Finite Group Databasebeta

← #29714794

Subgroups — #29714794

Order 15616.

1394 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840433C211yesmaximal, characteristic
2C2×#16839962C211yesmaximal, characteristic, complemented
3#21451189C211yesmaximal, characteristic, complemented
4#21447051C212yesmaximal, complemented
5#21446905C212yesmaximal, complemented
6#21446755C211yesmaximal, characteristic, complemented
7#21446625C211yesmaximal, characteristic, complemented
8#21446150C212yesmaximal, complemented
9#21446077C212yesmaximal, complemented
10#21445819C211yesmaximal, characteristic, complemented
11C2×C4×SD16×C61C211yesmaximal, characteristic, complemented
12C22×#11758222—21no—
13C22×#11758201C2212yes—
14#16840433—24no—
15#16840433—24no—
16C2×#14519774C2211yescharacteristic
17C2×#14519773C2212yes—
18C2×#14519751—21no—
19C2×#14519750—22no—
20C2×C4×Dic122C2211yescharacteristic
21C2×#14519739C2212yes—
22#16839962—24no—
23C2×C4⋊#92418C2211yescharacteristic
24C2×#14519699C2212yes—
25C2×#14519699C2212yes—
26C2×#14519698C2212yes—
27C2×#14519697C2212yes—
28#16839695—22no—
29#16839669C2212yes—
30#16839668—21no—
31#16839660—21no—
32#16839658C2212yes—
33#16839632C2211yescharacteristic
34#16839524C2212yes—
35#16839516C2212yes—
36#16839516C2212yes—
37C22×SD16×C61C2211yescharacteristic
38C2×C4×Q8×C61C2211yescharacteristic
39C2×C4×D4×C61C2211yescharacteristic
40C4×SD16×C61—24no—
41C2×#106×C61C2211yescharacteristic
42C2×#103×C61C2211yescharacteristic
43C2×#102×C61C2211yescharacteristic
44C2×C4×C8×C61C2211yescharacteristic
45C2×#11758222—28no—
46C2×#11758222—44no—
47C2×#11758222—44no—
48C22×Dic122D411yescharacteristic
49C22×D244—21no—
50C22×D244—21no—
51C22×D244C2312yes—
52C22×C22⋊D61—22no—
53C22×C22⋊D61—22no—
54C22×C4×D61—22no—
55C2×#11758201—24no—
56C2×#11758201—24no—
57C2×#11758201—42no—
58C22×#92418C2312yes—
59#14519774—22no—
60#14519774D414yes—
61#14519773—24no—
62#14519773—24no—
63#14519751—28no—
64#14519750—216no—
65C4×Dic122—22no—
66C4×Dic122—22no—
67#14519739—24no—
68#14519739—24no—
69C2×C4⋊Dic61—21no—
70C2×C4⋊Dic61—21no—
71C2×C4⋊Dic61D411yescharacteristic
72C2×C4⋊Dic61C2312yes—
73C2×#11758196—22no—
74C2×#11758196D412yes—
75C2×#11758198—22no—
76C2×C4×Dic61C2312yes—
77C4⋊#92418—22no—
78#14519699—24no—
79#14519699—42no—
80#14519698—42no—
81#14519697—24no—
82#14519684—22no—
83#14519684—22no—
84#14519684—22no—
85C22×Q8×C61C2311yescharacteristic
86C22×D4×C61C2311yescharacteristic
87C23×C4×C61—21no—
88C2×SD16×C61—22no—
89C2×SD16×C61—22no—
90C2×SD16×C61—22no—
91C22×C8×C61C2311yescharacteristic
92C22×C8×C61C2311yescharacteristic
93C4×Q8×C61—22no—
94C4×Q8×C61—22no—
95C4×D4×C61—22no—
96C4×D4×C61D414yes—
97C2×C4⋊C4×C61—21no—
98C2×C4⋊C4×C61C2311yescharacteristic
