Finite Group Databasebeta

← #29655247

Subgroups — #29655247

Order 15616.

848 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#21457037C211yesmaximal, characteristic
2#21456960C211yesmaximal, characteristic, complemented
3#21451940C211yesmaximal, characteristic, complemented
4#21444827C214yesmaximal, complemented
5#21444816C214yesmaximal
6#21444331C211yesmaximal, characteristic, complemented
7#21444305C211yesmaximal, characteristic
8#21444175C211yesmaximal, characteristic, complemented
9C61×#3563C211yesmaximal, characteristic
10C2×#14519777C2211yescharacteristic
11#16840172—21no—
12#16840158—21no—
13#16840121—21no—
14#16840112—21no—
15#16840090C2214yes—
16#16840090C2212yes—
17#16840073C2212yes—
18#16840057—21no—
19#16840052—21no—
20#16840047—22no—
21#16839819C2211yescharacteristic
22#16839815C2211yescharacteristic
23#16839692C2214yes—
24#16839682—22no—
25#16839631—21no—
26#16839627—21no—
27#16839608C2211yescharacteristic
28#16839499C2214yes—
29#16839499C2212yes—
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×#11758196—24no—
55C2×C4×Dic61—21no—
56#14519710—22no—
57#14519708—24no—
58#14519708—22no—
59C4⋊#92418—21no—
60C4⋊#92418—21no—
61#14519701—24no—
62#14519701—22no—
63#14519698—22no—
64C22⋊#92418C2312yes—
65C22⋊#92418C2312yes—
66#14519688—21no—
67C4×#92418—21no—
68#14519684—21no—
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—41no—
146C2×Dic122—41no—
147C2×Dic122—82no—
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—
198#11758196—84no—
199C4×Dic61—24no—
200C2×#92418—22no—
201C2×#92418—22no—
202C23×C61—21no—
203C23×C61C22×D411yescharacteristic
204Q8×C61—41no—
205Q8×C61—42no—
206Q8×C61—42no—
207Q8×C61—82no—
208D4×C61—41no—
209D4×C61—42no—
210C2×C4×C61—22no—
211C2×C4×C61—22no—
212C2×C4×C61—22no—
213C2×C4×C61—21no—
214C2×C4×C61—21no—
215C2×C4×C61—22no—
216C2×C4×C61—21no—
217C2×C4×C61—22no—
218C2×C4×C61—22no—
219C2×C4×C61—24no—
220C2×C4×C61—41no—
221C2×C4×C61—42no—
222C2×C4×C61—42no—
223C2×C4×C61—82no—
224C2×C4×C61#12011yescharacteristic
225C2×C4×C61#12011yescharacteristic
226C2×C4×C61#12011yescharacteristic
227C2×C4×C61#12011yescharacteristic
228C2×C4×C61C22×D411yescharacteristic
229C2×C4×C61C22×D411yescharacteristic
230C2×Dic61—24no—
231C2×Dic61—42no—
232C2×Dic61—42no—
233C2×Dic61—42no—
234C2×Dic61—44no—
235Dic122—82no—
236Dic122—82no—
237#92418—42no—
238#92418—42no—
239#18442—611nomaximal, Sylow
240C22×C61—21no—
241C22×C61—21no—
242C22×C61—41no—
243C22×C61—41no—
244C22×C61—41no—
245C22×C61C2×#12011yescharacteristic
246C22×C61#57712yes—
247C4×C61—21no—
248C4×C61—21no—
249C4×C61—21no—
250C4×C61—21no—
251C4×C61—21no—
252C4×C61—42no—
253C4×C61—41no—
254C4×C61—41no—
255C4×C61—42no—
256C4×C61—42no—
257C4×C61—42no—
258C4×C61—41no—
259C4×C61#45612yes—
260Dic61—44no—
261Dic61—82no—
