Finite Group Databasebeta

← #29707743

Subgroups — #29707743

Order 15616.

1414 conjugacy classes of subgroups.

The subgroup lattice has 550 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839763C211yesmaximal, characteristic
2C2×#16839771C211yesmaximal, characteristic, complemented
3C2×#16839758C211yesmaximal, characteristic
4C22×#14519701C211yesmaximal, characteristic, complemented
5#21448439C211yesmaximal, characteristic, complemented
6#21447062C214yesmaximal, complemented
7#21447055C214yesmaximal, complemented
8#21445516C211yesmaximal, characteristic, complemented
9C2×C61×#536C211yesmaximal, characteristic
10C22×#11758194C2211yescharacteristic
11C23×#92418C2211yescharacteristic
12C2×#14519736C2211yescharacteristic
13C22×C4⋊Dic61C2211yescharacteristic
14#16839771—24no—
15#16839763—24no—
16#16839758—24no—
17C2×#14519701—24no—
18C2×#14519701C2214yes—
19C2×#14519701C2214yes—
20C2×#14519701C2212yes—
21C2×#14519701C2212yes—
22C2×#14519698C2212yes—
23C2×#14519698C2212yes—
24#16839660C2214yes—
25#16839634—22no—
26#16839631—21no—
27#16839627—21no—
28#16839517C2214yes—
29C22×#41×C61C2211yescharacteristic
30C23×Q8×C61C2211yescharacteristic
31C61×#536—24no—
32C61×#536—24no—
33C2×C4⋊Q8×C61C2212yes—
34C2×#124×C61C2212yes—
35C2×#123×C61—22no—
36C2×C22⋊Q8×C61—21no—
37C2×C22⋊Q8×C61—21no—
38C2×#117×C61C2211yescharacteristic
39C23×Dic61—21no—
40C2×#11758194—22no—
41C2×#11758194—22no—
42C2×#11758194—22no—
43C22×#92418—22no—
44C22×#92418—22no—
45C22×#92418—22no—
46C22×#92418C2312yes—
47C22×#92418C2312yes—
48#14519736—22no—
49#14519736D414yes—
50C2×C4⋊Dic61—22no—
51C2×C4⋊Dic61—22no—
52C2×C4⋊Dic61—21no—
53C2×C4⋊Dic61C2312yes—
54C2×C4⋊Dic61C2312yes—
55C2×#11758196—24no—
56C2×C22⋊Dic61—21no—
57C2×C4×Dic61—21no—
58#14519701—28no—
59#14519701—24no—
60#14519701—24no—
61#14519701—28no—
62#14519698—24no—
63#14519698—24no—
64#14519684—24no—
65C2×#41×C61—22no—
66C2×#41×C61—22no—
67C2×#41×C61—22no—
68C2×#41×C61—22no—
69C2×#41×C61—22no—
70C2×#41×C61D414yes—
71C22×Q8×C61—22no—
72C22×Q8×C61—21no—
73C22×Q8×C61—22no—
74C22×Q8×C61—22no—
75C22×Q8×C61—22no—
76C22×Q8×C61—22no—
77C22×Q8×C61D414yes—
78C22×Q8×C61C2311yescharacteristic
79C22×Q8×C61C2312yes—
80C22×D4×C61—21no—
81C22×D4×C61C2311yescharacteristic
82C23×C4×C61—21no—
83C23×C4×C61—21no—
84C23×C4×C61C2311yescharacteristic
85C4⋊Q8×C61—24no—
86C4⋊Q8×C61—24no—
87#124×C61—24no—
88#124×C61—24no—
89#123×C61—216no—
90C22⋊Q8×C61—28no—
91C22⋊Q8×C61—28no—
92#117×C61—22no—