99C2×C4⋊C4×C61C2311yescharacteristic
100C2×C22⋊C4×C61—21no—
101C2×C42×C61—21no—
102C2×C42×C61C2311yescharacteristic
103#106×C61—22no—
104#103×C61—22no—
105#102×C61—22no—
106C4×C8×C61—22no—
107C24×C61—21no—
108C23×D61—22no—
109C2×Q8×C61—21no—
110C2×Q8×C61—21no—
111C2×Q8×C61—21no—
112C2×Q8×C61—21no—
113C2×Q8×C61—21no—
114C2×Q8×C61#4112yescomplemented
115C2×D4×C61—21no—
116C2×D4×C61—21no—
117C2×D4×C61—21no—
118C2×D4×C61—21no—
119C2×D4×C61—21no—
120C2×D4×C61C2×D412yes—
121C22×C4×C61—21no—
122C22×C4×C61—21no—
123C22×C4×C61—24no—
124C22×C4×C61—21no—
125C22×C4×C61—42no—
126C22×C4×C61—42no—
127C22×C4×C61C2×D411yescharacteristic
128C22×C4×C61C2×D411yescharacteristic
129C22×C4×C61C2411yescharacteristic
130C2×C22⋊D61—28no—
131C2×C22⋊D61—28no—
132C2×C22⋊D61—44no—
133C2×C22⋊D61—44no—
134C2×C22⋊D61—82no—
135C2×C22⋊D61—82no—
136C22×Dic61—22no—
137C22×Dic61C2×D412yes—
138C22×Dic61C2×D412yes—
139#11758222—432no—
140C2×D244—22no—
141C2×D244—22no—
142C2×D244—22no—
143C2×D244—22no—
144C2×D244—24no—
145C2×D244—22no—
146C2×D244—24no—
147C2×D244—42no—
148C2×D244—42no—
149C2×D244—81no—
150C2×D244—81no—
151C2×D244C2×D414yes—
152C2×C4×D61—28no—
153C2×C4×D61—44no—
154C2×C4×D61—44no—
155C2×Dic122—22no—
156C2×Dic122—22no—
157C2×Dic122SD1614yescomplemented
158SD16×C61—28no—
159C2×C8×C61—21no—
160C2×C8×C61—21no—
161C2×C8×C61—21no—
162C2×C8×C61—21no—
163C2×C8×C61—21no—
164C2×C8×C61#4112yescomplemented
165C4⋊C4×C61—21no—
166C4⋊C4×C61—21no—
167C4⋊C4×C61—21no—
168C4⋊C4×C61—42no—
169C4⋊C4×C61C2×D412yes—
170C22⋊C4×C61—24no—
171C42×C61—42no—
172C42×C61C2×D412yes—
173C42×C61C2×D412yes—
174#11758201—48no—
175#11758198—28no—
176C4⋊Dic61—22no—
177C4⋊Dic61—22no—
178C4⋊Dic61—22no—
179C4⋊Dic61—42no—
180C4⋊Dic61—81no—
181#11758196—24no—
182#11758196—82no—
183C4×Dic61—22no—
184C4×Dic61—22no—
185C2×#92418—22no—
186C2×#92418—22no—
187C2×#92418—22no—
188C23×C61—22no—
189C23×C61—41no—
190C23×C61—41no—
191C23×C61—41no—
192C23×C61—41no—
193C23×C61—41no—
194C23×C61—41no—
195C23×C61C22×D411yescharacteristic
196C22×D61—24no—
197C22×D61—42no—
198C22×D61—42no—
199C22×D61—42no—
200C22×D61—42no—
201C22×D61—42no—
202C22×D61—42no—
203Q8×C61—22no—
204Q8×C61—22no—
205Q8×C61—41no—
206Q8×C61—41no—
207D4×C61—22no—
208D4×C61—22no—
209D4×C61—41no—
210D4×C61—41no—
211C2×C4×C61—21no—
212C2×C4×C61—21no—
213C2×C4×C61—22no—
214C2×C4×C61—22no—
215C2×C4×C61—41no—
216C2×C4×C61—41no—
217C2×C4×C61—41no—
218C2×C4×C61—41no—
219C2×C4×C61—48no—
220C2×C4×C61—41no—
221C2×C4×C61—41no—
222C2×C4×C61—41no—
223C2×C4×C61#12012yes—
224C2×C4×C61#12012yes—
225C2×C4×C61C2×SD1612yes—
226C2×C4×C61AGL(1,ℤ/8)12yes—
227C2×C4×C61C22×D411yescharacteristic
228C2×C4×C61C22×D411yescharacteristic
229C2×C4×C61C2×#4111yescharacteristic
230C2×C4×C61C2×#4111yescharacteristic
