Finite Group Databasebeta

← #29652929

Subgroups — #29652929

Order 15616.

1454 conjugacy classes of subgroups.

The subgroup lattice has 536 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839900C212yesmaximal, complemented
2#21449184C211yesmaximal, characteristic, complemented
3#21444788C214yesmaximal, complemented
4#21444771C214yesmaximal, complemented
5#21444387C212yesmaximal, complemented
6#21444137C211yesmaximal, characteristic, complemented
7C61×#3496C211yesmaximal, characteristic
8C23×Dic122C2211yescharacteristic, complemented
9C22×#11758200—22no—
10C2×#14519756—21no—
11#16839900—24no—
12#16839900C2218yescomplemented
13C2×#14519700—22no—
14C2×C22⋊#92418C2212yescomplemented
15#16839682C2212yescomplemented
16#16839674C2212yescomplemented
17#16839668—21no—
18#16839642—21no—
19#16839634—21no—
20#16839622C2211yescharacteristic, complemented
21#16839496C2218yescomplemented
22#16839488C2212yescomplemented
23#16839486C2212yescomplemented
24C2×H(5,𝔽2)×C61—21no—
25C61×#547—22no—
26C61×#546—24no—
27D42×C61—22no—
28C61×#537C2212yes—
29C61×#534—21no—
30C61×#534C2212yes—
31C61×#532—21no—
32C2×#121×C61—21no—
33C2×#121×C61C2212yes—
34C2×C4×D4×C61C2211yescharacteristic
35C22×Dic122—22no—
36C22×Dic122—21no—
37C22×Dic122—22no—
38C22×Dic122—44no—
39C22×Dic122C2312yes—
40C23×Dic61—21no—
41C2×#11758200—28no—
42C2×#11758200—48no—
43C22×#92418—22no—
44#14519756—44no—
45C2×C4⋊Dic61—21no—
46C2×#11758196—22no—
47C2×C22⋊Dic61—22no—
48C2×C22⋊Dic61—21no—
49#14519700—28no—
50C22⋊#92418—22no—
51C22⋊#92418C2314yes—
52#14519684—22no—
53#14519684—22no—
54#14519684C2312yes—
55H(5,𝔽2)×C61—44no—
56H(5,𝔽2)×C61—44no—
57C2×#41×C61—24no—
58C2×#41×C61—22no—
59C22×D4×C61—24no—
60C22×D4×C61—22no—
61C22×D4×C61—22no—
62C22×D4×C61—21no—
63C22×D4×C61D411yescharacteristic
64C22×D4×C61D412yes—
65C23×C4×C61D411yescharacteristic
66C23×C4×C61C2311yescharacteristic
67C4⋊Q8×C61—21no—
68#127×C61—21no—
69#124×C61—22no—
70#123×C61—24no—
71#123×C61—24no—
72C22⋊Q8×C61—24no—
73#121×C61—22no—
74#121×C61—22no—
75#121×C61—24no—
76#121×C61—22no—
77#121×C61—22no—
78#121×C61—22no—
79#121×C61—22no—
80#121×C61D414yes—
81#121×C61C2314yes—
82#120×C61—24no—
83#120×C61—24no—
84C4×D4×C61—22no—
85C4×D4×C61—22no—
86C4×D4×C61—22no—
87C4×D4×C61—22no—
88C2×C4⋊C4×C61—21no—
89C2×C4⋊C4×C61D412yes—
90C2×C4⋊C4×C61C2311yescharacteristic
91C2×C22⋊C4×C61—22no—
92C2×C22⋊C4×C61—22no—
93C2×C22⋊C4×C61D411yescharacteristic
94C2×C22⋊C4×C61C2311yescharacteristic
95C2×C42×C61D411yescharacteristic
96C24×C61—22no—
97C24×C61—22no—
98C24×C61C2×D411yescharacteristic
99C24×C61C2×D411yescharacteristic
100#41×C61—44no—
101#41×C61—416no—
102#41×C61—44no—
103C2×Q8×C61—22no—
104C2×D4×C61—22no—
105C2×D4×C61—22no—
106C2×D4×C61—24no—
107C2×D4×C61—22no—
108C2×D4×C61—24no—
109C2×D4×C61—24no—
110C2×D4×C61—22no—
111C2×D4×C61—22no—
112C2×D4×C61—22no—
113C2×D4×C61—44no—
114C2×D4×C61—416no—
115C2×D4×C61—44no—
116C2×D4×C61—44no—
117C2×D4×C61—44no—
118C2×D4×C61—44no—
119C2×D4×C61—82no—
120C2×D4×C61SD1614yescomplemented
