Finite Group Databasebeta

← #30544581

Subgroups — #30544581

Order 15616.

1878 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21454824C218yesmaximal, complemented
2#21454815C218yesmaximal, complemented
3#21454760C211yesmaximal, characteristic, complemented
4#21454754C211yesmaximal, characteristic, complemented
5#21454537C211yesmaximal, characteristic, complemented
6#21454531C211yesmaximal, characteristic, complemented
7#21454523C212yesmaximal, complemented
8#21454514C212yesmaximal, complemented
9#21454463C212yesmaximal, complemented
10#21454457C211yesmaximal, characteristic, complemented
11#21454247C212yesmaximal, complemented
12C2×#16839718C211yesmaximal, characteristic, complemented
13C61×#3455C211yesmaximal, characteristic
14C2×#14519784C2212yes—
15C2×#14519784C2212yes—
16C22×#11758194C2211yescharacteristic
17C23×#92418C2211yescharacteristic
18#16839893C2214yescomplemented
19#16839893C2218yescomplemented
20Q8×#92418C2214yescomplemented
21#16839840C2218yes—
22#16839840C2214yescomplemented
23#16839834C2218yes—
24#16839834C2218yes—
25#16839834C2218yescomplemented
26D4×#92418C2218yes—
27D4×#92418C2214yescomplemented
28#16839823C2218yes—
29#16839823C2218yescomplemented
30#16839815C2218yescomplemented
31#16839789C2212yes—
32#16839789C2212yes—
33#16839786C2211yescharacteristic
34#16839786C2211yescharacteristic
35#16839729C2212yes—
36#16839729C2212yes—
37#16839726C2212yes—
38#16839718C2214yes—
39#16839718C2214yes—
40#16839718C2212yes—
41#16839718C2212yes—
42#16839718C2214yescomplemented
43#16839718C2214yescomplemented
44C4×#11758194C2212yes—
45C4×#11758194C2214yescomplemented
46C2×#14519688C2211yescharacteristic
47C2×#14519688C2211yescharacteristic
48C2×#14519688C2214yescomplemented
49C2×C4×#92418C2211yescharacteristic
50C2×C4×#92418C2211yescharacteristic
51C61×#542C414yes—
52C61×#541C412yes—
53C61×#540C412yes—
54C61×#539C414yes—
55C61×#538C414yes—
56C61×#533C2211yescharacteristic
57C61×#532C2211yescharacteristic
58C61×#529C2218yes—
59C61×#525C2212yes—
60C4×#41×C61C2212yes—
61C2×#117×C61C2211yescharacteristic
62#14519784—28no—
63#14519784—28no—
64C2×#11758194—24no—
65C2×#11758194C2312yes—
66C2×#11758194C2312yes—
67C2×#11758194C2314yes—
68C2×#11758194C2314yes—
69C2×#11758194C2312yes—
70C2×#11758194C2312yes—
71C22×#92418—24no—
72C22×#92418C2311yescharacteristic
73C22×#92418C2311yescharacteristic
74C22×#92418C2314yes—
75C22×#92418C2314yes—
76C22×#92418C2314yes—
77C22×#92418C2311yescharacteristic
78C22×#92418C2311yescharacteristic
79C4⋊#92418C2314yes—
80C4⋊#92418C2314yes—
81C4⋊#92418C2318yes—
82C4⋊#92418C2318yes—