262#3563D6111yescharacteristic
263#3238—611no—
264#3229—611no—
265#2660—611no—
266#1611—614no—
267#1609—614no—
268#1506—611no—
269#1500—611no—
270#1470—611no—
271C2×C61—22no—
272C2×C61—81no—
273C2×C61#301011yescharacteristic
274C2×C61C2×#45611yescharacteristic
275C2×C61#302811yescharacteristic
276#564—21no—
277#563—21no—
278#557—21no—
279#554—21no—
280#552—21no—
281#552—21no—
282#552—22no—
283#544—21no—
284#544C2×D6112yes—
285#541—21no—
286#541—22no—
287#539C2×D6112yes—
288#533C2×D6111yescharacteristic
289C2×C4⋊Q8—611no—
290#529C2×D6111yescharacteristic
291C2×#125C2×D6111yescharacteristic
292#501—1221no—
293#498—1221no—
294#489—1221no—
295#487—1221no—
296#483—614no—
297#483—612no—
298#480—612no—
299#477—1221no—
300#476—1221no—
301#475—1222no—
302#432—611no—
303#431—611no—
304#398—614no—
305#395—1222no—
306#384—1221no—
307#382—1221no—
308#376—611no—
309#328—614no—
310#328—612no—
311#327—612no—
312#324—614no—
313C61#1844211yescharacteristic, complemented, Sylow
314C22×Q8—1221no—
315C4⋊Q8—21no—
316C4⋊Q8—22no—
317C4⋊Q8—22no—
318C4⋊Q8—1221no—
319C4⋊Q8—1221no—
320C4⋊Q8—1221no—
321C4⋊Q8—1221no—
322C4⋊Q8C22⋊D6111yescharacteristic
323C4⋊Q8C22⋊D6111yescharacteristic
324#126—22no—
325#126—22no—
326#126—22no—
327#126—22no—
328#126—21no—
329#125—22no—
330#125—22no—
331#125—22no—
332#125—22no—
333#125—21no—
334#125—22no—
335#125—22no—
336#125—22no—
337#125—22no—
338#125—21no—
339#125—21no—
340#125—21no—
341#125—21no—
342#125—21no—
343#125C22⋊D6111yescharacteristic
344#124C22⋊D6111yescharacteristic
345#123—22no—
346#123—21no—
347#123—22no—
348C22⋊Q8—21no—
349C22⋊Q8—21no—
350C22⋊Q8—22no—
351C22⋊Q8—22no—
352C22⋊Q8—22no—
353C22⋊Q8—21no—
354C22⋊Q8—22no—
355C22⋊Q8C22×D6111yescharacteristic
356C22⋊Q8C22×D6112yes—
357#121C22×D6111yescharacteristic
358C4×Q8—21no—
359C4×Q8—22no—
360C4×Q8—21no—
361C4×Q8—21no—
362C4×Q8C22⋊D6111yescharacteristic
363C4×Q8C22⋊D6112yes—
364C4×D4—21no—
365C4×D4—22no—
366C4×D4C22⋊D6111yescharacteristic
367#117—22no—
368#117—21no—
369#117—21no—
370#117—22no—
371#117—21no—
372C2×C4⋊C4—21no—
373C2×C4⋊C4—612no—
374C2×C4⋊C4—1224no—
375C2×C4⋊C4—1221no—
376C2×C4⋊C4C22⋊D6111yescharacteristic
377C2×C4⋊C4C22⋊D6112yes—
378C2×C4⋊C4C22⋊D6111yescharacteristic
379C2×C4⋊C4C22×D6112yes—
380C2×C42—1221no—
381C2×C42C22×D6111yescharacteristic
382#107—1222no—
383#106—1224no—
384#106—1222no—
385C4⋊C8—1221no—
386C4⋊C8—1221no—
387#103—1224no—
388#103—1222no—
389#102—1222no—
390C22⋊C8—612no—
391C22⋊C8—612no—
392#97—1221no—
393C4×C8—1221no—
394#95—612no—
395#95—1221no—
396#95—1224no—
397C2×Q8—22no—
398C2×Q8—21no—
399C2×Q8—4882no—
400C2×Q8—2441no—
401C2×Q8—2441no—
402C2×Q8C2×C22⋊D6112yes—
403C2×Q8C2×C22⋊D6111yescharacteristic
404C2×D4—21no—
405C2×D4C2×C22⋊D6111yescharacteristic
406C22×C4—22no—
407C22×C4—21no—