93#117×C61—22no—
94C2×C4⋊C4×C61—22no—
95C2×C4⋊C4×C61—22no—
96C2×C4⋊C4×C61D411yescharacteristic
97C2×C4⋊C4×C61D411yescharacteristic
98C2×C22⋊C4×C61—22no—
99C2×C22⋊C4×C61—22no—
100C2×C22⋊C4×C61D412yes—
101C2×C42×C61C2312yes—
102C24×C61—21no—
103C24×C61C2×D411yescharacteristic
104#41×C61—28no—
105#41×C61—28no—
106#41×C61—44no—
107#41×C61—44no—
108C2×Q8×C61—24no—
109C2×Q8×C61—22no—
110C2×Q8×C61—21no—
111C2×Q8×C61—28no—
112C2×Q8×C61—28no—
113C2×Q8×C61—22no—
114C2×Q8×C61—21no—
115C2×Q8×C61—22no—
116C2×Q8×C61—22no—
117C2×Q8×C61—22no—
118C2×Q8×C61—21no—
119C2×Q8×C61—21no—
120C2×Q8×C61—21no—
121C2×Q8×C61—42no—
122C2×Q8×C61—44no—
123C2×Q8×C61—44no—
124C2×Q8×C61—42no—
125C2×Q8×C61C2×D414yes—
126C2×Q8×C61C2×D412yes—
127C2×D4×C61—24no—
128C2×D4×C61—21no—
129C2×D4×C61—21no—
130C2×D4×C61—21no—
131C2×D4×C61—21no—
132C2×D4×C61—21no—
133C2×D4×C61—42no—
134C2×D4×C61—42no—
135C2×D4×C61C2×D412yes—
136C22×C4×C61—24no—
137C22×C4×C61—22no—
138C22×C4×C61—21no—
139C22×C4×C61—24no—
140C22×C4×C61—42no—
141C22×C4×C61—42no—
142C22×C4×C61—42no—
143C22×C4×C61—42no—
144C22×C4×C61C2×D412yes—
145C22×C4×C61C2×D412yes—
146C22×C4×C61C2×D411yescharacteristic
147C22×C4×C61C2×D412yes—
148C22×C4×C61C2×D412yes—
149C22×C4×C61C2×D412yes—
150C22×C4×C61#4112yes—
151C22×C4×C61#4112yes—
152C22×C4×C61#4112yes—
153C22×C4×C61C2411yescharacteristic
154C22×Dic61—22no—
155C22×Dic61—22no—
156C22×Dic61—42no—
157C22×Dic61—82no—
158C4⋊C4×C61—28no—
159C4⋊C4×C61—28no—
160C4⋊C4×C61—22no—
161C4⋊C4×C61—22no—
162C22⋊C4×C61—28no—
163C22⋊C4×C61—28no—
164C22⋊C4×C61—24no—
165C42×C61—22no—
166C42×C61—22no—
167C22⋊Dic61—24no—
168C4⋊Dic61—22no—
169C4⋊Dic61—22no—
170C4⋊Dic61—22no—
171C4⋊Dic61—42no—
172C4⋊Dic61C2×D414yes—
173#11758196—84no—
174C4×Dic61—24no—
175#11758194—44no—
176C2×#92418—22no—
177C2×#92418—22no—
178C2×#92418—22no—
179C2×#92418—22no—
180C2×#92418—22no—
181C2×#92418—22no—
182C2×#92418—44no—
183C23×C61—22no—
184C23×C61—22no—
185C23×C61—22no—
186C23×C61—41no—
187C23×C61—41no—
188C23×C61—42no—
189C23×C61—42no—
190C23×C61#12012yes—
191C23×C61#12112yes—
192C23×C61#12412yes—
193C23×C61C22×D411yescharacteristic
194Q8×C61—24no—
195Q8×C61—22no—
196Q8×C61—24no—
197Q8×C61—22no—
198Q8×C61—42no—
199Q8×C61—41no—
200Q8×C61—42no—
201Q8×C61—41no—
202Q8×C61—48no—
203D4×C61—22no—
204D4×C61—22no—
205D4×C61—48no—
206D4×C61—41no—
207D4×C61—41no—
208C2×C4×C61—22no—
209C2×C4×C61—22no—
210C2×C4×C61—24no—
211C2×C4×C61—22no—
212C2×C4×C61—24no—
213C2×C4×C61—24no—
