Finite Group Databasebeta

← #29736170

Subgroups — #29736170

Order 15616.

1094 conjugacy classes of subgroups.

The subgroup lattice has 534 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21451200C212yesmaximal, complemented
2#21451162C211yesmaximal, characteristic
3#21447171C212yesmaximal, complemented
4#21447171C212yesmaximal, complemented
5#21446805C212yesmaximal, complemented
6#21446100C212yesmaximal
7#21446100C212yesmaximal
8C4×#14519692C211yesmaximal, characteristic
9C61×#3501C211yesmaximal, characteristic
10C22×#11758196C2211yescharacteristic
11C2×#14519692C2212yes—
12#16839695—21no—
13#16839693—22no—
14#16839685—21no—
15#16839681—22no—
16#16839657—24no—
17#16839646—21no—
18#16839642C2211yescharacteristic
19#16839642C2211yescharacteristic
20#16839635—21no—
21#16839634C2211yescharacteristic
22#16839634C2211yescharacteristic
23#16839634C2214yes—
24#16839634C2214yes—
25#16839622C2211yescharacteristic
26C4×C22⋊Dic61C2212yes—
27#16839529C2212yes—
28#16839529C2214yes—
29#16839529C2214yes—
30C61×#554—22no—
31C61×#554—24no—
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
43C2×C4⋊Dic61—21no—
44C2×#11758196—22no—
45C2×#11758196—21no—
46C2×#11758196—21no—
47C2×#11758196—24no—
48C2×#11758196—21no—
49C2×#11758196—22no—
50C2×#11758196—24no—
51C2×#11758196—21no—
52C2×C22⋊Dic61—21no—
53C2×C22⋊Dic61—21no—
54C2×C22⋊Dic61C2311yescharacteristic
55C2×C22⋊Dic61C2311yescharacteristic
56C2×C22⋊Dic61C2314yes—
57C2×C4×Dic61—22no—
58#14519692Q818yescomplemented
59#14519684—22no—
60#14519684—24no—
61#14519684—21no—
62#14519684—24no—
63#14519684—21no—
64#14519684—21no—
65#14519684—21no—
66#14519684—21no—
67C22×Q8×C61—21no—
68C22×Q8×C61D412yes—
69C22×D4×C61C2311yescharacteristic
70C23×C4×C61D411yescharacteristic
71C23×C4×C61C2311yescharacteristic
72C4⋊Q8×C61—21no—
73C4⋊Q8×C61—21no—
74C4⋊Q8×C61—24no—
75#125×C61—22no—
76#125×C61—24no—
77C22⋊Q8×C61—24no—
78C22⋊Q8×C61—24no—
79C22⋊Q8×C61—22no—
80C22⋊Q8×C61—22no—
81C22⋊Q8×C61—22no—
82C22⋊Q8×C61—22no—
83C22⋊Q8×C61—22no—
84C22⋊Q8×C61—22no—
85C22⋊Q8×C61—24no—
86C22⋊Q8×C61—24no—
87C22⋊Q8×C61—22no—
88C22⋊Q8×C61—22no—
89C4×Q8×C61—22no—
90C4×Q8×C61—22no—
91C4×D4×C61—22no—
92C4×D4×C61—24no—
93C4×D4×C61—22no—
94C4×D4×C61—24no—
95#117×C61—22no—
96C2×C4⋊C4×C61—21no—
97C2×C4⋊C4×C61—22no—
98C2×C4⋊C4×C61D412yes—
99C2×C4⋊C4×C61D411yescharacteristic
100C2×C4⋊C4×C61D414yes—
101C2×C22⋊C4×C61—22no—
102C2×C22⋊C4×C61D411yescharacteristic
103C2×C22⋊C4×C61C2311yescharacteristic
104C2×C22⋊C4×C61C2314yes—
105C2×C42×C61D411yescharacteristic
106C24×C61C2×D411yescharacteristic
107C24×C61C2411yescharacteristic
108C2×Q8×C61—22no—
109C2×Q8×C61—22no—
110C2×Q8×C61—22no—
111C2×Q8×C61—22no—
112C2×Q8×C61—82no—
113C2×Q8×C61—82no—
114C2×D4×C61—22no—
115C2×D4×C61—22no—
116C2×D4×C61C2×Q814yes—
117C22×C4×C61—22no—
118C22×C4×C61—22no—
119C22×C4×C61—21no—
