Finite Group Databasebeta

← #29716683

Subgroups — #29716683

Order 15616.

1122 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
1#21454153C212yesmaximal, complemented
2C4×#14519758C211yesmaximal, characteristic
3#21447171C212yesmaximal, complemented
4#21446985C212yesmaximal, complemented
5#21446805C212yesmaximal, complemented
6#21446232C212yesmaximal
7#21446100C212yesmaximal
8#21445807C211yesmaximal, characteristic
9C61×#3501C211yesmaximal, characteristic
10C2×#14519758C2211yescharacteristic
11C22×C4×Dic61C2211yescharacteristic
12C2×#14519692C2212yes—
13#16839669—22no—
14#16839658—22no—
15#16839650—21no—
16#16839643—24no—
17#16839643—21no—
18#16839642C2211yescharacteristic
19#16839634C2211yescharacteristic
20#16839634C2214yes—
21#16839623C2211yescharacteristic
22#16839623C2214yes—
23#16839622C2212yes—
24C4×#11758196—21no—
25C4×C22⋊Dic61—21no—
26C4×C22⋊Dic61C2211yescharacteristic
27#16839529C2212yes—
28#16839529C2214yes—
29#16839529C2214yes—
30C61×#554—24no—
31C61×#554—22no—
32D4×Q8×C61—22no—
33C61×#544—21no—
34C61×#543—21no—
35C61×#543C2212yes—
36C61×#533—21no—
37C2×C22⋊Q8×C61—21no—
38C2×C22⋊Q8×C61C2212yes—
39C61×#518C2212yes—
40C2×C4×D4×C61C2211yescharacteristic
41C23×Dic61C2311yescharacteristic
42C23×Dic61C2311yescharacteristic
43#14519758—22no—
44#14519758Q814yescomplemented
45C2×C4⋊Dic61—22no—
46C2×#11758196—21no—
47C2×#11758196—21no—
48C2×#11758196—24no—
49C2×C22⋊Dic61—21no—
50C2×C22⋊Dic61—21no—
51C2×C22⋊Dic61C2314yes—
52C2×C22⋊Dic61C2311yescharacteristic
53C2×C22⋊Dic61C2311yescharacteristic
54C2×C4×Dic61—21no—
55C2×C4×Dic61—22no—
56C2×C4×Dic61—21no—
57C2×C4×Dic61—22no—
58C2×C4×Dic61—21no—
59#14519692—24no—
60#14519684—21no—
61#14519684—21no—
62#14519684—21no—
63#14519684—21no—
64#14519684—21no—
65#14519684—22no—
66#14519684—24no—
67#14519684—24no—
68#14519684—24no—
69C22×Q8×C61—21no—
70C22×Q8×C61D412yes—
71C22×D4×C61C2311yescharacteristic
72C23×C4×C61D411yescharacteristic
73C23×C4×C61C2311yescharacteristic
74C4⋊Q8×C61—21no—
75C4⋊Q8×C61—21no—
76C4⋊Q8×C61—24no—
77#125×C61—22no—
78#125×C61—24no—
79C22⋊Q8×C61—24no—
80C22⋊Q8×C61—24no—
81C22⋊Q8×C61—22no—
82C22⋊Q8×C61—22no—
83C22⋊Q8×C61—22no—
84C22⋊Q8×C61—22no—
85C22⋊Q8×C61—22no—
86C22⋊Q8×C61—22no—
87C22⋊Q8×C61—24no—
88C22⋊Q8×C61—24no—
89C22⋊Q8×C61—22no—
90C22⋊Q8×C61—22no—
91C4×Q8×C61—22no—
92C4×Q8×C61—22no—
93C4×D4×C61—22no—
94C4×D4×C61—24no—
95C4×D4×C61—22no—
96C4×D4×C61—24no—
97#117×C61—22no—
