Finite Group Databasebeta

← #29646084

Subgroups — #29646084

Order 15616.

1314 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14519763C211yesmaximal, characteristic
2#21454448C212yesmaximal, complemented
3#21444169C218yesmaximal, complemented
4#21444146C211yesmaximal, characteristic, complemented
5#21444145C212yesmaximal, complemented
6C61×#1462C211yesmaximal, characteristic, complemented
7C23×Dic122C411yescharacteristic
8C23×C22⋊D61C411yescharacteristic
9C2×#14519763—22no—
10C2×#14519763—22no—
11C2×#14519763—22no—
12C2×#14519763—22no—
13C2×#14519763C414yes—
14C2×#14519763C2214yes—
15C22×C22⋊Dic61C411yescharacteristic
16C22×#11758198C411yescharacteristic
17C22×#11758198C2211yescharacteristic
18C22×C4×Dic61C2211yescharacteristic
19#16839816—22no—
20#16839816C2214yescomplemented
21#16839812—22no—
22#16839812C2214yescomplemented
23#16839785—22no—
24#16839726C2212yes—
25M16×Dic61—22no—
26#16839486C2214yescomplemented
27#16839485C2218yescomplemented
28C22×C22⋊C4×C61C2211yescharacteristic
29C61×#407C2212yes—
30C61×#323C2214yescomplemented
31C24×D61C2×C411yescharacteristic
32C22×Dic122—21no—
33C22×Dic122—22no—
34C22×Dic122—21no—
35C22×Dic122—44no—
36C22×Dic122C2×C414yes—
37C22×C22⋊D61—21no—
38C22×C22⋊D61—22no—
39C22×C22⋊D61—21no—
40C22×C22⋊D61—44no—
41C22×C22⋊D61C2×C414yes—
42C23×Dic61C2×C411yescharacteristic
43C23×Dic61C2×C411yescharacteristic
44C23×Dic61C2×C411yescharacteristic
45C2×#11758194—22no—
46#14519763—24no—
47#14519763—24no—
48#14519763—24no—
49#14519763—24no—
50#14519763—24no—
51#14519763—24no—
52#14519763—24no—
53#14519763—24no—
54C2×C22⋊Dic61—21no—
55C2×C22⋊Dic61—21no—
56C2×C22⋊Dic61—22no—
57C2×C22⋊Dic61C2×C414yes—
58C2×#11758198—22no—
59C2×#11758198—22no—
60C2×#11758198—21no—
61C2×#11758198—21no—
62C2×#11758198—22no—
63C2×#11758198C2×C414yes—
64C2×#11758198C2×C412yes—
65C2×#11758198C2312yes—
66C2×C4×Dic61C2×C412yes—
67C2×C4×Dic61D412yes—
68C2×C4×Dic61D412yes—
69C2×C4×Dic61D412yes—
70C2×C4×Dic61D412yes—
71C2×C4×Dic61C2312yes—
72#14519706—24no—
73#14519706—24no—
74C22⋊#92418—22no—
75C22⋊#92418C2314yes—
76#14519687—24no—
77C8×Dic61—24no—
78C25×C61C2×C411yescharacteristic
79C23×C4×C61C2×C411yescharacteristic
80C23×C4×C61C2311yescharacteristic
81C2×M16×C61—22no—
82C2×C22⋊C4×C61—22no—
83C2×C22⋊C4×C61—22no—
84C2×C22⋊C4×C61C2×C412yes—
85C2×C22⋊C4×C61C2312yes—
86C22⋊C8×C61—22no—
87C22⋊C8×C61C2314yes—
88C24×C61—21no—
89C24×C61—21no—
90C24×C61—42no—
91C24×C61—42no—
92C24×C61—42no—
93C24×C61C22×C411yescharacteristic
94C24×C61C22×C412yes—
95C23×D61—21no—
96C23×D61—21no—
97C23×D61—42no—
98C23×D61—42no—
99C23×D61—42no—
100C23×D61C22×C412yes—
101C22×C4×C61—21no—
102C22×C4×C61—21no—
103C22×C4×C61—42no—
104C22×C4×C61—42no—
105C22×C4×C61C22×C412yes—
106C22×C4×C61C22×C411yescharacteristic
107C22×C4×C61C2×D412yes—
108C22×C4×C61C2×D412yes—
