Finite Group Databasebeta

← #29689895

Subgroups — #29689895

Order 15616.

1162 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840314C211yesmaximal, characteristic
2C22×#14519706C211yesmaximal, characteristic
3#21448221C211yesmaximal, characteristic
4#21446765C212yesmaximal, complemented
5#21446740C212yesmaximal, complemented
6#21446243C211yesmaximal, characteristic
7#21446242C211yesmaximal, characteristic
8#21446240C211yesmaximal, characteristic
9#21445823C212yesmaximal, complemented
10#21445822C212yesmaximal, complemented
11C61×#1892C211yesmaximal, characteristic
12C23×#92418C2211yescharacteristic
13#16840314—24no—
14#16840314C418yescomplemented
15C2×#14519776C2211yescharacteristic
16C2×#14519775C2211yescharacteristic
17C2×#14519767—21no—
18C2×#14519756C2211yescharacteristic
19C2×#14519754—21no—
20C2×Q8×Dic61C2211yescharacteristic
21C2×#14519742—21no—
22C2×#14519742C2211yescharacteristic
23C2×#14519733—21no—
24C2×#14519733—21no—
25C22×C4×Dic61C2211yescharacteristic
26C2×#14519706—24no—
27C2×#14519706C2212yes—
28C2×#14519706C2212yes—
29C2×#14519702C2212yes—
30C2×#14519701C2212yes—
31C2×#14519700C2212yes—
32#16839631C2212yes—
33#16839626—21no—
34#16839622C2211yescharacteristic
35#16839620—21no—
36C4⋊C4×Dic61C2212yes—
37#16839613—21no—
38#16839525C2212yes—
39#16839524C2212yes—
40#16839524C2212yes—
41C23×C8×C61C2211yescharacteristic
42C2×C22⋊Q8×C61C2211yescharacteristic
43C2×#103×C61C2212yes—
44C61×#380C2211yescharacteristic
45C61×#340C2212yes—
46C22×Dic122—21no—
47C22×Dic122C2311yescharacteristic
48C23×Dic61D411yescharacteristic
49C23×Dic61D411yescharacteristic
50C22×#92418—22no—
51C22×#92418—42no—
52C22×#92418C2312yes—
53#14519776—22no—
54#14519776C2×C414yes—
55#14519775—22no—
56#14519775C2×C414yes—
57#14519767—28no—
58#14519756—22no—
59#14519756C2×C414yes—
60#14519754—28no—
61Q8×Dic61—22no—
62Q8×Dic61C2×C414yes—
63#14519742—28no—
64#14519742—22no—
65#14519742C2×C414yes—
66#14519733—28no—
67#14519733—44no—
68C2×C4⋊Dic61—21no—
69C2×C4⋊Dic61C2312yes—
70C2×C4⋊Dic61C2311yescharacteristic
71C2×#11758196—21no—
72C2×#11758196—21no—
73C2×#11758196—21no—
74C2×#11758196—21no—
75C2×#11758196—21no—
76C2×#11758196—22no—
77C2×#11758196—22no—
78C2×C22⋊Dic61—21no—
79C2×C22⋊Dic61—21no—
80C2×C4×Dic61—21no—
81C2×C4×Dic61—21no—
82C2×C4×Dic61—22no—
83C2×C4×Dic61C2×C414yes—
84C2×C4×Dic61D411yescharacteristic
85C2×C4×Dic61D411yescharacteristic
86C2×C4×Dic61D414yes—
87C2×C4×Dic61C2311yescharacteristic
88C2×C4×Dic61C2311yescharacteristic
89#14519706—24no—
90#14519706—24no—
91#14519702—42no—
92#14519701—42no—
93#14519700—42no—
94#14519684—21no—
95#14519684—22no—
96C22×Q8×C61C2311yescharacteristic
97C23×C4×C61C2311yescharacteristic
98C22×C8×C61—22no—
99C22×C8×C61—42no—
100C22×C8×C61C2312yes—
101C22⋊Q8×C61—22no—
102C22⋊Q8×C61C2×C414yes—
103C2×C4⋊C4×C61—21no—
104C2×C4⋊C4×C61C2312yes—
105C2×C4⋊C4×C61C2311yescharacteristic
106C2×C22⋊C4×C61—21no—
107C2×C22⋊C4×C61—21no—
