Finite Group Databasebeta

← #29689975

Subgroups — #29689975

Order 15616.

1118 conjugacy classes of subgroups.

The subgroup lattice has 538 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840291C211yesmaximal, characteristic
2C2×#16839784C211yesmaximal, characteristic, complemented
3#21448215C211yesmaximal, characteristic, complemented
4#21446776C211yesmaximal, characteristic, complemented
5#21446774C211yesmaximal, characteristic, complemented
6#21446752C211yesmaximal, characteristic, complemented
7#21446254C212yesmaximal
8#21446249C212yesmaximal, complemented
9#21445832C212yesmaximal, complemented
10#21445827C212yesmaximal, complemented
11C61×#1983C211yesmaximal, characteristic, complemented
12C22×#11758189C2212yes—
13#16840291—24no—
14#16840291C418yescomplemented
15C2×#14519770—21no—
16C2×#14519769C2211yescharacteristic
17C2×#14519759—21no—
18C2×#14519753C2211yescharacteristic
19C2×#14519748C2211yescharacteristic
20C2×Q8×Dic61C2211yescharacteristic
21C2×D4×Dic61C2211yescharacteristic
22C2×#14519737—21no—
23C2×#14519733—21no—
24C2×C42×D61C2211yescharacteristic
25#16839784—24no—
26C2×C4⋊#92418C2211yescharacteristic
27C2×#14519706C2212yes—
28C2×#14519701C2212yes—
29C2×#14519698C2212yes—
30#16839629—21no—
31#16839627C2212yes—
32#16839624—21no—
33#16839619C2212yes—
34C4×C4⋊Dic61C2211yescharacteristic
35C4×#11758198—21no—
36#16839608—21no—
37#16839525C2212yes—
38#16839525C2212yes—
39#16839524C2212yes—
40#16839524C2212yes—
41C2×#124×C61C2211yescharacteristic
42C2×C4⋊C8×C61C2211yescharacteristic
43C2×#103×C61C2212yes—
44C2×#102×C61C2212yes—
45C4×C4⋊C4×C61C2211yescharacteristic
46C22×C22⋊D61—21no—
47C23×Dic61—21no—
48C22×C4×D61D411yescharacteristic
49C22×C4×D61D411yescharacteristic
50C22×C4×D61C2311yescharacteristic
51C2×#11758189—24no—
52C2×#11758189—42no—
53C2×#11758189—42no—
54C22×#92418C2312yes—
55#14519770—28no—
56#14519769—22no—
57#14519769C2×C414yes—
58#14519759—28no—
59#14519753—22no—
60#14519753C2×C414yes—
61#14519748—22no—
62#14519748C2×C414yes—
63Q8×Dic61—22no—
64Q8×Dic61C2×C414yes—
65D4×Dic61—22no—
66D4×Dic61C2×C414yes—
67#14519737—28no—
68#14519733—28no—
69C2×C4⋊Dic61—21no—
70C2×C4⋊Dic61C2312yes—
71C2×C4⋊Dic61C2311yescharacteristic
72C2×C4⋊Dic61C2311yescharacteristic
73C2×#11758196—21no—
74C2×#11758196—21no—
75C2×#11758196—21no—
76C2×C22⋊Dic61—21no—
77C2×C22⋊Dic61—21no—
78C2×#11758198—21no—
79C2×#11758198—22no—
80C2×C4×Dic61—21no—
81C2×C4×Dic61—21no—
82C2×C4×Dic61D411yescharacteristic
83C2×C4×Dic61D411yescharacteristic
84C2×C4×Dic61C2311yescharacteristic
85C42×D61C2×C414yes—
86C42×D61D414yes—
87C4⋊#92418—22no—
88#14519706—24no—
89#14519701—42no—
90#14519698—42no—
91#14519684—21no—
92#14519684—22no—
93#14519684—22no—
94#14519684—22no—
95C22×Q8×C61C2311yescharacteristic
96C22×D4×C61C2311yescharacteristic
97C22×C8×C61C2312yes—
98#124×C61—22no—
99#124×C61C2×C414yes—
100C2×C4⋊C4×C61C2312yes—
101C2×C22⋊C4×C61—21no—
102C2×C22⋊C4×C61—21no—
103C2×C42×C61—21no—
104C2×C42×C61C2311yescharacteristic
105C4⋊C8×C61—22no—
106#103×C61—42no—
107#102×C61—42no—
108#95×C61—21no—