98C2×C4⋊C4×C61—21no—
99C2×C4⋊C4×C61—22no—
100C2×C4⋊C4×C61D411yescharacteristic
101C2×C4⋊C4×C61D414yes—
102C2×C4⋊C4×C61D412yes—
103C2×C22⋊C4×C61—22no—
104C2×C22⋊C4×C61D411yescharacteristic
105C2×C22⋊C4×C61C2311yescharacteristic
106C2×C22⋊C4×C61C2314yes—
107C2×C42×C61D411yescharacteristic
108C24×C61C2×D411yescharacteristic
109C24×C61C2411yescharacteristic
110C2×Q8×C61—22no—
111C2×Q8×C61—22no—
112C2×Q8×C61—22no—
113C2×Q8×C61—22no—
114C2×Q8×C61—44no—
115C2×Q8×C61—82no—
116C2×D4×C61—22no—
117C2×D4×C61—22no—
118C2×D4×C61C2×Q812yes—
119C2×D4×C61#4112yes—
120C22×C4×C61—22no—
121C22×C4×C61—22no—
122C22×C4×C61—21no—
123C22×C4×C61—21no—
124C22×C4×C61—42no—
125C22×C4×C61—42no—
126C22×C4×C61C2×D411yescharacteristic
127C22×C4×C61C2×D412yes—
128C22×C4×C61C2×D411yescharacteristic
129C22×C4×C61C2×D411yescharacteristic
130C22×C4×C61C2×D414yes—
131C22×C4×C61C2×D414yes—
132C22×C4×C61C2×D411yescharacteristic
133C22×C4×C61C2×D411yescharacteristic
134C22×C4×C61C2×D412yes—
135C22×Dic61—22no—
136C22×Dic61—21no—
137C22×Dic61—21no—
138C22×Dic61—22no—
139C22×Dic61—21no—
140C22×Dic61—21no—
141C22×Dic61—22no—
142C22×Dic61—21no—
143C22×Dic61—24no—
144C22×Dic61—42no—
145C22×Dic61C2×Q812yescomplemented
146C4⋊C4×C61—22no—
147C4⋊C4×C61—24no—
148C4⋊C4×C61—22no—
149C4⋊C4×C61—22no—
150C4⋊C4×C61—24no—
151C4⋊C4×C61—22no—
152C4⋊C4×C61—24no—
153C4⋊C4×C61—22no—
154C4⋊C4×C61—22no—
155C4⋊C4×C61—22no—
156C4⋊C4×C61—24no—
157C22⋊C4×C61—24no—
158C22⋊C4×C61—24no—
159C22⋊C4×C61—22no—
160C22⋊C4×C61—24no—
161C22⋊C4×C61—22no—
162C42×C61—24no—
163C42×C61—21no—
164C42×C61—21no—
165C22⋊Dic61—24no—
166C22⋊Dic61—21no—
167C22⋊Dic61—24no—
168C22⋊Dic61—22no—
169C22⋊Dic61—24no—
170C22⋊Dic61—42no—
171C22⋊Dic61C2×Q812yescomplemented
172C4⋊Dic61—44no—
173#11758196—24no—
174#11758196—48no—
175#11758196—42no—
176C4×Dic61—42no—
177C4×Dic61—41no—
178C4×Dic61—42no—
179C4×Dic61—41no—
180C23×C61—21no—
181C23×C61—21no—
182C23×C61—41no—
183C23×C61—41no—
184C23×C61—41no—
185C23×C61—41no—
186C23×C61C22⋊Q812yes—
187C23×C61C22×D411yescharacteristic
188C23×C61C22×D412yes—
189C23×C61C22×Q811yescharacteristic
190C23×C61C2×#4111yescharacteristic
191C23×C612−1+412yes—
192Q8×C61—44no—
193Q8×C61—82no—
194Q8×C61—82no—
195Q8×C61—82no—
196D4×C61—24no—
197D4×C61—24no—
198C2×C4×C61—21no—
199C2×C4×C61—21no—
200C2×C4×C61—22no—
201C2×C4×C61—24no—
202C2×C4×C61—21no—
203C2×C4×C61—24no—
204C2×C4×C61—24no—
205C2×C4×C61—21no—
206C2×C4×C61—42no—
207C2×C4×C61—41no—