108#103×C61—42no—
109#95×C61—21no—
110C24×C61C2×D411yescharacteristic
111C2×Q8×C61—21no—
112C2×Q8×C61—22no—
113C2×Q8×C61—22no—
114C2×Q8×C61C22×C412yes—
115C22×C4×C61—22no—
116C22×C4×C61—22no—
117C22×C4×C61—21no—
118C22×C4×C61—21no—
119C22×C4×C61—22no—
120C22×C4×C61C22×C412yes—
121C22×C4×C61C2×D411yescharacteristic
122C22×C4×C61C2×D412yes—
123C22×C4×C61C2411yescharacteristic
124C22×Dic61—21no—
125C22×Dic61—21no—
126C22×Dic61—22no—
127C22×Dic61—42no—
128C22×Dic61C22⋊C414yes—
129C22×Dic61C22⋊C414yes—
130C22×Dic61SD1614yes—
131C22×Dic61Q1614yes—
132C22×Dic61C2×D411yescharacteristic
133C22×Dic61C2×D411yescharacteristic
134C22×Dic61C2×D411yescharacteristic
135C22×Dic61C2×D411yescharacteristic
136C2×Dic122—22no—
137C2×Dic122—22no—
138C2×Dic122—21no—
139C2×Dic122—24no—
140C2×Dic122—44no—
141C2×Dic122C22×C412yes—
142C2×C8×C61—22no—
143C2×C8×C61—22no—
144C2×C8×C61—22no—
145C2×C8×C61—44no—
146C4⋊C4×C61—24no—
147C4⋊C4×C61—21no—
148C4⋊C4×C61—22no—
149C4⋊C4×C61—22no—
150C4⋊C4×C61C22×C412yes—
151C22⋊C4×C61—24no—
152C22⋊C4×C61—42no—
153C22⋊Dic61—24no—
154C22⋊Dic61—42no—
155C4⋊Dic61—21no—
156C4⋊Dic61—24no—
157C4⋊Dic61—22no—
158C4⋊Dic61—22no—
159C4⋊Dic61C22×C412yes—
160#11758196—24no—
161#11758196—24no—
162#11758196—24no—
163#11758196—24no—
164#11758196—24no—
165#11758196—44no—
166#11758196—44no—
167C4×Dic61—24no—
168C4×Dic61—24no—
169C4×Dic61—44no—
170C4×Dic61C22⋊C414yes—
171C4×Dic61C22⋊C414yes—
172C4×Dic61C22×C412yes—
173C4×Dic61C22×C412yes—
174C4×Dic61C2×D412yes—
175C4×Dic61C2×D412yes—
176C2×#92418—22no—
177C2×#92418—22no—
178C2×#92418—22no—
179C2×#92418—44no—
180C23×C61—22no—
181C23×C61—22no—
182C23×C61C2×C22⋊C412yes—
183C23×C61C2×SD1612yes—
184C23×C61C2×Q1612yes—
185C23×C61C22×D411yescharacteristic
186Q8×C61—24no—
187Q8×C61—41no—
188Q8×C61—41no—
189C2×C4×C61—22no—
190C2×C4×C61—24no—
191C2×C4×C61—22no—
192C2×C4×C61—22no—
193C2×C4×C61—22no—
194C2×C4×C61—22no—
195C2×C4×C61—42no—
196C2×C4×C61—42no—
197C2×C4×C61—42no—
198C2×C4×C61—42no—
199C2×C4×C61—42no—
200C2×C4×C61C2×C22⋊C412yes—
201C2×C4×C61C23×C411yescharacteristic
202C2×C4×C61C22×D411yescharacteristic
203C2×C4×C61H(5,𝔽2)12yes—
204C2×C4×C612−1+412yes—
205C2×Dic61—22no—
206C2×Dic61—22no—
207C2×Dic61—22no—
208C2×Dic61—22no—
209C2×Dic61—44no—
210C2×Dic61—42no—
211C2×Dic61—42no—
212C2×Dic61—42no—
213C2×Dic61—42no—
214C2×Dic61—42no—
215C2×Dic61#103116yescomplemented
216C2×Dic61C2×C22⋊C412yes—
217C2×Dic61C2×C22⋊C412yes—
218C2×Dic61C2×C22⋊C412yes—
219C2×Dic61C2×C22⋊C412yes—
220C2×Dic61C2×SD1612yes—
221C2×Dic61C2×SD1612yes—
222C2×Dic61C2×Q1612yes—
223C2×Dic61C2×Q1612yes—
224Dic122—24no—
225Dic122—48no—
226Dic122—41no—
227Dic122—41no—
228C8×C61—42no—
229#92418—42no—
230#21885—611nomaximal, Sylow
231C22×C61—22no—
232C22×C61—22no—
233C22×C61—24no—
234C22×C61—24no—
235C22×C61C2×#10314yes—
236C22×C61C22×C22⋊C411yescharacteristic
237C22×C61#53712yes—
238C22×C61C22×SD1611yescharacteristic
239C22×C61C22×Q1611yescharacteristic
