Finite Group Databasebeta

← #29657630

Subgroups — #29657630

Order 15616.

882 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#21457161C211yesmaximal, characteristic, complemented
2#21456374C211yesmaximal, characteristic
3#21455624C211yesmaximal, characteristic, complemented
4#21445096C212yesmaximal, complemented
5#21445005C212yesmaximal, complemented
6#21444977C212yesmaximal, complemented
7#21444940C212yesmaximal, complemented
8#21444496C211yesmaximal, characteristic
9#21444274C211yesmaximal, characteristic, complemented
10C2×Q8⋊#92418C211yesmaximal, characteristic, complemented
11C61×#3507C211yesmaximal, characteristic
12#16840319C2211yescharacteristic
13#16840158C2211yescharacteristic
14#16840134C2211yescharacteristic
15#16840114C2211yescharacteristic
16#16840108C2211yescharacteristic
17#16840069—22no—
18#16840057C2212yes—
19#16840057C2212yes—
20#16840054—21no—
21#16840047C2212yes—
22#16840001—22no—
23#16839989—22no—
24#16839972C2212yes—
25#16839969—21no—
26C4×#11758200—22no—
27#16839802—21no—
28#16839786C2211yescharacteristic
29C2×C4⋊#92418C2211yescharacteristic
30#16839771—21no—
31C2×C4×#92418C2211yescharacteristic
32#16839508C2212yes—
33#16839506C2212yes—
34Q8⋊#92418C2212yes—
35Q8⋊#92418C2212yes—
36Q8⋊#92418C2212yes—
37Q8⋊#92418C2212yes—
38C61×#559C2212yes—
39C61×#557C2212yes—
40C61×#556—21no—
41C61×#556—22no—
42C61×#554—21no—
43C61×#552—22no—
44C61×#539—22no—
45C61×#536—21no—
46C61×#533C2211yescharacteristic
47C61×#532C2211yescharacteristic
48C61×#529—22no—
49C61×#520—21no—
50C2×C4×Q8×C61C2211yescharacteristic
51C2×#11758202—22no—
52C2×#11758200—22no—
53C2×#11758194—21no—
54C22×#92418—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#14519710—22no—
66#14519708—22no—
67#14519708—22no—
68C4⋊#92418—21no—
69C4⋊#92418—21no—
70C4⋊#92418C2312yes—
71C4⋊#92418C2312yes—
72#14519701—22no—
73#14519701—22no—
74#14519701—22no—
75#14519700—22no—
76#14519700—22no—
77#14519698—22no—
78C4×#92418—21no—
79C4×#92418C2311yescharacteristic
80C4×#92418C2311yescharacteristic
81C2×#41×C61—21no—
82C22×Q8×C61D411yescharacteristic
83C4⋊Q8×C61—22no—
84C4⋊Q8×C61D411yescharacteristic
85C4⋊Q8×C61D411yescharacteristic
86C4⋊Q8×C61C2311yescharacteristic
87#127×C61C2311yescharacteristic
88#126×C61—24no—
89#126×C61—24no—
90#125×C61—24no—
91#125×C61—22no—
92#125×C61—21no—
93#124×C61—22no—
94#124×C61—22no—
95#124×C61—21no—
96#123×C61—24no—
97#123×C61—24no—
98C22⋊Q8×C61—22no—
99C22⋊Q8×C61—22no—
100C22⋊Q8×C61—24no—
101C22⋊Q8×C61—21no—
102C22⋊Q8×C61D412yes—
103#121×C61—21no—
104#121×C61D412yes—
105C4×Q8×C61—22no—
106C4×Q8×C61—22no—
107C4×Q8×C61—21no—
108C4×Q8×C61—21no—
109C4×Q8×C61—21no—
110C4×Q8×C61D412yes—
111C4×Q8×C61C2312yes—
112C4×Q8×C61C2312yes—
113C4×D4×C61—24no—
114C4×D4×C61—22no—
115C4×D4×C61—22no—
116C4×D4×C61—21no—
117C4×D4×C61D412yes—
118#117×C61—22no—
119#117×C61—24no—
120#117×C61—21no—
121C2×C4⋊C4×C61—22no—
122C2×C4⋊C4×C61—21no—
