C2×D244—82no—
133C2×D244—82no—
134C2×D244C22⋊C412yes—
135C2×D244C22⋊C414yes—
136C2×C4×D61—28no—
137C2×C4×D61—22no—
138C2×C4×D61—22no—
139C2×C4×D61—44no—
140C2×C4×D61—44no—
141C2×C4×D61—41no—
142C2×C4×D61—41no—
143C2×C4×D61—41no—
144C2×C4×D61—41no—
145C2×C8×C61—41no—
146C2×C8×C61—41no—
147C2×C8×C61—41no—
148C4⋊C4×C61—22no—
149C4⋊C4×C61—22no—
150C4⋊C4×C61—21no—
151C4⋊C4×C61—42no—
152C4⋊C4×C61—41no—
153C4⋊C4×C61—41no—
154C4⋊C4×C61—41no—
155C4⋊C4×C61—41no—
156C4⋊C4×C61C22⋊C412yes—
157C42×C61—22no—
158C42×C61—22no—
159C42×C61C22×C411yescharacteristic
160C42×C61C2×D411yescharacteristic
161C42×C61C2×D411yescharacteristic
162C42×C61C2×D411yescharacteristic
163#11758198—28no—
164#11758198—22no—
165#11758198—22no—
166C4⋊Dic61—21no—
167C4⋊Dic61C22⋊C412yes—
168C2×#92418—22no—
169C2×#92418—41no—
170C2×#92418—41no—
171C2×#92418—41no—
172C2×#92418—41no—
173C2×#92418—41no—
174C23×C61#12011yescharacteristic
175C22×D61—24no—
176C22×D61—22no—
177C22×D61—42no—
178C22×D61—41no—
179C22×D61—41no—
180C22×D61—42no—
181C22×D61—42no—
182C22×D61—41no—
183C22×D61—41no—
184C22×D61—42no—
185C22×D61—42no—
186C22×D61—41no—
187C22×D61—42no—
188C22×D61—41no—
189Q8×C61—22no—
190Q8×C61—22no—
191Q8×C61—22no—
192Q8×C61#10214yescomplemented
193C2×C4×C61—22no—
194C2×C4×C61—22no—
195C2×C4×C61—22no—
196C2×C4×C61—21no—
197C2×C4×C61—21no—
198C2×C4×C61—22no—
199C2×C4×C61—41no—
200C2×C4×C61—41no—
201C2×C4×C61—81no—
202C2×C4×C61—81no—
203C2×C4×C61—81no—
204C2×C4×C61C2×C22⋊C411yescharacteristic
205C2×C4×C61C2×C22⋊C411yescharacteristic
206C2×C4×C61C2×C22⋊C411yescharacteristic
207C2×C4×C61#12011yescharacteristic
208C2×C4×C61#12011yescharacteristic
209C2×C4×C61#12011yescharacteristic
210C2×C4×C61C2×D811yescharacteristic
211C2×C4×C61C2×SD1611yescharacteristic
212C2×C4×C61#13511yescharacteristic
213C2×C4×C61#13511yescharacteristic
214C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
215C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
216C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
217C2×C4×C61#13711yescharacteristic
218C2×Dic61—22no—
219C2×Dic61—41no—
220C2×Dic61—41no—
221D244—22no—
222D244—22no—
223D244—44no—
224D244—416no—
225D244—42no—
226D244—42no—
227D244—42no—
228D244—42no—
229D244—42no—
230D244—41no—
231D244—42no—
232D244—42no—
233D244—41no—
234D244—42no—
235D244—82no—
236D244—88no—
237C4×D61—416no—
238C4×D61—42no—
239C4×D61—42no—
240C4×D61—42no—
241C4×D61—42no—
242C8×C61—81no—
243#92418—42no—
244#92418—81no—
245#8444—611nomaximal, Sylow
246C22×C61#37911yescharacteristic
247C22×C61#44711yescharacteristic
248C22×C61#44911yescharacteristic
249C22×C61#45011yescharacteristic
250C22×C61#45211yescharacteristic
251C22×C61#45311yescharacteristic
252C22×C61#45511yescharacteristic
253C2×D61—44no—
254C2×D61—42no—
255C2×D61—44no—
256C2×D61—42no—
257C2×D61—42no—
258C2×D61—44no—
259C2×D61—81no—
260C2×D61—82no—
261C2×D61—81no—
262C2×D61—82no—
263C2×D61—81no—
264C2×D61—82no—
265C2×D61—81no—
266C2×D61—82no—
267C4×C61—21no—
268C4×C61—21no—
269C4×C61—24no—
270C4×C61—42no—
271C4×C61—161no—
272C4×C61C2×#10211yescharacteristic
273C4×C61#41611yescharacteristic
274C4×C61#41711yescharacteristic
275C4×C61#41811yescharacteristic
276C4×C61#44711yescharacteristic
277C4×C61#44911yescharacteristic
278C4×C61#45011yescharacteristic
279C4×C61#45211yescharacteristic
280Dic61—42no—
281C2×#550—611no—
282#1957—611no—
283#1803D6111yescharacteristic, complemented
284#1802—611no—
