Finite Group Databasebeta

← #29707514

Subgroups — #29707514

Order 15616.

1558 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14519756C211yesmaximal, characteristic
2C2×#16839805C211yesmaximal, characteristic, complemented
3C2×#16839770C211yesmaximal, characteristic, complemented
4C2×#16839770C211yesmaximal, characteristic, complemented
5C2×#16839726C211yesmaximal, characteristic
6#21447022C214yesmaximal, complemented
7#21447015C214yesmaximal, complemented
8#21445540C211yesmaximal, characteristic, complemented
9C61×#2280C211yesmaximal, characteristic
10C23×Dic122C2211yescharacteristic
11C22×#11758194C2211yescharacteristic
12C22×#11758194C2211yescharacteristic
13C2×#14519756—24no—
14C2×#14519756—22no—
15C2×#14519756—24no—
16C2×#14519756C2214yes—
17C22×C4⋊Dic61C2211yescharacteristic
18C22×#11758196—21no—
19C22×C22⋊Dic61—21no—
20#16839805—24no—
21C2×#14519710C2212yescomplemented
22C2×#14519708C2212yescomplemented
23#16839770—24no—
24#16839770—24no—
25C2×#14519700C2212yescomplemented
26C2×#14519700C2214yescomplemented
27C2×#14519700C2212yescomplemented
28#16839726—24no—
29C2×C22⋊#92418C2214yes—
30#16839524C2214yescomplemented
31C24×C4×C61C2211yescharacteristic
32C2×#117×C61C2211yescharacteristic
33C2×#117×C61C2211yescharacteristic
34C61×#381—21no—
35C61×#380C2212yes—
36C61×#380C2212yes—
37C61×#379—21no—
38C4×C22⋊C4×C61—22no—
39C22×Dic122—22no—
40C22×Dic122—22no—
41C22×Dic122—22no—
42C22×Dic122—22no—
43C22×Dic122—21no—
44C22×Dic122—22no—
45C22×Dic122D414yes—
46C22×Dic122C2312yes—
47C23×Dic61—21no—
48C23×Dic61—21no—
49C2×#11758194—22no—
50C2×#11758194—22no—
51C2×#11758194—22no—
52C2×#11758194—22no—
53C2×#11758194—22no—
54C2×#11758194D414yes—
55C22×#92418C2312yes—
56C22×#92418C2312yes—
57#14519756—216no—
58#14519756—28no—
59#14519756—28no—
60C2×C4⋊Dic61—22no—
61C2×C4⋊Dic61—21no—
62C2×C4⋊Dic61D414yes—
63C2×C4⋊Dic61C2312yes—
64C2×#11758196—28no—
65C2×#11758196—24no—
66C2×C22⋊Dic61—28no—
67C2×C22⋊Dic61—24no—
68#14519710—24no—
69#14519708—24no—
70#14519700—24no—
71#14519700—24no—
72#14519700—28no—
73C22⋊#92418—28no—
74C25×C61D411yescharacteristic
75C23×C4×C61—22no—
76C23×C4×C61—22no—
77C23×C4×C61—22no—
78C23×C4×C61—22no—
79C23×C4×C61—22no—
80C23×C4×C61—22no—
81C23×C4×C61D411yescharacteristic
82C23×C4×C61D412yes—
83C23×C4×C61C2312yes—
84C23×C4×C61C2311yescharacteristic
85#117×C61—22no—
86#117×C61—22no—
87#117×C61—22no—
88#117×C61—22no—
89C2×C4⋊C4×C61C2312yes—
90C2×C4⋊C4×C61C2311yescharacteristic
91C2×C4⋊C4×C61C2311yescharacteristic
92C2×C22⋊C4×C61—22no—
93C2×C22⋊C4×C61—22no—
94C2×C22⋊C4×C61—21no—
95C2×C22⋊C4×C61—21no—
96C2×C42×C61—21no—
97C2×C42×C61—21no—
98#95×C61—22no—
99#95×C61—22no—
100C24×C61—22no—
101C24×C61—24no—
102C24×C61—22no—
103C24×C61—22no—
104C24×C61—42no—
105C24×C61C2×D412yes—
106C24×C61C2×D411yescharacteristic
107C22×C4×C61—22no—
108C22×C4×C61—22no—
109C22×C4×C61—22no—
110C22×C4×C61—24no—
111C22×C4×C61—24no—
112C22×C4×C61—24no—
113C22×C4×C61—22no—
114C22×C4×C61—22no—
115C22×C4×C61—22no—
116C22×C4×C61—22no—
117C22×C4×C61—22no—
118C22×C4×C61—22no—
119C22×C4×C61—21no—
120C22×C4×C61—21no—
121C22×C4×C61—22no—
