Finite Group Databasebeta

← #29708050

Subgroups — #29708050

Order 15616.

1154 conjugacy classes of subgroups.

The subgroup lattice has 508 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840317C211yesmaximal, characteristic
2C2×#16839799C212yesmaximal
3#21448300C211yesmaximal, characteristic
4#21447160C212yesmaximal, complemented
5#21447155C212yesmaximal, complemented
6#21447152C211yesmaximal, characteristic, complemented
7#21447149C211yesmaximal, characteristic, complemented
8#21447148C211yesmaximal, characteristic, complemented
9#21445534C212yesmaximal
10#336×D61C211yesmaximal, characteristic
11C61×#2057C211yesmaximal, characteristic, complemented
12C22×C8×D61C2212yes—
13#16840317—28no—
14C2×#14519777C2211yescharacteristic
15C2×#14519768C2212yes—
16C2×#14519757—21no—
17C2×#14519753—21no—
18C2×Q8×Dic61—21no—
19C2×#14519742—21no—
20C2×#14519735C2211yescharacteristic
21C2×#14519735C2212yes—
22C2×#14519735C2211yescharacteristic
23C2×C42×D61C2211yescharacteristic
24#16839799—28no—
25C2×#14519711C2212yes—
26C2×#14519708C2212yes—
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×C4×D61D411yescharacteristic
47C22×C4×D61D411yescharacteristic
48C22×C4×D61C2311yescharacteristic
49C22×#92418C2312yes—
50C2×C8×D61—24no—
51C2×C8×D61—24no—
52C2×C8×D61Q818yescomplemented
53#14519777—24no—
54#14519768—24no—
55#14519768—24no—
56#14519757—28no—
57#14519753—28no—
58Q8×Dic61—28no—
59#14519742—28no—
60#14519735—24no—
61#14519735—22no—
62#14519735—24no—
63#14519735D418yes—
64#14519735Q814yescomplemented
65C2×C4⋊Dic61—21no—
66C2×C4⋊Dic61—22no—
67C2×C4⋊Dic61—21no—
68C2×C4⋊Dic61C2311yescharacteristic
69C2×C4⋊Dic61C2312yes—
70C2×C4⋊Dic61C2311yescharacteristic
71C2×#11758196—22no—
72C2×#11758196—21no—
73C2×#11758196—21no—
74C2×#11758198—22no—
75C2×#11758198—21no—
76C2×#11758198—21no—
77C2×C4×Dic61—22no—
78C2×C4×Dic61—21no—
79C2×C4×Dic61—21no—
80C2×C4×Dic61D411yescharacteristic
81C2×C4×Dic61D411yescharacteristic
82C2×C4×Dic61C2311yescharacteristic
83C42×D61—24no—
84#14519711—24no—
85#14519708—24no—
86#14519708—24no—
87#14519684—21no—
88#14519684—21no—
89#14519684—21no—
90C22×Q8×C61—21no—
91C22×C8×C61C2312yes—
92C4⋊Q8×C61—24no—
93C2×C4⋊C4×C61—21no—
94C2×C4⋊C4×C61—21no—
95C2×C4⋊C4×C61C2311yescharacteristic
96C2×C4⋊C4×C61C2312yes—
97C2×C4⋊C4×C61C2311yescharacteristic
98C2×C42×C61C2311yescharacteristic
99#107×C61—24no—
100#95×C61—21no—
101C23×D61C2×D411yescharacteristic
102C2×Q8×C61—24no—
103C2×Q8×C61—42no—
104C2×Q8×C61—42no—
105C22×C4×C61—21no—
106C22×C4×C61—22no—
107C22×C4×C61—21no—
108C22×C4×C61C2×D411yescharacteristic
109C22×C4×C61C2×D411yescharacteristic
110C22×C4×C61C2411yescharacteristic
111C22×Dic61—22no—
112C22×Dic61—21no—
113C22×Dic61—21no—
114C22×Dic61C2×D411yescharacteristic
115C22×Dic61C2×D411yescharacteristic
