Finite Group Databasebeta

← #29734516

Subgroups — #29734516

Order 15616.

1094 conjugacy classes of subgroups.

The subgroup lattice has 526 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456366C211yesmaximal, characteristic, complemented
2#21456212C211yesmaximal, characteristic, complemented
3#21450772C211yesmaximal, characteristic, complemented
4#21447026C214yesmaximal, complemented
5#21446764C211yesmaximal, characteristic, complemented
6#21446739C211yesmaximal, characteristic, complemented
7#21446045C214yesmaximal, complemented
8C4×#14519694C211yesmaximal, characteristic, complemented
9C61×#3506C211yesmaximal, characteristic
10C2×#14519777C2211yescharacteristic
11C2×#14519756—22no—
12#16840172—22no—
13#16840165—21no—
14#16840158—21no—
15#16840121—22no—
16#16840112—21no—
17#16840108—21no—
18#16839805C2214yes—
19C2×#14519704C2214yes—
20#16839763C2214yes—
21C2×#14519694C2212yes—
22C4×#11758194C2211yescharacteristic
23C4×#11758194C2211yescharacteristic
24#16839698—22no—
25#16839665C2212yes—
26#16839631C2212yes—
27C4×C4⋊Dic61C2211yescharacteristic
28#16839531C2212yes—
29#16839521C2214yes—
30C61×#559—22no—
31C61×#556—21no—
32C61×#555—21no—
33C61×#553—22no—
34C61×#550—21no—
35C61×#547—21no—
36C61×#538—22no—
37C61×#535C2212yes—
38C61×#533C2211yescharacteristic
39C61×#532C2211yescharacteristic
40C61×#525—22no—
41C61×#520C2212yes—
42C2×C4×D4×C61C2211yescharacteristic
43C22×Dic122—21no—
44C2×#11758194C2312yes—
45C2×#11758194C2312yes—
46#14519777—21no—
47#14519777—21no—
48#14519777—21no—
49#14519777—21no—
50#14519756—48no—
51C2×C4⋊Dic61—21no—
52C2×C4⋊Dic61—21no—
53C2×C4⋊Dic61C2312yes—
54C2×#11758196—24no—
55C2×C22⋊Dic61—24no—
56C2×C4×Dic61C2312yes—
57#14519710—24no—
58#14519708—24no—
59#14519704—28no—
60#14519701—24no—
61#14519698—24no—
62#14519694—24no—
63#14519688—21no—
64#14519688—21no—
65C4×#92418—21no—
66C4×#92418—21no—
67#14519684—21no—
68C2×#41×C61—22no—
69C2×#41×C61C2312yes—
70C22×D4×C61C2311yescharacteristic
71C23×C4×C61D412yes—
72C4⋊Q8×C61D411yescharacteristic
73C4⋊Q8×C61D411yescharacteristic
74C4⋊Q8×C61D411yescharacteristic
75#127×C61—22no—
76#127×C61D411yescharacteristic
77#126×C61—24no—
78#125×C61D412yes—
79#124×C61—24no—
80#124×C61—22no—
81#124×C61D412yes—
82#123×C61—24no—
83#123×C61—24no—
84C22⋊Q8×C61—22no—
85C22⋊Q8×C61—21no—
86C22⋊Q8×C61—21no—
87#121×C61—24no—
88#121×C61—22no—
89#121×C61—21no—
90#121×C61—21no—
91#121×C61—22no—
92#121×C61—22no—
93#120×C61—24no—
94C4×Q8×C61—22no—
95C4×Q8×C61—21no—
96C4×Q8×C61—21no—
97C4×D4×C61—24no—
98C4×D4×C61—22no—
99C4×D4×C61—24no—
100C4×D4×C61—21no—
101C4×D4×C61—21no—
102C4×D4×C61—21no—
103C4×D4×C61—21no—
104C4×D4×C61—21no—
105C4×D4×C61—21no—
106#117×C61—22no—
107#117×C61—22no—
108#117×C61C2312yes—
109C2×C4⋊C4×C61—22no—
110C2×C4⋊C4×C61C2311yescharacteristic
111C2×C22⋊C4×C61D412yes—
112C2×C42×C61C2311yescharacteristic
113C24×C61C2×D412yes—
114#41×C61—24no—
