Finite Group Databasebeta

← #29689047

Subgroups — #29689047

Order 15616.

1498 conjugacy classes of subgroups.

The subgroup lattice has 506 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14519757C211yesmaximal, characteristic
2C2×C22⋊C8×D61C211yesmaximal, characteristic
3#21448201C211yesmaximal, characteristic
4#21446241C214yesmaximal, complemented
5#21446130C214yesmaximal, complemented
6#21445905C211yesmaximal, characteristic, complemented
7#21445755C211yesmaximal, characteristic, complemented
8#21445753C211yesmaximal, characteristic, complemented
9C61×#1823C211yesmaximal, characteristic, complemented
10C23×Dic122C2211yescharacteristic
11C23×C4×D61C2211yescharacteristic
12C22×C8×D61C2212yes—
13C2×#14519757—21no—
14C2×#14519757—21no—
15C2×#14519757—24no—
16C2×#14519757C418yescomplemented
17C2×#14519757C2212yes—
18C2×#14519757C2212yes—
19C2×#14519735C2212yes—
20C22×C4⋊Dic61—21no—
21C22×#11758198—21no—
22C2×#14519702C2214yescomplemented
23C2×#14519700C2214yescomplemented
24C2×C22⋊#92418C2211yescharacteristic
25C2×#14519689C2212yes—
26C22⋊C8×D61—24no—
27#16839624—21no—
28#16839622C2211yescharacteristic
29#16839619C2212yes—
30#16839617—21no—
31C22⋊C4×Dic61—21no—
32#16839609—21no—
33#16839524C2214yescomplemented
34C22×C4⋊C4×C61C2211yescharacteristic
35C2×C22⋊C8×C61C2211yescharacteristic
36C61×#380C2211yescharacteristic
37C61×#340C2214yescomplemented
38C24×D61D411yescharacteristic
39C22×Dic122—21no—
40C22×Dic122—22no—
41C22×Dic122—44no—
42C22×Dic122C2×C414yes—
43C22×Dic122C2312yes—
44C23×Dic61—21no—
45C23×Dic61D411yescharacteristic
46C22×C4×D61—44no—
47C22×C4×D61C2×C414yes—
48C22×C4×D61D411yescharacteristic
49C22×C4×D61D411yescharacteristic
50C22×C4×D61D414yes—
51C22×C4×D61C2311yescharacteristic
52C22×C4×D61C2311yescharacteristic
53C22×#92418C2312yes—
54C2×C8×D61—24no—
55C2×C8×D61—44no—
56#14519757—28no—
57#14519757—28no—
58#14519757—24no—
59#14519757—24no—
60#14519757C2×C418yes—
61#14519757C2×C418yes—
62#14519735—44no—
63C2×C4⋊Dic61—28no—
64C2×C4⋊Dic61—22no—
65C2×C4⋊Dic61—22no—
66C2×C4⋊Dic61C2312yes—
67C2×C22⋊Dic61—21no—
68C2×#11758198—28no—
69C2×#11758198—22no—
70C2×#11758198—22no—
71C2×#11758198—22no—
72C2×#11758198—22no—
73C2×C4×Dic61—22no—
74#14519702—44no—
75#14519700—44no—
76C22⋊#92418—22no—
77#14519689—24no—
78#14519684—21no—
79#14519684—22no—
80C23×C4×C61—21no—
81C23×C4×C61C2311yescharacteristic
82C22×C8×C61C2312yes—
83C2×C4⋊C4×C61—21no—
84C2×C4⋊C4×C61—22no—
85C2×C4⋊C4×C61C2×C414yes—
86C2×C4⋊C4×C61C2312yes—
87C2×C4⋊C4×C61C2312yes—
88C2×C22⋊C4×C61—21no—
89C22⋊C8×C61—22no—
90#95×C61—21no—
91C24×C61C2×D411yescharacteristic
92C23×D61—22no—
93C23×D61—22no—
94C23×D61—24no—
95C23×D61—42no—
96C23×D61C22⋊C414yes—
97C23×D61C2×D411yescharacteristic
98C23×D61C2×D411yescharacteristic
99C22×C4×C61—24no—
100C22×C4×C61—24no—
101C22×C4×C61—22no—
102C22×C4×C61—22no—
103C22×C4×C61—42no—
104C22×C4×C61—42no—