231C2×C4×C61H(5,𝔽2)11yescharacteristic
232C2×C4×C612−1+411yescharacteristic
233C22⋊D61—416no—
234C22⋊D61—88no—
235C2×Dic61—22no—
236C2×Dic61—22no—
237C2×Dic61—22no—
238C2×Dic61—22no—
239C2×Dic61—22no—
240C2×Dic61—42no—
241C2×Dic61—42no—
242C2×Dic61—42no—
243C2×Dic61C2×SD1614yes—
244D244—24no—
245D244—24no—
246D244—42no—
247D244—42no—
248D244—48no—
249D244—84no—
250C4×D61—416no—
251Dic122—28no—
252C8×C61—22no—
253C8×C61—41no—
254#92418—42no—
255#24188—611nomaximal, Sylow
256C22×C61—42no—
257C22×C61—42no—
258C22×C61—42no—
259C22×C61—81no—
260C22×C61—81no—
261C22×C61—81no—
262C22×C61—81no—
263C22×C61C2×#12011yescharacteristic
264C22×C61#54811yescharacteristic
265C22×C61#55011yescharacteristic
266C22×C61C22×SD1611yescharacteristic
267C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
268C22×C61#57611yescharacteristic
269C22×C61#57711yescharacteristic
270C2×D61—44no—
271C2×D61—44no—
272C2×D61—44no—
273C2×D61—82no—
274C2×D61—82no—
275C2×D61—82no—
276C2×D61—82no—
277C4×C61—22no—
278C4×C61—42no—
279C4×C61—42no—
280C4×C61—81no—
281C4×C61#45014yes—
282C4×C61#52511yescharacteristic
283C4×C61#53711yescharacteristic
284C4×C61#54611yescharacteristic
285C4×C61#54711yescharacteristic
286Dic61—24no—
287Dic61—42no—
288Dic61—82no—
289C2×#550—611no—
290C2×#460—611no—
291C2×C4×SD16D6111yescharacteristic, complemented
292#2582—611no—
293#2041—612no—
294#2014—612no—
295#1984—611no—
296#1957—611no—
297#1877—612no—
298#1864—612no—
299#1802—611no—
300C2×C61—82no—
301C2×C61#240711yescharacteristic
302C2×C61C2×#45011yescharacteristic
303C2×C61#301211yescharacteristic
304C2×C61#336211yescharacteristic
305C2×C61#336811yescharacteristic
306C2×C61#339211yescharacteristic
307C2×C61#339911yescharacteristic
308D61—84no—
309C22×#41—1221no—
310C22×SD16—612no—
311C22×SD16C2×D6111yescharacteristic
312#550—1224no—
313#550—1224no—
314C2×#127—612no—
315C2×#127—611no—
316C2×#121—1222no—
317C2×#121—1221no—
318C2×C4×Q8—611no—
319C2×C4×Q8C2×D6111yescharacteristic
320C2×C4×D4—612no—
321C2×C4×D4C2×D6111yescharacteristic
322#460—1224no—
323C4×SD16—24no—
324C2×#106C2×D6111yescharacteristic
325C2×C4⋊C8—611no—
326C2×#103C2×D6111yescharacteristic
327C2×#102—612no—
328C2×#102—612no—
329C2×#102—612no—
330C2×#102—612no—
331C2×#102C2×D6111yescharacteristic
332C2×C4×C8C2×D6111yescharacteristic
333#398—1222no—
334#391—612no—
335#391—1221no—
336#389—612no—
337#389—1221no—
338#384—611no—
339#340—612no—
340#336—612no—
341#336—612no—
342C61#2418811yescharacteristic, complemented, Sylow
343C2×#41—1228no—
344C2×#41—2444no—
345C2×#41—2444no—
346C22×Q8—611no—
347C22×Q8C22×D6111yescharacteristic
348C22×D4—612no—
349C22×D4—1222no—
350C22×D4—1222no—
351C22×D4—1221no—
352C22×D4—1221no—
353C22×D4C22×D6111yescharacteristic
354C23×C4—21no—