83C22⋊#92418C2318yes—
84C22⋊#92418C2318yes—
85C22⋊#92418C2318yes—
86#14519688C2312yes—
87#14519688C2312yes—
88#14519688C2314yes—
89#14519688C2314yes—
90#14519688C2314yes—
91#14519688C2314yes—
92C4×#92418C2314yes—
93C4×#92418C2312yes—
94C4×#92418C2312yes—
95C4×#92418C2314yes—
96C4×#92418C2314yes—
97C4×#92418C2314yes—
98C2×#41×C61C2312yes—
99C23×C4×C61C2311yescharacteristic
100C4⋊Q8×C61C2×C411yescharacteristic
101C4⋊Q8×C61C2×C412yes—
102#127×C61C2×C411yescharacteristic
103#126×C61C2×C418yes—
104#126×C61C2×C418yes—
105#125×C61C2×C412yes—
106#125×C61C2×C418yes—
107#124×C61C2×C412yes—
108#124×C61C2×C418yes—
109#123×C61C2×C418yes—
110#123×C61C2×C414yes—
111#123×C61C2×C418yes—
112C22⋊Q8×C61C2×C414yes—
113C22⋊Q8×C61C2×C418yes—
114C22⋊Q8×C61C2×C414yes—
115#121×C61C2×C414yes—
116#121×C61C2×C418yes—
117#121×C61C2×C414yes—
118#120×C61C2×C414yes—
119C4×Q8×C61C2×C414yes—
120C4×Q8×C61C2314yes—
121C4×D4×C61C2×C418yes—
122C4×D4×C61C2×C414yes—
123C4×D4×C61C2314yes—
124C4×D4×C61C2318yes—
125#117×C61C2×C412yes—
126#117×C61C2×C412yes—
127#117×C61C2312yes—
128#117×C61C2314yes—
129#117×C61C2314yes—
130C2×C4⋊C4×C61C2×C412yes—
131C2×C22⋊C4×C61C2×C412yes—
132C2×C42×C61C2311yescharacteristic
133C2×C42×C61C2311yescharacteristic
134C2×C42×C61C2314yes—
135C24×C61C22×C411yescharacteristic
136#41×C61—24no—
137#41×C61—24no—
138C2×Q8×C61C22×C412yes—
139C2×Q8×C61C22×C414yes—
140C2×D4×C61C22×C414yes—
141C2×D4×C61C22×C412yes—
142C2×D4×C61C22×C418yes—
143C2×D4×C61C22×C414yes—
144C22×C4×C61—22no—
145C22×C4×C61—22no—
146C22×C4×C61C22×C411yescharacteristic
147C22×C4×C61C22×C412yes—
148C22×C4×C61C22×C412yes—
149C22×C4×C61C22×C412yes—
150C22×C4×C61C22×C418yes—
151C22×C4×C61C2414yes—
152C22×C4×C61C2412yes—
153C22×C4×C61C2411yescharacteristic
154C4⋊C4×C61C22×C414yes—
155C4⋊C4×C61C22×C414yes—
156C4⋊C4×C61C22×C414yes—
157C4⋊C4×C61C22×C418yes—
158C4⋊C4×C61C22×C414yes—
159C4⋊C4×C61C22×C418yes—
160C4⋊C4×C61C22×C418yes—
161C4⋊C4×C61C22×C414yes—
162C22⋊C4×C61C22×C414yes—
163C22⋊C4×C61C22×C414yes—
164C22⋊C4×C61C22×C414yes—
165C22⋊C4×C61C22×C418yes—
166C22⋊C4×C61C22×C418yes—
167C22⋊C4×C61C22×C418yes—
168C22⋊C4×C61C22×C418yes—
169C42×C61C22×C412yes—
170C42×C61C22×C411yescharacteristic
171C42×C61C22×C411yescharacteristic
172C42×C61C22×C418yes—
173C42×C61C2412yes—
174C42×C61C2412yes—
175C42×C61C2414yes—
176#11758194—24no—
177#11758194—24no—
178#11758194—24no—
179#11758194—24no—
180#11758194—28no—
181#11758194—28no—
182C2×#92418—22no—
183C2×#92418—22no—
184C2×#92418—28no—
185C2×#92418—28no—
186C2×#92418—28no—
187C2×#92418#4114yes—
188C2×#92418#4114yes—