123C2×C4⋊C4×C61D411yescharacteristic
124C2×C42×C61—21no—
125C2×C42×C61C2311yescharacteristic
126#41×C61—42no—
127#41×C61—42no—
128C2×Q8×C61—21no—
129C2×Q8×C61—21no—
130C2×Q8×C61—21no—
131C2×Q8×C61—22no—
132C2×Q8×C61—22no—
133C2×Q8×C61C2×D412yes—
134C2×Q8×C61C2×D412yes—
135C2×D4×C61—22no—
136C2×D4×C61—21no—
137C2×D4×C61—21no—
138C2×D4×C61C2×D412yes—
139C22×C4×C61—21no—
140C22×C4×C61—24no—
141C22×C4×C61—21no—
142C22×C4×C61—22no—
143C22×C4×C61C2×D411yescharacteristic
144C22×C4×C61C2×D411yescharacteristic
145C22×C4×C61C2×D411yescharacteristic
146C2×Dic122—21no—
147C2×Dic122—21no—
148C4⋊C4×C61—24no—
149C4⋊C4×C61—24no—
150C4⋊C4×C61—24no—
151C4⋊C4×C61—21no—
152C4⋊C4×C61—21no—
153C4⋊C4×C61—22no—
154C4⋊C4×C61—21no—
155C4⋊C4×C61—22no—
156C4⋊C4×C61—21no—
157C4⋊C4×C61—22no—
158C4⋊C4×C61—21no—
159C4⋊C4×C61—22no—
160C4⋊C4×C61C2×D412yes—
161C4⋊C4×C61C2×D412yes—
162C4⋊C4×C61C2×D412yes—
163C22⋊C4×C61—24no—
164C22⋊C4×C61—24no—
165C22⋊C4×C61—24no—
166C22⋊C4×C61—22no—
167C22⋊C4×C61—22no—
168C22⋊C4×C61—22no—
169C42×C61—24no—
170C42×C61—22no—
171C42×C61—22no—
172C42×C61—22no—
173C42×C61C2×D411yescharacteristic
174C42×C61C2×D411yescharacteristic
175C42×C61C2×D411yescharacteristic
176C42×C61C2411yescharacteristic
177C22⋊Dic61—41no—
178#11758202—82no—
179#11758200—44no—
180C4⋊Dic61—21no—
181C4⋊Dic61—21no—
182C4⋊Dic61—21no—
183C4⋊Dic61—21no—
184#11758196—41no—
185#11758196—41no—
186C4×Dic61—41no—
187#11758194—42no—
188C2×#92418—21no—
189C2×#92418—21no—
190C2×#92418—22no—
191C2×#92418—22no—
192C2×#92418—42no—
193C2×#92418—81no—
194C23×C61—22no—
195C23×C61#12011yescharacteristic
196Q8×C61—22no—
197Q8×C61—22no—
198Q8×C61—22no—
199Q8×C61—22no—
200Q8×C61—41no—
201Q8×C61—41no—
202Q8×C61—42no—
203Q8×C61—81no—
204D4×C61—42no—
205D4×C61—44no—
206D4×C61—41no—
207D4×C61—41no—
208D4×C61—41no—
209D4×C61—41no—
210C2×C4×C61—21no—
211C2×C4×C61—21no—
212C2×C4×C61—22no—
213C2×C4×C61—22no—
214C2×C4×C61—22no—
215C2×C4×C61—22no—
216C2×C4×C61—22no—
217C2×C4×C61—41no—
218C2×C4×C61—42no—
219C2×C4×C61—44no—
220C2×C4×C61—81no—
221C2×C4×C61—84no—
222C2×C4×C61#12011yescharacteristic
223C2×C4×C61#12011yescharacteristic
224C2×C4×C61#12011yescharacteristic
225C2×C4×C61#13512yes—
226C2×C4×C61#13712yes—
227C2×C4×C61C22×D411yescharacteristic
228C2×C4×C61C22×D411yescharacteristic
229C2×C4×C61C22×D411yescharacteristic
230C2×Dic61—41no—
231C2×Dic61—41no—
232Dic122—42no—
233Dic122—41no—
234Dic122—41no—
235#92418—22no—
236#92418—41no—
237#92418—42no—
238#92418—42no—
239#18667—611nomaximal, Sylow
240C22×C61—21no—
241C22×C61—21no—
242C22×C61—42no—
243C22×C61—42no—
244C22×C61—42no—
245C22×C61#45212yes—
246C22×C61C2×#12011yescharacteristic
247C4×C61—21no—
248C4×C61—21no—
249C4×C61—21no—
250C4×C61—42no—
251C4×C61—42no—
252C4×C61—42no—
253C4×C61—42no—
254C4×C61—42no—
255C4×C61#45212yes—
256C4×C61C2×#13511yescharacteristic
257C4×C61C2×#13711yescharacteristic
258C4×C61#57711yescharacteristic