355C23×C4—1222no—
356C2×SD16—22no—
357C2×SD16—22no—
358C2×SD16—22no—
359C2×SD16—1224no—
360C2×SD16—1224no—
361C2×SD16—2442no—
362C22×C8—612no—
363C22×C8C22×D6111yescharacteristic
364C22×C8C22×D6111yescharacteristic
365#127—614no—
366#127—1224no—
367#127—1224no—
368#127—1222no—
369#121—12216no—
370#121—1228no—
371C4×Q8—22no—
372C4×Q8—22no—
373C4×Q8—1222no—
374C4×Q8—1222no—
375C4×D4—22no—
376C4×D4—1224no—
377C4×D4—1224no—
378C4×D4C22⋊D6114yes—
379C2×C4⋊C4—21no—
380C2×C4⋊C4—611no—
381C2×C4⋊C4—612no—
382C2×C4⋊C4—612no—
383C2×C4⋊C4—1222no—
384C2×C4⋊C4—1221no—
385C2×C4⋊C4—1221no—
386C2×C4⋊C4C22×D6111yescharacteristic
387C2×C4⋊C4C22×D6111yescharacteristic
388C2×C22⋊C4—21no—
389C2×C22⋊C4—1222no—
390C2×C42—21no—
391C2×C42—612no—
392C2×C42C22×D6111yescharacteristic
393#106—22no—
394C4⋊C8—1222no—
395#103—22no—
396#102—22no—
397#102—1224no—
398#102—1224no—
399#102—2442no—
400#102—2442no—
401C4×C8—22no—
402#95—1222no—
403#95—1222no—
404#95—1222no—
405C24—21no—
406C24—1222no—
407#41—24432no—
408C2×Q8—21no—
409C2×Q8—21no—
410C2×Q8—21no—
411C2×Q8—21no—
412C2×Q8—21no—
413C2×Q8—614no—
414C2×Q8—1222no—
415C2×Q8—1222no—
416C2×Q8#1175822212yescomplemented
417C2×D4—21no—
418C2×D4—21no—
419C2×D4—21no—
420C2×D4—21no—
421C2×D4—21no—
422C2×D4—614no—
423C2×D4—1228no—
424C2×D4—1228no—
425C2×D4—1222no—
426C2×D4—1222no—
427C2×D4—1222no—
428C2×D4—1222no—
429C2×D4—1222no—
430C2×D4—1224no—
431C2×D4—1224no—
432C2×D4—4881no—
433C2×D4—4881no—
434C2×D4—4882no—
435C2×D4—4882no—
436C2×D4—2444no—
437C2×D4—2444no—
438C2×D4—2442no—
439C2×D4—2442no—
440C2×D4C2×C22⋊D6112yes—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—24no—
444C22×C4—21no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—612no—
448C22×C4—612no—
449C22×C4—1228no—
450C22×C4—1222no—
451C22×C4—2444no—
452C22×C4—2444no—
453C22×C4D4×D6111yescharacteristic
454C22×C4D4×D6111yescharacteristic
455C22×C4C23×D6111yescharacteristic
456SD16—28no—
457SD16—2448no—
458C2×C8—21no—
459C2×C8—21no—
460C2×C8—21no—
461C2×C8—21no—
462C2×C8—21no—
463C2×C8—1222no—
464C2×C8—1222no—
465C2×C8—1222no—
466C2×C8#1175822212yescomplemented
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—42no—
471C4⋊C4—1224no—
472C4⋊C4—1222no—
473C4⋊C4—1222no—
474C4⋊C4—1222no—
475C4⋊C4—4881no—
476C4⋊C4—4882no—
477C4⋊C4—2442no—
478C4⋊C4C2×C22⋊D6112yes—
479C22⋊C4—24no—
480C22⋊C4—1228no—
481C42—42no—
482C42—1222no—
483C42—1222no—
484C42D4×D6112yes—
485C42C2×C22⋊D6112yes—
486C23—22no—
487C23—41no—
488C23—41no—
489C23—41no—
490C23—41no—
491C23—41no—
492C23—41no—
493C23—1224no—
494C23—2442no—
495C23—2442no—
496C23—2442no—
497C23—2442no—
498C23—2442no—
499C23—2442no—
500C23C2×D4×D6111yescharacteristic
501Q8—22no—
502Q8—22no—
503Q8—41no—
504Q8—41no—
505Q8—1228no—
506D4—22no—
507D4—22no—
