Finite Group Databasebeta

← #29657901

Subgroups — #29657901

Order 15616.

924 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#21457210C211yesmaximal, characteristic, complemented
2#21457206C211yesmaximal, characteristic, complemented
3#21456141C211yesmaximal, characteristic, complemented
4#21445033C212yesmaximal, complemented
5#21445026C212yesmaximal, complemented
6#21445019C212yesmaximal, complemented
7#21445012C212yesmaximal, complemented
8#21444554C211yesmaximal, characteristic, complemented
9#21444274C211yesmaximal, characteristic, complemented
10#21444270C211yesmaximal, characteristic, complemented
11C61×#3548C211yesmaximal, characteristic
12#16840319C2211yescharacteristic
13#16840116—22no—
14#16840112C2211yescharacteristic
15#16840112C2211yescharacteristic
16#16840108C2211yescharacteristic
17#16840108C2211yescharacteristic
18#16840054—21no—
19#16840047C2212yes—
20#16840044—21no—
21#16840042C2212yes—
22#16840018—22no—
23#16840006—22no—
24#16839972C2212yes—
25#16839969—21no—
26#16839967C2212yes—
27#16839959—21no—
28#16839786C2211yescharacteristic
29#16839786C2211yescharacteristic
30#16839758—22no—
31C2×C4×#92418C2211yescharacteristic
32#16839511C2212yes—
33#16839506C2212yes—
34Q8⋊#92418C2212yes—
35Q8⋊#92418C2212yes—
36D4⋊#92418C2212yes—
37D4⋊#92418C2212yes—
38C61×#562—22no—
39C61×#554—21no—
40C61×#554C2212yes—
41C61×#550—21no—
42C61×#550C2212yes—
43C61×#547—21no—
44C61×#547—21no—
45C61×#540—22no—
46C61×#537—22no—
47C61×#537—22no—
48C61×#533C2211yescharacteristic
49C61×#532C2211yescharacteristic
50C61×#532C2211yescharacteristic
51C2×#11758202—22no—
52C2×#11758200—22no—
53C2×#11758194—21no—
54C2×#11758194—21no—
55C22×#92418—21no—
56#14519777C2311yescharacteristic
57#14519777C2311yescharacteristic
58#14519769—21no—
59#14519756—21no—
60#14519756—21no—
61C4×Dic122—21no—
62C4×Dic122—21no—
63#14519736—21no—
64#14519710—22no—
65#14519708—22no—
66C4⋊#92418—21no—
67C4⋊#92418—21no—
68C4⋊#92418C2312yes—
69C4⋊#92418C2312yes—
70#14519701—22no—
71#14519701—22no—
72#14519701—22no—
73#14519700—22no—
74#14519700—22no—
75#14519698—22no—
76#14519698—22no—
77#14519698—22no—
78C4×#92418—21no—
79C4×#92418C2311yescharacteristic
80C4×#92418C2311yescharacteristic
81C2×#41×C61—22no—
82C2×#41×C61—22no—
83C2×#41×C61—21no—
84C2×#41×C61D412yes—
85C4⋊Q8×C61D411yescharacteristic
86C4⋊Q8×C61D411yescharacteristic
87C4⋊Q8×C61D411yescharacteristic
88C4⋊Q8×C61C2311yescharacteristic
89#127×C61—22no—
90#127×C61D411yescharacteristic
91#127×C61C2311yescharacteristic
92#125×C61—22no—
93#125×C61—21no—
94#124×C61—21no—
95#124×C61D412yes—
96#123×C61—24no—
97#123×C61—24no—
98C22⋊Q8×C61—22no—
99C22⋊Q8×C61—24no—
100C22⋊Q8×C61—22no—
101C22⋊Q8×C61—24no—
102C22⋊Q8×C61—21no—
103C22⋊Q8×C61D412yes—
104#121×C61—24no—
105#121×C61—24no—
106#121×C61—22no—
107#121×C61—24no—
108#121×C61—24no—
109#121×C61—21no—
110#121×C61—21no—
111#121×C61—21no—
112#121×C61—22no—
113#121×C61D412yes—
114C4×Q8×C61—21no—
115C4×Q8×C61C2312yes—
116C4×D4×C61—22no—
117C4×D4×C61—22no—
118C4×D4×C61—21no—