259C4×C61#57811yescharacteristic
260Dic61—81no—
261Dic61—81no—
262#3507D6111yescharacteristic
263#3249—611no—
264#3167—611no—
265#3078—611no—
266#1653—612no—
267#1640—612no—
268#1636—612no—
269#1630—612no—
270#1547—611no—
271#1493—611no—
272C2×Q8⋊C8—611no—
273C2×C61—22no—
274C2×C61—82no—
275C2×C61C2×#45211yescharacteristic
276C2×C61#301311yescharacteristic
277C2×C61#301811yescharacteristic
278#559C2×D6112yes—
279#557C2×D6112yes—
280#556—21no—
281#556—22no—
282#554—21no—
283#552—22no—
284#539—22no—
285#536—21no—
286#533—611no—
287#533C2×D6111yescharacteristic
288#532C2×D6111yescharacteristic
289#529—22no—
290#520—21no—
291C2×C4×Q8C2×D6111yescharacteristic
292#498—611no—
293#492—611no—
294#487—611no—
295#486—611no—
296#479—1222no—
297#477—612no—
298#477—612no—
299#477—1221no—
300#475—612no—
301#467—1222no—
302#465—1222no—
303#462—612no—
304#462—1221no—
305C4×SD16—1222no—
306#428—1221no—
307#423—611no—
308C2×C4⋊C8—611no—
309#419—1221no—
310C2×C4×C8—611no—
311#332—612no—
312#331—612no—
313Q8⋊C8—612no—
314Q8⋊C8—612no—
315Q8⋊C8—612no—
316Q8⋊C8—612no—
317C61#1866711yescharacteristic, complemented, Sylow
318C2×#41—21no—
319C22×Q8C22⋊D6111yescharacteristic
320C2×Q16—1222no—
321C2×SD16—1222no—
322C2×M16—1221no—
323C22×C8—1221no—
324C22×C8—1221no—
325C4⋊Q8—22no—
326C4⋊Q8—611no—
327C4⋊Q8—611no—
328C4⋊Q8C22⋊D6111yescharacteristic
329C4⋊Q8C22⋊D6111yescharacteristic
330C4⋊Q8C22×D6111yescharacteristic
331#127C22×D6111yescharacteristic
332#126—24no—
333#126—24no—
334#125—22no—
335#125—24no—
336#125—21no—
337#125—1221no—
338#124—22no—
339#124—22no—
340#124—21no—
341#123—24no—
342#123—24no—
343C22⋊Q8—22no—
344C22⋊Q8—22no—
345C22⋊Q8—24no—
346C22⋊Q8—21no—
347C22⋊Q8—1221no—
348C22⋊Q8—1221no—
349C22⋊Q8C22⋊D6112yes—
350#121—21no—
351#121C22⋊D6112yes—
352C4×Q8—22no—
353C4×Q8—22no—
354C4×Q8—21no—
355C4×Q8—21no—
356C4×Q8—21no—
357C4×Q8—1221no—
358C4×Q8—1221no—
359C4×Q8C22⋊D6112yes—
360C4×Q8C22×D6112yes—
361C4×Q8C22×D6112yes—
362C4×D4—24no—
363C4×D4—22no—
364C4×D4—22no—
365C4×D4—21no—
366C4×D4C22⋊D6112yes—
367#117—22no—
368#117—24no—
369#117—21no—
370#117—1221no—
371C2×C4⋊C4—22no—
372C2×C4⋊C4—21no—
373C2×C4⋊C4C22⋊D6111yescharacteristic
374C2×C42—21no—
375C2×C42C22×D6111yescharacteristic
376#107—1222no—
377#107—1222no—
378#106—1222no—
379#106—1222no—
380C4⋊C8—612no—
381C4⋊C8—612no—
382C4⋊C8—1221no—
383C4⋊C8—1221no—
384#103—1222no—
385#103—1222no—
386#103—1222no—
387#103—1222no—
388#103—1222no—
389#102—1222no—
390C4×C8—611no—
391C4×C8—611no—
392C4×C8—1221no—
393#41—42no—
394#41—42no—
395C2×Q8—21no—
396C2×Q8—21no—
397C2×Q8—21no—
398C2×Q8—22no—
399C2×Q8—22no—
400C2×Q8—1221no—
401C2×Q8—1221no—
402C2×Q8C2×C22⋊D6112yes—
403C2×Q8C2×C22⋊D6112yes—
404C2×D4—22no—
405C2×D4—21no—
406C2×D4—21no—
407C2×D4C2×C22⋊D6112yes—
408C22×C4—21no—
409C22×C4—24no—
410C22×C4—21no—
411C22×C4—22no—
412C22×C4C2×C22⋊D6111yescharacteristic
413C22×C4C2×C22⋊D6111yescharacteristic