120C22×C4×C61—21no—
121C22×C4×C61—81no—
122C22×C4×C61—81no—
123C22×C4×C61C2×D411yescharacteristic
124C22×C4×C61C2×D411yescharacteristic
125C22×C4×C61C2×D414yes—
126C22×C4×C61C2×D414yes—
127C22×C4×C61C2×D411yescharacteristic
128C22×C4×C61C2×D411yescharacteristic
129C22×C4×C61C2×D411yescharacteristic
130C22×C4×C61C2×D412yes—
131C22×C4×C61C2×D412yes—
132C22×Dic61—21no—
133C22×Dic61—22no—
134C22×Dic61—24no—
135C22×Dic61—21no—
136C22×Dic61—22no—
137C22×Dic61—21no—
138C22×Dic61—21no—
139C22×Dic61—42no—
140C22×Dic61—81no—
141C4⋊C4×C61—22no—
142C4⋊C4×C61—24no—
143C4⋊C4×C61—22no—
144C4⋊C4×C61—22no—
145C4⋊C4×C61—24no—
146C4⋊C4×C61—22no—
147C4⋊C4×C61—24no—
148C4⋊C4×C61—22no—
149C4⋊C4×C61—22no—
150C4⋊C4×C61—22no—
151C4⋊C4×C61—24no—
152C22⋊C4×C61—24no—
153C22⋊C4×C61—24no—
154C22⋊C4×C61—22no—
155C22⋊C4×C61—24no—
156C22⋊C4×C61—22no—
157C42×C61—24no—
158C42×C61—21no—
159C42×C61—21no—
160C22⋊Dic61—24no—
161C22⋊Dic61—22no—
162C22⋊Dic61—22no—
163C22⋊Dic61—42no—
164C22⋊Dic61—81no—
165C22⋊Dic61C2×Q818yescomplemented
166C4⋊Dic61—81no—
167#11758196—48no—
168#11758196—42no—
169#11758196—41no—
170#11758196—41no—
171#11758196—42no—
172#11758196—84no—
173#11758196—81no—
174C4×Dic61—82no—
175C23×C61—21no—
176C23×C61—21no—
177C23×C61—41no—
178C23×C61—41no—
179C23×C61—41no—
180C23×C61—41no—
181C23×C61C4⋊Q812yes—
182C23×C61C22×D411yescharacteristic
183C23×C61C22×D412yes—
184C23×C61C22×Q812yes—
185C23×C612−1+412yes—
186Q8×C61—44no—
187Q8×C61—82no—
188Q8×C61—82no—
189Q8×C61—82no—
190D4×C61—28no—
191C2×C4×C61—21no—
192C2×C4×C61—21no—
193C2×C4×C61—21no—
194C2×C4×C61—24no—
195C2×C4×C61—24no—
196C2×C4×C61—21no—
197C2×C4×C61—42no—
198C2×C4×C61—41no—
199C2×C4×C61—41no—
200C2×C4×C61—42no—
201C2×C4×C61—42no—
202C2×C4×C61—44no—
203C2×C4×C61—41no—
204C2×C4×C61—41no—
205C2×C4×C61—44no—
206C2×C4×C61—81no—
207C2×C4×C61—82no—
208C2×C4×C61—82no—
209C2×C4×C61#12011yescharacteristic
210C2×C4×C61#12011yescharacteristic
211C2×C4×C61#12012yes—
212C2×C4×C61C4⋊Q812yes—
213C2×Dic61—24no—
214C2×Dic61—22no—
215C2×Dic61—28no—
216C2×Dic61—44no—
217C2×Dic61—44no—
218C2×Dic61—42no—
219C2×Dic61—41no—
220C2×Dic61—41no—
221C2×Dic61—41no—
222C2×Dic61—41no—
223C2×Dic61—44no—
224C2×Dic61—41no—
225C2×Dic61—41no—
226C2×Dic61—41no—
227C2×Dic61—41no—
228C2×Dic61—41no—
229C2×Dic61—41no—
230C2×Dic61—41no—
231#25854—611nomaximal, Sylow
232C22×C61—22no—
233C22×C61—21no—
234C22×C61—22no—
235C22×C61—21no—
236C22×C61—22no—
237C22×C61—42no—
238C22×C61—42no—
239C22×C61—42no—
240C22×C61—41no—
241C22×C61—41no—
242C22×C61—81no—
243C22×C61C2×#12011yescharacteristic
244C22×C61C2×C4⋊Q811yescharacteristic
245C22×C61#53611yescharacteristic
246C22×C61D4×Q814yes—
247C4×C61—22no—
248C4×C61—21no—
249C4×C61—21no—
250C4×C61—22no—
251C4×C61—44no—
252C4×C61—41no—
253C4×C61—41no—
254C4×C61—44no—
255Dic61—48no—
256Dic61—42no—
257Dic61—81no—
