Finite Group Databasebeta

← #29707510

Subgroups — #29707510

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
1C2×#16839802C211yesmaximal, characteristic, complemented
2C2×#16839771C211yesmaximal, characteristic, complemented
3C2×#16839771C211yesmaximal, characteristic, complemented
4C2×#16839726C211yesmaximal, characteristic
5#21448202C211yesmaximal, characteristic
6#21447019C214yesmaximal, complemented
7#21447012C214yesmaximal, complemented
8#21445538C211yesmaximal, characteristic, complemented
9C22×C22⋊Q8×C61C211yesmaximal, characteristic
10C22×#11758194C2211yescharacteristic
11C22×#11758194C2211yescharacteristic
12C2×#14519736C2211yescharacteristic
13C2×#14519736C2211yescharacteristic
14#16839802—24no—
15C2×#14519710C2212yescomplemented
16C2×#14519708C2212yescomplemented
17#16839771—24no—
18#16839771—24no—
19C2×#14519701C2212yescomplemented
20C2×#14519701C2214yescomplemented
21C2×#14519701C2212yescomplemented
22#16839726—24no—
23C2×C22⋊#92418C2214yes—
24#16839625—21no—
25#16839622C2212yes—
26#16839622C2212yes—
27#16839618—21no—
28C4×C22⋊Dic61—22no—
29#16839524C2214yescomplemented
30C23×Q8×C61C2211yescharacteristic
31C24×C4×C61C2211yescharacteristic
32C2×C22⋊Q8×C61—24no—
33C2×C22⋊Q8×C61—22no—
34C2×C22⋊Q8×C61—24no—
35C2×C22⋊Q8×C61C2214yes—
36C22×C4⋊C4×C61—21no—
37C22×C4⋊C4×C61C2211yescharacteristic
38C22×C22⋊C4×C61—21no—
39C2×#11758194—22no—
40C2×#11758194—22no—
41C2×#11758194—22no—
42C2×#11758194—22no—
43C2×#11758194—22no—
44C2×#11758194D414yes—
45C22×#92418C2312yes—
46C22×#92418C2312yes—
47#14519736—22no—
48#14519736—22no—
49#14519736—22no—
50#14519736—22no—
51C2×C4⋊Dic61C2312yes—
52C2×C4⋊Dic61C2311yescharacteristic
53C2×C4⋊Dic61C2311yescharacteristic
54C2×C22⋊Dic61—21no—
55C2×C22⋊Dic61—21no—
56C2×C22⋊Dic61—22no—
57C2×C22⋊Dic61—22no—
58C2×C4×Dic61—21no—
59C2×C4×Dic61—21no—
60#14519710—24no—
61#14519708—24no—
62#14519701—24no—
63#14519701—24no—
64#14519701—28no—
65C22⋊#92418—28no—
66#14519684—22no—
67#14519684—22no—
68C25×C61D411yescharacteristic
69C22×Q8×C61—22no—
70C22×Q8×C61—21no—
71C22×Q8×C61—22no—
72C22×Q8×C61—22no—
73C22×Q8×C61—22no—
74C22×Q8×C61—22no—
75C22×Q8×C61D414yes—
76C22×Q8×C61C2312yes—
77C23×C4×C61—21no—
78C23×C4×C61—21no—
79C23×C4×C61—22no—
80C23×C4×C61—22no—
81C23×C4×C61—22no—
82C23×C4×C61—22no—
83C23×C4×C61—22no—
84C23×C4×C61—22no—
85C23×C4×C61D412yes—
86C23×C4×C61D411yescharacteristic
87C23×C4×C61C2312yes—
88C23×C4×C61C2311yescharacteristic
89C22⋊Q8×C61—216no—
90C22⋊Q8×C61—28no—
91C22⋊Q8×C61—28no—
92C2×C4⋊C4×C61—28no—
93C2×C4⋊C4×C61—24no—
94C2×C4⋊C4×C61—21no—
95C2×C4⋊C4×C61—22no—
96C2×C4⋊C4×C61D414yes—
97C2×C4⋊C4×C61C2312yes—
98C2×C22⋊C4×C61—28no—