115#41×C61—24no—
116#41×C61—84no—
117C2×Q8×C61—21no—
118C2×Q8×C61—21no—
119C2×Q8×C61C2×D412yes—
120C2×D4×C61—22no—
121C2×D4×C61—24no—
122C2×D4×C61—24no—
123C2×D4×C61—21no—
124C2×D4×C61—21no—
125C2×D4×C61—21no—
126C2×D4×C61—21no—
127C2×D4×C61—22no—
128C2×D4×C61—22no—
129C2×D4×C61C2×D412yes—
130C22×C4×C61—24no—
131C22×C4×C61—22no—
132C22×C4×C61—22no—
133C22×C4×C61—22no—
134C22×C4×C61—44no—
135C22×C4×C61C2×D411yescharacteristic
136C22×C4×C61C2×D412yes—
137C22×C4×C61C2×D411yescharacteristic
138C22×C4×C61C2411yescharacteristic
139C22×Dic61—22no—
140C22×Dic61—22no—
141C2×Dic122—44no—
142C2×Dic122—41no—
143C2×Dic122—41no—
144C4⋊C4×C61—24no—
145C4⋊C4×C61—21no—
146C4⋊C4×C61—21no—
147C4⋊C4×C61—22no—
148C4⋊C4×C61—21no—
149C4⋊C4×C61—22no—
150C4⋊C4×C61—21no—
151C4⋊C4×C61—21no—
152C4⋊C4×C61—21no—
153C4⋊C4×C61C2×D412yes—
154C4⋊C4×C61C2×D412yes—
155C22⋊C4×C61—24no—
156C22⋊C4×C61—24no—
157C22⋊C4×C61—24no—
158C22⋊C4×C61—22no—
159C22⋊C4×C61—22no—
160C22⋊C4×C61—22no—
161C22⋊C4×C61—22no—
162C22⋊C4×C61—22no—
163C42×C61—24no—
164C42×C61—22no—
165C42×C61C2×D411yescharacteristic
166C42×C61C2×D411yescharacteristic
167C42×C61C2×D411yescharacteristic
168C42×C61C2×D411yescharacteristic
169C22⋊Dic61—48no—
170C4⋊Dic61—24no—
171C4⋊Dic61—22no—
172C4⋊Dic61—22no—
173C4⋊Dic61—41no—
174C4⋊Dic61—41no—
175#11758196—48no—
176C4×Dic61—24no—
177#11758194—22no—
178#11758194—22no—
179#11758194—22no—
180#11758194#4114yescomplemented
181C2×#92418—22no—
182C2×#92418—22no—
183C23×C61—22no—
184C23×C61—22no—
185C23×C61—22no—
186C23×C61—22no—
187C23×C61—42no—
188C23×C61—42no—
189C23×C61C22×D411yescharacteristic
190Q8×C61—24no—
191Q8×C61—42no—
192Q8×C61—81no—
193D4×C61—24no—
194D4×C61—42no—
195D4×C61—42no—
196D4×C61—42no—
197D4×C61—42no—
198D4×C61—42no—
199D4×C61—81no—
200D4×C61—84no—
201D4×C61—84no—
202C2×C4×C61—22no—
203C2×C4×C61—22no—
204C2×C4×C61—22no—
205C2×C4×C61—22no—
206C2×C4×C61—22no—
207C2×C4×C61—41no—
208C2×C4×C61—44no—
209C2×C4×C61—44no—
210C2×C4×C61—42no—
211C2×C4×C61—82no—
212C2×C4×C61—84no—
213C2×C4×C61#12011yescharacteristic
214C2×C4×C61#12011yescharacteristic
215C2×C4×C61#12011yescharacteristic
216C2×C4×C61#12011yescharacteristic
217C2×C4×C61C22×D411yescharacteristic
218C2×C4×C61C22×D411yescharacteristic
219C2×C4×C61C2×#4112yes—
220C2×C4×C612−1+412yes—
221C2×Dic61—24no—
222C2×Dic61—42no—
223C2×Dic61—42no—
224C2×Dic61—44no—
225C2×Dic61—42no—
226Dic122—82no—
227Dic122—82no—
228#92418—24no—
229#92418—42no—
230#25694—611nomaximal, Sylow
231C22×C61—21no—
232C22×C61—21no—
233C22×C61—44no—
234C22×C61—42no—
235C22×C61—42no—
236C22×C61—42no—
237C22×C61—42no—
238C22×C61—42no—
239C22×C61—42no—
240C22×C61—82no—
241C22×C61C2×#12011yescharacteristic
242C22×C61#53211yescharacteristic
243C22×C61#53611yescharacteristic
244C4×C61—21no—
245C4×C61—21no—
246C4×C61—21no—
247C4×C61—42no—
248C4×C61—42no—
249C4×C61—42no—
250C4×C61—42no—
251C4×C61#45312yes—