109C24×C61—21no—
110C23×D61C2×D411yescharacteristic
111C2×Q8×C61—21no—
112C2×Q8×C61—22no—
113C2×Q8×C61—22no—
114C2×Q8×C61C22×C412yes—
115C2×D4×C61—21no—
116C2×D4×C61—22no—
117C2×D4×C61—22no—
118C2×D4×C61C22×C412yes—
119C22×C4×C61—21no—
120C22×C4×C61—22no—
121C22×C4×C61C2×D411yescharacteristic
122C22×C4×C61C2×D411yescharacteristic
123C22×C4×C61C2411yescharacteristic
124C2×C22⋊D61—24no—
125C2×C22⋊D61—42no—
126C2×C22⋊D61—42no—
127C22×Dic61—21no—
128C22×Dic61—21no—
129C22×Dic61—24no—
130C22×Dic61—22no—
131C22×Dic61—44no—
132C22×Dic61C2×D411yescharacteristic
133C22×Dic61C2×D411yescharacteristic
134C22×Dic61C2×D411yescharacteristic
135C2×C4×D61—22no—
136C2×C4×D61—22no—
137C2×C4×D61—42no—
138C2×C4×D61—41no—
139C2×C4×D61—41no—
140C2×C4×D61C22⋊C414yes—
141C2×C4×D61C22⋊C414yes—
142C2×C4×D61C22×C412yes—
143C2×C4×D61C2×D412yes—
144C2×C8×C61—22no—
145C2×C8×C61—22no—
146C2×C8×C61—22no—
147C4⋊C4×C61—22no—
148C4⋊C4×C61—22no—
149C22⋊C4×C61—24no—
150C22⋊C4×C61—24no—
151C42×C61—42no—
152C42×C61C22×C412yes—
153C42×C61C2×D412yes—
154C22⋊Dic61—24no—
155C22⋊Dic61—24no—
156#11758198—24no—
157#11758198—44no—
158C4⋊Dic61—21no—
159C4⋊Dic61—21no—
160C4⋊Dic61—24no—
161C4⋊Dic61—22no—
162C4⋊Dic61—22no—
163C4⋊Dic61C22×C412yes—
164C4⋊Dic61C22×C412yes—
165#11758196—24no—
166#11758196—24no—
167#11758196—24no—
168C4×Dic61—24no—
169C4×Dic61—42no—
170C4×Dic61C22⋊C414yes—
171C4×Dic61C22⋊C414yes—
172C4×Dic61C22×C412yes—
173C4×Dic61C2×D412yes—
174C2×#92418—22no—
175C2×#92418—22no—
176C2×#92418—22no—
177#11758189—48no—
178C23×C61—22no—
179C23×C61—42no—
180C23×C61—42no—
181C23×C61—41no—
182C23×C61—41no—
183C23×C61C22×D411yescharacteristic
184C22×D61—22no—
185C22×D61—22no—
186C22×D61—21no—
187C22×D61—21no—
188C22×D61C2×C22⋊C412yes—
189Q8×C61—24no—
190Q8×C61—41no—
191Q8×C61—41no—
192D4×C61—24no—
193D4×C61—41no—
194D4×C61—41no—
195C2×C4×C61—21no—
196C2×C4×C61—21no—
197C2×C4×C61—22no—
198C2×C4×C61—22no—
199C2×C4×C61—42no—
200C2×C4×C61—42no—
201C2×C4×C61—42no—
202C2×C4×C61—42no—
203C2×C4×C61C2×C22⋊C412yes—
204C2×C4×C61C2×C22⋊C412yes—
205C2×C4×C61C23×C411yescharacteristic
206C2×C4×C61C22×D411yescharacteristic
207C2×C4×C61H(5,𝔽2)12yes—
208C2×C4×C612−1+412yes—
209C22⋊D61—48no—
210C2×Dic61—22no—
211C2×Dic61—22no—
212C2×Dic61—21no—
213C2×Dic61—21no—
214C2×Dic61—22no—
215C2×Dic61—22no—
216C2×Dic61—48no—
217C2×Dic61—42no—
218C2×Dic61—41no—
219C2×Dic61—41no—
220C2×Dic61—42no—
221C2×Dic61—42no—
222C2×Dic61—42no—
223C2×Dic61C2×C22⋊C412yes—
224C2×Dic61C2×C22⋊C412yes—
225C2×Dic61C2×C22⋊C412yes—
226C4×D61—24no—
227C4×D61—22no—
228C4×D61—22no—
229C4×D61—44no—
230C4×D61—44no—
231C8×C61—42no—
232#92418—42no—
233#21895—611nomaximal, Sylow
234C22×C61—42no—
235C22×C61—42no—
236C22×C61—42no—
237C22×C61—82no—
238C22×C61—82no—
239C22×C61C22×C22⋊C411yescharacteristic
240C22×C61#53712yes—
241C22×C61#57511yescharacteristic
242C22×C61#57512yes—
243C22×C61#57511yescharacteristic
244C2×D61—22no—
245C2×D61—22no—
246C2×D61—22no—
247C2×D61—42no—
248C2×D61—42no—
249C4×C61—22no—
250C4×C61—22no—
251C4×C61—42no—
252C4×C61—82no—
253C4×C61#52012yes—
