Finite Group Databasebeta

← #29708058

Subgroups — #29708058

Order 15616.

1358 conjugacy classes of subgroups.

The subgroup lattice has 526 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4⋊Q8×D61C211yesmaximal, characteristic
2C2×#16839803C212yesmaximal
3#21448300C211yesmaximal, characteristic
4#21447153C211yesmaximal, characteristic, complemented
5#21447151C212yesmaximal, complemented
6#21447149C211yesmaximal, characteristic, complemented
7#21447147C211yesmaximal, characteristic, complemented
8#21447145C212yesmaximal, complemented
9#21445537C212yesmaximal
10#21445511C211yesmaximal, characteristic
11C61×#2057C211yesmaximal, characteristic, complemented
12C22×Q8×D61—21no—
13C22×#11758189C2212yes—
14C2×#14519777C2211yescharacteristic
15C2×#14519776—21no—
16C2×#14519776C2212yes—
17C2×#14519775—21no—
18C4⋊Q8×D61—28no—
19C2×#14519735C2211yescharacteristic
20C2×#14519735C2211yescharacteristic
21C2×C4⋊C4×D61—21no—
22C2×C4⋊C4×D61C2212yes—
23C2×C42×D61C2211yescharacteristic
24#16839803—28no—
25C2×#14519710C2212yes—
26C2×#14519709C2212yes—
27C2×#14519708C2212yes—
28#16839661C2211yescharacteristic
29#16839660C2211yescharacteristic
30#16839658C2211yescharacteristic
31#16839636—21no—
32#16839633—21no—
33C4⋊C4×Dic61—21no—
34#16839613—21no—
35#16839525C2212yes—
36#16839524C2212yes—
37#16839524C2212yes—
38#16839517C2211yescharacteristic
39#16839516C2212yes—
40C2×C4⋊Q8×C61C2211yescharacteristic
41C2×#107×C61C2212yes—
42C61×#389C2211yescharacteristic
43C61×#340C2212yes—
44C61×#336C2211yescharacteristic
45C22×Dic122—21no—
46C22×Dic122—22no—
47C2×Q8×D61—28no—
48C2×Q8×D61—44no—
49C2×Q8×D61—44no—
50C22×C4×D61—22no—
51C22×C4×D61D411yescharacteristic
52C22×C4×D61D411yescharacteristic
53C22×C4×D61C2311yescharacteristic
54C2×#11758189—24no—
55C2×#11758189—24no—
56C2×#11758189Q818yescomplemented
57C22×#92418C2312yes—
58#14519777—24no—
59#14519776—28no—
60#14519776—24no—
61#14519776—24no—
62#14519775—28no—
63#14519735—22no—
64#14519735—24no—
65#14519735Q814yescomplemented
66C2×C4⋊Dic61—21no—
67C2×C4⋊Dic61—21no—
68C2×C4⋊Dic61C2311yescharacteristic
69C2×C4⋊Dic61C2312yes—
70C2×C4⋊Dic61C2311yescharacteristic
71C2×#11758196—22no—
72C2×#11758196—22no—
73C2×#11758196—21no—
74C2×#11758196—21no—
75C4⋊C4×D61—28no—
76C4⋊C4×D61—24no—
77C4⋊C4×D61D418yes—
78C2×#11758198—21no—
79C2×#11758198—21no—
80C2×C4×Dic61—21no—
81C2×C4×Dic61—21no—
82C2×C4×Dic61D411yescharacteristic
83C2×C4×Dic61D411yescharacteristic
84C2×C4×Dic61C2311yescharacteristic
85C42×D61—24no—
86#14519710—24no—
87#14519709—24no—
88#14519708—24no—
89#14519684—21no—
90#14519684—21no—
91#14519684—21no—
92C22×Q8×C61—21no—
93C22×C8×C61C2312yes—
94C4⋊Q8×C61—24no—
95C2×C4⋊C4×C61—21no—
96C2×C4⋊C4×C61—21no—
97C2×C4⋊C4×C61C2311yescharacteristic
98C2×C4⋊C4×C61C2312yes—
99C2×C4⋊C4×C61C2311yescharacteristic
100C2×C42×C61C2311yescharacteristic
101#107×C61—24no—
102#95×C61—21no—
103C23×D61C2×D411yescharacteristic
104C2×Q8×C61—24no—
105C2×Q8×C61—42no—
106C2×Q8×C61—42no—
107C22×C4×C61—22no—
108C22×C4×C61—21no—
109C22×C4×C61—21no—
110C22×C4×C61C2×D411yescharacteristic
111C22×C4×C61C2×D411yescharacteristic