122C22×C4×C61—44no—
123C22×C4×C61—44no—
124C22×C4×C61—44no—
125C22×C4×C61—44no—
126C22×C4×C61C2×D412yes—
127C22×C4×C61C2×D411yescharacteristic
128C22×C4×C61C2×D412yes—
129C22×C4×C61C2×D412yes—
130C22×C4×C61C2×D412yes—
131C22×C4×C61#4112yes—
132C22×C4×C61#4112yes—
133C22×C4×C61#4112yes—
134C22×C4×C61C2411yescharacteristic
135C22×Dic61—24no—
136C22×Dic61—24no—
137C22×Dic61—22no—
138C22×Dic61—22no—
139C22×Dic61—44no—
140C22×Dic61—42no—
141C22×Dic61—42no—
142C2×Dic122—28no—
143C2×Dic122—28no—
144C2×Dic122—22no—
145C2×Dic122—22no—
146C2×Dic122—22no—
147C2×Dic122—22no—
148C2×Dic122—24no—
149C2×Dic122—22no—
150C2×Dic122—44no—
151C2×Dic122—42no—
152C2×Dic122—42no—
153C2×Dic122—44no—
154C2×Dic122C2×D414yes—
155C4⋊C4×C61—21no—
156C4⋊C4×C61—21no—
157C4⋊C4×C61—22no—
158C4⋊C4×C61—21no—
159C4⋊C4×C61—21no—
160C4⋊C4×C61—22no—
161C22⋊C4×C61—44no—
162C22⋊C4×C61—42no—
163C22⋊C4×C61—42no—
164C22⋊C4×C61—44no—
165C42×C61—42no—
166C42×C61—42no—
167C22⋊Dic61—216no—
168C4⋊Dic61—24no—
169C4⋊Dic61—22no—
170C4⋊Dic61C2×D414yes—
171#11758196—216no—
172#11758194—28no—
173#11758194—44no—
174C2×#92418—22no—
175C2×#92418—22no—
176C2×#92418—22no—
177C2×#92418—22no—
178C2×#92418—22no—
179C2×#92418C2×D414yes—
180C23×C61—22no—
181C23×C61—22no—
182C23×C61—28no—
183C23×C61—22no—
184C23×C61—22no—
185C23×C61—22no—
186C23×C61—22no—
187C23×C61—22no—
188C23×C61—22no—
189C23×C61—48no—
190C23×C61—48no—
191C23×C61—44no—
192C23×C61—44no—
193C23×C61#12012yes—
194C23×C61#12112yes—
195C23×C61#12312yes—
196C23×C61#13714yes—
197C23×C61C22×D411yescharacteristic
198C2×C4×C61—22no—
199C2×C4×C61—22no—
200C2×C4×C61—22no—
201C2×C4×C61—44no—
202C2×C4×C61—42no—
203C2×C4×C61—42no—
204C2×C4×C61—42no—
205C2×C4×C61—42no—
206C2×C4×C61—48no—
207C2×C4×C61—41no—
208C2×C4×C61—41no—
209C2×C4×C61—42no—
210C2×C4×C61—42no—
211C2×C4×C61—42no—
212C2×C4×C61—42no—
213C2×C4×C61—42no—
214C2×C4×C61#12012yes—
215C2×C4×C61#12114yes—
216C2×C4×C61#12114yes—
217C2×C4×C61#12112yes—
218C2×C4×C61#12314yes—
219C2×C4×C61#12312yes—
220C2×C4×C61#12414yes—
221C2×C4×C61C22×D411yescharacteristic
222C2×C4×C61C22×D411yescharacteristic
223C2×C4×C61C22×D411yescharacteristic
224C2×C4×C61C2×#4111yescharacteristic
225C2×C4×C61C2×#4111yescharacteristic
226C2×C4×C61C2×#4111yescharacteristic
227C2×Dic61—24no—
228C2×Dic61—24no—
229C2×Dic61—48no—
230C2×Dic61—42no—
231C2×Dic61—42no—
232C2×Dic61—44no—
233C2×Dic61—48no—
234Dic122—24no—
235Dic122—24no—
236Dic122—42no—
237Dic122—42no—
238Dic122—48no—
239#92418—24no—
240#92418—42no—
241#23481—611nomaximal, Sylow
242C22×C61—24no—
243C22×C61—22no—
244C22×C61—22no—
245C22×C61—24no—
246C22×C61—24no—
247C22×C61—24no—
248C22×C61—24no—
249C22×C61—416no—
250C22×C61—42no—
251C22×C61—42no—
252C22×C61—42no—
253C22×C61—42no—
254C22×C61#39414yes—
255C22×C61C2×#12011yescharacteristic
256C22×C61C2×#12111yescharacteristic
257C22×C61C2×#12311yescharacteristic
258C22×C61C2×#13712yes—
259C22×C61#57512yes—
260C4×C61—42no—
261C4×C61—81no—
262C4×C61—81no—
263C4×C61—82no—
264C4×C61#39414yes—
265C4×C61C2×#12111yescharacteristic
266C4×C61C2×#12111yescharacteristic
267C4×C61C2×#12311yescharacteristic
