Finite Group Databasebeta

← #29689754

Subgroups — #29689754

Order 15616.

1118 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840293C211yesmaximal, characteristic
2C2×#16839816C211yesmaximal, characteristic, complemented
3#21448222C211yesmaximal, characteristic, complemented
4#21446786C212yesmaximal, complemented
5#21446634C212yesmaximal, complemented
6#21446142C211yesmaximal, characteristic
7#21446131C211yesmaximal, characteristic, complemented
8#21446129C211yesmaximal, characteristic, complemented
9#21445833C212yesmaximal, complemented
10#21445827C212yesmaximal, complemented
11C61×#1875C211yesmaximal, characteristic
12#16840293—24no—
13#16840293C418yescomplemented
14C2×#14519770—21no—
15C2×#14519768C2211yescharacteristic
16C2×#14519764C2211yescharacteristic
17C2×#14519758—21no—
18C2×#14519756C2211yescharacteristic
19C2×#14519742C2211yescharacteristic
20C2×D4×Dic61—21no—
21C2×D4×Dic61C2211yescharacteristic
22C2×#14519736C2211yescharacteristic
23C2×#14519733—21no—
24C22×C4×Dic61C2211yescharacteristic
25#16839816—24no—
26C2×#14519706C2212yes—
27C2×#14519702C2212yes—
28C2×#14519700C2212yes—
29C2×#14519698C2212yes—
30C2×C22⋊#92418C2211yescharacteristic
31C2×#14519687C2212yes—
32#16839631C2212yes—
33#16839627C2212yes—
34#16839620—21no—
35#16839620—21no—
36C4×#11758196—21no—
37C4×#11758196—21no—
38#16839525C2212yes—
39#16839524C2212yes—
40#16839524C2212yes—
41C2×#121×C61C2211yescharacteristic
42C2×#117×C61C2211yescharacteristic
43C2×#102×C61C2212yes—
44C2×C22⋊C8×C61C2211yescharacteristic
45C61×#340C2212yes—
46C22×Dic122C2311yescharacteristic
47C23×Dic61—21no—
48C23×Dic61D411yescharacteristic
49C23×Dic61D411yescharacteristic
50C22×#92418C2312yes—
51#14519770—28no—
52#14519768—22no—
53#14519768C2×C414yes—
54#14519764—22no—
55#14519764C2×C414yes—
56#14519758—28no—
57#14519756—22no—
58#14519756C2×C414yes—
59#14519742—22no—
60#14519742C2×C414yes—
61D4×Dic61—28no—
62D4×Dic61—22no—
63D4×Dic61C2×C414yes—
64#14519736—42no—
65#14519733—28no—
66C2×C4⋊Dic61—21no—
67C2×C4⋊Dic61C2312yes—
68C2×C4⋊Dic61C2311yescharacteristic
69C2×#11758196—21no—
70C2×#11758196—21no—
71C2×#11758196—21no—
72C2×#11758196—22no—
73C2×#11758196—22no—
74C2×C22⋊Dic61—21no—
75C2×C22⋊Dic61—21no—
76C2×C22⋊Dic61—21no—
77C2×C22⋊Dic61—21no—
78C2×C22⋊Dic61—21no—
79C2×C4×Dic61—21no—
80C2×C4×Dic61—21no—
81C2×C4×Dic61C2×C414yes—
82C2×C4×Dic61D411yescharacteristic
83C2×C4×Dic61D411yescharacteristic
84C2×C4×Dic61D414yes—
85C2×C4×Dic61C2311yescharacteristic
86C2×C4×Dic61C2311yescharacteristic
87#14519706—24no—
88#14519702—42no—
89#14519700—42no—
90#14519698—42no—
91C22⋊#92418—22no—
92#14519687—24no—
93#14519684—22no—
94#14519684—22no—
95C22×D4×C61—21no—
96C22×D4×C61C2311yescharacteristic
97C23×C4×C61C2311yescharacteristic
98C22×C8×C61C2312yes—
99#121×C61—22no—
100#121×C61C2×C414yes—
101#117×C61—42no—
102C2×C4⋊C4×C61C2312yes—
103C2×C4⋊C4×C61C2311yescharacteristic
104C2×C22⋊C4×C61—21no—
105C2×C22⋊C4×C61—21no—
106C2×C42×C61—21no—
107#102×C61—42no—