258Dic61—81no—
259Dic61—81no—
260#3501D6111yescharacteristic
261#2583—612no—
262#2579—611no—
263#2062—612no—
264#2062—612no—
265#1996—612no—
266#1867—612no—
267#1867—612no—
268C4×#99—611no—
269C2×C61—24no—
270C2×C61—21no—
271C2×C61—21no—
272C2×C61—42no—
273C2×C61#243411yescharacteristic
274C2×C61#302511yescharacteristic
275C2×C61#302611yescharacteristic
276#554—24no—
277#554—22no—
278D4×Q8—22no—
279#544—21no—
280#543—21no—
281#543C2×D6112yes—
282#533—21no—
283C2×C22⋊Q8—21no—
284C2×C22⋊Q8C2×D6112yes—
285#518C2×D6112yes—
286C2×C4×D4C2×D6111yescharacteristic
287C22×C4⋊C4—611no—
288C2×#99—612no—
289#398—1222no—
290#398—1221no—
291#396—1221no—
292#395—1222no—
293#389—1224no—
294#387—1221no—
295#386—611no—
296#386—611no—
297#385—611no—
298#385—611no—
299#385—614no—
300#385—614no—
301#385—1221no—
302#380—611no—
303C4×C22⋊C4—612no—
304#342—612no—
305#342—614no—
306#342—614no—
307C61#2585411yescharacteristic, complemented, Sylow
308C22×Q8—21no—
309C22×Q8C22⋊D6112yes—
310C22×D4C22×D6111yescharacteristic
311C23×C4—611no—
312C23×C4—611no—
313C23×C4C22⋊D6111yescharacteristic
314C23×C4C22×D6111yescharacteristic
315C4⋊Q8—21no—
316C4⋊Q8—21no—
317C4⋊Q8—24no—
318#125—22no—
319#125—24no—
320C22⋊Q8—22no—
321C22⋊Q8—22no—
322C22⋊Q8—24no—
323C22⋊Q8—24no—
324C22⋊Q8—22no—
325C22⋊Q8—22no—
326C22⋊Q8—22no—
327C22⋊Q8—22no—
328C22⋊Q8—22no—
329C22⋊Q8—22no—
330C22⋊Q8—24no—
331C22⋊Q8—24no—
332C4×Q8—22no—
333C4×Q8—22no—
334C4×D4—22no—
335C4×D4—22no—
336C4×D4—24no—
337C4×D4—24no—
338#117—22no—
339C2×C4⋊C4—21no—
340C2×C4⋊C4—22no—
341C2×C4⋊C4—1221no—
342C2×C4⋊C4—1221no—
343C2×C4⋊C4—1224no—
344C2×C4⋊C4—1221no—
345C2×C4⋊C4—1222no—
346C2×C4⋊C4—1222no—
347C2×C4⋊C4—1221no—
348C2×C4⋊C4—1221no—
349C2×C4⋊C4—1224no—
350C2×C4⋊C4C22⋊D6111yescharacteristic
351C2×C4⋊C4C22⋊D6114yes—
352C2×C4⋊C4C22⋊D6112yes—
353C2×C22⋊C4—22no—
354C2×C22⋊C4—614no—
355C2×C22⋊C4—611no—
356C2×C22⋊C4—611no—
357C2×C22⋊C4—1221no—
358C2×C22⋊C4—1221no—
359C2×C22⋊C4C22⋊D6111yescharacteristic
360C2×C22⋊C4C22×D6111yescharacteristic
361C2×C22⋊C4C22×D6114yes—
362C2×C42—1222no—
363C2×C42C22⋊D6111yescharacteristic
364#99—618no—
365#95—1222no—
366#95—1221no—
367#95—1221no—
368#95—1221no—
369#95—1224no—
370#95—1224no—
371#95—1221no—
372#95—1221no—
373C24C2×C22⋊D6111yescharacteristic
374C24C23×D6111yescharacteristic
375C2×Q8—22no—
376C2×Q8—22no—
377C2×Q8—22no—
378C2×Q8—22no—
379C2×Q8—82no—
380C2×Q8—82no—
381C2×D4—22no—
382C2×D4—22no—
383C2×D4Q8×D6114yes—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—81no—
389C22×C4—81no—
390C22×C4—1224no—
391C22×C4—1221no—
392C22×C4—1222no—
393C22×C4—1221no—
394C22×C4—1221no—
395C22×C4—1221no—
396C22×C4—1222no—
397C22×C4—4881no—
398C22×C4—2442no—
399C22×C4C2×C22⋊D6111yescharacteristic
400C22×C4C2×C22⋊D6111yescharacteristic
401C22×C4C2×C22⋊D6114yes—
402C22×C4C2×C22⋊D6114yes—
403C22×C4C2×C22⋊D6111yescharacteristic