268C4×C61C2×#12411yescharacteristic
269Dic61—44no—
270Dic61—44no—
271C22×C22⋊Q8—611no—
272C2×#428—611no—
273C2×#419—611no—
274C2×#419—611no—
275C2×#407—611no—
276#2280D6111yescharacteristic
277#2036—614no—
278#2035—614no—
279#1736—611no—
280C2×C61—24no—
281C2×C61—44no—
282C2×C61C2×#39411yescharacteristic
283C2×C61#300411yescharacteristic
284C2×C61#300511yescharacteristic
285C2×C61#305612yes—
286C2×C61#309512yes—
287C23×Q8—611no—
288C24×C4C2×D6111yescharacteristic
289C22×M16—611no—
290C22×M16—611no—
291C2×C22⋊Q8—614no—
292C2×C22⋊Q8—1224no—
293C2×C22⋊Q8—1224no—
294C2×C22⋊Q8—1222no—
295C2×#117C2×D6111yescharacteristic
296C2×#117C2×D6111yescharacteristic
297C22×C4⋊C4—611no—
298C22×C4⋊C4—1221no—
299C22×C22⋊C4—1221no—
300#428—1224no—
301C2×#107—612no—
302C2×#106—612no—
303#419—1224no—
304#419—1224no—
305C2×#103—612no—
306C2×#103—614no—
307C2×#103—612no—
308#407—1224no—
309C2×C22⋊C8—614no—
310#381—21no—
311#380C2×D6112yes—
312#380C2×D6112yes—
313#379—21no—
314C4×C22⋊C4—22no—
315#340—614no—
316C61#2348111yescharacteristic, complemented, Sylow
317C25C22⋊D6111yescharacteristic
318C22×Q8—614no—
319C22×Q8—612no—
320C22×Q8—1222no—
321C22×Q8—1221no—
322C22×Q8—1222no—
323C22×Q8—1222no—
324C22×Q8—1222no—
325C22×Q8—1222no—
326C23×C4—22no—
327C23×C4—22no—
328C23×C4—22no—
329C23×C4—22no—
330C23×C4—22no—
331C23×C4—22no—
332C23×C4—1221no—
333C23×C4—1221no—
334C23×C4C22⋊D6111yescharacteristic
335C23×C4C22⋊D6112yes—
336C23×C4C22×D6112yes—
337C23×C4C22×D6111yescharacteristic
338C2×M16—614no—
339C2×M16—1222no—
340C2×M16—1222no—
341C2×M16—1222no—
342C2×M16—1222no—
343C2×M16—1222no—
344C22×C8—612no—
345C22×C8—612no—
346C22⋊Q8—1228no—
347C22⋊Q8—1228no—
348C22⋊Q8—12216no—
349#117—22no—
350#117—22no—
351#117—22no—
352#117—22no—
353C2×C4⋊C4—614no—
354C2×C4⋊C4—612no—
355C2×C4⋊C4—1228no—
356C2×C4⋊C4—1222no—
357C2×C4⋊C4—1221no—
358C2×C4⋊C4—1224no—
359C2×C4⋊C4C22×D6112yes—
360C2×C4⋊C4C22×D6111yescharacteristic
361C2×C4⋊C4C22×D6111yescharacteristic
362C2×C22⋊C4—21no—
363C2×C22⋊C4—21no—
364C2×C22⋊C4—22no—
365C2×C22⋊C4—22no—
366C2×C22⋊C4—1228no—
367C2×C22⋊C4—1224no—
368C2×C42—21no—
369C2×C42—21no—
370#107—1224no—
371#106—1224no—
372#103—1228no—
373#103—1224no—
374#103—1224no—
375C22⋊C8—1228no—
376#95—22no—
377#95—22no—
378C24—22no—
379C24—24no—
380C24—22no—
381C24—22no—
382C24—42no—
383C24C2×C22⋊D6112yes—
384C24C2×C22⋊D6111yescharacteristic
385C2×Q8—614no—
386C2×Q8—1222no—
387C2×Q8—1224no—
388C2×Q8—1222no—
389C2×Q8—1222no—
390C2×Q8—1222no—
391C2×Q8—1222no—
392C2×Q8—1228no—
393C2×Q8—1228no—
394C2×Q8—2442no—
395C2×Q8—2444no—
396C2×Q8—2442no—
397C2×Q8—2444no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—22no—
413C22×C4—44no—
414C22×C4—44no—
415C22×C4—44no—
416C22×C4—44no—
417C22×C4—1222no—
418C22×C4—1222no—
419C22×C4—1224no—
420C22×C4—1224no—
421C22×C4—2444no—
422C22×C4—2442no—
423C22×C4—2442no—
424C22×C4#1175822212yes—
425C22×C4#1175822212yes—
426C22×C4D4×D6112yes—
427C22×C4D4×D6112yes—
428C22×C4D4×D6112yes—
429C22×C4#1175822412yes—
430C22×C4C2×C22⋊D6112yes—
431C22×C4C2×C22⋊D6111yescharacteristic
432C22×C4C23×D6111yescharacteristic
433M16—1228no—