109C22×C4×C61C2411yescharacteristic
110C2×C22⋊D61—24no—
111C2×C22⋊D61—24no—
112C2×C22⋊D61—44no—
113C2×C22⋊D61—48no—
114C2×C22⋊D61—48no—
115C2×C22⋊D61—44no—
116C22×Dic61—24no—
117C22×Dic61—21no—
118C22×Dic61—21no—
119C22×Dic61—42no—
120C22×Dic61—42no—
121C22×Dic61C22⋊C411yescharacteristic
122C22×Dic61C22⋊C412yes—
123C22×Dic61C22⋊C411yescharacteristic
124C22×Dic61C22⋊C411yescharacteristic
125C22×Dic61C22⋊C412yes—
126C22×Dic61C22⋊C411yescharacteristic
127C22×Dic61C22×C412yes—
128C22×Dic61C22×C412yes—
129C22×Dic61C22×C412yes—
130C2×Dic122—24no—
131C2×Dic122—24no—
132C2×Dic122—44no—
133C2×Dic122—48no—
134C2×Dic122—44no—
135C2×Dic122—48no—
136M16×C61—44no—
137C2×C8×C61—24no—
138C22⋊C4×C61—24no—
139C22⋊C4×C61—24no—
140C22⋊Dic61—24no—
141C22⋊Dic61—24no—
142#11758198—24no—
143#11758198—24no—
144#11758198—24no—
145#11758198—24no—
146C4×Dic61C22⋊C412yes—
147C4×Dic61C22⋊C412yes—
148C4×Dic61C22⋊C412yes—
149C4×Dic61C22⋊C412yes—
150C4×Dic61C2×D412yes—
151C4×Dic61C2×D412yes—
152C4×Dic61C2×D412yes—
153C4×Dic61C2×D412yes—
154#11758194—44no—
155C2×#92418—24no—
156C23×C61—22no—
157C23×C61—22no—
158C23×C61—22no—
159C23×C61—22no—
160C23×C61—24no—
161C23×C61—42no—
162C23×C61—42no—
163C23×C61—42no—
164C23×C61—42no—
165C23×C61—42no—
166C23×C61—42no—
167C23×C61—42no—
168C23×C61—42no—
169C23×C61—88no—
170C23×C61C2×C22⋊C411yescharacteristic
171C23×C61C2×C22⋊C411yescharacteristic
172C23×C61C23×C411yescharacteristic
173C22×D61—22no—
174C22×D61—24no—
175C22×D61—42no—
176C22×D61—42no—
177C22×D61—42no—
178C22×D61—42no—
179C22×D61—42no—
180C22×D61—42no—
181C22×D61—42no—
182C22×D61—42no—
183C22×D61—88no—
184C2×C4×C61—24no—
185C2×C4×C61—22no—
186C2×C4×C61—41no—
187C2×C4×C61—42no—
188C2×C4×C61—41no—
189C2×C4×C61—42no—
190C2×C4×C61—42no—
191C2×C4×C61—42no—
192C2×C4×C61C2×C22⋊C412yes—
193C2×C4×C61C22×D412yes—
194C22⋊D61—48no—
195C22⋊D61—48no—
196C2×Dic61—24no—
197C2×Dic61—22no—
198C2×Dic61—24no—
199C2×Dic61—24no—
200C2×Dic61—22no—
201C2×Dic61—41no—
202C2×Dic61—41no—
203C2×Dic61—42no—
204C2×Dic61—42no—
205C2×Dic61—42no—
206C2×Dic61—42no—
207C2×Dic61#10014yes—
208C2×Dic61C2×C22⋊C412yes—
209C2×Dic61C2×C22⋊C412yes—
210C2×Dic61C2×C22⋊C412yes—
211C2×Dic61C2×C22⋊C412yes—
212Dic122—48no—
213Dic122—48no—
214C8×C61—44no—
215#92418—44no—
216#17393—611nomaximal, Sylow
217C22×C61—28no—
218C22×C61—21no—
219C22×C61—21no—
220C22×C61—21no—
221C22×C61—21no—
222C22×C61—21no—
223C22×C61—21no—
224C22×C61—42no—
225C22×C61—42no—
226C22×C61—42no—
227C22×C61—84no—
228C22×C61—84no—
229C22×C61—84no—
230C22×C61#41012yes—
231C22×C61C2×#10012yes—
232C22×C61C22×C22⋊C411yescharacteristic
233C22×C61#56812yes—
234C2×D61—42no—
235C2×D61—42no—
236C2×D61—42no—
237C2×D61—84no—
238C2×D61—84no—
239C2×D61—84no—
240C4×C61—22no—
241C4×C61—22no—
242C4×C61—42no—
243Dic61—22no—
244Dic61—22no—
245Dic61—22no—
246Dic61—42no—