119C4×D4×C61—21no—
120C4×D4×C61—21no—
121C4×D4×C61C2312yes—
122#117×C61—21no—
123C2×C4⋊C4×C61—22no—
124C2×C4⋊C4×C61—22no—
125C2×C42×C61C2311yescharacteristic
126#41×C61—44no—
127#41×C61—81no—
128#41×C61—81no—
129#41×C61—82no—
130#41×C61—82no—
131#41×C61—84no—
132C2×Q8×C61—21no—
133C2×Q8×C61—21no—
134C2×Q8×C61C2×D412yes—
135C2×Q8×C61C2×D412yes—
136C2×D4×C61—24no—
137C2×D4×C61—24no—
138C2×D4×C61—22no—
139C2×D4×C61—22no—
140C2×D4×C61—21no—
141C2×D4×C61—21no—
142C2×D4×C61—21no—
143C2×D4×C61—21no—
144C2×D4×C61C2×D412yes—
145C2×D4×C61C2×D412yes—
146C22×C4×C61—24no—
147C22×C4×C61—24no—
148C22×C4×C61—22no—
149C22×C4×C61—22no—
150C22×C4×C61C2×D411yescharacteristic
151C22×C4×C61C2×D411yescharacteristic
152C22×C4×C61C2×D411yescharacteristic
153C2×Dic122—21no—
154C2×Dic122—21no—
155C4⋊C4×C61—24no—
156C4⋊C4×C61—24no—
157C4⋊C4×C61—21no—
158C4⋊C4×C61—21no—
159C4⋊C4×C61—21no—
160C4⋊C4×C61—21no—
161C4⋊C4×C61—22no—
162C4⋊C4×C61—21no—
163C4⋊C4×C61—21no—
164C4⋊C4×C61—22no—
165C4⋊C4×C61C2×D412yes—
166C4⋊C4×C61C2×D412yes—
167C22⋊C4×C61—24no—
168C22⋊C4×C61—24no—
169C22⋊C4×C61—22no—
170C22⋊C4×C61—22no—
171C22⋊C4×C61—22no—
172C22⋊C4×C61—22no—
173C22⋊C4×C61—22no—
174C42×C61—22no—
175C42×C61C2×D411yescharacteristic
176C42×C61C2×D411yescharacteristic
177C42×C61C2×D411yescharacteristic
178C42×C61C2411yescharacteristic
179C22⋊Dic61—41no—
180#11758202—82no—
181#11758200—82no—
182C4⋊Dic61—21no—
183C4⋊Dic61—21no—
184C4⋊Dic61—21no—
185C4⋊Dic61—21no—
186#11758196—41no—
187#11758196—41no—
188C4×Dic61—41no—
189#11758194—81no—
190#11758194—81no—
191C2×#92418—21no—
192C2×#92418—21no—
193C2×#92418—22no—
194C2×#92418—22no—
195C2×#92418—42no—
196C23×C61—22no—
197C23×C61—22no—
198C23×C61#12011yescharacteristic
199Q8×C61—22no—
200Q8×C61—22no—
201Q8×C61—42no—
202Q8×C61—42no—
203Q8×C61—81no—
204D4×C61—22no—
205D4×C61—22no—
206D4×C61—42no—
207D4×C61—41no—
208D4×C61—41no—
209D4×C61—41no—
210D4×C61—41no—
211D4×C61—42no—
212D4×C61—81no—
213D4×C61—82no—
214D4×C61—82no—
215D4×C61—84no—
216D4×C61—84no—
217C2×C4×C61—21no—
218C2×C4×C61—21no—
219C2×C4×C61—21no—
220C2×C4×C61—21no—
221C2×C4×C61—22no—
222C2×C4×C61—22no—
223C2×C4×C61—22no—
224C2×C4×C61—22no—
225C2×C4×C61—22no—
226C2×C4×C61—42no—
227C2×C4×C61—42no—
228C2×C4×C61—44no—
229C2×C4×C61—84no—
230C2×C4×C61—84no—
231C2×C4×C61#12011yescharacteristic
232C2×C4×C61#12011yescharacteristic
233C2×C4×C61#12011yescharacteristic
234C2×C4×C61C22×D411yescharacteristic
235C2×C4×C61C22×D411yescharacteristic
236C2×C4×C61C22×D411yescharacteristic
237C2×Dic61—41no—
238C2×Dic61—41no—
239Dic122—41no—
240Dic122—41no—
241Dic122—41no—
242Dic122—41no—
243#92418—41no—
244#92418—41no—
245#92418—42no—
246#92418—42no—
247#18692—611nomaximal, Sylow
248C22×C61—21no—
249C22×C61—21no—
250C22×C61—21no—
251C22×C61—42no—
252C22×C61—42no—
253C22×C61—42no—
254C22×C61—42no—
255C22×C61—42no—
256C22×C61—42no—
257C22×C61C2×#12011yescharacteristic
258C4×C61—21no—
259C4×C61—21no—
260C4×C61—21no—
261C4×C61—42no—
262C4×C61—42no—
263C4×C61—42no—
264C4×C61—42no—
265C4×C61#45512yes—