252C4×C61#54712yes—
253C4×C61#54812yes—
254Dic61—44no—
255Dic61—82no—
256#3506D6111yescharacteristic
257#3165—611no—
258#3141—611no—
259#2542—611no—
260#2037—614no—
261#1986—611no—
262#1980—611no—
263#1859—614no—
264C4×#100—611no—
265C2×C61—21no—
266C2×C61—21no—
267C2×C61—82no—
268C2×C61—82no—
269C2×C61#243211yescharacteristic
270C2×C61C2×#45311yescharacteristic
271C2×C61#302211yescharacteristic
272#559—22no—
273#556—21no—
274#555—21no—
275#553—22no—
276#550—21no—
277#547—21no—
278#538—22no—
279#535C2×D6112yes—
280#533C2×D6111yescharacteristic
281#532C2×D6111yescharacteristic
282C2×C4⋊Q8—611no—
283#525—22no—
284C2×C22⋊Q8—1222no—
285#520C2×D6112yes—
286C2×C4×D4C2×D6111yescharacteristic
287#501—1222no—
288#500—1221no—
289#498—1221no—
290#489—1222no—
291#487—1221no—
292#486—1221no—
293#428—614no—
294C2×#104—614no—
295#417—614no—
296C2×#100—612no—
297C4×M16—611no—
298C4×M16—611no—
299#399—1222no—
300#390—612no—
301#384—612no—
302C4×C4⋊C4—611no—
303#343—612no—
304#339—614no—
305C61#2569411yescharacteristic, complemented, Sylow
306C2×#41—22no—
307C2×#41C22×D6112yes—
308C22×Q8—1221no—
309C22×D4C22×D6111yescharacteristic
310C23×C4C22⋊D6112yes—
311C2×M16—612no—
312C2×M16—612no—
313C4⋊Q8—1221no—
314C4⋊Q8—1221no—
315C4⋊Q8—1221no—
316C4⋊Q8—1221no—
317C4⋊Q8C22⋊D6111yescharacteristic
318C4⋊Q8C22⋊D6111yescharacteristic
319C4⋊Q8C22⋊D6111yescharacteristic
320#127—22no—
321#127C22⋊D6111yescharacteristic
322#126—24no—
323#125C22⋊D6112yes—
324#124—24no—
325#124—22no—
326#124C22⋊D6112yes—
327#123—24no—
328#123—24no—
329C22⋊Q8—22no—
330C22⋊Q8—21no—
331C22⋊Q8—21no—
332C22⋊Q8—2448no—
333#121—22no—
334#121—24no—
335#121—21no—
336#121—21no—
337#121—22no—
338#121—22no—
339#120—24no—
340C4×Q8—22no—
341C4×Q8—21no—
342C4×Q8—21no—
343C4×D4—24no—
344C4×D4—22no—
345C4×D4—24no—
346C4×D4—21no—
347C4×D4—21no—
348C4×D4—21no—
349C4×D4—21no—
350C4×D4—21no—
351C4×D4—21no—
352#117—22no—
353#117—22no—
354#117C22×D6112yes—
355C2×C4⋊C4—22no—
356C2×C4⋊C4—612no—
357C2×C4⋊C4—1221no—
358C2×C4⋊C4—1224no—
359C2×C4⋊C4—1221no—
360C2×C4⋊C4C22×D6111yescharacteristic
361C2×C22⋊C4—1224no—
362C2×C22⋊C4C22⋊D6112yes—
363C2×C42—612no—
364C2×C42C22×D6111yescharacteristic
365#107—1224no—
366#106—1224no—
367#104—1228no—
368#103—1224no—
369#102—1224no—
370#100—1224no—
371#97—1221no—
372#97—1221no—
373C4×C8—1221no—
374C4×C8—1221no—
375#95—1221no—
376C24C2×C22⋊D6112yes—
377#41—24no—
378#41—24no—
379#41—84no—
380C2×Q8—21no—
381C2×Q8—21no—
382C2×Q8—2444no—
383C2×Q8—2441no—
384C2×Q8—2441no—
385C2×Q8C2×C22⋊D6112yes—
386C2×D4—22no—
387C2×D4—24no—
388C2×D4—24no—
389C2×D4—21no—
390C2×D4—21no—
391C2×D4—22no—
392C2×D4—22no—
393C2×D4—21no—
394C2×D4—21no—
395C2×D4C2×C22⋊D6112yes—
396C22×C4—24no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—44no—
401C22×C4—1222no—
402C22×C4—1222no—
403C22×C4C2×C22⋊D6111yescharacteristic
404C22×C4C2×C22⋊D6112yes—
405C22×C4C2×C22⋊D6111yescharacteristic