108C22⋊C8×C61—22no—
109C24×C61—21no—
110C24×C61C2×D411yescharacteristic
111C2×D4×C61—24no—
112C2×D4×C61—21no—
113C2×D4×C61—22no—
114C2×D4×C61—22no—
115C2×D4×C61—44no—
116C2×D4×C61C22×C412yes—
117C22×C4×C61—22no—
118C22×C4×C61—22no—
119C22×C4×C61—21no—
120C22×C4×C61—22no—
121C22×C4×C61C22×C412yes—
122C22×C4×C61C2×D411yescharacteristic
123C22×C4×C61C2×D412yes—
124C22×C4×C61C2411yescharacteristic
125C22×Dic61—21no—
126C22×Dic61—21no—
127C22×Dic61—24no—
128C22×Dic61—22no—
129C22×Dic61—42no—
130C22×Dic61—42no—
131C22×Dic61—44no—
132C22×Dic61C22⋊C414yes—
133C22×Dic61C22⋊C414yes—
134C22×Dic61C2×D411yescharacteristic
135C22×Dic61C2×D411yescharacteristic
136C22×Dic61C2×D411yescharacteristic
137C22×Dic61C2×D411yescharacteristic
138C2×Dic122—22no—
139C2×Dic122—22no—
140C2×Dic122—21no—
141C2×Dic122C22×C412yes—
142C2×C8×C61—22no—
143C2×C8×C61—22no—
144C2×C8×C61—22no—
145C4⋊C4×C61—21no—
146C4⋊C4×C61—22no—
147C4⋊C4×C61—22no—
148C4⋊C4×C61C22×C412yes—
149C22⋊C4×C61—24no—
150C22⋊C4×C61—42no—
151C42×C61—42no—
152C22⋊Dic61—24no—
153C22⋊Dic61—24no—
154C22⋊Dic61—24no—
155C22⋊Dic61—24no—
156C22⋊Dic61—42no—
157C4⋊Dic61—21no—
158C4⋊Dic61—24no—
159C4⋊Dic61—22no—
160C4⋊Dic61—22no—
161C4⋊Dic61C22×C412yes—
162#11758196—24no—
163#11758196—24no—
164#11758196—24no—
165#11758196—44no—
166#11758196—44no—
167C4×Dic61—24no—
168C4×Dic61—42no—
169C4×Dic61C22⋊C414yes—
170C4×Dic61C22⋊C414yes—
171C4×Dic61C22×C412yes—
172C4×Dic61C22×C412yes—
173C4×Dic61C2×D412yes—
174C4×Dic61C2×D412yes—
175C2×#92418—22no—
176C2×#92418—22no—
177C2×#92418—22no—
178C23×C61—21no—
179C23×C61—21no—
180C23×C61—22no—
181C23×C61—22no—
182C23×C61—22no—
183C23×C61—41no—
184C23×C61—41no—
185C23×C61—42no—
186C23×C61—42no—
187C23×C61C2×C22⋊C412yes—
188C23×C61C22×D411yescharacteristic
189D4×C61—24no—
190D4×C61—48no—
191D4×C61—42no—
192C2×C4×C61—21no—
193C2×C4×C61—21no—
194C2×C4×C61—24no—
195C2×C4×C61—24no—
196C2×C4×C61—22no—
197C2×C4×C61—42no—
198C2×C4×C61—42no—
199C2×C4×C61—42no—
200C2×C4×C61—42no—
201C2×C4×C61C2×C22⋊C412yes—
202C2×C4×C61C23×C411yescharacteristic
203C2×C4×C61C22×D411yescharacteristic
204C2×C4×C61H(5,𝔽2)12yes—
205C2×C4×C612−1+412yes—
206C2×Dic61—22no—
207C2×Dic61—22no—
208C2×Dic61—22no—
209C2×Dic61—22no—
210C2×Dic61—22no—
211C2×Dic61—22no—
212C2×Dic61—44no—
213C2×Dic61—44no—
214C2×Dic61—48no—
215C2×Dic61—41no—
216C2×Dic61—41no—
217C2×Dic61—42no—
218C2×Dic61—42no—
219C2×Dic61—42no—
220C2×Dic61—42no—
221C2×Dic61C2×C22⋊C412yes—
222C2×Dic61C2×C22⋊C412yes—
223C2×Dic61C2×C22⋊C412yes—
224C2×Dic61C2×C22⋊C412yes—
225Dic122—24no—
226Dic122—42no—
227C8×C61—42no—
228#92418—42no—
229#21868—611nomaximal, Sylow
230C22×C61—22no—
231C22×C61—22no—
232C22×C61—22no—
233C22×C61—42no—
234C22×C61—42no—
235C22×C61—42no—
236C22×C61—42no—
237C22×C61—42no—
238C22×C61—82no—
239C22×C61—82no—
240C22×C61C22×C22⋊C411yescharacteristic
241C22×C61#53511yescharacteristic
242C22×C61#53611yescharacteristic
243C22×C61#57512yes—
244C22×C61#57512yes—
245C4×C61—22no—
246C4×C61—42no—
247C4×C61—82no—
