Finite Group Databasebeta

← #29606993

Subgroups — #29606993

Order 15616.

1254 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839702C211yesmaximal, characteristic
2C2×#16839658C211yesmaximal, characteristic
3C2×#16839613C211yesmaximal, characteristic
4#21444088C211yesmaximal, characteristic
5#21444082C212yesmaximal
6#21444077C212yesmaximal
7#21444069C211yesmaximal, characteristic
8#21444051C212yesmaximal
9#21444045C212yesmaximal
10#21444044C211yesmaximal, characteristic
11C61×#1444C211yesmaximal, characteristic
12C22×C4⋊Dic61C2211yescharacteristic
13C22×#11758196C2211yescharacteristic
14C22×#11758196C2211yescharacteristic
15C22×#11758196C2212yes—
16C22×C4×Dic61C2211yescharacteristic
17C22×C4×Dic61C2211yescharacteristic
18C22×C4×Dic61C2211yescharacteristic
19#16839702C418yescomplemented
20#16839658—24no—
21#16839613—24no—
22C2×#14519684C2212yes—
23C2×#14519684C2212yes—
24C2×#14519684C2211yescharacteristic
25C2×#14519684C2212yes—
26C2×#14519684C2211yescharacteristic
27C2×#14519684C2212yes—
28C2×#14519684C2212yes—
29C2×#14519684C2212yes—
30C2×#14519684C2212yes—
31C2×#14519684C2212yes—
32C2×#14519684C2212yes—
33C22×C4⋊C4×C61C2211yescharacteristic
34C22×C4⋊C4×C61C2212yes—
35C2×#95×C61C2211yescharacteristic
36C2×#95×C61C2212yes—
37C2×#95×C61C2211yescharacteristic
38C23×Dic61C2312yes—
39C23×Dic61C2311yescharacteristic
40C23×Dic61C2312yes—
41C23×Dic61C2311yescharacteristic
42C23×Dic61C2311yescharacteristic
43C23×Dic61C2311yescharacteristic
44C2×C4⋊Dic61—22no—
45C2×C4⋊Dic61—22no—
46C2×C4⋊Dic61C2×C414yes—
47C2×#11758196—22no—
48C2×#11758196—22no—
49C2×#11758196—22no—
50C2×#11758196—22no—
51C2×#11758196—22no—
52C2×#11758196—24no—
53C2×#11758196—42no—
54C2×#11758196—42no—
55C2×#11758196C2×C414yes—
56C2×C4×Dic61—22no—
57C2×C4×Dic61—22no—
58C2×C4×Dic61—22no—
59C2×C4×Dic61—42no—
60C2×C4×Dic61—42no—
61C2×C4×Dic61D414yes—
62C2×C4×Dic61Q814yes—
63#14519684—22no—
64#14519684—22no—
65#14519684—24no—
66#14519684—24no—
67#14519684—24no—
68#14519684—42no—
69#14519684—42no—
70#14519684—42no—
71#14519684—42no—
72#14519684C2×C418yes—
73#14519684C2×C418yes—
74C23×C4×C61C2312yes—
75C23×C4×C61C2312yes—
76C23×C4×C61C2311yescharacteristic
77C23×C4×C61C2311yescharacteristic
78C23×C4×C61C2311yescharacteristic
79C2×C4⋊C4×C61—22no—
80C2×C4⋊C4×C61—24no—
81C2×C4⋊C4×C61—42no—
82C2×C4⋊C4×C61—42no—
83C2×C4⋊C4×C61—42no—
84#95×C61—22no—
85#95×C61—42no—
86#95×C61C2×C414yes—
87C24×C61C2411yescharacteristic
88C22×C4×C61—22no—
89C22×C4×C61—22no—
90C22×C4×C61—21no—
91C22×C4×C61—21no—
92C22×C4×C61—22no—
93C22×C4×C61—22no—
94C22×C4×C61—21no—
95C22×C4×C61—21no—
96C22×C4×C61—22no—
97C22×C4×C61—22no—
98C22×C4×C61—22no—
99C22×C4×C61—22no—
100C22×C4×C61—22no—
101C22×C4×C61—22no—
102C22×C4×C61—22no—
103C22×C4×C61—22no—
104C22×C4×C61—22no—
105C22×C4×C61—22no—
106C22×C4×C61—22no—
107C22×C4×C61—22no—
108C22×C4×C61—22no—
109C22×C4×C61—22no—
110C22×C4×C61—22no—
111C22×C4×C61C22×C414yes—
112C22×C4×C61C22×C412yes—
113C22×C4×C61C2×D412yes—
114C22×C4×C61C2×Q812yes—
115C22×C4×C61#4112yes—
116C22×C4×C61#4112yes—
117C22×Dic61—21no—
118C22×Dic61—21no—
119C22×Dic61—21no—
