Finite Group Databasebeta

← #29716689

Subgroups — #29716689

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#21459622C211yesmaximal, characteristic
2#21454158C212yesmaximal, complemented
3#21447172C212yesmaximal, complemented
4#21446986C212yesmaximal, complemented
5#21446804C212yesmaximal, complemented
6#21446233C212yesmaximal
7#21446099C212yesmaximal
8#21445807C211yesmaximal, characteristic
9C4×#123×C61C211yesmaximal, characteristic
10#16840471—24no—
11#16840469—22no—
12D4×Dic122—22no—
13#16840393—21no—
14#16840388—21no—
15#16840388C2212yes—
16#16840319—21no—
17C2×#14519756—21no—
18C2×#14519756C2212yes—
19#16840237C2212yes—
20C2×#14519692C2212yes—
21#16839668—22no—
22#16839660—22no—
23#16839646—24no—
24#16839635C2214yes—
25#16839625C2214yes—
26#16839622C2212yes—
27#16839529C2212yes—
28#16839528C2214yes—
29#16839528C2214yes—
30C2×#123×C61C2211yescharacteristic
31C2×C4×D4×C61C2211yescharacteristic
32C22×C42×C61C2211yescharacteristic
33C61×#388—21no—
34C61×#387—21no—
35C61×#386C2211yescharacteristic
36C61×#385C2211yescharacteristic
37C61×#381C2211yescharacteristic
38C4×C4⋊C4×C61—21no—
39C4×C22⋊C4×C61—21no—
40C4×C22⋊C4×C61C2211yescharacteristic
41C22×Dic122—21no—
42C22×Dic122D412yes—
43#14519777—21no—
44#14519777—21no—
45#14519776—24no—
46#14519769—22no—
47#14519768—24no—
48#14519756—22no—
49#14519756—22no—
50#14519756—22no—
51#14519756—22no—
52#14519756—22no—
53#14519756—22no—
54#14519756—22no—
55#14519756—22no—
56#14519754—24no—
57#14519754—24no—
58#14519754—24no—
59#14519754—24no—
60C4×Dic122—22no—
61C4×Dic122—22no—
62D4×Dic61—24no—
63D4×Dic61—24no—
64#14519736—22no—
65C2×C4⋊Dic61—21no—
66C2×C4⋊Dic61—22no—
67C2×C4⋊Dic61D412yes—
68C2×#11758196—22no—
69C2×#11758196—24no—
70C2×#11758196D414yes—
71C2×C22⋊Dic61—22no—
72C2×C22⋊Dic61C2314yes—
73C2×C22⋊Dic61C2314yes—
74#14519692—24no—
75#14519684—22no—
76#14519684—24no—
77#14519684—24no—
78#14519684—24no—
79C22×D4×C61C2311yescharacteristic
80C23×C4×C61D411yescharacteristic
81C23×C4×C61C2311yescharacteristic
82C23×C4×C61C2311yescharacteristic
83C23×C4×C61C2311yescharacteristic
84#123×C61—22no—
85#123×C61Q814yescomplemented
86C4×D4×C61—22no—
87C4×D4×C61—22no—
88C2×C4⋊C4×C61—21no—
89C2×C4⋊C4×C61—21no—
90C2×C4⋊C4×C61D411yescharacteristic
91C2×C22⋊C4×C61—21no—
92C2×C22⋊C4×C61—21no—
93C2×C22⋊C4×C61D411yescharacteristic
94C2×C22⋊C4×C61C2311yescharacteristic
95C2×C22⋊C4×C61C2311yescharacteristic
96C2×C22⋊C4×C61C2311yescharacteristic
97C2×C42×C61—21no—
98C2×C42×C61—21no—
99C2×C42×C61—21no—
100C2×C42×C61—22no—
101C2×C42×C61—22no—
102C2×C42×C61D411yescharacteristic
103#95×C61—21no—
104#95×C61—21no—
105#95×C61—21no—
106#95×C61—21no—
107#95×C61—21no—
108C24×C61C2×D411yescharacteristic
109C24×C61C2411yescharacteristic
110C2×D4×C61—22no—
111C2×D4×C61—22no—
112C2×D4×C61C2×Q812yes—
113C2×D4×C61#4112yes—
114C22×C4×C61—21no—
115C22×C4×C61—21no—
116C22×C4×C61—21no—
117C22×C4×C61—22no—