189C2×#92418C2412yes—
190C2×#92418C2412yes—
191C2×#92418C2414yes—
192C2×#92418C2414yes—
193C2×#92418C2411yescharacteristic
194C2×#92418C2411yescharacteristic
195C2×#92418C2412yes—
196C23×C61—22no—
197C23×C61—42no—
198C23×C61C23×C412yes—
199C23×C61C23×C414yes—
200C23×C61C23×C411yescharacteristic
201Q8×C61—24no—
202Q8×C61—44no—
203D4×C61—28no—
204D4×C61—24no—
205D4×C61—44no—
206D4×C61—48no—
207C2×C4×C61—22no—
208C2×C4×C61—24no—
209C2×C4×C61—22no—
210C2×C4×C61—24no—
211C2×C4×C61—24no—
212C2×C4×C61—22no—
213C2×C4×C61—42no—
214C2×C4×C61—48no—
215C2×C4×C61—42no—
216C2×C4×C61C23×C414yes—
217C2×C4×C61C23×C414yes—
218C2×C4×C61C23×C412yes—
219C2×C4×C61C23×C418yes—
220C2×C4×C61C23×C412yes—
221C2×C4×C61C23×C412yes—
222C2×C4×C61C23×C411yescharacteristic
223C2×C4×C61C2×#4112yes—
224C2×C4×C61C2×#4112yes—
225C2×C4×C61C2511yescharacteristic
226#92418—22no—
227#92418—22no—
228#92418—24no—
229#92418—24no—
230#92418C2×#4114yes—
231#92418C2×#4112yes—
232#92418C2×#4112yes—
233#60209—611nomaximal, Sylow
234C22×C61—22no—
235C22×C61—22no—
236C22×C61—24no—
237C22×C61—28no—
238C22×C61—24no—
239C22×C61—44no—
240C22×C61C4×#4112yes—
241C22×C61C24×C411yescharacteristic
242C4×C61—22no—
243C4×C61—24no—
244C4×C61—24no—
245C4×C61—22no—
246C4×C61—28no—
247C4×C61—22no—
248C4×C61C4×#4112yes—
249C4×C61C22×#4111yescharacteristic
250C4×C61C22×#4111yescharacteristic
251#3455D6111yescharacteristic
252#2969—618no—
253#2968—618no—
254#2961—611no—
255#2960—611no—
256#2932—611no—
257#2931—611no—
258#2930—612no—
259#2929—612no—
260#2922—612no—
261#2921—611no—
262#2875—612no—
263C2×#405—611no—
264C2×C61—22no—
265C2×C61—44no—
266C2×C61—42no—
267C2×C61C2×C4×#4111yescharacteristic
268C2×C61#357312yes—
269C2×#131—612no—
270C2×#131—612no—
271C22×M16—611no—
272C23×C8—611no—
273#542Dic6114yes—
274#541Dic6112yes—
275#540Dic6112yes—
276#539Dic6114yes—
277#538Dic6114yes—
278#533C2×D6111yescharacteristic
279#532C2×D6111yescharacteristic
280#529C2×D6118yes—
281#525C2×D6112yes—
282C4×#41C2×D6112yes—
283C2×#117C2×D6111yescharacteristic
284#446—614no—
285#446—618no—
286C8×Q8—614no—
287#436—618no—
288#436—614no—
289#435—618no—
290#435—618no—
291#435—618no—
292C8×D4—618no—
293C8×D4—614no—
294#433—618no—
295#433—618no—
296#431—618no—
297#424—612no—
298#424—612no—
299#423—611no—
300#423—611no—
301#408—612no—
302#408—612no—
303#407—612no—
304#405—614no—
305#405—614no—
306#405—612no—
307#405—612no—
308#405—614no—
309#405—614no—
310C4×M16—612no—
311C4×M16—614no—
312C2×#97—611no—
313C2×#97—611no—
314C2×#97—614no—
315C2×C4×C8—611no—
316C2×C4×C8—611no—
317C61#6020911yescharacteristic, complemented, Sylow