285C2×#331—611no—
286C2×Q8⋊C8—611no—
287C2×D4⋊C8—611no—
288C2×C61#162111yescharacteristic
289C2×C61#163111yescharacteristic
290C2×C61#164111yescharacteristic
291C2×C61#164211yescharacteristic
292C2×C61#178611yescharacteristic
293C2×C61#178811yescharacteristic
294C2×C61#179211yescharacteristic
295D61—84no—
296D61—82no—
297C22×#41—1221no—
298#550—614no—
299#550—1222no—
300#550—1222no—
301#550—1222no—
302C2×#127—611no—
303C2×#127—1221no—
304C2×#121—1222no—
305C2×#121—1221no—
306C2×C4×Q8C2×D6111yescharacteristic
307C2×C4×D4—611no—
308C2×C4×D4—1221no—
309C2×C4⋊C8—611no—
310C2×C4⋊C8C2×D6111yescharacteristic
311C2×#103—21no—
312C2×#102—1221no—
313C2×#102—1222no—
314C2×C4×C8—611no—
315#384C2×D6111yescharacteristic
316#340—21no—
317#340—1221no—
318#331—1222no—
319Q8⋊C8—1222no—
320D4⋊C8—1222no—
321C61#844411yescharacteristic, complemented, Sylow
322C2×#41—1224no—
323C2×#41—1224no—
324C2×#41—2442no—
325C2×#41—2442no—
326C2×#41—2442no—
327C2×#41—2442no—
328C22×Q8C22⋊D6111yescharacteristic
329C22×D4—611no—
330C22×D4—612no—
331C22×D4—1222no—
332C22×D4—1222no—
333C22×D4—1221no—
334C22×D4—1221no—
335C23×C4—1222no—
336C23×C4—1221no—
337C22×C8—21no—
338C22×C8—1221no—
339C22×C8—1221no—
340#127—612no—
341#127—612no—
342#127—1224no—
343#127—1224no—
344#127—1221no—
345#127—1221no—
346#121—1228no—
347#121—1224no—
348#121—1228no—
349#121—1224no—
350C4×Q8—21no—
351C4×Q8—21no—
352C4×Q8C4×D6112yes—
353C4×Q8D24412yes—
354C4×D4—612no—
355C4×D4—612no—
356C4×D4—1224no—
357C4×D4—1224no—
358C4×D4—1221no—
359C4×D4—1221no—
360C2×C4⋊C4—21no—
361C2×C4⋊C4—21no—
362C2×C4⋊C4—21no—
363C2×C4⋊C4—21no—
364C2×C4⋊C4—611no—
365C2×C4⋊C4C22⋊D6111yescharacteristic
366C2×C22⋊C4—1222no—
367C2×C22⋊C4—1221no—
368C2×C42—21no—
369C2×C42C22×D6111yescharacteristic
370C4⋊C8—21no—
371C4⋊C8—21no—
372C4⋊C8—1221no—
373C4⋊C8—1221no—
374#103—42no—
375#102—2444no—
376#102—2442no—
377C4×C8—1221no—
378C4×C8—1221no—
379C24—1221no—
380C24—1222no—
381#41—2448no—
382#41—2448no—
383#41—2448no—
384#41—2448no—
385C2×Q8—21no—
386C2×Q8—21no—
387C2×Q8—21no—
388C2×Q8#1175819812yes—
389C2×Q8#1175819912yes—
390C2×Q8#1175820112yes—
391C2×D4—612no—
392C2×D4—614no—
393C2×D4—1228no—
394C2×D4—1228no—
395C2×D4—1221no—
396C2×D4—1222no—
397C2×D4—1222no—
398C2×D4—1222no—
399C2×D4—1222no—
400C2×D4—1222no—
401C2×D4—1221no—
402C2×D4—1221no—
403C2×D4—1221no—
404C2×D4—1221no—
405C2×D4—4882no—
406C2×D4—4882no—
407C2×D4—2442no—
408C2×D4—2442no—
409C2×D4—2444no—
410C2×D4—2444no—
411C2×D4—2441no—
412C2×D4—2441no—
413C2×D4—2441no—
414C2×D4—2441no—
415C2×D4—2441no—
416C2×D4—2441no—
417C2×D4—2441no—
418C2×D4—2441no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—41no—
422C22×C4—1228no—
423C22×C4—1221no—
424C22×C4—1222no—
425C22×C4—1222no—
426C22×C4—2444no—
427C22×C4—2444no—
428C22×C4—2441no—
429C22×C4—2441no—
430C22×C4—2441no—
431C22×C4—2441no—
432C22×C4D4×D6111yescharacteristic
433C22×C4D4×D6111yescharacteristic
434C22×C4C2×C22⋊D6111yescharacteristic
435C2×C8—41no—
436C2×C8—41no—
437C2×C8—41no—
438C2×C8—1222no—
439C2×C8—2441no—
440C2×C8—2441no—
441C2×C8—2441no—
442C2×C8—2441no—
443C2×C8—2441no—
444C4⋊C4—22no—
445C4⋊C4—21no—
446C4⋊C4—22no—
447C4⋊C4—42no—
448C4⋊C4—41no—
449C4⋊C4—41no—
450C4⋊C4—41no—
451C4⋊C4—41no—
452C4⋊C4—612no—
453C4⋊C4—1221no—
454C4⋊C4#1175819812yes—
455C22⋊C4—1228no—
456C22⋊C4—1222no—
457C22⋊C4—1222no—
458C42—22no—
459C42—22no—
460C42C2×C4×D6111yescharacteristic
461C42C2×D24411yescharacteristic