118C22×C4×C61—22no—
119C22×C4×C61—21no—
120C22×C4×C61—22no—
121C22×C4×C61—21no—
122C22×C4×C61—21no—
123C22×C4×C61—21no—
124C22×C4×C61—42no—
125C22×C4×C61—42no—
126C22×C4×C61—42no—
127C22×C4×C61C2×D412yes—
128C22×C4×C61C2×D411yescharacteristic
129C22×C4×C61C2×D411yescharacteristic
130C22×C4×C61C2×D411yescharacteristic
131C22×C4×C61C2×D411yescharacteristic
132C22×C4×C61C2×D412yes—
133C22×C4×C61C2×D411yescharacteristic
134C22×C4×C61C2×Q812yescomplemented
135C22×Dic61—22no—
136C22×Dic61—22no—
137C22×Dic61—24no—
138C22×Dic61C2×D414yes—
139C22×Dic61C2×D414yes—
140C2×Dic122—22no—
141C2×Dic122—22no—
142C2×Dic122—22no—
143C2×Dic122—22no—
144C2×Dic122—44no—
145C2×Dic122—82no—
146C4⋊C4×C61—22no—
147C4⋊C4×C61—24no—
148C4⋊C4×C61—42no—
149C22⋊C4×C61—24no—
150C22⋊C4×C61—22no—
151C22⋊C4×C61—21no—
152C22⋊C4×C61—22no—
153C22⋊C4×C61—22no—
154C22⋊C4×C61—42no—
155C22⋊C4×C61C2×Q812yescomplemented
156C42×C61—21no—
157C42×C61—21no—
158C42×C61—42no—
159C42×C61—41no—
160C42×C61—42no—
161C42×C61—41no—
162C22⋊Dic61—24no—
163C22⋊Dic61—24no—
164C22⋊Dic61—24no—
165C22⋊Dic61—24no—
166C22⋊Dic61—24no—
167C4⋊Dic61—22no—
168C4⋊Dic61—22no—
169C4⋊Dic61—22no—
170C4⋊Dic61—22no—
171C4⋊Dic61—22no—
172C4⋊Dic61—22no—
173C4⋊Dic61—44no—
174#11758196—24no—
175#11758196—24no—
176#11758196—24no—
177#11758196—24no—
178#11758196—48no—
179C4×Dic61—24no—
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—
192D4×C61—24no—
193D4×C61—24no—
194C2×C4×C61—21no—
195C2×C4×C61—21no—
196C2×C4×C61—24no—
197C2×C4×C61—22no—
198C2×C4×C61—24no—
199C2×C4×C61—22no—
200C2×C4×C61—21no—
201C2×C4×C61—21no—
202C2×C4×C61—22no—
203C2×C4×C61—22no—
204C2×C4×C61—24no—
205C2×C4×C61—41no—
206C2×C4×C61—42no—
207C2×C4×C61—41no—
208C2×C4×C61—41no—
209C2×C4×C61—42no—
210C2×C4×C61—41no—
211C2×C4×C61—41no—
212C2×C4×C61—41no—
213C2×C4×C61—41no—
214C2×C4×C61—42no—
215C2×C4×C61—42no—
216C2×C4×C61—41no—
217C2×C4×C61—41no—
218C2×C4×C61—41no—
219C2×C4×C61—41no—
220C2×C4×C61—41no—
221C2×C4×C61—81no—
222C2×C4×C61#12011yescharacteristic
223C2×C4×C61#12012yes—
224C2×C4×C61#12011yescharacteristic
225C2×C4×C61C22⋊Q812yes—
226C2×Dic61—24no—
227C2×Dic61—24no—
228C2×Dic61—44no—
229C2×Dic61—44no—
230C2×Dic61—44no—
231C2×Dic61—44no—
232C2×Dic61—44no—
233C2×Dic61—82no—
234C2×Dic61—82no—
235Dic122—44no—
236Dic122—82no—
237Dic122—82no—
238Dic122—82no—
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—22no—
259C4×C61—21no—
260C4×C61—42no—
261C4×C61—42no—
262C4×C61—41no—
263C4×C61—41no—
264C4×C61—42no—
265C4×C61—81no—
266C4×C61#45812yes—
267Dic61—44no—
268Dic61—44no—
269Dic61—84no—
270#3501—611no—
271#2855—612no—
272C4×#123D6111yescharacteristic
273#2062—612no—
274#2030—612no—
275#1996—612no—
276#1890—612no—
277#1867—612no—
278#1799—611no—
279C2×C61—22no—
280C2×C61—21no—
281C2×C61—21no—
282C2×C61—22no—
283C2×C61—42no—
284C2×C61#240911yescharacteristic
285C2×C61C2×#45811yescharacteristic
286C2×C61#302511yescharacteristic