248C4×C61#52012yes—
249C4×C61#53712yes—
250Dic61—22no—
251Dic61—22no—
252Dic61—22no—
253Dic61—22no—
254Dic61—42no—
255Dic61—42no—
256Dic61—82no—
257C2×#529—611no—
258C2×#432—611no—
259#2285—611no—
260#1992—612no—
261#1959—612no—
262#1875—611no—
263#1875D6111yescharacteristic
264#1873—611no—
265#1873—611no—
266#1805—612no—
267#1804—612no—
268C2×C61—42no—
269C2×C61—82no—
270C2×C61#231111yescharacteristic
271C2×C61#289611yescharacteristic
272C2×C61#289611yescharacteristic
273C2×C61#321912yes—
274C2×C61#324012yes—
275#529—618no—
276#529—1224no—
277C2×#126—1221no—
278C2×#125—611no—
279C2×#124—611no—
280C2×#123—1221no—
281C2×C22⋊Q8—611no—
282C2×#121C2×D6111yescharacteristic
283C2×C4×Q8—611no—
284C2×C4×D4—611no—
285C2×C4×D4—1221no—
286C2×#117—611no—
287C2×#117—1221no—
288C2×#117C2×D6111yescharacteristic
289C22×C42—611no—
290#432—1224no—
291C2×C4⋊C8—612no—
292C2×#103—612no—
293C2×#103—612no—
294C2×#102—612no—
295C2×#102C2×D6112yes—
296C2×C22⋊C8—611no—
297C2×C22⋊C8C2×D6111yescharacteristic
298C2×#97—612no—
299#384—612no—
300#382—612no—
301#380—1221no—
302#380—1221no—
303C4×C4⋊C4—1221no—
304C4×C4⋊C4—1221no—
305#340—612no—
306#340—612no—
307#340—612no—
308#340C2×D6112yes—
309C61#2186811yescharacteristic, complemented, Sylow
310C22×Q8—611no—
311C22×D4—21no—
312C22×D4C22×D6111yescharacteristic
313C23×C4—611no—
314C23×C4—611no—
315C23×C4—1221no—
316C23×C4C22×D6111yescharacteristic
317C22×C8—612no—
318C22×C8C22×D6112yes—
319#126—1228no—
320#125—614no—
321#125—1222no—
322#124—614no—
323#124—1222no—
324#123—1228no—
325C22⋊Q8—614no—
326C22⋊Q8—1222no—
327#121—22no—
328#121C4×D6114yes—
329C4×Q8—614no—
330C4×Q8—1222no—
331C4×D4—614no—
332C4×D4—1222no—
333C4×D4—1228no—
334#117—42no—
335#117—1228no—
336#117—2442no—
337C2×C4⋊C4—611no—
338C2×C4⋊C4—612no—
339C2×C4⋊C4—1221no—
340C2×C4⋊C4—1221no—
341C2×C4⋊C4—1221no—
342C2×C4⋊C4—1221no—
343C2×C4⋊C4—1222no—
344C2×C4⋊C4—1222no—
345C2×C4⋊C4C22×D6112yes—
346C2×C4⋊C4C22×D6111yescharacteristic
347C2×C22⋊C4—21no—
348C2×C22⋊C4—21no—
349C2×C22⋊C4—1221no—
350C2×C22⋊C4—1221no—
351C2×C22⋊C4—1221no—
352C2×C22⋊C4—1221no—
353C2×C22⋊C4—1221no—
354C2×C42—21no—
355C2×C42—614no—
356C2×C42—611no—
357C2×C42—611no—
358C2×C42—614no—
359C2×C42—611no—
360C2×C42—611no—
361C2×C42—1221no—
362C2×C42—1221no—
363C4⋊C8—1224no—
364#103—2442no—
365#103—2442no—
366#102—42no—
367#102—2442no—
368C22⋊C8—22no—
369C22⋊C8—1222no—
370#97—1224no—
371#95—1222no—
372#95—1222no—
373C24—21no—
374C24D4×D6111yescharacteristic
375C2×Q8—612no—
376C2×Q8—1221no—
377C2×Q8—1222no—
378C2×Q8—1222no—
379C2×D4—24no—
380C2×D4—21no—
381C2×D4—22no—
382C2×D4—22no—
383C2×D4—44no—
384C2×D4C2×C4×D6112yes—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—21no—
388C22×C4—22no—
389C22×C4—611no—
390C22×C4—611no—
391C22×C4—611no—
392C22×C4—611no—
393C22×C4—614no—
394C22×C4—614no—
395C22×C4—1224no—
396C22×C4—1221no—
397C22×C4—1221no—
398C22×C4—1222no—
399C22×C4—2442no—
400C22×C4—2442no—
401C22×C4—2444no—
402C22×C4C2×C4×D6112yes—