208C2×C4×C61—42no—
209C2×C4×C61—44no—
210C2×C4×C61—41no—
211C2×C4×C61—41no—
212C2×C4×C61—44no—
213C2×C4×C61—81no—
214C2×C4×C61—82no—
215C2×C4×C61—82no—
216C2×C4×C61#12011yescharacteristic
217C2×C4×C61#12012yes—
218C2×C4×C61#12011yescharacteristic
219C2×C4×C61C22⋊Q812yes—
220C2×Dic61—24no—
221C2×Dic61—22no—
222C2×Dic61—22no—
223C2×Dic61—22no—
224C2×Dic61—24no—
225C2×Dic61—41no—
226C2×Dic61—41no—
227C2×Dic61—42no—
228C2×Dic61—41no—
229C2×Dic61—41no—
230C2×Dic61—41no—
231C2×Dic61—41no—
232C2×Dic61—42no—
233C2×Dic61—41no—
234C2×Dic61—44no—
235C2×Dic61—44no—
236C2×Dic61—41no—
237C2×Dic61—41no—
238C2×Dic61—44no—
239#24351—611nomaximal, Sylow
240C22×C61—22no—
241C22×C61—21no—
242C22×C61—22no—
243C22×C61—21no—
244C22×C61—22no—
245C22×C61—41no—
246C22×C61—41no—
247C22×C61—42no—
248C22×C61—42no—
249C22×C61—41no—
250C22×C61—41no—
251C22×C61—81no—
252C22×C61C2×#12011yescharacteristic
253C22×C61C2×C22⋊Q811yescharacteristic
254C22×C61#53611yescharacteristic
255C22×C61#54812yes—
256C22×C61D4×Q812yes—
257C4×C61—22no—
258C4×C61—21no—
259C4×C61—22no—
260C4×C61—41no—
261C4×C61—44no—
262C4×C61—44no—
263C4×C61—41no—
264C4×C61#45812yes—
265Dic61—42no—
266Dic61—42no—
267Dic61—42no—
268Dic61—84no—
269Dic61—81no—
270#3501D6111yescharacteristic
271#2855—612no—
272C4×#123—611no—
273#2062—612no—
274#2030—612no—
275#1996—612no—
276#1890—612no—
277#1867—612no—
278#1799—611no—
279C2×C61—21no—
280C2×C61—22no—
281C2×C61—21no—
282C2×C61—22no—
283C2×C61—42no—
284C2×C61#240911yescharacteristic
285C2×C61C2×#45811yescharacteristic
286C2×C61#302511yescharacteristic
287#554—24no—
288#554—22no—
289D4×Q8—22no—
290#544—21no—
291#543—21no—
292#543C2×D6112yes—
293#533—21no—
294C2×#123—611no—
295C2×C22⋊Q8—21no—
296C2×C22⋊Q8C2×D6112yes—
297#518C2×D6112yes—
298C2×C4×D4C2×D6111yescharacteristic
299C22×C42—611no—
300C2×#99—612no—
301#391—1222no—
302#389—1222no—
303#388—1221no—
304#387—1224no—
305#387—1221no—
306#386—611no—
307#385—611no—
308#385—614no—
309#381—614no—
310#381—611no—
311#380—612no—
312C4×C4⋊C4—1221no—
313C4×C22⋊C4—611no—
314C4×C22⋊C4—1221no—
315#342—612no—
316#342—614no—
317#342—614no—
318C61#2435111yescharacteristic, complemented, Sylow
319C22×Q8—21no—
320C22×Q8C22⋊D6112yes—
321C22×D4C22×D6111yescharacteristic
322C23×C4—611no—
323C23×C4—611no—
324C23×C4C22⋊D6111yescharacteristic
325C23×C4C22×D6111yescharacteristic
326C4⋊Q8—21no—
327C4⋊Q8—21no—
328C4⋊Q8—24no—
329#125—22no—
330#125—24no—