406C22×C4C23×D6111yescharacteristic
407M16—614no—
408M16—1222no—
409M16—1222no—
410M16—1222no—
411C2×C8—1222no—
412C2×C8—1222no—
413C4⋊C4—24no—
414C4⋊C4—21no—
415C4⋊C4—21no—
416C4⋊C4—21no—
417C4⋊C4—21no—
418C4⋊C4—22no—
419C4⋊C4—21no—
420C4⋊C4—22no—
421C4⋊C4—21no—
422C4⋊C4—1224no—
423C4⋊C4—1222no—
424C4⋊C4—1222no—
425C4⋊C4—2448no—
426C4⋊C4—2441no—
427C4⋊C4—2441no—
428C4⋊C4C2×C22⋊D6112yes—
429C4⋊C4C2×C22⋊D6112yes—
430C22⋊C4—24no—
431C22⋊C4—24no—
432C22⋊C4—24no—
433C22⋊C4—22no—
434C22⋊C4—22no—
435C22⋊C4—22no—
436C22⋊C4—22no—
437C22⋊C4—22no—
438C22⋊C4—2448no—
439C42—24no—
440C42—22no—
441C42—1224no—
442C42C2×C22⋊D6111yescharacteristic
443C42C2×C22⋊D6111yescharacteristic
444C42C2×C22⋊D6111yescharacteristic
445C42C2×C22⋊D6111yescharacteristic
446C23—22no—
447C23—22no—
448C23—22no—
449C23—22no—
450C23—42no—
451C23—42no—
452C23C22×C22⋊D6111yescharacteristic
453Q8—24no—
454Q8—42no—
455Q8—81no—
456Q8—4882no—
457Q8—4882no—
458D4—24no—
459D4—42no—
460D4—42no—
461D4—42no—
462D4—42no—
463D4—42no—
464D4—81no—
465D4—84no—
466D4—84no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—41no—
473C2×C4—44no—
474C2×C4—44no—
475C2×C4—42no—
476C2×C4—82no—
477C2×C4—84no—
478C2×C4—1224no—
479C2×C4—2444no—
480C2×C4—2442no—
481C2×C4—2442no—
482C2×C4—2442no—
483C2×C4#1451974611yescharacteristic
484C2×C4#1451974611yescharacteristic
485C2×C4#1451974611yescharacteristic
486C2×C4#1451974611yescharacteristic
487C2×C4C22×C22⋊D6111yescharacteristic
488C2×C4C22×C22⋊D6111yescharacteristic
489C2×C4#41×D6112yes—
490C2×C4#1451982412yes—
491C8—1224no—
492C8—2442no—
493C22—21no—
494C22—21no—
495C22—44no—
496C22—42no—
497C22—42no—
498C22—42no—
499C22—42no—
500C22—42no—
501C22—42no—
502C22—82no—
503C22C2×#1451974611yescharacteristic
504C22#1684030811yescharacteristic
505C22#1684033311yescharacteristic
506C4—21no—
507C4—21no—
508C4—21no—
509C4—42no—
510C4—42no—
511C4—42no—
512C4—42no—
513C4—4882no—
514C4—2444no—
515C4#1683993012yes—
516C4#1684041112yes—
517C4#1684041912yes—
518C2—21no—
519C2—21no—
520C2—82no—
521C2—82no—
522C2#2144944311yescharacteristic
523C2C2×#1683993011yescharacteristic
524C2#2145519811yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
Fitting, prime radicalC61×#3506C2
socleC22×C61C2×#120
hypercenter#3506D61
hypocenterC61#25694
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3506

Complements

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

NH ≅ G/Ntype
1. #21456366C2semidirect
2. #21456212C2semidirect
3. #21450772C2semidirect
4. #21447026C2semidirect
5. #21446764C2semidirect
6. #21446739C2semidirect
7. #21446045C2semidirect
8. C4×#14519694C2semidirect
180. #11758194#41semidirect
305. C61#25694semidirect

Maximal subgroups

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

subgroupindex
#214563662
#214562122
#214507722
#214470262
#214467642
#214467392
#214460452
C4×#145196942
C61×#35062
#2569461