508D4—41no—
509D4—41no—
510D4—1224no—
511D4—1224no—
512D4—4884no—
513D4—4888no—
514D4—24416no—
515D4—2442no—
516D4—2442no—
517D4—2448no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—48no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—614no—
531C2×C4—1222no—
532C2×C4—1222no—
533C2×C4—1222no—
534C2×C4—1222no—
535C2×C4—1222no—
536C2×C4—24416no—
537C2×C4—2442no—
538C2×C4—2442no—
539C2×C4—2442no—
540C2×C4#1451974512yes—
541C2×C4#1451974512yes—
542C2×C4SD16×D6112yes—
543C2×C4#1451980412yes—
544C2×C4C2×D4×D6111yescharacteristic
545C2×C4C22×C22⋊D6111yescharacteristic
546C2×C4C2×#1175822211yescharacteristic
547C2×C4C2×#1175822211yescharacteristic
548C2×C4#1451982311yescharacteristic
549C2×C4#1451982411yescharacteristic
550C8—22no—
551C8—41no—
552C8—2442no—
553C22—42no—
554C22—42no—
555C22—42no—
556C22—81no—
557C22—81no—
558C22—81no—
559C22—81no—
560C22—4882no—
561C22—4882no—
562C22—4882no—
563C22—4882no—
564C22—2444no—
565C22—2444no—
566C22—2444no—
567C22C2×#1451974511yescharacteristic
568C22#1684042111yescharacteristic
569C22#1684043311yescharacteristic
570C22C2×SD16×D6111yescharacteristic
571C22C2×#1451980411yescharacteristic
572C22#1684056411yescharacteristic
573C22#1684056911yescharacteristic
574C4—22no—
575C4—42no—
576C4—42no—
577C4—81no—
578C4—1224no—
579C4—4882no—
580C4—2442no—
581C4#1683991114yes—
582C4#1684027311yescharacteristic
583C4#1684034011yescharacteristic
584C4#1684040811yescharacteristic
585C4#1684041711yescharacteristic
586C2—82no—
587C2—4884no—
588C2#2144916411yescharacteristic
589C2C2×#1683991111yescharacteristic
590C2#2145511911yescharacteristic
591C2#2145848811yescharacteristic
592C2#2145854611yescharacteristic
593C2#2145879811yescharacteristic
594C2#2145885911yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C4×SD16×C61C2
socleC23×C61C22×D4
prime radicalC2×#16840433C2
hypercenterC2×C4×SD16D61
hypocenterC61#24188
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C4×SD16

Complements

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

NH ≅ G/Ntype
2. C2×#16839962C2semidirect
3. #21451189C2semidirect
4. #21447051C2semidirect
5. #21446905C2semidirect
6. #21446755C2semidirect
7. #21446625C2semidirect
8. #21446150C2semidirect
9. #21446077C2semidirect
10. #21445819C2semidirect
11. C2×C4×SD16×C61C2semidirect
114. C2×Q8×C61#41semidirect
157. C2×Dic122SD16semidirect
164. C2×C8×C61#41semidirect
291. C2×C4×SD16D61semidirect
342. C61#24188semidirect
416. C2×Q8#11758222semidirect
466. C2×C8#11758222semidirect

Maximal subgroups

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

subgroupindex
C2×#168404332
C2×#168399622
#214511892
#214470512
#214469052
#214467552
#214466252
#214461502
#214460772
#214458192
C2×C4×SD16×C612
#2418861