254C4×C61#53511yescharacteristic
255C4×C61#53611yescharacteristic
256Dic61—22no—
257Dic61—22no—
258Dic61—22no—
259Dic61—42no—
260Dic61—42no—
261Dic61—82no—
262C2×#529—611no—
263C2×#423—611no—
264#2283—611no—
265#1989—611no—
266#1989—611no—
267#1983—611no—
268#1983D6111yescharacteristic, complemented
269#1895—612no—
270#1894—612no—
271#1805—612no—
272#1804—612no—
273C2×C61—82no—
274C2×C61#231111yescharacteristic
275C2×C61#289611yescharacteristic
276C2×C61#289611yescharacteristic
277C2×C61#325612yes—
278C2×C61#325612yes—
279D61—42no—
280C22×M16—612no—
281#529—618no—
282#529—1224no—
283C2×#126—1221no—
284C2×#125—611no—
285C2×#124C2×D6111yescharacteristic
286C2×#123—1221no—
287C2×C22⋊Q8—611no—
288C2×#121—611no—
289C2×C4×Q8—611no—
290C2×C4×D4—611no—
291C2×C4×D4—1221no—
292C2×#117—1221no—
293C22×C42—611no—
294#423—1224no—
295C2×C4⋊C8—612no—
296C2×C4⋊C8—611no—
297C2×C4⋊C8C2×D6111yescharacteristic
298C2×#103—612no—
299C2×#103C2×D6112yes—
300C2×#102—612no—
301C2×#102C2×D6112yes—
302#383—1221no—
303#382—612no—
304#381—1221no—
305#380—612no—
306C4×C4⋊C4—611no—
307C4×C4⋊C4C2×D6111yescharacteristic
308C4×C22⋊C4—1221no—
309#376—1221no—
310#340—612no—
311#340—612no—
312#340—612no—
313#340—612no—
314C61#2189511yescharacteristic, complemented, Sylow
315C22×Q8C22×D6111yescharacteristic
316C22×D4—1221no—
317C22×D4C22×D6111yescharacteristic
318C23×C4—611no—
319C23×C4—611no—
320C23×C4—611no—
321C23×C4—1221no—
322C2×M16—1224no—
323C2×M16—2442no—
324C2×M16—2442no—
325C22×C8—612no—
326C22×C8C22×D6112yes—
327#126—1228no—
328#125—614no—
329#125—1222no—
330#124—22no—
331#124C4×D6114yes—
332#123—1228no—
333C22⋊Q8—614no—
334C22⋊Q8—1222no—
335#121—614no—
336#121—1222no—
337C4×Q8—614no—
338C4×Q8—1222no—
339C4×D4—614no—
340C4×D4—1222no—
341C4×D4—1228no—
342#117—1228no—
343C2×C4⋊C4—611no—
344C2×C4⋊C4—611no—
345C2×C4⋊C4—612no—
346C2×C4⋊C4—1221no—
347C2×C4⋊C4—1221no—
348C2×C4⋊C4—1221no—
349C2×C4⋊C4—1221no—
350C2×C4⋊C4C22×D6112yes—
351C2×C22⋊C4—21no—
352C2×C22⋊C4—21no—
353C2×C22⋊C4—1221no—
354C2×C22⋊C4—1221no—
355C2×C22⋊C4—1221no—
356C2×C22⋊C4—1222no—
357C2×C42—21no—
358C2×C42—611no—
359C2×C42—611no—
360C2×C42—611no—
361C2×C42—614no—
362C2×C42—614no—
363C2×C42—1221no—
364C2×C42—1221no—
365C2×C42C22×D6111yescharacteristic
366C4⋊C8—22no—
367C4⋊C8—1224no—
368C4⋊C8—1222no—
369#103—42no—
370#103—2442no—
371#102—42no—
372#102—2442no—
373#95—21no—
374#95—1222no—
375#95—1222no—
376#95—1222no—
377#95—1221no—
378C24—21no—
379C24—611no—
380C2×Q8—21no—
381C2×Q8—22no—
382C2×Q8—22no—
383C2×Q8C2×C4×D6112yes—
384C2×D4—21no—
385C2×D4—22no—
386C2×D4—22no—
387C2×D4—1224no—
388C2×D4—2442no—
389C2×D4—2442no—
390C2×D4C2×C4×D6112yes—
391C22×C4—21no—
392C22×C4—22no—
393C22×C4—611no—
394C22×C4—611no—
395C22×C4—611no—
396C22×C4—612no—
397C22×C4—612no—
398C22×C4—614no—
399C22×C4—614no—
400C22×C4—1224no—
401C22×C4—1222no—
402C22×C4—1222no—
403C22×C4—1221no—
404C22×C4—1221no—
405C22×C4—1222no—
406C22×C4—2441no—
407C22×C4—2442no—
408C22×C4—2441no—
409C22×C4—2444no—
410C22×C4D4×D6111yescharacteristic
411C22×C4D4×D6111yescharacteristic