120C22×Dic61—21no—
121C22×Dic61—22no—
122C22×Dic61—22no—
123C22×Dic61—21no—
124C22×Dic61—22no—
125C22×Dic61—21no—
126C22×Dic61—21no—
127C22×Dic61—21no—
128C22×Dic61—22no—
129C22×Dic61—22no—
130C22×Dic61—22no—
131C22×Dic61—22no—
132C22×Dic61—22no—
133C22×Dic61—22no—
134C22×Dic61—22no—
135C22×Dic61—22no—
136C22×Dic61—22no—
137C22×Dic61—22no—
138C22×Dic61—22no—
139C22×Dic61—22no—
140C22×Dic61—22no—
141C22×Dic61—22no—
142C22×Dic61—22no—
143C22×Dic61—22no—
144C22×Dic61C22×C412yes—
145C22×Dic61C22×C414yes—
146C22×Dic61C22×C412yes—
147C22×Dic61C2×D412yes—
148C22×Dic61C2×D412yes—
149C22×Dic61C2×Q812yes—
150C22×Dic61C2×Q812yes—
151C22×Dic61#4112yes—
152C22×Dic61#4112yes—
153C4⋊C4×C61—44no—
154C4⋊C4×C61—48no—
155C4⋊Dic61—28no—
156#11758196—28no—
157#11758196—28no—
158#11758196—48no—
159C4×Dic61—28no—
160C4×Dic61—44no—
161C4×Dic61—44no—
162C23×C61C23×C411yescharacteristic
163C23×C61C22×D411yescharacteristic
164C23×C61C22×Q811yescharacteristic
165C23×C61C2×#4111yescharacteristic
166C23×C61C2×#4111yescharacteristic
167C23×C61C2×#4111yescharacteristic
168C23×C61C2×#4111yescharacteristic
169C23×C61H(5,𝔽2)12yes—
170C23×C61H(5,𝔽2)12yes—
171C23×C612−1+412yes—
172C23×C612−1+412yes—
173C2×C4×C61—24no—
174C2×C4×C61—22no—
175C2×C4×C61—22no—
176C2×C4×C61—24no—
177C2×C4×C61—24no—
178C2×C4×C61—24no—
179C2×C4×C61—24no—
180C2×C4×C61—22no—
181C2×C4×C61—22no—
182C2×C4×C61—24no—
183C2×C4×C61—24no—
184C2×C4×C61—41no—
185C2×C4×C61—41no—
186C2×C4×C61—41no—
187C2×C4×C61—41no—
188C2×C4×C61—42no—
189C2×C4×C61—42no—
190C2×C4×C61—42no—
191C2×C4×C61—42no—
192C2×C4×C61—42no—
193C2×C4×C61—42no—
194C2×C4×C61—42no—
195C2×C4×C61—42no—
196C2×C4×C61—42no—
197C2×C4×C61—42no—
198C2×C4×C61—42no—
199C2×C4×C61—42no—
200C2×C4×C61—42no—
201C2×C4×C61C4×D414yes—
202C2×Dic61—22no—
203C2×Dic61—22no—
204C2×Dic61—22no—
205C2×Dic61—22no—
206C2×Dic61—22no—
207C2×Dic61—22no—
208C2×Dic61—22no—
209C2×Dic61—22no—
210C2×Dic61—24no—
211C2×Dic61—24no—
212C2×Dic61—22no—
213C2×Dic61—24no—
214C2×Dic61—24no—
215C2×Dic61—24no—
216C2×Dic61—24no—
217C2×Dic61—22no—
218C2×Dic61—22no—
219C2×Dic61—22no—
220C2×Dic61—41no—
221C2×Dic61—42no—
222C2×Dic61—41no—
223C2×Dic61—42no—
224C2×Dic61—42no—
225C2×Dic61—42no—
226C2×Dic61—42no—
227C2×Dic61—42no—
228C2×Dic61—42no—
229C2×Dic61—42no—
230C2×Dic61—42no—
231C2×Dic61—42no—
232C2×Dic61—42no—
233C2×Dic61—42no—
234C2×Dic61—42no—
235C2×Dic61—42no—
236C2×Dic61C4×Q814yes—
237C2×Dic61C22⋊Q814yes—
238#14318—611nomaximal, Sylow
239C22×C61C2×C4×D411yescharacteristic
240C22×C61C2×C4×Q811yescharacteristic
241C22×C61C4×#4111yescharacteristic
242C22×C61#52012yes—
243C22×C61#52012yes—
244C22×C61C2×C22⋊Q811yescharacteristic
245C22×C61#52511yescharacteristic
246C22×C61#52911yescharacteristic
247C22×C61#53311yescharacteristic
248C22×C61#53411yescharacteristic
249C22×C61#53511yescharacteristic
250C22×C61#53611yescharacteristic
251C22×C61#53611yescharacteristic
252C22×C61#53812yes—
253C22×C61#53912yes—
254C22×C61#53912yes—
255C22×C61#54012yes—
256C22×C61#54112yes—
257C22×C61#54112yes—
258C22×C61#54212yes—
259C22×C61#54212yes—
260C22×C61#54412yes—
261C22×C61#54412yes—