266C4×C61#57611yescharacteristic
267C4×C61#57711yescharacteristic
268C4×C61#57711yescharacteristic
269C4×C61#57811yescharacteristic
270Dic61—81no—
271Dic61—81no—
272#3548D6111yescharacteristic
273#3255—611no—
274#3254—611no—
275#3131—611no—
276#1644—612no—
277#1643—612no—
278#1642—612no—
279#1641—612no—
280#1560—611no—
281#1493—611no—
282#1492—611no—
283C2×C61—41no—
284C2×C61—82no—
285C2×C61—82no—
286C2×C61#301211yescharacteristic
287C2×C61#301311yescharacteristic
288C2×C61C2×#45511yescharacteristic
289#562—22no—
290#554—21no—
291#554C2×D6112yes—
292#550—21no—
293#550C2×D6112yes—
294#547—21no—
295#547—21no—
296#540—22no—
297#537—22no—
298#537—22no—
299#533—611no—
300#533C2×D6111yescharacteristic
301#532C2×D6111yescharacteristic
302#532C2×D6111yescharacteristic
303#488—1222no—
304#487—611no—
305#487—611no—
306#486—611no—
307#486—611no—
308#477—1221no—
309#475—612no—
310#475—1221no—
311#474—612no—
312#470—1222no—
313#468—1222no—
314#462—612no—
315#462—1221no—
316#461—612no—
317#460—1221no—
318#423—611no—
319#423—611no—
320#416—1222no—
321C2×C4×C8—611no—
322#333—612no—
323#331—612no—
324Q8⋊C8—612no—
325Q8⋊C8—612no—
326D4⋊C8—612no—
327D4⋊C8—612no—
328C61#1869211yescharacteristic, complemented, Sylow
329C2×#41—22no—
330C2×#41—22no—
331C2×#41—21no—
332C2×#41C22⋊D6112yes—
333C2×Q16—1222no—
334C2×SD16—1222no—
335C2×M16—1221no—
336C2×M16—1221no—
337C22×C8—1221no—
338C4⋊Q8—611no—
339C4⋊Q8—611no—
340C4⋊Q8C22⋊D6111yescharacteristic
341C4⋊Q8C22⋊D6111yescharacteristic
342C4⋊Q8C22⋊D6111yescharacteristic
343C4⋊Q8C22×D6111yescharacteristic
344#127—22no—
345#127C22⋊D6111yescharacteristic
346#127C22×D6111yescharacteristic
347#125—22no—
348#125—21no—
349#125—1221no—
350#124—21no—
351#124C22⋊D6112yes—
352#123—24no—
353#123—24no—
354C22⋊Q8—22no—
355C22⋊Q8—24no—
356C22⋊Q8—22no—
357C22⋊Q8—24no—
358C22⋊Q8—21no—
359C22⋊Q8—1221no—
360C22⋊Q8—1221no—
361C22⋊Q8C22⋊D6112yes—
362#121—24no—
363#121—24no—
364#121—22no—
365#121—24no—
366#121—24no—
367#121—21no—
368#121—21no—
369#121—21no—
370#121—22no—
371#121C22⋊D6112yes—
372C4×Q8—21no—
373C4×Q8—1221no—
374C4×Q8—1221no—
375C4×Q8C22×D6112yes—
376C4×D4—22no—
377C4×D4—22no—
378C4×D4—21no—
379C4×D4—21no—
380C4×D4—21no—
381C4×D4C22×D6112yes—
382#117—21no—
383#117—1221no—
384C2×C4⋊C4—22no—
385C2×C4⋊C4—22no—
386C2×C42C22×D6111yescharacteristic
387#107—1222no—
388#106—1222no—
389C4⋊C8—612no—
390C4⋊C8—612no—
391C4⋊C8—1221no—
392C4⋊C8—1221no—
393#103—1222no—
394#103—1222no—
395#103—1222no—
396#103—1222no—
397#103—1222no—
398#102—1222no—
399#102—1222no—
400#102—1222no—
401C4×C8—611no—
402C4×C8—611no—
403C4×C8—1221no—
404#41—44no—
405#41—81no—
406#41—81no—
407#41—82no—
408#41—82no—
409#41—84no—
410C2×Q8—21no—
411C2×Q8—21no—
412C2×Q8—1221no—
413C2×Q8—1221no—
414C2×Q8C2×C22⋊D6112yes—
415C2×Q8C2×C22⋊D6112yes—
416C2×D4—24no—
417C2×D4—24no—
418C2×D4—22no—
419C2×D4—22no—
420C2×D4—21no—
421C2×D4—21no—
422C2×D4—21no—
423C2×D4—21no—
424C2×D4C2×C22⋊D6112yes—
425C2×D4C2×C22⋊D6112yes—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4C2×C22⋊D6111yescharacteristic