112C22×C4×C61C2411yescharacteristic
113C22×Dic61—22no—
114C22×Dic61—21no—
115C22×Dic61—21no—
116C22×Dic61C2×D411yescharacteristic
117C22×Dic61C2×D411yescharacteristic
118C22×Dic61C2×D411yescharacteristic
119Q8×D61—432no—
120C2×C4×D61—22no—
121C2×C4×D61—22no—
122C2×C4×D61—22no—
123C2×C4×D61—22no—
124C2×C4×D61—22no—
125C2×C4×D61—22no—
126C2×C4×D61—28no—
127C2×C4×D61—44no—
128C2×C4×D61—44no—
129C2×C4×D61C2×D414yes—
130C2×C4×D61C2×Q814yes—
131C2×C4×D61C2×Q812yes—
132C2×C4×D61#4112yes—
133C2×Dic122—24no—
134C2×Dic122—28no—
135C2×Dic122—44no—
136C2×Dic122—42no—
137C2×Dic122—42no—
138C2×Dic122—44no—
139C2×C8×C61—22no—
140C2×C8×C61—22no—
141C2×C8×C61C2×Q814yescomplemented
142C4⋊C4×C61—24no—
143C4⋊C4×C61—22no—
144C4⋊C4×C61—21no—
145C4⋊C4×C61—22no—
146C4⋊C4×C61—42no—
147C4⋊C4×C61C2×D414yes—
148C4⋊C4×C61C2×Q812yescomplemented
149C42×C61—22no—
150#11758198—24no—
151#11758198—42no—
152C4⋊Dic61—24no—
153C4⋊Dic61—22no—
154C4⋊Dic61—21no—
155C4⋊Dic61—22no—
156C4⋊Dic61—42no—
157C4⋊Dic61C2×D414yes—
158C4⋊Dic61C2×Q812yescomplemented
159#11758196—28no—
160#11758196—28no—
161#11758196—42no—
162#11758196—42no—
163C4×Dic61—22no—
164C4×Dic61—22no—
165C4×Dic61—22no—
166C4×Dic61—24no—
167C4×Dic61—42no—
168C2×#92418—22no—
169C2×#92418—22no—
170C2×#92418C2×Q814yescomplemented
171#11758189—216no—
172C23×C61C22×D411yescharacteristic
173C22×D61—21no—
174C22×D61—21no—
175C22×D61—21no—
176C22×D61—21no—
177C22×D61#12012yes—
178C22×D61C22⋊Q812yes—
179C22×D61C4⋊Q812yes—
180Q8×C61—48no—
181C2×C4×C61—21no—
182C2×C4×C61—21no—
183C2×C4×C61—21no—
184C2×C4×C61—21no—
185C2×C4×C61—24no—
186C2×C4×C61—22no—
187C2×C4×C61—41no—
188C2×C4×C61—42no—
189C2×C4×C61—42no—
190C2×C4×C61—41no—
191C2×C4×C61—41no—
192C2×C4×C61—41no—
193C2×C4×C61—41no—
194C2×C4×C61AGL(1,ℤ/8)12yes—
195C2×C4×C61#13712yes—
196C2×C4×C61C22×D412yes—
197C2×C4×C61C22×Q812yes—
198C2×C4×C61C22×Q811yescharacteristic
199C2×C4×C61C2×#4111yescharacteristic
200C2×Dic61—21no—
201C2×Dic61—21no—
202C2×Dic61—21no—
203C2×Dic61—21no—
204C2×Dic61—24no—
205C2×Dic61—22no—
206C2×Dic61—41no—
207C2×Dic61—42no—
208C2×Dic61—42no—
209C2×Dic61—41no—
210C2×Dic61—41no—
211C2×Dic61—41no—
212C2×Dic61—41no—
213C2×Dic61#12012yes—
214C2×Dic61C22⋊Q812yes—
215C2×Dic61C4⋊Q812yes—
216C2×Dic61AGL(1,ℤ/8)12yes—
217C2×Dic61#13712yes—
218C4×D61—28no—
219C4×D61—28no—
220C4×D61—416no—
221C4×D61C22⋊Q818yes—
222C4×D61C4⋊Q818yes—
223Dic122—416no—
224Dic122—48no—
225C8×C61—24no—
226#92418—24no—
227#23552—611nomaximal, Sylow
228C22×C61C2×#12011yescharacteristic
229C22×C61C2×C22⋊Q811yescharacteristic
230C22×C61C2×C4⋊Q811yescharacteristic
231C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
232C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
233C22×C61C2×#13711yescharacteristic
234C22×C61C2×#13711yescharacteristic
235C2×D61—22no—
236C2×D61—22no—
237C2×D61—22no—
238C2×D61—22no—
239C2×D61—22no—
240C2×D61—22no—
241C2×D61#39514yes—
242C4×C61—22no—
243C4×C61—22no—
244C4×C61—42no—
245C4×C61—44no—
246C4×C61—81no—
247C4×C61C2×C22⋊Q812yes—
248C4×C61C2×C4⋊Q812yes—
249Dic61—22no—
250Dic61—22no—
251Dic61—42no—