262C4×C61—24no—
263C4×C61—44no—
264C4×C61—42no—
265C4×C61—42no—
266C4×C61—82no—
267Dic61—24no—
268Dic61—24no—
269Dic61—42no—
270Dic61—44no—
271Dic61—42no—
272Dic61—82no—
273C2×#400—611no—
274C2×#389—611no—
275C2×C4×C4⋊C4—611no—
276#1445—611no—
277#1444—612no—
278#1444D6111yescharacteristic
279#1442—612no—
280#1441—611no—
281#1436—612no—
282#1434—611no—
283#1434—612no—
284C2×C61C4×C22⋊Q811yescharacteristic
285C2×C61#232211yescharacteristic
286C2×C61#232611yescharacteristic
287C2×C61#233012yes—
288C2×C61#233412yes—
289C2×C61#262711yescharacteristic
290C2×C61#262811yescharacteristic
291C2×C61#263411yescharacteristic
292C2×C61#263511yescharacteristic
293C2×C61#265412yes—
294C2×C61#266612yes—
295C22×C4⋊C4—611no—
296C22×C4⋊C4—611no—
297C22×C4⋊C4—611no—
298C22×C4⋊C4—612no—
299C22×C4⋊C4C2×D6112yes—
300C22×C4⋊C4C2×D6111yescharacteristic
301C22×C42—611no—
302C22×C42—611no—
303C22×C42—611no—
304#400—618no—
305#389—1224no—
306C4×C4⋊C4—1224no—
307C2×#95—612no—
308C2×#95—612no—
309C2×#95—612no—
310C2×#95—612no—
311C2×#95—612no—
312C2×#95—611no—
313C2×#95—611no—
314C2×#95—612no—
315C2×#95—612no—
316C2×#95—612no—
317C2×#95—612no—
318C2×#95C2×D6111yescharacteristic
319C2×#95C2×D6112yes—
320C2×#95C2×D6111yescharacteristic
321C61#1431811yescharacteristic, complemented, Sylow
322C23×C4—611no—
323C23×C4—611no—
324C23×C4—611no—
325C23×C4—612no—
326C23×C4—611no—
327C23×C4—612no—
328C23×C4C22×D6112yes—
329C23×C4C22×D6112yes—
330C23×C4C22×D6111yescharacteristic
331C23×C4C22×D6111yescharacteristic
332C23×C4C22×D6111yescharacteristic
333C2×C4⋊C4—22no—
334C2×C4⋊C4—24no—
335C2×C4⋊C4—42no—
336C2×C4⋊C4—42no—
337C2×C4⋊C4—42no—
338C2×C4⋊C4—614no—
339C2×C4⋊C4—614no—
340C2×C4⋊C4—1222no—
341C2×C4⋊C4—1222no—
342C2×C4⋊C4—1222no—
343C2×C4⋊C4—1222no—
344C2×C4⋊C4—1222no—
345C2×C4⋊C4—1222no—
346C2×C4⋊C4—1222no—
347C2×C4⋊C4—1224no—
348C2×C4⋊C4—2442no—
349C2×C4⋊C4—2442no—
350C2×C42—614no—
351C2×C42—614no—
352C2×C42—1222no—
353C2×C42—1222no—
354C2×C42—1222no—
355C2×C42—2442no—
356C2×C42—2442no—
357#95—22no—
358#95—42no—
359#95—618no—
360#95—618no—
361#95—1222no—
362#95—1224no—
363#95—1224no—
364#95—1224no—
365#95—1222no—
366#95—2442no—
367#95—2442no—
368#95—2442no—
369#95—2442no—
370#95C4×D6114yes—
371C24C23×D6111yescharacteristic
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—22no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—612no—
396C22×C4—612no—
397C22×C4—612no—
398C22×C4—612no—
399C22×C4—612no—
400C22×C4—612no—
401C22×C4—612no—
402C22×C4—614no—
403C22×C4—612no—
404C22×C4—1221no—
405C22×C4—1221no—
406C22×C4—1221no—
407C22×C4—1221no—
408C22×C4—1222no—
409C22×C4—1222no—
410C22×C4—1221no—
411C22×C4—1222no—
412C22×C4—1221no—
413C22×C4—1221no—
414C22×C4—1221no—
415C22×C4—1222no—
416C22×C4—1222no—
417C22×C4—1222no—
418C22×C4—1222no—
419C22×C4—1222no—
420C22×C4—1222no—
421C22×C4—1222no—
422C22×C4—1222no—
423C22×C4—1222no—
424C22×C4—1222no—
425C22×C4—1222no—
426C22×C4—1222no—
427C22×C4—1222no—
428C22×C4—1222no—
429C22×C4—1222no—
430C22×C4—1222no—
431C22×C4C2×C4×D6114yes—
432C22×C4C2×C4×D6112yes—
433C22×C4D4×D6112yes—
434C22×C4#1175822412yes—