121C2×D4×C61C2×D414yes—
122C22×C4×C61—22no—
123C22×C4×C61—24no—
124C22×C4×C61—22no—
125C22×C4×C61—21no—
126C22×C4×C61—21no—
127C22×C4×C61—42no—
128C22×C4×C61—42no—
129C22×C4×C61C2×D411yescharacteristic
130C22×C4×C61C2×D412yes—
131C22×C4×C61C2×D411yescharacteristic
132C22×C4×C61C2×D411yescharacteristic
133C22×C4×C61C2×D411yescharacteristic
134C22×C4×C61C2×D412yes—
135C22×C4×C61C2411yescharacteristic
136C22×Dic61—22no—
137C22×Dic61—22no—
138C22×Dic61—44no—
139C22×Dic61—42no—
140C2×Dic122—28no—
141C2×Dic122—24no—
142C2×Dic122—24no—
143C2×Dic122—44no—
144C2×Dic122—48no—
145C2×Dic122—44no—
146C2×Dic122—42no—
147C2×Dic122—41no—
148C2×Dic122—41no—
149C4⋊C4×C61—22no—
150C4⋊C4×C61—22no—
151C4⋊C4×C61—22no—
152C4⋊C4×C61—22no—
153C4⋊C4×C61C2×D414yes—
154C22⋊C4×C61—24no—
155C22⋊C4×C61—24no—
156C22⋊C4×C61—22no—
157C22⋊C4×C61—24no—
158C22⋊C4×C61—24no—
159C22⋊C4×C61—22no—
160C42×C61—21no—
161C42×C61—21no—
162C22⋊Dic61—44no—
163C22⋊Dic61—42no—
164#11758200—416no—
165C4⋊Dic61—42no—
166#11758196—44no—
167C2×#92418—24no—
168C2×#92418—44no—
169C23×C61—21no—
170C23×C61—21no—
171C23×C61—24no—
172C23×C61—41no—
173C23×C61—41no—
174C23×C61—41no—
175C23×C61—41no—
176C23×C61—44no—
177C23×C61—44no—
178C23×C61—82no—
179C23×C61—82no—
180C23×C61—82no—
181C23×C61—82no—
182C23×C61#12012yes—
183C23×C61C2×SD1612yes—
184C23×C61C2×SD1612yescomplemented
185C23×C61AGL(1,ℤ/8)12yes—
186C23×C61C22×D411yescharacteristic
187Q8×C61—44no—
188D4×C61—28no—
189D4×C61—44no—
190D4×C61—44no—
191D4×C61—48no—
192D4×C61—44no—
193D4×C61—44no—
194D4×C61—48no—
195D4×C61—44no—
196D4×C61—82no—
197D4×C61—82no—
198C2×C4×C61—21no—
199C2×C4×C61—21no—
200C2×C4×C61—24no—
201C2×C4×C61—22no—
202C2×C4×C61—21no—
203C2×C4×C61—21no—
204C2×C4×C61—24no—
205C2×C4×C61—42no—
206C2×C4×C61—41no—
207C2×C4×C61—48no—
208C2×C4×C61—44no—
209C2×C4×C61—44no—
210C2×C4×C61—42no—
211C2×C4×C61—41no—
212C2×C4×C61—41no—
213C2×C4×C61—81no—
214C2×C4×C61#12011yescharacteristic
215C2×C4×C61#12011yescharacteristic
216C2×C4×C61C2×SD1612yescomplemented
217C2×C4×C61C22×D412yes—
218C2×Dic61—24no—
219C2×Dic61—48no—
220C2×Dic61—42no—
221C2×Dic61—42no—
222C2×Dic61—42no—
223C2×Dic61—42no—
224C2×Dic61—42no—
225Dic122—28no—
226Dic122—44no—
227Dic122—44no—
228Dic122—44no—
229#92418—44no—
230#18195—611nomaximal, Sylow
231C22×C61—21no—
232C22×C61—22no—
233C22×C61—21no—
234C22×C61—22no—
235C22×C61—22no—
236C22×C61—42no—
237C22×C61—42no—
238C22×C61—42no—
239C22×C61—41no—
240C22×C61—41no—
241C22×C61—44no—
242C22×C61—44no—
243C22×C61—44no—
244C22×C61—81no—
245C22×C61—164no—
246C22×C61—164no—
247C22×C61#45014yes—
248C22×C61C2×#12011yescharacteristic
249C22×C61C22×SD1611yescharacteristic
250C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
251C4×C61—22no—
252C4×C61—22no—
253C4×C61—21no—
254C4×C61—44no—
255C4×C61—41no—
256C4×C61—41no—
257C4×C61B(4,𝔽2)12yes—
258Dic61—44no—
259Dic61—82no—
260#3496D6111yescharacteristic
261C2×#448—612no—
262#2409—611no—
263#1604—614no—
264#1601—614no—
265#1519—612no—
266#1459—611no—
267C2×C61—21no—