240C22×C61#57512yes—
241C4×C61—22no—
242C4×C61—42no—
243C4×C61—42no—
244C4×C61—82no—
245C4×C61#52012yes—
246C4×C61#53511yescharacteristic
247C4×C61#53611yescharacteristic
248Dic61—22no—
249Dic61—22no—
250Dic61—42no—
251Dic61—42no—
252Dic61—82no—
253Dic61C2×#10314yes—
254Dic61C2×#10314yes—
255C2×#533—611no—
256C22×C4⋊C8—611no—
257#2285—611no—
258#1987—612no—
259#1981—612no—
260#1892—611no—
261#1892—611no—
262#1892—611no—
263#1892D6111yescharacteristic
264#1803—612no—
265#1803—612no—
266C2×C61—24no—
267C2×C61#231111yescharacteristic
268C2×C61C22×#10311yescharacteristic
269C2×C61#289611yescharacteristic
270C2×C61#324612yes—
271C2×C61#324912yes—
272C23×C8—611no—
273C23×C8C2×D6111yescharacteristic
274#533—618no—
275#533—1224no—
276C2×C4⋊Q8—611no—
277C2×C4⋊Q8—611no—
278C2×#125—1221no—
279C2×C22⋊Q8—611no—
280C2×C22⋊Q8—1221no—
281C2×C22⋊Q8C2×D6111yescharacteristic
282C2×C4×Q8—611no—
283C2×C4×Q8—611no—
284C2×C4×Q8—1221no—
285C2×#117—1221no—
286C2×#117—1221no—
287C22×C42—611no—
288C2×C4⋊C8—612no—
289C2×C4⋊C8—612no—
290C2×C4⋊C8—1224no—
291C2×#103—612no—
292C2×#103—612no—
293C2×#103—612no—
294C2×#103C2×D6112yes—
295#384—612no—
296#382—1221no—
297#380—611no—
298#380—1221no—
299#380C2×D6111yescharacteristic
300C4×C4⋊C4—612no—
301C4×C4⋊C4—1221no—
302#340—612no—
303#340—612no—
304#340—612no—
305#340C2×D6112yes—
306C61#2188511yescharacteristic, complemented, Sylow
307C22×Q8—611no—
308C22×Q8—1221no—
309C22×Q8C22×D6111yescharacteristic
310C23×C4—611no—
311C23×C4—611no—
312C23×C4C22×D6111yescharacteristic
313C22×C8—22no—
314C22×C8—42no—
315C22×C8—612no—
316C22×C8—1222no—
317C22×C8—2442no—
318C22×C8C22×D6112yes—
319C4⋊Q8—614no—
320C4⋊Q8—614no—
321C4⋊Q8—1222no—
322C4⋊Q8—1222no—
323#125—1228no—
324C22⋊Q8—22no—
325C22⋊Q8—614no—
326C22⋊Q8—1228no—
327C22⋊Q8—1222no—
328C22⋊Q8C4×D6114yes—
329C4×Q8—614no—
330C4×Q8—614no—
331C4×Q8—1222no—
332C4×Q8—1228no—
333C4×Q8—1222no—
334#117—1228no—
335#117—2444no—
336C2×C4⋊C4—21no—
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—1221no—
344C2×C4⋊C4—1221no—
345C2×C4⋊C4—1222no—
346C2×C4⋊C4—1222no—
347C2×C4⋊C4C22×D6112yes—
348C2×C4⋊C4C22×D6111yescharacteristic
349C2×C22⋊C4—21no—
350C2×C22⋊C4—21no—
351C2×C22⋊C4—1221no—
352C2×C22⋊C4—1221no—
353C2×C42—614no—
354C2×C42—611no—
355C2×C42—611no—
356C2×C42—614no—
357C2×C42—611no—
358C2×C42—611no—
359C2×C42—1221no—
360C2×C42—1221no—
361C2×C42—1222no—
362C4⋊C8—1224no—
363C4⋊C8—1224no—
364#103—42no—
365#103—2442no—
366#103—2442no—
367#103—2442no—
368#95—21no—
369#95—1222no—
370#95—1221no—
371C24D4×D6111yescharacteristic
372C2×Q8—21no—
373C2×Q8—22no—
374C2×Q8—22no—
375C2×Q8—612no—
376C2×Q8—1224no—
377C2×Q8—1221no—
378C2×Q8—1222no—
379C2×Q8—1222no—
380C2×Q8—2444no—
381C2×Q8C2×C4×D6112yes—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—22no—
387C22×C4—611no—
388C22×C4—611no—
389C22×C4—614no—
390C22×C4—614no—
391C22×C4—614no—
392C22×C4—611no—
393C22×C4—611no—
394C22×C4—614no—
395C22×C4—1221no—