105C22×C4×C61C22×C412yes—
106C22×C4×C61C2×D411yescharacteristic
107C22×C4×C61C2×D412yes—
108C22×C4×C61C2411yescharacteristic
109C22×Dic61—24no—
110C22×Dic61—22no—
111C22×Dic61—22no—
112C22×Dic61—42no—
113C22×Dic61—44no—
114C22×Dic61C22⋊C414yes—
115C22×Dic61C2×D411yescharacteristic
116C22×Dic61C2×D411yescharacteristic
117C2×C4×D61—24no—
118C2×C4×D61—24no—
119C2×C4×D61—48no—
120C2×C4×D61—48no—
121C2×C4×D61—42no—
122C2×C4×D61—42no—
123C2×C4×D61C22⋊C414yes—
124C2×C4×D61C22⋊C414yes—
125C2×C4×D61C22×C412yes—
126C2×C4×D61C22×C412yes—
127C2×C4×D61C2×D412yes—
128C2×C4×D61C2×D412yes—
129C2×Dic122—24no—
130C2×Dic122—24no—
131C2×Dic122—24no—
132C2×Dic122—22no—
133C2×Dic122—48no—
134C2×Dic122—44no—
135C2×Dic122—48no—
136C2×Dic122C22×C414yes—
137C2×C8×C61—22no—
138C2×C8×C61—24no—
139C4⋊C4×C61—24no—
140C4⋊C4×C61—22no—
141C4⋊C4×C61—24no—
142C4⋊C4×C61C22×C414yes—
143C22⋊C4×C61—42no—
144C22⋊Dic61—42no—
145#11758198—28no—
146#11758198—28no—
147#11758198—44no—
148#11758198—44no—
149C4⋊Dic61—28no—
150C4⋊Dic61—28no—
151C4⋊Dic61—24no—
152C4×Dic61—44no—
153C2×#92418—22no—
154C2×#92418—24no—
155C8×D61—48no—
156C23×C61—22no—
157C23×C61—22no—
158C23×C61C2×C22⋊C412yes—
159C23×C61#13512yes—
160C23×C61#13512yes—
161C23×C61C22×D411yescharacteristic
162C22×D61—22no—
163C22×D61—21no—
164C22×D61—21no—
165C22×D61—22no—
166C22×D61—21no—
167C22×D61—21no—
168C22×D61—22no—
169C22×D61—28no—
170C22×D61—28no—
171C22×D61—44no—
172C22×D61—48no—
173C22×D61—48no—
174C22×D61C2×C22⋊C412yes—
175C22×D61C2×C22⋊C412yes—
176C22×D61#13714yes—
177C2×C4×C61—22no—
178C2×C4×C61—24no—
179C2×C4×C61—24no—
180C2×C4×C61—24no—
181C2×C4×C61—42no—
182C2×C4×C61—42no—
183C2×C4×C61—48no—
184C2×C4×C61—42no—
185C2×C4×C61—42no—
186C2×C4×C61—42no—
187C2×C4×C61C2×C22⋊C412yes—
188C2×C4×C61C23×C411yescharacteristic
189C2×C4×C61C22×D411yescharacteristic
190C2×C4×C61H(5,𝔽2)14yes—
191C2×Dic61—22no—
192C2×Dic61—22no—
193C2×Dic61—24no—
194C2×Dic61—44no—
195C2×Dic61—48no—
196C2×Dic61—44no—
197C2×Dic61—42no—
198C2×Dic61—42no—
199C2×Dic61—42no—
200C2×Dic61C2×C22⋊C412yes—
201C2×Dic61C2×C22⋊C412yes—
202C4×D61—24no—
203C4×D61—24no—
204C4×D61—24no—
205C4×D61—24no—
206C4×D61—44no—
207C4×D61—44no—
208Dic122—24no—
209Dic122—24no—
210Dic122—48no—
211Dic122—44no—
212C8×C61—42no—
213#92418—42no—
214#21762—611nomaximal, Sylow
215C22×C61—22no—
216C22×C61—22no—
217C22×C61—24no—
218C22×C61—24no—
219C22×C61#41614yes—
220C22×C61C22×C22⋊C411yescharacteristic
221C22×C61#53412yes—
222C22×C61C2×#13511yescharacteristic
223C22×C61C2×#13511yescharacteristic
224C22×C61C2×#13712yes—
225C2×D61—22no—
226C2×D61—22no—
227C2×D61—24no—
228C2×D61—24no—
229C2×D61—22no—
230C2×D61—22no—
231C2×D61—22no—
232C2×D61—416no—
233C2×D61—42no—
234C2×D61—42no—
235C2×D61#41914yes—
236C4×C61—22no—
237C4×C61—44no—
238C4×C61—82no—
239C4×C61#51812yes—
240C4×C61#53412yes—
241Dic61—22no—
242Dic61—22no—
243Dic61—44no—
244Dic61—82no—
245C22×C22⋊Q8—611no—