252Dic61—44no—
253Dic61—81no—
254Dic61#39514yes—
255C22×C4⋊Q8—611no—
256C2×#428—612no—
257#2296—611no—
258#2058—612no—
259#2058—611no—
260#2058—611no—
261#2057—612no—
262#2057—611no—
263#2057D6111yescharacteristic, complemented
264#1736—612no—
265#1728—611no—
266C2×C61C2×#39511yescharacteristic
267C2×C61#300311yescharacteristic
268C2×C61#300411yescharacteristic
269C2×C61#307911yescharacteristic
270C2×C61#307911yescharacteristic
271C2×C61#316511yescharacteristic
272C2×C61#316511yescharacteristic
273D61—24no—
274C23×Q8—1221no—
275C22×M16—612no—
276C2×C4⋊Q8—612no—
277C2×C4⋊Q8—611no—
278C2×C4⋊Q8—1221no—
279C2×C4⋊Q8—1221no—
280C2×C4⋊Q8—1228no—
281C2×C4⋊Q8C2×D6111yescharacteristic
282C2×#117—611no—
283C2×#117—611no—
284C22×C4⋊C4—612no—
285C22×C4⋊C4—1221no—
286C22×C42—611no—
287#428—1228no—
288C2×#107—612no—
289C2×#107C2×D6112yes—
290C2×#106—612no—
291C2×#106—612no—
292#389—611no—
293#389—611no—
294#389—611no—
295#389C2×D6111yescharacteristic
296#385—1221no—
297#385—1221no—
298C4×C4⋊C4—1221no—
299C4×C4⋊C4—1221no—
300#340—612no—
301#340—612no—
302#340—612no—
303#340C2×D6112yes—
304#336—612no—
305#336—611no—
306#336C2×D6111yescharacteristic
307C61#2355211yescharacteristic, complemented, Sylow
308C22×Q8—21no—
309C22×Q8—1222no—
310C22×Q8—1228no—
311C22×Q8—1221no—
312C22×Q8—2444no—
313C22×Q8—2444no—
314C23×C4—611no—
315C23×C4—611no—
316C23×C4—611no—
317C23×C4—1222no—
318C2×M16—618no—
319C2×M16—1224no—
320C2×M16—1224no—
321C22×C8—612no—
322C22×C8C22×D6112yes—
323C4⋊Q8—24no—
324C4⋊Q8—1224no—
325C4⋊Q8—1224no—
326C4⋊Q8—1228no—
327C4⋊Q8—1228no—
328C4⋊Q8—1224no—
329#117—614no—
330#117—1222no—
331#117—1224no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—612no—
335C2×C4⋊C4—611no—
336C2×C4⋊C4—618no—
337C2×C4⋊C4—611no—
338C2×C4⋊C4—1224no—
339C2×C4⋊C4—1228no—
340C2×C4⋊C4—1222no—
341C2×C4⋊C4—1222no—
342C2×C4⋊C4—1221no—
343C2×C4⋊C4—1221no—
344C2×C4⋊C4—1221no—
345C2×C4⋊C4—1221no—
346C2×C4⋊C4C22×D6111yescharacteristic
347C2×C4⋊C4C22×D6112yes—
348C2×C4⋊C4C22×D6111yescharacteristic
349C2×C22⋊C4—1221no—
350C2×C22⋊C4—1221no—
351C2×C42—611no—
352C2×C42—611no—
353C2×C42—611no—
354C2×C42—1224no—
355C2×C42—1221no—
356C2×C42—1221no—
357C2×C42C22×D6111yescharacteristic
358#107—24no—
359#107—1224no—
360#106—1224no—
361#106—1224no—
362#95—21no—
363#95—1221no—
364#95—1221no—
365#95—1221no—
366C24—611no—
367C2×Q8—24no—
368C2×Q8—42no—
369C2×Q8—42no—
370C2×Q8—1224no—
371C2×Q8—1228no—
372C2×Q8—2442no—
373C2×Q8—2444no—
374C2×Q8—2444no—
375C2×Q8—24432no—
376C2×Q8—2442no—
377C22×C4—21no—
378C22×C4—22no—
379C22×C4—21no—
380C22×C4—611no—
381C22×C4—614no—
382C22×C4—612no—
383C22×C4—614no—
384C22×C4—612no—
385C22×C4—611no—
386C22×C4—611no—
387C22×C4—1222no—
388C22×C4—1222no—
389C22×C4—1222no—
390C22×C4—1222no—
391C22×C4—1228no—
392C22×C4—1222no—
393C22×C4—1222no—
394C22×C4—1221no—
395C22×C4—1222no—
396C22×C4—1221no—
397C22×C4—2444no—
398C22×C4—2444no—
399C22×C4D4×D6111yescharacteristic
400C22×C4D4×D6111yescharacteristic
401C22×C4C23×D6111yescharacteristic
402M16—12216no—
403C2×C8—22no—
404C2×C8—22no—
405C2×C8—614no—
406C2×C8—1222no—
407C2×C8—1222no—