268C2×C61—21no—
269C2×C61—24no—
270C2×C61—42no—
271C2×C61—84no—
272C2×C61C2×#45011yescharacteristic
273C2×C61#302211yescharacteristic
274C2×C61C2×B(4,𝔽2)11yescharacteristic
275C2×H(5,𝔽2)—21no—
276C23×Q8—611no—
277C22×SD16—1222no—
278#547—22no—
279#546—24no—
280D42—22no—
281#537C2×D6112yes—
282#534—21no—
283#534C2×D6112yes—
284#532—21no—
285C2×C22⋊Q8—1221no—
286C2×#121—21no—
287C2×#121C2×D6112yes—
288C2×C4×D4C2×D6111yescharacteristic
289#448—618no—
290#448—1224no—
291C2×#103—1222no—
292C2×C22⋊C8—612no—
293#395—612no—
294#393—612no—
295#391—1221no—
296#386—1221no—
297#385—1221no—
298#380—611no—
299#327—618no—
300#324—612no—
301#323—612no—
302C61#1819511yescharacteristic, complemented, Sylow
303H(5,𝔽2)—44no—
304H(5,𝔽2)—44no—
305C2×#41—24no—
306C2×#41—22no—
307C22×Q8—612no—
308C22×Q8—1221no—
309C22×Q8—1222no—
310C22×Q8—1222no—
311C22×Q8—2444no—
312C22×D4—24no—
313C22×D4—22no—
314C22×D4—22no—
315C22×D4—21no—
316C22×D4C22⋊D6111yescharacteristic
317C22×D4C22⋊D6112yes—
318C23×C4—1221no—
319C23×C4C22⋊D6111yescharacteristic
320C23×C4C22×D6111yescharacteristic
321C2×SD16—1228no—
322C2×SD16—2448no—
323C22×C8—1222no—
324C4⋊Q8—21no—
325#127—21no—
326#124—22no—
327#123—24no—
328#123—24no—
329C22⋊Q8—24no—
330C22⋊Q8—2444no—
331#121—22no—
332#121—22no—
333#121—24no—
334#121—22no—
335#121—22no—
336#121—22no—
337#121—22no—
338#121C22⋊D6114yes—
339#121C22×D6114yes—
340#120—24no—
341#120—24no—
342C4×D4—22no—
343C4×D4—22no—
344C4×D4—22no—
345C4×D4—22no—
346C2×C4⋊C4—21no—
347C2×C4⋊C4—1221no—
348C2×C4⋊C4—1222no—
349C2×C4⋊C4C22⋊D6112yes—
350C2×C4⋊C4C22×D6111yescharacteristic
351C2×C22⋊C4—22no—
352C2×C22⋊C4—22no—
353C2×C22⋊C4—1222no—
354C2×C22⋊C4—1221no—
355C2×C22⋊C4C22⋊D6111yescharacteristic
356C2×C22⋊C4C22×D6111yescharacteristic
357C2×C42C22⋊D6111yescharacteristic
358#103—1228no—
359C22⋊C8—614no—
360C22⋊C8—1222no—
361#95—612no—
362#95—1222no—
363#95—1222no—
364C24—22no—
365C24—22no—
366C24C2×C22⋊D6111yescharacteristic
367C24C2×C22⋊D6111yescharacteristic
368#41—44no—
369#41—416no—
370#41—44no—
371C2×Q8—22no—
372C2×Q8—1224no—
373C2×Q8—1228no—
374C2×Q8—1224no—
375C2×Q8—2442no—
376C2×Q8—2444no—
377C2×Q8—2444no—
378C2×Q8—2448no—
379C2×Q8—2441no—
380C2×Q8—2441no—
381C2×D4—22no—
382C2×D4—22no—
383C2×D4—24no—
384C2×D4—22no—
385C2×D4—24no—
386C2×D4—24no—
387C2×D4—22no—
388C2×D4—22no—
389C2×D4—22no—
390C2×D4—44no—
391C2×D4—416no—
392C2×D4—44no—
393C2×D4—44no—
394C2×D4—44no—
395C2×D4—44no—
396C2×D4—82no—
397C2×D4#1175820014yescomplemented
398C2×D4C2×C22⋊D6114yes—
399C22×C4—22no—
400C22×C4—24no—
401C22×C4—22no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—1222no—
407C22×C4—1222no—
408C22×C4—2444no—
409C22×C4—2442no—
410C22×C4C2×C22⋊D6112yes—
411C22×C4C2×C22⋊D6111yescharacteristic
412C22×C4C2×C22⋊D6111yescharacteristic
413C22×C4C2×C22⋊D6111yescharacteristic
414C22×C4C2×C22⋊D6111yescharacteristic
415C22×C4C2×C22⋊D6112yes—
416C22×C4C23×D6111yescharacteristic
417SD16—24416no—
418C2×C8—1224no—
419C2×C8—2444no—
420C4⋊C4—22no—
421C4⋊C4—22no—
422C4⋊C4—22no—