214C2×C4×C61—41no—
215C2×C4×C61—42no—
216C2×C4×C61—41no—
217C2×C4×C61—42no—
218C2×C4×C61—48no—
219C2×C4×C61—48no—
220C2×C4×C61#12012yes—
221C2×C4×C61#12114yes—
222C2×C4×C61#12114yes—
223C2×C4×C61#12112yes—
224C2×C4×C61#12314yes—
225C2×C4×C61#12314yes—
226C2×C4×C61#12412yes—
227C2×C4×C61C22×D411yescharacteristic
228C2×C4×C61C22×D411yescharacteristic
229C2×C4×C61C22×D411yescharacteristic
230C2×C4×C61C2×#4111yescharacteristic
231C2×C4×C61C2×#4111yescharacteristic
232C2×C4×C61C2×#4111yescharacteristic
233C2×Dic61—24no—
234C2×Dic61—42no—
235C2×Dic61—42no—
236C2×Dic61—42no—
237C2×Dic61—44no—
238C2×Dic61—84no—
239#92418—42no—
240#92418—42no—
241#23510—611nomaximal, Sylow
242C22×C61—24no—
243C22×C61—22no—
244C22×C61—22no—
245C22×C61—24no—
246C22×C61—42no—
247C22×C61—42no—
248C22×C61—42no—
249C22×C61—82no—
250C22×C61—82no—
251C22×C61#39414yes—
252C22×C61C2×#12011yescharacteristic
253C22×C61C2×#12111yescharacteristic
254C22×C61C2×#12411yescharacteristic
255C22×C61#57712yes—
256C22×C61#57812yes—
257C4×C61—42no—
258C4×C61—44no—
259C4×C61—42no—
260C4×C61—42no—
261C4×C61#39414yes—
262C4×C61C2×#12111yescharacteristic
263C4×C61C2×#12111yescharacteristic
264C4×C61C2×#12311yescharacteristic
265C4×C61C2×#12311yescharacteristic
266Dic61—44no—
267Dic61—82no—
268C2×#536D6111yescharacteristic
269C2×#417—611no—
270C2×#419—611no—
271C2×#416—611no—
272C22×#103—611no—
273#2317—611no—
274#2043—614no—
275#2042—614no—
276#1729—611no—
277C2×C61—24no—
278C2×C61—82no—
279C2×C61C2×#39411yescharacteristic
280C2×C61#301011yescharacteristic
281C2×C61#301111yescharacteristic
282C2×C61#304612yes—
283C2×C61#313712yes—
284C22×#41C2×D6111yescharacteristic
285C23×Q8C2×D6111yescharacteristic
286C22×M16—611no—
287C23×C8—611no—
288#536—24no—
289#536—24no—
290C2×C4⋊Q8C2×D6112yes—
291C2×#124C2×D6112yes—
292C2×#123—22no—
293C2×C22⋊Q8—21no—
294C2×C22⋊Q8—21no—
295C2×#117—611no—
296C2×#117C2×D6111yescharacteristic
297C22×C4⋊C4—611no—
298#419—1224no—
299#417—1224no—
300#416—1224no—
301C2×#103—614no—
302C2×#103—612no—
303C2×#103—614no—
304C2×#103—612no—
305C2×#103—1224no—
306C2×#102—612no—
307C2×#102—612no—
308#389—614no—
309#385—1222no—
310#384—1221no—
311#382—1221no—
312#336—614no—
313C61#2351011yescharacteristic, complemented, Sylow
314C2×#41—22no—
315C2×#41—22no—
316C2×#41—22no—
317C2×#41—22no—
318C2×#41—22no—
319C2×#41C22⋊D6114yes—
320C22×Q8—21no—
321C22×Q8—22no—
322C22×Q8—22no—
323C22×Q8—22no—
324C22×Q8—22no—
325C22×Q8—22no—
