Finite Group Databasebeta

← #29655256

Subgroups — #29655256

Order 15616.

832 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21457046C211yesmaximal, characteristic
2#21456967C211yesmaximal, characteristic, complemented
3#21451770C211yesmaximal, characteristic, complemented
4#21444861C214yesmaximal, complemented
5#21444850C214yesmaximal
6#21444331C211yesmaximal, characteristic, complemented
7#21444305C211yesmaximal, characteristic
8#21444175C211yesmaximal, characteristic, complemented
9C61×#3563C211yesmaximal, characteristic
10C2×#14519777C2211yescharacteristic
11#16840172—21no—
12#16840165—21no—
13#16840121—21no—
14#16840112—21no—
15#16840095C2214yes—
16#16840095C2212yes—
17#16840078C2212yes—
18#16840057—22no—
19#16840047—21no—
20#16840042—21no—
21#16839819C2211yescharacteristic
22#16839815C2211yescharacteristic
23#16839701—22no—
24#16839701C2214yes—
25#16839667—21no—
26#16839667—21no—
27#16839608C2211yescharacteristic
28#16839499C2212yes—
29#16839499C2214yes—
30#16839496C2212yes—
31#16839488C2214yes—
32C61×#564—21no—
33C61×#563—21no—
34C61×#557—21no—
35C61×#554—21no—
36C61×#552—21no—
37C61×#552—22no—
38C61×#552—21no—
39C61×#544—21no—
40C61×#544C2212yes—
41C61×#541—21no—
42C61×#541—22no—
43C61×#539C2212yes—
44C61×#533C2211yescharacteristic
45C61×#529C2211yescharacteristic
46C2×#125×C61C2211yescharacteristic
47C22×Dic122—21no—
48#14519777—21no—
49#14519777—21no—
50#14519777—21no—
51#14519777—21no—
52C2×C4⋊Dic61—21no—
53C2×C4⋊Dic61C2312yes—
54C2×C4×Dic61—21no—
55#14519710—22no—
56#14519710—24no—
57#14519708—22no—
58C4⋊#92418—21no—
59C4⋊#92418—21no—
60#14519701—22no—
61#14519701—24no—
62#14519698—22no—
63C22⋊#92418C2312yes—
64C22⋊#92418C2312yes—
65#14519688—21no—
66C4×#92418—21no—
67#14519684—21no—
68#14519684—24no—
69#14519684—24no—
70#14519684C2312yes—
71C4⋊Q8×C61—21no—
72C4⋊Q8×C61—22no—
73C4⋊Q8×C61—22no—
74C4⋊Q8×C61D411yescharacteristic
75C4⋊Q8×C61D411yescharacteristic
76#126×C61—22no—
77#126×C61—22no—
78#126×C61—22no—
79#126×C61—22no—
80#126×C61—21no—
81#125×C61—22no—
82#125×C61—22no—
83#125×C61—22no—
84#125×C61—22no—
85#125×C61—21no—
86#125×C61—22no—
87#125×C61—22no—
88#125×C61—22no—
89#125×C61—22no—
90#125×C61—21no—
91#125×C61—21no—
92#125×C61—21no—
93#125×C61—21no—
94#125×C61—21no—
95#125×C61D411yescharacteristic
96#124×C61D411yescharacteristic
97#123×C61—22no—
98#123×C61—22no—
99#123×C61—21no—
100C22⋊Q8×C61—21no—
101C22⋊Q8×C61—21no—
102C22⋊Q8×C61—22no—
103C22⋊Q8×C61—22no—
104C22⋊Q8×C61—22no—
105C22⋊Q8×C61—21no—
106C22⋊Q8×C61—22no—
107C22⋊Q8×C61C2312yes—
108C22⋊Q8×C61C2311yescharacteristic
109#121×C61C2311yescharacteristic
110C4×Q8×C61—21no—
111C4×Q8×C61—22no—
112C4×Q8×C61—21no—
113C4×Q8×C61—21no—
114C4×Q8×C61D412yes—
115C4×Q8×C61D411yescharacteristic
116C4×D4×C61—22no—
117C4×D4×C61—21no—
118C4×D4×C61D411yescharacteristic
119#117×C61—22no—
120#117×C61—21no—
121#117×C61—21no—
122#117×C61—22no—