403C22×C4D4×D6112yes—
404C22×C4D4×D6111yescharacteristic
405C22×C4C23×D6111yescharacteristic
406C2×C8—22no—
407C2×C8—22no—
408C2×C8—22no—
409C2×C8—1222no—
410C2×C8—1222no—
411C2×C8—1222no—
412C4⋊C4—21no—
413C4⋊C4—22no—
414C4⋊C4—22no—
415C4⋊C4—612no—
416C4⋊C4—1224no—
417C4⋊C4—1224no—
418C4⋊C4—1224no—
419C4⋊C4—1221no—
420C4⋊C4—1224no—
421C4⋊C4—1222no—
422C4⋊C4—1222no—
423C4⋊C4—2444no—
424C4⋊C4—2444no—
425C4⋊C4C2×C4×D6112yes—
426C22⋊C4—24no—
427C22⋊C4—42no—
428C22⋊C4—1224no—
429C22⋊C4—1224no—
430C22⋊C4—1224no—
431C22⋊C4—1224no—
432C22⋊C4—2442no—
433C42—42no—
434C42—612no—
435C42—612no—
436C42—612no—
437C42—612no—
438C42—614no—
439C42—614no—
440C42—1224no—
441C42—2442no—
442C23—22no—
443C23—22no—
444C23—21no—
445C23—21no—
446C23—22no—
447C23—42no—
448C23—42no—
449C23—41no—
450C23—41no—
451C23C22⋊C4×D6112yes—
452C23C2×D4×D6111yescharacteristic
453Q8—1224no—
454Q8—2442no—
455D4—24no—
456D4—48no—
457D4—42no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—24no—
461C2×C4—24no—
462C2×C4—22no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—612no—
468C2×C4—612no—
469C2×C4—612no—
470C2×C4—612no—
471C2×C4—1222no—
472C2×C4—1222no—
473C2×C4—1222no—
474C2×C4—1222no—
475C2×C4—1222no—
476C2×C4—1222no—
477C2×C4—2444no—
478C2×C4—2444no—
479C2×C4—2448no—
480C2×C4—2442no—
481C2×C4—2442no—
482C2×C4—2442no—
483C2×C4—2442no—
484C2×C4—2441no—
485C2×C4—2441no—
486C2×C4C22⋊C4×D6112yes—
487C2×C4C22×C4×D6111yescharacteristic
488C2×C4C2×D4×D6111yescharacteristic
489C2×C4#1451982212yes—
490C2×C4#1451982412yes—
491C8—42no—
492C8—2442no—
493C22—22no—
494C22—22no—
495C22—22no—
496C22—42no—
497C22—42no—
498C22—42no—
499C22—42no—
500C22—42no—
501C22—82no—
502C22—82no—
503C22C2×C22⋊C4×D6111yescharacteristic
504C22#1684032711yescharacteristic
505C22#1684033411yescharacteristic
506C22#1684055812yes—
507C22#1684055912yes—
508C4—22no—
509C4—42no—
510C4—82no—
511C4—1222no—
512C4—1222no—
513C4—1222no—
514C4—1222no—
515C4—4882no—
516C4—2442no—
517C4—2442no—
518C4#1684024112yes—
519C4#1684033912yes—
520C2—42no—
521C2—82no—
522C2#2144840611yescharacteristic
523C2#2145433311yescharacteristic
524C2#2145433411yescharacteristic
525C2#2145685412yes—
526C2#2145705712yes—
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1875C2
socleC23×C61C22×D4
prime radicalC2×#121×C61C22
cosocleC22×C4×C61C24
hypercenter#1875D61
hypocenterC61#21868
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1875

Complements

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

NH ≅ G/Ntype
2. C2×#16839816C2semidirect
3. #21448222C2semidirect
4. #21446786C2semidirect
5. #21446634C2semidirect
7. #21446131C2semidirect
8. #21446129C2semidirect
9. #21445833C2semidirect
10. #21445827C2semidirect
13. #16840293C4semidirect
309. C61#21868semidirect

Maximal subgroups

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

subgroupindex
C2×#168402932
C2×#168398162
#214482222
#214467862
#214466342
#214461422
#214461312
#214461292
#214458332
#214458272
C61×#18752
#2186861