434M16—2444no—
435C2×C8—614no—
436C2×C8—1222no—
437C2×C8—1222no—
438C2×C8—1222no—
439C2×C8—1222no—
440C2×C8—1222no—
441C4⋊C4—21no—
442C4⋊C4—21no—
443C4⋊C4—22no—
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—22no—
447C4⋊C4—614no—
448C4⋊C4—12216no—
449C4⋊C4—1222no—
450C4⋊C4—1224no—
451C22⋊C4—44no—
452C22⋊C4—42no—
453C22⋊C4—42no—
454C22⋊C4—44no—
455C22⋊C4—12216no—
456C42—42no—
457C42—42no—
458C23—22no—
459C23—22no—
460C23—28no—
461C23—22no—
462C23—22no—
463C23—22no—
464C23—22no—
465C23—22no—
466C23—22no—
467C23—48no—
468C23—48no—
469C23—44no—
470C23—44no—
471C23#1451974512yes—
472C23#1451974912yes—
473C23#1451976012yes—
474C23#1451980714yes—
475C23C22×C22⋊D6111yescharacteristic
476Q8—1224no—
477Q8—1224no—
478Q8—2442no—
479Q8—2442no—
480Q8—2448no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—44no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—48no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—1224no—
498C2×C4—1224no—
499C2×C4—2448no—
500C2×C4—2442no—
501C2×C4—2442no—
502C2×C4—2444no—
503C2×C4—2448no—
504C2×C4#1451974512yes—
505C2×C4#1451974912yes—
506C2×C4#1451975014yes—
507C2×C4#1451975014yes—
508C2×C4#1451975914yes—
509C2×C4#1451976012yes—
510C2×C4#1451976314yes—
511C2×C4C2×D4×D6111yescharacteristic
512C2×C4C2×D4×D6111yescharacteristic
513C2×C4C2×D4×D6111yescharacteristic
514C2×C4C2×#1175822411yescharacteristic
515C2×C4C2×#1175822211yescharacteristic
516C2×C4C2×#1175822211yescharacteristic
517C8—1224no—
518C8—2442no—
519C22—24no—
520C22—24no—
521C22—22no—
522C22—22no—
523C22—24no—
524C22—24no—
525C22—24no—
526C22—416no—
527C22—42no—
528C22—42no—
529C22—42no—
530C22—42no—
531C22#1683967714yes—
532C22C2×#1451974511yescharacteristic
533C22C2×#1451974911yescharacteristic
534C22C2×#1451976011yescharacteristic
535C22C2×#1451980712yes—
536C22#1684056012yes—
537C4—42no—
538C4—82no—
539C4—81no—
540C4—81no—
541C4—2444no—
542C4—2444no—
543C4#1683967714yes—
544C4C2×#1451975011yescharacteristic
545C4C2×#1451975011yescharacteristic
546C4C2×#1451975911yescharacteristic
547C4C2×#1451976311yescharacteristic
548C2—24no—
549C2—44no—
550C2C2×#1683967711yescharacteristic
551C2#2145505211yescharacteristic
552C2#2145505911yescharacteristic
553C2#2145545112yes—
554C2#2145576712yes—
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2280C2
socleC23×C61C22×D4
prime radicalC25×C61D4
hypercenter#2280D61
hypocenterC61#23481
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C24×C4 ◁ #2280

Complements

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

NH ≅ G/Ntype
2. C2×#16839805C2semidirect
3. C2×#16839770C2semidirect
4. C2×#16839770C2semidirect
6. #21447022C2semidirect
7. #21447015C2semidirect
8. #21445540C2semidirect
21. C2×#14519710C22semidirect
22. C2×#14519708C22semidirect
25. C2×#14519700C22semidirect
26. C2×#14519700C22semidirect
27. C2×#14519700C22semidirect
30. #16839524C22semidirect
316. C61#23481semidirect

Maximal subgroups

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

subgroupindex
C22×#145197562
C2×#168398052
C2×#168397702
C2×#168397262
#214470222
#214470152
#214455402
C61×#22802
#2348161