396C22×C4—1221no—
397C22×C4—1222no—
398C22×C4—2442no—
399C22×C4C2×C4×D6112yes—
400C22×C4D4×D6112yes—
401C22×C4D4×D6111yescharacteristic
402C22×C4C23×D6111yescharacteristic
403C2×C8—22no—
404C2×C8—22no—
405C2×C8—22no—
406C2×C8—44no—
407C2×C8—1222no—
408C2×C8—1222no—
409C2×C8—1222no—
410C2×C8—2444no—
411C4⋊C4—21no—
412C4⋊C4—24no—
413C4⋊C4—22no—
414C4⋊C4—22no—
415C4⋊C4—612no—
416C4⋊C4—1224no—
417C4⋊C4—1224no—
418C4⋊C4—1224no—
419C4⋊C4—1224no—
420C4⋊C4—1224no—
421C4⋊C4—1221no—
422C4⋊C4—1224no—
423C4⋊C4—1222no—
424C4⋊C4—1222no—
425C4⋊C4—2444no—
426C4⋊C4—2444no—
427C4⋊C4C2×C4×D6112yes—
428C22⋊C4—24no—
429C22⋊C4—42no—
430C22⋊C4—1224no—
431C22⋊C4—2442no—
432C42—612no—
433C42—612no—
434C42—612no—
435C42—612no—
436C42—614no—
437C42—614no—
438C42—1224no—
439C42—1224no—
440C42—2444no—
441C23—22no—
442C23—22no—
443C23C22⋊C4×D6112yes—
444C23SD16×D6112yes—
445C23Q16×D6112yes—
446C23C2×D4×D6111yescharacteristic
447Q8—24no—
448Q8—41no—
449Q8—41no—
450Q8—1224no—
451Q8—2448no—
452Q8—2441no—
453Q8—2441no—
454C2×C4—22no—
455C2×C4—24no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—612no—
466C2×C4—612no—
467C2×C4—6116no—
468C2×C4—612no—
469C2×C4—612no—
470C2×C4—612no—
471C2×C4—612no—
472C2×C4—612no—
473C2×C4—612no—
474C2×C4—1222no—
475C2×C4—1222no—
476C2×C4—1222no—
477C2×C4—1222no—
478C2×C4—2444no—
479C2×C4—2442no—
480C2×C4—2442no—
481C2×C4—2442no—
482C2×C4—2442no—
483C2×C4—2442no—
484C2×C4C22⋊C4×D6112yes—
485C2×C4C22×C4×D6111yescharacteristic
486C2×C4C2×D4×D6111yescharacteristic
487C2×C4#1451982212yes—
488C2×C4#1451982412yes—
489C8—42no—
490C8—2442no—
491C22—24no—
492C22—22no—
493C22—24no—
494C22—22no—
495C22#103×D6114yes—
496C22C2×C22⋊C4×D6111yescharacteristic
497C22#1684033912yes—
498C22C2×SD16×D6111yescharacteristic
499C22C2×Q16×D6111yescharacteristic
500C22#1684055912yes—
501C4—22no—
502C4—42no—
503C4—42no—
504C4—82no—
505C4—614no—
506C4—614no—
507C4—1222no—
508C4—1222no—
509C4—4882no—
510C4—2442no—
511C4—2442no—
512C4#1684024112yes—
513C4#1684032711yescharacteristic
514C4#1684033411yescharacteristic
515C2—24no—
516C2#2144840611yescharacteristic
517C2C2×#103×D6111yescharacteristic
518C2#2145433411yescharacteristic
519C2#2145713012yes—
520C2#2145716012yes—
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1892C2
socleC23×C61C22×D4
prime radicalC24×C61C2×D4
cosocleC22×C4×C61C24
hypercenter#1892D61
hypocenterC61#21885
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1892

Complements

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

NH ≅ G/Ntype
4. #21446765C2semidirect
5. #21446740C2semidirect
9. #21445823C2semidirect
10. #21445822C2semidirect
14. #16840314C4semidirect
215. C2×Dic61#103semidirect
306. C61#21885semidirect

Maximal subgroups

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

subgroupindex
C2×#168403142
C22×#145197062
#214482212
#214467652
#214467402
#214462432
#214462422
#214462402
#214458232
#214458222
C61×#18922
#2188561