326C22×Q8C22⋊D6114yes—
327C22×Q8C22×D6112yes—
328C22×Q8C22×D6111yescharacteristic
329C22×D4—21no—
330C22×D4C22×D6111yescharacteristic
331C23×C4—21no—
332C23×C4—21no—
333C23×C4—1221no—
334C23×C4C22×D6111yescharacteristic
335C2×M16—1222no—
336C2×M16—1222no—
337C2×M16—1222no—
338C22×C8—612no—
339C22×C8—612no—
340C22×C8—1222no—
341C22×C8—1222no—
342C22×C8—1222no—
343C4⋊Q8—24no—
344C4⋊Q8—24no—
345#124—24no—
346#124—24no—
347#123—216no—
348C22⋊Q8—28no—
349C22⋊Q8—28no—
350#117—22no—
351#117—22no—
352#117—614no—
353#117—1222no—
354C2×C4⋊C4—22no—
355C2×C4⋊C4—22no—
356C2×C4⋊C4—612no—
357C2×C4⋊C4—612no—
358C2×C4⋊C4—1222no—
359C2×C4⋊C4—1222no—
360C2×C4⋊C4—1224no—
361C2×C4⋊C4—1221no—
362C2×C4⋊C4C22⋊D6111yescharacteristic
363C2×C4⋊C4C22⋊D6111yescharacteristic
364C2×C22⋊C4—22no—
365C2×C22⋊C4—22no—
366C2×C22⋊C4—1221no—
367C2×C22⋊C4C22⋊D6112yes—
368C2×C42—1221no—
369C2×C42C22×D6112yes—
370#103—1224no—
371#103—1228no—
372#103—1228no—
373#103—1224no—
374#102—1224no—
375#102—1224no—
376#95—1224no—
377C24—21no—
378C24C2×C22⋊D6111yescharacteristic
379#41—28no—
380#41—28no—
381#41—44no—
382#41—44no—
383C2×Q8—22no—
384C2×Q8—21no—
385C2×Q8—24no—
386C2×Q8—22no—
387C2×Q8—21no—
388C2×Q8—22no—
389C2×Q8—22no—
390C2×Q8—22no—
391C2×Q8—21no—
392C2×Q8—21no—
393C2×Q8—21no—
394C2×Q8—28no—
395C2×Q8—28no—
396C2×Q8—42no—
397C2×Q8—44no—
398C2×Q8—44no—
399C2×Q8—42no—
400C2×Q8C2×C22⋊D6114yes—
401C2×Q8C2×C22⋊D6112yes—
402C2×D4—24no—
403C2×D4—21no—
404C2×D4—21no—
405C2×D4—21no—
406C2×D4—21no—
407C2×D4—21no—
408C2×D4—42no—
409C2×D4—42no—
410C2×D4C2×C22⋊D6112yes—
411C22×C4—22no—
412C22×C4—21no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—1222no—
420C22×C4—1222no—
421C22×C4—4882no—
422C22×C4—2442no—
423C22×C4D4×D6112yes—
424C22×C4#1175822412yes—
425C22×C4#1175822412yes—
426C22×C4#1175822412yes—
427C22×C4C2×C22⋊D6112yes—
428C22×C4C2×C22⋊D6112yes—
429C22×C4C2×C22⋊D6112yes—
430C22×C4C2×C22⋊D6112yes—
431C22×C4C2×C22⋊D6111yescharacteristic
432C22×C4C23×D6111yescharacteristic
433M16—2444no—
434C2×C8—1222no—
435C2×C8—1222no—
436C2×C8—1222no—
437C2×C8—1222no—
438C2×C8—1222no—
439C2×C8—1222no—
440C2×C8—2444no—
441C4⋊C4—28no—
442C4⋊C4—28no—
443C4⋊C4—22no—
444C4⋊C4—22no—
445C4⋊C4—614no—
446C4⋊C4—1222no—
447C4⋊C4—1222no—
448C4⋊C4—1222no—
449C4⋊C4—4884no—
450C4⋊C4—2442no—
451C22⋊C4—28no—
452C22⋊C4—28no—
453C22⋊C4—24no—
454C22⋊C4—1224no—
455C42—22no—
456C42—22no—
457C42—1224no—
458C23—22no—
459C23—22no—
460C23—22no—
461C23—41no—