404C22×C4C2×C22⋊D6111yescharacteristic
405C22×C4C2×C22⋊D6112yes—
406C22×C4C2×C22⋊D6112yes—
407C22×C4C2×C22⋊D6111yescharacteristic
408C4⋊C4—22no—
409C4⋊C4—24no—
410C4⋊C4—22no—
411C4⋊C4—22no—
412C4⋊C4—24no—
413C4⋊C4—22no—
414C4⋊C4—24no—
415C4⋊C4—22no—
416C4⋊C4—22no—
417C4⋊C4—22no—
418C4⋊C4—24no—
419C4⋊C4—4881no—
420C4⋊C4—4884no—
421C4⋊C4—4881no—
422C4⋊C4—2442no—
423C4⋊C4—2448no—
424C4⋊C4—2441no—
425C4⋊C4—2441no—
426C4⋊C4—2442no—
427C22⋊C4—24no—
428C22⋊C4—24no—
429C22⋊C4—22no—
430C22⋊C4—24no—
431C22⋊C4—22no—
432C22⋊C4—618no—
433C22⋊C4—1224no—
434C22⋊C4—1222no—
435C22⋊C4—1222no—
436C22⋊C4—4881no—
437C22⋊C4—2442no—
438C42—24no—
439C42—21no—
440C42—21no—
441C42—4882no—
442C23—21no—
443C23—21no—
444C23—41no—
445C23—41no—
446C23—41no—
447C23—41no—
448C23#1451977512yes—
449C23C22×C22⋊D6111yescharacteristic
450C23C22×C22⋊D6112yes—
451C23C2×Q8×D6112yes—
452C23#1451982512yes—
453Q8—44no—
454Q8—82no—
455Q8—82no—
456Q8—82no—
457D4—28no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—24no—
462C2×C4—24no—
463C2×C4—21no—
464C2×C4—42no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—41no—
470C2×C4—44no—
471C2×C4—44no—
472C2×C4—41no—
473C2×C4—81no—
474C2×C4—82no—
475C2×C4—82no—
476C2×C4—1222no—
477C2×C4—1224no—
478C2×C4—1228no—
479C2×C4—2441no—
480C2×C4—2441no—
481C2×C4—2441no—
482C2×C4—2441no—
483C2×C4—2444no—
484C2×C4—2441no—
485C2×C4—2441no—
486C2×C4—2441no—
487C2×C4—2441no—
488C2×C4—2441no—
489C2×C4—2444no—
490C2×C4—2444no—
491C2×C4—2442no—
492C2×C4—2441no—
493C2×C4—2441no—
494C2×C4#1451974611yescharacteristic
495C2×C4#1451974611yescharacteristic
496C2×C4#1451974612yes—
497C2×C4#1451977512yes—
498C22—22no—
499C22—21no—
500C22—22no—
501C22—21no—
502C22—22no—
503C22—42no—
504C22—42no—
505C22—42no—
506C22—41no—
507C22—41no—
508C22—81no—
509C22C2×#1451974611yescharacteristic
510C22C2×#1451977511yescharacteristic
511C22#1684033611yescharacteristic
512C22Q8×C22⋊D6114yes—
513C4—21no—
514C4—21no—
515C4—22no—
516C4—22no—
517C4—44no—
518C4—41no—
519C4—41no—
520C4—44no—
521C4—4881no—
522C4—4881no—
523C4—4881no—
524C4—2448no—
525C4—2442no—
526C2—21no—
527C2—24no—
528C2—21no—
529C2—42no—
530C2#2144945911yescharacteristic
531C2#2145521111yescharacteristic
532C2#2145521611yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC24×C61C24
FrattiniC24C23×D61
FittingC61×#3501C2
socleC22×C61C2×#120
prime radicalC22×D4×C61C23
hypercenter#3501D61
hypocenterC61#25854
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. #21451200C2semidirect
3. #21447171C2semidirect
4. #21447171C2semidirect
5. #21446805C2semidirect
58. #14519692Q8semidirect
165. C22⋊Dic61C2×Q8semidirect
307. C61#25854semidirect

Maximal subgroups

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

subgroupindex
#214512002
#214511622
#214471712
#214468052
#214461002
C4×#145196922
C61×#35012
#2585461