246C22×C22⋊C8—611no—
247#2280—611no—
248#1892—614no—
249#1873—614no—
250#1823—611no—
251#1823D6111yescharacteristic, complemented
252#1787—611no—
253#1787—611no—
254C2×C61—24no—
255C2×C61#230811yescharacteristic
256C2×C61C2×#41611yescharacteristic
257C2×C61C2×#41911yescharacteristic
258C2×C61#319012yes—
259C2×C61#319012yes—
260D61—24no—
261D61—42no—
262C23×Q8—611no—
263C24×C4—611no—
264C23×C8—612no—
265C2×C22⋊Q8—618no—
266C2×C22⋊Q8—612no—
267C2×C22⋊Q8—612no—
268C2×C22⋊Q8—1221no—
269C2×C22⋊Q8—1221no—
270C2×C22⋊Q8—1224no—
271C2×#117—612no—
272C22×C4⋊C4—1221no—
273C22×C4⋊C4C2×D6111yescharacteristic
274C22×C22⋊C4—1221no—
275C2×#103—614no—
276C2×#103—614no—
277C2×C22⋊C8—611no—
278C2×C22⋊C8—612no—
279C2×C22⋊C8—1224no—
280C2×C22⋊C8C2×D6111yescharacteristic
281#381—1221no—
282#380—611no—
283#380—612no—
284#380C2×D6111yescharacteristic
285#379—1221no—
286C4×C22⋊C4—1221no—
287C4×C22⋊C4—1221no—
288#340—614no—
289#340C2×D6114yescomplemented
290C61#2176211yescharacteristic, complemented, Sylow
291C25—611no—
292C22×Q8—614no—
293C22×Q8—612no—
294C22×Q8—1221no—
295C22×Q8—1222no—
296C22×Q8—2444no—
297C23×C4—21no—
298C23×C4—611no—
299C23×C4—611no—
300C23×C4—611no—
301C23×C4—614no—
302C23×C4—611no—
303C23×C4—611no—
304C23×C4—614no—
305C23×C4—1221no—
306C23×C4—2444no—
307C23×C4C22×D6111yescharacteristic
308C22×C8—612no—
309C22×C8—1224no—
310C22×C8—2444no—
311C22×C8C22×D6112yes—
312C22⋊Q8—618no—
313C22⋊Q8—618no—
314C22⋊Q8—1224no—
315C22⋊Q8—1224no—
316C22⋊Q8—1228no—
317C22⋊Q8—1228no—
318#117—2444no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—612no—
322C2×C4⋊C4—1222no—
323C2×C4⋊C4—1222no—
324C2×C4⋊C4—1228no—
325C2×C4⋊C4C4×D6114yes—
326C2×C4⋊C4C22×D6112yes—
327C2×C4⋊C4C22×D6112yes—
328C2×C22⋊C4—21no—
329C2×C22⋊C4—1222no—
330C2×C22⋊C4—1222no—
331C2×C22⋊C4—1228no—
332C2×C22⋊C4—1222no—
333C2×C22⋊C4—1222no—
334C2×C22⋊C4—1221no—
335C2×C42—1222no—
336#103—2444no—
337#103—2444no—
338C22⋊C8—22no—
339C22⋊C8—1224no—
340C22⋊C8—1222no—
341#95—21no—
342#95—1222no—
343#95—1221no—
344C24—611no—
345C24—611no—
346C24—614no—
347C24—1224no—
348C24—1222no—
349C24—1222no—
350C24—2442no—
351C24D4×D6111yescharacteristic
352C2×Q8—614no—
353C2×Q8—1222no—
354C2×Q8—1224no—
355C2×Q8—1224no—
356C2×Q8—1224no—
357C2×Q8—2448no—
358C2×Q8—2444no—
359C2×Q8—2448no—
360C22×C4—24no—
361C22×C4—22no—
362C22×C4—24no—
363C22×C4—22no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—611no—
367C22×C4—611no—
368C22×C4—614no—
369C22×C4—612no—
370C22×C4—612no—
371C22×C4—612no—
372C22×C4—612no—
373C22×C4—614no—
374C22×C4—614no—
375C22×C4—1224no—
376C22×C4—1224no—
377C22×C4—1222no—
378C22×C4—1224no—
379C22×C4—1222no—
380C22×C4—2448no—
381C22×C4—2448no—
382C22×C4—2442no—
383C22×C4—2442no—
384C22×C4—2442no—
385C22×C4—2444no—
386C22×C4C2×C4×D6112yes—
387C22×C4D4×D6112yes—
388C22×C4D4×D6111yescharacteristic
389C22×C4C23×D6111yescharacteristic
390C2×C8—22no—