412C22×C4C23×D6111yescharacteristic
413M16—2448no—
414C2×C8—22no—
415C2×C8—22no—
416C2×C8—22no—
417C2×C8—1222no—
418C2×C8—1222no—
419C2×C8—1222no—
420C4⋊C4—22no—
421C4⋊C4—22no—
422C4⋊C4—612no—
423C4⋊C4—612no—
424C4⋊C4—1224no—
425C4⋊C4—1224no—
426C4⋊C4—1221no—
427C4⋊C4—1221no—
428C4⋊C4—1224no—
429C4⋊C4—1224no—
430C4⋊C4—1222no—
431C4⋊C4—1222no—
432C22⋊C4—24no—
433C22⋊C4—24no—
434C22⋊C4—1224no—
435C22⋊C4—1224no—
436C22⋊C4—1224no—
437C22⋊C4—2444no—
438C42—42no—
439C42—612no—
440C42—612no—
441C42—614no—
442C42—614no—
443C42—1224no—
444C42—2442no—
445C42C2×C4×D6112yes—
446C42D4×D6112yes—
447C23—22no—
448C23—41no—
449C23—42no—
450C23—41no—
451C23—42no—
452C23—612no—
453C23—1221no—
454C23—1222no—
455C23—1221no—
456C23—1222no—
457C23C2×D4×D6111yescharacteristic
458Q8—24no—
459Q8—41no—
460Q8—41no—
461D4—24no—
462D4—41no—
463D4—41no—
464D4—2448no—
465C2×C4—21no—
466C2×C4—22no—
467C2×C4—21no—
468C2×C4—22no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—612no—
474C2×C4—612no—
475C2×C4—612no—
476C2×C4—1221no—
477C2×C4—1222no—
478C2×C4—1221no—
479C2×C4—1222no—
480C2×C4—1224no—
481C2×C4—1222no—
482C2×C4—1222no—
483C2×C4—1222no—
484C2×C4—1222no—
485C2×C4—2444no—
486C2×C4—2444no—
487C2×C4—2448no—
488C2×C4—2442no—
489C2×C4—2442no—
490C2×C4—2442no—
491C2×C4—2441no—
492C2×C4—2442no—
493C2×C4—2441no—
494C2×C4C22⋊C4×D6112yes—
495C2×C4C22⋊C4×D6112yes—
496C2×C4C22×C4×D6111yescharacteristic
497C2×C4C2×D4×D6111yescharacteristic
498C2×C4#1451982212yes—
499C2×C4#1451982412yes—
500C8—42no—
501C8—2442no—
502C22—42no—
503C22—42no—
504C22—42no—
505C22—82no—
506C22—82no—
507C22—1222no—
508C22—1222no—
509C22—1222no—
510C22—2442no—
511C22—2442no—
512C22C2×C22⋊C4×D6111yescharacteristic
513C22#1684034112yes—
514C22#1684055811yescharacteristic
515C22#1684055912yes—
516C22#1684056111yescharacteristic
517C4—22no—
518C4—22no—
519C4—42no—
520C4—82no—
521C4—1222no—
522C4—1222no—
523C4—1222no—
524C4—4882no—
525C4—2442no—
526C4—2442no—
527C4#1684024112yes—
528C4#1684032511yescharacteristic
529C4#1684033211yescharacteristic
530C2—82no—
531C2—2442no—
532C2#2144840911yescharacteristic
533C2#2145433411yescharacteristic
534C2#2145433511yescharacteristic
535C2#2145721712yes—
536C2#2145721812yes—
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1983C2
socleC23×C61C22×D4
prime radicalC2×#14519748C22
cosocleC22×C4×C61C24
hypercenter#1983D61
hypocenterC61#21895
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1983

Complements

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

NH ≅ G/Ntype
2. C2×#16839784C2semidirect
3. #21448215C2semidirect
4. #21446776C2semidirect
5. #21446774C2semidirect
6. #21446752C2semidirect
8. #21446249C2semidirect
9. #21445832C2semidirect
10. #21445827C2semidirect
11. C61×#1983C2semidirect
14. #16840291C4semidirect
268. #1983D61semidirect
314. C61#21895semidirect

Maximal subgroups

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

subgroupindex
C2×#168402912
C2×#168397842
#214482152
#214467762
#214467742
#214467522
#214462542
#214462492
#214458322
#214458272
C61×#19832
#2189561