414C22×C4C2×C22⋊D6111yescharacteristic
415Q16—4882no—
416SD16—2444no—
417M16—2442no—
418C2×C8—1221no—
419C2×C8—1221no—
420C2×C8—1222no—
421C2×C8—1222no—
422C2×C8—4881no—
423C2×C8—2442no—
424C4⋊C4—24no—
425C4⋊C4—24no—
426C4⋊C4—24no—
427C4⋊C4—22no—
428C4⋊C4—21no—
429C4⋊C4—22no—
430C4⋊C4—21no—
431C4⋊C4—21no—
432C4⋊C4—22no—
433C4⋊C4—21no—
434C4⋊C4—22no—
435C4⋊C4—21no—
436C4⋊C4—1221no—
437C4⋊C4—1221no—
438C4⋊C4—1221no—
439C4⋊C4—1221no—
440C4⋊C4—2441no—
441C4⋊C4—2441no—
442C4⋊C4C2×C22⋊D6112yes—
443C4⋊C4C2×C22⋊D6112yes—
444C4⋊C4C2×C22⋊D6112yes—
445C22⋊C4—24no—
446C22⋊C4—24no—
447C22⋊C4—24no—
448C22⋊C4—22no—
449C22⋊C4—22no—
450C22⋊C4—22no—
451C22⋊C4—2441no—
452C42—24no—
453C42—22no—
454C42—22no—
455C42—22no—
456C42—2441no—
457C42C2×C22⋊D6111yescharacteristic
458C42C2×C22⋊D6111yescharacteristic
459C42C2×C22⋊D6111yescharacteristic
460C42C23×D6111yescharacteristic
461C23—22no—
462C23#1451974611yescharacteristic
463Q8—22no—
464Q8—22no—
465Q8—22no—
466Q8—22no—
467Q8—41no—
468Q8—41no—
469Q8—42no—
470Q8—81no—
471Q8—2442no—
472Q8—2441no—
473Q8—2441no—
474D4—42no—
475D4—44no—
476D4—41no—
477D4—41no—
478D4—41no—
479D4—41no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—41no—
488C2×C4—42no—
489C2×C4—44no—
490C2×C4—81no—
491C2×C4—84no—
492C2×C4—2441no—
493C2×C4—2441no—
494C2×C4#1451974611yescharacteristic
495C2×C4#1451974611yescharacteristic
496C2×C4#1451974611yescharacteristic
497C2×C4#1451979712yes—
498C2×C4#1451980712yes—
499C2×C4C22×C22⋊D6111yescharacteristic
500C2×C4C22×C22⋊D6111yescharacteristic
501C2×C4C22×C22⋊D6111yescharacteristic
502C8—1222no—
503C8—2441no—
504C8—2442no—
505C8—2442no—
506C22—21no—
507C22—21no—
508C22—42no—
509C22—42no—
510C22—42no—
511C22#1683992412yes—
512C22C2×#1451974611yescharacteristic
513C4—21no—
514C4—21no—
515C4—21no—
516C4—42no—
517C4—42no—
518C4—42no—
519C4—42no—
520C4—42no—
521C4—4881no—
522C4—4881no—
523C4#1683992412yes—
524C4C2×#1451979711yescharacteristic
525C4C2×#1451980711yescharacteristic
526C4#1684057011yescharacteristic
527C4#1684057711yescharacteristic
528C2—22no—
529C2—82no—
530C2C2×#1683992411yescharacteristic
531C2#2145513411yescharacteristic
532C2#2145516811yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3507C2
socleC22×C61C2×#120
prime radicalC61×#532C22
hypercenter#3507D61
hypocenterC61#18667
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3507

Complements

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

NH ≅ G/Ntype
1. #21457161C2semidirect
3. #21455624C2semidirect
4. #21445096C2semidirect
5. #21445005C2semidirect
6. #21444977C2semidirect
7. #21444940C2semidirect
9. #21444274C2semidirect
10. C2×Q8⋊#92418C2semidirect
317. C61#18667semidirect

Maximal subgroups

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

subgroupindex
#214571612
#214563742
#214556242
#214450962
#214450052
#214449772
#214449402
#214444962
#214442742
C2×Q8⋊#924182
C61×#35072
#1866761