391C2×C8—24no—
392C2×C8—1222no—
393C2×C8—1224no—
394C2×C8—2448no—
395C4⋊C4—22no—
396C4⋊C4—24no—
397C4⋊C4—24no—
398C4⋊C4—1228no—
399C4⋊C4—1228no—
400C4⋊C4—1224no—
401C4⋊C4C2×C4×D6114yes—
402C22⋊C4—42no—
403C22⋊C4—1228no—
404C22⋊C4—1228no—
405C22⋊C4—2444no—
406C22⋊C4—2444no—
407C22⋊C4—2442no—
408C42—2444no—
409C23—22no—
410C23—22no—
411C23—612no—
412C23—614no—
413C23—612no—
414C23—1228no—
415C23—1221no—
416C23—1221no—
417C23—1222no—
418C23—1228no—
419C23—1221no—
420C23—1221no—
421C23—1222no—
422C23—1222no—
423C23—2444no—
424C23—2448no—
425C23—2448no—
426C23C22⋊C4×D6112yes—
427C23#1451979612yes—
428C23#1451979812yes—
429C23C2×D4×D6111yescharacteristic
430Q8—1224no—
431Q8—1224no—
432Q8—2448no—
433Q8—2444no—
434C2×C4—22no—
435C2×C4—24no—
436C2×C4—24no—
437C2×C4—24no—
438C2×C4—48no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—612no—
445C2×C4—612no—
446C2×C4—1222no—
447C2×C4—1222no—
448C2×C4—1224no—
449C2×C4—1224no—
450C2×C4—1224no—
451C2×C4—1224no—
452C2×C4—1224no—
453C2×C4—2444no—
454C2×C4—2444no—
455C2×C4—2444no—
456C2×C4—2442no—
457C2×C4—2442no—
458C2×C4—2442no—
459C2×C4—2448no—
460C2×C4—2444no—
461C2×C4C22⋊C4×D6112yes—
462C2×C4C22×C4×D6111yescharacteristic
463C2×C4C2×D4×D6111yescharacteristic
464C2×C4#1451982314yes—
465C8—42no—
466C8—2442no—
467C22—22no—
468C22—24no—
469C22—24no—
470C22—22no—
471C22—614no—
472C22—1222no—
473C22—1224no—
474C22—1224no—
475C22—1222no—
476C22—1222no—
477C22—1222no—
478C22—1222no—
479C22—24416no—
480C22—2442no—
481C22—2442no—
482C22#1683975514yes—
483C22C2×C22⋊C4×D6111yescharacteristic
484C22#1684032312yes—
485C22C2×#1451979611yescharacteristic
486C22C2×#1451979811yescharacteristic
487C22#137×D6112yes—
488C4—22no—
489C4—44no—
490C4—82no—
491C4—1222no—
492C4—1222no—
493C4—4882no—
494C4—2444no—
495C4#1684023612yes—
496C4#1684032312yes—
497C2—24no—
498C2—1224no—
499C2—2442no—
500C2#2144839611yescharacteristic
501C2C2×#1683975511yescharacteristic
502C2#419×D6111yescharacteristic
503C2#2145659912yes—
504C2#2145660512yes—
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1823C2
socleC23×C61C22×D4
prime radicalC24×D61D4
cosocleC22×C4×C61C24
hypercenter#1823D61
hypocenterC61#21762
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1823

Complements

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

NH ≅ G/Ntype
4. #21446241C2semidirect
5. #21446130C2semidirect
6. #21445905C2semidirect
7. #21445755C2semidirect
8. #21445753C2semidirect
9. C61×#1823C2semidirect
16. C2×#14519757C4semidirect
22. C2×#14519702C22semidirect
23. C2×#14519700C22semidirect
33. #16839524C22semidirect
37. C61×#340C22semidirect
251. #1823D61semidirect
289. #340C2×D61semidirect
290. C61#21762semidirect

Maximal subgroups

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

subgroupindex
C22×#145197572
C2×C22⋊C8×D612
#214482012
#214462412
#214461302
#214459052
#214457552
#214457532
C61×#18232
#2176261