247Dic61C2×#10014yes—
248C22×#124—611no—
249#2920—612no—
250#1469—618no—
251#1462—612no—
252#1462—611no—
253#1462D6111yescharacteristic, complemented
254C2×C61—22no—
255C2×C61—22no—
256C2×C61—22no—
257C2×C61—82no—
258C2×C61#287911yescharacteristic
259C2×C61C2×#41011yescharacteristic
260C2×C61C22×#10011yescharacteristic
261D61—82no—
262C23×Q8—611no—
263C23×D4—611no—
264C2×#124—614no—
265C2×#124—614no—
266C2×#124—1222no—
267C2×#124—1222no—
268C2×#124—1222no—
269C2×#124—1222no—
270C22×C22⋊C4—611no—
271C22×C22⋊C4—611no—
272C22×C22⋊C4—611no—
273C22×C22⋊C4C2×D6111yescharacteristic
274C22×C42—611no—
275#432—614no—
276#432—1222no—
277#431—614no—
278#431—1222no—
279#423—1222no—
280#407—612no—
281#407C2×D6112yes—
282C4×M16—1222no—
283#323—614no—
284#323—618no—
285#323C2×D6114yescomplemented
286C61#1739311yescharacteristic, complemented, Sylow
287C25—611no—
288C25C4×D6111yescharacteristic
289C22×Q8—614no—
290C22×Q8—1221no—
291C22×Q8—1222no—
292C22×Q8—1221no—
293C22×Q8—2444no—
294C22×D4—614no—
295C22×D4—1221no—
296C22×D4—1222no—
297C22×D4—1221no—
298C22×D4—2444no—
299C23×C4—611no—
300C23×C4—611no—
301C23×C4—611no—
302C23×C4C4×D6111yescharacteristic
303C23×C4C22×D6111yescharacteristic
304C2×M16—22no—
305C2×M16—1222no—
306#124—1224no—
307#124—1224no—
308#124—1224no—
309#124—1224no—
310#124—1224no—
311#124—1224no—
312#124—1224no—
313#124—1224no—
314C2×C22⋊C4—22no—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—614no—
317C2×C22⋊C4—614no—
318C2×C22⋊C4—612no—
319C2×C22⋊C4—612no—
320C2×C22⋊C4—1221no—
321C2×C22⋊C4—1221no—
322C2×C22⋊C4—1222no—
323C2×C22⋊C4—1221no—
324C2×C22⋊C4—1221no—
325C2×C22⋊C4—1222no—
326C2×C22⋊C4—1222no—
327C2×C22⋊C4—1222no—
328C2×C22⋊C4C4×D6112yes—
329C2×C22⋊C4C22×D6112yes—
330C2×C42—612no—
331C2×C42—612no—
332C2×C42—612no—
333C2×C42—612no—
334C2×C42—612no—
335C2×C42—612no—
336C4⋊C8—1224no—
337C4⋊C8—1224no—
338C22⋊C8—22no—
339C22⋊C8—614no—
340C22⋊C8—1222no—
341C22⋊C8C22×D6114yes—
342#97—1224no—
343C4×C8—1224no—
344C24—21no—
345C24—21no—
346C24—42no—
347C24—42no—
348C24—42no—
349C24—612no—
350C24—1221no—
351C24—1221no—
352C24—2442no—
353C24—2442no—
354C24—2442no—
355C24C2×C4×D6111yescharacteristic
356C24C2×C4×D6112yes—
357C2×Q8—1224no—
358C2×Q8—1224no—
359C2×Q8—2444no—
360C2×Q8—2448no—
361C2×Q8—2448no—
362C2×Q8—2444no—
363C2×D4—1224no—
364C2×D4—1224no—
365C2×D4—2444no—
366C2×D4—2448no—
367C2×D4—2444no—
368C2×D4—2448no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—611no—
374C22×C4—612no—
375C22×C4—612no—
376C22×C4—612no—
377C22×C4—611no—
378C22×C4—611no—
379C22×C4—611no—
380C22×C4—612no—
381C22×C4—612no—
382C22×C4—1224no—
383C22×C4—1221no—
384C22×C4—1221no—
385C22×C4—2442no—
386C22×C4—2442no—
387C22×C4C2×C4×D6112yes—
388C22×C4C2×C4×D6111yescharacteristic
389C22×C4D4×D6112yes—
390C22×C4D4×D6112yes—
391C22×C4C23×D6111yescharacteristic