116C22×Dic61C2×D411yescharacteristic
117C2×C4×D61—22no—
118C2×C4×D61—22no—
119C2×C4×D61—22no—
120C2×C4×D61—22no—
121C2×C4×D61—41no—
122C2×C4×D61—41no—
123C2×C4×D61C2×D414yes—
124C2×C4×D61C2×Q814yes—
125C2×C4×D61C2×Q812yes—
126C2×C4×D61#4112yes—
127C2×Dic122—24no—
128C2×Dic122—42no—
129C2×Dic122—42no—
130C2×C8×C61—22no—
131C2×C8×C61—22no—
132C2×C8×C61C2×Q814yes—
133C4⋊C4×C61—24no—
134C4⋊C4×C61—22no—
135C4⋊C4×C61—21no—
136C4⋊C4×C61—22no—
137C4⋊C4×C61—42no—
138C4⋊C4×C61C2×D414yes—
139C4⋊C4×C61C2×Q812yes—
140C42×C61—22no—
141#11758198—28no—
142#11758198—24no—
143#11758198—42no—
144C4⋊Dic61—24no—
145C4⋊Dic61—28no—
146C4⋊Dic61—22no—
147C4⋊Dic61—21no—
148C4⋊Dic61—22no—
149C4⋊Dic61—42no—
150C4⋊Dic61C2×D414yes—
151C4⋊Dic61C2×Q812yes—
152#11758196—28no—
153#11758196—42no—
154#11758196—42no—
155C4×Dic61—22no—
156C4×Dic61—22no—
157C4×Dic61—22no—
158C4×Dic61—28no—
159C4×Dic61—24no—
160C4×Dic61—42no—
161C2×#92418—22no—
162C2×#92418—22no—
163C2×#92418C2×Q814yes—
164C8×D61—216no—
165C23×C61C22×D411yescharacteristic
166C22×D61—21no—
167C22×D61—21no—
168C22×D61—21no—
169C22×D61—21no—
170C22×D61#12012yes—
171C22×D61C22⋊Q812yes—
172C22×D61C4⋊Q812yes—
173Q8×C61—48no—
174C2×C4×C61—21no—
175C2×C4×C61—21no—
176C2×C4×C61—21no—
177C2×C4×C61—21no—
178C2×C4×C61—24no—
179C2×C4×C61—22no—
180C2×C4×C61—41no—
181C2×C4×C61—42no—
182C2×C4×C61—41no—
183C2×C4×C61—42no—
184C2×C4×C61—41no—
185C2×C4×C61—41no—
186C2×C4×C61—41no—
187C2×C4×C61#13512yes—
188C2×C4×C61#13512yes—
189C2×C4×C61C22×D412yes—
190C2×C4×C61C22×Q812yes—
191C2×C4×C61C22×Q811yescharacteristic
192C2×C4×C61C2×#4111yescharacteristic
193C2×Dic61—21no—
194C2×Dic61—21no—
195C2×Dic61—24no—
196C2×Dic61—22no—
197C2×Dic61—21no—
198C2×Dic61—21no—
199C2×Dic61—21no—
200C2×Dic61—21no—
201C2×Dic61—41no—
202C2×Dic61—42no—
203C2×Dic61—41no—
204C2×Dic61—42no—
205C2×Dic61—41no—
206C2×Dic61—41no—
207C2×Dic61—41no—
208C2×Dic61#12012yes—
209C2×Dic61C22⋊Q812yes—
210C2×Dic61C4⋊Q812yes—
211C4×D61—44no—
212C4×D61—44no—
213C4×D61C22⋊Q818yes—
214C4×D61C4⋊Q818yes—
215Dic122—48no—
216C8×C61—24no—
217#92418—24no—
218#23551—611nomaximal, Sylow
219C22×C61C2×#12011yescharacteristic
220C22×C61C2×C22⋊Q811yescharacteristic
221C22×C61C2×C4⋊Q811yescharacteristic
222C22×C61C2×#13511yescharacteristic
223C22×C61C2×#13511yescharacteristic
224C22×C61#57511yescharacteristic
225C22×C61#57511yescharacteristic
226C2×D61—22no—
227C2×D61—22no—
228C2×D61—22no—
229C2×D61—22no—
230C2×D61—22no—
231C2×D61—22no—
232C2×D61#39514yes—
233C4×C61—22no—
234C4×C61—22no—
235C4×C61—42no—
236C4×C61—44no—
237C4×C61—81no—
238C4×C61C2×C22⋊Q812yes—
239C4×C61C2×C4⋊Q812yes—
240Dic61—42no—
241Dic61—44no—
242Dic61—41no—
243Dic61—41no—
244Dic61—81no—
245Dic61#39514yes—
246C2×#533—611no—
247C2×#427—612no—
248#2296—611no—
249#2059—612no—
250#2059—612no—
251#2058—611no—