318C2×#41C22×D6112yes—
319C23×C4C22×D6111yescharacteristic
320#131—1228no—
321#131—1228no—
322C2×M16—612no—
323C2×M16—612no—
324C2×M16—612no—
325C2×M16—612no—
326C2×M16—614no—
327C2×M16—614no—
328C2×M16—1224no—
329C22×C8—611no—
330C22×C8—611no—
331C22×C8—611no—
332C22×C8—614no—
333C22×C8—611no—
334C22×C8—614no—
335C22×C8—614no—
336C22×C8—1224no—
337C4⋊Q8C2×Dic6111yescharacteristic
338C4⋊Q8C2×Dic6112yes—
339#127C2×Dic6111yescharacteristic
340#126C2×Dic6118yes—
341#126C2×Dic6118yes—
342#125C2×Dic6118yes—
343#125C2×Dic6112yes—
344#124C2×Dic6118yes—
345#124C2×Dic6112yes—
346#123C2×Dic6114yes—
347#123C2×Dic6118yes—
348#123C2×Dic6118yes—
349C22⋊Q8C2×Dic6114yes—
350C22⋊Q8C2×Dic6118yes—
351C22⋊Q8C2×Dic6114yes—
352#121C2×Dic6114yes—
353#121C2×Dic6118yes—
354#121C2×Dic6114yes—
355#120C2×Dic6114yes—
356C4×Q8C2×Dic6114yes—
357C4×Q8C22×D6114yes—
358C4×D4C2×Dic6118yes—
359C4×D4C2×Dic6114yes—
360C4×D4C22×D6118yes—
361C4×D4C22×D6114yes—
362#117C2×Dic6112yes—
363#117C2×Dic6112yes—
364#117C22×D6114yes—
365#117C22×D6112yes—
366#117C22×D6114yes—
367C2×C4⋊C4C2×Dic6112yes—
368C2×C22⋊C4C2×Dic6112yes—
369C2×C42C22×D6111yescharacteristic
370C2×C42C22×D6111yescharacteristic
371C2×C42C22×D6114yes—
372C4⋊C8—614no—
373C4⋊C8—614no—
374C4⋊C8—618no—
375C4⋊C8—618no—
376C22⋊C8—618no—
377C22⋊C8—618no—
378C22⋊C8—618no—
379#97—612no—
380#97—612no—
381#97—614no—
382#97—614no—
383#97—614no—
384#97—614no—
385C4×C8—614no—
386C4×C8—612no—
387C4×C8—612no—
388C4×C8—614no—
389C4×C8—614no—
390C4×C8—614no—
391C24C22×Dic6111yescharacteristic
392#41—24no—
393#41—24no—
394C2×Q8C22×Dic6112yes—
395C2×Q8C22×Dic6114yes—
396C2×D4C22×Dic6114yes—
397C2×D4C22×Dic6112yes—
398C2×D4C22×Dic6118yes—
399C2×D4C22×Dic6114yes—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4C22×Dic6111yescharacteristic
403C22×C4C22×Dic6112yes—
404C22×C4C22×Dic6112yes—
405C22×C4C22×Dic6112yes—
406C22×C4C22×Dic6118yes—
407C22×C4C23×D6114yes—
408C22×C4C23×D6112yes—
409C22×C4C23×D6111yescharacteristic
410M16—1224no—
411M16—1224no—
412M16—1228no—
413M16—1228no—
414M16—1224no—
415M16—1224no—
416C2×C8—614no—
417C2×C8—614no—
418C2×C8—611no—
419C2×C8—611no—
420C2×C8—612no—
421C2×C8—612no—
422C2×C8—612no—
423C2×C8—614no—
424C2×C8—614no—
425C2×C8—1228no—
426C2×C8—1228no—
427C2×C8—1228no—
428C2×C8—1222no—
429C2×C8—1222no—
430C4⋊C4C22×Dic6114yes—
431C4⋊C4C22×Dic6114yes—
432C4⋊C4C22×Dic6114yes—
433C4⋊C4C22×Dic6118yes—
434C4⋊C4C22×Dic6114yes—
435C4⋊C4C22×Dic6118yes—
436C4⋊C4C22×Dic6118yes—
437C4⋊C4C22×Dic6114yes—
438C22⋊C4C22×Dic6114yes—
439C22⋊C4C22×Dic6114yes—
440C22⋊C4C22×Dic6114yes—