408C2×C8Q8×D6114yescomplemented
409C4⋊C4—24no—
410C4⋊C4—22no—
411C4⋊C4—21no—
412C4⋊C4—22no—
413C4⋊C4—42no—
414C4⋊C4—614no—
415C4⋊C4—612no—
416C4⋊C4—1228no—
417C4⋊C4—1228no—
418C4⋊C4—1224no—
419C4⋊C4—1222no—
420C4⋊C4—1221no—
421C4⋊C4—1222no—
422C4⋊C4—2442no—
423C4⋊C4—2442no—
424C4⋊C4—2442no—
425C4⋊C4D4×D6114yes—
426C4⋊C4Q8×D6112yescomplemented
427C22⋊C4—1224no—
428C22⋊C4—2442no—
429C42—22no—
430C42—1222no—
431C42—1222no—
432C42—1224no—
433C42—1222no—
434C42—2442no—
435C23—612no—
436C23—612no—
437C23—612no—
438C23—1221no—
439C23—1221no—
440C23—1221no—
441C23—1221no—
442C23C2×D4×D6111yescharacteristic
443Q8—48no—
444Q8—2448no—
445Q8—24416no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—24no—
451C2×C4—22no—
452C2×C4—42no—
453C2×C4—41no—
454C2×C4—42no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—612no—
460C2×C4—612no—
461C2×C4—612no—
462C2×C4—618no—
463C2×C4—618no—
464C2×C4—612no—
465C2×C4—612no—
466C2×C4—1228no—
467C2×C4—1228no—
468C2×C4—1221no—
469C2×C4—1221no—
470C2×C4—1221no—
471C2×C4—1221no—
472C2×C4—1224no—
473C2×C4—1222no—
474C2×C4—24416no—
475C2×C4—2441no—
476C2×C4—2441no—
477C2×C4—2441no—
478C2×C4—2441no—
479C2×C4—2442no—
480C2×C4—2442no—
481C2×C4—2441no—
482C2×C4#1451980112yes—
483C2×C4#1451980912yes—
484C2×C4C2×D4×D6112yes—
485C2×C4C2×Q8×D6112yes—
486C2×C4C2×Q8×D6111yescharacteristic
487C2×C4#41×D6111yescharacteristic
488C8—24no—
489C8—1224no—
490C22—614no—
491C22—1222no—
492C22—1222no—
493C22—1222no—
494C22—1222no—
495C22—1222no—
496C22—1222no—
497C22#120×D6111yescharacteristic
498C22C22⋊Q8×D6111yescharacteristic
499C22C4⋊Q8×D6111yescharacteristic
500C22AGL(1,ℤ/8)×D6111yescharacteristic
501C22C2×#1451980111yescharacteristic
502C22#137×D6111yescharacteristic
503C22C2×#1451980911yescharacteristic
504C4—22no—
505C4—22no—
506C4—42no—
507C4—44no—
508C4—81no—
509C4—614no—
510C4—1222no—
511C4—1222no—
512C4—4881no—
513C4—2442no—
514C4—2444no—
515C4C22⋊Q8×D6112yes—
516C4C4⋊Q8×D6112yes—
517C2—1224no—
518C2#395×D6111yescharacteristic
519C2#2145504311yescharacteristic
520C2#2145505411yescharacteristic
521C2#2145562711yescharacteristic
522C2#2145563311yescharacteristic
523C2#2145636211yescharacteristic
524C2#2145636411yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2057C2
socleC23×C61C22×D4
prime radicalC23×D61C2×D4
hypercenter#2057D61
hypocenterC61#23552
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2057

Complements

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

NH ≅ G/Ntype
4. #21447153C2semidirect
5. #21447151C2semidirect
6. #21447149C2semidirect
7. #21447147C2semidirect
8. #21447145C2semidirect
11. C61×#2057C2semidirect
56. C2×#11758189Q8semidirect
65. #14519735Q8semidirect
141. C2×C8×C61C2×Q8semidirect
148. C4⋊C4×C61C2×Q8semidirect
158. C4⋊Dic61C2×Q8semidirect
170. C2×#92418C2×Q8semidirect
263. #2057D61semidirect
307. C61#23552semidirect
408. C2×C8Q8×D61semidirect
426. C4⋊C4Q8×D61semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊Q8×D612
C2×#168398032
#214483002
#214471532
#214471512
#214471492
#214471472
#214471452
#214455372
#214455112
C61×#20572
#2355261