99C2×C22⋊C4×C61—24no—
100C24×C61—22no—
101C24×C61—24no—
102C24×C61—22no—
103C24×C61—22no—
104C24×C61—42no—
105C24×C61C2×D412yes—
106C24×C61C2×D411yescharacteristic
107C2×Q8×C61—24no—
108C2×Q8×C61—22no—
109C2×Q8×C61—28no—
110C2×Q8×C61—28no—
111C2×Q8×C61—22no—
112C2×Q8×C61—22no—
113C2×Q8×C61—22no—
114C2×Q8×C61—22no—
115C2×Q8×C61—42no—
116C2×Q8×C61—44no—
117C2×Q8×C61—42no—
118C2×Q8×C61—44no—
119C2×Q8×C61C2×D414yes—
120C22×C4×C61—24no—
121C22×C4×C61—24no—
122C22×C4×C61—22no—
123C22×C4×C61—22no—
124C22×C4×C61—22no—
125C22×C4×C61—22no—
126C22×C4×C61—22no—
127C22×C4×C61—24no—
128C22×C4×C61—24no—
129C22×C4×C61—24no—
130C22×C4×C61—22no—
131C22×C4×C61—22no—
132C22×C4×C61—22no—
133C22×C4×C61—22no—
134C22×C4×C61—22no—
135C22×C4×C61—22no—
136C22×C4×C61—44no—
137C22×C4×C61—42no—
138C22×C4×C61—42no—
139C22×C4×C61—44no—
140C22×C4×C61—44no—
141C22×C4×C61—44no—
142C22×C4×C61—44no—
143C22×C4×C61C2×D412yes—
144C22×C4×C61C2×D411yescharacteristic
145C22×C4×C61C2×D412yes—
146C22×C4×C61C2×D412yes—
147C22×C4×C61C2×D412yes—
148C22×C4×C61#4112yes—
149C22×C4×C61#4112yes—
150C22×C4×C61#4112yes—
151C22×C4×C61C2411yescharacteristic
152C22×Dic61—21no—
153C22×Dic61—21no—
154C22×Dic61—22no—
155C4⋊C4×C61—216no—
156C4⋊C4×C61—24no—
157C4⋊C4×C61—22no—
158C4⋊C4×C61C2×D414yes—
159C22⋊C4×C61—216no—
160C22⋊Dic61—44no—
161C22⋊Dic61—42no—
162C22⋊Dic61—42no—
163C22⋊Dic61—44no—
164C4⋊Dic61—21no—
165C4⋊Dic61—21no—
166C4⋊Dic61—22no—
167C4⋊Dic61—21no—
168C4⋊Dic61—21no—
169C4⋊Dic61—22no—
170C4×Dic61—42no—
171C4×Dic61—42no—
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
198Q8×C61—24no—
199Q8×C61—24no—
200Q8×C61—42no—
201Q8×C61—42no—
202Q8×C61—48no—
203C2×C4×C61—24no—
204C2×C4×C61—24no—
205C2×C4×C61—22no—
206C2×C4×C61—22no—
207C2×C4×C61—22no—
208C2×C4×C61—48no—
209C2×C4×C61—42no—
210C2×C4×C61—42no—
211C2×C4×C61—44no—
212C2×C4×C61—44no—
213C2×C4×C61—48no—
214C2×C4×C61—42no—
215C2×C4×C61—42no—
216C2×C4×C61—42no—
217C2×C4×C61—42no—
218C2×C4×C61—48no—
219C2×C4×C61#12012yes—
220C2×C4×C61#12114yes—
221C2×C4×C61#12114yes—
222C2×C4×C61#12112yes—
223C2×C4×C61#12314yes—
224C2×C4×C61#12312yes—
225C2×C4×C61#12414yes—
226C2×C4×C61C22×D411yescharacteristic
227C2×C4×C61C22×D411yescharacteristic
228C2×C4×C61C22×D411yescharacteristic
229C2×C4×C61C2×#4111yescharacteristic
230C2×C4×C61C2×#4111yescharacteristic
231C2×C4×C61C2×#4111yescharacteristic
232C2×Dic61—41no—
233C2×Dic61—41no—
234C2×Dic61—42no—
235C2×Dic61—42no—
236C2×Dic61—42no—
237C2×Dic61—42no—
238C2×Dic61—42no—
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—44no—
262C4×C61—44no—
263C4×C61#39414yes—
264C4×C61C2×#12111yescharacteristic
265C4×C61C2×#12111yescharacteristic