123#117×C61—21no—
124C2×C4⋊C4×C61—21no—
125C2×C4⋊C4×C61D412yes—
126C2×C4⋊C4×C61D411yescharacteristic
127C2×C4⋊C4×C61D411yescharacteristic
128C2×C4⋊C4×C61C2312yes—
129C2×C42×C61C2311yescharacteristic
130C2×Q8×C61—22no—
131C2×Q8×C61—21no—
132C2×Q8×C61C2×D412yes—
133C2×Q8×C61C2×D411yescharacteristic
134C2×D4×C61—21no—
135C2×D4×C61C2×D411yescharacteristic
136C22×C4×C61—22no—
137C22×C4×C61—21no—
138C22×C4×C61C2×D412yes—
139C22×C4×C61C2×D411yescharacteristic
140C22×C4×C61C2×D411yescharacteristic
141C22×C4×C61C2×D412yes—
142C22×C4×C61C2411yescharacteristic
143C22×Dic61—22no—
144C22×Dic61—22no—
145C2×Dic122—44no—
146C2×Dic122—41no—
147C2×Dic122—41no—
148C4⋊C4×C61—22no—
149C4⋊C4×C61—22no—
150C4⋊C4×C61—22no—
151C4⋊C4×C61—22no—
152C4⋊C4×C61—22no—
153C4⋊C4×C61—22no—
154C4⋊C4×C61—22no—
155C4⋊C4×C61—22no—
156C4⋊C4×C61—21no—
157C4⋊C4×C61—21no—
158C4⋊C4×C61—21no—
159C4⋊C4×C61—22no—
160C4⋊C4×C61—21no—
161C4⋊C4×C61—22no—
162C4⋊C4×C61—21no—
163C4⋊C4×C61—21no—
164C4⋊C4×C61—21no—
165C4⋊C4×C61—21no—
166C4⋊C4×C61—21no—
167C4⋊C4×C61—21no—
168C4⋊C4×C61—22no—
169C4⋊C4×C61—22no—
170C4⋊C4×C61—22no—
171C4⋊C4×C61—21no—
172C4⋊C4×C61—22no—
173C4⋊C4×C61C2×D411yescharacteristic
174C4⋊C4×C61C2×D411yescharacteristic
175C4⋊C4×C61C2×D412yes—
176C22⋊C4×C61—22no—
177C22⋊C4×C61—22no—
178C22⋊C4×C61—22no—
179C22⋊C4×C61—22no—
180C22⋊C4×C61—21no—
181C22⋊C4×C61—21no—
182C22⋊C4×C61—21no—
183C22⋊C4×C61—21no—
184C22⋊C4×C61—21no—
185C42×C61—22no—
186C42×C61—22no—
187C42×C61—22no—
188C42×C61—22no—
189C42×C61—21no—
190C42×C61C2×D411yescharacteristic
191C42×C61C2×D411yescharacteristic
192C42×C61C2×D411yescharacteristic
193C42×C61C2×D411yescharacteristic
194C4⋊Dic61—22no—
195C4⋊Dic61—22no—
196C4⋊Dic61—41no—
197C4⋊Dic61—41no—
198C4×Dic61—42no—
199C2×#92418—22no—
200C2×#92418—22no—
201C23×C61—21no—
202C23×C61C22×D411yescharacteristic
203Q8×C61—41no—
204Q8×C61—42no—
205Q8×C61—42no—
206Q8×C61—82no—
207D4×C61—41no—
208D4×C61—42no—
209C2×C4×C61—22no—
210C2×C4×C61—22no—
211C2×C4×C61—22no—
212C2×C4×C61—21no—
213C2×C4×C61—21no—
214C2×C4×C61—22no—
215C2×C4×C61—21no—
216C2×C4×C61—22no—
217C2×C4×C61—22no—
218C2×C4×C61—24no—
219C2×C4×C61—41no—
220C2×C4×C61—42no—
221C2×C4×C61—42no—
222C2×C4×C61—82no—
223C2×C4×C61#12011yescharacteristic
224C2×C4×C61#12011yescharacteristic
225C2×C4×C61#12011yescharacteristic
226C2×C4×C61#12011yescharacteristic
227C2×C4×C61C22×D411yescharacteristic
228C2×C4×C61C22×D411yescharacteristic
229C2×Dic61—42no—
230C2×Dic61—44no—
231C2×Dic61—42no—
232C2×Dic61—42no—
233C2×Dic61—42no—
234Dic122—82no—
235Dic122—82no—
236#92418—42no—
237#92418—42no—
238#18443—611nomaximal, Sylow
239C22×C61—21no—
240C22×C61—21no—
241C22×C61—41no—
242C22×C61—41no—
243C22×C61—41no—
244C22×C61C2×#12011yescharacteristic
245C22×C61#57812yes—
246C4×C61—21no—
247C4×C61—21no—
248C4×C61—21no—
249C4×C61—21no—
250C4×C61—21no—
251C4×C61—42no—
252C4×C61—41no—
253C4×C61—41no—
254C4×C61—42no—