331#123—614no—
332#123—1222no—
333C22⋊Q8—22no—
334C22⋊Q8—22no—
335C22⋊Q8—24no—
336C22⋊Q8—24no—
337C22⋊Q8—22no—
338C22⋊Q8—22no—
339C22⋊Q8—22no—
340C22⋊Q8—22no—
341C22⋊Q8—22no—
342C22⋊Q8—22no—
343C22⋊Q8—24no—
344C22⋊Q8—24no—
345C4×Q8—22no—
346C4×Q8—22no—
347C4×D4—22no—
348C4×D4—22no—
349C4×D4—24no—
350C4×D4—24no—
351#117—22no—
352C2×C4⋊C4—21no—
353C2×C4⋊C4—22no—
354C2×C4⋊C4—1221no—
355C2×C4⋊C4—1221no—
356C2×C4⋊C4—1222no—
357C2×C4⋊C4—1224no—
358C2×C4⋊C4C22⋊D6111yescharacteristic
359C2×C4⋊C4C22⋊D6114yes—
360C2×C4⋊C4C22⋊D6112yes—
361C2×C22⋊C4—22no—
362C2×C22⋊C4—611no—
363C2×C22⋊C4—611no—
364C2×C22⋊C4—614no—
365C2×C22⋊C4—1221no—
366C2×C22⋊C4—1221no—
367C2×C22⋊C4C22⋊D6111yescharacteristic
368C2×C22⋊C4C22×D6111yescharacteristic
369C2×C22⋊C4C22×D6114yes—
370C2×C42—1221no—
371C2×C42—1221no—
372C2×C42—1222no—
373C2×C42—1221no—
374C2×C42—1222no—
375C2×C42C22⋊D6111yescharacteristic
376#99—1224no—
377#95—1221no—
378#95—1221no—
379#95—1221no—
380#95—1221no—
381#95—1221no—
382#95—1222no—
383#95—1224no—
384#95—1224no—
385#95—1224no—
386C24C2×C22⋊D6111yescharacteristic
387C24C23×D6111yescharacteristic
388C2×Q8—22no—
389C2×Q8—22no—
390C2×Q8—22no—
391C2×Q8—22no—
392C2×Q8—44no—
393C2×Q8—82no—
394C2×D4—22no—
395C2×D4—22no—
396C2×D4Q8×D6112yes—
397C2×D4#1175822612yes—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—612no—
405C22×C4—1222no—
406C22×C4—1221no—
407C22×C4—1221no—
408C22×C4—1222no—
409C22×C4—1221no—
410C22×C4—1221no—
411C22×C4—1222no—
412C22×C4—1221no—
413C22×C4—1224no—
414C22×C4—2442no—
415C22×C4C2×C22⋊D6111yescharacteristic
416C22×C4C2×C22⋊D6112yes—
417C22×C4C2×C22⋊D6111yescharacteristic
418C22×C4C2×C22⋊D6114yes—
419C22×C4C2×C22⋊D6111yescharacteristic
420C22×C4C2×C22⋊D6111yescharacteristic
421C22×C4C2×C22⋊D6114yes—
422C22×C4C2×C22⋊D6111yescharacteristic
423C22×C4C2×C22⋊D6112yes—
424C4⋊C4—22no—
425C4⋊C4—24no—
426C4⋊C4—22no—
427C4⋊C4—22no—
428C4⋊C4—24no—
429C4⋊C4—22no—
430C4⋊C4—24no—
431C4⋊C4—22no—
432C4⋊C4—22no—
433C4⋊C4—22no—
434C4⋊C4—24no—
435C4⋊C4—1224no—
436C4⋊C4—2444no—
437C4⋊C4—2442no—
438C4⋊C4—2448no—
439C22⋊C4—24no—
440C22⋊C4—24no—
441C22⋊C4—22no—
442C22⋊C4—24no—
443C22⋊C4—22no—
444C22⋊C4—612no—
445C22⋊C4—1221no—
446C22⋊C4—1224no—
447C22⋊C4—1224no—
448C22⋊C4—1222no—
449C22⋊C4—1224no—
450C22⋊C4—2442no—
451C42—24no—
452C42—21no—
453C42—21no—
454C42—2442no—
455C42—2441no—
456C42—2442no—
457C42—2441no—
458C23—21no—
459C23—21no—
460C23—41no—
461C23—41no—