392M16—44no—
393M16—2444no—
394C2×C8—24no—
395C2×C8—1224no—
396C22⋊C4—24no—
397C22⋊C4—24no—
398C22⋊C4—1224no—
399C22⋊C4—1224no—
400C22⋊C4—1224no—
401C22⋊C4—1224no—
402C22⋊C4—1224no—
403C22⋊C4—1224no—
404C42—612no—
405C42—612no—
406C42—612no—
407C42—612no—
408C42—612no—
409C42—612no—
410C42—612no—
411C42—612no—
412C23—22no—
413C23—22no—
414C23—22no—
415C23—22no—
416C23—24no—
417C23—42no—
418C23—42no—
419C23—42no—
420C23—42no—
421C23—42no—
422C23—42no—
423C23—42no—
424C23—42no—
425C23—88no—
426C23—1224no—
427C23—1222no—
428C23—4888no—
429C23—2442no—
430C23—2442no—
431C23—2442no—
432C23—2442no—
433C23—2442no—
434C23—2442no—
435C23—2442no—
436C23—2442no—
437C23C22⋊C4×D6111yescharacteristic
438C23C22⋊C4×D6111yescharacteristic
439C23C22×C4×D6111yescharacteristic
440Q8—2448no—
441Q8—2448no—
442D4—2448no—
443D4—2448no—
444C2×C4—24no—
445C2×C4—22no—
446C2×C4—41no—
447C2×C4—42no—
448C2×C4—41no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—612no—
453C2×C4—614no—
454C2×C4—612no—
455C2×C4—612no—
456C2×C4—612no—
457C2×C4—1224no—
458C2×C4—1224no—
459C2×C4—1222no—
460C2×C4—1224no—
461C2×C4—1222no—
462C2×C4—2441no—
463C2×C4—2441no—
464C2×C4—2442no—
465C2×C4—2442no—
466C2×C4—2442no—
467C2×C4—2442no—
468C2×C4C22⋊C4×D6112yes—
469C2×C4C2×D4×D6112yes—
470C8—44no—
471C8—2444no—
472C22—28no—
473C22—21no—
474C22—21no—
475C22—21no—
476C22—21no—
477C22—21no—
478C22—21no—
479C22—42no—
480C22—42no—
481C22—42no—
482C22—84no—
483C22—84no—
484C22—84no—
485C22—4884no—
486C22—4884no—
487C22—4884no—
488C22—2442no—
489C22—2442no—
490C22—2442no—
491C22#1683973412yes—
492C22#100×D6112yes—
493C22C2×C22⋊C4×D6111yescharacteristic
494C22#1684052812yes—
495C4—22no—
496C4—22no—
497C4—42no—
498C4—614no—
499C4—1222no—
500C4—1222no—
501C4—1222no—
502C4—2442no—
503C2—22no—
504C2—22no—
505C2—22no—
506C2—82no—
507C2—4882no—
508C2#2145426211yescharacteristic
509C2C2×#1683973411yescharacteristic
510C2C2×#100×D6111yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D61
commutatorC23×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1462C2
socleC22×C61C22×C22⋊C4
prime radicalC23×C22⋊D61C4
cosocleC22×C4×C61C24
hypercenter#1462D61
hypocenterC61#17393
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1462

Complements

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

NH ≅ G/Ntype
2. #21454448C2semidirect
3. #21444169C2semidirect
4. #21444146C2semidirect
5. #21444145C2semidirect
6. C61×#1462C2semidirect
20. #16839816C22semidirect
22. #16839812C22semidirect
26. #16839486C22semidirect
27. #16839485C22semidirect
30. C61×#323C22semidirect
253. #1462D61semidirect
285. #323C2×D61semidirect
286. C61#17393semidirect

Maximal subgroups

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

subgroupindex
C22×#145197632
#214544482
#214441692
#214441462
#214441452
C61×#14622
#1739361