266C4×C61C2×#12311yescharacteristic
267C4×C61C2×#12411yescharacteristic
268Dic61—81no—
269Dic61—81no—
270Dic61—82no—
271C22×C22⋊Q8D6111yescharacteristic
272C2×#428—611no—
273C2×#419—611no—
274C2×#419—611no—
275C2×#407—611no—
276#2280—611no—
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×Q8C2×D6111yescharacteristic
288C24×C4C2×D6111yescharacteristic
289C22×M16—611no—
290C22×M16—611no—
291C2×C22⋊Q8—24no—
292C2×C22⋊Q8—22no—
293C2×C22⋊Q8—24no—
294C2×C22⋊Q8C2×D6114yes—
295C2×#117—611no—
296C2×#117—611no—
297C22×C4⋊C4—21no—
298C22×C4⋊C4C2×D6111yescharacteristic
299C22×C22⋊C4—21no—
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—1221no—
311#380—612no—
312#380—612no—
313#379—1221no—
314C4×C22⋊C4—1222no—
315#340—614no—
316C61#2348111yescharacteristic, complemented, Sylow
317C25C22⋊D6111yescharacteristic
318C22×Q8—21no—
319C22×Q8—22no—
320C22×Q8—22no—
321C22×Q8—22no—
322C22×Q8—22no—
323C22×Q8—22no—
324C22×Q8C22⋊D6114yes—
325C22×Q8C22×D6112yes—
326C23×C4—21no—
327C23×C4—21no—
328C23×C4—22no—
329C23×C4—22no—
330C23×C4—22no—
331C23×C4—22no—
332C23×C4—22no—
333C23×C4—22no—
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—216no—
347C22⋊Q8—28no—
348C22⋊Q8—28no—
349#117—1222no—
350#117—1222no—
351#117—1222no—
352#117—1222no—
353C2×C4⋊C4—28no—
354C2×C4⋊C4—24no—
355C2×C4⋊C4—22no—
356C2×C4⋊C4—21no—
357C2×C4⋊C4—612no—
358C2×C4⋊C4—611no—
359C2×C4⋊C4—611no—
360C2×C4⋊C4C22⋊D6114yes—
361C2×C4⋊C4C22×D6112yes—
362C2×C22⋊C4—28no—
363C2×C22⋊C4—24no—
364C2×C22⋊C4—1221no—
365C2×C22⋊C4—1221no—
366C2×C22⋊C4—1222no—
367C2×C22⋊C4—1222no—
368C2×C42—1221no—
369C2×C42—1221no—
370#107—1224no—
371#106—1224no—
372#103—1228no—
373#103—1224no—
374#103—1224no—
375C22⋊C8—1228no—
376#95—1222no—
377#95—1222no—
378C24—24no—
379C24—22no—
380C24—22no—
381C24—22no—
382C24—42no—
383C24C2×C22⋊D6112yes—
384C24C2×C22⋊D6111yescharacteristic
385C2×Q8—22no—
386C2×Q8—24no—
387C2×Q8—22no—
388C2×Q8—22no—
389C2×Q8—22no—
390C2×Q8—22no—
391C2×Q8—28no—
392C2×Q8—28no—
393C2×Q8—44no—
394C2×Q8—42no—
395C2×Q8—42no—
396C2×Q8—44no—
397C2×Q8C2×C22⋊D6114yes—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—44no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—44no—
418C22×C4—44no—
419C22×C4—44no—
420C22×C4—44no—
421C22×C4—1221no—
422C22×C4—1221no—
423C22×C4—1222no—
424C22×C4D4×D6112yes—
425C22×C4#1175822412yes—
426C22×C4#1175822412yes—
427C22×C4#1175822412yes—
428C22×C4C2×C22⋊D6112yes—
429C22×C4C2×C22⋊D6112yes—
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—216no—
442C4⋊C4—22no—
443C4⋊C4—24no—
444C4⋊C4—1222no—
445C4⋊C4—1221no—
446C4⋊C4—1221no—
447C4⋊C4—1221no—
448C4⋊C4—1221no—
449C4⋊C4—1222no—