435C22×C4Q8×D6112yes—
436C22×C4#1175822612yes—
437C4⋊C4—44no—
438C4⋊C4—48no—
439C4⋊C4—1228no—
440C4⋊C4—1228no—
441C4⋊C4—1228no—
442C4⋊C4—2448no—
443C42—1228no—
444C42—2444no—
445C42—2444no—
446C23C22×C4×D6111yescharacteristic
447C23C2×D4×D6111yescharacteristic
448C23C2×Q8×D6111yescharacteristic
449C23#41×D6111yescharacteristic
450C23#41×D6111yescharacteristic
451C23C2×#1175822411yescharacteristic
452C23C2×#1175822611yescharacteristic
453C23#1451982212yes—
454C23#1451982212yes—
455C23#1451982412yes—
456C23#1451982512yes—
457C2×C4—24no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—24no—
461C2×C4—24no—
462C2×C4—24no—
463C2×C4—24no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—24no—
467C2×C4—24no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—614no—
486C2×C4—614no—
487C2×C4—1222no—
488C2×C4—1222no—
489C2×C4—1222no—
490C2×C4—1222no—
491C2×C4—1222no—
492C2×C4—1222no—
493C2×C4—1222no—
494C2×C4—1222no—
495C2×C4—1224no—
496C2×C4—1224no—
497C2×C4—1222no—
498C2×C4—1224no—
499C2×C4—1224no—
500C2×C4—1224no—
501C2×C4—1224no—
502C2×C4—1222no—
503C2×C4—1222no—
504C2×C4—1222no—
505C2×C4—2442no—
506C2×C4—2442no—
507C2×C4—2442no—
508C2×C4—2442no—
509C2×C4—2442no—
510C2×C4—2442no—
511C2×C4—2442no—
512C2×C4—2441no—
513C2×C4—2442no—
514C2×C4—2441no—
515C2×C4—2442no—
516C2×C4—2442no—
517C2×C4—2442no—
518C2×C4—2442no—
519C2×C4—2442no—
520C2×C4—2442no—
521C2×C4#1451973914yes—
522C22C2×#1451973911yescharacteristic
523C22C4×Q8×D6111yescharacteristic
524C22C4×#1175822411yescharacteristic
525C22#1684024112yes—
526C22#1684024312yes—
527C22C22⋊Q8×D6111yescharacteristic
528C22#1684027211yescharacteristic
529C22#1684029711yescharacteristic
530C22#1684031711yescharacteristic
531C22#1684032211yescharacteristic
532C22#1684032511yescharacteristic
533C22#1684033411yescharacteristic
534C22#1684033711yescharacteristic
535C22#1684034712yes—
536C22#1684035812yes—
537C22#1684036112yes—
538C22#1684036212yes—
539C22#1684037112yes—
540C22#1684037212yes—
541C22#1684038012yes—
542C22#1684038612yes—
543C22#1684038912yes—
544C22#1684039112yes—
545C4—24no—
546C4—42no—
547C4—44no—
548C4—42no—
549C4—82no—
550C4—1224no—
551C4—1224no—
552C4—4882no—
553C4—2442no—
554C4—2444no—
555C4—2442no—
556C2#2144834911yescharacteristic
557C2#2144848711yescharacteristic
558C2#2144852711yescharacteristic
559C2#2144855712yes—
560C2#2144859512yes—
561C2#2145163111yescharacteristic
562C2#2145164511yescharacteristic
563C2#2145170011yescharacteristic
564C2#2145170811yescharacteristic
565C2#2145189612yes—
566C2#2145200512yes—
subgroupG/H
center, FrattiniC24C23×D61
commutatorC23×C61C23×C4
FittingC61×#1444C2
socle, prime radical, cosocleC24×C61C24
hypercenter#1444D61
hypocenterC61#14318
LCSG ▷ C23×C61 ▷ C61
UCSC1 ◁ C24 ◁ #1444

Complements

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

NH ≅ G/Ntype
19. #16839702C4semidirect
321. C61#14318semidirect

Maximal subgroups

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

subgroupindex
C2×#168397022
C2×#168396582
C2×#168396132
#214440882
#214440822
#214440772
#214440692
#214440512
#214440452
#214440442
C61×#14442
#1431861