423C4⋊C4—22no—
424C4⋊C4—2442no—
425C4⋊C4—2444no—
426C4⋊C4C2×C22⋊D6114yes—
427C22⋊C4—24no—
428C22⋊C4—24no—
429C22⋊C4—22no—
430C22⋊C4—24no—
431C22⋊C4—22no—
432C22⋊C4—24no—
433C22⋊C4—2444no—
434C22⋊C4—2442no—
435C42—21no—
436C42—21no—
437C23—21no—
438C23—21no—
439C23—24no—
440C23—41no—
441C23—41no—
442C23—41no—
443C23—41no—
444C23—44no—
445C23—44no—
446C23—82no—
447C23—82no—
448C23—82no—
449C23—82no—
450C23#1451974612yes—
451C23C2×#1175820012yes—
452C23C2×#1175820012yescomplemented
453C23#1451980312yes—
454C23C22×C22⋊D6111yescharacteristic
455Q8—44no—
456Q8—1228no—
457Q8—2444no—
458Q8—2444no—
459Q8—2444no—
460D4—28no—
461D4—44no—
462D4—44no—
463D4—48no—
464D4—44no—
465D4—44no—
466D4—48no—
467D4—44no—
468D4—82no—
469D4—82no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—22no—
473C2×C4—24no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—24no—
477C2×C4—42no—
478C2×C4—41no—
479C2×C4—48no—
480C2×C4—44no—
481C2×C4—44no—
482C2×C4—42no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—81no—
486C2×C4—1224no—
487C2×C4—2442no—
488C2×C4—2442no—
489C2×C4—2442no—
490C2×C4—2442no—
491C2×C4—2448no—
492C2×C4—2442no—
493C2×C4#1451974611yescharacteristic
494C2×C4#1451974611yescharacteristic
495C2×C4C2×#1175820012yescomplemented
496C2×C4C22×C22⋊D6112yes—
497C8—2444no—
498C22—21no—
499C22—22no—
500C22—21no—
501C22—22no—
502C22—22no—
503C22—42no—
504C22—42no—
505C22—42no—
506C22—41no—
507C22—41no—
508C22—44no—
509C22—44no—
510C22—44no—
511C22—81no—
512C22—164no—
513C22—164no—
514C22#1683991314yes—
515C22C2×#1451974611yescharacteristic
516C22C22×#1175820011yescharacteristic
517C22C2×#1451980311yescharacteristic
518C4—22no—
519C4—21no—
520C4—22no—
521C4—41no—
522C4—41no—
523C4—44no—
524C4—4882no—
525C4—2444no—
526C4#1683994912yes—
527C2—21no—
528C2—21no—
529C2—24no—
530C2—42no—
531C2—84no—
532C2C2×#1683991311yescharacteristic
533C2#2145519811yescharacteristic
534C2C2×#1683994911yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
Fitting, prime radicalC61×#3496C2
socleC22×C61C2×#120
hypercenter#3496D61
hypocenterC61#18195
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3496

Complements

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

NH ≅ G/Ntype
1. C2×#16839900C2semidirect
2. #21449184C2semidirect
3. #21444788C2semidirect
4. #21444771C2semidirect
5. #21444387C2semidirect
6. #21444137C2semidirect
8. C23×Dic122C22semidirect
12. #16839900C22semidirect
14. C2×C22⋊#92418C22semidirect
15. #16839682C22semidirect
16. #16839674C22semidirect
20. #16839622C22semidirect
21. #16839496C22semidirect
22. #16839488C22semidirect
23. #16839486C22semidirect
120. C2×D4×C61SD16semidirect
184. C23×C61C2×SD16semidirect
216. C2×C4×C61C2×SD16semidirect
302. C61#18195semidirect
397. C2×D4#11758200semidirect
452. C23C2×#11758200semidirect
495. C2×C4C2×#11758200semidirect

Maximal subgroups

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

subgroupindex
C2×#168399002
#214491842
#214447882
#214447712
#214443872
#214441372
C61×#34962
#1819561