462C23—41no—
463C23—41no—
464C23#1451975312yes—
465C23C22×C22⋊D6111yescharacteristic
466C23C22×C22⋊D6112yes—
467C23C2×Q8×D6111yescharacteristic
468C23C2×#1175822611yescharacteristic
469C23#1451982512yes—
470Q8—44no—
471Q8—82no—
472Q8—82no—
473Q8—82no—
474D4—24no—
475D4—24no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—22no—
479C2×C4—24no—
480C2×C4—21no—
481C2×C4—24no—
482C2×C4—24no—
483C2×C4—21no—
484C2×C4—42no—
485C2×C4—41no—
486C2×C4—42no—
487C2×C4—44no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—44no—
491C2×C4—81no—
492C2×C4—82no—
493C2×C4—82no—
494C2×C4—1222no—
495C2×C4—1224no—
496C2×C4—1222no—
497C2×C4—1222no—
498C2×C4—1224no—
499C2×C4—2441no—
500C2×C4—2441no—
501C2×C4—2444no—
502C2×C4—2444no—
503C2×C4—2441no—
504C2×C4—2444no—
505C2×C4—2441no—
506C2×C4—2441no—
507C2×C4—2442no—
508C2×C4—2441no—
509C2×C4—2441no—
510C2×C4—2441no—
511C2×C4—2441no—
512C2×C4—2442no—
513C2×C4#1451974611yescharacteristic
514C2×C4#1451974612yes—
515C2×C4#1451974611yescharacteristic
516C2×C4#1451975312yes—
517C22—22no—
518C22—21no—
519C22—22no—
520C22—21no—
521C22—22no—
522C22—41no—
523C22—41no—
524C22—42no—
525C22—42no—
526C22—41no—
527C22—41no—
528C22—81no—
529C22C2×#1451974611yescharacteristic
530C22C2×#1451975311yescharacteristic
531C22#1684033611yescharacteristic
532C22#1684042312yes—
533C22Q8×C22⋊D6112yes—
534C4—21no—
535C4—22no—
536C4—22no—
537C4—41no—
538C4—44no—
539C4—44no—
540C4—41no—
541C4—4884no—
542C4—4881no—
543C4—2442no—
544C4—2442no—
545C4—2442no—
546C4#1683995212yes—
547C2—22no—
548C2—21no—
549C2—21no—
550C2—22no—
551C2—42no—
552C2#2144917811yescharacteristic
553C2C2×#1683995211yescharacteristic
554C2#2145521111yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC24×C61C24
FrattiniC24C23×D61
FittingC61×#3501C2
socleC22×C61C2×#120
prime radicalC22×D4×C61C23
hypercenter#3501D61
hypocenterC61#24351
LCSG ▷ C24×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3501

Complements

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

NH ≅ G/Ntype
1. #21454153C2semidirect
3. #21447171C2semidirect
4. #21446985C2semidirect
5. #21446805C2semidirect
44. #14519758Q8semidirect
145. C22×Dic61C2×Q8semidirect
171. C22⋊Dic61C2×Q8semidirect
318. C61#24351semidirect

Maximal subgroups

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

subgroupindex
#214541532
C4×#145197582
#214471712
#214469852
#214468052
#214462322
#214461002
#214458072
C61×#35012
#2435161