255C4×C61—42no—
256C4×C61—42no—
257C4×C61—41no—
258C4×C61#45412yes—
259Dic61—82no—
260Dic61—82no—
261#3563D6111yescharacteristic
262#3239—611no—
263#3230—611no—
264#2642—611no—
265#1617—614no—
266#1615—614no—
267#1506—611no—
268#1500—611no—
269#1470—611no—
270C2×C61—22no—
271C2×C61—81no—
272C2×C61#301111yescharacteristic
273C2×C61C2×#45411yescharacteristic
274C2×C61#302811yescharacteristic
275#564—21no—
276#563—21no—
277#557—21no—
278#554—21no—
279#552—21no—
280#552—21no—
281#552—22no—
282#544—21no—
283#544C2×D6112yes—
284#541—21no—
285#541—22no—
286#539C2×D6112yes—
287#533C2×D6111yescharacteristic
288C2×C4⋊Q8—611no—
289#529C2×D6111yescharacteristic
290C2×#125C2×D6111yescharacteristic
291#501—1221no—
292#500—1221no—
293#489—1221no—
294#487—1221no—
295#484—614no—
296#484—612no—
297#481—612no—
298#477—1222no—
299#475—1221no—
300#474—1221no—
301#432—611no—
302#431—611no—
303#400—614no—
304#400—1222no—
305#391—1221no—
306#391—1221no—
307#376—611no—
308#328—614no—
309#328—612no—
310#327—612no—
311#324—614no—
312C61#1844311yescharacteristic, complemented, Sylow
313C22×Q8—1221no—
314C4⋊Q8—21no—
315C4⋊Q8—22no—
316C4⋊Q8—22no—
317C4⋊Q8—1221no—
318C4⋊Q8—1221no—
319C4⋊Q8—1221no—
320C4⋊Q8—1221no—
321C4⋊Q8C22⋊D6111yescharacteristic
322C4⋊Q8C22⋊D6111yescharacteristic
323#126—22no—
324#126—22no—
325#126—22no—
326#126—22no—
327#126—21no—
328#125—22no—
329#125—22no—
330#125—22no—
331#125—22no—
332#125—21no—
333#125—22no—
334#125—22no—
335#125—22no—
336#125—22no—
337#125—21no—
338#125—21no—
339#125—21no—
340#125—21no—
341#125—21no—
342#125C22⋊D6111yescharacteristic
343#124C22⋊D6111yescharacteristic
344#123—22no—
345#123—21no—
346#123—22no—
347C22⋊Q8—21no—
348C22⋊Q8—21no—
349C22⋊Q8—22no—
350C22⋊Q8—22no—
351C22⋊Q8—22no—
352C22⋊Q8—21no—
353C22⋊Q8—22no—
354C22⋊Q8C22×D6111yescharacteristic
355C22⋊Q8C22×D6112yes—
356#121C22×D6111yescharacteristic
357C4×Q8—21no—
358C4×Q8—22no—
359C4×Q8—21no—
360C4×Q8—21no—
361C4×Q8C22⋊D6111yescharacteristic
362C4×Q8C22⋊D6112yes—
363C4×D4—21no—
364C4×D4—22no—
365C4×D4C22⋊D6111yescharacteristic
366#117—22no—
367#117—21no—
368#117—21no—
369#117—22no—
370#117—21no—
371C2×C4⋊C4—21no—
372C2×C4⋊C4—612no—
373C2×C4⋊C4—1221no—
374C2×C4⋊C4C22⋊D6111yescharacteristic
375C2×C4⋊C4C22⋊D6112yes—
376C2×C4⋊C4C22⋊D6111yescharacteristic
377C2×C4⋊C4C22×D6112yes—
378C2×C42—1221no—
379C2×C42C22×D6111yescharacteristic
380#107—1222no—
381#107—1224no—
382#106—1222no—
383C4⋊C8—1221no—
384C4⋊C8—1221no—
385#103—1222no—
386#103—1224no—
387#102—1222no—
388C22⋊C8—612no—
389C22⋊C8—612no—
390#97—1221no—
391C4×C8—1221no—
392#95—612no—
393#95—1221no—
394#95—1224no—
395#95—1224no—
396C2×Q8—22no—
397C2×Q8—21no—
398C2×Q8—2444no—
399C2×Q8—2441no—
400C2×Q8—2441no—
401C2×Q8C2×C22⋊D6112yes—
402C2×Q8C2×C22⋊D6111yescharacteristic
403C2×D4—21no—
404C2×D4C2×C22⋊D6111yescharacteristic
405C22×C4—22no—
406C22×C4—21no—
407C22×C4—1222no—
408C22×C4—1222no—