462C42D4×D6111yescharacteristic
463C42D4×D6111yescharacteristic
464C23—1222no—
465C23—1224no—
466C23—2441no—
467C23—2442no—
468C23—2442no—
469C23—2441no—
470C23—2441no—
471C23—2442no—
472C23—2442no—
473C23—2441no—
474C23—2441no—
475C23—2442no—
476C23—2441no—
477C23—2442no—
478C23#1451974511yescharacteristic
479Q8—22no—
480Q8—22no—
481Q8—22no—
482Q8#1451969714yescomplemented
483D4—1222no—
484D4—1222no—
485D4—4882no—
486D4—4888no—
487D4—2444no—
488D4—24416no—
489D4—2442no—
490D4—2442no—
491D4—2442no—
492D4—2442no—
493D4—2442no—
494D4—2441no—
495D4—2442no—
496D4—2442no—
497D4—2441no—
498D4—2442no—
499C2×C4—21no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—21no—
504C2×C4—22no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—81no—
508C2×C4—81no—
509C2×C4—81no—
510C2×C4—1222no—
511C2×C4—24416no—
512C2×C4—2441no—
513C2×C4—2441no—
514C2×C4—2442no—
515C2×C4—2442no—
516C2×C4—2442no—
517C2×C4—2442no—
518C2×C4C22⋊C4×D6111yescharacteristic
519C2×C4C22⋊C4×D6111yescharacteristic
520C2×C4C2×#1175819811yescharacteristic
521C2×C4#1451974511yescharacteristic
522C2×C4#1451974711yescharacteristic
523C2×C4#1451974711yescharacteristic
524C2×C4C2×#1175819911yescharacteristic
525C2×C4C2×#1175820111yescharacteristic
526C2×C4#1451979811yescharacteristic
527C2×C4#1451980011yescharacteristic
528C2×C4#1451980211yescharacteristic
529C2×C4#1451980211yescharacteristic
530C2×C4#1451980411yescharacteristic
531C2×C4#1451980911yescharacteristic
532C8—81no—
533C8—4881no—
534C8—2442no—
535C22—4881no—
536C22—4882no—
537C22—4881no—
538C22—4882no—
539C22—4881no—
540C22—4882no—
541C22—4881no—
542C22—4882no—
543C22—2444no—
544C22—2442no—
545C22—2442no—
546C22—2444no—
547C22—2442no—
548C22—2444no—
549C22#1683961711yescharacteristic
550C22#1683989511yescharacteristic
551C22#1683990611yescharacteristic
552C22#1683991211yescharacteristic
553C22#1683992311yescharacteristic
554C22#1683992811yescharacteristic
555C22#1683993811yescharacteristic
556C4—24no—
557C4—21no—
558C4—21no—
559C4—42no—
560C4—161no—
561C4—2442no—
562C4C2×#1451969711yescharacteristic
563C4#1683975711yescharacteristic
564C4#1683976211yescharacteristic
565C4#1683976611yescharacteristic
566C4#1683989511yescharacteristic
567C4#1683991011yescharacteristic
568C4#1683991211yescharacteristic
569C4#1683992711yescharacteristic
570C2—4882no—
571C2—4884no—
572C2#2144488211yescharacteristic
573C2#2144494411yescharacteristic
574C2#2144501011yescharacteristic
575C2#2144501811yescharacteristic
576C2#2144574911yescharacteristic
577C2#2144575811yescharacteristic
578C2#2144577111yescharacteristic
subgroupG/H
centerC23#14519745
commutatorC42×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#1803C2
socleC23×C61#120
prime radicalC2×#16840434C2
cosocleC2×C42×C61C23
hypercenter#1803D61
hypocenterC61#8444
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1803

Complements

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

NH ≅ G/Ntype
2. #21446624C2semidirect
3. #21445816C2semidirect
4. C2×#16839505C2semidirect
5. C2×Q8⋊#92418C2semidirect
6. C2×#16839489C2semidirect
7. C61×#1803C2semidirect
12. #16840434C4semidirect
192. Q8×C61#102semidirect
283. #1803D61semidirect
321. C61#8444semidirect
482. Q8#14519697semidirect

Maximal subgroups

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

subgroupindex
C2×#168404342
#214466242
#214458162
C2×#168395052
C2×Q8⋊#924182
C2×#168394892
C61×#18032
#844461