462C23—41no—
463C23—42no—
464C23—42no—
465C23#1451974612yes—
466C23#1451974812yes—
467C23#1451976412yes—
468C23C22×C22⋊D6111yescharacteristic
469Q8—24no—
470Q8—22no—
471Q8—24no—
472Q8—22no—
473Q8—42no—
474Q8—41no—
475Q8—42no—
476Q8—41no—
477Q8—48no—
478D4—22no—
479D4—22no—
480D4—48no—
481D4—41no—
482D4—41no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—24no—
486C2×C4—22no—
487C2×C4—24no—
488C2×C4—24no—
489C2×C4—41no—
490C2×C4—42no—
491C2×C4—41no—
492C2×C4—42no—
493C2×C4—48no—
494C2×C4—48no—
495C2×C4—1224no—
496C2×C4—4884no—
497C2×C4—2442no—
498C2×C4—2444no—
499C2×C4—2442no—
500C2×C4—2442no—
501C2×C4#1451974612yes—
502C2×C4#1451974812yes—
503C2×C4#1451974914yes—
504C2×C4#1451974914yes—
505C2×C4#1451975814yes—
506C2×C4#1451975814yes—
507C2×C4#1451976412yes—
508C2×C4C2×D4×D6111yescharacteristic
509C2×C4C22×C22⋊D6111yescharacteristic
510C2×C4C22×C22⋊D6111yescharacteristic
511C2×C4C2×#1175822411yescharacteristic
512C2×C4C2×#1175822411yescharacteristic
513C2×C4C2×#1175822411yescharacteristic
514C8—2442no—
515C8—2442no—
516C22—22no—
517C22—24no—
518C22—22no—
519C22—24no—
520C22—42no—
521C22—42no—
522C22—42no—
523C22—82no—
524C22—82no—
525C22#1683967614yes—
526C22C2×#1451974611yescharacteristic
527C22C2×#1451974811yescharacteristic
528C22C2×#1451976411yescharacteristic
529C22#1684057012yes—
530C22#1684057712yes—
531C4—42no—
532C4—42no—
533C4—42no—
534C4—44no—
535C4—4882no—
536C4—2444no—
537C4#1683967614yes—
538C4C2×#1451974911yescharacteristic
539C4C2×#1451974911yescharacteristic
540C4C2×#1451975811yescharacteristic
541C4C2×#1451975811yescharacteristic
542C2—24no—
543C2—82no—
544C2C2×#1683967611yescharacteristic
545C2#2145510511yescharacteristic
546C2#2145511411yescharacteristic
547C2#2145536512yes—
548C2#2145618412yes—
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#536C2
socleC23×C61C22×D4
prime radicalC22×#41×C61C22
hypercenterC2×#536D61
hypocenterC61#23510
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#536

Complements

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

NH ≅ G/Ntype
2. C2×#16839771C2semidirect
4. C22×#14519701C2semidirect
5. #21448439C2semidirect
6. #21447062C2semidirect
7. #21447055C2semidirect
8. #21445516C2semidirect
313. C61#23510semidirect

Maximal subgroups

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

subgroupindex
C2×#168397632
C2×#168397712
C2×#168397582
C22×#145197012
#214484392
#214470622
#214470552
#214455162
C2×C61×#5362
#2351061