287#554—1222no—
288#554—1224no—
289D4×Q8—1222no—
290#544—1221no—
291#543—612no—
292#543—1221no—
293#533—1221no—
294C2×#123C2×D6111yescharacteristic
295C2×C22⋊Q8—612no—
296C2×C22⋊Q8—1221no—
297#518—612no—
298C2×C4×D4C2×D6111yescharacteristic
299C22×C42C2×D6111yescharacteristic
300C2×#99—612no—
301#391—1222no—
302#389—1222no—
303#388—21no—
304#387—21no—
305#387—1224no—
306#386C2×D6111yescharacteristic
307#385—614no—
308#385C2×D6111yescharacteristic
309#381—614no—
310#381C2×D6111yescharacteristic
311#380—612no—
312C4×C4⋊C4—21no—
313C4×C22⋊C4—21no—
314C4×C22⋊C4C2×D6111yescharacteristic
315#342—612no—
316#342—614no—
317#342—614no—
318C61#2435111yescharacteristic, complemented, Sylow
319C22×Q8—612no—
320C22×Q8—1221no—
321C22×D4C22×D6111yescharacteristic
322C23×C4C22⋊D6111yescharacteristic
323C23×C4C22×D6111yescharacteristic
324C23×C4C22×D6111yescharacteristic
325C23×C4C22×D6111yescharacteristic
326C4⋊Q8—1221no—
327C4⋊Q8—1221no—
328C4⋊Q8—1224no—
329#125—1222no—
330#125—1224no—
331#123—22no—
332#123Dic12214yescomplemented
333C22⋊Q8—1224no—
334C22⋊Q8—1224no—
335C22⋊Q8—1224no—
336C22⋊Q8—1224no—
337C22⋊Q8—1222no—
338C22⋊Q8—1222no—
339C22⋊Q8—1222no—
340C22⋊Q8—1222no—
341C22⋊Q8—1222no—
342C22⋊Q8—1222no—
343C22⋊Q8—1222no—
344C22⋊Q8—1222no—
345C4×Q8—1222no—
346C4×Q8—1222no—
347C4×D4—22no—
348C4×D4—22no—
349C4×D4—1224no—
350C4×D4—1224no—
351#117—1222no—
352C2×C4⋊C4—21no—
353C2×C4⋊C4—21no—
354C2×C4⋊C4—614no—
355C2×C4⋊C4—612no—
356C2×C4⋊C4—1222no—
357C2×C4⋊C4—1221no—
358C2×C4⋊C4—1224no—
359C2×C4⋊C4—1222no—
360C2×C4⋊C4C22⋊D6111yescharacteristic
361C2×C22⋊C4—21no—
362C2×C22⋊C4—21no—
363C2×C22⋊C4—614no—
364C2×C22⋊C4—614no—
365C2×C22⋊C4—1222no—
366C2×C22⋊C4C22⋊D6111yescharacteristic
367C2×C22⋊C4C22×D6111yescharacteristic
368C2×C22⋊C4C22×D6111yescharacteristic
369C2×C22⋊C4C22×D6111yescharacteristic
370C2×C42—22no—
371C2×C42—22no—
372C2×C42—21no—
373C2×C42—21no—
374C2×C42—21no—
375C2×C42C22⋊D6111yescharacteristic
376#99—1224no—
377#95—21no—
378#95—21no—
379#95—21no—
380#95—21no—
381#95—21no—
382#95—1224no—
383#95—1224no—
384#95—1224no—
385#95—1222no—
386C24C2×C22⋊D6111yescharacteristic
387C24C23×D6111yescharacteristic
388C2×Q8—1222no—
389C2×Q8—1222no—
390C2×Q8—1222no—
391C2×Q8—1222no—
392C2×Q8—4882no—
393C2×Q8—2444no—
394C2×D4—22no—
395C2×D4—22no—
396C2×D4C2×Dic12212yes—
397C2×D4#1175822212yes—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—22no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4—42no—
411C22×C4—614no—
412C22×C4—614no—
413C22×C4—1222no—
414C22×C4—1222no—
415C22×C4—1224no—
416C22×C4C2×Dic12212yescomplemented
417C22×C4D4×D6112yes—
418C22×C4D4×D6112yes—
419C22×C4C2×C22⋊D6111yescharacteristic
420C22×C4C2×C22⋊D6111yescharacteristic
421C22×C4C2×C22⋊D6111yescharacteristic
422C22×C4C2×C22⋊D6111yescharacteristic
423C22×C4C2×C22⋊D6111yescharacteristic
424C4⋊C4—22no—
425C4⋊C4—24no—
426C4⋊C4—42no—
427C4⋊C4—1224no—
428C4⋊C4—1222no—