409C22×C4C2×C22⋊D6112yes—
410C22×C4C2×C22⋊D6111yescharacteristic
411C22×C4C2×C22⋊D6111yescharacteristic
412C22×C4C2×C22⋊D6112yes—
413C22×C4C23×D6111yescharacteristic
414C2×C8—1222no—
415C2×C8—1222no—
416C4⋊C4—22no—
417C4⋊C4—22no—
418C4⋊C4—22no—
419C4⋊C4—22no—
420C4⋊C4—22no—
421C4⋊C4—22no—
422C4⋊C4—22no—
423C4⋊C4—22no—
424C4⋊C4—22no—
425C4⋊C4—22no—
426C4⋊C4—21no—
427C4⋊C4—22no—
428C4⋊C4—21no—
429C4⋊C4—21no—
430C4⋊C4—21no—
431C4⋊C4—22no—
432C4⋊C4—21no—
433C4⋊C4—22no—
434C4⋊C4—21no—
435C4⋊C4—21no—
436C4⋊C4—21no—
437C4⋊C4—21no—
438C4⋊C4—21no—
439C4⋊C4—21no—
440C4⋊C4—22no—
441C4⋊C4—1222no—
442C4⋊C4—1222no—
443C4⋊C4—2441no—
444C4⋊C4—2441no—
445C4⋊C4C2×C22⋊D6111yescharacteristic
446C4⋊C4C2×C22⋊D6111yescharacteristic
447C4⋊C4C2×C22⋊D6112yes—
448C22⋊C4—22no—
449C22⋊C4—22no—
450C22⋊C4—22no—
451C22⋊C4—22no—
452C22⋊C4—21no—
453C22⋊C4—21no—
454C22⋊C4—21no—
455C22⋊C4—21no—
456C22⋊C4—21no—
457C42—22no—
458C42—22no—
459C42—22no—
460C42—22no—
461C42—21no—
462C42—2442no—
463C42C2×C22⋊D6111yescharacteristic
464C42C2×C22⋊D6111yescharacteristic
465C42C2×C22⋊D6111yescharacteristic
466C42C2×C22⋊D6111yescharacteristic
467C23—21no—
468C23C22×C22⋊D6111yescharacteristic
469Q8—41no—
470Q8—42no—
471Q8—42no—
472Q8—82no—
473Q8—4882no—
474Q8—4882no—
475D4—41no—
476D4—42no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—22no—
483C2×C4—21no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—24no—
487C2×C4—41no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—82no—
491C2×C4—2442no—
492C2×C4—2442no—
493C2×C4—2444no—
494C2×C4—2442no—
495C2×C4—2442no—
496C2×C4#1451974611yescharacteristic
497C2×C4#1451974611yescharacteristic
498C2×C4#1451974611yescharacteristic
499C2×C4#1451974611yescharacteristic
500C2×C4C22×C22⋊D6111yescharacteristic
501C2×C4C22×C22⋊D6111yescharacteristic
502C8—2442no—
503C8—2442no—
504C22—21no—
505C22—21no—
506C22—41no—
507C22—41no—
508C22—41no—
509C22C2×#1451974611yescharacteristic
510C22#1684057712yes—
511C4—21no—
512C4—21no—
513C4—21no—
514C4—21no—
515C4—21no—
516C4—42no—
517C4—41no—
518C4—41no—
519C4—42no—
520C4—42no—
521C4—42no—
522C4—41no—
523C4—4882no—
524C4—4882no—
525C4#1683993512yes—
526C2—22no—
527C2—81no—
528C2#2145511411yescharacteristic
529C2C2×#1683993511yescharacteristic
530C2#2145522511yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#3563C2
socleC22×C61C2×#120
prime radical#121×C61C23
hypercenter#3563D61
hypocenterC61#18443
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3563

Complements

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

NH ≅ G/Ntype
2. #21456967C2semidirect
3. #21451770C2semidirect
4. #21444861C2semidirect
6. #21444331C2semidirect
8. #21444175C2semidirect
312. C61#18443semidirect

Maximal subgroups

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

subgroupindex
#214570462
#214569672
#214517702
#214448612
#214448502
#214443312
#214443052
#214441752
C61×#35632
#1844361