408C22×C4—1222no—
409C22×C4—1222no—
410C22×C4C2×C22⋊D6112yes—
411C22×C4C2×C22⋊D6111yescharacteristic
412C22×C4C2×C22⋊D6111yescharacteristic
413C22×C4C2×C22⋊D6112yes—
414C22×C4C23×D6111yescharacteristic
415C2×C8—1222no—
416C2×C8—1222no—
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—22no—
427C4⋊C4—21no—
428C4⋊C4—22no—
429C4⋊C4—21no—
430C4⋊C4—21no—
431C4⋊C4—21no—
432C4⋊C4—22no—
433C4⋊C4—21no—
434C4⋊C4—22no—
435C4⋊C4—21no—
436C4⋊C4—21no—
437C4⋊C4—21no—
438C4⋊C4—21no—
439C4⋊C4—21no—
440C4⋊C4—21no—
441C4⋊C4—22no—
442C4⋊C4—1222no—
443C4⋊C4—1222no—
444C4⋊C4—4884no—
445C4⋊C4—2441no—
446C4⋊C4—2441no—
447C4⋊C4C2×C22⋊D6111yescharacteristic
448C4⋊C4C2×C22⋊D6112yes—
449C4⋊C4C2×C22⋊D6111yescharacteristic
450C22⋊C4—22no—
451C22⋊C4—22no—
452C22⋊C4—22no—
453C22⋊C4—22no—
454C22⋊C4—21no—
455C22⋊C4—21no—
456C22⋊C4—21no—
457C22⋊C4—21no—
458C22⋊C4—21no—
459C42—22no—
460C42—22no—
461C42—22no—
462C42—22no—
463C42—21no—
464C42—1224no—
465C42C2×C22⋊D6111yescharacteristic
466C42C2×C22⋊D6111yescharacteristic
467C42C2×C22⋊D6111yescharacteristic
468C42C2×C22⋊D6111yescharacteristic
469C23—21no—
470C23C22×C22⋊D6111yescharacteristic
471Q8—41no—
472Q8—42no—
473Q8—42no—
474Q8—82no—
475Q8—4882no—
476Q8—4882no—
477D4—41no—
478D4—42no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—22no—
485C2×C4—21no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—24no—
489C2×C4—41no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—82no—
493C2×C4—1224no—
494C2×C4—2442no—
495C2×C4—2444no—
496C2×C4—2442no—
497C2×C4—2442no—
498C2×C4#1451974611yescharacteristic
499C2×C4#1451974611yescharacteristic
500C2×C4#1451974611yescharacteristic
501C2×C4#1451974611yescharacteristic
502C2×C4C22×C22⋊D6111yescharacteristic
503C2×C4C22×C22⋊D6111yescharacteristic
504C8—2442no—
505C8—2442no—
506C22—21no—
507C22—21no—
508C22—41no—
509C22—41no—
510C22—41no—
511C22C2×#1451974611yescharacteristic
512C22#1684057012yes—
513C4—21no—
514C4—21no—
515C4—21no—
516C4—21no—
517C4—21no—
518C4—42no—
519C4—41no—
520C4—41no—
521C4—42no—
522C4—42no—
523C4—42no—
524C4—41no—
525C4—4882no—
526C4—2444no—
527C4#1683994512yes—
528C2—22no—
529C2—81no—
530C2#2145510511yescharacteristic
531C2C2×#1683994511yescharacteristic
532C2#2145522511yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#3563C2
socleC22×C61C2×#120
prime radical#121×C61C23
hypercenter#3563D61
hypocenterC61#18442
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. #21456960C2semidirect
3. #21451940C2semidirect
4. #21444827C2semidirect
6. #21444331C2semidirect
8. #21444175C2semidirect
313. C61#18442semidirect

Maximal subgroups

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

subgroupindex
#214570372
#214569602
#214519402
#214448272
#214448162
#214443312
#214443052
#214441752
C61×#35632
#1844261