429C4⋊C4—1224no—
430C4⋊C4—1222no—
431C4⋊C4—1222no—
432C4⋊C4—1222no—
433C4⋊C4—1224no—
434C4⋊C4—1222no—
435C4⋊C4—1224no—
436C4⋊C4—1222no—
437C4⋊C4—2448no—
438C4⋊C4—2444no—
439C22⋊C4—22no—
440C22⋊C4—21no—
441C22⋊C4—24no—
442C22⋊C4—22no—
443C22⋊C4—22no—
444C22⋊C4—42no—
445C22⋊C4—1224no—
446C22⋊C4—1224no—
447C22⋊C4—1224no—
448C22⋊C4—1224no—
449C22⋊C4—1224no—
450C22⋊C4C2×Dic12212yescomplemented
451C42—21no—
452C42—21no—
453C42—42no—
454C42—41no—
455C42—42no—
456C42—41no—
457C42—1224no—
458C23—21no—
459C23—21no—
460C23—41no—
461C23—41no—
462C23—41no—
463C23—41no—
464C23#1451975612yes—
465C23C2×D4×D6112yes—
466C23C22×C22⋊D6111yescharacteristic
467C23C22×Dic12211yescharacteristic
468C23C2×#1175822211yescharacteristic
469C23#1451982412yes—
470Q8—4882no—
471Q8—4882no—
472Q8—4882no—
473Q8—2444no—
474D4—24no—
475D4—24no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—22no—
479C2×C4—24no—
480C2×C4—22no—
481C2×C4—24no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—24no—
487C2×C4—42no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—42no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—81no—
504C2×C4—1224no—
505C2×C4—1224no—
506C2×C4—4882no—
507C2×C4—4882no—
508C2×C4—2444no—
509C2×C4—2444no—
510C2×C4—2444no—
511C2×C4—2444no—
512C2×C4—2444no—
513C2×C4#1451974511yescharacteristic
514C2×C4#1451974512yes—
515C2×C4#1451974511yescharacteristic
516C2×C4#1451975612yes—
517C22—21no—
518C22—22no—
519C22—21no—
520C22—22no—
521C22—22no—
522C22—41no—
523C22—41no—
524C22—42no—
525C22—42no—
526C22—41no—
527C22—41no—
528C22—81no—
529C22C2×#1451974511yescharacteristic
530C22C2×#1451975611yescharacteristic
531C22#1684033311yescharacteristic
532C22#1684042112yes—
533C22D4×Dic12212yes—
534C4—21no—
535C4—22no—
536C4—22no—
537C4—42no—
538C4—41no—
539C4—42no—
540C4—41no—
541C4—42no—
542C4—81no—
543C4—4884no—
544C4—2444no—
545C4—2444no—
546C4#1683995112yes—
547C2—21no—
548C2—21no—
549C2—22no—
550C2—22no—
551C2—42no—
552C2#2144918411yescharacteristic
553C2C2×#1683995111yescharacteristic
554C2#2145521011yescharacteristic
subgroupG/H
centerC22C2×#14519745
commutator, cosocleC24×C61C24
FrattiniC24C23×D61
FittingC4×#123×C61C2
socleC22×C61C2×#120
prime radicalC22×D4×C61C23
hypercenterC4×#123D61
hypocenterC61#24351
LCSG ▷ C24×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ C4×#123

Complements

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

NH ≅ G/Ntype
2. #21454158C2semidirect
3. #21447172C2semidirect
4. #21446986C2semidirect
5. #21446804C2semidirect
85. #123×C61Q8semidirect
134. C22×C4×C61C2×Q8semidirect
155. C22⋊C4×C61C2×Q8semidirect
318. C61#24351semidirect
332. #123Dic122semidirect
416. C22×C4C2×Dic122semidirect
450. C22⋊C4C2×Dic122semidirect

Maximal subgroups

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

subgroupindex
#214596222
#214541582
#214471722
#214469862
#214468042
#214462332
#214460992
#214458072
C4×#123×C612
#2435161