450C4⋊C4C2×C22⋊D6114yes—
451C22⋊C4—216no—
452C22⋊C4—2444no—
453C22⋊C4—2444no—
454C22⋊C4—2442no—
455C22⋊C4—2442no—
456C42—2442no—
457C42—2442no—
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#1451974612yes—
472C23#1451974912yes—
473C23#1451975812yes—
474C23#1451980814yes—
475C23C22×C22⋊D6111yescharacteristic
476Q8—24no—
477Q8—24no—
478Q8—42no—
479Q8—42no—
480Q8—48no—
481C2×C4—24no—
482C2×C4—24no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—48no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—44no—
490C2×C4—44no—
491C2×C4—48no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—48no—
497C2×C4—2441no—
498C2×C4—2441no—
499C2×C4—2442no—
500C2×C4—2442no—
501C2×C4—2442no—
502C2×C4—2442no—
503C2×C4—2442no—
504C2×C4#1451974612yes—
505C2×C4#1451974814yes—
506C2×C4#1451974914yes—
507C2×C4#1451974912yes—
508C2×C4#1451975814yes—
509C2×C4#1451975812yes—
510C2×C4#1451976414yes—
511C2×C4C2×D4×D6111yescharacteristic
512C2×C4C22×C22⋊D6111yescharacteristic
513C2×C4C22×C22⋊D6111yescharacteristic
514C2×C4C2×#1175822411yescharacteristic
515C2×C4C2×#1175822411yescharacteristic
516C2×C4C2×#1175822411yescharacteristic
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#1683967614yes—
532C22C2×#1451974611yescharacteristic
533C22C2×#1451974911yescharacteristic
534C22C2×#1451975811yescharacteristic
535C22C2×#1451980812yes—
536C22#1684056012yes—
537C4—42no—
538C4—44no—
539C4—44no—
540C4—4882no—
541C4—4881no—
542C4—4881no—
543C4#1683967614yes—
544C4C2×#1451974811yescharacteristic
545C4C2×#1451974911yescharacteristic
546C4C2×#1451975811yescharacteristic
547C4C2×#1451976411yescharacteristic
548C2—24no—
549C2—44no—
550C2C2×#1683967611yescharacteristic
551C2#2145505311yescharacteristic
552C2#2145506111yescharacteristic
553C2#2145544512yes—
554C2#2145576312yes—
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC22×C22⋊Q8×C61C2
socleC23×C61C22×D4
prime radicalC25×C61D4
hypercenterC22×C22⋊Q8D61
hypocenterC61#23481
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C24×C4 ◁ C22×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C2×#16839802C2semidirect
2. C2×#16839771C2semidirect
3. C2×#16839771C2semidirect
6. #21447019C2semidirect
7. #21447012C2semidirect
8. #21445538C2semidirect
15. C2×#14519710C22semidirect
16. C2×#14519708C22semidirect
19. C2×#14519701C22semidirect
20. C2×#14519701C22semidirect
21. C2×#14519701C22semidirect
29. #16839524C22semidirect
316. C61#23481semidirect

Maximal subgroups

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

subgroupindex
C2×#168398022
C2×#168397712
C2×#168397262
#214482022
#214470192
#214470122
#214455382
C22×C22⋊Q8×C612
#2348161