441C22⋊C4C22×Dic6118yes—
442C22⋊C4C22×Dic6118yes—
443C22⋊C4C22×Dic6118yes—
444C22⋊C4C22×Dic6118yes—
445C42C22×Dic6112yes—
446C42C22×Dic6111yescharacteristic
447C42C22×Dic6111yescharacteristic
448C42C22×Dic6118yes—
449C42C23×D6112yes—
450C42C23×D6112yes—
451C42C23×D6114yes—
452C23—22no—
453C23—42no—
454C23C23×Dic6112yes—
455C23C23×Dic6114yes—
456C23C23×Dic6111yescharacteristic
457Q8—24no—
458Q8—44no—
459D4—28no—
460D4—24no—
461D4—44no—
462D4—48no—
463C2×C4—22no—
464C2×C4—24no—
465C2×C4—22no—
466C2×C4—24no—
467C2×C4—24no—
468C2×C4—22no—
469C2×C4—42no—
470C2×C4—48no—
471C2×C4—42no—
472C2×C4C23×Dic6114yes—
473C2×C4C23×Dic6114yes—
474C2×C4C23×Dic6112yes—
475C2×C4C23×Dic6118yes—
476C2×C4C23×Dic6112yes—
477C2×C4C23×Dic6112yes—
478C2×C4C23×Dic6111yescharacteristic
479C2×C4#41×D6112yes—
480C2×C4#41×D6112yes—
481C2×C4C24×D6111yescharacteristic
482C8—614no—
483C8—612no—
484C8—612no—
485C8—1222no—
486C8—1224no—
487C8—1224no—
488C8—1222no—
489C22—22no—
490C22—28no—
491C22—24no—
492C22—22no—
493C22—24no—
494C22—44no—
495C22#41×Dic6112yes—
496C22C24×Dic6111yescharacteristic
497C4—24no—
498C4—22no—
499C4—28no—
500C4—22no—
501C4—22no—
502C4—24no—
503C4#41×Dic6112yes—
504C4C2×#41×D6111yescharacteristic
505C4C2×#41×D6111yescharacteristic
506C2—22no—
507C2—42no—
508C2—44no—
509C2C2×#41×Dic6111yescharacteristic
510C2#2146053112yes—
subgroupG/H
center, FrattiniC2×C4C24×D61
commutator, socleC22×C61C24×C4
Fitting, prime radicalC61×#3455C2
cosocleC2×C4×C61C25
hypercenter#3455D61
hypocenterC61#60209
LCSG ▷ C22×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ #3455

Complements

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

NH ≅ G/Ntype
1. #21454824C2semidirect
2. #21454815C2semidirect
3. #21454760C2semidirect
4. #21454754C2semidirect
5. #21454537C2semidirect
6. #21454531C2semidirect
7. #21454523C2semidirect
8. #21454514C2semidirect
9. #21454463C2semidirect
10. #21454457C2semidirect
11. #21454247C2semidirect
12. C2×#16839718C2semidirect
18. #16839893C22semidirect
19. #16839893C22semidirect
20. Q8×#92418C22semidirect
22. #16839840C22semidirect
25. #16839834C22semidirect
27. D4×#92418C22semidirect
29. #16839823C22semidirect
30. #16839815C22semidirect
42. #16839718C22semidirect
43. #16839718C22semidirect
45. C4×#11758194C22semidirect
48. C2×#14519688C22semidirect
317. C61#60209semidirect

Maximal subgroups

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

subgroupindex
#214548242
#214548152
#214547602
#214547542
#214545372
#214545312
#214545232
#214545142
#214544632
#214544572
#214542472
C2×#168397182
C61×#34552
#6020961