252#2058—611no—
253#2057—611no—
254#2057D6111yescharacteristic, complemented
255#1735—612no—
256C2×#336—611no—
257C2×C61C2×#39511yescharacteristic
258C2×C61#300211yescharacteristic
259C2×C61#300211yescharacteristic
260C2×C61#307611yescharacteristic
261C2×C61#307611yescharacteristic
262C2×C61#316011yescharacteristic
263C2×C61#316211yescharacteristic
264D61—24no—
265C23×C8—612no—
266#533—1228no—
267C2×C4⋊Q8—611no—
268C2×C4⋊Q8C2×D6111yescharacteristic
269C2×#125—612no—
270C2×C22⋊Q8—1221no—
271C2×C22⋊Q8—1221no—
272C2×C4×Q8—1221no—
273C2×C4×Q8—1221no—
274C2×#117—612no—
275C2×#117—611no—
276C2×#117—611no—
277C22×C42—611no—
278#427—1228no—
279C2×#107—612no—
280C2×#107C2×D6112yes—
281C2×#106—612no—
282C2×#106—612no—
283#389—611no—
284#389—611no—
285#389—611no—
286#389C2×D6111yescharacteristic
287#385—1221no—
288#385—1221no—
289C4×C4⋊C4—1221no—
290C4×C4⋊C4—1221no—
291#340—612no—
292#340—612no—
293#340—612no—
294#340C2×D6112yes—
295#336—612no—
296#336—611no—
297#336C2×D6111yescharacteristic
298C61#2355111yescharacteristic, complemented, Sylow
299C22×Q8—21no—
300C22×Q8—1221no—
301C23×C4—611no—
302C23×C4—611no—
303C23×C4—611no—
304C22×C8—612no—
305C22×C8—618no—
306C22×C8—1224no—
307C22×C8—1224no—
308C22×C8C22×D6112yes—
309C4⋊Q8—24no—
310C4⋊Q8—1224no—
311#125—1224no—
312#125—1224no—
313C22⋊Q8—1228no—
314C22⋊Q8—1228no—
315C4×Q8—1228no—
316C4×Q8—1228no—
317#117—618no—
318#117—614no—
319#117—1224no—
320#117—1222no—
321#117—1224no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—612no—
325C2×C4⋊C4—611no—
326C2×C4⋊C4—611no—
327C2×C4⋊C4—1222no—
328C2×C4⋊C4—1222no—
329C2×C4⋊C4—1221no—
330C2×C4⋊C4—1221no—
331C2×C4⋊C4—1221no—
332C2×C4⋊C4—1221no—
333C2×C4⋊C4C22×D6111yescharacteristic
334C2×C4⋊C4C22×D6112yes—
335C2×C4⋊C4C22×D6111yescharacteristic
336C2×C22⋊C4—1222no—
337C2×C22⋊C4—1221no—
338C2×C22⋊C4—1221no—
339C2×C42—611no—
340C2×C42—611no—
341C2×C42—611no—
342C2×C42—1224no—
343C2×C42—1222no—
344C2×C42—1221no—
345C2×C42—1221no—
346C2×C42C22×D6111yescharacteristic
347#107—24no—
348#107—1224no—
349#106—1224no—
350#106—1224no—
351#95—21no—
352#95—1221no—
353#95—1221no—
354#95—1221no—
355C24—611no—
356C2×Q8—24no—
357C2×Q8—42no—
358C2×Q8—42no—
359C2×Q8—1224no—
360C2×Q8—2442no—
361C2×Q8—2442no—
362C22×C4—21no—
363C22×C4—22no—
364C22×C4—21no—
365C22×C4—611no—
366C22×C4—614no—
367C22×C4—612no—
368C22×C4—614no—
369C22×C4—612no—
370C22×C4—611no—
371C22×C4—611no—
372C22×C4—1222no—
373C22×C4—1222no—
374C22×C4—1221no—
375C22×C4—1222no—
376C22×C4—1222no—
377C22×C4—1222no—
378C22×C4—1221no—
379C22×C4—2441no—
380C22×C4—2441no—
381C22×C4D4×D6111yescharacteristic
382C22×C4D4×D6111yescharacteristic
383C22×C4C23×D6111yescharacteristic
384C2×C8—22no—
385C2×C8—22no—
386C2×C8—614no—
387C2×C8—12216no—
388C2×C8—1222no—
389C2×C8—1222no—
390C2×C8Q8×D6114yes—