431C22×C4C2×C22⋊D6111yescharacteristic
432C22×C4C2×C22⋊D6111yescharacteristic
433Q16—4882no—
434SD16—4882no—
435M16—4881no—
436M16—4881no—
437C2×C8—1221no—
438C2×C8—1221no—
439C2×C8—1222no—
440C2×C8—1222no—
441C2×C8—2442no—
442C4⋊C4—24no—
443C4⋊C4—24no—
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—22no—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—22no—
452C4⋊C4—1221no—
453C4⋊C4—1221no—
454C4⋊C4—1221no—
455C4⋊C4—1221no—
456C4⋊C4—2441no—
457C4⋊C4—2441no—
458C4⋊C4C2×C22⋊D6112yes—
459C4⋊C4C2×C22⋊D6112yes—
460C22⋊C4—24no—
461C22⋊C4—24no—
462C22⋊C4—22no—
463C22⋊C4—22no—
464C22⋊C4—22no—
465C22⋊C4—22no—
466C22⋊C4—22no—
467C22⋊C4—2441no—
468C42—22no—
469C42—2441no—
470C42C2×C22⋊D6111yescharacteristic
471C42C2×C22⋊D6111yescharacteristic
472C42C2×C22⋊D6111yescharacteristic
473C42C23×D6111yescharacteristic
474C23—22no—
475C23—22no—
476C23#1451974611yescharacteristic
477Q8—22no—
478Q8—22no—
479Q8—42no—
480Q8—42no—
481Q8—81no—
482Q8—2441no—
483Q8—2441no—
484Q8—2441no—
485Q8—2441no—
486D4—22no—
487D4—22no—
488D4—42no—
489D4—42no—
490D4—41no—
491D4—41no—
492D4—41no—
493D4—41no—
494D4—81no—
495D4—82no—
496D4—82no—
497D4—84no—
498D4—84no—
499C2×C4—22no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—44no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—84no—
512C2×C4—84no—
513C2×C4—2441no—
514C2×C4—2441no—
515C2×C4#1451974611yescharacteristic
516C2×C4#1451974611yescharacteristic
517C2×C4#1451974611yescharacteristic
518C2×C4C22×C22⋊D6111yescharacteristic
519C2×C4C22×C22⋊D6111yescharacteristic
520C2×C4C22×C22⋊D6111yescharacteristic
521C8—2441no—
522C8—2441no—
523C8—2442no—
524C8—2442no—
525C22—21no—
526C22—21no—
527C22—21no—
528C22—42no—
529C22—42no—
530C22—42no—
531C22—42no—
532C22—42no—
533C22—42no—
534C22C2×#1451974611yescharacteristic
535C4—21no—
536C4—21no—
537C4—21no—
538C4—42no—
539C4—42no—
540C4—42no—
541C4—42no—
542C4—4881no—
543C4—4881no—
544C4#1683994012yes—
545C4#1684056511yescharacteristic
546C4#1684057011yescharacteristic
547C4#1684057311yescharacteristic
548C4#1684057711yescharacteristic
549C2—41no—
550C2—82no—
551C2—82no—
552C2#2145512311yescharacteristic
553C2#2145513411yescharacteristic
554C2C2×#1683994011yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
Fitting, prime radicalC61×#3548C2
socleC22×C61C2×#120
hypercenter#3548D61
hypocenterC61#18692
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3548

Complements

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

NH ≅ G/Ntype
1. #21457210C2semidirect
2. #21457206C2semidirect
3. #21456141C2semidirect
4. #21445033C2semidirect
5. #21445026C2semidirect
6. #21445019C2semidirect
7. #21445012C2semidirect
8. #21444554C2semidirect
9. #21444274C2semidirect
10. #21444270C2semidirect
328. C61#18692semidirect

Maximal subgroups

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

subgroupindex
#214572102
#214572062
#214561412
#214450332
#214450262
#214450192
#214450122
#214445542
#214442742
#214442702
C61×#35482
#1869261