391C4⋊C4—24no—
392C4⋊C4—22no—
393C4⋊C4—21no—
394C4⋊C4—22no—
395C4⋊C4—42no—
396C4⋊C4—614no—
397C4⋊C4—612no—
398C4⋊C4—1228no—
399C4⋊C4—1228no—
400C4⋊C4—1224no—
401C4⋊C4—1222no—
402C4⋊C4—1221no—
403C4⋊C4—1222no—
404C4⋊C4—2442no—
405C4⋊C4—2442no—
406C4⋊C4—2442no—
407C4⋊C4D4×D6114yes—
408C4⋊C4Q8×D6112yes—
409C22⋊C4—1224no—
410C22⋊C4—1228no—
411C22⋊C4—2442no—
412C42—22no—
413C42—1222no—
414C42—1222no—
415C42—1224no—
416C42—1228no—
417C42—1222no—
418C42—2442no—
419C23—612no—
420C23—612no—
421C23—612no—
422C23—1221no—
423C23—1221no—
424C23—1221no—
425C23—1221no—
426C23C2×D4×D6111yescharacteristic
427Q8—48no—
428Q8—2448no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—24no—
434C2×C4—22no—
435C2×C4—41no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—612no—
443C2×C4—612no—
444C2×C4—612no—
445C2×C4—618no—
446C2×C4—618no—
447C2×C4—1221no—
448C2×C4—1221no—
449C2×C4—1224no—
450C2×C4—1222no—
451C2×C4—1221no—
452C2×C4—1221no—
453C2×C4—1221no—
454C2×C4—1221no—
455C2×C4—2444no—
456C2×C4—2444no—
457C2×C4—2441no—
458C2×C4—2441no—
459C2×C4—2441no—
460C2×C4—2441no—
461C2×C4—2442no—
462C2×C4—2441no—
463C2×C4—2442no—
464C2×C4#1451979612yes—
465C2×C4#1451980012yes—
466C2×C4C2×D4×D6112yes—
467C2×C4C2×Q8×D6112yes—
468C2×C4C2×Q8×D6111yescharacteristic
469C2×C4#41×D6111yescharacteristic
470C8—24no—
471C8—1224no—
472C22—614no—
473C22—1222no—
474C22—1222no—
475C22—1222no—
476C22—1222no—
477C22—1222no—
478C22—1222no—
479C22#120×D6111yescharacteristic
480C22C22⋊Q8×D6111yescharacteristic
481C22C4⋊Q8×D6111yescharacteristic
482C22C2×#1451979611yescharacteristic
483C22C2×#1451980011yescharacteristic
484C22#1684055811yescharacteristic
485C22#1684056111yescharacteristic
486C4—22no—
487C4—22no—
488C4—44no—
489C4—42no—
490C4—81no—
491C4—614no—
492C4—4881no—
493C4—2441no—
494C4—2441no—
495C4—2444no—
496C4—2442no—
497C4C22⋊Q8×D6112yes—
498C4C4⋊Q8×D6112yes—
499C2—1224no—
500C2#395×D6111yescharacteristic
501C2#2145503211yescharacteristic
502C2#2145504211yescharacteristic
503C2#2145560011yescharacteristic
504C2#2145561011yescharacteristic
505C2#2145633011yescharacteristic
506C2#2145634011yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2057C2
socleC23×C61C22×D4
prime radicalC23×D61C2×D4
hypercenter#2057D61
hypocenterC61#23551
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. #21447160C2semidirect
5. #21447155C2semidirect
6. #21447152C2semidirect
7. #21447149C2semidirect
8. #21447148C2semidirect
11. C61×#2057C2semidirect
52. C2×C8×D61Q8semidirect
64. #14519735Q8semidirect
254. #2057D61semidirect
298. C61#23551semidirect

Maximal subgroups

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

subgroupindex
C2×#168403172
C2×#168397992
#214483002
#214471602
#214471552
#214471522
#214471492
#214471482
#214455342
#336×D612
C61×#20572
#2355161